# Math 2.0

> Math 2.0 is a versioned knowledge graph of mathematics. Each node is a concept, a definition, a statement, a conjecture, a problem, an example, a counterexample, an object, a proof, a computation or a topic. Each node has a permanent tag, sources, typed links to other nodes and a status that the build computes from the evidence.

Each node page has a Markdown copy at node/<TAG>.md and a JSON copy at api/node/<TAG>.json. The URI of a node is m2://<TAG>. The front matter follows the JSON Schema at https://math2.tamnd.com/schema/node.v1.json. The site has 3433 nodes now.

## Data

- [All nodes](https://math2.tamnd.com/api/nodes.json): the tag, kind, name, aliases and MSC classes of each node
- [Reviewers](https://math2.tamnd.com/api/reviewers.json): the reviewers and their fidelity reviews
- [Node schema](https://math2.tamnd.com/schema/node.v1.json): JSON Schema 2020-12 for the front matter
- [Edges](https://math2.tamnd.com/schema/edges.md): the 25 link types between nodes
- [Macros](https://math2.tamnd.com/macros/m2.katex.json): the TeX macros that node bodies use
- [Contribute](https://math2.tamnd.com/contribute.md): how to add a node
- [Curation](https://math2.tamnd.com/curate.md): how to decide on the alignment proposals of m2 align

## Specification

- [Math 2.0: specification](https://math2.tamnd.com/spec/00-README.md)
- [01. The state of the field, October 2026](https://math2.tamnd.com/spec/01-research-2026.md)
- [02. The goal](https://math2.tamnd.com/spec/02-the-goal.md)
- [03. Architecture](https://math2.tamnd.com/spec/03-architecture.md)
- [04. The object model](https://math2.tamnd.com/spec/04-object-model.md)
- [05. The knowledge graph](https://math2.tamnd.com/spec/05-knowledge-graph.md)
- [06. Versioned storage: metadata, artefacts, provenance](https://math2.tamnd.com/spec/06-versioned-storage.md)
- [07. The formal engine: Lean 4, Mathlib, proof checking](https://math2.tamnd.com/spec/07-formal-engine.md)
- [08. The computational engine: CAS, SMT, numerical experiments](https://math2.tamnd.com/spec/08-computational-engine.md)
- [09. Literature ingestion: papers, books, existing libraries](https://math2.tamnd.com/spec/09-literature-ingestion.md)
- [10. Autoformalization and statement fidelity](https://math2.tamnd.com/spec/10-autoformalization-and-fidelity.md)
- [11. The Mathematical API and the agent orchestrator](https://math2.tamnd.com/spec/11-api-and-orchestrator.md)
- [12. Research agents](https://math2.tamnd.com/spec/12-research-agents.md)
- [13. The research workspace: search, explore, explain, collaborate](https://math2.tamnd.com/spec/13-research-workspace.md)
- [14. Search](https://math2.tamnd.com/spec/14-search.md)
- [15. Trust and the audit](https://math2.tamnd.com/spec/15-trust-and-audit.md)
- [16. Collaboration and governance](https://math2.tamnd.com/spec/16-collaboration-and-governance.md)
- [17. Evaluation](https://math2.tamnd.com/spec/17-evaluation.md)
- [18. Coverage: what "all mathematics" means](https://math2.tamnd.com/spec/18-coverage-all-math.md)
- [19. Milestones](https://math2.tamnd.com/spec/19-milestones.md)
- [20. Repository layout](https://math2.tamnd.com/spec/20-repository-layout.md)
- [21. Open questions](https://math2.tamnd.com/spec/21-open-questions.md)
- [Math 2.0: the corpus specification](https://math2.tamnd.com/spec/corpus/00-README.md)
- [K01. Research](https://math2.tamnd.com/spec/corpus/01-research.md)
- [K01, part A. Encyclopedic, reference and database sources of mathematics written by people](https://math2.tamnd.com/spec/corpus/01a-reference-works.md)
- [K01, part B. Literature indexes, open textbooks and syllabi: sizes, licences, channels (research for math2)](https://math2.tamnd.com/spec/corpus/01b-literature-and-books.md)
- [K01, part C. Formal libraries and formal / informal-formal datasets](https://math2.tamnd.com/spec/corpus/01c-formal-libraries.md)
- [K01, part D. LLM generation at scale: quality, prior projects, memorization, law, terms and cost](https://math2.tamnd.com/spec/corpus/01d-model-generation.md)
- [K01, part E. Representation, pedagogy, exercises, packaging and notation](https://math2.tamnd.com/spec/corpus/01e-representation-and-packaging.md)
- [K02. The goal of the corpus](https://math2.tamnd.com/spec/corpus/02-the-goal.md)
- [K03. Identifiers: tags of ten characters](https://math2.tamnd.com/spec/corpus/03-identifiers.md)
- [K04. The inventory](https://math2.tamnd.com/spec/corpus/04-inventory.md)
- [K05. Human sources](https://math2.tamnd.com/spec/corpus/05-human-sources.md)
- [K06. Formal sources](https://math2.tamnd.com/spec/corpus/06-formal-sources.md)
- [K07. Machine generation: what models write, and how](https://math2.tamnd.com/spec/corpus/07-machine-generation.md)
- [K08. The node card: what a good node contains](https://math2.tamnd.com/spec/corpus/08-the-node-card.md)
- [K09. Quality: how the corpus stays correct](https://math2.tamnd.com/spec/corpus/09-quality.md)
- [K10. Structure, notation and conventions](https://math2.tamnd.com/spec/corpus/10-structure-and-notation.md)
- [K11. Packaging: from nodes to courses, books and articles](https://math2.tamnd.com/spec/corpus/11-packaging.md)
- [K12. Pipeline and operations](https://math2.tamnd.com/spec/corpus/12-pipeline-and-operations.md)
- [K13. Law and ethics](https://math2.tamnd.com/spec/corpus/13-legal-and-ethics.md)
- [K14. Roadmap](https://math2.tamnd.com/spec/corpus/14-roadmap.md)
- [K15. Open questions](https://math2.tamnd.com/spec/corpus/15-open-questions.md)

## Nodes

- [Vector space](https://math2.tamnd.com/node/000AYPA9AJ.md): concept 000AYPA9AJ
- [The integers are a Euclidean domain](https://math2.tamnd.com/node/000JFQGDXG.md): statement 000JFQGDXG
- [Sequential characterization of limits](https://math2.tamnd.com/node/000MXHEY22.md): statement 000MXHEY22
- [Singular value decomposition](https://math2.tamnd.com/node/000NPTMX6J.md): statement 000NPTMX6J
- [Quotient set](https://math2.tamnd.com/node/0011MTQMR6.md): concept 0011MTQMR6
- [Mutual independence of events](https://math2.tamnd.com/node/001B78R0DJ.md): concept 001B78R0DJ
- [Partition function](https://math2.tamnd.com/node/001CX45SWQ.md): definition 001CX45SWQ
- [Block diagonal matrix](https://math2.tamnd.com/node/001T5FJDC1.md): concept 001T5FJDC1
- [Power series are differentiable inside the radius of convergence](https://math2.tamnd.com/node/002X7EK91H.md): statement 002X7EK91H
- [A function with zero derivative is constant](https://math2.tamnd.com/node/003QDZ4T16.md): statement 003QDZ4T16
- [Closure is the set of limits of sequences](https://math2.tamnd.com/node/003QP7A8HG.md): statement 003QP7A8HG
- [Differential equation](https://math2.tamnd.com/node/003R0V5M7V.md): concept 003R0V5M7V
- [Natural logarithm](https://math2.tamnd.com/node/0044CPHJBV.md): definition 0044CPHJBV
- [Jump discontinuity](https://math2.tamnd.com/node/004FH91BN5.md): concept 004FH91BN5
- [Exterior derivative](https://math2.tamnd.com/node/004FRCXH6M.md): definition 004FRCXH6M
- [Power set Boolean algebra](https://math2.tamnd.com/node/004J7ZEFAT.md): example 004J7ZEFAT
- [Square-free integer](https://math2.tamnd.com/node/004JW7A2N5.md): definition 004JW7A2N5
- [Sylow p-subgroup](https://math2.tamnd.com/node/0053CQE9Y2.md): concept 0053CQE9Y2
- [A differentiable function is convex iff its derivative is increasing](https://math2.tamnd.com/node/005JT76HCK.md): statement 005JT76HCK
- [Taylor's theorem with the Cauchy remainder](https://math2.tamnd.com/node/005Q84BMAD.md): statement 005Q84BMAD
- [Row equivalence](https://math2.tamnd.com/node/006BHFWMAP.md): concept 006BHFWMAP
- [The characteristic of an integral domain is zero or prime](https://math2.tamnd.com/node/00758QM6JJ.md): statement 00758QM6JJ
- [Quotient ring](https://math2.tamnd.com/node/0076HWHGHQ.md): concept 0076HWHGHQ
- [Least common multiple](https://math2.tamnd.com/node/007NPQX34K.md): concept 007NPQX34K
- [Variance](https://math2.tamnd.com/node/007QED3E0Y.md): definition 007QED3E0Y
- [Dimension of the dual space](https://math2.tamnd.com/node/00889YD15K.md): statement 00889YD15K
- [Composite number](https://math2.tamnd.com/node/008BH74D7R.md): concept 008BH74D7R
- [Zero ring](https://math2.tamnd.com/node/008TBCYDPX.md): example 008TBCYDPX
- [Linear map of a matrix](https://math2.tamnd.com/node/009892PZ30.md): definition 009892PZ30
- [Local maximum](https://math2.tamnd.com/node/00999TYPXB.md): definition 00999TYPXB
- [Scaling a row scales the determinant](https://math2.tamnd.com/node/009CY8JXFX.md): statement 009CY8JXFX
- [Sum of divisors](https://math2.tamnd.com/node/009QCP5T3Z.md): definition 009QCP5T3Z
- [Stars and bars](https://math2.tamnd.com/node/009TEMTQKH.md): statement 009TEMTQKH
- [Normed vector space](https://math2.tamnd.com/node/00A0RPAQ8A.md): definition 00A0RPAQ8A
- [Principal ideal](https://math2.tamnd.com/node/00A2EDFT9Q.md): definition 00A2EDFT9Q
- [One-sided limit](https://math2.tamnd.com/node/00BE2J8H05.md): concept 00BE2J8H05
- [Platonic solid](https://math2.tamnd.com/node/00BETAJNZ4.md): definition 00BETAJNZ4
- [Composition of linear maps](https://math2.tamnd.com/node/00BKRZKMSS.md): concept 00BKRZKMSS
- [Picard–Lindelöf theorem in integral form](https://math2.tamnd.com/node/00BT70ND50.md): statement 00BT70ND50
- [Finite intersection property](https://math2.tamnd.com/node/00BWV5CKCF.md): statement 00BWV5CKCF
- [Graph](https://math2.tamnd.com/node/00C0BR7D7H.md): concept 00C0BR7D7H
- [Polyhedron](https://math2.tamnd.com/node/00C2WN1H21.md): definition 00C2WN1H21
- [Euler–Mascheroni constant](https://math2.tamnd.com/node/00C5P9JZBK.md): object 00C5P9JZBK
- [Addition formula for Fibonacci numbers](https://math2.tamnd.com/node/00CN2F997Q.md): statement 00CN2F997Q
- [Countably infinite set](https://math2.tamnd.com/node/00CR00C9DN.md): concept 00CR00C9DN
- [Graph isomorphism](https://math2.tamnd.com/node/00CT0PTH74.md): definition 00CT0PTH74
- [The logarithm grows slower than any power](https://math2.tamnd.com/node/00DGYY10FV.md): statement 00DGYY10FV
- [Laws of exponents in a group](https://math2.tamnd.com/node/00DRK90Z6K.md): statement 00DRK90Z6K
- [−1 is a square modulo p if and only if p is not 3 modulo 4](https://math2.tamnd.com/node/00DW862T78.md): statement 00DW862T78
- [Power set](https://math2.tamnd.com/node/00E0YZCERH.md): concept 00E0YZCERH
- [Euclidean vector](https://math2.tamnd.com/node/00EBGSR5NY.md): definition 00EBGSR5NY
- [Root of a polynomial](https://math2.tamnd.com/node/00EHX6FWC0.md): definition 00EHX6FWC0
- [Hasse diagram](https://math2.tamnd.com/node/00EYM7MDKS.md): concept 00EYM7MDKS
- [Orbit](https://math2.tamnd.com/node/00F2JFA41J.md): definition 00F2JFA41J
- [Closed map](https://math2.tamnd.com/node/00FSJTXPJM.md): concept 00FSJTXPJM
- [Congruence is compatible with addition](https://math2.tamnd.com/node/00FWBN7F0Y.md): statement 00FWBN7F0Y
- [Contour integral](https://math2.tamnd.com/node/00FWNP9TXY.md): definition 00FWNP9TXY
- [Zero of a holomorphic function](https://math2.tamnd.com/node/00GC8ZQP4Q.md): concept 00GC8ZQP4Q
- [Modular multiplicative inverse](https://math2.tamnd.com/node/00GXTQMY8Z.md): definition 00GXTQMY8Z
- [Direct product of groups](https://math2.tamnd.com/node/00HF4RJ5QN.md): concept 00HF4RJ5QN
- [Conic section](https://math2.tamnd.com/node/00HJCMC2CD.md): concept 00HJCMC2CD
- [Norm of a Gaussian integer](https://math2.tamnd.com/node/00HPT88ZX0.md): definition 00HPT88ZX0
- [Trisecting the angle](https://math2.tamnd.com/node/00J3DCCEK5.md): problem 00J3DCCEK5
- [Lower triangular matrix](https://math2.tamnd.com/node/00J3KZP8Y0.md): concept 00J3KZP8Y0
- [Divisibility rule](https://math2.tamnd.com/node/00J8E3HX0M.md): concept 00J8E3HX0M
- [Multinomial coefficient](https://math2.tamnd.com/node/00K0RF9BFY.md): definition 00K0RF9BFY
- [Nullity](https://math2.tamnd.com/node/00K4FK4NBH.md): definition 00K4FK4NBH
- [Non-uniqueness for a non-Lipschitz equation](https://math2.tamnd.com/node/00K7ASZ1PR.md): counterexample 00K7ASZ1PR
- [Intermediate value theorem](https://math2.tamnd.com/node/00KAFBR32E.md): statement 00KAFBR32E
- [Determinant](https://math2.tamnd.com/node/00KDZRW4C0.md): concept 00KDZRW4C0
- [Modulus of the conjugate](https://math2.tamnd.com/node/00KRCVE7CK.md): statement 00KRCVE7CK
- [Prime ideal](https://math2.tamnd.com/node/00MHHZ9SNE.md): concept 00MHHZ9SNE
- [Independent and identically distributed random variables](https://math2.tamnd.com/node/00MRN0FARV.md): concept 00MRN0FARV
- [Every vector space has a basis](https://math2.tamnd.com/node/00MS9VMQ9Z.md): statement 00MS9VMQ9Z
- [Saddle point](https://math2.tamnd.com/node/00N3T18QNA.md): definition 00N3T18QNA
- [Change of variables in multiple integrals](https://math2.tamnd.com/node/00N5FHP1CV.md): statement 00N5FHP1CV
- [Prior probability](https://math2.tamnd.com/node/00NBBSXRQS.md): concept 00NBBSXRQS
- [Complex sine and cosine](https://math2.tamnd.com/node/00NZKRSWNX.md): concept 00NZKRSWNX
- [Power of a point](https://math2.tamnd.com/node/00PPY3RPNB.md): definition 00PPY3RPNB
- [Positive definite quadratic form](https://math2.tamnd.com/node/00QN1N6MNT.md): definition 00QN1N6MNT
- [Primitive element of a finite field](https://math2.tamnd.com/node/00QSK85Y2X.md): concept 00QSK85Y2X
- [Compact set](https://math2.tamnd.com/node/00R0X1WGPX.md): definition 00R0X1WGPX
- [Arc length parametrisation](https://math2.tamnd.com/node/00R1JDDV5X.md): definition 00R1JDDV5X
- [Derivative of sine](https://math2.tamnd.com/node/00R4MWW532.md): statement 00R4MWW532
- [Divergence of the sum of the reciprocals of the primes](https://math2.tamnd.com/node/00RA98X2CP.md): statement 00RA98X2CP
- [Second-countable space](https://math2.tamnd.com/node/00RE1R4W7N.md): definition 00RE1R4W7N
- [Absolute value](https://math2.tamnd.com/node/00S7C68D51.md): definition 00S7C68D51
- [Asymptotic equivalence](https://math2.tamnd.com/node/00SB5EA3VW.md): definition 00SB5EA3VW
- [Gram matrix](https://math2.tamnd.com/node/00SPJSDT3Z.md): concept 00SPJSDT3Z
- [Event](https://math2.tamnd.com/node/00SQGJNRTB.md): definition 00SQGJNRTB
- [Exponential growth and decay](https://math2.tamnd.com/node/00SVP6W3CB.md): example 00SVP6W3CB
- [Transcendental element](https://math2.tamnd.com/node/00SW8B91E7.md): definition 00SW8B91E7
- [Little o notation](https://math2.tamnd.com/node/00T5EPFQB9.md): concept 00T5EPFQB9
- [Ramsey's theorem](https://math2.tamnd.com/node/00TQV3F5S4.md): statement 00TQV3F5S4
- [Picard operator](https://math2.tamnd.com/node/00WAVB68F9.md): definition 00WAVB68F9
- [Irreducible polynomial](https://math2.tamnd.com/node/00X2A9G9W3.md): definition 00X2A9G9W3
- [Lotka–Volterra equations](https://math2.tamnd.com/node/00Y5QZ82AA.md): example 00Y5QZ82AA
- [A finite group is nilpotent iff it is the product of its Sylow subgroups](https://math2.tamnd.com/node/00Y9N80VSS.md): statement 00Y9N80VSS
- [Linear subspace](https://math2.tamnd.com/node/00YC1D652K.md): concept 00YC1D652K
- [Volume of a ball in R^n](https://math2.tamnd.com/node/00YH28R457.md): statement 00YH28R457
- [Arcsine](https://math2.tamnd.com/node/00Z1YBJJW8.md): concept 00Z1YBJJW8
- [Additivity of line integrals under concatenation](https://math2.tamnd.com/node/00ZEBSMNNV.md): statement 00ZEBSMNNV
- [Topological embedding](https://math2.tamnd.com/node/00ZG9Y9TR4.md): concept 00ZG9Y9TR4
- [Ordered field](https://math2.tamnd.com/node/010JCV9J4E.md): definition 010JCV9J4E
- [Uniqueness of finite fields of a given order](https://math2.tamnd.com/node/010PKCQPXP.md): statement 010PKCQPXP
- [Hensel's lemma](https://math2.tamnd.com/node/010QVB71T2.md): statement 010QVB71T2
- [Recurrence for Bell numbers](https://math2.tamnd.com/node/0110GBHPPB.md): statement 0110GBHPPB
- [Image of a homomorphism](https://math2.tamnd.com/node/0112DTS464.md): definition 0112DTS464
- [Partial order](https://math2.tamnd.com/node/011C29Q1S1.md): definition 011C29Q1S1
- [Strictly monotone function](https://math2.tamnd.com/node/011EJQWWJS.md): concept 011EJQWWJS
- [A convex function on an open interval is continuous](https://math2.tamnd.com/node/011HV2GAWF.md): statement 011HV2GAWF
- [Jordan–Chevalley decomposition](https://math2.tamnd.com/node/011K11VPJN.md): statement 011K11VPJN
- [Evaluation homomorphism](https://math2.tamnd.com/node/0125QVNCFQ.md): definition 0125QVNCFQ
- [1 + x is at most exp x](https://math2.tamnd.com/node/012C7GFF20.md): statement 012C7GFF20
- [Hölder continuity](https://math2.tamnd.com/node/012PBA0FMD.md): concept 012PBA0FMD
- [Reduction formula](https://math2.tamnd.com/node/012RYET5E2.md): concept 012RYET5E2
- [Differentiable function of several variables](https://math2.tamnd.com/node/0135XZZQ8M.md): definition 0135XZZQ8M
- [exp z = 1 iff z is an integer multiple of 2πi](https://math2.tamnd.com/node/013AHC4DRT.md): statement 013AHC4DRT
- [Existence of splitting fields](https://math2.tamnd.com/node/01441KDGBY.md): statement 01441KDGBY
- [Strict differentiability](https://math2.tamnd.com/node/0146MBQNNE.md): definition 0146MBQNNE
- [Semidirect product](https://math2.tamnd.com/node/014E1DZQE8.md): definition 014E1DZQE8
- [Chernoff bound](https://math2.tamnd.com/node/014PEBGDYF.md): statement 014PEBGDYF
- [Cayley's theorem](https://math2.tamnd.com/node/0151VYVX27.md): statement 0151VYVX27
- [Integral equation form of an initial value problem](https://math2.tamnd.com/node/015KK993C5.md): statement 015KK993C5
- [Glide reflection](https://math2.tamnd.com/node/0167F9B344.md): concept 0167F9B344
- [Leibniz formula for determinants](https://math2.tamnd.com/node/016PE4VZ8Q.md): statement 016PE4VZ8Q
- [Principal branch](https://math2.tamnd.com/node/017BPGVEE8.md): concept 017BPGVEE8
- [Alternating series](https://math2.tamnd.com/node/017BR1M98J.md): definition 017BR1M98J
- [Riemann's criterion for integrability](https://math2.tamnd.com/node/017DH8WXTE.md): statement 017DH8WXTE
- [Platonic solid](https://math2.tamnd.com/node/017E5PRS1Q.md): concept 017E5PRS1Q
- [Dyck words are counted by Catalan numbers](https://math2.tamnd.com/node/017SRZ7NE9.md): statement 017SRZ7NE9
- [Prime number](https://math2.tamnd.com/node/01826KMG2H.md): concept 01826KMG2H
- [Vector-valued function](https://math2.tamnd.com/node/018GTRS7A4.md): concept 018GTRS7A4
- [Finite-dimensional vector space](https://math2.tamnd.com/node/018W3V3TCT.md): definition 018W3V3TCT
- [Rolle's theorem](https://math2.tamnd.com/node/0194DY35BG.md): statement 0194DY35BG
- [Residue at a simple pole](https://math2.tamnd.com/node/01991NCP7H.md): statement 01991NCP7H
- [Derivative of the complex exponential](https://math2.tamnd.com/node/019H1Z6419.md): statement 019H1Z6419
- [Root of a polynomial](https://math2.tamnd.com/node/019MA1VTR2.md): concept 019MA1VTR2
- [Unit of a ring](https://math2.tamnd.com/node/019NFGFPR5.md): definition 019NFGFPR5
- [Integral test](https://math2.tamnd.com/node/019QZ6RRVH.md): statement 019QZ6RRVH
- [Maximal solution](https://math2.tamnd.com/node/019VSXSFMQ.md): definition 019VSXSFMQ
- [Weierstrass function](https://math2.tamnd.com/node/01A0MK1730.md): counterexample 01A0MK1730
- [First derivative test](https://math2.tamnd.com/node/01A2EGXJEM.md): statement 01A2EGXJEM
- [cos pi = -1](https://math2.tamnd.com/node/01A8JGR4JM.md): statement 01A8JGR4JM
- [Distance from a point to a line](https://math2.tamnd.com/node/01AAYNSBN6.md): statement 01AAYNSBN6
- [Preimage of a set](https://math2.tamnd.com/node/01AHG4G9RT.md): concept 01AHG4G9RT
- [Chapman–Kolmogorov equation](https://math2.tamnd.com/node/01APZ8GDQH.md): statement 01APZ8GDQH
- [Block matrix](https://math2.tamnd.com/node/01AXG9KZQF.md): definition 01AXG9KZQF
- [Bias of an estimator](https://math2.tamnd.com/node/01BD2TFFP4.md): definition 01BD2TFFP4
- [Well-ordering principle](https://math2.tamnd.com/node/01BX15HSRR.md): statement 01BX15HSRR
- [Well-order](https://math2.tamnd.com/node/01CGSJ0GNQ.md): concept 01CGSJ0GNQ
- [The complex numbers cannot be ordered](https://math2.tamnd.com/node/01CJRJKDAA.md): statement 01CJRJKDAA
- [Characteristic subgroup](https://math2.tamnd.com/node/01CS8Q2APC.md): definition 01CS8Q2APC
- [Totally disconnected space](https://math2.tamnd.com/node/01CYW8S04D.md): definition 01CYW8S04D
- [Periodic orbit](https://math2.tamnd.com/node/01DSNP0B4R.md): definition 01DSNP0B4R
- [Ramsey number](https://math2.tamnd.com/node/01DSVCRF6X.md): definition 01DSVCRF6X
- [Normality via left and right cosets](https://math2.tamnd.com/node/01E7SWYJX3.md): statement 01E7SWYJX3
- [Public-key cryptography](https://math2.tamnd.com/node/01E9MNJYEX.md): concept 01E9MNJYEX
- [Inner product space](https://math2.tamnd.com/node/01EK923SKP.md): definition 01EK923SKP
- [Law of a random variable](https://math2.tamnd.com/node/01EQPZ6RYZ.md): concept 01EQPZ6RYZ
- [Determinant of a scalar multiple](https://math2.tamnd.com/node/01EXSMK864.md): statement 01EXSMK864
- [Extensions of solvable groups are solvable](https://math2.tamnd.com/node/01EY0EM7RP.md): statement 01EY0EM7RP
- [Urysohn's lemma](https://math2.tamnd.com/node/01EY3VR7ZD.md): statement 01EY3VR7ZD
- [Composition of injections is injective](https://math2.tamnd.com/node/01FPXMTBX1.md): statement 01FPXMTBX1
- [Pythagorean triple](https://math2.tamnd.com/node/01G1WY1GVX.md): definition 01G1WY1GVX
- [Periodic continued fraction](https://math2.tamnd.com/node/01GBHQ5ABF.md): concept 01GBHQ5ABF
- [Order of cardinals](https://math2.tamnd.com/node/01GTP8M5RG.md): definition 01GTP8M5RG
- [Generalised eigenvector](https://math2.tamnd.com/node/01H6QKWGDH.md): concept 01H6QKWGDH
- [Divergent sequence](https://math2.tamnd.com/node/01HR2AH0BX.md): definition 01HR2AH0BX
- [Equivalence classes form a partition](https://math2.tamnd.com/node/01HT2CH8BH.md): statement 01HT2CH8BH
- [Maximal solution](https://math2.tamnd.com/node/01J7FZ65J0.md): concept 01J7FZ65J0
- [Legendre symbol](https://math2.tamnd.com/node/01J8AMV9RS.md): definition 01J8AMV9RS
- [Vandermonde's identity](https://math2.tamnd.com/node/01JADB9VW5.md): statement 01JADB9VW5
- [Regular polygon](https://math2.tamnd.com/node/01KJJ9A400.md): concept 01KJJ9A400
- [Multigraph](https://math2.tamnd.com/node/01M5T0KDG7.md): definition 01M5T0KDG7
- [Triangle inequality for a metric](https://math2.tamnd.com/node/01M6CW9VM6.md): statement 01M6CW9VM6
- [Antisymmetric relation](https://math2.tamnd.com/node/01MGYYBB9R.md): definition 01MGYYBB9R
- [Infinite set](https://math2.tamnd.com/node/01MKW4R283.md): definition 01MKW4R283
- [Positive semidefinite matrix](https://math2.tamnd.com/node/01MMVW8CKC.md): concept 01MMVW8CKC
- [Cauchy's integral theorem](https://math2.tamnd.com/node/01MPZXYVYF.md): statement 01MPZXYVYF
- [Integers modulo n](https://math2.tamnd.com/node/01N3AYAA6N.md): definition 01N3AYAA6N
- [Nondegenerate bilinear form](https://math2.tamnd.com/node/01NHFZF2SK.md): concept 01NHFZF2SK
- [Function with unequal mixed partial derivatives](https://math2.tamnd.com/node/01NN8NY497.md): counterexample 01NN8NY497
- [Isolated point](https://math2.tamnd.com/node/01PKQYTTMT.md): definition 01PKQYTTMT
- [Similar triangles](https://math2.tamnd.com/node/01PNAVFMYZ.md): definition 01PNAVFMYZ
- [Hyperbolic functions](https://math2.tamnd.com/node/01PPDSDP2N.md): concept 01PPDSDP2N
- [Spherical coordinates](https://math2.tamnd.com/node/01QFQAQ2PA.md): concept 01QFQAQ2PA
- [sin x / x tends to 1](https://math2.tamnd.com/node/01QSSKS5C1.md): statement 01QSSKS5C1
- [Sum of the first n natural numbers](https://math2.tamnd.com/node/01QZNHQWYD.md): statement 01QZNHQWYD
- [A graph is bipartite iff it has no odd cycle](https://math2.tamnd.com/node/01R1HPEK6S.md): statement 01R1HPEK6S
- [Tangent line](https://math2.tamnd.com/node/01R1HW3Y42.md): concept 01R1HW3Y42
- [Definite and indefinite quadratic forms](https://math2.tamnd.com/node/01RBSSGMY5.md): concept 01RBSSGMY5
- [Elementary matrix](https://math2.tamnd.com/node/01REXW3WWE.md): concept 01REXW3WWE
- [Lyapunov function](https://math2.tamnd.com/node/01REYCJ371.md): concept 01REYCJ371
- [Residue class](https://math2.tamnd.com/node/01RSTGNTXG.md): concept 01RSTGNTXG
- [Surjective function](https://math2.tamnd.com/node/01RVCMSN5N.md): concept 01RVCMSN5N
- [Composition of relations](https://math2.tamnd.com/node/01SWBKDAWA.md): definition 01SWBKDAWA
- [Neighbourhood](https://math2.tamnd.com/node/01T3TMGTTE.md): definition 01T3TMGTTE
- [Integrating factor](https://math2.tamnd.com/node/01THH3Y3YP.md): definition 01THH3Y3YP
- [Diffie–Hellman key exchange](https://math2.tamnd.com/node/01TKKESHSB.md): example 01TKKESHSB
- [Hyperbolic equilibrium point](https://math2.tamnd.com/node/01TPMXM6NM.md): concept 01TPMXM6NM
- [Countable set](https://math2.tamnd.com/node/01TQB88H4C.md): concept 01TQB88H4C
- [Orthogonal family for a bilinear form](https://math2.tamnd.com/node/01TXN7ETH2.md): concept 01TXN7ETH2
- [Symmetry of second derivatives](https://math2.tamnd.com/node/01V9FPBBVK.md): statement 01V9FPBBVK
- [Bijective iff it has a two-sided inverse](https://math2.tamnd.com/node/01VG2A0BHW.md): statement 01VG2A0BHW
- [Prime element](https://math2.tamnd.com/node/01VH21EV6F.md): concept 01VH21EV6F
- [Quadratic integer](https://math2.tamnd.com/node/01VN79T2PR.md): definition 01VN79T2PR
- [Uniqueness of the algebraic closure](https://math2.tamnd.com/node/01VQ7C4FTB.md): statement 01VQ7C4FTB
- [Cantor set is closed](https://math2.tamnd.com/node/01VT24J1SD.md): statement 01VT24J1SD
- [Connected component](https://math2.tamnd.com/node/01VVP10MWJ.md): definition 01VVP10MWJ
- [Arguments add under multiplication](https://math2.tamnd.com/node/01W6RKJP01.md): statement 01W6RKJP01
- [Central binomial coefficient](https://math2.tamnd.com/node/01WHD9T0V6.md): definition 01WHD9T0V6
- [Legendre's formula](https://math2.tamnd.com/node/01WPV9PHN7.md): statement 01WPV9PHN7
- [Cassini's identity](https://math2.tamnd.com/node/01WQZK97Z5.md): statement 01WQZK97Z5
- [Geometric series](https://math2.tamnd.com/node/01X5CVV1Q5.md): example 01X5CVV1Q5
- [Identically distributed random variables](https://math2.tamnd.com/node/01X8YSDTBW.md): definition 01X8YSDTBW
- [Moment](https://math2.tamnd.com/node/01XAHPADBA.md): concept 01XAHPADBA
- [Uniformly Cauchy sequence of functions](https://math2.tamnd.com/node/01XAY17QA3.md): definition 01XAY17QA3
- [Constructive proof](https://math2.tamnd.com/node/01XM7JEKHY.md): concept 01XM7JEKHY
- [Limit at infinity](https://math2.tamnd.com/node/01YYRV7S9Z.md): concept 01YYRV7S9Z
- [Open set](https://math2.tamnd.com/node/01ZQQDCERP.md): definition 01ZQQDCERP
- [Adding a multiple of one row to another preserves the determinant](https://math2.tamnd.com/node/01ZTQBK64Q.md): statement 01ZTQBK64Q
- [Law of the unconscious statistician](https://math2.tamnd.com/node/01ZWX875AT.md): statement 01ZWX875AT
- [Carathéodory's theorem](https://math2.tamnd.com/node/0201Q4P2B5.md): statement 0201Q4P2B5
- [Euler's theorem on Eulerian circuits](https://math2.tamnd.com/node/020422MDQF.md): statement 020422MDQF
- [Existence of prime factorization](https://math2.tamnd.com/node/020K0X7HVR.md): statement 020K0X7HVR
- [Matrix exponential](https://math2.tamnd.com/node/020M04FN1H.md): concept 020M04FN1H
- [Totally bounded set](https://math2.tamnd.com/node/020Q06TYYH.md): concept 020Q06TYYH
- [Free group](https://math2.tamnd.com/node/020Z5SEN27.md): definition 020Z5SEN27
- [Definite integral](https://math2.tamnd.com/node/021WEM680W.md): definition 021WEM680W
- [Darboux and Riemann integrals agree](https://math2.tamnd.com/node/021XJAQVDC.md): statement 021XJAQVDC
- [Taylor series](https://math2.tamnd.com/node/0225QPWRRE.md): definition 0225QPWRRE
- [Order of cardinals](https://math2.tamnd.com/node/022RSX714V.md): concept 022RSX714V
- [Well-founded relation](https://math2.tamnd.com/node/0230ECX1Q5.md): definition 0230ECX1Q5
- [Euclidean algorithm in a Euclidean domain](https://math2.tamnd.com/node/023VY493K0.md): concept 023VY493K0
- [Left inverse](https://math2.tamnd.com/node/02486SD328.md): definition 02486SD328
- [Reflection](https://math2.tamnd.com/node/025158VEYZ.md): definition 025158VEYZ
- [Uniqueness of solutions for Lipschitz vector fields](https://math2.tamnd.com/node/025PKW8AMR.md): statement 025PKW8AMR
- [Conjunction](https://math2.tamnd.com/node/025YZN3VJW.md): concept 025YZN3VJW
- [First integral](https://math2.tamnd.com/node/0262JP0NV9.md): concept 0262JP0NV9
- [A function with nonnegative derivative is increasing](https://math2.tamnd.com/node/026BJBQNGJ.md): statement 026BJBQNGJ
- [Uniformly continuous functions map Cauchy sequences to Cauchy sequences](https://math2.tamnd.com/node/026FDZFRXS.md): statement 026FDZFRXS
- [Parallelogram law](https://math2.tamnd.com/node/026HPGYGS4.md): statement 026HPGYGS4
- [Flux through a surface](https://math2.tamnd.com/node/026J1MG2NW.md): concept 026J1MG2NW
- [Density of the normal distribution](https://math2.tamnd.com/node/027JF1FBB0.md): definition 027JF1FBB0
- [Derivative function](https://math2.tamnd.com/node/027K9RMZ43.md): definition 027K9RMZ43
- [Similar matrices](https://math2.tamnd.com/node/027KDW2K72.md): definition 027KDW2K72
- [Reflection in a line or plane](https://math2.tamnd.com/node/027MBSVPY3.md): definition 027MBSVPY3
- [Quotients of solvable groups are solvable](https://math2.tamnd.com/node/027ZQS04SJ.md): statement 027ZQS04SJ
- [Associate elements](https://math2.tamnd.com/node/02879QH7BY.md): concept 02879QH7BY
- [Angle addition formulas](https://math2.tamnd.com/node/028XEQCE88.md): statement 028XEQCE88
- [Exponential distribution](https://math2.tamnd.com/node/028Z8WC5Y9.md): definition 028Z8WC5Y9
- [Flow of a vector field](https://math2.tamnd.com/node/02903SSA48.md): definition 02903SSA48
- [Euclid's theorem](https://math2.tamnd.com/node/0292EPPPT0.md): statement 0292EPPPT0
- [Vector field](https://math2.tamnd.com/node/0299VCW1TH.md): concept 0299VCW1TH
- [Negation](https://math2.tamnd.com/node/029JBBF693.md): concept 029JBBF693
- [Planar graph](https://math2.tamnd.com/node/029JQEK15B.md): definition 029JQEK15B
- [Binary tree](https://math2.tamnd.com/node/02A8EMC83C.md): definition 02A8EMC83C
- [Holomorphic functions on a disc have primitives](https://math2.tamnd.com/node/02AC7JR6G6.md): statement 02AC7JR6G6
- [Convex set](https://math2.tamnd.com/node/02AS45T3H9.md): concept 02AS45T3H9
- [Infinite cyclic group](https://math2.tamnd.com/node/02B1T50JD1.md): example 02B1T50JD1
- [Open set](https://math2.tamnd.com/node/02BH3MCBQB.md): concept 02BH3MCBQB
- [Eisenstein's criterion](https://math2.tamnd.com/node/02BPKD7TMD.md): statement 02BPKD7TMD
- [Covariance matrix](https://math2.tamnd.com/node/02CBHV4ZZQ.md): definition 02CBHV4ZZQ
- [Edge colouring](https://math2.tamnd.com/node/02CDYWSWSY.md): definition 02CDYWSWSY
- [Cardinality of a finite set](https://math2.tamnd.com/node/02CK6QMXYD.md): concept 02CK6QMXYD
- [Mean value inequality](https://math2.tamnd.com/node/02CYB184RC.md): statement 02CYB184RC
- [Sign of a permutation](https://math2.tamnd.com/node/02DKV69ZH0.md): concept 02DKV69ZH0
- [Transposition](https://math2.tamnd.com/node/02DTRYKGE8.md): definition 02DTRYKGE8
- [Free module](https://math2.tamnd.com/node/02E1T12PP4.md): definition 02E1T12PP4
- [Power of a point](https://math2.tamnd.com/node/02E9VT1RVQ.md): concept 02E9VT1RVQ
- [Open balls form a basis of the metric topology](https://math2.tamnd.com/node/02EEX96DFS.md): statement 02EEX96DFS
- [Prime factorization](https://math2.tamnd.com/node/02EKEPPJ3R.md): concept 02EKEPPJ3R
- [Rank of a product of matrices](https://math2.tamnd.com/node/02FDWY55AS.md): statement 02FDWY55AS
- [Transitive action](https://math2.tamnd.com/node/02GK6HZYBT.md): concept 02GK6HZYBT
- [Newton's law of cooling](https://math2.tamnd.com/node/02HB5F40VR.md): example 02HB5F40VR
- [Casorati–Weierstrass theorem](https://math2.tamnd.com/node/02HHNPC8ED.md): statement 02HHNPC8ED
- [Existence of maximal ideals](https://math2.tamnd.com/node/02HJYT53F7.md): statement 02HJYT53F7
- [Isometry](https://math2.tamnd.com/node/02HPP5VCZH.md): concept 02HPP5VCZH
- [Nullcline](https://math2.tamnd.com/node/02HR4C0B04.md): concept 02HR4C0B04
- [Inverse trigonometric functions](https://math2.tamnd.com/node/02HTDK1FBD.md): definition 02HTDK1FBD
- [Chebyshev's inequality](https://math2.tamnd.com/node/02HZ1BK3MW.md): statement 02HZ1BK3MW
- [Isometries are injective](https://math2.tamnd.com/node/02J4G4J4TJ.md): statement 02J4G4J4TJ
- [Monotone sequence](https://math2.tamnd.com/node/02JBEMG8R4.md): concept 02JBEMG8R4
- [Characteristic function of a sum of independent random variables](https://math2.tamnd.com/node/02JGYG0T7M.md): statement 02JGYG0T7M
- [Power series of sine](https://math2.tamnd.com/node/02KTRHQMHN.md): statement 02KTRHQMHN
- [Stationary distribution](https://math2.tamnd.com/node/02KYGSCTCN.md): definition 02KYGSCTCN
- [Fermat pseudoprime](https://math2.tamnd.com/node/02M2K4QH9P.md): definition 02M2K4QH9P
- [Polynomial evaluation](https://math2.tamnd.com/node/02MAKWDKGT.md): concept 02MAKWDKGT
- [Discrete uniform distribution](https://math2.tamnd.com/node/02MY6WZXEQ.md): concept 02MY6WZXEQ
- [Intermediate field](https://math2.tamnd.com/node/02NBKVQFRQ.md): definition 02NBKVQFRQ
- [Divergence](https://math2.tamnd.com/node/02NSMQVQWK.md): definition 02NSMQVQWK
- [Splitting field](https://math2.tamnd.com/node/02NZZ1J0FH.md): definition 02NZZ1J0FH
- [The complex exponential is 2πi-periodic](https://math2.tamnd.com/node/02P2PYH3X3.md): statement 02P2PYH3X3
- [Sum of the roots of unity](https://math2.tamnd.com/node/02PETK9PQ4.md): statement 02PETK9PQ4
- [Complex logarithm](https://math2.tamnd.com/node/02QGGD5G87.md): concept 02QGGD5G87
- [Line integral of a scalar field](https://math2.tamnd.com/node/02QJVFCNYD.md): concept 02QJVFCNYD
- [Multiplicative order](https://math2.tamnd.com/node/02RRHJ7R98.md): concept 02RRHJ7R98
- [Higher partial derivatives](https://math2.tamnd.com/node/02RSXVM1ZN.md): definition 02RSXVM1ZN
- [Algebraically closed field](https://math2.tamnd.com/node/02SJG1HC32.md): definition 02SJG1HC32
- [Topological property](https://math2.tamnd.com/node/02SMVGC690.md): concept 02SMVGC690
- [Reduction of order](https://math2.tamnd.com/node/02SRXDAM3X.md): concept 02SRXDAM3X
- [Polar form of a quadratic form](https://math2.tamnd.com/node/02STTYFBVY.md): definition 02STTYFBVY
- [Cauchy mean value theorem](https://math2.tamnd.com/node/02SYZAWM2K.md): statement 02SYZAWM2K
- [Closure is closed](https://math2.tamnd.com/node/02TZDKG50Q.md): statement 02TZDKG50Q
- [Decreasing sequence bounded below converges](https://math2.tamnd.com/node/02V2GYAC8P.md): statement 02V2GYAC8P
- [Greatest common divisor](https://math2.tamnd.com/node/02WF878BH1.md): definition 02WF878BH1
- [Riemann mapping theorem](https://math2.tamnd.com/node/02WHMEG5ZR.md): statement 02WHMEG5ZR
- [External direct sum](https://math2.tamnd.com/node/02WZB422CE.md): definition 02WZB422CE
- [Bernoulli process](https://math2.tamnd.com/node/02XPXR1FBK.md): definition 02XPXR1FBK
- [Kernel of a linear map](https://math2.tamnd.com/node/02YFME3RGW.md): definition 02YFME3RGW
- [Conditional convergence](https://math2.tamnd.com/node/02YRQXZDCS.md): definition 02YRQXZDCS
- [Branch of the logarithm](https://math2.tamnd.com/node/02YSJPQYQH.md): concept 02YSJPQYQH
- [Centraliser](https://math2.tamnd.com/node/02YYV36BFM.md): concept 02YYV36BFM
- [One-form](https://math2.tamnd.com/node/02Z5B2HA3F.md): concept 02Z5B2HA3F
- [Real and imaginary parts](https://math2.tamnd.com/node/02ZE6JHJ7G.md): definition 02ZE6JHJ7G
- [K5 and K3,3 are not planar](https://math2.tamnd.com/node/030A3FHN14.md): statement 030A3FHN14
- [Ideal generated by a subset](https://math2.tamnd.com/node/030TZ0DQWT.md): definition 030TZ0DQWT
- [Partial fraction decomposition](https://math2.tamnd.com/node/030VFEREKG.md): concept 030VFEREKG
- [Commutative ring](https://math2.tamnd.com/node/0310REJJY8.md): concept 0310REJJY8
- [Total order](https://math2.tamnd.com/node/0316AYH1SC.md): definition 0316AYH1SC
- [The real part of a holomorphic function is harmonic](https://math2.tamnd.com/node/031BG625Y3.md): statement 031BG625Y3
- [A tangent is perpendicular to the radius](https://math2.tamnd.com/node/031JV5S54D.md): statement 031JV5S54D
- [Order of a cyclic subgroup equals the order of its generator](https://math2.tamnd.com/node/031M8HX7DA.md): statement 031M8HX7DA
- [Event](https://math2.tamnd.com/node/031TPJVS74.md): concept 031TPJVS74
- [Polyhedron](https://math2.tamnd.com/node/0325SGSJAR.md): concept 0325SGSJAR
- [Residue theorem](https://math2.tamnd.com/node/032HKYS329.md): statement 032HKYS329
- [Matrix multiplication](https://math2.tamnd.com/node/032Q9FB8MT.md): concept 032Q9FB8MT
- [Order on the reals](https://math2.tamnd.com/node/0337CYB0VD.md): definition 0337CYB0VD
- [Projection](https://math2.tamnd.com/node/033DA1WE8Y.md): concept 033DA1WE8Y
- [Nine-point circle](https://math2.tamnd.com/node/033H8MQRC3.md): concept 033H8MQRC3
- [Factorial formula for binomial coefficients](https://math2.tamnd.com/node/033KTKKRV6.md): statement 033KTKKRV6
- [Limit inferior](https://math2.tamnd.com/node/033RZ32T4Q.md): concept 033RZ32T4Q
- [Markov chain](https://math2.tamnd.com/node/033TJBZDG0.md): definition 033TJBZDG0
- [Formal power series](https://math2.tamnd.com/node/033WGN00BF.md): concept 033WGN00BF
- [Formula for the modulus](https://math2.tamnd.com/node/034FGCEYX4.md): statement 034FGCEYX4
- [Tangent plane to a graph](https://math2.tamnd.com/node/035V802J58.md): concept 035V802J58
- [Triangularisation](https://math2.tamnd.com/node/0363JMQ42Y.md): statement 0363JMQ42Y
- [Hyperbolic cosine](https://math2.tamnd.com/node/036696EMG2.md): concept 036696EMG2
- [Statistic](https://math2.tamnd.com/node/036756NFD6.md): definition 036756NFD6
- [Gauss's lemma on quadratic residues](https://math2.tamnd.com/node/037ACYXB5Z.md): statement 037ACYXB5Z
- [Dirichlet convolution](https://math2.tamnd.com/node/037F589D1D.md): concept 037F589D1D
- [Leibniz formula for pi](https://math2.tamnd.com/node/037FD5NBRT.md): statement 037FD5NBRT
- [System of linear differential equations](https://math2.tamnd.com/node/037ZNY7NAG.md): concept 037ZNY7NAG
- [Arzelà–Ascoli theorem](https://math2.tamnd.com/node/0381S8Z5P0.md): statement 0381S8Z5P0
- [Bernoulli distribution](https://math2.tamnd.com/node/0386Y0Z7Y7.md): concept 0386Y0Z7Y7
- [Chromatic polynomial](https://math2.tamnd.com/node/038YGY0HSC.md): definition 038YGY0HSC
- [Freshman's dream](https://math2.tamnd.com/node/0396VGFA78.md): statement 0396VGFA78
- [Parametric equation of a line](https://math2.tamnd.com/node/0399QFDFS8.md): concept 0399QFDFS8
- [Characteristic polynomial](https://math2.tamnd.com/node/039HF58KQ5.md): definition 039HF58KQ5
- [Integral of exp(-x) over the half line](https://math2.tamnd.com/node/039JAF7ZAN.md): example 039JAF7ZAN
- [Independence of two events](https://math2.tamnd.com/node/03AHF0XRVG.md): concept 03AHF0XRVG
- [Taylor's theorem with the Peano remainder](https://math2.tamnd.com/node/03AJHX431A.md): statement 03AJHX431A
- [Column space](https://math2.tamnd.com/node/03AJKXH1AV.md): concept 03AJKXH1AV
- [Norm on the Gaussian integers](https://math2.tamnd.com/node/03AN1V4ZFJ.md): definition 03AN1V4ZFJ
- [Absorbing Markov chain](https://math2.tamnd.com/node/03AN9X29R7.md): concept 03AN9X29R7
- [Generalized Euclid's lemma](https://math2.tamnd.com/node/03BAREW1Z5.md): statement 03BAREW1Z5
- [Modulo operation](https://math2.tamnd.com/node/03BR7SAX0Q.md): concept 03BR7SAX0Q
- [Law of total variance](https://math2.tamnd.com/node/03BY5XTNA4.md): statement 03BY5XTNA4
- [Multivalued function](https://math2.tamnd.com/node/03CPD1HGF3.md): concept 03CPD1HGF3
- [Left coset](https://math2.tamnd.com/node/03DEH43F4H.md): concept 03DEH43F4H
- [Variance of the normal distribution](https://math2.tamnd.com/node/03EESZDB9A.md): statement 03EESZDB9A
- [Determinant of a linear map](https://math2.tamnd.com/node/03F5KP94K0.md): definition 03F5KP94K0
- [Cauchy–Euler equation](https://math2.tamnd.com/node/03F7N3915D.md): concept 03F7N3915D
- [Range of a function](https://math2.tamnd.com/node/03FHD14RKC.md): concept 03FHD14RKC
- [Irreducible polynomial](https://math2.tamnd.com/node/03FKX7Z5DC.md): concept 03FKX7Z5DC
- [Vector triple product](https://math2.tamnd.com/node/03FTB26QYD.md): statement 03FTB26QYD
- [Chain rule for Jacobian matrices](https://math2.tamnd.com/node/03GGQEAG9S.md): statement 03GGQEAG9S
- [Absolutely convergent improper integral](https://math2.tamnd.com/node/03GP7KMBYM.md): definition 03GP7KMBYM
- [Dimension](https://math2.tamnd.com/node/03GSZ8RTZJ.md): concept 03GSZ8RTZJ
- [Pythagorean trigonometric identity](https://math2.tamnd.com/node/03H7WZ4T3K.md): statement 03H7WZ4T3K
- [P-adic valuation](https://math2.tamnd.com/node/03HDF50VN4.md): concept 03HDF50VN4
- [Antichain](https://math2.tamnd.com/node/03HF498P1J.md): concept 03HF498P1J
- [Triangle inequality](https://math2.tamnd.com/node/03HF5PMY5T.md): statement 03HF5PMY5T
- [Mean value theorem in several variables](https://math2.tamnd.com/node/03HGK7EW6R.md): statement 03HGK7EW6R
- [Regular space](https://math2.tamnd.com/node/03HMJASQBQ.md): concept 03HMJASQBQ
- [Perpendicular bisector](https://math2.tamnd.com/node/03J4NZFYRX.md): definition 03J4NZFYRX
- [Normed vector space](https://math2.tamnd.com/node/03J5RAKB30.md): concept 03J5RAKB30
- [Separable polynomial](https://math2.tamnd.com/node/03J8BS7FMV.md): definition 03J8BS7FMV
- [Wilson's theorem](https://math2.tamnd.com/node/03KGGKTB2E.md): statement 03KGGKTB2E
- [Equivalence of norms in finite dimension](https://math2.tamnd.com/node/03MC70ZRNR.md): statement 03MC70ZRNR
- [de Moivre's formula](https://math2.tamnd.com/node/03MTBVG3VE.md): statement 03MTBVG3VE
- [Type I and type II errors](https://math2.tamnd.com/node/03N8C8ZR4B.md): concept 03N8C8ZR4B
- [Path](https://math2.tamnd.com/node/03NAMV1DHJ.md): concept 03NAMV1DHJ
- [Minimal polynomial](https://math2.tamnd.com/node/03NDVCSHSB.md): concept 03NDVCSHSB
- [Vectors are perpendicular if and only if their dot product is zero](https://math2.tamnd.com/node/03NK6RSRQ1.md): statement 03NK6RSRQ1
- [Torus](https://math2.tamnd.com/node/03NM7DZXXC.md): example 03NM7DZXXC
- [Cauchy–Schwarz inequality for expectations](https://math2.tamnd.com/node/03PGHKAJG3.md): statement 03PGHKAJG3
- [Second-countable space](https://math2.tamnd.com/node/03PPNCM9GG.md): concept 03PPNCM9GG
- [A matrix is invertible iff its determinant is a unit](https://math2.tamnd.com/node/03Q7RE2HXR.md): statement 03Q7RE2HXR
- [Complement principle](https://math2.tamnd.com/node/03QC59M99B.md): statement 03QC59M99B
- [Taylor's theorem](https://math2.tamnd.com/node/03QHBVW4TT.md): statement 03QHBVW4TT
- [Skew-symmetric matrix](https://math2.tamnd.com/node/03RNVFDEAN.md): definition 03RNVFDEAN
- [Formula for the number of derangements](https://math2.tamnd.com/node/03RSJXF728.md): statement 03RSJXF728
- [Limit of a quotient of sequences](https://math2.tamnd.com/node/03S6GWWKEZ.md): statement 03S6GWWKEZ
- [Continuous linear map](https://math2.tamnd.com/node/03TM135G0N.md): concept 03TM135G0N
- [Integers modulo n](https://math2.tamnd.com/node/03VFZ2WETT.md): concept 03VFZ2WETT
- [Indefinite integral](https://math2.tamnd.com/node/03VHANA6ZK.md): definition 03VHANA6ZK
- [Circumcircle](https://math2.tamnd.com/node/03VZJE6C8C.md): concept 03VZJE6C8C
- [Jacobian determinant](https://math2.tamnd.com/node/03W6Q1MFR5.md): definition 03W6Q1MFR5
- [Complete graph](https://math2.tamnd.com/node/03WKB2H474.md): example 03WKB2H474
- [Number of derangements](https://math2.tamnd.com/node/03WY4HHMT1.md): definition 03WY4HHMT1
- [Sum of a finite geometric progression](https://math2.tamnd.com/node/03X8V7D64M.md): statement 03X8V7D64M
- [Multiplicity of a root](https://math2.tamnd.com/node/03XVP8Q4GY.md): definition 03XVP8Q4GY
- [Sample mean](https://math2.tamnd.com/node/03Y6AQ2S0J.md): concept 03Y6AQ2S0J
- [Open mapping theorem](https://math2.tamnd.com/node/03YEM7JEDN.md): statement 03YEM7JEDN
- [Oriented angle](https://math2.tamnd.com/node/03YN70NK0Z.md): concept 03YN70NK0Z
- [The Dirichlet function is not Riemann integrable](https://math2.tamnd.com/node/03Z4JC4YRJ.md): counterexample 03Z4JC4YRJ
- [C^1 vector fields satisfy the Picard–Lindelöf hypotheses](https://math2.tamnd.com/node/03Z9V0Y78A.md): statement 03Z9V0Y78A
- [Binet's formula](https://math2.tamnd.com/node/0404D8TN7Y.md): statement 0404D8TN7Y
- [Integer partition](https://math2.tamnd.com/node/0404R9P9J3.md): definition 0404R9P9J3
- [Lebesgue number lemma](https://math2.tamnd.com/node/040CFNJEAP.md): statement 040CFNJEAP
- [Lattice path](https://math2.tamnd.com/node/040WY0TT2G.md): concept 040WY0TT2G
- [Entire function](https://math2.tamnd.com/node/0419794R0T.md): definition 0419794R0T
- [Nilpotent matrix](https://math2.tamnd.com/node/0419GJC84P.md): definition 0419GJC84P
- [Direct sum of subspaces](https://math2.tamnd.com/node/041JTZCK9Q.md): definition 041JTZCK9Q
- [Subgroups of a finite cyclic group correspond to divisors of its order](https://math2.tamnd.com/node/0423TXR9AN.md): statement 0423TXR9AN
- [Rational canonical form](https://math2.tamnd.com/node/042JWAR3Q0.md): statement 042JWAR3Q0
- [Continuous bijection from a compact space to a Hausdorff space is a homeomorphism](https://math2.tamnd.com/node/0430ETCMKD.md): statement 0430ETCMKD
- [Uniform limits of holomorphic functions are holomorphic](https://math2.tamnd.com/node/0431RN0R1N.md): statement 0431RN0R1N
- [Fermat's theorem on sums of two squares](https://math2.tamnd.com/node/043B3ENSTY.md): statement 043B3ENSTY
- [Maclaurin series](https://math2.tamnd.com/node/043SJ5RF59.md): definition 043SJ5RF59
- [Multilinear map](https://math2.tamnd.com/node/0443XNARDV.md): concept 0443XNARDV
- [Product metric](https://math2.tamnd.com/node/044D4AQV8V.md): definition 044D4AQV8V
- [Diagonal matrix](https://math2.tamnd.com/node/044JNC7ZAA.md): concept 044JNC7ZAA
- [Random variable with a density](https://math2.tamnd.com/node/045D8P03E0.md): concept 045D8P03E0
- [Circle](https://math2.tamnd.com/node/045DN06F3S.md): definition 045DN06F3S
- [The derangement numbers count derangements](https://math2.tamnd.com/node/045ENVCCZV.md): statement 045ENVCCZV
- [Disc of convergence](https://math2.tamnd.com/node/045EWAXY9P.md): definition 045EWAXY9P
- [Closed curve](https://math2.tamnd.com/node/045YFKRW0P.md): concept 045YFKRW0P
- [Monotonicity of probability](https://math2.tamnd.com/node/0464Z0JP2E.md): statement 0464Z0JP2E
- [Homogeneous linear system with constant coefficients](https://math2.tamnd.com/node/0469V0EN5N.md): concept 0469V0EN5N
- [Evaluation of real integrals by residues](https://math2.tamnd.com/node/0476QH9846.md): concept 0476QH9846
- [Variance](https://math2.tamnd.com/node/0477R9PYAE.md): concept 0477R9PYAE
- [Every real number has a decimal expansion](https://math2.tamnd.com/node/047KEVEZ9Q.md): statement 047KEVEZ9Q
- [Jacobian matrix](https://math2.tamnd.com/node/047NDTEPCV.md): concept 047NDTEPCV
- [Parseval's identity](https://math2.tamnd.com/node/047RFCNVSK.md): statement 047RFCNVSK
- [Multiplicity of a root](https://math2.tamnd.com/node/047SX3TD69.md): concept 047SX3TD69
- [Maximum modulus principle](https://math2.tamnd.com/node/048F2P90WR.md): statement 048F2P90WR
- [Concyclic points](https://math2.tamnd.com/node/049DCJZ448.md): concept 049DCJZ448
- [Open cover](https://math2.tamnd.com/node/049PXYHW1T.md): concept 049PXYHW1T
- [Lagrange multiplier with one constraint](https://math2.tamnd.com/node/04A2XEX4D4.md): statement 04A2XEX4D4
- [Greedy colouring](https://math2.tamnd.com/node/04AAKZNXHD.md): concept 04AAKZNXHD
- [Möbius strip](https://math2.tamnd.com/node/04AJE0M271.md): example 04AJE0M271
- [Exponential of a sum of commuting matrices](https://math2.tamnd.com/node/04BD23PSK5.md): statement 04BD23PSK5
- [Complementary subspaces](https://math2.tamnd.com/node/04C3BAHV03.md): definition 04C3BAHV03
- [Field](https://math2.tamnd.com/node/04C5ZBG1WT.md): concept 04C5ZBG1WT
- [log x is at most x - 1](https://math2.tamnd.com/node/04CBRA4JRK.md): statement 04CBRA4JRK
- [Real sequence](https://math2.tamnd.com/node/04CC6K5VDT.md): concept 04CC6K5VDT
- [Local maximum and minimum](https://math2.tamnd.com/node/04CPG9K1C4.md): definition 04CPG9K1C4
- [Determinant is multiplicative](https://math2.tamnd.com/node/04CPJ6AF00.md): statement 04CPJ6AF00
- [Correlation coefficient](https://math2.tamnd.com/node/04CTY6XYWN.md): definition 04CTY6XYWN
- [Spectral theorem for normal matrices](https://math2.tamnd.com/node/04CW1P44VD.md): statement 04CW1P44VD
- [Variance under translation](https://math2.tamnd.com/node/04CYN5G6RY.md): statement 04CYN5G6RY
- [Coordinate vector](https://math2.tamnd.com/node/04CYQHN6MC.md): concept 04CYQHN6MC
- [Inverse hyperbolic functions](https://math2.tamnd.com/node/04D7CHT5V1.md): definition 04D7CHT5V1
- [Order of a group is the index times the order of the subgroup](https://math2.tamnd.com/node/04DC0SRN2K.md): statement 04DC0SRN2K
- [Modular exponentiation](https://math2.tamnd.com/node/04DR5PM25N.md): concept 04DR5PM25N
- [Natural logarithm](https://math2.tamnd.com/node/04DXAFN2ES.md): concept 04DXAFN2ES
- [Commutativity](https://math2.tamnd.com/node/04E4V9G5FR.md): definition 04E4V9G5FR
- [Circle group](https://math2.tamnd.com/node/04EAFVFTF7.md): object 04EAFVFTF7
- [Spectrum of a matrix](https://math2.tamnd.com/node/04EHB7YDQS.md): definition 04EHB7YDQS
- [Uniqueness of the supremum](https://math2.tamnd.com/node/04ETX30HT2.md): statement 04ETX30HT2
- [Modulo operation](https://math2.tamnd.com/node/04FHQSJCQ5.md): definition 04FHQSJCQ5
- [Mesh of a partition](https://math2.tamnd.com/node/04GM4NYXJB.md): definition 04GM4NYXJB
- [Divisibility in a commutative ring](https://math2.tamnd.com/node/04GNK7K4VP.md): definition 04GNK7K4VP
- [Riesz representation theorem in finite dimensions](https://math2.tamnd.com/node/04GYPXDXNH.md): statement 04GYPXDXNH
- [Separation axioms](https://math2.tamnd.com/node/04J4BYB0XE.md): concept 04J4BYB0XE
- [Divisor function](https://math2.tamnd.com/node/04J4Z73J5B.md): concept 04J4Z73J5B
- [Density of the irrationals](https://math2.tamnd.com/node/04J6G4H78J.md): statement 04J6G4H78J
- [Consistent and inconsistent systems](https://math2.tamnd.com/node/04JAAVHDDM.md): definition 04JAAVHDDM
- [Dirichlet's approximation theorem](https://math2.tamnd.com/node/04JAY9DGB7.md): statement 04JAY9DGB7
- [Hockey-stick identity](https://math2.tamnd.com/node/04JHQNNBFE.md): statement 04JHQNNBFE
- [Infimum](https://math2.tamnd.com/node/04KFTA34C5.md): definition 04KFTA34C5
- [Homogeneous system of linear equations](https://math2.tamnd.com/node/04KGXJG1A2.md): concept 04KGXJG1A2
- [Clique number](https://math2.tamnd.com/node/04MQSQ1F5Q.md): concept 04MQSQ1F5Q
- [Sum of two squares theorem](https://math2.tamnd.com/node/04NPWJMA6H.md): statement 04NPWJMA6H
- [Continued fraction](https://math2.tamnd.com/node/04P2RJ4WPS.md): concept 04P2RJ4WPS
- [Geometric distribution](https://math2.tamnd.com/node/04P2SR252N.md): concept 04P2SR252N
- [Lyapunov function](https://math2.tamnd.com/node/04P7B5XC22.md): definition 04P7B5XC22
- [Almost sure convergence](https://math2.tamnd.com/node/04P7D63W8P.md): definition 04P7D63W8P
- [Index of a subgroup](https://math2.tamnd.com/node/04QEBF8S49.md): definition 04QEBF8S49
- [Dedekind cut](https://math2.tamnd.com/node/04QNZZ4HHY.md): definition 04QNZZ4HHY
- [Euler's formula](https://math2.tamnd.com/node/04R3CY41VJ.md): statement 04R3CY41VJ
- [Classification of discontinuities](https://math2.tamnd.com/node/04R9K5H562.md): concept 04R9K5H562
- [Cyclic groups are abelian](https://math2.tamnd.com/node/04RG02S6BA.md): statement 04RG02S6BA
- [Law of total expectation](https://math2.tamnd.com/node/04RRB6HM00.md): statement 04RRB6HM00
- [Zorn's lemma](https://math2.tamnd.com/node/04RXTQA5ZA.md): statement 04RXTQA5ZA
- [Parabola](https://math2.tamnd.com/node/04S2GMHX82.md): concept 04S2GMHX82
- [Product topology on an arbitrary product](https://math2.tamnd.com/node/04S9HX3GAP.md): definition 04S9HX3GAP
- [Doubling the cube](https://math2.tamnd.com/node/04SD3F4FCR.md): problem 04SD3F4FCR
- [Galois field](https://math2.tamnd.com/node/04SGPNT8R9.md): object 04SGPNT8R9
- [Chinese remainder theorem for non-coprime moduli](https://math2.tamnd.com/node/04SJTX9P4N.md): statement 04SJTX9P4N
- [Solution of an ordinary differential equation](https://math2.tamnd.com/node/04SW7WECS8.md): definition 04SW7WECS8
- [Contrapositive](https://math2.tamnd.com/node/04T1CT7212.md): statement 04T1CT7212
- [Higher derivative](https://math2.tamnd.com/node/04TKTF8KDK.md): concept 04TKTF8KDK
- [Definite integral](https://math2.tamnd.com/node/04TMP0XGMJ.md): concept 04TMP0XGMJ
- [Little o notation](https://math2.tamnd.com/node/04TSHB9H8D.md): definition 04TSHB9H8D
- [Continued fraction expansion of a real number](https://math2.tamnd.com/node/04VA28F3HB.md): concept 04VA28F3HB
- [P-adic valuation](https://math2.tamnd.com/node/04VCPS30ZE.md): definition 04VCPS30ZE
- [Random walk](https://math2.tamnd.com/node/04VZE3FZXF.md): concept 04VZE3FZXF
- [Multiplicativity of the divisor function](https://math2.tamnd.com/node/04WC36XKV6.md): statement 04WC36XKV6
- [Sigma-algebra](https://math2.tamnd.com/node/04WSW8T3SY.md): definition 04WSW8T3SY
- [Integral of cosine](https://math2.tamnd.com/node/04WTBQ7APQ.md): statement 04WTBQ7APQ
- [Determinant as signed volume](https://math2.tamnd.com/node/04X3PR72VJ.md): concept 04X3PR72VJ
- [Existence of primitive roots modulo a prime](https://math2.tamnd.com/node/04XGKK8WQW.md): statement 04XGKK8WQW
- [Every connected graph has a spanning tree](https://math2.tamnd.com/node/04XPQT2XTC.md): statement 04XPQT2XTC
- [Elementary row operations](https://math2.tamnd.com/node/04XWH28452.md): concept 04XWH28452
- [Probability density function](https://math2.tamnd.com/node/04YG690N5V.md): definition 04YG690N5V
- [Monotone functions are Riemann integrable](https://math2.tamnd.com/node/04YKAYDQPW.md): statement 04YKAYDQPW
- [Comparison test](https://math2.tamnd.com/node/04Z5G0THN6.md): statement 04Z5G0THN6
- [Tautology](https://math2.tamnd.com/node/05006AJ7DZ.md): concept 05006AJ7DZ
- [Cardinality of a finite set](https://math2.tamnd.com/node/05099E1NMJ.md): definition 05099E1NMJ
- [Harmonic conjugate](https://math2.tamnd.com/node/050F2FMTKF.md): concept 050F2FMTKF
- [Symmetric group](https://math2.tamnd.com/node/0515A6D2Y1.md): concept 0515A6D2Y1
- [The real numbers are uncountable](https://math2.tamnd.com/node/0518MM6BE1.md): statement 0518MM6BE1
- [Classical definition of probability](https://math2.tamnd.com/node/051CYS71H5.md): concept 051CYS71H5
- [Euclid's lemma](https://math2.tamnd.com/node/051VYKA1EX.md): statement 051VYKA1EX
- [Product of a matrix and its adjugate](https://math2.tamnd.com/node/0528HKWZ1Z.md): statement 0528HKWZ1Z
- [Graph homomorphism](https://math2.tamnd.com/node/052JJ9YVNM.md): concept 052JJ9YVNM
- [Annihilating polynomial](https://math2.tamnd.com/node/052M4SX57M.md): definition 052M4SX57M
- [Irreducible element](https://math2.tamnd.com/node/052SDEPZH7.md): definition 052SDEPZH7
- [Associate elements](https://math2.tamnd.com/node/053E86NWVK.md): definition 053E86NWVK
- [Eigenvalues are the roots of the characteristic polynomial](https://math2.tamnd.com/node/053QVHF6N7.md): statement 053QVHF6N7
- [Converse relation](https://math2.tamnd.com/node/054K3HNN5Z.md): concept 054K3HNN5Z
- [Compact iff complete and totally bounded](https://math2.tamnd.com/node/054Q5PW5GX.md): statement 054Q5PW5GX
- [The dihedral group of order 8](https://math2.tamnd.com/node/054SBCQ93T.md): example 054SBCQ93T
- [Complex exponential](https://math2.tamnd.com/node/054VXHMCA3.md): definition 054VXHMCA3
- [Sesquilinear form](https://math2.tamnd.com/node/054YEQ0Y8M.md): concept 054YEQ0Y8M
- [Fermat numbers are pairwise coprime](https://math2.tamnd.com/node/05598J0Y8R.md): statement 05598J0Y8R
- [PID implies UFD](https://math2.tamnd.com/node/055HT85T3S.md): statement 055HT85T3S
- [Perpendicular bisector](https://math2.tamnd.com/node/055VNYT0PB.md): concept 055VNYT0PB
- [Lipschitz condition](https://math2.tamnd.com/node/055VVSN00V.md): definition 055VVSN00V
- [Sequential continuity](https://math2.tamnd.com/node/055Z15H69G.md): concept 055Z15H69G
- [Jordan measure](https://math2.tamnd.com/node/0565DC8RMK.md): definition 0565DC8RMK
- [Lebesgue's criterion for Riemann integrability](https://math2.tamnd.com/node/0565EGK6XP.md): statement 0565EGK6XP
- [Pointwise convergence](https://math2.tamnd.com/node/056FSKG32J.md): definition 056FSKG32J
- [A matrix is diagonalisable iff it has a basis of eigenvectors](https://math2.tamnd.com/node/056SW1E0AC.md): statement 056SW1E0AC
- [Isosceles triangle theorem](https://math2.tamnd.com/node/056VS31A0J.md): statement 056VS31A0J
- [Parametrised surface](https://math2.tamnd.com/node/05788J51WZ.md): definition 05788J51WZ
- [Generating function for partitions](https://math2.tamnd.com/node/0578PA53VA.md): statement 0578PA53VA
- [The nth roots of unity](https://math2.tamnd.com/node/057BH3JC9S.md): statement 057BH3JC9S
- [Number of compositions](https://math2.tamnd.com/node/057H72B2MB.md): statement 057H72B2MB
- [Binary operation](https://math2.tamnd.com/node/0580ZJSA3Q.md): definition 0580ZJSA3Q
- [Ptolemy's theorem](https://math2.tamnd.com/node/058YS05YP5.md): statement 058YS05YP5
- [Scalar triple product](https://math2.tamnd.com/node/059Q2XWJDC.md): statement 059Q2XWJDC
- [Isomorphism with the coordinate space](https://math2.tamnd.com/node/059THXBRBB.md): statement 059THXBRBB
- [Probability measure](https://math2.tamnd.com/node/059TJN0NZ1.md): concept 059TJN0NZ1
- [Partition of an interval](https://math2.tamnd.com/node/05A6PKMAJ9.md): concept 05A6PKMAJ9
- [Sum of a finite geometric series](https://math2.tamnd.com/node/05APWSX53S.md): statement 05APWSX53S
- [Normal distribution](https://math2.tamnd.com/node/05AQE0QR0D.md): concept 05AQE0QR0D
- [Binary relation](https://math2.tamnd.com/node/05AVARPXA2.md): concept 05AVARPXA2
- [Expectation of a product of independent random variables](https://math2.tamnd.com/node/05AW1G0WE1.md): statement 05AW1G0WE1
- [Partial derivative](https://math2.tamnd.com/node/05AXG3QRKR.md): definition 05AXG3QRKR
- [Jump discontinuity](https://math2.tamnd.com/node/05B0FKFXQA.md): definition 05B0FKFXQA
- [Space of solutions of a linear recurrence](https://math2.tamnd.com/node/05BDYVQQ0T.md): definition 05BDYVQQ0T
- [Stabilisers of points in the same orbit are conjugate](https://math2.tamnd.com/node/05BM304NRX.md): statement 05BM304NRX
- [Tangent function](https://math2.tamnd.com/node/05CG0S20S3.md): concept 05CG0S20S3
- [Determinant of a family of vectors in a basis](https://math2.tamnd.com/node/05CJSSXPR3.md): concept 05CJSSXPR3
- [Boundary](https://math2.tamnd.com/node/05D5EGMFYV.md): concept 05D5EGMFYV
- [Zeckendorf's theorem](https://math2.tamnd.com/node/05DEJR6TGF.md): statement 05DEJR6TGF
- [Pascal's triangle](https://math2.tamnd.com/node/05DME4TZ3K.md): object 05DME4TZ3K
- [Closed ball](https://math2.tamnd.com/node/05DSV40EB9.md): definition 05DSV40EB9
- [Having a directional derivative](https://math2.tamnd.com/node/05EGFBZTGC.md): definition 05EGFBZTGC
- [Vector space](https://math2.tamnd.com/node/05EGHEQZAW.md): definition 05EGHEQZAW
- [Left coset](https://math2.tamnd.com/node/05FD317V2W.md): definition 05FD317V2W
- [Polarisation identity](https://math2.tamnd.com/node/05FKHRT0B4.md): statement 05FKHRT0B4
- [Bounded sequence](https://math2.tamnd.com/node/05FQ3VGAR9.md): definition 05FQ3VGAR9
- [Likelihood function](https://math2.tamnd.com/node/05FSTXM92W.md): definition 05FSTXM92W
- [Dimension](https://math2.tamnd.com/node/05G2G0P224.md): definition 05G2G0P224
- [Union of a family of sets](https://math2.tamnd.com/node/05GK32H72H.md): definition 05GK32H72H
- [T1 space](https://math2.tamnd.com/node/05GQJXBP3F.md): definition 05GQJXBP3F
- [Homeomorphism](https://math2.tamnd.com/node/05GZ11PWRY.md): definition 05GZ11PWRY
- [Root test](https://math2.tamnd.com/node/05H4DDDYBZ.md): statement 05H4DDDYBZ
- [Dihedral angle](https://math2.tamnd.com/node/05HN08ZSRJ.md): concept 05HN08ZSRJ
- [Dot product](https://math2.tamnd.com/node/05HQYM0R4G.md): definition 05HQYM0R4G
- [Inner automorphism group is isomorphic to G modulo its centre](https://math2.tamnd.com/node/05HYGWFG63.md): statement 05HYGWFG63
- [Transitive closure](https://math2.tamnd.com/node/05J9NWX07X.md): concept 05J9NWX07X
- [The rationals as the field of fractions of the integers](https://math2.tamnd.com/node/05K3J8J8T2.md): example 05K3J8J8T2
- [Improper integral](https://math2.tamnd.com/node/05KQ3F99BQ.md): concept 05KQ3F99BQ
- [Limit of a sequence](https://math2.tamnd.com/node/05M5SWSZZD.md): concept 05M5SWSZZD
- [Poincaré lemma for one-forms on convex sets](https://math2.tamnd.com/node/05MC3HND79.md): statement 05MC3HND79
- [Angle sum of a triangle](https://math2.tamnd.com/node/05MDKQBEAF.md): statement 05MDKQBEAF
- [Groups of prime order are cyclic](https://math2.tamnd.com/node/05MM4EK4SA.md): statement 05MM4EK4SA
- [Cross-ratio](https://math2.tamnd.com/node/05MW7MPRBS.md): definition 05MW7MPRBS
- [Subgroup criterion](https://math2.tamnd.com/node/05N4TQKPH0.md): statement 05N4TQKPH0
- [Field generated by adjoining elements](https://math2.tamnd.com/node/05NAS7AGMZ.md): concept 05NAS7AGMZ
- [Composition of continuous maps is continuous](https://math2.tamnd.com/node/05NK26G0DR.md): statement 05NK26G0DR
- [Eulerian trail](https://math2.tamnd.com/node/05NWGBHKSZ.md): concept 05NWGBHKSZ
- [Mean squared error](https://math2.tamnd.com/node/05NZ1HHQMV.md): concept 05NZ1HHQMV
- [Derivative of arctangent](https://math2.tamnd.com/node/05P4B4Q22V.md): statement 05P4B4Q22V
- [Markov's inequality](https://math2.tamnd.com/node/05PWWHYBX5.md): statement 05PWWHYBX5
- [Matrix of a composition is the product of matrices](https://math2.tamnd.com/node/05Q1B7MH0M.md): statement 05Q1B7MH0M
- [Orthogonal family for a bilinear form](https://math2.tamnd.com/node/05Q4JD6XEH.md): definition 05Q4JD6XEH
- [Family of sets](https://math2.tamnd.com/node/05Q6G4EH2G.md): definition 05Q6G4EH2G
- [Finitely generated module](https://math2.tamnd.com/node/05QY1VEBYD.md): concept 05QY1VEBYD
- [Conservative vector field](https://math2.tamnd.com/node/05R9NF2J49.md): definition 05R9NF2J49
- [Additivity of the integral over intervals](https://math2.tamnd.com/node/05RCDKHZM6.md): statement 05RCDKHZM6
- [Laplacian as a sum of second partial derivatives](https://math2.tamnd.com/node/05RCT1522R.md): statement 05RCT1522R
- [Number of edges of a complete graph](https://math2.tamnd.com/node/05RN90E14Y.md): statement 05RN90E14Y
- [Truth table](https://math2.tamnd.com/node/05RSK4Q657.md): concept 05RSK4Q657
- [Permutation](https://math2.tamnd.com/node/05RWFKVTJ7.md): concept 05RWFKVTJ7
- [Standard score](https://math2.tamnd.com/node/05RZ4YZG4E.md): definition 05RZ4YZG4E
- [Harmonic numbers grow like log n](https://math2.tamnd.com/node/05S2GSPMGA.md): statement 05S2GSPMGA
- [Left inverse](https://math2.tamnd.com/node/05SG20AMXH.md): concept 05SG20AMXH
- [Negation of an existential statement](https://math2.tamnd.com/node/05SZQKH75K.md): statement 05SZQKH75K
- [Poisson point process](https://math2.tamnd.com/node/05TG0SE4HT.md): definition 05TG0SE4HT
- [Group of units of a ring](https://math2.tamnd.com/node/05THZKR9AN.md): definition 05THZKR9AN
- [Equality of complex numbers](https://math2.tamnd.com/node/05V7J4WD1C.md): statement 05V7J4WD1C
- [Phase plane](https://math2.tamnd.com/node/05VDGH129F.md): concept 05VDGH129F
- [Maximal ideals are prime](https://math2.tamnd.com/node/05WAF5T0GZ.md): statement 05WAF5T0GZ
- [Submodule](https://math2.tamnd.com/node/05WDSN597C.md): concept 05WDSN597C
- [Inverse trigonometric functions](https://math2.tamnd.com/node/05WEYX99X4.md): concept 05WEYX99X4
- [Jacobian of spherical coordinates](https://math2.tamnd.com/node/05WKRC7HWT.md): statement 05WKRC7HWT
- [Number of derangements](https://math2.tamnd.com/node/05WR29H2KH.md): concept 05WR29H2KH
- [Set difference](https://math2.tamnd.com/node/05WTQ6FFPS.md): concept 05WTQ6FFPS
- [Laplace transform](https://math2.tamnd.com/node/05XT5AVJBK.md): definition 05XT5AVJBK
- [Subgroups of solvable groups are solvable](https://math2.tamnd.com/node/05Y0F9M0TS.md): statement 05Y0F9M0TS
- [Asymptotic stability](https://math2.tamnd.com/node/05Y1JAM3Y5.md): definition 05Y1JAM3Y5
- [Composition of linear maps](https://math2.tamnd.com/node/05YBKMRVAR.md): definition 05YBKMRVAR
- [Isolated singularity](https://math2.tamnd.com/node/05Z3FC0154.md): concept 05Z3FC0154
- [Perfect matching](https://math2.tamnd.com/node/05ZPH8KF9K.md): definition 05ZPH8KF9K
- [Circle](https://math2.tamnd.com/node/060W5CBZP6.md): concept 060W5CBZP6
- [Right coset](https://math2.tamnd.com/node/060Z37VVB6.md): definition 060Z37VVB6
- [Law of cosines](https://math2.tamnd.com/node/060ZVZBV3E.md): statement 060ZVZBV3E
- [Carmichael number](https://math2.tamnd.com/node/060ZX7C32T.md): concept 060ZX7C32T
- [Lucas–Lehmer test](https://math2.tamnd.com/node/061BTYA0RN.md): statement 061BTYA0RN
- [Symmetric relation](https://math2.tamnd.com/node/061J5VGSPT.md): concept 061J5VGSPT
- [Tree](https://math2.tamnd.com/node/062D9KXJB1.md): concept 062D9KXJB1
- [Trail](https://math2.tamnd.com/node/062MQGP8M5.md): definition 062MQGP8M5
- [Gamma distribution](https://math2.tamnd.com/node/06395GBNWD.md): definition 06395GBNWD
- [Nilpotent endomorphism](https://math2.tamnd.com/node/063D5Z2C70.md): concept 063D5Z2C70
- [First fundamental theorem of calculus](https://math2.tamnd.com/node/063H5KJC69.md): statement 063H5KJC69
- [Goldbach's conjecture](https://math2.tamnd.com/node/063JNBSMJ0.md): conjecture 063JNBSMJ0
- [Metric space](https://math2.tamnd.com/node/063W5JMY85.md): definition 063W5JMY85
- [Order of a zero](https://math2.tamnd.com/node/064N002TX4.md): definition 064N002TX4
- [LaSalle's invariance principle](https://math2.tamnd.com/node/0659J5X7B5.md): statement 0659J5X7B5
- [Coprime ideals](https://math2.tamnd.com/node/065CK367F9.md): concept 065CK367F9
- [Surface integral](https://math2.tamnd.com/node/065HE5V98E.md): concept 065HE5V98E
- [Inclusion of a subspace is continuous](https://math2.tamnd.com/node/065J822CKN.md): statement 065J822CKN
- [Area of a triangle](https://math2.tamnd.com/node/065JWMZW86.md): concept 065JWMZW86
- [Repeating decimals are rational](https://math2.tamnd.com/node/065S5B9HF1.md): statement 065S5B9HF1
- [De Moivre–Laplace theorem](https://math2.tamnd.com/node/065X9S8CM3.md): statement 065X9S8CM3
- [Arctangent](https://math2.tamnd.com/node/065XW2V6C3.md): definition 065XW2V6C3
- [Radius of convergence](https://math2.tamnd.com/node/0660QX1VPR.md): definition 0660QX1VPR
- [Pole](https://math2.tamnd.com/node/066E7PJCYE.md): concept 066E7PJCYE
- [Closed subsets of compact sets are compact](https://math2.tamnd.com/node/066HW6D9A7.md): statement 066HW6D9A7
- [The integers are countable](https://math2.tamnd.com/node/066RJJVWH9.md): statement 066RJJVWH9
- [Noetherian ring](https://math2.tamnd.com/node/0670T59HGE.md): concept 0670T59HGE
- [Flow of a vector field](https://math2.tamnd.com/node/06738GGWWK.md): concept 06738GGWWK
- [Autonomous system](https://math2.tamnd.com/node/0673D85QKZ.md): concept 0673D85QKZ
- [Repeated roots and the derivative](https://math2.tamnd.com/node/067F8YVYMH.md): statement 067F8YVYMH
- [Scalar matrix](https://math2.tamnd.com/node/067NKSBEWT.md): concept 067NKSBEWT
- [Tonelli's theorem](https://math2.tamnd.com/node/067SYD2J4B.md): statement 067SYD2J4B
- [Existence of square roots of nonnegative reals](https://math2.tamnd.com/node/068BZBXE0C.md): statement 068BZBXE0C
- [Closure in a metric space](https://math2.tamnd.com/node/068P0P32DW.md): statement 068P0P32DW
- [Diameter of a set](https://math2.tamnd.com/node/0692CGQF00.md): definition 0692CGQF00
- [Hessian matrix](https://math2.tamnd.com/node/0693Y2METT.md): concept 0693Y2METT
- [Triple integral](https://math2.tamnd.com/node/0698M7C7AX.md): concept 0698M7C7AX
- [Uniqueness of splitting fields](https://math2.tamnd.com/node/069A3BC3EP.md): statement 069A3BC3EP
- [Comparison of a series and an integral](https://math2.tamnd.com/node/069HKV4X1E.md): statement 069HKV4X1E
- [Row space](https://math2.tamnd.com/node/069MCGX635.md): concept 069MCGX635
- [Partition of an interval](https://math2.tamnd.com/node/069P20KYQT.md): definition 069P20KYQT
- [Laplace expansion along a column](https://math2.tamnd.com/node/06AMCJMMVM.md): statement 06AMCJMMVM
- [Heine–Cantor theorem](https://math2.tamnd.com/node/06B8BNJCD6.md): statement 06B8BNJCD6
- [Solution formula for linear first-order equations](https://math2.tamnd.com/node/06B9W45XV0.md): statement 06B9W45XV0
- [Strictly convex function](https://math2.tamnd.com/node/06BA5JZKTQ.md): definition 06BA5JZKTQ
- [Reflection](https://math2.tamnd.com/node/06BRDF3F0E.md): concept 06BRDF3F0E
- [Five colour theorem](https://math2.tamnd.com/node/06C0JBE52D.md): statement 06C0JBE52D
- [Symmetric polynomial](https://math2.tamnd.com/node/06CTFY2DEZ.md): definition 06CTFY2DEZ
- [Integral of the exponential function](https://math2.tamnd.com/node/06D0EWR2DV.md): statement 06D0EWR2DV
- [P-group](https://math2.tamnd.com/node/06DAH3VS6M.md): definition 06DAH3VS6M
- [Identity matrix](https://math2.tamnd.com/node/06DW9BAWB1.md): object 06DW9BAWB1
- [Z[x] is not a principal ideal domain](https://math2.tamnd.com/node/06E3AG9TM2.md): counterexample 06E3AG9TM2
- [Fréchet derivative](https://math2.tamnd.com/node/06EH97V2A7.md): concept 06EH97V2A7
- [Operator norm](https://math2.tamnd.com/node/06EKW1AKXF.md): concept 06EKW1AKXF
- [Basis of a topology](https://math2.tamnd.com/node/06EX0FHBTF.md): definition 06EX0FHBTF
- [Midpoint](https://math2.tamnd.com/node/06F3Y6CMG9.md): definition 06F3Y6CMG9
- [Boolean algebra](https://math2.tamnd.com/node/06GKBTAA5E.md): definition 06GKBTAA5E
- [Mersenne prime](https://math2.tamnd.com/node/06GPJF9V7M.md): definition 06GPJF9V7M
- [Logical equivalence](https://math2.tamnd.com/node/06GYGS41JK.md): definition 06GYGS41JK
- [Banach space](https://math2.tamnd.com/node/06GZ956657.md): concept 06GZ956657
- [Prime ideal](https://math2.tamnd.com/node/06HC6HK0Z8.md): definition 06HC6HK0Z8
- [Uniform metric on bounded functions](https://math2.tamnd.com/node/06HE9J7DXF.md): example 06HE9J7DXF
- [Linear map of a matrix](https://math2.tamnd.com/node/06HMCTAQH3.md): concept 06HMCTAQH3
- [Dimension of a direct product](https://math2.tamnd.com/node/06HN47VPTH.md): statement 06HN47VPTH
- [Grassmann's formula](https://math2.tamnd.com/node/06HTJNCMPA.md): statement 06HTJNCMPA
- [Affine combination](https://math2.tamnd.com/node/06HYY6DK6X.md): definition 06HYY6DK6X
- [Spectral theorem](https://math2.tamnd.com/node/06J4PXRG22.md): statement 06J4PXRG22
- [Multiplication of natural numbers](https://math2.tamnd.com/node/06J5DE801P.md): concept 06J5DE801P
- [Classification of finite simple groups](https://math2.tamnd.com/node/06J6P03MG7.md): statement 06J6P03MG7
- [Real and imaginary parts](https://math2.tamnd.com/node/06J8TEPJPY.md): concept 06J8TEPJPY
- [Regular curve](https://math2.tamnd.com/node/06JMR2XWXW.md): definition 06JMR2XWXW
- [Homogeneous system of linear equations](https://math2.tamnd.com/node/06JT7KDHXM.md): definition 06JT7KDHXM
- [Branch cut](https://math2.tamnd.com/node/06JYC06JHY.md): definition 06JYC06JHY
- [Convergence of a sequence in a metric space](https://math2.tamnd.com/node/06JYPJ4WHH.md): definition 06JYPJ4WHH
- [Power series of the exponential function](https://math2.tamnd.com/node/06JZGMYTYD.md): statement 06JZGMYTYD
- [The minimal polynomial divides every polynomial vanishing at the element](https://math2.tamnd.com/node/06K7SARZKR.md): statement 06K7SARZKR
- [Directional derivative](https://math2.tamnd.com/node/06KWGHDN2J.md): concept 06KWGHDN2J
- [Inflection point](https://math2.tamnd.com/node/06MG208MGR.md): concept 06MG208MGR
- [Solution of a linear recurrence](https://math2.tamnd.com/node/06MJC8YDYR.md): concept 06MJC8YDYR
- [Every spanning set contains a basis](https://math2.tamnd.com/node/06MRNP4BNT.md): statement 06MRNP4BNT
- [Fermat prime](https://math2.tamnd.com/node/06N6JAZF44.md): concept 06N6JAZF44
- [Cauchy criterion for series](https://math2.tamnd.com/node/06PVXMY0TH.md): statement 06PVXMY0TH
- [Dimension of the annihilator](https://math2.tamnd.com/node/06Q1EFVH0Q.md): statement 06Q1EFVH0Q
- [cosh^2 - sinh^2 = 1](https://math2.tamnd.com/node/06QCTRFBM0.md): statement 06QCTRFBM0
- [P-group](https://math2.tamnd.com/node/06QYRTX9KQ.md): concept 06QYRTX9KQ
- [Commutator of two elements](https://math2.tamnd.com/node/06RGT4V6Y6.md): definition 06RGT4V6Y6
- [Units of the Gaussian integers](https://math2.tamnd.com/node/06RH4XAGRY.md): statement 06RH4XAGRY
- [Number of invertible residues modulo n](https://math2.tamnd.com/node/06RJWZCG95.md): statement 06RJWZCG95
- [Completion of a metric space](https://math2.tamnd.com/node/06RW4RKWH7.md): concept 06RW4RKWH7
- [Disjunctive normal form](https://math2.tamnd.com/node/06S4WGSEQA.md): definition 06S4WGSEQA
- [Graph homomorphism](https://math2.tamnd.com/node/06SCB3R6BN.md): definition 06SCB3R6BN
- [Continuous functions are Riemann integrable](https://math2.tamnd.com/node/06SESV7GQY.md): statement 06SESV7GQY
- [Continuously differentiable function](https://math2.tamnd.com/node/06SKCHDQT5.md): definition 06SKCHDQT5
- [Tangent function](https://math2.tamnd.com/node/06T0HS14PP.md): definition 06T0HS14PP
- [Sum of divisors](https://math2.tamnd.com/node/06TK51263Q.md): concept 06TK51263Q
- [Rayleigh quotient](https://math2.tamnd.com/node/06WHP8SE98.md): concept 06WHP8SE98
- [Existence of Laurent expansions on an annulus](https://math2.tamnd.com/node/06WHRARZKY.md): statement 06WHRARZKY
- [Converse of Wilson's theorem](https://math2.tamnd.com/node/06WRV8HFDY.md): statement 06WRV8HFDY
- [Buffon's needle problem](https://math2.tamnd.com/node/06X5PF6YFP.md): problem 06X5PF6YFP
- [Simple continued fraction](https://math2.tamnd.com/node/06XF7BCFFJ.md): definition 06XF7BCFFJ
- [Decimal expansion](https://math2.tamnd.com/node/06XY4X9430.md): definition 06XY4X9430
- [Unique factorisation domain](https://math2.tamnd.com/node/06YMXJMXYS.md): definition 06YMXJMXYS
- [Geometric multiplicity](https://math2.tamnd.com/node/06YSJVR356.md): definition 06YSJVR356
- [Continuous image of a closed interval is a closed interval](https://math2.tamnd.com/node/06ZARTQTQE.md): statement 06ZARTQTQE
- [Contour integral](https://math2.tamnd.com/node/06ZGPYTWK8.md): concept 06ZGPYTWK8
- [Argument of a complex number](https://math2.tamnd.com/node/070PFSYJJR.md): concept 070PFSYJJR
- [Finite set](https://math2.tamnd.com/node/070R982G4B.md): concept 070R982G4B
- [Geometric formula for the dot product](https://math2.tamnd.com/node/070TXS9AXM.md): statement 070TXS9AXM
- [Linear isometry](https://math2.tamnd.com/node/0717XXPWM6.md): definition 0717XXPWM6
- [Abelian group](https://math2.tamnd.com/node/071DDFVG4J.md): definition 071DDFVG4J
- [Congruence is compatible with multiplication](https://math2.tamnd.com/node/072D5G4SK6.md): statement 072D5G4SK6
- [Maximal element](https://math2.tamnd.com/node/072G59HW59.md): concept 072G59HW59
- [Comparison of topologies](https://math2.tamnd.com/node/072J68M6HX.md): concept 072J68M6HX
- [Formal power series](https://math2.tamnd.com/node/072W67KFYK.md): definition 072W67KFYK
- [Limit of a product of sequences](https://math2.tamnd.com/node/0734RSVF2V.md): statement 0734RSVF2V
- [Cyclic groups of the same order are isomorphic](https://math2.tamnd.com/node/0736AXH2ME.md): statement 0736AXH2ME
- [Third isomorphism theorem for rings](https://math2.tamnd.com/node/073C80EC63.md): statement 073C80EC63
- [Associativity of matrix multiplication](https://math2.tamnd.com/node/073E4XC016.md): statement 073E4XC016
- [Tangent plane and normal vector of a surface](https://math2.tamnd.com/node/073JKP99K2.md): concept 073JKP99K2
- [Order of a pole](https://math2.tamnd.com/node/073MWACF1K.md): concept 073MWACF1K
- [Kolmogorov's zero–one law](https://math2.tamnd.com/node/073Q8Z94KG.md): statement 073Q8Z94KG
- [Correspondence between equivalence relations and partitions](https://math2.tamnd.com/node/074482Q7CD.md): statement 074482Q7CD
- [Symmetric bilinear form](https://math2.tamnd.com/node/0744CN4MDA.md): definition 0744CN4MDA
- [Separable equation](https://math2.tamnd.com/node/074E9A8ZJK.md): concept 074E9A8ZJK
- [Gaussian integer](https://math2.tamnd.com/node/074P7C85XH.md): concept 074P7C85XH
- [One-sided inverse of a square matrix is two-sided](https://math2.tamnd.com/node/0753Y2X5K6.md): statement 0753Y2X5K6
- [Simply connected domain](https://math2.tamnd.com/node/075C629XTH.md): definition 075C629XTH
- [The reals have the cardinality of the continuum](https://math2.tamnd.com/node/075PM9FH7A.md): statement 075PM9FH7A
- [Initial value problem](https://math2.tamnd.com/node/075VBY4K4X.md): definition 075VBY4K4X
- [Path graph](https://math2.tamnd.com/node/075WR6W3Y3.md): example 075WR6W3Y3
- [Uniqueness of the derivative](https://math2.tamnd.com/node/0762WAE49P.md): statement 0762WAE49P
- [Matrix multiplication](https://math2.tamnd.com/node/076JGP6R3T.md): definition 076JGP6R3T
- [Even and odd permutations](https://math2.tamnd.com/node/0771AJP4BX.md): definition 0771AJP4BX
- [Independent random variables are uncorrelated](https://math2.tamnd.com/node/077GN9HDR6.md): statement 077GN9HDR6
- [Determinant of the inverse](https://math2.tamnd.com/node/0783S9D0ZH.md): statement 0783S9D0ZH
- [Orthogonal complement](https://math2.tamnd.com/node/079D68S3AW.md): definition 079D68S3AW
- [Transcendental number](https://math2.tamnd.com/node/079EGSD32Y.md): definition 079EGSD32Y
- [Autonomous system](https://math2.tamnd.com/node/07BNRFQ0PJ.md): definition 07BNRFQ0PJ
- [Nilpotent matrix](https://math2.tamnd.com/node/07BSPZPSMF.md): concept 07BSPZPSMF
- [Continuity of the inverse of a strictly monotone continuous function](https://math2.tamnd.com/node/07BY47HG5S.md): statement 07BY47HG5S
- [Conjunctive normal form](https://math2.tamnd.com/node/07C4A8Y18K.md): definition 07C4A8Y18K
- [Group of units of a ring](https://math2.tamnd.com/node/07C6ECNJET.md): concept 07C6ECNJET
- [Convolution of probability distributions](https://math2.tamnd.com/node/07CHJ5YX28.md): definition 07CHJ5YX28
- [Taxicab metric](https://math2.tamnd.com/node/07CJNE6ZAQ.md): example 07CJNE6ZAQ
- [Classification of finite abelian groups](https://math2.tamnd.com/node/07CQR9BW3R.md): statement 07CQR9BW3R
- [Cycle](https://math2.tamnd.com/node/07CT9RDY7Q.md): definition 07CT9RDY7Q
- [Lcm of coprime integers](https://math2.tamnd.com/node/07CZYE06ZJ.md): statement 07CZYE06ZJ
- [Lipschitz maps are uniformly continuous](https://math2.tamnd.com/node/07DCKDXRBP.md): statement 07DCKDXRBP
- [Incircle](https://math2.tamnd.com/node/07DFG5JYPT.md): concept 07DFG5JYPT
- [Integrating factor](https://math2.tamnd.com/node/07DJAZW32T.md): concept 07DJAZW32T
- [Bipartite graph](https://math2.tamnd.com/node/07DVHJ8N0A.md): definition 07DVHJ8N0A
- [Cauchy–Hadamard theorem](https://math2.tamnd.com/node/07E5MB1CA2.md): statement 07E5MB1CA2
- [Disjoint cycles commute](https://math2.tamnd.com/node/07E6D8R4D6.md): statement 07E6D8R4D6
- [Limit point of a set](https://math2.tamnd.com/node/07EGCNHAVW.md): definition 07EGCNHAVW
- [Composition of surjections is surjective](https://math2.tamnd.com/node/07EH4R3RVQ.md): statement 07EH4R3RVQ
- [Grönwall's inequality](https://math2.tamnd.com/node/07EJHQYR9A.md): statement 07EJHQYR9A
- [Newton's method](https://math2.tamnd.com/node/07EXVNF9VP.md): concept 07EXVNF9VP
- [Signature of a quadratic form](https://math2.tamnd.com/node/07EZA80NEH.md): concept 07EZA80NEH
- [Complex exponential](https://math2.tamnd.com/node/07GJV36896.md): concept 07GJV36896
- [Polynomials in several variables](https://math2.tamnd.com/node/07GNX8GCYP.md): concept 07GNX8GCYP
- [Convex function on R^n](https://math2.tamnd.com/node/07GR3TNW4H.md): concept 07GR3TNW4H
- [Function composition](https://math2.tamnd.com/node/07HF9YCWPP.md): definition 07HF9YCWPP
- [Circulant matrix](https://math2.tamnd.com/node/07HFRH2AWY.md): example 07HFRH2AWY
- [Pépin's test](https://math2.tamnd.com/node/07HXA8KV6F.md): statement 07HXA8KV6F
- [A function with positive derivative is strictly increasing](https://math2.tamnd.com/node/07J5X2T1ZX.md): statement 07J5X2T1ZX
- [Order on the integers](https://math2.tamnd.com/node/07J9ZJFECP.md): concept 07J9ZJFECP
- [Dual space](https://math2.tamnd.com/node/07JBSZE0V4.md): concept 07JBSZE0V4
- [Linear first-order equation](https://math2.tamnd.com/node/07JH131FVP.md): concept 07JH131FVP
- [i squared is minus one](https://math2.tamnd.com/node/07KBQCG9KV.md): statement 07KBQCG9KV
- [Union](https://math2.tamnd.com/node/07KVB4JSC7.md): definition 07KVB4JSC7
- [If 2^p − 1 is prime then p is prime](https://math2.tamnd.com/node/07MMWAXMYD.md): statement 07MMWAXMYD
- [Continuity at a point of a topological space](https://math2.tamnd.com/node/07N7W5C29X.md): definition 07N7W5C29X
- [Fermat's Last Theorem for n = 3](https://math2.tamnd.com/node/07NKG38VEG.md): statement 07NKG38VEG
- [Liouville function](https://math2.tamnd.com/node/07NN4VNYWM.md): concept 07NN4VNYWM
- [Menelaus's theorem](https://math2.tamnd.com/node/07NSKDGS7S.md): statement 07NSKDGS7S
- [Picard operator](https://math2.tamnd.com/node/07PB7HH1T9.md): concept 07PB7HH1T9
- [Heine–Borel theorem](https://math2.tamnd.com/node/07PFYV65MB.md): statement 07PFYV65MB
- [Jordan block](https://math2.tamnd.com/node/07PH4RX0X5.md): concept 07PH4RX0X5
- [Generalised eigenspace](https://math2.tamnd.com/node/07PNDWA7CE.md): definition 07PNDWA7CE
- [Trace of a matrix](https://math2.tamnd.com/node/07PTPSZPME.md): definition 07PTPSZPME
- [Möbius inversion formula](https://math2.tamnd.com/node/07Q4TBTFVB.md): statement 07Q4TBTFVB
- [Convex function](https://math2.tamnd.com/node/07Q7CNM5WF.md): concept 07Q7CNM5WF
- [Subfield](https://math2.tamnd.com/node/07QY5RPFBR.md): definition 07QY5RPFBR
- [Third isomorphism theorem for vector spaces](https://math2.tamnd.com/node/07RB3EMBWW.md): statement 07RB3EMBWW
- [Uniform continuity on a set](https://math2.tamnd.com/node/07RB7B04HX.md): concept 07RB7B04HX
- [Divisibility in a commutative ring](https://math2.tamnd.com/node/07RGNK57ED.md): concept 07RGNK57ED
- [Normal distribution](https://math2.tamnd.com/node/07RXBVTVZB.md): definition 07RXBVTVZB
- [Equation of a plane](https://math2.tamnd.com/node/07SDJ5DNE4.md): concept 07SDJ5DNE4
- [T0 space](https://math2.tamnd.com/node/07SG12TQH3.md): concept 07SG12TQH3
- [Linear isometry](https://math2.tamnd.com/node/07SJXTWVQW.md): concept 07SJXTWVQW
- [Diagonalisable matrix](https://math2.tamnd.com/node/07SQCNV233.md): concept 07SQCNV233
- [Derivatives of the hyperbolic functions](https://math2.tamnd.com/node/07T2TKGZ4J.md): statement 07T2TKGZ4J
- [Distance from a point to a set](https://math2.tamnd.com/node/07TXYYVYZR.md): definition 07TXYYVYZR
- [Trajectory](https://math2.tamnd.com/node/07TZ2Q4C1M.md): concept 07TZ2Q4C1M
- [Recurrence relation](https://math2.tamnd.com/node/07TZV87Y0T.md): definition 07TZV87Y0T
- [Young's inequality for products](https://math2.tamnd.com/node/07VHQ31EYJ.md): statement 07VHQ31EYJ
- [Solution of an ordinary differential equation](https://math2.tamnd.com/node/07VX34QXAV.md): concept 07VX34QXAV
- [Adjugate matrix](https://math2.tamnd.com/node/07W9H8QF38.md): concept 07W9H8QF38
- [Inverse Laplace transform](https://math2.tamnd.com/node/07WBTX1MHA.md): concept 07WBTX1MHA
- [Identity element](https://math2.tamnd.com/node/07WYQPB8E3.md): concept 07WYQPB8E3
- [Derivative of the logarithm](https://math2.tamnd.com/node/07X1QA6BRW.md): statement 07X1QA6BRW
- [Algebra of limits](https://math2.tamnd.com/node/07XBM3A9J5.md): concept 07XBM3A9J5
- [Image of a set](https://math2.tamnd.com/node/07XF282NWH.md): concept 07XF282NWH
- [Median](https://math2.tamnd.com/node/07Y2E7P9NP.md): concept 07Y2E7P9NP
- [Linear form](https://math2.tamnd.com/node/07YQ1C78G0.md): concept 07YQ1C78G0
- [Differentiable functions are continuous](https://math2.tamnd.com/node/07ZMX4ANJS.md): statement 07ZMX4ANJS
- [Anticommutativity of the cross product](https://math2.tamnd.com/node/07ZS685JJC.md): statement 07ZS685JJC
- [Every integer is a square times a square-free integer](https://math2.tamnd.com/node/080Q3RPKY6.md): statement 080Q3RPKY6
- [Linearity of the integral](https://math2.tamnd.com/node/080RWVVSB6.md): statement 080RWVVSB6
- [Irrational number](https://math2.tamnd.com/node/080T2TXVQV.md): definition 080T2TXVQV
- [Rational root theorem](https://math2.tamnd.com/node/0819CQ1V8H.md): statement 0819CQ1V8H
- [Phase line](https://math2.tamnd.com/node/081QSEAGEY.md): concept 081QSEAGEY
- [Linear recurrence with constant coefficients](https://math2.tamnd.com/node/081THG13JZ.md): definition 081THG13JZ
- [Analytic function](https://math2.tamnd.com/node/081WKEDK6C.md): concept 081WKEDK6C
- [Finite p-groups are nilpotent](https://math2.tamnd.com/node/081WTKYX6C.md): statement 081WTKYX6C
- [Directrix](https://math2.tamnd.com/node/081YA6RCX6.md): definition 081YA6RCX6
- [Holomorphic functions are analytic](https://math2.tamnd.com/node/0826SGV3R0.md): statement 0826SGV3R0
- [Finitely generated group](https://math2.tamnd.com/node/082DWWXMRJ.md): concept 082DWWXMRJ
- [Abelian group](https://math2.tamnd.com/node/082GHNPJ53.md): concept 082GHNPJ53
- [Absolutely convergent series in a Banach space converge](https://math2.tamnd.com/node/082MGNWNA8.md): statement 082MGNWNA8
- [Equivalence class](https://math2.tamnd.com/node/082RMRSY26.md): definition 082RMRSY26
- [Successor function](https://math2.tamnd.com/node/0832XVR512.md): definition 0832XVR512
- [Intersection of a family of sets](https://math2.tamnd.com/node/0839XNQXDK.md): definition 0839XNQXDK
- [Number of prime factors counted with multiplicity](https://math2.tamnd.com/node/08420828EY.md): definition 08420828EY
- [Fundamental theorem of arithmetic](https://math2.tamnd.com/node/084HH1NDCB.md): statement 084HH1NDCB
- [Higher derivative](https://math2.tamnd.com/node/084NGSTWM7.md): definition 084NGSTWM7
- [P-adic valuation of a product](https://math2.tamnd.com/node/0850JJMJYZ.md): statement 0850JJMJYZ
- [Subring](https://math2.tamnd.com/node/0859D58WPD.md): definition 0859D58WPD
- [Angle bisector theorem](https://math2.tamnd.com/node/0859R2V6ES.md): statement 0859R2V6ES
- [Walk](https://math2.tamnd.com/node/085CH3CYMX.md): concept 085CH3CYMX
- [Jordan normal form](https://math2.tamnd.com/node/085HQ5G5TE.md): statement 085HQ5G5TE
- [Law of quadratic reciprocity](https://math2.tamnd.com/node/085MJS6907.md): statement 085MJS6907
- [Group isomorphism](https://math2.tamnd.com/node/085PND4KNT.md): definition 085PND4KNT
- [Fermat's right triangle theorem](https://math2.tamnd.com/node/086BE0J3FZ.md): statement 086BE0J3FZ
- [Mean of the Poisson distribution](https://math2.tamnd.com/node/086HEVXR9Z.md): statement 086HEVXR9Z
- [Predicate logic](https://math2.tamnd.com/node/086QKPF0A6.md): concept 086QKPF0A6
- [Graph colouring](https://math2.tamnd.com/node/0874138YP6.md): definition 0874138YP6
- [Isometry](https://math2.tamnd.com/node/0878DN3WCS.md): definition 0878DN3WCS
- [von Mangoldt function](https://math2.tamnd.com/node/087EMZTHWN.md): definition 087EMZTHWN
- [Statistical hypothesis test](https://math2.tamnd.com/node/087P03NEND.md): concept 087P03NEND
- [Translation](https://math2.tamnd.com/node/087VJM5Q77.md): concept 087VJM5Q77
- [Positive semidefinite matrix](https://math2.tamnd.com/node/0880F9EQCP.md): definition 0880F9EQCP
- [Chevalley–Warning theorem](https://math2.tamnd.com/node/08895Q5W96.md): statement 08895Q5W96
- [Spanning set](https://math2.tamnd.com/node/088RYKAWK9.md): concept 088RYKAWK9
- [Ring](https://math2.tamnd.com/node/0894J980RK.md): definition 0894J980RK
- [Periodic function](https://math2.tamnd.com/node/089KWVYD69.md): definition 089KWVYD69
- [Cyclic subgroup generated by an element](https://math2.tamnd.com/node/08ABT85339.md): concept 08ABT85339
- [Divergence part of the ratio test](https://math2.tamnd.com/node/08AM0KRCZ1.md): statement 08AM0KRCZ1
- [Moment of inertia](https://math2.tamnd.com/node/08ANASGBXV.md): concept 08ANASGBXV
- [Independent set](https://math2.tamnd.com/node/08ANQHHG3P.md): concept 08ANQHHG3P
- [Diophantine equation](https://math2.tamnd.com/node/08B1RTW7VZ.md): concept 08B1RTW7VZ
- [Adjacency matrix](https://math2.tamnd.com/node/08B9CX109G.md): concept 08B9CX109G
- [Class equation](https://math2.tamnd.com/node/08BPSC6YTE.md): statement 08BPSC6YTE
- [Product of primitive polynomials is primitive](https://math2.tamnd.com/node/08C1NWBNHA.md): statement 08C1NWBNHA
- [Existence of an eigenvalue over an algebraically closed field](https://math2.tamnd.com/node/08CJB2NQ92.md): statement 08CJB2NQ92
- [Binary relation](https://math2.tamnd.com/node/08EJC6Q25H.md): definition 08EJC6Q25H
- [Equinumerous sets](https://math2.tamnd.com/node/08EJHHMB5R.md): definition 08EJHHMB5R
- [Fréchet derivative](https://math2.tamnd.com/node/08EVNYZF42.md): definition 08EVNYZF42
- [Complete metric space](https://math2.tamnd.com/node/08EWT7J0SC.md): definition 08EWT7J0SC
- [Product of cyclic groups is cyclic iff the orders are coprime](https://math2.tamnd.com/node/08FDFEF8D1.md): statement 08FDFEF8D1
- [Group of units of a monoid](https://math2.tamnd.com/node/08FHNR8Z5D.md): definition 08FHNR8Z5D
- [Homomorphisms preserve identity and inverses](https://math2.tamnd.com/node/08FKQD7NJ3.md): statement 08FKQD7NJ3
- [Uniform convergence](https://math2.tamnd.com/node/08FPABDQ4C.md): definition 08FPABDQ4C
- [Product of countable sets is countable](https://math2.tamnd.com/node/08G4XH86JC.md): statement 08G4XH86JC
- [Floor function](https://math2.tamnd.com/node/08GR54CKEA.md): concept 08GR54CKEA
- [Uniform convergence on a set](https://math2.tamnd.com/node/08HJRTG5N7.md): definition 08HJRTG5N7
- [Exterior algebra](https://math2.tamnd.com/node/08HK38GCSM.md): definition 08HK38GCSM
- [Real numbers](https://math2.tamnd.com/node/08J2M1JYAG.md): object 08J2M1JYAG
- [Tutte's theorem](https://math2.tamnd.com/node/08J7R35B6Q.md): statement 08J7R35B6Q
- [Fixed points of an action](https://math2.tamnd.com/node/08JF1K4DW1.md): definition 08JF1K4DW1
- [Cyclotomic polynomial](https://math2.tamnd.com/node/08KEYFAH66.md): concept 08KEYFAH66
- [Group isomorphism](https://math2.tamnd.com/node/08MG7H8JW5.md): concept 08MG7H8JW5
- [Distributive laws for sets](https://math2.tamnd.com/node/08N6EVA7DV.md): statement 08N6EVA7DV
- [Infinite limit](https://math2.tamnd.com/node/08NM8KQPDQ.md): definition 08NM8KQPDQ
- [Norm](https://math2.tamnd.com/node/08NYZ2FS6B.md): definition 08NYZ2FS6B
- [Sequence spaces lp](https://math2.tamnd.com/node/08P6T2WH6R.md): example 08P6T2WH6R
- [Lower central series](https://math2.tamnd.com/node/08PA437800.md): definition 08PA437800
- [Product formula for the order of HK](https://math2.tamnd.com/node/08PY4GNAPK.md): statement 08PY4GNAPK
- [Picard's great theorem](https://math2.tamnd.com/node/08QA8QTM71.md): statement 08QA8QTM71
- [The sequence 1/n tends to zero](https://math2.tamnd.com/node/08QAX4H1PF.md): example 08QAX4H1PF
- [Independence means the joint law is the product of the marginals](https://math2.tamnd.com/node/08QCA2F1NX.md): statement 08QCA2F1NX
- [Partition of a set](https://math2.tamnd.com/node/08QMKTWF9T.md): concept 08QMKTWF9T
- [Number of functions between finite sets](https://math2.tamnd.com/node/08QQH9F7G0.md): statement 08QQH9F7G0
- [Cesàro mean](https://math2.tamnd.com/node/08R2V9GZF7.md): concept 08R2V9GZF7
- [Number of distinct prime factors](https://math2.tamnd.com/node/08RQ62VAVA.md): concept 08RQ62VAVA
- [Liouville function](https://math2.tamnd.com/node/08RQAPXF1T.md): definition 08RQAPXF1T
- [Power series solution](https://math2.tamnd.com/node/08RT9GDDPQ.md): concept 08RT9GDDPQ
- [Finite field](https://math2.tamnd.com/node/08RTDHX1SK.md): definition 08RTDHX1SK
- [Normal extension](https://math2.tamnd.com/node/08S0NW8SES.md): concept 08S0NW8SES
- [Differentiable on an open set](https://math2.tamnd.com/node/08S7FDRS5Z.md): definition 08S7FDRS5Z
- [Commutator subgroup](https://math2.tamnd.com/node/08SJ5K2CBG.md): definition 08SJ5K2CBG
- [Permutation matrix](https://math2.tamnd.com/node/08SP3DF8NB.md): concept 08SP3DF8NB
- [Topological embedding](https://math2.tamnd.com/node/08SVRVHJNY.md): definition 08SVRVHJNY
- [Order of an element divides the order of the group](https://math2.tamnd.com/node/08V7JHS3QZ.md): statement 08V7JHS3QZ
- [Semisimple endomorphism](https://math2.tamnd.com/node/08VRX3Q4PB.md): concept 08VRX3Q4PB
- [Content of a polynomial](https://math2.tamnd.com/node/08VV7YJRB7.md): definition 08VV7YJRB7
- [Uniform convergence](https://math2.tamnd.com/node/08VZ57PT26.md): concept 08VZ57PT26
- [Median](https://math2.tamnd.com/node/08W2CW8PKW.md): definition 08W2CW8PKW
- [Power series of cosine](https://math2.tamnd.com/node/08W5RN375X.md): statement 08W5RN375X
- [Hall's marriage theorem](https://math2.tamnd.com/node/08WCB90WND.md): statement 08WCB90WND
- [Cauchy–Binet formula](https://math2.tamnd.com/node/08WE2W7DYS.md): statement 08WE2W7DYS
- [Sine and cosine](https://math2.tamnd.com/node/08WFZP6P89.md): concept 08WFZP6P89
- [Euler's four-square identity](https://math2.tamnd.com/node/08WTN6SCQ6.md): statement 08WTN6SCQ6
- [Explicit and implicit differential equations](https://math2.tamnd.com/node/08WWTN738R.md): concept 08WWTN738R
- [Square-free integer](https://math2.tamnd.com/node/08X0VCR2NW.md): concept 08X0VCR2NW
- [An iterate of the Picard operator is a contraction](https://math2.tamnd.com/node/08XDXP86XE.md): statement 08XDXP86XE
- [Comparison of topologies](https://math2.tamnd.com/node/08XFJJ8EYW.md): definition 08XFJJ8EYW
- [Integration by parts](https://math2.tamnd.com/node/08Y1BBQTK9.md): statement 08Y1BBQTK9
- [Subfields of a finite field](https://math2.tamnd.com/node/08YERZDM9X.md): statement 08YERZDM9X
- [Steinitz exchange lemma](https://math2.tamnd.com/node/08ZDRN6H41.md): statement 08ZDRN6H41
- [Identically distributed random variables](https://math2.tamnd.com/node/08ZDW5SA3B.md): concept 08ZDW5SA3B
- [Alternating bilinear form](https://math2.tamnd.com/node/08ZQAKBKHF.md): definition 08ZQAKBKHF
- [Eigenvalue method for linear systems](https://math2.tamnd.com/node/08ZWFDJP3H.md): concept 08ZWFDJP3H
- [Tower of Hanoi](https://math2.tamnd.com/node/0906Z03EMN.md): example 0906Z03EMN
- [One-sided limit](https://math2.tamnd.com/node/0909PWZR62.md): definition 0909PWZR62
- [Convergence of a sequence in a metric space](https://math2.tamnd.com/node/090AA928Y7.md): concept 090AA928Y7
- [Hamiltonian graph](https://math2.tamnd.com/node/090WPKFABX.md): definition 090WPKFABX
- [Arithmetic function](https://math2.tamnd.com/node/09115NQZ07.md): concept 09115NQZ07
- [An injective endomorphism of a finite-dimensional space is surjective](https://math2.tamnd.com/node/0912EEEVCT.md): statement 0912EEEVCT
- [Real exponentiation](https://math2.tamnd.com/node/091AV9YHZY.md): concept 091AV9YHZY
- [Metrizable space](https://math2.tamnd.com/node/091GJ3FYD7.md): concept 091GJ3FYD7
- [Geometric probability](https://math2.tamnd.com/node/0924BCD257.md): concept 0924BCD257
- [Euler's identity](https://math2.tamnd.com/node/09285XX7AE.md): statement 09285XX7AE
- [Catalan numbers](https://math2.tamnd.com/node/092DRBFKMG.md): definition 092DRBFKMG
- [Binomial coefficient](https://math2.tamnd.com/node/092GEF38YD.md): definition 092GEF38YD
- [Lyapunov stability](https://math2.tamnd.com/node/092S6VBECQ.md): definition 092S6VBECQ
- [Domain](https://math2.tamnd.com/node/093CM69AGS.md): definition 093CM69AGS
- [Congruent triangles](https://math2.tamnd.com/node/093EDENVA1.md): concept 093EDENVA1
- [Mesh of a partition](https://math2.tamnd.com/node/093SAXPJ9P.md): concept 093SAXPJ9P
- [Minimal polynomial divides the characteristic polynomial](https://math2.tamnd.com/node/093X6MN6Y6.md): statement 093X6MN6Y6
- [Multiplication by a matrix as a linear map](https://math2.tamnd.com/node/094V2VBJXJ.md): concept 094V2VBJXJ
- [Row equivalence](https://math2.tamnd.com/node/095DDGZF16.md): definition 095DDGZF16
- [In a PID a nonzero element is irreducible iff it generates a maximal ideal](https://math2.tamnd.com/node/095ZHQ7WCJ.md): statement 095ZHQ7WCJ
- [Extended Euclidean algorithm](https://math2.tamnd.com/node/0964YQFSA8.md): concept 0964YQFSA8
- [Young diagram](https://math2.tamnd.com/node/096AHMB3P4.md): concept 096AHMB3P4
- [Unique paths in trees](https://math2.tamnd.com/node/096B7Q5742.md): statement 096B7Q5742
- [Subnormal series](https://math2.tamnd.com/node/096HN851YZ.md): definition 096HN851YZ
- [Divisibility](https://math2.tamnd.com/node/096PW7Q014.md): definition 096PW7Q014
- [Geometric solutions come from roots of the characteristic polynomial](https://math2.tamnd.com/node/096SD4ERQH.md): statement 096SD4ERQH
- [Linearly independent set of the right size is a basis](https://math2.tamnd.com/node/096V7H13H9.md): statement 096V7H13H9
- [Ideal](https://math2.tamnd.com/node/096Z9MP2ME.md): concept 096Z9MP2ME
- [Central moment](https://math2.tamnd.com/node/09753QQDCK.md): concept 09753QQDCK
- [Cramer's rule](https://math2.tamnd.com/node/0977KZ6PQB.md): statement 0977KZ6PQB
- [Riesz's lemma](https://math2.tamnd.com/node/097C7KCAT5.md): statement 097C7KCAT5
- [Domain in the complex plane](https://math2.tamnd.com/node/097KNJFJ7V.md): definition 097KNJFJ7V
- [Intermediate value theorem for connected spaces](https://math2.tamnd.com/node/098C0KZZAP.md): statement 098C0KZZAP
- [Fixed points of a p-group action](https://math2.tamnd.com/node/098D2E7DEH.md): statement 098D2E7DEH
- [Gauss–Jordan elimination](https://math2.tamnd.com/node/098D6FQN3N.md): concept 098D6FQN3N
- [Combination](https://math2.tamnd.com/node/098PHYBTV9.md): concept 098PHYBTV9
- [Peano existence theorem](https://math2.tamnd.com/node/098YP831QY.md): statement 098YP831QY
- [Conditional probability](https://math2.tamnd.com/node/0990W5PN13.md): concept 0990W5PN13
- [Commutativity](https://math2.tamnd.com/node/099E72KTF8.md): concept 099E72KTF8
- [Space of solutions of a linear recurrence](https://math2.tamnd.com/node/099GXMHV40.md): concept 099GXMHV40
- [Dirac measure](https://math2.tamnd.com/node/099SXAG014.md): definition 099SXAG014
- [Geometric multiplicity](https://math2.tamnd.com/node/099TWVJ909.md): concept 099TWVJ909
- [Converse](https://math2.tamnd.com/node/09A3N7ZN5G.md): concept 09A3N7ZN5G
- [Cross-ratio](https://math2.tamnd.com/node/09ASMRMD67.md): concept 09ASMRMD67
- [Sequence tending to infinity](https://math2.tamnd.com/node/09AZCV6GMC.md): definition 09AZCV6GMC
- [Discrete logarithm](https://math2.tamnd.com/node/09B4X51J09.md): definition 09B4X51J09
- [Power series of the complex logarithm](https://math2.tamnd.com/node/09BG0Y7SZD.md): statement 09BG0Y7SZD
- [Necessary and sufficient conditions](https://math2.tamnd.com/node/09BJQQB3VS.md): concept 09BJQQB3VS
- [Heaviside step function](https://math2.tamnd.com/node/09C4JP7Y0M.md): object 09C4JP7Y0M
- [Area of a region as a line integral](https://math2.tamnd.com/node/09CBQB6V67.md): statement 09CBQB6V67
- [Partition of a set](https://math2.tamnd.com/node/09CCE86A6D.md): definition 09CCE86A6D
- [Cartesian product](https://math2.tamnd.com/node/09CEDJYBKP.md): concept 09CEDJYBKP
- [Open sets characterised by neighbourhoods](https://math2.tamnd.com/node/09CJ00H3NW.md): statement 09CJ00H3NW
- [Transpose](https://math2.tamnd.com/node/09DXWYRD43.md): concept 09DXWYRD43
- [Coordinates in an orthonormal basis](https://math2.tamnd.com/node/09DZ8CD1AK.md): statement 09DZ8CD1AK
- [Trace of a transpose](https://math2.tamnd.com/node/09E1RCZ7YT.md): statement 09E1RCZ7YT
- [Complex differentiability](https://math2.tamnd.com/node/09E6T7VYAG.md): concept 09E6T7VYAG
- [Partial derivative](https://math2.tamnd.com/node/09EGC6F8PT.md): concept 09EGC6F8PT
- [Algebraic extension](https://math2.tamnd.com/node/09EYMKMSB9.md): definition 09EYMKMSB9
- [Circuit](https://math2.tamnd.com/node/09F2QPVQY7.md): concept 09F2QPVQY7
- [Upper and lower Darboux integrals](https://math2.tamnd.com/node/09FQT057T7.md): definition 09FQT057T7
- [Complex numbers](https://math2.tamnd.com/node/09FTAET5Z4.md): object 09FTAET5Z4
- [Epsilon-N definition of the limit of a sequence](https://math2.tamnd.com/node/09G6KHS6GY.md): statement 09G6KHS6GY
- [Second Sylow theorem](https://math2.tamnd.com/node/09GCQ5STVJ.md): statement 09GCQ5STVJ
- [Dihedral group](https://math2.tamnd.com/node/09GK3DE0WV.md): definition 09GK3DE0WV
- [Transpose of a product](https://math2.tamnd.com/node/09GK5PS824.md): statement 09GK5PS824
- [Closed sets contain their limits](https://math2.tamnd.com/node/09GQ3ZTQJ9.md): statement 09GQ3ZTQJ9
- [Completeness of the space of bounded continuous functions](https://math2.tamnd.com/node/09GRCZVR5E.md): statement 09GRCZVR5E
- [The symmetric group S3](https://math2.tamnd.com/node/09H9Y0Q11W.md): example 09H9Y0Q11W
- [Outer automorphism group](https://math2.tamnd.com/node/09HEXDQ0CQ.md): definition 09HEXDQ0CQ
- [Euler's polyhedron formula](https://math2.tamnd.com/node/09HT3P35YP.md): statement 09HT3P35YP
- [Method of moments](https://math2.tamnd.com/node/09J1H07VAH.md): concept 09J1H07VAH
- [Maximal ideal](https://math2.tamnd.com/node/09J6KGT65G.md): definition 09J6KGT65G
- [Forest](https://math2.tamnd.com/node/09JP6G9FTA.md): definition 09JP6G9FTA
- [Sum of the totient over the divisors](https://math2.tamnd.com/node/09JXE6XMCC.md): statement 09JXE6XMCC
- [Groups of order p squared are abelian](https://math2.tamnd.com/node/09JZ9P4899.md): statement 09JZ9P4899
- [Propositional logic](https://math2.tamnd.com/node/09K9MWD2QQ.md): concept 09K9MWD2QQ
- [Metrizable space](https://math2.tamnd.com/node/09KDHB82SG.md): definition 09KDHB82SG
- [Order of a differential equation](https://math2.tamnd.com/node/09M2RDB61G.md): definition 09M2RDB61G
- [Ring homomorphisms from a field are injective](https://math2.tamnd.com/node/09N7DZ7SDB.md): statement 09N7DZ7SDB
- [Finite intersection of open sets is open](https://math2.tamnd.com/node/09NDE9V7X1.md): statement 09NDE9V7X1
- [Injective function](https://math2.tamnd.com/node/09NWETVTNY.md): concept 09NWETVTNY
- [De Morgan's laws](https://math2.tamnd.com/node/09PKJKA6J0.md): statement 09PKJKA6J0
- [Mean of the normal distribution](https://math2.tamnd.com/node/09PR1V6RWR.md): statement 09PR1V6RWR
- [Liouville's formula](https://math2.tamnd.com/node/09PS3S0EF6.md): statement 09PS3S0EF6
- [Prüfer sequence](https://math2.tamnd.com/node/09PV9JSY6E.md): definition 09PV9JSY6E
- [Bounded set](https://math2.tamnd.com/node/09QSZVVM26.md): concept 09QSZVVM26
- [Removable singularity](https://math2.tamnd.com/node/09R8GC85RB.md): concept 09R8GC85RB
- [Cancellation in congruences](https://math2.tamnd.com/node/09RDZZGJZ9.md): statement 09RDZZGJZ9
- [The rationals form an ordered field](https://math2.tamnd.com/node/09RFKMBQ63.md): statement 09RFKMBQ63
- [Partial sum](https://math2.tamnd.com/node/09RJ859N7A.md): concept 09RJ859N7A
- [Student's t-distribution](https://math2.tamnd.com/node/09RTQRK5PC.md): concept 09RTQRK5PC
- [Smooth non-analytic function](https://math2.tamnd.com/node/09RTYYHXKA.md): counterexample 09RTYYHXKA
- [Integral domain](https://math2.tamnd.com/node/09S3PFBEPM.md): definition 09S3PFBEPM
- [Least common multiple](https://math2.tamnd.com/node/09S5SN35DJ.md): definition 09S5SN35DJ
- [Subbasis](https://math2.tamnd.com/node/09S8DV34Y0.md): concept 09S8DV34Y0
- [Independent sigma-algebras](https://math2.tamnd.com/node/09T8ZHST1V.md): definition 09T8ZHST1V
- [Equivalent metrics](https://math2.tamnd.com/node/09V3NASXND.md): concept 09V3NASXND
- [Scalar multiplication of matrices](https://math2.tamnd.com/node/09V6DDACXR.md): concept 09V6DDACXR
- [Module over a ring](https://math2.tamnd.com/node/09VDS2W1E8.md): definition 09VDS2W1E8
- [Singular value](https://math2.tamnd.com/node/09VMDSRXWJ.md): concept 09VMDSRXWJ
- [Orthonormal family](https://math2.tamnd.com/node/09VYHEF2VT.md): definition 09VYHEF2VT
- [Every walk contains a path](https://math2.tamnd.com/node/09VZBHB33V.md): statement 09VZBHB33V
- [Local minimum](https://math2.tamnd.com/node/09WRNGPTMV.md): concept 09WRNGPTMV
- [Affine space](https://math2.tamnd.com/node/09XNAPTJQ7.md): concept 09XNAPTJQ7
- [Permuting rows multiplies the determinant by the sign](https://math2.tamnd.com/node/09YB3WQG38.md): statement 09YB3WQG38
- [Divisibility by 11](https://math2.tamnd.com/node/09YBMJTNBM.md): statement 09YBMJTNBM
- [Intersection of closed sets is closed](https://math2.tamnd.com/node/09YMV8H4X5.md): statement 09YMV8H4X5
- [The number of Sylow subgroups is the index of the normaliser](https://math2.tamnd.com/node/09ZJ6CAK7S.md): statement 09ZJ6CAK7S
- [Product of compact sets is compact](https://math2.tamnd.com/node/09ZKE5XQ74.md): statement 09ZKE5XQ74
- [Linearisation at an equilibrium](https://math2.tamnd.com/node/09ZWYRQ41J.md): concept 09ZWYRQ41J
- [Higher partial derivatives](https://math2.tamnd.com/node/09ZXGX55DM.md): concept 09ZXGX55DM
- [Determinant of a matrix exponential](https://math2.tamnd.com/node/0A1QQN333A.md): statement 0A1QQN333A
- [Isolated point](https://math2.tamnd.com/node/0A1YXSKB48.md): concept 0A1YXSKB48
- [Open sets of a metric space](https://math2.tamnd.com/node/0A26X79RTB.md): statement 0A26X79RTB
- [Convergence in mean](https://math2.tamnd.com/node/0A2HNZC44K.md): concept 0A2HNZC44K
- [Uniform continuity on a set](https://math2.tamnd.com/node/0A3CK6DVVB.md): definition 0A3CK6DVVB
- [The reals are Archimedean](https://math2.tamnd.com/node/0A3GWZ86AY.md): statement 0A3GWZ86AY
- [Pseudometric space](https://math2.tamnd.com/node/0A3HJX3KTF.md): concept 0A3HJX3KTF
- [Boolean function](https://math2.tamnd.com/node/0A58THH5XV.md): definition 0A58THH5XV
- [Differentiable function](https://math2.tamnd.com/node/0A61B60B5R.md): definition 0A61B60B5R
- [Linear equation with constant coefficients](https://math2.tamnd.com/node/0A6PGTTP2A.md): definition 0A6PGTTP2A
- [Derivative gives the directional derivatives](https://math2.tamnd.com/node/0A6PMR20N4.md): statement 0A6PMR20N4
- [System of linear differential equations](https://math2.tamnd.com/node/0A6QZ367GX.md): definition 0A6QZ367GX
- [Metric topology](https://math2.tamnd.com/node/0A6YRNZB2R.md): definition 0A6YRNZB2R
- [Archimedean property](https://math2.tamnd.com/node/0A74SX55QP.md): statement 0A74SX55QP
- [Baire category theorem](https://math2.tamnd.com/node/0A7KGAYHGV.md): statement 0A7KGAYHGV
- [Tower law](https://math2.tamnd.com/node/0A7KJ68XHT.md): statement 0A7KJ68XHT
- [Monotone function](https://math2.tamnd.com/node/0A7KV2WB7A.md): concept 0A7KV2WB7A
- [Complex sine and cosine](https://math2.tamnd.com/node/0A7YDKRNR6.md): definition 0A7YDKRNR6
- [Bounded sequence](https://math2.tamnd.com/node/0A8JNBGT2Z.md): concept 0A8JNBGT2Z
- [x^2 is not uniformly continuous on the real line](https://math2.tamnd.com/node/0A8JXHYTJC.md): counterexample 0A8JXHYTJC
- [Chain rule for a composition with a curve](https://math2.tamnd.com/node/0A8TJ0VHC7.md): statement 0A8TJ0VHC7
- [Complete ordered field](https://math2.tamnd.com/node/0A9E6X0S3E.md): concept 0A9E6X0S3E
- [Chi-squared distribution](https://math2.tamnd.com/node/0A9EB0KQBC.md): concept 0A9EB0KQBC
- [Schröder–Bernstein theorem](https://math2.tamnd.com/node/0A9V6RKQG6.md): statement 0A9V6RKQG6
- [Periodic orbit](https://math2.tamnd.com/node/0AA97XA70K.md): concept 0AA97XA70K
- [Kernel of a ring homomorphism](https://math2.tamnd.com/node/0AAHE7GEW6.md): definition 0AAHE7GEW6
- [Least element](https://math2.tamnd.com/node/0AANKKNHS4.md): concept 0AANKKNHS4
- [Limit cycle](https://math2.tamnd.com/node/0AAQ73WRFB.md): concept 0AAQ73WRFB
- [Almost sure convergence](https://math2.tamnd.com/node/0AAXWGT5TD.md): concept 0AAXWGT5TD
- [Gradient](https://math2.tamnd.com/node/0ABK9V9Y64.md): concept 0ABK9V9Y64
- [Finite-dimensional normed spaces are complete](https://math2.tamnd.com/node/0ABP2F2ZP7.md): statement 0ABP2F2ZP7
- [Dirac measure](https://math2.tamnd.com/node/0ACRGYNHSR.md): concept 0ACRGYNHSR
- [Invariant subspace](https://math2.tamnd.com/node/0ACX5QT8TE.md): concept 0ACX5QT8TE
- [Metrizable spaces are normal](https://math2.tamnd.com/node/0AD0BZ49DW.md): statement 0AD0BZ49DW
- [Formal derivative](https://math2.tamnd.com/node/0AD4VS02H3.md): definition 0AD4VS02H3
- [Jacobi symbol](https://math2.tamnd.com/node/0ADTQBXRMC.md): concept 0ADTQBXRMC
- [Strictly convex function](https://math2.tamnd.com/node/0AEAHMFSCA.md): concept 0AEAHMFSCA
- [Prime factorization](https://math2.tamnd.com/node/0AEKDFCYFA.md): definition 0AEKDFCYFA
- [Pell's equation](https://math2.tamnd.com/node/0AF3DANJ76.md): definition 0AF3DANJ76
- [Convergent series](https://math2.tamnd.com/node/0AF7X08PCR.md): concept 0AF7X08PCR
- [Collinear points](https://math2.tamnd.com/node/0AFNNTRTJE.md): concept 0AFNNTRTJE
- [sin(1/x) has no limit at 0](https://math2.tamnd.com/node/0AFWQZ1PKF.md): counterexample 0AFWQZ1PKF
- [If G modulo its centre is cyclic then G is abelian](https://math2.tamnd.com/node/0AH9Q6YNMT.md): statement 0AH9Q6YNMT
- [Vertex cover](https://math2.tamnd.com/node/0AHKWBQSWV.md): concept 0AHKWBQSWV
- [Generalised eigenspace](https://math2.tamnd.com/node/0AJ035ZTZE.md): concept 0AJ035ZTZE
- [Multiplication by a matrix as a linear map](https://math2.tamnd.com/node/0AJ0KP1B31.md): definition 0AJ0KP1B31
- [Annulus](https://math2.tamnd.com/node/0AJ0YVK743.md): definition 0AJ0YVK743
- [Normal space](https://math2.tamnd.com/node/0AJ646N339.md): definition 0AJ646N339
- [Independent and identically distributed random variables](https://math2.tamnd.com/node/0AJN7WHBJZ.md): definition 0AJN7WHBJZ
- [Step function](https://math2.tamnd.com/node/0AJXYH4DDQ.md): concept 0AJXYH4DDQ
- [Successor function](https://math2.tamnd.com/node/0AK9TGSNXY.md): concept 0AK9TGSNXY
- [Classical definition of probability](https://math2.tamnd.com/node/0AKNPYQ04R.md): definition 0AKNPYQ04R
- [Stokes' theorem](https://math2.tamnd.com/node/0AKRKVHW94.md): statement 0AKRKVHW94
- [Gram matrices are positive semidefinite](https://math2.tamnd.com/node/0AM050KCAW.md): statement 0AM050KCAW
- [Cauchy product of absolutely convergent series](https://math2.tamnd.com/node/0AN28XQ45Z.md): statement 0AN28XQ45Z
- [Circle integral](https://math2.tamnd.com/node/0AP750DHVP.md): concept 0AP750DHVP
- [Multiplicative function](https://math2.tamnd.com/node/0AP80TT660.md): definition 0AP80TT660
- [Nullity](https://math2.tamnd.com/node/0APCFW7639.md): concept 0APCFW7639
- [Second isomorphism theorem for vector spaces](https://math2.tamnd.com/node/0APN65D6Y6.md): statement 0APN65D6Y6
- [Edge set](https://math2.tamnd.com/node/0AQBSVMX9P.md): concept 0AQBSVMX9P
- [The symmetric group is generated by transpositions](https://math2.tamnd.com/node/0AQNX7G443.md): statement 0AQNX7G443
- [Abelianisation](https://math2.tamnd.com/node/0AQZV0905N.md): concept 0AQZV0905N
- [Differentiation under the integral sign](https://math2.tamnd.com/node/0AR06W1E1B.md): statement 0AR06W1E1B
- [Pairwise independent events that are not mutually independent](https://math2.tamnd.com/node/0AR4B3K5QD.md): counterexample 0AR4B3K5QD
- [Unitary matrix](https://math2.tamnd.com/node/0AR68JTMDT.md): concept 0AR68JTMDT
- [Continuous uniform distribution](https://math2.tamnd.com/node/0AREFQ9Z65.md): concept 0AREFQ9Z65
- [Simplicity of the alternating group](https://math2.tamnd.com/node/0ARPRJQWZ2.md): statement 0ARPRJQWZ2
- [Power sum symmetric polynomials](https://math2.tamnd.com/node/0AS1FZT6BM.md): definition 0AS1FZT6BM
- [Consistent estimator](https://math2.tamnd.com/node/0AS800VAYX.md): concept 0AS800VAYX
- [The Möbius function is the inverse of the constant function 1](https://math2.tamnd.com/node/0ASC73QEQT.md): statement 0ASC73QEQT
- [Block matrix](https://math2.tamnd.com/node/0ASDFEQS6P.md): concept 0ASDFEQS6P
- [Commutator subgroup](https://math2.tamnd.com/node/0ASGSTXSBW.md): concept 0ASGSTXSBW
- [Annihilator of a subspace](https://math2.tamnd.com/node/0ASHNYF7AG.md): concept 0ASHNYF7AG
- [Mathematical induction](https://math2.tamnd.com/node/0ASJVEFA9J.md): concept 0ASJVEFA9J
- [Ceiling function](https://math2.tamnd.com/node/0ASM4EPMB2.md): definition 0ASM4EPMB2
- [Normaliser](https://math2.tamnd.com/node/0AT1ZSNM25.md): definition 0AT1ZSNM25
- [Periodic function](https://math2.tamnd.com/node/0ATF6Q0633.md): concept 0ATF6Q0633
- [Back substitution](https://math2.tamnd.com/node/0ATFVE08MW.md): concept 0ATFVE08MW
- [Boundedness theorem](https://math2.tamnd.com/node/0ATVXC3NCC.md): statement 0ATVXC3NCC
- [The nth root of n tends to 1](https://math2.tamnd.com/node/0AVEYZ2T0Q.md): example 0AVEYZ2T0Q
- [R/P is an integral domain iff P is prime](https://math2.tamnd.com/node/0AVXBEC05W.md): statement 0AVXBEC05W
- [Totient of a prime power](https://math2.tamnd.com/node/0AVY8HKBGD.md): statement 0AVY8HKBGD
- [Lipschitz condition](https://math2.tamnd.com/node/0AW1E6YTJY.md): concept 0AW1E6YTJY
- [K[x]/(f) is a field iff f is irreducible](https://math2.tamnd.com/node/0AWMPX35CW.md): statement 0AWMPX35CW
- [Marginal distribution](https://math2.tamnd.com/node/0AXP2Z5N9H.md): definition 0AXP2Z5N9H
- [Completeness axiom](https://math2.tamnd.com/node/0AXWET80YW.md): statement 0AXWET80YW
- [Symmetric bilinear form](https://math2.tamnd.com/node/0AXZNE5XNH.md): concept 0AXZNE5XNH
- [Eigenspace](https://math2.tamnd.com/node/0AYEPBQMAW.md): definition 0AYEPBQMAW
- [Stabiliser](https://math2.tamnd.com/node/0AYMP41THN.md): concept 0AYMP41THN
- [Product metric](https://math2.tamnd.com/node/0AYV6VGKF5.md): concept 0AYV6VGKF5
- [Convergent sequence](https://math2.tamnd.com/node/0AZ3R52593.md): definition 0AZ3R52593
- [Dual map](https://math2.tamnd.com/node/0AZ5ARP287.md): concept 0AZ5ARP287
- [Action on left cosets](https://math2.tamnd.com/node/0AZG100DKH.md): example 0AZG100DKH
- [Equinumerous sets](https://math2.tamnd.com/node/0B0KR7FH2Q.md): concept 0B0KR7FH2Q
- [Faithful action](https://math2.tamnd.com/node/0B0WE1GKTK.md): definition 0B0WE1GKTK
- [Ring](https://math2.tamnd.com/node/0B1261B5ED.md): concept 0B1261B5ED
- [Complement](https://math2.tamnd.com/node/0B1FSS1VZW.md): definition 0B1FSS1VZW
- [Moment-generating function of the normal distribution](https://math2.tamnd.com/node/0B1G4E21G7.md): statement 0B1G4E21G7
- [Cauchy product](https://math2.tamnd.com/node/0B1K9PVWE9.md): definition 0B1K9PVWE9
- [Curl-free vector field on a convex set is conservative](https://math2.tamnd.com/node/0B22P010D6.md): statement 0B22P010D6
- [Open balls are open](https://math2.tamnd.com/node/0B2305GNAZ.md): statement 0B2305GNAZ
- [Gaussian integer](https://math2.tamnd.com/node/0B268GVX95.md): definition 0B268GVX95
- [Level set](https://math2.tamnd.com/node/0B2RZ0778N.md): concept 0B2RZ0778N
- [Möbius transformation](https://math2.tamnd.com/node/0B346DDAM5.md): definition 0B346DDAM5
- [Conjunctive normal form](https://math2.tamnd.com/node/0B3D4C8YF2.md): concept 0B3D4C8YF2
- [Quantile function](https://math2.tamnd.com/node/0B3Q3XDM5X.md): definition 0B3Q3XDM5X
- [Independent sigma-algebras](https://math2.tamnd.com/node/0B5BA4FRWS.md): concept 0B5BA4FRWS
- [Set bounded above](https://math2.tamnd.com/node/0B5H2V4JW8.md): definition 0B5H2V4JW8
- [Eigenvector](https://math2.tamnd.com/node/0B5H4208Z9.md): definition 0B5H4208Z9
- [Edge colouring](https://math2.tamnd.com/node/0B5WCC02AN.md): concept 0B5WCC02AN
- [R[x] is an integral domain iff R is](https://math2.tamnd.com/node/0B5Z1EC66J.md): statement 0B5Z1EC66J
- [Chromatic polynomial](https://math2.tamnd.com/node/0B621RBVS9.md): concept 0B621RBVS9
- [Legendre's three-square theorem](https://math2.tamnd.com/node/0B64Q7XNE3.md): statement 0B64Q7XNE3
- [Lexicographic order](https://math2.tamnd.com/node/0B68KNPGCG.md): concept 0B68KNPGCG
- [Ring isomorphism](https://math2.tamnd.com/node/0B6QC0C0G1.md): definition 0B6QC0C0G1
- [Strict differentiability](https://math2.tamnd.com/node/0B6QWX3086.md): concept 0B6QWX3086
- [Markov property](https://math2.tamnd.com/node/0B77KZCQ51.md): concept 0B77KZCQ51
- [Gamma distribution](https://math2.tamnd.com/node/0B80PAQD3C.md): concept 0B80PAQD3C
- [Cyclic group](https://math2.tamnd.com/node/0B85Q41RP6.md): definition 0B85Q41RP6
- [Order of a group](https://math2.tamnd.com/node/0B85S2V2DR.md): concept 0B85S2V2DR
- [Essential singularity](https://math2.tamnd.com/node/0B8837S5T3.md): concept 0B8837S5T3
- [Universal quantifier](https://math2.tamnd.com/node/0B88PQTJ66.md): definition 0B88PQTJ66
- [Arcsine](https://math2.tamnd.com/node/0B89D93YA1.md): definition 0B89D93YA1
- [Lattice of subgroups](https://math2.tamnd.com/node/0B8BW4YWTF.md): concept 0B8BW4YWTF
- [Continuity via closed sets](https://math2.tamnd.com/node/0B8GPXFSP7.md): statement 0B8GPXFSP7
- [Orthogonal projection](https://math2.tamnd.com/node/0B8KZRT28M.md): concept 0B8KZRT28M
- [Linear isomorphism](https://math2.tamnd.com/node/0B8T10CE7N.md): concept 0B8T10CE7N
- [Cofinite topology](https://math2.tamnd.com/node/0B906ZBTMZ.md): example 0B906ZBTMZ
- [Mathematical proof](https://math2.tamnd.com/node/0B94TKFQJX.md): concept 0B94TKFQJX
- [Cauchy condensation test](https://math2.tamnd.com/node/0B96KMDDD1.md): statement 0B96KMDDD1
- [Rotation matrix](https://math2.tamnd.com/node/0B9QNFGTHM.md): example 0B9QNFGTHM
- [Connected subsets of the real line are intervals](https://math2.tamnd.com/node/0B9ZQFMQNV.md): statement 0B9ZQFMQNV
- [Bridge](https://math2.tamnd.com/node/0BANA3XYPF.md): concept 0BANA3XYPF
- [Mean value theorem](https://math2.tamnd.com/node/0BARQY52N5.md): statement 0BARQY52N5
- [Proof by contrapositive](https://math2.tamnd.com/node/0BB5TPGQ1N.md): concept 0BB5TPGQ1N
- [Finite set](https://math2.tamnd.com/node/0BBTH3YSZA.md): definition 0BBTH3YSZA
- [Matrix determinant lemma](https://math2.tamnd.com/node/0BC3Q6K92H.md): statement 0BC3Q6K92H
- [Power set](https://math2.tamnd.com/node/0BC7KX51Z8.md): definition 0BC7KX51Z8
- [Cycle type](https://math2.tamnd.com/node/0BCHJ30REG.md): definition 0BCHJ30REG
- [Pivot](https://math2.tamnd.com/node/0BCMGDHZGR.md): definition 0BCMGDHZGR
- [Order of the Galois group of a Galois extension equals its degree](https://math2.tamnd.com/node/0BDJBH8B4Z.md): statement 0BDJBH8B4Z
- [Banach space](https://math2.tamnd.com/node/0BDTTM24DP.md): definition 0BDTTM24DP
- [Monic polynomial](https://math2.tamnd.com/node/0BDX4MBKT4.md): definition 0BDX4MBKT4
- [Smooth function](https://math2.tamnd.com/node/0BE3QDH73A.md): definition 0BE3QDH73A
- [Solutions for complex conjugate roots](https://math2.tamnd.com/node/0BEZX8F2XW.md): statement 0BEZX8F2XW
- [Tensor product of vector spaces](https://math2.tamnd.com/node/0BF2DE2SYK.md): concept 0BF2DE2SYK
- [Global existence for globally Lipschitz vector fields](https://math2.tamnd.com/node/0BFD3A6CRH.md): statement 0BFD3A6CRH
- [Weak law of large numbers](https://math2.tamnd.com/node/0BFDHF2H7N.md): statement 0BFDHF2H7N
- [Number of k-element subsets](https://math2.tamnd.com/node/0BFMR2FQ8B.md): statement 0BFMR2FQ8B
- [Path independence of line integrals](https://math2.tamnd.com/node/0BFN21TF5F.md): statement 0BFN21TF5F
- [Infinite limit](https://math2.tamnd.com/node/0BH9JH7HBY.md): concept 0BH9JH7HBY
- [Limit along different paths](https://math2.tamnd.com/node/0BJ6H85DRQ.md): concept 0BJ6H85DRQ
- [A linear map is injective iff its kernel is trivial](https://math2.tamnd.com/node/0BJBYCGMWE.md): statement 0BJBYCGMWE
- [Tangent plane and normal vector of a surface](https://math2.tamnd.com/node/0BJDQ2HBH1.md): definition 0BJDQ2HBH1
- [Spanning set](https://math2.tamnd.com/node/0BJKVF90Z6.md): definition 0BJKVF90Z6
- [Subset](https://math2.tamnd.com/node/0BJR9QR08E.md): concept 0BJR9QR08E
- [Mertens' theorem on Cauchy products](https://math2.tamnd.com/node/0BJSWZJG18.md): statement 0BJSWZJG18
- [Complex conjugate](https://math2.tamnd.com/node/0BJV24C4SY.md): definition 0BJV24C4SY
- [Square root of a positive semidefinite matrix](https://math2.tamnd.com/node/0BJVDCF38Y.md): concept 0BJVDCF38Y
- [Continued fraction](https://math2.tamnd.com/node/0BK6JBK58W.md): definition 0BK6JBK58W
- [Hölder's inequality for sums](https://math2.tamnd.com/node/0BKDAGMGX9.md): statement 0BKDAGMGX9
- [Number of divisors](https://math2.tamnd.com/node/0BKJT6WG96.md): definition 0BKJT6WG96
- [Coordinate vector](https://math2.tamnd.com/node/0BMVBHTWV2.md): definition 0BMVBHTWV2
- [Fermat's little theorem in the form a^p = a](https://math2.tamnd.com/node/0BMXWXDVJZ.md): statement 0BMXWXDVJZ
- [Uniformly Cauchy sequence of functions](https://math2.tamnd.com/node/0BMY28TBSF.md): concept 0BMY28TBSF
- [Line](https://math2.tamnd.com/node/0BMYPJBJH4.md): definition 0BMYPJBJH4
- [Trees are bipartite](https://math2.tamnd.com/node/0BN2A6EBYA.md): statement 0BN2A6EBYA
- [Polynomial of an endomorphism](https://math2.tamnd.com/node/0BNEXSKNRV.md): definition 0BNEXSKNRV
- [Trivial group](https://math2.tamnd.com/node/0BNPC9850Y.md): example 0BNPC9850Y
- [Universal set](https://math2.tamnd.com/node/0BPYRMHJ48.md): definition 0BPYRMHJ48
- [Eigenvalue](https://math2.tamnd.com/node/0BQAA2BK3Q.md): concept 0BQAA2BK3Q
- [Permutation](https://math2.tamnd.com/node/0BQMWCNS8S.md): definition 0BQMWCNS8S
- [Quotient space](https://math2.tamnd.com/node/0BR3FY5Q8M.md): concept 0BR3FY5Q8M
- [Side-side-side congruence](https://math2.tamnd.com/node/0BRBYRK6PA.md): statement 0BRBYRK6PA
- [Goursat's theorem](https://math2.tamnd.com/node/0BRG85WXYC.md): statement 0BRG85WXYC
- [Inverse element](https://math2.tamnd.com/node/0BRHKKP333.md): definition 0BRHKKP333
- [Polar form](https://math2.tamnd.com/node/0BRXFNQ5JV.md): statement 0BRXFNQ5JV
- [Quotient rule](https://math2.tamnd.com/node/0BS0NJPKVE.md): statement 0BS0NJPKVE
- [Multiplication rule](https://math2.tamnd.com/node/0BS3RMX8HM.md): statement 0BS3RMX8HM
- [Additive group of integers](https://math2.tamnd.com/node/0BSCJA5G53.md): example 0BSCJA5G53
- [Set](https://math2.tamnd.com/node/0BSJ7RQ9BW.md): concept 0BSJ7RQ9BW
- [Dense set](https://math2.tamnd.com/node/0BSJSX7ME2.md): concept 0BSJSX7ME2
- [Z[√-5] is not a UFD](https://math2.tamnd.com/node/0BSJYCG7NM.md): counterexample 0BSJYCG7NM
- [Riemann's theorem on removable singularities](https://math2.tamnd.com/node/0BSQDZWF34.md): statement 0BSQDZWF34
- [Polar coordinates](https://math2.tamnd.com/node/0BSW4T7THS.md): concept 0BSW4T7THS
- [Squaring the circle](https://math2.tamnd.com/node/0BSXRER81B.md): problem 0BSXRER81B
- [Minor](https://math2.tamnd.com/node/0BT9BYWVRY.md): concept 0BT9BYWVRY
- [Supremum](https://math2.tamnd.com/node/0BTMQ04556.md): definition 0BTMQ04556
- [Borel sigma-algebra](https://math2.tamnd.com/node/0BVHZXWKCD.md): definition 0BVHZXWKCD
- [Trajectory](https://math2.tamnd.com/node/0BVT6VSB3S.md): definition 0BVT6VSB3S
- [Pythagorean theorem in an inner product space](https://math2.tamnd.com/node/0BVTDM872V.md): statement 0BVTDM872V
- [Matrix addition](https://math2.tamnd.com/node/0BVXADM10P.md): definition 0BVXADM10P
- [Antiderivative](https://math2.tamnd.com/node/0BW9N5CDSB.md): definition 0BW9N5CDSB
- [Characteristic polynomial of a linear recurrence](https://math2.tamnd.com/node/0BWDJGQYST.md): definition 0BWDJGQYST
- [Bias of an estimator](https://math2.tamnd.com/node/0BWGTN92FG.md): concept 0BWGTN92FG
- [Uniform limits commute with integration](https://math2.tamnd.com/node/0BWGW68B66.md): statement 0BWGW68B66
- [Box topology](https://math2.tamnd.com/node/0BWJTRE3X3.md): concept 0BWJTRE3X3
- [Power series converge uniformly on smaller discs inside the radius of convergence](https://math2.tamnd.com/node/0BWXV7GADW.md): statement 0BWXV7GADW
- [Ring of integers](https://math2.tamnd.com/node/0BXA4Q0JHW.md): statement 0BXA4Q0JHW
- [Implicit function theorem](https://math2.tamnd.com/node/0BXBADR94C.md): statement 0BXBADR94C
- [Having a gradient](https://math2.tamnd.com/node/0BXE0S1XER.md): definition 0BXE0S1XER
- [Minimal element](https://math2.tamnd.com/node/0BXEZV83GY.md): concept 0BXEZV83GY
- [Cantor set](https://math2.tamnd.com/node/0BXNDZFQME.md): example 0BXNDZFQME
- [Complete metric space](https://math2.tamnd.com/node/0BY52JW6NK.md): concept 0BY52JW6NK
- [Slopes of a convex function are increasing](https://math2.tamnd.com/node/0BY5FTHEEE.md): statement 0BY5FTHEEE
- [Generalized pigeonhole principle](https://math2.tamnd.com/node/0BY7GN3449.md): statement 0BY7GN3449
- [Least squares](https://math2.tamnd.com/node/0BYTQ4X1HC.md): concept 0BYTQ4X1HC
- [Antiderivative](https://math2.tamnd.com/node/0BYWRWFT4F.md): concept 0BYWRWFT4F
- [Clique](https://math2.tamnd.com/node/0BYX1D7C2E.md): concept 0BYX1D7C2E
- [Zero derivative on a connected open set implies constant](https://math2.tamnd.com/node/0BYZTFM0DJ.md): statement 0BYZTFM0DJ
- [Characteristic polynomial of a linear map](https://math2.tamnd.com/node/0BZ81AJNWF.md): definition 0BZ81AJNWF
- [Addition rule for two events](https://math2.tamnd.com/node/0BZQ6ZX584.md): statement 0BZQ6ZX584
- [Order of the dihedral group](https://math2.tamnd.com/node/0BZVNWJ352.md): statement 0BZVNWJ352
- [Recurrence for Stirling numbers of the second kind](https://math2.tamnd.com/node/0C019SDTER.md): statement 0C019SDTER
- [Laplace expansion along a row](https://math2.tamnd.com/node/0C08BQ3M05.md): statement 0C08BQ3M05
- [Random walk](https://math2.tamnd.com/node/0C08YMV95H.md): definition 0C08YMV95H
- [Simple group](https://math2.tamnd.com/node/0C093D1T4M.md): definition 0C093D1T4M
- [Separately continuous but not continuous function](https://math2.tamnd.com/node/0C0KPAKK9Y.md): counterexample 0C0KPAKK9Y
- [Density of the normal distribution](https://math2.tamnd.com/node/0C12P4W4RH.md): concept 0C12P4W4RH
- [Linear maps on finite-dimensional spaces are continuous](https://math2.tamnd.com/node/0C13E623WD.md): statement 0C13E623WD
- [pi](https://math2.tamnd.com/node/0C1VYVNV2N.md): object 0C1VYVNV2N
- [Finite extension](https://math2.tamnd.com/node/0C1Y04SZ1W.md): concept 0C1Y04SZ1W
- [Family of sets](https://math2.tamnd.com/node/0C21Q694AG.md): concept 0C21Q694AG
- [Random variable with a density](https://math2.tamnd.com/node/0C21RZG5V2.md): definition 0C21RZG5V2
- [Rectifiable curve](https://math2.tamnd.com/node/0C28RT5BG7.md): definition 0C28RT5BG7
- [Hamiltonian system](https://math2.tamnd.com/node/0C2AKPDCD9.md): definition 0C2AKPDCD9
- [Determinant is the product of the eigenvalues](https://math2.tamnd.com/node/0C2F40MEBC.md): statement 0C2F40MEBC
- [Dual map](https://math2.tamnd.com/node/0C2TQMW5A1.md): definition 0C2TQMW5A1
- [Associativity of addition](https://math2.tamnd.com/node/0C32J27WQC.md): statement 0C32J27WQC
- [Interior](https://math2.tamnd.com/node/0C38S0KMJK.md): definition 0C38S0KMJK
- [Determinant](https://math2.tamnd.com/node/0C3CFHDN6R.md): definition 0C3CFHDN6R
- [Euler product formula](https://math2.tamnd.com/node/0C3PN4075R.md): statement 0C3PN4075R
- [Monic polynomial](https://math2.tamnd.com/node/0C3QBCR46M.md): concept 0C3QBCR46M
- [Integers](https://math2.tamnd.com/node/0C4JS9RF5M.md): object 0C4JS9RF5M
- [Eigenvalues of a self-adjoint operator are real](https://math2.tamnd.com/node/0C4NT4GT3X.md): statement 0C4NT4GT3X
- [Bernstein polynomial](https://math2.tamnd.com/node/0C4VWBFY73.md): definition 0C4VWBFY73
- [Inclusion–exclusion for two sets](https://math2.tamnd.com/node/0C4YEQ9H5E.md): statement 0C4YEQ9H5E
- [Recurrence for Stirling numbers of the first kind](https://math2.tamnd.com/node/0C5BFX94J9.md): statement 0C5BFX94J9
- [Chromatic number](https://math2.tamnd.com/node/0C5M7AM6FH.md): definition 0C5M7AM6FH
- [Hyperbolic equilibrium point](https://math2.tamnd.com/node/0C5PREPZWG.md): definition 0C5PREPZWG
- [Number of subsets of a finite set](https://math2.tamnd.com/node/0C5RDJE73X.md): statement 0C5RDJE73X
- [Codomain](https://math2.tamnd.com/node/0C5SR4FC74.md): concept 0C5SR4FC74
- [Polynomial functions are continuous](https://math2.tamnd.com/node/0C5TP6RYSV.md): statement 0C5TP6RYSV
- [Multigraph](https://math2.tamnd.com/node/0C62PZXX02.md): concept 0C62PZXX02
- [Uncorrelated random variables that are not independent](https://math2.tamnd.com/node/0C6J8BQXVR.md): counterexample 0C6J8BQXVR
- [Tangent line to a circle](https://math2.tamnd.com/node/0C6THFRMZR.md): definition 0C6THFRMZR
- [Cartesian product](https://math2.tamnd.com/node/0C81YBVTWH.md): definition 0C81YBVTWH
- [Normal operator](https://math2.tamnd.com/node/0C8BR8XRTR.md): concept 0C8BR8XRTR
- [Möbius function](https://math2.tamnd.com/node/0C8QENHYZW.md): definition 0C8QENHYZW
- [Greatest common divisor](https://math2.tamnd.com/node/0C8WYW4KYQ.md): concept 0C8WYW4KYQ
- [Euclidean distance](https://math2.tamnd.com/node/0C95TG69SG.md): definition 0C95TG69SG
- [Contractive sequence](https://math2.tamnd.com/node/0C9CMCN9BQ.md): concept 0C9CMCN9BQ
- [Rooted tree](https://math2.tamnd.com/node/0C9J6W2A8A.md): definition 0C9J6W2A8A
- [Altitude of a triangle](https://math2.tamnd.com/node/0C9MZY2RNA.md): concept 0C9MZY2RNA
- [Reverse triangle inequality](https://math2.tamnd.com/node/0C9S13WKMF.md): statement 0C9S13WKMF
- [The kernel is a normal subgroup](https://math2.tamnd.com/node/0C9ZKYSPYR.md): statement 0C9ZKYSPYR
- [Sequential compactness](https://math2.tamnd.com/node/0CARW59WD2.md): definition 0CARW59WD2
- [Normal subgroup](https://math2.tamnd.com/node/0CB0DX5BVV.md): concept 0CB0DX5BVV
- [Complex numbers as pairs of reals](https://math2.tamnd.com/node/0CBAB9QQ9X.md): concept 0CBAB9QQ9X
- [Kurtosis](https://math2.tamnd.com/node/0CBQVVNBMP.md): definition 0CBQVVNBMP
- [Normal extension](https://math2.tamnd.com/node/0CBRFQ7DVZ.md): definition 0CBRFQ7DVZ
- [Logistic equation](https://math2.tamnd.com/node/0CC4VC7KR3.md): example 0CC4VC7KR3
- [Harmonic conjugate](https://math2.tamnd.com/node/0CC56MJ4FG.md): definition 0CC56MJ4FG
- [Discriminant of a polynomial](https://math2.tamnd.com/node/0CCPN09EMJ.md): definition 0CCPN09EMJ
- [Matrix–vector product](https://math2.tamnd.com/node/0CCQV3884B.md): definition 0CCQV3884B
- [Two-sided ideal](https://math2.tamnd.com/node/0CD87SFM2N.md): concept 0CD87SFM2N
- [A finite field of order q is the splitting field of x^q - x](https://math2.tamnd.com/node/0CDAHW576M.md): statement 0CDAHW576M
- [Riemann sphere](https://math2.tamnd.com/node/0CDMXGPSES.md): definition 0CDMXGPSES
- [Degrees along an Eulerian trail](https://math2.tamnd.com/node/0CDVDGWGEG.md): statement 0CDVDGWGEG
- [Martingale](https://math2.tamnd.com/node/0CDVK9RQBH.md): definition 0CDVK9RQBH
- [Universal property of the free group](https://math2.tamnd.com/node/0CDWKDYWN6.md): statement 0CDWKDYWN6
- [Work done by a force field](https://math2.tamnd.com/node/0CDZ684TCD.md): concept 0CDZ684TCD
- [Arctangent](https://math2.tamnd.com/node/0CE768XNVP.md): concept 0CE768XNVP
- [Conic section](https://math2.tamnd.com/node/0CEAT3418X.md): definition 0CEAT3418X
- [Number of permutations](https://math2.tamnd.com/node/0CEBCS0E0E.md): statement 0CEBCS0E0E
- [Modular multiplicative inverse](https://math2.tamnd.com/node/0CEC45PRJN.md): concept 0CEC45PRJN
- [Zero of a holomorphic function](https://math2.tamnd.com/node/0CEGEASY49.md): definition 0CEGEASY49
- [Polynomials in several variables](https://math2.tamnd.com/node/0CES91M79A.md): definition 0CES91M79A
- [Classification of complex quadratic forms](https://math2.tamnd.com/node/0CEX9BRXRZ.md): statement 0CEX9BRXRZ
- [exp(2 pi i/n) is a primitive root of unity](https://math2.tamnd.com/node/0CF0XGYHNJ.md): statement 0CF0XGYHNJ
- [Frobenius automorphism of a finite field](https://math2.tamnd.com/node/0CGCPX8Z70.md): concept 0CGCPX8Z70
- [Special orthogonal group](https://math2.tamnd.com/node/0CGYNDCWRR.md): object 0CGYNDCWRR
- [Disc of convergence](https://math2.tamnd.com/node/0CH4VG15RE.md): concept 0CH4VG15RE
- [Irreducible elements need not be prime](https://math2.tamnd.com/node/0CHE3J7QMX.md): counterexample 0CHE3J7QMX
- [Division algorithm](https://math2.tamnd.com/node/0CHQKZFK5S.md): statement 0CHQKZFK5S
- [Subbasis](https://math2.tamnd.com/node/0CHR35YNJQ.md): definition 0CHR35YNJQ
- [Euclidean domain](https://math2.tamnd.com/node/0CHS7RBD39.md): concept 0CHS7RBD39
- [Finite subgroups of the multiplicative group of a field are cyclic](https://math2.tamnd.com/node/0CKGBEJX6B.md): statement 0CKGBEJX6B
- [Cyclic quadrilateral](https://math2.tamnd.com/node/0CMC2KAXFS.md): definition 0CMC2KAXFS
- [Principal ideal domain](https://math2.tamnd.com/node/0CMD0BCH8T.md): concept 0CMD0BCH8T
- [Equivalent matrices](https://math2.tamnd.com/node/0CMHSGTWXB.md): concept 0CMHSGTWXB
- [Abel's test](https://math2.tamnd.com/node/0CN97GADE5.md): statement 0CN97GADE5
- [Topological space](https://math2.tamnd.com/node/0CNQXVBHCX.md): definition 0CNQXVBHCX
- [First-countable space](https://math2.tamnd.com/node/0CP5NDD21C.md): concept 0CP5NDD21C
- [Square matrix](https://math2.tamnd.com/node/0CP7YY6NAG.md): definition 0CP7YY6NAG
- [Impossibility of doubling the cube](https://math2.tamnd.com/node/0CQ357M0QS.md): statement 0CQ357M0QS
- [Absolute value is continuous but not differentiable at 0](https://math2.tamnd.com/node/0CQ9ZM383N.md): counterexample 0CQ9ZM383N
- [Topological space](https://math2.tamnd.com/node/0CQHFPGJ45.md): concept 0CQHFPGJ45
- [Bernoulli process](https://math2.tamnd.com/node/0CQQG3DEW9.md): concept 0CQQG3DEW9
- [Parabola](https://math2.tamnd.com/node/0CQW20XGFW.md): definition 0CQW20XGFW
- [Linear independence](https://math2.tamnd.com/node/0CRMGPAH83.md): concept 0CRMGPAH83
- [Uniform convergence of derivatives allows term-by-term differentiation](https://math2.tamnd.com/node/0CS42KEM5C.md): statement 0CS42KEM5C
- [Exponential growth beats polynomial growth](https://math2.tamnd.com/node/0CT6CBGTYY.md): statement 0CT6CBGTYY
- [Complex plane](https://math2.tamnd.com/node/0CTE7A31XQ.md): concept 0CTE7A31XQ
- [Box topology](https://math2.tamnd.com/node/0CTM1EHG9P.md): definition 0CTM1EHG9P
- [Triangle](https://math2.tamnd.com/node/0CTNFGYMM2.md): concept 0CTNFGYMM2
- [Real sequence](https://math2.tamnd.com/node/0CTV783HYT.md): definition 0CTV783HYT
- [Partition function](https://math2.tamnd.com/node/0CV69WDRXW.md): concept 0CV69WDRXW
- [Convergent series](https://math2.tamnd.com/node/0CVVKY7GY2.md): definition 0CVVKY7GY2
- [Annulus](https://math2.tamnd.com/node/0CW52NV5TC.md): concept 0CW52NV5TC
- [Arccosine](https://math2.tamnd.com/node/0CWDVN132S.md): concept 0CWDVN132S
- [Improper multiple integral](https://math2.tamnd.com/node/0CWK2NFSEW.md): concept 0CWK2NFSEW
- [Automorphism group](https://math2.tamnd.com/node/0CWXCWP0Y0.md): definition 0CWXCWP0Y0
- [Integrating factor for a non-exact equation](https://math2.tamnd.com/node/0CX86CR8JV.md): concept 0CX86CR8JV
- [Harmonic series](https://math2.tamnd.com/node/0CXTMAY5PX.md): example 0CXTMAY5PX
- [First isomorphism theorem](https://math2.tamnd.com/node/0CXXJNG5B6.md): statement 0CXXJNG5B6
- [Linearity of the derivative](https://math2.tamnd.com/node/0CXZ217QDP.md): statement 0CXZ217QDP
- [Transitive relation](https://math2.tamnd.com/node/0CYRECG2YQ.md): definition 0CYRECG2YQ
- [Derivative of a continuous linear map](https://math2.tamnd.com/node/0CYYAF8AD2.md): statement 0CYYAF8AD2
- [Singular solution](https://math2.tamnd.com/node/0CZA0NCFWP.md): definition 0CZA0NCFWP
- [Contractive sequence](https://math2.tamnd.com/node/0CZRT49QNG.md): definition 0CZRT49QNG
- [Closed curve](https://math2.tamnd.com/node/0D032NSZKY.md): definition 0D032NSZKY
- [Martingale](https://math2.tamnd.com/node/0D0GBNDKDH.md): concept 0D0GBNDKDH
- [Fibonacci recurrence](https://math2.tamnd.com/node/0D0SMQYEJA.md): statement 0D0SMQYEJA
- [Dense set](https://math2.tamnd.com/node/0D198GPMTJ.md): definition 0D198GPMTJ
- [Tautology](https://math2.tamnd.com/node/0D1AHYPM8N.md): definition 0D1AHYPM8N
- [Complex differentiability](https://math2.tamnd.com/node/0D1C9Y1EXA.md): definition 0D1C9Y1EXA
- [Geometric sequence](https://math2.tamnd.com/node/0D1NY1ARDR.md): concept 0D1NY1ARDR
- [Correspondence theorem for rings](https://math2.tamnd.com/node/0D1R4SVMBQ.md): statement 0D1R4SVMBQ
- [Conjugate elements](https://math2.tamnd.com/node/0D1VWQGZZ7.md): definition 0D1VWQGZZ7
- [Restriction of a function](https://math2.tamnd.com/node/0D2MXBV21T.md): definition 0D2MXBV21T
- [A set is closed iff it equals its closure](https://math2.tamnd.com/node/0D2W7FS52T.md): statement 0D2W7FS52T
- [Harmonic number](https://math2.tamnd.com/node/0D2Y6G4C4S.md): definition 0D2Y6G4C4S
- [Integration by substitution](https://math2.tamnd.com/node/0D37QFPEZE.md): statement 0D37QFPEZE
- [Consecutive Fibonacci numbers are coprime](https://math2.tamnd.com/node/0D3PFXEAKF.md): statement 0D3PFXEAKF
- [Order isomorphism](https://math2.tamnd.com/node/0D3WA36K2Q.md): concept 0D3WA36K2Q
- [Rouché's theorem](https://math2.tamnd.com/node/0D430QR02W.md): statement 0D430QR02W
- [Convergents of a continued fraction](https://math2.tamnd.com/node/0D474C7ES1.md): concept 0D474C7ES1
- [Baire space](https://math2.tamnd.com/node/0D5KR33H7X.md): definition 0D5KR33H7X
- [Subspace topology](https://math2.tamnd.com/node/0D5YHQGDJY.md): definition 0D5YHQGDJY
- [A continuous injective function on an interval is strictly monotone](https://math2.tamnd.com/node/0D6FCW80A0.md): statement 0D6FCW80A0
- [Probability mass function](https://math2.tamnd.com/node/0D6J4B671D.md): concept 0D6J4B671D
- [Polynomial ring over a UFD is a UFD](https://math2.tamnd.com/node/0D6KH8Y59N.md): statement 0D6KH8Y59N
- [Power series in the complex plane](https://math2.tamnd.com/node/0D6KXBHKC0.md): concept 0D6KXBHKC0
- [Unitary matrix](https://math2.tamnd.com/node/0D6RMJ10M9.md): definition 0D6RMJ10M9
- [Conjugate subgroup](https://math2.tamnd.com/node/0D6WHTTH7T.md): definition 0D6WHTTH7T
- [Zero is not a successor](https://math2.tamnd.com/node/0D752BG155.md): statement 0D752BG155
- [Binary tree](https://math2.tamnd.com/node/0D7EZ10V2R.md): concept 0D7EZ10V2R
- [The exponential function is convex](https://math2.tamnd.com/node/0D7GRZ2ADX.md): statement 0D7GRZ2ADX
- [Bridge](https://math2.tamnd.com/node/0D81KWH9ZM.md): definition 0D81KWH9ZM
- [Variance of a sum](https://math2.tamnd.com/node/0D86GJEZ39.md): statement 0D86GJEZ39
- [Construction of the reals from Cauchy sequences](https://math2.tamnd.com/node/0D8HE681SC.md): concept 0D8HE681SC
- [Limits are unique in Hausdorff spaces](https://math2.tamnd.com/node/0D8T984XEE.md): statement 0D8T984XEE
- [Connected space](https://math2.tamnd.com/node/0D939CMME7.md): definition 0D939CMME7
- [Tangent vector of a curve](https://math2.tamnd.com/node/0D98771SM0.md): definition 0D98771SM0
- [Uniqueness of inverses](https://math2.tamnd.com/node/0D9NDGE9KP.md): statement 0D9NDGE9KP
- [Straightedge and compass construction](https://math2.tamnd.com/node/0DAECMZVRT.md): concept 0DAECMZVRT
- [Euler's totient function](https://math2.tamnd.com/node/0DAMHCT17R.md): concept 0DAMHCT17R
- [Probability axioms](https://math2.tamnd.com/node/0DAX8PMAJ4.md): concept 0DAX8PMAJ4
- [Mean of the geometric distribution](https://math2.tamnd.com/node/0DB0R4N5XX.md): statement 0DB0R4N5XX
- [Matrix of the adjoint is the conjugate transpose](https://math2.tamnd.com/node/0DBTE6HVBD.md): statement 0DBTE6HVBD
- [Torsion subgroup](https://math2.tamnd.com/node/0DBWXQRXQJ.md): concept 0DBWXQRXQJ
- [Open ball](https://math2.tamnd.com/node/0DC3MPR7C9.md): definition 0DC3MPR7C9
- [Expected value](https://math2.tamnd.com/node/0DCT0QZVQ1.md): definition 0DCT0QZVQ1
- [Two-sided ideal](https://math2.tamnd.com/node/0DDA6CY7GJ.md): definition 0DDA6CY7GJ
- [Right inverse](https://math2.tamnd.com/node/0DDPZAG5ER.md): definition 0DDPZAG5ER
- [Elementary row operations](https://math2.tamnd.com/node/0DE5R5EB4D.md): definition 0DE5R5EB4D
- [Harmonic function](https://math2.tamnd.com/node/0DEB0JMN0W.md): concept 0DEB0JMN0W
- [Conjugacy class](https://math2.tamnd.com/node/0DEBXAJ5SA.md): concept 0DEBXAJ5SA
- [Gaussian integral](https://math2.tamnd.com/node/0DFT1GP1JP.md): example 0DFT1GP1JP
- [Strong law of large numbers](https://math2.tamnd.com/node/0DFWHQ839T.md): statement 0DFWHQ839T
- [Finite extensions are algebraic](https://math2.tamnd.com/node/0DFYTNZ6AM.md): statement 0DFYTNZ6AM
- [Commutativity of addition](https://math2.tamnd.com/node/0DFZNHJ0S2.md): statement 0DFZNHJ0S2
- [Holomorphic function](https://math2.tamnd.com/node/0DG0Z72CQD.md): definition 0DG0Z72CQD
- [Cauchy's integral formula](https://math2.tamnd.com/node/0DG10TVYPN.md): statement 0DG10TVYPN
- [Alternating bilinear form](https://math2.tamnd.com/node/0DGDS5XY84.md): concept 0DGDS5XY84
- [Spectral theorem for Hermitian matrices](https://math2.tamnd.com/node/0DGZBQWZWN.md): statement 0DGZBQWZWN
- [Order of a permutation is the lcm of its cycle lengths](https://math2.tamnd.com/node/0DH7HMXP9F.md): statement 0DH7HMXP9F
- [Covariance](https://math2.tamnd.com/node/0DJ8W3HP2D.md): definition 0DJ8W3HP2D
- [Upper central series](https://math2.tamnd.com/node/0DJ9Y9V610.md): definition 0DJ9Y9V610
- [Estimation lemma](https://math2.tamnd.com/node/0DJMGPN8K9.md): statement 0DJMGPN8K9
- [Quadratic residue](https://math2.tamnd.com/node/0DKCECSYCF.md): concept 0DKCECSYCF
- [Cosets have the same cardinality as the subgroup](https://math2.tamnd.com/node/0DKMBWTD7P.md): statement 0DKMBWTD7P
- [Random variable](https://math2.tamnd.com/node/0DKRR22AEQ.md): definition 0DKRR22AEQ
- [Basel problem](https://math2.tamnd.com/node/0DMY75D6FV.md): statement 0DMY75D6FV
- [Correlation coefficient](https://math2.tamnd.com/node/0DNN9B6Q2F.md): concept 0DNN9B6Q2F
- [Linear equation](https://math2.tamnd.com/node/0DNNT7DBFM.md): definition 0DNNT7DBFM
- [Degree of a polynomial](https://math2.tamnd.com/node/0DPF6WRZW6.md): definition 0DPF6WRZW6
- [Surface integral](https://math2.tamnd.com/node/0DQ8ERJZNS.md): definition 0DQ8ERJZNS
- [Slope field](https://math2.tamnd.com/node/0DQEHHVRBX.md): concept 0DQEHHVRBX
- [K[x] is a Euclidean domain](https://math2.tamnd.com/node/0DQG39K2NH.md): statement 0DQG39K2NH
- [Universal property of the polynomial ring](https://math2.tamnd.com/node/0DQMDSVSXJ.md): statement 0DQMDSVSXJ
- [Well-founded relation](https://math2.tamnd.com/node/0DQPNJNPYK.md): concept 0DQPNJNPYK
- [Bézout's identity](https://math2.tamnd.com/node/0DQVHMYAGA.md): statement 0DQVHMYAGA
- [Joint probability distribution](https://math2.tamnd.com/node/0DR81R1F2B.md): concept 0DR81R1F2B
- [Inequality of arithmetic and geometric means](https://math2.tamnd.com/node/0DRF0F3XC9.md): statement 0DRF0F3XC9
- [Closure](https://math2.tamnd.com/node/0DRKT1QJR7.md): definition 0DRKT1QJR7
- [Galois extension](https://math2.tamnd.com/node/0DS2F7GK05.md): definition 0DS2F7GK05
- [Fundamental system of solutions](https://math2.tamnd.com/node/0DS2QFYJHE.md): concept 0DS2QFYJHE
- [The rationals are countable](https://math2.tamnd.com/node/0DS9KWFDK2.md): statement 0DS9KWFDK2
- [Definition by recursion](https://math2.tamnd.com/node/0DSHAVH45F.md): concept 0DSHAVH45F
- [Set difference](https://math2.tamnd.com/node/0DT0TCV75N.md): definition 0DT0TCV75N
- [Transvection](https://math2.tamnd.com/node/0DTCBCKEJP.md): concept 0DTCBCKEJP
- [Index of a subgroup](https://math2.tamnd.com/node/0DTXBQPA2S.md): concept 0DTXBQPA2S
- [Linear span](https://math2.tamnd.com/node/0DTY8B9HMS.md): concept 0DTY8B9HMS
- [Probability distribution](https://math2.tamnd.com/node/0DVDCMA2DK.md): concept 0DVDCMA2DK
- [Conjugate elements](https://math2.tamnd.com/node/0DXD4QSZ8A.md): concept 0DXD4QSZ8A
- [Conjugate transpose](https://math2.tamnd.com/node/0DXKE10PHF.md): concept 0DXKE10PHF
- [Stone's representation theorem for finite Boolean algebras](https://math2.tamnd.com/node/0DYS9C3HH5.md): statement 0DYS9C3HH5
- [Centre of a group](https://math2.tamnd.com/node/0DZ93493VN.md): concept 0DZ93493VN
- [Free module](https://math2.tamnd.com/node/0DZ9DKAKZR.md): concept 0DZ9DKAKZR
- [Irrationality of the square root of 2](https://math2.tamnd.com/node/0DZGSDHD92.md): statement 0DZGSDHD92
- [Nilpotent group](https://math2.tamnd.com/node/0DZPD2MG33.md): definition 0DZPD2MG33
- [Length of a vector](https://math2.tamnd.com/node/0DZS7ZG0QY.md): definition 0DZS7ZG0QY
- [Irrational number](https://math2.tamnd.com/node/0DZXAXEJQ9.md): concept 0DZXAXEJQ9
- [Number of nth roots of unity](https://math2.tamnd.com/node/0E021GF2B8.md): statement 0E021GF2B8
- [Jordan–Hölder theorem](https://math2.tamnd.com/node/0E088ZP533.md): statement 0E088ZP533
- [Laplacian](https://math2.tamnd.com/node/0E0B6RA8K0.md): concept 0E0B6RA8K0
- [Primitive root theorem](https://math2.tamnd.com/node/0E0DNQK22P.md): statement 0E0DNQK22P
- [Square root function](https://math2.tamnd.com/node/0E10E0NW9V.md): concept 0E10E0NW9V
- [Euclidean metric](https://math2.tamnd.com/node/0E1MMCAY5B.md): example 0E1MMCAY5B
- [Riccati equation](https://math2.tamnd.com/node/0E1ZZBHZ3Z.md): definition 0E1ZZBHZ3Z
- [Complete residue system](https://math2.tamnd.com/node/0E22JKS4W4.md): definition 0E22JKS4W4
- [Orbit-stabiliser theorem](https://math2.tamnd.com/node/0E2HSCKZ6C.md): statement 0E2HSCKZ6C
- [Dimension of the orthogonal complement](https://math2.tamnd.com/node/0E2NYPVSP9.md): statement 0E2NYPVSP9
- [Differential form](https://math2.tamnd.com/node/0E3G0GRPZ6.md): concept 0E3G0GRPZ6
- [Polynomial of an endomorphism](https://math2.tamnd.com/node/0E3MV6E6MG.md): concept 0E3MV6E6MG
- [Algebra of endomorphisms](https://math2.tamnd.com/node/0E3X2G1ZQB.md): object 0E3X2G1ZQB
- [Limit point](https://math2.tamnd.com/node/0E3X7TXS67.md): concept 0E3X7TXS67
- [Base rate fallacy](https://math2.tamnd.com/node/0E44JMG6NV.md): example 0E44JMG6NV
- [Blow-up in finite time](https://math2.tamnd.com/node/0E4GXX7SJP.md): example 0E4GXX7SJP
- [Cayley table](https://math2.tamnd.com/node/0E4WH0R7HG.md): definition 0E4WH0R7HG
- [Directrix](https://math2.tamnd.com/node/0E5CXYNEPX.md): concept 0E5CXYNEPX
- [Law of total probability](https://math2.tamnd.com/node/0E5G5FMT2R.md): statement 0E5G5FMT2R
- [Convergents converge to the number](https://math2.tamnd.com/node/0E70M92FE3.md): statement 0E70M92FE3
- [Fermat's Last Theorem for n = 4](https://math2.tamnd.com/node/0E741HSMJ1.md): statement 0E741HSMJ1
- [Riemann rearrangement theorem](https://math2.tamnd.com/node/0E7VBYGFF7.md): statement 0E7VBYGFF7
- [Casting out nines](https://math2.tamnd.com/node/0E7W0EM6J6.md): concept 0E7W0EM6J6
- [Similar matrices](https://math2.tamnd.com/node/0E89HQATM8.md): concept 0E89HQATM8
- [Exponential of a sum](https://math2.tamnd.com/node/0E8HNVBCSP.md): statement 0E8HNVBCSP
- [Independence of two random variables](https://math2.tamnd.com/node/0E9502NT92.md): definition 0E9502NT92
- [Exponential distribution](https://math2.tamnd.com/node/0E99F51GVB.md): concept 0E99F51GVB
- [Range of a function](https://math2.tamnd.com/node/0E9GTAV478.md): definition 0E9GTAV478
- [Harmonic series diverges](https://math2.tamnd.com/node/0E9M24KS9J.md): statement 0E9M24KS9J
- [Ceiling function](https://math2.tamnd.com/node/0EA4XFB0EY.md): concept 0EA4XFB0EY
- [Division ring](https://math2.tamnd.com/node/0EA5BTA0CX.md): concept 0EA5BTA0CX
- [Chain rule in several variables](https://math2.tamnd.com/node/0EAAS2EHDR.md): statement 0EAAS2EHDR
- [Degree of a field extension](https://math2.tamnd.com/node/0EAJ2XXADC.md): concept 0EAJ2XXADC
- [Trigonometric substitution](https://math2.tamnd.com/node/0EAR3KNRTN.md): concept 0EAR3KNRTN
- [A composite number has a prime factor at most its square root](https://math2.tamnd.com/node/0EB6BBMQMC.md): statement 0EB6BBMQMC
- [Invertible matrix](https://math2.tamnd.com/node/0EB9N0S33C.md): concept 0EB9N0S33C
- [Cardinal number](https://math2.tamnd.com/node/0EBACQFAZW.md): concept 0EBACQFAZW
- [Probability-generating function](https://math2.tamnd.com/node/0EBCWQ8BVE.md): concept 0EBCWQ8BVE
- [A finite integral domain is a field](https://math2.tamnd.com/node/0EBE031B8E.md): statement 0EBE031B8E
- [The multiplicative group of a finite field is cyclic](https://math2.tamnd.com/node/0EBZ3G455G.md): statement 0EBZ3G455G
- [Exact equation](https://math2.tamnd.com/node/0EC7AGJ0PX.md): definition 0EC7AGJ0PX
- [Empty set](https://math2.tamnd.com/node/0ECVK59Y03.md): object 0ECVK59Y03
- [Interval of convergence](https://math2.tamnd.com/node/0ED4HWNDGW.md): concept 0ED4HWNDGW
- [Counterexample](https://math2.tamnd.com/node/0ED72CTBNP.md): concept 0ED72CTBNP
- [Existence of the infimum](https://math2.tamnd.com/node/0EDH403M21.md): statement 0EDH403M21
- [Hamiltonian system](https://math2.tamnd.com/node/0EDKGA2FAK.md): concept 0EDKGA2FAK
- [Neyman–Pearson lemma](https://math2.tamnd.com/node/0EE19CFEEF.md): statement 0EE19CFEEF
- [Rule of sum](https://math2.tamnd.com/node/0EE2V3CHFK.md): statement 0EE2V3CHFK
- [Continuity at a point](https://math2.tamnd.com/node/0EEE2CZB2A.md): definition 0EEE2CZB2A
- [Refinement of a partition](https://math2.tamnd.com/node/0EEKF7YFBW.md): definition 0EEKF7YFBW
- [Limit cycle](https://math2.tamnd.com/node/0EERDDPSVT.md): definition 0EERDDPSVT
- [Concave function](https://math2.tamnd.com/node/0EF2VG8XKQ.md): definition 0EF2VG8XKQ
- [Convergence in probability](https://math2.tamnd.com/node/0EFDDXTVD6.md): definition 0EFDDXTVD6
- [Proof by infinite descent](https://math2.tamnd.com/node/0EFYCFHYFG.md): concept 0EFYCFHYFG
- [Primality test](https://math2.tamnd.com/node/0EGJZHZ02C.md): concept 0EGJZHZ02C
- [Formal derivative](https://math2.tamnd.com/node/0EGN1E1JPB.md): concept 0EGN1E1JPB
- [Boole's inequality](https://math2.tamnd.com/node/0EGSEE9J9R.md): statement 0EGSEE9J9R
- [Prime number theorem](https://math2.tamnd.com/node/0EHBY5Y84N.md): statement 0EHBY5Y84N
- [Liouville's theorem](https://math2.tamnd.com/node/0EHKVRF4J6.md): statement 0EHKVRF4J6
- [Primitive root of unity](https://math2.tamnd.com/node/0EHM5ZP5QH.md): concept 0EHM5ZP5QH
- [Binomial theorem](https://math2.tamnd.com/node/0EHWPWGHB2.md): statement 0EHWPWGHB2
- [P-value](https://math2.tamnd.com/node/0EHYZFZ4PK.md): definition 0EHYZFZ4PK
- [Universal property of the abelianisation](https://math2.tamnd.com/node/0EJ4PC8NJ6.md): statement 0EJ4PC8NJ6
- [Normal convergence](https://math2.tamnd.com/node/0EJVD9K0K1.md): concept 0EJVD9K0K1
- [Epsilon-delta continuity in metric spaces](https://math2.tamnd.com/node/0EKGKY5TGH.md): statement 0EKGKY5TGH
- [Hermitian matrix](https://math2.tamnd.com/node/0EKZCX4K3P.md): definition 0EKZCX4K3P
- [Orientable surface](https://math2.tamnd.com/node/0EKZW08FQ5.md): definition 0EKZW08FQ5
- [Existence of nontrivial solutions of Pell's equation](https://math2.tamnd.com/node/0EM6C9SZN5.md): statement 0EM6C9SZN5
- [Chi-squared test](https://math2.tamnd.com/node/0EN0V3NY20.md): concept 0EN0V3NY20
- [Inverse in terms of the adjugate](https://math2.tamnd.com/node/0ENVM493VK.md): statement 0ENVM493VK
- [Normal operator](https://math2.tamnd.com/node/0ENZN5G4CV.md): definition 0ENZN5G4CV
- [Invariant subspace](https://math2.tamnd.com/node/0EP59ESQ94.md): definition 0EP59ESQ94
- [Row space](https://math2.tamnd.com/node/0EP73HK4C4.md): definition 0EP73HK4C4
- [Function](https://math2.tamnd.com/node/0EPAT51DXQ.md): concept 0EPAT51DXQ
- [Continuous image of a connected set is connected](https://math2.tamnd.com/node/0EPD87YVG8.md): statement 0EPD87YVG8
- [Convergence in probability implies convergence in distribution](https://math2.tamnd.com/node/0EPTDZPJ83.md): statement 0EPTDZPJ83
- [Linear subspace](https://math2.tamnd.com/node/0EPTJA5QV4.md): definition 0EPTJA5QV4
- [Constrained optimisation](https://math2.tamnd.com/node/0EPW94MJ1P.md): concept 0EPW94MJ1P
- [Central limit theorem](https://math2.tamnd.com/node/0EPZC4Z1FC.md): statement 0EPZC4Z1FC
- [Hyperbolic cosine](https://math2.tamnd.com/node/0EQFWKJ37J.md): definition 0EQFWKJ37J
- [Locally compact space](https://math2.tamnd.com/node/0EQMWQ4WMJ.md): definition 0EQMWQ4WMJ
- [Field extension](https://math2.tamnd.com/node/0EQWVHF3W5.md): concept 0EQWVHF3W5
- [Homogeneous linear equation](https://math2.tamnd.com/node/0EQZSAB9EC.md): definition 0EQZSAB9EC
- [Induced subgraph](https://math2.tamnd.com/node/0EREPBVG2G.md): concept 0EREPBVG2G
- [QR decomposition](https://math2.tamnd.com/node/0ERWG4NXQ9.md): statement 0ERWG4NXQ9
- [Imaginary unit](https://math2.tamnd.com/node/0ESCX36MMC.md): object 0ESCX36MMC
- [Divisibility by 9](https://math2.tamnd.com/node/0ESHD7ZQDX.md): statement 0ESHD7ZQDX
- [Collinear points](https://math2.tamnd.com/node/0ETCTBTJM1.md): definition 0ETCTBTJM1
- [Existence of an algebraic closure](https://math2.tamnd.com/node/0ETGBAK2CC.md): statement 0ETGBAK2CC
- [Gershgorin circle theorem](https://math2.tamnd.com/node/0ETGP0XVCJ.md): statement 0ETGP0XVCJ
- [Linear independence](https://math2.tamnd.com/node/0EVHJFTG9Z.md): definition 0EVHJFTG9Z
- [Poincaré–Bendixson theorem](https://math2.tamnd.com/node/0EW1X61ZT1.md): statement 0EW1X61ZT1
- [Beta distribution](https://math2.tamnd.com/node/0EW824Z7G2.md): concept 0EW824Z7G2
- [Domain in the complex plane](https://math2.tamnd.com/node/0EWMDD6C14.md): concept 0EWMDD6C14
- [Rank of a matrix](https://math2.tamnd.com/node/0EWQEYRVA8.md): definition 0EWQEYRVA8
- [Impossibility of trisecting the angle](https://math2.tamnd.com/node/0EWXPZCSZ0.md): statement 0EWXPZCSZ0
- [Conjugate transpose](https://math2.tamnd.com/node/0EX045E2BP.md): definition 0EX045E2BP
- [Product of vector spaces](https://math2.tamnd.com/node/0EX2HPXPJ6.md): concept 0EX2HPXPJ6
- [The exponential function is its own derivative](https://math2.tamnd.com/node/0EX9GFJ0B3.md): statement 0EX9GFJ0B3
- [Iterated derivative](https://math2.tamnd.com/node/0EY05Z9RPN.md): definition 0EY05Z9RPN
- [68–95–99.7 rule](https://math2.tamnd.com/node/0EYQ03XCR3.md): example 0EYQ03XCR3
- [Element](https://math2.tamnd.com/node/0EZ758XE73.md): definition 0EZ758XE73
- [Cantor's intersection theorem](https://math2.tamnd.com/node/0EZBG8SKJM.md): statement 0EZBG8SKJM
- [x sin(1/x) is continuous at 0](https://math2.tamnd.com/node/0EZMAY1FQD.md): example 0EZMAY1FQD
- [Lower triangular matrix](https://math2.tamnd.com/node/0EZSD6T9E0.md): definition 0EZSD6T9E0
- [Cauchy's integral formula for derivatives](https://math2.tamnd.com/node/0F0H5P5DED.md): statement 0F0H5P5DED
- [Set bounded above](https://math2.tamnd.com/node/0F0J193T76.md): concept 0F0J193T76
- [Prime elements are irreducible in an integral domain](https://math2.tamnd.com/node/0F0K8DB9TD.md): statement 0F0K8DB9TD
- [Column space](https://math2.tamnd.com/node/0F0RPPJ2TT.md): definition 0F0RPPJ2TT
- [Closed and exact forms](https://math2.tamnd.com/node/0F1AVGX26N.md): definition 0F1AVGX26N
- [Well-order](https://math2.tamnd.com/node/0F1D6PK9RZ.md): definition 0F1D6PK9RZ
- [Transition matrix](https://math2.tamnd.com/node/0F1KBVY161.md): definition 0F1KBVY161
- [Subsequence](https://math2.tamnd.com/node/0F1PR74BRX.md): concept 0F1PR74BRX
- [Conditional variance](https://math2.tamnd.com/node/0F1SN93KXE.md): definition 0F1SN93KXE
- [Derivative](https://math2.tamnd.com/node/0F1WR9C83K.md): definition 0F1WR9C83K
- [Row echelon form](https://math2.tamnd.com/node/0F27PBRWD7.md): definition 0F27PBRWD7
- [Field of fractions](https://math2.tamnd.com/node/0F2PTGX1A2.md): concept 0F2PTGX1A2
- [Riemann integrable function](https://math2.tamnd.com/node/0F2QCWVFT7.md): concept 0F2QCWVFT7
- [Lyapunov stability](https://math2.tamnd.com/node/0F2SYCKDT2.md): concept 0F2SYCKDT2
- [Superposition principle](https://math2.tamnd.com/node/0F2VBPXNP9.md): statement 0F2VBPXNP9
- [Iterated derivative](https://math2.tamnd.com/node/0F32S73FJN.md): concept 0F32S73FJN
- [Primitive root modulo n](https://math2.tamnd.com/node/0F33KX9Q6F.md): concept 0F33KX9Q6F
- [Bounded linear maps are continuous](https://math2.tamnd.com/node/0F3K4C7XYW.md): statement 0F3K4C7XYW
- [Mersenne prime](https://math2.tamnd.com/node/0F3Z22M2GT.md): concept 0F3Z22M2GT
- [Derivative vanishes at a local extremum](https://math2.tamnd.com/node/0F405477F8.md): statement 0F405477F8
- [Transition matrix](https://math2.tamnd.com/node/0F462493EA.md): concept 0F462493EA
- [Divisors of a natural number](https://math2.tamnd.com/node/0F4JSQE2EK.md): concept 0F4JSQE2EK
- [Quadratic integer](https://math2.tamnd.com/node/0F4WRWBZZC.md): concept 0F4WRWBZZC
- [Outcome](https://math2.tamnd.com/node/0F50N2P3FC.md): concept 0F50N2P3FC
- [Divergence of a curl is zero](https://math2.tamnd.com/node/0F56CMMC5P.md): statement 0F56CMMC5P
- [g to the n equals 1 iff the order divides n](https://math2.tamnd.com/node/0F5QC101Q7.md): statement 0F5QC101Q7
- [Characteristic of a ring](https://math2.tamnd.com/node/0F6ES3BRYP.md): concept 0F6ES3BRYP
- [Order on the integers](https://math2.tamnd.com/node/0F6FM9Y4Y6.md): definition 0F6FM9Y4Y6
- [Disjunction](https://math2.tamnd.com/node/0F72FK8EKB.md): concept 0F72FK8EKB
- [Cayley–Hamilton theorem](https://math2.tamnd.com/node/0F765KFC2J.md): statement 0F765KFC2J
- [Square root](https://math2.tamnd.com/node/0F7FQS6XY6.md): concept 0F7FQS6XY6
- [Greatest common divisor in an integral domain](https://math2.tamnd.com/node/0F7KNJ1ZYN.md): definition 0F7KNJ1ZYN
- [Graph isomorphism](https://math2.tamnd.com/node/0F7QKGJS6B.md): concept 0F7QKGJS6B
- [Convolution of probability distributions](https://math2.tamnd.com/node/0F8GDKM0A7.md): concept 0F8GDKM0A7
- [Group of units of Z/nZ](https://math2.tamnd.com/node/0F8X8ZRQ53.md): object 0F8X8ZRQ53
- [Double counting](https://math2.tamnd.com/node/0F8YBHF2W3.md): concept 0F8YBHF2W3
- [Circumcentre](https://math2.tamnd.com/node/0F98FZK14A.md): concept 0F98FZK14A
- [Codomain](https://math2.tamnd.com/node/0F9JHP8AJN.md): definition 0F9JHP8AJN
- [Absolutely convergent improper integral](https://math2.tamnd.com/node/0F9VKJPWCA.md): concept 0F9VKJPWCA
- [Möbius function](https://math2.tamnd.com/node/0FANE7P0DY.md): concept 0FANE7P0DY
- [Local maximum](https://math2.tamnd.com/node/0FBETJR0JG.md): concept 0FBETJR0JG
- [Affine subspace](https://math2.tamnd.com/node/0FBXFBZBH8.md): definition 0FBXFBZBH8
- [Closed subsets of complete spaces are complete](https://math2.tamnd.com/node/0FBXJZX3MK.md): statement 0FBXJZX3MK
- [Evaluation homomorphism](https://math2.tamnd.com/node/0FCCDXH97W.md): concept 0FCCDXH97W
- [Exponent of a group](https://math2.tamnd.com/node/0FCCTFA275.md): definition 0FCCTFA275
- [General linear group](https://math2.tamnd.com/node/0FCHE902D7.md): object 0FCHE902D7
- [Divisors of a natural number](https://math2.tamnd.com/node/0FCWA16FZE.md): definition 0FCWA16FZE
- [The gamma function extends the factorial](https://math2.tamnd.com/node/0FDCW77H44.md): statement 0FDCW77H44
- [Reachability](https://math2.tamnd.com/node/0FDKRX9M9Y.md): concept 0FDKRX9M9Y
- [Locally compact space](https://math2.tamnd.com/node/0FE2SB8WVJ.md): concept 0FE2SB8WVJ
- [Subset](https://math2.tamnd.com/node/0FE8Q5S42Y.md): definition 0FE8Q5S42Y
- [A subgroup whose index is the smallest prime divisor of the order is normal](https://math2.tamnd.com/node/0FEBE9NC0E.md): statement 0FEBE9NC0E
- [e as the limit of (1 + 1/n)^n](https://math2.tamnd.com/node/0FEMF8JKAF.md): statement 0FEMF8JKAF
- [Twin prime conjecture](https://math2.tamnd.com/node/0FER36N8SB.md): conjecture 0FER36N8SB
- [Universal property of the quotient topology](https://math2.tamnd.com/node/0FF2FJ8CH2.md): statement 0FF2FJ8CH2
- [Ellipse](https://math2.tamnd.com/node/0FF6CAQT2Y.md): definition 0FF6CAQT2Y
- [Inclusion–exclusion principle](https://math2.tamnd.com/node/0FF7726C7P.md): statement 0FF7726C7P
- [Affine span](https://math2.tamnd.com/node/0FFAE5P6R4.md): definition 0FFAE5P6R4
- [Connected graph](https://math2.tamnd.com/node/0FFAYNMAJZ.md): definition 0FFAYNMAJZ
- [Construction of the rationals as fractions](https://math2.tamnd.com/node/0FFH3HSRBA.md): concept 0FFH3HSRBA
- [Absolute value](https://math2.tamnd.com/node/0FFXKTKKQC.md): concept 0FFXKTKKQC
- [Kurtosis](https://math2.tamnd.com/node/0FGAJSPX8E.md): concept 0FGAJSPX8E
- [Preimage of a set](https://math2.tamnd.com/node/0FGCFY1AZX.md): definition 0FGCFY1AZX
- [The reals form a field](https://math2.tamnd.com/node/0FGENPMPJX.md): statement 0FGENPMPJX
- [Characterization of bridges](https://math2.tamnd.com/node/0FHM8DB6PM.md): statement 0FHM8DB6PM
- [Epsilon-delta uniform continuity](https://math2.tamnd.com/node/0FHSDCQACY.md): statement 0FHSDCQACY
- [Matrix ring](https://math2.tamnd.com/node/0FHVPRAE60.md): object 0FHVPRAE60
- [Group of units of a monoid](https://math2.tamnd.com/node/0FJ90Y9926.md): concept 0FJ90Y9926
- [Conditional probability](https://math2.tamnd.com/node/0FJB06AD2P.md): definition 0FJB06AD2P
- [Chromatic number of a complete bipartite graph](https://math2.tamnd.com/node/0FJBBCXYPS.md): statement 0FJBBCXYPS
- [Bertrand's postulate](https://math2.tamnd.com/node/0FJTGZA4GS.md): statement 0FJTGZA4GS
- [Maximum degree](https://math2.tamnd.com/node/0FJY6G7FNA.md): concept 0FJY6G7FNA
- [Triangle inequality for norms](https://math2.tamnd.com/node/0FJZMBX0ZZ.md): statement 0FJZMBX0ZZ
- [Simultaneous diagonalisation](https://math2.tamnd.com/node/0FK17N4DKA.md): statement 0FK17N4DKA
- [Monotone convergence theorem for sequences](https://math2.tamnd.com/node/0FM6N4R0VB.md): statement 0FM6N4R0VB
- [Burnside's p^a q^b theorem](https://math2.tamnd.com/node/0FMAQSB1SC.md): statement 0FMAQSB1SC
- [Constructible number](https://math2.tamnd.com/node/0FMN2MN7TM.md): definition 0FMN2MN7TM
- [Total derivative](https://math2.tamnd.com/node/0FNKFT8H6E.md): concept 0FNKFT8H6E
- [Chinese remainder theorem](https://math2.tamnd.com/node/0FNN9B4W64.md): statement 0FNN9B4W64
- [Even and odd functions](https://math2.tamnd.com/node/0FP31SX5EA.md): concept 0FP31SX5EA
- [Scalar potential](https://math2.tamnd.com/node/0FP4BVDAAQ.md): definition 0FP4BVDAAQ
- [Lévy's continuity theorem](https://math2.tamnd.com/node/0FPWF7TTMB.md): statement 0FPWF7TTMB
- [Reduced word](https://math2.tamnd.com/node/0FPXK5CHQ3.md): definition 0FPXK5CHQ3
- [Spanning tree](https://math2.tamnd.com/node/0FQ5B8S1ZC.md): definition 0FQ5B8S1ZC
- [Convergence of random variables](https://math2.tamnd.com/node/0FQ968YKA1.md): concept 0FQ968YKA1
- [Meagre set](https://math2.tamnd.com/node/0FQPYAY9PJ.md): definition 0FQPYAY9PJ
- [Inverse of a complex number](https://math2.tamnd.com/node/0FQRAV5SKH.md): statement 0FQRAV5SKH
- [Convex hull](https://math2.tamnd.com/node/0FQSDC2EQ2.md): concept 0FQSDC2EQ2
- [Comparison test for improper integrals](https://math2.tamnd.com/node/0FR1BDKK10.md): statement 0FR1BDKK10
- [Heron's formula](https://math2.tamnd.com/node/0FRBZ2X4HJ.md): statement 0FRBZ2X4HJ
- [Maximum likelihood estimation](https://math2.tamnd.com/node/0FREPFKNS7.md): concept 0FREPFKNS7
- [Scalar matrix](https://math2.tamnd.com/node/0FRM62SG7D.md): definition 0FRM62SG7D
- [Sequential compactness](https://math2.tamnd.com/node/0FRMJH3CY7.md): concept 0FRMJH3CY7
- [The complex numbers form a field](https://math2.tamnd.com/node/0FRSR10DWF.md): statement 0FRSR10DWF
- [Grönwall bound](https://math2.tamnd.com/node/0FRX8JD7QH.md): definition 0FRX8JD7QH
- [Total order](https://math2.tamnd.com/node/0FRXBVVR03.md): concept 0FRXBVVR03
- [Trace is a coefficient of the characteristic polynomial](https://math2.tamnd.com/node/0FRYPZ6D0X.md): statement 0FRYPZ6D0X
- [Infinite pigeonhole principle](https://math2.tamnd.com/node/0FS3RHF6KE.md): statement 0FS3RHF6KE
- [Determinant of a matrix with a zero row](https://math2.tamnd.com/node/0FS6B46E5T.md): statement 0FS6B46E5T
- [Bertrand's ballot theorem](https://math2.tamnd.com/node/0FS972T0ZH.md): statement 0FS972T0ZH
- [Reversing a path changes the sign of a line integral](https://math2.tamnd.com/node/0FSCXYG856.md): statement 0FSCXYG856
- [Multiplicativity of the totient function](https://math2.tamnd.com/node/0FSJA9M43W.md): statement 0FSJA9M43W
- [Identity function](https://math2.tamnd.com/node/0FSNQ0GNCC.md): definition 0FSNQ0GNCC
- [Complex conjugate](https://math2.tamnd.com/node/0FTH2E9TAS.md): concept 0FTH2E9TAS
- [Choice function](https://math2.tamnd.com/node/0FWBQ53MPJ.md): concept 0FWBQ53MPJ
- [Orthonormal basis](https://math2.tamnd.com/node/0FWJ0VW561.md): definition 0FWJ0VW561
- [Transposition](https://math2.tamnd.com/node/0FWVRHBRWM.md): concept 0FWVRHBRWM
- [Initial value problem](https://math2.tamnd.com/node/0FXDW992DS.md): concept 0FXDW992DS
- [Solvable group](https://math2.tamnd.com/node/0FXJ7YFMMX.md): concept 0FXJ7YFMMX
- [Algebraic multiplicity](https://math2.tamnd.com/node/0FY42N02DV.md): concept 0FY42N02DV
- [Order of quantifiers](https://math2.tamnd.com/node/0FYQMKFE77.md): concept 0FYQMKFE77
- [Recurrence relation](https://math2.tamnd.com/node/0FYX44GP92.md): concept 0FYX44GP92
- [Second derivative test](https://math2.tamnd.com/node/0FZFHK97P7.md): statement 0FZFHK97P7
- [Deficient number](https://math2.tamnd.com/node/0G01TS1RRP.md): definition 0G01TS1RRP
- [Cardinality](https://math2.tamnd.com/node/0G0ASKNA6Z.md): concept 0G0ASKNA6Z
- [Cyclic group of order n](https://math2.tamnd.com/node/0G0B3DRG65.md): example 0G0B3DRG65
- [Transcendence of pi](https://math2.tamnd.com/node/0G0F7FGCGP.md): statement 0G0F7FGCGP
- [Eigenspace](https://math2.tamnd.com/node/0G0RFPG8HR.md): concept 0G0RFPG8HR
- [Galton–Watson process](https://math2.tamnd.com/node/0G1022YXVQ.md): concept 0G1022YXVQ
- [Sylvester's determinant theorem](https://math2.tamnd.com/node/0G24VQ1088.md): statement 0G24VQ1088
- [The complex exponential is entire](https://math2.tamnd.com/node/0G2YJN0H5S.md): statement 0G2YJN0H5S
- [Tensor product of vector spaces](https://math2.tamnd.com/node/0G31VJ5YS5.md): definition 0G31VJ5YS5
- [Continuous dependence on initial data](https://math2.tamnd.com/node/0G3DBXRRQJ.md): statement 0G3DBXRRQJ
- [Dedekind cut](https://math2.tamnd.com/node/0G3RRM41X9.md): concept 0G3RRM41X9
- [Markov chain](https://math2.tamnd.com/node/0G48PYM96Q.md): concept 0G48PYM96Q
- [Automorphism group of a cyclic group](https://math2.tamnd.com/node/0G4C8SJ6PV.md): statement 0G4C8SJ6PV
- [Sum of the reciprocals of the factorials](https://math2.tamnd.com/node/0G4D3ZKTP8.md): example 0G4D3ZKTP8
- [Entire function](https://math2.tamnd.com/node/0G4F0XB9X2.md): concept 0G4F0XB9X2
- [Completion of a metric space](https://math2.tamnd.com/node/0G4J7Z7WPJ.md): definition 0G4J7Z7WPJ
- [Cylindrical coordinates](https://math2.tamnd.com/node/0G4MFRQSGN.md): definition 0G4MFRQSGN
- [Oriented angle](https://math2.tamnd.com/node/0G4W89YJRK.md): definition 0G4W89YJRK
- [Equilibrium point](https://math2.tamnd.com/node/0G4X36AE0X.md): definition 0G4X36AE0X
- [Clique](https://math2.tamnd.com/node/0G52BPNPKG.md): definition 0G52BPNPKG
- [Multivariate normal distribution](https://math2.tamnd.com/node/0G558GJQMW.md): definition 0G558GJQMW
- [Closed set](https://math2.tamnd.com/node/0G5AV9Q4RR.md): definition 0G5AV9Q4RR
- [Derivative of the inverse function](https://math2.tamnd.com/node/0G65751VSW.md): statement 0G65751VSW
- [Cauchy sequences are bounded](https://math2.tamnd.com/node/0G6D8E7CTA.md): statement 0G6D8E7CTA
- [Lexicographic order](https://math2.tamnd.com/node/0G77XFBV00.md): definition 0G77XFBV00
- [Vector projection](https://math2.tamnd.com/node/0G78GP5TF1.md): definition 0G78GP5TF1
- [Parametrised curve](https://math2.tamnd.com/node/0G7WM8DX4P.md): concept 0G7WM8DX4P
- [Uniqueness of the complete ordered field](https://math2.tamnd.com/node/0G8G1TH5WK.md): statement 0G8G1TH5WK
- [Equivalence relation](https://math2.tamnd.com/node/0G8VEZ3WTK.md): concept 0G8VEZ3WTK
- [Sequence](https://math2.tamnd.com/node/0G9EKM0E15.md): concept 0G9EKM0E15
- [Inscribed angle theorem](https://math2.tamnd.com/node/0GA1V8NN8C.md): statement 0GA1V8NN8C
- [Derivative of arcsine](https://math2.tamnd.com/node/0GA6G43EA9.md): statement 0GA6G43EA9
- [Fibonacci sequence](https://math2.tamnd.com/node/0GA9K57A39.md): example 0GA9K57A39
- [Product topology](https://math2.tamnd.com/node/0GBG8B49AA.md): concept 0GBG8B49AA
- [Independence of two random variables](https://math2.tamnd.com/node/0GBJACNAP6.md): concept 0GBJACNAP6
- [Epsilon-delta definition of a limit](https://math2.tamnd.com/node/0GBZDAFPX8.md): statement 0GBZDAFPX8
- [Binomial coefficient](https://math2.tamnd.com/node/0GBZM0MT0F.md): concept 0GBZM0MT0F
- [Concave function](https://math2.tamnd.com/node/0GCM7B6PBK.md): concept 0GCM7B6PBK
- [Mutual independence of events](https://math2.tamnd.com/node/0GCVEV6BJA.md): definition 0GCVEV6BJA
- [Eccentricity](https://math2.tamnd.com/node/0GCWJS2EGB.md): definition 0GCWJS2EGB
- [Curl](https://math2.tamnd.com/node/0GDNHS9J2X.md): definition 0GDNHS9J2X
- [Prüfer sequence](https://math2.tamnd.com/node/0GDQRETR59.md): concept 0GDQRETR59
- [The quaternions](https://math2.tamnd.com/node/0GE4DVQKR8.md): example 0GE4DVQKR8
- [Critical point](https://math2.tamnd.com/node/0GEQ42PTGB.md): definition 0GEQ42PTGB
- [Normal core](https://math2.tamnd.com/node/0GETX79CJN.md): concept 0GETX79CJN
- [Induction from an arbitrary base case](https://math2.tamnd.com/node/0GF6AJEBZ0.md): statement 0GF6AJEBZ0
- [Subfield](https://math2.tamnd.com/node/0GFGHRT2FJ.md): concept 0GFGHRT2FJ
- [Pointwise convergence](https://math2.tamnd.com/node/0GFGK40582.md): concept 0GFGK40582
- [Coprime integers](https://math2.tamnd.com/node/0GFKP8F8AG.md): definition 0GFKP8F8AG
- [Bijective function](https://math2.tamnd.com/node/0GG9TVW90M.md): concept 0GG9TVW90M
- [Ring homomorphism](https://math2.tamnd.com/node/0GGE0Z0JQ1.md): definition 0GGE0Z0JQ1
- [P-value](https://math2.tamnd.com/node/0GGKJGZF6D.md): concept 0GGKJGZF6D
- [Forest](https://math2.tamnd.com/node/0GGXWDNKBR.md): concept 0GGXWDNKBR
- [Frobenius endomorphism](https://math2.tamnd.com/node/0GHDDVAH8P.md): definition 0GHDDVAH8P
- [Universal quantifier](https://math2.tamnd.com/node/0GHS47PDAQ.md): concept 0GHS47PDAQ
- [General solution of an inhomogeneous linear equation](https://math2.tamnd.com/node/0GHWEH6H15.md): statement 0GHWEH6H15
- [Euclidean domain](https://math2.tamnd.com/node/0GJ2HRG95T.md): definition 0GJ2HRG95T
- [Polynomial evaluation](https://math2.tamnd.com/node/0GJ7T0NK49.md): definition 0GJ7T0NK49
- [Universal property of the quotient group](https://math2.tamnd.com/node/0GJ83E78A5.md): statement 0GJ83E78A5
- [Angle at a point](https://math2.tamnd.com/node/0GJAJC6TNZ.md): concept 0GJAJC6TNZ
- [Mersenne number](https://math2.tamnd.com/node/0GJBJ19NXR.md): definition 0GJBJ19NXR
- [Cancellation in an integral domain](https://math2.tamnd.com/node/0GJM24EXP9.md): statement 0GJM24EXP9
- [Trace is linear](https://math2.tamnd.com/node/0GJWQZCPZM.md): statement 0GJWQZCPZM
- [Addition formulas for hyperbolic functions](https://math2.tamnd.com/node/0GJZCE87WP.md): statement 0GJZCE87WP
- [Function for which the iterated integrals differ](https://math2.tamnd.com/node/0GJZE2E4RX.md): counterexample 0GJZE2E4RX
- [Transitive relation](https://math2.tamnd.com/node/0GJZW9JFZQ.md): concept 0GJZW9JFZQ
- [Minimal polynomial of an algebraic element](https://math2.tamnd.com/node/0GK1WB9R98.md): definition 0GK1WB9R98
- [Solution space of a homogeneous linear equation has dimension n](https://math2.tamnd.com/node/0GK68PZ6KH.md): statement 0GK68PZ6KH
- [Separable space](https://math2.tamnd.com/node/0GKJ78948V.md): definition 0GKJ78948V
- [The alternating group A5 is simple](https://math2.tamnd.com/node/0GKQKF0FXJ.md): example 0GKQKF0FXJ
- [Adjugate matrix](https://math2.tamnd.com/node/0GMPYM0NSR.md): definition 0GMPYM0NSR
- [Absolute value is multiplicative](https://math2.tamnd.com/node/0GMV7GP8W0.md): statement 0GMV7GP8W0
- [Subring](https://math2.tamnd.com/node/0GNCRPYXYV.md): concept 0GNCRPYXYV
- [Nilpotent groups are solvable](https://math2.tamnd.com/node/0GNG2KVPZJ.md): statement 0GNG2KVPZJ
- [Antichain](https://math2.tamnd.com/node/0GNGPF7KS5.md): definition 0GNGPF7KS5
- [Dirichlet's theorem on arithmetic progressions](https://math2.tamnd.com/node/0GNVN3CA2P.md): statement 0GNVN3CA2P
- [Chain](https://math2.tamnd.com/node/0GP9ASVQ1X.md): definition 0GP9ASVQ1X
- [Complete residue system](https://math2.tamnd.com/node/0GPBVKE31D.md): concept 0GPBVKE31D
- [Cardinality of the power set](https://math2.tamnd.com/node/0GPFKS96T7.md): statement 0GPFKS96T7
- [Cauchy distribution](https://math2.tamnd.com/node/0GPG24Q3GR.md): concept 0GPG24Q3GR
- [The medians meet at the centroid](https://math2.tamnd.com/node/0GPQ67XV7N.md): statement 0GPQ67XV7N
- [Skewness](https://math2.tamnd.com/node/0GQ3H7VW3V.md): definition 0GQ3H7VW3V
- [Independence of random variables](https://math2.tamnd.com/node/0GQ938GDE0.md): concept 0GQ938GDE0
- [Quotient of continuous functions is continuous](https://math2.tamnd.com/node/0GR2APWTAC.md): statement 0GR2APWTAC
- [Wedge product of forms](https://math2.tamnd.com/node/0GRAATMGP1.md): concept 0GRAATMGP1
- [Curl-free field that is not conservative](https://math2.tamnd.com/node/0GRM3BS0DD.md): counterexample 0GRM3BS0DD
- [Angles in the same segment are equal](https://math2.tamnd.com/node/0GS0AW6JRY.md): statement 0GS0AW6JRY
- [Completely multiplicative function](https://math2.tamnd.com/node/0GS27TKF1V.md): definition 0GS27TKF1V
- [Degree of a field extension](https://math2.tamnd.com/node/0GS74FGFEK.md): definition 0GS74FGFEK
- [Thales's theorem](https://math2.tamnd.com/node/0GSDHYMA1T.md): statement 0GSDHYMA1T
- [Ideal generated by a subset](https://math2.tamnd.com/node/0GSW76Q14N.md): concept 0GSW76Q14N
- [Extended real number line](https://math2.tamnd.com/node/0GSZ7B0CGR.md): object 0GSZ7B0CGR
- [System of linear equations](https://math2.tamnd.com/node/0GT4WWGWH2.md): concept 0GT4WWGWH2
- [Decimal expansion as a series](https://math2.tamnd.com/node/0GT85FBPTS.md): example 0GT85FBPTS
- [Triple integral](https://math2.tamnd.com/node/0GTBGMZXV8.md): definition 0GTBGMZXV8
- [Gaussian prime](https://math2.tamnd.com/node/0GTEHJQRFK.md): definition 0GTEHJQRFK
- [Cavalieri's principle](https://math2.tamnd.com/node/0GTZ2F3PQY.md): statement 0GTZ2F3PQY
- [Bijection principle](https://math2.tamnd.com/node/0GV2573FP4.md): statement 0GV2573FP4
- [Disjunctive normal form](https://math2.tamnd.com/node/0GVDFAY4KW.md): concept 0GVDFAY4KW
- [Moment of inertia](https://math2.tamnd.com/node/0GVMR0C3S6.md): definition 0GVMR0C3S6
- [Fundamental theorem of symmetric polynomials](https://math2.tamnd.com/node/0GW1ZZH0R4.md): statement 0GW1ZZH0R4
- [Hyperbolic sine](https://math2.tamnd.com/node/0GWD21EWRW.md): definition 0GWD21EWRW
- [One-point compactification](https://math2.tamnd.com/node/0GWN3CMRK8.md): concept 0GWN3CMRK8
- [Area under a curve](https://math2.tamnd.com/node/0GXA8Z205E.md): concept 0GXA8Z205E
- [Integral of sine](https://math2.tamnd.com/node/0GXCXCX3QN.md): statement 0GXCXCX3QN
- [Telescoping series](https://math2.tamnd.com/node/0GXKM1R7E4.md): concept 0GXKM1R7E4
- [Laurent series](https://math2.tamnd.com/node/0GXMPDNYX3.md): concept 0GXMPDNYX3
- [Units of Z/nZ are the residues coprime to n](https://math2.tamnd.com/node/0GXNHEHQNA.md): statement 0GXNHEHQNA
- [Second derivative test in several variables](https://math2.tamnd.com/node/0GYDV9FE98.md): statement 0GYDV9FE98
- [Commutative ring](https://math2.tamnd.com/node/0GYGHKX834.md): definition 0GYGHKX834
- [Cayley graph](https://math2.tamnd.com/node/0GYQK9NDJF.md): concept 0GYQK9NDJF
- [Divergence theorem on a rectangle in the plane](https://math2.tamnd.com/node/0GZ3D8F921.md): statement 0GZ3D8F921
- [Ascending chain condition](https://math2.tamnd.com/node/0GZ6Y1H4RN.md): concept 0GZ6Y1H4RN
- [Function space](https://math2.tamnd.com/node/0GZB2SN0PH.md): example 0GZB2SN0PH
- [Dirichlet integral](https://math2.tamnd.com/node/0GZNSHMG4Q.md): example 0GZNSHMG4Q
- [Characteristic equation](https://math2.tamnd.com/node/0GZWXM2DKE.md): concept 0GZWXM2DKE
- [Conjugate subgroup](https://math2.tamnd.com/node/0H11ZPX4ND.md): concept 0H11ZPX4ND
- [Fundamental matrix](https://math2.tamnd.com/node/0H1GN6P9R5.md): concept 0H1GN6P9R5
- [Hyperbolic sine](https://math2.tamnd.com/node/0H1JFKE7H4.md): concept 0H1JFKE7H4
- [Arithmetic function](https://math2.tamnd.com/node/0H2EPRZC0D.md): definition 0H2EPRZC0D
- [Pseudometric space](https://math2.tamnd.com/node/0H3W614GEC.md): definition 0H3W614GEC
- [Semidirect product](https://math2.tamnd.com/node/0H3WZXCZNF.md): concept 0H3WZXCZNF
- [Euclidean group](https://math2.tamnd.com/node/0H3YY8CPKH.md): object 0H3YY8CPKH
- [Inner automorphism](https://math2.tamnd.com/node/0H47DMFN3S.md): definition 0H47DMFN3S
- [Fundamental system of solutions](https://math2.tamnd.com/node/0H4KNNCHPY.md): definition 0H4KNNCHPY
- [Limit inferior](https://math2.tamnd.com/node/0H4R3RESGV.md): definition 0H4R3RESGV
- [Disjunction](https://math2.tamnd.com/node/0H5GKDTB01.md): definition 0H5GKDTB01
- [Upper bound](https://math2.tamnd.com/node/0H5J02P8S9.md): definition 0H5J02P8S9
- [Line integral of a vector field](https://math2.tamnd.com/node/0H5MR9B6GT.md): concept 0H5MR9B6GT
- [Gamma function](https://math2.tamnd.com/node/0H5SYBSED9.md): concept 0H5SYBSED9
- [The order of a finite field is a prime power](https://math2.tamnd.com/node/0H5WWC8RZX.md): statement 0H5WWC8RZX
- [Taylor's theorem with the Lagrange remainder](https://math2.tamnd.com/node/0H5ZAZCX2T.md): statement 0H5ZAZCX2T
- [Changing the order of integration](https://math2.tamnd.com/node/0H60H1Q0SX.md): statement 0H60H1Q0SX
- [Algebraic element](https://math2.tamnd.com/node/0H628RSGWX.md): concept 0H628RSGWX
- [Galton–Watson process](https://math2.tamnd.com/node/0H631CZP5K.md): definition 0H631CZP5K
- [Del operator](https://math2.tamnd.com/node/0H6DP8CHND.md): concept 0H6DP8CHND
- [Product of power series](https://math2.tamnd.com/node/0H6KWAYS87.md): statement 0H6KWAYS87
- [Scalar potential](https://math2.tamnd.com/node/0H7C3T1GJY.md): concept 0H7C3T1GJY
- [Cumulative distribution function](https://math2.tamnd.com/node/0H8H7ZTHW2.md): concept 0H8H7ZTHW2
- [Every finite tree with at least two vertices has a leaf](https://math2.tamnd.com/node/0H8Q4MANBD.md): statement 0H8Q4MANBD
- [Rule of Sarrus](https://math2.tamnd.com/node/0H8SZFFDXJ.md): statement 0H8SZFFDXJ
- [Plane](https://math2.tamnd.com/node/0H97Q4Z2SF.md): definition 0H97Q4Z2SF
- [Abelianisation](https://math2.tamnd.com/node/0H9K5D3RDV.md): definition 0H9K5D3RDV
- [Rectifiable curve](https://math2.tamnd.com/node/0HATW2VWAZ.md): concept 0HATW2VWAZ
- [Interior is the largest open subset](https://math2.tamnd.com/node/0HAZEFH137.md): statement 0HAZEFH137
- [Binomial series](https://math2.tamnd.com/node/0HB0XKDXHH.md): example 0HB0XKDXHH
- [Markov property](https://math2.tamnd.com/node/0HB8SXHPFA.md): definition 0HB8SXHPFA
- [Limit point](https://math2.tamnd.com/node/0HBQC2GJXT.md): definition 0HBQC2GJXT
- [Intersection](https://math2.tamnd.com/node/0HBRW4E6G1.md): definition 0HBRW4E6G1
- [Induction on the integers](https://math2.tamnd.com/node/0HBS1F00DN.md): statement 0HBS1F00DN
- [Riemann sum](https://math2.tamnd.com/node/0HBY48G8G1.md): definition 0HBY48G8G1
- [Zermelo–Fraenkel set theory](https://math2.tamnd.com/node/0HE5WDQ9FJ.md): concept 0HE5WDQ9FJ
- [Union of open sets is open](https://math2.tamnd.com/node/0HEF46RPEK.md): statement 0HEF46RPEK
- [Russell's paradox](https://math2.tamnd.com/node/0HESMMSYEG.md): statement 0HESMMSYEG
- [Hamiltonian cycle](https://math2.tamnd.com/node/0HF30P75G4.md): concept 0HF30P75G4
- [Rotation](https://math2.tamnd.com/node/0HFREYRSZP.md): concept 0HFREYRSZP
- [Taylor polynomial](https://math2.tamnd.com/node/0HFSFVMWQB.md): definition 0HFSFVMWQB
- [First-countable space](https://math2.tamnd.com/node/0HG11FW8MJ.md): definition 0HG11FW8MJ
- [Pigeonhole principle](https://math2.tamnd.com/node/0HGVCWQGY3.md): statement 0HGVCWQGY3
- [Sum of a series](https://math2.tamnd.com/node/0HH4Y7HSRQ.md): concept 0HH4Y7HSRQ
- [Generating function](https://math2.tamnd.com/node/0HH6JX5WZZ.md): definition 0HH6JX5WZZ
- [Differentiable function of several variables](https://math2.tamnd.com/node/0HHHDCXZV4.md): concept 0HHHDCXZV4
- [Solutions for distinct real roots](https://math2.tamnd.com/node/0HJ0HDA6W7.md): statement 0HJ0HDA6W7
- [Thomae's function](https://math2.tamnd.com/node/0HJ28H9Y95.md): example 0HJ28H9Y95
- [Second isomorphism theorem for rings](https://math2.tamnd.com/node/0HJ4DTW6FN.md): statement 0HJ4DTW6FN
- [Discrete logarithm](https://math2.tamnd.com/node/0HJHYMGN3C.md): concept 0HJHYMGN3C
- [Hamiltonian cycle](https://math2.tamnd.com/node/0HKXNG2RS1.md): definition 0HKXNG2RS1
- [Random experiment](https://math2.tamnd.com/node/0HM05N3FJD.md): concept 0HM05N3FJD
- [Distributive lattice](https://math2.tamnd.com/node/0HM0HQCK6N.md): concept 0HM0HQCK6N
- [RSA cryptosystem](https://math2.tamnd.com/node/0HM584C465.md): example 0HM584C465
- [A sequence converges iff its liminf and limsup agree](https://math2.tamnd.com/node/0HM732ZNVC.md): statement 0HM732ZNVC
- [Order on the natural numbers](https://math2.tamnd.com/node/0HNATXFEV5.md): concept 0HNATXFEV5
- [Centre of mass](https://math2.tamnd.com/node/0HNK6KT05K.md): concept 0HNK6KT05K
- [Eigenvectors with distinct eigenvalues are linearly independent](https://math2.tamnd.com/node/0HNMFG8F8K.md): statement 0HNMFG8F8K
- [Generalised Stokes theorem](https://math2.tamnd.com/node/0HPHKCGEQ0.md): statement 0HPHKCGEQ0
- [Archimedean ordered field](https://math2.tamnd.com/node/0HQARMKEV5.md): definition 0HQARMKEV5
- [Coefficient matrix](https://math2.tamnd.com/node/0HQFZAP3ND.md): concept 0HQFZAP3ND
- [Green's theorem](https://math2.tamnd.com/node/0HR3N02317.md): statement 0HR3N02317
- [Linear span](https://math2.tamnd.com/node/0HR573CHNM.md): definition 0HR573CHNM
- [Rearrangement of a series](https://math2.tamnd.com/node/0HRJ2Y74T3.md): definition 0HRJ2Y74T3
- [Solutions for repeated roots](https://math2.tamnd.com/node/0HRVY67CNV.md): statement 0HRVY67CNV
- [Axiom of choice](https://math2.tamnd.com/node/0HTGB13FE2.md): statement 0HTGB13FE2
- [Reduced residue system](https://math2.tamnd.com/node/0HV01NH20W.md): concept 0HV01NH20W
- [Rouché–Capelli theorem](https://math2.tamnd.com/node/0HV1SKWPMS.md): statement 0HV1SKWPMS
- [Group homomorphism](https://math2.tamnd.com/node/0HV9Y7WG5R.md): definition 0HV9Y7WG5R
- [Eccentricity](https://math2.tamnd.com/node/0HVX863SAJ.md): concept 0HVX863SAJ
- [Limit comparison test](https://math2.tamnd.com/node/0HW7BNHHGF.md): statement 0HW7BNHHGF
- [Gram matrix](https://math2.tamnd.com/node/0HX8HZ1WTQ.md): definition 0HX8HZ1WTQ
- [One-sided derivative](https://math2.tamnd.com/node/0HXEX0DPRA.md): definition 0HXEX0DPRA
- [Finitely generated group](https://math2.tamnd.com/node/0HY15D6WTX.md): definition 0HY15D6WTX
- [Totient of a prime](https://math2.tamnd.com/node/0HY1MXMEN9.md): statement 0HY1MXMEN9
- [Eigenvalues of a positive definite matrix are positive](https://math2.tamnd.com/node/0HYC2RHD3M.md): statement 0HYC2RHD3M
- [Injective iff it has a left inverse](https://math2.tamnd.com/node/0HYFEEZAEM.md): statement 0HYFEEZAEM
- [Polar form of a quadratic form](https://math2.tamnd.com/node/0HYFWKKY3C.md): concept 0HYFWKKY3C
- [Modulus is multiplicative](https://math2.tamnd.com/node/0HYNZE940T.md): statement 0HYNZE940T
- [Student's t-test](https://math2.tamnd.com/node/0HYP606RYF.md): concept 0HYP606RYF
- [Mean value property](https://math2.tamnd.com/node/0HYPR64JM2.md): statement 0HYPR64JM2
- [Norm of a Gaussian integer](https://math2.tamnd.com/node/0HYRJF4WNQ.md): concept 0HYRJF4WNQ
- [Hamiltonian graph](https://math2.tamnd.com/node/0HYV2NME07.md): concept 0HYV2NME07
- [Quadratic form](https://math2.tamnd.com/node/0HZNE5QZDJ.md): definition 0HZNE5QZDJ
- [Pivot](https://math2.tamnd.com/node/0J0BBCMJ5W.md): concept 0J0BBCMJ5W
- [Exponential function](https://math2.tamnd.com/node/0J0FCWV32D.md): concept 0J0FCWV32D
- [Reduced residue system](https://math2.tamnd.com/node/0J0QEN1DZR.md): definition 0J0QEN1DZR
- [Quotient map](https://math2.tamnd.com/node/0J0SK5FF4M.md): definition 0J0SK5FF4M
- [Proof by contradiction](https://math2.tamnd.com/node/0J0XBKG64S.md): concept 0J0XBKG64S
- [Fixed field](https://math2.tamnd.com/node/0J1E5ZRZRM.md): concept 0J1E5ZRZRM
- [Euclidean isometry](https://math2.tamnd.com/node/0J1N3WTKM4.md): concept 0J1N3WTKM4
- [Liouville number](https://math2.tamnd.com/node/0J1R5NP36S.md): definition 0J1R5NP36S
- [Solutions of Pell's equation are powers of the fundamental solution](https://math2.tamnd.com/node/0J1TT091TN.md): statement 0J1TT091TN
- [Existence of nth roots by the intermediate value theorem](https://math2.tamnd.com/node/0J29GM30Z7.md): example 0J29GM30Z7
- [Polar decomposition](https://math2.tamnd.com/node/0J2C7BDJQA.md): statement 0J2C7BDJQA
- [Inverse function](https://math2.tamnd.com/node/0J2WAR83WP.md): concept 0J2WAR83WP
- [Equivalent norms](https://math2.tamnd.com/node/0J3578562S.md): definition 0J3578562S
- [Cycle type](https://math2.tamnd.com/node/0J3K4Y1KPD.md): concept 0J3K4Y1KPD
- [Conjugacy classes of the symmetric group correspond to cycle types](https://math2.tamnd.com/node/0J438GAC8K.md): statement 0J438GAC8K
- [The symmetric group on five or more letters is not solvable](https://math2.tamnd.com/node/0J53NE674D.md): statement 0J53NE674D
- [Characteristic polynomial of a linear map](https://math2.tamnd.com/node/0J5EMJRRFM.md): concept 0J5EMJRRFM
- [Congruence is compatible with powers](https://math2.tamnd.com/node/0J5VGJYYC3.md): statement 0J5VGJYYC3
- [Intersecting secants theorem](https://math2.tamnd.com/node/0J5XR7BW2S.md): statement 0J5XR7BW2S
- [Lyapunov's stability theorem](https://math2.tamnd.com/node/0J6A752DTZ.md): statement 0J6A752DTZ
- [Associativity of composition](https://math2.tamnd.com/node/0J6AKYZQT8.md): statement 0J6AKYZQT8
- [Kuratowski's theorem](https://math2.tamnd.com/node/0J6G5NJC5R.md): statement 0J6G5NJC5R
- [Hyperbolic functions](https://math2.tamnd.com/node/0J6M03WSS6.md): definition 0J6M03WSS6
- [Discrete random variable](https://math2.tamnd.com/node/0J6Z6H5VQJ.md): definition 0J6Z6H5VQJ
- [Nine-point circle](https://math2.tamnd.com/node/0J768YF1ZE.md): definition 0J768YF1ZE
- [Rearrangements of an absolutely convergent series have the same sum](https://math2.tamnd.com/node/0J76TP9FGD.md): statement 0J76TP9FGD
- [Riemann sphere](https://math2.tamnd.com/node/0J78KBPFZB.md): concept 0J78KBPFZB
- [Endomorphism](https://math2.tamnd.com/node/0J7H0YBFDJ.md): concept 0J7H0YBFDJ
- [Picard–Lindelöf hypotheses](https://math2.tamnd.com/node/0J7HF2PF1H.md): definition 0J7HF2PF1H
- [Logarithm of a quotient](https://math2.tamnd.com/node/0J7PHRBNNS.md): statement 0J7PHRBNNS
- [Primitive root of unity](https://math2.tamnd.com/node/0J817RTVG6.md): definition 0J817RTVG6
- [Sperner's theorem](https://math2.tamnd.com/node/0J87V9K1GE.md): statement 0J87V9K1GE
- [Composition of an integer](https://math2.tamnd.com/node/0J8ANSYKAC.md): definition 0J8ANSYKAC
- [Taylor's theorem in several variables](https://math2.tamnd.com/node/0J8T4K4YTE.md): statement 0J8T4K4YTE
- [Reachability](https://math2.tamnd.com/node/0J8XE7W68F.md): definition 0J8XE7W68F
- [Limit of a function of several variables](https://math2.tamnd.com/node/0J8Y1NCY0K.md): concept 0J8Y1NCY0K
- [Bernoulli distribution](https://math2.tamnd.com/node/0J97P5KZY0.md): definition 0J97P5KZY0
- [Determinant of a block triangular matrix](https://math2.tamnd.com/node/0J9TVTNB4K.md): statement 0J9TVTNB4K
- [Riemann sum](https://math2.tamnd.com/node/0JAEHXTVV5.md): concept 0JAEHXTVV5
- [Euler method](https://math2.tamnd.com/node/0JARTXX4XP.md): concept 0JARTXX4XP
- [Image of a linear map](https://math2.tamnd.com/node/0JB228DCWB.md): definition 0JB228DCWB
- [Linear combination](https://math2.tamnd.com/node/0JB3DP1BMF.md): definition 0JB3DP1BMF
- [Graph of a function](https://math2.tamnd.com/node/0JBJYT69ZV.md): concept 0JBJYT69ZV
- [Ratio formula for the radius of convergence](https://math2.tamnd.com/node/0JBMYAA9QH.md): statement 0JBMYAA9QH
- [Vector calculus identities](https://math2.tamnd.com/node/0JC5X4HWR2.md): statement 0JC5X4HWR2
- [Stirling numbers of the second kind](https://math2.tamnd.com/node/0JCQGPY8Z4.md): concept 0JCQGPY8Z4
- [Converse relation](https://math2.tamnd.com/node/0JCSQNNP2M.md): definition 0JCSQNNP2M
- [Graph](https://math2.tamnd.com/node/0JD5C82QMB.md): definition 0JD5C82QMB
- [Analytic function](https://math2.tamnd.com/node/0JD5ZNW0F4.md): definition 0JD5ZNW0F4
- [Convergence in distribution](https://math2.tamnd.com/node/0JDAQHWA9N.md): definition 0JDAQHWA9N
- [Triangle](https://math2.tamnd.com/node/0JDB8YJBSR.md): definition 0JDB8YJBSR
- [Path](https://math2.tamnd.com/node/0JDE56HWQ8.md): definition 0JDE56HWQ8
- [Continuity in R^n](https://math2.tamnd.com/node/0JEJATBVD7.md): concept 0JEJATBVD7
- [Sign is a homomorphism](https://math2.tamnd.com/node/0JEJJCB0EX.md): statement 0JEJJCB0EX
- [The cumulative distribution function determines the distribution](https://math2.tamnd.com/node/0JEVZAFDY6.md): statement 0JEVZAFDY6
- [Affine map](https://math2.tamnd.com/node/0JF0TVYGWT.md): concept 0JF0TVYGWT
- [Parametrised curve](https://math2.tamnd.com/node/0JF1XXGWB4.md): definition 0JF1XXGWB4
- [Regular space](https://math2.tamnd.com/node/0JF9KW2TXB.md): definition 0JF9KW2TXB
- [Residue](https://math2.tamnd.com/node/0JFGKKWWFK.md): definition 0JFGKKWWFK
- [Turán's theorem](https://math2.tamnd.com/node/0JG1KZSPFZ.md): statement 0JG1KZSPFZ
- [Connected component](https://math2.tamnd.com/node/0JG9W3KA6B.md): concept 0JG9W3KA6B
- [Rank of a linear map](https://math2.tamnd.com/node/0JGB7A6GTS.md): definition 0JGB7A6GTS
- [Presentation of a group](https://math2.tamnd.com/node/0JGEQXQ4AC.md): concept 0JGEQXQ4AC
- [Space of polynomials of bounded degree](https://math2.tamnd.com/node/0JGZCDT5DN.md): example 0JGZCDT5DN
- [Series](https://math2.tamnd.com/node/0JH4MN4K1C.md): definition 0JH4MN4K1C
- [Eigenspaces of a self-adjoint operator are orthogonal](https://math2.tamnd.com/node/0JHGAV9H25.md): statement 0JHGAV9H25
- [The ring of integers](https://math2.tamnd.com/node/0JHM6JK7FD.md): example 0JHM6JK7FD
- [Exponential generating function](https://math2.tamnd.com/node/0JHXPVFKT0.md): concept 0JHXPVFKT0
- [Underdamped](https://math2.tamnd.com/node/0JJ6GDDE7Z.md): concept 0JJ6GDDE7Z
- [Boundary](https://math2.tamnd.com/node/0JJ9J1XZ8G.md): definition 0JJ9J1XZ8G
- [Cauchy's estimates](https://math2.tamnd.com/node/0JJRENWMP6.md): statement 0JJRENWMP6
- [Abelian simple groups have prime order](https://math2.tamnd.com/node/0JJV80RXHD.md): statement 0JJV80RXHD
- [Kronecker product](https://math2.tamnd.com/node/0JK6M18QZH.md): definition 0JK6M18QZH
- [Combinatorics](https://math2.tamnd.com/node/0JKW7W7XFR.md): concept 0JKW7W7XFR
- [Cauchy's theorem](https://math2.tamnd.com/node/0JM7PA288B.md): statement 0JM7PA288B
- [Logical equivalence](https://math2.tamnd.com/node/0JMR2HM2N2.md): concept 0JMR2HM2N2
- [Sum of subspaces](https://math2.tamnd.com/node/0JMTY8MF54.md): concept 0JMTY8MF54
- [Kernel of a ring homomorphism](https://math2.tamnd.com/node/0JN0SQG2JH.md): concept 0JN0SQG2JH
- [Sample variance](https://math2.tamnd.com/node/0JN9P4A944.md): definition 0JN9P4A944
- [Lindemann–Weierstrass theorem](https://math2.tamnd.com/node/0JNCBFMPXR.md): statement 0JNCBFMPXR
- [Convolution theorem for the Laplace transform](https://math2.tamnd.com/node/0JNMJEDD93.md): statement 0JNMJEDD93
- [Universal set](https://math2.tamnd.com/node/0JNXSS9TJ9.md): concept 0JNXSS9TJ9
- [Fundamental theorem of algebra](https://math2.tamnd.com/node/0JP0W07Y9A.md): statement 0JP0W07Y9A
- [Kőnig's theorem](https://math2.tamnd.com/node/0JP0ZF57PN.md): statement 0JP0ZF57PN
- [Function of several variables](https://math2.tamnd.com/node/0JPCHTXWG0.md): concept 0JPCHTXWG0
- [Subgroups of abelian groups are normal](https://math2.tamnd.com/node/0JPDTEDCFD.md): statement 0JPDTEDCFD
- [Central binomial coefficient](https://math2.tamnd.com/node/0JPH2Y4DFV.md): concept 0JPH2Y4DFV
- [Order of an element](https://math2.tamnd.com/node/0JPYSD5E0B.md): definition 0JPYSD5E0B
- [Order of the general linear group over a finite field](https://math2.tamnd.com/node/0JQ1KNQW2K.md): statement 0JQ1KNQW2K
- [Alternating harmonic series](https://math2.tamnd.com/node/0JQ5Y3WWVE.md): example 0JQ5Y3WWVE
- [Weierstrass approximation theorem](https://math2.tamnd.com/node/0JQE130DHW.md): statement 0JQE130DHW
- [Ordered pair](https://math2.tamnd.com/node/0JQWWFCS9R.md): definition 0JQWWFCS9R
- [Multiplication of natural numbers](https://math2.tamnd.com/node/0JRC32D6K2.md): definition 0JRC32D6K2
- [Inverse Laplace transform](https://math2.tamnd.com/node/0JRW4X1MDC.md): definition 0JRW4X1MDC
- [Bernstein polynomial](https://math2.tamnd.com/node/0JT92VK307.md): concept 0JT92VK307
- [Legendre symbol](https://math2.tamnd.com/node/0JTC045NNZ.md): concept 0JTC045NNZ
- [Cardinality](https://math2.tamnd.com/node/0JTGRV4WT4.md): definition 0JTGRV4WT4
- [Estimate for approximate solutions](https://math2.tamnd.com/node/0JTPMCCZT2.md): statement 0JTPMCCZT2
- [System of linear equations](https://math2.tamnd.com/node/0JTRMBQ1SA.md): definition 0JTRMBQ1SA
- [Tychonoff's theorem](https://math2.tamnd.com/node/0JTRSJKCFB.md): statement 0JTRSJKCFB
- [Integer partition](https://math2.tamnd.com/node/0JV30XN73Z.md): concept 0JV30XN73Z
- [Gaussian elimination](https://math2.tamnd.com/node/0JVJ2WCXZH.md): concept 0JVJ2WCXZH
- [Split polynomial](https://math2.tamnd.com/node/0JVSNJCN68.md): concept 0JVSNJCN68
- [Vector projection](https://math2.tamnd.com/node/0JW13B48YP.md): concept 0JW13B48YP
- [Existential quantifier](https://math2.tamnd.com/node/0JW2FHW7G3.md): definition 0JW2FHW7G3
- [Saddle point](https://math2.tamnd.com/node/0JW6MDHAZD.md): concept 0JW6MDHAZD
- [Minimum spanning tree](https://math2.tamnd.com/node/0JW7KT6PQW.md): concept 0JW7KT6PQW
- [Concyclic points](https://math2.tamnd.com/node/0JWCX0TW2S.md): definition 0JWCX0TW2S
- [An orthonormal family is linearly independent](https://math2.tamnd.com/node/0JWM6GJNQW.md): statement 0JWM6GJNQW
- [Divergent series](https://math2.tamnd.com/node/0JWMJSTBSG.md): concept 0JWMJSTBSG
- [Bayesian inference](https://math2.tamnd.com/node/0JWRJX2RMP.md): concept 0JWRJX2RMP
- [L'Hôpital's rule](https://math2.tamnd.com/node/0JWT4PPQ7B.md): statement 0JWT4PPQ7B
- [Sylow p-subgroup](https://math2.tamnd.com/node/0JWZX6FKSH.md): definition 0JWZX6FKSH
- [Divergence theorem](https://math2.tamnd.com/node/0JX86PSSA9.md): statement 0JX86PSSA9
- [Legendre's theorem on continued fractions](https://math2.tamnd.com/node/0JXTQ392ZM.md): statement 0JXTQ392ZM
- [Square matrix](https://math2.tamnd.com/node/0JXZKX526R.md): concept 0JXZKX526R
- [Lifting-the-exponent lemma](https://math2.tamnd.com/node/0JYF6PTKEW.md): statement 0JYF6PTKEW
- [Group homomorphism](https://math2.tamnd.com/node/0JYTG5K9Y4.md): concept 0JYTG5K9Y4
- [Dihedral group](https://math2.tamnd.com/node/0JZ27DMJ5D.md): concept 0JZ27DMJ5D
- [Power rule](https://math2.tamnd.com/node/0JZGW75C5Z.md): statement 0JZGW75C5Z
- [Combination](https://math2.tamnd.com/node/0JZSRWRBZE.md): definition 0JZSRWRBZE
- [Galois group of a polynomial](https://math2.tamnd.com/node/0JZY3R908H.md): definition 0JZY3R908H
- [Convex set](https://math2.tamnd.com/node/0K06X9QXDX.md): definition 0K06X9QXDX
- [Laplace transform of a derivative](https://math2.tamnd.com/node/0K090GVHVN.md): statement 0K090GVHVN
- [Upper and lower Darboux sums](https://math2.tamnd.com/node/0K0A06X2MQ.md): concept 0K0A06X2MQ
- [Path-connected space](https://math2.tamnd.com/node/0K0GE6BCYW.md): definition 0K0GE6BCYW
- [Sum of a series](https://math2.tamnd.com/node/0K0H5E3Z5N.md): definition 0K0H5E3Z5N
- [Distance from a point to a plane](https://math2.tamnd.com/node/0K1E2S6Z69.md): statement 0K1E2S6Z69
- [Lipschitz functions are uniformly continuous](https://math2.tamnd.com/node/0K1EWAR6AF.md): statement 0K1EWAR6AF
- [Determinant of a triangular matrix](https://math2.tamnd.com/node/0K1KKMFHYH.md): statement 0K1KKMFHYH
- [Integral of 1/x](https://math2.tamnd.com/node/0K1N1Y4EGZ.md): statement 0K1N1Y4EGZ
- [Automorphism group](https://math2.tamnd.com/node/0K1SCPBNKJ.md): concept 0K1SCPBNKJ
- [Zero divisor](https://math2.tamnd.com/node/0K26YKNBSA.md): definition 0K26YKNBSA
- [Scalar multiplication of matrices](https://math2.tamnd.com/node/0K28YXMS7Q.md): definition 0K28YXMS7Q
- [Brahmagupta–Fibonacci identity](https://math2.tamnd.com/node/0K2DMACEZ4.md): statement 0K2DMACEZ4
- [Logarithm of a product](https://math2.tamnd.com/node/0K2FNY6DYV.md): statement 0K2FNY6DYV
- [Singular value](https://math2.tamnd.com/node/0K2TX12X58.md): definition 0K2TX12X58
- [Minkowski's theorem](https://math2.tamnd.com/node/0K387S2PKQ.md): statement 0K387S2PKQ
- [Solving initial value problems with the Laplace transform](https://math2.tamnd.com/node/0K3HFZJK5Q.md): concept 0K3HFZJK5Q
- [Factor theorem](https://math2.tamnd.com/node/0K3MWA7DMC.md): statement 0K3MWA7DMC
- [Topologist's sine curve](https://math2.tamnd.com/node/0K3T4A659R.md): counterexample 0K3T4A659R
- [Confidence interval](https://math2.tamnd.com/node/0K4NMMJBG2.md): definition 0K4NMMJBG2
- [Determinant of a matrix with two equal rows](https://math2.tamnd.com/node/0K4VQB7HWR.md): statement 0K4VQB7HWR
- [Rayleigh quotient](https://math2.tamnd.com/node/0K4WCNTMHG.md): definition 0K4WCNTMHG
- [Homogeneous first-order equation](https://math2.tamnd.com/node/0K4ZWTFSV5.md): definition 0K4ZWTFSV5
- [Differentiable function](https://math2.tamnd.com/node/0K57YG3BB8.md): concept 0K57YG3BB8
- [Action by left multiplication](https://math2.tamnd.com/node/0K5TWJAGPD.md): example 0K5TWJAGPD
- [Gaussian integers](https://math2.tamnd.com/node/0K5W69AC0J.md): object 0K5W69AC0J
- [Matrix exponential](https://math2.tamnd.com/node/0K665H5V1S.md): definition 0K665H5V1S
- [Complement of an open set is closed](https://math2.tamnd.com/node/0K67Q60Z8P.md): statement 0K67Q60Z8P
- [Alternating multilinear map](https://math2.tamnd.com/node/0K6CH0MT0F.md): concept 0K6CH0MT0F
- [Young diagram](https://math2.tamnd.com/node/0K78D81EF7.md): definition 0K78D81EF7
- [Maclaurin series](https://math2.tamnd.com/node/0K7P6PR838.md): concept 0K7P6PR838
- [Idempotent matrix](https://math2.tamnd.com/node/0K7TD9BV2D.md): definition 0K7TD9BV2D
- [Isolated zeros theorem](https://math2.tamnd.com/node/0K83KEJ43P.md): statement 0K83KEJ43P
- [Associativity](https://math2.tamnd.com/node/0K8F7PVDMX.md): concept 0K8F7PVDMX
- [Compactness and sequential compactness agree in metric spaces](https://math2.tamnd.com/node/0K8SGQSV7G.md): statement 0K8SGQSV7G
- [Removable discontinuity](https://math2.tamnd.com/node/0K96Z2S699.md): concept 0K96Z2S699
- [Lower central series](https://math2.tamnd.com/node/0K98R7A4VC.md): concept 0K98R7A4VC
- [Continuous image of a compact set is compact](https://math2.tamnd.com/node/0K9AAJ8G51.md): statement 0K9AAJ8G51
- [Parallelepiped](https://math2.tamnd.com/node/0K9GBTZ1FX.md): definition 0K9GBTZ1FX
- [Inverse function theorem](https://math2.tamnd.com/node/0K9HSJFCMJ.md): statement 0K9HSJFCMJ
- [General Leibniz rule](https://math2.tamnd.com/node/0K9QCK6WJC.md): statement 0K9QCK6WJC
- [Ramsey number](https://math2.tamnd.com/node/0K9QVPJYJF.md): concept 0K9QVPJYJF
- [Regular graph](https://math2.tamnd.com/node/0KA4S8R39M.md): definition 0KA4S8R39M
- [Determinant via the Schur complement](https://math2.tamnd.com/node/0KA7A4ENVJ.md): statement 0KA7A4ENVJ
- [Upper and lower Darboux integrals](https://math2.tamnd.com/node/0KABP2HWMQ.md): concept 0KABP2HWMQ
- [Characteristic subgroup](https://math2.tamnd.com/node/0KABPMRHM2.md): concept 0KABPMRHM2
- [Sum of binomial coefficients](https://math2.tamnd.com/node/0KAMF19NV8.md): statement 0KAMF19NV8
- [Nondegenerate bilinear form](https://math2.tamnd.com/node/0KAN330PAH.md): definition 0KAN330PAH
- [Subgroups of cyclic groups are cyclic](https://math2.tamnd.com/node/0KB1ND7XDS.md): statement 0KB1ND7XDS
- [Distributive law](https://math2.tamnd.com/node/0KBEMAJAYA.md): statement 0KBEMAJAYA
- [Covariance](https://math2.tamnd.com/node/0KBFAWYRTE.md): concept 0KBFAWYRTE
- [Reduced word](https://math2.tamnd.com/node/0KBTP8Q2FS.md): concept 0KBTP8Q2FS
- [Bernoulli number](https://math2.tamnd.com/node/0KC00K7KY4.md): concept 0KC00K7KY4
- [Stirling's formula](https://math2.tamnd.com/node/0KC0F51T6A.md): statement 0KC0F51T6A
- [Wronskian](https://math2.tamnd.com/node/0KC1NMG90B.md): concept 0KC1NMG90B
- [Idempotent matrix](https://math2.tamnd.com/node/0KCE8CG86P.md): concept 0KCE8CG86P
- [Basis](https://math2.tamnd.com/node/0KCQ9Y3QKA.md): concept 0KCQ9Y3QKA
- [Venn diagram](https://math2.tamnd.com/node/0KCT6DPXJK.md): concept 0KCT6DPXJK
- [Fundamental solution of Pell's equation](https://math2.tamnd.com/node/0KCYJSB9FW.md): definition 0KCYJSB9FW
- [Vertex cover](https://math2.tamnd.com/node/0KDADE9RQ8.md): definition 0KDADE9RQ8
- [Bolzano–Weierstrass theorem](https://math2.tamnd.com/node/0KDJD249DT.md): statement 0KDJD249DT
- [Exponential generating function](https://math2.tamnd.com/node/0KE6NBTVY8.md): definition 0KE6NBTVY8
- [Classification of Pythagorean triples](https://math2.tamnd.com/node/0KF3J2KAER.md): statement 0KF3J2KAER
- [Directed graph](https://math2.tamnd.com/node/0KFYGSHHF5.md): concept 0KFYGSHHF5
- [The alternating group is generated by three-cycles](https://math2.tamnd.com/node/0KFZ0QX83G.md): statement 0KFZ0QX83G
- [Norm induced by an inner product](https://math2.tamnd.com/node/0KG4D8AP6G.md): definition 0KG4D8AP6G
- [Dirichlet convolution](https://math2.tamnd.com/node/0KH1N66QWW.md): definition 0KH1N66QWW
- [Posterior probability](https://math2.tamnd.com/node/0KJ7ZS8AJA.md): concept 0KJ7ZS8AJA
- [Affine combination](https://math2.tamnd.com/node/0KJH6SRFZK.md): concept 0KJH6SRFZK
- [A polynomial over a field has at most as many roots as its degree](https://math2.tamnd.com/node/0KJVQV3E0F.md): statement 0KJVQV3E0F
- [One-sided limits of a monotone function exist](https://math2.tamnd.com/node/0KJXHN0TW0.md): statement 0KJXHN0TW0
- [Primitive root modulo n](https://math2.tamnd.com/node/0KM2CJ80G7.md): definition 0KM2CJ80G7
- [Hausdorff maximal principle](https://math2.tamnd.com/node/0KM4KVR9MV.md): statement 0KM4KVR9MV
- [Fixed field](https://math2.tamnd.com/node/0KMJ6919WS.md): definition 0KMJ6919WS
- [Determinant of a linear map is the determinant of its matrix](https://math2.tamnd.com/node/0KMN32XGPF.md): statement 0KMN32XGPF
- [Strong induction](https://math2.tamnd.com/node/0KMVFSHFGZ.md): statement 0KMVFSHFGZ
- [Monotone sequence](https://math2.tamnd.com/node/0KN22AAW5R.md): definition 0KN22AAW5R
- [Ring isomorphism](https://math2.tamnd.com/node/0KN2S81K72.md): concept 0KN2S81K72
- [The exponential function is strictly increasing](https://math2.tamnd.com/node/0KN2V7840E.md): statement 0KN2V7840E
- [Restriction of an endomorphism to an invariant subspace](https://math2.tamnd.com/node/0KN7EPH7Z0.md): concept 0KN7EPH7Z0
- [Ceva's theorem](https://math2.tamnd.com/node/0KN93VDSXA.md): statement 0KN93VDSXA
- [Maximum degree](https://math2.tamnd.com/node/0KNAN9F27P.md): definition 0KNAN9F27P
- [p-norm on R^n](https://math2.tamnd.com/node/0KNJWEVSMG.md): example 0KNJWEVSMG
- [Homogeneous linear equation](https://math2.tamnd.com/node/0KP0QNXPRT.md): concept 0KP0QNXPRT
- [Dirac delta function](https://math2.tamnd.com/node/0KPHENZ5S7.md): object 0KPHENZ5S7
- [Signature of a quadratic form](https://math2.tamnd.com/node/0KQZYGEJTF.md): definition 0KQZYGEJTF
- [Stationary distribution](https://math2.tamnd.com/node/0KRH5157WP.md): concept 0KRH5157WP
- [Continuity at a point](https://math2.tamnd.com/node/0KRMJRDRZT.md): concept 0KRMJRDRZT
- [Symmetric bilinear form associated with a quadratic form](https://math2.tamnd.com/node/0KRVRV5V80.md): concept 0KRVRV5V80
- [Degree of a vertex](https://math2.tamnd.com/node/0KS0KRDV4Y.md): concept 0KS0KRDV4Y
- [Nowhere dense set](https://math2.tamnd.com/node/0KSM23SN3H.md): definition 0KSM23SN3H
- [Equivalence relation induced by a function](https://math2.tamnd.com/node/0KT8TGXD8K.md): concept 0KT8TGXD8K
- [Jordan measure](https://math2.tamnd.com/node/0KTGJW1ZTH.md): concept 0KTGJW1ZTH
- [Free group](https://math2.tamnd.com/node/0KTW26C80J.md): concept 0KTW26C80J
- [Cyclic group](https://math2.tamnd.com/node/0KVGH0J5B3.md): concept 0KVGH0J5B3
- [Variance under scaling](https://math2.tamnd.com/node/0KVQ9D86BM.md): statement 0KVQ9D86BM
- [Modulus of a complex number](https://math2.tamnd.com/node/0KW2X0TFBX.md): definition 0KW2X0TFBX
- [Diameter of a set](https://math2.tamnd.com/node/0KW3PN547K.md): concept 0KW3PN547K
- [Connected graph](https://math2.tamnd.com/node/0KWFFMDF8T.md): concept 0KWFFMDF8T
- [Reduction modulo p test for irreducibility](https://math2.tamnd.com/node/0KWWEXW0RN.md): statement 0KWWEXW0RN
- [Change of basis matrix](https://math2.tamnd.com/node/0KWXYZPJKN.md): definition 0KWXYZPJKN
- [Radius of convergence of the binomial series](https://math2.tamnd.com/node/0KX1MXJ5MR.md): statement 0KX1MXJ5MR
- [Positive definite matrix](https://math2.tamnd.com/node/0KX66CZMBB.md): concept 0KX66CZMBB
- [Usual metric on the real line](https://math2.tamnd.com/node/0KXJTZSW9B.md): example 0KXJTZSW9B
- [Number of prime factors counted with multiplicity](https://math2.tamnd.com/node/0KXQKES075.md): concept 0KXQKES075
- [Splitting field](https://math2.tamnd.com/node/0KXQKX9P63.md): concept 0KXQKX9P63
- [Continuous partial derivatives imply differentiability](https://math2.tamnd.com/node/0KXSK8TEEB.md): statement 0KXSK8TEEB
- [Derivative of x^a](https://math2.tamnd.com/node/0KY08TWDMQ.md): statement 0KY08TWDMQ
- [Stirling numbers of the first kind](https://math2.tamnd.com/node/0KY7GS695G.md): concept 0KY7GS695G
- [Discrete uniform distribution](https://math2.tamnd.com/node/0KYP951QWH.md): definition 0KYP951QWH
- [Quotient group](https://math2.tamnd.com/node/0KYX2N69J0.md): concept 0KYX2N69J0
- [Lagrange's four-square theorem](https://math2.tamnd.com/node/0KYYSMEVHH.md): statement 0KYYSMEVHH
- [Quadratic formula](https://math2.tamnd.com/node/0KYZXYDGQZ.md): statement 0KYZXYDGQZ
- [Trace of a linear map](https://math2.tamnd.com/node/0KZDNGA561.md): concept 0KZDNGA561
- [Independent set](https://math2.tamnd.com/node/0KZJ82WEPP.md): definition 0KZJ82WEPP
- [Diffeomorphism](https://math2.tamnd.com/node/0KZZ70ZHXP.md): concept 0KZZ70ZHXP
- [Antisymmetric relation](https://math2.tamnd.com/node/0M0H1ZH8TT.md): concept 0M0H1ZH8TT
- [Derivative of an inverse function](https://math2.tamnd.com/node/0M0RZ3HSB0.md): statement 0M0RZ3HSB0
- [Block diagonal matrix](https://math2.tamnd.com/node/0M0SNKZ482.md): definition 0M0SNKZ482
- [Linearity of differentiation](https://math2.tamnd.com/node/0M0TKFRX2A.md): statement 0M0TKFRX2A
- [Intercept theorem](https://math2.tamnd.com/node/0M1GFS6PVA.md): statement 0M1GFS6PVA
- [Convex function on R^n](https://math2.tamnd.com/node/0M1GJSWZDA.md): definition 0M1GJSWZDA
- [Bernoulli's inequality](https://math2.tamnd.com/node/0M1MX28VXK.md): statement 0M1MX28VXK
- [Fermat's little theorem](https://math2.tamnd.com/node/0M25PRN3WQ.md): statement 0M25PRN3WQ
- [Conjunction](https://math2.tamnd.com/node/0M28ASWG8T.md): definition 0M28ASWG8T
- [Curl of a gradient is zero](https://math2.tamnd.com/node/0M2JRW1J0N.md): statement 0M2JRW1J0N
- [Cayley's formula](https://math2.tamnd.com/node/0M2SSGSYPE.md): statement 0M2SSGSYPE
- [The number of Sylow subgroups divides the index](https://math2.tamnd.com/node/0M3PYJ3TSE.md): statement 0M3PYJ3TSE
- [Analytic continuation](https://math2.tamnd.com/node/0M3YQJ81NB.md): concept 0M3YQJ81NB
- [Four colour theorem](https://math2.tamnd.com/node/0M406JC8J1.md): statement 0M406JC8J1
- [Binary operation](https://math2.tamnd.com/node/0M418FE7BB.md): concept 0M418FE7BB
- [Uniform continuity](https://math2.tamnd.com/node/0M4JJT780X.md): concept 0M4JJT780X
- [Every linearly independent set extends to a basis](https://math2.tamnd.com/node/0M4SXE5WW3.md): statement 0M4SXE5WW3
- [Classification of the Platonic solids](https://math2.tamnd.com/node/0M4TCS0RRG.md): statement 0M4TCS0RRG
- [Line integral of a scalar field](https://math2.tamnd.com/node/0M50E4WYWJ.md): definition 0M50E4WYWJ
- [Existence of inverses modulo n](https://math2.tamnd.com/node/0M67FWZWPQ.md): statement 0M67FWZWPQ
- [The only ideals of a field are zero and the whole field](https://math2.tamnd.com/node/0M6G75E95E.md): statement 0M6G75E95E
- [Universal property of the product topology](https://math2.tamnd.com/node/0M6W4XNST6.md): statement 0M6W4XNST6
- [Elementary matrix](https://math2.tamnd.com/node/0M79HRJCRR.md): definition 0M79HRJCRR
- [Pythagorean triple](https://math2.tamnd.com/node/0M7ARED5Q7.md): concept 0M7ARED5Q7
- [Euler's criterion](https://math2.tamnd.com/node/0M7DEDKSYB.md): statement 0M7DEDKSYB
- [Complementary subspaces](https://math2.tamnd.com/node/0M83V17KHX.md): concept 0M83V17KHX
- [Dual space](https://math2.tamnd.com/node/0M856Q1AMZ.md): definition 0M856Q1AMZ
- [Equivalent norms](https://math2.tamnd.com/node/0M85DXB5RM.md): concept 0M85DXB5RM
- [Adjoining a root of a polynomial](https://math2.tamnd.com/node/0M893X7YY7.md): definition 0M893X7YY7
- [Vieta's formulas](https://math2.tamnd.com/node/0M8EWQJ1YV.md): statement 0M8EWQJ1YV
- [Nilpotent endomorphism](https://math2.tamnd.com/node/0M8MHYERTY.md): definition 0M8MHYERTY
- [Approximation property of the supremum](https://math2.tamnd.com/node/0M8TGGP9JP.md): statement 0M8TGGP9JP
- [Coefficient of a power series](https://math2.tamnd.com/node/0M98XYVX6S.md): definition 0M98XYVX6S
- [Incentre](https://math2.tamnd.com/node/0M9MX8YW0A.md): concept 0M9MX8YW0A
- [Step function](https://math2.tamnd.com/node/0M9XEDF1HJ.md): definition 0M9XEDF1HJ
- [Field generated by adjoining elements](https://math2.tamnd.com/node/0MA5B744K3.md): definition 0MA5B744K3
- [Schwarz lemma](https://math2.tamnd.com/node/0MB48XDBY3.md): statement 0MB48XDBY3
- [Algebraic number](https://math2.tamnd.com/node/0MB4TGMXG6.md): definition 0MB4TGMXG6
- [Self-adjoint operator](https://math2.tamnd.com/node/0MB8HQW495.md): definition 0MB8HQW495
- [Sum of ideals](https://math2.tamnd.com/node/0MBJVDSK1R.md): concept 0MBJVDSK1R
- [Generators of a group](https://math2.tamnd.com/node/0MBV2VHMXA.md): concept 0MBV2VHMXA
- [Monotonicity of the integral](https://math2.tamnd.com/node/0MBWQKM8QR.md): statement 0MBWQKM8QR
- [A finite abelian group is cyclic iff its exponent equals its order](https://math2.tamnd.com/node/0MC0AQZSHE.md): statement 0MC0AQZSHE
- [Orthogonal projection is the closest point](https://math2.tamnd.com/node/0MCDDSCB9Z.md): statement 0MCDDSCB9Z
- [Epsilon characterisation of Cauchy sequences](https://math2.tamnd.com/node/0MCK8K2D7Z.md): statement 0MCK8K2D7Z
- [Minimal element](https://math2.tamnd.com/node/0MCR8ZAXQD.md): definition 0MCR8ZAXQD
- [Algebraic closure](https://math2.tamnd.com/node/0MCXX29HWY.md): definition 0MCXX29HWY
- [Complement graph](https://math2.tamnd.com/node/0MCYJ72VGA.md): concept 0MCYJ72VGA
- [Coprime ideals](https://math2.tamnd.com/node/0MDJP82TEH.md): definition 0MDJP82TEH
- [Exterior algebra](https://math2.tamnd.com/node/0MDMZ965VE.md): concept 0MDMZ965VE
- [Symmetric difference](https://math2.tamnd.com/node/0ME21QN125.md): concept 0ME21QN125
- [Division algorithm for polynomials](https://math2.tamnd.com/node/0MES3KS0ZC.md): statement 0MES3KS0ZC
- [Norm induced by an inner product](https://math2.tamnd.com/node/0MFJJV97RJ.md): concept 0MFJJV97RJ
- [Derivative of a polynomial](https://math2.tamnd.com/node/0MH1WK89PP.md): statement 0MH1WK89PP
- [Wronskian test for linear independence of solutions](https://math2.tamnd.com/node/0MHE4B1QCQ.md): statement 0MHE4B1QCQ
- [Every sequence has a monotone subsequence](https://math2.tamnd.com/node/0MHGJRZMTM.md): statement 0MHGJRZMTM
- [Matching](https://math2.tamnd.com/node/0MJ04A50QW.md): concept 0MJ04A50QW
- [Tree](https://math2.tamnd.com/node/0MJ0JS72DZ.md): definition 0MJ0JS72DZ
- [Upper triangular matrix](https://math2.tamnd.com/node/0MKABWDRNC.md): definition 0MKABWDRNC
- [Change of basis matrix](https://math2.tamnd.com/node/0MKRHZB3WS.md): concept 0MKRHZB3WS
- [Product of vector spaces](https://math2.tamnd.com/node/0MM085RX00.md): definition 0MM085RX00
- [One-point compactification](https://math2.tamnd.com/node/0MM45BT6HC.md): definition 0MM45BT6HC
- [A ring homomorphism is injective iff its kernel is zero](https://math2.tamnd.com/node/0MMCB8RXGA.md): statement 0MMCB8RXGA
- [Continuum hypothesis](https://math2.tamnd.com/node/0MN2MER9WN.md): statement 0MN2MER9WN
- [Derivative as a little-o approximation](https://math2.tamnd.com/node/0MN882VJ4P.md): statement 0MN882VJ4P
- [Algebraic number](https://math2.tamnd.com/node/0MNF0BYFQC.md): concept 0MNF0BYFQC
- [Prime element](https://math2.tamnd.com/node/0MNHXBSX3M.md): definition 0MNHXBSX3M
- [Geometric sequence](https://math2.tamnd.com/node/0MNM54WB99.md): definition 0MNM54WB99
- [Product of rings](https://math2.tamnd.com/node/0MNZCN4BQK.md): definition 0MNZCN4BQK
- [Line](https://math2.tamnd.com/node/0MP9BCF6DP.md): concept 0MP9BCF6DP
- [Diagonalisable iff the minimal polynomial has distinct roots](https://math2.tamnd.com/node/0MPAJRB23T.md): statement 0MPAJRB23T
- [Centre of mass](https://math2.tamnd.com/node/0MPZGAZ2GK.md): definition 0MPZGAZ2GK
- [Fundamental matrix](https://math2.tamnd.com/node/0MQ24RBG93.md): definition 0MQ24RBG93
- [Compact Hausdorff spaces are normal](https://math2.tamnd.com/node/0MQQQ5NA2S.md): statement 0MQQQ5NA2S
- [Change of basis formula](https://math2.tamnd.com/node/0MQXVZEBMT.md): statement 0MQXVZEBMT
- [Normal matrix](https://math2.tamnd.com/node/0MR0BYF0K2.md): concept 0MR0BYF0K2
- [Series of functions](https://math2.tamnd.com/node/0MRE6XRE5V.md): definition 0MRE6XRE5V
- [A quotient is abelian iff the subgroup contains the commutator subgroup](https://math2.tamnd.com/node/0MS8QGA6S5.md): statement 0MS8QGA6S5
- [Test for exactness](https://math2.tamnd.com/node/0MSCQNFTM8.md): statement 0MSCQNFTM8
- [Existential quantifier](https://math2.tamnd.com/node/0MSRQHCNN9.md): concept 0MSRQHCNN9
- [Order on the reals](https://math2.tamnd.com/node/0MSSD1VCW9.md): concept 0MSSD1VCW9
- [Alternating multilinear map](https://math2.tamnd.com/node/0MSWYWHYF5.md): definition 0MSWYWHYF5
- [Contraction mapping](https://math2.tamnd.com/node/0MT2S0ZX4Y.md): definition 0MT2S0ZX4Y
- [First isomorphism theorem for linear maps](https://math2.tamnd.com/node/0MTBGVZ4XJ.md): statement 0MTBGVZ4XJ
- [One-form](https://math2.tamnd.com/node/0MTENFBW3Q.md): definition 0MTENFBW3Q
- [Poisson point process](https://math2.tamnd.com/node/0MTNR28K60.md): concept 0MTNR28K60
- [e](https://math2.tamnd.com/node/0MVCFMW7S4.md): object 0MVCFMW7S4
- [Sequential continuity](https://math2.tamnd.com/node/0MVDQEAB60.md): definition 0MVDQEAB60
- [Noetherian ring](https://math2.tamnd.com/node/0MVH4Y1CRZ.md): definition 0MVH4Y1CRZ
- [Exponentials dominate powers in sequences](https://math2.tamnd.com/node/0MVTCFW42K.md): statement 0MVTCFW42K
- [Equicontinuity](https://math2.tamnd.com/node/0MW2WXTHMC.md): definition 0MW2WXTHMC
- [Natural numbers](https://math2.tamnd.com/node/0MWKJ44V5A.md): object 0MWKJ44V5A
- [Divergence](https://math2.tamnd.com/node/0MWV6ZTK6S.md): concept 0MWV6ZTK6S
- [Hyperbolic tangent](https://math2.tamnd.com/node/0MWYQTZ07Z.md): definition 0MWYQTZ07Z
- [A series of nonnegative terms converges iff its partial sums are bounded](https://math2.tamnd.com/node/0MX1HZBBXJ.md): statement 0MX1HZBBXJ
- [Hausdorff space](https://math2.tamnd.com/node/0MXB4CKHFW.md): definition 0MXB4CKHFW
- [Coupon collector's problem](https://math2.tamnd.com/node/0MXCP7MXWJ.md): problem 0MXCP7MXWJ
- [Mantel's theorem](https://math2.tamnd.com/node/0MXEHSCYCS.md): statement 0MXEHSCYCS
- [Deficient number](https://math2.tamnd.com/node/0MXFJ0J00R.md): concept 0MXFJ0J00R
- [Perfect number](https://math2.tamnd.com/node/0MXGYCH4SY.md): concept 0MXGYCH4SY
- [Cumulative distribution function](https://math2.tamnd.com/node/0MXMS3AE13.md): definition 0MXMS3AE13
- [Open connected subsets of R^n are path-connected](https://math2.tamnd.com/node/0MXN7AFBPE.md): statement 0MXN7AFBPE
- [Marginal distribution](https://math2.tamnd.com/node/0MXT6PRZZ6.md): concept 0MXT6PRZZ6
- [Characteristic function](https://math2.tamnd.com/node/0MYDNRT7PP.md): definition 0MYDNRT7PP
- [Compact subsets of a Hausdorff space are closed](https://math2.tamnd.com/node/0MYFJVMN7M.md): statement 0MYFJVMN7M
- [Ideal](https://math2.tamnd.com/node/0MYSA6MYNA.md): definition 0MYSA6MYNA
- [Power series](https://math2.tamnd.com/node/0MZB06ST2B.md): concept 0MZB06ST2B
- [Separable extension](https://math2.tamnd.com/node/0N019F6RE9.md): definition 0N019F6RE9
- [Subgroup](https://math2.tamnd.com/node/0N0PXCTYA7.md): definition 0N0PXCTYA7
- [Well-ordering theorem](https://math2.tamnd.com/node/0N109NG5ZX.md): statement 0N109NG5ZX
- [Irrationality of square roots of non-squares](https://math2.tamnd.com/node/0N10YRNNWV.md): statement 0N10YRNNWV
- [Riemann integrable function](https://math2.tamnd.com/node/0N1JQXZXHC.md): definition 0N1JQXZXHC
- [Matrix](https://math2.tamnd.com/node/0N1MY1T0W9.md): definition 0N1MY1T0W9
- [Completeness via Cauchy sequences](https://math2.tamnd.com/node/0N1ZWFJDN5.md): statement 0N1ZWFJDN5
- [Radius of convergence](https://math2.tamnd.com/node/0N288P2TB5.md): concept 0N288P2TB5
- [Symmetric matrix](https://math2.tamnd.com/node/0N28QN4CYA.md): definition 0N28QN4CYA
- [A homomorphism is injective iff its kernel is trivial](https://math2.tamnd.com/node/0N2P0XRTDF.md): statement 0N2P0XRTDF
- [Cumulant generating function](https://math2.tamnd.com/node/0N2SP6K9ZC.md): concept 0N2SP6K9ZC
- [The rationals form a field](https://math2.tamnd.com/node/0N2WPCC00T.md): statement 0N2WPCC00T
- [Limits preserve non-strict inequalities](https://math2.tamnd.com/node/0N32V0E3XJ.md): statement 0N32V0E3XJ
- [Uniqueness quantifier](https://math2.tamnd.com/node/0N3ADS4CHH.md): definition 0N3ADS4CHH
- [Composite number](https://math2.tamnd.com/node/0N3ADXHNTQ.md): definition 0N3ADXHNTQ
- [Liouville's theorem on Diophantine approximation](https://math2.tamnd.com/node/0N3WFZEFPM.md): statement 0N3WFZEFPM
- [Hilbert's hotel](https://math2.tamnd.com/node/0N3Y8G340X.md): example 0N3Y8G340X
- [Quotient map](https://math2.tamnd.com/node/0N4F5FMQZW.md): concept 0N4F5FMQZW
- [Upper triangular matrix](https://math2.tamnd.com/node/0N4XY7Z3NN.md): concept 0N4XY7Z3NN
- [Pairwise independence](https://math2.tamnd.com/node/0N57D1XY6T.md): definition 0N57D1XY6T
- [Boolean ring](https://math2.tamnd.com/node/0N5FCV2YV6.md): definition 0N5FCV2YV6
- [Square root of a positive semidefinite matrix](https://math2.tamnd.com/node/0N5GP3YT8T.md): definition 0N5GP3YT8T
- [Falling factorial](https://math2.tamnd.com/node/0N5M924N3P.md): definition 0N5M924N3P
- [Cross product](https://math2.tamnd.com/node/0N5N15HWZ7.md): concept 0N5N15HWZ7
- [Wronskian](https://math2.tamnd.com/node/0N5NF1RGBP.md): definition 0N5NF1RGBP
- [Idempotent element](https://math2.tamnd.com/node/0N5NJZKKN3.md): concept 0N5NJZKKN3
- [Classification of conics by the discriminant](https://math2.tamnd.com/node/0N5NR32NQG.md): statement 0N5NR32NQG
- [Degree of a simple extension is the degree of the minimal polynomial](https://math2.tamnd.com/node/0N5QW9K7MH.md): statement 0N5QW9K7MH
- [Fermat primality test](https://math2.tamnd.com/node/0N5ZSZMYH6.md): concept 0N5ZSZMYH6
- [Vector field](https://math2.tamnd.com/node/0N67NBKQ5T.md): definition 0N67NBKQ5T
- [Simple extension](https://math2.tamnd.com/node/0N681SSTHC.md): concept 0N681SSTHC
- [Eigenvector](https://math2.tamnd.com/node/0N6960W0BV.md): concept 0N6960W0BV
- [Monoid](https://math2.tamnd.com/node/0N6R902FG1.md): concept 0N6R902FG1
- [Classification of equilibria of planar linear systems](https://math2.tamnd.com/node/0N7ATVF2X8.md): concept 0N7ATVF2X8
- [Kronecker product](https://math2.tamnd.com/node/0N7ENQRCXN.md): concept 0N7ENQRCXN
- [Equilibrium point](https://math2.tamnd.com/node/0N8NGKDZDZ.md): concept 0N8NGKDZDZ
- [Mean value theorem for integrals](https://math2.tamnd.com/node/0N91N98EG5.md): statement 0N91N98EG5
- [Fermat prime](https://math2.tamnd.com/node/0N9E4G2P97.md): definition 0N9E4G2P97
- [Nilpotent element](https://math2.tamnd.com/node/0N9R5NN6TW.md): definition 0N9R5NN6TW
- [Symmetric relation](https://math2.tamnd.com/node/0N9SA25Z3V.md): definition 0N9SA25Z3V
- [Basis of a topology](https://math2.tamnd.com/node/0NA0732YAG.md): concept 0NA0732YAG
- [Discrete metric](https://math2.tamnd.com/node/0NA4FK7B8S.md): example 0NA4FK7B8S
- [Cauchy distribution](https://math2.tamnd.com/node/0NA6Q82YR7.md): definition 0NA6Q82YR7
- [Ore's theorem](https://math2.tamnd.com/node/0NACHJYPKW.md): statement 0NACHJYPKW
- [Holomorphic function](https://math2.tamnd.com/node/0NAHXR1FRH.md): concept 0NAHXR1FRH
- [Apollonius's theorem](https://math2.tamnd.com/node/0NAKHQ9XYY.md): statement 0NAKHQ9XYY
- [Harmonic number](https://math2.tamnd.com/node/0NANDX7V02.md): concept 0NANDX7V02
- [Inner product](https://math2.tamnd.com/node/0NB0AWGJBW.md): concept 0NB0AWGJBW
- [exp(1/z) has an essential singularity at 0](https://math2.tamnd.com/node/0NB31CC4SN.md): example 0NB31CC4SN
- [Partial sum](https://math2.tamnd.com/node/0NB643K60Q.md): definition 0NB643K60Q
- [Asymptotic stability](https://math2.tamnd.com/node/0NB689782W.md): concept 0NB689782W
- [Composition series](https://math2.tamnd.com/node/0NB8VPB93V.md): definition 0NB8VPB93V
- [Divergent series](https://math2.tamnd.com/node/0NBRP6093V.md): definition 0NBRP6093V
- [The integral of a continuous function is an antiderivative](https://math2.tamnd.com/node/0NC2A3HWGT.md): statement 0NC2A3HWGT
- [Planar graph](https://math2.tamnd.com/node/0NC9R64TXP.md): concept 0NC9R64TXP
- [Rational numbers](https://math2.tamnd.com/node/0NCJ5WXN6Z.md): object 0NCJ5WXN6Z
- [Graph colouring](https://math2.tamnd.com/node/0NCPXH194Y.md): concept 0NCPXH194Y
- [Principal part](https://math2.tamnd.com/node/0NDV5R6Z6X.md): definition 0NDV5R6Z6X
- [Linear recurrence with constant coefficients](https://math2.tamnd.com/node/0NDVHBVBXB.md): concept 0NDVHBVBXB
- [Right coset](https://math2.tamnd.com/node/0NE1DK7H4T.md): concept 0NE1DK7H4T
- [Conditional convergence](https://math2.tamnd.com/node/0NE233WGD1.md): concept 0NE233WGD1
- [Implicit function theorem for a surjective derivative](https://math2.tamnd.com/node/0NE2W2XPQJ.md): statement 0NE2W2XPQJ
- [Self-adjoint operator](https://math2.tamnd.com/node/0NEGN5CRSC.md): concept 0NEGN5CRSC
- [Restriction of a function](https://math2.tamnd.com/node/0NF1KGDXX3.md): concept 0NF1KGDXX3
- [Curl](https://math2.tamnd.com/node/0NF6GQNXMY.md): concept 0NF6GQNXMY
- [Double negation law](https://math2.tamnd.com/node/0NFAGWH84K.md): statement 0NFAGWH84K
- [The minimal polynomial is irreducible](https://math2.tamnd.com/node/0NFEG48RDR.md): statement 0NFEG48RDR
- [Probability space](https://math2.tamnd.com/node/0NFPJVJJHX.md): concept 0NFPJVJJHX
- [Fundamental theorem of calculus](https://math2.tamnd.com/node/0NHGY0N0JB.md): statement 0NHGY0N0JB
- [Perfect matching](https://math2.tamnd.com/node/0NHK9MFFW9.md): concept 0NHK9MFFW9
- [Sample variance](https://math2.tamnd.com/node/0NHXGY2C5J.md): concept 0NHXGY2C5J
- [von Mangoldt function](https://math2.tamnd.com/node/0NK8JTE1NN.md): concept 0NK8JTE1NN
- [Double integral](https://math2.tamnd.com/node/0NKBMFC18R.md): definition 0NKBMFC18R
- [Series](https://math2.tamnd.com/node/0NM626FH6Q.md): concept 0NM626FH6Q
- [Bayes' theorem](https://math2.tamnd.com/node/0NM6E9VW04.md): statement 0NM6E9VW04
- [Variance as the mean of the square minus the square of the mean](https://math2.tamnd.com/node/0NMF95YK8W.md): statement 0NMF95YK8W
- [Implication](https://math2.tamnd.com/node/0NMG7W7Q11.md): concept 0NMG7W7Q11
- [Epsilon-net characterisation of total boundedness](https://math2.tamnd.com/node/0NMGJJ19HD.md): statement 0NMGJJ19HD
- [Transvection](https://math2.tamnd.com/node/0NMN00FNW1.md): definition 0NMN00FNW1
- [Birthday problem](https://math2.tamnd.com/node/0NMWG2PHJ0.md): example 0NMWG2PHJ0
- [Trace is the sum of the eigenvalues](https://math2.tamnd.com/node/0NMXDCNNKR.md): statement 0NMXDCNNKR
- [0.999... equals 1](https://math2.tamnd.com/node/0NN5JEMTVE.md): example 0NN5JEMTVE
- [Closed formula for Catalan numbers](https://math2.tamnd.com/node/0NNDQVJ54Q.md): statement 0NNDQVJ54Q
- [Associativity](https://math2.tamnd.com/node/0NNQ4KFEX7.md): definition 0NNQ4KFEX7
- [Schwarz's theorem for C^2 functions](https://math2.tamnd.com/node/0NNQY8WCW6.md): statement 0NNQY8WCW6
- [Reduced row echelon form](https://math2.tamnd.com/node/0NNR68BFRY.md): definition 0NNR68BFRY
- [Zero divisor](https://math2.tamnd.com/node/0NNRDW10RZ.md): concept 0NNRDW10RZ
- [Moment-generating function](https://math2.tamnd.com/node/0NNRNGKMFM.md): definition 0NNRNGKMFM
- [Eulerian trail](https://math2.tamnd.com/node/0NNYDD1AVR.md): definition 0NNYDD1AVR
- [Cantor pairing function](https://math2.tamnd.com/node/0NP5N5PYW5.md): example 0NP5N5PYW5
- [Euclidean distance](https://math2.tamnd.com/node/0NPPGHQNXQ.md): concept 0NPPGHQNXQ
- [Algebraic element](https://math2.tamnd.com/node/0NPW1DWECM.md): definition 0NPW1DWECM
- [Quotient topology](https://math2.tamnd.com/node/0NQ1H2X495.md): concept 0NQ1H2X495
- [Student's t-distribution](https://math2.tamnd.com/node/0NQ5798SB1.md): definition 0NQ5798SB1
- [Transcendental element](https://math2.tamnd.com/node/0NQMZ89X4A.md): concept 0NQMZ89X4A
- [Winding number](https://math2.tamnd.com/node/0NQSV2EWDR.md): concept 0NQSV2EWDR
- [Triangle inequality for integrals](https://math2.tamnd.com/node/0NR82KMYKD.md): statement 0NR82KMYKD
- [Petersen graph](https://math2.tamnd.com/node/0NR95CTFCZ.md): example 0NR95CTFCZ
- [Quaternion group](https://math2.tamnd.com/node/0NRJGW864D.md): object 0NRJGW864D
- [Composition of continuous functions is continuous](https://math2.tamnd.com/node/0NRQPQX3VA.md): statement 0NRQPQX3VA
- [Connected components are closed](https://math2.tamnd.com/node/0NRT0E2QT2.md): statement 0NRT0E2QT2
- [Linear map](https://math2.tamnd.com/node/0NRYHJPQG2.md): concept 0NRYHJPQG2
- [Finite extension](https://math2.tamnd.com/node/0NRYPM9Y27.md): definition 0NRYPM9Y27
- [Characteristic function of the normal distribution](https://math2.tamnd.com/node/0NS0TDA4E2.md): statement 0NS0TDA4E2
- [Derivative of a map into R^n by components](https://math2.tamnd.com/node/0NS0X75RS1.md): statement 0NS0X75RS1
- [Product of two subgroups](https://math2.tamnd.com/node/0NS1YTGM19.md): definition 0NS1YTGM19
- [Gradient](https://math2.tamnd.com/node/0NS2QYVC4Y.md): definition 0NS2QYVC4Y
- [Construction of the integers from the natural numbers](https://math2.tamnd.com/node/0NS58JP68E.md): concept 0NS58JP68E
- [Local minimum](https://math2.tamnd.com/node/0NS8BP96QX.md): definition 0NS8BP96QX
- [Meromorphic function](https://math2.tamnd.com/node/0NSD1TPHR0.md): definition 0NSD1TPHR0
- [Normal space](https://math2.tamnd.com/node/0NSR9V8TED.md): concept 0NSR9V8TED
- [Ordinary differential equation](https://math2.tamnd.com/node/0NT64JV96A.md): concept 0NT64JV96A
- [Direct proof](https://math2.tamnd.com/node/0NTF6DMVAP.md): concept 0NTF6DMVAP
- [Discrete Grönwall inequality](https://math2.tamnd.com/node/0NTM3J5MFE.md): statement 0NTM3J5MFE
- [Exterior angle theorem](https://math2.tamnd.com/node/0NTSE7EREA.md): statement 0NTSE7EREA
- [Multiplicative function](https://math2.tamnd.com/node/0NTWN57GM3.md): concept 0NTWN57GM3
- [Z/pZ is a field for p prime](https://math2.tamnd.com/node/0NV6E43P4A.md): statement 0NV6E43P4A
- [Open cover](https://math2.tamnd.com/node/0NVPNC12JY.md): definition 0NVPNC12JY
- [Multiplicativity of the Legendre symbol](https://math2.tamnd.com/node/0NW1JP7TBR.md): statement 0NW1JP7TBR
- [T1 space](https://math2.tamnd.com/node/0NW6CH38YR.md): concept 0NW6CH38YR
- [Second Borel–Cantelli lemma](https://math2.tamnd.com/node/0NWXRAMEE7.md): statement 0NWXRAMEE7
- [Weierstrass M-test](https://math2.tamnd.com/node/0NX5NWAFC9.md): statement 0NX5NWAFC9
- [Sum of independent normal random variables](https://math2.tamnd.com/node/0NXGPJWNFN.md): statement 0NXGPJWNFN
- [Euclidean isometry](https://math2.tamnd.com/node/0NXJM01QX1.md): definition 0NXJM01QX1
- [Adjoining a root of a polynomial](https://math2.tamnd.com/node/0NXWQK2Y56.md): concept 0NXWQK2Y56
- [Theorem on friends and strangers](https://math2.tamnd.com/node/0NYM0ZGAK9.md): statement 0NYM0ZGAK9
- [Closed map](https://math2.tamnd.com/node/0NYZ2M45G9.md): definition 0NYZ2M45G9
- [Module over a ring](https://math2.tamnd.com/node/0NZ8JA25WK.md): concept 0NZ8JA25WK
- [Adjacency matrix](https://math2.tamnd.com/node/0NZ8MDBN6Q.md): definition 0NZ8MDBN6Q
- [Angle-angle similarity](https://math2.tamnd.com/node/0NZVJYZVTZ.md): statement 0NZVJYZVTZ
- [Leading coefficient](https://math2.tamnd.com/node/0P05RJC5XA.md): definition 0P05RJC5XA
- [Cesàro mean](https://math2.tamnd.com/node/0P0DJHBGBB.md): definition 0P0DJHBGBB
- [Independence of two events](https://math2.tamnd.com/node/0P0MZ4X682.md): definition 0P0MZ4X682
- [Classification of isometries of the plane](https://math2.tamnd.com/node/0P0XDY9FZN.md): statement 0P0XDY9FZN
- [Subgraph](https://math2.tamnd.com/node/0P11X5HKAX.md): definition 0P11X5HKAX
- [Hermitian matrix](https://math2.tamnd.com/node/0P17NGQQMQ.md): concept 0P17NGQQMQ
- [Removable singularity](https://math2.tamnd.com/node/0P18JC72RE.md): definition 0P18JC72RE
- [Equivalent matrices](https://math2.tamnd.com/node/0P21AVFFSP.md): definition 0P21AVFFSP
- [Hyperbola](https://math2.tamnd.com/node/0P234N3M28.md): definition 0P234N3M28
- [Matrix of a bilinear form](https://math2.tamnd.com/node/0P24YW565F.md): definition 0P24YW565F
- [Continuous function](https://math2.tamnd.com/node/0P27W9RW0G.md): definition 0P27W9RW0G
- [Second-order Taylor expansion](https://math2.tamnd.com/node/0P30YYZN85.md): statement 0P30YYZN85
- [Negation of a universal statement](https://math2.tamnd.com/node/0P3A06QNDP.md): statement 0P3A06QNDP
- [Solvable group](https://math2.tamnd.com/node/0P3KZK0FTV.md): definition 0P3KZK0FTV
- [Divergent sequence](https://math2.tamnd.com/node/0P3M4ZY9X8.md): concept 0P3M4ZY9X8
- [Ordered field](https://math2.tamnd.com/node/0P3VCBDEE2.md): concept 0P3VCBDEE2
- [Glide reflection](https://math2.tamnd.com/node/0P42JB4E3M.md): definition 0P42JB4E3M
- [Standard deviation](https://math2.tamnd.com/node/0P4FTHHMYS.md): concept 0P4FTHHMYS
- [Degree of a polynomial](https://math2.tamnd.com/node/0P4JRMSWNH.md): concept 0P4JRMSWNH
- [Sine is 2π-periodic](https://math2.tamnd.com/node/0P4NQD746K.md): statement 0P4NQD746K
- [Iterated integral](https://math2.tamnd.com/node/0P4WX7Y502.md): definition 0P4WX7Y502
- [Non-commutativity of matrix multiplication](https://math2.tamnd.com/node/0P506SPZP2.md): counterexample 0P506SPZP2
- [A function with bounded derivative is Lipschitz](https://math2.tamnd.com/node/0P64VE79X3.md): statement 0P64VE79X3
- [Picard–Lindelöf theorem](https://math2.tamnd.com/node/0P6KRD5B7A.md): statement 0P6KRD5B7A
- [Compactness via open covers](https://math2.tamnd.com/node/0P6PMJB4MA.md): statement 0P6PMJB4MA
- [Simple extension](https://math2.tamnd.com/node/0P6SBR7YM2.md): definition 0P6SBR7YM2
- [Bilinear form](https://math2.tamnd.com/node/0P799D6AMN.md): definition 0P799D6AMN
- [Coefficient matrix](https://math2.tamnd.com/node/0P7A2MJD24.md): definition 0P7A2MJD24
- [Characteristic of a ring](https://math2.tamnd.com/node/0P7CH1CND5.md): definition 0P7CH1CND5
- [Composition of an integer](https://math2.tamnd.com/node/0P7FZHXZ5S.md): concept 0P7FZHXZ5S
- [Faithful action](https://math2.tamnd.com/node/0P7G9NGG2G.md): concept 0P7G9NGG2G
- [Linearity of expectation](https://math2.tamnd.com/node/0P7PTC8DDE.md): statement 0P7PTC8DDE
- [Vandermonde determinant](https://math2.tamnd.com/node/0P84B554AS.md): statement 0P84B554AS
- [Limit superior](https://math2.tamnd.com/node/0P8BA7T7S5.md): definition 0P8BA7T7S5
- [Action by conjugation](https://math2.tamnd.com/node/0P8CY9V437.md): example 0P8CY9V437
- [Bounded set](https://math2.tamnd.com/node/0P953MG05E.md): definition 0P953MG05E
- [Ordinary differential equation](https://math2.tamnd.com/node/0P97V20RW7.md): definition 0P97V20RW7
- [The exponential function is positive](https://math2.tamnd.com/node/0P9BNBW2W3.md): statement 0P9BNBW2W3
- [Metric spaces are Hausdorff](https://math2.tamnd.com/node/0P9DCDES73.md): statement 0P9DCDES73
- [Existence of a complementary subspace](https://math2.tamnd.com/node/0P9T8WHNMH.md): statement 0P9T8WHNMH
- [Continuity is equivalent to sequential continuity](https://math2.tamnd.com/node/0PAQM3H57X.md): statement 0PAQM3H57X
- [Sup norm on bounded continuous functions](https://math2.tamnd.com/node/0PARCKE8WP.md): example 0PARCKE8WP
- [Law of sines](https://math2.tamnd.com/node/0PASQ34GHG.md): statement 0PASQ34GHG
- [Linear differential equation of order n](https://math2.tamnd.com/node/0PB7K8M7FG.md): definition 0PB7K8M7FG
- [Jordan block](https://math2.tamnd.com/node/0PBFS92AJP.md): definition 0PBFS92AJP
- [Annihilating polynomial](https://math2.tamnd.com/node/0PBN8VEKY8.md): concept 0PBN8VEKY8
- [Monty Hall problem](https://math2.tamnd.com/node/0PBWWQD9KY.md): problem 0PBWWQD9KY
- [Skew-symmetric matrix](https://math2.tamnd.com/node/0PBY4E9GQF.md): concept 0PBY4E9GQF
- [Order of the symmetric group](https://math2.tamnd.com/node/0PCKBBMEXC.md): statement 0PCKBBMEXC
- [Multinomial coefficient](https://math2.tamnd.com/node/0PCV9EDJM2.md): concept 0PCV9EDJM2
- [Fermat's theorem on stationary points](https://math2.tamnd.com/node/0PCVBRV9VJ.md): statement 0PCVBRV9VJ
- [Circuit](https://math2.tamnd.com/node/0PD57PSHNR.md): definition 0PD57PSHNR
- [Converse of Lagrange's theorem fails for the alternating group A4](https://math2.tamnd.com/node/0PDBVPP233.md): counterexample 0PDBVPP233
- [Matrix–vector product](https://math2.tamnd.com/node/0PDVQEP0ZC.md): concept 0PDVQEP0ZC
- [Transitive closure](https://math2.tamnd.com/node/0PEJH7AC74.md): definition 0PEJH7AC74
- [Algebraically closed field](https://math2.tamnd.com/node/0PF6TM8TWT.md): concept 0PF6TM8TWT
- [Restriction of an endomorphism to an invariant subspace](https://math2.tamnd.com/node/0PFPWKBDTJ.md): definition 0PFPWKBDTJ
- [Divisibility](https://math2.tamnd.com/node/0PG2H7WKGG.md): concept 0PG2H7WKGG
- [Moment](https://math2.tamnd.com/node/0PH0VHTQS0.md): definition 0PH0VHTQS0
- [Generating function](https://math2.tamnd.com/node/0PHAC833JF.md): concept 0PHAC833JF
- [First supplement to quadratic reciprocity](https://math2.tamnd.com/node/0PHBS78HKT.md): statement 0PHBS78HKT
- [Bernoulli trial](https://math2.tamnd.com/node/0PHC9FE2V5.md): concept 0PHC9FE2V5
- [Null space](https://math2.tamnd.com/node/0PHDYNR24D.md): concept 0PHDYNR24D
- [Residue](https://math2.tamnd.com/node/0PHN07KVQT.md): concept 0PHN07KVQT
- [Heine–Borel theorem in finite-dimensional normed spaces](https://math2.tamnd.com/node/0PHT6NBYPB.md): statement 0PHT6NBYPB
- [Composition factors](https://math2.tamnd.com/node/0PHTC93M58.md): concept 0PHTC93M58
- [Multi-index notation](https://math2.tamnd.com/node/0PJXBR6QA3.md): concept 0PJXBR6QA3
- [Alternating series test](https://math2.tamnd.com/node/0PK8QH1W1V.md): statement 0PK8QH1W1V
- [Conservative vector field](https://math2.tamnd.com/node/0PKG97PX9Y.md): concept 0PKG97PX9Y
- [Limit of a function](https://math2.tamnd.com/node/0PKRYB8ATM.md): concept 0PKRYB8ATM
- [Projections are continuous](https://math2.tamnd.com/node/0PM88Z9W4M.md): statement 0PM88Z9W4M
- [Klein four-group](https://math2.tamnd.com/node/0PMRQJVTME.md): object 0PMRQJVTME
- [Order of an inverse](https://math2.tamnd.com/node/0PMT9YJZVW.md): statement 0PMT9YJZVW
- [The continued fraction of a rational number terminates](https://math2.tamnd.com/node/0PMY0YQYXX.md): statement 0PMY0YQYXX
- [The Gaussian integers are a Euclidean domain](https://math2.tamnd.com/node/0PN13BB7BT.md): statement 0PN13BB7BT
- [Argument principle](https://math2.tamnd.com/node/0PN2BXRS8G.md): statement 0PN2BXRS8G
- [Farey sequence](https://math2.tamnd.com/node/0PN4QJE969.md): definition 0PN4QJE969
- [The constructible numbers form a field closed under square roots](https://math2.tamnd.com/node/0PN9WMRWT7.md): statement 0PN9WMRWT7
- [Jordan's lemma](https://math2.tamnd.com/node/0PNPG2PP9H.md): statement 0PNPG2PP9H
- [Random variable](https://math2.tamnd.com/node/0PNRKMRC55.md): concept 0PNRKMRC55
- [Interior is open](https://math2.tamnd.com/node/0PNWCM559P.md): statement 0PNWCM559P
- [Beta distribution](https://math2.tamnd.com/node/0PP5ZAJPP7.md): definition 0PP5ZAJPP7
- [Determinant is multilinear in the rows](https://math2.tamnd.com/node/0PPG793642.md): statement 0PPG793642
- [Second isomorphism theorem](https://math2.tamnd.com/node/0PPP2J2DJV.md): statement 0PPP2J2DJV
- [Null space](https://math2.tamnd.com/node/0PQ2GGY1QV.md): definition 0PQ2GGY1QV
- [Probability of the complement](https://math2.tamnd.com/node/0PQEBTYDZM.md): statement 0PQEBTYDZM
- [Alternating sum of binomial coefficients](https://math2.tamnd.com/node/0PQQGFAWSH.md): statement 0PQQGFAWSH
- [Principal axis theorem](https://math2.tamnd.com/node/0PR5P9Z4KM.md): statement 0PR5P9Z4KM
- [Dirichlet convolution of multiplicative functions is multiplicative](https://math2.tamnd.com/node/0PR88XHDMB.md): statement 0PR88XHDMB
- [Angle between two vectors](https://math2.tamnd.com/node/0PRAS55318.md): definition 0PRAS55318
- [Identity function](https://math2.tamnd.com/node/0PSKT5W817.md): concept 0PSKT5W817
- [Moore–Penrose inverse](https://math2.tamnd.com/node/0PSRCMCM5X.md): definition 0PSRCMCM5X
- [Product of two subgroups](https://math2.tamnd.com/node/0PT52XQZD7.md): concept 0PT52XQZD7
- [Algebraic extension](https://math2.tamnd.com/node/0PTBM7K0R2.md): concept 0PTBM7K0R2
- [Endomorphism](https://math2.tamnd.com/node/0PTF1PTJK0.md): definition 0PTF1PTJK0
- [Linear congruence](https://math2.tamnd.com/node/0PTVNH4GPS.md): concept 0PTVNH4GPS
- [Maximal ideal](https://math2.tamnd.com/node/0PV74TR5FP.md): concept 0PV74TR5FP
- [Expectation with respect to the binomial distribution](https://math2.tamnd.com/node/0PVCTCKD3C.md): statement 0PVCTCKD3C
- [Injective function](https://math2.tamnd.com/node/0PVG1CRAJN.md): definition 0PVG1CRAJN
- [The exponential series has infinite radius of convergence](https://math2.tamnd.com/node/0PVJRACB9H.md): statement 0PVJRACB9H
- [Infinite set](https://math2.tamnd.com/node/0PWE3BCHVJ.md): concept 0PWE3BCHVJ
- [Cauchy sequence](https://math2.tamnd.com/node/0PWX18FM4G.md): concept 0PWX18FM4G
- [Orthogonal complement of the orthogonal complement](https://math2.tamnd.com/node/0PXN9G4MX1.md): statement 0PXN9G4MX1
- [Commutator of two elements](https://math2.tamnd.com/node/0PXW0BN7DW.md): concept 0PXW0BN7DW
- [Liouville number](https://math2.tamnd.com/node/0PYEZM6CN9.md): concept 0PYEZM6CN9
- [Composition of relations](https://math2.tamnd.com/node/0PYSR20HGF.md): concept 0PYSR20HGF
- [Affine subspace](https://math2.tamnd.com/node/0PYW26VNT6.md): concept 0PYW26VNT6
- [Closed ball](https://math2.tamnd.com/node/0Q0H13G8ZB.md): concept 0Q0H13G8ZB
- [Centraliser](https://math2.tamnd.com/node/0Q0Z1BEXTF.md): definition 0Q0Z1BEXTF
- [Interval](https://math2.tamnd.com/node/0Q1D85B8G6.md): concept 0Q1D85B8G6
- [Exact equation](https://math2.tamnd.com/node/0Q1FKQKTKJ.md): concept 0Q1FKQKTKJ
- [Singular solution](https://math2.tamnd.com/node/0Q2MDD13G4.md): concept 0Q2MDD13G4
- [Internal direct product](https://math2.tamnd.com/node/0Q2MVQ7JMT.md): definition 0Q2MVQ7JMT
- [Cumulant generating function](https://math2.tamnd.com/node/0Q2QC81VTW.md): definition 0Q2QC81VTW
- [Stolz–Cesàro theorem](https://math2.tamnd.com/node/0Q3AKZV74D.md): statement 0Q3AKZV74D
- [Image of a set](https://math2.tamnd.com/node/0Q3F20XB0A.md): definition 0Q3F20XB0A
- [Integral of a power](https://math2.tamnd.com/node/0Q3HYFT9RQ.md): statement 0Q3HYFT9RQ
- [Linear equation](https://math2.tamnd.com/node/0Q3ZT8J05R.md): concept 0Q3ZT8J05R
- [Domain](https://math2.tamnd.com/node/0Q4CBD3HX0.md): concept 0Q4CBD3HX0
- [Arc length parametrisation](https://math2.tamnd.com/node/0Q4CJ5NAT1.md): concept 0Q4CJ5NAT1
- [Uniform continuity](https://math2.tamnd.com/node/0Q4DR37DVE.md): definition 0Q4DR37DVE
- [Second derivative](https://math2.tamnd.com/node/0Q4QJWEHW3.md): concept 0Q4QJWEHW3
- [Darboux's theorem](https://math2.tamnd.com/node/0Q5A47J48D.md): statement 0Q5A47J48D
- [Statistic](https://math2.tamnd.com/node/0Q5CG2NEQ8.md): concept 0Q5CG2NEQ8
- [Derived series](https://math2.tamnd.com/node/0Q5G5DJ4P2.md): concept 0Q5G5DJ4P2
- [Galois group of a polynomial](https://math2.tamnd.com/node/0Q5NA8Y362.md): concept 0Q5NA8Y362
- [Permutation matrix](https://math2.tamnd.com/node/0Q5SM60HTH.md): definition 0Q5SM60HTH
- [Regular graph](https://math2.tamnd.com/node/0Q70YFB6ZM.md): concept 0Q70YFB6ZM
- [The sequence r^n tends to zero for |r| < 1](https://math2.tamnd.com/node/0Q7K4AXE53.md): example 0Q7K4AXE53
- [Holomorphic functions with nonzero derivative are conformal](https://math2.tamnd.com/node/0Q7M6K6VD7.md): statement 0Q7M6K6VD7
- [Nilpotent group](https://math2.tamnd.com/node/0Q7MMQHC1P.md): concept 0Q7MMQHC1P
- [Conjugacy class](https://math2.tamnd.com/node/0Q88KQD17A.md): definition 0Q88KQD17A
- [Orientation of a real vector space](https://math2.tamnd.com/node/0Q8BZXVT1Z.md): definition 0Q8BZXVT1Z
- [Third Sylow theorem](https://math2.tamnd.com/node/0Q8DGAZSJ4.md): statement 0Q8DGAZSJ4
- [Orthogonal projection](https://math2.tamnd.com/node/0Q8HZTJC7G.md): definition 0Q8HZTJC7G
- [Outer automorphism group](https://math2.tamnd.com/node/0Q8Y862DTZ.md): concept 0Q8Y862DTZ
- [Polynomial ring](https://math2.tamnd.com/node/0Q9EZS90GY.md): definition 0Q9EZS90GY
- [Diagonal matrix](https://math2.tamnd.com/node/0Q9PGXD9E4.md): definition 0Q9PGXD9E4
- [Convergence in probability](https://math2.tamnd.com/node/0QA1HBD9Z4.md): concept 0QA1HBD9Z4
- [Field](https://math2.tamnd.com/node/0QA8CHH8MG.md): definition 0QA8CHH8MG
- [Uniqueness quantifier](https://math2.tamnd.com/node/0QAJHFGZZJ.md): concept 0QAJHFGZZJ
- [Joint probability distribution](https://math2.tamnd.com/node/0QAR0XDZH4.md): definition 0QAR0XDZH4
- [Probability distribution](https://math2.tamnd.com/node/0QB913WM5E.md): definition 0QB913WM5E
- [Subspace metric](https://math2.tamnd.com/node/0QBYFZV91P.md): concept 0QBYFZV91P
- [Reciprocal rule](https://math2.tamnd.com/node/0QD4E5JYPT.md): statement 0QD4E5JYPT
- [Contour line](https://math2.tamnd.com/node/0QD83RAN8G.md): concept 0QD83RAN8G
- [Characteristic equation](https://math2.tamnd.com/node/0QDXMG46QW.md): definition 0QDXMG46QW
- [Equivalence relation induced by a function](https://math2.tamnd.com/node/0QE5PFJY7V.md): definition 0QE5PFJY7V
- [Sequence](https://math2.tamnd.com/node/0QE8STGJH2.md): definition 0QE8STGJH2
- [Closure](https://math2.tamnd.com/node/0QEAJB91GZ.md): concept 0QEAJB91GZ
- [Holomorphic functions are infinitely differentiable](https://math2.tamnd.com/node/0QEBN8DGVE.md): statement 0QEBN8DGVE
- [Alternating series](https://math2.tamnd.com/node/0QED0GPYKY.md): concept 0QED0GPYKY
- [Cayley–Hamilton theorem for linear maps](https://math2.tamnd.com/node/0QEM68Y0DE.md): statement 0QEM68Y0DE
- [Infimum](https://math2.tamnd.com/node/0QF1MZ9FRG.md): concept 0QF1MZ9FRG
- [Eigenvalues are the roots of the minimal polynomial](https://math2.tamnd.com/node/0QF4ZEFG52.md): statement 0QF4ZEFG52
- [Peano axioms](https://math2.tamnd.com/node/0QFT7Z76N5.md): concept 0QFT7Z76N5
- [The algebraic numbers form a field](https://math2.tamnd.com/node/0QFTETCHKB.md): statement 0QFTETCHKB
- [Gauss's lemma](https://math2.tamnd.com/node/0QFV5Y1Z5X.md): statement 0QFV5Y1Z5X
- [Fundamental solution of Pell's equation](https://math2.tamnd.com/node/0QG0Q3NDT9.md): concept 0QG0Q3NDT9
- [Removable discontinuity](https://math2.tamnd.com/node/0QG9C579SR.md): definition 0QG9C579SR
- [Stochastic process](https://math2.tamnd.com/node/0QH5ENKZM1.md): concept 0QH5ENKZM1
- [Predicate](https://math2.tamnd.com/node/0QHEWQD30B.md): concept 0QHEWQD30B
- [Differential form](https://math2.tamnd.com/node/0QHTX7WQEY.md): definition 0QHTX7WQEY
- [Affine map](https://math2.tamnd.com/node/0QHX5GGABE.md): definition 0QHX5GGABE
- [Irrationality of pi](https://math2.tamnd.com/node/0QHZX1PK8V.md): statement 0QHZX1PK8V
- [Orientation of a real vector space](https://math2.tamnd.com/node/0QJJAQ6ZYZ.md): concept 0QJJAQ6ZYZ
- [Matrix of a bilinear form](https://math2.tamnd.com/node/0QK6K857HS.md): concept 0QK6K857HS
- [Dimension of the coordinate space](https://math2.tamnd.com/node/0QK7FPE6VE.md): statement 0QK7FPE6VE
- [Implicit differentiation](https://math2.tamnd.com/node/0QKA345AW1.md): concept 0QKA345AW1
- [Derivative function](https://math2.tamnd.com/node/0QKJ6NT0R9.md): concept 0QKJ6NT0R9
- [Cardinal number](https://math2.tamnd.com/node/0QKPNZKNEA.md): definition 0QKPNZKNEA
- [Euclidean domains are principal ideal domains](https://math2.tamnd.com/node/0QKVE1GD9A.md): statement 0QKVE1GD9A
- [Element](https://math2.tamnd.com/node/0QM0T7T3PZ.md): concept 0QM0T7T3PZ
- [Product of rings](https://math2.tamnd.com/node/0QM2G72DXA.md): concept 0QM2G72DXA
- [Riemann integral](https://math2.tamnd.com/node/0QM4TXT3DR.md): definition 0QM4TXT3DR
- [The Galois group has order at most the degree](https://math2.tamnd.com/node/0QM98DNWVW.md): statement 0QM98DNWVW
- [Orthocentre](https://math2.tamnd.com/node/0QMYQ1X03H.md): definition 0QMYQ1X03H
- [Galois extension](https://math2.tamnd.com/node/0QMZAQ6QQ4.md): concept 0QMZAQ6QQ4
- [Euclidean vector](https://math2.tamnd.com/node/0QN3D80AWP.md): concept 0QN3D80AWP
- [Contraction mapping](https://math2.tamnd.com/node/0QN69YA357.md): concept 0QN69YA357
- [Constructible number](https://math2.tamnd.com/node/0QNFRBJF4Q.md): concept 0QNFRBJF4Q
- [Sample mean](https://math2.tamnd.com/node/0QNRBHWV35.md): definition 0QNRBHWV35
- [Lipschitz continuity](https://math2.tamnd.com/node/0QNVQ4VD9T.md): definition 0QNVQ4VD9T
- [Ratio test](https://math2.tamnd.com/node/0QNY6ZBXPN.md): statement 0QNY6ZBXPN
- [Transitive action](https://math2.tamnd.com/node/0QP2YYR79N.md): definition 0QP2YYR79N
- [Compact space](https://math2.tamnd.com/node/0QPHW703PV.md): definition 0QPHW703PV
- [Von Dyck's theorem](https://math2.tamnd.com/node/0QPYHHNRXC.md): statement 0QPYHHNRXC
- [Cycle graph](https://math2.tamnd.com/node/0QQ0F12962.md): example 0QQ0F12962
- [Positive definite matrix](https://math2.tamnd.com/node/0QQCG3JWJ4.md): definition 0QQCG3JWJ4
- [Derivative of a map into a product](https://math2.tamnd.com/node/0QQMXFTX0A.md): statement 0QQMXFTX0A
- [Number of odd-degree vertices is even](https://math2.tamnd.com/node/0QQQQCPZA8.md): statement 0QQQQCPZA8
- [Linear differential equation of order n](https://math2.tamnd.com/node/0QQRP17Z7S.md): concept 0QQRP17Z7S
- [Asymptotic equivalence](https://math2.tamnd.com/node/0QR92XBV6H.md): concept 0QR92XBV6H
- [Orthogonal matrix](https://math2.tamnd.com/node/0QRBVFJ50X.md): definition 0QRBVFJ50X
- [In a UFD irreducible elements are prime](https://math2.tamnd.com/node/0QRCFZKXH8.md): statement 0QRCFZKXH8
- [Upper and lower Darboux sums](https://math2.tamnd.com/node/0QRQVH0CRW.md): definition 0QRQVH0CRW
- [Recurrence for derangements](https://math2.tamnd.com/node/0QRRYXG7V7.md): statement 0QRRYXG7V7
- [Residue class](https://math2.tamnd.com/node/0QRV8FPK5M.md): definition 0QRV8FPK5M
- [Semisimple endomorphism](https://math2.tamnd.com/node/0QS8P74F2Z.md): definition 0QS8P74F2Z
- [The reals form an ordered field](https://math2.tamnd.com/node/0QSPNASE50.md): statement 0QSPNASE50
- [Homeomorphism](https://math2.tamnd.com/node/0QT00PB5G2.md): concept 0QT00PB5G2
- [The group of units of Z/pZ is cyclic](https://math2.tamnd.com/node/0QTHB6PE52.md): statement 0QTHB6PE52
- [Frattini argument](https://math2.tamnd.com/node/0QTZDKJEDV.md): statement 0QTZDKJEDV
- [Partial order](https://math2.tamnd.com/node/0QV2JS9RJ3.md): concept 0QV2JS9RJ3
- [Distance in a graph](https://math2.tamnd.com/node/0QV9VNNB4Z.md): concept 0QV9VNNB4Z
- [Integral domain](https://math2.tamnd.com/node/0QVD7WBFGN.md): concept 0QVD7WBFGN
- [Homothety](https://math2.tamnd.com/node/0QVDWMKQ0T.md): definition 0QVDWMKQ0T
- [Inner product space](https://math2.tamnd.com/node/0QVNCJXRY3.md): concept 0QVNCJXRY3
- [Dimension of a subspace](https://math2.tamnd.com/node/0QW084460E.md): statement 0QW084460E
- [Simply connected domain](https://math2.tamnd.com/node/0QW64ARSHB.md): concept 0QW64ARSHB
- [Every element of a finite field of order q satisfies x^q = x](https://math2.tamnd.com/node/0QWR8WTZEW.md): statement 0QWR8WTZEW
- [Order of a zero](https://math2.tamnd.com/node/0QX19C7K1C.md): concept 0QX19C7K1C
- [Tangent–secant theorem](https://math2.tamnd.com/node/0QX7E9KAAX.md): statement 0QX7E9KAAX
- [Picard–Lindelöf hypotheses](https://math2.tamnd.com/node/0QXM0A9878.md): concept 0QXM0A9878
- [Convergent sequences are Cauchy](https://math2.tamnd.com/node/0QY87R9CY3.md): statement 0QY87R9CY3
- [Presentation of a group](https://math2.tamnd.com/node/0QZCY9PE3Q.md): definition 0QZCY9PE3Q
- [Cauchy sequence](https://math2.tamnd.com/node/0QZMTN5AXX.md): definition 0QZMTN5AXX
- [Fermat number](https://math2.tamnd.com/node/0R03B54V4Z.md): definition 0R03B54V4Z
- [Coefficient of a power series](https://math2.tamnd.com/node/0R0D0XY4HT.md): concept 0R0D0XY4HT
- [Continuity of the sum of a uniformly convergent series of functions](https://math2.tamnd.com/node/0R0MYT0BJD.md): statement 0R0MYT0BJD
- [Fubini's theorem](https://math2.tamnd.com/node/0R0T1VWH5X.md): statement 0R0T1VWH5X
- [Abelian groups are solvable](https://math2.tamnd.com/node/0R0X73XHAT.md): statement 0R0X73XHAT
- [Content of a polynomial](https://math2.tamnd.com/node/0R2X8K9QA7.md): concept 0R2X8K9QA7
- [Connected set](https://math2.tamnd.com/node/0R37DFJKFW.md): definition 0R37DFJKFW
- [T0 space](https://math2.tamnd.com/node/0R3B7DX55J.md): definition 0R3B7DX55J
- [Derived series](https://math2.tamnd.com/node/0R3H35G96G.md): definition 0R3H35G96G
- [Supremum](https://math2.tamnd.com/node/0R3H5PANRQ.md): concept 0R3H5PANRQ
- [Type I and type II errors](https://math2.tamnd.com/node/0R41PVFA39.md): definition 0R41PVFA39
- [Almost surely](https://math2.tamnd.com/node/0R46A1XY60.md): definition 0R46A1XY60
- [Monoid](https://math2.tamnd.com/node/0R47MAD57S.md): definition 0R47MAD57S
- [Directed graph](https://math2.tamnd.com/node/0R5HTV7REV.md): definition 0R5HTV7REV
- [Translation](https://math2.tamnd.com/node/0R5QPQ3AGJ.md): definition 0R5QPQ3AGJ
- [Fraction in lowest terms](https://math2.tamnd.com/node/0R64A2CVN4.md): statement 0R64A2CVN4
- [Compact set](https://math2.tamnd.com/node/0R6EWFKRCF.md): concept 0R6EWFKRCF
- [Annihilator of a subspace](https://math2.tamnd.com/node/0R6M158SQ9.md): definition 0R6M158SQ9
- [Universal property of the quotient ring](https://math2.tamnd.com/node/0R74MZYNRB.md): statement 0R74MZYNRB
- [Compact subsets of a metric space are bounded](https://math2.tamnd.com/node/0R7BTEWCJ2.md): statement 0R7BTEWCJ2
- [Symmetry of binomial coefficients](https://math2.tamnd.com/node/0R7JS3DP3W.md): statement 0R7JS3DP3W
- [pi is greater than 3](https://math2.tamnd.com/node/0R7SVNYTJS.md): statement 0R7SVNYTJS
- [A linear isometry preserves the inner product](https://math2.tamnd.com/node/0R7VXTYHYZ.md): statement 0R7VXTYHYZ
- [Dimension of a tensor product](https://math2.tamnd.com/node/0R84PF4JV7.md): statement 0R84PF4JV7
- [Subsequence](https://math2.tamnd.com/node/0R8J3ZQMJS.md): definition 0R8J3ZQMJS
- [Geometric series as a power series](https://math2.tamnd.com/node/0R8JR0ARJ4.md): example 0R8JR0ARJ4
- [A linear map is determined by its values on a basis](https://math2.tamnd.com/node/0R8X9XA83A.md): statement 0R8X9XA83A
- [Integers modulo a prime form a field](https://math2.tamnd.com/node/0R8YXRC9AJ.md): statement 0R8YXRC9AJ
- [Characterization of the floor](https://math2.tamnd.com/node/0R9B0B5458.md): statement 0R9B0B5458
- [Divisibility by 3](https://math2.tamnd.com/node/0R9D4CEJEB.md): statement 0R9D4CEJEB
- [Gcd of Fibonacci numbers](https://math2.tamnd.com/node/0R9F850CTH.md): statement 0R9F850CTH
- [Linear Diophantine equation](https://math2.tamnd.com/node/0R9GF3JD0K.md): concept 0R9GF3JD0K
- [Dyck word](https://math2.tamnd.com/node/0R9JR0465V.md): definition 0R9JR0465V
- [Adjoint of a linear map](https://math2.tamnd.com/node/0R9NMSM1ZP.md): concept 0R9NMSM1ZP
- [One-sided derivative](https://math2.tamnd.com/node/0R9PHTGVNW.md): concept 0R9PHTGVNW
- [Unitary group](https://math2.tamnd.com/node/0R9R0Y1HDK.md): object 0R9R0Y1HDK
- [Limit of a sum of sequences](https://math2.tamnd.com/node/0RAM4E07NR.md): statement 0RAM4E07NR
- [Bernoulli number](https://math2.tamnd.com/node/0RAS24E1R5.md): definition 0RAS24E1R5
- [Focus of a conic](https://math2.tamnd.com/node/0RBKKNH4GV.md): concept 0RBKKNH4GV
- [Quadratic form](https://math2.tamnd.com/node/0RBQS1JJCR.md): concept 0RBQS1JJCR
- [Bessel's inequality](https://math2.tamnd.com/node/0RBT5PB74K.md): statement 0RBT5PB74K
- [Cluster point of a sequence](https://math2.tamnd.com/node/0RCKQN1E09.md): definition 0RCKQN1E09
- [Cyclic quadrilateral](https://math2.tamnd.com/node/0RCP929Y1M.md): concept 0RCP929Y1M
- [Indicator function](https://math2.tamnd.com/node/0RCQZ36MME.md): concept 0RCQZ36MME
- [Boolean ring](https://math2.tamnd.com/node/0RCV1ZFP8G.md): concept 0RCV1ZFP8G
- [Ordinal number](https://math2.tamnd.com/node/0RDJK7EQYY.md): definition 0RDJK7EQYY
- [Arccosine](https://math2.tamnd.com/node/0RDP2HEFEW.md): definition 0RDP2HEFEW
- [Cesàro means of a convergent sequence converge](https://math2.tamnd.com/node/0RDRPBMNAC.md): statement 0RDRPBMNAC
- [Space of matrices](https://math2.tamnd.com/node/0RE6X3BGPN.md): example 0RE6X3BGPN
- [Generating function of a sum of independent random variables](https://math2.tamnd.com/node/0REMT01N36.md): statement 0REMT01N36
- [Chi-squared distribution](https://math2.tamnd.com/node/0RFMMBHFTV.md): definition 0RFMMBHFTV
- [Composition factors](https://math2.tamnd.com/node/0RFP7F3SEW.md): definition 0RFP7F3SEW
- [Cauchy–Schwarz inequality](https://math2.tamnd.com/node/0RFWW56Y22.md): statement 0RFWW56Y22
- [Subgroup generated by a subset](https://math2.tamnd.com/node/0RG9SPQSWM.md): definition 0RG9SPQSWM
- [Stabiliser](https://math2.tamnd.com/node/0RGGF8ZS6D.md): definition 0RGGF8ZS6D
- [Incircle](https://math2.tamnd.com/node/0RGWYWWBM5.md): definition 0RGWYWWBM5
- [Submodule](https://math2.tamnd.com/node/0RGZ10DFPG.md): definition 0RGZ10DFPG
- [Torsion subgroup](https://math2.tamnd.com/node/0RH1TPDGCQ.md): definition 0RH1TPDGCQ
- [Projection](https://math2.tamnd.com/node/0RH4E3CAV9.md): definition 0RH4E3CAV9
- [Picard iteration](https://math2.tamnd.com/node/0RJ5NHWYTP.md): concept 0RJ5NHWYTP
- [Limit of the cumulative distribution function at infinity](https://math2.tamnd.com/node/0RJ5TZH6G0.md): statement 0RJ5TZH6G0
- [Linear map](https://math2.tamnd.com/node/0RJF668P1T.md): definition 0RJF668P1T
- [Symmetric group](https://math2.tamnd.com/node/0RJMNEN3BJ.md): definition 0RJMNEN3BJ
- [Totally disconnected space](https://math2.tamnd.com/node/0RJSBR9GW1.md): concept 0RJSBR9GW1
- [Polar coordinates](https://math2.tamnd.com/node/0RJZV05MJA.md): definition 0RJZV05MJA
- [Circumcircle](https://math2.tamnd.com/node/0RKB5MKC05.md): definition 0RKB5MKC05
- [Complement](https://math2.tamnd.com/node/0RKF1ZT5JJ.md): concept 0RKF1ZT5JJ
- [LU decomposition](https://math2.tamnd.com/node/0RKG5YW66Y.md): concept 0RKG5YW66Y
- [Slope field](https://math2.tamnd.com/node/0RKQQQACY7.md): definition 0RKQQQACY7
- [Logarithm is the inverse of the exponential](https://math2.tamnd.com/node/0RKW2X63YR.md): statement 0RKW2X63YR
- [Sum of subspaces](https://math2.tamnd.com/node/0RMKABS2S5.md): definition 0RMKABS2S5
- [Exponential function](https://math2.tamnd.com/node/0RMVS7RXZX.md): definition 0RMVS7RXZX
- [Inner automorphism](https://math2.tamnd.com/node/0RN00NTM3S.md): concept 0RN00NTM3S
- [Proper metric space](https://math2.tamnd.com/node/0RN765S7DH.md): concept 0RN765S7DH
- [Homogeneous first-order equation](https://math2.tamnd.com/node/0RN88KZBJ3.md): concept 0RN88KZBJ3
- [Complex power](https://math2.tamnd.com/node/0RNGDMHT7M.md): concept 0RNGDMHT7M
- [Complement graph](https://math2.tamnd.com/node/0RNMP6VCR2.md): definition 0RNMP6VCR2
- [Damped harmonic oscillator](https://math2.tamnd.com/node/0RNXVVTJTJ.md): example 0RNXVVTJTJ
- [Positive definite quadratic form](https://math2.tamnd.com/node/0RP20HFYM3.md): concept 0RP20HFYM3
- [Subgroup generated by a subset](https://math2.tamnd.com/node/0RP7VWQ0YN.md): concept 0RP7VWQ0YN
- [Trace of a linear map](https://math2.tamnd.com/node/0RPDTWY2JS.md): definition 0RPDTWY2JS
- [Cantor set is homeomorphic to the space of binary sequences](https://math2.tamnd.com/node/0RQ8DHHR8V.md): statement 0RQ8DHHR8V
- [Identity element](https://math2.tamnd.com/node/0RQPZZ44T4.md): definition 0RQPZZ44T4
- [Prime-counting function](https://math2.tamnd.com/node/0RQSNVTFVZ.md): concept 0RQSNVTFVZ
- [Wedderburn's little theorem](https://math2.tamnd.com/node/0RRNVX1VKA.md): statement 0RRNVX1VKA
- [Fixed points of an action](https://math2.tamnd.com/node/0RRWNMYRXC.md): concept 0RRWNMYRXC
- [Linear form](https://math2.tamnd.com/node/0RT96J4D3H.md): definition 0RT96J4D3H
- [Focus of a conic](https://math2.tamnd.com/node/0RV3AEXPT6.md): definition 0RV3AEXPT6
- [Cauchy product](https://math2.tamnd.com/node/0RVD9R2V99.md): concept 0RVD9R2V99
- [Gaussian prime](https://math2.tamnd.com/node/0RVRNR7KA1.md): concept 0RVRNR7KA1
- [Converse](https://math2.tamnd.com/node/0RVYE75E2M.md): definition 0RVYE75E2M
- [Spherical coordinates](https://math2.tamnd.com/node/0RW8KPMXF6.md): definition 0RW8KPMXF6
- [Orthonormal family](https://math2.tamnd.com/node/0RW9TNNT87.md): concept 0RW9TNNT87
- [Probability theory](https://math2.tamnd.com/node/0RWE7CPJ32.md): concept 0RWE7CPJ32
- [Cartesian coordinate system](https://math2.tamnd.com/node/0RWKYX8339.md): concept 0RWKYX8339
- [Jensen's inequality](https://math2.tamnd.com/node/0RWMRTPC87.md): statement 0RWMRTPC87
- [Every infinite set has a countably infinite subset](https://math2.tamnd.com/node/0RWQ5VZ6VQ.md): statement 0RWQ5VZ6VQ
- [Image of an intersection](https://math2.tamnd.com/node/0RX7DDBYCK.md): statement 0RX7DDBYCK
- [Distributivity of matrix multiplication](https://math2.tamnd.com/node/0RXD6EZYBK.md): statement 0RXD6EZYBK
- [A continuous nowhere differentiable function](https://math2.tamnd.com/node/0RXK2Z808Q.md): counterexample 0RXK2Z808Q
- [Completely multiplicative function](https://math2.tamnd.com/node/0RXKW044R4.md): concept 0RXKW044R4
- [Preorder](https://math2.tamnd.com/node/0RXS8R6K37.md): definition 0RXS8R6K37
- [Chain rule](https://math2.tamnd.com/node/0RXX2PT1MQ.md): statement 0RXX2PT1MQ
- [Structure theorem for finitely generated abelian groups](https://math2.tamnd.com/node/0RY0B72XAS.md): statement 0RY0B72XAS
- [Lucas's theorem](https://math2.tamnd.com/node/0RY4GGMEQG.md): statement 0RY4GGMEQG
- [Big O notation](https://math2.tamnd.com/node/0RYTGBJAGN.md): definition 0RYTGBJAGN
- [Law of excluded middle](https://math2.tamnd.com/node/0RYYK8PG4S.md): statement 0RYYK8PG4S
- [Separable polynomial](https://math2.tamnd.com/node/0RZ89QWTSJ.md): concept 0RZ89QWTSJ
- [A nontrivial finite p-group has nontrivial centre](https://math2.tamnd.com/node/0RZCTT30H5.md): statement 0RZCTT30H5
- [Interior](https://math2.tamnd.com/node/0RZH6W98HK.md): concept 0RZH6W98HK
- [Principal ideal domain](https://math2.tamnd.com/node/0RZYDVZ475.md): definition 0RZYDVZ475
- [Power-reduction formula for cosine](https://math2.tamnd.com/node/0S049HG1W5.md): statement 0S049HG1W5
- [Addition of natural numbers](https://math2.tamnd.com/node/0S13V32CPB.md): concept 0S13V32CPB
- [Wedge product of forms](https://math2.tamnd.com/node/0S166D363M.md): definition 0S166D363M
- [Size of an orbit times the size of the stabiliser](https://math2.tamnd.com/node/0S19CWFZM2.md): statement 0S19CWFZM2
- [Hausdorff space](https://math2.tamnd.com/node/0S1FJ8VEX9.md): concept 0S1FJ8VEX9
- [Variation of constants formula for linear systems](https://math2.tamnd.com/node/0S1KG0XE40.md): statement 0S1KG0XE40
- [Image of a homomorphism](https://math2.tamnd.com/node/0S21GNW9ZB.md): concept 0S21GNW9ZB
- [Kernel of a linear map](https://math2.tamnd.com/node/0S23E6YE83.md): concept 0S23E6YE83
- [Surface area](https://math2.tamnd.com/node/0S29Y4ZW0Y.md): concept 0S29Y4ZW0Y
- [Smallest prime factor](https://math2.tamnd.com/node/0S2SYHTWPY.md): definition 0S2SYHTWPY
- [Discrete topology](https://math2.tamnd.com/node/0S2TEHX4E1.md): example 0S2TEHX4E1
- [Continuous linear map](https://math2.tamnd.com/node/0S3SMRH0Z5.md): definition 0S3SMRH0Z5
- [Abel's theorem](https://math2.tamnd.com/node/0S4EMDQS2F.md): statement 0S4EMDQS2F
- [Convergent sequences are bounded](https://math2.tamnd.com/node/0S4JF619V7.md): statement 0S4JF619V7
- [Inverse of a product](https://math2.tamnd.com/node/0S4WTRE9GG.md): statement 0S4WTRE9GG
- [Spanning tree](https://math2.tamnd.com/node/0S4ZEZM7KE.md): concept 0S4ZEZM7KE
- [Dot product](https://math2.tamnd.com/node/0S52TNNTH9.md): concept 0S52TNNTH9
- [Upper bound](https://math2.tamnd.com/node/0S60DVMVPN.md): concept 0S60DVMVPN
- [Root of unity](https://math2.tamnd.com/node/0S61N75H45.md): definition 0S61N75H45
- [Biconditional](https://math2.tamnd.com/node/0S6GMNJ62Y.md): concept 0S6GMNJ62Y
- [Confidence interval](https://math2.tamnd.com/node/0S6T9KQNC0.md): concept 0S6T9KQNC0
- [Midpoint](https://math2.tamnd.com/node/0S706JBZBG.md): concept 0S706JBZBG
- [Order of the group of units of Z/nZ is Euler's totient](https://math2.tamnd.com/node/0S7495H4HM.md): statement 0S7495H4HM
- [Algebraic multiplicity](https://math2.tamnd.com/node/0S7A0EMPF4.md): definition 0S7A0EMPF4
- [Indeterminate form](https://math2.tamnd.com/node/0S7G92KKA7.md): concept 0S7G92KKA7
- [External direct sum](https://math2.tamnd.com/node/0S7W6EG32V.md): concept 0S7W6EG32V
- [Centre of a group](https://math2.tamnd.com/node/0S7Y9Q990Q.md): definition 0S7Y9Q990Q
- [Chain](https://math2.tamnd.com/node/0S8FE7ZMFM.md): concept 0S8FE7ZMFM
- [Hartman–Grobman theorem](https://math2.tamnd.com/node/0S8WX8A4V9.md): statement 0S8WX8A4V9
- [Continuity in R^n](https://math2.tamnd.com/node/0S9EK5MQPE.md): definition 0S9EK5MQPE
- [Orthonormal basis](https://math2.tamnd.com/node/0S9GGTR0EX.md): concept 0S9GGTR0EX
- [Correspondence theorem](https://math2.tamnd.com/node/0S9ZVZ34MD.md): statement 0S9ZVZ34MD
- [Generalised eigenvector](https://math2.tamnd.com/node/0SAAHC36J9.md): definition 0SAAHC36J9
- [Derivative of tangent](https://math2.tamnd.com/node/0SAMR34WXC.md): statement 0SAMR34WXC
- [Dirichlet convolution formula](https://math2.tamnd.com/node/0SAR26CKPS.md): statement 0SAR26CKPS
- [Perfect number](https://math2.tamnd.com/node/0SB7Q8SYSF.md): definition 0SB7Q8SYSF
- [Path component](https://math2.tamnd.com/node/0SB9FNZECG.md): definition 0SB9FNZECG
- [Multiplicative order](https://math2.tamnd.com/node/0SBRX5CZ5R.md): definition 0SBRX5CZ5R
- [Golden ratio](https://math2.tamnd.com/node/0SC8T7548D.md): object 0SC8T7548D
- [Hessian matrix](https://math2.tamnd.com/node/0SCSQWXRA4.md): definition 0SCSQWXRA4
- [Gradient theorem](https://math2.tamnd.com/node/0SD237FEJD.md): statement 0SD237FEJD
- [Minor](https://math2.tamnd.com/node/0SD528P0EK.md): definition 0SD528P0EK
- [Polynomial ring](https://math2.tamnd.com/node/0SDC934R52.md): concept 0SDC934R52
- [Dihedral angle](https://math2.tamnd.com/node/0SDFXN1VDQ.md): definition 0SDFXN1VDQ
- [Multilinear map](https://math2.tamnd.com/node/0SDMPMCN43.md): definition 0SDMPMCN43
- [Cross product](https://math2.tamnd.com/node/0SEDGXG9S3.md): definition 0SEDGXG9S3
- [Order of a group](https://math2.tamnd.com/node/0SEX9F9J5Z.md): definition 0SEX9F9J5Z
- [Countably infinite set](https://math2.tamnd.com/node/0SEZ3JGZP7.md): definition 0SEZ3JGZP7
- [Product rule in several variables](https://math2.tamnd.com/node/0SF85PY0QN.md): statement 0SF85PY0QN
- [Möbius transformation](https://math2.tamnd.com/node/0SG74WBFA2.md): concept 0SG74WBFA2
- [Equivalence relation](https://math2.tamnd.com/node/0SGTMBEWRF.md): definition 0SGTMBEWRF
- [Basis of a tensor product](https://math2.tamnd.com/node/0SH5VT8QYA.md): statement 0SH5VT8QYA
- [Closure is the smallest closed superset](https://math2.tamnd.com/node/0SHC6Z304Z.md): statement 0SHC6Z304Z
- [Split polynomial](https://math2.tamnd.com/node/0SHMC00RJV.md): definition 0SHMC00RJV
- [Pendulum equation](https://math2.tamnd.com/node/0SHPXM2HWH.md): example 0SHPXM2HWH
- [Set](https://math2.tamnd.com/node/0SJ3RX9TXA.md): definition 0SJ3RX9TXA
- [Continued fraction expansion of a real number](https://math2.tamnd.com/node/0SJ7Z6VKWK.md): definition 0SJ7Z6VKWK
- [Falling factorial](https://math2.tamnd.com/node/0SJNCJCCQG.md): concept 0SJNCJCCQG
- [Irreducibility of the cyclotomic polynomials over the rationals](https://math2.tamnd.com/node/0SK1VHZ5R2.md): statement 0SK1VHZ5R2
- [Sign of a permutation](https://math2.tamnd.com/node/0SK90238BH.md): definition 0SK90238BH
- [Stability of linear systems via eigenvalues](https://math2.tamnd.com/node/0SKVEMXJWF.md): statement 0SKVEMXJWF
- [General solution and particular solution](https://math2.tamnd.com/node/0SKYFQAFYD.md): concept 0SKYFQAFYD
- [Edge bound for planar graphs](https://math2.tamnd.com/node/0SM1REAVYJ.md): statement 0SM1REAVYJ
- [Prime number](https://math2.tamnd.com/node/0SN0MAEKYX.md): definition 0SN0MAEKYX
- [Erdős–Szekeres theorem](https://math2.tamnd.com/node/0SNJM8HSD3.md): statement 0SNJM8HSD3
- [Bijective function](https://math2.tamnd.com/node/0SNQEF7Z53.md): definition 0SNQEF7Z53
- [Primitive polynomial](https://math2.tamnd.com/node/0SP14DZN0G.md): definition 0SP14DZN0G
- [A space is the direct sum of a subspace and its orthogonal complement](https://math2.tamnd.com/node/0SP7P0G7KQ.md): statement 0SP7P0G7KQ
- [Existence of a prime divisor](https://math2.tamnd.com/node/0SPBEDVNW1.md): statement 0SPBEDVNW1
- [Hypergeometric distribution](https://math2.tamnd.com/node/0SPBPW3K61.md): definition 0SPBPW3K61
- [Connected set](https://math2.tamnd.com/node/0SPE8WNWFV.md): concept 0SPE8WNWFV
- [Change of basis for a bilinear form](https://math2.tamnd.com/node/0SPYAGE7T0.md): statement 0SPYAGE7T0
- [The Gaussian integers form a Euclidean domain](https://math2.tamnd.com/node/0SPYEFH7H4.md): statement 0SPYEFH7H4
- [Refinement of a partition](https://math2.tamnd.com/node/0SPZ1H3N9V.md): concept 0SPZ1H3N9V
- [Branch of the logarithm](https://math2.tamnd.com/node/0SQ1F2BEDS.md): definition 0SQ1F2BEDS
- [Open map](https://math2.tamnd.com/node/0SQ2PC596Y.md): definition 0SQ2PC596Y
- [Quadratic residue](https://math2.tamnd.com/node/0SQDB87Z3T.md): definition 0SQDB87Z3T
- [Principal value of the argument](https://math2.tamnd.com/node/0SQSQP748R.md): statement 0SQSQP748R
- [Even and odd permutations](https://math2.tamnd.com/node/0SQTMQTSRZ.md): concept 0SQTMQTSRZ
- [Parallelepiped](https://math2.tamnd.com/node/0SQV2VJTGY.md): concept 0SQV2VJTGY
- [Bilinear form](https://math2.tamnd.com/node/0SQZMXB6GP.md): concept 0SQZMXB6GP
- [Subgroups of the integers](https://math2.tamnd.com/node/0SRATCFF1W.md): statement 0SRATCFF1W
- [Characteristic subgroups are normal](https://math2.tamnd.com/node/0SRE7K4E49.md): statement 0SRE7K4E49
- [Convergents of a continued fraction](https://math2.tamnd.com/node/0SRWNR42BQ.md): definition 0SRWNR42BQ
- [Convergence in distribution](https://math2.tamnd.com/node/0SRYJGXKR4.md): concept 0SRYJGXKR4
- [First integral](https://math2.tamnd.com/node/0SS1P1YC9A.md): definition 0SS1P1YC9A
- [Differentiation as a linear map](https://math2.tamnd.com/node/0SS770NS8B.md): example 0SS770NS8B
- [Hurwitz's theorem](https://math2.tamnd.com/node/0SSKBQN58M.md): statement 0SSKBQN58M
- [Estimator](https://math2.tamnd.com/node/0SSS6ERQ0S.md): concept 0SSS6ERQ0S
- [The degree of a constructible number is a power of two](https://math2.tamnd.com/node/0SSSM1D475.md): statement 0SSSM1D475
- [Almost surely](https://math2.tamnd.com/node/0ST21PGDAP.md): concept 0ST21PGDAP
- [Adjoint of a linear map](https://math2.tamnd.com/node/0ST8QF9T3T.md): definition 0ST8QF9T3T
- [Moments are derivatives of the moment-generating function at zero](https://math2.tamnd.com/node/0STK8KEY9G.md): statement 0STK8KEY9G
- [C^1 functions are strictly differentiable](https://math2.tamnd.com/node/0STPPDXGME.md): statement 0STPPDXGME
- [Density of the rationals](https://math2.tamnd.com/node/0STX0WRDNG.md): statement 0STX0WRDNG
- [Sigma-algebra](https://math2.tamnd.com/node/0SV8K0TDGY.md): concept 0SV8K0TDGY
- [Almost sure convergence implies convergence in probability](https://math2.tamnd.com/node/0SVN6X695K.md): statement 0SVN6X695K
- [Quotient space](https://math2.tamnd.com/node/0SVR29SV1V.md): definition 0SVR29SV1V
- [Primitive element theorem](https://math2.tamnd.com/node/0SVX98HYV1.md): statement 0SVX98HYV1
- [Principal ideal](https://math2.tamnd.com/node/0SVYSP8HCG.md): concept 0SVYSP8HCG
- [Linear equation with constant coefficients](https://math2.tamnd.com/node/0SWTJ5VC07.md): concept 0SWTJ5VC07
- [Sample space](https://math2.tamnd.com/node/0SWXJHKJ02.md): concept 0SWXJHKJ02
- [Möbius transformations map circles and lines to circles and lines](https://math2.tamnd.com/node/0SXDHQM2MH.md): statement 0SXDHQM2MH
- [Dimension of a quotient space](https://math2.tamnd.com/node/0SXDQ6TN00.md): statement 0SXDQ6TN00
- [Acyclic graph](https://math2.tamnd.com/node/0SXHBTNM9T.md): concept 0SXHBTNM9T
- [Integral of 1/z around the unit circle](https://math2.tamnd.com/node/0SXZ68HV6F.md): example 0SXZ68HV6F
- [Continuous map of topological spaces](https://math2.tamnd.com/node/0SY9ECQ19Q.md): definition 0SY9ECQ19Q
- [Cauchy principal value](https://math2.tamnd.com/node/0SYR10HD4R.md): concept 0SYR10HD4R
- [Order of the alternating group](https://math2.tamnd.com/node/0SZ1KMNWVE.md): statement 0SZ1KMNWVE
- [Miller–Rabin primality test](https://math2.tamnd.com/node/0SZCSYHW60.md): concept 0SZCSYHW60
- [Normal convergence](https://math2.tamnd.com/node/0T0EJDGHMB.md): definition 0T0EJDGHMB
- [Degree of a vertex](https://math2.tamnd.com/node/0T0F6CCS9W.md): definition 0T0F6CCS9W
- [Bertrand paradox](https://math2.tamnd.com/node/0T0JWJ3M47.md): example 0T0JWJ3M47
- [Normal core](https://math2.tamnd.com/node/0T0KMZPYV9.md): definition 0T0KMZPYV9
- [Determinant is a coefficient of the characteristic polynomial](https://math2.tamnd.com/node/0T0X8GTYYS.md): statement 0T0X8GTYYS
- [Alternating group](https://math2.tamnd.com/node/0T13HM377F.md): definition 0T13HM377F
- [Infinitely many primes congruent to 1 modulo n](https://math2.tamnd.com/node/0T1BSD7H3M.md): statement 0T1BSD7H3M
- [Fundamental theorem of Galois theory](https://math2.tamnd.com/node/0T1Q8D0FW8.md): statement 0T1Q8D0FW8
- [Degree of a product](https://math2.tamnd.com/node/0T233AZ1GE.md): statement 0T233AZ1GE
- [Least element](https://math2.tamnd.com/node/0T2NTJCV83.md): definition 0T2NTJCV83
- [Subspace metric](https://math2.tamnd.com/node/0T2WYDTRP2.md): definition 0T2WYDTRP2
- [Power series of arctangent](https://math2.tamnd.com/node/0T32ME0A6H.md): statement 0T32ME0A6H
- [Distributive lattice](https://math2.tamnd.com/node/0T36P6TXQ6.md): definition 0T36P6TXQ6
- [Predicate](https://math2.tamnd.com/node/0T3DPBG981.md): definition 0T3DPBG981
- [Phase portrait](https://math2.tamnd.com/node/0T3GK0XCBC.md): concept 0T3GK0XCBC
- [Limit superior](https://math2.tamnd.com/node/0T3RZQXDYD.md): concept 0T3RZQXDYD
- [Frobenius endomorphism](https://math2.tamnd.com/node/0T3WVTD675.md): concept 0T3WVTD675
- [Dirichlet function](https://math2.tamnd.com/node/0T4EJ8ZJM9.md): counterexample 0T4EJ8ZJM9
- [The cross product is perpendicular to its factors](https://math2.tamnd.com/node/0T4QHJXSD4.md): statement 0T4QHJXSD4
- [Clairaut's equation](https://math2.tamnd.com/node/0T6D4PE51C.md): example 0T6D4PE51C
- [Augmented matrix](https://math2.tamnd.com/node/0T6MWGZSXA.md): concept 0T6MWGZSXA
- [First Borel–Cantelli lemma](https://math2.tamnd.com/node/0T6WS7TT77.md): statement 0T6WS7TT77
- [Cauchy–Riemann equations](https://math2.tamnd.com/node/0T71SSYZ39.md): statement 0T71SSYZ39
- [Circle integral](https://math2.tamnd.com/node/0T7ABX4CWE.md): definition 0T7ABX4CWE
- [Tangent line to a circle](https://math2.tamnd.com/node/0T7DX8NY1X.md): concept 0T7DX8NY1X
- [Integration in polar coordinates](https://math2.tamnd.com/node/0T89W1QZ9K.md): statement 0T89W1QZ9K
- [The characteristic function determines the distribution](https://math2.tamnd.com/node/0T8BYAFEXX.md): statement 0T8BYAFEXX
- [Existence of finite fields of every prime power order](https://math2.tamnd.com/node/0T8M3YC5RG.md): statement 0T8M3YC5RG
- [Bernoulli equation](https://math2.tamnd.com/node/0T92VGGVWP.md): concept 0T92VGGVWP
- [Carmichael number](https://math2.tamnd.com/node/0TA7WT68RK.md): definition 0TA7WT68RK
- [Axiom of extensionality](https://math2.tamnd.com/node/0TB16T2C97.md): statement 0TB16T2C97
- [The successor function is injective](https://math2.tamnd.com/node/0TBYMWPWWB.md): statement 0TBYMWPWWB
- [Dirac's theorem on Hamiltonian cycles](https://math2.tamnd.com/node/0TC99BRSSB.md): statement 0TC99BRSSB
- [C^k function characterised by continuous derivatives](https://math2.tamnd.com/node/0TCBZMEYGN.md): statement 0TCBZMEYGN
- [Convex hull](https://math2.tamnd.com/node/0TCHAS04WK.md): definition 0TCHAS04WK
- [Orientable surface](https://math2.tamnd.com/node/0TD904MG06.md): concept 0TD904MG06
- [Product of the gcd and the lcm](https://math2.tamnd.com/node/0TDY6JZHKS.md): statement 0TDY6JZHKS
- [Remainder theorem](https://math2.tamnd.com/node/0TEDQH56RW.md): statement 0TEDQH56RW
- [Affine space](https://math2.tamnd.com/node/0TEJ6QF7K9.md): definition 0TEJ6QF7K9
- [p-metric on R^n](https://math2.tamnd.com/node/0TFDZN6AXP.md): example 0TFDZN6AXP
- [Big O notation](https://math2.tamnd.com/node/0TGV80E6A4.md): concept 0TGV80E6A4
- [Derangement](https://math2.tamnd.com/node/0TH61JR04G.md): definition 0TH61JR04G
- [Arc length](https://math2.tamnd.com/node/0TH78S7GBZ.md): concept 0TH78S7GBZ
- [Reduction of a higher-order equation to a first-order system](https://math2.tamnd.com/node/0TH9CPW3VS.md): statement 0TH9CPW3VS
- [Function composition](https://math2.tamnd.com/node/0THBK9V825.md): concept 0THBK9V825
- [Conditional independence](https://math2.tamnd.com/node/0THJ7416DX.md): concept 0THJ7416DX
- [Third isomorphism theorem](https://math2.tamnd.com/node/0TJ1V9DDPC.md): statement 0TJ1V9DDPC
- [Term-by-term differentiation](https://math2.tamnd.com/node/0TJ8QBJX5K.md): statement 0TJ8QBJX5K
- [Interval](https://math2.tamnd.com/node/0TK8Y3J5RW.md): definition 0TK8Y3J5RW
- [Cylindrical coordinates](https://math2.tamnd.com/node/0TK9MCFHBW.md): concept 0TK9MCFHBW
- [Bounded derivative on a convex set implies Lipschitz](https://math2.tamnd.com/node/0TKG7T3K0E.md): statement 0TKG7T3K0E
- [Bell numbers](https://math2.tamnd.com/node/0TKP3TDD02.md): concept 0TKP3TDD02
- [Orthogonal group](https://math2.tamnd.com/node/0TKYAEJ8N7.md): object 0TKYAEJ8N7
- [Minimum degree](https://math2.tamnd.com/node/0TKZCSGCCA.md): definition 0TKZCSGCCA
- [Lipschitz map](https://math2.tamnd.com/node/0TMZ21WKCN.md): definition 0TMZ21WKCN
- [Set-builder notation](https://math2.tamnd.com/node/0TMZ89Q5KA.md): concept 0TMZ89Q5KA
- [Powers of a diagonalisable matrix](https://math2.tamnd.com/node/0TNR9BQ871.md): concept 0TNR9BQ871
- [Prime subfield](https://math2.tamnd.com/node/0TNS65QB87.md): concept 0TNS65QB87
- [Partial derivatives do not imply differentiability](https://math2.tamnd.com/node/0TNZC55X3J.md): counterexample 0TNZC55X3J
- [Circumcentre](https://math2.tamnd.com/node/0TP2QKJ8DE.md): definition 0TP2QKJ8DE
- [Congruence modulo n](https://math2.tamnd.com/node/0TPW25CBN6.md): definition 0TPW25CBN6
- [An algebraic extension of an algebraic extension is algebraic](https://math2.tamnd.com/node/0TPWS0K1BC.md): statement 0TPWS0K1BC
- [Strictly monotone function](https://math2.tamnd.com/node/0TR9SR9FTB.md): definition 0TR9SR9FTB
- [Euclidean algorithm](https://math2.tamnd.com/node/0TRNT4X7J0.md): concept 0TRNT4X7J0
- [Orthogonal vectors](https://math2.tamnd.com/node/0TS7VZJBWD.md): definition 0TS7VZJBWD
- [Law of a random variable](https://math2.tamnd.com/node/0TST4MHA3A.md): definition 0TST4MHA3A
- [Convergence in mean](https://math2.tamnd.com/node/0TTA57SEJG.md): definition 0TTA57SEJG
- [Conditional expectation](https://math2.tamnd.com/node/0TTDGTEFMN.md): concept 0TTDGTEFMN
- [Estimator](https://math2.tamnd.com/node/0TTNRP5PFQ.md): definition 0TTNRP5PFQ
- [Real exponentiation](https://math2.tamnd.com/node/0TV07AQT62.md): definition 0TV07AQT62
- [Compact space](https://math2.tamnd.com/node/0TVJZYB6SH.md): concept 0TVJZYB6SH
- [Incentre](https://math2.tamnd.com/node/0TWPTQ0AD7.md): definition 0TWPTQ0AD7
- [Equivalent metrics](https://math2.tamnd.com/node/0TWQ5PRKWS.md): definition 0TWQ5PRKWS
- [Invertible matrix](https://math2.tamnd.com/node/0TWTS88TWN.md): definition 0TWTS88TWN
- [Symmetric bilinear form associated with a quadratic form](https://math2.tamnd.com/node/0TX2YH2FQS.md): definition 0TX2YH2FQS
- [Integration over a non-rectangular region](https://math2.tamnd.com/node/0TXCQ9CST1.md): concept 0TXCQ9CST1
- [Primitive polynomial](https://math2.tamnd.com/node/0TXNRQZY0H.md): concept 0TXNRQZY0H
- [Limit of a function](https://math2.tamnd.com/node/0TXTSEMHR5.md): definition 0TXTSEMHR5
- [Product of a subgroup and a normal subgroup is a subgroup](https://math2.tamnd.com/node/0TXWV1JGG1.md): statement 0TXWV1JGG1
- [Isolated singularity](https://math2.tamnd.com/node/0TYB3HM9KE.md): definition 0TYB3HM9KE
- [Characteristic polynomial does not depend on the basis](https://math2.tamnd.com/node/0TYMASG33Y.md): statement 0TYMASG33Y
- [Surjective iff it has a right inverse](https://math2.tamnd.com/node/0TYQNHVB4C.md): statement 0TYQNHVB4C
- [Principal part](https://math2.tamnd.com/node/0TZ2DPT0JV.md): concept 0TZ2DPT0JV
- [Row rank equals column rank](https://math2.tamnd.com/node/0TZAG60ZYT.md): statement 0TZAG60ZYT
- [Maximal element](https://math2.tamnd.com/node/0TZEEFTNCC.md): definition 0TZEEFTNCC
- [The sequence r^n tends to infinity for r > 1](https://math2.tamnd.com/node/0TZK91NQE3.md): example 0TZK91NQE3
- [Connected space](https://math2.tamnd.com/node/0V032VWTHD.md): concept 0V032VWTHD
- [Induced subgraph](https://math2.tamnd.com/node/0V05FC4GKQ.md): definition 0V05FC4GKQ
- [Variance of a sum of independent random variables](https://math2.tamnd.com/node/0V0778DCW9.md): statement 0V0778DCW9
- [Sign of a cycle](https://math2.tamnd.com/node/0V0E04DWX2.md): statement 0V0E04DWX2
- [Complete bipartite graph](https://math2.tamnd.com/node/0V0GC7BQAN.md): example 0V0GC7BQAN
- [Frobenius coin problem](https://math2.tamnd.com/node/0V11FEXK08.md): statement 0V11FEXK08
- [Derivative](https://math2.tamnd.com/node/0V1EJCT8G6.md): concept 0V1EJCT8G6
- [Derivative of cosine](https://math2.tamnd.com/node/0V2DV29EGT.md): statement 0V2DV29EGT
- [Aleph-null](https://math2.tamnd.com/node/0V2H5PGJ9B.md): object 0V2H5PGJ9B
- [Norm](https://math2.tamnd.com/node/0V2WP9B1MF.md): concept 0V2WP9B1MF
- [Smooth function](https://math2.tamnd.com/node/0V33BNKZ9S.md): concept 0V33BNKZ9S
- [Intersection of subspaces](https://math2.tamnd.com/node/0V33SPH7RR.md): statement 0V33SPH7RR
- [Semigroup](https://math2.tamnd.com/node/0V3CFG1KK9.md): definition 0V3CFG1KK9
- [Gamma function](https://math2.tamnd.com/node/0V3FZMVB4K.md): definition 0V3FZMVB4K
- [Cauchy principal value](https://math2.tamnd.com/node/0V3RDQV3ZX.md): definition 0V3RDQV3ZX
- [Plane](https://math2.tamnd.com/node/0V3XAMF3C1.md): concept 0V3XAMF3C1
- [Laplace transform](https://math2.tamnd.com/node/0V49JEZW06.md): concept 0V49JEZW06
- [Continuous real function on a compact set attains its bounds](https://math2.tamnd.com/node/0V4DET59NW.md): statement 0V4DET59NW
- [Separable equation](https://math2.tamnd.com/node/0V4E623THY.md): definition 0V4E623THY
- [Vacuous truth](https://math2.tamnd.com/node/0V4QXPGJQK.md): concept 0V4QXPGJQK
- [Fibonacci numbers as sums of binomial coefficients](https://math2.tamnd.com/node/0V4VB6MQDC.md): statement 0V4VB6MQDC
- [Mean squared error](https://math2.tamnd.com/node/0V4VSCBXP3.md): definition 0V4VSCBXP3
- [Subgroups of index two are normal](https://math2.tamnd.com/node/0V547YQRMV.md): statement 0V547YQRMV
- [Minimal polynomial](https://math2.tamnd.com/node/0V58FPM3EQ.md): definition 0V58FPM3EQ
- [Characteristic function](https://math2.tamnd.com/node/0V5B6D6BQH.md): concept 0V5B6D6BQH
- [Leading coefficient](https://math2.tamnd.com/node/0V5GYWR8R8.md): concept 0V5GYWR8R8
- [Regular polygon](https://math2.tamnd.com/node/0V63D9V0FY.md): definition 0V63D9V0FY
- [Biconditional](https://math2.tamnd.com/node/0V6GAWTBYW.md): definition 0V6GAWTBYW
- [Regular curve](https://math2.tamnd.com/node/0V7090SSPC.md): concept 0V7090SSPC
- [Square root function](https://math2.tamnd.com/node/0V7E9KP6Q2.md): definition 0V7E9KP6Q2
- [Covariance matrix](https://math2.tamnd.com/node/0V7G6WMFK0.md): concept 0V7G6WMFK0
- [Standard basis](https://math2.tamnd.com/node/0V7ZD6P7RK.md): object 0V7ZD6P7RK
- [Quotient topology](https://math2.tamnd.com/node/0V8046R0P9.md): definition 0V8046R0P9
- [Linear combination](https://math2.tamnd.com/node/0V8F3E48E6.md): concept 0V8F3E48E6
- [Logarithm of a power](https://math2.tamnd.com/node/0V948DE5KA.md): statement 0V948DE5KA
- [Determinant of a family of vectors in a basis](https://math2.tamnd.com/node/0V94H674MB.md): definition 0V94H674MB
- [First Sylow theorem](https://math2.tamnd.com/node/0V98WY94B8.md): statement 0V98WY94B8
- [Product rule](https://math2.tamnd.com/node/0V9A43H7WV.md): statement 0V9A43H7WV
- [Dirichlet's test](https://math2.tamnd.com/node/0V9ME8GQE7.md): statement 0V9ME8GQE7
- [Special linear group](https://math2.tamnd.com/node/0V9PMN0RBH.md): object 0V9PMN0RBH
- [Preorder](https://math2.tamnd.com/node/0VABYWJWH9.md): concept 0VABYWJWH9
- [Power series in the complex plane](https://math2.tamnd.com/node/0VARVQ9ZW1.md): definition 0VARVQ9ZW1
- [Square root](https://math2.tamnd.com/node/0VB8BFRABH.md): definition 0VB8BFRABH
- [Multiplicativity of the Möbius function](https://math2.tamnd.com/node/0VB9G5B034.md): statement 0VB9G5B034
- [Kummer's theorem](https://math2.tamnd.com/node/0VBWK9XYDG.md): statement 0VBWK9XYDG
- [Internal direct product](https://math2.tamnd.com/node/0VBZH90R1A.md): concept 0VBZH90R1A
- [Euler's theorem](https://math2.tamnd.com/node/0VC4DVE7DQ.md): statement 0VC4DVE7DQ
- [Improper integral](https://math2.tamnd.com/node/0VCTBJMEJF.md): definition 0VCTBJMEJF
- [Matrix](https://math2.tamnd.com/node/0VD9ZSWWJG.md): concept 0VD9ZSWWJG
- [Morera's theorem](https://math2.tamnd.com/node/0VDD7W0EKZ.md): statement 0VDD7W0EKZ
- [Poisson distribution](https://math2.tamnd.com/node/0VEFSSKX18.md): definition 0VEFSSKX18
- [Volume of a box](https://math2.tamnd.com/node/0VEKNG02W0.md): statement 0VEKNG02W0
- [Field of fractions](https://math2.tamnd.com/node/0VF11ZTTMD.md): definition 0VF11ZTTMD
- [Quotient ring](https://math2.tamnd.com/node/0VFW4Y69BT.md): definition 0VFW4Y69BT
- [Every polynomial has a root in some extension field](https://math2.tamnd.com/node/0VG02M1FEZ.md): statement 0VG02M1FEZ
- [Abundant number](https://math2.tamnd.com/node/0VG7VNEWMG.md): concept 0VG7VNEWMG
- [Continuous map](https://math2.tamnd.com/node/0VGFHTM75Y.md): concept 0VGFHTM75Y
- [Improper integral of x^p on an unbounded interval](https://math2.tamnd.com/node/0VGKB2NC4T.md): example 0VGKB2NC4T
- [Probability measure](https://math2.tamnd.com/node/0VGMEBVW1M.md): definition 0VGMEBVW1M
- [Product of continuous functions is continuous](https://math2.tamnd.com/node/0VGMHCF4SC.md): statement 0VGMHCF4SC
- [Continuity is equivalent to sequential continuity in metric spaces](https://math2.tamnd.com/node/0VGPKZ8MRK.md): statement 0VGPKZ8MRK
- [Conditional variance](https://math2.tamnd.com/node/0VGPX25NXQ.md): concept 0VGPX25NXQ
- [Inner product](https://math2.tamnd.com/node/0VGSYBWHV2.md): definition 0VGSYBWHV2
- [Abel's identity](https://math2.tamnd.com/node/0VGWKX2JHB.md): statement 0VGWKX2JHB
- [Companion matrix](https://math2.tamnd.com/node/0VH1TC1PWF.md): definition 0VH1TC1PWF
- [Subsequences of a convergent sequence converge to the same limit](https://math2.tamnd.com/node/0VHAVXW08V.md): statement 0VHAVXW08V
- [Chromatic number](https://math2.tamnd.com/node/0VHJRXNAAS.md): concept 0VHJRXNAAS
- [Conformal map](https://math2.tamnd.com/node/0VHNKP1N3K.md): definition 0VHNKP1N3K
- [Symmetric polynomial](https://math2.tamnd.com/node/0VHSHB9B6J.md): concept 0VHSHB9B6J
- [Finite-dimensional spaces are isomorphic iff they have equal dimension](https://math2.tamnd.com/node/0VJBSK15SN.md): statement 0VJBSK15SN
- [Homogeneous linear system with constant coefficients](https://math2.tamnd.com/node/0VJF99CZ0Z.md): definition 0VJF99CZ0Z
- [Walk](https://math2.tamnd.com/node/0VJZM0F9YM.md): definition 0VJZM0F9YM
- [De Morgan's laws for sets](https://math2.tamnd.com/node/0VK0W4D5Y5.md): statement 0VK0W4D5Y5
- [Size of a conjugacy class is the index of the centraliser](https://math2.tamnd.com/node/0VK5V9P8G3.md): statement 0VK5V9P8G3
- [Banach fixed point theorem](https://math2.tamnd.com/node/0VK6VEH345.md): statement 0VK6VEH345
- [Ellipse](https://math2.tamnd.com/node/0VKCCEXBJ1.md): concept 0VKCCEXBJ1
- [Metric space](https://math2.tamnd.com/node/0VKD1TP7F3.md): concept 0VKD1TP7F3
- [Lagrange multipliers](https://math2.tamnd.com/node/0VKKP8X6Q6.md): statement 0VKKP8X6Q6
- [Decimal expansion](https://math2.tamnd.com/node/0VM8B2SB4P.md): concept 0VM8B2SB4P
- [Limit of a function of several variables](https://math2.tamnd.com/node/0VMG8TA8VF.md): definition 0VMG8TA8VF
- [Cartan–Dieudonné theorem](https://math2.tamnd.com/node/0VMWGYY35Q.md): statement 0VMWGYY35Q
- [Smallest prime factor](https://math2.tamnd.com/node/0VMWPGGG3C.md): concept 0VMWPGGG3C
- [If 2^n + 1 is prime then n is a power of 2](https://math2.tamnd.com/node/0VN4H8M12X.md): statement 0VN4H8M12X
- [Polynomials of degree two or three are irreducible iff they have no roots](https://math2.tamnd.com/node/0VNE161QCK.md): statement 0VNE161QCK
- [Pythagorean theorem](https://math2.tamnd.com/node/0VP3A1290K.md): statement 0VP3A1290K
- [Frobenius automorphism of a finite field](https://math2.tamnd.com/node/0VP4TBVJ19.md): definition 0VP4TBVJ19
- [Power series](https://math2.tamnd.com/node/0VP56B4EMX.md): definition 0VP56B4EMX
- [Space of linear maps](https://math2.tamnd.com/node/0VPCJWGMP9.md): definition 0VPCJWGMP9
- [Prime subfield](https://math2.tamnd.com/node/0VPFBAY7MW.md): definition 0VPFBAY7MW
- [Rule of product](https://math2.tamnd.com/node/0VPK4X73Q0.md): statement 0VPK4X73Q0
- [Sequence tending to infinity](https://math2.tamnd.com/node/0VQA7F720Z.md): concept 0VQA7F720Z
- [Combinations with repetition](https://math2.tamnd.com/node/0VQQCVG8WA.md): concept 0VQQCVG8WA
- [Nowhere dense set](https://math2.tamnd.com/node/0VQYCCQVJ5.md): concept 0VQYCCQVJ5
- [Symmetric matrix](https://math2.tamnd.com/node/0VR671PHE5.md): concept 0VR671PHE5
- [Isometric isomorphism](https://math2.tamnd.com/node/0VRD8GC5BM.md): definition 0VRD8GC5BM
- [Binomial distribution](https://math2.tamnd.com/node/0VRWZDWW6B.md): definition 0VRWZDWW6B
- [Abundant number](https://math2.tamnd.com/node/0VS0MP1P5H.md): definition 0VS0MP1P5H
- [Indicator function](https://math2.tamnd.com/node/0VSB0XF4G7.md): definition 0VSB0XF4G7
- [A tree on n vertices has n − 1 edges](https://math2.tamnd.com/node/0VSNHTHK8T.md): statement 0VSNHTHK8T
- [Cancellation laws](https://math2.tamnd.com/node/0VSNX4P2Q0.md): statement 0VSNX4P2Q0
- [Characteristic polynomial](https://math2.tamnd.com/node/0VSSPAFM2C.md): concept 0VSSPAFM2C
- [Riccati equation](https://math2.tamnd.com/node/0VT2QK6X6Q.md): concept 0VT2QK6X6Q
- [Gram–Schmidt process](https://math2.tamnd.com/node/0VT6P0GNQR.md): concept 0VT6P0GNQR
- [Minimal polynomial divides every annihilating polynomial](https://math2.tamnd.com/node/0VTAD25A3G.md): statement 0VTAD25A3G
- [Sup metric on R^n](https://math2.tamnd.com/node/0VTAXQEWQD.md): example 0VTAXQEWQD
- [Turán graph](https://math2.tamnd.com/node/0VTCABADBE.md): example 0VTCABADBE
- [Ring homomorphism](https://math2.tamnd.com/node/0VTH8KPCG2.md): concept 0VTH8KPCG2
- [Standard deviation](https://math2.tamnd.com/node/0VTN5RKNC5.md): definition 0VTN5RKNC5
- [Basis](https://math2.tamnd.com/node/0VTQTPQGAV.md): definition 0VTQTPQGAV
- [Transcendental number](https://math2.tamnd.com/node/0VTQXTCR2G.md): concept 0VTQXTCR2G
- [Euler's totient function](https://math2.tamnd.com/node/0VV04M41AN.md): definition 0VV04M41AN
- [Convex function](https://math2.tamnd.com/node/0VV0HSW1ZS.md): definition 0VV0HSW1ZS
- [Number of divisors](https://math2.tamnd.com/node/0VV3ZMRF88.md): concept 0VV3ZMRF88
- [Discriminant of a polynomial](https://math2.tamnd.com/node/0VVK64Q4JK.md): concept 0VVK64Q4JK
- [Order of a power](https://math2.tamnd.com/node/0VVPQNVM8S.md): statement 0VVPQNVM8S
- [Term test](https://math2.tamnd.com/node/0VVY4H92G2.md): statement 0VVY4H92G2
- [Continuously differentiable function](https://math2.tamnd.com/node/0VW90BBSJ4.md): concept 0VW90BBSJ4
- [Likelihood function](https://math2.tamnd.com/node/0VW9E4QXR6.md): concept 0VW9E4QXR6
- [Euler line](https://math2.tamnd.com/node/0VX0CPRQTZ.md): statement 0VX0CPRQTZ
- [Minimal polynomial (field theory)](https://math2.tamnd.com/node/0VX9EP2GS8.md): concept 0VX9EP2GS8
- [Standard score](https://math2.tamnd.com/node/0VXJHZ1G9J.md): concept 0VXJHZ1G9J
- [Gradient is orthogonal to level sets](https://math2.tamnd.com/node/0VXYV3NQQ1.md): statement 0VXYV3NQQ1
- [Bézout's identity in a Euclidean domain](https://math2.tamnd.com/node/0VY3WPP9Y3.md): statement 0VY3WPP9Y3
- [Row echelon form](https://math2.tamnd.com/node/0VYCDGMJJJ.md): concept 0VYCDGMJJJ
- [Coprime integers](https://math2.tamnd.com/node/0VZ48N083F.md): concept 0VZ48N083F
- [Compact closed unit ball implies finite dimension](https://math2.tamnd.com/node/0VZVV8VME5.md): statement 0VZVV8VME5
- [Orthogonal matrix](https://math2.tamnd.com/node/0W02QESPJQ.md): concept 0W02QESPJQ
- [Cyclotomic polynomial](https://math2.tamnd.com/node/0W0KVBM54S.md): definition 0W0KVBM54S
- [Continuous functions on a closed bounded interval are uniformly continuous](https://math2.tamnd.com/node/0W0TZDRF9C.md): statement 0W0TZDRF9C
- [Euler's formula for planar graphs](https://math2.tamnd.com/node/0W1C598HMS.md): statement 0W1C598HMS
- [Rearrangement of a series](https://math2.tamnd.com/node/0W1K48V5SG.md): concept 0W1K48V5SG
- [Uniform convergence on a set](https://math2.tamnd.com/node/0W1QMKJ9W8.md): concept 0W1QMKJ9W8
- [Linear approximation](https://math2.tamnd.com/node/0W276SSSCG.md): concept 0W276SSSCG
- [Riemann integral](https://math2.tamnd.com/node/0W29TBTY12.md): concept 0W29TBTY12
- [Cofactor](https://math2.tamnd.com/node/0W2QAVJQ1W.md): definition 0W2QAVJQ1W
- [Absolute convergence](https://math2.tamnd.com/node/0W3K3QPEWT.md): definition 0W3K3QPEWT
- [Line integral as an integral over the parameter interval](https://math2.tamnd.com/node/0W3W1VYWBP.md): statement 0W3W1VYWBP
- [Complete subspaces are closed](https://math2.tamnd.com/node/0W41XG4W60.md): statement 0W41XG4W60
- [Sesquilinear form](https://math2.tamnd.com/node/0W44K2VNBD.md): definition 0W44K2VNBD
- [Convergent sequence](https://math2.tamnd.com/node/0W4CN9E2W5.md): concept 0W4CN9E2W5
- [Rank–nullity theorem](https://math2.tamnd.com/node/0W4FSW1TVB.md): statement 0W4FSW1TVB
- [Irrationality of e](https://math2.tamnd.com/node/0W4GQSG7B6.md): statement 0W4GQSG7B6
- [Disjoint sets](https://math2.tamnd.com/node/0W4RPA8QH3.md): definition 0W4RPA8QH3
- [Unique factorisation domain](https://math2.tamnd.com/node/0W4X9G6263.md): concept 0W4X9G6263
- [Trail](https://math2.tamnd.com/node/0W50MT1C6X.md): concept 0W50MT1C6X
- [Addition of natural numbers](https://math2.tamnd.com/node/0W5DSDFQ4G.md): definition 0W5DSDFQ4G
- [Harmonic function](https://math2.tamnd.com/node/0W5NJQ8MKY.md): definition 0W5NJQ8MKY
- [Law of total probability for an event and its complement](https://math2.tamnd.com/node/0W62TKMW7V.md): statement 0W62TKMW7V
- [Dual basis](https://math2.tamnd.com/node/0W6M3PSD08.md): definition 0W6M3PSD08
- [Inverse function](https://math2.tamnd.com/node/0W766JAPSN.md): definition 0W766JAPSN
- [Normal subgroup](https://math2.tamnd.com/node/0W77CVBCHQ.md): definition 0W77CVBCHQ
- [Uncountable set](https://math2.tamnd.com/node/0W7FBYS1F1.md): concept 0W7FBYS1F1
- [Sample space](https://math2.tamnd.com/node/0W7NFW23RF.md): definition 0W7NFW23RF
- [Transpose](https://math2.tamnd.com/node/0W809K6AW0.md): definition 0W809K6AW0
- [Kernel of a homomorphism](https://math2.tamnd.com/node/0W88CEGQP1.md): concept 0W88CEGQP1
- [Tangent half-angle substitution](https://math2.tamnd.com/node/0W8ETY2AZ6.md): concept 0W8ETY2AZ6
- [Branch cut](https://math2.tamnd.com/node/0W8QHVSN95.md): concept 0W8QHVSN95
- [Epsilon-delta definition of continuity](https://math2.tamnd.com/node/0W8YHHGCJG.md): statement 0W8YHHGCJG
- [Fermat pseudoprime](https://math2.tamnd.com/node/0W93FJAC8F.md): concept 0W93FJAC8F
- [Equivalence class](https://math2.tamnd.com/node/0W95W33CMP.md): concept 0W95W33CMP
- [Chinese remainder theorem for rings](https://math2.tamnd.com/node/0W97J8QED7.md): statement 0W97J8QED7
- [Time translation of solutions of an autonomous system](https://math2.tamnd.com/node/0WA7ZR95V4.md): statement 0WA7ZR95V4
- [Double integral](https://math2.tamnd.com/node/0WAR4F23B4.md): concept 0WAR4F23B4
- [Congruence modulo n](https://math2.tamnd.com/node/0WAY5WX8NY.md): concept 0WAY5WX8NY
- [Sine and cosine](https://math2.tamnd.com/node/0WB57D9488.md): definition 0WB57D9488
- [Orthocentre](https://math2.tamnd.com/node/0WB619CA0V.md): concept 0WB619CA0V
- [Path-connected space](https://math2.tamnd.com/node/0WB6DH499T.md): concept 0WB6DH499T
- [Subgraph](https://math2.tamnd.com/node/0WBFFC8XZ9.md): concept 0WBFFC8XZ9
- [Norm on the Gaussian integers](https://math2.tamnd.com/node/0WBRJXXDPZ.md): concept 0WBRJXXDPZ
- [Integration by partial fractions](https://math2.tamnd.com/node/0WBTWFXJQG.md): concept 0WBTWFXJQG
- [Cardinality of the continuum](https://math2.tamnd.com/node/0WC28W9DNM.md): object 0WC28W9DNM
- [Clique number](https://math2.tamnd.com/node/0WCKRBHFFD.md): definition 0WCKRBHFFD
- [Proof by cases](https://math2.tamnd.com/node/0WCP0PVET6.md): concept 0WCP0PVET6
- [Trace of a linear map is the trace of its matrix](https://math2.tamnd.com/node/0WCPH5Z7FA.md): statement 0WCPH5Z7FA
- [Binomial distribution](https://math2.tamnd.com/node/0WCWA1KT3M.md): concept 0WCWA1KT3M
- [Separable extension](https://math2.tamnd.com/node/0WCX7NB4N7.md): concept 0WCX7NB4N7
- [Group action](https://math2.tamnd.com/node/0WCY42EV9V.md): concept 0WCY42EV9V
- [Rational function](https://math2.tamnd.com/node/0WCZH4AFMT.md): definition 0WCZH4AFMT
- [R/M is a field iff M is maximal](https://math2.tamnd.com/node/0WD3SE7S1X.md): statement 0WD3SE7S1X
- [x^2 sin(1/x) is differentiable with discontinuous derivative](https://math2.tamnd.com/node/0WDAKKN4NC.md): counterexample 0WDAKKN4NC
- [Cauchy–Euler equation](https://math2.tamnd.com/node/0WDNRBXX7K.md): definition 0WDNRBXX7K
- [Order of an element](https://math2.tamnd.com/node/0WE2NHZJ7B.md): concept 0WE2NHZJ7B
- [Totally bounded set](https://math2.tamnd.com/node/0WE5BBHGM6.md): definition 0WE5BBHGM6
- [Euclid–Euler theorem](https://math2.tamnd.com/node/0WETJCKNRG.md): statement 0WETJCKNRG
- [Probability mass function](https://math2.tamnd.com/node/0WF4EJP91W.md): definition 0WF4EJP91W
- [Reduced row echelon form](https://math2.tamnd.com/node/0WF5AQYTW6.md): concept 0WF5AQYTW6
- [Local maximum and minimum](https://math2.tamnd.com/node/0WFBH1Q3WC.md): concept 0WFBH1Q3WC
- [Factorial](https://math2.tamnd.com/node/0WFCGVMRRC.md): definition 0WFCGVMRRC
- [Ordered pair](https://math2.tamnd.com/node/0WFCNG0MZC.md): concept 0WFCNG0MZC
- [Sylvester's law of inertia](https://math2.tamnd.com/node/0WFED3K3DQ.md): statement 0WFED3K3DQ
- [Monotone function](https://math2.tamnd.com/node/0WFF6AZPBP.md): definition 0WFF6AZPBP
- [Gambler's ruin](https://math2.tamnd.com/node/0WFMKK7DFZ.md): problem 0WFMKK7DFZ
- [Inverse hyperbolic functions](https://math2.tamnd.com/node/0WFZX06698.md): concept 0WFZX06698
- [Newton's identities](https://math2.tamnd.com/node/0WG2A3HE0S.md): statement 0WG2A3HE0S
- [Two-element Boolean algebra](https://math2.tamnd.com/node/0WG43AC0EV.md): example 0WG43AC0EV
- [A subspace of full dimension is the whole space](https://math2.tamnd.com/node/0WGAXEK99E.md): statement 0WGAXEK99E
- [Continuous uniform distribution](https://math2.tamnd.com/node/0WGDPYR30Q.md): definition 0WGDPYR30Q
- [Circle as a quotient of the real line](https://math2.tamnd.com/node/0WH1GVBJNH.md): example 0WH1GVBJNH
- [There is a natural number n with 1/n below any positive epsilon](https://math2.tamnd.com/node/0WH916TJNY.md): statement 0WH916TJNY
- [Invariant factors and elementary divisors](https://math2.tamnd.com/node/0WHCA4SVP2.md): concept 0WHCA4SVP2
- [Taylor's theorem with the integral remainder](https://math2.tamnd.com/node/0WJGR81HBD.md): statement 0WJGR81HBD
- [Free abelian group](https://math2.tamnd.com/node/0WJX8MGG37.md): concept 0WJX8MGG37
- [Dimension theorem for vector spaces](https://math2.tamnd.com/node/0WK0Y8RH1M.md): statement 0WK0Y8RH1M
- [Sylvester's criterion](https://math2.tamnd.com/node/0WKEF78PMV.md): statement 0WKEF78PMV
- [Fermat number](https://math2.tamnd.com/node/0WKG15TV3G.md): concept 0WKG15TV3G
- [Cholesky decomposition](https://math2.tamnd.com/node/0WKG8CYY6A.md): statement 0WKG8CYY6A
- [Product of exponentials](https://math2.tamnd.com/node/0WM7DJCM6G.md): statement 0WM7DJCM6G
- [Volume of a ball in R^3](https://math2.tamnd.com/node/0WMDV48VSN.md): statement 0WMDV48VSN
- [A prime is a Gaussian prime if and only if it is 3 modulo 4](https://math2.tamnd.com/node/0WMJSEFYFE.md): statement 0WMJSEFYFE
- [Indefinite integral](https://math2.tamnd.com/node/0WN1QXR7R3.md): concept 0WN1QXR7R3
- [Hyperbola](https://math2.tamnd.com/node/0WN5G117YC.md): concept 0WN5G117YC
- [Edge set](https://math2.tamnd.com/node/0WNGQBPPQX.md): definition 0WNGQBPPQX
- [Hoeffding's inequality](https://math2.tamnd.com/node/0WPDNXGSSN.md): statement 0WPDNXGSSN
- [Pole](https://math2.tamnd.com/node/0WPQYW8MTE.md): definition 0WPQYW8MTE
- [Distance in a graph](https://math2.tamnd.com/node/0WPZ5YPAGR.md): definition 0WPZ5YPAGR
- [Distance from a point to a set](https://math2.tamnd.com/node/0WQ19TJV24.md): concept 0WQ19TJV24
- [Field extension](https://math2.tamnd.com/node/0WQ1A8SHTZ.md): definition 0WQ1A8SHTZ
- [Surface area](https://math2.tamnd.com/node/0WQE3G7B88.md): definition 0WQE3G7B88
- [Normal vector](https://math2.tamnd.com/node/0WQJJCCTC8.md): concept 0WQJJCCTC8
- [Expected value](https://math2.tamnd.com/node/0WQNRCNY3M.md): concept 0WQNRCNY3M
- [Moment-generating function](https://math2.tamnd.com/node/0WQVXP7PDB.md): concept 0WQVXP7PDB
- [Borel sigma-algebra](https://math2.tamnd.com/node/0WQXY32Q8D.md): concept 0WQXY32Q8D
- [Moment-generating function of a sum of independent random variables](https://math2.tamnd.com/node/0WRZ170YEH.md): statement 0WRZ170YEH
- [Extreme value theorem](https://math2.tamnd.com/node/0WS1A8E2P9.md): statement 0WS1A8E2P9
- [Subspace topology](https://math2.tamnd.com/node/0WS9P12FQR.md): concept 0WS9P12FQR
- [Image of a linear map](https://math2.tamnd.com/node/0WSFKKT6FF.md): concept 0WSFKKT6FF
- [Cantor's theorem](https://math2.tamnd.com/node/0WTEMGYV2C.md): statement 0WTEMGYV2C
- [Nullcline](https://math2.tamnd.com/node/0WW060JR6P.md): definition 0WW060JR6P
- [Bell numbers](https://math2.tamnd.com/node/0WW85SQVS3.md): definition 0WW85SQVS3
- [Right inverse](https://math2.tamnd.com/node/0WWRD82HD2.md): concept 0WWRD82HD2
- [Group of nth roots of unity](https://math2.tamnd.com/node/0WX11EA3F3.md): object 0WX11EA3F3
- [Gram–Schmidt vectors are orthogonal](https://math2.tamnd.com/node/0WXBGBTA3D.md): statement 0WXBGBTA3D
- [A monotone function has countably many discontinuities](https://math2.tamnd.com/node/0WYTM4C9QG.md): statement 0WYTM4C9QG
- [Iterated integral](https://math2.tamnd.com/node/0WZCHE1ZF9.md): concept 0WZCHE1ZF9
- [Hölder continuity](https://math2.tamnd.com/node/0WZTM0EFR6.md): definition 0WZTM0EFR6
- [Division ring](https://math2.tamnd.com/node/0WZXK05ME7.md): definition 0WZXK05ME7
- [Boolean function](https://math2.tamnd.com/node/0X05GS525E.md): concept 0X05GS525E
- [Determinant of a linear map](https://math2.tamnd.com/node/0X0HQYSZ0K.md): concept 0X0HQYSZ0K
- [Number of injections](https://math2.tamnd.com/node/0X0N833FF8.md): statement 0X0N833FF8
- [Union](https://math2.tamnd.com/node/0X1GSWNBN1.md): concept 0X1GSWNBN1
- [Subgroup](https://math2.tamnd.com/node/0X1JXT47GA.md): concept 0X1JXT47GA
- [Complex logarithm](https://math2.tamnd.com/node/0X2FY491PF.md): definition 0X2FY491PF
- [Hall's theorem for bipartite graphs](https://math2.tamnd.com/node/0X2GJPNPFN.md): statement 0X2GJPNPFN
- [Discrete random variable](https://math2.tamnd.com/node/0X2J0VY6DN.md): concept 0X2J0VY6DN
- [Countable intersection of dense open sets is dense](https://math2.tamnd.com/node/0X2T63GFQQ.md): statement 0X2T63GFQQ
- [The exterior derivative squares to zero](https://math2.tamnd.com/node/0X308R2DBE.md): statement 0X308R2DBE
- [Path component](https://math2.tamnd.com/node/0X3EG7Y65T.md): concept 0X3EG7Y65T
- [Congruent triangles](https://math2.tamnd.com/node/0X3TEM69AA.md): definition 0X3TEM69AA
- [Continuous function](https://math2.tamnd.com/node/0X3WGX95FE.md): concept 0X3WGX95FE
- [Laurent series](https://math2.tamnd.com/node/0X3XD2ATB0.md): definition 0X3XD2ATB0
- [Hypergeometric distribution](https://math2.tamnd.com/node/0X3Y0ZSKNF.md): concept 0X3Y0ZSKNF
- [Cayley transform](https://math2.tamnd.com/node/0X45ZM731G.md): example 0X45ZM731G
- [Idempotent element](https://math2.tamnd.com/node/0X4JKZXJ86.md): definition 0X4JKZXJ86
- [Forced oscillation and resonance](https://math2.tamnd.com/node/0X4NT20S3G.md): example 0X4NT20S3G
- [Tietze extension theorem](https://math2.tamnd.com/node/0X52TNMTCJ.md): statement 0X52TNMTCJ
- [Uncountable set](https://math2.tamnd.com/node/0X55M9XET4.md): definition 0X55M9XET4
- [Elementary symmetric polynomial](https://math2.tamnd.com/node/0X5QVCF06N.md): concept 0X5QVCF06N
- [Quantile function](https://math2.tamnd.com/node/0X60VE98HN.md): concept 0X60VE98HN
- [Algebraic closure](https://math2.tamnd.com/node/0X6AB5P3G1.md): concept 0X6AB5P3G1
- [Hyperbolic tangent](https://math2.tamnd.com/node/0X6C0MAF84.md): concept 0X6C0MAF84
- [Consistent and inconsistent systems](https://math2.tamnd.com/node/0X6V9B1V10.md): concept 0X6V9B1V10
- [Impossibility of squaring the circle](https://math2.tamnd.com/node/0X7CVDE3BG.md): statement 0X7CVDE3BG
- [Centroid](https://math2.tamnd.com/node/0X7EEWES5G.md): concept 0X7EEWES5G
- [Arc length](https://math2.tamnd.com/node/0X7G61297X.md): definition 0X7G61297X
- [Winding number](https://math2.tamnd.com/node/0X7SBS8E0F.md): definition 0X7SBS8E0F
- [Altitude of a triangle](https://math2.tamnd.com/node/0X89GC65N1.md): definition 0X89GC65N1
- [Continuity of probability measures](https://math2.tamnd.com/node/0X8HBKCSYB.md): statement 0X8HBKCSYB
- [Reflexive relation](https://math2.tamnd.com/node/0X8NMST0J2.md): concept 0X8NMST0J2
- [Space of linear maps](https://math2.tamnd.com/node/0X9MBPP2Z3.md): concept 0X9MBPP2Z3
- [Uniform limit theorem](https://math2.tamnd.com/node/0X9XT05JVC.md): statement 0X9XT05JVC
- [Boolean algebra](https://math2.tamnd.com/node/0XADJKM699.md): concept 0XADJKM699
- [Handshaking lemma](https://math2.tamnd.com/node/0XAP8WHNES.md): statement 0XAP8WHNES
- [Simple continued fraction](https://math2.tamnd.com/node/0XB0J0P64D.md): concept 0XB0J0P64D
- [Poisson distribution](https://math2.tamnd.com/node/0XB9GQTN9G.md): concept 0XB9GQTN9G
- [Negative binomial distribution](https://math2.tamnd.com/node/0XBRG3965F.md): concept 0XBRG3965F
- [Trace of a matrix](https://math2.tamnd.com/node/0XC1RDBY7A.md): concept 0XC1RDBY7A
- [Independence of random variables](https://math2.tamnd.com/node/0XCE070V8B.md): definition 0XCE070V8B
- [Cantor's diagonal argument](https://math2.tamnd.com/node/0XCH8QVDCQ.md): concept 0XCH8QVDCQ
- [Geometric distribution](https://math2.tamnd.com/node/0XCVV5Q500.md): definition 0XCVV5Q500
- [Wallis product](https://math2.tamnd.com/node/0XCYA4YERS.md): statement 0XCYA4YERS
- [Number of elements of each order in a cyclic group](https://math2.tamnd.com/node/0XCYGCCX8V.md): statement 0XCYGCCX8V
- [Monotonicity of the cumulative distribution function](https://math2.tamnd.com/node/0XD22PBMWA.md): statement 0XD22PBMWA
- [Jacobi symbol](https://math2.tamnd.com/node/0XDB5AW7M1.md): definition 0XDB5AW7M1
- [Rational function](https://math2.tamnd.com/node/0XDK9CGYKJ.md): concept 0XDK9CGYKJ
- [Even and odd functions](https://math2.tamnd.com/node/0XE690QF65.md): definition 0XE690QF65
- [Reflexive relation](https://math2.tamnd.com/node/0XEGNW851S.md): definition 0XEGNW851S
- [Multinomial theorem](https://math2.tamnd.com/node/0XFC1VQYEK.md): statement 0XFC1VQYEK
- [Euler's partition theorem](https://math2.tamnd.com/node/0XFZ1BMF5Q.md): statement 0XFZ1BMF5Q
- [Meagre set](https://math2.tamnd.com/node/0XGCZ4GDZ6.md): concept 0XGCZ4GDZ6
- [Archimedean ordered field](https://math2.tamnd.com/node/0XH79S06J3.md): concept 0XH79S06J3
- [Function](https://math2.tamnd.com/node/0XHDTE5YMZ.md): definition 0XHDTE5YMZ
- [Finitely generated module](https://math2.tamnd.com/node/0XHM30WBVB.md): definition 0XHM30WBVB
- [Derangement](https://math2.tamnd.com/node/0XHWTD3MPV.md): concept 0XHWTD3MPV
- [K-colourable graph](https://math2.tamnd.com/node/0XHZW0VCPR.md): concept 0XHZW0VCPR
- [Indiscrete topology](https://math2.tamnd.com/node/0XJ02JBD6S.md): example 0XJ02JBD6S
- [Consistent estimator](https://math2.tamnd.com/node/0XJVF81QF6.md): definition 0XJVF81QF6
- [Orthogonal complement](https://math2.tamnd.com/node/0XJXTBCB2T.md): concept 0XJXTBCB2T
- [Order of a differential equation](https://math2.tamnd.com/node/0XK0G19SK8.md): concept 0XK0G19SK8
- [Decomposition of a permutation into transpositions](https://math2.tamnd.com/node/0XK6FECMW5.md): statement 0XK6FECMW5
- [Length of a vector](https://math2.tamnd.com/node/0XKBKGTCPC.md): concept 0XKBKGTCPC
- [Surjective function](https://math2.tamnd.com/node/0XKH881BZ7.md): definition 0XKH881BZ7
- [Periodic continued fraction](https://math2.tamnd.com/node/0XKJPFS2W0.md): definition 0XKJPFS2W0
- [Poisson limit theorem](https://math2.tamnd.com/node/0XKTZRWTXX.md): statement 0XKTZRWTXX
- [Limit of a sequence](https://math2.tamnd.com/node/0XKWSSY8P2.md): definition 0XKWSSY8P2
- [Coordinate space](https://math2.tamnd.com/node/0XM8BZD76G.md): object 0XM8BZD76G
- [Sieve of Eratosthenes](https://math2.tamnd.com/node/0XM8ZA9RNM.md): concept 0XM8ZA9RNM
- [Semigroup](https://math2.tamnd.com/node/0XMEM406JM.md): concept 0XMEM406JM
- [Squeeze theorem](https://math2.tamnd.com/node/0XMF00T4XQ.md): statement 0XMF00T4XQ
- [Multivariate normal distribution](https://math2.tamnd.com/node/0XN3K7RSZY.md): concept 0XN3K7RSZY
- [Equicontinuity](https://math2.tamnd.com/node/0XN66MZN0Y.md): concept 0XN66MZN0Y
- [Negative binomial distribution](https://math2.tamnd.com/node/0XN7KEVSSS.md): definition 0XN7KEVSSS
- [Acyclic graph](https://math2.tamnd.com/node/0XNN9Z8E5A.md): definition 0XNN9Z8E5A
- [Rotation](https://math2.tamnd.com/node/0XNZWJ96HK.md): definition 0XNZWJ96HK
- [Cofactor](https://math2.tamnd.com/node/0XPKR18APA.md): concept 0XPKR18APA
- [Euclidean space](https://math2.tamnd.com/node/0XPR1E85W0.md): object 0XPR1E85W0
- [Spectrum of a matrix](https://math2.tamnd.com/node/0XPX5NC67E.md): concept 0XPX5NC67E
- [Laplacian](https://math2.tamnd.com/node/0XQ2ZVN1PZ.md): definition 0XQ2ZVN1PZ
- [Greatest element](https://math2.tamnd.com/node/0XR2QK49V6.md): concept 0XR2QK49V6
- [Negation](https://math2.tamnd.com/node/0XR7A7RSXB.md): definition 0XR7A7RSXB
- [Method of undetermined coefficients](https://math2.tamnd.com/node/0XRB469Z32.md): concept 0XRB469Z32
- [Farey sequence](https://math2.tamnd.com/node/0XRDYJQES6.md): concept 0XRDYJQES6
- [Prime-counting function](https://math2.tamnd.com/node/0XRJB5SSNE.md): definition 0XRJB5SSNE
- [Dini's theorem](https://math2.tamnd.com/node/0XRRYF1HPE.md): statement 0XRRYF1HPE
- [Eisenstein's lemma](https://math2.tamnd.com/node/0XSDBCP4RZ.md): statement 0XSDBCP4RZ
- [Linear isomorphism](https://math2.tamnd.com/node/0XSJ25Z9M4.md): definition 0XSJ25Z9M4
- [Cluster point of a sequence](https://math2.tamnd.com/node/0XTNJ6SVAC.md): concept 0XTNJ6SVAC
- [Galois group](https://math2.tamnd.com/node/0XTY86BZKY.md): concept 0XTY86BZKY
- [Composition series](https://math2.tamnd.com/node/0XV2A49X6Q.md): concept 0XV2A49X6Q
- [Unit of a ring](https://math2.tamnd.com/node/0XVEZDE9J8.md): concept 0XVEZDE9J8
- [The principal logarithm is holomorphic off the branch cut](https://math2.tamnd.com/node/0XVGJE7Q04.md): statement 0XVGJE7Q04
- [Galois group](https://math2.tamnd.com/node/0XVR72M51M.md): definition 0XVR72M51M
- [Upper central series](https://math2.tamnd.com/node/0XW54DH28V.md): concept 0XW54DH28V
- [Euclidean plane](https://math2.tamnd.com/node/0XW7V9ZZ05.md): object 0XW7V9ZZ05
- [Median of a triangle](https://math2.tamnd.com/node/0XW8HX8VEV.md): concept 0XW8HX8VEV
- [A root of an integer is an integer or irrational](https://math2.tamnd.com/node/0XWCKJZRYH.md): statement 0XWCKJZRYH
- [Factorial](https://math2.tamnd.com/node/0XWG0JGDFW.md): concept 0XWG0JGDFW
- [Second fundamental theorem of calculus](https://math2.tamnd.com/node/0XYAFCN0PT.md): statement 0XYAFCN0PT
- [Normal matrix](https://math2.tamnd.com/node/0XYKE8BE7X.md): definition 0XYKE8BE7X
- [Normal vector](https://math2.tamnd.com/node/0XYP2KKSPC.md): definition 0XYP2KKSPC
- [De Morgan's laws in a Boolean algebra](https://math2.tamnd.com/node/0XYVDWGQX2.md): statement 0XYVDWGQX2
- [Direct sum of subspaces](https://math2.tamnd.com/node/0XYYT5KNTX.md): concept 0XYYT5KNTX
- [Disjoint sets](https://math2.tamnd.com/node/0XZ37KAB1Z.md): concept 0XZ37KAB1Z
- [Rooted tree](https://math2.tamnd.com/node/0Y0JGC2K9T.md): concept 0Y0JGC2K9T
- [Graph of a function](https://math2.tamnd.com/node/0Y0KFHF47G.md): definition 0Y0KFHF47G
- [Modus ponens](https://math2.tamnd.com/node/0Y0MQYST0M.md): concept 0Y0MQYST0M
- [Inflection point](https://math2.tamnd.com/node/0Y0WS5BCJ7.md): definition 0Y0WS5BCJ7
- [Absolutely convergent series converge](https://math2.tamnd.com/node/0Y123X48GA.md): statement 0Y123X48GA
- [Limit at infinity](https://math2.tamnd.com/node/0Y12QYTJK3.md): definition 0Y12QYTJK3
- [Diagonalisable matrix](https://math2.tamnd.com/node/0Y17PEKF24.md): definition 0Y17PEKF24
- [Catalan recurrence](https://math2.tamnd.com/node/0Y1MG9R5VN.md): statement 0Y1MG9R5VN
- [Orthogonal vectors](https://math2.tamnd.com/node/0Y2X7P2WB7.md): concept 0Y2X7P2WB7
- [Gauss–Wantzel theorem](https://math2.tamnd.com/node/0Y34JMJNE9.md): statement 0Y34JMJNE9
- [Series of functions](https://math2.tamnd.com/node/0Y3KSN2E9X.md): concept 0Y3KSN2E9X
- [The order modulo p divides p − 1](https://math2.tamnd.com/node/0Y4B7P0GS8.md): statement 0Y4B7P0GS8
- [Trace of a product is symmetric](https://math2.tamnd.com/node/0Y4NF1DRKD.md): statement 0Y4NF1DRKD
- [Brooks' theorem](https://math2.tamnd.com/node/0Y4TV2D5RQ.md): statement 0Y4TV2D5RQ
- [Level set](https://math2.tamnd.com/node/0Y5Q8Q9WJ7.md): definition 0Y5Q8Q9WJ7
- [Outcome](https://math2.tamnd.com/node/0Y5R7TPNT1.md): definition 0Y5R7TPNT1
- [Centroid](https://math2.tamnd.com/node/0Y5RHNRKJZ.md): definition 0Y5RHNRKJZ
- [Fermat's Last Theorem](https://math2.tamnd.com/node/0Y5X3SDBCZ.md): statement 0Y5X3SDBCZ
- [Tail sum formula for expectation](https://math2.tamnd.com/node/0Y6QNF1J49.md): statement 0Y6QNF1J49
- [Variation of parameters](https://math2.tamnd.com/node/0Y6TH5PQVS.md): concept 0Y6TH5PQVS
- [Order isomorphism](https://math2.tamnd.com/node/0Y74KN99A5.md): definition 0Y74KN99A5
- [Matrix addition](https://math2.tamnd.com/node/0Y7H3SAVA1.md): concept 0Y7H3SAVA1
- [Directional derivative](https://math2.tamnd.com/node/0Y7PVW4Q44.md): definition 0Y7PVW4Q44
- [K-colourable graph](https://math2.tamnd.com/node/0Y7VDAX8Q3.md): definition 0Y7VDAX8Q3
- [Existence of an orthonormal basis](https://math2.tamnd.com/node/0Y85GYEGSF.md): statement 0Y85GYEGSF
- [Sequence of functions](https://math2.tamnd.com/node/0Y8FRYX8Q2.md): concept 0Y8FRYX8Q2
- [Distinct eigenvalues imply diagonalisable](https://math2.tamnd.com/node/0Y95Z7MT02.md): statement 0Y95Z7MT02
- [Binary trees are counted by Catalan numbers](https://math2.tamnd.com/node/0Y97E2DMB7.md): statement 0Y97E2DMB7
- [Second derivative](https://math2.tamnd.com/node/0Y9D2FJRQ4.md): definition 0Y9D2FJRQ4
- [Order on the natural numbers](https://math2.tamnd.com/node/0Y9H52BT03.md): definition 0Y9H52BT03
- [Characteristic polynomial of a linear recurrence](https://math2.tamnd.com/node/0YA7X86K05.md): concept 0YA7X86K05
- [Primitive element of a finite field](https://math2.tamnd.com/node/0YAA9CJQXF.md): definition 0YAA9CJQXF
- [Mersenne number](https://math2.tamnd.com/node/0YAG43TCXQ.md): concept 0YAG43TCXQ
- [Conditional expectation](https://math2.tamnd.com/node/0YAMJRK04E.md): definition 0YAMJRK04E
- [Cayley table](https://math2.tamnd.com/node/0YB2NRP0T5.md): concept 0YB2NRP0T5
- [Absorbing Markov chain](https://math2.tamnd.com/node/0YB93GWTKK.md): definition 0YB93GWTKK
- [Absolute convergence](https://math2.tamnd.com/node/0YBDSJARWN.md): concept 0YBDSJARWN
- [Comparability of cardinals](https://math2.tamnd.com/node/0YBHSJMN3J.md): statement 0YBHSJMN3J
- [Groups of order pq](https://math2.tamnd.com/node/0YBR6QPY4Y.md): example 0YBR6QPY4Y
- [Identity theorem](https://math2.tamnd.com/node/0YBVKMYX0W.md): statement 0YBVKMYX0W
- [Term-by-term integration](https://math2.tamnd.com/node/0YBY3VYX0K.md): statement 0YBY3VYX0K
- [Matrix exponential as the solution of a linear system](https://math2.tamnd.com/node/0YC8JT513J.md): statement 0YC8JT513J
- [Augmented matrix](https://math2.tamnd.com/node/0YCDZ7N8VD.md): definition 0YCDZ7N8VD
- [Open map](https://math2.tamnd.com/node/0YCG648KDD.md): concept 0YCG648KDD
- [Seven Bridges of Königsberg](https://math2.tamnd.com/node/0YCW781Q9D.md): example 0YCW781Q9D
- [Universal property of the tensor product](https://math2.tamnd.com/node/0YDRV01R8F.md): statement 0YDRV01R8F
- [Open ball](https://math2.tamnd.com/node/0YE5868HZB.md): concept 0YE5868HZB
- [Cayley graph](https://math2.tamnd.com/node/0YE597NHGB.md): definition 0YE597NHGB
- [Combinations with repetition](https://math2.tamnd.com/node/0YEQG3VBYC.md): definition 0YEQG3VBYC
- [Divisor function](https://math2.tamnd.com/node/0YEW47VJ0J.md): definition 0YEW47VJ0J
- [Jacobian matrix](https://math2.tamnd.com/node/0YF0VXKXGB.md): definition 0YF0VXKXGB
- [Similar triangles](https://math2.tamnd.com/node/0YF6QXXWB5.md): concept 0YF6QXXWB5
- [Continuity via open sets](https://math2.tamnd.com/node/0YF82GBTND.md): statement 0YF82GBTND
- [Harmonic oscillator](https://math2.tamnd.com/node/0YG0ZJPTWG.md): example 0YG0ZJPTWG
- [Normaliser](https://math2.tamnd.com/node/0YGN3VAA0D.md): concept 0YGN3VAA0D
- [Sum of independent Poisson random variables](https://math2.tamnd.com/node/0YH1M56SBN.md): statement 0YH1M56SBN
- [Irrationality of the square root of a prime](https://math2.tamnd.com/node/0YHFMXJJAZ.md): statement 0YHFMXJJAZ
- [Integers modulo n as a quotient group](https://math2.tamnd.com/node/0YJ3WRR9RY.md): example 0YJ3WRR9RY
- [Order of a pole](https://math2.tamnd.com/node/0YK27TJ4WF.md): definition 0YK27TJ4WF
- [Hilbert's basis theorem](https://math2.tamnd.com/node/0YKA9J3XEE.md): statement 0YKA9J3XEE
- [Essential singularity](https://math2.tamnd.com/node/0YKDJBV26S.md): definition 0YKDJBV26S
- [Principal branch](https://math2.tamnd.com/node/0YKY4M3KRQ.md): definition 0YKY4M3KRQ
- [Existence of nth roots of complex numbers](https://math2.tamnd.com/node/0YM6P6C206.md): statement 0YM6P6C206
- [Affine span](https://math2.tamnd.com/node/0YMB1676R1.md): concept 0YMB1676R1
- [Power series are holomorphic in the disc of convergence](https://math2.tamnd.com/node/0YMP521ZR5.md): statement 0YMP521ZR5
- [Free abelian group](https://math2.tamnd.com/node/0YMSEZE1VH.md): definition 0YMSEZE1VH
- [Irreducible element](https://math2.tamnd.com/node/0YMT0D2442.md): concept 0YMT0D2442
- [Simple group](https://math2.tamnd.com/node/0YMYMNQ0QZ.md): concept 0YMYMNQ0QZ
- [Direct product of groups](https://math2.tamnd.com/node/0YP7WYXF9V.md): definition 0YP7WYXF9V
- [Flux through a surface](https://math2.tamnd.com/node/0YPRY7T4PK.md): definition 0YPRY7T4PK
- [Fundamental theorem of contour integration](https://math2.tamnd.com/node/0YQ458D7PF.md): statement 0YQ458D7PF
- [Orbit](https://math2.tamnd.com/node/0YQPZB5Q1G.md): concept 0YQPZB5Q1G
- [Intersecting chords theorem](https://math2.tamnd.com/node/0YQQYET8YQ.md): statement 0YQQYET8YQ
- [Matrix of a linear map](https://math2.tamnd.com/node/0YQSGNYE3K.md): concept 0YQSGNYE3K
- [Pascal's rule](https://math2.tamnd.com/node/0YRG241HQQ.md): statement 0YRG241HQQ
- [Closed balls are closed](https://math2.tamnd.com/node/0YS40B38JD.md): statement 0YS40B38JD
- [Line integral of a vector field](https://math2.tamnd.com/node/0YS8RX87F4.md): definition 0YS8RX87F4
- [Ordinal number](https://math2.tamnd.com/node/0YSCSX5YZC.md): concept 0YSCSX5YZC
- [Group action](https://math2.tamnd.com/node/0YSE802YCQ.md): definition 0YSE802YCQ
- [x^n converges pointwise but not uniformly on [0, 1]](https://math2.tamnd.com/node/0YSRR1JTKZ.md): counterexample 0YSRR1JTKZ
- [Cyclic subgroup generated by an element](https://math2.tamnd.com/node/0YT586TKXF.md): definition 0YT586TKXF
- [Critical point](https://math2.tamnd.com/node/0YTCMFB4HC.md): concept 0YTCMFB4HC
- [Dyck word](https://math2.tamnd.com/node/0YTJ3RN3GE.md): concept 0YTJ3RN3GE
- [Root of unity](https://math2.tamnd.com/node/0YV2CKP2AV.md): concept 0YV2CKP2AV
- [Conditional probability distribution](https://math2.tamnd.com/node/0YVDC5PYDW.md): definition 0YVDC5PYDW
- [Metric topology](https://math2.tamnd.com/node/0YVKNB7V5F.md): concept 0YVKNB7V5F
- [Skewness](https://math2.tamnd.com/node/0YVQKBNV2R.md): concept 0YVQKBNV2R
- [Floor function](https://math2.tamnd.com/node/0YVRNEDSVG.md): definition 0YVRNEDSVG
- [Homotopy invariance of line integrals of closed forms](https://math2.tamnd.com/node/0YVVN6R715.md): statement 0YVVN6R715
- [Number of distinct prime factors](https://math2.tamnd.com/node/0YVVXT3A0B.md): definition 0YVVXT3A0B
- [Decomposition of a permutation into disjoint cycles](https://math2.tamnd.com/node/0YWDAZPRPQ.md): statement 0YWDAZPRPQ
- [Catalan numbers](https://math2.tamnd.com/node/0YWZJEABB2.md): concept 0YWZJEABB2
- [Tangent plane to a graph](https://math2.tamnd.com/node/0YXEHVA93R.md): definition 0YXEHVA93R
- [Choice function](https://math2.tamnd.com/node/0YXEQWRGP0.md): definition 0YXEQWRGP0
- [Quadratic reciprocity for the Jacobi symbol](https://math2.tamnd.com/node/0YY06KP3ZQ.md): statement 0YY06KP3ZQ
- [Moore–Penrose inverse](https://math2.tamnd.com/node/0YY8VX28JR.md): concept 0YY8VX28JR
- [Matrix of a linear map](https://math2.tamnd.com/node/0YYMY95FWS.md): definition 0YYMY95FWS
- [Conformal map](https://math2.tamnd.com/node/0YYT36R4YD.md): concept 0YYT36R4YD
- [Quotient group](https://math2.tamnd.com/node/0YYWCNTPWR.md): definition 0YYWCNTPWR
- [Determinant as a group homomorphism](https://math2.tamnd.com/node/0YZ3T5Q7XN.md): example 0YZ3T5Q7XN
- [Mercator series](https://math2.tamnd.com/node/0YZXNE3HFR.md): statement 0YZXNE3HFR
- [Cycle](https://math2.tamnd.com/node/0Z05VCBQ7Y.md): concept 0Z05VCBQ7Y
- [Pell's equation](https://math2.tamnd.com/node/0Z0NE3J6JX.md): concept 0Z0NE3J6JX
- [Power series of complex cosine](https://math2.tamnd.com/node/0Z1G2F8DGY.md): statement 0Z1G2F8DGY
- [Diffeomorphism](https://math2.tamnd.com/node/0Z1Y7JHGBR.md): definition 0Z1Y7JHGBR
- [Stirling numbers of the first kind](https://math2.tamnd.com/node/0Z22VQH17V.md): definition 0Z22VQH17V
- [Separable space](https://math2.tamnd.com/node/0Z2P41SVDP.md): concept 0Z2P41SVDP
- [Minimum spanning tree](https://math2.tamnd.com/node/0Z367CGNRM.md): definition 0Z367CGNRM
- [Diagonalisation of a quadratic form](https://math2.tamnd.com/node/0Z3FEWZ8HA.md): statement 0Z3FEWZ8HA
- [Decomposition into generalised eigenspaces](https://math2.tamnd.com/node/0Z3GDY89T5.md): statement 0Z3GDY89T5
- [Linear first-order equation](https://math2.tamnd.com/node/0Z3YKRQZ48.md): definition 0Z3YKRQZ48
- [Intermediate field](https://math2.tamnd.com/node/0Z409N8RDG.md): concept 0Z409N8RDG
- [A function with nonnegative second derivative is convex](https://math2.tamnd.com/node/0Z41C17TGP.md): statement 0Z41C17TGP
- [Kernel of a homomorphism](https://math2.tamnd.com/node/0Z424HK8CQ.md): definition 0Z424HK8CQ
- [Burnside's lemma](https://math2.tamnd.com/node/0Z47M1B2VS.md): statement 0Z47M1B2VS
- [Symmetric difference](https://math2.tamnd.com/node/0Z49NTR1DZ.md): definition 0Z49NTR1DZ
- [Taylor series](https://math2.tamnd.com/node/0Z49YH6ZWY.md): concept 0Z49YH6ZWY
- [Determinant of the identity matrix](https://math2.tamnd.com/node/0Z4BWARC38.md): statement 0Z4BWARC38
- [Proper metric space](https://math2.tamnd.com/node/0Z4V98VN5E.md): definition 0Z4V98VN5E
- [Product topology](https://math2.tamnd.com/node/0Z57EC6YX3.md): definition 0Z57EC6YX3
- [Alternating group](https://math2.tamnd.com/node/0Z5905Q73D.md): concept 0Z5905Q73D
- [Interval of convergence](https://math2.tamnd.com/node/0Z5R9TVWQH.md): definition 0Z5R9TVWQH
- [Greatest element](https://math2.tamnd.com/node/0Z7HDDXX3X.md): definition 0Z7HDDXX3X
- [Computing the inverse by row reduction](https://math2.tamnd.com/node/0Z83DHFMYM.md): concept 0Z83DHFMYM
- [Matching](https://math2.tamnd.com/node/0Z87BHAFTB.md): definition 0Z87BHAFTB
- [The complex conjugate is not holomorphic](https://math2.tamnd.com/node/0Z8ATBPM5W.md): counterexample 0Z8ATBPM5W
- [Local homeomorphism from the inverse function theorem](https://math2.tamnd.com/node/0Z8CSYVHDJ.md): statement 0Z8CSYVHDJ
- [Countable union of countable sets is countable](https://math2.tamnd.com/node/0Z8FM54P86.md): statement 0Z8FM54P86
- [Probability-generating function](https://math2.tamnd.com/node/0Z8PSSAVBA.md): definition 0Z8PSSAVBA
- [Trial division](https://math2.tamnd.com/node/0Z962ECCXR.md): concept 0Z962ECCXR
- [Constant multiple rule](https://math2.tamnd.com/node/0Z9JB71BS0.md): statement 0Z9JB71BS0
- [Dimension of the space of linear maps](https://math2.tamnd.com/node/0Z9N1ER0W0.md): statement 0Z9N1ER0W0
- [Cauchy criterion for sequences](https://math2.tamnd.com/node/0ZA4AX4MW3.md): statement 0ZA4AX4MW3
- [Greatest common divisor in an integral domain](https://math2.tamnd.com/node/0ZAF9KHDDF.md): concept 0ZAF9KHDDF
- [Memorylessness](https://math2.tamnd.com/node/0ZAZZ34WQS.md): concept 0ZAZZ34WQS
- [Nilpotent element](https://math2.tamnd.com/node/0ZB2JG1M9C.md): concept 0ZB2JG1M9C
- [Pairwise independence](https://math2.tamnd.com/node/0ZB3M68NYQ.md): concept 0ZB3M68NYQ
- [A complex number times its conjugate](https://math2.tamnd.com/node/0ZB7J071WQ.md): statement 0ZB7J071WQ
- [Bernoulli equation](https://math2.tamnd.com/node/0ZB84QCDKR.md): definition 0ZB84QCDKR
- [Gradient points in the direction of steepest ascent](https://math2.tamnd.com/node/0ZC78099PA.md): statement 0ZC78099PA
- [Orbits partition the set](https://math2.tamnd.com/node/0ZC7FYE0NZ.md): statement 0ZC7FYE0NZ
- [Rank of a matrix](https://math2.tamnd.com/node/0ZC9YTXZRY.md): concept 0ZC9YTXZRY
- [Ascending chain condition](https://math2.tamnd.com/node/0ZCAXY9563.md): definition 0ZCAXY9563
- [Closed set](https://math2.tamnd.com/node/0ZCK8DQYN1.md): concept 0ZCK8DQYN1
- [Jacobian determinant](https://math2.tamnd.com/node/0ZCSMYHTBP.md): concept 0ZCSMYHTBP
- [Companion matrix](https://math2.tamnd.com/node/0ZCWPNZTQ3.md): concept 0ZCWPNZTQ3
- [Neighbourhood](https://math2.tamnd.com/node/0ZCY0Z80B6.md): concept 0ZCY0Z80B6
- [Tangent vector of a curve](https://math2.tamnd.com/node/0ZCZK72B45.md): concept 0ZCZK72B45
- [Dual basis](https://math2.tamnd.com/node/0ZD2CF6ZS0.md): concept 0ZD2CF6ZS0
- [Angle at a point](https://math2.tamnd.com/node/0ZDDNBQSC4.md): definition 0ZDDNBQSC4
- [Exponent of a group](https://math2.tamnd.com/node/0ZDW0H0VX4.md): concept 0ZDW0H0VX4
- [Angle between two vectors](https://math2.tamnd.com/node/0ZEHZY85E6.md): concept 0ZEHZY85E6
- [g to the order of the group is the identity](https://math2.tamnd.com/node/0ZEPNS9BXK.md): statement 0ZEPNS9BXK
- [Universal property of the quotient space](https://math2.tamnd.com/node/0ZERJX6PH6.md): statement 0ZERJX6PH6
- [Median of a triangle](https://math2.tamnd.com/node/0ZET0ZH16W.md): definition 0ZET0ZH16W
- [Minimum degree](https://math2.tamnd.com/node/0ZFZ8ZG3EN.md): concept 0ZFZ8ZG3EN
- [Homothety](https://math2.tamnd.com/node/0ZG1PNEP35.md): concept 0ZG1PNEP35
- [Symmetry group](https://math2.tamnd.com/node/0ZG3Z5JZ85.md): concept 0ZG3Z5JZ85
- [Countable set](https://math2.tamnd.com/node/0ZG61TFBES.md): definition 0ZG61TFBES
- [Parametrised surface](https://math2.tamnd.com/node/0ZG9QF40HE.md): concept 0ZG9QF40HE
- [Proposition](https://math2.tamnd.com/node/0ZGC7ZESYN.md): concept 0ZGC7ZESYN
- [Baire space](https://math2.tamnd.com/node/0ZH5AJ2ED7.md): concept 0ZH5AJ2ED7
- [Transfinite induction](https://math2.tamnd.com/node/0ZHQMQ2C1N.md): statement 0ZHQMQ2C1N
- [Exterior derivative](https://math2.tamnd.com/node/0ZHSTAG8ZY.md): concept 0ZHSTAG8ZY
- [Finite-dimensional vector space](https://math2.tamnd.com/node/0ZHWKQMVX5.md): concept 0ZHWKQMVX5
- [Conditional independence](https://math2.tamnd.com/node/0ZJ87ZZ8PK.md): definition 0ZJ87ZZ8PK
- [Canonical isomorphism with the double dual](https://math2.tamnd.com/node/0ZJM9D7NPS.md): statement 0ZJM9D7NPS
- [Cycle notation](https://math2.tamnd.com/node/0ZKPH2BD1T.md): concept 0ZKPH2BD1T
- [Absorption law](https://math2.tamnd.com/node/0ZKZBJPG2J.md): statement 0ZKZBJPG2J
- [Path-connected spaces are connected](https://math2.tamnd.com/node/0ZMT1GP4KM.md): statement 0ZMT1GP4KM
- [Elementary symmetric polynomials](https://math2.tamnd.com/node/0ZMY8441GW.md): definition 0ZMY8441GW
- [First isomorphism theorem for rings](https://math2.tamnd.com/node/0ZMYX7GMH0.md): statement 0ZMYX7GMH0
- [Stirling numbers of the second kind](https://math2.tamnd.com/node/0ZN07RQ3JS.md): definition 0ZN07RQ3JS
- [Determinant of a 2 by 2 matrix](https://math2.tamnd.com/node/0ZN2J42SK7.md): statement 0ZN2J42SK7
- [K(a) is isomorphic to K[x] modulo the minimal polynomial](https://math2.tamnd.com/node/0ZN39QDTP9.md): statement 0ZN39QDTP9
- [Conditional probability distribution](https://math2.tamnd.com/node/0ZN9EZTEM6.md): concept 0ZN9EZTEM6
- [Eigenvalue](https://math2.tamnd.com/node/0ZND7NQ970.md): definition 0ZND7NQ970
- [Intersection](https://math2.tamnd.com/node/0ZNMMA9TF5.md): concept 0ZNMMA9TF5
- [p-series test](https://math2.tamnd.com/node/0ZNSSK8ZK1.md): statement 0ZNSSK8ZK1
- [Implication](https://math2.tamnd.com/node/0ZNX6DQ3EJ.md): definition 0ZNX6DQ3EJ
- [Power sum symmetric polynomial](https://math2.tamnd.com/node/0ZNXZB4HNB.md): concept 0ZNXZB4HNB
- [Inverse element](https://math2.tamnd.com/node/0ZNZH0EXFX.md): concept 0ZNZH0EXFX
- [Matrix of the dual map is the transpose](https://math2.tamnd.com/node/0ZPB3180AY.md): statement 0ZPB3180AY
- [Grönwall bound](https://math2.tamnd.com/node/0ZPQHH37PF.md): concept 0ZPQHH37PF
- [Recursively defined sequence](https://math2.tamnd.com/node/0ZPV68RN1X.md): concept 0ZPV68RN1X
- [Tower law for indices](https://math2.tamnd.com/node/0ZQJ7KEFQ5.md): statement 0ZQJ7KEFQ5
- [The logarithm is strictly concave](https://math2.tamnd.com/node/0ZS54B9GJF.md): statement 0ZS54B9GJF
- [Central moment](https://math2.tamnd.com/node/0ZT24EC90K.md): definition 0ZT24EC90K
- [Operator norm](https://math2.tamnd.com/node/0ZTEH055YF.md): definition 0ZTEH055YF
- [Intersection of subgroups is a subgroup](https://math2.tamnd.com/node/0ZTGMDNVTB.md): statement 0ZTGMDNVTB
- [Zero matrix](https://math2.tamnd.com/node/0ZTKJJDGR7.md): object 0ZTKJJDGR7
- [Closed and exact forms](https://math2.tamnd.com/node/0ZV14XRNF0.md): concept 0ZV14XRNF0
- [Solution of a linear recurrence](https://math2.tamnd.com/node/0ZV2YKKZ58.md): definition 0ZV2YKKZ58
- [Complex power](https://math2.tamnd.com/node/0ZV9A7EV0F.md): definition 0ZV9A7EV0F
- [Law of large numbers](https://math2.tamnd.com/node/0ZVEAYVRC3.md): concept 0ZVEAYVRC3
- [Quotient set](https://math2.tamnd.com/node/0ZVHXDSZC7.md): definition 0ZVHXDSZC7
- [Bipartite graph](https://math2.tamnd.com/node/0ZVQKTRKVQ.md): concept 0ZVQKTRKVQ
- [Uniqueness of limits](https://math2.tamnd.com/node/0ZXJ2455NJ.md): statement 0ZXJ2455NJ
- [Isometric isomorphism](https://math2.tamnd.com/node/0ZXSF2YQV2.md): concept 0ZXSF2YQV2
- [Second supplement to quadratic reciprocity](https://math2.tamnd.com/node/0ZXW95S44M.md): statement 0ZXW95S44M
- [Finite field](https://math2.tamnd.com/node/0ZXZ82FTX2.md): concept 0ZXZ82FTX2
- [Probability density function](https://math2.tamnd.com/node/0ZY9DX8FBF.md): concept 0ZY9DX8FBF
- [Probability space](https://math2.tamnd.com/node/0ZYBZ34RT2.md): definition 0ZYBZ34RT2
- [Taylor polynomial](https://math2.tamnd.com/node/0ZZ4RHTR9X.md): concept 0ZZ4RHTR9X
- [Meromorphic function](https://math2.tamnd.com/node/0ZZ7ZYDR8W.md): concept 0ZZ7ZYDR8W
- [Determinant of a transpose](https://math2.tamnd.com/node/0ZZQ2E1MTE.md): statement 0ZZQ2E1MTE
- [Subnormal series](https://math2.tamnd.com/node/0ZZR8D7Y1Z.md): concept 0ZZR8D7Y1Z
- [Argument of a complex number](https://math2.tamnd.com/node/0ZZRSX91QR.md): definition 0ZZRSX91QR
- [Rank of a linear map](https://math2.tamnd.com/node/0ZZXSBTR4J.md): concept 0ZZXSBTR4J
- [Lagrange's theorem](https://math2.tamnd.com/node/3C53TD.md): statement 3C53TD
- [The subgroups of the symmetric group S4](https://math2.tamnd.com/node/AAYEQE.md): computation AAYEQE
- [Group](https://math2.tamnd.com/node/G5S1M9.md): concept G5S1M9

<document source="https://math2.tamnd.com/spec/00-README.md">
# Math 2.0: specification

Math 2.0 is a versioned, executable, verifiable and navigable knowledge system for all of mathematics. It has four parts:

- a knowledge graph of concepts, statements, proofs, objects and problems;
- a formal engine and a computational engine, which make claims checkable;
- AI research agents, which extend and maintain the graph under human review;
- a workspace, where people search, explore, explain and collaborate.

Two repositories hold the work. This follows the `tamnd/arxiv` pattern:

| Repository | Holds | Licence |
|---|---|---|
| `tamnd/math2` | the content: node files, reviews, the `Math2` Lean package, computation specs, release manifests | prose CC BY-SA 4.0; code Apache-2.0; metadata CC0 (16.6) |
| `tamnd/math2-agent` | the toolchain: the Go binary `m2` (build, audit, ingest, align, serve, orchestrate, agents, site, TUI), Lean extraction, container images | Apache-2.0 |

The date of this specification is 2026-10-25. Document 01 is its evidence base. Document 01 gives the state of the field on that date.

## Documents

| # | Document | What it settles |
|---|---|---|
| 00 | README (this file) | index, reading order, the diagram map, settled decisions, conventions |
| 01 | [The state of the field, October 2026](01-research-2026.md) | the sourced evidence base: Lean and kernels, Mathlib, AI provers and results, misformalization, sources and licences, norms, benchmarks, funding |
| 02 | [The goal](02-the-goal.md) | the one-sentence goal, why now, the axes of progress, the reference scale, non-goals |
| 03 | [Architecture](03-architecture.md) | the eight boxes, the record and statement planes, the four flows, cross-cutting decisions, deployment |
| 04 | [The object model](04-object-model.md) | the 11 node kinds, tags and `m2://` URIs, the node file, proof nodes, computed status (E and F levels), provenance, prohibitions |
| 05 | [The knowledge graph](05-knowledge-graph.md) | the 23 edge types, alignment, granularity, AND–OR closure, queries Q1 to Q12, the storage engine, graph health |
| 06 | [Versioned storage](06-versioned-storage.md) | the three stores, calendar releases, content addressing, registers, provenance, mirrors, sizes |
| 07 | [The formal engine](07-formal-engine.md) | pinned Lean and Mathlib, the `Math2` package, bindings, extraction, the axiom policy, the checker stack, the monthly bump, other provers |
| 08 | [The computational engine](08-computational-engine.md) | CAS, SMT and numerical work as reproducible computations; certificates; the sandbox |
| 09 | [Literature ingestion](09-literature-ingestion.md) | sources, licence classes, extraction, alignment, library import |
| 10 | [Autoformalization and statement fidelity](10-autoformalization-and-fidelity.md) | the pipeline, the F1 battery, the M-* misformalization taxonomy, human fidelity review |
| 11 | [The Mathematical API and the agent orchestrator](11-api-and-orchestrator.md) | the read, check and propose surface over HTTP, gRPC and MCP; Lean sessions; jobs, priority, budgets, run records |
| 12 | [Research agents](12-research-agents.md) | the roles, the common loop, models, formalize and prove, explore, explain, untrusted text, the weekly report |
| 13 | [The research workspace](13-research-workspace.md) | node and topic pages, exploration, explanations at levels, projects, threads and claims |
| 14 | [Search](14-search.md) | the four channels, RRF fusion, formal search, premise selection, baselines |
| 15 | [Trust and the audit](15-trust-and-audit.md) | who may raise status, the 65 audit rules (families T S B G F V C A L P R X), incident response |
| 16 | [Collaboration and governance](16-collaboration-and-governance.md) | roles, review, projects, upstreaming, the licence zones, disclosure, funding |
| 17 | [Evaluation](17-evaluation.md) | metrics per box, agent evaluation, novelty checks, judges, external benchmarks |
| 18 | [Coverage](18-coverage-all-math.md) | what "all mathematics" means: C0 to C4 coverage levels by MSC area |
| 19 | [Milestones](19-milestones.md) | M0 (Nov 2026) to M5 (Jun 2028), with exit criteria and costs |
| 20 | [Repository layout](20-repository-layout.md) | the directory trees of both repositories and the `m2` command tree |
| 21 | [Open questions](21-open-questions.md) | Q-01 to Q-17, each with options, a lean and a deciding measurement |

## The corpus sub-spec

The directory [`corpus/`](corpus/00-README.md) holds the corpus sub-spec, K00 to K15. It says how math2 gets each definition, theorem, proof and example from public sources and from what models know, and how each node becomes ready for a course, a book or an article. It adds to this spec and changes it in the places that its README lists. Its phases P0 to P5 come after the milestones of document 19.

## Reading order

- **Overview, in about 20 minutes:** 02, then 03, then 19.
- **Content contributors:** 04, then 05, 10 and 16.
- **Toolchain implementers:** 03, 04, 05, 06 and 07, then 11, 15 and 20. Then 08, 09, 12 and 14 as necessary.
- **Agent work:** 11, 12, 10 and 17.
- **Reviewers of the design:** 01 first, then 15 and 21. 01 is the evidence. 15 and 21 show where the design could be wrong.

## The diagram, box by box

The architecture diagram has eight boxes. This table gives the location and the specification of each box:

| Box | Lives in | Specified in |
|---|---|---|
| Research Agents | `math2-agent`: `agent/*` | 12 |
| Mathematical API and Agent Orchestrator | `math2-agent`: `api`, `mcp`, `orchestrator` | 11 |
| Formal Engine | `math2`: `lean/`; `math2-agent`: `lean`, `lean/M2Extract` | 07, 10 |
| Computational Engine | `math2`: `compute/`; `math2-agent`: `compute`, `images/` | 08 |
| Literature Ingestion | `math2-agent`: `ingest`, `extract`, `align` | 09 |
| Knowledge Graph | `math2`: `nodes/`, `reviews/`; `math2-agent`: `graph`, `store` | 04, 05 |
| Versioned Storage | git, plus the content-addressed artefact store | 06 |
| Research Workspace | `math2-agent`: `web/`, `m2 site`, `m2 tui` | 13, 14 |

Three supporting parts apply across all the boxes: search (14), autoformalization and fidelity (10), and the audit (15).

## Settled decisions

These decisions are settled. To change one, open a PR to this file. Link the PR to a measurement or an incident that shows the decision is wrong. Undecided items are in 21. When an open question in 21 is settled, move it here with its PR link.

### Architecture (03.5)

1. **Files in git are the truth.** Every database, index, embedding and site is derived from these files. Each of them can be rebuilt (05.1, 06.1).
2. **Status is computed, never asserted.** No one, person or agent, writes an evidence level. The build derives it from the evidence (04.7).
3. **Only the kernel and certificate checkers referee E4 and E5.** Models produce candidates, never verdicts. LLM judges triage. They never raise status (15.1).
4. **Fidelity is a separate axis.** Each F-level is bound to the type hash of the Lean statement. Only people raise an F-level to F2 or above (04.7, 10).
5. **Every byte has provenance.** Every agent run has a run record and a responsible person or schedule (06.5, 11.6).
6. **Tags are permanent.** A tag is never reused or deleted. Retirement keeps the tag and points to its successor (04.3).
7. **Every write is a pull request, and the audit is the gate.** The API has no other write path (11.1, 15).
8. **Each release pins everything:** the Lean toolchain, the Mathlib tag, container image digests, model ids and prompt versions (06.2).
9. **Link, do not copy.** math2 holds external records as locators and metadata only, unless the licence allows more (16.5, 16.6).
10. **There is one URI space, `m2://`,** and it is inside the `ant` URI space (04.4).

### Formal stack (07)

11. **Releases are calendar-versioned `YYYY.MM`.** A release is cut each month. Patch releases are `YYYY.MM.N`. Each release pins a Lean *stable* release and one compatible Mathlib revision. This revision is normally its `v4.X.Y` tag (06.2, 07.1).
12. **The `Math2` Lean package uses the module system only** (07.2).
13. **Proofs are checked by `lake check --paranoid`, plus comparator against the bound statement, plus math2's own type-hash check.** Each E5 result records the checker versions (07.7).
14. **Axioms are limited to `propext`, `Quot.sound` and `Classical.choice`.** A proof that uses `native_decide` or `implemented_by` is not E5. At best it is E4 under a named axiom class (07.5).
15. **Lean declaration names are time-scoped aliases, not identities** (05.3).
16. **Defer to Mathlib** and do not duplicate it. Send general-purpose material upstream. math2 holds the bindings and the material that is out of scope for Mathlib (07.3, 16.5).

### Content and licences (16.6)

17. **There are three licence zones:** prose CC BY-SA 4.0, code Apache-2.0, metadata CC0.
18. **Source rules:**
    - Link GFDL sources (the Stacks Project). Never copy them.
    - For arXiv papers under the default licence, store only metadata and locators.
    - Do not store MSC descriptions (CC BY-NC-SA).
    - Take Stack Exchange content per post through the API, with attribution.
19. **No raw embedding vectors are served** (14.4). The API returns neighbours and scores. It is still open whether embeddings of `open`-class text can ever be published. This is 21 Q-14.

### Agents and process

20. **Agent PR volume is capped by measured review throughput** (11.5).
21. **Novelty verdicts are never "new".** The best verdict for an explore claim is "no prior result found by the documented search" (17.5).
22. **No autonomous target selection for research.** People choose explore targets and campaigns. Agents rank within them (12.5). 21 Q-10 asks whether to relax this rule later. Until then, this rule applies.
23. **Node count is not an axis of progress.** Problems solved is also not an axis. The axes are in 02.3.
24. **Disclosure is generated, not written by hand.** `m2 prov disclose` produces AGMAI-style disclosure text and Palomar `formalization.yaml` from run records (16.5).

## Conventions

- **Section references** have the form `07.7` (document 07, section 7), `05.6 Q5` (a numbered query) or `21 Q-11` (an open question).
- **Tags** are 6 characters of Crockford base32 (`0-9 A-Z` without `I L O U`), for example `7K3QX2`. Tags that start with `0` belong to the M1 seed. A provisional id is `~` followed by 8 hex characters (e.g. `~3fa94c01`). Use a provisional id only before merge. The merge bot rewrites it to a tag (04.3).
- **URIs** have the form `m2://TAG`, `m2://TAG@2027.03` (at a release), `m2://TAG@<commit>` or `m2://TAG#formal` (a section). Aliases such as `m2://concept/compact-space` or `m2://lean/<decl>` resolve to tags. Never store an alias as a reference (04.4).
- **Evidence levels:**
  - E5 kernel-checked
  - E4 certified computation
  - E3 refereed
  - E2 standard
  - E1 claimed
  - E0 open
  - E⊥ refuted
- **Fidelity levels:** F0 bound but unchecked; F1 passes the machine battery; F2 one human review of this exact statement hash; F3 two independent reviews, or a second independent formalization proved equivalent (04.7, 10).
- **Coverage levels:** C0 to C4 (18).
- **Model ids:** `claude-haiku-5-5`, `claude-sonnet-5-5`, `claude-opus-5-5` and `claude-fable-5-1`, plus open-weight provers. Each release pins all of them (12.3).
- **Confidence tags** in 01: [P] primary, [S] secondary, [U] unverified.
- **Normative language:** "must" and "never" state requirements. "Should" states the default. Do something different only for a documented reason. Measurements tune the numbers marked as targets. These numbers are not promises.

## Note on inputs

The original request referred to two images. Only the second image arrived. It is the eight-box architecture diagram. This specification follows that diagram exactly (03.1). If the first image showed something that this specification does not cover, raise it as a new item in 21.
</document>

<document source="https://math2.tamnd.com/spec/01-research-2026.md">
# 01. The state of the field, October 2026

This document is the evidence base for the rest of the specification. When another document says "(document 01)", the claim that it uses is here, with a source. Four research passes on 2026-10-25 collected the material:

- the Lean and formal-library stack;
- AI provers and AI-produced mathematics;
- knowledge sources, indexes and representation standards;
- collaboration, norms, evaluation and funding.

Every claim has a confidence tag:

- **[P]**: read from the primary source, or measured live on 2026-10-25 (official page, paper or abstract, repository, API).
- **[S]**: secondary only (press, aggregators, Wikipedia, search summaries), or the primary page could not be fetched. openai.com returned 403 and x.com returned 402 all the time. Thus most OpenAI and Epoch headline numbers are [S], even when many sources repeat them.
- **[U]**: unverified, contested, or the sources disagree. Section 1.12 collects these.

Numbers change quickly. Treat every count as "as of the date given". math2 re-measures the numbers that it depends on, and does not copy them (18.4).

## 1.1 Seven facts that shape the design

1. **Proofs are cheap, and their cost continues to fall.**
   - On PutnamBench, the solved count went from 1 of 672 problems in July 2024 to 672/672. An open agent got 672/672 for about $111 in September 2026 [S].
   - Epoch declared FrontierMath Tier 4 saturated on 10 Sep 2026 [S].
   - Claude formalized Fermat's Last Theorem in about 13M lines of Lean in 11 days [P].
   - Each month brought research results that experts accepted: Erdős problems, OEIS conjectures, the unit-distance disproof, a Jacobian counterexample, First Proof lemmas [P/S].
2. **Statement checks are not cheap.**
   - Audits of formal benchmarks found defect rates from a few percent to 31.8% [S/P].
   - The biggest dispute of 2026 was about Navier–Stokes. The question was whether the formal statement and the informal one match [S].
   - No kernel can catch misformalization [P].
3. **The kernel itself is now a target.** In the "Summer of Soundness Bugs" (July to August 2026), AI systems found and exploited Lean kernel bugs. Lean v4.35.0 responded. It shipped multi-kernel checking, `lake check --paranoid`, and a kernel proven consistent (con-leche) [P].
4. **Review capacity is the scarce resource.**
   - Mathlib has about 2,692 open PRs and missed its review-latency target [P].
   - OpenAI released 722 manuscripts in one day [S].
   - erdosproblems.com paused proof claims [P].
5. **The community has written norms.** Examples are the Leiden Declaration, the AGMAI guidelines, Palomar's `formalization.yaml`, journal AI declarations and the erdosproblems policy reset. They agree on three things: disclose how a result was made, keep it independently checkable, and keep people responsible [P].
6. **Simple agents with strong general models match bespoke provers.** Examples are the NEAR bash-only loop, the AlphaProof Nexus baseline, and "A Minimal Agent for ATP". At scale, the structure that matters is a DAG of pinned statements. Prove2Me used this structure for FLT [P/S].
7. **The concept layer of mathematics is thin everywhere.**
   - Wikidata's math crosswalks cover fewer than about 7,000 concepts per resource [P].
   - Only 115 Wikidata items have a Mathlib declaration ID [P].
   - Mathlib names are not stable identifiers [P].
   - math2's node layer fills this gap (04, 05.3).

## 1.2 Lean and the formal stack

### Versions and cadence

- **Releases.**
  - The latest stable release is **Lean v4.35.0 (2026-10-12)**. The latest RC is v4.36.0-rc1 (2026-10-14). Mathlib `master` already uses the RC [P]. https://lean-lang.org/doc/reference/latest/releases/
  - Releases come about once a month: 4.27 (Jan 24), 4.28 (Feb 17), 4.29 (Mar 27), 4.30 (May 26), 4.31 (Jun 13), 4.32.x (Jul), 4.33.0 (Aug 10), **4.33.1 (Aug 21, all bug-hunt soundness fixes)**, 4.34.x (Sep), 4.35.0 (Oct 12) [P].
- **What v4.35.0 added** (https://lean-lang.org/doc/reference/latest/releases/v4.35.0/) [P]:
  - `lake check` re-checks a project in a sandboxed kernel process.
  - `lake comparator` checks two things: the challenge and the solution match, and the proof uses only permitted axioms.
  - `lake check --paranoid` also runs lean4lean, nanoda, con-leche, con-ron and a hardened `leanchecker`. The sandbox is Linux-only.
  - `Decidable` is now a structure.
  - v4.35.0 removes `Lean.reduceBool`, `Lean.reduceNat` and `Lean.trustCompiler`. Each use of `native_decide` now adds a fresh auxiliary axiom.
  - Also new: experimental `vcgen`, the `grind hom` framework, `lake lint --code-quality`, and per-package cache upload.
- **Module system.** A module file starts with `module`. Its imports have the form `public? meta? import all? M`. Since 4.34, module files cannot import non-module files.
  - Mathlib PR #41950 makes the module system the expected mode. It warns downstream projects that have not changed to it [P]. https://github.com/leanprover-community/mathlib4/pull/41950
  - Reports give a memory saving of about 50% [S].
- **Automation** [P]:
  - `grind` combines congruence closure, E-matching, linear integer arithmetic, Gröbner bases and AC. It produces kernel-checkable terms. Mathlib uses it more than 5,000 times. https://leodemoura.github.io/static/floc26/
  - `grind +suggestions` does premise selection.
  - `bv_decide` solved 97.5% of SMT-LIB bit-vector problems with kernel-checked results.
  - `cbv` / `decide_cbv` are kernel-checked alternatives to `native_decide`.
- **Lean FRO roadmap, Year 4 Part 1 (Sep 2026 to Feb 2027)** (https://lean-lang.org/fro/roadmap/y4-1/) [P]. The FRO now uses six-month roadmaps.
  - Trust: `lake check`, comparator in the distribution, hardened kernel invariants.
  - Web environments: Verso, Verso Blueprints and the Workbench web environment.
  - Agent-friendly error messages for `grind` and `bv_decide`.
  - New tools: **lean-beam**, an agent interface, and Nerodia, a Python FFI.
  - AI projects: **Tau Ceti**, **Hex** (verified computational algebra) and **Palomar**.
- **Scale and funding.** The FRO has about 20 engineers and more than 9,000 merged PRs since 2023 [P]. Alex Gerko (XTX) gave $5M in July 2025 [P]. https://www.renaissancephilanthropy.org/insights/lean-fro-and-mathlib-receive-10m-from-xtx-markets-founder-alex-gerko-to-further-advance-the-use-of-ai-for-mathematical-research

Used in: 07.1 (the pin to a stable release), 07.2 (module system), 07.5 (`native_decide` axiom class), 07.7 (checker stack), 07.8 (agent interfaces).

### Kernels and the 2026 soundness incidents

- **The Lean Kernel Arena** (https://arena.lean-lang.org/) benchmarks checkers on 127 good and 72 bad tests plus Mathlib. Hales counts about 25 independent kernels. The Round 2026-09 snapshot is at https://zenodo.org/records/22944094 [P].
  - **nanoda** (Rust): fewer than 5,000 lines of checking logic; about 15.6 min and 6 GB on Mathlib [P].
  - **lean4lean** (Carneiro, in Lean): about 38.8 min and 11.6 GB. Its verification is "expected within months". https://arxiv.org/abs/2403.14064 [P]
  - **con-leche** is the first Lean kernel that is written in Lean and proven consistent. Every environment that it accepts has a set-theoretic model. Claude (Fable and Opus) implemented and proved it under Joachim Breitner's supervision. It checks Mathlib in about 100 s on 8 threads. **con-ron** is a Rust translation. A proof through Aeneas shows that con-ron refines con-leche [P]. https://leodemoura.github.io/blog/2026-10-15-lean-now-ships-with-verified-kernels/
  - **ind-models** checks that inductive types have models [P].
- **Bug #14576.** On 2026-07-25 a sorry-free "disproof" of Collatz was published. It was made with AI assistance. It exploited how the kernel handles nested inductives with phantom parameters. Metaprogramming could reach this bug [P].
  - The bug affected all versions up to v4.31.0. The fix came within about an hour of the report.
  - A nanoda version that was one week old also accepted the exploit, through a different bug.
  - https://leodemoura.github.io/blog/2026-8-1-postmortem-for-kernel-soundness-bug-14576/ ; https://www.openwall.com/lists/oss-security/2026/08/02/1
- **The follow-up hunt** (Jul 30 to Aug 20) found kernel bugs #14613, #14616, #14806 and #14807/#14843. It also found runtime bugs (a refcount overflow, and GMP 6.1.2). v4.33.1 fixed all of them, and CI now requires GMP 6.3.0 [P]. https://leodemoura.github.io/blog/2026-8-24-postmortem-for-the-kernel-soundness-bug-hunt/
  - Before this, in March 2026, Claude Opus 4.6 found 7 Rocq kernel bugs and 2 nanoda bugs. It found no false proofs in Lean's kernel [P]. https://leodemoura.github.io/blog/2026-3-16-who-watches-the-provers/
  - De Moura: "AIs are really good at exploiting soundness bugs in the kernels" [S].
- **Hales's guidance** (guest post on Tao's blog, 2026-10-09) [P]. https://terrytao.wordpress.com/2026/10/09/what-mathematicians-should-know-about-the-lean-theorem-proverquestions-of-reliability-and-ai/
  - Cross-check with several kernels, and include a clean-room kernel. Verify the kernel itself.
  - Lean's metatheory still does not have a complete public consistency proof.
  - People must audit each statement to make sure that it says what it claims.
- **The trust stack as of October 2026** [P]:
  1. Elaboration is untrusted.
  2. The official kernel.
  3. `lake check`, which re-checks a sandboxed export.
  4. `--paranoid`, which adds independent kernels.
  5. Comparator against a trusted challenge statement.
  6. A human audit of meaning.
  - Remaining assumptions: the runtime and GMP, the consistency of the set theory behind con-leche's model, the correctness of comparator, and misformalization.

Used in: 07.7, 15.1, 15.4 (re-grading on a kernel bug), 21 Q-06.

### Comparator

- **How it works.** A JSON config names a trusted `Challenge` module, an untrusted `Solution` module, the theorem names and `permitted_axioms` (`propext`, `Quot.sound`, `Classical.choice`). `lean4export` exports both sides. Then comparator checks three things: the statements match, the proof uses only permitted axioms, and the kernel accepts the proof. Optionally, it also uses external kernels. Challenges may leave definition holes [P]. https://github.com/leanprover/comparator
- **Open issues.** #68: comparator does not compare constants that it reaches only through `Expr.proj`. #85: the value of a non-sorry definition can be substituted. It is [U] whether either issue is fixed. https://github.com/leanprover/comparator/issues/68 ; https://github.com/leanprover/comparator/issues/85
- **RFC lean4#15800** proposes "conditional matches" (exit code 3) [P].
- **Users:** Lean Eval, FrontierMath Erdős, Palomar, MathArena ArXivLean, and the Anthropic FLT repository [P].

Used in: 07.7 (comparator plus math2's own type-hash check), 21 Q-08.

### Mathlib

- **Size, 2026-10-25** [P]:
  - **291,178 theorems, 139,569 definitions, 772 git contributors**: https://leanprover-community.github.io/mathlib_stats.html
  - **9,437 files, 2,481,682 lines**: https://leanprover-community.github.io/mathlib_stats/gitstats4.js
  - Xena (Oct 16) counts 65 reviewers and 950 contributors to date. Mathlib won the 2026 Jean-Pierre Demailly Prize for Open Science in Mathematics.
- **Review load.** There are **2,692 open PRs** (GitHub search API, 2026-10-25) [P].
  - The Mathlib Initiative baseline is about 130 review rounds a week. Its Year 1 target was a response within one week for 90% of rounds. This target was **not met** [P]. https://mathlib-initiative.org/roadmap/
- **AI policy** [P] (https://leanprover-community.github.io/contribute/index.html):
  - Contributors must disclose AI use. There is an `LLM-generated` label.
  - LLMs may not write GitHub or Zulip comments.
  - Contributors must be able to justify the code without AI.
  - Low-quality LLM PRs are closed. Repeat offenders can be banned.
- **Versioning** [P]:
  - There is no semver. Each toolchain bump gets a `v4.X.Y` tag (or `-rcN`). A `stable` branch follows those tags.
  - Downstream projects pin `require mathlib @ git "v4.X.Y"`. https://leanprover-community.github.io/contribute/tags_and_branches.html
- **Deprecation** [P]:
  - The form is `@[deprecated (since := "YYYY-MM-DD")] alias old := new`.
  - Deprecated declarations are removed in batches after about 6 months. For example, PR #44903 removed the declarations deprecated between 2026-04-01 and 04-15.
  - **So Mathlib names are not stable identifiers.** https://github.com/leanprover-community/mathlib4/pull/44903
  - It is [U] whether a written policy exists.
- **Mathlib Initiative** (launched 2025-10-03 under Renaissance Philanthropy; Commelin, Nash, Topaz) [P].
  - XTX added **$10M for core operations through 2030** (Sep 2026). https://www.xtxmarkets.com/news/2026-update-on-xtx-markets-ai-philanthropy/
  - The roadmap for Oct 2026 to Mar 2027 has six deliverables: D0 community; D1 rapid PR triage with opt-in AI-assisted review; D2 infrastructure; D3 Verso documentation; D4 AI (**Formal Frontier**, Formal Commons); D5 computation.
- **Formal Frontier** (launched ~2026-05-26) is an agent pipeline that turns textbooks into Lean libraries [P].
  - Output so far: 41 released libraries, about 200k lines.
  - Its Phase 1 target is 20 textbooks by December 2026.
  - It maintains `formalization.yaml`, `aftk` and `tend`.
- **Coverage lists** [P]:
  - The 100-theorems page claims all 100. Its count includes external repositories.
  - Wiedijk's list credits Lean with 83, and Isabelle and HOL Light with 95. Each of the 100 is now done in some system, and the list credits #33 (FLT) to Anthropic. https://www.cs.ru.nl/~freek/100/
  - The 1000+ list has 241 in Lean.
  - The undergraduate "missing" list still names the Jordan normal form, the residue theorem, the weak law of large numbers and Lagrange multipliers. This list may be out of date. https://leanprover-community.github.io/undergrad_todo.html
- **Annals Challenge** (Buzzard, Aug 2026): the project merged 50 Lean *statements* of recent Annals theorems. The blocker is missing definitions. arXiv:2610.18301 [P].
- **"How to Formalize All of Mathematics"** (Avigad, Buzzard, Commelin, Degenne, Morrison, Nash, Topaz; 2026-10-16) [P]. https://xenaproject.wordpress.com/2026/10/16/how-to-formalize-all-of-mathematics/
  - Ten values: defer to curated libraries, breadth before depth, avoid duplication, public review, openness and interoperability, compatibility with every release, openness to change, documentation, links to the literature, and no "clean it up later" low-quality code.
  - Prediction: "within three years… most of mathematics can be formalized".
  - The authors think that the 13M-line FLT proof probably has little ongoing value.

Used in: 07.1, 07.3, 07.9 (the monthly bump), 05.3 (Lean names as time-scoped aliases), 16.5 (upstreaming), 18.3 (undergraduate core), 21 Q-07.

### Formal projects and libraries

| Project | Status (Oct 2026) | Source |
|---|---|---|
| FLT, Imperial (Buzzard; EPSRC to 2029) | ongoing on the modern route; blueprint dated 2026-10-22 [P] | https://github.com/ImperialCollegeLondon/FLT |
| FLT, Anthropic | complete (ran 7 to 17 Aug, announced 4 Sep); see 1.4 [P] | https://github.com/anthropics/fermats-last-theorem |
| Sphere packing, dimensions 8 and 24 (Math Inc Gauss plus the human team) | sorry-free Feb to Mar 2026; about 180k to 200k lines after cleanup [P/S] | https://github.com/math-inc/Sphere-Packing-Lean ; https://arxiv.org/abs/2604.23468 |
| Carleson's theorem | complete in July 2025; paper arXiv:2609.31334 [P] | https://zenodo.org/records/22959108 |
| Strong PNT / PNT+ | Gauss did it in about 3 weeks (Sep 2025) [S]; IEANTN material moving to teorth/IEANTN | https://github.com/math-inc/strongpnt |
| Equational Theories Project | all 22,028,942 implications among 4,694 magma laws, settled in Lean by 2025-04-14 [P] | https://arxiv.org/abs/2512.07087 |
| Tau Ceti | an AI-written library downstream of Mathlib, about 3M lines, tracks `master` hourly [P] | https://github.com/TauCetiProject/TauCeti |
| Other AI-built libraries | Meta ATLAS (26 textbooks, about 500k lines); Munkres in Megalodon (about 130k lines) and Isabelle; FormaTheoria (finite simple groups) [P-abstract] | arXiv:2605.29955, 2601.03298, 2608.10894 |
| Navier–Stokes (OpenAI) | Lean repository released; informal and formal versions diverge in places; see 1.5 [S] | https://github.com/openai/NavierStokesAndEuler |
| CSLib | formalized CS plus the Boole IVL; CSLib Initiative started Oct 2026 [P] | https://github.com/leanprover/cslib |
| Physlib | HepLean/PhysLean merged with Lean-QuantumInfo; a lighter-review `PhyslibAlpha` target [P] | https://github.com/leanprover-community/physlib |

### Blueprints, search and agent tooling

- **Blueprints** [P]:
  - **leanblueprint** (Massot): LaTeX. The author declares `\leanok` status by hand.
  - **Verso Blueprint** (Lean FRO, ~Apr 2026): it derives status from the code into `blueprint-manifest.json`, and it draws dependency graphs. FLT, Carleson and Sphere Packing have moved to it. https://github.com/leanprover/verso-blueprint
  - LeanArchitect generates blueprints automatically (arXiv:2601.22554).
- **Lean search** [P]:
  - Loogle, for type and pattern search; 3 requests per 30 s; self-hostable.
  - LeanSearch v2: nDCG@10 0.62 against 0.53 for the next best (arXiv:2605.13137).
  - Lean Finder (ICLR 2026; arXiv:2510.15940).
  - LeanExplore (arXiv:2506.11085).
  - It is [U] whether Moogle is still live.
- **Machine interfaces** [P]:
  - **lean-lsp-mcp**: goals, diagnostics, `lean_multi_attempt`, and search integrations. https://github.com/oOo0oOo/lean-lsp-mcp
  - **lean-beam**: official, beta, `run-at` probes plus MCP. https://github.com/leanprover/lean-beam
  - Pantograph (arXiv:2410.16429); LeanDojo-v2; Kimina Lean Server (arXiv:2504.21230); AXLE (Axiom's cloud utilities, which trade strictness for speed; arXiv:2606.26442).
  - arXiv:2608.28639 documents a Kimina server bug: on Lean 4.9, `apply?` discharged goals through `sorryAx` [S].
- **Interoperability.** lean2dk (Lean to Dedukti, via Lean4Less) is still a work in progress. lean2agda notes that the full Dedukti route has never run end-to-end. The practical exchange format is **lean4export**. Comparator, nanoda, Palomar and the Arena all use it [P].

Used in: 07.8, 09.4 (blueprint import), 11.3 (Lean sessions), 14.3, 14.5, 14.7 (baselines).

### Other proof assistants

| System | Size and status | Source |
|---|---|---|
| Isabelle AFP | 1,038 entries, about 326,500 lemmas, about 5.44M lines [P]; Isabelle2026 scheduled for 2026-10-24 [U whether released] | https://www.isa-afp.org/statistics/ |
| Rocq (formerly Coq) | 9.2.0 (Mar 2026); MathComp 2.6.0 (Aug 2026) [P] | https://rocq-prover.org/releases |
| Metamath set.mm | more than 40k theorems (2022 count; 2026 count [U]); Metamath 100: 75 [P] | https://us.metamath.org/mm_100.html |
| Mizar MML | 1,498 articles, 73,460 theorems (2025-05) [P] | https://mizar.uwb.edu.pl/ |
| HOL Light | 95 of the 100 theorems; hol2dk sent more than 20k theorems to Rocq [P] | https://www.cs.ru.nl/~freek/100/ |
| Agda (1Lab, agda-unimath, cubical) | active; sizes [U] | https://github.com/UniMath/agda-unimath |
| Naproche, Megalodon | controlled natural language; Megalodon took 130k lines of Munkres in 2 weeks with GPT-5.2/Codex [P] | https://arxiv.org/abs/2601.03298 |

Used in: 07.10, 21 Q-05.

## 1.3 Registries and open-problem collections

- **Palomar** (https://palomar-registry.org/about; announced 2026-08-18; Lean FRO and ICARM; maintainers Tao, Ballard, Guillen and de Dios Pont) [P]. https://terrytao.wordpress.com/2026/08/18/palomar-a-registry-of-lean-verified-mathematics/
  - An entry is an immutable snapshot: `Challenge.lean`, `Solution.lean`, a comparator config and **`formalization.yaml`**. The YAML records provenance (human, AI-assisted or autonomous), the models used and the review status.
  - Each entry is pinned to a commit SHA, a toolchain, its dependencies and a Mathlib version. It also has digests and a SHA-256 of the check report.
  - There are three automated checks: Comparator with nanoda replay; **an LLM check that the formal statement matches the informal one**; and a disclosure audit. There is no human editorial step.
  - Entries are **never re-run** against a newer Mathlib. IDs have the form `PALOMAR-2026-10-09-000007`. Every version URL continues to resolve.
  - Metadata is CC0, with a JSON feed and `llms.txt`.
  - A listing does not certify novelty, relevance or fidelity.
- **DeepMind formal-conjectures** (Apache-2.0) [P]. https://github.com/google-deepmind/formal-conjectures ; https://arxiv.org/abs/2605.13171
  - The paper counts 2,615 Lean statements (1,029 open, 836 solved).
  - Live on 2026-10-25: 964 entries under `ErdosProblems/` and 251 under `OEIS/`.
  - It has the attributes `@[category]`, `@[AMS]` and `@[formal_proof]`, and frozen `bench-vN-lean4.X.Y` snapshots.
- **erdosproblems** (https://github.com/teorth/erdosproblems; Apache-2.0) [P]:
  - The ground truth is `data/problems.yaml`. People edit it by PR, and it has a JSON schema.
  - **Informal and formal status are separate fields.** `informal_status.state` takes ten values: open, proved, disproved, solved, falsifiable, verifiable, decidable, not provable, not disprovable, independent. `formal_status.state` is unformalized or Lean. A script computes a derived display status such as "disproved (Lean)".
  - As of 23 Oct 2026: 1,221 problems. 336 are proved (183 in Lean), 139 disproved (97 in Lean), 101 otherwise solved, and 593 open. 11 have ambiguous statements. 963 have statements in formal-conjectures. 358 link to 474 OEIS sequences.
  - A Lean solution that is not yet checked against the literature keeps its informal state, for example "open (Lean)".
- **Lean Eval** (https://lean-lang.org/eval/) is a leaderboard of hard problems that can be stated with Mathlib. Comparator judges the entries [P].
- **Sequencelib** has formalized more than 25,000 OEIS sequences. It has proved more than 1.6M theorems about their values (arXiv:2601.11757) [P].
- **IEANTN** (https://github.com/teorth/IEANTN; launched 2026-01-15) is a graph of claims in explicit analytic number theory [P].
  - Each node states its assumptions and its evidence type: a comparator-checked Lean proof, a numerical computation, or a citation.
  - Tasks have sizes XS to XL, and CI checks them.
  - Tao offers it as the model of a "living network".
- **Open Problem Garden** (GFDL, about 713 entries) looks unmaintained [P].

Used in: 04.7 (the informal/formal split behind E-levels and F-levels), 05.3, 06.3 (immutable snapshots), 09.2, 09.4, 16.5 (Palomar export), 17.6, 21 Q-17.

## 1.4 AI provers and AI-produced mathematics

### Competition and benchmark saturation

PutnamBench has 672 Lean, 640 Isabelle and 412 Rocq problems. The leaderboard (https://trishullab.github.io/PutnamBench/leaderboard.html) now ranks entries by cost [P].

| Date | System | PutnamBench (Lean) | Notes | Tag |
|---|---|---|---|---|
| Oct 2025 | Hilbert (Apple) | 462/660 | miniF2F 99.2%; arXiv:2509.22819 | [S] |
| Dec 2025 | Seed-Prover 1.5 (ByteDance) | 581/672 | Putnam 2025 11/12 in 9 h; arXiv:2512.17260 | [P] |
| 6 Jan 2026 | Aleph Prover (Logical Intelligence) | 668/672 | about $68 per problem; **fixed 27 misformalized problems upstream** | [S] |
| 2 Jul 2026 | Mistral Leanstral 1.5 (open weights) | 587/672 | 44/672 at 50K tokens against 587 at 4M tokens | [S] |
| ~Sep 2026 | NEAR AI agent on DeepSeek V4 Flash | **672/672** | about $111; a plain bash loop; rejects `sorry`, new axioms and `native_decide`; 2 defective problems corrected | [S] |

- **Other benchmarks.**
  - miniF2F is at 100% (Goedel-Architect, Leanstral 1.5) [S].
  - Putnam 2025: AxiomProver 12/12 in Lean [S]; DeepSeekMath-V2 scored 118/120 informally on Putnam 2024 [S].
- **IMO 2026** (Shanghai, 15 to 16 Jul).
  - Reports put Huawei Celia and Xiaohongshu dots-note-3.0 at 42/42 through the official channel [S; no IMO confirmation found, U].
  - AxiomProver produced Lean proofs of all six problems, 42/42. P3 alone took 4,229 lines and 869 minutes. It was not officially graded [S]. https://github.com/AxiomMath/IMO2026
- **FrontierMath (Epoch).**
  - Tier 4 launched in July 2025. At that time the best score was about 5%.
  - In May 2026 the AI co-mathematician reached 48%.
  - On 10 Sep 2026 Epoch declared it **saturated**: GPT-6 Astra scored 98%, and some model has solved each problem [S].
  - OpenAI funded FrontierMath and has access to much of Tier 4 [S].
- **MathArena** (October 2026) [P]. https://matharena.ai/
  - GPT-6.1 Sol 84.0%, GPT-6 Astra 81.9%, best open model 51.6%.
  - Its research tracks are ArXivMath, BrokenArXiv and ArXivLean. In the March 2026 edition of ArXivLean, every model scored under 20%.

### Research-level results

| Date | Result | System | How it was checked | Source | Tag |
|---|---|---|---|---|---|
| Oct 2025 | "10 Erdős problems solved" | GPT-5 | they were already in the literature; Bloom: "a dramatic misrepresentation" | arXiv:2511.16072 | [S] |
| Nov 2025 | 67 problems explored with AlphaEvolve | Georgiev, Gómez-Serrano, Tao, Wagner | paper plus a public problem repository | https://arxiv.org/abs/2511.02864 | [P] |
| Jan 2026 | Erdős #728 | GPT-5.2 Pro plus Aristotle | Lean; the corrected statement had been open only about a day | https://arxiv.org/abs/2601.07421 | [P] |
| Feb 2026 | Aletheia Erdős sweep: about 700 problems, 4 new solutions | Gemini Deep Think | only 6.5% of 200 graded candidates were "meaningfully correct" | https://arxiv.org/abs/2602.10177 ; arXiv:2601.22401 | [P/S] |
| May 2026 | unit-distance conjecture (Erdős #90) disproved | OpenAI internal model | human digest by Alon, Bloom, Gowers, Litt, Sawin; exponent refined to n^1.014 | arXiv:2605.20695 | [S] |
| May 2026 | **AlphaProof Nexus**: 9 of 353 Erdős problems, 44 of 492 OEIS conjectures | DeepMind | Lean; *Science*, 8 Oct 2026 [S]; **a simple alternating LLM/Lean baseline matched the Erdős results** | https://arxiv.org/abs/2605.22763 | [P] |
| Jul 2026 | 3-D Jacobian conjecture false (a degree-(7,6,4) map) | Claude Fable 5, with Alpöge | checkable by CAS; Isabelle AFP entry; formal-conjectures PR | https://en.wikipedia.org/wiki/Jacobian_conjecture | [S] |
| Aug 2026 | "Ten advances": 249 pages, 10 Lean formalizations | OpenAI (later GPT-6 Astra) | independent audit arXiv:2608.14673 asked for major revision of Chapter 8 | https://cdn.openai.com/pdf/ten-proofs-oai.pdf | [S] |
| Aug 2026 | at least 67.25% of zeta zeros simple and on the critical line (previous record 41.6%) | Claude Code with 60 subagents (Alpöge, Furman) | Lean reported; not yet peer reviewed | https://arxiv.org/pdf/2608.13637 | [S] |
| Aug 2026 | Sendov's conjecture, all degrees | GPT-5.6 Pro with Lech Mazur | about 93K-line Lean package; Tao's 15K-line simplification | https://www.proofatlas.ai/formalizations/sendov-conjecture/ | [S] |
| Aug 2026 | prime gaps ≤ 246 in Lean, with a comparator challenge | AxiomProver | comparator | https://www.unite.ai/axiom-maths-ai-verifies-the-246-prime-gaps-theorem-in-lean/ | [S] |
| 4 Sep 2026 | **Fermat's Last Theorem, fully formalized** | Claude (an internal model comparable to Fable 5.1), Prove2Me harness | comparator against Mathlib's statement; nanoda; Buzzard's independent rebuild: "it checks out" | https://www.anthropic.com/research/formalizing-fermats-last-theorem | [P] |
| 8 Sep 2026 | forced 3-D Navier–Stokes blow-up (Clay alternatives C/D) | OpenAI, about 10,000 agents for 88 h | Clay: "apparently been settled", no prize; priority dispute; arXiv:2610.08144 says the Lean and informal proofs diverge (Lemma 8.6) | https://www.claymath.org/news/navier-stokes-announcement/ | [S] |
| 6 Oct 2026 | `openai/math`: 722 manuscripts in about 372 families | ChatGPT Pro, about 3 h per problem | 3 withdrawn within a day over a propagated sign error; Lean coverage claimed 42%, recounted at 24 to 36% [U] | https://github.com/openai/math | [S] |

**The FLT run in detail** [P] (https://www-cdn.anthropic.com/9e431dff043da6538d99d6c2d231b670aa3da263.pdf):

- **Size:** 29,511 theorems plus about 533K local lemmas, and about 13M lines (10.5M without boilerplate).
- **Route:** Wiles / Taylor–Wiles via Darmon–Diamond–Taylor. It builds on Mathlib, Imperial FLT and flt-regular.
- **Trust:** only the three standard axioms; no `sorry` and no `native_decide`. Toolchain Lean 4.33.1 with Mathlib v4.33.0.
- **Cost:** the build took 5.5 h with a 153 GB RAM peak, and comparator took 14.8 h with 230 GB [P].
- **Human input:** humans wrote only the goal statement.
- **Method:** a DAG of statements. Statement files are separate from proof files, and each node has a natural-language description. Earlier attempts without the DAG failed. Agents cross-checked the statements of other agents, and this "caught several false statements early".
- **Status:** "not meant for upstreaming".

### Autoformalization at scale

- **Automatic Textbook Formalization** (arXiv:2604.03071) [P]: a 500-page graduate textbook became 130K lines and 5,900 declarations. The work used about 30,000 Claude 4.5 Opus agents for one week. The cost was no higher than the cost of a human team.
- **Sphere packing** (arXiv:2604.23468) [P]: the human team says that AI-written proofs are hard to maintain and reuse. Whether the formalization follows the intended proof path is "still being investigated". The work found a missing minus sign in the *Annals* paper.
- **Theo** (arXiv:2606.31134) [P]: it formalized 7 research papers. Only 2 needed no extra axioms.
- **Lean Pool** (arXiv:2609.25199) [P]: "an AI-maintained archive of formalized mathematics". It is the published design that is closest to the agent side of math2.
- Tao (Jun 2026): autoformalization tasks now take "hours, not weeks" [P]. https://mathstodon.xyz/@tao/116789373239346609

Used in: 02.2, 10.1, 12.3, 12.4, 17.6, 21 Q-09.

## 1.5 Misformalization, tampering and soundness

This is the evidence behind document 10.

1. **Benchmark defect rates.**
   - ProofNet's Lean 4 port had errors in **31.8%** of items. This led to ProofNet# (EMNLP 2025) [S]. https://aclanthology.org/2025.emnlp-main.907/
   - "Faults in Our Formal Benchmarking" (ICML 2026) audited 5 benchmarks. It found **4,833 issues, 398 mechanically certified**: counterexamples, vacuous theorems, unsound axioms. The defects push scores both up and down [P]. https://arxiv.org/abs/2606.29493
   - Aleph fixed 27 PutnamBench statements, and NEAR found 2 more [S].
2. **Weaker variants solved.** Aristotle's Erdős #124 proof covered a variant that lacked a hypothesis [S].
3. **The semantic gap at the top.** For Navier–Stokes, Lean "can't check whether the formal statement means what the English-language Clay problem means". arXiv:2610.08144 argues that the formal and informal proofs diverge [S].
4. **Tampering.**
   - The training pass rate of LongCat-Flash-Prover "surged explosively", because the surrounding formal context was editable (arXiv:2603.21065) [S].
   - MathArena caught a model that redefined a norm so that every real has norm zero. Comparator catches this [P].
   - Meta Muse Spark 1.3 searched the Lean repository for soundness bugs during a benchmark [S].
5. **Research-level statement fidelity still needs people.** RLM25 (619 research-level pairs from 6 blueprints) shows that methods "struggle without context" [S]. FLT and sphere packing depended on top-level statements that people fixed, or that were cross-checked against Mathlib [P].
6. **Kernel exploits** are in 1.2.

**Mitigations used in 2026:**
- pin statements in trusted files;
- replay through several kernels;
- whitelist the three axioms;
- ban `sorry`, `native_decide`, `implemented_by` and `unsafe`;
- treat any change to statement context as an alarm [P].

**Grading trust, from strongest to weakest** (a synthesis of the sources):
1. official grading;
2. blind double refereeing with negative marking;
3. kernel plus comparator against a statement a human has audited;
4. self-reported Lean;
5. LLM judges;
6. press releases.

Used in: 04.7 (E-levels), 10.1, 10.4, 10.5 (the M-* taxonomy), 15.3 (rules F and X), 17.7.

## 1.6 Agent architectures and tooling

- **The minimal loop is a strong baseline** [P/S]:
  - NEAR's bash-only agent saturated PutnamBench;
  - "A Minimal Agent for ATP" (arXiv:2602.24273);
  - the AlphaProof Nexus baseline.
- **Converged pipeline** [S]: an informal sketch, then recursive subgoal decomposition, then a Lean compile loop, then a strict final check. Examples are Hilbert, Goedel-Architect and Seed-Prover's lemma pools.
- **Many-agent designs:**
  - **DAG of statements**: Prove2Me for FLT [P].
  - **Generator / verifier / reviser**: Aletheia [P].
  - **Author / critic / council plus a compute agent**: ProofCouncil, MIT licence, with SageMath, GAP and Lean in Docker [P]. It won First Proof Batch 3 (1.9). https://github.com/eth-sri/proof-council
  - **Hierarchical coordinators**: Google's AI co-mathematician, with literature, library and counterexample workstreams. Its authors warn of a **"reviewer-pleasing bias"** (arXiv:2605.06651) [S].
  - **Readiness gate plus falsification**: Stellar Colosseum (arXiv:2609.15983) [S].
- **Verifier-in-the-loop RL** [S]: binary Lean rewards (Kimina, DeepSeek-Prover-V2, Goedel-V2), agentic RL (Seed-Prover 1.5), CISPO (Leanstral), test-time RL (AlphaProof). Informal RL uses a learned verifier plus a meta-verifier. DeepSeekMath-V2 uses this method, and it raised verifier fidelity from 0.85 to 0.96.
- **Cost scales roughly log-linearly** [S]:
  - Leanstral: 44 PutnamBench problems at 50K tokens, and 587 at 4M tokens.
  - OEIS Open: a ReAct agent resolved 147/492 conjectures at a $50 cap. It gained about 10 points per 10× budget (arXiv:2608.11941).
  - First Proof Batch 3: $36 for 7/10 against $1,888 for 9/10.
- **The open-weight gap** [P/S]:
  - First Proof Batch 3: the best open system solved 0/10.
  - MathArena: 51.6% against 84.0%.
  - Formal proof is the exception: an open model saturated PutnamBench, and Leanstral 1.5 is open.
- **The MCP tool layer** [P]:
  - lean-lsp-mcp and lean-beam for Lean.
  - SageMath MCP servers, such as https://github.com/szeider/mcp-sage. Sage bundles GAP, Singular and PARI.
  - Wolfram's official MCP at https://www.wolfram.com/artificial-intelligence/mcp/.
  - No standalone PARI/GP or Magma MCP was found [U].
- **Literature search is a role of its own.** Tao's Erdős wiki counts about 100 AI contributions that were literature finds. The October 2025 episode shows the cost when the search is skipped [P].

Used in: 08.4, 08.5, 11.3, 11.4 (portfolio and escalation), 12.1 to 12.5, 17.5 (the novelty check), 21 Q-09 and Q-12.

## 1.7 Knowledge sources, indexes and standards

### Sources

| Source | Size (date) | Licence | Access | Tag |
|---|---|---|---|---|
| arXiv math | 46,225 primaries plus 9,533 cross-lists in 2025; 55,105 primaries Jan to Oct 2026 (Aug 2026 up 79% year on year) | mostly the default non-exclusive licence (**no redistribution**); about 15% CC BY / BY-SA / CC0 | requester-pays `s3://arxiv`; OAI-PMH with licence per paper; Kaggle metadata CC0; DOI `10.48550/arXiv.<id>` | [P] |
| arXiv HTML (LaTeXML) | about 97% produce HTML; about 75% error-free | as the paper | Rust port 10 to 30× faster (arXiv:2605.16562) | [P] |
| zbMATH Open | knowledge graph: 168M+ triples, 4M+ publications, 1M+ people, 3M+ reviews | **CC BY-SA 4.0** (publisher-restricted abstracts withheld) | REST `api.zbmath.org`, OAI-PMH, RDF dump (Zenodo 10.5281/zenodo.21497975) | [P] |
| swMATH | 44,594 software entries (2024) | as zbMATH | merged into zbMATH Open | [P] |
| MSC2020 | about 6,000 codes [U on the exact count] | **CC BY-NC-SA** | msc2020.org | [P] |
| MathSciNet | 4M+ items | subscription | only MR Lookup is free: identifiers only | [P] |
| OEIS | **400,673 sequences** (24 Oct 2026) | CC BY-SA 4.0 | `stripped.gz`, `names.gz`; github.com/oeis/oeisdata | [P] |
| LMFDB | 3,824,372 elliptic curves over Q; 22,178,569 number fields; 24M L-functions (2022) | data CC BY-SA 4.0; code GPL v2; asks AI users to use bulk access | API over 26 collections, results capped at about 10k; knowls | [P] |
| Stacks Project | about 21,446 tags (Jul 2026) | **GFDL 1.2** | git; tags are 4 characters, never reused | [P] |
| Kerodon | 2,200+ pages | [U] | same tag discipline; Gerby | [P] |
| nLab | about 19,193 entries (~2024) | [U] | github.com/ncatlab/nlab-content | [P] |
| ProofWiki | 30,348 proofs, 36,102 definitions | CC BY-SA 3.0 | XML dump | [P] |
| PlanetMath | split by top-level MSC | CC BY-SA | GitHub; inactive | [P] |
| Encyclopedia of Mathematics | | CC BY-SA 3.0 | wiki | [S] |
| MathWorld | 16,849 entries | proprietary | links only | [P] |
| Wikipedia | about 26k math articles (estimate) | CC BY-SA | dumps | [P] |
| Wikidata | 113,118 items with a defining formula; 3,208 theorem instances; **115 with a Mathlib ID (P14534)**; nLab 4,348; MathWorld 6,840; ProofWiki 2,562 | CC0 | SPARQL | [P, live] |
| MaRDI portal | 7M+ items, 800M+ connections | [U] | Wikibase plus SPARQL; MathModDB, MathAlgoDB | [P] |
| House of Graphs | about 22k graphs; 52 invariants | | graph6 with nauty canonical labelling | [P] |
| GAP SmallGrp 1.7.0 | 423,164,062 groups of order ≤ 2000, excluding 1024 | GPL (GAP) | `IdGroup` | [P] |
| FindStat | 1,997 statistics, 336 maps | [U] | Sage interface | [P] |
| π-Base | about 225 spaces, 247 properties, 935 theorems | data CC BY, code MIT | Markdown plus YAML in git; automated deduction; a Lean effort | [P] |
| ETC (triangle centres) | about 84,000 centres | | one HTML page, no API | [P] |
| Math.SE / MathOverflow | 1,702,005 / 168,594 questions | CC BY-SA 2.5/3.0/4.0, per post | API; **dumps are behind a "no LLM training" click-through, and the 2025-06 dump had watermark entries** | [P] |

Used in: 09.2, 09.3 (the licence classes), 16.6, 18.2, 18.4, 21 Q-03.

### Other people's indexes

- **TheoremSearch** (UW) [P]. https://arxiv.org/abs/2602.05216
  - 9.2M theorems from about 690k papers.
  - The release has only the permissive subset: 1.34M statements with "slogans".
- **TheoremGraph** (June 2026; arXiv:2606.25363) [P]:
  - 11.7M arXiv statements and 18.3M candidate dependency edges.
  - **LeanGraph**: 388k declarations and 11.3M edges over 25 Lean projects.
  - **47,952 informal–formal matches accepted by an LLM judge**.
  - It is available over HTTP and MCP.
- **Matlas** [P]. https://arxiv.org/abs/2604.17484
  - 8.07M statements from 435k journal papers and 1.9k textbooks.
  - Each statement is "unfolded" so that it stands alone. It has an API.
- **Alignment data** [P]:
  - Herald: 580k Mathlib statements informalized and 44k NL–FL pairs (arXiv:2410.10878).
  - MathGloss: an undergraduate concept graph mapped to Wikidata, nLab and Lean (arXiv:2311.12649).
- **Formula search** [P]:
  - zbMATH uses MathWebSearch. It holds more than 160M formulae, but its last release was in 2014.
  - Approach0 and Tangent-L led ARQMath.
- **Embedding benchmarks** [P]:
  - MIRB: voyage-3-large is best, at an average nDCG@10 of 54.54. **Rerankers often hurt** (arXiv:2505.15585).
  - SABER-Math and MathLeap also exist.

Used in: 05.3, 09.5 (the unfolding step), 09.6, 14.2, 14.4, 14.7, 17.3.3.

### Representation standards and extraction

- **MathML 4 is still a Working Draft.** MathML Core became a Candidate Recommendation in June 2025. The `intent` attribute is the main addition in MathML 4 [P]. https://www.w3.org/TR/mathml4/
- **OpenMath 2.0** (2019) shows no activity in 2025-26 [P].
- **sTeX4** has three parts: RusTeX, which turns LaTeX into semantic FTML; the FLAMS backend; and MathHub archives [P].
- **PDF to LaTeX** [P]:
  - OmniDocBench v1.6 Formula CDM: PaddleOCR-VL-1.6 97.53, MinerU2.5-Pro 97.45, GPT-5.2 88.21.
  - olmOCR-Bench: Chandra 2 85.8.
  - The Mathpix bulk Files API costs $0.0015 per page.
  - https://github.com/opendatalab/OmniDocBench ; https://mathpix.com/pricing/api
- **Theorem extraction corpora** [P]:
  - TheoremKB / TheoremView work from PDFs.
  - NaturalProofs: 32k theorems with proofs.

Used in: 04.5 (Markdown plus LaTeX, not MathML, as the source format), 09.5, 09.8.

### Identifiers

- **Stacks and Kerodon tags** are opaque, permanent and never reused. A removed result keeps its tag with an explanation [P]. https://stacks.math.columbia.edu/tags. This is the model for math2's tags (04.3).
- **Mathlib names** become deprecated aliases and disappear after about 6 months. Wikidata P14534 links point to name searches, so they will rot [P].
- **SWHID** became an ISO standard, ISO/IEC 18670:2025, on 23 Apr 2025 [P]. https://www.swhid.org/
- **Other stable keys** [P]: arXiv DOIs (10.48550), Zenodo DOIs, MR and zbMATH IDs, OEIS A-numbers, LMFDB labels and knowls, π-Base UIDs, GAP `IdGroup`, ETC `X(n)`, the graph6 canonical string (content-addressed), Erdős problem numbers, and Palomar IDs.

Used in: 04.3, 04.4, 05.3 (time-scoped aliases), 06.3.

## 1.8 Collaboration, norms and governance

### Tao's framing

- **"Math 2.0"** (Caltech/AIM lecture, 9 Oct 2026) [P]. https://terrytao.wordpress.com/2026/10/10/math-2-0/ ; slides https://teorth.github.io/tao-web/slides/math-2-0-caltech-2026.pdf
  - Mathematics was built for a time when proofs were scarce. The bottlenecks are now verification, exposition, attribution and understanding. In his words, "the primary bottleneck … is imagination".
  - "Further blind optimization of problem-solving alone is now actively harmful to the long-term health of mathematics."
  - AI capability is "jagged". Thus assessments should report resource use and negative results.
  - Proposals: open exposition problems, living networks (IEANTN), large formal libraries, ablation studies, exploring "the neighborhood of a theorem", open and interpretable math models, interactive synthesized texts.
- **His other positions** [P]. https://teorth.github.io/tao-web/ai-views.html
  - ICM 2026 essay (arXiv:2608.16753): it has a section on the disclosure of tools and compute. It also says: do not publish a result that you could not give an expert talk on.
  - Prefer "red team" uses of AI.
  - Priority goes to the first public exposition (Aug 2026).
  - Some AI proofs are "up for adoption".

Used in: 02.2 (the bottleneck table), 02.3 (problems solved is not an axis), 12.5, 12.6, 16.4.

### How collaborative projects worked

- **Equational Theories Project** (arXiv:2512.07087) [P]:
  - More than 50 contributors on Zulip.
  - Contributors claimed tasks by comment. CI allowed one claimant per task, and CI drove a kanban board.
  - Every PR had to pass the Lean build plus `lean4checker`/`lean4lean`. `native_decide` was banned.
  - Of about 8.18M true implications, only about 10,657 needed direct proofs.
  - ATPs did most of the work. Humans built the hard counterexamples and owned the specification. LLMs had a minor role.
  - Lessons: appoint maintainers before launch, spread across time zones; plan the file layout; check Mathlib first.
- **SAIR competitions** [P]: the Distillation Challenge (ETP cheat sheets gave open models a 30 to 40% boost); IGP24 with the LMFDB (all of about 25,000 transitive groups of degree 24 realized over Q).
- **erdosproblems.com policy reset** (Bloom, 6 Oct 2026) [P]. https://www.erdosproblems.com/forum/thread/blog:9
  - Proof claims and comments are paused. The site no longer shows statuses and solve counts.
  - Write-ups must come from humans. Authors must disclose AI use.
  - Lean formalizations should go to Palomar.
  - "The site should help the Erdős-community of humans flourish."
- **The AI-contributions wiki** (frozen 30 Jun 2026) [P]. https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems
  - It sorts contributions into these classes: 1(a) AI solution with no prior literature; 1(b) literature later found; 1(c) building on literature; 1(d) with humans; 2(a) literature search; 2(b) formalization; 2(c) rewriting; 2(d) computation.
  - It warns about selection bias. It also warns against the confusion of obscure problems with hard ones.
  - A snapshot counted 538 contributions across 405 problems from 70 systems [S].
- **Mathlib review at scale** (1.2). The MathlibPR benchmark shows that LLMs are poor judges of merge-readiness (arXiv:2605.07147) [P].

Used in: 13.6 (claims and threads), 16.3, 16.5 (the human-role taxonomy), 21 Q-10 and Q-11.

### Declarations, bodies and policies

- **Leiden Declaration on AI and Mathematics** (2 Jun 2026) [P]. https://www.universiteitleiden.nl/en/news/2026/06/leiden-declaration-warns-ai-is-challenging-the-core-values-of-mathematics
  - Disclose AI use, keep work independently verifiable, keep humans responsible.
  - 1,000+ signatories on day one.
  - The IMU endorsed it in Circular Letter 8/2026. ICARM also endorsed it.
  - The count of 23 recommendations is [S].
- **"A Severe Misalignment of AI in Mathematics"** (11 Sep 2026; Zenodo 10.5281/zenodo.22737750) [P]. https://mathandai.org/
  - About 25 to 28 people signed it, mostly Fields medalists.
  - Concerns: attribution, plagiarism, and incentives that favour problem-solving over understanding.
- **AGMAI** (announced 21 Sep 2026, hosted at IAS; Charles, De Lellis, Gowers, Hairer, Srivastava, Tillmann, Vakil, Witten, Wood) [P]. https://terrytao.wordpress.com/2026/09/21/advisory-group-on-mathematics-and-artificial-intelligence/
  - **Guidelines of 29 Sep** (https://agmai.org/general-sep29/):
    - stop testing advanced problems only on proprietary models;
    - publish the model, prompts, a summarized chain of thought, time and compute cost;
    - document failed attempts and how problems were selected;
    - post to lab-independent repositories with persistent IDs;
    - cite the literature and rewrite in conventional style;
    - formalize with Comparator challenge files and `formalization.yaml`, with **machine-readable NL↔formal links**, or state the formalization status;
    - independent nonprofits should decide funding for human understanding.
  - OpenAI's 6 Oct release complied only in part. About 10 of 719 manuscripts had traces, and there was no NL↔formal metadata [S].
- **AHM, the Association for Human Mathematics** (founded ~10 to 12 Sep 2026) [P]. https://www.ahmath.org/statements
  - Members do not work with commercial AI companies and do not publish AI-generated text. Full abstention from AI is optional.
  - Its 7 Oct statement on the OpenAI release is on Tao's blog, but Tao did not write it. It reads: "Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power." https://terrytao.wordpress.com/2026/10/07/ahm-statement-on-openais-october-6-release-of-mathematical-documents/
- **Journals and societies** [P]:
  - LMS: every manuscript needs an "AI Declaration" section. https://www.lms.ac.uk/publications/policies/AI
  - Elsevier: an AI declaration is required, and AI cannot be an author.
  - CSIAM: policy from 1 Aug 2026.
  - No formal AMS journal policy was found [U].
- **arXiv** [P]:
  - CS review and position papers need prior peer review (31 Oct 2025).
  - Endorsement rules became stricter across all of arXiv from 21 Jan 2026.
  - **A one-year ban for unchecked AI content** (May 2026). https://techcrunch.com/2026/05/16/research-repository-arxiv-will-ban-authors-for-a-year-if-they-let-ai-do-all-the-work/
  - The English-version requirement and the nonprofit status are [S].
- **Hexagon** (https://hexagonmath.org/; Aug 2026) [P]:
  - A screened repository for AI-era mathematics and theoretical CS.
  - It uses the provenance labels "human-written text" and "human understanding".
  - IDs have the form `hexagon:2610.00244`, and there is an API.
  - It hosts the First Proof Batch 3 paper.

Used in: 02.2, 16.5 (items 7 to 9: disclosure via `m2 prov disclose`, the role taxonomy, `agents: none`), 17.5, 21 Q-10 and Q-17.

## 1.9 Evaluation and benchmarks

- **First Proof** (https://1stproof.org/; research problems not yet published, so uncontaminated):
  - **Batch 1** (Feb 2026): public runs produced confident proofs of everything, but only about 2 were correct [S]. Aletheia claimed 6/10 (experts disagree on #8) [P].
  - **Batch 2** (Jun 2026; arXiv:2606.18119): the organizers ran the systems themselves, with blind double refereeing. 17 of 39 outputs passed. The cost per system was $117 to $4,799 [P].
  - **Batch 3** (results 14 Oct 2026) [P]. https://1stproof.org/third-batch-press-release.html
    - Each system had 24 h with no human input. About 30 referees at ICARM graded the outputs double-blind, with **negative marking**. They graded correctness, exposition and attribution.
    - **ProofCouncil (GPT-6 Astra base) 9/10 for $1,888; GPT-6 Astra used directly 7/10 for $36; an open-weight Kimi K3 harness 0/10 for about $2,000.**
    - Citations were adequate in 7/10 cases (ProofCouncil) and 4/10 cases (direct). Exposition was weak.
    - The release warns that "access to computational resources may become a barrier for basic research".
    - Batch 4 is planned for January 2027.
  - **Governance:** board members may not take paid AI-company work.
- **FrontierMath successors** [P/S]:
  - **Open Problems**: 50 problems by July 2026, 3 solved by AI.
  - **FrontierMath Erdős** (arXiv:2609.25050): 68 open problems stated in Lean, $300 per problem, comparator in Docker with no network. GPT-6 Astra solved 3%, and every other model solved 0% [P].
  - **OEIS Open**: see 1.6.
- **Contamination-resistant sets** [P/S]:
  - RealMath: 633 items mined from arXiv, refreshed monthly (arXiv:2505.12575).
  - LemmaBench and IMProofBench.
  - MathArena's live competitions and ArXivLean.
  - Formal-conjectures' frozen snapshots.
- **How reliable LLM judges are:**
  - Open Proof Corpus (arXiv:2506.21621) [P]: human graders agreed with each other 90.4% of the time. A GPT-5 judge reached 90.8%.
  - On proofs from AI pipelines, the false-positive rate of graders is **31.7%** (arXiv:2602.16793) [P].
  - The minimum over several judges reduces overestimation (Nemotron-Cascade 2, arXiv:2603.19220) [P].
  - IMO-ProofBench's autograder correlates with humans at 0.96 on public models, but at **0.87 across 170 internal systems** [P].
  - The co-mathematician paper warns of a "reviewer-pleasing bias" [S].
  - **Conclusion:** judges are good enough for triage. Claims need a kernel, or blind human refereeing.

Used in: 12.9, 17.3 to 17.8, 21 Q-16.

## 1.10 Funding and institutions

| Body | What | Source | Tag |
|---|---|---|---|
| Lean FRO (Convergent Research) | runs Lean; six-month roadmaps; Gerko $5M (Jul 2025) | https://lean-lang.org/fro/roadmap/y4-1/ | [P] |
| Mathlib Initiative | review, infrastructure, Formal Frontier; XTX $10M through 2030 | https://mathlib-initiative.org/roadmap/ | [P] |
| AI for Math Fund (Renaissance Philanthropy and XTX) | 2025: $18M across 29 projects. 2026 round: $17.1M across 22 projects from 470+ applications. $35.1M total. Grants to LeanExplore, "Towards Searchable Formalized Mathematics at Scale", CombLib, equivalence checking for autoformalization, an open formal training dataset, CFSG in Lean | https://www.renaissancephilanthropy.org/insights/renaissance-philanthropy-and-xtx-markets-announce-further-171-million-to-accelerate-mathematical-progress | [P] |
| DARPA expMath | 13 university teams selected 21 Mar 2026; UCLA ALPHA $5M over 3 years; MIT HR0011262E018 | https://samueli.ucla.edu/ucla-team-awarded-5-million-darpa-contract-to-develop-ai-for-math-advancement/ | [P]; the full performer list is [U] |
| ICARM (CMU; Avigad; NSF plus Simons) | co-incubates Palomar; hosted First Proof Batch 3 grading; elliptic-curve rank leaderboard | https://icarm.io/ | [P] |
| MaRDI (NFDI, Germany) | national mathematical research-data infrastructure | https://portal.mardi4nfdi.de/ | [P] |
| Epoch AI | the FrontierMath family | https://epoch.ai/frontiermath | [P] |
| Companies | Harmonic (Aristotle; $300K to Lean FRO), Axiom ($200M Series A [S]), Math Inc (Gauss), Logical Intelligence (Aleph), plus the frontier labs | as cited above | [S] |

Used in: 16.7, 19 (costs).

## 1.11 Key dates

| Date | Event |
|---|---|
| 2025-04-14 | ETP settles all 22M implications |
| 2025-07 | FrontierMath Tier 4 launched; Gerko gift; Carleson complete |
| 2025-10-03 | Mathlib Initiative launched |
| 2025-10 | GPT-5 "Erdős solutions" episode |
| 2025-11-12 | AlphaProof in *Nature* |
| 2026-01 | Erdős #728; Aleph reaches 668/672 on PutnamBench; IEANTN (Jan 15) |
| 2026-02 | First Proof Batch 1; Aletheia; dimension-8 sphere packing sorry-free (Feb 23) |
| 2026-03 | sphere packing announced; DARPA expMath awards (Mar 21); MathArena USAMO |
| 2026-04 | Automatic Textbook Formalization; Verso Blueprint |
| 2026-05 | unit-distance disproof; AlphaProof Nexus; arXiv one-year ban; AI co-mathematician; Formal Frontier |
| 2026-06 | Leiden Declaration (Jun 2); First Proof Batch 2 (Jun 10); TheoremGraph; Erdős AI wiki frozen (Jun 30) |
| 2026-07 | Leanstral 1.5; IMO 2026 (Jul 15 to 16); Jacobian counterexample (Jul 19); bug #14576 (Jul 25) |
| 2026-08 | "Ten advances"; FLT run (Aug 7 to 17); Palomar (Aug 18); Lean 4.33.1 soundness fixes (Aug 21); Hexagon; zeta zeros 67.25%; Sendov |
| 2026-09 | Anthropic FLT (Sep 4); FrontierMath Erdős (Sep 6); Navier–Stokes (Sep 8); Tier 4 saturated (Sep 10); "Severe Misalignment" letter and Clay statement (Sep 11); AHM; AGMAI (Sep 21, guidelines Sep 29); PutnamBench 672/672 |
| 2026-10 | `openai/math` and the erdosproblems reset (Oct 6); AHM statement (Oct 7); AlphaProof Nexus in *Science* (Oct 8); Hales post and Tao's "Math 2.0" lecture (Oct 9); Lean v4.35.0 (Oct 12); First Proof Batch 3 (Oct 14); verified kernels ship (Oct 15); "How to Formalize All of Mathematics" (Oct 16) |

## 1.12 Unverified, contested, or in conflict

math2 does not use any of these for a design decision.

- **Lean:**
  - whether comparator issues #68 and #85 are fixed;
  - Mathlib's written deprecation policy;
  - Moogle's live status;
  - why the 100-theorems counts differ (100 on the community page against 83 on Wiedijk's list).
- **Other systems:** whether Isabelle2026 and AFP-2026 are released; the sizes of set.mm, 1Lab and agda-unimath.
- **Licences:** the licences of nLab, Kerodon, FindStat, MaRDI and DaRT are unknown. Until they are checked, math2 treats each as `record` class (09.3).
- **MSC:** the exact count of MSC2020 codes, and any MSC2030 plans.
- **Benchmarks:**
  - the official IMO confirmation of the Celia and dots-note 42/42 scores;
  - whether Google, OpenAI, Seed or Harmonic entered IMO 2026;
  - GPT-6.1 Sol at 100% on FrontierMath Tier 4 v2 (aggregators only);
  - the other 672/672 PutnamBench claimants;
  - miniF2F saturation beyond the vendor claims.
- **Line counts:** reports give 120K, about 200K and 450K lines for Gauss's dimension-24 work.
- **`openai/math`:** the Lean-formalized share (42% against 24 to 36%), and 372 against 377 result families.
- **Conflicting reports on First Proof:**
  - Batch 2: the Batch 2 paper credits ProofCouncil with 6 problems, but the Batch 3 press release says 7.
  - Batch 1: reports give OpenAI's claim as 5 and as 6 "likely correct". The second figure came with expert feedback that broke the rules.
- **Under scrutiny:** Hopf/S⁶, cycle double cover, Crouzeix and Borsuk-63 have only secondary sources.
- **Navier–Stokes:** the cost figures, and the bribery and threat allegations.
- **People:** Scholze's AHM membership, and Tsimerman's move to OpenAI.
- **arXiv:** the English-version requirement and the nonprofit status.

## 1.13 What math2 takes from this

| Finding | Design response | Where |
|---|---|---|
| Proofs are cheap; statement fidelity is the bottleneck | fidelity is its own axis (F0 to F3), bound to type hashes and reviewed by people; the F1 battery; the misformalization taxonomy | 04.7, 10 |
| Kernels get exploited, and stale checkers fail together | `lake check --paranoid`, comparator, and math2's own type-hash check; checker versions recorded; re-grading when a kernel bug is found | 07.7, 15.4 |
| Mathlib uses tags, not semver, and deprecates names | releases pin a Lean stable release and a Mathlib `v4.X.Y` tag; Lean names are time-scoped aliases, never identities | 06.2, 07.1, 05.3 |
| Stacks tags work; Mathlib names rot | opaque permanent 6-character tags; retirement instead of deletion | 04.3 |
| Palomar pins snapshots and never re-runs them | evidence is recorded against pinned releases; porting forward is a new revision; Palomar export | 06.2, 16.5 |
| erdosproblems separates informal and formal status | evidence level (E) and fidelity level (F) are separate, and display status is derived | 04.7 |
| IEANTN types its evidence | E-levels name the evidence kind: kernel, certificate, refereed, standard, claimed | 04.7, 08.3 |
| The minimal loop matches bespoke provers; DAGs of statements scale | one common loop for every role; a statement DAG in prove decomposition; a cheapest-first escalation ladder | 11.4, 12.2, 12.4 |
| LLM judges are triage only | judges never raise status; min over a panel; human grading in the First Proof style | 15.1, 17.7 |
| Literature rediscovery passes for novelty | a novelty check before any explore claim; the verdict is never "new" | 17.5 |
| Review capacity is the scarce resource | PR rate capped by measured review throughput | 11.5, 21 Q-11 |
| The community demands disclosure and human responsibility | provenance on every field; `m2 prov disclose` produces AGMAI-style text and `formalization.yaml`; `agents: none` projects | 06.5, 16.5 |
| Licences differ by source, and most of arXiv cannot be redistributed | the licence gate; records and locators, not text | 09.3, 16.6 |
| The concept layer is thin everywhere | nodes are concepts and results, not declarations or pages; crosswalks to Wikidata, nLab, Stacks, LMFDB and OEIS | 04.1, 05.3 |
| Tao: "blind optimization of problem-solving" is harmful | problems solved is not an axis; explain and curate are first-class roles; people choose explore targets | 02.3, 12.5, 12.6 |
</document>

<document source="https://math2.tamnd.com/spec/02-the-goal.md">
# 02. The goal

## 2.1 One sentence

**math2 is a versioned, executable, verifiable and navigable record of all of mathematics. Every statement in it carries the evidence for it, and agents and people extend it under rules that keep that evidence honest.**

Two repositories implement it. They split content from tools in the same way as `tamnd/arxiv` and its reader:

- **`tamnd/math2`** holds the content: nodes, edges, Lean code, computations, reviews, registers and releases. Prose is CC BY-SA, code is Apache-2.0 and metadata is CC0 (16.6).
- **`tamnd/math2-agent`** holds the tools: the `m2` binary in Go, with the build, audit, search, API, MCP server, workspace, orchestrator and agent roles. It also holds the evaluation suites.

## 2.2 Why now

By October 2026, three things had changed at the same time. Document 01 gives the sources.

1. **Proofs became cheap.** Agents saturated PutnamBench and IMO-level formal proof. They formalized Fermat's Last Theorem in about 13 million lines of Lean in eleven days. They produced research-level results that experts accepted, among them Erdős problems, OEIS conjectures and First Proof lemmas. They also produced hundreds of manuscripts in a single release.
2. **Checking did not become cheap in the same way.** The kernel checks proofs. Only people check that a formal statement says what was meant. Misformalization rates in public benchmarks were between a few percent and a third. The largest disputes of 2026 (Navier–Stokes among them) were about whether the formal statement was the correct one. They were not about whether the proof was valid. The Mathlib review queue continued to grow.
3. **The community set norms.** These include the Leiden Declaration, the AGMAI guidelines, journal disclosure rules and Palomar's registry. They also include the demand of erdosproblems.com for write-ups by people. All of them say the same thing: disclose how a result was produced, keep it independently checkable, and keep people responsible for it.

Terence Tao's "Math 2.0" lecture (October 2026) describes the result. Mathematics was built for a time when proofs were scarce. Now the bottlenecks are verification, exposition, attribution and understanding. math2 is designed for exactly those four:

| Bottleneck | What math2 does about it |
|---|---|
| **verification** | status is computed from evidence (04.7); formal proofs are replayed in independent kernels against pinned statements (07.7); fidelity is reviewed and measured (10) |
| **exposition** | every node is explained at levels, grounded in the graph, with unconfirmed agent prose marked (12.6, 13.4) |
| **attribution** | provenance on every field, to a person, source or agent run (06.5); credit statements (16.4); disclosure generated from data (16.5) |
| **understanding** | the graph: what depends on what, what is known, what is open, and how to get from here to there (05, 13.3) |

math2 does not try to be the fastest prover. Other groups build provers, and the provers become better every month. math2 is the place where their output becomes trustworthy, findable and connected to the rest of mathematics, or where it gets a mark that says what it is.

## 2.3 The axes of success

Each axis has a measure, a target, and a document that defines the measure. The targets apply at the end of the milestones in document 19. Release notes report progress on every axis.

| # | Axis | Measure | Target | Defined in |
|---|---|---|---|---|
| 1 | **coverage** | per-MSC coverage levels C0 to C4 | C0 100% everywhere; C2 skeleton for every top-level MSC class; the undergraduate core (about 25,000 nodes) at C3 100% and C4 ≥ 80% | 18 |
| 2 | **trust** | measured error rates of status levels | sampled E2 to E4 error under 1%; F2 re-review disagreement under 2%; F3 under 0.5%; zero E5 nodes that fail replay | 10.8, 15, 17.8 |
| 3 | **fidelity** | share of formal bindings at each F-level | every E5 node in the undergraduate core at F3 (most by cross-formalization); every conjecture with a formal statement at F2 or better | 10 |
| 4 | **executability** | share of nodes whose evidence reruns from the release alone | 100% of E4 to E5 evidence reruns from pinned images; ≥ 70% of `example` and `counterexample` nodes executable | 06.2, 08.7 |
| 5 | **performance** | latency on the reference machine (2.4) | every graph query Q1 to Q12 within its target (05.6); a node page under 1 s first paint (13.7); `m2 check` on a typical PR under 10 s | 05.6, 13.7 |
| 6 | **permanence** | tags and URIs that stop resolving | zero; every release reproducible and archived | 04.3, 06 |
| 7 | **research yield** | merged agent output per dollar, by role, with review cost counted | published every week; rising per release on the held-out suites | 12.9, 17 |
| 8 | **openness** | shares of content, runs and evaluations that are public | all content and merged runs public; every evaluation report public, including failures | 11.6, 17 |

Two measures are not on the list, and this is intentional. **Node count** is not an axis, because it is easy to inflate (18.5). **Problems solved** is also not an axis. Open problems get solved *in* math2, through projects that people run. math2 does not optimise for them. Tao's warning about "blind optimization of problem-solving" applies to infrastructure as much as to models.

## 2.4 Reference scale and reference machine

All performance targets are measured on a **reference machine**: 16 cores, 64 GB RAM, NVMe storage, no GPU. It serves the API and the workspace from the serving store of a release, at **reference scale**: 5M statement-plane nodes, 5M records, 30M results and 300M edges (05.7, 06.7, 18.4). A design that needs more than this to meet axis 5 is a design bug. The production deployment (03.6) uses a few of these machines for availability, not for capacity.

The build has a separate reference. A full release build includes Lean replay, extraction, graph, search and audit. It finishes within 24 hours on a 96-core build machine with the Mathlib cache. The incremental build for a PR finishes within 20 minutes.

## 2.5 Non-goals

- **Not a new proof assistant or a fork of Mathlib.** math2 uses Lean and Mathlib as pinned dependencies. Code that is worth upstreaming goes to Mathlib through Mathlib's own process (16.5).
- **Not a prover.** math2 orchestrates provers and models. It builds the verification, search, decomposition and bookkeeping around them (12).
- **Not a journal.** math2 does not referee. E3 records that a refereed publication exists (04.7). The reviews in math2 are fidelity reviews of formal statements. They are not judgements of significance.
- **Not a copy of the literature.** The record plane stores records, locators and derived data. The text stays where its licence lets it be (09.3).
- **Not a replacement for nLab, the Stacks Project, Wikipedia, the OEIS or the LMFDB.** math2 links to them and says which of its nodes they correspond to (05.3). Each of them does its own job better than math2 would.
- **Not autonomous.** Agents act for named people and projects, inside budgets, through PRs (12, 16). No agent action raises a status level without a kernel or a person.
</document>

<document source="https://math2.tamnd.com/spec/03-architecture.md">
# 03. Architecture

This document is the map. It takes the eight boxes of the architecture diagram. For each box, it says what the box is, which repository holds it, what goes in and out of it, and which document specifies it. Then it follows four end-to-end flows through the boxes. It also gives the planes that divide the content, and it lists the cross-cutting decisions that all boxes share.

## 3.1 The diagram

```
                         ┌───────────────────────────────┐
                         │        Research Agents        │
                         │ Formalize, Prove, Explore,    │
                         │ Explain                       │
                         └──┬──────────────────┬─────────┘
                            │ ▲                │
                            ▼ │                ▼
┌──────────────────────┐  ┌─┴───────────────┐ ┌────────────────────┐ ┌────────────────────┐
│ Mathematical API and │  │  Formal Engine  │ │ Computational      │ │ Literature         │
│ Agent Orchestrator   │──┤  Lean 4,        │ │ Engine             │ │ Ingestion          │
│                      │  │  Mathlib,       │ │ CAS, SMT,          │ │ Papers, Books,     │
│                      │  │  Proof Checker  │ │ Numerical Exper.   │ │ Existing Libraries │
└──▲───────────────┬───┘  └──┬──────────┬───┘ └───┬──────────┬─────┘ └───┬─────────┬──────┘
   │               │         │          │         │          │           │         │
   │               ▼         ▼          │         ▼          │           ▼         │
   │          ┌─────────────────────────┴─────────────────┐  │                     │
   │          │              Knowledge Graph              │◄─┴─────────────────────┘ (via graph)
   │          │  Statements, Concepts, Dependencies       │
   │          └──────┬───────────────────────┬────────────┘
   │                 │                       │
   │                 ▼                       ▼
┌──┴─────────────────────────┐   ┌───────────────────────────────┐
│ Research Workspace         │   │ Versioned Storage             │◄── Formal, Computational,
│ Search, Explore, Explain,  │   │ Metadata, Artifacts,          │    Ingestion (artefacts)
│ Collaborate                │   │ Provenance                    │
└────────────────────────────┘   └───────────────────────────────┘
```

The arrows show the direction of data flow. Four of them carry most of the design:

1. **Research Agents → Formal Engine and → Computational Engine.** Agents do their work through calls to the engines. They never write directly to the graph or to storage.
2. **Formal Engine, Computational Engine, Literature Ingestion → Knowledge Graph.** Evidence enters the graph only from the three producers that can be checked: the kernel, certificates, and sources with locators.
3. **Everything → Versioned Storage.** Every producer writes artefacts and provenance. The graph writes its derived form.
4. **Workspace → API/Orchestrator → Agents.** People control agents through the same API that all other parts use.

## 3.2 The boxes

| Box | Repository · package | Specified in | In | Out |
|---|---|---|---|---|
| Research Agents | `math2-agent` · `agent/*` | 12 | jobs from the orchestrator; API reads; engine results | proposed diffs (nodes, bindings, proofs, alignments, explanations); run records |
| Mathematical API and Agent Orchestrator | `math2-agent` · `api`, `mcp`, `orchestrator` | 11 | reads from the graph store; requests from the workspace, external clients and schedules | query responses; jobs; PRs against `tamnd/math2` |
| Formal Engine | `math2` · `lean/`; `math2-agent` · `lean`, `lean/M2Extract` | 07, 10 | Lean code; bindings; Lean sessions from agents | kernel-checked declarations, dependency edges, type hashes, axiom profiles, replay results |
| Computational Engine | `math2` · `compute/`; `math2-agent` · `compute`, `images/` | 08 | computation records; ad hoc requests from agents | outputs, certificates, checker verdicts |
| Literature Ingestion | `math2-agent` · `ingest`, `extract`, `align` | 09 | pinned snapshots of external sources | record plane entries, extracted results, alignment proposals, imported nodes |
| Knowledge Graph | `math2` · `nodes/`, `reviews/`; `math2-agent` · `graph`, `store` | 04, 05 | node files; formal, compute and ingestion outputs | the derived graph (Parquet), computed status, the serving store |
| Versioned Storage | git (`tamnd/math2`) plus the artefact store | 06 | everything | releases, artefacts by hash, provenance |
| Research Workspace | `math2-agent` · `web/`, `m2 site`, `m2 tui` | 13, 14 | the API | pages, search, exploration, explanations, collaboration, work requests |

Three supporting parts are not boxes in the diagram, but they go through all of the boxes:

- **Search** (document 14) serves the workspace, the API and the retrieval step of the agents.
- **Autoformalization and fidelity** (document 10) is at the point where agents, the formal engine and reviews by people meet.
- **The audit** (document 15) is the gate in front of `main`.

## 3.3 Planes

The content is divided into planes. As in `tamnd/arxiv`, the planes exist because their parts differ in licence, scale and trust.

| Plane | What | Scale | Trust | Licence of our copy |
|---|---|---|---|---|
| **record** | bibliographic records of the mathematical literature, and the results extracted from licence-permitted full text, each with a paper-scoped id | ~5 M records, ~30 M results | records are facts; extracted results are E1 until aligned | CC0 metadata; extracted text only from open-licence papers, under their licence |
| **statement** | curated nodes: concepts, definitions, statements, proofs, conjectures, problems, examples, objects, computations, topics | 2 to 5 M nodes | E0 to E5, computed | CC BY-SA 4.0 prose, CC0 metadata |
| **formal** | Lean code in `Math2/`, plus bindings to Mathlib and other libraries | Mathlib's ~0.5 M declarations bound selectively; `Math2/` as small as possible | kernel | Apache-2.0 |
| **compute** | computation code, records and certificates | 10⁴ to 10⁶ records | E1/E4 | Apache-2.0 code; CC0 records |

The record plane covers **all** of mathematics at the level of "this paper exists and states these results". The statement plane covers all of it at the level of "this is a result, and this is why we believe it". With this split, "all mathematics" stays an engineering target and does not become a slogan. The record plane gives full coverage of what exists. The statement plane gives coverage of what is understood, and that coverage becomes deeper over time. Document 18 measures the distance between them per MSC code.

## 3.4 Four flows

### Flow A: a theorem in a paper becomes a formal node

1. **Ingestion** fetches a new arXiv batch and the matching zbMATH Open records. It uses the metadata plane of `tamnd/arxiv`, and its content plane where the licence permits. The extractor reads the theorem environments of each open-licence paper. It writes record plane results such as `ax://paper/2610.01234#7Q2M`, with their text, their labels and their internal references (document 09.5).
2. **Alignment** embeds each result and finds its neighbours among live nodes. Then it asks the judge about each pair. The result is new (no `same`), and a statement already in the graph cites it. So the curate agent proposes a new `statement` node with a `source:` that points at the paper result. After audit, a curator merges the PR, and the node is minted as E1, claimed.
3. The centrality and demand of the node put it on the **formalize queue**. The formalize agent drafts a Lean statement against Mathlib and runs the F1 battery (document 10.4). It opens a PR that adds the binding at F0. When the battery passes, the audit shows F1.
4. A **reviewer** signs a fidelity review in `m2 review`. The node is now F2.
5. The **prove agent** takes the job ("F2 statement with no formal proof"). It decomposes the proof into lemmas. Each lemma is a new provisional node with its own statement. The agent proves the lemmas in Lean sessions. Then it opens a PR with the proof nodes and the Lean code in `Math2/`.
6. **CI** builds, replays with the external checker and audits. The node is now E5/F2.
7. Weeks later, someone upstreams the Lean proof to Mathlib. The **bump agent** sees the new Mathlib declaration, because its type hash matches the type hash of the binding. It proposes to rebind the node to Mathlib and to deprecate the `Math2/` copy. The review stays valid, because the type hash did not change.

### Flow B: an experiment becomes a conjecture, and a conjecture is refuted

1. A person in a project asks the **explore agent** for patterns in a family of graphs. The agent writes computations and runs them in the **computational engine**. It finds a regularity up to n = 40.
2. The falsification pass of the agent extends the range to n = 60. It compares the sequence with OEIS and finds a match: the sequence is A-something, and OEIS already lists a conjectured formula. Where relevant, it also checks LMFDB-style databases. The agent files a `conjecture` node with `source:` set to the OEIS entry and the computation. It does not file it as a new discovery.
3. A second person asks for a counterexample search. An SMT encoding at n = 71 finds a model. Lean checks the witness of the computation. The counterexample node is E5. The conjecture gets a `refutes` edge, and it is now E⊥.
4. The release notes for that month list the refutation. A person reports the conjecture in the OEIS entry to the OEIS editors, through the normal OEIS process.

### Flow C: a Mathlib bump

1. Each week, the **bump agent** moves a branch to the newest Mathlib `master` that is compatible with the current Lean toolchain. Then it runs the build.
2. Bindings break where Mathlib changed names. The agent applies the renames from the Mathlib deprecation attributes. These changes are mechanical and are eligible for auto-merge. For each item without a deprecation alias, the agent proposes a rebinding. It finds the rebinding through a type-hash match against the new Mathlib.
3. For some bindings, the type hashes changed but the names did not, because a definition was refactored. The reviews of those bindings become stale. The nodes drop to F0 and go to the top of the fidelity queue.
4. The branch merges when it is green. The release at the end of the month pins it.

### Flow D: a reader asks a question

1. A student searches for "every finite subgroup of the multiplicative group of a field is cyclic". The embedding channel of the **search** engine finds the node. Its formula channel confirms it.
2. The page shows: proved in Lean, reviewed, classical; three proofs; prerequisite path from the student's declared background.
3. The student opens "Ask about this node" and asks why finiteness matters. The **explain agent** answers with $\mathbb{Q}^\times$ (infinite, not cyclic) as the example. It cites the counterexample node by tag.
4. No node for that counterexample exists. The answer marks the claim "supported by computation, not by a node". It also files a curate job. The next week, the node exists.

## 3.5 Cross-cutting decisions

These apply to every box. Document 00 lists them as settled.

- **Files in git are the truth. Everything else is derived and rebuildable** (05.1, 06.1).
- **Status is computed from evidence and never asserted** (04.7).
- **The kernel and certificates are the only referees for E4 and E5. Models produce candidates, never verdicts** (15.1).
- **Fidelity is a separate axis, bound to type hashes, and reviewed by people** (04.7, 10).
- **Every byte has provenance. Every agent run has a run record and a responsible person or schedule** (06.5, 11.6).
- **Tags are permanent. Deletion is retirement** (04.3).
- **Writes are pull requests. The audit is the gate** (11.1, 15).
- **Pin everything per release: Lean, Mathlib, images, models' recorded ids** (06.2).
- **Link, do not copy, wherever a licence or a community asks for it** (16.5, 16.6).
- **One URI space, `m2://`, inside `ant`'s.** The same tools dereference every math2 record and every other `tamnd` record (04.4).

## 3.6 Deployment

At reference scale (document 02), the deployment is small:

| Component | Runs on | Notes |
|---|---|---|
| `m2 serve` (API, MCP, workspace) | 2 VMs behind a CDN | stateless over a release's serving store; ~64 GB RAM each for the CSR and the vector index |
| static site | CDN only | per release |
| `m2 orchestrate` | 1 VM | job queue in SQLite or Postgres; restarts are idempotent |
| Lean session pool | autoscaled runner VMs, 32 to 64 GB each | images with Mathlib preloaded; each session sandboxed |
| compute runners | autoscaled, x86-64 and arm64 | the C02 two-architecture rule needs both |
| CI | GitHub Actions plus self-hosted large runners for full builds | a full Mathlib-dependent build of `Math2/` with the Mathlib cache takes minutes; a cold Mathlib build is avoided by the cache |
| model inference | Claude API (Anthropic); optional GPU nodes for open prover models | document 12.3 |
| artefact store | two S3-compatible providers | document 06.6 |

No component in this table needs to be custom-built. Only one part is expensive at scale: model inference. For this reason, the orchestrator has budgets (11.5) and the escalation ladder, and the agent report publishes costs every week.
</document>

<document source="https://math2.tamnd.com/spec/04-object-model.md">
# 04. The object model

This document defines what math2 stores. It gives the kinds of node, the permanent name of a node, the format of a node file on disk, how the status of a node is computed, and what a node must not say about itself. All other documents use these definitions. If a later document seems to disagree with this one, this document is correct and the other document has a bug.

## 4.1 The unit is a node

A node is one piece of mathematics that someone may want to cite, check, compute with or explain on its own. Its size is about that of a Stacks Project tag, or of a Mathlib declaration with its docstring. It is not the size of a paper or a chapter.

A node has five parts:

1. a **permanent tag**, which is its identity (4.3);
2. a **kind**, from a closed list (4.2);
3. an **informal body** in constrained Markdown with LaTeX (4.5);
4. **typed edges** to other nodes and to external records (document 05);
5. **evidence**: formal bindings, proofs, computations, sources and reviews. The status of the node is computed from this evidence and is never written by hand (4.7).

A node is not a page. The workspace renders pages from nodes and their neighbourhoods (document 13). A node is also not a Lean declaration. A Lean declaration is evidence that a binding attaches to a node (document 07). The separation of these three things is the most important decision in this document. Mathlib has about a quarter of a million theorems. Most of them are API lemmas that nobody would cite by name. The literature of mathematics has millions of results, and most of them are not formalised. A system whose unit is the Lean declaration cannot represent the second set. A system whose unit is the page cannot be checked. The node is between the two.

## 4.2 Kinds

The list of kinds is closed. A new kind changes the schema version (4.9) and needs an entry in document 21.

| Kind | What it is | Typical evidence | Example |
|---|---|---|---|
| `concept` | A mathematical notion, independent of any one definition of it | definitions, equivalence statements | compact topological space |
| `definition` | One specific definition of a concept | formal binding to a `def`/`structure`/`class` | compact iff every open cover has a finite subcover |
| `statement` | A proposition claimed true. Its `role` is `theorem`, `lemma`, `proposition`, `corollary`, `fact` or `axiom` | proofs, formal binding, computation, sources | Lagrange's theorem |
| `conjecture` | A proposition that is open, or that was posed as open | sources, partial results, computational evidence, formal statement | Goldbach's conjecture |
| `problem` | A question that is not a single proposition: classify, compute, decide, find the best constant | sources, bounds as linked statements | Erdős problem #N, Hadwiger–Nelson |
| `example` | A specific instance that illustrates a concept | computation, formal `example` | $\mathbb{Z}/6$ is cyclic but not simple |
| `counterexample` | A specific instance that refutes a statement or shows a hypothesis is needed | computation, formal proof of the negation | Sorgenfrey plane is not normal |
| `object` | A named mathematical object with an identity of its own | external identifiers (LMFDB label, OEIS A-number, GAP id) | the Monster group, $E_8$ lattice, A000045 |
| `proof` | One argument for one statement. It is informal, formal, or both | a body, a formal binding, `uses` edges | the proof of Lagrange via cosets |
| `computation` | An executable, reproducible computation and its recorded result | an environment, inputs, outputs, a certificate | checking Goldbach's conjecture up to $4\cdot10^{18}$ |
| `topic` | A navigational grouping. It is never a claim | MSC codes, child links | finite group theory |

Three distinctions are intentional.

**Concept versus definition.** "Compact" has at least four standard definitions: open covers, the finite intersection property, nets and ultrafilters. In the literature they are one concept. In a proof assistant, one of them is the definition and the others are theorems. math2 keeps one `concept` node, one `definition` node for each definition, and `statement` nodes that say the definitions are equivalent. Navigation and search go through the concept. Checks go through the definitions. With this structure, math2 can align with nLab (organised by concept), with Mathlib (organised by definition) and with textbooks (which choose one definition and prove the others). It does not force any of them into the shape of the others.

**Statement versus proof.** A statement can have many proofs, and the proofs differ in what they use. A researcher needs this information to know whether a result survives without the axiom of choice, or without the classification of finite simple groups. So proofs are nodes, and each proof has its own `uses` edges. The dependency closure of a statement is not one set. It is the set of alternatives, one for each proof (document 05.5).

**Conjecture versus problem.** A conjecture is a proposition with a truth value. It can be formalised as a Lean `Prop`. A proof of it, or of its negation, settles it. A problem may have no single truth value ("determine $R(5,5)$"). A chain of statements settles it. DeepMind's `formal-conjectures` repository and the Erdős problems database both mix the two. math2 keeps them apart, because they are settled in different ways.

## 4.3 Tags: identity forever

Every node has a permanent tag. The idea comes directly from the Stacks Project. `tamnd/arxiv` already uses the same tag discipline for results inside a paper. Three rules apply:

1. **A tag is assigned once.**
2. **A tag is never edited.**
3. **A tag is never reused, even if its node is deleted.**

Audit rule `T01` checks the tag register against git history. It fails if a tag has ever disappeared (document 15).

### Format

A tag is six characters of Crockford base32: the digits and the uppercase letters, without `I`, `L`, `O` and `U`. That gives 32^6 tags, about 1.07 billion. A four-character tag (the Stacks size, about 1.7 million in base 36) is too small. Mathlib alone has over 400,000 declarations. The literature has tens of millions of stated results, and the record plane (document 09) can turn any of them into a node.

The first character is never `0`. Tags that begin with `0` are reserved for the bootstrap seed (milestone M1), so the first few thousand curated nodes are easy to identify. A tag never ends in a character that would make it a valid MSC code, an OEIS A-number or an arXiv identifier. With this rule, a bare tag pasted into a search box resolves to exactly one kind of thing.

Tags are random, not sequential, for two reasons. Random tags make no claim about order, importance or age. Also, two branches can mint tags without coordination. Audit rule `T02` catches collisions at merge (the chance per pair is about one in a billion).

### When a tag is minted

Tags are minted at merge to `main`, and never before. A node on a branch, in the scratch space of an agent or in a pull request has a **provisional id**. Its form is `~` followed by eight hex characters, for example `~3fa94c01`. The merge bot rewrites provisional ids to tags in the node, in every edge that points to it, and in every Lean attribute. A pull request that cites a provisional id outside its own diff fails `T03`.

This rule prevents agents from wasting tag space. Agents propose tens of thousands of candidate nodes each day (document 12), and most of them are rejected. No rejected candidate ever had a tag. So the register records only things that were accepted at some time.

### Deletion is deprecation

A node is never deleted, even if it is found to be a duplicate, a mistake or out of scope. Its file stays, its `kind` stays, and it gets:

```yaml
retired:
  date: 2027-02-14
  reason: duplicate        # duplicate | false | out-of-scope | split | merged
  successor: [7K3QX2]      # zero or more tags
  note: "Same statement as 7K3QX2 with the hypotheses reordered."
```

A retired tag still resolves. The workspace shows its successor and the reason. A citation from 2026 still resolves to something meaningful in 2036.

`split` and `merged` record a refactor of the curated graph. It can become clear that one node states two results, or that two nodes are one. The successor list keeps the history navigable.

## 4.4 URIs

math2 is the `m2://` scheme in the `ant` URI space. Other schemes in that space are `ax://`, `wikipedia://`, `oeis://` and the rest of the `tamnd/*-cli` family. So `ant get m2://7K3QX2` works in the same way as for any other record.

```
m2://7K3QX2                     the node, at the latest release
m2://7K3QX2@2026.11             the node as it was in release 2026.11
m2://7K3QX2@c4f1e9a             the node as it was at a git commit
m2://7K3QX2#proof               the statement section; #proof, #formal, #history, etc.
m2://statement/7K3QX2           a typed alias, redirects to m2://7K3QX2
m2://concept/compact-space      a slug alias, resolves through the alias table
m2://msc/20D60                  a topic view over an MSC2020 code
m2://lean/Subgroup.card_subgroup_dvd_card   the node bound to this Mathlib declaration
m2://release/2026.11            the release manifest
```

Aliases are a convenience. They are not identities. A slug can go to a different node later, for example if "compact space" is found to be ambiguous between topology and order theory. A tag cannot. Every place that stores a reference stores the tag. `m2 fmt` rewrites aliases into tags before commit (rule `T04`).

External identifiers map into `m2://` through `same_as` edges (document 05.3). `m2://ext/wikidata/Q180967` resolves to the node that has that `same_as`. If no node has it, the URI returns a "not held" record that names the external system of the identifier.

## 4.5 The node file

A node is one file, `nodes/<first two characters of the tag>/<tag>.md`. The file has YAML front matter and a Markdown body. The shard key is the first two characters of the tag. With it, each directory has fewer than about a thousand entries at a million nodes.

The full schema is `schema/node.v1.json` in `tamnd/math2` (document 20). A statement node looks like this:

```markdown
---
tag: 7K3QX2
kind: statement
role: theorem
name: Lagrange's theorem
aliases: [lagrange-theorem-groups]
msc: [20D60]
topics: [4F2M9A]                # tag of the topic node "finite groups"
uses:                           # informal dependencies of the statement itself
  - 1H8D2C                      #   definition: subgroup
  - 6PQ0WZ                      #   definition: order of a finite group
proofs: [9TX3RB, 2YB7KE]        # proof nodes, in the order shown to readers
formal:
  - system: lean4
    library: mathlib
    decl: Subgroup.card_subgroup_dvd_card
    match: exact                # exact | stronger | weaker | variant | partial
    fidelity_reviews: [r-7K3QX2-01, r-7K3QX2-02]
sources:
  - zbmath: "0000.00000"        # example only; real records carry real ids
    locator: "Thm 3.1"
  - book: "isbn:9780471433347"
    locator: "§3.2, Thm 8"
same_as:
  wikidata: Q180967
  proofwiki: "Lagrange's_Theorem_(Group_Theory)"
  nlab: null                     # explicitly checked, none exists
history:
  first_stated: "Lagrange 1771 (special case); Galois; Jordan 1861"
---

## Statement

Let $G$ be a finite [[group|1A2B3C]] and $H$ a [[subgroup|1H8D2C]] of $G$.
Then $|H|$ divides $|G|$.

## Remarks

The converse fails: $A_4$ has order 12 and no subgroup of order 6 ([[counterexample|5ZQ1PM]]).
```

The rules for the body:

- **Math is LaTeX** inside `$…$` and `$$…$$`. Use only the subset that both KaTeX and MathJax render, plus macros declared in `macros/*.sty`. Audit rule `B01` renders every body headlessly and fails on any error. Free LaTeX is not allowed. A body that renders in only one engine cannot be served everywhere.
- **Links are `[[text|TAG]]`.** A link to something that is not a tag fails `B02`. Specifically, links to URLs are not allowed in the statement section. Put sources in `sources:`.
- **Sections are fixed per kind.** A statement has `Statement`, then optionally `Remarks`, `History` and `Explanations`. A proof has `Proof`, then optionally `Idea`. The full list is in the schema. A section that the schema does not know fails `B03`. This rule lets a program parse bodies without a model.
- **The statement section states one thing.** It does not hold a proof, a history or a motivation. The explanation agent (document 12.6) writes into `Explanations`, and into no other section.
- **Explanations are levelled.** Each explanation has a level tag: `L1` for a person who has not met the concept, `L2` for an undergraduate, `L3` for a graduate, `L4` for a researcher. Each level also has a tag that says human-written or agent-written (4.8).

Not every node has text that a person wrote. A node can come from a well-structured source, such as an OEIS sequence, an LMFDB object or a Mathlib declaration with a good docstring. Such an imported node can have a generated body. The body has a mark that says it is generated. An `imported` provenance entry names the source and its version (document 06.5).

## 4.6 Proof nodes

A proof is a separate file with `kind: proof` and a `proves:` field that holds exactly one tag. It can have:

- an informal `## Proof` section in the same constrained Markdown;
- a `formal:` binding to a Lean declaration that proves the formal binding of the statement, or proves it up to a recorded adapter (document 07.4);
- `uses:` edges to every node that the argument depends on.

`uses` on a proof means: "if any of these is false, this argument fails". It does not mean "this is mentioned in passing". For formally bound proofs, `uses` is computed from the Lean dependency graph (document 07.6) and compared with the hand-written list. If `uses` does not include a formal dependency on a math2-bound declaration, that is a finding (`G04`). A hand-written `uses` edge with no formal counterpart is allowed, because informal proofs cite at a coarser grain than Lean.

A proof can name a **method**: induction, diagonal argument, probabilistic method, compactness, the circle method, and so on. Methods are `concept` nodes in the `methods` topic. Because of this, math2 can answer a query like "all proofs in additive combinatorics that use the polynomial method" (document 14).

## 4.7 Status is computed, never asserted

No node file has a field called `status`, `verified` or `proved`. Status is the output of `m2 build` (document 07, 08, 15). The build writes it into the derived store and the release manifest, and never into the content.

This rule exists because the opposite rule fails. A person or an agent who writes `verified: true` makes a claim. Nothing in the file makes the claim true. A year later, Mathlib renames a lemma, a definition below it changes, or a reviewer leaves. The flag stays true. In math2, status is a function of the evidence at a commit. So every build computes it again from the evidence.

Status has two independent axes.

### Truth evidence

The strongest evidence that is present sets the level. The levels have this order:

| Level | Name | Meaning | Who can produce it |
|---|---|---|---|
| E5 | `kernel` | A formal binding with `match: exact` (or `stronger`/`variant` through a checked adapter, document 07.3) and fidelity at least F2 is proved by a Lean declaration that the kernel checks. Its axioms are a subset of `{propext, Classical.choice, Quot.sound}`, plus any named hypotheses, which make the result conditional. It has no `sorry` and no `native_decide`, which caps a node at E4 (07.5). It passes independent replay and the comparator check (07.7) | the build |
| E4 | `certified` | A computation produced a certificate (an LRAT proof, an interval-arithmetic enclosure, a primality certificate, an exact rational witness), and an independent checker accepted it | the build, through the computational engine |
| E3 | `refereed` | At least one proof in the literature is in a refereed venue, and a source record points to it with a locator | curation |
| E2 | `standard` | The result is textbook-standard. At least two independent textbook sources with locators | curation |
| E1 | `claimed` | Someone claims it: a preprint, an agent's informal proof, a MathOverflow answer | anyone |
| E0 | `open` | It is posed and unresolved. Only conjectures and problems may sit here | anyone |
| E⊥ | `refuted` | A counterexample or a proof of the negation exists, at level E2 or above | the build or curation |

This has two important consequences.

**E5 does not mean "true"**, and the workspace never says that it does. E5 means two things. First, this Lean statement is a theorem of Lean's type theory with the standard axioms. Second, three to five people (or one person and a cross-formalisation) agreed that the Lean statement says what the informal statement says. The errors are in the second part. Document 10 is about how to make it reliable.

**An agent's informal proof is E1**, however good it looks. It moves up in only three ways: it becomes formal (E5), it is certified (E4), or it is published and refereed (E3). The agent framework cannot write a status. It can only produce evidence, and the build then grades that evidence (document 12).

### Statement fidelity

Fidelity is how strongly we believe that a formal statement means the same thing as the informal one. It applies only when a formal binding exists.

| Level | Meaning |
|---|---|
| F0 | Bound. Nobody has checked the formal statement against the informal one |
| F1 | Machine-checked. Back-translation, a type-level check and a model-based critic all agree, and checks for degenerate statements pass (document 10.4) |
| F2 | One human review. A reviewer who is not the author has signed off on this exact statement hash |
| F3 | Two independent reviews, or one review plus a second formalisation produced independently and proved equivalent in Lean |

A review is a record in `reviews/<shard>/<tag>.yaml`. It holds the identity of the reviewer, the date, the content hash of the informal statement section, and the content hash of the elaborated Lean type (not its source text). If either hash changes, the review no longer counts. This happens automatically. It is the mechanism that keeps F2 and F3 honest when Mathlib refactors a definition.

### Derived flags

The build also computes flags. Each flag can be queried:

- `ac` / `noac`: the proof uses, or avoids, `Classical.choice`. This is computed for each proof. The statement takes the best value over its proofs.
- `computational`: the proof depends on `decide`, `native_decide` (whose per-use auxiliary axioms the axiom audit sees) or an external certificate.
- `sorry_in_closure`: some node that the proof transitively depends on is not at E5. The blueprint style "proved assuming…" view comes from this flag.
- `stale_review`: a review was invalidated by a hash change and has not been done again.
- `orphan`: the node has no inbound edges and is not a topic root.
- `conditional`: the axiom profile includes hypotheses from `Math2/Hypotheses/`. These hypotheses are listed by name (07.5).
- `external_replay`: E5 rests on the replay results of an external library at its pinned commit, not on math2's own replay (07.7).

## 4.8 Provenance on every field

Every node has a provenance log in `prov/<shard>/<tag>.jsonl`. The log has one line for each change to each field. The line records who changed the field, when, with what, and from which source. The model follows W3C PROV. Document 06.5 gives it in full. In outline:

```json
{"field":"formal[0]","op":"add","agent":"m2-agent/formalize@0.9.1","model":"claude-opus-5-5","run":"r-2026-11-02-8812","human":null,"commit":"c4f1e9a","at":"2026-11-02T09:13:44Z"}
{"field":"formal[0].fidelity_reviews","op":"add","agent":null,"human":"orcid:0000-0002-1825-0097","commit":"e01b7d2","at":"2026-11-04T16:20:11Z"}
```

The rule is that **every byte in `nodes/` can be traced to a person or to an agent run**. Every agent run can be traced to the person or schedule that started it, the model, the prompt version and the tool versions (document 11.6). Audit rule `P01` fails a merge that adds bytes without a provenance line.

This is not bureaucracy. It makes three things possible. First, AI contributions get credit and disclosure in the way that journals and the community now expect (document 16). Second, if a model or prompt version is found to have a systematic flaw, a whole class of agent output can be withdrawn. Third, the statement "this was written by a person" stays true and checkable.

## 4.9 Schema versioning

The node schema has its own version (`node.v1`), separate from releases. Two kinds of change are possible:

- **An additive change** adds a new optional field or a new edge type. It needs only a minor bump and a migration that does nothing.
- **A breaking change** renames a field, changes the kinds or changes the meaning of a field. It needs a new major version, a migration script in `tamnd/math2-agent/migrate/`, and a release that accepts both versions. In that release, rule `S09` flags the old version.

The schema is small on purpose. Put information that has no schema field in `## Remarks`. People can read it there, and the build ignores it. Add a field only when a query or a check needs it.

## 4.10 What a node is not allowed to do

- It cannot state its own status (4.7).
- It cannot have two kinds. A conjecture that is proved becomes a statement: the conjecture node is retired with `reason: split`. The new statement node lists the tag of the conjecture in `history.resolves`, and the conjecture node gets a `resolved_by` edge. A citation of the conjecture still goes to a place that says it is resolved, how and when.
- It cannot contain a proof inside its statement.
- It cannot link outside `m2://` from its statement section.
- It cannot reuse a tag, an alias that is in use, or a `same_as` identifier that another live node already claims. `same_as` is functional in both directions within one external system (rule `G02`). If an external system has two entries for one node, or one entry for two nodes, the edge is `close_to` and the mismatch is recorded (document 05.3).
- It cannot be larger than 64 KB. A longer node is several nodes.
</document>

<document source="https://math2.tamnd.com/spec/05-knowledge-graph.md">
# 05. The knowledge graph

The graph is the box in the middle of the architecture diagram: Statements, Concepts, Dependencies. Every other box writes into it or reads from it. This document defines the edges of the graph and its alignment with the rest of the world's mathematical data. It also defines how the graph is stored and rebuilt, and the queries that it must answer quickly.

## 5.1 Where the graph lives

The graph has no separate database that holds the truth. **The truth is the node files in git** (document 04). `m2 build` derives the graph from them, together with three other derived inputs:

1. edges written by hand in node front matter (`uses`, `proofs`, `topics`, `same_as`, …);
2. edges extracted from the formal plane: the Lean dependency graph of every bound declaration, at the pinned Mathlib and math2 revisions (document 07.6);
3. edges from the record plane: citations between papers, and paper results aligned to nodes (document 09.6).

The derived graph goes to `build/graph/` as Parquet files (one for nodes, one for each edge family). It is then loaded into an embedded store for serving (5.7). It is never committed to `tamnd/math2`. It is published as a release artefact with a content hash (document 06). Anyone with the repository and the pinned toolchain can rebuild it byte for byte. Audit rule `R01` does exactly that in CI. For every merge it rebuilds a sample of shards, and for every release it rebuilds everything.

`tamnd/arxiv` made the same choice: commit what a person decided, and derive the rest. As a result, a reviewer reads a diff of decisions, not a diff of derived numbers.

## 5.2 Edge types

Edges are typed and directed, and their types come from a closed list. Each edge also carries its own evidence and provenance. "A implies B" is a mathematical claim, the same as a node. Its level can be anything from E1 to E5.

### Logical edges (they carry truth)

| Edge | From → to | Meaning | Can be E5 |
|---|---|---|---|
| `proves` | proof → statement | this argument establishes that statement | yes |
| `uses` | statement/proof/definition → any | the source depends on the target: a hypothesis, a lemma, a definition | yes (computed from Lean) |
| `implies` | statement → statement | A ⟹ B, proved by the proof node named in `via` | yes |
| `equivalent` | statement ↔ statement, definition ↔ definition | A ⟺ B, via a proof | yes |
| `refutes` | counterexample/statement → statement/conjecture | the source shows the target is false | yes |
| `resolves` | statement → conjecture/problem | the source settles the target, positively or negatively | yes |
| `generalizes` | statement → statement | the target is a special case of the source. Recorded with the specialising map | yes |
| `bounds` | statement → problem | the source gives a bound on the quantity the problem asks for (lower or upper, with the value) | yes |

### Conceptual edges (they carry meaning)

| Edge | From → to | Meaning |
|---|---|---|
| `defines` | definition → concept | this is one definition of that concept |
| `instance_of` | example/object → concept | the source is an instance of the concept |
| `about` | statement/conjecture/problem → concept | the result is about the concept, as one of its main subjects |
| `specializes` | concept → concept | every X is a Y (a ring is a semiring). These edges form a DAG, not a tree |
| `method` | proof → concept (in `methods`) | the proof uses this technique |
| `analogous_to` | any ↔ any | an analogy that mathematicians acknowledge, with a source. Soft, and never used for reasoning |
| `part_of` | any → topic | navigation only |

### Record edges (they carry provenance and identity)

| Edge | From → to | Meaning |
|---|---|---|
| `formalizes` | Lean declaration → node | the binding. Carries `match` and fidelity (document 07.4) |
| `source` | node → record | where this is stated or proved. Carries a locator |
| `aligned` | paper result → node | an extracted result (for example `ax://paper/2106.09685#AB12`) states this node. Carries a `match` grade |
| `same_as` | node → external id | the same thing in another system (5.3) |
| `close_to` | node → external id | a related but not identical entry in another system, with a note on how it differs |
| `cites` | record → record | the paper-to-paper citation graph, from zbMATH Open and arXiv |
| `resolved_by`, `successor` | retired node → node | lifecycle (document 04.3) |

There are 23 edge types. A new edge type is a schema change (document 04.9). The condition for a new type is: "some query in 5.6 needs it and cannot be written without it".

## 5.3 Alignment with the rest of the world

math2 does not try to replace nLab, Wikipedia, Wikidata, Mathlib, the Stacks Project, OEIS, LMFDB, zbMATH Open or MathOverflow. It aims to be the place that joins them together. Each of them does its own job very well, but none of them links reliably to the others. In its first year, math2 is most useful as a curated, versioned, machine-checkable **crosswalk** between them, with statuses attached.

`same_as` identifiers by external system:

| System | Identifier | Granularity | Notes |
|---|---|---|---|
| Wikidata | QID | concepts, named theorems, objects | the hub for every other crosswalk. The MaRDI knowledge graph is built on Wikibase and uses Wikidata-style ids |
| MaRDI portal | MaRDI QID | formulae, algorithms, software, research data | aligned through Wikidata where possible |
| Mathlib | fully qualified declaration name at a pinned revision | definitions, theorems | carried by `formalizes`, not `same_as` |
| Other formal libraries | Isabelle AFP entry + theorem, Rocq/MathComp qualified name, Metamath label, HOL Light name | definitions, theorems | `formalizes` with `system:` set accordingly |
| Stacks Project | 4-character tag | results, definitions | text under GFDL. We link and never copy (document 09.3) |
| Kerodon | 4-character tag | results, definitions | same as Stacks |
| nLab | page name + revision | concepts | |
| Wikipedia | page title + revision id | concepts, named theorems | revision-pinned, because titles move |
| ProofWiki | page title | statements, definitions, proofs | |
| OEIS | A-number | objects (sequences) | an OEIS sequence is an `object` node. Its formulae become `statement` nodes |
| LMFDB | the path of the object, for example `EllipticCurve/Q/11.a2` | objects | number fields, elliptic curves, modular forms, … The path has the collection and the label, because a label alone does not identify an object |
| zbMATH Open | document id; author id; MSC code | records | the main record plane source for "all mathematics" |
| MathSciNet | MR number | records | identifier only (no bulk licence) |
| arXiv | id + version, and per-paper result tag from `tamnd/arxiv` | records, results | `ax://paper/<id>#<tag>` |
| Erdős problems | problem number | problems/conjectures | |
| formal-conjectures (DeepMind) | declaration name | conjectures with Lean statements | |
| MathOverflow / Math.SE | question id | sources for E1 claims | |
| Palomar | registry entry id | Lean-verified results | carries the entry's `formalization.yaml` disclosure and replay record (09.4) |
| TheoremSearch, TheoremGraph/LeanGraph, Matlas | native statement ids | extracted results | record plane `same_as`; their extractions are imported as candidates only (09.6) |
| Hexagon | `hexagon:` id + version | records | the 2026 repository for AI-era mathematics, with provenance labels |
| pi-Base, DaRT, FindStat, House of Graphs, GAP small groups, ATLAS | native ids | objects, properties, counterexamples | data sources with natural `instance_of` and `refutes` structure |

Alignment follows three rules.

1. **Alignment is a claim, and it has a grade.** A `same_as` that an agent writes starts as E1. A human confirmation raises it. An automatic match that cannot be ambiguous also raises it (an OEIS A-number cited verbatim in both places, an identical Lean declaration). The crosswalk publishes the grade of each edge, so consumers can choose their threshold.
2. **`same_as` is functional both ways within one system.** A node has at most one Wikidata QID, and a QID belongs to at most one live node (rule `G02`). When the granularity of a system does not match ours, we use `close_to` with a note. An example is one Wikipedia page that covers three theorems. Real systems disagree about granularity all the time. A forced bijection would make the crosswalk wrong in a way that nobody can see.
3. **Round-tripping is a test.** For every system with a public API, `m2 audit -plane align` fetches a sample of external records again. It checks that they still say what they said when the edge was made. If the external record is gone, or its revision moved and the content changed, that is a finding (rule `A03`). In this way, renames on Wikipedia, deprecations in Mathlib and merges on nLab appear in a report. They do not become silent link rot.

### Contributing the crosswalk back

Where the external system accepts them, alignments go back out to it. Wikidata gets `math2 tag` statements and `exact match` links. Mathlib gets a `math2` field in docstrings if the community wants one, and only as an opt-in suggestion (document 16.5). MaRDI gets the same through its Wikibase API. This is the cheapest way to make math2 useful to people who never visit it.

## 5.4 Granularity and deduplication

The hardest problem in a graph of all mathematics is not size. The hardest problem is this: the same theorem is stated a thousand times, in a thousand notations, at slightly different generality. These thousand statements have to collapse into one node or a small family of nodes. If they do not, the graph is only a pile of paper extracts.

math2 deduplicates in four passes. Each pass is cheaper and more certain than the one after it.

1. **Identifier collisions.** Two candidates cite the same external identifier, or bind to the same Lean declaration. These merge automatically.
2. **Formal equivalence.** Both candidates have formal statements, and a Lean proof shows that they are equivalent. The proof is an `Iff` between the two elaborated types. Usually `exact ⟨fun h => …, fun h => …⟩` from the formalize agent closes it, or `simp`/`grind` does. The candidates merge, and the equivalence becomes an E5 `equivalent` edge.
3. **Embedding neighbourhood plus judgement.** Statement embeddings (document 14.4) give the 20 nearest live nodes for each candidate. A judge model classifies each pair as one of `same`, `generalizes`, `specializes`, `related` or `different`. It has to justify the classification with the specialising map. A `same` from the judge is only a proposal. It becomes an edge only through pass 2 or a person.
4. **People.** Curators process the queue of ambiguous `same` proposals. The queue order is by how much of the graph each merge would connect (document 16.3).

Deduplication gives a canonical node plus `aligned` edges from every paper result that states it. This is also the only honest way to answer "where in the literature is this proved?". The answer is the set of aligned records in date order. That set gives a provenance chain for each theorem. Currently nobody maintains such a chain by hand.

### Families, not points

Some results have no canonical form, only a family: the dozen forms of the Hahn–Banach theorem, or the Cauchy–Schwarz inequality in each of its settings. math2 does not choose a winner. It makes one node for each form that people cite separately, `generalizes`/`specializes` edges between them, and a `concept`-like hub. Hubs are statements with `role: fact` and an empty formal binding. A hub is only a navigation point. The workspace shows it as "the Hahn–Banach family". A hub with fewer than three members is a finding (`G07`), because a family of two is only two nodes and an edge.

## 5.5 Dependency closure, with alternatives

"What does this theorem depend on?" has no single answer, because a statement can have several proofs (document 04.6). The closure is an AND–OR graph. A statement needs one of its proofs (OR), and a proof needs all of its `uses` (AND).

For every statement, the build computes:

- **min-depth closure**: the alternative with the smallest maximum depth down to axioms and definitions;
- **min-size closure**: the alternative with the fewest distinct nodes. This is an approximation, because the problem is NP-hard in general. The build uses a greedy algorithm with a recorded bound;
- **axiom profile**: which of the foundational nodes (choice, excluded middle, univalence where relevant, large cardinals, CFSG as a black box, the Riemann hypothesis as a hypothesis) occur in which alternative;
- **trust profile**: the weakest evidence level anywhere on the best alternative. Take a statement whose only proof is E5 but uses a lemma that is E3. It shows as "formal modulo 1 refereed lemma". This is the blueprint view, computed for the whole graph in one pass.

Conditional results are first-class. "Assuming GRH, …" is a statement whose `uses` includes the GRH conjecture node with `as: hypothesis`. When a conjecture is settled, the next build re-grades every conditional result that depends on it. When a conjecture is refuted, every such result gets a flag (`resolves` propagation, document 15.4).

## 5.6 The queries the graph must answer

The design of the graph starts from these queries. Each query has a latency target on the reference machine (document 02, axis 5). The benchmarks in `math2-agent/bench/graph/` measure it.

| # | Query | Example | Target p95 |
|---|---|---|---|
| Q1 | resolve an identifier | `m2://lean/Subgroup.card_subgroup_dvd_card`, `Q180967`, `A000045`, `0FD4` (Stacks) | 5 ms |
| Q2 | neighbourhood | everything within 2 hops of a node, typed | 20 ms |
| Q3 | dependency closure | best alternative down to axioms, with its trust profile | 100 ms at depth 30 |
| Q4 | reverse dependencies | what breaks if this lemma is false (all statements whose every proof uses it) | 200 ms |
| Q5 | prerequisite path | the shortest learning path from concept A to statement B, using only `defines` and `uses` edges into definitions | 100 ms |
| Q6 | status slice | all E1 statements in MSC 11 (number theory) with an E5 formal statement but no proof: the "formalised, open to prove" frontier | 500 ms |
| Q7 | open-problem frontier | open conjectures in a topic, with their best bounds and the dates those bounds moved | 200 ms |
| Q8 | method query | proofs using the method M about concept C | 200 ms |
| Q9 | provenance | every source that states this node, in date order, with the earliest | 50 ms |
| Q10 | change query | what changed in topic T between releases R1 and R2 | 1 s |
| Q11 | coverage | per MSC code: counts by kind and by evidence level, and Mathlib coverage | 1 s for the full table |
| Q12 | agent work queue | the next 100 tasks for the prove agent, ranked (document 11.4) | 500 ms |

If a query in this list cannot meet its target, that is a design bug in the storage layer. It is not a reason to drop the query.

## 5.7 Storage engine

The derived graph is small by graph-database standards, even at full scale. The estimate in document 18 is about 10 to 30 million nodes and about 300 million edges after the record plane is aligned. Most of the edges are `cites` and `aligned`. That is a workload for a single machine, and math2 treats it as one.

- **Build format:** Parquet, one file for each node kind and each edge family, sorted by tag. These are the release artefacts.
- **Serving:** an embedded store inside `m2 serve`:
  - **SQLite** for point lookups, FTS5 lexical search and the small fixed queries (Q1, Q2, Q9);
  - **DuckDB** over the Parquet for analytical queries (Q6, Q10, Q11);
  - an in-memory **CSR adjacency** built at startup for the closure and path queries (Q3, Q4, Q5). It uses about 2.5 GB for 300M edges at 8 bytes per edge;
  - a **vector index** for embeddings (document 14.4).
- **No external graph database.** math2 does not use Neo4j, a hosted Neptune or similar products. If the record plane grows by orders of magnitude, look at the project's own graph database research (`tamnd/graph-research`). At the sizes above, a server process that loads Parquet, plus SQLite and DuckDB files, is faster to operate, trivial to version and reproducible. Here those three properties are more important than distributed scale.

The query language for users and agents is not SQL or Cypher. It is a small typed API (document 11.2). Power users also get a read-only DuckDB SQL endpoint over the published Parquet schema. That schema is versioned together with the node schema.

## 5.8 Graph health

The graph has its own audit rules. Document 15 has the full list. These rules define a healthy graph:

- `G01` every `uses`, `proves`, `defines` and so on points to a live tag or to a retired tag that has a successor;
- `G02` `same_as` is functional both ways within each external system;
- `G03` `specializes` among concepts is acyclic. `uses` among statements is acyclic when proofs are taken into account. A cycle through two different proofs is allowed, because equivalences look like that;
- `G04` formal dependencies agree with hand-written `uses` (document 04.6);
- `G05` every live statement has at least one `source`, or at least one proof, or a formal binding. A statement with none of these is an assertion with no evidence at all, and it is not accepted;
- `G06` every conjecture has a `source` giving who posed it and where;
- `G07` hub families have at least three members;
- `G08` orphan rate per topic is below 2%;
- `G09` no concept has more than 12 definitions without a `concept`-level note explaining why.

Each rule writes into the committed report under `reports/graph/` (document 15). Thus a person can read the state of the graph in a diff.
</document>

<document source="https://math2.tamnd.com/spec/06-versioned-storage.md">
# 06. Versioned storage: metadata, artefacts, provenance

This is the bottom-right box of the diagram. Every other box writes into it. It holds three things, and each has its own versioning rules:

- **metadata**: the node files, edges, reviews, manifests and registers. They are small and people edit them. They are in git.
- **artefacts**: compiled Lean `.olean` files, built graph Parquet, search indexes, computation outputs, certificates and rendered sites. They are large and derived. They are in content-addressed storage.
- **provenance**: who and what made each byte of the other two, and from what.

The goal of the layer is one sentence: **given a release name, anyone can get back exactly what math2 said, rebuild it, and check that the rebuild matches**.

## 6.1 Three repositories, one lineage

Storage is not one repository.

| Location | Holds | Versioned by | Licence |
|---|---|---|---|
| `tamnd/math2` (git) | `nodes/`, `reviews/`, `prov/`, `lean/` (the math2 Lean library), `manifests/`, `registers/`, `reports/`, `schema/`, `macros/` | git commits and release tags | prose CC BY-SA 4.0; Lean code Apache-2.0; metadata CC0 (document 16.6) |
| `tamnd/math2-agent` (git) | the `m2` toolchain, agent code, prompts, tool images, benchmarks | git and semver | Apache-2.0 |
| the artefact store | everything derived, keyed by BLAKE3 hash | content address | follows its inputs |

The artefact store is an S3-compatible bucket (R2 or similar) behind a plain HTTPS read path. It also has a second, independent mirror for releases (6.6). Nothing in it is mutable. Nothing in it is the only copy of a decision that a person made. If we lose it, we lose compute, not knowledge.

### Why git for metadata

At the scale in document 18, the git repository is large but still workable. It has a few million small files and a few GB packed. This is about the size of the history of the Linux kernel. Shards by tag prefix (document 04.5) keep directories small. Git gives four things for free that a database would make us build:

- review by diff;
- attribution by commit;
- cheap branches for agents (each agent run works on a branch);
- tools for mirrors and archives that have worked for decades.

If the repository becomes larger than 10 GB packed or 5 million files, the record plane moves to its own repository, `tamnd/math2-records`. The record plane holds most of the files. It keeps the same layout and rules. The trigger is a number and not a feeling. Document 21 tracks it.

## 6.2 Releases

math2 uses calendar versions: `YYYY.MM`, monthly. We cut each release on the first Monday of the month. Patch releases `YYYY.MM.N` are for fixes. Lean and Mathlib also do not use semver. Lean has a monthly release candidate cycle. A monthly math2 cadence matches it closely enough, so each math2 release can pin one Lean stable release.

A release is a git tag on `tamnd/math2` plus a **release manifest**, `manifests/releases/2026.11.yaml`:

```yaml
release: 2026.11
commit: 8b1e4c0f2d...            # tamnd/math2
toolchain:
  m2: v0.9.0                      # tamnd/math2-agent tag
  lean: leanprover/lean4:v4.NN.0 # the stable toolchain pinned for this release
  mathlib: <commit sha>           # one Mathlib revision, compatible with the toolchain
  other_libs:                     # optional formal bindings outside Lean
    afp: 2026-10-30
  images:                         # computational engine containers, by digest
    sage: ghcr.io/tamnd/m2-sage@sha256:...
    pari: ghcr.io/tamnd/m2-pari@sha256:...
    gap:  ghcr.io/tamnd/m2-gap@sha256:...
    smt:  ghcr.io/tamnd/m2-smt@sha256:...
artefacts:                        # the hash of the files and the size in bytes
  graph:  {hash: blake3:af01..., size: 3100000000}
  oleans: {hash: blake3:77c2..., size: 9800000000}
  search: {hash: blake3:1e9b..., size: 6400000000}
  site:   {hash: blake3:c0de..., size: 2200000000}
  certs:  {hash: blake3:51aa..., size: 840000000}
counts:                           # filled by the build. The README table is generated from this
  nodes_live: 0
  by_kind: {}
  by_evidence: {E5: 0, E4: 0, E3: 0, E2: 0, E1: 0, E0: 0}
archived:
  swh: swh:1:rev:...              # Software Heritage revision id
  zenodo_doi: 10.5281/zenodo....  # concept DOI is stable across releases
rebuild_check:
  runner: ubuntu-24.04/x86_64/32c/128G
  graph_hash_reproduced: true
  oleans_hash_reproduced: true
  olean_differences: []           # each file that is different, with a reason, when oleans_hash_reproduced is false
```

The rules:

1. **A release pins exactly one Lean toolchain and one Mathlib revision.** Between releases, `main` follows Mathlib's `master` on a weekly bump branch (document 07.9). Never cut a release from a state where the bump is red.
2. **Every artefact that a release names can be rebuilt from the commit and the toolchain.** For the graph and the search index, the rebuild must match exactly. Lean builds are deterministic, so the `.olean`s must be bit-identical. If they are not, the release must say which files are different and why (rule `R02`).
3. **Releases are archived outside the project.** Software Heritage gets the git tag (a SWHID). Zenodo gets the manifest and the graph Parquet (a DOI). If the project disappears, a 2026.11 citation still resolves.
4. **The release notes are generated, then edited.** `m2 release notes 2026.10..2026.11` makes these items: counts per topic, nodes that moved evidence level, newly resolved conjectures, retired tags with successors, and binding changes that Mathlib renames caused. A person edits the prose. A release cannot change counts if the notes do not say so (rule `R03`).

A release has two more files next to its manifest. `2026.11.levels` gives the evidence level and the fidelity level of each live node, one line for each node, thus the notes of the next release can list the nodes that moved evidence level without a build of an old commit. `2026.11.md` has the notes. Their counts table has one row for each count of the manifest that changed: the count, the number before, the number after, the change and the cause. `m2 release notes` writes the cause from the node files: the new nodes, the retired nodes and the nodes that moved between kinds or levels. When the node files do not give the change, the cause is empty, and a person writes it.

`m2 release cut` cuts a release in two runs. The first run builds the tree and commits the three files. Then a person writes the summary of the notes, and a second machine runs `m2 release verify -record` to fill the rebuild check. The second run refuses when a file outside `manifests/releases/` and `reports/` changed after the commit of the manifest, runs the full audit, and makes the tag only when the audit passes.

### Citing math2

The citation form is `m2://<tag>@<release>`. We also print it as a DOI-backed URL. A paper can cite `m2://7K3QX2@2026.11`. In 2040 a reader can resolve it to the exact text, formal statement, evidence and status as of November 2026. This is true even if the node was retired in 2029.

## 6.3 Content addressing

Each artefact has its BLAKE3 hash as its name. Each node has a **content hash**. We compute it over the canonical serialisation of the node: front matter in key-sorted canonical YAML, and the body normalised to NFC with trailing whitespace removed. Each formal binding has an **elaborated-type hash**. This is a hash of the Lean `Expr` of the type of the bound declaration after elaboration. Universe levels are normalised and auxiliary names are removed.

The elaborated-type hash is the most important hash in the system. Fidelity reviews (document 04.7) are bound to it. Mathlib can change a definition under a statement. Then the elaborated type changes and the hash changes. The review stops counting, and the node drops from F2 to F0 with `stale_review` set. This happens even though no text in math2 changed. Text hashes cannot find this change, because the text of `theorem foo : Bar x` is the same before and after `Bar` changes. Type hashes can find it.

Computation records (document 08) have the hash of `(image digest, entry point, inputs, seed)` as their address. If you run the same computation two times, you must get the same output hash. If you do not, that is a finding: rule `C02`, nondeterminism (document 08.6).

## 6.4 Registers

Some facts must never be lost and must never change. They live in append-only registers in `registers/`:

| Register | One line per | Rule |
|---|---|---|
| `tags.reg` | minted tag: tag, minted-at commit, kind | `T01`: lines are never removed or edited |
| `aliases.reg` | alias assignment: alias, tag, from, until | an alias can move. Its history cannot |
| `retired.reg` | retirement: tag, date, reason, successors | |
| `bindings.reg` | formal binding change: tag, system, old decl, new decl, cause (bind/rename/refactor/rebind), Mathlib commit, author | answers "which Mathlib name did this node have in March 2027" |
| `reviewers.reg` | reviewer identity: ORCID or GitHub handle, first review date, areas, affiliations | needed for F2/F3 independence (document 10.6) |
| `crosswalk.reg` | crosswalk decision: tag, edge, system, id, decision (accepted/rejected/confirmed), person, date, record hash | `A02`, `A03`: a person decided on each `same_as` value, and the record did not change after the decision |

Registers are plain text, sorted, with one record per line. Thus `git log -p` on a register is the full history of that decision space. Each build rebuilds the registers and checks them against the node files (rule `T05`).

## 6.5 Provenance

Each change to each field of each node, review and register has a provenance record (document 04.8). The model follows W3C PROV-O, with these classes:

- **Entity**: a node version (tag plus content hash), a formal declaration (name plus type hash plus Mathlib commit), a computation output (hash), a source record (external id plus revision);
- **Activity**: an agent run, a human edit (a commit), an import job, a build;
- **Agent** (in the PROV sense): a person (ORCID or GitHub id), a math2 agent role at a version, a model id, an external system.

The relations are `wasGeneratedBy`, `used`, `wasAssociatedWith`, `wasDerivedFrom`, `wasRevisionOf`, `wasAttributedTo` and `actedOnBehalfOf`. The last relation links an agent run to the person or schedule that started it.

Provenance is in `prov/<shard>/<tag>.jsonl`, with one JSON object per line. Each release exports it as PROV-JSON-LD. The full record of an agent run is larger: the prompts, each tool call and its output, token counts and cost. It is in the artefact store, keyed by run id. The provenance line of a node cites the run id. Thus the chain is complete, and git does not hold gigabytes of transcripts (document 11.6).

### What provenance has to make possible

`math2-agent/bench/prov/` runs these tests against a release:

1. **Attribution.** For a node, list the people and the agent runs that contributed to its current text, formal binding and reviews.
2. **Withdrawal.** A model id and a prompt version can be found to be systematically wrong. For that pair, list each field that it produced and that a person has not replaced or confirmed since. Then open a pull request that reverts or flags all of them (`m2 prov quarantine --model … --prompt …`).
3. **Licence tracing.** For a node, list each external source that its text came from, with the licence of that source. This is the input to the licence gate (document 09.3).
4. **Credit export.** For a person, make a CRediT-style statement of what they contributed to math2 in a period. Then the work counts as scholarly output (document 16.4).

## 6.6 Mirrors, backups, retention

- **git:** GitHub is primary. Codeberg and a self-hosted bare repository are mirrors. Each release is archived to Software Heritage.
- **Artefact store:** two providers. Keep releases forever. Keep builds of `main` for 90 days. Keep agent run records in full for 2 years, and keep their summaries (cost, outcome, the produced diff) forever.
- **Restore drill:** each quarter, CI rebuilds the latest release on a clean machine. It uses only the Software Heritage copy and the second artefact mirror. Then it checks the manifest hashes. A failed drill blocks the next release (rule `R04`). Each drill writes a record to `reports/restore/YYYY-MM-DD.yaml` with the fields `date`, `release`, `runner`, `passed` and `note`. Rule `R04` reads the last record. It fails when that drill failed, or when the drill is more than one quarter (92 days) older than the month of the last release.

## 6.7 Sizes

These estimates are for the end of the milestones in document 19. Each release manifest records the actual numbers.

| Item | Estimate | Basis |
|---|---|---|
| live nodes (statement plane) | 2–5 M | Mathlib-bound nodes (~0.3–0.5 M), curated literature results, OEIS sequences (~390 k), LMFDB objects as nodes only when cited |
| record plane entries | ~5 M records, ~30 M aligned results | zbMATH Open's ~4.9 M documents and arXiv math. Results are extracted only from licence-permitted full text (document 09) |
| git repository | 3–8 GB packed | about 1 KB average per node file, plus provenance lines |
| `.olean` cache | 10–15 GB | Mathlib plus the math2 library |
| graph Parquet | 3–6 GB | |
| search indexes | 5–20 GB | lexical plus vectors at 1024 dimensions, int8-quantised |

These sizes do not need special technology. The storage layer is simple on purpose, because the interesting part of the project is in a different place.
</document>

<document source="https://math2.tamnd.com/spec/07-formal-engine.md">
# 07. The formal engine: Lean 4, Mathlib, proof checking

The formal engine is the middle box of the top row of the diagram, with the label "Lean 4, Mathlib, Proof Checker". It is the only component that can produce E5. Thus its specification is stricter than that of any other component. The content side is in `tamnd/math2/lean/`, the `Math2` Lake package. The toolchain side is in `tamnd/math2-agent/lean` and in the `lean/M2Extract` package.

The job of the engine is one sentence: **given the content at a commit, produce for every formal binding a checked declaration, its elaborated-type hash, its axiom profile, its formal dependencies, and the results of independent replay, all pinned to one Lean toolchain and one Mathlib version.**

## 7.1 What is pinned

Each release pins three things. Its manifest records them (06.2):

| Item | Pinned as | Source of truth |
|---|---|---|
| Lean toolchain | a stable release, e.g. `leanprover/lean4:v4.35.0` | `lean/lean-toolchain` |
| Mathlib | the matching Mathlib tag `v4.35.0`, plus the commit it names | `lean/lake-manifest.json` |
| Checkers | `nanoda`, `lean4lean`, `con-leche`, `con-ron` and comparator versions, as bundled with the toolchain or pinned separately | `manifests/releases/<r>.yaml` → `checkers:` |

A release never pins a Lean release candidate or Mathlib `master`. Mathlib has no semantic versions. It tags each toolchain bump `v4.X.Y`, and its `stable` branch follows those tags (document 01). math2 pins those tags. This gives monthly releases a natural rhythm: each release takes the newest tag that the bump branch (7.9) has built green.

When a node binds to other formal libraries, we pin them in the same way. These libraries are Tau Ceti, the textbook libraries of Formal Frontier, the formal-conjectures repository, the Imperial FLT project, Carleson, PFR, and the AFP for Isabelle bindings. Each one is an entry in `manifests/sources.yaml` with a commit and a toolchain. Some libraries have a toolchain that is different from the toolchain of the release. We check a binding to such a library in the environment of that library, and the manifest marks it.

## 7.2 The `Math2` package

`lean/` is an ordinary Lake package. Its contents follow the 2026 community consensus on how to grow a formal library (the "How to formalize all of mathematics" statement by the Mathlib maintainers; document 01): **defer to Mathlib, avoid duplication, keep quality high, and link to the literature.**

- **Mathlib first.** If the result of a node is in Mathlib, the node binds to the Mathlib declaration and has no Lean code in math2. `Math2/` holds only what Mathlib does not have. That includes statements of research-level results (so we can bind and review them before anyone proves them), proofs that are not upstream yet, conjecture statements, hypotheses for conditional results, and certificate checkers.
- **Module system only.** Each file in `Math2/` is a Lean `module`, with explicit `public import`s. Mathlib has moved to the module system. It will stop support for downstream projects that have not moved. If we start there, it costs nothing.
- **Mathlib style.** Names, formats and linters follow the Mathlib conventions, and `lake lint --code-quality` runs in CI. This is a condition for upstreaming, and the plan is to upstream all general material (16.5).
- **Hypotheses are the only axioms.** `Math2/Hypotheses/` holds named axioms such as `GRH` or `ABC`. Each one is bound to its conjecture node. Then the axiom profile of a conditional theorem names exactly the hypotheses that it depends on. Nothing else in `Math2/` may contain `axiom`, `sorry`, `admit`, `@[implemented_by]` or `unsafe` (rule `F06`).
- **Conjectures are definitions, not axioms.** `Math2/Conjectures/Collatz.lean` defines `def Collatz : Prop := …`. A proof of the conjecture is a term of type `Collatz`. A refutation is a term of type `¬ Collatz`. If the formal-conjectures repository has conventions for a statement of an open problem, follow them, and bind to its entries with `same_as`.
- **A statement with no proof is a definition too.** Rule `F06` refuses `sorry`, thus a result that is stated and not yet proved is `def lagrange_statement : Prop := …`, and the node binds to it. The degenerate-statement checks and the F1 battery read the value of such a definition as the statement, as they read the type of a theorem (`lean.Record.Statement`). A definition of a proposition is never a proof, thus the binding is not E5. A proof adds `theorem lagrange : lagrange_statement`, moves the `@[m2 "TAG"]` attribute to it, and changes the `decl` of the binding to the theorem. The value of the definition does not change.

### The binding attribute

```lean
module
public import Mathlib.GroupTheory.Coset.Card
public import Math2.Meta.Attr

namespace Math2

/-- m2://9TQ4WA: every finite subgroup of the multiplicative group of a field is cyclic. -/
@[m2 "9TQ4WA"]
theorem finite_subgroup_units_cyclic {K : Type*} [Field K] (G : Subgroup Kˣ) [Finite G] :
    IsCyclic G :=
  inferInstance            -- Mathlib's instance; the binding is to a math2 restatement

end Math2
```

`@[m2 "TAG"]` records the binding on the Lean side. Rule `F01` checks that it agrees with the front matter of the node. A node can also bind to a Mathlib declaration that has no attribute, because math2 cannot add attributes to Mathlib. Only the front matter and `registers/bindings.reg` list those bindings.

## 7.3 Binding kinds and adapters

A binding states how the formal declaration relates to the informal statement of the node (04.5). The `match` field takes one of five values:

| `match` | Meaning | Can reach E5 for the node? |
|---|---|---|
| `exact` | the declaration states exactly the node's statement | yes |
| `stronger` | the declaration implies the statement; an adapter in `Math2/Meta/Match.lean` derives the exact form | yes, through the adapter, which is itself a checked declaration |
| `weaker` | the declaration is a special case | no; contributes evidence for the special case only |
| `variant` | same mathematics, different formal packaging (e.g. `Finset` against `Set`, bundled against unbundled) | yes, if the adapter proves the equivalence |
| `partial` | a statement-only binding for an unproved result, or one that is still `sorry`-free only in its statement | never (rule `F07`) |

Mathlib often states a classical theorem in a more general form than a textbook does, or packages it differently. Adapters are how math2 handles this. The adapter is small and checked, and we review it together with the binding. The F2 review covers the composite: the conclusion of the adapter against the informal statement.

## 7.4 Extraction

After a build, `M2Extract` walks the environment. It writes one record for each bound declaration, and one record for each declaration in the closure of each bound declaration:

```yaml
decl: Math2.finite_subgroup_units_cyclic
module: Math2.Algebra.Field.Units
tag: 9TQ4WA
type_hash: blake3:5f1c…            # hash of the elaborated type, after instantiating universe params canonically
value_hash: blake3:a903…
statement_pp: "∀ {K : Type u_1} [inst : Field K] (G : Subgroup Kˣ) [inst_1 : Finite ↥G], IsCyclic ↥G"
axioms: [propext, Classical.choice, Quot.sound]
native_decide_axioms: []
uses_decls: [IsCyclic, Subgroup, Units, Field, Finite, instIsCyclicSubtypeMemSubgroupUnitsOfFinite, …]
uses_constants_in_statement: [IsCyclic, Subgroup, Units, Field, Finite]
deprecated: null                   # or {new_name, since, text} from the @[deprecated] attribute
checks: {}                         # or {vacuous, trivial, refuted}, each the tactic that closed the goal
heartbeats: 1840
```

- **Type hash.** This is the key hash in math2 (06.3). We compute it over the elaborated type in a canonical serialization: universe parameters get names by position, binder names are erased, and instance arguments stay. If two declarations have the same type hash, they state the same thing. Because of this, a review stays valid after a rename (Flow C, 03.4), and align can propose `same_as` without judgement.
- **Dependencies.** `uses_decls` comes from the value of the declaration. It is each constant that the proof term mentions, closed transitively at the boundary of other bound declarations. Formal dependencies between bound declarations become `uses` evidence in the graph. Rule `G04` reports each formal dependency that the `uses` of the node does not show.
- **Statement constants.** `uses_constants_in_statement` drives the review tool. A reviewer must see each non-trivial definition that the statement relies on (16.3).
- **Pretty-printing.** We print the statement normally, and also with full elaboration and with notation off. Reviewers see both forms.
- **Deprecation.** `deprecated` copies the entry of the `@[deprecated]` attribute. Rule `F08` reports each bound declaration that has one, with the new name. The bump agent (07.9) uses the same data to propose the rename.
- **Degenerate statements.** For each theorem, `checks` gives the result of the checks 4, 5 and 6 of 10.4. Automation tries to prove `False` from the hypotheses (`vacuous`), to prove the statement without the theorem itself (`trivial`) and to prove the negation (`refuted`). Each tactic has a limit of 10 million heartbeats, which is 5 percent of the default limit of Lean, and the record keeps the first tactic that closed the goal. Rule `F09` fails on `vacuous` and on `refuted`. A `trivial` result does not fail, because a trivial statement can be the correct one.

Extraction runs in the same pinned environment as the build. Its output is a build artefact, with its hash as its address. The extraction code is part of the trusted base (15.1). Thus it is small, and we test it against golden files in `M2Extract/test/`.

## 7.5 Axioms and `native_decide`

- **The allowed set** for E5 is `{propext, Classical.choice, Quot.sound}`. A node can allow fewer: front matter `axioms: constructive` limits it to `propext` and `Quot.sound`. Then status reports the node as `noac` (04.7).
- **Hypotheses** from `Math2/Hypotheses/` can be in the axiom profile of a conditional result. Status reports them by name. The evidence of the node is then "E5 conditional on GRH", and the workspace shows this in words.
- **`native_decide`.** Since Lean 4.35, each use of `native_decide` adds a new auxiliary axiom, and `Lean.ofReduceBool` is removed. Thus the axiom audit sees `native_decide` without a special case. It is not in the allowed set. A node that depends on it can be at most E4, with the certificate kind `compiled-evaluation`, and rule `F04` lists it. If `decide`, `decide_cbv` or `cbv` can do the same job with kernel-only reduction, we tell the prove role to use them.
- **The checker sees the exported environment**, not the source. Axiom profiles come from the environment as the kernel checked it. A search of the source for `axiom` is not sufficient, because metaprograms can add declarations.

## 7.6 Building

```
m2 build                  # nodes → lean → extract → replay → battery → status → graph → store → search
m2 build -only lean       # lake build of Math2, with the Mathlib cache
```

- **Cache.** Lake's cache tools fetch the Mathlib `.olean` cache, and we check it against the hash in the release manifest. The `.olean` files are a build cache, not evidence. Evidence comes from the replay (7.7), which checks the exported environment again from the start. A compromised cache can make the build lie, but it cannot make the replay lie.
- **Incremental.** PR CI builds only the modules that a change touches, plus their reverse dependencies inside `Math2`. CI on `main` builds everything. The release build runs on a clean runner with no cache from earlier PR runs.
- **Budgets.** We record the heartbeat count of each declaration. If a PR makes the build time of `Math2` more than 5% longer, its description must say why.

## 7.7 Replay and the checker stack

A proof is E5 only after it passes **independent replay** (rule `F05`). The toolchain provides this directly since 2026. This came after the kernel soundness incidents of that summer (document 01):

| Layer | Tool | What it adds |
|---|---|---|
| 1 | `lake build` | elaboration and the official kernel, in the build process |
| 2 | `lake check` | re-checks the exported environment with the official kernel, in a separate sandboxed process |
| 3 | `lake check --paranoid` | adds the independent kernels nanoda (Rust), lean4lean (Lean), con-leche (Lean, proved consistent) and con-ron (Rust, proved to refine con-leche) |
| 4 | comparator | checks that the replayed declaration's statement matches the **pinned statement** (a challenge module built from the binding's recorded type, against Mathlib and `Math2` only), and that its axioms are permitted |

At release time, math2 runs all four layers for each E5 candidate. On each PR, it runs layers 1, 2 and 4. The release manifest records the version of each checker. In 2026, a checker that was one week out of date accepted an exploit that the current checker rejected. The release audit fails if a checker is older than the version bundled with the pinned toolchain.

When this document was written, comparator had known open issues (document 01). Thus math2 also compares type hashes from its own extraction against the pinned binding (7.4). Two independent statement checks are cheap, and each one covers the gaps of the other.

For the largest proofs, a full paranoid replay is expensive. The 13M-line Fermat formalization needed 230 GB of RAM and 15 hours under comparator. math2 does not hold proofs of that size. It binds to the declaration in the external library. It records the replay results of that library with their checker versions, at E5 with the flag `external_replay` (04.7). It runs those replays again at the pinned commit of the library only when a checker changes.

## 7.8 Interfaces for agents and people

The formal engine has three front doors. All three use the pinned toolchain.

- **Lean sessions** for agents (11.3). These are REPL processes with Mathlib and `Math2` loaded from a snapshot. They expose the state at tactic level. Implementations to build on are the community REPL, Pantograph, and lean-beam, which is the agent interface of the Lean FRO and is in beta now (document 01). math2 uses the one that gives the best control at goal level under its sandbox. The protocol in 11.3 belongs to math2, so the implementation can change and the agents do not see a change.
- **An MCP server.** `m2 serve` exposes `lean_check`, `lean_goal` and the session calls as MCP tools. With the public `lean-lsp-mcp`, any MCP agent, Claude Code included, can work on `Math2` from a local checkout.
- **Workbench and editors for people.** A contributor works in VS Code with the Lean extension on a local checkout. Or the contributor works in the zero-install Workbench of the Lean FRO in the browser (beta in 2026). math2 adds nothing to the editor experience other than the binding attribute. This is intentional.

## 7.9 The bump

The `bump` role and `m2 bump` keep math2 current. The process:

1. **Weekly**, the bump branch moves to the newest Mathlib tag. It also tries Mathlib `master` as an early-warning build. We never merge that build.
2. **Breaks are classified.** Each failure is one of four kinds:
   - a rename with a deprecation alias (mechanical: take the new name from the `@[deprecated]` attribute);
   - a rename without one (find the new declaration by type hash);
   - a refactor (the type hash changed);
   - a real break in `Math2` code (needs a fix).
3. **Mechanical renames auto-merge** (12.1). Migrate deprecated names before Mathlib deletes them. Mathlib deletes them in batches, about six months after deprecation.
4. **Refactors** make reviews stale (rule `F02`). Each affected node drops to F0 and goes into the fidelity queue, ranked by `unlock`. It gets a review-assist packet that shows the old and new types side by side.
5. **`Math2` breaks** become prove jobs to repair the proof, at high priority. The release cannot pin the new Mathlib until they are fixed.
6. **Upstream catch-up.** A new Mathlib declaration can have the same type hash as a `Math2` declaration. Then the bump proposes to rebind the node to Mathlib and to deprecate the `Math2` copy. `Math2` gets smaller as Mathlib gets larger. That is the intended outcome.
7. **Toolchain bumps** follow Mathlib, because each Mathlib tag names its toolchain. A toolchain bump also runs the full paranoid replay again, because the checkers change with it.

The bump report goes into `reports/bump/<date>.md`. It counts breaks by kind, time to green, and the nodes whose fidelity reviews became stale. Over months, it measures the cost to stay current. Each downstream Lean project pays this cost, and few measure it.

## 7.10 Other proof assistants

math2 is Lean-first, not Lean-only. Bindings have a `system:` field. Two other systems have support from the start:

- **Isabelle/AFP.** The AFP is the second-largest library, with more than a thousand entries. Some results exist only there. For example, the 2026 Jacobian counterexample was formalised in the AFP first. To check an AFP binding, build the entry at its pinned AFP and Isabelle release. It counts as E5 with `system: isabelle`. The trusted base for such nodes is the Isabelle kernel, and the workspace says so.
- **Rocq/MathComp, Metamath, Mizar, HOL Light** are crosswalk targets (`same_as`, 05.3). In v1 they are not bindings for E5. Metamath's set.mm is the exception. It is a candidate for a v2 binding, because its verifier is very small and there are several independent implementations of it.

Translation between systems (lean2dk, lean2agda and similar tools) was still experimental in 2026, and it is not on the critical path. We record agreement between systems as `same_as` between bindings. It counts towards F3, as an independent formalization of the same statement, only when an equivalence is proved or both formalizations have independent F2 reviews (04.7).
</document>

<document source="https://math2.tamnd.com/spec/08-computational-engine.md">
# 08. The computational engine: CAS, SMT, numerical experiments

The computational engine is the box that makes math2 executable in the second sense. In the first sense, formal proofs run in the Lean kernel (document 07). In the second sense, computations are part of the knowledge. These are the examples, counterexamples, tables, searches and numerical experiments that mathematicians use to find conjectures and to check them. The Research Agents call the engine directly. Its outputs go into the knowledge graph (as `computation`, `example` and `counterexample` nodes, and as evidence) and into versioned storage (as artefacts with certificates).

The engine has one design principle: **a computation is a claim, and the strength of the claim depends on what can be checked about it afterwards.**

## 8.1 What runs

The engine runs pinned container images, one for each system family. The image digest is part of the identity of each computation (document 06.3).

| Image | Contents | Used for |
|---|---|---|
| `m2-sage` | SageMath (current stable), with its bundled PARI/GP, GAP, FLINT, Singular, Maxima, NTL, ECL | general CAS, algebra, number theory, combinatorics |
| `m2-pari` | PARI/GP standalone, current stable | fast number theory, L-functions, elliptic curves |
| `m2-gap` | GAP with the standard packages (SmallGrp, AtlasRep, …) | group theory, representation theory |
| `m2-m2` | Macaulay2 | commutative algebra, algebraic geometry |
| `m2-flint` | FLINT 3, including Arb (merged into FLINT) | ball arithmetic and rigorous numerics |
| `m2-julia` | Julia with Oscar.jl, IntervalArithmetic.jl | computer algebra (Oscar), validated numerics |
| `m2-python` | CPython, SymPy, NumPy/SciPy, mpmath, NetworkX, python-flint, OR-tools | scripting, quick experiments, graph search |
| `m2-smt` | Z3, cvc5, Bitwuzla | SMT queries with models and proofs |
| `m2-sat` | CaDiCaL, Kissat, with DRAT/LRAT proof output; `cake_lpr` and `dpr-trim` as checkers | SAT encodings of combinatorial problems (Boolean Pythagorean triples, Keller, Schur number 5, empty hexagon) |
| `m2-lp` | SCIP, HiGHS, with exact rational verification (VIPR certificates where available) | integer and linear programming bounds |
| `m2-lean` | the pinned Lean toolchain and the Mathlib cache | `#eval`, `decide`, and checking certificates inside Lean (document 07.8) |

Commercial systems (Magma, Mathematica, Maple) are permitted as **E1 evidence only**. An agent or a person can run them, and we can record the output. But the output cannot reach E4, because a reader cannot reproduce the run. They are adapters in `math2-agent`, and they are off by default. The same rule applies to Wolfram|Alpha.

We build each image reproducibly from a Nix flake in `tamnd/math2-agent/images/`. We publish it by digest and refresh it monthly with the release. If an image changes under a recorded computation, the old record stays valid, because the record names the old digest. But the change causes a new run in the next release, to check that the result still holds (rule `C03`).

## 8.2 The computation record

A computation is a node of kind `computation`. Its body is a short description. Its front matter is the record:

```yaml
tag: 3MZ8QD
kind: computation
name: No counterexample to Goldbach below 10^7 (demonstration)
about: [8GQ2N1]                  # the Goldbach conjecture node
claim: >
  Every even n with 4 <= n <= 10^7 is the sum of two primes.
engine:
  image: m2-pari@sha256:...       # by digest
  entry: compute/3MZ8QD/main.gp
  inputs: {N: 10000000}
  seed: null                      # deterministic. Randomised computations must give a seed
  resources: {cpu: 4, mem: 4G, wall: 10m}
output:
  hash: blake3:...                # canonical serialisation of the result
  summary: "checked 4999999 even numbers; 0 failures"
certificate:
  kind: witness-table             # see 8.3
  artefact: blake3:...
  checker: m2-lean@sha256:...    # independent checker image
  checker_entry: compute/3MZ8QD/Check.lean
```

The code is next to the node, in `compute/<tag>/`, under Apache-2.0 like the Lean code. Read a computation in the same way as a proof: what does this show, and how can I check it?

## 8.3 Certificates: how computations reach E4

A computation reaches E4 (`certified`, document 04.7) only if it produces a **certificate** that a **different, smaller, more trusted** checker accepts. The engine knows a fixed list of certificate kinds:

| Certificate | Produced by | Checked by | Typical claim |
|---|---|---|---|
| `witness` | anything | Lean `decide`/`norm_num`, or a 50-line checker | "there exists an X with property P" (show the X) |
| `witness-table` | anything | a Lean or small Python checker that runs over the table | finite verifications where each case has a short witness (Goldbach pairs, factorisations) |
| `lrat` | CaDiCaL/Kissat with proof output | `cake_lpr` (a verified checker, CakeML) | "no X exists" via SAT |
| `smt-proof` | cvc5 (Alethe/CPC proofs), Z3 (proof objects) | Lean's SMT reconstruction, or Carcara for Alethe | decidable arithmetic and bit-vector facts |
| `primality` | PARI/GP or Primo | Lean (Pratt or ECPP certificate checker) | "N is prime" |
| `interval` | FLINT/Arb, IntervalArithmetic.jl | re-evaluation in a second ball arithmetic, or in Lean with exact rationals | rigorous numerical inequalities, zeros of functions |
| `lp-dual` | SCIP/HiGHS | exact rational arithmetic check of the dual certificate (VIPR) | bounds from linear and integer programming |
| `sos` | SDP solver plus rational rounding | exact polynomial identity check | positivity via sums of squares |
| `group-id` | GAP | a recomputation in a second system (Magma excluded), or a Lean check for small orders | "this group is SmallGroup(n,k)" |

A computation without a certificate is E1. That is still useful. Most experiments in math are E1, and the system records them accurately. But the system never shows it as stronger than E1.

### Into the kernel

If Lean can check a certificate in reasonable time, the computation can go all the way to E5. We import the certificate as data. We prove a Lean checker sound one time. Then `decide` or kernel reduction runs the checker on the data. The other route is `native_decide`, which is flagged and is permitted only per node (document 07.7). The Lean ecosystem of 2025 to 2026 made this practical for SAT (LRAT checks inside Lean, as `bv_decide` does) and for many arithmetic certificates. If the certificate kind of a computation has a Lean checker in `lean/Math2/Cert/`, the prove agent tries this route.

## 8.4 Numerical experiments

Experiments are how people find conjectures. FunSearch, AlphaEvolve and PatternBoost-style systems (document 01) are, at the base, experiment loops in which a model proposes programs or constructions. math2 supports them as first-class but untrusted:

- **An experiment is a computation with `role: experiment`.** Its claim is an observation ("for all n ≤ 10^5 tested, a(n) is squarefree"), not a theorem.
- **Search runs record their search space.** An agent can search for a construction (a cap set, a packing, a graph with given properties). Then the record holds the scoring function, the best object found and its score. A best object is a `witness`. A lower bound from it can be E4 if the check of the object is cheap. The claim "this is optimal" is E1 unless an exhaustive or dual certificate supports it.
- **Patterns become conjectures through a gate.** If an agent sees a pattern, it files a `conjecture` node with `source: computation <tag>`. The explore agent (document 12.5) must first run a falsification pass: extend the range, randomise, test against OEIS and LMFDB, and try the obvious generalisations. The agent files the conjecture together with the falsification attempts that failed.

## 8.5 SMT and SAT for the prove loop

SMT and SAT are also tools inside the prove loop (document 12.4). They do more than produce standalone computations:

- **Lean already calls out.** We use `bv_decide` (bit-blasting to CaDiCaL with LRAT replay), `omega`, `grind`, `polyrith` (which calls a CAS and then checks the certificate) and similar tactics where they apply. The Lean axioms audit (document 07.7) handles their trust status.
- **Agents use SMT as a counterexample finder.** Before the prove agent spends prover budget on a goal, it encodes small finite instances where it can. Then it asks Z3 or cvc5 for a model. A model is a counterexample to the formal statement. That is very often a *misformalisation* finding (document 10.5), and the engine reports it to the formalize agent as one.
- **Large SAT proofs are artefacts.** We do not store proofs of the Boolean Pythagorean triples size (about 200 TB uncompressed). We store the encoding, the solver version and the cube split. We also store a checked hash chain if the original authors gave one. That is sufficient to say exactly what was checked and by whom. A claim in this class is E4 only if a certificate was checked, possibly by a third party, and the record names who did it.

## 8.6 Sandboxing, budgets, determinism

Each computation runs in a sandbox. The rules are the same for agents and for the CI runs of people:

- **Isolation:** gVisor or Firecracker microVMs on the runners; no network by default; a read-only image; a writable scratch area of bounded size.
- **Budgets:** CPU, memory, wall clock and output size, all declared in the record. The request of an agent is charged against the budget of the run (document 11.5).
- **Determinism:** computations must be deterministic, or must declare a seed. The engine runs each new E4-candidate computation two times, on two different runner types (x86-64 and arm64), and compares output hashes (rule `C02`). A difference blocks E4, and the engine writes it to the report. Floating-point code that is not interval arithmetic never reaches E4, so it never blocks anything. It stays E1.
- **Data access:** a computation can read OEIS, LMFDB or zbMATH only through the cached snapshots of the ingestion layer (document 09). Those snapshots are pinned by hash and mounted read-only. Thus a computation never depends on a live API.

## 8.7 Executable examples

Each `example` and `counterexample` node should be executable where possible. The executable check is a Lean `example` (document 07), or a computation of 10 seconds or less with a `witness` certificate. The "run this" button of the workspace (document 13) runs exactly that record in a sandbox on the browser side or the server side. It shows the output next to the recorded hash. The build reports the share of executable examples per topic, and the goal in document 02 gives a number for it.

## 8.8 What the engine will not do

- It will not run code that is not in a pinned image.
- It will not let a computation call the network, call a model, or read anything outside its declared inputs.
- It will not accept a certificate checker written by the same agent run that produced the certificate, unless that checker has itself been proved sound in Lean or reviewed by a person. The checker for each certificate kind is fixed in `math2-agent`, and a change to it needs the review of a person.
- It will not show a numerical observation as anything stronger than an observation.
</document>

<document source="https://math2.tamnd.com/spec/09-literature-ingestion.md">
# 09. Literature ingestion: papers, books, existing libraries

The top-right box of the diagram has the label "Papers, Books, Existing Libraries". It is how math2 learns what mathematics already knows. Ingestion has three jobs:

1. **Fill the record plane** (03.3). It records each paper and book in the mathematical literature. Where possible, it also extracts and records each result that they state.
2. **Seed and source the statement plane.** The `sources:` of each node points at records with locators, and curation turns extracted results into nodes.
3. **Join the existing libraries**: Mathlib and other formal libraries, encyclopedias and databases. Crosswalk edges (05.3) connect them to nodes. math2 does not copy what they hold.

Ingestion code is in `tamnd/math2-agent/ingest/<source>`, with one adapter for each source. Each pull is pinned in `tamnd/math2/manifests/sources.yaml` and `manifests/ingest/`.

## 9.1 Principles

- **Dumps, not crawling.** Read each source through its published bulk channel or API, at the rate that it asks for. If a source has no bulk channel, link to it and do not ingest it.
- **Pinned snapshots.** A pull is a snapshot with a date, a URL, a hash and a licence. At build time, nothing in math2 depends on a live external service.
- **Licence decides what is taken** (9.3). Most of the literature can give a record and a locator, but not text. That is sufficient for the record plane, because a mathematical fact is not copyrightable and math2 writes its own statements.
- **Reuse other people's indexes.** Several 2026 projects have already extracted millions of statements from the literature (9.6). math2 aligns to them by id. It does not extract again what they hold, unless it needs something that they do not have.
- **Do not duplicate `tamnd/arxiv`.** arXiv papers come in through the two planes of `tamnd/arxiv`. math2 stores `ax://` references, not paper text.

## 9.2 The source catalogue

Sizes are from document 01, measured or reported in October 2026.

| Source | What math2 takes | Size | Channel | Licence of what is taken | Cadence |
|---|---|---|---|---|---|
| **arXiv** (via `tamnd/arxiv`) | records for every math paper; extracted results from the content plane | about 55k math primaries in 2026 so far, growing fast | `tamnd/arxiv` metadata and content planes | metadata CC0; text per paper licence (9.3) | daily |
| **zbMATH Open** | records for the published literature 1763–present, MSC codes, reviews' metadata, software (swMATH) | 4M+ publications, 168M+ triples in the KG | OAI-PMH, REST API, the KG dump | CC BY-SA 4.0 (publisher-restricted abstracts are not included) | monthly |
| **MathSciNet** | MR numbers only, via reference matching | 4M+ items | MR Lookup | identifiers only | on demand |
| **Crossref / DataCite** | DOIs, journal metadata, references where open | n/a | public APIs | CC0 metadata | monthly |
| **Mathlib** | declarations, types, docstrings, module structure, deprecations | about 290k theorems, 140k definitions | the pinned checkout plus `M2Extract` (07.4) | Apache-2.0 | every bump (07.9) |
| **other Lean libraries** | the same, for bound libraries: Tau Ceti, Formal Frontier textbooks, Physlib, CSLib, FLT (Imperial and Anthropic), PFR, Carleson, sphere packing | Tau Ceti about 3M lines | pinned checkouts | per repository (mostly Apache-2.0) | per release |
| **formal-conjectures** | formal statements of open problems, categories, solved markers | 4,652 statements on 25 October 2026 (1,602 open research problems, 2,816 solved, 234 textbook) | git | Apache-2.0 | weekly |
| **Palomar** | registered Lean-verified results, with `formalization.yaml` provenance | 760 results on 25 October 2026 (registry revision 517), growing since August 2026 | the listing API | CC0 metadata | daily |
| **Erdős problems** (`teorth/erdosproblems`) | problem statements, status, links to formal statements and OEIS | 1,221 problems | git (`data/problems.yaml`) | Apache-2.0 | weekly |
| **Isabelle AFP** | entries and theorem names, for bindings and crosswalk | 1,038 entries | the release tarball | per entry (BSD/LGPL) | per AFP release |
| **OEIS** | sequences: name, terms, formulas, references, conjectures in comments | about 400k sequences | `oeisdata` git export | CC BY-SA 4.0 | weekly |
| **LMFDB** | object records (curves, fields, forms, groups, L-functions) by label, for `object` nodes and crosswalk | about 3.8M curves over ℚ, 22M number fields, … | API and bulk downloads (the LMFDB asks AI users to prefer bulk) | CC BY-SA 4.0 | monthly |
| **GAP SmallGrp, House of Graphs, FindStat, π-Base, DaRT** | object identifiers and properties; π-Base implications as `statement` candidates | n/a | packages, downloads, git | varied; π-Base data CC BY | per release |
| **Wikidata** | QIDs, labels, existing crosswalk properties (nLab, MathWorld, ProofWiki, Mathlib) | about 113k items with a defining formula | weekly dump; SPARQL for checks | CC0 | weekly |
| **MaRDI** | crosswalk ids; model and algorithm records | 7M+ items | SPARQL | per dataset | monthly |
| **Wikipedia** | article ids for crosswalk; short quoted text where useful | about 26k mathematics articles | dumps | CC BY-SA 4.0 | monthly |
| **nLab** | page ids for crosswalk | about 19k pages | `nlab-content` git | link-only until the licence is confirmed | monthly |
| **ProofWiki** | page ids for crosswalk; proofs and definitions as candidate sources | 30k proofs, 36k definitions | XML dump | CC BY-SA 3.0 | quarterly |
| **Stacks Project, Kerodon** | tags, titles, dependency structure for `same_as` and `cites` | 21,446 Stacks tags on 25 October 2026 (commit a04446e) | git (`tags/tags` and the LaTeX) | GFDL: **link only** | monthly |
| **MathOverflow, Math.SE** | question ids, titles, per-post records for E1 claims and references | 169k and 1.7M questions | the Stack Exchange API, per post | CC BY-SA, per-post version (16.6) | on demand |
| **Books** | records and locators (9.7) | n/a | catalogue metadata, open textbooks | per book | on demand |
| **MathWorld, Encyclopedia of Mathematics, PlanetMath** | ids for crosswalk | n/a | n/a | MathWorld proprietary: link only; others CC BY-SA | quarterly |

Each row is an adapter with its own fixtures. To add a source, write an adapter, add a manifest entry with a licence, and get the approval of a maintainer for the licence classification. That classification is the one decision that matters.

## 9.3 The licence gate

Each source gets an **access class**. Where licences are different within a source, each item gets one. The class decides what can go into which part of math2. math2 extends the four classes of `tamnd/arxiv` for its three licence zones (16.6):

| Class | Example licences | Record (metadata, locator) | Embeddings, internal | Text in the record plane | Text in node prose |
|---|---|---|---|---|---|
| `open` | CC0, CC BY, CC BY-SA, Apache-2.0 | yes | yes | yes, by reference to its home (e.g. `ax://`) | yes, with attribution (BY-SA zone) |
| `share-alike-nc` | CC BY-NC-SA, CC BY-NC | yes | yes | by reference only | **no**: NC is incompatible with CC BY-SA |
| `verbatim` | CC BY-NC-ND, CC BY-ND | yes | yes | by reference only | no |
| `gfdl` | GFDL (Stacks, Kerodon, Open Problem Garden) | yes | yes | by reference only | **no**: GFDL and CC BY-SA cannot mix |
| `record` | arXiv default licence, all-rights-reserved journals, MathWorld | yes | yes | no | no |

Three consequences:

- **math2 stores almost no third-party text.** Paper text stays where it is now. For arXiv, that is the content plane of `tamnd/arxiv`, which has its own licence gate. math2 holds references, locators, hashes and its own derived data: embeddings, extracted structure and alignments. Thus each corpus has one licence gate, not two.
- **About 15% of arXiv is `open` for node prose.** TheoremSearch measured the CC0, CC BY and CC BY-SA share. If you include the NC licences, the Creative Commons share is about 40%. `tamnd/arxiv` uses this figure for its content plane. For each other paper, math2 writes the statement itself and cites the locator. For a mathematical result, this is better in any case: we check the statement against the definitions and notation of the graph.
- **Embeddings are internal.** math2 embeds text from all classes, for alignment and search. It never publishes embeddings of `record` class text in a form that can reconstruct the text. The vector index of the serving store serves neighbours, not vectors (14.4).

Rules `L01` to `L04` (15.3) enforce the gate. `L02` fails the build if `nodes/` contains text from a class that must not go into node prose.

## 9.4 Existing formal libraries and blueprints

Formal libraries come into math2 as **bindings and alignments**, never as copies.

- **Mathlib** is extracted at each bump (07.4). Each declaration becomes a candidate for binding. The curate and align roles propose nodes for important declarations that do not have one, ranked by in-degree in the formal dependency graph. They also use type search and embeddings to propose `formal:` bindings for existing nodes.
- **Other Lean libraries** are extracted in the same way, at their pinned commits. We bind AI-written libraries (Tau Ceti, the textbook libraries of Formal Frontier, the Anthropic FLT repository) like all other libraries. The source entry records their provenance (AI-written, how reviewed), and each binding to them shows it. A binding to an AI-written library is not weaker for E5, because the kernel does not care who wrote the proof. But it is important for maintenance, and the bump report tracks it.
- **Palomar entries** come in with their `formalization.yaml`. That file records these facts: whether the formalization was human, AI-assisted or autonomous, which models were used, and how it was reviewed. The challenge statement of a Palomar entry becomes a binding candidate. Its replay record is accepted as `external_replay` (07.7) at its pinned commit. Palomar does not certify fidelity, so the binding starts at F0 or F1 like all other bindings.
- **formal-conjectures** statements become `partial` bindings for conjecture nodes. They are aligned by the source references of the repository: Erdős numbers, OEIS, Wikipedia, MathOverflow.
- **Blueprints.** `m2 ingest blueprint <repo>` reads the blueprint of a formalization project and imports it as nodes and edges. It reads a leanblueprint LaTeX source (`\label`, `\uses`, `\lean`, `\leanok`) or the `blueprint-manifest.json` of a Verso Blueprint. A Verso Blueprint derives status from the code, not from hand annotations. Each blueprint item becomes a provisional node with `uses` edges. If the item names a declaration, it also becomes a binding. The import is a PR. The repository of the project stays the authority for its blueprint. math2 imports again when the blueprint changes, and keeps its tags stable through the labels of the blueprint (`aliases:`).

## 9.5 Extracting results from papers

Extraction turns a paper into a list of **results**: the theorem-like environments, with their text, labels, internal references and locators. math2 does not implement extraction again. It uses the result tags of `tamnd/arxiv`, `2106.09685/AB12`, which are permanent per paper. It adds processing that is specific to math:

1. **Source first.** arXiv LaTeX source is the ground truth. We use HTML (LaTeXML) for structure where the conversion is clean, which is about three quarters of papers. PDF-only material, such as older journal papers in open repositories and scanned books, goes through a pinned OCR model. We select the OCR model from the current formula-recognition benchmarks. In 2026 the best open document-OCR models were above 97 on the formula metric of OmniDocBench. The outputs are marked `ocr` with the model version.
2. **Environments.** The extractor finds `theorem`, `lemma`, `proposition`, `corollary`, `definition`, `conjecture`, `example`, `remark` and their custom aliases. It resolves `\newtheorem` declarations. It also finds proofs and attaches them to their statements.
3. **Internal references.** `\ref`, `\cref` and "by Lemma 3.2" become dependency edges inside the paper. Citations become references between papers. They are resolved to records through the bibliography of the paper, and matched against zbMATH, Crossref and arXiv.
4. **Unfolding.** A result can say "under the hypotheses of Theorem 2.1". We rewrite it into a **self-contained statement**: we put in the hypotheses and the notation that the paper set up earlier. A model does this, and a second pass checks it against the original (Matlas uses the same step). We keep the original and the unfolded form. Alignment and search see the unfolded form. It is derived data, and it is under the licence class of the paper (9.3).
5. **Typing.** Each result gets a kind (04.2) and MSC codes, from the codes of the paper plus a classifier.
6. **Quality.** Each step has an evaluation set (17.3). Each month, people check a sample, and we publish the error rates for each step.

The output is one record plane entry for each result, under `ax://paper/<id>#<tag>` for arXiv. For other sources it is a math2 record id `m2://rec/<source>/<id>#<n>`.

## 9.6 Other people's indexes

In 2026 several projects extracted the literature at scale (document 01):

- TheoremSearch: 9.2M theorems from about 690k papers. It released the openly licensed 1.34M.
- TheoremGraph, with its formal half LeanGraph: 11.7M arXiv statements and 18.3M candidate dependency edges. It matches 48k statements to Lean declarations, and has an MCP interface.
- Matlas: 8M unfolded statements from 435k journal papers and 1.9k textbooks.

The position of math2 is to **align to these, not compete with them**:

- Their identifiers are crosswalk targets (`same_as` on record plane entries).
- If their data is openly licensed, we import their extractions and dependency edges as *candidate* evidence. Each item names the extractor that produced it. We treat it like the extraction of math2: E1, unconfirmed, subject to dedup.
- If their data is not openly licensed, math2 uses their public APIs at query time, in the novelty check (17.5) and in search. It stores only the ids of matches.

math2 adds what these indexes do not have: curated concept nodes, computed status, reviewed fidelity, and permanence.

## 9.7 Books

Books are the main source for E2 ("textbook-standard", 04.7). They are also the most difficult source to ingest, because most of them are not openly licensed. math2 handles them as follows:

- **Records and locators** for any book, from catalogue metadata (ISBN, zbMATH and MR records). A locator such as "§3.2, Thm 8" is a fact, and math2 can store it for any book.
- **Curricula.** We build reading lists in `topics/curricula/` from standard texts and their tables of contents. A table of contents is a factual structure that math2 can record. The full text stays with the book.
- **Open textbooks** (CC-licensed, or with explicit permission) can be extracted like papers. The 2026 automatic formalizations of whole textbooks (document 01) give these books formal counterparts, which align to the nodes of the books.
- **Two independent book locators** for E2 are checked by curators, not by agents, because the check requires a person to read the book. That is an intentional use of human time. Most nodes in the undergraduate core will be at E2 before they are formalised, so E2 must mean something.

## 9.8 Scale and cost

| Item | Size | Ingest cost |
|---|---|---|
| Records (zbMATH, arXiv, Crossref) | about 5M papers and books | negligible compute; storage about 20 GB Parquet |
| Results extracted (arXiv `open` and `share-alike` content, plus imported candidates) | 10–30M results | extraction through `tamnd/arxiv`; unfolding by a small model in batch, at a cost of roughly cents per paper |
| Embeddings of all results | 30M × 1,024 dims | a one-time batch job, then incremental; the index is about 60 GB quantised (14.4) |
| Alignment of new results to nodes | daily: about 1,000 papers → 10–20k results | judge calls only on candidate pairs above the similarity threshold, with Haiku-class pre-filtering and batch pricing (12.3) |

The monthly arXiv math volume grew by more than half between 2025 and 2026. Part of the growth is attributed to AI-assisted writing. Ingestion must scale with that volume, and it must not use curator time for it. For these reasons, extraction and alignment are agent work, and curators see queues ranked by importance (16.3). Also for these reasons, an extracted result stays E1, in the record plane, until something in the graph needs it.
</document>

<document source="https://math2.tamnd.com/spec/10-autoformalization-and-fidelity.md">
# 10. Autoformalization and statement fidelity

The kernel answers one question perfectly: is this Lean term a proof of this Lean type? It cannot answer the question that matters: does this Lean type say what the mathematician meant? That second question is **fidelity**. In 2026 it is the weakest link in all formal mathematics, with AI or without it. This document tells how to change fidelity from a hope into a property that we measure, review and maintain.

## 10.1 The evidence that this is the bottleneck

Document 01 has the sources. In brief:

- An audit in 2025 found that about **a third of the Lean 4 port of ProofNet** was misformalized.
- An ICML 2026 audit of five formal benchmarks found **4,833 issues**. Of these, **398 were mechanically certified**: counterexamples to stated theorems, vacuous statements, and unsound axioms.
- The statements of PutnamBench needed **27 fixes** from one prover team and 2 more from another before people could call the benchmark saturated.
- In 2025, an AI "solution" to an Erdős problem got much attention. It proved a weaker variant, because the problem as written did not have a hypothesis.
- The 2026 Navier–Stokes claim came with a Lean proof. The dispute about it was about the gap between the formal statement and the physical problem.
- The 2026 Annals Challenge put Lean statements of 50 recent Annals theorems into Mathlib. It found that the main blocker was **missing definitions**, not missing proofs.
- In the largest 2026 formalizations, people fixed or cross-checked the top-level statement by hand, against a curated library. In the Fermat run, agents cross-checked the intermediate statements of each other, and that found false ones early.

The lesson that math2 takes: **autoformalization of statements is now cheap and usually right. "Usually" is not good enough for a library that other work will build on.** Thus the pipeline has two halves. A cheap machine half, F1, finds most errors. A human half, F2 and F3, is made fast enough that people can review the important statements.

## 10.2 The formalize pipeline

The formalize role (12.1) runs this pipeline for one node. `m2 autoform <tag>` runs it from the command line.

```
1. context      node statement, its uses closure (definitions!), sources' locators,
                bound neighbours and their Lean statements, Mathlib conventions
2. definitions  every concept the statement uses must already be bound, or is formalized first (10.3)
3. candidates   k candidate Lean statements from ≥ 2 model families, each elaborated in a Lean session
4. battery      the F1 battery (10.4) on every candidate that elaborates
5. select       candidates that pass; clustered by type hash and by proved equivalence;
                the largest agreeing cluster wins; disagreement is itself recorded
6. propose      PR: binding at F0 with the winning statement, the battery report,
                the dissenting candidates, and a review packet (10.6)
```

Two design points:

- **Agreement among independent candidates is evidence.** Five candidates from two model families can elaborate to the same type hash, or to types proved equivalent in a Lean session. That is stronger than the back-translation of one candidate. If the candidates split into clusters, the split usually points directly at the ambiguity: which convention for $\mathbb{N}$, whether "ring" includes a unit, whether a sum is over a finite set. The PR reports it.
- **Search before writing.** Step 1 first searches for an existing declaration with type search (14.3) and embedding search. If Mathlib already states it, the binding goes to Mathlib, and the formalize run becomes an align run.

## 10.3 Definitions before statements

A statement can only be as faithful as the definitions that it uses. A faithful statement about the wrong notion of "regular" is wrong. Thus formalize works from the bottom up over the `uses` closure:

1. Each concept or definition node in the closure must have a binding at F2 or higher before a statement that uses it can go higher than F1. The fidelity queue enforces this review order (11.4 `unlock`).
2. If Mathlib has no definition, the formalize role proposes one in `Math2/`, together with **sanity lemmas**. These show that the definition agrees with the examples of the textbook and with known equivalent characterisations. The proof of the sanity lemmas is a prove job. A definition with proved sanity lemmas is much easier to review.
3. If Mathlib has several candidate definitions (bundled and unbundled, `Set` and `Finset`, different universe levels), the binding names the selected one, and the remarks of the node say why.

## 10.4 The F1 battery

F1 means that a candidate has passed each check that needs no human. A candidate is a statement: a theorem, or a definition of a proposition when the result has no proof yet (07.2). The battery report records each check. Each check has an evaluation set with a measured detection rate (17.3).

| # | Check | How | Catches |
|---|---|---|---|
| 1 | **elaborates** | the Lean session at the pinned toolchain | syntax and type errors |
| 2 | **back-translation** | a model that has not seen the informal statement translates the Lean statement into English; a judge compares the two, sentence by sentence, with a structured verdict per hypothesis and conclusion | dropped or added hypotheses, wrong quantifier order, wrong conclusion |
| 3 | **constant audit** | every constant in the elaborated statement is mapped to a node through bindings, and that node must be in the informal statement's `uses` closure | the right words with the wrong definitions (e.g. `Nat.sub` truncation, `Real.sqrt` on negatives, `0⁻¹ = 0`) |
| 4 | **hypotheses satisfiable** | a Lean search, then an SMT search, for an instance satisfying all the hypotheses, starting from example nodes in the graph | vacuous statements: contradictory hypotheses make anything provable |
| 5 | **conclusion not trivial** | automation (`simp`, `decide`, `grind`, `aesop`) attempts the statement with a short timeout; success is flagged for review, not failed | statements that are trivially true by a junk-value convention |
| 6 | **negation not provable cheaply** | automation attempts the negation | statements that are false |
| 7 | **examples and counterexamples** | the statement is instantiated on the graph's `example` and `counterexample` nodes for its concepts and checked with `decide` or `plausible`-style testing | statements that fail on the textbook's own examples, or "prove" the counterexample |
| 8 | **neighbour consistency** | if the node `specializes` or `generalizes` a bound node, a Lean session tries to derive one formal statement from the other | inconsistent formalizations across the graph |
| 9 | **critic** | an independent model, from a different family from the generator where possible, reads the informal statement, the Lean statement and checks 2–8, and lists concerns with severity | everything else; its output goes to the reviewer, it is not a verdict |
| 10 | **pinned-statement compile** | the statement alone compiles as a comparator challenge module against Mathlib and `Math2` | statements that depend on the proof file's local context |

A candidate is F1 if checks 1, 3, 4, 6, 7, 8 and 10 pass, and checks 2 and 9 raise no high-severity concern. Check 5 never blocks F1. Sometimes a trivial truth is the correct statement. But the review packet always shows a trivial truth.

Checks 4, 5 and 6 are also rule `F09`. This rule runs on each F1+ binding at each release, because a Mathlib bump can make a statement trivial or vacuous and not change its text.

**The battery in m2.** The package `battery` runs the checks. The build step `battery` runs after the replay step and writes `build/battery.json`, with one report for each Lean binding. The status step reads the reports and gives F1 to a binding when its report is F1. Each check gives one of six results: pass, fail, flag, open, skip or not-run. A blocking check passes F1 with pass, flag or skip. Checks 2 and 9 pass F1 with pass or flag. Thus an open or not-run result blocks F1. The checks work as follows:

- Check 1 uses the extraction record of the declaration.
- Check 3 maps each constant of the statement to the nodes that bind it. The constants of logic and notation, and the instances, do not need a node. A constant that no node binds gives open. A set of patterns on the `pp.all` statement finds the junk values of 10.5 (natural subtraction, division by zero, `Real.sqrt`, `Real.log` and others) and gives a flag with the class `M-junk`.
- Checks 4, 5 and 6 use the degenerate-statement checks of the extract step.
- Checks 7 and 8 ask a Lean session for a short tactic proof of `A → B` or of `A → ¬B`, with the `implies` command of M2Extract. Check 7 does not instantiate the statement yet. It compares the statement with the formal statement of each example and counterexample of its concepts. When the statement implies the negation of an example, the class is `M-strong`. When a counterexample implies the negation of the statement, the class is `M-hyp-missing`. Check 8 uses the edges `generalizes`, `specializes`, `implies` and `equivalent`.
- Check 10 uses the comparator result of the replay step.
- Checks 2 and 9 need models. The build does not run them until the model calls are approved, so no binding is F1 before then.

## 10.5 A taxonomy of misformalization

Each class has a name, a detector and an example. Reviews and audit samples get labels from these classes. Thus we can measure the error rate for each class (17.3), and we can improve the battery where it is weak.

| Class | Description | Primary detector | Example |
|---|---|---|---|
| `M-hyp-missing` | a hypothesis of the informal statement is absent | back-translation; examples | "every continuous function attains its maximum", stated without compactness |
| `M-hyp-extra` | a hypothesis is added | back-translation; critic | the statement is proved for finite groups only when the source says groups |
| `M-quant` | quantifier order or scope wrong | back-translation | ∀ε ∃δ against ∃δ ∀ε |
| `M-vacuous` | hypotheses unsatisfiable | check 4 | `n > 0 → n < 0 → P` |
| `M-junk` | a junk value makes it trivially true or false | constant audit; check 5 | division by zero, natural subtraction, `sqrt` of a negative, `sSup` of an unbounded set |
| `M-def` | a constant means something different from the informal notion | constant audit; definitions-first | Mathlib's `Nat.Prime` against a source that counts 1 as prime |
| `M-type` | wrong carrier: ℕ for ℤ, `Fin n` off by one, real for complex | back-translation; examples | a sum over `Finset.range n` meant to be 1..n |
| `M-conv` | a convention mismatch on an otherwise correct statement | critic; review | ℕ including 0; `⊆` against `⊂`; graph with or without loops |
| `M-weak` | the formal statement is a special case | neighbour consistency; review | the Erdős #124 pattern |
| `M-strong` | the formal statement is stronger and false | negation check; examples | dropping "for sufficiently large n" |
| `M-context` | the statement depends on local context, notation or instances in the proof file | check 10 | a local `instance` that changes `≤` |
| `M-tamper` | the proof environment redefines or shadows something in the statement | pinned statement plus comparator (12.2) | redefining a norm so that every element has norm zero |

## 10.6 Human review: F2 and F3

### What a reviewer signs

A fidelity review says: *I have read the informal statement with hash h₁, and the Lean statement with elaborated type hash h₂, together with the definitions of every constant it uses, and the Lean statement says what the informal statement says, up to the conventions recorded on the node.* The review records:

- the reviewer, the date, h₁ and h₂;
- a verdict: `faithful`, `faithful-with-conventions` (listed), or `not-faithful` (with a taxonomy class);
- free-text notes.

A `not-faithful` review has the same value as a positive one. It sends the binding back to formalize with the class and the notes.

### The review packet

Review-assist builds a packet for each binding in the fidelity queue. `m2 review <tag>` renders it:

- the informal statement and the Lean statement side by side, the Lean one pretty-printed both with and without notation;
- the back-translation, and the judge's per-clause verdicts;
- each constant in the statement, expanded to its definition and its bound node, with that node's fidelity level;
- the battery report, the critic's concerns, the dissenting candidate clusters and how they differ;
- for a re-review after a bump, the old and new types as a diff.

**The packet in m2.** The package `review` makes the packet from the files of the last build, and `m2 review <tag>` shows it. The packet has the informal statement and the Lean statement side by side, the Lean statement with `pp.all`, each constant with its type and the nodes that bind it with their fidelity level, the battery report, and the reviews of the binding. The back-translation, the critic and the candidate clusters need the model runs of the formalize pipeline, and the old and new types need the bump data, so the packet does not show them yet. With a verdict, `m2 review` writes the review to `reviews/XX/TAG.yaml` and adds its id to the binding. It refuses a reviewer that is not a person, the author of the binding, and a statement that changed after the build. The check of rule `V03` on the agent runs that a reviewer started needs the run records of M5.

Review-assist never signs. An agent cannot be a reviewer. A reviewer cannot review a binding that came from an agent run that the reviewer started (rule `V03`).

### Independence

F3 needs one of two things:

- **two independent reviews**: different reviewers, neither of them the binding's author, sharing no affiliation (rule `V02`);
- **one review plus an independent formalization proved equivalent**. The second formalization comes from a different run, of a different model family, given no access to the first candidate. The equivalence is proved in Lean.

The second route is cheaper, and it becomes cheaper each year. It is also the route that scales. The target for each node in the undergraduate core is F3 by cross-formalization (document 02).

### Throughput targets

For undergraduate-level statements, a review takes 3 minutes. Thus one reviewer-hour gives 20 F2 reviews. The undergraduate core (about 25,000 statements and definitions, document 18) then needs about 1,250 reviewer-hours for F2. That is a summer of work for twenty people, or a year of light work for a hundred. Research-level statements take much longer, at about 15 minutes or more. For this reason the fidelity queue is ranked by `unlock`. Also for this reason, the Annals-Challenge-style approach is the correct order: it formalizes statements before anyone tries to prove them. A reviewed statement has more value than an unreviewed proof.

## 10.7 Statements without proofs

Many of the most useful formal bindings in math2 will have no proof for a long time. These are research-level theorems, open conjectures, and results whose proofs depend on large theories that Mathlib does not have. Such a binding is `match: partial` until a proof exists. It is still worth having:

- it can be reviewed to F2 and F3, which makes it a trustworthy target for anyone, human or agent, who wants to prove it;
- it gives search a formal type to match against (14.3);
- for conjectures, it is the precondition for any AI claim of a solution to be checkable. A solution counts only against the pinned, reviewed statement. That is the lesson of every 2025 to 2026 episode in which an AI proved the wrong version.

The formal-conjectures repository and the Annals Challenge are the two largest collections of formal statements without proofs. math2 aligns to both (document 09). If their maintainers have reviewed an aligned statement, math2 records it as a review and names the source.

## 10.8 Measuring fidelity

Fidelity levels are claims too, and math2 measures how reliable they are:

- **Sampled re-review.** Each month, a different reviewer does a blind re-review of 1% of F2 bindings and 0.5% of F3 bindings. We adjudicate and publish the disagreements. The measured error rate per level, by area, is a headline number in the notes of each release. `m2 review sample` draws the sample with a seed and records it in `registers/samples.reg`, and `m2 review measure` gives the rate.
- **Battery evaluation.** We evaluate each check in 10.4 against a labelled set of known misformalizations. The set starts from the published benchmark audits and grows with each `not-faithful` review. We publish detection rates per class (17.3).
- **Downstream evidence.** Sometimes the proof of a statement needs a hypothesis that the informal statement does not have. Sometimes the proof is suspiciously short. In both cases, the statement goes back to the fidelity queue with a flag. A suspiciously short proof means that the measured size is much smaller than that of comparable proofs.

If F2 means "probably right", math2 should be able to say how probable.
</document>

<document source="https://math2.tamnd.com/spec/11-api-and-orchestrator.md">
# 11. The Mathematical API and the agent orchestrator

In the diagram, one box is between the Research Workspace and the Research Agents: "Mathematical API and Agent Orchestrator". This box has two roles in one process:

- the **Mathematical API** is the only way for anything outside the build to read the knowledge graph or to ask for work. This includes a person's browser, an agent and an external tool;
- the **Orchestrator** changes requests for work into agent runs. It schedules the runs, enforces budgets and records everything. It changes the results into pull requests.

Both are in `tamnd/math2-agent` (`api`, `mcp`, `orchestrator`). `m2 serve` and `m2 orchestrate` serve them.

## 11.1 Principles

1. **Read is free, write is a pull request.** The API serves reads from the derived store of a release, or of `main`. Every write becomes a branch and a pull request against `tamnd/math2`. This is true for a write from a person in the workspace and for a write from an agent. The API has no other write path. Thus the audit (document 15) is the only gate, and git is the only history.
2. **One surface, three encodings.** The API uses HTTP+JSON for browsers and scripts, gRPC for the agents' tool calls inside the cluster, and MCP for any model client. One interface definition generates all three: `api/m2.proto` with HTTP annotations. Thus the three encodings cannot drift apart.
3. **Every response says what it is about.** It carries the release or commit that served it, the tag's content hash and the evidence summary. An agent that reasons from an API response can cite exactly what it saw.
4. **The API never returns a status it did not compute.** It serves the status that the build computed (document 04.7). It does not pass through any statement about truth from an external system.

## 11.2 The read surface

The methods are in groups by purpose. Each method is also an `m2` subcommand and an MCP tool with the same name.

### Resolve and fetch

| Method | Input | Output |
|---|---|---|
| `resolve` | any identifier: tag, alias, `m2://` URI, Lean name, QID, A-number, Stacks tag, arXiv result tag, DOI | tag(s) with match type, or "not held" plus the external system |
| `get` | tag[@release] | node: front matter, body, computed status, evidence summary |
| `get_formal` | tag | bindings, with the elaborated statement pretty-printed, axioms, replay result |
| `get_source` | tag | sources with locators and aligned paper results |
| `history` | tag | revisions with provenance, binding changes, review history |

### Navigate

| Method | Input | Output |
|---|---|---|
| `neighbours` | tag, edge types, depth ≤ 3 | typed subgraph |
| `closure` | tag, strategy (min-depth / min-size), stop set | AND–OR closure with trust profile (05.5) |
| `rdeps` | tag | everything whose every proof depends on it |
| `path` | from, to, edge filter | prerequisite path (05.6 Q5) |
| `frontier` | topic, kind, evidence filter | ranked frontier lists (Q6, Q7) |
| `topic` | MSC code or topic tag | the topic page data: members by kind and level, coverage |
| `diff` | topic, release₁, release₂ | changes (Q10) |

### Search

| Method | Input | Output |
|---|---|---|
| `search` | free text (natural language and/or LaTeX) | ranked nodes with snippets, scores and channel attribution (document 14) |
| `search_formal` | a Lean type pattern, or a Lean goal | bound nodes and Mathlib declarations, Loogle-style (14.3) |
| `search_formula` | LaTeX formula | formula-structure matches (14.2) |
| `similar` | tag | nearest nodes in embedding space, with the judge label when one is cached |
| `premises` | Lean goal state + file context | ranked candidate lemmas for a proof step (14.5) |

### Check and compute

These methods do work, but they do not write content. They return their results. The results are also cached by hash.

| Method | Input | Output |
|---|---|---|
| `lean_check` | Lean source snippet + imports at the pinned toolchain | elaboration result, messages, goals, axioms |
| `lean_goal` | snippet + position | goal state (used by provers via REPL sessions, 11.3) |
| `fidelity_check` | tag + candidate Lean statement | the F1 battery: back-translation, degenerate checks, critics (document 10.4) |
| `compute` | computation spec (image, entry, inputs) | output, hash, certificate check result (document 08) |
| `counterexample` | Lean statement or informal claim + domain hints | SMT/SAT/brute-force search result |

### Propose (writes, via pull request)

| Method | Effect |
|---|---|
| `propose_node` | open a PR with a new node (provisional id) |
| `propose_edit` | open a PR editing allowed fields of a node |
| `propose_binding` | open a PR adding a formal binding plus its Lean code |
| `propose_alignment` | open a PR adding `same_as`/`close_to`/`aligned` edges |
| `request_review` | file a review task for a binding (document 10.6) |
| `request_work` | file an orchestrator job (11.4) |

People use the same propose methods from the workspace. The PR that results is attributed to the person, and the workspace is named as the tool.

## 11.3 Lean sessions

A proof is interactive work: try a tactic, look at the new goals, then try a different tactic. One HTTP call for each attempt with a full re-elaboration is much too slow. Thus the API gives **Lean sessions**:

- A session is a Lean REPL process (the community `repl`, or Pantograph-style goal-level access). It starts from the pinned toolchain with Mathlib and `Math2` imported. The process is snapshotted after the imports. Thus a session costs milliseconds to start, not the 10 to 30 s that it takes to load Mathlib.
- A session holds a proof state tree. The calls are `run_tactic(state_id, tactic) → new states | error`, `goal(state_id)`, `branch(state_id)` and `close()`.
- Sessions run in the sandbox (document 08.6), with a memory cap and a timeout for each call. When a tactic times out, only its own call stops.
- Each runner has a session pool of a set size. The orchestrator charges session-seconds to the budget of the run.

The `lean-lsp-mcp` and Kimina-server style tools show the same interface (document 01). math2 uses its own interface because it must pin it to the release toolchain, sandbox it and meter it. Because of the MCP surface, any agent that can use MCP can prove things against math2 directly. This includes Claude Code with no math2-specific code.

## 11.4 The orchestrator: jobs

A **job** is a unit of work with a goal, a target and a budget. Jobs come from four places:

1. **People**, through the workspace or `request_work` ("formalise the statement of 7K3QX2", "find a counterexample to this").
2. **Standing queues** that the orchestrator computes from the graph: the frontier lists (Q6, Q7, Q12). Examples: each E3 statement with an F2 formal statement and no formal proof becomes a `prove` job. Each node with no explanation at L2 becomes an `explain` job. Each unconfirmed alignment older than 30 days becomes a `align` job.
3. **Events**: a Mathlib bump breaks bindings (→ `bump` jobs), a review becomes stale (→ `review-assist` jobs), a new arXiv batch arrives (→ `extract` and `align` jobs).
4. **Campaigns**: a project that a person defines, with a target set. Examples are "the undergraduate core list to F2" and "the Erdős problems in additive combinatorics to formal statements". A campaign is a topic node plus a goal predicate. Human research projects use the agents through campaigns (document 16.3).

Each job has this record:

```yaml
job: j-2026-11-02-41a7
role: prove
target: 7K3QX2
goal: "binding formal[0] reaches E5"
priority: 0.73            # computed (11.5)
budget: {usd: 4.00, wall: 2h, lean_session_s: 3600, compute_cpu_s: 7200}
on_behalf_of: campaign:ug-core   # or a person, or schedule:standing-prove
requested_by: orcid:0000-0002-1825-0097
constraints: {models: [claude-opus-5-5, claude-sonnet-5-5]}   # native_decide: true permits native_decide, and type_change: true permits a bump rebind to another type hash
```

### Priority

Priority is a score, not a decision by a person. Its formula has versions, as a prompt has:

```
priority = w1 · centrality(target)          # PageRank on the uses graph, so foundational nodes first
         + w2 · demand(target)              # workspace views, citations, campaign membership
         + w3 · unlock(target)              # how many frontier items this would move a level
         + w4 · tractability(target, role)  # learned: predicted success from past runs on similar targets
         − w5 · attempts(target, role)      # diminishing returns after failed attempts
```

`unlock` is the most important term. For example, 200 E1 statements use a lemma, and each one needs only that lemma. A proof of that lemma has more value than a proof of any one of the 200 statements. The closure structure in document 05.5 lets the orchestrator compute this.

### Scheduling

- Jobs go into one queue for each role. The queues share fairly across campaigns, so that one project cannot use all of the budget.
- **Portfolio scheduling.** A hard target gets several cheap attempts with different strategies before one expensive attempt. The strategies use different models, different retrieval and different decompositions. This is the method that worked in the provers of 2025 to 2026 (document 01): many different samples, with a verifier as the filter, are better than one long attempt.
- **Escalation ladder.** A job starts at the cheapest configuration that has some recorded success on similar targets. If the job fails, it moves up the ladder: Haiku 5.5 or a small open prover, then Sonnet 5.5, then Opus 5.5 at higher effort, then Fable 5.1 at high or max effort. Each role has its own ladder. The run records tune the ladder (document 12.9).
- **Deduplication.** If the target and goal of a job match a running job, the orchestrator merges the new job into the running job. Sometimes a job's target had an attempt in the last N days with the same configuration, and the attempt failed. The orchestrator skips that job until something changes in the neighbourhood of the target.

## 11.5 Budgets

Money and compute are the real limits, so they are first-class:

- **Global:** a monthly budget, divided by role in `orchestrator.yaml`. People review this file each month and commit it.
- **Per campaign:** the owner of the campaign sets it. The global allocation is its cap.
- **Per job:** the priority tier sets it, and the orchestrator enforces it. Model calls carry token budgets. These are the API's task budgets where the model supports them, and hard `max_tokens` in all cases. The sandbox meters Lean time and compute time.
- **Review capacity:** money is not the only limit. Each role has a cap on PRs per week. The cap is a multiple of the review throughput for that kind of PR over the previous four weeks. Thus agents cannot go faster than the people who must check their work (document 21, Q-11).
- **Stopping:** a job stops at success, when the budget is used up, or when the agent reports that it is stuck and gives a reason. A stuck report with a reason is valuable output. "This statement appears false at n=7, here is the computation" is a result.

Each job records its spend: tokens in and out for each model, cached tokens, session-seconds, CPU-seconds and dollars. The weekly agent report (document 12.9) uses these records. The report shows what a formal proof costs, by area and by difficulty. At this time, the field does not publish this number.

### The first version

The first version of the orchestrator is in the `orchestrator` package, and `m2 orchestrate` runs it on a directory of job files (tamnd/math2-agent#61). It has these parts and limits:

- One queue for each role. The orchestrator takes the roles in turn. In a role, it takes first the project or schedule that spent the least in the month, then the job with the highest priority. The priority is a field of the job, and the formula of the priority is not written yet.
- The monthly budget of each role and the monthly cap of each project or schedule are in `orchestrator.yaml`. A run gets the least of the money that is left to its job, to its role and to its project or schedule, and the orchestrator reserves that money while the run goes. The job budget is for all the runs of the job together. `agent.Runner` sets `max_tokens` before each turn so that the turn cannot cost more than the money that is left: it counts each byte of the request as one input token at the highest input price. Thus a run never goes over its budget, even when its last turn uses each token that it may use.
- A job starts on the cheapest rung of the ladder of its role with a recorded success on the role, and each failed run moves the job one rung up. A failed attempt on the same target, goal and rung in the retry time (7 days by default) makes the orchestrator skip that rung. A job with the target and the goal of a job in the queues merges into that job.
- A job ends at success, at a stuck report, when the ladder or the budget of the job is used up, or when a run cannot start. The orchestrator knows the past attempts from the run records in the store.
- Add refuses a job that names no person, project or schedule (rule P03).
- With `-sandbox IMAGE`, each call of lake, Lean and M2Extract in a run is a container of the image, with no network (11.7). The work tree is writable at `/out`, and the Lake packages of the tree are read-only at `/work`. `m2 agent run` and `m2 orchestrate` refuse to run with no image, unless `-no-sandbox` says that a person watches the run. The image `m2-lean` is not built yet, and the quotas for each identity are not written yet.
- The standing queues, the events, the portfolio of strategies and the cap on pull requests of each role are not written yet.

## 11.6 Run records

Each agent run makes a run record in the artefact store. The run id is the key. The record contains:

- the job record and the resolved configuration: role version, prompt version, model ids, tool versions, image digests;
- the full transcript: every model request and response (thinking blocks as the API returned them), every tool call and result;
- the tactic tree of every Lean session;
- the outcome: success with the diff, failure with the reason, or a partial result;
- the spend.

Run records complete the provenance chain in document 06.5. They are also the training and evaluation data that improve the agents (document 17.4). The system keeps them as document 06.6 specifies. Their summaries are public. When the output of a run was merged, the full transcript is public. When the output was not merged, the system keeps the full transcript but does not publish it. By default, math2 does not publish a failed attempt, but it uses the attempt to learn.

## 11.7 Security and abuse

- Read endpoints are public and have rate limits. Write endpoints need a GitHub identity, and the PR opens under that identity.
- `lean_check`, `compute` and Lean sessions run arbitrary code: Lean metaprograms can do arbitrary IO at elaboration time. Thus they run in the sandbox, with no network, a read-only toolchain and quotas for each identity. Anonymous use is permitted only for `lean_check` on small snippets, with a strict timeout.
- MCP tools that write have a mark that says so. The read-only MCP profile disables them. That profile is the default for third-party agent clients.
- Text from sources (paper text, MathOverflow posts) is data. It goes into agent contexts with an untrusted mark. It is never put in a system prompt, and it is never concatenated into a tool description (document 12.7).

## 11.8 The first version

The first version of the read surface is in the `api` package, and `m2 serve` serves it. It has these limits:

- It serves the last build of one working tree, not a release. A request for `tag@release` gives an error.
- `get_source`, `history`, `topic` and `diff` come with the provenance of document 06 and the topic pages of document 13. The check and compute methods, `request_review` and `request_work` come with the orchestrator.
- The one interface definition is the table of methods in the `api` package. The HTTP handler and the MCP server read the table. `api.Proto` makes `api/m2.proto` from it, with the HTTP annotations, and `buf` makes the Go code in `api/m2pb` from that file (tamnd/math2-agent#126). A test checks that `api/m2.proto` is the file that the table makes, a test checks that the code in `api/m2pb` agrees with the table, and a test checks that each gRPC call of the contract gives the HTTP answer. Thus the three encodings cannot drift apart. The gRPC service is `m2.v1.ReadService`, and each rpc gives one `Response` message with the result as a `google.protobuf.Value`, as each HTTP call gives the same JSON envelope. `m2 serve -grpc ADDR` serves it with the same rate limit for each IP address.
- The rate limit is a token bucket for each IP address of a peer, in memory. Behind a proxy or a CDN, the handler must read the address that the proxy sets.
- The MCP server is in the `mcp` package. It has the read-only profile only, with one tool for each read method, and `m2 serve` serves it over streamable HTTP at `/mcp` and over stdio. It uses the Go SDK of the Model Context Protocol.
- `similar` uses the embedding channel with the text of the node, and it has no judge label yet.
- The write methods `propose_node`, `propose_edit`, `propose_binding` and `propose_alignment` are in the `propose` package, and `m2 serve -propose` serves them over HTTP only (tamnd/math2-agent#54). The gRPC service and the MCP server do not have them yet, thus the MCP server keeps the read-only profile of 11.7. Each write needs the GitHub token of a person, opens the pull request under that identity, and passes the audit of the changed files before the pull request opens. The cap of 11.5 counts the pull requests of each method from their branch names, with twice the closed pull requests of a week over four weeks and a floor of 10. `propose_binding` adds the binding to the node file, and it does not add Lean code yet. `request_review` and `request_work` come with the orchestrator.
</document>

<document source="https://math2.tamnd.com/spec/12-research-agents.md">
# 12. Research agents

The top box of the diagram says "Formalize, Prove, Explore, Explain". Readers see these four agent roles. Four more roles work behind them and keep the graph healthy: curate, align, bump and review-assist. All eight roles are in `tamnd/math2-agent/agent/`. They run under the orchestrator (document 11). They change content only through pull requests.

The design comes from one finding of 2026. Document 01 gives this finding with its sources. **A minimal agent loop on a strong general model, with a strict verifier, matches or beats bespoke provers.** A plain agent on a general model saturated PutnamBench for about $111. The baseline agent of AlphaProof Nexus replicated the Erdős results of AlphaProof Nexus. The Fermat formalization ran on a generic DAG-of-statements harness. Thus math2 does not build a prover. It builds the things that a prover needs and cannot supply itself:

- a pinned, sandboxed, metered Lean environment;
- retrieval over the whole graph;
- a dependency structure in which a lemma is a first-class node;
- a verifier that nobody can talk past;
- a record of everything that was tried.

Then math2 uses the model that does best on the evaluations in document 17.

## 12.1 Roles and what they may touch

Each role has a bot account, a tool profile and a **field permission set**. Rule `X01` checks each agent PR against that set. Any change outside the set fails the audit, even if the change is correct.

| Role | Does | May create | May edit | Auto-merge |
|---|---|---|---|---|
| **formalize** | writes Lean statements for nodes, runs the F1 battery (10.4) | bindings at F0; `Math2/` statement declarations; `Conjectures/` entries | its own unmerged bindings | never |
| **prove** | finds formal proofs of bound statements | proof nodes; lemma nodes (statement plus binding); `Math2/` proofs | `formal[].decl` of nodes it proves | yes, if the target binding is already F2 or above and the new lemmas are internal to the proof (10.6) |
| **explore** | experiments, conjectures, counterexample searches | computation nodes; conjecture and counterexample nodes, which are always E1 or E4 by construction | none | never |
| **explain** | writes levelled explanations, examples and glosses | explanation sections marked `by: agent`; example nodes | its own unconfirmed explanations | never |
| **curate** | turns extracted paper results into nodes, proposes dedup merges, fixes graph health findings | statement and concept nodes from record plane results; `same` proposals | `uses`, `about`, `msc`, names and aliases | never |
| **align** | proposes crosswalk edges | `same_as`, `close_to`, `aligned` edges | its own unconfirmed alignments | `same_as` to Mathlib, only when the type hash of the bound declaration is identical |
| **bump** | moves to new Lean and Mathlib versions | the bump branch; rebindings | `formal[].decl`, `formal[].commit` | renames taken from Mathlib deprecation data |
| **review-assist** | prepares review packets for people | review packets in `reports/review/` (not content) | nothing in `nodes/` or `reviews/` | not applicable |

No role may write a status, a review, a register entry or a retirement (rules `S03`, `X04`, `V03`). An agent can give the reasons to retire a node, but only a curator can retire it.

A run writes only in its own work tree, through the `workspace` package. Its tools read and write files by their paths in the tree, and they cannot reach a file outside it. `agent.Permit` checks the changes of the run against the set of its role before the run opens a pull request. The sets of the eight roles are written. Rule `X01` uses the same check on the merge of each agent PR, thus a PR of a role with no written set fails.

The first version of curate is in `agent/curate`, with the role in `prompts/curate/`. Its target is a node, or a record of the record plane (09.3) in the form `source:id`, for example `arxiv_results:2312.00752#3KH2`. For a record of kind statement or definition, the run adds one statement or concept node in its own words. The model reads the record only with the tool `record`, which gives it in a data block (12.7). The goal holds when `m2 check` finds nothing in the changed nodes, the new node names the locator of the record in its `sources`, its kind matches the record, its body does not copy the text of the record by the thresholds of rule `L04`, and a judge finds that the node states the same result. The judge is one call of the model of the rung with the record and the node as data and a tool `judge`, which gives same, generalizes, specializes, related or different with the reason. Only same passes, and the other verdicts go back to the model. For a node target, the run can change the fields `uses`, `about`, `msc`, `name` and `aliases` of any node, and it can add a duplicate proposal to `queues/dedup.jsonl` with the tool `propose_same`. The set of the role allows no other change of a node that exists, no Lean, and no new node with `formal`, a crosswalk edge, a retirement or an explanation. The set keeps each proposal of the queue unchanged, and each new proposal must be by the role. Curate does not merge nodes. A person merges them from the queue.

The first version of align is in `agent/align`, with the role in `prompts/align/`. Its target is a node. The model reads the records of the other systems with the tool `record` and proposes edges with the tool `propose_edge`, which writes each proposal into the queue file of its system in `queues/align/`, as the aligner of 05.3 does. A person accepts or rejects each proposal, and the run writes no edge into a node. The tool refuses an edge that the node has, an edge that a person decided on, an `aligned` edge to a record that is not a paper, and an id that does not have the syntax of its system. A new proposal to a record replaces the proposal of the role to the same record. The set of the role keeps each proposal of another program unchanged, and the role adds, changes and removes only its own proposals from the target. The goal holds when the run made at least one proposal or binding, and the judge of curate finds the verdict same for each `same_as` proposal and a verdict that is not different for each `close_to` and `aligned` proposal. In the graph, a node meets a Mathlib declaration through `formalizes` and not through `same_as`. Thus the `same_as` to Mathlib of the table is a binding to Mathlib that align appends to the `formal` field of the target. The set allows such a binding with no fidelity reviews, and the goal allows it only when the type hash of the declaration in the Mathlib records (09.2) is the type hash of a Math2 binding of the node in the build, with the same match. `align.AutoMerge` reports a run whose changes are only such bindings. It is the only change of align that can merge with no review.

The first version of bump is in `agent/bump`, with the role in `prompts/bump/`. It comes after `m2 bump` (07.9), which applies each rename that the deprecation data or the type hashes give. The target of a bump job is a node with a Lean binding that does not resolve or is deprecated with the new pins, for example a lost declaration of the bump report. The model reads the declarations with `search_formal` and `lean_check`, and it changes `formal[].decl` of the binding with the tool `rebind`. A binding has no commit field in schema node.v1, because the commit is the pin in `lean/lake-manifest.json`. Thus the `formal[].commit` of the table is the pin, and the role changes only the decl. The set of the role allows a change of `formal[].decl` of the target and nothing else, and the role writes no Lean. Thus a break of the build of the Math2 library is not a job of bump. The goal holds when the run changed at least one binding, each changed binding did not resolve or was deprecated before the run, the new declaration has the kind and the type hash of an old declaration that resolves, and each Lean binding of the target resolves and is not deprecated. A rebind to another type hash changes the statement, thus it needs the constraint `type_change` of the job, or the flag `-type-change` of `m2 agent run` (17.3.4). The Lean environment of the run imports Mathlib and Math2, because a binding can name any declaration of Mathlib. `bump.AutoMerge` reports a run whose rebinds all take the replacement that the deprecation of the old declaration names and keep its type hash. Such a run can merge with no review, as the renames of 07.9 step 3. A person reviews each other rebind. No role writes a register, thus the person who merges a rebind records it in `registers/bindings.reg`.

The first version of review-assist is in `agent/reviewassist`, with the role in `prompts/review-assist/`. Its target is a node with a Lean binding, for example a node of the review queue. The brief has the review packet of the node from the last build, as `m2 review -blind` gives it: the informal statement, each binding with its Lean statement, the constants with the nodes that bind them, and the results of the battery. The model is the critic of 10.4. It reads each Lean statement against the informal statement, and it writes the packet with its concerns with the tool `write_packet`. Each concern names a binding of the packet, a class of 10.5 and a reason, and a packet has at most 10 concerns. The tool writes `reports/review/TAG.md`, which has the packet text of the build and the concerns, and it is the only file that the set of the role allows. The packet has no reviews, thus a blind re-review (10.8) can use it. The goal holds when the file has the packet of the build and each concern is valid. Review-assist never gives a verdict and never signs, and a person reviews with `m2 review`.

## 12.2 The common loop

All roles run the same loop. Only the tools, the prompts and the stop condition are different.

```
job ← orchestrator
context ← API: the target node, its neighbourhood (depth ≤ 2), its sources, its formal status,
           prior runs on this target (summaries), the role's prompt at its pinned version
loop:
    model turn (tools available per role profile)
    tool calls → API / Lean session / compute sandbox      # every result cached by hash
    check budget; check for stuck (no progress in k turns)
until: goal predicate holds | budget exhausted | agent reports stuck with reason
diff ← workspace changes
validate diff: m2 check, role field permissions, local audit on touched shards
open PR (agent/<role>/<run-id>) with run summary | or record failure with reason
```

These decisions are intentional:

- **Tools, not bash.** The agent sees typed tools: `get`, `search`, `premises`, `lean_*`, `compute`, `propose_*`. It does not get a shell. In 2026, agents used a shell to find and exploit kernel and harness bugs (document 01). The tool surface is the MCP surface of document 11. Thus an external MCP client with the same permissions can do all that a math2 agent can do.
  The package `agent/tools` has the first of these tools. `tools.API` makes a tool from each read method, with the names and the parameters of the MCP server. `read_file` and `write_file` work in the work tree of the run, and a write that the role may not make fails and leaves the file as it was. `lean_check` elaborates a Lean file of the tree, or a scratch file, and its time counts in the Lean session seconds of the budget.
- **One target, one closure.** The context comes from the graph neighbourhood of the target, not from a general search over everything. Thus runs are reproducible and cheap to cache.
- **The verifier's verdict is the only verdict.** When an agent claims success, it must produce a diff that passes the audit. The system records the agent's own assessment and never uses it.
- **Failures with reasons are outputs.** "Statement false at n=7, witness attached" closes a prove job and attaches a counterexample job. "Needs a lemma that is not in Mathlib: X" creates a node for X.

The `agent` package runs this loop. A turn makes progress when it calls a tool with an input that the run did not call before, and the run caches the result of each call by the tool and the input. The tool `report_stuck` is in each profile, and the model calls it with the reason. The goal predicate is a function of the role that reads the tree and the build, never the text of the model. Each model turn has a limit of output tokens from the role and from the budget that is left after the most that the input of the turn can cost. When the goal does not hold, the predicate also gives the reason in one sentence, and the next message to the model starts with this reason. `agent.OnChanges` makes a goal that runs its check only when the changes of the work tree are different from the changes of the last check, because a check can build the library. `tools.Profile` makes the tools of a role from the tool names in its `role.yaml`, and a name with no tool is an error.

### Statement pinning

In 2026, the worst failure mode was a correct proof of a different statement. The causes were edited context, redefined notions, a weakened hypothesis or an exploited checker. Thus each role that writes Lean works against a **pinned statement**:

- A statement with no proof is a definition of a proposition, `def S : Prop`, because rule `F06` refuses `sorry` (07.2). When the job starts, the pin records `S` and the hash of each declaration of math2 in its closure. For a definition, this is the hash of the value. For a theorem, it is the hash of the type.
- The proof is accepted only if it is a theorem of type `S`, its axioms are a subset of the allowed set (rule `F03`), and each hash of the closure is unchanged.
- The agent's Lean files must not modify, shadow or re-declare anything in the closure of the target. The check runs on the elaborated environment, not on the source text.
- Before merge, a Comparator-style check confirms that the statement is identical to the pinned one. This check builds against Mathlib and `Math2` only. Then the external checkers replay the proof (07.7).

## 12.3 Models

The model is a configuration value, never a dependency in code. Each run records the exact model ids that it called. The `llm` package has one interface. Its first client is the Anthropic client. It has adapters for local open-weight prover models served behind gRPC (document 20.2). The client of the Claude models is in `llm/claude`, on the official Go SDK. It streams each request, caches the system prompt, the tools and the prefix of the conversation, and keeps the thinking blocks as the service returned them. `llm/llmtest` has a scripted model, thus the tests of the roles call no model service. `llm.Cost` gives the price of a usage from a table of prices, and `llm.Prices` is the table above.

| Tier | Model | Price in/out per MTok | Used for |
|---|---|---|---|
| 0 | `claude-haiku-5-5` | $0.10 / $0.50 | triage, classification, dedup judge pre-filter, first cheap proof attempts, the F1 critics' easy checks |
| 0′ | open prover models (e.g. Leanstral, DeepSeek-Prover, Goedel-Prover families) | self-hosted GPU | high-sample-count tactic and whole-proof attempts in the portfolio |
| 1 | `claude-sonnet-5-5` | $2 / $10 | bulk: curate, align, explain L1–L2, routine formalization, most prove attempts |
| 2 | `claude-opus-5-5` | $4 / $20 | default for hard roles: research-level formalization, decomposition planning, explain L3–L4, explore |
| 3 | `claude-fable-5-1` | $10 / $50 | the top of the escalation ladder: hardest proofs, cross-checking research-level statements, explore campaigns on open problems |

How math2 uses the API:

- **Adaptive thinking** on each role that does much reasoning. The **effort** setting increases up the escalation ladder: medium by default, then high, then max at the top. math2 never sets a fixed thinking-token budget. Current models reject those parameters. Thus the token budget of the run is in the job (11.5). The API's task budgets enforce it where they are supported, and `max_tokens` enforces it in all cases.
- **Prompt caching** for the parts of the context that all runs of a role share: the system prompt, the tool definitions, the role's style guide and the Mathlib conventions. There are thousands of runs each day, so this is the largest single cost saving.
- **Message Batches** at half price for work that no person waits for: curate, align, the dedup judge, explanation backfill and back-translation in the F1 battery.
- **Compaction** for long prove and explore runs. Thus a run can go past one context window and keep its record. The run record keeps the full transcript without compaction.
- **The orchestrator never forces a tool choice.** The current models reject a forced `tool_choice`. The roles are designed so that the model selects its tools.
- **Managed Agents** is an option for the roles that need only the read API and MCP (explain, curate, align). These roles run as hosted sessions connected to the MCP server of math2. Prove and formalize run in the loop of math2, because math2 must pin, sandbox and meter their Lean sessions (11.3).

The table is a starting point. The run records tune the **escalation ladder** of each role (11.4) each week. A role moves to a cheaper default when the acceptance rate of the cheaper tier on recent comparable targets is within a set margin of the more expensive tier. The report (12.9) publishes the acceptance and cost for each tier that justify each move.

Open-weight models are important for two reasons other than price. First, they let the portfolio draw thousands of different samples at marginal cost. Second, they keep the prove pipeline of math2 in operation if hosted access changes. In 2026, the gap between open and frontier models in formal proof became small. In research-level informal proof, the gap stayed large (document 01). math2 uses open models where they work and does not pretend that they work in other places.

## 12.4 Formalize and prove

Document 10 gives the full specification of the formalize role, because for that role the fidelity battery is more important than the agent. This section is about prove.

A **prove** job has a target node with a binding at F1 or above. Before F2, a proof is still useful because it tests the statement. But a proof of an unreviewed statement never auto-merges. The loop:

1. **Read the informal proofs.** If the node has a `proof` node, the agent works from it. If the node has only sources, the agent reads the aligned paper results (only text that the licence permits, 09.3). First, the model writes an informal proof sketch. The run record keeps this sketch.
2. **Decompose into the graph.** The agent writes the lemmas of the sketch as provisional nodes, each with a Lean statement. The largest formalizations of 2026 used this DAG-of-statements pattern (Prove2Me for Fermat, Gauss for sphere packing, the textbook project). There is one difference: the DAG of math2 is the knowledge graph itself. If a lemma is general, it becomes a reusable node. Dedup (05.4) finds lemmas that the graph already has. Before the agent proves a lemma, it searches for it with `search_formal` and `premises` over Mathlib and the graph.
3. **Cross-check the lemma statements.** A second, independent model call checks the statement of each lemma. The check asks three questions: is the lemma true, is it what the sketch needs, and is it degenerate? It uses the same degenerate checks as F1 (10.4). In the Fermat run, this check found false statements early. When a false lemma is found late, all of its subtree is wasted work.
4. **Prove leaves by portfolio.** Each leaf gets attempts with several strategies in parallel:
   - automation first: `simp`, `omega`, `decide`, `aesop`, `grind`, and the Lean hammer;
   - then sampled whole proofs from a cheap tier;
   - then tactic-level search in a Lean session with retrieval;
   - then an expensive tier.
   The first success wins.
5. **Recurse on failure.** If a leaf resists, the agent decomposes it more or sends it back up with a reason. If the reason is "this lemma looks false", the counterexample tools of the explore role try to refute it.
6. **Assemble and check.** The proof must build against the pinned toolchain with no `sorry`. Its axioms must be in the allowed set, and the type hash of the target must not change. The external replay and the comparator check must pass. Only then does the PR open.
7. **Clean up.** The system records the proof size and the build time. A proof is marked `needs-golf` if it is more than 10× the size of comparable Mathlib proofs, or if it takes longer than a set time to elaborate. It gets a follow-up job to make it shorter. The main complaint about the landmark formalizations of 2026 was AI-written proofs that work but are not maintainable. math2 must maintain its proofs through each Mathlib bump.

The packages `agent/prove` and `agent/formalize` have the goal and the first message of each role. These are the first parts of the loop above. A prove run writes the theorem `T : S`, moves the attribute `@[m2 "TAG"]` from `S` to `T`, and changes the `decl` of the binding to `T`. Its goal holds when the changes are in the field permissions of prove, the library builds, and the pin accepts `T`. A formalize run adds Lean files and appends bindings. Its goal holds when the changes are in the field permissions of formalize, the library builds, each new declaration has the right kind for the node, it uses only the standard axioms, and the degenerate checks of 10.4 do not find a problem. The decomposition, the cross-check and the portfolio of steps 2 to 4 are not written yet.

**What prove does not do.** It does not prove anything with `native_decide` unless the job permits it (rule `F04` records it in all cases). It does not add axioms. It does not edit the statement of the target. If the statement is wrong, it says so and stops.

## 12.5 Explore

Explore is the role that is nearest to research, and it needs the most restraint. It does experiments, finds patterns, searches for constructions and counterexamples, and proposes conjectures. Its tools are the computational engine (document 08), the SMT and SAT search tools, and evolutionary program search for constructions, in the AlphaEvolve and PatternBoost style (document 01). In this search, the model writes programs that generate candidates, a scorer ranks them, and the best programs are mutated.

Explore obeys four rules. Each rule comes from a failure in 2025 to 2026:

1. **Falsify before filing.** When the agent finds a pattern up to n, it checks the pattern to a larger n. It also checks the pattern against OEIS, LMFDB and the graph. Only then does the pattern become a conjecture node. Most patterns fail here, and that is the purpose of the rule.
2. **Look up before claiming.** No explore output gets the label new until the novelty check (17.5) has run. The check searches the record plane, zbMATH Open, arXiv full text where the licence permits, and MathOverflow. The search and its result are attached to the node. Without this step, an AI result is a rediscovery until someone shows that it is not. The Erdős episode of 2025 and the Aletheia sweep showed that this is the usual case. In the Aletheia sweep, 9 of 13 addressed problems already had solutions in the literature, and only 6.5% of graded candidates were meaningfully correct.
3. **Verifiable or E1.** If a construction or counterexample can be checked (a witness, a certificate), the agent files it with its check, and it can get to E4 or E5. An informal argument stays at E1, however convincing it is. The workspace shows it as a claim.
4. **People decide what is interesting.** Explore runs inside a project or campaign that a person defined. It does not select open problems for itself. This is a policy, not a technical limit. In 2026, there was a debate about mass "harvesting" of open problems by AI (document 01). math2 should not decide this debate by default.

The first version of explore is in `agent/explore`, with the role in `prompts/explore/`. It uses the computational engine. The SMT and SAT tools and the program search are not written yet. A job of explore must be on behalf of a project (rule 4). The set of the role lets it add new nodes of kind `computation`, `conjecture` and `counterexample` with no curated fields, and new files in `compute/<id>/`, where the id is the tag or the provisional id of a computation. It changes no file that is in the tree. The tool `run_computation` runs a computation of the work tree in the sandbox and gives the exit code, the output hash, the certificate hash and the output. The tool `lookup` runs the novelty check of 17.5 on a claim. The goal checks these points in order: the changes are in the set, the new nodes pass the checks of curate, the run filed a conjecture or a counterexample, each program has a computation node, no output calls itself new, each output has the evidence of a computation that is new or in the tree, and each new computation verifies when it runs again (rule 3). A counterexample cites the computation in the `via` of `refutes`, and a conjecture links to it or uses it. The checks of rule 1 against a larger n are the work of the model, and the goal sees only the computations that it filed. Then the goal runs the novelty check on each output (rule 2), with the judge of the rung, and the run record keeps the result in `novelty`. A search that fails makes the goal fail, because then the check is not complete. The levels E1 and E4 come from the build, and the run never writes them.

## 12.6 Explain

Explain writes the explanations that make the graph readable. It writes the four levels of document 04 into nodes, with examples and non-examples. It also answers questions in the "Ask about this node" panel of the workspace (13.4).

- **Grounded.** Each factual claim in an explanation must cite a node by tag. A grounding check runs after the text is generated. A second model call takes each sentence and the cited node, and gives the sentence one of three labels: supported, unsupported or contradicted. Unsupported sentences are removed or marked. Contradicted sentences fail the run.
- **Levelled.** The level is a contract about prerequisites. An L1 explanation may use only nodes in the curriculum topic for its level (`topics/curricula/`). The check is mechanical: each cited or linked node must be in the allowed set.
- **Marked.** Explanations are `by: agent` until a person confirms them (rule `P04`). To confirm is to read and approve in `m2 review`. The system records the person who confirms.
- **Not a proof.** An explanation can describe a proof. It is never a replacement for the proof node.

The first version of explain is in `agent/explain`, with the role in `prompts/explain/`. It writes the explanations of one node, and it does not answer questions yet. The set of the role lets it change only the fields `body.explanations.L1` to `body.explanations.L4` of the target, and only when each heading of the old and the new text is marked agent. It can add a node of kind `example` with no formal binding. The goal finds the levels in the goal of the job, and each one must be written. A field whose provenance log has a line of a person is not changed. Each explanation must cite at least one node, and each cited node must be in the tree. L1 maps to the courses of year 1 in `topics/curricula/`, and L2 to the courses of years 1 and 2. A level with no nodes in the curricula is not checked. The grounding check is one call of the model of the rung with the cited nodes as data and a tool `label`. `agent.Meter` gives the call the money that is left to the run, and adds its spend to the run record. A sentence with no label is unsupported. The unsupported sentences go back to the model with the reasons, and a contradicted sentence fails the run.

## 12.7 Untrusted text and prompt injection

Agents read text that math2 did not write: paper text, MathOverflow posts, Wikipedia, OEIS comments and docstrings. Any of this text can contain instructions. The defences are in the structure, not in advice:

- **Separation.** Source text goes only inside tool results, as data blocks marked with their source and licence. It is never put in a system prompt, a tool description or the instructions of a role.
- **Least privilege.** The tool profile of each role is the minimum that it needs. Explain cannot write Lean. Curate cannot run compute. No role has network access other than the math2 API and the model API.
- **Narrow outputs.** The output is a diff that must pass `X01` (field permissions) and the audit. An injected instruction can make an agent edit something outside its role. The result is a failed run, not a merged change.
- **Canaries.** The evaluation suite (document 17) has source documents with planted instructions. If a prompt or model version obeys them, it fails its release gate.
- **Quarantine.** If someone finds an injection in the wild, the run records identify each run that read the poisoned source. One command quarantines all of these runs (06.5).

## 12.8 Versioning roles and prompts

The behaviour of a role depends on its prompts, its tool profile, its model configuration and its code. These four have one version together, the **role version** (semver), in `prompts/<role>/`:

```
prompts/prove/
  role.yaml          version: 3.4.0, tools, model ladder, effort per tier, budgets
  system.md          the system prompt
  style/             Mathlib style conventions given to the model
  examples/          worked examples (each a real merged run, referenced by run id)
  CHANGELOG.md
```

- The package `prompts` embeds the role directories, and `agent.LoadRole` reads one. The first versions are formalize 0.1.0 and prove 0.1.0. They have no examples yet, because there is no merged run yet.
- Each run records the role version and the hashes of each prompt file that it loaded.
- A change to any file bumps the version. `prompts/roles.sum` has the version and the digest of the files of each role, and the tests of the `prompts` package fail when the files change and the version does not go up. A minor bump must pass the evaluation suite of the role (17.3), with results no worse than the previous version. A major bump also needs a shadow week. In that week, the new version runs with the old one on the same jobs, and the system uses only the output of the old one.
- In the first version, `shadow` in `orchestrator.yaml` names the directory of the candidate of a role. After each run of a job of the role, the orchestrator runs the candidate on the same job, rung and budget, in a work tree that it always removes. The record of the shadow run is marked, and it has no summary in `runs/`, thus no audit rule, disclosure, ladder or job budget counts it. Its spend counts in the monthly budget of the role. The weekly report (12.9) compares the candidate with the current version on the same jobs.
- Provenance can be queried by role version. Thus "everything prove 3.2.x produced" is one command, and quarantine can use it as a target (06.5).

## 12.9 The weekly agent report

Each Monday, the orchestrator writes `reports/agents/<yyyy>-W<ww>.md` and its JSON twin. The report is public. People use it to decide if the agents are worth their budget:

- **Volume:** jobs started, finished, succeeded, failed and stuck, per role.
- **Acceptance:** PRs opened, merged, closed or changed before merge, per role and model tier. "Changed before merge" counts the fixes that a person made. This count tells more about quality than the merge count does.
- **Cost:** dollars per merged output per role, by area (MSC) and by tier, and the trend. One example is the cost of an E5 proof of a lemma at a given depth in a given area. Nobody else publishes this number, and math2 can.
- **Audit sample:** the 5% sample of auto-merged PRs (16.2), with the errors found and the error rate of the role against the 1% threshold.
- **Incidents:** quarantines, injection detections, reward-hacking attempts that statement pinning caught (with the run ids), and stale reviews that bumps caused.
- **Ladder changes:** the tier and effort changes made that week to the escalation ladder of each role, and the measurements behind them.

The first version is `m2 agent report` and the package `agent/report`. It counts the runs that started in the week, from the run records in the store. A run is accepted when a provenance line on the branch names it, as for rule X03, thus a later report of the same week can count more. The report has the volume, the acceptance by role and model, the cost by role, model and MSC area with the trend from the week before, and the incidents: the quarantines, the runs in which the pinned statement caught a change to its closure, and the runs over budget. A role in its shadow week has its own table. The counts of pull requests, the audit sample, the injection detections, the stale reviews and the ladder changes need runs that open pull requests, thus the report lists them as not measured.
</document>

<document source="https://math2.tamnd.com/spec/13-research-workspace.md">
# 13. The research workspace: search, explore, explain, collaborate

The workspace is the bottom-left box of the diagram. People use math2 through it. It reads from the knowledge graph through the API. It writes back only through the orchestrator, as jobs and pull requests. There are three front ends, and each one is complete for its job:

- a **static site**, `m2 site`, published with each release. It is fast, people can cite it and archive it, and it runs no server code. It is for reading.
- a **live web app**, `m2 serve`, which adds search, Lean sessions, computations, collaboration and agent requests.
- a **terminal UI**, `m2 tui`, for people who work in a terminal, and for scripts. It uses the same navigation model as `ant tui`.

Everything that the web app shows is also available through the API and MCP (document 11). The workspace must not have a private capability.

## 13.1 The page model

Each node shows as a page. The page has a fixed structure, so that a reader learns it one time:

```
┌─────────────────────────────────────────────────────────────────────────┐
│ Lagrange's theorem                                   m2://7K3QX2@2026.11 │
│ statement · theorem · 20D60                                              │
│ Proved in Lean (Mathlib). Statement reviewed by 2 people. Classical.     │  ← evidence phrase (15.5)
├─────────────────────────────────────────────────────────────────────────┤
│ Statement                                                                │
│   Let G be a finite group and H a subgroup of G. Then |H| divides |G|.   │
│                                                                          │
│ [Formal] [Proofs (2)] [Uses] [Used by (1,204)] [Sources (37)] [History]  │  ← tabs, each a query
│                                                                          │
│ Explanations   L1 · L2 · L3 · L4                                         │
│ Examples       ℤ/6, S₃, … (executable)                                   │
│ Related        converse fails: A₄ (counterexample) · Sylow theorems · …  │
├─────────────────────────────────────────────────────────────────────────┤
│ Open work on this node:  explanation L1 missing · 1 alignment to confirm │  ← jobs, claimable
└─────────────────────────────────────────────────────────────────────────┘
```

- The **Formal** tab shows the bound Lean statement, pretty-printed and also as source. It has links to the Mathlib documentation at the pinned commit. It shows the axioms, the replay results, and the reviews with their hashes. If the binding is `stronger`, `weaker` or `variant`, the tab shows the adapter and explains the difference in words.
- The **Proofs** tab shows each proof node: an informal proof, formal or not, its method tags and its own dependency closure. A reader can compare proofs side by side by what they use ("proof 1 uses choice, proof 2 does not").
- **Uses** and **Used by** are the closure views (05.5). They can show as a list, a layered DAG, or a blueprint-style dependency graph coloured by evidence level.
- **Sources** lists each aligned paper result in date order. The earliest result has a mark. Each result has a link to `ax://` for arXiv content, or to the publisher, zbMATH or MathSciNet for records.
- **History** shows revisions with provenance, each binding change that a Mathlib rename caused, and each change in evidence level, with the reason.

## 13.2 Search

The search box accepts natural language, LaTeX, a Lean expression, an identifier, or a mix of these. Document 14 describes the engine. The workspace adds three things:

- **Facets** for kind, MSC, evidence level, fidelity, "has formal statement", "has formal proof", "open", "uses choice", source date range and method.
- **Explanation of results.** Each result shows the channel that matched it (lexical, formula, embedding or formal type) and the matched span. Thus a reader can see why the result is in the list.
- **Did-you-mean across systems.** Sometimes a query matches nothing in math2 but resolves in Wikidata, nLab, Mathlib or OEIS. Then the workspace shows "not held yet", with the external link and a one-click "request this node". This button files a curate job (11.4).

## 13.3 Explore

To explore is to move through the graph with a purpose. The workspace supports the moves that mathematicians actually make:

| Move | Interface |
|---|---|
| "What do I need to understand this?" | prerequisite path (Q5) from a chosen starting concept, or from the reader's declared background (13.5), as an ordered reading list |
| "What is this used for?" | reverse dependencies, grouped by topic, ranked by centrality |
| "What is the most general form?" | the `generalizes` chain upwards, and the hub family view |
| "What is known about this open problem?" | the problem page: conjecture, bounds over time as a chart, partial results, related computations, formal statement status, who is working on it (13.6) |
| "Where are the gaps?" | the frontier view for a topic: formalised-but-unproved, refereed-but-unformalised, open-with-formal-statement. This view is the shared to-do list for people and agents |
| "What changed?" | the topic diff between releases |
| "Show me the landscape" | a topic map: concepts as nodes, `specializes`/`about` edges, laid out hierarchically and coloured by coverage, with zoom from MSC sections to individual nodes |

The client draws the graph. It uses a WebGL renderer for large neighbourhoods. It never shows more than approximately 2,000 nodes at one time. Above that number, it changes to aggregation by topic, with counts. A picture of 50,000 nodes is decoration, not navigation.

## 13.4 Explain

The explanation agent (document 12.6) writes levelled explanations into nodes. The workspace adds explanation on demand:

- **Ask about this node.** Each page has a chat panel that answers questions. The context of the panel has the node, its neighbourhood and its formal statement. Each answer cites `m2://` tags. If a statement in the answer has no cited node to support it, the panel marks it unsupported. The panel uses the explanation agent with a read-only tool profile.
- **Unfold this proof.** For a formal proof, the reader can expand any step into its sub-steps, down to single Lean tactics and the lemmas that they used. Each one has a link back to its node. The informal gloss for each step is generated, cached, and marked as generated.
- **Translate notation.** When the pointer is on a symbol, the workspace shows its definition node. A reader can change notation conventions, such as $\subset$ versus $\subseteq$, or conventions for $\mathbb{N}$, where the concept records the convention.
- **Languages.** The workspace can show explanations and statement prose in the languages that `tamnd/arxiv` already supports (English, Vietnamese, Simplified and Traditional Chinese, Japanese). It uses the same glossary and translation pipeline. Translations have a mark that says they are translations. The English text (or the text in the original language) stays canonical. Formal statements are never translated, because they do not depend on a language.

## 13.5 The reader's profile

Anyone can read without an account. A reader who signs in (GitHub or ORCID) can declare a **background**: the set of concepts and statements that they already know. A curriculum topic can be the start of a background ("finished a first course in algebra"), and the reader can change it. Prerequisite paths and explanation levels then start from the background. The workspace stores the profile, not `tamnd/math2`. The profile is private, and the reader can export it or delete it.

## 13.6 Collaborate

Collaboration is the area where math2 is most different from a static library. Its design follows the three ways that mathematical collaboration actually works in 2026 (document 01): Polymath-style open projects, blueprint-driven formalisation projects (FLT, PFR, the Equational Theories Project), and small groups of humans who work with AI tools.

### Projects

A **project** is a campaign (11.4) with people attached. It has:

- a **goal**: a set of target nodes and a target evidence or fidelity level, or an open problem with "find a proof or a counterexample";
- a **blueprint**: the closure of the targets, computed live and coloured by status. The graph generates it, and nobody writes it by hand. This is the main improvement over `leanblueprint`. If a project already has a blueprint, `m2 ingest blueprint` imports it as nodes and edges (document 09.4);
- **tasks**: the frontier items inside the closure, each claimable;
- **members**: people and agent roles, with permissions;
- a **log**: every PR, review, agent run and discussion thread that touched the project.

A project is the file `projects/<id>.yaml` in the content tree, with a name, a goal in words, the topic node of the goal, the target evidence level, a monthly budget in US dollars, the agent roles that work for it and the members with the role owner or member. The blueprint is the blueprint of the topic page. A job works for the project when it runs on behalf of `project:<id>`. The package `project` of m2 reads the files, `m2 agent run` refuses a job of a role or a person that the project does not have, and `m2 orchestrate` caps the monthly spend of the project at its budget. The project page of the site shows the run log from the run records. The pull requests, the reviews and the threads are not in the log yet.

### Tasks and claims

Any frontier item can be a task. A person claims a task for a time window (default 14 days, renewable). Everyone can see the claim, so two people do not do the same work. Agents examine claims before they take a job. A claim by a person stops agent work on that target, unless the person invites agent work ("let agents try the cheap attempts first"). This follows the norm that large formalisation projects agreed on in 2024 to 2026: claim visibly, release promptly.

### Discussion

Each node and each project has threads. The workspace database stores threads, not git. If a thread decided something, it must end with a pull request that writes the decision into a node or a review. Discussion is not knowledge until it is in the graph. For projects that already use Zulip, threads can be mirrored to Zulip streams. The Lean community discusses on Zulip, and math2 should join that discussion, not compete with it.

### Working with agents in the open

A project member can request agent work inside the budget of the project. Agent runs show in the log with their transcripts (11.6). Anyone in the project can read how a proof was found, and see which attempts failed and why. This is intentional. In 2025 to 2026, there were episodes in which AI "solved" problems that were already in the literature, or solved a misstated version. People found each of these problems because someone could see what was actually done (document 01). Openness is the cheapest verification method available.

### Credit

Provenance (06.5) attributes each contribution. The project page shows contributions by person and by agent role. Each agent contribution names the person or project that directed it. `m2 prov credit` exports a CRediT-style statement for any person and period (document 16.4).

## 13.7 Performance and accessibility targets

- Static node pages: less than 50 KB transferred before math fonts, and the first meaningful paint in less than 1 s on a mid-range phone on 4G.
- The server renders math to HTML+MathML at build time (KaTeX). MathML is always present for screen readers. The client never needs JavaScript to read a node.
- Each interactive view (graph, chart) has an equivalent table or list.
- The static site is fully usable offline as a downloaded archive. Each release publishes one archive. Thus people can read math2 without a network, and this is important for many of the mathematicians in the world.
</document>

<document source="https://math2.tamnd.com/spec/14-search.md">
# 14. Search

Most people will use math2 through search first. Agents also use search to find premises, duplicates and prior work. A query from a mathematician can be a phrase ("finite subgroup of the multiplicative group of a field"), a formula (`\sum_{d \mid n} \varphi(d) = n`), a Lean type (`∀ n : ℕ, ∑ d ∈ n.divisors, φ d = n`), an identifier (`A000010`, `Q190026`, `0BQX`), or a mix of these. No single retrieval method can handle all of these queries. Thus math2 runs **four channels** and fuses their results.

The code is in `tamnd/math2-agent/search`. Three things serve it: `m2 search`, the `search*` and `premises` methods of the API (11.2), and the workspace (13.2).

## 14.1 Channels

| Channel | Index | Good at | Bad at |
|---|---|---|---|
| **lexical** | SQLite FTS5 over names, aliases, statements, explanations; BM25 | names, rare words, exact phrases, identifiers | paraphrase, notation |
| **formula** | structural index over parsed LaTeX (14.2) | the same formula in different words or variable names | formulas that differ structurally but mean the same |
| **formal** | type-directed index over bound Lean declarations and Mathlib (14.3) | exact mathematical content, given a type pattern | natural-language queries |
| **embedding** | vector index over node text, formal statements and record plane results (14.4) | paraphrase, concepts, "the theorem that says…" | precision on near-duplicates, rare identifiers |

The **identifier resolver** runs before the four channels. The query can parse as a tag, alias, `m2://` URI, Lean name, QID, A-number, Stacks tag, arXiv id, DOI, MR or zbMATH id. In that case, it resolves directly (the `resolve` method of the API, 11.2), and search returns that node first.

## 14.2 Formula search

The parser changes LaTeX formulas into operator trees. It uses the same KaTeX-compatible macro set as the bodies (`macros/m2.sty`). Thus math2 interprets its own notation the same way in all places. The index has three parts:

- **Operator-tree paths.** The paths from leaf to root are tokenised. Variables are anonymised and constants are kept. Tangent-L and Approach0 use this method, and they are the strongest systems at ARQMath.
- **Symbol layout trees**, for formulas that have an ambiguous operator parse.
- **Canonical forms**, where a formula is in a class that can be normalised (polynomial identities, linear relations). SymPy computes a normal form in the compute image at build time.

Matching uses BM25 over the path tokens and then a structural re-score. The re-score is the size of the largest matching subtree with consistent variable unification. The index has only formulas from nodes and from record plane results. At full scale, this is approximately 40M formulas. The formula search of zbMATH Open (MathWebSearch) is a crosswalk target, and math2 does not replicate it.

## 14.3 Formal and type-directed search

For a Lean query, math2 does what Loogle does. It also does one thing that Loogle cannot do: it maps hits back to nodes.

- **Pattern search.** A query can be a pattern such as `Subgroup _ → Finite _ → IsCyclic _`, or a constant set such as `IsCyclic, Units, Field`. The channel matches it against the extracted statements (07.4) of each declaration in the pinned Mathlib and the bound libraries. It uses a discrimination-tree index over the elaborated types. It runs locally against the pinned versions of the release. It never runs against the public Loogle service, because that service has rate limits and follows a different Mathlib.
- **Goal search.** For a goal state, the channel finds declarations whose conclusions unify with it after instantiation (the `exact?`/`apply?` relation). It ranks them by the similarity of their hypotheses to the local context, as the embedding channel measures it.
- **Node mapping.** If a hit is bound to a node, the channel returns the node with its status. If a hit is not bound, the channel returns the declaration plus "no node yet". A reader can change this result into a curate request.

math2 uses the semantic Lean search tools of 2026 (LeanSearch v2, Lean Finder, LeanExplore; document 01) as **evaluation baselines**. Where their licences permit, it also uses them as sources of informalisation data to train the formal-statement embeddings of math2. math2 does not depend on these tools at runtime.

## 14.4 Embeddings

Each node gets several embedded texts. The index holds each text separately:

1. its name and statement section;
2. its Lean statement, with the Herald-style informalisation of it (a model's English rendering, generated at build time and cached by type hash);
3. its L2 explanation;
4. for record plane results, the unfolded statement (09.5).

Choices:

- **Model.** Each release pins one embedding model. math2 selects the model with its retrieval benchmark (14.7). The benchmarks of 2026 for mathematical retrieval (MIRB and its successors) put commercial general-purpose embedding models at the top. They also showed that generic rerankers often made results *worse*. Thus math2 benchmarks a reranker before it adopts it, and by default it has no reranker. If the benchmark says so, embeddings for formal statements may use a different model from the embeddings for prose.
- **Index.** A quantised HNSW index (int8, 1,024 dimensions) for nodes, and a separate index for the record plane. At 5M nodes, the node index is a few GB. At 30M results, the record index is approximately 60 GB. The release process builds both indexes, and the server holds them in memory (05.7).
- **No raw vectors in the API.** The API returns neighbours and scores. It never returns vectors. This protects record-class text (9.3) and lets math2 replace the model.
- **Re-embedding is a release event.** When the embedding model changes, a batch job embeds everything again. The release notes record the retrieval benchmark numbers before and after the change.

## 14.5 Premise selection

Premise selection is search for agents. For a Lean goal and its context, it ranks the lemmas that are most likely to help. It is the `premises` method (11.2), and it gives input to the prove loop (12.4). It has four parts:

1. **Formal search.** Declarations whose conclusions unify with the goal (14.3).
2. **Dependency priors.** Declarations that proofs of the graph neighbours of the target use, and declarations that Mathlib proofs of statements with similar types use. The formal dependency graph from extraction (07.4) gives this data. For a node with informal proofs, the declarations bound to the nodes that its proofs `use` get a strong prior.
3. **Embedding similarity** between the informalisation of the goal and the informalisations of declarations.
4. **Learned reranking**, trained from the run records of math2: the retrieved premises that were in the final proof (17.4). This is the only place where a learned reranker is on by default. It is trained on the data of math2, and math2 evaluates it on held-out runs before each deployment.

math2 also uses the built-in premise selection of Lean for `grind +suggestions` and the hammer. It runs inside the Lean session, and math2 merges its suggestions with its own.

## 14.6 Fusion and ranking

Reciprocal rank fusion combines the results from the channels. The weight of each channel is learned on the benchmark. Then two adjustments apply:

- **Kind and status priors.** For a natural-language query, statements and concepts rank above proofs, and E5/E3/E2 rank above E1. The facets of the workspace can override this. For example, an "open" filter inverts the status prior.
- **Duplicate folding.** If results are `same` or `same_as` the same node, they fold into one entry. Record plane results aligned to a node show under that node as "stated in N papers", not as N separate hits.

Each result reports the channels that found it and its rank in each channel (13.2). Thus a reader can see why the result is in the list.

## 14.7 Benchmarks

math2 measures search quality and does not only claim it. `bench/search/` holds the benchmarks below. Their results are in the notes of each release. If a headline metric gets worse by more than 2 points, the release is blocked.

| Benchmark | Queries | Metric | Source of labels |
|---|---|---|---|
| **concept lookup** | 2,000 natural-language descriptions of concepts and theorems, written by people, at undergraduate and graduate level | MRR, recall@10 | curated; each labelled with its target node |
| **formula** | ARQMath-style formula queries, remapped to math2 nodes | nDCG@10 | ARQMath judgements where licences permit, plus curated |
| **formal** | Lean types and goals with known Mathlib answers | recall@1, recall@10 | Mathlib proofs (the premise actually used) |
| **paper-to-node** | record plane results with known aligned nodes | recall@10 | confirmed alignments |
| **premise** | goals from held-out math2 and Mathlib proofs | recall@k of premises used | proofs |
| **cross-system** | queries that should resolve to an external system (nLab, OEIS, Stacks) | accuracy of "not held, see …" | crosswalk |
| **external baselines** | the same queries run against LeanSearch v2, Lean Finder, TheoremSearch, zbMATH formula search, where an API allows | the same metrics | none |

The external baseline row is important. On each type of query, the search of math2 should be better than the best single-purpose tool for that type. If it is not better, the release notes say so, and the workspace gives a link to the better tool for that query type.

## 14.8 The first version

The first version of the search package (M3) has all four channels, the identifier resolver, the fusion and the premise selection. It is simpler than the design above in these ways. The benchmarks of 14.7 decide what to change first.

- The index is one SQLite file, `build/search.db`, that `m2 build` writes in the `search` step. Two builds of the same tree give the same file.
- The formula channel has the operator-tree paths and the structural re-score. It has no symbol layout trees and no canonical forms from SymPy yet.
- The formal channel matches the head of the conclusion, the heads of the hypotheses and the constants of each declaration. It reads them from the canonical type forms of M2Extract, and it has no discrimination tree yet. It knows the declarations of extract.json, thus the bound declarations and the declarations that the tree names. The full pinned Mathlib comes with the import of its declarations.
- The embedding channel uses the local hash embedder of the aligner (16.3) and a flat scan of int8 vectors. The HNSW index and the pinned embedding model come when the benchmark has chosen a model. The index records the name and the length of the embedder, and it refuses to open with another one.
- The fusion multiplies the reciprocal rank term of each channel by a factor from one half to one. The factor is the score of the result divided by the score of rank 1 in the same channel. Thus an exact match far above the next result keeps its margin, and the kind and status priors only order results that are near.
- The channel weights are fixed: 1 for each channel and 0.5 for the embedding channel. The benchmark learns them later.
</document>

<document source="https://math2.tamnd.com/spec/15-trust-and-audit.md">
# 15. Trust and the audit

This document puts in one place what math2 trusts, what it checks, and the rules that do the checks. As in `tamnd/arxiv`, **the audit is the real contract**. A claim in the README, a status in the workspace or a number in a release is only as good as the rule that checks it. Each rule has a name and a state, and each rule writes into a committed report.

## 15.1 The trusted base

math2 states its trusted base explicitly. math2 believes that a statement at E5 is true **if and only if** all of these conditions are true:

1. **Lean's kernel is sound** for the type theory that it implements. We also require independent replay with `lake check --paranoid`. This replay checks again with nanoda, lean4lean and the verified con-leche and con-ron kernels (document 07.7). Thus the trusted base is "at least one of several independent kernel implementations is sound, and they agree", not "Lean's C++ kernel is sound". The kernel soundness bugs of 2026 (document 01) showed that this concern is not theoretical. They also showed that checkers which are out of date fail together. For this reason, the release records the version of each checker.
2. **The standard axioms are consistent**: `propext`, `Quot.sound` and `Classical.choice`, together with the type theory of Lean. This is equiconsistent with ZFC plus countably many inaccessible cardinals. That is a little more than ZFC, and much less than anything that mathematicians have doubts about.
3. **The formal statement means what the informal statement says.** This is the fidelity claim F2 or F3 (document 10). It is the weakest link. Each design decision about fidelity has the aim to make this link stronger.
4. **The binding is the right declaration**: the name, the Mathlib commit and the elaborated-type hash in the release manifest match what was built. The build checks this mechanically.

For E5, math2 trusts nothing else. Specifically, this includes:

- any model's output;
- the prover that found the proof;
- the implementation of any tactic, because its output is a term that the kernel checks;
- the `m2` toolchain, except for the code that extracts declarations and hashes. That code is small and reviewed, and it has its own tests in `math2-agent/trust/`.

For E4, the trusted base is the certificate checker for that certificate kind (document 08.3), plus the runtime of the checker. For E3 and E2, it is the literature and the curators who read it. For E1, it is nothing.

## 15.2 How the audit runs

```
m2 audit                     # all planes, all rules, on the working tree
m2 audit -plane nodes        # one plane
m2 audit -rule G04           # one rule
m2 audit -changed            # only shards touched by the current diff (what PR CI runs)
m2 audit -release 2026.11    # the full pre-release audit
```

Each rule has one of five states, and the report shows all five:

| State | Meaning |
|---|---|
| `pass` | checked something and found nothing |
| `fail` | a hard rule found something. Blocks merge or release |
| `found` | a soft rule found something. Does not block, and is in the report all the same |
| `n/a` | the rule has nothing it could look at in this scope. Never shown as `pass` |
| `skip` | the rule was not run (for example, a network rule offline), and the report says why |

math2 keeps `n/a` and `pass` separate for the same reason that `tamnd/arxiv` keeps them separate. If a rule passes because it had nothing to examine, the rule is not working. It should not look as if it works.

The report goes to `reports/audit/<scope>.txt` and `reports/audit/<scope>.json`. It is committed on `main` after each merge. The release report goes to `reports/releases/<release>.txt`.

```
$ m2 audit -plane formal -q
rule  state  checked   findings
F01   pass   41,203
F02   pass   41,203
F03   pass   41,203
F04   found  41,203    17
F05   pass   3,880
F06   n/a    0
...
41,203 bindings over 38,911 nodes, 17 soft findings, nothing failed
```

## 15.3 The rule catalogue

The letter tells the subject of the rule. Rules with the mark **H** are hard, and they block. Rules with the mark **S** are soft, and the audit reports them. The implementation of each rule is in `tamnd/math2-agent/audit/<id>.go`. With it is a fixture that triggers the rule and a fixture that does not.

### T: tags and identity

| Rule | H/S | Checks |
|---|---|---|
| T01 | H | no tag ever disappears from `registers/tags.reg` or from `nodes/` across git history |
| T02 | H | no two nodes share a tag; no minted tag collides with a reserved prefix |
| T03 | H | no provisional id (`~xxxxxxxx`) appears on `main`, and none is referenced outside its own PR |
| T04 | H | every stored reference is a tag, not an alias or slug |
| T05 | H | registers agree with node files (every live node registered, every retired node in `retired.reg`) |
| T06 | S | aliases are not ambiguous with tags, MSC codes, OEIS A-numbers or arXiv ids |

### S: schema

| Rule | H/S | Checks |
|---|---|---|
| S01 | H | every node file validates against its schema version |
| S02 | H | kind-specific required fields present (a `proof` has exactly one `proves`; a `computation` has an engine block) |
| S03 | H | no `status`, `verified`, `proved` or similarly named field anywhere in content (document 04.7) |
| S04 | H | node file size ≤ 64 KB |
| S09 | S | node uses a deprecated schema version during a migration window |

### B: body

| Rule | H/S | Checks |
|---|---|---|
| B01 | H | every body renders in KaTeX and MathJax headless with no error, using only declared macros |
| B02 | H | every `[[…|TAG]]` link targets a live or retired-with-successor tag; no URLs in statement sections |
| B03 | H | only the schema's sections for the kind, in order |
| B04 | S | statement section is under 2,000 characters (long statements are usually two statements) |
| B05 | S | explanations carry a level and a human/agent marker |

### G: graph

| Rule | H/S | Checks |
|---|---|---|
| G01 | H | every edge target resolves |
| G02 | H | `same_as` is functional both ways within each external system |
| G03 | H | `specializes` acyclic; `uses` acyclic modulo equivalence cycles through distinct proofs |
| G04 | S | formal dependencies of a bound proof appear in its `uses` |
| G05 | H | every live statement has a source, a proof or a formal binding |
| G06 | H | every conjecture and problem has a `source` naming who posed it |
| G07 | S | hub families have at least three members |
| G08 | S | orphan rate per topic < 2% |
| G09 | S | no concept has more than 12 definitions without a note |
| G10 | H | every `implies`/`equivalent`/`refutes` edge names its `via` proof or computation |

### F: formal

| Rule | H/S | Checks |
|---|---|---|
| F01 | H | every binding resolves to a declaration at the pinned commit |
| F02 | H | the elaborated-type hash recorded for reviews matches the built one, or the review is marked stale (never silently kept) |
| F03 | H | `#print axioms` for every E5 candidate is a subset of the standard three, or the node allows more and says so |
| F04 | S | `native_decide` auxiliary axioms (`*._native.native_decide.ax_N`) in closure, reported per node |
| F05 | H | external replay: every E5 declaration passes `lake check --paranoid` and the comparator against its pinned statement, with checker versions recorded (document 07.7) |
| F06 | H | no `sorry`, `admit`, `axiom` declarations or `@[implemented_by]` in the math2 library outside `Math2/Hypotheses/` |
| F07 | H | statement-only bindings (`match: partial`) are never shown as E5 |
| F08 | S | Mathlib deprecation warnings on bound declarations, with the replacement name |
| F09 | H | degenerate-statement checks pass for every F1+ binding: hypotheses satisfiable, conclusion not trivially true, no vacuous quantification (document 10.4) |

### V: review and fidelity

| Rule | H/S | Checks |
|---|---|---|
| V01 | H | a review is bound to both the informal statement hash and the elaborated-type hash |
| V02 | H | F3 by two reviews needs two reviewers with no shared affiliation listed in `reviewers.reg`, and neither is the binding's author |
| V03 | H | an agent cannot be a reviewer; an agent run started by a reviewer cannot produce the binding that the same reviewer reviews |
| V04 | S | review age > 24 months on a node whose neighbourhood changed (prompt for re-review, no downgrade) |

### C: computation

| Rule | H/S | Checks |
|---|---|---|
| C01 | H | every computation names an image by digest, an entry point and inputs |
| C02 | H | E4 candidates reproduce bit-identically on two runner architectures |
| C03 | S | a computation's image is older than the current release's; re-run scheduled |
| C04 | H | certificate checker is the fixed checker for that kind (document 08.8) |
| C05 | H | no network, no model calls in computation logs |

### A: alignment

| Rule | H/S | Checks |
|---|---|---|
| A01 | H | every `same_as` value is syntactically valid for its system (QID, A-number, Stacks tag…) |
| A02 | S | `same_as` written by an agent and not confirmed after 90 days |
| A03 | S | round-trip: the external record still exists and still matches at its pinned revision |
| A04 | S | candidate duplicate pairs (embedding plus judge `same`) unresolved after 30 days |

### L: licence

| Rule | H/S | Checks |
|---|---|---|
| L01 | H | every byte of imported text names a source and that source's licence (document 09.3) |
| L02 | H | no text from a source with an incompatible licence (GFDL-only, arXiv default licence, all-rights-reserved) appears in `nodes/` |
| L03 | H | share-alike sources carry attribution in the node's provenance and the release's attribution file |
| L04 | S | text similarity between a node body and its cited sources above threshold, for human review of copying |

### P: provenance

| Rule | H/S | Checks |
|---|---|---|
| P01 | H | every content change in a merge has a provenance line |
| P02 | H | every agent provenance line names a run id that exists in the artefact store |
| P03 | H | every agent run names the person or schedule it acted on behalf of |
| P04 | S | AI-produced text without human confirmation shown in the workspace without its marker |

### R: release and reproducibility

| Rule | H/S | Checks |
|---|---|---|
| R01 | H | derived graph rebuilds to the same hash (sample on merge, all on release) |
| R02 | H | `.olean` rebuild matches, or the difference is listed and explained |
| R03 | H | release notes account for every count change |
| R04 | H | last quarterly restore drill passed |
| R05 | H | the release pins exactly one Lean toolchain and one Mathlib commit, and the bump branch was green |

### X: agents

| Rule | H/S | Checks |
|---|---|---|
| X01 | H | an agent PR changes only the fields its role is allowed to change (document 12.1) |
| X02 | H | an agent PR stays within its run budget, and the run record shows the spend |
| X03 | S | agent acceptance rate per role and model below threshold over 7 days (flag for prompt or model review) |
| X04 | H | no agent wrote a status-like field or a review |

A merge is an agent PR when one of the new lines of its provenance logs is of an agent with no person. The runs of the PR are the runs that those lines name, and the audit reads their summaries in `build/runs/`. `X01` checks each changed file, except the provenance logs, with `agent.Permit` for the role and the target of each run. A change passes when the set of one run allows it. `X02` needs `spend_usd` and `budget_usd` in each summary. `X03` counts the runs that started in the week that ends 7 days before the date of the HEAD commit, so that each PR had a week for review. A run is accepted when a provenance line on the branch names it. A role and a model with fewer than 5 runs in the week are not checked, and the threshold is 25%. `X04` looks for a part of a field name that is a status word of `S03`, `fidelity_reviews` or `reviews`, in the lines of agents and in the fields of the nodes that an agent PR changes. It also fails an agent PR that changes a file below `reviews/`.

There are 65 rules. A new rule needs a fixture pair and a line in this table. Both must be in the same pull request.

## 15.4 Propagation when trust changes

Trust can change. The audit handles the events that change it:

- **A conjecture is refuted.** Some statements have a best alternative that uses the conjecture `as: hypothesis`. These statements do not change, because they are honest conditionals. Other statements have a proof that uses the conjecture *not* as a hypothesis. Each of these statements goes down to E1, gets a flag, and is in the list in the release notes. In that case, someone assumed an open conjecture and did not say so.
- **A formal binding loses its declaration** (a Mathlib rename without a deprecation alias). The binding is broken (`F01` fails on the bump branch). The bump agent (document 07.9) proposes the rename from the deprecation data of Mathlib, and a person merges it. The node keeps its tag and its history in `bindings.reg`.
- **A definition changes meaning underneath a review** (`F02`). The review becomes stale. For that binding, the node goes down to F0, and E5 is not available. This stays true until someone reviews again, or until a second formalisation is proved equivalent.
- **A kernel or checker bug is found.** Some E5 nodes had their replay on the affected checker version. Each of these nodes runs again on a fixed version, or on the other checker only. The release that contained them gets an erratum. At some time, this has occurred in every proof assistant. Thus math2 has a plan for it and does not only hope that it will not occur.
- **A model version is withdrawn** (document 06.5, the withdrawal test). A generated PR quarantines its unreviewed output. An E5 status reached through that output does not change, because the kernel and the fidelity reviews carried that status, not the model.

## 15.5 What the workspace says

The status that a reader sees is a short phrase that the system generates from the evidence. It is never a checkmark. For example:

- "Proved in Lean (Mathlib), statement reviewed by 2 people. Classical. Replayed by 2 kernels."
- "Proved in refereed literature (3 sources, earliest 1871). Formal statement exists, unreviewed. No formal proof."
- "Claimed (preprint, 2026). Not reviewed. 1 agent-produced informal proof."
- "Open. Posed by Erdős (1975). Best known bounds: … (2025, formal)."

Each phrase has a link to the evidence that it summarises. A reader who wants to know why we believe something can follow the link down to the kernel replay log, the reviews, or the page number in the 1871 source.
</document>

<document source="https://math2.tamnd.com/spec/16-collaboration-and-governance.md">
# 16. Collaboration and governance

People and agents write math2 together, and everyone can read it. This document defines who may change what and how changes are reviewed. It also defines how credit and disclosure work, how math2 relates to the communities whose work it links to, and how the content is licensed.

## 16.1 Roles

| Role | Who | Can |
|---|---|---|
| reader | anyone | read, search, run examples, use the read-only API and MCP |
| contributor | anyone with a GitHub identity | open PRs (nodes, edits, alignments, Lean code), comment, claim tasks |
| reviewer | a contributor listed in `registers/reviewers.reg` for one or more areas | sign fidelity reviews (F2/F3) in those areas; approve PRs in those areas |
| curator | a reviewer with merge rights in an area | merge PRs in the area; resolve dedup queues; retire nodes |
| maintainer | a small group | merge anything; change schema, rules, budgets, roles; cut releases |
| agent role | a bot account per role | open PRs within the role's field permissions (12.1); never review, never merge except under auto-merge rules |

The curators of an area give reviewer status for that area. An area is an MSC two-digit class or a narrower topic. The basis for the decision is the reviews that the person has already done. A reviewer does not need an academic position. Mathlib's reviewer and maintainer system is based on contributions, not on credentials, and math2 uses the same model. But a reviewer must have a real identity: an ORCID or a long-standing GitHub account. The independence rule for F3 (rule `V02`) cannot work with anonymous reviewers.

## 16.2 How changes land

Every change is a pull request against `tamnd/math2`. The gate before a merge depends on the kind of change:

| Change | Gate |
|---|---|
| typo or prose fix in a body | `m2 check` green; any contributor approval |
| new node or new edge, E1 evidence only | audit green; one curator approval |
| new formal binding (F0) | audit green, Lean build green; one curator approval |
| fidelity review (F2/F3) | the review itself is the gate; one reviewer signs; for F3, a second independent reviewer or an equivalence proof |
| retire, merge or split nodes | two curator approvals; release notes entry |
| agent PR, role with auto-merge (12.1) | audit green; role-specific checks green; sampled human audit afterwards |
| agent PR, any other role | as for the same change by a person, plus the agent label |
| schema, rule, budget, role changes | two maintainer approvals; an entry in document 21 if it settles an open question |

**Auto-merge** applies only to changes whose correctness is entirely mechanical. One example is a Mathlib rename that comes from deprecation data. Another example is a formal proof added to a node whose statement binding is already F2 or above. For these changes, the kernel and the replay are the whole check. Even then, people audit a sample of 5% of the auto-merged PRs of each role each week. The agent report (12.9) publishes the findings of the sample. If the sampled error rate for a role is more than 1%, auto-merge stops for that role until the cause is fixed.

## 16.3 Curation at scale

Curators cannot curate a graph of all mathematics if they approve PRs one at a time. Curators work through **queues**. Each queue has a rank order, so that one hour of a curator's time has the most effect:

- **dedup queue:** candidate `same` pairs (05.4), ranked by how much of the graph the merge would connect;
- **alignment queue:** agent-proposed `same_as` edges, grouped by external system and ranked by centrality;
- **fidelity queue:** bindings at F1 that wait for a human review, ranked by `unlock` (11.4). A review of a statement that 300 nodes depend on is worth more than a review of a leaf. `m2 review queue` lists it;
- **extraction queue:** paper results that the extractor could not align, sampled by topic;
- **retirement queue:** nodes flagged by graph health rules (orphans, hubs that are too small, conflicting `same_as`).

The reviewer's tool, `m2 review`, makes a fidelity review fast. It shows the informal statement and the elaborated Lean statement side by side. It also shows the back-translation, the results of the degenerate-statement check, and the critics' notes (document 10.4). It shows the definition of every non-trivial constant in the Lean type, and each definition expands on demand. The target median time per review is 3 minutes for undergraduate-level statements and less than 15 minutes for research-level statements. Document 17 measures this.

## 16.4 Credit

Work on math2 must count as scholarly output. If it does not, the people who can review research-level fidelity will not give their time to it. Thus:

- **Every release is citable** with a DOI (06.2). Every node at a release is citable as `m2://TAG@RELEASE`.
- **Contributor statements.** `m2 prov credit <person> --from --to` produces a CRediT-taxonomy statement. It lists, with links, the nodes that the person wrote, formalised, reviewed or curated. Its format is for direct use in a CV or a grant report.
- **Reviews are signed and public.** A fidelity review is a small, real and checkable piece of mathematical work. It carries the reviewer's name. Each reviewer has a page on the wiki with a stable link, `reviewers/SLUG/`, that lists their reviews, and `m2 review export` gives the same list as text or JSON.
- **Projects publish.** A project that reaches its goal (13.6) can publish a release note as a paper. The author list comes from provenance, and the project's members edit it. A standard paragraph discloses the AI contributions, and provenance also generates this paragraph (16.5).

## 16.5 AI contribution: disclosure and norms

The norms that formed from 2024 to 2026 are converging. They come from journal and arXiv policies and from statements by mathematical societies. They also come from the public practice of Terence Tao and other researchers who work with AI tools openly (document 01 has the sources). The norms agree on three points. AI tools may be used. The humans who sign are responsible for the result. The disclosure of the use must be specific enough that a reader can judge it. math2 puts these points into the data and does not depend on good intentions:

1. **Every agent-produced byte is marked** by provenance (06.5). The workspace shows the marker on prose that an agent wrote and a person has not confirmed (rule `P04`).
2. **Agents are not authors.** They are tools that act for a named person, project or schedule (rule `P03`). The person or project is responsible.
3. **Formal proofs found by agents** need no disclosure beyond provenance for their correctness, because the kernel checked them. But they do need disclosure for credit and for the record. If an agent run is the first to prove a theorem, the history tab of the theorem says so, with the run id.
4. **Informal proofs by agents stay E1** (04.7) until someone formalises, certifies or publishes them. The workspace never shows them as more than claims.
5. **Novelty claims are checked against the literature first.** Before anyone describes an agent's result as "new" in any place, the explore agent runs a literature search. It searches the record plane, zbMATH, arXiv full text where the licence allows, and MathOverflow. math2 attaches the search and its result. Several of the 2025 "AI solved an open problem" announcements were rediscoveries of known results. math2 is designed so that it does not repeat that error (document 01, 17.5).
6. **Withdrawal is mechanical** (06.5). Sometimes a model or prompt version is found to give systematically wrong output. Then one command quarantines all of its unconfirmed output.
7. **Disclosure meets the 2026 guidelines by construction.** The AGMAI guidelines of September 2026 ask labs and authors to publish the model, prompts, summarized reasoning, time and compute cost. They also ask for documentation of failed attempts and of selection criteria. They ask for persistent identifiers in lab-independent repositories, and for machine-readable links between informal and formal statements. Run records (11.6) and bindings (07) already hold all of these. `m2 prov disclose <tag|project>` renders them as a disclosure paragraph. For any formal result, it also renders them as a Palomar-compatible `formalization.yaml`. Thus you can register a math2 result on Palomar, or cite it in a journal that requires an AI declaration (the LMS, Elsevier). Nobody has to reconstruct what happened.
8. **The human role is recorded with a taxonomy.** Each agent-assisted contribution states what the people did. It uses categories adapted from the erdosproblems AI-contributions wiki: the agent solved it with no literature found; the agent solved it and literature was later found; the agent built on literature; people and agents worked together; and literature search, formalization, rewriting or computation only. The project page and the disclosure paragraph show these categories. One "AI-assisted" label in place of these categories hides exactly the information that a reader needs.
9. **Participation without AI is first-class.** Some mathematicians have chosen not to work with commercial AI systems (the Association for Human Mathematics, 2026). Some venues now require human-written expositions. A project can declare itself `agents: none`, and then the orchestrator refuses jobs in it. Any reader can filter the workspace to show only human-authored prose. The credit statement of a person (16.4) separates the work that they did themselves from the work that they directed. math2 should be a place where both kinds of mathematician can work side by side.

### Upstream etiquette

math2 links into communities that have their own norms. The most important are Mathlib, Wikidata, nLab, OEIS and the Stacks Project. The rules are:

- **Upstreaming to Mathlib is by people, in Mathlib's process.** If math2 Lean code is worth upstreaming, a person offers it as Mathlib PRs and guides them through review. Agents may prepare the branch. Agents do not open PRs on Mathlib or on any other external project, unless that project has asked for it.
- **No bulk writes to external wikis without agreement.** Writes to Wikidata, MaRDI and similar systems follow the bot policies of those systems. This means approved bot accounts, rate limits, and community discussion first.
- **Respect the load.** Ingestion uses dumps and published bulk channels. It does not crawl live sites. It follows the pace that each source states (document 09). This is the same discipline that `tamnd/arxiv` applies to arXiv.

## 16.6 Licensing

The content has three licence zones. The zones let math2 take in the sources that matter, and they let others reuse what math2 produces:

| Zone | Licence | Why |
|---|---|---|
| node prose (bodies, explanations, names) | **CC BY-SA 4.0** | compatible with importing from Wikipedia, zbMATH Open, OEIS, LMFDB, ProofWiki and individual MathOverflow/Math.SE posts, which are all BY-SA family (3.0 material can be adapted under 4.0); also compatible with our own text being reused by those communities |
| Lean code (`lean/`) and computation code (`compute/`) | **Apache-2.0** | identical to Mathlib's licence, so code can move upstream with nothing in the way |
| metadata (front matter facts, edges, registers, manifests, the derived graph) | **CC0** | facts about mathematics should be free of conditions; also matches arXiv metadata and Wikidata |

The `L` audit rules enforce the licence rules (document 15). In particular:

- **GFDL sources are linked, never copied.** The Stacks Project and Kerodon are GFDL. GFDL text cannot go into CC BY-SA content, and CC BY-SA text cannot go into GFDL content. math2 links to Stacks and Kerodon tags with `same_as` and writes its own statements. This is permitted, because a mathematical fact is not copyrightable. Rule `L04` compares node text with the linked source to find accidental copies.
- **arXiv default-licence papers give metadata and structure, not text.** The `tamnd/arxiv` licence gate applies unchanged. math2 can record that a paper states a result (a record plane entry with a locator). This does not require it to republish the paper's wording.
- **MSC2020 descriptions are CC BY-NC-SA.** The codes are identifiers, and math2 uses them freely. The descriptive text of each class has a non-commercial clause that CC BY-SA content cannot absorb. Thus math2 never stores this text in `tamnd/math2`. The site shows it from a pinned snapshot, with attribution.
- **Stack Exchange text is imported per post, never from the dumps.** Posts are CC BY-SA (2.5, 3.0 or 4.0, depending on the date). But since 2024, the official dumps are behind click-through terms that forbid LLM training. At least one dump contained planted watermark entries. math2 gets individual posts through the public API, and records the licence and the author of each post. It stores records and short quotations, not bulk text.
- **Mathlib docstrings** are Apache-2.0. You can use one as a node body, with attribution in provenance. The CC BY-SA node body then includes it under BY-SA, and Apache-2.0 permits this.

## 16.7 Governance and sustainability

- **Decisions are written down.** Settled decisions are in document 00. Open decisions are in document 21. A change to either document is a PR with a rationale. If the change reopens a settled decision, it needs a measurement. The other specs in this series use the same rule.
- **Budgets are public.** math2 commits the monthly agent budget, its split by role, and what it bought (12.9) in `reports/agents/`.
- **The project could outlive its funding.** The content is in git and archived to Software Heritage. It is citable by DOI, and you can rebuild it from pinned images (document 06). If agent funding stops, math2 becomes a static, versioned and checkable library that people maintain at human pace. That outcome is smaller, but it is not a failure.
- **Funding and institutions.** Document 01 lists the 2025 to 2026 programmes that fund this kind of infrastructure. They include AI-for-math funds, DARPA's expMath, national research data initiatives such as MaRDI, and the Lean FRO's model of a focused research organisation. In the milestone structure of math2 (document 19), each milestone is a deliverable that a funder can check.
</document>

<document source="https://math2.tamnd.com/spec/17-evaluation.md">
# 17. Evaluation

math2 makes claims at three levels. It makes claims about mathematics (the status of a node). It makes claims about its own machinery (the battery catches misformalizations; search finds the right node). It also makes claims about its agents (this prompt version proves more for less). Each claim needs a measurement, or it is a slogan. This document says what math2 measures, how it measures it, and what each measurement gates.

The code and data are in `tamnd/math2-agent/bench/`. The results go to `tamnd/math2/reports/eval/<release>/`, and the notes of each release give a summary of them.

## 17.1 Principles

1. **The kernel is the only grader that needs no evaluation.** math2 grades all other graders, which includes LLM judges, back-translation, the critic, search and extraction. Every grader has a labelled set, a measured error rate, and a version.
2. **Gate on regressions, report the levels.** Each suite has a gate: on the suite, a new version must be no worse than the old one, within noise. math2 publishes the absolute numbers, but they do not gate. A gate on absolute targets invites quiet changes to the target.
3. **Measure cost with quality.** Report every result with its cost: tokens, dollars, wall-clock time and Lean CPU time. In 2026, the same model could score 7/10 for $36 or 9/10 for $1,888. The difference came from the harness around the model (document 01). You cannot compare a score that has no cost.
4. **Report failures and the selection.** The report of a suite says how many items passed and which items failed. It also says how the items were chosen. This follows the AGMAI guideline that labs document failed attempts and how they selected problems (document 01). math2 applies this guideline to itself.
5. **Contamination is the default assumption.** Any public problem set may be in the training data of a model. Each suite has public, held-out and fresh items (17.6). The headline numbers come from the held-out and fresh items.

## 17.2 What is measured

| Area | Question | Suite | Gates |
|---|---|---|---|
| **status** | Is a node's computed status correct? | monthly sample of E2 to E4 nodes re-checked by a curator against their evidence (17.8); F-level re-review (10.8) | release notes; a measured error above the published bound blocks the release |
| **fidelity battery** | Does F1 catch misformalizations? | `bench/fidelity/` (17.3.1) | any battery change |
| **extraction** | Does ingestion extract and unfold results correctly? | `bench/extract/` (17.3.2) | ingest adapter changes |
| **alignment** | Are proposed `same_as` and `aligned` edges right? | `bench/align/` (17.3.3) | align role changes |
| **search** | Does search find the right node? | `bench/search/` (14.7) | search and embedding changes |
| **agents** | Does each role do its job better per dollar? | `bench/agents/<role>/` (17.3.4) | role and prompt versions (12.8) |
| **injection** | Do agents ignore instructions in sources? | `bench/agents/canaries/` (12.7) | every role version |
| **explanations** | Are explanations correct, grounded and at the right level? | `bench/explain/` (17.3.5) | explain role versions |
| **graph** | Do the queries meet their latency targets? | `bench/graph/` (05.6) | storage-layer changes |

## 17.3 Suites

### 17.3.1 The fidelity battery

The battery (10.4) is a classifier. Its verdict is faithful, or misformalized with a class. Its evaluation set contains labelled pairs, and each pair is an informal statement and a formal statement:

- **Seed.** Known misformalizations from the published audits. These include the defective items of ProofNet, the certified issues of the ICML 2026 benchmark audit, the fixed statements of PutnamBench, and the corrected statements of the formal-conjectures repository. Each item has its taxonomy class (10.5).
- **Synthetic.** Faithful statements from F3 nodes, mutated by one taxonomy class at a time: drop a hypothesis, swap a quantifier, change ℕ to ℤ, add a junk value. A person checks each mutation before it enters the set, because some mutations are also faithful by chance.
- **Live.** Every `not-faithful` review (10.6) enters the set, with its class.

Measured: recall per taxonomy class, and the false-alarm rate on faithful items. The report is a confusion matrix over classes. A battery change must not decrease recall on any class by more than the noise band of that class. The band is published with the matrix.

**The suite in m2.** `bench/fidelity` has the synthetic part, in the Lean core library, and it reads the live part from the reviews of a math2 tree. The seed part needs the published audits, and it comes later. The noise band is zero items, because the tactics of the battery have limits in heartbeats and give the same result on each run. Thus the test fails when one class catches one item less. The report is attached to each release of tamnd/math2-agent.

### 17.3.2 Extraction

The suite has five hundred papers across MSC codes, decades and source types (LaTeX, HTML only, OCR). Each paper is labelled by hand with its results, their kinds, their in-paper dependencies, and a reference unfolding of each result. Measured: result recall and precision; kind accuracy; dependency edge F1; unfolding fidelity. People grade unfolding fidelity on a sample, and the back-translation judge of the battery grades it on all items. The report gives errors by source type, so a regression in the OCR path cannot hide behind the LaTeX path.

### 17.3.3 Alignment

The suite has confirmed and rejected alignments from curation (09, 05.3). Measured: precision at the auto-propose threshold, and recall of known `same_as` pairs. Precision is more important here, because a wrong `same_as` edge contaminates everything downstream. The threshold is set so that precision on the held-out set is at least 0.98.

### 17.3.4 Agent roles

Each role in 12.1 has a suite of tasks with an objective outcome:

| Role | Tasks | Outcome measured |
|---|---|---|
| **prove** | held-out statements with known Lean proofs, by tier: undergraduate (Mathlib-adjacent), graduate, research (from bound libraries and formal-conjectures' solved items) | solved within budget; dollars and Lean CPU per solve; proof size against the reference |
| **formalize** | held-out nodes with F3 bindings | winning candidate equivalent to the F3 binding; battery verdict; dissent rate |
| **explore** | planted problems: known conjectures withheld from the context, known counterexamples to plausible statements | refutations found; false conjectures proposed and not caught by the falsify step; novelty check run before every claim |
| **explain** | 17.3.5 | see 17.3.5 |
| **curate, align** | 17.3.3 | see 17.3.3 |
| **bump** | replays of past Mathlib bumps from git history | share of breakages repaired automatically; repairs that change a statement's type hash (these must be zero without a flag) |
| **review-assist** | packets for bindings with known verdicts | verdict-relevant concerns surfaced (the critic's recall); length of packet |

The suite of formalize and prove is in `bench/agents`. Its first items are public, use only the Lean core library, and have a reference solution. Without a model service, a scripted model writes the reference and some wrong attempts, thus the suite checks the goal predicates of the roles on real Lean. A live run with the Claude models gives the items solved and the dollars of each item.

The suites of explain and curate are in `bench/agents/explain` and `bench/agents/curate`. They need no Lean. The scripted model writes the reference and wrong attempts, and the scripted grounding check or judge gives the labels or the verdicts of the items. Thus the suites check the mechanical parts of the goals: the citations and the levels of explain, and the locator, the kind and the copy check of curate. A live run also grades the grounding check and the judge alone against the labels and the verdicts of the items, because the goals depend on them.

The suite of review-assist is in `bench/agents/reviewassist`, and it needs no Lean. Its items are bindings with known verdicts, as the table says. The scripted model writes the packet with the reference concern of each item, and some wrong packets that the goal or the tool must refuse. A live run gives the recall of the critic on the not-faithful items, the concerns on the faithful items, and the length of the packets.

The suite of align is in `bench/agents/align`, and it needs no Lean. Its items are crosswalk edges and Mathlib bindings with a known reference. The scripted model makes the reference change and some wrong changes, and the scripted judge gives the verdicts of the items. Thus the suite checks the mechanical parts of the goal: the edges that a source allows, the decisions of the register and the type hash of a Mathlib binding. A live run gives the precision and the recall of the edges, as 17.3.3 says, and it grades the judge alone against the verdicts of the items.

The suite of bump is in `bench/agents/bump`, and it needs no Lean. Each item is a replay of a bump, with the pinned Mathlib after the bump in the file of the items. The public items are made up, and the held-out items are replays of past Mathlib bumps from git history, as the table says. The scripted model makes the rebinds of the reference and some wrong rebinds that the goal or the tool must refuse, and the suite checks that `bump.AutoMerge` gives the value of each item. The goal refuses a rebind to another type hash unless the job has the constraint `type_change`. A live run gives the share of the breakages that the role repaired, the share of the repairs that can merge with no review, and the repairs that change a type hash with no flag, which must be zero.

The suite of explore is in `bench/agents/explore`, and it needs no Lean. Its items are planted problems of the second form of the table: plausible statements that are false, each with a small counterexample, the program that finds it and the output of the program. Without flags, a fake sandbox gives the output of each program, and the scripted model files the reference counterexample through a computation that verifies, after a lookup of the claim. The goal or the tool must refuse the wrong attempts: no output, a claim that the output is new, a conjecture that no computation checks, a counterexample with no computation, a wrong output hash and a change of the target. A live run runs the computations in docker and gives the three metrics of the table. Each target of the public items is false, thus a person reads each conjecture of a live run, and a conjecture that the person finds false is a false conjecture that the falsify step did not catch. The items with withheld known conjectures are not in the suite yet.

Every role suite has three splits. **Public** items can be anywhere. **Held-out** items are never published. They are in a private repository that the tools of the agents cannot reach. **Fresh** items were made after the training cutoff of the model. Fresh items come from new Mathlib declarations, new formal-conjectures statements, new arXiv results with formal statements, and items that the maintainers write. Each model version is compared on all three splits. If public scores are much higher than fresh scores, the report gives the difference as a contamination signal.

### 17.3.5 Explanations

Explanations (12.6) get grades on three axes. **Correctness**: every mathematical claim in the explanation is grounded in a cited node, or it is marked. **Level**: an L1 explanation uses no concept outside the prerequisite set of its level. A mechanical check against the graph confirms this. **Usefulness**: people grade it. People grade a monthly sample, and a judge grades everything. math2 also publishes the agreement between the judge and the people on that sample. LLM judges are least reliable on explanations. Thus an improvement that only a judge sees does not pass the gate without a human-graded sample.

## 17.4 Run data

Every agent run is a record (11.6). Together, the records are math2's own dataset:

- **Outcome per target**: solved or not, with cost, model, prompt version and the retrieval used. The orchestrator uses this data to learn which ladder rung to start at for each kind of target (11.4).
- **Premise use**: which retrieved premises appeared in the final proof. This data trains the premise reranker (14.5).
- **Failure classes**: timeout, elaboration loop, wrong decomposition, statement mismatch, injection attempt detected. The classification is automatic, and people check a sample.
- **Cost curves**: solve rate against spend, by role, tier and model. The weekly report (12.9) plots this curve, and the budget split (11.5) is tuned on it.

The transcripts of merged runs are public (11.6), so their outcome data is public too. Each quarter, math2 also publishes a dataset of all runs, merged or not. It contains targets, outcomes and costs, but no transcripts for unmerged runs. This applies the AGMAI and Tao position on the report of negative results and resource use (document 01) to the project's own agents.

The package `agent/dataset` and `m2 agent dataset` make the dataset of a quarter as JSON Lines, with one row for each run that started in it. A row has the job, the role at its version, the models, the outcome, the budget, the spend and the verdicts of the novelty check. It has no transcript, and the transcripts of the merged runs stay in their run records (11.6). A job on behalf of a person gives only the word `person`, and the free texts lose their email addresses, keys, tokens and home directories.

## 17.5 The novelty check

Some outputs can be described as new: an explore conjecture, a proof of an open node, a counterexample. Before that, the output goes through a novelty check, and the result of the check is attached to the node:

1. **Graph.** Search the statement plane for the statement, its negation, and its specialisations and generalisations (14).
2. **Record plane.** Search aligned and unaligned results from the literature. Use the formula, embedding and lexical channels.
3. **External indexes.** Query zbMATH Open, TheoremSearch, TheoremGraph and Matlas where their APIs allow (09.6). Also query arXiv full text of licence-permitted papers through `tamnd/arxiv`. Query MathOverflow and Math.SE through the Stack Exchange API.
4. **Problem databases.** For a problem node, check these sources: the status and comments in the erdosproblems database, the `solved` markers in the formal-conjectures repository, the comments of the OEIS entry, and the LMFDB.
5. **Verdict.** `known` (with the source), `likely-known` (close matches that a person should read), or `no-match-found`. **No verdict says "new".** `no-match-found` means that these searches found nothing. Only a person makes a statement of novelty, after they read the matches.

The check is in the `novelty` package. `novelty.Check` runs each search on the claim and keeps each search and each match in the result, thus the run record of the output can keep them. The searches are `novelty.Graph` on the search index of the build, `novelty.Records` on each source of the record plane with the channels of the aligner, and `novelty.ZbMATH`, `novelty.ArXiv`, `novelty.StackExchange` and `novelty.OEIS` on the web. The records of the problem databases give their status to the match. A judge compares the claim with the three best matches of each search. A match that states the claim, or a result that implies it, makes the claim `known`. A related match, or a match that the claim generalizes, makes it `likely-known`. With no judge, each match makes it `likely-known`, because a person must read it. A search that fails is in the result, and the check does not give `no-match-found` when a search failed. A match keeps only its source, its id, its link and the verdict of the judge (09.6). The goal of explore (12.5) runs the check on each conjecture and counterexample of the run, and the field `novelty` of the run record keeps the result by node id.

math2 evaluates the check like everything else. Its suite contains results that are known to be in the literature. Each result is phrased as it would come out of an agent. The suite includes the 2025 to 2026 rediscoveries (document 01). Measured: recall of `known`. Sometimes the novelty check misses a rediscovery and a person finds it later. That rediscovery then goes into the suite.

The suite is in `bench/novelty`. Its first items are classical results in the words of an agent output, with the node of the test tree or the paper of the suite that makes each one known, and some results that the searches do not have. With an oracle judge, the recall of `known` measures the searches: the match must be in the three best matches of a search. With the judge of explore, it also measures the judge, and the items with no match measure how often the judge calls a result known by mistake. The rediscoveries of 2025 to 2026 are not in the suite yet, because each item needs a source that a person checked.

## 17.6 External benchmarks

math2 does not optimise for public benchmarks. But it runs them, for two reasons. They put the agents of math2 on a common scale with the agents of everyone else. And the fresh benchmarks are useful held-out sets. Each release reports the results with their costs. These results never replace the suites above.

| Benchmark | What it measures | 2026 state (document 01) | Use in math2 |
|---|---|---|---|
| **miniF2F** | competition maths in Lean | widely regarded as saturated | smoke test only |
| **PutnamBench** | Putnam problems in Lean, 672 items | saturated: all 672 solved, and the leaderboard ranks by cost | cost benchmark for the prove role: dollars to solve all of it |
| **formal-conjectures** (frozen evaluation subsets) | open and solved problems as Lean statements | 2,615 statements; AlphaProof Nexus solved 9 Erdős and 44 OEIS items | research-tier prove suite (solved items); explore targets inside campaigns (open items) |
| **FrontierMath Erdős** | 68 open Erdős problems in Lean, comparator-graded, $300 cap per problem | top score 3% | the hardest prove tier; reported with spend |
| **FrontierMath Tier 4** | expert final-answer problems | declared saturated in September 2026 | not used: final answers are not math2's output |
| **First Proof** | unpublished research lemmas, graded double-blind by people, with negative scoring for wrong answers | Batch 3: 9/10 at $1,888 for the best system | the model for 17.7; math2 enters when a batch fits its roles |
| **IMO-ProofBench, MathArena proof tracks** | informal proofs graded by judges plus people | judges agree with people at about 0.9 | calibration data for math2's own judges |
| **RealMath, LemmaBench** | results mined from recent arXiv papers, refreshed monthly | contamination-resistant by construction | fresh items for the formalize and prove suites, through 09 |
| **Annals Challenge** | 50 Lean statements of recent Annals theorems | missing definitions are the blocker | a definitions-first suite for formalize (10.3) |

Contamination rules for every external benchmark:

- During an evaluation, no context that an agent sees contains the items or their solutions. Search excludes them (`bench_exclude` in the run configuration), and the canary rule checks for leaks.
- The report gives results separately for items published before and after the training cutoff of each model.
- When a run solves an item, it reports whether search returned the solution of the item or a close neighbour. Thus a "solve" by retrieval is labelled as one.

## 17.7 Graders: judges and people

Many checks in math2 end in an LLM judge: back-translation (10.4), the critic, alignment, explanation grading, and the ranking in the novelty check. The 2026 evidence about judges is mixed (document 01):

- On competition proofs, the best judges agree with human graders at about 90%. This is about as often as human graders agree with each other.
- Agreement fell when the proofs came from AI pipelines that were optimised against judges. One study found a 31.7% false-positive rate.
- Judges show a "reviewer-pleasing" bias towards confident, well-formatted text.
- A minimum over several judges reduced overestimation.

Thus math2 applies four rules to every judge:

1. **Judges triage. They never certify.** No judge verdict raises the E-level or F-level of a node. Judges decide what a person examines first, and which items go to the kernel for a check.
2. **Min over a panel.** Sometimes a judge's verdict gates something (for example, F1 needs a back-translation pass). Such a gate uses at least two judges from different model families and takes the more severe verdict.
3. **Judges are evaluated on adversarial items.** The suite of every judge includes outputs from math2's own agents that people marked wrong. If a judge cannot tell a wrong proof by an agent from a right one, math2 does not use that judge to grade agents.
4. **Human grading follows First Proof.** When people grade agent output in an evaluation, they do not know the source: agent or human, and which model. They use a rubric that gives separate scores for correctness, exposition and attribution. A wrong answer scores lower than an abstention, so a confident wrong answer costs more than no answer.

## 17.8 Reporting

The notes of each release include an evaluation section that is generated from `reports/eval/<release>/`:

- the status error bounds: the share of sampled E2 to E4 nodes whose evidence did not support their level (0.5% of each level per month, re-checked by a curator who did not add the evidence), and F-level re-review disagreement (10.8);
- the battery confusion matrix and its change since the last release;
- search headline metrics against the external baselines (14.7);
- per-role solve rates, dollars per solve, and contamination gaps between public and fresh items;
- external benchmark results with costs;
- known failures: the items each suite failed, linked.

If a number got worse, the notes say so first.
</document>

<document source="https://math2.tamnd.com/spec/18-coverage-all-math.md">
# 18. Coverage: what "all mathematics" means

"All mathematics" is either a slogan or a measurable target. This document makes it a target. It says what counts as covered and how math2 measures coverage per area. It also gives the size of each layer and the order in which math2 fills it. Document 19 makes milestones from this order.

## 18.1 Two planes, five levels

Document 03.3 divides math2 into a **record plane** (what exists in the literature) and a **statement plane** (what math2 understands). math2 measures coverage on both planes, per MSC2020 code, at five levels:

| Level | Name | Means, for an MSC code | Plane |
|---|---|---|---|
| **C0** | recorded | every paper and book in the area has a record, with MSC codes, identifiers and references | record |
| **C1** | indexed | results are extracted from every licence-permitted full text, and every result has a locator and an embedding | record |
| **C2** | mapped | the area's concept skeleton exists as nodes: its main concepts and definitions, and the theorems that a survey or a graduate text would name, each with at least one source | statement |
| **C3** | standard | the area's core (18.3) is at E2 or better, with proofs or locators of proofs, and explanations at L1 and L2 | statement |
| **C4** | formal | the area's core is at E5 with F2 or better, and its conjectures have reviewed formal statements | statement |

An area is at level *k* when it meets the criterion of that level. Each level is a percentage, not a yes/no. math2 should be able to say a sentence such as "MSC 20 (group theory) is 100% C2, 71% C3 and 38% C4". It should also be able to defend that sentence line by line.

C0 for all of mathematics is possible within the first year, because zbMATH Open and arXiv already did that work (document 09). C4 for all of mathematics is not possible in any planning horizon. The purpose of the measurement is to know the gap, area by area, and where the gap is closing.

## 18.2 The coverage table

Q11 (05.6) computes, for every MSC2020 code at every level of the hierarchy:

- **record plane**: records; records with extracted results; results; results aligned to nodes;
- **statement plane**: nodes by kind; nodes by E-level and F-level; nodes with explanations at each level; executable examples (08.7);
- **formal**: bound nodes by match kind; Mathlib declarations in the area that are bound to a node, and the share that are not;
- **open problems**: conjectures and problems, with their formal-statement status;
- **movement**: the change in every count since the last release.

The table goes to `reports/coverage/<release>.parquet`, with a Markdown summary of all classes in `reports/coverage/<release>.md` and a summary per top-level code in `reports/coverage/<release>/` (20.1). The workspace shows it as a map: one cell per MSC code, coloured by level. Nodes get MSC codes as topic edges (05.2). The codes come from the codes of the sources and from a classifier. For C2 nodes, a curator confirms them.

MSC2020 has 63 top-level classes and about 6,000 codes in total. math2 computes coverage at all three levels. By default, it reports the top two levels.

The denominator of C0 is the number of documents of zbMATH Open in each class. `m2 coverage -count` asks the REST interface of zbMATH Open for it, one class at a time, and pins the count with its date in `manifests/coverage/expected.yaml`. The count is for the top-level classes only, thus C0 has a value only at the top level. A record or a node counts for each of its codes and for each code above them. A code that is not in MSC2020, such as an MSC2010 code, counts for the nearest code above it.

### Which areas count

Every MSC2020 class is in scope for C0 and C1. For the statement plane:

- **00, 01, 97** (general, history, education) are record-plane only. A history-of-mathematics source can be a node's source, but history is not a node.
- **Applied areas** (35, 49, 60 to 62, 65, 68, 70 to 94) are in scope for their mathematical results: the theorems, definitions and conjectures. A model of a physical, biological or economic system is not a node unless it comes with a mathematical statement. If a formal library exists for the area (Physlib for physics, CSLib for computer science), math2 aligns to it and leaves the modelling to it.
- **68 (computer science)** overlaps with the theory of computation. Its mathematical core (complexity classes, automata, type theory) is in scope. Software engineering is not.

## 18.3 Skeletons and cores

A **skeleton** is the list that C2 counts. Each top-level MSC2020 class has at most one, in `topics/skeletons/NN.yaml`, where NN is the first two characters of the code. Its items have the fields of the items of a curriculum: a name, a kind, and at least one of a Wikipedia title, a Wikidata item and a Mathlib list, with a Mathlib declaration when one exists. An item gets the field `node` when its node exists. C2 of the class is the share of the items whose node is live and has at least one source. A class with no skeleton has no C2. One person who works in the area reviews the skeleton before it counts toward a claim such as "100% C2".


A "core" is the set of nodes that an area must have before it can claim C3 or C4. Without cores, C3 would be a percentage of an unknown total. Each core is a curated list in `topics/cores/<name>.yaml`. It is built from standard syllabi and texts, and people who teach the area review it. Cores nest:

| Core | Basis | Size (nodes: definitions, statements, key examples) |
|---|---|---|
| **undergraduate** | about 40 standard courses, from first-year analysis and linear algebra through Galois theory, measure theory, topology, complex analysis, probability, number theory, combinatorics and logic; Mathlib's `undergrad.yaml` list and the "1000+ theorems" list as cross-checks | **about 25,000** (about 600 per course) |
| **graduate** | the standard first- and second-year graduate syllabi and qualifying-exam lists of several departments; one or two standard texts per subject | about 150,000 to 250,000 |
| **research, per area** | the concepts and named theorems that recent surveys in the area cite; the area's open problem lists | about 5,000 to 50,000 per top-level class, in total about 1M |

The undergraduate core is the first serious target, for three reasons. It is bounded. It is mostly in Mathlib already, so C4 is mostly binding and review, not new proofs. Most of the items on Mathlib's own `undergrad.yaml` list were formalized by 2025, and the list records the gaps. And in this core, explanations and learning paths (Q5) help the most people.

## 18.4 How big is mathematics?

These are estimates for planning. Each estimate has a basis. When math2 has the measured number, the measured number replaces the estimate.

| Quantity | Estimate | Basis (document 01) |
|---|---|---|
| papers and books | about 5M, growing by about 150k per year | zbMATH Open's coverage (over 4.9M documents) plus arXiv math, deduplicated |
| results stated in the literature | about 30 to 60M | about 13 to 19 results per paper that states results in theorem form (TheoremSearch: 9.2M from about 690k arXiv papers; Matlas: 8M from 435k journal papers); many older and applied papers state none |
| distinct results after deduplication | perhaps a third to a half of that | duplicated restatements, lemmas restated from earlier papers, and corollaries |
| concepts | about 100,000 to 300,000 | Wikidata items with a defining formula (about 113k), nLab (about 19k pages), Wikipedia mathematics (about 26k articles) and the Stacks Project (about 21k tags) overlap heavily, and each covers part of it |
| formal declarations | about 430k in Mathlib, and several million across all Lean libraries | Mathlib's own statistics; Tau Ceti is about 3M lines alone |
| open problems with a formal statement | about 1,000 open in formal-conjectures, plus FrontierMath Erdős and Annals-Challenge-style collections | those repositories |

Thus the statement plane at full graduate coverage, plus research cores, is about 1 to 2M nodes. If OEIS sequences and the most-cited objects are also nodes, it is 2 to 5M (06.7). The record plane is about 5M records and 30M results. Both fit the storage design without change.

## 18.5 The order of work

Breadth first on the record plane, depth first on the statement plane:

1. **C0 everywhere.** It is cheap. It makes everything else possible, because the source of every node needs a record to point at.
2. **C2 skeletons everywhere.** For each top-level MSC class: the 200 to 2,000 concepts and named theorems that define the area. These skeletons make the graph navigable across all of mathematics, even where the graph is shallow. This is curated work with much agent help. The proposals come from surveys, the tables of contents of textbooks, and Wikipedia's category structure.
3. **C3, then C4, for the undergraduate core.** Then the same for graduate cores, in order of Mathlib coverage, because a binding costs less than a proof.
4. **C1 everywhere,** as `tamnd/arxiv`'s content plane and the licence gate allow.
5. **Research cores on demand.** math2 builds the research core of an area when a project (13.6) or a campaign asks for it. The reason is that research-level curation needs the people who work in the area.

Two rules keep this honest:

- **The table is published every release, including the regressions.** Coverage can decrease. A Mathlib bump can break bindings, or a re-review can lower an F-level. The table shows that.
- **Counts are not the target.** People who teach the area review the composition of each core. Sometimes the node count of an area grows while its core coverage stays flat. That is a curation problem, and the coverage report is designed to show exactly that.
</document>

<document source="https://math2.tamnd.com/spec/19-milestones.md">
# 19. Milestones

There are six milestones, M0 to M5. Each one ends in a release. Someone outside the project can check that release against written exit criteria. The `m2` command table (20.2) names the milestone in which each command arrives. A command that has not arrived exits non-zero and says which milestone delivers it.

The dates assume that work starts in November 2026 with a small core team. The team is two to three engineers, one Lean lead, and a curator group that grows from five to about thirty people. Each milestone has a short "out of scope" list. Thus work that slips is visible and does not move quietly to the next milestone.

## M0: the skeleton (Nov to Dec 2026)

**Deliverables**

- Both repositories, with the layout in document 20, the licence files, and contribution guides.
- The node schema v1 (04.5) as JSON Schema, with fixtures.
- Tag minting and the provisional-id flow (04.3).
- `m2 new`, `m2 fmt`, `m2 check`: schema, body grammar and link checks on a working tree, with no Lean.
- CI for both repositories. The `lean/` package pins a Lean stable release and a Mathlib tag (07.1), and builds an empty `Math2` with the module system.
- This specification, committed as `tamnd/math2-agent/spec/`.
- The public wiki at https://tamnd.github.io/math2. `m2 site build` makes it from `main`, and each push to `main` updates it. It has a page for each node and each spec document, and JSON, Markdown and `llms.txt` for agents. The plan moved this from M3 to M0, so that readers and agents can use the graph from the start.

**Exit criteria**

- `m2 check` runs on 100 hand-written fixture nodes that cover all 11 kinds. It catches every planted error in a fixture set of 200 broken nodes.
- A contributor who is new to the project uses only the README. With it, they can add a node with a provisional id, open a PR, and have the PR pass CI.
- The wiki shows the spec, the milestones and each node on `main`, with no manual step.

**Out of scope:** status computation, graph, any agent.

## M1: the seed and the trust core (Jan to Mar 2027)

**Deliverables**

- **The bootstrap seed**: about 2,000 curated nodes, with tags that begin with `0` (04.3). They cover the concept skeleton of first- and second-year undergraduate mathematics, with sources, edges and Mathlib bindings.
- `m2 build`: Lean build, extraction (07.4), status computation (04.7) and the derived graph.
- `m2 audit` with every T, S, B, G, F and V rule (15.3), and `m2 status`.
- The checker stack: `lake check --paranoid` and the comparator against pinned statements (07.7), with checker versions recorded.
- Fidelity reviews as data (10.6), signed by hand. There is no review tool yet.
- **Release 2027.03**: the first numbered release, with a DOI, archived to Software Heritage and Zenodo.

**Exit criteria**

- Every E5 node in the seed passes replay in at least two independent kernels.
- Every seed node is at E2 or better, and at least 30% are at E5 with F2.
- The release rebuilds bit-for-bit from its manifest on a second machine (rule `R01`).
- An external Lean user reviews 50 random F2 bindings and agrees with 49 or more.

**Out of scope:** ingestion beyond Mathlib, search, the workspace.

## M2: the graph meets the world (Apr to Jun 2027)

**Deliverables**

- `m2 graph`: queries Q3 to Q7 (05.6).
- `m2 ingest` for the first sources:
  - arXiv metadata through `tamnd/arxiv`;
  - zbMATH Open;
  - Mathlib (full extraction);
  - Wikidata;
  - OEIS;
  - formal-conjectures;
  - erdosproblems;
  - Palomar;
  - the Stacks Project (tags only);
  - LMFDB (labels).
- `m2 align`, with crosswalk edges and the curation queue (16.3).
- `m2 bump`: the monthly Mathlib bump, with its report (07.9).
- `m2 release`: monthly releases from this point. Each release pins a new Mathlib tag when the bump is green.

**Exit criteria**

- **C0 at 100%** for every MSC class: every zbMATH Open and arXiv math record is in the record plane.
- Three consecutive monthly bumps complete with no hand edits outside `Math2/` repairs.
- Alignment precision at least 0.98 on the held-out set (17.3.3).
- Q1 to Q7 meet their p95 targets at reference scale. Where the real graph is still smaller, the test uses synthetic nodes (02.4).

**Out of scope:** extraction of results from full text, public search.

## M3: open to readers (Jul to Sep 2027)

**Deliverables**

- `m2 search` with all four channels (14), and the benchmark in 14.7.
- `m2 serve`: the read API, the read-only MCP profile, and the workspace's node pages, search, explore views and graph (13).
- `m2 compute`: the computational engine with certificates (08). Executable examples have a "run this" button.
- `m2 prov`: show, credit and quarantine.
- Result extraction from licence-permitted arXiv full text (09.5), for C1.
- The offline archive of each release. The public site starts in M0. M3 adds the explore views, the graph view and the API to it.

**Exit criteria**

- The search benchmark beats the best external baseline on concept lookup and paper-to-node. The release notes report every metric on which it does not.
- The workspace meets the performance and accessibility targets of 13.7.
- Any MCP client, Claude Code included, can use the read-only profile against the public endpoint with no math2-specific code.
- C2 skeletons exist for 30 top-level MSC classes.

**Out of scope:** write access through the API, autoformalization.

## M4: fidelity at scale (Oct to Dec 2027)

**Deliverables**

- `m2 review`: the reviewer's tool and review packets (10.6).
- `m2 autoform`: the formalize pipeline with the full F1 battery (10.2 to 10.4).
- Fidelity evaluation: the battery suite with its confusion matrix (17.3.1), and the monthly re-review sample (10.8).
- Write access through the API, which opens PRs (11).
- Reviewer onboarding and the credit export (16.4).

**Exit criteria**

- **The undergraduate core (18.3) is at C3 100% and C4 50%.**
- Measured F2 disagreement is under 2%, from at least 500 re-reviews.
- The battery's recall is at least 0.9 on every misformalization class with 30 or more labelled items.
- At least 50 active reviewers, none of whom is a project employee.

**Out of scope:** unattended agent runs.

## M5: the research engine (Jan to Jun 2028)

**Deliverables**

- `m2 agent run` for every role in 12.1, and `m2 orchestrate` with budgets, the escalation ladder and the work queues (11.4, 11.5).
- Role versioning, shadow weeks and the weekly agent report (12.8, 12.9).
- Projects in the workspace, with project budgets for agent work (13.6).
- The novelty check (17.5) for every explore output.
- The quarterly public run dataset (17.4).

**Exit criteria**

- **Undergraduate core at C4 80%**, and every E5 node in it at F3. Most of the F3s come from cross-formalization.
- C2 skeletons for every top-level MSC class in scope (18.2).
- People outside the core team run at least three research projects. Each project produces work that they publish elsewhere.
- Per-role solve rates and costs are published for 12 consecutive weeks.
- No agent output raised a status level without a kernel check or a person (rule audit `X`, zero violations).

## After M5

The milestones stop at M5, because at M5 math2 is a complete system. After M5, the work is about depth: graduate cores, research cores on demand, C1 everywhere that licences allow, and more formal libraries bound. Releases continue monthly. Every release reports the axes in 02.3. The targets for 2029 will come from the measurements, and math2 does not set them now.

## Costs, roughly

These are planning numbers. math2 publishes them so that a funder can compare them with the weekly reports.

| Item | M0 to M2 | M3 to M5 |
|---|---|---|
| people | 3 to 4 FTE | 5 to 6 FTE plus paid reviewer stipends |
| model inference | under $5k per month, mainly alignment and extraction in batch | $20k to 60k per month, set by budget, not demand (11.5) |
| compute (CI, Lean runners, serving) | about $3k per month | about $10k per month |
| storage and archiving | small | under $1k per month |

The model line is the only line that can grow without limit. It is also the line with the clearest controls. Batch pricing, prompt caching and the escalation ladder (12.3) keep most calls on the cheapest model that passes the suite of the role. The weekly report shows what each dollar bought.
</document>

<document source="https://math2.tamnd.com/spec/20-repository-layout.md">
# 20. Repository layout

There are two repositories. They use the same split as `tamnd/arxiv` and `tamnd/arxiv-reader`:

- `tamnd/math2` is the **content**. It is the thing that must be right.
- `tamnd/math2-agent` is the **toolchain and the agents** that build, check, grow and serve the content. Its binary is `m2`.

Content never depends on the toolchain's code. A person with a text editor, `lake` and Python can read, edit and check the core of `tamnd/math2`. For this, they never need to install `m2`. The full audit needs `m2`, in the same way as the full audit of `tamnd/arxiv` needs `ax`.

## 20.1 `tamnd/math2`

```
math2/
  README.md                      what it is, the counts table (generated), how to check it
  LICENCE                        CC BY-SA 4.0 (prose), Apache-2.0 (code), CC0 (metadata). See 16.6
  CONTRIBUTING.md                how people and agents contribute; disclosure; review
  CITATION.cff                   how to cite a release and a node

  schema/
    node.v1.json                 JSON Schema for node front matter, per kind
    review.v1.json
    computation.v1.json
    manifest.v1.json
    edges.md                     the 23 edge types (document 05.2), generated from the schema

  macros/
    m2.sty                       the only LaTeX macros a body may use, KaTeX/MathJax compatible
    m2.katex.json                the same macros for KaTeX

  nodes/
    7K/7K3QX2.md                 one file per node, sharded by the first two tag characters
    ...

  compute/
    3MZ8QD/main.gp               code for computation nodes, next to the tag
    3MZ8QD/Check.lean

  reviews/
    7K/7K3QX2.yaml               fidelity reviews, bound to statement and type hashes

  prov/
    7K/7K3QX2.jsonl              provenance, one line per field change

  registers/
    tags.reg  aliases.reg  retired.reg  bindings.reg  reviewers.reg  crosswalk.reg

  queues/
    align/wikidata.jsonl         proposed same_as, close_to and aligned edges, one file for each external system (16.3)
    dedup.jsonl                  proposed duplicate pairs (05.4)

  lean/                          the math2 Lean library (a Lake package)
    lakefile.toml                requires mathlib at the pinned commit
    lean-toolchain               the pinned Lean release
    Math2.lean
    Math2/
      Meta/Attr.lean             @[m2 "TAG"] attribute, the binding marker
      Meta/Match.lean            adapters: stating that a Mathlib decl matches a node up to a recorded transformation
      Algebra/ …                 statements and proofs math2 holds that Mathlib does not (yet)
      NumberTheory/ …
      Conjectures/ …             formal statements of open conjectures (Prop-valued defs, never axioms)
      Hypotheses/ …              the only place `axiom` may appear: named hypotheses such as GRH, for conditional results. Each is a node
      Cert/ …                    certificate checkers proved sound (LRAT, Pratt, interval)
    test/                        golden tests for the attribute and the adapters

  topics/
    msc2020.yaml                 MSC2020 codes → topic tags, with our own short labels. The official class descriptions are CC BY-NC-SA, so they are
                                 not stored here; the site fetches them from a pinned snapshot and shows them with attribution (16.6)
    curricula/                   reading lists (undergraduate core, qualifying exams). Each one is a topic with ordered members
    skeletons/                   the C2 skeleton of each top-level MSC2020 class, one file NN.yaml for each class (18.3)

  manifests/
    releases/2026.11.yaml        release manifests (document 06.2)
    releases/2026.11.levels      the evidence and fidelity level of each live node at the release
    releases/2026.11.md          the release notes: a summary by a person, and the sections that m2 release notes writes
    sources.yaml                 every external snapshot used: URL, hash, date, licence (document 09)
    images.yaml                  computational images by digest
    ingest/                      per-source import manifests
    coverage/expected.yaml       the pinned number of documents in each MSC2020 class, the denominator of C0 (18.2)

  reports/
    audit/                       committed audit reports (document 15)
    bump/                        one report for each Mathlib bump: the breaks by kind, the time to green, the stale reviews (07.9)
    graph/                       graph health
    coverage/                    per-MSC coverage tables (document 18)
    releases/                    the full pre-release report per release
    restore/                     one record for each quarterly restore drill (06.6, rule R04)
    agents/                      weekly agent acceptance and cost reports (document 12.9)
    eval/                        per-release evaluation reports (document 17.8)

  .github/
    workflows/                   check.yml (schema+body+Lean build on PR), audit.yml (full on main), bump.yml (weekly Mathlib bump), release.yml
    scripts/check_nodes.py       the Python-only check that needs no m2: schema, tags, links, sections
```

### The Lean package

`lean/` is an ordinary Lake package named `Math2`. It depends on Mathlib at the pinned commit. It holds only what Mathlib does not have. A node bound to a Mathlib declaration has no Lean code in math2 at all. It has only the binding in its front matter.

Sometimes math2 holds a theorem and Mathlib gains it later. Then the bump agent proposes two changes: the binding moves to Mathlib, and the math2 copy becomes deprecated (document 07.9). The aim is that `Math2/` gets smaller in each area as Mathlib grows. The aim is also that math2's Lean contributions move upstream (document 16.5).

`@[m2 "7K3QX2"]` on a declaration is the binding seen from the Lean side. The build checks that the attribute and the front matter agree (rule `F01`). The attribute is the only Lean metaprogramming in the content repository. Everything else that must inspect Lean (dependency extraction, type hashing, axiom audit) is in the toolchain.

## 20.2 `tamnd/math2-agent`

It is a Go module for the toolchain and the agents. It also has a small Lean package for extraction and a web front end for the workspace.

```
math2-agent/
  README.md
  go.mod                         module github.com/tamnd/math2-agent
  cmd/
    m2/                          the binary

  check/                         the fast rules of m2 check: S01 to S04, B01 to B03, G01 (15.3)
  cli/                           the command table and the flags of each command
  format/                        m2 fmt: canonical front matter, aliases to tags (04.4, 04.5)
  node/                          parse, canonicalise, hash node files (document 04)
  schema/                        schema validation, migrations
  tag/                           minting, provisional id rewrite, registers
  graph/                         derive edges, closure (AND–OR), CSR, Parquet writer (document 05)
  store/                         SQLite + DuckDB + CSR + vector serving store (05.7)
  build/                         the build graph: nodes → Lean → compute → graph → search → site
  bump/                          the Mathlib bump: pins, breaks, renames and the bump report (07.9)
  compute/                       image runner, sandbox, certificate dispatch (document 08)
  ingest/                        source adapters (document 09), one subpackage per source
    arxiv/  zbmath/  crossref/  oeis/  lmfdb/  wikidata/  mardi/  nlab/  stacks/  proofwiki/  mathlib/  leanlibs/
    afp/  palomar/  blueprint/  erdos/  formalconj/  mo/  objects/  theoremidx/
  extract/                       statement extraction from papers (09.5)
  align/                         dedup and crosswalk (05.3, 05.4)
  autoform/                      statement autoformalization and fidelity pipeline (document 10)
  search/                        lexical, formula, embedding, type-directed search (document 14)
  novelty/                       the novelty check: the searches, the judge and the verdict (17.5)
  audit/                         one file per rule, with fixtures (document 15)
  prov/                          provenance writing, PROV export, quarantine (06.5)
  release/                       release manifest, notes, archive (06.2)
  api/                           the Mathematical API: HTTP+JSON, gRPC (document 11)
  mcp/                           the MCP server over the API
  orchestrator/                  job queue, scheduler, budgets, run records (document 11)
  agent/                         the research agents (document 12)
    formalize/  prove/  explore/  explain/  curate/  align/  bump/  review-assist/
  llm/                           model clients (Anthropic SDK first; adapters for local prover models)
  sandbox/                       gVisor/Firecracker wrappers shared by compute and agents
  uri/                           m2:// parser and the ant driver
  site/                          the wiki: a static site for people and agents

  lean/                          drive Lake, the Lean REPL and the extraction package; axiom audit; replay (document 07)
    M2Extract/                   Lean package: dependency graph, elaborated-type hash, axioms, statement pretty-printing, degenerate-statement checks
    M2Extract/lakefile.toml      depends on Mathlib and on tamnd/math2's Math2 package at the pinned commits

  prompts/                       versioned prompts per agent role; each change bumps the role version (12.8)
  images/                        Nix flakes for every computational image (08.1)
  web/                           the research workspace front end (document 13)
  bench/
    graph/                       Q1 to Q12 latency benchmarks (05.6)
    search/                      retrieval benchmarks (14.7)
    fidelity/                    the F1 battery suite (17.3.1)
    extract/                     extraction and unfolding (17.3.2)
    align/                       alignment precision and recall (17.3.3)
    agents/                      per-role suites (public and fresh splits; held-out items live in a private repository) and canaries (17.3.4, 12.7)
    explain/                     explanation grading (17.3.5)
    novelty/                     the novelty check's recall suite (17.5)
    external/                    runners for external benchmarks (17.6)
    prov/                        provenance tests (06.5)
  migrate/                       schema migrations (04.9)
  docs/                          command reference, generated
  spec/                          this specification (documents 00 to 21)
```

The module has no `internal/` directory. Each Go directory is a package that other programs can import, for example a bot that reads node files with `node` and checks them with `schema`. Thus each package follows the conventions of the Go standard library. It has a package comment, a doc comment on each exported name and testable examples. It exports only what other programs need. A change to an exported name is a breaking change, and the release notes say so.

### The command set

Each command names the milestone in which it arrives (document 19). If the milestone of a command has not arrived, the command exits non-zero and says which milestone delivers it. It does not pretend to succeed. That is the `ax` convention.

| Command | What it does | Milestone |
|---|---|---|
| `m2 new <kind>` | scaffold a node with a provisional id | M0 |
| `m2 fmt` | canonicalise front matter, rewrite aliases to tags | M0 |
| `m2 check` | schema, body, links on the working tree (fast, no Lean) | M0 |
| `m2 mint [-commit C]` | change provisional ids to tags before merge, and add them to `registers/tags.reg` | M0 |
| `m2 site build` | the static wiki for GitHub Pages: HTML for people, JSON, Markdown and `llms.txt` for agents | M0 |
| `m2 build` | full build: Lean, extraction, status, graph, search | M1 |
| `m2 audit [-plane X] [-rule R] [-changed] [-release R]` | the audit (document 15) | M1 |
| `m2 status <tag>` | the computed status and the evidence behind it | M1 |
| `m2 graph closure|rdeps|path|frontier <tag>` | graph queries Q3 to Q7 | M2 |
| `m2 ingest <source> [--snapshot]` | pull a pinned snapshot of a source (document 09) | M2 |
| `m2 align <source>` | propose crosswalk edges | M2 |
| `m2 search <query>` | the search engine (document 14) | M3 |
| `m2 serve [--addr]` | the API, the MCP server and the workspace | M3 |
| `m2 compute run|verify <tag>` | the computational engine (document 08) | M3 |
| `m2 review <tag>` | the reviewer's tool: side-by-side informal and formal, back-translation, checks, sign | M4 |
| `m2 autoform <tag>` | statement autoformalization for a node | M4 |
| `m2 agent run <role> [--target T] [--budget B]` | one agent run | M5 |
| `m2 orchestrate` | the scheduler daemon | M5 |
| `m2 bump` | Mathlib/Lean bump branch (document 07.9) | M2 |
| `m2 prov show|quarantine|credit|disclose` | provenance operations; `disclose` renders a disclosure paragraph and `formalization.yaml` (16.5) | M3 |
| `m2 release cut|notes|verify` | releases | M2 |

### Why Go for the toolchain

The rest of the `tamnd` toolchain is Go: `ax`, `ant`, and the `*-cli` family. `m2 ingest` links the packages of that family as libraries. Go builds one static binary and cross-compiles easily. It has the official Anthropic SDK, DuckDB and SQLite bindings, and good sandbox control. Lean-specific work happens in Lean (`lean/M2Extract`), driven over the Lean REPL protocol. Python appears only inside computational images and in the no-dependency check script in `tamnd/math2`.

One exception may come later: a Python sidecar for local open-weight prover models (document 12.3). The reason is that the inference stacks of these models are in Python. If the sidecar exists, it sits behind a gRPC interface in `llm`. Nothing else in the toolchain knows about it.

## 20.3 How the two repositories move together

- `tamnd/math2-agent` is versioned by semver. `tamnd/math2` is versioned by calendar (document 06.2).
- Each math2 release names the `m2` version that built it. CI in `tamnd/math2` uses the `m2` version pinned in `manifests/releases/next.yaml`. A person bumps that version in a PR.
- A schema change lands in `math2-agent` first, behind a flag. Then a migration PR lands in `math2`. Then the flag is removed. The two repositories never change in a single step.
- Agents open PRs against `tamnd/math2` from a bot account (document 16.2), with branch names `agent/<role>/<run-id>`. A person merges them, or for some roles a rule-limited auto-merge does (document 12.1).
</document>

<document source="https://math2.tamnd.com/spec/21-open-questions.md">
# 21. Open questions

This document lists the decisions that are not yet settled. Each entry gives the question, the options, and the current default (what the code does until the question is settled). It also gives **what would settle it**, which is usually a measurement. When a question is settled, its entry moves to the decision list in document 00, with a link to the PR and the measurement (16.7).

## 21.1 Content and model

**Q-01. Is `concept` separate from `definition` worth its cost?**
- Options: keep both kinds (04.2); or merge them, so that a concept is just its preferred definition plus equivalences.
- Default: keep both.
- Settles it: after M2, measure how many concepts have more than one definition node. Also measure how often curators make the split incorrectly (audit `S` findings on concept/definition misuse). If fewer than 10% of concepts have a second definition and the error rate is high, merge.

**Q-02. How should proof steps be represented?**
- Options: proofs as nodes with `uses` edges only (current); or a finer structure, with steps as sub-nodes, as in Lamport-style structured proofs or the Fermat run's DAG of intermediate statements.
- Default: coarse. An intermediate statement that is worth citing becomes its own `statement` node.
- Settles it: whether explain and the prove decomposition (12.4) produce many orphan intermediate statements (flag `orphan`). If they do, a lighter sub-node kind may be cheaper than full nodes.

**Q-03. How much of the OEIS and the LMFDB becomes nodes?**
- Options: every sequence and object as a node; only those cited by another node (current for the LMFDB); or all OEIS sequences but only cited LMFDB objects (current for the OEIS).
- Default: as stated in 06.7.
- Settles it: the repository size trigger (Q-13), and whether uncited object nodes ever receive edges.

**Q-04. Should math2 adopt a second informal format besides Markdown with LaTeX?**
- Options: stay with constrained Markdown (04.5); or allow Verso documents for nodes that are mostly Lean.
- Default: Markdown only.
- Settles it: the share of blueprint imports (09.4) that come from Verso, and whether the round trip loses information.

## 21.2 Formal

**Q-05. Isabelle, Rocq and Metamath bindings at E5?**
The AFP binding path exists (07.10). Rocq and HOL Light are crosswalk only. Metamath is a candidate for v2: its verifier is tiny and has independent implementations.
- Default: Lean and AFP only.
- Settles it: demand from projects, and whether the checker of a second system can go through the same replay discipline as the Lean checker (07.7).

**Q-06. What fraction of the build should go to replay?**
`lake check --paranoid` across several kernels is slow at FLT scale. 07.7 allows `external_replay` for proofs above a size threshold.
- Default: full replay for everything under 1M lines, and an external replay record for the rest.
- Settles it: measured replay time per release. If it fits the 24-hour build budget (02.4), the threshold is removed.

**Q-07. When should Math2 code move to Mathlib?**
- Options: as soon as it is stable; only on a Mathlib maintainer's request; or never move it, and let Mathlib cherry-pick.
- Default: a person offers it when its sanity lemmas and review are complete (16.5).
- Settles it: Mathlib's own policy on AI-assisted contributions, which is evolving, and the review-queue data from the Mathlib Initiative.

**Q-08. How should math2 handle Comparator's open issues?**
Comparator had known gaps in October 2026. 07.7 adds math2's own type-hash check as a second statement check.
- Default: both checks.
- Settles it: when Comparator ships in the Lean distribution with those issues closed. Then decide whether the type-hash check is still necessary. Measure whether it ever caught anything that Comparator missed.

## 21.3 Agents

**Q-09. Should math2 run its own fine-tuned prover models?**
- Options: hosted frontier models plus open prover models as they are released (current, 12.3); or fine-tune an open model on math2's run data (17.4).
- Default: no fine-tuning.
- Settles it: whether the quarterly run dataset shows a solve-rate gap on math2's own targets. It would have to be a gap that a fine-tune could close at a lower cost per solve than the ladder. A 2026 finding makes it likely that the default holds: general models with a minimal loop match bespoke provers (12).

**Q-10. Should agents be allowed to choose explore targets?**
12.5 says no: people define campaigns. Some open problems would plausibly benefit from wide automatic sweeps. For others, there are requests that they not be "harvested" (the 2026 debates around erdosproblems.com and the `openai/math` release).
- Default: no autonomous target selection.
- Settles it: this is a community decision, not a measurement. The proposal would need maintainer approval plus an RFC open for a month.

**Q-11. How should agent output be paced to match review capacity?**
Agents can produce much more than people can review. That is the Mathlib queue problem, and the lesson of the 719-manuscript release.
- Default: the orchestrator sets a weekly limit on the PR rate of each role. The limit is a multiple of the review throughput of the previous four weeks (11.5).
- Settles it: the multiple is tuned on the weekly report (12.9). The open question is whether review-queue age is a better control signal than throughput.

**Q-12. Should Managed Agents run the explain, curate and align roles?**
12.3 lists hosted agent infrastructure as an option for the long-running roles that need no Lean session.
- Default: self-hosted loop for every role.
- Settles it: a month of shadow runs that compare cost, latency and suite scores.

## 21.4 Storage and scale

**Q-13. When does the record plane move to its own repository?**
06.1 sets the trigger: more than 10 GB packed, or more than 5 million files, in `tamnd/math2`.
- Default: one repository until the trigger.
- Settles it: the trigger. The open part is whether to split by plane (`tamnd/math2-records`) or by MSC range. Split by plane, unless clone-time measurements say otherwise.

**Q-14. Should math2 serve vectors?**
14.4 forbids the return of raw embeddings. This protects record-class text and keeps the model swappable. Some research users want vectors for their own analysis.
- Default: no vectors.
- Settles it: a licence review of whether embeddings of `open` class text could be published, and whether anyone asks for them through a project.

## 21.5 Governance

**Q-15. Who are the maintainers, and how are they chosen?**
16.7 assumes a maintainer group with two-approval rules. It does not say how the group is formed after the founding team.
- Default: founding maintainers, then nominations by existing maintainers with a public comment period.
- Settles it: a governance document, before M3. The models to consider are Mathlib's maintainer process, the OEIS editors and the Stacks Project's single-editor model.

**Q-16. Should reviewers be paid?**
19 budgets reviewer stipends from M3. Payment for review could compromise independence. Or it could be the only way that research-level review happens at all.
- Default: stipends for undergraduate and graduate core review only. Research-level review is volunteer and credited.
- Settles it: measured re-review disagreement for paid against unpaid reviews (10.8).

**Q-17. Relationship with Palomar and erdosproblems.com.**
Both are community registries with their own norms. math2 imports from both and can export to Palomar (16.5).
- Default: import and link; never write to them except by people following their processes.
- Settles it: a conversation with their maintainers, before M2.
</document>

<document source="https://math2.tamnd.com/spec/corpus/00-README.md">
# Math 2.0: the corpus specification

The main spec (the files `00` to `21` in the parent directory) says how math2 holds, checks and serves mathematics. This sub-spec says how math2 gets the mathematics: each definition, theorem, proof and example that a person can find in a public source, and what models know, as nodes that people can read and machines can check, ready to go into a course, a book or an article (K2.1).

The corpus sub-spec adds to the main spec and changes it in a small number of named places. Where the two disagree, this sub-spec is the newer text, and the change is listed in "Changes to the main spec" below. The main spec files are not edited by this sub-spec. A change becomes part of them in the same pull request that implements it.

This sub-spec was written on 2026-10-26. Its evidence base is K01, which reflects the sources as of that date.

## Documents

| # | Document | What it settles |
|---|---|---|
| 00 | README (this file) | index, reading order, settled corpus decisions, conventions, the rules that the corpus adds |
| 01 | [Research](01-research.md) | the method, the findings that change the plan, the corrections to the main spec, what could not be verified |
| 01a | [Part A: reference works](01a-reference-works.md) | encyclopedias, wikis, databases, problem lists and Q&A sites: sizes, licences, channels |
| 01b | [Part B: literature and books](01b-literature-and-books.md) | literature indexes, full text, extraction projects, open textbooks, syllabi, theorem lists |
| 01c | [Part C: formal libraries](01c-formal-libraries.md) | formal libraries, translation between systems, formal and informal datasets |
| 01d | [Part D: model generation](01d-model-generation.md) | error rates, projects at scale, memorization, the law on model output, provider terms, cost |
| 01e | [Part E: representation and packaging](01e-representation-and-packaging.md) | formats, levels of explanation, exercises, book and course formats, notation |
| 02 | [The goal](02-the-goal.md) | the goal in one sentence, items, writing levels W0 to W3, formal outcomes, what the corpus must not do, the reference scale |
| 03 | [Identifiers](03-identifiers.md) | tags of ten characters, the check character, sharding, one space for nodes and inventory rows |
| 04 | [The inventory](04-inventory.md) | mentions and rows, clustering, outcomes, which declarations and objects become nodes, measuring what is missing |
| 05 | [Human sources](05-human-sources.md) | the source document, licence reading, public domain, reference works, textbooks, literature, curricula, exercise banks, adapters, import order |
| 06 | [Formal sources](06-formal-sources.md) | intake classes, the libraries, Mathlib, libraries that AI wrote, other systems, crosswalks, datasets, informalization |
| 07 | [Machine generation](07-machine-generation.md) | what models are for, the four new roles, generation from what models know, the write job, the model register, configurations, provider terms |
| 08 | [The node card](08-the-node-card.md) | the house style, front matter, the card of each kind, proofs in steps, explanations at levels, exercises, schema changes |
| 09 | [Quality](09-quality.md) | the gates before merge, sanity checks, the copy check, source confirmation, what people check, sample audits, reader reports |
| 10 | [Structure and notation](10-structure-and-notation.md) | the taxonomy, curricula, the notation register, conventions, names and languages |
| 11 | [Packaging](11-packaging.md) | the volume file, linearization, editorial text, outputs, the pack check, licence, editions, the Lean companion |
| 12 | [Pipeline and operations](12-pipeline-and-operations.md) | the stages of an item, jobs, priority, batches, review capacity, cost, measurement, commands and libraries |
| 13 | [Law and ethics](13-legal-and-ethics.md) | the three zones, copyright in model text, copies, human sources, the EU AI Act, provider terms, disclosure, questions for a lawyer |
| 14 | [Roadmap](14-roadmap.md) | phases 0 to 5 with entry and exit criteria, what runs all the time, the people that the phases need |
| 15 | [Open questions](15-open-questions.md) | OQ01 to OQ60, each with the way to settle it and the phase that it blocks |

## Reading order

- **Overview:** K02, then K14, then the settled decisions below.
- **Before an importer is written:** K05 or K06, then K04, K13 and K15.
- **Before agents write nodes:** K07, K08 and K09, then K12.
- **Before a volume is built:** K08, K10 and K11.
- **Reviewers of the design:** K01 first, then K09, K13 and K15. K01 is the evidence; K09, K13 and K15 are where the design could be wrong.

## How the corpus relates to the main spec

The corpus uses the objects of the main spec and adds no new box to its architecture:

| The corpus needs | The main spec gives | The corpus adds |
|---|---|---|
| a place for each item | nodes and kinds (spec 04) | the inventory as a register of rows (K04), the exercise node (K8.7) |
| sources | the record plane and licence classes (spec 09) | the source document, licence reading, the public-domain rule (K05), intake classes (K06) |
| writers | agent roles and run records (spec 11, 12) | four new roles, the model register, generation from what models know (K07) |
| checks | the audit and the status levels (spec 15) | the copy check, source confirmation, sample audits, new rules (K09) |
| readers | the workspace and explanations at levels (spec 13) | the node card (K08), the notation register (K10), volumes and the pack check (K11) |

## Settled corpus decisions

These decisions are settled. A change to one of them is a change to this file and to the document that it names, in one commit, with the reason.

1. **Tags have ten characters.** Nine random characters and one check character, from Crockford base32. The tags of six characters that exist are frozen and stay valid (K03).
2. **Nodes and inventory rows share one identifier space.** A row identifier is never a node tag (K3.6, rule T08).
3. **The inventory is the denominator.** The corpus is complete when each row has an outcome. The open count is the number of rows with no outcome, and its target is zero (K04).
4. **No row from models alone.** A mention that a model gave becomes a row only when a source that people wrote confirms it (rule I03).
5. **The licence class decides what the corpus takes from a source.** Only `open` text enters the prose zone. `share-alike-nc`, `gfdl` and `record` give records, locators and links. `verbatim` gives short quotations only (K5.1, K13.4).
6. **The licence is read at the source.** An aggregator does not change a licence. The licence is read again at each snapshot, and the reading has a date and a person (K5.2.1).
7. **Public domain means the US and the EU.** A work is `open` as public domain only if it is in the public domain in both (K5.2.3).
8. **Each formal library has one intake class:** `bind`, `crosswalk`, `link`, `evaluate` or `exclude` (K6.2). Benchmarks evaluate roles; they are not sources (K6.8).
9. **Models write; they never decide what is true.** Models never raise a status, and a judge is of a family other than the writer's (K07, rule X09). This keeps settled decision 3 of the main spec.
10. **The copy check runs against a reference corpus,** not only against the cited sources. No restricted text is in the context of a write job (K9.4, K7.5).
11. **The prose zone stays CC BY-SA 4.0,** with a notice that the licence applies only where copyright subsists. The provenance says which sections software wrote (K13.2). This keeps settled decision 17 of the main spec.
12. **The exercise is a node.** Its text is in the prose zone. Its Lean statement and its reference solution are in the code zone, Apache-2.0 (K8.7, K13.1).
13. **One notation register.** Each macro has one meaning in its scope, and the LaTeX macros and the KaTeX map are generated from the register (K10.4).
14. **A volume adds no mathematics.** Each definition, statement, proof and example in a volume is a node, and glue states no mathematics (K11).
15. **Review capacity sets the speed.** Agent work is capped by what people can review, not by the cost of the models (K12.5).
16. **Phases have entry and exit criteria, not dates** (K14).

## Conventions

- **Numbering.** Corpus documents are K01 to K15, and their sections are K2.1, K2.2 and so on. A reference to "spec 09" or "spec 9.3" is to the main spec. A reference to "settled decision 17" is to the list in the main README.
- **Rules.** A rule has an id in the families of the main spec 15, and **H** (hard: the audit fails) or **S** (soft: the audit reports). The rules in this sub-spec are proposed. They become part of the audit when `m2 audit` implements them, and the main spec 15 lists them in the same pull request.
- **Not settled.** Where the sources disagree or a fact is not known, the text says "not settled" or names the question in K15, and it takes the safe course (K15.1).
- **Facts and dates.** Each fact from the research has a URL and the date of reading in K01. A figure that a later reading changes is changed in the corpus document and in the part of K01, in the same commit (K1.6).
- **Style.** The text follows ASD-STE100 Simplified Technical English.

## Rules that the corpus adds

| Rule | H/S | Checks | Where |
|---|---|---|---|
| T06 | H | changed: a tag has six or ten characters | K3.8 |
| T08 | H | the tag register and the inventory register have no value in common | K3.6, K3.7 |
| T09 | H | each tag of ten characters has a valid check character | K3.7 |
| T10 | H | no tag of six characters is minted after the cut-over commit | K3.7 |
| I01 | H | each `inventory` value in a node points to a row whose outcome is `node` or `member` with the tag of that node | K4.9 |
| I02 | H | each row with the outcome `node` or `member` points to a live node, or to a retired node with a successor | K4.9 |
| I03 | H | no row has only mentions from models | K4.9 |
| I04 | H | each outcome that a rule gave names a rule in `schema/outcomes.v1.yaml` | K4.9 |
| I05 | S | a `deferred` row whose date has passed | K4.9 |
| I06 | S | an MSC code at two digits whose estimated recall is below 90% | K4.9 |
| L05 | H | each agent-written body has a clean copy check result. Soft until the calibration is done | K9.9 |
| L06 | H | each item from a source with a licence for each item records the licence of the item | K13.9 |
| L07 | S | a node with a takedown report older than two working days that is not quarantined | K13.9 |
| L08 | H | each source document has the required fields, and each mention names a source | K5.14 |
| L09 | H | a source with a choice of licences has an `election` | K5.14 |
| L10 | H | a public-domain source of the `open` class is public domain in the US and the EU | K5.14 |
| L11 | H | a mention with text comes from a snapshot whose licence was read on or before the snapshot date | K5.14 |
| L12 | S | a source whose licence was last read more than twelve months ago | K5.14 |
| F10 | H | each mention from a library records the library, the pin, the declaration, the intake class and the licence of the item | K6.12 |
| F11 | H | no section holds text from a library of class `link`, `evaluate` or `exclude`, or from an item whose licence is not open | K6.12 |
| F12 | H | each library source document has `authored`, and each mention copies it | K6.12 |
| F13 | S | a binding to an `ai` or `mixed` declaration whose F2 review does not cover the definitions that its statement uses | K6.12 |
| P05 | H | each source that an agent added resolves to a record, and the record matches the claim | K9.9 |
| P06 | S | a source with `locator_unconfirmed` older than 180 days | K9.9 |
| X05 | H | an agent-written statement or proof has the results of the sanity checks in its run record | K9.9 |
| X06 | S | a stratum whose error rate is near its threshold | K9.9 |
| X07 | S | a theorem that a body names that resolves to no node and has no source | K9.9 |
| X08 | H | each field that an agent wrote names a model in the model register | K7.10 |
| X09 | H | each judge verdict is from a model of a family other than the writer's | K7.10 |
| N01 | H | each macro in a body has a register entry in scope, or the node defines it | K10.7 |
| N02 | H | two entries with the same glyph in overlapping scopes are the same macro, or list each other in `conflicts` | K10.7 |
| N03 | S | a curriculum puts a member before one of its prerequisites | K10.7 |
| N04 | S | a node uses a non-default option of a convention and does not say so | K10.7 |
| N05 | S | a concept with no `specializes` edge and no topic other than its MSC code | K10.7 |
| N06 | H | each macro has one `denotes`, and the macro files are the output of the register | K10.7 |
| N07 | S | a symbol in a `notation` field that no register entry has as `introduced_by` | K10.7 |

The pack check of a volume (PC1 to PC10, K11.6) is not an audit rule. It runs when a volume is built, and a volume that fails it is not published.

## Changes to the main spec

| Main spec | Change | Where |
|---|---|---|
| 00, 04.3, 04.4, 04.5, 05.1, 06.4, 06.7, 15.3, 20.1 | tags of ten characters, the shard key, the inventory register; rules T08 to T10 | K3.7 |
| 04.9 | the new edges, sections and fields of node.v1; the exercise kind in node.v2 | K8.9 |
| 1.7, 9.2, 9.3, 9.6, 16.6 | figures and licences of sources, including Kerodon as `record` | K1.4, K5.13 |
| 1.7, 9.4 | figures of formal libraries | K6.11 |
| 9.1 | adapters are `importer/<source>` | K5.11 |
| 12 | the roles `enumerate`, `write`, `informalize` and `exercise` | K7.3 |
| 15 | the rules above | this file |
| 16.6 | the zone of the Lean part of an exercise is the code zone | K13.1 |
</document>

<document source="https://math2.tamnd.com/spec/corpus/01-research.md">
# K01. Research

The corpus spec rests on five research notes, written on 2026-10-26. They are parts A to E of this document, each in its own file. This file says how the research was done, what each part covers, which findings change the plan, which corrections they make to the main spec, and what the research could not verify.

The main spec has its own research document (spec 01). The parts here extend it for the corpus. They do not repeat it, except where a figure has changed or a statement in spec 01 is wrong.

## K1.1 Method

- **One date.** Each fact was read on 2026-10-26, unless the part gives another date. Each fact has its URL next to it.
- **Polite reading.** The research made single requests, with the User-Agent `m2/research (+https://github.com/tamnd/math2-agent)`. It did not crawl and it downloaded no dump. Dump sizes come from HEAD requests and directory listings. The largest samples were three days of arXiv OAI-PMH for mathematics (1,421 records), the mathematics list of the Open Textbook Library, four Mathlib YAML files and some tree-only listings of git repositories.
- **Measured and reported.** A count that the research made itself, for example from a Wikidata query or a repository tree, is marked **[measured]**. Other counts are as the source reports them.
- **Primary and secondary.** Part D marks each fact **[P]** (paper, official page, licence text, court docket), **[S]** (a law-firm note, a tracker, news) or **[snippet]** (seen only in a search excerpt). A spec section may quote a [snippet] fact only after a person reads the primary source.
- **UNVERIFIED** marks a fact that the research could not confirm, with the reason. These facts are in K1.5 and in K15.
- **No model calls.** The research called no model API, and it used no account or key.

## K1.2 The parts

| Part | File | Scope | Used by |
|---|---|---|---|
| A | `01a-reference-works.md` | encyclopedias, wikis and databases that people wrote: Wikipedia, Wikidata, nLab, ProofWiki, PlanetMath, the EoM, MathWorld, OEIS, DLMF, LMFDB, the subwikis, π-Base, the graph and knot databases, problem lists, MathOverflow and Math.SE | K05, K13 |
| B | `01b-literature-and-books.md` | literature indexes and full text, extraction projects, open textbooks, syllabi and theorem lists | K05, K10, K13 |
| C | `01c-formal-libraries.md` | formal libraries, translation between systems, formal and informal datasets, informalization | K06 |
| D | `01d-model-generation.md` | the error rates of model-written mathematics, projects at scale, memorization and copy detection, the law on model output, provider terms, cost | K07, K09, K12, K13 |
| E | `01e-representation-and-packaging.md` | representation formats, levels of explanation, exercises, book and course formats, notation | K08, K10, K11 |

## K1.3 Findings that change the plan

**Sources (parts A and B).**

1. **Most of the Wikidata "formulas" are primes.** Of the 113,118 items with a defining formula, 105,345 are prime numbers. Only 6,560 are other formulas. Wikidata is still the best crosswalk: 6,840 MathWorld ids, 4,348 nLab ids, 2,562 ProofWiki ids, 1,224 EoM ids, 438 Metamath ids and 115 Mathlib ids.
2. **Some large sources give no text.** The nLab (20,778 pages) has no licence. The originals of the EoM are © Springer. The DLMF is © NIST and not in the public domain. Kerodon states no licence, so it is `record` and not GFDL as the main spec says. These sources give mentions, locators and links (K5.3).
3. **Some large sources give text with attribution.** Wikipedia (28,196 mathematics articles), ProofWiki (47,254 content pages, a daily dump), PlanetMath (9,477 entries), OEIS (400,709 sequences), groupprops, π-Base, DaRT and Fungrim are `open`.
4. **Access is closing.** Wikimedia set API rate limits in 2026 and rewrote its robot policy. The LMFDB has a JavaScript gate and forbids `/api/` to robots. Stack Exchange put its dumps behind a click-through that excludes the training of language models. The importers use dumps and bulk downloads, and keep each limit (K5.10).
5. **New arXiv mathematics is more open than the stock.** In a 2026 sample, 36.9% of new papers were `open`, against about 15% for all papers.
6. **Open textbooks are fewer than they look.** About 48% of the mathematics books of the Open Textbook Library are `open`, and its licence data is sometimes stale. Some well-known free books (Hatcher, *Book of Proof*, Axler) are not open for the prose zone. Table A of K5.4 lists the books that are.
7. **Extraction projects mix licences.** The CC BY-SA release of TheoremSearch includes GFDL text from the Stacks Project. Its `-permissive` release is clean. Matlas states no licence.

**Formal libraries (part C).**

8. **Mathlib has three different counts.** 430,853 theorems and definitions (the statistics page), 483,112 declarations that a user can complete, and 797,294 constants. The corpus counts the second (K6.4).
9. **Translation between systems is weak.** No tool translates the large libraries of other systems into Lean. Model translation succeeds in a minority of cases (K6.6).
10. **Benchmarks have errors.** Several formal benchmarks have measured error rates in their statements, so the corpus uses them to evaluate roles, not as sources (K6.8).

**Model generation (part D).**

11. **Checks by models alone are not enough.** Model judges accept many wrong steps, strong models "prove" false statements, and between 11.4% and 56.8% of generated citations do not exist (part D, §1.6). The checks of K09 follow from these numbers.
12. **Models reproduce text.** In ordinary use, encyclopedia-style output had more than 27% of its characters in verbatim matches of 50 characters or more with public web text. The human median is 0%. So the copy check compares with a reference corpus (K9.4).
13. **The law protects the human part.** In the United States, text that a model wrote with no human control has no copyright. The output cases are not decided. K13 gives the decisions that follow.
14. **The cost of model text is small next to review.** One million nodes cost about $4,000 to $400,000 in model calls, by configuration, before the acceptance rate. Review capacity is the limit (K12.5).

**Representation and packaging (part E).**

15. **MathML 4 is still a working draft.** Its charter target of a Candidate Recommendation in May 2026 was missed. QTI 3.0 supports MathML 3 only. So the corpus stores LaTeX with its notation register, and the packager generates MathML (K10.4, K11).
16. **No system makes a textbook from a knowledge graph.** The packaging of K11 is new and has no prior evaluation.
17. **Exercise banks are mostly non-commercial.** The WeBWorK OPL, OpenStax and APEX are NC. Exercises are math2's own text, with links to the banks (K8.7).

## K1.4 Corrections to the main spec

The research corrects the main spec in these places. The detail is in the section named:

| Main spec | Correction | Detail |
|---|---|---|
| 1.7, 9.2 | Wikidata formulas, Wikipedia, nLab, ProofWiki, EoM, Erdős problems, LMFDB, zbMATH, MathWorld and OEIS figures and licences | K5.13 |
| 9.3, 16.6 | Kerodon is `record`, not GFDL | K5.13 |
| 9.3 | the `open` share of arXiv: two figures, the stock and the 2026 flow | K5.6 |
| 9.6 | TheoremSearch: 8.4M live; take the `-permissive` release | K5.7 |
| 1.7, 9.4 | Mathlib, AFP, Agda, Megalodon, Mizar, Metamath and Tau Ceti figures | K6.11 |
| 1.2 | FLT keeps both a leanblueprint and a Verso blueprint. Only verso-flt and verso-sphere-packing are in the Verso Blueprint catalogue, and Carleson waits for a toolchain bump. "Moved to it" is too strong | part E, §1.7 |
| 16.6 | the zone of the Lean statement and the reference solution of an exercise is not stated. It is the code zone, Apache-2.0 | K13.1 |
| 9.1 | adapters are `importer/<source>`, not `internal/ingest/<source>` | K5.11 |

## K1.5 What the research could not verify

Each item below is in K15 with the question that settles it.

- **Sites that blocked or failed:** the Encyclopedia of Mathematics (HTTP 502), EuDML, the zbMATH API terms, the HathiTrust terms, CORE, the Unpaywall terms, the Internet Archive terms, GroupNames, and the current page of LMFDB downloads.
- **Terms that could not be read:** the Stack Exchange Public Network Terms of 13 November 2025 (loaded by JavaScript), and the Semantic Scholar licence of abstracts.
- **Licences that sources leave open:** the nLab, the Manifold Atlas, FindStat, House of Graphs, KnotInfo, the knowls of the LMFDB, the zbMATH knowledge graph (CC BY or CC BY-SA), the 14 AIM titles, the US port of CC BY-SA 3.0, Coquelicot, the remaining 20 Formal Frontier repositories, Verso Blueprint, LaTeXML and MMT.
- **Counts that do not agree:** the Mizar articles (1,493 or 1,498), the Tau Ceti size, the Metamath counts, the TheoremSearch rows, the ProofWiki Main Page counters, the Mathlib `100.yaml` and Wiedijk's Lean count, and the 1000+ counts.
- **Facts seen only in search excerpts:** marked [snippet] in part D and "from search" in parts A and B. Before a spec section quotes one of them as a rule, a person reads the primary source.
- **The law outside the United States** on the copyright of model output (part D, §3b).

## K1.6 Facts age

The research is a snapshot. Counts grow, licences change, and terms change without notice (the OEIS licence page says so). So:

- Each source document (K5.2) holds its own date of reading, and rule L12 reports a licence that is older than twelve months.
- Before each phase of K14, a person reads again the licences and terms of the sources that the phase imports, and updates the parts of this document that changed.
- A figure in a corpus document that comes from this research names the part and the date. A figure that a later reading changes is changed in the corpus document and in the part, in the same commit.
</document>

<document source="https://math2.tamnd.com/spec/corpus/01a-reference-works.md">
# K01, part A. Encyclopedic, reference and database sources of mathematics written by people

Research for the math2 spec. This file extends `spec/01-research-2026.md` §1.7, `spec/09-literature-ingestion.md` §9.2 and §9.3, and `spec/16-collaboration-and-governance.md` §16.6. It does not repeat what those sections already say, except where a figure has changed or a statement in the spec is wrong.

- **Read date.** Every fact below was read on **2026-10-26**, unless another date is given. Each fact has its URL.
- **Fetches.** These were polite, single-page fetches. Where a fetch tool was used, the User-Agent was `m2/research (+https://github.com/tamnd/math2-agent)`.
- **Own counts.** Some counts are mine, not the source's. They are marked **[measured]**. They come from one of these:
  - a Wikidata SPARQL query (WDQS, 2026-10-26 08:25 to 08:47 UTC);
  - a MediaWiki `siteinfo` API call;
  - a tree-only listing of a git repository (`git clone --filter=blob:none --no-checkout`, then `git ls-tree`; no file contents were downloaded).
- **No dumps.** No dump was downloaded. Dump sizes come from HEAD requests or directory listings.
- **UNVERIFIED** marks a fact that I could not confirm. The reason is given each time.
- **Licence gate.** "What math2 can take" uses the classes in spec 9.3: `open`, `share-alike-nc`, `verbatim`, `gfdl` and `record`. The three practical outcomes are:
  - **text with attribution**: `open`, so text can go into node prose in the BY-SA zone;
  - **link only**: `gfdl`, or no grant of rights; math2 stores the locator and writes its own statement;
  - **record only**: `record`; metadata, identifiers and facts, but no text.

  Facts, identifiers and numerical data are not copyrightable, so math2 can take them as CC0 metadata (16.6) from any source. The one exception is database rights in the EU sui generis sense. Rows below say so where a source asserts this.

---

## A.1 Wikipedia (mathematics)

**What it holds and its size**

English Wikipedia:
- The WikiProject Mathematics assessment table counts **33,957 pages** in the project. 33,913 are assessed and 44 unassessed.
  https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Wikipedia_1.0/Assessment
  - The page gives no date for the counts. Its last edit was 28 Sep 2026 14:10 UTC.
- The table includes non-article pages: Category 2,302, Redirect 1,658, Template 1,012, Draft 172, and 503 others.
- The article classes are FA 32, FL 3, GA 231, B 1,446, C 4,459, Start 13,847, Stub 7,741 and List 437. They sum to **28,196 article pages** [measured from the table].
  - This replaces the "about 26k (estimate)" in spec 1.7.
- By importance: Top 208, High 955, Mid 4,363, Low 15,324.

Other languages. Each wiki counts in its own way, so the figures are not comparable:
- **German**: Portal:Mathematik says it is the central page for "20.732 mathematischen Artikeln". The figure is undated; the portal was last edited 7 May 2026. https://de.wikipedia.org/wiki/Portal:Mathematik
- **French**: Projet:Mathématiques has 12,192 project articles, of which 6,825 are evaluated (56%). It was "mis à jour 25 octobre 2026". https://fr.wikipedia.org/wiki/Projet:Mathématiques/Évaluation
- **Japanese**: Portal:数学 shows 8,865 articles. The page was last updated 1 Apr 2026, and the count is undated. https://ja.wikipedia.org/wiki/Portal:数学
- **Russian**: Проект:Математика lists 3,928 articles. This is the *project-tagged* set, not all mathematics articles. The figure is undated and "обновляется автоматически". https://ru.wikipedia.org/wiki/Проект:Математика
- **Chinese**: Wikipedia:数学专题 says "数学专题目前已有5,682篇条目". Its assessment tables sum to about 6,033. https://zh.wikipedia.org/wiki/Wikipedia:数学专题
  - The timestamp on the statistics equals the fetch time, so it is probably a live magic word, not a date of the count.

**Licence of the text**
- CC BY-SA 4.0. Text is also under the unversioned GFDL "unless otherwise noted". A page that includes text available *only* under CC BY-SA is not available under the GFDL. https://en.wikipedia.org/wiki/Wikipedia:Copyrights
- The dumps page says the same. It also says Wikidata structured data is CC0. https://dumps.wikimedia.org/legal.html

**Bulk channels and their terms**
- **XML dumps.**
  - In 2026 the `enwiki` dump directory has runs only on the 1st of each month: 20260401 to 20261001. https://dumps.wikimedia.org/enwiki/
    - Wikimedia used to run twice a month. Whether that changed formally is UNVERIFIED: I saw only the listing, not an announcement.
  - `enwiki-latest-pages-articles-multistream.xml.bz2` is 26,917,225,724 bytes, dated 08 Oct 2026. https://dumps.wikimedia.org/enwiki/latest/
- **HTML ("enterprise_html") dumps** on dumps.wikimedia.org. The last run listed is **20250320**. https://dumps.wikimedia.org/other/enterprise_html/runs/
  - The free public HTML snapshots look stopped, or moved to Wikimedia Enterprise. The reason is UNVERIFIED.
- **Wikimedia Enterprise.** https://enterprise.wikimedia.com/pricing/
  - Free account: Snapshot API with monthly updates, "up to 30/mo" requests and "up to 1,500/mo" snapshot chunks; On-demand API up to 50,000 requests a month.
  - Paid account: daily snapshots, the Realtime API, and bespoke pricing.
  - "Every response includes machine-readable license information."
- **API rate limits.** These are new in 2026 and "subject to experimentation and change". https://www.mediawiki.org/wiki/Wikimedia_APIs/Rate_limits (revision 8654799, edited 15 Oct 2026)
  - The limits are per minute, per user:

    | Client | Limit |
    |---|---|
    | unidentified (IP only) | **10** |
    | compliant User-Agent only | 200 |
    | authenticated, new user | 200 |
    | established editor | 2,000 |
    | bot flag, WMCS or known client | exempt |

  - Rollout: low limits from early March 2026, identified tiers from early April 2026. https://www.mail-archive.com/wikitech-l@lists.wikimedia.org/msg97202.html (2 Mar 2026; summary from search, not read in full)
- **Robot policy.** This rewrites the 2009 policy; the page was last edited 16 Mar 2026. https://wikitech.wikimedia.org/wiki/Robot_policy
  - Crawl only `/wiki/Article_name`, with fewer than 10 concurrent requests and fewer than 20 requests per second on average.
  - REST API: unauthenticated, concurrency 3 and fewer than 5 requests per second; authenticated, up to 10 per second.
  - Action API: unauthenticated, concurrency 1 and fewer than 5 requests per second.
  - "Consider dumps before making live requests."
  - There is no AI-specific clause.

**AI terms and 2024–2026 changes**
- **10 Nov 2025.** The Wikimedia Foundation asked AI developers to attribute the human contributors and to use Wikimedia Enterprise instead of scraping. It made no legal threat. https://wikimediafoundation.org/news/2025/11/10/in-the-ai-era-wikipedia-has-never-been-more-valuable/ (the post itself was not fetched; content from https://techcrunch.com/2025/11/10/wikipedia-urges-ai-companies-to-use-its-paid-api-and-stop-scraping/)
- **15 Jan 2026.** Enterprise announced Amazon, Meta, Microsoft, Mistral AI and Perplexity as partners, alongside Google and others. https://enterprise.wikimedia.com/blog/wikipedia-25-enterprise-partners/ (from a search summary)
- **No licence change.** The text stays CC BY-SA 4.0. What these deals buy is volume and an SLA, not rights.
- **Rogue OpenAI agents.** On 5 to 6 Oct 2026 the Foundation reported that rogue OpenAI agents made millions of API requests and that this may have contributed to the WDQS outage of 7 to 11 May 2026. https://www.theregister.com/ai-and-ml/2026/10/06/wikimedia-foundation-comes-forward-as-latest-openai-agent-assault-victim/5301400 ; https://thehackernews.com/2026/10/wikimedia-says-openai-agents-tried-to.html (from search summaries)
  - This is a likely driver of the stricter limits.
  - Wikipedia's own article on the Foundation states the dates in a confusing way.

**What math2 can take:** `open`.
- Short text with attribution can go into node prose, because CC BY-SA 4.0 equals the node prose zone. Each item needs the page and revision ID, and attribution through the history URL.
- Get articles from the monthly dump or the free Enterprise tier, never by crawling.
- An imported page may contain CC BY-SA-only text. That text cannot be used under the GFDL, but math2 does not need the GFDL.

---

## A.2 Wikidata

**What it holds** [measured, WDQS, 2026-10-26 ~08:25 UTC]
- **Defining formula (P2534): 113,118 items.** But **105,345 of these are instances of *prime number* (Q49008)**. Megaprimes, gigantic primes and Mersenne numbers add more.
  - Only **6,560** items with P2534 are *not* primes or prime-like. That is the real pool of mathematical formulas.
  - Only **5,492** P2534 items have an English Wikipedia article.
  - The top non-prime classes are formula 865, mathematical concept 458, theorem 436, function 95 and physical law 74.
  - **This changes the meaning of the "about 113k items with a defining formula" figure in spec 1.7 and 9.2.**
- **Theorem.** Instances of theorem (Q65943), counting subclasses: **3,208**. This matches spec 1.7.
- Crosswalk properties, as `P-id`: label, number of items:

  | Property | Label | Items |
  |---|---|---|
  | P2812 | MathWorld ID | 6,840 |
  | P4215 | nLab ID | 4,348 |
  | P8427 | KnotInfo ID | 2,978 |
  | P6781 | ProofWiki ID | 2,562 |
  | P8426 | Knot Atlas ID | 2,228 |
  | P7554 | Encyclopedia of Mathematics article ID | 1,224 |
  | P7726 | PlanetMath ID | 951 |
  | P12975 | Lexikon der Mathematik entry ID | 603 |
  | P829 | OEIS ID | 469 |
  | P12888 | Metamath statement ID | 438 |
  | P3285 | Mathematics Subject Classification ID | 408 |
  | P12934 | The House of Graphs ID | 213 |
  | P11497 | Digital Library of Mathematical Functions ID | 199 |
  | P14534 | Mathlib Declaration ID | 115 |
  | P12987 | LMFDB knowl ID | 109 |
  | P5610 | OpenMath ID | 54 |
  | P14106 | Erdős Problem number | 3 |

  Other relevant properties:

  | Property | Label | Items |
  |---|---|---|
  | P7263 | Prime Pages ID | 126,184 (the source of the prime items) |
  | P4252 | All-Russian Mathematical Portal ID | 5,090 |
  | P1482 | Stack Exchange tag | 7,215 (all sites) |
  | P894 | zbMATH Open document ID | 1,888 |
  | P889 | Mathematical Reviews ID | 141 |
  | P6830 | swMATH work ID | 125 |

  The query that found these matched property labels, so a property with an unusual label may be missing.
- **Gaps.** There is no Wikidata property for the Stacks Project, Kerodon, groupprops, Topospaces, π-Base, ISGCI, DaRT, FindStat, Fungrim or the Wolfram Functions Site. My label search found none.
  - P14106 (Erdős Problem number) is used on only **3** items, against 1,221 problems.

**Licence:** structured data CC0. Unstructured content, such as talk pages, is CC BY-SA 4.0. https://dumps.wikimedia.org/legal.html

**Bulk channels and their terms**
- **JSON dump.** `latest-all.json.bz2` is 103,488,697,311 bytes, dated 21 Oct 2026. There are also `.nt` and `.ttl` dumps. https://dumps.wikimedia.org/wikidatawiki/entities/
- **WDQS.** https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual
  - The hard timeout is 60 s.
  - One client, defined as User-Agent plus IP, gets 60 s of processing time per 60 s.
  - The limit is 5 parallel queries per IP and 30 error queries per minute. Clients over a limit get HTTP 429.
  - The manual says WDQS is "migrating to a new backend in 2026–2027" with new v2 endpoint URLs.
  - The details of the scholarly graph split were UNVERIFIED: the manual page does not describe them.
- The general API rate limits (A.1) apply.

**AI terms:** none specific; CC0. See A.1 for the May 2026 WDQS outage.

**What math2 can take:** `open`, CC0. All of it can go into the metadata zone.
- Use the weekly dump, not WDQS, for anything bulk.
- Treat P2534 as a source of formulas only after removing the prime-number items.

---

## A.3 nLab

**What it holds and its size**
- **20,778 pages** [measured]: the count of `content.md` files in `ncatlab/nlab-content` at the last commit, 2026-10-25T23:35:34Z ("Revision ID: 236426 … Page name: category theory").
  - The tree is not truncated: 73,134 entries in all. https://api.github.com/repos/ncatlab/nlab-content/git/trees/master?recursive=1
  - The repository has 208,017 commits. Each wiki edit is mirrored as a commit, in Markdown plus itex2MML. https://github.com/ncatlab/nlab-content
  - This replaces "about 19,193 entries (~2024)" in spec 1.7.

**Licence:** **none.**
- The HomePage says: "Using and distributing content obtained from the nLab is free and encouraged if you acknowledge the source" and "There is currently no consensus on a more formal license statement". It suggests checking the nLab homepages of individual contributors. https://ncatlab.org/nlab/show/HomePage
- The GitHub repository has no licence file (`license: null`). https://api.github.com/repos/ncatlab/nlab-content
- The history of the question is in a 2013 nForum thread, where the CC link was removed from the HomePage. https://nforum.ncatlab.org/discussion/3545/
  - In that thread Toby Bartels says a licence cannot be retroactive, and that copying nLab text into Wikipedia would be illegal even with a licence statement.
- There was no change in 2024 to 2026 that I could find.

**Bulk channel:** the git mirror, updated within minutes of each edit. No API terms are stated.

**AI terms:** none stated.

**What math2 can take:** **link only**, with records and identifiers.
- The "free … if you acknowledge the source" permission covers using and distributing the text. It grants no clear right to adapt and relicense under CC BY-SA, and it has not been agreed by the many authors. So the gate's `record`/link-only class still applies.
- Page names and the link graph are facts, so math2 can take them as CC0 metadata.
- To lift the restriction, math2 would need an explicit statement from the nLab steering committee. Even then, it could only cover authors who agree.

---

## A.4 ProofWiki

**What it holds and its size**
- Special:Statistics: **47,254 content pages**, 128,658 pages in all, 821,010 edits, 5,412 registered users, **12 active users** in the last 30 days. https://proofwiki.org/wiki/Special:Statistics
- The Main Page reports "30,348 Proofs" and "36,102 Definitions". https://proofwiki.org/wiki/Main_Page
  - These equal the figures in spec 1.7. Whether the Main Page counters are live is UNVERIFIED.

**Licence:** **CC BY-SA 3.0 and the GFDL** for all content except talk pages and user pages: "All content, except for that on Discusson pages and user home pages, is subject to the CC-by-SA 3.0 licence" and the GNU Free Documentation Licence. Talk pages are exempt from both. https://proofwiki.org/wiki/ProofWiki:Copyrights

**Bulk channel:** **a daily XML dump is live.**
- `https://proofwiki.org/xmldump/latest.xml.gz` returned HTTP 200, `content-length: 36599568` (about 36.6 MB gzipped), `last-modified: Mon, 26 Oct 2026 04:48:09 GMT`. This was a HEAD request only.
- The old URL `latest.xml` returns 301.
- Historical copies exist on the Internet Archive (WikiTeam 2017). https://archive.org/details/proofwikiorg_w-20170808-wikidump

**AI terms:** none stated. NaturalProofs (2021) and MathPile (2023) built training sets from the dump. https://arxiv.org/abs/2104.01112 ; https://arxiv.org/pdf/2312.17120

**What math2 can take:** `open`. Proofs and definitions can go into node prose with attribution.
- CC BY-SA 3.0 adaptations may be licensed under BY-SA 4.0, as spec 16.6 already says.
- Use the daily dump, not page crawls.
- Record the page title and revision ID for each item.

---

## A.5 PlanetMath

**What it holds and its size**
- **9,477 `.tex` entries** [measured]: tree-only clones of all 64 repositories in the `planetmath` GitHub organisation. Each repository is one top-level MSC class, plus `fem2016`. https://github.com/planetmath
  - For example: 03 Mathematical logic 906, 11 Number theory 1,071, 54 General topology 483.
- **The project is inactive.** The most recent push to any repository was 2024-02-19 (`16_Associative_rings_and_algebras`), and most last changed in March 2018.
  - The `planetmath.org` server's front page has `Last-Modified: Sat, 10 Mar 2018`.
  - Wikipedia's infobox says "Inactive". https://en.wikipedia.org/wiki/PlanetMath

**Licence:** **CC BY-SA 3.0 Unported**. The `LICENSE` file in each repository is the CC BY-SA 3.0 legal code. https://raw.githubusercontent.com/planetmath/11_Number_theory/master/LICENSE

**Bulk channel:** git. No API.

**AI terms:** none.

**What math2 can take:** `open`. Text can go in with attribution. Record the repository and the file path.

---

## A.6 Encyclopedia of Mathematics (encyclopediaofmath.org) and the Springer *Encyclopaedia of Mathematics*

**History**
- The original is the Soviet *Matematicheskaya entsiklopediya* (1977, ed. I. M. Vinogradov).
- The English edition, edited by Michiel Hazewinkel, came out from Kluwer in 10 volumes (1987–1994), with Supplements I to III (1997, 2000, 2002) and a CD-ROM (1998).
- Kluwer became part of Springer in 2003. "The 2002 version contains more than 8,000 entries".
- The free static online version redirected to the wiki from 29 Nov 2011. The wiki was "a collaboration between Springer and the European Mathematical Society". https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics
- An undated notice on the wiki's main page says the EoM "has moved from Springer Verlag to EMS Press". From search snippets, the snapshot is about mid-2020. https://encyclopediaofmath.org/wiki/Main_Page

**Licence: mixed, and mostly NOT open.**
- The main page says: "The original articles from the Encyclopaedia of Mathematics remain copyrighted to Springer but any new articles added and any changes made to existing articles within encyclopediaofmath.org will come under the Creative Commons Attribution Share-Alike License".
- The License Notice says new text is co-licensed under CC BY-SA 3.0 Unported and the GFDL. https://encyclopediaofmath.org/wiki/Encyclopedia_of_Mathematics:License_Notice
- Both statements were read through search-engine snippets. **The live site returned HTTP 502** on every attempt on 2026-10-26, between about 08:30 and 08:45 UTC, so the current wording is **UNVERIFIED**.
- Whether the move to EMS Press transferred or relicensed Springer's copyright is **UNVERIFIED**. I found no statement either way.
- A Creative Commons wiki case page from 2010 lists the Springer EoM as "copyright None" and "Open or Free Statement: no". https://wiki.creativecommons.org/wiki/Springer_-_Springer_Link_-_Encyclopaedia_of_Mathematics
- An EMS Magazine article (2025) by FIZ Karlsruhe staff says EoM runs the same wiki software as Wikipedia "albeit under a different license and editorial policies". zbMATH Open entity linking now links to EoM, nLab, MathWorld and Wikipedia. https://euromathsoc.org/magazine/articles/252 (from search summary)

**Size:** UNVERIFIED (the site was down). Wikidata has 1,224 items with an EoM ID (P7554) [measured].

**Bulk channel:** none known. There is a MediaWiki API (`api.php`); it was also 502 on the read date. No dump.

**AI terms:** none found.

**What math2 can take:** **record only, by default.**
- Most articles are derived from Springer-copyrighted originals. An edited article is a derivative of that text, so its BY-SA grant covers only the changes.
- Only an article that the page history shows to be wholly new on the wiki is `open`.
- Spec 1.7 and 9.2 list EoM as "CC BY-SA 3.0". **That is wrong for most of the text and should change.**

---

## A.7 Wolfram MathWorld

**Size:** **16,939 entries**, "Last Updated: Sun Oct 25 2026". https://mathworld.wolfram.com/ (Spec 1.7 says 16,849.)

**Licence and terms:** proprietary. https://www.wolfram.com/legal/terms/mathworld.html
- "these pages may not be copied, mirrored, redistributed, printed, or reproduced in bulk without permission."
- "Reproduction for commercial purposes is not permitted, nor is use of robots to create archival copies."
- Prefetching "may violate the above proscription against bulk downloading."
- "Limited use of material for academic research and teaching is permitted, providing conventional citation standards are followed" (quoted from the search snippet of the same page).
- The terms say nothing about AI.
- `robots.txt` says "Bulk downloading of these pages is not permitted for the purposes of making archival copies" and disallows only `/random.html`. It links a sitemap. https://mathworld.wolfram.com/robots.txt

**Bulk channel:** none. Wikidata P2812 holds 6,840 MathWorld IDs [measured], so the crosswalk does not need the site.

**What math2 can take:** **record only**: entry IDs from Wikidata, plus links. This confirms the spec.

---

## A.8 OEIS

**Size:** "Contains **400709** sequences". https://oeis.org/ (Spec 1.7 has 400,673 on 24 Oct.)

**Licence:** CC BY-SA 4.0. https://oeis.org/wiki/The_OEIS_End-User_License_Agreement (revision approved 24 Feb 2023)
- Attribution "should credit The Online Encyclopedia of Integer Sequences and provide a URL to the main page https://oeis.org/ or to a specific sequence".
- The EULA "may change … from time to time, without advance notice".

**Bulk channels**
- The download options are `stripped.gz`, `names.gz` and the git repository `oeis/oeisdata`, which has "incremental updates". https://oeis.org/wiki/Download
- `oeisdata` was last pushed 2026-10-26T07:03:53Z (commit "oeis data export"). GitHub reports its licence as "Other". https://api.github.com/repos/oeis/oeisdata
- `robots.txt` has `Crawl-Delay: 10`, disallows `/search`, `/edit` and `/history`, and fully blocks one named bot ("ShapBot"). https://oeis.org/robots.txt

**AI terms:**
- None in the EULA or the Download page.
- The Terms of Use text contains no clause on robots, automation, scraping, bulk use, machine learning or crawling. I checked by keyword search over https://oeis.org/OEISTermsOfUse.odt.txt.

**What math2 can take:** `open`, with text and attribution. This confirms the spec. Use `oeisdata` only.

---

## A.9 DLMF (NIST Digital Library of Mathematical Functions)

**Version:** 1.2.8, released 2026-09-15. https://dlmf.nist.gov/about/notices

**Licence:** **NIST-owned copyright; it is not public domain.**
- "All materials on this website are owned by NIST". NIST holds copyrights assigned by the DLMF authors.
- "Limited copying and internal distribution of the content of these pages is permitted for research and teaching."
- "Reproduction, copying, or distribution for any commercial purpose is strictly prohibited."
- "Bulk copying, reproduction, or redistribution in any form is not permitted."
- Excessive downloading may get the offending subnet blocked.
- There is nothing on AI.
- This is a surprise, because works of the US government are usually public domain. The DLMF authors were mostly not federal employees, and they assigned copyright to NIST.

**Bulk channel:** none. Wikidata P11497 holds 199 DLMF IDs [measured].

**What math2 can take:** **record only**.
- Equation numbers such as "DLMF 5.2.1" are stable locators.
- Each formula is a fact, so math2 can restate it in its own words or LaTeX, with a citation. It must not copy the text or the commentary.

---

## A.10 Fungrim (Mathematical Functions Grimoire)

**Size:** "457 symbols, 3130 entries (definitions, formulas, tables, plots)", 82 topics. https://fungrim.org/
- The page build timestamp is 2021-03-15. The repository was last pushed 2022-02-24. https://api.github.com/repos/fredrik-johansson/fungrim
- **It is dormant.**

**Licence:** **MIT** ("Fungrim is provided under the MIT license"). The repository `LICENSE` reads "Copyright (c) 2019 Fredrik Johansson".

**Bulk channel:** git. The entries are machine-readable Python/Grim expressions, with symbolic formulas and conditions.

**What math2 can take:** `open`. MIT is permissive: keep the notice in provenance.
- Fungrim is unusually valuable, because each formula has explicit assumptions. That makes it a good source for `statement` nodes, and later for Lean.

---

## A.11 Wolfram Functions Site

**Size:** **307,409 formulas** "as of October 26, 2026". https://functions.wolfram.com/

**Licence:** "© 1998–2026 Wolfram Research, Inc.".
- Neither the home page nor the About page states any terms of use.
- The About page says formulas are downloadable as notebooks and PDFs, and in MathML and ASCII. https://functions.wolfram.com/About/
- `robots.txt` allows crawling except `/webMathematica`. It lists 15 sitemaps.
- The general Wolfram terms that apply are UNVERIFIED (not found).

**What math2 can take:** **record only**: links, and formula IDs as locators. The formulas are facts, so math2 restates them in its own words.

---

## A.12 LMFDB

**Licence** (from the source template, because the site gates non-browser clients). https://raw.githubusercontent.com/LMFDB/lmfdb/main/lmfdb/templates/license.html
- "The database underpinning the LMFDB is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC-BY-SA)".
- The code is GPL v2 or later.
- The page says copyright is in the collection: "completeness claims are protected … as are search results for queries. Individual facts … are not".
- On AI: "How does this apply to training an AI model using this data? See this guidance from Creative Commons. We also encourage you to use more efficient access options rather than scraping the website directly, and to rate limit your website requests".

**2025–2026 changes in access**
- **`www.lmfdb.org` now redirects requests to `beta.lmfdb.org/gate.html`.**
  - `gate.html` is a JavaScript page that sets a `human=1` cookie. It has `Last-Modified: Thu, 14 Aug 2025`.
  - A client without JavaScript gets "Enable JavaScript, then reload."
  - I observed this on 2026-10-26 with curl. I did not try to bypass it.
- **`robots.txt` disallows `/api/`** for all agents. It also has `Crawl-delay: 30` and fully blocks two named bots. https://www.lmfdb.org/robots.txt
  - So the "API over 26 collections" in spec 1.7 is off-limits to robots.
  - The homepage footer reads "LMFDB Release 1.2.1 · SageMath 10.5". https://www.lmfdb.org/

**Size:** not re-measured, because of the gate. The spec 1.7 figures are from 2022.

**What math2 can take:** `open` (CC BY-SA 4.0).
- Get the data from LMFDB's bulk downloads, or by agreement. Do not use the API or scrape the site.
- Knowl text, the LMFDB's short explanations, is part of "the database", so presumably CC BY-SA too. That knowls fall under the data licence is UNVERIFIED: the licence page does not mention them.

---

## A.13 The subwikis: groupprops, Topospaces, Commalg, Diffgeom, Market

These are MediaWiki `siteinfo` figures [measured].

| Wiki | Articles | Pages | Edits | Active users | Licence |
|---|---|---|---|---|---|
| groupprops.subwiki.org | 8,076 | 15,469 | 51,383 | 1 | CC BY-SA 3.0 |
| topospaces.subwiki.org | 825 | 1,450 | 3,862 | 1 | CC BY-SA 3.0 |
| commalg.subwiki.org | 429 | 708 | 1,630 | 1 | CC BY-SA 3.0 |
| diffgeom.subwiki.org | 435 | 716 | 1,521 | 1 | CC BY-SA 3.0 |
| market.subwiki.org | 223 | 464 | 1,638 | 1 | CC BY-SA 3.0 |

- **Source:** `https://<wiki>.subwiki.org/w/api.php?action=query&meta=siteinfo&siprop=statistics|rightsinfo`. The `rightsinfo` field is "https://creativecommons.org/licenses/by-sa/3.0/".
- **Bulk channel:** the MediaWiki API (Special:Export). No published dump was found.
- **AI terms:** none.
- **What math2 can take:** `open`.
  - Groupprops is the most useful: it has structured "property implications" and "facts" pages about groups, and it suits `statement` candidates.
  - Fetch it page by page and slowly. The site is small and maintained by one person.

---

## A.14 Manifold Atlas (MPIM Bonn)

**Size:** **215 articles**, 2,721 pages, 17,516 edits, 0 active users [measured, `siteinfo`].
- It runs MediaWiki 1.18.4 and is reachable over HTTP; HTTPS refused the connection on the read date. http://www.map.mpim-bonn.mpg.de/api.php
- It has an editorial board and refereed pages.

**Licence: self-contradictory.** The Manifold Atlas:Copyrights page says "the copyright rules are the same as for Wikipedia, the GNU Free Documentation License 1.3", and then that "The content of the Manifold Atlas is released into the public domain". The `rightsinfo` field is empty.

**What math2 can take:** treat it as `gfdl`, **link only**, until the editors clarify. The public-domain sentence is not a clean dedication: there is no CC0 and no waiver text. Records and links are fine.

---

## A.15 π-Base

**Size** [measured, `pi-base/data` tree at commit 2026-10-24T21:57:19+02:00, "Torus initial PR (#1777)"]: **225 spaces, 247 properties, 935 theorems, and 2,147 space–property assertions**. This matches spec 1.7.

**Licence:** **CC BY 4.0**. The GitHub licence of the `pi-base/data` repository is `CC-BY-4.0`. https://api.github.com/repos/pi-base/data

**Bulk channel:** git (Markdown plus YAML). It was pushed 2026-10-26.

**What math2 can take:** `open`. Theorems are implications between properties, and they are good `statement` candidates.

---

## A.16 House of Graphs

**Size:** "nearly 22 000 graphs" in the 2.0 paper (2022). https://arxiv.org/abs/2210.17253
- The site is a JavaScript application.
- The API endpoints I tried (`/api/statistics`, `/api/enums/invariants`) return `{"error":true,"message":"Unauthorized"}`.
- The news entry of 23 Apr 2026 added the invariant "independent domination number" (from a search summary of https://houseofgraphs.org/).
- **The current count of graphs and of invariants is UNVERIFIED.** The site shows nothing without JavaScript, and the API needs authentication. The 2022 paper says 44 invariants; spec 1.7 says 52.

**Licence:** none found. Users are asked to cite Coolsaet, D'hondt and Goedgebeur, *Discrete Appl. Math.* 325 (2023) 97–107.

**What math2 can take:** **record only**.
- graph6 strings and invariant values are facts, so they are fine as CC0 metadata.
- Wikidata has 213 House of Graphs IDs (P12934) [measured].

---

## A.17 ISGCI (Information System on Graph Classes and their Inclusions)

**Size:** **1,695 graph classes, 242,975 inclusions, 30,147 complexities, 52,029 bounds**, "updated 2026-10-22". The latest news item is dated 2025-11-16. https://www.graphclasses.org/

**Licence:** none stated. The site shows only "by H.N. de Ridder et al. 2001-2014" and an Impressum.

**Bulk channel:** none listed on the page. The accompanying Java application loads the data; the format and URL are UNVERIFIED.

**What math2 can take:** **record only**.
- Class IDs (`gc_N`) are stable locators.
- Inclusion facts are facts and can be CC0 metadata. The collection as a whole may still carry database rights in the EU (ISGCI is maintained in Germany).

---

## A.18 Database of Ring Theory (DaRT)

**Size** [measured, `rschwiebert/dart_data` tree at commit 2026-10-24]:
- **218 rings, 178 ring properties, 72 theorems, 46 modules, 94 module properties.**
- It also holds logic (implication) files, 574 ring citation files and 269 publication files.
- The repository was pushed 2026-10-25.

**Licence:** **CC BY 4.0** (`dart_data` repository licence). https://api.github.com/repos/rschwiebert/dart_data
- MathBases lists DaRT's licence as MIT, which probably refers to the app code. https://mathbases.org/d/ring-theory (from a search summary)

**Bulk channel:** git (YAML). The site is https://ringtheory.herokuapp.com/

**What math2 can take:** `open`.

---

## A.19 KnotInfo / LinkInfo and the Knot Atlas

**KnotInfo** (Livingston and Moore)
- It covers knots of crossing number 3 to 13. The site's option list goes up to "3-13".
- I counted about 116 invariant fields on the search page, which is my own tally.
- Citation form: "KnotInfo: Table of Knot Invariants, https://knotinfo.org/ (May 16, 2026)" (from search).
- The old host `knotinfo.math.indiana.edu` no longer resolves (DNS failure on 2026-10-26).
- **Licence:** none on the site. https://knotinfo.org/
- **Download:** the full invariant table is an Excel spreadsheet, about 16 MB zipped. There is also a polynomial file (8 MB), 3D coordinates (46 MB) and diagrams (about 9,000 files). https://knotinfo.org/homelinks/database_download.php
- **A licence surprise.** The `database_knotinfo` Python package (version 2026.10.5) redistributes "Complete contents of the KnotInfo and LinkInfo databases in CSV format" under **GPL-3.0**. https://api.github.com/repos/soehms/database_knotinfo ; https://pypi.org/pypi/database-knotinfo/json

**Knot Atlas** (Bar-Natan and Morrison)
- **2,710 articles** but 1,651,548 pages in all, because data pages are generated. 1,729,610 edits, 1 active user [measured, `siteinfo`]. https://katlas.org/api.php
- **Licence:** `rightsinfo` is empty. No licence page was found: `Knot_Atlas:General_disclaimer` returns 404.
- The data is produced by the `KnotTheory`` Mathematica package. https://katlas.org/wiki/Main_Page (last edited 6 May 2026)

**What math2 can take:** **record only** for both.
- Knot names (Rolfson notation, `11n_34` and similar) and invariant values are facts. Wikidata has 2,978 KnotInfo IDs and 2,228 Knot Atlas IDs [measured].
- Do not import the GPL-3.0 CSV package as text. GPL does not fit into the CC BY-SA or CC0 zones, and the facts can be taken without it.

---

## A.20 ATLAS of Finite Group Representations (v3)

**Size:** "information (including 5215 representations) on about 716 groups", version 3.004. https://brauer.maths.qmul.ac.uk/Atlas/v3/

**Licence:** none stated on the site.
- The GAP interface `AtlasRep` is GPL-3.0-or-later. Version 2.1.12 came out in August 2026 (from search). https://docs.gap-system.org/pkg/atlasrep/doc/chap0.html
- The package archives do **not** contain the data; it is fetched on demand (from search, AtlasRep documentation).
- The mirror is http://atlas.math.rwth-aachen.de/Atlas/v3 (from search).

**What math2 can take:** **record only**: group names, representation identifiers and facts.

---

## A.21 Open Problem Garden

- **Licence:** GFDL. The footer says "Content distributed under" with a GNU FDL badge whose image file is named `gnu-fdl3`, so probably version 1.3. The exact version is UNVERIFIED. http://www.openproblemgarden.org/
- **Size:** the subject counts sum to 713. Problems can appear in several subjects, so this is an upper bound on distinct problems. The Recent Activity list shows no dates. This matches spec 1.3.
- **What math2 can take:** `gfdl`, **link only**. This confirms the spec.

---

## A.22 Erdős problems

- **Size:** "There are 1221 problems in the database." https://www.erdosproblems.com/
- **Licence of the problem text: none stated on erdosproblems.com.** The site has no licence or terms of reuse.
- **The GitHub repository `teorth/erdosproblems` is Apache-2.0, but `data/problems.yaml` contains no problem statements.** It holds only these fields: number, prize, `informal_status`, `formal_status`, status, oeis, formalized and tags. Its 1,221 entries have zero `statement` fields [measured]. https://raw.githubusercontent.com/teorth/erdosproblems/main/data/problems.yaml (repository pushed 2026-10-25)
  - **Spec 9.2 says math2 takes "problem statements" from `teorth/erdosproblems`. That is wrong.** The statements are on Bloom's site with no licence. The only Apache-2.0 statements are the *formal* ones in DeepMind's formal-conjectures.
- **What math2 can take:**
  - from the repository: status and links, as `open` metadata;
  - from erdosproblems.com: the informal statements, as **record only**, by link and problem number;
  - from formal-conjectures: the formal statements, `open` (Apache-2.0).

  math2 writes its own informal statement for each problem.

---

## A.23 Kourovka Notebook (Unsolved Problems in Group Theory)

- It is on arXiv as **arXiv:1401.0300**. The latest version, **v49, is dated 20 Oct 2026**. It is the **21st edition**, "which contains 150 new problems". https://arxiv.org/abs/1401.0300 ; https://export.arxiv.org/api/query?id_list=1401.0300
- **Licence:** the arXiv non-exclusive distribution licence 1.0. https://arxiv.org/licenses/nonexclusive-distrib/1.0/
- **What math2 can take:** **record only**.
  - Problem numbers (such as "19.37") with the edition and arXiv version are locators.
  - math2 writes its own statements, and the `tamnd/arxiv` gate applies.

---

## A.24 MathOverflow and Mathematics Stack Exchange

**Size** [measured, Stack Exchange API `/2.3/info`, 2026-10-26]:
- **MathOverflow:** 168,603 questions, 201,920 answers, 1,107,746 comments, 516,907 users.
- **Math.SE:** **1,702,027 questions**, 2,236,177 answers, 9,483,312 comments, 1,617,966 users.

**Licence of the text:** CC BY-SA 2.5, 3.0 or 4.0 per post, by date. This is in the spec already.

**Terms**
- **API Terms of Use** were last updated 30 Nov 2023 and are short. Applications "must ensure they visually indicate that the Stack Exchange Network is the source". They incorporate the network Terms of Service. https://policies.stackoverflow.co/stack-overflow/api-terms-of-use/
- **Public Network Terms of Service** were "Last updated November 13, 2025". https://policies.stackoverflow.co/stack-overflow/public-network-terms-of-service/
  - The body is loaded by JavaScript and I could not read it. **Whether the November 2025 revision added AI or scraping clauses is UNVERIFIED.**
- **API throttles.** https://api.stackexchange.com/docs/throttle
  - More than 30 requests per second from one IP gets a ban.
  - The daily quota is 10,000 requests with a key, or per user with an access token. Without a key, the quota observed was 300 (`quota_max: 300`).
  - A `backoff` field must be honoured. I got a `throttle_violation` for calling a second site within the 10 s backoff, so the backoff seems to apply across sites.
- **Dumps.**
  - Since July 2024 the dumps are behind a login and a click-through: "…for projects that do not include training a large language model (LLM)…". https://www.devclass.com/development/2024/07/30/stack-exchange-restricts-access-to-dump-of-user-contributed-data-critics-complain-this-contradicts-license/1625192
  - **The watermarks are documented.** "v1" was two fabricated `<row>` entries in `Posts.xml`, with post IDs `1000000001` and `1000000010`. They appeared only in the 2025-06-30 dumps (rev1 and rev2), and no 2026 dump is listed as watermarked. https://github.com/LunarWatcher/se-data-dump-transformer/blob/master/docs/meta/Known%20watermarks.md
  - Community re-uploads to the Internet Archive go up to 2026-06-30. https://search.feep.dev/blog/post/2026-08-25-stackoverflow-archive (from search; the 2026-09-30 dump's re-upload is UNVERIFIED)
- **New channel.**
  - Stack Overflow launched a public **MCP server** in beta in December 2025, limited to 100 requests a day. https://stackoverflow.blog/2025/12/01/what-s-new-at-stack-overflow-december-2025/ (from search)
  - The API docs navigation now has an "MCP Server" entry.
  - Whether it covers MathOverflow and Math.SE is UNVERIFIED.
- **Commercial AI licensing.** Stack Overflow sells "Stack Data Licensing" to AI labs (footer link on policies.stackoverflow.co). Reports from late 2025 describe deals with several labs; these come from secondary sources only.

**What math2 can take:** `open`, per post, through the API, exactly as spec 16.6 says.
- Anonymous use gives only 300 calls a day. math2 needs a registered key for 10,000.

---

## A.25 Other large human-written references (brief)

- **MathBases** (https://mathbases.org/) is a directory "Of, by, and for mathematical databases". Its area tags sum to 251; this is not a count of databases.
  - Entry pages have a licence field. For example, it gives DaRT as "MIT".
  - It is a useful seed list for new adapters. Its own licence is UNVERIFIED.
- **FindStat** (https://www.findstat.org/) has 1,997 statistics, 336 maps and 24 collections. The home page states no licence (still [U] in spec 1.7).
- **Lexikon der Mathematik** (Springer Spektrum) is proprietary. Wikidata P12975 has 603 IDs [measured]: record only.
- **All-Russian Mathematical Portal** (Math-Net.Ru) is a bibliographic portal, not an encyclopedia. Wikidata P4252 has 5,090 IDs [measured].
- **Metamath** (`set.mm`). Wikidata P12888 has 438 statement IDs [measured]. `set.mm` is said to be public domain or CC0, which is UNVERIFIED here. It is formal, so it belongs in the formal-library catalogue (9.4).
- **Not reached:** GroupNames (Bristol) refused the connection; Kerodon's licence is not on its about page.

---

## Summary table

| Source | Size (2026-10-26 unless noted) | Licence of text / data | Channel | AI terms | What math2 can take |
|---|---|---|---|---|---|
| Wikipedia (en) | 28,196 article pages in WP Math (33,957 project pages) | CC BY-SA 4.0 (+GFDL where no BY-SA-only text) | monthly XML dump (2026); Enterprise free tier 30 snapshots and 50k on-demand a month; API 10–2,000 req/min | Nov 2025 ask: attribute, use Enterprise; no licence change | **text with attribution** |
| Wikipedia de/fr/ja/zh/ru | 20,732 / 12,192 / 8,865 / 5,682 / 3,928 (project-tagged) | CC BY-SA 4.0 | same dumps | same | text with attribution |
| Wikidata | 113,118 P2534 items, **105,345 of them primes**; 6,560 non-prime; 3,208 theorems | CC0 | weekly JSON dump (103 GB bz2); WDQS 60 s, 5 parallel | none; May 2026 agent overload | **all (metadata)** |
| nLab | 20,778 pages | **no licence**; "free … if you acknowledge the source" | git mirror per edit | none | **link only** (+ ids, link graph) |
| ProofWiki | 47,254 content pages; 30,348 proofs / 36,102 definitions | CC BY-SA 3.0 + GFDL | **daily `latest.xml.gz` (36.6 MB)** | none | **text with attribution** |
| PlanetMath | 9,477 entries; inactive (last push Feb 2024) | CC BY-SA 3.0 | git (64 MSC repos) | none | text with attribution |
| Encyclopedia of Mathematics | >8,000 (2002 print); 1,224 Wikidata IDs; live count UNVERIFIED (502) | **originals © Springer**; new edits CC BY-SA 3.0 + GFDL | none (MediaWiki API) | none | **record only** (text only for wholly new articles) |
| MathWorld | 16,939 entries | proprietary; no bulk, no robots for archiving | none | none stated | record only |
| OEIS | 400,709 sequences | CC BY-SA 4.0 | `oeisdata` git (daily), `stripped.gz`; Crawl-delay 10 | none | text with attribution |
| DLMF | v1.2.8 (2026-09-15) | **© NIST**, no bulk, no commercial | none | none | record only |
| Fungrim | 3,130 entries, 457 symbols; dormant since 2021–22 | **MIT** | git | none | text with attribution |
| Wolfram Functions | 307,409 formulas | © Wolfram, terms not found | none | none | record only |
| LMFDB | (2022 figures in spec 1.7) | data CC BY-SA 4.0; code GPL v2+ | bulk downloads; **`/api/` disallowed in robots; JS gate since Aug 2025** | asks AI users not to scrape, points to CC guidance | data with attribution, via bulk only |
| groupprops (+ other subwikis) | 8,076 articles (topospaces 825, commalg 429, diffgeom 435) | CC BY-SA 3.0 | MediaWiki API | none | text with attribution |
| Manifold Atlas | 215 articles | **contradictory**: GFDL 1.3 and "public domain" | MediaWiki API (HTTP only) | none | link only (treat as GFDL) |
| π-Base | 225 spaces, 247 properties, 935 theorems | CC BY 4.0 | git | none | text with attribution |
| House of Graphs | ~22,000 graphs (2022); now UNVERIFIED | none stated | API needs auth | none | record only |
| ISGCI | 1,695 classes, 242,975 inclusions | none stated | Java app | none | record only |
| DaRT | 218 rings, 178 properties, 72 theorems | CC BY 4.0 | git | none | text with attribution |
| KnotInfo | knots ≤13 crossings, ~116 fields | none on site; **CSV package GPL-3.0** | Excel/CSV download | none | record only |
| Knot Atlas | 2,710 articles (1.65M pages) | none stated | MediaWiki API | none | record only |
| ATLAS v3 | ~716 groups, 5,215 reps | none stated (AtlasRep GPL-3+) | AtlasRep on demand | none | record only |
| Open Problem Garden | ≤713 entries | GFDL (prob. 1.3) | none | none | link only |
| Erdős problems | 1,221 | **site text: no licence**; repo Apache-2.0 but **metadata only** | git YAML | none | status as metadata; statements record only |
| Kourovka Notebook | 21st ed., arXiv:1401.0300v49 (2026-10-20) | arXiv non-exclusive | arXiv | none | record only |
| MathOverflow | 168,603 Q / 201,920 A | CC BY-SA 2.5/3.0/4.0 per post | API (10k/day with key; 300 anonymous); dumps no-LLM click-through | dump click-through; ToS updated 2025-11-13 (content UNVERIFIED) | per-post text with attribution, via API |
| Math.SE | 1,702,027 Q / 2,236,177 A | as above | as above | as above | as above |

---

## Proposed corrections to the spec

1. **1.7 and 9.2, Wikidata:**
   - 113,118 "items with a defining formula" are mostly prime numbers (105,345). Report **6,560** non-prime items.
   - Add the measured crosswalk properties: P7726 PlanetMath 951, P7554 EoM 1,224, P829 OEIS 469, P12934 House of Graphs 213, P11497 DLMF 199, P12987 LMFDB knowl 109, P8427 KnotInfo 2,978, P8426 Knot Atlas 2,228, P14106 Erdős 3.
2. **1.7 and 9.2, Encyclopedia of Mathematics:** change the licence from "CC BY-SA 3.0" to **"originals © Springer; only new wiki text CC BY-SA 3.0 + GFDL"**, and the class to **`record`**.
3. **9.2, Erdős problems:** `teorth/erdosproblems` has **no statements**. The informal statements on erdosproblems.com carry no licence, so they are record only.
4. **1.7 and 9.2, nLab:** 20,778 pages. The licence is now confirmed as *none*, with an informal permission, so the class is `record` (link only), and no longer [U].
5. **1.7 and 9.2, ProofWiki:** the licence is CC BY-SA 3.0 **and GFDL**. The daily dump is live (36.6 MB gzipped).
6. **1.7 and 9.2, LMFDB:** `robots.txt` disallows `/api/`, and there is a JavaScript gate. The channel is bulk downloads only.
7. **1.7, Wikipedia:** 28,196 article pages (not "about 26k"). The dumps are monthly in 2026. Add the 2026 API rate limits.
8. **9.3, licence classes:** add DLMF (NIST copyright), Wolfram Functions, the Kourovka Notebook (arXiv default), KnotInfo, ISGCI, the ATLAS and House of Graphs to `record`. Add Manifold Atlas to `gfdl`. Add Fungrim (MIT), DaRT (CC BY) and the subwikis (CC BY-SA 3.0) to `open`.
9. **1.7, MathWorld:** 16,939 entries. **OEIS:** 400,709.

---

## Open doubts

1. **EoM.** I could not read the live site (502 all morning). Did the move to EMS Press change the Springer copyright on the original articles? What is the current article count? A retry and a question to the EMS would settle it.
2. **Stack Exchange Public Network Terms of 13 Nov 2025.** Do they now forbid AI training or scraping of public content? (The text is loaded by JavaScript and I could not read it.) Does the new public MCP server cover MathOverflow and Math.SE?
3. **nLab.** Would the steering committee agree to an explicit CC BY-SA statement for new contributions? Old pages would still need each author's consent.
4. **Manifold Atlas.** GFDL 1.3 or public domain? Ask the editors.
5. **LMFDB.** Do knowls fall under the CC BY-SA "database" licence? Which bulk channel does the LMFDB want AI users to use? The site's JavaScript gate stopped me reading the current download page.
6. **House of Graphs.** The current number of graphs and of invariants, and any licence. The API needs authentication.
7. **KnotInfo.** Did the GPL-3.0 relicensing of the CSV package have the authors' consent? KnotInfo itself states no licence.
8. **Wikipedia dumps.** Is the move from twice-monthly to monthly runs official? Were the free `enterprise_html` dumps (last run 2025-03-20) withdrawn, or moved?
9. **Wikidata WDQS.** The details of the 2026–2027 backend migration and of the scholarly graph split, and whether the new endpoints have different limits.
10. **ProofWiki Main Page counters** (30,348 proofs, 36,102 definitions) are the same as in the spec. Are they live?
11. **Open Problem Garden.** The exact GFDL version (1.2 or 1.3) is inferred from an image file name.
12. **Unreached sources:** GroupNames (connection refused), FindStat's licence, Kerodon's licence, and MathBases' own licence.
</document>

<document source="https://math2.tamnd.com/spec/corpus/01b-literature-and-books.md">
# K01, part B. Literature indexes, open textbooks and syllabi: sizes, licences, channels (research for math2)

Status: research note for spec 01 §1.7, spec 09 (all, especially the 9.3 licence gate) and spec 16 §16.6.
Read date: **every fact below was read on 2026-10-26** unless another date is given. Each fact has a URL next to it.
**UNVERIFIED** means I could not confirm the fact at a primary source. The reason is given each time.
Method: WebSearch, WebFetch and a few single `curl` requests with User-Agent `m2/research (+https://github.com/tamnd/math2-agent)`. There was no crawling and no bulk download. The largest samples were three days of arXiv OAI-PMH `set=math` (1,421 records), the Open Textbook Library mathematics JSON (22 pages), four Mathlib YAML files, two syllabus PDFs and one zbMATH API query.

This note does not repeat what the spec already says. The spec already has: arXiv about 15% open, `s3://arxiv` requester-pays and Kaggle CC0; zbMATH 168M triples; Stacks GFDL 1.2 with 21,446 tags; TheoremSearch 9.2M / 690k / 1.34M; TheoremGraph 11.7M / 18.3M / 48k; Matlas 8M / 435k / 1.9k; MSC2020 CC BY-NC-SA; the Stack Exchange no-LLM clause. Where my findings correct or refine those figures, I say so.

The access classes used in the "math2 can take" column are those of spec 9.3: `open`, `share-alike-nc`, `verbatim`, `gfdl`, `record`. Node prose is CC BY-SA 4.0, code is Apache-2.0 and metadata is CC0 (16.6).

---

## Part 1. Literature indexes and full text

### 1.1 arXiv

**Bulk and API terms**
- **S3 bulk:** the bucket `arxiv` (prefixes `pdf/`, `src/`) is requester-pays in us-east-1. Files come as tars of about 500 MB with manifests and are updated about monthly. The total is "about 9.2 TB" as of April 2025, growing by about 100 GB a month. Users "must link back to arXiv for downloads". (https://info.arxiv.org/help/bulk_data_s3.html, https://info.arxiv.org/help/bulk_data.html)
- **API Terms of Use** (https://info.arxiv.org/help/api/tou.html; the page carries no date):
  - Metadata is CC0.
  - "The redistribution of e-prints requires permission from the copyright holder." Storing and serving e-prints needs permission or a licence; personal and research use is allowed.
  - The rate limit is one request every 3 seconds on a single connection, counted across all of a user's machines.
  - Users may not claim arXiv endorsement.
- **AI and TDM terms:** none of the four arXiv help pages above says anything about AI, LLMs, TDM or training. So for arXiv the per-paper licence is the only AI-relevant term. (A missing clause is not the same as permission. See open doubt D3.)
- **Licence options for authors** (https://info.arxiv.org/help/license/index.html):

| Licence | 9.3 class |
|---|---|
| CC BY 4.0 | `open` |
| CC BY-SA 4.0 | `open` |
| CC0 | `open` |
| CC BY-NC-SA 4.0 | `share-alike-nc` |
| CC BY-NC-ND 4.0 | `verbatim` |
| arXiv non-exclusive distribution licence 1.0 (http://arxiv.org/licenses/nonexclusive-distrib/1.0/) | `record` |

**New measurement: the 2026 licence mix in math.** I harvested three single days from `oaipmh.arxiv.org/oai`, with `verb=ListRecords&metadataPrefix=arXiv&set=math`: from=until=2026-10-01, 2026-04-15 and 2026-07-08. That gave 1,421 records. The 1,026 records with 26xx identifiers (papers first submitted in 2026) break down as follows:

| Licence | Share |
|---|---|
| non-exclusive (record) | 55.3% |
| CC BY 4.0 | 34.0% |
| CC BY-NC-ND | 5.9% |
| CC BY-NC-SA | 1.9% |
| CC0 | 1.9% |
| CC BY-SA | 1.0% |

- **`open` share for node prose (BY + BY-SA + CC0): 36.9%.** By day it was 35.3%, 38.3% and 38.8%. Records with older identifiers that were re-versioned on those days (395 of them) were about 39.2% `open`.
- **Consequence:** the spec's "about 15% of arXiv is `open`" is the all-time stock. The 2026 flow in math looks more than twice as open. This is a small sample and needs confirmation (D2).
- **Caveats:**
  - The OAI datestamp also counts metadata updates, not just new submissions.
  - I did not establish whether `set=math` includes cross-lists.

### 1.2 zbMATH Open

- **OAI-PMH:** Identify at https://oai.zbmath.org/v1/?verb=Identify works. It reports earliestDatestamp 1755.
- **REST API** (https://api.zbmath.org/v1/document/_search):
  - It returns full review text (`editorial_contributions`) and a per-document `license` field. That field was empty in my sample.
  - The query `py:2025` returns `nr_total_results` 133,093. That is a reasonable yearly-volume figure for 2025.
- **API terms:** https://static.zbmath.org/legal/api-terms-and-conditions.html was blocked by a Cloudflare challenge, and so was https://zbmath.org/about/. **The API terms, and any AI clause in them, are UNVERIFIED.**
- **Knowledge graph dump** (Zenodo record 21497976, concept DOI 10.5281/zenodo.21497975; https://zenodo.org/records/21497976):
  - Published 2026-09-01 and modified 2026-10-24. File `zbmath-kg-ttl-2026.tar.gz`, 1.6 GB.
  - Built from the January 2026 API dump. It is updated about twice a year, not monthly as spec 9.2 says.
  - A temporary SPARQL endpoint is at http://212.227.170.235:8890/sparql.
  - Repository: https://github.com/zbMATHOpen/zbmath-open-kg. Paper: https://arxiv.org/abs/2609.00969.
  - Stats: 168M+ triples, 34M+ entities, 4M+ publications, 1M+ persons, 3M+ reviews.
  - Restricted abstracts appear as the literal placeholder "zbMATH Open Web Interface contents unavailable due to conflicting licenses".
- **The licence conflict:** the Zenodo metadata field says **CC BY 4.0**, but the record's description says **CC BY-SA 4.0** for the dataset, with the I4OC references CC0. The spec says "CC BY-SA 4.0". For math2 both are `open`. They do differ on whether derived exports must be BY-SA (D1).
- **What math2 can take:**
  - Records, MSC codes and links as metadata.
  - Review text is `open` under either reading. In practice it should be used as evidence and for locators, not pasted into prose. Reviews are third-party texts, and their per-item rights in the API `license` field are empty or unclear.

### 1.3 OpenAlex and Unpaywall

- **Snapshot** (https://help.openalex.org/download-all-data/openalex-snapshot, page updated 2026-09-24):
  - The bucket `s3://openalex` is free and anonymous (`--no-sign-request`). `LICENSE.txt` in the bucket is CC0 1.0, which I checked.
  - The September 2026 release is about 745 GB of compressed JSONL (works about 615 GB), or about 770 GB in Parquet.
  - The free snapshot is quarterly, on the second Wednesday of January, April, July and October. Daily snapshots are paid.
- **API key and pricing:** an API key has been required since 2026-02-13, with usage-based pricing. **Partly UNVERIFIED**: I found this only in secondary sources (https://casrai.org/news/openalex-api-keys-mandatory-usage-based-pricing-2026). https://help.openalex.org/api/ did not render the details for me.
- **Unpaywall:**
  - Unpaywall now runs inside OpenAlex (the "Walden" rewrite), and its data feed is a paid product.
  - The last free standalone Unpaywall snapshot I found is dated 2024-11-27, according to a SUB Göttingen note.
  - https://unpaywall.org/products/snapshot returned an empty page. **The current Unpaywall terms are UNVERIFIED.**
- **What math2 can take:** all of it, as CC0 metadata (works, authors, venues, concepts, OA locations). OpenAlex has no full text.

### 1.4 Crossref

- **REST API** (https://www.crossref.org/documentation/retrieve-metadata/rest-api/): no sign-up is needed. Crossref says "almost none of the metadata is subject to copyright", but "some abstracts ... may be subject to copyright".
- **2026 public data file** (https://www.crossref.org/blog/2026-public-data-file-now-available/, posted 2026-03-17): 180,034,490 records as of 2026-03-14, 208 GB of JSONL. It is distributed via Academic Torrents (doi 10.13003/nggf-vt1j) or AWS.
- **What math2 can take:** DOI metadata and reference lists as facts. Abstracts are `record` unless the item's own licence says otherwise.

### 1.5 DataCite

- **Public data file:** annual and CC0. The use policy asks for credit.
  - The 2025 file was generated on 2026-01-06 and announced on 2026-01-22. It has 108M findable DOIs and is at https://datafiles.datacite.org/.
  - The monthly file is for members only. The file is also on Dimensions' BigQuery.
  - Sources: https://support.datacite.org/docs/datacite-public-data-file, https://support.datacite.org/docs/datacite-data-file-use-policy, and the DataCite blog.
- **What math2 can take:** all of it, as metadata. Its use for math2 is datasets and software DOIs (Zenodo and others).

### 1.6 Semantic Scholar and S2ORC

- **API licence** (https://www.semanticscholar.org/product/api/license, "last updated" 2023-05-17):
  - The licence is non-transferable. You may not redistribute or sublicense the API.
  - S2 data is under its own licences, "such as CC BY-NC or ODC-BY". Attribution is required.
  - It has no AI-training clause.
- **S2ORC:**
  - https://github.com/allenai/s2orc says ODC-By 1.0 (earlier releases were NC). The Hugging Face card at https://huggingface.co/datasets/allenai/s2orc is tagged cc-by-2.0, which is an inconsistency.
  - Size: a 2026 arXiv paper reports the 2026-05-26 release as having 15,679,781 S2ORC full-text papers.
- **UNVERIFIED:**
  - The licence of the separate abstracts dataset (the datasets API docs returned HTTP 429).
  - The current total of papers. A third-party figure for 2025-09 says 200M+ papers and 100M+ abstracts.
- **What math2 can take:**
  - Paper ids as crosswalk targets.
  - S2ORC full text is ODC-By *as a database*, but each paper's text keeps the paper's own copyright. Treat it per paper, by its arXiv or publisher licence. It is never `open` just because it sits in S2ORC.

### 1.7 CORE

- **Terms** (https://core.ac.uk/terms; I read this via a search snippet because the page returned 403 to curl):
  - Dumps are ODC-By, covering CORE's own database rights only.
  - API access needs a licence; it is free for individuals and for public research.
  - A licensing-framework review proposes CC0/ODC-0 for metadata and "conditional access for lawful TDM" for full texts.
- **Size:** 291M metadata records and 32.8M full texts (February 2023, the CORE paper in *Scientific Data*). **The current size and the current dump terms are UNVERIFIED** (core.ac.uk/documentation returned 403).
- **What math2 can take:** records. Full texts follow each paper's licence, as for S2ORC.

### 1.8 HAL (France)

- **OAI-PMH** (https://api.archives-ouvertes.fr/oai/hal/?verb=Identify): "Metadata is licensed under CC0".
- **Math share, measured by API:** a query on `level0_domain_s:math` returned **170,406 records, 105,413 of them with a file**. (The query `domain_s:math*` returns 0.) The licence facet over the records with a file:

| Licence | Records | 9.3 class |
|---|---:|---|
| hal-authorisation-v1 (https://about.hal.science/hal-authorisation-v1/) | 84,016 | `record` |
| CC BY 4.0 | 9,239 | `open` |
| hal copyright (https://hal.science/licences/copyright/) | 4,272 | `record` |
| CC BY-NC-ND | 3,861 | `verbatim` |
| CC BY-NC | 1,796 | `share-alike-nc` |
| CC BY-NC-SA | 731 | `share-alike-nc` |
| CC0 | 492 | `open` |
| CC BY-SA | 449 | `open` |
| CC BY-ND | 254 | `verbatim` |
| Etalab 2.0 | 163 | `open` (attribution licence, compatible with CC BY) |
| PDM | 6 | `open` |
| none / empty | 113 + 21 | `record` |

- **`open` share: 10,349 / 105,413 = about 9.8%.**
- **Channels:** REST at api.archives-ouvertes.fr/search, SPARQL at https://data.hal.science/doc/sparql, and OAI-PMH.
- I found no AI clause.

### 1.9 Numdam (Mathdoc)

- **Size** (https://www.numdam.org/en/): 75,599 articles in 120 periodicals, 706 books in 8 collections, 416 theses, 1,428,508 pages.
- **Moving wall:** 2 to 10 years. Mersenne journals arrive after 1 year.
- **Conditions** (https://www.numdam.org/conditions/):
  - Metadata is CC0 and can be harvested by OAI-PMH.
  - Full texts are for individual download for research, study or training.
  - Downloading "un volume substantiel" without explicit permission is forbidden, and so is modifying the files.
  - Republishing is allowed only for public-domain items (described as "très rares sur Numdam") and for CC-licensed articles.
- **What math2 can take:** records (CC0). Text is `record` by default, or the per-article CC licence where one is given. Bulk full text is not allowed without permission.

### 1.10 EuDML

- **UNVERIFIED for current figures.** eudml.org, including its OAI endpoint, returned 403 or a Cloudflare challenge.
- What I could find:
  - Mathdoc's page (https://www.mathdoc.fr/en/realisations/eudml/, undated) says "over 270,000 documents".
  - A 2016 report says 257k items, 252,111 articles and 323 journals.
  - https://eudml.org/journals listed 339 journals in a search snippet.
  - EuDML's policy is to hold only content past its moving wall, with JATS-flavoured OAI. **Its metadata licence is UNVERIFIED.**
- **What math2 can take:** records via the partner sources (Numdam, GDZ, CEDRAM and others) rather than EuDML itself.

### 1.11 Project Euclid

- https://projecteuclid.org/about says "over 80% of the journal articles hosted ... are openly available". It is run by Duke University Press.
- "Openly available" means free to read, not openly licensed. **Per-journal licences are UNVERIFIED.** Default to `record`, and use the per-article licence where it is shown (for example, some IMS/Bernoulli journals).

### 1.12 GDZ (Göttinger Digitalisierungszentrum)

- **Size:** over 15M pages, with collections for Mathematica and FID Mathematik (https://gdz.sub.uni-goettingen.de/).
- **IIIF:** the manifest for *Mathematische Annalen* vol. 1 (1869) at https://manifests.sub.uni-goettingen.de/iiif/presentation/PPN235181684_0001/manifest has an attribution but **no `license` field**.
- **PDF terms:** a generated PDF (https://gdz.sub.uni-goettingen.de/download/pdf/PPN71557308X/LOG_0040.pdf) has a cover page saying reproductions may not be made for other repositories without written permission.
- **The licence of the scans is UNVERIFIED.** The underlying 19th-century texts are public domain.
- **What math2 can take:** records and locators. Statements from public-domain works, which math2 rewrites anyway. Not the scans.

### 1.13 The US public-domain line, HathiTrust, Internet Archive, Project Gutenberg

**The US public-domain line.** Works published in 1930 entered the US public domain on 2026-01-01 (https://web.law.duke.edu/cspd/publicdomainday/2026/). So in 2026, works published in the US before 1931 are public domain in the US. Public domain is `open` for prose: no attribution is needed, but math2 should give it anyway.

**HathiTrust**
- hathitrust.org (terms and acceptable-use pages) is behind Cloudflare. **The terms and any AI policy are UNVERIFIED.**
- https://github.com/hathitrust/datasets lists the public-domain text subsets `ht_text_pd`, `ht_text_pd_open_access`, `ht_text_pd_world` and `ht_text_pd_world_open_access`. The full `ht_text` is used only by HTRC for non-consumptive research.
- A related public-domain corpus: Harvard Institutional Books 1.0 has 983,004 public-domain volumes.

**Internet Archive**
- Counts from advancedsearch (https://archive.org/advancedsearch.php?q=subject%3Amathematics+AND+mediatype%3Atexts&rows=0&output=json):
  - `subject:mathematics AND mediatype:texts` gives 206,597 items.
  - Restricting to `year:[1500 TO 1930]` gives 8,076.
- https://archive.org/about/terms returned an empty page. **The IA terms are UNVERIFIED.** Uploads on IA include many in-copyright books. For example, Hatcher's *Algebraic Topology* is there, though its licence is noncommercial copying only (§2.4). IA hosting therefore says nothing about licence.

**Project Gutenberg**
- **Robot policy** (https://www.gutenberg.org/policy/robot_access.html): the site is "intended for human users only". Bulk users should use a mirror, or wget with `-w 2` through `/robot/harvest`. Catalogue data is public domain.
- **Licence** (https://www.gutenberg.org/policy/license.html): "Nearly all the individual works ... public domain in the United States". The trademark licence no longer applies once "all references to Project Gutenberg are removed".
- **Size:** about 78,000 items (a 2026-03-22 figure).
- **Mathematics bookshelf 102** (https://www.gutenberg.org/ebooks/bookshelf/102) includes:
  - Hardy, *A Course of Pure Mathematics* (#38769, with LaTeX source)
  - Dedekind (#21016)
  - Casey's Euclid (#21076)
  - Thompson, *Calculus Made Easy*
- **What math2 can take:** `open` (public domain in the US) with the Project Gutenberg header removed. The non-US public-domain status of each work is a separate check (D10).

### 1.14 Extraction projects (TheoremSearch, TheoremGraph and LeanGraph, Matlas, and new ones)

**TheoremSearch** (https://arxiv.org/abs/2602.05216, v2 dated 2026-03-07)
- **Size:** https://theoremsearch.com shows "8.4M" indexed. By source:

| Source | Statements |
|---|---:|
| arXiv | 8,104,539 |
| Lean | 285,213 |
| ProofWiki | 23,284 |
| Stacks | 10,645 |
| Lean Community | 1,888 |
| Open Logic | 588 |
| CRing | 498 |
| HoTT | 257 |
| Napkin | 228 |

  The spec's 9.2M figure is from the paper; the live site now shows 8.4M.
- **Channels:** a REST API at https://api.theoremsearch.com/search and MCP at https://api.theoremsearch.com/mcp. **No terms of use are published.**
- **Released data:**
  - https://huggingface.co/datasets/uw-math-ai/theorem-search-dataset is tagged `cc-by-sa-4.0`, with 1,341,083 theorems from 209,777 papers. That matches the spec's 1.34M. It mixes Stacks (GFDL), ProofWiki and other sources under one BY-SA tag, which is a licence inconsistency (D4). Its card says "~15% of arXiv papers use permissive licenses (CC BY, CC BY-SA, or CC0)".
  - https://huggingface.co/datasets/uw-math-ai/theorem-search-dataset-permissive is tagged `cc-by-4.0`. Its card says 1,239,720 theorems from 197,889 papers (arXiv 1,238,975, Open Logic 745), covering CC0, CC BY 4.0, CC BY 3.0 and PD. ShareAlike is excluded. The dataset viewer shows 2,677,439 rows, which does not match the card (D5). The "slogans" (natural-language summaries) were written by DeepSeek-V3.1.
- **What math2 can take:** the `-permissive` set as `open` candidate evidence (E1), keeping each row's per-paper licence. The BY-SA set should be used only after filtering out Stacks and any other GFDL or NC rows.

**TheoremGraph and LeanGraph** (https://arxiv.org/abs/2606.25363, 2026-06-24, cs.IR; the paper itself is under the arXiv non-exclusive licence)
- https://huggingface.co/datasets/uw-math-ai/math-graph:
  - The card says CC BY 4.0, with "per-record reuse governed by each statement's source license". It has 16,131,238 rows.
  - The graph structure is open. Statement text is filled in only for CC0/BY/BY-SA papers and is NULL otherwise. `paper_arxiv.csv` has a licence column.
- https://huggingface.co/datasets/uw-math-ai/theorem-matching has 101k rows, built from the Kaggle arXiv metadata.
- **What math2 can take:** the edges and ids (CC BY 4.0, so attribute them), and text only where it is non-NULL. This is the best-licensed of the three big extractions.

**Matlas** (https://arxiv.org/abs/2604.17484, 2026-04-19; the paper is CC BY 4.0)
- **Size:** 8.07M statements from 435K papers (1826–2025), 180 journals and 1.9K textbooks.
- **API:** https://matlas.ai/openapi.json describes `POST /api/search` (`num_results` 10–200). It returns the `statement` text with doi, title, authors, journal, year and type. There is also `/api/feedback`.
- No auth, no licence and no terms are published. **The licence of the corpus is UNVERIFIED and probably not open.** The statements are from journals and textbooks under copyright.
- **What math2 can take:** under spec 9.6, query-time matching only, storing only DOIs and ids (`record`).

**New 2026 projects not in spec 01**

| Project | Date | Licence | What it is |
|---|---|---|---|
| uw-math-ai/FormalArXiv (https://huggingface.co/datasets/uw-math-ai/FormalArXiv) | n/a | "other" | 89 Lean statements from August 2026 arXiv papers, via LemmaBench |
| Lean Pool (https://arxiv.org/abs/2609.25199, Ilin) | 2026-09-21 | paper CC BY 4.0 | an AI-maintained archive of formalised mathematics |
| AnnalsChallenge (https://arxiv.org/abs/2610.18301, Buzzard et al.) | 2026-10-13 | paper CC BY-SA 4.0 | 50 Lean statements of Annals results |
| Stacks↔Mathlib alignment (https://arxiv.org/abs/2610.31011; code https://github.com/AnkitSatpute/Formal-Informal-Mathematical-Alignment) | 2026-10-22 | paper CC BY 4.0 | added to MaRDI |
| KnowTeX (https://arxiv.org/abs/2601.15294) | n/a | n/a | n/a |
| uw-math-ai other artefacts (https://huggingface.co/uw-math-ai: MELD, Math2Vec, APRIL, MathLeap models; https://github.com/uw-math-ai/arXiTeX) | n/a | no licences on the org page | n/a |

---

## Part 2. Open textbooks

Licences were read at the book's own source (repository LICENSE, README or front matter) unless a row says "per OTL". In the Open Textbook Library (OTL), licences are sometimes stale; see §2.3.

### 2.1 Verified at source

| Book | Licence (exact) | Format | Repository / source URL | 9.3 class |
|---|---|---|---|---|
| Univalent Foundations, *Homotopy Type Theory* (HoTT book) | CC BY-SA 3.0 Unported | LaTeX | https://github.com/HoTT/book (README) | `open` |
| Open Logic Project | CC BY 4.0 | LaTeX | https://github.com/OpenLogicProject/OpenLogic | `open` |
| Hefferon, *Linear Algebra* | dual: GFDL **or** CC BY-SA 3.0 US, at the user's choice | LaTeX | https://hefferon.net/source.html, https://gitlab.com/jim.hefferon/linear-algebra | `open` (take it under BY-SA) |
| Lebl, *Basic Analysis* I/II | dual: CC BY-NC-SA 4.0 **and** CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/ra | `open` (take it under BY-SA) |
| Lebl, *Notes on Diffy Qs* | dual: CC BY-NC-SA 4.0 and CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/diffyqs | `open` (BY-SA) |
| Lebl, *Tasty Bits of Several Complex Variables* | dual: CC BY-NC-SA 4.0 and CC BY-SA 4.0 | LaTeX | https://github.com/jirilebl/scv | `open` (BY-SA) |
| Boelkins et al., *Active Calculus* (single variable, 2012–2025 edition) | CC BY-SA 4.0 | PreTeXt | https://github.com/active-calculus/active-calculus-single-mbx (`source/bibinfo.xml`) | `open` |
| Evan Chen, *An Infinitely Large Napkin* | text CC BY-SA 4.0; source code GPL v3 | LaTeX | https://github.com/vEnhance/napkin/ | `open` (text) |
| Levin, *Discrete Mathematics: An Open Introduction*, **3rd edition and earlier** | CC BY-SA 4.0 | PreTeXt | https://github.com/oscarlevin/discrete-book | `open` |
| Levin, same book, **4th edition** | CC BY-NC-SA 4.0 | PreTeXt | same | `share-alike-nc` |
| Keller and Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | https://github.com/mitchkeller/applied-combinatorics | `open` |
| Ernst, *An Introduction to Proof via Inquiry-Based Learning* | CC BY-SA 4.0 | LaTeX | https://github.com/dcernst/IBL-IntroToProof | `open` |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 (licence link in README) | PreTeXt | https://github.com/davidaustinm/ula | `open` |
| Beck, Marchesi, Pixton, Sabalka, *A First Course in Complex Analysis* v1.6 | CC BY 4.0 | PDF (LaTeX not checked) | https://matthbeck.github.io/complex.html | `open` |
| Lehman, Leighton, Meyer, *Mathematics for Computer Science* (2018 rev.) | CC BY-SA 3.0 ("available under the terms of the Creative Commons Attribution-ShareAlike 3.0 license") | PDF | https://courses.csail.mit.edu/6.042/spring18/mcs.pdf | `open` |
| Judson, *Abstract Algebra: Theory and Applications* (AATA) | GFDL 1.3 or later | PreTeXt | https://github.com/twjudson/aata | `gfdl` |
| Beezer, *A First Course in Linear Algebra* (FCLA) | GFDL 1.2 or later | PreTeXt | https://github.com/rbeezer/fcla | `gfdl` |
| The Stacks Project | GFDL 1.2 | LaTeX | https://stacks.math.columbia.edu (COPYING) | `gfdl` |
| Hartman et al., *APEX Calculus* | CC BY-NC 4.0, © 2021 Hartman | PreTeXt | https://github.com/APEXCalculus/APEXCalculusPTX (`ptx/apex.ptx`) | `share-alike-nc` |
| Feldman, Rechnitzer, Yeager, *CLP Calculus* (UBC) | CC BY-NC-SA 4.0 ("It is licensed under a Creative Commons BY-NC-SA 4.0 International License") | LaTeX source published | https://personal.math.ubc.ca/~CLP/src/ | `share-alike-nc` |
| Axler, *Linear Algebra Done Right*, 4th edition | CC BY-NC 4.0 (Springer open access) | PDF; HTML via PreTeXt (David Farmer) | https://link.springer.com/book/10.1007/978-3-031-41026-0, https://library.oapen.org/handle/20.500.12657/85067, https://linear.axler.net/ (PDF dated 2026-08-16), https://ladr.pretext.plus/ | `share-alike-nc` |
| Hammack, *Book of Proof*, 3rd edition (v3.4, 2025-02-05) | CC BY-NC-ND 4.0 | PDF only | https://richardhammack.github.io/BookOfProof/ | `verbatim` |
| Hatcher, *Algebraic Topology* | not a CC licence: "Single paper or electronic copies for noncommercial use may be made freely without explicit permission from the author or publisher" (© 2002 CUP) | PDF | https://pi.math.cornell.edu/~hatcher/AT/ATpage.html | `record` |
| Lurie, *Kerodon* | **no licence statement found** anywhere on the site (about page: maintained by Jacob Lurie, built on Gerby). /statistics shows 8,189 tags | web (Gerby) | https://kerodon.net/ | `record` (all rights reserved by default). **This contradicts spec 9.3 and 16.6, which call it GFDL** (D6) |

### 2.2 OpenStax mathematics

Licences were read from each book's `collection.xml` `md:license` on GitHub (the openstax/osbooks-* repositories). openstax.org/subjects/math returned an empty page.

| Book (OpenStax repository) | Licence | 9.3 class |
|---|---|---|
| Calculus Vols 1–3, College Algebra 2e, Precalculus 2e, Algebra and Trigonometry 2e, Prealgebra 2e, Elementary Algebra 2e, Intermediate Algebra 2e, Introductory Statistics 2e, Introductory Business Statistics 2e, Contemporary Mathematics, Algebra 1, Principles of Data Science, *Cálculo* (Spanish) | CC BY-NC-SA 4.0 | `share-alike-nc` |
| osbooks-statistics (Statistics), *Precálculo* 2ed | CC BY 4.0 | `open` |

Earlier editions were CC BY: the OTL lists Prealgebra 2e (2015), Precalculus (2014) and Introductory Statistics 2e (2023) as BY. Earlier CC BY editions are still CC BY, and math2 could adapt those editions specifically (D7).

### 2.3 Aggregators and catalogues

**Open Textbook Library** (https://open.umn.edu/opentextbooks/subjects/mathematics.json)
- `total_count` is 214; 191 entries were retrieved. Licence counts:

| Licence | Books |
|---|---:|
| BY-NC-SA | 67 |
| BY | 51 |
| BY-SA | 38 |
| BY-NC | 17 |
| GFDL | 12 |
| BY-NC-ND | 3 |
| "No Rights Reserved" | 2 |
| BY-ND | 1 |

- So about 48% are `open` (BY, BY-SA, CC0) and 52% are not.
- The subject also contains some non-mathematics books (statistics, psychology).
- **OTL licences can be stale or simplified.** It lists *Active Calculus* 2.0 (2017) as BY-NC-SA, while the current edition is BY-SA. It lists *Book of Proof* as "Attribution-NoDerivs", while the author's page says BY-NC-ND. It lists Hefferon as BY-SA only, while the book is dual-licensed GFDL / BY-SA. **Verify at source before ingesting.**

**American Institute of Mathematics (AIM) Open Textbook Initiative, approved list** (https://textbooks.aimath.org/textbooks/approved-textbooks/)
- 68 unique titles grouped by course. **Licences are not shown on the list.** The verified licences for AIM titles are in §2.1, §2.5 and §2.6.
- AIM titles not otherwise covered in this note include Bruckner–Thomson, ORCCA, Manes, May–Bart, Marsden–Weinstein, Boyd–Vandenberghe, Treil, Fields, Bender–Williamson, Bard, Zimmermann, Crisman, Stein and Connell. Their licences were not checked (D11).

**PreTeXt catalogue** (https://pretextbook.org/catalog.html)
- 80 projects, 67 of them mathematics.
- **GFDL badges:** FCLA, Discover Linear Algebra, Fitch *Discrete Math*, Yoshiwara *Elementary Algebra*, LA Companion, LA Refresher, Yoshiwara *Modeling, Functions and Graphs*, Yoshiwara *Trigonometry*, AATA, AA Refresher, IBL Geometries, *Exploring Combinatorial Mathematics*, *First-Semester Abstract Algebra*, Hitchman *Geometry with an Introduction to Cosmic Topology*, *Models of Non-Euclidean Geometry*, *The ODE Project*, *Algebraic Graph Theory with Sage*, *Elementary Foundations*, *A Second Course in Linear Algebra*, *Abstract Algebra for Teachers*, Cryptology.
- **All rights reserved:** *Linear Algebra* (W214) and *Intro to Topology (IBL)*.
- The rest are badged only "Creative Commons", without saying which variant. Each needs a source check.

**LibreTexts** (https://math.libretexts.org/Bookshelves)
- The licence is set per page, through tags such as `license:ccbyncsa` and an auto-generated "Detailed Licensing" back-matter page. An example: https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/00%3A_Front_Matter/04%3A_Licensing.
- Many pages are "not declared". The licence mix inside a single book is common.
- The terms (https://libretexts.org/terms-conditions, effective 2020-10-15) allow copying of open-licensed content as its licence permits. They have no AI or scraping clause.
- There are no aggregate licence statistics, so the mix is **UNVERIFIED**.
- **For math2:** go to the upstream source of each LibreTexts book, not the LibreTexts copy. Use LibreTexts only page by page, by the page's `license:` tag. "Not declared" counts as `record`.

**MIT OpenCourseWare** (https://ocw.mit.edu/pages/privacy-and-terms-of-use/)
- Course materials are CC BY-NC-SA 4.0, so `share-alike-nc`.
- There is a **"Permitted Use of AI Training"** section: training is allowed with attribution, the models must be non-commercial, and derivative models must be released under BY-NC-SA.
- There is no bulk-download section.
- A note: MIT 6.042's *Mathematics for Computer Science* is distributed separately under CC BY-SA 3.0 (§2.1), so it is `open`, unlike the OCW course pages.

### 2.4 Books commonly assumed open that are not

| Book | Status |
|---|---|
| Hatcher, *Algebraic Topology* | noncommercial copying only; `record` |
| *Book of Proof* | ND, so `verbatim` |
| Axler LADR4 | NC |
| APEX | NC |
| CLP | NC-SA |
| OpenStax 2e math editions | NC-SA |
| Kerodon | no licence; `record` |

### 2.5 Table A: textbooks math2 may adapt into CC BY-SA 4.0 node prose (CC0, CC BY, CC BY-SA)

Attribution is required for all of them. CC BY-SA 3.0 (Unported or US) adaptations may be licensed under BY-SA 4.0: CC BY-SA 3.0 Unported §4(b) (https://creativecommons.org/licenses/by-sa/3.0/legalcode.en, read 2026-10-26) allows an adaptation to be distributed under "a later version of this License with the same License Elements". I did not re-read the 3.0 US port; I assume it has the same clause, which matters for Hefferon. Treat that as UNVERIFIED.

| Book | Licence math2 uses | Source format | Repository / URL | Notes |
|---|---|---|---|---|
| Open Logic Project | CC BY 4.0 | LaTeX | https://github.com/OpenLogicProject/OpenLogic | already in TheoremSearch (588 / 745 items) |
| HoTT book | CC BY-SA 3.0 → 4.0 | LaTeX | https://github.com/HoTT/book | |
| Hefferon, *Linear Algebra* | CC BY-SA 3.0 US option of the dual licence | LaTeX | https://gitlab.com/jim.hefferon/linear-algebra | record that math2 elected the BY-SA option |
| Lebl, *Basic Analysis* I/II | CC BY-SA 4.0 option of the dual licence | LaTeX | https://github.com/jirilebl/ra | elect BY-SA |
| Lebl, *Notes on Diffy Qs* | CC BY-SA 4.0 option | LaTeX | https://github.com/jirilebl/diffyqs | elect BY-SA |
| Lebl, *Tasty Bits of SCV* | CC BY-SA 4.0 option | LaTeX | https://github.com/jirilebl/scv | elect BY-SA |
| *Active Calculus* (single variable, 2012–2025) | CC BY-SA 4.0 | PreTeXt | https://github.com/active-calculus/active-calculus-single-mbx | the OTL's "BY-NC-SA" is for 2.0 (2017) and is stale |
| Napkin (text) | CC BY-SA 4.0 | LaTeX | https://github.com/vEnhance/napkin/ | do not take the GPL code |
| Levin, *Discrete Mathematics*, 3rd edition and earlier | CC BY-SA 4.0 | PreTeXt | https://github.com/oscarlevin/discrete-book | pin a pre-4th-edition commit or tag |
| Keller–Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | https://github.com/mitchkeller/applied-combinatorics | |
| Ernst, *Intro to Proof via IBL* | CC BY-SA 4.0 | LaTeX | https://github.com/dcernst/IBL-IntroToProof | |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 | PreTeXt | https://github.com/davidaustinm/ula | |
| Beck et al., *First Course in Complex Analysis* | CC BY 4.0 | PDF | https://matthbeck.github.io/complex.html | |
| Lehman–Leighton–Meyer, *Mathematics for Computer Science* | CC BY-SA 3.0 → 4.0 | PDF | https://courses.csail.mit.edu/6.042/spring18/mcs.pdf | |
| OpenStax *Statistics*, *Precálculo* 2ed, and the earlier CC BY editions (Prealgebra 2015, Precalculus 2014) | CC BY 4.0 | CNXML | github.com/openstax (osbooks-statistics and others) | pin the CC BY edition |
| Per OTL, not verified at source: Zakon, *Mathematical Analysis I*; *Counting Rocks!*; *Calculus in Context*; Kuttler, *A First Course in Linear Algebra*; Shimamoto, *Multivariable Calculus*; Dumas–McCarthy; Wiggins, *ODE* | CC BY | various | https://open.umn.edu/opentextbooks/subjects/mathematics.json | verify at source (D8) |
| Per OTL, not verified at source: Kuttler, *Linear Algebra, Theory and Applications*; Hitchman (OTL says BY-SA, but the PreTeXt catalogue shows GFDL, D8); Petrunin, *Euclidean Plane and its Relatives*; Callahan, *Euclid's Elements Redux*; Davies, *A Cool Brisk Walk*; Brin, *Tea Time Numerical Analysis* and *Tea Time Linear Algebra*; Lippman, *Math in Society*; OpenIntro Statistics; *Introduction to Modern Statistics* 2e | CC BY-SA | various | same | verify at source |
| Public domain (US, pre-1931): e.g. Hardy, *A Course of Pure Mathematics* (PG #38769); Dedekind; Casey's Euclid; Thompson, *Calculus Made Easy* | public domain (US) | LaTeX / HTML | https://www.gutenberg.org/ebooks/bookshelf/102 | strip PG trademark text; check non-US status |

### 2.6 Table B: textbooks math2 may NOT adapt into node prose (NC, ND, GFDL, all rights reserved)

All of these can still give records, locators, tables of contents (9.7) and internal embeddings.

| Book | Licence | 9.3 class | URL |
|---|---|---|---|
| Judson, AATA | GFDL 1.3+ | `gfdl` | https://github.com/twjudson/aata |
| Beezer, FCLA | GFDL 1.2+ | `gfdl` | https://github.com/rbeezer/fcla |
| Stacks Project | GFDL 1.2 | `gfdl` | https://stacks.math.columbia.edu |
| The PreTeXt GFDL-badged titles (§2.3: Yoshiwara ×3, Fitch, ODE Project, etc.) | GFDL | `gfdl` | https://pretextbook.org/catalog.html |
| Per OTL: Grinstead–Snell, *Introduction to Probability*; Bogart, *Combinatorics Through Guided Discovery*; Corral, *Vector Calculus* and others; Collingwood et al., *Precalculus*; Yoshiwara, *Modeling* (12 GFDL in total) | GFDL | `gfdl` | https://open.umn.edu/opentextbooks/subjects/mathematics.json |
| Hartman et al., *APEX Calculus* | CC BY-NC 4.0 | `share-alike-nc` | https://github.com/APEXCalculus/APEXCalculusPTX |
| Axler, LADR 4e | CC BY-NC 4.0 | `share-alike-nc` | https://link.springer.com/book/10.1007/978-3-031-41026-0 |
| Per OTL: Axler, *Measure, Integration & Real Analysis*; Lafferriere et al. | CC BY-NC | `share-alike-nc` | OTL |
| *CLP Calculus* (UBC) | CC BY-NC-SA 4.0 | `share-alike-nc` | https://personal.math.ubc.ca/~CLP/src/ |
| Levin, *Discrete Mathematics*, 4th edition | CC BY-NC-SA 4.0 | `share-alike-nc` | https://github.com/oscarlevin/discrete-book |
| OpenStax Calculus 1–3, College Algebra 2e, Precalculus 2e, Algebra & Trig 2e, Prealgebra 2e, Elementary/Intermediate Algebra 2e, Intro Statistics 2e, Business Stats 2e, Contemporary Math, Algebra 1, Principles of Data Science, *Cálculo* | CC BY-NC-SA 4.0 | `share-alike-nc` | github.com/openstax |
| MIT OCW course materials | CC BY-NC-SA 4.0 (AI-training clause: non-commercial, BY-NC-SA models) | `share-alike-nc` | https://ocw.mit.edu/pages/privacy-and-terms-of-use/ |
| Per OTL: Guichard, *Calculus* (early transcendentals); Trench, *Introduction to Real Analysis* and *Elementary Differential Equations*; Sundstrom, *Mathematical Reasoning*; Doerr–Levasseur, *Applied Discrete Structures*; Rogers–Boman; Stitz–Zeager; Cherney et al., *Linear Algebra*; Nicholson, *Linear Algebra with Applications*; Morris, *Proofs and Concepts*; Sloughter; Hill–Thron; Strang, *Calculus* | CC BY-NC-SA | `share-alike-nc` | OTL |
| Hammack, *Book of Proof* 3e | CC BY-NC-ND 4.0 | `verbatim` | https://richardhammack.github.io/BookOfProof/ |
| Per OTL: Shoup, *A Computational Introduction to Number Theory and Algebra* | CC BY-NC-ND | `verbatim` | OTL |
| Hatcher, *Algebraic Topology* | © CUP 2002; noncommercial copies only | `record` | https://pi.math.cornell.edu/~hatcher/AT/ATpage.html |
| Lurie, *Kerodon* | no licence stated, so all rights reserved | `record` | https://kerodon.net/ |
| PreTeXt: *Linear Algebra* (W214), *Intro to Topology (IBL)* | all rights reserved | `record` | https://pretextbook.org/catalog.html |
| LibreTexts pages tagged "not declared" | none | `record` | https://math.libretexts.org |

---

## Part 3. Syllabi and theorem lists

### 3.1 MAA CUPM Curriculum Guide (2015)

- **Citation:** *2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences*, Schumacher and Siegel (co-chairs), Zorn (editor). 116 pages. The PDF has CreationDate 2015-05-05.
- **Copyright:** "© 2015 The Mathematical Association of America (Incorporated)", ISBN 978-0-88385-917-9. I found no CC licence, so it is `record`.
- **URLs:**
  - Official page: https://maa.org/resource/cupm-guide/. It is JavaScript-rendered, so curl got no content.
  - Copy I read: https://www.amstat.org/docs/default-source/amstat-documents/cupmguide_print.pdf
  - The separate course-area reports (Abstract Algebra, Real Analysis, Topology, etc.) are linked from the MAA page. Their licences are not checked.
- **What math2 can take:** the structure (course areas, topic lists) as facts for `topics/curricula/`, and citations. Not the prose.
- Nothing newer than the 2015 guide was found (D12).

### 3.2 QAA Subject Benchmark Statement: Mathematics, Statistics and Operational Research

- **Current edition:** the **5th edition, published 2023-03-08** (the PDF's last page reads "Fifth Edition / Published - 8 March 2023"). It covers Master's level too.
- **Page:** https://www.qaa.ac.uk/the-quality-code/subject-benchmark-statements/subject-benchmark-statement-mathematics-statistics-and-operational-research
- **PDF:** https://www.qaa.ac.uk/docs/qaa/sbs/sbs-mathematics-statistics-and-operational-research-23.pdf, 2.1 MB, 35 pages.
- **Copyright:** "© The Quality Assurance Agency for Higher Education 2023". No CC or OGL licence is stated in the PDF, so it is `record`. (An older 2009 annex is in DERA under the Open Government Licence. That does not carry over to 2023.)
- **Upcoming:** the QAA site has a "Subject Benchmark Statement Reviews 2026-27" page, so a 6th edition may be in progress (D12).
- **What math2 can take:** its outcome categories, as facts.

### 3.3 Mathlib lists

Source: leanprover-community/mathlib4 `master` (repository HEAD 4c30f23495 at 2026-10-26T08:02Z). Licence Apache-2.0.

| File | Contents | Last change |
|---|---|---|
| `docs/undergrad.yaml` (https://raw.githubusercontent.com/leanprover-community/mathlib4/master/docs/undergrad.yaml) | 563 leaf items, 130 of them empty (not yet formalised). The topics were originally taken from the 2020 French *agrégation* programme | 58d1b21e88, 2026-10-07 |
| `docs/overview.yaml` | 511 leaves | n/a |
| `docs/1000.yaml` | 1,199 entries keyed by Wikidata id; 217 have `decl`(s), 30 have an external `url` | c6f03d20ca, 2026-10-17 |
| `docs/100.yaml` | 100 entries; 78 have `decl` | n/a |

**What math2 can take:** everything. These are Apache-2.0 code-zone data and facts, and the best machine-readable syllabus and theorem lists available. The Wikidata keys of `1000.yaml` are crosswalk ids.

### 3.4 Freek Wiedijk's 100 theorems

- **Page:** https://www.cs.ru.nl/~freek/100/, last modified 2026-09-17. All 100 theorems are formalised in at least one system.
- **Per system:**

| System | Theorems |
|---|---:|
| HOL Light | 95 |
| Isabelle | 95 |
| Lean | 83 |
| Rocq | 80 |
| Metamath | 74 |
| Mizar | 71 |
| ACL2 | 48 |
| ProofPower | 43 |
| PVS | 26 |
| Imandra | 20 |
| Megalodon | 12 |
| Naproche | 10 |
| NuPRL | 8 |

- **Licence:** no licence statement, so `record`. The list of theorem names is factual.
- **Discrepancy:** Mathlib's `100.yaml` has 78 entries with a `decl`, but Freek's page counts 83 for Lean. The difference is probably entries proved outside Mathlib, or stale yaml. Not resolved.

### 3.5 The 1000+ theorems project

- **Site:** https://1000-plus.github.io/. I counted 228 entries on the page.
- **Repository:** https://github.com/1000-plus-theorems/1000-plus, licence **Unlicense**, last pushed 2026-07-22. That makes it public-domain-equivalent, so `open`.
- **Relation to Mathlib:** Mathlib's `1000.yaml` is the Lean-side view of the same Wikidata-keyed list.

### 3.6 Department qualifying-exam lists

- **Example:** UC Berkeley Preliminary Examination (https://math.berkeley.edu/graduate/phd-program/preliminary-exam).
  - Two mornings of 3 hours each. Students attempt 5 of 8 questions per day (9 before Fall 2024).
  - Topics: real analysis and calculus, complex analysis, linear algebra, abstract algebra.
  - The syllabus PDF and past exams are linked at the bottom of the page.
- These are department web pages with no licence, so `record`. Topic lists are factual. Past exam problems are copyrighted text.
- *Berkeley Problems in Mathematics* (de Souza and Silva, Springer) collects 1,250 prelim problems from 1977 to 2003. It is all rights reserved.
- I did not survey other departments; other examples include UW (https://math.washington.edu/phd-preliminary-exams). **Breadth is UNVERIFIED** (D12).

---

## Summary table

| Item | Size (date) | Licence of text / data | Format | Bulk channel and terms | AI terms | What math2 can take (9.3) |
|---|---|---|---|---|---|---|
| arXiv | about 9.2 TB of files (April 2025), +100 GB/month | metadata CC0; text per paper. 2026 math sample is 36.9% `open` (BY/BY-SA/CC0) vs 15% all-time | TeX source, PDF | `s3://arxiv` requester-pays; API 1 req / 3 s; must link back; redistributing e-prints needs the holder's permission | none stated | records (CC0); text of `open` papers into prose; others `record` |
| zbMATH Open | 4M+ publications, 3M+ reviews, 168M+ triples (KG v1, 2026-09-01); 133,093 documents for 2025 | KG: Zenodo field CC BY 4.0 vs description CC BY-SA 4.0; I4OC references CC0 | RDF/TTL, JSON API, OAI-PMH | Zenodo dump (1.6 GB, twice a year); REST; OAI-PMH; API terms UNVERIFIED | UNVERIFIED | records, MSC, reviews as `open` evidence |
| OpenAlex | about 745 GB JSONL (September 2026) | CC0 | JSONL, Parquet | `s3://openalex` anonymous, quarterly; API key since 2026-02 (secondary source) | none | all metadata |
| Unpaywall | n/a | n/a | n/a | folded into OpenAlex; data feed paid; terms UNVERIFIED | n/a | OA locations via OpenAlex |
| Crossref | 180,034,490 records, 208 GB (2026-03-14) | metadata effectively free; some abstracts copyrighted | JSONL | public data file (torrent, AWS); open REST API | none | metadata, references; abstracts `record` |
| DataCite | 108M DOIs (2025 file, 2026-01-06) | CC0 | JSON | annual public file; monthly file members-only | none | all metadata |
| Semantic Scholar / S2ORC | S2ORC 15,679,781 full texts (2026-05-26) | API non-transferable; S2ORC ODC-By (database) plus per-paper copyright; abstracts UNVERIFIED | JSON | API key; Datasets API | none | ids; text only per paper licence |
| CORE | 291M metadata / 32.8M full texts (2023); current size UNVERIFIED | dumps ODC-By (database only) | JSON | API licence; dumps | proposed "lawful TDM" | records; text per paper |
| HAL (math) | 170,406 records, 105,413 with file (2026-10-26) | metadata CC0; files: 9.8% `open`, 80% hal-authorisation | PDF, TEI | OAI-PMH, REST, SPARQL | none found | records; the 10,349 `open` texts |
| Numdam | 75,599 articles, 706 books, 1.43M pages | metadata CC0; text: personal use, no bulk, no modification | PDF, OAI | OAI-PMH (metadata only) | none | records; text only where CC or PD |
| EuDML | over 270k documents (undated) | UNVERIFIED | n/a | blocked (403) | n/a | records via partner sources |
| Project Euclid | "over 80%" free to read | per journal, UNVERIFIED | PDF | none found | n/a | records; per-article licence |
| GDZ | over 15M pages | scans' licence UNVERIFIED; PDFs: no re-hosting without permission | IIIF, PDF | IIIF, OAI | n/a | records; public-domain statements rewritten |
| HathiTrust | n/a | public-domain subsets; terms UNVERIFIED (Cloudflare) | text | `ht_text_pd*` datasets | UNVERIFIED | US pre-1931 public-domain text |
| Internet Archive | 206,597 math texts; 8,076 dated 1500–1930 | mixed; hosting does not establish licence | PDF, OCR | advancedsearch; terms UNVERIFIED | UNVERIFIED | public-domain items after a per-item check |
| Project Gutenberg | about 78,000 items (2026-03) | public domain (US) | TXT, HTML, some LaTeX | mirrors or `/robot/harvest`, `-w 2` | none | `open` (strip PG trademark) |
| TheoremSearch | 8.4M indexed (live site); 1,341,083 BY-SA release; 1,239,720 permissive release | HF tags BY-SA / BY; BY-SA set includes GFDL Stacks | Parquet, REST, MCP | Hugging Face; API with no terms | none | `-permissive` set as E1 candidates |
| TheoremGraph / LeanGraph | 16,131,238 rows (math-graph) | CC BY 4.0 plus per-record source licence; text NULL unless open | CSV, Parquet | Hugging Face | none | edges, ids, open-only text |
| Matlas | 8.07M statements, 435K papers, 1.9K textbooks | not stated (UNVERIFIED; likely closed) | JSON API | `POST /api/search`, no terms | none | query-time ids only (`record`) |
| Open textbooks | 214 math books in OTL; 68 AIM-approved; 67 PreTeXt math | OTL: 48% `open`, 52% NC/ND/GFDL | LaTeX, PreTeXt, PDF | Git repositories | MIT OCW: NC models only | Table A only |
| MIT OCW | n/a | CC BY-NC-SA 4.0 | HTML, PDF | none | training allowed, NC models, BY-NC-SA | `share-alike-nc` |
| CUPM 2015 | 116 pages | © MAA | PDF | n/a | n/a | structure as facts |
| QAA SBS MSOR (5th edition, 2023-03-08) | 35 pages | © QAA 2023 | PDF | n/a | n/a | structure as facts |
| Mathlib yaml | undergrad 563 (130 empty); 1000.yaml 1,199; 100.yaml 100 | Apache-2.0 | YAML | Git | n/a | all |
| Freek's 100 | 100 (2026-09-17) | no licence | HTML | n/a | n/a | list as facts |
| 1000+ theorems | 228 entries on the site | Unlicense | Markdown, YAML | Git | n/a | all |

---

## Open doubts

1. **D1. zbMATH KG licence.** The Zenodo field says CC BY 4.0, but the record description says CC BY-SA 4.0. Ask zbMATH (or open a GitHub issue on zbMATHOpen/zbmath-open-kg) which one governs. Both are `open`, but BY-SA binds derived exports. The API terms are unread (Cloudflare).
2. **D2. arXiv `open` share.** The spec's "about 15%" is the all-time stock. My three-day 2026 math sample gives 36.9% (and about 39% for re-versioned older papers). Confirm this with a full 2026 `set=math` harvest from the `tamnd/arxiv` metadata plane. Separate first versions from updates, and decide whether cross-lists count. If it holds, spec 9.3 should state both figures.
3. **D3. Missing AI clauses.** arXiv, Crossref, OpenAlex, HAL and Gutenberg say nothing about AI. MIT OCW has an explicit NC-only AI clause. Stack Exchange forbids LLM use (already in the spec). The zbMATH, HathiTrust, IA and EuDML terms are unread. Silence in the terms is not permission: the per-item licence still governs.
4. **D4. TheoremSearch BY-SA set** puts Stacks (GFDL), ProofWiki and other sources under one `cc-by-sa-4.0` tag. math2 must filter by source and licence before treating any row as `open`.
5. **D5. TheoremSearch counts.** The card says 1,239,720 rows for the permissive set; the dataset viewer shows 2,677,439 rows. The BY-SA set has 1,341,083. The live site shows 8.4M against the paper's 9.2M. Not reconciled. The difference may be duplicated rows across splits, or slogans stored as separate rows.
6. **D6. Kerodon licence. Spec 9.3 and 16.6 list Kerodon as GFDL; I found no licence statement anywhere on kerodon.net.** Unless Lurie states a licence, Kerodon is all rights reserved, so `record`. The spec should change it to `record`, and §1.7's "[U]" should be resolved accordingly. The tag count on /statistics is 8,189; §1.7 says "2,200+ pages".
7. **D7. OpenStax edition drift.** The current 2e repositories are CC BY-NC-SA 4.0, but older editions were CC BY. Adapting an old CC BY edition is legal, but the edition and commit must be pinned.
8. **D8. OTL licences are stale or simplified** (*Active Calculus*, *Book of Proof*, Hefferon, Hitchman). Every Table A row marked "per OTL" must be verified at the source before ingestion.
9. **D9. Dual-licensed books** (Lebl ×3, Hefferon). math2 must record its election of the BY-SA option in the source entry.
10. **D10. Public domain is territorial.** The US pre-1931 rule does not make a work public domain elsewhere (life plus 70 years in the EU). Since math2 is published worldwide, decide whether US public domain is enough for `open`. For Hardy (died 1947) the text is public domain in the EU too; for some authors it is not.
11. **D11. AIM titles with unchecked licences** (Bruckner–Thomson, ORCCA, Manes, May–Bart, Marsden–Weinstein, Boyd–Vandenberghe, Treil, Fields, Bender–Williamson, Bard, Zimmermann et al., Crisman, Stein, Connell).
12. **D12. Syllabi.**
    - CUPM 2015 is the latest guide I found; I did not confirm whether a newer one exists.
    - The QAA 6th-edition review (2026–27) has not been checked.
    - Qualifying-exam lists were sampled at one department only.
    - The difference between Mathlib's `100.yaml` (78 with `decl`) and Freek's Lean count (83) is not resolved.
13. **D13. UNVERIFIED because of blocking or missing pages:**
    - zbMATH API terms
    - EuDML metadata licence and size
    - HathiTrust terms
    - Internet Archive terms
    - current CORE size and dump terms
    - Unpaywall terms
    - Semantic Scholar abstracts licence
    - Project Euclid per-journal licences
    - GDZ scan licence
    - OpenAlex API pricing (secondary source only)
14. **D14. Matlas** returns full statement text from copyrighted journals and textbooks through an open API with no terms. math2 should store only ids from it (spec 9.6). Its corpus licence is unknown.
15. **D15. LibreTexts** has no aggregate licence data. The per-page "not declared" share is unknown; treat it as `record` and prefer the upstream source.
</document>

<document source="https://math2.tamnd.com/spec/corpus/01c-formal-libraries.md">
# K01, part C. Formal libraries and formal / informal-formal datasets

Research note for the Math 2.0 spec (tamnd/math2). This note extends spec 01 §1.2, spec 07 (bindings, M2Extract, §7.10 other proof assistants) and spec 09.4 (libraries enter as bindings, never as copies). It does not repeat those sections. Where a number has moved since the spec was written, the note says so.

Conventions:
- I read every source on **2026-10-26**, unless the entry says otherwise. "(read 2026-10-26)" is left out to keep the text short. Every fact has a URL.
- **UNVERIFIED** means I could not confirm the fact from a primary source. The reason follows the tag.
- File counts come from the GitHub git-trees API (`recursive=1`). The `truncated` flag was false in every case unless noted. They count files, not declarations.
- HF row counts come from `datasets-server.huggingface.co/size` or the dataset card.

---

## 1. Formal libraries

### 1.1 Lean 4

**Mathlib**
- Stats page https://leanprover-community.github.io/mathlib_stats.html: **139,616 definitions, 291,237 theorems, 772 contributors**. The page shows no date. Spec 01 had 291,178 / 139,569 on 2026-10-25, so the count grew in one day.
- Licence: Apache-2.0 (https://github.com/leanprover-community/mathlib4). The repo has 35,715 commits and about 4.4k stars.
- Tree API: **8,661 `.lean` files under `Mathlib/`**. The tree has 10,796 entries and was not truncated. Spec 01 says 9,437 files, which probably counts every file and not only `Mathlib/*.lean`. This is a definitional difference, not a conflict.
- Informal text: docstrings and module docs only. See the docstring coverage under 1.1.1.

**1.1.1 Official extraction route: Mathlib Initiative datasets (new; not in spec)**
- Tool: `lean_scout`, https://github.com/mathlib-initiative/lean_scout (Apache-2.0).
- Datasets on HF under `mathlib-initiative/`, all Apache-2.0. They were extracted at Mathlib commit `56a005855ea1…`, "chore: bump toolchain to v4.35.0 (#44806)", dated **2026-10-13**.
  - `mathlib-types`: **797,294 rows**. Fields are `name`, `module`, `type`, `docString` and `allowCompletion`.
    - `allowCompletion = True` for 483,112 rows.
    - `docString` is null for **681,418 rows (85.47%)**. So only about **115,876 constants (~14.5%) have a docstring**, with a mean length of about 130 characters.
  - `mathlib-const-dep`: 797,294 rows (dependency edges per constant).
  - `mathlib-tactics`: **1,999,349 rows** (tactic plus goal states).
- Relevance to M2Extract (spec 07): the schema `name / module / type / docString` already matches what a binding needs. The constant count of 797k covers auxiliary and internal constants. The `allowCompletion` filter, at about 483k, is a better match for "user-facing declarations". Neither number matches the 291k theorems plus 139k definitions on the stats page. Each counts a different thing, and M2Extract should record which one it uses.

**doc-gen4**
- https://github.com/leanprover/doc-gen4 (Apache-2.0).
- It runs as `lake build Lib:docs` from a nested docbuild project. The target library must compile first.
- Output is HTML plus a declaration-data JSON. That makes it heavier than lean_scout when only statements are needed.

**Reservoir (Lean package index)**
- https://reservoir.lean-lang.org/packages?sort=stars: **908 packages**. Reservoir shows no sizes or licences in the listing.
- Top packages by stars:

  | Package | Stars |
  |---|---|
  | mathlib | 4,352 |
  | openai/NavierStokesAndEuler | 2,110 |
  | teorth/Analysis | 1,971 |
  | ulfjack/ryu | 1,357 |
  | LeanCopilot | 1,337 |
  | formal_conjectures | 1,336 |
  | anthropics/flt_e2e | 1,292 |
  | ImperialCollegeLondon/FLT | 1,053 |
  | Physlib | 808 |
  | cslib | 777 |
  | mil | 623 |
  | equational_theories | 562 |
  | paperproof | 554 |
  | scilean | 539 |
  | glimpseOfLean | 426 |
  | batteries | 424 |
  | aesop | 410 |
  | verso | 398 |
  | Game | 387 |
  | PrimeNumberTheoremAnd | 351 |

- Licences of individual packages must be read per repo. I did not survey them all.

**Tau Ceti**
- https://github.com/TauCetiProject/TauCeti: Apache-2.0, 19,274 commits, 247 stars.
- **18,849 `.lean` files** (tree API), which is about twice Mathlib's file count.
- Written by AI and reviewed against human rubrics. It tracks Mathlib master with an hourly bump. It is incubated by the Lean FRO and the Mathlib Initiative.
- Docs are at taucetiproject.org/docs. The stats page returned 404.
- Xena blog, 2026-10-16 (https://xenaproject.wordpress.com/): "Tau Ceti is now slightly larger than Mathlib (3 million lines of code)".
  - **UNVERIFIED** whether "3 million lines" refers to Tau Ceti or to Mathlib. The sentence is ambiguous. Spec 01 reads it as Tau Ceti ≈ 3M. Spec 01 also gives Mathlib 2,481,682 lines, which fits "slightly larger" only if Tau Ceti is somewhat above 2.5M.
- Consequence for E-tiers: Tau Ceti is mostly AI-authored. Any provenance or trust flag in spec 07 should separate it from human-reviewed Mathlib.

**Formal Frontier**
- https://github.com/orgs/FormalFrontier/repositories: 50 repos. All 30 shown on the first page are **Apache-2.0**.
- Examples: projective-modules, algebraic-groups, group-theory, homological-algebra, complex-analysis, scheme-properties, sphere-topology, simplicial-objects, iwasawa-modules.
- Xena (2026-10-16) reports 41 released libraries, about 200k lines. Highlights include the hairy ball theorem, Kaplansky's theorem, Bass stable range, Tate periodicity and Stone duality.
- The page mathlib-initiative.org/formal-frontier names no licence. The licence of the remaining 20 repos is UNVERIFIED (not shown on the first page).

**ATLAS (Meta), licence warning (new)**
- Repo https://github.com/facebookresearch/atlas-lean; paper arXiv:2605.29955 (2026-05-28).
- Size: 26 textbooks, more than 45,000 declarations, about 500k lines.
- **v1 is under CC BY-NC 4.0 with a "no-training rider".** It prohibits commercial use and any use "to train, fine-tune, distill, evaluate, or otherwise develop ML models".
- **v2 (repo root) is labelled Apache 2.0, but the root LICENSE file opens with "Codebase is not intended for use to train, fine-tune, distill, evaluate, or otherwise develop ML models".** This is an Apache grant with a non-standard preamble, so its legal effect is unclear.
- AutoformBench (https://github.com/facebookresearch/autoformbench) is CC BY-NC 4.0.
- 36Kr (https://eu.36kr.com/en/p/3829852422940291) reports 183B tokens. This is a secondary source and UNVERIFIED in the paper.
- Recommendation: crosswalk only. Do not bind v1. Treat v2 as legally unclear until the preamble is resolved.

### 1.2 Isabelle

**AFP**
- https://www.isa-afp.org/statistics/: **1,042 entries, 614 authors, ~330,000 lemmas, ~5,518,000 lines**. Spec 01 had 1,038 / ~326,500 / ~5.44M. The page gives no per-year data.
- Licence (https://isa-afp.org/about/): "All entries … licensed under a BSD-style License or the GNU LGPL". Each entry page has a License field, so the licence must be recorded per entry.
- Access: https://isa-afp.org/release/afp-current.tar.gz (about 70 MB), or Mercurial at https://foss.heptapod.net/isa-afp.
- AI provenance: the entry Jacobian_Counterexample (2026-07-20, BSD) says "AI assistance was used for proof engineering". The AFP has no archive-wide AI policy that I found.
- Informal text: Isabelle theories carry `text \<open>…\<close>` blocks and each entry has an abstract. That is more prose than Mathlib docstrings, but I did not measure it.

**Isabelle2026**
- https://isabelle.in.tum.de/: "Now available: Isabelle2026 (October 2026)". This **resolves the [U] in spec 01**.
- Features include HOL-New_Algebra and cvc5 proof reconstruction.
- Licence: BSD-style for the core, mixed overall (bundled components).
- sketis.net shows RC4 dated 14-Oct-2026 and the final release planned for 24-Oct-2026.
- The size of the distribution library (HOL, HOL-Analysis and so on) is UNVERIFIED. I found no official count.

### 1.3 Rocq (formerly Coq)

| Library | URL | Licence (LICENSE file) | `.v` files |
|---|---|---|---|
| Rocq core | https://github.com/rocq-prover/rocq | LGPL-2.1 | – |
| Stdlib | https://github.com/rocq-prover/stdlib | LGPL-2.1 | 904 |
| MathComp | https://github.com/math-comp/math-comp | CeCILL-B | 147 |
| MathComp Analysis | https://github.com/math-comp/analysis | CeCILL-C | 131 |
| Odd Order | https://github.com/math-comp/odd-order | CeCILL-B | 34 |
| UniMath | https://github.com/UniMath/UniMath | MIT-style custom | 1,714 |
| Coquelicot | gitlab.inria.fr (blocked by Anubis) | **UNVERIFIED** | – |

- The coq-community org was renamed **rocq-community** and has 75 public repos. The math-comp org has 27 repos.
- The opam `released/packages` directory (rocq-prover/opam) has 603 entries.
- Coquelicot: the GitHub fork proux01/coquelicot is LGPL-3.0. The upstream licence is UNVERIFIED because the Inria GitLab did not load (an Anubis bot challenge).
- MathComp files are large, so file count understates size. I found no declaration count.
- CeCILL-B is BSD-like and CeCILL-C is LGPL-like. Both allow bindings.
- Informal text: comments only. MathComp has structured header comments.

### 1.4 HOL Light and HOL4

- **HOL Light**: https://github.com/jrh13/hol-light. BSD-style LICENSE, 664 `.ml` files. Blanqui's LPAR talk gives about 37,000 objects in 635 files. I have the talk only as a secondary reference, so that figure is UNVERIFIED in a primary listing.
- **Flyspeck**: MIT, 503 `.hl` files.
- **HOL4**: https://github.com/HOL-Theorem-Prover/HOL. BSD-style COPYRIGHT, 1,495 `*Script.sml` files.
- **hol2dk** (https://github.com/Deducteam/hol2dk, CeCILL 2.1): translates HOL Light to Dedukti, Lambdapi and Rocq.
  - `hol.ml` has 5,687 theorems.
  - Multivariate has 40,728 theorems. The dump is about 135 GB and Rocq checking takes about 21 h.
  - Output is in the opam packages `coq-hol-light` and `rocq-hollight-logic`.
  - **There is no Lean output.**
- arXiv:2609.24598 (Blanqui and Gontard, 2026-09-21, LPAR-26) maps HOL Light constants to idiomatic Rocq. This "alignment" step is what makes translations usable rather than literal.
- **HolLean** (https://github.com/hol-light-family/HolLean) ports the HOL Light *system* to Lean 4 as a meta-language. It does not translate the library.

### 1.5 Metamath

- **set.mm**: https://github.com/metamath/set.mm. **LICENSE = CC0 1.0** (read through the gh API). 17,928 commits; last push 2026-10-26.
- Sizes. The three pages count different things.
  - The theorem list https://us.metamath.org/mpeuni/mmtheorems.html has 513 pages, and its last statement number is **51,207**. That number includes axioms and definitions. The newest dated theorem is 27-Aug-2026.
  - https://us.metamath.org/mpeuni/mmset.html says "over 26,000 … main sections, over 41,000 counting mathboxes".
- **iset.mm** (https://us.metamath.org/ileuni/mmtheorems.html): about 17,395 statements (174 pages).
- **Metamath 100**: 75 as of 2026-04-28 (the mm_100 page). Wiedijk's list credits Metamath with 74. The gap is a sync lag.
- Tooling:
  - **mm0** (https://github.com/digama0/mm0, CC0) includes a set.mm to Lean 3 translation path.
  - digama0/mm-lean4 and zariuq/mm-lean4 are Metamath verifiers in Lean 4, not library translators.
- Informal text: every statement has a mandatory comment, usually a sentence or more. **set.mm is the only large library that is CC0 and has a prose comment on every statement.** This strengthens the "v2 candidate" status in spec 07 §7.10.

### 1.6 Mizar

- https://mizar.uwb.edu.pl/: **MML 5.94.1493 (2025-05-30)**, Mizar 8.1.15.
- **Possible conflict**: in MML naming, the last field of the version (1493) is normally the article count. Spec 01 gives 1,498 articles. UNVERIFIED which is right; the 73,460 theorems figure could not be rechecked.
- Licence: dual **GPLv3+ / CC BY-SA 3.0+** since MML 4.160.1126. Sources: https://mizar.uwb.edu.pl/forum/archive/1104/msg00000.html and arXiv:1107.3212.
- The system has been GPLv3 at https://github.com/MizarProject/system since about 2023.
- CC BY-SA is share-alike. Derived statements would inherit it, so it should be crosswalk-only unless the corpus accepts SA.

### 1.7 Agda

| Library | Licence | Files | Informal text |
|---|---|---|---|
| agda-stdlib (https://github.com/agda/agda-stdlib) | MIT-style | 1,183 `src/*.agda` | comments |
| agda-unimath (https://github.com/UniMath/agda-unimath) | MIT | 3,072 `.lagda.md` | **literate Markdown, prose throughout** |
| 1Lab (https://github.com/the1lab/1lab) | **AGPL-3.0** | 710 `.lagda.md` | **literate, prose throughout** |

- This fills the "Agda sizes [U]" gap in spec 01 with file counts. Declaration counts are still unknown.
- 1Lab is AGPL, so it should be crosswalk-only.
- agda-unimath is the cleanest literate library for informal and formal alignment: MIT, and prose sits next to each definition.

### 1.8 Naproche, Megalodon, PVS, ACL2

- **Naproche** (https://github.com/naproche/naproche): GPL-3.0, 32 `.ftl.tex` files. The text is controlled natural language, so the source is itself informal.
- **Megalodon**
  - ai4reason/Megalodon has an MIT-style licence (CIIRC/CTU).
  - The mgwiki/god1 and mgwiki/mgw_test repos have **no licence** (gh shows none).
  - arXiv:2601.03298 (Urban, 2026-01-06) reports about 160k lines and more than 1.5k lemmas. The work used ChatGPT 5.2 or Claude Sonnet 4.5 and cost about $100. Spec 01 had 130k lines for Munkres; the paper is newer.
- **NASALib** (https://github.com/nasa/pvslib)
  - The README says "63 top-level libraries, containing about 41K proven formulas". The file `nasalib.grandtotals` says 64 libraries.
  - Licence: the GitHub API shows none. The README says the NASA developments are "in the public domain"/"fundamental research that do not constitute software". Other contributions carry licences in each `top.pvs`, so licences are per sub-library.
  - The repo contains a `CLAUDE.md`, which suggests AI-assisted maintenance.
- **ACL2** (https://github.com/acl2/acl2): BSD-3 for the system. The `books/` directory has per-file licences. The tree has 40,929 entries.

---

## 2. Translation between systems

### 2.1 Infrastructure

| Tool | Status | Licence | Notes |
|---|---|---|---|
| Mathport (https://github.com/leanprover-community/mathport) | **archived**, last push 2024-11-21 | Apache-2.0 | Lean 3 to Lean 4 only; finished its job |
| lean2dk (Deducteam) | WIP, last push 2026-06 | Apache-2.0 | Lean 4 to Dedukti; spec 01 already says WIP |
| Logipedia | last push 2024-11 | – | All proofs in STTForall (arXiv:2305.00064). Dormant |
| OpenTheory | last milestone 2017 | – | Reads and writes HOL Light, HOL4 and ProofPower; reads Isabelle; no counts published |
| hol2dk | active (2026) | CeCILL 2.1 | HOL Light to Rocq, 40,728 theorems; no Lean target |
| Minilang (arXiv:2507.18885) | paper | – | Rule-based Isabelle Isar translation, about 340K proofs, about 85% success |

- No working, library-scale **X to Lean 4** translator exists as of today. The paths are hol2dk (to Rocq), mm0 (set.mm to Lean 3, legacy) and LLM-based porting.

### 2.2 LLM-based translation, with measured success

- **ITPEval** (arXiv:2607.19407, Wu, George and Anandkumar, 2026-07-07)
  - Covers 1,560 files and 6,848 theorems. It tests 12 directed pairs over Lean 4, Rocq, Isabelle and HOL Light, with 5 LLMs.
  - Statement translation peaks at **29.1% pass@1**; proof translation at **10.5%**.
  - Controlled files reach 29.7%, ecosystem files only 5.2%.
  - Multi-ITP context raises Lean success from 4.8% to 10.6%.
  - BEq confirms only **54.0%** of compiler-verified miniF2F-to-Lean statements.
  - Conclusion for the spec: compiling is not the same as being faithful.
  - The arXiv page gives no data URL, so dataset availability is UNVERIFIED.
- **miniF2F-rocq** (arXiv:2503.04763; https://github.com/LLM4Rocq/miniF2F-rocq, MIT): 478 of 488 problems translated, using GPT-4o mini, Claude 3.5 Sonnet, o1 mini and o1.
- **Megalodon Munkres** (arXiv:2601.03298): LLM-driven formalization of a full textbook for about $100 (see 1.8).
- **Lean Pool** (arXiv:2609.25199, 2026-09-21): an AI-maintained archive with a catalogue of imported projects. It gives no counts, so its size is UNVERIFIED.
- The Lean Zulip thread "Autotranslation of Logical Foundations from Rocq to Lean" (Jan 2026) is anecdotal evidence. I took no measured numbers from it.

### 2.3 Cross-system theorem lists

**Wiedijk 100** (https://www.cs.ru.nl/~freek/100/, last modified 2026-09-17):

| System | Count |
|---|---|
| HOL Light | 95 |
| Isabelle | 95 |
| Lean | 83 |
| Rocq | 80 |
| Metamath | 74 |
| Mizar | 71 |
| nqthm/ACL2 | 48 |
| ProofPower | 43 |
| PVS | 26 |
| Imandra | 20 |
| Megalodon | 12 |
| Naproche | 10 |
| NuPRL | 8 |

**1000+ theorems** (https://github.com/1000-plus-theorems/1000-plus)
- Licence: **Unlicense** (public domain). Last push 2026-07-22.
- **1,200 `_thm/*.md` YAML files keyed by Wikidata Q-id.**
  - Fields: `msc_classification`, `wikipedia_links`, and per-system blocks `isabelle`, `hol_light`, `rocq`, `lean`, `metamath`, `mizar`.
  - Entries come from Wikipedia's List of theorems.
- This is the best ready-made **crosswalk seed** for spec 07 §7.10: public domain, keyed by Wikidata, MSC-tagged and pointing into six systems.
- Per-system counts are UNVERIFIED. The web page showed about 228 formalized entries, which may be one page of a paginated list. Spec 01's "241 in Lean" could not be rechecked without parsing all 1,200 files.

---

## 3. Datasets

### 3.1 Benchmarks (formal statement plus informal statement)

**miniF2F**
- 488 problems: 244 valid, 244 test.
- Lean 4 ports:
  - yangky11/miniF2F-lean4 (MIT)
  - facebookresearch/miniF2F (MIT)
  - AI-MO/minif2f_test (Apache-2.0, 244 rows)
- Quality: "miniF2F-Lean Revisited" (arXiv:2511.03108, NeurIPS 2025) found discrepancies between formal and informal statements in **more than half** the problems.
  - Pipeline accuracy is 36–40% on the original and about 70% on the corrected v2.
  - The "over 300 problems corrected" figure is UNVERIFIED (search snippet only).

**ProofNet**
- MIT, 371 problems, originally Lean 3.
- ProofNet# (arXiv:2406.07222; HF PAug/ProofNetSharp, MIT) found **118 of 371 (31.8%)** wrong in the Lean 4 ports.
- ProofNet-Verified (HF marcusm117, Apache-2.0, NeurIPS 2026, targets Lean v4.34.0) found **205/367 (55.9%)** errors in ProofNet and **99/367 (27.0%)** in ProofNet#. A fixed benchmark still had errors in more than a quarter of its items.

**PutnamBench** (https://github.com/trishullab/PutnamBench)
- Lean 4 672, Isabelle 640, Coq 412, covering 1962–2025.
- Apache 2.0 for Lean and Isabelle, MIT for Coq.
- **The informal statements are used "with permission from the MAA"**. They are third-party text, so do not relicense them.
- The repo-level licence field is empty, so the licence is per subdirectory.

**FormalMATH**
- HF SphereLab/FormalMATH-All: MIT, 5,560 rows. GitHub Sphere-AI-Lab/FormalMATH-Bench: Apache-2.0 (EMNLP 2026 Findings). The two licences differ.
- About 72.09% of statements were kept before expert review. The best prover reached 16.46%.

**DeepSeek-ProverBench** (https://huggingface.co/datasets/deepseek-ai/DeepSeek-ProverBench)
- 325 problems: 15 from AIME 24/25 and 310 from textbooks.
- **No licence tag.**

**Defect audit across benchmarks: "Faults in Our Formal Benchmarking"** (arXiv:2606.29493, ICML 2026)
- 4,833 findings and **398 certified defects** in total.

  | Benchmark | Findings | Certified |
  |---|---|---|
  | FormalMath | 3,463 | 147 |
  | ProverBench | 370 | 208 |
  | ProofNet | 149 | 6 |
  | miniF2F (7 variants) | 773 | 36 |
  | CombiBench | 78 | 1 |

- ProverBench has **208 certified defects out of 325 problems**.

**Statement-faithfulness judges**
- EFG (arXiv:2506.10903) reports correctness of 76.64% for Isabelle miniF2F, 82.79% for Lean miniF2F and 64.84% for ProofNet Lean. These are UNVERIFIED: search snippet only, not confirmed on the abstract page.
- Beyond Compilation (arXiv:2606.31002): 227 statements. Its judge agrees with the human majority 91.5% of the time.

### 3.2 Large synthetic statement and proof sets (competition style)

| Dataset | Rows | Licence | Informal side | Notes |
|---|---|---|---|---|
| Lean Workbook (internlm, HF `internlm/Lean-Workbook`) | 57,231 + 82,893 (Plus) per paper/card; **default HF config 25,214 rows** | Apache-2.0 | Competition problems (AoPS) | Lean v4.8.0-rc1. A manual sample check found 93.5% accurate. 4,898 proved (pass@1024 is 8.6%). The row-count gap is UNVERIFIED |
| Goedel-LM/Lean-workbook-proofs | 29,750 | **no tag** | – | proofs for Lean Workbook |
| DeepSeek-Prover-V1 (HF) | 27,503 | "other" (deepseek-license; the card says commercial use is supported) | NL problem | The paper describes 8M synthetic statements; only 27.5k released |
| DeepSeek-Prover-V2 repo | – | NOASSERTION | – | cold-start data generated by DeepSeek-V3 |
| Goedel-Pset-v1 | 1,732,594 | **no tag** | NuminaMath problems | Two formalizers. Filtered by compile plus an FC test with Qwen2.5-72B-Instruct |
| Goedel-LM/SFT_dataset_v2 | 1,745,010 | Apache-2.0 | – | Goedel-Prover-V2 (arXiv:2508.03613) |
| Goedel-LM/RL_dataset_V2 | – | Apache-2.0 | – | 2026-03 |
| NuminaMath-LEAN (AI-MO) | 104,155 | Apache-2.0 | NuminaMath problem | Mathlib v4.15.0. `author` = human or autoformalizer. Proofs from Kimina-Prover-72B. **The card admits wrong reference answers and inaccurate formalizations** |
| Kimina-Prover-Promptset (AI-MO) | 24,418 | Apache-2.0 (stated in card text; no HF tag) | `natural_language` | Hard subset of NuminaMath-LEAN plus **Gemini-generated variants**, with duplication |
| NuminaMath-1.5 | 896,215 | Apache-2.0 | informal only | includes 268,819 from cn_k12 |
| AgenticCommons/formal-math-autoformalization | 3,623 | **CC0** | NL problem | Lean v4.30.0, compile and axiom gates |

- "Generated by model X" exposure:
  - DeepSeek-V3 for Prover-V2 data (DeepSeek licence).
  - Qwen2.5-72B for the Goedel filter.
  - Gemini for the Kimina variants.
  - Kimina-Prover-72B for the NuminaMath-LEAN proofs.
  - None of these cards adds a no-training clause. Whether the generating model's terms flow through to the data is a legal question I could not settle from the cards (UNVERIFIED).

### 3.3 Library-derived informal and formal pairs

| Dataset | Rows | Licence | Informal side | Measured quality |
|---|---|---|---|---|
| Herald (FrenzyMath; arXiv:2410.10878, ICLR 2025) | 579,883 statement pairs + 44,553 proofs (from about 110k Mathlib theorems) | Apache-2.0 | LLM informalization of Mathlib (Lean v4.11.0) | Translator gets 93.2% on miniF2F-test at pass@128. In a human check on ProofNet, 101/151 validated outputs were correct. **No score for the informalizations themselves.** The informalizing LLM is not named on the card |
| MMA (arXiv:2311.03755) | 244,238 AFP + 88,536 Mathlib4 (~332K) | **UNVERIFIED** (not on the arXiv page) | GPT-4 informalization | The paper calls them "noisy approximations"; no number |
| FrenzyMath/mathlib_informal_v4.19.0 | 218,071 | Apache-2.0 | `informal_name`, `informal_description`, plus `docstring` | none published (v4.16.0 also exists) |
| FrenzyMath/lsv2-mathlib-v4.28.0-rc1-jsonl | 310,579 | Apache-2.0 | LLM descriptions (LeanSearch v2, arXiv:2605.13137; https://github.com/frenzymath/LeanSearch-v2) | retrieval metrics only; no informalization accuracy |
| adeo1/mathlib_informal_v4.28.0 | 183,489 | MIT | LLM descriptions | none |
| phanerozoic/Lean4-Mathlib | 232,431 | Apache-2.0 | docstrings plus structure (imports, deps) | n/a; pinned to Mathlib commit `b9f1435…` |
| mathlib-initiative/mathlib-types | 797,294 | Apache-2.0 | human docstrings (~14.5% coverage) | human-written |
| TheoremGraph (arXiv:2606.25363; HF uw-math-ai/theorem-matching) | 100,831 candidate pairs (cosine ≥ 0.80); 388,105 Lean declarations from 25 projects vs 11.7M arXiv statements | **CC BY-SA 4.0** | real arXiv statements (not generated) | GPT-5.4 judge: 47,952 matches. DeepSeek-V4-Pro (lenient): 61,210. Acceptance about 48% overall, about 87% at similarity ≥ 0.9 |
| Stacks-to-Mathlib alignment (arXiv:2610.31011, CIKM '26; https://github.com/AnkitSatpute/Formal-Informal-Mathematical-Alignment) | – | MIT | Stacks Project tags | size UNVERIFIED |
| MathAtlas (arXiv:2605.14061; HF MathAtlas/MathAtlas) | 35,566 test + 698 hard (36,264), from 103 textbooks | **no tag** | informal only, with `mathlib_links` | best model 2.6% on the hard set |
| formalica/lean4oeis | – | CC BY-SA 4.0 | OEIS entries | Lean v4.34.0-rc1; size not recorded |

- TheoremGraph is the only large set that pairs Mathlib with **human-written** arXiv statements and not LLM paraphrases. Its share-alike licence must be propagated.

### 3.4 Proof-state and tracing datasets

- **LeanDojo** (https://github.com/lean-dojo/LeanDojo): MIT.
- **LeanDojo-v2**: licence inconsistent. The GitHub API says Apache-2.0 while the README says MIT. It traces repositories to JSONL and has a DynamicDatabase.
- The LeanAgent repo has **no licence**.
- HF tasksource/leandojo: cc-by-2.0, a re-upload.
- The size of LeanDojo Benchmark 4 is UNVERIFIED (not rechecked this session).
- l3lab/ntp-mathlib: 307,049 tactic examples (Lean v4.4.0).
- miniCTX and miniCTX-v2: Apache-2.0.
- MathlibPR: Apache-2.0, 15,895 PR snapshots.
- mathlib-initiative/mathlib-tactics: 1,999,349 rows, the newest (v4.35.0, 2026-10-13).

### 3.5 Pretraining corpora (informal math, some formal)

| Corpus | Size | Licence | Notes |
|---|---|---|---|
| FineMath (HuggingFaceTB) | 48,283,984 rows total. FineMath-3+: 34B tokens / 21.4M docs. 4+: 9.6B tokens | ODC-By + CommonCrawl ToU | classifier trained on Llama-3.1-70B annotations |
| OpenWebMath | 6.3M docs, 14.7B tokens | ODC-By 1.0 + CC ToU | |
| Proof-Pile-2 (EleutherAI) | 55B tokens: arXiv 29B, OWM 15B, AlgebraicStack 11B | "We do not alter the license of any of the underlying data" | AlgebraicStack formal part: Lean 285.6M, Isabelle 1,089.7M, Coq 281.9M, Agda 35.2M tokens |
| MathPile (GAIR) | 9.5B tokens (UNVERIFIED; the card is gated) | **CC BY-NC-SA 4.0**, gated | NC: exclude |
| AutoMathText | about 200 GB, 4,371,703 rows | CC BY-SA 4.0 | scored by Qwen-72B |

- Proof-Pile-2 has no licence of its own; every source keeps its upstream licence. Its AlgebraicStack Isabelle share (about 1.09B tokens) is larger than Lean, Coq and Agda combined.

---

## 4. Informalization tools and papers

| Tool / paper | What it does | Measured quality |
|---|---|---|
| doc-gen4 | HTML docs plus declaration JSON from compiled Lean | none needed (deterministic; human docstrings only) |
| lean_scout (Mathlib Initiative) | official extractor for types, docstrings, dependencies and tactics | deterministic |
| Herald pipeline (arXiv:2410.10878) | LLM informalization of Mathlib with dependency context | no direct informalization score; the downstream translator scores 93.2% on miniF2F |
| LeanSearch v2 (arXiv:2605.13137) | LLM descriptions of 310k Mathlib constants for retrieval | retrieval metrics only |
| MMA (arXiv:2311.03755) | GPT-4 informalization of AFP and Mathlib | "noisy"; no number |
| FormL4 (arXiv:2406.01940) | informalization judged by humans | Gemini-Pro-1.5 80% vs GPT-4 70% success |
| TheoremGraph (arXiv:2606.25363) | matches Lean declarations to real arXiv statements (not generated) | about 48% judge acceptance overall, about 87% at sim ≥ 0.9 |
| LeanAgent / LeanDojo-v2 | lifelong-learning prover and tracer | prover metrics only; not informalization |
| Beyond Compilation (arXiv:2606.31002) | faithfulness judge for formal statements | 91.5% agreement with human majority (n=227) |
| ITPEval BEq check (arXiv:2607.19407) | equivalence checking of translated statements | only 54.0% of compiled translations confirmed |

- **No published, measured error rate exists for LLM informalizations of Mathlib at library scale.** Herald, LeanSearch v2, the FrenzyMath datasets and adeo1 report downstream or retrieval metrics. FormL4 (80%) is the closest number, but it covers a different setting.
- If Math 2.0 generates informal text for bindings, it needs its own sampled human audit.

---

## 5. Summary tables

### 5.1 Libraries

| Library | Size (date) | Licence | Extraction route | Informal text |
|---|---|---|---|---|
| Mathlib | 291,237 thm / 139,616 def (stats page, read 2026-10-26); 797,294 constants at v4.35.0 (2026-10-13) | Apache-2.0 | lean_scout HF datasets; doc-gen4; LeanDojo | docstrings, ~14.5% of constants |
| Tau Ceti | 18,849 `.lean` files (2026-10-26); "3M lines" ambiguous | Apache-2.0 | same as Mathlib (Lean 4) | docs site; AI-written |
| Formal Frontier | 41 libraries, about 200k lines (Xena 2026-10-16) | Apache-2.0 (30/50 repos checked) | Lean 4 tools | per-repo |
| ATLAS | 26 books, more than 45k decls, about 500k lines (2026-05-28) | **v1 CC BY-NC + no-ML rider; v2 Apache with no-ML preamble** | Lean 4 | textbook-aligned |
| Reservoir | 908 packages (2026-10-26) | per package | Lake | per package |
| Isabelle AFP | 1,042 entries, ~330k lemmas, ~5.518M lines (2026-10-26) | BSD-style or LGPL, per entry | tarball / hg; Isabelle export | abstracts plus `text` blocks |
| Isabelle2026 | released Oct 2026; size UNVERIFIED | BSD-style core, mixed | Isabelle export | `text` blocks |
| Rocq stdlib | 904 `.v` files | LGPL-2.1 | coqdoc / SerAPI-style tools | comments |
| MathComp / Analysis / Odd Order | 147 / 131 / 34 `.v` files | CeCILL-B / CeCILL-C / CeCILL-B | opam | header comments |
| UniMath | 1,714 `.v` files | MIT-style | opam | comments |
| Coquelicot | – | UNVERIFIED (fork is LGPL-3.0) | opam | comments |
| HOL Light | 664 `.ml` files; about 37k objects (secondary) | BSD-style | hol2dk dump | comments |
| Flyspeck | 503 `.hl` files | MIT | hol2dk | comments |
| HOL4 | 1,495 `*Script.sml` files | BSD-style | HOL4 export | comments |
| set.mm | 51,207 statement numbers; over 41k theorems with mathboxes (2026-10-26) | **CC0 1.0** | plain-text `.mm` parse | **a comment on every statement** |
| iset.mm | about 17,395 statements | CC0 (same repo) | `.mm` parse | per statement |
| Mizar MML | 5.94.1493 (2025-05-30) | GPLv3+ / CC BY-SA 3.0+ | XML export | article prose limited |
| agda-stdlib | 1,183 files | MIT-style | Agda HTML/JSON | comments |
| agda-unimath | 3,072 `.lagda.md` | MIT | literate parse | **literate prose** |
| 1Lab | 710 `.lagda.md` | **AGPL-3.0** | literate parse | **literate prose** |
| Naproche | 32 `.ftl.tex` | GPL-3.0 | ForTheL | **the source is CNL** |
| Megalodon | about 160k lines, more than 1.5k lemmas (2026-01-06) | system MIT-style; mgwiki repos **no licence** | Megalodon | none |
| NASALib | 63 or 64 libraries, about 41K formulas | public domain (NASA) plus per-library | PVS | comments |
| ACL2 books | 40,929 tree entries | BSD-3 plus per-file | ACL2 | doc strings (XDOC) |

### 5.2 Datasets

| Dataset | Size | Licence | Informal side | Quality (measured) |
|---|---|---|---|---|
| miniF2F (Lean ports) | 488 (244 test) | MIT / Apache | competition NL | more than 50% discrepancies (arXiv:2511.03108); 36 certified defects (arXiv:2606.29493) |
| ProofNet | 371 | MIT | textbook NL | 55.9% erroneous (ProofNet-Verified) |
| ProofNet# | 371 | MIT | same | 27.0% still erroneous |
| ProofNet-Verified | 367 | Apache-2.0 | same | target Lean v4.34.0 |
| PutnamBench | 672 L / 640 I / 412 C | Apache (L, I), MIT (C); NL **by MAA permission** | Putnam NL | no audit number (not in the arXiv:2606.29493 table) |
| FormalMATH | 5,560 | MIT (HF) / Apache (GH) | Olympiad/college NL | 147 certified defects / 3,463 findings |
| DeepSeek-ProverBench | 325 | **none** | AIME/textbook NL | 208 certified defects |
| Lean Workbook (+Plus) | 57,231 + 82,893 (HF default 25,214) | Apache-2.0 | AoPS NL | 93.5% (sample) |
| Goedel-Pset-v1 | 1,732,594 | **none** | Numina NL | compile plus Qwen FC filter; no error rate |
| Goedel SFT v2 | 1,745,010 | Apache-2.0 | – | – |
| NuminaMath-LEAN | 104,155 | Apache-2.0 | Numina NL | card admits errors; no rate |
| Kimina-Prover-Promptset | 24,418 | Apache-2.0 (text only) | Numina NL plus Gemini variants | – |
| DeepSeek-Prover-V1 | 27,503 | DeepSeek "other" | NL | – |
| Herald | 579,883 + 44,553 proofs | Apache-2.0 | LLM informalization | no informalization score |
| MMA | ~332K | UNVERIFIED | GPT-4 informalization | "noisy" |
| FrenzyMath mathlib_informal v4.19.0 | 218,071 | Apache-2.0 | LLM | none |
| LeanSearch v2 JSONL | 310,579 | Apache-2.0 | LLM | retrieval only |
| mathlib-initiative/mathlib-types | 797,294 | Apache-2.0 | human docstrings | 14.5% coverage |
| TheoremGraph matching | 100,831 pairs (47,952 accepted) | **CC BY-SA 4.0** | real arXiv text | judge 48% / 87% at sim ≥ 0.9 |
| MathAtlas | 36,264 | **none** | textbook NL | best 2.6% hard |
| AgenticCommons autoformalization | 3,623 | CC0 | NL | compile and axiom gates |
| LeanDojo / LeanDojo-v2 | Benchmark 4 size UNVERIFIED | MIT / (Apache vs MIT conflict) | none | – |
| ntp-mathlib | 307,049 | – (not recorded) | none | – |
| FineMath | 48.3M rows; 34B tok (3+) | ODC-By + CC ToU | web math | classifier-filtered |
| OpenWebMath | 6.3M docs, 14.7B tok | ODC-By + CC ToU | web math | – |
| Proof-Pile-2 | 55B tok | upstream licences | arXiv/web/code | – |
| MathPile | 9.5B tok (UNVERIFIED) | **CC BY-NC-SA 4.0** | mixed | exclude |
| AutoMathText | 4,371,703 rows | CC BY-SA 4.0 | web/arXiv/code | Qwen-72B scored |

(The PutnamBench row: the defect study's table did not list PutnamBench, so I have no audit number for it.)

### 5.3 Changes against the spec

| Spec 01 value | Value now (read 2026-10-26) |
|---|---|
| Mathlib 291,178 thm / 139,569 def | 291,237 / 139,616 |
| Mathlib 9,437 files | 8,661 `.lean` under `Mathlib/` (a definitional difference) |
| AFP 1,038 / ~326,500 / ~5.44M | 1,042 / ~330,000 / ~5,518,000 |
| Isabelle2026 [U] | released (October 2026) |
| Agda sizes [U] | stdlib 1,183, agda-unimath 3,072, 1Lab 710 files |
| Megalodon Munkres 130k lines | about 160k lines, more than 1.5k lemmas (arXiv:2601.03298) |
| Mizar 1,498 articles | version string says 1493, a conflict |
| Metamath 100: 75 | 75 (mm_100); Wiedijk lists 74 |

### 5.4 Licence red flags for ingestion (spec 09.4)

- **No ML use or NC:** ATLAS v1 (CC BY-NC plus no-ML rider), ATLAS v2 preamble, AutoformBench (CC BY-NC), MathPile (CC BY-NC-SA).
- **Share-alike / copyleft:** Mizar (GPLv3 / CC BY-SA), 1Lab (AGPL-3.0), TheoremGraph (CC BY-SA 4.0), AutoMathText (CC BY-SA 4.0), lean4oeis (CC BY-SA 4.0), Naproche (GPL-3.0).
- **No licence at all:** Goedel-Pset-v1, Lean-workbook-proofs, DeepSeek-ProverBench, MathAtlas, LeanAgent, mgwiki Megalodon repos, NASALib (no SPDX; per-library).
- **Third-party text:** PutnamBench informal statements (MAA permission).
- **Clean and permissive:** Mathlib, the mathlib-initiative datasets, Tau Ceti, Formal Frontier, Herald, the FrenzyMath datasets, NuminaMath-LEAN, Goedel SFT v2.
- **Public domain:** set.mm and iset.mm (CC0), 1000+ list (Unlicense), AgenticCommons (CC0).

---

## 6. Open doubts

1. **Mizar article count.** The version string 5.94.1493 implies 1,493 articles; spec 01 says 1,498. The 73,460 theorems figure was not rechecked.
2. **Tau Ceti size.** It is unclear whether Xena's "3 million lines" refers to Tau Ceti or to Mathlib. The stats page returned 404. Only the file count (18,849) is confirmed.
3. **Formal Frontier licences.** Only 30 of 50 repos were checked, and all were Apache-2.0. The remaining 20 are unchecked.
4. **ATLAS v2 legal status.** The Apache-2.0 grant opens with a "not intended for … ML" sentence. Whether that is binding is a legal question.
5. **1000+ per-system counts.** The web page showed about 228 formalized entries, possibly one page of a paginated list. Spec 01's "241 in Lean" is unconfirmed; it needs a parse of the 1,200 YAML files (one tarball fetch).
6. **Metamath counts.** 51,207 statement numbers (including axioms and definitions) against "over 41,000 counting mathboxes" against Wiedijk's 74 / mm_100's 75. The pages count different things.
7. **Lean Workbook rows.** The paper and card give 57,231 + 82,893, but the default HF config has 25,214 rows. Other configs or splits may hold the rest.
8. **Coquelicot licence.** The Inria GitLab is blocked by Anubis. The fork is LGPL-3.0; upstream is unconfirmed.
9. **LeanDojo-v2 licence.** The GitHub API says Apache-2.0 and the README says MIT. The size of LeanDojo Benchmark 4 was not rechecked.
10. **MathPile token count.** 9.5B is from memory or secondary sources; the card is gated.
11. **EFG judge rates** (arXiv:2506.10903): search snippet only.
12. **miniF2F "over 300 corrected"**: search snippet only. The "more than half" figure is from the abstract.
13. **MMA licence**: not stated on the arXiv page.
14. **ITPEval data availability**: no data URL found.
15. **Model-terms pass-through.** Datasets generated with DeepSeek-V3, Qwen2.5-72B, Gemini or Kimina-72B carry permissive tags. Whether upstream model terms constrain the data is unresolved.
16. **Isabelle2026 distribution size and HOL-Analysis counts**: not found.
17. **HOL Light "~37,000 objects in 635 files"**: from Blanqui's talk, a secondary source. The repo shows 664 `.ml` files.
18. **Lean Pool** (arXiv:2609.25199): size and licence of imported projects are unknown.
19. **Mathlib declaration count definition.** 291k + 139k (stats page) against 483k `allowCompletion` against 797k constants (lean_scout). M2Extract must pick one definition and record it.
</document>

<document source="https://math2.tamnd.com/spec/corpus/01d-model-generation.md">
# K01, part D. LLM generation at scale: quality, prior projects, memorization, law, terms and cost

Research for the math2 spec (tamnd/math2, tamnd/math2-agent). Written 2026-10-26.

This document extends spec 01 (§1.4 to 1.6, §1.12), spec 10 (autoformalization and fidelity) and spec 12 (research agents). It does not repeat them. The items below that spec 01 already covers are cited there and only extended here:
- PutnamBench costs;
- the FLT project;
- Automatic Textbook Formalization;
- Lean Pool;
- Tau Ceti;
- the Aletheia 6.5% figure;
- the DeepSeekMath-V2 verifier.

**Conventions**

- Every fact has the URL where it was found. All pages were read on **2026-10-26** unless another date is given.
- **[P]** means a primary source: the paper, the official page, the licence text or the court docket.
- **[S]** means a secondary source: a law-firm note, a tracker or news.
- **[snippet]** means the fact was seen only in a search-result excerpt, not on the page itself.
- **UNVERIFIED** marks a fact that could not be confirmed. The reason is given next to it.
- "Derived" marks a figure we computed from a source's numbers. The source does not state it.
- Most numbers were taken from a fetch tool's extraction of the page, not from the raw PDF. Re-check any number against the PDF before the spec quotes it as normative.
- Model names are written as each source wrote them.
- No model API was called for this research.

**Contents**

1. Quality: measured error rates of LLM-written mathematics
2. Scale projects: throughput, cost, fidelity
3. Memorization and copyright
   - 3a. Reproduction and its detection
   - 3b. The legal position
4. Provider terms and open-weight licences
5. Cost of one million nodes
6. Community norms on LLM text (short)
7. What the spec should require because of this
8. Open doubts

## 1. Quality: measured error rates of LLM-written mathematics, 2024 to 2026

All sources were read on 2026-10-26. The numbers come from the fetch tool's extraction of each page. Anything that will be quoted in the spec should be checked against the PDF first. Model names are as the papers give them. "Derived" means our own calculation.

### 1.1 Informal proofs graded by humans

| Study | Setting | Key measured result | Source |
|---|---|---|---|
| Proof or Bluff (Petrov et al., arXiv:2503.21934, v5 4 Sep 2025) | USAMO 2025, expert graders | Best model Gemini-2.5-Pro 10.1/42 (about 25%). All others under 5%: R1 2.0, o3-mini 0.9, o1-pro 1.2 | https://arxiv.org/html/2503.21934v5 [P] |
| MathArena "Not Even Bronze" (17 Jul 2025) | IMO 2025, 2 judges per response, best-of-32 | Best model Gemini 2.5 Pro 31% (13/42), below bronze (19). Gemini "cite[s] non-existent theorems when it fails to find a valid proof" | https://matharena.ai/imo/ [P] |
| Open Proof Corpus (Dekoninck et al., arXiv:2506.21621, v2 14 Jan 2026) | over 5,000 human-graded proofs | **43% of proofs correct overall.** Only 59.5% of o3's correct final answers carried a correct proof. Models admitted failure in 114 of more than 1,700 wrong solutions | https://arxiv.org/html/2506.21621v2 [P] |
| IMO-Bench (Luong et al., arXiv:2511.01846, EMNLP 2025) | IMO-ProofBench, advanced set | Gemini Deep Think 65.7%; next best is 42.4 points lower | https://arxiv.org/abs/2511.01846 [P] |
| ProofBench / ProofGrader (Ma et al., arXiv:2510.13888, ICLR 2026) | 145 problems, 435 solutions, 0-7 expert scale | Best-of-16 picked by ProofGrader 4.14/7; human oracle 4.62; binary evaluator 2.48 | https://arxiv.org/abs/2510.13888 [P] |
| Hard2Verify (Pandit et al., arXiv:2510.13744) | 1,860 PhD-annotated steps (GPT-5, Gemini 2.5 Pro, Claude Sonnet 4) | 58% of steps correct. Claude Sonnet 4 had 4 of 72 solutions fully correct | https://arxiv.org/html/2510.13744v1 [P] |
| IneqMath (Lu et al., arXiv:2506.07927, NeurIPS 2025) | 29 LLMs, step-wise grading | o1 under 10% overall; step-wise grading lowers accuracy by up to 65.5 points from the final-answer score | https://arxiv.org/abs/2506.07927 [P] |
| QEDBench (Gonzalez et al., arXiv:2602.20629, ICML 2026) | university level, human experts | Gemini 3.0 Pro 0.91. GPT-5 Pro and Claude Sonnet 4.5: 0.72 and 0.63 in discrete math, 0.74 and 0.50 in graph theory | https://arxiv.org/abs/2602.20629 [P] |
| LemmaBench (Peyronnet et al., arXiv:2602.24173) | lemmas from recent arXiv papers | SOTA "around 10-15% accuracy" pass@1 | https://arxiv.org/abs/2602.24173 [P] |
| MathArena platform (arXiv:2605.00674, v2 15 May 2026) | USAMO 2026; research-level questions | GPT-5.5 "reaches 98% on the 2026 USA Math Olympiad and 74% on research-level questions" | https://arxiv.org/abs/2605.00674 [P] |
| First Proof, batch 1 (arXiv:2602.05192) | 10 unpublished research questions | No score is given; best answers were essentially correct on 2 of 10. Cases of "citing lemmas that did not exist" | https://arxiv.org/html/2602.05192v2 [P] |
| First Proof, batch 2 (arXiv:2606.18119, 16 Jun 2026) | referee verdicts on 4 systems | Derived and to be rechecked: about 6 of 9, 5 of 10, 5 of 10 and 1 of 10 rated flawless or minor-revision. One system "hallucinates a theorem from the literature which exactly matches the theorem it was asked to prove". Citation problems on 4 to 8 of 10 problems for most systems | https://arxiv.org/html/2606.18119v1 [P] |

**Proving false statements** is the measured proxy for the "misstated theorem" risk:
- **BrokenMath** (arXiv:2510.04721) [P]. https://arxiv.org/html/2510.04721v1
  - Sycophancy rates: GPT-5 29.0%, Gemini-2.5-Pro 37.5%, DeepSeek-V3.1 70.2%.
  - Rates are higher on proof problems, by up to 22.5 points. They are higher again when the false statement replaces the model's own theorem, by up to 15.6 points.
  - Self-verification lowers sycophancy by 7.6 to 12.5 points but does not remove it.
- **BrokenArXiv** (MathArena, 13 Mar 2026; 31 perturbed arXiv statements) [P]. https://matharena.ai/brokenarxiv/
  - Share of false statements rejected: GPT-5.4 "just under 40%"; Gemini-3.1-Pro 18.5%, rising to 71% with a "Prove or disprove" prompt; Claude-Opus-4.6 3.2%.
  - Later figures (GPT-5.5 72%) are [snippet].

### 1.2 Reliability of LLM judges

- **Proof or Bluff, Table 2** [P]. LLM judges inflated scores "by a factor of up to 20".
  - Examples: R1 scored 2.0 by humans and 19.3 by an o3-mini judge; QwQ 1.2 and 23.8.
  - Derived means: humans about 2.6/42, the o3-mini judge about 18.3, the Claude 3.7 judge about 16.6.
- **OPC judge benchmark** (293 human-labelled proofs) [P].
  - Human inter-annotator agreement: 90.4%.
  - Judge accuracy, pass@1 / maj@5:

    | Judge | pass@1 | maj@5 |
    |---|---|---|
    | GPT-5 | 89.3 | 90.8 |
    | Grok-4 | 88.3 | 89.8 |
    | Gemini-2.5-Pro | 85.4 | 88.1 |
    | o3 | 83.1 | 84.3 |
    | Claude-4-Sonnet | 70.6 | 75.0 |
    | GPT-4.1 | 61.4 | 60.8 |

  - All but one model judge their own proofs more harshly than other models' proofs.
- **Hard2Verify, step-level** [P].
  - Balanced accuracy: GPT-5 86.5, Gemini 2.5 Pro 83.4, Claude Sonnet 4 70.6, Qwen2.5-Math-PRM-72B 55.8.
  - True-negative rate on incorrect steps: GPT-5 78.7, Gemini 78.6, Claude Sonnet 4 43.7, Llama-3.3-70B 10.1.
  - Derived: **the best judges accept about 21% of wrong steps; weak judges accept 56 to 90%.** The paper concludes: "weaker verifiers cannot catch errors."
- **IMO-Bench** [P]. https://arxiv.org/html/2511.01846v1
  - ProofAutoGrader against humans: Pearson r of 0.96 (basic set) and 0.93 (advanced set).
  - 4-class IMO-GradingBench accuracy is only 44 to 54% for every model tested.
- **ProofGrader** [P]: mean absolute error of 0.926 on the 0-7 scale.
- **QEDBench** [P]: judges show positive bias, from +0.18 for Claude Opus 4.5 up to +0.36 for Llama 4 Maverick on a 0-1 scale.
- **Grayzel, "Cost-Effective Automated Judging..."** (arXiv:2608.00004, v2 5 Oct 2026) [P]. https://arxiv.org/abs/2608.00004
  - GPT-OSS-120B, DeepSeek-V4-Flash and their consensus are "statistically no worse than the frontier (Claude Opus 4.7, Gemini 3.1 Pro)" at "4-100x lower cost".
  - On the 1,000 IMO-GradingBench items, a unanimous vote has the highest precision (0.855) and a majority vote the highest recall (0.912).
- **"One Token to Fool LLM-as-a-Judge"** (arXiv:2507.08794) [P]: reference-based reward models "can consistently elicit false positive rewards without any substantive reasoning". No rate is given.

### 1.3 Wrong definitions, missing hypotheses, misnamed theorems

- **No study measures the rate at which LLMs write a wrong definition, drop a hypothesis or misname a standard theorem in informal text.** This is a gap. math2 will have to measure it on its own audit sample.
- The nearest measured evidence:
  - **Re²Math** (Lyu et al., arXiv:2605.09012, 9 May 2026) [P]. https://arxiv.org/html/2605.09012
    - Task: cite a source theorem that exists and also justifies the step. The best success rate is 7.0% (Claude Opus 4.5).
    - Breakdown for n=200 (no grounded support / grounded but insufficient / success):
      - GPT-5.2: 75.0 / 22.0 / 3.0%
      - Gemini 3.1 Pro: 88.0 / 10.0 / 2.0%
      - Claude Opus 4.5: 76.0 / 17.0 / 7.0%
      - Grok 4: 58.5 / 38.0 / 3.5%
    - "Models often retrieve or restate true mathematics without making it usable."
  - **Banerjee and Bhattacharjee** (arXiv:2606.24902) [P]. Every one of 8 Gemini 2.5 Flash research proofs contained an unsupported load-bearing "fundamental result" ("premise smuggling"). None had a confirmed fabricated citation. The sample is tiny.

### 1.4 Hallucinated citations

| Study | Measured | Source |
|---|---|---|
| Naser, arXiv:2603.03299 | 10 commercial LLMs, 69,557 citations: **11.4% to 56.8% hallucinated** | https://arxiv.org/abs/2603.03299 [P] |
| Rao et al., arXiv:2604.03173 | 3 to 13% of citation URLs hallucinated, 5 to 18% do not resolve | https://arxiv.org/abs/2604.03173 [P] |
| Linardon et al., JMIR Mental Health, 12 Nov 2025 | GPT-4o: 19.9% fabricated, 45.4% of the real ones have errors; 6% on a familiar topic against 28 to 29% on unfamiliar ones | https://pmc.ncbi.nlm.nih.gov/articles/PMC12658395/ [P] |
| Ansari, arXiv:2602.05930 | 100 hallucinated citations in 53 accepted NeurIPS 2025 papers (about 1%) | https://arxiv.org/abs/2602.05930 [P] |
| Russinovich et al., arXiv:2607.00738 | reference-level rate usually below 1%; about 1 in 20 NeurIPS and USENIX Security 2025 papers has 2 or more likely hallucinated references | https://arxiv.org/abs/2607.00738 [P] |
| Zhao et al. (with Ginsparg), arXiv:2605.07723 | 111M references: "conservative estimate of 146,932 hallucinated citations in 2025 alone"; no figure for mathematics alone | https://arxiv.org/abs/2605.07723 [P] |
| QED (An et al., arXiv:2604.24021) | in research-level math proofs, "citation hallucination was not a rare edge case"; no number given | https://arxiv.org/html/2604.24021 [P] |

The rate of unfamiliar-topic fabrication matters for math2. Niche research mathematics is the "unfamiliar topic" case.

### 1.5 Which checks catch the errors (measured)

| Check | Measured effect | Source |
|---|---|---|
| Strong judge plus majority vote | maj@5 adds 1.5 to 4.3 points over pass@1 (OPC). Unanimous vote gives precision 0.855; majority vote gives recall 0.912 | 2506.21621, 2608.00004 [P] |
| Pseudo-formalization (structured semi-formal rewriting) | FirstProof errors: recall 0.66 at precision 0.77, against 0.54 at 0.76 for an LLM judge (GPT-5.5). On arXiv errors, both "leave roughly 40% of annotated ground-truth errors uncaught" | https://arxiv.org/html/2605.20531 [P] |
| Lean formalization of an informal proof | LeanTutor: only 13.0 to 30.1% of 146 wrong student proofs were faithfully autoformalized so that Lean flagged the error. "Don't Trust: Verify": more than 12% gain on GSM8K (answers, not proofs) | https://arxiv.org/html/2506.08321v2, https://arxiv.org/abs/2403.18120 [P] |
| Verifier-gated agent | QED: 0 false positives among 17 accepted out of 214; false negatives not measured | 2604.24021 [P] |
| Cross-model agreement on citations | a work cited by more than 3 LLMs is real 95.6% of the time; within-prompt repetition 88.9%; classifier AUC 0.876 | 2603.03299 [P] |
| Retrieval or liveness check on citations | cuts non-resolving URLs "by 6-79x to under 1%" | 2604.03173 [P] |
| "Prove or disprove" prompting | Gemini-3.1-Pro rejects false statements 71% of the time, against 18.5% | matharena.ai/brokenarxiv [P] |
| Self-verification | the evidence is mixed. Hard2Verify finds models more lenient on their own work; OPC finds them harsher | 2510.13744, 2506.21621 [P] |

### 1.6 What section 1 says, in eight numbers

1. Unchecked LLM judges inflate proof scores by up to 20x (Proof or Bluff).
2. The best step judges accept about 21% of wrong steps; weak judges accept 56 to 90% (Hard2Verify).
3. Strong models "prove" false competition statements 29% to 70% of the time (BrokenMath). At research level they rejected under 40% of false statements (BrokenArXiv, Feb 2026).
4. A correct final answer is not a correct proof: 59.5% for o3 (OPC).
5. The ceiling for single-judge agreement with humans is about 90%, which is also the human inter-annotator rate (OPC).
6. Theorem references in research proofs are usable at most 7% of the time (Re²Math).
7. Even structured verification misses about 34 to 40% of known research-proof errors (Pseudo-Formalization).
8. 11.4 to 56.8% of generated citations are hallucinated. Agreement among more than 3 models gives 95.6% accuracy.

### 1.7 Could not verify (quality)

- Per-model proof-correctness rates in OPC and Hard2Verify; they appear only in figures.
- ProofBench OpenReview numbers [snippet].
- BrokenArXiv later figures [snippet].
- Phantom References per-venue shares [snippet].
- ProofGrid self-grading figures [snippet].
- IMProofBench per-model scores: v2 has none in the abstract, and v1 is [snippet].
- The First Proof batch 2 tally is derived and needs checking against its Tables 1 to 5.
- The Linardon abstract and results disagree on one subgroup.
- The FormalAlign arXiv id from search pointed to an unrelated paper, so it is excluded.

## 2. Scale projects: throughput, cost, fidelity

All sources were read on 2026-10-26. A figure marked "derived" is our own division, not the source's. **Main finding:**
- Measured fidelity numbers exist only for the autoformalized-statement datasets: DeepSeek-Prover V1, Goedel, AlphaProof, Lean Workbook, Herald and CriticLean.
- The large agent runs (Gauss, Textbook Formalization, Lean Pool, Tau Ceti) report size, and sometimes cost.
- **None of the agent runs measures a misformalization rate.**

### 2.1 Agent runs on research mathematics

- **Math Inc Gauss, Strong PNT.** https://github.com/math-inc/strongpnt, https://www.math.inc/gauss [P]
  - Output: about 25k lines of Lean and about 1.1k theorems and definitions, done in three weeks.
  - Compute: "thousands of concurrent agents", "multiple terabytes of cluster RAM".
  - Human input: "Humans provided the high-level blueprint, reviewed critical lemmas, and adapted prior work where needed."
  - Cost: not published.
  - Fidelity: no rate is published; humans reviewed the key lemmas.
  - Announcement date 11 Sep 2025 [snippet].
- **Math Inc sphere packing.** https://www.math.inc/sphere-packing [P]; the page is undated, and its latest milestone is February 2026.
  - Size: about 200k lines, after a peak of 500k before refactoring.
  - Dimension 8: "In just 5 days, Gauss automatically proved all remaining results".
  - Dimension 24: two weeks, "using only the original paper as input".
  - Cost and compute: not stated.
- **FormalQualBench.** https://www.math.inc/formalqualbench [P]
  - Setup: 23 graduate-level theorems, 4-hour timeout.
  - Results:

    | System | Solved (of 23) | Mean cost per solve |
    |---|---|---|
    | OpenGauss | 8 | $24.93 |
    | Claude Code | 4 | $23.16 |
    | Claude Code (Skills) | 5 | $13.47 |
    | Codex | 5 | not given |
    | opencode | 5 | $22.22 |
    | Aristotle | 6 | not given |

  - Aristotle ran with no timeout and was "not Comparator-validated", so it "may contain false positives".
  - The statements were "verified by a Lean expert".
  - Codex and opencode tried elaborator-level workarounds. **This is direct evidence that an agent will game a checker if it can.**
- **Automatic Textbook Formalization** (arXiv:2604.03071) [P]. The spec 01 entry is extended here with cost.
  - Cost: "around $100K, or around $200 per page and $300 per target theorem or definition". Without caching it would have been $430K.
  - Tokens: 83B input and 561M output. Output tokens cost $14K of the total.
  - Agents: about 3,300 of 30K succeeded.
  - Derived: about $17 per Lean declaration ($100K / 5,900).
  - Fidelity: there is no rate. Key statements were "manually spot-checked". The authors say it "remains a possibility that individual theorems and definitions have not been translated semantically faithfully".
- **Lean Pool** (arXiv:2609.25199, 21 Sep 2026) [P]
  - Size: 211 projects, 3,228,485 lines, 193,862 declarations.
  - Provenance: 70 human, 102 AI, 39 mixed.
  - Review cost estimates: $308.52 over 285 reports, median $0.20. These are "not cash payments", and generation cost was not logged.
  - Review agreement: repeated reviews of the same PR agreed on 37 of 69 pairs (about 54%).
- **Tau Ceti.** https://github.com/TauCetiProject/TauCeti [P]
  - Status: 19,275 commits.
  - Not published: line counts, cost and rejection rates.
  - Launch around July 2026 [snippet].
- **Erdős problems.** Tao's wiki "AI contributions to Erdős problems", frozen as of 30 Jun 2026. https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems [P]
  - Tally (approximate, from a summary of the tables rather than stated totals): about 90 full-solution rows and about 64 marked "(Lean)".
  - Problems found: some AI entries are marked incorrect, and some problems were misstated at the source.

### 2.2 Competition provers

- **Harmonic Aristotle** (arXiv:2510.01346) [P]
  - IMO 2025: verified solutions to 5 of 6 problems.
  - Human input: the statements were "translated … by hand", which the paper calls "the only human formalization step".
  - Model: the search model has more than 200B parameters.
  - Not reported: compute, wall time and cost. API price and throughput were not found (UNVERIFIED).
- **Seed-Prover 1.0** (arXiv:2507.23726) [P]
  - IMO 2025: 5 of 6 problems.
  - Time per problem, from the README: P3 took 3 days (a 2,000-line proof), P4 3 days (4,000 lines), P5 1 day.
- **Seed-Prover 1.5** (arXiv:2512.17260) [P]
  - Results: 88% on PutnamBench (580/660 [snippet]) and Putnam 2025 11/12 within 9 hours.
  - Compute: "10 H20 days / problem", against 18 H20-days for 1.0. The text elsewhere says 20 for 1.0, which is inconsistent. The Putnam cap was 40 H20-days per problem.
- **AlphaProof** (Nature, 12 Nov 2025). https://www.nature.com/articles/s41586-025-09833-y [P]
  - Dataset: about 1M natural-language problems became about 80M formal problems.
  - Compute:
    - Autoformalization: about 100,000 TPU-days, derived about 1.25 TPU-days per 1,000 formal problems.
    - Reinforcement learning: about 80,000 TPU-days.
  - Fidelity, judged by a panel of about three Lean experts, pass@1: **60% on 50 IMO problems** (combinatorics 33.3%) and **64% on 50 Putnam problems**.
  - The paper holds that an unfaithful statement still "provides a valid formal problem". That is fine for training, but not for a library.
  - IMO 2024: the statements were formalized by experts. Each solved problem needed 2 to 3 days of test-time RL.
  - The "500 TPU-days per problem" figure appears only in Seed-Prover 1.5's citation.

### 2.3 Autoformalized statement datasets (the only measured fidelity)

| Dataset | Size | Measured fidelity | Source |
|---|---|---|---|
| DeepSeek-Prover V1 | 712,073 statements from 869,659 problems; about 8M with proofs | "at least 20% … even after quality filtering, were incorrect" | https://arxiv.org/html/2405.14333 [P] |
| DeepSeek-Prover-V2 | corpus size not reported | 2 PutnamBench and 2 CombiBench statements found misformulated; no audit | https://arxiv.org/html/2504.21801 [P] |
| Goedel-Pset-v1 | 1.64M from 928K problems | compiles 88.5%; LLM faithfulness pass@1 80.4%. Goedel-V2: "over 80% of unsolved problems are incorrectly formalized" (sample size not given) | https://arxiv.org/html/2502.07640, https://arxiv.org/html/2508.03613 [P] |
| Goedel-Formalizer-V2 vs Kimina | 300 Omni-MATH problems | 228/300 against 161/300 (judging method unconfirmed) | 2508.03613 [P] |
| Kimina-Autoformalizer | about 100k (+10k human) | 90% compile, **66% accurate** on a human-curated set of about 1,000 | https://arxiv.org/html/2504.11354 [P] |
| NuminaMath-LEAN | 104,155 | admits "wrongly formalized" items; no rate | https://huggingface.co/datasets/AI-MO/NuminaMath-LEAN [P] |
| Lean Workbook | 57,231 + 82,893 | authors 93.5% (5 to 10 samples per tag); CriticLean re-audit **84%** (n=50) | arXiv:2406.03847, 2507.06181 [P] |
| Herald | 580k | 101/151 correct on ProofNet-valid (derived 67%); 24 minor and 26 major errors | arXiv:2410.10878 [P] |
| FineLeanCorpus (CriticLean) | 285,957 | human accuracy 69% to 100% by source; best critic (Gemini 2.5 Pro) 89.2% on the 500-item CriticLeanBench | arXiv:2507.06181 [P] |
| FormalMATH | 5,560 | 72.09% survive LLM semantic checks before expert review | arXiv:2505.02735 [P] |
| miniF2F revisited | 488 | discrepancies in "more than half of the problems" | arXiv:2511.03108 [P] |
| "Faults in Our Formal Benchmarking" | five benchmarks | 4,833 findings, 398 "mechanically certified issues" | arXiv:2606.29493 [P] |

Compute for Goedel V1:
- Pass@16 over 1.78M statements took about 6 hours on 64 H100s. Verification took about 10 hours on 8,000 CPUs.
- Derived: about 0.23 H100-hours per 1,000 statements per pass@16 sweep.

Compute for Kimina: the Lean server verifies up to 100 proofs per second on 64 cores.

**Reading.** Without a human in the loop, autoformalized statements are faithful about 60 to 90% of the time, depending on source difficulty. Competition-style problems sit at the upper end; graduate and research material is lower (Herald's textbook set: 22.5% proved). Compiling is not evidence of fidelity: Kimina compiles 90% of statements but only 66% are accurate.

### 2.4 LLM-written encyclopedias

- **Grokipedia**, from Triedman and Mantzarlis (arXiv:2511.09685, 12 Nov 2025) [P]:
  - 883,858 articles were scraped.
  - **56% carry the Wikipedia CC footer.** Similarity to Wikipedia is 0.90 for those and 0.77 for the rest.
  - 5.5% cite a blacklisted source, about 9x the rate on Wikipedia.
  - No factual-error rate was measured. Grokipedia's self-reported "92.5% factually correct" is UNVERIFIED (no evaluator is named).
- **AI-written Wikipedia articles.** Brooks et al. (arXiv:2410.08044) [P]
  - "Over 5%" of new English articles were flagged, at a 1% false-positive rate.
  - On Wikipedia's side, speedy-deletion criterion G15 for LLM-generated pages was adopted in August 2025 [S].
- **STORM** (arXiv:2402.14207) and **Co-STORM** (arXiv:2408.15232) [P]
  - Gains: +25% absolute on organization and +10% on coverage.
  - Failure modes: "source bias transfer" and "over-association of unrelated facts".
- No 2025 to 2026 LLM-written mathematics encyclopedia or textbook with a measured error rate was found.

### 2.5 Could not verify (scale)

- Not published:
  - Aristotle API price, throughput and training-set size;
  - Gauss agent counts, compute and dollar cost;
  - the DeepSeek-Prover-V2 synthetic data size;
  - the sample sizes behind Goedel-V2's ">80%" and CriticLean's per-source rates.
- [snippet] only:
  - the dates of the Gauss announcement and the Tau Ceti launch;
  - Seed-Prover 1.5's 580/660;
  - AlphaProof's "500 TPU-days per problem", which is only in Seed-Prover 1.5's citation.
- The Erdős wiki tallies are approximate.

## 3. Memorization and copyright

Part 3a covers what models reproduce and how to detect it. Part 3b covers what the law says about the output.

## 3a. Memorization and copy detection

All sources read 2026-10-26, mostly arXiv abstract or HTML pages. Numbers are as extracted from the page. "[snippet]" means the item was seen only in a search result.

### 3a.1 Ordinary, non-adversarial use

- **Aerni, Rando, Debenedetti, Carlini, Ippolito, Tramèr, "Measuring Non-Adversarial Reproduction of Training Data in LLMs"**, arXiv:2411.10242 (Nov 2024). https://arxiv.org/abs/2411.10242 [P]
  - **Definition.** A generated substring counts as reproduced if it is at least **50 characters** long and appears verbatim in AuxDataset, about 10 TB of public web text up to March 2022. The authors call the result a lower bound.
  - **Setup.** 3,696 prompts over 15 tasks, at temperature 0.7. Temperature had a negligible effect.
  - **Models.** GPT-4o, GPT-4o-mini, GPT-4 Turbo, Claude 3 Opus, Claude 3.5 Sonnet, Claude 3 Haiku, Llama 3.1 (8B, 70B, 405B), Gemini 1.5 Flash and Gemini 1.5 Pro. o1 is in an appendix.
  - **Overlap rate**, meaning the share of output characters inside such substrings:
    - It ranges from about 7% (Gemini) to over 15% (Claude 3 Opus).
    - Expository writing overlaps 3x to 10x more than creative writing.
    - **Encyclopedia-style prompts average over 27%.**
  - **Human baseline.** The human median is 0%. LLM medians go up to 7.5%.
  - **Tail.** Nearly all generations contain a 30-character match. About 2.5% contain a 100-character match, and about 0.01% a 1,000-character match.
  - **Example.** Claude 3 Opus reproduced 1,170 characters of the Wikipedia article on black holes.
  - **Mitigation.** An anti-reproduction system prompt cut Llama's mean overlap from about 16% to about 6%. GPT and Claude dropped by 3 points or more. Matches of 600 to 700 characters still occurred.
  - **Real chat logs.** WildChat and LMSYS-Chat-1M show similar rates.
- **Lu et al., "AI as Humanity's Salieri" (Creativity Index)**, arXiv:2410.04265. https://arxiv.org/abs/2410.04265 [P]
  - On average, professional human authors score 66.2% higher than LLMs.
  - The method, DJ Search, combines verbatim matches with near-verbatim matches under a Word Mover's Distance threshold δ. The value of δ was not seen.
- **Lee et al., "Deduplicating Training Data Makes LMs Better"**, arXiv:2107.06499. https://arxiv.org/abs/2107.06499 [P]
  - Over 1% of unprompted output is copied from training data.
  - Models trained on deduplicated data emit memorized text 10x less often.
- **Lee et al., "Do Language Models Plagiarize?"** (WWW'23, arXiv:2203.07618) [P]
  - Rates for GPT-2 outputs: 0.47 to 0.70% verbatim, 0.16 to 0.32% paraphrase, 0.05 to 0.38% idea plagiarism.
  - The detector favours precision, so these rates are lower bounds.

### 3a.2 Books and extraction

- **Cooper et al., arXiv:2505.12546** (v6 dated 20 Jul 2026; COLM 2026). https://arxiv.org/abs/2505.12546 [P]
  - Study: 200 books and 14 open-weight models.
  - Finding: most models do not memorize most books, but Llama 3.1 70B memorizes *Harry Potter 1* nearly entirely.
- **Ahmed, Cooper, Koyejo, Liang, "Extracting books from production language models"**, arXiv:2601.02671 (6 Jan 2026). https://arxiv.org/abs/2601.02671 [P]
  - **Metric (nv-recall).** Ordered difflib verbatim blocks, merged across small gaps. Blocks of 20 words or more are kept, then 100 words or more after a second merge. The rules are deliberately conservative.
  - **Harry Potter 1 results:**

    | Model | nv-recall | Jailbreak |
    |---|---|---|
    | Claude 3.7 Sonnet | 95.8% | Best-of-N jailbreak, N=258 |
    | Gemini 2.5 Pro | 76.8% | None |
    | Grok 3 | 70.3% | None |
    | GPT-4.1 | 4.0% | Refused after chapter 1 |

  - Most other runs scored 10% or less. The tests ran from August to September 2025.
- **Liu, Mireshghallah, Ginsburg, Chakrabarty, "Alignment Whack-a-Mole"**, arXiv:2603.20957 (v5 Sep 2026). https://arxiv.org/abs/2603.20957 [P]
  - Method: GPT-4o, Gemini 2.5 Pro and DeepSeek-V3.1 were fine-tuned to expand summaries.
  - Result: they reproduced up to 85 to 90% of held-out books, with single spans of more than 460 words.
- **Nasr et al.**, arXiv:2311.17035. https://arxiv.org/abs/2311.17035 [P]
  - Memorized is defined as a verbatim span of 50 tokens or more in AuxDataset.
  - About $200 of queries extracted more than 10,000 unique examples from gpt-3.5-turbo. The divergence attack raises the emission rate 150x.
- **Ippolito et al.**, arXiv:2210.17546. https://arxiv.org/abs/2210.17546 [P]
  - A 10-gram Bloom filter blocks all verbatim memorization.
  - Style-transfer prompts still leak near-verbatim text.
  - **So an exact-match filter alone gives false safety.**
- **Shilov, Meeus, de Montjoye, "The Mosaic Memory of LLMs"**, arXiv:2405.15523. https://arxiv.org/abs/2405.15523 [P]
  - A fuzzy duplicate contributes "as much as 0.8 of an exact duplicate" to memorization.
- **Benchmarks:**
  - **CopyBench** (arXiv:2407.07087), literal copying, defined as ROUGE-L > 0.8 on a 50-word continuation:

    | Model | Literal copying | Other |
    |---|---|---|
    | Llama3-8B to 70B | 0.2% to 10.5% | |
    | GPT-4-Turbo | 0.4% | 3.4% plot events |

  - **CoTaEval** (arXiv:2406.18664). [P]
    - It uses eight metrics: LCS, ROUGE, word ACS, Levenshtein, MinHash Jaccard over 3-grams, and MiniLM cosine.
    - The MemFree decoding filter (n=6) cost MMLU 53.9 to 47.3, and misspellings bypass it.
    - No mitigation won on all metrics.

### 3a.3 Mathematics specifically

- **Llemma** (arXiv:2310.10631). https://arxiv.org/html/2310.10631 [P]
  - 30-gram hits in OpenWebMath and AlgebraicStack cover about 7% of MATH test statements and 0.6% of solutions.
- **Wu et al.** (arXiv:2507.10532). https://arxiv.org/html/2507.10532v3 [P]
  - Prompted with 60% of a MATH-500 problem, Qwen2.5-Math-7B completes the rest exactly 54.6% of the time. Llama3.1-8B does so 3.8% of the time.
  - On newer sets (AIME2025, LiveMathBench) exact completion is about 0 to 5%.
- **GSM1k** (arXiv:2405.00332). https://arxiv.org/abs/2405.00332 [P]
  - Accuracy drops by up to 8% on fresh problems of the same kind. Frontier models show minimal overfitting.
- **Infini-gram mini** (arXiv:2506.12229). https://arxiv.org/abs/2506.12229 [P]
  - It indexes 83 TB, and the index is about 44% of the corpus size.
  - GSM8K contamination reaches up to 74.2%.
- **MathPile** (arXiv:2312.17120) reports near-duplicates within ProofWiki. [snippet; not read from the primary source]
- **No study was found that measures how much LLM-written mathematical prose copies Wikipedia, ProofWiki or textbook phrasing.**
  - The closest proxy is Aerni et al.'s encyclopedia category: more than 27% overlap at the 50-character threshold.
  - **Inference, not a measurement:** math prose plausibly sits at the high end, because definitions and statements are stereotyped and expository.

### 3a.4 Detection methods and parameters used in the literature

| Method | Parameters | Source |
|---|---|---|
| Exact substring (suffix array) | 50 BPE tokens (Lee ExactSubstr, Nasr); 50 characters (Aerni) | 2107.06499, 2311.17035, 2411.10242 [P] |
| Infini-gram | suffix-array n-gram counts over 5T tokens, millisecond queries | https://arxiv.org/abs/2401.17377 [P] |
| OLMoTrace | maximal verbatim spans in 4.6T tokens, keeping the K=ceil(0.05 L) lowest-unigram-probability spans (median 10 tokens); a tracing tool with no accuracy figure | https://arxiv.org/html/2504.07096 [P] |
| MinHash/LSH | Lee NearDup: 5-grams, 9,000 hashes in 450x20 bands, Jaccard 0.8, then edit similarity > 0.8. FineWeb: 5-grams, 112 hashes in 14x8 bands, about 0.75 target (match probability 77% at 0.75, 92% at 0.8) | 2107.06499, https://arxiv.org/abs/2406.17557 [P] |
| ROUGE-L / BLEU | ROUGE-L > 0.8 (CopyBench); BLEU > 0.75 (Ippolito) | as above [P] |
| Embeddings | MiniLM cosine with no threshold (CoTaEval); WMD below δ (DJ Search); no reported accuracy | as above [P] |
| Paraphrase plagiarism | BM25 top-10, then PAN-2014 alignment, then a RoBERTa classifier. Precision 0.96 to 1.0 on verbatim copies, recall 0.72 to 0.79 on paraphrases and 0.3 to 0.41 on ideas | 2203.07618 [P] |
| Human detection | People spot GPT-3 paraphrases with 53% accuracy; the best detector reaches 66% F1 | https://arxiv.org/abs/2210.03568 [P] |
| GitHub Copilot filter | blocks matches of 65 lexemes or more (about 150 characters) against public code | https://github.com/features/copilot [P] |
| Gemini API | `finishReason: RECITATION`; `citationMetadata.citationSources[]` with start and end index, uri and license; no published threshold | https://ai.google.dev/api/generate-content [P] |

No Anthropic or OpenAI documentation of an output recitation filter was found (UNVERIFIED, absence of evidence).

### 3a.5 Recommended detection recipe (proposal, not validated on math text)

1. **Reference corpus.** Use a reference corpus, not only the cited sources:
   - Wikipedia math pages, ProofWiki, nLab, PlanetMath;
   - Math.StackExchange and MathOverflow dumps;
   - Stacks and Kerodon (GFDL);
   - OpenWebMath or arXiv;
   - licensed textbooks where indexing is permitted;
   - optionally, the infini-gram or infini-gram mini web-scale API.
2. **Normalize.** Lowercase, collapse whitespace and canonicalize LaTeX. Mask display and inline formulas as a placeholder, so that standard formulas do not trigger matches.
3. **Exact check (suffix array).**
   - Flag any normalized prose span of **50 characters or more**.
   - Block for rewrite at **100 characters or more**, or when the 50-character overlap rate exceeds about 10% of the node.
   - Use an allowlist of stock phrases ("let G be a group"), calibrated on human-written math.
4. **Near-duplicate check.** Run 5-gram MinHash per paragraph, with FineWeb-style 112 hashes in 14x8 bands, taking candidates at Jaccard 0.5 or more. Confirm with either of these:
   - ROUGE-L of 0.8 or more;
   - difflib blocks of 20 words or more (nv-recall-style rules).
5. **Paraphrase check.** Retrieve the top-k candidates by embedding or BM25. Send pairs above a high cosine threshold, initially 0.9 and uncalibrated, to human review. Expect low recall.
6. **Calibrate** every threshold by running the same pipeline over human-written Wikipedia and ProofWiki pages checked against each other.
7. **Record match spans** (source, uri, licence, offsets), as Gemini's citation metadata does. When a match is to a CC BY-SA source, attribute it and keep the share-alike terms. Never keep a GFDL match.

### 3a.6 Could not verify (memorization)

- How ParaPO's regurgitation metric is defined (arXiv:2504.14452 reports 17.3 to 12.9 for Llama3.1-8B creative writing).
- The value of δ in DJ Search.
- Cooper et al.'s coverage figure of 90.89% [snippet, older version].
- The GPT-4 50-character and Llama 3 8-gram contamination rules [secondary snippet only].
- MathPile's ProofWiki duplication [snippet].
- BloomScrub numbers (arXiv:2504.16046) [snippet].
- Gemini's recitation threshold.
- The Nature Communications version of Mosaic Memory, which needs a login.

## 3b. The legal position of LLM output

### United States

- **Copyright Office, Part 2: Copyrightability (29 Jan 2025)** [P]. https://www.copyright.gov/ai/Copyright-and-Artificial-Intelligence-Part-2-Copyrightability-Report.pdf (read 2026-10-26). Its summarised conclusions:
  - "Copyright does not extend to purely AI-generated material, or material where there is insufficient human control over the expressive elements."
  - "Based on the functioning of current generally available technology, prompts do not alone provide sufficient control."
  - "Human authors are entitled to copyright in their works of authorship that are perceptible in AI-generated outputs, as well as the creative selection, coordination, or arrangement of material in the outputs, or creative modifications of the outputs."
  - "The case has not been made for additional copyright or sui generis protection for AI-generated content."
  - Whether a human contribution is enough is decided "on a case-by-case basis".
- **Copyright Office, Part 3: Generative AI Training** [P]. The pre-publication version came out on 9 May 2025. On 2026-10-26, https://www.copyright.gov/ai/ still lists it as pre-publication: "A final version of Part 3 will be published in the future, without any substantive changes expected." Two of its points matter here (PDF read 2026-10-26):
  - When a model "can generate verbatim or substantially similar copies of a training example, without that expression being provided externally in the form of a prompt or other input, it must exist in some form in the model's weights". There is then "a strong argument that copying the model's weights implicates the right of reproduction for the memorized examples."
  - "copying the resulting weights will only infringe where there is substantial similarity."
- No Part 4 and no 2026 report is listed on https://www.copyright.gov/ai/ (read 2026-10-26).
- **Thaler v. Perlmutter.** The D.C. Circuit held on 18 Mar 2025 (No. 23-5233) that the Copyright Act requires a human author. The Supreme Court denied certiorari on **2 Mar 2026** (No. 25-449; docket entry "Petition DENIED") [P]. https://www.supremecourt.gov/docket/docketfiles/html/public/25-449.html (read 2026-10-26). A denial is not a ruling on the merits. The human-authorship rule of the D.C. Circuit stands.
- **Training cases.** All of these concern inputs, not the copyrightability of outputs.
  - *Bartz v. Anthropic* (N.D. Cal.): Judge Alsup's order of 23 Jun 2025 held that training on lawfully acquired books was fair use ("exceedingly transformative"). Building a library of pirated copies was not fair use. The case settled for $1.5B. Final approval came on **20 Jul 2026** (3:24-cv-05417-AMO; about 482,460 works) [S: Authors Alliance https://www.authorsalliance.org/2026/07/21/bartz-v-anthropic-settlement-receives-final-approval/ ; vLex https://case-law.vlex.com/vid/bartz-v-anthropic-pbc-1129578441 ; read 2026-10-26].
  - *Kadrey v. Meta* (N.D. Cal., 25 Jun 2025, Chhabria J.): fair use on that record, because the plaintiffs showed no market dilution. The judge warned that "in many circumstances it will be illegal to copy copyright-protected works to train generative AI models without permission" [S: https://chatgptiseatingtheworld.com/2025/06/25/meta-prevails-on-fair-use-in-ai-training-in-kadrey-v-meta-but-judge-chhabria-cautions-a-better-record-of-dilution-or-market-harm-could-prevail-in-other-cases/ ; read 2026-10-26].
  - *Thomson Reuters v. Ross* (3d Cir., No. 25-2153, decided 29 Sep 2026; the opinion was first issued under seal): this is the first appellate ruling on fair use in AI training. Copying Westlaw headnotes to train a non-generative legal search tool was **not** fair use, and the headnotes were held original. The court said that generative AI is a different case [S: https://www.ballardspahr.com/insights/alerts-and-articles/2026/10/third-circuit-addresses-fair-use-in-ai-training-but-leaves-generative-ai-questions-unresolved ; https://patentlyo.com/patent/2026/10/ease-is-not-necessity-third-circuit-affirms-no-fair-use-in-thomson-reuters-v-ross.html ; read 2026-10-26].
  - *NYT v. OpenAI* (S.D.N.Y.): this is the main US case about regurgitated **outputs**. It had no decision on the merits by late October 2026 [S: https://www.nortonrosefulbright.com/en/knowledge/publications/ce8eaa5f/ai-in-litigation-series-an-update-on-ai-copyright-cases-in-2026 ; read 2026-10-26]. Search summaries said that the DOJ filed in support of OpenAI on 28 Sep 2026. UNVERIFIED: the filing was not seen.

### European Union, United Kingdom, Germany

- **GEMA v. OpenAI**, LG München I, 42 O 14139/24, 11 Nov 2025 [S]. The court held that memorisation of song lyrics in the model is a reproduction, and that so is the output of the lyrics in response to prompts. It held that the TDM exception covers making the training corpus but not "permanent embodiment of works in trained models". OpenAI was held liable for the outputs. OpenAI appealed, and no appellate decision was found [S: https://www.williamfry.com/knowledge/gema-v-openai-europes-direction-on-ai-infringement/ ; https://ailawsuittracker.com/cases/gema-v-openai/ ; read 2026-10-26]. The primary judgment text was not read (merlin.obs.coe.int and cms.law returned 403).
- **CJEU C-250/25 *Like Company v. Google*** is the first reference to the CJEU on generative AI. It asks whether chatbot output that partly matches press articles is a reproduction or a communication to the public. The Grand Chamber heard it on 10 Mar 2026. No Advocate General's opinion had been found by late September 2026, and a judgment is expected in 2027 [S: https://www.create.ac.uk/blog/2026/03/03/copyright-and-ai-case-c-250-25-like-company-before-the-grand-chamber-of-the-court-of-justice-of-the-european-union-opinion-by-the-european-copyright-society/ ; https://www.stibbe.com/publications-and-insights/ai-copyright-claims-in-europe-key-considerations ; read 2026-10-26]. UNVERIFIED: whether the opinion came out in October 2026. The CURIA page was not checked.
- **Getty Images v. Stability AI**, [2025] EWHC 2863 (Ch), 4 Nov 2025: the model weights "do not store an infringing copy". Permission to appeal was granted on 16 Dec 2025. No Court of Appeal judgment was found [S: https://www.taylorwessing.com/en/insights-and-events/insights/2026/01/next-steps-for-getty-v-stability-why-has-permission-to-appeal-been-granted ; read 2026-10-26].
- **EU AI Act, duties for GPAI providers (Art. 53 to 55).** They apply from **2 Aug 2025**. The Commission's powers to enforce and fine apply from **2 Aug 2026**. Models placed on the market before 2 Aug 2025 have until 2 Aug 2027 [P: https://digital-strategy.ec.europa.eu/en/policies/guidelines-gpai-providers ; read 2026-10-26]. The duties fall on **providers** of models: technical documentation, a copyright policy and a public summary of the training content. A user of a model's outputs has none of these duties. A "downstream modifier" becomes a provider only if its modification uses more than one third of the original model's training compute (Commission guidelines, July 2025, para 63) [S: https://digital-strategy.ec.europa.eu/en/faqs/guidelines-obligations-general-purpose-ai-providers via search summary; read 2026-10-26]. A math2 fine-tune of an open prover would be far below that threshold.
- **EU AI Act, Art. 50, transparency.** It applies from **2 Aug 2026** [S: https://artificialintelligenceact.eu/article/50/ ; read 2026-10-26].
  - Art. 50(2): providers of systems that generate text must mark outputs "in a machine-readable format ... detectable as artificially generated". This duty falls on the model provider.
  - Art. 50(4): **deployers** that publish AI-generated text "with the purpose of informing the public on matters of public interest" must disclose that the text is artificially generated. This duty does not apply "where the AI-generated content has undergone a process of human review or editorial control and where a natural or legal person holds editorial responsibility for the publication".
  - The Digital Omnibus on AI (Regulation (EU) 2026/1744, in force 27 Jul 2026) reportedly gives a grace period to 2 Dec 2026 for Art. 50(2) marking. Sources disagree on its scope [S: https://fpf.org/blog/the-ai-act-implementation-timeline-what-changes-under-the-ai-omnibus/ ; https://cdp.cooley.com/digital-ai-omnibus-delays-key-deadlines-introduces-new-rules/ ; read 2026-10-26].
  - The Commission's Art. 50 guidelines (20 Jul 2026) and the code of practice on transparency of AI-generated content (10 Jun 2026) are reported but were not read. UNVERIFIED.
  - Whether a mathematics library publishes text "on matters of public interest" is not settled. math2 already discloses AI use per field, so it meets Art. 50(4) whichever way the question is decided.

### What this means for CC BY-SA 4.0 on LLM-written text

1. **Text generated purely by an LLM has, in the US, no copyright for math2 to license.** Prompts alone are not enough (Copyright Office, Part 2). Mathematical facts, definitions and short functional statements have thin protection in any case. Applying CC BY-SA 4.0 to such text is not wrong, but it may be **inoperative**. CC BY-SA 4.0 §8(a) says that the licence does not "reduce, limit, restrict, or impose conditions on any use ... that could lawfully be made without permission", and §2(a)(2) says that "where Exceptions and Limitations apply to Your use, this Public License does not apply" [P: https://creativecommons.org/licenses/by-sa/4.0/legalcode.en ; read 2026-10-26]. Thus share-alike cannot be enforced on unedited machine text.
2. **Creative Commons' own position** [P: https://creativecommons.org/2023/08/18/understanding-cc-licenses-and-generative-ai/ ; read 2026-10-26]: "The CC license you choose will apply to the creative work that you contribute to the final product, even if the portion produced by the generative AI system itself may be uncopyrightable." It also says: "We encourage the use of CC0 for those works that do not involve a significant degree of human creativity." The 2026 CC FAQ does not answer the question for outputs (https://creativecommons.org/faq/ ; read 2026-10-26).
3. **What human work adds.** Human selection, arrangement and creative modification of the corpus are protectable. So are the human-written parts. The graph structure, the curation and the edited nodes can thus carry CC BY-SA. Unedited agent text cannot, at least in the US. In other jurisdictions the answer differs and is unsettled (e.g. the UK CDPA s.9(3) "computer-generated works" rule; not researched here).
4. **The real legal risk is regurgitation, not ownership.**
   - If an LLM reproduces a protected source (a textbook, or GFDL text from Stacks or Kerodon), the third party's copyright attaches to the output.
   - The GEMA judgment held the model provider liable for such output. Whoever republishes the output is also exposed.
   - CC BY-SA text that is reproduced closely needs attribution and share-alike. This is compatible with math2 only when the attribution is recorded.
   - GFDL and NC text are incompatible with math2.
   - Thus every generated node needs a copy check against a reference corpus. The check must cover more than the cited sources, because a node that a model writes from memory may have no cited source at all (see section 3a).
5. **Provider terms do not change this picture.** Each major provider assigns to the customer whatever rights it has in the outputs (section 4). An assignment cannot create a copyright that the law does not recognise.

## 4. Provider terms

### Hosted APIs

| Provider | Ownership of outputs | Publishing outputs | Using outputs to train other models | Source (read 2026-10-26) |
|---|---|---|---|---|
| Anthropic (Commercial Terms, effective 17 Jun 2025) | Customer "owns its Outputs". "Anthropic hereby assigns to Customer its right, title and interest (if any) in and to Outputs." "Anthropic may not train models on Customer Content from Services." | no restriction in the Commercial Terms. The AUP bans "Plagiarize or submit AI-assisted work without proper permission or attribution" and bans infringing third-party IP. | Commercial Terms D.4: may not "access the Services to build a competing product or service, including to train competing AI models". **AUP (effective 12 Nov 2026): may not "Use outputs to train an AI model (e.g., "model scraping" or "model distillation") without prior authorization from Anthropic".** | https://www.anthropic.com/legal/commercial-terms ; https://www.anthropic.com/legal/aup |
| OpenAI (Services Agreement ONLINE v.010126) | §4.1: Customer "owns all Output. OpenAI hereby assigns to Customer all OpenAI's right, title, and interest, if any, in and to Output." §4.4: "Output may not be unique, and other users may receive similar content". | no restriction found in the agreement | §3.3(e): may not, "except for a Permitted Exception, use Output to develop artificial intelligence models that compete with OpenAI's products and services". The Permitted Exceptions are (a) classifiers and embeddings that are "not distributed or made commercially available to third parties", and (b) fine-tuning on OpenAI's own services. | https://cdn.openai.com/osa/openai-services-agreement.pdf (openai.com returned 403; the CDN PDF was read with pdftotext) |
| Google (Gemini API Additional Terms, last updated 2026-04-28) | "Google won't claim ownership over that content." "Google may generate the same or similar content for others." | "You're responsible for your use of generated content, and for the use of that content by anyone you share it with." Applicable law "may require the provision of attribution to your users when returned as part of an API call". | "You may not use the Services to develop models that compete with the Services (e.g., Gemini API or Google AI Studio)." | https://ai.google.dev/gemini-api/terms |

The consequences for math2:

- **Publishing is allowed by all three providers.** None of them claims the outputs, and none forbids publication under an open licence.
- **The training clauses bind math2, and math2's downstream users, differently.**
  - math2 itself may not use Claude outputs to train or fine-tune a model without Anthropic's authorisation, under the AUP that takes effect on 12 Nov 2026. This rules out a math2 "fine-tune an open prover on our Claude-written proofs" plan unless it is authorised. The same holds for OpenAI and Google outputs in any model that "competes".
  - Third parties who later download math2 under CC BY-SA have no contract with Anthropic. Whether the AUP reaches them is a contract question, not a copyright question. The licence of math2 cannot pass a training ban on to them, and it should not try to.
- **Dataset releases need provenance.** The model id recorded for each field (06.5) lets a downstream user filter the corpus by model family when that user's own terms require it.
- **Indemnities** apply only to the customer's paid use. They are not a reason to skip copy checks.
  - Anthropic Commercial Terms K.1: Anthropic defends claims "alleging that Customer's paid use of the Services" violates third-party IP.
  - OpenAI reportedly extends its indemnity to output in its Service Terms of 12 Jun 2026 [S, not read].

### Open-weight model licences, checked on Hugging Face and GitHub on 2026-10-26

| Family (newest seen) | Licence | Extra duties on outputs or on models trained with outputs |
|---|---|---|
| Llama 4 (Scout/Maverick; no newer Meta LLM on HF) | Llama 4 Community License | **Yes.** "If you use the Llama Materials or any outputs or results of the Llama Materials to create, train, fine tune, or otherwise improve an AI model, which is distributed or made available, you shall also include "Llama" at the beginning of any such AI model name." "Built with Llama" must be displayed when such a model or derivative is distributed. Over 700M MAU needs a separate licence. The licence ends if you sue Meta claiming that Llama outputs infringe. https://raw.githubusercontent.com/meta-llama/llama-models/main/models/llama4/LICENSE |
| DeepSeek V4 / V4.1 (Flash, Pro; Sep to Oct 2026) | MIT (HF tag `license:mit`) | none. |
| DeepSeek-Prover-V2 (Apr 2025) | **conflicting.** The HF LICENSE file is MIT, but the README says "subject to the Model License" (DeepSeek Model Agreement). | The DeepSeek Model Agreement defines "Derivatives of the Model" to include any model trained on "synthetic data [generated] by the Model". Use restrictions (Attachment A) flow down to derivatives. §6: "DeepSeek claims no rights in the Output". https://raw.githubusercontent.com/deepseek-ai/DeepSeek-V3/main/LICENSE-MODEL |
| DeepSeek-Math-V2 (Nov 2025) | Apache-2.0 | none |
| Qwen3.5 (Feb 2026) | Apache-2.0 | none |
| **Qwen3.8 (Aug 2026)** | **Qwen Community License 1.0** (new in 2026) | Commercial products with more than 100M MAU or more than US$20M monthly revenue must display the model name. A "Model as a Service" or "AI Work Assistant" business needs a separate licence. "Internal Use" is exempt "provided that such Use does not make the Software, its outputs, or its underlying model capabilities available to any third party". It is unclear whether publishing a corpus of outputs is a non-internal use. https://huggingface.co/Qwen/Qwen3.8-Flash-Next-FP8/raw/main/LICENSE |
| Kimi K2 (2025) | Modified MIT | display "Kimi K2" above 100M MAU or $20M monthly revenue |
| **Kimi K3 (Jun 2026)** | Kimi K3 License | "Model as a Service" businesses above $20M revenue in 12 months need an agreement. Display "Kimi K3" above the MAU or revenue thresholds. The exemption for internal use is again tied to not making "the Software, its outputs, or its underlying capabilities available to third parties". https://huggingface.co/moonshotai/Kimi-K3/raw/main/LICENSE |
| GLM-4.6 / GLM-5 (Feb 2026) | MIT | none |
| Mistral Large 3 (Dec 2025), Leanstral 1.5 (Jul 2026) | Apache-2.0 | none |
| gpt-oss-120b/20b (Aug 2025) | Apache-2.0 plus a one-line usage policy ("comply with all applicable law") | none |
| Goedel-Prover-V2 | Apache-2.0 | none |
| Kimina-Prover-72B | MIT | none |

- In 2026, the Chinese labs moved from plain MIT or Apache to custom licences: Qwen 3.8 and Kimi K3.
- For a corpus that math2 publishes, the duties that matter are these. Llama's naming duty applies only if someone trains a distributed model on Llama outputs. The DeepSeek Model Agreement flows its restrictions down to any model trained on the outputs. The internal-use exemptions of Qwen 3.8 and Kimi K3 do not cover outputs made available to third parties, so the MaaS and display thresholds apply. math2 is far below the thresholds, but the terms should be recorded per model.
- MIT, Apache-2.0 and the gpt-oss terms add nothing.

## 5. Cost

### List prices on 2026-10-26 (USD per million tokens; official pages only)

Anthropic, from https://platform.claude.com/docs/en/about-claude/pricing (read 2026-10-26):

| Model | Input | Cache hit | Output | Batch input | Batch output |
|---|---|---|---|---|---|
| Claude Fable 5.1 | $10 | $0.25 (0.025×) | $50 | $5 | $25 |
| Claude Opus 5.5 | $4 | $0.20 (0.05×) | $20 | $2 | $10 |
| Claude Sonnet 5.5 | $2 | $0.10 (0.05×) | $10 | $1 | $5 |
| Claude Haiku 5.5, prompt ≤100k tokens | $0.10 | $0.01 | $0.50 | $0.05 | $0.25 |
| Claude Haiku 5.5, prompt >100k tokens | $0.50 | $0.05 | $2.50 | $0.25 | $1.25 |

- **Cache writes:** 1.25× input for a 5-minute TTL and 2× for 1 hour. "These multipliers stack with other pricing modifiers, including the Batch API discount".
- **Long context:** "Claude 4.6 and later models (except Claude Haiku 5.5) ... include the full 1M token context window at standard pricing."
- **Data residency:** `inference_geo: "us"` costs 1.1×.
- **Fast mode** for Opus 5.5 costs $8/$40, and it is not available with Batch.
- **Batch API** (https://platform.claude.com/docs/en/build-with-claude/batch-processing ; read 2026-10-26):
  - a 50% discount;
  - up to 100,000 requests or 256 MB per batch, "most batches completing within 1 hour", expiry at 24 h, results kept 29 days;
  - "cache hits are provided on a best-effort basis. Users typically experience cache hit rates ranging from 30 percent to 98 percent".
- The prices in spec 12.3 match this page.

OpenAI, from https://developers.openai.com/api/docs/pricing (platform.openai.com redirects there; read 2026-10-26), short context ≤272K input:

| Model | Input | Cached | Output | Batch in / cached / out |
|---|---|---|---|---|
| gpt-6-astra | $10 | $1.00 | $50 | $5 / $0.50 / $25 |
| gpt-6.1-sol | $2 | $0.10 | $10 | $1 / $0.05 / $5 |
| gpt-6-luna | $0.10 | $0.01 | $0.50 | $0.05 / $0.005 / $0.25 |

- Long context costs more: gpt-6-astra $20/$75, gpt-6.1-sol $4/$15.
- Flex rates equal the Batch rates.

Google, from https://ai.google.dev/gemini-api/docs/pricing ("Last updated 2026-10-22 UTC"; read 2026-10-26):

| Model | Input | Output | Batch |
|---|---|---|---|
| Gemini 4 Argon | $2.00 until 30 Nov 2026, then **$4.00** from 1 Dec 2026 | $10.00, then **$20.00** | 50% |
| Gemini 3.8 Flash | $0.75 until 31 Dec 2026, then $1.50 | $3.75, then $7.50 | 50% |
| Gemini 3.5 Flash-Lite | $0.30 | $2.50 | 50% |

Google's prices for cached input were not read.

### An estimate for one million nodes

The token counts below are **assumptions, not measurements**. Calibrate them on the first 1,000 real runs (the cost per merged output in 12.9). Tokenizers differ between providers, so the same text costs a different number of tokens.

- **Generation call:** 10k input, of which 8k is a cached shared prefix (system prompt, style guide, neighbourhood) and 2k is unique. 6k output, including thinking.
- **Three check calls** (grounding or judge, critic, back-translation or consistency), each with 8k input (2k cached, 6k unique) and 2k output.
- All calls use Batch. Cache reads are assumed to hit.

| Configuration | $ per node | $ per 1M nodes |
|---|---|---|
| all Claude Haiku 5.5 | 0.004 | ~$4k |
| all Claude Sonnet 5.5 (≈ all gpt-6.1-sol) | 0.081 | ~$81k |
| all Claude Opus 5.5 | 0.161 | ~$161k |
| all Claude Fable 5.1 (≈ all gpt-6-astra) | 0.40 | ~$400k |
| generate on Opus 5.5; check with 2 Sonnet 5.5 calls and 1 gpt-6.1-sol call (cross-family) | 0.113 | ~$113k |
| generate on Fable 5.1; checks as above | 0.209 | ~$209k |
| all Sonnet 5.5, zero cache hits | 0.094 | ~$94k |
| all Gemini 4 Argon, before / after 1 Dec 2026 (no cache price assumed) | 0.094 / 0.188 | ~$94k / ~$188k |

- **Divide by the acceptance rate.** At a 50% first-pass acceptance rate, the cost doubles.
- **Output tokens dominate:** 70 to 85% of the cost in every configuration. The thinking budget is the lever that matters most, not the size of the context.
- **Formal work is not in this table.** A Lean statement with its F1 battery, and above all a Lean proof, costs orders of magnitude more per item (section 2). Price it separately.
- At one million nodes, informal generation costs tens to hundreds of thousands of dollars. The cost is much smaller than the review capacity that math2 would need for the output (10.6), and much smaller than formal proof.

### 5.1 Model facts that bear on scale

Source: https://platform.claude.com/docs/en/about-claude/models/overview [P].

- **Model ids:**
  - `claude-fable-5-1`
  - `claude-opus-5-5`
  - `claude-sonnet-5-5`
  - `claude-haiku-5-5`
- **Context and output:** a 1M-token context window, and up to 128K output tokens per request. On the Batch API, output can go to 300K with the beta header `output-300k-2026-03-24`.
- **Training cutoff:** reliable knowledge and training data go to **June 2026**. Mathematics published after that date is unknown to the models unless it is retrieved into the prompt. This bears on explore and novelty checks (16.5 item 5): a model's "this is new" is worth nothing for results after its cutoff.
- **Retirement:** each model is retired no sooner than 1 Sep, 22 Sep, 28 Sep and 7 Oct 2027 respectively. A million-node campaign that runs past those dates will cross a model change. This is a further reason to record exact model ids (12.3) and to keep `m2 prov quarantine` working.
- **Size rule of thumb:** 1M tokens is about 555k words, or about 2.5M characters.
  - So a node of 1,500 words of prose is about 2.7k tokens, before LaTeX. LaTeX tokenizes worse than prose.
  - The token assumptions in 5 are plausible, but they still need calibration.

## 6. Community norms on LLM text (short)

- **Wikipedia.** Speedy-deletion criterion G15 for unreviewed LLM-generated pages was adopted in August 2025, and later renamed "Unambiguously LLM-generated pages". https://en.wikipedia.org/wiki/Wikipedia:Speedy_deletion [S, the policy page as summarised; the adoption discussion is in the talk archives]. WikiProject AI Cleanup exists to find and fix such text (https://en.wikipedia.org/wiki/Wikipedia:WikiProject_AI_Cleanup) [S].
  - Wikipedia is a source that math2 imports under CC BY-SA (16.6). math2 must not push unreviewed agent text back into it (16.5 "Upstream etiquette" already forbids bulk writes).
- **arXiv.** On 31 Oct 2025, arXiv said that review articles and position papers in the CS category must have been peer-reviewed before submission, citing the flood of LLM-written submissions. https://blog.arxiv.org/?p=1740 [S, read via summary].
- **Grokipedia** (section 2.4) is the large-scale counter-example: 56% of its articles carry Wikipedia's CC footer, and no error rate is published.

These norms agree with spec 16.5. They add one point: **communities sanction unreviewed machine text in bulk, not the use of AI as such.** math2's protection is the marker plus human confirmation (P04). It should not ship unconfirmed prose as if it were reviewed.

## 7. What the spec should require because of this

The numbers in brackets point to the evidence above. Each item names the spec section it changes.

### 7.1 Quality and checking (spec 10, 12, 15)

1. **No LLM verdict is evidence of correctness for an informal proof (12.6, 16.5 item 4).**
   - Evidence: LLM judges inflate scores up to 20x [1.2]. The best step judges accept about 21% of wrong steps, and weaker judges 56 to 90% [1.2].
   - Requirement: agent-written informal proofs stay E1, and a judge pass is recorded as triage only. The spec should say this explicitly for judges, as it already does for proofs.
2. **No Haiku-tier or weak open-model judge for proofs or statements (12.3).**
   - Evidence: Hard2Verify finds that "weaker verifiers cannot catch errors". Claude Sonnet 4-class judges flag only 43.7% of wrong steps [1.2].
   - Requirement: limit tier 0 to triage, classification and dedup pre-filters. Proof and statement judging starts at the tier that the run records show reaches a measured true-negative rate, for example 75% or more on a labelled set.
   - Grayzel's result supports a cheap cross-family consensus [1.2]. Treat that as something to measure, not to assume.
3. **Judges vote across families; accept by unanimity, reject by majority (10.4 check 9, 12.6).**
   - Evidence: a unanimous vote gives the best precision, and a majority vote the best recall [1.2, 1.5]. Cross-model agreement is the best citation check that has been measured [1.5].
   - Requirement: record each judge's verdict and model id in the run record. Disagreement goes to a human, and is never settled by a tiebreak call.
4. **An anti-sycophancy check in prove and formalize (10.4, 12.4).**
   - Evidence: strong models "prove" false statements 29 to 97% of the time [1.1]. A "prove or disprove" framing raised the rejection rate of false statements from 18.5% to 71% [1.5].
   - Requirements:
     - (a) The prompt of every prove and formalize job is framed as "prove or disprove".
     - (b) A periodic canary: inject perturbed false statements into the queue, and measure how many the agents "prove". Publish the rate in the weekly report (12.9).
     - Check 6 of the F1 battery (cheap negation) only catches cheaply false statements. It does not catch the subtle ones that BrokenMath and BrokenArXiv use.
5. **Compiling and back-translation are not fidelity (10.4).**
   - Evidence: Kimina compiles 90% of statements but only 66% are accurate. Unsupervised autoformalization is faithful 60 to 90% of the time [2.3], and less often on graduate material.
   - Requirements:
     - Set the prior error rate for F1 at 10 to 40% until math2's own sampled re-review (10.8) measures it.
     - Do not treat F1 as "probably right" in the workspace.
6. **Measure the informal error classes that no one has measured (10.8, 17.3).**
   - Evidence: there is no published rate for wrong definitions, missing hypotheses or misnamed theorems in informal LLM text [1.3].
   - Requirement: add an informal audit beside the F2 re-review. Each month, a blind expert audit of a seeded sample of agent-written definitions and statements, classified with M-* style classes for informal text:
     - wrong definition;
     - dropped hypothesis;
     - misattributed name;
     - non-standard convention unstated.
   - Without this, the published error rates in release notes cover only formal bindings.
7. **Source records written by agents must be resolved and checked (04 `sources:`, a new rule in G or P).**
   - Evidence: 11.4 to 56.8% of generated citations are hallucinated [1.4]. Usable theorem references in research proofs run at 7% or less [1.3]. Fabrication is worst on unfamiliar topics [1.4].
   - Requirement: a hard rule. A `sources:` entry added by an agent must resolve to a real record (DOI, zbMATH, arXiv id, ISBN, or a record-plane entry), and the resolution is logged. A locator ("Thm 3.1") written by an agent counts toward E2 or E3 only after it is checked against retrievable text, or after a person confirms it.
   - Also: theorem names in prose ("by the X–Y theorem") must link to a node or carry a source. A name that resolves to nothing fails.
8. **Prefer pseudo-formal structure for informal proofs (12.4, 04).**
   - Evidence: pseudo-formalization raised error recall from 0.54 to 0.66, but still missed about 40% of errors on arXiv proofs [1.5].
   - Requirement: agent-written informal proofs use a step-structured format with each step's justification as a link or a citation. This helps the judge, but it does not change E1.
9. **Watch for checker gaming (11.3, X rules).**
   - Evidence: in FormalQualBench, agents tried elaborator-level workarounds [2.1]. Aristotle's results were flagged as "not Comparator-validated" [2.1].
   - Requirement: the comparator-pinned compile (check 10) and the axiom audit are mandatory for every accepted proof, with no exception for "trusted" provers.

### 7.2 Scale planning (spec 12.3, 12.9, 10.6)

10. **Budget review, not generation.**
    - Evidence: a million informal nodes costs about $4k to $400k in model calls, depending on the tier mix [5]. The published formal campaigns cost about $17 per declaration, or $300 per target theorem [2.1]. No large agent run has published a misformalization rate [2].
    - Requirements:
      - The spec's throughput targets (10.6) must be set against review capacity.
      - The weekly report should publish **cost per accepted node and per F2 binding**, and not only spend.
11. **Publish a fidelity rate before claiming scale.** No prior project publishes a misformalization rate for its agent-written library [2]. math2's sampled re-review (10.8) is a differentiator. The spec should require that a release with more than N agent-written bindings is not tagged until the sample has been drawn and measured.
12. **Plan for model churn.**
    - Facts: the current models retire from Sep 2027, and their knowledge stops at June 2026 [5.1].
    - Requirements:
      - Novelty checks (16.5 item 5) must retrieve, and not rely on model memory.
      - Every campaign record states the model ids, so that `m2 prov quarantine` can act on them.

### 7.3 Copying and licence (spec 15 L-rules, 16.6)

13. **Extend L04: compare with a reference corpus, not only with cited sources (15 L04).**
    - Evidence: LLMs reproduce text without being asked. Over 27% of encyclopedia-style output sits in verbatim spans of 50 characters or more. One case reproduced 1,170 characters of Wikipedia [3a.1]. Text a model writes from memory has no cited source to compare against.
    - Requirement: a new rule `L05` (S, rising to H above a hard threshold), run on every agent-written prose field. Check it against an indexed reference corpus:
      - Wikipedia math, ProofWiki, nLab and PlanetMath;
      - Math.SE and MathOverflow, under the per-post rules of 16.6;
      - **Stacks and Kerodon (GFDL)**;
      - OpenWebMath or arXiv text, for detection only;
      - any licensed textbook index.
    - Parameters to start with, and then to calibrate [3a.5]:
      - Text normalization: LaTeX normalized and formulas masked.
      - Exact matching: flag a span of 50 characters or more; hard-fail at 100 characters or more against a GFDL, NC or all-rights-reserved source; flag at 10% or more covered.
      - Near-duplicate matching: 5-gram MinHash with Jaccard 0.5 as the candidate threshold, confirmed with ROUGE-L 0.8 or more, or with difflib blocks of 20 words or more.
      - A stock-phrase allowlist.
    - Every threshold is calibrated on human-written math pages, whose median overlap is 0% [3a.1].
14. **An exact-match filter is not enough.**
    - Evidence: a 10-gram filter blocks verbatim copying while near-verbatim text gets through, and fuzzy duplicates count about 0.8 of exact ones [3a.2].
    - Requirement: L05 needs a near-duplicate stage, and a paraphrase stage that is sampled and sent to humans, not blocking.
15. **Record match spans.** For each L05 hit, record the source, URI, licence and offsets in provenance. This follows Gemini's citation metadata [3a.4]. A hit against a CC BY-SA source becomes an attributed import (L03). A hit against GFDL, NC or arXiv-default text is rewritten (L02).
16. **Anti-reproduction instructions in every prose prompt (12.8 prompts).**
    - Evidence: such a system prompt cut mean overlap by more than half in one model, but long matches remained [3a.1].
    - This is a cheap first layer, not a control.
17. **Licence the unedited agent text honestly (16.6).**
    - Evidence: in the US, purely AI-generated text has no copyright, and CC BY-SA cannot bind uses that need no permission (§8(a), §2(a)(2)) [3b]. CC itself suggests CC0 for works without significant human creativity [3b].
    - The spec should choose one of these:
      - **(a)** Keep CC BY-SA for all prose. Add a notice that the licence applies to the extent that copyright subsists, and that agent-written, unconfirmed fields may be in the public domain in some jurisdictions. Provenance (P04) shows which fields these are.
      - **(b)** Release unconfirmed agent prose under CC0 until a person edits or confirms it. It then becomes CC BY-SA.
    - Option (a) is simpler and keeps one prose zone. Option (b) is more honest, but makes licence zones depend on provenance. **Recommendation: (a) with the notice.** The collection, its selection and arrangement, and human-edited text stay under CC BY-SA in any case.
    - Text an LLM regurgitates from a CC BY-SA source keeps that source's terms whatever math2 says. This is why item 15 matters.
18. **Art. 50(4) disclosure is satisfied by provenance (16.5).** EU deployer disclosure for published AI text applies from 2 Aug 2026, unless a human reviewed the text and someone holds editorial responsibility [3b]. math2's per-field markers (P04) and named responsible persons (P03) cover both routes. The spec should say this in 16.5 item 7, and the site should carry a machine-readable AI-generated flag on unconfirmed fields.

### 7.4 Provider and model terms (spec 12.3, 20.2, 06.5)

19. **No training on hosted-model outputs without authorisation (12.3, 20.2).**
    - Facts:
      - Anthropic's AUP, in force from 12 Nov 2026, forbids using outputs to train an AI model without prior authorisation.
      - OpenAI and Google forbid using outputs to build "competing" models [4].
    - Requirement: any plan to fine-tune the open provers of tier 0′ on math2 data must exclude, or get authorisation for, fields generated by hosted models. Provenance must make that filter mechanical. Add it to the provenance tests of 06.5: "list each field produced by model family X".
20. **Record each model's licence and its output terms (12.3 price table, a new `models.reg`).**
    - Several open-weight licences have output clauses:
      - Llama 4 naming;
      - the DeepSeek Model Agreement derivatives clause, and the LICENSE file that conflicts with the README for DeepSeek-Prover-V2;
      - the Qwen 3.8 and Kimi K3 internal-use exemptions, which exclude outputs made available to third parties [4].
    - Requirement: a register of models with these fields: licence, URL, date read, output clauses, and thresholds. The orchestrator refuses a model with no entry.
    - Prefer MIT or Apache models for tier 0′: DeepSeek V4, DeepSeek-Math-V2, Goedel-Prover-V2, Kimina, Leanstral, GLM, gpt-oss.
21. **math2's licence does not pass provider training bans on to users (16.6).** A CC BY-SA licence cannot add a no-training term, and math2 should not try. The release notes can state which fields came from which provider. Downstream users with their own contracts can then filter.

### 7.5 Cost accounting (12.3, 12.9)

22. **Price table and prices in `llm.Prices` must be dated, and they must model batch, cache read, cache write (1.25x / 2x), the Haiku >100k tier, and residency (1.1x)** [5].
    - Google's Gemini 4 Argon price doubles on 1 Dec 2026, and Gemini 3.8 Flash on 1 Jan 2027 [5]. A static table goes stale within weeks.
23. **Thinking tokens are the cost lever.** Output, including thinking, is 70 to 85% of the modelled cost [5]. The weekly report should give output tokens per accepted node for each role and tier. Effort settings (12.3) should be tuned on that number.

## 8. Open doubts

1. **No measured error rate for informal LLM definitions and statements.** Section 1.3 found none. The prior in 7.1 item 5 is borrowed from formal autoformalization, and may be wrong in either direction for prose.
2. **The mathematical reproduction rate is unmeasured.**
   - The figure of over 27% at 50 characters is for encyclopedia prose in general, and was measured on 2024 models against a 2022 web snapshot.
   - Math prose may be higher, because of stereotyped wording, or lower, because formulas are masked.
   - The L05 thresholds in 7.3 are untested on mathematics.
3. **What "copying" means for mathematical statements.** Standard definitions have few reasonable wordings, and the merger doctrine and thin protection may apply. A strict L05 could flood review with false positives. Calibration on human-written pages is required before L05 can block.
4. **Copyright in AI output outside the US.**
   - The UK CDPA s.9(3) "computer-generated works" rule and EU originality law were not researched.
   - The CJEU judgment in C-250/25 (expected 2027) and the GEMA appeal may change the EU position on memorized output.
5. **Whether the Anthropic AUP's training ban reaches third parties** who download math2. Their only relation is through CC BY-SA, and not through any contract. Legal advice is needed before math2 makes any claim either way.
6. **DeepSeek-Prover-V2's licence conflict** (an MIT LICENSE file, against a README pointing to the Model Agreement). Which one governs is unclear. Ask DeepSeek, or avoid using its outputs for training.
7. **Qwen 3.8 and Kimi K3 "internal use".** It is not clear whether publishing a corpus of outputs is a use that "makes outputs available to third parties" in a way that triggers the MaaS clauses. math2 is far below the revenue thresholds, but the reading is uncertain.
8. **Art. 50(4) scope.** Whether a mathematics library publishes "matters of public interest" is unsettled. The Commission's Art. 50 guidelines (20 Jul 2026) and the transparency code of practice were reported but not read (UNVERIFIED).
9. **Cost model.**
   - The token counts are assumptions, and cache-hit rates on Batch are "best-effort", between 30 and 98%.
   - Acceptance rates are unknown until math2 runs. The true cost per accepted node could be 2 to 5 times the table.
   - Formal work is priced only by analogy, at $17 per declaration in Automatic Textbook Formalization.
10. **Judge evidence comes from older models.** Most measurements are of 2025 models (GPT-5, Gemini 2.5, Claude Sonnet 4 and 4.5, Opus 4.5 and 4.6). Opus 5.5 and Fable 5.1 have no published proof-judging measurements that we found. math2 must measure its own judges on a labelled set before item 2 of 7.1 can be settled.
11. **Self-verification evidence conflicts.** Hard2Verify finds models more lenient on their own work, and OPC finds them harsher. Whether the generator may also act as a judge must be measured, not assumed. The safe default is a different family, as 10.4 check 9 already says.
12. **Snippet-only and derived figures** throughout sections 1 and 2 must be re-checked against the primary PDFs before any is quoted in the spec. This applies above all to the First Proof batch 2 tally, BrokenArXiv's later figures, Seed-Prover 1.5's 580/660, and the "500 TPU-days per problem" attributed to AlphaProof.
13. **No Anthropic or OpenAI recitation filter is documented.** We cannot rely on the provider to block regurgitation. Gemini has a RECITATION finish reason, but its threshold is not published.
</document>

<document source="https://math2.tamnd.com/spec/corpus/01e-representation-and-packaging.md">
# K01, part E. Representation, pedagogy, exercises, packaging and notation

Research for the Math 2.0 spec (tamnd/math2). This file builds on spec 01 §1.7 (semantic formats), 01 §1.2 (blueprints), 04 (object model) and 13 (research workspace), and tries not to repeat them. Facts those documents already state are only referred to.

Rules for this file:
- I read every source on **2026-10-26**, unless the entry says otherwise.
- A fact I could not confirm is marked **UNVERIFIED**.
- "From search" means I saw the claim only in a search-result excerpt, not on the page itself.

---

## 0. What the spec already says (so it is not repeated)

**Spec 04:**
- **Node kinds:** concept, definition, statement, conjecture, problem, example, counterexample, object, proof, computation, topic.
- **Tags:** 6-character Crockford base32, permanent.
- **Body sections:** fixed per kind.
- **Explanations:** levels L1 to L4, each marked human-written or agent-written.
- **Math and links:** math is LaTeX limited to the macros in `macros/m2.sty` and `m2.katex.json`; links are `[[text|TAG]]`.
- **Status:** computed, never written by hand.

**Spec 13:**
- a fixed page model;
- prerequisite paths that start from the reader's background;
- a "Translate notation" feature;
- project blueprints generated from the graph.

**Spec 01 §1.7 already covers:**
- MathML 4 is still a Working Draft and has `intent`;
- MathML Core is a CR (June 2025);
- OpenMath 2.0 is inactive;
- sTeX4 = RusTeX + FLAMS + MathHub;
- arXiv HTML is made with LaTeXML;
- Wikidata math counts and P14534;
- MathGloss.

**Spec 01 §1.2 already covers:** leanblueprint, Verso Blueprint, LeanArchitect, Naproche and Megalodon.

What follows is new detail, plus a decision for each item.

---

## 1. Representation

### 1.1 OMDoc and MMT

**What it is**
- OMDoc is an XML format for mathematical documents. It has three levels:
  - object: formulas, in OpenMath or Content MathML;
  - statement: definitions, theorems, proofs, examples;
  - theory: theories, imports and views.
- The project started in 1998.
- KWARC says "the formal part of OMDoc has been refined into the MMT format", with the "MMT System as the reference implementation". KWARC is "extending MMT to cover the full flexiformal coverage of OMDoc".
- Source: https://kwarc.info/systems/omdoc/

**MMT in 2026**
- MMT is written in Scala and stores content in "archives".
- Repo: https://github.com/UniFormal/MMT. It has 7,458 commits and no releases listed on GitHub.
- The licence is in `COPYING.txt`. I did not read it, so the licence is **UNVERIFIED**.
- KWARC is now at FAU Erlangen-Nürnberg.

**Take for math2**
- The **theory/view** idea: content lives in theories, and a view maps one theory into another (§4.6, Realms).
- The **object/statement/theory** split matches math2's formula / node / topic split. math2 needs no XML for this.
- Do not adopt OMDoc as a storage format. Its tooling is research-grade.

### 1.2 sTeX and SMGloM

**sTeX 4**
- Version 4.1.0, dated 2026-06-24, under LPPL. Maintainers: M. Kohlhase and D. Müller. https://www.ctan.org/pkg/stex
- The toolchain (https://kwarc.info/systems/sTeX/):
  - RusTeX converts LaTeX to FTML (HTML with semantic annotations);
  - FLAMS, the "Flexiformal Annotation Management System", is GPL-3.0 (https://github.com/FlexiFormal/flams);
  - MathHub hosts the archives;
  - there is a VS Code IDE.
- The older LaTeXML plugin is deprecated.
- mathhub.info now shows a FLAMS-branded interface.

**Syntax**

From the manual (https://ctan.math.illinois.edu/macros/latex/contrib/stex/doc/stex-manual.pdf). I saw these in excerpts, not by reading the whole PDF.
- **Modules:** `\begin{smodule}{Name}` opens a module. A symbol must belong to a module.
- **Symbols and notations:**
  - `\symdecl{name}` declares a symbol, which is an identity with no notation. `\symdecl*` declares one without a semantic macro.
  - `\notation{name}[variant]{…}` attaches a notation to a symbol. A symbol can have several notations.
  - `\symdef` is `\symdecl` plus `\notation` in one command. Example:
    `\symdef{plus}[name=addition,args=2,type=\funspace{\Nat,\Nat}{\Nat}]{#1 \mathbin{\maincomp{+}} #2}`
  - `prec=` sets precedence for bracketing. `args=a` marks a flexary (associative) argument.
- **Variables:** `\vardef` declares a variable.
- **Imports:**
  - `\importmodule` is transitive. Example: `\importmodule[sTeX/MathBase/Functions]{mod?Function}`.
  - `\requiremodule` is not transitive.
  - `\usemodule` is for use outside a module.
- **Definitions:** `sdefinition` with `for=`, plus `\definiendum` / `\definame`. `\defnotation` marks where a notation is introduced.
- The syntax changed incompatibly on 2023-04-25. This shows the cost of tying content to a LaTeX package's macro API.

**SMGloM**
- The CICM 2022 sTeX3 paper (https://kwarc.info/people/dmueller/pubs/cicm22stexsd.pdf) reports:
  - at least 2,250 concepts;
  - verbalisations in English for 93% of them, German 71%, Chinese 11%.
- **Signature modules** (language-independent symbols, "modsig") are separate from **natural-language modules** ("mhmodnl"), one per language.

**SMGloM licences**
- The early SMGloM Public License (SPL 0.1) "protects symbols against non-conservative changes while allowing derived works". https://kwarc.info/people/mkohlhase/submit/cicm14-smglom-system.pdf
- The current `smglom/mathsoft` archive is CC BY-SA 4.0: https://gl.mathhub.info/smglom/mathsoft/-/blob/main/LICENSE
- I did not check the other SMGloM archives. That they share this licence is **UNVERIFIED**.

**ALeA (Adaptive Learning Assistant, KWARC)**
- ALeA is built on sTeX content. It was funded through the VoLL-KI project (BMBF, 2021 to 2025).
- It keeps a learner model with values M(C,D) in [0,1] for each concept C and each cognitive dimension D (Bloom-like).
- It has been used in 6 courses with more than 1,000 FAU students. The platform is at https://courses.voll-ki.fau.de/.
- **Guided tours:** the system lists all prerequisites of a target, removes the ones the learner already knows, and shows the rest as a sequence of definitions. It also generates flashcards and quizzes.
- Sources: https://sme.uni-bamberg.de/ai4ai/p5.pdf ; CICM 2023 paper, https://www.researchgate.net/publication/373539971

**Take for math2**
1. **Separate a symbol's identity from its notations.** math2 macros are currently only render macros. Each macro should declare the concept tag it denotes, its arity and its precedence. One concept may have several notations, and one of them is the default.
2. **Keep the language-independent part apart from the prose part.** Spec 13 §13.4 already keeps formal statements untranslated. The SMGloM split suggests also keeping a per-concept "verbalisation" record per language: the noun phrase, plural and gender.
3. **ALeA's guided tour** is close to the spec 13 prerequisite path. ALeA also tracks a cognitive dimension per concept, and math2 can do the same with the MATH-taxonomy groups (§2.2).
4. Do not depend on the sTeX macros themselves (LPPL, API churn). Export to sTeX/FTML only if someone asks for it.

### 1.3 OpenMath Content Dictionaries

**What it is**
- 218 Content Dictionaries (CDs) are listed at https://openmath.org/cdnames/. Each has a status: official, experimental, private or obsolete.
- `arith1` (version 3, 2004-03-30) defines abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times and unary_minus. https://openmath.org/cd/arith1
- `transc1` follows Abramowitz and Stegun for branch cuts. https://openmath.org/cd/transc1
- The repo https://github.com/OpenMath/CDs was last updated 2022-06-28.
- Content MathML's `cd` / `cdgroup` mechanism refers to OpenMath CDs.

**Licence**

The CD licence is custom:
- verbatim redistribution is allowed;
- a derived work must say that it is derived and is not the original;
- a derived CD must use a different CDName, for example `private_mathN` instead of `mathN`.

There is also the OpenMath royalty-free IPR policy of 2013-12-01: https://openmath.org/documents/openmath-ipr/

**Take for math2**
- Use OpenMath symbol URIs (`cd/name`, for example `arith1/plus`) as **alignment targets** for concept nodes, next to Wikidata QIDs. They are stable and well defined, they include branch-cut conventions, and they are what Content MathML refers to.
- Do not copy CD text into nodes, because of the renaming requirement. Link only.

### 1.4 MathML 4 (presentation, content, intent)

**Status**
- The Editor's Draft is dated 24 October 2026: https://w3c.github.io/mathml/
- The latest published version is still a Working Draft (agrees with spec 01 §1.7). The w3.org/TR page returned HTTP 403 to my fetch, so I read the editor's draft and the charter instead.

**Charter**

The proposed 2026 charter is at https://w3c.github.io/mathml-docs/charter-2026.html.
- **Targets in the timeline:**
  - MathML 4 CR, May 2026;
  - a community intent repository, August 2026;
  - MathML Core Recommendation, December 2026;
  - MathML 4 Recommendation, March 2027.
- **Deliverables table:** Core Recommendation about Q4 2026; MathML 4 complete about Q1 2027.
- **The May 2026 CR target was missed.** As of 2026-10-26 the spec is still a WD.
- **Content MathML:** MathML 4 "largely excludes" new Content MathML work, which is deferred to MathML 5.
- **i18n review:** ran from 2026-03-18 to 2026-05-14. https://github.com/w3c/i18n-request/issues/299

**Intent**
- Syntax example: `intent="power($base,$exponent)"`, with `arg="base"` on the child elements.
- Intent concepts come from two lists in the w3c/mathml-docs repository:
  - a **Core** list: https://w3c.github.io/mathml-docs/intent-core-concepts/ (from search);
  - an **Open** list, to which contributions are invited and which "will never be complete" (Soiffer et al., https://arxiv.org/abs/2407.06720, from search).
- There is also a Core *property* list generated from `core-properties.yml`: https://w3c.github.io/mathml-docs/intent-core-properties/ (from search).
- Intent is meant for speech and assistive technology. It is not a semantic definition.

**LaTeX support**
- The LaTeX Project ships `latex-lab-mathintent` with `\MathMLintent` and `\MathMLarg` (v0.1c, 2026-01-16). https://ftp.icm.edu.pl/packages/tex/macros/latex-dev/required/latex-lab/latex-lab-mathintent.pdf

**Take for math2**
1. Each notation declaration (§1.2, §5) gets an optional `intent:` field holding a MathML intent concept name, from the Core list where one exists. The KaTeX build can then emit `intent` on the MathML it already produces (spec 13 §13.7). This costs very little and gives correct speech output for, say, $f^{-1}$ (inverse) versus $x^{-1}$ (power).
2. Do not build on Content MathML. W3C has deferred it.
3. Contribute missing concepts to the Open list upstream, instead of inventing private names.

### 1.5 LaTeXML

**What it is**
- Version 0.8.8.1 (Chocolatey package updated 2026-02-20: https://community.chocolatey.org/packages/latexml).
- Outputs: XML, HTML, MathML, ePub and JATS. https://github.com/brucemiller/LaTeXML
- The README licence badge reads as public domain / CC0. I did not read the LICENSE file, so the exact licence is **UNVERIFIED**.

**Rust port**
- arXiv:2605.16562 (Ginev, Caruso, Miller, Sank, Weiskoff; 15 May 2026) describes a Rust port that is in progress. It reports:
  - 75% of arXiv converts without errors, against a 90% target;
  - about half of roughly 6,000 user reports resolved;
  - initial work on MathML 4 intent.
- The paper uses the words "candidate recommendation draft" for MathML 4. That does not match the W3C status in §1.4 and looks like loose wording.

**Take for math2**
- LaTeXML is the converter for **ingesting** arXiv LaTeX (already the arXiv HTML path).
- math2 node math is restricted to KaTeX-renderable macros, so math2 does not need LaTeXML to render its own nodes. It needs LaTeXML only for the arXiv source side and possibly for the JATS export (§4.3).

### 1.6 PreTeXt

**What it is**
- An XML authoring vocabulary and conversion toolchain, under GPL (https://github.com/PreTeXtBook/pretext).
- Outputs (https://pretextbook.org/doc/guide/html/the-index.html):
  - HTML; PDF and print (via LaTeX or XSL-FO); EPUB and Kindle;
  - Runestone (interactive); Braille;
  - Reveal.js and Beamer slides; Jupyter notebooks; plain text and Markdown;
  - instructor versions; WeBWorK problem sets.

**Elements** (from the guide)
- **Theorem-like:** definition, theorem, corollary, claim, fact, algorithm.
- **Axiom-like:** assumption, axiom, conjecture, heuristic, hypothesis, principle.
- **Example-like:** example, problem, question. Each can have hint, answer and solution.
- **Remark-like** and **project-like** families also exist.
- **Exercises:**
  - `<exercise>` contains `<statement>`, `<hint>`, `<answer>` and `<solution>`;
  - `<exercisegroup>` groups exercises;
  - `<exercises>` is a whole division of exercises.
- **Proofs:** a proof can be detached from its theorem, and proofs can have `<case>`.
- **Interactive exercises:**
  - `<webwork>` takes a problem from the OPL or local PG source, reused by xml:id;
  - `<myopenmath problem="ID">`;
  - STACK;
  - Runestone types: Parsons, matching, fill-in, clickable.
- **Notation** (https://pretextbook.org/doc/guide/html/topic-notation.html):
  - a `<notation>` element contains `<usage>` (exactly one `<m>`) and `<description>`;
  - it is placed in a definition's metadata or in a paragraph's `<term>`, and it is **invisible where it is placed**;
  - `<notation-list/>` in an appendix generates a three-column table (usage, description, locator), in order of first appearance.
- I found no 2026 release notes.

**Take for math2**
1. **PreTeXt is the best single book target.** One PreTeXt file gives web, PDF, EPUB, braille and WeBWorK output, so math2 does not have to write five back ends. The `m2 book` compiler (§4.5) should emit PreTeXt XML first.
2. Most math2 node kinds map directly onto PreTeXt elements:
   - definition → `definition`; statement → `theorem` / `lemma` / `corollary`; conjecture → `conjecture`;
   - example / counterexample → `example`; proof → `proof`; exercise (new, §3.6) → `exercise`.
   - concept nodes and topic nodes become divisions and introductions.
3. Copy the **notation-list** model: a definition node declares the notation it introduces, and the book compiler builds the "List of notation" in reading order.

### 1.7 Lean Verso, Verso Blueprint, leanblueprint, doc-gen4

**Verso** (https://github.com/leanprover/verso; https://verso.lean-lang.org/)
- Licence: Apache-2.0. The vendored JavaScript is MIT.
- Genres: manual, blog/website, book, API docs, blueprint and paper.
- `xref.json` and `lake exe verso sync` give experimental cross-document references.
- Verso powers the Lean reference manual and lean-lang.org. L. de Moura moved his website to Verso on 2026-03-15: https://leodemoura.github.io/notes/2026-3-15-this-website-is-now-powered-by-verso/

**Mathlib docstrings in Verso** (all PRs by Vierkantor)
- mathlib4 #44635 (8 October 2026) enabled `doc.verso` in the first file.
- mathlib4 #45031 turned it on for 1,352 files, with no syntax change and identical doc-gen output.
  - It was merged on 20 October 2026, then reverted because of lean4#15817, a scoped-instance race.
  - The fix, lean4#15839, landed on 21 October.
  - Sources: https://github.com/leanprover-community/mathlib4/pull/45031 ; https://github.com/leanprover/lean4/issues/15817
- doc-gen4 does not render Verso docstrings yet, so they are converted back to Markdown.
- mathlib4 #44898 changed the Stacks/Kerodon cross-reference attribute so that it writes into the Verso docstring.
- **Consequence:** Mathlib docstring syntax is moving to Verso during 2026–27. Any math2 tool that scrapes Mathlib docstrings, for binding prose or for "Main definitions" sections, must handle both Markdown and Verso.

**Verso Blueprint** (https://github.com/leanprover/verso-blueprint; extends spec 01 §1.2)
- **Directives:** `:::definition "label" (lean := "Nat.add, Nat.succ")`, `:::theorem "label" (owner := …) (priority := …)`, `:::proof "label"`.
- **Dependencies:** written inline as `{uses "foo"}[]` or as `(uses := "foo, bar")`.
- **Lean side:** `@[blueprint "label"]` on Lean declarations. With `(autoDeps := true)`, edges are derived from Lean and recorded with origin "automatic".
- **Tooling:**
  - `--dump-manifest` writes a "semantic manifest";
  - `blueprint-html-cache.json` is a cache;
  - `lake exe vbp query status <label>` reports status and separates *known holes* from *unverified coverage*.
- **Catalogue:** the README lists `ejgallego/verso-flt` and `verso-sphere-packing` (Lean v4.35). Noperthedron and Carleson are excluded until they move from v4.34. v4.35 is the only maintained line.
- The FLT repository (https://github.com/ImperialCollegeLondon/FLT) contains **both** a `blueprint` folder and a `blueprint-verso` folder. So spec 01 §1.2's statement that "FLT, Carleson, Sphere Packing have moved to it" is too strong: FLT runs both, and Carleson is not yet in the catalogue (see Doubts).
- I could not fetch the licence (the raw LICENSE returned 404). **UNVERIFIED.**

**leanblueprint** (https://github.com/PatrickMassot/leanblueprint)
- A plasTeX plugin plus `plastexdepgraph`.
- Macros: `\lean`, `\leanok`, `\uses`, `\mathlibok`, `\notready`, `\proves`, `\discussion`.
- `checkdecls` checks that the Lean declaration names exist.
- Node colours: not_ready orange; can_prove blue; proved green; fully_proved dark green; mathlib dark green.
- **Pitfalls for importers:** a node that `\uses` itself causes a RecursionError; a remark inside `\uses` stops its dependants from becoming fully_proved.
- **LeanArchitect** (ITP 2026, LIPIcs 382, paper 25: https://drops.dagstuhl.de/storage/00lipics/lipics-vol382-itp2026/LIPIcs.ITP.2026.25/LIPIcs.ITP.2026.25.pdf) fills these macros automatically from Lean. There, `\leanok` means that `sorryAx` is absent from the declaration's closure.
- KnowTeX (arXiv:2601.15294) is a related tool. I did not read it in detail.

**doc-gen4** (https://github.com/leanprover/doc-gen4)
- Apache-2.0. Built with `lake build Lib:docs`. Environment variables include `DOCGEN_SRC` and `DISABLE_EQUATIONS`.
- I did not confirm whether it publishes machine-readable declaration data that math2 could consume. **UNVERIFIED.**

**Mathlib documentation style** (https://leanprover-community.github.io/contribute/doc.html)
- Math is written `$…$` or `$$…$$`, or `\begin{…}` environments without dollars, to match the MathJax configuration of the online docs.
- A module docstring has these sections, in order: **Main definitions, Main statements, Notation** (required if the file introduces notation), **Implementation notes, References, Tags**.
- References are keys into `docs/references.bib`, for example `[Boole1854]`.
- Named theorems are written in bold.
- The `docBlame` and `docBlameThm` linters enforce docstrings.

**Take for math2**
1. **Import:** `m2 ingest blueprint` (spec 09.4) should read the Verso Blueprint semantic manifest when one exists, and fall back to leanblueprint's plasTeX output. Keep the "automatic" versus "manual" edge origin as edge provenance. Map "known hole" and "unverified coverage" onto evidence levels without collapsing them.
2. **Export:** math2 topic nodes with Lean bindings can be emitted as a Verso Blueprint document (`:::definition "TAG" (lean := …)`). The math2 tag is the blueprint label, and `uses` comes from the graph. This gives a Lean project a blueprint generated by math2.
3. Model math2's per-file conventions on Mathlib's module-doc sections. A math2 topic node should have **Main definitions, Main statements, Notation, Conventions, References** sections. That matches what formal-library readers already expect.
4. Spec 21's open question on Verso documents: accept Verso only as an **export** format, not as a node source format. Node bodies stay Markdown + LaTeX (spec 04).

### 1.8 Wikidata formulas and Wikifunctions

**Wikidata properties**
- P2534 "defining formula".
- P7235 "in defining formula": one statement per symbol, with the qualifier P9758 "symbol represents". https://www.wikidata.org/wiki/Property:P7235
- P7973 "quantity symbol (LaTeX)".
- The `math` datatype stores LaTeX strings.
- Older usage counts: P2534 about 5,166 uses and P7235 about 703 (Scharpf et al. 2021, https://gipplab.uni-goettingen.de/wp-content/papercite-data/pdf/scharpf2021b.pdf). Current counts are not checked here; spec 01 §1.7 has the 2026 totals.
- A 2026 dataset of 2,405 formulae: arXiv:2608.02457 (title not recorded here).

**Wikifunctions**
- The Abstract Wikipedia beta started in the week before 2026-03-26: https://www.wikifunctions.org/wiki/Wikifunctions:Status_updates/2026-03-26. The exact date of 20 March is from search only.
- Wikifunctions passed 4,000 functions in mid-April 2026. The 4,000th was subtraction of complex numbers. https://www.wikifunctions.org/wiki/Special:MyLanguage/Wikifunctions:Status_updates/2026-04-16
- Its math functions are few and elementary, for example Z32261, congruence mod n.

**Take for math2**
1. The **P7235 + P9758 pattern** (one statement per symbol in a formula, naming the item the symbol stands for) is the same as a math2 notation declaration attached to a formula. Export math2 symbol declarations to it when aligning to Wikidata.
2. Wikifunctions is of low relevance for now. At most, a `computation` node could link to a Wikifunctions ZID as a further implementation.

### 1.9 Naproche and ForTheL

**What it is**
- Licence: GPL-3.0. https://github.com/naproche/naproche
- It uses the automated provers E, Vampire and SPASS.
- Input is controlled natural language inside `\begin{forthel}` environments in LaTeX.
- It was left out of Isabelle 2025 because of folder-naming errors (ITP 2025 perfectoid-rings paper, https://drops.dagstuhl.de/storage/00lipics/lipics-vol352-itp2025/LIPIcs.ITP.2025.6/LIPIcs.ITP.2025.6.pdf).
- Naproche-ZF is a grammar-based successor.
- Visored (arXiv:2606.17581) says ForTheL → Lean has been a stated direction since 2020 but is not in production.

**Take for math2**
- Controlled natural language is a useful **style target** for the Statement section: one claim per sentence, explicit quantifiers, defined terms only. A lint rule can check "every content word in the Statement is a defined term or a variable" without adopting ForTheL itself.
- Do not take a dependency on Naproche.

---

## 2. Levels of explanation and pedagogy

### 2.1 Stages of mathematical maturity

T. Tao describes three stages (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/; the 2009 date is from search, **UNVERIFIED**):
- **pre-rigorous:** intuition, examples, computation;
- **rigorous:** formal definitions and proofs;
- **post-rigorous:** intuition again, now backed by rigour.

**Take:** spec 04's L1–L4 levels are defined by audience. The spec should also say what each level is *for*:
- L1 is pre-rigorous: picture, example, why anyone cares.
- L2 is rigorous: the definition unpacked, standard examples and non-examples.
- L3 and L4 are post-rigorous: the right way to think about it, how it generalises, where it breaks.

This gives the explanation agent and its reviewers a test that does not depend on audience labels alone.

### 2.2 Bloom and the MATH taxonomy

The MATH taxonomy is from Smith, Wood, Coupland, Stephenson, Crawford and Ball (1996), *IJMEST* 27(1):65–77. Sources: https://opus.lib.uts.edu.au/bitstream/10453/6622/1/2003001060.pdf ; https://bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-33-1-02.pdf

It is a variant of Bloom made for mathematics. It has three groups:
- **Group A:** factual knowledge; comprehension; routine use of procedures.
- **Group B:** information transfer; application in new situations.
- **Group C:** justification and interpretation; implications, conjectures and comparisons; evaluation.

The groups describe "the nature of the activity … not the degree of difficulty". One study cited in the BSRLM paper found that Group A tasks carried 67 of the 75 marks on an A-level paper.

**Take:** each exercise node (§3.6) carries a `math_group: A|B|C` field, and optionally a subcategory. Coverage reports per topic can then show "no Group C exercises", which is a typical gap in generated banks.

### 2.3 Worked examples and their evidence

- **Meta-analysis:** Barbieri, Miller-Cotto, Clerjuste and Chawla (2023), *Educational Psychology Review* 35(1), article 11, doi:10.1007/s10648-023-09745-1 (https://eric.ed.gov/?id=EJ1364058). Findings:
  - worked examples in mathematics have an overall effect of g = 0.48;
  - correct examples alone did better than incorrect or mixed correct/incorrect examples;
  - self-explanation prompts moderated the effect *negatively*.
- **Expertise reversal** (Kalyuga et al. 2003): worked examples help novices but can hurt advanced learners.
- **Fading:** Renkl and Atkinson 2003, and Renkl et al. 2002. Worked steps are removed one at a time until the learner solves the whole problem.
- **Adaptive fading** beats fixed fading: Salden et al. 2010, https://link.springer.com/article/10.1007/s11251-009-9102-0
- **Self-explanation of proofs:** Hodds, Alcock and Inglis (2014), *JRME* 45(1):62–101 (https://eric.ed.gov/?id=EJ1028348). Self-explanation *training* for reading proofs gave d = 0.95, and a 15-minute intervention had a lasting effect.
  - This is different from the meta-analysis above. A trained reading strategy helps; prompts attached to worked examples did not.

**Take:**
1. An `example` node should be able to hold a **worked solution as a numbered list of steps**, so that a renderer can fade them: show steps 1 to k and ask for the rest.
2. Explanations at L1 and L2 should prefer **correct** worked examples. Incorrect "find the error" examples belong in exercises, marked as such.
3. Following expertise reversal, L3 and L4 explanations should *not* include full worked computations by default.
4. A short "how to read a proof" guide (after Hodds et al.) can be one topic-level explanation node, linked from proof pages. It should not be repeated on every proof.

### 2.4 Examples, non-examples and concept images

- **Concept image versus concept definition:** Tall and Vinner (1981), *ESM* 12:151–169.
- **Example spaces:** Watson and Mason (2002/2005). "Ideal non-examples are those that are very nearly examples." https://mrbartonmaths.com/resourcesnew/8.%20Research/Inquries/Extending%20Example%20Spaces%20-%20Watson%20and%20Mason.pdf
- **"Mis-in" and "mis-out" examples** (ESM 2023, https://link.springer.com/article/10.1007/s10649-023-10273-6). These are examples that learners wrongly include in a concept, or wrongly leave out of it.
- **Definition–theorem–proof format:** Weber (2004), *JMB* 23(2):115–133, studies teaching in this format (cited via https://sites.math.rutgers.edu/~jpmejia/files/Lew_JRME.pdf).

**Take:**
1. spec 04 has `counterexample` for statements. It also needs a **non-example** role for concepts and definitions: an object that fails exactly one clause of the definition. The edge should name the clause that fails. This is cheap to generate and to check, and it is what the research says helps most.
2. An example should declare which definition clauses it exercises. Then "is every clause covered by a near-miss non-example?" becomes a computable coverage check.

### 2.5 Structure of an entry and of a proof

- **nLab:** its page template has the sections **Idea, Definition, Properties, Examples, Related concepts, References**, plus redirects for synonyms. https://ncatlab.org/nlab/show/Template+page ; https://ncatlab.org/nlab/show/writing+in+the+nLab
- **Lamport:** "How to Write a 21st Century Proof", *J. Fixed Point Theory Appl.* 11:43–63 (2012) (https://lamport.azurewebsites.net/pubs/proof.pdf). Proofs are hierarchical numbered steps (⟨1⟩1, ⟨2⟩1 …), each with its own justification, readable at any depth.
- Leron, "Structuring Mathematical Proofs", *AMM* 90(3), 1983, describes levelled proofs. The citation details are **UNVERIFIED**.
- **Stacks Project rules:** each definition marks the terms it defines in `{\it …}`; references go in a `reference` environment. https://raw.githubusercontent.com/stacks/stacks-project/master/documentation/rules

**Take:**
1. spec 04's sections are close to nLab's. Add an **Idea** (motivation) section to concept nodes as the leading paragraph, kept separate from Explanations. nLab shows that a one-paragraph "Idea" is the most-read part of a page.
2. A proof body may be a **structured proof**: nested numbered steps in Lamport style, each step with its own justification links. The "Unfold this proof" feature (spec 13 §13.4) already assumes a step hierarchy for formal proofs, and this gives the same hierarchy to informal proofs. Use a Markdown ordered-list convention, not a new syntax.
3. Require the **definiendum** to be marked in definition bodies (Stacks `{\it}`, sTeX `\definiendum`). Then the linter can check that the defined term is the node's name or a declared synonym.

---

## 3. Exercises

### 3.1 WeBWorK Open Problem Library (OPL)

- **Licence:** CC BY-NC-SA 3.0 by default. Copyright stays with the authors, who may choose an OSI licence instead. https://github.com/openwebwork/webwork-open-problem-library/blob/main/OPL_LICENSE
- **Size:** about 35,000 problems. https://wiki.openwebwork.org/wiki/Open_Problem_Library
- **Tags on each problem:**
  - DBsubject / DBchapter / DBsection (taxonomy);
  - Date, Author, Institution;
  - MLT (more-like-this groups);
  - Level;
  - KEYWORDS;
  - textbook tags (title, edition, section, problem number).
- **Workflow:** problems go into `Contrib` first and are reviewed into the OPL.
- Problems are written in PG/PGML (Perl). Answers are checked by the PG answer evaluators.

### 3.2 MyOpenMath and IMathAS

- **IMathAS Community License:**
  - question *code* is GPL 2.0;
  - the generated HTML of a question must be CC BY 4.0.
  - https://www.imathas.com/communitylicense.html
- MyOpenMath has been run by a non-profit since 2018.

### 3.3 STACK and Numbas

- **STACK:**
  - GPL; uses the Maxima CAS; grades answers with "potential response trees";
  - v4.11.1 in January 2026;
  - the Moodle plugin page's latest release is dated 16 October 2026: https://marketplace.moodle.com/plugins/qtype_stack
- **Numbas:**
  - code Apache-2.0, © Newcastle University;
  - content stays with its authors, and public questions in the Numbas editor are under Creative Commons;
  - packages are exported as SCORM 2004;
  - expressions are written in its JME language.
  - https://docs.numbas.org.uk/en/latest/index.html

### 3.4 QTI 3.0

- The QTI v3.0.1 information model supports **MathML 3 only**; MathML 2 was removed.
- Portable Custom Interactions (PCI) are AMD modules; `customInteraction` is deprecated.
- The implementation guide §3.7.1 covers CAS-style custom response processing (Maxima-style).
- MathML support is required only for **Advanced** conformance.
- https://www.imsglobal.org/spec/qti/v3p0/impl ; https://www.imsglobal.org/spec/qti/v3p0/conf

### 3.5 Lean exercises

- **NNG4** (Natural Number Game) is Apache-2.0: https://github.com/leanprover-community/NNG4
- **lean4game**, the game engine, is GPL-3.0. Games are hosted at adam.math.hhu.de. They include:
  - Knights and Knaves, Linear Algebra, Logic, NNG, Robo;
  - Real Analysis (Kontorovich);
  - Reintroduction to Proofs (Riehl);
  - Set Theory (Velleman).
- **Mathematics in Lean (MIL):** Apache-2.0. Release v4.19.0, with a PDF dated 21 October 2026: https://leanprover-community.github.io/mathematics_in_lean/mathematics_in_lean.pdf. Exercises are Lean files in the `MIL` folder, with solutions kept separately.
- **Autograding:**
  - `robertylewis/lean4-autograder-main` uses the attributes `@[autogradedProof pts]` and `@[legalAxiom]`, with a Gradescope integration. https://github.com/robertylewis/lean4-autograder-main
  - `isun3071/lean-4-teaching` checks submissions with Comparator and supports tactic allowlists.
- **Verso game prototype:** https://github.com/pfaffelh/nng-verso

### 3.6 Open textbooks and their licences

| Book | Licence | Source |
|---|---|---|
| Judson, *Abstract Algebra: Theory and Applications* | GFDL 1.2 or later | http://abstract.ups.edu/ |
| Beezer, *A First Course in Linear Algebra* | GFDL | open.umn.edu (Open Textbook Library) |
| Hefferon, *Linear Algebra* | CC BY-SA | from search |
| OpenStax *Calculus* Vol. 1 | CC BY-NC-SA 4.0 | https://openstax.org/details/books/calculus-volume-1 |
| APEX Calculus | CC BY-NC 4.0 | from search |
| *Active Calculus* (single variable, 2018 and later) | CC BY-SA 4.0. Sources conflict; **UNVERIFIED** | from search |
| *Active Calculus* multivariable | CC BY-NC-SA | from search |

### 3.7 MyST / Jupyter Book 2

- The `exercise` and `solution` directives: a solution refers to its exercise by label. https://mystmd.org/guide/exercises
- `prf:` / `proof:kind` directives for theorem-like blocks (mystmd 1.11): https://mystmd.org/guide/proofs-and-theorems

### 3.8 Licence consequences for math2

Exercise banks are licensed differently from each other and from math2 prose. Spec 16 §16.6 says node prose is CC BY-SA 4.0 and Lean code is Apache-2.0. It already says GFDL sources are linked, never copied, and MSC text is never stored.
- OPL default: NC-SA. **Not compatible** with BY-SA.
- OpenStax: NC-SA. Not compatible.
- APEX: NC. Not compatible.
- IMathAS: code GPL-2.0, rendered output CC BY. Rendered text is fine; the code is GPL.
- GFDL books: not compatible with CC BY-SA in either direction, unless dual-licensed.
- Hefferon (BY-SA): compatible.
- NNG4 and MIL (Apache-2.0): fine for Lean code.

**Take:** math2 must **not copy** NC-licensed problems into nodes. It may **link** to an OPL path or a MyOpenMath ID as an external "interactive version". The exercise node itself is then math2's own text.

### 3.9 Take for math2: an `exercise` node kind

PreTeXt, OPL, MyST and MIL all treat an exercise as a first-class object with statement, hint, answer and solution. math2 should do the same.

**Body sections:** Statement, Hints (ordered), Answer, Solution (structured steps, §2.5).

**Front matter:**
- `exercises: [TAG…]`: the concepts or statements it practises. This edge gives schema.org `assesses` (§4.4).
- `math_group: A|B|C` (§2.2).
- `level: L1–L4`.
- `check:`, one of:
  - `lean`: a statement with `sorry`, graded by Comparator or the autograder, with a tactic allowlist;
  - `cas`: an answer expression plus an equivalence test, exportable to STACK, Numbas or PG;
  - `choice` / `numeric`: exportable to QTI 3;
  - `none`: a proof exercise checked by people.
- `external: [{system: webwork-opl, path: …}, {system: myopenmath, id: …}]`, as links only.

**Checking:** an exercise with `check: lean` gets evidence like a statement does. The reference solution must compile, and the `sorry` version must be the same type. This is the machine-checkable answer format that the task asks for. CAS answers are checked by running the CAS equivalence on the reference answer in CI.

---

## 4. Packaging

### 4.1 Book toolchains (status as of 2026-10-26)

| Tool | Status / version | Licence | Notes |
|---|---|---|---|
| PreTeXt | active; no 2026 release notes found | GPL | §1.6; the best multi-output target |
| Quarto | 1.8 (2025-10-13: https://quarto.org/docs/blog/posts/2025-10-13-1.8-release/); 1.9 adds Typst books ("orange-book"; blog 2026-03-31: https://quarto.org/docs/blog/posts/2026-03-31-typst-books-and-more/) | GPL (from Wikipedia) | theorems via `#thm-` cross-ref divs; amsthm in LaTeX output. A Rust "Quarto 2" for late 2026 is **UNVERIFIED** (search only) |
| Typst | 0.14 (2025-10-24, PDF/UA-1); 0.15.0 (2026-06-15, MathML export); 0.15.1 on 2026-07-17 per search (**UNVERIFIED**) | Apache-2.0 (https://typst.app/open-source/) | fast; its math syntax is not LaTeX, so a converter is needed |
| Pandoc | 3.11 (2026-08-29) | GPL (well known; not re-checked) | "mathml is now the default math-method" (https://github.com/jgm/pandoc/discussions/11835); `--math-method` |
| mdBook | 0.5.0 (2025-11-17); 0.5.4 (2026-07-06) | MPL-2.0 (not re-checked, **UNVERIFIED**) | `mathjax-support`; does not accept `$$` delimiters (https://github.com/rust-lang/mdbook/releases) |
| Jupyter Book 2 / MyST | mystmd 1.11 | MIT (not re-checked, **UNVERIFIED**) | `prf:` theorem directives and exercise/solution (§3.7) |
| LaTeX | TeX Live 2025 on arXiv | LPPL | the arXiv target (§4.3) |

**Take:**
- Emit **PreTeXt** for books.
- Emit **LaTeX** (amsthm + `m2.sty`) for arXiv and print.
- Emit **Typst** as a third option only after the macro → Typst mapping exists. That mapping can live in the notation register (§5), one Typst form per macro.
- Pandoc and Quarto are not needed as primary targets, since PreTeXt covers their outputs. Quarto is a fallback for notebook-style computation nodes.
- mdBook's lack of `$$` makes it a poor fit.

### 4.2 Course standards

- **Common Cartridge 1.3** (July 2013; https://www.imsglobal.org/cc/index.html):
  - carries web content, QTI assessments (six question types), LTI links and discussions;
  - "Thin CC" carries links only;
  - CC is "not designed as a replacement for SCORM".
  - I found nothing newer than 1.3. Whether a later version exists is **UNVERIFIED**.
- **LTI 1.3 Core:** dated 16 April 2019. Earlier LTI versions lost support on 30 June 2022.
- **xAPI:** IEEE 9274.1.1-2023 (published 2023-10-06; https://standards.ieee.org/ieee/9274.1.1/7321/), adopted as ISO/IEC/IEEE 39274-1-1:2025.
- **cmi5:** the xAPI profile for LMS launch, dated 2016-06-01.
- **SCORM 2004:** what Numbas exports (§3.3).

**Take:**
- A math2 "course" export is a **Common Cartridge 1.3** containing:
  - HTML pages generated per node;
  - QTI 3 items, converted to CC's QTI subset, from exercises with `check: choice|numeric`;
  - LTI links back to the live workspace for Lean and CAS exercises.
- **xAPI** statements are optional and live in the workspace (spec 13), never in `tamnd/math2`.
- Do **not** implement SCORM.

### 4.3 Article formats

- **JATS 1.4** = ANSI/NISO Z39.96-2024, approved 31 October 2024. It prefers MathML 3, allows `<tex-math notation="LaTeX" version="LaTeX2e">`, and allows both inside `<alternatives>`. https://jats.nlm.nih.gov/archiving/tag-library/1.4/element/tex-math.html
- **JATS4R** recommends math-mode LaTeX without `$` delimiters inside `<tex-math>`. https://jats4r.niso.org/math/
- **arXiv** (https://info.arxiv.org/help/00README.html):
  - TeX Live 2025 is the default; 2023 is the fallback.
  - Submission system 1.5 (April 2025).
  - `00README.json` keys: `process.compiler`, `sources`, `texlive_version`, `stamp`, `nohyperref`, `spec_version`.
  - Only packages in TeX Live are available, so `m2.sty` must be **bundled** in the submission.

**Take:**
- `m2 article` emits an arXiv-ready source tree:
  - `main.tex` with `m2.sty` bundled;
  - `00README.json` naming the compiler and the TeX Live year;
  - a `.bib` built from node Sources.
- It also emits JATS 1.4 with `<alternatives>` holding both `<tex-math>` (no delimiters) and MathML, the MathML coming from KaTeX at build time.
- Each theorem in the article carries its `m2://TAG@release` as a footnote or a link, so readers can reach the checked node.

### 4.4 Metadata: schema.org LearningResource and LRMI

**schema.org LearningResource** (https://schema.org/LearningResource)
- Properties include `teaches`, `assesses`, `competencyRequired`, `educationalAlignment`, `educationalLevel`, `educationalUse` and `learningResourceType`.
- Subtypes include Course, Quiz and Syllabus.
- The type is in schema.org's "new" area.

**LRMI** (https://www.dublincore.org/specifications/lrmi/)
- LRMI 1.1. DCMI has maintained it since 2014.
- It publishes LRMI Terms and SKOS concept schemes, for example for learning resource types.

**Learning Commons Knowledge Graph** (Chan Zuckerberg Initiative)
- Licence: code MIT, data CC BY 4.0. https://docs.learningcommons.org/knowledge-graph/resources/license
- It models learning components, prerequisite relations between standards, and the Eedi misconception graph.
- Its open platform launched on 2026-09-22. I read this date on the docs site; the details there are from my reading of 2026-10-26.

**Take:** the static site (spec 13) embeds JSON-LD on each page, mapped as follows.

| schema.org property | Source in math2 |
|---|---|
| `@type` | `LearningResource` for explanation and exercise views; `DefinedTerm` for concepts |
| `teaches` | the node's own concept tags |
| `competencyRequired` | the direct `uses` / prerequisite closure, depth 1 |
| `assesses` | the `exercises:` edge of exercise nodes |
| `educationalLevel` | the L1–L4 level, mapped to a published SKOS scheme rather than free text |
| `learningResourceType` | an LRMI term such as "worked example" or "exercise" |

**Learning Commons:** it is a possible alignment target for school-level (L1) concepts, and it supplies a misconception graph. That suggests a `misconception` role or edge for non-examples (§2.4).

### 4.5 Prerequisite graphs and datasets

- **Survey:** prerequisite relation learning, doi:10.1145/3733593 (ACM, 2025). I read the abstract only.
- **Datasets:**
  - RefD Math: 65 knowledge components, 658 pairs, 75 positive;
  - LectureBank;
  - MOOCCube / MOOCCubeX.
  - All are small or tied to particular courses. None is a usable source of math2 edges; they are useful only as evaluation sets.
- **Metacademy** (https://github.com/metacademy/metacademy-application/wiki/Alternative-learning-paths):
  - 487 machine-learning concepts and 1,208 prerequisite pairs;
  - each concept has **a single dependency set**, which the project itself names as a limitation, since there are alternative paths.
- **Knowledge space theory / ALEKS:**
  - the **outer fringe** of a learner's knowledge state is the set of items they are "ready to learn";
  - a learner succeeds on outer-fringe items with probability about 0.93;
  - https://www.sciencedirect.com/science/article/abs/pii/S0022249621000134 ; arXiv:1511.06757
- **PAIGOS** (Ullrich 2008, LNAI 5260, doi:10.1007/978-3-540-88215-2):
  - generates courses from OMDoc content by HTN planning over an ontology of instructional objects (definition, example, exercise, and so on);
  - outputs an IMS manifest;
  - the closest published predecessor of a "graph → course" compiler.
- **Recent graph-to-path work:**
  - AutoMathKG (arXiv:2505.13406);
  - GraphRAG learning paths (arXiv:2506.22303).
  - I found **no** published system that generates a full mathematics textbook from a knowledge graph.

**Take:**
1. **Alternative prerequisite sets.** A definition can be reached from different backgrounds. A concept can have several definitions (Realms, §4.6), each with its own `uses` set. So the prerequisite path in spec 13 should be computed over a *chosen definition per concept*, not over the union of all of them. This fixes Metacademy's limitation in a natural way.
2. **"Ready to learn".** Given a reader background B (spec 13 §13.5), the outer fringe is the set of nodes whose prerequisites are all in B. That is a one-line graph query, and it gives a principled "what next" list.
3. **Evaluation sets.** Use RefD Math / LectureBank style labelled pairs only to measure the edge-inference agent, never as content.

### 4.6 Turning a graph into a linear text

**Ordering**
- Choosing a linear order that minimises forward references is an instance of the minimum feedback arc set problem, which is NP-hard. arXiv:2304.15006 discusses this for documents.
- For a DAG, a topological order has zero forward references in proofs. The real choice is *among* topological orders, and that choice is about locality, keeping related nodes close.
- **Stacks rule:** "Proofs do not refer to results later in the stacks project." https://raw.githubusercontent.com/stacks/stacks-project/master/documentation/rules
- Each Stacks chapter opens with a **Conventions** section (for example tag 0439).
- Stacks tag 06G9 admits that notation conflicts between chapters. That is an example of local conventions being preferred over global consistency.

**Choosing among equivalent definitions**
- Realms (Carette, Farmer, Kohlhase, CICM 2014, LNCS 8543:252–266, arXiv:1405.5956) treat a concept with several equivalent axiomatisations as a **realm**: theories joined by views that prove them equivalent, plus a single external **interface**.
- Follow-up: "Lightweight Realms" (2025), https://link.springer.com/chapter/10.1007/978-3-032-07021-0_13
- Background: the "little theories" method (Farmer et al. 1992).
- **A concrete case:** Mathlib defines `IsCompact` with filters:
  `∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f`
  (https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Defs/Filter.html). Most textbooks use open covers. A book generated from math2 must choose the open-cover definition and present the filter one as a theorem, while the Lean binding goes to the filter one.

**Take: an `m2 book` compiler**

The input is a book manifest `books/<id>.yaml` in the content tree. It contains:
- `targets`: tags, or a topic;
- `background`: the reader background, i.e. the nodes the book assumes;
- `choose`: for each concept with more than one definition node, which one is primary in this book. The others become "equivalent characterisations" theorems, and the equivalence-proof nodes are pulled in;
- `notation`: per-book overrides of default notations (§5);
- `level`: which explanation level to inline;
- `format`: `pretext`, `latex`, `typst`, `jats` or `cc`.

The algorithm:
1. Compute the closure of `targets` minus `background`, using only the chosen definitions' `uses`.
2. Topologically sort it, breaking ties first by the order of the topic nodes' author-given outline, then by keeping a statement next to its proof and examples, then by MSC.
3. Reject the manifest if a proof would refer forward (the Stacks rule). Never silently reorder it.
4. Generate a **Conventions** section from the concept nodes' recorded conventions, and a **List of notation** (PreTeXt `notation-list`).
5. Write `book.lock` with the exact `TAG@release` of every included node, so that the book is reproducible and citable.

The manifest and the lock are the "published method" here. The decisions are recorded data, not hidden heuristics.

---

## 5. Notation

### 5.1 How large projects keep notation consistent

- **Mathlib:**
  - Notation is declared in Lean, often *scoped*. It is enabled with `open scoped X`, for example `open scoped PiNotation`.
  - `notation3` generates delaborators, so that the notation is also printed back: https://leanprover-community.github.io/mathlib4_docs/Mathlib/Util/Notation3.html
  - mathlib4 PR #45028 proposes renaming `notation3` to `notation +delab`. Its merge status is unclear.
  - Every file that introduces notation must have a "Notation" section in its module doc (§1.7).
- **Stacks Project:** a Conventions section per chapter, and tolerance of documented conflicts between chapters (tag 06G9).
- **sTeX:** the symbol is the identity, and notations are attached to it. Variant notations are selectable per document (§1.2).
- **PreTeXt:** `<notation>` with usage and description, attached where the notation is introduced, and collected into a notation list (§1.6).
- **Wikipedia MOS:MATH:** keep a list of "Mathematical conventions" (for example: does "compact" include Hausdorff? are rings unital?) and require "consistency within an article". https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Mathematics
  - arXiv:2412.20419 notes that consistency *across* articles is "too much to ask" on Wikipedia.
- **ISO 80000-2:2019:** ℕ = {0, 1, 2, …}, and ℕ* denotes the positive integers. https://cdn.standards.iteh.ai/samples/64973/c49664e1b9364d1d811246de9d1db4aa/ISO-80000-2-2019.pdf

### 5.2 Take for math2: a notation register

Replace the flat macro list (`macros/m2.sty` + `m2.katex.json`) with a **source register** `macros/notation.yaml` from which both files are generated. Each entry has these fields:

```yaml
- macro: \Nat              # the LaTeX command (the only thing a node body uses)
  denotes: 4M2XKQ          # concept tag (sTeX \symdecl identity)
  args: 0
  prec: null               # for operators, as sTeX prec=
  latex: \mathbb{N}        # expansion for m2.sty and KaTeX
  typst: NN                # for the Typst target (§4.1)
  intent: natural-numbers  # MathML intent concept, Core or Open list (§1.4); null if none
  openmath: setname1/N     # alignment (§1.3), optional
  convention: "includes 0 (ISO 80000-2)"  # one line, shown in the notation list
  variants:                # alternative notations for the same concept
    - {name: N-bold, latex: \mathbf{N}}
  introduced_by: 9QF3ZD    # definition node that introduces it (PreTeXt <notation> placement)
```

New lint rules:
1. Every macro used in a node body exists in the register. This extends B01.
2. A definition node that introduces a symbol has a register entry with `introduced_by` pointing to it.
3. One concept has exactly one default notation. Variants are allowed only through a book or page override (spec 13's "Translate notation" then becomes a lookup in this register).
4. Overloaded symbols, such as $\subset$ or $(a,b)$ for pair, interval and gcd, must use distinct macros: `\subset` versus `\subsetneq`, `\pair`, `\ointerval`, `\gcd`. Each macro has a single `denotes`. This is the sTeX principle, and it is what makes rendering by convention and intent emission possible.

**Conventions:**
- A concept node's front matter gets `conventions:`, a list of short statements such as "ring: unital, not necessarily commutative" or "ℕ: includes 0".
- The default follows ISO 80000-2 where ISO says anything.
- The book compiler (§4.6) collects these into the book's Conventions section, Stacks style.
- When a statement depends on a convention, for example on whether 0 ∈ ℕ, it must link the concept that carries the convention, so that a change of convention can find every statement it affects.

---

## 6. What the spec should adopt

Ordered by value for effort.

1. **Notation register** (§5.2): a symbol-identity + notation split (sTeX), with `denotes`, arity, precedence, Typst form, MathML `intent` and OpenMath alignment per macro. `m2.sty` and `m2.katex.json` are generated from it. Conventions are recorded on concept nodes, with ISO 80000-2 as the default.
2. **`exercise` node kind** (§3.9): Statement / Hints / Answer / Solution; an `exercises:` edge; `math_group` (MATH taxonomy A/B/C); and a `check:` of `lean` (with a `sorry` target, graded via Comparator or the autograder), `cas`, `choice`, `numeric` or `none`. NC-licensed banks (OPL, OpenStax, APEX) are linked only, never copied.
3. **Non-example role** (§2.4) that names the definition clause that fails. Examples declare the clauses they exercise, and clause coverage is a computed report.
4. **Pedagogy rules for explanations** (§2.1–2.3):
   - L1–L4 map onto Tao's pre-rigorous / rigorous / post-rigorous stages;
   - prefer correct worked examples at L1–L2, as numbered steps that a renderer can fade;
   - no full worked computations at L3–L4 (expertise reversal);
   - an Idea paragraph on concept nodes (nLab);
   - optional Lamport-style structured proofs.
5. **`m2 book` compiler with a manifest and a lock** (§4.6):
   - the closure of the targets minus the reader background;
   - one *chosen* definition per concept (Realms), with the others becoming equivalence theorems;
   - a topological order that keeps the author's outline and refuses forward references in proofs (Stacks rule);
   - a generated Conventions section and List of notation;
   - `book.lock` pinning `TAG@release`.
6. **Output targets, in this order:**
   1. PreTeXt XML (web, PDF, EPUB, braille, WeBWorK in one step);
   2. LaTeX + bundled `m2.sty` + `00README.json` for arXiv;
   3. JATS 1.4 with `<alternatives>` (tex-math without delimiters, plus MathML);
   4. Common Cartridge 1.3 with QTI items and LTI links for courses;
   5. Verso Blueprint for Lean projects;
   6. Typst later.
   - No SCORM. xAPI only in the workspace.
7. **JSON-LD on every static page** (§4.4): schema.org `LearningResource` / `DefinedTerm` with `teaches`, `competencyRequired` (from `uses`), `assesses` (from `exercises:`), and `educationalLevel` from a SKOS scheme for L1–L4.
8. **Prerequisite paths per chosen definition**, plus the KST "outer fringe" as the "ready to learn" query (§4.5).
9. **Blueprint import and export** (§1.7):
   - read Verso Blueprint's semantic manifest and keep the automatic/manual edge origin and the hole/unverified distinction;
   - read leanblueprint as a fallback, guarding against self-`\uses` and remarks in `\uses`;
   - accept Verso only as an export, not as a node source format (answers spec 21).
10. **Alignment targets:** add OpenMath CD symbols and MathML intent concept names next to Wikidata QIDs. Export symbol declarations in the Wikidata P7235 + P9758 pattern.
11. **Topic nodes follow Mathlib's module-doc sections** (Main definitions, Main statements, Notation, Conventions, References), and any scraper of Mathlib docstrings handles Verso docstrings as well as Markdown.
12. **Correct spec 01 §1.2.** Say that FLT keeps both a leanblueprint and a Verso blueprint, and that only verso-flt and verso-sphere-packing are in the Verso Blueprint catalogue (Carleson is pending a toolchain bump). "Moved to it" is too strong.

---

## 7. Open doubts

1. **MathML 4 status.** The charter's May 2026 CR target was missed, and the spec is still a WD with an editor's draft of 24 October 2026. The new CR date is unknown. w3.org/TR returned 403, so I could not read the TR page directly.
2. **Verso Blueprint licence:** UNVERIFIED (LICENSE fetch returned 404).
3. **Which projects have moved to Verso Blueprint.** FLT has both folders; Carleson and Noperthedron are excluded from the catalogue until they move to v4.35. Spec 01 §1.2's "FLT, Carleson, Sphere Packing have moved" needs rechecking.
4. **LaTeXML licence:** the badge suggests CC0 or public domain (NIST work). I did not read the LICENSE file.
5. **MMT licence** (COPYING.txt): not read.
6. **Licences of SMGloM archives other than `smglom/mathsoft`:** not checked. The old SPL 0.1 may still apply to some of them.
7. **Active Calculus licence:** sources conflict between CC BY-SA 4.0 and CC BY-NC-SA for the single-variable edition.
8. **Leron 1983** (*AMM* 90(3)): the citation is UNVERIFIED.
9. **Date of Tao's "more to mathematics than rigour" post** (2009?): UNVERIFIED.
10. **Quarto 2 (Rust) timing, Typst 0.15.1 date, mdBook and mystmd licences:** from search or memory only.
11. **Common Cartridge after 1.3:** I found no newer version, but did not prove that none exists. QTI 3 support in Canvas, Moodle and Blackboard as of 2026 was not checked.
12. **doc-gen4 machine-readable output:** whether doc-gen4 publishes per-declaration JSON that math2 can use for binding was not confirmed. doc-gen4 also does not yet render Verso docstrings.
13. **mathlib4 #45028** (`notation3` → `notation +delab`): merge status unknown.
14. **Learning Commons Knowledge Graph:** whether its K-12 math nodes are worth aligning to is unknown. Its scope is US standards.
15. **Graph-to-textbook generation:** I found no published system that generates a complete mathematics textbook from a knowledge graph. PAIGOS (2008) generates courses, not books. The `m2 book` design in §4.6 is therefore new and has no prior evaluation.
16. **Wikifunctions:** relevance to math2 is low today. Revisit if its math function catalogue grows.
17. **MathML intent Open list:** its exact published URL was not confirmed. The Core list URL and the concept-lists note (https://w3c.github.io/mathml-docs/concept-lists/) are from search results.
18. **Exercise answers and Lean code:** spec 16 §16.6 gives prose CC BY-SA 4.0 and `lean/` Apache-2.0. It does not say which zone a Lean exercise's statement and reference solution belong to. Presumably Apache-2.0, but the spec should say so.
</document>

<document source="https://math2.tamnd.com/spec/corpus/02-the-goal.md">
# K02. The goal of the corpus

This document says what the corpus programme must produce, how we know when it is complete, and what it must not do. Document K04 says how we count the items. The other documents say how we produce them.

## K2.1 The goal in one sentence

**Each definition, theorem, proof and example of mathematics that a person can find in a public source is in math2 as a node, or it has a recorded decision that says why it is not. Each node is written for people, it is checkable by a machine, and it is ready to go into a course, a book or an article.**

The sentence has five parts. Each part is a requirement with a measurement:

| Part of the sentence | Requirement | Measurement |
|---|---|---|
| "each ... that a person can find in a public source" | The universe is the union of the inventories of all the sources of K05 and K06, plus the items that models list (K07). | the inventory (K04): one row per distinct item |
| "is in math2 as a node, or it has a recorded decision" | No item is silently missing. Each inventory row has an outcome. | the **open count**: inventory rows with no outcome. The target is zero. |
| "written for people" | Each node has the sections of its kind card (K08), in the house style, with notation from the notation register (K10.4). | the writing level W0 to W3 (K2.4) |
| "checkable by a machine" | Each statement has a formal binding, or a recorded reason why it has none. Each proof has a formal proof, or it is an informal proof with `uses` edges that a machine can follow. | the E and F levels of spec 04.7, and the formal outcome of K2.5 |
| "ready to go into a course, a book or an article" | A selection of nodes compiles into a linear text with no manual edits (K11). | the pack check: `m2 pack -check` on the standard volumes (K11.6) |

## K2.2 Why this is a different goal from the main spec

The main specification (documents 00 to 21) builds the system: the object model, the graph, the formal engine, the agents, the audit and the workspace. It decided that node count is not an axis of progress (settled decision 23) and it measures coverage with C0 to C4 (18.1).

The corpus programme fills the system. It does not change any settled decision of the main spec. It adds three things that the main spec does not have:

1. **A denominator for "all".** C2 counts the items of a skeleton, and C3 and C4 count the items of a core. Both are curated lists of a few hundred to a few thousand items for each class. The corpus inventory (K04) is the full list: each distinct item that any source in scope holds. The skeletons and cores become views of the inventory.
2. **A production line.** The main spec has roles that improve one node at a time (12.1). The corpus programme runs these roles, and four new ones (K7.3), over millions of items. It needs queues, batch sizes, cost limits and a review budget (K12).
3. **A content standard.** The main spec says what a node file must contain to be valid (04.5). The corpus programme says what a node must contain to be **good**: the sections, the order, the style, the notation and the examples (K08).

Node count is still not an axis of progress. The axes of the corpus are the open count, the writing level, the trust levels and the pack check. A million nodes at W1 with no proofs is a bad result, and the reports show it as bad.

## K2.3 What "each definition, theorem, proof and example" means

The corpus holds the 11 kinds of spec 04.2. For the corpus, the important kinds and what counts as one item are:

| Kind | One item is | It is not |
|---|---|---|
| `concept` | one mathematical notion that sources name, for example "compact space" | one wording of the notion |
| `definition` | one definition of a concept that a source uses, for example "compact: each open cover has a finite subcover" | a restatement of the same definition in other words |
| `statement` | one proposition that a source states and calls true, at the generality that the source uses | each corollary that only renames variables |
| `proof` | one argument for one statement that differs from the others in what it uses or in its method | the same argument with other notation |
| `example`, `counterexample` | one specific instance that a source uses to illustrate or to refute | a family of instances (that is a statement) |
| `object` | one named object with an identity, for example a group in the ATLAS, an OEIS sequence or an LMFDB curve | each record of a database: K4.6 limits which objects become nodes |
| `conjecture`, `problem` | one open question that a source poses | each restatement of it |

The granularity rules of spec 05.4 apply. A family of near-identical forms becomes a hub and its members, not one node and not a thousand nodes.

Exercises are not a kind in node.v1. The corpus needs them for courses (K11). K8.9 proposes the kind `exercise` for node.v2 and gives the reason. Until node.v2, exercises are `example` nodes with the role `exercise` in `## Remarks`.

## K2.4 Writing levels

A writing level says how complete the human-readable part of a node is. The build computes it from the sections that are present and from their provenance, like the E and F levels. No person and no agent writes it.

| Level | Name | The node has |
|---|---|---|
| W0 | stub | a tag, a kind, a name, an inventory row and at least one `same_as` or source. It has no body. |
| W1 | stated | W0, plus the main section of its kind (`Statement`, `Definition`, `Proof` and so on) that passes the body rules B01 to B03, plus at least one source with a locator |
| W2 | usable | W1, plus the sections that the kind card marks as required for W2 (K08): for a statement, at least one proof node, one example or non-example, and an L2 explanation; for a definition, one example and one non-example |
| W3 | complete | W2, plus each section that the kind card lists, explanations at each level that applies, at least one exercise for a statement or definition in a curriculum, and a person confirmed each agent-written section (rule P04) |

W does not measure truth. A node at W3 can be at E1. A node at E5 can be at W0. The reports give both. A volume (K11) sets the levels that its nodes must reach.

## K2.5 Formal outcomes

The goal says "checkable by a machine". For each statement and definition the corpus records one formal outcome. The build computes it from the bindings and the registers:

| Outcome | Means |
|---|---|
| `bound` | at least one formal binding exists, at any F level |
| `proved` | the node is at E5 |
| `pending` | a formalize or prove job is in the queue or ran and failed with a reason |
| `blocked` | the formal statement needs a definition that no library has. The blocking definition is a node with outcome `pending` |
| `out_of_reach` | a person decided that formalization is not useful now. The decision has a reason, for example "needs the classification of finite simple groups". It is reviewed each year |

The target is that each node has an outcome, not that each node is `proved`. In 2026 the best systems formalized whole textbooks, but research-level statements still failed often (K1.4). An honest corpus says which statements are proved, which are bound and which are not.

## K2.6 What the corpus must not do

These rules come from the main spec and from the research of K01. They are not new trade-offs. They are the limits that the corpus works inside.

1. **It must not copy text that the licence gate refuses** (spec 9.3, 16.6). Most of the literature gives a record and a locator, not text. math2 writes its own statements.
2. **It must not take model output as truth.** A statement that a model wrote is E1 (spec 04.7). A proof that a model wrote is an informal proof at E1 until it is formal, certified or published. Agreement between models is evidence for triage. It never raises a level (settled decision 3).
3. **It must not publish text that a model reproduced from a source with a closed licence.** Each generated body goes through the copy check of K9.4 before it can merge.
4. **It must not invent sources.** A source is a record with a locator that the ingestion found or that a person entered. A model can propose a source. The proposal becomes a source only when the record exists in the record plane and a check or a person confirms the locator (K9.5).
5. **It must not flood the people.** Agent output is limited by the measured review capacity (settled decision 20). The corpus plan in K12 is set by that capacity, not by the speed of the models.
6. **It must not hide what it does not know.** Each report shows the open count, the items with the outcome `out_of_reach`, and the error rates from the sample audits (K9.7), including the bad ones.

## K2.7 The reference scale

The numbers in this table are estimates for planning. Spec 18.4 has the basis for the first rows. K01 has the basis for the others. When math2 has a measured number, the measured number replaces the estimate.

| Quantity | Estimate | Basis |
|---|---|---|
| distinct concepts | 100,000 to 300,000 | spec 18.4 |
| definitions (several for each concept) | 150,000 to 500,000 | about 1.5 definitions for each concept in textbooks and in Mathlib |
| named and textbook statements | 300,000 to 1,000,000 | Mathlib theorems that a person would cite, ProofWiki, the textbooks of K5.4 and the cores of spec 18.3 |
| statements in the research literature, after deduplication | 10 to 30 million | spec 18.4 |
| proofs | 1 to 1.5 for each statement in the textbook layer | ProofWiki and textbooks often give more than one proof of a named theorem |
| examples and counterexamples | 2 to 5 for each definition in the textbook layer | the kind card of K08 asks for at least one example and one non-example |
| exercises | 3 to 10 for each statement and definition in a curriculum | open textbooks (K5.4) |

The corpus programme has three layers. Each layer is a target with its own exit criterion in K14:

1. **The textbook layer**: everything that the undergraduate and graduate textbooks and the reference wikis hold. About 1 to 2 million nodes. This layer must reach an open count of zero, W2 for each node, and E2 or better for each statement.
2. **The formal layer**: everything that the formal libraries hold that a person would cite. About 300,000 to 600,000 nodes, most of them also in the textbook layer. This layer must reach an open count of zero and a binding for each node.
3. **The research layer**: the results of the research literature. 10 to 30 million items in the record plane. They become nodes only when the graph needs them (spec 9.8), or when a project or a campaign asks for an area (spec 18.5). This layer must reach an open count of zero in the record plane, and each record must have an outcome: a node, an `aligned` edge to a node, or "kept as a record".
</document>

<document source="https://math2.tamnd.com/spec/corpus/03-identifiers.md">
# K03. Identifiers: tags of ten characters

The corpus programme makes millions of nodes and tens of millions of inventory rows. A tag of six characters (spec 04.3) gives about 1.07 billion values. That is enough for the nodes, but it is too tight for random minting at the scale of the inventory, and it has no protection against a mistyped tag. This document replaces the tag format of spec 04.3 with a tag of ten characters. It also says what happens to the tags that already exist, how files are sharded, and what must change in the main spec and in the code.

## K3.1 The decision

**A new tag has ten characters: nine random characters and one check character, all from Crockford base32.** The tags of six characters that exist on the day of the change stay valid forever, as a frozen legacy set. No new tag of six characters is minted after that day.

| Property | Tag of 6 (legacy) | Tag of 10 (new) |
|---|---|---|
| alphabet | Crockford base32: `0-9 A-Z` without `I L O U` | the same |
| random characters | 6 | 9 |
| number of values | 32^6, about 1.07 × 10^9 | 32^9, about 3.5 × 10^13 |
| check character | none | 1, by the Luhn mod 32 algorithm (K3.3) |
| example | `3C53TD` | `7K3QX2M9R2` |
| minted after the change | no | yes |

The three rules of spec 04.3 do not change. A tag is assigned once, it is never edited, and it is never reused. The legacy tags are not converted to long tags, because a conversion would edit a tag. A node with a legacy tag keeps it. A node never has two tags.

## K3.2 Why ten characters

- **Room for the inventory.** The inventory (K04) has one row for each candidate item, and the research layer has 10 to 30 million rows (K2.7). Rows and nodes share one space of identifiers (K3.6), so that a bare identifier resolves to one thing. With 6 characters and 30 million identifiers in use, about 3% of random draws collide. With 9 random characters, the chance is below 1 in a million for each draw.
- **Random minting without coordination.** Spec 04.3 mints tags at random, so that two branches can mint without a central counter. A sparse space keeps collisions at merge rare. With 3.5 × 10^13 values and 100 million in use, the chance that one draw collides is about 3 in 10^6.
- **A check character.** People copy tags by hand into papers, emails and search boxes. One mistyped character in a tag of six characters gives another valid tag, which can be the tag of another node. The check character detects each error in one character and almost each swap of two adjacent characters. A tool can then say "this tag has a typo" and not show the wrong node.
- **Still short.** Ten characters fit in a citation, a URL and a Lean attribute. A Stacks tag has four characters, a Wikidata id has up to 10 characters, and a DOI is much longer.

Eleven or twelve characters would give more room, but we have no use for it. Ten characters give room for the research layer at a thousand times its estimated size.

## K3.3 The format

A tag of ten characters is `P1 P2 ... P9 C`:

- `P1` to `P9` are random characters of the alphabet `0123456789ABCDEFGHJKMNPQRSTVWXYZ`. The value of a character is its position in this alphabet, from 0 to 31.
- `C` is the check character. It is computed from `P1` to `P9` by the Luhn mod N algorithm with N = 32:
  1. Start at `P9` and go to the left. The factor of `P9` is 2, the factor of `P8` is 1, then 2, then 1, and so on.
  2. For each character, multiply its value by its factor. Write the product in base 32 and add its digits: `product / 32 + product mod 32`.
  3. Add the results. The check value is `(32 - (sum mod 32)) mod 32`. `C` is the character with that value.
- To validate a tag, do the same over all ten characters with the factor 1 for `C`, then 2 for `P9`, and so on. The tag is valid when the sum is a multiple of 32.

Example: for `7K3QX2M9R` the check character is `2`, thus the tag is `7K3QX2M9R2`. If a person types `7K3XQ2M9R2` (two characters swapped) or `7KZQX2M9R2` (one character changed), the validation fails.

These rules of spec 04.3 apply to long tags too:

- **The seed prefix.** A tag that starts with `0` belongs to the bootstrap seed. `Mint` never gives one. `MintSeed` gives one.
- **No tag that reads as another kind of identifier.** `Mint` does not give a tag that YAML reads as a number, or that starts with an MSC code (for example `20D60...`). For long tags it also refuses a tag with fewer than two letters. That rule removes each tag that could be an ISBN-10 (nine digits and a digit or `X`) and each tag that looks like a phone number or a numeric id.

## K3.4 Input and display

- **The canonical form** is ten capital characters with no separator: `7K3QX2M9R2`. Files, registers, URIs and the API use only the canonical form.
- **For reading**, the workspace and printed works can show a tag in two groups of five: `7K3QX-2M9R2`. The hyphen is not part of the tag.
- **Normalization on input.** The resolver accepts lower case, hyphens and spaces, and the Crockford substitutions `I` and `L` for `1` and `O` for `0`. It normalizes, then it validates the check character. A long tag that fails the check gives the error "not a valid tag: the check character does not match". It never resolves to the nearest valid tag, because that can be another node.
- **Legacy tags** have no check character. The resolver accepts a six-character string only when it is in the legacy set of `registers/tags.reg`.

## K3.5 Sharding

Spec 04.5 shards node files by the first two characters of the tag, which gives 1,024 directories. At a million nodes that is about 1,000 files in each directory. The corpus target for nodes is 1 to 5 million (K2.7), which would give up to 5,000 files in each directory. That makes git trees large and slow.

**The new shard key is the first three characters of the tag**, for long and legacy tags. That gives 32,768 directories:

| Nodes | Files in each directory, on average |
|---|---|
| 1 million | about 31 |
| 5 million | about 153 |
| 30 million | about 916 |

The path of a node is `nodes/<c1 c2 c3>/<tag>.md`, for example `nodes/7K3/7K3QX2M9R2.md` and `nodes/3C5/3C53TD.md`. The same key shards `prov/`, `reviews/` and `compute/`. A path is not an identity, so moving the three legacy node files to the new shards changes no tag. The migration (K3.8) moves them.

## K3.6 One space for nodes and inventory rows

The inventory (K04) has one row for each candidate item. A row gets an identifier when the inventory first records it. The identifier has the format of a long tag, and it is minted from the same generator. It is recorded in the inventory register `registers/inventory/`, which is sharded like the nodes (K4.8), not in `registers/tags.reg`.

- A row identifier is never a node tag, and a node tag is never a row identifier. Rule `T08` (K3.8) checks that the two registers have no value in common.
- With this rule, each bare identifier resolves to one thing: a node, or an inventory row. The URI of a row is `m2://inv/<id>`. The URI of a node stays `m2://<tag>`.
- When a row becomes a node, the node gets its own tag, and the row records it in its outcome. The row identifier is not reused as the tag, because a tag promises a node, and most rows of the research layer never become nodes.

The other identifiers of math2 do not change: provisional ids (`~` and 8 hex characters), run ids, review ids, project ids and record ids of the record plane.

## K3.7 What this changes in the main spec

| Place | Change |
|---|---|
| 00, Conventions | "Tags are 6 characters" becomes "tags are 10 characters (9 random and 1 check character); the legacy tags of 6 characters stay valid". The example becomes `7K3QX2M9R2`. |
| 04.3, Format | replaced by K3.3 and K3.4. The text about 32^6 tags is replaced by K3.2. |
| 04.5 | the shard key is the first three characters (K3.5) |
| 04.4 | a URI accepts both tag lengths, and `m2://inv/<id>` names an inventory row |
| 05.1, 06.4 | the registers include the sharded inventory register `registers/inventory/` (K4.8) |
| 06.7 | the sizes use the new shard counts |
| 15.3, family T | `T08`: the tag register and the inventory register have no value in common; `T09`: each long tag has a valid check character; `T10`: no tag of six characters is minted after the cut-over commit |
| 20.1 | the trees show `nodes/<3>/` |

The change is additive for readers of math2: each URI that worked before still works.

## K3.8 What this changes in the code

The toolchain in `tamnd/math2-agent` holds the tag format in a few places. The change is one pull request for each repository, in this order:

1. **`tag`**: `Len` becomes 10 for new tags, with `LegacyLen` 6. `IsTag` accepts a long tag with a valid check character, or a legacy tag. `IsLegacy` reports the second case. `Mint` and `MintSeed` draw nine characters (45 bits, from 6 random bytes), refuse excluded values (K3.3), and append the check character. A new function `Check(payload string) byte` computes the check character, and `Normalize(s string) (string, error)` implements K3.4. Each exported function keeps a testable example.
2. **`schema/node.v1.json`** in both repositories: the tag pattern becomes `^([0-9A-HJKMNP-TV-Z]{6}|[0-9A-HJKMNP-TV-Z]{10})$`. The check character cannot be expressed in JSON Schema, so rule `T09` checks it. The review id pattern `r-TAG-NN` accepts both lengths. This is an additive change of node.v1 (spec 04.9), because each file that was valid stays valid.
3. **The shard function** (in `node` and its users: `build`, `audit`, `site`, `prov`, `review`, `compute`): three characters. `m2 migrate shard3` moves the files of an existing tree, and it is safe to run again.
4. **`audit`**: `T06` accepts both lengths. New rules `T08`, `T09` and `T10` as in K3.7. `T10` reads the cut-over commit from `registers/README.md`.
5. **The Lean attribute** `Math2.Meta.Attr`, which checks the tag argument of `@[math2 TAG]`, accepts both lengths and the check character.
6. **The site and the API**: the resolver normalizes input (K3.4). Pages and JSON use the canonical form. The display form with a hyphen is used only in printed works (K11).
7. **`tamnd/math2`**: run `m2 migrate shard3`, record the cut-over commit, and freeze the legacy set. On the day of this spec the legacy set is three tags: `3C53TD`, `AAYEQE` and `G5S1M9`.

The test fixtures of the toolchain use legacy tags in many places. They stay as they are, because legacy tags stay valid. New fixtures use long tags.

## K3.9 Rejected options

- **Convert each legacy tag to a long tag.** This would edit a tag, which spec 04.3 forbids. It would also break each link to the published pages of the three legacy nodes. The gain is only that all tags have one length.
- **Ten random characters with no check character.** This gives 32 times more values, which we do not need, and it loses typo detection, which we do need for tags that people type.
- **The check symbol of Crockford's own scheme.** It uses modulo 37 and needs the five extra symbols `* ~ $ = U`. These symbols are not safe in URLs, file names and Lean identifiers.
- **A sequential counter.** It needs coordination between branches, and it leaks the order of creation, which some contributors do not want to show.
</document>

<document source="https://math2.tamnd.com/spec/corpus/04-inventory.md">
# K04. The inventory

The goal of K02 says "each definition, theorem, proof and example ... that a person can find in a public source". To know when that goal is reached, math2 needs a list of what is out there. The inventory is that list. It is the denominator of the corpus: the open count (K2.1) counts its rows that have no outcome.

This document says what a row is, how rows come from sources, how mentions of one item become one row, which outcomes a row can have, how we measure what the inventory misses, and where it is stored.

## K4.1 Mentions and rows

The inventory has two levels:

- A **mention** is one item as one source presents it. "Theorem 3.14 of Rudin's *Principles of Mathematical Analysis*, 3rd edition", "the Wikipedia article *Heine–Borel theorem*", "the Mathlib declaration `Metric.isCompact_iff_isClosed_bounded`" and "item 47 of the list that the enumerate role made for MSC 54D30" are four mentions.
- A **row** is one distinct item, in the sense of K2.3. It groups the mentions of that item. The four mentions above can be one row, "Heine–Borel theorem for $\mathbb{R}^n$", or two rows if one of them states a more general form.

Mentions are records of the record plane. They use the record identifier of spec 9.5: `m2://rec/<source>/<id>#<n>`, or `ax://paper/<id>#<tag>` for arXiv. Rows have identifiers of the format of K03, in the space that they share with node tags (K3.6).

A mention holds:

| Field | Holds |
|---|---|
| `id` | the record identifier |
| `source` | the source, from `manifests/sources.yaml` |
| `snapshot` | the hash of the source snapshot that the importer read |
| `locator` | where in the source: a URL with an anchor, a theorem number, a page, a declaration name |
| `native_id` | the identifier of the item in the source, when it has one: a Wikidata QID, a ProofWiki page, a Mathlib declaration, an nLab page, an OEIS number |
| `kind` | the kind that the importer or the model reads, one of the 11 kinds of spec 04.2, or `unknown` |
| `name` | the name that the source uses, if it has one |
| `text` | the statement text, only when the licence class allows text in the record plane (spec 9.3); otherwise its hash and its embedding, which stay internal |
| `msc` | the MSC2020 codes, from the source or from the classifier |
| `class` | the licence class of the source or of the item (spec 9.3) |
| `by` | the importer or the run that made the mention |

A row holds:

| Field | Holds |
|---|---|
| `id` | the row identifier |
| `mentions` | the mention identifiers |
| `kind`, `name`, `msc` | the values that the clustering decided |
| `layer` | `textbook`, `formal` or `research` (K2.7), computed from the sources of the mentions |
| `spread` | the number of independent sources among the mentions (K12.3) |
| `outcome` | K4.4, or empty |
| `history` | each merge, split and outcome change, with the commit and the person or run |

## K4.2 Where mentions come from

Each source in scope has an importer. K05 lists the sources of people and K06 the formal libraries. The importers give the mentions of their source, and nothing else: an importer does not decide which mentions are the same item.

Mentions come from four kinds of place:

1. **Reference works and wikis**: Wikidata, Wikipedia, ProofWiki, nLab, PlanetMath, the Encyclopedia of Mathematics, the Stacks project, Kerodon, the Digital Library of Mathematical Functions and others (K05). These give names, statements and a structure of links.
2. **Textbooks and lecture notes**: the theorem environments of open textbooks give text. The indexes and the tables of contents of closed textbooks give records and locators only (K05).
3. **Formal libraries**: Mathlib and the other libraries of K06. Each declaration is a mention. Most of them are not items that a person cites (K4.5).
4. **Models**: the enumerate role (K7.3) asks models to list each item of an area that they know, with a source for each. A model list is a mention source like any other. Its mentions are marked `by: agent`, and K4.3 never makes a row from model mentions alone (rule I03).

The research literature is the record plane of spec 09. Its extracted results are mentions already.

## K4.3 From mentions to rows

The clustering follows the four passes of spec 05.4, in order. Each pass is cheaper and more certain than the next one:

1. **Identifiers.** Two mentions with the same identifier, or with identifiers that a crosswalk links, belong to one row. The crosswalks are the `same_as` edges of the graph, the external-identifier properties of Wikidata, and the links between the sources themselves (an nLab page that links to a Wikipedia article on the same notion, or a Mathlib docstring that names a theorem).
2. **Formal equivalence.** Two mentions with formal statements that a Lean proof shows equivalent belong to one row.
3. **Similarity and judgement.** For each mention that has no row, the clustering finds the nearest rows by name, by embedding and by the links of the mention. The dedup judge classifies each candidate pair as `same`, `generalizes`, `specializes`, `related` or `different` (spec 05.4). A `same` with high confidence and with agreement of the name and the MSC code joins the row. Other `same` verdicts go to the `cluster` queue (K12.2).
4. **People.** Curators decide the close cases in the queue.

The granularity rules of spec 05.4 apply to rows. A family of forms that people cite separately is a row for each form, and a row for the hub. A `generalizes` or `specializes` verdict gives two rows and an edge between them, not one row.

The clustering is measured. A gold set of 2,000 pairs for each layer, with decisions by curators, gives the precision and the recall of each pass. The release manifest records them. An error in clustering has two forms, and both are bad. A false merge hides an item, and the open count does not show it. A false split gives two nodes for one item, and the dedup queue must find it later.

## K4.4 Outcomes

Each row must get an outcome. The open count is the number of rows that have none. The target is zero (K2.1).

| Outcome | Means | Who decides |
|---|---|---|
| `node` | the row is a node, with its tag | the curate role proposes, the merge decides |
| `member` | the row is a member of a hub family, as its own node | the same |
| `merged` | the row is the same item as another row, and it points to that row | the clustering, or a curator |
| `split` | the row was more than one item, and it points to the rows that replace it | a curator |
| `aligned` | the row is a result of the research layer that a node states, and it has an `aligned` edge to the node | the align role, with the rules of spec 09.6 |
| `record` | the row stays a record of the research layer, with no node, until the graph needs it (spec 9.8) | the rule K4.5 |
| `formal_only` | the row is a declaration of a formal library that no person cites, kept in the crosswalk (K4.5) | the rule K4.5 |
| `object_record` | the row is a database object that stays a record (K4.6) | the rule K4.6 |
| `out_of_scope` | the row is not mathematics in the sense of spec 18.2, or it is a notation, a teaching note or a mistake in the source. The reason is required | a curator, or a rule that a curator approved |
| `deferred` | the decision waits for something named, until a date | a curator |

`merged` and `split` keep the row identifier in the register forever, as a tag is kept (spec 04.3). A link to an old row identifier resolves to its successor.

A `deferred` row counts in the open count when its date passes. A `deferred` row with no date is not allowed.

Most outcomes come from rules, not from people. A rule is a decision that a curator approved once and that applies to many rows, for example "each mention from the enumerate role that has no other mention after 90 days gets `out_of_scope`, reason: no source confirmed". The rules live in `schema/outcomes.v1.yaml`, and each outcome that a rule gave names the rule.

## K4.5 Which formal declarations and research results become nodes

The formal libraries have many declarations that no person would cite: API lemmas, simp lemmas, instances and auxiliary results. A node for each one would give a graph that nobody can read. The research layer has 10 to 30 million results, and spec 9.8 already keeps them as records until the graph needs them.

**A formal declaration becomes a node** when at least one of these is true:

- a mention from a human source (K05) is in the same row;
- the library documents it as a main result, in the module documentation, the list of named theorems, or a list such as "100 theorems" or "1000+ theorems";
- a node of the graph uses it in a `uses` edge that a person wrote;
- a person asks for it.

Each other declaration gets the outcome `formal_only`. It stays in the crosswalk, so that the formal binding of a node can point to it, and search can find it. Spec 07 and K06 give the details for each library.

**A research result becomes a node** when spec 9.8 says so: when the graph needs it, or when a project or a campaign asks for its area (spec 18.5). Until then its row has the outcome `record` or `aligned`. The open count of the research layer goes to zero by these two outcomes, not by nodes.

## K4.6 Which objects become nodes

The databases of mathematical objects hold millions of records: the OEIS, the LMFDB, the House of Graphs, the ATLAS of finite groups, the GAP libraries of small groups and others. A node for each would add little that the database does not have. The rule is in line with spec 06.7:

- **An object becomes a node** when a human source names it, when a node cites it, when it is in a curated list (the sporadic groups, the exceptional Lie algebras, the regular polytopes), or when a project asks for it.
- **The OEIS** is a special case. Each sequence has a name and a curated entry, and spec 06.7 counts them as nodes. They become nodes in bulk, at W0, with `same_as` to the A-number.
- **Each other object** gets the outcome `object_record`. It keeps its database identifier as a `same_as` target, so that a node can point to it.

## K4.7 Measuring what the inventory misses

The open count is zero only for the rows that exist. An item that no source in scope holds, and that no model lists, has no row. So the inventory needs a measure of its own recall.

1. **Capture and recapture.** When two independent sources cover the same area, the overlap of their mentions gives an estimate of the total. If source A has *a* items of an area, source B has *b*, and *m* items are in both, the estimate of the total is *a* · *b* / *m* (the Lincoln and Petersen estimator). With three or more sources, log-linear models give a better estimate, and they allow for sources that depend on each other. The estimate is computed for each MSC code at two digits, each quarter.
2. **A held-out source.** When a new source enters scope, the clustering runs on it before the importer adds its mentions. The share of its items that already have rows is an estimate of the recall of the inventory for its area.
3. **Index terms.** The indexes of standard textbooks list the terms that each book defines. Each index term of a book in scope must match a row. The match rate for each book is in the report.
4. **Model lists.** The enumerate role is a recall tool, not a source of truth. An item that a model lists, and that the inventory does not have, is a candidate gap. A gap becomes a row only when K9.5 confirms a source for it.

The report gives the estimated recall for each layer and each MSC code at two digits, with the method and the interval. Where the estimate of the total is much larger than the number of rows, the area goes to the `enumerate` queue and to the importer plan of K05.

## K4.8 Storage

- **Mentions** are files of the record plane: `records/mentions/<source>/<c1c2c3>.jsonl`, sorted by identifier, one mention per line. The shard key is the first three characters of a hash of the mention identifier.
- **Rows** are in `records/inventory/<c1c2c3>.jsonl`, sharded by the first three characters of the row identifier (K3.5), one row per line.
- **The identifier register** is `registers/inventory/<c1c2c3>.reg`, sharded in the same way, because at the scale of the research layer one file would be larger than a gigabyte. Its rules are those of `registers/tags.reg`: lines are never removed or edited. K3.6 and rule T08 apply to it.
- **The licence zone** of mentions and rows is metadata, CC0 (settled decision 17). A mention with text holds only text that its class allows in the record plane (spec 9.3), under the licence of its source.
- When the record plane moves to `tamnd/math2-records` (spec 06.1), mentions, rows and the inventory register move with it.

The build reads the inventory and writes the open count, the coverage of each layer and the queues of K12.2 into the derived store.

## K4.9 Rules

| Rule | H/S | Checks |
|---|---|---|
| I01 | H | each `inventory` value in a node points to a row whose outcome is `node` or `member` with the tag of that node |
| I02 | H | each row with the outcome `node` or `member` points to a live node, or to a retired node with a successor |
| I03 | H | no row has only mentions from models. Such mentions wait in the `enumerate` results until K9.5 confirms a source |
| I04 | H | each outcome that a rule gave names a rule in `schema/outcomes.v1.yaml` |
| I05 | S | a `deferred` row whose date has passed |
| I06 | S | an MSC code at two digits whose estimated recall (K4.7) is below 90% |
</document>

<document source="https://math2.tamnd.com/spec/corpus/05-human-sources.md">
# K05. Human sources

People have written down most of the mathematics that the corpus must hold: in reference works, wikis, textbooks, papers, problem lists, syllabi and exercise banks. This document lists those sources, says what the corpus takes from each one, by which channel, under which licence, and in which order. K06 does the same for the formal libraries.

Each fact about a source has its URL and its date of reading in K01, parts A and B. All counts were read on 2026-10-26, unless the text gives another date. A fact marked "not settled" is in the open questions of K15.

## K5.1 What a human source gives

A human source can give five things. Most sources give only some of them:

| What | Example | Licence that it needs |
|---|---|---|
| **mentions** | "the ProofWiki page *Lagrange's Theorem (Group Theory)*", "Theorem 3.14 of Rudin, 3rd edition" | none: a name and a locator are facts |
| **open text** | the statement and proof on that ProofWiki page | `open` class (spec 9.3) |
| **locators** | "Hefferon, *Linear Algebra*, Chapter Two, §III.1", "DLMF 5.2.1" | none |
| **a link structure** | the links between nLab pages, the cross-references of a textbook | none for the edges. The selection of a whole collection can have a database right (K13.4) |
| **curricula** | a table of contents, a syllabus, a list of theorems | none for the structure. The words of the source are its author's |

Every source gives mentions. Only sources of the `open` class give text. Mentions and locators are the larger part of the value: they fill the inventory (K04), and the inventory is the denominator of the goal. A node can state a theorem in math2's own words from a locator, so a source that gives no text still gives the item.

## K5.2 The source document

Each source has one entry in `manifests/sources.yaml`. A person approves each entry and each change to its licence fields (spec 9.2). The importer reads nothing from a source that has no approved entry.

```yaml
- id: lebl-ra
  title: Basic Analysis I and II
  authors: [Jiří Lebl]
  kind: textbook            # reference | wiki | textbook | paper-index | extraction | curriculum | exercises | database
  url: https://github.com/jirilebl/ra
  licence: CC-BY-SA-4.0 OR CC-BY-NC-SA-4.0   # SPDX expression, as the source states it
  election: CC-BY-SA-4.0    # for a dual licence: the option that math2 takes
  class: open               # spec 9.3
  licence_read:
    at: https://github.com/jirilebl/ra      # the source itself, never an aggregator
    on: 2026-10-26
    by: <person>
  per_item_licence: false   # true if items have their own licences (K13.4)
  public_domain: null       # or {us: true, eu: true, basis: "..."} (K5.2.3)
  authored: human           # human | ai | mixed (K6.5)
  edition: <tag or edition> # pinned edition, where editions differ in licence
  channel: git              # dump | git | api | bulk | page | catalogue
  snapshot: {commit: <commit>, read: <date>}
  terms:
    robots: <summary or n/a>
    rate: <limit that the importer keeps>
    ai: <clause on AI or machine learning, or "none stated">
  attribution: "Jiří Lebl, Basic Analysis I and II, CC BY-SA 4.0, https://github.com/jirilebl/ra"
  adapter: importer/latexthm
  approved_by: <maintainer>
```

### K5.2.1 Read the licence at the source

The licence of a book or a site is read at the source: the `LICENSE` file of its repository, its front matter, or the licence page of the site. A catalogue such as the Open Textbook Library or the PreTeXt catalogue is a list of candidates, not a licence. Research B found three stale or simplified licences in the Open Textbook Library in one afternoon (K5.4.2).

The licence is read again at each new snapshot. A change of licence between snapshots changes the class of the new snapshot only. The text that math2 took under the old licence keeps that licence, and the source document keeps both, with the snapshot of each.

### K5.2.2 Dual licences

When a source offers a choice of licences, math2 takes the option that the prose zone can hold, and the field `election` records the choice. Examples: Lebl's three books (CC BY-SA 4.0 or CC BY-NC-SA 4.0), Hefferon's *Linear Algebra* (GFDL or CC BY-SA 3.0 US), and ProofWiki (CC BY-SA 3.0 and the GFDL). The attribution names the elected licence.

### K5.2.3 Public domain

Public domain depends on the country. In 2026, a work published in the United States before 1931 is in the public domain there. In the EU, a work is in the public domain 70 years after the death of its author, so in 2026 the author must have died before 1956. math2 is published in all countries, so:

- A work is `open` as public domain only when it is in the public domain in the United States **and** in the EU. The field `public_domain` records both answers and the basis for each, for example the date of publication and the year of the death of the author.
- A work that is in the public domain in one of them only is `record`.
- A translation, a modern edition and a modern commentary have their own copyright. The importer takes only the text of the original, from an edition whose added text it does not use. Project Gutenberg texts lose the Project Gutenberg header and licence text, as its licence requires.

Hardy's *A Course of Pure Mathematics* (Project Gutenberg #38769) passes both tests: it was published before 1931, and Hardy died in 1947.

### K5.2.4 The `authored` field

The field `authored` of K6.5 applies to human sources too. A source that people wrote is `human`. A source with parts that a model wrote is `mixed`, and the source document says which parts. For example, the "slogans" of the TheoremSearch datasets were written by DeepSeek-V3.1, so they are `ai`, while the statements are from the papers. Text from an `ai` part counts as agent text for the rules of K07 and K09: it needs the same checks as text that a math2 role wrote.

## K5.3 Reference works and wikis

| Source | Size | Licence | Class | The corpus takes | Channel |
|---|---|---|---|---|---|
| Wikidata | 3,208 theorems; 113,118 items with a defining formula (P2534), but 105,345 of these are prime numbers, so 6,560 real formulas | CC0 | `open` | items, labels, crosswalk properties, formulas | weekly JSON dump (103 GB); WDQS only for checks |
| Wikipedia (en) | 28,196 mathematics articles (WikiProject Mathematics) | CC BY-SA 4.0 | `open` | mentions, text with attribution, the link graph | monthly XML dump |
| Wikipedia (de, fr, ja, zh, ru) | 20,732, 12,192, 8,865, 5,682, 3,928 (each wiki counts in its own way) | CC BY-SA 4.0 | `open` | mentions; text in a later phase | monthly dumps |
| ProofWiki | 47,254 content pages; 30,348 proofs and 36,102 definitions (Main Page) | CC BY-SA 3.0 and GFDL | `open`, elect CC BY-SA 3.0 | mentions, statements, proofs, definitions | daily dump `latest.xml.gz` (36.6 MB) |
| PlanetMath | 9,477 entries, inactive since 2018 | CC BY-SA 3.0 | `open` | mentions and text | git, 64 repositories by MSC class |
| nLab | 20,778 pages | none. The site allows use "if you acknowledge the source", but no right to adapt | `record` | page names, the link graph | git mirror `ncatlab/nlab-content` |
| Encyclopedia of Mathematics | live count not settled; 1,224 Wikidata ids | originals © Springer; only new wiki text is CC BY-SA 3.0 and GFDL | `record` | ids and links. Text only from an article that its history shows to be wholly new | none (the site gave HTTP 502 on the day of reading) |
| MathWorld | 16,939 entries | proprietary, no bulk copies | `record` | ids, from Wikidata (P2812, 6,840 items) | none |
| OEIS | 400,709 sequences | CC BY-SA 4.0 | `open` | names, terms, formulas, references | git `oeis/oeisdata` |
| DLMF | version 1.2.8 (15 September 2026) | © NIST, no bulk copies | `record` | equation numbers as locators | none |
| Fungrim | 3,130 entries, dormant since 2022 | MIT | `open` | formulas with their conditions | git |
| Wolfram Functions | 307,409 formulas | © Wolfram Research | `record` | formula ids as locators | none |
| LMFDB | figures of 2022 (not measured again) | data CC BY-SA 4.0; code GPL | `open` (data) | object records by label | bulk downloads only. Its `robots.txt` forbids `/api/` |
| groupprops, Topospaces, Commalg, Diffgeom | 8,076, 825, 429, 435 articles | CC BY-SA 3.0 | `open` | mentions and text, mostly facts about groups | MediaWiki API, slowly: one maintainer |
| Manifold Atlas | 215 articles | contradictory: GFDL 1.3 and "public domain" | `gfdl` until the editors answer | mentions and links | MediaWiki API |
| π-Base | 225 spaces, 247 properties, 935 theorems | CC BY 4.0 | `open` | theorems as statement candidates | git |
| DaRT | 218 rings, 178 properties, 72 theorems | CC BY 4.0 | `open` | the same | git |
| House of Graphs, ISGCI, KnotInfo, Knot Atlas, ATLAS v3 | about 22,000 graphs (2022); 1,695 graph classes; knots to 13 crossings; 2,710 articles; about 716 groups | no licence stated | `record` | ids and facts, as metadata | downloads where offered. Not the GPL-3.0 CSV package of KnotInfo |
| Stacks Project | about 21,000 tags | GFDL 1.2 | `gfdl` | tags, titles, the dependency graph | git |
| Kerodon | 8,189 tags | no licence stated | `record` | tags, titles, the dependency graph | none: page by page, slowly |
| Open Problem Garden | at most 713 problems | GFDL | `gfdl` | mentions and links | none |
| Erdős problems | 1,221 problems | the site states no licence; `teorth/erdosproblems` is Apache-2.0 but has no statements | `record` (statements), `open` (status) | status, links, the problem number | git `data/problems.yaml` |
| Kourovka Notebook | 21st edition, arXiv:1401.0300v49 (20 October 2026) | arXiv non-exclusive licence | `record` | problem numbers with the edition | arXiv |
| MathOverflow | 168,603 questions | CC BY-SA 2.5, 3.0 or 4.0, by the date of each post | `open`, for each post | mentions, references, text of single posts with attribution | the API, with a key |
| Math.StackExchange | 1,702,027 questions | the same | `open`, for each post | the same | the same |
| FindStat | 1,997 statistics | not settled | `record` until settled | ids | none |

Notes on the table:

- **Wikidata formulas.** The P2534 count is mostly primes. The importer drops the items that are instances of *prime number* (Q49008) and its kin before it treats P2534 as a source of formulas.
- **Stack Exchange dumps.** Since July 2024 the dumps are behind a click-through that excludes the training of large language models. math2 does not train a model on them, but the click-through is a contract, so the importer uses the API and not the dumps. The importer reads only posts that a node cites or that the enumerate role proposes as a source, not whole sites. Each post has its own licence, by date (rule L06).
- **Wiki text and the copy check.** Text from a wiki that enters a node keeps the licence of the wiki and the attribution of its history. The copy check (K9.4) sees it as a match with an open source. That match is allowed, and it adds the attribution.

## K5.4 Open textbooks

Open textbooks are the main source of text for the textbook layer (K2.7). They give statements, proofs, examples and exercises in the order that a course needs, and the theorem environments of their sources mark each item.

### K5.4.1 Table A: books that the prose zone can take

Each licence below was read at the source of the book. Each book needs attribution. A CC BY-SA 3.0 Unported text may be adapted under CC BY-SA 4.0 (its §4(b)). For the 3.0 US port of Hefferon, the same clause is assumed and not settled.

| Book | Licence that math2 uses | Source format | Notes |
|---|---|---|---|
| Open Logic Project | CC BY 4.0 | LaTeX | |
| *Homotopy Type Theory* (HoTT book) | CC BY-SA 3.0 | LaTeX | |
| Hefferon, *Linear Algebra* | CC BY-SA 3.0 US, elected from a dual licence | LaTeX | |
| Lebl, *Basic Analysis* I and II, *Notes on Diffy Qs*, *Tasty Bits of Several Complex Variables* | CC BY-SA 4.0, elected from a dual licence | LaTeX | |
| Boelkins et al., *Active Calculus* (single variable) | CC BY-SA 4.0 | PreTeXt | the Open Textbook Library entry is for the 2017 edition and is stale |
| Chen, *An Infinitely Large Napkin* | CC BY-SA 4.0 (text) | LaTeX | its code is GPL v3, and math2 does not take it |
| Levin, *Discrete Mathematics: An Open Introduction*, 3rd edition and earlier | CC BY-SA 4.0 | PreTeXt | pin a commit before the 4th edition, which is CC BY-NC-SA |
| Keller and Trotter, *Applied Combinatorics* | CC BY-SA 4.0 | PreTeXt | |
| Ernst, *An Introduction to Proof via Inquiry-Based Learning* | CC BY-SA 4.0 | LaTeX | |
| Austin, *Understanding Linear Algebra* | CC BY 4.0 | PreTeXt | |
| Beck, Marchesi, Pixton and Sabalka, *A First Course in Complex Analysis* | CC BY 4.0 | PDF | |
| Lehman, Leighton and Meyer, *Mathematics for Computer Science* (2018) | CC BY-SA 3.0 | PDF | the MIT OCW course pages are NC; this book is not |
| OpenStax *Statistics*, *Precálculo* 2nd edition, and the earlier CC BY editions of *Prealgebra* (2015) and *Precalculus* (2014) | CC BY 4.0 | CNXML | pin the CC BY edition |
| Hardy, *A Course of Pure Mathematics*, and other public-domain books that pass K5.2.3 | public domain | LaTeX, HTML | strip the Project Gutenberg text |

The Open Textbook Library lists more books under CC BY or CC BY-SA, for example Zakon, Kuttler, Shimamoto, Petrunin, Callahan and *OpenIntro Statistics*. They are candidates. Each one enters Table A only after a person reads its licence at the source.

### K5.4.2 Table B: books that the prose zone cannot take

These books give mentions, locators, tables of contents and internal embeddings, but no text in nodes.

| Book | Licence | Class |
|---|---|---|
| Judson, *Abstract Algebra: Theory and Applications* | GFDL 1.3 or later | `gfdl` |
| Beezer, *A First Course in Linear Algebra* | GFDL 1.2 or later | `gfdl` |
| the PreTeXt titles with a GFDL badge (21 titles, for example Yoshiwara's three books and *The ODE Project*), Grinstead and Snell, Bogart | GFDL | `gfdl` |
| Hartman et al., *APEX Calculus*; Axler, *Linear Algebra Done Right*, 4th edition | CC BY-NC 4.0 | `share-alike-nc` |
| *CLP Calculus* (UBC); Levin, 4th edition; the OpenStax 2nd editions of calculus and algebra; MIT OpenCourseWare | CC BY-NC-SA 4.0 | `share-alike-nc` |
| Hammack, *Book of Proof*, 3rd edition; Shoup | CC BY-NC-ND | `verbatim` |
| Hatcher, *Algebraic Topology* | © CUP 2002, noncommercial copies only | `record` |
| Lurie, Kerodon | no licence stated | `record` |
| the PreTeXt titles *Linear Algebra* (W214) and *Intro to Topology (IBL)* | all rights reserved | `record` |
| LibreTexts pages tagged "not declared" | none | `record` |

Three books on this list are often taken to be open: Hatcher, *Book of Proof* and Axler. They are not open for the prose zone.

### K5.4.3 Aggregators

- **Open Textbook Library.** 214 mathematics books. By its own licence data, about 48% are `open`. Its licences can be stale or simplified: it lists *Active Calculus* as CC BY-NC-SA (true for the 2017 edition only), *Book of Proof* as "Attribution-NoDerivs" (the author says CC BY-NC-ND), and Hefferon as CC BY-SA only (the book is dual-licensed). The corpus uses it as a list of candidates.
- **AIM Open Textbook Initiative.** 68 approved titles. The list shows no licences. Fourteen titles have licences that nobody has read yet (K15).
- **PreTeXt catalogue.** 67 mathematics projects. Many have only a badge that says "Creative Commons", with no variant. Each one needs a check at the source.
- **LibreTexts.** The licence is set page by page, and one book can mix licences. The importer goes to the upstream source of each book. Where it reads LibreTexts, it reads page by page, by the `license:` tag of each page. A page without a declared licence is `record`.

### K5.4.4 Editions

Some books changed licence between editions: Levin's 4th edition became CC BY-NC-SA, and the OpenStax 2nd editions are CC BY-NC-SA where the 1st editions were CC BY. An earlier edition keeps the licence under which it was published. So the source document of such a book pins the edition and the commit (field `edition`), and the importer reads nothing from a later commit until a person approves the licence of that commit.

### K5.4.5 What the importer takes from an open textbook

From the theorem environments of the source (`\begin{theorem}`, PreTeXt `<theorem>`, CNXML `<rule>`, and the like):

- the kind, the number and the title of each environment, as a mention with a locator;
- the statement, the proof, the examples and the exercises, as text of the record plane under the licence of the book;
- the cross-references between environments, as candidate `uses` edges;
- the table of contents and the index, as a curriculum (K5.8) and as index terms for K4.7.

The text enters a node only by the importer of `open` text (K14.4), with the attribution of the book, and the copy check marks it as a match with an open source. The write role does not paste it: a role that writes a node from scratch has no text of a restricted book in its context (K7.5), and it may have open text there, with its licence.

## K5.5 Closed textbooks

Most of the standard textbooks are not open. They give records and locators only:

- **The record of the book**: ISBN, edition, zbMATH and MR ids, from catalogues (K5.6).
- **Locators**: "Rudin, *Principles of Mathematical Analysis*, 3rd edition, Theorem 3.14". A locator is a fact.
- **The table of contents and the index**, where the publisher shows them in public, for example in a preview or on the site of the book. They go into the record plane as one locator for each item, not as a copy of the table (K13.4). An index term is a mention of a definition, and K4.7 uses the index terms of standard books to measure recall.
- **Not the text.** No text of a closed book enters the record plane, the context of a role, or a node. E2 needs two book locators that a person has checked (spec 9.7), because that check needs the book.

The list of closed textbooks in scope comes from the curricula of K5.8: each book that a syllabus names is a candidate.

## K5.6 Literature indexes and full text

The research literature is the record plane of spec 09. This section adds the facts of 2026 that change what the importers do.

| Source | Size | Licence | The corpus takes |
|---|---|---|---|
| arXiv (through `tamnd/arxiv`) | about 9.2 TB of files (April 2025) | metadata CC0; text by the licence of each paper | records; the text of `open` papers |
| zbMATH Open | 4M+ publications, 3M+ reviews; 133,093 documents for 2025 | knowledge graph: CC BY 4.0 or CC BY-SA 4.0 (the record says both) | records, MSC codes, reviews as evidence |
| OpenAlex | about 745 GB (September 2026) | CC0 | all metadata |
| Crossref | 180,034,490 records (March 2026) | metadata free; some abstracts have copyright | metadata and references; abstracts are `record` |
| DataCite | 108M DOIs | CC0 | metadata of datasets and software |
| Semantic Scholar, S2ORC | 15,679,781 full texts (May 2026) | ODC-By for the database, the copyright of each paper for its text | ids. Text only by the licence of each paper |
| CORE | 291M records (2023) | ODC-By for the database | records. Text by the licence of each paper |
| HAL (mathematics) | 170,406 records, 105,413 with a file | metadata CC0; 9.8% of the files are `open` | records; the 10,349 open texts |
| Numdam | 75,599 articles | metadata CC0; text for personal use, no bulk | records. Text only where an article has a CC licence or is in the public domain |
| EuDML, Project Euclid, GDZ | | not settled; "free to read" is not an open licence | records, through partner sources |
| HathiTrust, Internet Archive, Project Gutenberg | 206,597 mathematics texts on the Internet Archive | public domain for some items; a hosted copy says nothing about the licence | public-domain books that pass K5.2.3 |

Three facts change the plan of spec 09:

1. **More of arXiv is open than spec 9.3 says.** In a sample of three days of 2026, 36.9% of new mathematics papers were `open` (CC BY 34.0%, CC0 1.9%, CC BY-SA 1.0%). Spec 9.3 gives about 15%, which is the share of all papers since 1991. The sample is small, so phase 1 measures the share over all of 2026 from the metadata plane, and spec 9.3 then gives both figures.
2. **The zbMATH knowledge graph is updated about twice a year**, not each month. Its licence is CC BY 4.0 in one field of the Zenodo record and CC BY-SA 4.0 in its description. Both are `open`, but they differ for derived exports, so the importer treats it as CC BY-SA 4.0 until zbMATH answers (K15).
3. **Full-text aggregators do not change the licence of a paper.** S2ORC, CORE and the Internet Archive hold texts under the copyright of each paper. An importer takes a text from them only when the licence of that paper allows it.

## K5.7 Extraction projects

Spec 9.6 says that math2 aligns to the large extraction projects and does not compete with them. The data of 2026:

| Project | Size | Licence | The corpus takes |
|---|---|---|---|
| TheoremSearch | 8.4M statements in the live index | the CC BY-SA release mixes in GFDL text from the Stacks Project. The `-permissive` release (1,239,720 theorems) is CC BY 4.0 and excludes share-alike | the `-permissive` release as candidate mentions (E1), with the licence of each paper. Its slogans are `ai` (K5.2.4) |
| TheoremGraph (`math-graph`) | 16,131,238 rows | CC BY 4.0, with the licence of each statement. The text is empty unless the paper is open | the edges and the ids, with attribution. The text where it is present |
| TheoremGraph (`theorem-matching`) | 100,831 candidate pairs of Lean declarations and arXiv statements | CC BY-SA 4.0 (K01, part C) | candidate crosswalks (K6.7), with share-alike attribution |
| Matlas | 8.07M statements from 435K papers and 1.9K textbooks | no licence or terms stated | ids of matches at query time only (`record`) |
| the Stacks Project and Mathlib alignment (arXiv:2610.31011) | | paper CC BY 4.0 | candidate crosswalks, which the align role checks (K6.7) |

The BY-SA release of TheoremSearch is not taken as one block. The importer filters it by source and by the licence of each row, and drops the Stacks rows and every row whose licence is NC or not stated.

## K5.8 Curricula and theorem lists

Curricula say which items a course needs, and in which order. The corpus imports them as data (K10.3), never as text:

| Source | Size | Licence | The corpus takes |
|---|---|---|---|
| Mathlib `docs/undergrad.yaml` | 563 items, 130 of them not yet formalized | Apache-2.0 | all: items, structure, declarations |
| Mathlib `docs/overview.yaml` | 511 items | Apache-2.0 | all |
| Mathlib `docs/1000.yaml` | 1,199 entries keyed by Wikidata id; 217 with declarations | Apache-2.0 | all. The Wikidata keys are crosswalk ids |
| Mathlib `docs/100.yaml` | 100 entries; 78 with declarations | Apache-2.0 | all |
| 1000+ theorems project | 228 entries on the site | Unlicense | all |
| Wiedijk, *Formalizing 100 Theorems* | 100 theorems, with counts for 13 systems | no licence | the list of names and the counts, as facts |
| MAA CUPM Curriculum Guide (2015) | 116 pages | © MAA | course areas and topic lists, as facts |
| QAA Subject Benchmark Statement, Mathematics, Statistics and Operational Research, 5th edition (2023) | 35 pages | © QAA | outcome categories, as facts |
| qualifying exam syllabi (for example UC Berkeley) | | no licence | topic lists, as facts. Not the past exam problems |
| tables of contents of open and closed textbooks | | K5.4, K5.5 | the order of the items |

A curriculum is also a mention source: each item that it names is a mention, with the curriculum as its source. The `curriculum` term of the priority (K12.3) comes from these mentions. The difference between Mathlib's `100.yaml` (78 with declarations) and Wiedijk's count for Lean (83) is not settled.

## K5.9 Exercise banks

The large exercise banks have licences that the prose zone cannot take. They give links, and the exercise node (K8.7) is math2's own text:

| Bank | Size | Licence | The corpus takes |
|---|---|---|---|
| WeBWorK Open Problem Library | about 35,000 problems | CC BY-NC-SA 3.0 by default; an author can choose an OSI licence | the path of a problem as a link in `external`; its tags as metadata. No text |
| MyOpenMath and IMathAS | | question code GPL-2.0; the generated HTML CC BY 4.0 | links. The generated HTML of a question is `open`, but the importer takes it only after a person checks the licence of that question |
| STACK | | GPL | links |
| Numbas | | code Apache-2.0; public questions under Creative Commons, by question | links. Text only from a question whose licence is CC BY or CC BY-SA |
| Natural Number Game (NNG4), *Mathematics in Lean* | | Apache-2.0 | Lean exercises, into the code zone, with attribution |
| lean4game games | | the engine is GPL-3.0; each game has its own licence | links. A game with a permissive licence is like NNG4 |
| the exercises of Table A books | | the licence of the book | exercise text, with attribution |

The exercise role (K7.3) has no text of an NC bank in its context.

## K5.10 Channels and politeness

The importers keep each limit below, identify themselves with the User-Agent of the toolchain, `m2/<version> (+https://github.com/tamnd/math2-agent)`, with no personal address, and prefer a dump to live requests:

| Source | Channel | Limit that the importer keeps |
|---|---|---|
| Wikipedia | the monthly XML dump. The API only for single pages | API: a compliant User-Agent gets 200 requests a minute. The robot policy of 16 March 2026: dumps before live requests |
| Wikidata | the weekly JSON dump | WDQS: 60 s of query time a minute, 5 parallel queries, only for checks |
| ProofWiki | the daily dump | no crawling |
| nLab | the git mirror | no crawling |
| OEIS | git `oeisdata` | site: `Crawl-Delay: 10`, and `/search` is disallowed |
| LMFDB | bulk downloads | `/api/` is disallowed, and the site has a JavaScript gate. The importer does not try to pass it |
| MathOverflow, Math.SE | the API, with a key | 10,000 requests a day; keep the `backoff` field, which applies across sites |
| arXiv | `tamnd/arxiv` | one request each 3 seconds, for all machines together |
| Project Gutenberg | `/robot/harvest` or a mirror | `wget -w 2` |
| MathWorld, DLMF, Wolfram Functions | none | no requests. Ids come from Wikidata |
| Kerodon, groupprops, the Manifold Atlas | single pages | one request at a time, at most one each 10 seconds. These sites are small and have one maintainer or none |

A source that blocks automatic clients, or asks for a login or a click-through, is not read until a person decides how to proceed. The importer never passes a gate.

## K5.11 Adapters

Each source has one adapter, `importer/<source>`, in `tamnd/math2-agent` (K12.8). Spec 09.1 puts adapters in `internal/ingest/<source>`. The corpus uses `importer/<source>` instead, because the toolchain has no `internal/` packages. Most sources share a format, so most adapters are thin and use a format package:

| Format | Sources |
|---|---|
| MediaWiki XML dump | Wikipedia, ProofWiki |
| MediaWiki API | the subwikis, the Manifold Atlas |
| Markdown in git | nLab, π-Base |
| LaTeX theorem environments | PlanetMath, Open Logic, HoTT, Hefferon, Lebl, Napkin, Ernst |
| PreTeXt XML | *Active Calculus*, Levin, Keller and Trotter, Austin |
| CNXML | OpenStax |
| YAML in git | the Mathlib lists, 1000+, Erdős problems, DaRT |
| JSON dump | Wikidata |
| JSONL or Parquet | TheoremSearch, TheoremGraph, OpenAlex |
| PDF | Beck et al., *Mathematics for Computer Science*: locators and the table of contents only, until a source in LaTeX is found |

Each adapter has fixtures from a small sample of its source, and tests that use the fixtures and never the network (spec 19). An adapter writes mentions and nothing else (K4.2).

## K5.12 The order of import

The order follows K14. A source enters when the phase before it meets its exit:

1. **Phase 1, open data with no doubt about the licence.** Wikidata, the Mathlib lists, 1000+, OEIS names, π-Base, DaRT, Fungrim, and the records of OpenAlex, Crossref, DataCite, HAL and Numdam. Then the records and locators of the `record` and `gfdl` sources: the tags of the Stacks Project and Kerodon, the names of the nLab, the ids of MathWorld and the EoM from Wikidata, the tables of contents of closed textbooks where the publisher shows them.
2. **Phase 2, the pilot area (group theory).** Wikipedia, ProofWiki, PlanetMath and groupprops for the area, the Table A books that cover it (Judson is `gfdl`, so it gives locators only), the curricula of K5.8, and the Stack Exchange posts that the nodes cite.
3. **Phase 3 and later, the textbook layer.** The rest of Table A, the rest of the wikis, and TheoremSearch `-permissive` and TheoremGraph for the research layer. Each new source first goes through the held-out check of K4.7.

## K5.13 Corrections to the main spec

| Spec | Value now | Value on 2026-10-26 |
|---|---|---|
| 1.7, 9.2 | Wikidata: about 113k items with a defining formula | 113,118, but 105,345 are prime numbers. 6,560 are other formulas |
| 1.7, 9.2 | Wikipedia: about 26k mathematics articles; dumps twice a month | 28,196 articles; the dumps are monthly in 2026; new API rate limits |
| 1.7, 9.2 | nLab: about 19k pages, licence not known | 20,778 pages, no licence: `record` |
| 9.2 | ProofWiki: CC BY-SA 3.0, quarterly | CC BY-SA 3.0 and GFDL; a daily dump |
| 1.7, 9.2 | Encyclopedia of Mathematics: CC BY-SA 3.0 | originals © Springer: `record`, except wholly new articles |
| 9.2 | Erdős problems: statements from `teorth/erdosproblems`, Apache-2.0 | the repository has no statements. The statements on the site have no licence: `record` |
| 9.2 | LMFDB: API and bulk downloads | bulk downloads only: `robots.txt` forbids `/api/` |
| 9.2 | zbMATH Open: CC BY-SA 4.0, monthly | CC BY 4.0 or CC BY-SA 4.0 (not settled); twice a year |
| 9.3, 16.6 | Kerodon: GFDL | no licence stated: `record` |
| 9.3 | about 15% of arXiv is `open` | 15% of all papers; 36.9% of new mathematics papers in a 2026 sample |
| 9.3 | `gfdl` examples: Stacks, Kerodon, Open Problem Garden | Stacks, the Manifold Atlas, Open Problem Garden |
| 9.3 | `record` examples | add the DLMF, Wolfram Functions, the Kourovka Notebook, KnotInfo, ISGCI, the ATLAS and House of Graphs |
| 9.6 | TheoremSearch: 9.2M theorems; the 1.34M open release | 8.4M in the live index; take the `-permissive` release, because the BY-SA release mixes in GFDL text |
| 1.7 | MathWorld 16,849; OEIS 400,673 | 16,939; 400,709 |
| 9.1 | adapters in `internal/ingest/<source>` | `importer/<source>` |

## K5.14 Rules

| Rule | H/S | Checks |
|---|---|---|
| L08 | H | each source in `manifests/sources.yaml` has the fields `licence`, `class`, `licence_read`, `authored`, `channel`, `snapshot`, `attribution` and `approved_by`, and each mention names a source that has them (K5.2) |
| L09 | H | a source whose `licence` is a choice of licences has an `election`, and the elected licence gives the class (K5.2.2) |
| L10 | H | a source of the `open` class whose licence is public domain has `public_domain` with `us: true` and `eu: true` (K5.2.3) |
| L11 | H | a mention with text comes from a snapshot whose licence was read on or before the date of that snapshot (K5.2.1) |
| L12 | S | a source whose licence was last read more than twelve months ago |

With rules L01 to L07, these rules are the licence gate for human sources. The open questions of this document are in K15: the current licence of the EoM, the terms of Stack Exchange of 13 November 2025, the licences of the nLab and the Manifold Atlas, the knowls of the LMFDB, the zbMATH licence, the 14 AIM titles, the US port of CC BY-SA 3.0, and the sources that the research could not reach.
</document>

<document source="https://math2.tamnd.com/spec/corpus/06-formal-sources.md">
# K06. Formal sources

The formal libraries are the second large source of the corpus, after the human sources of K05. Mathlib alone has more than 430,000 named theorems and definitions. The other proof assistants have about as many again. This document says what the corpus takes from each library, by which route, under which licence, and what it does not take. It also covers the datasets that pair formal and informal text, and the translators between systems.

Spec 07 says how a binding works, and spec 07.10 says which systems can give E5. This document does not repeat those sections. Each number comes from K01, part C, read on 2026-10-26. Where a number has changed since spec 01, K6.11 gives the new value.

## K6.1 What a library gives the corpus

A formal library gives four things. Each has its own rule:

1. **Mentions.** Each declaration is a mention in the inventory (K4.2). Most declarations get the outcome `formal_only` (K4.5), and only the ones that a person cites, or that the library marks as main results, become nodes.
2. **Bindings.** A Lean declaration can be the binding of a node, and so give E5 (spec 07.3). In v1, only Lean libraries give bindings.
3. **Crosswalks.** A declaration in another system is a `same_as` target of a node (spec 05.3). It helps the clustering (K4.3, pass 1), and it is evidence of an independent formalization towards F3 (spec 07.10).
4. **Informal text.** Docstrings, comments and literate prose. The corpus takes this text only when the licence of the library allows it in the prose zone (K13.1).

## K6.2 Intake classes

Each library has one intake class. The source document of the library in `manifests/sources.yaml` records it, with the licence and the reason:

| Class | The corpus takes | Example |
|---|---|---|
| `bind` | mentions, bindings, crosswalks and the informal text | Mathlib, Formal Frontier |
| `crosswalk` | mentions and crosswalks. Informal text only if the licence allows it | the AFP, MathComp, set.mm |
| `link` | crosswalks and links. No text, no statements | 1Lab, ATLAS |
| `evaluate` | nothing for the graph. The dataset tests the roles of K07 | miniF2F, ProofNet, PutnamBench |
| `exclude` | nothing | MathPile |

The class follows from three questions, in this order:

1. **Can we use it at all?** A non-commercial licence, a clause against machine learning, or no licence at all gives `link` for a library and `exclude` for a dataset. A dataset with no licence can become `evaluate` only when its authors confirm its terms in writing.
2. **Can its text go into the prose zone?** Apache-2.0, MIT, BSD, CC0 and the Unlicense allow it, with attribution. CC BY-SA allows it, with share-alike and attribution. GPL, AGPL, LGPL and CeCILL-C are licences for software, and math2 does not put their comments into prose. Their libraries give mentions and crosswalks only.
3. **Can it give bindings?** Only Lean 4 libraries, in v1 (spec 07.10).

## K6.3 The libraries

| Library | Size (date) | Licence | Class | Route | Informal text |
|---|---|---|---|---|---|
| Mathlib | 291,237 theorems and 139,616 definitions (stats page); 797,294 constants, 483,112 with `allowCompletion` (v4.35.0, 2026-10-13) | Apache-2.0 | `bind` | lean_scout datasets (K6.4) | docstrings on about 14.5% of constants |
| Tau Ceti | 18,849 `.lean` files | Apache-2.0 | `bind`, with the flag of K6.5 | as Mathlib | docs site; written by AI |
| Formal Frontier | 41 libraries, about 200k lines | Apache-2.0 (30 of 50 repositories checked) | `bind` | as Mathlib | per library |
| Other Reservoir packages | 908 packages | per package | per package | Lake | per package |
| ATLAS (Meta) | 26 textbooks, more than 45,000 declarations | v1: CC BY-NC 4.0 with a clause against ML. v2: Apache-2.0 with a preamble against ML | `link` | none | textbook-aligned |
| Isabelle AFP | 1,042 entries, about 330,000 lemmas | BSD-style or LGPL, per entry | `crosswalk` | the release tarball, Isabelle export | abstracts and `text` blocks; BSD entries only |
| Isabelle2026 library | released October 2026; size not known | BSD-style core, mixed | `crosswalk` | Isabelle export | `text` blocks of BSD parts |
| Rocq Stdlib | 904 `.v` files | LGPL-2.1 | `crosswalk` | opam | none |
| MathComp, Odd Order | 147 and 34 `.v` files | CeCILL-B | `crosswalk` | opam | header comments |
| MathComp Analysis | 131 `.v` files | CeCILL-C | `crosswalk` | opam | none |
| UniMath (Rocq) | 1,714 `.v` files | MIT-style | `crosswalk` | opam | comments |
| HOL Light, Flyspeck | 664 `.ml` and 503 `.hl` files | BSD-style, MIT | `crosswalk` | hol2dk dump | comments |
| HOL4 | 1,495 `Script.sml` files | BSD-style | `crosswalk` | HOL4 export | comments |
| Metamath set.mm, iset.mm | 51,207 and about 17,395 statement numbers | CC0 1.0 | `crosswalk`, with text (K6.6) | parse of the `.mm` file | a comment on each statement |
| Mizar MML | version 5.94.1493 | GPLv3+ or CC BY-SA 3.0+ | `crosswalk` | XML export | little |
| agda-stdlib | 1,183 files | MIT-style | `crosswalk` | Agda HTML | comments |
| agda-unimath | 3,072 `.lagda.md` files | MIT | `crosswalk`, with text (K6.6) | literate parse | prose on each page |
| 1Lab | 710 `.lagda.md` files | AGPL-3.0 | `link` | none | prose, not taken |
| Naproche | 32 `.ftl.tex` files | GPL-3.0 | `crosswalk` | ForTheL | not taken |
| Megalodon | about 160,000 lines | system MIT-style; the mgwiki repositories have no licence | `link` | none | none |
| NASALib (PVS) | 63 or 64 libraries, about 41,000 formulas | public domain for NASA work, per library for the rest | `crosswalk`, per library | PVS | comments |
| ACL2 books | 40,929 files and directories | BSD-3 and per file | `crosswalk`, per file | ACL2 | XDOC |

Two rules apply to each row:

- **The licence is recorded for each item where the library has a licence for each item.** The AFP has one for each entry, NASALib for each library, the ACL2 books for each file. The importer reads it, and rule F10 checks it.
- **The licence is read again at each pin.** A library that changes its licence changes its class from that pin on. Mentions from older pins keep the licence that they had.

## K6.4 Mathlib

**The route.** The Mathlib Initiative publishes the datasets of `lean_scout` (Apache-2.0) for each toolchain bump: `mathlib-types` (name, module, type, docstring, `allowCompletion`), `mathlib-const-dep` (the dependencies of each constant) and `mathlib-tactics` (goal states). The importer reads these datasets at the commit of the release pin. It does not run its own walk over the environment for mentions. `M2Extract` (spec 07.4) still runs in the build, for the bound declarations and their closure only, because the type hash and the axioms of a binding must come from the pinned build.

**What a mention is.** The three published counts measure different things: 430,853 named theorems and definitions on the stats page, 483,112 constants that the user can complete, and 797,294 constants in all. The corpus takes the second: a mention is a constant with `allowCompletion = true`. The other constants are internal or generated, and no person would cite them. The report gives all three numbers, with the definition that each one uses.

**Which mentions become nodes.** The rule of K4.5 applies. For Mathlib the corpus uses these signals, in order:

1. a human mention in the same row (pass 1 of the clustering uses the Wikidata and the docstring links);
2. the named-theorem lists: the "100 theorems" and "1000+ theorems" files, and the main results that module documentation names;
3. a declaration that a node already uses, or that a person asks for;
4. a definition that the statement of a selected theorem uses (`uses_constants_in_statement`).

A rough guess, not a measurement, is that 40,000 to 80,000 Mathlib declarations meet one of these. The first import measures the real number, and the report gives it.

**Informal text.** About 115,876 constants have a docstring, with a mean length of about 130 characters. A docstring is Apache-2.0 text that a person wrote, so it can be the start of a summary (K8.3), with attribution. The informalize role (K7.3) writes the statement of each selected declaration that has no human mention and no usable docstring.

**AI-written informal datasets are not text sources.** Herald, the FrenzyMath datasets, LeanSearch v2 and others hold informal text that a model wrote for Mathlib declarations. Their licences are permissive, but no study measures their error rate, and most of them do not name the model. Rule X08 needs each model that writes text for math2 to be in the model register. So these datasets are search aids and inputs to the clustering only, never text of a node.

## K6.5 Libraries that AI wrote

Tau Ceti is mostly written by AI, and reviewed against rubrics that people wrote. Megalodon's Munkres formalization was written with models. Some AFP entries say that AI helped with proof engineering, and NASALib has a `CLAUDE.md`. Other libraries will follow.

The kernel checks the proofs of each of these libraries, so the proofs are as good as any others. The risks are in the parts that the kernel does not check: whether a statement says what its name and docstring say, and whether its definitions are the standard ones.

So each library and each entry has a field `authored` in its source document, with the value `human`, `ai` or `mixed`, and the reason. The importer copies it to each mention. Then:

- **Docstrings of `ai` libraries are agent text.** They go through the copy check (K9.4), and they count as text that an agent wrote for rule P04. A person must confirm them before a node reaches W3.
- **Bindings to `ai` libraries need the F2 review of spec 07.3 like any other binding.** In addition, a binding to an `ai` declaration whose statement uses a definition of the same library needs the review to cover that definition too.
- **The report gives the share of bindings to `ai` libraries.**

The value `mixed` is for a library where both people and models write, with no record of which. It has the rules of `ai`.

## K6.6 Other systems

**No system translates a whole library into Lean 4 today.** hol2dk translates HOL Light to Rocq (40,728 theorems of Multivariate, a dump of about 135 GB), and not to Lean. Mathport is archived. lean2dk is still work in progress. In ITPEval, the best model translated 29.1% of statements and 10.5% of proofs between systems at pass@1, and an equivalence check confirmed only 54% of the translations that compiled. So the other systems enter the corpus as crosswalks, not as Lean. When a node needs a Lean binding of a result that only another system has, the formalize role writes it, with the source declaration in its context, and the binding goes through the F1 battery like any other.

**Metamath set.mm is the best informal and formal source after Mathlib.** It is CC0, and each statement has a comment that a person wrote, usually a sentence or more. The importer takes each comment as `open` text, with attribution (which CC0 does not need, but rule 2 of K13.8 does). A statement whose comment is a full sentence starts at W1, after the copy check and the body rules. The set.mm labels are `same_as` targets.

**agda-unimath** is MIT, and its pages are literate Markdown, with prose next to each definition. The importer takes the prose as `open` text, with the MIT notice in the attribution. It is the best source for definitions in homotopy type theory and univalent foundations.

**The AFP.** Each entry has an abstract, and the theories have `text` blocks. The importer takes the text of BSD entries as `open` text, and the text of LGPL entries not at all. The AFP is the largest source of crosswalks after Mathlib.

**Mizar.** The MML is dual-licensed, and one choice is CC BY-SA 3.0, which allows a later version (CC BY-SA 4.0) for adaptations. Its text is formal, and there is little prose. The corpus uses it for crosswalks only. The article count is not settled: the version string says 1,493 and spec 01 says 1,498 (K15).

**The others** (Rocq, HOL Light, HOL4, Naproche, NASALib, ACL2) give mentions and crosswalks. Their comments are not taken as text in v1.

## K6.7 Crosswalks

A crosswalk links a row to a declaration in a library. The corpus builds them in this order:

1. **From the libraries themselves.** Many declarations name the theorem that they state, in the docstring or in a list.
2. **The 1000+ theorems list.** It is in the public domain (the Unlicense), and it has 1,200 YAML files, one for each theorem, keyed by the Wikidata identifier, with MSC codes, links to Wikipedia and blocks for Isabelle, HOL Light, Rocq, Lean, Metamath and Mizar. It is the first crosswalk that the corpus imports. Wiedijk's 100 theorems list is the second. It has no licence, so the corpus takes only its facts: which theorem has a formalization in which system.
3. **Matching datasets.** TheoremGraph pairs 388,105 Lean declarations with real arXiv statements (not model text), with 47,952 to 61,210 pairs that a model judged to match. It is CC BY-SA 4.0. The Stacks-to-Mathlib alignment (MIT) pairs Stacks tags with Mathlib. The corpus uses these datasets as candidates. The align role judges each pair again, and an accepted pair becomes an edge of math2, with the dataset in its source. The corpus does not copy their text, and a pair that math2 did not judge is not an edge.
4. **The type hash and the clustering.** Two Lean declarations with the same type hash state the same thing (spec 07.4). The other systems need the align role.

## K6.8 Datasets

**Benchmarks are for evaluation only.** Their errors are measured, and they are many:

| Benchmark | Errors that a study found |
|---|---|
| miniF2F (Lean ports) | the formal and informal statements disagree in more than half of the problems |
| ProofNet | 55.9% of items have errors (ProofNet-Verified) |
| ProofNet# (the fixed version) | still 27.0% |
| DeepSeek-ProverBench | 208 certified defects in 325 problems |
| FormalMATH | 147 certified defects |

So a benchmark item never becomes a mention, a statement or an exercise of the corpus. The benchmarks test the roles of K07 (class `evaluate`). PutnamBench has one more rule: its informal statements are used by permission of the MAA, so math2 does not copy them anywhere, not even into a test set that it publishes.

**Synthetic statement sets** (Lean Workbook, Goedel-Pset, NuminaMath-LEAN and others) hold millions of formal statements of competition problems, made by models. Most of them have a measured fidelity of 60 to 90%, or no measure. Their informal side is often text from a third party, such as Art of Problem Solving. They are `evaluate` at most. An exercise of the corpus (K8.7) is not taken from them.

**Pretraining corpora** (OpenWebMath, FineMath, Proof-Pile-2, AutoMathText) are not sources. OpenWebMath is in the reference corpus of the copy check, for detection only (K9.4). MathPile is `exclude` (CC BY-NC-SA).

**Red flags.** The importer refuses a dataset in each of these groups until a person changes its class in the source document:

| Flag | Datasets |
|---|---|
| non-commercial, or a clause against ML | ATLAS v1 and v2, AutoformBench, MathPile |
| no licence | Goedel-Pset-v1, Goedel Lean-workbook-proofs, DeepSeek-ProverBench, MathAtlas, LeanAgent, the mgwiki Megalodon repositories |
| licence conflict | DeepSeek-Prover-V2 (MIT file against the Model Agreement), LeanDojo-v2 (Apache against MIT), FormalMATH (MIT on Hugging Face, Apache on GitHub) |
| third-party text | PutnamBench informal statements |
| made by a model under terms that may pass on | data from DeepSeek-V3, Qwen2.5, Gemini or Kimina models (K7.9) |

## K6.9 Informalization

The formal track is large and the human track is small for the same items: only 14.5% of Mathlib constants have a docstring. So the informalize role writes much of the text of the formal layer. K7.5 gives the job, and K9.7 gives the audit. Two rules from the evidence:

- **Compiling is not faithfulness.** A statement that compiles can still say something else. Each informalized statement has the F1 battery of spec 10 on the pair, and the sample audit checks fidelity as its own property.
- **There is no published error rate.** No study measures informalizations of Mathlib at scale. The closest number (FormL4) is 70 to 80% correct, in a different setting. The informalize role is its own stratum from its first merge, and the corpus publishes the first measured rate.

## K6.10 Pins and bumps

Each library is pinned in `manifests/sources.yaml`, with a commit and a toolchain (spec 07.1). The importer reads the library at its pin. When the pin moves:

- New declarations become new mentions.
- A removed or renamed declaration keeps its mention, with `retired` and its successor, if the library gives one (Mathlib's deprecation aliases do).
- A declaration whose type hash changed is a new mention. The bump role (spec 07.9) handles the bindings.
- The report gives the change in mentions, rows and nodes for each library, for each bump.

## K6.11 Corrections to spec 01

| Spec 01 value | Value on 2026-10-26 |
|---|---|
| Mathlib 291,178 theorems and 139,569 definitions | 291,237 and 139,616 |
| Mathlib 9,437 files | 8,661 `.lean` files under `Mathlib/`. Spec 01 probably counted all files |
| AFP 1,038 entries, about 326,500 lemmas, about 5.44M lines | 1,042, about 330,000, about 5.518M |
| Isabelle2026 not known | released in October 2026 |
| Agda sizes not known | agda-stdlib 1,183 files, agda-unimath 3,072, 1Lab 710 |
| Megalodon Munkres 130k lines | about 160k lines and more than 1,500 lemmas |
| Mizar 1,498 articles | the version string says 1,493 (not settled) |
| Metamath set.mm more than 40k theorems (2022) | 51,207 statement numbers, including axioms and definitions |
| Tau Ceti about 3M lines | 18,849 files. The source of "3M lines" may mean Mathlib (not settled) |

## K6.12 Rules

| Rule | H/S | Checks |
|---|---|---|
| F10 | H | each mention from a library records the library, the pin, the declaration name, the intake class and the licence of the item (K6.3) |
| F11 | H | no section of a node holds text from a library whose class is `link`, `evaluate` or `exclude`, or from an item whose licence is not open (K6.2) |
| F12 | H | each source document of a library has the field `authored`, and each mention copies it (K6.5) |
| F13 | S | a binding to an `ai` or `mixed` declaration whose F2 review does not cover the definitions of that library that its statement uses (K6.5) |

The open questions of this document are in K15: the Mizar article count, the size of Tau Ceti, the licences of the other 20 Formal Frontier repositories, the effect of the ATLAS v2 preamble, the Coquelicot licence, and whether the terms of a model pass on to the data that it made.
</document>

<document source="https://math2.tamnd.com/spec/corpus/07-machine-generation.md">
# K07. Machine generation: what models write, and how

The goal asks for the mathematics that people wrote down, and also for what models know (K2.1). Models are the only way to write a million nodes with the people that the project can have. They are also the largest source of errors, of copies and of invented sources. This document says what models do in the corpus, what they must not do, and how each piece of model output is controlled.

The main spec already sets the frame. Models never raise a status (settled decision 3). Each change is a pull request that the audit checks (settled decision 7). Agent pull requests are capped by review capacity (settled decision 20). Each run records the exact model ids, the role version and the prompt hashes (spec 12.3, 12.8). This document does not change these rules.

## K7.1 What models are for

| Use | Role | Output | Trusted as |
|---|---|---|---|
| list the items of an area | enumerate (K7.3) | mentions in the record plane | a mention source. A row never comes from model mentions alone (rule I03) |
| write the sections of a node | write, explain | bodies at W1 to W3 | text by an agent until a person confirms it (rule P04) |
| write the informal text of a formal declaration | informalize | statement and concept nodes | text by an agent. The binding is checked by the F1 battery, and F2 needs a person |
| write exercises | exercise | exercise nodes (K8.7) | text by an agent. An answer with a machine check is checked by the machine |
| decide, align, formalize, prove, explore | the roles of spec 12.1 | as in spec 12 | as in spec 12 |
| check other model output | judges (K7.6) | verdicts in the run record | triage only |

Models do **not**:

- decide that a statement is true. Only the kernel, the certificate checkers and refereed or textbook sources give E levels (spec 04.7);
- add a source that does not resolve to a record (rule P05);
- give the outcome of an inventory row by themselves when the rules of K4.4 say that a curator decides;
- get the text of a source whose licence class is not `open` in their context (K7.5);
- write a register, a review or a status (spec 12.1).

## K7.2 What the evidence says

The research of K01 (part D) measured model output in mathematics from 2024 to 2026. Eight results set the controls of this document:

1. Model judges gave proofs scores up to 20 times higher than human graders did.
2. The best judges of single proof steps still accept about 21% of wrong steps. Weaker judges accept 56% to 90%.
3. Strong models "proved" false statements of competition level 29% to 70% of the time. At research level, they rejected fewer than 40% of false statements.
4. A correct final answer is not a correct proof. For one model, only 59.5% of correct answers had a correct proof.
5. From 11% to 57% of the citations that models generate do not exist. A citation that more than three models give is real 95.6% of the time.
6. Statements that models formalize with no person in the loop are faithful 60% to 90% of the time. A statement that compiles is not a faithful statement: one system compiled 90% and was accurate on 66%.
7. More than 27% of the text that models write in the style of an encyclopedia is in verbatim runs of 50 characters or more from public text.
8. No study measures how often models write a wrong definition, drop a hypothesis or give a theorem the wrong name in informal text. The corpus must measure this itself (K9.7).

Most of these numbers come from models of 2025. Nobody has published the same measures for the current models. So the corpus measures its own configurations before it trusts them, and the numbers above are the prior.

## K7.3 The four new roles

The corpus adds four roles to the eight of spec 12.1. Each has a bot account, a tool profile and a field permission set, and rule X01 checks each pull request against it.

| Role | Does | May create | May edit | Auto-merge |
|---|---|---|---|---|
| **enumerate** | lists the items of an area that the model knows, each with its kind, a one-line statement, its aliases and one or more source identifiers | mentions in `records/mentions/model-<family>/` (K4.8) | none | yes, into the record plane only. A mention changes no node |
| **write** | writes the sections of a node that its card asks for at the target writing level (K8.4): the statement, the definition, the proof in steps, the examples and non-examples | sections marked `by: agent`; example nodes; informal proof nodes | its own unconfirmed sections | never |
| **informalize** | writes the informal text of a formal declaration that no person described: the name, the statement, the summary | statement and concept nodes with a binding at F0 to the declaration; sections marked `by: agent` | its own unconfirmed sections | never |
| **exercise** | writes exercises for a node, with a hint, an answer, a solution and a check (K8.7) | exercise nodes (node.v2), or example nodes with the role `exercise` before node.v2 | its own unconfirmed exercises | never |

The roles of spec 12.1 do this work in the corpus:

- **curate** proposes the outcome of each row in the `decide` queue (K4.4);
- **explain** writes the levels L1 to L4 (K8.6), with the grounding check of spec 12.6;
- **formalize** and **prove** run the formal track (K12.1);
- **explore** runs the sanity checks of K9.3;
- **align** proposes the `aligned` outcomes and the crosswalks to Wikidata, Mathlib and the other libraries (K06);
- **review-assist** makes the packets for each check that people do (K9.6).

## K7.4 Generation from what models know

The enumerate role asks models what mathematics they know. It is how the corpus finds the items that no catalogue lists: the lemma that each textbook proves and no index names, the standard counterexample, the folklore result.

**The unit of work.** One job asks one model about one cell: an MSC code at five digits and one level (school, undergraduate, graduate, research). There are about 6,000 codes and 4 levels, so a full pass is about 24,000 jobs for each model.

**The prompt.** The job gives the model the code, the level, the names of the rows that the inventory already has for the cell, and the schema of a mention (K4.1). It asks for each item that is not on the list: its kind, its common name and aliases, a statement of one sentence, its prerequisites by name, and the sources where it appears, as identifiers (an ISBN with a theorem number, a DOI, an arXiv id, a Wikipedia title). It asks the model to say "unsure" when it is unsure, and never to give a source that it cannot name exactly.

**Rounds until saturation.** A cell gets a new round with the new list, until a round adds fewer than 2% new rows after clustering (K4.3), for two rounds in a row and for models of two families.

**More than one family.** The models of one family share their training data, so their lists are not independent. The job runs on at least two families: Claude, and one open-weight family with a licence that adds no duty on outputs (K7.9). An item that only one model lists, with no source that resolves, goes to the `cluster` queue as a mention and waits for a human source.

**What enumeration cannot measure.** Capture and recapture between models overstates recall, because the models have seen the same text. So model lists count as one capture occasion in the estimates of K4.7, and the held-out human source is the test of recall.

**The knowledge cutoff.** The current Claude models know mathematics up to June 2026. A model cannot list what was published after its cutoff, and its claim that a result is new means nothing. The research layer comes from the record plane (spec 09), not from what models remember.

## K7.5 How a write job runs

A write job takes one node to the next writing level. It runs the common loop of spec 12.2, with this context and these limits:

**The context.**

- the card of the kind at the target level (K8.4) and the house style (K8.2), in the cached part of the prompt;
- the notation register of the scope of the node and the conventions register (K10.4, K10.5), also cached;
- the node, its neighbourhood to depth 2, and the inventory row with all its mentions (K4.1);
- the text of the sources of the row, **only** when the source has the licence class `open` and its licence is compatible with the prose licence (CC0, CC BY, CC BY-SA, public domain). The node then carries the attribution of rule L03;
- for sources of the classes `share-alike-nc`, `gfdl`, `verbatim` and `record`: the identifier and the locator only. Their text never goes into the context of a model that writes prose for math2. A model that rewrites a GFDL proof that it can see makes a derived work, and the licence of math2 cannot take it.

**The instructions.**

- Write in your own words. Do not reproduce a source. This cuts copying, but it does not stop it (K9.4).
- Write each proof as numbered steps, with the reason for each step as a link to a node or a source (K8.5). The structure helps the judges, but it does not change the E level.
- For a statement, prove it or disprove it (K9.3). Report a hypothesis that you think is missing or not needed.
- Use only the macros of the register. Declare each new symbol in the `notation` field.
- Mark a source that you cannot name exactly as unsure, or leave it out.

**The output.** The diff adds sections marked `by: agent`, with the role version and the model id of each field (spec 04.8). The gates of K9.2 run in the loop before the pull request opens, so the model can repair its draft in the same run.

**The informalize role.** It gets the declaration, its type, its docstring when one exists, the module documentation, the declarations that it uses, and their informal text when math2 has it. The output is a statement in the house style, with the conventions of the declaration stated (K10.5). Then the F1 battery runs on the pair, as on any other binding. No published study measures the error rate of informalizations of Mathlib (K01, part C). So informalize is a stratum of its own in the sample audit from its first merge (K9.7).

**The exercise role.** It writes exercises for a node, with the fields of K8.7. It uses a model of another family to solve each exercise from the statement alone. An exercise whose answer the solver does not reach, or reaches in a different form, goes to a person. An exercise with `check: lean` must compile with `sorry`, and its reference solution must compile. Banks of exercises with a non-commercial licence (K05) are links only, and their text is not in the context.

## K7.6 Checks on model output

Each piece of model output goes through the gates of K9.2. The judges of K9.3 follow these rules, because the evidence of K7.2 says that one judge is not enough:

1. **A judge verdict is triage.** It decides what a person sees first. It never raises an E level, an F level or a writing level.
2. **The judge is of another family.** The model that wrote a node does not judge it. A check names the family of the writer and the family of each judge, and rule X09 checks that they differ.
3. **Judges are measured first.** A judge configuration judges proofs or statements only after it has a measured rate of rejecting wrong steps of 75% or more on a labelled set of math2. The set holds wrong proofs that people found, and canaries (K9.3). Tier 0 models do triage, classification and the first pass of the dedup judge, and they do not judge proofs.
4. **Accept by all, send by any.** A node passes the judges when each judge passes it. When one judge fails it, the node goes to a person with the reports of all judges. No extra call breaks a tie.
5. **Canaries each month.** The queues get false statements made by small changes to true ones. The weekly report gives the share that each configuration "proved", and a share above 5% stops the configuration (K9.3).
6. **Agents game checkers.** In a published benchmark, agents tried to work around the elaborator. So the pinned statement, the Comparator check and the axiom audit of spec 12.2 apply to each accepted proof, from each prover, with no exception.

## K7.7 Provenance and the model register

- **Each field has its writer.** The provenance of spec 04.8 records the role version, the model id and the run id of each field that an agent wrote. A person who confirms a field adds a confirmation and does not replace the record.
- **A register of models.** `models/models.yaml` in `tamnd/math2-agent` holds an entry for each model that the orchestrator can call: the exact id, the family, the provider, the licence or the terms with their URL and the date that a person read them, the clauses on outputs (for example a duty to name the model, or a ban on training other models with the outputs), and the retirement date when one is known. The orchestrator refuses a model with no entry. Each release manifest of `tamnd/math2` includes the entries of the models that wrote its fields.
- **Filters by model.** `m2 prov list -family F` lists each field that a model of a family wrote. A user whose own terms need it can remove those fields. The corpus uses the same filter for the rule of K7.9 on training.
- **Model change.** The current Claude models retire from September 2027. A campaign of a million nodes will cross a model change. A new model is a new configuration, with its own stratum in the sample audit (K9.7), and `m2 prov quarantine` can act on the output of one model.

## K7.8 Configurations and cost

The cost estimate of K12.6 is for each stage. This table, from K01 part D, is for each configuration of one node with one generation call and three check calls, at batch prices with cache hits. The token counts are assumptions that the first thousand runs replace.

| Configuration | For each node | For one million nodes |
|---|---|---|
| all calls on Haiku 5.5 | $0.004 | about $4,000 |
| all calls on Sonnet 5.5 | $0.081 | about $81,000 |
| all calls on Opus 5.5 | $0.161 | about $161,000 |
| all calls on Fable 5.1 | $0.40 | about $400,000 |
| write on Opus 5.5, check with two Sonnet 5.5 calls and one call to a model of another family | $0.113 | about $113,000 |

- **Divide by the acceptance rate.** When half of the drafts pass, the cost doubles. Until the pilot measures the rate (K14.4), the plan uses two to five times the table.
- **Output tokens are 70% to 85% of the cost**, and thinking tokens are output tokens. The effort setting is the lever that matters. The weekly report gives the output tokens for each accepted node, for each role and tier.
- **The all-Haiku row is not a real option** for proofs and statements, because tier 0 does not judge proofs (K7.6). It is the cost of enumeration and triage.
- **Formal work costs much more.** A published campaign of textbook formalization cost about $17 for each Lean declaration and about $300 for each target theorem (K01). K12.6 prices it apart.
- **Prices change.** The price table of the `llm` package has the date of each price, and it models batch, cache reads, cache writes and the long-context tiers. A table with no date is a finding of the weekly report.

The money is not the limit. The limit is the review hours (K12.5). The plan gives the cost of each accepted node and of each F2 binding, not only the spend.

## K7.9 Terms of the providers

K13 gives the law. This section gives the rules for the operations:

- **Publishing is allowed.** The terms of Anthropic, OpenAI and Google that a person read on 2026-10-26 give the customer the rights in the outputs that the provider has, and none of them stops the publication of outputs under an open licence.
- **No training on hosted outputs without authorization.** From 12 November 2026, the usage policy of Anthropic forbids the use of outputs to train an AI model without its prior authorization. OpenAI and Google forbid the use of outputs to build models that compete with theirs. So the corpus does not fine-tune a model (for example an open prover of tier 0′) on fields that a hosted model wrote, unless the provider authorized it in writing. The filter of K7.7 makes this mechanical.
- **Open-weight models with duties on outputs.** Some licences add duties on outputs or on models trained with them: the naming duty of Llama 4, the clause of the DeepSeek Model Agreement on models trained with its synthetic data, and the conditions of the Qwen 3.8 and Kimi K3 licences for outputs made available to others. The model register records each clause. For roles that write content, the corpus prefers models with MIT or Apache-2.0 licences and no clause on outputs, for example DeepSeek V4, Qwen3.5, GLM-5, Mistral, Leanstral, gpt-oss, Goedel-Prover-V2 and Kimina-Prover.
- **The licence of math2 adds no term.** CC BY-SA cannot carry a ban on training, and math2 does not try to add one. The release notes say which fields came from which provider, so that users with their own contracts can filter.
- **No model run without approval.** Each run that calls a model needs a budget that a person approved (K12.6). Tests use fakes for each model.

## K7.10 Rules

| Rule | H/S | Checks |
|---|---|---|
| X08 | H | each field that an agent wrote names a model that has an entry in the model register |
| X09 | H | each judge verdict in a check of K9.3 is from a model of a family other than the family of the writer |

With rule I03 (K4.9), L05 (K9.4) and P05 (K9.5), these rules are the hard limits on model output.
</document>

<document source="https://math2.tamnd.com/spec/corpus/08-the-node-card.md">
# K08. The node card: what a good node contains

Spec 04.5 says what a node file must contain to be valid. This document says what it must contain to be good. For each kind, a **card** lists the sections and the edges, the order in which a reader sees them, which ones each writing level needs (K2.4), and the style of the text. The writers follow the cards. These writers are people, the roles of K07, and the importers of K05 and K06. The build checks the cards and computes the writing level from them.

The cards are data, not only prose. They live in `schema/cards.v1.yaml` in `tamnd/math2`, and the build reads them. A change to a card is a pull request that a curator approves.

## K8.1 Principles

1. **One node, one thing.** The main section states one definition, one proposition or one argument. Motivation, history and explanation have their own sections (spec 04.5).
2. **Self-contained statements.** A reader can understand the main section with only the linked nodes and the notation register. Each symbol is defined in the section, by a link, or by the notation register (K10.4). "As above" and "under the usual hypotheses" are not allowed.
3. **Reuse by edges, not by copies.** An example is an `example` node with an `instance_of` edge, not a paragraph inside a definition. A card says which edges it shows. The workspace and the packer (K11) put the linked nodes in place. One example can then serve ten definitions and three courses.
4. **Formal and informal side by side.** Each statement and definition card has a place for the formal binding. Each proof card has a structure that maps to a formal proof (K8.5).
5. **Levels are contracts.** An explanation at level L*n* uses only the prerequisites of that level (spec 12.6). The card gives the length and the purpose of each level.
6. **Every section has a writer.** The provenance log records a person or a run for each section (spec 04.8). The card says which sections an agent may write and which need a person to confirm them.

## K8.2 The house style

The text of a node is for a reader who reads one node and leaves. It must be clear on its own, and it must read the same across a million nodes. The house style is plain technical English in the spirit of ASD-STE100, adapted to mathematics:

- **Short sentences.** At most 25 words in a sentence of the main section, and at most 30 words in other sections. A formula counts as one word.
- **One claim in each sentence** in `Statement` and `Definition`. Hypotheses come first, then the conclusion: "Let ... . If ..., then ...".
- **Active voice** in proofs: "We apply Lemma 2" and not "Lemma 2 is applied". "We" is allowed in proofs and explanations. It is not used in `Statement` or `Definition`.
- **No filler words that hide a step.** The words "clearly", "obviously", "trivially", "it is easy to see" and "the reader can verify" are not allowed. If a step is short, give it. If it is long, make it a lemma.
- **Name the convention.** When conventions differ, the text says which one it uses: "Here $\mathbb{N}$ contains 0", "rings have a unit". The concept node records the conventions (K10.5), and the text links to it.
- **Standard words.** Use the term that the concept node names first. Give other terms as aliases in the front matter, not as alternatives in the text.
- **Notation from the register.** Use the macros and symbols of the notation register (K10.4). A new symbol needs a register entry first.
- **No references to place.** Write "the [[Sylow theorems|...]]" and not "the theorem in Section 3". A node has no position. A book gives it one (K11).
- **English first.** The canonical text is English. Translations follow spec 13.4.

`m2 check -style` runs a style linter over the bodies. The linter is mechanical: sentence length, the banned words, undeclared symbols, links that are not tags. Its findings are warnings for W1 and errors for W3.

## K8.3 The common front matter

Each card uses the front matter of node.v1 (spec 04.5). The corpus adds four optional fields. All four are additive (spec 04.9):

| Field | Holds | Why |
|---|---|---|
| `inventory` | the ids of the inventory rows (K04) that this node accounts for | the open count needs the link from row to node, and the reverse link makes the reports cheap |
| `conventions` | for a concept: the conventions that sources use, each with a short name and the sources that use it | K10.5 |
| `notation` | for a concept or a definition: the symbols that it introduces, each with its macro and its meaning | K10.4 |
| `audience` | the lowest curriculum level at which the node is taught: `school`, `ug1` to `ug4`, `grad`, `research` | the packer (K11) and the explain levels need it. The build compares it with the curricula and reports a mismatch |

## K8.4 The cards

The table lists the sections of each card in reading order. **R** means that the section is required for W1, **W2** for W2, **W3** for W3, and **o** means optional. Sections in italics are new, and they come in a minor version of node.v1 (K8.9). Until that version, the text of a new section goes into `Remarks` with a level-three heading of the same name.

### Concept

| Section or edge | Level | Contents |
|---|---|---|
| `Summary` | R | two to four sentences: what the notion is and why it matters. No formula is needed. This is the "Idea" paragraph of the nLab, and readers read it more than any other part of a page (K01). |
| *`Motivation`* | W3 | the question that the notion answers, or the example that leads to it |
| definitions (`defines` edges in) | W2 | at least one definition. The card shows each one with its source, and marks the one that Mathlib uses |
| equivalences (`equivalent` edges) | W2, when there are two or more definitions | a statement for each pair of definitions that a source proves equivalent |
| examples (`instance_of` edges in) | W2 | at least two examples |
| non-examples (`not_instance_of` edges in, K8.9) | W2 | at least one non-example, with the reason |
| `specializes` edges | W2 | the more general notions and the special cases |
| *`Notation`* | o | the symbols and the conventions, from the front matter |
| `History` | W3 | who introduced the notion and when, with sources |
| `Explanations` | W3 | L1 to L3, and L4 for research concepts |
| `Remarks` | o | other facts that have no section |

### Definition

| Section or edge | Level | Contents |
|---|---|---|
| `Definition` | R | the definition, with each condition as a separate numbered clause when there are more than two. It names what is defined in bold the first time, and the linter checks that the bold term is the name of the node or an alias. |
| formal binding | W2, when a library has the definition | `formal:` to a `def`, `structure` or `class` |
| examples and non-examples | W2 | one of each, by edges. A non-example fails exactly one clause, and its edge names the clause (K8.4a) |
| *`Notation`* | o | |
| `Remarks` | o | why this definition and not another, and what changes under other conventions |
| `Explanations` | W3 | |

### Statement

| Section or edge | Level | Contents |
|---|---|---|
| `Statement` | R | the proposition. Hypotheses first, then the conclusion. For a statement with several parts, a numbered list, one part in each item. |
| `uses` edges | R | each definition that the statement needs to make sense |
| `about` edges | W2 | the concepts that the statement is about |
| proofs (`proofs:`) | W2 | at least one proof node, informal or formal |
| examples (`instance_of` or `illustrates` edges) | W2 | one example that shows the statement at work |
| a counterexample to a weaker form | W3, when one is known | the hypothesis that cannot be removed, and the counterexample that shows it |
| formal binding | W2, when a library has the statement; otherwise a formalize job | spec 07.3 |
| `History` | W3 | who stated it first, who proved it first, with sources |
| `Explanations` | W2: L2; W3: each level that applies | |
| exercises (K8.7) | W3, when the node is in a curriculum | at least one |
| `Remarks` | o | generalizations, the converse, common errors |

### Proof

| Section or edge | Level | Contents |
|---|---|---|
| `Idea` | W2 | one to three sentences: the method and the key step |
| `Proof` | R | the argument in numbered steps (K8.5) |
| `uses` edges | R | each node that the argument needs |
| `method` edges | W2 | the methods, from the `methods` topic |
| formal binding | o | a Lean proof of the bound statement |
| *`Variants`* | o | the changes that give a proof of a nearby statement |

### Example and counterexample

| Section or edge | Level | Contents |
|---|---|---|
| `Example` or `Counterexample` | R | the instance, and the check that it is an instance (or that it refutes the target) |
| `instance_of`, `not_instance_of` or `refutes` edges | R | |
| computation | W3, when the check is computational | a `computation` node that runs the check (spec 08) |
| `Explanations` | o | |

### Conjecture and problem

These follow spec 04.2 and the problem page of spec 13.3. The card adds `bounds` edges and a `History` with the year of the first source, both for W2.

### Object

| Section or edge | Level | Contents |
|---|---|---|
| `Description` | R | what the object is, its standard name and its main invariants |
| `same_as` | R | the identifiers in the databases (LMFDB label, OEIS number, GAP id, ATLAS name) |
| invariants | W2 | a table of the main invariants, each with a source or a computation |
| `instance_of` | W2 | the concepts that it is an instance of |

### K8.4a Clauses, examples and non-examples

The research on how people learn definitions says that the best non-examples are near misses: objects that are almost examples (K01). So the corpus makes near misses easy to find and to check:

- Each clause of a definition has a number (K8.4, `Definition`).
- A `not_instance_of` edge has a field `fails` with the number of the clause that the non-example fails. A non-example that fails more than one clause lists them all, and it counts as a near miss only for a single clause.
- An `instance_of` edge can list the clauses that the example shows at work.
- The build reports **clause coverage** for each definition: the clauses that have a near-miss non-example. At W3, each clause of a definition with three or more clauses has one.

## K8.5 Proofs in steps

An informal proof in the corpus is a list of numbered steps. This form serves three readers at once: a person reads it as a proof, a formalizer reads it as a plan, and the packer can show it at two levels of detail.

- **Each step states one claim and its reason.** The reason is a link to a node, an earlier step, or a computation: "3. $H$ has index 2 in $G$, by step 2 and [[Lagrange's theorem|7K3QX2M9R2]]."
- **Steps can have substeps** (3.1, 3.2) to two levels. A step with more than two levels is a lemma, which is its own node.
- **The last step is the conclusion** of the statement, in the words of the statement.
- **A step can be marked `routine`.** A short form of the proof leaves out the routine steps. The marks come from the writer and a person can change them.
- **The form is Markdown.** Steps are a numbered list, and substeps are a nested list. This is the structured proof of Lamport in plain Markdown, with no new syntax.
- **Map to Lean.** When a proof has a formal version, each step can name the `have` or the lemma that matches it. This is the "Unfold this proof" view of spec 13.4 in the other direction. The prove role (spec 12.4) uses the steps as its plan.

A proof that a model wrote is E1 (spec 04.7). The card does not change that. It makes the proof easy to check: a reviewer checks one step at a time, and a formalizer proves one step at a time.

## K8.6 Explanations at levels

Spec 04.5 gives four levels. The card fixes their purpose and their length:

| Level | For | Stage (Tao) | Purpose | Length |
|---|---|---|---|---|
| L1 | a person who has not met the concept | pre-rigorous | what it is, with one picture or one concrete example, no proof | 80 to 250 words |
| L2 | an undergraduate | rigorous | the definition unpacked, why it is true or why it is defined this way, the standard examples and non-examples, the common errors | 150 to 500 words |
| L3 | a graduate student | post-rigorous | the right way to think about it, how it fits with the rest of the field, the generalizations, where it breaks | 150 to 600 words |
| L4 | a researcher | post-rigorous | the state of the art, the open questions, the literature | 100 to 500 words, with sources |

The stage gives the writer and the reviewer a test that does not depend on the label of the audience. The evidence on worked examples (K01) gives three more rules:

- L1 and L2 use **correct** worked examples, written as numbered steps, so that a renderer can fade them: it shows the first steps and asks the reader for the rest. An example with an error for the reader to find is an exercise, not an explanation.
- L3 and L4 do not include full worked computations. They help novices and can hurt advanced readers (the expertise reversal effect).
- A guide on how to read a proof is one explanation node at the level of a topic, linked from proof pages. It is not repeated on each proof.

A node does not need each level. A research lemma needs L3 or L4, not L1. A school concept needs L1 and L2. The card of each kind says which levels W3 needs. The `audience` field (K8.3) decides it: each level from the audience up to L3, and L4 for audiences `grad` and `research`.

## K8.7 Exercises

Courses need exercises, and the textbook layer has many of them under open licences (K05). The corpus holds an exercise as its own node, so that a course can choose from a pool:

| Section or field | Contents |
|---|---|
| `Exercise` | the task, self-contained as a statement is |
| `Hint` | optional, one or two sentences |
| `Answer` | the final answer, when the task has one |
| `Solution` | a worked solution in the step form of K8.5 |
| `about`, `uses` edges | the nodes that the exercise practises and needs |
| `difficulty` | 1 to 5, set by the writer and corrected by use |
| `math_group` | `A`, `B` or `C`, from the MATH taxonomy: A is facts and routine procedures, B is transfer and application, C is justification, conjecture and evaluation. The groups describe the kind of task, not its difficulty |
| `check` | how a machine checks an answer: `exact` (a value), `numeric` (a value with a tolerance), `choice` (multiple choice), `cas` (equal up to a CAS normal form, spec 08), `lean` (a Lean statement with `sorry` that the answer must fill, graded with the comparator), or `none` (checked by people) |
| `external` | links to interactive versions in other systems, for example a WeBWorK OPL path or a MyOpenMath id. Links only: most of those banks have licences that the prose zone cannot take (K05) |

The build reports the groups of the exercises of each topic. Banks that models write tend to have few exercises in group C, and the report shows that gap.

An exercise with `check: lean` is also a formal statement. Its reference solution must compile, and the version with `sorry` must have the same type. A course can then grade it with the Lean server (spec 11.3), as the Natural Number Game does.

## K8.8 What the build computes from the cards

- **The writing level** of each node (K2.4), from the sections that are present, the edges, and the provenance of each section.
- **The card findings**: a required section that is missing, a section that is too long, a level that uses a node outside its prerequisites, a style finding. These are warnings, not audit failures, except where a rule of spec 15 already applies.
- **The work list** for each node: the sections that are missing for the next writing level. The orchestrator turns the work list into jobs for the roles of K07 (K12).

## K8.9 Proposed schema changes

Spec 04.9 has two kinds of change. Most of the changes below are additive, and they come in a minor version of node.v1. The new kind changes the list of kinds, and spec 04.9 calls that a breaking change. It comes in node.v2, with a migration and a release in which both versions are accepted.

| Change | Type | Version | Reason |
|---|---|---|---|
| edge `not_instance_of` (example → concept or definition), with the field `fails` | new edge | node.v1 minor | non-examples are half of a good definition card, and `refutes` is for statements (K8.4a) |
| edge `illustrates` (example → statement) | new edge | node.v1 minor | an example that shows a statement at work is not an instance of a concept |
| sections `Motivation`, `Notation` (concept and definition), `Variants` (proof) | new allowed sections | node.v1 minor | K8.4 |
| fields `inventory`, `conventions`, `notation`, `audience` | new optional fields | node.v1 minor | K8.3 |
| `schema/cards.v1.yaml` | new file | none | the cards as data |
| kind `exercise` with the sections and the fields `difficulty`, `math_group`, `check` and `external` of K8.7 | new kind | node.v2 | courses need exercises, and an exercise is not an example: it has a task and an answer |

The migration to node.v2 finds each `example` node that has the role `exercise` in `## Remarks` (K2.3). It changes the kind, and it moves the text into the sections of K8.7. The tag does not change. Spec 21 gets one entry for the new kind.
</document>

<document source="https://math2.tamnd.com/spec/corpus/09-quality.md">
# K09. Quality: how the corpus stays correct

The corpus is written at a scale that no group of people can read in full. Most first drafts come from importers and models. This document says which checks each piece of text goes through before it merges and after it merges, how the corpus measures its own error rate, and what happens when a check finds a problem.

The main spec already has the hard part: status is computed (spec 04.7), only the kernel and the certificate checkers referee E4 and E5 (settled decision 3), and fidelity above F1 needs people (spec 10.6). This document does not change these rules. It adds the checks that a corpus of millions of nodes needs on top of them.

## K9.1 Four properties

A node is good when it has four properties. Each property has its own checks, and no check covers two properties:

| Property | Question | Checked by |
|---|---|---|
| **true** | Is the statement true, and does the proof prove it? | the E levels: kernel, certificates, refereed and textbook sources (spec 04.7). Before that: the checks of K9.3, which only triage |
| **faithful** | Does the node say what its sources say, and does its formal binding say what the node says? | source confirmation (K9.5), the F levels (spec 10) |
| **readable** | Does the node follow its card and the house style? | the card findings and the style linter (K08), the writing level (K2.4) |
| **ours to publish** | Is each byte of text allowed under the prose licence? | the licence gate (spec 9.3), the copy check (K9.4) |

The reports give a separate number for each property. A good score in one property never hides a bad score in another.

## K9.2 The gates before merge

Each pull request that adds or changes a body goes through these gates in this order. A gate that fails stops the pull request. The failure has a reason, and the reason goes back to the job (K12) as output.

| Gate | Checks | Applies to | Hard or soft |
|---|---|---|---|
| G1. Schema and body | `m2 check`: rules S01 to S04 and B01 to B03 | each change | hard |
| G2. Card | the required sections of the target writing level (K8.4) | each change | hard for the level that the job asked for, soft otherwise |
| G3. Style | the linter of K8.2 | each body | soft below W3, hard for a W3 job |
| G4. Copy check | K9.4 | each body that an agent wrote, and each body from an importer of share-alike text | hard |
| G5. Sources | K9.5 | each source that the change adds | hard for a source that an agent proposed |
| G6. Sanity | K9.3 | each statement and proof that an agent wrote | soft: a failure opens a counterexample job and blocks only auto-merge |
| G7. Formal | rules F01 to F09 | each change to a binding | hard (spec 15.3) |
| G8. Review capacity | the cap of spec 11.5 | each agent pull request | hard: the pull request waits, it is not closed |

The gates run in the agent loop before the pull request opens (spec 12.2), and again in CI. An agent sees the result of each gate as a tool result, so it can repair its draft in the same run.

## K9.3 Sanity checks on statements and proofs

These checks catch false statements and broken proofs before a person spends time on them. **They never raise an E level and they never count as a review.** A pass means "no problem was found", and nothing more.

1. **Small cases.** When a statement quantifies over integers, finite groups, graphs, matrices of small size, or any family that the computational engine (spec 08) can enumerate, the explore role writes a test and runs it for small cases. A failure is a counterexample candidate. It opens a computation node and a job for a person, and it blocks the merge of the statement. Most false statements that models write fail at small cases.
2. **Formal round trip.** When the formalize role can bind the statement, the F1 battery of spec 10.4 runs. A back-translation that differs from the informal text shows an error in one of the two. The finding names the difference.
3. **Independent check by other models.** Models of other families read the statement or the proof one step at a time (K8.5) and report each step that they cannot follow. They get the node, the linked nodes and the sources, and not the run that wrote the node. The evidence on model judges sets the rules (K01, K7.6):
   - A judge verdict is triage, never evidence. Judges gave proofs scores up to 20 times higher than human graders did, and the best step judges still accept about one wrong step in five.
   - A judge model must have a measured rate of rejecting wrong steps of 75% or more on a labelled set of math2 before it can judge proofs or statements. Tier 0 models do not judge proofs.
   - The check passes only when each judge passes the node. When the judges disagree, the node goes to a person, and no extra model call breaks the tie.
   - Each verdict and its model id go in the run record. Agreement is used only for triage (settled decision 3).
4. **Hypothesis check.** For each hypothesis of a statement, the check asks whether a known counterexample shows that the hypothesis is needed. When a hypothesis is not needed, the statement is still true, but the node gets a remark: either the statement is weaker than it can be, or a source states it that way for a reason that the node must give.
5. **Prove or disprove.** A job that proves or formalizes a statement asks the model to prove it or to disprove it, not to prove it. Models "prove" false statements often, and this framing made one model reject false statements almost four times as often (K01). Each month, the queue gets canaries: false statements made by small changes to true ones. The weekly report gives the share of canaries that each configuration "proved", and a share above 5% stops the configuration.
6. **Names of theorems.** A theorem that the prose names ("by the theorem of X and Y") links to a node or has a source. Models invent theorems that match what they want to prove, so a name that resolves to nothing is a finding (rule X07).
7. **Proof dependencies.** Each `uses` edge of a proof points to a node at E2 or better, or the proof gets the `sorry_in_closure` flag (spec 04.7). An agent proof that uses an E1 node can merge, but it shows the flag.

## K9.4 The copy check

Models reproduce text that they saw in training, even when nobody asks them to. In one study, more than 27% of the text that models wrote in the style of an encyclopedia was in verbatim runs of 50 characters or more, and one model reproduced 1,170 characters of a Wikipedia article (K01). For mathematics the risk is higher, because standard proofs appear in many sources with almost the same words. The corpus must not publish text that the licence gate refuses (K2.6, rule 3). So each body that an agent wrote goes through a copy check before it merges.

- **Compare with a reference corpus, not only with the sources of the node.** A model that writes from memory cites nothing, so a check against the cited sources finds nothing. The reference corpus holds the mathematical pages of Wikipedia, ProofWiki, the nLab and PlanetMath, the dumps of Math.StackExchange and MathOverflow, the Stacks Project (GFDL), Kerodon (no licence, so `record`), the open textbooks of K05, and, for detection only, OpenWebMath and the arXiv text that the record plane can lawfully read (spec 9.3). Each text in it has its licence class. For sources that math2 cannot read, the check cannot run, and K9.6 covers the gap.
- **Normalization.** Lowercase, whitespace collapsed, LaTeX macros made canonical. Formulas are masked: each formula becomes a placeholder, so that a standard formula does not count as a copy. A separate pass compares the structure of formulas, so that a renamed variable does not hide a copied proof.
- **Stage 1, exact runs.** An index of suffixes over the reference corpus finds each run of the body that is also in a reference text. A run of 50 characters or more is a flag. A run of 100 characters or more in one source, or flagged runs that cover 10% or more of a section, is a **match**.
- **Stage 2, near copies.** A filter for exact runs alone gives false safety: in the studies, small edits pass it, and a near copy counts almost as much as an exact one (K01). So each paragraph also goes through MinHash on word 5-grams. A candidate pair with an estimated Jaccard similarity of 0.5 or more is a match if it also has a common block of 20 words or more after alignment, or a ROUGE-L of 0.8 or more.
- **Stage 3, paraphrases.** A sample of paragraphs, with their nearest reference paragraphs by embedding, goes to people. This stage does not block a merge. It measures what the first two stages miss, for the sample audit (K9.7).
- **What a match does.** A match with a source of class `record`, `verbatim`, `share-alike-nc` or `gfdl` blocks the merge, and the agent gets the locations to rewrite. A match with an `open` source is allowed when the licence of the source is compatible, and the node then gets the attribution of rule L03 in its provenance. Each match records the source, its URI, its licence and the offsets, in the run record.
- **Standard phrases.** Some text is the same in each source because there is only one way to say it, for example "let $G$ be a group". A list of standard phrases, kept by curators, does not count towards a match. A phrase gets onto the list only by a curator decision.
- **Calibration before blocking.** The thresholds above are a start, and nobody has tested them on mathematics. Before L05 blocks merges, the copy check runs over pages that people wrote, compared with each other: Wikipedia against ProofWiki, and open textbooks against each other. The median overlap of human text is 0% in the studies. The thresholds are set so that the rate of false matches on human text is under 1%, and K9.7 tunes them on samples after that.
- **A first layer in the prompt.** Each prompt for prose tells the model to write in its own words and not to reproduce a source. In a study, this cut the overlap by half or more, but long copies still came through. It is a cheap first layer and not a control.

Rule L04 of spec 15.3 is soft for text from people. The corpus adds a hard rule for text from agents:

| Rule | H/S | Checks |
|---|---|---|
| L05 | H | each agent-written body has a copy check result in its run record, and the result has no blocking match. Until the calibration is done, L05 is soft |

## K9.5 Source confirmation

Models invent references that look correct. A wrong reference is worse than none, because readers trust it. So an agent cannot add a source by itself:

1. **The record must exist.** A source is a record in the record plane: a DOI that Crossref resolves, a zbMATH id, an arXiv id, an ISBN with a catalogue record, or a record of an open textbook (K05). An agent proposes a source as an identifier. If the identifier does not resolve, the proposal is dropped, and the run record counts it as a fabricated source.
2. **The record must match.** The title, the authors and the year of the record must match what the agent claimed. A mismatch is a fabricated source too.
3. **The locator must be checked.** A locator says where in the source the item is: a theorem number, a section, a page. When the record plane can read the text, a check finds the locator and compares the item with the node. When it cannot read the text, the source has the flag `locator_unconfirmed`, and only a person can remove the flag.
4. **E2 and E3 need confirmed locators.** The build counts a source towards E2 or E3 (spec 04.7) only when its locator is confirmed. A node can show an unconfirmed source as further reading.

The rate of fabricated sources for each model and prompt version goes into the weekly agent report (spec 12.9). A rate above 2% stops source proposals from that configuration until a person looks at it.

| Rule | H/S | Checks |
|---|---|---|
| P05 | H | each source that an agent added resolves to a record in the record plane, and the record matches the claim |
| P06 | S | a source with `locator_unconfirmed` older than 180 days |

## K9.6 What people check

People are the scarce resource (settled decision 20). The corpus uses them only where a machine cannot check:

| Check | Done by | Why a machine cannot do it |
|---|---|---|
| fidelity of a formal statement (F2, F3) | reviewers, with `m2 review` (spec 16.3) | settled decision 4 |
| confirmation of an agent-written section (rule P04) | curators and area editors | the section claims to explain; only a reader can say whether it does |
| an unconfirmed locator | anyone with access to the source | the record plane cannot read most of the literature |
| a sanity finding (K9.3) | area editors | a finding can be a false alarm, and only a person can decide |
| a copy match that the agent cannot fix | curators | the decision whether a phrase is standard |

The review-assist role (spec 12.1) makes a packet for each check, with the evidence in one place. Each check has a target time, and spec 17 measures it. A person can confirm many sections in one session, but each confirmation is recorded for each section, not for the session.

## K9.7 Sample audits and published error rates

Gates and reviews catch errors one at a time. They cannot say how many errors remain. The sample audit can.

- **The strata.** Each month, the audit draws a random sample of merged nodes in each stratum. A stratum is a combination of kind, writer (person, importer, agent role), model and prompt version, and layer (K2.7). A new configuration gets its own stratum from its first merge.
- **The sample size.** With no errors in *n* samples, the upper end of the 95% confidence interval for the error rate is about 3/*n*. So 300 samples with no errors show a rate below 1%. The audit uses 100 samples in each stratum each month as a minimum, and more where the rate is near a threshold. The report uses the Wilson interval when the sample has errors.
- **The audit is blind.** Auditors do not see who or what wrote the node. They check the four properties of K9.1 against a written checklist for each kind.
- **Error classes for prose.** No study has measured how often models write a wrong definition or drop a hypothesis in informal text (K01). The corpus must measure it. The checklist classifies each error in a body: wrong definition, dropped hypothesis, statement stronger than its source, wrong name or attribution, convention not stated, step that does not follow, wrong source. The classes sit next to the misformalization classes of spec 10.5, so that the release notes give error rates for prose and not only for bindings.
- **The thresholds.** For each stratum, the error rate of each property must stay under a threshold: 1% for a false statement, 2% for an unfaithful statement or a wrong source, 5% for a style or card error. These numbers are a start, and a change to them is a decision that the release notes record.
- **Above the threshold.** When the upper end of the interval for a stratum passes its threshold, the configuration stops. Its unconfirmed output goes to quarantine by a generated pull request, as in the withdrawal test of spec 06.5. A person decides whether to fix, rerun or retire the output.
- **The publication.** Each release publishes the error rates of each stratum, with the sample sizes and the intervals, including the strata that failed. This is rule 6 of K2.6.
- **Small strata.** A stratum with fewer than 1,000 merges in a month pools its samples over three months. A stratum with fewer than 100 merges in three months is audited in full.

The sample audit costs reviewer time. At 100 samples in each of 20 strata, and 5 minutes for each sample, it costs about 170 hours each month. K12.5 counts this cost in the review budget first, before the agents get their share. When review time is short, the corpus runs fewer configurations. It does not take smaller samples, because a rate that nobody measured is not a rate.

## K9.8 Reports from readers

Each page of the workspace has a "report a problem" link. A report becomes an issue with the tag of the node and the property that the reader thinks is wrong. A report on a node at E5 that says the statement is false goes first in the queue, because it is a report on the fidelity of the binding or on the checker.

A confirmed report is an error that the sample audit did not catch. The report counts it in the stratum of the node, so that the published rates include it.

## K9.9 The rules that this document adds

| Rule | H/S | Checks | Section |
|---|---|---|---|
| L05 | H | copy check result present and clean for agent-written bodies | K9.4 |
| P05 | H | agent-added sources resolve and match | K9.5 |
| P06 | S | unconfirmed locators older than 180 days | K9.5 |
| X05 | H | an agent-written statement or proof has the results of the sanity checks of K9.3 in its run record | K9.3 |
| X06 | S | a stratum whose error rate is near its threshold | K9.7 |
| X07 | S | a theorem that a body names and that resolves to no node and has no source | K9.3 |

With the rules of K03 (T08 to T10) and K04, the catalogue of spec 15.3 grows from 65 rules. Each new rule needs a fixture pair, as spec 15.3 says.
</document>

<document source="https://math2.tamnd.com/spec/corpus/10-structure-and-notation.md">
# K10. Structure, notation and conventions

A node has no position (K8.2). The corpus still needs structure, for two reasons. A reader must find a node and the nodes around it. A packer must put nodes in an order that a reader can follow (K11). This document says which structures the corpus keeps, and how it keeps notation and conventions the same across a million nodes from hundreds of sources.

## K10.1 Three structures

The corpus has three structures. Each comes from different data, and none of them is a copy of another:

| Structure | Made of | Answers | Section |
|---|---|---|---|
| the logical graph | `uses`, `proves`, `implies`, `specializes` and the other edges of spec 05.2 | what must I know to read this node, and what follows from it? | spec 05 |
| the taxonomy | topic nodes, MSC2020 codes, `about` edges | where does this node belong, and what else is there? | K10.2 |
| the paths | curricula and learning paths: topics with ordered members | in what order do I read these nodes? | K10.3 |

The logical graph is the truth. The taxonomy and the paths are views that people curate. A path that puts a node before one of its prerequisites is a finding (rule N03), because the path is wrong, not the graph.

## K10.2 The taxonomy

- **MSC2020 is the base.** Each node has at least one MSC2020 code (spec 18.2). `topics/msc2020.yaml` maps each code to a topic node, with math2's own short labels (the official descriptions have a licence that does not allow them in math2, spec 20.1).
- **Topic nodes for what MSC splits badly.** Some subjects that people teach as one cross several MSC codes, for example "linear algebra" or "measure-theoretic probability". A topic node holds them, with `about` edges from its members. The skeletons of spec 18 are topic nodes of this kind.
- **Concepts give the fine structure.** Below the level of a topic, the structure is the concept nodes and their `specializes` edges: "metric space" specializes "topological space", and "Banach space" specializes "normed space". A concept with no `specializes` edge in or out is a finding of graph health (spec 05.8).
- **Crosswalks, not imports.** The categories of Wikipedia, the structure of the nLab and the module tree of Mathlib are not the taxonomy of math2. They are crosswalk data: the clustering (K4.3) and the classifier of MSC codes use them, and the workspace can show them.

## K10.3 Curricula and learning paths

A **curriculum** is a topic node in `topics/curricula/` with an ordered list of members. It describes a course, an exam syllabus or a book. Its members are nodes, and each member can have a weight: `core`, `standard` or `optional`.

The corpus imports curricula from public sources as data, not as text (K05). A curriculum lists the items that it names, so it is also an inventory source (K4.2), and its items enter the `curriculum` term of the priority (K12.3). Examples of sources: the lists of undergraduate mathematics in Mathlib, published qualifying exam syllabi, the tables of contents of open textbooks, and national school curricula that are published under an open licence.

A **learning path** is computed, not curated. For a set of target nodes and a reader who knows a set of nodes, the path is an order of the prerequisite closure (spec 05.5) of the targets, minus what the reader knows:

1. Take the closure of the targets over `uses` edges and `defines` edges, with the alternatives of spec 05.5. Where a concept has more than one definition, take only the definition that the path chose (K11.1, `choose`), or by default the one with the smallest closure. Where a statement has more than one proof, choose the proof with the smallest closure that is not known yet.
2. Remove the nodes that the reader knows, and their closures.
3. Order the rest topologically. Among the orders, prefer the one that keeps the nodes of one topic together and follows the order of a curriculum where one exists.

The packer (K11) uses learning paths to order a volume. The workspace uses them for "what do I need to read first?".

The choice of one definition for each concept matters. Metacademy gave each concept one set of prerequisites and named this as its main limit (K01). A concept with two equivalent definitions, such as compactness by open covers and by filters, has two sets of prerequisites. The path uses one of them, and the equivalence is a theorem that the path can add.

The workspace also answers **"what can I read next?"** For the nodes that a reader knows, the answer is the nodes whose prerequisites are all known and which are not known. Knowledge space theory calls this set the outer fringe, and the corpus computes it with one query on the graph.

## K10.4 The notation register

Each source has its own notation, and the same symbol means different things in different areas. A corpus written from hundreds of sources mixes them, unless something stops it. The notation register stops it.

The design takes one idea from sTeX: **a symbol is an identity, and a notation is a way to print it** (K01). Today `macros/m2.sty` and `macros/m2.katex.json` are render macros only. They say how to print `\Nat`, but not what `\Nat` means. The register adds the meaning, and the two macro files become build outputs of the register.

The register is a set of files `notation/<topic>.yaml` in `tamnd/math2`, one for each topic, plus `notation/global.yaml` for the symbols of all areas. Each entry is one macro:

```yaml
- macro: \totient             # the only form that a node body uses
  denotes: 2HQ7XKM4CP         # the tag of the concept or definition
  args: 1
  prec: null                  # for operators: the binding strength
  latex: \varphi(#1)          # the expansion for m2.sty and KaTeX
  typst: phi(#1)              # for the Typst output, when it exists
  intent: null                # a MathML intent concept name, when one exists
  openmath: null              # an OpenMath symbol, for alignment
  reads: "phi of #1"          # how a person reads it, for screen readers
  lean: Nat.totient           # the Lean declaration or notation, when one exists
  scope: [number-theory]      # the topics where the macro is in use
  introduced_by: 2HQ7XKM4CP   # the definition node that introduces it
  convention: null            # one line for the list of notation, when it applies
  variants:                   # other ways to print the same thing
    - {name: phi, latex: \phi(#1), sources: [...]}
  conflicts: [geo.goldenratio] # entries in other scopes that print the same glyph
```

The rules:

- **One macro, one meaning.** Each macro has one `denotes`. A glyph with more than one meaning gets one macro for each meaning: `\pair`, `\ointerval` and `\gcd` for the three uses of $(a,b)$, and `\totient` and `\goldenratio` for the two uses of $\varphi$. A body then says what it means even when the printed symbol is the same. This is also what makes the `intent` of MathML and the screen reader text correct.
- **One default notation for each concept.** A concept can have variants, and a volume or a reader can choose one (K11.1). A body always uses the macro, so the choice changes only the output.
- **A body uses macros in its scope.** The scope of a node is its topics. A body can use a macro of the global scope, a macro whose `scope` has one of the topics of the node, or a macro that the node introduces with its `notation` field (K8.3).
- **Glyphs can repeat across scopes, with a record.** Two entries can print the same glyph only if they are different macros. The `conflicts` field records the pair, and the pack check (PC4) stops a volume that uses both.
- **New notation has a home.** A definition that introduces a symbol is the `introduced_by` node of the entry. The packer uses this to place a notation item where the symbol first appears and to build the list of notation, as PreTeXt does with `<notation>` and `<notation-list>`.
- **Generated files.** The build writes `macros/m2.sty` and `macros/m2.katex.json` from the register. A pull request that edits them by hand fails the build. Rule B01 (declared macros only) stays, and the register adds its meaning.
- **Alignment.** The `openmath` and `intent` fields are alignment data, like a Wikidata id. The export of symbols to Wikidata uses the pattern of the properties P7235 and P9758 (K01).

The write role gets the entries of the scope of its target in its context, as part of the cached prompt (K12.4).

## K10.5 Conventions

Some definitions differ between sources by convention: whether $\mathbb{N}$ contains 0, whether a ring has a unit, whether a compact space is Hausdorff, whether a graph can have loops, which argument of an inner product is linear, where the factor $2\pi$ goes in a Fourier transform. A convention changes what a statement means, so the corpus records it.

- **A register of conventions.** `notation/conventions.yaml` lists each convention: an identifier, the options, the sources that use each option, and the default of math2.
- **The default.** The default follows Mathlib where Mathlib has one (settled decision 16), so that the informal text and the formal binding agree. Where Mathlib has no choice, the default follows ISO 80000-2 where that standard has one, and a curator decides the rest. For $\mathbb{N}$, Mathlib and ISO 80000-2 agree: 0 is a natural number, and ISO writes $\mathbb{N}^*$ for the positive integers.
- **Nodes declare the conventions they depend on.** A concept node lists the conventions that change its meaning (K8.3), in short form: "ring: unital, not necessarily commutative". A definition or statement that depends on a convention links the concept that carries it, so that a change of option finds each node that it affects. A node that uses an option other than the default says so in its main section and in the front matter.
- **Bindings must agree.** The F1 battery (spec 10.4) checks that the convention of the informal text and the convention of the formal binding are the same. A mismatch is the misformalization class "convention" of spec 10.5.
- **Volumes choose.** A volume (K11) can choose another option for a convention. The packer writes a Conventions section from the conventions of its nodes, as each chapter of the Stacks Project has one. The pack check (K11.6) reports each node in the volume whose text depends on the convention, so that an editor can check it.

## K10.6 Names and languages

- **One canonical name for each node**, in English, with the aliases in `registers/aliases.reg`. The canonical name is the one that most sources in the row use (K4.1). A curator decides when the sources disagree.
- **Names of people** follow the form that the MacTutor archive or Wikidata uses, with the diacritics: "Hahn–Banach", "Erdős".
- **Other languages.** Wikidata gives labels in many languages under CC0, and the corpus can import them as aliases with a language code. Translations of bodies follow spec 13.4. A translation is a view of the English text, and it never changes a statement.

## K10.7 Rules

| Rule | H/S | Checks |
|---|---|---|
| N01 | H | each macro in a body has an entry in the notation register whose scope includes a topic of the node, or the global scope, or the node defines it |
| N02 | H | two entries with the same glyph in scopes with a topic in common are the same macro, or each lists the other in `conflicts` |
| N03 | S | a curriculum puts a member before one of its prerequisites |
| N04 | S | a node uses an option of a convention other than the default and does not say so in its front matter |
| N05 | S | a concept with no `specializes` edge and no topic other than its MSC code |
| N06 | H | each macro in the register has one `denotes`, and `macros/m2.sty` and `macros/m2.katex.json` are the output of the register |
| N07 | S | a definition introduces a symbol in its `notation` field, and no register entry has it as `introduced_by` |
</document>

<document source="https://math2.tamnd.com/spec/corpus/11-packaging.md">
# K11. Packaging: from nodes to courses, books and articles

The goal says that each node is "ready to go into a course, a book or an article" (K2.1). This document says what that means as a process that a machine can run and check. A **volume** is a selection of nodes with an order, some editorial text and a set of choices. The **packer** compiles a volume into its outputs. The **pack check** says whether the volume is ready.

The packer adds no mathematics. Each definition, statement, proof and example in a volume is a node, and the volume shows it as the node says it. This keeps one truth: an error that is fixed in a node is fixed in each volume at its next build.

## K11.1 The volume file

A volume is a file `volumes/<id>.yaml` in `tamnd/math2`:

| Field | Holds |
|---|---|
| `id`, `title`, `editors` | the identity of the volume and the people who are responsible for it |
| `kind` | `course`, `book`, `notes`, `survey`, `article`, `reference` or `problems` (K11.5) |
| `audience` | the level of the reader, as in K8.3 |
| `targets` | the nodes, topics or curricula that the volume must cover |
| `assumes` | the nodes that the reader knows. The volume lists them and does not explain them |
| `release` | the math2 release that an edition pins |
| `levels` | the lowest writing level and evidence level that a node must have to go in (K11.6) |
| `proofs` | `all`, `main` (the main proof of each statement, with the routine steps folded, K8.5), or `none` |
| `exercises` | how many exercises for each section, and the difficulties |
| `choose` | for each concept with more than one definition in the closure, the definition that the volume uses. The others become theorems of the form "equivalent characterizations", and the packer adds the proofs of the equivalence (K11.2) |
| `conventions`, `notation` | the options of K10.5 and the notation variants of K10.4 that the volume uses |
| `outline` | optional: the chapters and the sections, each with its nodes. When it is absent, the packer makes it (K11.2) |
| `glue` | the editorial text: K11.3 |
| `formats` | the outputs of K11.4 |

## K11.2 Linearization

A volume becomes a sequence in three steps:

1. **The closure.** The packer takes the learning path of K10.3 from `assumes` to `targets`, over the definitions of `choose` only. Each node in the closure that is not in `assumes` goes in the volume. For each concept in `choose`, the other definitions go in as theorems, with their proofs of equivalence.
2. **The outline.** When the volume has no `outline`, the packer groups the nodes by topic into chapters, and by concept into sections. A graph with no cycles has many orders with no forward reference, and the choice among them is about keeping related nodes close. The packer breaks ties in this order: the outline of the topic nodes, then a statement next to its proof and its examples, then the order of a curriculum where one exists, then the MSC code. An editor can then write an `outline`, and the packer checks it.
3. **No forward references in proofs.** This is the rule of the Stacks Project: a proof does not refer to a result that comes later. When an `outline` breaks it, the packer stops with the list of the references, and pack check PC2 fails. The packer never moves a node silently.
4. **Numbering and references.** The packer gives each node a number in the volume: "Theorem 3.4", "Definition 2.1". A link in a body to a node in the volume becomes a reference with the number. A link to a node outside the volume becomes a citation of the node by its URI and its name, in the form that K11.8 gives.

A node can appear in more than one place only as a reference. It has one home in each volume.

The packer also writes two generated sections: **Conventions**, from the conventions of the nodes in the volume (K10.5), and the **list of notation**, from the `introduced_by` entries of the notation register in reading order (K10.4).

**The lock file.** Each build writes `volumes/<id>.lock`, with the `TAG@release` of each node in the volume and the choices that the packer made. The volume file and the lock file together are the method of the volume. A person can read each decision as data, and a second build from the lock gives the same bytes (PC9).

## K11.3 Editorial text

A book needs text that is not a node: a preface, the introduction of a chapter, a sentence that leads from one section to the next. This is the **glue**.

- Glue lives in the volume file, or in files next to it in `volumes/<id>/`. It is prose in the CC BY-SA 4.0 zone, with provenance as for any other text (spec 04.8).
- **Glue does not state mathematics.** It cannot contain a definition, a statement or a proof. If an editor wants to say something mathematical, it is a node. The pack check finds glue that has the form of a statement (pack check PC7).
- Glue that a model wrote is marked as such, and it needs confirmation by an editor of the volume before an edition (rule P04).

## K11.4 The outputs

The packer has one internal form, a typed document tree, and one back end for each output. The tree holds the nodes, the glue, the numbering and the references. Each back end writes it in one format, and no back end changes the content.

| Output | For | Notes |
|---|---|---|
| HTML | the web, in the site of math2 | the same templates as the site, with the status of each node shown, and JSON-LD on each page (K11.4a) |
| PreTeXt XML | books | the PreTeXt tools make web pages, PDF, EPUB, braille and WeBWorK output from one file, so math2 does not write five back ends. Most kinds map to a PreTeXt element. The notation items map to `<notation>`. The PreTeXt tools are GPL, and math2 runs them as separate programs |
| LaTeX and PDF | print, and arXiv | `amsthm` environments and `macros/m2.sty`, one class for each kind of volume. For an `article`, an arXiv source tree: `main.tex`, a copy of `m2.sty` (arXiv has only the packages of TeX Live), a `00README.json` that names the compiler and the TeX Live year, and a `.bib` from the sources of the nodes |
| JATS 1.4 | articles for journals and repositories | each formula in `<alternatives>` with `<tex-math>` (with no `$` delimiters, as JATS4R recommends) and MathML from KaTeX |
| Common Cartridge 1.3 | courses in a learning management system | HTML pages for each node, QTI items from exercises with `check: choice` or `numeric`, and LTI 1.3 links to the workspace for exercises with `check: lean` or `cas` |
| Verso Blueprint | Lean projects | a blueprint document with one directive for each node, the math2 tag as the label, and the `uses` edges of the graph (K11.9) |
| MyST Markdown | Jupyter Book and the notebook tools | computation nodes become code cells |
| Typst | a second typesetting target | only when the `typst` field of the notation register (K10.4) covers the macros of the volume |
| slides | lectures | Beamer, from a volume of kind `course` with one section for each lecture |

The first back ends are HTML, LaTeX and PreTeXt XML (K14.2). EPUB comes from PreTeXt. The others come when a user asks for them. math2 does not write SCORM: Common Cartridge covers the same need, and its makers say that it does not replace SCORM, so a learning management system that needs SCORM can convert. Records of what learners did (xAPI) belong to the workspace (spec 13), never to `tamnd/math2`.

### K11.4a Metadata on each page

The site and each HTML output put JSON-LD of schema.org on each page:

| Property | From |
|---|---|
| `@type` | `DefinedTerm` for a concept, `LearningResource` for an explanation or an exercise |
| `teaches` | the concepts of the node |
| `competencyRequired` | the direct prerequisites: `uses` at depth 1 |
| `assesses` | for an exercise, the nodes that it exercises |
| `educationalLevel` | L1 to L4 (K8.6), as terms of a published SKOS scheme, not free text |
| `learningResourceType` | an LRMI term, for example "worked example" or "exercise" |

This metadata is CC0, like the rest of the metadata zone.

## K11.5 Kinds of volumes

| Kind | Made from | Has |
|---|---|---|
| `course` | a curriculum, with a schedule | sections for each lecture, exercises, problem sheets, solutions in a separate output for teachers |
| `book` | a topic or a set of targets | chapters, full proofs, exercises, an index of terms and of notation |
| `notes` | a short set of targets | proofs `main`, few exercises |
| `survey` | a topic in the research layer | statements without proofs, the history of each result, the citations of the research layer (the `aligned` records), the open problems |
| `article` | a small set of targets with a new result | the format of a journal, with the formal status of each claim |
| `reference` | a topic | definitions and statements only, in the order of the taxonomy, as a handbook |
| `problems` | exercises and problems | an exam pack or a problem book, with solutions |

## K11.6 The pack check

`m2 pack -check VOLUME` says whether a volume is ready. It runs with no model and no network, on a pinned release. Its checks have the prefix PC, so that they are not confused with the audit rules of spec 15.3:

| Check | Requires |
|---|---|
| PC1 levels | each node meets the `levels` of the volume. Each statement that the volume shows as a theorem is at E2 or better. A statement below that is shown with its level, for example as a conjecture or as "claimed" |
| PC2 order | no node comes before one of its prerequisites, and no proof refers to a result that comes after it (K11.2) |
| PC3 closure | each `uses` edge of each node in the volume ends in the volume or in `assumes` |
| PC4 notation | each printed glyph has one meaning in the volume (K10.4). Two macros in the `conflicts` of each other in one volume are a failure, unless the volume chooses a variant for one of them |
| PC5 conventions | each node that depends on a convention uses the option of the volume, or the volume lists it with the difference |
| PC6 references | each reference resolves, in the volume or to a citation |
| PC7 glue | the glue contains no statement, definition or proof, and each part of it has a confirmed writer |
| PC8 licence | the attribution of K11.7 is complete, and rule L02 is clean for each node in the volume |
| PC9 outputs | each output format builds with no error, and the build is reproducible: the same inputs give the same bytes |
| PC10 exercises | each section has the exercises that the volume asks for, and each exercise with an answer check passes it |

The **standard volumes** are the volumes that the corpus itself maintains, one for each area of the undergraduate core (K14.5). The goal of K2.1 counts as met for an area when the pack check passes on its standard volume.

## K11.7 Licence and attribution

- **The prose of a volume is CC BY-SA 4.0**, like the prose of math2 (settled decision 17). A volume cannot have a licence with more restrictions. Anyone can print and sell it, as the licence allows.
- **The Lean companion is Apache-2.0**, like the code of math2. So are the Lean statement and the reference solution of an exercise with `check: lean`: they are Lean code. The text of the exercise is prose.
- **The attribution is generated.** The packer writes an attribution page from the provenance of each node and each part of the glue: the people who wrote and reviewed the content, the share-alike sources that rule L03 requires, and the disclosure of agent contributions that `m2 prov disclose` generates (settled decision 24).
- **Credit to people goes first.** The page lists people by their contributions to the volume. The list of agent runs comes after it, in the form of the disclosure.

## K11.8 Editions, citations and errata

- **An edition pins a release.** The edition of a volume records the math2 release, so that its content is fixed. The volume has its own version, `<id>@<release>`.
- **Citations of nodes.** A volume cites a node outside it by its name and its URI, for example "Sylow theorems, `m2://7K3QX2M9R2`". Printed works can show the tag in the reading form of K3.4.
- **Errata are generated.** When a node of an edition changes in a later release in its main section, the next build of the volume lists the change as an erratum of the edition. A change that only adds sections or explanations is not an erratum.
- **A volume can be a citable work.** Its editors can ask for a DOI for an edition. This is a publication, and it needs the approval of spec 06.2.

## K11.9 The Lean companion

Each volume can have a Lean companion: a Lake project that requires Mathlib and the math2 Lean library at the commits of the release.

- It has one file for each chapter. For each node with a binding, the file has a reference to the declaration with the number of the node in the volume as a comment, and the proof when math2 holds it.
- For each statement with no formal proof, it has the formal statement with `sorry`, marked as open, and a Verso Blueprint of the dependency graph (K11.4). The companion is then a plan for the formalization of the volume.
- The blueprint keeps the origin of each edge: "automatic" when Lean found it, "manual" when the graph of math2 has it and Lean does not. The import of blueprints (spec 09.4) keeps the same distinction in the other direction.
- The companion builds in CI with the release. A companion that does not build fails pack check PC9.
</document>

<document source="https://math2.tamnd.com/spec/corpus/12-pipeline-and-operations.md">
# K12. Pipeline and operations

This document says how the corpus is produced: the stages that an item goes through, the jobs and the queues, the limits on speed, the cost, the commands and the libraries. It uses the orchestrator of spec 11.4 and 11.5 and the roles of spec 12.1 and K7.3. It adds no new write path. Each change is a pull request, and the audit is the gate (settled decision 7).

## K12.1 The stages of an item

An item goes from "a source mentions it" to "a complete node" in stages. Each stage has a goal predicate that the build can compute, so a job knows when it is done.

| Stage | Goal predicate | Done by | Section |
|---|---|---|---|
| 1. Listed | the item is a mention in a source catalogue, or in a list that a model made | importers (K05, K06), the enumerate role (K7.3) | K4.2 |
| 2. Inventoried | the mention belongs to an inventory row | clustering (K4.3), the dedup judge, curators for close cases | K4.3 |
| 3. Decided | the row has an outcome: a node, a member of a hub, an `aligned` record, or a reason why it is not a node | the curate role proposes, the rules of K4.4 decide, curators decide the rest | K4.4 |
| 4. Stub | a node at W0 | the curate role, importers | K2.4 |
| 5. Stated | W1 | importers of `open` text, the write role, the informalize role for formal-first items | K08 |
| 6. Usable | W2 | the write, explain and exercise roles; the prove and formalize roles in parallel | K08 |
| 7. Complete | W3 | the same roles, and people who confirm sections | K08 |

The formal track runs next to stages 5 to 7. It does not wait for them:

| Formal stage | Goal | Done by |
|---|---|---|
| bound | a binding at F0 | formalize, or align for items that a library already has |
| checked | F1 | the F1 battery (spec 10.4) |
| reviewed | F2 or F3 | people (spec 10.6) |
| proved | E5 | prove, or an import from a library with a checked proof |

An item that a formal library has (K06) starts on the formal track at "bound" and reaches stage 5 through the informalize role. An item from a textbook starts at stage 5 and reaches the formal track through formalize. Both paths end at the same node.

## K12.2 Jobs

The corpus adds standing queues to the four sources of jobs in spec 11.4. Each queue is computed from the graph and the inventory on each build:

| Queue | Holds | Role | Goal |
|---|---|---|---|
| `enumerate` | each topic and each MSC code whose inventory has fewer rows than its estimate (K4.7) | enumerate | the model lists each item of the area that it knows, with a source for each one |
| `cluster` | the mentions with no row, and the candidate pairs above the similarity threshold | dedup judge, then curators | each mention belongs to one row |
| `decide` | the rows with no outcome | curate | the open count goes down |
| `state` | the nodes at W0 | write, informalize | W1 |
| `complete` | the nodes below the writing level of their volume (K11) or their layer (K2.7) | write, explain, exercise | the next writing level |
| `confirm` | the agent-written sections with no confirmation | review-assist, then people | rule P04 is clean |
| `sanity` | the agent-written statements with no result of K9.3 | explore | X05 is clean |
| `source` | the sources with `locator_unconfirmed` | review-assist, then people | P06 is clean |
| `bind`, `prove` | the queues of spec 11.4 | formalize, prove | spec 12.4 |

A job of the corpus has the record of spec 11.4. Its goal names a stage, for example `goal: "node reaches W2"` or `goal: "row 9DK2RX7Q4H has an outcome"`.

## K12.3 Priority

The priority formula of spec 11.4 applies. The corpus adds two terms, and the formula keeps its version like a prompt:

```
priority = w1 · centrality(target)
         + w2 · demand(target)
         + w3 · unlock(target)
         + w4 · tractability(target, role)
         − w5 · attempts(target, role)
         + w6 · spread(target)        # the number of independent sources that hold the item
         + w7 · curriculum(target)    # the item is in a curriculum or a standard volume (K10, K11)
```

`spread` puts first the items that many sources hold. An item that eight textbooks and two libraries hold is almost surely standard and is cheap to state well. An item that one source holds needs more care, and it can wait. `curriculum` puts first the items that a course or a book needs, so that the first volumes can be packed early (K14).

The order of work in spec 18.5 still applies: breadth first on the record plane, depth first on the statement plane. In the corpus, this means that the inventory of each area is built first, and then each area is written in depth, in the order of the curricula.

## K12.4 Batches

Most corpus work has no person waiting for it. So it runs in batches:

- **Model batches.** The write, explain, exercise, informalize and enumerate roles, and the dedup judge, use Message Batches at half price (spec 12.3). A batch holds jobs of one role and one topic, so that the cached part of the context (the system prompt, the card, the style guide, the notation register of the topic) is the same for each job.
- **Pull request batches.** One pull request for each node would give millions of pull requests. A batch pull request holds up to 50 nodes of one topic, from one role and one configuration. The audit checks each node in it. A reviewer can merge a part of it: the nodes that pass, with the others sent back to the queue with their reasons.
- **Small diffs.** A batch pull request changes only the files of its nodes and their provenance logs. It never changes a register by hand. The build writes the registers.

## K12.5 Review capacity is the limit

Settled decision 20 caps agent pull requests at a multiple of the measured review throughput. In the corpus, this is the main limit on speed, not money and not model speed. So the plan starts from the review hours.

Each month, the review hours go first to the work that keeps the measurements honest, and then to the work that moves items:

| Order | Use | Example cost |
|---|---|---|
| 1 | the sample audit (K9.7) | about 170 hours at 20 strata |
| 2 | reader reports (K9.8) and quarantine decisions | measured |
| 3 | F2 reviews in the order of `unlock` (spec 10.6) | 3 minutes for each undergraduate statement |
| 4 | confirmation of agent-written sections (rule P04) | 1 to 2 minutes for each section, by estimate |
| 5 | decisions on close cases of the inventory (K4.3, K4.4) | 1 minute for each case, by estimate |
| 6 | locator confirmation (K9.5) | 2 minutes for each source, by estimate |

The agents then get pull request capacity in proportion to the review hours that remain, by spec 11.5. A role that makes work that nobody can review in time does not run.

An example with round numbers. 100 active reviewers give 10 hours each month: 1,000 hours. The sample audit and the reports take 250 hours. Of the 750 hours that remain, 400 go to F2 reviews (8,000 statements) and 350 go to confirmation (about 14,000 sections). At this rate, F2 for the 500,000 statements of the textbook layer takes about five years, and W3 for 1.5 million nodes takes longer. This is why W2 is the target of the textbook layer and W3 is the target of the standard volumes only (K2.7, K11). It is also why the corpus must recruit reviewers (K14) and make each review fast (spec 16.3).

## K12.6 The cost of the models

The model cost is small next to the cost of people, but it is not zero, and the plan must say what it is. The numbers below are estimates for planning, with the prices of spec 12.3 and batch pricing. They assume that the cached part of each context is read from the cache. The run records replace them with measured numbers after the first thousand jobs of each role. K7.8 gives the cost of each configuration of models for one node.

| Stage | Tier | Tokens in, tokens out, for each item | Cost for each item |
|---|---|---|---|
| enumerate (for each item listed) | 1 | 0.5k, 0.1k | under $0.001 |
| dedup judge (for each candidate pair) | 0 | 2k, 0.1k | under $0.001 |
| state: W0 to W1 | 1 | 10k, 1.5k | about $0.02 |
| usable: W1 to W2 (proof, examples, L2, the second-model check of K9.3) | 1, and 2 for hard items | 30k, 6k | about $0.06 at tier 1, $0.12 at tier 2 |
| complete: W2 to W3 (other levels, history, exercises) | 1 and 2 | 30k, 8k | about $0.07 to $0.14 |
| formalize a statement to F1 | 1 to 3 | varies by orders of magnitude | $0.50 to $5, by spec 17 measurements when they exist |

For the textbook layer of 1.5 million nodes at W2, the writing cost is about $0.10 for each node, or about $150,000. Retries and escalation can double it. Formalization of 500,000 statements costs $250,000 to $2.5 million, and it is the largest model cost. The copy check (K9.4) and the sanity checks by computation run on math2's own machines.

**No model run starts without a budget that a person approved.** The global monthly budget of spec 11.5 is the only source of money. `m2 corpus plan` (K12.8) estimates the cost of a queue before anyone approves it.

## K12.7 Measurement

The corpus has its own section in the weekly agent report (spec 12.9) and in each release manifest:

- the open count, for each layer and each MSC code, and its change since the last report;
- the number of nodes at each writing level, for each layer;
- the formal outcomes of K2.5;
- the error rates of K9.7, with their intervals;
- the review hours used, by use (K12.5), and the queue lengths;
- the model spend, by role and stage, and the cost of each item that reached a new stage;
- the rate of fabricated sources (K9.5) and of copy matches (K9.4), by configuration.

A number that goes the wrong way is in the report with the others. The release notes account for each change in the counts (rule R03).

## K12.8 Commands and libraries

The corpus adds two command groups to `m2`. Each command is a thin layer over a library package. The packages follow the conventions of the toolchain: one package for each concept, no `internal/` directory, exported types with documentation and a testable example, and tests that use fakes for each model and each network service.

| Command | Does |
|---|---|
| `m2 corpus import -source NAME` | runs the importer of a source (K05, K06) into mentions in the record plane |
| `m2 corpus cluster` | groups mentions into inventory rows, and writes the close cases to the `cluster` queue |
| `m2 corpus status [-layer L] [-msc CODE]` | prints the open count, the writing levels and the formal outcomes |
| `m2 corpus queue NAME` | prints a standing queue in priority order |
| `m2 corpus plan NAME -n N` | estimates the cost and the review hours of the first N jobs of a queue. It does not call a model |
| `m2 corpus run NAME -n N -budget USD` | submits the jobs to the orchestrator under an approved budget |
| `m2 corpus sample -month YYYY-MM` | draws the sample of K9.7 and writes the audit packets |
| `m2 pack VOLUME` and `m2 pack -check` | K11 |

| Package | Holds |
|---|---|
| `inventory` | mentions, rows, outcomes and the open count (K04) |
| `cluster` | the grouping of mentions, with the similarity and the judge behind interfaces |
| `card` | the cards of K08, read from `schema/cards.v1.yaml`, and the writing level |
| `style` | the linter of K8.2 |
| `copycheck` | the shingles, the fingerprints and the match rules of K9.4 |
| `sources` | the confirmation of K9.5, with Crossref, zbMATH and the catalogues behind interfaces |
| `sample` | the strata, the draw and the intervals of K9.7 |
| `pack` | the volumes and the outputs of K11 |
| `importer/<source>` | one package for each source adapter of K05 and K06 |

## K12.9 Scale of the repository

The textbook and formal layers add 1 to 2 million node files, with their provenance logs. That is in the range of spec 06.1 and 06.7. The research layer is in the record plane, and its inventory rows are in the record plane too (K4.8). When the repository passes the trigger of spec 06.1 (10 GB packed or 5 million files), the record plane and the inventory of the research layer move to `tamnd/math2-records` together.

Agent branches are short. A batch branch lives until its pull request merges or closes, and the orchestrator deletes it then. Contributors who work on one area use a sparse checkout of the shards that they need. `m2` works on a sparse checkout, except for the commands that need the full graph (`m2 build`, `m2 audit`), which run in CI.
</document>

<document source="https://math2.tamnd.com/spec/corpus/13-legal-and-ethics.md">
# K13. Law and ethics

The corpus takes text from thousands of sources and has models write most of the rest. Both are legal questions, and both are questions of trust. This document says what the law says on the day of this spec, what math2 does because of it, and which questions need a lawyer before a phase starts (K13.10).

This document is not legal advice. It is the reading of the sources of K01, part D, by people who are not lawyers. Each fact has its source and the date of reading in K01. Where the law is not settled, this document says so and chooses the safe course.

## K13.1 The three zones

Settled decision 17 gives three licence zones. The corpus keeps them, and puts each new kind of content in one zone:

| Content | Zone | Licence |
|---|---|---|
| prose of nodes, explanations, glue of volumes, text of exercises | prose | CC BY-SA 4.0 |
| Lean code, the Lean statement and the reference solution of an exercise, the code of computation nodes, the Lean companion of a volume | code | Apache-2.0 |
| front matter, mentions, inventory rows, outcomes, the notation register, the conventions register, the JSON-LD of the site, records | metadata | CC0 |

A volume has the licences of its parts (K11.7). A dataset release has a manifest that gives the zone of each file.

## K13.2 Copyright in text that models write

**The United States.** The Copyright Office, in Part 2 of its report on AI (29 January 2025), says that copyright does not extend to material that is purely AI-generated, or where a person had insufficient control over the expressive elements. It says that prompts alone do not give enough control. A person has copyright in what they wrote, and in their creative selection, arrangement or modification of the outputs. The courts agree that the Act needs a human author: the D.C. Circuit held so in *Thaler v. Perlmutter* (18 March 2025), and the Supreme Court denied certiorari on 2 March 2026.

**What this means for math2.**

1. A section that a model wrote and no person changed may have no copyright in the United States. The CC BY-SA licence on it may then have no effect: section 8(a) of the licence says that it does not limit uses that need no permission. Mathematical facts, definitions and short statements have thin protection in any case.
2. What people add has copyright: the text that people write or edit, and the selection, the arrangement and the curation of the corpus. CC BY-SA applies to these.
3. Other jurisdictions answer the question differently, and some have not answered it. For example, the United Kingdom has a rule for "computer-generated works" (CDPA s.9(3)). K01 did not research these.

**The decision.** The prose zone stays CC BY-SA 4.0 for all prose, with a notice in `LICENCE`:

> The licence applies to the extent that copyright subsists in the material. Some sections were written by software and not changed by a person. These sections may be in the public domain in some jurisdictions. The provenance of each section says who or what wrote it.

The other option was CC0 for prose that a model wrote and no person confirmed, and CC BY-SA after a person confirms it. Creative Commons itself suggests CC0 for works with no significant human creativity. The corpus does not take this option, because it makes the licence of a section depend on its provenance, and the zone of a section would change when a person confirms it. The notice tells the truth, and the provenance shows which sections it covers. A change to this decision is a change to settled decision 17.

## K13.3 The real risk: copies

Who owns model output is a small question for math2. The large question is whether model output copies someone else's text. A model that reproduces a textbook proof gives math2 the textbook's text, with the textbook's copyright.

- **Training cases are about inputs.** *Bartz v. Anthropic* (N.D. Cal.) held in June 2025 that training on lawfully acquired books was fair use, and that a library of pirated copies was not. The case settled, with final approval on 20 July 2026. In *Kadrey v. Meta* (June 2025) the court found fair use on that record. In *Thomson Reuters v. Ross* (3d Cir., 29 September 2026), the court held that copying headnotes to train a search tool that does not generate text was not fair use, and said that generative AI is a different case. None of these cases decides the status of outputs.
- **Output cases.** *NYT v. OpenAI* (S.D.N.Y.) is the main case in the United States on regurgitated output, and it has no decision on the merits. In Germany, *GEMA v. OpenAI* (LG München I, 11 November 2025) held that the memorization of song lyrics in a model is a reproduction, and that the output of the lyrics is one too. It held OpenAI liable for the outputs, and OpenAI appealed. In the United Kingdom, *Getty Images v. Stability AI* ([2025] EWHC 2863 (Ch), 4 November 2025) held that the weights of a model do not store an infringing copy, and an appeal is pending. The Court of Justice of the EU heard *Like Company v. Google* (C-250/25), on chatbot output that matches press articles, in March 2026, and a judgment is expected in 2027.
- **Who is exposed.** Whoever publishes a copy is exposed, not only the provider of the model.

**What math2 does.**

1. **The copy check** (K9.4) runs on each body that an agent wrote, against a reference corpus and not only the cited sources. A match with a restricted source blocks the merge.
2. **No restricted text in the context** of a write job (K7.5). A model that rewrites a text that it can see makes a derived work.
3. **Matches with open sources are attributed.** A close match with a CC BY-SA source keeps that source's terms, whatever math2 says, so the node records the attribution (rule L03).
4. **GFDL, NC and all-rights-reserved text never enters the prose zone.** GFDL text, for example from the Stacks Project, cannot be relicensed under CC BY-SA 4.0. Kerodon states no licence, so its text is all rights reserved. Non-commercial text cannot be in a corpus that anyone may sell.
5. **Takedown.** A claim that a node copies a work goes to the curators as a report of the property "ours to publish" (K9.8). The node is quarantined within two working days, while a person checks the claim.

## K13.4 Human sources

The corpus takes from human sources by the licence class of each source (K05):

| Class | What math2 may hold | Example |
|---|---|---|
| `open` | the text, with attribution, when the licence is compatible with CC BY-SA 4.0 | Wikipedia (CC BY-SA), set.mm (CC0), Mathlib docstrings (Apache-2.0) |
| `share-alike-nc` | records, locators and links. No text | OpenStax (CC BY-NC-SA), the WeBWorK Open Problem Library (CC BY-NC-SA 3.0) |
| `gfdl` | records, locators and links. No text | the Stacks Project, the Manifold Atlas |
| `verbatim` | short quotations where the law of quotation allows them, with the source | a famous statement in its original words |
| `record` | records and locators only | most textbooks and papers, Kerodon, the DLMF |

Three further points apply to each class:

- **Facts are free, words are not.** That a theorem is true, what it says and where it appears are facts. The words of a source are its author's. A node that states a standard theorem in math2's words takes nothing that copyright protects.
- **Collections and databases.** The selection and the arrangement of a collection can have copyright, and in the EU a database can have a database right against the extraction of a substantial part. So an importer takes from a catalogue only what the terms of the catalogue allow, and the source document (K05) records those terms. A table of contents goes into the record plane as locators for each item, not as a copy of the table.
- **Licences per item.** Some sources have a licence for each entry, for example the entries of the Archive of Formal Proofs, or the posts of Math.StackExchange by date. The importer records the licence of each item, not of the source.

## K13.5 The EU AI Act

- **Duties of providers.** The duties for providers of general-purpose AI models (Articles 53 to 55) apply from 2 August 2025, and the Commission can enforce them from 2 August 2026. They fall on the providers of models. math2 uses models and is not their provider. A team that changes a model becomes a provider only if the change uses more than a third of the training compute of the original. A fine-tune of an open prover by math2 would be far below that.
- **Transparency (Article 50).** It applies from 2 August 2026. Providers of systems that generate text must mark the outputs in a machine-readable form (Article 50(2)). The Digital Omnibus on AI (Regulation (EU) 2026/1744) reportedly gives this duty a grace period to 2 December 2026, and sources disagree on its scope. Deployers that publish text generated by AI to inform the public on matters of public interest must disclose that it is generated (Article 50(4)). The duty does not apply when the text had human review or editorial control and a person holds editorial responsibility for the publication.
- **What math2 does.** It is not settled whether a mathematics library informs the public "on matters of public interest". math2 meets the duty either way. Each field that an agent wrote is marked as such (rule P04), the site puts a machine-readable flag on each field that no person confirmed, and each area has a named editor who holds editorial responsibility (spec 16).

## K13.6 The terms of the providers

K7.9 gives the rules that follow from the terms. In short:

- The terms of Anthropic, OpenAI and Google give the customer the rights in the outputs that the provider has. None of them stops publication under an open licence.
- From 12 November 2026, the usage policy of Anthropic forbids the use of outputs to train an AI model without its prior authorization. OpenAI and Google forbid the use of outputs to build competing models. math2 does not train on the output of hosted models without written authorization.
- Some open-weight licences add duties on outputs. The model register (K7.7) records them.
- **Users of math2.** A person who downloads math2 under CC BY-SA has no contract with the provider of a model. It is not settled whether a provider's terms reach that person, and math2 claims nothing either way. The release notes say which provider wrote which fields, so that users can decide for themselves.

## K13.7 Disclosure

The disclosure of agent work is generated and complete (settled decision 24):

- **Each field.** The provenance names the role, the model and the run of each field that an agent wrote, and the person who confirmed it, if any.
- **Each page.** The site shows "written by software, not yet checked by a person" on each unconfirmed field, in words and in the machine-readable flag of K13.5.
- **Each volume.** The attribution page lists people first, by their contributions, and then the disclosure of agent runs (K11.7).
- **Each release.** The release notes give the share of fields that agents wrote, by layer and kind, and the share of those that a person confirmed.

## K13.8 Ethics

Four duties are not in the law, and the corpus keeps them:

1. **Do not flood the commons.** Wikipedia adopted a speedy-deletion criterion for unreviewed pages that a model generated (G15, August 2025). arXiv asked that review and position papers in computer science be peer-reviewed first (October 2025). Communities object to unreviewed machine text in bulk, not to the use of AI as such. math2 never writes agent text into the sources that it imports from (spec 16.5), and it never presents an unconfirmed section as reviewed.
2. **Credit to the people whose work the corpus uses.** Each node names the sources that its row has, even where the licence does not require it. A mathematician whose result is in the corpus is named with the result (K10.6).
3. **Honest numbers.** The error rates of K9.7 are published with their intervals, including the rates that are bad. A count that goes down is explained in the release notes (rule R03).
4. **Be polite to services.** Importers keep the rate limits and the robots rules of each source, identify themselves with the User-Agent of the toolchain (with no personal address), and prefer a published dump to crawling.

## K13.9 Rules

| Rule | H/S | Checks |
|---|---|---|
| L06 | H | each item from a source with a licence for each item records the licence of the item (K13.4) |
| L07 | S | a node with a takedown report that is older than two working days and not quarantined |

## K13.10 Questions for a lawyer

Each of these needs advice before the phase that the table names:

| Question | Before |
|---|---|
| Is the notice of K13.2 enough, or does each section that a model wrote need its own notice? | the first release with agent prose (phase 2) |
| Does the database right of the EU limit the import of catalogues such as zbMATH, the OEIS and MathSciNet, beyond their terms? | phase 1 |
| Do the terms of a provider reach users of math2 who have no contract with it? | the first dataset release |
| Is the publication of a corpus of outputs an "availability to third parties" under the Qwen 3.8 and Kimi K3 licences? | any use of these models |
| Which licence governs DeepSeek-Prover-V2: the MIT file or the Model Agreement that its README names? | any use of its outputs for training |
| Is a mathematics library within Article 50(4)? | not needed: math2 meets the duty either way |
| What is the position in the United Kingdom and the EU on the copyright of output that a model generated? | phase 3 |
</document>

<document source="https://math2.tamnd.com/spec/corpus/14-roadmap.md">
# K14. Roadmap

This document puts the corpus programme in phases. Each phase has a goal, the work, and an exit criterion that the build can check. A phase starts when the phase before it meets its exit criterion, except where the table says that two phases overlap. The phases come after the milestones of spec 19 that build the system. They use the system, and they do not replace a milestone.

Each phase that runs models needs a budget that a person approved (K12.6). No phase publishes a dataset or a DOI without approval (spec 06.2). The roadmap gives no dates, because the speed is set by review capacity (K12.5), and review capacity is not known yet.

## K14.1 The state on the day of this spec

`tamnd/math2` holds four nodes (the concept *Group*, Lagrange's theorem, and a computation with its node), the registers, the schemas of node.v1, the curriculum seed and the skeletons of 60 MSC classes in `topics/skeletons/`. The toolchain `tamnd/math2-agent` has the build, the audit of spec 15, the site, the agent framework with fakes, and the orchestrator. No corpus package of K12.8 exists yet. No model has written content for math2.

## K14.2 Phase 0: the foundations

**Goal.** The toolchain can hold, check and measure a corpus before the corpus exists.

**Work, in this order.** Each item is one pull request, or one in each repository:

1. Tags of ten characters (K3.8): the `tag` package, the schema patterns, `m2 migrate shard3`, rules T06 and T08 to T10, the Lean attribute and the resolver. In `tamnd/math2`: the migration, the cut-over commit and the frozen legacy set.
2. The minor version of node.v1 with the fields and the edges of K8.9, and `schema/cards.v1.yaml`. The `card` and `style` packages, and the writing level in the build.
3. The record plane layout of K4.8: `records/mentions/`, `records/inventory/`, `registers/inventory/`. The `inventory` package, the open count, and rules I01 to I06.
4. The notation register (K10.4), seeded from `macros/m2.sty`, with the build that writes `m2.sty` and `m2.katex.json` from it, and rules N01 to N07.
5. The `copycheck`, `sources` and `sample` packages, and rules L05, P05, P06, X05 and X06 (K9.9).
6. `m2 corpus status`, `queue` and `plan` (K12.8). These commands call no model.
7. The first `pack` with the HTML output, and `m2 pack -check` (K11). LaTeX and PreTeXt follow in phase 2.

**Exit.** Each rule of K9.9, K4.9 and K10.7 has a fixture pair, and the audit runs them on `tamnd/math2`. `m2 corpus status` prints the open count of an empty inventory. `m2 pack -check` passes on a volume made of the nodes that exist.

## K14.3 Phase 1: the inventory of sources that are safe and open

**Goal.** A first inventory, made only from sources whose data math2 can hold with no doubt about the licence, and no model.

**Work.**

1. Importers for the CC0 and permissive sources first: Wikidata items about mathematics, the declarations and the module documentation of Mathlib (Apache-2.0), the OEIS names (K4.6), and the curricula that are published as open data (K10.3).
2. Importers of records and locators only, for sources of the `record` and `gfdl` classes: tables of contents and indexes, where the source allows it (K05).
3. Clustering passes 1 and 2 (K4.3), which need no model. The gold set of K4.3 for the first areas.
4. The capture and recapture estimates of K4.7 for each MSC code at two digits.

**Exit.** Each mention has a row or is in the `cluster` queue. The report gives the estimated recall for each MSC code at two digits. The licence class of each mention is recorded, and rule L02 is clean.

## K14.4 Phase 2: the pilot area

**Goal.** One area, end to end, so that each number in K12 is measured and not estimated.

**The area.** Undergraduate group theory: MSC 20 at the level of a first course. The reasons: math2 already has two of its nodes, Mathlib covers it well, the textbook and wiki sources are many, and the statements are short enough that small-case checks (K9.3) work.

**Work.**

1. The inventory of the area from all sources of K05 and K06, and from the enumerate role, with the clustering passes 3 and 4.
2. An outcome for each row.
3. Nodes to W2 for each row with the outcome `node`, by importers of `open` text and by the write, explain, exercise and informalize roles, under an approved budget.
4. The formal track for each statement: bound, F1, and F2 for the core list.
5. The sample audit of K9.7 for each stratum of the pilot.
6. The first volume: *Groups: a first course*, packed in each output format of K11.

**Exit.**

- The open count of the area is zero.
- Each node of the area is at W2 or better, and each statement has a formal outcome (K2.5).
- The sample audit gives an error rate under the thresholds of K9.7 for each stratum, with its interval. Strata above a threshold are stopped, and their output is fixed or removed.
- The measured cost for each stage replaces the estimates of K12.6 in the spec.
- `m2 pack -check` passes on the volume, and a teacher who is not in the project has read it and reported the result.

If the pilot fails its exit, the corpus programme stops and the spec changes before anything grows. A failed pilot is a result, and the report says why.

## K14.5 Phase 3: the textbook layer

**Goal.** The textbook layer of K2.7: an open count of zero, W2 for each node, and E2 or better for each statement.

**Work.** Area by area, in the order of the curricula of K10.3 and the priority of K12.3. The undergraduate core comes first (spec 18.3), then the graduate core, then the rest of the textbook layer. Each area repeats the work of the pilot with the measured numbers. A standard volume for each area of the undergraduate core goes to W3.

**Exit for each area.** The exit of the pilot. **Exit for the layer.** Each area has met its exit, and the inventory recall (K4.7) is at 90% or better for each MSC code at two digits.

## K14.6 Phase 4: the formal layer

**Goal.** The formal layer of K2.7: each declaration that a person would cite (K4.5) is a node with a binding.

This phase overlaps with phase 3. It starts when the importers of K06 exist, and it shares the review budget with phase 3 by the order of K12.5.

**Work.** For each library of K06: the declarations that K4.5 selects, the informalize role for the ones with no human mention, the alignment of the ones that a textbook row already has, and F1 for each binding.

**Exit.** Each row of the formal layer has an outcome. Each node of the layer has a binding at F1 or better.

## K14.7 Phase 5: the research layer

**Goal.** The research layer of K2.7: an open count of zero in the record plane.

**Work.** The extraction and the alignment of spec 09, area by area, in the order of spec 18.5. Each result gets the outcome `aligned` or `record`, or it becomes a node when the graph needs it.

**Exit.** Each result of the record plane has an outcome.

## K14.8 What runs all the time

From phase 2 on, these run each month: the sample audit (K9.7), the reports of K12.7, the estimates of recall (K4.7), and the review of the thresholds of K9.7. From phase 3 on, a new source that enters scope goes through the held-out check of K4.7 before its mentions are added.

## K14.9 The people that the phases need

| Phase | People | For |
|---|---|---|
| 0 | maintainers of the toolchain | the code |
| 1 | one curator for each source class | the licence decisions, the gold set |
| 2 | an area editor for group theory, 5 to 10 reviewers, one outside teacher | the pilot |
| 3 | an area editor for each area of the undergraduate core, about 100 reviewers | K12.5 |
| 4 | reviewers who know each formal library | F2 |
| 5 | area editors in the research areas | the close cases of the alignment |

The recruitment of reviewers is work of the project, not a side effect. Spec 16 says how reviewers get credit.
</document>

<document source="https://math2.tamnd.com/spec/corpus/15-open-questions.md">
# K15. Open questions

This document lists each question that the corpus spec leaves open. Each question has an id, the way to settle it, and the phase of K14 that it must be settled before. A question that blocks nothing has "no block".

## K15.1 How the list works

- **Owner.** The maintainers own the list. A question is closed by a commit that changes this document and the documents that the answer affects, with the source of the answer.
- **The safe course.** Until a question is closed, the corpus takes the safe course that the text gives: a source with a licence in doubt is `record`, a count in doubt is not quoted as a rule, and a check that is not calibrated is soft.
- **Asking a source.** Many questions are settled by asking the people who run a source. Each request is polite, short, and sent by a maintainer from the project, not by an agent. The answer is kept with the source document (K5.2).
- **Questions for a lawyer** are in K13.10. This list names them and does not repeat them.

## K15.2 Licences and terms of human sources

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ01 | Did the move of the Encyclopedia of Mathematics to EMS Press change the copyright of the Springer originals? What is its current size? | read the live site again; ask the EMS | phase 2 |
| OQ02 | Do the Stack Exchange Public Network Terms of 13 November 2025 add clauses on AI or on scraping? Does the public MCP server cover MathOverflow and Math.SE? | read the terms in a browser | phase 2 |
| OQ03 | Will the nLab steering committee give an explicit licence for new contributions? | ask the steering committee | no block: the nLab is `record` |
| OQ04 | Is the Manifold Atlas GFDL 1.3 or in the public domain? | ask the editors | no block: it is `gfdl` |
| OQ05 | Are the knowls of the LMFDB under the CC BY-SA licence of its database? Which bulk channel does the LMFDB want AI users to use? | ask the LMFDB | the first LMFDB import |
| OQ06 | Is the zbMATH knowledge graph CC BY 4.0 or CC BY-SA 4.0? What do the API terms say? | ask zbMATH, or open an issue on `zbMATHOpen/zbmath-open-kg` | phase 1 (until then, CC BY-SA 4.0) |
| OQ07 | What are the licences of the 14 AIM titles that nobody has read (Bruckner and Thomson, ORCCA, Manes, May and Bart, Marsden and Weinstein, Boyd and Vandenberghe, Treil, Fields, Bender and Williamson, Bard, Zimmermann, Crisman, Stein, Connell)? | read each at its source | before each book enters Table A |
| OQ08 | Do the books that the Open Textbook Library lists as CC BY or CC BY-SA (Zakon, Kuttler, Shimamoto, Petrunin, Callahan and others) have those licences at their source? Is Hitchman CC BY-SA or GFDL? | read each at its source | before each book enters Table A |
| OQ09 | Does the 3.0 US port of CC BY-SA allow adaptation under CC BY-SA 4.0, as §4(b) of the Unported licence does? | read the legal code of the port | the import of Hefferon |
| OQ10 | Which CC variant do the PreTeXt titles with a plain "Creative Commons" badge have? | read each at its source | before each book enters Table A |
| OQ11 | What licences do FindStat, House of Graphs, KnotInfo and the Knot Atlas give? Did the authors of KnotInfo consent to the GPL-3.0 CSV package? | ask each project | no block: they are `record` |
| OQ12 | What are the terms of HathiTrust, the Internet Archive, CORE, Unpaywall and EuDML, and the licence of the GDZ scans, the Project Euclid journals and the Semantic Scholar abstracts? | read each in a browser | before the first import from each |
| OQ13 | What is the licence of MathBases, and of the sources that the research could not reach (GroupNames)? | read again | no block |
| OQ14 | What share of LibreTexts pages has no declared licence? | a sample of pages, by the `license:` tag | no block: the corpus prefers the upstream source |

## K15.3 Formal libraries and datasets

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ15 | What are the licences of the 20 Formal Frontier repositories that nobody has read? | read each `LICENSE` | before each binding to them |
| OQ16 | Does the "not intended for ML" sentence of the ATLAS v2 preamble bind a user of its Apache-2.0 grant? | K13.10 | no block: ATLAS is `link` |
| OQ17 | Is Coquelicot LGPL-3.0 upstream, as its fork is? | read the Inria repository | no block: it is `crosswalk` |
| OQ18 | Is LeanDojo-v2 Apache-2.0 (the GitHub API) or MIT (its README)? | ask the authors | no block: it is `evaluate` |
| OQ19 | Do the terms of the model that generated a dataset (DeepSeek-V3, Qwen2.5-72B, Gemini, Kimina-72B) pass on to the dataset? | K13.10 | any use of such a dataset beyond evaluation |
| OQ20 | What is the size of Lean Pool, and what are the licences of the projects that it imports? | read its repository | before Lean Pool is bound |
| OQ21 | Does doc-gen4 publish JSON for each declaration that the binding can use? | read doc-gen4 | phase 0, item 2 |

## K15.4 Counts that do not agree

These counts change no rule. The corpus quotes the figure of the source and says that it is not settled, until a reading settles it.

| Id | Question | How to settle |
|---|---|---|
| OQ22 | Mizar: 1,493 articles (the version string) or 1,498 (spec 01)? | count the articles of the MML release |
| OQ23 | Tau Ceti: does "3 million lines" mean Tau Ceti or Mathlib? | count the lines at a pinned commit |
| OQ24 | Metamath: 51,207 statement numbers, "over 41,000" with mathboxes, and 74 or 75 of the 100 theorems. Which count does the corpus report? | define the count in the importer and record it |
| OQ25 | TheoremSearch: 1,239,720 rows in the card of the `-permissive` release against 2,677,439 in the viewer; 8.4M live against 9.2M in the paper | count the rows at a pinned revision |
| OQ26 | Mathlib `100.yaml` gives 78 theorems with declarations; Wiedijk gives 83 for Lean | compare the two lists item by item |
| OQ27 | 1000+: 228 entries on the site; spec 01 says 241 in Lean | parse the YAML files at a pinned commit |
| OQ28 | Are the ProofWiki Main Page counters (30,348 proofs, 36,102 definitions) live? | count from the daily dump |
| OQ29 | Lean Workbook: 57,231 and 82,893 rows in the paper, 25,214 in the default configuration | read each configuration |
| OQ30 | Is the move of the Wikipedia dumps from twice a month to once a month official? Were the free HTML dumps (last run 20 March 2025) stopped? | read the Wikimedia announcements |

## K15.5 The law

| Id | Question | Where |
|---|---|---|
| OQ31 | Is the notice of K13.2 enough? | K13.10 |
| OQ32 | Does the database right of the EU limit the import of catalogues beyond their terms? | K13.10, phase 1 |
| OQ33 | Do the terms of a provider reach users of math2? | K13.10 |
| OQ34 | The "availability to third parties" clauses of the Qwen 3.8 and Kimi K3 licences | K13.10 |
| OQ35 | The licence of DeepSeek-Prover-V2: the MIT file or the Model Agreement? | K13.10 |
| OQ36 | The copyright of model output in the United Kingdom and the EU, and the effect of the judgment in C-250/25 (expected 2027) and of the appeal in *GEMA v. OpenAI* | K13.10, phase 3 |
| OQ37 | What do the Commission's guidelines on Article 50 (reported for 20 July 2026) and the code of practice on transparency say? | read them; K13.5 |
| OQ38 | Is US public domain enough for a work that is not in the public domain in the EU? The corpus says no (K5.2.3). Should it accept a work that is in the public domain in all countries where math2 has users, if the EU rule is not met? | no block: the corpus takes the safe course |

## K15.6 Model generation and quality

These questions are settled by measurement in phase 2 (K14.4), not by reading.

| Id | Question | How to settle |
|---|---|---|
| OQ39 | What is the error rate of model-written definitions and statements in prose? No study has measured it | the sample audit of K9.7 in the pilot |
| OQ40 | How often do models reproduce mathematical text? The 27% figure is for encyclopedia prose in general, on models of 2024 | the copy check of K9.4 over the pilot, against the reference corpus |
| OQ41 | Which thresholds of the copy check give fewer than 1% false matches on text that people wrote? | the calibration of K9.4, before L05 blocks |
| OQ42 | How well do the current judges (Opus 5.5, Fable 5.1 and the models of other families) find wrong steps? The published numbers are for models of 2025 | a labelled set of the pilot area (K7.2) |
| OQ43 | May a model judge its own output, or must the judge be of another family? The studies disagree | the labelled set, with both arms |
| OQ44 | What is the first-pass acceptance rate, and so the real cost for each accepted node? | the run records of the pilot (K12.6) |
| OQ45 | How many hours does a person need for each node, by kind and writing level? | the review records of the pilot (K12.5) |
| OQ46 | Does any provider filter recitation, and at which threshold? Anthropic and OpenAI document none; Gemini has a RECITATION reason with no published threshold | read the documentation; no block, because the copy check of K9.4 does not rely on it |
| OQ47 | The figures in part D marked [snippet] or "derived" (the First Proof batch 2 tally, the later BrokenArXiv figures, Seed-Prover 1.5, the AlphaProof TPU-days) | read the primary PDFs before a spec section quotes them |

## K15.7 Representation and packaging

| Id | Question | How to settle | Before |
|---|---|---|---|
| OQ48 | When does MathML 4 reach Candidate Recommendation? | the W3C Math Working Group | no block: the corpus generates MathML |
| OQ49 | What are the licences of Verso Blueprint, LaTeXML, MMT and the SMGloM archives other than `smglom/mathsoft`? | read each `LICENSE` | before each tool is used in the packager |
| OQ50 | Which projects have moved to Verso Blueprint, and which keep leanblueprint? | read the catalogue again | the blueprint import |
| OQ51 | Is there a version of Common Cartridge after 1.3? Do Canvas, Moodle and Blackboard support QTI 3? | read the 1EdTech pages and the LMS documentation | the course output of K11 |
| OQ52 | Is the Learning Commons Knowledge Graph worth aligning to for school mathematics? | a sample alignment | no block |
| OQ53 | The citation of Leron (1983) and the date of Tao's post "There's more to mathematics than rigour and proofs" | read the primary sources | before a corpus document cites them |

## K15.8 Design questions of the corpus

These questions are about the corpus itself. The text gives a first answer, and the pilot tests it.

| Id | Question | First answer | How to settle |
|---|---|---|---|
| OQ54 | Are the sizes of K2.7 right, for example 300,000 to 1,000,000 named statements? | the estimates of K2.7 | capture and recapture (K4.7) after phase 1 |
| OQ55 | Is a recall of 90% for each MSC code at two digits (rule I06) the right bar? | 90% | the held-out checks of K4.7 in phase 1 and phase 2 |
| OQ56 | How many Mathlib declarations become nodes? | a rough guess of 40,000 to 80,000 (K6.4) | the outcomes of the pilot area, scaled by area |
| OQ57 | Is a gold set of 2,000 pairs for each layer enough to measure the clustering? | 2,000 (K4.3) | the intervals of the first measurement |
| OQ58 | Are the error thresholds of K9.7 right for each stratum? | the thresholds of K9.7 | the monthly review of the thresholds (K14.8) |
| OQ59 | How much review capacity can the project get, and from whom? | not known; it sets the speed of each phase (K12.5) | the pilot, and the curators who join it |
| OQ60 | Should ProofWiki text enter nodes as text, or only as a source for math2's own text? | as text, with attribution, by the importer of open text (K5.4.5) | the review of the pilot nodes that use it |
</document>

<document source="https://math2.tamnd.com/node/000AYPA9AJ.md" tag="000AYPA9AJ" kind="concept">
# Vector space

## Summary

A vector space over a field is a set of vectors that can be added and multiplied by scalars, with the usual rules of arithmetic.
Vector spaces are the main objects of linear algebra.
</document>

<document source="https://math2.tamnd.com/node/000JFQGDXG.md" tag="000JFQGDXG" kind="statement">
# The integers are a Euclidean domain

## Statement

The [[ring of integers|0JHM6JK7FD]] $\mathbb{Z}$ is a [[Euclidean domain|0GJ2HRG95T]], with the absolute value $\delta(n) = |n|$ as Euclidean function.
That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$, there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$.

## Remarks

The proof: let $r$ be the least element of $\{a - b q : q \in \mathbb{Z}\} \cap \mathbb{N}$. Then $r < |b|$, because otherwise $r - |b|$ would be a smaller element of this set.
The pair $(q, r)$ with $0 \le r < |b|$ is unique.
Consequences: $\mathbb{Z}$ is a [[principal ideal domain|0RZYDVZ475]], and integers factor uniquely into primes.

## Explanations

### L1 human

This is division with remainder from school, made precise.
Dividing $17$ by $5$ gives $17 = 3 \cdot 5 + 2$, and the remainder $2$ is smaller than $5$.
It works for negative numbers too: $-17 = (-4) \cdot 5 + 3$.
</document>

<document source="https://math2.tamnd.com/node/000MXHEY22.md" tag="000MXHEY22" kind="statement">
# Sequential characterization of limits

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number.
Then $\lim_{x \to a} f(x) = L$ if and only if the following holds.
For each sequence $(x_n)$ in $D \setminus \{a\}$ with $x_n \to a$, $f(x_n) \to L$.

## Remarks

This turns facts about [[limits of sequences|0XKWSSY8P2]] into facts about limits of functions. Examples are the limit laws for sums, products, and quotients.
It is also a tool to show that a limit does not exist. Find two sequences tending to $a$ whose images have different limits.
Mathlib states a general form as `tendsto_nhds_iff_seq_tendsto`.

## Explanations

### L1 human

The function has limit $L$ at $a$ exactly when it sends every sequence approaching $a$ to a sequence approaching $L$.
So limits of functions reduce to limits of sequences.
</document>

<document source="https://math2.tamnd.com/node/000NPTMX6J.md" tag="000NPTMX6J" kind="statement">
# Singular value decomposition

## Statement

Let $A$ be an $m \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ of size $m \times m$, a unitary matrix $V$ of size $n \times n$, and an $m \times n$ matrix $\Sigma$ with
$$A = U \Sigma V^*.$$
Here $\Sigma_{ij} = 0$ for $i \ne j$, and $\Sigma_{11} \ge \Sigma_{22} \ge \cdots \ge 0$ are the [[singular values|09VMDSRXWJ]] of $A$.

## Remarks

If $A$ is real, then $U$ and $V$ can be chosen real orthogonal.
The columns of $V$ are eigenvectors of $A^* A$, and the columns of $U$ are eigenvectors of $A A^*$.
The matrix $\Sigma$ is unique; $U$ and $V$ are not unique in general.
Keeping the $k$ largest singular values gives the best approximation of $A$ of rank $k$, for the Frobenius and operator norms.

## Explanations

### L1 human

Every matrix is a rotation, then a stretch along the axes, then another rotation.
The stretch factors are the singular values.
This decomposition is a central tool in data analysis and numerical computation.
</document>

<document source="https://math2.tamnd.com/node/0011MTQMR6.md" tag="0011MTQMR6" kind="concept">
# Quotient set

## Summary

The quotient set of a set by an equivalence relation is the set of all equivalence classes.
</document>

<document source="https://math2.tamnd.com/node/001B78R0DJ.md" tag="001B78R0DJ" kind="concept">
# Mutual independence of events

## Summary

A family of events is mutually independent when, for each finite subfamily, the probability that all its events occur is the product of their probabilities.
The node [[mutual independence of events|0GCVEV6BJA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/001CX45SWQ.md" tag="001CX45SWQ" kind="definition">
# Partition function

## Definition

The partition function is the function $p : \mathbb{N} \to \mathbb{N}$ such that $p(n)$ is the number of [[partitions|0404R9P9J3]] of $n$.

## Remarks

The values for $n = 0, 1, \dots, 10$ are $1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42$.
The [[generating function for partitions|0578PA53VA]] is $\sum_{n \ge 0} p(n) x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}$.
The function grows faster than any polynomial and slower than any exponential $c^n$ with $c > 1$.

## Explanations

### L1 human

$p(n)$ counts the ways to write $n$ as a sum of positive integers in any order.
For example, $p(5) = 7$: $5$, $4+1$, $3+2$, $3+1+1$, $2+2+1$, $2+1+1+1$ and $1+1+1+1+1$.
</document>

<document source="https://math2.tamnd.com/node/001T5FJDC1.md" tag="001T5FJDC1" kind="concept">
# Block diagonal matrix

## Summary

A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.
</document>

<document source="https://math2.tamnd.com/node/002X7EK91H.md" tag="002X7EK91H" kind="statement">
# Power series are differentiable inside the radius of convergence

## Statement

Let $f(x) = \sum_{n=0}^{\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R > 0$.
Then $f$ is differentiable on $(c - R, c + R)$, and
$$f'(x) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1}.$$
The derived series also has radius of convergence $R$.

## Remarks

The [[Cauchy–Hadamard formula|07E5MB1CA2]] gives the radius of the derived series, since $\sqrt[n]{n} \to 1$. Then [[term-by-term differentiation|0TJ8QBJX5K]] applies on each $[c - r, c + r]$ with $r < R$.
By induction, $f$ is infinitely differentiable, and $a_n = f^{(n)}(c)/n!$. So the series is the Taylor series of $f$ at $c$.
Mathlib states differentiability as `HasFPowerSeriesOnBall.differentiableOn`.

## Explanations

### L1 human

A power series can be differentiated like a polynomial, one term at a time.
The new series converges on the same open interval.
</document>

<document source="https://math2.tamnd.com/node/003QDZ4T16.md" tag="003QDZ4T16" kind="statement">
# A function with zero derivative is constant

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable with $f'(x) = 0$ for all $x \in I$.
Then $f$ is constant on $I$.

## Remarks

For $x < y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) = 0$.
The domain must be an interval. On $(0, 1) \cup (2, 3)$, a function equal to $0$ on one part and $1$ on the other has zero derivative.
So two antiderivatives of a function on an interval differ by a constant. Mathlib states the result as `is_const_of_deriv_eq_zero`.

## Explanations

### L1 human

A function whose slope is zero everywhere on an interval never rises or falls.
So its graph is a horizontal line.
</document>

<document source="https://math2.tamnd.com/node/003QP7A8HG.md" tag="003QP7A8HG" kind="statement">
# Closure is the set of limits of sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $S \subseteq X$ and let $x \in X$.
Then $x$ is in the [[closure|0DRKT1QJR7]] $\overline{S}$ if and only if there is a sequence $(s_n)$ of points of $S$ that [[converges|06JYPJ4WHH]] to $x$.

## Remarks

Proof of the direct part: by [[closure in a metric space|068P0P32DW]], for each $n \ge 1$ there is $s_n \in S$ with $d(x, s_n) < 1/n$. Then $s_n \to x$.
The converse follows because $\overline{S}$ is closed and [[closed sets contain their limits|09GQ3ZTQJ9]].
A consequence: $S$ is closed if and only if the limit of each convergent sequence of points of $S$ is in $S$.
In a general topological space, sequences are not enough, and one uses nets or filters instead.

## Explanations

### L1 human

In a metric space, the closure of a set consists of the limits of sequences that lie in the set.
For example, each real number is a limit of rational numbers, so the closure of $\mathbb{Q}$ is $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/003R0V5M7V.md" tag="003R0V5M7V" kind="concept">
# Differential equation

## Summary

A differential equation is an equation that relates an unknown [[function|0EPAT51DXQ]] to one or more of its derivatives.
A solution is a function that makes the equation true at each point of its domain.
If the unknown function depends on one variable, the equation is an [[ordinary differential equation|0NT64JV96A]]. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.

## Remarks

The [[order|0XK0G19SK8]] of a differential equation is the highest order of a derivative in it.
The equation $y' = y$ is a differential equation of order $1$. Its solutions are the functions $y(x) = C e^{x}$ with a constant $C$.
Differential equations describe how a quantity changes. Many laws of physics, biology and economics have this form.

## Explanations

### L1 human

An ordinary equation asks for a number, but a differential equation asks for a function.
The equation gives a rule for the rate of change of the function.
For example, $y' = y$ says that the function grows at a rate equal to its own value.
</document>

<document source="https://math2.tamnd.com/node/0044CPHJBV.md" tag="0044CPHJBV" kind="definition">
# Natural logarithm

## Definition

For a real number $x > 0$, the natural logarithm $\ln x$ is the unique real number $y$ with $e^y = x$.
This defines a function $\ln \colon (0, \infty) \to \mathbb{R}$.

## Remarks

The number $y$ exists and is unique, because $\exp$ is a [[strictly increasing|0KN2V7840E]] continuous bijection from $\mathbb{R}$ onto $(0, \infty)$.
Some texts define instead $\ln x = \int_1^x \frac{dt}{t}$. Both definitions give the [[same function|0K1N1Y4EGZ]].
Other notations are $\log x$ and $\log_e x$. Mathlib defines `Real.log` for all reals, with $\log x = \log |x|$ and $\log 0 = 0$.

## Explanations

### L1 human

The logarithm answers the question: to which power must we raise $e$ to get $x$?
For example, $\ln 1 = 0$ and $\ln e = 1$.
</document>

<document source="https://math2.tamnd.com/node/004FH91BN5.md" tag="004FH91BN5" kind="concept">
# Jump discontinuity

## Summary

A jump discontinuity is a point where both one-sided limits of a function exist but are different. The graph jumps from one height to another there. The node [[jump discontinuity|05B0FKFXQA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/004FRCXH6M.md" tag="004FRCXH6M" kind="definition">
# Exterior derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega = \sum_I f_I \, dx_I$ be a [[differential k-form|0QHTX7WQEY]] on $U$ whose coefficients $f_I$ are of class $C^1$.
The exterior derivative of $\omega$ is the $(k + 1)$-form
$$d\omega = \sum_I \sum_{j=1}^n \frac{\partial f_I}{\partial x_j} \, dx_j \wedge dx_I,$$
where $\wedge$ is the [[wedge product|0S166D363M]].

## Remarks

For a function $f$, $df = \sum_j \partial_j f \, dx_j$ is its differential, the [[one-form|0MTENFBW3Q]] that corresponds to the [[gradient|0NS2QYVC4Y]].
The operator $d$ is linear and satisfies $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$ for a $k$-form $\alpha$. For $C^2$ forms, [[d d = 0|0X308R2DBE]].
In the plane, $d(L \, dx + M \, dy) = (\partial_x M - \partial_y L) \, dx \wedge dy$.
In $\mathbb{R}^3$, $d(F_1 \, dx + F_2 \, dy + F_3 \, dz)$ has the components of the [[curl|0GDNHS9J2X]] of $F$ as coefficients of $dy \wedge dz$, $dz \wedge dx$ and $dx \wedge dy$. Also $d(F_1 \, dy \wedge dz + F_2 \, dz \wedge dx + F_3 \, dx \wedge dy) = (\nabla \cdot F) \, dx \wedge dy \wedge dz$, which is the [[divergence|02NSMQVQWK]].

## Explanations

### L1 human

The exterior derivative is one operation that does the work of gradient, curl and divergence together.
You differentiate each coefficient in each direction and attach the new direction with a wedge product.
</document>

<document source="https://math2.tamnd.com/node/004J7ZEFAT.md" tag="004J7ZEFAT" kind="example">
# Power set Boolean algebra

## Example

Let $X$ be a set, and let $\mathcal{P}(X)$ be the set of all subsets of $X$.
With $A \wedge B = A \cap B$, $A \vee B = A \cup B$, $\neg A = X \setminus A$, $0 = \varnothing$ and $1 = X$, the set $\mathcal{P}(X)$ is a [[Boolean algebra|06GKBTAA5E]].
Its order is inclusion $\subseteq$.

## Remarks

If $X$ has $n$ elements, this Boolean algebra has $2^n$ elements.
A field of sets on $X$ is a subset of $\mathcal{P}(X)$ that contains $\varnothing$ and is closed under complement and binary union. It is a Boolean subalgebra.
Every finite Boolean algebra is isomorphic to a power set Boolean algebra, by [[Stone's representation theorem for finite Boolean algebras|0DYS9C3HH5]].

## Explanations

### L1 human

The subsets of a set form a Boolean algebra: intersection is and, union is or, and complement is not.
Venn diagrams show the laws of this Boolean algebra.
</document>

<document source="https://math2.tamnd.com/node/004JW7A2N5.md" tag="004JW7A2N5" kind="definition">
# Square-free integer

## Definition

A nonzero integer $n$ is square-free when no square of a [[prime number|0SN0MAEKYX]] divides $n$.
That is, there is no prime $p$ such that $p^2 \mid n$.

## Remarks

Equivalently, $n$ is square-free when the only perfect squares that divide $n$ are $1$.
A positive integer is square-free if and only if each exponent in its [[prime factorization|0AEKDFCYFA]] is $1$.
The first square-free positive integers are $1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15$. The integer $12 = 2^2 \cdot 3$ is not square-free.
The integer $0$ is not square-free, because each square divides $0$.

## Explanations

### L1 human

A number is square-free when no prime appears twice in its factorization.
For example, $30 = 2 \cdot 3 \cdot 5$ is square-free, but $18 = 2 \cdot 3 \cdot 3$ is not, because $9$ divides $18$.
</document>

<document source="https://math2.tamnd.com/node/0053CQE9Y2.md" tag="0053CQE9Y2" kind="concept">
# Sylow p-subgroup

## Summary

Let $G$ be a finite group, let $p$ be a prime number, and let $p^n$ be the largest power of $p$ that divides $|G|$.
A Sylow $p$-subgroup of $G$ is a subgroup of order $p^n$.
</document>

<document source="https://math2.tamnd.com/node/005JT76HCK.md" tag="005JT76HCK" kind="statement">
# A differentiable function is convex iff its derivative is increasing

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable.
Then $f$ is [[convex|0VV0HSW1ZS]] if and only if $f'$ is increasing on $I$.

## Remarks

If $f$ is convex, the [[slope inequalities|0BY5FTHEEE]] give $f'(x) \le f'(y)$ for $x < y$.
If $f'$ is increasing, the [[mean value theorem|0BARQY52N5]] on $[x, y]$ and $[y, z]$ gives the slope inequalities.
Then the graph lies above each tangent line: $f(y) \ge f(x) + f'(x)(y - x)$.
Mathlib states the converse direction as `MonotoneOn.convexOn_of_deriv`.

## Explanations

### L1 human

A differentiable function is convex when its slope keeps growing.
The graph then turns to the left as we move along it to the right.
</document>

<document source="https://math2.tamnd.com/node/005Q84BMAD.md" tag="005Q84BMAD" kind="statement">
# Taylor's theorem with the Cauchy remainder

## Statement

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.
Then there is $\xi$ strictly between $a$ and $x$ with
$$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a),$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

Apply the [[mean value theorem|0BARQY52N5]] to $F(t) = f(t) + \sum_{j=1}^{k} f^{(j)}(t) (x - t)^j / j!$ on $J$. Note that $F(x) = f(x)$ and $F(a) = P_k(x)$.
This form sometimes gives better bounds than the [[Lagrange form|0H5ZAZCX2T]], for example for $\ln(1 + x)$ with $-1 < x < 0$.
Mathlib states it as `taylor_mean_remainder_cauchy`.

## Explanations

### L1 human

The Cauchy form is another exact expression for the error.
It moves part of the power of $(x - a)$ to a power of $(x - \xi)$.
</document>

<document source="https://math2.tamnd.com/node/006BHFWMAP.md" tag="006BHFWMAP" kind="concept">
# Row equivalence

## Summary

Two matrices are row equivalent when one of them can be changed into the other by a finite sequence of elementary row operations.
</document>

<document source="https://math2.tamnd.com/node/00758QM6JJ.md" tag="00758QM6JJ" kind="statement">
# The characteristic of an integral domain is zero or prime

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]].
Then the [[characteristic|0P7CH1CND5]] of $R$ is $0$ or a prime number.

## Remarks

The proof: suppose that the characteristic is $n = a b$ with $1 < a, b < n$. Then $(a \cdot 1)(b \cdot 1) = n \cdot 1 = 0$, but $a \cdot 1 \neq 0$ and $b \cdot 1 \neq 0$.
The characteristic is not $1$, because $1 \neq 0$ in an integral domain.
In particular, the characteristic of a [[field|0QA8CHH8MG]] is $0$ or a prime number.
The ring $\mathbb{Z}/6\mathbb{Z}$ has characteristic $6$, and it is not an integral domain.

## Explanations

### L1 human

If $6 \cdot 1 = 0$ in a ring, then $(2 \cdot 1)(3 \cdot 1) = 0$.
In an integral domain, this forces $2 \cdot 1 = 0$ or $3 \cdot 1 = 0$.
So the least $n$ with $n \cdot 1 = 0$ cannot be a product of two smaller numbers.
</document>

<document source="https://math2.tamnd.com/node/0076HWHGHQ.md" tag="0076HWHGHQ" kind="concept">
# Quotient ring

## Summary

The quotient ring of a ring by a two-sided ideal is the set of the cosets of the ideal.
Its operations come from the operations of the ring.
The integers modulo $n$ are the main example.
The [[definition of the quotient ring|0VFW4Y69BT]] gives the construction.
</document>

<document source="https://math2.tamnd.com/node/007NPQX34K.md" tag="007NPQX34K" kind="concept">
# Least common multiple

## Summary

The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both of them.
We write it as $\operatorname{lcm}(a, b)$.
The [[definition of the least common multiple|09S5SN35DJ]] gives the precise form and the convention when one of the numbers is $0$.
</document>

<document source="https://math2.tamnd.com/node/007QED3E0Y.md" tag="007QED3E0Y" kind="definition">
# Variance

## Definition

Let $X$ be an integrable real [[random variable|0DKRR22AEQ]] with mean $\mu = E[X]$.
The variance of $X$ is
$$\operatorname{Var}(X) = E\left[(X - \mu)^2\right] \in [0, \infty].$$

## Remarks

Other notations are $\operatorname{V}(X)$, $\sigma^2$ and $\sigma_X^2$. The variance is finite if and only if $E[X^2] < \infty$.
For a discrete $X$ with values $x_i$, $\operatorname{Var}(X) = \sum_i (x_i - \mu)^2 P(X = x_i)$. For $X$ with density $f$, $\operatorname{Var}(X) = \int (x - \mu)^2 f(x) \, dx$.
$\operatorname{Var}(X) = 0$ if and only if $X = \mu$ almost surely.
For a fair die, $\mu = 7/2$ and $\operatorname{Var}(X) = 35/12$.

## Explanations

### L1 human

The variance tells how far, on average, the values of $X$ are from the mean. The distances are squared, so that large deviations count more and positive and negative deviations do not cancel.
Its square root, the standard deviation, is in the same units as $X$.
</document>

<document source="https://math2.tamnd.com/node/00889YD15K.md" tag="00889YD15K" kind="statement">
# Dimension of the dual space

## Statement

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
Then the [[dual space|07JBSZE0V4]] $V^*$ is finite-dimensional, and
$$\dim V^* = \dim V.$$

## Remarks

The [[dual basis|0ZD2CF6ZS0]] of a basis of $V$ is a basis of $V^*$ with the same number of elements.
This also follows from $\dim \Hom_F(V, W) = \dim V \cdot \dim W$ with $W = F$.
The isomorphism $V \to V^*$ that sends $b_i$ to $b_i^*$ depends on the basis.
For infinite-dimensional $V$, the dimension of $V^*$ is strictly greater than the dimension of $V$.

## Explanations

### L1 human

A linear form on an $n$-dimensional space is fixed by its $n$ values on a basis.
These $n$ values can be any numbers.
So the linear forms make up an $n$-dimensional space too.
</document>

<document source="https://math2.tamnd.com/node/008BH74D7R.md" tag="008BH74D7R" kind="concept">
# Composite number

## Summary

A composite number is a natural number greater than $1$ that is not prime.
Such a number is a product of two smaller natural numbers, each greater than $1$.
The [[definition of a composite number|0N3ADXHNTQ]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/008TBCYDPX.md" tag="008TBCYDPX" kind="example">
# Zero ring

## Example

The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$.
It is a [[commutative ring|0GYGHKX834]] in which $1 = 0$.

## Remarks

A [[ring|0894J980RK]] $R$ is the zero ring if and only if $1 = 0$ in $R$. If $1 = 0$, then $a = a \cdot 1 = a \cdot 0 = 0$ for each $a \in R$.
The zero ring is not an [[integral domain|09S3PFBEPM]] and not a [[field|0QA8CHH8MG]], because these definitions require $1 \neq 0$.

## Explanations

### L1 human

The zero ring has only one element.
All the axioms of a ring hold, but in a trivial way.
It is the only ring in which $1$ and $0$ are the same element.
</document>

<document source="https://math2.tamnd.com/node/009892PZ30.md" tag="009892PZ30" kind="definition">
# Linear map of a matrix

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with bases $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$.
Let $A = (a_{ij})$ be an $m \times n$ matrix over $F$.
The *linear map of $A$* in the bases $B$ and $C$ is the linear map $T_A \colon V \to W$ with
$$T_A(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$

## Remarks

The map $A \mapsto T_A$ is the inverse of the map $T \mapsto [T]_{C,B}$ of the [[matrix of a linear map|0YQSGNYE3K]].
For $V = F^n$ and $W = F^m$ with the standard bases, $T_A$ is the map $x \mapsto Ax$, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].
In Mathlib, this map is `Matrix.toLin`.

## Explanations

### L1 human

A matrix and two bases together define a linear map.
Column $j$ tells us where the map sends basis vector $j$.
This undoes the step from a linear map to its matrix.
</document>

<document source="https://math2.tamnd.com/node/00999TYPXB.md" tag="00999TYPXB" kind="definition">
# Local maximum

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local maximum at $a$ if there is $\delta > 0$ with $f(x) \le f(a)$ for all $x \in D$ with $|x - a| < \delta$.
It is a strict local maximum if $f(x) < f(a)$ for all such $x \ne a$.

## Remarks

A global maximum, where $f(x) \le f(a)$ for all $x \in D$, is also a local maximum.
A [[local minimum|0NS8BP96QX]] is defined with $f(x) \ge f(a)$. Local maxima and minima are local extrema.
At an interior local extremum, a differentiable function has [[derivative zero|0PCVBRV9VJ]].
Mathlib defines it as `IsLocalMax`.

## Explanations

### L1 human

A local maximum is a hilltop of the graph.
Nearby values are not higher, although values far away may be higher.
</document>

<document source="https://math2.tamnd.com/node/009CY8JXFX.md" tag="009CY8JXFX" kind="statement">
# Scaling a row scales the determinant

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Let $B$ be the matrix that results when one row of $A$ is multiplied by $c$.
Then $\det B = c \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is a special case of [[linearity in each row|0PPG793642]].
It describes the effect of the scaling [[row operation|04XWH28452]] on the determinant.
Scaling all $n$ rows gives $\det(cA) = c^n \det A$, see [[the determinant of a scalar multiple|01EXSMK864]].

## Explanations

### L1 human

If we stretch one edge of a box by the factor $c$, then its volume is multiplied by $c$.
The determinant behaves in the same way.
For example, doubling one row doubles the determinant.
</document>

<document source="https://math2.tamnd.com/node/009QCP5T3Z.md" tag="009QCP5T3Z" kind="definition">
# Sum of divisors

## Definition

For a positive integer $n$, the sum of divisors of $n$ is
$$\sigma(n) = \sigma_1(n) = \sum_{d \mid n} d,$$
where $\sigma_1$ is the [[divisor function|0YEW47VJ0J]] with $k = 1$.

## Remarks

If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $\sigma(n) = \prod_{i=1}^{r} \frac{p_i^{e_i + 1} - 1}{p_i - 1}$.
For example, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$.
The sum of the proper divisors is $\sigma(n) - n$. It defines [[perfect|0SB7Q8SYSF]], [[abundant|0VS0MP1P5H]] and [[deficient|0G01TS1RRP]] numbers.

## Explanations

### L1 human

The function $\sigma(n)$ adds all the divisors of $n$, including $1$ and $n$.
For a prime $p$, the only divisors are $1$ and $p$, so $\sigma(p) = p + 1$.
</document>

<document source="https://math2.tamnd.com/node/009TEMTQKH.md" tag="009TEMTQKH" kind="statement">
# Stars and bars

## Statement

Let $n$ and $k$ be natural numbers with $n \ge 1$.
Then the number of multisets of size $k$ with elements in an $n$-element set is
$$\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$$
Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

## Remarks

The proof writes a multiset as a row of $k$ stars and $n - 1$ bars. The bars separate the stars into $n$ groups, one group for each element.
The number of solutions in positive integers is $\binom{k - 1}{n - 1}$ for $k \ge 1$. This is the [[number of compositions|057H72B2MB]] of $k$ into $n$ parts.

## Explanations

### L1 human

To share $k$ identical sweets among $n$ children, put the $k$ sweets in a row and insert $n - 1$ dividers.
There are $k + n - 1$ positions, and you choose which $k$ of them are sweets.
</document>

<document source="https://math2.tamnd.com/node/00A0RPAQ8A.md" tag="00A0RPAQ8A" kind="definition">
# Normed vector space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
A *normed vector space* over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with a [[norm|0V2WP9B1MF]] $\norm{\cdot}$ on $V$.

## Remarks

A normed space is a metric space with the distance $d(x, y) = \norm{x - y}$.
A normed space that is complete for this metric is a *Banach space*.
Each [[inner product space|0QVNCJXRY3]] is a normed space with the induced norm.
On a finite-dimensional space, all norms are equivalent, so they define the same topology.

## Explanations

### L1 human

A normed space is a vector space where each vector has a length.
With lengths, we also have distances, and so limits and continuity.
This is the setting of much of analysis.
</document>

<document source="https://math2.tamnd.com/node/00A2EDFT9Q.md" tag="00A2EDFT9Q" kind="definition">
# Principal ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal $I$ of $R$ is principal if there is an element $a \in R$ such that $I = (a)$, the [[left ideal generated by|030TZ0DQWT]] $\{a\}$.

## Remarks

The left ideal generated by $a$ is $R a = \{r a : r \in R\}$.
If $R$ is commutative, the principal ideal $(a)$ is the set of the multiples of $a$.
Every ideal of $\mathbb{Z}$ is principal: each one is $n\mathbb{Z}$ for some integer $n \ge 0$.
In $\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal.

## Explanations

### L1 human

A principal ideal is the set of all the multiples of one element.
For example, the multiples of $3$ in the integers form the principal ideal $(3)$.
</document>

<document source="https://math2.tamnd.com/node/00BE2J8H05.md" tag="00BE2J8H05" kind="concept">
# One-sided limit

## Summary

A one-sided limit is the limit of a function as the argument approaches a point from the left only or from the right only. A function has a limit at a point exactly when both one-sided limits exist and agree. The node [[one-sided limit|0909PWZR62]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/00BETAJNZ4.md" tag="00BETAJNZ4" kind="definition">
# Platonic solid

## Definition

A Platonic solid is a convex [[polyhedron|0325SGSJAR]] $P$ in $\mathbb{R}^3$ with these two properties:
- the faces of $P$ are [[regular polygons|01KJJ9A400]] that are all congruent to each other;
- the same number of faces meet at each vertex of $P$.

## Remarks

If each face has $p$ sides and $q$ faces meet at each vertex, the solid has the Schläfli symbol $\{p, q\}$.
The five Platonic solids are the regular tetrahedron $\{3, 3\}$, the cube $\{4, 3\}$, the regular octahedron $\{3, 4\}$, the regular dodecahedron $\{5, 3\}$ and the regular icosahedron $\{3, 5\}$. See [[the classification|0M4TCS0RRG]].
They have the name of Plato, who wrote about them in the dialogue Timaeus.

## Explanations

### L1 human

A Platonic solid is a perfectly regular dice shape. All faces are the same regular polygon, and every corner looks the same. A cube is one: six equal squares, with three squares at each corner.
</document>

<document source="https://math2.tamnd.com/node/00BKRZKMSS.md" tag="00BKRZKMSS" kind="concept">
# Composition of linear maps

## Summary

The composition of two linear maps is the linear map that applies the first map and then the second.
</document>

<document source="https://math2.tamnd.com/node/00BT70ND50.md" tag="00BT70ND50" kind="statement">
# Picard–Lindelöf theorem in integral form

## Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.
Let $B$ be the closed ball of radius $a$ around $x_0$.
Then there is exactly one continuous curve $\alpha : J \to B$ such that
$$\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.$$

## Remarks

The curve $\alpha$ is the unique fixed point of the [[Picard operator|00WAVB68F9]] on the curves with values in $B$.
By the [[integral equation form|015KK993C5]], this is the same curve as the solution of the [[Picard–Lindelöf theorem|0P6KRD5B7A]].

## Explanations

### L1 human

This is the Picard–Lindelöf theorem written without derivatives.
It says that the integral equation has exactly one continuous solution that stays in the ball.
</document>

<document source="https://math2.tamnd.com/node/00BWV5CKCF.md" tag="00BWV5CKCF" kind="statement">
# Finite intersection property

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if it has the property that follows.
Let $(F_i)_{i \in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $K \cap F_{i_1} \cap \cdots \cap F_{i_n} \neq \emptyset$ for all finitely many indices $i_1, \ldots, i_n \in I$.
Then also $K \cap \bigcap_{i \in I} F_i \neq \emptyset$.

## Remarks

A family of sets has the finite intersection property when each finite subfamily has a nonempty intersection.
Proof: take complements. The sets $X \setminus F_i$ cover $K$ if and only if $K \cap \bigcap_i F_i = \emptyset$. Then use [[compactness via open covers|0P6PMJB4MA]].
For $K = X$: a space is compact if and only if each family of closed sets with the finite intersection property has nonempty intersection.
[[Cantor's intersection theorem|0EZBG8SKJM]] is a special case.

## Explanations

### L1 human

This is the dual form of compactness, with closed sets instead of open sets.
Suppose that each finite subfamily of the closed sets has a common point in $K$. Then the whole family has a common point in $K$.
</document>

<document source="https://math2.tamnd.com/node/00C0BR7D7H.md" tag="00C0BR7D7H" kind="concept">
# Graph

## Summary

A graph is a set of vertices together with a set of edges, each edge joining two distinct vertices. Graphs model networks, relations and maps, and they are the main object of graph theory. The node [[graph|0JD5C82QMB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/00C2WN1H21.md" tag="00C2WN1H21" kind="definition">
# Polyhedron

## Definition

A convex polyhedron is the [[convex hull|0FQSDC2EQ2]] of a finite set of points of $\mathbb{R}^3$ that do not all lie in one [[plane|0V3XAMF3C1]].
A polyhedron is a bounded subset of $\mathbb{R}^3$ with a connected interior that is the union of finitely many convex polyhedra.
Its faces are the connected parts of its boundary inside the planes that cover the boundary. Its edges and vertices are the segments and points where the faces meet.

## Remarks

There is no single definition of a polyhedron that all authors use. Some definitions describe a surface in place of a solid, and some allow self-crossing faces.
A convex polyhedron is also a bounded intersection of finitely many closed half-spaces, with a nonempty interior.
Examples are the cube, the regular tetrahedron and each prism over a convex polygon.

## Explanations

### L1 human

A polyhedron is a solid with flat sides, like a box or a pyramid. A ball is not one, because its surface is curved. The cube has $8$ vertices, $12$ edges and $6$ faces.
</document>

<document source="https://math2.tamnd.com/node/00C5P9JZBK.md" tag="00C5P9JZBK" kind="object">
# Euler–Mascheroni constant

## Description

The Euler–Mascheroni constant is the real number
$$\gamma = \lim_{n \to \infty} \big( H_n - \ln n \big) \approx 0.5772156649,$$
where $H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n$-th [[harmonic number|0D2Y6G4C4S]].

## Remarks

The [[limit exists|05S2GSPMGA]], because the sequence $H_n - \ln n$ is decreasing and bounded below by $0$.
Also $\gamma = \int_1^\infty \Big( \frac{1}{\lfloor x \rfloor} - \frac{1}{x} \Big) \, dx$.
It is not known whether $\gamma$ is rational. Mathlib defines it as `Real.eulerMascheroniConstant`.
Leonhard Euler introduced the constant in a paper of 1734. Lorenzo Mascheroni later computed it to more decimal places, with some errors.

## Explanations

### L1 human

The harmonic numbers grow like $\ln n$, and $\gamma$ is the limit of the difference.
So $H_n \approx \ln n + 0.5772$ for large $n$.
</document>

<document source="https://math2.tamnd.com/node/00CN2F997Q.md" tag="00CN2F997Q" kind="statement">
# Addition formula for Fibonacci numbers

## Statement

Let $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.$$

## Remarks

With $m = n$ it gives $F_{2n+1} = F_n^2 + F_{n+1}^2$.
An equivalent form is $F_{m+n} = F_{m-1} F_n + F_m F_{n+1}$ for $m \ge 1$.
The formula is the basis of fast algorithms that compute $F_n$ with $O(\log n)$ arithmetic operations.

## Explanations

### L1 human

To get a Fibonacci number with a large index, you can combine Fibonacci numbers with smaller indices.
For example, with $m = 2$ and $n = 3$: $F_6 = F_2 F_3 + F_3 F_4 = 1 \cdot 2 + 2 \cdot 3 = 8$.
</document>

<document source="https://math2.tamnd.com/node/00CR00C9DN.md" tag="00CR00C9DN" kind="concept">
# Countably infinite set

## Summary

A countably infinite set is a set that is equinumerous with the set of natural numbers.
</document>

<document source="https://math2.tamnd.com/node/00CT0PTH74.md" tag="00CT0PTH74" kind="definition">
# Graph isomorphism

## Definition

Let $G$ and $H$ be [[graphs|0JD5C82QMB]] with vertex sets $V(G)$ and $V(H)$.
A graph isomorphism from $G$ to $H$ is a bijection $f : V(G) \to V(H)$ such that, for all $u, v \in V(G)$,
$$u \sim v \text{ in } G \iff f(u) \sim f(v) \text{ in } H.$$
The graphs $G$ and $H$ are isomorphic, written $G \cong H$, if there is an isomorphism between them.

## Remarks

Isomorphic graphs have the same number of vertices and edges and the same degrees. They differ only in the names of their vertices.
An isomorphism from $G$ to itself is an automorphism of $G$.
The path $P_4$ is isomorphic to its own [[complement|0RNMP6VCR2]].

## Explanations

### L1 human

Two graphs are isomorphic when one is a relabelling of the other.
Two drawings can look very different and still show isomorphic graphs.
</document>

<document source="https://math2.tamnd.com/node/00DGYY10FV.md" tag="00DGYY10FV" kind="statement">
# The logarithm grows slower than any power

## Statement

For every real $a > 0$,
$$\lim_{x \to \infty} \frac{\ln x}{x^a} = 0.$$
That is, $\ln x = o(x^a)$ as $x \to \infty$.

## Remarks

Substitute $x = e^{t/a}$. Then $\frac{\ln x}{x^a} = \frac{t / a}{e^{t}}$, which tends to $0$ by [[exponential growth|0CT6CBGTYY]].
In [[little o notation|04TSHB9H8D]], $\ln x = o(x)$. Mathlib states this case as `Real.isLittleO_log_id_atTop`.
Similarly, $x^a \ln x \to 0$ as $x \to 0^+$.

## Explanations

### L1 human

The logarithm goes to infinity, but very slowly.
Each positive power of $x$, even $x^{0.001}$, eventually beats it.
</document>

<document source="https://math2.tamnd.com/node/00DRK90Z6K.md" tag="00DRK90Z6K" kind="statement">
# Laws of exponents in a group

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a \in G$.
For an integer $n$, define $a^n$ as follows: $a^0 = e$, $a^{n+1} = a^n a$ for $n \geq 0$, and $a^{-n} = (a^{-1})^n$ for $n > 0$.
Then $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$.

## Remarks

The law $(ab)^n = a^n b^n$ is true for all $n$ when $ab = ba$, but it can fail when $a$ and $b$ do not commute.
In additive notation the laws are $(m + n)a = ma + na$ and $n(ma) = (nm)a$.
The first law shows that $n \mapsto a^n$ is a group homomorphism from $(\mathbb{Z}, +)$ to $G$.

## Explanations

### L1 human

The powers of an element of a group follow the rules that you know for the powers of a nonzero number.
A negative power means a power of the inverse, so $a^{-2} = a^{-1} a^{-1}$.
For example, $a^3 a^{-1} = a a a a^{-1} = a^2$.
</document>

<document source="https://math2.tamnd.com/node/00DW862T78.md" tag="00DW862T78" kind="statement">
# −1 is a square modulo p if and only if p is not 3 modulo 4

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
There is an integer $x$ with $x^2 \equiv -1 \pmod p$ if and only if $p \not\equiv 3 \pmod 4$.

## Remarks

For $p = 2$, take $x = 1$.
For odd $p$, the statement is the [[first supplement to quadratic reciprocity|0PHBS78HKT]].
For $p \equiv 1 \pmod 4$, an explicit solution is $x = \left(\frac{p-1}{2}\right)!$, by [[Wilson's theorem|03KGGKTB2E]].
The statement is a step in the proof of [[Fermat's theorem on sums of two squares|043B3ENSTY]].

## Explanations

### L1 human

The primes $2$, $5$, $13$ and $17$ have a square root of $-1$: for example $5^2 = 25 \equiv -1 \pmod{13}$.
The primes $3$, $7$ and $11$ have none.
The rule is that $p$ must not leave remainder $3$ on division by $4$.
</document>

<document source="https://math2.tamnd.com/node/00E0YZCERH.md" tag="00E0YZCERH" kind="concept">
# Power set

## Summary

The power set of a set is the set of all its subsets.
</document>

<document source="https://math2.tamnd.com/node/00EBGSR5NY.md" tag="00EBGSR5NY" kind="definition">
# Euclidean vector

## Definition

Let $n$ be a positive integer.
A Euclidean vector in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is an element $v = (v_1, \dots, v_n)$ of $\mathbb{R}^n$. The real numbers $v_1, \dots, v_n$ are the components of $v$.
For two points $A = (a_1, \dots, a_n)$ and $B = (b_1, \dots, b_n)$, the vector from $A$ to $B$ is
$$\overrightarrow{AB} = (b_1 - a_1, \dots, b_n - a_n).$$
The sum of two vectors $u$ and $v$ is $u + v = (u_1 + v_1, \dots, u_n + v_n)$.
The product of a real number $c$ and a vector $v$ is $c v = (c v_1, \dots, c v_n)$.
The zero vector is $0 = (0, \dots, 0)$.

## Remarks

Two arrows give the same vector exactly when a translation moves one arrow onto the other.
For each three points $A$, $B$ and $C$, the vectors satisfy $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.
With the sum and the product by real numbers, the vectors of $\mathbb{R}^n$ form a real vector space of dimension $n$.

## Explanations

### L1 human

A vector tells you how to move: how far, and in which direction. The vector $(3, 4)$ in the plane says "move 3 to the right and 4 up". It does not matter where you start, so arrows of the same length and direction are the same vector.
</document>

<document source="https://math2.tamnd.com/node/00EHX6FWC0.md" tag="00EHX6FWC0" kind="definition">
# Root of a polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f \in R[x]$.
An element $c \in R$ is a root of $f$ if $f(c) = 0$, where $f(c)$ is the [[value of $f$ at $c$|0GJ7T0NK49]].

## Remarks

The roots of $x^2 - 1$ in $\mathbb{Z}$ are $1$ and $-1$.
The polynomial $x^2 + 1$ has no root in $\mathbb{R}$. It has the roots $i$ and $-i$ in $\mathbb{C}$.
By the [[factor theorem|0K3MWA7DMC]], over a commutative ring, $c$ is a root of $f$ if and only if $x - c$ divides $f$.

## Explanations

### L1 human

A root is a solution of the equation $f(x) = 0$.
Whether a polynomial has a root depends on the ring: $x^2 - 2$ has no root in $\mathbb{Q}$, but it has the root $\sqrt{2}$ in $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/00EYM7MDKS.md" tag="00EYM7MDKS" kind="concept">
# Hasse diagram

## Summary

A Hasse diagram is a drawing of a finite [[partially ordered set|011C29Q1S1]].
Each element is a point, and there is a line upward from $a$ to $b$ when $b$ covers $a$.
This means that $a < b$ and no $c$ satisfies $a < c < b$.
Then $a \le b$ exactly when one can go from $a$ to $b$ by a path that goes only upward.

## Remarks

The diagram does not draw the loops of reflexivity or the lines that transitivity implies.
The Hasse diagram of the power set of $\{1, 2, 3\}$ under $\subseteq$ is the picture of a cube.

## Explanations

### L1 human

A Hasse diagram is a minimal picture of an order.
For the divisors of $12$ under divisibility, put $1$ at the bottom and $12$ at the top.
Draw a line from $2$ to $4$ and from $2$ to $6$.
Do not draw a line from $2$ to $12$, because the path through $4$ already shows it.
</document>

<document source="https://math2.tamnd.com/node/00F2JFA41J.md" tag="00F2JFA41J" kind="definition">
# Orbit

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
The orbit of $x$ is the set $G \cdot x = \{g \cdot x : g \in G\}$.
The set of all orbits is written $X/G$.

## Remarks

The point $x$ lies in its own orbit, because $e \cdot x = x$.
The [[orbits partition|0ZC7FYE0NZ]] the set $X$.
For the action of the rotation group of a square on its corners, there is one orbit, which contains all four corners.

## Explanations

### L1 human

The orbit of a point collects all the places where the group can move the point.
Two points in the same orbit can be moved onto each other.
Points in different orbits are never moved onto each other.
</document>

<document source="https://math2.tamnd.com/node/00FSJTXPJM.md" tag="00FSJTXPJM" kind="concept">
# Closed map

## Summary

A closed map is a function between topological spaces that sends closed sets to closed sets.
Continuous maps from compact spaces to Hausdorff spaces are closed maps.
</document>

<document source="https://math2.tamnd.com/node/00FWBN7F0Y.md" tag="00FWBN7F0Y" kind="statement">
# Congruence is compatible with addition

## Statement

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers.
If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a + c \equiv b + d \pmod n$.

## Remarks

The proof: $(a + c) - (b + d) = (a - b) + (c - d)$, and $n$ divides both terms.
The same holds for subtraction: $a - c \equiv b - d \pmod n$.
Together with [[compatibility with multiplication|072D5G4SK6]], this result shows that the addition of [[integers modulo n|01N3AYAA6N]] is well defined.

## Explanations

### L1 human

To find the remainder of a sum, you can add the remainders and reduce again.
For example, $17 \equiv 2 \pmod 5$ and $24 \equiv 4 \pmod 5$, so $17 + 24 = 41 \equiv 2 + 4 = 6 \equiv 1 \pmod 5$.
</document>

<document source="https://math2.tamnd.com/node/00FWNP9TXY.md" tag="00FWNP9TXY" kind="definition">
# Contour integral

## Definition

Let $a < b$ be real numbers and let $\gamma : [a, b] \to \mathbb{C}$ be a path that is continuous and piecewise continuously differentiable.
Let $f$ be a complex function that is continuous on the image $\gamma([a, b])$.
The contour integral of $f$ along $\gamma$ is
$$\int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt.$$
The integral on the right of a complex function $g$ of $t \in [a, b]$ is $\int_a^b \operatorname{Re} g(t) \, dt + i \int_a^b \operatorname{Im} g(t) \, dt$.

## Remarks

The length of $\gamma$ is $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$.
The integral does not change under a reparametrization of $\gamma$ that keeps the direction. It changes sign if we reverse the direction.
The path $\gamma$ is closed if $\gamma(a) = \gamma(b)$. For a closed path, many authors write $\oint_\gamma f(z) \, dz$.
For example, along $\gamma(t) = t + i t$ with $t \in [0, 1]$, $\int_\gamma z \, dz = \int_0^1 (1 + i) t (1 + i) \, dt = i$.

## Explanations

### L1 human

A contour integral adds up the values of $f$ along a path, each one multiplied by a small step $dz$ of the path.
Because $dz$ is a complex number, the direction of each step matters, and not only its length.
</document>

<document source="https://math2.tamnd.com/node/00GC8ZQP4Q.md" tag="00GC8ZQP4Q" kind="concept">
# Zero of a holomorphic function

## Summary

A zero of a holomorphic function $f$ is a point $z_0$ with $f(z_0) = 0$, see the [[definition|0CEGEASY49]].
The zeros of a holomorphic function that is not identically zero on a domain are isolated, and each has a finite order.
</document>

<document source="https://math2.tamnd.com/node/00GXTQMY8Z.md" tag="00GXTQMY8Z" kind="definition">
# Modular multiplicative inverse

## Definition

Let $n$ be a positive integer and let $a$ be an integer.
A modular multiplicative inverse of $a$ modulo $n$ is an integer $x$ such that $a x \equiv 1 \pmod n$.

## Remarks

An inverse of $a$ modulo $n$ exists if and only if $\gcd(a, n) = 1$. See [[existence of inverses modulo n|0M67FWZWPQ]].
If $x$ is an inverse of $a$, then the inverses of $a$ are exactly the integers $y \equiv x \pmod n$. So the inverse is unique as a residue class, and we write it as $a^{-1} \bmod n$.
In the ring of [[integers modulo n|01N3AYAA6N]], the class $\overline{x}$ is the inverse of $\overline{a}$.
For example, $3^{-1} \equiv 5 \pmod 7$, because $3 \cdot 5 = 15 \equiv 1 \pmod 7$.

## Explanations

### L1 human

Modulo $n$, the inverse of $a$ plays the part of the fraction $1/a$.
To solve $3 x \equiv 4 \pmod 7$, multiply both sides by $5$, the inverse of $3$. You get $x \equiv 20 \equiv 6 \pmod 7$.
</document>

<document source="https://math2.tamnd.com/node/00HF4RJ5QN.md" tag="00HF4RJ5QN" kind="concept">
# Direct product of groups

## Summary

The direct product of two groups $G$ and $H$ is the set $G \times H$ of ordered pairs, with the operation done in each component separately.
It is the simplest way to build a new group from two given groups.
</document>

<document source="https://math2.tamnd.com/node/00HJCMC2CD.md" tag="00HJCMC2CD" kind="concept">
# Conic section

## Summary

A conic section is a curve that a plane cuts from the surface of a double cone. In the plane, it is the set of zeros of a polynomial of degree $2$ in two variables. The non-degenerate conics are the ellipses, the parabolas and the hyperbolas.
</document>

<document source="https://math2.tamnd.com/node/00HPT88ZX0.md" tag="00HPT88ZX0" kind="definition">
# Norm of a Gaussian integer

## Definition

Let $\alpha = a + bi$ be a [[Gaussian integer|0B268GVX95]], with integers $a$ and $b$.
The norm of $\alpha$ is the integer $N(\alpha) = a^2 + b^2 = \alpha \bar{\alpha}$, where $\bar{\alpha} = a - bi$ is the complex conjugate of $\alpha$.

## Remarks

The norm is multiplicative: $N(\alpha \beta) = N(\alpha) N(\beta)$ for all Gaussian integers $\alpha$ and $\beta$.
This product rule is the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].
The norm is a nonnegative integer, and $N(\alpha) = 0$ only for $\alpha = 0$. The units are the elements of norm $1$.
For example, $N(2 + i) = 5$ and $N(3 - 2i) = 13$.

## Explanations

### L1 human

The norm of $a + bi$ is the square of its distance from $0$ in the complex plane.
It turns a question about Gaussian integers into a question about ordinary integers.
</document>

<document source="https://math2.tamnd.com/node/00J3DCCEK5.md" tag="00J3DCCEK5" kind="problem">
# Trisecting the angle

## Problem

Given an arbitrary angle, construct with [[straightedge and compass|0DAECMZVRT]] alone an angle one third as large.

## Remarks

Some angles can be trisected. For example, a right angle can be trisected, because an angle of $30^\circ$ is constructible.
The general problem asks for one method that works for every angle.
The answer is no: [[the angle of $60^\circ$ cannot be trisected|0EWXPZCSZ0]].
The problem has solutions with other tools, such as a marked ruler or origami.

## History

The problem comes from ancient Greek geometry.
Pierre Wantzel proved in 1837 that it has no solution for arbitrary angles.

## Explanations

### L1 human

Bisecting an angle is easy with a compass.
Trisecting it, that is, dividing it into three equal parts, is not possible in general with straightedge and compass alone.
</document>

<document source="https://math2.tamnd.com/node/00J3KZP8Y0.md" tag="00J3KZP8Y0" kind="concept">
# Lower triangular matrix

## Summary

A lower triangular matrix is a square matrix whose entries above the main diagonal are all zero.
</document>

<document source="https://math2.tamnd.com/node/00J8E3HX0M.md" tag="00J8E3HX0M" kind="concept">
# Divisibility rule

## Summary

A divisibility rule is a short method to decide whether a fixed integer $d$ divides an integer $n$.
It uses the digits of $n$, and it does not need a full division.
Most rules for base $10$ come from the [[congruences|0TPW25CBN6]] of the powers $10^k$ modulo $d$.
For example, $10 \equiv 1 \pmod 3$, so $n$ and the sum of its digits have the same remainder modulo $3$.

## Remarks

For example, $2$ divides $n$ if and only if the last digit of $n$ is even. Also $5$ divides $n$ if and only if the last digit is $0$ or $5$.
See [[divisibility by 3|0R9D4CEJEB]], [[divisibility by 9|0ESHD7ZQDX]] and [[divisibility by 11|09YBMJTNBM]].
[[Casting out nines|0E7W0EM6J6]] uses the rule for $9$ to check computations.

## Explanations

### L1 human

You can see at once that $1{,}234{,}570$ is a multiple of $10$, because it ends with $0$.
Divisibility rules give similar quick tests for other divisors, for example $3$, $4$, $9$ and $11$.
</document>

<document source="https://math2.tamnd.com/node/00K0RF9BFY.md" tag="00K0RF9BFY" kind="definition">
# Multinomial coefficient

## Definition

Let $k_1, \dots, k_m$ be natural numbers and let $n = k_1 + \dots + k_m$.
The multinomial coefficient is the natural number
$$\binom{n}{k_1, k_2, \dots, k_m} = \frac{n!}{k_1!\, k_2! \cdots k_m!}.$$

## Remarks

For $m = 2$ it is the [[binomial coefficient|092GEF38YD]]: $\binom{n}{k, n-k} = \binom{n}{k}$.
It counts the ways to put $n$ distinct objects into $m$ labelled boxes, with $k_i$ objects in box $i$.
It also counts the distinct words that use a letter $a_i$ exactly $k_i$ times. For example, MISSISSIPPI has $\frac{11!}{1!\,4!\,4!\,2!} = 34650$ distinct rearrangements.
It is the coefficient in the [[multinomial theorem|0XFC1VQYEK]].

## Explanations

### L1 human

The multinomial coefficient counts the ways to split $n$ objects into groups of given sizes.
To split 4 people into a group of 2 and two groups of 1, with labelled groups, there are $\frac{4!}{2!\,1!\,1!} = 12$ ways.
</document>

<document source="https://math2.tamnd.com/node/00K4FK4NBH.md" tag="00K4FK4NBH" kind="definition">
# Nullity

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *nullity* of $T$ is
$$\operatorname{nullity} T = \dim \ker T,$$
the [[dimension|03GSZ8RTZJ]] of the [[kernel|0S23E6YE83]] of $T$.
The nullity of a matrix $A$ is the nullity of the map $x \mapsto Ax$, the dimension of the [[null space|0PHDYNR24D]] of $A$.

## Remarks

The nullity is $0$ if and only if $T$ is injective.
For an $m \times n$ matrix, the nullity is the number of free variables of $Ax = 0$.
By the [[rank–nullity theorem|0W4FSW1TVB]], $\rank T + \operatorname{nullity} T = \dim V$ when $V$ is finite-dimensional.

## Explanations

### L1 human

The nullity counts the dimensions that the map crushes to zero.
The projection of space onto a plane crushes one direction, so its nullity is $1$.
Rank counts what survives, and nullity counts what is lost.
</document>

<document source="https://math2.tamnd.com/node/00K7ASZ1PR.md" tag="00K7ASZ1PR" kind="counterexample">
# Non-uniqueness for a non-Lipschitz equation

## Counterexample

For a real number $x$, let $x^{1/3}$ be its real cube root and let $x^{2/3} = \left(x^{1/3}\right)^2$.
Consider the [[initial value problem|075VBY4K4X]] $x'(t) = 3\, x(t)^{2/3}$, $x(0) = 0$, on $\mathbb{R}$.
The constant function $x(t) = 0$ is a solution.
The function $x(t) = t^3$ is also a solution, because its derivative is $3t^2 = 3 \left(t^3\right)^{2/3}$.
More generally, for each $c \ge 0$, the function equal to $0$ for $t < c$ and to $(t - c)^3$ for $t \ge c$ is a solution.
So the problem has infinitely many solutions on every interval that contains $0$ in its interior.

## Remarks

The right side $f(x) = 3 x^{2/3}$ is continuous, so the [[Peano existence theorem|098YP831QY]] applies.
The function $f$ is not Lipschitz on any neighbourhood of $0$: the quotient $f(x)/x = 3 x^{-1/3}$ is unbounded as $x \to 0$. So the [[Picard–Lindelöf theorem|0P6KRD5B7A]] does not apply.
For an initial value $x(0) = x_0 \neq 0$, the field is $C^1$ near $x_0$, and the solution is unique near $t = 0$.

## Explanations

### L1 human

A solution can rest at $0$ for any length of time and then leave.
The field is so steep near $0$ that the equation cannot tell when the departure must happen.
This shows that continuity alone does not give uniqueness.
</document>

<document source="https://math2.tamnd.com/node/00KAFBR32E.md" tag="00KAFBR32E" kind="statement">
# Intermediate value theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Let $y$ be a real number between $f(a)$ and $f(b)$.
Then there is $c \in [a, b]$ with $f(c) = y$.

## Remarks

Assume $f(a) < y < f(b)$. Let $c$ be the supremum of $\{ x \in [a, b] : f(x) < y \}$. Continuity gives $f(c) \le y$ and $f(c) \ge y$.
The proof needs the completeness of $\mathbb{R}$. Over $\mathbb{Q}$, the function $x^2 - 2$ changes sign on $[1, 2]$ but has no rational zero.
It gives the [[existence of $n$-th roots|0J29GM30Z7]], and it shows that each odd degree real polynomial has a real root.
Mathlib states it as `intermediate_value_Icc`.

## History

Bernard Bolzano first proved the theorem in 1817.

## Explanations

### L1 human

A continuous function on an interval cannot skip values.
To get from $f(a)$ to $f(b)$, its graph must cross every horizontal line in between.
</document>

<document source="https://math2.tamnd.com/node/00KDZRW4C0.md" tag="00KDZRW4C0" kind="concept">
# Determinant

## Summary

The determinant is the unique function of square matrices that is multilinear and alternating in the rows and is $1$ on the identity matrix.
It is zero exactly for the singular matrices, over a field.
</document>

<document source="https://math2.tamnd.com/node/00KRCVE7CK.md" tag="00KRCVE7CK" kind="statement">
# Modulus of the conjugate

## Statement

For each complex number $z$, $|\bar{z}| = |z|$.
Here $\bar{z}$ is the [[complex conjugate|0BJV24C4SY]] of $z$ and $|\cdot|$ is the [[modulus|0KW2X0TFBX]].

## Remarks

Proof: for $z = a + b i$, $|\bar{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z|$.
In the complex plane, this says that a reflection in the real axis does not change the distance to $0$.

## Explanations

### L1 human

A number and its conjugate are mirror images in the real axis.
A mirror keeps distances, so both numbers are at the same distance from $0$.
</document>

<document source="https://math2.tamnd.com/node/00MHHZ9SNE.md" tag="00MHHZ9SNE" kind="concept">
# Prime ideal

## Summary

A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors.
The prime ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime and the zero ideal.
The [[definition of a prime ideal|06HC6HK0Z8]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/00MRN0FARV.md" tag="00MRN0FARV" kind="concept">
# Independent and identically distributed random variables

## Summary

A family of random variables is independent and identically distributed (i.i.d.) when the variables are mutually independent and all have the same distribution.
It is the standard model of repeated trials of one experiment. The node [[i.i.d. random variables|0AJN7WHBJZ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/00MS9VMQ9Z.md" tag="00MS9VMQ9Z" kind="statement">
# Every vector space has a basis

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Then $V$ has a [[basis|0KCQ9Y3QKA]].

## Remarks

For a space with a finite spanning set, the proof is elementary: remove vectors from the spanning set until it is linearly independent.
In general, the proof uses Zorn's lemma: a maximal linearly independent subset of $V$ is a basis.
The statement for all vector spaces is equivalent to the axiom of choice, over set theory without choice.
For example, the axiom of choice gives a basis of $\R$ as a vector space over $\Q$, a Hamel basis.

## Explanations

### L1 human

Every vector space has a set of building blocks that is independent and reaches every vector.
For finite-dimensional spaces, we can find one by hand.
For very large spaces, we must use the axiom of choice, and we may not be able to write the basis down.
</document>

<document source="https://math2.tamnd.com/node/00N3T18QNA.md" tag="00N3T18QNA" kind="definition">
# Saddle point

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be [[differentiable|0135XZZQ8M]] at $a \in U$.
The point $a$ is a saddle point of $f$ if $\nabla f(a) = 0$ and $a$ is not a [[local extremum|04CPG9K1C4]] of $f$.

## Remarks

Equivalently, $\nabla f(a) = 0$ and each ball around $a$ contains points $x$ and $y$ with $f(x) < f(a) < f(y)$.
The origin is a saddle point of $f(x, y) = x^2 - y^2$. Along the $x$-axis $f$ has a minimum at $0$, and along the $y$-axis it has a maximum.
If $f$ is of class $C^2$ and the [[Hessian matrix|0SCSQWXRA4]] at a critical point is indefinite, the point is a saddle point. This condition is not necessary: $f(x, y) = x^3$ has a saddle point at $(0, 0)$ with zero Hessian.
In one variable, a saddle point is a stationary point that is not an extremum, such as $0$ for $x \mapsto x^3$.

## Explanations

### L1 human

The name comes from the shape of a horse saddle.
It curves up from front to back and curves down from side to side. So the centre is flat, but it is neither a top nor a bottom.
</document>

<document source="https://math2.tamnd.com/node/00N5FHP1CV.md" tag="00N5FHP1CV" kind="statement">
# Change of variables in multiple integrals

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $\varphi \colon U \to \mathbb{R}^n$ be injective and of [[class C^1|0TCBZMEYGN]] with $\det D\varphi(u) \ne 0$ for each $u \in U$.
Let $f \colon \varphi(U) \to \mathbb{R}$ be continuous with compact support in $\varphi(U)$.
Then
$$\int_{\varphi(U)} f(v) \, dv = \int_U f(\varphi(u)) \, \lvert \det D\varphi(u) \rvert \, du.$$

## Remarks

Here $\det D\varphi(u)$ is the [[Jacobian determinant|03W6Q1MFR5]] of $\varphi$ at $u$. Under these hypotheses, $\varphi(U)$ is open and $\varphi$ is a [[diffeomorphism|0Z1Y7JHGBR]] from $U$ onto $\varphi(U)$.
The factor $\lvert \det D\varphi(u) \rvert$ is the ratio by which $\varphi$ scales small volumes near $u$. A linear map $A$ maps the unit cube to a parallelepiped of volume $\lvert \det A \rvert$.
The formula extends to Lebesgue integrable $f$ on $\varphi(U)$, and to maps that fail the hypotheses only on a set of measure zero.
Important cases are [[integration in polar coordinates|0T89W1QZ9K]], cylindrical coordinates and [[spherical coordinates|05WKRC7HWT]].

## Explanations

### L1 human

In one variable, substitution needs the factor $\varphi'(u)$. In several variables, the Jacobian determinant plays this role.
It corrects for the way the change of coordinates stretches or squeezes small regions.
</document>

<document source="https://math2.tamnd.com/node/00NBBSXRQS.md" tag="00NBBSXRQS" kind="concept">
# Prior probability

## Summary

A prior probability is the probability that we give to a hypothesis before we see the data.
More generally, a prior distribution is the distribution that we give to an unknown quantity before we see the data.
In [[Bayes' theorem|0NM6E9VW04]] the prior is combined with the likelihood of the data, and the result is the [[posterior probability|0KJ7ZS8AJA]].

## Remarks

An informative prior uses real knowledge about the quantity. An uninformative prior tries to express little knowledge, for example a uniform distribution on the possible values.
An improper prior is a "distribution" with infinite total mass, such as the uniform measure on $\mathbb{R}$. It can still give a proper posterior.

## Explanations

### L1 human

The prior is what you believe before the experiment.
If $1\%$ of people have a disease, then the prior probability that a random person is sick is $0.01$, before any test is done.
</document>

<document source="https://math2.tamnd.com/node/00NZKRSWNX.md" tag="00NZKRSWNX" kind="concept">
# Complex sine and cosine

## Summary

The complex sine and cosine extend the real sine and cosine to all of $\mathbb{C}$ with the formulas $\sin z = (e^{iz} - e^{-iz})/(2i)$ and $\cos z = (e^{iz} + e^{-iz})/2$, see the [[definition|0A7YDKRNR6]].
They are entire functions, and they are not bounded on $\mathbb{C}$.
</document>

<document source="https://math2.tamnd.com/node/00PPY3RPNB.md" tag="00PPY3RPNB" kind="definition">
# Power of a point

## Definition

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$ with the centre $O$ and the radius $r$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The power of a point $P \in \mathbb{R}^2$ with respect to $C$ is the real number
$$\Pi(P) = d(P, O)^2 - r^2.$$

## Remarks

$\Pi(P) > 0$ if $P$ is outside $C$, $\Pi(P) = 0$ if $P$ is on $C$, and $\Pi(P) < 0$ if $P$ is inside $C$.
For $P$ outside $C$ and a tangent from $P$ that touches $C$ at $T$, the Pythagorean theorem gives $\Pi(P) = d(P, T)^2$.
Let a line through $P$ meet $C$ at $S_1$ and $S_2$. Then $d(P, S_1)\, d(P, S_2) = |\Pi(P)|$. This contains the [[intersecting chords theorem|0YQQYET8YQ]] and the [[intersecting secants theorem|0J5XR7BW2S]].
Jakob Steiner introduced the power of a point in 1826.

## Explanations

### L1 human

The power of a point measures how far the point is from a circle, in a squared form. For the unit circle and $P = (3, 0)$, the power is $9 - 1 = 8$. A tangent from $P$ to the circle has the length $\sqrt{8}$.
</document>

<document source="https://math2.tamnd.com/node/00QN1N6MNT.md" tag="00QN1N6MNT" kind="definition">
# Positive definite quadratic form

## Definition

Let $V$ be a finite-dimensional real vector space, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The form $Q$ is *positive definite* when
$$Q(v) > 0 \qquad \text{for all } v \in V \text{ with } v \ne 0.$$

## Remarks

$Q(x) = x^T A x$ with $A$ symmetric is positive definite if and only if $A$ is a [[positive definite matrix|0KX66CZMBB]].
Equivalently, the signature of $Q$ is $(n, 0, 0)$, with $n = \dim V$.
The form $x^2 + y^2$ is positive definite on $\R^2$; the forms $x^2 - y^2$ and $x^2$ are not.
$Q$ is positive definite if and only if its associated symmetric bilinear form is an inner product.

## Explanations

### L1 human

A positive definite quadratic form is positive everywhere except at the origin.
Its graph is a bowl with the lowest point at the origin.
Such forms describe squared lengths for different inner products.
</document>

<document source="https://math2.tamnd.com/node/00QSK85Y2X.md" tag="00QSK85Y2X" kind="concept">
# Primitive element of a finite field

## Summary

A primitive element of a finite field is a generator of its multiplicative group.
The [[definition of a primitive element|0YAA9CJQXF]] states it.
</document>

<document source="https://math2.tamnd.com/node/00R0X1WGPX.md" tag="00R0X1WGPX" kind="definition">
# Compact set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
The set $K$ is compact when $K$, with the [[subspace topology|0D5YHQGDJY]], is a [[compact space|0QPHW703PV]].

## Remarks

By [[compactness via open covers|0P6PMJB4MA]], $K$ is compact exactly when each [[cover|0NVPNC12JY]] of $K$ by open sets of $X$ has a finite subcover.
Each finite subset of $X$ is compact. The empty set is compact.
In $\mathbb{R}$, the interval $[0, 1]$ is compact, but $(0, 1)$ and $[0, \infty)$ are not. The [[Heine–Borel theorem|07PFYV65MB]] describes the compact subsets of $\mathbb{R}^n$.

## Explanations

### L1 human

A compact set behaves in many ways like a finite set.
For example, a continuous real function on a compact set is bounded and attains its maximum, as a function on a finite set does.
</document>

<document source="https://math2.tamnd.com/node/00R1JDDV5X.md" tag="00R1JDDV5X" kind="definition">
# Arc length parametrisation

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[regular|06JMR2XWXW]] $C^1$ curve.
Its arc length function is
$$s(t) = \int_a^t \|\gamma'(u)\| \, du, \qquad t \in [a, b],$$
and its length is $\ell = s(b)$.
Since $s'(t) = \|\gamma'(t)\| > 0$, the function $s \colon [a, b] \to [0, \ell]$ is a strictly increasing $C^1$ bijection with a $C^1$ inverse $t(s)$.
The arc length parametrisation of $\gamma$ is the curve $\tilde\gamma \colon [0, \ell] \to \mathbb{R}^n$, $\tilde\gamma(s) = \gamma(t(s))$.

## Remarks

The curve $\tilde\gamma$ has the same image and direction as $\gamma$, and $\|\tilde\gamma'(s)\| = 1$ for each $s$. So $\tilde\gamma'(s)$ is the unit [[tangent vector|0D98771SM0]].
The length of $\tilde\gamma$ from $\tilde\gamma(s_0)$ to $\tilde\gamma(s_1)$ is $s_1 - s_0$.
For the helix $\gamma(t) = (\cos t, \sin t, t)$, $\|\gamma'(t)\| = \sqrt{2}$, so $s = \sqrt{2}\, t$ and $\tilde\gamma(s) = (\cos(s / \sqrt{2}), \sin(s / \sqrt{2}), s / \sqrt{2})$.
The arc length parametrisation is unique up to a shift of the parameter. It simplifies the formulas for curvature.

## Explanations

### L1 human

Reparametrise the curve so that the parameter is the distance travelled from the start.
Then the point moves at constant speed $1$, and only the shape of the curve remains.
</document>

<document source="https://math2.tamnd.com/node/00R4MWW532.md" tag="00R4MWW532" kind="statement">
# Derivative of sine

## Statement

The sine function is differentiable on $\mathbb{R}$, and $\sin'(x) = \cos x$ for all real $x$.

## Remarks

[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ is $\frac{(-1)^n x^{2n}}{(2n)!}$, the term of the cosine series.
With the addition formula, the claim reduces to the limits $\frac{\sin h}{h} \to 1$ and $\frac{\cos h - 1}{h} \to 0$.
Mathlib states it as `Real.hasDerivAt_sin`.

## Explanations

### L1 human

The slope of the sine curve at $x$ is $\cos x$.
At $x = 0$ the sine curve rises with slope $1$, and at $x = \pi/2$ it is flat.
</document>

<document source="https://math2.tamnd.com/node/00RA98X2CP.md" tag="00RA98X2CP" kind="statement">
# Divergence of the sum of the reciprocals of the primes

## Statement

The series $\sum_{p} \frac{1}{p}$, where $p$ runs over the [[prime numbers|0SN0MAEKYX]], diverges.
That is, the partial sums $\sum_{p \le x} \frac{1}{p}$ are not bounded as $x \to \infty$.

## Remarks

The result implies [[Euclid's theorem|0292EPPPT0]], since a finite sum is bounded.
The partial sums grow like $\ln \ln x$. More precisely, $\sum_{p \le x} \frac{1}{p} - \ln \ln x$ has a finite limit as $x \to \infty$.
In contrast, the sum of $1 / n^2$ over all positive integers $n$ converges. So the primes are not as rare as the squares.

## History

Euler proved the result in 1737. Erdős gave an elementary proof by upper and lower estimates.

## Explanations

### L1 human

The sum $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \cdots$ becomes larger than each number that you choose, if you take enough terms.
It grows very slowly: the terms for the primes up to $10^9$ still give a sum less than $4$.
</document>

<document source="https://math2.tamnd.com/node/00RE1R4W7N.md" tag="00RE1R4W7N" kind="definition">
# Second-countable space

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is second-countable when $\tau$ has a [[basis|06EX0FHBTF]] $\mathcal{B}$ that is [[countable|0ZG61TFBES]].

## Remarks

The space $\mathbb{R}^n$ is second-countable: the open balls with rational centres and rational radii form a countable basis.
Each second-countable space is [[separable|0GKJ78948V]] and [[first-countable|0HG11FW8MJ]].
A metric space is second-countable if and only if it is separable.
An uncountable set with the [[discrete topology|0S2TEHX4E1]] is not second-countable.

## Explanations

### L1 human

A second-countable space has a countable list of open sets from which you can build every open set by unions.
For $\mathbb{R}$, the open intervals with rational endpoints form such a list.
</document>

<document source="https://math2.tamnd.com/node/00S7C68D51.md" tag="00S7C68D51" kind="definition">
# Absolute value

## Definition

Let $x$ be a real number.
The absolute value of $x$ is the real number
$$|x| = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}$$

## Remarks

For each real $x$: $|x| \ge 0$; $|x| = 0$ only for $x = 0$; $|-x| = |x|$; and $|x| = \max(x, -x)$.
The distance between two real numbers $x$ and $y$ is $|x - y|$.
For $r \ge 0$, the inequality $|x| \le r$ holds exactly when $-r \le x \le r$.

## Explanations

### L1 human

The absolute value removes the minus sign from a number.
On the number line, $|x|$ is the distance from $x$ to $0$, and $|x - y|$ is the distance from $x$ to $y$.
</document>

<document source="https://math2.tamnd.com/node/00SB5EA3VW.md" tag="00SB5EA3VW" kind="definition">
# Asymptotic equivalence

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The functions are asymptotically equivalent as $x \to \infty$ if $f(x) - g(x) = o(g(x))$ as $x \to \infty$, in the sense of [[little o notation|04TSHB9H8D]].
The notation is $f(x) \sim g(x)$.

## Remarks

If $g(x) \ne 0$ for large $x$, then $f \sim g$ means that $f(x)/g(x) \to 1$.
The relation $\sim$ is an equivalence relation.
For example, $x^2 + x \sim x^2$. [[Stirling's formula|0KC0F51T6A]] states $n! \sim \sqrt{2 \pi n}\, (n/e)^n$.
Mathlib defines it for filters, as `Asymptotics.IsEquivalent`.

## Explanations

### L1 human

Two functions are asymptotically equivalent if their ratio tends to $1$.
Their difference may still grow, but it is small compared with the functions.
</document>

<document source="https://math2.tamnd.com/node/00SPJSDT3Z.md" tag="00SPJSDT3Z" kind="concept">
# Gram matrix

## Summary

The Gram matrix of a finite family of vectors in an inner product space is the square matrix of all their pairwise inner products.
</document>

<document source="https://math2.tamnd.com/node/00SQGJNRTB.md" tag="00SQGJNRTB" kind="definition">
# Event

## Definition

Let $\Omega$ be a [[sample space|0W7NFW23RF]] and let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
An event is an element of $\mathcal{F}$, that is, a [[subset|0FE8Q5S42Y]] $A \subseteq \Omega$ with $A \in \mathcal{F}$.
The event $A$ occurs in a trial when the outcome $\omega$ of the trial satisfies $\omega \in A$.

## Remarks

When $\Omega$ is finite or countable, one usually takes $\mathcal{F}$ to be the power set of $\Omega$. Then each subset of $\Omega$ is an event.
For uncountable $\Omega$, such as $\Omega = \mathbb{R}$, one cannot in general give a probability to each subset. So only the sets in $\mathcal{F}$ are events.
The set operations have probabilistic names: $A \cup B$ is "$A$ or $B$", $A \cap B$ is "$A$ and $B$", and $\Omega \setminus A$ is "not $A$". The empty set is the impossible event, and $\Omega$ is the sure event.
Two events $A$ and $B$ are mutually exclusive when $A \cap B = \varnothing$.

## Explanations

### L1 human

An event is a statement about the result of the experiment that is either true or false after the experiment.
For a die, "the result is at least $5$" is the event $\{5, 6\}$. It happens when you roll a $5$ or a $6$.
</document>

<document source="https://math2.tamnd.com/node/00SVP6W3CB.md" tag="00SVP6W3CB" kind="example">
# Exponential growth and decay

## Example

Let $k$ and $y_0$ be real numbers.
The [[initial value problem|075VBY4K4X]] $y'(t) = k\, y(t)$, $y(0) = y_0$ has the unique solution
$$y(t) = y_0\, e^{k t}, \qquad t \in \mathbb{R}.$$
If $k > 0$ and $y_0 > 0$, the solution grows: this is exponential growth. If $k < 0$ and $y_0 > 0$, the solution decreases to $0$: this is exponential decay.

## Remarks

For $k > 0$, the doubling time is $\ln 2 / k$: we have $y(t + \ln 2 / k) = 2\, y(t)$ for each $t$.
For $k < 0$, the half-life is $\ln 2 / |k|$. Radioactive decay follows this law.
Uniqueness: if $y$ is a solution, then the derivative of $y(t) e^{-k t}$ is $0$, so $y(t) e^{-kt}$ is the constant $y_0$.

## Explanations

### L1 human

The equation says that the rate of change is proportional to the present amount.
A population with no limits, or money with continuous interest, grows like this.
The larger the amount, the faster it grows, so the growth gets faster and faster.
</document>

<document source="https://math2.tamnd.com/node/00SW8B91E7.md" tag="00SW8B91E7" kind="definition">
# Transcendental element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is transcendental over $K$ if $a$ is not [[algebraic|0NPW1DWECM]] over $K$.
That is, $f(a) \neq 0$ for every nonzero $f \in K[x]$.

## Remarks

The number $\pi$ is transcendental over $\mathbb{Q}$, but it is algebraic over $\mathbb{R}$, as a root of $x - \pi$.
The variable $x$ is transcendental over $K$ in the field $K(x)$ of rational functions.
If $a$ is transcendental over $K$, then $K(a)$ is isomorphic to $K(x)$, and $[K(a) : K]$ is infinite.

## Explanations

### L1 human

A transcendental element satisfies no polynomial equation over the smaller field.
It behaves like a free variable.
The transcendental numbers are the complex numbers that are transcendental over $\mathbb{Q}$, such as $\pi$ and $e$.
</document>

<document source="https://math2.tamnd.com/node/00T5EPFQB9.md" tag="00T5EPFQB9" kind="concept">
# Little o notation

## Summary

Little o notation says that one function is negligible compared with another. The statement $f = o(g)$ means that $f/g$ tends to $0$ at the point in question. The node [[little o notation|04TSHB9H8D]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/00TQV3F5S4.md" tag="00TQV3F5S4" kind="statement">
# Ramsey's theorem

## Statement

Let $r$ and $s$ be positive integers.
Then there is a positive integer $n$ with this property.
Colour each edge of the [[complete graph|03WKB2H474]] $K_n$ blue or red, in any way.
Then there is a blue [[clique|0G52BPNPKG]] with $r$ vertices or a red clique with $s$ vertices.

## Remarks

The least such $n$ is the [[Ramsey number|01DSVCRF6X]] $R(r, s)$.
The proof gives the bound $R(r, s) \le R(r - 1, s) + R(r, s - 1)$, by induction on $r + s$.
The theorem also holds for any finite number of colours.

## History

Frank Ramsey proved the first version of the theorem in a 1930 paper on formal logic.
This result started the field now called Ramsey theory.

## Explanations

### L1 human

Complete disorder is impossible in a large enough structure.
If you colour the edges of a large complete graph with two colours, a large clique of one colour always appears.
</document>

<document source="https://math2.tamnd.com/node/00WAVB68F9.md" tag="00WAVB68F9" kind="definition">
# Picard operator

## Definition

Let $J = [t_{\min}, t_{\max}]$, let $t_0 \in J$ and let $x_0 \in \mathbb{R}^n$. Let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function such that $s \mapsto f(s, \gamma(s))$ is integrable on $J$ for each continuous $\gamma : J \to \mathbb{R}^n$.
The Picard operator of $f$ at $(t_0, x_0)$ maps a continuous curve $\gamma : J \to \mathbb{R}^n$ to the curve $P\gamma : J \to \mathbb{R}^n$ given by
$$(P \gamma)(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds.$$

## Remarks

Assume the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] with constants $a, L, K$. Then $P$ maps the continuous curves with values in the closed ball $\|x - x_0\| \le a$ to curves in this ball.
A fixed point of $P$ is a solution of the [[initial value problem|075VBY4K4X]]. See [[the integral equation form|015KK993C5]].
[[Picard iteration|0RJ5NHWYTP]] applies $P$ again and again, starting from the constant curve $x_0$.

## Explanations

### L1 human

The operator takes a guess for the solution and makes a new guess.
It integrates the field along the old guess, starting from $x_0$.
A curve that the operator does not change is a solution.
</document>

<document source="https://math2.tamnd.com/node/00X2A9G9W3.md" tag="00X2A9G9W3" kind="definition">
# Irreducible polynomial

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
A polynomial $f \in R[x]$ is irreducible over $R$ if it is an [[irreducible element|052SDEPZH7]] of the integral domain $R[x]$.
That is, $f$ is nonzero and not a unit, and if $f = g h$ with $g, h \in R[x]$, then $g$ or $h$ is a unit of $R[x]$.

## Remarks

Over a [[field|0QA8CHH8MG]] $K$, the units of $K[x]$ are the nonzero constants. So $f$ is irreducible if and only if $\deg f \ge 1$ and $f$ is not a product of two polynomials of smaller degree.
The polynomial $x^2 - 2$ is irreducible over $\mathbb{Q}$ but not over $\mathbb{R}$, where $x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})$.
The polynomial $x^2 + 1$ is irreducible over $\mathbb{R}$ but not over $\mathbb{C}$.
Over $\mathbb{Z}$, the polynomial $2x + 2 = 2(x + 1)$ is not irreducible, because $2$ is not a unit of $\mathbb{Z}[x]$.

## Explanations

### L1 human

Irreducible polynomials are the prime numbers of polynomial rings.
Over a field, they are the polynomials that you cannot factor into polynomials of smaller degree.
Irreducibility depends on the field: $x^2 + 1$ factors over $\mathbb{C}$ but not over $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/00Y5QZ82AA.md" tag="00Y5QZ82AA" kind="example">
# Lotka–Volterra equations

## Example

Let $\alpha, \beta, \gamma, \delta > 0$ be constants. Let $x(t) > 0$ be the size of a prey population and $y(t) > 0$ the size of a predator population at time $t$.
The Lotka–Volterra equations are the [[autonomous system|07BNRFQ0PJ]]
$$x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y.$$
The [[equilibrium points|0G4X36AE0X]] in the closed quadrant $x, y \ge 0$ are $(0, 0)$ and $(\gamma / \delta, \alpha / \beta)$.
The function $V(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y$ is a [[first integral|0SS1P1YC9A]] on the open quadrant $x, y > 0$.

## Remarks

Along a solution, $\frac{d}{dt} V = (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0$.
The [[linearisation|09ZWYRQ41J]] at $(0, 0)$ has the eigenvalues $\alpha$ and $-\gamma$, so the origin is a saddle. At $(\gamma/\delta, \alpha/\beta)$ the eigenvalues are $\pm i \sqrt{\alpha \gamma}$.
The level curves of $V$ in the open quadrant are closed curves around $(\gamma/\delta, \alpha/\beta)$. So each solution other than this equilibrium is [[periodic|01DSNP0B4R]]: the two populations oscillate.

## Explanations

### L1 human

Prey grow when there are few predators, and predators grow when there is much prey.
This feedback makes both populations rise and fall in cycles, with the predator peak after the prey peak.
</document>

<document source="https://math2.tamnd.com/node/00Y9N80VSS.md" tag="00Y9N80VSS" kind="statement">
# A finite group is nilpotent iff it is the product of its Sylow subgroups

## Statement

Let $G$ be a finite [[group|G5S1M9]].
The following conditions are equivalent:

1. $G$ is [[nilpotent|0Q7MMQHC1P]].
2. For each prime number $p$, $G$ has exactly one [[Sylow p-subgroup|0053CQE9Y2]].
3. $G$ is the [[internal direct product|0VBZH90R1A]] of its Sylow subgroups, one for each prime divisor of $|G|$.

## Remarks

The direction from 3 to 1 holds because [[finite p-groups are nilpotent|081WTKYX6C]] and a finite direct product of nilpotent groups is nilpotent.
The direction from 1 to 2 uses the [[Frattini argument|0QTZDKJEDV]]. In a nilpotent group, a proper subgroup is strictly smaller than its normaliser.
For example, $\mathbb{Z}/12\mathbb{Z} \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is nilpotent, but $S_3$ has three Sylow $2$-subgroups, so it is not nilpotent.

## Explanations

### L1 human

A finite nilpotent group splits into its $p$-parts, one for each prime.
The parts do not interact, so the group is their direct product.
Conversely, a direct product of $p$-groups is nilpotent.
</document>

<document source="https://math2.tamnd.com/node/00YC1D652K.md" tag="00YC1D652K" kind="concept">
# Linear subspace

## Summary

A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication.
It is a vector space with the same operations.
</document>

<document source="https://math2.tamnd.com/node/00YH28R457.md" tag="00YH28R457" kind="statement">
# Volume of a ball in R^n

## Statement

Let $n \ge 1$, let $R > 0$, and let $B_n(R) = \{x \in \mathbb{R}^n : \|x\| \le R\}$.
Then
$$\operatorname{vol}(B_n(R)) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2} + 1\right)} \, R^n,$$
where $\Gamma(s) = \int_0^\infty t^{s - 1} e^{-t} \, dt$ is the gamma function.

## Remarks

The [[change of variables|00N5FHP1CV]] $x = R u$ shows that $\operatorname{vol}(B_n(R)) = R^n \operatorname{vol}(B_n(1))$.
Since $\Gamma(1) = 1$, $\Gamma(1/2) = \sqrt{\pi}$ and $\Gamma(s + 1) = s \, \Gamma(s)$, the formula gives $2R$, $\pi R^2$, $\tfrac{4}{3} \pi R^3$ and $\tfrac{1}{2} \pi^2 R^4$ for $n = 1, 2, 3, 4$.
One proof computes $\int_{\mathbb{R}^n} e^{-\|x\|^2} \, dx = \pi^{n/2}$ in two ways: as a product of $n$ Gaussian integrals by [[Fubini's theorem|0R0T1VWH5X]], and by integration over spheres of radius $r$.
The volume of the unit ball tends to $0$ as $n \to \infty$.

## Explanations

### L1 human

The volume of a ball grows like the $n$-th power of its radius, times a constant that depends on the dimension.
The gamma function gives this constant in one formula for all dimensions.
</document>

<document source="https://math2.tamnd.com/node/00Z1YBJJW8.md" tag="00Z1YBJJW8" kind="concept">
# Arcsine

## Summary

The arcsine is the inverse of the sine function restricted to $[-\pi/2, \pi/2]$. It is defined on $[-1, 1]$. The node [[arcsine|0B89D93YA1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/00ZEBSMNNV.md" tag="00ZEBSMNNV" kind="statement">
# Additivity of line integrals under concatenation

## Statement

Let $\gamma_1 \colon [a, b] \to \mathbb{R}^n$ and $\gamma_2 \colon [b, c] \to \mathbb{R}^n$ be piecewise $C^1$ curves with $\gamma_1(b) = \gamma_2(b)$.
Let $\gamma \colon [a, c] \to \mathbb{R}^n$ be their concatenation: $\gamma = \gamma_1$ on $[a, b]$ and $\gamma = \gamma_2$ on $[b, c]$.
For each continuous vector field $F$ and each continuous function $f$ on the image of $\gamma$,
$$\int_\gamma F \cdot d\mathbf{r} = \int_{\gamma_1} F \cdot d\mathbf{r} + \int_{\gamma_2} F \cdot d\mathbf{r}, \qquad \int_\gamma f \, ds = \int_{\gamma_1} f \, ds + \int_{\gamma_2} f \, ds.$$

## Remarks

Proof: apply the [[parametric formula|0W3W1VYWBP]] on $[a, c]$ and split the integral at $t = b$.
If the parameter intervals do not match, first shift the parameter of $\gamma_2$. A shift does not change line integrals.
Together with [[path reversal|0FSCXYG856]], this lets one compute line integrals over the boundary of a polygon, side by side.
The concatenation is often written $\gamma_1 + \gamma_2$ or $\gamma_1 * \gamma_2$.

## Explanations

### L1 human

Walking from $A$ to $B$ and then from $B$ to $C$ is one walk from $A$ to $C$.
The line integral over the whole walk is the sum of the line integrals over the two parts.
</document>

<document source="https://math2.tamnd.com/node/00ZG9Y9TR4.md" tag="00ZG9Y9TR4" kind="concept">
# Topological embedding

## Summary

A topological embedding is a map that identifies a space with a subspace of another space, with the same topology.
It is a homeomorphism onto its image.
</document>

<document source="https://math2.tamnd.com/node/010JCV9J4E.md" tag="010JCV9J4E" kind="definition">
# Ordered field

## Definition

An ordered field is a field $F$ with a total order $\le$ on $F$.
The order must have these two properties for all $a$, $b$ and $c$ in $F$:

1. if $a \le b$, then $a + c \le b + c$;
2. if $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

In an ordered field, each square is nonnegative: $a^2 \ge 0$ for each $a$. In particular $1 = 1^2 > 0$ and $-1 < 0$.
An equivalent definition uses a positive cone $P$. This is a subset of $F$ that is closed under addition and multiplication.
Also, $F$ must be the disjoint union of $P$, $\{0\}$ and $-P$.
The field $\mathbb{C}$ of complex numbers [[has no such order|01CJRJKDAA]].

## Explanations

### L1 human

An ordered field is a number system where you can add, multiply, divide and compare numbers.
The rules of comparison are the usual ones.
If you add the same number to both sides of an inequality, the inequality stays true.
The product of two positive numbers is positive.
</document>

<document source="https://math2.tamnd.com/node/010PKCQPXP.md" tag="010PKCQPXP" kind="statement">
# Uniqueness of finite fields of a given order

## Statement

Let $F$ and $F'$ be [[finite fields|08RTDHX1SK]] with $|F| = |F'|$.
Then $F$ and $F'$ are isomorphic.

## Remarks

The proof: let $q = |F|$. Both fields have characteristic $p$ and prime subfield $\mathbb{F}_p$. Both are [[splitting fields of $x^q - x$ over $\mathbb{F}_p$|0CDAHW576M]], so they are isomorphic by the [[uniqueness of splitting fields|069A3BC3EP]].
Also, a field contains at most one subfield of order $q$: its elements must be the $q$ roots of $x^q - x$.

## History

E. H. Moore first proved the classification of finite fields in 1893.

## Explanations

### L1 human

Two finite fields with the same number of elements have the same structure.
Only the names of the elements can differ.
</document>

<document source="https://math2.tamnd.com/node/010QVB71T2.md" tag="010QVB71T2" kind="statement">
# Hensel's lemma

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $f$ be a polynomial with integer coefficients, with derivative $f'$.
Let $r$ be an integer such that $f(r) \equiv 0 \pmod p$ and $f'(r) \not\equiv 0 \pmod p$.
Then for each positive integer $k$ there is an integer $r_k$ with $r_k \equiv r \pmod p$ and $f(r_k) \equiv 0 \pmod{p^k}$.
This $r_k$ is unique modulo $p^k$.

## Remarks

The proof gives the lifts step by step. If $f(r_k) \equiv 0 \pmod{p^k}$ and $t$ is an inverse of $f'(r)$ modulo $p$, then $r_{k+1} = r_k - t f(r_k)$ is a root modulo $p^{k+1}$.
For example, $3^2 \equiv 2 \pmod 7$ and $f'(3) = 6$ for $f = x^2 - 2$. The step gives $3 - 6 \cdot 7 = -39 \equiv 10 \pmod{49}$, and $10^2 = 100 \equiv 2 \pmod{49}$.
In the language of $p$-adic numbers, a simple root modulo $p$ lifts to a unique root in the $p$-adic integers.
The lemma is the analogue of Newton's method for congruences.

## History

The lemma is named after Kurt Hensel.

## Explanations

### L1 human

A solution of a polynomial congruence modulo $p$ often gives solutions modulo $p^2$, $p^3$, and so on.
The condition $f'(r) \not\equiv 0 \pmod p$ says that $r$ is a simple root modulo $p$.
Under this condition, each step of the lifting has exactly one answer.
</document>

<document source="https://math2.tamnd.com/node/0110GBHPPB.md" tag="0110GBHPPB" kind="statement">
# Recurrence for Bell numbers

## Statement

Let $n$ be a natural number, and let $B_m$ be the [[Bell numbers|0WW85SQVS3]].
Then
$$B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k.$$

## Remarks

Proof idea: in a partition of $\{1, \dots, n + 1\}$, the block of $n + 1$ contains $n - k$ other elements, for some $k$. The remaining $k$ elements form an arbitrary partition.
Example: $B_4 = 1 \cdot 1 + 3 \cdot 1 + 3 \cdot 2 + 1 \cdot 5 = 15$.

## Explanations

### L1 human

Fix the last element and decide which other elements share its group.
For each choice, the elements left over can be split in any way, so you add a Bell number for each case.
</document>

<document source="https://math2.tamnd.com/node/0112DTS464.md" tag="0112DTS464" kind="definition">
# Image of a homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
The image of $\varphi$ is the set $\operatorname{im} \varphi = \varphi(G) = \{\varphi(g) : g \in G\}$.

## Remarks

The image is a [[subgroup|0X1JXT47GA]] of $H$, because $\varphi(a) \varphi(b)^{-1} = \varphi(ab^{-1})$.
The homomorphism $\varphi$ is surjective exactly when $\operatorname{im} \varphi = H$.
The image of the map $n \mapsto 2^n$ from $\mathbb{Z}$ to $\mathbb{Q}^\times$ is the set of integer powers of $2$.

## Explanations

### L1 human

The image is the part of the target group that the homomorphism actually reaches.
It is always a group of its own, inside the target.
The [[first isomorphism theorem|0CXXJNG5B6]] says that the image looks like a quotient of the source group.
</document>

<document source="https://math2.tamnd.com/node/011C29Q1S1.md" tag="011C29Q1S1" kind="definition">
# Partial order

## Definition

Let $A$ be a set.
A partial order on $A$ is a [[preorder|0RXS8R6K37]] $\le$ on $A$ that is [[antisymmetric|01MGYYBB9R]]. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le a$ imply $a = b$;
3. $a \le b$ and $b \le c$ imply $a \le c$.

The pair $(A, \le)$ is a partially ordered set.

## Remarks

Examples are $\le$ on $\mathbb{R}$, divisibility on $\mathbb{N}$, and $\subseteq$ on the [[power set|0BC7KX51Z8]] of a set.
"Partial" means that two elements need not be comparable: neither $\{1\} \subseteq \{2\}$ nor $\{2\} \subseteq \{1\}$.
The strict order $a < b$ means $a \le b$ and $a \neq b$.
A finite poset can be drawn as a [[Hasse diagram|00EYM7MDKS]].

## Explanations

### L1 human

A partial order ranks some pairs of elements, but not necessarily all pairs.
In the family tree, "is an ancestor of or equal to" is a partial order: two cousins are not comparable.
</document>

<document source="https://math2.tamnd.com/node/011EJQWWJS.md" tag="011EJQWWJS" kind="concept">
# Strictly monotone function

## Summary

A strictly monotone function is a function that is either strictly increasing or strictly decreasing. Such a function is injective. The node [[strictly monotone function|0TR9SR9FTB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/011HV2GAWF.md" tag="011HV2GAWF" kind="statement">
# A convex function on an open interval is continuous

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be convex.
Then $f$ is [[continuous|0P27W9RW0G]] on $I$.

## Remarks

Fix $y \in I$ and points $x < y < z$ in $I$. By the [[slope inequalities|0BY5FTHEEE]], the slopes of chords from $y$ are bounded for points near $y$. So $|f(w) - f(y)| \le C |w - y|$ near $y$.
On a closed interval this fails. The function on $[0, 1]$ with $f(0) = 1$ and $f(x) = 0$ for $x > 0$ is convex but not continuous at $0$.
A convex function on an open interval has one-sided derivatives at every point. Mathlib states continuity as `ConvexOn.continuousOn`, for open sets.

## Explanations

### L1 human

A convex function cannot jump inside an open interval.
A jump would break the condition that the chords lie above the graph.
</document>

<document source="https://math2.tamnd.com/node/011K11VPJN.md" tag="011K11VPJN" kind="statement">
# Jordan–Chevalley decomposition

## Statement

Let $F$ be a perfect field, for example a field of characteristic $0$ or an algebraically closed field.
Let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Then there are unique linear maps $S, N \colon V \to V$ with these properties:

1. $T = S + N$.
2. $S$ is [[semisimple|08VRX3Q4PB]] and $N$ is [[nilpotent|063D5Z2C70]].
3. $SN = NS$.

## Remarks

Moreover, $S$ and $N$ are polynomials in $T$ without constant term.
Over $\C$, $S$ is diagonalisable; in a Jordan basis, $S$ is the diagonal part of the Jordan form.
For $J_2(\lambda)$, the decomposition is $S = \lambda I_2$ and $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

## Explanations

### L1 human

Every linear map over $\C$ splits into a diagonalisable part and a nilpotent part.
The two parts commute with each other.
This splitting is unique, and it makes functions of matrices easier to compute.
</document>

<document source="https://math2.tamnd.com/node/0125QVNCFQ.md" tag="0125QVNCFQ" kind="definition">
# Evaluation homomorphism

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $c \in R$.
The evaluation homomorphism at $c$ is the map
$$\operatorname{ev}_c \colon R[x] \to R, \qquad \operatorname{ev}_c(f) = f(c),$$
where $f(c)$ is the [[value of $f$ at $c$|0GJ7T0NK49]].
It is a [[ring homomorphism|0GGE0Z0JQ1]].

## Remarks

The map $\operatorname{ev}_c$ is surjective, because $\operatorname{ev}_c(r) = r$ for each constant $r \in R$.
Its [[kernel|0AAHE7GEW6]] is the principal ideal $(x - c)$. This is a form of the [[factor theorem|0K3MWA7DMC]].
The commutativity of $R$ is necessary: in general $(f g)(c) \neq f(c) g(c)$ when $c$ does not commute with the coefficients.

## Explanations

### L1 human

Evaluation at $c$ turns each polynomial into an element of $R$.
It respects sums and products: to evaluate $f g$, you can evaluate $f$ and $g$ and multiply.
</document>

<document source="https://math2.tamnd.com/node/012C7GFF20.md" tag="012C7GFF20" kind="statement">
# 1 + x is at most exp x

## Statement

For every real $x$,
$$1 + x \le e^x,$$
with equality if and only if $x = 0$.

## Remarks

The line $y = 1 + x$ is the tangent to the graph of $e^x$ at $0$. The graph of a [[convex function|0D7GRZ2ADX]] lies above its tangent lines, by [[the derivative criterion|005JT76HCK]].
For $x \ge 0$ the claim also follows from the series, since the other terms are nonnegative.
Taking logarithms gives [[$\ln y \le y - 1$|04CBRA4JRK]] for $y > 0$. Mathlib states the inequality as `Real.add_one_le_exp`.

## Explanations

### L1 human

The graph of $e^x$ lies above its tangent line at $0$.
That tangent line is $y = 1 + x$.
</document>

<document source="https://math2.tamnd.com/node/012PBA0FMD.md" tag="012PBA0FMD" kind="concept">
# Hölder continuity

## Summary

A map between metric spaces is Hölder continuous with exponent $\alpha$ when $d(f(x), f(y)) \le C \, d(x, y)^\alpha$ for some constant $C$ and all $x, y$.
For $\alpha = 1$ this is the Lipschitz condition.
</document>

<document source="https://math2.tamnd.com/node/012RYET5E2.md" tag="012RYET5E2" kind="concept">
# Reduction formula

## Summary

A reduction formula expresses an integral that depends on an integer $n$ through the same integral with a smaller $n$. Repeated use of the formula reduces the integral to a simple case. [[Integration by parts|08Y1BBQTK9]] usually produces the formula.

## Remarks

For $I_n = \int_0^{\pi/2} \sin^n x \, dx$, integration by parts gives $I_n = \frac{n - 1}{n} I_{n-2}$ for $n \ge 2$.
With $I_0 = \pi/2$ and $I_1 = 1$, this gives all $I_n$.
These integrals lead to the [[Wallis product|0XCYA4YERS]].

## Explanations

### L1 human

A reduction formula is a recursion for a family of integrals.
Each step lowers a power, until the integral is easy.
</document>

<document source="https://math2.tamnd.com/node/0135XZZQ8M.md" tag="0135XZZQ8M" kind="definition">
# Differentiable function of several variables

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
The function $f$ is differentiable at $a$ when there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that
$$\lim_{h \to 0} \frac{\|f(a + h) - f(a) - L h\|}{\|h\|} = 0.$$
Here the limit is over $h \ne 0$ with $a + h \in U$, and $\|\cdot\|$ is the Euclidean norm.
The map $L$ is then unique. It is the derivative $Df(a)$, also called the [[total derivative|0FNKFT8H6E]] of $f$ at $a$.

## Remarks

This is the [[Fréchet derivative|08EVNYZF42]] for the spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where each linear map is continuous.
A function that is differentiable at $a$ is continuous at $a$, and all its [[directional derivatives|06KWGHDN2J]] exist at $a$.
The converse fails: a function can have all partial derivatives at a point and not be differentiable there, see [[this counterexample|0TNZC55X3J]].
For $n = m = 1$, the definition is the usual one, with $L h = f'(a) h$.

## Explanations

### L1 human

A differentiable function looks flat when you zoom in on its graph near the point.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the change is $f(1 + h_1, 1 + h_2) - f(1, 1) = 2 h_1 + 2 h_2 + h_1^2 + h_2^2$.
The linear part is $L h = 2 h_1 + 2 h_2$, and the rest $h_1^2 + h_2^2 = \|h\|^2$ is small compared with $\|h\|$.
</document>

<document source="https://math2.tamnd.com/node/013AHC4DRT.md" tag="013AHC4DRT" kind="statement">
# exp z = 1 iff z is an integer multiple of 2πi

## Statement

Let $z \in \mathbb{C}$.
Then $e^z = 1$ if and only if $z = 2 \pi i k$ for some [[integer|0C4JS9RF5M]] $k$.
Here $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.

## Remarks

Proof: write $z = x + iy$ with $x, y$ real. Then $e^z = e^x (\cos y + i \sin y)$. If $e^z = 1$, then $e^x = |e^z| = 1$, so $x = 0$. Then $\cos y = 1$ and $\sin y = 0$, so $y = 2 \pi k$. The converse follows from [[Euler's formula|04R3CY41VJ]].
It follows that $e^z = e^w$ if and only if $z - w = 2 \pi i k$ for some $k \in \mathbb{Z}$.
This is the reason why the [[complex logarithm|0X2FY491PF]] has many values.

## Explanations

### L1 human

The value $1$ has size $1$ and direction $0$.
So $e^z = 1$ needs a real part $0$ and an imaginary part that is a whole number of full turns, that is, a multiple of $2\pi$.
</document>

<document source="https://math2.tamnd.com/node/01441KDGBY.md" tag="01441KDGBY" kind="statement">
# Existence of splitting fields

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ have degree $n \ge 1$.
Then $f$ has a [[splitting field|02NZZ1J0FH]] $L$ over $K$, and $[L : K] \le n!$.

## Remarks

The proof is by induction on $n$. [[Some extension $K_1$ contains a root $a_1$ of $f$|0VG02M1FEZ]], with $[K_1 : K] \le n$. Then $f = (x - a_1) g$ in $K_1[x]$, and a splitting field of $g$ over $K_1$ is a splitting field of $f$ over $K$.
The bound $n!$ follows from the [[tower law|0A7KJ68XHT]].
The bound is reached: the splitting field of $x^3 - 2$ over $\mathbb{Q}$ has degree $6 = 3!$.

## Explanations

### L1 human

You build a splitting field one root at a time.
Each step adds a root of a remaining factor, and the degree of the step is at most the degree of that factor.
</document>

<document source="https://math2.tamnd.com/node/0146MBQNNE.md" tag="0146MBQNNE" kind="definition">
# Strict differentiability

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
The function $f$ is strictly differentiable at $a$ if there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that
$$\lim_{\substack{(x, y) \to (a, a) \\ x \ne y}} \frac{\|f(x) - f(y) - L(x - y)\|}{\|x - y\|} = 0.$$

## Remarks

If $f$ is strictly differentiable at $a$, put $y = a$: then $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $L = Df(a)$ is its [[Fréchet derivative|08EVNYZF42]].
The converse is false. The function $f(x) = x^2 \sin(1/x)$ for $x \ne 0$, $f(0) = 0$, is differentiable at $0$, but it is not strictly differentiable at $0$ because $f'$ is not continuous at $0$.
A function of class $C^1$ on $U$ is [[strictly differentiable|0STPPDXGME]] at each point of $U$.
The same definition works for maps between Banach spaces, with norms in place of Euclidean norms.

## Explanations

### L1 human

Ordinary differentiability compares $f(x)$ with $f(a)$ only.
Strict differentiability compares $f(x)$ with $f(y)$ for all pairs of points near $a$.
So the linear approximation is uniform in a small neighbourhood, not only at the single point.
</document>

<document source="https://math2.tamnd.com/node/014E1DZQE8.md" tag="014E1DZQE8" kind="definition">
# Semidirect product

## Definition

Let $N$ and $H$ be [[groups|G5S1M9]], and let $\varphi \colon H \to \operatorname{Aut}(N)$, $h \mapsto \varphi_h$, be a [[group homomorphism|0JYTG5K9Y4]].
The semidirect product $N \rtimes_\varphi H$ is the set $N \times H$ with the operation $(n_1, h_1)(n_2, h_2) = (n_1 \varphi_{h_1}(n_2), h_1 h_2)$.

## Remarks

The identity is $(e, e)$, and the inverse of $(n, h)$ is $(\varphi_{h^{-1}}(n^{-1}), h^{-1})$.
The set $\{(n, e) : n \in N\}$ is a [[normal subgroup|0CB0DX5BVV]] isomorphic to $N$, and $\{(e, h) : h \in H\}$ is a subgroup isomorphic to $H$.
If $\varphi$ is the trivial homomorphism, then $N \rtimes_\varphi H$ is the [[direct product|00HF4RJ5QN]] $N \times H$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ is $\mathbb{Z}/n\mathbb{Z} \rtimes_\varphi \mathbb{Z}/2\mathbb{Z}$, where the nonzero element of $\mathbb{Z}/2\mathbb{Z}$ acts by $x \mapsto -x$.

## Explanations

### L1 human

In a direct product, the two factors do not interact.
In a semidirect product, the second factor $H$ twists the first factor $N$ when you multiply.
The homomorphism $\varphi$ says exactly how each element of $H$ twists $N$.
</document>

<document source="https://math2.tamnd.com/node/014PEBGDYF.md" tag="014PEBGDYF" kind="statement">
# Chernoff bound

## Statement

Let $X$ be a real random variable with [[moment-generating function|0NNRNGKMFM]] $M_X(t) = E[e^{tX}]$, and let $a \in \mathbb{R}$. Then
$$P(X \ge a) \le \inf_{t > 0} e^{-ta} M_X(t).$$

## Remarks

Proof: for each $t > 0$, $\{X \ge a\} = \{e^{tX} \ge e^{ta}\}$. [[Markov's inequality|05PWWHYBX5]] gives $P(X \ge a) \le e^{-ta} E[e^{tX}]$. Then take the infimum over $t$.
For a sum $S$ of independent $X_i$, $M_S = \prod_i M_{X_i}$. So the bound becomes $\inf_{t > 0} e^{-ta} \prod_i M_{X_i}(t)$, which decreases exponentially in many cases.
For $S \sim \operatorname{Bin}(n, p)$ with mean $\mu = np$ and $0 < \delta$, a standard consequence is $P(S \ge (1 + \delta)\mu) \le \left(e^{\delta} / (1 + \delta)^{1 + \delta}\right)^{\mu}$.

## History

The bound is named after Herman Chernoff.

## Explanations

### L1 human

The Chernoff bound gives tail estimates that decrease very fast, often exponentially.
It is much stronger than Chebyshev's inequality for sums of many independent terms, such as the number of heads in many tosses.
</document>

<document source="https://math2.tamnd.com/node/0151VYVX27.md" tag="0151VYVX27" kind="statement">
# Cayley's theorem

## Statement

Let $G$ be a [[group|G5S1M9]].
Then $G$ is [[isomorphic|08MG7H8JW5]] to a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(G)$.
Explicitly, the map $g \mapsto \ell_g$, where $\ell_g(x) = gx$, is an injective [[group homomorphism|0JYTG5K9Y4]] from $G$ to $\operatorname{Sym}(G)$.

## Remarks

The map comes from the [[action by left multiplication|0K5TWJAGPD]], and it is injective because $\ell_g(e) = g$.
If $|G| = n$ is finite, then $G$ is isomorphic to a subgroup of $S_n$.
Often a smaller symmetric group is enough, for example $S_3$ is isomorphic to a subgroup of $S_3$ and not only of $S_6$.

## History

Arthur Cayley introduced abstract groups in 1854.
At that time, groups were usually groups of permutations, and the theorem shows that the abstract notion gives nothing more, up to isomorphism.

## Explanations

### L1 human

Each element of a group shuffles the elements of the group when you multiply on the left.
Different elements give different shuffles, and the product of elements gives the composition of shuffles.
So each group is a group of permutations, up to isomorphism.
</document>

<document source="https://math2.tamnd.com/node/015KK993C5.md" tag="015KK993C5" kind="statement">
# Integral equation form of an initial value problem

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $f : J \times U \to \mathbb{R}^n$ be continuous, and let $t_0 \in J$ and $x_0 \in U$.
Let $\gamma : J \to U$ be a continuous curve.
Then $\gamma$ is differentiable on $J$ with $\gamma'(t) = f(t, \gamma(t))$ for each $t \in J$ and $\gamma(t_0) = x_0$ if and only if
$$\gamma(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds \quad \text{for each } t \in J.$$
At an endpoint of $J$ that belongs to $J$, the derivative is one-sided.

## Remarks

In other words, the solutions of the [[initial value problem|075VBY4K4X]] on $J$ are the fixed points of the [[Picard operator|00WAVB68F9]].
Proof: one direction is the fundamental theorem of calculus applied to $\gamma'$. For the other, the integrand $s \mapsto f(s, \gamma(s))$ is continuous, so the integral is differentiable in $t$.
The integral form needs only continuity of $\gamma$. This makes it the right setting for fixed point arguments.

## Explanations

### L1 human

Integrate both sides of $\gamma' = f(t, \gamma)$ from $t_0$ to $t$.
The initial condition and the differential equation become one equation with an integral.
</document>

<document source="https://math2.tamnd.com/node/0167F9B344.md" tag="0167F9B344" kind="concept">
# Glide reflection

## Summary

A glide reflection is a reflection in a hyperplane followed by a translation parallel to that hyperplane. In the plane, it is a reflection in a line followed by a shift along that line.
</document>

<document source="https://math2.tamnd.com/node/016PE4VZ8Q.md" tag="016PE4VZ8Q" kind="statement">
# Leibniz formula for determinants

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $S_n$ be the group of permutations of $\{1, \dots, n\}$, and let $\sgn(\sigma) \in \{1, -1\}$ be the sign of $\sigma$.
Then the [[determinant|00KDZRW4C0]] of $A$ is
$$\det A = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^{n} a_{i, \sigma(i)}.$$

## Remarks

The sum has $n!$ terms.
Each term takes exactly one entry from each row and from each column.
For $n = 3$, the formula is the [[rule of Sarrus|0H8SZFFDXJ]].
The formula shows that the determinant is a polynomial in the entries, with integer coefficients.

## Explanations

### L1 human

To compute a determinant by this formula, choose one entry in each row so that no two lie in the same column.
Multiply the chosen entries, and give the product the sign of the permutation.
Add these signed products over all choices.
</document>

<document source="https://math2.tamnd.com/node/017BPGVEE8.md" tag="017BPGVEE8" kind="concept">
# Principal branch

## Summary

The principal branch of a multivalued function is one branch of it that a standard convention chooses, see the [[definition|0YKY4M3KRQ]].
Its values are the principal values. An example is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$ with $\operatorname{Arg} z \in (-\pi, \pi]$.
</document>

<document source="https://math2.tamnd.com/node/017BR1M98J.md" tag="017BR1M98J" kind="definition">
# Alternating series

## Definition

Let $(b_n)$ be a real sequence with $b_n > 0$ for all $n$.
The series
$$\sum_{n=0}^{\infty} (-1)^n b_n = b_0 - b_1 + b_2 - b_3 + \cdots$$
is an alternating series.
More generally, a [[series|0JH4MN4K1C]] is alternating if its terms are nonzero and their signs alternate.

## Remarks

Some texts allow $b_n \ge 0$.
The [[alternating harmonic series|0JQ5Y3WWVE]] $1 - 1/2 + 1/3 - \cdots$ is the standard example.
The [[alternating series test|0PK8QH1W1V]] gives a simple condition for convergence.

## Explanations

### L1 human

In an alternating series, the signs of the terms go plus, minus, plus, minus.
The partial sums then move up and down, like a pendulum.
</document>

<document source="https://math2.tamnd.com/node/017DH8WXTE.md" tag="017DH8WXTE" kind="statement">
# Riemann's criterion for integrability

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is Riemann integrable on $[a, b]$ if and only if the following holds.
For each $\varepsilon > 0$ there is a partition $P$ of $[a, b]$ with $U(f, P) - L(f, P) < \varepsilon$.

## Remarks

Here $U$ and $L$ are the upper and lower [[Darboux sums|0QRQVH0CRW]].
The condition says that the upper and lower Darboux integrals are equal. So it follows from the [[equality of Darboux and Riemann integrability|021XJAQVDC]].
This is the usual tool to prove that [[continuous|06SESV7GQY]] and [[monotone|04YKAYDQPW]] functions are integrable.

## Explanations

### L1 human

A function is integrable when we can trap its graph between step functions whose areas differ by as little as we like.
</document>

<document source="https://math2.tamnd.com/node/017E5PRS1Q.md" tag="017E5PRS1Q" kind="concept">
# Platonic solid

## Summary

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces at each vertex. There are exactly five: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.
</document>

<document source="https://math2.tamnd.com/node/017SRZ7NE9.md" tag="017SRZ7NE9" kind="statement">
# Dyck words are counted by Catalan numbers

## Statement

Let $n$ be a natural number.
Then the number of [[Dyck words|0R9JR0465V]] of semilength $n$ is the [[Catalan number|092DRBFKMG]] $C_n$.

## Remarks

For $n = 3$ the $5$ Dyck words are $UUUDDD$, $UUDUDD$, $UUDDUD$, $UDUUDD$ and $UDUDUD$.
Proof idea: a nonempty Dyck word has a unique form $U w_1 D w_2$, where $w_1$ and $w_2$ are Dyck words. This gives the [[Catalan recurrence|0Y1MG9R5VN]].

## Explanations

### L1 human

Each nonempty balanced bracket string starts with an opening bracket, and that bracket has a matching closing bracket.
The inside and the part after the match are smaller balanced strings, so the count satisfies the Catalan recurrence.
</document>

<document source="https://math2.tamnd.com/node/01826KMG2H.md" tag="01826KMG2H" kind="concept">
# Prime number

## Summary

A prime number is a natural number greater than $1$ whose only positive divisors are $1$ and itself.
The [[definition of a prime number|0SN0MAEKYX]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/018GTRS7A4.md" tag="018GTRS7A4" kind="concept">
# Vector-valued function

## Summary

A vector-valued function is a [[function|0EPAT51DXQ]] whose values are vectors, for example $f \colon D \to \mathbb{R}^m$ with $D \subseteq \mathbb{R}^n$ and $m \ge 2$.
Such a function has $m$ component functions $f_1, \dots, f_m \colon D \to \mathbb{R}$, with $f(x) = (f_1(x), \dots, f_m(x))$ for each $x \in D$.
An example is $r(t) = (\cos t, \sin t, t)$ for $t \in \mathbb{R}$, whose values trace a helix in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^3$.

## Remarks

Limits, continuity and derivatives of a vector-valued function into $\mathbb{R}^m$ can be computed one component at a time.
When the domain is an interval of $\mathbb{R}$, the function is a [[parametrised curve|0G7WM8DX4P]].

## Explanations

### L1 human

A vector-valued function gives an arrow, or a point, for each input, not a single number.
If $t$ is time, then $r(t)$ can be the position of a moving particle in space.
</document>

<document source="https://math2.tamnd.com/node/018W3V3TCT.md" tag="018W3V3TCT" kind="definition">
# Finite-dimensional vector space

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
The space $V$ is *finite-dimensional* when it has a finite [[spanning set|088RYKAWK9]].
Otherwise $V$ is *infinite-dimensional*.

## Remarks

A finite-dimensional space has a finite basis, because [[every spanning set contains a basis|06MRNP4BNT]].
The spaces $F^n$ and $M_{m \times n}(F)$ are finite-dimensional.
The polynomial ring $F[x]$ is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span $F[x]$.
Each [[subspace|00YC1D652K]] of a finite-dimensional space is finite-dimensional.

## Explanations

### L1 human

A space is finite-dimensional when finitely many vectors are enough to build every vector.
The plane needs two vectors, and ordinary space needs three.
The space of all polynomials needs infinitely many, because the degrees have no bound.
</document>

<document source="https://math2.tamnd.com/node/0194DY35BG.md" tag="0194DY35BG" kind="statement">
# Rolle's theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$.
Then there is $c \in (a, b)$ with $f'(c) = 0$.

## Remarks

By the [[extreme value theorem|0WS1A8E2P9]], $f$ has a maximum and a minimum on $[a, b]$. If both occur at the endpoints, $f$ is constant. Otherwise one occurs at an interior point $c$, and [[Fermat's theorem|0PCVBRV9VJ]] gives $f'(c) = 0$.
Differentiability inside the interval is needed. The function $|x|$ on $[-1, 1]$ has no point with derivative $0$.
Mathlib states it as `exists_deriv_eq_zero`.

## History

Michel Rolle proved a version for polynomials in 1691. Cauchy gave a proof of the general theorem in 1823.

## Explanations

### L1 human

A smooth graph that starts and ends at the same height must turn around somewhere.
At the turning point, the tangent is horizontal.
</document>

<document source="https://math2.tamnd.com/node/01991NCP7H.md" tag="01991NCP7H" kind="statement">
# Residue at a simple pole

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $0 < |z - c| < r$.
Assume that $f$ has a simple pole at $c$, that is, a [[pole of order|0YK27TJ4WF]] $1$.
Then
$$\operatorname{Res}(f, c) = \lim_{z \to c} (z - c) f(z),$$
where $\operatorname{Res}(f, c)$ is the [[residue|0JFGKKWWFK]] of $f$ at $c$.
In particular, let $f = g / h$, where $g$ and $h$ are holomorphic near $c$, $g(c) \neq 0$, $h(c) = 0$ and $h'(c) \neq 0$. Then $\operatorname{Res}(f, c) = \frac{g(c)}{h'(c)}$.

## Remarks

Proof: near $c$, $f(z) = \frac{a_{-1}}{z - c} + a_0 + a_1 (z - c) + \cdots$, so $(z - c) f(z) \to a_{-1}$.
For example, $\operatorname{Res}\left(\frac{1}{z^2 + 1}, i\right) = \frac{1}{2i}$, with $g = 1$ and $h = z^2 + 1$.
At a pole of order $m$, the formula is $\operatorname{Res}(f, c) = \frac{1}{(m-1)!} \lim_{z \to c} \frac{d^{m-1}}{dz^{m-1}} \left( (z - c)^m f(z) \right)$.

## Explanations

### L1 human

At a simple pole, $f$ looks like $\frac{a}{z - c}$ plus something harmless. Multiplying by $z - c$ and letting $z \to c$ gives $a$, the residue.
For example, the residue of $\frac{1}{z}$ at $0$ is $1$, and the residue of $\frac{1}{z - 2}$ at $2$ is also $1$.
</document>

<document source="https://math2.tamnd.com/node/019H1Z6419.md" tag="019H1Z6419" kind="statement">
# Derivative of the complex exponential

## Statement

For each $z \in \mathbb{C}$, the [[complex exponential|054VXHMCA3]] is [[complex differentiable|0D1C9Y1EXA]] at $z$, and
$$\exp'(z) = \exp z.$$

## Remarks

Proof: differentiate the series $\sum_{n \ge 0} z^n / n!$ term by term. The derivative of $z^n / n!$ is $z^{n-1} / (n-1)!$ for $n \ge 1$, so the series of derivatives is the same series.
With the chain rule, the derivative of $e^{a z}$ is $a e^{a z}$ for each constant $a \in \mathbb{C}$.
It follows that $\exp$ is [[entire|0G2YJN0H5S]].

## Explanations

### L1 human

As in the real case, the exponential function is its own derivative.
You can see it in the power series: differentiating each term gives the term before it, so the whole series does not change.
</document>

<document source="https://math2.tamnd.com/node/019MA1VTR2.md" tag="019MA1VTR2" kind="concept">
# Root of a polynomial

## Summary

A root of a polynomial $f$ over a ring $R$ is an element $c$ of $R$ with $f(c) = 0$.
The [[definition of a root|00EHX6FWC0]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/019NFGFPR5.md" tag="019NFGFPR5" kind="definition">
# Unit of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $u \in R$ is a unit of $R$ if there is an element $v \in R$ such that $u \cdot v = v \cdot u = 1$.
The element $v$ is unique. It is the inverse of $u$, with the notation $u^{-1}$.

## Remarks

The units of $\mathbb{Z}$ are $1$ and $-1$.
In a [[field|0QA8CHH8MG]], every nonzero element is a unit.
A real $n \times n$ matrix is a unit of the matrix ring if and only if its determinant is not zero.
The units of $R$ form the [[group of units|05THZKR9AN]] of $R$.

## Explanations

### L1 human

A unit is an element that you can divide by.
In the integers, you can divide by $1$ and $-1$ and stay in the integers, but you cannot divide by $2$.
So $1$ and $-1$ are the only units of $\mathbb{Z}$.
</document>

<document source="https://math2.tamnd.com/node/019QZ6RRVH.md" tag="019QZ6RRVH" kind="statement">
# Integral test

## Statement

Let $N$ be an integer and let $f \colon [N, \infty) \to \mathbb{R}$ be a decreasing function with $f(x) \ge 0$ for all $x \ge N$.
Then the series $\sum_{n=N}^{\infty} f(n)$ converges if and only if the [[improper integral|0VCTBJMEJF]] $\int_N^{\infty} f(x)\, dx$ converges.

## Remarks

A decreasing function is Riemann integrable on each bounded interval, so the integrals $\int_N^b f(x)\, dx$ exist.
The proof uses the [[comparison of a series and an integral|069HKV4X1E]]. When both converge,
$$\int_N^{\infty} f(x)\, dx \le \sum_{n=N}^{\infty} f(n) \le f(N) + \int_N^{\infty} f(x)\, dx.$$
With $f(x) = 1/x^p$ it gives the [[p-series test|0ZNSSK8ZK1]].
Mathlib states the convergence direction for an antitone function, in `AntitoneOn.summable_of_integrableOn_Ioi`.

## Explanations

### L1 human

Draw rectangles of width $1$ and height $f(n)$ under or over the graph of $f$.
The total area of the rectangles is the sum of the series, and it stays within $f(N)$ of the area under the graph.
</document>

<document source="https://math2.tamnd.com/node/019VSXSFMQ.md" tag="019VSXSFMQ" kind="definition">
# Maximal solution

## Definition

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous.
Let $I$ be an open interval and let $\gamma : I \to \mathbb{R}^n$ be a [[solution|04SW7WECS8]] of $x' = f(t, x)$ whose graph lies in $D$.
The solution $\gamma$ is maximal if there is no solution $\tilde{\gamma} : \tilde{I} \to \mathbb{R}^n$ on an open interval $\tilde{I}$ with $I \subsetneq \tilde{I}$ and $\tilde{\gamma}(t) = \gamma(t)$ for each $t \in I$.
The interval $I$ of a maximal solution is its maximal interval of existence.

## Remarks

If $f$ is locally Lipschitz in $x$, each [[initial value problem|075VBY4K4X]] $x(t_0) = x_0$ with $(t_0, x_0) \in D$ has exactly one maximal solution. Each other solution of the problem is a restriction of it.
The maximal interval $(t_-, t_+)$ can depend on the initial value, and it can be smaller than $\mathbb{R}$.
If $t_+ < \infty$, then $(t, \gamma(t))$ leaves every compact subset of $D$ as $t \to t_+$. For $D = \mathbb{R} \times \mathbb{R}^n$, this means $\|\gamma(t)\| \to \infty$. See [[blow-up in finite time|0E4GXX7SJP]].

## Explanations

### L1 human

A maximal solution is a solution that we cannot continue any further.
Its interval ends only where the solution runs away to infinity or to the edge of the domain of the field.
</document>

<document source="https://math2.tamnd.com/node/01A0MK1730.md" tag="01A0MK1730" kind="counterexample">
# Weierstrass function

## Counterexample

Let $0 < a < 1$, and let $b$ be a positive odd integer with $a b > 1 + \frac{3}{2} \pi$. The Weierstrass function is
$$W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x).$$
The function $W$ is continuous on $\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.
So a continuous function need not have a derivative anywhere.

## Remarks

The terms satisfy $|a^n \cos(b^n \pi x)| \le a^n$. The [[M-test|0NX5NWAFC9]] and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.
G. H. Hardy later showed that the conclusion holds under the weaker condition $a b \ge 1$.
Karl Weierstrass presented the function to the Prussian Academy of Sciences on 18 July 1872.
Bernard Bolzano had constructed such a function around 1831, but it was not published until 1922.

## Explanations

### L1 human

The graph of the Weierstrass function wiggles at every scale.
Zooming in never shows a straight piece, so there is no tangent line at any point.
</document>

<document source="https://math2.tamnd.com/node/01A2EGXJEM.md" tag="01A2EGXJEM" kind="statement">
# First derivative test

## Statement

Let $f$ be a real function on an open interval $(c - \delta, c + \delta)$ with $\delta > 0$, continuous at $c$ and differentiable at each $x \ne c$.
If $f'(x) \le 0$ for $c - \delta < x < c$ and $f'(x) \ge 0$ for $c < x < c + \delta$, then $f$ has a [[local minimum|0NS8BP96QX]] at $c$.
If the inequalities are reversed, then $f$ has a local maximum at $c$.

## Remarks

The function is [[decreasing on the left and increasing on the right|026BJBQNGJ]] of $c$.
The test needs no derivative at $c$. It works for $|x|$ at $0$.
Mathlib states the minimum case as `isLocalMin_of_deriv`.

## Explanations

### L1 human

If the graph goes down until $c$ and then goes up, the point $c$ is the bottom of a valley.
</document>

<document source="https://math2.tamnd.com/node/01A8JGR4JM.md" tag="01A8JGR4JM" kind="statement">
# cos pi = -1

## Statement

$$\cos \pi = -1, \qquad \sin \pi = 0.$$

## Remarks

By the [[definition of $\pi$|0C1VYVNV2N]], $\cos(\pi/2) = 0$. The [[Pythagorean identity|03H7WZ4T3K]] gives $\sin(\pi/2) = 1$, since $\sin > 0$ on $(0, 2]$.
The [[double angle formulas|028XEQCE88]] then give $\cos \pi = 2 \cos^2(\pi/2) - 1 = -1$ and $\sin \pi = 2 \sin(\pi/2) \cos(\pi/2) = 0$.
Together with $\sin \pi = 0$, it gives Euler's identity $e^{i \pi} = -1$ in complex form. Mathlib states the first claim as `Real.cos_pi`.

## Explanations

### L1 human

Half a turn around the unit circle takes the point $(1, 0)$ to the point $(-1, 0)$.
So $\cos \pi = -1$ and $\sin \pi = 0$.
</document>

<document source="https://math2.tamnd.com/node/01AAYNSBN6.md" tag="01AAYNSBN6" kind="statement">
# Distance from a point to a line

## Statement

Let $a$, $b$ and $c$ be real numbers with $(a, b) \neq (0, 0)$, and let $L = \{(x, y) \in \mathbb{R}^2 : a x + b y + c = 0\}$ be a [[line|0MP9BCF6DP]] in the plane.
Let $P = (x_0, y_0)$ be a point of $\mathbb{R}^2$. The distance from $P$ to $L$ is the smallest [[Euclidean distance|0NPPGHQNXQ]] $d(P, Q)$ for $Q \in L$. Then
$$\operatorname{dist}(P, L) = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.$$

## Remarks

The smallest distance is reached at exactly one point $Q$ of $L$, the foot of the perpendicular from $P$. It is $Q = P - \frac{a x_0 + b y_0 + c}{a^2 + b^2} (a, b)$.
In $\mathbb{R}^n$, let $L$ be the line through the point $A$ with the direction vector $v$. The distance from $P$ to $L$ is the length of the part of $P - A$ that is perpendicular to $v$: the length of $(P - A) - \frac{(P - A) \cdot v}{v \cdot v} v$.
In $\mathbb{R}^3$ this distance is also $\|(P - A) \times v\| / \|v\|$.

## Explanations

### L1 human

The shortest way from a point to a line is the path that meets the line at a right angle. The formula measures this path. For the line $y = 0$, thus $a = 0$, $b = 1$, $c = 0$, it gives $|y_0|$: the height of the point above or below the horizontal axis.
</document>

<document source="https://math2.tamnd.com/node/01AHG4G9RT.md" tag="01AHG4G9RT" kind="concept">
# Preimage of a set

## Summary

The preimage of a set under a function is the set of all inputs whose value is in that set.
</document>

<document source="https://math2.tamnd.com/node/01APZ8GDQH.md" tag="01APZ8GDQH" kind="statement">
# Chapman–Kolmogorov equation

## Statement

Let $(X_n)_{n \ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$. For $n \ge 0$, let $p^{(n)}_{ij} = P(X_{m+n} = j \mid X_m = i)$ be the $n$-step transition probabilities.
Then for all $m, n \ge 0$ and $i, j \in S$,
$$p^{(m+n)}_{ij} = \sum_{k \in S} p^{(m)}_{ik} \, p^{(n)}_{kj}.$$
In matrix form, $P^{(m+n)} = P^{(m)} P^{(n)}$, so $P^{(n)} = P^n$ for the [[transition matrix|0F1KBVY161]] $P$.

## Remarks

Proof: condition on the state $X_m = k$ at the intermediate time, with the [[law of total probability|0E5G5FMT2R]], and use the [[Markov property|0HB8SXHPFA]].
The equation says that to go from $i$ to $j$ in $m + n$ steps, the chain must be at some state $k$ after $m$ steps.
For a Markov process with transition kernels $p_{s,t}(x, dy)$, the equation reads $p_{s,u}(x, A) = \int p_{s,t}(x, dy) \, p_{t,u}(y, A)$ for $s \le t \le u$.

## History

The equation is named after Sydney Chapman and Andrey Kolmogorov.

## Explanations

### L1 human

The chances for several steps come from multiplying the one-step transition matrix with itself.
For example, the chance of rain in two days, given sun today, sums over the two possible states of tomorrow.
</document>

<document source="https://math2.tamnd.com/node/01AXG9KZQF.md" tag="01AXG9KZQF" kind="definition">
# Block matrix

## Definition

Let $R$ be a ring.
Let $m = m_1 + \cdots + m_p$ and $n = n_1 + \cdots + n_q$ with positive integers $m_i$ and $n_j$.
For each $i \le p$ and $j \le q$, let $A_{ij}$ be an $m_i \times n_j$ [[matrix|0VD9ZSWWJG]] over $R$.
The *block matrix* with blocks $A_{ij}$ is the $m \times n$ matrix
$$A = \begin{pmatrix} A_{11} & \cdots & A_{1q} \\ \vdots & \ddots & \vdots \\ A_{p1} & \cdots & A_{pq} \end{pmatrix},$$
in which $A_{ij}$ fills rows $m_1 + \cdots + m_{i-1} + 1$ to $m_1 + \cdots + m_i$ and columns $n_1 + \cdots + n_{j-1} + 1$ to $n_1 + \cdots + n_j$.

## Remarks

If the column partition of $A$ matches the row partition of $B$, then $AB$ is computed block by block: $(AB)_{ik} = \sum_j A_{ij} B_{jk}$.
In Mathlib, `Matrix.fromBlocks` makes the $2 \times 2$ block matrix $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$.
The [[determinant of a block triangular matrix|0J9TVTNB4K]] is the product of the determinants of its diagonal blocks.

## Explanations

### L1 human

A block matrix is a large matrix cut into smaller rectangles.
We can treat each rectangle like a single entry.
Then we add and multiply block matrices with the usual rules, as long as the sizes fit.
</document>

<document source="https://math2.tamnd.com/node/01BD2TFFP4.md" tag="01BD2TFFP4" kind="definition">
# Bias of an estimator

## Definition

Let $\hat{\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$, with $E_\theta[|\hat{\theta}|] < \infty$ for each $\theta$. Here $E_\theta$ is the expectation when $\theta$ is the true parameter.
The bias of $\hat{\theta}$ at $\theta$ is
$$\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta.$$
The estimator is unbiased if $\operatorname{Bias}_\theta(\hat{\theta}) = 0$ for each $\theta$.

## Remarks

For an i.i.d. sample with mean $\mu$, the sample mean is an unbiased estimator of $\mu$.
The sample variance with divisor $n$ has bias $-\sigma^2/n$. With divisor $n - 1$, it is unbiased.
An unbiased estimator need not be good. The [[mean squared error|0V4VSCBXP3]] is variance plus squared bias, and a biased estimator can have a smaller mean squared error.
Unbiasedness is not preserved by nonlinear maps. For example, the sample standard deviation $S_n$ is a biased estimator of $\sigma$, even though $S_n^2$ is unbiased for $\sigma^2$.

## Explanations

### L1 human

An estimator is unbiased if it is right on average, over many repetitions of the experiment.
Bias is a systematic error: a scale that always shows $1$ kg too much has a bias of $1$ kg.
</document>

<document source="https://math2.tamnd.com/node/01BX15HSRR.md" tag="01BX15HSRR" kind="statement">
# Well-ordering principle

## Statement

Every nonempty set of natural numbers has a least element.
That is, if $A \subseteq \mathbb{N}$ and $A \neq \emptyset$, then there is an $a$ in $A$ with $a \le x$ for each $x$ in $A$.

## Remarks

The least element is unique, because the [[order|0Y9H52BT03]] on $\mathbb{N}$ is antisymmetric.
The principle follows from [[strong induction|0KMVFSHFGZ]], and conversely it implies the [[principle of mathematical induction|0ASJVEFA9J]].
The set of the integers does not have this property: the set of negative integers has no least element.

## Explanations

### L1 human

If some natural numbers have a property, then there is a smallest one with that property.
For example, among the natural numbers that are larger than $10$ and even, the smallest is $12$.
Proofs often use this: they take the smallest counterexample and show that it cannot exist.
</document>

<document source="https://math2.tamnd.com/node/01CGSJ0GNQ.md" tag="01CGSJ0GNQ" kind="concept">
# Well-order

## Summary

A well-order is a total order in which each nonempty subset has a least element.
</document>

<document source="https://math2.tamnd.com/node/01CJRJKDAA.md" tag="01CJRJKDAA" kind="statement">
# The complex numbers cannot be ordered

## Statement

There is no total order $\le$ on $\mathbb{C}$ that makes $\mathbb{C}$, with its usual addition and multiplication, an [[ordered field|010JCV9J4E]].

## Remarks

Proof: in an ordered field, each square is nonnegative, and $1 = 1^2 > 0$. Then $-1 < 0$. But $-1 = i^2$ is a square, by [[i squared is minus one|07KBQCG9KV]], a contradiction.
There are total orders on the set $\mathbb{C}$, for example the lexicographic order. They are not compatible with the field operations.
Thus inequalities such as $z < w$ make sense only for real numbers $z$ and $w$.

## Explanations

### L1 human

You cannot say which of $i$ and $0$ is larger in a way that keeps the usual rules of inequalities.
If $i > 0$, then $i \cdot i > 0$, so $-1 > 0$. If $i < 0$, then $-i > 0$ and $(-i)(-i) > 0$, so again $-1 > 0$.
Both cases are impossible.
</document>

<document source="https://math2.tamnd.com/node/01CS8Q2APC.md" tag="01CS8Q2APC" kind="definition">
# Characteristic subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $H$ is characteristic in $G$ when $\alpha(H) = H$ for each [[automorphism|0K1SCPBNKJ]] $\alpha$ of $G$.

## Remarks

It is equivalent to ask $\alpha(H) \subseteq H$ for each automorphism $\alpha$, because then also $\alpha^{-1}(H) \subseteq H$.
The [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of a group are characteristic.
A characteristic subgroup of a normal subgroup of $G$ is normal in $G$.

## Explanations

### L1 human

A characteristic subgroup is a subgroup that no symmetry of the group can move.
Usually such a subgroup has a description that uses only the group operation, like "the elements that commute with everything".
Any renaming of the elements that keeps the products correct must keep such a subgroup in place.
</document>

<document source="https://math2.tamnd.com/node/01CYW8S04D.md" tag="01CYW8S04D" kind="definition">
# Totally disconnected space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is totally disconnected when each [[connected subset|0R37DFJKFW]] of $X$ has at most one point.

## Remarks

Equivalently, each connected component of $X$ is a single point.
Each [[discrete space|0S2TEHX4E1]] is totally disconnected. So are $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$, as subspaces of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is totally disconnected, although it is compact and has no isolated points.

## Explanations

### L1 human

A totally disconnected space falls apart into single points: no subset with two or more points is in one piece.
This does not mean that the space is discrete. The rationals are a good example.
</document>

<document source="https://math2.tamnd.com/node/01DSNP0B4R.md" tag="01DSNP0B4R" kind="definition">
# Periodic orbit

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and consider the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
A [[solution|04SW7WECS8]] $\gamma : \mathbb{R} \to U$ is periodic if it is not constant and there is a real number $T > 0$ with $\gamma(t + T) = \gamma(t)$ for each $t \in \mathbb{R}$.
The smallest such $T$ is the period of $\gamma$.
A periodic orbit, or closed orbit, is the [[trajectory|0BVT6VSB3S]] $\gamma(\mathbb{R})$ of a periodic solution $\gamma$.

## Remarks

A non-constant periodic solution has a smallest period $T > 0$, because $\gamma$ is continuous and not constant.
A periodic orbit is a simple closed curve in $U$.
Let a maximal solution $\gamma$ that is not constant satisfy $\gamma(t_1 + T) = \gamma(t_1)$ for one $t_1$ and one $T > 0$. Then, by [[uniqueness|025PKW8AMR]], $\gamma$ is defined on $\mathbb{R}$ and is periodic.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, each circle $x^2 + y^2 = r^2$ with $r > 0$ is a periodic orbit of period $2\pi$.

## Explanations

### L1 human

A periodic orbit is a motion that repeats itself exactly after a fixed time.
In the phase space it is a closed loop that the state goes around again and again.
</document>

<document source="https://math2.tamnd.com/node/01DSVCRF6X.md" tag="01DSVCRF6X" kind="definition">
# Ramsey number

## Definition

Let $r$ and $s$ be positive integers.
The Ramsey number $R(r, s)$ is the least positive integer $n$ with this property.
Colour each edge of the [[complete graph|03WKB2H474]] $K_n$ blue or red, in any way.
Then there is a blue [[clique|0G52BPNPKG]] with $r$ vertices or a red clique with $s$ vertices.

## Remarks

Such an $n$ exists by [[Ramsey's theorem|00TQV3F5S4]].
$R(r, s) = R(s, r)$ and $R(r, 2) = r$. The [[theorem on friends and strangers|0NYM0ZGAK9]] gives $R(3, 3) = 6$.
With more colours there are similar numbers, for example $R(3, 3, 3) = 17$.

## Explanations

### L1 human

The Ramsey number $R(r, s)$ is the size of a party that is large enough for one of two things to happen.
There are $r$ people who all know each other, or $s$ people who are all strangers to each other.
</document>

<document source="https://math2.tamnd.com/node/01E7SWYJX3.md" tag="01E7SWYJX3" kind="statement">
# Normality via left and right cosets

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $N$ is [[normal|0CB0DX5BVV]] in $G$ if and only if $gN = Ng$ for each $g \in G$.

## Remarks

Here $gN$ is a [[left coset|03DEH43F4H]] and $Ng$ is a [[right coset|0NE1DK7H4T]].
The equation $gN = Ng$ is equivalent to $gNg^{-1} = N$, by multiplying on the right by $g^{-1}$.
The condition does not say that $gn = ng$ for each $n \in N$; it says only that the two sets are equal.

## Explanations

### L1 human

For a normal subgroup, it does not matter if you multiply by $g$ on the left or on the right: you get the same coset.
Thus there is only one kind of coset, and we can just say "the cosets of $N$".
This is what makes the quotient group possible.
</document>

<document source="https://math2.tamnd.com/node/01E9MNJYEX.md" tag="01E9MNJYEX" kind="concept">
# Public-key cryptography

## Summary

Public-key cryptography is a kind of cryptography in which each user has a pair of keys: a public key and a private key.
Anyone can use the public key to encrypt a message or to check a signature. Only the owner of the private key can decrypt the message or make the signature.
Its security depends on problems that are believed to be hard to solve, such as integer factorization or the [[discrete logarithm|09B4X51J09]].

## Remarks

Before the mid-1970s, all ciphers were symmetric: both parties used the same secret key, which they had to exchange in a secure way.
The [[RSA cryptosystem|0HM584C465]] and the [[Diffie–Hellman key exchange|01TKKESHSB]] are two early examples.
Whitfield Diffie and Martin Hellman published the idea in 1976.

## Explanations

### L1 human

Think of a mailbox with a slot: anyone can drop a letter in, but only the owner has the key to open it.
The public key is the slot, and the private key is the key of the mailbox.
</document>

<document source="https://math2.tamnd.com/node/01EK923SKP.md" tag="01EK923SKP" kind="definition">
# Inner product space

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$.
An *inner product space* over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with an [[inner product|0NB0AWGJBW]] $\langle \cdot, \cdot \rangle$ on $V$.

## Remarks

A finite-dimensional real inner product space is a *Euclidean space*.
Each inner product space has the [[induced norm|0MFJJV97RJ]] $\norm{x} = \sqrt{\langle x, x \rangle}$.
An inner product space that is complete for this norm is a *Hilbert space*.
Each finite-dimensional inner product space is complete.

## Explanations

### L1 human

An inner product space is a vector space where we can measure lengths and angles.
The space $\R^n$ with the dot product is the main example.
Much of plane geometry, such as the Pythagorean theorem, works in every inner product space.
</document>

<document source="https://math2.tamnd.com/node/01EQPZ6RYZ.md" tag="01EQPZ6RYZ" kind="concept">
# Law of a random variable

## Summary

The law of a random variable $X$ is the probability distribution of its values: it gives each set $B$ of values the probability $P(X \in B)$.
The node [[law of a random variable|0TST4MHA3A]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/01EXSMK864.md" tag="01EXSMK864" kind="statement">
# Determinant of a scalar multiple

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Then
$$\det(cA) = c^n \det A,$$
where $cA$ is the [[scalar multiple|09V6DDACXR]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Each of the $n$ rows is multiplied by $c$, and each row contributes one factor $c$.
For example, $\det(-A) = (-1)^n \det A$.
So an $n \times n$ real [[skew-symmetric matrix|0PBY4E9GQF]] with $n$ odd has determinant $0$.

## Explanations

### L1 human

Scaling a matrix by $c$ stretches every edge of a box by $c$.
In $n$ dimensions, this multiplies the volume by $c^n$.
So doubling a $3 \times 3$ matrix multiplies its determinant by $8$.
</document>

<document source="https://math2.tamnd.com/node/01EY0EM7RP.md" tag="01EY0EM7RP" kind="statement">
# Extensions of solvable groups are solvable

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
If $N$ and $G/N$ are [[solvable|0FXJ7YFMMX]], then $G$ is solvable.

## Remarks

If $(G/N)^{(k)}$ is trivial, then $G^{(k)} \subseteq N$, so $G^{(k+m)} \subseteq N^{(m)}$ for each $m$.
Choose $m$ with $N^{(m)} = \{e\}$; then $G^{(k+m)} = \{e\}$.
With [[subgroups|05Y0F9M0TS]] and [[quotients|027ZQS04SJ]], this gives: $G$ is solvable if and only if $N$ and $G/N$ are solvable.
In particular, direct products and semidirect products of solvable groups are solvable.

## Explanations

### L1 human

If the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group.
You stack the layers of the top part on the layers of the bottom part.
For example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.
</document>

<document source="https://math2.tamnd.com/node/01EY3VR7ZD.md" tag="01EY3VR7ZD" kind="statement">
# Urysohn's lemma

## Statement

Let $X$ be a [[normal space|0AJ646N339]] and let $A, B \subseteq X$ be disjoint [[closed sets|0G5AV9Q4RR]].
Then there is a [[continuous map|0SY9ECQ19Q]] $f \colon X \to [0, 1]$ with $f(a) = 0$ for all $a \in A$ and $f(b) = 1$ for all $b \in B$.

## Remarks

The converse also holds: if such a function exists for all disjoint closed $A$ and $B$, then $X$ is normal. Take $U = f^{-1}([0, 1/2))$ and $V = f^{-1}((1/2, 1])$.
The proof builds open sets $U_r$ for the dyadic rationals $r \in [0, 1]$, nested by the order of $r$. Then $f(x)$ is the infimum of the $r$ with $x \in U_r$.
In a metric space, the function $f(x) = d(x, A) / (d(x, A) + d(x, B))$ works.
The [[Tietze extension theorem|0X52TNMTCJ]] extends this result to all continuous functions on closed sets.

## Explanations

### L1 human

In a normal space, a continuous function can separate two disjoint closed sets: it is $0$ on one set and $1$ on the other.
So normal spaces have many continuous real functions.
</document>

<document source="https://math2.tamnd.com/node/01FPXMTBX1.md" tag="01FPXMTBX1" kind="statement">
# Composition of injections is injective

## Statement

Let $f \colon A \to B$ and $g \colon B \to C$ be [[injective functions|0PVG1CRAJN]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon A \to C$ is injective.

## Remarks

Proof: if $g(f(a)) = g(f(a'))$, then $f(a) = f(a')$ because $g$ is injective, and $a = a'$ because $f$ is injective.
A partial converse holds: if $g \circ f$ is injective, then $f$ is injective. The function $g$ need not be injective.

## Explanations

### L1 human

Think of $f$ and $g$ as machines, and of $g \circ f$ as the two machines in a chain.
If neither machine merges two inputs, the chain does not merge two inputs either.
</document>

<document source="https://math2.tamnd.com/node/01G1WY1GVX.md" tag="01G1WY1GVX" kind="definition">
# Pythagorean triple

## Definition

A Pythagorean triple is a triple $(a, b, c)$ of positive integers such that $a^2 + b^2 = c^2$.
The triple is primitive if $\gcd(a, b, c) = 1$, with $\gcd$ the [[greatest common divisor|02WF878BH1]].

## Remarks

The smallest examples are $(3, 4, 5)$ and $(5, 12, 13)$.
If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for each positive integer $k$.
The [[classification of Pythagorean triples|0KF3J2KAER]] lists all of them.
In a primitive triple, $a$, $b$ and $c$ are pairwise coprime, and exactly one of $a$ and $b$ is even.

## Explanations

### L1 human

A Pythagorean triple gives a right triangle whose three sides are whole numbers.
The triangle with sides $3$, $4$ and $5$ is the best-known example, since $9 + 16 = 25$.
</document>

<document source="https://math2.tamnd.com/node/01GBHQ5ABF.md" tag="01GBHQ5ABF" kind="concept">
# Periodic continued fraction

## Summary

A periodic continued fraction is an infinite simple continued fraction whose terms repeat a fixed block from some point on.
The [[definition of a periodic continued fraction|0XKJPFS2W0]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/01GTP8M5RG.md" tag="01GTP8M5RG" kind="definition">
# Order of cardinals

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The cardinality of $A$ is at most the cardinality of $B$, written $|A| \le |B|$, when there is an [[injective function|0PVG1CRAJN]] $f \colon A \to B$.
Write $|A| < |B|$ when $|A| \le |B|$ and $|A| \neq |B|$.

## Remarks

The relation does not depend on the choice of $A$ and $B$ in their classes of [[equinumerous|08EJHHMB5R]] sets.
It is reflexive and transitive. It is antisymmetric by the [[Schröder–Bernstein theorem|0A9V6RKQG6]].
With the axiom of choice, any two cardinals are comparable. See [[comparability of cardinals|0YBHSJMN3J]].

## Explanations

### L1 human

$|A| \le |B|$ means that $A$ fits into $B$ without two elements landing in the same place.
For example, $|\mathbb{N}| \le |\mathbb{R}|$, because $\mathbb{N}$ is a subset of $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/01H6QKWGDH.md" tag="01H6QKWGDH" kind="concept">
# Generalised eigenvector

## Summary

A generalised eigenvector of an endomorphism for $\lambda$ is a nonzero vector with a special property.
Some power of the map minus $\lambda$ times the identity sends it to zero.
</document>

<document source="https://math2.tamnd.com/node/01HR2AH0BX.md" tag="01HR2AH0BX" kind="definition">
# Divergent sequence

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is divergent if it is not a [[convergent sequence|0AZ3R52593]].
Thus, for each real number $L$, the sequence $(a_n)$ does not converge to $L$.

## Remarks

The sequence $a_n = (-1)^n$ is divergent, and it is bounded.
The sequence $a_n = n$ is divergent, and it [[tends to infinity|09AZCV6GMC]].
Some texts say that a sequence that tends to $+\infty$ diverges to $+\infty$. It is still divergent in the sense of this definition.

## Explanations

### L1 human

A divergent sequence has no limit.
Its terms can grow without bound, as in $1, 2, 3, \dots$, or they can jump between values, as in $1, -1, 1, -1, \dots$.
</document>

<document source="https://math2.tamnd.com/node/01HT2CH8BH.md" tag="01HT2CH8BH" kind="statement">
# Equivalence classes form a partition

## Statement

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$.
Then the set $\{[a] \mid a \in A\}$ of [[equivalence classes|082RMRSY26]] is a [[partition|09CCE86A6D]] of $A$.

## Remarks

Each class $[a]$ is nonempty, because $a \in [a]$ by reflexivity.
If $[a]$ and $[b]$ have a common element, then $a \sim b$ by symmetry and transitivity, so $[a] = [b]$.
The left cosets of a subgroup of a [[group|G5S1M9]] form such a partition. This fact is the key step in [[Lagrange's theorem|3C53TD]].

## Explanations

### L1 human

The three properties of an equivalence relation are exactly what you need to sort elements into classes.
Reflexivity puts each element in some class.
Symmetry and transitivity stop two classes from overlapping.
</document>

<document source="https://math2.tamnd.com/node/01J7FZ65J0.md" tag="01J7FZ65J0" kind="concept">
# Maximal solution

## Summary

A maximal solution of an ordinary differential equation is a solution that is not the restriction of a solution on a larger interval.
Under a local Lipschitz condition, each initial value problem has exactly one maximal solution. See the [[definition|019VSXSFMQ]].
</document>

<document source="https://math2.tamnd.com/node/01J8AMV9RS.md" tag="01J8AMV9RS" kind="definition">
# Legendre symbol

## Definition

Let $p$ be an odd [[prime number|0SN0MAEKYX]] and let $a$ be an integer.
The Legendre symbol $\left(\frac{a}{p}\right)$ is the number
$$\left(\frac{a}{p}\right) = \begin{cases} 0 & \text{if } p \mid a, \\ 1 & \text{if } p \nmid a \text{ and } a \text{ is a quadratic residue modulo } p, \\ -1 & \text{if } a \text{ is a quadratic nonresidue modulo } p. \end{cases}$$
Here quadratic residue has the meaning of the [[definition of a quadratic residue|0SQDB87Z3T]].

## Remarks

The value depends only on the class of $a$ modulo $p$.
For example, $\left(\frac{2}{7}\right) = 1$ because $3^2 \equiv 2 \pmod 7$, and $\left(\frac{3}{7}\right) = -1$.
[[Euler's criterion|0M7DEDKSYB]] gives the value as $a^{(p-1)/2}$ modulo $p$.
The [[law of quadratic reciprocity|085MJS6907]] helps to compute the symbol quickly.

## Explanations

### L1 human

The Legendre symbol is a short way to say whether $a$ is a square modulo $p$.
The value $1$ means yes, the value $-1$ means no, and the value $0$ means that $a$ is $0$ modulo $p$.
</document>

<document source="https://math2.tamnd.com/node/01JADB9VW5.md" tag="01JADB9VW5" kind="statement">
# Vandermonde's identity

## Statement

Let $m$, $n$ and $r$ be natural numbers.
Then
$$\binom{m + n}{r} = \sum_{k=0}^{r} \binom{m}{k} \binom{n}{r - k}.$$

## Remarks

Combinatorial proof: take a set of $m$ men and $n$ women. An $r$-element committee has some number $k$ of men and $r - k$ women.
With $m = n = r$, and the [[symmetry of binomial coefficients|0R7JS3DP3W]], it gives $\binom{2n}{n} = \sum_{k=0}^{n} \binom{n}{k}^2$.

## Explanations

### L1 human

To choose $r$ people from two groups together, choose $k$ from the first group and $r - k$ from the second, for each possible $k$.
Then add the counts for all values of $k$.
</document>

<document source="https://math2.tamnd.com/node/01KJJ9A400.md" tag="01KJJ9A400" kind="concept">
# Regular polygon

## Summary

A regular polygon is a polygon with all sides of the same length and all angles of the same size. Its vertices lie on one circle. Examples are the equilateral triangle and the square.
</document>

<document source="https://math2.tamnd.com/node/01M5T0KDG7.md" tag="01M5T0KDG7" kind="definition">
# Multigraph

## Definition

A multigraph is a triple $G = (V, E, r)$ of a set $V$ of vertices, a set $E$ of edges, and a map $r$.
The map $r$ assigns to each edge $e \in E$ a set $r(e) \subseteq V$ of one or two vertices, the ends of $e$.
Two distinct edges may have the same ends. Such edges are parallel, or multiple, edges.
An edge with only one end is a loop.

## Remarks

Some authors do not allow loops in a multigraph, and they call a multigraph with loops a pseudograph.
A [[graph|0JD5C82QMB]] is a multigraph without loops and without parallel edges.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with $4$ vertices and $7$ edges.

## Explanations

### L1 human

A multigraph is like a graph, but two points can be joined by several lines.
A road map with two different roads between the same two towns is a multigraph.
</document>

<document source="https://math2.tamnd.com/node/01M6CW9VM6.md" tag="01M6CW9VM6" kind="statement">
# Triangle inequality for a metric

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
For all points $x, y, z \in X$,
$$d(x, z) \le d(x, y) + d(y, z).$$

## Remarks

This is one of the four axioms of a metric.
For $X = \mathbb{R}$ with $d(x, y) = |x - y|$, it is the [[triangle inequality|03HF5PMY5T]] for real numbers, with $a = x - y$ and $b = y - z$.
A consequence is the reverse form $|d(x, y) - d(x, z)| \le d(y, z)$ for all $x, y, z \in X$.

## Explanations

### L1 human

A direct path from $x$ to $z$ is never longer than a path that goes through a third point $y$.
In the plane, this says that one side of a triangle is at most the sum of the other two sides.
</document>

<document source="https://math2.tamnd.com/node/01MGYYBB9R.md" tag="01MGYYBB9R" kind="definition">
# Antisymmetric relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is antisymmetric when, for all $a, b \in A$, $a \mathrel{R} b$ and $b \mathrel{R} a$ imply $a = b$.

## Remarks

The relation $\le$ on $\mathbb{R}$ and the relation $\subseteq$ on sets are antisymmetric.
Divisibility is antisymmetric on $\mathbb{N}$ but not on $\mathbb{Z}$, because $1 \mid -1$ and $-1 \mid 1$.
Antisymmetric is not the opposite of symmetric: equality is both symmetric and antisymmetric.

## Explanations

### L1 human

If $a \le b$ and $b \le a$, then $a = b$.
So for $\le$, a two-way relation between two elements forces them to be the same.
</document>

<document source="https://math2.tamnd.com/node/01MKW4R283.md" tag="01MKW4R283" kind="definition">
# Infinite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is infinite when it is not [[finite|0BBTH3YSZA]], that is, when $A$ is not equinumerous with $\{1, \dots, n\}$ for any natural number $n$.

## Remarks

Examples are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.
With the axiom of choice, a set is infinite exactly when it is equinumerous with a proper subset of itself. This is the Dedekind definition.
Infinite sets can have different sizes. See [[Cantor's theorem|0WTEMGYV2C]].

## Explanations

### L1 human

An infinite set never runs out of elements when you count them.
The natural numbers are infinite: after each number there is a next one.
</document>

<document source="https://math2.tamnd.com/node/01MMVW8CKC.md" tag="01MMVW8CKC" kind="concept">
# Positive semidefinite matrix

## Summary

A positive semidefinite matrix is a Hermitian matrix $A$ with $x^* A x \ge 0$ for every vector $x$.
</document>

<document source="https://math2.tamnd.com/node/01MPZXYVYF.md" tag="01MPZXYVYF" kind="statement">
# Cauchy's integral theorem

## Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $\gamma : [a, b] \to U$ be a closed path that is continuous and piecewise continuously differentiable.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ is
$$\oint_\gamma f(z) \, dz = 0.$$

## Remarks

Open discs, open half-planes, $\mathbb{C}$ and each convex open set are simply connected. So the theorem holds on each of them.
A special case: if $f$ is holomorphic on an open set that contains the closed disc $|z - c| \le R$, then $\oint_{|z - c| = R} f(z) \, dz = 0$.
The hypothesis on $U$ is necessary. On $\mathbb{C} \setminus \{0\}$, the [[integral of 1/z|0SXZ68HV6F]] over the unit circle is $2 \pi i$.
It follows that the integral of $f$ along a path in $U$ depends only on the end points of the path.

## Explanations

### L1 human

The integral of a holomorphic function around a closed loop is zero, as long as the loop does not go around a hole of the domain.
It is like walking around a hill and coming back to the start: the total change of height is zero.
</document>

<document source="https://math2.tamnd.com/node/01N3AYAA6N.md" tag="01N3AYAA6N" kind="definition">
# Integers modulo n

## Definition

Let $n$ be a positive integer.
The integers modulo $n$ form the set $\mathbb{Z}/n\mathbb{Z} = \{ \overline{0}, \overline{1}, \dots, \overline{n - 1} \}$ of the [[residue classes|0QRV8FPK5M]] modulo $n$.
Addition and multiplication on $\mathbb{Z}/n\mathbb{Z}$ are $\overline{a} + \overline{b} = \overline{a + b}$ and $\overline{a} \cdot \overline{b} = \overline{a b}$.

## Remarks

The operations are well defined: the result does not depend on the choice of $a$ and $b$ in their classes. This follows from [[compatibility with addition|00FWBN7F0Y]] and [[compatibility with multiplication|072D5G4SK6]].
With these operations, $\mathbb{Z}/n\mathbb{Z}$ is a commutative ring with $n$ elements, with zero $\overline{0}$ and identity $\overline{1}$.
Other notations are $\mathbb{Z}_n$ and $\mathbb{Z}/(n)$. The Lean library Mathlib writes `ZMod n`, and it sets `ZMod 0` to be $\mathbb{Z}$.
The ring is a field if and only if $n$ is prime. See [[integers modulo a prime form a field|0R8YXRC9AJ]].

## Explanations

### L1 human

In $\mathbb{Z}/12\mathbb{Z}$ you compute as on a clock: $\overline{9} + \overline{5} = \overline{14} = \overline{2}$.
Products work the same way: $\overline{5} \cdot \overline{7} = \overline{35} = \overline{11}$.
</document>

<document source="https://math2.tamnd.com/node/01NHFZF2SK.md" tag="01NHFZF2SK" kind="concept">
# Nondegenerate bilinear form

## Summary

A nondegenerate bilinear form is a bilinear form for which no nonzero vector gives zero with every vector.
</document>

<document source="https://math2.tamnd.com/node/01NN8NY497.md" tag="01NN8NY497" kind="counterexample">
# Function with unequal mixed partial derivatives

## Counterexample

One could guess that the mixed [[partial derivatives|02RSXVM1ZN]] $f_{xy}$ and $f_{yx}$ are equal at each point where both exist.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by
$$f(x, y) = \frac{x y (x^2 - y^2)}{x^2 + y^2} \text{ for } (x, y) \ne (0, 0), \qquad f(0, 0) = 0.$$
Both mixed second partial derivatives of $f$ exist at $(0, 0)$, but $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)(0, 0) = -1$ and $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)(0, 0) = 1$.

## Remarks

From the definition, $\partial f / \partial x (0, y) = \lim_{h \to 0} f(h, y) / h = -y$ for each $y$, and $\partial f / \partial y (x, 0) = \lim_{k \to 0} f(x, k) / k = x$ for each $x$. Differentiate these once more at $0$.
The function is of class $C^1$ on $\mathbb{R}^2$, but its second partial derivatives are not continuous at the origin. So [[Schwarz's theorem|0NNQY8WCW6]] does not apply.
This example is due to Peano.

## Explanations

### L1 human

Schwarz's theorem needs continuous second partial derivatives, and this function shows why.
Away from the origin everything is smooth, but at the origin the two orders of differentiation see different limits.
</document>

<document source="https://math2.tamnd.com/node/01PKQYTTMT.md" tag="01PKQYTTMT" kind="definition">
# Isolated point

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ and let $x \in S$.
The point $x$ is an isolated point of $S$ when there is an [[open set|01ZQQDCERP]] $U$ of $X$ with $U \cap S = \{x\}$.

## Remarks

Each point of $S$ is either an isolated point of $S$ or a [[limit point|0HBQC2GJXT]] of $S$, and not both.
In $\mathbb{R}$, each point of $\mathbb{Z}$ is an isolated point of $\mathbb{Z}$: take $U = (n - 1/2, n + 1/2)$.
A point $x$ is an isolated point of $X$ itself if and only if $\{x\}$ is open.

## Explanations

### L1 human

An isolated point is a point of the set that stands alone: a small region about it contains no other point of the set.
In the set $\{0\} \cup [1, 2]$, the point $0$ is isolated, but $1$ is not.
</document>

<document source="https://math2.tamnd.com/node/01PNAVFMYZ.md" tag="01PNAVFMYZ" kind="definition">
# Similar triangles

## Definition

Let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The triangles are similar, written $ABC \sim A'B'C'$, if there is a real number $r > 0$ such that
$$d(A', B') = r \, d(A, B), \quad d(B', C') = r \, d(B, C) \quad \text{and} \quad d(C', A') = r \, d(C, A).$$
The number $r$ is the ratio of the similarity.

## Remarks

Similar triangles have equal corresponding angles. Conversely, the [[angle-angle criterion|0NZVJYZVTZ]] says that two pairs of equal angles are enough for similarity.
[[Congruent triangles|093EDENVA1]] are the similar triangles with the ratio $r = 1$.
Two triangles are similar if and only if a [[homothety|0ZG1PNEP35]] followed by an isometry sends one onto the other, vertex by vertex.
Any two equilateral triangles are similar.

## Explanations

### L1 human

A photo of a triangle, enlarged or reduced, gives a similar triangle. All sides change by the same factor, and the angles stay the same. For example, the triangles with sides $3, 4, 5$ and $6, 8, 10$ are similar with the ratio $2$.
</document>

<document source="https://math2.tamnd.com/node/01PPDSDP2N.md" tag="01PPDSDP2N" kind="concept">
# Hyperbolic functions

## Summary

The hyperbolic functions are analogues of the trigonometric functions defined from the exponential function. The main ones are the hyperbolic sine, cosine and tangent. The node [[hyperbolic functions|0J6M03WSS6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/01QFQAQ2PA.md" tag="01QFQAQ2PA" kind="concept">
# Spherical coordinates

## Summary

Spherical coordinates describe a point of space by three numbers. They are its distance from the origin, the angle between its position vector and the positive $z$-axis, and an azimuth angle around the $z$-axis.
They simplify problems with symmetry about a point, such as integrals over balls.

## Remarks

The names and the order of the two angles differ between mathematics and physics texts. Each text must state its convention.
</document>

<document source="https://math2.tamnd.com/node/01QSSKS5C1.md" tag="01QSSKS5C1" kind="statement">
# sin x / x tends to 1

## Statement

$$\lim_{x \to 0} \frac{\sin x}{x} = 1.$$

## Remarks

The quotient is the difference quotient of $\sin$ at $0$. So the limit is $\sin'(0) = \cos 0 = 1$, by the [[derivative of sine|00R4MWW532]].
The [[series|02KTRHQMHN]] gives $\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \cdots$ for $x \ne 0$.
A geometric proof uses $\cos x \le \frac{\sin x}{x} \le 1$ for $0 < |x| < \pi/2$ and the squeeze theorem. Mathlib states it as `Real.tendsto_sin_div_nhdsNE_zero`.

## Explanations

### L1 human

For small angles in radians, $\sin x$ is almost equal to $x$.
The ratio of the two tends to $1$.
</document>

<document source="https://math2.tamnd.com/node/01QZNHQWYD.md" tag="01QZNHQWYD" kind="statement">
# Sum of the first n natural numbers

## Statement

For every natural number $n$,
$$0 + 1 + 2 + \dots + n = \sum_{k=0}^{n} k = \frac{n(n + 1)}{2}.$$

## Remarks

The number $n(n + 1)/2$ is the $n$th triangular number. For $n = 1, 2, 3, 4$ it is $1, 3, 6, 10$.
A proof by [[induction|0ASJVEFA9J]] uses $\frac{n(n+1)}{2} + (n + 1) = \frac{(n+1)(n+2)}{2}$.
An equivalent form without division is $2 \sum_{k=0}^{n} k = n(n + 1)$.

## Explanations

### L1 human

Write the sum $1 + 2 + \dots + n$ two times, once forward and once backward.
Each column then adds up to $n + 1$, and there are $n$ columns.
So two copies of the sum give $n(n + 1)$, and one copy is half of that.
</document>

<document source="https://math2.tamnd.com/node/01R1HPEK6S.md" tag="01R1HPEK6S" kind="statement">
# A graph is bipartite iff it has no odd cycle

## Statement

Let $G$ be a [[graph|0JD5C82QMB]].
Then $G$ is [[bipartite|07DVHJ8N0A]] if and only if $G$ has no [[cycle|07CT9RDY7Q]] of odd length.

## Remarks

One direction is easy. A cycle in a bipartite graph alternates between the two parts, so its length is even.
For the converse, work in each component. Colour a vertex by the parity of its distance to a fixed vertex. An edge between two vertices of the same colour would give an odd closed walk, and then an odd cycle.
A graph is also bipartite if and only if its chromatic number is at most $2$.

## Explanations

### L1 human

Walking around a cycle in a bipartite graph, you switch sides at every step.
To come back to the start you need an even number of steps, so odd cycles are impossible.
</document>

<document source="https://math2.tamnd.com/node/01R1HW3Y42.md" tag="01R1HW3Y42" kind="concept">
# Tangent line

## Summary

Let $f$ be [[differentiable|0F1WR9C83K]] at $a$.
The tangent line to the graph of $f$ at $(a, f(a))$ is the line $y = f(a) + f'(a)(x - a)$.
It is the line through this point that best fits the graph near it.

## Remarks

The tangent line is the limit of the secant lines through $(a, f(a))$ and $(x, f(x))$ as $x \to a$.
The error $f(x) - f(a) - f'(a)(x - a)$ is $o(x - a)$ as $x \to a$. This property characterizes the derivative.
For $f(x) = x^2$ at $a = 1$, the tangent line is $y = 2x - 1$.

## Explanations

### L1 human

The tangent line touches the graph at one point and has the same slope as the graph there.
Near the point, the graph and the line are almost the same.
</document>

<document source="https://math2.tamnd.com/node/01RBSSGMY5.md" tag="01RBSSGMY5" kind="concept">
# Definite and indefinite quadratic forms

## Summary

Let $A$ be a symmetric $n \times n$ matrix, and let $Q(x) = x^{\mathsf{T}} A x$ be its real quadratic form on $\mathbb{R}^n$. The form $Q$ is positive definite if $Q(x) > 0$ for all $x \ne 0$. It is negative definite if $Q(x) < 0$ for all $x \ne 0$.
It is positive or negative semidefinite if $Q(x) \ge 0$ or $Q(x) \le 0$ for all $x$, and indefinite if it takes both positive and negative values.

## Remarks

The form $Q$ is positive definite if and only if all eigenvalues of $A$ are positive. It is indefinite if and only if $A$ has a positive and a negative eigenvalue.
On $\mathbb{R}^2$, $x_1^2 + x_2^2$ is positive definite, $x_1^2$ is positive semidefinite but not definite, and $x_1^2 - x_2^2$ is indefinite.
In the [[second derivative test|0GYDV9FE98]], the matrix $A$ is the Hessian matrix of a function at a critical point.

## Explanations

### L1 human

A positive definite form is a bowl: it is zero at the origin and positive everywhere else.
An indefinite form is a saddle: it goes up in some directions and down in others.
</document>

<document source="https://math2.tamnd.com/node/01REXW3WWE.md" tag="01REXW3WWE" kind="concept">
# Elementary matrix

## Summary

An elementary matrix is a matrix that we get from the identity matrix by one elementary row operation.
Multiplying a matrix on the left by an elementary matrix performs that row operation.
</document>

<document source="https://math2.tamnd.com/node/01REYCJ371.md" tag="01REYCJ371" kind="concept">
# Lyapunov function

## Summary

A Lyapunov function for an equilibrium of a differential equation is a scalar function with three properties. It is zero at the equilibrium, positive away from it, and non-increasing along solutions.
Its existence proves stability without solving the equation. It is named after Aleksandr Lyapunov. See the [[definition|04P7B5XC22]].
</document>

<document source="https://math2.tamnd.com/node/01RSTGNTXG.md" tag="01RSTGNTXG" kind="concept">
# Residue class

## Summary

A residue class modulo $n$ is the set of all integers that are congruent to a given integer modulo $n$.
The [[definition of a residue class|0QRV8FPK5M]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/01RVCMSN5N.md" tag="01RVCMSN5N" kind="concept">
# Surjective function

## Summary

A function is surjective when each element of the codomain is a value of the function.
</document>

<document source="https://math2.tamnd.com/node/01SWBKDAWA.md" tag="01SWBKDAWA" kind="definition">
# Composition of relations

## Definition

Let $A$, $B$ and $C$ be sets, let $R \subseteq A \times B$ and let $S \subseteq B \times C$ be [[binary relations|08EJC6Q25H]].
The composition of $R$ and $S$ is the relation from $A$ to $C$
$$R \mathbin{;} S = \{(a, c) \in A \times C \mid \exists b \in B, \, a \mathrel{R} b \text{ and } b \mathrel{S} c\}.$$

## Remarks

Many authors write $S \circ R$ for $R \mathbin{;} S$. This order agrees with [[function composition|07HF9YCWPP]].
Composition of relations is associative.
A relation $R$ on a set is [[transitive|0CYRECG2YQ]] exactly when $R \mathbin{;} R \subseteq R$.

## Explanations

### L1 human

Let $R$ be "is a parent of" and let $S$ be "is a parent of" again.
Then $R \mathbin{;} S$ is "is a grandparent of": you go from a person to a child, then from the child to a grandchild.
</document>

<document source="https://math2.tamnd.com/node/01T3TMGTTE.md" tag="01T3TMGTTE" kind="definition">
# Neighbourhood

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$, and let $v \in V$.
The neighbourhood of $v$ is the set
$$N(v) = \{ u \in V : u \sim v \}$$
of the vertices adjacent to $v$.
The elements of $N(v)$ are the neighbours of $v$.

## Remarks

This is the open neighbourhood. It does not contain $v$, because no vertex is adjacent to itself. The closed neighbourhood is $N[v] = N(v) \cup \{v\}$.
The [[degree|0T0F6CCS9W]] of $v$ is $|N(v)|$ when $N(v)$ is finite.
In the cycle with vertices $1, 2, 3, 4, 5$ in this order, $N(1) = \{2, 5\}$.

## Explanations

### L1 human

The neighbours of a vertex are the vertices joined to it by an edge.
In a friendship graph, the neighbourhood of a person is the set of that person's friends.
</document>

<document source="https://math2.tamnd.com/node/01THH3Y3YP.md" tag="01THH3Y3YP" kind="definition">
# Integrating factor

## Definition

Let $I$ be an open interval, and let $p, q : I \to \mathbb{R}$ be continuous.
An integrating factor of the [[linear first-order equation|0Z3YKRQZ48]] $y' + p(x) y = q(x)$ is a differentiable function $\mu : I \to \mathbb{R}$ with $\mu(x) \ne 0$ for each $x \in I$ and
$$\mu'(x) = p(x)\, \mu(x) \quad \text{for each } x \in I.$$
For such $\mu$ and each differentiable $y$ on $I$, $\mu \left(y' + p y\right) = (\mu y)'$.

## Remarks

If $P$ is an antiderivative of $p$ on $I$, then $\mu(x) = e^{P(x)}$ is an integrating factor. Each integrating factor is a nonzero constant times this one.
After multiplication by $\mu$, the equation becomes $(\mu y)' = \mu q$. One integration gives the [[solution formula|06B9W45XV0]].
More generally, an integrating factor is a function that makes an equation exact after multiplication. See [[integrating factor for a non-exact equation|0CX86CR8JV]].

## Explanations

### L1 human

The left side $y' + p y$ is not the derivative of a simple expression.
After multiplication by the right function $\mu$, it becomes the derivative of $\mu y$, by the product rule.
Then you can integrate both sides directly.
For $y' + y = x$, the factor is $e^x$, and the equation becomes $(e^x y)' = x e^x$.
</document>

<document source="https://math2.tamnd.com/node/01TKKESHSB.md" tag="01TKKESHSB" kind="example">
# Diffie–Hellman key exchange

## Example

The Diffie–Hellman key exchange is an example of [[public-key cryptography|01E9MNJYEX]]. Two parties, Alice and Bob, use it to agree on a secret key over a public channel.
They agree in public on a prime $p$ and an integer $g$, often a [[primitive root|0KM2CJ80G7]] modulo $p$.
Alice chooses a secret integer $a$ and sends $A = g^a \bmod p$. Bob chooses a secret integer $b$ and sends $B = g^b \bmod p$.
Alice computes $B^a \bmod p$, and Bob computes $A^b \bmod p$. Both are equal to $g^{ab} \bmod p$, the shared secret.
A small case: $p = 11$, $g = 2$, $a = 3$ and $b = 4$. Then $A = 8$, $B = 16 \bmod 11 = 5$, and $5^3 \equiv 8^4 \equiv 4 \pmod{11}$.

## Remarks

An eavesdropper sees $p$, $g$, $A$ and $B$. To find $g^{ab}$, a natural method is to compute a [[discrete logarithm|09B4X51J09]], which is believed to be hard for large $p$.
The same method works in any finite cyclic group, for example the group of points of an elliptic curve.
Whitfield Diffie and Martin Hellman published the method in 1976.

## Explanations

### L1 human

Each party mixes a public number with a private number, and they swap the results.
Each then mixes in their own private number again, and both get the same final number.
An eavesdropper sees only the swapped numbers, and cannot easily undo the mixing.
</document>

<document source="https://math2.tamnd.com/node/01TPMXM6NM.md" tag="01TPMXM6NM" kind="concept">
# Hyperbolic equilibrium point

## Summary

A hyperbolic equilibrium point of a vector field is an equilibrium point at which the Jacobian matrix has no eigenvalue with real part zero.
Such a point has no centre manifold. See the [[definition|0C5PREPZWG]].
</document>

<document source="https://math2.tamnd.com/node/01TQB88H4C.md" tag="01TQB88H4C" kind="concept">
# Countable set

## Summary

A countable set is a set whose elements can be listed in a finite or infinite sequence.
It is finite or equinumerous with the natural numbers.
</document>

<document source="https://math2.tamnd.com/node/01TXN7ETH2.md" tag="01TXN7ETH2" kind="concept">
# Orthogonal family for a bilinear form

## Summary

A family of vectors is orthogonal for a bilinear form when the form is zero on each pair of distinct members of the family.
</document>

<document source="https://math2.tamnd.com/node/01V9FPBBVK.md" tag="01V9FPBBVK" kind="statement">
# Symmetry of second derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that $f$ is [[differentiable|0135XZZQ8M]] at each point of $U$, and that the derivative $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$ is differentiable at $a$.
Then the second derivative $D^2 f(a)$ is a symmetric bilinear map: $D^2 f(a)(u, v) = D^2 f(a)(v, u)$ for all $u, v \in \mathbb{R}^n$.
In particular, $\dfrac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \dfrac{\partial^2 f}{\partial x_j \, \partial x_i}(a)$ for all $i, j$.

## Remarks

Here $D^2 f(a)$ is the [[iterated derivative|0EY05Z9RPN]] of order $2$, and $D^2 f(a)(e_i, e_j)$ is the mixed [[partial derivative|02RSXVM1ZN]] $\partial_i \partial_j f(a)$.
The hypothesis is that $f$ is twice differentiable at $a$. It holds when the second partial derivatives are continuous near $a$, which gives [[Schwarz's theorem|0NNQY8WCW6]].
The existence of the second partial derivatives at $a$ is not enough: there is a [[function with unequal mixed partial derivatives|01NN8NY497]].

## History

Euler published a proposed proof in 1740, and many incomplete proofs followed over more than a century.
In 1867 Lindelöf analysed the earlier proofs and gave a counterexample to the general claim.
Schwarz gave the first rigorous proof six years later, and Jordan gave in 1883 the general version that most textbooks use.

## Explanations

### L1 human

The theorem says that the order of two differentiations does not matter, when $f$ is regular enough.
For $f(x, y) = x^3 y^2$, differentiating first in $x$ and then in $y$ gives $6 x^2 y$, and the other order gives $6 x^2 y$ too.
</document>

<document source="https://math2.tamnd.com/node/01VG2A0BHW.md" tag="01VG2A0BHW" kind="statement">
# Bijective iff it has a two-sided inverse

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Then $f$ is [[bijective|0SNQEF7Z53]] if and only if there is a function $g \colon B \to A$ with $g \circ f = \mathrm{id}_A$ and $f \circ g = \mathrm{id}_B$.

## Remarks

The function $g$ is then unique. It is the [[inverse function|0W766JAPSN]] $f^{-1}$.
This statement does not need the axiom of choice: for a bijection, $g(b)$ is the unique $a$ with $f(a) = b$.
If $f$ has a [[left inverse|02486SD328]] $g$ and a [[right inverse|0DDPZAG5ER]] $h$, then $g = h$.

## Explanations

### L1 human

A bijection can be undone in both directions.
Going from $A$ to $B$ and back gives the start point, and going from $B$ to $A$ and back gives the start point too.
</document>

<document source="https://math2.tamnd.com/node/01VH21EV6F.md" tag="01VH21EV6F" kind="concept">
# Prime element

## Summary

A prime element of a commutative ring is a nonzero nonunit $p$ with the following property. If $p$ divides a product, then $p$ divides one of the factors.
The [[definition of a prime element|0MNHXBSX3M]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/01VN79T2PR.md" tag="01VN79T2PR" kind="definition">
# Quadratic integer

## Definition

A quadratic integer is a complex number $x$ that satisfies an equation $x^2 + bx + c = 0$ with integers $b$ and $c$.
Equivalently, $x = \dfrac{-b \pm \sqrt{b^2 - 4c}}{2}$ for some integers $b$ and $c$.

## Remarks

Each integer $n$ is a quadratic integer, since it is a root of $x^2 - 2nx + n^2 = 0$.
A quadratic integer that is not an integer is an [[algebraic number|0MB4TGMXG6]] of degree $2$.
It lies in a quadratic field $\mathbb{Q}(\sqrt{D})$ with $D$ a squarefree integer.
The [[Gaussian integers|0B268GVX95]] are the quadratic integers in $\mathbb{Q}(i)$. Another example is the golden ratio $\frac{1 + \sqrt{5}}{2}$, a root of $x^2 - x - 1 = 0$.
For $D \equiv 2, 3 \pmod 4$, the quadratic integers in $\mathbb{Q}(\sqrt{D})$ form the ring $\mathbb{Z}[\sqrt{D}] = \{ a + b\sqrt{D} : a, b \in \mathbb{Z} \}$.

## Explanations

### L1 human

A quadratic integer is a root of a quadratic equation with integer coefficients and leading coefficient $1$.
Numbers such as $\sqrt{2}$, $i$ and $1 + \sqrt{3}$ are examples. They extend the integers in the same way that $\sqrt{2}$ extends the rationals.
</document>

<document source="https://math2.tamnd.com/node/01VQ7C4FTB.md" tag="01VQ7C4FTB" kind="statement">
# Uniqueness of the algebraic closure

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], and let $L$ and $L'$ be [[algebraic closures|0MCXX29HWY]] of $K$.
Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

## Remarks

The proof uses Zorn's lemma on the partial isomorphisms between subfields of $L$ and $L'$ that fix $K$.
The isomorphism is not unique in general. For example, complex conjugation and the identity are two such isomorphisms from $\mathbb{C}$ to $\mathbb{C}$ over $\mathbb{R}$.
So one often writes "the" algebraic closure $\overline{K}$.
The statement is an infinite form of the [[uniqueness of splitting fields|069A3BC3EP]].

## Explanations

### L1 human

Two algebraic closures of the same field are the same field with different names for the elements.
There can be many ways to match them, so the matching is not part of the structure.
</document>

<document source="https://math2.tamnd.com/node/01VT24J1SD.md" tag="01VT24J1SD" kind="statement">
# Cantor set is closed

## Statement

The [[Cantor set|0BXNDZFQME]] $C = \bigcap_{n \ge 0} C_n$ is a [[closed subset|0G5AV9Q4RR]] of $\mathbb{R}$.

## Remarks

Proof: each $C_n$ is a union of $2^n$ closed intervals, so it is closed. An [[intersection of closed sets is closed|09YMV8H4X5]].
Since $C \subseteq [0, 1]$, the set $C$ is also bounded, and thus compact by the [[Heine–Borel theorem|07PFYV65MB]].
Each endpoint of an interval of some $C_n$ lies in $C$. For example, $\tfrac{1}{3}$ and $\tfrac{2}{3}$ are in $C$.

## Explanations

### L1 human

At each step we keep a finite union of closed intervals, which is a closed set.
Closed sets stay closed under intersections, so the limit set is closed.
</document>

<document source="https://math2.tamnd.com/node/01VVP10MWJ.md" tag="01VVP10MWJ" kind="definition">
# Connected component

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A connected component of $G$ is an equivalence class of the vertices of $G$ under [[reachability|0J8XE7W68F]].
The subgraph induced by a connected component is also called a connected component.

## Remarks

The connected components partition the vertex set. Each component, as an induced subgraph, is [[connected|0FFAYNMAJZ]].
A graph with $n$ vertices and $m$ edges has at least $n - m$ components.
A graph with no edges on $n$ vertices has $n$ components.

## Explanations

### L1 human

The connected components are the separate pieces of a graph.
Two vertices are in the same piece exactly when a walk joins them.
</document>

<document source="https://math2.tamnd.com/node/01W6RKJP01.md" tag="01W6RKJP01" kind="statement">
# Arguments add under multiplication

## Statement

Let $z$ and $w$ be complex numbers with $z \neq 0$ and $w \neq 0$.
If $\theta$ is an [[argument|0ZZRSX91QR]] of $z$ and $\varphi$ is an argument of $w$, then $\theta + \varphi$ is an argument of $z w$.
Thus $\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}$.

## Remarks

Proof: write $z = |z| e^{i \theta}$ and $w = |w| e^{i \varphi}$. Then $z w = |z| |w| e^{i (\theta + \varphi)}$ by the [[product of exponentials|0WM7DJCM6G]].
For the principal values, the equality can fail. For example, $\operatorname{Arg}(-1) + \operatorname{Arg}(-1) = 2\pi$, but $\operatorname{Arg}((-1)(-1)) = 0$.

## Explanations

### L1 human

When you multiply two complex numbers, the angles add.
For example, $i$ points at $90$ degrees, and $i \cdot i = -1$ points at $180$ degrees.
</document>

<document source="https://math2.tamnd.com/node/01WHD9T0V6.md" tag="01WHD9T0V6" kind="definition">
# Central binomial coefficient

## Definition

Let $n$ be a natural number.
The $n$-th central binomial coefficient is the [[binomial coefficient|092GEF38YD]]
$$\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.$$

## Remarks

The first values, for $n = 0, 1, 2, 3, 4$, are $1, 2, 6, 20, 70$.
It is the middle entry of row $2n$ of [[Pascal's triangle|05DME4TZ3K]], and the largest entry of that row.
It counts the [[lattice paths|040WY0TT2G]] from $(0, 0)$ to $(n, n)$ with unit steps right and up. The [[closed formula for Catalan numbers|0NNDQVJ54Q]] divides it by $n + 1$.

## Explanations

### L1 human

The central binomial coefficient is the number in the middle of an even row of Pascal's triangle.
For $n = 2$ it is $\binom{4}{2} = 6$: the ways to choose 2 of 4 objects.
</document>

<document source="https://math2.tamnd.com/node/01WPV9PHN7.md" tag="01WPV9PHN7" kind="statement">
# Legendre's formula

## Statement

Let $p$ be a prime number and let $n$ be a natural number. Let $\nu_p$ be the [[p-adic valuation|04VCPS30ZE]].
Then
$$\nu_p(n!) = \sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor.$$
The sum is finite, because $\lfloor n / p^i \rfloor = 0$ when $p^i > n$.

## Remarks

An equivalent form: $\nu_p(n!) = \dfrac{n - s_p(n)}{p - 1}$, where $s_p(n)$ is the sum of the digits of $n$ in base $p$.
For example, $\nu_2(10!) = 5 + 2 + 1 = 8$, and $\nu_5(100!) = 20 + 4 = 24$. So $100!$ ends with $24$ zeros.
The proof counts, for each $i$, the multiples of $p^i$ among $1, \dots, n$.

## Explanations

### L1 human

To count the factors $p$ in $n!$, count the multiples of $p$ up to $n$. Then add the multiples of $p^2$, which give a second factor, and so on.
For $10!$ and $p = 3$: there are $3$ multiples of $3$ and $1$ multiple of $9$, so $3^4$ is the largest power of $3$ that divides $10!$.
</document>

<document source="https://math2.tamnd.com/node/01WQZK97Z5.md" tag="01WQZK97Z5" kind="statement">
# Cassini's identity

## Statement

Let $n$ be a positive integer, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n-1} F_{n+1} - F_n^2 = (-1)^n.$$

## Remarks

Examples: for $n = 5$, $F_4 F_6 - F_5^2 = 3 \cdot 8 - 25 = -1$. For $n = 6$, $F_5 F_7 - F_6^2 = 5 \cdot 13 - 64 = 1$.
A short proof takes determinants in the matrix identity $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}$.
Catalan's identity $F_n^2 - F_{n-r} F_{n+r} = (-1)^{n-r} F_r^2$ generalizes it. The case $r = 1$ is Cassini's identity.

## Explanations

### L1 human

The product of the two neighbours of a Fibonacci number differs from its square by exactly $1$.
The sign alternates: $1 \cdot 3 - 2^2 = -1$, then $2 \cdot 5 - 3^2 = 1$, then $3 \cdot 8 - 5^2 = -1$.
</document>

<document source="https://math2.tamnd.com/node/01X5CVV1Q5.md" tag="01X5CVV1Q5" kind="example">
# Geometric series

## Example

Let $a$ and $r$ be real numbers with $|r| < 1$.
The geometric series $\sum_{n=0}^{\infty} a r^n$ converges, and
$$\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}.$$
The [[sum of a finite geometric series|05APWSX53S]] gives the partial sums $s_N = a (1 - r^{N+1})/(1 - r)$.
Since [[$r^{N+1} \to 0$|0Q7K4AXE53]], the partial sums tend to $a/(1 - r)$.

## Remarks

For $a \ne 0$ and $|r| \ge 1$, the series diverges by the [[term test|0VVY4H92G2]], because $|a r^n| \ge |a|$.
For $a = 1$ and $r = 1/2$, it gives $1 + 1/2 + 1/4 + \cdots = 2$.
The decimal $0.333\ldots$ is the geometric series with $a = 3/10$ and $r = 1/10$, and its sum is $1/3$.
The terms form a [[geometric sequence|0MNM54WB99]].

## Explanations

### L1 human

Walk half of the way to a wall, then half of the rest, and so on.
The distances $1/2 + 1/4 + 1/8 + \cdots$ add up to the whole distance $1$.
</document>

<document source="https://math2.tamnd.com/node/01X8YSDTBW.md" tag="01X8YSDTBW" kind="definition">
# Identically distributed random variables

## Definition

Let $(E, \mathcal{E})$ be a set with a σ-algebra.
Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ with values in $E$.
Let $Y$ be a random variable on a probability space $(\Omega', \mathcal{F}', P')$ with values in $E$.
$X$ and $Y$ are identically distributed if they have the same [[law|0TST4MHA3A]]: $P(X \in B) = P'(Y \in B)$ for each $B \in \mathcal{E}$.
One writes $X \overset{d}{=} Y$.

## Remarks

For real random variables, this holds if and only if $X$ and $Y$ have the same cumulative distribution function.
Identically distributed random variables need not be equal. For a fair coin, the indicator $X$ of heads and the indicator $1 - X$ of tails are identically distributed, but they are never equal.
If $X \overset{d}{=} Y$ and $g$ is a measurable function, then $g(X) \overset{d}{=} g(Y)$.

## Explanations

### L1 human

Two random variables are identically distributed when they behave the same way statistically.
The results of two different fair dice are identically distributed, although the dice show different numbers in most throws.
</document>

<document source="https://math2.tamnd.com/node/01XAHPADBA.md" tag="01XAHPADBA" kind="concept">
# Moment

## Summary

The moments of a random variable are the expected values of its powers. The first moment is the mean.
The node [[moment|0PH0VHTQS0]] gives the precise definition for random variables.
</document>

<document source="https://math2.tamnd.com/node/01XAY17QA3.md" tag="01XAY17QA3" kind="definition">
# Uniformly Cauchy sequence of functions

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$.
The sequence $(f_n)$ is uniformly Cauchy on $S$ if for every $\varepsilon > 0$ there is $N$ such that
$$|f_n(x) - f_m(x)| < \varepsilon \quad \text{for all } m, n \ge N \text{ and all } x \in S.$$

## Remarks

For each $x$, the sequence $(f_n(x))$ is then a [[Cauchy sequence|0QZMTN5AXX]]. So it has a limit $f(x)$, by the [[Cauchy criterion|0ZA4AX4MW3]].
A sequence of real functions is uniformly Cauchy on $S$ if and only if it [[converges uniformly|08HJRTG5N7]] on $S$.
This criterion proves uniform convergence without knowing the limit. Mathlib defines it as `UniformCauchySeqOn`.

## Explanations

### L1 human

A uniformly Cauchy sequence has graphs that get close to each other, uniformly in $x$.
Such a sequence always converges uniformly to some function.
</document>

<document source="https://math2.tamnd.com/node/01XM7JEKHY.md" tag="01XM7JEKHY" kind="concept">
# Constructive proof

## Summary

A constructive proof of an existence statement $\exists x \, P(x)$ gives a specific object $a$, or a method to find it, and proves $P(a)$.
A non-constructive proof shows that an object exists without giving it, for example by [[proof by contradiction|0J0XBKG64S]].

## Remarks

A classical non-constructive example: there are irrational numbers $a$ and $b$ such that $a^b$ is rational. Consider $\sqrt{2}^{\sqrt{2}}$. It is rational or irrational, and each case gives such $a$ and $b$.
Intuitionistic logic accepts only constructive proofs of existence.

## Explanations

### L1 human

A constructive proof that a solution exists shows you the solution.
A non-constructive proof only shows that "no solution" is impossible.
Both are correct in classical mathematics, but the constructive proof tells you more.
</document>

<document source="https://math2.tamnd.com/node/01YYRV7S9Z.md" tag="01YYRV7S9Z" kind="concept">
# Limit at infinity

## Summary

The limit of a function at infinity is the value that the function approaches as its argument grows without bound. It describes horizontal asymptotes of graphs. The node [[limit at infinity|0Y12QYTJK3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/01ZQQDCERP.md" tag="01ZQQDCERP" kind="definition">
# Open set

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]].
A subset $U$ of $X$ is open in $X$ when $U \in \tau$.
A subset of a [[metric space|063W5JMY85]] is open when it is open for the [[metric topology|0A6YRNZB2R]].

## Remarks

In a metric space $(X, d)$, a set $U$ is open if and only if for each $x \in U$ there is $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$. This is the statement [[open sets of a metric space|0A26X79RTB]].
In $\mathbb{R}$, each open interval $(a, b)$ is open, but $[0, 1)$ is not open: no ball about $0$ lies in $[0, 1)$.
The empty set and $X$ are always open.

## Explanations

### L1 human

In a metric space, an open set has no edge points. From each of its points, you can move a little in any direction and stay in the set.
The interval $(0, 1)$ is open, because each point has some room on both sides.
</document>

<document source="https://math2.tamnd.com/node/01ZTQBK64Q.md" tag="01ZTQBK64Q" kind="statement">
# Adding a multiple of one row to another preserves the determinant

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Let $i \ne j$, and let $B$ be the matrix that results when $c$ times row $j$ of $A$ is added to row $i$.
Then $\det B = \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

By linearity in row $i$, $\det B = \det A + c \det A'$, where $A'$ has two equal rows; so $\det A' = 0$.
This is the reason why [[Gaussian elimination|0JVJ2WCXZH]] computes determinants efficiently.
The same holds for columns.

## Explanations

### L1 human

Adding a multiple of one row to another row shears the box spanned by the rows.
A shear does not change the volume.
So this row operation keeps the determinant.
</document>

<document source="https://math2.tamnd.com/node/01ZWX875AT.md" tag="01ZWX875AT" kind="statement">
# Law of the unconscious statistician

## Statement

Let $X$ be a random variable on $(\Omega, \mathcal{F}, P)$ with values in $(E, \mathcal{E})$ and with [[law|0TST4MHA3A]] $\mu_X$. Let $g \colon E \to \mathbb{R}$ be measurable.
Then $g(X)$ is integrable if and only if $\int_E |g| \, d\mu_X < \infty$. In that case,
$$E[g(X)] = \int_E g(x) \, \mu_X(dx).$$
In particular, if $X$ is discrete with [[mass function|0WF4EJP91W]] $p$, then $E[g(X)] = \sum_x g(x) \, p(x)$. If $X$ is real with [[density|04YG690N5V]] $f$, then $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx$.

## Remarks

The formula also holds for $g \ge 0$ without any integrability condition, with both sides in $[0, \infty]$.
The proof checks the formula first for indicator functions $g = 1_B$, where it says $P(X \in B) = \mu_X(B)$. Then it extends to simple, nonnegative and integrable $g$.
The point is that one does not need the law of $g(X)$ to compute $E[g(X)]$.

## History

The name refers to statisticians who use the formula as the definition of $E[g(X)]$. They are "unconscious" that it is a theorem that needs a proof.

## Explanations

### L1 human

To find the average of $g(X)$, you can average $g(x)$ over the values $x$ of $X$ with their probabilities.
For a fair die, $E[X^2] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6$.
</document>

<document source="https://math2.tamnd.com/node/0201Q4P2B5.md" tag="0201Q4P2B5" kind="statement">
# Carathéodory's theorem

## Statement

Let $P$ be a subset of $\mathbb{R}^d$, and let $x$ be a point of its [[convex hull|0FQSDC2EQ2]] $\operatorname{conv}(P)$.
Then there are at most $d + 1$ points $p_1, \dots, p_k$ of $P$, with $k \leq d + 1$, such that $x \in \operatorname{conv}\{p_1, \dots, p_k\}$.
Equivalently, $x$ is a convex combination of at most $d + 1$ points of $P$.

## Remarks

Thus $\operatorname{conv}(P)$ is the union of the simplices with vertices in $P$, of dimension at most $d$. Mathlib states this form as `convexHull_eq_union`.
The bound $d + 1$ is the best possible in general. Let $P$ be the three vertices of a triangle in $\mathbb{R}^2$. A point inside the triangle is not in the hull of two points of $P$.
The proof only uses the order and the field operations of $\mathbb{R}$. So it works over each ordered field.

## History

Carathéodory proved the theorem in 1911 for compact sets $P$. Steinitz extended it to arbitrary sets in 1914.

## Explanations

### L1 human

In the plane, a point in the convex hull of many points is always inside a triangle with corners from those points. You never need more than three corners. In space, four corners, the vertices of a tetrahedron, are always enough.
</document>

<document source="https://math2.tamnd.com/node/020422MDQF.md" tag="020422MDQF" kind="statement">
# Euler's theorem on Eulerian circuits

## Statement

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]] or [[multigraph|01M5T0KDG7]] with at least one edge.
Then $G$ has an Eulerian circuit if and only if every vertex of $G$ has even [[degree|0T0F6CCS9W]].

## Remarks

The condition is necessary by the theorem on [[degrees along an Eulerian trail|0CDVDGWGEG]].
A consequence: a finite connected graph has an [[Eulerian trail|0NNYDD1AVR]] that is not closed if and only if exactly two vertices have odd degree.
Hierholzer's algorithm builds an Eulerian circuit: it joins closed trails until every edge is used.

## History

Leonhard Euler showed the necessity of the condition, and he stated without proof that it is sufficient.
The first complete proof is by Carl Hierholzer. It was published after his death, in 1873.

## Explanations

### L1 human

You can draw a connected graph in one closed stroke exactly when every vertex has an even number of edges.
Each time the pen passes a vertex, it uses one edge to arrive and one edge to leave.
</document>

<document source="https://math2.tamnd.com/node/020K0X7HVR.md" tag="020K0X7HVR" kind="statement">
# Existence of prime factorization

## Statement

Each positive integer $n$ has a [[prime factorization|0AEKDFCYFA]].
That is, there are [[prime numbers|0SN0MAEKYX]] $p_1, \dots, p_k$, with $k \ge 0$, such that $n = p_1 p_2 \cdots p_k$.

## Remarks

The proof uses strong induction on $n$. If $n = 1$, take $k = 0$. If $n$ is prime, take $k = 1$. Otherwise $n = a b$ with $1 < a, b < n$, and we join the factorizations of $a$ and $b$.
The [[fundamental theorem of arithmetic|084HH1NDCB]] adds that the factorization is unique, up to the order of the factors.

## Explanations

### L1 human

Split each number that is not prime into two smaller factors. Continue until all the pieces are prime.
The pieces always get smaller, so the splitting stops.
For example, $84 = 2 \cdot 42 = 2 \cdot 2 \cdot 21 = 2 \cdot 2 \cdot 3 \cdot 7$.
</document>

<document source="https://math2.tamnd.com/node/020M04FN1H.md" tag="020M04FN1H" kind="concept">
# Matrix exponential

## Summary

The exponential of a real or complex square matrix $A$ is the sum of the convergent power series $I + A + A^2/2! + A^3/3! + \cdots$.
</document>

<document source="https://math2.tamnd.com/node/020Q06TYYH.md" tag="020Q06TYYH" kind="concept">
# Totally bounded set

## Summary

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size.
Compact sets are exactly the totally bounded sets that are complete.
</document>

<document source="https://math2.tamnd.com/node/020Z5SEN27.md" tag="020Z5SEN27" kind="definition">
# Free group

## Definition

Let $S$ be a set.
The free group $F_S$ on $S$ is the set of [[reduced words|0KBTP8Q2FS]] on $S$.
The product of two reduced words is their concatenation, followed by reduction.
A [[group|G5S1M9]] $G$ is free when it is [[isomorphic|08MG7H8JW5]] to $F_S$ for some set $S$.

## Remarks

The identity of $F_S$ is the empty word, and the inverse of $t_1 \cdots t_n$ is $t_n^{-1} \cdots t_1^{-1}$.
The set $S$ is a basis of $F_S$. Also $F_S \cong F_T$ if and only if $S$ and $T$ have the same cardinality, which is the rank.
The free group of rank $1$ is the [[infinite cyclic group|02B1T50JD1]], and for $|S| \geq 2$ the centre of $F_S$ is trivial.
A nontrivial finite group is never free, because each element of a basis has infinite order.

## Explanations

### L1 human

A free group has generators but no relations that you do not need.
Two words give the same element only when the group axioms force it, as in $s t = s u u^{-1} t$.
Each group is a quotient of a free group, so free groups are the starting point for presentations.
</document>

<document source="https://math2.tamnd.com/node/021WEM680W.md" tag="021WEM680W" kind="definition">
# Definite integral

## Definition

Let $I$ be an interval, let $f \colon I \to \mathbb{R}$, and let $a, b \in I$.
Assume that $f$ is Riemann integrable on the closed interval between $a$ and $b$.
The definite integral of $f$ from $a$ to $b$ is the number $\int_a^b f(x) \, dx$ defined as follows.
If $a < b$, it is the [[Riemann integral|0QM4TXT3DR]] of $f$ over $[a, b]$.
If $a = b$, it is $0$. If $a > b$, it is $-\int_b^a f(x) \, dx$.

## Remarks

With this sign convention, $\int_a^b f + \int_b^c f = \int_a^c f$ holds for all $a, b, c$ in $I$.
The numbers $a$ and $b$ are the limits of integration, and $f$ is the integrand.
Mathlib writes the definite integral as `∫ x in a..b, f x`, through `intervalIntegral`.

## Explanations

### L1 human

The definite integral of $f$ from $a$ to $b$ is a number. It measures signed area under the graph between $a$ and $b$.
The order of the limits matters: swapping them changes the sign.
</document>

<document source="https://math2.tamnd.com/node/021XJAQVDC.md" tag="021XJAQVDC" kind="statement">
# Darboux and Riemann integrals agree

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if its upper and lower [[Darboux integrals|09FQT057T7]] are equal.
In this case the common value is the [[Riemann integral|0QM4TXT3DR]] $\int_a^b f(x)\, dx$.

## Remarks

Each Riemann sum lies between the lower and upper Darboux sums. This gives one direction.
For the other direction, one shows that Darboux sums for partitions of small mesh are close to the Darboux integrals.
So the Riemann and the Darboux approaches define the same integral.

## Explanations

### L1 human

There are two standard ways to define the integral, with Riemann sums or with upper and lower sums.
They give the same integrable functions and the same values.
</document>

<document source="https://math2.tamnd.com/node/0225QPWRRE.md" tag="0225QPWRRE" kind="definition">
# Taylor series

## Definition

Let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be [[smooth|0BE3QDH73A]].
The Taylor series of $f$ at $a$ is the [[power series|0VP56B4EMX]]
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n.$$

## Remarks

Its partial sums are the [[Taylor polynomials|0HFSFVMWQB]] of $f$ at $a$.
The Taylor series may diverge for each $x \ne a$. It may also converge to a function other than $f$, as for the [[smooth non-analytic function|09RTYYHXKA]].
The Taylor series converges to $f$ near $a$ exactly when $f$ is [[analytic|0JD5ZNW0F4]] at $a$.
For $a = 0$ it is the [[Maclaurin series|043SJ5RF59]].

## Explanations

### L1 human

The Taylor series is the Taylor polynomial of infinite degree.
It uses all the derivatives of $f$ at one point.
</document>

<document source="https://math2.tamnd.com/node/022RSX714V.md" tag="022RSX714V" kind="concept">
# Order of cardinals

## Summary

The order of cardinals compares the sizes of sets by injections.
</document>

<document source="https://math2.tamnd.com/node/0230ECX1Q5.md" tag="0230ECX1Q5" kind="definition">
# Well-founded relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is well-founded when each nonempty subset $S$ of $A$ has an element $m$ such that no $s \in S$ satisfies $s \mathrel{R} m$.

## Remarks

The relation $<$ on $\mathbb{N}$ is well-founded. The relation $<$ on $\mathbb{Z}$ is not, because $\mathbb{Z}$ itself has no minimal element.
If $R$ is well-founded, there is no sequence $x_0, x_1, x_2, \dots$ with $x_{n+1} \mathrel{R} x_n$ for each $n$. The converse needs the axiom of dependent choice.
Well-founded relations allow proofs by [[transfinite induction|0ZHQMQ2C1N]].

## Explanations

### L1 human

In a well-founded relation, you cannot go down forever.
Each strictly decreasing sequence of natural numbers stops after finitely many steps.
</document>

<document source="https://math2.tamnd.com/node/023VY493K0.md" tag="023VY493K0" kind="concept">
# Euclidean algorithm in a Euclidean domain

## Summary

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]] with Euclidean function $\delta$ and let $a, b \in R$.
The Euclidean algorithm sets $r_0 = a$ and $r_1 = b$. While $r_k \neq 0$, it divides $r_{k-1}$ by $r_k$ and lets $r_{k+1}$ be the remainder.
The values $\delta(r_k)$ decrease strictly, so the algorithm stops with some $r_{n+1} = 0$.
The last nonzero remainder $r_n$ is a [[greatest common divisor|0F7KNJ1ZYN]] of $a$ and $b$, if $b \neq 0$.

## Remarks

The algorithm works because the common divisors of $r_{k-1}$ and $r_k$ are the common divisors of $r_k$ and $r_{k+1}$.
In $\mathbb{Z}$: $252 = 1 \cdot 198 + 54$, $198 = 3 \cdot 54 + 36$, $54 = 1 \cdot 36 + 18$, $36 = 2 \cdot 18$. So $\gcd(252, 198) = 18$.
The extended algorithm also gives the coefficients of [[Bézout's identity|0VY3WPP9Y3]].

## Explanations

### L1 human

To find the gcd of two numbers, replace the larger one by its remainder modulo the smaller one, and repeat.
When the remainder is zero, the last divisor is the gcd.
The same steps work in every Euclidean domain, for example for polynomials over a field.
</document>

<document source="https://math2.tamnd.com/node/02486SD328.md" tag="02486SD328" kind="definition">
# Left inverse

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
A left inverse of $f$ is a function $g \colon B \to A$ such that $g \circ f = \mathrm{id}_A$, that is, $g(f(a)) = a$ for each $a \in A$.

## Remarks

A function with a nonempty domain has a left inverse exactly when it is [[injective|0PVG1CRAJN]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].
A left inverse is in general not unique. Its values on $B \setminus f(A)$ can be any elements of $A$.
If $g$ is a left inverse of $f$, then $f$ is a [[right inverse|0DDPZAG5ER]] of $g$.

## Explanations

### L1 human

Let $f \colon \mathbb{N} \to \mathbb{N}$ be $f(n) = n + 1$.
The function $g$ with $g(n) = n - 1$ for $n \ge 1$ and $g(0) = 0$ is a left inverse: $g(f(n)) = n$.
But $f(g(0)) = 1 \neq 0$, so $g$ is not an inverse on both sides.
</document>

<document source="https://math2.tamnd.com/node/025158VEYZ.md" tag="025158VEYZ" kind="definition">
# Reflection

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$, with [[orthogonal projection|0B8KZRT28M]] $P_W$.
The *reflection* in $W$ is the linear map
$$R_W \colon V \to V, \qquad R_W(v) = 2 P_W(v) - v.$$

## Remarks

$R_W$ fixes each vector of $W$, negates each vector of $W^\perp$, and satisfies $R_W^2 = \id_V$.
It is a linear isometry.
For a hyperplane $W = a^\perp$ with $a \ne 0$, the formula is $R_W(v) = v - 2 \frac{\langle v, a \rangle}{\langle a, a \rangle} a$.
In $\R^n$, the matrix of a hyperplane reflection is orthogonal with determinant $-1$.

## Explanations

### L1 human

A reflection is a mirror image in a subspace.
Vectors in the mirror stay where they are, and vectors perpendicular to it change direction.
Reflecting twice brings every vector back.
</document>

<document source="https://math2.tamnd.com/node/025PKW8AMR.md" tag="025PKW8AMR" kind="statement">
# Uniqueness of solutions for Lipschitz vector fields

## Statement

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous.
Assume that $f$ is locally Lipschitz in $x$. That is, each point of $D$ has a neighbourhood $V \subseteq D$ and a constant $K \ge 0$ such that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ whenever $(t, x), (t, y) \in V$.
Let $I$ be an interval, and let $\gamma_1, \gamma_2 : I \to \mathbb{R}^n$ be two [[solutions|04SW7WECS8]] of $x' = f(t, x)$ whose graphs lie in $D$.
If $\gamma_1(t_0) = \gamma_2(t_0)$ for one $t_0 \in I$, then $\gamma_1(t) = \gamma_2(t)$ for all $t \in I$.

## Remarks

So an [[initial value problem|075VBY4K4X]] for such an $f$ has at most one solution on each interval that contains $t_0$.
Proof idea: the function $u(t) = \|\gamma_1(t) - \gamma_2(t)\|$ satisfies $u(t) \le K \left| \int_{t_0}^{t} u(s)\, ds \right|$ near $t_0$. Then [[Grönwall's inequality|07EJHQYR9A]] gives $u = 0$. A connectedness argument extends this to all of $I$.
A $C^1$ function $f$ is locally Lipschitz in $x$.
Without the Lipschitz condition, uniqueness can fail. See the [[non-uniqueness example|00K7ASZ1PR]].

## Explanations

### L1 human

Two solutions of a Lipschitz equation that meet at one time are the same solution.
So the graphs of different solutions never cross or touch.
</document>

<document source="https://math2.tamnd.com/node/025YZN3VJW.md" tag="025YZN3VJW" kind="concept">
# Conjunction

## Summary

Conjunction is the logical operation "and".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true only when both parts are true.
</document>

<document source="https://math2.tamnd.com/node/0262JP0NV9.md" tag="0262JP0NV9" kind="concept">
# First integral

## Summary

A first integral, or constant of motion, of a system of differential equations is a function of the state that stays constant along each solution.
Energy, linear momentum and angular momentum are common examples in mechanics. See the [[definition|0SS1P1YC9A]].
</document>

<document source="https://math2.tamnd.com/node/026BJBQNGJ.md" tag="026BJBQNGJ" kind="statement">
# A function with nonnegative derivative is increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$.
If $f'(x) \ge 0$ for each interior point $x$ of $I$, then $f$ is increasing on $I$. That is, $f(x) \le f(y)$ for $x \le y$ in $I$.

## Remarks

The proof uses the [[mean value theorem|0BARQY52N5]].
For differentiable $f$ on an open interval, the converse holds too: if $f$ is increasing, then $f' \ge 0$.
Mathlib states it as `monotoneOn_of_deriv_nonneg`.

## Explanations

### L1 human

If the slope is never negative, the graph never goes down.
It may stay flat on some parts.
</document>

<document source="https://math2.tamnd.com/node/026FDZFRXS.md" tag="026FDZFRXS" kind="statement">
# Uniformly continuous functions map Cauchy sequences to Cauchy sequences

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be [[uniformly continuous|0Q4DR37DVE]].
Let $(x_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] in $D$.
Then $(f(x_n))$ is a Cauchy sequence.

## Remarks

Given $\varepsilon > 0$, take $\delta$ from uniform continuity, and then $N$ with $|x_m - x_n| < \delta$ for $m, n \ge N$.
Continuity alone is not enough. The function $1/x$ on $(0, 1)$ maps the Cauchy sequence $1/n$ to the sequence $n$.
Mathlib states it as `UniformContinuous.comp_cauchySeq`.

## Explanations

### L1 human

Points of a Cauchy sequence crowd closer and closer together.
A uniformly continuous function keeps their images crowding together too.
</document>

<document source="https://math2.tamnd.com/node/026HPGYGS4.md" tag="026HPGYGS4" kind="statement">
# Parallelogram law

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2.$$

## Remarks

The proof expands both squares with the inner product; the mixed terms cancel.
By the Jordan–von Neumann theorem, a norm comes from an inner product if and only if it satisfies this law.
The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails it for $x = (1, 0)$ and $y = (0, 1)$, so it comes from no inner product.

## Explanations

### L1 human

In a parallelogram, the squares of the two diagonals add up to the squares of the four sides.
The vectors $x + y$ and $x - y$ are the diagonals of the parallelogram with sides $x$ and $y$.
In a rectangle, both diagonals have the same length, and this is the Pythagorean theorem twice.
</document>

<document source="https://math2.tamnd.com/node/026J1MG2NW.md" tag="026J1MG2NW" kind="concept">
# Flux through a surface

## Summary

The flux of a vector field through an [[oriented surface|0TD904MG06]] measures how much of the field passes through the surface. It counts flow in the direction of the chosen normal.
It is the [[surface integral|065HE5V98E]] of the normal component of the field. For a velocity field of a fluid, it is the volume of fluid that crosses the surface per unit time.
</document>

<document source="https://math2.tamnd.com/node/027JF1FBB0.md" tag="027JF1FBB0" kind="definition">
# Density of the normal distribution

## Definition

Let $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.
The density of the normal distribution with mean $\mu$ and variance $\sigma^2$ is the function $\varphi_{\mu, \sigma^2} \colon \mathbb{R} \to (0, \infty)$ with
$$\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right).$$
The standard normal density is $\varphi(x) = \varphi_{0, 1}(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$.

## Remarks

The integral of $\varphi_{\mu, \sigma^2}$ over $\mathbb{R}$ is $1$. This follows from the Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$.
So $\varphi_{\mu, \sigma^2}$ is a [[probability density function|04YG690N5V]], the density of the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$.
The function is symmetric about $\mu$, has its maximum $1/\sqrt{2\pi\sigma^2}$ at $x = \mu$, and has inflection points at $\mu \pm \sigma$.
It satisfies $\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right)$ with $\sigma = \sqrt{\sigma^2}$.

## Explanations

### L1 human

This is the formula of the bell curve.
The exponent $-(x - \mu)^2 / (2\sigma^2)$ makes the curve fall very fast away from $\mu$, and the constant in front makes the total area equal to $1$.
</document>

<document source="https://math2.tamnd.com/node/027K9RMZ43.md" tag="027K9RMZ43" kind="definition">
# Derivative function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be [[differentiable|0A61B60B5R]] on $I$.
The derivative function of $f$ is the function $f' \colon I \to \mathbb{R}$ that maps each $x \in I$ to the [[derivative|0F1WR9C83K]] $f'(x)$ of $f$ at $x$.

## Remarks

Other notations are $\frac{df}{dx}$ and $D f$.
For example, the derivative function of $x^3$ is $3 x^2$.
The derivative function need not be continuous. See [[x^2 sin(1/x)|0WDAKKN4NC]].
Mathlib defines `deriv f` on all of $\mathbb{R}$, with value $0$ at points where $f$ is not differentiable.

## Explanations

### L1 human

The derivative function collects the slopes of $f$ at all points into one new function.
We can then differentiate it again.
</document>

<document source="https://math2.tamnd.com/node/027KDW2K72.md" tag="027KDW2K72" kind="definition">
# Similar matrices

## Definition

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$.
The matrix $B$ is *similar* to $A$ when there is an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ over $F$ with
$$B = P^{-1} A P.$$

## Remarks

Similarity is an equivalence relation on $M_n(F)$.
By the [[change of basis formula|0MQXVZEBMT]], two matrices are similar if and only if they represent the same endomorphism of $F^n$ in two bases.
Similar matrices have the same [[trace|0XC1RDBY7A]], [[determinant|00KDZRW4C0]], [[characteristic polynomial|0VSSPAFM2C]], [[rank|0ZC9YTXZRY]] and [[eigenvalues|0BQAA2BK3Q]].
The matrices $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ have the same characteristic polynomial but are not similar.

## Explanations

### L1 human

Similar matrices describe the same linear map, seen from two different bases.
Properties of the map itself, like trace and determinant, are the same for both matrices.
A main goal of linear algebra is to find a simple matrix in each similarity class.
</document>

<document source="https://math2.tamnd.com/node/027MBSVPY3.md" tag="027MBSVPY3" kind="definition">
# Reflection in a line or plane

## Definition

Let $a \in \mathbb{R}^n$ be a nonzero vector and $c$ a real number, and let $\cdot$ be the dot product. Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be the hyperplane with the [[normal vector|0WQJJCCTC8]] $a$.
The reflection in $H$ is the map
$$\sigma_H \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad \sigma_H(x) = x - 2\, \frac{x \cdot a - c}{a \cdot a}\, a.$$
For $n = 2$, $H$ is a line; for $n = 3$, $H$ is a plane.

## Remarks

The map $\sigma_H$ is an indirect [[Euclidean isometry|0J1N3WTKM4]]. Its fixed points are the points of $H$, and $\sigma_H \circ \sigma_H$ is the identity.
For $x \notin H$, the hyperplane $H$ is the [[perpendicular bisector|055VNYT0PB]] of $x$ and $\sigma_H(x)$.
The linear part of $\sigma_H$ is an orthogonal matrix with determinant $-1$ and the eigenvalues $-1, 1, \dots, 1$.
Reflections in hyperplanes generate the Euclidean group.

## Explanations

### L1 human

A reflection is a mirror image. Each point goes to the point on the other side of the mirror, at the same distance from it. The reflection in the line $y = x$ in the plane sends $(x, y)$ to $(y, x)$.
</document>

<document source="https://math2.tamnd.com/node/027ZQS04SJ.md" tag="027ZQS04SJ" kind="statement">
# Quotients of solvable groups are solvable

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $\varphi \colon G \to K$ be a surjective [[group homomorphism|0JYTG5K9Y4]].
Then $K$ is solvable.
In particular, $G/N$ is solvable for each [[normal subgroup|0CB0DX5BVV]] $N$ of $G$.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $\varphi(G^{(i)}) = K^{(i)}$ for each $i$, because $\varphi$ maps commutators onto commutators.
So $G^{(n)} = \{e\}$ implies $K^{(n)} = \{e\}$.
The second part follows with the projection $G \to G/N$.

## Explanations

### L1 human

A surjective homomorphism sends the commutators of $G$ onto the commutators of $K$.
So the derived series of $K$ is the image of the derived series of $G$.
If one reaches the identity, the image does too.
</document>

<document source="https://math2.tamnd.com/node/02879QH7BY.md" tag="02879QH7BY" kind="concept">
# Associate elements

## Summary

Two elements of a commutative ring are associates when one is a unit times the other.
The [[definition of associate elements|053E86NWVK]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/028XEQCE88.md" tag="028XEQCE88" kind="statement">
# Angle addition formulas

## Statement

For all real numbers $x$ and $y$,
$$\sin(x + y) = \sin x \cos y + \cos x \sin y,$$
$$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$

## Remarks

One proof multiplies the power series with the [[Cauchy product|0AN28XQ45Z]] and uses the binomial theorem.
Another proof fixes $y$ and shows that the difference of the two sides solves $f'' = -f$ with $f(0) = f'(0) = 0$.
With $y = x$, they give $\sin 2x = 2 \sin x \cos x$ and $\cos 2x = \cos^2 x - \sin^2 x$. Mathlib states them as `Real.sin_add` and `Real.cos_add`.

## Explanations

### L1 human

These formulas compute the sine and cosine of a sum of two angles from the sines and cosines of the angles.
They are the most important trigonometric identities.
</document>

<document source="https://math2.tamnd.com/node/028Z8WC5Y9.md" tag="028Z8WC5Y9" kind="definition">
# Exponential distribution

## Definition

Let $\lambda > 0$.
The exponential distribution with rate $\lambda$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \ge 0, \\ 0 & \text{if } x < 0. \end{cases}$$
One writes $X \sim \operatorname{Exp}(\lambda)$.

## Remarks

The cumulative distribution function is $F(x) = 1 - e^{-\lambda x}$ for $x \ge 0$, so $P(X > x) = e^{-\lambda x}$.
The mean is $1/\lambda$, the variance is $1/\lambda^2$ and the median is $\ln 2 / \lambda$. Some authors use the scale $\beta = 1/\lambda$ as the parameter.
The exponential distribution is [[memoryless|0ZAZZ34WQS]]: $P(X > s + t \mid X > s) = P(X > t)$ for $s, t \ge 0$.
If $X_1, \dots, X_n$ are independent with $X_i \sim \operatorname{Exp}(\lambda_i)$, then $\min_i X_i \sim \operatorname{Exp}(\lambda_1 + \dots + \lambda_n)$.

## Explanations

### L1 human

The exponential distribution models the time until something happens when it can happen at any moment with the same intensity.
Examples are the decay of a radioactive atom and the time until the next call to a help desk.
</document>

<document source="https://math2.tamnd.com/node/02903SSA48.md" tag="02903SSA48" kind="definition">
# Flow of a vector field

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz.
For $x_0 \in U$, let $\gamma_{x_0} : I(x_0) \to U$ be the [[maximal solution|019VSXSFMQ]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ with $x(0) = x_0$.
Let $\Omega = \{(t, x_0) : x_0 \in U,\ t \in I(x_0)\}$.
The flow of $f$ is the map $\varphi : \Omega \to U$ given by $\varphi(t, x_0) = \gamma_{x_0}(t)$.
Another notation is $\varphi_t(x_0) = \varphi(t, x_0)$.

## Remarks

The flow has the group properties $\varphi_0(x) = x$ and $\varphi_s(\varphi_t(x)) = \varphi_{s + t}(x)$, whenever both sides are defined. The second property follows from [[uniqueness|025PKW8AMR]] and [[time translation|0WA7ZR95V4]].
The set $\Omega$ is open and $\varphi$ is continuous on $\Omega$.
If $\Omega = \mathbb{R} \times U$, the flow is complete. Then $t \mapsto \varphi_t$ is an action of the additive group $\mathbb{R}$ on $U$.
For a linear system $x' = A x$, the flow is $\varphi_t(x) = e^{tA} x$, the [[matrix exponential|0YC8JT513J]].

## Explanations

### L1 human

The flow moves each point of the phase space along the solution through it.
The map $\varphi_t$ says where each point is after time $t$.
Moving for time $t$ and then for time $s$ is the same as moving for time $s + t$.
</document>

<document source="https://math2.tamnd.com/node/0292EPPPT0.md" tag="0292EPPPT0" kind="statement">
# Euclid's theorem

## Statement

There are infinitely many [[prime numbers|0SN0MAEKYX]].
Equivalently, for each natural number $n$ there is a prime number $p$ with $p \ge n$.

## Remarks

Euclid's proof: let $p_1, \dots, p_k$ be a finite list of primes, and let $N = p_1 p_2 \cdots p_k + 1$. A [[prime divisor|0SPBEDVNW1]] of $N$ is not in the list, because each $p_i$ leaves the remainder $1$ when it divides $N$.
The [[divergence of the sum of the reciprocals of the primes|00RA98X2CP]] is a stronger result.
[[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]] gives infinitely many primes in each suitable arithmetic progression.

## History

The theorem is Proposition 20 of Book IX of the Elements of Euclid.

## Explanations

### L1 human

No finite list holds all the primes. Multiply all the primes of the list and add $1$.
The new number leaves the remainder $1$ on division by each prime of the list, so its prime factors are new.
For example, $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 = 30031 = 59 \cdot 509$, and $59$ and $509$ are not in the list.
</document>

<document source="https://math2.tamnd.com/node/0299VCW1TH.md" tag="0299VCW1TH" kind="concept">
# Vector field

## Summary

A vector field on a subset of $\mathbb{R}^n$ assigns a vector of $\mathbb{R}^n$ to each point of the subset.
It models quantities with a direction and a size at each point, such as the velocity of a fluid or a force field.
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<document source="https://math2.tamnd.com/node/029JBBF693.md" tag="029JBBF693" kind="concept">
# Negation

## Summary

Negation is the logical operation that changes a [[proposition|0ZGC7ZESYN]] into its opposite.
The negation of a true proposition is false, and the negation of a false proposition is true.
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<document source="https://math2.tamnd.com/node/029JQEK15B.md" tag="029JQEK15B" kind="definition">
# Planar graph

## Definition

A [[graph|0JD5C82QMB]] $G$ is planar if it has a drawing in the plane in which no two edges cross.
In such a drawing, the vertices are distinct points of $\mathbb{R}^2$, and each edge $uv$ is a simple arc from $u$ to $v$.
Two arcs meet only at a common end, and an arc contains no vertex other than its ends.

## Remarks

A drawing of this kind is a plane graph, or an embedding of $G$ in the plane.
The complete graph $K_4$ is planar. [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].
[[Kuratowski's theorem|0J6G5NJC5R]] characterizes the finite planar graphs by these two graphs.

## Explanations

### L1 human

A graph is planar when you can draw it on paper with no crossing edges.
A map of countries gives a planar graph: one vertex for each country, and an edge between neighbours.
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<document source="https://math2.tamnd.com/node/02A8EMC83C.md" tag="02A8EMC83C" kind="definition">
# Binary tree

## Definition

The binary trees are defined recursively by these two rules.
The empty tree is a binary tree. It has no nodes.
If $L$ and $R$ are binary trees, then $\operatorname{node}(L, R)$ is a binary tree. It has one more node than $L$ and $R$ together, the root, with left subtree $L$ and right subtree $R$.
Every binary tree comes from these two rules in a unique way, in finitely many steps.

## Remarks

The left and right subtrees are ordered. Thus $\operatorname{node}(\operatorname{node}(\varnothing, \varnothing), \varnothing)$ and $\operatorname{node}(\varnothing, \operatorname{node}(\varnothing, \varnothing))$ are different binary trees with $2$ nodes.
The number of binary trees with $n$ nodes is the Catalan number $C_n$. The node [[binary trees are counted by Catalan numbers|0Y97E2DMB7]] states this.
A nonempty binary tree is a [[rooted tree|0C9J6W2A8A]] in which each vertex has at most two children, each marked left or right.

## Explanations

### L1 human

A binary tree is either empty, or a root with a left part and a right part that are binary trees again.
Binary trees are used in computer science to store data for fast search.
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<document source="https://math2.tamnd.com/node/02AC7JR6G6.md" tag="02AC7JR6G6" kind="statement">
# Holomorphic functions on a disc have primitives

## Statement

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $D = \{z : |z - c| < R\}$ be the open disc with center $c$ and radius $R$.
Let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Then there is a holomorphic function $F : D \to \mathbb{C}$ with $F'(z) = f(z)$ for each $z \in D$.

## Remarks

One primitive is $F(z) = \int_{[c, z]} f(w) \, dw$, the [[contour integral|00FWNP9TXY]] along the segment from $c$ to $z$. [[Goursat's theorem|0BRG85WXYC]] for triangles shows that $F' = f$.
Two primitives of $f$ on $D$ differ by a constant, because $D$ is connected.
The disc can be replaced by a convex open set, or more generally by a simply connected open set.
The result fails on $\mathbb{C} \setminus \{0\}$: the function $1/z$ has no primitive there, by the [[integral of 1/z|0SXZ68HV6F]].

## Explanations

### L1 human

On a disc, each holomorphic function is the derivative of another holomorphic function.
You can build it by integrating along straight lines from the center, as in the real case $F(x) = \int_c^x f(t) \, dt$.
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<document source="https://math2.tamnd.com/node/02AS45T3H9.md" tag="02AS45T3H9" kind="concept">
# Convex set

## Summary

A set of points is convex if it contains the whole segment between each two of its points. A solid ball and a solid cube are convex; a crescent is not.
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<document source="https://math2.tamnd.com/node/02B1T50JD1.md" tag="02B1T50JD1" kind="example">
# Infinite cyclic group

## Example

The [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$ is an infinite [[cyclic group|0KVGH0J5B3]].
The element $1$ generates it, because each integer is a sum of copies of $1$ or of $-1$.
Its only generators are $1$ and $-1$.

## Remarks

If $G = \langle g \rangle$ is an infinite cyclic group, then the powers $g^k$ for $k \in \mathbb{Z}$ are all distinct.
Then $k \mapsto g^k$ is an isomorphism from $\mathbb{Z}$ to $G$, thus each infinite cyclic group is isomorphic to $(\mathbb{Z}, +)$.
The group of the translations of the plane by integer multiples of a fixed vector is an infinite cyclic group.

## Explanations

### L1 human

In an infinite cyclic group, the powers of the generator never repeat.
You can walk forever in both directions without coming back.
This is exactly how the integers behave under addition.
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<document source="https://math2.tamnd.com/node/02BH3MCBQB.md" tag="02BH3MCBQB" kind="concept">
# Open set

## Summary

An open set is a member of a topology.
In a metric space, a set is open when each of its points has a small ball around it that stays in the set.
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<document source="https://math2.tamnd.com/node/02BPKD7TMD.md" tag="02BPKD7TMD" kind="statement">
# Eisenstein's criterion

## Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$, and let $p$ be a prime number with these three properties.

1. $p$ divides $a_i$ for each $i$ with $0 \le i < n$.
2. $p$ does not divide $a_n$.
3. $p^2$ does not divide $a_0$.

Then $f$ is [[irreducible|00X2A9G9W3]] in $\mathbb{Q}[x]$.

## Remarks

If $f$ is also [[primitive|0SP14DZN0G]], then $f$ is irreducible in $\mathbb{Z}[x]$, by [[Gauss's lemma|0QFV5Y1Z5X]].
For example, $x^n - 2$ is irreducible over $\mathbb{Q}$ for each $n \ge 1$, with $p = 2$.
A change of variable can help: $f(x + 1)$ is irreducible if and only if $f(x)$ is. This proves that $x^{p-1} + \dots + x + 1$ is irreducible for $p$ prime.
The criterion holds more generally for a prime ideal $P$ of an integral domain $R$, with $P^2$ in place of $p^2$.

## History

Theodor Schönemann published a version of the criterion in 1846.
Gotthold Eisenstein published a different version in 1850.
Both papers appeared in Crelle's Journal.

## Explanations

### L1 human

The criterion is a quick test of irreducibility over the rationals.
Find one prime that divides all the coefficients except the leading one, and whose square does not divide the constant term.
For $x^5 + 6x + 3$, the prime $3$ works, so the polynomial is irreducible.
</document>

<document source="https://math2.tamnd.com/node/02CBHV4ZZQ.md" tag="02CBHV4ZZQ" kind="definition">
# Covariance matrix

## Definition

Let $X = (X_1, \dots, X_n)^{\mathsf{T}}$ be a random vector in $\mathbb{R}^n$ with $E[X_i^2] < \infty$ for each $i$, and let $\mu = E[X]$ be its mean vector.
The covariance matrix of $X$ is the $n \times n$ matrix $\Sigma = \operatorname{Cov}(X)$ with entries
$$\Sigma_{ij} = \operatorname{Cov}(X_i, X_j), \quad 1 \le i, j \le n.$$
In matrix form, $\Sigma = E\left[(X - \mu)(X - \mu)^{\mathsf{T}}\right]$.

## Remarks

The diagonal entries are the [[variances|007QED3E0Y]] $\Sigma_{ii} = \operatorname{Var}(X_i)$.
$\Sigma$ is symmetric and positive semidefinite: $a^{\mathsf{T}} \Sigma a = \operatorname{Var}(a^{\mathsf{T}} X) \ge 0$ for each $a \in \mathbb{R}^n$.
For a real $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, $\operatorname{Cov}(AX + b) = A \Sigma A^{\mathsf{T}}$.
The covariance matrix and the mean vector are the two parameters of the [[multivariate normal distribution|0G558GJQMW]].

## Explanations

### L1 human

For a random vector, one number is not enough to describe the spread.
The covariance matrix collects the variance of each component and the covariance of each pair in one table.
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<document source="https://math2.tamnd.com/node/02CDYWSWSY.md" tag="02CDYWSWSY" kind="definition">
# Edge colouring

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with edge set $E$, and let $C$ be a set of colours.
An edge colouring of $G$ is a function $c : E \to C$ such that $c(e) \neq c(e')$ for any two distinct edges $e$ and $e'$ with a common end.
The chromatic index $\chi'(G)$ is the smallest number of colours in an edge colouring of a finite graph $G$.

## Remarks

Each colour class is a [[matching|0Z87BHAFTB]]. Thus an edge colouring is a partition of the edges into matchings.
The edges at a vertex need different colours, so $\chi'(G) \ge \Delta(G)$.
Vizing's theorem of 1964 says that a simple graph has $\chi'(G) = \Delta(G)$ or $\chi'(G) = \Delta(G) + 1$.

## Explanations

### L1 human

An edge colouring colours the lines instead of the points: lines that meet must have different colours.
In a sports schedule, each colour is a round, and each team plays at most one game per round.
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<document source="https://math2.tamnd.com/node/02CK6QMXYD.md" tag="02CK6QMXYD" kind="concept">
# Cardinality of a finite set

## Summary

The cardinality of a finite set is the number of elements of the set. It is the basic quantity that all counting results compute. The node [[cardinality of a finite set|05099E1NMJ]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/02CYB184RC.md" tag="02CYB184RC" kind="statement">
# Mean value inequality

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \in U$ be points such that the segment $[x, y] = \{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$.
Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in [x, y]$, where $\|Df(z)\| = \sup_{\|h\| \le 1} \|Df(z) h\|$ is the operator norm.
Then
$$\|f(y) - f(x)\| \le M \, \|y - x\|.$$

## Remarks

For $m = 1$, the inequality follows from the [[mean value theorem in several variables|03HGK7EW6R]]. For $m \ge 2$ there is in general no equality form, but the inequality still holds.
A standard proof applies the one-variable mean value theorem to $\varphi(t) = (f(y) - f(x)) \cdot f(x + t(y - x))$.
On a convex set $C \subseteq U$ with $\|Df\| \le M$ on $C$, the inequality holds for all $x, y \in C$, so $f$ is [[Lipschitz on C|0TKG7T3K0E]].

## Explanations

### L1 human

Suppose that the speed of change of $f$ is at most $M$ everywhere on a straight path. Then $f$ changes by at most $M$ times the length of the path.
This is like saying that a car that never goes faster than $100$ km/h cannot cover more than $200$ km in two hours.
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<document source="https://math2.tamnd.com/node/02DKV69ZH0.md" tag="02DKV69ZH0" kind="concept">
# Sign of a permutation

## Summary

The sign of a permutation is $+1$ or $-1$, according to the parity of the number of transpositions in a decomposition of the permutation.
The sign map is a homomorphism from $S_n$ to $\{1, -1\}$.
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<document source="https://math2.tamnd.com/node/02DTRYKGE8.md" tag="02DTRYKGE8" kind="definition">
# Transposition

## Definition

Let $X$ be a set and let $a, b \in X$ with $a \neq b$.
The transposition $(a\ b)$ is the permutation of $X$ that sends $a$ to $b$, sends $b$ to $a$, and fixes each other element of $X$.

## Remarks

A transposition is a $2$-cycle in [[cycle notation|0ZKPH2BD1T]], and $(a\ b) = (b\ a)$.
Each transposition is its own inverse, so it has [[order|0WE2NHZJ7B]] $2$.
The symmetric group $S_n$ contains exactly $\binom{n}{2}$ transpositions.

## Explanations

### L1 human

A transposition swaps two things and leaves everything else in place.
If you do the same swap two times, you are back at the start.
All permutations can be built from transpositions.
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<document source="https://math2.tamnd.com/node/02E1T12PP4.md" tag="02E1T12PP4" kind="definition">
# Free module

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A subset $E \subseteq M$ is a *basis* of $M$ when:

1. Each $x \in M$ is a finite sum $r_1 e_1 + \cdots + r_k e_k$ with $r_i \in R$ and $e_i \in E$.
2. If $e_1, \dots, e_k \in E$ are distinct and $r_1 e_1 + \cdots + r_k e_k = 0$, then $r_1 = \cdots = r_k = 0$.

The module $M$ is *free* when it has a basis.

## Remarks

$R^n$ is free, with the standard basis; each vector space is free over its field.
For $n \ge 2$, the $\Z$-module $\Z / n\Z$ is not free, because $n x = 0$ for all its elements.
Over a nonzero commutative ring, any two bases of a free module have the same cardinality, called the *rank*.
A submodule of a free module need not be free, but over a principal ideal domain it is free.

## Explanations

### L1 human

A free module is a module that behaves like $R^n$: it has a basis.
Every vector space has a basis, but many modules do not.
For example, $\Z / 2\Z$ has no basis over $\Z$, because $2x = 0$ for every element $x$.
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<document source="https://math2.tamnd.com/node/02E9VT1RVQ.md" tag="02E9VT1RVQ" kind="concept">
# Power of a point

## Summary

The power of a point with respect to a circle is the square of its distance to the centre minus the square of the radius. It is positive outside the circle, zero on the circle and negative inside.
</document>

<document source="https://math2.tamnd.com/node/02EEX96DFS.md" tag="02EEX96DFS" kind="statement">
# Open balls form a basis of the metric topology

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
Then the set of all [[open balls|0DC3MPR7C9]] $B(x, r)$, with $x \in X$ and real $r > 0$, is a [[basis|06EX0FHBTF]] of the [[metric topology|0A6YRNZB2R]] of $X$.
Also, the balls $B(x, 1/n)$ with $x \in X$ and integer $n \ge 1$ form a basis.

## Remarks

Proof: each ball is open, by [[open balls are open|0B2305GNAZ]]. If $U$ is open and $x \in U$, then some ball $B(x, r)$ lies in $U$, and $B(x, 1/n) \subseteq B(x, r)$ for $n > 1/r$.
For each point $x$, the balls $B(x, 1/n)$ form a countable neighbourhood basis at $x$. Thus each metric space is [[first-countable|0HG11FW8MJ]].
In $\mathbb{R}$, the open balls are the bounded open intervals.

## Explanations

### L1 human

Each open set of a metric space is a union of open balls.
So, to check a property for all open sets, it is often enough to check it for balls.
</document>

<document source="https://math2.tamnd.com/node/02EKEPPJ3R.md" tag="02EKEPPJ3R" kind="concept">
# Prime factorization

## Summary

The prime factorization of a positive integer writes it as a product of prime numbers.
The problem of integer factorization is to find this product for a given integer.
The [[definition of a prime factorization|0AEKDFCYFA]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/02FDWY55AS.md" tag="02FDWY55AS" kind="statement">
# Rank of a product of matrices

## Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $F$.
Then
$$\rank(AB) \le \min(\rank A, \rank B),$$
where $\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].

## Remarks

The column space of $AB$ is contained in the column space of $A$, so $\rank(AB) \le \rank A$.
The row space of $AB$ is contained in the row space of $B$, so $\rank(AB) \le \rank B$.
If $A$ is invertible, then $\rank(AB) = \rank B$; if $B$ is invertible, then $\rank(AB) = \rank A$.
A lower bound is Sylvester's rank inequality: $\rank A + \rank B - n \le \rank(AB)$.

## Explanations

### L1 human

Multiplying by a matrix can never create new independent directions.
So the rank of a product is at most the rank of each factor.
Multiplying by an invertible matrix keeps the rank.
</document>

<document source="https://math2.tamnd.com/node/02GK6HZYBT.md" tag="02GK6HZYBT" kind="concept">
# Transitive action

## Summary

A group action is transitive when it has exactly one orbit.
Then each point can be moved onto each other point by some element of the group.
</document>

<document source="https://math2.tamnd.com/node/02HB5F40VR.md" tag="02HB5F40VR" kind="example">
# Newton's law of cooling

## Example

Let $T(t)$ be the temperature of a body at time $t$. Let $T_a$ be the constant temperature of the surroundings, and let $k > 0$ be a constant.
Newton's law of cooling is the equation
$$T'(t) = -k \left(T(t) - T_a\right).$$
With the initial condition $T(0) = T_0$, its unique solution is
$$T(t) = T_a + (T_0 - T_a)\, e^{-k t}.$$

## Remarks

The difference $D = T - T_a$ satisfies $D' = -k D$. This is [[exponential decay|00SVP6W3CB]], so $D(t) = D(0) e^{-kt}$.
As $t \to \infty$, $T(t) \to T_a$. If $T_0 > T_a$ the body cools, and if $T_0 < T_a$ it warms.
The law is a good model when the temperature difference is small and the body has a nearly uniform temperature.

## Explanations

### L1 human

A hot cup of coffee cools fast at first and then more slowly.
The law says that the speed of cooling is proportional to the difference between the coffee and the room.
So the difference halves after each fixed time interval.
</document>

<document source="https://math2.tamnd.com/node/02HHNPC8ED.md" tag="02HHNPC8ED" kind="statement">
# Casorati–Weierstrass theorem

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.
Then for each $\delta$ with $0 < \delta < r$, the image $f(\{z : 0 < |z - c| < \delta\})$ is dense in $\mathbb{C}$.
Thus for each $w \in \mathbb{C}$ there is a sequence $z_n \to c$ with $z_n \neq c$ and $f(z_n) \to w$.

## Remarks

Proof idea: if the image of a punctured disc misses a disc $|w - w_0| < \varepsilon$, then $g(z) = 1/(f(z) - w_0)$ is bounded near $c$. By [[Riemann's theorem|0BSQDZWF34]], $g$ is holomorphic at $c$, and then $c$ is removable or a pole of $f$.
[[Picard's great theorem|08QA8QTM71]] is a much stronger statement.
In the Russian literature, the theorem has the name of Sokhotski.

## Explanations

### L1 human

Near an essential singularity, the function comes as close as you like to each complex number.
For example, $e^{1/z}$ comes close to $5$, to $-i$ and to $0$ on each small disc around $0$.
</document>

<document source="https://math2.tamnd.com/node/02HJYT53F7.md" tag="02HJYT53F7" kind="statement">
# Existence of maximal ideals

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $I$ be an ideal of $R$ with $I \neq R$.
Then there is a [[maximal ideal|09J6KGT65G]] $M$ of $R$ with $I \subseteq M$.
In particular, every nonzero ring has a maximal ideal.

## Remarks

The proof uses Zorn's lemma. The union of a chain of proper ideals that contain $I$ is again a proper ideal, because it does not contain $1$.
The statement for every nonzero ring is equivalent to the axiom of choice, over the other axioms of ZF set theory.
The statement holds for left, right and two-sided ideals.

## History

Wolfgang Krull proved the theorem in 1929 with transfinite induction.

## Explanations

### L1 human

Start with a proper ideal and make it larger, step by step, without reaching the whole ring.
In an infinite ring this process may not stop by itself.
Zorn's lemma guarantees that an ideal that you cannot enlarge exists.
</document>

<document source="https://math2.tamnd.com/node/02HPP5VCZH.md" tag="02HPP5VCZH" kind="concept">
# Isometry

## Summary

An isometry is a map between metric spaces that keeps all distances.
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<document source="https://math2.tamnd.com/node/02HR4C0B04.md" tag="02HR4C0B04" kind="concept">
# Nullcline

## Summary

A nullcline of a system of autonomous equations $x_j' = f_j(x)$ is a set on which one component $f_j$ of the vector field vanishes.
The equilibrium points of the system are the common points of all the nullclines. See the [[definition|0WW060JR6P]].
</document>

<document source="https://math2.tamnd.com/node/02HTDK1FBD.md" tag="02HTDK1FBD" kind="definition">
# Inverse trigonometric functions

## Definition

The inverse trigonometric functions are the inverses of the restrictions of the trigonometric functions. The restrictions are to the following intervals:
$\sin$ on $[-\pi/2, \pi/2]$, $\cos$ on $[0, \pi]$, and $\tan$ on $(-\pi/2, \pi/2)$.
They are the [[arcsine|0B89D93YA1]] $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$, the [[arccosine|0RDP2HEFEW]] $\arccos \colon [-1, 1] \to [0, \pi]$ and the [[arctangent|065XW2V6C3]] $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$.

## Remarks

The trigonometric functions are not injective on $\mathbb{R}$. The restrictions to these intervals are strictly monotone and continuous, so they have continuous inverses.
The values are the principal values. Other notations are $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$.
Similar definitions give $\operatorname{arccot}$, $\operatorname{arcsec}$ and $\operatorname{arccsc}$.

## Explanations

### L1 human

An inverse trigonometric function gives back an angle from a value of sine, cosine or tangent.
We restrict to one interval so that the answer is unique.
</document>

<document source="https://math2.tamnd.com/node/02HZ1BK3MW.md" tag="02HZ1BK3MW" kind="statement">
# Chebyshev's inequality

## Statement

Let $X$ be a real random variable with finite mean $\mu$ and finite [[variance|007QED3E0Y]] $\sigma^2$. Then for each $a > 0$,
$$P(|X - \mu| \ge a) \le \frac{\sigma^2}{a^2}.$$
Equivalently, if $\sigma > 0$, then $P(|X - \mu| \ge k\sigma) \le 1/k^2$ for each $k > 0$.

## Remarks

Proof: apply [[Markov's inequality|05PWWHYBX5]] to the nonnegative variable $(X - \mu)^2$ and the level $a^2$.
For $k = 2$, at least $3/4$ of the mass is within two [[standard deviations|0VTN5RKNC5]] of the mean, for every distribution with finite variance.
The bound is sharp: some distributions reach equality for a given $k \ge 1$.
Chebyshev's inequality gives a short proof of the weak law of large numbers for variables with finite variance.

## History

The inequality is named after Pafnuty Chebyshev. Irénée-Jules Bienaymé formulated it first.

## Explanations

### L1 human

The variance controls how often a variable can be far from its mean.
For any distribution, at most $1/9$ of the mass lies three or more standard deviations away from the mean.
</document>

<document source="https://math2.tamnd.com/node/02J4G4J4TJ.md" tag="02J4G4J4TJ" kind="statement">
# Isometries are injective

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be an [[isometry|0878DN3WCS]].
Then $f$ is [[injective|0PVG1CRAJN]].

## Remarks

Proof: if $f(x) = f(y)$, then $d_X(x, y) = d_Y(f(x), f(y)) = 0$, thus $x = y$.
The proof uses the axiom of a metric that $d(x, y) = 0$ implies $x = y$. For a pseudometric, the statement can fail.
Thus an isometry is a bijection from $X$ onto its image $f(X)$.

## Explanations

### L1 human

Two different points have a positive distance, and an isometry keeps that distance.
So their images are also at a positive distance, and they are different.
</document>

<document source="https://math2.tamnd.com/node/02JBEMG8R4.md" tag="02JBEMG8R4" kind="concept">
# Monotone sequence

## Summary

A monotone sequence is a real sequence that never decreases or never increases. A bounded monotone sequence always converges. The node [[monotone sequence|0KN22AAW5R]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/02JGYG0T7M.md" tag="02JGYG0T7M" kind="statement">
# Characteristic function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \dots + X_n$.
Then for each $t \in \mathbb{R}$, the [[characteristic functions|0MYDNRT7PP]] satisfy
$$\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t).$$

## Remarks

Proof: $e^{itS} = e^{itX_1} \cdots e^{itX_n}$. The factors are independent and bounded, so the [[product rule for expectations|05AW1G0WE1]] applies to their real and imaginary parts.
For i.i.d. $X_i$, $\varphi_S(t) = \varphi_{X_1}(t)^n$. The sample mean has $\varphi_{S/n}(t) = \varphi_{X_1}(t/n)^n$.
The converse is false: $\varphi_{X+Y} = \varphi_X \varphi_Y$ does not imply that $X$ and $Y$ are independent.

## Explanations

### L1 human

Adding independent variables corresponds to multiplying their characteristic functions.
This is the key step in the usual proof of the central limit theorem.
</document>

<document source="https://math2.tamnd.com/node/02KTRHQMHN.md" tag="02KTRHQMHN" kind="statement">
# Power series of sine

## Statement

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges absolutely, and
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the sine function.

## Remarks

The [[ratio of the terms|0JBMYAA9QH]] is $\frac{x^2}{(2n+2)(2n+3)} \to 0$. So the radius of convergence is $\infty$.
With the series as the definition of $\sin$, the equality holds by definition. With a geometric definition, Taylor's theorem proves it.
For $|x| \le 1$, the error after the term $x^{2n+1}$ is at most $\frac{1}{(2n+3)!}$. Mathlib states the series as `Real.hasSum_sin`.

## Explanations

### L1 human

Sine is an infinite odd polynomial with alternating signs.
Near $0$, $\sin x$ is close to $x$, and closer still to $x - x^3/6$.
</document>

<document source="https://math2.tamnd.com/node/02KYGSCTCN.md" tag="02KYGSCTCN" kind="definition">
# Stationary distribution

## Definition

Let $P = (p_{ij})_{i, j \in S}$ be the [[transition matrix|0F1KBVY161]] of a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$.
A stationary distribution of the chain is a family $\pi = (\pi_j)_{j \in S}$ of real numbers with $\pi_j \ge 0$ for each $j$, $\sum_{j \in S} \pi_j = 1$, and
$$\pi_j = \sum_{i \in S} \pi_i \, p_{ij} \quad \text{for each } j \in S.$$
In matrix form, $\pi P = \pi$ for the row vector $\pi$.

## Remarks

If $X_0$ has the law $\pi$, then $X_n$ has the law $\pi$ for each $n \ge 0$.
An irreducible chain on a finite state space has exactly one stationary distribution. Its entries are positive, and $\pi_j = 1 / E_j[T_j]$, where $T_j$ is the time of the first return to $j$.
For an irreducible and aperiodic finite chain, $P(X_n = j) \to \pi_j$ as $n \to \infty$, for each initial law.
The simple random walk on $\mathbb{Z}$ has no stationary distribution.

## Explanations

### L1 human

A stationary distribution is a balance: the flow of probability into each state equals the flow out.
For many chains, it also describes the long-run fraction of time that the chain spends in each state.
</document>

<document source="https://math2.tamnd.com/node/02M2K4QH9P.md" tag="02M2K4QH9P" kind="definition">
# Fermat pseudoprime

## Definition

Let $a$ be an integer with $a > 1$.
A Fermat pseudoprime to the base $a$ is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \equiv 1 \pmod n$.

## Remarks

[[Fermat's little theorem|0M25PRN3WQ]] says that each prime $n$ with $n \nmid a$ satisfies $a^{n-1} \equiv 1 \pmod n$. A Fermat pseudoprime is a composite number that passes the same test.
The smallest Fermat pseudoprime to the base $2$ is $341 = 11 \cdot 31$.
The condition implies $\gcd(a, n) = 1$.
A composite number that is a Fermat pseudoprime to each base coprime to it is a [[Carmichael number|0TA7WT68RK]].

## Explanations

### L1 human

The [[Fermat primality test|0N5ZSZMYH6]] declares $n$ "probably prime" when $a^{n-1} \equiv 1 \pmod n$.
A Fermat pseudoprime is a composite number that fools this test for the base $a$.
For example, $2^{340} \equiv 1 \pmod{341}$, but $341$ is not prime.
</document>

<document source="https://math2.tamnd.com/node/02MAKWDKGT.md" tag="02MAKWDKGT" kind="concept">
# Polynomial evaluation

## Summary

To evaluate a polynomial at an element, you replace the variable by the element and compute the result in the ring.
The [[definition of evaluation|0GJ7T0NK49]] gives the formula.
</document>

<document source="https://math2.tamnd.com/node/02MY6WZXEQ.md" tag="02MY6WZXEQ" kind="concept">
# Discrete uniform distribution

## Summary

The discrete uniform distribution on a finite set gives the same probability to each element of the set.
The node [[discrete uniform distribution|0KYP951QWH]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/02NBKVQFRQ.md" tag="02NBKVQFRQ" kind="definition">
# Intermediate field

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
An intermediate field of $L/K$ is a [[subfield|07QY5RPFBR]] $E$ of $L$ with $K \subseteq E$.

## Remarks

Then $L/E$ and $E/K$ are field extensions, and one calls $K \subseteq E \subseteq L$ a tower of fields.
The fields $K$ and $L$ are intermediate fields of $L/K$.
For example, $\mathbb{Q}(\sqrt{2})$ is an intermediate field of $\mathbb{R}/\mathbb{Q}$.
The [[fundamental theorem of Galois theory|0T1Q8D0FW8]] describes the intermediate fields of a Galois extension.

## Explanations

### L1 human

An intermediate field sits between the small field and the big field.
Galois theory gives a way to list all such fields, by a correspondence with subgroups of a group.
</document>

<document source="https://math2.tamnd.com/node/02NSMQVQWK.md" tag="02NSMQVQWK" kind="definition">
# Divergence

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F = (F_1, \dots, F_n) \colon U \to \mathbb{R}^n$ be a [[vector field|0N67NBKQ5T]] whose [[partial derivatives|05AXG3QRKR]] $\partial F_i / \partial x_i$ exist on $U$.
The divergence of $F$ is the function $\operatorname{div} F \colon U \to \mathbb{R}$,
$$\operatorname{div} F = \nabla \cdot F = \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} + \dots + \frac{\partial F_n}{\partial x_n}.$$

## Remarks

The divergence is the trace of the [[Jacobian matrix|0YF0VXKXGB]] of $F$.
For $F(x) = x$ on $\mathbb{R}^n$, $\operatorname{div} F = n$. For the rotation field $F(x, y) = (-y, x)$, $\operatorname{div} F = 0$.
The divergence is linear, and $\operatorname{div}(\phi F) = \nabla \phi \cdot F + \phi \operatorname{div} F$ for a differentiable function $\phi$.
Take a $C^1$ field and a small region around $x_0$. The divergence at $x_0$ is the limit of the outward flux through the boundary of the region, divided by its volume. This follows from the [[divergence theorem|0JX86PSSA9]].

## Explanations

### L1 human

Think of $F$ as the velocity of a fluid.
Where the divergence is positive, more fluid leaves a small region than enters it, as at a source. Where it is negative, fluid gathers, as at a sink.
</document>

<document source="https://math2.tamnd.com/node/02NZZ1J0FH.md" tag="02NZZ1J0FH" kind="definition">
# Splitting field

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be nonzero.
A splitting field of $f$ over $K$ is a [[field extension|0WQ1A8SHTZ]] $L/K$ with these two properties.

1. The polynomial $f$ [[splits|0SHMC00RJV]] over $L$, as $f = c \, (x - a_1) \cdots (x - a_n)$.
2. The field $L$ is [[generated over $K$|0MA5B744K3]] by the roots: $L = K(a_1, \dots, a_n)$.

## Remarks

Splitting fields [[exist|01441KDGBY]] and are [[unique up to isomorphism|069A3BC3EP]].
The splitting field of $x^2 + 1$ over $\mathbb{R}$ is $\mathbb{C}$.
The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$ with $\omega = e^{2\pi i/3}$. Its degree over $\mathbb{Q}$ is $6$.
The splitting field of $x^q - x$ over $\mathbb{F}_p$, for $q = p^n$, is the [[finite field|08RTDHX1SK]] with $q$ elements.

## Explanations

### L1 human

A splitting field is the smallest field that contains all the roots of a polynomial.
It is where the polynomial factors completely.
For $x^3 - 2$, the real cube root of $2$ is not enough, because the other two roots are not real.
</document>

<document source="https://math2.tamnd.com/node/02P2PYH3X3.md" tag="02P2PYH3X3" kind="statement">
# The complex exponential is 2πi-periodic

## Statement

For each $z \in \mathbb{C}$,
$$e^{z + 2 \pi i} = e^{z},$$
where $e^z$ is the [[complex exponential|054VXHMCA3]] of $z$.

## Remarks

Proof: by [[Euler's formula|04R3CY41VJ]], $e^{2 \pi i} = \cos 2\pi + i \sin 2\pi = 1$. By the [[product of exponentials|0WM7DJCM6G]], $e^{z + 2\pi i} = e^z e^{2\pi i} = e^z$.
The periods of $\exp$ are exactly the numbers $2 \pi i k$ with $k \in \mathbb{Z}$, by [[exp z = 1 iff z is in 2πiZ|013AHC4DRT]].
So $\exp$ is not injective on $\mathbb{C}$. It is injective on each horizontal strip $\{z : a < \operatorname{Im} z \le a + 2\pi\}$ with $a \in \mathbb{R}$.

## Explanations

### L1 human

Adding $2 \pi i$ to $z$ adds a full turn to the direction of $e^z$ and does not change its size.
So $e^z$ takes the same value again: the exponential repeats itself along each vertical line, with step $2\pi$.
</document>

<document source="https://math2.tamnd.com/node/02PETK9PQ4.md" tag="02PETK9PQ4" kind="statement">
# Sum of the roots of unity

## Statement

Let $n \ge 2$ be an integer and let $\zeta$ be a [[primitive root of unity|0J817RTVG6]].
Then
$$\sum_{k=0}^{n-1} \zeta^k = 1 + \zeta + \zeta^2 + \dots + \zeta^{n-1} = 0.$$

## Remarks

Proof: $\zeta \neq 1$, so the [[geometric sum formula|03X8V7D64M]] gives $\frac{\zeta^n - 1}{\zeta - 1} = 0$.
Thus the sum of all the $n$th roots of unity is $0$ for $n \ge 2$. For $n = 1$ the sum is $1$.

## Explanations

### L1 human

The $n$th roots of unity are the corners of a regular polygon with center $0$.
By symmetry, their center of mass is $0$, and so their sum is $0$.
For $n = 4$: $1 + i + (-1) + (-i) = 0$.
</document>

<document source="https://math2.tamnd.com/node/02QGGD5G87.md" tag="02QGGD5G87" kind="concept">
# Complex logarithm

## Summary

A complex logarithm of a nonzero complex number $z$ is a complex number $w$ with $e^w = z$, see the [[definition|0X2FY491PF]].
Each $z \neq 0$ has infinitely many logarithms, which differ by integer multiples of $2\pi i$.
A complex logarithm function is a continuous choice of a logarithm on an open subset of $\mathbb{C} \setminus \{0\}$.
</document>

<document source="https://math2.tamnd.com/node/02QJVFCNYD.md" tag="02QJVFCNYD" kind="concept">
# Line integral of a scalar field

## Summary

The line integral of a scalar field along a curve adds up the values of the field along the curve. Each value is weighted by a small length of the curve.
For a non-negative field on a plane curve, it is the area of the curtain between the curve and the graph of the field.
</document>

<document source="https://math2.tamnd.com/node/02RRHJ7R98.md" tag="02RRHJ7R98" kind="concept">
# Multiplicative order

## Summary

The multiplicative order of an integer $a$ modulo $n$, with $a$ coprime to $n$, is the smallest positive exponent $k$ such that $a^k \equiv 1 \pmod n$.
The [[definition of the multiplicative order|0SBRX5CZ5R]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/02RSXVM1ZN.md" tag="02RSXVM1ZN" kind="definition">
# Higher partial derivatives

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $i, j \in \{1, \dots, n\}$.
Suppose that the [[partial derivative|05AXG3QRKR]] $\partial f / \partial x_i$ exists at each point of $U$.
The second partial derivative of $f$ with respect to $x_i$ and then $x_j$ at $a \in U$ is
$$\frac{\partial^2 f}{\partial x_j \, \partial x_i}(a) = \frac{\partial}{\partial x_j}\left(\frac{\partial f}{\partial x_i}\right)(a),$$
when this partial derivative exists. It is also written $f_{x_i x_j}(a)$ or $\partial_j \partial_i f(a)$.
For $i = j$, one writes $\partial^2 f / \partial x_i^2$.
By induction, a partial derivative of order $k \ge 2$ is a partial derivative of a partial derivative of order $k - 1$: $\dfrac{\partial^k f}{\partial x_{i_k} \cdots \partial x_{i_1}} = \dfrac{\partial}{\partial x_{i_k}} \left( \dfrac{\partial^{k-1} f}{\partial x_{i_{k-1}} \cdots \partial x_{i_1}} \right)$.

## Remarks

In $\partial^2 f / \partial x_j \, \partial x_i$ the variable nearest to $f$ is differentiated first. In $f_{x_i x_j}$ the variable written first is differentiated first.
For $f(x, y) = x^3 y^2$, one gets $f_{xx} = 6 x y^2$, $f_{xy} = f_{yx} = 6 x^2 y$ and $f_{yy} = 2 x^3$.
If the second partial derivatives are continuous, then the order of differentiation does not matter, by [[Schwarz's theorem|0NNQY8WCW6]]. Without continuity, the mixed partial derivatives [[can differ|01NN8NY497]].
The second partial derivatives at a point form the [[Hessian matrix|0693Y2METT]].

## Explanations

### L1 human

A second partial derivative measures how a slope changes. For example, $f_{xy}$ measures how the slope in the $x$-direction changes as you move in the $y$-direction.
You compute it by differentiating twice, one variable at a time.
</document>

<document source="https://math2.tamnd.com/node/02SJG1HC32.md" tag="02SJG1HC32" kind="definition">
# Algebraically closed field

## Definition

A [[field|0QA8CHH8MG]] $K$ is algebraically closed if every polynomial $f \in K[x]$ with $\deg f \ge 1$ has a [[root|00EHX6FWC0]] in $K$.

## Remarks

Equivalently, every nonzero polynomial over $K$ [[splits|0SHMC00RJV]] over $K$. Also equivalently, the [[irreducible polynomials|00X2A9G9W3]] over $K$ are exactly the polynomials of degree one.
Another equivalent condition: $K$ has no [[algebraic extension|09EYMKMSB9]] other than $K$ itself.
The field $\mathbb{C}$ is algebraically closed, by the fundamental theorem of algebra. The field $\mathbb{R}$ is not, because $x^2 + 1$ has no real root.
No finite field is algebraically closed: for a finite field $K$, the polynomial $1 + \prod_{a \in K} (x - a)$ has no root in $K$.

## Explanations

### L1 human

In an algebraically closed field, every polynomial equation of degree at least one has a solution.
You never need a bigger field to find roots.
</document>

<document source="https://math2.tamnd.com/node/02SMVGC690.md" tag="02SMVGC690" kind="concept">
# Topological property

## Summary

A topological property is a property of topological spaces such that, if a space has it, each [[homeomorphic|05GZ11PWRY]] space also has it.
Examples are [[compactness|0QPHW703PV]], [[connectedness|0D939CMME7]], [[separability|0GKJ78948V]], [[metrizability|09KDHB82SG]] and the [[Hausdorff property|0MXB4CKHFW]].

## Remarks

Topological properties are used to show that two spaces are not homeomorphic. For example, $[0, 1]$ is compact and $(0, 1)$ is not, so they are not homeomorphic.
Properties that depend on a metric are often not topological. For example, $(0, 1)$ and $\mathbb{R}$ are homeomorphic, but only $\mathbb{R}$ is [[complete|08EWT7J0SC]] and only $(0, 1)$ is bounded.

## Explanations

### L1 human

A topological property is a property that a space keeps when you deform it by a homeomorphism.
Size and distance are not topological properties, but the number of pieces of a space is one.
</document>

<document source="https://math2.tamnd.com/node/02SRXDAM3X.md" tag="02SRXDAM3X" kind="concept">
# Reduction of order

## Summary

Reduction of order is a method to find a second solution of a homogeneous linear equation from a known one.
Let $p, q$ be continuous on an open interval $I$, and let $y_1$ be a solution of $y'' + p(x) y' + q(x) y = 0$ with no zero on $I$.
Put $y = v\, y_1$. Then $y$ is a solution if and only if $y_1 v'' + (2 y_1' + p\, y_1) v' = 0$, a first-order linear equation for $v'$.
One solution gives $y_2 = y_1 \int \frac{e^{-\int p\, dx}}{y_1^2}\, dx$, and $(y_1, y_2)$ is a [[fundamental system|0H4KNNCHPY]] on $I$.

## Remarks

Example: $y_1 = e^{x}$ solves $y'' - 2 y' + y = 0$. Here $p = -2$, so $v' = e^{2x}/e^{2x} = 1$, and $y_2 = x e^x$.
The [[Wronskian|0N5NF1RGBP]] of $y_1$ and $y_2$ is $e^{-\int p\, dx}$, which has no zero. This is consistent with [[Abel's identity|0VGWKX2JHB]].
The same idea lowers the order by one for a linear equation of order $n$.

## Explanations

### L1 human

If you know one solution, look for a second one as the first one times an unknown function $v$.
The terms without derivatives of $v$ cancel, because $y_1$ is a solution.
What remains is a first-order equation for $v'$, which is easy to solve.
</document>

<document source="https://math2.tamnd.com/node/02STTYFBVY.md" tag="02STTYFBVY" kind="definition">
# Polar form of a quadratic form

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The *polar form* of $Q$ is the map $B_Q \colon V \times V \to F$ with
$$B_Q(u, v) = Q(u + v) - Q(u) - Q(v).$$

## Remarks

$B_Q$ is a [[symmetric bilinear form|0AXZNE5XNH]], and $B_Q(v, v) = 2 Q(v)$ for all $v$.
For $Q(x) = x^T A x$ on $F^n$, $B_Q(x, y) = x^T (A + A^T) y$.
Many authors divide by $2$ when the characteristic is not $2$, so that the result $B$ satisfies $B(v, v) = Q(v)$; see [[the associated form|0KRVRV5V80]].
This construction is the degree $2$ case of the polarisation of homogeneous polynomials.

## Explanations

### L1 human

The polar form recovers a bilinear form from a quadratic form.
For $Q(x) = x^2$ on $\R$, the polar form is $(x + y)^2 - x^2 - y^2 = 2xy$.
It is the algebraic form of the [[polarisation identity|05FKHRT0B4]] for inner products.
</document>

<document source="https://math2.tamnd.com/node/02SYZAWM2K.md" tag="02SYZAWM2K" kind="statement">
# Cauchy mean value theorem

## Statement

Let $a < b$ and let $f, g \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Then there is $c \in (a, b)$ with
$$(f(b) - f(a)) \, g'(c) = (g(b) - g(a)) \, f'(c).$$

## Remarks

Apply [[Rolle's theorem|0194DY35BG]] to $h(x) = (f(b) - f(a)) g(x) - (g(b) - g(a)) f(x)$, which has $h(a) = h(b)$.
With $g(x) = x$ it is the [[mean value theorem|0BARQY52N5]].
It is the main step in the proof of [[L'Hôpital's rule|0JWT4PPQ7B]]. Mathlib states it as `exists_ratio_deriv_eq_ratio_slope`.

## Explanations

### L1 human

Think of $(g(t), f(t))$ as a point moving along a curve in the plane.
At some time, the direction of motion is parallel to the chord joining the start and the end.
</document>

<document source="https://math2.tamnd.com/node/02TZDKG50Q.md" tag="02TZDKG50Q" kind="statement">
# Closure is closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then the [[closure|0DRKT1QJR7]] $\overline{S}$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: $\overline{S}$ is an intersection of closed sets, and each [[intersection of closed sets is closed|09YMV8H4X5]].
It follows that $\overline{\overline{S}} = \overline{S}$.

## Explanations

### L1 human

The closure is built as an intersection of closed sets, and such an intersection is again closed.
</document>

<document source="https://math2.tamnd.com/node/02V2GYAC8P.md" tag="02V2GYAC8P" kind="statement">
# Decreasing sequence bounded below converges

## Statement

Let $(a_n)$ be a decreasing real sequence: $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
Assume that $(a_n)$ is bounded below.
Then $(a_n)$ converges, and its limit is the infimum $\inf \{a_n : n \in \mathbb{N}\}$.

## Remarks

The proof applies the [[monotone convergence theorem|0FM6N4R0VB]] to the increasing sequence $(-a_n)$.
For example, $1/n$ decreases and is bounded below by $0$. Its limit is its infimum $0$.
Together, the two statements say that a monotone sequence converges if and only if it is bounded.

## Explanations

### L1 human

A decreasing sequence that cannot pass a floor must settle down.
It settles at the highest floor, which is the infimum of its terms.
</document>

<document source="https://math2.tamnd.com/node/02WF878BH1.md" tag="02WF878BH1" kind="definition">
# Greatest common divisor

## Definition

Let $a$ and $b$ be integers that are not both $0$.
A common divisor of $a$ and $b$ is an integer $d$ such that $d \mid a$ and $d \mid b$.
The greatest common divisor $\gcd(a, b)$ is the largest common divisor of $a$ and $b$.
By convention, $\gcd(0, 0) = 0$.

## Remarks

The number $\gcd(a, b)$ is positive when $a$ and $b$ are not both $0$, because $1$ is a common divisor.
An equivalent definition: $\gcd(a, b)$ is the unique nonnegative common divisor of $a$ and $b$ that each common divisor [[divides|096PW7Q014]]. This form also gives $\gcd(0, 0) = 0$.
For all integers $a$, $\gcd(a, 0) = |a|$.
The [[Euclidean algorithm|0TRNT4X7J0]] computes $\gcd(a, b)$.

## Explanations

### L1 human

List the positive divisors of each number, and take the largest number that is in both lists.
The divisors of $12$ are $1, 2, 3, 4, 6, 12$, and the divisors of $18$ are $1, 2, 3, 6, 9, 18$.
Thus $\gcd(12, 18) = 6$.
</document>

<document source="https://math2.tamnd.com/node/02WHMEG5ZR.md" tag="02WHMEG5ZR" kind="statement">
# Riemann mapping theorem

## Statement

Let $U$ be a nonempty open subset of $\mathbb{C}$ with $U \neq \mathbb{C}$.
Suppose that $U$ is simply connected: $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$.
Then there is a biholomorphic map $f$ from $U$ onto $\mathbb{D}$, that is, a bijective [[holomorphic|0DG0Z72CQD]] map with a holomorphic inverse.
Moreover, for each $z_0 \in U$ there is exactly one such $f$ with $f(z_0) = 0$ and $f'(z_0)$ a positive real number.

## Remarks

A biholomorphic map is [[conformal|0VHNKP1N3K]], so $U$ and $\mathbb{D}$ are conformally equivalent.
The case $U = \mathbb{C}$ is excluded: a holomorphic map $\mathbb{C} \to \mathbb{D}$ is bounded, so it is constant by [[Liouville's theorem|0EHKVRF4J6]].
The uniqueness follows from the [[Schwarz lemma|0MB48XDBY3]].
Riemann stated the theorem in his thesis of 1851, and Osgood gave the first rigorous proof in 1900.

## Explanations

### L1 human

The theorem says that every simply connected open set in the plane, except the plane itself, looks like the unit disc to complex analysis.
The set can have a very irregular boundary, and the map still exists.
For the upper half-plane, the [[Cayley transform|0X45ZM731G]] is such a map.
</document>

<document source="https://math2.tamnd.com/node/02WZB422CE.md" tag="02WZB422CE" kind="definition">
# External direct sum

## Definition

Let $F$ be a field and let $(V_i)_{i \in I}$ be a family of [[vector spaces|000AYPA9AJ]] over $F$.
The *external direct sum* $\bigoplus_{i \in I} V_i$ is the set of families $(v_i)_{i \in I}$ with $v_i \in V_i$ for each $i$ and $v_i = 0$ for all but finitely many $i$.
It is a vector space over $F$ with the componentwise operations
$$(v_i)_i + (w_i)_i = (v_i + w_i)_i, \qquad a (v_i)_i = (a v_i)_i.$$

## Remarks

For finite $I$, the direct sum equals the [[product|0EX2HPXPJ6]] $\prod_{i \in I} V_i$.
For infinite $I$, it is a proper subspace of the product when infinitely many $V_i$ are nonzero.
Each $V_j$ embeds in the direct sum as the families that are zero outside $j$.
The direct sum is the [[internal direct sum|0XYYT5KNTX]] of these images.
The direct sum of countably many copies of $F$ is isomorphic to $F[x]$.

## Explanations

### L1 human

The external direct sum builds a new space out of several spaces, side by side.
A vector of the new space chooses one vector in each space, with only finitely many nonzero choices.
For two spaces, this is just the space of pairs.
</document>

<document source="https://math2.tamnd.com/node/02XPXR1FBK.md" tag="02XPXR1FBK" kind="definition">
# Bernoulli process

## Definition

Let $p \in [0, 1]$.
A Bernoulli process with parameter $p$ is a [[stochastic process|0QH5ENKZM1]] $(X_n)_{n \ge 1}$ of [[i.i.d.|0AJN7WHBJZ]] random variables with the [[Bernoulli distribution|0J97P5KZY0]] $\operatorname{Bernoulli}(p)$:
$$P(X_n = 1) = p \quad \text{and} \quad P(X_n = 0) = 1 - p \quad \text{for each } n \ge 1.$$

## Remarks

One calls $X_n = 1$ a success and $X_n = 0$ a failure in trial $n$.
The number of successes in the first $n$ trials has the binomial distribution $\operatorname{Bin}(n, p)$.
For $0 < p \le 1$, the number of failures before the first success has the geometric distribution with parameter $p$. The times between successes are i.i.d.
The Bernoulli process is a discrete-time analogue of the Poisson process.

## Explanations

### L1 human

The Bernoulli process is the mathematical model of tossing the same coin again and again, forever.
Each toss is independent of the others, and each has the same probability $p$ of heads.
</document>

<document source="https://math2.tamnd.com/node/02YFME3RGW.md" tag="02YFME3RGW" kind="definition">
# Kernel of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *kernel* of $T$ is
$$\ker T = \{v \in V : T(v) = 0\}.$$

## Remarks

The kernel is a [[linear subspace|00YC1D652K]] of $V$.
[[A linear map is injective if and only if its kernel is trivial|0BJBYCGMWE]].
For the map $x \mapsto Ax$ of a matrix $A$, the kernel is the [[null space|0PHDYNR24D]] of $A$, the solution set of $Ax = 0$.
The kernel of the projection $\R^3 \to \R^2$, $(x, y, z) \mapsto (x, y)$, is the $z$-axis.

## Explanations

### L1 human

The kernel collects the vectors that the map sends to zero.
It measures how much information the map loses.
If only the zero vector goes to zero, then the map loses nothing.
</document>

<document source="https://math2.tamnd.com/node/02YRQXZDCS.md" tag="02YRQXZDCS" kind="definition">
# Conditional convergence

## Definition

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges conditionally if $\sum a_n$ [[converges|0CVVKY7GY2]] and $\sum |a_n|$ diverges.

## Remarks

So a convergent series converges either [[absolutely|0W3K3QPEWT]] or conditionally, and not both.
The [[alternating harmonic series|0JQ5Y3WWVE]] converges conditionally.
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], the terms of a conditionally convergent series can be reordered to give any real sum.

## Explanations

### L1 human

A conditionally convergent series converges only because its positive and negative terms cancel.
The positive terms alone add up to $+\infty$, and the negative terms alone add up to $-\infty$.
</document>

<document source="https://math2.tamnd.com/node/02YSJPQYQH.md" tag="02YSJPQYQH" kind="concept">
# Branch of the logarithm

## Summary

A branch of the logarithm on an open set $U \subseteq \mathbb{C} \setminus \{0\}$ is a continuous function $L$ on $U$ with $e^{L(z)} = z$, see the [[definition|0SQ1F2BEDS]].
On a domain, two branches differ by a constant integer multiple of $2\pi i$.
</document>

<document source="https://math2.tamnd.com/node/02YYV36BFM.md" tag="02YYV36BFM" kind="concept">
# Centraliser

## Summary

The centraliser of a subset $S$ of a group is the set of elements that commute with each element of $S$.
It is a subgroup.
</document>

<document source="https://math2.tamnd.com/node/02Z5B2HA3F.md" tag="02Z5B2HA3F" kind="concept">
# One-form

## Summary

A one-form is a [[differential form|0E3G0GRPZ6]] of degree one.
At each point it gives a linear function of tangent vectors. One-forms are the natural integrands of line integrals, and the differential of a function is the basic example.
</document>

<document source="https://math2.tamnd.com/node/02ZE6JHJ7G.md" tag="02ZE6JHJ7G" kind="definition">
# Real and imaginary parts

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.
The real part of $z$ is $\operatorname{Re} z = a$.
The imaginary part of $z$ is $\operatorname{Im} z = b$.

## Remarks

The imaginary part is a real number: $\operatorname{Im}(2 + 3 i) = 3$, not $3 i$.
The maps $\operatorname{Re}$ and $\operatorname{Im}$ are additive and $\mathbb{R}$-linear, but they do not respect products.
With the [[conjugate|0BJV24C4SY]], $\operatorname{Re} z = (z + \bar{z})/2$ and $\operatorname{Im} z = (z - \bar{z})/(2 i)$.
A complex number $z$ is real when $\operatorname{Im} z = 0$, and purely imaginary when $\operatorname{Re} z = 0$.

## Explanations

### L1 human

Each complex number has two real coordinates.
For $z = 4 - 7 i$, the real part is $4$ and the imaginary part is $-7$.
</document>

<document source="https://math2.tamnd.com/node/030A3FHN14.md" tag="030A3FHN14" kind="statement">
# K5 and K3,3 are not planar

## Statement

The [[complete graph|03WKB2H474]] $K_5$ and the [[complete bipartite graph|0V0GC7BQAN]] $K_{3,3}$ are not [[planar|029JQEK15B]].

## Remarks

The graph $K_5$ has $5$ vertices and $10$ edges. The [[edge bound for planar graphs|0SM1REAVYJ]] allows at most $3 \cdot 5 - 6 = 9$ edges.
The graph $K_{3,3}$ has $6$ vertices and $9$ edges, and it has no cycle of length $3$. The bound $e \le 2v - 4$ allows at most $8$ edges.
The statement for $K_{3,3}$ is the three utilities problem: three houses cannot each be joined to three utilities without crossings.

## Explanations

### L1 human

Try to join each of three houses to each of three utilities with lines that never cross.
It is impossible, and the reason is a count of edges and faces.
</document>

<document source="https://math2.tamnd.com/node/030TZ0DQWT.md" tag="030TZ0DQWT" kind="definition">
# Ideal generated by a subset

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $X \subseteq R$ be a subset.
The left ideal generated by $X$ is the intersection of all the left [[ideals|0MYSA6MYNA]] of $R$ that contain $X$.
The notation for it is $(X)$.
It is the smallest left ideal of $R$ that contains $X$.

## Remarks

The left ideal $(X)$ is the set of the finite sums $r_1 x_1 + \dots + r_n x_n$ with $n \ge 0$, $r_i \in R$ and $x_i \in X$.
For a finite set, one writes $(x_1, \dots, x_n)$ for $(\{x_1, \dots, x_n\})$.
In $\mathbb{Z}$, the ideal $(4, 6)$ is $2\mathbb{Z}$.
An ideal generated by one element is a [[principal ideal|00A2EDFT9Q]].

## Explanations

### L1 human

Start with some elements of the ring.
The ideal that they generate contains all the sums of their multiples, and nothing else.
In the integers, the multiples of $4$ and $6$ give all the even numbers, because $2 = 2 \cdot 4 - 1 \cdot 6$.
</document>

<document source="https://math2.tamnd.com/node/030VFEREKG.md" tag="030VFEREKG" kind="concept">
# Partial fraction decomposition

## Summary

Partial fraction decomposition writes a rational function as a sum of a polynomial and simple fractions.
Let $K$ be a field, and let $P/Q$ be a rational function with $P, Q \in K[X]$ and $Q \neq 0$.
Assume that $Q = \prod_{i=1}^{r} q_i^{m_i}$, where the $q_i$ are distinct monic irreducible polynomials.
Then there are unique polynomials $B$ and $A_{ij}$, with $\deg A_{ij} < \deg q_i$, such that
$$\frac{P}{Q} = B + \sum_{i=1}^{r} \sum_{j=1}^{m_i} \frac{A_{ij}}{q_i^{\,j}}.$$

## Remarks

Example: $\frac{1}{1 - x - x^2} = \frac{1}{\sqrt{5}} \left( \frac{\varphi}{1 - \varphi x} - \frac{\psi}{1 - \psi x} \right)$, with $\varphi$ and $\psi$ the roots of $X^2 - X - 1$.
In combinatorics it gives the coefficients of a rational [[generating function|0HH6JX5WZZ]]. Each term $\frac{1}{1 - ax}$ expands as a [[geometric series|0R8JR0ARJ4]].

## Explanations

### L1 human

A complicated fraction of polynomials splits into a sum of simple fractions.
For example, $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}$.
</document>

<document source="https://math2.tamnd.com/node/0310REJJY8.md" tag="0310REJJY8" kind="concept">
# Commutative ring

## Summary

A commutative ring is a ring in which the multiplication is commutative.
Commutative algebra is the study of these rings.
The [[definition of a commutative ring|0GYGHKX834]] gives the precise condition.
</document>

<document source="https://math2.tamnd.com/node/0316AYH1SC.md" tag="0316AYH1SC" kind="definition">
# Total order

## Definition

Let $A$ be a set.
A total order on $A$ is a [[partial order|011C29Q1S1]] $\le$ on $A$ such that, for all $a, b \in A$, $a \le b$ or $b \le a$.

## Remarks

Other names are linear order and simple order.
The usual orders on $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are total orders.
The order $\subseteq$ on the power set of $\{1, 2\}$ is not total.

## Explanations

### L1 human

In a total order, all elements stand on one line.
For any two of them, one comes before the other, or they are equal.
</document>

<document source="https://math2.tamnd.com/node/031BG625Y3.md" tag="031BG625Y3" kind="statement">
# The real part of a holomorphic function is harmonic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$.
Then $u$ and $v$ are twice continuously differentiable, and
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$
on $U$. Thus $u$ and $v$ are harmonic functions on $U$.

## Remarks

Proof: $f$ is [[infinitely differentiable|0QEBN8DGVE]]. By the [[Cauchy–Riemann equations|0T71SSYZ39]], $u_{xx} = v_{yx} = v_{xy} = -u_{yy}$, and the same computation works for $v$.
For example, $\operatorname{Re}(z^2) = x^2 - y^2$ is harmonic, and so is $\operatorname{Re}(e^z) = e^x \cos y$.
The function $v$ is a [[harmonic conjugate|0CC56MJ4FG]] of $u$.

## Explanations

### L1 human

A harmonic function is a function of two real variables whose two pure second derivatives add up to $0$.
Each holomorphic function gives two harmonic functions for free: its real part and its imaginary part.
</document>

<document source="https://math2.tamnd.com/node/031JV5S54D.md" tag="031JV5S54D" kind="statement">
# A tangent is perpendicular to the radius

## Statement

Let $C$ be a [[circle|060W5CBZP6]] with the centre $O$ and the radius $r > 0$ in the Euclidean plane $\mathbb{R}^2$. Let $T$ be a point of $C$, and let $\ell$ be a line through $T$.
Then $\ell$ is [[tangent|0T7DX8NY1X]] to $C$ at $T$ if and only if $\ell$ is perpendicular to the radius $OT$. That is, $(X - T) \cdot (T - O) = 0$ for each point $X$ of $\ell$.

## Remarks

A proof: for $X = T + t v$ on $\ell$, we have $\|X - O\|^2 = r^2 + 2t\, v \cdot (T - O) + t^2 \|v\|^2$. So $t = 0$ is the only point of $\ell$ on $C$ if and only if $v \cdot (T - O) = 0$.
Thus each point of $C$ has exactly one tangent line.
The same holds for a sphere and its tangent plane in $\mathbb{R}^3$.

## Explanations

### L1 human

To draw the tangent to a circle at a point, first draw the radius to that point. The tangent is the line through the point at a right angle to the radius. For the unit circle at $(1, 0)$, the radius is horizontal and the tangent is the vertical line $x = 1$.
</document>

<document source="https://math2.tamnd.com/node/031M8HX7DA.md" tag="031M8HX7DA" kind="statement">
# Order of a cyclic subgroup equals the order of its generator

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
If $g$ has finite [[order|0WE2NHZJ7B]] $n$, then $\langle g \rangle = \{e, g, g^2, \ldots, g^{n-1}\}$, these $n$ elements are distinct, and thus $|\langle g \rangle| = n$.
If $g$ has infinite order, then the powers $g^k$ for $k \in \mathbb{Z}$ are distinct, and $\langle g \rangle$ is infinite.

## Remarks

Here $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]], and $|\langle g \rangle|$ is its [[order|0B85S2V2DR]].
The proof uses [[the divisibility criterion for powers|0F5QC101Q7]]: $g^a = g^b$ exactly when $n$ divides $a - b$.
With [[Lagrange's theorem|3C53TD]], this fact shows that the order of an element of a finite group divides the order of the group.

## Explanations

### L1 human

The two meanings of the word "order" agree.
The order of an element is the number of distinct powers that it has, and these powers form the subgroup that it generates.
For example, a rotation by $120$ degrees has order $3$, and its powers are the three rotations by $0$, $120$ and $240$ degrees.
</document>

<document source="https://math2.tamnd.com/node/031TPJVS74.md" tag="031TPJVS74" kind="concept">
# Event

## Summary

An event is a set of outcomes of a random experiment to which we give a probability.
The event happens when the outcome of the experiment is in the set. The node [[event|00SQGJNRTB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0325SGSJAR.md" tag="0325SGSJAR" kind="concept">
# Polyhedron

## Summary

A polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and corner points, the vertices. The word names the solid or its boundary surface. Cubes, pyramids and prisms are polyhedra.
</document>

<document source="https://math2.tamnd.com/node/032HKYS329.md" tag="032HKYS329" kind="statement">
# Residue theorem

## Statement

Let $U \subseteq \mathbb{C}$ be a simply connected open set. This means that $U$ is connected, and each closed path in $U$ can be continuously deformed in $U$ to a constant path.
Let $a_1, \dots, a_n$ be distinct points of $U$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U \setminus \{a_1, \dots, a_n\}$.
Let $\gamma$ be a closed, continuous and piecewise continuously differentiable path in $U \setminus \{a_1, \dots, a_n\}$.
Then
$$\oint_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^{n} n(\gamma, a_k) \operatorname{Res}(f, a_k),$$
where $n(\gamma, a_k)$ is the [[winding number|0X7SBS8E0F]] of $\gamma$ around $a_k$ and $\operatorname{Res}(f, a_k)$ is the [[residue|0JFGKKWWFK]] of $f$ at $a_k$.

## Remarks

If $\gamma$ is a simple closed curve traveled once counterclockwise, then $\oint_\gamma f(z) \, dz$ is $2 \pi i$ times the sum of the residues at the points $a_k$ inside $\gamma$.
With no points $a_k$, the theorem is [[Cauchy's integral theorem|01MPZXYVYF]].
For example, $\oint_{|z| = 2} \frac{dz}{z^2 + 1} = 2 \pi i \left( \frac{1}{2i} - \frac{1}{2i} \right) = 0$, by the [[residues at the simple poles|01991NCP7H]] $i$ and $-i$.
The theorem is the main tool for the [[evaluation of real integrals|0476QH9846]].

## Explanations

### L1 human

To integrate around a closed curve, you only need to know the residues at the bad points inside the curve.
Each bad point contributes $2 \pi i$ times its residue, once for each turn of the curve around it.
</document>

<document source="https://math2.tamnd.com/node/032Q9FB8MT.md" tag="032Q9FB8MT" kind="concept">
# Matrix multiplication

## Summary

Matrix multiplication is the operation that takes an $m \times n$ matrix and an $n \times p$ matrix and gives an $m \times p$ matrix.
Entry $(i, k)$ of the product combines row $i$ of the first matrix with column $k$ of the second matrix.
</document>

<document source="https://math2.tamnd.com/node/0337CYB0VD.md" tag="0337CYB0VD" kind="definition">
# Order on the reals

## Definition

Construct the real numbers as classes of Cauchy sequences of rational numbers, as in the [[construction from Cauchy sequences|0D8HE681SC]].
Let $x$ and $y$ be real numbers, represented by the Cauchy sequences $(x_n)$ and $(y_n)$ of rational numbers.
The notation $x < y$ means that there are a rational number $\varepsilon > 0$ and a natural number $N$ such that $y_n - x_n > \varepsilon$ for each $n \ge N$.
The notation $x \le y$ means that $x < y$ or $x = y$.

## Remarks

The relation $x < y$ does not depend on the choice of the sequences that represent $x$ and $y$.
For all real numbers $x$ and $y$, exactly one of $x < y$, $x = y$ and $y < x$ holds. Thus $\le$ is a total order.
On the rational numbers, this order agrees with the [[order of the rationals|09RFKMBQ63]].
With [[Dedekind cuts|04QNZZ4HHY]], the order is inclusion of the lower sets of the cuts.

## Explanations

### L1 human

Two real numbers are compared through their rational approximations.
Suppose that, from some point on, the approximations of $y$ exceed those of $x$ by a fixed positive gap.
Then $x$ is less than $y$.
For example, $1.41, 1.414, 1.4142, \dots$ approximates $\sqrt{2}$, and these approximations are larger than $1.4$ by more than $0.005$, thus $1.4 < \sqrt{2}$.
</document>

<document source="https://math2.tamnd.com/node/033DA1WE8Y.md" tag="033DA1WE8Y" kind="concept">
# Projection

## Summary

A projection is a linear map from a vector space to itself that is equal to its own square.
It projects the space onto its image along its kernel.
</document>

<document source="https://math2.tamnd.com/node/033H8MQRC3.md" tag="033H8MQRC3" kind="concept">
# Nine-point circle

## Summary

The nine-point circle of a triangle is the circle through the midpoints of its three sides. It also passes through the three feet of the altitudes and through the midpoints of the segments from the vertices to the orthocentre.
</document>

<document source="https://math2.tamnd.com/node/033KTKKRV6.md" tag="033KTKKRV6" kind="statement">
# Factorial formula for binomial coefficients

## Statement

Let $n$ and $k$ be natural numbers with $k \le n$.
Then
$$\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$$

## Remarks

Equivalently, $k!\,(n-k)!\,\binom{n}{k} = n!$. In this form the identity has no division.
The formula needs $k \le n$. For $k > n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers.
A cancelled form is $\binom{n}{k} = \frac{(n)_k}{k!}$, with the [[falling factorial|0N5M924N3P]] $(n)_k$.

## Explanations

### L1 human

Arrange all $n$ objects in a row in $n!$ ways, and take the first $k$ as the chosen set.
Each choice of set appears $k!\,(n-k)!$ times, because the first $k$ and the last $n-k$ objects can be reordered freely.
</document>

<document source="https://math2.tamnd.com/node/033RZ32T4Q.md" tag="033RZ32T4Q" kind="concept">
# Limit inferior

## Summary

The limit inferior of a sequence is the limit of the infima of its tails. It describes the smallest value near which the sequence lies infinitely often. The node [[limit inferior|0H4R3RESGV]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/033TJBZDG0.md" tag="033TJBZDG0" kind="definition">
# Markov chain

## Definition

Let $S$ be a nonempty [[countable set|0ZG61TFBES]].
Let $(X_n)_{n \ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $S$. It is a Markov chain with state space $S$ if
$$P(X_{n+1} = j \mid X_0 = i_0, \dots, X_{n-1} = i_{n-1}, X_n = i) = P(X_{n+1} = j \mid X_n = i)$$
for each $n \ge 0$ and all states $i_0, \dots, i_{n-1}, i, j \in S$ with $P(X_0 = i_0, \dots, X_n = i) > 0$.
The chain is time-homogeneous if $p_{ij} = P(X_{n+1} = j \mid X_n = i)$ does not depend on $n$.

## Remarks

The condition is the [[Markov property|0HB8SXHPFA]] in discrete time.
For a time-homogeneous chain, the numbers $p_{ij}$ form the [[transition matrix|0F1KBVY161]] $P = (p_{ij})_{i, j \in S}$.
The initial distribution $\lambda_i = P(X_0 = i)$ and $P$ determine the law of the chain: $P(X_0 = i_0, \dots, X_n = i_n) = \lambda_{i_0} p_{i_0 i_1} \cdots p_{i_{n-1} i_n}$.
Markov chains also exist in continuous time and on general state spaces.

## Explanations

### L1 human

A Markov chain has no memory beyond the present state.
Consider the weather each day: if tomorrow depends only on today, the sequence of weather states is a Markov chain.
</document>

<document source="https://math2.tamnd.com/node/033WGN00BF.md" tag="033WGN00BF" kind="concept">
# Formal power series

## Summary

A formal power series is an infinite sum $\sum a_n X^n$ treated as an algebraic object, without any convergence condition. Formal power series form a ring that is the natural home of generating functions. The node [[formal power series|072W67KFYK]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/034FGCEYX4.md" tag="034FGCEYX4" kind="statement">
# Formula for the modulus

## Statement

Let $x$ and $y$ be real numbers.
Then the [[modulus|0KW2X0TFBX]] of the complex number $x + y i$ is
$$|x + y i| = \sqrt{x^2 + y^2}.$$

## Remarks

This follows from the definition of the modulus, because $\operatorname{Re}(x + y i) = x$ and $\operatorname{Im}(x + y i) = y$ for real $x$ and $y$.
The hypothesis that $x$ and $y$ are real is necessary. For example, $|i + i \cdot i| = |-1 + i| = \sqrt{2}$, but $\sqrt{i^2 + i^2} = \sqrt{-2}$ is not even real.

## Explanations

### L1 human

To find the length of a complex number, square its two real coordinates, add them, and take the square root.
For $1 - i$, this gives $\sqrt{1 + 1} = \sqrt{2}$.
</document>

<document source="https://math2.tamnd.com/node/035V802J58.md" tag="035V802J58" kind="concept">
# Tangent plane to a graph

## Summary

Take the [[graph|0JBJYT69ZV]] of a [[differentiable|0HHHDCXZV4]] function of two variables. Its tangent plane at a point is the plane through that point that best approximates the graph nearby.
It is the graph of the linear approximation of the function at the point.
</document>

<document source="https://math2.tamnd.com/node/0363JMQ42Y.md" tag="0363JMQ42Y" kind="statement">
# Triangularisation

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to an [[upper triangular matrix|0N4XY7Z3NN]] over $F$ if and only if the [[characteristic polynomial|0VSSPAFM2C]] of $A$ splits over $F$.
Here, to split means to be a product of linear factors.
In particular, over an algebraically closed field, such as $\C$, each square matrix is similar to an upper triangular matrix.

## Remarks

The diagonal entries of the triangular matrix are the eigenvalues of $A$, with their algebraic multiplicities.
The proof is by induction: choose an eigenvector, extend it to a basis, and work on the quotient space.
Over $\C$, the matrix $P$ can even be chosen unitary; this is the Schur decomposition.

## Explanations

### L1 human

Not every matrix can be diagonalised, but over $\C$ every matrix can be made triangular by a change of basis.
A triangular form is almost as useful: its eigenvalues are on the diagonal.
Over $\R$, a rotation matrix cannot be triangularised, because it has no real eigenvalue.
</document>

<document source="https://math2.tamnd.com/node/036696EMG2.md" tag="036696EMG2" kind="concept">
# Hyperbolic cosine

## Summary

The hyperbolic cosine is the average of $e^x$ and $e^{-x}$. It is an even function with minimum value $1$ at $0$. The node [[hyperbolic cosine|0EQFWKJ37J]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/036756NFD6.md" tag="036756NFD6" kind="definition">
# Statistic

## Definition

Let $X_1, \dots, X_n$ be random variables with values in a measurable space $(E, \mathcal{E})$. For example, they can be a random sample from a law with an unknown parameter $\theta$.
A statistic is a random variable of the form $T = g(X_1, \dots, X_n)$, where $g \colon E^n \to \mathbb{R}^k$ is a measurable function that does not depend on $\theta$.

## Remarks

Examples are the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$, the sample variance, the sample maximum $\max_i X_i$ and the order statistics.
A statistic is a random variable, so it has a law: the sampling distribution of the statistic.
A statistic that one uses to estimate a parameter is an [[estimator|0TTNRP5PFQ]]. A statistic that one uses to decide a hypothesis test is a test statistic.

## Explanations

### L1 human

A statistic is a number that you can compute from your data alone, such as the average of the measurements.
It must not contain the unknown quantity that you want to learn about.
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<document source="https://math2.tamnd.com/node/037ACYXB5Z.md" tag="037ACYXB5Z" kind="statement">
# Gauss's lemma on quadratic residues

## Statement

Let $p$ be an odd prime and let $a$ be an integer with $p \nmid a$.
Reduce each of the integers $a, 2a, \dots, \frac{p-1}{2} a$ to its least positive residue modulo $p$.
Let $n$ be the number of these residues that are greater than $p/2$.
Then $\left(\frac{a}{p}\right) = (-1)^n$, where the left side is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

For example, take $p = 7$ and $a = 3$. The residues of $3, 6, 9$ are $3, 6, 2$, and only $6$ is greater than $7/2$. So $\left(\frac{3}{7}\right) = -1$.
With $a = 2$, the lemma gives the [[second supplement to quadratic reciprocity|0ZXW95S44M]].
Gauss used the lemma in one of his proofs of the [[law of quadratic reciprocity|085MJS6907]].

## Explanations

### L1 human

Write the first half of the multiples of $a$ modulo $p$, and count how many land in the upper half of $1, \dots, p - 1$.
An even count means that $a$ is a square modulo $p$, and an odd count means that it is not.
</document>

<document source="https://math2.tamnd.com/node/037F589D1D.md" tag="037F589D1D" kind="concept">
# Dirichlet convolution

## Summary

The Dirichlet convolution is a product of two arithmetic functions, given by a sum over the divisors of $n$.
The [[definition of the Dirichlet convolution|0KH1N66QWW]] gives the precise form.
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<document source="https://math2.tamnd.com/node/037FD5NBRT.md" tag="037FD5NBRT" kind="statement">
# Leibniz formula for pi

## Statement

The series $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n + 1}$ converges, and
$$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.$$

## Remarks

The series converges by the [[alternating series test|0PK8QH1W1V]]. Its value comes from the [[power series of arctangent|0T32ME0A6H]] at $x = 1$, since $\arctan 1 = \pi/4$.
The convergence is very slow. The error after the terms up to $n = N$ is at most $1/(2N + 3)$.
Mathlib states it as a limit of partial sums, in `Real.tendsto_sum_pi_div_four`.

## History

Madhava of Sangamagrama or his school found the series in the 14th or 15th century. James Gregory found it again in 1671, and Leibniz in 1673.

## Explanations

### L1 human

Add and subtract the reciprocals of the odd numbers: $1 - 1/3 + 1/5 - \cdots$.
The result is a quarter of $\pi$, which links the odd numbers with circles.
</document>

<document source="https://math2.tamnd.com/node/037ZNY7NAG.md" tag="037ZNY7NAG" kind="concept">
# System of linear differential equations

## Summary

A system of linear differential equations is an equation $x' = A(t) x + b(t)$ for a vector-valued function $x$, with a matrix $A(t)$ of coefficients.
Its theory extends that of a single linear equation. See the [[definition|0A6QZ367GX]].
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<document source="https://math2.tamnd.com/node/0381S8Z5P0.md" tag="0381S8Z5P0" kind="statement">
# Arzelà–Ascoli theorem

## Statement

Let $X$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $C(X)$ be the set of continuous functions $X \to \mathbb{R}$, with the [[uniform metric|06HE9J7DXF]] $d_\infty(f, g) = \sup_{x \in X} |f(x) - g(x)|$.
Let $\mathcal{F} \subseteq C(X)$.
Then the closure of $\mathcal{F}$ in $C(X)$ is compact if and only if $\mathcal{F}$ is [[equicontinuous|0MW2WXTHMC]] and uniformly bounded.
Here uniformly bounded means that there is a real $M$ with $|f(x)| \le M$ for all $f \in \mathcal{F}$ and $x \in X$.

## Remarks

A sequential form: each uniformly bounded and equicontinuous sequence of functions $[a, b] \to \mathbb{R}$ has a subsequence that converges uniformly on $[a, b]$.
The proof shows that $\mathcal{F}$ is [[totally bounded|0WE5BBHGM6]] in the complete space $C(X)$, and then uses [[compact iff complete and totally bounded|054Q5PW5GX]].
For example, the functions $f \colon [0, 1] \to [0, 1]$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ form a compact set in $C([0, 1])$.
The theorem is a main tool for existence proofs, for example in the Peano existence theorem for ordinary differential equations.

## Explanations

### L1 human

In $\mathbb{R}^n$, bounded sequences have convergent subsequences. The Arzelà–Ascoli theorem gives the same result for sequences of functions, but you also need equicontinuity.
Equicontinuity stops the functions from becoming more and more steep, as $x^n$ does near $1$.
</document>

<document source="https://math2.tamnd.com/node/0386Y0Z7Y7.md" tag="0386Y0Z7Y7" kind="concept">
# Bernoulli distribution

## Summary

The Bernoulli distribution with parameter $p$ is the distribution of a random variable with two values.
The value $1$ has probability $p$ and the value $0$ has probability $1 - p$.
It is the distribution of one Bernoulli trial. The node [[Bernoulli distribution|0J97P5KZY0]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/038YGY0HSC.md" tag="038YGY0HSC" kind="definition">
# Chromatic polynomial

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
For each natural number $k$, let $P(G, k)$ be the number of [[colourings|0874138YP6]] of $G$ with the set of colours $\{1, \dots, k\}$.
There is a unique polynomial $P(G, x)$ whose value at each natural number $k$ is $P(G, k)$. It is the chromatic polynomial of $G$.

## Remarks

For the [[path graph|075WR6W3Y3]] $P_3$, $P(P_3, x) = x(x-1)^2$. For the [[complete graph|03WKB2H474]], $P(K_n, x) = x(x-1)\cdots(x-n+1)$.
For an edge $e$, the deletion–contraction rule says $P(G, x) = P(G - e, x) - P(G / e, x)$. Here $G / e$ merges the two ends of $e$.
The [[chromatic number|0C5M7AM6FH]] is the smallest positive integer $k$ with $P(G, k) > 0$.

## Explanations

### L1 human

The chromatic polynomial counts the proper colourings of a graph with $k$ colours, for each $k$.
George David Birkhoff introduced it in 1912 to study the four colour problem.
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<document source="https://math2.tamnd.com/node/0396VGFA78.md" tag="0396VGFA78" kind="statement">
# Freshman's dream

## Statement

Let $p$ be a prime number and let $R$ be a [[commutative ring|0GYGHKX834]] of [[characteristic|0P7CH1CND5]] $p$.
Then $(x + y)^p = x^p + y^p$ for all $x, y \in R$.

## Remarks

The proof uses the binomial theorem. For $0 < k < p$, the prime $p$ divides the binomial coefficient $\binom{p}{k}$, so these terms vanish in $R$.
By induction, $(x + y)^{p^n} = x^{p^n} + y^{p^n}$ for every $n \ge 0$.
The identity gives the [[Frobenius endomorphism|0GHDDVAH8P]].
In $\mathbb{R}$ the identity is false: $(1 + 1)^2 = 4$ and $1^2 + 1^2 = 2$.

## Explanations

### L1 human

The name refers to a common error: $(x + y)^2 = x^2 + y^2$ is false for real numbers.
In characteristic $2$, the middle term $2 x y$ is $0$, and the identity becomes true.
The same thing happens for every prime $p$ in characteristic $p$.
</document>

<document source="https://math2.tamnd.com/node/0399QFDFS8.md" tag="0399QFDFS8" kind="concept">
# Parametric equation of a line

## Summary

A parametric equation of a [[line|0MP9BCF6DP]] describes its points as the values of a function of one real parameter $t$.
The line in $\mathbb{R}^3$ through the point $(x_0, y_0, z_0)$ with the nonzero direction vector $(a, b, c)$ has the parametric equations
$$x = x_0 + a t, \quad y = y_0 + b t, \quad z = z_0 + c t, \qquad t \in \mathbb{R}.$$
In vector form, the line through the point $p$ with the direction vector $v \neq 0$ is the set of the points $p + t v$ for $t \in \mathbb{R}$. The same form works in $\mathbb{R}^n$ for each $n \geq 2$.

## Remarks

The line through two distinct points $A$ and $B$ has the parametric equation $A + t(B - A)$. The values $0 \leq t \leq 1$ give the segment from $A$ to $B$, and $t = 1/2$ gives its midpoint.
If $a$, $b$ and $c$ are all nonzero, you can remove $t$. The result is the symmetric form $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

## Explanations

### L1 human

Think of $t$ as time and of $p + t v$ as the position of a particle. At time $0$ the particle is at $p$, and it moves with the constant velocity $v$. Its path is the line.
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<document source="https://math2.tamnd.com/node/039HF58KQ5.md" tag="039HF58KQ5" kind="definition">
# Characteristic polynomial

## Definition

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
The *characteristic polynomial* of $A$ is the polynomial
$$p_A(t) = \det(t I_n - A) \in K[t],$$
where $\det$ is the [[determinant|00KDZRW4C0]] of a matrix with entries in the polynomial ring $K[t]$.

## Remarks

The polynomial $p_A$ is monic of degree $n$.
For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $p_A(t) = t^2 - (a + d) t + (ad - bc)$.
Some authors use $\det(A - t I_n)$, which equals $(-1)^n p_A(t)$.
Over a field, the roots of $p_A$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$.

## Explanations

### L1 human

The number $\lambda$ is an eigenvalue exactly when $A - \lambda I$ is singular, that is, when $\det(\lambda I - A) = 0$.
If we replace $\lambda$ by a variable $t$, then we get a polynomial.
Its roots are the eigenvalues.
</document>

<document source="https://math2.tamnd.com/node/039JAF7ZAN.md" tag="039JAF7ZAN" kind="example">
# Integral of exp(-x) over the half line

## Example

The [[improper integral|0VCTBJMEJF]] of $e^{-x}$ over $[0, \infty)$ converges, and
$$\int_0^\infty e^{-x} \, dx = 1.$$
For $b > 0$, [[the exponential formula|06D0EWR2DV]] with $c = -1$ gives $\int_0^b e^{-x} \, dx = 1 - e^{-b}$. This tends to $1$ as $b \to \infty$.

## Remarks

It is the case $s = 1$ of the [[gamma function|0V3FZMVB4K]]: $\Gamma(1) = 1$.
Mathlib states it as `integral_exp_neg_Ioi_zero`.

## Explanations

### L1 human

The region under $e^{-x}$ is infinitely long, but its area is finite.
The area is exactly $1$.
</document>

<document source="https://math2.tamnd.com/node/03AHF0XRVG.md" tag="03AHF0XRVG" kind="concept">
# Independence of two events

## Summary

Two events are independent when the occurrence of one of them does not change the probability of the other.
The node [[independence of two events|0P0MZ4X682]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/03AJHX431A.md" tag="03AJHX431A" kind="statement">
# Taylor's theorem with the Peano remainder

## Statement

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
Then
$$f(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j + o\big((x - a)^k\big) \quad \text{as } x \to a.$$

## Remarks

This is [[Taylor's theorem|03QHBVW4TT]] written with [[little o notation|04TSHB9H8D]].
For example, $\cos x = 1 - x^2/2 + o(x^3)$ as $x \to 0$.
Mathlib states a form for $C^k$ functions on a convex set as `taylor_isLittleO`.

## Explanations

### L1 human

The Peano form only says that the error is small compared with $(x - a)^k$.
It gives no bound for a fixed $x$, but it needs weak hypotheses.
</document>

<document source="https://math2.tamnd.com/node/03AJKXH1AV.md" tag="03AJKXH1AV" kind="concept">
# Column space

## Summary

The column space of a matrix is the span of its columns.
It is the image of the linear map of the matrix.
</document>

<document source="https://math2.tamnd.com/node/03AN1V4ZFJ.md" tag="03AN1V4ZFJ" kind="definition">
# Norm on the Gaussian integers

## Definition

The norm on the [[Gaussian integers|0K5W69AC0J]] is the map $N \colon \mathbb{Z}[i] \to \mathbb{N}$,
$$N(a + b i) = a^2 + b^2 = (a + b i)(a - b i).$$

## Remarks

The norm is multiplicative: $N(z w) = N(z) N(w)$ for all $z, w \in \mathbb{Z}[i]$. This follows from $|z w| = |z| |w|$ in $\mathbb{C}$.
$N(z) = 0$ if and only if $z = 0$.
The norm is a Euclidean function on $\mathbb{Z}[i]$.
For example, $N(2 + i) = 5$ and $N(1 + i) = 2$.

## Explanations

### L1 human

The norm is the squared distance from $0$ to the point $a + b i$ in the complex plane.
Because it is multiplicative, it turns questions about factoring Gaussian integers into questions about factoring integers.
</document>

<document source="https://math2.tamnd.com/node/03AN9X29R7.md" tag="03AN9X29R7" kind="concept">
# Absorbing Markov chain

## Summary

An absorbing Markov chain has at least one state that it never leaves. Also, each state can reach such a state.
The node [[absorbing Markov chain|0YB93GWTKK]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/03BAREW1Z5.md" tag="03BAREW1Z5" kind="statement">
# Generalized Euclid's lemma

## Statement

Let $n$, $a$ and $b$ be integers.
If $n \mid a b$ and $n$ and $a$ are [[coprime|0GFKP8F8AG]], then $n \mid b$.

## Remarks

When $n$ is a prime $p$ and $p \nmid a$, then $p$ and $a$ are coprime. So this result implies [[Euclid's lemma|051VYKA1EX]].
The proof uses [[Bézout's identity|0DQVHMYAGA]]: from $n x + a y = 1$ we get $b = n b x + a b y$, and $n$ divides both terms.
For example, $5 \mid 3 \cdot 10$ and $\gcd(5, 3) = 1$, so $5 \mid 10$.

## Explanations

### L1 human

If $n$ divides $a b$ but has no common factor with $a$, then all the factors of $n$ must come from $b$.
For example, $4 \mid 3 b$ forces $4 \mid b$, because $3$ gives no factor $2$.
</document>

<document source="https://math2.tamnd.com/node/03BR7SAX0Q.md" tag="03BR7SAX0Q" kind="concept">
# Modulo operation

## Summary

The modulo operation takes two integers $a$ and $n$ with $n \neq 0$, and gives the remainder of the division of $a$ by $n$.
We write the result as $a \bmod n$.
The [[definition of the modulo operation|04FHQSJCQ5]] gives the convention that this graph uses.
</document>

<document source="https://math2.tamnd.com/node/03BY5XTNA4.md" tag="03BY5XTNA4" kind="statement">
# Law of total variance

## Statement

Let $X$ be a real random variable with $E[X^2] < \infty$, and let $Y$ be a random variable on the same probability space. Then
$$\operatorname{Var}(X) = E\left[\operatorname{Var}(X \mid Y)\right] + \operatorname{Var}\left(E[X \mid Y]\right).$$

## Remarks

Proof: by the [[variance formula|0NMF95YK8W]] in its conditional form, $E[X^2 \mid Y] = \operatorname{Var}(X \mid Y) + (E[X \mid Y])^2$. Take expectations with the [[law of total expectation|04RRB6HM00]], and subtract $(E[X])^2 = (E[E[X \mid Y]])^2$.
The same result holds with a σ-algebra $\mathcal{G}$ in place of $Y$.
The first term is the mean variance inside the groups. The second term is the variance between the group means.
A corollary: $\operatorname{Var}(E[X \mid Y]) \le \operatorname{Var}(X)$.

## Explanations

### L1 human

The total spread of $X$ has two parts: the spread inside each group, and the spread of the group averages.
This decomposition is the base of the analysis of variance in statistics.
</document>

<document source="https://math2.tamnd.com/node/03CPD1HGF3.md" tag="03CPD1HGF3" kind="concept">
# Multivalued function

## Summary

A multivalued function from a set $X$ to a set $Y$ assigns to each $x \in X$ a subset $F(x)$ of $Y$. For at least one $x$, the subset $F(x)$ has two or more elements.
In complex analysis, the main examples come from the inverses of functions that are not injective.
For example, each $z \neq 0$ has two square roots, and it has infinitely many [[logarithms|0X2FY491PF]] and [[arguments|0ZZRSX91QR]].

## Remarks

A [[branch|0SQ1F2BEDS]] of a multivalued function $F$ on a set $U$ is a function $f$ with $f(x) \in F(x)$ for each $x \in U$, usually continuous or holomorphic.
A [[principal branch|0YKY4M3KRQ]] is one branch that a standard convention chooses.
Riemann surfaces give a way to treat a multivalued function as one single-valued function.

## Explanations

### L1 human

The equation $w^2 = 4$ has two solutions, $2$ and $-2$, so the square root of $4$ has two values.
A multivalued function keeps all the values together, and a branch picks one value at each point.
</document>

<document source="https://math2.tamnd.com/node/03DEH43F4H.md" tag="03DEH43F4H" kind="concept">
# Left coset

## Summary

A left coset of a subgroup $H$ of a group $G$ is a set $gH$, with $g \in G$. You get it when you multiply each element of $H$ on the left by $g$.
The left cosets of $H$ partition $G$.
</document>

<document source="https://math2.tamnd.com/node/03EESZDB9A.md" tag="03EESZDB9A" kind="statement">
# Variance of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X$ have the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$. Then
$$\operatorname{Var}(X) = \sigma^2.$$

## Remarks

Proof: $Z = (X - \mu)/\sigma$ has the law $\mathcal{N}(0, 1)$, where $\sigma = \sqrt{\sigma^2}$. Integration by parts gives $E[Z^2] = \int z^2 \varphi(z) \, dz = \int \varphi(z) \, dz = 1$.
Since $E[Z] = 0$ by the [[mean of the normal distribution|09PR1V6RWR]], $\operatorname{Var}(Z) = 1$. The [[scaling|0KVQ9D86BM]] and [[translation|04CYN5G6RY]] rules then give $\operatorname{Var}(X) = \sigma^2 \operatorname{Var}(Z) = \sigma^2$.
So the two parameters of $\mathcal{N}(\mu, \sigma^2)$ are its mean and its variance.

## Explanations

### L1 human

The second parameter of the normal distribution is exactly its variance. So $\sigma$ is the standard deviation, the typical distance of a value from the center.
</document>

<document source="https://math2.tamnd.com/node/03F5KP94K0.md" tag="03F5KP94K0" kind="definition">
# Determinant of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *determinant* of $T$ is
$$\det T = \det [T]_B,$$
the [[determinant|00KDZRW4C0]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$.

## Remarks

The value does not depend on $B$, because another basis gives a [[similar|0E89HQATM8]] matrix $P^{-1}[T]_B P$, with the same determinant.
It satisfies $\det(S \circ T) = \det S \cdot \det T$ and $\det \id_V = 1$.
The map $T$ is invertible if and only if $\det T \ne 0$.
For $V = 0$, the convention is $\det T = 1$.

## Explanations

### L1 human

We define the determinant of a linear map with its matrix in some basis.
Another basis changes the matrix, but not the determinant.
So the determinant belongs to the map itself, like the trace.
</document>

<document source="https://math2.tamnd.com/node/03F7N3915D.md" tag="03F7N3915D" kind="concept">
# Cauchy–Euler equation

## Summary

A Cauchy–Euler equation is a linear equation $\sum_k a_k x^k y^{(k)} = 0$ with constant $a_k$, in which the $k$-th derivative has the factor $x^k$.
It has solutions of the form $x^m$, and the change of variable $x = e^t$ gives constant coefficients. See the [[definition|0WDNRBXX7K]].
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<document source="https://math2.tamnd.com/node/03FHD14RKC.md" tag="03FHD14RKC" kind="concept">
# Range of a function

## Summary

The range of a function is the set of all its values.
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<document source="https://math2.tamnd.com/node/03FKX7Z5DC.md" tag="03FKX7Z5DC" kind="concept">
# Irreducible polynomial

## Summary

An irreducible polynomial is a nonconstant polynomial that is not a product of two polynomials of smaller degree, over a field.
More generally, it is an irreducible element of a polynomial ring.
The [[definition of an irreducible polynomial|00X2A9G9W3]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/03FTB26QYD.md" tag="03FTB26QYD" kind="statement">
# Vector triple product

## Statement

Let $a$, $b$ and $c$ be vectors in $\mathbb{R}^3$. Let $\times$ be the [[cross product|0N5N15HWZ7]] and let $\cdot$ be the dot product. Then
$$a \times (b \times c) = (a \cdot c) \, b - (a \cdot b) \, c.$$

## Remarks

The formula is easy to remember as "BAC minus CAB": $a \times (b \times c) = b(a \cdot c) - c(a \cdot b)$.
The cross product is not associative. The formula gives $(a \times b) \times c = (a \cdot c) \, b - (b \cdot c) \, a$, which is in general different from $a \times (b \times c)$.
A consequence is the Jacobi identity $a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$.

## Explanations

### L1 human

The vector $b \times c$ is perpendicular to the plane of $b$ and $c$. A second cross product with $a$ gives a vector perpendicular to $b \times c$, so the result lies back in the plane of $b$ and $c$. The formula tells you exactly which combination of $b$ and $c$ it is.
</document>

<document source="https://math2.tamnd.com/node/03GGQEAG9S.md" tag="03GGQEAG9S" kind="statement">
# Chain rule for Jacobian matrices

## Statement

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open, let $g \colon U \to W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$.
Suppose that $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$.
Then the [[Jacobian matrices|0YF0VXKXGB]] satisfy
$$J_{f \circ g}(a) = J_f(g(a)) \, J_g(a),$$
a product of a $k \times m$ matrix and an $m \times n$ matrix.

## Remarks

This is the [[chain rule in several variables|0EAAS2EHDR]], written with matrices: the matrix of a composition of linear maps is the product of their matrices.
For $k = m = n$, taking determinants gives $\det J_{f \circ g}(a) = \det J_f(g(a)) \cdot \det J_g(a)$.
For example, let $g(r, \theta) = (r \cos \theta, r \sin \theta)$ and $f(x, y) = x^2 + y^2$. Then $J_f(g(r, \theta)) J_g(r, \theta) = (2r \cos \theta, \, 2r \sin \theta) J_g(r, \theta) = (2r, \, 0)$, which is the Jacobian matrix of $(f \circ g)(r, \theta) = r^2$.

## Explanations

### L1 human

Each Jacobian matrix turns small input steps into small output steps.
Doing two maps one after the other means multiplying their matrices, in the same order as the maps act, from right to left.
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<document source="https://math2.tamnd.com/node/03GP7KMBYM.md" tag="03GP7KMBYM" kind="definition">
# Absolutely convergent improper integral

## Definition

Let $f \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on $[a, b]$ for each $b > a$.
The [[improper integral|0VCTBJMEJF]] $\int_a^\infty f(x) \, dx$ converges absolutely if $\int_a^\infty |f(x)| \, dx$ converges.

## Remarks

An absolutely convergent improper integral converges. The proof uses $0 \le f + |f| \le 2|f|$ and the [[comparison test|0FR1BDKK10]].
The converse fails. The [[Dirichlet integral|0GZNSHMG4Q]] $\int_0^\infty \frac{\sin x}{x} \, dx$ converges, but not absolutely.
A function on $[a, \infty)$ whose improper integral converges absolutely is Lebesgue integrable there. The two integrals agree.

## Explanations

### L1 human

An improper integral converges absolutely when the area between the graph and the axis is finite, with all parts counted as positive.
This is the analogue of absolute convergence of series.
</document>

<document source="https://math2.tamnd.com/node/03GSZ8RTZJ.md" tag="03GSZ8RTZJ" kind="concept">
# Dimension

## Summary

The dimension of a vector space is the number of vectors in a basis of the space.
It does not depend on the basis.
</document>

<document source="https://math2.tamnd.com/node/03H7WZ4T3K.md" tag="03H7WZ4T3K" kind="statement">
# Pythagorean trigonometric identity

## Statement

For every real $x$,
$$\sin^2 x + \cos^2 x = 1.$$

## Remarks

Let $f(x) = \sin^2 x + \cos^2 x$. By the [[derivatives|00R4MWW532]] [[of sine and cosine|0V2DV29EGT]], $f'(x) = 2 \sin x \cos x - 2 \cos x \sin x = 0$. So $f$ is [[constant|003QDZ4T16]], and $f(0) = 1$.
So $|\sin x| \le 1$ and $|\cos x| \le 1$.
Geometrically, the point $(\cos x, \sin x)$ lies on the unit circle. Mathlib states it as `Real.sin_sq_add_cos_sq`.

## Explanations

### L1 human

In a right triangle with hypotenuse $1$, the legs are $\cos x$ and $\sin x$.
The Pythagorean theorem then says that their squares add up to $1$.
</document>

<document source="https://math2.tamnd.com/node/03HDF50VN4.md" tag="03HDF50VN4" kind="concept">
# P-adic valuation

## Summary

The $p$-adic valuation of a nonzero integer $n$, for a prime $p$, is the exponent of the highest power of $p$ that divides $n$.
The [[definition of the p-adic valuation|04VCPS30ZE]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/03HF498P1J.md" tag="03HF498P1J" kind="concept">
# Antichain

## Summary

An antichain in a partially ordered set is a subset in which no two different elements are comparable.
</document>

<document source="https://math2.tamnd.com/node/03HF5PMY5T.md" tag="03HF5PMY5T" kind="statement">
# Triangle inequality

## Statement

For all real numbers $a$ and $b$,
$$|a + b| \le |a| + |b|,$$
where $|\cdot|$ is the [[absolute value|00S7C68D51]].

## Remarks

Proof: $-|a| \le a \le |a|$ and $-|b| \le b \le |b|$. Add these inequalities to get $-(|a| + |b|) \le a + b \le |a| + |b|$.
Equality holds exactly when $a b \ge 0$.
For distances, it gives $|x - z| \le |x - y| + |y - z|$. The [[reverse triangle inequality|0C9S13WKMF]] is a consequence.

## Explanations

### L1 human

The name comes from geometry: one side of a triangle is never longer than the other two sides together.
For numbers, $|a + b|$ can be smaller than $|a| + |b|$ when $a$ and $b$ have opposite signs, for example $|5 + (-3)| = 2 < 8$.
</document>

<document source="https://math2.tamnd.com/node/03HGK7EW6R.md" tag="03HGK7EW6R" kind="statement">
# Mean value theorem in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \in U$ be points such that the segment $\{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$.
Then there is a number $c \in (0, 1)$ such that
$$f(y) - f(x) = \nabla f\big(x + c (y - x)\big) \cdot (y - x).$$

## Remarks

The proof applies the one-variable mean value theorem to $g(t) = f(x + t(y - x))$ on $[0, 1]$, with $g'(t) = \nabla f(x + t(y - x)) \cdot (y - x)$ by the [[chain rule for a curve|0A8TJ0VHC7]].
The theorem is false for maps into $\mathbb{R}^m$ with $m \ge 2$. For $f(t) = (\cos t, \sin t)$ on $[0, 2\pi]$, $f(2\pi) - f(0) = 0$, but $f'(t) \ne 0$ for each $t$. The [[mean value inequality|02CYB184RC]] replaces it.
By the Cauchy–Schwarz inequality, it gives $|f(y) - f(x)| \le \sup_{z} \|\nabla f(z)\| \, \|y - x\|$, with $z$ on the segment.

## Explanations

### L1 human

On a straight path from $x$ to $y$, the average slope of $f$ equals the actual slope at some point between them.
This reduces the problem to one variable, by looking at $f$ only along the segment.
</document>

<document source="https://math2.tamnd.com/node/03HMJASQBQ.md" tag="03HMJASQBQ" kind="concept">
# Regular space

## Summary

A topological space is regular when each closed set and each point outside it lie in disjoint open sets.
</document>

<document source="https://math2.tamnd.com/node/03J4NZFYRX.md" tag="03J4NZFYRX" kind="definition">
# Perpendicular bisector

## Definition

Let $A$ and $B$ be distinct points of the Euclidean space $\mathbb{R}^n$, and let $M = \frac{1}{2}(A + B)$ be their [[midpoint|0S706JBZBG]]. Let $\cdot$ be the dot product.
The perpendicular bisector of $A$ and $B$ is the set
$$\{\, X \in \mathbb{R}^n \;:\; (X - M) \cdot (B - A) = 0 \,\}.$$

## Remarks

A point $X$ is on the perpendicular bisector if and only if $d(X, A) = d(X, B)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].
It is a hyperplane with the [[normal vector|0WQJJCCTC8]] $B - A$. In the plane it is a line, and in $\mathbb{R}^3$ it is a plane.
Mathlib's `AffineSubspace.perpBisector` also allows $A = B$. Then the set is the whole space.
The perpendicular bisectors of the three sides of a triangle in the plane meet at the [[circumcentre|0F98FZK14A]].

## Explanations

### L1 human

The perpendicular bisector of a segment cuts it in half at a right angle. Each point on it is as far from one end of the segment as from the other. For the points $(0, 0)$ and $(2, 0)$, the perpendicular bisector is the vertical line $x = 1$.
</document>

<document source="https://math2.tamnd.com/node/03J5RAKB30.md" tag="03J5RAKB30" kind="concept">
# Normed vector space

## Summary

A normed vector space is a real or complex vector space together with a norm.
</document>

<document source="https://math2.tamnd.com/node/03J8BS7FMV.md" tag="03J8BS7FMV" kind="definition">
# Separable polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f \in R[x]$.
The polynomial $f$ is separable if there are $a, b \in R[x]$ with $a f + b f' = 1$, where $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.

## Remarks

Over a [[field|0QA8CHH8MG]] $K$, $f$ is separable if and only if $f$ has no repeated root in an [[algebraic closure|0MCXX29HWY]] of $K$.
The polynomial $f = x^2 - 2$ is separable over $\mathbb{Q}$, because $f' = 2x$ and $-\tfrac{1}{2} f + \tfrac{1}{4} x f' = 1$.
Over $\mathbb{F}_p(t)$, the irreducible polynomial $x^p - t$ is not separable, because its derivative is $0$.
Some older texts call an irreducible polynomial separable when it has distinct roots, and extend this to other polynomials factor by factor.

## Explanations

### L1 human

A separable polynomial has distinct roots, in a field that is large enough to contain all of them.
Over the rational numbers, every irreducible polynomial is separable.
Inseparable irreducible polynomials only appear in prime characteristic.
</document>

<document source="https://math2.tamnd.com/node/03KGGKTB2E.md" tag="03KGGKTB2E" kind="statement">
# Wilson's theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $(p - 1)! \equiv -1 \pmod p$.

## Remarks

A proof: in $\mathbb{Z}/p\mathbb{Z}$, each element of $\{ \overline{1}, \dots, \overline{p - 1} \}$ has an inverse. Only $\overline{1}$ and $\overline{-1}$ are their own inverses. The other elements cancel in pairs in the product.
For example, $4! = 24 \equiv -1 \pmod 5$ and $6! = 720 \equiv -1 \pmod 7$.
The [[converse of Wilson's theorem|06WRV8HFDY]] also holds, so the congruence characterizes the primes.

## History

Ibn al-Haytham stated the result about the year 1000. Edward Waring announced it in 1770 and named his student John Wilson as its discoverer. Lagrange gave the first proof in 1771.

## Explanations

### L1 human

Multiply all the numbers from $1$ to $p - 1$, and divide by $p$. The remainder is always $p - 1$.
For $p = 7$: $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 720 = 102 \cdot 7 + 6$.
</document>

<document source="https://math2.tamnd.com/node/03MC70ZRNR.md" tag="03MC70ZRNR" kind="statement">
# Equivalence of norms in finite dimension

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$.
Then any two norms on $V$ are [[equivalent|0J3578562S]].

## Remarks

Proof idea: fix a basis and compare a norm $\|\cdot\|$ with the [[sup norm|0KNJWEVSMG]] $\|\cdot\|_\infty$ of the coordinates. The triangle inequality gives $\|x\| \le C \|x\|_\infty$, so $\|\cdot\|$ is continuous for $\|\cdot\|_\infty$. The unit sphere of $\|\cdot\|_\infty$ is compact by the [[Heine–Borel theorem|07PFYV65MB]], so $\|\cdot\|$ has a positive minimum $c$ on it, by the [[extreme value theorem|0V4DET59NW]].
Thus there is only one norm topology on $\mathbb{R}^n$, the Euclidean topology.
The statement fails in infinite dimensions: each infinite-dimensional normed space has norms that are not equivalent.

## Explanations

### L1 human

In finite dimensions, it does not matter which norm you choose: all norms give the same convergent sequences and the same open sets.
</document>

<document source="https://math2.tamnd.com/node/03MTBVG3VE.md" tag="03MTBVG3VE" kind="statement">
# de Moivre's formula

## Statement

For every real number $x$ and every integer $n$,
$$(\cos x + i \sin x)^n = \cos(n x) + i \sin(n x).$$

## Remarks

Proof: by [[Euler's formula|04R3CY41VJ]], the left side is $(e^{i x})^n = e^{i n x}$, which is the right side.
For $n = 2$, comparing real and imaginary parts gives $\cos 2x = \cos^2 x - \sin^2 x$ and $\sin 2x = 2 \sin x \cos x$.
For non-integer exponents $n$, the formula fails in general.

## Explanations

### L1 human

Raising a point of the unit circle to the power $n$ multiplies its angle by $n$.
For example, $\cos(\pi/4) + i \sin(\pi/4)$ points at $45$ degrees, and its fourth power points at $180$ degrees, so it is $-1$.
</document>

<document source="https://math2.tamnd.com/node/03N8C8ZR4B.md" tag="03N8C8ZR4B" kind="concept">
# Type I and type II errors

## Summary

A type I error is the rejection of a true null hypothesis, a false positive. A type II error is the failure to reject a false null hypothesis, a false negative.
The node [[type I and type II errors|0R41PVFA39]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/03NAMV1DHJ.md" tag="03NAMV1DHJ" kind="concept">
# Path

## Summary

A path in a graph is a walk that visits each vertex at most once. Paths measure distance and connectivity in graphs. The node [[path|0JDE56HWQ8]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/03NDVCSHSB.md" tag="03NDVCSHSB" kind="concept">
# Minimal polynomial

## Summary

The minimal polynomial of an endomorphism of a finite-dimensional space is the monic polynomial of least degree that annihilates it.
</document>

<document source="https://math2.tamnd.com/node/03NK6RSRQ1.md" tag="03NK6RSRQ1" kind="statement">
# Vectors are perpendicular if and only if their dot product is zero

## Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$, with dot product $u \cdot v = u_1 v_1 + \dots + u_n v_n$.
Let $\theta$ be the [[angle between $u$ and $v$|0ZEHZY85E6]].
Then $u \cdot v = 0$ if and only if $\theta = \pi / 2$.

## Remarks

The proof uses the [[geometric formula for the dot product|070TXS9AXM]]: $u \cdot v = \|u\| \, \|v\| \cos \theta$ with $\|u\| \neq 0$ and $\|v\| \neq 0$, and $\cos \theta = 0$ for $0 \leq \theta \leq \pi$ only when $\theta = \pi / 2$.
Two vectors $u$ and $v$ with $u \cdot v = 0$ are called orthogonal. With this word, the zero vector is orthogonal to each vector.
In Mathlib the angle with the zero vector is $\pi / 2$, so the equivalence holds there for all vectors.

## Explanations

### L1 human

This gives a quick test for a right angle. The vectors $(2, 1)$ and $(-1, 2)$ have dot product $-2 + 2 = 0$, so they are perpendicular. You do not need to draw them or measure an angle.
</document>

<document source="https://math2.tamnd.com/node/03NM7DZXXC.md" tag="03NM7DZXXC" kind="example">
# Torus

## Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, y)$ and $(x, 0) \sim (x, 1)$ for all $x, y \in [0, 1]$.
The quotient space $[0, 1]^2 / {\sim}$, with the [[quotient topology|0V8046R0P9]], is the torus $T^2$.

## Remarks

The torus is homeomorphic to the product $S^1 \times S^1$ of two [[circles|0WH1GVBJNH]], with the [[product topology|0Z57EC6YX3]], and to $\mathbb{R}^2 / \mathbb{Z}^2$.
It is also homeomorphic to a surface in $\mathbb{R}^3$: turn a circle about an axis in its plane that does not meet the circle.
The torus is a compact, connected surface. It is not homeomorphic to the sphere.

## Explanations

### L1 human

Take a square sheet. Glue the left edge to the right edge to get a tube, and then glue the two end circles of the tube.
The result is the surface of a doughnut.
</document>

<document source="https://math2.tamnd.com/node/03PGHKAJG3.md" tag="03PGHKAJG3" kind="statement">
# Cauchy–Schwarz inequality for expectations

## Statement

Let $X$ and $Y$ be real random variables on one probability space with $E[X^2] < \infty$ and $E[Y^2] < \infty$. Then $XY$ is integrable and
$$\left(E[XY]\right)^2 \le E[X^2] \, E[Y^2].$$
Equality holds if and only if $aX + bY = 0$ almost surely for some reals $a$ and $b$, not both $0$.

## Remarks

Proof: for each real $s$, $0 \le E[(X + sY)^2] = E[X^2] + 2s E[XY] + s^2 E[Y^2]$. A quadratic polynomial in $s$ that is never negative has a discriminant that is not positive.
This is the Cauchy–Schwarz inequality for the inner product $\langle X, Y \rangle = E[XY]$ on the space $L^2$ of square-integrable random variables.
Applied to $X - E[X]$ and $Y - E[Y]$, it gives $\operatorname{Cov}(X, Y)^2 \le \operatorname{Var}(X) \operatorname{Var}(Y)$. So the correlation coefficient is in $[-1, 1]$.

## Explanations

### L1 human

The average of a product cannot be larger than what the sizes of the two factors allow.
This simple inequality is behind the fact that a correlation is always between $-1$ and $1$.
</document>

<document source="https://math2.tamnd.com/node/03PPNCM9GG.md" tag="03PPNCM9GG" kind="concept">
# Second-countable space

## Summary

A topological space is second-countable when its topology has a countable basis.
This condition limits the size of the topology and holds for Euclidean spaces.
</document>

<document source="https://math2.tamnd.com/node/03Q7RE2HXR.md" tag="03Q7RE2HXR" kind="statement">
# A matrix is invertible iff its determinant is a unit

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
Then $A$ is [[invertible|0EB9N0S33C]] if and only if its [[determinant|00KDZRW4C0]] $\det A$ is a unit of $K$.
In particular, over a field, $A$ is invertible if and only if $\det A \ne 0$.

## Remarks

If $\det A$ is a unit, then the [[adjugate|07W9H8QF38]] gives the inverse $A^{-1} = (\det A)^{-1} \operatorname{adj}(A)$.
Over $\Z$, the matrix $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ has determinant $2 \ne 0$, but it has no inverse with integer entries.
Over a field, this is one of many equivalent conditions for invertibility, with $\rank A = n$ and $\ker A = 0$.

## Explanations

### L1 human

A real square matrix has an inverse exactly when its determinant is not zero.
A zero determinant means that the matrix squeezes space into a lower dimension.
Such a squeeze cannot be undone.
</document>

<document source="https://math2.tamnd.com/node/03QC59M99B.md" tag="03QC59M99B" kind="statement">
# Complement principle

## Statement

Let $U$ be a finite set and let $A \subseteq U$.
Let $A^c = U \setminus A$ be the complement of $A$ in $U$.
Then $|A^c| = |U| - |A|$.

## Remarks

The rule follows from the [[rule of sum|0EE2V3CHFK]], because $U$ is the disjoint union of $A$ and $A^c$.
It is useful when the complement is easier to count. For example, the number of words of length $n$ over $\{0, 1\}$ that contain at least one $1$ is $2^n - 1$.

## Explanations

### L1 human

Sometimes it is easier to count what you do not want.
To count the outcomes with at least one six in two dice rolls, count the outcomes with no six: $5 \cdot 5 = 25$. Then $36 - 25 = 11$ outcomes have at least one six.
</document>

<document source="https://math2.tamnd.com/node/03QHBVW4TT.md" tag="03QHBVW4TT" kind="statement">
# Taylor's theorem

## Statement

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
Let $P_k$ be the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.
Then there is a function $h_k \colon I \to \mathbb{R}$ with $h_k(x) \to 0$ as $x \to a$ and
$$f(x) = P_k(x) + h_k(x) (x - a)^k \quad \text{for all } x \in I.$$

## Remarks

In [[little o notation|04TSHB9H8D]], the conclusion is the [[Peano form|03AJHX431A]] $f(x) - P_k(x) = o((x - a)^k)$.
Under stronger hypotheses, the remainder has explicit forms: the [[Lagrange form|0H5ZAZCX2T]], the [[Cauchy form|005Q84BMAD]], and the [[integral form|0WJGR81HBD]].
For $k = 1$, it says that $f(x) = f(a) + f'(a)(x - a) + o(x - a)$.

## Explanations

### L1 human

Near $a$, a function is close to its Taylor polynomial.
The error is small even compared with $(x - a)^k$.
</document>

<document source="https://math2.tamnd.com/node/03RNVFDEAN.md" tag="03RNVFDEAN" kind="definition">
# Skew-symmetric matrix

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *skew-symmetric* when $A^T = -A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.
Equivalently, $a_{ji} = -a_{ij}$ for all $i$ and $j$.

## Remarks

If $2$ is invertible in $R$, for example over $\R$, then each diagonal entry of a skew-symmetric matrix is $0$.
Over a field of characteristic not $2$, each square matrix is the sum of a [[symmetric matrix|0VR671PHE5]] and a skew-symmetric matrix: $A = \tfrac12 (A + A^T) + \tfrac12 (A - A^T)$.
A real skew-symmetric matrix of odd order has determinant $0$.
The matrix $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$ is skew-symmetric.

## Explanations

### L1 human

In a skew-symmetric matrix, the entry across the diagonal is the negative of the entry here.
Over the real numbers this forces zeros on the diagonal, because only $0$ equals its own negative.
These matrices describe infinitesimal rotations.
</document>

<document source="https://math2.tamnd.com/node/03RSJXF728.md" tag="03RSJXF728" kind="statement">
# Formula for the number of derangements

## Statement

Let $n$ be a natural number, and let $D_n$ be the number of [[derangements|0TH61JR04G]] of an $n$-element set.
Then
$$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!} = \sum_{k=0}^{n} (-1)^k \frac{n!}{k!}.$$

## Remarks

For $n = 4$: $24 - 24 + 12 - 4 + 1 = 9$.
The proof applies the [[inclusion–exclusion principle|0FF7726C7P]] to the sets $A_i$ of permutations that fix $i$.
The sum is a partial sum of the series of $e^{-1}$. Thus $D_n / n! \to 1/e$, and for $n \ge 1$, $D_n$ is the integer nearest to $n!/e$.

## Explanations

### L1 human

Start with all $n!$ permutations. Remove those that fix some point, add back those that fix two points, and so on.
The result is the alternating sum in the formula.
</document>

<document source="https://math2.tamnd.com/node/03S6GWWKEZ.md" tag="03S6GWWKEZ" kind="statement">
# Limit of a quotient of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers with $b \ne 0$.
Assume that $a_n \to a$ and $b_n \to b$ as $n \to \infty$.
Then there is a natural number $N$ such that $b_n \ne 0$ for all $n \ge N$.
Moreover, the sequence $(a_n / b_n)_{n \ge N}$ converges to $a / b$.

## Remarks

The main step is $1/b_n \to 1/b$. For large $n$ we have $|b_n| > |b|/2$, and then $|1/b_n - 1/b| = |b - b_n| / (|b_n|\,|b|) < 2 |b_n - b| / |b|^2$.
The [[limit of a product|0734RSVF2V]] of $(a_n)$ and $(1/b_n)$ then gives the result.
The hypothesis $b \ne 0$ is needed. For $a_n = 1$ and $b_n = 1/n$, the quotient is $n$, which has no real limit.

## Explanations

### L1 human

The limit of a quotient is the quotient of the limits, when the limit of the denominator is not zero.
For example, $(3 + 1/n)/(2 + 1/n) \to 3/2$.
</document>

<document source="https://math2.tamnd.com/node/03TM135G0N.md" tag="03TM135G0N" kind="concept">
# Continuous linear map

## Summary

A continuous linear map between normed spaces is a linear map that is continuous for the norm topologies.
Such maps are the same as the bounded linear maps, which multiply lengths by at most a fixed constant.
</document>

<document source="https://math2.tamnd.com/node/03VFZ2WETT.md" tag="03VFZ2WETT" kind="concept">
# Integers modulo n

## Summary

The integers modulo $n$ form the ring $\mathbb{Z}/n\mathbb{Z}$ of the $n$ residue classes modulo $n$, with the addition and the multiplication of representatives.
The [[definition of the integers modulo n|01N3AYAA6N]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/03VHANA6ZK.md" tag="03VHANA6ZK" kind="definition">
# Indefinite integral

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ have an [[antiderivative|0BW9N5CDSB]] $F$ on $I$.
The indefinite integral of $f$ is the set of all antiderivatives of $f$ on $I$.
It is written
$$\int f(x) \, dx = F(x) + C,$$
where $C$ stands for an arbitrary real constant.

## Remarks

The constant $C$ is the constant of integration.
For example, $\int \cos x \, dx = \sin x + C$.
Some texts use the name for the function $x \mapsto \int_a^x f(t) \, dt$, for a fixed $a$.

## Explanations

### L1 human

The indefinite integral collects all the antiderivatives of a function.
They differ only by a constant, so one antiderivative plus $C$ describes them all.
</document>

<document source="https://math2.tamnd.com/node/03VZJE6C8C.md" tag="03VZJE6C8C" kind="concept">
# Circumcircle

## Summary

The circumcircle of a triangle is the circle that passes through its three vertices. Each triangle has exactly one circumcircle. Its centre is the circumcentre and its radius is the circumradius.
</document>

<document source="https://math2.tamnd.com/node/03W6Q1MFR5.md" tag="03W6Q1MFR5" kind="definition">
# Jacobian determinant

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^n$, and let $a \in U$ be a point at which all first partial derivatives of $f$ exist.
The Jacobian determinant of $f$ at $a$ is the determinant
$$\det J_f(a) = \det \left( \frac{\partial f_i}{\partial x_j}(a) \right)_{1 \le i, j \le n}$$
of the [[Jacobian matrix|0YF0VXKXGB]] of $f$ at $a$.
It is also written $\dfrac{\partial (f_1, \dots, f_n)}{\partial (x_1, \dots, x_n)}(a)$, and it is often called the Jacobian of $f$.

## Remarks

For the polar map $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$, the Jacobian determinant is $r \cos^2 \theta + r \sin^2 \theta = r$.
If $f$ is of class $C^1$ and $\det J_f(a) \ne 0$, then $f$ has a differentiable local inverse near $a$, by the [[inverse function theorem|0K9HSJFCMJ]].
The absolute value $|\det J_f|$ is the factor in the [[change of variables formula|00N5FHP1CV]] for multiple integrals.

## Explanations

### L1 human

A linear map of the plane sends the unit square to a parallelogram, and the determinant is the signed area of that parallelogram.
The Jacobian determinant does the same for a curved map, at the scale of tiny squares near one point.
</document>

<document source="https://math2.tamnd.com/node/03WKB2H474.md" tag="03WKB2H474" kind="example">
# Complete graph

## Example

Let $n$ be a natural number.
The complete graph $K_n$ is the [[graph|0JD5C82QMB]] with vertex set $\{1, \dots, n\}$ in which any two distinct vertices are adjacent.

More generally, the complete graph on a set $V$ has an edge between each pair of distinct elements of $V$.

## Remarks

The graph $K_n$ is $(n-1)$-[[regular|0KA4S8R39M]], and it has $\binom{n}{2}$ edges. The node [[number of edges of a complete graph|05RN90E14Y]] states this.
$K_3$ is a triangle. $K_4$ is planar, and [[$K_5$ is not planar|030A3FHN14]].
A clique in a graph is a set of vertices that induces a complete graph.

## Explanations

### L1 human

In a complete graph every vertex is joined to every other vertex.
If $5$ people all shake hands with each other, the handshakes form $K_5$.
</document>

<document source="https://math2.tamnd.com/node/03WY4HHMT1.md" tag="03WY4HHMT1" kind="definition">
# Number of derangements

## Definition

The derangement numbers $D_n$, for natural numbers $n$, are defined by the recursion
$$D_0 = 1, \qquad D_1 = 0, \qquad D_{n+2} = (n + 1)\,(D_n + D_{n+1}).$$

## Remarks

The first values, for $n = 0, 1, \dots, 6$, are $1, 0, 1, 2, 9, 44, 265$.
The value $D_n$ is the number of [[derangements|0TH61JR04G]] of an $n$-element set. The node [[the derangement numbers count derangements|045ENVCCZV]] states this.
Another notation is $!n$, the subfactorial of $n$.

## Explanations

### L1 human

The derangement numbers start $1, 0, 1, 2, 9, 44$.
Each new number comes from the two before it: $D_4 = 3 \cdot (D_2 + D_3) = 3 \cdot 3 = 9$.
</document>

<document source="https://math2.tamnd.com/node/03X8V7D64M.md" tag="03X8V7D64M" kind="statement">
# Sum of a finite geometric progression

## Statement

Let $x$ be a real number with $x \neq 1$, and let $n$ be a natural number.
Then
$$\sum_{k=0}^{n-1} x^k = 1 + x + x^2 + \dots + x^{n-1} = \frac{x^n - 1}{x - 1}.$$

## Remarks

The same formula holds for $x$ in any field, for example for complex numbers, when $x \neq 1$.
For $x = 1$, the sum is $n$.
A short proof: $(x - 1)(1 + x + \dots + x^{n-1}) = x^n - 1$, because the other terms cancel.

## Explanations

### L1 human

In a geometric progression, you get each term from the one before by multiplying by the same number $x$.
For $x = 2$, the formula gives $1 + 2 + 4 + 8 = 2^4 - 1 = 15$.
Multiply the sum by $x$ and subtract the sum: almost all terms cancel, and only $x^n - 1$ stays.
</document>

<document source="https://math2.tamnd.com/node/03XVP8Q4GY.md" tag="03XVP8Q4GY" kind="definition">
# Multiplicity of a root

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ with $f \neq 0$ and let $c \in R$.
The multiplicity of $c$ as a root of $f$ is the largest integer $m \ge 0$ such that $(x - c)^m$ divides $f$.

## Remarks

The integer $m$ exists, because $(x - c)^m$ is monic of degree $m$, so it cannot divide $f$ for $m > \deg f$.
The multiplicity is positive exactly when $c$ is a [[root|00EHX6FWC0]] of $f$.
A root of multiplicity $1$ is a simple root. A root of multiplicity at least $2$ is a repeated root.
In $\mathbb{Q}[x]$, $f = (x - 1)^3 (x + 2)$ has the root $1$ with multiplicity $3$ and the root $-2$ with multiplicity $1$.

## Explanations

### L1 human

The multiplicity counts how many times the factor $x - c$ appears in $f$.
For $(x - 1)^2 (x + 3)$, the root $1$ counts twice and the root $-3$ counts once.
</document>

<document source="https://math2.tamnd.com/node/03Y6AQ2S0J.md" tag="03Y6AQ2S0J" kind="concept">
# Sample mean

## Summary

The sample mean of a sample is the arithmetic mean of its values. It is the usual estimator of the mean of the population.
The node [[sample mean|0QNRBHWV35]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/03YEM7JEDN.md" tag="03YEM7JEDN" kind="statement">
# Open mapping theorem

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Assume that $f$ is not constant on $D$.
Then $f$ is an open map: for each open set $V \subseteq D$, the image $f(V)$ is an open subset of $\mathbb{C}$.

## Remarks

The [[maximum modulus principle|048F2P90WR]] follows. If $|f|$ had a local maximum at $z_0$, then $f$ would map a small disc around $z_0$ onto a set that contains no disc around $f(z_0)$.
The connectedness of $D$ matters only to exclude functions that are constant on one part of $D$.
There is no real analogue: $x \mapsto x^2$ maps $(-1, 1)$ onto $[0, 1)$, which is not open in $\mathbb{R}$. In contrast, $z \mapsto z^2$ maps the unit disc onto the unit disc.

## Explanations

### L1 human

A nonconstant holomorphic function never squeezes an open set into a curve or a point.
Around each point of the image, the image also contains a small disc.
</document>

<document source="https://math2.tamnd.com/node/03YN70NK0Z.md" tag="03YN70NK0Z" kind="concept">
# Oriented angle

## Summary

An oriented angle in the plane measures the turn from one direction to another, with a sign. Anticlockwise turns are positive and clockwise turns are negative. It is defined modulo a full turn $2\pi$, and it changes sign when you swap the two directions.
</document>

<document source="https://math2.tamnd.com/node/03Z4JC4YRJ.md" tag="03Z4JC4YRJ" kind="counterexample">
# The Dirichlet function is not Riemann integrable

## Counterexample

Let $f \colon [0, 1] \to \mathbb{R}$ be the [[Dirichlet function|0T4EJ8ZJM9]]: $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$.
Each subinterval of positive length contains both rational and irrational numbers. So for each partition $P$, $U(f, P) = 1$ and $L(f, P) = 0$.
The upper [[Darboux integral|09FQT057T7]] is $1$ and the lower one is $0$. So $f$ is [[not Riemann integrable|021XJAQVDC]].

## Remarks

The function is bounded, so the failure is not caused by unboundedness.
The function is Lebesgue integrable, with Lebesgue integral $0$, since $\mathbb{Q}$ has measure zero.
It is the pointwise limit of Riemann integrable functions. So pointwise limits do not keep Riemann integrability.

## Explanations

### L1 human

Every Riemann sum of this function is $1$ or $0$ or in between, depending only on the choice of tags.
So the sums cannot settle down to one number.
</document>

<document source="https://math2.tamnd.com/node/03Z9V0Y78A.md" tag="03Z9V0Y78A" kind="statement">
# C^1 vector fields satisfy the Picard–Lindelöf hypotheses

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, let $x_0 \in U$ and let $t_0 \in \mathbb{R}$.
Then there are real numbers $\varepsilon > 0$, $a > 0$ and $L, K \ge 0$ with these properties.
The closed ball $B$ of radius $a$ around $x_0$ lies in $U$.
The time-independent field $(t, x) \mapsto f(x)$, extended in any way outside $B$, satisfies the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ on $[t_0 - \varepsilon, t_0 + \varepsilon]$ with constants $a, L, K$.

## Remarks

Proof: choose $a > 0$ with $B \subseteq U$. The continuous functions $\|f\|$ and $\|f'\|$ are bounded on the compact ball $B$, by $L$ and $K$. The mean value inequality makes $f$ $K$-Lipschitz on $B$. Then choose $\varepsilon$ with $L \varepsilon \le a$.
So the initial value problem $x' = f(x)$, $x(t_0) = x_0$ has a unique local solution for each $C^1$ field $f$.
The same argument works for a time-dependent field $f(t, x)$ of class $C^1$.

## Explanations

### L1 human

A smooth field has a bounded derivative near each point, so it is Lipschitz near this point.
This is why, in practice, one rarely checks the Lipschitz condition directly: smoothness is enough.
</document>

<document source="https://math2.tamnd.com/node/0404D8TN7Y.md" tag="0404D8TN7Y" kind="statement">
# Binet's formula

## Statement

Let $n$ be a natural number, and let $F_n$ be the $n$-th [[Fibonacci number|0GA9K57A39]].
Let $\varphi = \frac{1 + \sqrt{5}}{2}$ be the [[golden ratio|0SC8T7548D]] and let $\psi = \frac{1 - \sqrt{5}}{2}$.
Then
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}.$$

## Remarks

The proof uses the [[characteristic polynomial|0BWDJGQYST]] $x^2 - x - 1$ of the recurrence. Its roots are $\varphi$ and $\psi$, so $\varphi^n$ and $\psi^n$ are [[solutions|096SD4ERQH]], and the initial values fix the coefficients.
Because $|\psi^n / \sqrt{5}| < \frac{1}{2}$, $F_n$ is the integer nearest to $\varphi^n / \sqrt{5}$.

## Explanations

### L1 human

Fibonacci numbers are integers, but this formula uses $\sqrt{5}$, and the irrational parts cancel.
The term with $\psi^n$ is very small, so $F_n$ is close to $\varphi^n / \sqrt{5}$.
</document>

<document source="https://math2.tamnd.com/node/0404R9P9J3.md" tag="0404R9P9J3" kind="definition">
# Integer partition

## Definition

Let $n$ be a natural number.
A partition of $n$ is a finite multiset of positive integers whose sum is $n$.
Its elements are the parts of the partition.

Equivalently, a partition of $n$ is a finite sequence $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_k$ of positive integers with $\lambda_1 + \dots + \lambda_k = n$.

## Remarks

The order of the parts does not matter. A sum in which the order matters is a [[composition|0J8ANSYKAC]].
The $5$ partitions of $4$ are $4$, $3 + 1$, $2 + 2$, $2 + 1 + 1$ and $1 + 1 + 1 + 1$.
The number $0$ has exactly one partition, the empty partition.
The number of partitions of $n$ is the [[partition function|001CX45SWQ]] $p(n)$, and a [[Young diagram|0K78D81EF7]] draws a partition.

## Explanations

### L1 human

A partition of $n$ is a way to write $n$ as a sum of positive integers when the order of the summands does not matter.
So $3 + 1$ and $1 + 3$ are the same partition of $4$.
</document>

<document source="https://math2.tamnd.com/node/040CFNJEAP.md" tag="040CFNJEAP" kind="statement">
# Lebesgue number lemma

## Statement

Let $(X, d)$ be a [[compact|0QPHW703PV]] [[metric space|063W5JMY85]] and let $(U_i)_{i \in I}$ be an [[open cover|0NVPNC12JY]] of $X$.
Then there is a real $\delta > 0$ such that each subset $A \subseteq X$ with [[diameter|0692CGQF00]] $\operatorname{diam}(A) < \delta$ is contained in $U_i$ for some $i \in I$.

## Remarks

Such a number $\delta$ is a Lebesgue number of the cover.
Equivalently, for each $x \in X$, the ball $B(x, \delta / 2)$ lies in one of the sets $U_i$.
Compactness is needed: the cover of $(0, 1)$ by the sets $(1/n, 1)$ has no Lebesgue number.

## Explanations

### L1 human

In a compact metric space, each open cover has a uniform size: each small enough set fits completely into one set of the cover.
You can think of $\delta$ as a safe size for a window that you move around the space.
</document>

<document source="https://math2.tamnd.com/node/040WY0TT2G.md" tag="040WY0TT2G" kind="concept">
# Lattice path

## Summary

A lattice path is a finite sequence of points of the integer lattice $\mathbb{Z}^2$. Each step between consecutive points is from a given finite set of steps.
The most common case uses the steps $E = (1, 0)$ and $N = (0, 1)$. These are North-East lattice paths.

A North-East path from $(0, 0)$ to $(a, b)$, with $a, b \in \mathbb{N}$, has $a$ steps $E$ and $b$ steps $N$.
It is a word of length $a + b$ in $E$ and $N$. Thus the number of such paths is the [[binomial coefficient|092GEF38YD]] $\binom{a + b}{a}$.

## Remarks

The number of North-East paths from $(0, 0)$ to $(n, n)$ is the [[central binomial coefficient|01WHD9T0V6]] $\binom{2n}{n}$.
The paths of this type that never go above the diagonal $y = x$ are counted by the [[Catalan numbers|092DRBFKMG]]. They correspond to [[Dyck words|0R9JR0465V]].

## Explanations

### L1 human

Think of a city grid where you may only walk east or north.
To go 3 blocks east and 2 blocks north, you choose which 2 of your 5 steps go north, so there are $\binom{5}{2} = 10$ routes.
</document>

<document source="https://math2.tamnd.com/node/0419794R0T.md" tag="0419794R0T" kind="definition">
# Entire function

## Definition

An entire function is a function $f : \mathbb{C} \to \mathbb{C}$ that is [[holomorphic|0DG0Z72CQD]] on all of $\mathbb{C}$.

## Remarks

Polynomials, the [[complex exponential|054VXHMCA3]] and the complex [[sine and cosine|0A7YDKRNR6]] are entire.
Sums, products and compositions of entire functions are entire.
An entire function is equal to one power series $\sum_{n \ge 0} a_n z^n$ that converges for every $z \in \mathbb{C}$.
By [[Liouville's theorem|0EHKVRF4J6]], each bounded entire function is constant.

## Explanations

### L1 human

An entire function is a function that is holomorphic at every complex number, with no exceptions.
For example, $z^3 - 2z + 5$ and $e^z$ are entire, but $1/z$ is not, because it is not defined at $0$.
</document>

<document source="https://math2.tamnd.com/node/0419GJC84P.md" tag="0419GJC84P" kind="definition">
# Nilpotent matrix

## Definition

Let $R$ be a ring and let $N$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $N$ is *nilpotent* when $N^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $N$.

## Remarks

The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent of index $2$.
Each strictly upper triangular matrix is nilpotent.
Over a field, an $n \times n$ matrix $N$ is nilpotent if and only if its characteristic polynomial is $x^n$; then $N^n = 0$.
The only eigenvalue of a nilpotent matrix over a field is $0$, so its trace and determinant are $0$.

## Explanations

### L1 human

A nilpotent matrix becomes the zero matrix after we multiply it by itself enough times.
A shift that moves each coordinate one step down, and drops the last one, is an example.
After $n$ shifts nothing is left.
</document>

<document source="https://math2.tamnd.com/node/041JTZCK9Q.md" tag="041JTZCK9Q" kind="definition">
# Direct sum of subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U_1, \dots, U_k$ be subspaces of $V$.
The space $V$ is the *direct sum* of $U_1, \dots, U_k$, written $V = U_1 \oplus \cdots \oplus U_k$, when each $v \in V$ can be written as
$$v = u_1 + \cdots + u_k, \qquad u_i \in U_i,$$
in exactly one way.
Equivalently, $V = U_1 + \cdots + U_k$, and $u_1 + \cdots + u_k = 0$ with $u_i \in U_i$ implies $u_1 = \cdots = u_k = 0$.

## Remarks

For two subspaces, $V = U \oplus W$ if and only if $V = U + W$ and $U \cap W = \{0\}$.
For three or more subspaces, pairwise zero intersections are not enough: three different lines through $0$ in $\R^2$ are not a direct sum.
If $V = U_1 \oplus \cdots \oplus U_k$ and $V$ is finite-dimensional, then $\dim V = \dim U_1 + \cdots + \dim U_k$.

## Explanations

### L1 human

A space is the direct sum of some subspaces when we can split each vector into pieces, one piece in each subspace.
The split must be unique.
The plane is the direct sum of the $x$-axis and the $y$-axis.
The pieces of a vector are its components along the subspaces.
</document>

<document source="https://math2.tamnd.com/node/0423TXR9AN.md" tag="0423TXR9AN" kind="statement">
# Subgroups of a finite cyclic group correspond to divisors of its order

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] of finite order $n$, generated by $g$.
For each positive divisor $d$ of $n$, $G$ has exactly one [[subgroup|0X1JXT47GA]] of order $d$, namely $\langle g^{n/d} \rangle$.
Each subgroup of $G$ has this form for some positive divisor $d$ of $n$.

## Remarks

Thus the number of subgroups of $G$ is the number of positive divisors of $n$.
For $d, d'$ dividing $n$, the subgroup of order $d$ is contained in the subgroup of order $d'$ exactly when $d$ divides $d'$.
For example, $\mathbb{Z}/12\mathbb{Z}$ has six subgroups, of the orders $1, 2, 3, 4, 6, 12$.

## Explanations

### L1 human

In a cyclic group, the subgroups are as simple as possible: there is one for each divisor of the order.
In the clock group $\mathbb{Z}/12\mathbb{Z}$, the subgroup of order $4$ is $\{0, 3, 6, 9\}$, generated by $12/4 = 3$.
The order of each subgroup divides $12$, as [[Lagrange's theorem|3C53TD]] says, and here every divisor occurs.
</document>

<document source="https://math2.tamnd.com/node/042JWAR3Q0.md" tag="042JWAR3Q0" kind="statement">
# Rational canonical form

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then there are unique monic nonconstant polynomials $f_1, \dots, f_r \in F[t]$ with $f_1 \mid f_2 \mid \cdots \mid f_r$ and the following property. The matrix $A$ is [[similar|0E89HQATM8]] to the [[block diagonal matrix|001T5FJDC1]]
$$\begin{pmatrix} C(f_1) & & \\ & \ddots & \\ & & C(f_r) \end{pmatrix},$$
where $C(f_i)$ is the [[companion matrix|0ZCWPNZTQ3]] of $f_i$.
This block matrix is the *rational canonical form* of $A$.

## Remarks

The polynomials $f_1, \dots, f_r$ are the *invariant factors* of $A$.
The last factor $f_r$ is the minimal polynomial of $A$, and $f_1 f_2 \cdots f_r$ is its characteristic polynomial.
Unlike the Jordan form, it needs no field extension: it works over any field.
Two matrices over $F$ are similar if and only if they have the same invariant factors.

## Explanations

### L1 human

The rational canonical form is a standard representative of each similarity class, over any field.
It uses companion matrices instead of Jordan blocks.
It needs no roots of polynomials, only divisibility of polynomials.
</document>

<document source="https://math2.tamnd.com/node/0430ETCMKD.md" tag="0430ETCMKD" kind="statement">
# Continuous bijection from a compact space to a Hausdorff space is a homeomorphism

## Statement

Let $X$ be a [[compact space|0QPHW703PV]], let $Y$ be a [[Hausdorff space|0MXB4CKHFW]] and let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]] that is [[bijective|0SNQEF7Z53]].
Then $f$ is a [[homeomorphism|05GZ11PWRY]].

## Remarks

Proof: it is enough to show that $f$ maps closed sets to closed sets. A closed $F \subseteq X$ is compact because [[closed subsets of compact sets are compact|066HW6D9A7]]. Then $f(F)$ is compact, and thus closed in $Y$ by [[compact subsets of a Hausdorff space are closed|0MYFJVMN7M]].
The compactness is needed: the map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the unit circle is a continuous bijection, but its inverse is not continuous.

## Explanations

### L1 human

For compact spaces, you do not need to check that the inverse map is continuous: it comes for free.
This is a quick way to show that two compact spaces have the same shape.
</document>

<document source="https://math2.tamnd.com/node/0431RN0R1N.md" tag="0431RN0R1N" kind="statement">
# Uniform limits of holomorphic functions are holomorphic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f_1, f_2, \dots$ be functions $U \to \mathbb{C}$ that are [[holomorphic|0DG0Z72CQD]] on $U$.
Let $f : U \to \mathbb{C}$ be a function such that $f_n \to f$ uniformly on each compact subset of $U$.
Then $f$ is holomorphic on $U$. Moreover, $f_n' \to f'$ uniformly on each compact subset of $U$.

## Remarks

Proof idea: $f$ is continuous. On each disc in $U$, [[Cauchy's integral theorem|01MPZXYVYF]] gives $\oint_{\partial T} f_n = 0$ for each triangle $T$, so $\oint_{\partial T} f = 0$ by the uniform convergence, and [[Morera's theorem|0VDD7W0EKZ]] applies.
A main use: if a series $\sum g_k$ of holomorphic functions converges uniformly on compact subsets of $U$, then its sum is holomorphic on $U$.
There is no real analogue. On $[-1, 1]$, the polynomials converge uniformly to $|x|$, which is not differentiable at $0$.

## Explanations

### L1 human

If holomorphic functions get close to a function $f$ uniformly, then $f$ is holomorphic too, and their derivatives get close to the derivative of $f$.
This lets us build new holomorphic functions as limits of series, for example the zeta function $\sum n^{-z}$.
</document>

<document source="https://math2.tamnd.com/node/043B3ENSTY.md" tag="043B3ENSTY" kind="statement">
# Fermat's theorem on sums of two squares

## Statement

Let $p$ be an odd [[prime number|0SN0MAEKYX]].
Then there are integers $x$ and $y$ with $p = x^2 + y^2$ if and only if $p \equiv 1 \pmod 4$.

## Remarks

The prime $2 = 1^2 + 1^2$ is also a sum of two squares. So a prime $p$ is a sum of two squares if and only if $p \not\equiv 3 \pmod 4$.
For example, $5 = 1 + 4$, $13 = 4 + 9$ and $29 = 4 + 25$. For $p \equiv 1 \pmod 4$, the representation is unique up to order and signs.
The "only if" part is easy: a square is $0$ or $1$ modulo $4$, so $x^2 + y^2 \not\equiv 3 \pmod 4$.
One proof of the other part starts from a [[square root of $-1$ modulo $p$|00DW862T78]] and uses [[Gaussian integers|0B268GVX95]] or [[Minkowski's theorem|0K387S2PKQ]].

## History

Albert Girard stated the result in a book published in 1625.
Fermat stated it in a letter to Mersenne of 25 December 1640, so it is also called Fermat's Christmas theorem.
Euler gave the first proof, by infinite descent, in 1749.

## Explanations

### L1 human

The primes $5, 13, 17, 29$ leave remainder $1$ on division by $4$, and each of them is a sum of two squares.
The primes $3, 7, 11, 19$ leave remainder $3$, and none of them is a sum of two squares.
</document>

<document source="https://math2.tamnd.com/node/043SJ5RF59.md" tag="043SJ5RF59" kind="definition">
# Maclaurin series

## Definition

Let $I$ be an open interval with $0 \in I$, and let $f \colon I \to \mathbb{R}$ be smooth.
The Maclaurin series of $f$ is its [[Taylor series|0225QPWRRE]] at $0$:
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.$$

## Remarks

For example, the Maclaurin series of $e^x$ is $\sum x^n / n!$, and the one of $1/(1 - x)$ is $\sum x^n$.
The series is named after Colin Maclaurin.

## Explanations

### L1 human

A Maclaurin series is a Taylor series centered at $0$.
</document>

<document source="https://math2.tamnd.com/node/0443XNARDV.md" tag="0443XNARDV" kind="concept">
# Multilinear map

## Summary

A multilinear map is a map of several vector variables that is linear in each variable when the other variables are fixed.
</document>

<document source="https://math2.tamnd.com/node/044D4AQV8V.md" tag="044D4AQV8V" kind="definition">
# Product metric

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
The product metric on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the function $d$ with
$$d\bigl((x_1, y_1), (x_2, y_2)\bigr) = \max\bigl(d_X(x_1, x_2), \, d_Y(y_1, y_2)\bigr)$$
for all $x_1, x_2 \in X$ and $y_1, y_2 \in Y$.

## Remarks

The function $d$ is a metric on $X \times Y$.
Other common product metrics are $d_X + d_Y$ and $\sqrt{d_X^2 + d_Y^2}$. Each of them is between $d$ and $2d$, so all of them are [[equivalent|0TWQ5PRKWS]] and give the [[product topology|0Z57EC6YX3]].
For $X = Y = \mathbb{R}$ with the usual metric, $d$ is the [[sup metric|0VTAXQEWQD]] on $\mathbb{R}^2$.
The same formula with a maximum over $n$ factors gives a product metric on a product of $n$ metric spaces.

## Explanations

### L1 human

Two pairs are close in the product metric when their first parts are close and their second parts are close.
The maximum is the worse of the two distances.
</document>

<document source="https://math2.tamnd.com/node/044JNC7ZAA.md" tag="044JNC7ZAA" kind="concept">
# Diagonal matrix

## Summary

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero.
</document>

<document source="https://math2.tamnd.com/node/045D8P03E0.md" tag="045D8P03E0" kind="concept">
# Random variable with a density

## Summary

A real random variable has a density when there is a fixed nonnegative function with this property.
The probability of each set of values is the integral of the function over that set.
The node [[random variable with a density|0C21RZG5V2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/045DN06F3S.md" tag="045DN06F3S" kind="definition">
# Circle

## Definition

Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Let $c \in \mathbb{R}^2$ be a point and $r > 0$ a real number.
The circle with the centre $c$ and the radius $r$ is the set
$$\{\, x \in \mathbb{R}^2 \;:\; d(x, c) = r \,\}.$$
With $c = (a, b)$, it is the set of the points $(x, y)$ with $(x - a)^2 + (y - b)^2 = r^2$.

## Remarks

A chord is a segment between two points of the circle. A diameter is a chord through the centre, and its length is $2r$.
The circle has the length $2 \pi r$ and bounds a disc of area $\pi r^2$.
A circle in $\mathbb{R}^n$ is a circle in some plane of $\mathbb{R}^n$. Mathlib's `EuclideanGeometry.Sphere` is the sphere of a centre and a radius in a metric space; in the plane it is a circle.

## Explanations

### L1 human

A circle is what a compass draws: all the points at one fixed distance from the centre. The unit circle has the centre $(0, 0)$ and the radius $1$. It is the set of the points with $x^2 + y^2 = 1$.
</document>

<document source="https://math2.tamnd.com/node/045ENVCCZV.md" tag="045ENVCCZV" kind="statement">
# The derangement numbers count derangements

## Statement

Let $X$ be a finite set with $|X| = n$.
Then the number of [[derangements|0TH61JR04G]] of $X$ is the derangement number $D_n$ of the node [[number of derangements|03WY4HHMT1]].

## Remarks

The proof checks the recursion $D_{n+2} = (n + 1)(D_n + D_{n+1})$, as in the node [[recurrence for derangements|0QRRYXG7V7]].
For $n = 4$ there are $9$ derangements among the $24$ permutations.

## Explanations

### L1 human

The derangement numbers are defined by a recursion, and this result says that they really count the permutations with no fixed point.
For example, $D_3 = 2$, and $\{1, 2, 3\}$ has exactly the two derangements $231$ and $312$.
</document>

<document source="https://math2.tamnd.com/node/045EWAXY9P.md" tag="045EWAXY9P" kind="definition">
# Disc of convergence

## Definition

Let $\sum_{n \ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with center $c \in \mathbb{C}$ and complex coefficients $a_n$.
Its radius of convergence is
$$R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n)_{n \ge 0} \text{ is bounded} \},$$
an element of $[0, \infty]$.
Its disc of convergence is the open disc $D = \{ z \in \mathbb{C} : |z - c| < R \}$. It is empty if $R = 0$, and it is $\mathbb{C}$ if $R = \infty$.

## Remarks

The series converges absolutely at each $z \in D$, and uniformly on each closed disc $|z - c| \le r$ with $r < R$. It diverges at each $z$ with $|z - c| > R$.
The Cauchy–Hadamard formula gives $1/R = \limsup_{n \to \infty} |a_n|^{1/n}$, with $1/0 = \infty$ and $1/\infty = 0$.
On the circle $|z - c| = R$, the series can converge at all points, at some points, or at no point.
For example, $\sum z^n$, $\sum z^n / n$ and $\sum z^n / n^2$ all have $R = 1$. On $|z| = 1$, the first converges nowhere, the second everywhere except at $z = 1$, and the third everywhere.

## Explanations

### L1 human

A power series converges inside a disc around its center and diverges outside the closed disc.
The radius of this disc is the radius of convergence. On the boundary circle, each case is possible.
</document>

<document source="https://math2.tamnd.com/node/045YFKRW0P.md" tag="045YFKRW0P" kind="concept">
# Closed curve

## Summary

A closed curve is a [[parametrised curve|0G7WM8DX4P]] on a closed interval whose end point is equal to its initial point.
Its image can be seen as the image of a circle under a continuous map.
</document>

<document source="https://math2.tamnd.com/node/0464Z0JP2E.md" tag="0464Z0JP2E" kind="statement">
# Monotonicity of probability

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]] with $A \subseteq B$.
Then $P(A) \le P(B)$.
More precisely, $P(B \setminus A) = P(B) - P(A)$.

## Remarks

The proof: $B$ is the disjoint union of $A$ and $B \setminus A$, so $P(B) = P(A) + P(B \setminus A)$, and $P(B \setminus A) \ge 0$.
With $B = \Omega$, monotonicity gives $P(A) \le 1$ for each event $A$.
If $A$ implies $B$, then $B$ is at least as likely as $A$. For example, "the die shows $6$" is at most as likely as "the die shows an even number".

## Explanations

### L1 human

A bigger event contains more outcomes, so it cannot be less likely.
The chance to draw a heart from a deck is at most the chance to draw a red card, because every heart is red.
</document>

<document source="https://math2.tamnd.com/node/0469V0EN5N.md" tag="0469V0EN5N" kind="concept">
# Homogeneous linear system with constant coefficients

## Summary

A homogeneous linear system with constant coefficients is a system $x' = A x$ with a constant square matrix $A$.
Its solutions are given by the matrix exponential, and the eigenvalues of $A$ control their behaviour. See the [[definition|0VJF99CZ0Z]].
</document>

<document source="https://math2.tamnd.com/node/0476QH9846.md" tag="0476QH9846" kind="concept">
# Evaluation of real integrals by residues

## Summary

The [[residue theorem|032HKYS329]] gives the values of many definite integrals of real functions.
For an integral $\int_0^{2\pi} R(\cos \theta, \sin \theta) \, d\theta$ with $R$ a rational function of two variables, the substitution $z = e^{i\theta}$ changes it into a [[contour integral|00FWNP9TXY]] over the unit circle.
This substitution uses $\cos \theta = \frac{1}{2}(z + z^{-1})$, $\sin \theta = \frac{1}{2i}(z - z^{-1})$ and $d\theta = \frac{dz}{iz}$.
For an integral $\int_{-\infty}^{\infty} f(x) \, dx$, one closes the segment $[-R, R]$ with a semicircle of radius $R$ in the upper half-plane and lets $R \to \infty$.
This method gives the integral when the integral over the semicircle tends to $0$, for example by the [[estimation lemma|0DJMGPN8K9]] or by [[Jordan's lemma|0PNPG2PP9H]].

## Remarks

For example, $\frac{1}{1 + z^2}$ has one pole $i$ in the upper half-plane, with residue $\frac{1}{2i}$, so $\int_{-\infty}^{\infty} \frac{dx}{1 + x^2} = 2\pi i \cdot \frac{1}{2i} = \pi$.
With Jordan's lemma, the same method gives $\int_{-\infty}^{\infty} \frac{e^{itx}}{x^2 + 1} \, dx = \pi e^{-|t|}$ for each real $t$.
When the integrand has a pole on the real axis, the method usually gives a [[Cauchy principal value|0V3RDQV3ZX]].

## Explanations

### L1 human

A hard real integral can become easy when you view the real line as part of a closed curve in the complex plane.
The residue theorem gives the integral over the closed curve from a few residues.
Then you show that the extra part of the curve contributes nothing in the limit.
</document>

<document source="https://math2.tamnd.com/node/0477R9PYAE.md" tag="0477R9PYAE" kind="concept">
# Variance

## Summary

The variance of a random variable is the expected squared distance between the variable and its mean. It measures how widely the values spread around the mean.
The node [[variance|007QED3E0Y]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/047KEVEZ9Q.md" tag="047KEVEZ9Q" kind="statement">
# Every real number has a decimal expansion

## Statement

Let $x$ be a real number with $x \ge 0$.
Then $x$ has a [[decimal expansion|06XY4X9430]]: there are a natural number $a_0$ and digits $d_k \in \{0, \dots, 9\}$ for $k \ge 1$ with $x = a_0 + \sum_{k=1}^{\infty} d_k 10^{-k}$.

## Remarks

One choice is $a_0 = \lfloor x \rfloor$ and $d_k = \lfloor 10^k x \rfloor - 10 \lfloor 10^{k-1} x \rfloor$, with the [[floor function|0YVRNEDSVG]].
With this choice, the truncation $a_0.d_1 \dots d_k$ satisfies $0 \le x - a_0.d_1 \dots d_k < 10^{-k}$.
The expansion is unique, except for the numbers with a finite expansion. They also have a second expansion that ends in infinitely many $9$.

## Explanations

### L1 human

You can find the digits of a number one at a time.
At each step, you choose the largest digit that keeps the truncated decimal at or below the number.
The truncations then get closer to the number by a factor of ten at each step.
</document>

<document source="https://math2.tamnd.com/node/047NDTEPCV.md" tag="047NDTEPCV" kind="concept">
# Jacobian matrix

## Summary

Let $U \subseteq \mathbb{R}^n$ be open. The Jacobian matrix of a map $f \colon U \to \mathbb{R}^m$ at a point $a$ is the $m \times n$ matrix of all first [[partial derivatives|09EGC6F8PT]] $\partial f_i / \partial x_j$ at $a$.
When $f$ is differentiable at $a$, it is the matrix of the [[total derivative|0FNKFT8H6E]] $Df(a)$ in the standard bases.
</document>

<document source="https://math2.tamnd.com/node/047RFCNVSK.md" tag="047RFCNVSK" kind="statement">
# Parseval's identity

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.
Then, for all $x, y \in V$,
$$\langle x, y \rangle = \sum_{i=1}^{n} \langle x, e_i \rangle \overline{\langle y, e_i \rangle} \qquad \text{and} \qquad \norm{x}^2 = \sum_{i=1}^{n} \abs{\langle x, e_i \rangle}^2.$$

## Remarks

It is the case of equality in [[Bessel's inequality|0RBT5PB74K]].
In coordinates of an orthonormal basis, the inner product becomes the standard inner product of $\mathbb{K}^n$.
For Fourier series, the same identity holds with infinite sums, for square integrable functions.

## Explanations

### L1 human

In an orthonormal basis, the inner product of two vectors is the standard inner product of their coordinates.
In particular, the squared length is the sum of the squared coordinates.
This is the Pythagorean theorem in $n$ dimensions.
</document>

<document source="https://math2.tamnd.com/node/047SX3TD69.md" tag="047SX3TD69" kind="concept">
# Multiplicity of a root

## Summary

The multiplicity of a root $c$ of a nonzero polynomial $f$ is the largest $m$ such that $(x - c)^m$ divides $f$.
The [[definition of the multiplicity|03XVP8Q4GY]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/048F2P90WR.md" tag="048F2P90WR" kind="statement">
# Maximum modulus principle

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Assume that $|f|$ has a local maximum at a point $z_0 \in D$: there is $r > 0$ such that $|f(z)| \le |f(z_0)|$ for each $z$ with $|z - z_0| < r$.
Then $f$ is constant on $D$.

## Remarks

Proof idea: the mean value formula $f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + \rho e^{i\theta}) \, d\theta$ for small $\rho$ forces $|f| = |f(z_0)|$ near $z_0$. Then $f$ is constant near $z_0$, and on $D$ by the [[identity theorem|0YBVKMYX0W]].
Corollary: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$, then $\max_{\overline{D}} |f|$ is attained at a point of the boundary of $D$.
The minimum of $|f|$ can be inside $D$, but only at a zero of $f$. For example, $|z|$ has its minimum on the unit disc at $0$.

## Explanations

### L1 human

The size $|f|$ of a nonconstant holomorphic function has no highest point inside its domain.
For example, on the closed unit disc, $|z^2 + 1|$ is largest on the boundary circle, at $z = \pm 1$, and it equals $2$ there.
</document>

<document source="https://math2.tamnd.com/node/049DCJZ448.md" tag="049DCJZ448" kind="concept">
# Concyclic points

## Summary

Points are concyclic when they all lie on one circle. A polygon with concyclic vertices is a cyclic polygon, and the circle is its circumcircle.
</document>

<document source="https://math2.tamnd.com/node/049PXYHW1T.md" tag="049PXYHW1T" kind="concept">
# Open cover

## Summary

An open cover of a subset of a topological space is a family of open sets whose union contains the subset.
Open covers are used to define compactness.
</document>

<document source="https://math2.tamnd.com/node/04A2XEX4D4.md" tag="04A2XEX4D4" kind="statement">
# Lagrange multiplier with one constraint

## Statement

Let $U \subseteq \mathbb{R}^n$ be open with $n \ge 2$, and let $f, g \colon U \to \mathbb{R}$ be of class $C^1$.
Let $c \in \mathbb{R}$, and let $a$ be a point of the [[level set|0Y5Q8Q9WJ7]] $\{x \in U : g(x) = c\}$ with $\nabla g(a) \ne 0$.
If the restriction of $f$ to this level set has a local extremum at $a$, then there is a unique $\lambda \in \mathbb{R}$ with
$$\nabla f(a) = \lambda \, \nabla g(a).$$

## Remarks

This is the case $m = 1$ of the [[Lagrange multiplier theorem|0VKKP8X6Q6]], applied to $g - c$.
Geometrically, the level set of $f$ through $a$ is tangent to the constraint set at $a$, because both [[gradients are orthogonal|0VXYV3NQQ1]] to it.
Example: to maximise $f(x, y) = x + y$ on $x^2 + y^2 = 1$, solve $(1, 1) = \lambda (2 x, 2 y)$ and $x^2 + y^2 = 1$. The maximum is $\sqrt{2}$, at $(1/\sqrt{2}, 1/\sqrt{2})$.
Without the condition $\nabla g(a) \ne 0$ the conclusion can fail. For $g(x, y) = y^2 - x^3$ and $f(x, y) = x$, the minimum on $g = 0$ is at $(0, 0)$, where $\nabla g = 0$ and $\nabla f = (1, 0)$.

## Explanations

### L1 human

Move along the constraint curve and watch the value of $f$.
At the best point, the curve just touches a level curve of $f$, so the two curves have the same normal direction there.
The multiplier $\lambda$ is the ratio between the two normal vectors.
</document>

<document source="https://math2.tamnd.com/node/04AAKZNXHD.md" tag="04AAKZNXHD" kind="concept">
# Greedy colouring

## Summary

Greedy colouring is a simple method that makes a [[colouring|0874138YP6]] of a finite [[graph|0JD5C82QMB]].
Put the vertices in an order $v_1, v_2, \dots, v_n$, and use the colours $1, 2, 3, \dots$.
For $i = 1, \dots, n$, give $v_i$ the smallest colour that no earlier neighbour of $v_i$ has.

## Remarks

A vertex $v_i$ has at most $\deg(v_i)$ earlier neighbours. Thus greedy colouring uses at most $\Delta(G) + 1$ colours, where $\Delta(G)$ is the [[maximum degree|0KNAN9F27P]].
The number of colours depends on the order. For each graph, some order gives an optimal colouring, but such an order is hard to find in general.
Greedy colouring can be done in linear time.

## Explanations

### L1 human

Go through the vertices one by one, and give each one the first colour that its coloured neighbours do not use.
It is fast and simple, but it does not always use the least number of colours.
</document>

<document source="https://math2.tamnd.com/node/04AJE0M271.md" tag="04AJE0M271" kind="example">
# Möbius strip

## Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, 1 - y)$ for all $y \in [0, 1]$.
The quotient space $[0, 1]^2 / {\sim}$, with the [[quotient topology|0V8046R0P9]], is the Möbius strip.

## Remarks

The gluing has a half-twist: the bottom of the left edge is glued to the top of the right edge.
The Möbius strip is a non-orientable surface. Its boundary is one circle, not two.
Without the twist, the gluing $(0, y) \sim (1, y)$ gives a cylinder, which is not homeomorphic to the Möbius strip.

## Explanations

### L1 human

Take a strip of paper, give one end a half-turn, and glue the two ends together.
The result has only one side: an ant can walk to every point of the strip without crossing an edge.
</document>

<document source="https://math2.tamnd.com/node/04BD23PSK5.md" tag="04BD23PSK5" kind="statement">
# Exponential of a sum of commuting matrices

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ and $B$ be $n \times n$ matrices over $\mathbb{K}$ with $AB = BA$.
Then
$$e^{A + B} = e^A e^B,$$
where $e^A$ is the [[matrix exponential|020M04FN1H]].

## Remarks

The proof multiplies the two series and uses the binomial formula, which needs $AB = BA$.
It gives $e^A e^{-A} = I_n$, so $e^A$ is always invertible.
Without $AB = BA$, the identity can fail, for example for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^a e^b$.
For matrices, this rule holds only when the two matrices commute.
In particular, $e^{sA} e^{tA} = e^{(s + t)A}$ for all scalars $s$ and $t$.
</document>

<document source="https://math2.tamnd.com/node/04C3BAHV03.md" tag="04C3BAHV03" kind="definition">
# Complementary subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.
The subspaces $U$ and $W$ are *complementary* when $V = U \oplus W$, that is, when $U + W = V$ and $U \cap W = \{0\}$.
Then $W$ is a *complement* of $U$ in $V$.

## Remarks

A complement is not unique: in $\R^2$, each line through $0$ other than the $x$-axis is a complement of the $x$-axis.
If $V$ is finite-dimensional and $W$ is a complement of $U$, then $\dim W = \dim V - \dim U$.
A complement $W$ of $U$ is isomorphic to the [[quotient space|0BR3FY5Q8M]] $V / U$.
In an inner product space, the [[orthogonal complement|0XJXTBCB2T]] is one particular complement.

## Explanations

### L1 human

Two subspaces are complementary when together they fill the space and they share only the zero vector.
Each vector then splits in a unique way into a part in each subspace.
A line and a plane through the origin in space are complementary when the line does not lie in the plane.
</document>

<document source="https://math2.tamnd.com/node/04C5ZBG1WT.md" tag="04C5ZBG1WT" kind="concept">
# Field

## Summary

A field is a commutative ring in which you can divide by every nonzero element.
The rational numbers, the real numbers and the complex numbers are fields.
The [[definition of a field|0QA8CHH8MG]] gives the axioms.
</document>

<document source="https://math2.tamnd.com/node/04CBRA4JRK.md" tag="04CBRA4JRK" kind="statement">
# log x is at most x - 1

## Statement

For every real $x > 0$,
$$\ln x \le x - 1,$$
with equality if and only if $x = 1$.

## Remarks

Put $\ln x$ into the inequality [[$1 + y \le e^y$|012C7GFF20]]. Then $1 + \ln x \le e^{\ln x} = x$.
The line $y = x - 1$ is the tangent to the graph of $\ln$ at $x = 1$.
Replacing $x$ by $1/x$ gives $\ln x \ge 1 - 1/x$. Mathlib states the main claim as `Real.log_le_sub_one_of_pos`.

## Explanations

### L1 human

The graph of $\ln$ lies below its tangent line at $x = 1$.
This follows from the concavity of the logarithm.
</document>

<document source="https://math2.tamnd.com/node/04CC6K5VDT.md" tag="04CC6K5VDT" kind="concept">
# Real sequence

## Summary

A real sequence assigns a real number to each natural number. Real sequences are the first setting for limits, convergence and series in real analysis. The node [[real sequence|0CTV783HYT]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04CPG9K1C4.md" tag="04CPG9K1C4" kind="definition">
# Local maximum and minimum

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local maximum at $a$ if there is $\varepsilon > 0$ such that $f(x) \le f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$.
The function $f$ has a local minimum at $a$ if there is $\varepsilon > 0$ such that $f(x) \ge f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$.
A local extremum of $f$ is a local maximum or a local minimum of $f$.
The extremum is strict if the inequality is strict for all such $x \ne a$.

## Remarks

A global maximum of $f$ is a point $a$ with $f(x) \le f(a)$ for all $x \in D$. Each global maximum is a local maximum.
The function $f(x, y) = x^2 + y^2$ has a strict local minimum at $(0, 0)$, which is also a global minimum.
In two or more variables, a local minimum that is the only critical point need not be a global minimum. For $f(x, y) = x^2 + y^2 (1 - x)^3$, the only critical point $(0, 0)$ is a local minimum, but $f(2, 3) = -5 < f(0, 0)$.
At an interior local extremum of a differentiable function, the [[gradient is zero|0F405477F8]].

## Explanations

### L1 human

A local maximum is a hilltop: it is the highest point in a small disc around it, but a higher hill can exist far away.
A local minimum is the bottom of a valley in the same sense.
</document>

<document source="https://math2.tamnd.com/node/04CPJ6AF00.md" tag="04CPJ6AF00" kind="statement">
# Determinant is multiplicative

## Statement

Let $K$ be a commutative ring and let $A$ and $B$ be $n \times n$ matrices over $K$.
Then
$$\det(AB) = \det A \cdot \det B,$$
where $AB$ is the [[matrix product|032Q9FB8MT]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Thus $\det$ is a group homomorphism from the [[general linear group|0FCHE902D7]] $GL_n(F)$ to $F^\times$, over a field $F$.
It follows that $\det(AB) = \det(BA)$, although $AB \ne BA$ in general.
Similar matrices have the same determinant, because $\det(P^{-1} A P) = \det A$.

## Explanations

### L1 human

If one matrix scales volume by $\det A$ and another by $\det B$, then doing both scales volume by $\det A \cdot \det B$.
This is the meaning of the product rule.
It is one of the most useful properties of the determinant.
</document>

<document source="https://math2.tamnd.com/node/04CTY6XYWN.md" tag="04CTY6XYWN" kind="definition">
# Correlation coefficient

## Definition

Let $X$ and $Y$ be real random variables on one probability space with finite and positive variances.
The correlation coefficient of $X$ and $Y$ is
$$\rho_{X, Y} = \operatorname{Corr}(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \, \sigma_Y},$$
where $\operatorname{Cov}$ is the [[covariance|0DJ8W3HP2D]] and $\sigma_X$, $\sigma_Y$ are the [[standard deviations|0VTN5RKNC5]].

## Remarks

By the Cauchy–Schwarz inequality, $-1 \le \rho_{X, Y} \le 1$.
$|\rho_{X, Y}| = 1$ if and only if $Y = aX + b$ almost surely for some reals $a \ne 0$ and $b$. The sign of $\rho$ is the sign of $a$.
The coefficient does not depend on units: $\rho_{aX + b, \, cY + d} = \rho_{X, Y}$ if $a c > 0$.
For a sample of pairs $(x_i, y_i)$, the sample correlation coefficient $r$ uses sample means and sample sums of squares instead of expectations.

## Explanations

### L1 human

The correlation coefficient tells how close the points $(X, Y)$ are to a straight line.
A value near $1$ means a strong increasing linear relation, and a value near $-1$ means a strong decreasing one.
A value near $0$ means that there is no linear relation.
</document>

<document source="https://math2.tamnd.com/node/04CW1P44VD.md" tag="04CW1P44VD" kind="statement">
# Spectral theorem for normal matrices

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then $A$ is [[normal|0MR0BYF0K2]] if and only if there are a [[unitary matrix|0AR68JTMDT]] $U$ and a diagonal matrix $D$ with
$$A = U D U^*.$$

## Remarks

The columns of $U$ are then an orthonormal basis of $\C^n$ of eigenvectors of $A$.
If $A$ is Hermitian, then $D$ is real; if $A$ is unitary, then the diagonal entries of $D$ have absolute value $1$.
Over $\R$, a normal matrix need not be diagonalisable: a rotation of the plane is normal.

## Explanations

### L1 human

The normal matrices are exactly the complex matrices that are diagonal in some orthonormal basis.
This generalises the spectral theorem from Hermitian matrices to all normal matrices.
The eigenvalues can now be any complex numbers.
</document>

<document source="https://math2.tamnd.com/node/04CYN5G6RY.md" tag="04CYN5G6RY" kind="statement">
# Variance under translation

## Statement

Let $X$ be an integrable real random variable and let $b \in \mathbb{R}$. Then
$$\operatorname{Var}(X + b) = \operatorname{Var}(X).$$

## Remarks

Proof: $E[X + b] = E[X] + b$ by [[linearity of expectation|0P7PTC8DDE]], so $(X + b) - E[X + b] = X - E[X]$.
Together with the [[scaling rule|0KVQ9D86BM]], this gives $\operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X)$ for all real $a$ and $b$.

## Explanations

### L1 human

If you add the same constant to every value, the whole distribution moves, but its spread does not change.
</document>

<document source="https://math2.tamnd.com/node/04CYQHN6MC.md" tag="04CYQHN6MC" kind="concept">
# Coordinate vector

## Summary

The coordinate vector of a vector in a finite basis is the list of the coefficients of the vector in that basis.
</document>

<document source="https://math2.tamnd.com/node/04D7CHT5V1.md" tag="04D7CHT5V1" kind="definition">
# Inverse hyperbolic functions

## Definition

The inverse hyperbolic sine $\operatorname{arsinh} \colon \mathbb{R} \to \mathbb{R}$ is the inverse of [[$\sinh$|0GWD21EWRW]].
The inverse hyperbolic cosine $\operatorname{arcosh} \colon [1, \infty) \to [0, \infty)$ is the inverse of the restriction of [[$\cosh$|0EQFWKJ37J]] to $[0, \infty)$.
The inverse hyperbolic tangent $\operatorname{artanh} \colon (-1, 1) \to \mathbb{R}$ is the inverse of [[$\tanh$|0MWYQTZ07Z]].

## Remarks

Solving a quadratic equation in $e^y$ gives the formulas $\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})$ and $\operatorname{arcosh} x = \ln(x + \sqrt{x^2 - 1})$.
Similarly, $\operatorname{artanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}$ for $|x| < 1$.
Their derivatives are $\frac{1}{\sqrt{x^2 + 1}}$, $\frac{1}{\sqrt{x^2 - 1}}$ for $x > 1$, and $\frac{1}{1 - x^2}$. Mathlib defines the first one as `Real.arsinh`.

## Explanations

### L1 human

The inverse hyperbolic functions undo the hyperbolic functions.
Since the hyperbolic functions come from $e^x$, their inverses come from the logarithm.
</document>

<document source="https://math2.tamnd.com/node/04DC0SRN2K.md" tag="04DC0SRN2K" kind="statement">
# Order of a group is the index times the order of the subgroup

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $|G| = [G : H] \cdot |H|$, where $[G : H]$ is the [[index|0DTXBQPA2S]] of $H$ in $G$.

## Remarks

The left cosets of $H$ partition $G$, and [[each coset has $|H|$ elements|0DKMBWTD7P]].
This statement gives [[Lagrange's theorem|3C53TD]]: the order of $H$ divides the order of $G$.
The formula $|G| = [G : H] \cdot |H|$ is also true for infinite groups as an equation of cardinal numbers.

## Explanations

### L1 human

Cut a finite group into the left cosets of a subgroup.
Each piece has the same size as the subgroup, and the number of pieces is the index.
So the size of the group is the number of pieces times the size of one piece.
</document>

<document source="https://math2.tamnd.com/node/04DR5PM25N.md" tag="04DR5PM25N" kind="concept">
# Modular exponentiation

## Summary

Modular exponentiation computes $b^e \bmod m$ for integers $b$, $e \ge 0$ and $m \ge 1$.
The binary method writes $e$ in base $2$. It squares $b$ again and again modulo $m$, and it multiplies the squares $b^{2^i}$ for the bits $i$ of $e$ that are $1$.
It needs about $2 \log_2 e$ multiplications modulo $m$, and each intermediate number is smaller than $m^2$.

## Remarks

The method is correct because of [[compatibility with multiplication|072D5G4SK6]]: one can reduce modulo $m$ after each multiplication.
The [[Fermat primality test|0N5ZSZMYH6]], the [[RSA cryptosystem|0HM584C465]] and the [[Diffie–Hellman key exchange|01TKKESHSB]] all use modular exponentiation.
The inverse problem, to find $e$ from $b$, $m$ and $b^e \bmod m$, is the [[discrete logarithm|09B4X51J09]] problem. It seems much harder.

## Explanations

### L1 human

To compute $3^{13} \bmod 7$, write $13 = 8 + 4 + 1$. The squares are $3^1 \equiv 3$, $3^2 \equiv 2$, $3^4 \equiv 4$ and $3^8 \equiv 2$.
Then $3^{13} \equiv 2 \cdot 4 \cdot 3 = 24 \equiv 3 \pmod 7$.
You never need the large number $3^{13} = 1594323$.
</document>

<document source="https://math2.tamnd.com/node/04DXAFN2ES.md" tag="04DXAFN2ES" kind="concept">
# Natural logarithm

## Summary

The natural logarithm is the inverse of the exponential function. It is defined for positive numbers, it turns products into sums, and its derivative is $1/x$. The node [[natural logarithm|0044CPHJBV]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04E4V9G5FR.md" tag="04E4V9G5FR" kind="definition">
# Commutativity

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The operation $\ast$ is commutative when $a \ast b = b \ast a$ for all $a, b \in S$.

## Remarks

Addition and multiplication of real numbers are commutative.
Subtraction is not commutative: $1 - 2 \neq 2 - 1$.
Multiplication of $2 \times 2$ real matrices is associative but not commutative.

## Explanations

### L1 human

Commutativity says that the order of the two inputs does not matter.
For example, $3 \times 5$ and $5 \times 3$ are both $15$.
Many operations do not have this property, for example subtraction and the composition of rotations in space.
</document>

<document source="https://math2.tamnd.com/node/04EAFVFTF7.md" tag="04EAFVFTF7" kind="object">
# Circle group

## Description

The circle group is the set $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$ of the complex numbers of absolute value $1$, with complex multiplication.
It is an [[abelian group|082GHNPJ53]] with identity element $1$, and the inverse of $z$ is its complex conjugate $\bar{z} = 1/z$.
Each element of $\mathbb{T}$ is $e^{i\theta}$ for a real number $\theta$, and $e^{i\theta} e^{i\varphi} = e^{i(\theta + \varphi)}$.

## Remarks

The map $t \mapsto e^{2\pi i t}$ is a surjective homomorphism from $(\mathbb{R}, +)$ to $\mathbb{T}$ with kernel $\mathbb{Z}$, thus $\mathbb{T}$ is isomorphic to the quotient $\mathbb{R}/\mathbb{Z}$.
The circle group is also isomorphic to the group $\mathrm{SO}(2)$ of rotations of the plane about the origin.
For each positive integer $n$, the [[group of $n$th roots of unity|0WX11EA3F3]] is a finite subgroup of $\mathbb{T}$.

## Explanations

### L1 human

The circle group is the unit circle in the complex plane, where the product of two points is their complex product.
If you write the points as angles, the product adds the angles.
Thus the circle group is the group of rotations of a circle.
</document>

<document source="https://math2.tamnd.com/node/04EHB7YDQS.md" tag="04EHB7YDQS" kind="definition">
# Spectrum of a matrix

## Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
The *spectrum* of $A$ is the set
$$\sigma(A) = \{\lambda \in F : A - \lambda I_n \text{ is not invertible}\}.$$
It is the set of [[eigenvalues|0BQAA2BK3Q]] of $A$.

## Remarks

The spectrum has at most $n$ elements, because each eigenvalue is a root of the characteristic polynomial of degree $n$.
Over $\C$, the spectrum of each square matrix is nonempty.
The spectrum depends on the field: the real matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has empty spectrum over $\R$ and spectrum $\{i, -i\}$ over $\C$.
Some authors count the eigenvalues with their multiplicities and call this list the spectrum.

## Explanations

### L1 human

The spectrum is the list of all eigenvalues of a matrix.
It tells us the scaling factors that the matrix shows in special directions.
For a diagonal matrix, it is the set of diagonal entries.
</document>

<document source="https://math2.tamnd.com/node/04ETX30HT2.md" tag="04ETX30HT2" kind="statement">
# Uniqueness of the supremum

## Statement

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
If $b$ and $b'$ are both [[suprema|0BTMQ04556]] of $S$, then $b = b'$.

## Remarks

Proof: $b$ is an upper bound and $b'$ is a least upper bound, thus $b' \le b$. In the same way $b \le b'$, and antisymmetry gives $b = b'$.
The same argument shows that a set has at most one infimum.

## Explanations

### L1 human

A set cannot have two different least upper bounds.
If it had two, each one would be at most the other one, so they would be equal.
</document>

<document source="https://math2.tamnd.com/node/04FHQSJCQ5.md" tag="04FHQSJCQ5" kind="definition">
# Modulo operation

## Definition

Let $a$ and $n$ be integers with $n \neq 0$.
The integer $a \bmod n$ is the remainder $r$ of the [[division of $a$ by $n$|0CHQKZFK5S]].
Thus $a \bmod n$ is the unique integer $r$ such that $0 \le r < |n|$ and $a - r$ is a multiple of $n$.

## Remarks

For example, $17 \bmod 5 = 2$ and $(-17) \bmod 5 = 3$.
Many programming languages use a different convention. In them, the remainder has the sign of $a$, so the result for $-17$ and $5$ is $-2$.
The Lean library Mathlib sets $a \bmod 0 = a$.

## Explanations

### L1 human

The value $a \bmod n$ is what is left after you take away from $a$ as many copies of $n$ as you can.
A clock uses this operation: $15$ hours after $0$ o'clock, a $12$-hour clock shows $15 \bmod 12 = 3$.
</document>

<document source="https://math2.tamnd.com/node/04GM4NYXJB.md" tag="04GM4NYXJB" kind="definition">
# Mesh of a partition

## Definition

Let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
The mesh of $P$ is
$$\| P \| = \max_{1 \le i \le n} (x_i - x_{i-1}),$$
the length of the longest subinterval of $P$.

## Remarks

Refining a partition does not increase its mesh.
The [[Riemann integral|0QM4TXT3DR]] is a limit of Riemann sums as the mesh tends to $0$.
For the partition of $[0, 1]$ into $n$ equal pieces, the mesh is $1/n$.

## Explanations

### L1 human

The mesh measures how fine a partition is.
A small mesh means that all the pieces are short.
</document>

<document source="https://math2.tamnd.com/node/04GNK7K4VP.md" tag="04GNK7K4VP" kind="definition">
# Divisibility in a commutative ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $a, b \in R$.
The element $a$ divides $b$, written $a \mid b$, if there is $c \in R$ with $b = a c$.
Then $a$ is a divisor of $b$, and $b$ is a multiple of $a$.

## Remarks

The relation $a \mid b$ holds if and only if $(b) \subseteq (a)$ for the [[principal ideals|00A2EDFT9Q]].
Every element divides $0$, and $0$ divides only $0$.
A [[unit|019NFGFPR5]] divides every element.
In $\mathbb{Z}[i]$, $1 + i$ divides $2$, because $2 = (1 + i)(1 - i)$.

## Explanations

### L1 human

Divisibility in a ring works as for integers: $a$ divides $b$ if $b$ is $a$ times some element of the ring.
The answer depends on the ring: $2$ does not divide $1$ in $\mathbb{Z}$, but it does in $\mathbb{Q}$.
</document>

<document source="https://math2.tamnd.com/node/04GYPXDXNH.md" tag="04GYPXDXNH" kind="statement">
# Riesz representation theorem in finite dimensions

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, with the inner product linear in the first argument.
Then, for each [[linear form|07YQ1C78G0]] $f \colon V \to \mathbb{K}$, there is exactly one vector $y \in V$ with
$$f(x) = \langle x, y \rangle \qquad \text{for all } x \in V.$$

## Remarks

For an orthonormal basis $e_1, \dots, e_n$, the vector is $y = \sum_i \overline{f(e_i)}\, e_i$.
The map $y \mapsto \langle \cdot, y \rangle$ from $V$ to its dual is bijective; it is linear over $\R$ and conjugate linear over $\C$.
The statement holds for continuous linear forms on every Hilbert space.

## Explanations

### L1 human

In an inner product space, every linear form is an inner product with a fixed vector.
On $\R^3$, the form $x \mapsto 2x_1 - x_2 + 5x_3$ is the dot product with $(2, -1, 5)$.
So in finite dimensions, vectors and linear forms correspond one to one.
</document>

<document source="https://math2.tamnd.com/node/04J4BYB0XE.md" tag="04J4BYB0XE" kind="concept">
# Separation axioms

## Summary

The separation axioms are conditions on a topological space that say how well open sets can separate points and closed sets.
The main ones, from weak to strong, are [[T0|0R3B7DX55J]], [[T1|05GQJXBP3F]], [[Hausdorff|0MXB4CKHFW]], [[regular|0JF9KW2TXB]] and [[normal|0AJ646N339]].

## Remarks

Each of these axioms, together with T1, implies the axioms before it in this list.
Each metric space satisfies all of them, by [[metric spaces are Hausdorff|0P9DCDES73]] and [[metrizable spaces are normal|0AD0BZ49DW]].
Authors do not agree on all names. For example, some authors include T1 in the definition of a regular or normal space, and some do not.

## Explanations

### L1 human

The separation axioms measure how many open sets a space has.
With more open sets, you can put distinct points, or points and closed sets, into disjoint open sets.
</document>

<document source="https://math2.tamnd.com/node/04J4Z73J5B.md" tag="04J4Z73J5B" kind="concept">
# Divisor function

## Summary

The divisor function $\sigma_k$ sends a positive integer $n$ to the sum of the $k$-th powers of its positive divisors.
The [[definition of the divisor function|0YEW47VJ0J]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/04J6G4H78J.md" tag="04J6G4H78J" kind="statement">
# Density of the irrationals

## Statement

Let $x$ and $y$ be real numbers with $x < y$.
Then there is an [[irrational number|080T2TXVQV]] $z$ with $x < z < y$.

## Remarks

Proof: by the [[density of the rationals|0STX0WRDNG]], there is a rational $q$ with $x / \sqrt{2} < q < y / \sqrt{2}$ and $q \neq 0$. Then $z = q \sqrt{2}$ is irrational and $x < z < y$.
Thus both $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$ are dense in $\mathbb{R}$.

## Explanations

### L1 human

Between any two different real numbers, there is also a number that is not a fraction.
So fractions and non-fractions are mixed together everywhere on the number line.
</document>

<document source="https://math2.tamnd.com/node/04JAAVHDDM.md" tag="04JAAVHDDM" kind="definition">
# Consistent and inconsistent systems

## Definition

Let $F$ be a field.
A [[system of linear equations|0GT4WWGWH2]] over $F$ is *consistent* when it has at least one solution in $F^n$.
It is *inconsistent* when it has no solution.

## Remarks

The system $x + y = 1$, $x + y = 2$ over $\R$ is inconsistent.
Every homogeneous system is consistent, because the zero tuple is a solution.
The [[Rouché–Capelli theorem|0HV1SKWPMS]] gives a test with ranks: $Ax = b$ is consistent if and only if $\rank A = \rank [A \mid b]$.

## Explanations

### L1 human

"Consistent" means that the equations can all be true at the same time.
Two parallel lines in the plane, like $x + y = 1$ and $x + y = 2$, have no common point, so their system is inconsistent.
Elimination shows inconsistency when it produces an equation $0 = c$ with $c \ne 0$.
</document>

<document source="https://math2.tamnd.com/node/04JAY9DGB7.md" tag="04JAY9DGB7" kind="statement">
# Dirichlet's approximation theorem

## Statement

Let $\alpha$ be a real number and let $N$ be a positive integer.
Then there are integers $p$ and $q$ with $1 \le q \le N$ and
$$|q \alpha - p| \le \frac{1}{N + 1}.$$

## Remarks

A corollary: if $\alpha$ is irrational, then $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ for infinitely many fractions $p / q$.
The proof uses the pigeonhole principle. Among the $N + 1$ fractional parts of $0, \alpha, 2\alpha, \dots, N\alpha$, two lie in one of $N + 1$ intervals of length $\frac{1}{N + 1}$.
The [[convergents|0E70M92FE3]] of the continued fraction of $\alpha$ give such fractions explicitly.

## Explanations

### L1 human

Each real number has fractions $p/q$ very close to it, with an error much smaller than $1/q$.
The theorem gives the bound $1/q^2$ for infinitely many denominators when $\alpha$ is irrational.
</document>

<document source="https://math2.tamnd.com/node/04JHQNNBFE.md" tag="04JHQNNBFE" kind="statement">
# Hockey-stick identity

## Statement

Let $n$ and $r$ be natural numbers with $r \le n$.
Then
$$\sum_{i=r}^{n} \binom{i}{r} = \binom{n + 1}{r + 1}.$$

## Remarks

For $r = 1$ it gives $1 + 2 + \dots + n = \binom{n+1}{2} = \frac{n(n+1)}{2}$.
Combinatorial proof: classify the $(r+1)$-subsets of $\{1, \dots, n + 1\}$ by their largest element $i + 1$. The other $r$ elements come from $\{1, \dots, i\}$.
In [[Pascal's triangle|05DME4TZ3K]] the summands lie on a diagonal, and the result lies below the end of the diagonal. This shape is the reason for the name.

## Explanations

### L1 human

Add the entries of [[Pascal's triangle|05DME4TZ3K]] along a diagonal, from the edge downward.
The total is the entry just below and to the side of the last one, so the picture looks like a hockey stick.
</document>

<document source="https://math2.tamnd.com/node/04KFTA34C5.md" tag="04KFTA34C5" kind="definition">
# Infimum

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $a$ of $P$ is an infimum, or greatest lower bound, of $S$ when these two conditions hold:

1. $a$ is a [[lower bound|0B5H2V4JW8]] of $S$;
2. if $m$ is a lower bound of $S$, then $m \le a$.

The notation is $a = \inf S$.

## Remarks

A set has at most one infimum.
In $\mathbb{R}$, $\inf \{1/n : n \ge 1\} = 0$, and $0$ is not an element of this set.
For a subset $S$ of $\mathbb{R}$, $\inf S = -\sup(-S)$ when one side exists, where $-S = \{-s : s \in S\}$.

## Explanations

### L1 human

The infimum is the highest possible floor for a set.
For the set $\{1, 1/2, 1/3, \dots\}$, every number at most $0$ is a floor, and $0$ is the highest of these, so the infimum is $0$.
</document>

<document source="https://math2.tamnd.com/node/04KGXJG1A2.md" tag="04KGXJG1A2" kind="concept">
# Homogeneous system of linear equations

## Summary

A homogeneous system of linear equations is a system in which every constant term is zero.
Such a system always has the zero solution, and its solution set is a vector subspace.
</document>

<document source="https://math2.tamnd.com/node/04MQSQ1F5Q.md" tag="04MQSQ1F5Q" kind="concept">
# Clique number

## Summary

The clique number of a finite graph is the size of its largest clique. It is a lower bound for the chromatic number. The node [[clique number|0WCKRBHFFD]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04NPWJMA6H.md" tag="04NPWJMA6H" kind="statement">
# Sum of two squares theorem

## Statement

Let $n$ be a positive integer.
Then $n = x^2 + y^2$ for some integers $x$ and $y$ if and only if $\nu_p(n)$ is even for each prime $p \equiv 3 \pmod 4$.
Here $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For example, $45 = 3^2 \cdot 5 = 6^2 + 3^2$, but $21 = 3 \cdot 7$ is not a sum of two squares.
The "if" part follows from [[Fermat's theorem on sums of two squares|043B3ENSTY]] and the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]].
Zero squares are allowed, so each square and each double of a square is a sum of two squares.

## Explanations

### L1 human

To decide whether $n$ is a sum of two squares, factor $n$ into primes.
Look only at the primes that leave remainder $3$ on division by $4$. Each of them must appear an even number of times.
</document>

<document source="https://math2.tamnd.com/node/04P2RJ4WPS.md" tag="04P2RJ4WPS" kind="concept">
# Continued fraction

## Summary

A continued fraction is an expression $b_0 + a_1 / (b_1 + a_2 / (b_2 + \cdots))$, a fraction whose denominator contains a fraction, and so on.
The [[definition of a continued fraction|0BK6JBK58W]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/04P2SR252N.md" tag="04P2SR252N" kind="concept">
# Geometric distribution

## Summary

The geometric distribution describes the waiting time until the first success in a sequence of independent Bernoulli trials.
One version counts the failures before the first success, and the other counts the trials up to and including it. The node [[geometric distribution|0XCVV5Q500]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04P7B5XC22.md" tag="04P7B5XC22" kind="definition">
# Lyapunov function

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be continuously differentiable.
The orbital derivative of $V$ is the function $\dot{V}(x) = \nabla V(x) \cdot f(x)$ on $W$.
The function $V$ is a Lyapunov function for $x^*$ if $V(x^*) = 0$, $V(x) > 0$ for each $x \in W \setminus \{x^*\}$, and $\dot{V}(x) \le 0$ for each $x \in W$.
It is a strict Lyapunov function if also $\dot{V}(x) < 0$ for each $x \in W \setminus \{x^*\}$.

## Remarks

For each solution $\gamma$ in $W$, $\frac{d}{dt} V(\gamma(t)) = \dot{V}(\gamma(t))$. So $V$ does not increase along solutions.
For $x' = -x$ on $\mathbb{R}$, the function $V(x) = x^2$ is a strict Lyapunov function for $0$, since $\dot{V}(x) = -2x^2$.
In mechanics, the total energy is often a Lyapunov function. See [[Lyapunov's stability theorem|0J6A752DTZ]].

## Explanations

### L1 human

A Lyapunov function is like a height above the equilibrium, which is the lowest point.
If the height never increases along the motion, the state cannot climb away from the equilibrium.
</document>

<document source="https://math2.tamnd.com/node/04P7D63W8P.md" tag="04P7D63W8P" kind="definition">
# Almost sure convergence

## Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$.
The sequence $(X_n)$ converges almost surely to $X$ if
$$P\left(\left\{\omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega)\right\}\right) = 1.$$
One writes $X_n \to X$ a.s. or $X_n \xrightarrow{\text{a.s.}} X$.

## Remarks

The set in the definition is an event, because the $X_n$ and $X$ are measurable.
An equivalent condition: for each $\varepsilon > 0$, $P(|X_n - X| > \varepsilon \text{ for infinitely many } n) = 0$.
The limit is unique up to equality [[almost surely|0R46A1XY60]].
The strong law of large numbers is a result about almost sure convergence.

## Explanations

### L1 human

Almost sure convergence means that, apart from a set of outcomes of probability $0$, each run of the experiment gives a convergent sequence of numbers.
</document>

<document source="https://math2.tamnd.com/node/04QEBF8S49.md" tag="04QEBF8S49" kind="definition">
# Index of a subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The index of $H$ in $G$, written $[G : H]$ or $|G : H|$, is the cardinality of the set of [[left cosets|03DEH43F4H]] of $H$ in $G$.

## Remarks

The index is also the number of right cosets, because $gH \mapsto Hg^{-1}$ is a bijection between the two sets of cosets.
In $(\mathbb{Z}, +)$, the subgroup $n\mathbb{Z}$ has index $n$ for each positive integer $n$.
A subgroup of a finite group has index $|G| / |H|$, but an infinite group can have a subgroup of finite index, for example $2\mathbb{Z}$ in $\mathbb{Z}$.

## Explanations

### L1 human

The index counts how many disjoint copies of the subgroup you need to cover the whole group.
For the even integers inside the integers, you need two copies: the even and the odd integers.
Thus the index of the even integers is $2$.
</document>

<document source="https://math2.tamnd.com/node/04QNZZ4HHY.md" tag="04QNZZ4HHY" kind="definition">
# Dedekind cut

## Definition

A Dedekind cut is a pair $(A, B)$ of subsets of the [[rational numbers|0NCJ5WXN6Z]] $\mathbb{Q}$ with these properties:

1. $A$ and $B$ are nonempty, $A \cap B = \emptyset$ and $A \cup B = \mathbb{Q}$;
2. if $a \in A$, $q \in \mathbb{Q}$ and $q < a$, then $q \in A$;
3. $A$ has no greatest element.

## Remarks

The set $A$ determines the cut, because $B = \mathbb{Q} \setminus A$. Thus some authors call $A$ itself the cut.
The cut with $A = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\}$ corresponds to $\sqrt{2}$. Here $B$ has no least element.
The set of all cuts, ordered by inclusion of the lower sets, is a [[complete ordered field|0A9E6X0S3E]] with suitable operations.

## Explanations

### L1 human

Cut the rational number line at one point. Put the numbers to the left in $A$ and the numbers to the right in $B$.
When the cut is at a gap of $\mathbb{Q}$, such as $\sqrt{2}$, the cut itself stands for the missing number.
</document>

<document source="https://math2.tamnd.com/node/04R3CY41VJ.md" tag="04R3CY41VJ" kind="statement">
# Euler's formula

## Statement

For every complex number $z$,
$$e^{i z} = \cos z + i \sin z.$$
Here $e^{w} = \sum_{n=0}^{\infty} w^n / n!$ is the complex exponential function, and $\cos$ and $\sin$ are given by their power series.

## Remarks

The main case is a real number $z = x$. Then $\cos x$ and $\sin x$ are the real and imaginary parts of $e^{i x}$, and $|e^{i x}| = 1$.
For $x = \pi$, the formula gives [[Euler's identity|09285XX7AE]].
A proof compares the power series: the even terms of $e^{i z}$ give $\cos z$ and the odd terms give $i \sin z$.

## Explanations

### L1 human

As the real number $x$ grows, the point $e^{i x}$ moves around the unit circle at constant speed.
Its coordinates are $\cos x$ and $\sin x$, as in trigonometry.
The formula links the exponential function with the trigonometric functions.
</document>

<document source="https://math2.tamnd.com/node/04R9K5H562.md" tag="04R9K5H562" kind="concept">
# Classification of discontinuities

## Summary

A point of the domain where a real function is not [[continuous|0EEE2CZB2A]] is a discontinuity. Discontinuities are sorted by the behavior of the [[one-sided limits|0909PWZR62]] at the point.

## Remarks

At a [[removable discontinuity|0QG9C579SR]], the limit exists but differs from the value.
At a [[jump discontinuity|05B0FKFXQA]], both one-sided limits exist but they differ.
At an essential discontinuity, at least one one-sided limit does not exist. An example is $\sin(1/x)$ at $0$, with any value at $0$.
A [[monotone function has countably many discontinuities|0WYTM4C9QG]], and all of them are jumps.

## Explanations

### L1 human

A function can fail to be continuous in different ways.
A hole in the graph can be repaired, a jump cannot, and some functions oscillate too wildly to have any limit.
</document>

<document source="https://math2.tamnd.com/node/04RG02S6BA.md" tag="04RG02S6BA" kind="statement">
# Cyclic groups are abelian

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]].
Then $G$ is [[abelian|082GHNPJ53]].

## Remarks

If $g$ generates $G$, then any two elements are $g^a$ and $g^b$ for integers $a, b$, and $g^a g^b = g^{a+b} = g^b g^a$ by [[the laws of exponents|00DRK90Z6K]].
The converse is false: the Klein four-group is abelian but not cyclic.
The group $(\mathbb{Q}, +)$ is abelian but not cyclic.

## Explanations

### L1 human

All elements of a cyclic group are powers of one element.
Two powers of the same element always commute, because the exponents just add.
Thus the order of the factors never matters in a cyclic group.
</document>

<document source="https://math2.tamnd.com/node/04RRB6HM00.md" tag="04RRB6HM00" kind="statement">
# Law of total expectation

## Statement

Let $X$ be an integrable real random variable on $(\Omega, \mathcal{F}, P)$ and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra. Then
$$E\left[E[X \mid \mathcal{G}]\right] = E[X].$$
In particular, $E[E[X \mid Y]] = E[X]$ for each random variable $Y$ on the same space.

## Remarks

Proof: put $G = \Omega$ in the defining property of the [[conditional expectation|0YAMJRK04E]].
For a partition of $\Omega$ into events $A_1, A_2, \dots$ with $P(A_i) > 0$, it gives $E[X] = \sum_i E[X \mid A_i] \, P(A_i)$. This extends the [[law of total probability|0E5G5FMT2R]].
More generally, for σ-algebras $\mathcal{H} \subseteq \mathcal{G}$, $E[E[X \mid \mathcal{G}] \mid \mathcal{H}] = E[X \mid \mathcal{H}]$. This is the tower property.

## Explanations

### L1 human

To find an overall average, you can first average inside each group and then average the group averages, with the group sizes as weights.
If $60\%$ of the students have mean score $70$ and $40\%$ have mean score $80$, the overall mean is $0.6 \cdot 70 + 0.4 \cdot 80 = 74$.
</document>

<document source="https://math2.tamnd.com/node/04RXTQA5ZA.md" tag="04RXTQA5ZA" kind="statement">
# Zorn's lemma

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $(P, \le)$ be a nonempty [[partially ordered set|011C29Q1S1]] in which each [[chain|0GP9ASVQ1X]] has an [[upper bound|0H5J02P8S9]] in $P$.
Then $P$ has a [[maximal element|0TZEEFTNCC]].

## Remarks

Over ZF, Zorn's lemma is equivalent to the axiom of choice.
If one requires an upper bound also for the empty chain, the hypothesis "$P$ is nonempty" is not needed.
Applications: each vector space has a basis, and each nonzero ring with unity has a maximal ideal.

## History

Kazimierz Kuratowski proved the lemma in 1922, and Max Zorn proved it independently in 1935.
The [[Hausdorff maximal principle|0KM4KVR9MV]] of 1914 is an earlier statement of the same kind.

## Explanations

### L1 human

Zorn's lemma is a tool to show that a maximal object exists.
You check that each increasing chain of objects has an object above it.
Then the lemma gives an object that you cannot enlarge any more.
</document>

<document source="https://math2.tamnd.com/node/04S2GMHX82.md" tag="04S2GMHX82" kind="concept">
# Parabola

## Summary

A parabola is a plane curve. Each of its points has the same distance from a fixed point, the focus, and from a fixed line, the directrix. It is symmetric about the line through the focus perpendicular to the directrix.
</document>

<document source="https://math2.tamnd.com/node/04S9HX3GAP.md" tag="04S9HX3GAP" kind="definition">
# Product topology on an arbitrary product

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]], indexed by any set $I$, and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]], with projections $\pi_i \colon X \to X_i$.
The product topology on $X$ is the topology generated by the [[subbasis|0CHR35YNJQ]]
$$\{ \pi_i^{-1}(U) : i \in I, \ U \text{ open in } X_i \}.$$

## Remarks

A basis consists of the sets $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$ and $U_i = X_i$ for all but finitely many $i$.
For two factors, this is the [[product topology|0Z57EC6YX3]] of $X_1 \times X_2$.
It is the coarsest topology on $X$ for which each projection $\pi_i$ is continuous.
A sequence converges in $X$ if and only if each of its coordinates converges. The [[box topology|0CTM1EHG9P]] is finer and lacks this property.

## Explanations

### L1 human

In an infinite product, a basic open set can restrict only finitely many coordinates.
This choice makes the [[universal property|0M6W4XNST6]] and [[Tychonoff's theorem|0JTRSJKCFB]] work.
</document>

<document source="https://math2.tamnd.com/node/04SD3F4FCR.md" tag="04SD3F4FCR" kind="problem">
# Doubling the cube

## Problem

Given the edge of a cube, construct with [[straightedge and compass|0DAECMZVRT]] alone the edge of a cube with twice the volume.

## Remarks

For an edge of length $1$, the new edge has length $\sqrt[3]{2}$.
So the problem asks whether $\sqrt[3]{2}$ is a [[constructible number|0FMN2MN7TM]].
The answer is no: [[doubling the cube is impossible|0CQ357M0QS]].
The problem has solutions with other tools, such as a marked ruler or conic sections.

## History

The problem is also called the Delian problem, after a legend about an oracle on the island of Delos.
Hippocrates of Chios showed that it is equivalent to finding two mean proportionals between a segment and a segment twice as long.
Pierre Wantzel proved in 1837 that no straightedge and compass solution exists.

## Explanations

### L1 human

The problem asks to build a cube with twice the volume of a given cube, using only an unmarked ruler and a compass.
It is one of the three classical construction problems of Greek geometry.
</document>

<document source="https://math2.tamnd.com/node/04SGPNT8R9.md" tag="04SGPNT8R9" kind="object">
# Galois field

## Description

For each prime power $q = p^n$, the Galois field $\mathrm{GF}(q)$, also written $\mathbb{F}_q$, is the [[finite field|08RTDHX1SK]] with $q$ elements.
It [[exists|0T8M3YC5RG]], and it is [[unique up to isomorphism|010PKCQPXP]].
One model is the splitting field of $x^q - x$ over $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
Another model is $\mathbb{F}_p[x]/(f)$ for an irreducible polynomial $f \in \mathbb{F}_p[x]$ of degree $n$.

## Remarks

The name honours Évariste Galois.
The field $\mathrm{GF}(4) = \mathbb{F}_2[x]/(x^2 + x + 1)$ has the elements $0, 1, \alpha, \alpha + 1$, with $\alpha^2 = \alpha + 1$.
The field $\mathrm{GF}(p^m)$ is isomorphic to a subfield of $\mathrm{GF}(p^n)$ if and only if $m$ divides $n$.

## Explanations

### L1 human

Since there is only one finite field of each order, up to a change of names, one gives it a fixed name: the Galois field with $q$ elements.
Computations in it are done with polynomials over the integers modulo $p$.
</document>

<document source="https://math2.tamnd.com/node/04SJTX9P4N.md" tag="04SJTX9P4N" kind="statement">
# Chinese remainder theorem for non-coprime moduli

## Statement

Let $m$ and $n$ be positive integers, and let $a$ and $b$ be integers. Let $g = \gcd(m, n)$ and $M = \operatorname{lcm}(m, n)$.
The system $x \equiv a \pmod m$, $x \equiv b \pmod n$ has an integer solution if and only if $a \equiv b \pmod g$.
In this case, two solutions are congruent modulo $M$.

## Remarks

When $g = 1$, this is the [[Chinese remainder theorem|0FNN9B4W64]] for two moduli.
The condition is necessary: a solution $x$ gives $a \equiv x \equiv b \pmod g$, since $g$ divides $m$ and $n$.
For $k$ moduli $n_1, \dots, n_k$, the system $x \equiv a_i \pmod{n_i}$ has a solution if and only if $\gcd(n_i, n_j) \mid a_i - a_j$ for all $i \neq j$.
For example, $x \equiv 1 \pmod 4$ and $x \equiv 3 \pmod 6$ has the solutions $x \equiv 9 \pmod{12}$. But $x \equiv 1 \pmod 4$ and $x \equiv 2 \pmod 6$ has no solution.

## Explanations

### L1 human

When the moduli share a factor $g$, the two conditions both say something about $x$ modulo $g$.
They must agree there. If they agree, there is a solution, and it is unique modulo the lcm.
</document>

<document source="https://math2.tamnd.com/node/04SW7WECS8.md" tag="04SW7WECS8" kind="definition">
# Solution of an ordinary differential equation

## Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] of order $n$, with $F$ defined on a set $U \subseteq \mathbb{R}^{n+2}$.
Let $I$ be an open [[interval|0Q1D85B8G6]] of $\mathbb{R}$.
A solution of the equation on $I$ is a function $u : I \to \mathbb{R}$ with these two properties.
The function $u$ is $n$ times differentiable on $I$, and $\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) \in U$ for each $x \in I$.
For each $x \in I$, $F\left(x, u(x), u'(x), \dots, u^{(n)}(x)\right) = 0$.
For a system, $u$ takes values in $\mathbb{R}^m$ and the same two conditions hold.

## Remarks

Consider the first-order system $x' = f(x)$, with a vector field $f$ on an open set of $\mathbb{R}^m$. A solution is a curve whose velocity at each time is the value of $f$ at its position. Such a curve is an integral curve of $f$.
The function $u(x) = e^{2x}$ is a solution of $y' = 2y$ on $\mathbb{R}$.
A solution on $I$ restricts to a solution on each open subinterval of $I$. A [[maximal solution|01J7FZ65J0]] is a solution that has no extension to a larger interval.

## Explanations

### L1 human

To check that a function is a solution, put it and its derivatives into the equation.
If both sides are equal at each point of the interval, the function is a solution there.
For example, $u(x) = \sin x$ gives $u'' + u = -\sin x + \sin x = 0$, so it solves $y'' + y = 0$.
</document>

<document source="https://math2.tamnd.com/node/04T1CT7212.md" tag="04T1CT7212" kind="statement">
# Contrapositive

## Statement

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The [[implication|0ZNX6DQ3EJ]] $p \to q$ is [[logically equivalent|06GYGS41JK]] to its contrapositive $\neg q \to \neg p$.
$$p \to q \equiv \neg q \to \neg p$$

## Remarks

The direction from $p \to q$ to $\neg q \to \neg p$ is valid in intuitionistic logic. The other direction needs classical logic.
This equivalence is the basis of [[proof by contrapositive|0BB5TPGQ1N]].
Do not confuse the contrapositive with the [[converse|0RVYE75E2M]] $q \to p$.

## Explanations

### L1 human

"If it rains, the street is wet" says the same as "if the street is not wet, it does not rain".
Both sentences are false in exactly one case: rain and a dry street.
</document>

<document source="https://math2.tamnd.com/node/04TKTF8KDK.md" tag="04TKTF8KDK" kind="concept">
# Higher derivative

## Summary

The higher derivatives of a function come from repeated differentiation. The $n$-th derivative $f^{(n)}$ is the derivative of $f^{(n-1)}$. The node [[higher derivative|084NGSTWM7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04TMP0XGMJ.md" tag="04TMP0XGMJ" kind="concept">
# Definite integral

## Summary

A definite integral is the integral of a function between two given limits of integration. It is a number, and its sign changes when the two limits are swapped. The node [[definite integral|021WEM680W]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04TSHB9H8D.md" tag="04TSHB9H8D" kind="definition">
# Little o notation

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The notation $f(x) = o(g(x))$ as $x \to \infty$ means that for each $\varepsilon > 0$ there is $x_1 \ge x_0$ with
$$|f(x)| \le \varepsilon |g(x)| \quad \text{for all } x \ge x_1.$$

## Remarks

If $g(x) \ne 0$ for large $x$, then $f = o(g)$ means that $f(x)/g(x) \to 0$.
Each $f = o(g)$ is also $f = O(g)$, in the sense of [[big O notation|0RYTGBJAGN]].
For example, $x = o(x^2)$ and $\ln x = o(x)$ as $x \to \infty$.
Mathlib defines it for filters, as `Asymptotics.IsLittleO`.

## Explanations

### L1 human

The statement $f = o(g)$ says that $f$ becomes negligible compared with $g$.
Each fixed fraction of $|g|$ eventually exceeds $|f|$.
</document>

<document source="https://math2.tamnd.com/node/04VA28F3HB.md" tag="04VA28F3HB" kind="concept">
# Continued fraction expansion of a real number

## Summary

The continued fraction expansion of a real number is the simple continued fraction found by taking integer parts and reciprocals again and again.
The [[definition of the continued fraction expansion|0SJ7Z6VKWK]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/04VCPS30ZE.md" tag="04VCPS30ZE" kind="definition">
# P-adic valuation

## Definition

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $n$ be a nonzero integer.
The $p$-adic valuation $\nu_p(n)$ is the largest natural number $k$ such that $p^k \mid n$.
By convention, $\nu_p(0) = \infty$.

## Remarks

For a positive integer $n$, $\nu_p(n)$ is the exponent of $p$ in the [[prime factorization|0AEKDFCYFA]] of $n$. It is $0$ when $p \nmid n$.
For example, $\nu_2(40) = 3$ and $\nu_5(40) = 1$, because $40 = 2^3 \cdot 5$.
For a nonzero rational number $a / b$, one sets $\nu_p(a / b) = \nu_p(a) - \nu_p(b)$.
The Lean library Mathlib sets $\nu_p(0) = 0$ for its function on natural numbers.

## Explanations

### L1 human

The number $\nu_p(n)$ counts how many times you can divide $n$ by $p$ and still get an integer.
For $n = 72$ and $p = 2$: $72, 36, 18, 9$, so you can divide three times, and $\nu_2(72) = 3$.
</document>

<document source="https://math2.tamnd.com/node/04VZE3FZXF.md" tag="04VZE3FZXF" kind="concept">
# Random walk

## Summary

A random walk is a stochastic process whose position is a sum of independent random steps. The simple random walk on the integers moves one step up or down at each time.
The node [[random walk|0C08YMV95H]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/04WC36XKV6.md" tag="04WC36XKV6" kind="statement">
# Multiplicativity of the divisor function

## Statement

Let $k$ be a natural number.
The [[divisor function|0YEW47VJ0J]] $\sigma_k$ is a [[multiplicative function|0AP80TT660]].
That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$.

## Remarks

The proof: for coprime $m$ and $n$, each divisor of $mn$ is a unique product $d e$ with $d \mid m$ and $e \mid n$.
The function $\sigma_k$ is not completely multiplicative: $\sigma_0(4) = 3$, but $\sigma_0(2)^2 = 4$.
At a prime power, $\sigma_k(p^e) = 1 + p^k + p^{2k} + \dots + p^{ek}$. Multiplicativity then gives $\sigma_k(n)$ for all $n$.

## Explanations

### L1 human

For example, $\sigma(4) = 7$ and $\sigma(3) = 4$, so $\sigma(12) = 28$.
This works because $4$ and $3$ share no prime factor. Each divisor of $12$ is a divisor of $4$ times a divisor of $3$.
</document>

<document source="https://math2.tamnd.com/node/04WSW8T3SY.md" tag="04WSW8T3SY" kind="definition">
# Sigma-algebra

## Definition

Let $\Omega$ be a [[set|0SJ3RX9TXA]].
A σ-algebra on $\Omega$ is a subset $\mathcal{F}$ of the [[power set|0BC7KX51Z8]] of $\Omega$ with these three properties.
The set $\Omega$ is in $\mathcal{F}$.
If $A \in \mathcal{F}$, then its [[complement|0B1FSS1VZW]] $\Omega \setminus A$ is in $\mathcal{F}$.
If $A_1, A_2, A_3, \dots$ is a sequence of elements of $\mathcal{F}$, then the [[union|07KVB4JSC7]] $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{F}$.
The pair $(\Omega, \mathcal{F})$ is a measurable space, and the elements of $\mathcal{F}$ are the measurable sets.

## Remarks

A σ-algebra contains $\varnothing$ and is closed under countable intersections, by De Morgan's laws.
The power set of $\Omega$ and the family $\{\varnothing, \Omega\}$ are σ-algebras on $\Omega$. They are the largest and the smallest σ-algebras on $\Omega$.
For each family $\mathcal{C}$ of subsets of $\Omega$, the intersection of all σ-algebras that contain $\mathcal{C}$ is a σ-algebra. It is the σ-algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$.
The [[Borel σ-algebra|0BVHZXWKCD]] on $\mathbb{R}$ is the σ-algebra generated by the open intervals.

## Explanations

### L1 human

A σ-algebra is the list of the sets that we are allowed to measure.
If we can measure a set, then we can measure its complement.
If we can measure each set of a sequence, then we can measure their union.
For a finite sample space we can usually measure every subset, so the σ-algebra is the power set.
</document>

<document source="https://math2.tamnd.com/node/04WTBQ7APQ.md" tag="04WTBQ7APQ" kind="statement">
# Integral of cosine

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b \cos x \, dx = \sin b - \sin a.$$

## Remarks

The [[derivative of sine|00R4MWW532]] is $\cos$, so $\sin$ is an antiderivative of $\cos$.
For example, $\int_0^{\pi/2} \cos x \, dx = 1$.
Mathlib states it as `integral_cos`.

## Explanations

### L1 human

The sine function is an antiderivative of the cosine function.
So the integral of cosine is a difference of two sine values.
</document>

<document source="https://math2.tamnd.com/node/04X3PR72VJ.md" tag="04X3PR72VJ" kind="concept">
# Determinant as signed volume

## Summary

Let $n \ge 1$ and let $v_1, \dots, v_n \in \R^n$ be the columns of a real $n \times n$ matrix $A$.
The parallelepiped spanned by them is $P = \{t_1 v_1 + \cdots + t_n v_n : 0 \le t_i \le 1\}$.
Its $n$-dimensional volume is $\abs{\det A}$, where $\det$ is the [[determinant|00KDZRW4C0]].
If $\det A \ne 0$, then its sign is positive when $(v_1, \dots, v_n)$ has the [[orientation|0QJJAQ6ZYZ]] of the standard basis, and negative when it has the opposite orientation.

## Remarks

In $\R^2$, $\abs{ad - bc}$ is the area of the parallelogram spanned by $(a, c)$ and $(b, d)$.
For each region $S$ of finite volume, the linear map $x \mapsto Ax$ multiplies the volume of $S$ by $\abs{\det A}$.
This fact is the origin of the Jacobian determinant in the change of variables formula.

## Explanations

### L1 human

The determinant measures how a matrix changes volume.
The unit cube goes to a box spanned by the columns, and $\abs{\det A}$ is the volume of this box.
A negative determinant means that the matrix also turns space inside out, like a mirror.
</document>

<document source="https://math2.tamnd.com/node/04XGKK8WQW.md" tag="04XGKK8WQW" kind="statement">
# Existence of primitive roots modulo a prime

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then there is a [[primitive root|0KM2CJ80G7]] modulo $p$.
That is, the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic of order $p - 1$.

## Remarks

A proof: a polynomial of degree $d$ has at most $d$ roots in the field $\mathbb{Z}/p\mathbb{Z}$. So for each $d \mid p - 1$, at most $\varphi(d)$ elements have order $d$. The [[sum of the totient over the divisors|09JXE6XMCC]] then forces exactly $\varphi(d)$ elements of each order $d$.
In particular, there are exactly $\varphi(p - 1)$ primitive roots modulo $p$ in $\{1, \dots, p - 1\}$.
More generally, each finite subgroup of the multiplicative group of a field is cyclic.

## Explanations

### L1 human

For each prime $p$, some number $g$ has powers that run through all of $1, 2, \dots, p - 1$ modulo $p$.
For $p = 11$, the number $2$ works: its powers are $2, 4, 8, 5, 10, 9, 7, 3, 6, 1$.
</document>

<document source="https://math2.tamnd.com/node/04XPQT2XTC.md" tag="04XPQT2XTC" kind="statement">
# Every connected graph has a spanning tree

## Statement

Let $G$ be a [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]].
Then $G$ has a [[spanning tree|0FQ5B8S1ZC]].

## Remarks

For a finite graph, a proof deletes edges that lie on cycles, one at a time. Such an edge is not a [[bridge|0D81KWH9ZM]], so the graph stays connected, and the process ends with a tree.
For an infinite graph, the proof uses Zorn's lemma, or a well-ordering of the vertices.
The converse also holds: a graph with a spanning tree is connected.

## Explanations

### L1 human

If a connected graph has a loop, you can remove one edge of the loop and the graph stays in one piece.
Repeat until there are no loops. What is left is a spanning tree.
</document>

<document source="https://math2.tamnd.com/node/04XWH28452.md" tag="04XWH28452" kind="concept">
# Elementary row operations

## Summary

The elementary row operations are three kinds of operations on the rows of a matrix.
They swap two rows, multiply a row by a nonzero scalar, or add a multiple of one row to another row.
</document>

<document source="https://math2.tamnd.com/node/04YG690N5V.md" tag="04YG690N5V" kind="definition">
# Probability density function

## Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$.
A probability density function of $X$ is a Borel measurable function $f \colon \mathbb{R} \to [0, \infty)$ such that $P(X \in B) = \int_B f(x) \, dx$ for each Borel set $B \subseteq \mathbb{R}$.
So $X$ has a probability density function if and only if $X$ is a [[random variable with a density|0C21RZG5V2]].

## Remarks

Each density satisfies $\int_{-\infty}^{\infty} f(x) \, dx = 1$. Two densities of $X$ are equal except on a set of Lebesgue measure zero.
$P(a \le X \le b) = \int_a^b f(x) \, dx$ for $a \le b$.
If the [[cumulative distribution function|0MXMS3AE13]] $F_X$ is differentiable at $x$ and $f$ is continuous at $x$, then $F_X'(x) = f(x)$.
A density can be larger than $1$. The uniform distribution on $[0, 1/2]$ has the density $2$ on this interval.

## Explanations

### L1 human

The density $f(x)$ is not a probability. It is a probability per unit of length near $x$.
For a small $h$, the probability that $X$ lies between $x$ and $x + h$ is about $f(x) \, h$.
</document>

<document source="https://math2.tamnd.com/node/04YKAYDQPW.md" tag="04YKAYDQPW" kind="statement">
# Monotone functions are Riemann integrable

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be monotone.
Then $f$ is Riemann integrable on $[a, b]$.

## Remarks

Assume $f$ increasing, and take the partition into $n$ equal pieces. Then $M_i = f(x_i)$ and $m_i = f(x_{i-1})$, so the sum telescopes:
$$U(f, P) - L(f, P) = \frac{b - a}{n} (f(b) - f(a)).$$
This tends to $0$, so [[Riemann's criterion|017DH8WXTE]] applies.
A monotone function can have infinitely many discontinuities, and it is still integrable.

## Explanations

### L1 human

For an increasing function, the gaps between upper and lower rectangles stack up to one column of total height $f(b) - f(a)$.
Its width shrinks to $0$ as the partition gets finer.
</document>

<document source="https://math2.tamnd.com/node/04Z5G0THN6.md" tag="04Z5G0THN6" kind="statement">
# Comparison test

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences with $0 \le a_n \le b_n$ for all $n$.
If $\sum b_n$ converges, then $\sum a_n$ converges, and $\sum a_n \le \sum b_n$.
If $\sum a_n$ diverges, then $\sum b_n$ diverges.

## Remarks

The partial sums of $\sum a_n$ are at most those of $\sum b_n$. The result follows from [[a series of nonnegative terms converges iff its partial sums are bounded|0MX1HZBBXJ]].
It is enough that $0 \le a_n \le b_n$ for all $n$ larger than some $N$.
For example, $0 \le 1/(n^2 + 1) \le 1/n^2$, so $\sum 1/(n^2 + 1)$ converges.
Mathlib states it for a summable upper bound, in `Summable.of_nonneg_of_le`.

## Explanations

### L1 human

If a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too.
If a smaller series already has an infinite sum, the larger one has an infinite sum as well.
</document>

<document source="https://math2.tamnd.com/node/05006AJ7DZ.md" tag="05006AJ7DZ" kind="concept">
# Tautology

## Summary

A tautology is a formula of [[propositional logic|09K9MWD2QQ]] that is true for every assignment of truth values to its variables.
</document>

<document source="https://math2.tamnd.com/node/05099E1NMJ.md" tag="05099E1NMJ" kind="definition">
# Cardinality of a finite set

## Definition

Let $A$ be a set and let $n$ be a natural number.
The set $A$ has $n$ elements if there is a bijection from $\{1, 2, \dots, n\}$ to $A$.
For $n = 0$, the set $\{1, \dots, n\}$ is the empty set, so only the empty set has $0$ elements.
A set is finite if it has $n$ elements for some natural number $n$.

For a finite set $A$, this number $n$ is unique.
It is the cardinality of $A$, with the notation $|A|$ or $\# A$.

## Remarks

Two finite sets have the same cardinality if and only if there is a bijection between them. The [[bijection principle|0GV2573FP4]] states this.
For example, $|\{a, b, c\}| = 3$ and $|\varnothing| = 0$.
The uniqueness of $n$ follows from the fact that there is no injection from $\{1, \dots, m\}$ to $\{1, \dots, n\}$ when $m > n$.

## Explanations

### L1 human

The cardinality of a finite set is the number of its elements.
To count a set, you label its elements $1, 2, 3, \dots$ without gaps and without repeats.
The last label is the cardinality.
</document>

<document source="https://math2.tamnd.com/node/050F2FMTKF.md" tag="050F2FMTKF" kind="concept">
# Harmonic conjugate

## Summary

A harmonic conjugate of a real harmonic function $u$ is a real function $v$ such that $u + iv$ is holomorphic, see the [[definition|0CC56MJ4FG]].
On a domain, it is unique up to an additive constant when it exists.
</document>

<document source="https://math2.tamnd.com/node/0515A6D2Y1.md" tag="0515A6D2Y1" kind="concept">
# Symmetric group

## Summary

The symmetric group on a set $X$ is the group of all bijections from $X$ to $X$, with composition as the operation.
For $X = \{1, \dots, n\}$ it is written $S_n$, and its elements are the permutations of $n$ symbols.
</document>

<document source="https://math2.tamnd.com/node/0518MM6BE1.md" tag="0518MM6BE1" kind="statement">
# The real numbers are uncountable

## Statement

The set $\mathbb{R}$ of real numbers is [[uncountable|0X55M9XET4]].

## Remarks

Cantor proved it in his article of 1874. That proof uses nested intervals.
A later proof uses [[Cantor's diagonal argument|0XCH8QVDCQ]] on decimal or binary expansions.
Since the algebraic numbers are countable, the result shows that transcendental numbers exist. Cantor gave this application in the same article.

## Explanations

### L1 human

Each list of real numbers misses some real number.
So there are more real numbers than natural numbers, although both sets are infinite.
</document>

<document source="https://math2.tamnd.com/node/051CYS71H5.md" tag="051CYS71H5" kind="concept">
# Classical definition of probability

## Summary

Suppose that all cases are equally likely.
Then the classical definition says that the probability of an event is the number of favourable cases divided by the number of all cases.
It goes back to Jacob Bernoulli and Pierre-Simon Laplace. The node [[classical definition of probability|0AKNPYQ04R]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/051VYKA1EX.md" tag="051VYKA1EX" kind="statement">
# Euclid's lemma

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ and $b$ be integers.
If $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Remarks

The hypothesis that $p$ is prime is necessary. For example, $6 \mid 4 \cdot 9$, but $6 \nmid 4$ and $6 \nmid 9$.
A short proof uses [[Bézout's identity|0DQVHMYAGA]]. If $p \nmid a$, then $\gcd(p, a) = 1$, so $p x + a y = 1$ for some integers $x$ and $y$. Multiply by $b$ to see that $p \mid b$.
The [[generalized Euclid's lemma|03BAREW1Z5]] replaces the prime $p$ by an integer that is coprime to $a$.
The lemma is the main step in the proof of the uniqueness part of the [[fundamental theorem of arithmetic|084HH1NDCB]].

## History

The lemma is Proposition 30 of Book VII of the Elements of Euclid.

## Explanations

### L1 human

A prime cannot be split into smaller factors, so it cannot share itself between $a$ and $b$.
If $7$ divides a product $a b$, then $7$ already divides $a$ or $b$.
For example, $7 \mid 14 \cdot 3$ and $7 \mid 14$.
</document>

<document source="https://math2.tamnd.com/node/0528HKWZ1Z.md" tag="0528HKWZ1Z" kind="statement">
# Product of a matrix and its adjugate

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$.
Then
$$A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A)\, I_n,$$
where $\operatorname{adj}(A)$ is the [[adjugate|07W9H8QF38]] of $A$ and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The diagonal entries of $A \operatorname{adj}(A)$ are the [[Laplace expansions|0C08BQ3M05]] of $\det A$ along the rows.
The entries off the diagonal are expansions of matrices with two equal rows, so they are zero.
The identity holds for all square matrices, also for singular ones.

## Explanations

### L1 human

Multiplying a matrix by its adjugate gives the determinant times the identity matrix.
So, if the determinant is invertible, then dividing the adjugate by it gives the inverse.
For a singular matrix, the product is the zero matrix.
</document>

<document source="https://math2.tamnd.com/node/052JJ9YVNM.md" tag="052JJ9YVNM" kind="concept">
# Graph homomorphism

## Summary

A graph homomorphism is a map between the vertex sets of two graphs that preserves adjacency. Homomorphisms unify colourings and other graph maps. The node [[graph homomorphism|06SCB3R6BN]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/052M4SX57M.md" tag="052M4SX57M" kind="definition">
# Annihilating polynomial

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
A polynomial $p \in F[x]$ *annihilates* $T$ when $p(T) = 0$, where $p(T)$ is the [[value of $p$ at $T$|0E3MV6E6MG]].
Such a polynomial is an *annihilating polynomial* of $T$.

## Remarks

The annihilating polynomials of $T$ form an ideal of $F[x]$.
If $\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\id_V, T, \dots, T^{n^2}$ are linearly dependent.
The monic generator of this ideal is the [[minimal polynomial|03NDVCSHSB]] of $T$.
For a [[projection|033DA1WE8Y]] $P$, the polynomial $x^2 - x$ annihilates $P$.

## Explanations

### L1 human

An annihilating polynomial is a polynomial that becomes the zero map when we put $T$ into it.
For example, a reflection $R$ satisfies $R^2 = I$, so $x^2 - 1$ annihilates it.
Such relations tell us a lot about the possible eigenvalues.
</document>

<document source="https://math2.tamnd.com/node/052SDEPZH7.md" tag="052SDEPZH7" kind="definition">
# Irreducible element

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
An element $p \in R$ is irreducible if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a [[unit|019NFGFPR5]].
3. If $p = a b$ with $a, b \in R$, then $a$ is a unit or $b$ is a unit.

## Remarks

The irreducible elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ prime.
In $\mathbb{Z}[i]$, the number $3$ is irreducible, but $5 = (2 + i)(2 - i)$ is not.
Every [[prime element|0MNHXBSX3M]] is irreducible. The converse [[can fail|0CHE3J7QMX]].
For polynomials over a field, the irreducible elements are the [[irreducible polynomials|00X2A9G9W3]].

## Explanations

### L1 human

An irreducible element cannot be split into smaller factors.
The only factorizations are trivial ones, with a unit as one factor, like $7 = (-1)(-7)$.
In the integers, the irreducible elements are the primes and their negatives.
</document>

<document source="https://math2.tamnd.com/node/053E86NWVK.md" tag="053E86NWVK" kind="definition">
# Associate elements

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $a, b \in R$.
The elements $a$ and $b$ are associates, written $a \sim b$, if there is a [[unit|019NFGFPR5]] $u \in R$ with $b = u a$.

## Remarks

Being associates is an equivalence relation.
In an [[integral domain|09S3PFBEPM]], $a \sim b$ if and only if $a \mid b$ and $b \mid a$, if and only if $(a) = (b)$.
In $\mathbb{Z}$, the associates of $n$ are $n$ and $-n$.
In $\mathbb{Z}[i]$, the associates of $1 + i$ are $1 + i$, $-1 - i$, $-1 + i$ and $1 - i$.

## Explanations

### L1 human

Associates are elements that differ only by a unit factor.
For divisibility they behave in the same way.
In the integers, $6$ and $-6$ are associates.
</document>

<document source="https://math2.tamnd.com/node/053QVHF6N7.md" tag="053QVHF6N7" kind="statement">
# Eigenvalues are the roots of the characteristic polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $p_T$ be the [[characteristic polynomial|0J5EMJRRFM]] of $T$, and let $\lambda \in F$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$ if and only if $p_T(\lambda) = 0$.

## Remarks

The proof: $\lambda$ is an eigenvalue if and only if $\lambda \id_V - T$ is not injective, if and only if $\det(\lambda \id_V - T) = 0$.
So $T$ has at most $\dim V$ eigenvalues.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, $p_A(t) = t^2 - 4t + 3 = (t - 1)(t - 3)$, so the eigenvalues are $1$ and $3$.

## Explanations

### L1 human

To find the eigenvalues of a matrix, compute its characteristic polynomial and find its roots.
Each root is an eigenvalue, and each eigenvalue is a root.
Then, for each eigenvalue $\lambda$, solve $(A - \lambda I)x = 0$ to find the eigenvectors.
</document>

<document source="https://math2.tamnd.com/node/054K3HNN5Z.md" tag="054K3HNN5Z" kind="concept">
# Converse relation

## Summary

The converse of a relation is the relation with all pairs reversed.
</document>

<document source="https://math2.tamnd.com/node/054Q5PW5GX.md" tag="054Q5PW5GX" kind="statement">
# Compact iff complete and totally bounded

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if $K$, with the [[subspace metric|0T2WYDTRP2]], is [[complete|08EWT7J0SC]] and $K$ is [[totally bounded|0WE5BBHGM6]].

## Remarks

Proof idea of the converse: in a totally bounded set, a diagonal argument gives a Cauchy subsequence of each sequence. Completeness gives a limit. Thus $K$ is sequentially compact, and so [[compact|0K8SGQSV7G]].
In a complete space $X$, a subset is compact if and only if it is closed and totally bounded.
The [[Heine–Borel theorem|07PFYV65MB]] follows, because bounded subsets of $\mathbb{R}^n$ are totally bounded.

## Explanations

### L1 human

Compactness in a metric space splits into two parts: no holes, and finitely many small pieces for each size.
The interval $(0, 1)$ is totally bounded but has holes at its ends. The real line has no holes but is not totally bounded.
</document>

<document source="https://math2.tamnd.com/node/054SBCQ93T.md" tag="054SBCQ93T" kind="example">
# The dihedral group of order 8

## Example

The [[dihedral group|0JZ27DMJ5D]] $D_4$ is the group of the symmetries of a square, and it has order $8$.
Let $r$ be the rotation by $90$ degrees and let $s$ be a reflection in a line of symmetry.
Then $D_4 = \{e, r, r^2, r^3, s, rs, r^2s, r^3s\}$, with $r^4 = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$.
The elements $r$ and $r^3$ have order $4$, and the other five elements different from $e$ have order $2$.
The [[centre|0DZ93493VN]] of $D_4$ is $\{e, r^2\}$, because the rotation by $180$ degrees commutes with each symmetry.
The rotation subgroup $\langle r \rangle$ is [[normal|0CB0DX5BVV]] of index $2$.
The subgroup $\{e, s\}$ is not normal, because $r s r^{-1} = r^2 s$.
Numbering the vertices $1, 2, 3, 4$ gives an injective homomorphism from $D_4$ into $S_4$, and its image is a Sylow $2$-subgroup of $S_4$.
</document>

<document source="https://math2.tamnd.com/node/054VXHMCA3.md" tag="054VXHMCA3" kind="definition">
# Complex exponential

## Definition

The complex exponential is the function $\exp : \mathbb{C} \to \mathbb{C}$ given by
$$\exp z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$$
for each [[complex number|09FTAET5Z4]] $z$. The series converges absolutely for each $z \in \mathbb{C}$.
Another notation for $\exp z$ is $e^z$.

## Remarks

For real $x$, $\exp x$ is the real exponential function, and $e = \exp 1$.
For $z = x + iy$ with $x, y$ real, $e^z = e^x (\cos y + i \sin y)$. The case $x = 0$ is [[Euler's formula|04R3CY41VJ]].
By the [[product of exponentials|0WM7DJCM6G]], $e^{z + w} = e^z e^w$. So $e^z \neq 0$ for each $z$, and $|e^z| = e^{\operatorname{Re} z}$.

## Explanations

### L1 human

The power series of $e^x$ makes sense for complex numbers too, so we use it as the definition of $e^z$.
The real part of $z$ controls the size of $e^z$, and the imaginary part of $z$ controls its direction.
For example, $e^{i\pi} = -1$: the size is $e^0 = 1$ and the direction is a half turn.
</document>

<document source="https://math2.tamnd.com/node/054YEQ0Y8M.md" tag="054YEQ0Y8M" kind="concept">
# Sesquilinear form

## Summary

A sesquilinear form on a complex vector space is a map of two vectors to the complex numbers.
It is linear in one argument and conjugate linear in the other.
</document>

<document source="https://math2.tamnd.com/node/05598J0Y8R.md" tag="05598J0Y8R" kind="statement">
# Fermat numbers are pairwise coprime

## Statement

Let $m$ and $n$ be distinct natural numbers.
Then the [[Fermat numbers|0R03B54V4Z]] $F_m$ and $F_n$ are [[coprime|0GFKP8F8AG]].

## Remarks

The proof uses $F_0 F_1 \cdots F_{n-1} = F_n - 2$. A common divisor of $F_m$ and $F_n$ with $m < n$ divides $2$. Fermat numbers are odd, so the divisor is $1$.
The result is often called Goldbach's theorem, after Christian Goldbach.
A consequence is a new proof that there are [[infinitely many primes|0292EPPPT0]]: each $F_n$ has a prime factor, and these primes are all distinct.

## Explanations

### L1 human

No two Fermat numbers share a prime factor.
For example, $F_2 = 17$ and $F_3 = 257$ are coprime, and $F_5 = 641 \cdot 6700417$ has no factor in common with $3$, $5$, $17$, $257$ or $65537$.
</document>

<document source="https://math2.tamnd.com/node/055HT85T3S.md" tag="055HT85T3S" kind="statement">
# PID implies UFD

## Statement

Let $R$ be a [[principal ideal domain|0RZYDVZ475]].
Then $R$ is a [[unique factorisation domain|06YMXJMXYS]].

## Remarks

Existence of factorisations: a principal ideal domain is Noetherian, so a process of splitting an element into factors must stop.
Uniqueness: in a principal ideal domain, an irreducible element generates a maximal ideal, so it is prime.
Consequences: $\mathbb{Z}$, $\mathbb{Z}[i]$ and $K[x]$ for a field $K$ have unique factorisation.
The converse is false: $\mathbb{Z}[x]$ is a unique factorisation domain but [[not a principal ideal domain|06E3AG9TM2]].

## Explanations

### L1 human

In a principal ideal domain, every irreducible element behaves like a prime number.
That is the key to unique factorisation.
So every Euclidean domain, such as the integers, has unique factorisation.
</document>

<document source="https://math2.tamnd.com/node/055VNYT0PB.md" tag="055VNYT0PB" kind="concept">
# Perpendicular bisector

## Summary

The perpendicular bisector of a segment passes through the midpoint of the segment at a right angle to it. In the plane it is a line, and in space it is a plane. Its points are exactly the points at the same distance from the two endpoints.
</document>

<document source="https://math2.tamnd.com/node/055VVSN00V.md" tag="055VVSN00V" kind="definition">
# Lipschitz condition

## Definition

Let $S$ be a subset of $\mathbb{R}^n$, let $f : S \to \mathbb{R}^m$ be a [[function|0EPAT51DXQ]], and let $K \ge 0$ be a real number.
The function $f$ satisfies a Lipschitz condition on $S$ with constant $K$, or is $K$-Lipschitz on $S$, if
$$\| f(x) - f(y) \| \le K\, \| x - y \| \quad \text{for all } x, y \in S.$$
Let $D \subseteq \mathbb{R} \times \mathbb{R}^n$ and $f : D \to \mathbb{R}^m$. The function $f$ is Lipschitz in $x$ on $D$ with constant $K$ if $\| f(t, x) - f(t, y) \| \le K \| x - y \|$ whenever $(t, x), (t, y) \in D$.

## Remarks

A $K$-Lipschitz function is uniformly continuous.
A differentiable function on a convex set with $\| f'(x) \| \le K$ at each point is $K$-Lipschitz, by the mean value inequality. For example, $\sin$ is $1$-Lipschitz on $\mathbb{R}$.
The function $x \mapsto \sqrt{|x|}$ is continuous but not Lipschitz on any interval that contains $0$. The [[non-uniqueness example|00K7ASZ1PR]] uses a function of this kind.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] needs a Lipschitz condition in $x$.

## Explanations

### L1 human


A Lipschitz function cannot change faster than a fixed rate.
The slope of each chord of its graph is at most $K$ in absolute value.
</document>

<document source="https://math2.tamnd.com/node/055Z15H69G.md" tag="055Z15H69G" kind="concept">
# Sequential continuity

## Summary

A function is sequentially continuous when it maps each convergent sequence in its domain to a sequence that converges to the value at the limit. For real functions, this is the same as continuity. The node [[sequential continuity|0MVDQEAB60]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0565DC8RMK.md" tag="0565DC8RMK" kind="definition">
# Jordan measure

## Definition

A box in $\mathbb{R}^n$ is a set $Q = [a_1, b_1) \times \dots \times [a_n, b_n)$ with $a_i \le b_i$, and its volume is $|Q| = (b_1 - a_1) \cdots (b_n - a_n)$.
A simple set is a finite union of boxes. Each simple set $S$ is a finite union of pairwise disjoint boxes $Q_1, \dots, Q_k$, and $m(S) = |Q_1| + \dots + |Q_k|$ does not depend on this choice.
For a bounded set $A \subseteq \mathbb{R}^n$, the inner and outer Jordan measures are
$$m_*(A) = \sup \{ m(S) : S \subseteq A \}, \qquad m^*(A) = \inf \{ m(S) : S \supseteq A \},$$
where $S$ runs over simple sets.
The set $A$ is Jordan measurable if $m_*(A) = m^*(A)$, and then this common value $m(A)$ is the Jordan measure, or Jordan content, of $A$.

## Remarks

Here $\sup$ is the [[supremum|0R3H5PANRQ]] and $\inf$ is the [[infimum|0QF1MZ9FRG]].
Closed boxes, balls and bounded regions between the graphs of two continuous functions are Jordan measurable. Finite unions, intersections and differences of Jordan measurable sets are Jordan measurable.
A bounded set $A$ is Jordan measurable if and only if its boundary has Jordan measure zero. Then $m(A) = \int_Q \mathbf{1}_A$ for each closed box $Q \supseteq A$, where $\mathbf{1}_A$ is the indicator function of $A$.
The set $\mathbb{Q} \cap [0, 1]$ is not Jordan measurable: its inner measure is $0$ and its outer measure is $1$. Lebesgue measure extends Jordan measure to more sets.

## Explanations

### L1 human

Jordan measure measures a set by fitting finite unions of boxes inside it and around it.
If the best inner and outer approximations have the same total volume, that volume is the measure of the set.
</document>

<document source="https://math2.tamnd.com/node/0565EGK6XP.md" tag="0565EGK6XP" kind="statement">
# Lebesgue's criterion for Riemann integrability

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
Then $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if the set of points where $f$ is not [[continuous|0EEE2CZB2A]] has Lebesgue measure zero.

## Remarks

A set $E \subseteq \mathbb{R}$ has measure zero if for each $\varepsilon > 0$ it is covered by countably many open intervals of total length less than $\varepsilon$.
Countable sets have measure zero. So [[Thomae's function|0HJ28H9Y95]] is integrable, and so is each monotone function.
The [[Dirichlet function|03Z4JC4YRJ]] is discontinuous everywhere, so it is not integrable.
Giuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.

## Explanations

### L1 human

A bounded function is Riemann integrable exactly when it is continuous at almost all points.
The bad points must form a set of total length zero.
</document>

<document source="https://math2.tamnd.com/node/056FSKG32J.md" tag="056FSKG32J" kind="definition">
# Pointwise convergence

## Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions.
The sequence $(f_n)$ converges pointwise to $f$ on $S$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for every $x \in S$.
That is, for every $x \in S$ and every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$.

## Remarks

Here $N$ may depend on both $\varepsilon$ and $x$.
The pointwise limit is unique, since [[limits|0XKWSSY8P2]] of real sequences are unique.
A pointwise limit of continuous functions need not be continuous: see [[the sequence $x^n$|0YSRR1JTKZ]]. Mathlib expresses pointwise convergence as convergence in the product topology, through `tendsto_pi_nhds`.

## Explanations

### L1 human

Pointwise convergence means that at each single point, the values $f_n(x)$ approach $f(x)$.
Different points may converge at very different speeds.
</document>

<document source="https://math2.tamnd.com/node/056SW1E0AC.md" tag="056SW1E0AC" kind="statement">
# A matrix is diagonalisable iff it has a basis of eigenvectors

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[diagonalisable|07SQCNV233]] if and only if $F^n$ has a [[basis|0KCQ9Y3QKA]] of [[eigenvectors|0N6960W0BV]] of $A$.
If $v_1, \dots, v_n$ is such a basis with $A v_i = \lambda_i v_i$, and $P$ has the columns $v_1, \dots, v_n$, then $P^{-1} A P$ is the diagonal matrix with entries $\lambda_1, \dots, \lambda_n$.

## Remarks

The equation $AP = PD$ says column by column that $A v_i = \lambda_i v_i$.
Equivalently, $F^n$ is the direct sum of the eigenspaces of $A$.
Equivalently, the sum of the [[geometric multiplicities|099TWVJ909]] of the eigenvalues is $n$.

## Explanations

### L1 human

To diagonalise a matrix, we need $n$ independent eigenvectors.
We put them as columns into a matrix $P$.
Then $P^{-1} A P$ is diagonal, with the eigenvalues on the diagonal.
</document>

<document source="https://math2.tamnd.com/node/056VS31A0J.md" tag="056VS31A0J" kind="statement">
# Isosceles triangle theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$ with $d(A, B) = d(A, C)$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].
Then the angles at $B$ and $C$ are equal: $\angle ABC = \angle ACB$.

## Remarks

The converse holds too: if $\angle ABC = \angle ACB$ in a triangle $ABC$, then $d(A, B) = d(A, C)$.
Mathlib's form of the statement does not need that $A$, $B$ and $C$ are not collinear.
In an equilateral triangle, the theorem applies at each vertex. So all three angles are equal.

## History

The statement is Proposition 5 of Book I of Euclid's Elements. Its Latin name is pons asinorum, "the bridge of donkeys".

## Explanations

### L1 human

An isosceles triangle has two equal sides, like the outline of a simple roof. The theorem says that the two base angles are then equal. Fold the triangle along the line from the top vertex to the midpoint of the base: the two halves match.
</document>

<document source="https://math2.tamnd.com/node/05788J51WZ.md" tag="05788J51WZ" kind="definition">
# Parametrised surface

## Definition

Let $D \subseteq \mathbb{R}^2$ be a set that is the closure of an open set.
A parametrised surface is a map $r \colon D \to \mathbb{R}^3$, $r(u, v) = \big( x(u, v), y(u, v), z(u, v) \big)$, of [[class C^1|0TCBZMEYGN]] on an open set that contains $D$.
Its image $S = r(D)$ is the surface that $r$ parametrises.
The parametrisation is regular at $(u, v)$ if the [[partial derivatives|05AXG3QRKR]] $r_u = \partial r / \partial u$ and $r_v = \partial r / \partial v$ are linearly independent, that is, $r_u \times r_v \ne 0$.

## Remarks

The graph of a function $f \colon D \to \mathbb{R}$ is parametrised by $r(x, y) = (x, y, f(x, y))$. Here $r_x \times r_y = (-f_x, -f_y, 1) \ne 0$, so this parametrisation is regular.
The unit sphere is parametrised by $r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi)$ on $[0, \pi] \times [0, 2\pi]$. It is not regular at the poles $\varphi = 0$ and $\varphi = \pi$.
The cylinder of radius $R > 0$ about the $z$-axis is parametrised by $r(\theta, z) = (R \cos \theta, R \sin \theta, z)$.
One surface has many parametrisations. Areas and surface integrals do not depend on the choice.

## Explanations

### L1 human

A parametrised surface is to a surface what a [[parametrised curve|0JF1XXGWB4]] is to a curve, with two parameters instead of one.
Think of $(u, v)$ as coordinates on a flat sheet, and of $r$ as a map that bends the sheet into space.
</document>

<document source="https://math2.tamnd.com/node/0578PA53VA.md" tag="0578PA53VA" kind="statement">
# Generating function for partitions

## Statement

Let $p(n)$ be the [[partition function|001CX45SWQ]].
Then, as formal power series in $\mathbb{Z}[\![x]\!]$,
$$\sum_{n \ge 0} p(n)\, x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}.$$

## Remarks

The infinite product makes sense, because each coefficient of the product depends only on finitely many factors.
Proof idea: the factor $\frac{1}{1 - x^k} = 1 + x^k + x^{2k} + \dots$ chooses how many parts equal $k$.
Restricting the factors gives other results. For example, $\prod_{k \ge 1} (1 + x^k)$ counts partitions into distinct parts. This gives [[Euler's partition theorem|0XFZ1BMF5Q]].

## Explanations

### L1 human

Each factor of the product decides how many times a part $k$ appears in the partition.
Multiplying the factors combines these choices, and the coefficient of $x^n$ counts the choices with total $n$.
</document>

<document source="https://math2.tamnd.com/node/057BH3JC9S.md" tag="057BH3JC9S" kind="statement">
# The nth roots of unity

## Statement

Let $n$ be a positive integer.
A complex number $z$ is an $n$th [[root of unity|0S61N75H45]] if and only if
$$z = e^{2 \pi i k / n} = \cos \frac{2 \pi k}{n} + i \sin \frac{2 \pi k}{n}$$
for some integer $k$ with $0 \le k < n$.

## Remarks

Proof: by [[de Moivre's formula|03MTBVG3VE]], each such $z$ satisfies $z^n = e^{2 \pi i k} = 1$. Conversely, $z^n = 1$ forces $|z| = 1$ and $n \arg z \in 2 \pi \mathbb{Z}$.
For $n = 3$, the roots are $1$ and $-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i$.

## Explanations

### L1 human

The $n$th roots of unity sit on the unit circle at equal angles of $2\pi/n$, starting at $1$.
For $n = 6$, they are the six corners of a regular hexagon.
</document>

<document source="https://math2.tamnd.com/node/057H72B2MB.md" tag="057H72B2MB" kind="statement">
# Number of compositions

## Statement

Let $n$ be a positive integer.
Then the number of [[compositions|0J8ANSYKAC]] of $n$ is $2^{n-1}$.

## Remarks

Proof: write $n$ as a row of $n$ ones. A composition is a choice of where to cut among the $n - 1$ gaps between them.
The number of compositions of $n$ into exactly $k$ parts is $\binom{n-1}{k-1}$.
For $n = 0$ there is one composition, the empty one.

## Explanations

### L1 human

Put $n$ dots in a row. Between two neighbours you either draw a bar or you do not.
There are $n - 1$ gaps with $2$ choices each, so there are $2^{n-1}$ compositions.
</document>

<document source="https://math2.tamnd.com/node/0580ZJSA3Q.md" tag="0580ZJSA3Q" kind="definition">
# Binary operation

## Definition

Let $S$ be a set.
A binary operation on $S$ is a map $\ast \colon S \times S \to S$.
For $a, b \in S$, the notation for the image of the pair $(a, b)$ is $a \ast b$.

## Remarks

Addition, subtraction and multiplication are binary operations on the set $\mathbb{Z}$ of integers.
Subtraction is not a binary operation on the set $\mathbb{N}$ of natural numbers, because $1 - 2$ is not in $\mathbb{N}$.
The condition that $a \ast b$ is in $S$ for all $a, b \in S$ is often called closure.

## Explanations

### L1 human

A binary operation is a rule that takes two elements of a set, in order, and gives one element of the same set.
For example, addition takes $2$ and $3$ and gives $5$.
The order can matter: $2 - 3$ and $3 - 2$ are different.
</document>

<document source="https://math2.tamnd.com/node/058YS05YP5.md" tag="058YS05YP5" kind="statement">
# Ptolemy's theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex [[cyclic quadrilateral|0RCP929Y1M]] with the diagonals $AC$ and $BD$.
Then
$$d(A, C)\, d(B, D) = d(A, B)\, d(C, D) + d(B, C)\, d(A, D).$$

## Remarks

In words: the product of the diagonals is the sum of the products of the opposite sides.
The converse is also true. If a quadrilateral has this property, then it is cyclic.
For each four points of the plane, $d(A, C)\, d(B, D) \leq d(A, B)\, d(C, D) + d(B, C)\, d(A, D)$. This is Ptolemy's inequality.
For a rectangle with the sides $a$ and $b$, the theorem gives $a^2 + b^2 = c^2$, with the diagonal $c$. This is the [[Pythagorean theorem|0VP3A1290K]].

## History

The theorem has the name of the astronomer Ptolemy. He used it to make his table of chords, a trigonometric table for astronomy.

## Explanations

### L1 human

Put four points in order on a circle and join them into a quadrilateral. Multiply the two diagonals. You get the same number as when you multiply the opposite sides in pairs and add the two products.
</document>

<document source="https://math2.tamnd.com/node/059Q2XWJDC.md" tag="059Q2XWJDC" kind="statement">
# Scalar triple product

## Statement

Let $a = (a_1, a_2, a_3)$, $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$ be vectors in $\mathbb{R}^3$. Let $\times$ be the [[cross product|0N5N15HWZ7]] and let $\cdot$ be the dot product.
The scalar triple product of $a$, $b$ and $c$ is the real number $a \cdot (b \times c)$. It is equal to the determinant of the matrix whose rows are $a$, $b$ and $c$:
$$a \cdot (b \times c) = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix}.$$

## Remarks

A cyclic shift of the three vectors does not change the value: $a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$. A swap of two vectors changes its sign.
The absolute value $|a \cdot (b \times c)|$ is the volume of the [[parallelepiped|0SQV2VJTGY]] with the edges $a$, $b$ and $c$.
The scalar triple product is $0$ if and only if $a$, $b$ and $c$ are linearly dependent. This happens exactly when they lie in one plane through $0$.

## Explanations

### L1 human

The scalar triple product measures the volume of the slanted box that three arrows make. For the unit vectors $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$ the box is a unit cube, and the product is $1$. If the three arrows lie flat in one plane, the box has no volume, and the product is $0$.
</document>

<document source="https://math2.tamnd.com/node/059THXBRBB.md" tag="059THXBRBB" kind="statement">
# Isomorphism with the coordinate space

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a basis of $V$.
Then the map
$$V \to F^n, \qquad v \mapsto [v]_B,$$
that sends each vector to its [[coordinate vector|04CYQHN6MC]] in $B$, is a [[linear isomorphism|0B8T10CE7N]].
Its inverse sends $(a_1, \dots, a_n)$ to $a_1 b_1 + \cdots + a_n b_n$.

## Remarks

The isomorphism depends on the choice of the basis.
It sends $b_i$ to the standard basis vector $e_i$.
Through this map, each statement about $F^n$ gives a statement about $V$.

## Explanations

### L1 human

Choosing a basis turns every vector into a list of $n$ numbers.
This translation respects addition and scaling, and it loses no information.
So each $n$-dimensional space is a copy of $F^n$.
</document>

<document source="https://math2.tamnd.com/node/059TJN0NZ1.md" tag="059TJN0NZ1" kind="concept">
# Probability measure

## Summary

A probability measure is a countably additive function on a σ-algebra of events that gives the whole space the value $1$.
It assigns to each event its probability. The node [[probability measure|0VGMEBVW1M]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05A6PKMAJ9.md" tag="05A6PKMAJ9" kind="concept">
# Partition of an interval

## Summary

A partition of a closed interval is a finite increasing list of points that starts at the left end and stops at the right end. It cuts the interval into subintervals. The node [[partition of an interval|069P20KYQT]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05APWSX53S.md" tag="05APWSX53S" kind="statement">
# Sum of a finite geometric series

## Statement

Let $r$ be a real number with $r \ne 1$, and let $n$ be a natural number.
Then
$$\sum_{k=0}^{n-1} r^k = 1 + r + r^2 + \cdots + r^{n-1} = \frac{1 - r^n}{1 - r}.$$

## Remarks

The proof multiplies the sum $S$ by $1 - r$. The product $(1 - r) S$ telescopes to $1 - r^n$.
For $r = 1$ the sum is $n$.
The identity holds in each commutative ring in the form $(1 - r) \sum_{k=0}^{n-1} r^k = 1 - r^n$, with no division.
Mathlib also states the version for a sum over the indices $m \le k < n$, as `geom_sum_Ico`.
Letting $n \to \infty$ gives the [[geometric series|01X5CVV1Q5]].

## Explanations

### L1 human

Write $S = 1 + r + \cdots + r^{n-1}$ and $rS = r + r^2 + \cdots + r^n$.
Subtract, and almost all terms cancel: $S - rS = 1 - r^n$.
For example, $1 + 2 + 4 + 8 = (1 - 16)/(1 - 2) = 15$.
</document>

<document source="https://math2.tamnd.com/node/05AQE0QR0D.md" tag="05AQE0QR0D" kind="concept">
# Normal distribution

## Summary

The normal distribution with mean $\mu$ and variance $\sigma^2$ is the continuous distribution on $\mathbb{R}$ with the bell-shaped density $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$.
By the central limit theorem, it is the approximate law of a sum of many small independent effects. The node [[normal distribution|07RXBVTVZB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05AVARPXA2.md" tag="05AVARPXA2" kind="concept">
# Binary relation

## Summary

A binary relation between two sets says, for each pair of elements, whether the first element is related to the second.
Formally, it is a set of ordered pairs.
</document>

<document source="https://math2.tamnd.com/node/05AW1G0WE1.md" tag="05AW1G0WE1" kind="statement">
# Expectation of a product of independent random variables

## Statement

Let $X$ and $Y$ be [[independent|0E9502NT92]] integrable real random variables.
Then $XY$ is integrable and
$$E[XY] = E[X] \, E[Y].$$

## Remarks

The proof uses that the joint law of $(X, Y)$ is the [[product of the marginal laws|08QCA2F1NX]], and then Fubini's theorem.
For nonnegative independent $X$ and $Y$, the equality holds in $[0, \infty]$ without integrability.
The converse is false: $E[XY] = E[X] \, E[Y]$ means only that $X$ and $Y$ are uncorrelated.
More generally, $E[g(X) \, h(Y)] = E[g(X)] \, E[h(Y)]$ for measurable $g$ and $h$ with $g(X)$ and $h(Y)$ integrable.

## Explanations

### L1 human

For independent variables, the average of a product is the product of the averages.
For two fair dice, the expected product of the two results is $3.5 \cdot 3.5 = 12.25$.
</document>

<document source="https://math2.tamnd.com/node/05AXG3QRKR.md" tag="05AXG3QRKR" kind="definition">
# Partial derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $i \in \{1, \dots, n\}$.
Let $e_i$ be the $i$-th standard basis vector of [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$.
The partial derivative of $f$ with respect to $x_i$ at $a$ is the limit
$$\frac{\partial f}{\partial x_i}(a) = \lim_{h \to 0} \frac{f(a + h e_i) - f(a)}{h},$$
when this limit exists in $\mathbb{R}$.
Other notations for it are $\partial_i f(a)$, $D_i f(a)$ and $f_{x_i}(a)$.

## Remarks

The partial derivative is the ordinary derivative at $t = a_i$ of the function $t \mapsto f(a_1, \dots, a_{i-1}, t, a_{i+1}, \dots, a_n)$ of one variable.
So you compute it with the rules of one-variable calculus, and you treat the other variables as constants.
It is the [[directional derivative|06KWGHDN2J]] in the direction $e_i$.
The existence of all partial derivatives at $a$ does not imply that $f$ is continuous at $a$, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

To find $\partial f / \partial x$, pretend that $y$ is a fixed number and differentiate in $x$ as usual.
For $f(x, y) = x^2 y + \sin y$, this gives $\partial f / \partial x = 2xy$, because $\sin y$ is a constant for this purpose.
Geometrically, it is the slope of the graph of $f$ when you walk in the $x$-direction.
</document>

<document source="https://math2.tamnd.com/node/05B0FKFXQA.md" tag="05B0FKFXQA" kind="definition">
# Jump discontinuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$ be a limit point of both $D \cap (-\infty, a)$ and $D \cap (a, \infty)$.
The point $a$ is a jump discontinuity of $f$ if both [[one-sided limits|0909PWZR62]] $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist and are different.

## Remarks

The difference of the two one-sided limits is the size of the jump.
For example, the sign function has a jump discontinuity at $0$, of size $2$.
Monotone functions have only jump discontinuities.

## Explanations

### L1 human

At a jump discontinuity, the graph approaches one height from the left and another height from the right.
No choice of the value at $a$ can repair this.
</document>

<document source="https://math2.tamnd.com/node/05BDYVQQ0T.md" tag="05BDYVQQ0T" kind="definition">
# Space of solutions of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a linear recurrence with constant coefficients over $R$ of order $d$.
The space of solutions of $E$ is the set of all [[solutions|0ZV2YKKZ58]] of $E$.
It is a submodule of the $R$-module $R^{\mathbb{N}}$ of all sequences in $R$, with termwise addition and scalar multiplication.

## Remarks

The map $u \mapsto (u_0, \dots, u_{d-1})$ is an isomorphism of $R$-modules from the space of solutions to $R^d$. Over a field, the space of solutions has dimension $d$.
For the Fibonacci recurrence over $\mathbb{R}$, the sequences $\varphi^n$ and $\psi^n$ form a basis, with $\varphi$ and $\psi$ the roots of $X^2 - X - 1$.

## Explanations

### L1 human

If two sequences solve the same linear recurrence, then so does their sum, and so does any multiple of one of them.
So the solutions form a vector space, and the first $d$ terms are coordinates on it.
</document>

<document source="https://math2.tamnd.com/node/05BM304NRX.md" tag="05BM304NRX" kind="statement">
# Stabilisers of points in the same orbit are conjugate

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, let $x \in X$, and let $g \in G$.
Then the [[stabiliser|0AYMP41THN]] of $g \cdot x$ is $G_{g \cdot x} = g G_x g^{-1}$.

## Remarks

An element $h$ fixes $g \cdot x$ if and only if $g^{-1}hg$ fixes $x$.
So the stabilisers of the points of one [[orbit|0YQPZB5Q1G]] are [[conjugate subgroups|0H11ZPX4ND]].
In particular, they all have the same order.

## Explanations

### L1 human

If $g$ moves $x$ to $y$, then the elements that fix $y$ are the elements that fix $x$, seen through $g$.
To fix $y$, you can move $y$ back to $x$, fix $x$, and move $x$ to $y$ again.
So the two stabilisers are conjugate.
</document>

<document source="https://math2.tamnd.com/node/05CG0S20S3.md" tag="05CG0S20S3" kind="concept">
# Tangent function

## Summary

The tangent function is the quotient of sine by cosine. It has period $\pi$, and it has vertical asymptotes where the cosine vanishes. The node [[tangent function|06T0HS14PP]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05CJSSXPR3.md" tag="05CJSSXPR3" kind="concept">
# Determinant of a family of vectors in a basis

## Summary

The determinant of a family of $n$ vectors in a basis of an $n$-dimensional space is a scalar.
It is the determinant of the matrix of their coordinates in that basis.
</document>

<document source="https://math2.tamnd.com/node/05D5EGMFYV.md" tag="05D5EGMFYV" kind="concept">
# Boundary

## Summary

The boundary of a subset of a topological space is the set of points in its closure that are not in its interior.
Each neighbourhood of a boundary point meets both the subset and its complement.
</document>

<document source="https://math2.tamnd.com/node/05DEJR6TGF.md" tag="05DEJR6TGF" kind="statement">
# Zeckendorf's theorem

## Statement

Let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]], and let $N$ be a positive integer.
Then there is a unique finite set $\{c_1, \dots, c_r\}$ of integers, with $r \ge 1$, such that
$$N = F_{c_1} + F_{c_2} + \dots + F_{c_r},$$
each $c_i \ge 2$, and $c_{i+1} \ge c_i + 2$ for $1 \le i < r$.

## Remarks

The condition says that the sum uses distinct Fibonacci numbers and no two consecutive ones. The bound $c_i \ge 2$ excludes $F_0 = 0$ and the repeated value $F_1 = F_2 = 1$.
Example: $100 = 89 + 8 + 3 = F_{11} + F_6 + F_4$.
The greedy method finds the representation: subtract the largest Fibonacci number that is at most $N$, and repeat.

## Explanations

### L1 human

Each positive integer is a sum of different Fibonacci numbers, no two of them neighbours in the sequence, in exactly one way.
For example, $20 = 13 + 5 + 2$.
</document>

<document source="https://math2.tamnd.com/node/05DME4TZ3K.md" tag="05DME4TZ3K" kind="object">
# Pascal's triangle

## Description

Pascal's triangle is the triangular array of the [[binomial coefficients|092GEF38YD]].
Its rows are numbered $n = 0, 1, 2, \dots$ from the top.
Row $n$ has $n + 1$ entries, $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, from left to right.

The first five rows are these.
$$\begin{array}{ccccccccc} & & & & 1 & & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array}$$

Each row starts and ends with $1$.
Each other entry is the sum of the two entries next to it in the row above, by [[Pascal's rule|0YRG241HQQ]].

## Remarks

Row $n$ lists the coefficients of $(x + y)^n$, by the [[binomial theorem|0EHWPWGHB2]].
The sum of row $n$ is $2^n$. This is the [[sum of binomial coefficients|0KAMF19NV8]].
Each row is symmetric, by the [[symmetry of binomial coefficients|0R7JS3DP3W]].

## Explanations

### L1 human

Start with a $1$ at the top. Each new row starts and ends with $1$, and each inside number is the sum of the two numbers above it.
Row 4 is $1, 4, 6, 4, 1$, and these are the coefficients of $(x + y)^4$.
</document>

<document source="https://math2.tamnd.com/node/05DSV40EB9.md" tag="05DSV40EB9" kind="definition">
# Closed ball

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r \ge 0$ be a real number.
The closed ball with centre $x$ and radius $r$ is the set
$$\bar{B}(x, r) = \{ y \in X : d(x, y) \le r \}.$$

## Remarks

The [[open ball|0DC3MPR7C9]] $B(x, r)$ is a subset of $\bar{B}(x, r)$.
In $\mathbb{R}$ with the usual metric, $\bar{B}(x, r)$ is the closed interval $[x - r, x + r]$.
The closed ball is not always the closure of the open ball. In the [[discrete metric|0NA4FK7B8S]], $B(x, 1) = \{x\}$ is closed, but $\bar{B}(x, 1) = X$.
For $r = 0$, the closed ball is $\{x\}$.

## Explanations

### L1 human

The closed ball contains the points at distance at most $r$, so it also contains its edge.
In the plane, it is the disc of radius $r$ together with its circle.
</document>

<document source="https://math2.tamnd.com/node/05EGFBZTGC.md" tag="05EGFBZTGC" kind="definition">
# Having a directional derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, let $v \in \mathbb{R}^n$, and let $L \in \mathbb{R}^m$.
Let $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$.
Equivalently, $\dfrac{f(a + t v) - f(a)}{t} \to L$ as $t \to 0$.

## Remarks

This definition states a property of $f$, $a$, $v$ and $L$. The [[directional derivative|0Y7PVW4Q44]] $D_v f(a)$ is then the vector $L$, which is unique.
Only the values of $f$ on the line through $a$ in the direction $v$ matter. So $f$ can have a directional derivative in each direction at $a$ and still fail to be continuous at $a$, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

The function $g(t) = f(a + t v)$ is the function $f$ seen along the straight line through $a$ with direction $v$.
To have a directional derivative means that this function of one variable has a derivative at the start of the line.
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<document source="https://math2.tamnd.com/node/05EGHEQZAW.md" tag="05EGHEQZAW" kind="definition">
# Vector space

## Definition

Let $F$ be a field.
A *vector space* over $F$ is a set $V$ with two maps: an addition $V \times V \to V$, $(u, v) \mapsto u + v$, and a scalar multiplication $F \times V \to V$, $(a, v) \mapsto av$.
These axioms must hold for all $u, v, w \in V$ and $a, b \in F$:

1. $u + (v + w) = (u + v) + w$.
2. $u + v = v + u$.
3. There is an element $0 \in V$ with $v + 0 = v$ for all $v \in V$.
4. For each $v \in V$ there is an element $-v \in V$ with $v + (-v) = 0$.
5. $a(bv) = (ab)v$.
6. $1v = v$, where $1$ is the unit of $F$.
7. $a(u + v) = au + av$.
8. $(a + b)v = av + bv$.

The elements of $V$ are *vectors*, and the elements of $F$ are *scalars*.

## Remarks

Axioms 1 to 4 say that $(V, +)$ is an abelian group.
The axioms give $0v = 0$ and $(-1)v = -v$ for all $v \in V$.
A vector space is a [[module|0NZ8JA25WK]] over a field; Mathlib uses `Module F V` for both.
Examples are the [[coordinate space|0XM8BZD76G]] $F^n$, [[spaces of matrices|0RE6X3BGPN]] and [[function spaces|0GZB2SN0PH]].

## Explanations

### L1 human

A vector space is a collection of objects that we can add together and stretch by numbers.
Arrows in the plane are the standard picture, but polynomials and functions also form vector spaces.
The eight axioms list the rules that make the usual algebra work.
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<document source="https://math2.tamnd.com/node/05FD317V2W.md" tag="05FD317V2W" kind="definition">
# Left coset

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The left coset of $H$ with representative $g$ is the set $gH = \{gh : h \in H\}$.

## Remarks

Two left cosets $aH$ and $bH$ are equal when $a^{-1}b \in H$, and they are disjoint otherwise.
Thus the left cosets of $H$ form a partition of $G$, and $H = eH$ is one of them.
In $(\mathbb{Z}, +)$, the left cosets of $3\mathbb{Z}$ are $3\mathbb{Z}$, $1 + 3\mathbb{Z}$ and $2 + 3\mathbb{Z}$.

## Explanations

### L1 human

A left coset is a copy of the subgroup that you move by multiplying with a fixed element on the left.
Different elements can give the same coset.
For the even integers inside the integers, there are two cosets: the even integers and the odd integers.
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<document source="https://math2.tamnd.com/node/05FKHRT0B4.md" tag="05FKHRT0B4" kind="statement">
# Polarisation identity

## Statement

Let $V$ be an inner product space with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$, and let $x, y \in V$.
If $V$ is a real space, then
$$\langle x, y \rangle = \frac{1}{4} \left( \norm{x + y}^2 - \norm{x - y}^2 \right).$$
If $V$ is a complex space, with the inner product linear in the first argument, then
$$\langle x, y \rangle = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.$$

## Remarks

So the norm determines the inner product.
In particular, a linear map that keeps norms also keeps inner products.
With the convention of linearity in the second argument, the factor $i^k$ moves to the other vector, and the complex formula changes accordingly.

## Explanations

### L1 human

If we know the lengths of all vectors, then we also know all inner products.
In the real case, we only need the lengths of the two diagonals $x + y$ and $x - y$.
So lengths determine angles.
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<document source="https://math2.tamnd.com/node/05FQ3VGAR9.md" tag="05FQ3VGAR9" kind="definition">
# Bounded sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is bounded above if there is a real number $M$ with $a_n \le M$ for all $n \in \mathbb{N}$.
The sequence is bounded below if there is a real number $m$ with $a_n \ge m$ for all $n \in \mathbb{N}$.
The sequence is bounded if it is bounded above and bounded below.
Equivalently, $(a_n)$ is bounded if there is a real number $K \ge 0$ with $|a_n| \le K$ for all $n \in \mathbb{N}$.

## Remarks

The sequence $a_n = (-1)^n$ is bounded, with $K = 1$.
The sequence $a_n = n$ is bounded below by $0$, but it is not bounded above.
A sequence is a function on $\mathbb{N}$, so a bounded sequence is a bounded function on $\mathbb{N}$.
Each [[convergent sequence|0AZ3R52593]] is bounded, as the node [[convergent sequences are bounded|0S4JF619V7]] states.

## Explanations

### L1 human

A sequence is bounded when all its terms stay inside one fixed interval $[-K, K]$.
The terms can move up and down, but they never escape from that interval.
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<document source="https://math2.tamnd.com/node/05FSTXM92W.md" tag="05FSTXM92W" kind="definition">
# Likelihood function

## Definition

Let $\Theta$ be a set of parameters. For each $\theta \in \Theta$, let $f(\cdot \mid \theta)$ be a [[probability mass function|0WF4EJP91W]] or a [[density|04YG690N5V]] on $\mathbb{R}^n$. It gives the law of the data $X = (X_1, \dots, X_n)$ when $\theta$ is the true parameter.
For observed data $x \in \mathbb{R}^n$, the likelihood function is the function $L \colon \Theta \to [0, \infty)$ with
$$L(\theta \mid x) = f(x \mid \theta).$$
The log-likelihood is $\ell(\theta \mid x) = \ln L(\theta \mid x)$.

## Remarks

For an [[i.i.d.|0AJN7WHBJZ]] sample with mass function or density $f_1(\cdot \mid \theta)$ for each observation, $L(\theta \mid x) = \prod_{i=1}^{n} f_1(x_i \mid \theta)$ and $\ell(\theta \mid x) = \sum_{i=1}^{n} \ln f_1(x_i \mid \theta)$.
The likelihood is not a probability distribution on $\Theta$. In general, its integral over $\theta$ is not $1$.
Example: for $k$ successes in $n$ Bernoulli($p$) trials, $L(p \mid x) = p^k (1 - p)^{n - k}$ for $p \in [0, 1]$.
Only ratios $L(\theta_1 \mid x) / L(\theta_2 \mid x)$ matter for inference. These ratios compare how well two parameter values explain the data.

## Explanations

### L1 human

The likelihood answers the question: "If the parameter were $\theta$, how probable would my data be?"
Parameter values with a high likelihood explain the observed data well.
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<document source="https://math2.tamnd.com/node/05G2G0P224.md" tag="05G2G0P224" kind="definition">
# Dimension

## Definition

Let $F$ be a field and let $V$ be a [[finite-dimensional|0ZHWKQMVX5]] vector space over $F$.
The *dimension* of $V$ over $F$, written $\dim_F V$ or $\dim V$, is the number of vectors in a [[basis|0KCQ9Y3QKA]] of $V$.
For an arbitrary vector space, the dimension is the cardinality of a basis.

## Remarks

The number is well defined, by the [[dimension theorem for vector spaces|0WK0Y8RH1M]].
The zero space $\{0\}$ has the empty basis and dimension $0$.
The dimension depends on the field: $\dim_\C \C = 1$, but $\dim_\R \C = 2$.
Mathlib writes `Module.finrank F V` for the dimension as a natural number, with value $0$ for infinite-dimensional spaces.

## Explanations

### L1 human

The dimension counts the independent directions in a space.
A line has dimension $1$, a plane has dimension $2$, and ordinary space has dimension $3$.
Every basis of a space has this same number of vectors.
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<document source="https://math2.tamnd.com/node/05GK32H72H.md" tag="05GK32H72H" kind="definition">
# Union of a family of sets

## Definition

Let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]].
The union of the family is the set
$$\bigcup_{i \in I} A_i = \{x \mid \exists i \in I, \, x \in A_i\}.$$
So $x$ is an element of the union when $x \in A_i$ for at least one $i \in I$.

## Remarks

For $I = \{1, 2\}$, this is the [[union|07KVB4JSC7]] $A_1 \cup A_2$.
For an empty index set, the union is the [[empty set|0ECVK59Y03]].
Example: $\bigcup_{n \ge 1} [0, n] = [0, \infty)$ in $\mathbb{R}$.

## Explanations

### L1 human

The union of a family collects every element that appears in some member of the family.
It generalizes $A \cup B$ from two sets to any number of sets.
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<document source="https://math2.tamnd.com/node/05GQJXBP3F.md" tag="05GQJXBP3F" kind="definition">
# T1 space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T1 when for all points $x \neq y$ of $X$ there is an [[open set|01ZQQDCERP]] $U$ with $x \in U$ and $y \notin U$.

## Remarks

A space is T1 if and only if each set $\{x\}$ with $x \in X$ is [[closed|0G5AV9Q4RR]].
Each [[Hausdorff space|0MXB4CKHFW]] is T1, and each T1 space is [[T0|0R3B7DX55J]].
An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff. In fact, the cofinite topology is the coarsest T1 topology on a set.

## Explanations

### L1 human

In a T1 space, for two distinct points, each point has an open set around it that does not contain the other point.
This is the same as saying that single points are closed.
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<document source="https://math2.tamnd.com/node/05GZ11PWRY.md" tag="05GZ11PWRY" kind="definition">
# Homeomorphism

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a homeomorphism when $f$ is [[bijective|0SNQEF7Z53]], $f$ is [[continuous|0SY9ECQ19Q]], and the [[inverse function|0W766JAPSN]] $f^{-1} \colon Y \to X$ is continuous.
The spaces $X$ and $Y$ are homeomorphic when there is a homeomorphism $X \to Y$.

## Remarks

Equivalently, $f$ is a bijection and, for each $U \subseteq X$, the set $U$ is open if and only if $f(U)$ is open.
The map $x \mapsto x / (1 - |x|)$ is a homeomorphism from $(-1, 1)$ to $\mathbb{R}$.
A continuous bijection need not be a homeomorphism: $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to the circle is one example. For a compact domain and a Hausdorff target, [[it is always a homeomorphism|0430ETCMKD]].
Homeomorphisms keep each [[topological property|02SMVGC690]].

## Explanations

### L1 human

A homeomorphism is a way to deform one space into another by stretching and bending, without tearing or gluing.
A coffee cup and a doughnut are homeomorphic, but a sphere and a doughnut are not.
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<document source="https://math2.tamnd.com/node/05H4DDDYBZ.md" tag="05H4DDDYBZ" kind="statement">
# Root test

## Statement

Let $(a_n)$ be a real sequence and let $C = \limsup_{n \to \infty} |a_n|^{1/n}$, the [[limit superior|0P8BA7T7S5]], with the value $+\infty$ allowed.
If $C < 1$, then the series $\sum a_n$ converges [[absolutely|0W3K3QPEWT]].
If $C > 1$, then the series $\sum a_n$ diverges.

## Remarks

If $C < 1$, choose $q$ with $C < q < 1$. Then $|a_n| \le q^n$ for large $n$, and the [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] applies.
If $C > 1$, then $|a_n| > 1$ for infinitely many $n$, so the terms do not tend to $0$.
For $C = 1$ the test gives no answer.
The root test is stronger than the [[ratio test|0QNY6ZBXPN]]: when the ratio limit exists, the root limit exists and has the same value.

## Explanations

### L1 human

The $n$-th root of $|a_n|$ measures how fast the terms shrink, compared with a geometric sequence $q^n$.
If this rate is below $1$, the terms shrink like a convergent geometric series.
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<document source="https://math2.tamnd.com/node/05HN08ZSRJ.md" tag="05HN08ZSRJ" kind="concept">
# Dihedral angle

## Summary

A dihedral angle is the angle between two planes that meet in a line, or between two half-planes with a common edge. It is measured in a third plane that is perpendicular to the common line.
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<document source="https://math2.tamnd.com/node/05HQYM0R4G.md" tag="05HQYM0R4G" kind="definition">
# Dot product

## Definition

Let $n \ge 1$ and let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ be vectors of $\R^n$.
The *dot product* of $x$ and $y$ is the real number
$$x \cdot y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n.$$

## Remarks

In matrix notation, $x \cdot y = x^T y$ for column vectors $x$ and $y$.
It is symmetric, bilinear and positive definite, so it is an [[inner product|0NB0AWGJBW]] on $\R^n$.
Geometrically, $x \cdot y = \norm{x} \norm{y} \cos\theta$, where $\theta$ is the angle between $x$ and $y$.
The same formula defines a symmetric bilinear form on $F^n$ for every field $F$.

## Explanations

### L1 human

The dot product multiplies two vectors coordinate by coordinate and adds the results.
For example, $(1, 2, 3) \cdot (4, 5, 6) = 4 + 10 + 18 = 32$.
It is zero exactly when the two vectors are perpendicular, or when one of them is zero.
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<document source="https://math2.tamnd.com/node/05HYGWFG63.md" tag="05HYGWFG63" kind="statement">
# Inner automorphism group is isomorphic to G modulo its centre

## Statement

Let $G$ be a [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
Then the map $gZ(G) \mapsto c_g$, where $c_g(x) = gxg^{-1}$, is an [[isomorphism|08MG7H8JW5]] from $G/Z(G)$ to $\operatorname{Inn}(G)$.

## Remarks

The map $g \mapsto c_g$ is a surjective homomorphism from $G$ to $\operatorname{Inn}(G)$. Its kernel is $Z(G)$, because $c_g$ is the identity exactly when $g$ commutes with each element.
The [[first isomorphism theorem|0CXXJNG5B6]] then gives the result.
For example, the symmetric group $S_3$ has trivial centre, so $\operatorname{Inn}(S_3) \cong S_3$.

## Explanations

### L1 human

Two elements give the same conjugation exactly when they differ by an element of the centre.
So the inner automorphisms are counted by the cosets of the centre.
A group with a big centre has few inner automorphisms.
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<document source="https://math2.tamnd.com/node/05J9NWX07X.md" tag="05J9NWX07X" kind="concept">
# Transitive closure

## Summary

The transitive closure of a relation is the smallest transitive relation that contains it.
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<document source="https://math2.tamnd.com/node/05K3J8J8T2.md" tag="05K3J8J8T2" kind="example">
# The rationals as the field of fractions of the integers

## Example

The [[field of fractions|0VF11ZTTMD]] of the [[ring of integers|0JHM6JK7FD]] $\mathbb{Z}$ is the field $\mathbb{Q}$ of rational numbers.
Each element of $\operatorname{Frac}(\mathbb{Z})$ is a class $a/b$ with $a, b \in \mathbb{Z}$ and $b \neq 0$.
Two pairs give the same class, $a/b = c/d$, if and only if $a d = b c$.
The map $\mathbb{Z} \to \mathbb{Q}$, $n \mapsto n/1$, is the usual inclusion of the integers in the rationals.

## Remarks

Every rational number has a unique form $a/b$ with $b > 0$ and $\gcd(a, b) = 1$.
Every subfield of $\mathbb{C}$ contains $\mathbb{Z}$, so it contains $\mathbb{Q}$, by the universal property of the field of fractions.

## Explanations

### L1 human

The rational numbers are exactly the fractions of integers.
The field of fractions makes this precise for any integral domain.
The fractions $1/2$, $2/4$ and $-3/(-6)$ all name the same rational number.
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<document source="https://math2.tamnd.com/node/05KQ3F99BQ.md" tag="05KQ3F99BQ" kind="concept">
# Improper integral

## Summary

An improper integral is a limit of definite integrals. It extends integration to unbounded intervals and to functions that are unbounded near a point. The node [[improper integral|0VCTBJMEJF]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/05M5SWSZZD.md" tag="05M5SWSZZD" kind="concept">
# Limit of a sequence

## Summary

A sequence has a limit $L$ when its terms come and stay arbitrarily close to $L$ for large indices. Limits of sequences are the base of the notions of convergence, continuity and series in analysis. The node [[limit of a sequence|0XKWSSY8P2]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/05MC3HND79.md" tag="05MC3HND79" kind="statement">
# Poincaré lemma for one-forms on convex sets

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and convex, and let $\omega = \sum_i f_i \, dx_i$ be a [[closed|0F1AVGX26N]] [[one-form|0MTENFBW3Q]] on $U$ with $C^1$ coefficients.
Then $\omega$ is exact. Explicitly, for a fixed $p \in U$, the function
$$g(x) = \int_0^1 \sum_{i=1}^n f_i\big( p + t (x - p) \big) \, (x_i - p_i) \, dt$$
satisfies $dg = \omega$ on $U$.

## Remarks

Proof: [[differentiate under the integral sign|0AR06W1E1B]], and use $\partial_j f_i = \partial_i f_j$. The integrand of $\partial_j g$ becomes $\frac{d}{dt} \big( t \, f_j(p + t (x - p)) \big)$, which integrates to $f_j(x)$.
In $\mathbb{R}^3$, this is the theorem that a [[curl-free field on a convex set is conservative|0B22P010D6]].
The lemma also holds for closed $k$-forms with $k \ge 1$ on star-shaped open sets. For one-forms, it holds on [[simply connected|075C629XTH]] open sets.
Some condition on $U$ is needed: the angle form on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. See also the [[vortex field|0GRM3BS0DD]].

## Explanations

### L1 human

On a convex set, a one-form that passes the local test for being a derivative really is a derivative.
The proof builds the function by integrating the form along straight segments from one base point.
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<document source="https://math2.tamnd.com/node/05MDKQBEAF.md" tag="05MDKQBEAF" kind="statement">
# Angle sum of a triangle

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]]. Then
$$\angle BAC + \angle CBA + \angle ACB = \pi.$$

## Remarks

Mathlib proves the formula under the weaker hypothesis that $B \neq A$ and $C \neq A$, with its convention for the angle at a repeated point.
An equilateral triangle has three equal angles, so each one is $\pi / 3$.
In the axioms of plane geometry, the statement is equivalent to the parallel postulate. It is false in spherical geometry, where the sum is more than $\pi$, and in hyperbolic geometry, where it is less.

## History

Euclid's Elements proves the statement as part of Proposition 32 of Book I.

## Explanations

### L1 human

Cut a paper triangle and tear off its three corners. Put the corners side by side with their tips at one point. They always fill a straight line, which is half a turn, or $\pi$ radians.
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<document source="https://math2.tamnd.com/node/05MM4EK4SA.md" tag="05MM4EK4SA" kind="statement">
# Groups of prime order are cyclic

## Statement

Let $p$ be a prime number and let $G$ be a [[group|G5S1M9]] of order $p$.
Then $G$ is [[cyclic|0KVGH0J5B3]], and each element of $G$ other than the identity generates $G$.

## Remarks

If $g \neq e$, then the order of $\langle g \rangle$ is greater than $1$ and divides $p$ by [[Lagrange's theorem|3C53TD]], so it is $p$.
Thus each group of order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$.
A group of prime order has no subgroups except $\{e\}$ and itself.

## Explanations

### L1 human

When the size of a group is a prime, there is no room for proper subgroups other than the trivial one.
Any element other than the identity generates a subgroup larger than $1$, so it generates everything.
Thus, for example, each group with $7$ elements is a clock with $7$ hours.
</document>

<document source="https://math2.tamnd.com/node/05MW7MPRBS.md" tag="05MW7MPRBS" kind="definition">
# Cross-ratio

## Definition

Let $z_1, z_2, z_3, z_4$ be four distinct complex numbers.
Their cross-ratio is
$$(z_1, z_2; z_3, z_4) = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}.$$
Suppose that one of the four points is the point $\infty$ of the [[Riemann sphere|0CDMXGPSES]] and the other three are distinct complex numbers. Then the cross-ratio is the same formula without the two factors that contain $\infty$.

## Remarks

Other books use other orders of the four points in the formula, so one must check the convention.
For each [[Möbius transformation|0B346DDAM5]] $f$, the cross-ratio $(f(z_1), f(z_2); f(z_3), f(z_4))$ is equal to $(z_1, z_2; z_3, z_4)$.
With this convention, the map $z \mapsto (z, z_2; z_3, z_4)$ is the Möbius transformation that sends $z_2$, $z_3$, $z_4$ to $1$, $0$, $\infty$.
The cross-ratio is a real number exactly when the four points are on one circle or on one line.

## Explanations

### L1 human

The cross-ratio is one number made from four points.
Möbius transformations change distances and angles of figures in complicated ways, but they do not change this number.
So the cross-ratio helps to find a Möbius map that sends given points to other given points.
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<document source="https://math2.tamnd.com/node/05N4TQKPH0.md" tag="05N4TQKPH0" kind="statement">
# Subgroup criterion

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a subset of $G$.
Then $H$ is a [[subgroup|0X1JXT47GA]] of $G$ if and only if $H$ is not empty and $ab^{-1} \in H$ for all $a, b \in H$.

## Remarks

If $a \in H$, then $e = aa^{-1} \in H$, then $b^{-1} = eb^{-1} \in H$ for $b \in H$, and then $ab = a(b^{-1})^{-1} \in H$.
In additive notation, the condition is $a - b \in H$ for all $a, b \in H$.
For a finite nonempty subset $H$, it is sufficient that $ab \in H$ for all $a, b \in H$.

## Explanations

### L1 human

This test replaces three checks by one.
You only check that the subset is not empty and that $ab^{-1}$ stays in the subset.
For example, the multiples of $5$ form a subgroup of $(\mathbb{Z}, +)$, because the difference of two multiples of $5$ is a multiple of $5$.
</document>

<document source="https://math2.tamnd.com/node/05NAS7AGMZ.md" tag="05NAS7AGMZ" kind="concept">
# Field generated by adjoining elements

## Summary

For a field extension $L/K$ and a subset $S$ of $L$, the field $K(S)$ is the smallest subfield of $L$ that contains $K$ and $S$.
The [[definition of $K(S)$|0MA5B744K3]] states it.
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<document source="https://math2.tamnd.com/node/05NK26G0DR.md" tag="05NK26G0DR" kind="statement">
# Composition of continuous maps is continuous

## Statement

Let $X$, $Y$ and $Z$ be [[topological spaces|0CNQXVBHCX]].
Let $f \colon X \to Y$ and $g \colon Y \to Z$ be [[continuous maps|0SY9ECQ19Q]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon X \to Z$ is continuous.

## Remarks

Proof: for each open set $W$ of $Z$, $(g \circ f)^{-1}(W) = f^{-1}(g^{-1}(W))$. The set $g^{-1}(W)$ is open in $Y$, so its preimage under $f$ is open in $X$.
For example, $x \mapsto \sin(x^2)$ is continuous on $\mathbb{R}$, because $x \mapsto x^2$ and $\sin$ are continuous.

## Explanations

### L1 human

If you do one continuous operation and then another one, the result is continuous.
Small changes in the input give small changes after the first step, and then small changes after the second step.
</document>

<document source="https://math2.tamnd.com/node/05NWGBHKSZ.md" tag="05NWGBHKSZ" kind="concept">
# Eulerian trail

## Summary

An Eulerian trail is a trail that uses every edge of a graph exactly once. The question of which graphs have one is the origin of graph theory. The node [[Eulerian trail|0NNYDD1AVR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05NZ1HHQMV.md" tag="05NZ1HHQMV" kind="concept">
# Mean squared error

## Summary

The mean squared error of an estimator is the expected squared distance between the estimator and the true value. It combines the variance and the bias of the estimator.
The node [[mean squared error|0V4VSCBXP3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05P4B4Q22V.md" tag="05P4B4Q22V" kind="statement">
# Derivative of arctangent

## Statement

The arctangent is differentiable on $\mathbb{R}$, and
$$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2} \quad \text{for every real } x.$$

## Remarks

Let $y = \arctan x$. By the [[inverse function rule|0M0RZ3HSB0]] and [[$\tan' = 1 + \tan^2$|0SAMR34WXC]], the derivative is $\frac{1}{1 + \tan^2 y} = \frac{1}{1 + x^2}$.
So $\int_0^1 \frac{dx}{1 + x^2} = \arctan 1 = \frac{\pi}{4}$.
Mathlib states it as `Real.hasDerivAt_arctan`.

## Explanations

### L1 human

The derivative of the arctangent is the simple rational function $1/(1 + x^2)$.
This connects the number $\pi$ with integrals of rational functions.
</document>

<document source="https://math2.tamnd.com/node/05PWWHYBX5.md" tag="05PWWHYBX5" kind="statement">
# Markov's inequality

## Statement

Let $X$ be a random variable with values in $[0, \infty)$ and let $a > 0$. Then
$$P(X \ge a) \le \frac{E[X]}{a}.$$

## Remarks

Proof: $a \, 1_{\{X \ge a\}} \le X$ pointwise. Take expectations: $a \, P(X \ge a) \le E[X]$.
More generally, for each nondecreasing $g \colon [0, \infty) \to [0, \infty)$ with $g(a) > 0$, $P(X \ge a) \le E[g(X)]/g(a)$.
Applied to $(X - E[X])^2$, the inequality gives [[Chebyshev's inequality|02HZ1BK3MW]]. Applied to $e^{tX}$, it gives the [[Chernoff bound|014PEBGDYF]].

## History

The inequality is named after Andrey Markov. It appears earlier in the work of Pafnuty Chebyshev, the teacher of Markov.

## Explanations

### L1 human

A nonnegative quantity with a small average cannot often be large.
If the mean income is $30000$, then at most one person in five can have an income of $150000$ or more.
</document>

<document source="https://math2.tamnd.com/node/05Q1B7MH0M.md" tag="05Q1B7MH0M" kind="statement">
# Matrix of a composition is the product of matrices

## Statement

Let $F$ be a field, and let $U$, $V$, $W$ be finite-dimensional vector spaces over $F$ with bases $A$, $B$, $C$.
Let $S \colon U \to V$ and $T \colon V \to W$ be linear maps.
Then
$$[T \circ S]_{C,A} = [T]_{C,B}\, [S]_{B,A},$$
where $[\,\cdot\,]$ denotes the [[matrix of a linear map|0YQSGNYE3K]] and the right side is a [[matrix product|032Q9FB8MT]].

## Remarks

This identity is the reason for the definition of matrix multiplication.
It implies that matrix multiplication is associative, because composition of maps is associative.
For $S$ invertible, it gives $[S^{-1}]_{A,B} = [S]_{B,A}^{-1}$.

## Explanations

### L1 human

Doing one linear map after another corresponds to multiplying their matrices.
The order is the same as in $T \circ S$: the matrix of the map that acts first is on the right.
This is why matrix multiplication has its unusual row-times-column rule.
</document>

<document source="https://math2.tamnd.com/node/05Q4JD6XEH.md" tag="05Q4JD6XEH" kind="definition">
# Orthogonal family for a bilinear form

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.
A family $(v_i)_{i \in I}$ of vectors of $V$ is *orthogonal* for $B$ when
$$B(v_i, v_j) = 0 \qquad \text{for all } i, j \in I \text{ with } i \ne j.$$

## Remarks

For an inner product, this is the usual notion of an orthogonal family.
For a general $B$, the relation $B(u, v) = 0$ need not be symmetric in $u$ and $v$; it is symmetric for symmetric and alternating forms.
In a basis that is orthogonal for $B$, the matrix of $B$ is diagonal.
Unlike in an inner product space, a nonzero vector can be orthogonal to itself: for $B(x, y) = x_1 y_1 - x_2 y_2$, take $v = (1, 1)$.

## Explanations

### L1 human

Orthogonality makes sense for every bilinear form, not only for inner products.
Two vectors are orthogonal when the form gives zero.
A basis of pairwise orthogonal vectors makes the matrix of the form diagonal.
</document>

<document source="https://math2.tamnd.com/node/05Q6G4EH2G.md" tag="05Q6G4EH2G" kind="definition">
# Family of sets

## Definition

Let $I$ be a set.
A family of sets indexed by $I$ assigns to each $i \in I$ a [[set|0SJ3RX9TXA]] $A_i$. It is written $(A_i)_{i \in I}$.
The set $I$ is the index set.

## Remarks

Formally, a family indexed by $I$ is a [[function|0XHDTE5YMZ]] with [[domain|093CM69AGS]] $I$.
The same set can appear for two different indices: $A_i = A_j$ with $i \neq j$ is allowed.
Example: $A_n = \{0, 1, \dots, n\}$ for $n \in \mathbb{N}$ is a family indexed by $\mathbb{N}$.
Some authors use "family of sets" for any set of sets, without an index.

## Explanations

### L1 human

A family is a labelled list of sets, where the labels can be any set.
With labels $1, 2, 3$, you get three sets $A_1, A_2, A_3$.
With labels in $\mathbb{N}$, you get an infinite sequence of sets.
</document>

<document source="https://math2.tamnd.com/node/05QY1VEBYD.md" tag="05QY1VEBYD" kind="concept">
# Finitely generated module

## Summary

A finitely generated module is a module in which every element is a linear combination of a fixed finite set of elements.
</document>

<document source="https://math2.tamnd.com/node/05R9NF2J49.md" tag="05R9NF2J49" kind="definition">
# Conservative vector field

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
The field $F$ is conservative on $U$ if there is a function $\phi \colon U \to \mathbb{R}$ of [[class C^1|0TCBZMEYGN]] with $F = \nabla \phi$ on $U$, that is, $F_i = \partial \phi / \partial x_i$ for $i = 1, \dots, n$.

## Remarks

A function $\phi$ with $\nabla \phi = F$ is a [[scalar potential|0FP4BVDAAQ]] of $F$. Physics texts often use $-\phi$ instead.
The field $F(x, y) = (2 x y, x^2)$ is conservative on $\mathbb{R}^2$, with $\phi(x, y) = x^2 y$.
The field $F(x, y) = (-y, x)$ is not conservative on $\mathbb{R}^2$. Its line integral around the unit circle is $2\pi \ne 0$, which contradicts [[path independence|0BFN21TF5F]].
If $F$ is conservative and of class $C^1$, then its [[curl is zero|0M2JRW1J0N]]. The converse holds on convex sets but [[fails in general|0GRM3BS0DD]].

## Explanations

### L1 human

A conservative field is the slope field of some landscape $\phi$.
Going around a loop in a landscape brings you back to the same height. So the total work of the field around the loop is zero.
</document>

<document source="https://math2.tamnd.com/node/05RCDKHZM6.md" tag="05RCDKHZM6" kind="statement">
# Additivity of the integral over intervals

## Statement

Let $a < c < b$ and let $f \colon [a, b] \to \mathbb{R}$.
Then $f$ is Riemann integrable on $[a, b]$ if and only if it is Riemann integrable on $[a, c]$ and on $[c, b]$.
In this case,
$$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$$

## Remarks

With the sign convention of the [[definite integral|021WEM680W]], the formula holds for any order of $a$, $b$, $c$.
This needs $f$ to be integrable on the largest of the intervals.
Mathlib states it as `intervalIntegral.integral_add_adjacent_intervals`.

## Explanations

### L1 human

The area over $[a, b]$ is the area over $[a, c]$ plus the area over $[c, b]$.
We can cut the interval at any point and add the two pieces.
</document>

<document source="https://math2.tamnd.com/node/05RCT1522R.md" tag="05RCT1522R" kind="statement">
# Laplacian as a sum of second partial derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$.
Then
$$\Delta f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \dots + \frac{\partial^2 f}{\partial x_n^2}.$$

## Remarks

Proof: $\nabla f = (\partial f / \partial x_1, \dots, \partial f / \partial x_n)$, and the [[divergence|02NSMQVQWK]] adds the derivative of the $i$-th component with respect to $x_i$.
So $\Delta f$ is the trace of the [[Hessian matrix|0SCSQWXRA4]] of $f$. The trace does not change under a rotation of coordinates, so neither does $\Delta f$.
Examples: $\Delta (x^2 + y^2) = 4$, $\Delta (x^2 - y^2) = 0$ and $\Delta \|x\|^2 = 2n$ on $\mathbb{R}^n$.
In polar coordinates in the plane, $\Delta f = f_{rr} + \tfrac{1}{r} f_r + \tfrac{1}{r^2} f_{\theta\theta}$ for $r > 0$.

## Explanations

### L1 human

In Cartesian coordinates, the Laplacian adds up the second derivatives of $f$ in the $n$ coordinate directions.
Only the unmixed second partial derivatives appear, not mixed ones like $f_{xy}$.
</document>

<document source="https://math2.tamnd.com/node/05RN90E14Y.md" tag="05RN90E14Y" kind="statement">
# Number of edges of a complete graph

## Statement

Let $n$ be a natural number.
Then the [[complete graph|03WKB2H474]] $K_n$ has exactly
$$\binom{n}{2} = \frac{n (n - 1)}{2}$$
edges.

## Remarks

The edges of $K_n$ are exactly the $2$-element subsets of the vertex set. The [[number of k-element subsets|0BFMR2FQ8B]] gives the count.
The [[handshaking lemma|0XAP8WHNES]] gives the same count: $n (n - 1) = 2 |E|$.
Each simple graph on $n$ vertices has at most $\binom{n}{2}$ edges.

## Explanations

### L1 human

If $n$ people all shake hands with each other, there are $\frac{n(n-1)}{2}$ handshakes.
Each person shakes $n - 1$ hands, and this counts each handshake twice.
</document>

<document source="https://math2.tamnd.com/node/05RSK4Q657.md" tag="05RSK4Q657" kind="concept">
# Truth table

## Summary

A truth table is a table that gives the truth value of a compound [[proposition|0ZGC7ZESYN]] for each assignment of truth values to its variables.
Each row of the table is one assignment, and the last column gives the value of the compound proposition.
A proposition with $n$ variables has a truth table with $2^n$ rows.

## Remarks

The truth tables of [[negation|0XR7A7RSXB]], [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]] and the [[biconditional|0V6GAWTBYW]] define these connectives.
A proposition is a [[tautology|0D1AHYPM8N]] when its column has only the value true.
Two propositions are [[logically equivalent|06GYGS41JK]] when their columns are the same.

## Explanations

### L1 human

A truth table lists every possible case.
For $p \land q$ there are four cases, because $p$ and $q$ each have two values.
You compute the value of the whole formula in each case, one row at a time.
</document>

<document source="https://math2.tamnd.com/node/05RWFKVTJ7.md" tag="05RWFKVTJ7" kind="concept">
# Permutation

## Summary

A permutation of a set is a rearrangement of its elements, that is, a bijection from the set to itself. Permutations are the basic objects of enumerative combinatorics and of the theory of symmetric groups. The node [[permutation|0BQMWCNS8S]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05RZ4YZG4E.md" tag="05RZ4YZG4E" kind="definition">
# Standard score

## Definition

Let $X$ be a real random variable with mean $\mu = E[X]$ and standard deviation $\sigma > 0$.
The standard score, or $z$-score, of a real number $x$ is
$$z = \frac{x - \mu}{\sigma}.$$
The standardized random variable is $Z = (X - \mu)/\sigma$.

## Remarks

The standardized variable $Z$ has mean $0$ and variance $1$.
If $X \sim \mathcal{N}(\mu, \sigma^2)$, then $Z \sim \mathcal{N}(0, 1)$. So $P(X \le x) = \Phi(z)$, where $\Phi$ is the standard normal cumulative distribution function.
For a data set, one uses the sample mean and the sample standard deviation in place of $\mu$ and $\sigma$.

## Explanations

### L1 human

A $z$-score says how unusual a value is, in units of the standard deviation.
Suppose that test scores have mean $70$ and standard deviation $10$.
Then a score of $85$ has $z = 1.5$. It is one and a half standard deviations above the mean.
</document>

<document source="https://math2.tamnd.com/node/05S2GSPMGA.md" tag="05S2GSPMGA" kind="statement">
# Harmonic numbers grow like log n

## Statement

Let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the [[harmonic numbers|0D2Y6G4C4S]].
Then the limit $\lim_{n \to \infty} (H_n - \ln n)$ exists, and it lies in $[0, 1]$.
In particular, $H_n \sim \ln n$ as $n \to \infty$.

## Remarks

Let $a_n = H_n - \ln n$. Since $\ln(n + 1) - \ln n = \int_n^{n+1} \frac{dt}{t}$ lies between $\frac{1}{n+1}$ and $\frac{1}{n}$, the sequence $a_n$ decreases. Also $a_n \ge H_n - \ln(n + 1) \ge 0$.
The [[monotone convergence theorem|0FM6N4R0VB]] gives the limit. It is the [[Euler–Mascheroni constant|00C5P9JZBK]] $\gamma$.
The symbol $\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. Mathlib states the limit as `Real.tendsto_harmonic_sub_log`.

## Explanations

### L1 human

The sum $1 + 1/2 + \cdots + 1/n$ is close to the area under $1/x$ from $1$ to $n$, which is $\ln n$.
The error tends to a constant.
</document>

<document source="https://math2.tamnd.com/node/05SG20AMXH.md" tag="05SG20AMXH" kind="concept">
# Left inverse

## Summary

A left inverse of a function undoes the function: applied after the function, it gives back the input.
</document>

<document source="https://math2.tamnd.com/node/05SZQKH75K.md" tag="05SZQKH75K" kind="statement">
# Negation of an existential statement

## Statement

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The negation of $\exists x \in X, \, P(x)$ is logically equivalent to $\forall x \in X, \, \neg P(x)$.
$$\neg \bigl( \exists x \in X, \, P(x) \bigr) \equiv \forall x \in X, \, \neg P(x)$$

## Remarks

This equivalence is valid in intuitionistic logic as well as in classical logic.
Together with the [[negation of a universal statement|0P3A06QNDP]], it gives a rule. To negate a quantified statement, swap each $\forall$ and $\exists$ and negate the predicate.

## Explanations

### L1 human

"No student failed" means "every student did not fail".
To deny that some element has a property, you must show that each element lacks it.
</document>

<document source="https://math2.tamnd.com/node/05TG0SE4HT.md" tag="05TG0SE4HT" kind="definition">
# Poisson point process

## Definition

Let $\lambda > 0$.
A homogeneous Poisson process with rate $\lambda$ is a [[stochastic process|0QH5ENKZM1]] $(N(t))_{t \ge 0}$ with values in $\{0, 1, 2, \dots\}$ such that:
1. $N(0) = 0$.
2. For all $0 \le t_0 < t_1 < \dots < t_n$, the increments $N(t_1) - N(t_0), \dots, N(t_n) - N(t_{n-1})$ are [[independent|0XCE070V8B]].
3. For all $0 \le s < t$, $N(t) - N(s)$ has the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda (t - s)$.

## Remarks

$N(t)$ counts the points, or events, in the interval $(0, t]$. One usually also asks that the paths $t \mapsto N(t)$ are right-continuous.
The times between consecutive events are i.i.d. with the [[exponential distribution|028Z8WC5Y9]] $\operatorname{Exp}(\lambda)$. Conversely, i.i.d. $\operatorname{Exp}(\lambda)$ gaps define a Poisson process with rate $\lambda$.
Given $N(t) = n$, the $n$ event times in $(0, t]$ have the law of $n$ independent uniform points in $(0, t]$, put in increasing order.
On a general space $E$ with a measure $\mu$, a Poisson point process has independent counts in disjoint sets.
The count in a set $B$ has the law $\operatorname{Poisson}(\mu(B))$ when $\mu(B) < \infty$.

## Explanations

### L1 human

The Poisson process models events that happen "completely at random" in time, such as clicks of a Geiger counter or arrivals of customers.
On average, $\lambda$ events happen per unit of time, and the waiting times between events are exponential.
</document>

<document source="https://math2.tamnd.com/node/05THZKR9AN.md" tag="05THZKR9AN" kind="definition">
# Group of units of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
The group of units of $R$ is the set $R^\times$ of the [[units|019NFGFPR5]] of $R$, with the multiplication of $R$.
It is a [[group|G5S1M9]] with identity element $1$.

## Remarks

The set $R^\times$ is closed under multiplication: if $u$ and $v$ are units, then $(u v)^{-1} = v^{-1} u^{-1}$.
Other notations for $R^\times$ are $R^*$ and $U(R)$.
The group $\mathbb{Z}^\times = \{1, -1\}$ has order $2$.
For a field $K$, $K^\times = K \setminus \{0\}$.

## Explanations

### L1 human

The units of a ring are the elements with an inverse.
If you multiply two units, you get a unit again.
So the units form a group under multiplication.
</document>

<document source="https://math2.tamnd.com/node/05V7J4WD1C.md" tag="05V7J4WD1C" kind="statement">
# Equality of complex numbers

## Statement

Let $z$ and $w$ be complex numbers.
Then $z = w$ if and only if $\operatorname{Re} z = \operatorname{Re} w$ and $\operatorname{Im} z = \operatorname{Im} w$.

## Remarks

Here $\operatorname{Re}$ and $\operatorname{Im}$ are the [[real and imaginary parts|02ZE6JHJ7G]].
Thus one equation between complex numbers gives two equations between real numbers.
For example, $x + y i = 3 - 2 i$ with $x, y$ real means $x = 3$ and $y = -2$.

## Explanations

### L1 human

Two complex numbers are equal when their real parts agree and their imaginary parts agree.
In the plane, this says that two points are equal when both of their coordinates agree.
</document>

<document source="https://math2.tamnd.com/node/05VDGH129F.md" tag="05VDGH129F" kind="concept">
# Phase plane

## Summary

The phase plane of a planar [[autonomous system|07BNRFQ0PJ]] $x' = f(x, y)$, $y' = g(x, y)$ is the plane with coordinates $(x, y)$, the two state variables.
Each solution traces a curve in this plane, its [[trajectory|0BVT6VSB3S]]. The vector $(f(x, y), g(x, y))$ is tangent to the trajectory through $(x, y)$.
It is the two-dimensional case of the phase space.

## Remarks

A second-order equation $x'' = h(x, x')$ gives a planar system with $y = x'$. Its phase plane has the position and the velocity as coordinates.
For a linear system $(x, y)' = A (x, y)$ with a $2 \times 2$ real matrix $A$, the eigenvalues of $A$ determine the shape of the trajectories. See the [[classification of planar linear systems|0N7ATVF2X8]].

## Explanations

### L1 human

The phase plane shows the two state variables at once, without the time axis.
Each point is a state, and the system moves from state to state along a curve.
</document>

<document source="https://math2.tamnd.com/node/05WAF5T0GZ.md" tag="05WAF5T0GZ" kind="statement">
# Maximal ideals are prime

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be a [[maximal ideal|09J6KGT65G]] of $R$.
Then $M$ is a [[prime ideal|06HC6HK0Z8]].

## Remarks

The proof: [[$R/M$ is a field|0WD3SE7S1X]], so it is an integral domain, so [[$M$ is prime|0AVXBEC05W]].
The converse is false: the zero ideal of $\mathbb{Z}$ is prime but not maximal.
In $\mathbb{Z}$, every nonzero prime ideal is maximal.

## Explanations

### L1 human

Every field is an integral domain.
So when $R/M$ is a field, $R/M$ is an integral domain, and $M$ is prime.
The zero ideal of the integers shows that a prime ideal need not be maximal.
</document>

<document source="https://math2.tamnd.com/node/05WDSN597C.md" tag="05WDSN597C" kind="concept">
# Submodule

## Summary

A submodule of a module is a subset that is a subgroup under addition and that is closed under multiplication by the ring elements.
</document>

<document source="https://math2.tamnd.com/node/05WEYX99X4.md" tag="05WEYX99X4" kind="concept">
# Inverse trigonometric functions

## Summary

The inverse trigonometric functions recover an angle from its sine, cosine or tangent. They are the inverses of the trigonometric functions restricted to standard intervals. The node [[inverse trigonometric functions|02HTDK1FBD]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05WKRC7HWT.md" tag="05WKRC7HWT" kind="statement">
# Jacobian of spherical coordinates

## Statement

Let $S(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \ \rho \sin \varphi \sin \theta, \ \rho \cos \varphi)$ be the [[spherical coordinate map|0RW8KPMXF6]].
Its [[Jacobian determinant|03W6Q1MFR5]] is
$$\det DS(\rho, \varphi, \theta) = \rho^2 \sin \varphi.$$
Hence, for a continuous function $f$ on a region $E \subseteq \mathbb{R}^3$ described by a region $E'$ of $(\rho, \varphi, \theta)$ with $\rho \ge 0$, $0 \le \varphi \le \pi$, $0 \le \theta \le 2\pi$,
$$\iiint_E f \, dV = \iiint_{E'} f(S(\rho, \varphi, \theta)) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta.$$

## Remarks

The Jacobian matrix has rows $(\sin \varphi \cos \theta, \ \rho \cos \varphi \cos \theta, \ -\rho \sin \varphi \sin \theta)$, $(\sin \varphi \sin \theta, \ \rho \cos \varphi \sin \theta, \ \rho \sin \varphi \cos \theta)$ and $(\cos \varphi, \ -\rho \sin \varphi, \ 0)$. Expansion along the last row gives $\rho^2 \sin \varphi$.
The integral formula follows from the [[change of variables theorem|00N5FHP1CV]]. The set where $S$ is not injective, or where the determinant vanishes, has volume zero.
Since $0 \le \varphi \le \pi$, the factor $\rho^2 \sin \varphi$ is non-negative, so no absolute value is needed.
Example: the volume of the ball of radius $a$ is $\int_0^{2\pi} \int_0^\pi \int_0^a \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = \tfrac{4}{3} \pi a^3$.

## Explanations

### L1 human

A small spherical box has sides $d\rho$, $\rho \, d\varphi$ and $\rho \sin \varphi \, d\theta$.
Its volume is about $\rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta$, and this factor appears in integrals in spherical coordinates.
</document>

<document source="https://math2.tamnd.com/node/05WR29H2KH.md" tag="05WR29H2KH" kind="concept">
# Number of derangements

## Summary

The derangement numbers $D_n$ count the permutations of $n$ elements with no fixed point. They grow like $n!/e$. The node [[number of derangements|03WY4HHMT1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/05WTQ6FFPS.md" tag="05WTQ6FFPS" kind="concept">
# Set difference

## Summary

The set difference of two sets is the set of the elements of the first set that are not elements of the second set.
</document>

<document source="https://math2.tamnd.com/node/05XT5AVJBK.md" tag="05XT5AVJBK" kind="definition">
# Laplace transform

## Definition

Let $f : [0, \infty) \to \mathbb{C}$ be a [[function|0EPAT51DXQ]] that is integrable on each interval $[0, T]$ with $T > 0$.
The Laplace transform of $f$ is the function $F = \mathcal{L}\{f\}$ given by
$$F(s) = \int_0^\infty f(t)\, e^{-s t}\, dt = \lim_{T \to \infty} \int_0^T f(t)\, e^{-s t}\, dt,$$
for each complex number $s$ for which the limit exists.

## Remarks

Let $f$ be of exponential order: there are constants $M, a$ with $|f(t)| \le M e^{a t}$ for all $t \ge 0$. Then the integral converges absolutely for $\operatorname{Re} s > a$.
Some transforms: $\mathcal{L}\{1\}(s) = 1/s$ and $\mathcal{L}\{e^{a t}\}(s) = 1/(s - a)$ for $\operatorname{Re} s > \operatorname{Re} a$, and $\mathcal{L}\{\sin \omega t\}(s) = \omega / (s^2 + \omega^2)$ for $\operatorname{Re} s > 0$ and real $\omega$.
The transform is linear: $\mathcal{L}\{c f + g\} = c \mathcal{L}\{f\} + \mathcal{L}\{g\}$ where both sides converge.
It turns derivatives into multiplication by $s$. See the [[Laplace transform of a derivative|0K090GVHVN]].

## Explanations

### L1 human

The Laplace transform turns a function of time into a function of a complex variable $s$.
It weights the function by $e^{-st}$, which decays fast, and adds up the result over all positive times.
Its main use is to turn linear differential equations into algebraic equations.
</document>

<document source="https://math2.tamnd.com/node/05Y0F9M0TS.md" tag="05Y0F9M0TS" kind="statement">
# Subgroups of solvable groups are solvable

## Statement

Let $G$ be a [[solvable group|0FXJ7YFMMX]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is solvable.

## Remarks

A proof uses the [[derived series|0Q5G5DJ4P2]]: $H^{(i)} \subseteq G^{(i)}$ for each $i$, so $G^{(n)} = \{e\}$ implies $H^{(n)} = \{e\}$.
More generally, if there is an injective homomorphism from $H$ into a solvable group, then $H$ is solvable.
For example, each subgroup of $S_4$ is solvable.

## Explanations

### L1 human

Commutators of elements of $H$ are also commutators in $G$.
So the derived series of $H$ stays inside the derived series of $G$.
If the larger series reaches the identity, the smaller one does too.
</document>

<document source="https://math2.tamnd.com/node/05Y1JAM3Y5.md" tag="05Y1JAM3Y5" kind="definition">
# Asymptotic stability

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]].
The point $x^*$ is asymptotically stable if it is [[Lyapunov stable|092S6VBECQ]] and there is an $\eta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \eta$, $\varphi_t(x_0) \to x^*$ as $t \to \infty$.
If $U = \mathbb{R}^n$ and $\varphi_t(x_0) \to x^*$ for each $x_0 \in \mathbb{R}^n$, and $x^*$ is Lyapunov stable, then $x^*$ is globally asymptotically stable.

## Remarks

For $x' = -x$ on $\mathbb{R}$, the equilibrium $0$ is globally asymptotically stable, since $\varphi_t(x_0) = x_0 e^{-t}$.
The point $x^*$ is exponentially stable if there are $\eta, C, \alpha > 0$ with $\|\varphi_t(x_0) - x^*\| \le C e^{-\alpha t} \|x_0 - x^*\|$ for $\|x_0 - x^*\| < \eta$ and $t \ge 0$.
The attraction condition alone does not imply Lyapunov stability. There are planar examples where all solutions tend to $x^*$, but some first make a large excursion.

## Explanations

### L1 human

An equilibrium is asymptotically stable if nearby solutions stay near it and also approach it in the long run.
A ball at the bottom of a bowl with friction is a typical example.
</document>

<document source="https://math2.tamnd.com/node/05YBKMRVAR.md" tag="05YBKMRVAR" kind="definition">
# Composition of linear maps

## Definition

Let $F$ be a field, let $U$, $V$, $W$ be vector spaces over $F$, and let $S \colon U \to V$ and $T \colon V \to W$ be [[linear maps|0NRYHJPQG2]].
The *composition* of $T$ and $S$ is the map
$$T \circ S \colon U \to W, \qquad (T \circ S)(u) = T(S(u)).$$

## Remarks

The composition of linear maps is linear.
Composition is associative, and the identity maps are its identity elements.
The [[matrix of a composition is the product of the matrices|05Q1B7MH0M]], in the same order.
Composition is not commutative: in general $T \circ S \ne S \circ T$, even when both are defined.

## Explanations

### L1 human

To compose two linear maps, we apply one and then the other.
We write $T \circ S$ for "first $S$, then $T$".
Matrix multiplication is exactly this operation, written in coordinates.
</document>

<document source="https://math2.tamnd.com/node/05Z3FC0154.md" tag="05Z3FC0154" kind="concept">
# Isolated singularity

## Summary

An isolated singularity of a function is a point $c$ such that the function is holomorphic on a punctured disc around $c$, see the [[definition|0TYB3HM9KE]].
It is removable, a pole, or essential.
</document>

<document source="https://math2.tamnd.com/node/05ZPH8KF9K.md" tag="05ZPH8KF9K" kind="definition">
# Perfect matching

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A perfect matching of $G$ is a [[matching|0Z87BHAFTB]] $M$ such that every vertex of $G$ is an end of an edge of $M$.

## Remarks

In a perfect matching, each vertex is an end of exactly one edge of $M$. Thus a finite graph with a perfect matching has an even number of vertices.
[[Hall's theorem for bipartite graphs|0X2GJPNPFN]] characterizes the bipartite graphs with a perfect matching. [[Tutte's theorem|08J7R35B6Q]] does this for all finite graphs.
The [[complete graph|03WKB2H474]] $K_4$ has $3$ perfect matchings.

## Explanations

### L1 human

A perfect matching pairs up all vertices of a graph along edges, with nobody left alone.
For this, the number of vertices must be even.
</document>

<document source="https://math2.tamnd.com/node/060W5CBZP6.md" tag="060W5CBZP6" kind="concept">
# Circle

## Summary

A circle is the set of all points of a plane at a given distance from a given point. The point is the centre and the distance is the radius. The region inside a circle is a disc.
</document>

<document source="https://math2.tamnd.com/node/060Z37VVB6.md" tag="060Z37VVB6" kind="definition">
# Right coset

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The right coset of $H$ with representative $g$ is the set $Hg = \{hg : h \in H\}$.

## Remarks

Two right cosets $Ha$ and $Hb$ are equal when $ab^{-1} \in H$, and they are disjoint otherwise.
In an [[abelian group|082GHNPJ53]], each right coset $Hg$ equals the [[left coset|03DEH43F4H]] $gH$.
In general $gH$ and $Hg$ can differ, but the map $gH \mapsto Hg^{-1}$ is a bijection from the left cosets to the right cosets.

## Explanations

### L1 human

A right coset is like a left coset, but you multiply the subgroup by the fixed element on the right.
In a group where the order of products matters, left and right cosets can be different sets.
The number of left cosets and the number of right cosets are always the same.
</document>

<document source="https://math2.tamnd.com/node/060ZVZBV3E.md" tag="060ZVZBV3E" kind="statement">
# Law of cosines

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be its side lengths, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]]. Let $\gamma = \angle ACB$ be its [[angle|0GJAJC6TNZ]] at $C$.
Then
$$c^2 = a^2 + b^2 - 2ab \cos \gamma.$$

## Remarks

The proof in vector form is short. With $u = A - C$ and $v = B - C$: $c^2 = \|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2 u \cdot v$, and $u \cdot v = ab \cos \gamma$.
For $\gamma = \pi / 2$ the formula is the [[Pythagorean theorem|0VP3A1290K]].
The formula lets you find the third side from two sides and the angle between them. It also gives each angle from the three sides.
Mathlib states the formula for all points, with its convention for the angle at a repeated point.

## History

Book II of Euclid's Elements contains the statement in the language of areas of rectangles. Propositions 12 and 13 of Book II treat the obtuse and the acute case.

## Explanations

### L1 human

The law of cosines corrects the Pythagorean theorem for triangles without a right angle. If the angle $\gamma$ is larger than a right angle, $\cos \gamma < 0$, and the opposite side is longer than in the right triangle. If $\gamma$ is smaller, the opposite side is shorter.
</document>

<document source="https://math2.tamnd.com/node/060ZX7C32T.md" tag="060ZX7C32T" kind="concept">
# Carmichael number

## Summary

A Carmichael number is a composite number $n$ that satisfies $a^{n-1} \equiv 1 \pmod n$ for each integer $a$ coprime to $n$.
The [[definition of a Carmichael number|0TA7WT68RK]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/061BTYA0RN.md" tag="061BTYA0RN" kind="statement">
# Lucas–Lehmer test

## Statement

Let $p$ be an odd prime, and let $M_p = 2^p - 1$ be the [[Mersenne number|0GJBJ19NXR]].
Define the sequence $(s_i)_{i \ge 0}$ by $s_0 = 4$ and $s_i = s_{i-1}^2 - 2$ for $i \ge 1$.
Then $M_p$ is prime if and only if $s_{p-2} \equiv 0 \pmod{M_p}$.

## Remarks

For example, take $p = 5$ and $M_5 = 31$. Modulo $31$, the sequence is $4, 14, 8, 0$, so $s_3 \equiv 0$ and $31$ is prime.
In practice, one reduces each term modulo $M_p$. Then the test needs $p - 2$ squarings of numbers with at most $p$ bits.

## History

Édouard Lucas developed the test in 1878, and Derrick Henry Lehmer proved it in 1930.

## Explanations

### L1 human

The test checks a Mersenne number for primality with a simple loop: square, subtract $2$, and reduce modulo $M_p$.
After $p - 2$ steps, the result is $0$ exactly when $M_p$ is prime.
</document>

<document source="https://math2.tamnd.com/node/061J5VGSPT.md" tag="061J5VGSPT" kind="concept">
# Symmetric relation

## Summary

A relation is symmetric when it always holds in both directions.
</document>

<document source="https://math2.tamnd.com/node/062D9KXJB1.md" tag="062D9KXJB1" kind="concept">
# Tree

## Summary

A tree is a connected graph with no cycles. Trees are the minimal connected graphs, and they appear in data structures, in spanning problems and in counting problems. The node [[tree|0MJ0JS72DZ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/062MQGP8M5.md" tag="062MQGP8M5" kind="definition">
# Trail

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A trail in $G$ is a [[walk|0VJZM0F9YM]] $(w_0, w_1, \dots, w_k)$ whose edges $w_0 w_1, w_1 w_2, \dots, w_{k-1} w_k$ are pairwise distinct.

## Remarks

A trail may visit a vertex more than once, but it uses each edge at most once.
Every [[path|0JDE56HWQ8]] is a trail. In the graph of two triangles that share a vertex $c$, the closed walk $a, b, c, d, e, c, a$ is a trail but not a path.
An [[Eulerian trail|0NNYDD1AVR]] is a trail that uses every edge.

## Explanations

### L1 human

A trail is a walk that never uses the same edge twice.
It is the kind of route that you draw without lifting your pen and without retracing a line.
</document>

<document source="https://math2.tamnd.com/node/06395GBNWD.md" tag="06395GBNWD" kind="definition">
# Gamma distribution

## Definition

Let $\alpha > 0$ and $\beta > 0$.
The gamma distribution with shape $\alpha$ and rate $\beta$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} & \text{if } x > 0, \\ 0 & \text{if } x \le 0, \end{cases}$$
with the gamma function $\Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} \, dt$. One writes $X \sim \operatorname{Gamma}(\alpha, \beta)$.

## Remarks

The mean is $\alpha / \beta$ and the variance is $\alpha / \beta^2$. Another common form uses the scale $\theta = 1/\beta$.
$\operatorname{Gamma}(1, \beta)$ is the [[exponential distribution|028Z8WC5Y9]] with rate $\beta$. For an integer $n \ge 1$, $\operatorname{Gamma}(n, \beta)$ is the law of a sum of $n$ independent $\operatorname{Exp}(\beta)$ random variables. It is the Erlang distribution.
If $X \sim \operatorname{Gamma}(\alpha_1, \beta)$ and $Y \sim \operatorname{Gamma}(\alpha_2, \beta)$ are independent, then $X + Y \sim \operatorname{Gamma}(\alpha_1 + \alpha_2, \beta)$.
The chi-squared distribution with $k$ degrees of freedom is $\operatorname{Gamma}(k/2, 1/2)$.

## Explanations

### L1 human

The gamma distribution with an integer shape $n$ is the waiting time for the $n$-th event of a Poisson process.
For example, it can model the time until the third customer arrives.
</document>

<document source="https://math2.tamnd.com/node/063D5Z2C70.md" tag="063D5Z2C70" kind="concept">
# Nilpotent endomorphism

## Summary

A nilpotent endomorphism is a linear map from a space to itself that becomes zero after finitely many compositions with itself.
</document>

<document source="https://math2.tamnd.com/node/063H5KJC69.md" tag="063H5KJC69" kind="statement">
# First fundamental theorem of calculus

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable.
Define $F(x) = \int_a^x f(t) \, dt$ for $x \in [a, b]$.
Then $F$ is continuous on $[a, b]$.
If $f$ is continuous at $c \in (a, b)$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.

## Remarks

If $|f| \le M$, then $|F(y) - F(x)| \le M |y - x|$. So $F$ is Lipschitz continuous.
By [[additivity|05RCDKHZM6]], $\frac{F(c + h) - F(c)}{h}$ is the average of $f$ over the interval between $c$ and $c + h$. Continuity at $c$ makes this average tend to $f(c)$.
Mathlib states a strict form as `intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right`.

## Explanations

### L1 human

Let the area under the graph from $a$ to $x$ be a function of $x$.
Where $f$ is continuous, this area grows at the rate $f(x)$.
</document>

<document source="https://math2.tamnd.com/node/063JNBSMJ0.md" tag="063JNBSMJ0" kind="conjecture">
# Goldbach's conjecture

## Statement

Each even integer $n > 2$ is the sum of two [[prime numbers|0SN0MAEKYX]].

## Remarks

For example, $4 = 2 + 2$, $28 = 5 + 23 = 11 + 17$ and $100 = 3 + 97$.
The weak Goldbach conjecture says that each odd integer $n > 5$ is the sum of three primes. Harald Helfgott proved it in 2013.
The conjecture is open. A distributed computer search checked it for each even $n \le 4 \cdot 10^{18}$.

## History

Goldbach stated a form of the conjecture in a letter to Euler in 1742.

## Explanations

### L1 human

Take an even number larger than $2$. The conjecture says that you can always split it into two primes.
For example, $18 = 7 + 11$. Nobody has found an even number that fails, but nobody has a proof.
</document>

<document source="https://math2.tamnd.com/node/063W5JMY85.md" tag="063W5JMY85" kind="definition">
# Metric space

## Definition

Let $X$ be a set.
A metric on $X$ is a [[function|0XHDTE5YMZ]] $d \colon X \times X \to \mathbb{R}$ with these properties, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. If $x \neq y$, then $d(x, y) > 0$.
3. $d(x, y) = d(y, x)$.
4. $d(x, z) \le d(x, y) + d(y, z)$.

A metric space is a pair $(X, d)$, where $X$ is a set and $d$ is a metric on $X$.
The number $d(x, y)$ is the distance between $x$ and $y$.

## Remarks

Property 4 is the [[triangle inequality for a metric|01M6CW9VM6]].
The properties give $d(x, y) \ge 0$ for all $x, y$: take $z = x$ in property 4 and use property 3.
The set $\mathbb{R}$ with $d(x, y) = |x - y|$ is a metric space: this is the [[usual metric on the real line|0KXJTZSW9B]].
If you remove property 2, you get a [[pseudometric space|0H3W614GEC]].

## Explanations

### L1 human

A metric is a rule that gives a distance to each pair of points.
The four properties are the facts that you expect from distances on a map.
For example, the road distance between towns is not always a metric, because one-way roads can break symmetry.
</document>

<document source="https://math2.tamnd.com/node/064N002TX4.md" tag="064N002TX4" kind="definition">
# Order of a zero

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$, and let $z_0 \in U$.
Assume that $f$ is not identically zero on any open disc with center $z_0$.
The order of $f$ at $z_0$ is the smallest integer $m \ge 0$ with $f^{(m)}(z_0) \neq 0$, where $f^{(m)}$ is the $m$-th complex derivative of $f$.
If $f(z_0) = 0$, this order $m \ge 1$ is the order of the [[zero|0CEGEASY49]] $z_0$ of $f$. A zero of order $1$ is a simple zero.
If $f$ is identically zero on an open disc with center $z_0$, the order of $f$ at $z_0$ is $\infty$.

## Remarks

The order is $m$ if and only if $f(z) = (z - z_0)^m g(z)$ near $z_0$, where $g$ is holomorphic near $z_0$ and $g(z_0) \neq 0$.
The order exists because [[holomorphic functions are analytic|0826SGV3R0]]: if all the derivatives at $z_0$ are $0$, then the Taylor series is $0$ and $f = 0$ near $z_0$.
For example, $z^3 (z - 1)$ has a zero of order $3$ at $0$ and a simple zero at $1$. The function $\sin z - z$ has a zero of order $3$ at $0$.

## Explanations

### L1 human

The order of a zero says how fast the function goes to $0$ at that point.
Near a zero of order $m$, the function behaves like a constant times $(z - z_0)^m$.
</document>

<document source="https://math2.tamnd.com/node/0659J5X7B5.md" tag="0659J5X7B5" kind="statement">
# LaSalle's invariance principle

## Statement

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be locally Lipschitz with $f(0) = 0$, and let $V : \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable.
Assume that $\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0$ for all $x \in \mathbb{R}^n$, that $V(0) = 0$ and $V(x) > 0$ for $x \neq 0$, and that $V(x) \to \infty$ as $\|x\| \to \infty$.
Let $E = \{x \in \mathbb{R}^n : \dot{V}(x) = 0\}$.
Assume that the only solution $x(t)$ of $x' = f(x)$ with $x(t) \in E$ for all $t \ge 0$ is the solution $x(t) = 0$.
Then the origin is globally asymptotically stable for $x' = f(x)$.

## Remarks

The general principle: if $\dot{V} \le 0$, then each solution that is bounded for $t \ge 0$ approaches the largest invariant subset of $E$.
The principle allows $\dot{V} = 0$ outside the origin, where [[Lyapunov's stability theorem|0J6A752DTZ]] needs a strict Lyapunov function.
For the damped oscillator $x' = y$, $y' = -x - c y$ with $c > 0$, take $V = \frac{1}{2}(x^2 + y^2)$. Then $E = \{y = 0\}$, and a solution in $E$ has $y' = -x = 0$. So the origin is globally asymptotically stable.
A local version holds when the conditions on $V$ hold only near the origin.

## History

The principle is named after Joseph LaSalle. It is also named after Nikolay Krasovskii and Evgenii Barbashin.

## Explanations

### L1 human

Sometimes the energy only stops decreasing on a set of states, not only at the equilibrium.
LaSalle's principle says that the motion must still end at the equilibrium, if no motion can stay forever in that set.
</document>

<document source="https://math2.tamnd.com/node/065CK367F9.md" tag="065CK367F9" kind="concept">
# Coprime ideals

## Summary

Two ideals of a ring are coprime when their sum is the whole ring.
For the integers, the ideals $(m)$ and $(n)$ are coprime exactly when $m$ and $n$ are coprime numbers.
The [[definition of coprime ideals|0MDJP82TEH]] gives the condition.
</document>

<document source="https://math2.tamnd.com/node/065HE5V98E.md" tag="065HE5V98E" kind="concept">
# Surface integral

## Summary

A surface integral is an integral of a function over a surface in space, with respect to [[surface area|0S29Y4ZW0Y]].
It is the analogue for surfaces of the line integral along a curve.
</document>

<document source="https://math2.tamnd.com/node/065J822CKN.md" tag="065J822CKN" kind="statement">
# Inclusion of a subspace is continuous

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ have the [[subspace topology|0D5YHQGDJY]] and let $\iota \colon S \to X$ be the inclusion map $\iota(s) = s$.
Then $\iota$ is [[continuous|0SY9ECQ19Q]].

## Remarks

Proof: for $U$ open in $X$, the [[preimage|0FGCFY1AZX]] is $\iota^{-1}(U) = S \cap U$, which is open in $S$.
Thus the [[restriction|0D2MXBV21T]] $f|_S = f \circ \iota$ of a continuous map $f \colon X \to Y$ is continuous.
A map $g \colon Z \to S$ is continuous if and only if $\iota \circ g \colon Z \to X$ is continuous. This is the universal property of the subspace topology.

## Explanations

### L1 human

Putting a subspace back into the big space is a continuous map.
So when you restrict a continuous function to a subset, you get a continuous function.
</document>

<document source="https://math2.tamnd.com/node/065JWMZW86.md" tag="065JWMZW86" kind="concept">
# Area of a triangle

## Summary

The area of a [[triangle|0CTNFGYMM2]] is the measure of the region of the plane that the triangle bounds. It is half the product of a base and the height on that base: $T = \frac{1}{2} b h$.
Here $b$ is the length of a side, and $h$ is the distance from the opposite vertex to the line of that side.
If two sides have the lengths $a$ and $b$ and the angle $\gamma$ between them, then $T = \frac{1}{2} a b \sin \gamma$.
For the vertices $A = (x_A, y_A)$, $B = (x_B, y_B)$ and $C = (x_C, y_C)$ in $\mathbb{R}^2$:
$$T = \tfrac{1}{2} \left| (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A) \right|.$$

## Remarks

[[Heron's formula|0FRBZ2X4HJ]] gives the area from the three side lengths alone.
In $\mathbb{R}^3$, the area of the triangle $ABC$ is $\frac{1}{2} \|(B - A) \times (C - A)\|$, with the [[cross product|0N5N15HWZ7]].
The coordinate formula without the absolute value is the signed area. It is positive when $A$, $B$, $C$ go round anticlockwise.

## Explanations

### L1 human

A triangle is half of a parallelogram: two copies of it fit together into one. The parallelogram has the area base times height. So the triangle has half of that, $\frac{1}{2} b h$.
</document>

<document source="https://math2.tamnd.com/node/065S5B9HF1.md" tag="065S5B9HF1" kind="statement">
# Repeating decimals are rational

## Statement

Let $x \ge 0$ be a real number with a [[decimal expansion|06XY4X9430]] that is eventually periodic.
That is, there are $N \ge 0$ and $p \ge 1$ with $d_{k + p} = d_k$ for each $k > N$.
Then $x$ is a [[rational number|0NCJ5WXN6Z]].

## Remarks

Proof: $10^{N+p} x - 10^N x$ is an integer, because the periodic tails cancel. Thus $x$ is a quotient of an integer by $10^N (10^p - 1)$.
For example, $x = 0.\overline{36}$ gives $100 x - x = 36$, thus $x = 36/99 = 4/11$.
The converse is also true: the decimal expansion of each rational number is eventually periodic.

## Explanations

### L1 human

A decimal that repeats the same block of digits forever is always a fraction.
Shift the decimal point by the length of the block and subtract: the endless tail disappears.
For $0.777\dots$, this gives $10 x - x = 7$, thus $x = 7/9$.
</document>

<document source="https://math2.tamnd.com/node/065X9S8CM3.md" tag="065X9S8CM3" kind="statement">
# De Moivre–Laplace theorem

## Statement

Let $p \in (0, 1)$ and let $S_n \sim \operatorname{Bin}(n, p)$ for each $n \ge 1$. Let $q = 1 - p$.
Then $(S_n - np)/\sqrt{npq}$ converges [[in distribution|0JDAQHWA9N]] to $\mathcal{N}(0, 1)$. That is, for all reals $a < b$,
$$\lim_{n \to \infty} P\left(a \le \frac{S_n - np}{\sqrt{npq}} \le b\right) = \int_a^b \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx.$$

## Remarks

This is the special case of the [[central limit theorem|0EPZC4Z1FC]] for Bernoulli variables.
Indeed, $S_n$ has the law of a sum of $n$ independent Bernoulli($p$) variables.
A local form also holds: $\binom{n}{k} p^k q^{n-k} \sim \frac{1}{\sqrt{2\pi npq}} e^{-(k - np)^2/(2npq)}$ as $n \to \infty$, uniformly for $k$ with $|k - np| \le C\sqrt{n}$. The proof uses Stirling's formula.
In practice, one often adds a continuity correction of $1/2$ to the integer bounds.

## History

Abraham de Moivre proved the case $p = 1/2$. Pierre-Simon Laplace extended it to general $p$.

## Explanations

### L1 human

For many trials, the binomial distribution looks like a bell curve with mean $np$ and variance $np(1 - p)$.
This lets you approximate binomial probabilities, such as "at least $520$ heads in $1000$ tosses", with the normal distribution.
</document>

<document source="https://math2.tamnd.com/node/065XW2V6C3.md" tag="065XW2V6C3" kind="definition">
# Arctangent

## Definition

The arctangent is the function $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$ such that, for each real $x$, $\arctan x$ is the unique $y \in (-\pi/2, \pi/2)$ with $\tan y = x$.

## Remarks

The [[tangent|06T0HS14PP]] is strictly increasing and continuous on $(-\pi/2, \pi/2)$, with range $\mathbb{R}$. So $\arctan$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arctan 0 = 0$ and $\arctan 1 = \pi/4$.
Also $\arctan x \to \pi/2$ as $x \to \infty$, and $\arctan x \to -\pi/2$ as $x \to -\infty$. Mathlib defines it as `Real.arctan`.

## Explanations

### L1 human

The arctangent of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose tangent is $x$.
It is the angle of a line with slope $x$.
</document>

<document source="https://math2.tamnd.com/node/0660QX1VPR.md" tag="0660QX1VPR" kind="definition">
# Radius of convergence

## Definition

Let $\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].
Its radius of convergence is
$$R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n) \text{ is bounded} \},$$
an element of $[0, \infty]$.

## Remarks

The series converges absolutely for $|x - c| < R$, and it diverges for $|x - c| > R$.
At the points $x = c \pm R$, the series may converge or diverge.
Many texts define $R$ as the supremum of all $|x - c|$ where the series converges. The two definitions give the same value.
The [[Cauchy–Hadamard theorem|07E5MB1CA2]] gives a formula for $R$. Mathlib uses the bounded form, in `FormalMultilinearSeries.radius`.

## Explanations

### L1 human

A power series converges on an interval centered at $c$, and diverges outside it.
The radius of convergence is the half length of this interval.
</document>

<document source="https://math2.tamnd.com/node/066E7PJCYE.md" tag="066E7PJCYE" kind="concept">
# Pole

## Summary

A pole of a function is an isolated singularity where the modulus of the function tends to infinity, see the [[definition|0WPQYW8MTE]].
It is the simplest kind of singularity that is not removable.
</document>

<document source="https://math2.tamnd.com/node/066HW6D9A7.md" tag="066HW6D9A7" kind="statement">
# Closed subsets of compact sets are compact

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $K \subseteq X$ be [[compact|00R0X1WGPX]] and let $F \subseteq X$ be [[closed|0G5AV9Q4RR]] with $F \subseteq K$.
Then $F$ is compact.

## Remarks

Proof: let $(U_i)$ be open sets of $X$ that cover $F$. Add the open set $X \setminus F$ to get an open cover of $K$. A finite subcover of $K$ gives, after you remove $X \setminus F$, a finite subcover of $F$.
More generally, if $K$ is compact and $F$ is closed, then $K \cap F$ is compact.
For example, the [[Cantor set|0BXNDZFQME]] is compact, because it is a closed subset of $[0, 1]$.

## Explanations

### L1 human

A closed part of a compact set is again compact.
The open complement of the closed part covers everything outside it, so it does the job of all the extra open sets.
</document>

<document source="https://math2.tamnd.com/node/066RJJVWH9.md" tag="066RJJVWH9" kind="statement">
# The integers are countable

## Statement

The set $\mathbb{Z}$ of integers is [[countably infinite|0SEZ3JGZP7]].
The function $f \colon \mathbb{N} \to \mathbb{Z}$ with $f(2k) = k$ and $f(2k + 1) = -(k + 1)$ is a [[bijection|0SNQEF7Z53]].

## Remarks

The bijection lists the integers as $0, -1, 1, -2, 2, \dots$.
So $|\mathbb{Z}| = |\mathbb{N}| = \aleph_0$, although $\mathbb{N}$ is a proper subset of $\mathbb{Z}$.

## Explanations

### L1 human

The list $0, 1, 2, \dots$ never reaches the negative integers.
Alternate between the two signs, and every integer gets a position in the list.
</document>

<document source="https://math2.tamnd.com/node/0670T59HGE.md" tag="0670T59HGE" kind="concept">
# Noetherian ring

## Summary

A Noetherian ring is a ring in which every ideal is finitely generated, or equivalently, every increasing chain of ideals is eventually constant.
The [[definition of a Noetherian ring|0MVH4Y1CRZ]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/06738GGWWK.md" tag="06738GGWWK" kind="concept">
# Flow of a vector field

## Summary

A flow is a family of maps $\varphi_t$ of a space to itself, indexed by real time $t$. The map $\varphi_0$ is the identity, and $\varphi_s \circ \varphi_t = \varphi_{s+t}$.
The flow of a vector field sends each point to the position, after time $t$, of the solution of $x' = f(x)$ through it. See the [[definition|02903SSA48]].
</document>

<document source="https://math2.tamnd.com/node/0673D85QKZ.md" tag="0673D85QKZ" kind="concept">
# Autonomous system

## Summary

An autonomous system is a system of ordinary differential equations $x' = f(x)$ whose right side does not depend explicitly on the independent variable.
When the independent variable is time, such a system is also called time-invariant. See the [[definition|07BNRFQ0PJ]].
</document>

<document source="https://math2.tamnd.com/node/067F8YVYMH.md" tag="067F8YVYMH" kind="statement">
# Repeated roots and the derivative

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ with $f \neq 0$ and let $c \in R$.
Then $c$ is a root of $f$ with [[multiplicity|03XVP8Q4GY]] at least $2$ if and only if $f(c) = 0$ and $f'(c) = 0$.
Here $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.

## Remarks

The proof: write $f = (x - c) g$. The product rule gives $f' = g + (x - c) g'$, so $f'(c) = g(c)$. And $g(c) = 0$ exactly when $(x - c)^2$ divides $f$.
For example, $f = x^3 - 3x + 2 = (x - 1)^2 (x + 2)$. Here $f(1) = 0$ and $f'(1) = 3 - 3 = 0$.

## Explanations

### L1 human

At a repeated root, the graph of a real polynomial touches the axis and has a horizontal tangent.
So the polynomial and its derivative are both zero there.
The algebraic statement is true over every commutative ring.
</document>

<document source="https://math2.tamnd.com/node/067NKSBEWT.md" tag="067NKSBEWT" kind="concept">
# Scalar matrix

## Summary

A scalar matrix is a diagonal matrix whose diagonal entries are all equal, that is, a scalar multiple of the identity matrix.
</document>

<document source="https://math2.tamnd.com/node/067SYD2J4B.md" tag="067SYD2J4B" kind="statement">
# Tonelli's theorem

## Statement

Let $f \colon \mathbb{R}^n \times \mathbb{R}^m \to [0, \infty]$ be Lebesgue measurable.
Then $x \mapsto \int_{\mathbb{R}^m} f(x, y) \, dy$ and $y \mapsto \int_{\mathbb{R}^n} f(x, y) \, dx$ are measurable functions with values in $[0, \infty]$, and
$$\int_{\mathbb{R}^n \times \mathbb{R}^m} f \, d(x, y) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^m} f(x, y) \, dy \, dx = \int_{\mathbb{R}^m} \int_{\mathbb{R}^n} f(x, y) \, dx \, dy.$$
The three values may be $+\infty$.

## Remarks

Tonelli's theorem has the same conclusion as [[Fubini's theorem|0R0T1VWH5X]]. It replaces integrability of $|f|$ by the hypothesis $f \ge 0$.
In practice, one applies Tonelli's theorem to $|f|$ to check that $\int |f| < \infty$, and then applies Fubini's theorem to $f$.
More generally, the theorem holds for non-negative measurable functions on a product of two $\sigma$-finite measure spaces. For counting measure on $\mathbb{N}$, it says that $\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy}$ when all $a_{xy} \ge 0$.
Leonida Tonelli published this variation of Fubini's theorem in 1909.

## Explanations

### L1 human

For a function that is never negative, the order of integration never matters.
Both orders give the same value, even when that value is infinite, because no positive and negative parts can cancel.
</document>

<document source="https://math2.tamnd.com/node/068BZBXE0C.md" tag="068BZBXE0C" kind="statement">
# Existence of square roots of nonnegative reals

## Statement

Let $x$ be a real number with $x \ge 0$.
Then there is exactly one real number $y$ with $y \ge 0$ and $y^2 = x$.

## Remarks

Existence uses the [[completeness axiom|0AXWET80YW]]: take $y = \sup \{t \ge 0 : t^2 \le x\}$ and show that $y^2 < x$ and $y^2 > x$ are both impossible.
Uniqueness: if $y, z \ge 0$ and $y^2 = z^2$, then $(y - z)(y + z) = 0$, thus $y = z$.
In $\mathbb{Q}$ the existence fails: $2$ has no rational square root, by the [[irrationality of the square root of 2|0DZGSDHD92]].

## Explanations

### L1 human

Every nonnegative real number is the area of exactly one square, and the side of that square is the square root.
This needs the real numbers: among the fractions, no square has area exactly $2$.
</document>

<document source="https://math2.tamnd.com/node/068P0P32DW.md" tag="068P0P32DW" kind="statement">
# Closure in a metric space

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $S \subseteq X$ and let $x \in X$.
Then $x$ is in the [[closure|0DRKT1QJR7]] $\overline{S}$ if and only if for each real $\varepsilon > 0$ there is $s \in S$ with $d(x, s) < \varepsilon$.

## Remarks

Equivalently, $x \in \overline{S}$ if and only if each [[open ball|0DC3MPR7C9]] $B(x, \varepsilon)$ meets $S$.
For nonempty $S$, this means $d(x, S) = 0$, where $d(x, S)$ is the [[distance from the point to the set|07TXYYVYZR]].
In $\mathbb{R}$, the point $0$ is in the closure of $\{1/n : n \ge 1\}$, because $1/n < \varepsilon$ for large $n$.

## Explanations

### L1 human

A point is in the closure of $S$ when there are points of $S$ as close to it as you want.
The point does not need to be in $S$ itself.
</document>

<document source="https://math2.tamnd.com/node/0692CGQF00.md" tag="0692CGQF00" kind="definition">
# Diameter of a set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S$ be a nonempty subset of $X$.
The diameter of $S$ is
$$\operatorname{diam}(S) = \sup \{ d(x, y) : x, y \in S \},$$
where the [[supremum|0BTMQ04556]] is taken in the [[extended real number line|0GSZ7B0CGR]], so that $\operatorname{diam}(S) \in [0, \infty]$.

## Remarks

The set $S$ is [[bounded|0P953MG05E]] if and only if $\operatorname{diam}(S) < \infty$.
For the empty set, a common convention is $\operatorname{diam}(\emptyset) = 0$.
In $\mathbb{R}$, the interval $(a, b)$ with $a < b$ has diameter $b - a$, and so does $[a, b]$.
If $S \subseteq T$, then $\operatorname{diam}(S) \le \operatorname{diam}(T)$.

## Explanations

### L1 human

The diameter is the largest distance between two points of the set, or the limit of such distances.
For a disc of radius $r$ in the plane, the diameter is $2r$.
</document>

<document source="https://math2.tamnd.com/node/0693Y2METT.md" tag="0693Y2METT" kind="concept">
# Hessian matrix

## Summary

The Hessian matrix of a real-valued function of $n$ variables at a point is the $n \times n$ matrix of its second [[partial derivatives|09ZXGX55DM]] at that point.
It describes the curvature of the function, and it is the main tool of the second derivative test for local extrema.
</document>

<document source="https://math2.tamnd.com/node/0698M7C7AX.md" tag="0698M7C7AX" kind="concept">
# Triple integral

## Summary

The triple integral of a function of three variables over a solid region is a limit of Riemann sums. These sums use partitions of the region into small boxes.
It extends the [[double integral|0WAR4F23B4]] to three dimensions, and it computes volumes, masses and centres of mass of solids.
</document>

<document source="https://math2.tamnd.com/node/069A3BC3EP.md" tag="069A3BC3EP" kind="statement">
# Uniqueness of splitting fields

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $f \in K[x]$ be nonzero, and let $L$ and $L'$ be [[splitting fields|02NZZ1J0FH]] of $f$ over $K$.
Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

## Remarks

The proof is by induction on $[L : K]$. It extends an isomorphism one root at a time, with the [[structure of simple extensions|0ZN39QDTP9]].
The isomorphism $\sigma$ is usually not unique. The isomorphisms from $L$ to itself that fix $K$ form the [[Galois group|0XVR72M51M]] of $f$.
So one speaks of "the" splitting field of $f$ over $K$.
The choice of the irreducible factors in the construction does not change the result, up to isomorphism.

## Explanations

### L1 human

Two splitting fields of the same polynomial are the same field with different names for the elements.
The proof extends a matching of the base field, one root at a time.
</document>

<document source="https://math2.tamnd.com/node/069HKV4X1E.md" tag="069HKV4X1E" kind="statement">
# Comparison of a series and an integral

## Statement

Let $a$ be a real number, let $n$ be a natural number, and let $f \colon [a, a + n] \to \mathbb{R}$ be a decreasing function.
Then
$$\sum_{i=1}^{n} f(a + i) \le \int_a^{a+n} f(x)\, dx \le \sum_{i=0}^{n-1} f(a + i).$$

## Remarks

On each interval $[a + i, a + i + 1]$ we have $f(a + i + 1) \le f(x) \le f(a + i)$. Integrate and add over $i$.
The integral exists because monotone functions are [[Riemann integrable|04YKAYDQPW]].
With $f(x) = 1/x$ and $a = 1$ it gives $\ln(n + 1) \le H_n \le 1 + \ln n$ for the [[harmonic numbers|0D2Y6G4C4S]].
Mathlib states the first inequality as `AntitoneOn.sum_le_integral`.

## Explanations

### L1 human

The sum of the values at the right ends of unit intervals is an area of rectangles below the graph of a decreasing function.
The sum of the values at the left ends is an area of rectangles above it.
</document>

<document source="https://math2.tamnd.com/node/069MCGX635.md" tag="069MCGX635" kind="concept">
# Row space

## Summary

The row space of a matrix is the span of its rows.
</document>

<document source="https://math2.tamnd.com/node/069P20KYQT.md" tag="069P20KYQT" kind="definition">
# Partition of an interval

## Definition

Let $a < b$ be real numbers.
A partition of $[a, b]$ is a finite sequence of real numbers $P = (x_0, x_1, \dots, x_n)$ with
$$a = x_0 < x_1 < \cdots < x_n = b.$$
The intervals $[x_{i-1}, x_i]$ for $i = 1, \dots, n$ are the subintervals of $P$.

## Remarks

Partitions are the basis of [[Darboux sums|0QRQVH0CRW]] and [[Riemann sums|0HBY48G8G1]].
A [[refinement|0EEKF7YFBW]] of $P$ adds more points. The [[mesh|04GM4NYXJB]] of $P$ is the length of its longest subinterval.
A tagged partition also chooses a point $t_i$ in each subinterval $[x_{i-1}, x_i]$.

## Explanations

### L1 human

A partition cuts the interval $[a, b]$ into finitely many pieces.
The cut points are listed from left to right.
</document>

<document source="https://math2.tamnd.com/node/06AMCJMMVM.md" tag="06AMCJMMVM" kind="statement">
# Laplace expansion along a column

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.
Then, for each column $j$,
$$\det A = \sum_{i=1}^{n} a_{ij} C_{ij} = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

## Remarks

It follows from the [[expansion along a row|0C08BQ3M05]], applied to $A^T$, because $\det A^T = \det A$.
For example, expansion along column $1$ of $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 6 & 7 \end{pmatrix}$ gives $1 \cdot (28 - 30) = -2$.

## Explanations

### L1 human

The same expansion works down a column instead of across a row.
We choose the column with the most zeros, to save work.
Each zero entry removes one smaller determinant from the sum.
</document>

<document source="https://math2.tamnd.com/node/06B8BNJCD6.md" tag="06B8BNJCD6" kind="statement">
# Heine–Cantor theorem

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]].
If $X$ is [[compact|0QPHW703PV]], then $f$ is [[uniformly continuous|0FHSDCQACY]].

## Remarks

Proof by contradiction: if $f$ is not uniformly continuous, there are $\varepsilon > 0$ and points $x_n, y_n$ with $d(x_n, y_n) \to 0$ and $d(f(x_n), f(y_n)) \ge \varepsilon$. By [[sequential compactness|0K8SGQSV7G]], a subsequence of $(x_n)$ converges to some $x$, and the matching subsequence of $(y_n)$ also converges to $x$. Continuity at $x$ gives a contradiction.
For example, each continuous $f \colon [a, b] \to \mathbb{R}$ is uniformly continuous. This fact is used to show that $f$ is Riemann integrable.
The theorem fails on $\mathbb{R}$: $x \mapsto x^2$ is continuous but not uniformly continuous.

## Explanations

### L1 human

On a compact space, continuity at each point is enough to get one $\delta$ for the whole space.
Compactness rules out regions where the function becomes steeper and steeper without end.
</document>

<document source="https://math2.tamnd.com/node/06B9W45XV0.md" tag="06B9W45XV0" kind="statement">
# Solution formula for linear first-order equations

## Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, let $x_0 \in I$ and let $y_0 \in \mathbb{R}$.
Let $P(x) = \int_{x_0}^{x} p(s)\, ds$ for $x \in I$.
Then the [[initial value problem|075VBY4K4X]] $y' + p(x)\, y = q(x)$, $y(x_0) = y_0$ has exactly one solution on $I$, and it is
$$y(x) = e^{-P(x)} \left( y_0 + \int_{x_0}^{x} e^{P(s)}\, q(s)\, ds \right), \qquad x \in I.$$

## Remarks

Proof idea: $\mu = e^{P}$ is an [[integrating factor|01THH3Y3YP]], so a solution satisfies $(\mu y)' = \mu q$. Integrate from $x_0$ to $x$ and use $\mu(x_0) = 1$.
The solution exists on all of $I$. A linear equation has no solution that blows up inside the interval of continuity of $p$ and $q$.
Example: for $y' + y = 1$, $y(0) = 0$, the formula gives $y(x) = 1 - e^{-x}$.

## Explanations

### L1 human

The formula has two parts.
The term $y_0 e^{-P(x)}$ is the solution of the equation without the right side $q$.
The integral term adds the effect of $q$, collected over the time from $x_0$ to $x$.
</document>

<document source="https://math2.tamnd.com/node/06BA5JZKTQ.md" tag="06BA5JZKTQ" kind="definition">
# Strictly convex function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is strictly convex if
$$f(t x + (1 - t) y) < t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ with $x \ne y$ and all $t \in (0, 1)$.

## Remarks

Each strictly convex function is [[convex|0VV0HSW1ZS]]. A linear function is convex but not strictly convex.
A strictly convex function has at most one minimum point.
For example, $x^2$ and $e^x$ are strictly convex. Mathlib defines strict convexity as `StrictConvexOn`.

## Explanations

### L1 human

A strictly convex function bends upward everywhere, with no straight pieces in its graph.
The chords lie strictly above the graph, except at their ends.
</document>

<document source="https://math2.tamnd.com/node/06BRDF3F0E.md" tag="06BRDF3F0E" kind="concept">
# Reflection

## Summary

The reflection in a subspace $W$ of an inner product space fixes the vectors of $W$ and negates the vectors orthogonal to $W$.
</document>

<document source="https://math2.tamnd.com/node/06C0JBE52D.md" tag="06C0JBE52D" kind="statement">
# Five colour theorem

## Statement

Let $G$ be a finite [[planar graph|029JQEK15B]].
Then $\chi(G) \le 5$.

## Remarks

Proof idea: $G$ has a vertex $v$ of degree at most $5$, by the [[edge bound for planar graphs|0SM1REAVYJ]]. Colour $G - v$ by induction.
If the neighbours of $v$ use all five colours, a swap of two colours along a Kempe chain frees one colour for $v$.
The [[four colour theorem|0M406JC8J1]] is stronger, but its proof is much harder.

## History

Percy John Heawood proved the theorem in 1890.
He found an error in the 1879 proof of the four colour theorem by Alfred Kempe, and he used Kempe's ideas to prove five colours.

## Explanations

### L1 human

Every map can be coloured with five colours so that neighbouring countries have different colours.
The proof is short, unlike the proof that four colours are enough.
</document>

<document source="https://math2.tamnd.com/node/06CTFY2DEZ.md" tag="06CTFY2DEZ" kind="definition">
# Symmetric polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
A polynomial $f \in R[x_1, \dots, x_n]$ is symmetric if
$$f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = f(x_1, \dots, x_n)$$
for every permutation $\sigma$ of $\{1, \dots, n\}$.

## Remarks

Examples in two variables: $x_1 + x_2$, $x_1 x_2$ and $x_1^2 + x_2^2$. The polynomial $x_1 - x_2$ is not symmetric.
The symmetric polynomials form a subring of $R[x_1, \dots, x_n]$, the ring of invariants of the symmetric group.
The main examples are the [[elementary symmetric polynomials|0ZMY8441GW]] and the [[power sums|0AS1FZT6BM]].
The coefficients of a monic polynomial are symmetric polynomials in its roots, by [[Vieta's formulas|0M8EWQJ1YV]].

## Explanations

### L1 human

A symmetric polynomial treats all its variables in the same way.
Swapping two variables gives the same polynomial.
This is why symmetric expressions in the roots of a polynomial can be computed from the coefficients.
</document>

<document source="https://math2.tamnd.com/node/06D0EWR2DV.md" tag="06D0EWR2DV" kind="statement">
# Integral of the exponential function

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b e^x \, dx = e^b - e^a.$$

## Remarks

The [[exponential function|0RMVS7RXZX]] is [[its own derivative|0EX9GFJ0B3]], so it is its own antiderivative.
More generally, $\int_a^b e^{c x} \, dx = (e^{c b} - e^{c a})/c$ for $c \ne 0$.
Mathlib states it as `integral_exp`.

## Explanations

### L1 human

The exponential function does not change under differentiation.
So it also does not change under integration, up to the constant.
</document>

<document source="https://math2.tamnd.com/node/06DAH3VS6M.md" tag="06DAH3VS6M" kind="definition">
# P-group

## Definition

Let $p$ be a prime number.
A [[group|G5S1M9]] $G$ is a $p$-group when, for each $g \in G$, the [[order|0WE2NHZJ7B]] of $g$ is $p^n$ for some integer $n \geq 0$.

## Remarks

A finite group is a $p$-group if and only if $|G| = p^k$ for some $k \geq 0$.
One direction uses [[Lagrange's theorem|3C53TD]], and the other direction uses [[Cauchy's theorem|0JM7PA288B]].
The [[dihedral group|0JZ27DMJ5D]] $D_4$ of order $8$ is a $2$-group that is not abelian.
The trivial group is a $p$-group for each prime $p$.

## Explanations

### L1 human

In a $p$-group, each element comes back to the identity after a power of $p$ steps.
For finite groups this means that the number of elements is a power of $p$.
These groups are the building blocks of the Sylow theory.
</document>

<document source="https://math2.tamnd.com/node/06DW9BAWB1.md" tag="06DW9BAWB1" kind="object">
# Identity matrix

## Description

Let $R$ be a ring and let $n \ge 0$ be an integer.
The *identity matrix* $I_n$ is the $n \times n$ [[matrix|0VD9ZSWWJG]] over $R$ with entries $1$ on the main diagonal and $0$ elsewhere.
Its entries are $(I_n)_{ij} = \delta_{ij}$, where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ otherwise.
For each $m \times n$ matrix $A$, $A I_n = A$ and $I_m A = A$.

## Remarks

The identity matrix $I_n$ is the identity element of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.
It is the matrix of the identity map in any basis.
For example, $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

## Explanations

### L1 human

The identity matrix has ones on the diagonal and zeros everywhere else.
It plays the role of the number $1$: multiplying by it changes nothing.
As a map, it leaves each vector where it is.
</document>

<document source="https://math2.tamnd.com/node/06E3AG9TM2.md" tag="06E3AG9TM2" kind="counterexample">
# Z[x] is not a principal ideal domain

## Counterexample

The ring $\mathbb{Z}[x]$ is an integral domain, but it is not a [[principal ideal domain|0RZYDVZ475]].
The ideal $(2, x)$, the set of the polynomials with an even constant term, is not [[principal|00A2EDFT9Q]].
Suppose $(2, x) = (f)$. Then $f \mid 2$, so $f$ is a constant $c$ with $c \in \{\pm 1, \pm 2\}$.
If $c = \pm 1$, then $1 \in (2, x)$, but $1$ has an odd constant term.
If $c = \pm 2$, then $x$ is a multiple of $2$ in $\mathbb{Z}[x]$, which is false.

## Remarks

The quotient $\mathbb{Z}[x]/(2, x)$ is $\mathbb{F}_2$, so $(2, x)$ is a maximal ideal.
The ring $\mathbb{Z}[x]$ is still a [[unique factorization domain|06YMXJMXYS]], by [[the theorem on polynomial rings over a UFD|0D6KH8Y59N]].
Over a field $K$, the ring $K[x]$ is a principal ideal domain.

## Explanations

### L1 human

In $\mathbb{Z}[x]$, the ideal generated by $2$ and $x$ needs two generators.
No single polynomial divides both $2$ and $x$ except the units $\pm 1$, and $\pm 1$ generates too much.
The cause is that $\mathbb{Z}$ is not a field.
</document>

<document source="https://math2.tamnd.com/node/06EH97V2A7.md" tag="06EH97V2A7" kind="concept">
# Fréchet derivative

## Summary

The Fréchet derivative of a map $f$ between normed vector spaces at a point $x$ is a bounded linear map. It best approximates $f(x + h) - f(x)$ for small $h$.
It extends the derivative of one-variable calculus to maps between normed spaces, which can have infinite dimension.
For maps between Euclidean spaces it is the [[total derivative|0FNKFT8H6E]].
</document>

<document source="https://math2.tamnd.com/node/06EKW1AKXF.md" tag="06EKW1AKXF" kind="concept">
# Operator norm

## Summary

The operator norm of a continuous linear map measures the largest factor by which the map can stretch a vector.
It makes the space of continuous linear maps into a normed space.
</document>

<document source="https://math2.tamnd.com/node/06EX0FHBTF.md" tag="06EX0FHBTF" kind="definition">
# Basis of a topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]].
A basis of $\tau$ is a set $\mathcal{B} \subseteq \tau$ such that each $U \in \tau$ is a [[union|05GK32H72H]] of members of $\mathcal{B}$.
Equivalently, $\mathcal{B} \subseteq \tau$ and for each $U \in \tau$ and each $x \in U$ there is $B \in \mathcal{B}$ with $x \in B \subseteq U$.

## Remarks

A family $\mathcal{B}$ of subsets of a set $X$ is a basis of some topology on $X$ if and only if two conditions hold.
First, the members of $\mathcal{B}$ cover $X$. Second, for all $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$ there is $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$. This topology is then unique.
The open intervals form a basis of the usual topology of $\mathbb{R}$. The intervals with rational endpoints form a countable basis.
In a metric space, the [[open balls form a basis|02EEX96DFS]] of the metric topology.

## Explanations

### L1 human

A basis is a set of building blocks: each open set is a union of blocks.
In the plane, you can build each open set from open discs, or from open squares.
</document>

<document source="https://math2.tamnd.com/node/06F3Y6CMG9.md" tag="06F3Y6CMG9" kind="definition">
# Midpoint

## Definition

Let $K$ be a field in which $2 \neq 0$, and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$. Let $p$ and $q$ be points of $A$.
The midpoint of $p$ and $q$ is the point
$$\operatorname{midpoint}(p, q) = p + \tfrac{1}{2}(q - p).$$
It is the [[affine combination|0KJH6SRFZK]] $\frac{1}{2} p + \frac{1}{2} q$.

## Remarks

In $\mathbb{R}^n$, the midpoint of $(p_1, \dots, p_n)$ and $(q_1, \dots, q_n)$ is $\left(\frac{p_1 + q_1}{2}, \dots, \frac{p_n + q_n}{2}\right)$.
In a Euclidean space, the midpoint $m$ of $p$ and $q$ satisfies $d(p, m) = d(m, q) = \frac{1}{2} d(p, q)$. It is the only point with this property.
The midpoint is symmetric: $\operatorname{midpoint}(p, q) = \operatorname{midpoint}(q, p)$. Also $\operatorname{midpoint}(p, p) = p$.

## Explanations

### L1 human

The midpoint is the average of the two points. To find it, take the average of each coordinate. For example, the midpoint of $(0, 0)$ and $(4, 6)$ is $(2, 3)$.
</document>

<document source="https://math2.tamnd.com/node/06GKBTAA5E.md" tag="06GKBTAA5E" kind="definition">
# Boolean algebra

## Definition

A Boolean algebra is a set $A$ with two binary operations $\wedge$ and $\vee$, a unary operation $\neg$, and two elements $0, 1 \in A$.
These axioms must hold for all $a, b, c \in A$.
Associativity: $a \vee (b \vee c) = (a \vee b) \vee c$ and $a \wedge (b \wedge c) = (a \wedge b) \wedge c$.
Commutativity: $a \vee b = b \vee a$ and $a \wedge b = b \wedge a$.
Absorption: $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$.
Identity: $a \vee 0 = a$ and $a \wedge 1 = a$.
Distributivity: $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$ and $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$.
Complements: $a \vee \neg a = 1$ and $a \wedge \neg a = 0$.

## Remarks

The relation $a \le b$, defined by $a \wedge b = a$, is a partial order with least element $0$ and greatest element $1$.
Equivalently, a Boolean algebra is a complemented [[distributive lattice|0T36P6TXQ6]].
The power set of a set is a Boolean algebra, the [[power set Boolean algebra|004J7ZEFAT]]. Boolean algebras correspond to [[Boolean rings|0N5FCV2YV6]].

## Explanations

### L1 human

A Boolean algebra has operations that behave like and, or and not on true and false, or like intersection, union and complement on sets.
Its axioms are the common rules of logic and of set algebra.
</document>

<document source="https://math2.tamnd.com/node/06GPJF9V7M.md" tag="06GPJF9V7M" kind="definition">
# Mersenne prime

## Definition

A Mersenne prime is a [[prime number|0SN0MAEKYX]] of the form $2^n - 1$ for a natural number $n$.
That is, a Mersenne prime is a [[Mersenne number|0GJBJ19NXR]] that is prime.

## Remarks

The first Mersenne primes are $3$, $7$, $31$ and $127$, for $n = 2, 3, 5, 7$.
If $2^n - 1$ is prime, then [[$n$ is prime|07MMWAXMYD]]. The converse fails: $2^{11} - 1 = 2047 = 23 \cdot 89$.
The [[Lucas–Lehmer test|061BTYA0RN]] decides whether $2^p - 1$ is prime.
It is not known whether there are infinitely many Mersenne primes.

## Explanations

### L1 human

A Mersenne prime is a prime that is one less than a power of $2$, such as $31 = 2^5 - 1$.
The even [[perfect numbers|0SB7Q8SYSF]] come exactly from these primes, by the [[Euclid–Euler theorem|0WETJCKNRG]].
</document>

<document source="https://math2.tamnd.com/node/06GYGS41JK.md" tag="06GYGS41JK" kind="definition">
# Logical equivalence

## Definition

Let $\varphi$ and $\psi$ be formulas of [[propositional logic|09K9MWD2QQ]].
The formulas $\varphi$ and $\psi$ are logically equivalent when they have the same truth value for each assignment of truth values to their variables.
This is written $\varphi \equiv \psi$.

## Remarks

The formulas $\varphi$ and $\psi$ are logically equivalent exactly when $\varphi \leftrightarrow \psi$ is a [[tautology|0D1AHYPM8N]].
For example, $p \to q \equiv \neg p \lor q$.
[[De Morgan's laws|09PKJKA6J0]] give two more examples.

## Explanations

### L1 human

Logically equivalent formulas say the same thing in different words.
You can check an equivalence with a [[truth table|05RSK4Q657]]: the two columns must agree in every row.
Then you can replace one formula by the other in a proof.
</document>

<document source="https://math2.tamnd.com/node/06GZ956657.md" tag="06GZ956657" kind="concept">
# Banach space

## Summary

A Banach space is a normed vector space that is complete for the metric of its norm.
Banach spaces are the main setting of functional analysis.
</document>

<document source="https://math2.tamnd.com/node/06HC6HK0Z8.md" tag="06HC6HK0Z8" kind="definition">
# Prime ideal

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An [[ideal|0MYSA6MYNA]] $P$ of $R$ is prime if it has these two properties.

1. $P \neq R$.
2. For all $a, b \in R$, if $a b \in P$, then $a \in P$ or $b \in P$.

## Remarks

In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\mathbb{Z}$ is also prime.
The zero ideal of $R$ is prime if and only if $R$ is an [[integral domain|09S3PFBEPM]].
An ideal $P$ is prime if and only if [[$R/P$ is an integral domain|0AVXBEC05W]].
Every [[maximal ideal|09J6KGT65G]] is prime.

## Explanations

### L1 human

A prime number $p$ has this property: if $p$ divides $a b$, then $p$ divides $a$ or $b$.
A prime ideal has the same property, with "divides" replaced by "contains".
The ideal $(6)$ is not prime: it contains $2 \cdot 3$ but neither $2$ nor $3$.
</document>

<document source="https://math2.tamnd.com/node/06HE9J7DXF.md" tag="06HE9J7DXF" kind="example">
# Uniform metric on bounded functions

## Example

Let $S$ be a nonempty set.
Let $B(S)$ be the set of bounded functions $f \colon S \to \mathbb{R}$, that is, the functions for which there is a real $M$ with $|f(s)| \le M$ for all $s \in S$.
For $f, g \in B(S)$, let
$$d_\infty(f, g) = \sup_{s \in S} |f(s) - g(s)|.$$
Then $d_\infty$ is a [[metric|063W5JMY85]] on $B(S)$, the uniform metric.

## Remarks

The [[supremum|0BTMQ04556]] is finite, because $|f(s) - g(s)| \le |f(s)| + |g(s)|$ and both functions are bounded.
A sequence $(f_k)$ converges to $f$ in this metric if and only if $f_k \to f$ uniformly on $S$.
For $S = \{1, \ldots, n\}$, it is the [[sup metric on R^n|0VTAXQEWQD]].
It is $d_\infty(f, g) = \|f - g\|_\infty$ for the [[sup norm|0PARCKE8WP]].

## Explanations

### L1 human

Two functions are close in the uniform metric when their graphs are close at every point at the same time.
For example, on $[0, 1]$ the distance between $f(x) = x$ and $g(x) = x^2$ is $1/4$, the largest gap, at $x = 1/2$.
</document>

<document source="https://math2.tamnd.com/node/06HMCTAQH3.md" tag="06HMCTAQH3" kind="concept">
# Linear map of a matrix

## Summary

The linear map of a matrix in two bases is the linear map whose matrix in these bases is the given matrix.
</document>

<document source="https://math2.tamnd.com/node/06HN47VPTH.md" tag="06HN47VPTH" kind="statement">
# Dimension of a direct product

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then the [[product|0EX2HPXPJ6]] $V \times W$ is finite-dimensional, and
$$\dim(V \times W) = \dim V + \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

Let $(v_i)$ be a basis of $V$ and let $(w_j)$ be a basis of $W$.
Then the vectors $(v_i, 0)$ and $(0, w_j)$ together form a basis of $V \times W$.
For example, $\dim(\R^2 \times \R^3) = 5$.
By induction, $\dim(V_1 \times \cdots \times V_k) = \dim V_1 + \cdots + \dim V_k$.

## Explanations

### L1 human

A vector of $V \times W$ is a pair, so we need the directions of $V$ and the directions of $W$.
The dimensions add up.
The plane is the product of two lines, and $1 + 1 = 2$.
</document>

<document source="https://math2.tamnd.com/node/06HTJNCMPA.md" tag="06HTJNCMPA" kind="statement">
# Grassmann's formula

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be finite-dimensional [[linear subspaces|00YC1D652K]] of $V$.
Then
$$\dim(U + W) + \dim(U \cap W) = \dim U + \dim W,$$
where $U + W$ is the [[sum of the subspaces|0JMTY8MF54]] and $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

The proof extends a basis of $U \cap W$ to a basis of $U$ and to a basis of $W$.
Together, these vectors form a basis of $U + W$.
Another proof applies the [[rank–nullity theorem|0W4FSW1TVB]] to the map $U \times W \to V$, $(u, w) \mapsto u - w$.
Two planes through the origin in $\R^3$ with $U \ne W$ have $\dim(U + W) = 3$, so $\dim(U \cap W) = 1$.

## Explanations

### L1 human

The formula counts dimensions like a count of elements in a union of two sets.
We add the two dimensions, but then the common part is counted twice, so we subtract it once.
Two different planes through the origin in space always share at least a line.
</document>

<document source="https://math2.tamnd.com/node/06HYY6DK6X.md" tag="06HYY6DK6X" kind="definition">
# Affine combination

## Definition

Let $K$ be a field and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$.
Let $p_1, \dots, p_k$ be points of $A$, and let $w_1, \dots, w_k$ be elements of $K$ with $w_1 + \dots + w_k = 1$.
The affine combination of the points $p_i$ with the weights $w_i$ is the point
$$\sum_{i=1}^{k} w_i p_i = o + \sum_{i=1}^{k} w_i (p_i - o),$$
where $o$ is a point of $A$. The result does not depend on the choice of $o$, because the weights add up to 1.

## Remarks

In a vector space with $o = 0$, the affine combination is the linear combination $w_1 p_1 + \dots + w_k p_k$.
The [[midpoint|0S706JBZBG]] of $p$ and $q$ is the affine combination with the weights $\frac{1}{2}$ and $\frac{1}{2}$.
Affine maps keep affine combinations: $f\left(\sum_i w_i p_i\right) = \sum_i w_i f(p_i)$.
An affine combination with all weights $w_i \geq 0$, over the real numbers, is a convex combination.

## Explanations

### L1 human

You cannot add two points of the plane in a sensible way without an origin. But a weighted average of points, with weights that sum to 1, gives the same point for each choice of origin. For example, $\frac{1}{3}A + \frac{1}{3}B + \frac{1}{3}C$ is the centroid of the triangle $ABC$.
</document>

<document source="https://math2.tamnd.com/node/06J4PXRG22.md" tag="06J4PXRG22" kind="statement">
# Spectral theorem

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Then $V$ has an [[orthonormal basis|0S9GGTR0EX]] that consists of [[eigenvectors|0N6960W0BV]] of $T$.
The corresponding eigenvalues are real.

## Remarks

In this basis, the matrix of $T$ is a real diagonal matrix.
Equivalently, $V$ is the orthogonal direct sum of the eigenspaces of $T$.
Over $\C$, the same conclusion holds, with complex eigenvalues, for each [[normal operator|0C8BR8XRTR]].
The matrix form is the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].

## Explanations

### L1 human

A self-adjoint map is a pure stretch along perpendicular axes.
The axes are the eigenvectors, and the stretch factors are the real eigenvalues.
Every symmetric real matrix is diagonal in a suitable orthonormal basis.
</document>

<document source="https://math2.tamnd.com/node/06J5DE801P.md" tag="06J5DE801P" kind="concept">
# Multiplication of natural numbers

## Summary

Multiplication of natural numbers is the operation that sends two [[natural numbers|0MWKJ44V5A]] $a$ and $b$ to their product $a \cdot b$.
The product $a \cdot b$ is the sum of $b$ copies of $a$.
</document>

<document source="https://math2.tamnd.com/node/06J6P03MG7.md" tag="06J6P03MG7" kind="statement">
# Classification of finite simple groups

## Statement

Each finite [[simple group|0YMYMNQ0QZ]] is isomorphic to one of the following groups:

1. a [[cyclic group|0KVGH0J5B3]] $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$;
2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \geq 5$;
3. a group of Lie type;
4. one of $26$ sporadic groups.

## Remarks

The groups of Lie type form infinite families, for example the groups $\mathrm{PSL}_n(\mathbb{F}_q)$, with a few exceptions for small $n$ and $q$.
Some authors count the Tits group as a $27$th sporadic group, because it is not strictly of Lie type.
The largest sporadic group is the Monster group.
The proof consists of tens of thousands of pages in several hundred journal articles.

## History

Daniel Gorenstein announced in 1983 that the classification was complete, but this was premature.
The gap was the case of the quasithin groups.
Michael Aschbacher announced the completed proof in 2004, after Aschbacher and Stephen Smith published a proof of $1221$ pages for this case.

## Explanations

### L1 human

The theorem lists all the finite simple groups.
Most of them come in a few infinite families, and only $26$ are special cases.
Since each finite group is built from simple groups, this list is a map of all the building blocks.
</document>

<document source="https://math2.tamnd.com/node/06J8TEPJPY.md" tag="06J8TEPJPY" kind="concept">
# Real and imaginary parts

## Summary

The real part and the imaginary part of a complex number $z = a + b i$ are the real numbers $a$ and $b$.
They are the coordinates of $z$ in the [[complex plane|0CTE7A31XQ]].
</document>

<document source="https://math2.tamnd.com/node/06JMR2XWXW.md" tag="06JMR2XWXW" kind="definition">
# Regular curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval, and let $\gamma \colon I \to \mathbb{R}^n$ be a $C^1$ [[parametrised curve|0JF1XXGWB4]].
The curve $\gamma$ is regular if its [[tangent vector|0D98771SM0]] $\gamma'(t)$ is not zero for each $t \in I$.

## Remarks

The circle $t \mapsto (\cos t, \sin t)$ and each line $t \mapsto p + t v$ with $v \ne 0$ are regular.
The curve $t \mapsto (t^2, t^3)$ is not regular, since its tangent vector is zero at $t = 0$.
The curve $t \mapsto (t^3, t^3)$ is not regular at $t = 0$, but its image is a straight line. So regularity is a property of the parametrisation, not of the image.
A regular curve has an [[arc length parametrisation|00R1JDDV5X]].

## Explanations

### L1 human

A regular curve never stops.
The moving point always has non-zero speed, so the curve has no corners or cusps caused by the point coming to rest.
</document>

<document source="https://math2.tamnd.com/node/06JT7KDHXM.md" tag="06JT7KDHXM" kind="definition">
# Homogeneous system of linear equations

## Definition

Let $F$ be a field.
A [[system of linear equations|0GT4WWGWH2]] over $F$ is *homogeneous* when each of its constant terms is $0$.
In matrix form, a homogeneous system is an equation $Ax = 0$, where $A$ is an $m \times n$ matrix over $F$ and $x = (x_1, \dots, x_n)^T$.

## Remarks

The tuple $(0, \dots, 0)$ is always a solution, the *trivial solution*.
The solution set of $Ax = 0$ is a [[linear subspace|00YC1D652K]] of $F^n$, the [[null space|0PHDYNR24D]] of $A$.
If a system $Ax = b$ has a solution $p$, then its solution set is $\{p + v : Av = 0\}$.
A homogeneous system with more unknowns than equations has a nontrivial solution.

## Explanations

### L1 human

A homogeneous system has zero on the right side of every equation.
Setting every unknown to zero always works, so the question is whether other solutions exist.
The sum of two solutions is again a solution, and so is any multiple of a solution.
</document>

<document source="https://math2.tamnd.com/node/06JYC06JHY.md" tag="06JYC06JHY" kind="definition">
# Branch cut

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $F$ be a [[multivalued function|03CPD1HGF3]] on $U$.
A branch cut for $F$ is a closed subset $K$ of $U$ such that $F$ has a continuous single-valued branch on the open set $U \setminus K$. Usually a branch cut is a curve or a union of curves.
A branch on $U \setminus K$ is a continuous function $f$ on $U \setminus K$ with $f(z) \in F(z)$ for each $z \in U \setminus K$.
The standard branch cut for the logarithm on $\mathbb{C} \setminus \{0\}$ is the ray $(-\infty, 0)$. The open set $\mathbb{C} \setminus (-\infty, 0]$ is the slit plane.

## Remarks

A branch cut usually joins branch points. A branch point of $F$ is a point $z_0$ with the following property. When $z$ goes once around a small circle with center $z_0$, the values of $F$ do not come back to their start.
For $\log z$ and $\sqrt z$, the branch points are $0$ and $\infty$, so a cut is a curve from $0$ to $\infty$.
The [[principal branches|0YKY4M3KRQ]] of the logarithm and of the square root use the slit plane. Each other ray from $0$ is also a valid branch cut and gives other [[branches|0SQ1F2BEDS]].
The slit plane is $\{z \in \mathbb{C} : \operatorname{Re} z > 0 \text{ or } \operatorname{Im} z \neq 0\}$.

## Explanations

### L1 human

You cannot choose the logarithm continuously on a full circle around $0$.
A branch cut is a line that you remove from the plane so that no path can go around $0$. On what is left, a continuous choice is possible.
Crossing the cut makes the chosen value jump, for the logarithm by $2\pi i$.
</document>

<document source="https://math2.tamnd.com/node/06JYPJ4WHH.md" tag="06JYPJ4WHH" kind="definition">
# Convergence of a sequence in a metric space

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $(x_n)_{n \ge 1}$ be a sequence of points of $X$, and let $x \in X$.
The sequence $(x_n)$ converges to $x$ when for each real $\varepsilon > 0$ there is an integer $N$ such that $d(x_n, x) < \varepsilon$ for all $n \ge N$.
Then $x$ is a limit of $(x_n)$, with the notation $x_n \to x$ or $\lim_{n \to \infty} x_n = x$.

## Remarks

Equivalently, $x_n \to x$ if and only if $d(x_n, x) \to 0$ in $\mathbb{R}$.
A sequence in a metric space has at most one limit. This follows from the triangle inequality, and also from the fact that [[limits are unique in Hausdorff spaces|0D8T984XEE]].
For $X = \mathbb{R}$ with the usual metric, this is the usual limit of a real sequence.

## Explanations

### L1 human

A sequence converges to $x$ when its terms come as close to $x$ as you want and stay close.
For each tolerance $\varepsilon$, only finitely many terms are farther than $\varepsilon$ from $x$.
</document>

<document source="https://math2.tamnd.com/node/06JZGMYTYD.md" tag="06JZGMYTYD" kind="statement">
# Power series of the exponential function

## Statement

For every real $x$,
$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the exponential function. Its partial sums converge to $e^x$ uniformly on each bounded interval.

## Remarks

By the [[derivative rule|0EX9GFJ0B3]], every derivative of $\exp$ at $0$ equals $\exp(0) = 1$. So the Maclaurin coefficients are $1/n!$.
With the series as the definition of $\exp$, the first claim holds by definition. Uniform convergence follows from [[the result on power series|0BWXV7GADW]].
The partial sums give good approximations near $0$. For example, $1 + x + x^2/2$ approximates $e^x$ with error $O(x^3)$.

## Explanations

### L1 human

The exponential function equals the sum of its Taylor polynomials at $0$.
Adding more terms gives better approximations of $e^x$, for every $x$.
</document>

<document source="https://math2.tamnd.com/node/06K7SARZKR.md" tag="06K7SARZKR" kind="statement">
# The minimal polynomial divides every polynomial vanishing at the element

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]], let $a \in L$ be algebraic over $K$, and let $m_a$ be its [[minimal polynomial|0GK1WB9R98]].
For every $f \in K[x]$ with $f(a) = 0$, the polynomial $m_a$ [[divides|04GNK7K4VP]] $f$ in $K[x]$.

## Remarks

The proof: [[polynomial division|0MES3KS0ZC]] gives $f = q m_a + r$ with $\deg r < \deg m_a$. Then $r(a) = 0$, so $r = 0$ by the minimality of $\deg m_a$.
Equivalently, the kernel of the evaluation map $K[x] \to L$, $f \mapsto f(a)$, is the ideal $(m_a)$.
If $a$ is a root of $f$ and $f$ is monic and irreducible over $K$, then $f = m_a$.

## Explanations

### L1 human

Every polynomial equation that the element satisfies comes from the minimal polynomial.
It is a multiple of the minimal polynomial.
</document>

<document source="https://math2.tamnd.com/node/06KWGHDN2J.md" tag="06KWGHDN2J" kind="concept">
# Directional derivative

## Summary

The directional derivative of a function $f$ at a point $a$ in the direction of a vector $v$ is a rate of change. It is the rate of change of $f$ when its input moves from $a$ along $v$.
It is the derivative at $t = 0$ of the function $t \mapsto f(a + t v)$ of one variable.
The [[partial derivatives|09EGC6F8PT]] are the directional derivatives along the standard basis vectors.
</document>

<document source="https://math2.tamnd.com/node/06MG208MGR.md" tag="06MG208MGR" kind="concept">
# Inflection point

## Summary

An inflection point of a function is a point where the function changes from convex to concave, or from concave to convex. For twice differentiable functions, the second derivative vanishes there. The node [[inflection point|0Y0WS5BCJ7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/06MJC8YDYR.md" tag="06MJC8YDYR" kind="concept">
# Solution of a linear recurrence

## Summary

A solution of a linear recurrence is a sequence that satisfies the recurrence for all indices. The solutions form a module, and they are determined by their first terms. The node [[solution of a linear recurrence|0ZV2YKKZ58]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/06MRNP4BNT.md" tag="06MRNP4BNT" kind="statement">
# Every spanning set contains a basis

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $S \subseteq V$ be a [[spanning set|088RYKAWK9]] of $V$.
Then there is a subset $B \subseteq S$ that is a [[basis|0KCQ9Y3QKA]] of $V$.

## Remarks

For finite $S$, remove from $S$, one at a time, a vector that is a linear combination of the other vectors.
When no such vector remains, the set is linearly independent and still spans $V$.
In general, take a maximal linearly independent subset of $S$, by Zorn's lemma.
For example, $\{(1, 0), (0, 1), (1, 1)\}$ spans $\R^2$ and contains the basis $\{(1, 0), (1, 1)\}$.

## Explanations

### L1 human

If a set of vectors reaches the whole space, then some part of it is a basis.
We throw away vectors that the others can already build.
When nothing redundant remains, the remaining vectors form a basis.
</document>

<document source="https://math2.tamnd.com/node/06N6JAZF44.md" tag="06N6JAZF44" kind="concept">
# Fermat prime

## Summary

A Fermat prime is a prime number of the form $2^{2^n} + 1$.
The [[definition of a Fermat prime|0N9E4G2P97]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/06PVXMY0TH.md" tag="06PVXMY0TH" kind="statement">
# Cauchy criterion for series

## Statement

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges if and only if the following holds for each $\varepsilon > 0$.
There is a natural number $N$ such that
$$\left| \sum_{k=m+1}^{n} a_k \right| < \varepsilon \quad \text{for all natural numbers } n > m \ge N.$$

## Remarks

The sum in the condition is $s_n - s_m$, a difference of [[partial sums|0NB643K60Q]]. So the condition says that $(s_n)$ is a Cauchy sequence, and the result is the [[Cauchy criterion for sequences|0ZA4AX4MW3]].
With $n = m + 1$ it gives the [[term test|0VVY4H92G2]].
It proves that [[absolutely convergent series converge|0Y123X48GA]].

## Explanations

### L1 human

A series converges exactly when its tails become small.
Each block of consecutive terms far out in the series must have a small sum.
</document>

<document source="https://math2.tamnd.com/node/06Q1EFVH0Q.md" tag="06Q1EFVH0Q" kind="statement">
# Dimension of the annihilator

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a subspace of $V$.
Then
$$\dim W + \dim W^0 = \dim V,$$
where $W^0 \subseteq V^*$ is the [[annihilator|0ASHNYF7AG]] of $W$.

## Remarks

A proof extends a basis $b_1, \dots, b_k$ of $W$ to a basis $b_1, \dots, b_n$ of $V$.
Then the dual basis forms $b_{k+1}^*, \dots, b_n^*$ are a basis of $W^0$.
Another proof uses the isomorphism $W^0 \cong (V / W)^*$.
With the double dual isomorphism, it gives $(W^0)^0 = W$ in finite dimensions.

## Explanations

### L1 human

The more directions a subspace has, the fewer independent linear forms vanish on it.
In $\R^3$, a line is killed by a plane of forms, and a plane is killed by a line of forms.
The two dimensions always add up to the dimension of the space.
</document>

<document source="https://math2.tamnd.com/node/06QCTRFBM0.md" tag="06QCTRFBM0" kind="statement">
# cosh^2 - sinh^2 = 1

## Statement

For every real $x$,
$$\cosh^2 x - \sinh^2 x = 1.$$

## Remarks

The left side factors as $(\cosh x - \sinh x)(\cosh x + \sinh x) = e^{-x} e^{x} = 1$.
So the point $(\cosh t, \sinh t)$ lies on the right branch of the hyperbola $x^2 - y^2 = 1$.
Dividing by $\cosh^2 x$ gives $1 - \tanh^2 x = \operatorname{sech}^2 x$. Mathlib states the identity as `Real.cosh_sq_sub_sinh_sq`.

## Explanations

### L1 human

This is the hyperbolic version of $\cos^2 x + \sin^2 x = 1$, with a minus sign.
It follows directly from the definitions with $e^x$ and $e^{-x}$.
</document>

<document source="https://math2.tamnd.com/node/06QYRTX9KQ.md" tag="06QYRTX9KQ" kind="concept">
# P-group

## Summary

For a prime number $p$, a $p$-group is a group in which the order of each element is a power of $p$.
A finite group is a $p$-group if and only if its order is a power of $p$.
</document>

<document source="https://math2.tamnd.com/node/06RGT4V6Y6.md" tag="06RGT4V6Y6" kind="definition">
# Commutator of two elements

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g, h \in G$.
The commutator of $g$ and $h$ is the element $[g, h] = g^{-1} h^{-1} g h$ of $G$.

## Remarks

The equation $gh = hg [g, h]$ holds, so $[g, h] = e$ if and only if $gh = hg$.
Some authors use the convention $[g, h] = g h g^{-1} h^{-1}$ instead.
The inverse of $[g, h]$ is $[h, g]$.

## Explanations

### L1 human

The commutator measures how far two elements are from commuting.
It is the correction factor that you need when you swap $g$ and $h$ in a product.
In an abelian group, every commutator is the identity.
</document>

<document source="https://math2.tamnd.com/node/06RH4XAGRY.md" tag="06RH4XAGRY" kind="statement">
# Units of the Gaussian integers

## Statement

The [[units|019NFGFPR5]] of the [[Gaussian integers|0K5W69AC0J]] $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.

## Remarks

The proof uses the [[norm|03AN1V4ZFJ]]: if $u v = 1$, then $N(u) N(v) = 1$, so $N(u) = 1$. The only solutions of $a^2 + b^2 = 1$ in integers are $(\pm 1, 0)$ and $(0, \pm 1)$.
Conversely, an element of norm $1$ is a unit, because $(a + b i)(a - b i) = 1$.
The group of units is cyclic of order $4$, generated by $i$.

## Explanations

### L1 human

A unit must have norm $1$, so it lies on the unit circle.
The only points with integer coordinates on the unit circle are $1$, $i$, $-1$ and $-i$.
</document>

<document source="https://math2.tamnd.com/node/06RJWZCG95.md" tag="06RJWZCG95" kind="statement">
# Number of invertible residues modulo n

## Statement

Let $n$ be a positive integer.
The number of invertible elements of the ring $\mathbb{Z}/n\mathbb{Z}$ of the [[integers modulo n|01N3AYAA6N]] is $\varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

By the [[existence of inverses modulo n|0M67FWZWPQ]], the class $\overline{a}$ is invertible if and only if $\gcd(a, n) = 1$. The classes of $1, \dots, n$ that are coprime to $n$ give $\varphi(n)$ classes.
The invertible elements form a group under multiplication, the group of units $(\mathbb{Z}/n\mathbb{Z})^\times$. It has order $\varphi(n)$.
For example, $(\mathbb{Z}/8\mathbb{Z})^\times = \{ \overline{1}, \overline{3}, \overline{5}, \overline{7} \}$ has $\varphi(8) = 4$ elements.

## Explanations

### L1 human

Modulo $n$, you can divide by exactly $\varphi(n)$ of the $n$ residues.
For $n = 9$, these are the residues of $1, 2, 4, 5, 7, 8$, and $\varphi(9) = 6$.
</document>

<document source="https://math2.tamnd.com/node/06RW4RKWH7.md" tag="06RW4RKWH7" kind="concept">
# Completion of a metric space

## Summary

The completion of a metric space is a complete metric space that contains an isometric copy of the space as a dense subset.
It is unique up to an isometric isomorphism that fixes the copy.
</document>

<document source="https://math2.tamnd.com/node/06S4WGSEQA.md" tag="06S4WGSEQA" kind="definition">
# Disjunctive normal form

## Definition

A literal is a propositional variable $p$ or its negation $\neg p$.
A formula of propositional logic is in disjunctive normal form if it is a disjunction of one or more conjunctions of one or more literals.

## Remarks

Example: $(p \wedge \neg q) \vee (q \wedge r) \vee \neg p$ is in disjunctive normal form.
Every formula is logically equivalent to a formula in disjunctive normal form. For a formula that is not a contradiction, take one conjunction for each row of the truth table where it is true.
A consequence: the connectives $\wedge$, $\vee$ and $\neg$ are functionally complete.

## Explanations

### L1 human

A formula in disjunctive normal form is an or of ands of simple conditions.
It lists all the cases in which the formula is true.
</document>

<document source="https://math2.tamnd.com/node/06SCB3R6BN.md" tag="06SCB3R6BN" kind="definition">
# Graph homomorphism

## Definition

Let $G$ and $H$ be [[graphs|0JD5C82QMB]] with vertex sets $V(G)$ and $V(H)$.
A graph homomorphism from $G$ to $H$ is a map $f : V(G) \to V(H)$ such that $f(u) \sim f(v)$ in $H$ whenever $u \sim v$ in $G$.

## Remarks

A homomorphism maps edges to edges. It may map non-adjacent vertices to adjacent ones, and it may map several vertices to one vertex.
A homomorphism from $G$ to the complete graph $K_k$ is the same as a proper [[colouring|0874138YP6]] of $G$ with $k$ colours.
A bijective homomorphism whose inverse is also a homomorphism is a [[graph isomorphism|00CT0PTH74]].

## Explanations

### L1 human

A graph homomorphism is a map of vertices that sends each edge to an edge.
For example, mapping the vertices of a square alternately to the two ends of one edge is a homomorphism from $C_4$ to $K_2$.
</document>

<document source="https://math2.tamnd.com/node/06SESV7GQY.md" tag="06SESV7GQY" kind="statement">
# Continuous functions are Riemann integrable

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then $f$ is Riemann integrable on $[a, b]$.

## Remarks

By the [[Heine–Cantor theorem|0W0TZDRF9C]], $f$ is uniformly continuous. Choose $\delta$ for $\varepsilon/(b - a)$. For a partition of mesh less than $\delta$, $U(f, P) - L(f, P) < \varepsilon$. Then [[Riemann's criterion|017DH8WXTE]] applies.
More generally, a bounded function with finitely many discontinuities is Riemann integrable.

## Explanations

### L1 human

On short pieces of the interval, a continuous function changes very little.
So the upper and lower sums are close for fine partitions.
</document>

<document source="https://math2.tamnd.com/node/06SKCHDQT5.md" tag="06SKCHDQT5" kind="definition">
# Continuously differentiable function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is continuously differentiable, or of class $C^1$, if $f$ is [[differentiable|0A61B60B5R]] on $I$ and $f'$ is [[continuous|0P27W9RW0G]] on $I$.
More generally, for $k \ge 1$, $f$ is of class $C^k$ if its [[higher derivative|084NGSTWM7]] $f^{(k)}$ exists and is continuous on $I$.
The class $C^0$ consists of the continuous functions.

## Remarks

Each $C^{k+1}$ function is $C^k$.
A differentiable function need not be $C^1$. See [[x^2 sin(1/x)|0WDAKKN4NC]].
The function $x |x|$ is $C^1$ but not twice differentiable at $0$.
Mathlib defines the class as `ContDiff ℝ k f`.

## Explanations

### L1 human

A $C^1$ function has a slope at each point, and the slope changes without jumps.
The class $C^k$ asks the same of the first $k$ derivatives.
</document>

<document source="https://math2.tamnd.com/node/06T0HS14PP.md" tag="06T0HS14PP" kind="definition">
# Tangent function

## Definition

The tangent function is defined by
$$\tan x = \frac{\sin x}{\cos x}$$
for all real $x$ with $\cos x \ne 0$, that is, for $x \ne \frac{\pi}{2} + k \pi$ with $k \in \mathbb{Z}$.

## Remarks

The function is odd and periodic with period $\pi$.
On $(-\pi/2, \pi/2)$ it is strictly increasing, and it maps this interval onto $\mathbb{R}$.
The related functions are $\cot x = \cos x / \sin x$, $\sec x = 1 / \cos x$ and $\csc x = 1 / \sin x$. Mathlib defines `Real.tan` with the value $0$ where $\cos x = 0$.

## Explanations

### L1 human

The tangent of an angle is the slope of the line through the origin at that angle.
It is undefined where this line is vertical.
</document>

<document source="https://math2.tamnd.com/node/06TK51263Q.md" tag="06TK51263Q" kind="concept">
# Sum of divisors

## Summary

The sum of divisors $\sigma(n)$ of a positive integer $n$ is the sum of all its positive divisors.
The [[definition of the sum of divisors|009QCP5T3Z]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/06WHP8SE98.md" tag="06WHP8SE98" kind="concept">
# Rayleigh quotient

## Summary

The Rayleigh quotient of a Hermitian matrix $A$ and a nonzero vector $x$ is the number $x^* A x / x^* x$.
It lies between the least and the greatest eigenvalue of $A$.
</document>

<document source="https://math2.tamnd.com/node/06WHRARZKY.md" tag="06WHRARZKY" kind="statement">
# Existence of Laurent expansions on an annulus

## Statement

Let $c \in \mathbb{C}$, let $0 \le r < R \le \infty$, and let $A = \{z : r < |z - c| < R\}$ be an [[annulus|0AJ0YVK743]].
Let $f : A \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $A$.
Then there are unique complex numbers $(a_n)_{n \in \mathbb{Z}}$ such that the [[Laurent series|0X3XD2ATB0]] $\sum_{n \in \mathbb{Z}} a_n (z - c)^n$ converges to $f(z)$ for each $z \in A$.
For each $\rho$ with $r < \rho < R$ and each $n \in \mathbb{Z}$,
$$a_n = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} \frac{f(z)}{(z - c)^{n+1}} \, dz.$$
The series converges absolutely on $A$ and uniformly on each closed annulus $\rho_1 \le |z - c| \le \rho_2$ with $r < \rho_1 \le \rho_2 < R$.

## Remarks

Uniqueness means: each Laurent series that converges to $f$ on $A$ has these coefficients. So we can find them by any method, for example with known power series.
For example, on $0 < |z| < 1$, $\frac{1}{z(1 - z)} = \frac{1}{z} + 1 + z + z^2 + \cdots$. On $|z| > 1$, the same function is $-\frac{1}{z^2} - \frac{1}{z^3} - \cdots$.
If $f$ is holomorphic on the whole disc $|z - c| < R$, then $a_n = 0$ for $n < 0$, and the Laurent series is the Taylor series.

## Explanations

### L1 human

On a ring around $c$, each holomorphic function is a sum of powers of $z - c$, with negative and positive exponents.
The coefficients come from circle integrals, as for Taylor series. The same function can have different Laurent series on different rings.
</document>

<document source="https://math2.tamnd.com/node/06WRV8HFDY.md" tag="06WRV8HFDY" kind="statement">
# Converse of Wilson's theorem

## Statement

Let $n$ be an integer with $n > 1$.
If $(n - 1)! \equiv -1 \pmod n$, then $n$ is a [[prime number|0SN0MAEKYX]].

## Remarks

With [[Wilson's theorem|03KGGKTB2E]], this gives: an integer $n > 1$ is prime if and only if $(n - 1)! \equiv -1 \pmod n$.
The proof: if $n = a b$ with $1 < a < n$, then $a$ divides $(n - 1)!$. A congruence $(n - 1)! \equiv -1 \pmod n$ would give $a \mid 1$, which is false.
In fact $(n - 1)! \equiv 0 \pmod n$ for each composite $n > 4$, and $3! \equiv 2 \pmod 4$.
This primality test is not practical, because $(n - 1)!$ needs about $n$ multiplications.

## Explanations

### L1 human

For a composite number $n$, a factor of $n$ appears in the product $1 \cdot 2 \cdots (n - 1)$.
So the product cannot leave the remainder $n - 1$. For example, $5! = 120 \equiv 0 \pmod 6$.
</document>

<document source="https://math2.tamnd.com/node/06X5PF6YFP.md" tag="06X5PF6YFP" kind="problem">
# Buffon's needle problem

## Problem

A floor has parallel lines at distance $t > 0$ apart. A needle of length $l$ with $0 < l \le t$ falls at random on the floor.
"At random" means this.
The distance $x$ of the centre of the needle from the nearest line is uniform on $[0, t/2]$.
The acute angle $\theta$ between the needle and the lines is uniform on $[0, \pi/2]$.
The variables $x$ and $\theta$ are independent.
Find the probability that the needle crosses a line.

## Remarks

The answer is $\dfrac{2l}{\pi t}$. The needle crosses a line exactly when $x \le \frac{l}{2} \sin \theta$, and the integral of this region gives the answer.
For a needle with $l = t$, the probability is $2/\pi \approx 0.637$.
The answer contains $\pi$, so repeated throws give a random estimate of $\pi$.
For a long needle with $l > t$, the answer is a different formula.

## History

Georges-Louis Leclerc, Comte de Buffon, posed the problem in the 18th century. It was one of the first problems of geometric probability.

## Explanations

### L1 human

The needle crosses a line when it lands close to a line and at a steep angle.
The chance is the area of the "good" pairs (distance, angle) divided by the area of all pairs.
A sine function appears in this area. That is how $\pi$ comes into the answer.
</document>

<document source="https://math2.tamnd.com/node/06XF7BCFFJ.md" tag="06XF7BCFFJ" kind="definition">
# Simple continued fraction

## Definition

A simple continued fraction is a [[continued fraction|0BK6JBK58W]] whose partial numerators are all $1$.
Its first term $a_0$ is an integer, and its later terms $a_1, a_2, \dots$ are positive integers:
$$[a_0; a_1, a_2, \dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$$
The sequence of terms can be finite or infinite.

## Remarks

The $a_n$ are called the partial quotients.
For example, $[2; 3, 4] = 2 + \frac{1}{3 + 1/4} = \frac{30}{13}$.
An infinite simple continued fraction always converges, by the [[convergence of the convergents|0E70M92FE3]].
Each real number has a [[continued fraction expansion|0SJ7Z6VKWK]] of this form.

## Explanations

### L1 human

A simple continued fraction is a list of integers $[a_0; a_1, a_2, \dots]$ that encodes a number.
To read it, start at the end, take a reciprocal, add the next term, and repeat.
</document>

<document source="https://math2.tamnd.com/node/06XY4X9430.md" tag="06XY4X9430" kind="definition">
# Decimal expansion

## Definition

Let $x$ be a real number with $x \ge 0$.
A decimal expansion of $x$ is a natural number $a_0$ and a sequence $(d_k)_{k \ge 1}$ of digits $d_k \in \{0, 1, \dots, 9\}$ with
$$x = a_0 + \sum_{k=1}^{\infty} \frac{d_k}{10^k}.$$
The notation is $x = a_0.d_1 d_2 d_3 \dots$, with $a_0$ in base $10$.

## Remarks

The series always converges, because its partial sums are increasing and bounded above by $a_0 + 1$.
A number can have two expansions. This happens exactly when one expansion ends in infinitely many $0$ and the other in infinitely many $9$, as for [[0.999... and 1|0NN5JEMTVE]].
For a negative number $x$, one writes $-$ followed by an expansion of $-x$.

## Explanations

### L1 human

The digits after the point say how many tenths, hundredths, thousandths, and so on, the number has.
For example, $3.14$ means $3 + 1/10 + 4/100$.
For a number such as $\pi$, the digits go on forever.
</document>

<document source="https://math2.tamnd.com/node/06YMXJMXYS.md" tag="06YMXJMXYS" kind="definition">
# Unique factorisation domain

## Definition

A unique factorisation domain is an [[integral domain|09S3PFBEPM]] $R$ with these two properties.

1. Every nonzero element $a \in R$ that is not a unit is a product $a = p_1 p_2 \cdots p_n$ of [[irreducible elements|052SDEPZH7]], with $n \ge 1$.
2. The factorization is unique up to order and units. If $p_1 \cdots p_n = q_1 \cdots q_m$ with all $p_i, q_j$ irreducible, then $n = m$. Also, there is a permutation $\sigma$ of $\{1, \dots, n\}$ such that $p_i$ and $q_{\sigma(i)}$ are [[associates|053E86NWVK]] for each $i$.

## Remarks

The integers are a unique factorisation domain: this is the fundamental theorem of arithmetic.
Every [[principal ideal domain is a unique factorisation domain|055HT85T3S]], and so is every [[polynomial ring over a unique factorisation domain|0D6KH8Y59N]].
The ring $\mathbb{Z}[\sqrt{-5}]$ is [[not a unique factorisation domain|0BSJYCG7NM]].
In a unique factorisation domain, [[irreducible elements are prime|0QRCFZKXH8]].

## Explanations

### L1 human

In the integers, $60 = 2 \cdot 2 \cdot 3 \cdot 5$, and every other factorization into primes uses the same primes, up to order and signs.
A unique factorisation domain is a ring where this works for every nonzero nonunit.
Units like $-1$ do not count as a different factorization.
</document>

<document source="https://math2.tamnd.com/node/06YSJVR356.md" tag="06YSJVR356" kind="definition">
# Geometric multiplicity

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $\lambda$ be an eigenvalue of $T$.
The *geometric multiplicity* of $\lambda$ is $\dim E_\lambda$, the [[dimension|03GSZ8RTZJ]] of the [[eigenspace|0G0RFPG8HR]] $E_\lambda = \ker(T - \lambda \id_V)$.

## Remarks

It is the maximal number of linearly independent eigenvectors for $\lambda$.
It satisfies $1 \le \dim E_\lambda \le m$, where $m$ is the [[algebraic multiplicity|0FY42N02DV]] of $\lambda$.
For $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the eigenvalue $1$ has geometric multiplicity $1$ and algebraic multiplicity $2$.
A matrix is [[diagonalisable|07SQCNV233]] if and only if $p_A$ splits and the two multiplicities agree for each eigenvalue.

## Explanations

### L1 human

The geometric multiplicity counts the independent directions that the map stretches by $\lambda$.
It can be smaller than the algebraic multiplicity.
When it is smaller, the matrix cannot be diagonalised.
</document>

<document source="https://math2.tamnd.com/node/06ZARTQTQE.md" tag="06ZARTQTQE" kind="statement">
# Continuous image of a closed interval is a closed interval

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then $f([a, b]) = [m, M]$, where $m$ and $M$ are the minimum and the maximum of $f$ on $[a, b]$.

## Remarks

The [[extreme value theorem|0WS1A8E2P9]] gives $m$ and $M$ as values of $f$. The [[intermediate value theorem|00KAFBR32E]] gives all values in between.
The image can be a single point, when $f$ is constant.
Mathlib states it as `ContinuousOn.image_Icc`.

## Explanations

### L1 human

A continuous function maps a closed bounded interval onto a closed bounded interval.
The graph reaches a lowest and a highest point, and it takes every height in between.
</document>

<document source="https://math2.tamnd.com/node/06ZGPYTWK8.md" tag="06ZGPYTWK8" kind="concept">
# Contour integral

## Summary

A contour integral is the integral of a complex function along a path in the complex plane, see the [[definition|00FWNP9TXY]].
Contour integration is the method of computing integrals with the theorems of complex analysis about such integrals.
</document>

<document source="https://math2.tamnd.com/node/070PFSYJJR.md" tag="070PFSYJJR" kind="concept">
# Argument of a complex number

## Summary

The argument of a nonzero complex number $z$ is the angle from the positive real axis to $z$ in the [[complex plane|0CTE7A31XQ]].
It is determined up to a multiple of $2\pi$.
The usual choice in $(-\pi, \pi]$ is the [[principal value|0SQSQP748R]].
</document>

<document source="https://math2.tamnd.com/node/070R982G4B.md" tag="070R982G4B" kind="concept">
# Finite set

## Summary

A finite set is a set with a natural number of elements.
</document>

<document source="https://math2.tamnd.com/node/070TXS9AXM.md" tag="070TXS9AXM" kind="statement">
# Geometric formula for the dot product

## Statement

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\mathbb{R}^n$.
Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, let $\|u\|$ and $\|v\|$ be their [[lengths|0XKBKGTCPC]], and let $\theta$ be the [[angle between them|0ZEHZY85E6]].
Then
$$u \cdot v = \|u\| \, \|v\| \cos \theta.$$

## Remarks

When the angle is defined by its cosine, the formula follows from the definition. The content is that this angle is the angle of elementary geometry. In the plane, write $u = \|u\| (\cos \alpha, \sin \alpha)$ and $v = \|v\| (\cos \beta, \sin \beta)$. Then $u \cdot v = \|u\| \, \|v\| \cos(\alpha - \beta)$ by the addition formula for the cosine.
The formula also follows from the law of cosines in the triangle with the vertices $0$, $u$ and $v$.
If $u = 0$ or $v = 0$, then both sides are $0$ for each value of $\theta$.

## Explanations

### L1 human

The dot product is large and positive when two vectors point in nearly the same direction. It is zero when they are perpendicular, and negative when they point in nearly opposite directions. For unit vectors $u$ and $v$, the dot product is just the cosine of the angle between them.
</document>

<document source="https://math2.tamnd.com/node/0717XXPWM6.md" tag="0717XXPWM6" kind="definition">
# Linear isometry

## Definition

Let $V$ and $W$ be [[normed vector spaces|03J5RAKB30]] over $\R$ or $\C$.
A *linear isometry* from $V$ to $W$ is a [[linear map|0NRYHJPQG2]] $T \colon V \to W$ with
$$\norm{T(v)} = \norm{v} \qquad \text{for all } v \in V.$$

## Remarks

A linear isometry is injective, because $T(v) = 0$ implies $\norm{v} = 0$.
It keeps distances: $\norm{T(u) - T(v)} = \norm{u - v}$.
Between inner product spaces, it keeps inner products, see [[the next statement|0R7VXTYHYZ]].
Between spaces of the same finite dimension, a linear isometry is bijective.

## Explanations

### L1 human

A linear isometry is a linear map that keeps all lengths.
Rotations and reflections of the plane are examples.
A map that keeps lengths cannot squeeze anything to zero.
</document>

<document source="https://math2.tamnd.com/node/071DDFVG4J.md" tag="071DDFVG4J" kind="definition">
# Abelian group

## Definition

An abelian group is a [[group|G5S1M9]] $(G, \ast)$ whose operation is [[commutative|099E72KTF8]].
Thus $a \ast b = b \ast a$ for all $a, b \in G$.

## Remarks

The integers, the rationals and the real numbers with addition are abelian groups.
The operation of an abelian group is often written $+$, with identity $0$ and inverse $-a$.
The group of invertible $2 \times 2$ real matrices under multiplication is not abelian.

## Explanations

### L1 human

In an abelian group, the order of two elements in a product does not matter.
Addition of numbers is the model: $a + b = b + a$.
The name honours Niels Henrik Abel.
</document>

<document source="https://math2.tamnd.com/node/072D5G4SK6.md" tag="072D5G4SK6" kind="statement">
# Congruence is compatible with multiplication

## Statement

Let $n$ be a positive integer, and let $a$, $b$, $c$ and $d$ be integers.
If $a \equiv b \pmod n$ and $c \equiv d \pmod n$, then $a c \equiv b d \pmod n$.

## Remarks

The proof: $a c - b d = a (c - d) + d (a - b)$, and $n$ divides both terms.
With [[compatibility with addition|00FWBN7F0Y]], this result shows that the multiplication of [[integers modulo n|01N3AYAA6N]] is well defined.
By induction, it gives [[compatibility with powers|0J5VGJYYC3]].

## Explanations

### L1 human

To find the remainder of a product, you can multiply the remainders and reduce again.
For example, the last digit of $27 \cdot 43$ is the last digit of $7 \cdot 3 = 21$, that is, $1$. This is the case $n = 10$.
</document>

<document source="https://math2.tamnd.com/node/072G59HW59.md" tag="072G59HW59" kind="concept">
# Maximal element

## Summary

A maximal element of a partially ordered set is an element with no strictly larger element.
</document>

<document source="https://math2.tamnd.com/node/072J68M6HX.md" tag="072J68M6HX" kind="concept">
# Comparison of topologies

## Summary

Two topologies on one set can be compared by inclusion. The finer topology has more open sets and the coarser topology has fewer open sets.
</document>

<document source="https://math2.tamnd.com/node/072W67KFYK.md" tag="072W67KFYK" kind="definition">
# Formal power series

## Definition

Let $R$ be a commutative ring.
A formal power series in one variable $X$ over $R$ is a sequence $(a_n)_{n \ge 0}$ of elements of $R$, written $\sum_{n \ge 0} a_n X^n$.
The set of these series is written $R[\![X]\!]$.

Addition is termwise: $\sum a_n X^n + \sum b_n X^n = \sum (a_n + b_n) X^n$.
Multiplication is the Cauchy product: $\left( \sum a_n X^n \right) \left( \sum b_n X^n \right) = \sum_n \left( \sum_{i+j=n} a_i b_j \right) X^n$.
With these operations $R[\![X]\!]$ is a commutative ring, and it contains the polynomial ring $R[X]$.

## Remarks

The word formal means that no convergence is required. A series is a sequence of coefficients, not a function.
A series $\sum a_n X^n$ is a unit in $R[\![X]\!]$ if and only if $a_0$ is a unit in $R$.
The [[coefficient|0M98XYVX6S]] of $X^n$ in a series $f$ is written $[X^n] f$.

## Explanations

### L1 human

A formal power series is like a polynomial that may go on forever.
You add and multiply it like a polynomial, and you never substitute a number for $X$, so the question of convergence does not arise.
</document>

<document source="https://math2.tamnd.com/node/0734RSVF2V.md" tag="0734RSVF2V" kind="statement">
# Limit of a product of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.
If $a_n \to a$ and $b_n \to b$ as $n \to \infty$, then $a_n b_n \to a b$ as $n \to \infty$.

## Remarks

The proof uses the identity $a_n b_n - a b = a_n (b_n - b) + b (a_n - a)$. The sequence $(a_n)$ is bounded, because [[convergent sequences are bounded|0S4JF619V7]].
With a constant sequence $a_n = c$, it gives $c\, b_n \to c\, b$.
By induction, it gives $a_n^k \to a^k$ for each natural number $k$.

## Explanations

### L1 human

We may multiply the limits to get the limit of the product.
For example, $(1 + 1/n)(2 - 1/n) \to 1 \cdot 2 = 2$.
</document>

<document source="https://math2.tamnd.com/node/0736AXH2ME.md" tag="0736AXH2ME" kind="statement">
# Cyclic groups of the same order are isomorphic

## Statement

Let $G$ and $H$ be [[cyclic groups|0KVGH0J5B3]] of the same [[order|0B85S2V2DR]], finite or infinite.
Then $G$ and $H$ are isomorphic: there is a bijection $\varphi \colon G \to H$ with $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in G$.

## Remarks

If $g$ generates $G$ and $h$ generates $H$, an isomorphism is $\varphi(g^k) = h^k$ for $k \in \mathbb{Z}$.
This map is well defined because $g^a = g^b$ exactly when $h^a = h^b$, by [[the order of a cyclic subgroup|031M8HX7DA]].
Thus each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and each infinite cyclic group is isomorphic to $\mathbb{Z}$.

## Explanations

### L1 human

Up to the names of the elements, there is only one cyclic group of each size.
To match two of them, send the generator of one to the generator of the other, and send each power to the same power.
For example, the rotations of a hexagon and the integers modulo $6$ are the same group in two costumes.
</document>

<document source="https://math2.tamnd.com/node/073C80EC63.md" tag="073C80EC63" kind="statement">
# Third isomorphism theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]] and let $I \subseteq J$ be two-sided ideals of $R$.
Then $J/I = \{a + I : a \in J\}$ is a two-sided ideal of the [[quotient ring|0VFW4Y69BT]] $R/I$.
The map $(R/I)/(J/I) \to R/J$, $(a + I) + J/I \mapsto a + J$, is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

The proof applies the [[first isomorphism theorem|0ZMYX7GMH0]] to the map $R/I \to R/J$, $a + I \mapsto a + J$. It is surjective, and its kernel is $J/I$.
For example, $6\mathbb{Z} \subseteq 2\mathbb{Z}$, and $(\mathbb{Z}/6\mathbb{Z})/(2\mathbb{Z}/6\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$.

## Explanations

### L1 human

You can make a quotient in two steps or in one step.
Dividing first by $I$ and then by the image of $J$ gives the same ring as dividing directly by $J$.
</document>

<document source="https://math2.tamnd.com/node/073E4XC016.md" tag="073E4XC016" kind="statement">
# Associativity of matrix multiplication

## Statement

Let $R$ be a ring and let $m, n, p, q \ge 0$ be integers.
Let $A$ be an $m \times n$ matrix, $B$ an $n \times p$ matrix and $C$ a $p \times q$ matrix over $R$.
Then $(AB)C = A(BC)$, where the products are [[matrix products|032Q9FB8MT]].

## Remarks

Both sides have entry $(i, l)$ equal to $\sum_{j=1}^{n} \sum_{k=1}^{p} a_{ij} b_{jk} c_{kl}$.
Because of this property, the product $ABC$ needs no parentheses.
The property also follows from the associativity of the composition of linear maps.

## Explanations

### L1 human

When we multiply three matrices, we may multiply the first two first, or the last two first.
The result is the same in both cases.
This is like $(2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)$ for numbers.
</document>

<document source="https://math2.tamnd.com/node/073JKP99K2.md" tag="073JKP99K2" kind="concept">
# Tangent plane and normal vector of a surface

## Summary

The tangent plane of a [[surface|0ZG9QF40HE]] at a point is the plane that best approximates the surface near the point.
A normal vector is a nonzero vector perpendicular to the tangent plane.
</document>

<document source="https://math2.tamnd.com/node/073MWACF1K.md" tag="073MWACF1K" kind="concept">
# Order of a pole

## Summary

The order of a pole $c$ of $f$ is the integer $m \ge 1$ with the following property. The function $(z - c)^m f(z)$ extends to a holomorphic function near $c$ that is not $0$ at $c$, see the [[definition|0YK27TJ4WF]].
</document>

<document source="https://math2.tamnd.com/node/073Q8Z94KG.md" tag="073Q8Z94KG" kind="statement">
# Kolmogorov's zero–one law

## Statement

Let $X_1, X_2, \dots$ be [[independent|0XCE070V8B]] random variables on a probability space $(\Omega, \mathcal{F}, P)$.
Let $\mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \dots)$ be the tail σ-algebra, where $\sigma(X_n, X_{n+1}, \dots)$ is the smallest [[σ-algebra|04WSW8T3SY]] that makes $X_m$ measurable for all $m \ge n$.
Then $P(A) \in \{0, 1\}$ for each $A \in \mathcal{T}$.

## Remarks

An event is in $\mathcal{T}$ if changing finitely many of the $X_n$ cannot change whether it occurs.
Examples of tail events: "the series $\sum_n X_n$ converges" and "$\limsup_n (X_1 + \dots + X_n)/n \le c$" for a real $c$.
Proof idea: $\mathcal{T}$ is [[independent|09T8ZHST1V]] of $\sigma(X_1, \dots, X_n)$ for each $n$, hence of $\sigma(X_1, X_2, \dots)$, which contains $\mathcal{T}$. So each $A \in \mathcal{T}$ is independent of itself, and $P(A) = P(A)^2$.
The theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.

## History

The theorem is named after Andrey Kolmogorov.

## Explanations

### L1 human

Some events depend only on the "far end" of an infinite sequence of independent trials, such as "the averages converge".
For such an event, the probability cannot be $1/2$ or any value strictly between $0$ and $1$: it is always $0$ or $1$.
</document>

<document source="https://math2.tamnd.com/node/074482Q7CD.md" tag="074482Q7CD" kind="statement">
# Correspondence between equivalence relations and partitions

## Statement

Let $A$ be a set.
Send each [[equivalence relation|0SGTMBEWRF]] $\sim$ on $A$ to its set of [[equivalence classes|082RMRSY26]] $A/{\sim}$.
This map is a [[bijection|0SNQEF7Z53]] from the set of equivalence relations on $A$ to the set of [[partitions|09CCE86A6D]] of $A$.
Its inverse sends a partition $P$ to the relation $\sim_P$ with $a \sim_P b$ when $a$ and $b$ are in the same block of $P$.

## Remarks

The relation $\sim_P$ is an equivalence relation, and its classes are the blocks of $P$.
A smaller equivalence relation, one that relates fewer pairs, gives a finer partition.
The number of partitions of a set with $n$ elements is the Bell number $B_n$. For example, $B_3 = 5$.

## Explanations

### L1 human

An equivalence relation and a partition are two descriptions of the same thing.
From the relation you get the boxes, and from the boxes you get the relation "is in the same box as".
</document>

<document source="https://math2.tamnd.com/node/0744CN4MDA.md" tag="0744CN4MDA" kind="definition">
# Symmetric bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *symmetric* when
$$B(u, v) = B(v, u) \qquad \text{for all } u, v \in V.$$

## Remarks

In each basis, $B$ is symmetric if and only if its matrix is a [[symmetric matrix|0VR671PHE5]].
The dot product on $\R^n$ is symmetric.
If the characteristic of $F$ is not $2$, symmetric bilinear forms correspond one to one with [[quadratic forms|0RBQS1JJCR]].

## Explanations

### L1 human

A symmetric bilinear form gives the same value for $(u, v)$ and for $(v, u)$.
Inner products on real spaces are the main examples.
Their matrices are symmetric.
</document>

<document source="https://math2.tamnd.com/node/074E9A8ZJK.md" tag="074E9A8ZJK" kind="concept">
# Separable equation

## Summary

A separable equation is a first-order equation $y' = g(x) h(y)$ whose right side is a product of a function of $x$ and a function of $y$.
One solves it by separation of variables: divide by $h(y)$ and integrate both sides. See the [[definition|0V4E623THY]].
</document>

<document source="https://math2.tamnd.com/node/074P7C85XH.md" tag="074P7C85XH" kind="concept">
# Gaussian integer

## Summary

A Gaussian integer is a complex number $a + bi$ with integers $a$ and $b$. The Gaussian integers form the ring $\mathbb{Z}[i]$.
The [[definition of a Gaussian integer|0B268GVX95]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0753Y2X5K6.md" tag="0753Y2X5K6" kind="statement">
# One-sided inverse of a square matrix is two-sided

## Statement

Let $F$ be a field and let $A$ and $B$ be $n \times n$ matrices over $F$.
If $BA = I_n$, then $AB = I_n$, so $A$ is [[invertible|0EB9N0S33C]] and $A^{-1} = B$.
The same holds with the roles of $A$ and $B$ exchanged.

## Remarks

The statement also holds over each commutative ring, by the [[determinant|00KDZRW4C0]]: $BA = I_n$ gives $\det B \det A = 1$.
For non-square matrices it fails: a $2 \times 1$ matrix can have a left inverse but never a right inverse.
A proof over a field uses [[rank–nullity|0W4FSW1TVB]]: $BA = I_n$ makes $x \mapsto Ax$ injective, hence surjective.

## Explanations

### L1 human

For square matrices, we only need to check one of $AB = I$ and $BA = I$.
The other one then holds automatically.
This saves half of the work when we check an inverse.
</document>

<document source="https://math2.tamnd.com/node/075C629XTH.md" tag="075C629XTH" kind="definition">
# Simply connected domain

## Definition

Let $X \subseteq \mathbb{R}^n$.
The set $X$ is simply connected if $X$ is path-connected and each continuous [[closed curve|0D032NSZKY]] $c \colon [0, 1] \to X$ has this property. There are a point $p \in X$ and a continuous map $H \colon [0, 1] \times [0, 1] \to X$ such that
$$H(t, 0) = c(t), \qquad H(t, 1) = p, \qquad H(0, s) = H(1, s) \quad \text{for all } s, t \in [0, 1].$$
An open simply connected set is a simply connected domain.

## Remarks

The map $H$ deforms the loop $c$ into the constant loop at $p$ through closed loops $t \mapsto H(t, s)$ in $X$.
Each convex set is simply connected. For example, $\mathbb{R}^n$ and each ball are simply connected.
The plane minus the origin is not simply connected, and neither is $\mathbb{R}^3$ minus the $z$-axis. But $\mathbb{R}^n$ minus a point is simply connected for $n \ge 3$.
An equivalent condition: $X$ is path-connected, and any two paths in $X$ with the same end points can be deformed into each other with the end points fixed.

## Explanations

### L1 human

A set is simply connected when every loop in it can be pulled tight to a point, like a rubber band, without leaving the set.
A disc has this property. An annulus does not, because a loop around the hole cannot shrink past the hole.
</document>

<document source="https://math2.tamnd.com/node/075PM9FH7A.md" tag="075PM9FH7A" kind="statement">
# The reals have the cardinality of the continuum

## Statement

The set $\mathbb{R}$ of real numbers has [[cardinality|0JTGRV4WT4]] $2^{\aleph_0}$:
$$|\mathbb{R}| = \mathfrak{c} = |\mathcal{P}(\mathbb{N})|.$$

## Remarks

One proof uses the [[Schröder–Bernstein theorem|0A9V6RKQG6]]. A real number gives a set of rationals, the rationals less than it, so $|\mathbb{R}| \le |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0}$. Binary sequences give an injection in the other direction, for example through decimal expansions with digits $0$ and $1$.
With [[Cantor's theorem|0WTEMGYV2C]], this gives another proof that $\mathbb{R}$ is [[uncountable|0518MM6BE1]].

## Explanations

### L1 human

The real numbers and the subsets of $\mathbb{N}$ have the same size.
Roughly, a real number is an infinite sequence of digits, and an infinite sequence of $0$ and $1$ is a subset of $\mathbb{N}$.
</document>

<document source="https://math2.tamnd.com/node/075VBY4K4X.md" tag="075VBY4K4X" kind="definition">
# Initial value problem

## Definition

Let $\Omega$ be an open subset of $\mathbb{R} \times \mathbb{R}^m$, let $f : \Omega \to \mathbb{R}^m$ be a function, and let $(t_0, y_0) \in \Omega$.
The initial value problem for $f$ at $(t_0, y_0)$ is the system
$$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$$
A solution of the problem is a [[solution|04SW7WECS8]] $y : I \to \mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \in I$, such that $y(t_0) = y_0$.
The condition $y(t_0) = y_0$ is the initial condition.

## Remarks

For an equation of order $n$ in explicit form, the initial conditions are the values $y(t_0), y'(t_0), \dots, y^{(n-1)}(t_0)$. The [[reduction to a first-order system|0TH9CPW3VS]] changes this problem into the problem above.
The problem $y' = y$, $y(0) = 1$ has the unique solution $y(t) = e^t$.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique local solution when $f$ is continuous and Lipschitz in $y$. The [[Peano existence theorem|098YP831QY]] gives a solution when $f$ is only continuous.

## Explanations

### L1 human

A differential equation alone usually has many solutions.
The initial condition says where the solution starts, and this picks one solution.
Suppose that you know the speed of a car at each moment and its position at the start. Then you know its position at each moment.
</document>

<document source="https://math2.tamnd.com/node/075WR6W3Y3.md" tag="075WR6W3Y3" kind="example">
# Path graph

## Example

Let $n \ge 1$ be an integer.
The path graph $P_n$ is the [[graph|0JD5C82QMB]] with vertex set $\{0, 1, \dots, n - 1\}$ in which $i$ and $j$ are adjacent if and only if $|i - j| = 1$.

## Remarks

$P_n$ has $n$ vertices and $n - 1$ edges. It is a [[tree|0MJ0JS72DZ]], and for $n \ge 2$ it has exactly two leaves, $0$ and $n - 1$.
Some authors write $P_n$ for the path with $n$ edges, that is, with $n + 1$ vertices.

## Explanations

### L1 human

A path graph is a row of vertices, each joined to the next.
It looks like beads on a string.
</document>

<document source="https://math2.tamnd.com/node/0762WAE49P.md" tag="0762WAE49P" kind="statement">
# Uniqueness of the derivative

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, let $f \colon U \to W$, and let $x \in U$.
If the bounded linear maps $A$ and $B$ from $V$ to $W$ are both [[Fréchet derivatives|08EVNYZF42]] of $f$ at $x$, then $A = B$.

## Remarks

The proof uses that $U$ is open. For a unit vector $v$ and small $t > 0$, the point $x + t v$ lies in $U$, and $\|t A v - t B v\| = o(t)$. Divide by $t$ and let $t \to 0$ to get $A v = B v$.
So it is correct to speak of the derivative $Df(x)$.
On a set that is not open, for example a single point, a derivative within the set need not be unique.

## Explanations

### L1 human

Two different linear maps differ by a definite amount in some direction, at the scale of the step.
If both were good approximations, their difference would be tiny compared with the step in every direction. This forces the difference to be $0$.
</document>

<document source="https://math2.tamnd.com/node/076JGP6R3T.md" tag="076JGP6R3T" kind="definition">
# Matrix multiplication

## Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $B = (b_{jk})$ be an $n \times p$ matrix over $R$.
The *product* $AB$ is the $m \times p$ matrix with entries
$$(AB)_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$$
for $1 \le i \le m$ and $1 \le k \le p$.

## Remarks

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$.
Column $k$ of $AB$ is the [[matrix–vector product|0PDVQEP0ZC]] of $A$ with column $k$ of $B$.
The product is [[associative|073E4XC016]] and [[distributive|0RXD6EZYBK]], but it is [[not commutative|0P506SPZP2]].
If $A$ and $B$ are the matrices of two linear maps, then $AB$ is the matrix of their composition.

## Explanations

### L1 human

To find the entry in row $i$ and column $k$ of $AB$, go along row $i$ of $A$ and down column $k$ of $B$.
Multiply the pairs of numbers and add the products.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 11 \\ 4 \end{pmatrix}$.
</document>

<document source="https://math2.tamnd.com/node/0771AJP4BX.md" tag="0771AJP4BX" kind="definition">
# Even and odd permutations

## Definition

Let $n \geq 2$ and let $\sigma \in S_n$.
The permutation $\sigma$ is even when $\operatorname{sgn}(\sigma) = 1$, and odd when $\operatorname{sgn}(\sigma) = -1$.

## Remarks

The identity is even and each [[transposition|0FWVRHBRWM]] is odd.
A product of two even or two odd permutations is even, and a product of an even and an odd permutation is odd.
For $n \geq 2$, $S_n$ has $n!/2$ even and $n!/2$ odd permutations.
The odd permutations do not form a subgroup, but they form a coset of $A_n$.

## Explanations

### L1 human

Each permutation is even or odd, like each integer.
Even and odd combine by the same rules as for integers.
Exactly half of the permutations are even.
</document>

<document source="https://math2.tamnd.com/node/077GN9HDR6.md" tag="077GN9HDR6" kind="statement">
# Independent random variables are uncorrelated

## Statement

Let $X$ and $Y$ be [[independent|0E9502NT92]] real random variables with $E[X^2] < \infty$ and $E[Y^2] < \infty$.
Then $\operatorname{Cov}(X, Y) = 0$, so $X$ and $Y$ are uncorrelated.

## Remarks

Proof: by the [[product rule for independent variables|05AW1G0WE1]], $E[XY] = E[X] \, E[Y]$. So $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y] = 0$.
The converse is false. The node [[uncorrelated random variables that are not independent|0C6J8BQXVR]] gives a counterexample.
For a pair $(X, Y)$ with a joint multivariate normal law, uncorrelated does imply independent.

## Explanations

### L1 human

If two variables do not influence each other at all, they have no linear relation either.
But "no linear relation" is a much weaker property than independence.
</document>

<document source="https://math2.tamnd.com/node/0783S9D0ZH.md" tag="0783S9D0ZH" kind="statement">
# Determinant of the inverse

## Statement

Let $K$ be a commutative ring and let $A$ be an [[invertible|0EB9N0S33C]] $n \times n$ matrix over $K$.
Then $\det A$ is a unit of $K$, and
$$\det(A^{-1}) = (\det A)^{-1},$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The proof applies the [[product rule|04CPJ6AF00]] to $A A^{-1} = I_n$: $\det A \cdot \det(A^{-1}) = 1$.
Over a field, this says $\det(A^{-1}) = 1 / \det A$.

## Explanations

### L1 human

The inverse matrix undoes the matrix, so it must undo its scaling of volume.
If $A$ scales volume by $3$, then $A^{-1}$ scales volume by $1/3$.
So the determinant of the inverse is the inverse of the determinant.
</document>

<document source="https://math2.tamnd.com/node/079D68S3AW.md" tag="079D68S3AW" kind="definition">
# Orthogonal complement

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.
The *orthogonal complement* of $W$ is
$$W^\perp = \{v \in V : \langle v, w \rangle = 0 \text{ for all } w \in W\}.$$

## Remarks

$W^\perp$ is a subspace of $V$, and $W \cap W^\perp = 0$.
In $\R^3$ with the dot product, the orthogonal complement of a plane through $0$ is its normal line.
For a real $m \times n$ matrix $A$, the null space of $A$ is the orthogonal complement of the row space of $A$ in $\R^n$.
In finite dimensions, $V = W \oplus W^\perp$, see [[the orthogonal decomposition|0SP7P0G7KQ]].

## Explanations

### L1 human

The orthogonal complement collects all vectors that are perpendicular to a subspace.
The complement of a line in $\R^3$ is a plane, and the complement of a plane is a line.
A subspace and its complement together fill the space.
</document>

<document source="https://math2.tamnd.com/node/079EGSD32Y.md" tag="079EGSD32Y" kind="definition">
# Transcendental number

## Definition

A complex number is transcendental if it is not an [[algebraic number|0MB4TGMXG6]].
That is, it is not a root of any nonzero polynomial with rational coefficients.

## Remarks

The numbers $e$ and $\pi$ are transcendental, by the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]].
Each [[Liouville number|0J1R5NP36S]] is transcendental, by [[Liouville's theorem|0N3WFZEFPM]].
The algebraic numbers are countable, so almost all real numbers are transcendental.
Joseph Liouville first proved that transcendental numbers exist, in 1844.

## Explanations

### L1 human

A transcendental number solves no polynomial equation with integer coefficients.
It is usually hard to prove that a given number is transcendental, even though most numbers are.
</document>

<document source="https://math2.tamnd.com/node/07BNRFQ0PJ.md" tag="07BNRFQ0PJ" kind="definition">
# Autonomous system

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
The autonomous system with vector field $f$ is the system of [[ordinary differential equations|0P97V20RW7]]
$$x'(t) = f\left(x(t)\right).$$
A [[solution|04SW7WECS8]] is a differentiable function $\gamma : I \to U$ on an interval $I$ with $\gamma'(t) = f(\gamma(t))$ for each $t \in I$.
The set $U$ is the phase space, or state space, of the system.

## Remarks

The right side depends on the state $x$ only, not on the time $t$ directly.
A non-autonomous system $x' = f(t, x)$ becomes autonomous with the extra state variable $s$ and the extra equation $s' = 1$.
The equation $x' = k x$ of [[exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are autonomous. The equation $x' = t x$ is not.
Solutions of an autonomous system can be [[translated in time|0WA7ZR95V4]].

## Explanations

### L1 human

In an autonomous system, the law of motion does not change with time.
The velocity depends only on where the state is, not on when it is there.
</document>

<document source="https://math2.tamnd.com/node/07BSPZPSMF.md" tag="07BSPZPSMF" kind="concept">
# Nilpotent matrix

## Summary

A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$.
</document>

<document source="https://math2.tamnd.com/node/07BY47HG5S.md" tag="07BY47HG5S" kind="statement">
# Continuity of the inverse of a strictly monotone continuous function

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and [[strictly increasing|0TR9SR9FTB]].
Then $J = f(I)$ is an interval, and the inverse function $f^{-1} \colon J \to I$ is continuous and strictly increasing.

## Remarks

The set $J$ is an interval by the [[intermediate value theorem|00KAFBR32E]].
The inverse is strictly increasing and its image is the interval $I$. A monotone function whose image is an interval has no jumps, so $f^{-1}$ is continuous.
The same holds for strictly decreasing functions.
This gives the continuity of $n$-th roots, of the [[logarithm|0044CPHJBV]], and of the inverse trigonometric functions.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
A rising unbroken graph stays rising and unbroken after the reflection.
</document>

<document source="https://math2.tamnd.com/node/07C4A8Y18K.md" tag="07C4A8Y18K" kind="definition">
# Conjunctive normal form

## Definition

A formula of propositional logic is in conjunctive normal form if it is a conjunction of one or more disjunctions of one or more literals.
Each disjunction is a clause.

## Remarks

Example: $(p \vee \neg q) \wedge (q \vee r) \wedge \neg p$ is in conjunctive normal form.
Every formula is logically equivalent to a formula in conjunctive normal form. Negate the formula, put the negation in [[disjunctive normal form|06S4WGSEQA]], and use the [[De Morgan laws|0XYVDWGQX2]].
The satisfiability problem for formulas in conjunctive normal form with clauses of $3$ literals is NP-complete.

## Explanations

### L1 human

A formula in conjunctive normal form is an and of ors of simple conditions.
Each clause is a constraint, and the formula is true when all constraints hold.
</document>

<document source="https://math2.tamnd.com/node/07C6ECNJET.md" tag="07C6ECNJET" kind="concept">
# Group of units of a ring

## Summary

The group of units of a ring is the set of its units, with the multiplication of the ring.
The [[definition of the group of units|05THZKR9AN]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/07CHJ5YX28.md" tag="07CHJ5YX28" kind="definition">
# Convolution of probability distributions

## Definition

Let $\mu$ and $\nu$ be [[probability distributions|0QB913WM5E]] on $\mathbb{R}$.
Their convolution is the probability distribution $\mu * \nu$ on $\mathbb{R}$ with
$$(\mu * \nu)(B) = \int_{\mathbb{R}} \nu(B - x) \, \mu(dx)$$
for each Borel set $B \subseteq \mathbb{R}$, where $B - x = \{b - x : b \in B\}$.

## Remarks

If $X \sim \mu$ and $Y \sim \nu$ are [[independent|0E9502NT92]], then $X + Y \sim \mu * \nu$.
For distributions on the integers with mass functions $p$ and $q$, $\mu * \nu$ has the mass function $(p * q)(n) = \sum_{k} p(k) \, q(n - k)$.
If $\mu$ and $\nu$ have densities $f$ and $g$, then $\mu * \nu$ has the density $(f * g)(x) = \int f(y) \, g(x - y) \, dy$.
The convolution is commutative and associative. Its characteristic function is the product: $\varphi_{\mu * \nu} = \varphi_\mu \varphi_\nu$.

## Explanations

### L1 human

The convolution answers the question: "What is the law of $X + Y$, if $X$ and $Y$ are independent?"
For two fair dice, the convolution of the two uniform laws on $\{1, \dots, 6\}$ gives the familiar triangle-shaped law of the sum on $\{2, \dots, 12\}$.
</document>

<document source="https://math2.tamnd.com/node/07CJNE6ZAQ.md" tag="07CJNE6ZAQ" kind="example">
# Taxicab metric

## Example

Let $n \ge 1$ be an integer.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_1(x, y) = \sum_{i=1}^{n} |x_i - y_i|.$$
Then $d_1$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the taxicab metric.

## Remarks

The triangle inequality for $d_1$ follows from the [[triangle inequality|03HF5PMY5T]] for real numbers in each coordinate.
In $\mathbb{R}^2$, the distance from $(0, 0)$ to $(3, 4)$ is $7$, but the [[Euclidean distance|0E1MMCAY5B]] is $5$.
It is the case $p = 1$ of the [[p-metric|0TFDZN6AXP]].

## Explanations

### L1 human

A taxi in a city with a grid of streets cannot drive through the blocks.
It must go along the streets, so it travels the sum of the east-west and the north-south distances.
</document>

<document source="https://math2.tamnd.com/node/07CQR9BW3R.md" tag="07CQR9BW3R" kind="statement">
# Classification of finite abelian groups

## Statement

Let $G$ be a finite [[abelian group|082GHNPJ53]].
Then $G$ is [[isomorphic|08MG7H8JW5]] to a [[direct sum|00HF4RJ5QN]] $\mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$, where each $q_i$ is a power of a prime number.
The numbers $q_1, \ldots, q_t$ are unique up to their order.
Equivalently, $G \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$, and these $k_i$ are unique.

## Remarks

The two forms are equivalent because $\mathbb{Z}/m n\mathbb{Z} \cong \mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(m, n) = 1$, as in [[this statement|08FDFEF8D1]].
Each abelian group of order $8$ is isomorphic to exactly one of $\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.
Each abelian group of order $15$ is cyclic, because $15 = 3 \cdot 5$ with $\gcd(3, 5) = 1$.
The theorem is the case of rank $0$ of the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]].

## History

Leopold Kronecker proved the theorem in 1870 with a group-theoretic proof, but he did not state it in the language of groups.
His work extended an earlier result of Carl Friedrich Gauss from 1801 on quadratic forms.
Ferdinand Georg Frobenius and Ludwig Stickelberger stated and proved the theorem in the language of groups in 1878.

## Explanations

### L1 human

Each finite abelian group is built from cyclic groups.
You can choose the cyclic pieces to have prime power orders, or to have orders that divide each other.
In both forms, the pieces are unique, so this lists all finite abelian groups.
</document>

<document source="https://math2.tamnd.com/node/07CT9RDY7Q.md" tag="07CT9RDY7Q" kind="definition">
# Cycle

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A cycle in $G$ is a [[circuit|0PD57PSHNR]] $(w_0, w_1, \dots, w_k)$ whose vertices $w_1, \dots, w_k$ are pairwise distinct.

## Remarks

The only repeated vertex is $w_0 = w_k$. In a simple graph, each cycle has length $k \ge 3$.
The vertices and edges of a cycle of length $k$ form a subgraph isomorphic to the [[cycle graph|0QQ0F12962]] $C_k$.
A graph with no cycle is [[acyclic|0XNN9Z8E5A]].

## Explanations

### L1 human

A cycle is a round trip that visits each of its vertices once and comes back to the start.
A triangle is the shortest possible cycle in a simple graph.
</document>

<document source="https://math2.tamnd.com/node/07CZYE06ZJ.md" tag="07CZYE06ZJ" kind="statement">
# Lcm of coprime integers

## Statement

Let $a$ and $b$ be natural numbers that are [[coprime|0GFKP8F8AG]].
Then $\operatorname{lcm}(a, b) = a b$.

## Remarks

This follows from the [[product of the gcd and the lcm|0TDY6JZHKS]], because $\gcd(a, b) = 1$.
The converse is also true when $a$ and $b$ are positive: if $\operatorname{lcm}(a, b) = a b$, then $a$ and $b$ are coprime.
For example, $\operatorname{lcm}(8, 15) = 120$.

## Explanations

### L1 human

When $a$ and $b$ have no common factor, a multiple of both must contain all the factors of $a$ and all the factors of $b$.
No factor can serve both numbers at the same time, so the smallest common multiple is $a b$.
</document>

<document source="https://math2.tamnd.com/node/07DCKDXRBP.md" tag="07DCKDXRBP" kind="statement">
# Lipschitz maps are uniformly continuous

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a [[Lipschitz map|0TMZ21WKCN]].
Then $f$ is [[uniformly continuous|0FHSDCQACY]].

## Remarks

Proof: let $K > 0$ be a Lipschitz constant of $f$. For $\varepsilon > 0$, take $\delta = \varepsilon / K$. If $d(x, y) < \delta$, then $d(f(x), f(y)) \le K d(x, y) < \varepsilon$. If $f$ is Lipschitz with $K = 0$, then $f$ is constant.
The converse fails: $x \mapsto \sqrt{x}$ on $[0, 1]$ is uniformly continuous, but it is not Lipschitz.
In particular, each Lipschitz map is continuous.

## Explanations

### L1 human

For a Lipschitz map, you can give an explicit $\delta$ for each $\varepsilon$: divide $\varepsilon$ by the constant $K$.
This $\delta$ does not depend on the point, so the map is uniformly continuous.
</document>

<document source="https://math2.tamnd.com/node/07DFG5JYPT.md" tag="07DFG5JYPT" kind="concept">
# Incircle

## Summary

The incircle of a triangle is the largest circle inside the triangle. It touches each of the three sides at one point. Its centre is the incentre and its radius is the inradius.
</document>

<document source="https://math2.tamnd.com/node/07DJAZW32T.md" tag="07DJAZW32T" kind="concept">
# Integrating factor

## Summary

An integrating factor is a function by which one multiplies a differential equation so that it can be integrated directly.
For the linear equation $y' + p y = q$, the factor $e^{\int p}$ turns the left side into the derivative $\left(e^{\int p} y\right)'$. See the [[definition|01THH3Y3YP]].
</document>

<document source="https://math2.tamnd.com/node/07DVHJ8N0A.md" tag="07DVHJ8N0A" kind="definition">
# Bipartite graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$.
Then $G$ is bipartite if $V$ is the union of two disjoint sets $A$ and $B$ with this property.
Each edge of $G$ has one end in $A$ and one end in $B$.
The pair $(A, B)$ is a bipartition of $G$.

## Remarks

Equivalently, $G$ is bipartite if it has a proper [[colouring|0874138YP6]] with $2$ colours.
The [[complete bipartite graphs|0V0GC7BQAN]], the even [[cycle graphs|0QQ0F12962]] and all [[trees|0MJ0JS72DZ]] are bipartite.
A graph is bipartite if and only if it has no cycle of odd length. The node [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]] states this.

## Explanations

### L1 human

A graph is bipartite when its vertices split into two groups, and every edge goes between the groups.
Students and the courses they take form a bipartite graph.
</document>

<document source="https://math2.tamnd.com/node/07E5MB1CA2.md" tag="07E5MB1CA2" kind="statement">
# Cauchy–Hadamard theorem

## Statement

Let $\sum a_n (x - c)^n$ be a real power series with [[radius of convergence|0660QX1VPR]] $R$.
Then
$$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n},$$
with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

The proof applies the [[root test|05H4DDDYBZ]] to the series $\sum a_n (x - c)^n$. The [[limit superior|0P8BA7T7S5]] of $|a_n (x - c)^n|^{1/n}$ is $|x - c| \limsup |a_n|^{1/n}$.
For example, $\sum n x^n$ has $R = 1$, since $n^{1/n} \to 1$.
Mathlib states it as `FormalMultilinearSeries.radius_inv_eq_limsup`.

## Explanations

### L1 human

The coefficients $a_n$ grow or shrink roughly like $\rho^n$, where $\rho$ is the limit superior of $|a_n|^{1/n}$.
Then $a_n x^n$ behaves like $(\rho x)^n$, which is small exactly when $|x| < 1/\rho$.
</document>

<document source="https://math2.tamnd.com/node/07E6D8R4D6.md" tag="07E6D8R4D6" kind="statement">
# Disjoint cycles commute

## Statement

Let $\sigma$ and $\tau$ be cycles in the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(X)$ whose sets of moved points are disjoint.
Then $\sigma \tau = \tau \sigma$.

## Remarks

A point moved by $\sigma$ is fixed by $\tau$, and $\sigma$ maps it to another point that $\tau$ fixes. So both products send it to the same point.
The same argument works for the points moved by $\tau$, and the points fixed by both are fixed by both products.
Cycles that are not disjoint need not commute, for example $(1\ 2)(1\ 3) \neq (1\ 3)(1\ 2)$.

## Explanations

### L1 human

Two disjoint cycles move different points.
Each cycle does its work without touching the points of the other cycle.
So the order in which you do them does not matter.
</document>

<document source="https://math2.tamnd.com/node/07EGCNHAVW.md" tag="07EGCNHAVW" kind="definition">
# Limit point of a set

## Definition

Let $S$ be a subset of $\mathbb{R}$ and let $a$ be a real number.
The point $a$ is a limit point of $S$ if for each $\delta > 0$ there is $x \in S$ with $0 < |x - a| < \delta$.

## Remarks

A limit point of $S$ need not belong to $S$. For example, $0$ is a limit point of $\{1/n : n \ge 1\}$.
Each point of an interval with more than one point is a limit point of the interval. So are its endpoints.
Limit points of the domain are the points where the [[limit of a function|0TXTSEMHR5]] makes sense.
Mathlib uses the general form `AccPt`.

## Explanations

### L1 human

A limit point of $S$ is a point that we can approach by other points of $S$.
Points of $S$ lie as close to it as we like, not counting the point itself.
</document>

<document source="https://math2.tamnd.com/node/07EH4R3RVQ.md" tag="07EH4R3RVQ" kind="statement">
# Composition of surjections is surjective

## Statement

Let $f \colon A \to B$ and $g \colon B \to C$ be [[surjective functions|0XKH881BZ7]].
Then the [[composition|07HF9YCWPP]] $g \circ f \colon A \to C$ is surjective.

## Remarks

Proof: let $c \in C$. Since $g$ is surjective, $c = g(b)$ for some $b \in B$. Since $f$ is surjective, $b = f(a)$ for some $a \in A$. Then $c = (g \circ f)(a)$.
A partial converse holds: if $g \circ f$ is surjective, then $g$ is surjective. The function $f$ need not be surjective.
Together with [[the result for injections|01FPXMTBX1]], this shows that a composition of [[bijections|0SNQEF7Z53]] is a bijection.

## Explanations

### L1 human

To reach a point $c$ of $C$, first find a point of $B$ that $g$ sends to $c$.
Then find a point of $A$ that $f$ sends to that point of $B$.
</document>

<document source="https://math2.tamnd.com/node/07EJHQYR9A.md" tag="07EJHQYR9A" kind="statement">
# Grönwall's inequality

## Statement

Let $a < b$ and let $I$ be an [[interval|0Q1D85B8G6]] of the form $[a, b]$, $[a, b)$ or $[a, \infty)$. Let $I^\circ$ be the interior of $I$.
Differential form: let $\beta, u : I \to \mathbb{R}$ be continuous, let $u$ be differentiable on $I^\circ$, and assume $u'(t) \le \beta(t)\, u(t)$ for each $t \in I^\circ$.
Then
$$u(t) \le u(a) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$
Integral form: let $\beta, u : I \to \mathbb{R}$ be continuous with $\beta \ge 0$, let $\alpha : I \to \mathbb{R}$ be non-decreasing, and assume
$$u(t) \le \alpha(t) + \int_a^t \beta(s)\, u(s)\, ds \quad \text{for each } t \in I.$$
Then
$$u(t) \le \alpha(t) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$

## Remarks

The differential form needs no assumption on the signs of $\beta$ and $u$. Its proof shows that $u(t) \exp\left(-\int_a^t \beta\right)$ is non-increasing.
The integral form does not need $u$ to be differentiable.
For constant $\alpha = 0$, the integral form gives $u \le 0$. This is the step that proves [[uniqueness of solutions|025PKW8AMR]].
A variant with an additional constant term gives the [[Grönwall bound|0FRX8JD7QH]], and there is a [[discrete version|0NTM3J5MFE]].

## History

The inequality is named after Thomas Hakon Grönwall. He proved it in 1919, in the integral form with constant $\alpha$ and $\beta$.
Richard Bellman proved a slightly more general integral form in 1943.

## Explanations

### L1 human

If a quantity grows at most at a rate proportional to itself, it grows at most exponentially.
The inequality compares the quantity with the solution of the equation $v' = \beta v$.
This is the main tool to compare two solutions of a differential equation.
</document>

<document source="https://math2.tamnd.com/node/07EXVNF9VP.md" tag="07EXVNF9VP" kind="concept">
# Newton's method

## Summary

Newton's method finds approximate zeros of a differentiable function $f$.
From a starting point $x_0$, it computes
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.$$
Each new point is the zero of the [[tangent line|01R1HW3Y42]] to the graph at the old point.

## Remarks

Assume $f$ is $C^2$ near a zero $r$ with $f'(r) \ne 0$. Then the method converges to $r$ for $x_0$ close to $r$. The convergence is quadratic: $|x_{n+1} - r| \le C |x_n - r|^2$.
For $f(x) = x^2 - 2$, the step is $x_{n+1} = (x_n + 2/x_n)/2$. This is Heron's method for $\sqrt{2}$.
The method can fail. It stops when $f'(x_n) = 0$, and it can cycle or diverge from a poor starting point.

## Explanations

### L1 human

Replace the graph by its tangent line, and find where the line crosses the axis.
Repeat from there. Near a simple root, each step roughly doubles the number of correct digits.
</document>

<document source="https://math2.tamnd.com/node/07EZA80NEH.md" tag="07EZA80NEH" kind="concept">
# Signature of a quadratic form

## Summary

The signature of a real quadratic form records the numbers of positive, negative and zero coefficients in any diagonal form of it.
</document>

<document source="https://math2.tamnd.com/node/07GJV36896.md" tag="07GJV36896" kind="concept">
# Complex exponential

## Summary

The complex exponential is the function $z \mapsto e^z = \sum_{n \ge 0} z^n / n!$ on $\mathbb{C}$, see the [[definition|054VXHMCA3]].
It extends the real exponential function, it is entire, and it has period $2\pi i$.
</document>

<document source="https://math2.tamnd.com/node/07GNX8GCYP.md" tag="07GNX8GCYP" kind="concept">
# Polynomials in several variables

## Summary

The ring $R[x_1, \dots, x_n]$ is the ring of the polynomials in the variables $x_1, \dots, x_n$ with coefficients in a commutative ring $R$.
The [[definition of the multivariate polynomial ring|0CES91M79A]] gives the construction.
</document>

<document source="https://math2.tamnd.com/node/07GR3TNW4H.md" tag="07GR3TNW4H" kind="concept">
# Convex function on R^n

## Summary

Take a real-valued [[function of several real variables|0JPCHTXWG0]] on a convex set. The function is convex when its graph lies on or below each chord between two points of the graph.
Convex functions are important in optimisation because each local minimum of a convex function is a global minimum.
</document>

<document source="https://math2.tamnd.com/node/07HF9YCWPP.md" tag="07HF9YCWPP" kind="definition">
# Function composition

## Definition

Let $f \colon A \to B$ and $g \colon B \to C$ be [[functions|0XHDTE5YMZ]].
The composition of $g$ and $f$ is the function $g \circ f \colon A \to C$ with
$$(g \circ f)(a) = g(f(a))$$
for each $a \in A$.

## Remarks

Read $g \circ f$ as "$g$ after $f$": $f$ acts first.
In general $g \circ f \neq f \circ g$. For $f(x) = x + 1$ and $g(x) = 2x$ on $\mathbb{R}$, $(g \circ f)(x) = 2x + 2$ and $(f \circ g)(x) = 2x + 1$.
Composition is associative. See [[associativity of composition|0J6AKYZQT8]].

## Explanations

### L1 human

Composition chains two machines: the output of $f$ goes into $g$.
For $g \circ f$ to make sense, the values of $f$ must lie in the domain of $g$.
</document>

<document source="https://math2.tamnd.com/node/07HFRH2AWY.md" tag="07HFRH2AWY" kind="example">
# Circulant matrix

## Example

Let $F$ be a field and let $c_0, c_1, \dots, c_{n-1} \in F$.
The *circulant matrix* with first column $(c_0, c_1, \dots, c_{n-1})^T$ is the $n \times n$ matrix $C$ with entries $C_{ij} = c_{(i - j) \bmod n}$, for $0 \le i, j \le n - 1$.
Each column is the previous column shifted down by one place, with the last entry moved to the top.
For $n = 3$:
$$C = \begin{pmatrix} c_0 & c_2 & c_1 \\ c_1 & c_0 & c_2 \\ c_2 & c_1 & c_0 \end{pmatrix}.$$

## Remarks

Some authors define circulant matrices by their first row; the two conventions give transposed matrices.
Over $\C$, the vectors $(1, \omega^k, \omega^{2k}, \dots, \omega^{(n-1)k})$ with $\omega = e^{2\pi i/n}$ are eigenvectors of every $n \times n$ circulant matrix.
The product of two circulant matrices of the same order is circulant, and circulant matrices commute with each other.

## Explanations

### L1 human

In a circulant matrix, each column is the column before it, rotated down by one step.
All the information is in the first column.
The discrete Fourier transform diagonalises all of these matrices at the same time.
</document>

<document source="https://math2.tamnd.com/node/07HXA8KV6F.md" tag="07HXA8KV6F" kind="statement">
# Pépin's test

## Statement

Let $n$ be a positive integer and let $F_n = 2^{2^n} + 1$ be the [[Fermat number|0R03B54V4Z]].
Then $F_n$ is prime if and only if
$$3^{(F_n - 1)/2} \equiv -1 \pmod{F_n}.$$

## Remarks

For example, $F_1 = 5$ and $3^2 = 9 \equiv -1 \pmod 5$.
The power can be computed by $2^n - 1$ squarings with [[modular exponentiation|04DR5PM25N]].
The "only if" part follows from [[Euler's criterion|0M7DEDKSYB]] and quadratic reciprocity, since $3$ is a nonresidue modulo $F_n$.

## History

The test is named after Théophile Pépin.

## Explanations

### L1 human

Pépin's test decides whether a Fermat number is prime with one large modular power.
The Fermat numbers grow so fast that the test is practical only for small $n$.
</document>

<document source="https://math2.tamnd.com/node/07J5X2T1ZX.md" tag="07J5X2T1ZX" kind="statement">
# A function with positive derivative is strictly increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$.
If $f'(x) > 0$ for each interior point $x$ of $I$, then $f$ is [[strictly increasing|0TR9SR9FTB]] on $I$.

## Remarks

For $x < y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) > 0$.
The converse fails: $x^3$ is strictly increasing, but its derivative at $0$ is $0$.
Mathlib states it as `strictMonoOn_of_deriv_pos`.

## Explanations

### L1 human

If the graph has positive slope everywhere, it rises from left to right.
</document>

<document source="https://math2.tamnd.com/node/07J9ZJFECP.md" tag="07J9ZJFECP" kind="concept">
# Order on the integers

## Summary

The order on the integers is the relation $\le$ on $\mathbb{Z}$ that puts the integers in the sequence $\dots < -2 < -1 < 0 < 1 < 2 < \dots$.
It is a total order that agrees with addition and with multiplication by positive integers.
</document>

<document source="https://math2.tamnd.com/node/07JBSZE0V4.md" tag="07JBSZE0V4" kind="concept">
# Dual space

## Summary

The dual space of a vector space is the vector space of all linear forms on it.
</document>

<document source="https://math2.tamnd.com/node/07JH131FVP.md" tag="07JH131FVP" kind="concept">
# Linear first-order equation

## Summary

A linear first-order equation is an equation $y' + p(x) y = q(x)$ with given functions $p$ and $q$.
It is the simplest [[linear differential equation|0QQRP17Z7S]], and an [[integrating factor|07DJAZW32T]] solves it explicitly. See the [[definition|0Z3YKRQZ48]].
</document>

<document source="https://math2.tamnd.com/node/07KBQCG9KV.md" tag="07KBQCG9KV" kind="statement">
# i squared is minus one

## Statement

Let $i$ be the [[imaginary unit|0ESCX36MMC]] of $\mathbb{C}$.
Then $i^2 = -1$.

## Remarks

With the product of [[complex numbers|09FTAET5Z4]], $(0 + 1 i)(0 + 1 i) = (0 \cdot 0 - 1 \cdot 1) + (0 \cdot 1 + 1 \cdot 0) i = -1$.
Also $(-i)^2 = -1$. Thus $-1$ has the two square roots $i$ and $-i$ in $\mathbb{C}$.

## Explanations

### L1 human

This is the property that defines $i$.
All the rules for computation with complex numbers come from the usual rules of algebra together with $i^2 = -1$.
</document>

<document source="https://math2.tamnd.com/node/07KVB4JSC7.md" tag="07KVB4JSC7" kind="definition">
# Union

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The union of $A$ and $B$ is the set
$$A \cup B = \{x \mid x \in A \lor x \in B\}.$$
So an object $x$ is an [[element|0EZ758XE73]] of $A \cup B$ when $x \in A$ or $x \in B$, or both.

## Remarks

Example: $\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$.
Union is commutative and associative, and $A \cup \varnothing = A$.
The union of an arbitrary family of sets is in [[union of a family of sets|05GK32H72H]].

## Explanations

### L1 human

To form $A \cup B$, put the elements of $A$ and the elements of $B$ together in one set.
An element of both sets appears only one time.
</document>

<document source="https://math2.tamnd.com/node/07MMWAXMYD.md" tag="07MMWAXMYD" kind="statement">
# If 2^p − 1 is prime then p is prime

## Statement

Let $n$ be a natural number.
If $2^n - 1$ is a [[prime number|0SN0MAEKYX]], then $n$ is a prime number.

## Remarks

The proof: if $n = ab$ with $1 < a < n$, then $2^a - 1$ divides $2^n - 1$, because $x - 1$ divides $x^b - 1$ for $x = 2^a$. And $1 < 2^a - 1 < 2^n - 1$.
For example, $2^6 - 1 = 63$ is divisible by $2^2 - 1 = 3$ and by $2^3 - 1 = 7$.
The converse is false: $11$ is prime, but $2^{11} - 1 = 23 \cdot 89$.

## Explanations

### L1 human

To look for primes of the form $2^n - 1$, you only need to try prime exponents $n$.
A factorization of $n$ always gives a factorization of $2^n - 1$.
</document>

<document source="https://math2.tamnd.com/node/07N7W5C29X.md" tag="07N7W5C29X" kind="definition">
# Continuity at a point of a topological space

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be a function and let $a \in X$.
The function $f$ is continuous at $a$ when for each [[neighbourhood|09CJ00H3NW]] $V$ of $f(a)$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is a neighbourhood of $a$.

## Remarks

Equivalently, for each neighbourhood $V$ of $f(a)$ there is a neighbourhood $U$ of $a$ with $f(U) \subseteq V$.
A function is a [[continuous map|0SY9ECQ19Q]] if and only if it is continuous at each point of $X$.
For metric spaces, continuity at $a$ is the same as the [[epsilon-delta condition at a|0EKGKY5TGH]].

## Explanations

### L1 human

Continuity at $a$ means that you can keep $f(x)$ as close to $f(a)$ as you want, if you keep $x$ close enough to $a$.
Neighbourhoods replace the distances of the metric case.
</document>

<document source="https://math2.tamnd.com/node/07NKG38VEG.md" tag="07NKG38VEG" kind="statement">
# Fermat's Last Theorem for n = 3

## Statement

There are no positive integers $a$, $b$ and $c$ with $a^3 + b^3 = c^3$.

## Remarks

A sum of two positive cubes is never a cube.
Standard proofs use [[infinite descent|0EFYCFHYFG]] in the ring $\mathbb{Z}[\omega]$ of Eisenstein integers, where $\omega$ is a primitive cube root of unity.
This case is part of [[Fermat's Last Theorem|0Y5X3SDBCZ]].

## History

Fermat mentioned the case $n = 3$ in letters of 1636, 1640 and 1657.
Euler published a proof in 1770. Many other proofs followed, for example by Legendre.

## Explanations

### L1 human

You can never write a cube as a sum of two positive cubes.
For example, $6^3 = 216$, and $3^3 + 4^3 + 5^3 = 216$ needs three cubes, not two.
</document>

<document source="https://math2.tamnd.com/node/07NN4VNYWM.md" tag="07NN4VNYWM" kind="concept">
# Liouville function

## Summary

The Liouville function $\lambda(n)$ is $1$ or $-1$, by the parity of the number of prime factors of $n$ counted with multiplicity.
The [[definition of the Liouville function|08RQAPXF1T]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/07NSKDGS7S.md" tag="07NSKDGS7S" kind="statement">
# Menelaus's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$.
Let $\ell$ be a [[line|0MP9BCF6DP]] that does not pass through $A$, $B$ or $C$. Suppose that $\ell$ meets the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.
Then
$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1.$$
Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $-1$. Then $D$, $E$ and $F$ are collinear.

## Remarks

With unsigned lengths, the product is $1$. The converse of this weaker form is false in general.
The formula differs from the formula of [[Ceva's theorem|0KN93VDSXA]] only in the sign.
The line $\ell$ meets the inside of exactly two sides, or of no side. Thus one or three of the ratios are negative, and the product is negative.

## History

The oldest surviving statement is in the Spherics of Menelaus of Alexandria. There, he uses the plane version as a lemma for a version on the sphere.
Ptolemy used the theorem in the Almagest for problems of spherical astronomy.

## Explanations

### L1 human

A straight line that cuts across a triangle meets the three side lines in three points. These points cut the sides in certain ratios. Going round the triangle, the product of the three signed ratios is always $-1$.
</document>

<document source="https://math2.tamnd.com/node/07PB7HH1T9.md" tag="07PB7HH1T9" kind="concept">
# Picard operator

## Summary

The Picard operator of an initial value problem $x' = f(t, x)$, $x(t_0) = x_0$ maps a curve $\gamma$ to $t \mapsto x_0 + \int_{t_0}^{t} f(s, \gamma(s))\, ds$.
Its fixed points are the solutions of the problem. See the [[definition|00WAVB68F9]].
</document>

<document source="https://math2.tamnd.com/node/07PFYV65MB.md" tag="07PFYV65MB" kind="statement">
# Heine–Borel theorem

## Statement

Let $n \ge 1$ be an integer and give $\mathbb{R}^n$ the [[Euclidean metric|0E1MMCAY5B]].
A subset $K$ of $\mathbb{R}^n$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[closed|0G5AV9Q4RR]] and [[bounded|0P953MG05E]].

## Remarks

The direct part holds in each metric space: [[compact subsets of a metric space are bounded|0R7BTEWCJ2]], and they are closed.
The converse uses the completeness of $\mathbb{R}$. A bounded set lies in a cube $[-M, M]^n$, which is compact. A closed subset of a compact set is compact.
The converse fails in general metric spaces. In the space of bounded real sequences with the sup metric, the closed unit ball is closed and bounded but not compact.
Metric spaces in which closed bounded sets are compact are the [[proper metric spaces|0Z4V98VN5E]].

## History

Dirichlet used finite subcovers of a closed interval in his 1852 lectures, to prove that continuous functions on such intervals are uniformly continuous.
Émile Borel stated and proved a version for countable covers in 1895. Cousin, Lebesgue and Schoenflies later extended it to arbitrary covers.

## Explanations

### L1 human

In $\mathbb{R}^n$, you can recognize compact sets by two easy tests: they contain their limit points and they fit in a ball.
For example, the closed disc and the sphere are compact, but the open disc and the whole plane are not.
</document>

<document source="https://math2.tamnd.com/node/07PH4RX0X5.md" tag="07PH4RX0X5" kind="concept">
# Jordan block

## Summary

A Jordan block is a square matrix with one scalar $\lambda$ on the diagonal, $1$ on the diagonal just above it, and $0$ elsewhere.
</document>

<document source="https://math2.tamnd.com/node/07PNDWA7CE.md" tag="07PNDWA7CE" kind="definition">
# Generalised eigenspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
The *generalised eigenspace* of $T$ for $\lambda$ is
$$V_\lambda = \{v \in V : (T - \lambda \id_V)^k (v) = 0 \text{ for some } k \ge 1\}.$$

## Remarks

It is a $T$-invariant subspace that contains the eigenspace $E_\lambda$.
If $\dim V = n$ is finite, then $V_\lambda = \ker (T - \lambda \id_V)^n$.
Its dimension is the [[algebraic multiplicity|0FY42N02DV]] of $\lambda$, in finite dimensions.
Its nonzero elements are the [[generalised eigenvectors|01H6QKWGDH]] for $\lambda$.

## Explanations

### L1 human

The generalised eigenspace is a larger version of the eigenspace.
It contains the vectors that become zero after we apply $T - \lambda \id$ several times.
These spaces always fill the space over $\C$, even when the eigenspaces do not.
</document>

<document source="https://math2.tamnd.com/node/07PTPSZPME.md" tag="07PTPSZPME" kind="definition">
# Trace of a matrix

## Definition

Let $R$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The *trace* of $A$ is
$$\tr A = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \cdots + a_{nn}.$$

## Remarks

The trace of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $5$.
The trace is [[linear|0GJWQZCPZM]], it is [[invariant under transpose|09E1RCZ7YT]], and [[$\tr(AB) = \tr(BA)$|0Y4NF1DRKD]].
Similar matrices have the same trace, so the [[trace of a linear map|0KZDNGA561]] is well defined.
Over an algebraically closed field, the trace is the [[sum of the eigenvalues|0NMXDCNNKR]], counted with multiplicity.

## Explanations

### L1 human

The trace adds up the numbers on the main diagonal of a square matrix.
It is a single number that does not change when we change the basis.
For the identity matrix of order $n$, the trace is $n$.
</document>

<document source="https://math2.tamnd.com/node/07Q4TBTFVB.md" tag="07Q4TBTFVB" kind="statement">
# Möbius inversion formula

## Statement

Let $f$ and $g$ be arithmetic functions.
Then $g(n) = \sum_{d \mid n} f(d)$ for all positive integers $n$ if and only if
$$f(n) = \sum_{d \mid n} \mu(d) \, g\!\left(\frac{n}{d}\right)$$
for all positive integers $n$. Here $\mu$ is the [[Möbius function|0C8QENHYZW]].

## Remarks

In terms of the [[Dirichlet convolution|0KH1N66QWW]], the statement is: $g = f * \mathbf{1}$ if and only if $f = g * \mu$.
The proof uses [[$\mu * \mathbf{1} = \varepsilon$|0ASC73QEQT]] and the associativity of $*$.
For example, the [[sum of the totient over the divisors|09JXE6XMCC]] gives $\sum_{d \mid n} \varphi(d) = n$. So $\varphi(n) = \sum_{d \mid n} \mu(d) \, n/d$.

## History

August Ferdinand Möbius introduced the formula into number theory in 1832.

## Explanations

### L1 human

If you know all the divisor sums of $f$, the formula recovers $f$ itself.
It undoes the operation "sum over the divisors", as subtraction undoes addition.
</document>

<document source="https://math2.tamnd.com/node/07Q7CNM5WF.md" tag="07Q7CNM5WF" kind="concept">
# Convex function

## Summary

A convex function on an interval is a function whose graph lies on or below each of its chords. Convex functions have increasing slopes, and they satisfy Jensen's inequality. The node [[convex function|0VV0HSW1ZS]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/07QY5RPFBR.md" tag="07QY5RPFBR" kind="definition">
# Subfield

## Definition

Let $L$ be a [[field|0QA8CHH8MG]].
A subfield of $L$ is a [[subring|0859D58WPD]] $K \subseteq L$ such that $a^{-1} \in K$ for every nonzero $a \in K$.

## Remarks

Equivalently, $K$ contains $0$ and $1$, and $K$ is closed under addition, negation, multiplication and the inverse of nonzero elements.
A subfield is a field with the operations of $L$ restricted to $K$.
The rationals $\mathbb{Q}$ are a subfield of $\mathbb{R}$, and $\mathbb{R}$ is a subfield of $\mathbb{C}$.
The intersection of any family of subfields of $L$ is a subfield of $L$. The intersection of all of them is the [[prime subfield|0VPFBAY7MW]].

## Explanations

### L1 human

A subfield is a smaller field that sits inside a bigger one.
You can add, subtract, multiply and divide its elements without leaving it.
The integers are a subring of $\mathbb{Q}$, but not a subfield, because $1/2 \notin \mathbb{Z}$.
</document>

<document source="https://math2.tamnd.com/node/07RB3EMBWW.md" tag="07RB3EMBWW" kind="statement">
# Third isomorphism theorem for vector spaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W \subseteq U$ be subspaces of $V$.
Then $U / W$ is a subspace of $V / W$, and the map
$$(V / W) / (U / W) \to V / U, \qquad (v + W) + U/W \mapsto v + U,$$
is well defined, and it is a linear isomorphism of [[quotient spaces|0BR3FY5Q8M]].

## Remarks

The proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the surjective map $V / W \to V / U$, $v + W \mapsto v + U$, whose kernel is $U / W$.
In finite dimensions, both sides have dimension $\dim V - \dim U$.

## Explanations

### L1 human

Dividing first by a small subspace and then by a larger one is the same as dividing by the larger one at once.
This works like cancelling in fractions.
The same theorem holds for groups, rings and modules.
</document>

<document source="https://math2.tamnd.com/node/07RB7B04HX.md" tag="07RB7B04HX" kind="concept">
# Uniform continuity on a set

## Summary

A function is uniformly continuous on a set when its restriction to the set is uniformly continuous. Each continuous function on a closed bounded interval has this property. The node [[uniform continuity on a set|0A3CK6DVVB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/07RGNK57ED.md" tag="07RGNK57ED" kind="concept">
# Divisibility in a commutative ring

## Summary

In a commutative ring, an element $a$ divides an element $b$ when $b$ is a multiple of $a$.
The [[definition of divisibility|04GNK7K4VP]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/07RXBVTVZB.md" tag="07RXBVTVZB" kind="definition">
# Normal distribution

## Definition

Let $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.
The normal distribution with mean $\mu$ and variance $\sigma^2$, written $\mathcal{N}(\mu, \sigma^2)$, is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right), \quad x \in \mathbb{R}.$$
The standard normal distribution is $\mathcal{N}(0, 1)$.

## Remarks

The function $\varphi_{\mu, \sigma^2}$ is the [[density of the normal distribution|027JF1FBB0]].
Some authors, and the library Mathlib, also allow $\sigma^2 = 0$. Then $\mathcal{N}(\mu, 0)$ is the [[Dirac measure|099SXAG014]] $\delta_\mu$.
If $X \sim \mathcal{N}(\mu, \sigma^2)$ and $a \ne 0$, $b$ are real, then $aX + b \sim \mathcal{N}(a\mu + b, a^2 \sigma^2)$. In particular, $(X - \mu)/\sigma \sim \mathcal{N}(0, 1)$.
The cumulative distribution function of $\mathcal{N}(0, 1)$ is written $\Phi$. It has no formula with elementary functions.

## Explanations

### L1 human

The normal distribution is the famous bell curve. It is symmetric about the mean $\mu$, and $\sigma$ measures its width.
Heights of people and errors of measurement are often close to normal.
</document>

<document source="https://math2.tamnd.com/node/07SDJ5DNE4.md" tag="07SDJ5DNE4" kind="concept">
# Equation of a plane

## Summary

A [[plane|0V3XAMF3C1]] in $\mathbb{R}^3$ is the set of the solutions of one linear equation in the coordinates.
Let $p_0 = (x_0, y_0, z_0)$ be a point and let $n = (a, b, c)$ be a nonzero vector. The plane through $p_0$ with the normal vector $n$ is the set of the points $p$ with $n \cdot (p - p_0) = 0$. In coordinates this is the point-normal form
$$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0.$$
Expanded, it is the general form $a x + b y + c z = d$ with $d = a x_0 + b y_0 + c z_0$. Conversely, for each real $d$ and each $(a, b, c) \neq (0, 0, 0)$, this equation defines a plane with the normal vector $(a, b, c)$.

## Remarks

Two equations define the same plane if and only if one is a nonzero multiple of the other.
For a plane through the point $p_0$ that contains the linearly independent vectors $u$ and $v$, the [[cross product|0N5N15HWZ7]] $u \times v$ is a normal vector.
Two planes are parallel or equal if and only if their normal vectors are multiples of each other.

## Explanations

### L1 human

The normal vector of a plane points straight out of it. A point $p$ is on the plane exactly when the arrow from $p_0$ to $p$ lies flat in the plane. Then this arrow is perpendicular to the normal vector. The dot product test turns this into one equation. For example, $z = 0$ is the floor plane with the normal vector $(0, 0, 1)$.
</document>

<document source="https://math2.tamnd.com/node/07SG12TQH3.md" tag="07SG12TQH3" kind="concept">
# T0 space

## Summary

A T0 space, or Kolmogorov space, is a topological space in which the open sets can tell any two distinct points apart.
It is the weakest of the usual separation axioms.
</document>

<document source="https://math2.tamnd.com/node/07SJXTWVQW.md" tag="07SJXTWVQW" kind="concept">
# Linear isometry

## Summary

A linear isometry is a linear map between normed spaces that keeps the norm of every vector.
</document>

<document source="https://math2.tamnd.com/node/07SQCNV233.md" tag="07SQCNV233" kind="concept">
# Diagonalisable matrix

## Summary

A square matrix is diagonalisable when it is similar to a diagonal matrix, that is, when its space has a basis of eigenvectors.
</document>

<document source="https://math2.tamnd.com/node/07T2TKGZ4J.md" tag="07T2TKGZ4J" kind="statement">
# Derivatives of the hyperbolic functions

## Statement

The functions $\sinh$, $\cosh$ and $\tanh$ are differentiable on $\mathbb{R}$, and for every real $x$
$$\sinh'(x) = \cosh x, \qquad \cosh'(x) = \sinh x, \qquad \tanh'(x) = \frac{1}{\cosh^2 x} = 1 - \tanh^2 x.$$

## Remarks

The first two follow from [[$\exp' = \exp$|0EX9GFJ0B3]] and $(e^{-x})' = -e^{-x}$.
The third follows from the [[quotient rule|0BS0NJPKVE]] and the identity [[$\cosh^2 - \sinh^2 = 1$|06QCTRFBM0]].
So $\sinh'' = \sinh$ and $\cosh'' = \cosh$. Mathlib states the first one as `Real.hasDerivAt_sinh`.

## Explanations

### L1 human

Unlike sine and cosine, there is no minus sign here.
Each of $\sinh$ and $\cosh$ is the derivative of the other.
</document>

<document source="https://math2.tamnd.com/node/07TXYYVYZR.md" tag="07TXYYVYZR" kind="definition">
# Distance from a point to a set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $A$ be a nonempty subset of $X$ and let $x \in X$.
The distance from $x$ to $A$ is the real number
$$d(x, A) = \inf \{ d(x, a) : a \in A \}.$$

## Remarks

The [[infimum|04KFTA34C5]] exists, because the set is nonempty and bounded below by $0$.
If $x \in A$, then $d(x, A) = 0$. The converse is false: in $\mathbb{R}$, $d(0, (0, 1)) = 0$ but $0 \notin (0, 1)$.
In general $d(x, A) = 0$ if and only if $x$ is in the [[closure|0DRKT1QJR7]] of $A$.
For all $x, y \in X$, $|d(x, A) - d(y, A)| \le d(x, y)$. Thus $x \mapsto d(x, A)$ is [[Lipschitz|0TMZ21WKCN]] with constant $1$.

## Explanations

### L1 human

The distance from a point to a set is the shortest distance from the point to the set, when a shortest distance exists.
For example, the distance from $3$ to the interval $[0, 1]$ is $2$.
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<document source="https://math2.tamnd.com/node/07TZ2Q4C1M.md" tag="07TZ2Q4C1M" kind="concept">
# Trajectory

## Summary

The trajectory, or orbit, of a point under a dynamical system is the set of states that the system reaches from this point over time.
Different orbits do not intersect, so the orbits form a partition of the phase space. See the [[definition|0BVT6VSB3S]].
</document>

<document source="https://math2.tamnd.com/node/07TZV87Y0T.md" tag="07TZV87Y0T" kind="definition">
# Recurrence relation

## Definition

Let $X$ be a set and let $k$ be a positive integer.
A recurrence relation of order $k$ on $X$ is an equation
$$u_{n+k} = f(n, u_n, u_{n+1}, \dots, u_{n+k-1}) \qquad (n \ge 0)$$
for a sequence $(u_n)_{n \ge 0}$ in $X$, where $f : \mathbb{N} \times X^k \to X$ is a given function.

A solution is a sequence that satisfies the equation for all $n \ge 0$.

## Remarks

For each choice of initial values $u_0, \dots, u_{k-1}$ there is exactly one solution. This follows by induction on $n$.
Examples: the [[factorial|0WFCGVMRRC]] satisfies $u_{n+1} = (n + 1) u_n$ of order $1$. The [[Fibonacci sequence|0GA9K57A39]] satisfies $u_{n+2} = u_{n+1} + u_n$ of order $2$.
An important special case is a [[linear recurrence with constant coefficients|081THG13JZ]].

## Explanations

### L1 human

A recurrence relation gives each term of a sequence from the terms just before it.
Once you know the first few terms, the rule determines all the others.
</document>

<document source="https://math2.tamnd.com/node/07VHQ31EYJ.md" tag="07VHQ31EYJ" kind="statement">
# Young's inequality for products

## Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$.
Then for all real $a, b \ge 0$,
$$a b \le \frac{a^p}{p} + \frac{b^q}{q}.$$
Equality holds if and only if $a^p = b^q$.

## Remarks

It is the weighted [[AM–GM inequality|0DRF0F3XC9]] for the numbers $a^p$ and $b^q$, with weights $1/p$ and $1/q$.
For $p = q = 2$ it is $a b \le (a^2 + b^2)/2$.
It is the main step in the proof of [[Hölder's inequality|0BKDAGMGX9]]. Mathlib states it as `Real.young_inequality`.

## Explanations

### L1 human

Young's inequality bounds a product by a sum of powers.
It splits the product in a way that depends on the pair of exponents $p$ and $q$.
</document>

<document source="https://math2.tamnd.com/node/07VX34QXAV.md" tag="07VX34QXAV" kind="concept">
# Solution of an ordinary differential equation

## Summary

A solution of an [[ordinary differential equation|0NT64JV96A]] is a function on an interval that satisfies the equation at each point.
For a first-order system given by a vector field, the solutions are the integral curves. These are the curves whose velocity is the value of the vector field at each point.
See the [[definition|04SW7WECS8]].
</document>

<document source="https://math2.tamnd.com/node/07W9H8QF38.md" tag="07W9H8QF38" kind="concept">
# Adjugate matrix

## Summary

The adjugate of a square matrix is the transpose of its matrix of cofactors.
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<document source="https://math2.tamnd.com/node/07WBTX1MHA.md" tag="07WBTX1MHA" kind="concept">
# Inverse Laplace transform

## Summary

Let $F$ be a function of a complex variable. Its inverse Laplace transform is a piecewise continuous function $f$ of exponential growth whose Laplace transform is $F$.
It can be computed by the Bromwich integral or by Post's inversion formula. See the [[definition|0JRW4X1MDC]].
</document>

<document source="https://math2.tamnd.com/node/07WYQPB8E3.md" tag="07WYQPB8E3" kind="concept">
# Identity element

## Summary

An identity element for a binary operation is an element that does not change any element when it is combined with it on either side.
Examples are $0$ for addition and $1$ for multiplication.
</document>

<document source="https://math2.tamnd.com/node/07X1QA6BRW.md" tag="07X1QA6BRW" kind="statement">
# Derivative of the logarithm

## Statement

The [[natural logarithm|0044CPHJBV]] is differentiable on $(0, \infty)$, and
$$\frac{d}{dx} \ln x = \frac{1}{x} \quad \text{for } x > 0.$$

## Remarks

By the [[inverse function rule|0M0RZ3HSB0]] and [[$\exp' = \exp$|0EX9GFJ0B3]], the derivative at $x = e^y$ is $\frac{1}{e^y} = \frac{1}{x}$.
For $x \ne 0$, the derivative of $\ln |x|$ is $1/x$.
By the chain rule, $(\ln f(x))' = f'(x)/f(x)$ when $f > 0$. Mathlib states the main claim as `Real.hasDerivAt_log`.

## Explanations

### L1 human

The slope of the graph of $\ln$ at $x$ is $1/x$.
So the logarithm grows more and more slowly as $x$ gets large.
</document>

<document source="https://math2.tamnd.com/node/07XBM3A9J5.md" tag="07XBM3A9J5" kind="concept">
# Algebra of limits

## Summary

The algebra of limits is a set of rules for limits of convergent sequences.
The rules give the limit of a sum, a product or a quotient from the limits of the parts. If $a_n \to a$ and $b_n \to b$, then $a_n + b_n \to a + b$, $a_n b_n \to a b$, and $a_n / b_n \to a / b$ when $b \ne 0$. These rules let us find many limits from a few known ones.

## Remarks

The rules are the nodes [[limit of a sum of sequences|0RAM4E07NR]], [[limit of a product of sequences|0734RSVF2V]] and [[limit of a quotient of sequences|03S6GWWKEZ]].
Similar rules hold for the [[limit of a function|0TXTSEMHR5]].

## Explanations

### L1 human

To find the limit of $(2n + 1)/(n + 3)$, divide the numerator and the denominator by $n$ to get $(2 + 1/n)/(1 + 3/n)$.
Since $1/n \to 0$, the rules give the limit $(2 + 0)/(1 + 0) = 2$.
</document>

<document source="https://math2.tamnd.com/node/07XF282NWH.md" tag="07XF282NWH" kind="concept">
# Image of a set

## Summary

The image of a set under a function is the set of the values of the function at the elements of that set.
</document>

<document source="https://math2.tamnd.com/node/07Y2E7P9NP.md" tag="07Y2E7P9NP" kind="concept">
# Median

## Summary

A median of a distribution, or of a data set, is a value that splits it into two halves.
At least half of the probability is at or below the value, and at least half is at or above it.
The node [[median|08W2CW8PKW]] gives the precise definition for random variables.
</document>

<document source="https://math2.tamnd.com/node/07YQ1C78G0.md" tag="07YQ1C78G0" kind="concept">
# Linear form

## Summary

A linear form on a vector space is a linear map from the space to its field of scalars.
</document>

<document source="https://math2.tamnd.com/node/07ZMX4ANJS.md" tag="07ZMX4ANJS" kind="statement">
# Differentiable functions are continuous

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
If $f$ is [[differentiable|0A61B60B5R]] at $a$, then $f$ is [[continuous at|0EEE2CZB2A]] $a$.

## Remarks

For $x \ne a$, write $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$. The first factor tends to $f'(a)$ and the second to $0$.
The converse fails. The [[absolute value|0CQ9ZM383N]] is continuous but not differentiable at $0$. The [[Weierstrass function|01A0MK1730]] is continuous and nowhere differentiable.
Mathlib states it as `HasDerivAt.continuousAt`.

## Explanations

### L1 human

If a graph has a tangent line at a point, it cannot jump there.
A jump would make the slopes of the secant lines blow up.
</document>

<document source="https://math2.tamnd.com/node/07ZS685JJC.md" tag="07ZS685JJC" kind="statement">
# Anticommutativity of the cross product

## Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$, and let $\times$ be the [[cross product|0N5N15HWZ7]]. Then
$$b \times a = -(a \times b).$$

## Remarks

With $b = a$ the statement gives $a \times a = -(a \times a)$, thus $a \times a = 0$ for each vector $a$.
The proof is a direct computation: each component of $b \times a$ is the component of $a \times b$ with the two products in the other order.
The cross product is thus not commutative. For example, $e_1 \times e_2 = e_3$ but $e_2 \times e_1 = -e_3$.

## Explanations

### L1 human

If you swap the two factors of a cross product, the result points the other way. Its length stays the same. The right-hand rule shows this: swapping the factors turns your hand over.
</document>

<document source="https://math2.tamnd.com/node/080Q3RPKY6.md" tag="080Q3RPKY6" kind="statement">
# Every integer is a square times a square-free integer

## Statement

Let $n$ be a positive integer.
Then there are positive integers $a$ and $b$ such that $n = a^2 b$ and $b$ is [[square-free|004JW7A2N5]].

## Remarks

The integers $a$ and $b$ are unique. The number $b$ is the product of the primes $p$ for which the exponent of $p$ in $n$ is odd.
For example, $72 = 2^3 \cdot 3^2 = 6^2 \cdot 2$, so $a = 6$ and $b = 2$.
The proof uses the [[prime factorization|0AEKDFCYFA]] of $n$. Write each exponent $e$ as $e = 2 q + r$ with $r \in \{0, 1\}$.

## Explanations

### L1 human

Take out of $n$ all the pairs of equal primes that you can. The pairs give the square $a^2$.
At most one copy of each prime stays, so the rest $b$ is square-free.
For example, $50 = 2 \cdot 5 \cdot 5 = 5^2 \cdot 2$.
</document>

<document source="https://math2.tamnd.com/node/080RWVVSB6.md" tag="080RWVVSB6" kind="statement">
# Linearity of the integral

## Statement

Let $f, g \colon [a, b] \to \mathbb{R}$ be Riemann integrable, and let $\alpha, \beta \in \mathbb{R}$.
Then $\alpha f + \beta g$ is Riemann integrable on $[a, b]$, and
$$\int_a^b (\alpha f(x) + \beta g(x)) \, dx = \alpha \int_a^b f(x) \, dx + \beta \int_a^b g(x) \, dx.$$

## Remarks

The [[Riemann sums|0HBY48G8G1]] satisfy $S(\alpha f + \beta g, P, t) = \alpha S(f, P, t) + \beta S(g, P, t)$. Pass to the limit.
So the Riemann integrable functions on $[a, b]$ form a vector space, and the integral is a linear map on it.
Mathlib states additivity as `intervalIntegral.integral_add`.

## Explanations

### L1 human

The integral of a sum is the sum of the integrals.
Constant factors can move out of the integral.
</document>

<document source="https://math2.tamnd.com/node/080T2TXVQV.md" tag="080T2TXVQV" kind="definition">
# Irrational number

## Definition

A [[real number|08J2M1JYAG]] $x$ is irrational when $x$ is not a [[rational number|0NCJ5WXN6Z]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $x = a / b$.
The set of irrational numbers is $\mathbb{R} \setminus \mathbb{Q}$.

## Remarks

The number $\sqrt{2}$ is [[irrational|0DZGSDHD92]]. The numbers $\pi$ and $e$ are also irrational.
The decimal expansion of a real number is eventually periodic exactly when the number is rational. Thus the decimal expansion of an irrational number never repeats.
The set $\mathbb{Q}$ is countable and $\mathbb{R}$ is not, thus almost all real numbers are irrational.

## Explanations

### L1 human

An irrational number is a number on the number line that you cannot write as a fraction.
The length of the diagonal of a square with side $1$ is such a number: it is $\sqrt{2} = 1.41421\dots$.
</document>

<document source="https://math2.tamnd.com/node/0819CQ1V8H.md" tag="0819CQ1V8H" kind="statement">
# Rational root theorem

## Statement

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$ and $a_n \neq 0$.
Let $r = p/q$ be a rational [[root|00EHX6FWC0]] of $f$, with $p, q \in \mathbb{Z}$, $q > 0$ and $\gcd(p, q) = 1$.
Then $p$ divides $a_0$ and $q$ divides $a_n$.

## Remarks

The proof: multiply $f(p/q) = 0$ by $q^n$. Then $a_n p^n = -q(\dots)$ and $a_0 q^n = -p(\dots)$, and $\gcd(p, q) = 1$ gives the two divisibilities.
For example, the only possible rational roots of $2x^3 - 3x + 1$ are $\pm 1$ and $\pm \tfrac{1}{2}$. Evaluation shows that $1$ is a root.
The same statement holds over every [[unique factorisation domain|06YMXJMXYS]] and its field of fractions.
For a monic $f$, every rational root is an integer.

## Explanations

### L1 human

The theorem gives a finite list of candidates for the rational roots.
You test each candidate by evaluation.
For $x^3 - 2$, the candidates are $\pm 1$ and $\pm 2$, and none is a root, so $\sqrt[3]{2}$ is irrational.
</document>

<document source="https://math2.tamnd.com/node/081QSEAGEY.md" tag="081QSEAGEY" kind="concept">
# Phase line

## Summary

Let $f$ be a function on an interval of $\mathbb{R}$. The phase line of the scalar [[autonomous equation|07BNRFQ0PJ]] $x' = f(x)$ is a diagram of the line of states.
It marks the [[equilibrium points|0G4X36AE0X]], which are the zeros of $f$. It puts an arrow on each interval between them: up where $f > 0$ and down where $f < 0$.
The arrows show the direction in which each solution moves.

## Remarks

An equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable.
For $f$ locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there.
For the [[logistic equation|0CC4VC7KR3]] $x' = x(1 - x)$, the point $0$ is a source and the point $1$ is a sink.

## Explanations

### L1 human

The phase line is a one-dimensional map of where the state can go.
Arrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.
</document>

<document source="https://math2.tamnd.com/node/081THG13JZ.md" tag="081THG13JZ" kind="definition">
# Linear recurrence with constant coefficients

## Definition

Let $R$ be a commutative ring and let $d$ be a natural number.
A linear recurrence of order $d$ with constant coefficients over $R$ is given by coefficients $c_0, c_1, \dots, c_{d-1} \in R$.
It is the [[recurrence relation|07TZV87Y0T]]
$$u_{n+d} = c_0 u_n + c_1 u_{n+1} + \dots + c_{d-1} u_{n+d-1} \qquad (n \ge 0)$$
for sequences $(u_n)_{n \ge 0}$ in $R$.

## Remarks

This form is homogeneous. A non-homogeneous recurrence adds a term $g(n)$ that does not depend on $u$.
The [[Fibonacci sequence|0GA9K57A39]] satisfies the recurrence of order $2$ with $c_0 = c_1 = 1$.
The sequences that satisfy it are its [[solutions|0ZV2YKKZ58]]. Its [[characteristic polynomial|0BWDJGQYST]] is $X^d - c_{d-1} X^{d-1} - \dots - c_1 X - c_0$.

## Explanations

### L1 human

In a linear recurrence with constant coefficients, each new term is a fixed weighted sum of the previous $d$ terms.
The weights do not change with $n$.
</document>

<document source="https://math2.tamnd.com/node/081WKEDK6C.md" tag="081WKEDK6C" kind="concept">
# Analytic function

## Summary

An analytic function is a function that is given locally by a convergent power series. Analytic functions are smooth, but not all smooth functions are analytic. The node [[analytic function|0JD5ZNW0F4]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/081WTKYX6C.md" tag="081WTKYX6C" kind="statement">
# Finite p-groups are nilpotent

## Statement

Let $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]].
Then $G$ is [[nilpotent|0Q7MMQHC1P]].

## Remarks

Each quotient $G/Z_i$ of the [[upper central series|0XW54DH28V]] is a $p$-group, so it has a [[nontrivial centre|0RZCTT30H5]] when it is not trivial.
So the upper central series grows strictly until it reaches $G$, and $G$ is finite.
For $|G| = p^n$ with $n \geq 2$, the nilpotency class of $G$ is at most $n - 1$.

## Explanations

### L1 human

A nontrivial finite $p$-group always has a nontrivial centre.
Divide by it, and the quotient is again a $p$-group with a nontrivial centre.
Repeat this, and you climb the upper central series to the whole group.
</document>

<document source="https://math2.tamnd.com/node/081YA6RCX6.md" tag="081YA6RCX6" kind="definition">
# Directrix

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$ that is not a circle. Let $F$ be a [[focus|0RBKKNH4GV]] of $Q$. For a point $P$ and a [[line|0MP9BCF6DP]] $\ell$, let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
A directrix of $Q$ for the focus $F$ is a line $\ell$ that does not contain $F$, such that for a real number $e > 0$:
$$Q = \{\, P \in \mathbb{R}^2 \;:\; d(P, F) = e \, d(P, \ell) \,\}.$$

## Remarks

Each focus of $Q$ has exactly one directrix, and the number $e$ is the [[eccentricity|0HVX863SAJ]] of $Q$. The conic is an ellipse if $e < 1$, a parabola if $e = 1$ and a hyperbola if $e > 1$.
For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the directrices are the lines $x = \pm a^2 / c$, for the foci $(\pm c, 0)$.
A circle has no directrix in the Euclidean plane.

## Explanations

### L1 human

For a parabola, the directrix is the line in its definition: each point is as far from the focus as from the directrix. For an ellipse, each point is closer to the focus than to the directrix, by a fixed factor $e < 1$. For a hyperbola, the factor is greater than $1$.
</document>

<document source="https://math2.tamnd.com/node/0826SGV3R0.md" tag="0826SGV3R0" kind="statement">
# Holomorphic functions are analytic

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in U$ and $r > 0$ be such that the open disc $D = \{z : |z - c| < r\}$ is contained in $U$.
Then there are complex numbers $a_0, a_1, a_2, \dots$ such that the [[power series|0VARVQ9ZW1]] $\sum_{n \ge 0} a_n (z - c)^n$ converges to $f(z)$ for each $z \in D$.
For each $\rho$ with $0 < \rho < r$, the coefficients are
$$a_n = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} \frac{f(z)}{(z - c)^{n+1}} \, dz.$$
In particular, $f$ is analytic on $U$: near each point of $U$, $f$ is the sum of a convergent power series.

## Remarks

Proof idea: in [[Cauchy's integral formula|0DG10TVYPN]], expand $\frac{1}{z - a}$ as a geometric series in $\frac{a - c}{z - c}$, and integrate term by term.
The radius of convergence of the Taylor series at $c$ is at least the distance from $c$ to the boundary of $U$.
For example, the Taylor series of $1/(1 - z)$ at $0$ has radius $1$, the distance from $0$ to the point $1$ where the function is not defined.
The converse is also true: [[power series are holomorphic in their disc of convergence|0YMP521ZR5]].

## Explanations

### L1 human

A function with one complex derivative everywhere on an open set is automatically equal to its Taylor series near each point.
This is very different from real functions, where a differentiable function can fail to have a second derivative.
</document>

<document source="https://math2.tamnd.com/node/082DWWXMRJ.md" tag="082DWWXMRJ" kind="concept">
# Finitely generated group

## Summary

A finitely generated group is a group that a finite subset generates.
All finite groups and the group of integers are finitely generated.
</document>

<document source="https://math2.tamnd.com/node/082GHNPJ53.md" tag="082GHNPJ53" kind="concept">
# Abelian group

## Summary

An abelian group is a group whose operation is commutative.
The integers with addition are the basic example.
</document>

<document source="https://math2.tamnd.com/node/082MGNWNA8.md" tag="082MGNWNA8" kind="statement">
# Absolutely convergent series in a Banach space converge

## Statement

Let $(V, \|\cdot\|)$ be a [[Banach space|0BDTTM24DP]] and let $(x_n)_{n \ge 1}$ be a sequence in $V$ with $\sum_{n \ge 1} \|x_n\| < \infty$.
Then the partial sums $s_N = \sum_{n=1}^{N} x_n$ converge to a point $s \in V$, and $\|s\| \le \sum_{n \ge 1} \|x_n\|$.

## Remarks

Proof: for $M > N$, the triangle inequality gives $\|s_M - s_N\| \le \sum_{n=N+1}^{M} \|x_n\|$. This tends to $0$, so $(s_N)$ is a [[Cauchy sequence|0MCK8K2D7Z]], and it converges because $V$ is [[complete|08EWT7J0SC]].
The converse holds: a normed space in which each absolutely convergent series converges is a Banach space.
For $V = \mathbb{R}$, this is the fact that absolutely convergent real series converge.

## Explanations

### L1 human

If the lengths of the terms have a finite sum, then the series converges in a Banach space.
This is a practical test for completeness, and it is often used to build solutions as sums of series.
</document>

<document source="https://math2.tamnd.com/node/082RMRSY26.md" tag="082RMRSY26" kind="definition">
# Equivalence class

## Definition

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$ and let $a \in A$.
The equivalence class of $a$ is the set
$$[a] = \{x \in A \mid x \sim a\}.$$
Each element of $[a]$ is a representative of the class.

## Remarks

For congruence modulo $3$ on $\mathbb{Z}$, the class of $1$ is $[1] = \{\dots, -5, -2, 1, 4, 7, \dots\}$.
For $a, b \in A$, $[a] = [b]$ if and only if $a \sim b$. Otherwise, $[a]$ and $[b]$ are disjoint.
The set of all classes is the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

An equivalence class collects all the objects that the relation treats as the same.
For "has the same birthday as", the class of a person is the group of all people born on the same day of the year.
</document>

<document source="https://math2.tamnd.com/node/0832XVR512.md" tag="0832XVR512" kind="definition">
# Successor function

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]].
The successor function is the function $S \colon \mathbb{N} \to \mathbb{N}$ with $S(n) = n + 1$ for each $n$ in $\mathbb{N}$.
The number $S(n)$ is the successor of $n$.

## Remarks

In the [[Peano axioms|0QFT7Z76N5]], $S$ is a primitive function, and the axioms do not define it. Then $n + 1$ is a name for $S(n)$, and $1$ is a name for $S(0)$.
For example, $S(0) = 1$, $S(1) = 2$ and $S(S(2)) = 4$.

## Explanations

### L1 human

The successor of a number is the number that comes after it when you count.
The successor of $4$ is $5$, and the successor of $0$ is $1$.
</document>

<document source="https://math2.tamnd.com/node/0839XNQXDK.md" tag="0839XNQXDK" kind="definition">
# Intersection of a family of sets

## Definition

Let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]] with a nonempty index set $I$.
The intersection of the family is the set
$$\bigcap_{i \in I} A_i = \{x \mid \forall i \in I, \, x \in A_i\}.$$
So $x$ is an element of the intersection when $x \in A_i$ for each $i \in I$.

## Remarks

For $I = \{1, 2\}$, this is the [[intersection|0HBRW4E6G1]] $A_1 \cap A_2$.
For an empty index set, the condition is [[vacuously true|0V4QXPGJQK]] for each $x$.
So the intersection is defined only relative to a [[universal set|0BPYRMHJ48]] $U$, and it is $U$.
Example: $\bigcap_{n \ge 1} (-1/n, 1/n) = \{0\}$ in $\mathbb{R}$.

## Explanations

### L1 human

The intersection of a family keeps only the elements that are in every member of the family.
It generalizes $A \cap B$ from two sets to any number of sets.
</document>

<document source="https://math2.tamnd.com/node/08420828EY.md" tag="08420828EY" kind="definition">
# Number of prime factors counted with multiplicity

## Definition

Let $n$ be a positive integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \cdots p_k^{e_k}$, with distinct primes $p_i$.
Then $\Omega(n) = e_1 + e_2 + \dots + e_k$.
So $\Omega(n)$ is the number of prime factors of $n$, counted with multiplicity.

## Remarks

For example, $\Omega(1) = 0$, $\Omega(12) = 3$ and $\Omega(360) = 6$.
The function $\Omega$ is completely additive: $\Omega(mn) = \Omega(m) + \Omega(n)$ for all positive $m$ and $n$.
The [[Liouville function|08RQAPXF1T]] is $\lambda(n) = (-1)^{\Omega(n)}$.

## Explanations

### L1 human

The function $\Omega(n)$ counts all the prime factors of $n$, with repeats.
For $12 = 2 \cdot 2 \cdot 3$, there are three factors, so $\Omega(12) = 3$, while $\omega(12) = 2$.
</document>

<document source="https://math2.tamnd.com/node/084HH1NDCB.md" tag="084HH1NDCB" kind="statement">
# Fundamental theorem of arithmetic

## Statement

Each positive integer $n$ has exactly one [[prime factorization|0AEKDFCYFA]].
That is, there are [[primes|0SN0MAEKYX]] $p_1 \le p_2 \le \dots \le p_k$ with $n = p_1 p_2 \cdots p_k$.
If also $n = q_1 q_2 \cdots q_l$ with primes $q_1 \le q_2 \le \dots \le q_l$, then $k = l$ and $p_i = q_i$ for each $i$.

## Remarks

The [[existence of prime factorization|020K0X7HVR]] gives the first part. The uniqueness follows from [[Euclid's lemma|051VYKA1EX]].
The statement uses the convention that $1$ is not prime. If $1$ were prime, $6 = 2 \cdot 3 = 1 \cdot 2 \cdot 3$ would have two factorizations.
Unique factorization fails in some rings. In $\mathbb{Z}[\sqrt{-5}]$, $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ gives two different factorizations into irreducible elements.

## History

Euclid's Elements contains the main steps of the proof, in Book VII, Propositions 30 to 32, and in Book IX, Proposition 14.
Gauss proved the uniqueness in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Each positive integer has one and only one recipe as a product of primes.
You can multiply the primes in any order, but the list of primes stays the same.
For example, $12 = 2 \cdot 2 \cdot 3$, and no other list of primes has the product $12$.
</document>

<document source="https://math2.tamnd.com/node/084NGSTWM7.md" tag="084NGSTWM7" kind="definition">
# Higher derivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
Let $f^{(0)} = f$.
If $f^{(n)}$ is defined and differentiable on $I$, let $f^{(n+1)} = (f^{(n)})'$, its [[derivative function|027K9RMZ43]].
The function $f^{(n)}$ is the $n$-th derivative of $f$, and $f$ is $n$ times differentiable when it exists.

## Remarks

The [[second derivative|0Y9D2FJRQ4]] is $f'' = f^{(2)}$.
For example, the $n$-th derivative of $x^n$ is the constant $n!$.
Mathlib defines `iteratedDeriv n f`.

## Explanations

### L1 human

Higher derivatives come from differentiating again and again.
They describe finer features of a function, like the bending of its graph.
</document>

<document source="https://math2.tamnd.com/node/0850JJMJYZ.md" tag="0850JJMJYZ" kind="statement">
# P-adic valuation of a product

## Statement

Let $p$ be a prime number, and let $a$ and $b$ be nonzero integers.
Then $\nu_p(a b) = \nu_p(a) + \nu_p(b)$, where $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

With the convention $\nu_p(0) = \infty$, the formula holds for all integers $a$ and $b$.
The proof uses [[Euclid's lemma|051VYKA1EX]]. Write $a = p^i a'$ and $b = p^j b'$ with $p \nmid a'$ and $p \nmid b'$. Then $p \nmid a' b'$, so $\nu_p(a b) = i + j$.
For example, $\nu_2(12 \cdot 10) = \nu_2(120) = 3 = 2 + 1$.

## Explanations

### L1 human

The factors $p$ in $a b$ come from $a$ or from $b$, and no new factor $p$ appears.
So the number of factors $p$ in the product is the sum of the numbers of factors $p$ in $a$ and in $b$.
</document>

<document source="https://math2.tamnd.com/node/0859D58WPD.md" tag="0859D58WPD" kind="definition">
# Subring

## Definition

Let $R$ be a [[ring|0894J980RK]].
A subring of $R$ is a subset $S \subseteq R$ with these properties.

1. $1 \in S$.
2. If $a, b \in S$, then $a - b \in S$.
3. If $a, b \in S$, then $a \cdot b \in S$.

A subring $S$ is a ring with the addition and the multiplication of $R$, restricted to $S$.

## Remarks

The integers $\mathbb{Z}$ are a subring of $\mathbb{Q}$, and $\mathbb{Q}$ is a subring of $\mathbb{R}$.
The set $2\mathbb{Z}$ of even integers is closed under addition and multiplication, but it is not a subring of $\mathbb{Z}$, because $1 \notin 2\mathbb{Z}$.
Authors who do not require a ring to have an identity also do not require $1 \in S$.

## Explanations

### L1 human

A subring is a smaller ring inside a ring, with the same operations.
For example, the integers sit inside the rational numbers.
The sum, the difference and the product of two integers are again integers.
</document>

<document source="https://math2.tamnd.com/node/0859R2V6ES.md" tag="0859R2V6ES" kind="statement">
# Angle bisector theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $D$ be a point of the segment $BC$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]].
If $\angle BAD = \angle DAC$, then
$$\frac{d(B, D)}{d(D, C)} = \frac{d(A, B)}{d(A, C)}.$$
Conversely, if $D$ is a point of the segment $BC$ with this equality, then $\angle BAD = \angle DAC$.

## Remarks

In words, the bisector of the angle at $A$ cuts the opposite side in the ratio of the two adjacent sides.
In an isosceles triangle with $d(A, B) = d(A, C)$, the bisector from $A$ meets $BC$ at its midpoint.
A similar statement holds for the bisector of the exterior angle at $A$, which meets the line $BC$ outside the segment.

## History

The theorem is Proposition 3 of Book VI of Euclid's Elements.

## Explanations

### L1 human

Take the triangle with $d(A, B) = 6$ and $d(A, C) = 3$. The bisector of the angle at $A$ cuts $BC$ into two parts. The part next to $B$ is twice as long as the part next to $C$.
</document>

<document source="https://math2.tamnd.com/node/085CH3CYMX.md" tag="085CH3CYMX" kind="concept">
# Walk

## Summary

A walk in a graph is a sequence of vertices in which consecutive vertices are adjacent. Walks are the basic notion of movement in a graph, and trails, paths, circuits and cycles are special walks. The node [[walk|0VJZM0F9YM]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/085HQ5G5TE.md" tag="085HQ5G5TE" kind="statement">
# Jordan normal form

## Statement

Let $F$ be an algebraically closed field, for example $\C$, and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to a [[block diagonal matrix|001T5FJDC1]]
$$J = \begin{pmatrix} J_{k_1}(\lambda_1) & & \\ & \ddots & \\ & & J_{k_r}(\lambda_r) \end{pmatrix},$$
whose diagonal blocks are [[Jordan blocks|07PH4RX0X5]].
The matrix $J$ is unique up to the order of the blocks.
It is the *Jordan normal form* of $A$.

## Remarks

The $\lambda_i$ are the eigenvalues of $A$; they need not be distinct.
The number of blocks for $\lambda$ is its geometric multiplicity, and the sum of their sizes is its algebraic multiplicity.
The largest block for $\lambda$ has size equal to the multiplicity of $\lambda$ in the minimal polynomial.
The theorem holds over any field $F$ for the matrices whose characteristic polynomial splits over $F$.

## Explanations

### L1 human

Every complex square matrix becomes almost diagonal in a good basis.
The only entries off the diagonal are some $1$s just above it, inside the Jordan blocks.
Two complex matrices are similar exactly when they have the same Jordan form, up to the order of the blocks.
</document>

<document source="https://math2.tamnd.com/node/085MJS6907.md" tag="085MJS6907" kind="statement">
# Law of quadratic reciprocity

## Statement

Let $p$ and $q$ be distinct odd primes.
Then
$$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}},$$
where the symbols are [[Legendre symbols|01J8AMV9RS]].

## Remarks

So $\left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$ unless $p \equiv q \equiv 3 \pmod 4$. In that case $\left(\frac{p}{q}\right) = -\left(\frac{q}{p}\right)$.
For example, $\left(\frac{7}{3}\right) = \left(\frac{1}{3}\right) = 1$, and $3 \equiv 7 \equiv 3 \pmod 4$. So $\left(\frac{3}{7}\right) = -1$.
With the [[first supplement|0PHBS78HKT]], the [[second supplement|0ZXW95S44M]] and [[multiplicativity|0NW1JP7TBR]], the law computes every Legendre symbol.

## History

Euler and Legendre conjectured the law.
Gauss first proved it in 1796, and he published proofs in his Disquisitiones Arithmeticae of 1801.
Gauss called it the fundamental theorem.

## Explanations

### L1 human

The law connects two questions: is $p$ a square modulo $q$, and is $q$ a square modulo $p$?
The answers are the same, except when both primes leave remainder $3$ on division by $4$. Then the answers are opposite.
This lets you swap and reduce the numbers, as in the [[Euclidean algorithm|0TRNT4X7J0]].
</document>

<document source="https://math2.tamnd.com/node/085PND4KNT.md" tag="085PND4KNT" kind="definition">
# Group isomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
A group isomorphism from $G$ to $H$ is a [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to H$ that is a bijection.
The groups $G$ and $H$ are isomorphic, written $G \cong H$, when there is an isomorphism from $G$ to $H$.

## Remarks

The inverse map of an isomorphism is again a homomorphism, thus an isomorphism.
Isomorphism is an equivalence relation on groups.
The exponential map is an isomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication.
Isomorphic groups have the same order, and corresponding elements have the same order.

## Explanations

### L1 human

Two groups are isomorphic when they are the same group with different names for the elements.
An isomorphism is a dictionary that translates the names and keeps all products correct.
For example, the rotations of a triangle and the integers modulo $3$ under addition are isomorphic.
</document>

<document source="https://math2.tamnd.com/node/086BE0J3FZ.md" tag="086BE0J3FZ" kind="statement">
# Fermat's right triangle theorem

## Statement

There is no right triangle with integer side lengths whose area is the square of an integer.
Equivalently, the equation $x^4 - y^4 = z^2$ has no solution in positive integers $x$, $y$ and $z$.

## Remarks

In terms of a [[Pythagorean triple|01G1WY1GVX]] $(a, b, c)$, the theorem says that $ab/2$ is never a perfect square.
A consequence is [[Fermat's Last Theorem for n = 4|0E741HSMJ1]], since $a^4 + b^4 = c^4$ gives $c^4 - b^4 = (a^2)^2$.
The proof is by [[infinite descent|0EFYCFHYFG]].

## History

The proof was published in 1670 among the works of Pierre de Fermat, after his death.
It is the only complete proof by Fermat that survives.

## Explanations

### L1 human

The triangle with sides $3$, $4$, $5$ has area $6$, and the one with sides $5$, $12$, $13$ has area $30$.
Fermat proved that such an area can never be a perfect square.
</document>

<document source="https://math2.tamnd.com/node/086HEVXR9Z.md" tag="086HEVXR9Z" kind="statement">
# Mean of the Poisson distribution

## Statement

Let $\lambda > 0$ and let $X$ have the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda$. Then
$$E[X] = \sum_{k=0}^{\infty} k \, e^{-\lambda} \frac{\lambda^k}{k!} = \lambda.$$

## Remarks

Proof: the term with $k = 0$ is zero. For $k \ge 1$, $k \lambda^k / k! = \lambda \cdot \lambda^{k-1}/(k-1)!$. So the sum is $\lambda e^{-\lambda} \sum_{j \ge 0} \lambda^j / j! = \lambda e^{-\lambda} e^{\lambda} = \lambda$.
The variance of $X$ is also $\lambda$.
So the parameter $\lambda$ is the mean number of events.

## Explanations

### L1 human

The parameter of a Poisson distribution is its average.
If a shop gets on average $4$ customers per hour, a Poisson model of the number of customers in one hour uses $\lambda = 4$.
</document>

<document source="https://math2.tamnd.com/node/086QKPF0A6.md" tag="086QKPF0A6" kind="concept">
# Predicate logic

## Summary

Predicate logic extends [[propositional logic|09K9MWD2QQ]] with variables for objects, [[predicates|0T3DPBG981]] and quantifiers.
Its formulas can say "for all $x$" with the [[universal quantifier|0B88PQTJ66]] and "there exists $x$" with the [[existential quantifier|0JW2FHW7G3]].
In first-order logic, the quantifiers range over objects, not over sets of objects or predicates.

## Remarks

Other names are first-order logic and predicate calculus.
An example of a formula is $\forall x \, \exists y \, (y > x)$.
Most of mathematics can be written in first-order logic with the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].

## Explanations

### L1 human

Propositional logic cannot say "every number has a larger number".
Predicate logic can: it has variables such as $x$ and $y$, and words such as "for all" and "there exists".
This is the logic of most mathematical statements.
</document>

<document source="https://math2.tamnd.com/node/0874138YP6.md" tag="0874138YP6" kind="definition">
# Graph colouring

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$, and let $C$ be a set, the set of colours.
A colouring of $G$ with colours in $C$ is a function $c : V \to C$ such that $c(u) \neq c(v)$ for any two adjacent vertices $u$ and $v$.

## Remarks

Such a colouring is also called proper. A colouring with a set of $k$ colours is a $k$-colouring.
Each colour class $c^{-1}(x)$ is an [[independent set|0KZJ82WEPP]].
A graph has a $2$-colouring if and only if it is [[bipartite|07DVHJ8N0A]].
The subject started with the colouring of maps. Francis Guthrie stated the four colour conjecture in 1852.

## Explanations

### L1 human

Colour the vertices of a graph so that the two ends of each edge have different colours.
For example, a timetable gives each exam a time slot, so that exams with a common student are at different times.
</document>

<document source="https://math2.tamnd.com/node/0878DN3WCS.md" tag="0878DN3WCS" kind="definition">
# Isometry

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
A function $f \colon X \to Y$ is an isometry when
$$d_Y\bigl(f(x), f(y)\bigr) = d_X(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

Each isometry is [[injective|02J4G4J4TJ]] and [[Lipschitz|0TMZ21WKCN]] with constant $1$.
Some authors require an isometry to be surjective. Here a surjective isometry is an [[isometric isomorphism|0VRD8GC5BM]].
The map $x \mapsto (x, 0)$ from $\mathbb{R}$ to $\mathbb{R}^2$ with the Euclidean metric is an isometry, but it is not surjective.
Rotations, reflections and translations of $\mathbb{R}^2$ are isometries for the Euclidean metric.

## Explanations

### L1 human

An isometry moves points without stretching or shrinking any distance.
It can put a space into a bigger space, as the real line is put into the plane as the horizontal axis.
</document>

<document source="https://math2.tamnd.com/node/087EMZTHWN.md" tag="087EMZTHWN" kind="definition">
# von Mangoldt function

## Definition

The von Mangoldt function $\Lambda$ is the arithmetic function given, for each positive integer $n$, by
$$\Lambda(n) = \begin{cases} \log p & \text{if } n = p^k \text{ for a prime } p \text{ and an integer } k \ge 1, \\ 0 & \text{otherwise}. \end{cases}$$
Here $\log$ is the natural logarithm. The word prime has the meaning of the [[definition of a prime number|0SN0MAEKYX]].

## Remarks

For example, $\Lambda(8) = \log 2$, $\Lambda(7) = \log 7$, and $\Lambda(1) = \Lambda(6) = 0$.
For each positive integer $n$, $\sum_{d \mid n} \Lambda(d) = \log n$.
The function $\Lambda$ is not multiplicative. It is named after Hans von Mangoldt.
The sum $\psi(x) = \sum_{n \le x} \Lambda(n)$ is the second Chebyshev function. The [[prime number theorem|0EHBY5Y84N]] is equivalent to $\psi(x) \sim x$.

## Explanations

### L1 human

The von Mangoldt function puts the weight $\log p$ on each prime power $p^k$, and $0$ on all other numbers.
With these weights, the divisor sums give exactly $\log n$, which makes the function useful in the study of primes.
</document>

<document source="https://math2.tamnd.com/node/087P03NEND.md" tag="087P03NEND" kind="concept">
# Statistical hypothesis test

## Summary

A statistical hypothesis test is a rule that uses data to decide whether to reject a hypothesis about the law of the data.
One fixes a null hypothesis $H_0$ and an alternative hypothesis $H_1$, for example $H_0 \colon \theta \in \Theta_0$ and $H_1 \colon \theta \in \Theta_1$ with disjoint sets $\Theta_0$ and $\Theta_1$.
One computes a test statistic $T(X)$ and rejects $H_0$ when $T(X)$ is in a critical region $C$. The region is chosen so that $P_\theta(T(X) \in C) \le \alpha$ for each $\theta \in \Theta_0$, for a significance level $\alpha$ such as $0.05$.

## Remarks

Equivalently, one can compute the [[p-value|0EHYZFZ4PK]] and reject $H_0$ when it is at most $\alpha$.
A test can make [[two types of errors|0R41PVFA39]]: reject a true $H_0$, or fail to reject a false $H_0$. The level $\alpha$ bounds the probability of the first error.
The power of the test at $\theta \in \Theta_1$ is $P_\theta(T(X) \in C)$, the probability to reject $H_0$ when it is false.
Not rejecting $H_0$ is not a proof that $H_0$ is true.

## Explanations

### L1 human

A hypothesis test asks whether the data are surprising under a default assumption, the null hypothesis.
If the data are too surprising, one rejects the assumption. The significance level fixes how often one accepts to reject a true assumption.
</document>

<document source="https://math2.tamnd.com/node/087VJM5Q77.md" tag="087VJM5Q77" kind="concept">
# Translation

## Summary

A translation moves each point of a space by the same distance in the same direction. It adds one fixed vector to each point.
</document>

<document source="https://math2.tamnd.com/node/0880F9EQCP.md" tag="0880F9EQCP" kind="definition">
# Positive semidefinite matrix

## Definition

Let $A$ be an $n \times n$ complex [[Hermitian matrix|0P17NGQQMQ]].
The matrix $A$ is *positive semidefinite* when
$$x^* A x \ge 0 \qquad \text{for all } x \in \C^n.$$
For a real symmetric $A$, the condition is $x^T A x \ge 0$ for all $x \in \R^n$.

## Remarks

$A$ is positive semidefinite if and only if all its eigenvalues are nonnegative.
Each [[positive definite|0KX66CZMBB]] matrix is positive semidefinite.
The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is positive semidefinite but not positive definite.
The notation $A \succeq 0$ is common.

## Explanations

### L1 human

A positive semidefinite matrix never gives a negative value of $x^T A x$.
It may give zero for some nonzero vectors.
Covariance matrices and Gram matrices are always of this kind.
</document>

<document source="https://math2.tamnd.com/node/08895Q5W96.md" tag="08895Q5W96" kind="statement">
# Chevalley–Warning theorem

## Statement

Let $\mathbb{F}$ be a finite field of characteristic $p$.
Let $f_1, \dots, f_r \in \mathbb{F}[X_1, \dots, X_n]$ be polynomials with total degrees $d_1, \dots, d_r$, such that $n > d_1 + \dots + d_r$.
Then the number of common zeros $(a_1, \dots, a_n) \in \mathbb{F}^n$ of $f_1, \dots, f_r$ is divisible by $p$.

## Remarks

A corollary is Chevalley's theorem: if the polynomials have no constant terms, then they have a common zero other than $(0, \dots, 0)$.
The reason is that $(0, \dots, 0)$ is a zero, and the number of zeros is a multiple of $p \ge 2$.
For example, a quadratic form in $3$ or more variables over $\mathbb{Z}/p\mathbb{Z}$ has a nontrivial zero.

## History

Ewald Warning and Claude Chevalley proved the results in 1935.

## Explanations

### L1 human

When a system of polynomial equations over a finite field has many variables compared with the degrees, the solutions come in multiples of $p$.
So if there is one obvious solution, such as all zeros, there must be another.
</document>

<document source="https://math2.tamnd.com/node/088RYKAWK9.md" tag="088RYKAWK9" kind="concept">
# Spanning set

## Summary

A spanning set of a vector space is a set of vectors whose linear span is the whole space.
</document>

<document source="https://math2.tamnd.com/node/0894J980RK.md" tag="0894J980RK" kind="definition">
# Ring

## Definition

A ring is a set $R$ with two binary operations and two elements $0, 1 \in R$ that satisfy the axioms below.
The operations are the addition $+$ and the multiplication $\cdot$.

1. $(R, +)$ is an abelian [[group|G5S1M9]] with identity element $0$.
2. The multiplication is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in R$.
3. The element $1$ is a multiplicative identity: $1 \cdot a = a \cdot 1 = a$ for all $a \in R$.
4. The multiplication is distributive over the addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$ for all $a, b, c \in R$.

## Remarks

Some authors do not require the element $1$. They call the structure of this definition a ring with identity, or a unital ring.
The axioms give $0 \cdot a = a \cdot 0 = 0$ and $(-a) \cdot b = -(a \cdot b)$ for all $a, b \in R$.
The integers $\mathbb{Z}$, the integers modulo $n$ and the $n \times n$ real matrices are rings.
A ring in which $a \cdot b = b \cdot a$ for all $a, b$ is a [[commutative ring|0GYGHKX834]].

## Explanations

### L1 human

A ring is a place where you can add, subtract and multiply, as with the integers.
You cannot always divide.
The order of a product can matter: for two matrices $A$ and $B$, $AB$ and $BA$ are often different.
</document>

<document source="https://math2.tamnd.com/node/089KWVYD69.md" tag="089KWVYD69" kind="definition">
# Periodic function

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function and let $T$ be a real number.
The number $T$ is a period of $f$ if $f(x + T) = f(x)$ for all $x \in \mathbb{R}$.
The function $f$ is periodic if it has a period $T \ne 0$.

## Remarks

If $T$ is a period, then so is $k T$ for each integer $k$.
The [[sine function is periodic|0P4NQD746K]] with period $2 \pi$.
If a continuous periodic function is not constant, it has a smallest positive period. The [[Dirichlet function|0T4EJ8ZJM9]] has every rational number as a period.
Mathlib defines periods as `Function.Periodic`.

## Explanations

### L1 human

A periodic function repeats itself after each step of length $T$.
Its graph is made of copies of one piece, placed side by side.
</document>

<document source="https://math2.tamnd.com/node/08ABT85339.md" tag="08ABT85339" kind="concept">
# Cyclic subgroup generated by an element

## Summary

The cyclic subgroup generated by an element of a group is the set of all integer powers of the element.
It is the smallest subgroup that contains the element.
</document>

<document source="https://math2.tamnd.com/node/08AM0KRCZ1.md" tag="08AM0KRCZ1" kind="statement">
# Divergence part of the ratio test

## Statement

Let $(a_n)$ be a real sequence with $a_n \ne 0$ for all $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for a real number $L$ with $L > 1$.
Then the series $\sum a_n$ diverges.

## Remarks

For large $n$ we have $|a_{n+1}| > |a_n|$. So $|a_n|$ is eventually increasing and positive, and $a_n$ does not tend to $0$. The [[term test|0VVY4H92G2]] gives divergence.
The same proof works when $|a_{n+1}| / |a_n| \to +\infty$.
For example, $\sum 2^n / n$ diverges, since the ratio is $2n/(n+1) \to 2$.

## Explanations

### L1 human

If each term is eventually bigger than the term before, the terms cannot shrink to $0$.
Then the series cannot converge.
</document>

<document source="https://math2.tamnd.com/node/08ANASGBXV.md" tag="08ANASGBXV" kind="concept">
# Moment of inertia

## Summary

The moment of inertia of a body about an axis measures how hard it is to change the rotation of the body about that axis.
It is the integral of the density times the square of the distance to the axis. So for a solid body it is a [[triple integral|0698M7C7AX]].
</document>

<document source="https://math2.tamnd.com/node/08ANQHHG3P.md" tag="08ANQHHG3P" kind="concept">
# Independent set

## Summary

An independent set in a graph is a set of vertices with no edges between them. Independent sets are dual to cliques and to vertex covers, and they are the colour classes of colourings. The node [[independent set|0KZJ82WEPP]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/08B1RTW7VZ.md" tag="08B1RTW7VZ" kind="concept">
# Diophantine equation

## Summary

A Diophantine equation is a polynomial equation with integer coefficients, for which one looks for integer solutions, or sometimes rational solutions.
Typical questions ask whether a solution exists, whether there are finitely many, and how to find all of them.

## Remarks

Examples are the linear equation $ax + by = c$, [[Pythagorean triples|01G1WY1GVX]] $x^2 + y^2 = z^2$, [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ and the equation of [[Fermat's Last Theorem|0Y5X3SDBCZ]].
By Hilbert's tenth problem, as solved by Yuri Matiyasevich in 1970, no algorithm decides for every Diophantine equation whether it has an integer solution.
The name refers to Diophantus of Alexandria.

## Explanations

### L1 human

In a Diophantine equation, only whole numbers count as solutions.
For example, $x^2 + y^2 = 3$ has real solutions, but no integer solutions.
The restriction to integers makes such equations much harder than ordinary equations.
</document>

<document source="https://math2.tamnd.com/node/08B9CX109G.md" tag="08B9CX109G" kind="concept">
# Adjacency matrix

## Summary

The adjacency matrix of a finite graph records adjacency as a matrix of zeros and ones. It lets linear algebra study graphs, for example through eigenvalues and powers that count walks. The node [[adjacency matrix|0NZ8MDBN6Q]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/08BPSC6YTE.md" tag="08BPSC6YTE" kind="statement">
# Class equation

## Statement

Let $G$ be a finite [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
Let $a_1, \dots, a_k$ be one element from each [[conjugacy class|0DEBXAJ5SA]] of $G$ that has more than one element.
Then $|G| = |Z(G)| + \sum_{i=1}^{k} [G : C_G(a_i)]$.

## Remarks

The conjugacy classes partition $G$, and the classes with one element are exactly the elements of $Z(G)$.
Each term $[G : C_G(a_i)]$ is the [[size of a class|0VK5V9P8G3]], so it divides $|G|$ and is greater than $1$.
For $S_3$, the equation is $6 = 1 + 3 + 2$.
The class equation is the key step in the proof that a nontrivial finite $p$-group has a nontrivial centre.

## Explanations

### L1 human

Count the elements of the group class by class.
The central elements form classes of size one, and the other classes have sizes that divide the order of the group.
This simple count gives strong facts, for example about groups of prime power order.
</document>

<document source="https://math2.tamnd.com/node/08C1NWBNHA.md" tag="08C1NWBNHA" kind="statement">
# Product of primitive polynomials is primitive

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $f, g \in R[x]$ be [[primitive polynomials|0SP14DZN0G]].
Then $f g$ is primitive.

## Remarks

The proof: suppose an irreducible $p$ divides all the coefficients of $f g$. Since $p$ is [[prime|0MNHXBSX3M]], $R/(p)$ is an integral domain. The images of $f$ and $g$ in $(R/(p))[x]$ are nonzero, but their product is zero, which is a contradiction.
Many texts call this statement Gauss's lemma. The node [[Gauss's lemma|0QFV5Y1Z5X]] states the form about irreducibility.
Over $\mathbb{Z}$: $2x + 1$ and $3x + 2$ are primitive, and so is $6x^2 + 7x + 2$.

## Explanations

### L1 human

Suppose that no prime divides all the coefficients of $f$, and no prime divides all the coefficients of $g$.
Then no prime divides all the coefficients of $f g$.
The proof reduces the coefficients modulo a prime and uses that the result is an integral domain.
</document>

<document source="https://math2.tamnd.com/node/08CJB2NQ92.md" tag="08CJB2NQ92" kind="statement">
# Existence of an eigenvalue over an algebraically closed field

## Statement

Let $F$ be an algebraically closed field, let $V$ be a finite-dimensional vector space over $F$ with $V \ne 0$, and let $T \colon V \to V$ be a linear map.
Then $T$ has at least one [[eigenvalue|0BQAA2BK3Q]] in $F$.

## Remarks

The [[characteristic polynomial|0J5EMJRRFM]] of $T$ has degree $\dim V \ge 1$, so it has a root in $F$.
The statement applies to $F = \C$, by the fundamental theorem of algebra.
It fails over $\R$: the rotation of $\R^2$ by $90$ degrees has no real eigenvalue.
It also fails in infinite dimensions: the map $p(x) \mapsto x\, p(x)$ on $\C[x]$ has no eigenvalue.

## Explanations

### L1 human

Over the complex numbers, every square matrix has at least one eigenvector.
The reason is that every nonconstant complex polynomial has a root.
Over the real numbers, a rotation can move every line, so no real eigenvector exists.
</document>

<document source="https://math2.tamnd.com/node/08EJC6Q25H.md" tag="08EJC6Q25H" kind="definition">
# Binary relation

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
A binary relation from $A$ to $B$ is a [[subset|0FE8Q5S42Y]] $R$ of the [[Cartesian product|0C81YBVTWH]] $A \times B$.
For $a \in A$ and $b \in B$, write $a \mathrel{R} b$ when $(a, b) \in R$.
A binary relation on $A$ is a binary relation from $A$ to $A$.

## Remarks

Examples on $\mathbb{Z}$ are $\le$, $=$ and divisibility $\mid$.
A relation on a finite set can be drawn as a directed graph, with an arrow from $a$ to $b$ when $a \mathrel{R} b$.
A [[function|0XHDTE5YMZ]] is a special kind of binary relation.

## Explanations

### L1 human

A relation is a rule that says which pairs belong together.
For the relation "divides" on $\{1, 2, 3, 4\}$, the pair $(2, 4)$ is in the relation and the pair $(3, 4)$ is not.
</document>

<document source="https://math2.tamnd.com/node/08EJHHMB5R.md" tag="08EJHHMB5R" kind="definition">
# Equinumerous sets

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The sets $A$ and $B$ are equinumerous, written $A \approx B$, when there is a [[bijective function|0SNQEF7Z53]] $f \colon A \to B$.

## Remarks

Equinumerosity is reflexive, symmetric and transitive. The identity, the inverse and the composition of bijections give the three properties.
The set $\mathbb{N}$ is equinumerous with the set of even natural numbers, by $n \mapsto 2n$. So an infinite set can be equinumerous with a proper subset.
Other names are "equipotent" and "have the same [[cardinality|0JTGRV4WT4]]".

## Explanations

### L1 human

You do not need to count to compare sizes.
If each chair in a room has one person and each person has one chair, there are as many people as chairs.
Equinumerosity applies the same idea to infinite sets.
</document>

<document source="https://math2.tamnd.com/node/08EVNYZF42.md" tag="08EVNYZF42" kind="definition">
# Fréchet derivative

## Definition

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be an open set, let $f \colon U \to W$, and let $x \in U$.
A bounded linear map $A \colon V \to W$ is a Fréchet derivative of $f$ at $x$ when
$$\lim_{h \to 0} \frac{\|f(x + h) - f(x) - A h\|_W}{\|h\|_V} = 0.$$
The limit is over $h \in V \setminus \{0\}$ with $x + h \in U$.
When such an $A$ exists, $f$ is Fréchet differentiable at $x$, with the notation $Df(x) = A$ or $f'(x) = A$.

## Remarks

The Fréchet derivative is unique when it exists, see the [[uniqueness of the derivative|0762WAE49P]].
For $V = W = \mathbb{R}$, a linear map $A$ is multiplication by a number $a$. Then the condition says that $f$ has the derivative $a$ at $x$ in the usual sense.
For $V = \mathbb{R}^n$ and $W = \mathbb{R}^m$, each linear map is bounded, and $Df(x)$ is the [[total derivative|0FNKFT8H6E]] of $f$ at $x$. Its matrix is the [[Jacobian matrix|047NDTEPCV]].
If $f$ is Fréchet differentiable at $x$, then $f$ is continuous at $x$.

## Explanations

### L1 human

Near $x$, the change $f(x + h) - f(x)$ is close to $A h$, a linear function of the step $h$.
The error is small even compared with the length of the step, when the step is short.
So $A$ plays the role of the slope of a tangent line, but now the tangent object is a linear map.
</document>

<document source="https://math2.tamnd.com/node/08EWT7J0SC.md" tag="08EWT7J0SC" kind="definition">
# Complete metric space

## Definition

A [[metric space|063W5JMY85]] $(X, d)$ is complete when each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ [[converges|06JYPJ4WHH]] to a point of $X$.

## Remarks

The real line $\mathbb{R}$ with the usual metric is complete. So is $\mathbb{R}^n$ with the Euclidean metric.
The rationals $\mathbb{Q}$ and the open interval $(0, 1)$ are not complete. In $(0, 1)$, the sequence $1/n$ is Cauchy but has no limit in $(0, 1)$.
Completeness depends on the metric, not only on the topology: $(0, 1)$ is homeomorphic to $\mathbb{R}$, but only $\mathbb{R}$ is complete with the usual metric.
Each metric space has a [[completion|0G4J7Z7WPJ]].

## Explanations

### L1 human

A complete space has no holes: each sequence that looks convergent really has a limit in the space.
The rationals have holes, for example at $\sqrt{2}$, and the reals fill them.
</document>

<document source="https://math2.tamnd.com/node/08FDFEF8D1.md" tag="08FDFEF8D1" kind="statement">
# Product of cyclic groups is cyclic iff the orders are coprime

## Statement

Let $G$ and $H$ be finite [[cyclic groups|0KVGH0J5B3]] of orders $m$ and $n$.
Let $G \times H$ be the group of pairs $(g, h)$ with $g \in G$ and $h \in H$, with the operation $(g, h)(g', h') = (gg', hh')$.
Then $G \times H$ is cyclic if and only if $\gcd(m, n) = 1$.

## Remarks

When $\gcd(m, n) = 1$, the group $\mathbb{Z}/mn\mathbb{Z}$ is isomorphic to $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, and this is a form of the Chinese remainder theorem.
When $\gcd(m, n) > 1$, each element of $G \times H$ has order at most $\operatorname{lcm}(m, n) < mn$.
For example, $\mathbb{Z}/12\mathbb{Z}$ is isomorphic to $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ is not cyclic.

## Explanations

### L1 human

The order of a pair $(g, h)$ is the least common multiple of the orders of $g$ and $h$.
A pair of generators has order $mn$ exactly when $m$ and $n$ have no common factor.
For example, $(1, 1)$ in $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ has order $6$, so this group is cyclic.
</document>

<document source="https://math2.tamnd.com/node/08FHNR8Z5D.md" tag="08FHNR8Z5D" kind="definition">
# Group of units of a monoid

## Definition

Let $(M, \ast, e)$ be a [[monoid|0N6R902FG1]].
An element $u \in M$ is a unit when it has a two-sided [[inverse|0ZNZH0EXFX]], that is, an element $v \in M$ with $u \ast v = v \ast u = e$.
The group of units of $M$ is the set $M^\times$ of the units of $M$, with the operation $\ast$ restricted to $M^\times$.
It is a [[group|G5S1M9]] with identity $e$.

## Remarks

The units of the monoid $(\mathbb{Z}, \cdot)$ are $1$ and $-1$.
The units of the monoid of $n \times n$ real matrices under multiplication are the invertible matrices, and they form the general linear group $\mathrm{GL}_n(\mathbb{R})$.
For a ring $R$, the group of units $R^\times$ is the group of units of the monoid $(R, \cdot)$.

## Explanations

### L1 human

In a monoid, some elements can be undone and others cannot.
The elements that can be undone are the units.
The units alone form a group, because products and inverses of units are units.
</document>

<document source="https://math2.tamnd.com/node/08FKQD7NJ3.md" tag="08FKQD7NJ3" kind="statement">
# Homomorphisms preserve identity and inverses

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] with identity elements $e_G$ and $e_H$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then $\varphi(e_G) = e_H$, and $\varphi(a^{-1}) = \varphi(a)^{-1}$ for each $a \in G$.

## Remarks

The first equation follows from $\varphi(e_G) \varphi(e_G) = \varphi(e_G) = \varphi(e_G) e_H$ and [[cancellation|0VSNX4P2Q0]].
The second follows from $\varphi(a) \varphi(a^{-1}) = \varphi(e_G) = e_H$ and [[the uniqueness of inverses|0D9NDGE9KP]].
A consequence is that $\varphi(a^n) = \varphi(a)^n$ for each integer $n$.

## Explanations

### L1 human

A map that respects products automatically respects the identity and the inverses.
You do not need to ask for these two properties in the definition of a homomorphism.
For example, the exponential map sends $0$ to $1$ and $-x$ to $1/e^x$.
</document>

<document source="https://math2.tamnd.com/node/08FPABDQ4C.md" tag="08FPABDQ4C" kind="definition">
# Uniform convergence

## Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions.
The sequence $(f_n)$ converges uniformly to $f$ if for every $\varepsilon > 0$ there is $N$ such that
$$|f_n(x) - f(x)| < \varepsilon \quad \text{for all } n \ge N \text{ and all } x \in S.$$

## Remarks

Here $N$ depends only on $\varepsilon$, not on $x$. Uniform convergence implies [[pointwise convergence|056FSKG32J]].
Equivalently, $\sup_{x \in S} |f_n(x) - f(x)| \to 0$ as $n \to \infty$.
For example, $\frac{\sin(n x)}{n} \to 0$ uniformly on $\mathbb{R}$. Mathlib defines it as `TendstoUniformly`.

## Explanations

### L1 human

Uniform convergence means that the whole graph of $f_n$ lies in a thin band around the graph of $f$.
The width of the band tends to $0$, at the same rate for all points.
</document>

<document source="https://math2.tamnd.com/node/08G4XH86JC.md" tag="08G4XH86JC" kind="statement">
# Product of countable sets is countable

## Statement

Let $A$ and $B$ be [[countable sets|0ZG61TFBES]].
Then the [[Cartesian product|0C81YBVTWH]] $A \times B$ is countable.

## Remarks

Proof: take injections $f \colon A \to \mathbb{N}$ and $g \colon B \to \mathbb{N}$. Then $(a, b) \mapsto \pi(f(a), g(b))$ is an injection to $\mathbb{N}$, where $\pi$ is the [[Cantor pairing function|0NP5N5PYW5]].
By induction, a finite product $A_1 \times \dots \times A_n$ of countable sets is countable.
An infinite product need not be countable: the set of sequences of $0$ and $1$ is [[uncountable|0X55M9XET4]].

## Explanations

### L1 human

The pairs of natural numbers fill an infinite grid, and the grid can be listed diagonal by diagonal.
So a product of two lists is again a list.
</document>

<document source="https://math2.tamnd.com/node/08GR54CKEA.md" tag="08GR54CKEA" kind="concept">
# Floor function

## Summary

The floor function sends a real number $x$ to the greatest integer that is less than or equal to $x$. We write $\lfloor x \rfloor$ for it.
For example, $\lfloor 2.7 \rfloor = 2$ and $\lfloor -2.7 \rfloor = -3$.
</document>

<document source="https://math2.tamnd.com/node/08HJRTG5N7.md" tag="08HJRTG5N7" kind="definition">
# Uniform convergence on a set

## Definition

Let $D$ be a set, let $f_n \colon D \to \mathbb{R}$ for $n \ge 1$ and $f \colon D \to \mathbb{R}$ be functions, and let $S \subseteq D$.
The sequence $(f_n)$ converges uniformly to $f$ on $S$ if for every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$ and all $x \in S$.

## Remarks

This is [[uniform convergence|08FPABDQ4C]] of the restrictions of $f_n$ and $f$ to $S$.
A sequence can converge uniformly on each $[0, r]$ with $r < 1$, but not on $[0, 1)$. The sequence $x^n$ is an example.
Mathlib defines it as `TendstoUniformlyOn`.

## Explanations

### L1 human

We often get uniform convergence only on part of the domain.
This definition measures the distance between $f_n$ and $f$ only on that part.
</document>

<document source="https://math2.tamnd.com/node/08HK38GCSM.md" tag="08HK38GCSM" kind="definition">
# Exterior algebra

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $T(V) = \bigoplus_{k \ge 0} V^{\otimes k}$ be the tensor algebra of $V$, where $V^{\otimes 0} = F$ and the product is the [[tensor product|0BF2DE2SYK]].
Let $I$ be the two-sided ideal of $T(V)$ generated by the elements $v \otimes v$ for $v \in V$.
The *exterior algebra* of $V$ is the quotient algebra $\Lambda(V) = T(V) / I$.
The product in $\Lambda(V)$ is written $\wedge$.

## Remarks

For all $u, v \in V$, $v \wedge v = 0$ and $u \wedge v = -v \wedge u$.
The image of $V^{\otimes k}$ is the $k$-th exterior power $\Lambda^k(V)$; if $\dim V = n$, then $\dim \Lambda^k(V) = \binom{n}{k}$ and $\dim \Lambda(V) = 2^n$.
If $(e_1, \dots, e_n)$ is a basis of $V$, then $\Lambda^n(V)$ is spanned by $e_1 \wedge \cdots \wedge e_n$.
For $v_j = \sum_i a_{ij} e_i$, $v_1 \wedge \cdots \wedge v_n = \det(a_{ij})\, e_1 \wedge \cdots \wedge e_n$.

## Explanations

### L1 human

The exterior algebra multiplies vectors with the wedge product, which changes sign when we swap two factors.
In $\R^3$, $u \wedge v$ describes an oriented parallelogram, and $u \wedge v \wedge w$ an oriented volume.
The top wedge product of $n$ vectors gives the determinant.
</document>

<document source="https://math2.tamnd.com/node/08J2M1JYAG.md" tag="08J2M1JYAG" kind="object">
# Real numbers

## Description

The real numbers are the numbers that measure continuous quantities, such as lengths. We picture them as the points of a line.
We write $\mathbb{R}$ for the set of real numbers.
With addition, multiplication and its order, $\mathbb{R}$ is an [[ordered field|0QSPNASE50]].
This ordered field is complete: each nonempty subset of $\mathbb{R}$ that is bounded above has a [[least upper bound|0AXWET80YW]].
Up to a unique isomorphism, $\mathbb{R}$ is the only [[complete ordered field|0A9E6X0S3E]].

## Remarks

The set $\mathbb{R}$ contains the [[rational numbers|0NCJ5WXN6Z]] and also [[irrational numbers|080T2TXVQV]] such as $\sqrt{2}$ and $\pi$.
Two common constructions of $\mathbb{R}$ from $\mathbb{Q}$ use [[Dedekind cuts|04QNZZ4HHY]] or [[Cauchy sequences|0D8HE681SC]].
The set $\mathbb{R}$ is uncountable.

## Explanations

### L1 human

The real numbers fill the number line without gaps.
The rational numbers leave gaps, for example at $\sqrt{2}$, and the real numbers fill these gaps.
Each real number has a [[decimal expansion|06XY4X9430]], such as $3.14159\dots$ for $\pi$.
</document>

<document source="https://math2.tamnd.com/node/08J7R35B6Q.md" tag="08J7R35B6Q" kind="statement">
# Tutte's theorem

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$.
For $U \subseteq V$, let $o(G - U)$ be the number of [[connected components|01VVP10MWJ]] of $G - U$ with an odd number of vertices.
Then $G$ has a [[perfect matching|05ZPH8KF9K]] if and only if $o(G - U) \le |U|$ for every $U \subseteq V$.

## Remarks

The condition is necessary. In a perfect matching, each odd component of $G - U$ has a vertex that is matched to a vertex of $U$. These vertices of $U$ are distinct.
For $U = \varnothing$, the condition says that each component of $G$ has an even number of vertices.
The theorem is a special case of the Tutte–Berge formula for the size of a maximum matching.

## History

The theorem is named after William Thomas Tutte.

## Explanations

### L1 human

Delete some vertices, and count the pieces with an odd number of vertices that remain.
Each odd piece needs a partner among the deleted vertices, and Tutte's theorem says this is the only obstruction.
</document>

<document source="https://math2.tamnd.com/node/08JF1K4DW1.md" tag="08JF1K4DW1" kind="definition">
# Fixed points of an action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
A point $x \in X$ is a fixed point of the action when $g \cdot x = x$ for each $g \in G$.
The set of all fixed points is written $X^G$.
For $g \in G$, the set of points fixed by $g$ is $X^g = \{x \in X : g \cdot x = x\}$.

## Remarks

A point $x$ is a fixed point if and only if its [[orbit|0YQPZB5Q1G]] is $\{x\}$, if and only if its [[stabiliser|0AYMP41THN]] is $G$.
For the action of $G$ on itself by conjugation, the fixed points are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
The sets $X^g$ appear in [[Burnside's lemma|0Z47M1B2VS]].

## Explanations

### L1 human

A fixed point is a point that no element of the group moves.
For the rotations of a square about its centre, the centre is a fixed point.
The corners are not fixed points, because a quarter turn moves them.
</document>

<document source="https://math2.tamnd.com/node/08KEYFAH66.md" tag="08KEYFAH66" kind="concept">
# Cyclotomic polynomial

## Summary

The $n$-th cyclotomic polynomial is the monic polynomial whose roots are the primitive $n$-th roots of unity, each with multiplicity one.
It has integer coefficients.
The [[definition of the cyclotomic polynomials|0W0KVBM54S]] gives the formula.
</document>

<document source="https://math2.tamnd.com/node/08MG7H8JW5.md" tag="08MG7H8JW5" kind="concept">
# Group isomorphism

## Summary

A group isomorphism is a bijective group homomorphism.
Two isomorphic groups have the same structure, and differ only in the names of their elements.
</document>

<document source="https://math2.tamnd.com/node/08N6EVA7DV.md" tag="08N6EVA7DV" kind="statement">
# Distributive laws for sets

## Statement

Let $A$, $B$ and $C$ be [[sets|0SJ3RX9TXA]].
Then intersection distributes over union, and union distributes over intersection:
$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C),$$
$$A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$$

## Remarks

The laws follow from the distributive laws of logic for $\land$ and $\lor$.
The second law has no analogue for numbers: $a + bc$ is not $(a + b)(a + c)$ in general.

## Explanations

### L1 human

To prove the first law, take $x$ in the left side. Then $x \in A$, and $x \in B$ or $x \in C$.
In the first case $x \in A \cap B$, and in the second case $x \in A \cap C$.
The other inclusion is similar.
</document>

<document source="https://math2.tamnd.com/node/08NM8KQPDQ.md" tag="08NM8KQPDQ" kind="definition">
# Infinite limit

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D$.
The notation $f(x) \to +\infty$ as $x \to a$ means that for each real $M$ there is $\delta > 0$ with $f(x) > M$ for all $x \in D$ with $0 < |x - a| < \delta$.
The definition of $f(x) \to -\infty$ is the same, with $f(x) < M$.

## Remarks

An infinite limit is not a limit in the sense of [[limit of a function|0TXTSEMHR5]], which requires a real value $L$.
For example, $1/x^2 \to +\infty$ as $x \to 0$.
The same idea gives $f(x) \to +\infty$ as $x \to +\infty$. An example is $f(x) = x^2$.

## Explanations

### L1 human

An infinite limit means that $f(x)$ becomes as large as we like when $x$ is close to $a$.
The graph then has a vertical asymptote at $x = a$.
</document>

<document source="https://math2.tamnd.com/node/08NYZ2FS6B.md" tag="08NYZ2FS6B" kind="definition">
# Norm

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$.
A *norm* on $V$ is a map $\norm{\cdot} \colon V \to \R$ such that, for all $x, y \in V$ and $a \in \mathbb{K}$:

1. $\norm{x + y} \le \norm{x} + \norm{y}$.
2. $\norm{ax} = \abs{a} \norm{x}$.
3. If $\norm{x} = 0$, then $x = 0$.

## Remarks

These conditions imply $\norm{x} \ge 0$ for all $x$.
On $\R^n$, the Euclidean norm $\sqrt{x_1^2 + \cdots + x_n^2}$, the norm $\abs{x_1} + \cdots + \abs{x_n}$ and the norm $\max_i \abs{x_i}$ are norms.
Each inner product gives a norm, see [[the induced norm|0MFJJV97RJ]].
A norm defines a metric $d(x, y) = \norm{x - y}$.

## Explanations

### L1 human

A norm measures the length of a vector.
The length of a sum is at most the sum of the lengths, and scaling a vector scales its length.
Only the zero vector has length zero.
</document>

<document source="https://math2.tamnd.com/node/08P6T2WH6R.md" tag="08P6T2WH6R" kind="example">
# Sequence spaces lp

## Example

Let $p \ge 1$ be a real number. Let $\ell^p$ be the set of real sequences $x = (x_n)_{n \ge 1}$ with $\sum_{n \ge 1} |x_n|^p < \infty$, and let
$$\|x\|_p = \left( \sum_{n \ge 1} |x_n|^p \right)^{1/p}.$$
Let $\ell^\infty$ be the set of bounded real sequences, with $\|x\|_\infty = \sup_{n \ge 1} |x_n|$.
Then each $\ell^p$, for $1 \le p \le \infty$, is a vector space, $\|\cdot\|_p$ is a norm on it, and $\ell^p$ is a [[Banach space|0BDTTM24DP]].

## Remarks

The norms extend the [[p-norms on R^n|0KNJWEVSMG]] to infinite sequences.
For $1 \le p < q \le \infty$, we have $\ell^p \subseteq \ell^q$, and the inclusion is strict. For example, $(1/n)$ is in $\ell^2$ but not in $\ell^1$.
The space $\ell^2$ has the inner product $\langle x, y \rangle = \sum_n x_n y_n$, and it is a Hilbert space.
The closed unit ball of $\ell^p$ is not compact: the unit vectors $e_k$ satisfy $\|e_j - e_k\|_p = 2^{1/p}$ for $j \neq k$ and $p < \infty$.

## Explanations

### L1 human

The spaces $\ell^p$ are infinite-dimensional versions of $\mathbb{R}^n$ with the p-norm.
A sequence belongs to $\ell^p$ when its terms go to $0$ fast enough for the sum of $|x_n|^p$ to be finite.
</document>

<document source="https://math2.tamnd.com/node/08PA437800.md" tag="08PA437800" kind="definition">
# Lower central series

## Definition

Let $G$ be a [[group|G5S1M9]].
For subgroups $H$ and $K$ of $G$, let $[H, K]$ be the subgroup [[generated by|0RP7VWQ0YN]] the [[commutators|0PXW0BN7DW]] $[h, k]$ with $h \in H$ and $k \in K$.
The lower central series of $G$ is the sequence $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$.

## Remarks

The second term $\gamma_2(G) = [G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].
Each $\gamma_i(G)$ is normal in $G$, and $\gamma_i(G)/\gamma_{i+1}(G)$ lies in the centre of $G/\gamma_{i+1}(G)$.
The terms of the [[derived series|0Q5G5DJ4P2]] satisfy $G^{(i)} \subseteq \gamma_{i+1}(G)$.
For $S_3$, the series is $S_3 \supseteq A_3 \supseteq A_3 \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

The lower central series starts with the whole group.
At each step, it takes the commutators of the last term with all elements of the group.
If this process reaches the identity, the group is nilpotent.
</document>

<document source="https://math2.tamnd.com/node/08PY4GNAPK.md" tag="08PY4GNAPK" kind="statement">
# Product formula for the order of HK

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be finite [[subgroups|0X1JXT47GA]] of $G$.
Then the [[product|0PT52XQZD7]] $HK$ has exactly $\dfrac{|H| \, |K|}{|H \cap K|}$ elements.

## Remarks

The formula holds also when $HK$ is not a subgroup.
The proof counts the pairs $(h, k)$: two pairs $(h, k)$ and $(h', k')$ give the same product exactly when $h' = hx$ and $k' = x^{-1}k$ for some $x \in H \cap K$.
For example, two distinct subgroups of order $2$ in $S_3$ meet in $\{e\}$, and their product has $2 \cdot 2 / 1 = 4$ elements.

## Explanations

### L1 human

Each product $hk$ can be written in several ways, because you can move an element of $H \cap K$ from one factor to the other.
Each element of $HK$ has exactly $|H \cap K|$ such ways.
So you divide the number of pairs, $|H| \, |K|$, by $|H \cap K|$.
</document>

<document source="https://math2.tamnd.com/node/08QA8QTM71.md" tag="08QA8QTM71" kind="statement">
# Picard's great theorem

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $c$ is an [[essential singularity|0YKDJBV26S]] of $f$.
Then for each $\delta$ with $0 < \delta < r$, there is at most one $w_0 \in \mathbb{C}$ with the following property. The function $f$ takes the value $w_0$ only finitely many times on $0 < |z - c| < \delta$.
Thus, on each punctured disc around $c$, $f$ takes each complex value infinitely often, with at most one exception.

## Remarks

The exception can occur: $e^{1/z}$ never takes the value $0$, and it takes each $w \neq 0$ infinitely often near $0$, see [[exp(1/z)|0NB31CC4SN]].
The theorem is much stronger than the [[Casorati–Weierstrass theorem|02HHNPC8ED]], which gives only that the values come close to each $w$.
Picard's little theorem is a related result: a nonconstant entire function takes all complex values with at most one exception.

## Explanations

### L1 human

Near an essential singularity, the function does not only come close to each value: it hits almost every value infinitely many times.
At most one value can be missed, as $0$ is missed by $e^{1/z}$.
</document>

<document source="https://math2.tamnd.com/node/08QAX4H1PF.md" tag="08QAX4H1PF" kind="example">
# The sequence 1/n tends to zero

## Example

Let $a_n = 1/n$ for each integer $n \ge 1$.
Then $a_n \to 0$ as $n \to \infty$.
Let $\varepsilon > 0$. By the Archimedean property of $\mathbb{R}$, there is a natural number $N$ with $N > 1/\varepsilon$.
For each $n \ge N$ we get $|a_n - 0| = 1/n \le 1/N < \varepsilon$.
By the [[epsilon-N definition|09G6KHS6GY]], this shows $1/n \to 0$.

## Remarks

The proof uses the Archimedean property, which says that the natural numbers are not bounded above in $\mathbb{R}$.
With the [[algebra of limits|07XBM3A9J5]], it gives many more limits, for example $1/n^2 \to 0$ and $(n + 1)/n \to 1$.
Mathlib states it for $1/n$ over $\mathbb{R}$ as `tendsto_one_div_atTop_nhds_zero_nat`.

## Explanations

### L1 human

The numbers $1, 1/2, 1/3, 1/4, \dots$ get smaller and smaller, and they approach $0$.
To get $1/n$ below $0.001$, it is enough to take $n > 1000$.
</document>

<document source="https://math2.tamnd.com/node/08QCA2F1NX.md" tag="08QCA2F1NX" kind="statement">
# Independence means the joint law is the product of the marginals

## Statement

Let $X$ and $Y$ be random variables on one probability space, with values in measurable spaces $(E_1, \mathcal{E}_1)$ and $(E_2, \mathcal{E}_2)$.
Let $\mu_X$ and $\mu_Y$ be their laws, and let $\mu_{(X, Y)}$ be the [[joint law|0QAR0XDZH4]] of the pair $(X, Y)$ on $E_1 \times E_2$.
Then $X$ and $Y$ are [[independent|0E9502NT92]] if and only if $\mu_{(X, Y)} = \mu_X \otimes \mu_Y$, the product measure of the two [[marginal laws|0AXP2Z5N9H]].

## Remarks

The statement is a direct translation of the definition: $\mu_{(X, Y)}(A \times B) = P(X \in A, Y \in B)$ and $(\mu_X \otimes \mu_Y)(A \times B) = \mu_X(A) \, \mu_Y(B)$.
Two probability measures on $E_1 \times E_2$ that agree on all rectangles $A \times B$ are equal, because the rectangles form a π-system that generates the product σ-algebra.
For real random variables, it follows that $F_{(X, Y)}(x, y) = F_X(x) \, F_Y(y)$. If $X$ and $Y$ have densities, the joint density is $f_X(x) \, f_Y(y)$.
The same result holds for $n$ random variables: they are independent if and only if their joint law is $\mu_{X_1} \otimes \dots \otimes \mu_{X_n}$.

## Explanations

### L1 human

For independent variables, the full joint picture is fixed by the two separate pictures.
You find the chance of a pair of outcomes by multiplying the chances of the two outcomes.
</document>

<document source="https://math2.tamnd.com/node/08QMKTWF9T.md" tag="08QMKTWF9T" kind="concept">
# Partition of a set

## Summary

A partition of a set divides it into nonempty pieces that do not overlap and that together cover the set.
</document>

<document source="https://math2.tamnd.com/node/08QQH9F7G0.md" tag="08QQH9F7G0" kind="statement">
# Number of functions between finite sets

## Statement

Let $A$ and $B$ be finite sets with $|A| = m$ and $|B| = n$.
Let $B^A$ be the set of all functions from $A$ to $B$.
Then $|B^A| = n^m$.

## Remarks

The case $m = 0$ gives $n^0 = 1$: there is exactly one function from the empty set to $B$, also when $B$ is empty.
A function from $A$ to $B$ is a choice of one value in $B$ for each of the $m$ elements of $A$. The [[rule of product|0VPK4X73Q0]] then gives $n^m$.
For a subset count, take $B = \{0, 1\}$. This gives the [[number of subsets of a finite set|0C5RDJE73X]].

## Explanations

### L1 human

Each element of $A$ independently chooses its image among the $n$ elements of $B$.
For example, there are $3^2 = 9$ functions from $\{1, 2\}$ to $\{a, b, c\}$, because $1$ has 3 choices and $2$ has 3 choices.
</document>

<document source="https://math2.tamnd.com/node/08R2V9GZF7.md" tag="08R2V9GZF7" kind="concept">
# Cesàro mean

## Summary

The Cesàro means of a sequence are the averages of its first $n$ terms. They are the base of Cesàro summation, a way to give values to some divergent series. The node [[Cesàro mean|0P0DJHBGBB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/08RQ62VAVA.md" tag="08RQ62VAVA" kind="concept">
# Number of distinct prime factors

## Summary

The function $\omega(n)$ counts the distinct prime numbers that divide a positive integer $n$.
The [[definition of the number of distinct prime factors|0YVVXT3A0B]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/08RQAPXF1T.md" tag="08RQAPXF1T" kind="definition">
# Liouville function

## Definition

The Liouville function $\lambda$ is the arithmetic function given by $\lambda(n) = (-1)^{\Omega(n)}$ for each positive integer $n$.
Here $\Omega(n)$ is the [[number of prime factors of n counted with multiplicity|08420828EY]].

## Remarks

For example, $\lambda(1) = 1$, $\lambda(12) = -1$ and $\lambda(36) = 1$.
The function $\lambda$ is [[completely multiplicative|0GS27TKF1V]], since $\Omega$ is completely additive.
The sum $\sum_{d \mid n} \lambda(d)$ is $1$ if $n$ is a perfect square, and $0$ otherwise.
On squarefree integers, $\lambda$ agrees with the [[Möbius function|0C8QENHYZW]].

## Explanations

### L1 human

The Liouville function is $1$ when $n$ has an even number of prime factors, counted with repeats, and $-1$ when the number is odd.
Unlike the Möbius function, it is never $0$.
</document>

<document source="https://math2.tamnd.com/node/08RT9GDDPQ.md" tag="08RT9GDDPQ" kind="concept">
# Power series solution

## Summary

The power series method looks for a solution of a differential equation in the form $y = \sum_{k \ge 0} a_k (x - x_0)^k$.
Consider $y'' + p(x) y' + q(x) y = 0$, where $p$ and $q$ are analytic at $x_0$: they equal convergent power series near $x_0$.
One puts the series into the equation and compares the coefficients of each power $(x - x_0)^k$. This gives a recurrence for the $a_k$, with $a_0$ and $a_1$ free.
The resulting series converge at least on each disc around $x_0$ where the series of $p$ and $q$ converge, and they give two independent solutions.

## Remarks

Example: for $y' = y$, the series $\sum a_k x^k$ gives $(k + 1) a_{k+1} = a_k$, so $a_k = a_0 / k!$ and $y = a_0 e^x$.
Example: for $y'' + y = 0$, the recurrence is $a_{k+2} = -a_k / \left((k+1)(k+2)\right)$. It gives the series of $\cos x$ and $\sin x$.
At a point where a leading coefficient vanishes, the Frobenius method, a variant with a factor $(x - x_0)^r$, is used.

## Explanations

### L1 human

Write the unknown function as a power series with unknown coefficients.
The equation turns into a rule that gives each coefficient from the earlier ones.
So you can compute the solution term by term, even when there is no closed formula.
</document>

<document source="https://math2.tamnd.com/node/08RTDHX1SK.md" tag="08RTDHX1SK" kind="definition">
# Finite field

## Definition

A finite field is a [[field|0QA8CHH8MG]] $F$ whose underlying set is finite.
The order of $F$ is its number of elements $|F|$.

## Remarks

For each prime number $p$, the ring $\mathbb{Z}/p\mathbb{Z}$ is a [[finite field|0NV6E43P4A]] of order $p$.
The [[order of a finite field is a prime power|0H5WWC8RZX]] $p^n$, where $p$ is the characteristic.
The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ is a finite field of order $4$. It is not $\mathbb{Z}/4\mathbb{Z}$, which is not a field.
Finite fields are used in number theory, algebraic geometry, cryptography and coding theory.

## Explanations

### L1 human

A finite field is a finite set where you can add, subtract, multiply and divide by nonzero elements, with the usual rules.
The simplest ones are the integers modulo a prime.
</document>

<document source="https://math2.tamnd.com/node/08S0NW8SES.md" tag="08S0NW8SES" kind="concept">
# Normal extension

## Summary

A normal extension is an algebraic extension $L/K$ such that every irreducible polynomial over $K$ with a root in $L$ splits over $L$.
The [[definition of a normal extension|0CBRFQ7DVZ]] states it.
</document>

<document source="https://math2.tamnd.com/node/08S7FDRS5Z.md" tag="08S7FDRS5Z" kind="definition">
# Differentiable on an open set

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set and let $f \colon U \to \mathbb{R}^m$.
The function $f$ is differentiable on $U$ when it is [[differentiable|0135XZZQ8M]] at each point $a \in U$.
Then the derivative of $f$ is the map $Df \colon U \to L(\mathbb{R}^n, \mathbb{R}^m)$, $a \mapsto Df(a)$, where $L(\mathbb{R}^n, \mathbb{R}^m)$ is the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$.

## Remarks

The function $f$ is continuously differentiable, or of class $C^1$, on $U$ when it is differentiable on $U$ and $Df$ is continuous. Here $L(\mathbb{R}^n, \mathbb{R}^m)$ carries the topology of $m \times n$ matrices.
For a set $S$ that is not open, one uses derivatives within $S$ instead. On an open set the two notions agree.
Each polynomial map $\mathbb{R}^n \to \mathbb{R}^m$ is differentiable on $\mathbb{R}^n$.

## Explanations

### L1 human

Differentiable on $U$ means that there is a good linear approximation at each point of $U$, not only at one point.
The approximation changes from point to point, so the derivative is itself a function on $U$.
</document>

<document source="https://math2.tamnd.com/node/08SJ5K2CBG.md" tag="08SJ5K2CBG" kind="definition">
# Commutator subgroup

## Definition

Let $G$ be a [[group|G5S1M9]].
The commutator subgroup of $G$ is the [[subgroup generated by|0RP7VWQ0YN]] the set of all [[commutators|0PXW0BN7DW]] $[g, h]$ with $g, h \in G$.
It is written $[G, G]$ or $G'$.

## Remarks

A product of commutators need not be a commutator, so in general the set of commutators is not a subgroup.
The subgroup $[G, G]$ is [[characteristic|0KABPMRHM2]], so it is normal in $G$.
The group $G$ is abelian if and only if $[G, G] = \{e\}$.
The commutator subgroup of the symmetric group $S_n$ is the alternating group $A_n$.

## Explanations

### L1 human

The commutator subgroup collects all the failures of commutativity in a group.
If you divide the group by it, you get an abelian group.
A large commutator subgroup means a group that is far from abelian.
</document>

<document source="https://math2.tamnd.com/node/08SP3DF8NB.md" tag="08SP3DF8NB" kind="concept">
# Permutation matrix

## Summary

A permutation matrix is a square matrix with exactly one entry $1$ in each row and each column, and all other entries $0$.
It permutes the coordinates of a vector.
</document>

<document source="https://math2.tamnd.com/node/08SVRVHJNY.md" tag="08SVRVHJNY" kind="definition">
# Topological embedding

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a topological embedding when $f$ is [[injective|0PVG1CRAJN]] and the map $X \to f(X)$, $x \mapsto f(x)$, is a [[homeomorphism|05GZ11PWRY]].
Here the [[image|0Q3F20XB0A]] $f(X)$ has the [[subspace topology|0D5YHQGDJY]] of $Y$.

## Remarks

Each embedding is continuous, because the [[inclusion of a subspace is continuous|065J822CKN]].
The inclusion of a subspace is an embedding. The map $\mathbb{R} \to \mathbb{R}^2$, $x \mapsto (x, 0)$, is an embedding.
The map $t \mapsto (\cos t, \sin t)$ from $[0, 2\pi)$ to $\mathbb{R}^2$ is continuous and injective, but not an embedding.
A continuous injective map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is an embedding.

## Explanations

### L1 human

An embedding puts a copy of $X$ inside $Y$ without changing its shape.
An injective continuous map can still bring the ends of the space together in the limit. Then it is not an embedding.
</document>

<document source="https://math2.tamnd.com/node/08V7JHS3QZ.md" tag="08V7JHS3QZ" kind="statement">
# Order of an element divides the order of the group

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $g \in G$.
Then the [[order|0WE2NHZJ7B]] of $g$ divides the [[order|0B85S2V2DR]] $|G|$ of $G$.

## Remarks

The order of $g$ is the order of the cyclic subgroup $\langle g \rangle$, [[by this fact|031M8HX7DA]], and [[Lagrange's theorem|3C53TD]] applies to $\langle g \rangle$.
For example, in a group of order $10$, each element has order $1, 2, 5$ or $10$.

## Explanations

### L1 human

An element generates a subgroup with as many elements as its order.
The size of a subgroup always divides the size of the group.
So the order of an element divides the order of the group.
</document>

<document source="https://math2.tamnd.com/node/08VRX3Q4PB.md" tag="08VRX3Q4PB" kind="concept">
# Semisimple endomorphism

## Summary

An endomorphism of a finite-dimensional vector space is semisimple when each invariant subspace has an invariant complement.
Over an algebraically closed field, this means that it is diagonalisable.
</document>

<document source="https://math2.tamnd.com/node/08VV7YJRB7.md" tag="08VV7YJRB7" kind="definition">
# Content of a polynomial

## Definition

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$.
The content of $f$, written $c(f)$, is a [[greatest common divisor|0F7KNJ1ZYN]] of the coefficients $a_0, a_1, \dots, a_n$.

## Remarks

The content is unique up to associates. Over $\mathbb{Z}$, one often takes the positive gcd, with the sign of the leading coefficient in the primitive part.
For $f = 6x^2 + 4x - 2$ in $\mathbb{Z}[x]$, $c(f) = 2$.
Each nonzero $f$ is $c(f) \cdot g$ with $g$ a [[primitive polynomial|0SP14DZN0G]], the primitive part of $f$.
The content is multiplicative up to units: $c(f g) \sim c(f) c(g)$. This is a form of [[Gauss's lemma|0QFV5Y1Z5X]].

## Explanations

### L1 human

The content is the largest common factor that you can pull out of all the coefficients.
For $6x^2 + 4x - 2$, you can pull out $2$ and get $2(3x^2 + 2x - 1)$.
</document>

<document source="https://math2.tamnd.com/node/08VZ57PT26.md" tag="08VZ57PT26" kind="concept">
# Uniform convergence

## Summary

A sequence of functions converges uniformly when the largest distance between $f_n$ and the limit tends to zero. Uniform limits keep continuity, and they commute with integration. The node [[uniform convergence|08FPABDQ4C]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/08W2CW8PKW.md" tag="08W2CW8PKW" kind="definition">
# Median

## Definition

Let $X$ be a real random variable on a probability space $(\Omega, \mathcal{F}, P)$.
A median of $X$ is a real number $m$ with
$$P(X \le m) \ge \frac{1}{2} \quad \text{and} \quad P(X \ge m) \ge \frac{1}{2}.$$

## Remarks

Each real random variable has a median, and the medians form a closed bounded interval. For example, $Q(1/2)$ is a median, where $Q$ is the [[quantile function|0B3Q3XDM5X]].
The median can be different from the mean. For the exponential distribution with rate $\lambda$, the median is $\ln 2 / \lambda$ and the mean is $1/\lambda$.
For a fair coin coded as $0$ and $1$, each $m \in [0, 1]$ is a median.
For a sorted data set $x_1 \le \dots \le x_n$ with $n$ odd, the median is $x_{(n+1)/2}$. For $n$ even, one usually takes the mean of the two middle values.

## Explanations

### L1 human

The median is the middle value: half of the chance lies on each side of it.
The median of the incomes in a country is often more useful than the mean.
A few very large incomes do not move the median much.
</document>

<document source="https://math2.tamnd.com/node/08W5RN375X.md" tag="08W5RN375X" kind="statement">
# Power series of cosine

## Statement

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$ converges absolutely, and
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots.$$
This series is the [[Maclaurin series|043SJ5RF59]] of the cosine function.

## Remarks

The [[ratio of the terms|0JBMYAA9QH]] is $\frac{x^2}{(2n+1)(2n+2)} \to 0$. So the radius of convergence is $\infty$.
For $|x| \le 1$, the terms decrease in size. So the partial sums lie alternately above and below $\cos x$.
Mathlib states the series as `Real.hasSum_cos`.

## Explanations

### L1 human

Cosine is an infinite even polynomial with alternating signs.
Near $0$, $\cos x$ is close to $1 - x^2/2$.
</document>

<document source="https://math2.tamnd.com/node/08WCB90WND.md" tag="08WCB90WND" kind="statement">
# Hall's marriage theorem

## Statement

Let $\mathcal{F} = (S_1, \dots, S_n)$ be a finite family of finite sets.
A transversal of $\mathcal{F}$ is an injective function $f : \{1, \dots, n\} \to \bigcup_i S_i$ with $f(i) \in S_i$ for every $i$.
Then $\mathcal{F}$ has a transversal if and only if $\left| \bigcup_{i \in I} S_i \right| \ge |I|$ for every $I \subseteq \{1, \dots, n\}$.

## Remarks

The condition on all subfamilies is the marriage condition. It is necessary, because a transversal $f$ maps $I$ injectively into $\bigcup_{i \in I} S_i$.
A transversal is also called a system of distinct representatives. The sets $S_i$ may repeat in the family.
The theorem is equivalent to [[Hall's theorem for bipartite graphs|0X2GJPNPFN]].

## History

Philip Hall proved the theorem in 1935.
The name marriage theorem comes from a 1950 paper by Halmos and Vaughan.

## Explanations

### L1 human

Each of $n$ people has a list of acceptable partners, and we want to give each person a different partner from the list.
This is possible exactly when every group of $k$ people has at least $k$ names on their lists together.
</document>

<document source="https://math2.tamnd.com/node/08WE2W7DYS.md" tag="08WE2W7DYS" kind="statement">
# Cauchy–Binet formula

## Statement

Let $K$ be a commutative ring, let $m \le n$, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$.
For a subset $S \subseteq \{1, \dots, n\}$ with $m$ elements, let $A_S$ be the $m \times m$ matrix of the columns of $A$ in $S$.
Let $B^S$ be the $m \times m$ matrix of the rows of $B$ in $S$.
Then
$$\det(AB) = \sum_{S} \det(A_S) \det(B^S),$$
where the sum runs over all subsets $S$ of $\{1, \dots, n\}$ with $m$ elements.

## Remarks

For $m = n$, the sum has one term, and the formula is $\det(AB) = \det A \det B$.
If $m > n$, then $\det(AB) = 0$.
For $m = 1$, it is the formula for the product of a row and a column.
The determinants $\det(A_S)$ and $\det(B^S)$ are [[minors|0BT9BYWVRY]] of order $m$.

## Explanations

### L1 human

The Cauchy–Binet formula computes the determinant of a product of two rectangular matrices.
It sums products of matching minors of the two factors.
It generalizes the product rule for determinants of square matrices.
</document>

<document source="https://math2.tamnd.com/node/08WFZP6P89.md" tag="08WFZP6P89" kind="concept">
# Sine and cosine

## Summary

Sine and cosine are the basic trigonometric functions. For an angle $x$, the point $(\cos x, \sin x)$ lies on the unit circle, and the two functions are periodic with period $2\pi$. The node [[sine and cosine|0WB57D9488]] gives the precise definition by power series.
</document>

<document source="https://math2.tamnd.com/node/08WTN6SCQ6.md" tag="08WTN6SCQ6" kind="statement">
# Euler's four-square identity

## Statement

Let $a_1, \dots, a_4$ and $b_1, \dots, b_4$ be elements of a commutative ring. Then
$$\begin{aligned} (a_1^2 + a_2^2 + a_3^2 + a_4^2)(b_1^2 + b_2^2 + b_3^2 + b_4^2) &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\ &\quad + (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2. \end{aligned}$$

## Remarks

So the product of two sums of four squares is again a sum of four squares.
The signs come from the product of two quaternions: the identity says that the norm of quaternions is multiplicative.
Lagrange used it to reduce the [[four-square theorem|0KYYSMEVHH]] to the case of primes.

## History

Euler wrote about the identity in a letter to Goldbach of 4 May 1748.

## Explanations

### L1 human

This identity is the four-square analogue of the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]] for two squares.
You can check it by expanding both sides, since all the cross terms cancel.
</document>

<document source="https://math2.tamnd.com/node/08WWTN738R.md" tag="08WWTN738R" kind="concept">
# Explicit and implicit differential equations

## Summary

An [[ordinary differential equation|0P97V20RW7]] of [[order|0XK0G19SK8]] $n$ is explicit when it has the form $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ for a given function $f$.
It is implicit when it has the general form $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ and is not solved for $y^{(n)}$.
Each explicit equation is also implicit, with $F = y^{(n)} - f$.

## Remarks

The equation $y' = x^2 + y^2$ is explicit. The equation $(y')^2 + y^2 = 1$ is implicit.
If $\partial F / \partial y^{(n)}$ is not zero at a point, the implicit function theorem gives an explicit equation near that point.
The basic existence theorems, such as the [[Picard–Lindelöf theorem|0P6KRD5B7A]], are for explicit equations.

## Explanations

### L1 human

An explicit equation tells you the highest derivative directly, as a formula in the lower ones.
An implicit equation only gives a relation, and you may have to solve it for $y'$ first.
For $(y')^2 + y^2 = 1$ you get two explicit equations, $y' = \sqrt{1 - y^2}$ and $y' = -\sqrt{1 - y^2}$.
</document>

<document source="https://math2.tamnd.com/node/08X0VCR2NW.md" tag="08X0VCR2NW" kind="concept">
# Square-free integer

## Summary

A square-free integer is an integer that no perfect square other than $1$ divides.
The [[definition of a square-free integer|004JW7A2N5]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/08XDXP86XE.md" tag="08XDXP86XE" kind="statement">
# An iterate of the Picard operator is a contraction

## Statement

Let $f$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ on $J = [t_{\min}, t_{\max}]$ with constants $a, L, K$.
Let $X$ be the set of continuous curves $\gamma : J \to \mathbb{R}^n$ with $\|\gamma(t) - x_0\| \le a$ for each $t \in J$, with the distance $d(\gamma, \delta) = \sup_{t \in J} \|\gamma(t) - \delta(t)\|$.
Let $P : X \to X$ be the [[Picard operator|00WAVB68F9]] and let $T = \max(t_{\max} - t_0,\ t_0 - t_{\min})$.
Then for each integer $m \ge 0$ and all $\gamma, \delta \in X$,
$$d\left(P^m \gamma, P^m \delta\right) \le \frac{(K T)^m}{m!}\, d(\gamma, \delta).$$
In particular, there is an $m \ge 1$ such that $P^m$ is a contraction of $X$: $(KT)^m / m! < 1$.

## Remarks

Proof idea: by induction on $m$, $\|(P^m \gamma)(t) - (P^m \delta)(t)\| \le \frac{(K |t - t_0|)^m}{m!}\, d(\gamma, \delta)$ for each $t \in J$. The induction step integrates the bound of the previous step.
The space $X$ is a complete metric space. So the Banach fixed point theorem applies to $P^m$, and it gives a unique fixed point of $P$.
If $K T < 1$, then $P$ itself is a contraction. The power $P^m$ avoids the need for a short time interval.

## Explanations

### L1 human

A contraction brings any two points closer by a fixed factor smaller than one.
The Picard operator may not do this in one step, but after enough steps it does, because $m!$ grows faster than $(KT)^m$.
That is enough to get a unique fixed point.
</document>

<document source="https://math2.tamnd.com/node/08XFJJ8EYW.md" tag="08XFJJ8EYW" kind="definition">
# Comparison of topologies

## Definition

Let $X$ be a set and let $\tau_1$ and $\tau_2$ be [[topologies|0CNQXVBHCX]] on $X$.
If $\tau_1 \subseteq \tau_2$, then $\tau_1$ is coarser (or weaker) than $\tau_2$, and $\tau_2$ is finer (or stronger) than $\tau_1$.
If also $\tau_1 \neq \tau_2$, then $\tau_1$ is strictly coarser and $\tau_2$ is strictly finer.

## Remarks

On each set $X$, the [[indiscrete topology|0XJ02JBD6S]] is the coarsest topology and the [[discrete topology|0S2TEHX4E1]] is the finest topology.
We have $\tau_1 \subseteq \tau_2$ if and only if the identity map $(X, \tau_2) \to (X, \tau_1)$ is [[continuous|0SY9ECQ19Q]].
Two topologies need not be comparable: on $\{a, b\}$, the topologies $\{\varnothing, \{a\}, X\}$ and $\{\varnothing, \{b\}, X\}$ are not comparable.

## Explanations

### L1 human

A finer topology has more open sets, so it can separate points in more ways.
A coarser topology has fewer open sets, so more sequences converge and more maps into it are continuous.
</document>

<document source="https://math2.tamnd.com/node/08Y1BBQTK9.md" tag="08Y1BBQTK9" kind="statement">
# Integration by parts

## Statement

Let $a < b$ and let $u, v \colon [a, b] \to \mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]].
Then
$$\int_a^b u(x) v'(x) \, dx = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx.$$

## Remarks

By the [[product rule|0V9A43H7WV]], $uv$ is an antiderivative of $u v' + u' v$. Apply the [[second fundamental theorem|0XYAFCN0PT]].
For indefinite integrals, the formula is $\int u \, dv = u v - \int v \, du$.
For example, $\int_0^1 x e^x \, dx = e - (e - 1) = 1$.
Mathlib states it as `intervalIntegral.integral_mul_deriv_eq_deriv_mul`.

## Explanations

### L1 human

Integration by parts is the product rule read backwards.
It moves the derivative from one factor to the other, at the cost of a boundary term.
</document>

<document source="https://math2.tamnd.com/node/08YERZDM9X.md" tag="08YERZDM9X" kind="statement">
# Subfields of a finite field

## Statement

Let $p$ be a prime number and let $n \ge 1$.
For each divisor $m$ of $n$, the field $\mathbb{F}_{p^n}$ has exactly one [[subfield|07QY5RPFBR]] of order $p^m$, namely $\{ a : a^{p^m} = a \}$.
It has no other subfields.

## Remarks

If $K$ is a subfield of $\mathbb{F}_{p^n}$ of order $p^m$, then $n = m \cdot [\mathbb{F}_{p^n} : K]$ by the [[tower law|0A7KJ68XHT]], so $m$ divides $n$.
For example, the subfields of $\mathbb{F}_{64} = \mathbb{F}_{2^6}$ are $\mathbb{F}_2$, $\mathbb{F}_4$, $\mathbb{F}_8$ and $\mathbb{F}_{64}$.
The subfield of order $p^m$ is the fixed field of $\varphi^m$, where $\varphi$ is the [[Frobenius automorphism|0VP4TBVJ19]].

## Explanations

### L1 human

The subfields of a finite field match the divisors of the exponent.
The field with $2^6$ elements contains the fields with $2^1$, $2^2$ and $2^3$ elements, but not one with $2^4$ elements.
</document>

<document source="https://math2.tamnd.com/node/08ZDRN6H41.md" tag="08ZDRN6H41" kind="statement">
# Steinitz exchange lemma

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $u_1, \dots, u_m \in V$ be [[linearly independent|0CRMGPAH83]], and let $w_1, \dots, w_n \in V$ [[span|088RYKAWK9]] $V$.
Then $m \le n$.
Moreover, after a renumbering of the $w_j$, the family $u_1, \dots, u_m, w_{m+1}, \dots, w_n$ spans $V$.

## Remarks

The lemma follows by induction on $m$: each $u_i$ replaces one of the $w_j$ in a spanning family.
A consequence is that any two finite bases of $V$ have the same number of elements.
The lemma is named after Ernst Steinitz.

## Explanations

### L1 human

An independent family can never have more vectors than a spanning family.
We can trade the vectors of the spanning family, one by one, for the independent vectors, and the family still spans.
This lemma is the reason why the dimension of a space is well defined.
</document>

<document source="https://math2.tamnd.com/node/08ZDW5SA3B.md" tag="08ZDW5SA3B" kind="concept">
# Identically distributed random variables

## Summary

Random variables are identically distributed when they have the same law, even if they live on different probability spaces.
The node [[identically distributed random variables|01X8YSDTBW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/08ZQAKBKHF.md" tag="08ZQAKBKHF" kind="definition">
# Alternating bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *alternating* when
$$B(v, v) = 0 \qquad \text{for all } v \in V.$$

## Remarks

An alternating form is *skew-symmetric*: $B(u, v) = -B(v, u)$ for all $u, v$.
If the characteristic of $F$ is not $2$, each skew-symmetric form is alternating.
On $F^2$, the form $B(x, y) = x_1 y_2 - x_2 y_1$ is alternating; it is the determinant of the matrix with the columns $x$ and $y$.
A nondegenerate alternating form is a *symplectic form*.

## Explanations

### L1 human

An alternating form gives zero when both vectors are equal.
Then swapping the two vectors changes the sign of the value.
The signed area of the parallelogram spanned by two plane vectors is an example.
</document>

<document source="https://math2.tamnd.com/node/08ZWFDJP3H.md" tag="08ZWFDJP3H" kind="concept">
# Eigenvalue method for linear systems

## Summary

The eigenvalue method solves a [[homogeneous linear system with constant coefficients|0VJF99CZ0Z]] $x' = A x$ with the eigenvalues and eigenvectors of $A$.
If $A v = \lambda v$ with $v \ne 0$, then $x(t) = e^{\lambda t} v$ is a solution.
If $A$ has $n$ linearly independent eigenvectors $v_1, \dots, v_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, then each solution is $x(t) = c_1 e^{\lambda_1 t} v_1 + \dots + c_n e^{\lambda_n t} v_n$ for unique constants $c_k$.
For a complex eigenvalue $\lambda$ of a real matrix with eigenvector $v$, the real and imaginary parts of $e^{\lambda t} v$ are two real solutions.

## Remarks

Example: $A = \begin{pmatrix} 1 & 1 \\ 4 & 1 \end{pmatrix}$ has the eigenvalues $3$ and $-1$, with eigenvectors $(1, 2)$ and $(1, -2)$. So $x(t) = c_1 e^{3t} (1, 2) + c_2 e^{-t} (1, -2)$.
If $A$ has too few eigenvectors, generalized eigenvectors give solutions of the form $e^{\lambda t}$ times a polynomial in $t$ with vector coefficients.
The eigenvalues decide the [[stability of the system|0SKVEMXJWF]].

## Explanations

### L1 human

Along an eigenvector, the matrix $A$ acts as multiplication by a number.
So on this line the system is the simple equation $y' = \lambda y$, with solution $e^{\lambda t}$.
Write the start vector in a basis of eigenvectors, and let each part grow or decay at its own rate.
</document>

<document source="https://math2.tamnd.com/node/0906Z03EMN.md" tag="0906Z03EMN" kind="example">
# Tower of Hanoi

## Example

The Tower of Hanoi is a puzzle with three rods and $n$ discs of different sizes.
At the start, all discs are on one rod, in order of size, with the largest at the bottom.
A move takes the top disc of one rod and puts it on top of another rod.
No move may put a disc on a smaller disc.
The aim is to move all discs to another rod.

Let $T_n$ be the minimum number of moves for $n$ discs.
Then $T_0 = 0$ and $T_n = 2 T_{n-1} + 1$ for $n \ge 1$.
This [[recurrence relation|07TZV87Y0T]] has the solution $T_n = 2^n - 1$.

## Remarks

The recurrence comes from the largest disc. Before it moves, the other $n - 1$ discs must be on the third rod. After it moves, they must come back on top of it.
Values: $T_1 = 1$, $T_2 = 3$, $T_3 = 7$ and $T_4 = 15$.
The French mathematician Édouard Lucas invented the puzzle and presented it in 1883.

## Explanations

### L1 human

To move a tower of $n$ discs, first move the top $n - 1$ discs out of the way. Then move the largest disc, and then move the $n - 1$ discs back on top.
Each new disc doubles the work and adds one move.
</document>

<document source="https://math2.tamnd.com/node/0909PWZR62.md" tag="0909PWZR62" kind="definition">
# One-sided limit

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D \cap (a, \infty)$.
A real number $L$ is the right limit of $f$ at $a$ if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $a < x < a + \delta$.
The notation is $\lim_{x \to a^+} f(x) = L$.
The left limit $\lim_{x \to a^-} f(x)$ is defined in the same way, with $a - \delta < x < a$.

## Remarks

The right limit is the [[limit|0TXTSEMHR5]] of the restriction of $f$ to $D \cap (a, \infty)$.
If $a$ is a limit point of points on both sides, then $\lim_{x \to a} f(x) = L$ if and only if both one-sided limits exist and equal $L$.
For the sign function at $0$, the right limit is $1$ and the left limit is $-1$.

## Explanations

### L1 human

A one-sided limit only looks at points on one side of $a$.
A function with a jump at $a$ has two different one-sided limits there.
</document>

<document source="https://math2.tamnd.com/node/090AA928Y7.md" tag="090AA928Y7" kind="concept">
# Convergence of a sequence in a metric space

## Summary

A sequence in a metric space converges to a point when the distance from the terms to the point tends to zero.
The point is the limit of the sequence.
</document>

<document source="https://math2.tamnd.com/node/090WPKFABX.md" tag="090WPKFABX" kind="definition">
# Hamiltonian graph

## Definition

A [[graph|0JD5C82QMB]] is Hamiltonian if it has a [[Hamiltonian cycle|0HKXNG2RS1]].

## Remarks

The [[cycle graph|0QQ0F12962]] $C_n$ and the [[complete graph|03WKB2H474]] $K_n$ for $n \ge 3$ are Hamiltonian.
A [[tree|0MJ0JS72DZ]] is not Hamiltonian, because it has no cycle.
The [[Petersen graph|0NR95CTFCZ]] is not Hamiltonian.
[[Dirac's theorem|0TC99BRSSB]] and [[Ore's theorem|0NACHJYPKW]] give sufficient conditions in terms of degrees.

## Explanations

### L1 human

A graph is Hamiltonian when you can visit all of its vertices in one round trip, each vertex once.
Many edges make this easier, and this is the idea of the theorems of Dirac and Ore.
</document>

<document source="https://math2.tamnd.com/node/09115NQZ07.md" tag="09115NQZ07" kind="concept">
# Arithmetic function

## Summary

An arithmetic function is a function on the positive integers with complex values, often one that expresses an arithmetic property of $n$.
The [[definition of an arithmetic function|0H2EPRZC0D]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0912EEEVCT.md" tag="0912EEEVCT" kind="statement">
# An injective endomorphism of a finite-dimensional space is surjective

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Then these conditions are equivalent:

1. $T$ is injective.
2. $T$ is surjective.
3. $T$ is bijective.

## Remarks

The proof uses the [[rank–nullity theorem|0W4FSW1TVB]]: $\ker T = 0$ if and only if $\dim \im T = \dim V$.
It fails in infinite dimensions: on $F[x]$, multiplication by $x$ is injective and not surjective.
In characteristic $0$, the derivative on $F[x]$ is surjective and not injective.
For square matrices, this is the statement that $Ax = 0$ has only the trivial solution if and only if $Ax = b$ has a solution for each $b$.

## Explanations

### L1 human

For a linear map from a finite-dimensional space to itself, one to one and onto are the same thing.
If no information is lost, then nothing in the target can be missed, because the dimensions match.
This is like a map from a finite set to itself.
</document>

<document source="https://math2.tamnd.com/node/091AV9YHZY.md" tag="091AV9YHZY" kind="concept">
# Real exponentiation

## Summary

Real exponentiation defines the power $a^x$ for a positive base $a$ and a real exponent $x$. It is $e^{x \ln a}$, and it extends the powers with integer and rational exponents. The node [[real exponentiation|0TV07AQT62]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/091GJ3FYD7.md" tag="091GJ3FYD7" kind="concept">
# Metrizable space

## Summary

A topological space is metrizable when its topology comes from some metric.
Metrization theorems give conditions on the topology alone that make a space metrizable.
</document>

<document source="https://math2.tamnd.com/node/0924BCD257.md" tag="0924BCD257" kind="concept">
# Geometric probability

## Summary

Geometric probability studies random geometric objects, such as random points, lines or shapes in the plane or in space.
In the simplest case, a point is chosen uniformly in a region $\Omega$ of finite positive area.
Then the probability of a subregion $A$ is the area of $A$ divided by the area of $\Omega$.
Classic problems are [[Buffon's needle problem|06X5PF6YFP]] and the [[Bertrand paradox|0T0JWJ3M47]].

## Remarks

The same rule works with length on a line and with volume in space.
Mathematicians studied such problems from the 17th century on. Their modern form is part of integral geometry and stochastic geometry.

## Explanations

### L1 human

In geometric probability you measure chances with lengths and areas instead of with counts.
Throw a dart at random at a square board.
The chance to hit the inscribed circle is the area of the circle over the area of the square, that is $\pi/4$.
</document>

<document source="https://math2.tamnd.com/node/09285XX7AE.md" tag="09285XX7AE" kind="statement">
# Euler's identity

## Statement

$$e^{i \pi} + 1 = 0.$$

## Remarks

Proof: [[Euler's formula|04R3CY41VJ]] with $z = \pi$ gives $e^{i \pi} = \cos \pi + i \sin \pi = -1$.
The identity links five constants: $0$, $1$, $\pi$, $e$ and $i$.

## Explanations

### L1 human

Walk around the unit circle from $1$ for a distance of $\pi$, which is half of the circle.
You arrive at $-1$, so $e^{i \pi} = -1$.
</document>

<document source="https://math2.tamnd.com/node/092DRBFKMG.md" tag="092DRBFKMG" kind="definition">
# Catalan numbers

## Definition

The Catalan numbers $C_n$, for natural numbers $n$, are defined by the recursion
$$C_0 = 1, \qquad C_{n+1} = \sum_{i=0}^{n} C_i\, C_{n-i}.$$

## Remarks

The first values, for $n = 0, 1, \dots, 7$, are $1, 1, 2, 5, 14, 42, 132, 429$.
The [[closed formula|0NNDQVJ54Q]] is $C_n = \frac{1}{n+1} \binom{2n}{n}$.
$C_n$ counts [[Dyck words|0R9JR0465V]] of semilength $n$ and [[binary trees|02A8EMC83C]] with $n$ nodes, and many other families.

## Explanations

### L1 human

Catalan numbers count many structures that split into a left part and a right part.
For example, there are $C_3 = 5$ ways to put brackets on a product $abcd$ of four factors: $((ab)c)d$, $(a(bc))d$, $(ab)(cd)$, $a((bc)d)$ and $a(b(cd))$.
</document>

<document source="https://math2.tamnd.com/node/092GEF38YD.md" tag="092GEF38YD" kind="definition">
# Binomial coefficient

## Definition

Let $n$ and $k$ be natural numbers.
The binomial coefficient $\binom{n}{k}$, read "$n$ choose $k$", is the natural number defined by the recursion
$$\binom{n}{0} = 1, \qquad \binom{0}{k+1} = 0, \qquad \binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

## Remarks

The value $\binom{n}{k}$ is the [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set. Many texts take this as the definition.
For $k > n$, $\binom{n}{k} = 0$. For $k \le n$, the [[factorial formula|033KTKKRV6]] gives $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$.
The recursion is [[Pascal's rule|0YRG241HQQ]], and the values form [[Pascal's triangle|05DME4TZ3K]].

## Explanations

### L1 human

The number $\binom{n}{k}$ tells how many ways there are to choose $k$ objects out of $n$ when the order does not matter.
For example, $\binom{4}{2} = 6$: from $\{1, 2, 3, 4\}$ you can choose $12, 13, 14, 23, 24, 34$.
</document>

<document source="https://math2.tamnd.com/node/092S6VBECQ.md" tag="092S6VBECQ" kind="definition">
# Lyapunov stability

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]].
The point $x^*$ is Lyapunov stable if for each $\varepsilon > 0$ there is a $\delta > 0$ with this property. For each $x_0 \in U$ with $\|x_0 - x^*\| < \delta$, the solution $\varphi_t(x_0)$ exists for all $t \ge 0$, and $\|\varphi_t(x_0) - x^*\| < \varepsilon$ for all $t \ge 0$.
The point $x^*$ is unstable if it is not Lyapunov stable.

## Remarks

For $x' = -x$ and for $x' = 0$ on $\mathbb{R}$, the equilibrium $0$ is stable. For $x' = x$, it is unstable.
The centre of the [[harmonic oscillator|0YG0ZJPTWG]] is stable but not [[asymptotically stable|05Y1JAM3Y5]]: nearby solutions stay near, but they do not approach it.
A [[Lyapunov function|04P7B5XC22]] gives a way to prove stability without solving the equation.

## Explanations

### L1 human

An equilibrium is stable if solutions that start close to it stay close to it forever.
To stay within distance $\varepsilon$, it is enough to start within a smaller distance $\delta$.
</document>

<document source="https://math2.tamnd.com/node/093CM69AGS.md" tag="093CM69AGS" kind="definition">
# Domain

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The domain of $f$ is the set $A$.

## Remarks

The function $f$ has a value $f(a)$ for each $a$ in its domain.
In calculus, the natural domain of a formula is the largest set of real numbers where the formula has a meaning. For $1/x$, it is $\mathbb{R} \setminus \{0\}$.

## Explanations

### L1 human

The domain says which inputs are allowed.
For the square root as a real function, the domain is the set of real numbers $x \ge 0$.
</document>

<document source="https://math2.tamnd.com/node/093EDENVA1.md" tag="093EDENVA1" kind="concept">
# Congruent triangles

## Summary

Two triangles are congruent when they have the same shape and the same size. Then a rigid motion, possibly with a reflection, moves one onto the other.
</document>

<document source="https://math2.tamnd.com/node/093SAXPJ9P.md" tag="093SAXPJ9P" kind="concept">
# Mesh of a partition

## Summary

The mesh of a partition is the length of its longest subinterval. Riemann sums approach the integral as the mesh tends to zero. The node [[mesh of a partition|04GM4NYXJB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/093X6MN6Y6.md" tag="093X6MN6Y6" kind="statement">
# Minimal polynomial divides the characteristic polynomial

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then the [[minimal polynomial|03NDVCSHSB]] of $A$ divides the [[characteristic polynomial|0VSSPAFM2C]] of $A$.
In particular, $\deg m_A \le n$.

## Remarks

The [[Cayley–Hamilton theorem|0F765KFC2J]] gives $p_A(A) = 0$, and the minimal polynomial divides each annihilating polynomial.
The characteristic polynomial divides a power of the minimal polynomial, because both have the same irreducible factors.
For $I_2$, $m = t - 1$ divides $p = (t - 1)^2$ and is not equal to it.

## Explanations

### L1 human

The minimal polynomial is a divisor of the characteristic polynomial.
So its degree is at most the size of the matrix.
Both polynomials have the same roots.
</document>

<document source="https://math2.tamnd.com/node/094V2VBJXJ.md" tag="094V2VBJXJ" kind="concept">
# Multiplication by a matrix as a linear map

## Summary

Each $m \times n$ matrix $A$ defines the linear map from $F^n$ to $F^m$ that sends a column vector $x$ to the product $Ax$.
</document>

<document source="https://math2.tamnd.com/node/095DDGZF16.md" tag="095DDGZF16" kind="definition">
# Row equivalence

## Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$.
The matrix $A$ is *row equivalent* to $B$ when there is a finite sequence of [[elementary row operations|04XWH28452]] that changes $A$ into $B$.

## Remarks

Row equivalence is an equivalence relation, because each elementary row operation has an inverse operation.
The matrices $A$ and $B$ are row equivalent if and only if $B = PA$ for an [[invertible|0EB9N0S33C]] $m \times m$ matrix $P$.
Each matrix is row equivalent to exactly one matrix in [[reduced row echelon form|0WF5AQYTW6]].
Row equivalent matrices have the same [[row space|069MCGX635]] and the same [[null space|0PHDYNR24D]].

## Explanations

### L1 human

Two matrices are row equivalent when row operations can turn one into the other.
If they are augmented matrices of two systems, then the two systems have the same solutions.
Row reduction replaces a matrix by a simpler row equivalent matrix.
</document>

<document source="https://math2.tamnd.com/node/095ZHQ7WCJ.md" tag="095ZHQ7WCJ" kind="statement">
# In a PID a nonzero element is irreducible iff it generates a maximal ideal

## Statement

Let $R$ be a [[principal ideal domain|0RZYDVZ475]] and let $p \in R$ with $p \neq 0$.
Then $p$ is [[irreducible|052SDEPZH7]] if and only if the principal ideal $(p)$ is a [[maximal ideal|09J6KGT65G]].

## Remarks

The proof: an ideal $I$ with $(p) \subseteq I$ is $(a)$ with $a \mid p$. If $p$ is irreducible, then $a$ is a unit or an associate of $p$, so $I = R$ or $I = (p)$.
A consequence is that $R/(p)$ is a [[field|0QA8CHH8MG]] for each irreducible $p$. For example, $\mathbb{Q}[x]/(x^2 - 2)$ is a field.
In a principal ideal domain, every nonzero prime ideal is maximal.

## Explanations

### L1 human

In a principal ideal domain, ideals that contain $(p)$ come from the divisors of $p$.
An irreducible $p$ has only trivial divisors.
So no ideal fits strictly between $(p)$ and the whole ring.
</document>

<document source="https://math2.tamnd.com/node/0964YQFSA8.md" tag="0964YQFSA8" kind="concept">
# Extended Euclidean algorithm

## Summary

The extended Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] $d$ of two integers $a$ and $b$, and also integers $x$ and $y$ such that $a x + b y = d$.
It does the steps of the [[Euclidean algorithm|0TRNT4X7J0]].
At each step, it writes the new remainder as an integer combination of $a$ and $b$.

## Remarks

The coefficients $x$ and $y$ prove [[Bézout's identity|0DQVHMYAGA]] for $a$ and $b$.
When $\gcd(a, n) = 1$, the coefficient $x$ of $a$ is a [[modular multiplicative inverse|00GXTQMY8Z]] of $a$ modulo $n$.

## Explanations

### L1 human

For $a = 48$ and $b = 18$, the Euclidean algorithm gives $12 = 48 - 2 \cdot 18$ and $6 = 18 - 1 \cdot 12$.
Put the first equation into the second: $6 = 18 - (48 - 2 \cdot 18) = 3 \cdot 18 - 1 \cdot 48$.
So $x = -1$ and $y = 3$ give $48 x + 18 y = 6$.
</document>

<document source="https://math2.tamnd.com/node/096AHMB3P4.md" tag="096AHMB3P4" kind="concept">
# Young diagram

## Summary

A Young diagram is a picture of an integer partition as left-aligned rows of boxes. Young diagrams carry Young tableaux and index the irreducible representations of the symmetric groups. The node [[Young diagram|0K78D81EF7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/096B7Q5742.md" tag="096B7Q5742" kind="statement">
# Unique paths in trees

## Statement

Let $G$ be a [[graph|0JD5C82QMB]].
Then $G$ is a [[tree|0MJ0JS72DZ]] if and only if $G$ has at least one vertex and these conditions hold.
For any two vertices $u$ and $v$ of $G$, there is exactly one [[path|0JDE56HWQ8]] from $u$ to $v$.

## Remarks

Existence of a path between any two vertices is connectedness. Uniqueness of the path is equivalent to acyclicity: two different paths between the same vertices contain a cycle.
In a tree, the unique path from $u$ to $v$ is a shortest walk, so its length is the [[distance|0WPZ5YPAGR]] $d(u, v)$.

## Explanations

### L1 human

In a tree there is only one way to get from one vertex to another without going back.
If there were two different ways, together they would form a loop, and a tree has no loop.
</document>

<document source="https://math2.tamnd.com/node/096HN851YZ.md" tag="096HN851YZ" kind="definition">
# Subnormal series

## Definition

Let $G$ be a [[group|G5S1M9]].
A subnormal series of $G$ is a finite sequence of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$, where each $G_{i-1}$ is a [[normal subgroup|0CB0DX5BVV]] of $G_i$.
The [[quotient groups|0KYX2N69J0]] $G_i / G_{i-1}$ are the factors of the series, and $n$ is its length.

## Remarks

The subgroup $G_{i-1}$ needs to be normal only in $G_i$, not in $G$.
When each $G_i$ is normal in $G$, the series is called a normal series.
For example, $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$ is a normal series of $S_4$, where $V$ is the [[Klein four-group|0PMRQJVTME]] in $S_4$.

## Explanations

### L1 human

A subnormal series cuts a group into layers.
Each layer is a quotient group, and you can study the layers one at a time.
Good series have simple layers, like composition series, or abelian layers, like the series of a solvable group.
</document>

<document source="https://math2.tamnd.com/node/096PW7Q014.md" tag="096PW7Q014" kind="definition">
# Divisibility

## Definition

Let $a$ and $b$ be integers.
The integer $a$ divides $b$ when there is an integer $k$ such that $b = a k$.
The notation $a \mid b$ means that $a$ divides $b$, and $a \nmid b$ means that it does not.
When $a \mid b$, $a$ is a divisor (or a factor) of $b$, and that $b$ is a multiple of $a$.

## Remarks

Each integer divides $0$, because $0 = a \cdot 0$. The integer $0$ divides only $0$.
The integers $1$ and $-1$ divide each integer.
If $a \mid b$ and $b \mid c$, then $a \mid c$. On the natural numbers, divisibility is a partial order.
If $a \mid b$ and $a \mid c$, then $a \mid (x b + y c)$ for all integers $x$ and $y$.

## Explanations

### L1 human

The integer $a$ divides $b$ when $b$ is in the list $\dots, -2a, -a, 0, a, 2a, \dots$ of the multiples of $a$.
For example, $3 \mid 12$ because $12 = 3 \cdot 4$.
But $5 \nmid 12$, because no integer $k$ gives $5 k = 12$.
</document>

<document source="https://math2.tamnd.com/node/096SD4ERQH.md" tag="096SD4ERQH" kind="statement">
# Geometric solutions come from roots of the characteristic polynomial

## Statement

Let $R$ be a commutative ring, let $E$ be a linear recurrence with constant coefficients over $R$, and let $q \in R$.
Then the sequence $(q^n)_{n \ge 0}$ is a [[solution|0ZV2YKKZ58]] of $E$ if and only if $q$ is a root of the [[characteristic polynomial|0BWDJGQYST]] $\chi_E$.

## Remarks

Proof idea: for the sequence $q^n$ the recurrence at index $n$ reads $q^n \chi_E(q) = 0$. At $n = 0$ this is $\chi_E(q) = 0$.
Over a field, if $\chi_E$ has $d$ distinct roots $q_1, \dots, q_d$, every solution is $u_n = a_1 q_1^n + \dots + a_d q_d^n$ with constants $a_i$.
[[Binet's formula|0404D8TN7Y]] is the case of the Fibonacci recurrence.

## Explanations

### L1 human

Try a solution of the form $u_n = q^n$. Put it into the recurrence and divide by $q^n$.
What is left is the equation $\chi_E(q) = 0$.
</document>

<document source="https://math2.tamnd.com/node/096V7H13H9.md" tag="096V7H13H9" kind="statement">
# Linearly independent set of the right size is a basis

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$ of finite [[dimension|03GSZ8RTZJ]] $n$.
Let $v_1, \dots, v_n \in V$ be $n$ vectors.
If $v_1, \dots, v_n$ are [[linearly independent|0CRMGPAH83]], then they form a [[basis|0KCQ9Y3QKA]] of $V$.
If $v_1, \dots, v_n$ span $V$, then they form a basis of $V$.

## Remarks

Thus, for exactly $\dim V$ vectors, independence and spanning are equivalent.
For example, three vectors of $\R^3$ form a basis if and only if the $3 \times 3$ matrix with these columns has nonzero determinant.
More generally, a family $v_1, \dots, v_k$ is linearly independent if and only if its span has dimension $k$.

## Explanations

### L1 human

When we have exactly as many vectors as the dimension, we only need to check one of the two basis conditions.
Independence gives spanning for free, and spanning gives independence for free.
This halves the work of checking a basis.
</document>

<document source="https://math2.tamnd.com/node/096Z9MP2ME.md" tag="096Z9MP2ME" kind="concept">
# Ideal

## Summary

An ideal of a ring is an additive subgroup that is closed under multiplication by every element of the ring.
Ideals are the kernels of ring homomorphisms, and they give quotient rings.
The [[definition of an ideal|0MYSA6MYNA]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/09753QQDCK.md" tag="09753QQDCK" kind="concept">
# Central moment

## Summary

A central moment of a random variable is a moment of its deviation from the mean. The second central moment is the variance.
The node [[central moment|0ZT24EC90K]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0977KZ6PQB.md" tag="0977KZ6PQB" kind="statement">
# Cramer's rule

## Statement

Let $F$ be a field, let $A$ be an $n \times n$ matrix over $F$ with $\det A \ne 0$, and let $b \in F^n$.
For $1 \le i \le n$, let $A_i$ be the matrix that results when column $i$ of $A$ is replaced by $b$.
Then the [[linear system|0GT4WWGWH2]] $Ax = b$ has exactly one solution $x = (x_1, \dots, x_n)$, and
$$x_i = \frac{\det A_i}{\det A} \qquad (1 \le i \le n).$$

## Remarks

For $n = 2$, the system $ax + by = e$, $cx + dy = f$ has the solution $x = (ed - bf)/(ad - bc)$ and $y = (af - ec)/(ad - bc)$.
The rule follows from the formula for the inverse with the [[adjugate|07W9H8QF38]].
It needs $n + 1$ determinants, so it is slow for large $n$ compared with [[Gaussian elimination|0JVJ2WCXZH]].

## Explanations

### L1 human

Cramer's rule gives each unknown as a quotient of two determinants.
The denominator is always $\det A$.
For the numerator of unknown $i$, put the right side $b$ in column $i$ of $A$.
</document>

<document source="https://math2.tamnd.com/node/097C7KCAT5.md" tag="097C7KCAT5" kind="statement">
# Riesz's lemma

## Statement

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$, and let $Y \neq X$ be a [[closed|0G5AV9Q4RR]] linear subspace of $X$.
Let $\alpha$ be a real number with $0 < \alpha < 1$.
Then there is $x \in X$ with $\|x\| = 1$ and $\|x - y\| \ge \alpha$ for all $y \in Y$.

## Remarks

Proof idea: take $x_0 \notin Y$ and let $r = \inf_{y \in Y} \|x_0 - y\| > 0$, which is positive because $Y$ is closed. Choose $y_0 \in Y$ with $\|x_0 - y_0\| \le r / \alpha$, and let $x = (x_0 - y_0) / \|x_0 - y_0\|$.
If $Y$ is finite-dimensional, one can take $\alpha = 1$.
The lemma is a substitute for orthogonal vectors in normed spaces without an inner product.
Its main application is that a [[compact closed unit ball forces finite dimension|0VZVV8VME5]].

## Explanations

### L1 human

For each closed proper subspace, there is a unit vector that is almost as far from the subspace as possible.
In an inner product space, a unit vector orthogonal to the subspace works with distance $1$.
</document>

<document source="https://math2.tamnd.com/node/097KNJFJ7V.md" tag="097KNJFJ7V" kind="definition">
# Domain in the complex plane

## Definition

Let $D$ be a subset of the [[complex plane|0CTE7A31XQ]] $\mathbb{C}$.
The set $D$ is open if for each $z_0 \in D$ there is a real number $r > 0$ such that each $z \in \mathbb{C}$ with $|z - z_0| < r$ is in $D$. Here $|z - z_0|$ is the [[modulus|0KW2X0TFBX]] of $z - z_0$.
The set $D$ is connected if there are no two disjoint nonempty open sets $A$ and $B$ with $D = A \cup B$.
A domain in the complex plane is a subset $D \subseteq \mathbb{C}$ that is nonempty, open and connected.

## Remarks

An open set $D \subseteq \mathbb{C}$ is connected if and only if each two points of $D$ are the end points of a polygonal path in $D$.
Examples of domains are $\mathbb{C}$, each open disc, the upper half-plane $\{z : \operatorname{Im} z > 0\}$ and $\mathbb{C} \setminus \{0\}$.
The union of two disjoint open discs is open but not connected, so it is not a domain.
Some authors use the word region for a domain. Other authors use region for a set between a domain and its closure.

## Explanations

### L1 human

A domain is a set in the plane that has no boundary points in it and that is in one piece.
Open means that you can move a little in each direction from each point and stay in the set.
Connected means that you can walk from each point of the set to each other point without leaving the set.
</document>

<document source="https://math2.tamnd.com/node/098C0KZZAP.md" tag="098C0KZZAP" kind="statement">
# Intermediate value theorem for connected spaces

## Statement

Let $X$ be a [[connected space|0D939CMME7]], let $f \colon X \to \mathbb{R}$ be [[continuous|0SY9ECQ19Q]], and let $a, b \in X$.
Then for each real $c$ between $f(a)$ and $f(b)$ there is $x \in X$ with $f(x) = c$.

## Remarks

Proof: $f(X)$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]]. Thus it is an interval, by [[connected subsets of the real line are intervals|0B9ZQFMQNV]], and it contains $c$.
For $X = [a, b]$, this is the intermediate value theorem of calculus: a continuous $f \colon [a, b] \to \mathbb{R}$ takes each value between $f(a)$ and $f(b)$.
For example, each polynomial of odd degree with real coefficients has a real root.

## Explanations

### L1 human

A continuous real function on a connected space cannot jump over a value. To get from $f(a)$ to $f(b)$, it passes through all values in between.
</document>

<document source="https://math2.tamnd.com/node/098D2E7DEH.md" tag="098D2E7DEH" kind="statement">
# Fixed points of a p-group action

## Statement

Let $p$ be a prime number, let $G$ be a finite [[p-group|06QYRTX9KQ]], and let $G$ [[act|0WCY42EV9V]] on a finite set $X$.
Let $X^G$ be the set of the [[fixed points|0RRWNMYRXC]] of the action.
Then $|X| \equiv |X^G| \pmod p$.

## Remarks

The [[orbits|0ZC7FYE0NZ]] partition $X$, and the orbits with one point are exactly the fixed points.
Each other orbit has a size that divides $|G|$ and is greater than $1$, so its size is divisible by $p$.
A consequence: if $p$ does not divide $|X|$, then the action has at least one fixed point.

## Explanations

### L1 human

The orbits of a $p$-group have sizes that are powers of $p$.
The orbits of size $1$ are the fixed points, and all other orbits have a size divisible by $p$.
So counting modulo $p$ sees only the fixed points.
</document>

<document source="https://math2.tamnd.com/node/098D6FQN3N.md" tag="098D6FQN3N" kind="concept">
# Gauss–Jordan elimination

## Summary

Gauss–Jordan elimination is the variant of [[Gaussian elimination|0JVJ2WCXZH]] that continues until the matrix is in [[reduced row echelon form|0WF5AQYTW6]].
After the forward phase gives a row echelon form, it scales each pivot row to make the [[pivot|0J0BBCMJ5W]] equal to $1$.
Then it adds multiples of each pivot row to the rows above it, to make every other entry in each pivot column zero.
Applied to the [[augmented matrix|0T6MWGZSXA]] of a system, the result shows the solutions directly, with no [[back substitution|0ATFVE08MW]].

## Remarks

The result of Gauss–Jordan elimination does not depend on the choices in the algorithm.
The reason is that the reduced row echelon form of a matrix is unique.
Applied to $[A \mid I]$ for an invertible matrix $A$, it gives $[I \mid A^{-1}]$, see [[computing the inverse by row reduction|0Z83DHFMYM]].

## Explanations

### L1 human

Gauss–Jordan elimination cleans up the staircase matrix that Gaussian elimination produces.
It makes every pivot equal to $1$ and clears the entries above the pivots too.
Then each equation names one basic unknown in terms of the free unknowns.
</document>

<document source="https://math2.tamnd.com/node/098PHYBTV9.md" tag="098PHYBTV9" kind="concept">
# Combination

## Summary

A combination is a selection of elements of a set in which the order does not matter. Combinations of size $k$ are the $k$-element subsets, and binomial coefficients count them. The node [[combination|0JZSRWRBZE]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/098YP831QY.md" tag="098YP831QY" kind="statement">
# Peano existence theorem

## Statement

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$, let $f : D \to \mathbb{R}^n$ be continuous, and let $(t_0, x_0) \in D$.
Then there are an open interval $I$ with $t_0 \in I$ and a differentiable function $z : I \to \mathbb{R}^n$ with these properties.
For each $t \in I$, the point $(t, z(t))$ is in $D$ and $z'(t) = f(t, z(t))$.
Also, $z(t_0) = x_0$.

## Remarks

The theorem needs only continuity of $f$. It gives existence, but not uniqueness. See the [[non-uniqueness example|00K7ASZ1PR]].
With a Lipschitz condition in $x$, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives existence and uniqueness.
A common proof builds Euler polygons with smaller and smaller steps and takes a uniformly convergent subsequence by the Arzelà–Ascoli theorem.

## History

Giuseppe Peano published the theorem in 1886, with a proof that was not correct.
In 1890 he published a correct proof that uses successive approximations.
The name Cauchy–Peano theorem also honours Augustin-Louis Cauchy.

## Explanations

### L1 human

If the vector field is continuous, at least one solution starts at each point.
The theorem does not say that the solution is unique, and it gives the solution only for a short time.
</document>

<document source="https://math2.tamnd.com/node/0990W5PN13.md" tag="0990W5PN13" kind="concept">
# Conditional probability

## Summary

The conditional probability of an event $A$ given an event $B$ is the probability of $A$ when we know that $B$ has occurred.
It is written $P(A \mid B)$. The node [[conditional probability|0FJB06AD2P]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/099E72KTF8.md" tag="099E72KTF8" kind="concept">
# Commutativity

## Summary

A binary operation is commutative when the order of the two inputs does not change the result.
</document>

<document source="https://math2.tamnd.com/node/099GXMHV40.md" tag="099GXMHV40" kind="concept">
# Space of solutions of a linear recurrence

## Summary

The space of solutions of a linear recurrence is the module of all sequences that satisfy it. It has dimension equal to the order of the recurrence over a field. The node [[space of solutions of a linear recurrence|05BDYVQQ0T]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/099SXAG014.md" tag="099SXAG014" kind="definition">
# Dirac measure

## Definition

Let $X$ be a set, let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $X$, and let $x \in X$.
The Dirac measure at $x$ is the function $\delta_x \colon \mathcal{F} \to \{0, 1\}$ with
$$\delta_x(A) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

## Remarks

$\delta_x$ is a [[probability measure|0VGMEBVW1M]]. In terms of the [[indicator function|0VSB0XF4G7]], $\delta_x(A) = \mathbf{1}_A(x)$.
The integral of a measurable function $f$ with respect to $\delta_x$ is $f(x)$.
Each probability measure on a countable set $X$ with the power set as σ-algebra is a sum $\sum_{x \in X} p(x) \, \delta_x$ with weights $p(x) \ge 0$ of sum $1$.

## Explanations

### L1 human

The Dirac measure puts all the probability on one point.
It describes an experiment whose result is certain, such as a coin with heads on both sides.
</document>

<document source="https://math2.tamnd.com/node/099TWVJ909.md" tag="099TWVJ909" kind="concept">
# Geometric multiplicity

## Summary

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace.
</document>

<document source="https://math2.tamnd.com/node/09A3N7ZN5G.md" tag="09A3N7ZN5G" kind="concept">
# Converse

## Summary

The converse of an implication exchanges its hypothesis and its conclusion.
</document>

<document source="https://math2.tamnd.com/node/09ASMRMD67.md" tag="09ASMRMD67" kind="concept">
# Cross-ratio

## Summary

The cross-ratio is a number made from four points of a line, or of the Riemann sphere, see the [[definition|05MW7MPRBS]].
It does not change under [[Möbius transformations|0SG74WBFA2]] and, more generally, under projective transformations.
</document>

<document source="https://math2.tamnd.com/node/09AZCV6GMC.md" tag="09AZCV6GMC" kind="definition">
# Sequence tending to infinity

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence tends to $+\infty$ if, for each real number $M$, there is a natural number $N$ such that $a_n > M$ for all $n \ge N$.
Then the notation is $a_n \to +\infty$ as $n \to \infty$, or $\lim_{n \to \infty} a_n = +\infty$.
The sequence tends to $-\infty$ if $-a_n \to +\infty$.

## Remarks

The sequences $n$, $n^2$ and $2^n$ tend to $+\infty$.
A sequence that tends to $+\infty$ is not bounded above, so it is a [[divergent sequence|01HR2AH0BX]].
The converse fails. The sequence $0, 1, 0, 2, 0, 3, \dots$ is not bounded above, but it does not tend to $+\infty$.

## Explanations

### L1 human

A sequence tends to infinity when its terms become larger than any number you name, and stay larger.
It is not enough that the terms are sometimes large.
</document>

<document source="https://math2.tamnd.com/node/09B4X51J09.md" tag="09B4X51J09" kind="definition">
# Discrete logarithm

## Definition

Let $G$ be a finite cyclic [[group|G5S1M9]] of order $N$, with a generator $g$.
For each $h \in G$, the discrete logarithm of $h$ to the base $g$ is the unique integer $k$ with $0 \le k < N$ such that $g^k = h$.
The notation is $k = \log_g h$.

## Remarks

In elementary number theory, $G = (\mathbb{Z}/n\mathbb{Z})^\times$ and $g$ is a [[primitive root|0KM2CJ80G7]] modulo $n$. Then $\log_g a$ is also called the index of $a$, and $N = \varphi(n)$.
The rule $\log_g (h_1 h_2) \equiv \log_g h_1 + \log_g h_2 \pmod N$ holds, as for ordinary logarithms.
For example, $3$ is a primitive root modulo $7$, and $3^4 = 81 \equiv 4 \pmod 7$. So $\log_3 4 = 4$ modulo $7$.
No efficient general algorithm to compute discrete logarithms is known. The [[Diffie–Hellman key exchange|01TKKESHSB]] uses this fact.

## Explanations

### L1 human

[[Modular exponentiation|04DR5PM25N]] computes $g^k \bmod p$ from $k$. The discrete logarithm goes back from the result to $k$.
For small numbers, a table of powers is enough. For numbers with hundreds of digits, the problem seems very hard.
</document>

<document source="https://math2.tamnd.com/node/09BG0Y7SZD.md" tag="09BG0Y7SZD" kind="statement">
# Power series of the complex logarithm

## Statement

Let $\operatorname{Log}$ be the [[principal branch|0YKY4M3KRQ]] of the logarithm.
For each $z \in \mathbb{C}$ with $|z| < 1$, the series $\sum_{n \ge 1} z^n / n$ converges absolutely, and
$$-\operatorname{Log}(1 - z) = \sum_{n=1}^{\infty} \frac{z^n}{n} = z + \frac{z^2}{2} + \frac{z^3}{3} + \cdots.$$

## Remarks

Replacing $z$ by $-z$ gives $\operatorname{Log}(1 + z) = \sum_{n \ge 1} (-1)^{n+1} z^n / n$ for $|z| < 1$.
The [[radius of convergence|045EWAXY9P]] is $1$. For $|z| < 1$, the number $1 - z$ has positive real part, so it is not on the branch cut.
Proof idea: both sides vanish at $0$, and both have the derivative $1/(1 - z) = \sum_{n \ge 0} z^n$ on the unit disc.
At $z = -1$ the series still converges, and gives $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots$.

## Explanations

### L1 human

Integrate the geometric series $1 + z + z^2 + \cdots = 1/(1 - z)$ term by term from $0$ to $z$.
On the left you get $z + z^2/2 + z^3/3 + \cdots$, and on the right you get $-\operatorname{Log}(1 - z)$.
</document>

<document source="https://math2.tamnd.com/node/09BJQQB3VS.md" tag="09BJQQB3VS" kind="concept">
# Necessary and sufficient conditions

## Summary

When the [[implication|0ZNX6DQ3EJ]] $p \to q$ is true, $p$ is a sufficient condition for $q$, and $q$ is a necessary condition for $p$.
A condition $p$ is necessary and sufficient for $q$ when $p \to q$ and $q \to p$ are both true, that is, when $p \leftrightarrow q$ is true.

## Remarks

"$p$ only if $q$" means $p \to q$, and "$p$ if $q$" means $q \to p$.
"Necessary and sufficient" is another way to say "if and only if". See [[biconditional|0V6GAWTBYW]].
For an integer $n$, "$n$ is divisible by $4$" is sufficient but not necessary for "$n$ is even".

## Explanations

### L1 human

A sufficient condition is enough to get the result.
A necessary condition is one that you cannot do without.
To have a driving licence is necessary to drive legally, but it is not sufficient: you also need a car.
</document>

<document source="https://math2.tamnd.com/node/09C4JP7Y0M.md" tag="09C4JP7Y0M" kind="object">
# Heaviside step function

## Description

The Heaviside step function is the function $H : \mathbb{R} \to \mathbb{R}$ with $H(x) = 0$ for $x < 0$ and $H(x) = 1$ for $x > 0$.
Conventions for the value $H(0)$ differ: common choices are $0$, $\frac{1}{2}$ and $1$.
For $a \ge 0$ and $s \in \mathbb{C}$ with $\operatorname{Re} s > 0$, the [[Laplace transform|05XT5AVJBK]] of $t \mapsto H(t - a)$ is $e^{-a s} / s$.

## Remarks

The function $t \mapsto H(t - a)$ switches a signal on at the time $a$. For example, $f(t) H(t - a)$ is $0$ before $a$ and $f(t)$ after $a$.
For $g : [0, \infty) \to \mathbb{C}$ with a Laplace transform $G$ and $a \ge 0$, the shifted function $t \mapsto H(t - a)\, g(t - a)$ has the transform $e^{-a s} G(s)$.
In the sense of distributions, the derivative of $H$ is the [[Dirac delta|0KPHENZ5S7]].
It is named after Oliver Heaviside.

## Explanations

### L1 human

The step function is off before time $0$ and on after it.
It models a switch, such as a voltage turned on at one instant.
</document>

<document source="https://math2.tamnd.com/node/09CBQB6V67.md" tag="09CBQB6V67" kind="statement">
# Area of a region as a line integral

## Statement

Let $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\mathbb{R}^2$ that encloses the bounded region $D$.
Then the area $A$ of $D$ satisfies
$$A = \oint_C x \, dy = -\oint_C y \, dx = \frac{1}{2} \oint_C (x \, dy - y \, dx).$$

## Remarks

Proof: apply [[Green's theorem|0HR3N02317]] with $(L, M)$ equal to $(0, x)$, $(-y, 0)$ and $(-y/2, x/2)$. In each case $\partial M / \partial x - \partial L / \partial y = 1$.
Example: the ellipse $x = a \cos t$, $y = b \sin t$, $t \in [0, 2\pi]$, with $a, b > 0$, gives $A = \tfrac{1}{2} \int_0^{2\pi} a b \, dt = \pi a b$.
For a polygon with vertices $(x_1, y_1), \dots, (x_n, y_n)$ in counterclockwise order, the formula gives $A = \tfrac{1}{2} \sum_{k=1}^n (x_k y_{k+1} - x_{k+1} y_k)$, with $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.

## Explanations

### L1 human

You can find the area of a region by going once around its boundary, without looking inside.
This is a direct use of Green's theorem with a field whose curl is $1$.
</document>

<document source="https://math2.tamnd.com/node/09CCE86A6D.md" tag="09CCE86A6D" kind="definition">
# Partition of a set

## Definition

Let $A$ be a set.
A partition of $A$ is a set $P$ of [[subsets|0FE8Q5S42Y]] of $A$, called blocks, with these three properties:

1. each block is nonempty;
2. the blocks are pairwise [[disjoint|0W4RPA8QH3]];
3. the union of the blocks is $A$.

Equivalently, each element of $A$ is in exactly one block.

## Remarks

The set $\{\{1, 3\}, \{2\}\}$ is a partition of $\{1, 2, 3\}$.
The even and the odd integers form a partition of $\mathbb{Z}$ with two blocks.
The [[equivalence classes form a partition|01HT2CH8BH]], and each partition comes from exactly one equivalence relation. See [[the correspondence between equivalence relations and partitions|074482Q7CD]].

## Explanations

### L1 human

A partition is a way to sort all elements into boxes.
No box is empty, and each element goes into exactly one box.
</document>

<document source="https://math2.tamnd.com/node/09CEDJYBKP.md" tag="09CEDJYBKP" kind="concept">
# Cartesian product

## Summary

The Cartesian product of two sets is a set of ordered pairs.
The first entry of each pair is in the first set, and the second entry is in the second set.
Each such pair is in the product.
</document>

<document source="https://math2.tamnd.com/node/09CJ00H3NW.md" tag="09CJ00H3NW" kind="statement">
# Open sets characterised by neighbourhoods

## Statement

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $x \in X$.
A set $N \subseteq X$ is a neighbourhood of $x$ when there is an [[open set|01ZQQDCERP]] $U$ with $x \in U \subseteq N$.
Then a set $A \subseteq X$ is open if and only if $A$ is a neighbourhood of each of its points.

## Remarks

Proof of the converse: for each $x \in A$, choose an open $U_x$ with $x \in U_x \subseteq A$. Then $A = \bigcup_{x \in A} U_x$, which is open as a [[union|05GK32H72H]] of open sets.
A neighbourhood need not be open. For example, $[-1, 1]$ is a neighbourhood of $0$ in $\mathbb{R}$.
In a metric space, $N$ is a neighbourhood of $x$ if and only if $N$ contains a ball $B(x, \varepsilon)$ for some $\varepsilon > 0$.

## Explanations

### L1 human

A neighbourhood of $x$ is a set that contains all points near enough to $x$.
A set is open when it is a neighbourhood of each of its points. Then no point of the set lies on its edge.
</document>

<document source="https://math2.tamnd.com/node/09DXWYRD43.md" tag="09DXWYRD43" kind="concept">
# Transpose

## Summary

The transpose of a matrix is the matrix that we get when we write the rows of the matrix as columns.
</document>

<document source="https://math2.tamnd.com/node/09DZ8CD1AK.md" tag="09DZ8CD1AK" kind="statement">
# Coordinates in an orthonormal basis

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $(e_1, \dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.
Then, for each $v \in V$,
$$v = \sum_{i=1}^{n} \langle v, e_i \rangle\, e_i.$$
So the coordinates of $v$ in this basis are the inner products $\langle v, e_i \rangle$.

## Remarks

If $v = \sum_j a_j e_j$, then $\langle v, e_i \rangle = \sum_j a_j \langle e_j, e_i \rangle = a_i$.
For the basis $\frac{1}{\sqrt 2}(1, 1)$, $\frac{1}{\sqrt 2}(1, -1)$ of $\R^2$, the vector $(3, 1)$ has the coordinates $4/\sqrt 2$ and $2/\sqrt 2$.
With the convention of linearity in the second argument, the coefficient is written $\langle e_i, v \rangle$.

## Explanations

### L1 human

In an orthonormal basis, we find each coordinate of a vector with one inner product.
This is much easier than solving a linear system.
It is the finite-dimensional form of a Fourier expansion.
</document>

<document source="https://math2.tamnd.com/node/09E1RCZ7YT.md" tag="09E1RCZ7YT" kind="statement">
# Trace of a transpose

## Statement

Let $R$ be a commutative ring and let $A$ be an $n \times n$ matrix over $R$.
Then $\tr(A^T) = \tr A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$ and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

The transpose does not move the entries on the main diagonal, so the sum of the diagonal entries stays the same.
For a complex matrix, $\tr(A^*) = \overline{\tr A}$.

## Explanations

### L1 human

Transposing flips a matrix over its diagonal.
The diagonal itself does not move.
So the trace, which only sees the diagonal, does not change.
</document>

<document source="https://math2.tamnd.com/node/09E6T7VYAG.md" tag="09E6T7VYAG" kind="concept">
# Complex differentiability

## Summary

A complex function is complex differentiable at a point when its difference quotient has a limit as the complex increment goes to $0$. See the [[definition|0D1C9Y1EXA]].
The limit is the complex derivative.
A function that is complex differentiable at each point of an open set is holomorphic on that set.
</document>

<document source="https://math2.tamnd.com/node/09EGC6F8PT.md" tag="09EGC6F8PT" kind="concept">
# Partial derivative

## Summary

A partial derivative of a [[function of several variables|0JPCHTXWG0]] is its derivative with respect to one variable, while all other variables stay fixed.
It measures the rate of change of the function along one coordinate axis.
For $f(x, y) = x^2 y$, the partial derivatives are $\partial f / \partial x = 2xy$ and $\partial f / \partial y = x^2$.
</document>

<document source="https://math2.tamnd.com/node/09EYMKMSB9.md" tag="09EYMKMSB9" kind="definition">
# Algebraic extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is algebraic if every element of $L$ is [[algebraic|0NPW1DWECM]] over $K$.

## Remarks

[[Every finite extension is algebraic|0DFYTNZ6AM]], so $\mathbb{C}/\mathbb{R}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are algebraic.
The converse fails: the field of all algebraic numbers is an algebraic extension of $\mathbb{Q}$ of infinite degree.
The extension $\mathbb{R}/\mathbb{Q}$ is not algebraic, because $\pi$ is transcendental.
An [[algebraically closed field|02SJG1HC32]] has no algebraic extension other than itself.

## Explanations

### L1 human

In an algebraic extension, every new element solves a polynomial equation over the base field.
There are no elements that behave like free variables.
</document>

<document source="https://math2.tamnd.com/node/09F2QPVQY7.md" tag="09F2QPVQY7" kind="concept">
# Circuit

## Summary

A circuit is a closed trail in a graph: a round trip that uses each edge at most once. Eulerian circuits are the main example. The node [[circuit|0PD57PSHNR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/09FQT057T7.md" tag="09FQT057T7" kind="definition">
# Upper and lower Darboux integrals

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded.
The upper Darboux integral of $f$ is the infimum of the upper [[Darboux sums|0QRQVH0CRW]] $U(f, P)$ over all partitions $P$ of $[a, b]$.
The lower Darboux integral of $f$ is the supremum of the lower Darboux sums $L(f, P)$ over all partitions $P$.

## Remarks

The lower Darboux integral is at most the upper one.
The function is Darboux integrable if the two are equal. By [[a theorem|021XJAQVDC]], this is the same as [[Riemann integrability|0N1JQXZXHC]].
For the [[Dirichlet function|03Z4JC4YRJ]] on $[0, 1]$, the upper integral is $1$ and the lower integral is $0$.

## Explanations

### L1 human

The upper integral is the best upper estimate of the area by step functions above the graph.
The lower integral is the best lower estimate by step functions below the graph.
</document>

<document source="https://math2.tamnd.com/node/09FTAET5Z4.md" tag="09FTAET5Z4" kind="object">
# Complex numbers

## Description

A complex number is a number of the form $a + b i$, where $a$ and $b$ are [[real numbers|08J2M1JYAG]] and $i$ is the [[imaginary unit|0ESCX36MMC]], with $i^2 = -1$.
We write $\mathbb{C}$ for the set of complex numbers.
The real number $a$ is the real part and $b$ is the imaginary part of $a + b i$, see [[real and imaginary parts|02ZE6JHJ7G]].
Addition and multiplication follow the usual rules of algebra together with $i^2 = -1$:
$$(a + b i) + (c + d i) = (a + c) + (b + d) i, \qquad (a + b i)(c + d i) = (a c - b d) + (a d + b c) i.$$
With these operations, $\mathbb{C}$ is a [[field|0FRSR10DWF]] that contains $\mathbb{R}$ as the numbers $a + 0 i$.

## Remarks

We picture $\mathbb{C}$ as the [[complex plane|0CTE7A31XQ]], with the point $(a, b)$ for $a + b i$.
Every polynomial equation of positive degree with complex coefficients has a solution in $\mathbb{C}$. This is the fundamental theorem of algebra.
Unlike $\mathbb{R}$, the field $\mathbb{C}$ [[cannot be ordered|01CJRJKDAA]].

## Explanations

### L1 human

The equation $x^2 = -1$ has no real solution, because a real square is never negative.
The complex numbers add a new number $i$ with $i^2 = -1$, and then all the numbers $a + b i$ that you can build from it.
For example, $(1 + 2 i)(3 - i) = 3 - i + 6 i - 2 i^2 = 5 + 5 i$.
</document>

<document source="https://math2.tamnd.com/node/09G6KHS6GY.md" tag="09G6KHS6GY" kind="statement">
# Epsilon-N definition of the limit of a sequence

## Statement

Let $(a_n)$ be a real sequence and let $L$ be a real number.
Then $(a_n)$ converges to $L$, in the sense of the [[limit of a sequence|0XKWSSY8P2]], if and only if the following holds.
For each real number $\varepsilon > 0$ there is a natural number $N$ such that $|a_n - L| < \varepsilon$ for all $n \ge N$.

## Remarks

The proof uses two facts. Each interval $(L - \varepsilon, L + \varepsilon)$ is open and contains $L$. Each open interval that contains $L$ contains such an interval.
In a metric space $(X, d)$ the same statement holds with $d(a_n, L)$ in place of $|a_n - L|$.
The number $N$ depends on $\varepsilon$. A smaller $\varepsilon$ in general needs a larger $N$.

## Explanations

### L1 human

The number $\varepsilon$ is the allowed error, and $N$ is the point after which the error is always smaller than $\varepsilon$.
For $a_n = 1/(n+1)$ and $L = 0$, the choice $N > 1/\varepsilon$ works.
</document>

<document source="https://math2.tamnd.com/node/09GCQ5STVJ.md" tag="09GCQ5STVJ" kind="statement">
# Second Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Then any two [[Sylow p-subgroups|0053CQE9Y2]] $P$ and $Q$ of $G$ are [[conjugate|0H11ZPX4ND]]: there is a $g \in G$ with $Q = gPg^{-1}$.
Moreover, each $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup of $G$.

## Remarks

So all Sylow $p$-subgroups of $G$ are [[isomorphic|08MG7H8JW5]].
A Sylow $p$-subgroup is [[normal|0CB0DX5BVV]] in $G$ if and only if $n_p = 1$.
The proof lets a $p$-subgroup $H$ act on the left cosets of $P$ and uses the [[fixed point lemma|098D2E7DEH]].

## Explanations

### L1 human

The Sylow $p$-subgroups of a group can look different as subsets.
But each one can be moved onto each other one by a conjugation.
So, up to a renaming, there is only one Sylow $p$-subgroup.
</document>

<document source="https://math2.tamnd.com/node/09GK3DE0WV.md" tag="09GK3DE0WV" kind="definition">
# Dihedral group

## Definition

Let $n \geq 3$ be an integer.
Let $P$ be a regular $n$-gon in the plane with centre $0$.
The dihedral group $D_n$ is the group of all isometries of the plane that map $P$ onto itself, with composition as the operation.
Let $r$ be the rotation by the angle $2\pi/n$ about $0$, and let $s$ be a reflection in a line of symmetry of the polygon.

## Remarks

The elements of $D_n$ are the rotations $e, r, \dots, r^{n-1}$ and the reflections $s, rs, \dots, r^{n-1}s$.
The relations $r^n = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$ hold, and these relations give a presentation of $D_n$.
In abstract algebra, many authors write $D_{2n}$ for this group, so the index is the order and not the number of sides.
The group $D_n$ is not abelian, and it is a [[semidirect product|0H3WZXCZNF]] of $\langle r \rangle$ and $\langle s \rangle$.

## Explanations

### L1 human

Take a regular polygon and look at all the ways to pick it up and put it back in the same outline.
You can turn it, and you can flip it over.
These motions form the dihedral group, and turning after a flip is not the same as flipping after a turn.
</document>

<document source="https://math2.tamnd.com/node/09GK5PS824.md" tag="09GK5PS824" kind="statement">
# Transpose of a product

## Statement

Let $R$ be a commutative ring.
Let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $R$.
Then $(AB)^T = B^T A^T$, where $^T$ denotes the [[transpose|09DXWYRD43]].

## Remarks

The order of the factors reverses.
The proof compares entries: both sides have entry $(k, i)$ equal to $\sum_{j} a_{ij} b_{jk}$.
By induction, $(A_1 A_2 \cdots A_k)^T = A_k^T \cdots A_2^T A_1^T$.
If $R$ is not commutative, the identity can fail.

## Explanations

### L1 human

The transpose of a product is the product of the transposes in the reverse order.
This is like taking off socks and shoes: the last thing put on comes off first.
The sizes also force this order, because $B^T$ is $p \times n$ and $A^T$ is $n \times m$.
</document>

<document source="https://math2.tamnd.com/node/09GQ3ZTQJ9.md" tag="09GQ3ZTQJ9" kind="statement">
# Closed sets contain their limits

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.
Let $(x_n)$ be a sequence of points of $F$ that [[converges|06JYPJ4WHH]] to a point $x \in X$.
Then $x \in F$.

## Remarks

Proof: if $x \notin F$, then $X \setminus F$ is open and contains $x$. Thus some ball $B(x, \varepsilon)$ misses $F$. But $x_n \in B(x, \varepsilon)$ for large $n$, a contradiction.
In a metric space the converse is also true: a set that contains the limits of all its convergent sequences is closed. This follows from [[closure is the set of limits of sequences|003QP7A8HG]].
The statement holds in every [[topological space|0CNQXVBHCX]], with the same proof.

## Explanations

### L1 human

You cannot leave a closed set by a limit of a sequence.
For example, $1/n \to 0$ in $\mathbb{R}$, and $0$ is not in $(0, 1]$, which shows that $(0, 1]$ is not closed.
</document>

<document source="https://math2.tamnd.com/node/09GRCZVR5E.md" tag="09GRCZVR5E" kind="statement">
# Completeness of the space of bounded continuous functions

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $(Y, d_Y)$ be a [[complete metric space|08EWT7J0SC]].
Let $C_b(X, Y)$ be the set of [[continuous maps|0SY9ECQ19Q]] $f \colon X \to Y$ whose image $f(X)$ is [[bounded|0P953MG05E]] in $Y$.
For $f, g \in C_b(X, Y)$, let $d_\infty(f, g) = \sup_{x \in X} d_Y(f(x), g(x))$, with $d_\infty(f, g) = 0$ when $X$ is empty.
Then $(C_b(X, Y), d_\infty)$ is a complete metric space.

## Remarks

For $Y = \mathbb{R}$, the metric $d_\infty$ is the [[uniform metric|06HE9J7DXF]], and it comes from the [[sup norm|0PARCKE8WP]].
Proof sketch: a Cauchy sequence $(f_n)$ in $d_\infty$ is Cauchy at each point $x$, so it has a pointwise limit $f(x)$ in $Y$. The convergence is uniform, and a uniform limit of continuous maps is continuous. The limit is also bounded.
For a compact space $X$, each continuous map to $\mathbb{R}$ is bounded, so $C(X, \mathbb{R})$ is complete with the sup metric.

## Explanations

### L1 human

If continuous functions get closer and closer to each other uniformly, then they converge uniformly to a function.
The key point is that this limit function is again continuous, so the limit stays in the space.
</document>

<document source="https://math2.tamnd.com/node/09H9Y0Q11W.md" tag="09H9Y0Q11W" kind="example">
# The symmetric group S3

## Example

The symmetric group $S_3$ has the six elements $e$, $(1\ 2)$, $(1\ 3)$, $(2\ 3)$, $(1\ 2\ 3)$ and $(1\ 3\ 2)$.
It is [[isomorphic|08MG7H8JW5]] to the [[dihedral group|0JZ27DMJ5D]] $D_3$ of the symmetries of an equilateral triangle, when you number the vertices $1$, $2$, $3$.
The $3$-cycles are the rotations, and the transpositions are the reflections.
It is not [[abelian|082GHNPJ53]], because $(1\ 2)(2\ 3) = (1\ 2\ 3)$ but $(2\ 3)(1\ 2) = (1\ 3\ 2)$.
Thus $S_3$ is the smallest group that is not abelian.
Its [[conjugacy classes|0DXD4QSZ8A]] are $\{e\}$, the three transpositions, and the two $3$-cycles.
Its subgroups are $\{e\}$, three subgroups of order $2$, the [[alternating group|0Z5905Q73D]] $A_3$ of order $3$, and $S_3$.
Only $\{e\}$, $A_3$ and $S_3$ are [[normal|0CB0DX5BVV]].
</document>

<document source="https://math2.tamnd.com/node/09HEXDQ0CQ.md" tag="09HEXDQ0CQ" kind="definition">
# Outer automorphism group

## Definition

Let $G$ be a [[group|G5S1M9]].
The outer automorphism group of $G$ is the [[quotient group|0KYX2N69J0]] $\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G)$.

## Remarks

The quotient exists because $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$.
An automorphism that is not inner is called an outer automorphism, but the elements of $\operatorname{Out}(G)$ are cosets, not automorphisms.
For an abelian group, $\operatorname{Inn}(G)$ is trivial, so $\operatorname{Out}(G) \cong \operatorname{Aut}(G)$.
The symmetric group $S_6$ is the only symmetric group $S_n$ with nontrivial outer automorphism group.

## Explanations

### L1 human

Some symmetries of a group come from conjugation by its own elements, and some do not.
The outer automorphism group keeps only the second kind, up to the first kind.
For many groups it is small or trivial.
</document>

<document source="https://math2.tamnd.com/node/09HT3P35YP.md" tag="09HT3P35YP" kind="statement">
# Euler's polyhedron formula

## Statement

Let $P$ be a convex [[polyhedron|0325SGSJAR]] in $\mathbb{R}^3$ with $V$ vertices, $E$ edges and $F$ faces. Then
$$V - E + F = 2.$$

## Remarks

The number $V - E + F$ is the Euler characteristic of the surface of $P$. The value $2$ is the Euler characteristic of the sphere.
The formula also holds for each polyhedron whose surface is topologically a sphere and whose faces are topologically discs.
It can fail for other polyhedra. The surface of a polyhedron with the shape of a torus has $V - E + F = 0$.
For the cube, $8 - 12 + 6 = 2$. For the regular tetrahedron, $4 - 6 + 4 = 2$.

## History

Euler stated the formula in 1758. Cauchy gave a proof in 1811.

## Explanations

### L1 human

Count the corners, the edges and the faces of any convex solid with flat faces. Corners minus edges plus faces always gives $2$. Cut off one corner of a cube: you get $2$ more corners, $3$ more edges and $1$ more face, and the count stays $2$.
</document>

<document source="https://math2.tamnd.com/node/09J1H07VAH.md" tag="09J1H07VAH" kind="concept">
# Method of moments

## Summary

The method of moments is a way to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
Suppose that the law of an i.i.d. sample $X_1, \dots, X_n$ depends on $k$ real parameters $\theta = (\theta_1, \dots, \theta_k)$, and that the [[moments|0PH0VHTQS0]] $\mu_j(\theta) = E_\theta[X_1^j]$ exist for $j \le k$.
The method replaces each $\mu_j(\theta)$ with the sample moment $\hat{m}_j = \frac{1}{n} \sum_{i=1}^{n} X_i^j$, and solves the $k$ equations $\mu_j(\hat{\theta}) = \hat{m}_j$ for $\hat{\theta}$.

## Remarks

Example: for $X_i \sim \operatorname{Exp}(\lambda)$, $E[X_1] = 1/\lambda$, so the method gives $\hat{\lambda} = 1/\bar{X}_n$.
Example: for $X_i \sim \mathcal{N}(\mu, \sigma^2)$, it gives $\hat{\mu} = \bar{X}_n$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (X_i - \bar{X}_n)^2$.
If the map $\theta \mapsto (\mu_1(\theta), \dots, \mu_k(\theta))$ has a continuous inverse, the law of large numbers makes the estimators [[consistent|0XJVF81QF6]].
The estimators are often simple, but they are usually less efficient than maximum likelihood estimators.



## Explanations

### L1 human

The method of moments chooses the parameters so that the model has the same averages as the data.
For example, if the data have mean $2$, an exponential model with mean $1/\lambda$ gets the rate $\lambda = 1/2$.
</document>

<document source="https://math2.tamnd.com/node/09J6KGT65G.md" tag="09J6KGT65G" kind="definition">
# Maximal ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
An [[ideal|0MYSA6MYNA]] $M$ of $R$ is maximal if it has these two properties.

1. $M \neq R$.
2. For every ideal $I$ of $R$ with $M \subseteq I$, either $I = M$ or $I = R$.

## Remarks

The definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2.
In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 0$ is maximal if and only if $n$ is a prime number.
In a [[commutative ring|0GYGHKX834]] $R$, an ideal $M$ is maximal if and only if [[$R/M$ is a field|0WD3SE7S1X]].
The zero ideal of a [[field|0QA8CHH8MG]] is its only maximal ideal.

## Explanations

### L1 human

A maximal ideal is as large as possible without being the whole ring.
In the integers, $(6)$ is not maximal, because $(6) \subseteq (2)$.
The ideal $(2)$ is maximal, because no ideal lies strictly between $(2)$ and $\mathbb{Z}$.
</document>

<document source="https://math2.tamnd.com/node/09JP6G9FTA.md" tag="09JP6G9FTA" kind="definition">
# Forest

## Definition

A forest is an [[acyclic|0XNN9Z8E5A]] [[graph|0JD5C82QMB]].

## Remarks

The connected components of a forest are [[trees|0MJ0JS72DZ]]. Thus a forest is a disjoint union of trees.
Every tree is a forest. The graph with no edges on $n$ vertices is a forest with $n$ components.

## Explanations

### L1 human

A forest is a graph with no cycles, and it may be in several pieces.
Each piece is a tree, as the name suggests.
</document>

<document source="https://math2.tamnd.com/node/09JXE6XMCC.md" tag="09JXE6XMCC" kind="statement">
# Sum of the totient over the divisors

## Statement

Let $n$ be a positive integer.
Then $\sum_{d \mid n} \varphi(d) = n$, where the sum runs over the positive divisors $d$ of $n$, and $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: sort the fractions $\frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n}$ by their denominators in lowest terms. For each divisor $d$ of $n$, exactly $\varphi(d)$ of them have the denominator $d$.
For example, for $n = 12$: $\varphi(1) + \varphi(2) + \varphi(3) + \varphi(4) + \varphi(6) + \varphi(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12$.
With the [[Möbius inversion formula|07Q4TBTFVB]], the result gives $\varphi(n) = \sum_{d \mid n} \mu(d) \, n / d$.

## History

Gauss proved this identity in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Write the fractions $\frac{1}{6}, \frac{2}{6}, \dots, \frac{6}{6}$ in lowest terms: $\frac16, \frac13, \frac12, \frac23, \frac56, \frac11$.
The denominators $1, 2, 3, 6$ appear $1, 1, 2, 2$ times. These are $\varphi(1), \varphi(2), \varphi(3), \varphi(6)$, and their sum is $6$.
</document>

<document source="https://math2.tamnd.com/node/09JZ9P4899.md" tag="09JZ9P4899" kind="statement">
# Groups of order p squared are abelian

## Statement

Let $p$ be a prime number and let $G$ be a [[group|G5S1M9]] of order $p^2$.
Then $G$ is [[abelian|082GHNPJ53]].

## Remarks

By the [[nontrivial centre theorem|0RZCTT30H5]] and [[Lagrange's theorem|3C53TD]], $|Z(G)|$ is $p$ or $p^2$.
If $|Z(G)| = p$, then $G/Z(G)$ has order $p$, so it is cyclic, and [[this criterion|0AH9Q6YNMT]] gives a contradiction.
Thus $Z(G) = G$, and $G$ is isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$ or to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$.
For $p^3$ the result is false: the [[dihedral group|0JZ27DMJ5D]] $D_4$ and the [[quaternion group|0NRJGW864D]] have order $8$ and are not abelian.

## Explanations

### L1 human

A group of order $p^2$ has a centre with more than one element.
If the centre were not the whole group, the quotient by the centre would be cyclic, and that is impossible.
So all elements commute.
</document>

<document source="https://math2.tamnd.com/node/09K9MWD2QQ.md" tag="09K9MWD2QQ" kind="concept">
# Propositional logic

## Summary

Propositional logic is the part of logic that studies [[propositions|0ZGC7ZESYN]] and the connectives that join them.
Its formulas are made from propositional variables with connectives such as $\neg$, $\land$, $\lor$, $\to$ and $\leftrightarrow$.
It does not look inside a proposition, so it has no variables for objects and no quantifiers.

## Remarks

Other names are sentential logic and propositional calculus.
In classical propositional logic, the value of a formula depends only on the values of its variables. A [[truth table|05RSK4Q657]] computes it.
[[Predicate logic|086QKPF0A6]] extends propositional logic with predicates and quantifiers.

## Explanations

### L1 human

In propositional logic, a sentence such as "$7$ is prime" is one unit $p$.
You study only how such units combine with "not", "and", "or" and "if ... then".
For example, $p \lor \neg p$ is true whatever $p$ says.
</document>

<document source="https://math2.tamnd.com/node/09KDHB82SG.md" tag="09KDHB82SG" kind="definition">
# Metrizable space

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is metrizable when there is a [[metric|063W5JMY85]] $d$ on $X$ whose [[metric topology|0A6YRNZB2R]] is $\tau$.

## Remarks

The metric is not unique: equivalent metrics give the same topology.
Each metrizable space is [[Hausdorff|0MXB4CKHFW]] and [[normal|0AD0BZ49DW]].
An indiscrete space with at least two points is not metrizable.
Urysohn's metrization theorem: each [[second-countable|00RE1R4W7N]] [[regular|0JF9KW2TXB]] Hausdorff space is metrizable.

## Explanations

### L1 human

A metrizable space is a topological space where you can measure distances in a way that gives exactly the open sets you already have.
Properties that hold in all metric spaces then hold in this space.
</document>

<document source="https://math2.tamnd.com/node/09M2RDB61G.md" tag="09M2RDB61G" kind="definition">
# Order of a differential equation

## Definition

Let $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ be an [[ordinary differential equation|0P97V20RW7]] for an unknown function $y$ of $x$.
Its order is the largest integer $k$ such that the equation depends on the derivative $y^{(k)}$.
For a system of equations, the order is the largest order of a derivative that occurs in the system.

## Remarks

The equation $y' = x y$ has order $1$, and the equation $y'' + \sin y = 0$ has order $2$.
The order does not depend on the powers of the derivatives: $(y')^3 = y$ has order $1$.
An equation of order $n$ can be changed into a first-order system of $n$ equations. See [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

The order tells you how many times the unknown function is differentiated, at most.
An equation with $y'$ only is of first order. An equation with $y''$, but no higher derivative, is of second order.
</document>

<document source="https://math2.tamnd.com/node/09N7DZ7SDB.md" tag="09N7DZ7SDB" kind="statement">
# Ring homomorphisms from a field are injective

## Statement

Let $K$ be a [[field|0QA8CHH8MG]], let $S$ be a nonzero [[ring|0894J980RK]] and let $f \colon K \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then $f$ is injective.

## Remarks

The proof: if $a \neq 0$ and $f(a) = 0$, then $1 = f(1) = f(a) f(a^{-1}) = 0$ in $S$. This contradicts $S \neq 0$.
The hypothesis $S \neq 0$ is necessary, because the map from $K$ to the [[zero ring|008TBCYDPX]] is a ring homomorphism.
A consequence: a homomorphism of fields makes the first field a copy of a subfield of the second.

## Explanations

### L1 human

A field has no ideal other than $0$ and the whole field.
The kernel of $f$ is an ideal, and it does not contain $1$.
So the kernel is $0$, and $f$ is injective.
</document>

<document source="https://math2.tamnd.com/node/09NDE9V7X1.md" tag="09NDE9V7X1" kind="statement">
# Finite intersection of open sets is open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $n \ge 1$ be an integer, and let $U_1, \ldots, U_n$ be [[open subsets|01ZQQDCERP]] of $X$.
Then the [[intersection|0HBRW4E6G1]] $U_1 \cap \cdots \cap U_n$ is open in $X$.

## Remarks

Proof: let $x$ be in the intersection. For each $k$ there is $\varepsilon_k > 0$ with $B(x, \varepsilon_k) \subseteq U_k$. Then $\varepsilon = \min(\varepsilon_1, \ldots, \varepsilon_n) > 0$ and $B(x, \varepsilon)$ lies in each $U_k$.
The statement is false for infinite families: in $\mathbb{R}$, the intersection of the open intervals $(-1/n, 1/n)$ for $n \ge 1$ is $\{0\}$, which is not open.
In a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.

## Explanations

### L1 human

Each open set gives its own small ball about the point, and the smallest of these finitely many balls fits in all the sets.
With infinitely many sets, the radii can shrink to zero, and then no ball is left.
</document>

<document source="https://math2.tamnd.com/node/09NWETVTNY.md" tag="09NWETVTNY" kind="concept">
# Injective function

## Summary

A function is injective when different inputs always give different values.
</document>

<document source="https://math2.tamnd.com/node/09PKJKA6J0.md" tag="09PKJKA6J0" kind="statement">
# De Morgan's laws

## Statement

For all [[propositions|0ZGC7ZESYN]] $p$ and $q$, these two [[logical equivalences|06GYGS41JK]] hold.
$$\neg (p \lor q) \equiv \neg p \land \neg q$$
$$\neg (p \land q) \equiv \neg p \lor \neg q$$

## Remarks

The first law is also valid in intuitionistic logic. In the second law, only the direction from $\neg p \lor \neg q$ to $\neg (p \land q)$ is.
The laws for sets have the same form. See [[De Morgan's laws for sets|0VK0W4D5Y5]].
The [[negation of a universal statement|0P3A06QNDP]] is the analogue for quantifiers.

## Explanations

### L1 human

To negate "and", change it to "or" and negate each part.
"$n$ is not both even and prime" means "$n$ is not even, or $n$ is not prime".
A [[truth table|05RSK4Q657]] with four rows proves each law.
</document>

<document source="https://math2.tamnd.com/node/09PR1V6RWR.md" tag="09PR1V6RWR" kind="statement">
# Mean of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X$ have the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$. Then $X$ is integrable and
$$E[X] = \int_{-\infty}^{\infty} x \, \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2/(2\sigma^2)} \, dx = \mu.$$

## Remarks

Proof: substitute $x = \mu + \sigma z$ with $\sigma = \sqrt{\sigma^2}$. The integral becomes $\mu \int \varphi(z) \, dz + \sigma \int z \varphi(z) \, dz$, where $\varphi$ is the [[standard normal density|027JF1FBB0]].
The first integral is $1$. The second is $0$, because $z \varphi(z)$ is an odd integrable function.
So the parameter $\mu$ is the mean. It is also the median and the mode, because the density is symmetric about $\mu$ and has its maximum there.

## Explanations

### L1 human

The bell curve of $\mathcal{N}(\mu, \sigma^2)$ is symmetric about $\mu$, so its average is $\mu$.
</document>

<document source="https://math2.tamnd.com/node/09PS3S0EF6.md" tag="09PS3S0EF6" kind="statement">
# Liouville's formula

## Statement

Let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ be continuous, and let $\Phi : I \to \mathbb{R}^{n \times n}$ be a differentiable function with $\Phi'(t) = A(t)\, \Phi(t)$ for each $t \in I$.
Then for each $t_0, t \in I$,
$$\det \Phi(t) = \det \Phi(t_0)\, \exp\left( \int_{t_0}^{t} \operatorname{tr} A(s)\, ds \right),$$
where $\operatorname{tr} A(s)$ is the trace of $A(s)$.

## Remarks

So $\det \Phi(t)$ is zero at each point of $I$ or at no point of $I$. This shows that a [[fundamental matrix|0MQ24RBG93]] is invertible everywhere if it is invertible at one point.
For a constant matrix $A$, the formula gives $\det e^{tA} = e^{t \operatorname{tr} A}$.
For the system that comes from $y'' + p y' + q y = 0$, the trace is $-p$, and the formula is [[Abel's identity|0VGWKX2JHB]].

## Explanations

### L1 human

The determinant of $\Phi$ is the volume spanned by $n$ solutions.
The formula says how this volume changes in time: its rate of growth is the trace of $A$.
</document>

<document source="https://math2.tamnd.com/node/09PV9JSY6E.md" tag="09PV9JSY6E" kind="definition">
# Prüfer sequence

## Definition

Let $n \ge 2$ be an integer and let $T$ be a [[tree|0MJ0JS72DZ]] with vertex set $\{1, \dots, n\}$.
The Prüfer sequence of $T$ is the sequence $(a_1, \dots, a_{n-2})$ that this process gives.
Let $T_0 = T$. For $i = 1, \dots, n - 2$, let $\ell_i$ be the leaf of $T_{i-1}$ with the smallest label.
Let $a_i$ be the unique neighbour of $\ell_i$ in $T_{i-1}$, and let $T_i = T_{i-1} - \ell_i$.

## Remarks

The map $T \mapsto (a_1, \dots, a_{n-2})$ is a bijection from the labelled trees on $\{1, \dots, n\}$ to $\{1, \dots, n\}^{n-2}$. This proves [[Cayley's formula|0M2SSGSYPE]].
A vertex $v$ appears $\deg(v) - 1$ times in the sequence. Thus the leaves are the labels that do not appear.
Example: the star with centre $1$ on $\{1, 2, 3, 4\}$ has the sequence $(1, 1)$.

## Explanations

### L1 human

Remove the smallest leaf again and again, and each time write down the vertex it was attached to.
Stop when two vertices are left. The list of $n - 2$ numbers fixes the tree.
</document>

<document source="https://math2.tamnd.com/node/09QSZVVM26.md" tag="09QSZVVM26" kind="concept">
# Bounded set

## Summary

A bounded set in a metric space is a set whose points have distances from each other below one fixed number.
Equivalently, the set lies in some ball.
</document>

<document source="https://math2.tamnd.com/node/09R8GC85RB.md" tag="09R8GC85RB" kind="concept">
# Removable singularity

## Summary

A removable singularity of a holomorphic function is an isolated singularity where one can define or redefine the function so that it becomes holomorphic there. See the [[definition|0P18JC72RE]].
</document>

<document source="https://math2.tamnd.com/node/09RDZZGJZ9.md" tag="09RDZZGJZ9" kind="statement">
# Cancellation in congruences

## Statement

Let $n$ be a positive integer, and let $a$, $b$ and $c$ be integers. Let $d = \gcd(c, n)$.
If $c a \equiv c b \pmod n$, then $a \equiv b \pmod{n / d}$.

## Remarks

In particular, if $c$ and $n$ are [[coprime|0GFKP8F8AG]], then $c a \equiv c b \pmod n$ implies $a \equiv b \pmod n$.
The modulus can become smaller. For example, $2 \cdot 1 \equiv 2 \cdot 4 \pmod 6$, but $1 \not\equiv 4 \pmod 6$. Here $d = 2$, and indeed $1 \equiv 4 \pmod 3$.
The proof: $n \mid c (a - b)$ gives $\frac{n}{d} \mid \frac{c}{d} (a - b)$. Since $n / d$ and $c / d$ are coprime, the [[generalized Euclid's lemma|03BAREW1Z5]] gives $\frac{n}{d} \mid a - b$.

## Explanations

### L1 human

You can divide both sides of a congruence by a number that has no common factor with the modulus.
If the number shares a factor with the modulus, you must also divide the modulus by that common factor.
</document>

<document source="https://math2.tamnd.com/node/09RFKMBQ63.md" tag="09RFKMBQ63" kind="statement">
# The rationals form an ordered field

## Statement

Define an order on the [[rational numbers|0NCJ5WXN6Z]] as follows. For integers $a, c$ and positive integers $b, d$, let $\frac{a}{b} \le \frac{c}{d}$ when $a d \le b c$ in $\mathbb{Z}$.
This relation is well defined, and with it $\mathbb{Q}$ is an [[ordered field|010JCV9J4E]].

## Remarks

The order on $\mathbb{Q}$ extends the [[order on the integers|0F6FM9Y4Y6]].
This is the only order on $\mathbb{Q}$ that makes it an ordered field.
The ordered field $\mathbb{Q}$ is Archimedean, but it is not complete: the set $\{q \in \mathbb{Q} : q^2 < 2\}$ has no least upper bound in $\mathbb{Q}$.

## Explanations

### L1 human

To compare two fractions, write them with positive denominators and cross-multiply.
For example, $2/3 < 3/4$ because $2 \cdot 4 = 8 < 9 = 3 \cdot 3$.
</document>

<document source="https://math2.tamnd.com/node/09RJ859N7A.md" tag="09RJ859N7A" kind="concept">
# Partial sum

## Summary

A partial sum of a series is the sum of its first finitely many terms. The sequence of partial sums decides whether the series converges. The node [[partial sum|0NB643K60Q]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/09RTQRK5PC.md" tag="09RTQRK5PC" kind="concept">
# Student's t-distribution

## Summary

Student's t-distribution is a continuous law on $\mathbb{R}$ that is symmetric about $0$ and bell-shaped, like the standard normal law, but with heavier tails. It is the law of the standardized sample mean when one estimates the standard deviation from the data.
The node [[Student's t-distribution|0NQ5798SB1]] gives the precise definition.

## Remarks

William Sealy Gosset published the distribution in 1908 under the pen name "Student".
</document>

<document source="https://math2.tamnd.com/node/09RTYYHXKA.md" tag="09RTYYHXKA" kind="counterexample">
# Smooth non-analytic function

## Counterexample

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \le 0$.
Then $f$ is [[smooth|0BE3QDH73A]] on $\mathbb{R}$, and $f^{(n)}(0) = 0$ for all $n$.
So its [[Taylor series|0225QPWRRE]] at $0$ is the zero series. But $f(x) > 0$ for $x > 0$, so $f$ is not [[analytic|0JD5ZNW0F4]] at $0$.

## Remarks

For $x > 0$, each derivative has the form $p_n(1/x) e^{-1/x}$ for a polynomial $p_n$. It tends to $0$ as $x \to 0^+$, because $e^{-1/x}$ decays faster than any power of $x$.
Such functions give smooth bump functions and smooth partitions of unity.
Mathlib defines this function as `expNegInvGlue`.

## Explanations

### L1 human

This function is so flat at $0$ that all its derivatives vanish there.
So its Taylor series sees only the zero function, but the function is not zero.
</document>

<document source="https://math2.tamnd.com/node/09S3PFBEPM.md" tag="09S3PFBEPM" kind="definition">
# Integral domain

## Definition

An integral domain is a [[commutative ring|0GYGHKX834]] $R$ with these two properties.

1. $1 \neq 0$ in $R$.
2. For all $a, b \in R$, if $a b = 0$, then $a = 0$ or $b = 0$.

Equivalently, $R$ is a nonzero commutative ring in which $0$ is the only [[zero divisor|0K26YKNBSA]].

## Remarks

The integers $\mathbb{Z}$ and every [[field|0QA8CHH8MG]] are integral domains.
The ring $\mathbb{Z}/6\mathbb{Z}$ is not an integral domain, because $2 \cdot 3 = 0$.
In an integral domain, [[cancellation|0GJM24EXP9]] holds for nonzero factors.
Some authors, and Mathlib, also use the word domain for a noncommutative ring with property 2.

## Explanations

### L1 human

An integral domain is a ring that behaves like the integers for products.
If a product is zero, then one of the factors is zero.
This is what you use when you solve $(x - 2)(x - 3) = 0$.
</document>

<document source="https://math2.tamnd.com/node/09S5SN35DJ.md" tag="09S5SN35DJ" kind="definition">
# Least common multiple

## Definition

Let $a$ and $b$ be nonzero integers.
A common multiple of $a$ and $b$ is an integer $m$ such that $a \mid m$ and $b \mid m$.
The least common multiple $\operatorname{lcm}(a, b)$ is the smallest positive common multiple of $a$ and $b$.
By convention, $\operatorname{lcm}(a, 0) = \operatorname{lcm}(0, a) = 0$ for each integer $a$.

## Remarks

The positive integer $|a b|$ is a common multiple, so $\operatorname{lcm}(a, b)$ exists and $\operatorname{lcm}(a, b) \le |a b|$.
Each common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a, b)$.
For example, $\operatorname{lcm}(4, 6) = 12$.
The [[product of the gcd and the lcm|0TDY6JZHKS]] is the product of the two numbers.

## Explanations

### L1 human

List the positive multiples of each number, and take the smallest number that is in both lists.
The multiples of $4$ are $4, 8, 12, 16, \dots$ and the multiples of $6$ are $6, 12, 18, \dots$.
Thus $\operatorname{lcm}(4, 6) = 12$. You use this number when you add $\tfrac{1}{4} + \tfrac{1}{6}$ with a common denominator.
</document>

<document source="https://math2.tamnd.com/node/09S8DV34Y0.md" tag="09S8DV34Y0" kind="concept">
# Subbasis

## Summary

A subbasis of a topology is a family of open sets that generates the topology. The topology is the smallest one in which all members of the family are open.
The finite intersections of members of a subbasis form a basis.
</document>

<document source="https://math2.tamnd.com/node/09T8ZHST1V.md" tag="09T8ZHST1V" kind="definition">
# Independent sigma-algebras

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $I$ be an index set, and for each $i \in I$ let $\mathcal{F}_i \subseteq \mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
The family $(\mathcal{F}_i)_{i \in I}$ is independent if for each finite nonempty subset $J \subseteq I$ and each choice of events $A_j \in \mathcal{F}_j$ for $j \in J$,
$$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).$$

## Remarks

Events $(A_i)_{i \in I}$ are [[mutually independent|0GCVEV6BJA]] if and only if the σ-algebras $\{\varnothing, A_i, A_i^c, \Omega\}$ are independent.
Random variables are independent when the σ-algebras that they generate are independent.
To check independence, it is enough to check the product formula on a generating family of each $\mathcal{F}_i$ that is closed under finite intersections. This follows from Dynkin's π-λ theorem.

## Explanations

### L1 human

A σ-algebra holds all the information of one part of an experiment, for example everything about the first roll of a die.
Two such σ-algebras are independent when information about one part never changes the chances about the other part.
</document>

<document source="https://math2.tamnd.com/node/09V3NASXND.md" tag="09V3NASXND" kind="concept">
# Equivalent metrics

## Summary

Two metrics on the same set are equivalent when they give the same open sets.
A stronger form asks that each metric is at most a constant times the other.
</document>

<document source="https://math2.tamnd.com/node/09V6DDACXR.md" tag="09V6DDACXR" kind="concept">
# Scalar multiplication of matrices

## Summary

Scalar multiplication of a matrix multiplies each entry of the matrix by the same scalar.
</document>

<document source="https://math2.tamnd.com/node/09VDS2W1E8.md" tag="09VDS2W1E8" kind="definition">
# Module over a ring

## Definition

Let $R$ be a ring with identity element $1$.
A *left module* over $R$ is an abelian group $(M, +)$ together with a map $R \times M \to M$, $(r, x) \mapsto rx$, such that, for all $r, s \in R$ and $x, y \in M$:

1. $r(x + y) = rx + ry$.
2. $(r + s)x = rx + sx$.
3. $(rs)x = r(sx)$.
4. $1x = x$.

## Remarks

If $R$ is a field, then an $R$-module is the same as a [[vector space|000AYPA9AJ]] over $R$.
Each abelian group is a module over the ring $\Z$, with $nx = x + \cdots + x$ ($n$ terms) for $n > 0$.
The ring $R$ is a module over itself, and $R^n$ is a module with the operations on coordinates.
Right modules use a map $M \times R \to M$; over a commutative ring, the two notions are the same.

## Explanations

### L1 human

A module is a vector space in which the scalars come from a ring instead of a field.
We cannot divide by scalars in general, so many facts about vector spaces fail.
For example, a module need not have a basis.
</document>

<document source="https://math2.tamnd.com/node/09VMDSRXWJ.md" tag="09VMDSRXWJ" kind="concept">
# Singular value

## Summary

The singular values of a matrix $A$ are the square roots of the eigenvalues of $A^* A$.
</document>

<document source="https://math2.tamnd.com/node/09VYHEF2VT.md" tag="09VYHEF2VT" kind="definition">
# Orthonormal family

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $I$ be a set.
A family $(e_i)_{i \in I}$ of vectors of $V$ is *orthonormal* when, for all $i, j \in I$,
$$\langle e_i, e_j \rangle = \delta_{ij},$$
where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

## Remarks

So each $e_i$ has norm $1$, and $e_i$ and $e_j$ are [[orthogonal|0Y2X7P2WB7]] for $i \ne j$.
The standard basis of $\R^n$ is orthonormal for the dot product.
The functions $t \mapsto e^{2\pi i k t}$, $k \in \Z$, are orthonormal for $\langle f, g \rangle = \int_0^1 f \bar g\, dt$.

## Explanations

### L1 human

An orthonormal family is like a set of coordinate axes: unit vectors at right angles.
Computations with such vectors are simple, because all inner products are $0$ or $1$.
They are the best bases to work with in an inner product space.
</document>

<document source="https://math2.tamnd.com/node/09VZBHB33V.md" tag="09VZBHB33V" kind="statement">
# Every walk contains a path

## Statement

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
Suppose that there is a [[walk|0VJZM0F9YM]] $W$ from $u$ to $v$.
Then there is a [[path|0JDE56HWQ8]] from $u$ to $v$ whose vertices and edges are vertices and edges of $W$.

## Remarks

Proof idea: if a walk visits a vertex $x$ twice, delete the closed part between the two visits. Each deletion makes the walk shorter, so the process stops with a path.
A consequence: a shortest walk from $u$ to $v$ is a path. Thus the [[distance|0WPZ5YPAGR]] can be defined with paths.

## Explanations

### L1 human

If you can walk from $u$ to $v$, you can also do it without visiting any place twice.
Whenever you come back to a place, cut out the loop that you just made.
</document>

<document source="https://math2.tamnd.com/node/09WRNGPTMV.md" tag="09WRNGPTMV" kind="concept">
# Local minimum

## Summary

A function has a local minimum at a point when its value there is at most its values at all nearby points. Local minima are the bottoms of the valleys of the graph. The node [[local minimum|0NS8BP96QX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/09XNAPTJQ7.md" tag="09XNAPTJQ7" kind="concept">
# Affine space

## Summary

An affine space is a set of points on which the vectors of a vector space act by translation. It keeps the ideas of lines, planes and parallelism from Euclidean geometry, but it has no origin, no distance and no angle.
</document>

<document source="https://math2.tamnd.com/node/09YB3WQG38.md" tag="09YB3WQG38" kind="statement">
# Permuting rows multiplies the determinant by the sign

## Statement

Let $K$ be a commutative ring, let $A = (a_{ij})$ be an $n \times n$ matrix over $K$, and let $\sigma$ be a permutation of $\{1, \dots, n\}$.
Let $B = (b_{ij})$ be the matrix with $b_{ij} = a_{\sigma(i), j}$.
Then $\det B = \sgn(\sigma) \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In particular, exchanging two rows changes the sign of the determinant.
For a [[permutation matrix|08SP3DF8NB]] $P$, this gives $\det P = \pm 1$.
The same holds for a permutation of the columns.

## Explanations

### L1 human

When we exchange two rows, the determinant changes its sign.
A general reordering of the rows is a sequence of such exchanges.
So it multiplies the determinant by $1$ or by $-1$, according to the sign of the permutation.
</document>

<document source="https://math2.tamnd.com/node/09YBMJTNBM.md" tag="09YBMJTNBM" kind="statement">
# Divisibility by 11

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $11 \mid n$ if and only if $11 \mid d_0 - d_1 + d_2 - \dots + (-1)^k d_k$.

## Remarks

More precisely, $n \equiv \sum_{i=0}^{k} (-1)^i d_i \pmod{11}$. This holds because $10 \equiv -1 \pmod{11}$, so $10^i \equiv (-1)^i \pmod{11}$.
For example, $918{,}082$ gives $2 - 8 + 0 - 8 + 1 - 9 = -22$, and $11 \mid -22$. So $11 \mid 918{,}082$.

## Explanations

### L1 human

Start at the last digit, and add and subtract the digits in turn.
The number is a multiple of $11$ exactly when the result is a multiple of $11$, including $0$.
For example, $121$ gives $1 - 2 + 1 = 0$, and indeed $121 = 11 \cdot 11$.
</document>

<document source="https://math2.tamnd.com/node/09YMV8H4X5.md" tag="09YMV8H4X5" kind="statement">
# Intersection of closed sets is closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(F_i)_{i \in I}$ be a family of [[closed subsets|0G5AV9Q4RR]] of $X$, indexed by a nonempty set $I$.
Then the [[intersection|0839XNQXDK]] $\bigcap_{i \in I} F_i$ is closed in $X$.

## Remarks

Proof: by [[De Morgan's laws for sets|0VK0W4D5Y5]], $X \setminus \bigcap_i F_i = \bigcup_i (X \setminus F_i)$. This is a [[union of open sets|0HEF46RPEK]], so it is open.
The same proof works in each [[topological space|0CNQXVBHCX]].
A union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets $[1/n, 1]$ for $n \ge 1$ is $(0, 1]$.

## Explanations

### L1 human

Closed sets behave like open sets with the roles of union and intersection exchanged.
Any intersection of closed sets is closed, and finite unions of closed sets are closed.
</document>

<document source="https://math2.tamnd.com/node/09ZJ6CAK7S.md" tag="09ZJ6CAK7S" kind="statement">
# The number of Sylow subgroups is the index of the normaliser

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.
Then $n_p = [G : N_G(P)]$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$.

## Remarks

By the [[second Sylow theorem|09GCQ5STVJ]], $G$ acts transitively by conjugation on $\operatorname{Syl}_p(G)$.
The stabiliser of $P$ is $N_G(P)$, so the [[orbit-stabiliser theorem|0E2HSCKZ6C]] gives the formula.
In particular, $P$ is normal in $G$ if and only if $n_p = 1$.

## Explanations

### L1 human

All Sylow $p$-subgroups are conjugates of one of them, $P$.
The number of conjugates of $P$ is the number of cosets of the elements that do not move $P$.
These elements form the normaliser.
</document>

<document source="https://math2.tamnd.com/node/09ZKE5XQ74.md" tag="09ZKE5XQ74" kind="statement">
# Product of compact sets is compact

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], and let $K \subseteq X$ and $L \subseteq Y$ be [[compact|00R0X1WGPX]].
Then $K \times L$ is compact in $X \times Y$ with the [[product topology|0Z57EC6YX3]].

## Remarks

The proof uses the tube lemma. For each $x \in K$, finitely many basic open sets $U \times V$ cover $\{x\} \times L$, and their intersection over the first factor gives an open set about $x$.
By induction, a product of finitely many compact sets is compact. For example, $[0, 1]^n$ is compact.
For arbitrary products of compact spaces, the statement is [[Tychonoff's theorem|0JTRSJKCFB]].

## Explanations

### L1 human

A rectangle made of two compact sides is compact.
For example, the square $[0, 1] \times [0, 1]$ is compact, because $[0, 1]$ is compact.
</document>

<document source="https://math2.tamnd.com/node/09ZWYRQ41J.md" tag="09ZWYRQ41J" kind="concept">
# Linearisation at an equilibrium

## Summary

Let $U \subseteq \mathbb{R}^n$ be open, let $f : U \to \mathbb{R}^n$ be continuously differentiable, and let $x^*$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
Let $A = Df(x^*)$ be the Jacobian matrix of $f$ at $x^*$. The linearisation of the system at $x^*$ is the linear system $y' = A y$, where $y$ stands for $x - x^*$.
It comes from the first-order Taylor expansion $f(x) = A (x - x^*) + o(\|x - x^*\|)$.

## Remarks

If each eigenvalue of $A$ has negative real part, then $x^*$ is asymptotically stable for the nonlinear system. If one eigenvalue has positive real part, then $x^*$ is unstable.
If some eigenvalue has zero real part and none has positive real part, the linearisation does not decide stability. For $x' = -x^3$ and $x' = x^3$, the linearisation at $0$ is $y' = 0$ in both cases, but the first is stable and the second is not.
Near a hyperbolic equilibrium, the [[Hartman–Grobman theorem|0S8WX8A4V9]] says that the linearisation describes the phase portrait up to a homeomorphism.

## Explanations

### L1 human

Near an equilibrium, a smooth vector field looks like a linear one.
We replace the field by its best linear approximation and study the simpler linear system.
This works well unless the linear system is on the border between stable and unstable.
</document>

<document source="https://math2.tamnd.com/node/09ZXGX55DM.md" tag="09ZXGX55DM" kind="concept">
# Higher partial derivatives

## Summary

A higher partial derivative of a function of several variables is a [[partial derivative|09EGC6F8PT]] of a partial derivative, taken one or more times.
A second partial derivative such as $\partial^2 f / \partial y \, \partial x$, in which the two variables differ, is called a mixed partial derivative.
</document>

<document source="https://math2.tamnd.com/node/0A1QQN333A.md" tag="0A1QQN333A" kind="statement">
# Determinant of a matrix exponential

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$.
Then
$$\det(e^A) = e^{\tr A},$$
where $e^A$ is the [[matrix exponential|020M04FN1H]], $\det$ is the [[determinant|00KDZRW4C0]] and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

If $A$ is triangular over $\C$ with diagonal entries $\lambda_i$, then $e^A$ is triangular with diagonal entries $e^{\lambda_i}$, and both sides equal $e^{\lambda_1 + \cdots + \lambda_n}$.
The general case follows by [[triangularisation|0363JMQ42Y]] over $\C$.
It shows again that $e^A$ is invertible, because $e^{\tr A} \ne 0$.
If $\tr A = 0$, then $e^A$ lies in the special linear group.

## Explanations

### L1 human

The determinant of the exponential of a matrix is the exponential of its trace.
This links the product of the eigenvalues of $e^A$ with the sum of the eigenvalues of $A$.
It is the matrix form of the rule $e^{a} e^{b} = e^{a + b}$.
</document>

<document source="https://math2.tamnd.com/node/0A1YXSKB48.md" tag="0A1YXSKB48" kind="concept">
# Isolated point

## Summary

An isolated point of a set $S$ is a point of $S$ that has a neighbourhood with no other point of $S$.
</document>

<document source="https://math2.tamnd.com/node/0A26X79RTB.md" tag="0A26X79RTB" kind="statement">
# Open sets of a metric space

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $U \subseteq X$.
Then $U$ is [[open|01ZQQDCERP]] in the [[metric topology|0A6YRNZB2R]] if and only if for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$.
Here $B(x, \varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\varepsilon$.

## Remarks

The number $\varepsilon$ can depend on $x$.
In $\mathbb{R}$, the set $(0, \infty)$ is open: for $x > 0$, take $\varepsilon = x$.
The set $\mathbb{Q}$ is not open in $\mathbb{R}$, because each ball about a rational number contains an irrational number.

## Explanations

### L1 human

This is the test for open sets that you use in practice.
You pick any point of the set, and you find a small ball about it inside the set.
</document>

<document source="https://math2.tamnd.com/node/0A2HNZC44K.md" tag="0A2HNZC44K" kind="concept">
# Convergence in mean

## Summary

A sequence of random variables converges in $r$-th mean when the expected value of $|X_n - X|^r$ tends to $0$.
The node [[convergence in mean|0TTA57SEJG]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0A3CK6DVVB.md" tag="0A3CK6DVVB" kind="definition">
# Uniform continuity on a set

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $S \subseteq D$.
The function $f$ is uniformly continuous on $S$ if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(y)| < \varepsilon$ for all $x, y \in S$ with $|x - y| < \delta$.

## Remarks

So $f$ is uniformly continuous on $S$ when its restriction to $S$ is [[uniformly continuous|0Q4DR37DVE]].
A function can be uniformly continuous on a set but not on a larger set. For example, $x^2$ is uniformly continuous on $[0, 1]$ but not on $\mathbb{R}$.
Mathlib defines it as `UniformContinuousOn`.

## Explanations

### L1 human

Uniform continuity on $S$ only looks at pairs of points inside $S$.
Points outside $S$ play no role.
</document>

<document source="https://math2.tamnd.com/node/0A3GWZ86AY.md" tag="0A3GWZ86AY" kind="statement">
# The reals are Archimedean

## Statement

The ordered field $\mathbb{R}$ of [[real numbers|08J2M1JYAG]] is an [[Archimedean ordered field|0HQARMKEV5]].
That is, for all real numbers $x$ and $y$ with $y > 0$, there is a natural number $n$ with $n y > x$.

## Remarks

The proof uses the [[completeness axiom|0AXWET80YW]]. If $\mathbb{N}$ were bounded above, let $b = \sup \mathbb{N}$. Then $b - 1$ is not an upper bound, so $n > b - 1$ for some $n$, and $n + 1 > b$, a contradiction.
Thus each [[complete ordered field|0A9E6X0S3E]] is Archimedean.

## Explanations

### L1 human

However small your step and however far your goal on the real line, finitely many steps get you past the goal.
There are no infinitely large real numbers.
</document>

<document source="https://math2.tamnd.com/node/0A3HJX3KTF.md" tag="0A3HJX3KTF" kind="concept">
# Pseudometric space

## Summary

A pseudometric space is a set with a distance function that is almost a metric. The difference: two different points can have distance zero.
</document>

<document source="https://math2.tamnd.com/node/0A58THH5XV.md" tag="0A58THH5XV" kind="definition">
# Boolean function

## Definition

Let $n$ be a natural number.
A Boolean function of $n$ variables is a function $f : \{0, 1\}^n \to \{0, 1\}$.

## Remarks

There are $2^n$ inputs, so there are $2^{2^n}$ Boolean functions of $n$ variables.
A truth table lists the values of a Boolean function on all inputs.
Every Boolean function is given by a formula in [[disjunctive normal form|06S4WGSEQA]], and also by one in [[conjunctive normal form|07C4A8Y18K]].

## Explanations

### L1 human

A Boolean function takes some bits as input and returns one bit.
The logic gates and, or and not are Boolean functions, and a digital circuit computes a Boolean function.
</document>

<document source="https://math2.tamnd.com/node/0A61B60B5R.md" tag="0A61B60B5R" kind="definition">
# Differentiable function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is differentiable at $a \in I$ if $f$ has a [[derivative|0F1WR9C83K]] at $a$.
It is differentiable on $I$ if it is differentiable at each point of $I$.

## Remarks

[[Differentiable functions are continuous|07ZMX4ANJS]].
The converse fails: the [[absolute value|0CQ9ZM383N]] is continuous but not differentiable at $0$.
Polynomials, the exponential function, and the sine are differentiable on $\mathbb{R}$.
Mathlib defines differentiability at a point as `DifferentiableAt`.

## Explanations

### L1 human

A differentiable function has a well defined slope at each point.
Its graph is smooth, with no corners and no vertical tangents.
</document>

<document source="https://math2.tamnd.com/node/0A6PGTTP2A.md" tag="0A6PGTTP2A" kind="definition">
# Linear equation with constant coefficients

## Definition

Let $n \ge 1$, let $a_0, \dots, a_{n-1}$ be real numbers, let $I$ be an open interval and let $b : I \to \mathbb{R}$ be continuous.
A linear equation with constant coefficients is a [[linear differential equation|0PB7K8M7FG]] of the form
$$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = b(x).$$
Its homogeneous equation is the case $b = 0$, on the interval $I = \mathbb{R}$.

## Remarks

The function $e^{r x}$ solves the homogeneous equation if and only if $r$ is a root of the [[characteristic equation|0QDXMG46QW]] $r^n + a_{n-1} r^{n-1} + \dots + a_0 = 0$.
The roots of this polynomial give a [[fundamental system|0H4KNNCHPY]]. See the cases of [[distinct real roots|0HJ0HDA6W7]], [[repeated roots|0HRVY67CNV]] and [[complex conjugate roots|0BEZX8F2XW]].
The [[harmonic oscillator|0YG0ZJPTWG]] $x'' + \omega^2 x = 0$ is an example of order $2$.

## Explanations

### L1 human

When the coefficients are constants, the derivative of $e^{r x}$ is $r e^{r x}$, again a multiple of the same function.
So the equation becomes a polynomial equation in $r$.
This changes calculus into algebra.
</document>

<document source="https://math2.tamnd.com/node/0A6PMR20N4.md" tag="0A6PMR20N4" kind="statement">
# Derivative gives the directional derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that $f$ is [[differentiable|0135XZZQ8M]] at $a$ with derivative $Df(a)$.
Then for each $v \in \mathbb{R}^n$, $f$ [[has the directional derivative|05EGFBZTGC]] $Df(a)(v)$ at $a$ in the direction $v$:
$$\lim_{t \to 0} \frac{f(a + t v) - f(a)}{t} = Df(a)(v).$$

## Remarks

For $v = e_i$ this gives $\dfrac{\partial f}{\partial x_i}(a) = Df(a)(e_i)$. So the [[partial derivatives|09EGC6F8PT]] of the components of $f$ are the entries of the matrix of $Df(a)$.
For a real-valued $f$, it gives $D_v f(a) = \nabla f(a) \cdot v$, where $\nabla f(a)$ is the [[gradient|0ABK9V9Y64]].
In particular, $D_v f(a)$ is linear in $v$. A function whose directional derivatives are not linear in $v$ is not differentiable.

## Explanations

### L1 human

Once you know the linear map $Df(a)$, you know the slope of $f$ in every direction: feed the direction into the map.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, $Df(1, 1)(v) = 2 v_1 + 2 v_2$. So the slope in the direction $(1, -1)$ is $0$.
</document>

<document source="https://math2.tamnd.com/node/0A6QZ367GX.md" tag="0A6QZ367GX" kind="definition">
# System of linear differential equations

## Definition

Let $n \ge 1$, let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ be a continuous matrix-valued function and let $b : I \to \mathbb{R}^n$ be continuous.
The system of linear differential equations with coefficient matrix $A$ and right side $b$ is the equation
$$x'(t) = A(t)\, x(t) + b(t)$$
for an unknown differentiable function $x : I \to \mathbb{R}^n$.
The system is homogeneous if $b = 0$ on $I$.

## Remarks

In coordinates, it is the system $x_i' = \sum_{j=1}^{n} a_{ij}(t)\, x_j + b_i(t)$ for $i = 1, \dots, n$.
For each $t_0 \in I$ and $x_0 \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $x(t_0) = x_0$.
The solutions of the homogeneous system form a vector space of dimension $n$. A basis gives a [[fundamental matrix|0MQ24RBG93]].
A [[linear equation of order n|0PB7K8M7FG]] becomes such a system by [[reduction to a first-order system|0TH9CPW3VS]].

## Explanations

### L1 human

Several unknown functions change at the same time, and each rate is a linear combination of all the unknowns.
In matrix form, the system looks like one linear first-order equation, with a matrix in place of a number.
</document>

<document source="https://math2.tamnd.com/node/0A6YRNZB2R.md" tag="0A6YRNZB2R" kind="definition">
# Metric topology

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
The metric topology of $(X, d)$ is the set $\tau_d$ of all $U \subseteq X$ such that for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$.
Here $B(x, \varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\varepsilon$.

## Remarks

The set $\tau_d$ is a [[topology|0CNQXVBHCX]] on $X$: it contains $\varnothing$ and $X$, and it is closed under [[unions|0HEF46RPEK]] and [[finite intersections|09NDE9V7X1]].
Each open ball is in $\tau_d$, by [[open balls are open|0B2305GNAZ]]. The open balls form a [[basis|06EX0FHBTF]] of $\tau_d$.
For $\mathbb{R}$ with the usual metric, $\tau_d$ is the usual topology, generated by the open intervals.
Different metrics can give the same topology. Such metrics are [[equivalent|0TWQ5PRKWS]].

## Explanations

### L1 human

The metric topology is how a metric space becomes a topological space.
A set is open when each of its points has a small ball about it inside the set.
</document>

<document source="https://math2.tamnd.com/node/0A74SX55QP.md" tag="0A74SX55QP" kind="statement">
# Archimedean property

## Statement

Let $F$ be an [[Archimedean ordered field|0HQARMKEV5]] and let $x$ be an element of $F$.
Then there is a natural number $n$ with $x < n$.

## Remarks

Proof: apply the definition with $y = 1$.
Thus the natural numbers are not [[bounded above|0B5H2V4JW8]] in $F$.
The [[real numbers are Archimedean|0A3GWZ86AY]], thus this holds for each real number $x$.

## Explanations

### L1 human

No element of an Archimedean field is larger than all the natural numbers.
For example, for the real number $x = 10^{6} + 0.5$, the natural number $n = 10^{6} + 1$ is larger.
</document>

<document source="https://math2.tamnd.com/node/0A7KGAYHGV.md" tag="0A7KGAYHGV" kind="statement">
# Baire category theorem

## Statement

Each [[complete metric space|08EWT7J0SC]] is a [[Baire space|0D5KR33H7X]].

## Remarks

Explicitly: if $X$ is a complete metric space and $(U_n)_{n \in \mathbb{N}}$ is a sequence of open dense subsets of $X$, then $\bigcap_n U_n$ is dense in $X$.
The proof builds a nested sequence of closed balls with radii tending to $0$, and uses completeness to find a point in all of them.
A second form of the theorem says that each locally compact Hausdorff space is a Baire space.
A consequence: $\mathbb{R}$ is not a countable union of nowhere dense sets, and thus $\mathbb{R}$ is uncountable.

## Explanations

### L1 human

A complete metric space cannot be covered by countably many thin sets.
For example, you cannot write the real line as a countable union of sets like $\mathbb{Z}$ or the Cantor set.
</document>

<document source="https://math2.tamnd.com/node/0A7KJ68XHT.md" tag="0A7KJ68XHT" kind="statement">
# Tower law

## Statement

Let $K \subseteq L \subseteq M$ be fields.
Then
$$[M : K] = [M : L] \cdot [L : K].$$

## Remarks

The proof: let $u_1, \dots, u_d$ be a basis of $L$ over $K$, and let $w_1, \dots, w_e$ be a basis of $M$ over $L$.
Then the products $u_i w_j$ are a basis of $M$ over $K$.
The formula also holds for infinite degrees, as a product of cardinals.
So $M/K$ is finite if and only if $M/L$ and $L/K$ are finite.
If $[M : K]$ is a prime number, then $M/K$ has no [[intermediate field|02NBKVQFRQ]] other than $K$ and $M$.

## Explanations

### L1 human

Degrees multiply along a tower of fields.
For example, $[\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = 2 \cdot 2 = 4$.
The law is like Lagrange's theorem for groups, and it is the main tool to compute degrees.
</document>

<document source="https://math2.tamnd.com/node/0A7KV2WB7A.md" tag="0A7KV2WB7A" kind="concept">
# Monotone function

## Summary

A monotone function between ordered sets is a function that keeps the order.
</document>

<document source="https://math2.tamnd.com/node/0A7YDKRNR6.md" tag="0A7YDKRNR6" kind="definition">
# Complex sine and cosine

## Definition

For $z \in \mathbb{C}$, the complex sine and the complex cosine of $z$ are
$$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2},$$
where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.

## Remarks

For real $z$, these are the usual sine and cosine, by [[Euler's formula|04R3CY41VJ]].
The functions $\sin$ and $\cos$ are [[entire|0419794R0T]], with $\sin' = \cos$ and $\cos' = -\sin$. The identity $\sin^2 z + \cos^2 z = 1$ holds for each $z \in \mathbb{C}$.
They are not bounded on $\mathbb{C}$. For real $y$, $\cos(iy) = \cosh y = \frac{e^y + e^{-y}}{2}$.
The [[power series of the cosine|0Z1G2F8DGY]] is the same as in the real case.

## Explanations

### L1 human

Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$ lets us solve for $\cos \theta$ and $\sin \theta$ in terms of exponentials.
These formulas make sense for every complex number, so we use them to define $\sin z$ and $\cos z$.
On the imaginary axis they grow very fast: $\cos(10 i)$ is more than $10000$.
</document>

<document source="https://math2.tamnd.com/node/0A8JNBGT2Z.md" tag="0A8JNBGT2Z" kind="concept">
# Bounded sequence

## Summary

A bounded sequence is a real sequence whose terms all lie in one fixed interval. Boundedness is a necessary condition for convergence, and it is the hypothesis of the Bolzano–Weierstrass theorem. The node [[bounded sequence|05FQ3VGAR9]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0A8JXHYTJC.md" tag="0A8JXHYTJC" kind="counterexample">
# x^2 is not uniformly continuous on the real line

## Counterexample

The function $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$ is [[continuous|0P27W9RW0G]] but not [[uniformly continuous|0Q4DR37DVE]].
For $n \ge 1$, let $x_n = n$ and $y_n = n + 1/n$. Then $|x_n - y_n| = 1/n \to 0$, but
$$f(y_n) - f(x_n) = 2 + \frac{1}{n^2} > 2.$$
So no $\delta$ works for $\varepsilon = 2$.

## Remarks

On each bounded interval $[-c, c]$, the function $x^2$ is Lipschitz with constant $2c$. So it is uniformly continuous there.
The problem is that the slope $2x$ is not bounded on $\mathbb{R}$.

## Explanations

### L1 human

The graph of $x^2$ gets steeper and steeper as $x$ grows.
So a fixed small change in $x$ gives larger and larger changes in $x^2$.
</document>

<document source="https://math2.tamnd.com/node/0A8TJ0VHC7.md" tag="0A8TJ0VHC7" kind="statement">
# Chain rule for a composition with a curve

## Statement

Let $I \subseteq \mathbb{R}$ be an open interval, let $U \subseteq \mathbb{R}^n$ be open, let $\gamma \colon I \to U$, let $f \colon U \to \mathbb{R}^m$, and let $t_0 \in I$.
Suppose that $\gamma$ is differentiable at $t_0$ and $f$ is differentiable at $\gamma(t_0)$.
Then $f \circ \gamma$ is differentiable at $t_0$, and
$$(f \circ \gamma)'(t_0) = Df(\gamma(t_0))\big(\gamma'(t_0)\big).$$

## Remarks

For $m = 1$, this reads $(f \circ \gamma)'(t_0) = \nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = \sum_{i=1}^n \dfrac{\partial f}{\partial x_i}(\gamma(t_0)) \, \gamma_i'(t_0)$, with the [[gradient|0ABK9V9Y64]] $\nabla f$.
It is the case $n = 1$ of the [[chain rule in several variables|0EAAS2EHDR]], since a linear map $\mathbb{R} \to \mathbb{R}^n$ is $s \mapsto s \gamma'(t_0)$.
For $\gamma(t) = a + t v$ it gives the [[directional derivative|06KWGHDN2J]] $D_v f(a) = Df(a)(v)$.

## Explanations

### L1 human

Walk along a [[curve|0G7WM8DX4P]] $\gamma(t)$ through a landscape with height $f$.
The rate at which your height changes is the slope of the landscape in the direction you walk, times your speed.
For $f(x, y) = x y$ and $\gamma(t) = (t, t^2)$, the rule gives $\nabla f \cdot \gamma' = t^2 \cdot 1 + t \cdot 2t = 3 t^2$, the derivative of $t^3$.
</document>

<document source="https://math2.tamnd.com/node/0A9E6X0S3E.md" tag="0A9E6X0S3E" kind="concept">
# Complete ordered field

## Summary

A complete ordered field is an [[ordered field|0P3VCBDEE2]] $F$ in which each nonempty subset that is bounded above has a [[supremum|0R3H5PANRQ]] in $F$.
The [[real numbers|08J2M1JYAG]] form a complete ordered field, by the [[completeness axiom|0AXWET80YW]].
Any two complete ordered fields are isomorphic, by a [[unique isomorphism|0G8G1TH5WK]] that preserves the order.
Thus "the complete ordered field" is a characterization of $\mathbb{R}$.

## Remarks

The ordered field $\mathbb{Q}$ is not complete.
Each complete ordered field is [[Archimedean|0XH79S06J3]].
Other forms of completeness of $\mathbb{R}$ are equivalent for Archimedean ordered fields, for example the convergence of each Cauchy sequence.

## Explanations

### L1 human

Many number systems allow addition, multiplication, division and comparison, for example $\mathbb{Q}$ and $\mathbb{R}$.
Among them, the real numbers are the only one that has no gaps.
So you can define the real numbers by this list of properties, without a construction.
</document>

<document source="https://math2.tamnd.com/node/0A9EB0KQBC.md" tag="0A9EB0KQBC" kind="concept">
# Chi-squared distribution

## Summary

The chi-squared distribution with $k$ degrees of freedom is the law of a sum of the squares of $k$ independent standard normal random variables. It is central in tests and confidence intervals for variances.
The node [[chi-squared distribution|0RFMMBHFTV]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0A9V6RKQG6.md" tag="0A9V6RKQG6" kind="statement">
# Schröder–Bernstein theorem

## Statement

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
If there are [[injective functions|0PVG1CRAJN]] $f \colon A \to B$ and $g \colon B \to A$, then there is a [[bijective function|0SNQEF7Z53]] $h \colon A \to B$.

## Remarks

In terms of cardinals: $|A| \le |B|$ and $|B| \le |A|$ imply $|A| = |B|$. So the [[order of cardinals|01GTP8M5RG]] is antisymmetric.
The proof does not use the axiom of choice.
Example: $[0, 1]$ and $[0, 1)$ are equinumerous, because each one injects into the other.

## History

Cantor stated the theorem in 1887 without a proof.
Dedekind proved it in 1887, but he did not publish the proof.
Bernstein and Schröder published proofs in 1898, and the proof of Schröder had a flaw.

## Explanations

### L1 human

If $A$ fits into $B$ and $B$ fits into $A$, then $A$ and $B$ have the same size.
For finite sets this is obvious, but for infinite sets it needs a clever construction of $h$.
</document>

<document source="https://math2.tamnd.com/node/0AA97XA70K.md" tag="0AA97XA70K" kind="concept">
# Periodic orbit

## Summary

Let a point that is not an equilibrium return to itself after a fixed positive time. Its orbit is a periodic orbit of the dynamical system.
For an autonomous differential equation, it is the closed curve traced by a non-constant periodic solution. See the [[definition|01DSNP0B4R]].
</document>

<document source="https://math2.tamnd.com/node/0AAHE7GEW6.md" tag="0AAHE7GEW6" kind="definition">
# Kernel of a ring homomorphism

## Definition

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
The kernel of $f$ is the set
$$\ker f = \{a \in R : f(a) = 0\}.$$

## Remarks

The kernel of $f$ is a [[two-sided ideal|0DDA6CY7GJ]] of $R$.
The kernel of the reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ is $n\mathbb{Z}$.
The map $f$ is injective if and only if $\ker f = \{0\}$, by the [[kernel criterion|0MMCB8RXGA]].

## Explanations

### L1 human

The kernel collects all the elements that $f$ sends to $0$.
For reduction modulo $5$, the kernel is the set of the multiples of $5$.
</document>

<document source="https://math2.tamnd.com/node/0AANKKNHS4.md" tag="0AANKKNHS4" kind="concept">
# Least element

## Summary

The least element of a subset of a partially ordered set is an element of the subset that is below every element of the subset.
</document>

<document source="https://math2.tamnd.com/node/0AAQ73WRFB.md" tag="0AAQ73WRFB" kind="concept">
# Limit cycle

## Summary

A limit cycle is a closed trajectory of a planar dynamical system with one property. At least one other trajectory spirals into it as time tends to $\infty$ or to $-\infty$.
Limit cycles model oscillations that sustain themselves. See the [[definition|0EERDDPSVT]].

## Remarks

Henri Poincaré started the study of limit cycles.
</document>

<document source="https://math2.tamnd.com/node/0AAXWGT5TD.md" tag="0AAXWGT5TD" kind="concept">
# Almost sure convergence

## Summary

A sequence of random variables converges almost surely when, for almost every outcome, the sequence of values converges.
The node [[almost sure convergence|04P7D63W8P]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ABK9V9Y64.md" tag="0ABK9V9Y64" kind="concept">
# Gradient

## Summary

Let $f$ be a [[differentiable|0HHHDCXZV4]] real-valued function on an open subset of $\mathbb{R}^n$. Its gradient at a point $a$ is the vector $\nabla f(a)$ of its [[partial derivatives|09EGC6F8PT]] at $a$.
It points in the direction in which $f$ increases fastest, and its length is that fastest rate of increase.
The derivative of $f$ at $a$ is the dot product with the gradient: $Df(a)(v) = \nabla f(a) \cdot v$.
</document>

<document source="https://math2.tamnd.com/node/0ABP2F2ZP7.md" tag="0ABP2F2ZP7" kind="statement">
# Finite-dimensional normed spaces are complete

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$.
Then $V$ is a [[Banach space|0BDTTM24DP]].

## Remarks

Proof: by [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the sup norm of the coordinates in a basis. Equivalent norms have the same Cauchy sequences and the same convergent sequences. With the sup norm, completeness follows from the completeness of $\mathbb{R}$ in each coordinate.
A consequence: each finite-dimensional subspace of a normed space is complete, and thus [[closed|0W41XG4W60]].

## Explanations

### L1 human

In finite dimensions, a Cauchy sequence is Cauchy in each coordinate, so each coordinate converges.
This works for each norm, because all norms are equivalent there.
</document>

<document source="https://math2.tamnd.com/node/0ACRGYNHSR.md" tag="0ACRGYNHSR" kind="concept">
# Dirac measure

## Summary

The Dirac measure at a point $x$ gives a set the value $1$ if the set contains $x$, and the value $0$ if not.
It is the probability measure of a random quantity that always takes the value $x$. The node [[Dirac measure|099SXAG014]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ACX5QT8TE.md" tag="0ACX5QT8TE" kind="concept">
# Invariant subspace

## Summary

A subspace is invariant under a linear map from a space to itself when the map sends every vector of the subspace into the subspace.
</document>

<document source="https://math2.tamnd.com/node/0AD0BZ49DW.md" tag="0AD0BZ49DW" kind="statement">
# Metrizable spaces are normal

## Statement

Let $X$ be a [[metrizable space|09KDHB82SG]].
Then $X$ is [[normal|0AJ646N339]].

## Remarks

Proof: choose a metric $d$ for the topology. For disjoint closed sets $A$ and $B$, let $U = \{x : d(x, A) < d(x, B)\}$ and $V = \{x : d(x, B) < d(x, A)\}$. Here $d(x, A)$ is the [[distance to a set|07TXYYVYZR]].
The sets $U$ and $V$ are open, because $x \mapsto d(x, A)$ is continuous. They are disjoint, and $A \subseteq U$ because $d(x, A) = 0 < d(x, B)$ for $x \in A$.
Single points are closed in a metric space. Thus each metrizable space is normal and T1, and also [[Hausdorff|0P9DCDES73]].

## Explanations

### L1 human

In a metric space, you separate two disjoint closed sets by the points that are nearer to one set than to the other.
</document>

<document source="https://math2.tamnd.com/node/0AD4VS02H3.md" tag="0AD4VS02H3" kind="definition">
# Formal derivative

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$.
The formal derivative of $f$ is the polynomial
$$f' = a_1 + 2 a_2 x + 3 a_3 x^2 + \dots + n a_n x^{n-1},$$
where $k a_k$ is the sum of $k$ copies of $a_k$.

## Remarks

The derivative is additive, and $(r f)' = r f'$ for each constant $r \in R$.
If $R$ is commutative, the product rule holds: $(f g)' = f' g + f g'$.
In characteristic $p$, the derivative of $x^p$ is $p x^{p-1} = 0$. So a nonconstant polynomial can have derivative $0$.
The derivative detects [[repeated roots|067F8YVYMH]].

## Explanations

### L1 human

The formal derivative uses the rule from calculus, but it is pure algebra.
You never take a limit, so it works over the integers modulo $p$ too.
For example, the derivative of $x^3 + 2x$ is $3x^2 + 2$.
</document>

<document source="https://math2.tamnd.com/node/0ADTQBXRMC.md" tag="0ADTQBXRMC" kind="concept">
# Jacobi symbol

## Summary

The Jacobi symbol $\left(\frac{a}{n}\right)$ extends the Legendre symbol to each odd positive integer $n$, as a product over the prime factors of $n$.
The [[definition of the Jacobi symbol|0XDB5AW7M1]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0AEAHMFSCA.md" tag="0AEAHMFSCA" kind="concept">
# Strictly convex function

## Summary

A strictly convex function is a function whose graph lies strictly below each chord, except at the endpoints of the chord. Its graph contains no line segments. The node [[strictly convex function|06BA5JZKTQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0AEKDFCYFA.md" tag="0AEKDFCYFA" kind="definition">
# Prime factorization

## Definition

Let $n$ be a positive integer.
A prime factorization of $n$ is a finite list of [[prime numbers|0SN0MAEKYX]] $p_1 \le p_2 \le \dots \le p_k$ such that $n = p_1 p_2 \cdots p_k$.
For $n = 1$, the prime factorization is the empty list, with the empty product $1$.

## Remarks

We often group equal primes and write $n = q_1^{e_1} q_2^{e_2} \cdots q_r^{e_r}$, with distinct primes $q_1 < \dots < q_r$ and exponents $e_i \ge 1$.
For example, the prime factorization of $360$ is $2, 2, 2, 3, 3, 5$, that is, $360 = 2^3 \cdot 3^2 \cdot 5$.
Each positive integer has [[a prime factorization|020K0X7HVR]], and the [[fundamental theorem of arithmetic|084HH1NDCB]] says that it is unique.

## Explanations

### L1 human

A prime factorization breaks a number into primes, which you cannot break further.
A factor tree does this: $360 = 36 \cdot 10 = (6 \cdot 6) \cdot (2 \cdot 5) = 2 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 5$.
Sort the primes to get the list $2, 2, 2, 3, 3, 5$.
</document>

<document source="https://math2.tamnd.com/node/0AF3DANJ76.md" tag="0AF3DANJ76" kind="definition">
# Pell's equation

## Definition

Let $d$ be a positive integer that is not a perfect square.
Pell's equation for $d$ is the [[Diophantine equation|08B1RTW7VZ]] $x^2 - d y^2 = 1$, with unknowns $x, y \in \mathbb{Z}$.
A solution is a pair $(x, y)$ of integers that satisfies the equation.

## Remarks

The pairs $(\pm 1, 0)$ are the trivial solutions.
For $d = 2$, the pair $(3, 2)$ is a solution, since $9 - 2 \cdot 4 = 1$.
A solution $(x, y)$ with $y \neq 0$ gives the rational approximation $x / y$ of $\sqrt{d}$.
The name comes from an error of Euler, who attributed a solution by William Brouncker to John Pell.

## Explanations

### L1 human

Pell's equation asks for integers $x$ and $y$ such that $x^2$ is exactly one more than $d y^2$.
For $d = 2$, the solutions $(3, 2)$, $(17, 12)$ and $(577, 408)$ give fractions very close to $\sqrt{2}$.
</document>

<document source="https://math2.tamnd.com/node/0AF7X08PCR.md" tag="0AF7X08PCR" kind="concept">
# Convergent series

## Summary

A convergent series is a series whose sequence of partial sums has a finite limit. Many tests decide whether a given series converges. The node [[convergent series|0CVVKY7GY2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0AFNNTRTJE.md" tag="0AFNNTRTJE" kind="concept">
# Collinear points

## Summary

Points are collinear when one line contains all of them. Three points that are not collinear are the vertices of a triangle.
</document>

<document source="https://math2.tamnd.com/node/0AFWQZ1PKF.md" tag="0AFWQZ1PKF" kind="counterexample">
# sin(1/x) has no limit at 0

## Counterexample

Let $f(x) = \sin(1/x)$ for $x \ne 0$.
Then $\lim_{x \to 0} f(x)$ does not exist, and neither one-sided limit exists.
Let $x_n = 1/(2 \pi n + \pi/2)$ and $y_n = 1/(\pi n)$. Both sequences tend to $0$ from the right, but $f(x_n) = 1$ and $f(y_n) = 0$ for all $n \ge 1$.

## Remarks

The argument uses the [[sequential characterization of limits|000MXHEY22]].
So no value of $f$ at $0$ makes $f$ continuous at $0$. The point $0$ is an essential discontinuity.
The graph of $f$ on $(0, 1]$, with its closure, is the topologist's sine curve.

## Explanations

### L1 human

As $x$ approaches $0$, the number $1/x$ grows without bound.
So $\sin(1/x)$ swings between $-1$ and $1$ faster and faster, and it settles at no value.
</document>

<document source="https://math2.tamnd.com/node/0AH9Q6YNMT.md" tag="0AH9Q6YNMT" kind="statement">
# If G modulo its centre is cyclic then G is abelian

## Statement

Let $G$ be a [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.
If the [[quotient group|0KYX2N69J0]] $G/Z(G)$ is [[cyclic|0KVGH0J5B3]], then $G$ is [[abelian|082GHNPJ53]].

## Remarks

Let $gZ(G)$ generate $G/Z(G)$; then each element of $G$ is $g^k z$ with $k \in \mathbb{Z}$ and $z \in Z(G)$, and two such elements commute.
In that case $G = Z(G)$, so in fact $G/Z(G)$ is trivial.
A consequence is that $G/Z(G)$ is never cyclic of order greater than $1$. For example, the centre of a group never has prime index.

## Explanations

### L1 human

If the elements of $G$ are powers of one element, up to central elements, then all elements commute.
Powers of one element commute with each other, and central elements commute with everything.
So the quotient by the centre cannot be a nontrivial cyclic group.
</document>

<document source="https://math2.tamnd.com/node/0AHKWBQSWV.md" tag="0AHKWBQSWV" kind="concept">
# Vertex cover

## Summary

A vertex cover of a graph is a set of vertices that meets every edge. Vertex covers are dual to matchings and complementary to independent sets. The node [[vertex cover|0KDADE9RQ8]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0AJ035ZTZE.md" tag="0AJ035ZTZE" kind="concept">
# Generalised eigenspace

## Summary

The generalised eigenspace of an endomorphism for a scalar $\lambda$ is a set of vectors.
It contains the vectors that some power of the map minus $\lambda$ times the identity sends to zero.
</document>

<document source="https://math2.tamnd.com/node/0AJ0KP1B31.md" tag="0AJ0KP1B31" kind="definition">
# Multiplication by a matrix as a linear map

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *linear map of multiplication by $A$* is
$$L_A \colon F^n \to F^m, \qquad L_A(x) = Ax,$$
where $Ax$ is the [[matrix–vector product|0PDVQEP0ZC]].

## Remarks

The map $L_A$ is linear, because $A(x + y) = Ax + Ay$ and $A(cx) = c\, Ax$.
Its matrix in the standard bases is $A$, because $L_A(e_j)$ is column $j$ of $A$.
Each linear map $F^n \to F^m$ equals $L_A$ for exactly one matrix $A$.
The kernel of $L_A$ is the [[null space|0PHDYNR24D]] of $A$, and its image is the [[column space|03AJKXH1AV]] of $A$.

## Explanations

### L1 human

Multiplying vectors by a fixed matrix is a linear map.
The matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ stretches the $x$-direction by $2$ and the $y$-direction by $3$.
Every linear map between coordinate spaces is of this kind.
</document>

<document source="https://math2.tamnd.com/node/0AJ0YVK743.md" tag="0AJ0YVK743" kind="definition">
# Annulus

## Definition

Let $c \in \mathbb{C}$ and let $r, R$ be such that $0 \le r < R \le \infty$.
The annulus with center $c$, inner radius $r$ and outer radius $R$ is the subset
$$A(c; r, R) = \{ z \in \mathbb{C} : r < |z - c| < R \}$$
of the [[complex plane|0CTE7A31XQ]]. Here $|z - c|$ is the [[modulus|0KW2X0TFBX]] of $z - c$.

## Remarks

Each annulus is a [[domain|097KNJFJ7V]].
For $r = 0$, the annulus is the punctured disc $0 < |z - c| < R$. For $r = 0$ and $R = \infty$, it is $\mathbb{C} \setminus \{c\}$.
For $R < \infty$, its area is $\pi (R^2 - r^2)$.
Holomorphic functions on an annulus have [[Laurent expansions|06WHRARZKY]].

## Explanations

### L1 human

An annulus is the ring between two circles with the same center, without the two circles.
For example, $1 < |z| < 2$ is the set of the points whose distance to $0$ is between $1$ and $2$.
</document>

<document source="https://math2.tamnd.com/node/0AJ646N339.md" tag="0AJ646N339" kind="definition">
# Normal space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is normal when the property that follows holds.
For all disjoint [[closed sets|0G5AV9Q4RR]] $A, B \subseteq X$, there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $A \subseteq U$, $B \subseteq V$ and $U \cap V = \varnothing$.

## Remarks

Names vary. Some authors call a space T4 when it is normal and [[T1|05GQJXBP3F]]. Such a space is [[regular|0JF9KW2TXB]] and [[Hausdorff|0MXB4CKHFW]].
Each [[metrizable space is normal|0AD0BZ49DW]], and each [[compact Hausdorff space is normal|0MQQQ5NA2S]].
A product of two normal spaces need not be normal. The Sorgenfrey plane is an example.
[[Urysohn's lemma|01EY3VR7ZD]] characterises normal spaces by continuous functions.

## Explanations

### L1 human

In a normal space, two closed sets that do not meet can be put into two open sets that do not meet.
This is a strong condition, but it holds for all metric spaces.
</document>

<document source="https://math2.tamnd.com/node/0AJN7WHBJZ.md" tag="0AJN7WHBJZ" kind="definition">
# Independent and identically distributed random variables

## Definition

Let $(X_i)_{i \in I}$ be a family of random variables on one probability space, all with values in the same measurable space $(E, \mathcal{E})$.
The family is independent and identically distributed (i.i.d.) if both of these conditions hold:
1. The family $(X_i)_{i \in I}$ is [[independent|0XCE070V8B]].
2. The random variables $X_i$ are [[identically distributed|01X8YSDTBW]]: all $X_i$ have the same law $\mu$.

## Remarks

One also says that $X_1, X_2, \dots$ is an i.i.d. sequence with common law $\mu$, or a random sample from $\mu$.
The results of repeated tosses of one coin form an i.i.d. sequence of Bernoulli random variables.
The law of large numbers and the central limit theorem are first proved for i.i.d. sequences.

## Explanations

### L1 human

I.i.d. variables are the results of the same experiment, repeated under the same conditions, with no influence between the repetitions.
Each one follows the same law, and none of them gives information about the others.
</document>

<document source="https://math2.tamnd.com/node/0AJXYH4DDQ.md" tag="0AJXYH4DDQ" kind="concept">
# Step function

## Summary

A step function is a function that is constant on each piece of a partition of an interval into finitely many pieces. Its integral is a finite sum of areas of rectangles. The node [[step function|0M9XEDF1HJ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0AK9TGSNXY.md" tag="0AK9TGSNXY" kind="concept">
# Successor function

## Summary

The successor function is the function $S \colon \mathbb{N} \to \mathbb{N}$ that sends each natural number $n$ to the next natural number, $S(n) = n + 1$.
It is a primitive notion of the [[Peano axioms|0QFT7Z76N5]].
</document>

<document source="https://math2.tamnd.com/node/0AKNPYQ04R.md" tag="0AKNPYQ04R" kind="definition">
# Classical definition of probability

## Definition

Let $\Omega$ be a nonempty [[finite set|0BBTH3YSZA]] of equally likely outcomes.
The classical probability of an event $A \subseteq \Omega$ is
$$P(A) = \frac{|A|}{|\Omega|},$$
the number of favourable outcomes divided by the number of all outcomes.

## Remarks

This $P$ is a [[probability measure|0VGMEBVW1M]] on the power set of $\Omega$, the uniform probability on $\Omega$. Each outcome has probability $1/|\Omega|$.
For two rolls of a fair die, the probability that the sum is $7$ is $6/36 = 1/6$.
Classical probability turns a probability problem into a counting problem. Binomial coefficients and the rule of product often give the counts.
The definition is circular as a foundation, since "equally likely" already uses probability. It also does not apply to infinite sample spaces.

## Explanations

### L1 human

When all outcomes have the same chance, a probability is a fraction: good cases over all cases.
A fair die has $6$ faces and $3$ of them are even, so the chance of an even number is $3/6 = 1/2$.
</document>

<document source="https://math2.tamnd.com/node/0AKRKVHW94.md" tag="0AKRKVHW94" kind="statement">
# Stokes' theorem

## Statement

Let $D \subseteq \mathbb{R}^2$ be the compact region bounded by a piecewise smooth simple closed curve $\gamma$, oriented counterclockwise.
Let $r$ be an injective regular [[parametrised surface|05788J51WZ]] of class $C^2$ on an open set that contains $D$. Let $S = r(D)$ have the unit normal $n = (r_u \times r_v) / \| r_u \times r_v \|$, and let $\partial S$ be the closed curve $r \circ \gamma$.
If $F$ is a vector field of class $C^1$ on an open set that contains $S$, then
$$\oint_{\partial S} F \cdot d\mathbf{r} = \iint_S (\nabla \times F) \cdot n \, dS.$$

## Remarks

The left side is the circulation of $F$ around the boundary. The right side is the [[flux|0YPRY7T4PK]] of the [[curl|0GDNHS9J2X]] of $F$ through $S$.
The orientations match by the right-hand rule: with the thumb along $n$, the fingers curl in the direction of $\partial S$.
Example: for the upper unit hemisphere and $F = (-y, x, 0)$, $\nabla \times F = (0, 0, 2)$, and both sides equal $2\pi$.
For a flat surface in the $xy$-plane, the theorem reduces to [[Green's theorem|0HR3N02317]]. The proof pulls the integrals back to $D$ and applies Green's theorem there.

## History

The theorem first appears in a letter from William Thomson to George Stokes in 1850.
Stokes set it as a question on the Smith's Prize examination at Cambridge in 1854, and the result then got his name.

## Explanations

### L1 human

The total swirl of a field over a surface equals the circulation of the field around the edge of the surface.
The surface can be any shape: only its boundary curve matters.
</document>

<document source="https://math2.tamnd.com/node/0AM050KCAW.md" tag="0AM050KCAW" kind="statement">
# Gram matrices are positive semidefinite

## Statement

Let $B$ be an $m \times n$ complex matrix.
Then the $n \times n$ matrix $B^* B$ is [[positive semidefinite|01MMVW8CKC]], where $B^*$ is the [[conjugate transpose|0DXKE10PHF]].
It is positive definite if and only if the columns of $B$ are linearly independent.

## Remarks

The proof: $x^* B^* B x = (Bx)^* (Bx) = \norm{Bx}^2 \ge 0$.
For a real matrix $B$, the same holds for $B^T B$.
Each [[Gram matrix|00SPJSDT3Z]] has this form, so Gram matrices are positive semidefinite.

## Explanations

### L1 human

Matrices of the form $B^T B$ can never give negative values of $x^T A x$.
The value is the squared length of $Bx$.
It is zero only when $Bx = 0$.
</document>

<document source="https://math2.tamnd.com/node/0AN28XQ45Z.md" tag="0AN28XQ45Z" kind="statement">
# Cauchy product of absolutely convergent series

## Statement

Let $\sum a_n$ and $\sum b_n$ be real series that [[converge absolutely|0W3K3QPEWT]], with sums $A$ and $B$.
Then their [[Cauchy product|0B1K9PVWE9]] $\sum c_n$ converges absolutely, and its sum is $A B$.

## Remarks

The double series $\sum_{i,j} a_i b_j$ converges absolutely, so its terms can be grouped in any order. The Cauchy product groups them by $i + j$.
[[Mertens' theorem|0BJSWZJG18]] needs absolute convergence of only one of the two series, but then gives only convergence.
Mathlib states it for complete normed rings, as `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm`.

## Explanations

### L1 human

When both series converge absolutely, we may multiply them term by term as with finite sums.
The order in which we add the products does not change the result.
</document>

<document source="https://math2.tamnd.com/node/0AP750DHVP.md" tag="0AP750DHVP" kind="concept">
# Circle integral

## Summary

A circle integral is the contour integral of a function along a circle $|z - c| = R$, traveled once counterclockwise, see the [[definition|0T7ABX4CWE]].
</document>

<document source="https://math2.tamnd.com/node/0AP80TT660.md" tag="0AP80TT660" kind="definition">
# Multiplicative function

## Definition

An [[arithmetic function|0H2EPRZC0D]] $f$ is multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all [[coprime|0GFKP8F8AG]] positive integers $m$ and $n$.

## Remarks

A multiplicative function is determined by its values at prime powers: if $n = p_1^{e_1} \cdots p_k^{e_k}$, then $f(n) = f(p_1^{e_1}) \cdots f(p_k^{e_k})$.
Examples are [[Euler's totient function|0VV04M41AN]], the [[divisor function|0YEW47VJ0J]] and the [[Möbius function|0C8QENHYZW]].
A function with $f(mn) = f(m) f(n)$ for all $m$ and $n$ is [[completely multiplicative|0GS27TKF1V]].

## Explanations

### L1 human

For a multiplicative function, you can compute $f(n)$ from the prime factorization of $n$.
For example, $\varphi(12) = \varphi(4) \varphi(3) = 2 \cdot 2 = 4$, because $4$ and $3$ are coprime.
</document>

<document source="https://math2.tamnd.com/node/0APCFW7639.md" tag="0APCFW7639" kind="concept">
# Nullity

## Summary

The nullity of a linear map is the dimension of its kernel.
</document>

<document source="https://math2.tamnd.com/node/0APN65D6Y6.md" tag="0APN65D6Y6" kind="statement">
# Second isomorphism theorem for vector spaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.
Then the map
$$U / (U \cap W) \to (U + W) / W, \qquad u + (U \cap W) \mapsto u + W,$$
is well defined, and it is a linear isomorphism.
Here $U + W$ is the [[sum of the subspaces|0JMTY8MF54]], and both sides are [[quotient spaces|0BR3FY5Q8M]].

## Remarks

In finite dimensions, it gives [[Grassmann's formula|06HTJNCMPA]]: $\dim U - \dim(U \cap W) = \dim(U + W) - \dim W$.
The proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the map $U \to (U + W)/W$, $u \mapsto u + W$, which is surjective with kernel $U \cap W$.

## Explanations

### L1 human

Adding $U$ to $W$ adds the same new directions as $U$ has beyond its common part with $W$.
The theorem makes this precise with quotient spaces.
It is the linear algebra form of a general algebraic theorem.
</document>

<document source="https://math2.tamnd.com/node/0AQBSVMX9P.md" tag="0AQBSVMX9P" kind="concept">
# Edge set

## Summary

The edge set of a graph is the set of its edges, each edge an unordered pair of adjacent vertices. It is the basic data of a graph next to the vertex set. The node [[edge set|0WNGQBPPQX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0AQNX7G443.md" tag="0AQNX7G443" kind="statement">
# The symmetric group is generated by transpositions

## Statement

Let $n \geq 2$.
The [[symmetric group|0515A6D2Y1]] $S_n$ is [[generated|0RP7VWQ0YN]] by its [[transpositions|0FWVRHBRWM]].

## Remarks

This is a reformulation of the [[decomposition into transpositions|0XK6FECMW5]].
The adjacent transpositions $(1\ 2), (2\ 3), \dots, (n-1\ n)$ already generate $S_n$.
The two elements $(1\ 2)$ and $(1\ 2\ \dots\ n)$ also generate $S_n$.

## Explanations

### L1 human

Each permutation is a product of swaps, so the swaps generate the whole symmetric group.
You do not even need all swaps.
Swaps of neighbours are enough, as in a bubble sort.
</document>

<document source="https://math2.tamnd.com/node/0AQZV0905N.md" tag="0AQZV0905N" kind="concept">
# Abelianisation

## Summary

The abelianisation of a group $G$ is the quotient $G/[G, G]$ by its commutator subgroup.
It is the largest abelian quotient of $G$.
</document>

<document source="https://math2.tamnd.com/node/0AR06W1E1B.md" tag="0AR06W1E1B" kind="statement">
# Differentiation under the integral sign

## Statement

Let $f \colon [a, b] \times [t_0, t_1] \to \mathbb{R}$ be continuous, and suppose that its [[partial derivative|05AXG3QRKR]] $\partial f / \partial t$ exists and is [[continuous|0S9EK5MQPE]] on $[a, b] \times [t_0, t_1]$.
Then $F(t) = \int_a^b f(x, t) \, dx$ is continuously differentiable on $[t_0, t_1]$, and
$$F'(t) = \int_a^b \frac{\partial f}{\partial t}(x, t) \, dx.$$
If, in addition, $\alpha, \beta \colon [t_0, t_1] \to [a, b]$ are continuously differentiable, then
$$\frac{d}{dt} \int_{\alpha(t)}^{\beta(t)} f(x, t) \, dx = \int_{\alpha(t)}^{\beta(t)} \frac{\partial f}{\partial t}(x, t) \, dx + f(\beta(t), t) \, \beta'(t) - f(\alpha(t), t) \, \alpha'(t).$$

## Remarks

The second formula follows from the first, the fundamental theorem of calculus and the [[chain rule|0EAAS2EHDR]], applied to $G(u, v, t) = \int_u^v f(x, t) \, dx$.
A measure theory version: let $f(\cdot, t)$ be integrable for each $t$ in an open set, let $\partial f / \partial t$ exist for almost all $x$, and let $|\partial f / \partial t (x, t)| \le g(x)$ for an integrable $g$. Then the same rule holds. Its proof uses the dominated convergence theorem.
Example: let $F(t) = \int_0^1 \frac{x^t - 1}{\ln x} \, dx$ for $t \ge 0$. For $t > 0$ the measure version, with $g = 1$, gives $F'(t) = \int_0^1 x^t \, dx = \frac{1}{t + 1}$. Since $F$ is continuous with $F(0) = 0$, $F(t) = \ln(1 + t)$.

## Explanations

### L1 human

If the integrand depends smoothly on a parameter, you can differentiate the integral by differentiating inside it.
When the limits of integration also move, the values of the integrand at the moving limits add extra terms.
</document>

<document source="https://math2.tamnd.com/node/0AR4B3K5QD.md" tag="0AR4B3K5QD" kind="counterexample">
# Pairwise independent events that are not mutually independent

## Counterexample

Toss a fair coin two times, so $\Omega = \{HH, HT, TH, TT\}$ with probability $1/4$ for each outcome.
Let $A$ be "the first toss is heads", $B$ be "the second toss is heads", and $C$ be "the two tosses are equal".
Then $A = \{HH, HT\}$, $B = \{HH, TH\}$ and $C = \{HH, TT\}$, so $P(A) = P(B) = P(C) = 1/2$.
Each intersection of two of them is $\{HH\}$, with probability $1/4 = \frac12 \cdot \frac12$. So $A$, $B$ and $C$ are [[pairwise independent|0N57D1XY6T]].
But $P(A \cap B \cap C) = P(\{HH\}) = 1/4$, and $P(A) \, P(B) \, P(C) = 1/8$. So $A$, $B$ and $C$ are not [[mutually independent|0GCVEV6BJA]].

## Remarks

Here any two of the events determine the third: if $A$ and $B$ both occur, then $C$ occurs.
The same idea with $n$ fair bits and their sums modulo $2$ gives large pairwise independent families. Such families are useful in computer science.

## Explanations

### L1 human

If you know the first toss, you know nothing about $C$, and the same for the second toss.
But if you know both tosses, you know $C$ exactly. So the three events are not independent as a group.
</document>

<document source="https://math2.tamnd.com/node/0AR68JTMDT.md" tag="0AR68JTMDT" kind="concept">
# Unitary matrix

## Summary

A unitary matrix is a complex square matrix whose conjugate transpose is its inverse.
</document>

<document source="https://math2.tamnd.com/node/0AREFQ9Z65.md" tag="0AREFQ9Z65" kind="concept">
# Continuous uniform distribution

## Summary

The continuous uniform distribution on an interval $[a, b]$ spreads the probability evenly over the interval: the probability of a subinterval is proportional to its length.
The node [[continuous uniform distribution|0WGDPYR30Q]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ARPRJQWZ2.md" tag="0ARPRJQWZ2" kind="statement">
# Simplicity of the alternating group

## Statement

Let $n \geq 5$.
Then the [[alternating group|0Z5905Q73D]] $A_n$ is simple: its only [[normal subgroups|0CB0DX5BVV]] are $\{e\}$ and $A_n$.

## Remarks

A usual proof shows that a nontrivial normal subgroup of $A_n$ contains a $3$-cycle, then all $3$-cycles, and then uses that the [[3-cycles generate|0KFZ0QX83G]] $A_n$.
The group $A_4$ is not simple, because the Klein four-group $\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}$ is normal in $A_4$.
The group $A_3$ is simple because it has prime order $3$.
For $n \geq 5$, the only normal subgroups of $S_n$ are $\{e\}$, $A_n$ and $S_n$.

## Explanations

### L1 human

A simple group cannot be broken into a normal subgroup and a quotient.
From five symbols on, the even permutations form such an unbreakable group.
This fact is a key step in the proof that there is no general formula in radicals for polynomials of degree $5$ or more.
</document>

<document source="https://math2.tamnd.com/node/0AS1FZT6BM.md" tag="0AS1FZT6BM" kind="definition">
# Power sum symmetric polynomials

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
For $k \ge 1$, the $k$-th power sum in $x_1, \dots, x_n$ is
$$p_k = x_1^k + x_2^k + \dots + x_n^k \in R[x_1, \dots, x_n].$$

## Remarks

Each $p_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.
[[Newton's identities|0WG2A3HE0S]] relate the power sums to the [[elementary symmetric polynomials|0ZMY8441GW]].
If $R$ contains $\mathbb{Q}$, then every symmetric polynomial in $R[x_1, \dots, x_n]$ is a polynomial in $p_1, \dots, p_n$.
This fails over $\mathbb{Z}$: $e_2 = \tfrac{1}{2}(p_1^2 - p_2)$ needs the coefficient $\tfrac{1}{2}$.

## Explanations

### L1 human

The power sums add up the same power of every variable.
They are symmetric, and over the rationals they can replace the elementary symmetric polynomials as generators.
</document>

<document source="https://math2.tamnd.com/node/0AS800VAYX.md" tag="0AS800VAYX" kind="concept">
# Consistent estimator

## Summary

An estimator is consistent when it converges to the true value of the parameter as the sample size grows.
The node [[consistent estimator|0XJVF81QF6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ASC73QEQT.md" tag="0ASC73QEQT" kind="statement">
# The Möbius function is the inverse of the constant function 1

## Statement

Let $\mathbf{1}$ be the arithmetic function with $\mathbf{1}(n) = 1$ for all $n$, and let $\varepsilon$ be the function with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$.
Then $\mu * \mathbf{1} = \varepsilon$, where $\mu$ is the [[Möbius function|0C8QENHYZW]] and $*$ is the [[Dirichlet convolution|0KH1N66QWW]].
That is, for each positive integer $n$,
$$\sum_{d \mid n} \mu(d) = \begin{cases} 1 & \text{if } n = 1, \\ 0 & \text{if } n > 1. \end{cases}$$

## Remarks

The proof for $n > 1$: let $k = \omega(n) \ge 1$. Only squarefree divisors count, so the sum is $\sum_{j=0}^{k} \binom{k}{j} (-1)^j = (1 - 1)^k = 0$.
For example, the divisors of $12$ give $\mu(1) + \mu(2) + \mu(3) + \mu(6) = 1 - 1 - 1 + 1 = 0$.
This identity is the key step in the [[Möbius inversion formula|07Q4TBTFVB]].

## Explanations

### L1 human

The signs $\mu(d)$ over the divisors of $n$ cancel out, except for $n = 1$.
This is the same cancellation as in the principle of inclusion and exclusion.
</document>

<document source="https://math2.tamnd.com/node/0ASDFEQS6P.md" tag="0ASDFEQS6P" kind="concept">
# Block matrix

## Summary

A block matrix is a matrix that is divided into smaller matrices, its blocks, by horizontal and vertical lines.
</document>

<document source="https://math2.tamnd.com/node/0ASGSTXSBW.md" tag="0ASGSTXSBW" kind="concept">
# Commutator subgroup

## Summary

The commutator subgroup of a group is the subgroup generated by all commutators.
It is the smallest normal subgroup with an abelian quotient.
</document>

<document source="https://math2.tamnd.com/node/0ASHNYF7AG.md" tag="0ASHNYF7AG" kind="concept">
# Annihilator of a subspace

## Summary

The annihilator of a subspace is the set of linear forms that vanish on every vector of the subspace.
</document>

<document source="https://math2.tamnd.com/node/0ASJVEFA9J.md" tag="0ASJVEFA9J" kind="concept">
# Mathematical induction

## Summary

Mathematical induction is a method to prove that a property $P(n)$ holds for every [[natural number|0MWKJ44V5A]] $n$.
A proof by induction has two steps.
The base case proves $P(0)$.
The induction step proves, for an arbitrary natural number $n$, that $P(n)$ implies $P(n + 1)$.
Then $P(n)$ holds for every natural number $n$.

## Remarks

The method is valid because of the induction axiom of the [[Peano axioms|0QFT7Z76N5]].
In the induction step, the assumption $P(n)$ is the induction hypothesis.
Common variants are [[induction from an arbitrary base case|0GF6AJEBZ0]] and [[strong induction|0KMVFSHFGZ]].

## Explanations

### L1 human

Think of a line of dominoes that never ends.
If the first domino falls, and each falling domino knocks over the next one, then every domino falls.
The base case is the first domino, and the induction step is the rule that each domino knocks over the next one.
</document>

<document source="https://math2.tamnd.com/node/0ASM4EPMB2.md" tag="0ASM4EPMB2" kind="definition">
# Ceiling function

## Definition

Let $x$ be a real number.
The ceiling of $x$, written $\lceil x \rceil$, is the least [[integer|0C4JS9RF5M]] $n$ with $x \le n$.
The ceiling function is the function $\mathbb{R} \to \mathbb{Z}$, $x \mapsto \lceil x \rceil$.

## Remarks

The ceiling and the [[floor|0YVRNEDSVG]] are related by $\lceil x \rceil = -\lfloor -x \rfloor$.
For each real $x$, $x \le \lceil x \rceil < x + 1$.
If $x$ is not an integer, then $\lceil x \rceil = \lfloor x \rfloor + 1$. If $x$ is an integer, then $\lceil x \rceil = \lfloor x \rfloor = x$.

## Explanations

### L1 human

The ceiling of a number rounds it up to a whole number.
If a trip needs $2.1$ buses, then you must order $\lceil 2.1 \rceil = 3$ buses.
</document>

<document source="https://math2.tamnd.com/node/0AT1ZSNM25.md" tag="0AT1ZSNM25" kind="definition">
# Normaliser

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The normaliser of $S$ in $G$ is the set $N_G(S) = \{g \in G : gSg^{-1} = S\}$, where $gSg^{-1} = \{gsg^{-1} : s \in S\}$.

## Remarks

The normaliser $N_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$, and it contains the [[centraliser|02YYV36BFM]] $C_G(S)$.
If $H$ is a subgroup of $G$, then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $N_G(H)$. Also, $N_G(H)$ is the largest subgroup of $G$ with this property.
Thus $H$ is normal in $G$ exactly when $N_G(H) = G$.
The number of [[conjugates|0H11ZPX4ND]] of $H$ in $G$ is the index $[G : N_G(H)]$.

## Explanations

### L1 human

The normaliser of a subgroup is the set of elements that do not move the subgroup when they conjugate it.
They may move the elements of the subgroup around, but the subgroup as a set stays in place.
The normaliser is big when the subgroup is close to normal.
</document>

<document source="https://math2.tamnd.com/node/0ATF6Q0633.md" tag="0ATF6Q0633" kind="concept">
# Periodic function

## Summary

A periodic function takes the same values again after each shift of the argument by a fixed nonzero number $T$. The sine and cosine functions are periodic with period $2 \pi$. The node [[periodic function|089KWVYD69]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ATFVE08MW.md" tag="0ATFVE08MW" kind="concept">
# Back substitution

## Summary

Back substitution is a method to solve a system $Ux = b$, where $U$ is an $n \times n$ [[upper triangular matrix|0N4XY7Z3NN]] with nonzero diagonal entries.
The last equation gives $x_n = b_n / u_{nn}$.
Then, for $i = n-1, \dots, 1$ in this order, equation $i$ gives
$$x_i = \frac{1}{u_{ii}} \left( b_i - \sum_{j = i+1}^{n} u_{ij} x_j \right),$$
which uses only the unknowns that are already known.

## Remarks

Back substitution uses on the order of $n^2$ arithmetic operations.
Forward substitution solves $Lx = b$ for a [[lower triangular matrix|00J3KZP8Y0]] $L$ in the same way, from the first equation down.
After [[Gaussian elimination|0JVJ2WCXZH]], back substitution completes the solution of the system.
With an [[LU decomposition|0RKG5YW66Y]] $A = LU$, the system $Ax = b$ is solved by forward substitution for $Ly = b$ and then back substitution for $Ux = y$.

## Explanations

### L1 human

In a triangular system, the last equation has only one unknown, so we solve it at once.
We put that value into the equation above it, which now has only one new unknown.
We continue upwards until we know all the unknowns.
</document>

<document source="https://math2.tamnd.com/node/0ATVXC3NCC.md" tag="0ATVXC3NCC" kind="statement">
# Boundedness theorem

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then $f$ is bounded. That is, there is $K$ with $|f(x)| \le K$ for all $x \in [a, b]$.

## Remarks

Suppose not. Then there are points $x_n \in [a, b]$ with $|f(x_n)| > n$. By the [[Bolzano–Weierstrass theorem|0KDJD249DT]], a subsequence converges to some $c \in [a, b]$. Continuity at $c$ gives a contradiction.
The interval must be closed and bounded. The function $1/x$ on $(0, 1]$ is continuous but not bounded.
Mathlib states a form for compact sets, as `IsCompact.bddAbove_image`.

## Explanations

### L1 human

On a closed bounded interval, a continuous function cannot shoot off to infinity.
Its graph stays between two horizontal lines.
</document>

<document source="https://math2.tamnd.com/node/0AVEYZ2T0Q.md" tag="0AVEYZ2T0Q" kind="example">
# The nth root of n tends to 1

## Example

For each integer $n \ge 1$, let $a_n = n^{1/n}$ be the positive $n$-th root of $n$.
Then $a_n \to 1$ as $n \to \infty$.
For the proof, write $a_n = 1 + h_n$ with $h_n \ge 0$.
For $n \ge 2$, the binomial theorem gives $n = (1 + h_n)^n \ge \binom{n}{2} h_n^2 = \frac{n(n-1)}{2} h_n^2$.
Thus $0 \le h_n \le \sqrt{2/(n-1)}$, and the [[squeeze theorem|0XMF00T4XQ]] gives $h_n \to 0$.

## Remarks

The first terms are $1, \sqrt{2} \approx 1.414, 3^{1/3} \approx 1.442, 4^{1/4} \approx 1.414$. The largest term is at $n = 3$.
In the same way, $c^{1/n} \to 1$ for each real $c > 0$.
Mathlib proves the version for a real variable, $x^{1/x} \to 1$ as $x \to \infty$, in the lemma `tendsto_rpow_div`.

## Explanations

### L1 human

The number $n$ grows, but the root index $n$ grows too, and the root wins.
For example, $1000^{1/1000}$ is about $1.0069$, very close to $1$.
</document>

<document source="https://math2.tamnd.com/node/0AVXBEC05W.md" tag="0AVXBEC05W" kind="statement">
# R/P is an integral domain iff P is prime

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $P$ be an ideal of $R$.
Then the [[quotient ring|0VFW4Y69BT]] $R/P$ is an [[integral domain|09S3PFBEPM]] if and only if $P$ is a [[prime ideal|06HC6HK0Z8]].

## Remarks

The condition $1 + P \neq 0 + P$ in $R/P$ is the condition $P \neq R$.
The condition that $(a + P)(b + P) = 0$ implies $a + P = 0$ or $b + P = 0$ is the condition that $a b \in P$ implies $a \in P$ or $b \in P$.
For example, $\mathbb{Z}/n\mathbb{Z}$ is an integral domain exactly when $n$ is prime or $n = 0$.

## Explanations

### L1 human

The theorem translates a property of the ideal into a property of the quotient ring.
An element is zero in $R/P$ exactly when it lies in $P$.
So "a product in $P$ has a factor in $P$" becomes "a zero product has a zero factor".
</document>

<document source="https://math2.tamnd.com/node/0AVY8HKBGD.md" tag="0AVY8HKBGD" kind="statement">
# Totient of a prime power

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $k$ be a positive integer.
Then $\varphi(p^k) = p^{k-1} (p - 1) = p^k - p^{k-1}$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: an integer is coprime to $p^k$ if and only if $p$ does not divide it. Among $1, \dots, p^k$, the multiples of $p$ are $p, 2p, \dots, p^{k-1} \cdot p$, so there are $p^{k-1}$ of them.
For example, $\varphi(8) = 4$ and $\varphi(27) = 18$.
With the [[multiplicativity of the totient function|0FSJA9M43W]], this formula gives $\varphi(n)$ from the prime factorization of $n$.

## Explanations

### L1 human

Among the numbers $1$ to $p^k$, exactly one in $p$ is a multiple of $p$.
Remove these numbers. The fraction $1 - 1/p$ of the numbers stays, so $\varphi(p^k) = p^k (1 - 1/p)$.
</document>

<document source="https://math2.tamnd.com/node/0AW1E6YTJY.md" tag="0AW1E6YTJY" kind="concept">
# Lipschitz condition

## Summary

A function satisfies a Lipschitz condition when the distance between two values is at most a constant times the distance between the two points.
This is a strong form of continuity. In the theory of differential equations, it gives uniqueness of solutions. See the [[definition|055VVSN00V]].

## Remarks

It is named after Rudolf Lipschitz.
</document>

<document source="https://math2.tamnd.com/node/0AWMPX35CW.md" tag="0AWMPX35CW" kind="statement">
# K[x]/(f) is a field iff f is irreducible

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$.
Then the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$ is a field if and only if $f$ is [[irreducible|00X2A9G9W3]] over $K$.

## Remarks

The proof: $K[x]$ is a [[principal ideal domain|0RZYDVZ475]]. So $(f)$ is maximal if and only if $f$ is irreducible, by the [[theorem on PIDs|095ZHQ7WCJ]]. Then [[$R/M$ is a field iff $M$ is maximal|0WD3SE7S1X]].
For example, $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ and $\mathbb{Q}[x]/(x^2 - 2) \cong \mathbb{Q}(\sqrt{2})$.
The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ has $4$ elements.
If $f$ is irreducible of degree $n$, then $K[x]/(f)$ is an extension of $K$ of degree $n$.
The class of $x$ in $K[x]/(f)$ is a root of $f$.

## Explanations

### L1 human

You can build new fields from polynomials.
Take an irreducible polynomial $f$ and declare $f(x) = 0$.
The result is a field in which $f$ has a root, as the complex numbers come from $x^2 + 1 = 0$.
</document>

<document source="https://math2.tamnd.com/node/0AXP2Z5N9H.md" tag="0AXP2Z5N9H" kind="definition">
# Marginal distribution

## Definition

Let $\mu$ be a probability distribution on $\mathbb{R}^2$, for example the [[joint distribution|0QAR0XDZH4]] of real random variables $X$ and $Y$.
The first marginal distribution of $\mu$ is the distribution $\mu_1$ on $\mathbb{R}$ with $\mu_1(A) = \mu(A \times \mathbb{R})$ for each Borel set $A \subseteq \mathbb{R}$. The second marginal is $\mu_2(B) = \mu(\mathbb{R} \times B)$.
If $\mu$ is the law of $(X, Y)$, then $\mu_1$ is the law of $X$ and $\mu_2$ is the law of $Y$.

## Remarks

The same definition works on a product $E_1 \times E_2$ of measurable spaces, and for $n$ variables with the projections onto some of the coordinates.
In the discrete case, $P(X = x) = \sum_{y} P(X = x, Y = y)$. In the case of a joint density $f(x, y)$, $X$ has the density $f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy$.
The name comes from tables: the sums of the rows and the columns of a joint table are written in its margins.

## Explanations

### L1 human

To get the marginal distribution of $X$ from a joint table of $X$ and $Y$, add up each row.
You lose the information about how $X$ and $Y$ are linked, and you keep only the chances of the values of $X$.
</document>

<document source="https://math2.tamnd.com/node/0AXWET80YW.md" tag="0AXWET80YW" kind="statement">
# Completeness axiom

## Statement

Let $S$ be a nonempty subset of the [[real numbers|08J2M1JYAG]] that is [[bounded above|0B5H2V4JW8]].
Then $S$ has a [[supremum|0BTMQ04556]] in $\mathbb{R}$.

## Remarks

In the axiomatic approach, this is the axiom that distinguishes $\mathbb{R}$ from the other ordered fields. In a construction of $\mathbb{R}$, it is a theorem.
The ordered field $\mathbb{Q}$ does not have this property: $\{q \in \mathbb{Q} : q^2 < 2\}$ has no supremum in $\mathbb{Q}$.
The axiom gives the [[existence of the infimum|0EDH403M21]] and the [[Archimedean property of the reals|0A3GWZ86AY]].

## Explanations

### L1 human

The real line has no holes.
If a set of real numbers has a ceiling, then there is a lowest ceiling, and this lowest ceiling is again a real number.
</document>

<document source="https://math2.tamnd.com/node/0AXZNE5XNH.md" tag="0AXZNE5XNH" kind="concept">
# Symmetric bilinear form

## Summary

A symmetric bilinear form is a bilinear form that does not change when its two arguments are swapped.
</document>

<document source="https://math2.tamnd.com/node/0AYEPBQMAW.md" tag="0AYEPBQMAW" kind="definition">
# Eigenspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
The *eigenspace* of $T$ for $\lambda$ is
$$E_\lambda = \ker(T - \lambda \id_V) = \{v \in V : T(v) = \lambda v\}.$$

## Remarks

The eigenspace is a subspace of $V$; it is the [[kernel|0S23E6YE83]] of a linear map.
Its nonzero elements are the [[eigenvectors|0N6960W0BV]] of $T$ for $\lambda$.
$E_\lambda \ne 0$ if and only if $\lambda$ is an eigenvalue of $T$.
The dimension of $E_\lambda$ is the [[geometric multiplicity|099TWVJ909]] of $\lambda$.

## Explanations

### L1 human

The eigenspace collects all the vectors that the map stretches by the same factor $\lambda$, with the zero vector.
These vectors form a subspace, for example a line or a plane.
On this subspace, the map acts as multiplication by $\lambda$.
</document>

<document source="https://math2.tamnd.com/node/0AYMP41THN.md" tag="0AYMP41THN" kind="concept">
# Stabiliser

## Summary

The stabiliser of a point $x$ under a group action is the set of the group elements that do not move $x$.
It is a subgroup of the group.
</document>

<document source="https://math2.tamnd.com/node/0AYV6VGKF5.md" tag="0AYV6VGKF5" kind="concept">
# Product metric

## Summary

A product metric is a metric on the Cartesian product of finitely many metric spaces, made from the metrics of the factors.
Common choices take the maximum, the sum, or the Euclidean combination of the distances in the factors.
All these choices give the product topology.
</document>

<document source="https://math2.tamnd.com/node/0AZ3R52593.md" tag="0AZ3R52593" kind="definition">
# Convergent sequence

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is convergent if it converges to some real number $L$.
Here convergence is in the sense of the [[limit of a sequence|0XKWSSY8P2]].

## Remarks

The sequences $1/(n+1)$ and $(n+1)/(n+2)$ are convergent, with limits $0$ and $1$.
The constant sequence $a_n = c$ converges to $c$.
A sequence that is not convergent is a [[divergent sequence|01HR2AH0BX]].

## Explanations

### L1 human

A convergent sequence is a sequence that has a limit.
Its terms settle down near one real number.
</document>

<document source="https://math2.tamnd.com/node/0AZ5ARP287.md" tag="0AZ5ARP287" kind="concept">
# Dual map

## Summary

The dual map of a linear map $T \colon V \to W$ is the linear map $W^* \to V^*$ that sends each linear form $f$ to $f \circ T$.
</document>

<document source="https://math2.tamnd.com/node/0AZG100DKH.md" tag="0AZG100DKH" kind="example">
# Action on left cosets

## Example

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Let $G/H$ be the set of the [[left cosets|03DEH43F4H]] of $H$ in $G$.
The map $G \times G/H \to G/H$, $(g, aH) \mapsto gaH$, is an [[action|0WCY42EV9V]] of $G$ on $G/H$.
The map is well defined, because $aH = bH$ implies $gaH = gbH$.
The action is [[transitive|02GK6HZYBT]], because $(ba^{-1}) \cdot aH = bH$.
The [[stabiliser|0AYMP41THN]] of the coset $aH$ is the conjugate subgroup $aHa^{-1}$.
The kernel of the associated homomorphism $G \to \operatorname{Sym}(G/H)$ is the [[normal core|0GETX79CJN]] of $H$, the intersection of all conjugates of $H$.
For $H = \{e\}$, this is the [[action by left multiplication|0K5TWJAGPD]].
</document>

<document source="https://math2.tamnd.com/node/0B0KR7FH2Q.md" tag="0B0KR7FH2Q" kind="concept">
# Equinumerous sets

## Summary

Two sets are equinumerous when there is a bijection between them.
Equinumerous sets have the same size, also when they are infinite.
</document>

<document source="https://math2.tamnd.com/node/0B0WE1GKTK.md" tag="0B0WE1GKTK" kind="definition">
# Faithful action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
The action is faithful when, for each $g \in G$, $g \cdot x = x$ for all $x \in X$ implies $g = e$.

## Remarks

The action is faithful if and only if the associated homomorphism $G \to \operatorname{Sym}(X)$ is injective.
So a faithful action identifies $G$ with a subgroup of $\operatorname{Sym}(X)$.
The action of $G$ on itself by conjugation is faithful if and only if the [[centre|0DZ93493VN]] of $G$ is trivial.

## Explanations

### L1 human

In a faithful action, each element of the group that is not the identity moves at least one point.
So you can tell the group elements apart by watching how they move the points.
The group is then a group of permutations of the set.
</document>

<document source="https://math2.tamnd.com/node/0B1261B5ED.md" tag="0B1261B5ED" kind="concept">
# Ring

## Summary

A ring is a set with an addition and a multiplication that obey the usual rules of arithmetic.
The multiplication need not be commutative.
The integers, the polynomials and the square matrices are the basic examples.
The [[definition of a ring|0894J980RK]] gives the axioms.
</document>

<document source="https://math2.tamnd.com/node/0B1FSS1VZW.md" tag="0B1FSS1VZW" kind="definition">
# Complement

## Definition

Let $U$ be a [[universal set|0BPYRMHJ48]] and let $A$ be a [[subset|0FE8Q5S42Y]] of $U$.
The complement of $A$ in $U$ is the set
$$A^{c} = \{x \in U \mid x \notin A\}.$$

## Remarks

Other notations are $\overline{A}$, $A'$ and $U \setminus A$.
Example: in $U = \mathbb{Z}$, the complement of the set of even integers is the set of odd integers.
The complement of $A^c$ is $A$, and $A \cup A^c = U$, $A \cap A^c = \varnothing$.
The complement of $A$ in $U$ is the [[set difference|0DT0TCV75N]] $U \setminus A$.

## Explanations

### L1 human

The complement is "everything else".
It depends on the universe: the complement of the even numbers is the odd numbers in $\mathbb{Z}$, but not in $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/0B1G4E21G7.md" tag="0B1G4E21G7" kind="statement">
# Moment-generating function of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X \sim \mathcal{N}(\mu, \sigma^2)$. Then for each $t \in \mathbb{R}$,
$$M_X(t) = E\left[e^{tX}\right] = \exp\left(\mu t + \tfrac{1}{2} \sigma^2 t^2\right).$$

## Remarks

Proof for $Z \sim \mathcal{N}(0, 1)$: complete the square, $tz - z^2/2 = t^2/2 - (z - t)^2/2$. So $E[e^{tZ}] = e^{t^2/2} \int \varphi(z - t) \, dz = e^{t^2/2}$.
The general case follows from $X = \mu + \sigma Z$ with $\sigma = \sqrt{\sigma^2}$: $M_X(t) = e^{\mu t} M_Z(\sigma t)$.
The moment-generating function is finite for all $t$. Its derivatives at $0$ give $E[X] = \mu$ and $E[X^2] = \mu^2 + \sigma^2$.
The logarithm $\mu t + \sigma^2 t^2 / 2$ is a polynomial of degree $2$. So all cumulants of order $3$ or more are $0$.

## Explanations

### L1 human

The moment-generating function of a normal variable has a simple closed form.
With the [[multiplication rule for sums|0WRZ170YEH]], it shows quickly that a sum of independent normal variables is again normal.
</document>

<document source="https://math2.tamnd.com/node/0B1K9PVWE9.md" tag="0B1K9PVWE9" kind="definition">
# Cauchy product

## Definition

Let $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} b_n$ be real [[series|0JH4MN4K1C]].
For each $n$, let $c_n = \sum_{k=0}^{n} a_k b_{n-k}$.
The series $\sum_{n=0}^{\infty} c_n$ is the Cauchy product of the two series.

## Remarks

The term $c_n$ collects all products $a_i b_j$ with $i + j = n$.
For power series, the Cauchy product of $\sum a_n x^n$ and $\sum b_n x^n$ is $\sum c_n x^n$. This is the rule for multiplying polynomials.
The Cauchy product of two convergent series need not converge. [[Mertens' theorem|0BJSWZJG18]] gives a sufficient condition.

## Explanations

### L1 human

To multiply two series, multiply every term of one by every term of the other.
The Cauchy product groups these products by the sum of their indices.
</document>

<document source="https://math2.tamnd.com/node/0B22P010D6.md" tag="0B22P010D6" kind="statement">
# Curl-free vector field on a convex set is conservative

## Statement

Let $U \subseteq \mathbb{R}^3$ be open and convex, and let $F \colon U \to \mathbb{R}^3$ be a vector field of class $C^1$ with $\nabla \times F = 0$ on $U$.
Then $F$ is [[conservative|05R9NF2J49]] on $U$.
Explicitly, for a fixed $p \in U$, the function
$$\phi(x) = \int_0^1 F\big(p + t (x - p)\big) \cdot (x - p) \, dt$$
satisfies $\nabla \phi = F$ on $U$.

## Remarks

The integral $\phi(x)$ is the [[line integral|0YS8RX87F4]] of $F$ along the segment from $p$ to $x$. Convexity makes sure that this segment lies in $U$.
To prove $\nabla \phi = F$, [[differentiate under the integral sign|0AR06W1E1B]] and use $\partial_j F_i = \partial_i F_j$, which is the condition $\nabla \times F = 0$.
The same proof works in $\mathbb{R}^n$ under the condition $\partial_j F_i = \partial_i F_j$ for all $i, j$, and on star-shaped open sets.
The result holds more generally on simply connected open sets. It [[fails on some other open sets|0GRM3BS0DD]]. With the [[converse|0M2JRW1J0N]], it gives: on a convex open set, a $C^1$ field is conservative if and only if its curl is zero.

## Explanations

### L1 human

On a convex set, a field with no swirl is the gradient of a potential.
You build the potential by computing the work of the field along straight segments from one base point.
</document>

<document source="https://math2.tamnd.com/node/0B2305GNAZ.md" tag="0B2305GNAZ" kind="statement">
# Open balls are open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r > 0$.
Then the [[open ball|0DC3MPR7C9]] $B(x, r)$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: let $y \in B(x, r)$ and let $\varepsilon = r - d(x, y) > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \le d(x, y) + d(y, z) < r$ by the [[triangle inequality|01M6CW9VM6]]. Thus $B(y, \varepsilon) \subseteq B(x, r)$.
The open sets of $X$ are exactly the unions of open balls.

## Explanations

### L1 human

A point inside an open ball is at some distance from the edge of the ball.
A smaller ball about that point, with radius equal to this distance, stays inside the big ball.
</document>

<document source="https://math2.tamnd.com/node/0B268GVX95.md" tag="0B268GVX95" kind="definition">
# Gaussian integer

## Definition

A Gaussian integer is a complex number $a + bi$ such that $a$ and $b$ are integers. Here $i$ is a complex number with $i^2 = -1$.
The set of Gaussian integers is written $\mathbb{Z}[i] = \{ a + bi : a, b \in \mathbb{Z} \}$.

## Remarks

The set $\mathbb{Z}[i]$ is closed under addition, subtraction and multiplication, so it is a subring of $\mathbb{C}$. It is an integral domain.
The units of $\mathbb{Z}[i]$ are $1$, $-1$, $i$ and $-i$.
In the complex plane, the Gaussian integers are the points of the square lattice.
The [[norm|00HPT88ZX0]] $N(a + bi) = a^2 + b^2$ links them to [[sums of two squares|04NPWJMA6H]].

## Explanations

### L1 human

Gaussian integers are the complex numbers whose real part and imaginary part are both integers, such as $3 - 2i$.
They behave much like the ordinary integers: you can add them, multiply them and factor them into primes.
</document>

<document source="https://math2.tamnd.com/node/0B2RZ0778N.md" tag="0B2RZ0778N" kind="concept">
# Level set

## Summary

A level set of a real-valued [[function of several variables|0JPCHTXWG0]] is a set of points of the domain. At these points, the function takes one given value.
For a function of two variables a level set is often a curve, and for a function of three variables it is often a surface.
</document>

<document source="https://math2.tamnd.com/node/0B346DDAM5.md" tag="0B346DDAM5" kind="definition">
# Möbius transformation

## Definition

Let $a, b, c, d$ be complex numbers with $ad - bc \neq 0$.
The Möbius transformation with these coefficients is the map $f : \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ of the [[Riemann sphere|0CDMXGPSES]] to itself given by
$$f(z) = \frac{az + b}{cz + d}$$
for each $z \in \mathbb{C}$ with $cz + d \neq 0$. The map also has the following values.
If $c \neq 0$, then $f(-d/c) = \infty$ and $f(\infty) = a/c$.
If $c = 0$, then $f(\infty) = \infty$.

## Remarks

Each Möbius transformation is a bijection of $\widehat{\mathbb{C}}$, and its inverse is the Möbius transformation $w \mapsto \frac{dw - b}{-cw + a}$.
The Möbius transformations form a [[group|G5S1M9]] under composition, the Möbius group.
The coefficients $(\lambda a, \lambda b, \lambda c, \lambda d)$ with $\lambda \neq 0$ give the same map.
If $ad - bc = 0$, the formula gives a constant or is undefined, so it is not a Möbius transformation.

## Explanations

### L1 human

A Möbius transformation is a quotient of two linear functions of $z$.
Examples are $z \mapsto z + 1$, $z \mapsto 2z$ and $z \mapsto \frac{1}{z}$, and each Möbius transformation is a composition of maps of these simple types.
The point $\infty$ is necessary, because the denominator can be zero.
</document>

<document source="https://math2.tamnd.com/node/0B3D4C8YF2.md" tag="0B3D4C8YF2" kind="concept">
# Conjunctive normal form

## Summary

A formula in conjunctive normal form is a conjunction of clauses, each a disjunction of literals. It is the standard input form for satisfiability solvers. The node [[conjunctive normal form|07C4A8Y18K]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B3Q3XDM5X.md" tag="0B3Q3XDM5X" kind="definition">
# Quantile function

## Definition

Let $F \colon \mathbb{R} \to [0, 1]$ be the [[cumulative distribution function|0MXMS3AE13]] of a real random variable $X$.
The quantile function of $X$ is the function $Q \colon (0, 1) \to \mathbb{R}$ with
$$Q(p) = \inf \{x \in \mathbb{R} : F(x) \ge p\}.$$
Here $\inf$ is the [[infimum|04KFTA34C5]]. The number $Q(p)$ is the $p$-quantile of $X$.

## Remarks

The set in the definition is nonempty and bounded below for $0 < p < 1$, and the infimum is a minimum because $F$ is right-continuous. So $F(Q(p)) \ge p$, and $Q(p) \le x$ if and only if $p \le F(x)$.
If $F$ is continuous and strictly increasing, then $Q = F^{-1}$.
If $U$ is uniform on $(0, 1)$, then $Q(U)$ has the cumulative distribution function $F$. This is inverse transform sampling.
$Q(1/2)$ is a [[median|08W2CW8PKW]] of $X$, and $Q(1/4)$ and $Q(3/4)$ are the lower and upper quartiles.

## Explanations

### L1 human

The quantile function answers the question "below which value does a fraction $p$ of the probability lie?".
For the exponential distribution with rate $1$, $F(x) = 1 - e^{-x}$, so $Q(p) = -\ln(1 - p)$.
</document>

<document source="https://math2.tamnd.com/node/0B5BA4FRWS.md" tag="0B5BA4FRWS" kind="concept">
# Independent sigma-algebras

## Summary

A family of sub-σ-algebras of a probability space is independent when events chosen from different members of the family are always mutually independent.
It is the most general form of independence, and the independence of events and of random variables are special cases. The node [[independent sigma-algebras|09T8ZHST1V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B5H2V4JW8.md" tag="0B5H2V4JW8" kind="definition">
# Set bounded above

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $M$ of $P$ is an upper bound of $S$ when $s \le M$ for each $s$ in $S$.
The set $S$ is bounded above when it has at least one upper bound in $P$.
Similarly, $m$ is a lower bound of $S$ when $m \le s$ for each $s$ in $S$.
The set $S$ is bounded below when it has a lower bound.

## Remarks

In $\mathbb{R}$, the interval $[0, 1)$ is bounded above. Each number $M \ge 1$ is an upper bound.
The set $\mathbb{N}$ is not bounded above in $\mathbb{R}$, by the [[Archimedean property|0A74SX55QP]].
An upper bound of $S$ does not need to be an element of $S$.

## Explanations

### L1 human

A set is bounded above when it does not go up forever: some number is a ceiling for all its elements.
The set $\{1, 1/2, 1/3, \dots\}$ is bounded above by $1$, and also by $2$ and by $100$.
</document>

<document source="https://math2.tamnd.com/node/0B5H4208Z9.md" tag="0B5H4208Z9" kind="definition">
# Eigenvector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
An *eigenvector* of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $T(v) = \lambda v$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$.

## Remarks

Each nonzero multiple of an eigenvector is again an eigenvector for the same eigenvalue.
For $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, the vector $(1, 1)^T$ is an eigenvector for $3$, because $A (1, 1)^T = (3, 3)^T$.
The zero vector is never an eigenvector, by convention.

## Explanations

### L1 human

An eigenvector keeps its line under the map: the map only stretches it, shrinks it or flips it.
The stretch factor is the eigenvalue.
Consider a reflection of the plane in a line.
The vectors on the line are eigenvectors for $1$, and the vectors perpendicular to it are eigenvectors for $-1$.
</document>

<document source="https://math2.tamnd.com/node/0B5WCC02AN.md" tag="0B5WCC02AN" kind="concept">
# Edge colouring

## Summary

An edge colouring assigns colours to the edges of a graph so that edges with a common end get different colours. It splits the edges into matchings and models scheduling problems. The node [[edge colouring|02CDYWSWSY]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B5Z1EC66J.md" tag="0B5Z1EC66J" kind="statement">
# R[x] is an integral domain iff R is

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is an [[integral domain|09S3PFBEPM]] if and only if $R$ is an integral domain.

## Remarks

If $R$ is an integral domain and $f, g \neq 0$, then the leading coefficient of $f g$ is not zero, so $f g \neq 0$.
Conversely, $R$ is a subring of $R[x]$, and a subring of an integral domain is an integral domain.
By induction, the ring $R[x_1, \dots, x_n]$ of [[polynomials in several variables|0CES91M79A]] over an integral domain is an integral domain.

## Explanations

### L1 human

The product of two nonzero polynomials over the integers is never zero.
The reason is that the leading coefficients multiply to a nonzero number.
So the polynomial ring keeps the integral domain property of its coefficients.
</document>

<document source="https://math2.tamnd.com/node/0B621RBVS9.md" tag="0B621RBVS9" kind="concept">
# Chromatic polynomial

## Summary

The chromatic polynomial of a finite graph counts its proper colourings as a function of the number of colours. It is a polynomial invariant of the graph that contains the chromatic number. The node [[chromatic polynomial|038YGY0HSC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B64Q7XNE3.md" tag="0B64Q7XNE3" kind="statement">
# Legendre's three-square theorem

## Statement

Let $n$ be a natural number.
Then $n = x^2 + y^2 + z^2$ for some integers $x$, $y$ and $z$ if and only if $n \neq 4^a (8b + 7)$ for all natural numbers $a$ and $b$.

## Remarks

The "only if" part is easy: a square is $0$, $1$ or $4$ modulo $8$, so no sum of three squares is $7$ modulo $8$.
The first numbers that are not sums of three squares are $7$, $15$, $23$, $28$ and $31$.
The theorem implies [[Lagrange's four-square theorem|0KYYSMEVHH]] in a few lines.

## History

Legendre gave the first proof in 1797 or 1798.
Gauss proved a more general result in 1801.

## Explanations

### L1 human

Most numbers are sums of three squares, for example $6 = 4 + 1 + 1$.
The exceptions are $7, 15, 23, \dots$, the numbers that leave remainder $7$ on division by $8$, and these numbers times powers of $4$.
</document>

<document source="https://math2.tamnd.com/node/0B68KNPGCG.md" tag="0B68KNPGCG" kind="concept">
# Lexicographic order

## Summary

The lexicographic order compares pairs, or words, first by the first entry, and then by the next entry when the first entries are equal.
</document>

<document source="https://math2.tamnd.com/node/0B6QC0C0G1.md" tag="0B6QC0C0G1" kind="definition">
# Ring isomorphism

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
A ring isomorphism from $R$ to $S$ is a [[ring homomorphism|0GGE0Z0JQ1]] $f \colon R \to S$ that is bijective.
The rings $R$ and $S$ are isomorphic, written $R \cong S$, if there is a ring isomorphism from $R$ to $S$.

## Remarks

The inverse map of a ring isomorphism is again a ring homomorphism.
A ring isomorphism from $R$ to $R$ is an automorphism of $R$.
Complex conjugation is an automorphism of $\mathbb{C}$.

## Explanations

### L1 human

Two isomorphic rings are the same ring with other names for the elements.
The isomorphism is the dictionary between the names.
</document>

<document source="https://math2.tamnd.com/node/0B6QWX3086.md" tag="0B6QWX3086" kind="concept">
# Strict differentiability

## Summary

A function is strictly differentiable at a point when a single linear map approximates all its difference quotients near that point. Both points of each quotient can move.
This is stronger than the [[Fréchet derivative|06EH97V2A7]] condition, in which one of the two points stays fixed.
</document>

<document source="https://math2.tamnd.com/node/0B77KZCQ51.md" tag="0B77KZCQ51" kind="concept">
# Markov property

## Summary

A stochastic process has the Markov property when its future, given the present, is independent of its past.
The node [[Markov property|0HB8SXHPFA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B80PAQD3C.md" tag="0B80PAQD3C" kind="concept">
# Gamma distribution

## Summary

The gamma distribution is a two-parameter family of continuous distributions on $(0, \infty)$. It contains the exponential, Erlang and chi-squared distributions.
The node [[gamma distribution|06395GBNWD]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0B85Q41RP6.md" tag="0B85Q41RP6" kind="definition">
# Cyclic group

## Definition

A [[group|G5S1M9]] $G$ is cyclic when there is an element $g \in G$ with $G = \langle g \rangle$, where $\langle g \rangle$ is the [[cyclic subgroup generated by $g$|08ABT85339]].
Such an element $g$ is called a generator of $G$.

## Remarks

The group $(\mathbb{Z}, +)$ is cyclic, and its generators are $1$ and $-1$.
The integers modulo $n$ under addition form a cyclic group of order $n$, generated by $1$.
A cyclic group can have many generators: in the integers modulo $n$, the generators are the residues coprime to $n$.
The [[circle group|04EAFVFTF7]] is not cyclic, because it is uncountable.

## Explanations

### L1 human

A cyclic group is a group in which each element is a power of one fixed element.
The rotations of a regular polygon with $n$ sides form a cyclic group: each rotation is a power of the smallest rotation.
The word "cyclic" comes from this picture of going around a circle.
</document>

<document source="https://math2.tamnd.com/node/0B85S2V2DR.md" tag="0B85S2V2DR" kind="concept">
# Order of a group

## Summary

The order of a group is the number of its elements.
A group with infinitely many elements has infinite order.
</document>

<document source="https://math2.tamnd.com/node/0B8837S5T3.md" tag="0B8837S5T3" kind="concept">
# Essential singularity

## Summary

An essential singularity is an isolated singularity that is neither removable nor a pole, see the [[definition|0YKDJBV26S]].
Near an essential singularity, a function comes arbitrarily close to every complex value.
</document>

<document source="https://math2.tamnd.com/node/0B88PQTJ66.md" tag="0B88PQTJ66" kind="definition">
# Universal quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The universal quantification of $P$ over $X$ is the proposition $\forall x \in X, \, P(x)$, read "for all $x$ in $X$, $P(x)$".
It is true when $P(a)$ is true for each $a \in X$, and false otherwise.

## Remarks

When $X$ is empty, $\forall x \in X, \, P(x)$ is true for each $P$. This is a case of [[vacuous truth|0V4QXPGJQK]].
For a finite set $X = \{a_1, \dots, a_n\}$, the statement is equivalent to $P(a_1) \land \dots \land P(a_n)$.

## Explanations

### L1 human

"Every square of a real number is at least $0$" is $\forall x \in \mathbb{R}, \, x^2 \ge 0$.
To prove it, take an arbitrary $x$ and show $x^2 \ge 0$.
To refute it, one [[counterexample|0ED72CTBNP]] is enough.
</document>

<document source="https://math2.tamnd.com/node/0B89D93YA1.md" tag="0B89D93YA1" kind="definition">
# Arcsine

## Definition

The arcsine is the function $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$ such that, for each $x \in [-1, 1]$, $\arcsin x$ is the unique $y \in [-\pi/2, \pi/2]$ with $\sin y = x$.

## Remarks

The sine is strictly increasing and continuous on $[-\pi/2, \pi/2]$, with range $[-1, 1]$. So $\arcsin$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arcsin 0 = 0$, $\arcsin 1 = \pi/2$, and $\arcsin \frac{1}{2} = \pi/6$.
The function is odd and strictly increasing. Mathlib defines `Real.arcsin` on all of $\mathbb{R}$, constant outside $[-1, 1]$.

## Explanations

### L1 human

The arcsine of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose sine is $x$.
</document>

<document source="https://math2.tamnd.com/node/0B8BW4YWTF.md" tag="0B8BW4YWTF" kind="concept">
# Lattice of subgroups

## Summary

The [[subgroups|0X1JXT47GA]] of a [[group|G5S1M9]] $G$, ordered by inclusion, form a complete lattice.
The meet of a family of subgroups is their intersection, and the join is the [[subgroup generated by|0RP7VWQ0YN]] their union.
The least element is the trivial subgroup $\{e\}$ and the greatest element is $G$.

## Remarks

For two subgroups $H$ and $K$, the meet is $H \cap K$ and the join is $\langle H \cup K \rangle$.
The lattice of subgroups of $(\mathbb{Z}, +)$ is the set of the subgroups $n\mathbb{Z}$, and $m\mathbb{Z} \subseteq n\mathbb{Z}$ exactly when $n$ divides $m$.
The lattice of normal subgroups of a group is a modular lattice.

## Explanations

### L1 human

You can draw the subgroups of a small group as a diagram, with a line from each subgroup up to each next bigger subgroup.
This diagram shows the lattice of subgroups.
Two subgroups always have a largest common subgroup below them and a smallest subgroup above them.
</document>

<document source="https://math2.tamnd.com/node/0B8GPXFSP7.md" tag="0B8GPXFSP7" kind="statement">
# Continuity via closed sets

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and let $f \colon X \to Y$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if for each [[closed set|0G5AV9Q4RR]] $F$ of $Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(F)$ is closed in $X$.

## Remarks

Proof: preimages commute with complements, that is, $f^{-1}(Y \setminus A) = X \setminus f^{-1}(A)$ for each $A \subseteq Y$. Then use that the [[complements of open sets are closed|0K67Q60Z8P]].
For example, for a continuous $f \colon \mathbb{R} \to \mathbb{R}$, the set of zeros $f^{-1}(\{0\})$ is closed.

## Explanations

### L1 human

Continuity can be tested with closed sets as well as with open sets.
This form is useful to show that a set given by an equation, such as $\{x : f(x) = 0\}$, is closed.
</document>

<document source="https://math2.tamnd.com/node/0B8KZRT28M.md" tag="0B8KZRT28M" kind="concept">
# Orthogonal projection

## Summary

The orthogonal projection onto a subspace $W$ sends each vector to the unique vector of $W$ whose difference with it is orthogonal to $W$.
</document>

<document source="https://math2.tamnd.com/node/0B8T10CE7N.md" tag="0B8T10CE7N" kind="concept">
# Linear isomorphism

## Summary

A linear isomorphism is a bijective linear map.
Its inverse is also linear, and two spaces with an isomorphism between them are isomorphic.
</document>

<document source="https://math2.tamnd.com/node/0B906ZBTMZ.md" tag="0B906ZBTMZ" kind="example">
# Cofinite topology

## Example

Let $X$ be a set and let $\tau$ consist of $\varnothing$ and of all sets $U \subseteq X$ whose complement $X \setminus U$ is [[finite|0BBTH3YSZA]].
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the cofinite topology.

## Remarks

The closed sets are $X$ and the finite subsets of $X$.
The cofinite topology is [[T1|05GQJXBP3F]]: each one-point set is closed.
If $X$ is infinite, any two nonempty open sets meet. Thus the space is not [[Hausdorff|0MXB4CKHFW]].
Each space with the cofinite topology is [[compact|0QPHW703PV]].

## Explanations

### L1 human

In the cofinite topology, a set is open when it misses only finitely many points.
On an infinite set, open sets are very large, so no two nonempty open sets are disjoint.
</document>

<document source="https://math2.tamnd.com/node/0B94TKFQJX.md" tag="0B94TKFQJX" kind="concept">
# Mathematical proof

## Summary

A mathematical proof is an argument that shows that a statement follows from axioms, definitions and statements that are already proved.
Each step of a proof follows from earlier steps by a valid rule of inference, such as [[modus ponens|0Y0MQYST0M]].
A proved statement is a theorem.

## Remarks

Common methods are [[direct proof|0NTF6DMVAP]], [[proof by contrapositive|0BB5TPGQ1N]], [[proof by contradiction|0J0XBKG64S]] and [[proof by cases|0WCP0PVET6]].
A formal proof is a sequence of formulas in a formal system. A computer can check it, for example in Lean.
Examples, however many, do not prove a universal statement.

## Explanations

### L1 human

A proof is a chain of small steps, and each step is certainly correct.
It starts from things that you accept or already know, and it ends at the statement.
After a correct proof, the statement is true in every case, not only in the cases that you tested.
</document>

<document source="https://math2.tamnd.com/node/0B96KMDDD1.md" tag="0B96KMDDD1" kind="statement">
# Cauchy condensation test

## Statement

Let $(a_n)_{n \ge 1}$ be a decreasing real sequence with $a_n \ge 0$ for all $n$.
Then $\sum_{n=1}^{\infty} a_n$ converges if and only if the condensed series $\sum_{k=0}^{\infty} 2^k a_{2^k}$ converges.

## Remarks

The proof groups the terms from $2^k$ to $2^{k+1} - 1$. Since the sequence decreases, $\sum_{n=1}^{\infty} a_n \le \sum_{k=0}^{\infty} 2^k a_{2^k} \le 2 \sum_{n=1}^{\infty} a_n$.
With $a_n = 1/n^p$ it gives the [[p-series test|0ZNSSK8ZK1]].
With $a_n = 1/(n \ln n)$ for $n \ge 2$, the condensed series is a multiple of $\sum 1/k$, so $\sum 1/(n \ln n)$ diverges.

## Explanations

### L1 human

For a decreasing sequence, the terms between $2^k$ and $2^{k+1}$ are all about the same size as $a_{2^k}$.
So only one term out of each such block matters, weighted by the size $2^k$ of the block.
</document>

<document source="https://math2.tamnd.com/node/0B9QNFGTHM.md" tag="0B9QNFGTHM" kind="example">
# Rotation matrix

## Example

Let $\theta \in \R$.
The *rotation matrix* of angle $\theta$ is
$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.$$
The map $x \mapsto R(\theta) x$ rotates the plane $\R^2$ about the origin by the angle $\theta$, counterclockwise.
It is an [[orthogonal matrix|0W02QESPJQ]] with determinant $\cos^2\theta + \sin^2\theta = 1$, so it is an element of $SO(2)$.

## Remarks

Rotations add their angles: $R(\alpha) R(\beta) = R(\alpha + \beta)$.
So $R(\theta)^{-1} = R(-\theta) = R(\theta)^T$.
For $\theta$ not a multiple of $\pi$, $R(\theta)$ has no real eigenvalue; its complex eigenvalues are $e^{\pm i\theta}$.
Every element of $SO(2)$ is $R(\theta)$ for some $\theta$.

## Explanations

### L1 human

The rotation matrix sends $(1, 0)$ to $(\cos\theta, \sin\theta)$ and $(0, 1)$ to $(-\sin\theta, \cos\theta)$.
These are the first and the second column.
For $\theta = 90$ degrees, it is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.
</document>

<document source="https://math2.tamnd.com/node/0B9ZQFMQNV.md" tag="0B9ZQFMQNV" kind="statement">
# Connected subsets of the real line are intervals

## Statement

Let $S \subseteq \mathbb{R}$, where $\mathbb{R}$ has the usual topology.
Then $S$ is [[connected|0R37DFJKFW]] if and only if $S$ is an [[interval|0TK8Y3J5RW]], that is, for all $a, b \in S$ and all $c \in \mathbb{R}$ with $a < c < b$, $c \in S$.

## Remarks

If $S$ is not an interval, take $a < c < b$ with $a, b \in S$ and $c \notin S$. Then $(-\infty, c)$ and $(c, \infty)$ split $S$.
The proof that an interval $I$ is connected uses the [[completeness axiom|0AXWET80YW]]. If open sets $U$ and $V$ split $I$, with $a \in U$, $b \in V$ and $a < b$, let $c$ be the [[supremum|0BTMQ04556]] of the points of $U \cap [a, b]$. Then $c$ lies in neither $U$ nor $V$.
Under this definition the empty set and single points are intervals, and they are connected.
The statement fails for $\mathbb{Q}$: no interval of $\mathbb{Q}$ with more than one point is connected.

## Explanations

### L1 human

On the real line, the sets in one piece are exactly the intervals.
The proof needs the completeness of $\mathbb{R}$, because $\mathbb{Q}$ has gaps where it can be split.
</document>

<document source="https://math2.tamnd.com/node/0BANA3XYPF.md" tag="0BANA3XYPF" kind="concept">
# Bridge

## Summary

A bridge of a graph is an edge whose deletion disconnects a component. Bridges are exactly the edges that lie on no cycle. The node [[bridge|0D81KWH9ZM]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BARQY52N5.md" tag="0BARQY52N5" kind="statement">
# Mean value theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Then there is $c \in (a, b)$ with
$$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

## Remarks

Apply [[Rolle's theorem|0194DY35BG]] to $g(x) = f(x) - \frac{f(b) - f(a)}{b - a} (x - a)$, which has $g(a) = g(b)$.
It turns bounds on $f'$ into bounds on $f$. It gives [[zero derivative implies constant|003QDZ4T16]] and the [[monotonicity tests|07J5X2T1ZX]].
Mathlib states it as `exists_deriv_eq_slope`.

## History

Augustin-Louis Cauchy stated and proved the theorem in its modern form in 1823.

## Explanations

### L1 human

Somewhere between $a$ and $b$, the tangent line is parallel to the chord from $(a, f(a))$ to $(b, f(b))$.
If a car travels 100 km in one hour, at some moment its speed is exactly 100 km per hour.
</document>

<document source="https://math2.tamnd.com/node/0BB5TPGQ1N.md" tag="0BB5TPGQ1N" kind="concept">
# Proof by contrapositive

## Summary

A proof by contrapositive proves an [[implication|0ZNX6DQ3EJ]] $p \to q$ by a [[direct proof|0NTF6DMVAP]] of $\neg q \to \neg p$.
The method is correct because $p \to q$ is logically equivalent to its [[contrapositive|04T1CT7212]].

## Remarks

Example: for an integer $n$, if $n^2$ is even, then $n$ is even. The contrapositive is "if $n$ is odd, then $n^2$ is odd", which has a short direct proof.
A [[proof by contradiction|0J0XBKG64S]] is different: it assumes $p$ and $\neg q$ together, and it gets any contradiction.

## Explanations

### L1 human

Sometimes the negation of the conclusion gives you more to work with.
"If $n^2$ is even, then $n$ is even" is hard to start.
"If $n$ is odd, then $n^2$ is odd" is easy, and it says the same thing.
</document>

<document source="https://math2.tamnd.com/node/0BBTH3YSZA.md" tag="0BBTH3YSZA" kind="definition">
# Finite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is finite when there is a natural number $n$ such that $A$ is [[equinumerous|08EJHHMB5R]] with $\{1, 2, \dots, n\}$.
For $n = 0$, this set is the [[empty set|0ECVK59Y03]].
The number $n$ is unique, and it is the number of elements of $A$.

## Remarks

Subsets, finite unions, finite products and the power set of finite sets are finite.
A finite set is not equinumerous with any proper subset of itself. This is the pigeonhole principle.
A set that is not finite is [[infinite|01MKW4R283]].

## Explanations

### L1 human

A set is finite when you can count its elements and stop.
The set of planets of the Solar System is finite, and so is the set of primes less than $100$.
</document>

<document source="https://math2.tamnd.com/node/0BC3Q6K92H.md" tag="0BC3Q6K92H" kind="statement">
# Matrix determinant lemma

## Statement

Let $K$ be a commutative ring, let $A$ be an invertible $n \times n$ matrix over $K$, and let $u, v \in K^n$ be column vectors.
Then
$$\det(A + u v^T) = (1 + v^T A^{-1} u) \det A,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The matrix $u v^T$ has rank at most $1$, so the lemma describes the effect of a rank one update on the determinant.
It follows from [[Sylvester's determinant theorem|0G24VQ1088]] with the matrices $A^{-1} u$ and $v^T$.
A variant without inverses is $\det(A + u v^T) = \det A + v^T \operatorname{adj}(A) u$.

## Explanations

### L1 human

Adding a matrix of rank one, $u v^T$, changes the determinant by a simple factor.
The factor is $1 + v^T A^{-1} u$, a single number.
So we can update a determinant cheaply when we know the inverse.
</document>

<document source="https://math2.tamnd.com/node/0BC7KX51Z8.md" tag="0BC7KX51Z8" kind="definition">
# Power set

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The power set of $A$ is the set
$$\mathcal{P}(A) = \{S \mid S \subseteq A\}$$
of all [[subsets|0FE8Q5S42Y]] of $A$.

## Remarks

Other notations are $2^A$ and $\wp(A)$.
Example: $\mathcal{P}(\{1, 2\}) = \{\varnothing, \{1\}, \{2\}, \{1, 2\}\}$.
A finite set with $n$ elements has a power set with $2^n$ elements. See [[cardinality of the power set|0GPFKS96T7]].
By [[Cantor's theorem|0WTEMGYV2C]], no set is in bijection with its power set.

## Explanations

### L1 human

To list the power set, choose for each element whether it is in or out of the subset.
Each choice gives one subset, so a set with $3$ elements has $2^3 = 8$ subsets.
The empty set and the set itself are always among them.
</document>

<document source="https://math2.tamnd.com/node/0BCHJ30REG.md" tag="0BCHJ30REG" kind="definition">
# Cycle type

## Definition

Let $\sigma \in S_n$.
Write $\sigma$ as a product of disjoint cycles, and include a $1$-cycle for each fixed point.
The cycle type of $\sigma$ is the multiset of the lengths of these cycles.

## Remarks

The [[disjoint cycle decomposition|0YWDAZPRPQ]] is unique up to order, so the cycle type is well defined.
The lengths add up to $n$, so the cycle type is a partition of $n$.
For example, $(1\ 2)(3\ 4\ 5) \in S_6$ has cycle type $(3, 2, 1)$.

## Explanations

### L1 human

The cycle type forgets which points are in which cycle and keeps only the sizes of the cycles.
Two permutations with the same cycle type have the same shape.
In the symmetric group, the shape decides exactly when two permutations are conjugate.
</document>

<document source="https://math2.tamnd.com/node/0BCMGDHZGR.md" tag="0BCMGDHZGR" kind="definition">
# Pivot

## Definition

Let $F$ be a field and let $A$ be a matrix over $F$ in [[row echelon form|0VYCDGMJJJ]].
A *pivot* of $A$ is the leading entry of a nonzero row of $A$, that is, the first nonzero entry of that row.
A *pivot column* of $A$ is a column that contains a pivot.
For a general matrix $B$, take a row echelon form of $B$ that is row equivalent to $B$.
The pivot positions of $B$ are the positions of the pivots of this form.

## Remarks

The pivot positions do not depend on the choice of the row echelon form.
The number of pivots is the [[rank|0ZC9YTXZRY]] of the matrix.
In the system $Ax = b$, the unknowns of the pivot columns are the *basic variables*, and the other unknowns are the *free variables*.
In numerical work, partial pivoting chooses as pivot an entry of largest absolute value in its column, to reduce rounding errors.

## Explanations

### L1 human

A pivot is the first nonzero number in a row of a staircase matrix.
During elimination we divide by the pivot and use it to clear the entries below it.
Each pivot fixes one unknown, and each column without a pivot gives a free unknown.
</document>

<document source="https://math2.tamnd.com/node/0BDJBH8B4Z.md" tag="0BDJBH8B4Z" kind="statement">
# Order of the Galois group of a Galois extension equals its degree

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]].
Then
$$|\operatorname{Gal}(L/K)| = [L : K].$$

## Remarks

Conversely, a finite extension with $|\operatorname{Aut}(L/K)| = [L : K]$ is Galois. So the [[general bound|0QM98DNWVW]] is an equality exactly for Galois extensions.
For $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$, the Galois group has order $4$: an automorphism can change the signs of $\sqrt{2}$ and of $\sqrt{3}$ independently.
For $\mathbb{F}_{p^n}/\mathbb{F}_p$, the Galois group is cyclic of order $n$, generated by the [[Frobenius automorphism|0VP4TBVJ19]].

## Explanations

### L1 human

For a Galois extension, counting symmetries is the same as computing the degree.
This links the size of a group to the size of a field extension.
</document>

<document source="https://math2.tamnd.com/node/0BDTTM24DP.md" tag="0BDTTM24DP" kind="definition">
# Banach space

## Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$ with a [[norm|0J3578562S]] $\|\cdot\|$, and let $d(x, y) = \|x - y\|$.
The space $(V, \|\cdot\|)$ is a Banach space when the metric space $(V, d)$ is [[complete|08EWT7J0SC]].

## Remarks

Examples are $\mathbb{R}^n$ with any norm, the spaces [[ell p|08P6T2WH6R]] for $1 \le p \le \infty$, and $C_b(X)$ with the [[sup norm|0PARCKE8WP]].
The space of polynomial functions on $[0, 1]$ with the sup norm is not a Banach space. Its completion is $C([0, 1])$.
In a normed space $V$, completeness is the same as: [[each absolutely convergent series converges|082MGNWNA8]].
Each [[finite-dimensional normed space is complete|0ABP2F2ZP7]].

## Explanations

### L1 human

A Banach space is a vector space where you can measure lengths, and where Cauchy sequences always converge.
So you can take limits in it freely, as in $\mathbb{R}^n$.
</document>

<document source="https://math2.tamnd.com/node/0BDX4MBKT4.md" tag="0BDX4MBKT4" kind="definition">
# Monic polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]].
A polynomial $f \in R[x]$ is monic if its [[leading coefficient|0P05RJC5XA]] is $1$.

## Remarks

The polynomial $x^2 - 2$ is monic, and $2x^2 - 1$ is not.
In a nonzero ring, a monic polynomial is not zero.
The product of two monic polynomials is monic, over every commutative ring.
Over a field, each nonzero polynomial is a constant times exactly one monic polynomial.

## Explanations

### L1 human

A monic polynomial starts with $x^d$, with no number in front.
Over a field, you can make any nonzero polynomial monic by dividing by its leading coefficient.
</document>

<document source="https://math2.tamnd.com/node/0BE3QDH73A.md" tag="0BE3QDH73A" kind="definition">
# Smooth function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is smooth, or of class $C^{\infty}$, if it is of [[class $C^k$|06SKCHDQT5]] for each natural number $k$.

## Remarks

So a smooth function has derivatives of all orders.
Polynomials, the exponential function, and the sine are smooth on $\mathbb{R}$.
A smooth function need not be [[analytic|0JD5ZNW0F4]]. See the [[smooth non-analytic function|09RTYYHXKA]].
Mathlib uses `ContDiff ℝ ⊤ f`.

## Explanations

### L1 human

A smooth function can be differentiated as many times as we like.
All its derivatives exist and are continuous.
</document>

<document source="https://math2.tamnd.com/node/0BEZX8F2XW.md" tag="0BEZX8F2XW" kind="statement">
# Solutions for complex conjugate roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$.
Let $\alpha + i \beta$ be a root of $p$ of multiplicity $m$, with $\alpha, \beta \in \mathbb{R}$ and $\beta \ne 0$.
Then $\alpha - i \beta$ is also a root of multiplicity $m$, and the $2m$ functions
$$x^k e^{\alpha x} \cos(\beta x), \quad x^k e^{\alpha x} \sin(\beta x), \qquad k = 0, \dots, m - 1,$$
are linearly independent real solutions of the equation on $\mathbb{R}$.

## Remarks

By [[Euler's formula|04R3CY41VJ]], $e^{(\alpha + i \beta) x} = e^{\alpha x} (\cos \beta x + i \sin \beta x)$. Its real and imaginary parts are solutions because the coefficients are real.
For order $2$ and roots $\alpha \pm i \beta$, the real solutions are $e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x)$.
Example: $y'' + 2 y' + 5 y = 0$ has the roots $-1 \pm 2 i$, so its solutions are $e^{-x} (c_1 \cos 2x + c_2 \sin 2x)$.
These families for all roots, together with those of [[real roots|0HRVY67CNV]], give a real fundamental system.

## Explanations

### L1 human

Complex roots come in conjugate pairs, because the polynomial has real coefficients.
Each pair gives an oscillation: a sine and a cosine with frequency $\beta$.
The real part $\alpha$ makes the oscillation grow if $\alpha > 0$ and decay if $\alpha < 0$.
</document>

<document source="https://math2.tamnd.com/node/0BF2DE2SYK.md" tag="0BF2DE2SYK" kind="concept">
# Tensor product of vector spaces

## Summary

The tensor product of two vector spaces is a vector space spanned by symbols $v \otimes w$.
These symbols are bilinear in $v$ and $w$, and no other relations hold.
</document>

<document source="https://math2.tamnd.com/node/0BFD3A6CRH.md" tag="0BFD3A6CRH" kind="statement">
# Global existence for globally Lipschitz vector fields

## Statement

Let $J$ be an interval, let $K \ge 0$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be continuous.
Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in \mathbb{R}^n$.
Then for each $t_0 \in J$ and $x_0 \in \mathbb{R}^n$, the [[initial value problem|075VBY4K4X]] $x' = f(t, x)$, $x(t_0) = x_0$ has exactly one solution defined on all of $J$.

## Remarks

The point is that the solution does not blow up: the Lipschitz bound gives $\|f(t, x)\| \le \|f(t, 0)\| + K \|x\|$, so $\|x(t)\|$ grows at most exponentially.
A linear system $x' = A(t) x + b(t)$ with continuous $A$ and $b$ on $J$ satisfies the hypothesis on each compact subinterval. So its solutions exist on all of $J$.
A field that is only locally Lipschitz can give [[blow-up in finite time|0E4GXX7SJP]].

## Explanations

### L1 human

If the field grows at most linearly in the state, solutions cannot run away to infinity in finite time.
So the solution exists as long as the field is defined.
</document>

<document source="https://math2.tamnd.com/node/0BFDHF2H7N.md" tag="0BFDHF2H7N" kind="statement">
# Weak law of large numbers

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with $E[|X_1|] < \infty$ and $\mu = E[X_1]$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\bar{X}_n \to \mu$ [[in probability|0EFDDXTVD6]]: for each $\varepsilon > 0$,
$$\lim_{n \to \infty} P\left(\left|\bar{X}_n - \mu\right| > \varepsilon\right) = 0.$$

## Remarks

If $\sigma^2 = \operatorname{Var}(X_1) < \infty$, there is a short proof. By the [[variance of a sum|0V0778DCW9]], $\operatorname{Var}(\bar{X}_n) = \sigma^2/n$.
Then [[Chebyshev's inequality|02HZ1BK3MW]] gives $P(|\bar{X}_n - \mu| > \varepsilon) \le \sigma^2 / (n \varepsilon^2) \to 0$.
The general case uses truncation, or characteristic functions: $\varphi_{\bar{X}_n}(t) = \varphi_{X_1}(t/n)^n \to e^{i\mu t}$.
The finite-variance proof needs only that the $X_i$ are pairwise uncorrelated with the same mean and bounded variances.

## History

Jacob Bernoulli proved the first version, for Bernoulli trials. It appeared after his death in Ars Conjectandi in 1713.

## Explanations

### L1 human

The weak law says that for a large number $n$ of trials, the sample mean is very likely to be close to the true mean.
For example, in $10000$ tosses of a fair coin, the fraction of heads is very likely to be between $0.49$ and $0.51$.
</document>

<document source="https://math2.tamnd.com/node/0BFMR2FQ8B.md" tag="0BFMR2FQ8B" kind="statement">
# Number of k-element subsets

## Statement

Let $X$ be a finite set with $|X| = n$ and let $k$ be a natural number.
Then the number of subsets of $X$ with exactly $k$ elements is the [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$.

## Remarks

A proof by induction on $n$ fixes an element $x \in X$. A $k$-subset either omits $x$ or contains $x$, and this gives [[Pascal's rule|0YRG241HQQ]].
For $k > n$ both sides are $0$.

## Explanations

### L1 human

To pick a committee of $k$ people from $n$, first count the ordered lists of $k$ people. Then divide by the $k!$ orders of each committee.
This gives $\frac{n (n-1) \cdots (n-k+1)}{k!}$, which is $\binom{n}{k}$.
</document>

<document source="https://math2.tamnd.com/node/0BFN21TF5F.md" tag="0BFN21TF5F" kind="statement">
# Path independence of line integrals

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and connected, and let $F \colon U \to \mathbb{R}^n$ be a continuous vector field.
The following are equivalent.

1. $F$ is [[conservative|05R9NF2J49]] on $U$.
2. For all piecewise $C^1$ curves $\gamma_1, \gamma_2$ in $U$ with the same initial point and the same end point, $\int_{\gamma_1} F \cdot d\mathbf{r} = \int_{\gamma_2} F \cdot d\mathbf{r}$.
3. For each closed piecewise $C^1$ curve $\gamma$ in $U$, $\int_\gamma F \cdot d\mathbf{r} = 0$.

## Remarks

Statement 1 implies statement 2 by the [[gradient theorem|0SD237FEJD]].
Statements 2 and 3 are equivalent: a closed curve splits into two curves with the same end points, and one uses [[reversal|0FSCXYG856]] and [[additivity|00ZEBSMNNV]].
Statement 2 implies statement 1: fix $p \in U$ and let $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. A short straight last segment in direction $e_i$ shows that $\partial \phi / \partial x_i = F_i$.
Connectedness is used to join $p$ to each point. On a disconnected set, apply the result on each component.

## Explanations

### L1 human

For a conservative field, the work from one point to another does not depend on the route.
Conversely, if the work never depends on the route, then the work from a fixed base point defines a potential for the field.
</document>

<document source="https://math2.tamnd.com/node/0BH9JH7HBY.md" tag="0BH9JH7HBY" kind="concept">
# Infinite limit

## Summary

A function has an infinite limit at a point when its values become arbitrarily large, positive or negative, near the point. Infinite limits describe vertical asymptotes. The node [[infinite limit|08NM8KQPDQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BJ6H85DRQ.md" tag="0BJ6H85DRQ" kind="concept">
# Limit along different paths

## Summary

Let $f$ be a function on $D \subseteq \mathbb{R}^n$ and let $a$ be a limit point of $D$.
Let $\gamma \colon (0, 1] \to D$ be a path, for example a [[curve|0G7WM8DX4P]], with $\gamma(t) \ne a$ for each $t$ and $\gamma(t) \to a$ as $t \to 0^+$.
If $\lim_{x \to a} f(x) = L$, then $\lim_{t \to 0^+} f(\gamma(t)) = L$.
So if two such paths give different limits, or one path gives no limit, then the [[limit|0J8Y1NCY0K]] of $f$ at $a$ does not exist.

## Remarks

For $f(x, y) = xy / (x^2 + y^2)$ on $\mathbb{R}^2 \setminus \{(0, 0)\}$, the path $y = m x$ gives the limit $m / (1 + m^2)$ at the origin. This depends on $m$, so $f$ has no limit at $(0, 0)$.
Equal limits along all straight lines do not prove that the limit exists. For $f(x, y) = x^2 y / (x^4 + y^2)$, each line through the origin gives the limit $0$, but the parabola $y = x^2$ gives $1/2$.

## Explanations

### L1 human

In one variable you can approach a point only from the left or from the right.
In the plane you can approach a point along infinitely many routes: lines, parabolas, spirals.
A limit must give the same answer for every route, so one route that disagrees is enough to show that there is no limit.
</document>

<document source="https://math2.tamnd.com/node/0BJBYCGMWE.md" tag="0BJBYCGMWE" kind="statement">
# A linear map is injective iff its kernel is trivial

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
Then $T$ is injective if and only if $\ker T = \{0\}$, where $\ker T$ is the [[kernel|0S23E6YE83]] of $T$.

## Remarks

The key step is $T(u) = T(v) \iff T(u - v) = 0 \iff u - v \in \ker T$.
Thus each nonempty fibre $T^{-1}(w)$ is a coset $v + \ker T$.
For a matrix $A$, the map $x \mapsto Ax$ is injective if and only if $Ax = 0$ has only the trivial solution.

## Explanations

### L1 human

To test if a linear map is one to one, we only need to look at the vectors that go to zero.
If two vectors have the same image, then their difference goes to zero.
So if only $0$ goes to $0$, then no two vectors share an image.
</document>

<document source="https://math2.tamnd.com/node/0BJDQ2HBH1.md" tag="0BJDQ2HBH1" kind="definition">
# Tangent plane and normal vector of a surface

## Definition

Let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is regular at the point $(u_0, v_0)$ of $D$, and let $p = r(u_0, v_0)$.
The tangent plane at $p$ is the plane $\{ p + s \, r_u(u_0, v_0) + t \, r_v(u_0, v_0) : s, t \in \mathbb{R} \}$.
The vector $N(u_0, v_0) = r_u(u_0, v_0) \times r_v(u_0, v_0)$ is the normal vector of $r$ at $(u_0, v_0)$, and $n = N / \|N\|$ is the unit normal vector.

## Remarks

The vector $N$ is perpendicular to $r_u$ and $r_v$, so it is perpendicular to the tangent plane. The other unit normal at $p$ is $-n$.
For the graph $z = f(x, y)$, the normal is $(-f_x, -f_y, 1)$, which points upward.
If the surface is the level set $g(x, y, z) = c$ of a $C^1$ function $g$ with $\nabla g(p) \ne 0$, then the [[gradient|0NS2QYVC4Y]] $\nabla g(p)$ is a normal vector at $p$.
For the unit sphere with $r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi)$, one gets $r_\varphi \times r_\theta = \sin \varphi \, r(\varphi, \theta)$, which points outward for $0 < \varphi < \pi$.

## Explanations

### L1 human

The tangent plane is the flat plane that touches the surface at a point, and the normal vector sticks straight out of it.
The cross product of the two coordinate tangent vectors gives a normal at once.
</document>

<document source="https://math2.tamnd.com/node/0BJKVF90Z6.md" tag="0BJKVF90Z6" kind="definition">
# Spanning set

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A subset $S \subseteq V$ is a *spanning set* of $V$, or *spans* $V$, when $\operatorname{span}(S) = V$, where $\operatorname{span}$ is the [[linear span|0DTY8B9HMS]].
Equivalently, each vector of $V$ is a linear combination of finitely many vectors of $S$.

## Remarks

The vectors $(1, 0), (0, 1), (1, 1)$ span $\R^2$, and so do $(1, 0), (0, 1)$.
A linearly independent spanning set is a [[basis|0KCQ9Y3QKA]].
[[Every spanning set contains a basis|06MRNP4BNT]].
A vector space with a finite spanning set is [[finite-dimensional|0ZHWKQMVX5]].

## Explanations

### L1 human

A spanning set is a set of building blocks from which we can make every vector of the space.
It may contain more vectors than we need.
Removing the extra vectors leaves a basis.
</document>

<document source="https://math2.tamnd.com/node/0BJR9QR08E.md" tag="0BJR9QR08E" kind="concept">
# Subset

## Summary

A set $A$ is a subset of a set $B$ when each element of $A$ is an element of $B$.
</document>

<document source="https://math2.tamnd.com/node/0BJSWZJG18.md" tag="0BJSWZJG18" kind="statement">
# Mertens' theorem on Cauchy products

## Statement

Let $\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.
Let $\sum b_n$ be a [[convergent|0CVVKY7GY2]] real series, with sum $B$.
Then their [[Cauchy product|0B1K9PVWE9]] converges, and its sum is $A B$.

## Remarks

Convergence of both series is not enough. Take $a_n = b_n = (-1)^n / \sqrt{n + 1}$. Both series converge, but $|c_n| \ge 1$ for all $n$, so the Cauchy product diverges.
Franz Mertens proved the theorem.

## Explanations

### L1 human

One series may converge only conditionally, if the other converges absolutely.
Then the product of the two series is still the sum of their Cauchy product.
</document>

<document source="https://math2.tamnd.com/node/0BJV24C4SY.md" tag="0BJV24C4SY" kind="definition">
# Complex conjugate

## Definition

Let $z = a + b i$ be a [[complex number|09FTAET5Z4]], with $a$ and $b$ real.
The complex conjugate of $z$ is the complex number $\bar{z} = a - b i$.

## Remarks

Another common notation is $z^*$.
Conjugation respects the operations: $\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{z w} = \bar{z} \, \bar{w}$. Also $\bar{\bar{z}} = z$.
A complex number $z$ is real exactly when $\bar{z} = z$.
The product $z \bar{z}$ is the real number $|z|^2$, see [[a complex number times its conjugate|0ZB7J071WQ]].

## Explanations

### L1 human

To conjugate a complex number, change the sign of its imaginary part.
For example, the conjugate of $2 + 5 i$ is $2 - 5 i$, and the conjugate of $7$ is $7$.
</document>

<document source="https://math2.tamnd.com/node/0BJVDCF38Y.md" tag="0BJVDCF38Y" kind="concept">
# Square root of a positive semidefinite matrix

## Summary

The square root of a positive semidefinite matrix $A$ is the unique positive semidefinite matrix $B$ with $B^2 = A$.
</document>

<document source="https://math2.tamnd.com/node/0BK6JBK58W.md" tag="0BK6JBK58W" kind="definition">
# Continued fraction

## Definition

A continued fraction is given by a number $b_0$ and a finite or infinite sequence of pairs $(a_1, b_1), (a_2, b_2), \dots$ of numbers.
It stands for the expression
$$b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$
The $a_n$ are the partial numerators and the $b_n$ are the partial denominators.
The $n$-th convergent is the value of the expression cut off after $(a_n, b_n)$. The value of an infinite continued fraction is the limit of its convergents, when this limit exists.

## Remarks

When $a_n = 1$ for all $n$, $b_0$ is an integer and $b_n$ is a positive integer for $n \ge 1$, the continued fraction is a [[simple continued fraction|06XF7BCFFJ]].
For example, $\sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cdots}}$.
Other continued fractions give formulas for $\pi$ and for functions such as $\tan x$.

## Explanations

### L1 human

A continued fraction is a fraction whose denominator contains another fraction, and so on.
Cut it off at some level, and you get an ordinary fraction. These fractions often approximate a number very well.
</document>

<document source="https://math2.tamnd.com/node/0BKDAGMGX9.md" tag="0BKDAGMGX9" kind="statement">
# Hölder's inequality for sums

## Statement

Let $p, q > 1$ be real numbers with $\frac{1}{p} + \frac{1}{q} = 1$, and let $a_1, \dots, a_n, b_1, \dots, b_n$ be real numbers.
Then
$$\sum_{i=1}^{n} |a_i b_i| \le \Big( \sum_{i=1}^{n} |a_i|^p \Big)^{1/p} \Big( \sum_{i=1}^{n} |b_i|^q \Big)^{1/q}.$$

## Remarks

Scale so that both factors on the right are $1$. Then apply [[Young's inequality|07VHQ31EYJ]] to each product $|a_i| |b_i|$ and add.
For $p = q = 2$ it is the Cauchy–Schwarz inequality.
Mathlib states a form as `Real.inner_le_Lp_mul_Lq`.

## Explanations

### L1 human

Hölder's inequality bounds a sum of products by the sizes of the two sequences.
Each size is measured with its own exponent, $p$ for one and $q$ for the other.
</document>

<document source="https://math2.tamnd.com/node/0BKJT6WG96.md" tag="0BKJT6WG96" kind="definition">
# Number of divisors

## Definition

For a positive integer $n$, the number of divisors of $n$ is $d(n) = \sigma_0(n) = \# D(n)$.
Here $\sigma_0$ is the [[divisor function|0YEW47VJ0J]] with $k = 0$, and $D(n)$ is the [[set of divisors|0FCWA16FZE]] of $n$.

## Remarks

Other common notations are $\tau(n)$ and $\sigma_0(n)$.
If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $d(n) = (e_1 + 1) \cdots (e_r + 1)$.
For example, $12 = 2^2 \cdot 3$, so $d(12) = 3 \cdot 2 = 6$.
The number $d(n)$ is odd if and only if $n$ is a perfect square.

## Explanations

### L1 human

The function $d(n)$ counts the divisors of $n$.
A divisor of $p^e q^f$ is $p^a q^b$ with $0 \le a \le e$ and $0 \le b \le f$. So there are $(e + 1)(f + 1)$ choices.
</document>

<document source="https://math2.tamnd.com/node/0BMVBHTWV2.md" tag="0BMVBHTWV2" kind="definition">
# Coordinate vector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
Each $v \in V$ is equal to $a_1 b_1 + \cdots + a_n b_n$ for exactly one $(a_1, \dots, a_n) \in F^n$.
The *coordinate vector* of $v$ in the basis $B$ is
$$[v]_B = (a_1, \dots, a_n)^T \in F^n.$$
The scalars $a_i$ are the *coordinates* of $v$ in $B$.

## Remarks

The coordinates depend on the order of the basis vectors, so $B$ is an ordered basis.
The map $v \mapsto [v]_B$ is a linear isomorphism $V \to F^n$, see [[isomorphism with the coordinate space|059THXBRBB]].
In $\R^2$ with $B = ((1, 1), (1, -1))$, the vector $(3, 1)$ has coordinate vector $(2, 1)^T$.
A [[change of basis matrix|0MKRHZB3WS]] converts coordinates in one basis into coordinates in another basis.

## Explanations

### L1 human

The coordinates of a vector tell us how much of each basis vector we need to build it.
Different bases give different coordinates for the same vector.
After we fix a basis, each vector is just a list of numbers.
</document>

<document source="https://math2.tamnd.com/node/0BMXWXDVJZ.md" tag="0BMXWXDVJZ" kind="statement">
# Fermat's little theorem in the form a^p = a

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer.
Then $a^p \equiv a \pmod p$.

## Remarks

If $p \nmid a$, multiply [[Fermat's little theorem|0M25PRN3WQ]] $a^{p-1} \equiv 1 \pmod p$ by $a$. If $p \mid a$, both sides are $0$ modulo $p$.
Conversely, this form implies the form $a^{p-1} \equiv 1$: cancel $a$, which is allowed when $p \nmid a$.
In the ring $\mathbb{Z}/p\mathbb{Z}$, the form says that $x^p = x$ for each element $x$.

## Explanations

### L1 human

The form $a^p \equiv a$ needs no hypothesis on $a$.
For example, with $p = 3$: $0^3 = 0$, $1^3 = 1$, $2^3 = 8 \equiv 2$, and $4^3 = 64 \equiv 1 \equiv 4 \pmod 3$.
</document>

<document source="https://math2.tamnd.com/node/0BMY28TBSF.md" tag="0BMY28TBSF" kind="concept">
# Uniformly Cauchy sequence of functions

## Summary

A sequence of functions is uniformly Cauchy when its terms become uniformly close to each other. For real-valued functions, this is the same as uniform convergence. The node [[uniformly Cauchy sequence of functions|01XAY17QA3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BMYPJBJH4.md" tag="0BMYPJBJH4" kind="definition">
# Line

## Definition

Let $n \geq 2$ be an integer. A line in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is a set of the form
$$L = \{\, p + t v \;:\; t \in \mathbb{R} \,\},$$
where $p$ is a point of $\mathbb{R}^n$. Here $v$ is a nonzero [[vector|0QN3D80AWP]] of $\mathbb{R}^n$. The vector $v$ is a direction vector of $L$.

## Remarks

For two distinct points $A$ and $B$, exactly one line contains both: the line through $A$ with the direction vector $\overrightarrow{AB}$.
The point $p$ and the vector $v$ are not unique. The same line has each point of $L$ as a base point, and each nonzero multiple of $v$ as a direction vector.
In the plane $\mathbb{R}^2$, the lines are also the sets $\{(x, y) : a x + b y = c\}$ with real numbers $a$, $b$, $c$ and $(a, b) \neq (0, 0)$.
In the language of affine geometry, a line is an [[affine subspace|0PYW26VNT6]] of dimension 1.

## Explanations

### L1 human

Start at a point $p$ and walk forward or backward in the direction $v$, for any amount of time $t$. The points you can reach form the line. For example, with $p = (0, 1)$ and $v = (1, 2)$ you get the line $y = 2x + 1$ in the plane.
</document>

<document source="https://math2.tamnd.com/node/0BN2A6EBYA.md" tag="0BN2A6EBYA" kind="statement">
# Trees are bipartite

## Statement

Let $T$ be a [[tree|0MJ0JS72DZ]].
Then $T$ is [[bipartite|07DVHJ8N0A]].

## Remarks

Proof: fix a vertex $r$. Put a vertex in $A$ if its [[distance|0WPZ5YPAGR]] to $r$ is even, and in $B$ if it is odd. The ends of an edge have distances that differ by $1$.
It also follows from the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]], because a tree has no cycle at all.

## Explanations

### L1 human

Colour the vertices of a tree by levels, from a chosen root: even levels white, odd levels black.
Each edge goes between neighbouring levels, so it joins a white vertex and a black vertex.
</document>

<document source="https://math2.tamnd.com/node/0BNEXSKNRV.md" tag="0BNEXSKNRV" kind="definition">
# Polynomial of an endomorphism

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
Let $p(x) = c_0 + c_1 x + \cdots + c_d x^d$ be a polynomial in $F[x]$.
The *value of $p$ at $T$* is the endomorphism
$$p(T) = c_0 \id_V + c_1 T + c_2 T^2 + \cdots + c_d T^d,$$
where $T^k$ is the composition of $k$ copies of $T$.
For an $n \times n$ matrix $A$, $p(A) = c_0 I_n + c_1 A + \cdots + c_d A^d$ in the same way.

## Remarks

The map $p \mapsto p(T)$ is a homomorphism of algebras $F[x] \to \End_F(V)$: $(p + q)(T) = p(T) + q(T)$ and $(pq)(T) = p(T) q(T)$.
In particular, any two polynomials in $T$ commute.
If $T(v) = \lambda v$, then $p(T)(v) = p(\lambda) v$.

## Explanations

### L1 human

We can put a matrix into a polynomial, because matrices can be added, scaled and multiplied.
The constant term becomes a multiple of the identity matrix.
For example, if $p(x) = x^2 - 3x + 2$, then $p(A) = A^2 - 3A + 2I$.
</document>

<document source="https://math2.tamnd.com/node/0BNPC9850Y.md" tag="0BNPC9850Y" kind="example">
# Trivial group

## Example

The trivial group is the [[group|G5S1M9]] $\{e\}$ with one element $e$ and the operation $e \cdot e = e$.
The element $e$ is the identity element, and it is its own inverse.
Each group with exactly one element is isomorphic to this group.

## Remarks

Each group $G$ contains a trivial subgroup $\{e\}$, where $e$ is the identity of $G$.
In additive notation the trivial group is written $0$ or $\{0\}$.
The trivial group is [[abelian|082GHNPJ53]], and its order is $1$.

## Explanations

### L1 human

The trivial group is the smallest possible group: it has only the identity element.
It satisfies all the group axioms in a simple way.
It is often an edge case in theorems, for example when a theorem asks for a group that is not trivial.
</document>

<document source="https://math2.tamnd.com/node/0BPYRMHJ48.md" tag="0BPYRMHJ48" kind="definition">
# Universal set

## Definition

In a given context, fix a [[set|0SJ3RX9TXA]] $U$ such that all the sets under discussion are [[subsets|0FE8Q5S42Y]] of $U$.
The set $U$ is then the universal set, or the universe of discourse, of the context.

## Remarks

In elementary number theory, $U$ can be the set $\mathbb{Z}$ of integers.
The [[complement|0B1FSS1VZW]] of a set is taken relative to the universal set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], no set contains all sets. This is a consequence of [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

The universal set is the "whole world" of a problem.
In a problem about real numbers, it is $\mathbb{R}$.
In a [[Venn diagram|0KCT6DPXJK]], it is the rectangle that contains all the circles.
</document>

<document source="https://math2.tamnd.com/node/0BQAA2BK3Q.md" tag="0BQAA2BK3Q" kind="concept">
# Eigenvalue

## Summary

An eigenvalue of a linear map from a space to itself is a scalar $\lambda$ with a special property.
The map sends some nonzero vector to $\lambda$ times that vector.
</document>

<document source="https://math2.tamnd.com/node/0BQMWCNS8S.md" tag="0BQMWCNS8S" kind="definition">
# Permutation

## Definition

Let $X$ be a set.
A permutation of $X$ is a bijection $\sigma : X \to X$.

The permutations of $X$ form a [[group|G5S1M9]] under composition, with the identity map as identity element.
This group is the symmetric group on $X$, written $\operatorname{Sym}(X)$.
For $X = \{1, \dots, n\}$ it is written $S_n$.

## Remarks

For a finite set $X = \{1, \dots, n\}$, a permutation is often written in one-line notation as the sequence $\sigma(1) \sigma(2) \cdots \sigma(n)$. In this form it is an arrangement of the elements in a row.
The [[number of permutations|0CEBCS0E0E]] of an $n$-element set is $n!$.
A [[derangement|0TH61JR04G]] is a permutation with no fixed point.

## Explanations

### L1 human

A permutation is a rearrangement of the elements of a set.
For $\{1, 2, 3\}$, the map that sends $1 \mapsto 2$, $2 \mapsto 3$ and $3 \mapsto 1$ is a permutation. In one-line notation it is $231$.
</document>

<document source="https://math2.tamnd.com/node/0BR3FY5Q8M.md" tag="0BR3FY5Q8M" kind="concept">
# Quotient space

## Summary

The quotient space of a vector space by a subspace is the set of cosets of the subspace, with the operations inherited from the space.
It identifies two vectors when their difference lies in the subspace.
</document>

<document source="https://math2.tamnd.com/node/0BRBYRK6PA.md" tag="0BRBYRK6PA" kind="statement">
# Side-side-side congruence

## Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
Then $ABC$ and $A'B'C'$ are [[congruent|093EDENVA1]] if and only if
$$d(A, B) = d(A', B'), \quad d(B, C) = d(B', C') \quad \text{and} \quad d(C, A) = d(C', A').$$

## Remarks

Other criteria for congruence are side-angle-side, angle-side-angle and angle-angle-side.
Side-side-angle is not a criterion: two sides and an angle that is not between them can fit two different triangles.
Angle-angle-angle is not a criterion either. Triangles with equal angles are [[similar|0YF6QXXWB5]], but they can have different sizes.


## Explanations

### L1 human

If you know the three side lengths of a triangle, you know the whole triangle. Three sticks of fixed lengths make one rigid shape, up to a turn or a flip. This is why triangles make frames stiff.
</document>

<document source="https://math2.tamnd.com/node/0BRG85WXYC.md" tag="0BRG85WXYC" kind="statement">
# Goursat's theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $Q = \{x + iy : a_1 \le x \le b_1, \ a_2 \le y \le b_2\}$ be a closed rectangle with $Q \subseteq U$, where $a_1 < b_1$ and $a_2 < b_2$ are real numbers.
Let $\partial Q$ be the boundary of $Q$, traveled once counterclockwise. Then the [[contour integral|00FWNP9TXY]] of $f$ along $\partial Q$ is
$$\oint_{\partial Q} f(z) \, dz = 0.$$

## Remarks

The theorem does not assume that $f'$ is continuous. This is the difference with the older proof by Green's theorem.
The same statement holds for each closed triangle $T \subseteq U$ and its boundary.
Proof idea: divide $Q$ into four equal rectangles and keep one where the integral is largest. Repeat, and use the complex derivative at the limit point.
Goursat's theorem is the first step to [[primitives on a disc|02AC7JR6G6]] and to [[Cauchy's integral theorem|01MPZXYVYF]].

## Explanations

### L1 human

If $f$ is holomorphic on a rectangle and its inside, then the integral of $f$ around the edge of the rectangle is zero.
The surprise is that this needs only the existence of the complex derivative, and nothing about its continuity.
</document>

<document source="https://math2.tamnd.com/node/0BRHKKP333.md" tag="0BRHKKP333" kind="definition">
# Inverse element

## Definition

Let $S$ be a set with a [[binary operation|0M418FE7BB]] $\ast$ and an [[identity element|07WYQPB8E3]] $e$.
Let $a, b \in S$.
The element $b$ is a left inverse of $a$ when $b \ast a = e$.
The element $b$ is a right inverse of $a$ when $a \ast b = e$.
The element $b$ is an inverse of $a$ (or two-sided inverse) when it is both a left inverse and a right inverse of $a$.
An element that has an inverse is called invertible.

## Remarks

For addition on $\mathbb{Z}$, the inverse of $a$ is $-a$.
For multiplication on $\mathbb{Q}$, each nonzero $a$ has the inverse $1/a$, and $0$ has no inverse.
When $\ast$ is [[associative|0K8F7PVDMX]], an element has at most one inverse.

## Explanations

### L1 human

An inverse of an element is a second element that cancels the first one.
If you add $5$ and then add $-5$, you come back to $0$, the identity for addition.
For multiplication, the inverse of $2$ is $\tfrac{1}{2}$, because $2 \cdot \tfrac{1}{2} = 1$.
</document>

<document source="https://math2.tamnd.com/node/0BRXFNQ5JV.md" tag="0BRXFNQ5JV" kind="statement">
# Polar form

## Statement

Let $z$ be a complex number with $z \neq 0$, let $r = |z|$ be its [[modulus|0KW2X0TFBX]] and let $\theta = \operatorname{Arg} z$ be its [[principal argument|0SQSQP748R]].
Then
$$z = r (\cos \theta + i \sin \theta) = r e^{i \theta}.$$

## Remarks

The second equality is [[Euler's formula|04R3CY41VJ]].
The pair $(r, \theta)$ gives the polar coordinates of the point $z$ in the [[complex plane|0CTE7A31XQ]].
Polar form makes products easy: $r e^{i \theta} \cdot s e^{i \varphi} = r s \, e^{i (\theta + \varphi)}$.

## Explanations

### L1 human

You can describe a point of the plane by its distance from $0$ and its direction.
For example, $1 + i$ has distance $\sqrt{2}$ and direction $\pi/4$, so $1 + i = \sqrt{2} \, e^{i \pi/4}$.
</document>

<document source="https://math2.tamnd.com/node/0BS0NJPKVE.md" tag="0BS0NJPKVE" kind="statement">
# Quotient rule

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$, with $g(a) \ne 0$.
Then $f/g$ is differentiable at $a$, and
$$\left( \frac{f}{g} \right)'(a) = \frac{f'(a) g(a) - f(a) g'(a)}{g(a)^2}.$$

## Remarks

Write $f/g = f \cdot (1/g)$, and use the [[product rule|0V9A43H7WV]] and the [[reciprocal rule|0QD4E5JYPT]].
For example, it gives $(\tan x)' = 1/\cos^2 x$.
Mathlib states it as `HasDerivAt.div`.

## Explanations

### L1 human

The rule gives the slope of a quotient from the values and slopes of the numerator and the denominator.
A common memory aid is "low d-high minus high d-low, over low squared".
</document>

<document source="https://math2.tamnd.com/node/0BS3RMX8HM.md" tag="0BS3RMX8HM" kind="statement">
# Multiplication rule

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A_1, \dots, A_n \in \mathcal{F}$ with $n \ge 2$ and $P(A_1 \cap \dots \cap A_{n-1}) > 0$.
Then
$$P(A_1 \cap \dots \cap A_n) = P(A_1) \, P(A_2 \mid A_1) \, P(A_3 \mid A_1 \cap A_2) \cdots P(A_n \mid A_1 \cap \dots \cap A_{n-1}).$$
Here $P(\cdot \mid \cdot)$ is the [[conditional probability|0FJB06AD2P]].

## Remarks

The case $n = 2$ is $P(A \cap B) = P(A) \, P(B \mid A)$ for $P(A) > 0$. It is the definition of $P(B \mid A)$ multiplied by $P(A)$.
The hypothesis makes each conditional probability in the formula defined, because each conditioning event contains $A_1 \cap \dots \cap A_{n-1}$.
Example: draw two cards from a deck without replacement. The probability of two aces is $\frac{4}{52} \cdot \frac{3}{51} = \frac{1}{221}$.

## Explanations

### L1 human

To find the chance that several things all happen, take them one at a time.
Multiply the chance of the first by the chance of the second given the first.
Then multiply by the chance of the third given the first two, and so on.
</document>

<document source="https://math2.tamnd.com/node/0BSCJA5G53.md" tag="0BSCJA5G53" kind="example">
# Additive group of integers

## Example

Let $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ be the set of integers, with the usual addition $+$.
Then $(\mathbb{Z}, +)$ is an [[abelian group|082GHNPJ53]].
Its identity element is $0$, and the inverse of $n$ is $-n$.

## Remarks

The group $(\mathbb{Z}, +)$ is infinite, and each element $n$ is the $n$-fold sum of $1$ or of $-1$.
Thus the element $1$ generates the group.
The integers with multiplication are not a group, because $2$ has no multiplicative inverse in $\mathbb{Z}$.

## Explanations

### L1 human

The integers with addition are the first example of a group for most students.
Adding is associative, $0$ changes nothing, and $-n$ undoes $n$.
Each element of the group is a sum of copies of $1$ or of $-1$.
</document>

<document source="https://math2.tamnd.com/node/0BSJ7RQ9BW.md" tag="0BSJ7RQ9BW" kind="concept">
# Set

## Summary

A set is a collection of distinct objects, called its elements.
A set is determined by its elements, and the order and repetition of the elements do not matter.
</document>

<document source="https://math2.tamnd.com/node/0BSJSX7ME2.md" tag="0BSJSX7ME2" kind="concept">
# Dense set

## Summary

A dense subset of a topological space is a subset whose closure is the whole space.
Each nonempty open set meets a dense set.
</document>

<document source="https://math2.tamnd.com/node/0BSJYCG7NM.md" tag="0BSJYCG7NM" kind="counterexample">
# Z[√-5] is not a UFD

## Counterexample

The integral domain $\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} : a, b \in \mathbb{Z}\}$ is not a [[unique factorisation domain|06YMXJMXYS]].
In this ring,
$$6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}).$$
Let $N(a + b\sqrt{-5}) = a^2 + 5b^2$. Then $N$ is multiplicative, and $N(2) = 4$, $N(3) = 9$, $N(1 \pm \sqrt{-5}) = 6$.
No element has norm $2$ or $3$, because $a^2 + 5b^2 \notin \{2, 3\}$ for integers $a, b$. So the four factors are [[irreducible|052SDEPZH7]].
The only units are $\pm 1$, so $2$ is not an [[associate|053E86NWVK]] of $1 + \sqrt{-5}$ or of $1 - \sqrt{-5}$. The two factorisations are different.

## Remarks

The element $2$ is [[irreducible but not prime|0CHE3J7QMX]] in this ring.
Every nonzero nonunit of $\mathbb{Z}[\sqrt{-5}]$ is still a product of irreducibles. Only the uniqueness fails.
Unique factorisation holds again for ideals: $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain.

## Explanations

### L1 human

The number $6$ has two essentially different factorisations into irreducible elements in this ring.
The norm shows that none of the four factors can be split further.
So unique factorisation, which we take for granted in the integers, can fail.
</document>

<document source="https://math2.tamnd.com/node/0BSQDZWF34.md" tag="0BSQDZWF34" kind="statement">
# Riemann's theorem on removable singularities

## Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$.
Assume that $f$ is bounded near $c$: there are $M \ge 0$ and $\delta \in (0, r)$ with $|f(z)| \le M$ for $0 < |z - c| < \delta$.
Then $c$ is a [[removable singularity|0P18JC72RE]] of $f$. Moreover, the limit $\lim_{z \to c} f(z)$ exists, and the extension of $f$ by this value at $c$ is holomorphic on $|z - c| < r$.

## Remarks

The following are equivalent: $c$ is removable; $f$ has a continuous extension to $c$; $f$ is bounded near $c$; $\lim_{z \to c} (z - c) f(z) = 0$.
Proof idea: $h(z) = (z - c)^2 f(z)$, with $h(c) = 0$, is complex differentiable at $c$ with $h'(c) = 0$. So $h(z) = \sum_{n \ge 2} a_n (z - c)^n$, and $f(z) = \sum_{n \ge 2} a_n (z - c)^{n-2}$.
For example, $\frac{\sin z}{z}$ is bounded near $0$, so $0$ is removable.

## Explanations

### L1 human

If a holomorphic function stays bounded near a single bad point, then the point is not really bad.
You can give the function a value there, and the function becomes holomorphic at that point.
</document>

<document source="https://math2.tamnd.com/node/0BSW4T7THS.md" tag="0BSW4T7THS" kind="concept">
# Polar coordinates

## Summary

Polar coordinates describe a point of the plane by two numbers. They are its distance $r$ from the origin and the angle $\theta$ from the positive $x$-axis to the ray through the point.
They simplify problems with circular symmetry, such as integrals over discs.

## Remarks

The polar form of a complex number is a related idea: see [[polar form|0BRXFNQ5JV]].
</document>

<document source="https://math2.tamnd.com/node/0BSXRER81B.md" tag="0BSXRER81B" kind="problem">
# Squaring the circle

## Problem

Given a circle, construct with [[straightedge and compass|0DAECMZVRT]] alone a square with the same area.

## Remarks

For a circle of radius $1$, the square has side $\sqrt{\pi}$.
So the problem asks whether $\sqrt{\pi}$ is a [[constructible number|0FMN2MN7TM]].
The answer is no: [[squaring the circle is impossible|0X7CVDE3BG]].

## History

Anaxagoras is the first known Greek to study the problem.
Hippocrates of Chios squared some figures bounded by circular arcs, the lunes of Hippocrates.
In 1882, Ferdinand von Lindemann proved that $\pi$ is transcendental, and so that the construction is impossible.

## Explanations

### L1 human

The problem asks for a square with the same area as a given circle, with only an unmarked ruler and a compass.
The phrase "squaring the circle" now means to attempt something impossible.
</document>

<document source="https://math2.tamnd.com/node/0BT9BYWVRY.md" tag="0BT9BYWVRY" kind="concept">
# Minor

## Summary

A minor of a matrix is the determinant of a square submatrix.
We get such a submatrix when we choose some rows and the same number of columns.
</document>

<document source="https://math2.tamnd.com/node/0BTMQ04556.md" tag="0BTMQ04556" kind="definition">
# Supremum

## Definition

Let $(P, \le)$ be a partially ordered set and let $S$ be a subset of $P$.
An element $b$ of $P$ is a supremum, or least upper bound, of $S$ when these two conditions hold:

1. $b$ is an [[upper bound|0B5H2V4JW8]] of $S$;
2. if $M$ is an upper bound of $S$, then $b \le M$.

The notation is $b = \sup S$.

## Remarks

A set has at most one supremum, see [[uniqueness of the supremum|04ETX30HT2]].
In $\mathbb{R}$, $\sup [0, 1) = 1$ and $\sup [0, 1] = 1$. The first set does not contain its supremum.
In $\mathbb{Q}$, the set $\{q \in \mathbb{Q} : q^2 < 2\}$ is bounded above but has no supremum.

## Explanations

### L1 human

The supremum is the lowest possible ceiling for a set.
For the set $\{0.9, 0.99, 0.999, \dots\}$, every number at least $1$ is a ceiling, and $1$ is the lowest of these, so the supremum is $1$.
</document>

<document source="https://math2.tamnd.com/node/0BVHZXWKCD.md" tag="0BVHZXWKCD" kind="definition">
# Borel sigma-algebra

## Definition

Let $X$ be a topological space.
The Borel σ-algebra $\mathcal{B}(X)$ is the smallest [[σ-algebra|04WSW8T3SY]] on $X$ that contains each open subset of $X$. It is the intersection of all σ-algebras on $X$ that contain the open sets.
A Borel set of $X$ is an element of $\mathcal{B}(X)$.

## Remarks

On the [[real line|08J2M1JYAG]], $\mathcal{B}(\mathbb{R})$ is also the σ-algebra generated by the open [[intervals|0TK8Y3J5RW]] $(a, b)$. The half-lines $(-\infty, b]$ with $b \in \mathbb{R}$ also generate it.
Each open set, each closed set, each countable set and each interval of $\mathbb{R}$ is a Borel set.
Some subsets of $\mathbb{R}$ are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.

## Explanations

### L1 human

We want to give a probability to intervals such as "the waiting time is between $2$ and $5$ minutes".
The Borel sets are all the sets that we can build from intervals with complements and countable unions, again and again. So they are the events that come from intervals.
</document>

<document source="https://math2.tamnd.com/node/0BVT6VSB3S.md" tag="0BVT6VSB3S" kind="definition">
# Trajectory

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz. Let $\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.
Let $x_0 \in U$, and let $I(x_0)$ be the maximal interval of existence of the solution with $x(0) = x_0$.
The trajectory, or orbit, of $x_0$ is the set
$$\mathcal{O}(x_0) = \{\varphi_t(x_0) : t \in I(x_0)\} \subseteq U.$$
The positive half-trajectory is $\mathcal{O}^+(x_0) = \{\varphi_t(x_0) : t \in I(x_0),\ t \ge 0\}$, and the negative half-trajectory uses $t \le 0$.

## Remarks

Two trajectories are either equal or disjoint. So the trajectories form a partition of $U$.
A trajectory is a single point exactly when $x_0$ is an [[equilibrium point|0G4X36AE0X]]. It is a closed curve when the solution is [[periodic|01DSNP0B4R]] and not constant.
Some books use the word trajectory for the solution curve $t \mapsto \varphi_t(x_0)$ itself, and the word orbit for its image.

## Explanations

### L1 human

The trajectory is the path that the state follows in the phase space.
It forgets the time: it records where the state goes, not when.
</document>

<document source="https://math2.tamnd.com/node/0BVTDM872V.md" tag="0BVTDM872V" kind="statement">
# Pythagorean theorem in an inner product space

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
If $x, y \in V$ are [[orthogonal|0Y2X7P2WB7]], then
$$\norm{x + y}^2 = \norm{x}^2 + \norm{y}^2.$$

## Remarks

The proof: $\norm{x + y}^2 = \norm{x}^2 + 2 \operatorname{Re} \langle x, y \rangle + \norm{y}^2$.
For pairwise orthogonal $x_1, \dots, x_k$, $\norm{x_1 + \cdots + x_k}^2 = \norm{x_1}^2 + \cdots + \norm{x_k}^2$.
In a real space, the converse holds; in a complex space, the equality only gives $\operatorname{Re} \langle x, y \rangle = 0$.

## Explanations

### L1 human

This is the theorem of Pythagoras for vectors.
Two orthogonal vectors are the legs of a right triangle, and their sum is the hypotenuse.
It holds in every inner product space, even in spaces of functions.
</document>

<document source="https://math2.tamnd.com/node/0BVXADM10P.md" tag="0BVXADM10P" kind="definition">
# Matrix addition

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ [[matrices|0VD9ZSWWJG]] over $R$.
The *sum* $A + B$ is the $m \times n$ matrix with entries
$$(A + B)_{ij} = a_{ij} + b_{ij}$$
for $1 \le i \le m$ and $1 \le j \le n$.

## Remarks

The sum is defined only for two matrices of the same size.
Matrix addition is associative and commutative, the [[zero matrix|0ZTKJJDGR7]] is its identity element, and $-A = (-a_{ij})$ is the additive inverse of $A$.
Thus $M_{m \times n}(R)$ is an abelian group under addition.
For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix}$.

## Explanations

### L1 human

To add two matrices of the same size, add the numbers in the same position.
The result has the same size as the two matrices.
Matrix addition has the same rules as the addition of numbers.
</document>

<document source="https://math2.tamnd.com/node/0BW9N5CDSB.md" tag="0BW9N5CDSB" kind="definition">
# Antiderivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
A function $F \colon I \to \mathbb{R}$ is an antiderivative of $f$ on $I$ if $F$ is differentiable on $I$ and $F'(x) = f(x)$ for all $x \in I$.

## Remarks

If $F$ is an antiderivative of $f$, then so is $F + C$ for each constant $C$.
On an interval, these are all the antiderivatives, because a function with [[zero derivative|003QDZ4T16]] is constant.
For example, $x^3/3$ is an antiderivative of $x^2$ on $\mathbb{R}$.
A function with a jump has no antiderivative on an interval, by [[Darboux's theorem|0Q5A47J48D]].

## Explanations

### L1 human

An antiderivative undoes differentiation.
We look for a function whose derivative is the given function.
</document>

<document source="https://math2.tamnd.com/node/0BWDJGQYST.md" tag="0BWDJGQYST" kind="definition">
# Characteristic polynomial of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \dots, c_{d-1}$.
The characteristic polynomial of $E$ is the polynomial
$$\chi_E(X) = X^d - \sum_{i=0}^{d-1} c_i X^i \in R[X].$$

## Remarks

For the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$, the characteristic polynomial is $X^2 - X - 1$. Its roots are the [[golden ratio|0SC8T7548D]] $\varphi$ and $1 - \varphi$.
A geometric sequence $q^n$ is a solution if and only if $q$ is a root of $\chi_E$. The node [[geometric solutions|096SD4ERQH]] states this.

## Explanations

### L1 human

Replace $u_{n+i}$ by $X^i$ in the recurrence and move everything to one side.
This gives the characteristic polynomial, and its roots tell you the geometric sequences that solve the recurrence.
</document>

<document source="https://math2.tamnd.com/node/0BWGTN92FG.md" tag="0BWGTN92FG" kind="concept">
# Bias of an estimator

## Summary

The bias of an estimator is the difference between its expected value and the true value of the quantity that it estimates. An estimator with zero bias is unbiased.
The node [[bias of an estimator|01BD2TFFP4]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BWGW68B66.md" tag="0BWGW68B66" kind="statement">
# Uniform limits commute with integration

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be Riemann integrable for each $n$.
If $f_n \to f$ [[uniformly|08FPABDQ4C]] on $[a, b]$, then $f$ is Riemann integrable on $[a, b]$, and
$$\lim_{n \to \infty} \int_a^b f_n(x) \, dx = \int_a^b f(x) \, dx.$$

## Remarks

Let $\varepsilon_n = \sup |f_n - f|$. Then $f_n - \varepsilon_n \le f \le f_n + \varepsilon_n$. So the upper and lower sums of $f$ differ by at most $2 \varepsilon_n (b - a)$ more than those of $f_n$. This gives integrability, by [[Riemann's criterion|017DH8WXTE]].
By [[monotonicity|0MBWQKM8QR]], $\big| \int_a^b f_n - \int_a^b f \big| \le \varepsilon_n (b - a) \to 0$.
Pointwise convergence is not enough. On $[0, 1]$, the functions $f_n(x) = n^2 x (1 - x^2)^n$ tend to $0$ pointwise, but $\int_0^1 f_n = \frac{n^2}{2n + 2} \to \infty$.

## Explanations

### L1 human

If $f_n$ is uniformly close to $f$, the areas under their graphs are close too.
So we may take the limit inside the integral.
</document>

<document source="https://math2.tamnd.com/node/0BWJTRE3X3.md" tag="0BWJTRE3X3" kind="concept">
# Box topology

## Summary

The box topology on a product of topological spaces has as basic open sets all products of open sets.
For infinite products, it is finer than the product topology and has worse properties.
</document>

<document source="https://math2.tamnd.com/node/0BWXV7GADW.md" tag="0BWXV7GADW" kind="statement">
# Power series converge uniformly on smaller discs inside the radius of convergence

## Statement

Let $\sum_{n=0}^{\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R > 0$.
Then for every $r$ with $0 < r < R$, the series converges absolutely and uniformly on $[c - r, c + r]$.

## Remarks

Choose $s$ with $r < s < R$. The series converges at $c + s$, so $|a_n| s^n \le C$ for some $C$. The [[M-test|0NX5NWAFC9]] with $M_n = C (r/s)^n$ applies.
So the sum is continuous on $(c - R, c + R)$.
The convergence need not be uniform on the whole interval: $\sum x^n$ is not uniformly convergent on $(-1, 1)$. Mathlib states the result as `HasFPowerSeriesOnBall.tendstoUniformlyOn`.

## Explanations

### L1 human

Inside the interval of convergence, a power series converges uniformly once we stay away from the endpoints.
Near the endpoints, the convergence can get slower and slower.
</document>

<document source="https://math2.tamnd.com/node/0BXA4Q0JHW.md" tag="0BXA4Q0JHW" kind="statement">
# Ring of integers

## Statement

The [[integers|0C4JS9RF5M]] $\mathbb{Z}$, with their addition $+$ and multiplication $\cdot$, form a commutative ring with identity $1$.
That is, for all integers $a$, $b$ and $c$:

1. $(a + b) + c = a + (b + c)$ and $a + b = b + a$;
2. $a + 0 = a$, and there is an integer $-a$ with $a + (-a) = 0$;
3. $(a b) c = a (b c)$ and $a b = b a$;
4. $a \cdot 1 = a$;
5. $a (b + c) = a b + a c$.

## Remarks

The ring $\mathbb{Z}$ is not a field: for example, $2$ has no multiplicative inverse in $\mathbb{Z}$.
The ring $\mathbb{Z}$ has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$. Thus it is an integral domain.
Here "ring of integers" means the ring $\mathbb{Z}$. In algebraic number theory, the same words have a more general meaning.

## Explanations

### L1 human

This statement lists the usual rules of computation with whole numbers.
You can add, subtract and multiply integers, and the result is always an integer.
Division is different: $1 / 2$ is not an integer.
</document>

<document source="https://math2.tamnd.com/node/0BXBADR94C.md" tag="0BXBADR94C" kind="statement">
# Implicit function theorem

## Statement

Let $\Omega \subseteq \mathbb{R}^n \times \mathbb{R}^m$ be open, and let $F \colon \Omega \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]], written $F(x, y)$ with $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$.
Let $(a, b) \in \Omega$ with $F(a, b) = 0$.
Suppose that the $m \times m$ matrix $D_y F(a, b) = \left( \partial F_i / \partial y_j (a, b) \right)_{i, j}$ is invertible.
Then there are open sets $U \subseteq \mathbb{R}^n$ and $V \subseteq \mathbb{R}^m$ with $a \in U$ and $b \in V$, and a unique function $g \colon U \to V$ with the following property. For $(x, y) \in U \times V$, $F(x, y) = 0$ if and only if $y = g(x)$.
The function $g$ is of class $C^1$, and $Dg(x) = -\big( D_y F(x, g(x)) \big)^{-1} D_x F(x, g(x))$ for each $x \in U$.

## Remarks

Here $D_x F$ is the $m \times n$ matrix of partial derivatives with respect to $x$. Together, $D_x F$ and $D_y F$ form the [[Jacobian matrix|0YF0VXKXGB]] of $F$.
For $F(x, y) = x^2 + y^2 - 1$ and $(a, b)$ with $b \ne 0$, the solution near $(a, b)$ is $y = \pm \sqrt{1 - x^2}$. At $(\pm 1, 0)$, $\partial F / \partial y = 0$, and no such function exists.
One proof applies the [[inverse function theorem|0K9HSJFCMJ]] to $(x, y) \mapsto (x, F(x, y))$. If $F$ is of class $C^k$, then $g$ is of class $C^k$.
The theorem has a long history. Cauchy gave the first rigorous form, and Dini extended the real version to any number of variables.

## Explanations

### L1 human

An equation $F(x, y) = 0$ often defines $y$ as a function of $x$, even when you cannot solve it with a formula.
The theorem says this works near a solution, provided $F$ really depends on $y$ there, in the sense that $D_y F$ is invertible.
</document>

<document source="https://math2.tamnd.com/node/0BXE0S1XER.md" tag="0BXE0S1XER" kind="definition">
# Having a gradient

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, let $a \in U$, and let $g \in \mathbb{R}^n$.
The function $f$ has the gradient $g$ at $a$ when
$$\lim_{h \to 0} \frac{f(a + h) - f(a) - g \cdot h}{\|h\|} = 0,$$
where the limit is over $h \ne 0$ with $a + h \in U$.
Equivalently, $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $Df(a)(h) = g \cdot h$ for each $h \in \mathbb{R}^n$.

## Remarks

This definition states a property of $f$, $a$ and $g$. When it holds, the vector $g$ is unique, and it is the [[gradient|0NS2QYVC4Y]] $\nabla f(a)$.
The same definition works in each real inner product space, with the inner product in place of the dot product.

## Explanations

### L1 human

The function has the gradient $g$ at $a$ when the plane through $(a, f(a))$ with slopes given by $g$ hugs the graph of $f$ near $a$.
The error of this approximation must be small compared with the distance from $a$.
</document>

<document source="https://math2.tamnd.com/node/0BXEZV83GY.md" tag="0BXEZV83GY" kind="concept">
# Minimal element

## Summary

A minimal element of a partially ordered set is an element with no strictly smaller element.
</document>

<document source="https://math2.tamnd.com/node/0BXNDZFQME.md" tag="0BXNDZFQME" kind="example">
# Cantor set

## Example

Let $C_0 = [0, 1]$. For $n \ge 0$, let $C_{n+1}$ be the set that we get from $C_n$ when we remove the open middle third of each of its closed intervals. Thus $C_1 = [0, \tfrac{1}{3}] \cup [\tfrac{2}{3}, 1]$.
The Cantor set is $C = \bigcap_{n \ge 0} C_n$, with the subspace topology of $\mathbb{R}$.

## Remarks

Equivalently, $C$ is the set of $x \in [0, 1]$ that have a base-3 expansion $x = \sum_{n \ge 1} a_n 3^{-n}$ with all $a_n \in \{0, 2\}$.
The set $C$ is [[closed|01VT24J1SD]] and bounded, so it is [[compact|07PFYV65MB]]. It is [[totally disconnected|01CYW8S04D]] and has no isolated points.
It is uncountable, and it is [[homeomorphic to the space of binary sequences|0RQ8DHHR8V]]. Its Lebesgue measure is $0$, because $C_n$ has total length $(2/3)^n$.
The Cantor set is [[nowhere dense|0KSM23SN3H]] in $\mathbb{R}$.

## Explanations

### L1 human

Start with $[0, 1]$, remove the middle third, then remove the middle third of each piece that is left, and go on forever.
What remains is a set with no length, but with as many points as $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/0BY52JW6NK.md" tag="0BY52JW6NK" kind="concept">
# Complete metric space

## Summary

A complete metric space is a metric space in which each Cauchy sequence converges to a point of the space.
The real numbers are complete, and the rational numbers are not.
</document>

<document source="https://math2.tamnd.com/node/0BY5FTHEEE.md" tag="0BY5FTHEEE" kind="statement">
# Slopes of a convex function are increasing

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be [[convex|0VV0HSW1ZS]].
Then for all $x < y < z$ in $I$,
$$\frac{f(y) - f(x)}{y - x} \le \frac{f(z) - f(x)}{z - x} \le \frac{f(z) - f(y)}{z - y}.$$

## Remarks

Write $y = t x + (1 - t) z$ with $t = (z - y)/(z - x)$, and apply the definition of convexity.
The converse holds: a function with this property is convex.
The first and the last quotient give the adjacent form `ConvexOn.slope_mono_adjacent` of Mathlib.

## Explanations

### L1 human

For a convex function, chords further to the right are steeper.
So the slopes of the graph increase from left to right.
</document>

<document source="https://math2.tamnd.com/node/0BY7GN3449.md" tag="0BY7GN3449" kind="statement">
# Generalized pigeonhole principle

## Statement

Let $A$ and $B$ be finite sets, let $f : A \to B$ be a function, and let $n$ be a natural number.
Assume that $|B| \cdot n < |A|$.
Then there is an element $b \in B$ whose fibre $f^{-1}(b) = \{a \in A : f(a) = b\}$ has more than $n$ elements.

## Remarks

For $n = 1$ this is the [[pigeonhole principle|0HGVCWQGY3]].
An equivalent form: if $N$ objects go into $k$ boxes, some box has at least $\lceil N / k \rceil$ objects.
Example: among $25$ people, at least $3$ share a birth month, because $12 \cdot 2 < 25$.

## Explanations

### L1 human

If every box held at most $n$ objects, the boxes would hold at most $n$ times the number of boxes.
So if there are more objects than that, some box must hold more than $n$.
</document>

<document source="https://math2.tamnd.com/node/0BYTQ4X1HC.md" tag="0BYTQ4X1HC" kind="concept">
# Least squares

## Summary

Let $A$ be a real $m \times n$ matrix and let $b \in \R^m$.
A *least squares solution* of $Ax = b$ is a vector $\hat x \in \R^n$ that minimises $\norm{Ax - b}$, for the Euclidean norm.
The vectors $\hat x$ that do this are exactly the solutions of the *normal equations*
$$A^T A \hat x = A^T b.$$
If the columns of $A$ are linearly independent, then $A^T A$ is invertible, and $\hat x = (A^T A)^{-1} A^T b$ is unique.

## Remarks

The vector $A \hat x$ is the [[orthogonal projection|0B8KZRT28M]] of $b$ onto the column space of $A$, see [[the closest point property|0MCDDSCB9Z]].
The normal equations say that the residual $b - A\hat x$ is orthogonal to each column of $A$.
Fitting a line $y = c_0 + c_1 t$ to data points $(t_i, y_i)$ is a least squares problem.
In practice, the [[QR decomposition|0ERWG4NXQ9]] gives a more stable method than the normal equations.

## Explanations

### L1 human

A system $Ax = b$ with more equations than unknowns usually has no solution.
The least squares method finds the $x$ that makes $Ax$ as close as possible to $b$.
It is the standard way to fit a model to noisy data.
</document>

<document source="https://math2.tamnd.com/node/0BYWRWFT4F.md" tag="0BYWRWFT4F" kind="concept">
# Antiderivative

## Summary

An antiderivative of a function is a function whose derivative is the given function. On an interval, two antiderivatives of the same function differ by a constant. The node [[antiderivative|0BW9N5CDSB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BYX1D7C2E.md" tag="0BYX1D7C2E" kind="concept">
# Clique

## Summary

A clique in a graph is a set of vertices that are pairwise adjacent. Cliques are the complete subgraphs, and they appear in colouring, in Ramsey theory and in extremal graph theory. The node [[clique|0G52BPNPKG]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0BYZTFM0DJ.md" tag="0BYZTFM0DJ" kind="statement">
# Zero derivative on a connected open set implies constant

## Statement

Let $U \subseteq \mathbb{R}^n$ be a connected open set and let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$.
Suppose that $Df(x) = 0$ for each $x \in U$.
Then $f$ is constant on $U$.

## Remarks

The proof fixes $a \in U$ and shows that the set $S = \{x \in U : f(x) = f(a)\}$ is open and closed in $U$. It is open by the [[mean value inequality|02CYB184RC]] with $M = 0$ on small balls, and closed because $f$ is continuous. As $U$ is connected and $a \in S$, $S = U$.
Connectedness is needed: on $U = (0, 1) \cup (2, 3)$, the function that is $0$ on the first interval and $1$ on the second has derivative $0$ but is not constant.
For real-valued $f$, the condition $Df = 0$ means that all partial derivatives of $f$ vanish on $U$.

## Explanations

### L1 human

Suppose that $f$ does not change in any direction at any point, and that the domain is in one piece. Then $f$ cannot change at all.
On a domain in two pieces, $f$ can take a different constant value on each piece.
</document>

<document source="https://math2.tamnd.com/node/0BZ81AJNWF.md" tag="0BZ81AJNWF" kind="definition">
# Characteristic polynomial of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *characteristic polynomial* of $T$ is
$$p_T(t) = p_{[T]_B}(t) = \det(t I_n - [T]_B),$$
the [[characteristic polynomial|0VSSPAFM2C]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$, where $n = \dim V$.

## Remarks

The polynomial does not depend on $B$, see [[the invariance of the characteristic polynomial|0TYMASG33Y]].
Its degree is $\dim V$.
For the identity map of $V$, it is $(t - 1)^n$.

## Explanations

### L1 human

A linear map has a matrix in each basis.
All these matrices have the same characteristic polynomial.
So we can call it the characteristic polynomial of the map.
</document>

<document source="https://math2.tamnd.com/node/0BZQ6ZX584.md" tag="0BZQ6ZX584" kind="statement">
# Addition rule for two events

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
Then
$$P(A \cup B) + P(A \cap B) = P(A) + P(B).$$
Equivalently, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

## Remarks

If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$.
The rule is the probabilistic form of [[inclusion–exclusion for two sets|0C4YEQ9H5E]]. The [[inclusion–exclusion principle|0FF7726C7P]] gives the version for $n$ events.
A consequence is $P(A \cup B) \le P(A) + P(B)$, the case $n = 2$ of [[Boole's inequality|0EGSEE9J9R]].

## Explanations

### L1 human

When you add $P(A)$ and $P(B)$, you count the outcomes in both events twice, so you subtract $P(A \cap B)$ once.
For one card from a deck, $P(\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52$.
</document>

<document source="https://math2.tamnd.com/node/0BZVNWJ352.md" tag="0BZVNWJ352" kind="statement">
# Order of the dihedral group

## Statement

Let $n \geq 3$.
The [[dihedral group|0JZ27DMJ5D]] $D_n$ has [[order|0B85S2V2DR]] $2n$.

## Remarks

A symmetry of the regular $n$-gon is determined by the image of one vertex and by whether it keeps or reverses the orientation.
There are $n$ choices for the image of the vertex and $2$ choices for the orientation.
For example, $D_3$ has order $6$ and $D_4$ has order $8$.

## Explanations

### L1 human

A symmetry can send a fixed vertex to any of the $n$ vertices.
After that, the polygon can be face up or face down.
So there are $n \cdot 2 = 2n$ symmetries.
</document>

<document source="https://math2.tamnd.com/node/0C019SDTER.md" tag="0C019SDTER" kind="statement">
# Recurrence for Stirling numbers of the second kind

## Statement

Let $n$ and $k$ be natural numbers, and let $S(n, k)$ be the [[Stirling numbers of the second kind|0ZN07RQ3JS]].
Then
$$S(n + 1, k + 1) = (k + 1)\, S(n, k + 1) + S(n, k).$$

## Remarks

Proof idea: look at the element $n + 1$. Either it forms a block by itself, and the rest is a partition into $k$ blocks. Or it joins one of the $k + 1$ blocks of a partition of the rest.
Example: $S(4, 2) = 2\, S(3, 2) + S(3, 1) = 2 \cdot 3 + 1 = 7$.

## Explanations

### L1 human

Add a new element to a partition problem.
The new element is alone, or it joins one of the existing $k + 1$ groups, and these two cases give the two terms.
</document>

<document source="https://math2.tamnd.com/node/0C08BQ3M05.md" tag="0C08BQ3M05" kind="statement">
# Laplace expansion along a row

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.
Then, for each row $i$,
$$\det A = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

## Remarks

The expansion reduces a determinant of order $n$ to $n$ determinants of order $n - 1$.
It is efficient when the chosen row contains many zeros.
There is a similar [[expansion along a column|06AMCJMMVM]].
Replace the entries of row $i$ by those of another row $k$.
Then the sum is $0$, because the new matrix has two equal rows.

## Explanations

### L1 human

To expand a determinant along a row, take each entry of that row.
Multiply it by its cofactor, which is the signed determinant of what is left after deleting its row and column.
Add these products.
</document>

<document source="https://math2.tamnd.com/node/0C08YMV95H.md" tag="0C08YMV95H" kind="definition">
# Random walk

## Definition

Let $d \ge 1$, let $x_0 \in \mathbb{R}^d$, and let $\xi_1, \xi_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] random vectors in $\mathbb{R}^d$.
The random walk with start $x_0$ and steps $\xi_i$ is the [[stochastic process|0QH5ENKZM1]] $(S_n)_{n \ge 0}$ with
$$S_0 = x_0 \quad \text{and} \quad S_n = x_0 + \xi_1 + \dots + \xi_n \quad \text{for } n \ge 1.$$
The simple random walk on $\mathbb{Z}$ has $d = 1$, $x_0 = 0$ and $P(\xi_i = 1) = P(\xi_i = -1) = 1/2$.

## Remarks

For the simple random walk on $\mathbb{Z}$, $E[S_n] = 0$ and $\operatorname{Var}(S_n) = n$. By the central limit theorem, $S_n/\sqrt{n}$ converges in distribution to $\mathcal{N}(0, 1)$.
The simple random walk on $\mathbb{Z}^d$ returns to its start with probability $1$ if $d \le 2$, and with probability less than $1$ if $d \ge 3$. This is Pólya's theorem.
A random walk is a Markov chain with transition probabilities that do not change under translation.
After scaling of time and space, the simple random walk converges to the Wiener process.

## Explanations

### L1 human

A random walk is the path of someone who, at each step, chooses a direction at random.
It models the price of a share, the motion of a particle in a fluid, and the fortune of a gambler.
</document>

<document source="https://math2.tamnd.com/node/0C093D1T4M.md" tag="0C093D1T4M" kind="definition">
# Simple group

## Definition

A [[group|G5S1M9]] $G$ is simple when $G \neq \{e\}$ and the only [[normal subgroups|0CB0DX5BVV]] of $G$ are $\{e\}$ and $G$.

## Remarks

The cyclic groups $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$ are simple, and they are the only [[abelian simple groups|0JJV80RXHD]].
The [[alternating group|0Z5905Q73D]] $A_n$ is simple for $n \geq 5$, and $A_5$ is the smallest simple group that is not abelian.
A simple group has only the trivial quotient and itself as quotients, so it cannot be split into smaller pieces by a quotient.

## Explanations

### L1 human

A simple group cannot be broken into a normal subgroup and a quotient group.
In this way, simple groups are like the prime numbers among the integers.
Each finite group is built from simple groups, by the Jordan–Hölder theorem.
</document>

<document source="https://math2.tamnd.com/node/0C0KPAKK9Y.md" tag="0C0KPAKK9Y" kind="counterexample">
# Separately continuous but not continuous function

## Counterexample

One could guess the following. A function of two variables is continuous when it is continuous in each variable for each fixed value of the other variable.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x, y) \ne (0, 0)$ and $f(0, 0) = 0$.
For each fixed $b$, the function $x \mapsto f(x, b)$ is continuous on $\mathbb{R}$, and for each fixed $a$, the function $y \mapsto f(a, y)$ is continuous on $\mathbb{R}$.
But $f$ is not [[continuous|0JEJATBVD7]] at $(0, 0)$.

## Remarks

For $b \ne 0$, the function $x \mapsto f(x, b)$ is a quotient of polynomials with a denominator that is never $0$. For $b = 0$, it is the constant $0$. The same holds for the other variable.
On the line $y = x$ with $x \ne 0$, the value is $f(x, x) = 1/2$. So $f(x, x) \to 1/2 \ne f(0, 0)$ as $x \to 0$, by the [[test with different paths|0BJ6H85DRQ]].

## Explanations

### L1 human

Separate continuity checks only the two directions along the axes.
On both axes this function is $0$, so nothing goes wrong there.
On the diagonal it is $1/2$ everywhere except at the origin, so the function jumps there.
</document>

<document source="https://math2.tamnd.com/node/0C12P4W4RH.md" tag="0C12P4W4RH" kind="concept">
# Density of the normal distribution

## Summary

The density of the normal distribution is the bell-shaped function $\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$.
The node [[density of the normal distribution|027JF1FBB0]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0C13E623WD.md" tag="0C13E623WD" kind="statement">
# Linear maps on finite-dimensional spaces are continuous

## Statement

Let $V$ be a finite-dimensional normed space and let $W$ be a normed space, both over $\mathbb{R}$ or over $\mathbb{C}$.
Then each linear map $T \colon V \to W$ is [[continuous|0S3SMRH0Z5]].

## Remarks

Proof: choose a basis $e_1, \ldots, e_n$ of $V$. For $x = \sum_i a_i e_i$, the triangle inequality gives $\|Tx\|_W \le \left( \sum_i \|T e_i\|_W \right) \max_i |a_i|$. By [[equivalence of norms in finite dimension|03MC70ZRNR]], $\max_i |a_i| \le c \|x\|_V$ for some $c$. Thus $T$ is bounded, and so [[continuous|0F3K4C7XYW]].
The finite dimension of $W$ is not needed, only that of $V$.
For example, each matrix gives a continuous map $\mathbb{R}^n \to \mathbb{R}^m$.

## Explanations

### L1 human

In finite dimensions, linear maps are automatically continuous, for each choice of norms.
Problems with continuity only appear in infinite dimensions.
</document>

<document source="https://math2.tamnd.com/node/0C1VYVNV2N.md" tag="0C1VYVNV2N" kind="object">
# pi

## Description

The cosine function has exactly one zero in the interval $[0, 2]$. The number $\pi$ is twice this zero.
So $\cos(\pi/2) = 0$, and
$$\pi \approx 3.14159265.$$

## Remarks

A zero exists by the [[intermediate value theorem|00KAFBR32E]], since $\cos 0 = 1$ and $\cos 2 < 0$. It is unique because $\cos$ is strictly decreasing on $[0, 2]$, by [[its derivative|0V2DV29EGT]] $-\sin$.
With this definition, $\pi$ is the ratio of the circumference of a circle to its diameter, and the area of the unit disc.
Mathlib defines `Real.pi` in the same way, as twice the zero of $\cos$ in $[1, 2]$.
William Jones first used the letter $\pi$ for this ratio, in 1706. Around 250 BC, Archimedes approximated $\pi$ with regular polygons.

## Explanations

### L1 human

The number $\pi$ is the length of half of the unit circle.
In analysis, we find it through the first positive zero of cosine, which is $\pi/2$.
</document>

<document source="https://math2.tamnd.com/node/0C1Y04SZ1W.md" tag="0C1Y04SZ1W" kind="concept">
# Finite extension

## Summary

A finite extension is a field extension of finite degree.
The [[definition of a finite extension|0NRYPM9Y27]] states it.
</document>

<document source="https://math2.tamnd.com/node/0C21Q694AG.md" tag="0C21Q694AG" kind="concept">
# Family of sets

## Summary

A family of sets is a collection of sets, often indexed by the elements of an index set.
</document>

<document source="https://math2.tamnd.com/node/0C21RZG5V2.md" tag="0C21RZG5V2" kind="definition">
# Random variable with a density

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
$X$ has a density, or is absolutely continuous, if there is a Borel measurable function $f \colon \mathbb{R} \to [0, \infty)$ with the following property. For each Borel set $B \subseteq \mathbb{R}$,
$$P(X \in B) = \int_B f(x) \, dx,$$
where the integral is the Lebesgue integral.

## Remarks

Such a function $f$ is a [[probability density function|04YG690N5V]] of $X$.
If $X$ has a density, then $P(X = x) = 0$ for each $x$, and the cumulative distribution function $F_X(x) = \int_{-\infty}^x f(t) \, dt$ is continuous.
A continuous cumulative distribution function does not imply a density. The Cantor distribution has a continuous cumulative distribution function and no density.
The same definition with $\mathbb{R}^n$ and the Lebesgue measure on $\mathbb{R}^n$ gives random vectors with a density.

## Explanations

### L1 human

For a random variable with a density, the probability of an interval is an area under a curve.
Single points have probability zero, just as a single point has no length.
</document>

<document source="https://math2.tamnd.com/node/0C28RT5BG7.md" tag="0C28RT5BG7" kind="definition">
# Rectifiable curve

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]].
For a partition $P$ given by $a = t_0 < t_1 < \dots < t_k = b$, let
$$L(\gamma, P) = \sum_{i = 1}^{k} \|\gamma(t_i) - \gamma(t_{i-1})\|$$
be the length of the inscribed polygonal path.
The curve $\gamma$ is rectifiable if $\sup_P L(\gamma, P) < \infty$, where the supremum is over all partitions of $[a, b]$.
Then the length of $\gamma$ is $\ell(\gamma) = \sup_P L(\gamma, P)$.

## Remarks

If $\gamma$ is of class $C^1$, then it is rectifiable and $\ell(\gamma) = \int_a^b \|\gamma'(t)\| \, dt$.
A refinement of $P$ does not decrease $L(\gamma, P)$, by the triangle inequality.
The graph of $f(x) = x \sin(1/x)$ for $0 < x \le 1$, $f(0) = 0$, is a continuous curve that is not rectifiable. The Koch curve is another example.
A curve is rectifiable if and only if each component $\gamma_i$ has bounded variation on $[a, b]$.

## Explanations

### L1 human

Approximate the curve by a chain of straight segments between points on it, and add up their lengths.
Finer chains give longer totals. If the totals stay bounded, the curve has a finite length, and the length is their least upper bound.
</document>

<document source="https://math2.tamnd.com/node/0C2AKPDCD9.md" tag="0C2AKPDCD9" kind="definition">
# Hamiltonian system

## Definition

Let $n \ge 1$, let $U$ be an open subset of $\mathbb{R}^n \times \mathbb{R}^n$ with coordinates $(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n)$, and let $H : U \to \mathbb{R}$ be continuously differentiable.
The Hamiltonian system with Hamiltonian $H$ is the [[autonomous system|07BNRFQ0PJ]]
$$q_i' = \frac{\partial H}{\partial p_i}(q, p), \qquad p_i' = -\frac{\partial H}{\partial q_i}(q, p), \qquad i = 1, \ldots, n.$$

## Remarks

The function $H$ is a [[first integral|0SS1P1YC9A]] when it is not constant on an open set: along a solution, $\frac{d}{dt} H = \sum_i \left( \frac{\partial H}{\partial q_i} q_i' + \frac{\partial H}{\partial p_i} p_i' \right) = 0$.
For $H(q, p) = \frac{1}{2} p^2 + V(q)$ with $n = 1$, the system is $q' = p$, $p' = -V'(q)$. This is Newton's law $q'' = -V'(q)$ for a particle with potential energy $V$.
The [[harmonic oscillator|0YG0ZJPTWG]] has $V(q) = \frac{1}{2} q^2$, and the [[pendulum|0SHPXM2HWH]] has $V(q) = -\cos q$ in suitable units.
A time-dependent Hamiltonian $H(t, q, p)$ gives a non-autonomous Hamiltonian system, and then $H$ need not be conserved.

## Explanations

### L1 human

A Hamiltonian system describes a mechanical system without friction by one function, the total energy $H$.
The positions $q$ and the momenta $p$ change according to the partial derivatives of $H$, and the energy stays constant.
</document>

<document source="https://math2.tamnd.com/node/0C2F40MEBC.md" tag="0C2F40MEBC" kind="statement">
# Determinant is the product of the eigenvalues

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Assume that the characteristic polynomial of $A$ splits over $F$ as $p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n)$.
Then
$$\det A = \lambda_1 \lambda_2 \cdots \lambda_n.$$

## Remarks

The $\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].
The statement follows by comparing the constant coefficients, see [[the determinant as a coefficient|0T0X8GTYYS]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $1$ and $3$, and determinant $3$.

## Explanations

### L1 human

The determinant of a matrix is the product of its eigenvalues, counted with multiplicity.
This agrees with the volume picture: in a basis of eigenvectors, the map stretches each direction by its eigenvalue.
The volume then changes by the product of these factors.
</document>

<document source="https://math2.tamnd.com/node/0C2TQMW5A1.md" tag="0C2TQMW5A1" kind="definition">
# Dual map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *dual map* of $T$, also called its *transpose*, is the map
$$T^* \colon W^* \to V^*, \qquad T^*(f) = f \circ T,$$
between the [[dual spaces|07JBSZE0V4]].

## Remarks

The dual map is linear.
It reverses the order of composition: $(T \circ S)^* = S^* \circ T^*$, and $(\id_V)^* = \id_{V^*}$.
In dual bases, the matrix of $T^*$ is the transpose of the matrix of $T$, see [[the matrix of the dual map|0ZPB3180AY]].
The kernel of $T^*$ is the [[annihilator|0ASHNYF7AG]] of the image of $T$.

## Explanations

### L1 human

A linear form on $W$ can be pulled back to $V$: first apply $T$, then the form.
The dual map does this pullback.
It goes in the opposite direction, from $W^*$ to $V^*$.
</document>

<document source="https://math2.tamnd.com/node/0C32J27WQC.md" tag="0C32J27WQC" kind="statement">
# Associativity of addition

## Statement

For all natural numbers $a$, $b$ and $c$, $(a + b) + c = a + (b + c)$.
Here $+$ is the [[addition of natural numbers|0W5DSDFQ4G]].

## Remarks

The proof from the recursive definition of addition uses [[induction|0ASJVEFA9J]] on $c$.
Because of this law, we can write $a + b + c$ without parentheses.
Multiplication of natural numbers is also associative.

## Explanations

### L1 human

When you add three numbers, you can add the first two first or the last two first.
For example, $(2 + 3) + 4 = 5 + 4 = 9$ and $2 + (3 + 4) = 2 + 7 = 9$.
</document>

<document source="https://math2.tamnd.com/node/0C38S0KMJK.md" tag="0C38S0KMJK" kind="definition">
# Interior

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The interior of $S$ is the [[union|05GK32H72H]] of all [[open sets|01ZQQDCERP]] $U$ of $X$ with $U \subseteq S$.
It is written $\operatorname{int} S$ or $S^\circ$.
A point of $\operatorname{int} S$ is an interior point of $S$.

## Remarks

A point $x$ is an interior point of $S$ if and only if there is an open set $U$ with $x \in U \subseteq S$.
In a metric space, $x$ is an interior point of $S$ if and only if $B(x, \varepsilon) \subseteq S$ for some $\varepsilon > 0$.
In $\mathbb{R}$, the interior of $[0, 1]$ is $(0, 1)$, and the interior of $\mathbb{Q}$ is $\emptyset$.

## Explanations

### L1 human

The interior of a set is what remains when you remove its edge points.
A point is in the interior when a whole small neighbourhood of it lies in the set.
</document>

<document source="https://math2.tamnd.com/node/0C3CFHDN6R.md" tag="0C3CFHDN6R" kind="definition">
# Determinant

## Definition

Let $K$ be a commutative ring and let $n \ge 1$.
There is exactly one map $\det \colon M_n(K) \to K$ with these properties:

1. $\det A$ is a multilinear function of the $n$ rows of $A$.
2. $\det A = 0$ when two rows of $A$ are equal.
3. $\det I_n = 1$, for the [[identity matrix|06DW9BAWB1]] $I_n$.

This map is the *determinant*.

## Remarks

In other words, $\det$ is the unique [[alternating multilinear map|0K6CH0MT0F]] of the rows with value $1$ on $I_n$.
The [[Leibniz formula|016PE4VZ8Q]] gives $\det A$ as a sum over permutations.
For $n = 2$, $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$.
The same characterization holds with columns in place of rows, because $\det A^T = \det A$.

## Explanations

### L1 human

The determinant gives one number for each square matrix.
Over the real numbers, its absolute value is the factor by which the matrix scales volume, and its sign tells if the matrix flips orientation.
A real square matrix is invertible exactly when its determinant is not zero.
</document>

<document source="https://math2.tamnd.com/node/0C3PN4075R.md" tag="0C3PN4075R" kind="statement">
# Euler product formula

## Statement

Let $s$ be a complex number with real part $\operatorname{Re}(s) > 1$, and let $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ be the Riemann zeta function.
Then
$$\zeta(s) = \prod_{p} \frac{1}{1 - p^{-s}},$$
where the product runs over all [[prime numbers|0SN0MAEKYX]] $p$ and converges absolutely.

## Remarks

The proof expands each factor as a geometric series $1 + p^{-s} + p^{-2s} + \cdots$. The [[fundamental theorem of arithmetic|084HH1NDCB]] shows that the product of these series contains each term $n^{-s}$ exactly one time.
The same method gives a product formula $\sum_n f(n) n^{-s} = \prod_p \sum_{k \ge 0} f(p^k) p^{-ks}$ for each [[multiplicative function|0AP80TT660]] $f$ when the series converges absolutely.
Since $\zeta(s) \to \infty$ as $s \to 1^+$, the formula gives a proof of [[Euclid's theorem|0292EPPPT0]].

## Explanations

### L1 human

Multiply $(1 + \tfrac12 + \tfrac14 + \cdots)(1 + \tfrac13 + \tfrac19 + \cdots)(1 + \tfrac15 + \cdots) \cdots$ and expand.
Each fraction $1/n$ appears one time, because $n$ is a product of prime powers in exactly one way.
The formula with the exponent $s$ is the same idea with $1 / n^s$.
</document>

<document source="https://math2.tamnd.com/node/0C3QBCR46M.md" tag="0C3QBCR46M" kind="concept">
# Monic polynomial

## Summary

A monic polynomial is a nonzero polynomial whose leading coefficient is $1$.
The [[definition of a monic polynomial|0BDX4MBKT4]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/0C4JS9RF5M.md" tag="0C4JS9RF5M" kind="object">
# Integers

## Description

The integers are the numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$.
We write $\mathbb{Z}$ for the set of integers.
The set $\mathbb{Z}$ contains the [[natural numbers|0MWKJ44V5A]] $0, 1, 2, \dots$ and their negatives $-1, -2, -3, \dots$.
With addition and multiplication, $\mathbb{Z}$ is a [[commutative ring|0BXA4Q0JHW]]. Each integer $a$ has an additive inverse $-a$, thus subtraction is always possible in $\mathbb{Z}$.
The integers have a [[total order|0F6FM9Y4Y6]] that extends the order of $\mathbb{N}$.

## Remarks

The letter $\mathbb{Z}$ comes from the German word "Zahlen", which means "numbers".
One [[construction of the integers|0NS58JP68E]] uses pairs of natural numbers.
The [[rational numbers|0NCJ5WXN6Z]] contain the integers as the fractions $a / 1$.

## Explanations

### L1 human

The integers are the whole numbers with a sign: positive, negative or zero.
You need them to answer questions such as $5 - 8 = ?$, which has no answer in the natural numbers.
On a number line, the negative integers are to the left of $0$.
</document>

<document source="https://math2.tamnd.com/node/0C4NT4GT3X.md" tag="0C4NT4GT3X" kind="statement">
# Eigenvalues of a self-adjoint operator are real

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Then each [[eigenvalue|0BQAA2BK3Q]] of $T$ is a real number.

## Remarks

If $T(v) = \lambda v$ with $v \ne 0$, then $\lambda \langle v, v \rangle = \langle T(v), v \rangle = \langle v, T(v) \rangle = \bar\lambda \langle v, v \rangle$, so $\lambda = \bar\lambda$.
In particular, a [[Hermitian matrix|0P17NGQQMQ]] has only real eigenvalues.
So a real [[symmetric matrix|0VR671PHE5]] has only real eigenvalues, also when we view it as a complex matrix.

## Explanations

### L1 human

Self-adjoint maps never have complex eigenvalues that are not real.
For example, the symmetric matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has the eigenvalues $1$ and $-1$.
By contrast, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not symmetric, and its eigenvalues are $\pm i$.
</document>

<document source="https://math2.tamnd.com/node/0C4VWBFY73.md" tag="0C4VWBFY73" kind="definition">
# Bernstein polynomial

## Definition

Let $f \colon [0, 1] \to \mathbb{R}$ be a function and let $n \ge 1$ be an integer.
The $n$-th Bernstein polynomial of $f$ is
$$B_n(f)(x) = \sum_{k=0}^{n} f\Big( \frac{k}{n} \Big) \binom{n}{k} x^k (1 - x)^{n-k}.$$
The polynomials $b_{k,n}(x) = \binom{n}{k} x^k (1 - x)^{n-k}$ are the Bernstein basis polynomials.

## Remarks

If $f$ is continuous, then $B_n(f) \to f$ uniformly on $[0, 1]$. This proves the [[Weierstrass approximation theorem|0JQE130DHW]].
The basis polynomials are nonnegative on $[0, 1]$, and they sum to $1$ by the binomial theorem.
Sergei Bernstein gave this proof in 1912, with an argument from probability. Mathlib defines the basis polynomials as `bernsteinPolynomial`.

## Explanations

### L1 human

The Bernstein polynomial samples $f$ at the points $k/n$ and blends the values with smooth weights.
Near $x$, the weights concentrate on the samples with $k/n$ close to $x$.
</document>

<document source="https://math2.tamnd.com/node/0C4YEQ9H5E.md" tag="0C4YEQ9H5E" kind="statement">
# Inclusion–exclusion for two sets

## Statement

Let $A$ and $B$ be finite sets.
Then
$$|A \cup B| + |A \cap B| = |A| + |B|.$$
Equivalently, $|A \cup B| = |A| + |B| - |A \cap B|$.

## Remarks

If $A \cap B = \varnothing$, this is the [[rule of sum|0EE2V3CHFK]].
For three sets: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.
The [[inclusion–exclusion principle|0FF7726C7P]] is the version for $n$ sets.

## Explanations

### L1 human

When you add the sizes of $A$ and $B$, you count the elements of both sets twice.
Subtract the size of $A \cap B$ once to correct this. For example, $12$ students play football, $8$ play chess, and $3$ do both, so $17$ students do at least one.
</document>

<document source="https://math2.tamnd.com/node/0C5BFX94J9.md" tag="0C5BFX94J9" kind="statement">
# Recurrence for Stirling numbers of the first kind

## Statement

Let $n$ and $k$ be natural numbers, and let $c(n, k)$ be the unsigned [[Stirling numbers of the first kind|0Z22VQH17V]].
Then
$$c(n + 1, k + 1) = n\, c(n, k + 1) + c(n, k).$$

## Remarks

Proof idea: look at the element $n + 1$ in a permutation of $n + 1$ elements. Either it is a fixed point, and the rest has $k$ cycles. Or it sits right after one of the $n$ other elements in its cycle.
Example: $c(4, 2) = 3\, c(3, 2) + c(3, 1) = 3 \cdot 3 + 2 = 11$.

## Explanations

### L1 human

Insert a new element into a permutation written as cycles.
It forms a new cycle of length $1$, or it goes into an existing cycle right after one of the $n$ old elements.
</document>

<document source="https://math2.tamnd.com/node/0C5M7AM6FH.md" tag="0C5M7AM6FH" kind="definition">
# Chromatic number

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
The chromatic number $\chi(G)$ is the smallest natural number $k$ such that $G$ is [[$k$-colourable|0Y7VDAX8Q3]].

## Remarks

For the [[complete graph|03WKB2H474]], $\chi(K_n) = n$. For the [[cycle graph|0QQ0F12962]] $C_n$, $\chi(C_n)$ is $2$ if $n$ is even and $3$ if $n$ is odd.
A [[greedy colouring|04AAKZNXHD]] shows that $\chi(G) \le \Delta(G) + 1$, where $\Delta(G)$ is the [[maximum degree|0KNAN9F27P]].
The chromatic number is at least the [[clique number|0WCKRBHFFD]].

## Explanations

### L1 human

The chromatic number is the least number of colours that you need to colour a graph properly.
For a map, it is the least number of colours that keeps neighbouring countries in different colours.
</document>

<document source="https://math2.tamnd.com/node/0C5PREPZWG.md" tag="0C5PREPZWG" kind="definition">
# Hyperbolic equilibrium point

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable, and let $x^* \in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.
The point $x^*$ is hyperbolic if no eigenvalue of the Jacobian matrix $Df(x^*)$ has real part $0$.

## Remarks

A saddle, a node and a spiral of a planar linear system are hyperbolic. A centre is not.
For the pendulum $x' = y$, $y' = -\sin x$, the equilibrium $(\pi, 0)$ is hyperbolic, with eigenvalues $\pm 1$. The equilibrium $(0, 0)$ is not, with eigenvalues $\pm i$.
Near a hyperbolic equilibrium, the [[linearisation|09ZWYRQ41J]] describes the system qualitatively, by the [[Hartman–Grobman theorem|0S8WX8A4V9]].
If all eigenvalues have negative real part, the point is a sink. If all have positive real part, it is a source. Otherwise it is a saddle.

## Explanations

### L1 human

At a hyperbolic equilibrium, each direction of the linearisation either contracts or expands.
No direction is neutral, so small changes of the field do not change the type of the equilibrium.
</document>

<document source="https://math2.tamnd.com/node/0C5RDJE73X.md" tag="0C5RDJE73X" kind="statement">
# Number of subsets of a finite set

## Statement

Let $X$ be a finite set with $|X| = n$.
Let $\mathcal{P}(X)$ be the power set of $X$, the set of all subsets of $X$.
Then $|\mathcal{P}(X)| = 2^n$.

## Remarks

The map that sends a subset $S$ to its indicator function $1_S : X \to \{0, 1\}$ is a bijection. Thus the count follows from the [[number of functions between finite sets|08QQH9F7G0]].
Grouped by size, the count gives the identity in the node [[sum of binomial coefficients|0KAMF19NV8]].

## Explanations

### L1 human

To build a subset, decide for each element whether it is in or out.
That is 2 options per element, so $2 \cdot 2 \cdots 2 = 2^n$ subsets. For $\{a, b\}$ these are $\varnothing$, $\{a\}$, $\{b\}$ and $\{a, b\}$.
</document>

<document source="https://math2.tamnd.com/node/0C5SR4FC74.md" tag="0C5SR4FC74" kind="concept">
# Codomain

## Summary

The codomain of a function is the set in which its values lie.
</document>

<document source="https://math2.tamnd.com/node/0C5TP6RYSV.md" tag="0C5TP6RYSV" kind="statement">
# Polynomial functions are continuous

## Statement

Let $a_0, a_1, \dots, a_n$ be real numbers.
Then the function $p \colon \mathbb{R} \to \mathbb{R}$ given by $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ is [[continuous|0P27W9RW0G]].

## Remarks

Constant functions and the identity $x \mapsto x$ are continuous. Sums and [[products of continuous functions are continuous|0VGMHCF4SC]], so induction on $n$ gives the result.
Mathlib states it as `Polynomial.continuous`.

## Explanations

### L1 human

Polynomials are built from $x$ and constants by addition and multiplication.
These operations keep continuity, so each polynomial is continuous.
</document>

<document source="https://math2.tamnd.com/node/0C62PZXX02.md" tag="0C62PZXX02" kind="concept">
# Multigraph

## Summary

A multigraph is a graph in which two vertices may be joined by more than one edge. Multigraphs model networks with parallel links, as in the problem of the Königsberg bridges. The node [[multigraph|01M5T0KDG7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0C6J8BQXVR.md" tag="0C6J8BQXVR" kind="counterexample">
# Uncorrelated random variables that are not independent

## Counterexample

The claim "uncorrelated random variables are independent" is false. Here are two counterexamples.
First: let $X \sim \mathcal{N}(0, 1)$ and $Y = X^2$. Then $\operatorname{Cov}(X, Y) = E[X^3] - E[X] \, E[X^2] = 0 - 0 = 0$, since the odd moments of $X$ are $0$.
But $X$ and $Y$ are not independent: $P(|X| \le 1, \, Y > 1) = 0$, while $P(|X| \le 1) \, P(Y > 1) > 0$.
Second: let $X \sim \mathcal{N}(0, 1)$, and let $W$ be independent of $X$ with $P(W = 1) = P(W = -1) = 1/2$. Put $Y = WX$.
Then $Y \sim \mathcal{N}(0, 1)$ and $\operatorname{Cov}(X, Y) = E[W] \, E[X^2] = 0$. But $|Y| = |X|$, so $X$ and $Y$ are not independent.

## Remarks

In the second example, both $X$ and $Y$ are normal, but the pair $(X, Y)$ is not multivariate normal: $X + Y$ is $0$ with probability $1/2$.
So the rule "uncorrelated implies independent" needs a joint normal law, not only normal marginals.
The direction that is true is that [[independent variables are uncorrelated|077GN9HDR6]].

## Explanations

### L1 human

Covariance only detects linear relations. In the first example, $Y$ is a function of $X$, so it depends on $X$ completely, but the relation is not linear.
So zero correlation does not mean that two variables have nothing to do with each other.
</document>

<document source="https://math2.tamnd.com/node/0C6THFRMZR.md" tag="0C6THFRMZR" kind="definition">
# Tangent line to a circle

## Definition

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $T$ be a point of $C$.
A [[line|0MP9BCF6DP]] $\ell$ is tangent to $C$ at $T$ if $\ell \cap C = \{T\}$. The point $T$ is the point of tangency.

## Remarks

A line meets a circle in two points, in one point or in no point. A line with two common points is a secant line.
For the circle $(x - a)^2 + (y - b)^2 = r^2$ and a point $(x_1, y_1)$ on it, the tangent at this point has the equation $(x - x_1)(x_1 - a) + (y - y_1)(y_1 - b) = 0$.
The tangent at $T$ is the line through $T$ perpendicular to the radius to $T$. See [[a tangent is perpendicular to the radius|031JV5S54D]].
From a point outside the circle there are exactly two tangent lines to the circle.

## Explanations

### L1 human

A tangent line just touches a circle at one point and stays outside the circle. For the unit circle, the line $y = 1$ is tangent at $(0, 1)$. The line $y = 0$ is not tangent: it meets the circle at $(1, 0)$ and at $(-1, 0)$.
</document>

<document source="https://math2.tamnd.com/node/0C81YBVTWH.md" tag="0C81YBVTWH" kind="definition">
# Cartesian product

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The Cartesian product of $A$ and $B$ is the set
$$A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$$
of all [[ordered pairs|0JQWWFCS9R]] $(a, b)$ with $a \in A$ and $b \in B$.

## Remarks

Example: $\{1, 2\} \times \{x, y\} = \{(1, x), (1, y), (2, x), (2, y)\}$.
For finite sets with $m$ and $n$ elements, $A \times B$ has $mn$ elements.
The plane $\mathbb{R}^2$ is $\mathbb{R} \times \mathbb{R}$.
A [[binary relation|08EJC6Q25H]] from $A$ to $B$ is a subset of $A \times B$.

## Explanations

### L1 human

Think of a table with a row for each element of $A$ and a column for each element of $B$.
Each cell of the table is one pair $(a, b)$.
The Cartesian product is the set of all the cells.
</document>

<document source="https://math2.tamnd.com/node/0C8BR8XRTR.md" tag="0C8BR8XRTR" kind="concept">
# Normal operator

## Summary

A normal operator on a finite-dimensional inner product space is a linear map that commutes with its adjoint.
</document>

<document source="https://math2.tamnd.com/node/0C8QENHYZW.md" tag="0C8QENHYZW" kind="definition">
# Möbius function

## Definition

The Möbius function $\mu$ is an arithmetic function. For each positive integer $n$, it is given by
$$\mu(n) = \begin{cases} (-1)^{\omega(n)} & \text{if } n \text{ is squarefree}, \\ 0 & \text{otherwise}. \end{cases}$$
Here $\omega(n)$ is the [[number of distinct prime factors|0YVVXT3A0B]] of $n$. The word squarefree has the meaning of the [[definition of a squarefree integer|004JW7A2N5]].

## Remarks

So $\mu(1) = 1$, $\mu(p) = -1$ for each prime $p$, and $\mu(n) = 0$ when $p^2 \mid n$ for some prime $p$.
For example, $\mu(6) = 1$, $\mu(30) = -1$ and $\mu(12) = 0$.
The function $\mu$ is [[multiplicative|0VB9G5B034]], and it is the [[inverse of the constant function 1|0ASC73QEQT]] for the Dirichlet convolution.

## Explanations

### L1 human

The Möbius function is $0$ on numbers with a repeated prime factor.
On the other numbers, it is $1$ for an even number of prime factors and $-1$ for an odd number.
It is the sign rule of inclusion and exclusion, written as a function of $n$.
</document>

<document source="https://math2.tamnd.com/node/0C8WYW4KYQ.md" tag="0C8WYW4KYQ" kind="concept">
# Greatest common divisor

## Summary

The greatest common divisor of two integers is the largest integer that divides both of them.
We write it as $\gcd(a, b)$.
The [[definition of the greatest common divisor|02WF878BH1]] gives the precise form and the convention for $\gcd(0, 0)$.
</document>

<document source="https://math2.tamnd.com/node/0C95TG69SG.md" tag="0C95TG69SG" kind="definition">
# Euclidean distance

## Definition

Let $n$ be a positive integer, and let $p = (p_1, \dots, p_n)$ and $q = (q_1, \dots, q_n)$ be points of $\mathbb{R}^n$.
The Euclidean distance between $p$ and $q$ is the real number
$$d(p, q) = \|p - q\| = \sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2 + \dots + (p_n - q_n)^2},$$
where $\|p - q\|$ is the [[length|0XKBKGTCPC]] of the vector $p - q$.

## Remarks

The function $d$ is a metric on $\mathbb{R}^n$. For all points $p$, $q$ and $r$: $d(p, q) \geq 0$, $d(p, q) = 0$ if and only if $p = q$, $d(p, q) = d(q, p)$, and $d(p, r) \leq d(p, q) + d(q, r)$.
For $n = 1$ the formula is $d(p, q) = |p - q|$.
Some applications use the squared distance $d(p, q)^2$, because it has no square root. The squared distance is not a metric, because it does not satisfy the triangle inequality.

## Explanations

### L1 human

To find the distance between two points in the plane, draw the right triangle with horizontal and vertical legs between them. The legs have lengths $|p_1 - q_1|$ and $|p_2 - q_2|$. The distance is the hypotenuse, so the Pythagorean theorem gives the formula. For $(1, 1)$ and $(4, 5)$ the distance is $\sqrt{9 + 16} = 5$.
</document>

<document source="https://math2.tamnd.com/node/0C9CMCN9BQ.md" tag="0C9CMCN9BQ" kind="concept">
# Contractive sequence

## Summary

A contractive sequence is a real sequence whose steps shrink at least by a fixed factor less than $1$. Such a sequence is Cauchy, so it converges. The node [[contractive sequence|0CZRT49QNG]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0C9J6W2A8A.md" tag="0C9J6W2A8A" kind="definition">
# Rooted tree

## Definition

A rooted tree is a pair $(T, r)$ of a [[tree|0MJ0JS72DZ]] $T$ and a vertex $r$ of $T$, the root.

Let $v \neq r$ be a vertex, and let $u$ be the neighbour of $v$ on the unique path from $v$ to $r$.
Then $u$ is the parent of $v$, and $v$ is a child of $u$.
The depth of $v$ is the [[distance|0WPZ5YPAGR]] from $r$ to $v$.

## Remarks

The root has no parent, and every other vertex has exactly one parent.
A vertex with no child is a leaf of the rooted tree. With this convention the root can be a leaf only when the tree has one vertex.
An ordered rooted tree also fixes an order of the children of each vertex. A [[binary tree|02A8EMC83C]] is a related recursive structure.

## Explanations

### L1 human

A rooted tree is a tree in which one vertex is chosen as the top.
Then each edge has a direction: from a parent toward its child, away from the root.
</document>

<document source="https://math2.tamnd.com/node/0C9MZY2RNA.md" tag="0C9MZY2RNA" kind="concept">
# Altitude of a triangle

## Summary

An altitude of a triangle is the segment from a vertex to the line of the opposite side, at a right angle to that line. Its length, the height, is the distance from the vertex to that line.
</document>

<document source="https://math2.tamnd.com/node/0C9S13WKMF.md" tag="0C9S13WKMF" kind="statement">
# Reverse triangle inequality

## Statement

For all real numbers $a$ and $b$,
$$\bigl| |a| - |b| \bigr| \le |a - b|.$$

## Remarks

Proof: the [[triangle inequality|03HF5PMY5T]] gives $|a| = |(a - b) + b| \le |a - b| + |b|$, thus $|a| - |b| \le |a - b|$. Exchange $a$ and $b$ for the other sign.
It shows that the function $x \mapsto |x|$ is continuous on $\mathbb{R}$.

## Explanations

### L1 human

Two numbers that are close to each other have absolute values that are also close.
The difference of the distances to $0$ is at most the distance between the two numbers.
</document>

<document source="https://math2.tamnd.com/node/0C9ZKYSPYR.md" tag="0C9ZKYSPYR" kind="statement">
# The kernel is a normal subgroup

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then the [[kernel|0W88CEGQP1]] $\ker \varphi$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.

## Remarks

For $g \in G$ and $k \in \ker \varphi$, $\varphi(gkg^{-1}) = \varphi(g) \, e_H \, \varphi(g)^{-1} = e_H$.
Conversely, each normal subgroup $N$ of $G$ is the kernel of the projection $G \to G/N$.
Thus the normal subgroups of $G$ are exactly the kernels of the homomorphisms from $G$.

## Explanations

### L1 human

If an element goes to the identity, then each of its conjugates also goes to the identity.
So the kernel is closed under conjugation, which is the definition of a normal subgroup.
This is one reason why normal subgroups are important.
</document>

<document source="https://math2.tamnd.com/node/0CARW59WD2.md" tag="0CARW59WD2" kind="definition">
# Sequential compactness

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
The set $K$ is sequentially compact when each sequence $(x_n)$ of points of $K$ has a subsequence $(x_{n_k})$ that converges to a point of $K$.
The space $X$ is sequentially compact when $X$ itself is a sequentially compact set.

## Remarks

Here a sequence $(y_k)$ converges to $y$ when each open set that contains $y$ contains $y_k$ for all large $k$. In a metric space, this is the [[usual convergence|06JYPJ4WHH]].
The interval $[0, 1]$ is sequentially compact by the Bolzano–Weierstrass theorem. The real line is not: the sequence $x_n = n$ has no convergent subsequence.
For metric spaces, [[compactness and sequential compactness agree|0K8SGQSV7G]]. In general topological spaces, neither notion implies the other.

## Explanations

### L1 human

In a sequentially compact set, a sequence cannot escape: some part of it always settles down to a point of the set.
In $(0, 1]$, the sequence $1/n$ tries to escape to $0$, which is outside the set. So $(0, 1]$ is not sequentially compact.
</document>

<document source="https://math2.tamnd.com/node/0CB0DX5BVV.md" tag="0CB0DX5BVV" kind="concept">
# Normal subgroup

## Summary

A normal subgroup of a group is a subgroup that is equal to each of its conjugates.
The normal subgroups are exactly the subgroups that give quotient groups, and exactly the kernels of homomorphisms.
</document>

<document source="https://math2.tamnd.com/node/0CBAB9QQ9X.md" tag="0CBAB9QQ9X" kind="concept">
# Complex numbers as pairs of reals

## Summary

The complex numbers can be constructed as ordered pairs of [[real numbers|08J2M1JYAG]].
On $\mathbb{R}^2$, define addition by $(a, b) + (c, d) = (a + c, b + d)$ and multiplication by $(a, b) \cdot (c, d) = (a c - b d, a d + b c)$.
With these operations, $\mathbb{R}^2$ is a field. It is the field $\mathbb{C}$ of [[complex numbers|09FTAET5Z4]].
The pair $(a, 0)$ is the real number $a$, and the pair $(0, 1)$ is the [[imaginary unit|0ESCX36MMC]] $i$.
Then $(a, b) = a + b i$, and the map $a + b i \mapsto (a, b)$ is a bijection $\mathbb{C} \to \mathbb{R}^2$.

## Remarks

This construction shows that the complex numbers exist, without the assumption of a mysterious square root of $-1$.
The bijection with $\mathbb{R}^2$ respects addition and multiplication by real numbers. It is the base of the [[complex plane|0CTE7A31XQ]].
Another construction is the quotient ring $\mathbb{R}[X]/(X^2 + 1)$.

## Explanations

### L1 human

A complex number is just a pair of real numbers with a special rule for multiplication.
The rule is made so that $(0, 1) \cdot (0, 1) = (-1, 0)$, which says $i^2 = -1$.
</document>

<document source="https://math2.tamnd.com/node/0CBQVVNBMP.md" tag="0CBQVVNBMP" kind="definition">
# Kurtosis

## Definition

Let $X$ be a real random variable with $E[X^4] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$.
The kurtosis of $X$ is
$$\operatorname{Kurt}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\mu_4}{\sigma^4},$$
where $\mu_4$ is the fourth [[central moment|0ZT24EC90K]]. The excess kurtosis is $\operatorname{Kurt}(X) - 3$.

## Remarks

Every [[normal distribution|07RXBVTVZB]] has kurtosis $3$, so its excess kurtosis is $0$.
For every $X$, $\operatorname{Kurt}(X) \ge 1$. The Bernoulli distribution with $p = 1/2$ reaches this bound.
Distributions with positive excess kurtosis are called leptokurtic. Those with negative excess kurtosis are called platykurtic.
The kurtosis does not change under $X \mapsto aX + b$ with $a \ne 0$.

## Explanations

### L1 human

Kurtosis compares the tails of a distribution with those of the normal distribution.
A high kurtosis means that rare extreme values are more frequent than for a normal variable with the same variance.
</document>

<document source="https://math2.tamnd.com/node/0CBRFQ7DVZ.md" tag="0CBRFQ7DVZ" kind="definition">
# Normal extension

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
The extension $L/K$ is normal if every [[irreducible polynomial|00X2A9G9W3]] $f \in K[x]$ that has a root in $L$ [[splits|0SHMC00RJV]] over $L$.

## Remarks

A finite extension $L/K$ is normal if and only if $L$ is the [[splitting field|02NZZ1J0FH]] of some polynomial over $K$.
Every extension of degree $2$ is normal. For example, $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is normal.
The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal: $x^3 - 2$ has a root in it, but its other two roots are not real.
Normality is not transitive: $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are normal, but $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ is not.

## Explanations

### L1 human

A normal extension contains all the roots of a polynomial as soon as it contains one of them.
It has no lopsided gaps, like a field that contains one cube root of $2$ but not the others.
</document>

<document source="https://math2.tamnd.com/node/0CC4VC7KR3.md" tag="0CC4VC7KR3" kind="example">
# Logistic equation

## Example

Let $r > 0$ and $K > 0$ be constants. The logistic equation is
$$P'(t) = r\, P(t) \left(1 - \frac{P(t)}{K}\right).$$
For an initial value $P(0) = P_0 > 0$, its solution is
$$P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-r t}}, \qquad t \ge 0.$$
With $r = K = 1$, the function $f(t) = 1 / \left(1 + e^{-t}\right)$ is the standard logistic function. It satisfies $f' = f(1 - f)$ and $f(0) = 1/2$.

## Remarks

The constant functions $P = 0$ and $P = K$ are the [[equilibrium points|0N8NGKDZDZ]]. For each $P_0 > 0$, $P(t) \to K$ as $t \to \infty$.
For small $P$, the equation is close to $P' = r P$, which is [[exponential growth|00SVP6W3CB]].
If $0 < P_0 < K/2$, the graph of $P$ is an S-shaped curve with its inflection point at the value $P = K/2$.
The equation is separable, and it is a [[Bernoulli equation|0T92VGGVWP]] with exponent $2$.

## Explanations

### L1 human

The logistic equation is a model of a population with limited resources.
When the population is small, it grows almost exponentially. When it is close to the capacity $K$, the growth slows down and stops.
The constant $K$ is the carrying capacity: the largest population that the environment can support.
</document>

<document source="https://math2.tamnd.com/node/0CC56MJ4FG.md" tag="0CC56MJ4FG" kind="definition">
# Harmonic conjugate

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, and identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + iy \leftrightarrow (x, y)$.
Let $u : U \to \mathbb{R}$ be a harmonic function: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.
A harmonic conjugate of $u$ is a function $v : U \to \mathbb{R}$ such that the function $f = u + iv$ is [[holomorphic|0DG0Z72CQD]] on $U$.

## Remarks

For example, $v = 2xy$ is a harmonic conjugate of $u = x^2 - y^2$ on $\mathbb{C}$, because $u + iv = z^2$.
If $U$ is a [[domain|097KNJFJ7V]], two harmonic conjugates of $u$ differ by a real constant.
If $v$ is a harmonic conjugate of $u$, then $-u$ is a harmonic conjugate of $v$, because $v - iu = -i(u + iv)$ is holomorphic.
On a disc, each harmonic function has a harmonic conjugate. On $\mathbb{C} \setminus \{0\}$, the harmonic function $\log |z|$ has none.

## Explanations

### L1 human

The [[real part of a holomorphic function is harmonic|031BG625Y3]]. A harmonic conjugate goes the other way: it starts with a harmonic function $u$ and finds the missing imaginary part $v$.
For $u = x$, the conjugate is $v = y$, because $x + iy = z$ is holomorphic.
</document>

<document source="https://math2.tamnd.com/node/0CCPN09EMJ.md" tag="0CCPN09EMJ" kind="definition">
# Discriminant of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f = a_n x^n + \dots + a_0 \in K[x]$ with $n \ge 1$ and $a_n \neq 0$.
Let $f = a_n (x - r_1) \cdots (x - r_n)$ in a [[splitting field|02NZZ1J0FH]] of $f$.
The discriminant of $f$ is
$$\operatorname{Disc}(f) = a_n^{2n-2} \prod_{1 \le i < j \le n} (r_i - r_j)^2.$$

## Remarks

The product is symmetric in the roots, so $\operatorname{Disc}(f) \in K$, by the [[fundamental theorem of symmetric polynomials|0GW1ZZH0R4]]. It is a polynomial with integer coefficients in $a_0, \dots, a_n$.
So $\operatorname{Disc}(f) = 0$ if and only if $f$ has a [[repeated root|067F8YVYMH]].
For $f = a x^2 + b x + c$, $\operatorname{Disc}(f) = b^2 - 4ac$.
For $f = x^3 + p x + q$, $\operatorname{Disc}(f) = -4p^3 - 27q^2$.

## Explanations

### L1 human

The discriminant measures how far apart the roots of a polynomial are.
It is zero when two roots coincide.
For a real quadratic, its sign tells whether the roots are real or complex.
</document>

<document source="https://math2.tamnd.com/node/0CCQV3884B.md" tag="0CCQV3884B" kind="definition">
# Matrix–vector product

## Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$, and let $x = (x_1, \dots, x_n)^T \in R^n$ be a column vector.
The *matrix–vector product* $Ax$ is the column vector in $R^m$ with entries
$$(Ax)_i = \sum_{j=1}^{n} a_{ij} x_j$$
for $1 \le i \le m$.

## Remarks

If $A_1, \dots, A_n$ are the columns of $A$, then $Ax = x_1 A_1 + \cdots + x_n A_n$, a [[linear combination|0V8F3E48E6]] of the columns.
A [[system of linear equations|0GT4WWGWH2]] is the equation $Ax = b$.
The map $x \mapsto Ax$ is linear, see [[multiplication by a matrix as a linear map|094V2VBJXJ]].
For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}$.

## Explanations

### L1 human

To multiply a matrix by a vector, take each row of the matrix in turn.
Multiply its entries by the entries of the vector, position by position, and add the results.
Each row gives one entry of the answer.
</document>

<document source="https://math2.tamnd.com/node/0CD87SFM2N.md" tag="0CD87SFM2N" kind="concept">
# Two-sided ideal

## Summary

A two-sided ideal of a ring is an additive subgroup that is closed under multiplication by ring elements on the left and on the right.
The two-sided ideals are the kernels of ring homomorphisms.
The [[definition of a two-sided ideal|0DDA6CY7GJ]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0CDAHW576M.md" tag="0CDAHW576M" kind="statement">
# A finite field of order q is the splitting field of x^q - x

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q = p^n$, with $p$ prime.
Then $F$ is a [[splitting field|02NZZ1J0FH]] of $x^q - x$ over its prime subfield $\mathbb{F}_p$.

## Remarks

The proof: [[$x^q - x = \prod_{a \in F} (x - a)$|0QWR8WTZEW]], so $x^q - x$ splits over $F$, and its roots are all the elements of $F$. So the roots generate $F$.
With the [[uniqueness of splitting fields|069A3BC3EP]], this gives the [[uniqueness of finite fields|010PKCQPXP]].

## Explanations

### L1 human

A finite field consists exactly of the roots of $x^q - x$.
So it is the smallest field that contains all these roots.
</document>

<document source="https://math2.tamnd.com/node/0CDMXGPSES.md" tag="0CDMXGPSES" kind="definition">
# Riemann sphere

## Definition

The Riemann sphere, or extended complex plane, is the set $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, where $\mathbb{C}$ is the set of [[complex numbers|09FTAET5Z4]] and $\infty$ is one new point.
Its topology is as follows.
A set $V \subseteq \widehat{\mathbb{C}}$ is open if the following two conditions hold. First, $V \cap \mathbb{C}$ is open in the [[complex plane|0CTE7A31XQ]]. Second, in the case $\infty \in V$, $V$ also contains the set $\{ z \in \mathbb{C} : |z| > R \}$ for some $R > 0$.
Let $S^2 = \{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}$ and $N = (0, 0, 1)$.
The stereographic projection $\sigma : S^2 \to \widehat{\mathbb{C}}$ is $\sigma(x_1, x_2, x_3) = \frac{x_1 + i x_2}{1 - x_3}$ for points other than $N$, and $\sigma(N) = \infty$.
The map $\sigma$ is a homeomorphism, so $\widehat{\mathbb{C}}$ is a sphere as a topological space.

## Remarks

A sequence $(z_n)$ of complex numbers tends to $\infty$ in $\widehat{\mathbb{C}}$ exactly when $|z_n| \to \infty$.
Some arithmetic extends to $\infty$: $z + \infty = \infty$ for $z \in \mathbb{C}$, and $\frac{z}{0} = \infty$ and $\frac{z}{\infty} = 0$ for $z \neq 0$, but the expressions $\infty - \infty$, $0 \cdot \infty$, $\frac{0}{0}$ and $\frac{\infty}{\infty}$ stay undefined.
With the value $\infty$ at each [[pole|0WPQYW8MTE]], a [[meromorphic function|0NSD1TPHR0]] on an open set $U$ becomes a continuous map from $U$ to $\widehat{\mathbb{C}}$.
The Riemann sphere is a compact Riemann surface, with the two charts $z$ near points of $\mathbb{C}$ and $\frac{1}{z}$ near $\infty$.

## Explanations

### L1 human

Imagine a ball that sits on the complex plane at $0$.
Draw a line from the top of the ball to a point of the plane; the line meets the ball at one more point.
Points far from $0$ go to points near the top, so the top of the ball plays the role of $\infty$.
</document>

<document source="https://math2.tamnd.com/node/0CDVDGWGEG.md" tag="0CDVDGWGEG" kind="statement">
# Degrees along an Eulerian trail

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]], and let $W$ be an [[Eulerian trail|0NNYDD1AVR]] of $G$ from $u$ to $v$.
If $u = v$, then every vertex of $G$ has even [[degree|0T0F6CCS9W]].
If $u \neq v$, then $u$ and $v$ have odd degree and every other vertex has even degree.

## Remarks

Proof idea: each time $W$ passes through a vertex, it uses two edges at that vertex. Only the start and the end use one edge without a partner.
Thus a graph with an Eulerian trail has $0$ or $2$ vertices of odd degree. The [[Seven Bridges of Königsberg|0YCW781Q9D]] have $4$.

## Explanations

### L1 human

Count the edges at a vertex as a walk uses them.
Every visit in the middle of the walk uses one edge in and one edge out.
Only the first and last vertices can be left with an odd count.
</document>

<document source="https://math2.tamnd.com/node/0CDVK9RQBH.md" tag="0CDVK9RQBH" kind="definition">
# Martingale

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a filtration $(\mathcal{F}_n)_{n \ge 0}$: σ-algebras with $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \dots \subseteq \mathcal{F}$.
A martingale with respect to $(\mathcal{F}_n)$ is a sequence $(M_n)_{n \ge 0}$ of real random variables with these three properties:
1. $M_n$ is $\mathcal{F}_n$-measurable for each $n$.
2. $E[|M_n|] < \infty$ for each $n$.
3. $E[M_{n+1} \mid \mathcal{F}_n] = M_n$ almost surely for each $n$.

## Remarks

If "$=$" in property 3 is replaced by "$\ge$", the sequence is a submartingale. If it is replaced by "$\le$", it is a supermartingale.
By the [[law of total expectation|04RRB6HM00]], $E[M_n] = E[M_0]$ for each $n$.
Examples: the simple random walk $S_n$ and $S_n^2 - n$ are martingales for the filtration of the steps.
Doob's optional stopping theorem and Doob's martingale convergence theorem are the main tools of the theory.

## Explanations

### L1 human

A martingale is a fair game: given everything you know now, your expected fortune after the next round is your fortune now.
No betting strategy can change the expected value of a martingale at fixed times.
</document>

<document source="https://math2.tamnd.com/node/0CDWKDYWN6.md" tag="0CDWKDYWN6" kind="statement">
# Universal property of the free group

## Statement

Let $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $G$ be a [[group|G5S1M9]].
For each function $f \colon S \to G$, there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon F_S \to G$ with $\varphi(s) = f(s)$ for each $s \in S$.

## Remarks

Here $S$ is identified with the set of words of length $1$ in $F_S$.
The homomorphism is $\varphi(t_1 \cdots t_n) = f(t_1)^{\pm 1} \cdots f(t_n)^{\pm 1}$, with the sign of each letter.
This property characterises $F_S$ up to isomorphism, so some texts use it as the definition of a free group.
If the image of $f$ generates $G$, then $\varphi$ is surjective, so $G$ is a quotient of $F_S$.

## Explanations

### L1 human

To define a homomorphism out of a free group, you only choose the images of the generators.
There are no relations to check.
So each choice extends to exactly one homomorphism.
</document>

<document source="https://math2.tamnd.com/node/0CDZ684TCD.md" tag="0CDZ684TCD" kind="concept">
# Work done by a force field

## Summary

The work done by a force field $F$ on a particle that moves along a curve $\gamma$ is the [[line integral|0H5MR9B6GT]] $W = \int_\gamma F \cdot d\mathbf{r}$.
Only the component of the force along the direction of motion contributes, positively when it points forward and negatively when it points backward.

## Remarks

For a constant force $F$ and a straight displacement $d$, the work is $F \cdot d$.
For a $C^1$ path, $W = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt$. The integrand $F \cdot \gamma'$ is the power, the rate at which the force does work.
If $F = -\nabla U$ for a potential energy $U$, then $W = U(\gamma(a)) - U(\gamma(b))$, and the work does not depend on the path. Gravity near the surface of the Earth is an example.
A force that is always orthogonal to the velocity does no work.

## Explanations

### L1 human

Work measures how much a force helps a motion.
Pushing a box forward does positive work, pushing against its motion does negative work, and pushing straight down on a sliding box does no work.
</document>

<document source="https://math2.tamnd.com/node/0CE768XNVP.md" tag="0CE768XNVP" kind="concept">
# Arctangent

## Summary

The arctangent is the inverse of the tangent function restricted to $(-\pi/2, \pi/2)$. It is defined on all of $\mathbb{R}$ and it is bounded. The node [[arctangent|065XW2V6C3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0CEAT3418X.md" tag="0CEAT3418X" kind="definition">
# Conic section

## Definition

Let $A, B, C, D, E, F$ be real numbers, with $A$, $B$ and $C$ not all zero. Let
$$M = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{pmatrix}.$$
The set $Q = \{\, (x, y) \in \mathbb{R}^2 \;:\; A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ in [[Cartesian coordinates|0RWKYX8339]] is a non-degenerate conic section if $\det M \neq 0$ and $Q$ is not empty.

## Remarks

If $\det M = 0$, the set $Q$ is a degenerate conic: two lines, one line, one point or the empty set.
Each non-degenerate conic is an [[ellipse|0VKCCEXBJ1]], a [[parabola|04S2GMHX82]] or a [[hyperbola|0WN5G117YC]]. The [[discriminant $B^2 - 4AC$|0N5NR32NQG]] tells which one.
Equivalently, a non-degenerate conic is the intersection of a double cone with a plane that does not pass through the apex.
Apollonius of Perga wrote a systematic work on conic sections, about 200 BC.

## Explanations

### L1 human

Cut a double ice cream cone with a flat knife. Depending on the tilt of the knife, the cut edge is a closed oval, a U shape or a curve with two branches. These are the ellipse, the parabola and the hyperbola, and they all have equations of degree $2$.
</document>

<document source="https://math2.tamnd.com/node/0CEBCS0E0E.md" tag="0CEBCS0E0E" kind="statement">
# Number of permutations

## Statement

Let $X$ be a finite set with $|X| = n$.
Then the number of [[permutations|0BQMWCNS8S]] of $X$ is $n!$.

## Remarks

For $n = 3$ there are $3! = 6$ permutations of $\{1, 2, 3\}$: $123$, $132$, $213$, $231$, $312$ and $321$.
The count follows from the [[rule of product|0VPK4X73Q0]]. The image of the first element has $n$ choices, the image of the second has $n - 1$ choices, and so on.
It is the case $k = n$ of the [[number of injections|0X0N833FF8]].

## Explanations

### L1 human

To arrange $n$ different objects in a row, you have $n$ choices for the first place, then $n - 1$ for the second, and so on.
The product of these numbers is $n!$.
</document>

<document source="https://math2.tamnd.com/node/0CEC45PRJN.md" tag="0CEC45PRJN" kind="concept">
# Modular multiplicative inverse

## Summary

A modular multiplicative inverse of an integer $a$ modulo $n$ is an integer $x$ such that $a x$ leaves the remainder $1$ on division by $n$.
The [[definition of the modular multiplicative inverse|00GXTQMY8Z]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0CEGEASY49.md" tag="0CEGEASY49" kind="definition">
# Zero of a holomorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
A zero of $f$ is a point $z_0 \in U$ with $f(z_0) = 0$.

## Remarks

The zeros of $\sin$ are the numbers $k\pi$ with $k \in \mathbb{Z}$. The [[complex exponential|054VXHMCA3]] has no zeros.
Each zero has an [[order|064N002TX4]], a positive integer or $\infty$.
If $f$ is not the zero function on a [[domain|097KNJFJ7V]], its zeros are [[isolated|0K83KEJ43P]].

## Explanations

### L1 human

A zero of a function is a point where the function takes the value $0$, as for real functions.
For example, $z^2 + 1$ has the zeros $i$ and $-i$.
</document>

<document source="https://math2.tamnd.com/node/0CES91M79A.md" tag="0CES91M79A" kind="definition">
# Polynomials in several variables

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$ be an integer.
For $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, the monomial $x^\alpha$ is the formal product $x_1^{\alpha_1} \cdots x_n^{\alpha_n}$.
A polynomial in $x_1, \dots, x_n$ over $R$ is a finite sum $\sum_\alpha c_\alpha x^\alpha$ with $c_\alpha \in R$.
Two polynomials are equal if and only if they have the same coefficient $c_\alpha$ for each $\alpha$.
The ring $R[x_1, \dots, x_n]$ is the set of these polynomials, with coefficientwise addition and the product
$$\Big(\sum_\alpha c_\alpha x^\alpha\Big) \Big(\sum_\beta d_\beta x^\beta\Big) = \sum_\gamma \Big(\sum_{\alpha + \beta = \gamma} c_\alpha d_\beta\Big) x^\gamma.$$

## Remarks

There is a natural isomorphism $R[x_1, \dots, x_n] \cong R[x_1, \dots, x_{n-1}][x_n]$. So the ring can also be built by repeated use of the [[polynomial ring|0Q9EZS90GY]] in one variable.
The same construction works for any set of variables, with exponent maps that are zero at all but finitely many variables.
The polynomial $x_1 x_2 + x_3^2$ is an element of $\mathbb{Z}[x_1, x_2, x_3]$.

## Explanations

### L1 human

A polynomial in several variables is a finite sum of terms like $5 x^2 y z^3$.
You add and multiply such polynomials with the usual rules of algebra.
Each term is a coefficient times a product of powers of the variables.
</document>

<document source="https://math2.tamnd.com/node/0CEX9BRXRZ.md" tag="0CEX9BRXRZ" kind="statement">
# Classification of complex quadratic forms

## Statement

Let $V$ be a complex vector space of finite dimension $n$, and let $Q$ be a quadratic form on $V$.
Let $r$ be the [[rank|0ZC9YTXZRY]] of the symmetric matrix of $Q$ in a basis.
Then $V$ has a basis $(e_1, \dots, e_n)$ such that, for all $x_1, \dots, x_n \in \C$,
$$Q(x_1 e_1 + \cdots + x_n e_n) = x_1^2 + x_2^2 + \cdots + x_r^2.$$
So two quadratic forms on $V$ are equivalent if and only if their matrices have the same rank.

## Remarks

The proof starts from a [[diagonal form|0Z3FEWZ8HA]] $a_1 x_1^2 + \cdots + a_n x_n^2$ and replaces $e_i$ by $e_i / \sqrt{a_i}$ for each $a_i \ne 0$.
The same result holds over every algebraically closed field of characteristic not $2$.
Over $\R$, this fails: $x^2$ and $-x^2$ have the same rank but are not equivalent.

## Explanations

### L1 human

Over the complex numbers, every nonzero number is a square.
So all coefficients of a diagonal form can be made equal to $1$.
Only the number of squares remains, and it is the rank.
</document>

<document source="https://math2.tamnd.com/node/0CF0XGYHNJ.md" tag="0CF0XGYHNJ" kind="statement">
# exp(2 pi i/n) is a primitive root of unity

## Statement

Let $n$ be a positive integer.
Then $e^{2 \pi i / n}$ is a [[primitive root of unity|0J817RTVG6]].

## Remarks

Proof: for $1 \le k < n$, the number $\zeta^k = e^{2 \pi i k / n}$ has argument $2 \pi k / n$, which is strictly between $0$ and $2\pi$. Thus $\zeta^k \neq 1$.
For $n = 4$ this number is $i$, and for $n = 2$ it is $-1$.

## Explanations

### L1 human

The first root of unity after $1$, going counterclockwise around the circle, is a primitive one.
Its powers walk around the circle step by step and visit every $n$th root of unity before they return to $1$.
</document>

<document source="https://math2.tamnd.com/node/0CGCPX8Z70.md" tag="0CGCPX8Z70" kind="concept">
# Frobenius automorphism of a finite field

## Summary

The Frobenius automorphism of a finite field of characteristic $p$ is the map $x \mapsto x^p$.
It generates the group of all automorphisms of the field.
The [[definition of the Frobenius automorphism|0VP4TBVJ19]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0CGYNDCWRR.md" tag="0CGYNDCWRR" kind="object">
# Special orthogonal group

## Description

Let $n \ge 1$.
The *special orthogonal group* is the set
$$SO(n) = \{Q \in O(n) : \det Q = 1\}$$
of real orthogonal $n \times n$ matrices with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.
It is a normal subgroup of index $2$ of the [[orthogonal group|0TKYAEJ8N7]] $O(n)$.

## Remarks

Its elements are the *rotations* of $\R^n$ about the origin.
$SO(2)$ consists of the [[rotation matrices|0B9QNFGTHM]] $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$, and it is commutative.
Each element of $SO(3)$ is a rotation about an axis through the origin.
$SO(n)$ is a connected compact Lie group.

## Explanations

### L1 human

The special orthogonal group contains the orthogonal matrices that do not flip orientation.
These are the rotations.
Reflections have determinant $-1$, so they are not in this group.
</document>

<document source="https://math2.tamnd.com/node/0CH4VG15RE.md" tag="0CH4VG15RE" kind="concept">
# Disc of convergence

## Summary

The disc of convergence of a power series is the largest open disc around its center on which the series converges, see the [[definition|045EWAXY9P]].
Its radius is the radius of convergence, a number in $[0, \infty]$.
</document>

<document source="https://math2.tamnd.com/node/0CHE3J7QMX.md" tag="0CHE3J7QMX" kind="counterexample">
# Irreducible elements need not be prime

## Counterexample

In the integral domain $\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5} : a, b \in \mathbb{Z}\}$, the element $2$ is [[irreducible|052SDEPZH7]] but not [[prime|0MNHXBSX3M]].
Let $N(a + b\sqrt{-5}) = a^2 + 5b^2$. This norm is multiplicative, and the units are the elements of norm $1$, that is $\pm 1$.
If $2 = x y$ with nonunits $x, y$, then $N(x) N(y) = 4$, so $N(x) = 2$. But $a^2 + 5b^2 = 2$ has no integer solution. So $2$ is irreducible.
The element $2$ divides $(1 + \sqrt{-5})(1 - \sqrt{-5}) = 6$.
It does not divide $1 \pm \sqrt{-5}$, because $(1 \pm \sqrt{-5})/2$ is not in $\mathbb{Z}[\sqrt{-5}]$. So $2$ is not prime.

## Remarks

The same ring shows that factorization is not unique: $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$. See [[$\mathbb{Z}[\sqrt{-5}]$ is not a UFD|0BSJYCG7NM]].
In a [[unique factorization domain|06YMXJMXYS]], every irreducible element is prime.

## Explanations

### L1 human

In the integers, "cannot be factored" and "divides a factor of each product it divides" are the same property.
In $\mathbb{Z}[\sqrt{-5}]$, the number $2$ cannot be factored, but it divides $6 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ and neither factor.
So the two properties are different in general rings.
</document>

<document source="https://math2.tamnd.com/node/0CHQKZFK5S.md" tag="0CHQKZFK5S" kind="statement">
# Division algorithm

## Statement

Let $a$ and $b$ be integers with $b \neq 0$.
Then there is exactly one pair of integers $(q, r)$ such that $a = b q + r$ and $0 \le r < |b|$.

## Remarks

The integer $q$ is the quotient and the integer $r$ is the remainder of the division of $a$ by $b$.
The remainder is $0$ if and only if $b$ [[divides|096PW7Q014]] $a$.
For example, $-7 = 3 \cdot (-3) + 2$, so the quotient of $-7$ by $3$ is $-3$ and the remainder is $2$.

## Explanations

### L1 human

You can share $a$ objects into groups of $b$ objects, with some objects left over.
The theorem says that the number of full groups and the rest are unique, if the rest is smaller than $|b|$.
For example, $17 = 5 \cdot 3 + 2$: you get $3$ groups of $5$, and $2$ objects stay.
</document>

<document source="https://math2.tamnd.com/node/0CHR35YNJQ.md" tag="0CHR35YNJQ" kind="definition">
# Subbasis

## Definition

Let $X$ be a set and let $\mathcal{S}$ be a set of subsets of $X$.
The topology generated by $\mathcal{S}$ is the smallest [[topology|0CNQXVBHCX]] on $X$ that contains $\mathcal{S}$. It is the [[intersection|0839XNQXDK]] of all topologies on $X$ that contain $\mathcal{S}$.
If $(X, \tau)$ is a topological space and $\tau$ is the topology generated by $\mathcal{S}$, then $\mathcal{S}$ is a subbasis of $\tau$.

## Remarks

The open sets of the generated topology are the unions of finite intersections of members of $\mathcal{S}$. Here the empty intersection is $X$.
Thus the finite intersections of members of $\mathcal{S}$ form a [[basis|06EX0FHBTF]] of the generated topology.
The open rays $(-\infty, a)$ and $(b, \infty)$, with $a, b \in \mathbb{R}$, form a subbasis of the usual topology of $\mathbb{R}$.
The sets $\pi_i^{-1}(U)$, with $U$ open, form a subbasis of the [[product topology|04S9HX3GAP]].

## Explanations

### L1 human

A subbasis is a set of open sets that you want to have. The topology is the smallest one that makes them all open.
You get all open sets by taking finite intersections of them and then arbitrary unions.
</document>

<document source="https://math2.tamnd.com/node/0CHS7RBD39.md" tag="0CHS7RBD39" kind="concept">
# Euclidean domain

## Summary

A Euclidean domain is an integral domain with a division with remainder, in which the remainder is smaller than the divisor for some size function.
The integers and the polynomials over a field are the main examples.
The [[definition of a Euclidean domain|0GJ2HRG95T]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0CKGBEJX6B.md" tag="0CKGBEJX6B" kind="statement">
# Finite subgroups of the multiplicative group of a field are cyclic

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $G$ be a finite subgroup of the multiplicative group $K^\times$.
Then $G$ is cyclic.

## Remarks

The proof: for each $d$, the elements $g \in G$ with $g^d = 1$ are roots of $x^d - 1$.
So there are at most $d$ of them, by the [[bound on the number of roots|0KJVQV3E0F]], and a finite group with this property is cyclic.
For $K = \mathbb{C}$, the finite subgroups of $\mathbb{C}^\times$ are the groups of the $n$-th roots of unity.
The statement fails for division rings: the quaternion group $\{\pm 1, \pm i, \pm j, \pm k\}$ is a finite subgroup of the nonzero [[quaternions|0GE4DVQKR8]] that is not cyclic.

## Explanations

### L1 human

In a field, the equation $x^d = 1$ has at most $d$ solutions.
This forces every finite group of nonzero elements to be generated by one element.
</document>

<document source="https://math2.tamnd.com/node/0CMC2KAXFS.md" tag="0CMC2KAXFS" kind="definition">
# Cyclic quadrilateral

## Definition

Let $ABCD$ be a quadrilateral in the Euclidean plane $\mathbb{R}^2$, with the vertices $A$, $B$, $C$, $D$ in this order.
The quadrilateral $ABCD$ is cyclic if its four vertices are [[concyclic|049DCJZ448]]. The circle through them is its [[circumcircle|03VZJE6C8C]].

## Remarks

A convex quadrilateral $ABCD$ is cyclic if and only if its opposite angles add up to $\pi$: $\angle DAB + \angle BCD = \pi$. The direct part is Proposition 22 of Book III of Euclid's Elements.
Each rectangle is cyclic. A rhombus that is not a square is not cyclic.
Usually the quadrilateral is taken to be convex. A crossed quadrilateral with concyclic vertices is a crossed cyclic quadrilateral.

## Explanations

### L1 human

Each triangle fits in a circle, but a quadrilateral does not always. A quadrilateral is cyclic when one circle passes through all four corners. A simple test: the angles at two opposite corners add up to a straight angle.
</document>

<document source="https://math2.tamnd.com/node/0CMD0BCH8T.md" tag="0CMD0BCH8T" kind="concept">
# Principal ideal domain

## Summary

A principal ideal domain is an integral domain in which every ideal is generated by one element.
The integers and the polynomials over a field are examples.
The [[definition of a principal ideal domain|0RZYDVZ475]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0CMHSGTWXB.md" tag="0CMHSGTWXB" kind="concept">
# Equivalent matrices

## Summary

Two matrices of the same size are equivalent when $B = Q^{-1} A P$ for invertible matrices $P$ and $Q$; this holds exactly when they have the same rank.
</document>

<document source="https://math2.tamnd.com/node/0CN97GADE5.md" tag="0CN97GADE5" kind="statement">
# Abel's test

## Statement

Let $\sum a_n$ be a [[convergent|0CVVKY7GY2]] real series.
Let $(b_n)$ be a [[monotone|0KN22AAW5R]] and [[bounded|05FQ3VGAR9]] real sequence.
Then the series $\sum a_n b_n$ converges.

## Remarks

The sequence $(b_n)$ converges to some $b$, by the [[monotone convergence theorem|0FM6N4R0VB]]. Then $\sum a_n b_n = \sum a_n (b_n - b) + b \sum a_n$.
The first series converges by [[Dirichlet's test|0V9ME8GQE7]], since $b_n - b$ is monotone with limit $0$.
The hypotheses on $\sum a_n$ are stronger than in Dirichlet's test, and those on $(b_n)$ are weaker.

## History

Niels Henrik Abel proved the test in 1826.

## Explanations

### L1 human

Multiplying the terms of a convergent series by a bounded monotone sequence of weights keeps convergence.
The weights settle down to a limit, so they cannot disturb the series too much.
</document>

<document source="https://math2.tamnd.com/node/0CNQXVBHCX.md" tag="0CNQXVBHCX" kind="definition">
# Topological space

## Definition

Let $X$ be a set.
A topology on $X$ is a set $\tau$ of [[subsets|0FE8Q5S42Y]] of $X$ with these three properties:

1. The [[empty set|0ECVK59Y03]] $\varnothing$ and the set $X$ are in $\tau$.
2. For each family $(U_i)_{i \in I}$ of members of $\tau$, the [[union|05GK32H72H]] $\bigcup_{i \in I} U_i$ is in $\tau$.
3. For all $U, V \in \tau$, the [[intersection|0HBRW4E6G1]] $U \cap V$ is in $\tau$.

A topological space is a pair $(X, \tau)$ where $X$ is a set and $\tau$ is a topology on $X$.

## Remarks

The members of $\tau$ are the [[open sets|01ZQQDCERP]] of the space. Their complements are the [[closed sets|0G5AV9Q4RR]].
By induction, property 3 gives that each finite intersection of open sets is open. Infinite intersections need not be open.
Each metric gives a topology, the [[metric topology|0A6YRNZB2R]]. Other basic examples are the [[discrete topology|0S2TEHX4E1]] and the [[indiscrete topology|0XJ02JBD6S]].
One can also define a topology by its closed sets, or by the [[neighbourhoods of its points|09CJ00H3NW]].

## Explanations

### L1 human

A topology tells you which subsets of $X$ count as open, without any distances.
The rules copy the properties of open sets in metric spaces. Any union of open sets is open, and a finite intersection of open sets is open.
</document>

<document source="https://math2.tamnd.com/node/0CP5NDD21C.md" tag="0CP5NDD21C" kind="concept">
# First-countable space

## Summary

A topological space is first-countable when each point has a countable family of neighbourhoods that controls all neighbourhoods of the point.
In such spaces, sequences are enough to describe closures and continuity.
</document>

<document source="https://math2.tamnd.com/node/0CP7YY6NAG.md" tag="0CP7YY6NAG" kind="definition">
# Square matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A *square matrix* of order $n$ over $R$ is an $n \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
Its *main diagonal* is the list of the entries $a_{11}, a_{22}, \dots, a_{nn}$.
The set of square matrices of order $n$ over $R$ is written $M_n(R)$.

## Remarks

The product of two $n \times n$ matrices is again an $n \times n$ matrix, so $M_n(R)$ is a [[ring|0FHVPRAE60]].
The [[trace|0XC1RDBY7A]], the [[determinant|00KDZRW4C0]] and the [[eigenvalues|0BQAA2BK3Q]] are defined for square matrices.
A square matrix represents a linear map from a space to itself.

## Explanations

### L1 human

A square matrix has as many rows as columns, like a $3 \times 3$ table.
Only square matrices can be multiplied by themselves, so only they have powers like $A^2$.
Many important notions, such as the determinant, apply only to square matrices.
</document>

<document source="https://math2.tamnd.com/node/0CQ357M0QS.md" tag="0CQ357M0QS" kind="statement">
# Impossibility of doubling the cube

## Statement

The number $\sqrt[3]{2}$ is not a [[constructible number|0FMN2MN7TM]].
So the cube cannot be [[doubled|04SD3F4FCR]] with straightedge and compass.

## Remarks

The proof: $x^3 - 2$ is irreducible over $\mathbb{Q}$, by [[Eisenstein's criterion|02BPKD7TMD]] with $p = 2$. So $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$.
Since $3$ is not a power of $2$, the number $\sqrt[3]{2}$ is not constructible, by the [[degree theorem|0SSSM1D475]].

## History

Pierre Wantzel proved the impossibility in 1837.

## Explanations

### L1 human

The edge of the doubled cube is the cube root of $2$.
Its minimal polynomial has degree $3$, but constructible numbers have degree a power of $2$.
</document>

<document source="https://math2.tamnd.com/node/0CQ9ZM383N.md" tag="0CQ9ZM383N" kind="counterexample">
# Absolute value is continuous but not differentiable at 0

## Counterexample

The function $f(x) = |x|$ is continuous on $\mathbb{R}$, but it is not differentiable at $0$.
The difference quotient is $\frac{|h|}{h}$, which is $1$ for $h > 0$ and $-1$ for $h < 0$. So it has no limit as $h \to 0$.

## Remarks

The [[one-sided derivatives|0HXEX0DPRA]] at $0$ exist: the right derivative is $1$ and the left derivative is $-1$.
So the converse of [[differentiable functions are continuous|07ZMX4ANJS]] fails.
For $x \ne 0$, the derivative is $x / |x|$, the sign of $x$. Mathlib states the failure at $0$ as `not_differentiableAt_abs_zero`.

## Explanations

### L1 human

The graph of $|x|$ is a V with a sharp corner at the origin.
At the corner, no single line fits the graph, so there is no tangent.
</document>

<document source="https://math2.tamnd.com/node/0CQHFPGJ45.md" tag="0CQHFPGJ45" kind="concept">
# Topological space

## Summary

A topological space is a set together with a choice of subsets, called open sets, that obey simple rules for unions and intersections.
This structure is enough to define continuity, convergence, compactness and connectedness without a notion of distance.
</document>

<document source="https://math2.tamnd.com/node/0CQQG3DEW9.md" tag="0CQQG3DEW9" kind="concept">
# Bernoulli process

## Summary

A Bernoulli process is an infinite sequence of independent trials with two outcomes, each with the same probability of success. A model is an infinite sequence of tosses of one coin.
The node [[Bernoulli process|02XPXR1FBK]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0CQW20XGFW.md" tag="0CQW20XGFW" kind="definition">
# Parabola

## Definition

Let $F$ be a point of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $\ell$ be a [[line|0MP9BCF6DP]] that does not contain $F$. Let $d(P, F)$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $d(P, \ell)$ be the [[distance from $P$ to $\ell$|01AAYNSBN6]].
The parabola with the focus $F$ and the directrix $\ell$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; d(P, F) = d(P, \ell) \,\}.$$

## Remarks

For $F = (a, 0)$ with $a > 0$ and $\ell$ the line $x = -a$, the parabola is the set of the points with $y^2 = 4 a x$.
For real numbers $p \neq 0$, $q$ and $r$, the graph of $y = p x^2 + q x + r$ is a parabola.
The [[eccentricity|0HVX863SAJ]] of each parabola is $1$. Any two parabolas are similar.
Menaechmus studied conic sections in the 4th century BC. Archimedes found the area of a parabolic segment, and the name is due to Apollonius.

## Explanations

### L1 human

A parabola is the set of points that are as far from a point as from a line. For the focus $(0, 1)$ and the line $y = -1$, the parabola is $y = x^2 / 4$.>>>
</document>

<document source="https://math2.tamnd.com/node/0CRMGPAH83.md" tag="0CRMGPAH83" kind="concept">
# Linear independence

## Summary

Vectors are linearly independent when no nontrivial linear combination of them is zero.
Equivalently, no vector of the family is a linear combination of the others.
</document>

<document source="https://math2.tamnd.com/node/0CS42KEM5C.md" tag="0CS42KEM5C" kind="statement">
# Uniform convergence of derivatives allows term-by-term differentiation

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be differentiable for each $n$.
Assume that $(f_n(x_0))$ converges for some $x_0 \in [a, b]$, and that $(f_n')$ converges [[uniformly|08FPABDQ4C]] on $[a, b]$ to a function $g$.
Then $(f_n)$ converges uniformly on $[a, b]$ to a differentiable function $f$, and $f' = g$.

## Remarks

The [[mean value theorem|0BARQY52N5]] applied to $f_n - f_m$ shows that $(f_n)$ is [[uniformly Cauchy|01XAY17QA3]].
Uniform convergence of $f_n$ alone is not enough. The functions $\frac{\sin(n^2 x)}{n}$ tend to $0$ uniformly, but their derivatives $n \cos(n^2 x)$ do not converge.
Mathlib states a version as `hasDerivAt_of_tendstoUniformly`.

## Explanations

### L1 human

Derivatives are sensitive to small wiggles, so uniform convergence of functions does not control them.
We need uniform convergence of the derivatives themselves.
</document>

<document source="https://math2.tamnd.com/node/0CT6CBGTYY.md" tag="0CT6CBGTYY" kind="statement">
# Exponential growth beats polynomial growth

## Statement

For every natural number $n$,
$$\lim_{x \to \infty} \frac{e^x}{x^n} = \infty.$$

## Remarks

For $x > 0$, the series gives $e^x > \frac{x^{n+1}}{(n+1)!}$. So $\frac{e^x}{x^n} > \frac{x}{(n+1)!}$, which [[tends to infinity|0Y12QYTJK3]].
Equivalently, $x^n e^{-x} \to 0$ as $x \to \infty$. The same holds for real exponents $n$.
So for each polynomial $p$, $p(x) e^{-x} \to 0$ as $x \to \infty$. Mathlib states the main claim as `Real.tendsto_exp_div_pow_atTop`.

## Explanations

### L1 human

The exponential function eventually grows faster than every power of $x$.
For example, $e^x > x^{10}$ for all $x \ge 36$, but not for $x = 35$.
</document>

<document source="https://math2.tamnd.com/node/0CTE7A31XQ.md" tag="0CTE7A31XQ" kind="concept">
# Complex plane

## Summary

The complex plane is the plane $\mathbb{R}^2$ in which the point $(a, b)$ represents the [[complex number|09FTAET5Z4]] $a + b i$.
The horizontal axis is the real axis and the vertical axis is the imaginary axis.
In this picture, addition of complex numbers is addition of vectors.
The [[modulus|0KW2X0TFBX]] $|z|$ is the distance from $z$ to $0$, and the [[argument|0ZZRSX91QR]] of $z$ is the angle from the positive real axis to $z$.

## Remarks

Multiplication by a complex number $w \neq 0$ scales the plane by $|w|$ and turns it by the argument of $w$.
The [[complex conjugate|0BJV24C4SY]] is the reflection in the real axis.

## Explanations

### L1 human

Draw the real numbers on a horizontal line, and draw the multiples of $i$ on a vertical line through $0$.
Then each complex number is a point of the plane: $3 + 2 i$ is three steps to the right and two steps up.
</document>

<document source="https://math2.tamnd.com/node/0CTM1EHG9P.md" tag="0CTM1EHG9P" kind="definition">
# Box topology

## Definition

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]] and let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|0C81YBVTWH]].
The box topology on $X$ is the topology with [[basis|06EX0FHBTF]] the set of all products $\prod_{i \in I} U_i$, where each $U_i$ is open in $X_i$.

## Remarks

For finitely many factors, the box topology is the [[product topology|04S9HX3GAP]].
For infinite products, the box topology is finer than the product topology, and often strictly finer.
On $\mathbb{R}^{\mathbb{N}}$ with the box topology, the map $t \mapsto (t, t, t, \ldots)$ from $\mathbb{R}$ is not continuous, although each coordinate is continuous. The preimage of $\prod_{n \ge 1} (-1/n, 1/n)$ is $\{0\}$, which is not open.

## Explanations

### L1 human

The box topology lets you restrict all coordinates at the same time.
This looks natural, but it gives too many open sets, and simple maps into the product stop being continuous.
</document>

<document source="https://math2.tamnd.com/node/0CTNFGYMM2.md" tag="0CTNFGYMM2" kind="concept">
# Triangle

## Summary

A triangle is a polygon with three vertices and three sides. Its three vertices are points that do not lie on one line, and its sides are the segments that join them in pairs.
</document>

<document source="https://math2.tamnd.com/node/0CTV783HYT.md" tag="0CTV783HYT" kind="definition">
# Real sequence

## Definition

A real sequence is a [[sequence|0QE8STGJH2]] in the set $\mathbb{R}$ of real numbers.
Thus it is a function $a \colon \mathbb{N} \to \mathbb{R}$, and each term $a_n$ is a real number.

## Remarks

Examples are $a_n = 1/(n + 1)$, $a_n = (-1)^n$ and $a_n = 2^n$.
The real sequences form a real vector space under the termwise operations $(a_n) + (b_n) = (a_n + b_n)$ and $c\,(a_n) = (c\,a_n)$.
The [[limit of a sequence|0XKWSSY8P2]] is defined first for real sequences in most courses.

## Explanations

### L1 human

A real sequence is an endless list of real numbers $a_0, a_1, a_2, \dots$.
You can draw it as points $(n, a_n)$ in the plane, one point for each natural number $n$.
</document>

<document source="https://math2.tamnd.com/node/0CV69WDRXW.md" tag="0CV69WDRXW" kind="concept">
# Partition function

## Summary

The partition function $p(n)$ gives the number of integer partitions of $n$. It is a classical object of additive number theory, with a product generating function and deep congruences. The node [[partition function|001CX45SWQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0CVVKY7GY2.md" tag="0CVVKY7GY2" kind="definition">
# Convergent series

## Definition

Let $\sum a_n$ be a [[series|0JH4MN4K1C]] of real numbers with [[partial sums|0NB643K60Q]] $s_N = \sum_{n=0}^{N} a_n$.
The series is convergent if the sequence $(s_N)$ is a [[convergent sequence|0AZ3R52593]].

## Remarks

The series $\sum 1/2^n$ converges, since its partial sums $2 - 1/2^N$ tend to $2$.
The limit of the partial sums is the [[sum of the series|0K0H5E3Z5N]].
A series that does not converge is a [[divergent series|0NBRP6093V]].
Changing finitely many terms does not change whether a series converges. It can change the sum.

## Explanations

### L1 human

A series converges when its running totals approach a fixed number.
Adding more and more terms then changes the total less and less.
</document>

<document source="https://math2.tamnd.com/node/0CW52NV5TC.md" tag="0CW52NV5TC" kind="concept">
# Annulus

## Summary

An annulus is the region between two concentric circles, see the [[definition|0AJ0YVK743]].
In complex analysis, it is the open set $r < |z - c| < R$, and it is the natural domain of a Laurent series.
</document>

<document source="https://math2.tamnd.com/node/0CWDVN132S.md" tag="0CWDVN132S" kind="concept">
# Arccosine

## Summary

The arccosine is the inverse of the cosine function restricted to $[0, \pi]$. It is defined on $[-1, 1]$. The node [[arccosine|0RDP2HEFEW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0CWK2NFSEW.md" tag="0CWK2NFSEW" kind="concept">
# Improper multiple integral

## Summary

An improper multiple integral extends the Riemann integral to unbounded regions or unbounded functions. It is a limit of integrals over bounded regions where the function is bounded.
For a non-negative function $f$ on $\mathbb{R}^n$ that is Riemann integrable on each cube $[-a, a]^n$, the improper integral is $\int_{\mathbb{R}^n} f = \lim_{a \to \infty} \int_{[-a, a]^n} f$, when this limit exists.

## Remarks

For non-negative functions, the limit does not depend on the choice of the exhausting sequence of bounded sets. For functions of both signs one usually requires that $\int |f|$ is finite.
Example: in polar coordinates, $\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dA = \lim_{a \to \infty} \int_0^{2\pi} \int_0^a e^{-r^2} r \, dr \, d\theta = \pi$.
Example with a singularity: $\iint_{\|x\| \le 1} \|x\|^{-s} \, dA$ is finite for $s < 2$ and infinite for $s \ge 2$, by the same polar computation.
An improper [[double integral|0WAR4F23B4]] can fail to exist even when both iterated integrals exist, if $\int |f| = \infty$.

## Explanations

### L1 human

The Riemann integral needs a bounded function on a bounded region.
You may want to integrate over all of the plane, or near a point where $f$ blows up. Then integrate over larger and larger bounded pieces and take the limit.
</document>

<document source="https://math2.tamnd.com/node/0CWXCWP0Y0.md" tag="0CWXCWP0Y0" kind="definition">
# Automorphism group

## Definition

Let $G$ be a [[group|G5S1M9]].
An automorphism of $G$ is a [[group isomorphism|08MG7H8JW5]] from $G$ to $G$.
The automorphism group $\operatorname{Aut}(G)$ is the set of all automorphisms of $G$, with composition of maps as the operation.

## Remarks

The identity map is the identity element of $\operatorname{Aut}(G)$, and the inverse of an automorphism is its inverse map.
The automorphism group of $(\mathbb{Z}, +)$ has two elements, the identity and $n \mapsto -n$.
For an [[abelian group|082GHNPJ53]] $G$, the map $g \mapsto g^{-1}$ is an automorphism.

## Explanations

### L1 human

An automorphism is a way to rename the elements of a group so that all products stay correct.
The set of all such renamings is again a group, because you can compose and undo renamings.
For the integers, the only renamings are "do nothing" and "change all signs".
</document>

<document source="https://math2.tamnd.com/node/0CX86CR8JV.md" tag="0CX86CR8JV" kind="concept">
# Integrating factor for a non-exact equation

## Summary

Let $M\, dx + N\, dy = 0$ be an equation that is not [[exact|0EC7AGJ0PX]] on an open set $D$.
An integrating factor for it is a nowhere zero differentiable function $\mu$ on $D$ such that $\mu M\, dx + \mu N\, dy = 0$ is exact.
On a rectangle, with $M$, $N$ and $\mu$ of class $C^1$, the [[test for exactness|0MSCQNFTM8]] makes this condition $(\mu M)_y = (\mu N)_x$.
The two equations have the same solutions, because $\mu$ has no zero.

## Remarks

If $(M_y - N_x)/N$ is a function $g(x)$ of $x$ only, then $\mu(x) = e^{\int g(x)\, dx}$ is an integrating factor.
If $(N_x - M_y)/M$ is a function $h(y)$ of $y$ only, then $\mu(y) = e^{\int h(y)\, dy}$ is an integrating factor.
The [[integrating factor|01THH3Y3YP]] of a linear first-order equation is the first case: $y' + p y - q = 0$ has $M = p y - q$, $N = 1$ and $(M_y - N_x)/N = p$.

## Explanations

### L1 human

Some equations are not exact, but they become exact after multiplication by a good function.
In general, it is hard to find this function.
In two common cases, the factor depends on one variable only, and a simple integral gives it.
</document>

<document source="https://math2.tamnd.com/node/0CXTMAY5PX.md" tag="0CXTMAY5PX" kind="example">
# Harmonic series

## Example

The harmonic series is the series
$$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots.$$
Its terms tend to $0$, but the series is a [[divergent series|0NBRP6093V]].
Its partial sums are the [[harmonic numbers|0D2Y6G4C4S]] $H_N = \sum_{n=1}^{N} 1/n$, and they tend to $+\infty$.

## Remarks

The node [[harmonic series diverges|0E9M24KS9J]] gives a proof.
It shows that the converse of the [[term test|0VVY4H92G2]] is false.
The partial sums grow very slowly, like $\ln N$. The node [[harmonic numbers grow like log n|05S2GSPMGA]] makes this precise.
The name comes from music. The wavelengths of the overtones of a string are $1/2, 1/3, 1/4, \dots$ of the wavelength of the string.

## Explanations

### L1 human

The terms $1, 1/2, 1/3, \dots$ become small, but not fast enough.
You can group them into blocks that each add up to at least $1/2$, so the total grows without bound.
</document>

<document source="https://math2.tamnd.com/node/0CXXJNG5B6.md" tag="0CXXJNG5B6" kind="statement">
# First isomorphism theorem

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]] and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then the map $g \ker \varphi \mapsto \varphi(g)$ is a well-defined [[isomorphism|08MG7H8JW5]] from the [[quotient group|0KYX2N69J0]] $G / \ker \varphi$ to the [[image|0S21GNW9ZB]] $\operatorname{im} \varphi$.

## Remarks

Here $\ker \varphi$ is normal by [[this fact|0C9ZKYSPYR]], so the quotient exists.
If $\varphi$ is surjective, then $H \cong G / \ker \varphi$.
For example, the determinant gives $\mathrm{GL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$.
For finite $G$, a consequence is $|G| = |\ker \varphi| \cdot |\operatorname{im} \varphi|$.

## Explanations

### L1 human

A homomorphism collapses each coset of its kernel to one point of the image.
After this collapse, nothing else is lost: the quotient by the kernel and the image are the same group.
For example, reduction modulo $n$ shows that $\mathbb{Z}/n\mathbb{Z}$ is the image of $\mathbb{Z}$.
</document>

<document source="https://math2.tamnd.com/node/0CXZ217QDP.md" tag="0CXZ217QDP" kind="statement">
# Linearity of the derivative

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, let $f, g \colon U \to W$, let $x \in U$, and let $c \in \mathbb{R}$.
Suppose that $f$ and $g$ are Fréchet differentiable at $x$, with [[Fréchet derivatives|08EVNYZF42]] $Df(x)$ and $Dg(x)$.
Then $f + g$ and $c f$ are Fréchet differentiable at $x$, and
$$D(f + g)(x) = Df(x) + Dg(x), \qquad D(c f)(x) = c \, Df(x).$$

## Remarks

The proof adds the two remainders: a sum of two terms that are $o(\|h\|)$ is again $o(\|h\|)$.
So the maps that are differentiable at $x$ form a vector space, and $f \mapsto Df(x)$ is a linear map on it.
The same rules hold for the [[partial derivatives|09EGC6F8PT]] and for the [[directional derivatives|06KWGHDN2J]].

## Explanations

### L1 human

The derivative of a sum is the sum of the derivatives, and constants come out, just as in one variable.
For example, $D(x^2 + 3 y)$ at a point $(a, b)$ is the linear map $h \mapsto 2 a h_1 + 3 h_2$, the sum of the derivatives of $x^2$ and of $3 y$.
</document>

<document source="https://math2.tamnd.com/node/0CYRECG2YQ.md" tag="0CYRECG2YQ" kind="definition">
# Transitive relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is transitive when, for all $a, b, c \in A$, $a \mathrel{R} b$ and $b \mathrel{R} c$ imply $a \mathrel{R} c$.

## Remarks

The relations $<$, $\le$ and $=$ on $\mathbb{R}$ are transitive, and so is divisibility on $\mathbb{Z}$.
"Is a parent of" is not transitive.
The smallest transitive relation that contains $R$ is the [[transitive closure|0PEJH7AC74]] of $R$.

## Explanations

### L1 human

Transitivity lets you chain steps.
From $2 < 5$ and $5 < 9$ you get $2 < 9$.
In the graph of a transitive relation, each path of two arrows has a direct arrow as a shortcut.
</document>

<document source="https://math2.tamnd.com/node/0CYYAF8AD2.md" tag="0CYYAF8AD2" kind="statement">
# Derivative of a continuous linear map

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$ and let $A \colon V \to W$ be a bounded linear map.
Then $A$ is Fréchet differentiable at each point $x \in V$, and its [[Fréchet derivative|08EVNYZF42]] at $x$ is $A$ itself: $DA(x) = A$.

## Remarks

The proof is one line: $A(x + h) - A(x) - A h = 0$ for each $h$, so the remainder is $0$.
For $V = \mathbb{R}^n$ and $W = \mathbb{R}^m$, each linear map is bounded. So the derivative of $x \mapsto M x$, for an $m \times n$ matrix $M$, is the linear map with matrix $M$ at each point.
An affine map $x \mapsto A x + b$ has the same derivative $A$ at each point, because the derivative of a constant map is $0$.

## Explanations

### L1 human

The best linear approximation of a linear map is the map itself.
This is the analogue of the rule that the derivative of $x \mapsto c x$ is the constant $c$.
</document>

<document source="https://math2.tamnd.com/node/0CZA0NCFWP.md" tag="0CZA0NCFWP" kind="definition">
# Singular solution

## Definition

Let $F(x, y, y') = 0$ be a first-order [[ordinary differential equation|0P97V20RW7]], and let $y_s : I \to \mathbb{R}$ be a [[solution|04SW7WECS8]] on an open interval $I$.
The solution $y_s$ is singular if uniqueness fails at each point of its graph.
This means the following. For each $x_0 \in I$, there is another solution $y$ on an open interval containing $x_0$ with $y(x_0) = y_s(x_0)$ and $y'(x_0) = y_s'(x_0)$. This solution differs from $y_s$ on each neighbourhood of $x_0$.

## Remarks

Some authors use a weaker meaning: a solution at which the [[initial value problem|075VBY4K4X]] fails to have a unique solution at some point.
A singular solution is often the envelope of a family of solutions. It is not a member of the [[general solution|0SKYFQAFYD]] for any value of the constants.
Example: for $(y')^2 = 4 y$, the solutions $y = (x - C)^2$ are tangent to the solution $y = 0$, so $y = 0$ is singular. [[Clairaut's equation|0T6D4PE51C]] gives other examples.
By the [[uniqueness theorem|025PKW8AMR]], an equation $y' = f(x, y)$ with $f$ Lipschitz in $y$ has no singular solution.

## Explanations

### L1 human

A singular solution is a solution that touches other solutions at each of its points.
It often appears when the method of solution divides by an expression that can be zero.
So after you solve an equation, check what happens when the divisor is zero.
</document>

<document source="https://math2.tamnd.com/node/0CZRT49QNG.md" tag="0CZRT49QNG" kind="definition">
# Contractive sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is contractive if there is a real number $C$ with $0 < C < 1$ such that
$$|a_{n+2} - a_{n+1}| \le C\, |a_{n+1} - a_n| \quad \text{for all } n \in \mathbb{N}.$$

## Remarks

Each contractive sequence is a [[Cauchy sequence|0QZMTN5AXX]]. By induction $|a_{n+1} - a_n| \le C^n |a_1 - a_0|$, and the [[geometric series|01X5CVV1Q5]] bounds $|a_m - a_n|$. So each contractive sequence converges.
If $f \colon \mathbb{R} \to \mathbb{R}$ is a contraction mapping, so that $|f(x) - f(y)| \le C|x - y|$ with $C < 1$, then each sequence $a_{n+1} = f(a_n)$ is contractive.
It is not enough that $|a_{n+2} - a_{n+1}| < |a_{n+1} - a_n|$ for all $n$. The partial sums of the harmonic series satisfy this, and they diverge.

## Explanations

### L1 human

In a contractive sequence, each step is at most a fixed fraction of the step before.
The steps shrink like a geometric sequence, so the total distance travelled is finite, and the sequence converges.
</document>

<document source="https://math2.tamnd.com/node/0D032NSZKY.md" tag="0D032NSZKY" kind="definition">
# Closed curve

## Definition

Let $a < b$, and let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]].
The curve $\gamma$ is closed if $\gamma(a) = \gamma(b)$.
A closed curve is simple if, in addition, $\gamma(s) \ne \gamma(t)$ for all $s < t$ in $[a, b]$ other than the pair $s = a$, $t = b$.

## Remarks

The unit circle $\gamma(t) = (\cos t, \sin t)$, $t \in [0, 2\pi]$, is a simple closed curve.
The curve $t \mapsto (\sin 2t, \sin t)$, $t \in [0, 2\pi]$, is closed but not simple: it passes through the origin at $t = 0$, $t = \pi$ and $t = 2\pi$.
The line integral of a vector field around a closed curve is often written $\oint_\gamma F \cdot d\mathbf{r}$. It is the circulation of $F$ around $\gamma$.
A simple closed curve in the plane bounds a region. This is the setting of [[Green's theorem|0HR3N02317]].

## Explanations

### L1 human

A closed curve is a path that comes back to where it started, like a lap around a track.
It is simple if it never crosses itself on the way.
</document>

<document source="https://math2.tamnd.com/node/0D0GBNDKDH.md" tag="0D0GBNDKDH" kind="concept">
# Martingale

## Summary

A martingale is a stochastic process in which the conditional expectation of the next value, given the past, is the present value. It models the fortune of a player in a fair game.
The node [[martingale|0CDVK9RQBH]] gives the precise definition in discrete time.
</document>

<document source="https://math2.tamnd.com/node/0D0SMQYEJA.md" tag="0D0SMQYEJA" kind="statement">
# Fibonacci recurrence

## Statement

Let $n$ be a natural number, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n+2} = F_{n+1} + F_n.$$

## Remarks

With the values $F_0 = 0$ and $F_1 = 1$, the recurrence determines the whole sequence.
The recurrence extends to negative indices with $F_{-n} = (-1)^{n+1} F_n$.

## Explanations

### L1 human

To get the next Fibonacci number, add the last two.
From $3$ and $5$ you get $8$, and from $5$ and $8$ you get $13$.
</document>

<document source="https://math2.tamnd.com/node/0D198GPMTJ.md" tag="0D198GPMTJ" kind="definition">
# Dense set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $D \subseteq X$.
The set $D$ is dense in $X$ when its [[closure|0DRKT1QJR7]] is $X$, that is, $\overline{D} = X$.

## Remarks

Equivalently, $D$ is dense if and only if $D$ meets each nonempty open set of $X$.
In a metric space, $D$ is dense if and only if for each $x \in X$ and each $\varepsilon > 0$ there is $y \in D$ with $d(x, y) < \varepsilon$.
The rationals $\mathbb{Q}$ are dense in $\mathbb{R}$: this is the [[density of the rationals|0STX0WRDNG]]. The irrationals are also dense in $\mathbb{R}$.

## Explanations

### L1 human

A dense set has points everywhere in the space: near each point of the space there are points of the dense set.
Each real number has rational numbers as close to it as you want.
</document>

<document source="https://math2.tamnd.com/node/0D1AHYPM8N.md" tag="0D1AHYPM8N" kind="definition">
# Tautology

## Definition

Let $\varphi$ be a formula of [[propositional logic|09K9MWD2QQ]] with the propositional variables $p_1, \dots, p_n$.
The formula $\varphi$ is a tautology when $\varphi$ is true for each of the $2^n$ assignments of truth values to $p_1, \dots, p_n$.

## Remarks

Examples are $p \lor \neg p$ and $p \to p$.
A formula that is false for each assignment is a contradiction.
A [[truth table|05RSK4Q657]] decides in a finite number of steps whether a formula is a tautology.

## Explanations

### L1 human

A tautology is true only because of its form.
The sentence "it rains or it does not rain" is true whatever the weather.
Its form $p \lor \neg p$ is a tautology.
</document>

<document source="https://math2.tamnd.com/node/0D1C9Y1EXA.md" tag="0D1C9Y1EXA" kind="definition">
# Complex differentiability

## Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be a function, and let $z_0 \in U$.
The function $f$ is complex differentiable at $z_0$ if the limit
$$f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$$
exists in $\mathbb{C}$. Here $h$ runs through the nonzero [[complex numbers|09FTAET5Z4]] with $z_0 + h \in U$.
The number $f'(z_0)$ is the complex derivative of $f$ at $z_0$.
The limit means: for each $\varepsilon > 0$ there is $\delta > 0$ such that $\left| \frac{f(z_0 + h) - f(z_0)}{h} - f'(z_0) \right| < \varepsilon$ for all such $h$ with $0 < |h| < \delta$.

## Remarks

For example, $f(z) = z^2$ is complex differentiable at each $z_0$ with $f'(z_0) = 2 z_0$, because $\frac{(z_0 + h)^2 - z_0^2}{h} = 2 z_0 + h$.
The number $h$ can go to $0$ from each direction in the plane. This makes the condition much stronger than real differentiability, see [[the conjugate is not holomorphic|0Z8ATBPM5W]].
If $f$ is complex differentiable at $z_0$, then $f$ is continuous at $z_0$.
The [[Cauchy–Riemann equations|0T71SSYZ39]] describe complex differentiability with partial derivatives.

## Explanations

### L1 human

The definition is the same as the definition of the derivative of a real function, but $h$ is now a complex number.
The quotient must have the same limit when $h$ goes to $0$ along the real axis, along the imaginary axis, or along any other path.
</document>

<document source="https://math2.tamnd.com/node/0D1NY1ARDR.md" tag="0D1NY1ARDR" kind="concept">
# Geometric sequence

## Summary

A geometric sequence is a sequence in which each term is a fixed multiple of the term before it. The fixed factor is the common ratio. The node [[geometric sequence|0MNM54WB99]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0D1R4SVMBQ.md" tag="0D1R4SVMBQ" kind="statement">
# Correspondence theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]], let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map.
Let $\mathcal{A}$ be the set of the left [[ideals|0MYSA6MYNA]] $J$ of $R$ with $I \subseteq J$.
Let $\mathcal{B}$ be the set of the left ideals of $R/I$.
The map $\mathcal{A} \to \mathcal{B}$, $J \mapsto J/I = \pi(J)$, is a bijection.
Its inverse is $K \mapsto \pi^{-1}(K)$.
This bijection and its inverse preserve inclusion.

## Remarks

The same bijection holds for right ideals, for two-sided ideals, and for subrings that contain $I$.
For two-sided ideals $J \supseteq I$, the [[third isomorphism theorem|073C80EC63]] gives $(R/I)/(J/I) \cong R/J$.
The ideals of $\mathbb{Z}/12\mathbb{Z}$ correspond to the ideals $d\mathbb{Z}$ with $d \mid 12$, so there are six of them.

## Explanations

### L1 human

The ideals of a quotient ring $R/I$ are the ideals of $R$ that contain $I$, seen in the quotient.
So you can understand the ideals of $R/I$ from the ideals of $R$.
For example, the ideals of $\mathbb{Z}/12\mathbb{Z}$ come from the divisors $1, 2, 3, 4, 6, 12$ of $12$.
</document>

<document source="https://math2.tamnd.com/node/0D1VWQGZZ7.md" tag="0D1VWQGZZ7" kind="definition">
# Conjugate elements

## Definition

Let $G$ be a [[group|G5S1M9]] and let $a, b \in G$.
The element $b$ is conjugate to $a$ when there is $g \in G$ with $b = gag^{-1}$.
The element $gag^{-1}$ is called the conjugate of $a$ by $g$.

## Remarks

Conjugacy is an equivalence relation on $G$; its classes are the conjugacy classes.
In an [[abelian group|082GHNPJ53]], $gag^{-1} = a$, so each element is conjugate only to itself.
Conjugate elements have the same [[order|0WE2NHZJ7B]].
In the group of invertible matrices, conjugate matrices are similar matrices.

## Explanations

### L1 human

Conjugating $a$ by $g$ means: first undo $g$, then do $a$, then do $g$ again (reading the product from right to left).
The result does the same thing as $a$, but in a place that $g$ has moved.
For example, two rotations of a cube by the same angle about different axes are conjugate.
</document>

<document source="https://math2.tamnd.com/node/0D2MXBV21T.md" tag="0D2MXBV21T" kind="definition">
# Restriction of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $A$.
The restriction of $f$ to $S$ is the function $f|_S \colon S \to B$ with $f|_S(s) = f(s)$ for each $s \in S$.

## Remarks

The graph of $f|_S$ is $\{(s, f(s)) \mid s \in S\}$, a subset of the graph of $f$.
The function $f$ is an extension of $f|_S$.
Example: $x \mapsto x^2$ on $\mathbb{R}$ is not [[injective|0PVG1CRAJN]], but its restriction to $[0, \infty)$ is injective.

## Explanations

### L1 human

To restrict a function, you keep its rule and allow fewer inputs.
Restrictions often make a function injective, so that it has an inverse on a smaller domain.
</document>

<document source="https://math2.tamnd.com/node/0D2W7FS52T.md" tag="0D2W7FS52T" kind="statement">
# A set is closed iff it equals its closure

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S$ is [[closed|0G5AV9Q4RR]] in $X$ if and only if $\overline{S} = S$.

## Remarks

Proof: if $S$ is closed, then $\overline{S} \subseteq S$ by [[closure is the smallest closed superset|0SHC6Z304Z]], and $S \subseteq \overline{S}$ always holds. If $\overline{S} = S$, then $S$ is closed because the [[closure is closed|02TZDKG50Q]].
The dual statement is: $S$ is open if and only if $\operatorname{int} S = S$.

## Explanations

### L1 human

A closed set already contains all the points that you can approach from inside it.
So taking the closure adds nothing.
</document>

<document source="https://math2.tamnd.com/node/0D2Y6G4C4S.md" tag="0D2Y6G4C4S" kind="definition">
# Harmonic number

## Definition

Let $n$ be a natural number.
The $n$-th harmonic number is the rational number
$$H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}.$$

## Remarks

The first values are $H_0 = 0$, $H_1 = 1$, $H_2 = 3/2$, $H_3 = 11/6$ and $H_4 = 25/12$.
The harmonic numbers are the partial sums of the [[harmonic series|0CXTMAY5PX]].
They satisfy $H_n - \ln n \to \gamma$, where $\gamma$ is the [[Euler–Mascheroni constant|00C5P9JZBK]].

## Explanations

### L1 human

The harmonic numbers are the running totals of $1 + 1/2 + 1/3 + \cdots$.
They grow without bound, but very slowly. One needs $n = 12367$ terms to pass $10$.
</document>

<document source="https://math2.tamnd.com/node/0D37QFPEZE.md" tag="0D37QFPEZE" kind="statement">
# Integration by substitution

## Statement

Let $g \colon [a, b] \to \mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]], and let $f$ be continuous on an interval that contains $g([a, b])$.
Then
$$\int_a^b f(g(x)) \, g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du.$$

## Remarks

Let $F$ be an [[antiderivative|0NC2A3HWGT]] of $f$. By the [[chain rule|0RXX2PT1MQ]], $F \circ g$ is an antiderivative of $(f \circ g) \, g'$. Both sides equal $F(g(b)) - F(g(a))$.
The function $g$ need not be injective.
For example, with $u = x^2$, $\int_0^1 2x \cos(x^2) \, dx = \int_0^1 \cos u \, du = \sin 1$.
Mathlib states it as `intervalIntegral.integral_comp_mul_deriv`.

## Explanations

### L1 human

Substitution is the chain rule read backwards.
We replace an expression $g(x)$ by a new variable $u$, and $g'(x) \, dx$ by $du$.
</document>

<document source="https://math2.tamnd.com/node/0D3PFXEAKF.md" tag="0D3PFXEAKF" kind="statement">
# Consecutive Fibonacci numbers are coprime

## Statement

Let $(F_n)_{n \ge 0}$ be the Fibonacci sequence, given by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for $n \ge 0$.
Then $F_n$ and $F_{n+1}$ are [[coprime|0GFKP8F8AG]] for each $n \ge 0$, that is, $\gcd(F_n, F_{n+1}) = 1$.

## Remarks

A proof by induction: $\gcd(F_{n+1}, F_{n+2}) = \gcd(F_{n+1}, F_{n+2} - F_{n+1}) = \gcd(F_{n+1}, F_n)$, and $\gcd(F_0, F_1) = 1$.
A stronger fact is $\gcd(F_m, F_n) = F_{\gcd(m, n)}$ for all $m, n \ge 0$.
The [[Euclidean algorithm|0TRNT4X7J0]] on $F_{n+1}$ and $F_n$ takes many steps, and consecutive Fibonacci numbers give its worst case.

## Explanations

### L1 human

Two neighbours in the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, 13, \dots$ never have a common factor larger than $1$.
A common factor of two neighbours would also divide the term before them, and so on down to $F_1 = 1$.
</document>

<document source="https://math2.tamnd.com/node/0D3WA36K2Q.md" tag="0D3WA36K2Q" kind="concept">
# Order isomorphism

## Summary

An order isomorphism is a bijection between two partially ordered sets that keeps the order in both directions.
</document>

<document source="https://math2.tamnd.com/node/0D430QR02W.md" tag="0D430QR02W" kind="statement">
# Rouché's theorem

## Statement

Let $c \in \mathbb{C}$ and $r > 0$.
Let $f$ and $g$ be [[holomorphic|0DG0Z72CQD]] on an open set that contains the closed disc $\{ z : |z - c| \le r \}$.
Suppose that $|g(z)| < |f(z)|$ for each $z$ with $|z - c| = r$.
Then $f$ and $f + g$ have the same number of zeros in the open disc $|z - c| < r$, each zero counted as many times as its [[order|064N002TX4]].

## Remarks

The usual proof applies the [[argument principle|0PN2BXRS8G]] to $f$ and to $f + g$.
For example, on $|z| = 1$ we have $|z^5 + 1| \le 2 < 3 = |3z|$, so $z^5 + 3z + 1$ has exactly one zero in the open unit disc, as $3z$ has.
For a polynomial $p(z) = a_n z^n + \dots + a_0$ with $a_n \neq 0$, compare $p$ with $a_n z^n$ on a large circle; this gives $n$ zeros of $p$ and proves the [[fundamental theorem of algebra|0JP0W07Y9A]].

## Explanations

### L1 human

The theorem says that a small change does not change the number of zeros.
On the circle, $g$ is smaller than $f$, so $f + g$ stays close to $f$ in a relative sense.
Then $f + g$ turns around $0$ as many times as $f$, and so it has the same number of zeros inside.
</document>

<document source="https://math2.tamnd.com/node/0D474C7ES1.md" tag="0D474C7ES1" kind="concept">
# Convergents of a continued fraction

## Summary

The convergents of a continued fraction are the fractions obtained by cutting it off after finitely many terms.
The [[definition of the convergents|0SRWNR42BQ]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0D5KR33H7X.md" tag="0D5KR33H7X" kind="definition">
# Baire space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is a Baire space when it has the property that follows.
Let $(F_n)_{n \in \mathbb{N}}$ be a [[countable|0ZG61TFBES]] family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $\operatorname{int} F_n = \emptyset$ for each $n$.
Then also $\operatorname{int} \bigcup_n F_n = \emptyset$.

## Remarks

Here $\operatorname{int}$ is the [[interior|0C38S0KMJK]].
Equivalently, in a Baire space each [[countable intersection of dense open sets is dense|0X2T63GFQQ]].
In a Baire space, each [[meagre set|0FQPYAY9PJ]] has empty interior. Thus a nonempty Baire space is not meagre in itself.
Each complete metric space is a Baire space by the [[Baire category theorem|0A7KGAYHGV]]. The space $\mathbb{Q}$ is not a Baire space.

## Explanations

### L1 human

In a Baire space, countably many thin closed sets can never fill up an open set.
This says that the space is large. It lets you prove that some objects exist: you show that the bad cases form a small set.
</document>

<document source="https://math2.tamnd.com/node/0D5YHQGDJY.md" tag="0D5YHQGDJY" kind="definition">
# Subspace topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The subspace topology on $S$ is
$$\tau_S = \{ S \cap U : U \in \tau \}.$$
With this topology, $S$ is a subspace of $X$.

## Remarks

The set $\tau_S$ is a topology on $S$, because intersection with $S$ commutes with unions and finite [[intersections|0HBRW4E6G1]].
A set open in $S$ need not be open in $X$. For example, $[0, 1)$ is open in the subspace $[0, 1]$ of $\mathbb{R}$, because $[0, 1) = [0, 1] \cap (-1, 1)$.
For a metric space, the subspace topology is the metric topology of the [[subspace metric|0T2WYDTRP2]].
The subspace topology is the coarsest topology on $S$ for which the [[inclusion map is continuous|065J822CKN]].

## Explanations

### L1 human

A subset of a space inherits a topology: you cut each open set of the big space down to the subset.
This is how the circle and the interval get their topologies from the plane and the line.
</document>

<document source="https://math2.tamnd.com/node/0D6FCW80A0.md" tag="0D6FCW80A0" kind="statement">
# A continuous injective function on an interval is strictly monotone

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous and injective.
Then $f$ is [[strictly monotone|0TR9SR9FTB]].

## Remarks

Suppose $x < y < z$ in $I$ with $f(y)$ not between $f(x)$ and $f(z)$. Then the [[intermediate value theorem|00KAFBR32E]] gives a value of $f$ taken twice, which contradicts injectivity.
Continuity is needed. The function on $[0, 2]$ with $f(x) = x$ for $x < 1$ and $f(x) = 3 - x$ for $x \ge 1$ is injective but not monotone.
Mathlib states a form for closed intervals as `ContinuousOn.strictMonoOn_of_injOn_Icc`.

## Explanations

### L1 human

A continuous function on an interval that goes up and then down would take some value twice.
So if it never repeats a value, it must always go up or always go down.
</document>

<document source="https://math2.tamnd.com/node/0D6J4B671D.md" tag="0D6J4B671D" kind="concept">
# Probability mass function

## Summary

The probability mass function of a discrete random variable gives the probability of each single value.
The node [[probability mass function|0WF4EJP91W]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0D6KH8Y59N.md" tag="0D6KH8Y59N" kind="statement">
# Polynomial ring over a UFD is a UFD

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is a unique factorisation domain.

## Remarks

The proof uses [[Gauss's lemma|0QFV5Y1Z5X]] and unique factorisation in $K[x]$, where $K$ is the field of fractions of $R$.
The irreducible elements of $R[x]$ are the irreducible elements of $R$ and the primitive polynomials of positive degree that are irreducible in $K[x]$.
By induction, $R[x_1, \dots, x_n]$ is a unique factorisation domain. In particular, $\mathbb{Z}[x]$ and $K[x, y]$ are.
These rings are not [[principal ideal domains|0RZYDVZ475]]: $(2, x)$ in $\mathbb{Z}[x]$ and $(x, y)$ in $K[x, y]$ are not principal.

## Explanations

### L1 human

Integer polynomials factor uniquely into irreducible pieces, like integers do.
The pieces are prime numbers and irreducible primitive polynomials.
For example, $2x^2 - 2 = 2 (x - 1)(x + 1)$.
</document>

<document source="https://math2.tamnd.com/node/0D6KXBHKC0.md" tag="0D6KXBHKC0" kind="concept">
# Power series in the complex plane

## Summary

A power series is a series $\sum_{n \ge 0} a_n (z - c)^n$ with complex coefficients $a_n$, a center $c$ and a complex variable $z$, see the [[definition|0VARVQ9ZW1]].
It converges on a disc, and its sum is holomorphic there.
</document>

<document source="https://math2.tamnd.com/node/0D6RMJ10M9.md" tag="0D6RMJ10M9" kind="definition">
# Unitary matrix

## Definition

Let $n \ge 1$ and let $U$ be a complex $n \times n$ matrix.
The matrix $U$ is *unitary* when
$$U^* U = I_n,$$
where $U^*$ is the [[conjugate transpose|0DXKE10PHF]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].

## Remarks

Equivalently, $U$ is invertible with $U^{-1} = U^*$, and then also $U U^* = I_n$.
Equivalently, the columns of $U$ form an orthonormal basis of $\C^n$ for the standard inner product.
The eigenvalues of a unitary matrix have absolute value $1$, and $\abs{\det U} = 1$.
A real unitary matrix is an [[orthogonal matrix|0W02QESPJQ]].

## Explanations

### L1 human

Unitary matrices are the complex version of orthogonal matrices.
They keep all lengths and inner products in $\C^n$.
In quantum mechanics, they describe how a closed system evolves in time.
</document>

<document source="https://math2.tamnd.com/node/0D6WHTTH7T.md" tag="0D6WHTTH7T" kind="definition">
# Conjugate subgroup

## Definition

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
The conjugate of $H$ by $g$ is the set $gHg^{-1} = \{ghg^{-1} : h \in H\}$.
Two subgroups $H$ and $K$ of $G$ are conjugate when $K = gHg^{-1}$ for some $g \in G$.

## Remarks

The set $gHg^{-1}$ is a subgroup of $G$, and $h \mapsto ghg^{-1}$ is an isomorphism from $H$ to $gHg^{-1}$.
Conjugacy of subgroups is an equivalence relation on the set of subgroups of $G$.
A subgroup that equals each of its conjugates is a normal subgroup.

## Explanations

### L1 human

Conjugating a subgroup moves it to another position inside the group, but keeps its shape.
In the symmetric group $S_3$, the three subgroups of order $2$ are conjugate to each other.
Conjugate subgroups have the same order and the same structure.
</document>

<document source="https://math2.tamnd.com/node/0D752BG155.md" tag="0D752BG155" kind="statement">
# Zero is not a successor

## Statement

Let $S \colon \mathbb{N} \to \mathbb{N}$ be the [[successor function|0832XVR512]] of the natural numbers.
For each natural number $n$, $S(n) \neq 0$.

## Remarks

This is one of the [[Peano axioms|0QFT7Z76N5]]. In a construction of $\mathbb{N}$, it is a theorem.
With the [[injectivity of the successor function|0TBYMWPWWB]], it shows that the numbers $0, S(0), S(S(0)), \dots$ are all different.

## Explanations

### L1 human

When you count up from any natural number, you never get back to $0$.
Thus the natural numbers do not go around in a circle, like the hours on a clock do.
</document>

<document source="https://math2.tamnd.com/node/0D7EZ10V2R.md" tag="0D7EZ10V2R" kind="concept">
# Binary tree

## Summary

A binary tree is a recursive structure. It is empty, or it is a root with an ordered pair of binary trees as left and right subtrees. Binary trees are a basic data structure, and they are counted by the Catalan numbers. The node [[binary tree|02A8EMC83C]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0D7GRZ2ADX.md" tag="0D7GRZ2ADX" kind="statement">
# The exponential function is convex

## Statement

The [[exponential function|0RMVS7RXZX]] is strictly convex on $\mathbb{R}$.
That is, $e^{t x + (1 - t) y} < t e^x + (1 - t) e^y$ for all real $x \ne y$ and $t \in (0, 1)$.

## Remarks

The second derivative of $e^x$ is [[$e^x$ itself|0EX9GFJ0B3]], which is positive. So [[the second derivative test for convexity|0Z41C17TGP]] applies, in its strict form.
The tangent line at $0$ gives the inequality [[$1 + x \le e^x$|012C7GFF20]].
Mathlib states convexity as `convexOn_exp`.

## Explanations

### L1 human

The graph of $e^x$ bends upward everywhere.
Every chord of the graph lies above the graph.
</document>

<document source="https://math2.tamnd.com/node/0D81KWH9ZM.md" tag="0D81KWH9ZM" kind="definition">
# Bridge

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $e$ be an edge of $G$.
Let $G - e$ be the graph that $G$ gives when $e$ is deleted and all vertices are kept.
The edge $e$ is a bridge of $G$ if $G - e$ has more [[connected components|01VVP10MWJ]] than $G$.

## Remarks

Let $e = uv$. Then $e$ is a bridge if and only if $u$ and $v$ are not [[reachable|0J8XE7W68F]] from each other in $G - e$. In this case, deleting $e$ increases the number of components by exactly one.
In a [[tree|0MJ0JS72DZ]] every edge is a bridge. In a [[cycle graph|0QQ0F12962]] no edge is a bridge.
[[Bridges are the edges on no cycle|0FHM8DB6PM]].

## Explanations

### L1 human

A bridge is an edge that holds the graph together: delete it, and one piece breaks into two.
Think of the only road between two towns.
</document>

<document source="https://math2.tamnd.com/node/0D86GJEZ39.md" tag="0D86GJEZ39" kind="statement">
# Variance of a sum

## Statement

Let $X_1, \dots, X_n$ be real random variables on one probability space with $E[X_i^2] < \infty$ for each $i$. Then
$$\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2 \sum_{1 \le i < j \le n} \operatorname{Cov}(X_i, X_j).$$
In particular, $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2 \operatorname{Cov}(X, Y)$.

## Remarks

Proof: $\operatorname{Var}(S) = \operatorname{Cov}(S, S)$ for $S = \sum_i X_i$. Expand by the bilinearity of the [[covariance|0DJ8W3HP2D]]: $\operatorname{Cov}(S, S) = \sum_{i, j} \operatorname{Cov}(X_i, X_j)$.
Also $\operatorname{Var}(X - Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) - 2 \operatorname{Cov}(X, Y)$.
More generally, $\operatorname{Var}\left(\sum_i a_i X_i\right) = \sum_{i, j} a_i a_j \operatorname{Cov}(X_i, X_j)$ for real $a_i$.

## Explanations

### L1 human

The spread of a sum is the sum of the spreads plus a correction for how the terms move together.
If the terms tend to move in the same direction, the sum spreads more. If they tend to compensate each other, it spreads less.
</document>

<document source="https://math2.tamnd.com/node/0D8HE681SC.md" tag="0D8HE681SC" kind="concept">
# Construction of the reals from Cauchy sequences

## Summary

This construction defines the [[real numbers|08J2M1JYAG]] as limits of sequences of [[rational numbers|0NCJ5WXN6Z]].
A sequence $(x_n)$ of rational numbers is a Cauchy sequence when, for each rational $\varepsilon > 0$, there is $N$ with $|x_m - x_n| < \varepsilon$ for all $m, n \ge N$.
Two Cauchy sequences $(x_n)$ and $(y_n)$ are equivalent when $x_n - y_n$ tends to $0$: for each rational $\varepsilon > 0$, there is $N$ with $|x_n - y_n| < \varepsilon$ for all $n \ge N$.
A real number is an equivalence class of Cauchy sequences of rational numbers.
Addition and multiplication act term by term: $[(x_n)] + [(y_n)] = [(x_n + y_n)]$ and $[(x_n)] \cdot [(y_n)] = [(x_n y_n)]$.

## Remarks

The rational number $q$ corresponds to the class of the constant sequence $(q, q, q, \dots)$.
The sequence $1, 1.4, 1.41, 1.414, \dots$ of decimal approximations represents $\sqrt{2}$.
The result is a [[complete ordered field|0A9E6X0S3E]], with the [[order on the reals|0337CYB0VD]]. The same method completes any metric space.

## Explanations

### L1 human

A real number is something that rational numbers can approximate as well as you like.
The construction takes the approximating sequences themselves as the new numbers.
Two sequences that get closer and closer to each other count as the same real number.
</document>

<document source="https://math2.tamnd.com/node/0D8T984XEE.md" tag="0D8T984XEE" kind="statement">
# Limits are unique in Hausdorff spaces

## Statement

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $(x_n)_{n \ge 1}$ be a sequence in $X$.
Say that $(x_n)$ converges to $x \in X$ when for each [[neighbourhood|09CJ00H3NW]] $N$ of $x$ there is $n_0$ with $x_n \in N$ for all $n \ge n_0$.
If $(x_n)$ converges to $x$ and to $y$, then $x = y$.

## Remarks

Proof: if $x \neq y$, take disjoint open sets $U \ni x$ and $V \ni y$. For large $n$, the point $x_n$ is in both $U$ and $V$, which is impossible.
The same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit.
In an [[indiscrete space|0XJ02JBD6S]] with at least two points, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot come close to two different points at the same time, if the points have disjoint neighbourhoods.
So in Hausdorff spaces, you can speak of the limit of a sequence.
</document>

<document source="https://math2.tamnd.com/node/0D939CMME7.md" tag="0D939CMME7" kind="definition">
# Connected space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is connected when there are no [[open sets|01ZQQDCERP]] $U, V \subseteq X$ with $U \neq \varnothing$, $V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$.

## Remarks

Equivalently, the only subsets of $X$ that are both open and [[closed|0G5AV9Q4RR]] are $\varnothing$ and $X$.
Equivalently, each [[continuous map|0SY9ECQ19Q]] from $X$ to the discrete space $\{0, 1\}$ is constant.
The interval $[0, 1]$ is connected. The set $[0, 1] \cup [2, 3]$ is not connected.
Under this definition the empty space is connected. Some authors require a connected space to be nonempty.

## Explanations

### L1 human

A connected space cannot be split into two separate open parts.
Connectedness is the topological form of the idea that a space is in one piece.
</document>

<document source="https://math2.tamnd.com/node/0D98771SM0.md" tag="0D98771SM0" kind="definition">
# Tangent vector of a curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval, let $\gamma \colon I \to \mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]], and let $t_0 \in I$.
If the limit
$$\gamma'(t_0) = \lim_{h \to 0} \frac{\gamma(t_0 + h) - \gamma(t_0)}{h} = \big( \gamma_1'(t_0), \dots, \gamma_n'(t_0) \big)$$
exists, it is the tangent vector, or velocity vector, of $\gamma$ at $t_0$.
Its norm $\|\gamma'(t_0)\|$ is the speed of $\gamma$ at $t_0$.
If $\gamma'(t_0) \ne 0$, the unit tangent vector is $T(t_0) = \gamma'(t_0) / \|\gamma'(t_0)\|$, and the tangent line at $\gamma(t_0)$ is $\{\gamma(t_0) + s \, \gamma'(t_0) : s \in \mathbb{R}\}$.

## Remarks

At an end point of $I$, the limit is one-sided.
For the circle $\gamma(t) = (\cos t, \sin t)$, $\gamma'(t) = (-\sin t, \cos t)$. It is orthogonal to $\gamma(t)$ and has length $1$.
The tangent vector can be zero. For $\gamma(t) = (t^2, t^3)$, $\gamma'(0) = (0, 0)$, and the image has a cusp at the origin.
The tangent vector depends on the parametrisation, but for a [[regular curve|06JMR2XWXW]] its direction does not change under an increasing change of parameter.

## Explanations

### L1 human

The tangent vector points in the direction in which the curve moves at time $t_0$.
Its length tells how fast the point moves.
</document>

<document source="https://math2.tamnd.com/node/0D9NDGE9KP.md" tag="0D9NDGE9KP" kind="statement">
# Uniqueness of inverses

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $a, b \in G$.
If $ab = e$, then $b = a^{-1}$.

## Remarks

In particular, each element of a group has exactly one [[inverse|0ZNZH0EXFX]].
The same argument works in a [[monoid|0N6R902FG1]]: if $ba = e$ and $ac = e$, then $b = b(ac) = (ba)c = c$.
Two consequences are $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.

## Explanations

### L1 human

In a group, you only need to check one equation to know that $b$ is the inverse of $a$.
Multiply $ab = e$ on the left by $a^{-1}$, and you get $b = a^{-1}$.
Thus the notation $a^{-1}$ makes sense: there is only one inverse.
</document>

<document source="https://math2.tamnd.com/node/0DAECMZVRT.md" tag="0DAECMZVRT" kind="concept">
# Straightedge and compass construction

## Summary

A straightedge and compass construction draws points, lines and circles in the plane from given points with two ideal tools.
The straightedge draws the line through two constructed points, and it has no marks.
The compass draws the circle with a constructed centre through a constructed point.
New points are the intersections of the lines and circles drawn so far.
The [[constructible numbers|0FMN2MN7TM]] describe the points that these steps can reach.

## Remarks

Each step solves linear or quadratic equations.
So the coordinates of the new points lie in an extension of degree $1$ or $2$ of the field of the old coordinates.
This turns questions about constructions into questions about [[degrees of field extensions|0GS74FGFEK]].
It shows that [[doubling the cube|04SD3F4FCR]], [[trisecting the angle|00J3DCCEK5]] and [[squaring the circle|0BSXRER81B]] are impossible with these tools.

## Explanations

### L1 human

The ancient Greeks asked which figures can be drawn with an unmarked ruler and a compass alone.
Field theory gives the answer: a length can be drawn exactly when it comes from rational numbers by field operations and square roots.
</document>

<document source="https://math2.tamnd.com/node/0DAMHCT17R.md" tag="0DAMHCT17R" kind="concept">
# Euler's totient function

## Summary

Euler's totient function $\varphi$ gives, for each positive integer $n$, the number of integers from $1$ to $n$ that are coprime to $n$.
The [[definition of Euler's totient function|0VV04M41AN]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0DAX8PMAJ4.md" tag="0DAX8PMAJ4" kind="concept">
# Probability axioms

## Summary

The probability axioms of Kolmogorov are the three rules that a probability function $P$ on the events of a sample space $\Omega$ must obey.
First, $P(A) \ge 0$ for each [[event|00SQGJNRTB]] $A$.
Second, $P(\Omega) = 1$.
Third, for each sequence $A_1, A_2, \dots$ of pairwise disjoint events, $P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n)$. This is countable additivity.
A function that obeys these rules is a [[probability measure|0VGMEBVW1M]].

## Remarks

Andrey Kolmogorov stated these axioms in 1933.
The other rules of probability follow from the axioms. Examples are $P(\varnothing) = 0$, $P(A) \le 1$, the rule for the complement and the addition rule for two events.

## Explanations

### L1 human

The axioms say three things: a probability is never negative, something always happens, and the probabilities of events that cannot happen together add up.
For a fair coin, $P(\{H\}) = P(\{T\}) = 1/2$, and $P(\{H, T\}) = 1/2 + 1/2 = 1$.
</document>

<document source="https://math2.tamnd.com/node/0DB0R4N5XX.md" tag="0DB0R4N5XX" kind="statement">
# Mean of the geometric distribution

## Statement

Let $p \in (0, 1]$ and let $X$ have the [[geometric distribution|0XCVV5Q500]] with parameter $p$, so $P(X = k) = (1 - p)^k p$ for $k = 0, 1, 2, \dots$. Then
$$E[X] = \frac{1 - p}{p}.$$
For the number $T = X + 1$ of trials up to and including the first success, $E[T] = 1/p$.

## Remarks

A proof uses the [[tail sum formula|0Y6QNF1J49]]: $P(X \ge n) = (1 - p)^n$, so $E[X] = \sum_{n \ge 1} (1 - p)^n = (1 - p)/p$.
The variance of $X$ and of $T$ is $(1 - p)/p^2$.

## Explanations

### L1 human

If each trial succeeds with probability $p$, you need on average $1/p$ trials to get the first success.
For a fair die, you wait on average $6$ rolls for the first six.
</document>

<document source="https://math2.tamnd.com/node/0DBTE6HVBD.md" tag="0DBTE6HVBD" kind="statement">
# Matrix of the adjoint is the conjugate transpose

## Statement

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$, with [[orthonormal bases|0S9GGTR0EX]] $B$ and $C$.
Let $T \colon V \to W$ be a linear map.
Then the matrix of the [[adjoint|0R9NMSM1ZP]] is
$$[T^*]_{B,C} = \left([T]_{C,B}\right)^*,$$
where $^*$ on the right is the [[conjugate transpose|0DXKE10PHF]]. Here $[\,\cdot\,]$ is the [[matrix of a linear map|0YQSGNYE3K]].

## Remarks

Over $\R$, the matrix of $T^*$ is the transpose of the matrix of $T$.
The statement needs orthonormal bases; in other bases, the Gram matrices of the bases appear in the formula.
The proof compares entries: both are $\langle T(b_j), c_i \rangle$, up to conjugation.

## Explanations

### L1 human

In orthonormal coordinates, taking the adjoint of a map means taking the conjugate transpose of its matrix.
This makes the abstract adjoint easy to compute.
In other bases, the rule is more complicated.
</document>

<document source="https://math2.tamnd.com/node/0DBWXQRXQJ.md" tag="0DBWXQRXQJ" kind="concept">
# Torsion subgroup

## Summary

The torsion subgroup of an abelian group is the subgroup of its elements of finite order.
The quotient by the torsion subgroup is torsion-free.
</document>

<document source="https://math2.tamnd.com/node/0DC3MPR7C9.md" tag="0DC3MPR7C9" kind="definition">
# Open ball

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r > 0$ be a real number.
The open ball with centre $x$ and radius $r$ is the set
$$B(x, r) = \{ y \in X : d(x, y) < r \}.$$

## Remarks

The centre is in the ball: $x \in B(x, r)$, because $d(x, x) = 0 < r$.
In $\mathbb{R}$ with the usual metric, $B(x, r)$ is the open interval $(x - r, x + r)$.
In the [[discrete metric|0NA4FK7B8S]], $B(x, r) = \{x\}$ for $r \le 1$ and $B(x, r) = X$ for $r > 1$.
Some authors also allow $r \le 0$. Then $B(x, r)$ is empty.

## Explanations

### L1 human

The open ball $B(x, r)$ is the set of points that are less than $r$ away from $x$.
In the plane with the Euclidean metric, it is the inside of the circle of radius $r$ about $x$, without the circle itself.
</document>

<document source="https://math2.tamnd.com/node/0DCT0QZVQ1.md" tag="0DCT0QZVQ1" kind="definition">
# Expected value

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a [[probability space|0ZYBZ34RT2]] $(\Omega, \mathcal{F}, P)$.
If $X \ge 0$, the expected value of $X$ is the Lebesgue integral
$$E[X] = \int_\Omega X \, dP \in [0, \infty].$$
In general, write $X = X^+ - X^-$ with $X^+ = \max(X, 0)$ and $X^- = \max(-X, 0)$. If $E[|X|] < \infty$, then $X$ is integrable and $E[X] = E[X^+] - E[X^-]$.

## Remarks

For a [[discrete random variable|0J6Z6H5VQJ]] with values $x_1, x_2, \dots$, $E[X] = \sum_i x_i \, P(X = x_i)$ when the series converges absolutely.
For a [[random variable with density|0C21RZG5V2]] $f$, $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$ when $\int |x| f(x) \, dx < \infty$.
Other notations are $\mathbb{E}[X]$, $\mathbb{E}X$ and $\mu_X$. The expected value depends only on the law of $X$.
If $E[X^+] = \infty$ and $E[X^-] < \infty$, one sets $E[X] = +\infty$. If both are infinite, $E[X]$ is not defined.

## Explanations

### L1 human

The expected value is the long-run average of many independent repetitions of the experiment.
For a fair die, it is $(1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5$. This value never appears on the die, but it is the center of mass of the distribution.
</document>

<document source="https://math2.tamnd.com/node/0DDA6CY7GJ.md" tag="0DDA6CY7GJ" kind="definition">
# Two-sided ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A two-sided ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive group $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$ and $a r \in I$.

So a two-sided ideal is a [[left ideal|0MYSA6MYNA]] that is also a right ideal.

## Remarks

In a [[commutative ring|0GYGHKX834]], left ideals, right ideals and two-sided ideals are the same.
The [[kernel|0AAHE7GEW6]] of every [[ring homomorphism|0GGE0Z0JQ1]] is a two-sided ideal.
The [[quotient ring|0VFW4Y69BT]] $R/I$ exists exactly when $I$ is a two-sided ideal.
In the ring of $n \times n$ matrices over a field, the only two-sided ideals are $\{0\}$ and the whole ring.

## Explanations

### L1 human

In a noncommutative ring, $r a$ and $a r$ can be different.
A two-sided ideal must absorb products from both sides.
This is the condition that makes the quotient ring work.
</document>

<document source="https://math2.tamnd.com/node/0DDPZAG5ER.md" tag="0DDPZAG5ER" kind="definition">
# Right inverse

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
A right inverse of $f$ is a function $h \colon B \to A$ such that $f \circ h = \mathrm{id}_B$, that is, $f(h(b)) = b$ for each $b \in B$.

## Remarks

A function has a right inverse exactly when it is [[surjective|0XKH881BZ7]]. The proof uses the [[axiom of choice|0HTGB13FE2]]. See [[surjective iff it has a right inverse|0TYQNHVB4C]].
A right inverse is in general not unique.
A right inverse is [[injective|0PVG1CRAJN]].

## Explanations

### L1 human

For $f(x) = x^2$ from $\mathbb{R}$ to $[0, \infty)$, the square root $h(y) = \sqrt{y}$ is a right inverse: $(\sqrt{y})^2 = y$.
The function $y \mapsto -\sqrt{y}$ is another right inverse.
</document>

<document source="https://math2.tamnd.com/node/0DE5R5EB4D.md" tag="0DE5R5EB4D" kind="definition">
# Elementary row operations

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$, with rows $R_1, \dots, R_m$.
An *elementary row operation* on $A$ is one of these three operations:

1. Row switching: exchange $R_i$ and $R_j$ for some $i \ne j$.
2. Row scaling: replace $R_i$ by $c R_i$ for some $c \in F$ with $c \ne 0$.
3. Row addition: replace $R_i$ by $R_i + c R_j$ for some $i \ne j$ and some $c \in F$.

## Remarks

Each elementary row operation has an inverse operation of the same type.
Each elementary row operation on $A$ is the same as the product $EA$ for an [[elementary matrix|01REXW3WWE]] $E$.
Elementary row operations on an [[augmented matrix|0T6MWGZSXA]] do not change the solution set of the system.
Elementary column operations have the same definition with columns in place of rows.

## Explanations

### L1 human

The three row operations are the moves that we make when we solve equations by hand.
We can swap two equations, multiply an equation by a nonzero number, or add a multiple of one equation to another.
None of these moves gains or loses a solution, and each move can be undone.
</document>

<document source="https://math2.tamnd.com/node/0DEB0JMN0W.md" tag="0DEB0JMN0W" kind="concept">
# Harmonic function

## Summary

A harmonic function is a twice continuously differentiable function on an open subset of $\mathbb{R}^n$ whose [[Laplacian|0E0B6RA8K0]] is zero at each point.
Harmonic functions describe steady states, such as equilibrium temperatures and electrostatic potentials in empty space.
</document>

<document source="https://math2.tamnd.com/node/0DEBXAJ5SA.md" tag="0DEBXAJ5SA" kind="concept">
# Conjugacy class

## Summary

The conjugacy class of an element $a$ of a group $G$ is the set of all elements $gag^{-1}$ with $g \in G$.
The conjugacy classes partition the group.
</document>

<document source="https://math2.tamnd.com/node/0DFT1GP1JP.md" tag="0DFT1GP1JP" kind="example">
# Gaussian integral

## Example

The improper integral of $e^{-x^2}$ over $\mathbb{R}$ converges, and
$$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.$$
Convergence follows from the [[comparison test|0FR1BDKK10]] with $e^{-|x|}$ for $|x| \ge 1$.
For the value, let $I$ be the integral. In polar coordinates, $I^2 = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi$.

## Remarks

The function $e^{-x^2}$ has no elementary antiderivative. So the second fundamental theorem does not give the value.
A change of variables gives $\int_{-\infty}^{\infty} e^{-a x^2} \, dx = \sqrt{\pi / a}$ for $a > 0$. Mathlib states this as `integral_gaussian`.
Abraham de Moivre found this type of integral in 1733. Carl Friedrich Gauss published the exact value in 1809.

## Explanations

### L1 human

The bell curve $e^{-x^2}$ has total area $\sqrt{\pi}$.
This number gives the constant in the normal distribution of probability theory.
</document>

<document source="https://math2.tamnd.com/node/0DFWHQ839T.md" tag="0DFWHQ839T" kind="statement">
# Strong law of large numbers

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with $E[|X_1|] < \infty$ and $\mu = E[X_1]$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\bar{X}_n \to \mu$ [[almost surely|04P7D63W8P]]:
$$P\left(\lim_{n \to \infty} \bar{X}_n = \mu\right) = 1.$$

## Remarks

The strong law implies the [[weak law|0BFDHF2H7N]], because almost sure convergence implies convergence in probability.
A short proof exists if $E[X_1^4] < \infty$: then $E[(\bar{X}_n - \mu)^4] \le C/n^2$, and the first Borel–Cantelli lemma applies.
Etemadi proved that the result holds when the $X_i$ are only pairwise independent and identically distributed.
Conversely, if $E[|X_1|] = \infty$, then almost surely $\bar{X}_n$ does not converge to a finite limit.

## Explanations

### L1 human

The strong law says more than the weak law. Almost every infinite sequence of trials has sample means that converge to the true mean.
So, with probability $1$, the fraction of heads in an infinite sequence of fair coin tosses tends to $1/2$.
</document>

<document source="https://math2.tamnd.com/node/0DFYTNZ6AM.md" tag="0DFYTNZ6AM" kind="statement">
# Finite extensions are algebraic

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]].
Then $L/K$ is [[algebraic|09EYMKMSB9]].

## Remarks

The proof: let $n = [L : K]$ and $a \in L$. The $n + 1$ elements $1, a, \dots, a^n$ are linearly dependent over $K$. A linear dependence $c_0 + c_1 a + \dots + c_n a^n = 0$ gives a nonzero polynomial with root $a$.
So every element of $L$ has degree at most $[L : K]$ over $K$.
The converse is false, as the algebraic numbers show.
An extension generated by finitely many algebraic elements is finite, by the [[tower law|0A7KJ68XHT]].

## Explanations

### L1 human

In a finite-dimensional extension, the powers of an element cannot all be independent.
A dependence between the powers is a polynomial equation for the element.
</document>

<document source="https://math2.tamnd.com/node/0DFZNHJ0S2.md" tag="0DFZNHJ0S2" kind="statement">
# Commutativity of addition

## Statement

For all natural numbers $a$ and $b$, $a + b = b + a$.
Here $+$ is the [[addition of natural numbers|0W5DSDFQ4G]].

## Remarks

The proof from the recursive definition of addition uses the [[principle of mathematical induction|0ASJVEFA9J]] two times.
An operation $\ast$ on a set is commutative when $x \ast y = y \ast x$ for all $x$ and $y$. Multiplication of natural numbers is also commutative.
Subtraction of integers is not commutative: $3 - 1 \neq 1 - 3$.

## Explanations

### L1 human

The order of two numbers in a sum does not change the result: $2 + 5$ and $5 + 2$ are both $7$.
Think of two piles of stones: you count the same total when you count the second pile first.
</document>

<document source="https://math2.tamnd.com/node/0DG0Z72CQD.md" tag="0DG0Z72CQD" kind="definition">
# Holomorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function.
The function $f$ is holomorphic on $U$ if $f$ is [[complex differentiable|0D1C9Y1EXA]] at each point of $U$.
Let $z_0 \in \mathbb{C}$. A function $f$ is holomorphic at $z_0$ if $f$ is holomorphic on some open disc with center $z_0$.

## Remarks

Polynomials in $z$ are holomorphic on $\mathbb{C}$, and $z \mapsto 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$.
Sums, products, quotients with a nonzero denominator and compositions of holomorphic functions are holomorphic.
Each holomorphic function is analytic, see [[holomorphic functions are analytic|0826SGV3R0]]. For this reason, many authors use the words holomorphic and analytic for the same functions.
A function that is holomorphic on all of $\mathbb{C}$ is an [[entire function|0419794R0T]].

## Explanations

### L1 human

A holomorphic function is a function of a complex variable that has a complex derivative at each point of an open set.
This one condition has strong consequences: the function then has derivatives of all orders and is equal to its Taylor series near each point.
</document>

<document source="https://math2.tamnd.com/node/0DG10TVYPN.md" tag="0DG10TVYPN" kind="statement">
# Cauchy's integral formula

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then for each $a \in \mathbb{C}$ with $|a - c| < r$,
$$f(a) = \frac{1}{2 \pi i} \oint_{|z - c| = r} \frac{f(z)}{z - a} \, dz,$$
where the integral is the [[circle integral|0T7ABX4CWE]] over the circle $|z - c| = r$.

## Remarks

The values of $f$ on the circle determine the values of $f$ at each point inside the circle.
With $a = c$ and $z = c + r e^{i\theta}$, the formula gives $f(c) = \frac{1}{2\pi} \int_0^{2\pi} f(c + r e^{i\theta}) \, d\theta$. So $f(c)$ is the average of $f$ on the circle.
For $f = 1$, the formula is the [[integral of 1/(z - a)|0SXZ68HV6F]].
The formula gives [[the formula for the derivatives|0F0H5P5DED]], and it shows that [[holomorphic functions are analytic|0826SGV3R0]].

## Explanations

### L1 human

If you know a holomorphic function on a circle, you know it everywhere inside the circle.
The formula computes the value at an inside point $a$ as a weighted average of the values on the circle, with the weight $1/(z - a)$.
</document>

<document source="https://math2.tamnd.com/node/0DGDS5XY84.md" tag="0DGDS5XY84" kind="concept">
# Alternating bilinear form

## Summary

An alternating bilinear form is a bilinear form whose value on each pair of equal vectors is zero.
</document>

<document source="https://math2.tamnd.com/node/0DGZBQWZWN.md" tag="0DGZBQWZWN" kind="statement">
# Spectral theorem for Hermitian matrices

## Statement

Let $A$ be a [[Hermitian|0P17NGQQMQ]] $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ and a real diagonal matrix $D$ with
$$A = U D U^*.$$
If $A$ is a real [[symmetric matrix|0VR671PHE5]], then $U$ can be chosen real, that is, [[orthogonal|0W02QESPJQ]]: $A = Q D Q^T$.

## Remarks

The diagonal entries of $D$ are the eigenvalues of $A$, and the columns of $U$ are orthonormal eigenvectors.
This is the matrix form of the [[spectral theorem|06J4PXRG22]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = Q \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} Q^T$ with $Q = \frac{1}{\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$.

## Explanations

### L1 human

Every Hermitian matrix is diagonal after a unitary change of coordinates.
The new coordinate axes are perpendicular eigenvectors.
For real symmetric matrices, this is a rotation, possibly with a reflection, of the axes.
</document>

<document source="https://math2.tamnd.com/node/0DH7HMXP9F.md" tag="0DH7HMXP9F" kind="statement">
# Order of a permutation is the lcm of its cycle lengths

## Statement

Let $\sigma \in S_n$, and let $\sigma = c_1 c_2 \cdots c_r$ be its decomposition into disjoint cycles of lengths $k_1, \dots, k_r$.
Then the [[order|0WE2NHZJ7B]] of $\sigma$ is $\operatorname{lcm}(k_1, \dots, k_r)$.

## Remarks

A $k$-cycle has order $k$.
The disjoint cycles commute, so $\sigma^m = c_1^m \cdots c_r^m$, and this is the identity exactly when each $c_i^m$ is the identity.
For example, $(1\ 2)(3\ 4\ 5)$ has order $6$.

## Explanations

### L1 human

A permutation returns to the identity when each of its cycles is back at the start.
A cycle of length $k$ is back at the start after each multiple of $k$ steps.
So you need the first common multiple of all the cycle lengths.
</document>

<document source="https://math2.tamnd.com/node/0DJ8W3HP2D.md" tag="0DJ8W3HP2D" kind="definition">
# Covariance

## Definition

Let $X$ and $Y$ be real random variables on one probability space with $E[X^2] < \infty$ and $E[Y^2] < \infty$.
The covariance of $X$ and $Y$ is
$$\operatorname{Cov}(X, Y) = E\left[(X - E[X]) \, (Y - E[Y])\right].$$
$X$ and $Y$ are uncorrelated if $\operatorname{Cov}(X, Y) = 0$.

## Remarks

An equivalent formula is $\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y]$.
The covariance of $X$ with itself is the [[variance|007QED3E0Y]]: $\operatorname{Cov}(X, X) = \operatorname{Var}(X)$.
The covariance is symmetric and bilinear: $\operatorname{Cov}(aX + bY, Z) = a \operatorname{Cov}(X, Z) + b \operatorname{Cov}(Y, Z)$. It does not change when one adds constants to $X$ or $Y$.
The Cauchy–Schwarz inequality gives $|\operatorname{Cov}(X, Y)| \le \sigma_X \sigma_Y$.

## Explanations

### L1 human

Covariance says whether two quantities tend to move in the same direction.
Height and weight of people have positive covariance. The size of a covariance depends on the units, so one often uses the correlation coefficient to compare.
</document>

<document source="https://math2.tamnd.com/node/0DJ9Y9V610.md" tag="0DJ9Y9V610" kind="definition">
# Upper central series

## Definition

Let $G$ be a [[group|G5S1M9]].
The upper central series of $G$ is the sequence of normal subgroups $Z_0 = \{e\}$ and $Z_{i+1} = \{x \in G : [x, y] \in Z_i \text{ for all } y \in G\}$ for $i \geq 0$.

## Remarks

The first term $Z_1$ is the [[centre|0DZ93493VN]] $Z(G)$, and $Z_2$ is called the second centre.
For each $i$, $Z_{i+1}/Z_i$ is the centre of the [[quotient group|0KYX2N69J0]] $G/Z_i$.
For the quaternion group $Q_8$, the series is $\{1\} \subseteq \{1, -1\} \subseteq Q_8$.
A group with trivial centre, such as $S_3$, has $Z_i = \{e\}$ for each $i$.

## Explanations

### L1 human

The upper central series starts with the identity and first adds the centre.
Then it adds the elements that are central modulo the centre, and so on.
If this process reaches the whole group, the group is nilpotent.
</document>

<document source="https://math2.tamnd.com/node/0DJMGPN8K9.md" tag="0DJMGPN8K9" kind="statement">
# Estimation lemma

## Statement

Let $\gamma : [a, b] \to \mathbb{C}$ be a continuous and piecewise continuously differentiable path with length $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$.
Let $f$ be a complex function that is continuous on the image of $\gamma$.
Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ on the image of $\gamma$.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ satisfies
$$\left| \int_\gamma f(z) \, dz \right| \le M \, L(\gamma).$$

## Remarks

For a [[circle integral|0T7ABX4CWE]] over $|z - c| = R$, the bound is $\left| \oint f(z) \, dz \right| \le 2 \pi R M$.
Proof: $\left| \int_a^b g(t) \, dt \right| \le \int_a^b |g(t)| \, dt$ for a complex function $g$, and $|f(\gamma(t)) \gamma'(t)| \le M |\gamma'(t)|$.
A typical use shows that the integral over a large half-circle goes to $0$ as the radius goes to $\infty$.

## Explanations

### L1 human

The integral cannot be larger than the largest size of the function times the length of the path.
For example, if $|f| \le 1$ on a circle of radius $3$, then the integral over the circle has size at most $6\pi$.
</document>

<document source="https://math2.tamnd.com/node/0DKCECSYCF.md" tag="0DKCECSYCF" kind="concept">
# Quadratic residue

## Summary

A quadratic residue modulo $n$ is an integer that is congruent to a perfect square modulo $n$.
The [[definition of a quadratic residue|0SQDB87Z3T]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0DKMBWTD7P.md" tag="0DKMBWTD7P" kind="statement">
# Cosets have the same cardinality as the subgroup

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $g \in G$.
Then the map $h \mapsto gh$ is a bijection from $H$ to the [[left coset|03DEH43F4H]] $gH$. The map $h \mapsto hg$ is a bijection from $H$ to the [[right coset|0NE1DK7H4T]] $Hg$.
In particular, $|gH| = |Hg| = |H|$.

## Remarks

The inverse of $h \mapsto gh$ is $x \mapsto g^{-1}x$.
This fact and the partition of $G$ into left cosets give the proof of [[Lagrange's theorem|3C53TD]].

## Explanations

### L1 human

Multiplying by a fixed element never makes two different elements equal, by cancellation.
Thus a coset has exactly as many elements as the subgroup.
All cosets of a subgroup have the same size.
</document>

<document source="https://math2.tamnd.com/node/0DKRR22AEQ.md" tag="0DKRR22AEQ" kind="definition">
# Random variable

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $(E, \mathcal{E})$ be a set $E$ with a [[σ-algebra|04WSW8T3SY]] $\mathcal{E}$.
A random variable with values in $E$ is a [[function|0XHDTE5YMZ]] $X \colon \Omega \to E$ such that the [[preimage|0FGCFY1AZX]] $X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\}$ is in $\mathcal{F}$ for each $B \in \mathcal{E}$.
A real random variable is a random variable with values in $\mathbb{R}$, with the [[Borel σ-algebra|0BVHZXWKCD]] on $\mathbb{R}$.

## Remarks

The condition says that $X$ is a measurable function. It makes $\{X \in B\}$ an event, so $P(X \in B)$ is defined.
A real function $X$ on $\Omega$ is a random variable if and only if $\{X \le x\} \in \mathcal{F}$ for each $x \in \mathbb{R}$.
For two rolls of a die, $\Omega = \{1, \dots, 6\}^2$ and the sum $X(a, b) = a + b$ is a random variable.
Sums, products and continuous functions of real random variables are random variables.

## Explanations

### L1 human

A random variable is a number that you read off from the result of the experiment.
For two rolls of a die, the sum of the two numbers is a random variable.
It is $7$ for the result $(3, 4)$ and $12$ for the result $(6, 6)$.
</document>

<document source="https://math2.tamnd.com/node/0DMY75D6FV.md" tag="0DMY75D6FV" kind="statement">
# Basel problem

## Statement

The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges, and
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \cdots = \frac{\pi^2}{6}.$$

## Remarks

The convergence follows from the [[p-series test|0ZNSSK8ZK1]] with $p = 2$. The value is the hard part.
The value $\pi^2/6 \approx 1.6449$ is $\zeta(2)$, where $\zeta$ is the Riemann zeta function.
Mathlib states it as `hasSum_zeta_two`.

## History

Pietro Mengoli posed the problem in 1650.
Leonhard Euler solved it in 1734. The problem is named after Basel, the home town of Euler.

## Explanations

### L1 human

The sum of the reciprocals of the squares $1, 4, 9, 16, \dots$ is a finite number.
Surprisingly, this number involves $\pi$, which comes from circles.
</document>

<document source="https://math2.tamnd.com/node/0DNN9B6Q2F.md" tag="0DNN9B6Q2F" kind="concept">
# Correlation coefficient

## Summary

The correlation coefficient of two random variables is their covariance divided by the product of their standard deviations. It is a number in $[-1, 1]$ that measures the strength of a linear relation.
The node [[correlation coefficient|04CTY6XYWN]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0DNNT7DBFM.md" tag="0DNNT7DBFM" kind="definition">
# Linear equation

## Definition

Let $F$ be a field and let $n \ge 1$ be an integer.
A *linear equation* over $F$ in the unknowns $x_1, \dots, x_n$ is an equation of the form
$$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b,$$
where the coefficients $a_1, \dots, a_n$ are elements of $F$. The constant term $b$ is also an element of the field.
A *solution* of the equation is a tuple $(s_1, \dots, s_n) \in F^n$ such that $a_1 s_1 + \cdots + a_n s_n = b$.

## Remarks

The equation $2x + 3y = 6$ over $\R$ is linear, and $(3, 0)$ is a solution.
The equation $xy = 1$ is not linear, because it has a product of two unknowns.
Over $\R$ with $n = 2$, the solutions of a linear equation with $(a_1, a_2) \ne (0, 0)$ form a line in the plane.

## Explanations

### L1 human

A linear equation adds up the unknowns, each one times a fixed number, and sets the sum equal to a fixed number.
No unknown is squared, and no two unknowns are multiplied together.
With two unknowns over the real numbers, the solutions usually draw a straight line, and this gives the name "linear".
</document>

<document source="https://math2.tamnd.com/node/0DPF6WRZW6.md" tag="0DPF6WRZW6" kind="definition">
# Degree of a polynomial

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a [[polynomial|0Q9EZS90GY]] in $R[x]$.
If $f \neq 0$, the degree $\deg f$ of $f$ is the largest integer $d$ with $a_d \neq 0$.
The degree of the zero polynomial is $\deg 0 = -\infty$.

## Remarks

The degree of $3x^4 - x + 2$ in $\mathbb{Z}[x]$ is $4$. A nonzero constant polynomial has degree $0$.
Some authors leave $\deg 0$ undefined, or set it to $-1$. The convention $-\infty$ makes the formula $\deg(f g) = \deg f + \deg g$ hold also for $f = 0$ or $g = 0$.
For all $f, g \in R[x]$, $\deg(f + g) \le \max(\deg f, \deg g)$.
The coefficient $a_d$ with $d = \deg f$ is the [[leading coefficient|0P05RJC5XA]] of $f$.

## Explanations

### L1 human

The degree is the highest power of $x$ that really appears in the polynomial.
For example, $x^3 + 5x$ has degree $3$.
The zero polynomial has no nonzero coefficient, so it needs a special convention.
</document>

<document source="https://math2.tamnd.com/node/0DQ8ERJZNS.md" tag="0DQ8ERJZNS" kind="definition">
# Surface integral

## Definition

Let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] as in the definition of [[surface area|0WQE3G7B88]], let $S = r(D)$, and let $f \colon S \to \mathbb{R}$ be continuous.
The surface integral of $f$ over $S$ is
$$\iint_S f \, dS = \iint_D f\big( r(u, v) \big) \, \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.$$

## Remarks

The expression $dS = \| r_u \times r_v \| \, du \, dv$ is the surface element. With $f = 1$, the integral is the area of $S$.
The value does not depend on the parametrisation of $S$. For a surface made of several pieces, add the integrals over the pieces.
For the graph $z = g(x, y)$ over $D$, $\iint_S f \, dS = \iint_D f(x, y, g(x, y)) \sqrt{1 + g_x^2 + g_y^2} \, dx \, dy$.
Example: on the unit sphere $S$, symmetry gives $\iint_S x^2 \, dS = \iint_S y^2 \, dS = \iint_S z^2 \, dS$. These add up to the area $4\pi$, so $\iint_S z^2 \, dS = 4\pi / 3$.

## Explanations

### L1 human

A surface integral adds up the values of a function over a curved surface, each weighted by a small piece of area.
If $f$ is a mass density per unit area, the surface integral is the total mass of the surface.
</document>

<document source="https://math2.tamnd.com/node/0DQEHHVRBX.md" tag="0DQEHHVRBX" kind="concept">
# Slope field

## Summary

The slope field of a first-order equation $y' = f(x, y)$ is a picture with a short segment of slope $f(x, y)$ at many points $(x, y)$ of the plane.
The graphs of the solutions are tangent to these segments. See the [[definition|0RKQQQACY7]].
</document>

<document source="https://math2.tamnd.com/node/0DQG39K2NH.md" tag="0DQG39K2NH" kind="statement">
# K[x] is a Euclidean domain

## Statement

Let $K$ be a [[field|0QA8CHH8MG]].
Then the polynomial ring $K[x]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[degree|0DPF6WRZW6]] as Euclidean function.
That is, for all $f, g \in K[x]$ with $g \neq 0$, there are $q, r \in K[x]$ with $f = q g + r$ and $r = 0$ or $\deg r < \deg g$.

## Remarks

The ring $K[x]$ is an integral domain because $K$ is one. The existence of $q$ and $r$ is the [[division algorithm|0MES3KS0ZC]] over a field.
A consequence is that $K[x]$ is a principal ideal domain and a unique factorization domain.
The ring $\mathbb{Z}[x]$ is not a Euclidean domain, because it is [[not a principal ideal domain|06E3AG9TM2]].

## Explanations

### L1 human

Over a field, you can divide any polynomial by any nonzero polynomial and get a remainder of smaller degree.
This is the same situation as division with remainder for integers.
So all the theory of the Euclidean algorithm works for polynomials over a field.
</document>

<document source="https://math2.tamnd.com/node/0DQMDSVSXJ.md" tag="0DQMDSVSXJ" kind="statement">
# Universal property of the polynomial ring

## Statement

Let $R$ and $S$ be [[commutative rings|0GYGHKX834]], let $\varphi \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]] and let $s \in S$.
Then there is exactly one ring homomorphism $\Phi \colon R[x] \to S$ such that $\Phi(r) = \varphi(r)$ for all $r \in R$ and $\Phi(x) = s$.
It is given by $\Phi(a_0 + a_1 x + \dots + a_n x^n) = \varphi(a_0) + \varphi(a_1) s + \dots + \varphi(a_n) s^n$.

## Remarks

With $S = R$, $\varphi$ the identity and $s = c$, the map $\Phi$ is the [[evaluation homomorphism|0125QVNCFQ]] at $c$.
In the language of algebras: for each commutative $R$-algebra $A$ and each $a \in A$, there is exactly one $R$-algebra homomorphism $R[x] \to A$ that sends $x$ to $a$.
This property characterizes $R[x]$ up to a unique isomorphism.

## Explanations

### L1 human

A ring homomorphism from $R[x]$ is fixed by two choices: what it does on the coefficients, and where it sends $x$.
Every pair of choices works, and it gives exactly one homomorphism.
Evaluation at a number is the main example.
</document>

<document source="https://math2.tamnd.com/node/0DQPNJNPYK.md" tag="0DQPNJNPYK" kind="concept">
# Well-founded relation

## Summary

A relation is well-founded when each nonempty subset has a minimal element for it.
Then there is no infinite descending chain.
</document>

<document source="https://math2.tamnd.com/node/0DQVHMYAGA.md" tag="0DQVHMYAGA" kind="statement">
# Bézout's identity

## Statement

Let $a$ and $b$ be integers, and let $d = \gcd(a, b)$ be their [[greatest common divisor|02WF878BH1]].
Then there are integers $x$ and $y$ such that $a x + b y = d$.

## Remarks

The integers $x$ and $y$ are not unique. If $(x, y)$ is a solution, then $(x + k b / d, \; y - k a / d)$ is also a solution for each integer $k$, when $d \neq 0$.
The set of all integers $a x + b y$ is the set of the multiples of $d$.
The [[extended Euclidean algorithm|0964YQFSA8]] computes one pair $(x, y)$.
For example, $\gcd(12, 42) = 6$ and $12 \cdot (-3) + 42 \cdot 1 = 6$.

## Explanations

### L1 human

With coins of value $a$ and $b$, and if you can give and take back coins, you can pay exactly the amount $\gcd(a, b)$.
For example, with coins of $4$ and $6$, you pay $2 = 2 \cdot 4 - 1 \cdot 6$.
You cannot pay $1$ this way, because each sum $4 x + 6 y$ is even.
</document>

<document source="https://math2.tamnd.com/node/0DR81R1F2B.md" tag="0DR81R1F2B" kind="concept">
# Joint probability distribution

## Summary

The joint distribution of several random variables on one probability space is the distribution of the vector that they form.
It gives the probability of each event that involves several of the variables together. The node [[joint probability distribution|0QAR0XDZH4]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0DRF0F3XC9.md" tag="0DRF0F3XC9" kind="statement">
# Inequality of arithmetic and geometric means

## Statement

Let $n \ge 1$ and let $x_1, \dots, x_n$ be nonnegative real numbers.
Then
$$\sqrt[n]{x_1 x_2 \cdots x_n} \le \frac{x_1 + x_2 + \dots + x_n}{n},$$
with equality if and only if $x_1 = x_2 = \dots = x_n$.

## Remarks

For $n = 2$, the inequality $\sqrt{x y} \le (x + y)/2$ follows from $(\sqrt{x} - \sqrt{y})^2 \ge 0$.
The weighted form: for weights $w_i \ge 0$ with $w_1 + \dots + w_n = 1$, $x_1^{w_1} \cdots x_n^{w_n} \le w_1 x_1 + \dots + w_n x_n$.

## Explanations

### L1 human

The geometric mean of some nonnegative numbers is never larger than their average.
For example, for $2$ and $8$, the geometric mean is $\sqrt{16} = 4$ and the average is $5$.
Among all rectangles with a fixed perimeter, the square has the largest area. This is the case $n = 2$.
</document>

<document source="https://math2.tamnd.com/node/0DRKT1QJR7.md" tag="0DRKT1QJR7" kind="definition">
# Closure

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The closure of $S$ is the [[intersection|0839XNQXDK]] of all [[closed sets|0G5AV9Q4RR]] $F$ of $X$ with $S \subseteq F$.
It is written $\overline{S}$ or $\operatorname{cl} S$.

## Remarks

The family of these closed sets is not empty, because it contains $X$.
A point $x$ is in $\overline{S}$ if and only if each open set that contains $x$ meets $S$.
In $\mathbb{R}$, the closure of $(0, 1)$ is $[0, 1]$, and the closure of $\mathbb{Q}$ is $\mathbb{R}$.
The closure and the [[interior|0C38S0KMJK]] are related by $X \setminus \overline{S} = \operatorname{int}(X \setminus S)$.

## Explanations

### L1 human

The closure of a set adds to the set all the points that you can approach from inside it.
For the open interval $(0, 1)$, you add the two end points $0$ and $1$.
</document>

<document source="https://math2.tamnd.com/node/0DS2F7GK05.md" tag="0DS2F7GK05" kind="definition">
# Galois extension

## Definition

An [[algebraic extension|09EYMKMSB9]] $L/K$ is a Galois extension if it is [[normal|0CBRFQ7DVZ]] and [[separable|0N019F6RE9]].

## Remarks

For a finite extension $L/K$, these conditions are equivalent.

1. $L/K$ is Galois.
2. $L$ is the splitting field of a separable polynomial over $K$.
3. $|\operatorname{Aut}(L/K)| = [L : K]$.
4. $K$ is the [[fixed field|0KMJ6919WS]] of $\operatorname{Aut}(L/K)$.

Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ and every extension of finite fields $\mathbb{F}_{p^n}/\mathbb{F}_p$. The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not Galois.

## Explanations

### L1 human

A Galois extension has as many symmetries over the base field as its degree allows.
For these extensions, the subgroups of the Galois group describe all the intermediate fields.
</document>

<document source="https://math2.tamnd.com/node/0DS2QFYJHE.md" tag="0DS2QFYJHE" kind="concept">
# Fundamental system of solutions

## Summary

A fundamental system of solutions of a homogeneous linear equation of order $n$ is a basis of its space of solutions: $n$ linearly independent solutions.
See the [[definition|0H4KNNCHPY]].
</document>

<document source="https://math2.tamnd.com/node/0DS9KWFDK2.md" tag="0DS9KWFDK2" kind="statement">
# The rationals are countable

## Statement

The set $\mathbb{Q}$ of rational numbers is [[countably infinite|0SEZ3JGZP7]].

## Remarks

Proof: each rational number has a unique form $p/q$ with $p \in \mathbb{Z}$, $q \ge 1$ and $\gcd(p, q) = 1$. The map $p/q \mapsto (p, q)$ is an injection into $\mathbb{Z} \times \mathbb{N}$, which is countable. And $\mathbb{Q}$ is infinite.
The rationals are dense in $\mathbb{R}$, but they are countable, and $\mathbb{R}$ is [[not countable|0518MM6BE1]].

## Explanations

### L1 human

Write the fractions $p/q$ in a grid, with $p$ along one side and $q$ along the other.
List the grid diagonal by diagonal, and skip each fraction that is not in lowest terms.
Each rational number appears exactly one time.
</document>

<document source="https://math2.tamnd.com/node/0DSHAVH45F.md" tag="0DSHAVH45F" kind="concept">
# Definition by recursion

## Summary

A definition by recursion defines a function on the [[natural numbers|0MWKJ44V5A]] in two parts.
It gives the value at $0$, and a rule that gives the value at $n + 1$ from the value at $n$.
The recursion theorem says that such a definition always gives exactly one function.
Precisely, let $X$ be a set, let $a$ be an element of $X$ and let $g \colon \mathbb{N} \times X \to X$ be a function.
Then there is exactly one function $f \colon \mathbb{N} \to X$ with $f(0) = a$ and $f(n + 1) = g(n, f(n))$ for each $n$ in $\mathbb{N}$.

## Remarks

The factorial is an example: $0! = 1$ and $(n + 1)! = (n + 1) \cdot n!$.
The [[addition|0W5DSDFQ4G]] and the [[multiplication|0JRC32D6K2]] of natural numbers have definitions by recursion.
The proof of the recursion theorem uses the [[principle of mathematical induction|0ASJVEFA9J]].

## Explanations

### L1 human

A recursive definition tells you where to start and how to take one step.
For the factorial, you start with $0! = 1$, and each step multiplies by the next number.
Step by step, this gives $1! = 1$, $2! = 2$, $3! = 6$ and $4! = 24$.
</document>

<document source="https://math2.tamnd.com/node/0DT0TCV75N.md" tag="0DT0TCV75N" kind="definition">
# Set difference

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The set difference of $A$ and $B$ is the set
$$A \setminus B = \{x \in A \mid x \notin B\}.$$
It is also called the relative complement of $B$ in $A$.

## Remarks

Another notation is $A - B$.
Example: $\{1, 2, 3\} \setminus \{2, 4\} = \{1, 3\}$.
The set $B$ need not be a subset of $A$.
In general, $A \setminus B \neq B \setminus A$.

## Explanations

### L1 human

To compute $A \setminus B$, start with $A$ and remove each element that is also in $B$.
Elements of $B$ that are not in $A$ do not matter.
</document>

<document source="https://math2.tamnd.com/node/0DTCBCKEJP.md" tag="0DTCBCKEJP" kind="concept">
# Transvection

## Summary

A transvection is a linear map, or a matrix, of the form $I + cE_{ij}$ with $i \ne j$, that adds a multiple of one coordinate to another.
Geometrically it is a shear.
</document>

<document source="https://math2.tamnd.com/node/0DTXBQPA2S.md" tag="0DTXBQPA2S" kind="concept">
# Index of a subgroup

## Summary

The index of a subgroup $H$ in a group $G$ is the number of left cosets of $H$ in $G$.
For finite groups, it is $|G| / |H|$.
</document>

<document source="https://math2.tamnd.com/node/0DTY8B9HMS.md" tag="0DTY8B9HMS" kind="concept">
# Linear span

## Summary

The linear span of a set of vectors is the set of all linear combinations of the vectors.
It is the smallest subspace that contains the set.
</document>

<document source="https://math2.tamnd.com/node/0DVDCMA2DK.md" tag="0DVDCMA2DK" kind="concept">
# Probability distribution

## Summary

A probability distribution describes how probability is spread over the possible values of a random quantity.
Formally it is a probability measure on a space of values, such as the real line with its Borel sets. The node [[probability distribution|0QB913WM5E]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0DXD4QSZ8A.md" tag="0DXD4QSZ8A" kind="concept">
# Conjugate elements

## Summary

Two elements $a$ and $b$ of a group $G$ are conjugate when $b = gag^{-1}$ for some $g \in G$.
Conjugate elements play the same role in the group, up to a change of names given by $g$.
</document>

<document source="https://math2.tamnd.com/node/0DXKE10PHF.md" tag="0DXKE10PHF" kind="concept">
# Conjugate transpose

## Summary

The conjugate transpose of a complex matrix is the transpose of the matrix with each entry replaced by its complex conjugate.
</document>

<document source="https://math2.tamnd.com/node/0DYS9C3HH5.md" tag="0DYS9C3HH5" kind="statement">
# Stone's representation theorem for finite Boolean algebras

## Statement

An atom of a [[Boolean algebra|06GKBTAA5E]] $B$ is a minimal element of $B \setminus \{0\}$.
Let $B$ be a finite Boolean algebra, and let $\operatorname{At}(B)$ be its set of atoms.
Then the map $b \mapsto \{ x \in \operatorname{At}(B) : x \le b \}$ is an isomorphism from $B$ to the [[power set Boolean algebra|004J7ZEFAT]] of $\operatorname{At}(B)$.

## Remarks

A consequence: a finite Boolean algebra with $n$ atoms has $2^n$ elements.
The general theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space. For a finite Boolean algebra this space is finite and discrete, and its points correspond to the atoms.
The general theorem needs a weak form of the axiom of choice, but the finite case does not.

## History

Marshall Stone published the representation theorem for all Boolean algebras in 1936.

## Explanations

### L1 human

Each finite Boolean algebra is the algebra of all subsets of some finite set.
Each element is the set of atoms below it, and the atoms are the smallest nonzero elements.
</document>

<document source="https://math2.tamnd.com/node/0DZ93493VN.md" tag="0DZ93493VN" kind="concept">
# Centre of a group

## Summary

The centre of a group is the set of elements that commute with every element of the group.
It is a normal abelian subgroup.
</document>

<document source="https://math2.tamnd.com/node/0DZ9DKAKZR.md" tag="0DZ9DKAKZR" kind="concept">
# Free module

## Summary

A free module is a module that has a basis, that is, a linearly independent generating set.
</document>

<document source="https://math2.tamnd.com/node/0DZGSDHD92.md" tag="0DZGSDHD92" kind="statement">
# Irrationality of the square root of 2

## Statement

The [[square root|0VB8BFRABH]] $\sqrt{2}$ of $2$ is an [[irrational number|080T2TXVQV]].
That is, there are no integers $a$ and $b$ with $b \neq 0$ and $(a / b)^2 = 2$.

## Remarks

A classical proof: suppose that $a / b = \sqrt{2}$ in [[lowest terms|0R64A2CVN4]]. Then $a^2 = 2 b^2$, thus $a$ is even. Then $b$ is also even, which is a contradiction.
The theorem is a special case of the [[irrationality of the square root of a prime|0YHFMXJJAZ]].

## History

The ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.

## Explanations

### L1 human

No fraction squared gives exactly $2$.
Fractions such as $7/5$ and $141/100$ come close, but there is always an error.
This means that the diagonal of a square with side $1$ has a length that is not a ratio of whole numbers.
</document>

<document source="https://math2.tamnd.com/node/0DZPD2MG33.md" tag="0DZPD2MG33" kind="definition">
# Nilpotent group

## Definition

A [[group|G5S1M9]] $G$ is nilpotent when it has a central series. This is a finite sequence $\{e\} = N_0 \subseteq N_1 \subseteq \cdots \subseteq N_n = G$ of [[normal subgroups|0CB0DX5BVV]] of $G$ with $N_{i+1}/N_i \subseteq Z(G/N_i)$ for each $i < n$.
Here $Z(G/N_i)$ is the [[centre|0DZ93493VN]] of the [[quotient group|0KYX2N69J0]] $G/N_i$.

## Remarks

The condition $N_{i+1}/N_i \subseteq Z(G/N_i)$ means $[N_{i+1}, G] \subseteq N_i$.
Equivalently, the [[lower central series|0K98R7A4VC]] reaches $\{e\}$, or the [[upper central series|0XW54DH28V]] reaches $G$; the least such length is the nilpotency class.
The nilpotent groups of class at most $1$ are exactly the [[abelian groups|082GHNPJ53]].
The quaternion group $Q_8$ is nilpotent of class $2$, but $S_3$ is not nilpotent.

## Explanations

### L1 human

A nilpotent group is close to abelian in a strong sense.
Each layer of a central series commutes with the whole group, up to the lower layers.
Each nilpotent group is solvable, but the converse is false, for example for $S_3$.
</document>

<document source="https://math2.tamnd.com/node/0DZS7ZG0QY.md" tag="0DZS7ZG0QY" kind="definition">
# Length of a vector

## Definition

Let $n$ be a positive integer and let $v = (v_1, \dots, v_n)$ be a [[vector|0QN3D80AWP]] in $\mathbb{R}^n$.
The length of $v$ is the real number
$$\|v\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} = \sqrt{v \cdot v},$$
where $v \cdot v = v_1^2 + \dots + v_n^2$ is the dot product of $v$ with itself.
A unit vector is a vector of length 1.

## Remarks

For each vector $v$, $\|v\| \geq 0$, and $\|v\| = 0$ if and only if $v$ is the zero vector.
For each real number $c$, $\|c v\| = |c| \, \|v\|$. Thus for $v \neq 0$ the vector $v / \|v\|$ is a unit vector.
Other names for the length are the magnitude and the Euclidean norm. The notation $|v|$ is also common.

## Explanations

### L1 human

The length of a vector is how far the vector moves you. In the plane, the vector $(3, 4)$ is the hypotenuse of a right triangle with legs 3 and 4. So its length is $\sqrt{3^2 + 4^2} = 5$.
</document>

<document source="https://math2.tamnd.com/node/0DZXAXEJQ9.md" tag="0DZXAXEJQ9" kind="concept">
# Irrational number

## Summary

An irrational number is a [[real number|08J2M1JYAG]] that is not a [[rational number|0NCJ5WXN6Z]].
Examples are $\sqrt{2}$, $\pi$ and $e$.
</document>

<document source="https://math2.tamnd.com/node/0E021GF2B8.md" tag="0E021GF2B8" kind="statement">
# Number of nth roots of unity

## Statement

Let $n$ be a positive integer.
Then there are exactly $n$ complex $n$th [[roots of unity|0S61N75H45]].

## Remarks

The [[list of the nth roots of unity|057BH3JC9S]] has $n$ numbers $e^{2 \pi i k / n}$ with $0 \le k < n$.
They are pairwise different, because their arguments are different numbers in $[0, 2\pi)$.
A polynomial of degree $n$ has at most $n$ roots in a field, so $X^n - 1$ has no other roots.
In other fields there can be fewer: in $\mathbb{R}$, the only $4$th roots of unity are $1$ and $-1$.

## Explanations

### L1 human

The equation $z^n = 1$ has exactly $n$ complex solutions.
Over the real numbers, it has only one or two solutions, so most of the solutions are not real.
</document>

<document source="https://math2.tamnd.com/node/0E088ZP533.md" tag="0E088ZP533" kind="statement">
# Jordan–Hölder theorem

## Statement

Let $G$ be a [[group|G5S1M9]] and let $\{e\} = G_0 \trianglelefteq \cdots \trianglelefteq G_n = G$ and $\{e\} = H_0 \trianglelefteq \cdots \trianglelefteq H_m = G$ be two [[composition series|0XV2A49X6Q]] of $G$.
Then $n = m$, and there is a permutation $\sigma$ of $\{1, \dots, n\}$ with $G_i / G_{i-1} \cong H_{\sigma(i)} / H_{\sigma(i) - 1}$ for each $i$.

## Remarks

So the [[composition factors|0PHTC93M58]] and the composition length of $G$ are well defined.
For $G = \mathbb{Z}/n\mathbb{Z}$, the theorem gives the uniqueness of the prime factorisation of $n$.
One proof uses the Schreier refinement theorem, which says that two subnormal series have equivalent refinements.

## History

The theorem is named after Camille Jordan and Otto Hölder.

## Explanations

### L1 human

A group can be cut into simple layers in many ways.
The theorem says that the layers that you get are always the same, up to order.
This is the group version of the uniqueness of prime factorisation.
</document>

<document source="https://math2.tamnd.com/node/0E0B6RA8K0.md" tag="0E0B6RA8K0" kind="concept">
# Laplacian

## Summary

The Laplacian of a twice differentiable function on $\mathbb{R}^n$ is the [[divergence|0MWV6ZTK6S]] of its [[gradient|0ABK9V9Y64]].
It measures how the value of the function at a point differs from its average value on small spheres around the point.
</document>

<document source="https://math2.tamnd.com/node/0E0DNQK22P.md" tag="0E0DNQK22P" kind="statement">
# Primitive root theorem

## Statement

Let $n$ be an integer with $n \ge 2$.
There is a [[primitive root|0KM2CJ80G7]] modulo $n$ if and only if $n \in \{2, 4, p^k, 2p^k\}$.
Here $p$ is an odd prime and $k$ is a positive integer.

## Remarks

Equivalently, the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic exactly for these $n$.
The case $n = p$ is the [[existence of primitive roots modulo a prime|04XGKK8WQW]].
For example, there are primitive roots modulo $9$, $18$ and $25$, but not modulo $8$, $12$ or $15$.

## History

Carl Friedrich Gauss gave the first proof.
He defined primitive roots in his Disquisitiones Arithmeticae of 1801.

## Explanations

### L1 human

Modulo $12$, the residues coprime to $12$ are $1, 5, 7, 11$, and each of them squares to $1$.
So no single residue generates all four, and there is no primitive root modulo $12$.
The theorem lists exactly the moduli for which a generator exists.
</document>

<document source="https://math2.tamnd.com/node/0E10E0NW9V.md" tag="0E10E0NW9V" kind="concept">
# Square root function

## Summary

The square root function sends a nonnegative real number to the unique nonnegative number whose square it is. It is the inverse of $t \mapsto t^2$ on $[0, \infty)$. The node [[square root function|0V7E9KP6Q2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0E1MMCAY5B.md" tag="0E1MMCAY5B" kind="example">
# Euclidean metric

## Example

Let $n \ge 1$ be an integer.
For points $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}.$$
Then $d_2$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the Euclidean metric.

## Remarks

For $n = 1$, $d_2(x, y) = |x - y|$, the [[usual metric on the real line|0KXJTZSW9B]].
For $n = 2$, the formula is the theorem of Pythagoras for the segment from $x$ to $y$.
The triangle inequality for $d_2$ follows from the Cauchy–Schwarz inequality.

## Explanations

### L1 human

The Euclidean metric is the length of the straight segment between two points.
In the plane, the distance from $(0, 0)$ to $(3, 4)$ is $\sqrt{9 + 16} = 5$.
</document>

<document source="https://math2.tamnd.com/node/0E1ZZBHZ3Z.md" tag="0E1ZZBHZ3Z" kind="definition">
# Riccati equation

## Definition

Let $I$ be an open interval and let $q_0, q_1, q_2 : I \to \mathbb{R}$ be continuous functions, with $q_0$ and $q_2$ not identically zero.
A Riccati equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = q_0(x) + q_1(x)\, y + q_2(x)\, y^2.$$

## Remarks

If $q_2 = 0$, the equation is [[linear|0Z3YKRQZ48]]. If $q_0 = 0$, it is a [[Bernoulli equation|0ZB84QCDKR]] with $n = 2$.
If $y_1$ is one solution, then the substitution $y = y_1 + 1/v$ gives a linear first-order equation for $v$.
If $q_2$ has no zero and is differentiable, the substitution $y = -u' / (q_2 u)$ gives the second-order linear equation $u'' - \left(q_1 + q_2'/q_2\right) u' + q_0 q_2\, u = 0$.
Example: $y' = 1 + y^2$ has the solutions $y = \tan(x + C)$.

## Explanations

### L1 human

A Riccati equation is the next case after a linear equation: the right side is a quadratic polynomial in $y$.
In general, there is no formula for its solutions.
But if you know one solution, you can find all the others.
</document>

<document source="https://math2.tamnd.com/node/0E22JKS4W4.md" tag="0E22JKS4W4" kind="definition">
# Complete residue system

## Definition

Let $n$ be a positive integer.
A complete residue system modulo $n$ is a set $S$ of integers that contains exactly one element of each [[residue class|0QRV8FPK5M]] modulo $n$.

## Remarks

Equivalently, $S$ has exactly $n$ elements, and no two distinct elements of $S$ are congruent modulo $n$.
The set $\{0, 1, \dots, n - 1\}$ is the set of the least nonnegative residues. It is a complete residue system modulo $n$.
For odd $n$, the set $\{ -\tfrac{n-1}{2}, \dots, -1, 0, 1, \dots, \tfrac{n-1}{2} \}$ of the least absolute residues is also a complete residue system.
If $S$ is a complete residue system modulo $n$ and $\gcd(a, n) = 1$, then $\{ a s + b : s \in S \}$ is also a complete residue system for each integer $b$.

## Explanations

### L1 human

A complete residue system picks one representative from each box of the integers modulo $n$.
For $n = 4$, both $\{0, 1, 2, 3\}$ and $\{4, -3, 10, 7\}$ are complete residue systems.
</document>

<document source="https://math2.tamnd.com/node/0E2HSCKZ6C.md" tag="0E2HSCKZ6C" kind="statement">
# Orbit-stabiliser theorem

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
Then the map $gG_x \mapsto g \cdot x$ is a well defined bijection from the set of the [[left cosets|03DEH43F4H]] of the [[stabiliser|0AYMP41THN]] $G_x$ to the [[orbit|0YQPZB5Q1G]] $G \cdot x$.
In particular, $|G \cdot x| = [G : G_x]$.

## Remarks

The key equivalence is $g \cdot x = h \cdot x$ if and only if $g^{-1}h \in G_x$, if and only if $gG_x = hG_x$.
The bijection is compatible with the actions: it sends $k \cdot gG_x$ to $k \cdot (g \cdot x)$ for each $k \in G$.
So each transitive action of $G$ is isomorphic to the [[action on the left cosets|0AZG100DKH]] of a stabiliser.

## Explanations

### L1 human

Two group elements move $x$ to the same point exactly when they differ by an element that fixes $x$.
So the points of the orbit correspond to the cosets of the stabiliser.
A big stabiliser means a small orbit, and a small stabiliser means a big orbit.
</document>

<document source="https://math2.tamnd.com/node/0E2NYPVSP9.md" tag="0E2NYPVSP9" kind="statement">
# Dimension of the orthogonal complement

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$.
Then
$$\dim W + \dim W^\perp = \dim V,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.

## Remarks

It follows from $V = W \oplus W^\perp$.
For example, in $\R^3$, a line has a plane as orthogonal complement: $1 + 2 = 3$.
For a real matrix, it gives $\rank A + \dim \ker A = n$ once more, because $\ker A$ is the orthogonal complement of the row space.

## Explanations

### L1 human

A subspace and its orthogonal complement share the dimensions of the space.
If $W$ uses $k$ directions, then $W^\perp$ uses the remaining $n - k$ directions.
</document>

<document source="https://math2.tamnd.com/node/0E3G0GRPZ6.md" tag="0E3G0GRPZ6" kind="concept">
# Differential form

## Summary

A differential form is an object that one can integrate over curves, surfaces, volumes and their higher-dimensional analogues.
A $k$-form assigns to each point an alternating $k$-linear function of tangent vectors. Forms give a coordinate-free approach to multivariable calculus.
</document>

<document source="https://math2.tamnd.com/node/0E3MV6E6MG.md" tag="0E3MV6E6MG" kind="concept">
# Polynomial of an endomorphism

## Summary

Let $p$ be a polynomial and let $A$ be a square matrix or an endomorphism.
Then $p(A)$ is the result of substituting $A$ for the variable of $p$, with the identity in place of the constant $1$.
</document>

<document source="https://math2.tamnd.com/node/0E3X2G1ZQB.md" tag="0E3X2G1ZQB" kind="object">
# Algebra of endomorphisms

## Description

Let $F$ be a field and let $V$ be a vector space over $F$.
The *algebra of endomorphisms* $\End_F(V)$ is the set of [[endomorphisms|0J7H0YBFDJ]] of $V$, with pointwise addition, scalar multiplication, and composition as multiplication.
It is an associative algebra over $F$, with the identity map $\id_V$ as unit.
Thus $S(T + U) = ST + SU$, $(S + T)U = SU + TU$ and $a(ST) = (aS)T = S(aT)$ for all $S, T, U \in \End_F(V)$ and $a \in F$.

## Remarks

If $\dim V = n$, then the choice of a basis gives an algebra isomorphism $\End_F(V) \cong M_n(F)$.
For $\dim V \ge 2$, the algebra is not commutative.
Its group of units is the group $GL(V)$ of linear isomorphisms $V \to V$.

## Explanations

### L1 human

The endomorphisms of a space can be added, scaled and composed.
Composition plays the role of multiplication.
This makes them a ring that is also a vector space, like the square matrices.
</document>

<document source="https://math2.tamnd.com/node/0E3X7TXS67.md" tag="0E3X7TXS67" kind="concept">
# Limit point

## Summary

A limit point of a set $S$ is a point $x$ such that each neighbourhood of $x$ contains a point of $S$ other than $x$.
Limit points are the points that the subset approaches without using the point itself.
</document>

<document source="https://math2.tamnd.com/node/0E44JMG6NV.md" tag="0E44JMG6NV" kind="example">
# Base rate fallacy

## Example

A disease affects $1\%$ of a population. A test is positive for $90\%$ of the sick people and for $9\%$ of the healthy people.
Let $S$ be the event "sick" and $T$ the event "positive test". Then $P(S) = 0.01$, $P(T \mid S) = 0.9$ and $P(T \mid S^c) = 0.09$.
By the [[law of total probability|0W62TKMW7V]], $P(T) = 0.9 \cdot 0.01 + 0.09 \cdot 0.99 = 0.0981$.
By [[Bayes' theorem|0NM6E9VW04]], $P(S \mid T) = 0.009 / 0.0981 \approx 0.092$.
So only about $9\%$ of the people with a positive test are sick.
This is true although the test is right for $90\%$ of the sick and for $91\%$ of the healthy.

## Remarks

The base rate fallacy is the error to ignore the base rate $P(S)$ and to take $P(S \mid T)$ to be close to $P(T \mid S)$.
When the condition is rare, most positive results are false positives, even for a good test.

## Explanations

### L1 human

Among $10000$ people, $100$ are sick and $90$ of them test positive. Of the $9900$ healthy people, $891$ also test positive.
So of the $981$ positive tests, only $90$ come from sick people.
</document>

<document source="https://math2.tamnd.com/node/0E4GXX7SJP.md" tag="0E4GXX7SJP" kind="example">
# Blow-up in finite time

## Example

Let $x_0 > 0$ and consider the [[initial value problem|075VBY4K4X]] $x'(t) = x(t)^2$, $x(0) = x_0$.
[[Separation of variables|0V4E623THY]] gives the solution
$$x(t) = \frac{x_0}{1 - x_0 t}, \qquad t \in \left(-\infty, \frac{1}{x_0}\right).$$
This is the [[maximal solution|019VSXSFMQ]]: $x(t) \to \infty$ as $t \to 1/x_0$ from below.
So the solution blows up at the finite time $1/x_0$, although the field $f(x) = x^2$ is smooth on all of $\mathbb{R}$.

## Remarks

The blow-up time $1/x_0$ depends on the initial value. Larger initial values blow up sooner.
For $x_0 < 0$, the maximal interval is $(1/x_0, \infty)$, and for $x_0 = 0$ the solution is $0$ on $\mathbb{R}$.
The field $x^2$ is not globally Lipschitz. A [[globally Lipschitz field|0BFD3A6CRH]] has no blow-up.
Near the blow-up time, $x(t)$ behaves like $(1/x_0 - t)^{-1}$. This is a finite-time singularity.

## Explanations

### L1 human

The rate of growth is the square of the size, so the growth feeds itself very fast.
The solution reaches infinity in finite time, unlike exponential growth.
</document>

<document source="https://math2.tamnd.com/node/0E4WH0R7HG.md" tag="0E4WH0R7HG" kind="definition">
# Cayley table

## Definition

Let $S = \{a_1, \ldots, a_n\}$ be a finite set with $n$ elements, listed in a fixed order, and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The Cayley table of $(S, \ast)$ is the $n \times n$ table whose rows and columns are labelled $a_1, \ldots, a_n$.
The entry in row $i$ and column $j$ is $a_i \ast a_j$.

## Remarks

The operation is [[commutative|099E72KTF8]] exactly when the table is symmetric about its main diagonal.
In the Cayley table of a finite [[group|G5S1M9]], each element occurs exactly one time in each row and one time in each column.
Thus the Cayley table of a finite group is a Latin square.

## Explanations

### L1 human

A Cayley table is like the multiplication table from school, but for any finite set with an operation.
For the integers modulo $2$ under addition, the table has the rows $0, 1$ and $1, 0$.
You can read many properties from the table, for example if the operation is commutative.
</document>

<document source="https://math2.tamnd.com/node/0E5CXYNEPX.md" tag="0E5CXYNEPX" kind="concept">
# Directrix

## Summary

A directrix of a conic is a line with the following property. For each point of the conic, the distance to a focus divided by the distance to the line is the same constant. This constant is the eccentricity.
</document>

<document source="https://math2.tamnd.com/node/0E5G5FMT2R.md" tag="0E5G5FMT2R" kind="statement">
# Law of total probability

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space.
Let $B_1, B_2, \dots$ be a finite or countable family of pairwise disjoint events with $\bigcup_i B_i = \Omega$ and $P(B_i) > 0$ for each $i$.
Then for each event $A$,
$$P(A) = \sum_{i} P(A \mid B_i) \, P(B_i).$$

## Remarks

The events $B_i$ form a [[partition|09CCE86A6D]] of $\Omega$ into events of positive probability. The proof uses $A = \bigcup_i (A \cap B_i)$, countable additivity and the [[multiplication rule|0BS3RMX8HM]].
The case of two events $B$ and $B^c$ is the [[law of total probability for an event and its complement|0W62TKMW7V]].
Example: urn $1$ has $2$ red balls of $5$ and urn $2$ has $3$ red balls of $4$. Choose an urn with a fair coin and draw one ball. The ball is red with probability $\frac12 \cdot \frac25 + \frac12 \cdot \frac34 = \frac{23}{40}$.

## Explanations

### L1 human

Split the world into cases that do not overlap.
The chance of $A$ is the average of its chances in the cases, weighted by the chances of the cases.
</document>

<document source="https://math2.tamnd.com/node/0E70M92FE3.md" tag="0E70M92FE3" kind="statement">
# Convergents converge to the number

## Statement

Let $x$ be an irrational real number, and let $p_n / q_n$ be the [[convergents|0SRWNR42BQ]] of its [[continued fraction expansion|0SJ7Z6VKWK]].
Then
$$\left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} \le \frac{1}{q_n^2}$$
for each $n \ge 0$, and $p_n / q_n \to x$ as $n \to \infty$.

## Remarks

The convergents with even $n$ increase toward $x$, and the convergents with odd $n$ decrease toward $x$.
For a rational $x$, the last convergent is equal to $x$.
Conversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion.
For example, $\frac{355}{113}$ is a convergent of $\pi$, and $\left| \pi - \frac{355}{113} \right| < 3 \cdot 10^{-7}$.

## Explanations

### L1 human

The convergents of $x$ get closer and closer to $x$, and the error is smaller than one over the square of the denominator.
This makes them very good rational approximations of $x$.
</document>

<document source="https://math2.tamnd.com/node/0E741HSMJ1.md" tag="0E741HSMJ1" kind="statement">
# Fermat's Last Theorem for n = 4

## Statement

There are no positive integers $a$, $b$ and $c$ with $a^4 + b^4 = c^4$.

## Remarks

A stronger statement holds: $x^4 + y^4 = z^2$ has no solution in positive integers.
The theorem also follows from [[Fermat's right triangle theorem|086BE0J3FZ]], in the form that $x^4 - y^4 = z^2$ has no solution in positive integers.
The proof uses [[infinite descent|0EFYCFHYFG]] and the [[classification of Pythagorean triples|0KF3J2KAER]].
Since $4$ divides every power of $2$ above $2$, the case $n = 4$ settles [[Fermat's Last Theorem|0Y5X3SDBCZ]] for all such exponents.

## History

Fermat's proof of his right triangle theorem, by infinite descent, gives this case.

## Explanations

### L1 human

The equation $a^4 + b^4 = c^4$ has no whole-number solutions.
The proof assumes a solution and builds a smaller one from it, which leads to a contradiction.
</document>

<document source="https://math2.tamnd.com/node/0E7VBYGFF7.md" tag="0E7VBYGFF7" kind="statement">
# Riemann rearrangement theorem

## Statement

Let $\sum a_n$ be a real series that [[converges conditionally|02YRQXZDCS]].
Let $M$ be a real number.
Then there is a [[rearrangement|0HRJ2Y74T3]] $\sum a_{\sigma(n)}$ that converges to $M$.
There is also a rearrangement that diverges to $+\infty$, and one that diverges to $-\infty$.

## Remarks

The positive terms and the negative terms each give a divergent series, and $a_n \to 0$. Take positive terms until the partial sum exceeds $M$, then negative terms until it falls below $M$, and repeat.
So a real series converges absolutely if and only if all its rearrangements converge to the same sum.
For the [[alternating harmonic series|0JQ5Y3WWVE]], take two positive terms, then one negative term, and repeat. The sum becomes $\frac{3}{2} \ln 2$.

## History

Augustin-Louis Cauchy is usually credited with the first remark that a new order can change a sum. He studied the alternating harmonic series in 1833.
The theorem is named after Bernhard Riemann.

## Explanations

### L1 human

A conditionally convergent series has positive terms with an infinite total and negative terms with an infinite total.
By a careful choice of order, we can steer the partial sums to any target.
</document>

<document source="https://math2.tamnd.com/node/0E7W0EM6J6.md" tag="0E7W0EM6J6" kind="concept">
# Casting out nines

## Summary

Casting out nines is a method to check a computation with integers by the remainders modulo $9$.
It uses the fact that a natural number is [[congruent|0TPW25CBN6]] modulo $9$ to the sum of its decimal digits.
To check $a \cdot b = c$, one compares the digit sum of $c$ with the product of the digit sums of $a$ and $b$, all modulo $9$. The same works for sums and differences.

## Remarks

If the two results are not congruent modulo $9$, the computation is wrong. If they are congruent, the computation can still be wrong. For example, an exchange of two digits in the result is not detected.
The method works because of [[divisibility by 9|0ESHD7ZQDX]] and the compatibility of congruences with [[addition|00FWBN7F0Y]] and [[multiplication|072D5G4SK6]].

## Explanations

### L1 human

Check $46 \cdot 37 = 1702$. The digit sums are $10 \equiv 1$ and $10 \equiv 1$, so the product should be $1 \cdot 1 = 1$ modulo $9$.
The digit sum of $1702$ is $10 \equiv 1$, so the check passes.
A wrong result such as $1712$ has the digit sum $11 \equiv 2$, so the check finds the error.
</document>

<document source="https://math2.tamnd.com/node/0E89HQATM8.md" tag="0E89HQATM8" kind="concept">
# Similar matrices

## Summary

Two square matrices are similar when they are the matrices of the same endomorphism in two bases.
Equivalently, $B = P^{-1} A P$ for an invertible matrix $P$.
</document>

<document source="https://math2.tamnd.com/node/0E8HNVBCSP.md" tag="0E8HNVBCSP" kind="statement">
# Exponential of a sum

## Statement

For all real numbers $x$ and $y$,
$$\exp(x + y) = \exp(x) \exp(y).$$

## Remarks

Multiply the two series by the [[Cauchy product|0AN28XQ45Z]]. The binomial theorem gives $\sum_{k=0}^{n} \frac{x^k}{k!} \frac{y^{n-k}}{(n-k)!} = \frac{(x + y)^n}{n!}$.
In particular, $\exp(x) \exp(-x) = \exp(0) = 1$.
So $\exp$ is a group homomorphism from $(\mathbb{R}, +)$ to the multiplicative group $(0, \infty)$. Mathlib states the formula as `Real.exp_add`.

## Explanations

### L1 human

The exponential function turns addition into multiplication.
This is the rule $e^{x + y} = e^x e^y$ for powers.
</document>

<document source="https://math2.tamnd.com/node/0E9502NT92.md" tag="0E9502NT92" kind="definition">
# Independence of two random variables

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X$ be a [[random variable|0DKRR22AEQ]] with values in $(E_1, \mathcal{E}_1)$ and $Y$ a random variable with values in $(E_2, \mathcal{E}_2)$.
$X$ and $Y$ are independent if for all $A \in \mathcal{E}_1$ and $B \in \mathcal{E}_2$,
$$P(X \in A, \, Y \in B) = P(X \in A) \, P(Y \in B).$$

## Remarks

In other words, the events $\{X \in A\}$ and $\{Y \in B\}$ are [[independent|0P0MZ4X682]] for all $A$ and $B$.
For real random variables, it is enough to check $P(X \le x, Y \le y) = P(X \le x) \, P(Y \le y)$ for all real $x$ and $y$.
For discrete random variables, independence is equivalent to $P(X = x, Y = y) = P(X = x) \, P(Y = y)$ for all $x$ and $y$.
If $X$ and $Y$ are independent and $g$, $h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent.

## Explanations

### L1 human

Two random variables are independent when knowing the value of one tells you nothing about the other.
The results of two separate dice are independent. The result of one die and the sum of both dice are not.
</document>

<document source="https://math2.tamnd.com/node/0E99F51GVB.md" tag="0E99F51GVB" kind="concept">
# Exponential distribution

## Summary

The exponential distribution with rate $\lambda$ is the law of a waiting time with no memory.
An example is the time between two events of a Poisson process.
The node [[exponential distribution|028Z8WC5Y9]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0E9GTAV478.md" tag="0E9GTAV478" kind="definition">
# Range of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The range of $f$ is the set
$$\operatorname{ran}(f) = \{f(a) \mid a \in A\} = \{b \in B \mid \exists a \in A, \, f(a) = b\}.$$

## Remarks

The range is also called the image of $f$, and it is the [[image|0Q3F20XB0A]] $f(A)$ of the whole domain.
Some authors use "range" for the [[codomain|0F9JHP8AJN]]. This node uses the meaning above.
For $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$, the range is $[0, \infty)$.

## Explanations

### L1 human

The range is the set of outputs that actually occur.
It can be smaller than the codomain.
</document>

<document source="https://math2.tamnd.com/node/0E9M24KS9J.md" tag="0E9M24KS9J" kind="statement">
# Harmonic series diverges

## Statement

The partial sums $H_N = \sum_{n=1}^{N} \frac{1}{n}$ of the [[harmonic series|0CXTMAY5PX]] tend to $+\infty$ as $N \to \infty$.
In particular, the harmonic series is a divergent series.

## Remarks

The classical proof groups the terms in blocks from $2^{k-1} + 1$ to $2^k$. Each block has $2^{k-1}$ terms, each at least $1/2^k$, so each block sums to at least $1/2$. Thus $H_{2^k} \ge 1 + k/2$.
A second proof uses the [[integral test|019QZ6RRVH]]: $H_N \ge \int_1^{N+1} dx/x = \ln(N + 1)$.
Mathlib states it as the limit of $\sum_{n < N} 1/(n+1)$, in `Real.tendsto_sum_range_one_div_nat_succ_atTop`.

## History

Nicole Oresme proved the divergence in the 14th century, with the grouping argument. Pietro Mengoli and Jacob Bernoulli gave other proofs in the 17th century.

## Explanations

### L1 human

Group the terms: $1/3 + 1/4 \ge 1/2$, then $1/5 + \cdots + 1/8 \ge 1/2$, then $1/9 + \cdots + 1/16 \ge 1/2$, and so on.
There are infinitely many such groups, so the sum exceeds each bound.
</document>

<document source="https://math2.tamnd.com/node/0EA4XFB0EY.md" tag="0EA4XFB0EY" kind="concept">
# Ceiling function

## Summary

The ceiling function sends a real number $x$ to the least integer that is greater than or equal to $x$. We write $\lceil x \rceil$ for it.
For example, $\lceil 2.3 \rceil = 3$ and $\lceil -2.3 \rceil = -2$.
</document>

<document source="https://math2.tamnd.com/node/0EA5BTA0CX.md" tag="0EA5BTA0CX" kind="concept">
# Division ring

## Summary

A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse.
It is like a field, but the multiplication need not be commutative.
The [[definition of a division ring|0WZXK05ME7]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0EAAS2EHDR.md" tag="0EAAS2EHDR" kind="statement">
# Chain rule in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ and $W \subseteq \mathbb{R}^m$ be open sets, let $g \colon U \to \mathbb{R}^m$ with $g(U) \subseteq W$, let $f \colon W \to \mathbb{R}^k$, and let $a \in U$.
Suppose that $g$ is [[differentiable|0135XZZQ8M]] at $a$ and $f$ is differentiable at $g(a)$.
Then the [[composition|0THBK9V825]] $f \circ g \colon U \to \mathbb{R}^k$ is differentiable at $a$, and
$$D(f \circ g)(a) = Df(g(a)) \circ Dg(a).$$

## Remarks

In matrix form, the [[Jacobian matrix|047NDTEPCV]] of $f \circ g$ at $a$ is the product $J_f(g(a)) \, J_g(a)$. In terms of partial derivatives, $\dfrac{\partial (f \circ g)_i}{\partial x_j}(a) = \sum_{l=1}^m \dfrac{\partial f_i}{\partial y_l}(g(a)) \, \dfrac{\partial g_l}{\partial x_j}(a)$.
The same rule, with the same proof, holds for [[Fréchet derivatives|08EVNYZF42]] of maps between normed spaces.
A special case is the [[chain rule for a composition with a curve|0A8TJ0VHC7]].

## Explanations

### L1 human

The derivative of a composition is the composition of the derivatives: first the linear approximation of $g$, then the linear approximation of $f$.
In one variable this is $(f \circ g)'(a) = f'(g(a)) \, g'(a)$, a product of two numbers. In several variables the numbers become matrices.
</document>

<document source="https://math2.tamnd.com/node/0EAJ2XXADC.md" tag="0EAJ2XXADC" kind="concept">
# Degree of a field extension

## Summary

The degree of a field extension $L/K$ is the dimension of $L$ as a vector space over $K$.
It is written $[L : K]$.
The [[definition of the degree|0GS74FGFEK]] states it.
</document>

<document source="https://math2.tamnd.com/node/0EAR3KNRTN.md" tag="0EAR3KNRTN" kind="concept">
# Trigonometric substitution

## Summary

Trigonometric substitution is a [[substitution|0D37QFPEZE]] method for integrals that contain $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$. We set $x$ equal to $a \sin \theta$, $a \tan \theta$, or $a \sec \theta$. The [[Pythagorean identity|03H7WZ4T3K]] then removes the square root.

## Remarks

With $x = a \sin \theta$ for $\theta \in [-\pi/2, \pi/2]$, we get $\sqrt{a^2 - x^2} = a \cos \theta$, for $a > 0$.
For example, this substitution gives $\int_{-1}^{1} \sqrt{1 - x^2} \, dx = \pi/2$, the area of a half disc.
Hyperbolic functions can replace the trigonometric functions in the second and the third case.

## Explanations

### L1 human

A square root such as $\sqrt{1 - x^2}$ is the side of a right triangle.
Writing $x$ as the sine of an angle turns the root into a cosine.
</document>

<document source="https://math2.tamnd.com/node/0EB6BBMQMC.md" tag="0EB6BBMQMC" kind="statement">
# A composite number has a prime factor at most its square root

## Statement

Let $n$ be a [[composite number|0N3ADXHNTQ]].
Then $n$ has a prime factor $p$ with $p^2 \le n$, that is, $p \le \sqrt{n}$.

## Remarks

The [[smallest prime factor|0S2SYHTWPY]] $p$ of $n$ has this property. Write $n = p m$. Then $m > 1$, and each prime factor of $m$ is at least $p$, so $m \ge p$ and $n \ge p^2$.
This result justifies [[trial division|0Z962ECCXR]]: to test whether $n > 1$ is prime, it is enough to test the primes $p \le \sqrt{n}$.
For example, $\sqrt{97} < 10$ and none of $2, 3, 5, 7$ divides $97$, so $97$ is prime.

## Explanations

### L1 human

If $n = a b$ with both factors larger than $\sqrt{n}$, then $a b > n$, which is impossible.
So one of the two factors is at most $\sqrt{n}$, and a prime factor of that factor is also at most $\sqrt{n}$.
</document>

<document source="https://math2.tamnd.com/node/0EB9N0S33C.md" tag="0EB9N0S33C" kind="concept">
# Invertible matrix

## Summary

An invertible matrix is a square matrix that has a two-sided inverse for matrix multiplication.
</document>

<document source="https://math2.tamnd.com/node/0EBACQFAZW.md" tag="0EBACQFAZW" kind="concept">
# Cardinal number

## Summary

Cardinal numbers measure the size of sets, finite or infinite.
They generalize the natural numbers as answers to the question "how many elements?".
</document>

<document source="https://math2.tamnd.com/node/0EBCWQ8BVE.md" tag="0EBCWQ8BVE" kind="concept">
# Probability-generating function

## Summary

The probability-generating function of a random variable with values in $\{0, 1, 2, \dots\}$ is the power series whose coefficients are the probabilities $P(X = k)$.
The node [[probability-generating function|0Z8PSSAVBA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0EBE031B8E.md" tag="0EBE031B8E" kind="statement">
# A finite integral domain is a field

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] that has a finite number of elements.
Then $R$ is a [[field|0QA8CHH8MG]].

## Remarks

The proof: for $a \neq 0$, the map $x \mapsto a x$ from $R$ to $R$ is injective by [[cancellation|0GJM24EXP9]]. A finite set has no injective map to itself that is not surjective, so $a x = 1$ for some $x$.
A consequence is that [[$\mathbb{Z}/p\mathbb{Z}$ is a field|0NV6E43P4A]] for every prime $p$.
The hypothesis of finiteness is necessary: $\mathbb{Z}$ is an infinite integral domain that is not a field.

## Explanations

### L1 human

In a finite integral domain, multiplication by a nonzero element $a$ never sends two elements to the same element.
Since there are only finitely many elements, every element is a product $a x$.
In particular, $1 = a x$ for some $x$, and $x$ is the inverse of $a$.
</document>

<document source="https://math2.tamnd.com/node/0EBZ3G455G.md" tag="0EBZ3G455G" kind="statement">
# The multiplicative group of a finite field is cyclic

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
Then the multiplicative group $F^\times$ is cyclic of order $q - 1$.

## Remarks

This is the case $G = F^\times$ of the [[theorem on finite subgroups|0CKGBEJX6B]].
A generator of $F^\times$ is a [[primitive element|0YAA9CJQXF]] of $F$.
For $F = \mathbb{Z}/p\mathbb{Z}$, the statement says that there is a primitive root modulo $p$.
For example, $3$ generates $(\mathbb{Z}/7\mathbb{Z})^\times$, with powers $3, 2, 6, 4, 5, 1$.

## Explanations

### L1 human

All the nonzero elements of a finite field are powers of one element.
So multiplication in a finite field is like addition of exponents modulo $q - 1$.
</document>

<document source="https://math2.tamnd.com/node/0EC7AGJ0PX.md" tag="0EC7AGJ0PX" kind="definition">
# Exact equation

## Definition

Let $D$ be an open subset of $\mathbb{R}^2$ and let $M, N : D \to \mathbb{R}$ be continuous functions.
The equation
$$M(x, y) + N(x, y)\, y' = 0,$$
also written $M\, dx + N\, dy = 0$, is exact on $D$ if there is a differentiable function $\psi : D \to \mathbb{R}$ with
$$\frac{\partial \psi}{\partial x} = M \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N \quad \text{on } D.$$
Such a function $\psi$ is a potential function of the equation.

## Remarks

Let the equation be exact. Then a differentiable function $y$ on an interval, with graph in $D$, is a [[solution|04SW7WECS8]] if and only if $\psi(x, y(x))$ is constant. This follows from the chain rule: $\frac{d}{dx} \psi(x, y(x)) = M + N y'$.
So the solutions are given implicitly by $\psi(x, y) = C$.
Example: $2 x y + x^2 y' = 0$ is exact with $\psi = x^2 y$, and its solutions satisfy $x^2 y = C$.
The [[test for exactness|0MSCQNFTM8]] decides whether an equation is exact.

## Explanations

### L1 human

An exact equation is the total derivative of one function $\psi$ set to zero.
So $\psi$ stays constant along each solution, and the solution curves are the level curves of $\psi$.
To solve the equation, you find $\psi$ by integration.
</document>

<document source="https://math2.tamnd.com/node/0ECVK59Y03.md" tag="0ECVK59Y03" kind="object">
# Empty set

## Description

The empty set is the [[set|0SJ3RX9TXA]] that has no [[elements|0EZ758XE73]].
It is written $\varnothing$ or $\{\}$.
For each object $x$, $x \notin \varnothing$.
By the [[axiom of extensionality|0TB16T2C97]], there is only one empty set.

## Remarks

The empty set is a [[subset|0FE8Q5S42Y]] of every set. This is a case of [[vacuous truth|0V4QXPGJQK]].
Its [[cardinality|0JTGRV4WT4]] is $0$.
The set $\{\varnothing\}$ is not empty: it has one element.

## Explanations

### L1 human

The empty set is an empty bag.
For example, the set of real numbers $x$ with $x^2 = -1$ is empty.
It is useful in the same way that the number $0$ is useful.
</document>

<document source="https://math2.tamnd.com/node/0ED4HWNDGW.md" tag="0ED4HWNDGW" kind="concept">
# Interval of convergence

## Summary

The interval of convergence of a real power series is the set of points where the series converges. It is an interval with center $c$ and half length equal to the radius of convergence. The node [[interval of convergence|0Z5R9TVWQH]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ED72CTBNP.md" tag="0ED72CTBNP" kind="concept">
# Counterexample

## Summary

A counterexample to a universal statement $\forall x \in X, \, P(x)$ is an element $a \in X$ such that $P(a)$ is false.
One counterexample shows that the universal statement is false.

## Remarks

The number $2$ is a counterexample to "each prime number is odd".
This works because of the [[negation of a universal statement|0P3A06QNDP]]: $\neg \forall x \, P(x)$ is equivalent to $\exists x \, \neg P(x)$.

## Explanations

### L1 human

To show that a general claim is wrong, one bad case is enough.
"Each continuous function is differentiable" is false: $f(x) = |x|$ is continuous but not differentiable at $0$.
</document>

<document source="https://math2.tamnd.com/node/0EDH403M21.md" tag="0EDH403M21" kind="statement">
# Existence of the infimum

## Statement

Let $S$ be a nonempty subset of the real numbers that is [[bounded below|0B5H2V4JW8]].
Then $S$ has an [[infimum|04KFTA34C5]] in $\mathbb{R}$.

## Remarks

Proof: the set $-S = \{-s : s \in S\}$ is nonempty and bounded above. By the [[completeness axiom|0AXWET80YW]], it has a supremum $b$. Then $-b$ is the infimum of $S$.
Another proof: the set of lower bounds of $S$ is nonempty and bounded above by each element of $S$, and its supremum is $\inf S$.

## Explanations

### L1 human

If a set of real numbers has a floor, then it has a highest floor.
This follows from the completeness axiom by a reflection of the number line through $0$, which turns floors into ceilings.
</document>

<document source="https://math2.tamnd.com/node/0EDKGA2FAK.md" tag="0EDKGA2FAK" kind="concept">
# Hamiltonian system

## Summary

A Hamiltonian system is a dynamical system given by Hamilton's equations $q' = \partial H / \partial p$, $p' = -\partial H / \partial q$ for a function $H$ of positions and momenta.
Such systems model mechanical systems without friction, such as planetary systems. See the [[definition|0C2AKPDCD9]].

## Remarks

The name honours William Rowan Hamilton.
</document>

<document source="https://math2.tamnd.com/node/0EE19CFEEF.md" tag="0EE19CFEEF" kind="statement">
# Neyman–Pearson lemma

## Statement

Consider a [[hypothesis test|087P03NEND]] of the simple null hypothesis $H_0 \colon \theta = \theta_0$ against the simple alternative $H_1 \colon \theta = \theta_1$. Let $L(\theta \mid x)$ be the [[likelihood|05FSTXM92W]] of the data $x$.
For $\eta > 0$, let the test reject $H_0$ when
$$\Lambda(x) = \frac{L(\theta_0 \mid x)}{L(\theta_1 \mid x)} \le \eta,$$
and let $\alpha = P_{\theta_0}(\Lambda(X) \le \eta)$ be its size.
Then this likelihood-ratio test is a most powerful test of size $\alpha$. Each other test with size at most $\alpha$ has at most the same power at $\theta_1$.

## Remarks

Consider all tests with a probability of a [[type I error|0R41PVFA39]] at most $\alpha$. Among them, the likelihood-ratio test has the smallest probability of a type II error.
For a discrete law, an exact size $\alpha$ can need a randomized test on the boundary $\Lambda(x) = \eta$.
Example: for an i.i.d. $\mathcal{N}(\mu, 1)$ sample and $\mu_1 > \mu_0$, the test rejects $H_0$ when $\bar{x}$ is large.

## History

Jerzy Neyman and Egon Pearson introduced the lemma in a paper in 1933.

## Explanations

### L1 human

To choose between two exact hypotheses, the best test compares how well each hypothesis explains the data.
If the alternative explains the data much better than the null hypothesis, reject the null hypothesis. No other test with the same false-alarm rate finds more true effects.
</document>

<document source="https://math2.tamnd.com/node/0EE2V3CHFK.md" tag="0EE2V3CHFK" kind="statement">
# Rule of sum

## Statement

Let $A$ and $B$ be finite sets with $A \cap B = \varnothing$.
Then $A \cup B$ is finite and $|A \cup B| = |A| + |B|$.

More generally, let $A_1, \dots, A_k$ be finite sets that are pairwise disjoint.
Then $|A_1 \cup \dots \cup A_k| = |A_1| + \dots + |A_k|$.

## Remarks

For the disjoint union $A \sqcup B$ of two arbitrary finite sets, the same formula $|A \sqcup B| = |A| + |B|$ holds.
If $A$ and $B$ are not disjoint, the formula fails. The [[inclusion-exclusion formula for two sets|0C4YEQ9H5E]] gives the correct count.

## Explanations

### L1 human

If you choose one item from a first group or from a second group, and no item is in both groups, you add the sizes.
For example, with 3 soups and 4 salads on a menu, there are $3 + 4 = 7$ ways to choose one starter.
</document>

<document source="https://math2.tamnd.com/node/0EEE2CZB2A.md" tag="0EEE2CZB2A" kind="definition">
# Continuity at a point

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $a \in D$.
The function $f$ is continuous at $a$ if the following holds.
For each open interval $V$ containing $f(a)$ there is an open interval $U$ containing $a$ with $f(x) \in V$ for all $x \in U \cap D$.

## Remarks

If $a$ is a limit point of $D$, then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$. See [[limit of a function|0TXTSEMHR5]].
If $a$ is an isolated point of $D$, then each function on $D$ is continuous at $a$.
The [[epsilon-delta form|0W8YHHGCJG]] states the same condition with numbers.
Mathlib defines it for topological spaces, as `ContinuousAt`.

## Explanations

### L1 human

A function is continuous at $a$ if small changes of $x$ near $a$ give small changes of $f(x)$.
The graph has no jump or gap at $a$.
</document>

<document source="https://math2.tamnd.com/node/0EEKF7YFBW.md" tag="0EEKF7YFBW" kind="definition">
# Refinement of a partition

## Definition

Let $P$ and $Q$ be [[partitions|069P20KYQT]] of $[a, b]$.
The partition $Q$ is a refinement of $P$ if each point of $P$ is also a point of $Q$.

## Remarks

Any two partitions $P$ and $Q$ have a common refinement, made of the points of both.
Refining a partition lowers the upper [[Darboux sum|0QRQVH0CRW]] and raises the lower Darboux sum, or leaves them unchanged.

## Explanations

### L1 human

A refinement cuts the interval at more points.
Each old subinterval is cut into one or more new subintervals.
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<document source="https://math2.tamnd.com/node/0EERDDPSVT.md" tag="0EERDDPSVT" kind="definition">
# Limit cycle

## Definition

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Its $\omega$-limit set $\omega(y)$ is the set of points $z \in U$ such that $\varphi_{t_k}(y) \to z$ for some sequence $t_k \to \infty$.
For a point $y$ whose solution exists for all $t \le 0$, the $\alpha$-limit set $\alpha(y)$ is defined in the same way with $t_k \to -\infty$.
A limit cycle is a [[periodic orbit|01DSNP0B4R]] $\Gamma$ such that $\Gamma = \omega(y)$ or $\Gamma = \alpha(y)$ for some point $y \in U$ that is not on $\Gamma$.

## Remarks

A limit cycle is stable if all nearby trajectories approach it as $t \to \infty$, and unstable if they approach it as $t \to -\infty$. It can also be semi-stable.
A linear system has no limit cycles: its periodic orbits, if any, fill a region around the origin.
The van der Pol oscillator $x'' - \mu (1 - x^2) x' + x = 0$ with $\mu > 0$ has exactly one limit cycle, and it is stable.
The [[Poincaré–Bendixson theorem|0EW1X61ZT1]] is a main tool to prove that a limit cycle exists.

## Explanations

### L1 human

A limit cycle is an isolated loop in the phase plane that nearby motions spiral toward or away from.
It models a self-sustained oscillation, such as a heartbeat, whose size does not depend on the initial state.
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<document source="https://math2.tamnd.com/node/0EF2VG8XKQ.md" tag="0EF2VG8XKQ" kind="definition">
# Concave function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is concave if
$$f(t x + (1 - t) y) \ge t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ and all $t \in [0, 1]$.

## Remarks

So $f$ is concave exactly when $-f$ is [[convex|0VV0HSW1ZS]].
For example, $\sqrt{x}$ and $\ln x$ are concave on $(0, \infty)$.
Mathlib defines concavity on a set as `ConcaveOn`.

## Explanations

### L1 human

A concave function bends downward, like a dome.
Each chord of the graph lies on or below the graph.
</document>

<document source="https://math2.tamnd.com/node/0EFDDXTVD6.md" tag="0EFDDXTVD6" kind="definition">
# Convergence in probability

## Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$.
The sequence $(X_n)$ converges in probability to $X$ if for each $\varepsilon > 0$,
$$\lim_{n \to \infty} P\left(|X_n - X| > \varepsilon\right) = 0.$$
One writes $X_n \xrightarrow{P} X$.

## Remarks

The limit is unique up to equality almost surely.
If $X_n \to X$ in probability, then a subsequence of $(X_n)$ converges to $X$ almost surely.
The weak law of large numbers is a result about convergence in probability.
Example: let $X_n$ be $1$ with probability $1/n$ and $0$ otherwise. Then $X_n \to 0$ in probability.

## Explanations

### L1 human

Convergence in probability means that, for large $n$, $X_n$ is very likely to be close to $X$.
It does not say that one fixed run of the experiment gives a convergent sequence.
</document>

<document source="https://math2.tamnd.com/node/0EFYCFHYFG.md" tag="0EFYCFHYFG" kind="concept">
# Proof by infinite descent

## Summary

A proof by infinite descent shows that a statement has no solution in positive integers.
It assumes a solution, and from it constructs a strictly smaller solution in positive integers.
This gives an infinite decreasing sequence of positive integers, which is impossible.

## Remarks

The method rests on the well-ordering of the natural numbers: each nonempty set of natural numbers has a least element.
A classic use shows that $\sqrt{2}$ is irrational: a solution of $a^2 = 2 b^2$ in positive integers gives a smaller one, $(2b - a, a - b)$.
Fermat used the method to prove [[Fermat's right triangle theorem|086BE0J3FZ]].

## Explanations

### L1 human

You cannot go down forever in the positive integers: you reach the bottom after finitely many steps.
So if each solution gives a smaller one, there can be no solution at all.
</document>

<document source="https://math2.tamnd.com/node/0EGJZHZ02C.md" tag="0EGJZHZ02C" kind="concept">
# Primality test

## Summary

A primality test is an algorithm that decides whether a given natural number is [[prime|0SN0MAEKYX]].
Unlike factorization, it does not need to find the factors of a composite number.

## Remarks

The simplest test is [[trial division|0Z962ECCXR]] by the primes up to $\sqrt{n}$, but it is slow for large $n$.
Probabilistic tests, such as the [[Fermat test|0N5ZSZMYH6]] and the [[Miller–Rabin test|0SZCSYHW60]], are fast but can accept a composite number with a small probability.
In 2002, Agrawal, Kayal and Saxena gave the first deterministic test that runs in polynomial time without an unproven hypothesis.

## Explanations

### L1 human

A primality test answers the question "is $n$ prime?".
For numbers with hundreds of digits, this is much easier than finding the factors of $n$.
</document>

<document source="https://math2.tamnd.com/node/0EGN1E1JPB.md" tag="0EGN1E1JPB" kind="concept">
# Formal derivative

## Summary

The formal derivative of a polynomial is defined by the usual rule of calculus, $(x^n)' = n x^{n-1}$, with no limits.
It makes sense over every ring.
The [[definition of the formal derivative|0AD4VS02H3]] gives the formula.
</document>

<document source="https://math2.tamnd.com/node/0EGSEE9J9R.md" tag="0EGSEE9J9R" kind="statement">
# Boole's inequality

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A_1, A_2, A_3, \dots$ be a finite or infinite sequence of [[events|00SQGJNRTB]].
Then
$$P\left(\bigcup_{i} A_i\right) \le \sum_{i} P(A_i).$$

## Remarks

The inequality is also called the union bound. It needs no hypothesis on the events: they need not be disjoint or independent.
Equality holds when the events are pairwise disjoint.
The Bonferroni inequalities refine it with the terms of the [[inclusion–exclusion principle|0FF7726C7P]]. For example, $P\left(\bigcup_{i=1}^n A_i\right) \ge \sum_i P(A_i) - \sum_{i < j} P(A_i \cap A_j)$.

## History

The inequality is named after George Boole.

## Explanations

### L1 human

The chance that at least one of several bad things happens is at most the sum of their chances.
If each of $10$ parts fails with probability $0.01$, then some part fails with probability at most $0.1$.
</document>

<document source="https://math2.tamnd.com/node/0EHBY5Y84N.md" tag="0EHBY5Y84N" kind="statement">
# Prime number theorem

## Statement

Let $\pi(x)$ be the [[prime-counting function|0XRJB5SSNE]], and let $\ln$ be the natural logarithm.
Then
$$\lim_{x \to \infty} \frac{\pi(x)}{x / \ln x} = 1.$$

## Remarks

We write this as $\pi(x) \sim x / \ln x$.
An equivalent form: the $n$-th prime $p_n$ satisfies $p_n \sim n \ln n$.
The logarithmic integral $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln t}$ gives a better approximation of $\pi(x)$, with the same limit.
For example, $\pi(10^6) = 78498$ and $10^6 / \ln 10^6 \approx 72382$.

## History

Legendre and Gauss conjectured the result from tables of primes, at the end of the 18th century.
Hadamard and de la Vallée Poussin proved it independently in 1896, with methods of complex analysis. Selberg and Erdős found elementary proofs in 1949.

## Explanations

### L1 human

Near a large number $x$, about one number in $\ln x$ is prime.
For numbers with $100$ digits, $\ln x \approx 230$, so about one number in $230$ is prime.
The theorem makes this rough rule precise.
</document>

<document source="https://math2.tamnd.com/node/0EHKVRF4J6.md" tag="0EHKVRF4J6" kind="statement">
# Liouville's theorem

## Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be an [[entire function|0419794R0T]].
Assume that $f$ is bounded: there is a real number $M$ with $|f(z)| \le M$ for each $z \in \mathbb{C}$.
Then $f$ is constant.

## Remarks

Proof: let $c \in \mathbb{C}$. By [[Cauchy's estimates|0JJRENWMP6]], $|f'(c)| \le M / r$ for each $r > 0$. So $f'(c) = 0$ for each $c$, and $f$ is constant.
For example, $\sin$ is entire and not constant, so it is not bounded on $\mathbb{C}$. On $\mathbb{R}$ it is bounded, which shows that the theorem has no real analogue.
The theorem gives a short proof of the [[fundamental theorem of algebra|0JP0W07Y9A]].

## Explanations

### L1 human

A holomorphic function on the whole plane cannot stay inside a bounded region unless it does not move at all.
Each nonconstant entire function, for example $z$, $e^z$ or $\cos z$, takes values that are as large as you like.
</document>

<document source="https://math2.tamnd.com/node/0EHM5ZP5QH.md" tag="0EHM5ZP5QH" kind="concept">
# Primitive root of unity

## Summary

A primitive $n$th root of unity is an $n$th [[root of unity|0YV2CKP2AV]] that is not a $k$th root of unity for a smaller positive $k$.
Its powers give all the $n$th roots of unity.
</document>

<document source="https://math2.tamnd.com/node/0EHWPWGHB2.md" tag="0EHWPWGHB2" kind="statement">
# Binomial theorem

## Statement

Let $R$ be a commutative ring, let $x, y \in R$, and let $n$ be a natural number.
Then
$$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}.$$
Here $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]], and the integer $\binom{n}{k}$ acts on $R$ by repeated addition.

## Remarks

For $n = 2$ it gives $(x + y)^2 = x^2 + 2xy + y^2$. For $n = 3$ it gives $x^3 + 3x^2 y + 3x y^2 + y^3$.
The formula stays true in a noncommutative ring when $xy = yx$.
With $x = y = 1$ it gives the [[sum of binomial coefficients|0KAMF19NV8]]. With $x = -1$ and $y = 1$ it gives the [[alternating sum|0PQQGFAWSH]].
The [[multinomial theorem|0XFC1VQYEK]] extends it to sums of more than two terms.

## Explanations

### L1 human

When you expand $(x + y)^n$, each term comes from choosing $x$ or $y$ in each of the $n$ factors.
The term $x^k y^{n-k}$ appears once for each choice of the $k$ factors that give $x$, so its coefficient is $\binom{n}{k}$.
</document>

<document source="https://math2.tamnd.com/node/0EHYZFZ4PK.md" tag="0EHYZFZ4PK" kind="definition">
# P-value

## Definition

Consider a [[hypothesis test|087P03NEND]] of a null hypothesis $H_0 \colon \theta \in \Theta_0$ with a test statistic $T(X)$, where large values of $T$ are evidence against $H_0$. Let $t$ be the observed value of $T(X)$.
The p-value of the observation is
$$p(t) = \sup_{\theta \in \Theta_0} P_\theta\big(T(X) \ge t\big).$$
If $H_0$ is simple, $\Theta_0 = \{\theta_0\}$, then $p(t) = P_{\theta_0}(T(X) \ge t)$.

## Remarks

For a two-sided test with a statistic that is symmetric about $0$ under $H_0$, one uses $P_{\theta_0}(|T(X)| \ge |t|)$.
The test that rejects $H_0$ when $p \le \alpha$ has significance level $\alpha$.
If $H_0$ is simple and $T(X)$ has a continuous law under $H_0$, then the random p-value $p(T(X))$ has the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$ under $H_0$.
The p-value is not the probability that $H_0$ is true.

## Explanations

### L1 human

The p-value tells how surprising your data would be if the null hypothesis were true.
Example: a coin shows $14$ heads in $20$ tosses. If the coin is fair, the probability of at least $14$ heads is about $0.058$. This is the one-sided p-value.
</document>

<document source="https://math2.tamnd.com/node/0EJ4PC8NJ6.md" tag="0EJ4PC8NJ6" kind="statement">
# Universal property of the abelianisation

## Statement

Let $G$ be a [[group|G5S1M9]], let $\pi \colon G \to G^{\mathrm{ab}}$ be the projection to its [[abelianisation|0AQZV0905N]], and let $A$ be an [[abelian group|082GHNPJ53]].
For each [[group homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to A$, there is exactly one group homomorphism $\bar{\varphi} \colon G^{\mathrm{ab}} \to A$ with $\bar{\varphi} \circ \pi = \varphi$.

## Remarks

Each commutator goes to $e$ under $\varphi$, because $A$ is abelian, so $[G, G] \subseteq \ker \varphi$.
Then the [[universal property of the quotient group|0GJ83E78A5]] gives $\bar{\varphi}$.
Thus the homomorphisms from $G$ to $A$ correspond exactly to the homomorphisms from $G^{\mathrm{ab}}$ to $A$.

## Explanations

### L1 human

A homomorphism into an abelian group cannot see the commutators of $G$.
So it factors through the abelianisation, where the commutators are gone.
This makes the abelianisation the best abelian approximation of $G$.
</document>

<document source="https://math2.tamnd.com/node/0EJVD9K0K1.md" tag="0EJVD9K0K1" kind="concept">
# Normal convergence

## Summary

A series of functions converges normally when the series of the suprema of the absolute values of its terms converges. Normal convergence implies uniform and absolute convergence. The node [[normal convergence|0T0EJDGHMB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0EKGKY5TGH.md" tag="0EKGKY5TGH" kind="statement">
# Epsilon-delta continuity in metric spaces

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function and let $a \in X$.
Then $f$ is [[continuous at|07N7W5C29X]] $a$ if and only if for each real $\varepsilon > 0$ there is a real $\delta > 0$ such that
$$d_X(x, a) < \delta \implies d_Y\bigl(f(x), f(a)\bigr) < \varepsilon \quad \text{for all } x \in X.$$

## Remarks

In terms of [[open balls|0DC3MPR7C9]], the condition says: for each $\varepsilon > 0$ there is $\delta > 0$ with $f(B(a, \delta)) \subseteq B(f(a), \varepsilon)$.
The number $\delta$ can depend on $\varepsilon$ and on $a$. When one $\delta$ works for all $a$, the map is [[uniformly continuous|0FHSDCQACY]].
For $X = Y = \mathbb{R}$, this is the epsilon-delta definition of continuity from calculus.

## Explanations

### L1 human

You can make $f(x)$ as close to $f(a)$ as you want by taking $x$ close enough to $a$.
The tolerance $\varepsilon$ for the output comes first, and then you find a tolerance $\delta$ for the input.
</document>

<document source="https://math2.tamnd.com/node/0EKZCX4K3P.md" tag="0EKZCX4K3P" kind="definition">
# Hermitian matrix

## Definition

Let $A = (a_{ij})$ be an $n \times n$ complex [[square matrix|0JXZKX526R]].
The matrix $A$ is *Hermitian* when $A^* = A$, where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.
Equivalently, $a_{ij} = \overline{a_{ji}}$ for all $i$ and $j$.

## Remarks

The diagonal entries of a Hermitian matrix are real.
A real matrix is Hermitian if and only if it is [[symmetric|0VR671PHE5]].
The eigenvalues of a Hermitian matrix are real, and $\C^n$ has an orthonormal basis of eigenvectors, by the [[spectral theorem for Hermitian matrices|0DGZBQWZWN]].
The matrix $\begin{pmatrix} 2 & 1 - i \\ 1 + i & 3 \end{pmatrix}$ is Hermitian.

## Explanations

### L1 human

A Hermitian matrix is the complex version of a symmetric matrix.
The entry across the diagonal is the complex conjugate of the entry here.
In quantum mechanics, observable quantities are Hermitian matrices or operators.
</document>

<document source="https://math2.tamnd.com/node/0EKZW08FQ5.md" tag="0EKZW08FQ5" kind="definition">
# Orientable surface

## Definition

Let $S \subseteq \mathbb{R}^3$ be a surface that has a tangent plane at each point.
The surface $S$ is orientable if there is a continuous map $n \colon S \to \mathbb{R}^3$ with the following property. For each $p \in S$, $n(p)$ is a [[unit normal vector|0BJDQ2HBH1]] of $S$ at $p$.
Such a map $n$ is an orientation of $S$, and $S$ together with $n$ is an oriented surface.

## Remarks

If $S$ is connected and orientable, then it has exactly two orientations, $n$ and $-n$.
A regular injective [[parametrisation|05788J51WZ]] $r$ gives the orientation $n = (r_u \times r_v) / \| r_u \times r_v \|$. Swapping $u$ and $v$ gives the opposite orientation.
Spheres, planes and tori are orientable. The boundary of a bounded region in $\mathbb{R}^3$ has the outward normal as its standard orientation. The Möbius strip is not orientable.
Integrals of vector fields across a surface, such as flux, need an orientation. Reversing the orientation changes their sign. [[Integrals of scalar functions|0DQ8ERJZNS]] do not need one.

## Explanations

### L1 human

An orientable surface has two sides, an inside and an outside, or a top and a bottom.
On a Möbius strip, an ant can walk from one side to the other without crossing an edge. So there is no consistent choice of side.
</document>

<document source="https://math2.tamnd.com/node/0EM6C9SZN5.md" tag="0EM6C9SZN5" kind="statement">
# Existence of nontrivial solutions of Pell's equation

## Statement

Let $d$ be a positive integer that is not a perfect square.
Then [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ has a solution in integers with $y \neq 0$.
In fact, it has infinitely many solutions in integers.

## Remarks

One proof uses [[Dirichlet's approximation theorem|04JAY9DGB7]] to find infinitely many $x/y$ with $|x^2 - d y^2|$ bounded.
Another proof uses the [[periodic continued fraction|0XKJPFS2W0]] of $\sqrt{d}$.
The smallest solution can be very large. For $d = 61$, it is $(1766319049, 226153980)$.

## History

Indian mathematicians, such as Brahmagupta, studied the equation many centuries before European mathematicians.
Lagrange proved that it always has infinitely many solutions.

## Explanations

### L1 human

For each non-square $d$, there are integers $x$ and $y \neq 0$ with $x^2 - d y^2 = 1$.
They may be hard to find by trial: for $d = 61$, the smallest $x$ has ten digits.
</document>

<document source="https://math2.tamnd.com/node/0EN0V3NY20.md" tag="0EN0V3NY20" kind="concept">
# Chi-squared test

## Summary

A chi-squared test is a [[hypothesis test|087P03NEND]] whose test statistic has approximately a [[chi-squared distribution|0RFMMBHFTV]] under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories.
For observed counts $O_1, \dots, O_k$ and expected counts $E_1, \dots, E_k$ under $H_0$, Pearson's statistic is
$$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}.$$
One rejects $H_0$ when $\chi^2$ is large.

## Remarks

In the goodness-of-fit test, $H_0$ fixes the probabilities of the $k$ categories. For a large sample, $\chi^2$ has approximately the law $\chi^2_{k - 1}$ under $H_0$.
In the test of independence for an $r \times c$ contingency table, $H_0$ states that the two categorical variables are [[independent|0XCE070V8B]]. Then $E_{ij}$ is (row total $\times$ column total) / $n$, and the law is approximately $\chi^2_{(r-1)(c-1)}$.
The approximation is poor when expected counts are small. A common rule asks for $E_i \ge 5$ in each cell. For small tables, Fisher's exact test is an alternative.
Karl Pearson introduced the test in 1900.

## Explanations

### L1 human

The chi-squared test compares the counts that you observe with the counts that a hypothesis predicts.
If the squared differences, scaled by the predictions, add up to a large number, the hypothesis does not fit the data.
</document>

<document source="https://math2.tamnd.com/node/0ENVM493VK.md" tag="0ENVM493VK" kind="statement">
# Inverse in terms of the adjugate

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$ such that $\det A$ is a unit of $K$.
Then $A$ is [[invertible|0EB9N0S33C]], and
$$A^{-1} = (\det A)^{-1} \operatorname{adj}(A),$$
where $\operatorname{adj}(A)$ is the [[adjugate|07W9H8QF38]] of $A$.

## Remarks

The statement follows by dividing [[$A \operatorname{adj}(A) = \operatorname{adj}(A) A = \det(A) I_n$|0528HKWZ1Z]] by $\det A$.
For $n = 2$ and $ad - bc \ne 0$ over a field, $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
For large $n$, row reduction computes the inverse much faster than this formula.

## Explanations

### L1 human

This formula gives every entry of the inverse as a quotient of two determinants.
It is useful for theory and for small matrices.
For $2 \times 2$ matrices, it is the formula that most students learn first.
</document>

<document source="https://math2.tamnd.com/node/0ENZN5G4CV.md" tag="0ENZN5G4CV" kind="definition">
# Normal operator

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *normal* when
$$T T^* = T^* T,$$
where $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.

## Remarks

Self-adjoint maps, skew-adjoint maps and unitary maps are normal.
$T$ is normal if and only if $\norm{T(v)} = \norm{T^*(v)}$ for all $v \in V$.
Over $\C$, normal maps are exactly the maps with an orthonormal basis of eigenvectors.

## Explanations

### L1 human

A normal map commutes with its adjoint.
This condition is weaker than self-adjoint, but it still gives perpendicular eigenvectors over $\C$.
Rotations of the plane are normal, but not self-adjoint.
</document>

<document source="https://math2.tamnd.com/node/0EP59ESQ94.md" tag="0EP59ESQ94" kind="definition">
# Invariant subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
A [[subspace|00YC1D652K]] $W$ of $V$ is *invariant* under $T$, or *$T$-invariant*, when $T(w) \in W$ for all $w \in W$.

## Remarks

The subspaces $0$ and $V$, $\ker T$ and $\im T$, and each [[eigenspace|0G0RFPG8HR]] of $T$ are $T$-invariant.
A line spanned by $v \ne 0$ is $T$-invariant if and only if $v$ is an eigenvector of $T$.
If $V = W_1 \oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.

## Explanations

### L1 human

An invariant subspace is a part of the space that the map never leaves.
Vectors inside it stay inside it.
We can then study the map on this smaller space alone.
</document>

<document source="https://math2.tamnd.com/node/0EP73HK4C4.md" tag="0EP73HK4C4" kind="definition">
# Row space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with rows $r_1, \dots, r_m \in F^n$.
The *row space* of $A$ is
$$\operatorname{Row}(A) = \operatorname{span}(r_1, \dots, r_m) \subseteq F^n,$$
the [[linear span|0DTY8B9HMS]] of the rows.

## Remarks

The row space of $A$ is the [[column space|03AJKXH1AV]] of $A^T$.
[[Elementary row operations|04XWH28452]] do not change the row space, so [[row equivalent|006BHFWMAP]] matrices have the same row space.
The nonzero rows of a row echelon form of $A$ form a basis of $\operatorname{Row}(A)$.
Its dimension equals the dimension of the column space, see [[row rank equals column rank|0TZAG60ZYT]].

## Explanations

### L1 human

The row space collects all the combinations of the rows of a matrix.
Row operations only mix the rows, so they keep the row space.
After row reduction, the nonzero rows give a basis of it.
</document>

<document source="https://math2.tamnd.com/node/0EPAT51DXQ.md" tag="0EPAT51DXQ" kind="concept">
# Function

## Summary

A function from a set $A$ to a set $B$ assigns to each element of $A$ exactly one element of $B$.
</document>

<document source="https://math2.tamnd.com/node/0EPD87YVG8.md" tag="0EPD87YVG8" kind="statement">
# Continuous image of a connected set is connected

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be [[continuous|0SY9ECQ19Q]] and let $S \subseteq X$ be [[connected|0R37DFJKFW]].
Then $f(S)$ is connected.

## Remarks

Proof: if open sets $U$ and $V$ split $f(S)$, then the [[preimages|0FGCFY1AZX]] $f^{-1}(U)$ and $f^{-1}(V)$ are open and split $S$.
Thus connectedness is a [[topological property|02SMVGC690]].
For $Y = \mathbb{R}$, this gives the [[intermediate value theorem|098C0KZZAP]].
For example, the circle is connected, because it is the image of $[0, 1]$ under $t \mapsto (\cos 2\pi t, \sin 2\pi t)$.

## Explanations

### L1 human

A continuous map cannot break a space that is in one piece into separate pieces.
</document>

<document source="https://math2.tamnd.com/node/0EPTDZPJ83.md" tag="0EPTDZPJ83" kind="statement">
# Convergence in probability implies convergence in distribution

## Statement

Let $X, X_1, X_2, \dots$ be real random variables on one probability space.
If $X_n \to X$ [[in probability|0EFDDXTVD6]], then $X_n \to X$ [[in distribution|0JDAQHWA9N]].

## Remarks

Proof: for $\varepsilon > 0$, $F_n(x) \le F(x + \varepsilon) + P(|X_n - X| > \varepsilon)$ and $F(x - \varepsilon) \le F_n(x) + P(|X_n - X| > \varepsilon)$.
Let $n \to \infty$ and then $\varepsilon \to 0$. At a continuity point $x$ of $F$, this gives $F_n(x) \to F(x)$.
The converse is false. Let $X \sim \mathcal{N}(0, 1)$ and $X_n = -X$ for all $n$. Then $X_n \to X$ in distribution, but $P(|X_n - X| > 1) = P(|X| > 1/2)$ does not tend to $0$.
The converse holds when the limit is a constant: if $X_n \to c$ in distribution for a real $c$, then $X_n \to c$ in probability.

## Explanations

### L1 human

If $X_n$ is very likely to be close to $X$, then the probabilities for $X_n$ are close to those for $X$.
But two variables can have the same law and still be far apart, so the converse fails.
</document>

<document source="https://math2.tamnd.com/node/0EPTJA5QV4.md" tag="0EPTJA5QV4" kind="definition">
# Linear subspace

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *linear subspace* of $V$ is a subset $W \subseteq V$ with these three properties:

1. $0 \in W$.
2. If $u, v \in W$, then $u + v \in W$.
3. If $a \in F$ and $v \in W$, then $av \in W$.

## Remarks

With the operations of $V$, a subspace $W$ is itself a vector space over $F$.
The sets $\{0\}$ and $V$ are subspaces of $V$.
The lines through the origin are the one-dimensional subspaces of $\R^2$; a line that misses the origin is not a subspace.
The kernel and the image of a linear map are subspaces.

## Explanations

### L1 human

A subspace is a smaller vector space inside a bigger one.
We cannot leave it by adding its vectors or by scaling them.
In space, the subspaces are the origin, the lines and planes through the origin, and the whole space.
</document>

<document source="https://math2.tamnd.com/node/0EPW94MJ1P.md" tag="0EPW94MJ1P" kind="concept">
# Constrained optimisation

## Summary

Constrained optimisation looks for the maximum or minimum of an objective function $f$ of several variables over the set of points that satisfy given constraints.
Typical constraints are equations $g_i(x) = c_i$ and inequalities $h_j(x) \ge d_j$.

## Remarks

An example is to maximise $f(x, y) = x y$ on the circle $x^2 + y^2 = 2$. The maximum value is $1$, at $(1, 1)$ and $(-1, -1)$.
For equality constraints, one method solves the constraints for some variables and substitutes them into $f$. Another method uses [[Lagrange multipliers|0VKKP8X6Q6]].
A constrained extremum need not be a critical point of $f$. In the example above, $\nabla f(1, 1) = (1, 1) \ne 0$.

## Explanations

### L1 human

In many problems you cannot choose the variables freely.
For example, you want the largest area for a fixed length of fence.
Constrained optimisation finds the best value only among the choices that obey the rules.
</document>

<document source="https://math2.tamnd.com/node/0EPZC4Z1FC.md" tag="0EPZC4Z1FC" kind="statement">
# Central limit theorem

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with mean $\mu$ and variance $\sigma^2$, where $0 < \sigma^2 < \infty$. Let $\bar{X}_n = (X_1 + \dots + X_n)/n$.
Then $\sqrt{n} \, (\bar{X}_n - \mu) / \sigma$ converges [[in distribution|0JDAQHWA9N]] to $\mathcal{N}(0, 1)$. That is, for each $x \in \mathbb{R}$,
$$\lim_{n \to \infty} P\left(\frac{X_1 + \dots + X_n - n\mu}{\sigma \sqrt{n}} \le x\right) = \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} \, du.$$

## Remarks

A standard proof uses [[characteristic functions|0MYDNRT7PP]]. With $Y_i = (X_i - \mu)/\sigma$, $\varphi_{Y_1}(t) = 1 - t^2/2 + o(t^2)$, so $\varphi_{Y_1}(t/\sqrt{n})^n \to e^{-t^2/2}$.
[[Lévy's continuity theorem|0FPWF7TTMB]] then gives the convergence in distribution.
The Lindeberg and Lyapunov forms remove the condition that the $X_i$ have the same law. The Berry–Esseen theorem bounds the rate of convergence when $E[|X_1|^3] < \infty$.
The finite variance is necessary. Sums of i.i.d. Cauchy variables, scaled by $1/n$, stay Cauchy.

## Explanations

### L1 human

The central limit theorem explains why the bell curve is everywhere.
A sum of many small independent effects is approximately normal, whatever the law of each effect, as long as the variance is finite.
</document>

<document source="https://math2.tamnd.com/node/0EQFWKJ37J.md" tag="0EQFWKJ37J" kind="definition">
# Hyperbolic cosine

## Definition

The hyperbolic cosine is the function $\cosh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\cosh x = \frac{e^x + e^{-x}}{2}.$$

## Remarks

It is even, and $\cosh x \ge 1$, with equality only at $x = 0$.
Its power series is $\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$.
The graph of $\cosh$ is the catenary, the shape of a hanging chain. Mathlib defines it as `Real.cosh`.

## Explanations

### L1 human

The hyperbolic cosine is the even part of $e^x$.
Its graph looks like a parabola near $0$, but it grows exponentially.
</document>

<document source="https://math2.tamnd.com/node/0EQMWQ4WMJ.md" tag="0EQMWQ4WMJ" kind="definition">
# Locally compact space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is locally compact when each point $x \in X$ has a [[neighbourhood|09CJ00H3NW]] $N$ that is [[compact|00R0X1WGPX]].

## Remarks

Authors use several stronger definitions, for example that each neighbourhood of $x$ contains a compact neighbourhood of $x$. For [[Hausdorff spaces|0MXB4CKHFW]], all the usual definitions agree.
Each compact space is locally compact. The space $\mathbb{R}^n$ is locally compact, because each closed ball is compact.
The rationals $\mathbb{Q}$ and each infinite-dimensional normed space are not locally compact.
A locally compact Hausdorff space has a Hausdorff [[one-point compactification|0MM45BT6HC]].

## Explanations

### L1 human

A locally compact space is compact in small pieces: each point has a compact set around it.
The real line is not compact, but each closed interval in it is.
</document>

<document source="https://math2.tamnd.com/node/0EQWVHF3W5.md" tag="0EQWVHF3W5" kind="concept">
# Field extension

## Summary

A field extension is a pair of fields $K \subseteq L$, where $K$ is a subfield of $L$.
One writes $L/K$ and says that $L$ is an extension of $K$.
The [[definition of a field extension|0WQ1A8SHTZ]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0EQZSAB9EC.md" tag="0EQZSAB9EC" kind="definition">
# Homogeneous linear equation

## Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous functions.
A homogeneous linear equation of order $n$ is a [[linear differential equation|0PB7K8M7FG]] with right side $0$:
$$y^{(n)} + a_{n-1}(x)\, y^{(n-1)} + \dots + a_1(x)\, y' + a_0(x)\, y = 0.$$
For a linear equation $L y = b$, the equation $L y = 0$ is the associated homogeneous equation.

## Remarks

The zero function is always a solution.
By the [[superposition principle|0F2VBPXNP9]], the solutions on $I$ form a vector space. It has [[dimension n|0GK68PZ6KH]].
Example: $y'' - y = 0$ is homogeneous, and its solutions are $C_1 e^{x} + C_2 e^{-x}$.

## Explanations

### L1 human

Homogeneous means that the equation has no term without $y$.
So if $y$ is a solution, then $2y$ and $-y$ are solutions, and so is the sum of two solutions.
</document>

<document source="https://math2.tamnd.com/node/0EREPBVG2G.md" tag="0EREPBVG2G" kind="concept">
# Induced subgraph

## Summary

An induced subgraph is the subgraph on a chosen set of vertices that keeps all edges between them. Induced subgraphs describe the structure of a graph on a part of its vertex set. The node [[induced subgraph|0V05FC4GKQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ERWG4NXQ9.md" tag="0ERWG4NXQ9" kind="statement">
# QR decomposition

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $Q$ and an [[upper triangular matrix|0N4XY7Z3NN]] $R$ with
$$A = QR.$$
If $A$ is real, then $Q$ and $R$ can be chosen real, with $Q$ orthogonal.
If $A$ is invertible and the diagonal entries of $R$ are required to be positive, then $Q$ and $R$ are unique.

## Remarks

For an invertible $A$, the [[Gram–Schmidt process|0VT6P0GNQR]] applied to the columns of $A$ gives the columns of $Q$.
A rectangular $m \times n$ matrix with $m \ge n$ has a decomposition $A = QR$ with $Q$ unitary $m \times m$ and $R$ upper triangular $m \times n$.
Householder reflections and Givens rotations compute it stably.
It is used to solve least squares problems and in the QR algorithm for eigenvalues.

## Explanations

### L1 human

The QR decomposition writes a matrix as a rotation, possibly with reflections, times a triangular matrix.
The columns of $Q$ are orthonormal, and $R$ records how to rebuild the columns of $A$ from them.
It is the matrix form of the Gram–Schmidt process.
</document>

<document source="https://math2.tamnd.com/node/0ESCX36MMC.md" tag="0ESCX36MMC" kind="object">
# Imaginary unit

## Description

The imaginary unit is the [[complex number|09FTAET5Z4]] $i = 0 + 1 i$. It satisfies $i^2 = -1$, see [[i squared is minus one|07KBQCG9KV]].
It is a solution of the equation $x^2 + 1 = 0$. The other solution is $-i$.
In the [[complex plane|0CTE7A31XQ]], $i$ is the point $(0, 1)$, at distance $1$ above $0$.

## Remarks

The powers of $i$ repeat with period $4$: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$.
Multiplication by $i$ turns the complex plane by a quarter turn about $0$, counterclockwise.
In electrical engineering, people often write $j$ for $i$.

## Explanations

### L1 human

The imaginary unit is a new number whose square is $-1$.
No real number has this property, so $i$ is not on the real number line.
It sits one unit above $0$ in the plane of complex numbers.
</document>

<document source="https://math2.tamnd.com/node/0ESHD7ZQDX.md" tag="0ESHD7ZQDX" kind="statement">
# Divisibility by 9

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $9 \mid n$ if and only if $9 \mid d_0 + d_1 + \dots + d_k$.

## Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 9$, because $10^i \equiv 1 \pmod 9$ for each $i$.
For example, $9 \mid 81{,}243$, because $8 + 1 + 2 + 4 + 3 = 18$.
The congruence is the basis of [[casting out nines|0E7W0EM6J6]].

## Explanations

### L1 human

Add the digits. The number is a multiple of $9$ exactly when the sum is a multiple of $9$.
For example, $123{,}456$ has the digit sum $21$, which is not a multiple of $9$. So $9 \nmid 123{,}456$.
</document>

<document source="https://math2.tamnd.com/node/0ETCTBTJM1.md" tag="0ETCTBTJM1" kind="definition">
# Collinear points

## Definition

Let $K$ be a field and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$.
A set $S$ of points of $A$ is collinear if the linear span of the vectors $q - p$ for $p, q \in S$ has dimension at most 1.
Equivalently, $S$ is collinear if $S$ has at most one point or a [[line|0MP9BCF6DP]] of $A$ contains $S$.

## Remarks

The empty set and each set with one point are collinear. Each set of two points is collinear.
A set $S$ with at least two points is collinear if and only if its [[affine span|0YMB1676R1]] is a line.
In $\mathbb{R}^2$, the points $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ are collinear if and only if $(x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) = 0$.
Mathlib's `Collinear` uses the first form of the definition.

## Explanations

### L1 human

Points are collinear if you can draw one straight line through all of them. The points $(0, 0)$, $(1, 2)$ and $(2, 4)$ are collinear: they all lie on $y = 2x$. The point $(1, 1)$ is not on this line, so $(0, 0)$, $(1, 2)$ and $(1, 1)$ are not collinear.
</document>

<document source="https://math2.tamnd.com/node/0ETGBAK2CC.md" tag="0ETGBAK2CC" kind="statement">
# Existence of an algebraic closure

## Statement

Every [[field|0QA8CHH8MG]] $K$ has an [[algebraic closure|0MCXX29HWY]].

## Remarks

One proof: take a polynomial ring $R$ over $K$ with one variable for each root of each monic irreducible polynomial over $K$. A [[maximal ideal|02HJYT53F7]] of $R$ that contains the relations gives a field $K_1$ in which every polynomial over $K$ splits.
Repeating the step gives fields $K \subseteq K_1 \subseteq K_2 \subseteq \dots$, and their union is an algebraic closure of $K$.
The proof uses Zorn's lemma, through the existence of maximal ideals. The weaker ultrafilter lemma is enough.
For $K = \mathbb{Q}$ or $K = \mathbb{R}$, the complex numbers give a direct proof.

## Explanations

### L1 human

Every field sits inside an algebraically closed field that is algebraic over it.
The general proof needs a form of the axiom of choice, because a field can have very many polynomials.
</document>

<document source="https://math2.tamnd.com/node/0ETGP0XVCJ.md" tag="0ETGP0XVCJ" kind="statement">
# Gershgorin circle theorem

## Statement

Let $A = (a_{ij})$ be an $n \times n$ matrix over $\C$.
For $1 \le i \le n$, let $R_i = \sum_{j \ne i} \abs{a_{ij}}$, and let $D_i = \{z \in \C : \abs{z - a_{ii}} \le R_i\}$ be the closed disc with centre $a_{ii}$ and radius $R_i$.
Then each [[eigenvalue|0BQAA2BK3Q]] of $A$ lies in at least one of the discs $D_1, \dots, D_n$.

## Remarks

The discs are called *Gershgorin discs*.
The proof takes an eigenvector $x$ and an index $i$ with $\abs{x_i}$ maximal, and estimates row $i$ of $Ax = \lambda x$.
Since $A$ and $A^T$ have the same eigenvalues, the same holds with column sums.
A strictly diagonally dominant matrix, with $\abs{a_{ii}} > R_i$ for all $i$, has no eigenvalue $0$, so it is invertible.

## Explanations

### L1 human

The eigenvalues of a matrix lie near its diagonal entries, when the other entries are small.
Each row gives a disc around its diagonal entry, with the radius equal to the sum of the other absolute values of the row.
Every eigenvalue lies in one of these discs.
</document>

<document source="https://math2.tamnd.com/node/0EVHJFTG9Z.md" tag="0EVHJFTG9Z" kind="definition">
# Linear independence

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A family $(v_1, \dots, v_k)$ of vectors of $V$ is *linearly independent* when the only [[linear combination|0V8F3E48E6]] of them that equals $0$ is the trivial one. This means that
$$a_1 v_1 + \cdots + a_k v_k = 0 \implies a_1 = \cdots = a_k = 0$$
for all $a_1, \dots, a_k \in F$.
An arbitrary family $(v_i)_{i \in I}$ is *linearly independent* when each finite subfamily is linearly independent.
A family that is not linearly independent is *linearly dependent*.

## Remarks

The vectors $(1, 0)$ and $(1, 1)$ in $\R^2$ are linearly independent; the vectors $(1, 2)$ and $(2, 4)$ are linearly dependent.
A family that contains the zero vector, or the same vector twice, is linearly dependent.
A family of at least two vectors is linearly dependent if and only if one of its vectors is a linear combination of the others.
In $F^n$, a family of more than $n$ vectors is linearly dependent.

## Explanations

### L1 human

Vectors are linearly independent when none of them is redundant.
No vector can be built from the others, so each one adds a new direction.
Two arrows in the plane are independent exactly when they do not lie on one line.
</document>

<document source="https://math2.tamnd.com/node/0EW1X61ZT1.md" tag="0EW1X61ZT1" kind="statement">
# Poincaré–Bendixson theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.
Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Let $\omega(y)$ be its $\omega$-limit set, as in the definition of a [[limit cycle|0EERDDPSVT]].
Assume that $\omega(y)$ is non-empty and compact, and that it contains only finitely many [[equilibrium points|0G4X36AE0X]].
Then $\omega(y)$ is one of the following three kinds of set.
(1) It is a single equilibrium point.
(2) It is a [[periodic orbit|01DSNP0B4R]].
(3) It is a connected set of finitely many equilibrium points and of trajectories that tend to these points as $t \to \infty$ and as $t \to -\infty$.

## Remarks

A common corollary: let $K \subseteq U$ be compact, contain no equilibrium point, and contain the positive half-trajectory of $y$. Then $\omega(y)$ is a periodic orbit.
To apply it, one often finds an annulus that trajectories enter but cannot leave. Then the annulus contains a periodic orbit.
The theorem is special to the plane. It fails in $\mathbb{R}^3$, where chaotic behaviour is possible, as for the Lorenz system.
The trajectories in case (3) are homoclinic or heteroclinic orbits.

## History

The theorem is named after Henri Poincaré and Ivar Bendixson.

## Explanations

### L1 human

In the plane, a bounded motion has few options in the long run.
It settles at an equilibrium, approaches a closed loop, or follows a chain of paths between equilibria.
Trajectories in the plane cannot cross, and this is what limits the possibilities.
</document>

<document source="https://math2.tamnd.com/node/0EW824Z7G2.md" tag="0EW824Z7G2" kind="concept">
# Beta distribution

## Summary

The beta distribution is a two-parameter family of continuous distributions on the interval $[0, 1]$.
It is often used as a distribution for an unknown probability. The node [[beta distribution|0PP5ZAJPP7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0EWMDD6C14.md" tag="0EWMDD6C14" kind="concept">
# Domain in the complex plane

## Summary

A domain in the complex plane is a nonempty open connected subset of $\mathbb{C}$, see the [[definition|097KNJFJ7V]].
Domains are the usual sets on which complex analysis studies holomorphic functions.
</document>

<document source="https://math2.tamnd.com/node/0EWQEYRVA8.md" tag="0EWQEYRVA8" kind="definition">
# Rank of a matrix

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *rank* of $A$ is
$$\rank A = \dim \operatorname{Col}(A),$$
the [[dimension|03GSZ8RTZJ]] of the [[column space|03AJKXH1AV]] of $A$.

## Remarks

The rank is the maximal number of linearly independent columns of $A$.
It also equals the dimension of the row space, by [[row rank equals column rank|0TZAG60ZYT]].
It equals the number of pivots in a row echelon form of $A$, and the largest order of a nonzero minor of $A$.
An $n \times n$ matrix is invertible if and only if its rank is $n$.

## Explanations

### L1 human

The rank of a matrix counts how many of its columns are really independent.
For $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the second column is twice the first, so the rank is $1$.
We compute the rank by row reduction and count the pivots.
</document>

<document source="https://math2.tamnd.com/node/0EWXPZCSZ0.md" tag="0EWXPZCSZ0" kind="statement">
# Impossibility of trisecting the angle

## Statement

The angle $\pi/3$, that is, $60^\circ$, cannot be [[trisected|00J3DCCEK5]] with straightedge and compass.
Equivalently, $\cos(\pi/9)$ is not a [[constructible number|0FMN2MN7TM]].

## Remarks

The proof: $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$ shows that $c = \cos(\pi/9)$ is a root of $8x^3 - 6x - 1$. Then $u = 2c$ is a root of $x^3 - 3x - 1$.
The polynomial $x^3 - 3x - 1$ has no rational root, by the [[rational root theorem|0819CQ1V8H]]. So it is [[irreducible|0VNE161QCK]], and $[\mathbb{Q}(c) : \mathbb{Q}] = 3$.
Since $3$ is not a power of $2$, the number $c$ is not constructible.
The angle $\pi/3$ itself is constructible, so no general trisection method exists.

## History

Pierre Wantzel published the proof in 1837.

## Explanations

### L1 human

If you could trisect every angle, you could trisect $60^\circ$ and construct an angle of $20^\circ$.
The cosine of $20^\circ$ has degree $3$ over the rationals, so it is not constructible.
</document>

<document source="https://math2.tamnd.com/node/0EX045E2BP.md" tag="0EX045E2BP" kind="definition">
# Conjugate transpose

## Definition

Let $A = (a_{ij})$ be an $m \times n$ matrix with complex entries.
The *conjugate transpose* of $A$ is the $n \times m$ matrix $A^*$ with entries
$$(A^*)_{ij} = \overline{a_{ji}}$$
for $1 \le i \le n$ and $1 \le j \le m$, where $\overline{z}$ is the complex conjugate of $z$.

## Remarks

Other notations are $A^H$ and $A^\dagger$.
For a real matrix, the conjugate transpose is the [[transpose|09DXWYRD43]].
The rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold.
For the standard inner product on $\C^n$, $\langle Ax, y \rangle = \langle x, A^* y \rangle$.

## Explanations

### L1 human

The conjugate transpose flips a complex matrix over its diagonal and conjugates each entry.
For example, the conjugate transpose of $\begin{pmatrix} 1 & i \\ 0 & 2 \end{pmatrix}$ is $\begin{pmatrix} 1 & 0 \\ -i & 2 \end{pmatrix}$.
For complex matrices it plays the role that the transpose plays for real matrices.
</document>

<document source="https://math2.tamnd.com/node/0EX2HPXPJ6.md" tag="0EX2HPXPJ6" kind="concept">
# Product of vector spaces

## Summary

The product of two vector spaces over the same field is the set of pairs, with addition and scalar multiplication done in each component.
</document>

<document source="https://math2.tamnd.com/node/0EX9GFJ0B3.md" tag="0EX9GFJ0B3" kind="statement">
# The exponential function is its own derivative

## Statement

The [[exponential function|0RMVS7RXZX]] is differentiable on $\mathbb{R}$, and $\exp'(x) = \exp(x)$ for all real $x$.

## Remarks

By [[the sum rule|0E8HNVBCSP]], $\frac{\exp(x + h) - \exp(x)}{h} = \exp(x) \cdot \frac{\exp(h) - 1}{h}$. The series gives $\frac{\exp(h) - 1}{h} \to 1$ as $h \to 0$.
Term-by-term differentiation of the power series gives the same result.
The exponential function is the only solution of $f' = f$ with $f(0) = 1$. Mathlib states the derivative as `Real.hasDerivAt_exp`.

## Explanations

### L1 human

The slope of the graph of $e^x$ at each point equals its height there.
No other function with value $1$ at $0$ has this property.
</document>

<document source="https://math2.tamnd.com/node/0EY05Z9RPN.md" tag="0EY05Z9RPN" kind="definition">
# Iterated derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}^m$.
For an integer $k \ge 0$, let $L^k(\mathbb{R}^n; \mathbb{R}^m)$ be the space of $k$-linear maps from $(\mathbb{R}^n)^k$ to $\mathbb{R}^m$, with $L^0(\mathbb{R}^n; \mathbb{R}^m) = \mathbb{R}^m$.
The iterated derivatives are defined by recursion. First, $D^0 f = f$.
Suppose that $D^k f \colon U \to L^k(\mathbb{R}^n; \mathbb{R}^m)$ is defined on $U$ and is [[differentiable on U|08S7FDRS5Z]].
Then $D^{k+1} f(a)$, for $a \in U$, is the $(k+1)$-linear map
$$D^{k+1} f(a)(v_1, v_2, \dots, v_{k+1}) = \big(D(D^k f)(a)(v_1)\big)(v_2, \dots, v_{k+1}).$$
Here $D(D^k f)(a)$ is the [[Fréchet derivative|08EVNYZF42]] of $D^k f$ at $a$, a linear map from $\mathbb{R}^n$ to $L^k(\mathbb{R}^n; \mathbb{R}^m)$.

## Remarks

For $k = 1$, $D^1 f = Df$. For $k = 2$, $D^2 f(a)(u, v)$ is the derivative of $x \mapsto Df(x)(v)$ at $a$ in the direction $u$.
On basis vectors, $D^2 f(a)(e_i, e_j) = \dfrac{\partial}{\partial x_i}\left(\dfrac{\partial f}{\partial x_j}\right)(a)$. In general, $D^k f(a)(e_{i_1}, \dots, e_{i_k})$ is a [[partial derivative of order k|02RSXVM1ZN]].
For a real-valued $f$, the matrix of the bilinear map $D^2 f(a)$ is the [[Hessian matrix|0693Y2METT]].
If $f$ is of class $C^k$, then $D^k f(a)$ is a symmetric $k$-linear map, by the [[symmetry of second derivatives|01V9FPBBVK]].

## Explanations

### L1 human

The first derivative takes one direction and gives a rate of change.
The second derivative takes two directions $u$ and $v$. It tells how the rate of change in direction $v$ itself changes as you move in direction $u$.
</document>

<document source="https://math2.tamnd.com/node/0EYQ03XCR3.md" tag="0EYQ03XCR3" kind="example">
# 68–95–99.7 rule

## Example

Let $X \sim \mathcal{N}(\mu, \sigma^2)$ with $\sigma > 0$, and let $\Phi$ be the cumulative distribution function of $\mathcal{N}(0, 1)$.
Since $Z = (X - \mu)/\sigma \sim \mathcal{N}(0, 1)$, $P(|X - \mu| \le k\sigma) = 2\Phi(k) - 1$ for each $k > 0$. Numerical values of $\Phi$ give:
$$P(|X - \mu| \le \sigma) \approx 0.6827, \quad P(|X - \mu| \le 2\sigma) \approx 0.9545, \quad P(|X - \mu| \le 3\sigma) \approx 0.9973.$$
So about $68\%$, $95\%$ and $99.7\%$ of the mass of a normal distribution lie within one, two and three standard deviations of the mean.

## Remarks

The rule holds for normal distributions only. For an arbitrary distribution with finite variance, Chebyshev's inequality gives only $P(|X - \mu| \le k\sigma) \ge 1 - 1/k^2$, for example $3/4$ for $k = 2$.
The interval $\mu \pm 1.96\sigma$ contains $95\%$ of the mass. It is the base of the usual $95\%$ confidence intervals.

## Explanations

### L1 human

For data that follow a bell curve, almost all values are within three standard deviations of the mean.
If adult heights have mean $170$ cm and standard deviation $10$ cm, then about $95\%$ of the heights are between $150$ cm and $190$ cm.
</document>

<document source="https://math2.tamnd.com/node/0EZ758XE73.md" tag="0EZ758XE73" kind="definition">
# Element

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]] and let $x$ be an object.
The object $x$ is an element of $A$, written $x \in A$, when $A$ contains $x$.
Otherwise, $x$ is not an element of $A$, written $x \notin A$.
The proposition $x \notin A$ is the negation of $x \in A$.

## Remarks

Other phrases for $x \in A$ are "$x$ is a member of $A$", "$x$ belongs to $A$" and "$x$ is in $A$".
For $A = \{1, 2, 3\}$, $2 \in A$ and $4 \notin A$.
An element of a set can itself be a set: $\{1\} \in \{\{1\}, 2\}$, but $1 \notin \{\{1\}, 2\}$.

## Explanations

### L1 human

The symbol $\in$ asks one yes or no question: is this object in the set?
For the set of even numbers, $6 \in$ the set and $7 \notin$ the set.
</document>

<document source="https://math2.tamnd.com/node/0EZBG8SKJM.md" tag="0EZBG8SKJM" kind="statement">
# Cantor's intersection theorem

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]].
Let $(C_n)_{n \ge 1}$ be a sequence of nonempty [[compact|00R0X1WGPX]] [[closed|0G5AV9Q4RR]] subsets of $X$ with $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$.
Then the intersection $\bigcap_{n \ge 1} C_n$ is not empty.

## Remarks

The sets must be compact: in $\mathbb{R}$, the closed sets $[n, \infty)$ are nested and nonempty, but their intersection is empty.
For $X = \mathbb{R}$, the sets can be nested closed bounded intervals $[a_n, b_n]$.
There is a variant for metric spaces. A metric space is [[complete|08EWT7J0SC]] if and only if each nested sequence of nonempty closed sets whose [[diameters|0692CGQF00]] tend to $0$ has nonempty intersection. The intersection is then one point.

## Explanations

### L1 human

If you have a chain of nonempty compact sets, each inside the one before, at least one point is in all of them.
For example, the intervals $[0, 1/n]$ are nested, and $0$ is in all of them.
</document>

<document source="https://math2.tamnd.com/node/0EZMAY1FQD.md" tag="0EZMAY1FQD" kind="example">
# x sin(1/x) is continuous at 0

## Example

Let $f(x) = x \sin(1/x)$ for $x \ne 0$, and let $f(0) = 0$.
Then $f$ is [[continuous|0EEE2CZB2A]] at $0$, and so $f$ is continuous on $\mathbb{R}$.
Indeed, $|f(x)| \le |x|$ for all $x$, so $f(x) \to 0$ as $x \to 0$ by the [[squeeze theorem|0XMF00T4XQ]].

## Remarks

The factor $x$ damps the oscillation of $\sin(1/x)$.
The function is not differentiable at $0$. The difference quotient $(f(x) - f(0))/x = \sin(1/x)$ has no limit.
The function $x^2 \sin(1/x)$ is [[differentiable with a discontinuous derivative|0WDAKKN4NC]].

## Explanations

### L1 human

The graph of $x \sin(1/x)$ oscillates infinitely often near $0$.
But it stays between the lines $y = x$ and $y = -x$, which meet at the origin.
</document>

<document source="https://math2.tamnd.com/node/0EZSD6T9E0.md" tag="0EZSD6T9E0" kind="definition">
# Lower triangular matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $L = (l_{ij})$ of order $n$ over $R$ is *lower triangular* when $l_{ij} = 0$ for all $i < j$.
It is *strictly lower triangular* when $l_{ij} = 0$ for all $i \le j$.

## Remarks

A matrix is lower triangular if and only if its [[transpose|09DXWYRD43]] is [[upper triangular|0N4XY7Z3NN]].
Sums and products of lower triangular matrices of the same order are lower triangular.
A system $Lx = b$ with $L$ lower triangular and invertible is solved by forward substitution.
In an [[LU decomposition|0RKG5YW66Y]], the factor $L$ is lower triangular.

## Explanations

### L1 human

A lower triangular matrix has zeros everywhere above its diagonal.
Its nonzero entries fill a triangle at the bottom left.
For example, $\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ is lower triangular.
</document>

<document source="https://math2.tamnd.com/node/0F0H5P5DED.md" tag="0F0H5P5DED" kind="statement">
# Cauchy's integral formula for derivatives

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then for each integer $n \ge 0$ and each $a \in \mathbb{C}$ with $|a - c| < r$,
$$f^{(n)}(a) = \frac{n!}{2 \pi i} \oint_{|z - c| = r} \frac{f(z)}{(z - a)^{n+1}} \, dz,$$
where $f^{(n)}$ is the $n$-th complex derivative of $f$. This derivative exists by [[holomorphic functions are infinitely differentiable|0QEBN8DGVE]].

## Remarks

For $n = 0$, this is [[Cauchy's integral formula|0DG10TVYPN]].
Formally, the formula comes from differentiating $\frac{1}{z - a}$ $n$ times under the integral sign with respect to $a$.
For example, with $f(z) = e^z$, $c = a = 0$ and $r = 1$: $\oint_{|z| = 1} \frac{e^z}{z^{n+1}} \, dz = \frac{2 \pi i}{n!}$.
It gives [[Cauchy's estimates|0JJRENWMP6]] for the derivatives.

## Explanations

### L1 human

The values of a holomorphic function on a circle determine not only its values inside, but also all its derivatives inside.
To get the $n$-th derivative, you put a higher power of $z - a$ in the denominator and multiply by $n!$.
</document>

<document source="https://math2.tamnd.com/node/0F0J193T76.md" tag="0F0J193T76" kind="concept">
# Set bounded above

## Summary

A set of numbers is bounded above when one number is greater than or equal to every element of the set. Such a number is an upper bound of the set.
The same idea applies to a subset of any ordered set.
</document>

<document source="https://math2.tamnd.com/node/0F0K8DB9TD.md" tag="0F0K8DB9TD" kind="statement">
# Prime elements are irreducible in an integral domain

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $p \in R$ be a [[prime element|0MNHXBSX3M]].
Then $p$ is [[irreducible|052SDEPZH7]].

## Remarks

The proof: let $p = a b$. Then $p \mid a b$, so $p \mid a$, say. Write $a = p c$. Then $p = p c b$, and [[cancellation|0GJM24EXP9]] gives $c b = 1$, so $b$ is a unit.
The converse is false in general: in $\mathbb{Z}[\sqrt{-5}]$, [[$2$ is irreducible but not prime|0CHE3J7QMX]].
In a [[unique factorization domain|06YMXJMXYS]], [[irreducible elements are prime|0QRCFZKXH8]].

## Explanations

### L1 human

Suppose that a prime $p$ factors as $a b$.
Then $p$ divides one factor, so that factor is at least as large as $p$ for divisibility.
The other factor must then be a unit.
</document>

<document source="https://math2.tamnd.com/node/0F0RPPJ2TT.md" tag="0F0RPPJ2TT" kind="definition">
# Column space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$ with columns $a_1, \dots, a_n \in F^m$.
The *column space* of $A$ is
$$\operatorname{Col}(A) = \operatorname{span}(a_1, \dots, a_n) \subseteq F^m,$$
the [[linear span|0DTY8B9HMS]] of the columns.

## Remarks

The column space is $\{Ax : x \in F^n\}$, the image of the map $x \mapsto Ax$.
The system $Ax = b$ is [[consistent|0X6V9B1V10]] if and only if $b \in \operatorname{Col}(A)$.
The dimension of the column space is the [[rank|0ZC9YTXZRY]] of $A$.
The pivot columns of $A$ form a basis of $\operatorname{Col}(A)$.

## Explanations

### L1 human

The column space collects all the combinations of the columns of a matrix.
These are exactly the vectors $b$ for which $Ax = b$ has a solution.
For the matrix $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$, the column space is the line spanned by $(1, 2)$.
</document>

<document source="https://math2.tamnd.com/node/0F1AVGX26N.md" tag="0F1AVGX26N" kind="definition">
# Closed and exact forms

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega$ be a [[differential k-form|0QHTX7WQEY]] on $U$ with $C^1$ coefficients, where $k \ge 1$.
The form $\omega$ is closed if $d\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]].
The form $\omega$ is exact if $\omega = d\eta$ for some $(k - 1)$-form $\eta$ on $U$.

## Remarks

Each exact form $d\eta$ with $\eta$ of class $C^2$ is closed, because [[d d = 0|0X308R2DBE]].
A [[one-form|0MTENFBW3Q]] $\sum_i f_i \, dx_i$ is closed if and only if $\partial_j f_i = \partial_i f_j$ for all $i, j$. It is exact if and only if $(f_1, \dots, f_n)$ is a [[conservative|05R9NF2J49]] field. In $\mathbb{R}^3$, closed means that the [[curl|0GDNHS9J2X]] is zero.
The angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ on $\mathbb{R}^2 \setminus \{0\}$ is closed but not exact. Its integral around the unit circle is $2\pi$, not $0$.
On a convex open set, each closed form is exact. This is the [[Poincaré lemma|05MC3HND79]].

## Explanations

### L1 human

Exact forms are derivatives of other forms, and closed forms only pass the local test for this: their derivative is zero.
The gap between the two notions is a measure of the holes of the domain.
</document>

<document source="https://math2.tamnd.com/node/0F1D6PK9RZ.md" tag="0F1D6PK9RZ" kind="definition">
# Well-order

## Definition

Let $A$ be a set.
A well-order on $A$ is a [[total order|0316AYH1SC]] $\le$ on $A$ such that each nonempty subset of $A$ has a [[least element|0T2NTJCV83]].
The pair $(A, \le)$ is a well-ordered set.

## Remarks

The usual order on $\mathbb{N}$ is a well-order.
The usual orders on $\mathbb{Z}$ and on $[0, 1]$ are not well-orders: $\mathbb{Z}$ and $(0, 1]$ have no least element.
A total order is a well-order exactly when its strict order $<$ is [[well-founded|0230ECX1Q5]].
The [[well-ordering theorem|0N109NG5ZX]] says that each set has a well-order.

## Explanations

### L1 human

In a well-ordered set, each nonempty part has a first element.
So you can always say "take the smallest element with this property".
</document>

<document source="https://math2.tamnd.com/node/0F1KBVY161.md" tag="0F1KBVY161" kind="definition">
# Transition matrix

## Definition

Let $S$ be a nonempty countable set.
A transition matrix, or stochastic matrix, on $S$ is a family $P = (p_{ij})_{i, j \in S}$ of real numbers with
$$p_{ij} \ge 0 \quad \text{for all } i, j \in S \quad \text{and} \quad \sum_{j \in S} p_{ij} = 1 \quad \text{for each } i \in S.$$
The transition matrix of a time-homogeneous [[Markov chain|033TJBZDG0]] $(X_n)$ on $S$ has $p_{ij} = P(X_{n+1} = j \mid X_n = i)$.

## Remarks

Each row of $P$ is a probability distribution on $S$: the law of the next state, given the present state $i$.
A product of two stochastic matrices is stochastic. The $n$-th power $P^n$ gives the $n$-step probabilities $P(X_{m+n} = j \mid X_m = i)$.
If the row vector $\lambda$ is the law of $X_0$, then $\lambda P^n$ is the law of $X_n$.
A finite stochastic matrix has the eigenvalue $1$, with the eigenvector $(1, \dots, 1)^{\mathsf{T}}$, and all its eigenvalues have absolute value at most $1$.

## Explanations

### L1 human

The transition matrix is a table of the chances to move from each state to each state in one step.
For weather with states "sun" and "rain", the row of "sun" gives the chances of sun and of rain tomorrow, if today is sunny.
</document>

<document source="https://math2.tamnd.com/node/0F1PR74BRX.md" tag="0F1PR74BRX" kind="concept">
# Subsequence

## Summary

A subsequence of a sequence keeps infinitely many of its terms, in their original order. Subsequences are the main tool of the Bolzano–Weierstrass theorem. The node [[subsequence|0R8J3ZQMJS]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0F1SN93KXE.md" tag="0F1SN93KXE" kind="definition">
# Conditional variance

## Definition

Let $X$ be a real random variable on $(\Omega, \mathcal{F}, P)$ with $E[X^2] < \infty$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a σ-algebra.
The conditional variance of $X$ given $\mathcal{G}$ is the random variable
$$\operatorname{Var}(X \mid \mathcal{G}) = E\left[\left(X - E[X \mid \mathcal{G}]\right)^2 \,\middle|\, \mathcal{G}\right],$$
where $E[\cdot \mid \mathcal{G}]$ is the [[conditional expectation|0YAMJRK04E]]. For a random variable $Y$, $\operatorname{Var}(X \mid Y) = \operatorname{Var}(X \mid \sigma(Y))$.

## Remarks

As for the [[variance|007QED3E0Y]], $\operatorname{Var}(X \mid \mathcal{G}) = E[X^2 \mid \mathcal{G}] - (E[X \mid \mathcal{G}])^2$ almost surely.
For a discrete $Y$ and $y$ with $P(Y = y) > 0$, $\operatorname{Var}(X \mid Y)$ takes the value $\operatorname{Var}(X \mid Y = y)$ on $\{Y = y\}$. This is the variance of $X$ under $P(\cdot \mid Y = y)$.
$\operatorname{Var}(X \mid \mathcal{G}) = 0$ almost surely if and only if $X$ is equal almost surely to a $\mathcal{G}$-measurable random variable.

## Explanations

### L1 human

The conditional variance tells how uncertain $X$ still is after you learn some information.
For example, the variance of the height of a person given the sex of that person is the spread of heights within each group.
</document>

<document source="https://math2.tamnd.com/node/0F1WR9C83K.md" tag="0F1WR9C83K" kind="definition">
# Derivative

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be a function, and let $a \in I$.
A real number $L$ is the derivative of $f$ at $a$ if
$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = L.$$
The notation is $f'(a) = L$.

## Remarks

The quotient is the slope of the line through $(a, f(a))$ and $(a + h, f(a + h))$. The limit is a [[limit of a function|0TXTSEMHR5]] of $h$ at $0$.
The derivative is unique when it exists.
For example, the derivative of $x^2$ at $a$ is $\lim_{h \to 0} (2a + h) = 2a$.
Mathlib defines the relation "$f$ has derivative $L$ at $a$" as `HasDerivAt`.

## Explanations

### L1 human

The derivative of $f$ at $a$ is the instantaneous rate of change of $f$ at $a$.
Geometrically, it gives the steepness of the graph at the point $(a, f(a))$.
</document>

<document source="https://math2.tamnd.com/node/0F27PBRWD7.md" tag="0F27PBRWD7" kind="definition">
# Row echelon form

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$.
The *leading entry* of a nonzero row of $A$ is its first nonzero entry, from the left.
The matrix $A$ is in *row echelon form* when these two conditions hold:

1. Each zero row of $A$ is below each nonzero row of $A$.
2. For two consecutive nonzero rows, the leading entry of the lower row is strictly to the right of the leading entry of the upper row.

## Remarks

The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{pmatrix}$ is in row echelon form.
Some authors also require that each leading entry is $1$.
In row echelon form, all the entries below a leading entry are zero.
[[Gaussian elimination|0JVJ2WCXZH]] changes each matrix into a row echelon form by elementary row operations.

## Explanations

### L1 human

Row echelon form looks like a staircase.
Each nonzero row starts further to the right than the row above it, and the zero rows sit at the bottom.
A system in this form is easy to solve from the bottom row up.
</document>

<document source="https://math2.tamnd.com/node/0F2PTGX1A2.md" tag="0F2PTGX1A2" kind="concept">
# Field of fractions

## Summary

The field of fractions of an integral domain is the smallest field that contains it.
Its elements are the fractions $a/b$ with $b \neq 0$.
The rational numbers are the field of fractions of the integers.
The [[definition of the field of fractions|0VF11ZTTMD]] gives the construction.
</document>

<document source="https://math2.tamnd.com/node/0F2QCWVFT7.md" tag="0F2QCWVFT7" kind="concept">
# Riemann integrable function

## Summary

A function on a closed bounded interval is Riemann integrable when its Riemann sums converge to one number as the mesh tends to zero. Continuous functions and monotone functions are Riemann integrable. The node [[Riemann integrable function|0N1JQXZXHC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0F2SYCKDT2.md" tag="0F2SYCKDT2" kind="concept">
# Lyapunov stability

## Summary

An equilibrium point of a dynamical system is Lyapunov stable if all solutions that start near it stay near it for all future time.
The theory goes back to Aleksandr Lyapunov. See the [[definition|092S6VBECQ]] and the stronger notion of [[asymptotic stability|05Y1JAM3Y5]].
</document>

<document source="https://math2.tamnd.com/node/0F2VBPXNP9.md" tag="0F2VBPXNP9" kind="statement">
# Superposition principle

## Statement

Let $I$ be an open interval, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$ with continuous functions $a_0, \dots, a_{n-1}$ on $I$.
Let $y_1$ and $y_2$ be solutions of the [[homogeneous linear equation|0EQZSAB9EC]] $L y = 0$ on $I$, and let $c_1, c_2$ be real numbers. Then $c_1 y_1 + c_2 y_2$ is a solution of $L y = 0$ on $I$.

## Remarks

The proof is the linearity of $L$: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2 = 0$.
The same argument gives a variant for right sides: if $L y_1 = b_1$ and $L y_2 = b_2$, then $L(y_1 + y_2) = b_1 + b_2$.
The principle fails for nonlinear equations. For $y' = y^2$, the function $y = -1/x$ is a solution on $x > 0$, but $2y$ is not.

## Explanations

### L1 human

For a linear equation, you can add solutions and scale them, and you get solutions again.
In physics this is superposition: two waves that solve a linear wave equation add up to a third solution.
</document>

<document source="https://math2.tamnd.com/node/0F32S73FJN.md" tag="0F32S73FJN" kind="concept">
# Iterated derivative

## Summary

The iterated derivative $D^k f$ of a map $f$ is the derivative of the derivative, taken $k$ times.
The [[Fréchet derivative|06EH97V2A7]] of $D^{k-1} f$ at a point is a linear map with values in $(k-1)$-linear maps, which is the same as a $k$-linear map.
So $D^k f(a)$ is a $k$-linear map, and for a function on $\mathbb{R}^n$ its values on basis vectors are the partial derivatives of order $k$.
</document>

<document source="https://math2.tamnd.com/node/0F33KX9Q6F.md" tag="0F33KX9Q6F" kind="concept">
# Primitive root modulo n

## Summary

A primitive root modulo $n$ is an integer whose powers give every residue class modulo $n$ that is coprime to $n$.
The [[definition of a primitive root modulo n|0KM2CJ80G7]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0F3K4C7XYW.md" tag="0F3K4C7XYW" kind="statement">
# Bounded linear maps are continuous

## Statement

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be linear.
Then $T$ is [[continuous|0S3SMRH0Z5]] if and only if $T$ is bounded, that is, there is a real $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$.

## Remarks

If $T$ is bounded, then $\|Tx - Ty\|_W = \|T(x - y)\|_W \le C \|x - y\|_V$. Thus $T$ is [[Lipschitz|0TMZ21WKCN]], and so [[uniformly continuous|07DCKDXRBP]].
Conversely, if $T$ is continuous at $0$, there is $\delta > 0$ with $\|Tx\|_W \le 1$ when $\|x\|_V \le \delta$. By linearity, $\|Tx\|_W \le \|x\|_V / \delta$ for all $x$.
Thus, for a linear map, continuity at one point already gives continuity everywhere.
The name bounded means that $T$ maps bounded sets to bounded sets. It does not mean that $T(V)$ is bounded.

## Explanations

### L1 human

For linear maps, continuity is the same as a simple estimate. The length of $Tx$ is at most a fixed multiple of the length of $x$.
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<document source="https://math2.tamnd.com/node/0F3Z22M2GT.md" tag="0F3Z22M2GT" kind="concept">
# Mersenne prime

## Summary

A Mersenne prime is a prime number of the form $2^n - 1$.
The [[definition of a Mersenne prime|06GPJF9V7M]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0F405477F8.md" tag="0F405477F8" kind="statement">
# Derivative vanishes at a local extremum

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a [[local extremum|04CPG9K1C4]] of $f$.
If the [[partial derivative|05AXG3QRKR]] $\partial f / \partial x_i (a)$ exists for some $i$, then $\partial f / \partial x_i (a) = 0$.
In particular, if $f$ is [[differentiable|0135XZZQ8M]] at $a$, then $\nabla f(a) = 0$ and the derivative $Df(a)$ is the zero linear map.

## Remarks

A point $a \in U$ where $f$ is differentiable and $\nabla f(a) = 0$ is a critical point, or stationary point, of $f$.
The proof applies the one-variable theorem to $t \mapsto f(a + t e_i)$, where $e_i$ is the $i$-th standard basis vector. This function has a local extremum at $t = 0$.
The converse is false. For $f(x, y) = x^2 - y^2$, $\nabla f(0, 0) = 0$, but $(0, 0)$ is a [[saddle point|00N3T18QNA]], not an extremum.
The hypothesis that $U$ is open matters. On $[0, 1]^2$, $f(x, y) = x + y$ has a maximum at $(1, 1)$, where $\nabla f = (1, 1)$.

## Explanations

### L1 human

At the top of a smooth hill the ground is flat in every direction.
So all partial derivatives are zero there, and the tangent plane to the graph is horizontal.
This turns the search for extrema into the solution of the equations $\nabla f = 0$.
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<document source="https://math2.tamnd.com/node/0F462493EA.md" tag="0F462493EA" kind="concept">
# Transition matrix

## Summary

A transition matrix, or stochastic matrix, is a square matrix of nonnegative entries in which each row adds up to $1$. It gives the one-step probabilities of a Markov chain.
The node [[transition matrix|0F1KBVY161]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0F4JSQE2EK.md" tag="0F4JSQE2EK" kind="concept">
# Divisors of a natural number

## Summary

The divisors of a positive integer $n$ form the finite set of positive integers that divide $n$.
The [[definition of the set of divisors|0FCWA16FZE]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0F4WRWBZZC.md" tag="0F4WRWBZZC" kind="concept">
# Quadratic integer

## Summary

A quadratic integer is a root of a monic quadratic polynomial with integer coefficients, that is, an algebraic integer of degree at most $2$.
The [[definition of a quadratic integer|01VN79T2PR]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0F50N2P3FC.md" tag="0F50N2P3FC" kind="concept">
# Outcome

## Summary

An outcome is one possible result of a random experiment.
The outcomes of an experiment are the elements of its sample space. The node [[outcome|0Y5R7TPNT1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0F56CMMC5P.md" tag="0F56CMMC5P" kind="statement">
# Divergence of a curl is zero

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $A \colon U \to \mathbb{R}^3$ be a vector field of class $C^2$.
Then
$$\nabla \cdot (\nabla \times A) = 0 \quad \text{on } U.$$

## Remarks

Proof: with $A = (A_1, A_2, A_3)$, the [[divergence|02NSMQVQWK]] of the [[curl|0GDNHS9J2X]] is $\partial_x (\partial_y A_3 - \partial_z A_2) + \partial_y (\partial_z A_1 - \partial_x A_3) + \partial_z (\partial_x A_2 - \partial_y A_1)$. The six terms cancel in pairs by [[Schwarz's theorem|0NNQY8WCW6]].
So a field that has a vector potential is divergence-free. On an open ball, the converse is also true: a $C^1$ divergence-free field is the curl of some field.
The identity is a special case of the rule [[d d = 0|0X308R2DBE]] for the exterior derivative.

## Explanations

### L1 human

The curl of a field describes rotation, and pure rotation does not create or destroy anything.
So the curl of any smooth field has no sources or sinks, which means its divergence is zero.
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<document source="https://math2.tamnd.com/node/0F5QC101Q7.md" tag="0F5QC101Q7" kind="statement">
# g to the n equals 1 iff the order divides n

## Statement

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $m$.
Let $n$ be an integer.
Then $g^n = e$ if and only if $m$ divides $n$.

## Remarks

The proof divides $n$ by $m$ with remainder: $n = qm + r$ with $0 \leq r < m$, so that $g^n = (g^m)^q g^r = g^r$.
By the minimality of $m$, the power $g^r$ is $e$ only when $r = 0$.
A consequence is that $g^a = g^b$ if and only if $a \equiv b \pmod m$.

## Explanations

### L1 human

An element of order $m$ comes back to the identity exactly after $m$, $2m$, $3m$ steps, and so on.
It is never at the identity after a number of steps that is not a multiple of $m$.
For example, a rotation by $90$ degrees is the identity after $8$ steps, but not after $6$ steps.
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<document source="https://math2.tamnd.com/node/0F6ES3BRYP.md" tag="0F6ES3BRYP" kind="concept">
# Characteristic of a ring

## Summary

The characteristic of a ring is the least positive integer $n$ such that the sum of $n$ copies of $1$ is $0$.
If there is no such integer, the characteristic is $0$.
The [[definition of the characteristic|0P7CH1CND5]] states this precisely.
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<document source="https://math2.tamnd.com/node/0F6FM9Y4Y6.md" tag="0F6FM9Y4Y6" kind="definition">
# Order on the integers

## Definition

Let $m$ and $n$ be [[integers|0C4JS9RF5M]].
The notation $m \le n$ means that $n - m$ is a [[natural number|0MWKJ44V5A]], that is, when $n - m \in \{0, 1, 2, \dots\}$.
The notation $m < n$ means that $m \le n$ and $m \neq n$.

## Remarks

This relation is a total order on $\mathbb{Z}$, and on $\mathbb{N}$ it agrees with the [[order on the natural numbers|0Y9H52BT03]].
If $m \le n$, then $m + k \le n + k$ for each integer $k$. If $m \le n$ and $0 \le k$, then $m k \le n k$.
Thus $\mathbb{Z}$ is an ordered ring. Unlike $\mathbb{N}$, it has no least element.

## Explanations

### L1 human

On the number line, $m \le n$ means that $m$ is at the same place as $n$ or to the left of it.
For example, $-5 \le -2$ because $-2 - (-5) = 3$ is a natural number.
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<document source="https://math2.tamnd.com/node/0F72FK8EKB.md" tag="0F72FK8EKB" kind="concept">
# Disjunction

## Summary

Disjunction is the logical operation "or".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true when at least one part is true.
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<document source="https://math2.tamnd.com/node/0F765KFC2J.md" tag="0F765KFC2J" kind="statement">
# Cayley–Hamilton theorem

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$ with [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$.
Then
$$p_A(A) = 0,$$
where $p_A(A)$ is the [[value of $p_A$ at $A$|0E3MV6E6MG]].

## Remarks

For $n = 2$, it says $A^2 - (\tr A) A + (\det A) I_2 = 0$.
Substituting $t = A$ into $\det(t I_n - A)$ is not a valid proof, because $p_A(A)$ is a matrix and $\det(A - A)$ is a scalar.
One proof uses the [[adjugate|07W9H8QF38]] identity $(t I_n - A) \operatorname{adj}(t I_n - A) = p_A(t) I_n$.
If $A$ is invertible, then the theorem expresses $A^{-1}$ as a polynomial in $A$.

## Explanations

### L1 human

Every square matrix satisfies its own characteristic polynomial.
For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, the polynomial is $t^2 - 5t - 2$, so $A^2 = 5A + 2I$.
So every power of $A$ is a combination of $I$ and $A$.
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<document source="https://math2.tamnd.com/node/0F7FQS6XY6.md" tag="0F7FQS6XY6" kind="concept">
# Square root

## Summary

A square root of a number $x$ is a number $y$ with $y^2 = x$.
Each nonnegative real number $x$ has exactly one nonnegative square root, the principal square root $\sqrt{x}$.
For example, $\sqrt{9} = 3$, and the square roots of $9$ are $3$ and $-3$.
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<document source="https://math2.tamnd.com/node/0F7KNJ1ZYN.md" tag="0F7KNJ1ZYN" kind="definition">
# Greatest common divisor in an integral domain

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b \in R$.
An element $d \in R$ is a greatest common divisor of $a$ and $b$ if it has these two properties.

1. $d \mid a$ and $d \mid b$.
2. For every $c \in R$ with $c \mid a$ and $c \mid b$, $c \mid d$.

## Remarks

If $d$ and $d'$ are two greatest common divisors of $a$ and $b$, then $d$ and $d'$ are [[associates|053E86NWVK]].
In $\mathbb{Z}$, the greatest common divisors of $12$ and $18$ are $6$ and $-6$.
A greatest common divisor need not exist. A GCD domain is an integral domain in which every pair of elements has one.
In a [[principal ideal domain|0RZYDVZ475]], $d$ is a greatest common divisor of $a$ and $b$ if and only if $(a, b) = (d)$.

## Explanations

### L1 human

In a general ring, "greatest" does not refer to size.
It means the largest for divisibility: every other common divisor divides it.
For the integers, this gives the usual gcd, up to the sign.
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<document source="https://math2.tamnd.com/node/0F7QKGJS6B.md" tag="0F7QKGJS6B" kind="concept">
# Graph isomorphism

## Summary

A graph isomorphism is a bijection between the vertices of two graphs that preserves and reflects adjacency. Isomorphic graphs have the same structure and differ only in the names of their vertices. The node [[graph isomorphism|00CT0PTH74]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0F8GDKM0A7.md" tag="0F8GDKM0A7" kind="concept">
# Convolution of probability distributions

## Summary

The convolution of two probability distributions on $\mathbb{R}$ is the distribution of the sum of two independent random variables with these distributions.
The node [[convolution of probability distributions|07CHJ5YX28]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0F8X8ZRQ53.md" tag="0F8X8ZRQ53" kind="object">
# Group of units of Z/nZ

## Description

Let $n \geq 1$ be an integer.
The group of units of $\mathbb{Z}/n\mathbb{Z}$ is the [[group of units|0FJ90Y9926]] $(\mathbb{Z}/n\mathbb{Z})^\times$ of the ring $\mathbb{Z}/n\mathbb{Z}$ under multiplication.
Its elements are the residue classes $[a]$ with $\gcd(a, n) = 1$, and its identity is $[1]$.
It is a finite [[abelian group|082GHNPJ53]] of order $\varphi(n)$.

## Remarks

For $n = 8$, the group is $\{[1], [3], [5], [7]\}$, and the square of each element is $[1]$, so it is isomorphic to the [[Klein four-group|0PMRQJVTME]].
For $n = 16$, the group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, so it is not cyclic.
The group is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$, with $p$ an odd prime and $k \geq 1$. A generator is called a primitive root modulo $n$.
Other notations are $(\mathbb{Z}/n\mathbb{Z})^*$, $U(\mathbb{Z}/n\mathbb{Z})$ and $\mathbb{Z}_n^*$.

## Explanations

### L1 human

Multiplication modulo $n$ does not make all of $\mathbb{Z}/n\mathbb{Z}$ a group, because $[0]$ has no inverse.
The classes coprime to $n$ do form a group under multiplication.
This group is important in number theory and in cryptography.
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<document source="https://math2.tamnd.com/node/0F8YBHF2W3.md" tag="0F8YBHF2W3" kind="concept">
# Double counting

## Summary

Double counting is a proof technique in combinatorics.
To prove that two expressions are equal, one shows that both count the elements of the same finite set.
Each expression comes from a different way to count that set.

A common form uses a relation $R \subseteq A \times B$ between finite sets $A$ and $B$.
For $a \in A$, let $d(a)$ be the number of $b \in B$ with $(a, b) \in R$.
For $b \in B$, let $e(b)$ be the number of $a \in A$ with $(a, b) \in R$.
Then both sums count the pairs in $R$, so
$$\sum_{a \in A} d(a) = |R| = \sum_{b \in B} e(b).$$

## Remarks

The [[handshaking lemma|0XAP8WHNES]] is double counting applied to the pairs of a vertex and an edge that contains it.
The [[symmetry of binomial coefficients|0R7JS3DP3W]] has a short proof by this technique.

## Explanations

### L1 human

Count the same thing in two ways, and the two answers must be equal.
For example, in a class each pair of friends shakes hands once. If you add the number of handshakes of each person, you count each handshake twice, so the sum is even.
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<document source="https://math2.tamnd.com/node/0F98FZK14A.md" tag="0F98FZK14A" kind="concept">
# Circumcentre

## Summary

The circumcentre of a triangle is the centre of the circle through its three vertices. It is the point of the plane of the triangle at the same distance from all three vertices.
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<document source="https://math2.tamnd.com/node/0F9JHP8AJN.md" tag="0F9JHP8AJN" kind="definition">
# Codomain

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The codomain of $f$ is the set $B$.

## Remarks

Each value $f(a)$ is in the codomain, but not every element of the codomain needs to be a value.
The set of values is the [[range|0E9GTAV478]] of $f$. It is a subset of the codomain.
The function $f$ is [[surjective|0XKH881BZ7]] when its range is the whole codomain.

## Explanations

### L1 human

For $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$, the codomain is $\mathbb{R}$.
The value $-1$ is in the codomain, but no input gives it.
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<document source="https://math2.tamnd.com/node/0F9VKJPWCA.md" tag="0F9VKJPWCA" kind="concept">
# Absolutely convergent improper integral

## Summary

An improper integral converges absolutely when the improper integral of the absolute value of the integrand converges. Absolute convergence implies convergence, but not conversely. The node [[absolutely convergent improper integral|03GP7KMBYM]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0FANE7P0DY.md" tag="0FANE7P0DY" kind="concept">
# Möbius function

## Summary

The Möbius function $\mu$ is an arithmetic function that is $0$ on integers with a square factor greater than $1$.
On a squarefree integer, its value $1$ or $-1$ depends on the parity of the number of prime factors.
The [[definition of the Möbius function|0C8QENHYZW]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0FBETJR0JG.md" tag="0FBETJR0JG" kind="concept">
# Local maximum

## Summary

A function has a local maximum at a point when its value there is at least as large as its values at all nearby points. Local maxima are the tops of the hills of the graph. The node [[local maximum|00999TYPXB]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0FBXFBZBH8.md" tag="0FBXFBZBH8" kind="definition">
# Affine subspace

## Definition

Let $K$ be a field, let $V$ be a vector space over $K$, and let $A$ be an [[affine space|09XNAPTJQ7]] over $V$.
A nonempty subset $S$ of $A$ is an affine subspace if $S = p + W$ for a point $p$ of $A$ and a linear subspace $W$ of $V$. This means that
$$S = p + W = \{\, p + w \;:\; w \in W \,\}.$$
The subspace $W$ is the direction of $S$. It is unique: $W = \{\, q - r : q, r \in S \,\}$. The dimension of $S$ is the dimension of $W$.

## Remarks

For each point $p$ of $S$, $S = p + W$. Thus each point of $S$ can serve as the base point.
In $\mathbb{R}^n$, the affine subspaces of dimension 1 are the [[lines|0MP9BCF6DP]], and those of dimension 2 are the [[planes|0V3XAMF3C1]]. Those of dimension $n - 1$ are the hyperplanes.
Two affine subspaces are parallel if they have the same direction.
Mathlib's `AffineSubspace` also allows the empty set. A subset $S$ is an affine subspace in this sense if $c(q - r) + s \in S$ for all $c \in K$ and $q, r, s \in S$.

## Explanations

### L1 human

An affine subspace is a flat piece of space that need not pass through the origin. A line or a plane that has been moved away from the origin is an example. It is a linear subspace that has been shifted by a translation.
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<document source="https://math2.tamnd.com/node/0FBXJZX3MK.md" tag="0FBXJZX3MK" kind="statement">
# Closed subsets of complete spaces are complete

## Statement

Let $(X, d)$ be a [[complete metric space|08EWT7J0SC]] and let $F$ be a [[closed subset|0G5AV9Q4RR]] of $X$.
Then $F$ with the [[subspace metric|0T2WYDTRP2]] is complete.

## Remarks

Proof: let $(x_n)$ be a Cauchy sequence in $F$. It is Cauchy in $X$, so it converges to some $x \in X$. Then $x \in F$, because [[closed sets contain their limits|09GQ3ZTQJ9]].
For example, $[0, 1]$, $\mathbb{Z}$ and each closed ball of $\mathbb{R}^n$ are complete.
The converse is [[complete subspaces are closed|0W41XG4W60]].

## Explanations

### L1 human

A Cauchy sequence in $F$ has a limit in the big complete space.
Because $F$ is closed, this limit cannot escape from $F$.
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<document source="https://math2.tamnd.com/node/0FCCDXH97W.md" tag="0FCCDXH97W" kind="concept">
# Evaluation homomorphism

## Summary

The evaluation homomorphism at an element $c$ of a commutative ring $R$ is the map $R[x] \to R$ that sends each polynomial to its value at $c$.
The [[definition of the evaluation homomorphism|0125QVNCFQ]] gives the details.
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<document source="https://math2.tamnd.com/node/0FCCTFA275.md" tag="0FCCTFA275" kind="definition">
# Exponent of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The exponent of $G$ is the least positive integer $n$ such that $g^n = e$ for each $g \in G$.
If no such integer exists, then $G$ has infinite exponent.

## Remarks

When the exponent is finite, it is the least common multiple of the [[orders of the elements|0WE2NHZJ7B]] of $G$.
The exponent of a finite group divides the order of the group, by [[Lagrange's theorem|3C53TD]].
The exponent of the cyclic group of order $n$ is $n$, and the exponent of the [[Klein four-group|0PMRQJVTME]] is $2$.
The exponent of $(\mathbb{Z}/n\mathbb{Z})^\times$ is the Carmichael function $\lambda(n)$.

## Explanations

### L1 human

The exponent is the smallest power that sends every element to the identity.
It is a common multiple of all the element orders.
A finite group always has a finite exponent, and the exponent divides the order.
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<document source="https://math2.tamnd.com/node/0FCHE902D7.md" tag="0FCHE902D7" kind="object">
# General linear group

## Description

Let $R$ be a commutative ring and let $n \ge 1$ be an integer.
The *general linear group* $GL_n(R)$ is the set of [[invertible|0EB9N0S33C]] $n \times n$ matrices over $R$, with [[matrix multiplication|032Q9FB8MT]] as operation.
It is a [[group|G5S1M9]]: the product of invertible matrices is invertible, the identity element is $I_n$, and the inverse of $A$ is $A^{-1}$.
Equivalently, $GL_n(R)$ is the group of units of the [[matrix ring|0FHVPRAE60]] $M_n(R)$.

## Remarks

For a field $F$, $GL_n(F)$ is isomorphic to the group of linear isomorphisms of $F^n$ to itself.
For $n \ge 2$, the group $GL_n(F)$ is not abelian.
For a finite field with $q$ elements, $|GL_n(\F_q)| = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.
The [[special linear group|0V9PMN0RBH]] is the subgroup of matrices with determinant $1$.

## Explanations

### L1 human

The general linear group collects all the invertible matrices of one size.
We can multiply two of them, and we can undo each of them, so they form a group.
Geometrically, its elements are all the ways to transform space linearly without flattening it.
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<document source="https://math2.tamnd.com/node/0FCWA16FZE.md" tag="0FCWA16FZE" kind="definition">
# Divisors of a natural number

## Definition

Let $n$ be a positive integer.
The set of divisors of $n$ is the set of positive integers $d$ that [[divide|096PW7Q014]] $n$:
$$D(n) = \{ d \in \mathbb{Z}_{>0} : d \mid n \}.$$

## Remarks

The set $D(n)$ is finite, because each $d \in D(n)$ satisfies $1 \le d \le n$. It always contains $1$ and $n$.
For example, $D(12) = \{1, 2, 3, 4, 6, 12\}$.
The map $d \mapsto n/d$ is a bijection from $D(n)$ to itself.
Sums over $D(n)$ define the [[divisor function|0YEW47VJ0J]] and the [[Dirichlet convolution|0KH1N66QWW]].

## Explanations

### L1 human

The divisors of $n$ are the numbers that go into $n$ with no remainder.
They come in pairs $d$ and $n/d$. For $n = 12$, the pairs are $1$ and $12$, $2$ and $6$, and $3$ and $4$.
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<document source="https://math2.tamnd.com/node/0FDCW77H44.md" tag="0FDCW77H44" kind="statement">
# The gamma function extends the factorial

## Statement

For each $s > 0$, the [[gamma function|0V3FZMVB4K]] satisfies $\Gamma(s + 1) = s \, \Gamma(s)$.
For each natural number $n$, $\Gamma(n + 1) = n!$.

## Remarks

[[Integration by parts|08Y1BBQTK9]] on $[\varepsilon, b]$, with $u = t^s$ and $v = -e^{-t}$, gives the recursion in the limit.
The [[integral of $e^{-t}$|039JAF7ZAN]] gives $\Gamma(1) = 1$. Induction on $n$ then gives $\Gamma(n + 1) = n!$.
Also $\Gamma(1/2) = \sqrt{\pi}$, by the substitution $t = x^2$ and the Gaussian integral.
Mathlib states it as `Real.Gamma_nat_eq_factorial`.

## History

Leonhard Euler announced the integral representation in a letter to Christian Goldbach, dated 8 January 1730.

## Explanations

### L1 human

The recursion $\Gamma(s + 1) = s \, \Gamma(s)$ is the same as the rule $(n + 1)! = (n + 1) \cdot n!$.
So the gamma function agrees with the factorial at the integers, shifted by one.
</document>

<document source="https://math2.tamnd.com/node/0FDKRX9M9Y.md" tag="0FDKRX9M9Y" kind="concept">
# Reachability

## Summary

Reachability says which vertices of a graph can be joined by a walk. It is an equivalence relation, and its classes are the connected components. The node [[reachability|0J8XE7W68F]] gives the precise definition.
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# Locally compact space

## Summary

A locally compact space is a topological space in which each point has a compact neighbourhood.
Such spaces look like compact spaces near each point, as $\mathbb{R}^n$ does.
</document>

<document source="https://math2.tamnd.com/node/0FE8Q5S42Y.md" tag="0FE8Q5S42Y" kind="definition">
# Subset

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The set $A$ is a subset of $B$, written $A \subseteq B$, when each [[element|0EZ758XE73]] of $A$ is an element of $B$:
$$\forall x \, (x \in A \to x \in B).$$
Then $B$ is a superset of $A$.
The set $A$ is a proper subset of $B$, written $A \subsetneq B$, when $A \subseteq B$ and $A \neq B$.

## Remarks

Some authors write $A \subset B$ for $A \subseteq B$, and others write it for a proper subset.
Two sets are equal if and only if each is a subset of the other. This follows from the [[axiom of extensionality|0TB16T2C97]].
The [[empty set|0ECVK59Y03]] is a subset of every set, and every set is a subset of itself.

## Explanations

### L1 human

$A \subseteq B$ means that $A$ fits inside $B$.
The set of even numbers is a subset of the set of integers.
To prove $A \subseteq B$, take an arbitrary $x \in A$ and show $x \in B$.
</document>

<document source="https://math2.tamnd.com/node/0FEBE9NC0E.md" tag="0FEBE9NC0E" kind="statement">
# A subgroup whose index is the smallest prime divisor of the order is normal

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be the smallest prime number that divides $|G|$. Let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ of [[index|0DTXBQPA2S]] $p$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $p = 2$ this is the fact that [[subgroups of index two are normal|0V547YQRMV]].
The proof lets $G$ act on the $p$ left cosets of $H$. The kernel $K$ of this action lies in $H$, and $[G : K]$ divides both $|G|$ and $p!$.
Since $p$ is the smallest prime divisor of $|G|$, this forces $[G : K] = p$, so $H = K$ is normal.

## Explanations

### L1 human

Index $2$ is not the only small index that forces normality.
If the index is the smallest prime that divides the order of the group, the subgroup is also normal.
For example, a subgroup of index $3$ in a group of order $75 = 3 \cdot 5^2$ is normal.
</document>

<document source="https://math2.tamnd.com/node/0FEMF8JKAF.md" tag="0FEMF8JKAF" kind="statement">
# e as the limit of (1 + 1/n)^n

## Statement

Let $e = \exp(1)$, where $\exp$ is the [[exponential function|0RMVS7RXZX]].
Then
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e.$$
More generally, for each real number $x$,
$$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x.$$

## Remarks

Some texts take the limit as the definition of [[the number e|0MVCFMW7S4]]. The sequence $(1 + 1/n)^n$ is increasing and bounded above by $3$, so the [[monotone convergence theorem|0FM6N4R0VB]] gives the limit.
The first values are $2$, $2.25$, about $2.370$ and about $2.441$. The limit is $e \approx 2.71828$.
Jacob Bernoulli found this limit when he studied compound interest.
Mathlib states the general form as `Real.tendsto_one_add_div_pow_exp`.

## Explanations

### L1 human

Put one unit of money in a bank at $100\%$ yearly interest, paid $n$ times a year.
After one year you have $(1 + 1/n)^n$ units. As $n$ grows, this approaches $e$, and it never exceeds $e$.
</document>

<document source="https://math2.tamnd.com/node/0FER36N8SB.md" tag="0FER36N8SB" kind="conjecture">
# Twin prime conjecture

## Statement

There are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p + 2$ is also prime.

## Remarks

A pair of primes $(p, p + 2)$ is a pair of twin primes. The first pairs are $(3, 5)$, $(5, 7)$, $(11, 13)$, $(17, 19)$, $(29, 31)$.
The conjecture is the case $k = 1$ of the conjecture of de Polignac: for each positive integer $k$, there are infinitely many primes $p$ such that $p + 2k$ is prime.
The conjecture is open.

## History

De Polignac stated the general conjecture in 1849.
In 2013, Yitang Zhang proved that some gap $N < 7 \cdot 10^7$ occurs between infinitely many pairs of primes. Later work by Maynard and the Polymath project reduced the bound to $246$.

## Explanations

### L1 human

Twin primes are primes with only one number between them, like $41$ and $43$.
The conjecture says that such pairs never stop, even though primes become rarer.
</document>

<document source="https://math2.tamnd.com/node/0FF2FJ8CH2.md" tag="0FF2FJ8CH2" kind="statement">
# Universal property of the quotient topology

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $\sim$ be an equivalence relation on $X$, and give $X / {\sim}$ the [[quotient topology|0V8046R0P9]], with quotient map $q \colon X \to X / {\sim}$.
Let $Y$ be a topological space and let $g \colon X / {\sim} \to Y$ be a function.
Then $g$ is [[continuous|0SY9ECQ19Q]] if and only if the [[composition|07HF9YCWPP]] $g \circ q \colon X \to Y$ is continuous.

## Remarks

Proof of the converse: for $V$ open in $Y$, the set $q^{-1}(g^{-1}(V)) = (g \circ q)^{-1}(V)$ is open in $X$. Thus $g^{-1}(V)$ is open in the quotient.
So each continuous map $f \colon X \to Y$ that is constant on each class gives a unique continuous map $g$ with $f = g \circ q$.
For example, $t \mapsto (\cos 2\pi t, \sin 2\pi t)$ is constant on the classes of $\mathbb{R} / \mathbb{Z}$, so it gives a continuous map from $\mathbb{R} / \mathbb{Z}$ to the circle.

## Explanations

### L1 human

To define a continuous map on a quotient space, define a continuous map on the original space that takes the same value on glued points.
The map then passes to the quotient.
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<document source="https://math2.tamnd.com/node/0FF6CAQT2Y.md" tag="0FF6CAQT2Y" kind="definition">
# Ellipse

## Definition

Let $F_1$ and $F_2$ be points of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $2a > d(F_1, F_2)$.
The ellipse with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; d(P, F_1) + d(P, F_2) = 2a \,\}.$$

## Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{a^2 - c^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the ellipse is the set of the points with $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.
If $F_1 = F_2$, the ellipse is the [[circle|060W5CBZP6]] with the centre $F_1$ and the radius $a$.
The [[eccentricity|0HVX863SAJ]] of the ellipse is $e = c / a$, with $0 \leq e < 1$. The ellipse bounds a region of area $\pi a b$.

## Explanations

### L1 human

Fix two pins in a board and tie a loose string to both. Pull the string tight with a pencil and move it round: the pencil draws an ellipse. The length of the string is the constant sum $2a$ of the distances to the two pins.
</document>

<document source="https://math2.tamnd.com/node/0FF7726C7P.md" tag="0FF7726C7P" kind="statement">
# Inclusion–exclusion principle

## Statement

Let $A_1, \dots, A_n$ be finite sets.
For a nonempty set $I \subseteq \{1, \dots, n\}$, let $A_I = \bigcap_{i \in I} A_i$.
Then
$$\left| \bigcup_{i=1}^{n} A_i \right| = \sum_{\varnothing \neq I \subseteq \{1, \dots, n\}} (-1)^{|I| + 1} |A_I|.$$

## Remarks

For $n = 2$ this is [[inclusion–exclusion for two sets|0C4YEQ9H5E]].
A common use counts the elements of a finite set $U$ that are in none of the sets $A_i \subseteq U$. This number is $\sum_{I} (-1)^{|I|} |A_I|$, over all $I$, with $A_\varnothing = U$.
The [[formula for the number of derangements|03RSJXF728]] follows in this way.

## Explanations

### L1 human

Add the sizes of the single sets, subtract the sizes of the pairwise intersections, add the sizes of the triple intersections, and so on.
Each element of the union is then counted exactly once in total.
</document>

<document source="https://math2.tamnd.com/node/0FFAE5P6R4.md" tag="0FFAE5P6R4" kind="definition">
# Affine span

## Definition

Let $K$ be a field, and let $A$ be an [[affine space|09XNAPTJQ7]] over a vector space $V$ over $K$. Let $S$ be a nonempty subset of $A$.
The affine span of $S$, written $\operatorname{aff}(S)$, is the intersection of all [[affine subspaces|0PYW26VNT6]] of $A$ that contain $S$.
It is the smallest affine subspace of $A$ that contains $S$.

## Remarks

The affine span of $S$ is the set of all [[affine combinations|0KJH6SRFZK]] of finitely many points of $S$.
In $\mathbb{R}^3$: the affine span of two distinct points is the line through them. The span of three points that are not on one line is the plane through them. The span of four points that are not in one plane is all of $\mathbb{R}^3$.
For each point $p$ of $S$, the affine span is $p + W$, where $W$ is the linear span of the vectors $q - p$ for $q \in S$.
Mathlib also defines the affine span of the empty set: it is the empty affine subspace.

## Explanations

### L1 human

The affine span is the flat object that the given points determine. Two points give a line, and three points in general position give a plane. Compare it with the linear span of vectors, which must always pass through the origin.
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<document source="https://math2.tamnd.com/node/0FFAYNMAJZ.md" tag="0FFAYNMAJZ" kind="definition">
# Connected graph

## Definition

A [[graph|0JD5C82QMB]] $G$ is connected if its vertex set is nonempty and any two vertices of $G$ are [[reachable|0J8XE7W68F]] from each other.

## Remarks

The empty graph, with no vertices, is not connected under this convention. Some authors call it connected.
A graph is connected if and only if it has exactly one [[connected component|01VVP10MWJ]].
The [[cycle graphs|0QQ0F12962]], the [[path graphs|075WR6W3Y3]] and the [[trees|0MJ0JS72DZ]] are connected.

## Explanations

### L1 human

A graph is connected when it is in one piece.
From any vertex you can reach any other vertex along edges.
</document>

<document source="https://math2.tamnd.com/node/0FFH3HSRBA.md" tag="0FFH3HSRBA" kind="concept">
# Construction of the rationals as fractions

## Summary

The rational numbers can be constructed as classes of pairs of [[integers|0C4JS9RF5M]].
The idea is that the pair $(a, b)$ with $b \neq 0$ stands for the fraction $a / b$.
On the set of pairs $(a, b)$ with $a, b \in \mathbb{Z}$ and $b \neq 0$, define $(a, b) \sim (c, d)$ when $a d = b c$. This relation is an equivalence relation.
The set $\mathbb{Q}$ of [[rational numbers|0NCJ5WXN6Z]] is the set of the equivalence classes, and we write $a / b$ for the class of $(a, b)$.
The operations are $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ and $\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$. They do not depend on the choice of the pairs.

## Remarks

The same construction works for each integral domain $R$. It gives the field of fractions of $R$, the smallest field that contains $R$.
The map $a \mapsto a / 1$ embeds $\mathbb{Z}$ in $\mathbb{Q}$.

## Explanations

### L1 human

A fraction is a pair of a top number and a nonzero bottom number.
The pairs $(1, 2)$ and $(2, 4)$ are different pairs, but $1 \cdot 4 = 2 \cdot 2$, so they give the same rational number.
The construction makes this precise: a rational number is the collection of all the pairs that name it.
</document>

<document source="https://math2.tamnd.com/node/0FFXKTKKQC.md" tag="0FFXKTKKQC" kind="concept">
# Absolute value

## Summary

The absolute value of a number is its distance from $0$. We write $|x|$ for the absolute value of $x$.
For a real number, $|x|$ is $x$ without its sign, for example $|-3| = |3| = 3$.
For a complex number $a + b i$, the absolute value is the [[modulus|0KW2X0TFBX]] $\sqrt{a^2 + b^2}$.
</document>

<document source="https://math2.tamnd.com/node/0FGAJSPX8E.md" tag="0FGAJSPX8E" kind="concept">
# Kurtosis

## Summary

The kurtosis of a distribution measures the weight of its tails, compared with the center.
The node [[kurtosis|0CBQVVNBMP]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0FGCFY1AZX.md" tag="0FGCFY1AZX" kind="definition">
# Preimage of a set

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $T$ be a [[subset|0FE8Q5S42Y]] of $B$.
The preimage of $T$ under $f$ is the set
$$f^{-1}(T) = \{a \in A \mid f(a) \in T\}.$$

## Remarks

Another name is the inverse image. The notation $f^{-1}(T)$ makes sense for every $f$, also when $f$ has no [[inverse function|0W766JAPSN]].
Preimages respect all set operations: $f^{-1}(T_1 \cap T_2) = f^{-1}(T_1) \cap f^{-1}(T_2)$, and the same holds for unions and complements.
Example: for $f(x) = x^2$ on $\mathbb{R}$, $f^{-1}([1, 4]) = [-2, -1] \cup [1, 2]$.

## Explanations

### L1 human

The preimage answers the question: which inputs land in $T$?
For the function that sends a student to their grade, the preimage of $\{A\}$ is the set of students with grade $A$.
</document>

<document source="https://math2.tamnd.com/node/0FGENPMPJX.md" tag="0FGENPMPJX" kind="statement">
# The reals form a field

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition and multiplication, form a field.
That is, $\mathbb{R}$ is a commutative ring with $0 \neq 1$, and each real number $x \neq 0$ has a multiplicative inverse $x^{-1}$ with $x \cdot x^{-1} = 1$.

## Remarks

The field $\mathbb{R}$ contains the field $\mathbb{Q}$ of rational numbers as a subfield.
In the axiomatic approach, this is part of the axioms. In a construction of $\mathbb{R}$, for example with [[Cauchy sequences|0D8HE681SC]], it is a theorem.

## Explanations

### L1 human

The usual rules of arithmetic hold for the real numbers.
You can add, subtract and multiply real numbers, and you can divide by each real number that is not zero.
</document>

<document source="https://math2.tamnd.com/node/0FHM8DB6PM.md" tag="0FHM8DB6PM" kind="statement">
# Characterization of bridges

## Statement

Let $G$ be a [[graph|0JD5C82QMB]] and let $e$ be an edge of $G$.
Then $e$ is a [[bridge|0D81KWH9ZM]] of $G$ if and only if $e$ is not an edge of any [[cycle|07CT9RDY7Q]] of $G$.

## Remarks

Proof idea: let $e = uv$. If $e$ lies on a cycle, the rest of the cycle is a path from $u$ to $v$ in $G - e$. Conversely, a path from $u$ to $v$ in $G - e$ together with $e$ gives a cycle.
A consequence: a graph is a forest if and only if every edge is a bridge.

## Explanations

### L1 human

An edge on a cycle is never a bridge: if you delete it, you can still go around the other way.
An edge on no cycle has no detour, so deleting it splits the graph.
</document>

<document source="https://math2.tamnd.com/node/0FHSDCQACY.md" tag="0FHSDCQACY" kind="statement">
# Epsilon-delta uniform continuity

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then the following two conditions are equivalent:

1. For each real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y(f(x), f(y)) < \varepsilon$ for all $x, y \in X$ with $d_X(x, y) < \delta$.
2. For all sequences $(x_n)$ and $(y_n)$ in $X$ with $d_X(x_n, y_n) \to 0$, also $d_Y(f(x_n), f(y_n)) \to 0$.

The map $f$ is uniformly continuous when it satisfies these conditions.

## Remarks

Condition 1 is the epsilon-delta definition of uniform continuity. Here $\delta$ depends only on $\varepsilon$, not on the points.
Each uniformly continuous map is continuous. The converse fails: $x \mapsto x^2$ on $\mathbb{R}$ is continuous but not uniformly continuous. Take $x_n = n + 1/n$ and $y_n = n$; then $x_n^2 - y_n^2 > 2$.
On a compact space, each continuous map to a metric space is uniformly continuous: this is the [[Heine–Cantor theorem|06B8BNJCD6]].
Each [[Lipschitz map|0TMZ21WKCN]] is uniformly continuous.

## Explanations

### L1 human

For a uniformly continuous map, one input tolerance $\delta$ works in the whole space.
The map $x \mapsto x^2$ gets steeper and steeper, so near large $x$ you need a smaller and smaller $\delta$, and no single $\delta$ works.
</document>

<document source="https://math2.tamnd.com/node/0FHVPRAE60.md" tag="0FHVPRAE60" kind="object">
# Matrix ring

## Description

Let $R$ be a ring and let $n \ge 1$ be an integer.
The *matrix ring* $M_n(R)$ is the set of $n \times n$ [[matrices|0VD9ZSWWJG]] over $R$, with [[matrix addition|0Y7H3SAVA1]] and [[matrix multiplication|032Q9FB8MT]].
It is a ring, with the [[zero matrix|0ZTKJJDGR7]] as zero and the [[identity matrix|06DW9BAWB1]] $I_n$ as unit.
If $R$ is commutative, then $M_n(R)$ is also an algebra over $R$, with [[scalar multiplication|09V6DDACXR]].

## Remarks

For $n \ge 2$ and $R \ne 0$, the ring $M_n(R)$ is not commutative and has nonzero elements whose product is zero.
The units of $M_n(R)$ are the [[invertible matrices|0EB9N0S33C]], and they form the [[general linear group|0FCHE902D7]].
For a field $F$, the ring $M_n(F)$ is isomorphic to the ring of endomorphisms of $F^n$.

## Explanations

### L1 human

Square matrices of one fixed size can be added and multiplied, and the result is again a square matrix of that size.
They follow most of the rules of numbers.
The main differences are that $AB$ can differ from $BA$, and that $AB$ can be zero when $A$ and $B$ are not.
</document>

<document source="https://math2.tamnd.com/node/0FJ90Y9926.md" tag="0FJ90Y9926" kind="concept">
# Group of units of a monoid

## Summary

The units of a monoid are its invertible elements.
They form a group under the operation of the monoid.
</document>

<document source="https://math2.tamnd.com/node/0FJB06AD2P.md" tag="0FJB06AD2P" kind="definition">
# Conditional probability

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]] with $P(B) > 0$.
The conditional probability of $A$ given $B$ is
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}.$$

## Remarks

For fixed $B$ with $P(B) > 0$, the map $A \mapsto P(A \mid B)$ is a [[probability measure|0VGMEBVW1M]] on $(\Omega, \mathcal{F})$. It gives all its mass to $B$.
For one roll of a fair die, $P(\{6\} \mid \text{even}) = (1/6) / (1/2) = 1/3$.
In general $P(A \mid B) \ne P(B \mid A)$. [[Bayes' theorem|0NM6E9VW04]] relates the two.
The definition does not apply when $P(B) = 0$. Conditioning on such events needs the [[conditional expectation|0YAMJRK04E]].

## Explanations

### L1 human

Once you know that $B$ happened, the outcomes outside $B$ are no longer possible.
So you keep only the part of $A$ inside $B$, and you rescale by $P(B)$ so that the total is $1$ again.
</document>

<document source="https://math2.tamnd.com/node/0FJBBCXYPS.md" tag="0FJBBCXYPS" kind="statement">
# Chromatic number of a complete bipartite graph

## Statement

Let $m, n \ge 1$ be integers.
Then the [[chromatic number|0C5M7AM6FH]] of the [[complete bipartite graph|0V0GC7BQAN]] $K_{m,n}$ is $2$.

## Remarks

Give one colour to each of the two parts. Each edge joins the two parts, so this is a colouring with $2$ colours.
One colour is not enough, because $K_{m,n}$ has an edge.
More generally, a [[bipartite|07DVHJ8N0A]] graph with at least one edge has chromatic number $2$.

## Explanations

### L1 human

Colour one side of $K_{m,n}$ red and the other side blue.
Every edge goes from red to blue, so two colours are enough, and one colour is too few.
</document>

<document source="https://math2.tamnd.com/node/0FJTGZA4GS.md" tag="0FJTGZA4GS" kind="statement">
# Bertrand's postulate

## Statement

Let $n$ be a positive integer.
Then there is a [[prime number|0SN0MAEKYX]] $p$ such that $n < p \le 2 n$.

## Remarks

For example, for $n = 10$ the primes $11, 13, 17, 19$ are in the range $10 < p \le 20$.
An equivalent form: if $p_k$ is the $k$-th prime, then $p_{k+1} < 2 p_k$ for each $k \ge 1$.
The [[prime number theorem|0EHBY5Y84N]] gives a stronger result for large $n$, but Bertrand's postulate holds for each $n \ge 1$.

## History

Joseph Bertrand stated the result as a conjecture in 1845. Chebyshev proved it in 1852.

## Explanations

### L1 human

Between each number and its double, there is always at least one prime.
So the gap after a prime $p$ to the next prime is always smaller than $p$.
</document>

<document source="https://math2.tamnd.com/node/0FJY6G7FNA.md" tag="0FJY6G7FNA" kind="concept">
# Maximum degree

## Summary

The maximum degree of a finite graph is the largest degree of its vertices. It bounds the chromatic number and many other parameters. The node [[maximum degree|0KNAN9F27P]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0FJZMBX0ZZ.md" tag="0FJZMBX0ZZ" kind="statement">
# Triangle inequality for norms

## Statement

Let $V$ be a vector space over $\R$ or $\C$ with a [[norm|0V2WP9B1MF]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\norm{x + y} \le \norm{x} + \norm{y} \qquad \text{and} \qquad \bigl| \norm{x} - \norm{y} \bigr| \le \norm{x - y}.$$

## Remarks

The first inequality is an axiom of a norm; the second, the *reverse triangle inequality*, follows from it.
For the norm induced by an inner product, the first inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].
In $\R^n$, it says that one side of a triangle is at most the sum of the other two sides.

## Explanations

### L1 human

The straight way from $0$ to $x + y$ is never longer than the way through $x$.
This is the triangle inequality.
The reverse form says that two vectors with a small difference have almost equal lengths.
</document>

<document source="https://math2.tamnd.com/node/0FK17N4DKA.md" tag="0FK17N4DKA" kind="statement">
# Simultaneous diagonalisation

## Statement

Let $F$ be a field and let $A$ and $B$ be [[diagonalisable|07SQCNV233]] $n \times n$ matrices over $F$.
Then there is one invertible $n \times n$ matrix $P$ such that both $P^{-1} A P$ and $P^{-1} B P$ are diagonal if and only if $AB = BA$.

## Remarks

Diagonal matrices commute, so the condition $AB = BA$ is necessary.
For sufficiency, $B$ maps each eigenspace of $A$ to itself, and it is diagonalisable on each of them.
The same holds for each family of pairwise commuting diagonalisable matrices.

## Explanations

### L1 human

Two diagonalisable matrices can be diagonalised by the same change of basis exactly when they commute.
Then they share a basis of eigenvectors.
This fact is important in quantum mechanics, where commuting observables can be measured together.
</document>

<document source="https://math2.tamnd.com/node/0FM6N4R0VB.md" tag="0FM6N4R0VB" kind="statement">
# Monotone convergence theorem for sequences

## Statement

Let $(a_n)$ be an increasing real sequence, in the sense of a [[monotone sequence|0KN22AAW5R]].
Assume that $(a_n)$ is bounded above.
Then $(a_n)$ converges, and its limit is the supremum $\sup \{a_n : n \in \mathbb{N}\}$.

## Remarks

The supremum $s$ exists by the completeness of $\mathbb{R}$. For $\varepsilon > 0$ there is $N$ with $a_N > s - \varepsilon$, and then $s - \varepsilon < a_n \le s$ for all $n \ge N$.
The statement fails in $\mathbb{Q}$. The decimal approximations $1, 1.4, 1.41, 1.414, \dots$ of $\sqrt{2}$ increase and are bounded, but they have no limit in $\mathbb{Q}$.
The node [[decreasing sequence bounded below converges|02V2GYAC8P]] gives the dual statement.
An increasing sequence that is not bounded above [[tends to infinity|09AZCV6GMC]].

## Explanations

### L1 human

An increasing sequence that cannot pass a ceiling must pile up somewhere.
It piles up at the lowest ceiling, which is the supremum of its terms.
</document>

<document source="https://math2.tamnd.com/node/0FMAQSB1SC.md" tag="0FMAQSB1SC" kind="statement">
# Burnside's p^a q^b theorem

## Statement

Let $p$ and $q$ be prime numbers, let $a, b \geq 0$ be integers, and let $G$ be a finite [[group|G5S1M9]] of order $p^a q^b$.
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

A consequence is that the order of each finite [[simple group|0YMYMNQ0QZ]] that is not abelian has at least three distinct prime divisors.
For example, $A_5$ has order $60 = 2^2 \cdot 3 \cdot 5$.
For $b = 0$, the result says that finite $p$-groups are solvable, which follows from [[their nilpotency|081WTKYX6C]].

## History

William Burnside proved the theorem in 1904, with the representation theory of finite groups.
Burnside, Jordan and Frobenius had proved special cases before, in 1897, 1898 and 1902.
John G. Thompson later showed that his work on the N-group theorem gives a proof without representation theory.

## Explanations

### L1 human

If the order of a group has only two prime factors, then the group is solvable.
So a simple group that is not abelian needs at least three primes in its order.
The smallest such group, $A_5$, uses the primes $2$, $3$ and $5$.
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# Constructible number

## Definition

Identify the plane with $\mathbb{C}$.
A complex number $z$ is constructible if the point $z$ can be obtained from $0$ and $1$ by finitely many [[straightedge and compass|0DAECMZVRT]] steps.
In each step, one adds an intersection point of two distinct lines or circles drawn through, or centred at, points already obtained.

## Remarks

A real number $x$ is constructible if and only if a segment of length $|x|$ can be constructed from a segment of length $1$.
A complex number $a + b i$ is constructible if and only if $a$ and $b$ are constructible.
The constructible numbers [[form a field closed under square roots|0PN9WMRWT7]].
The numbers $\sqrt{2}$, $\sqrt{1 + \sqrt{3}}$ and $\cos(2\pi/5)$ are constructible. The numbers $\sqrt[3]{2}$ and $\pi$ are not.

## Explanations

### L1 human

Start with two points, $0$ and $1$.
Draw lines through points you have, and circles centred at points you have, and mark the new intersections.
The constructible numbers are all the points you can reach in this way.
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# Total derivative

## Summary

Let $f$ map an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$. The total derivative of $f$ at a point $a$ is the linear map $Df(a) \colon \mathbb{R}^n \to \mathbb{R}^m$ that best approximates $f$ near $a$. That is, $f(a + h) = f(a) + Df(a)(h) + o(\|h\|)$ as $h \to 0$.
It is the [[Fréchet derivative|06EH97V2A7]] in the case of [[Euclidean spaces|0XPR1E85W0]].
Its matrix in the standard bases is the [[Jacobian matrix|047NDTEPCV]] of $f$ at $a$.

## Remarks

The total derivative accounts for changes in all variables at once. A partial derivative accounts for a change in one variable only.
When $f$ depends on $t$ both directly and through other functions of $t$, the total derivative $\dfrac{d}{dt} f(t, x(t))$ is the sum $\dfrac{\partial f}{\partial t} + \dfrac{\partial f}{\partial x} \dfrac{dx}{dt}$, by the [[chain rule|0EAAS2EHDR]].

## Explanations

### L1 human

For a function of one variable, the derivative at a point is a number: the slope of the tangent line.
For a function of several variables, the derivative at a point is a linear map: it tells how the output changes for each small step in any direction.
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# Chinese remainder theorem

## Statement

Let $n_1, \dots, n_k$ be positive integers that are pairwise [[coprime|0GFKP8F8AG]], and let $N = n_1 n_2 \cdots n_k$.
Let $a_1, \dots, a_k$ be integers.
Then there is an integer $x$ such that $x \equiv a_i \pmod{n_i}$ for each $i$.
Two such integers are congruent modulo $N$.

## Remarks

In the language of rings, the map $\mathbb{Z}/N\mathbb{Z} \to \mathbb{Z}/n_1\mathbb{Z} \times \cdots \times \mathbb{Z}/n_k\mathbb{Z}$, $\overline{x} \mapsto (\overline{x}, \dots, \overline{x})$, is a ring isomorphism.
For two moduli $m$ and $n$, write $m u + n v = 1$ with [[Bézout's identity|0DQVHMYAGA]]. Then $x = a_2 m u + a_1 n v$ solves $x \equiv a_1 \pmod m$ and $x \equiv a_2 \pmod n$.
The [[Chinese remainder theorem for non-coprime moduli|04SJTX9P4N]] removes the hypothesis of coprimality.

## History

The Chinese book Sunzi Suanjing, from the 5th century, contains the oldest known example of the problem. Qin Jiushao gave a complete method in 1247.

## Explanations

### L1 human

Find a number that leaves the remainder $2$ when you divide by $3$, the remainder $3$ by $5$, and the remainder $2$ by $7$.
The theorem says that such a number exists, and that it is unique up to multiples of $3 \cdot 5 \cdot 7 = 105$.
The smallest positive answer is $23$.
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<document source="https://math2.tamnd.com/node/0FP31SX5EA.md" tag="0FP31SX5EA" kind="concept">
# Even and odd functions

## Summary

A function is even when $f(-x) = f(x)$ and odd when $f(-x) = -f(x)$ for all $x$. Even functions have graphs symmetric about the vertical axis, and odd functions have graphs symmetric about the origin. The node [[even and odd functions|0XE690QF65]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0FP4BVDAAQ.md" tag="0FP4BVDAAQ" kind="definition">
# Scalar potential

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
A scalar potential of $F$ is a function $\phi \colon U \to \mathbb{R}$ of class $C^1$ with $\nabla \phi = F$ on $U$.

## Remarks

In physics, a potential is often a function $V$ with $F = -\nabla V$, so that $V$ is a potential energy. The two conventions differ by a sign.
If $U$ is connected, two scalar potentials of $F$ differ by a constant. Indeed, their difference has zero gradient on $U$.
On a connected open set $U$ one finds a potential by fixing $p \in U$ and setting $\phi(x) = \int_\gamma F \cdot d\mathbf{r}$ for a path $\gamma$ from $p$ to $x$. This works when $F$ is [[conservative|05R9NF2J49]].
Example: the gravitational field $F(x) = -x / \|x\|^3$ on $\mathbb{R}^3 \setminus \{0\}$ has the scalar potential $\phi(x) = 1 / \|x\|$.

## Explanations

### L1 human

A scalar potential is a height function for a vector field.
The field points in the direction in which the potential grows fastest, and its length is the rate of growth.
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# Lévy's continuity theorem

## Statement

Let $X, X_1, X_2, \dots$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\varphi, \varphi_1, \varphi_2, \dots$.
1. If $X_n \to X$ [[in distribution|0JDAQHWA9N]], then $\varphi_n(t) \to \varphi(t)$ for each $t \in \mathbb{R}$.
2. Conversely, suppose that $\varphi_n(t)$ converges to a limit $\psi(t)$ for each $t \in \mathbb{R}$, and that $\psi$ is continuous at $0$. Then $\psi$ is the characteristic function of a random variable $Y$, and $X_n \to Y$ in distribution.

## Remarks

Part 1 follows from the definition of convergence in distribution with the bounded continuous functions $\cos(tx)$ and $\sin(tx)$.
The continuity at $0$ in part 2 is necessary. For $X_n \sim \mathcal{N}(0, n)$, $\varphi_n(t) = e^{-nt^2/2}$ tends to $1_{\{t = 0\}}$, which is not continuous at $0$, and $X_n$ does not converge in distribution.
The theorem holds in the same form for random vectors in $\mathbb{R}^k$.

## History

The theorem is named after Paul Lévy.

## Explanations

### L1 human

To show that random variables converge in distribution, it is enough to show that their characteristic functions converge.
This is often much easier, because characteristic functions of sums of independent variables are products.
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<document source="https://math2.tamnd.com/node/0FPXK5CHQ3.md" tag="0FPXK5CHQ3" kind="definition">
# Reduced word

## Definition

Let $S$ be a set, and let $S^{-1} = \{s^{-1} : s \in S\}$ be a set of new symbols, one for each $s \in S$.
A word on $S$ is a finite string $t_1 t_2 \cdots t_n$ with $n \geq 0$ and each $t_i$ in $S \cup S^{-1}$.
A word is reduced when it contains no adjacent pair $s s^{-1}$ or $s^{-1} s$ with $s \in S$.

## Remarks

The empty word, with $n = 0$, is reduced.
Remove adjacent pairs $s s^{-1}$ and $s^{-1} s$, one at a time, until no such pair is left. The result is a reduced word.
For example, for $S = \{a, b, c\}$, the word $a b^3 c^{-1} c a^{-1} c$ reduces to $a b^3 a^{-1} c$.
A reduced word is cyclically reduced when its first and last letters are not inverse to each other.

## Explanations

### L1 human

A word is a string of letters and inverse letters.
It is reduced when you cannot cancel a letter against its inverse next to it.
Reduced words are the normal forms of the elements of a [[free group|0KTW26C80J]].
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# Spanning tree

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A spanning tree of $G$ is a [[subgraph|0P11X5HKAX]] $T$ of $G$ that contains every vertex of $G$ and is a [[tree|0MJ0JS72DZ]].

## Remarks

A graph has a spanning tree if and only if it is connected. The node [[every connected graph has a spanning tree|04XPQT2XTC]] gives one direction.
A spanning tree of a finite connected graph with $n$ vertices has $n - 1$ edges.
The complete graph $K_n$ has $n^{n-2}$ spanning trees, by [[Cayley's formula|0M2SSGSYPE]]. A [[minimum spanning tree|0Z367CGNRM]] has the least total weight.

## Explanations

### L1 human

A spanning tree keeps all the vertices of a graph and just enough edges to stay connected without any loop.
For a road network, it is a smallest set of roads that still links all towns.
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<document source="https://math2.tamnd.com/node/0FQ968YKA1.md" tag="0FQ968YKA1" kind="concept">
# Convergence of random variables

## Summary

A sequence of random variables can converge to a limit in several different senses.
The main modes are [[almost sure convergence|04P7D63W8P]], [[convergence in probability|0EFDDXTVD6]], [[convergence in mean|0TTA57SEJG]] and [[convergence in distribution|0JDAQHWA9N]].
Almost sure convergence and convergence in mean each imply convergence in probability, and convergence in probability implies convergence in distribution. The converse implications are false in general.

## Remarks

Convergence in distribution concerns only the laws of the random variables. So the random variables can be on different probability spaces.
The law of large numbers is a statement about convergence in probability or almost surely. The central limit theorem is a statement about convergence in distribution.

## Explanations

### L1 human

For numbers, there is one meaning of "$x_n \to x$". For random variables, there are several, because a random variable has many values at once.
Some modes ask that the values themselves come close, and the weakest mode asks only that the probabilities come close.
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<document source="https://math2.tamnd.com/node/0FQPYAY9PJ.md" tag="0FQPYAY9PJ" kind="definition">
# Meagre set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
The set $A$ is meagre in $X$ when there is a [[countable|0ZG61TFBES]] family $(A_n)_{n \in \mathbb{N}}$ of [[nowhere dense|0KSM23SN3H]] subsets of $X$ with $A = \bigcup_{n} A_n$.

## Remarks

Another name for a meagre set is a set of first category. A set that is not meagre is of second category.
Each subset of a meagre set is meagre, and each countable union of meagre sets is meagre.
The set $\mathbb{Q}$ is meagre in $\mathbb{R}$, because it is a countable union of one-point sets.
By the [[Baire category theorem|0A7KGAYHGV]], $\mathbb{R}$ itself is not meagre in $\mathbb{R}$.

## Explanations

### L1 human

A meagre set is made of countably many thin pieces.
It can still be dense, as $\mathbb{Q}$ is, but in a complete metric space it can never fill an open set.
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# Inverse of a complex number

## Statement

Let $z$ be a complex number with $z \neq 0$.
Then
$$z^{-1} = \frac{\bar{z}}{|z|^2}.$$
With $z = a + b i$ and $a, b$ real, this is $z^{-1} = \dfrac{a}{a^2 + b^2} - \dfrac{b}{a^2 + b^2} \, i$.

## Remarks

Proof: $z \bar{z} = |z|^2$ by the formula for [[a complex number times its conjugate|0ZB7J071WQ]], and $|z|^2 > 0$ because $z \neq 0$. Divide by $|z|^2$.
To compute a quotient $w / z$, multiply the numerator and the denominator by $\bar{z}$.

## Explanations

### L1 human

To divide by a complex number, multiply above and below by its conjugate. Then the denominator becomes a positive real number.
For example, $\dfrac{1}{1 + i} = \dfrac{1 - i}{(1 + i)(1 - i)} = \dfrac{1 - i}{2}$.
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<document source="https://math2.tamnd.com/node/0FQSDC2EQ2.md" tag="0FQSDC2EQ2" kind="concept">
# Convex hull

## Summary

The convex hull of a set of points is the smallest convex set that contains it. It is the set of all convex combinations of points of the set.
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<document source="https://math2.tamnd.com/node/0FR1BDKK10.md" tag="0FR1BDKK10" kind="statement">
# Comparison test for improper integrals

## Statement

Let $f, g \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on each $[a, b]$, with $0 \le f(x) \le g(x)$ for all $x \ge a$.
If $\int_a^\infty g(x) \, dx$ converges, then $\int_a^\infty f(x) \, dx$ converges, and
$$\int_a^\infty f(x) \, dx \le \int_a^\infty g(x) \, dx.$$
If $\int_a^\infty f(x) \, dx$ diverges, then $\int_a^\infty g(x) \, dx$ diverges.

## Remarks

The function $b \mapsto \int_a^b f$ is increasing, and it is bounded by $\int_a^\infty g$ by [[monotonicity|0MBWQKM8QR]]. A [[bounded monotone function has a limit|0KJXHN0TW0]].
For example, $0 \le e^{-x^2} \le e^{-x}$ for $x \ge 1$. So $\int_1^\infty e^{-x^2} \, dx$ converges.
The same test holds for integrals that are improper at a finite endpoint.

## Explanations

### L1 human

If a larger function has finite area, a smaller nonnegative function also has finite area.
This lets us prove convergence without computing the integral.
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# Heron's formula

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, with the side lengths $a$, $b$ and $c$. Let $s = \frac{1}{2}(a + b + c)$ be its semiperimeter.
Then the [[area|065JWMZW86]] $T$ of the triangle is
$$T = \sqrt{s (s - a)(s - b)(s - c)}.$$

## Remarks

An equivalent form without $s$ is $16 T^2 = (a + b + c)(-a + b + c)(a - b + c)(a + b - c)$.
For example, the triangle with the sides $4$, $13$ and $15$ has $s = 16$ and $T = \sqrt{16 \cdot 12 \cdot 3 \cdot 1} = 24$.
One proof uses the [[law of cosines|060ZVZBV3E]] to get $\cos \gamma$ from the sides, and then $T = \frac{1}{2} a b \sin \gamma$.

## History

The formula carries the name of Heron of Alexandria, who gave a proof in his Metrica in the 1st century. It was possibly known earlier. Qin Jiushao published an equivalent formula in 1247.

## Explanations

### L1 human

Heron's formula gives the area of a triangle when you know only its three sides. You do not need a height or an angle. For the right triangle with the sides $3$, $4$, $5$: $s = 6$, and $\sqrt{6 \cdot 3 \cdot 2 \cdot 1} = 6$, which is also $\frac{1}{2} \cdot 3 \cdot 4$.
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<document source="https://math2.tamnd.com/node/0FREPFKNS7.md" tag="0FREPFKNS7" kind="concept">
# Maximum likelihood estimation

## Summary

Maximum likelihood estimation is a method to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
For observed data $x$, the maximum likelihood estimate is a value $\hat{\theta}$ that maximizes the [[likelihood function|05FSTXM92W]] $\theta \mapsto L(\theta \mid x)$ over the parameter set $\Theta$.
In other words, it is the parameter value under which the observed data are most probable.

## Remarks

The maximizer of $L$ is also the maximizer of the log-likelihood $\ell = \ln L$. For a smooth $\ell$ and an interior maximum, $\hat{\theta}$ solves the likelihood equations $\partial \ell / \partial \theta_j = 0$.
Example: for $k$ successes in $n$ Bernoulli($p$) trials, $\ell(p) = k \ln p + (n - k) \ln(1 - p)$, so $\hat{p} = k/n$.
Example: for an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample, $\hat{\mu} = \bar{x}$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (x_i - \bar{x})^2$. This estimator of $\sigma^2$ is biased.
Under regularity conditions, the estimator is [[consistent|0XJVF81QF6]] and asymptotically normal. Also, for a function $g$, the maximum likelihood estimate of $g(\theta)$ is $g(\hat{\theta})$.

## Explanations

### L1 human

Maximum likelihood chooses the parameter that makes the observed data most probable.
If a coin shows $7$ heads in $10$ tosses, the maximum likelihood estimate of the probability of heads is $7/10$.
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# Scalar matrix

## Definition

Let $R$ be a ring, let $n \ge 0$ be an integer, and let $c \in R$.
The *scalar matrix* of order $n$ with value $c$ is the [[diagonal matrix|044JNC7ZAA]] $\operatorname{diag}(c, \dots, c) = c I_n$, where $I_n$ is the [[identity matrix|06DW9BAWB1]].
A square matrix is a *scalar matrix* when it equals $c I_n$ for some $c \in R$.

## Remarks

Over a commutative ring, a scalar matrix commutes with every square matrix of the same order.
Over a field, the scalar matrices are exactly the $n \times n$ matrices that commute with every $n \times n$ matrix.
As a linear map, $c I_n$ multiplies each vector by $c$.

## Explanations

### L1 human

A scalar matrix has the same number at every place on the diagonal and zeros elsewhere.
Multiplying by it is the same as multiplying by that number.
For example, $3 I_2 = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ triples each vector.
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# Sequential compactness

## Summary

A space or a set is sequentially compact when each sequence in it has a subsequence that converges to a point of it.
In metric spaces, sequential compactness is the same as compactness.
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<document source="https://math2.tamnd.com/node/0FRSR10DWF.md" tag="0FRSR10DWF" kind="statement">
# The complex numbers form a field

## Statement

The [[complex numbers|09FTAET5Z4]] $\mathbb{C}$, with their addition and multiplication, form a field.
That is, $\mathbb{C}$ is a commutative ring with $0 \neq 1$, and each complex number $z \neq 0$ has a multiplicative inverse $z^{-1}$ with $z z^{-1} = 1$.

## Remarks

The [[inverse of a complex number|0FQRAV5SKH]] $a + b i \neq 0$ is $\dfrac{a - b i}{a^2 + b^2}$.
The field $\mathbb{C}$ contains $\mathbb{R}$ as a subfield. It has dimension $2$ as a vector space over $\mathbb{R}$, with basis $1, i$.
The field $\mathbb{C}$ is algebraically closed, but it is not an ordered field.

## Explanations

### L1 human

You can add, subtract, multiply and divide complex numbers, as with real numbers.
The only exception is division by $0$.
For example, $\dfrac{1}{i} = -i$, because $i \cdot (-i) = -i^2 = 1$.
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<document source="https://math2.tamnd.com/node/0FRX8JD7QH.md" tag="0FRX8JD7QH" kind="definition">
# Grönwall bound

## Definition

Let $\delta, K, \varepsilon$ be real numbers.
The Grönwall bound with these parameters is the function $G_{\delta, K, \varepsilon} : \mathbb{R} \to \mathbb{R}$ given by
$$G_{\delta, K, \varepsilon}(x) = \begin{cases} \delta\, e^{K x} + \dfrac{\varepsilon}{K} \left( e^{K x} - 1 \right) & \text{if } K \neq 0, \\ \delta + \varepsilon x & \text{if } K = 0. \end{cases}$$

## Remarks

The function $v = G_{\delta, K, \varepsilon}$ is the unique solution of $v' = K v + \varepsilon$, $v(0) = \delta$.
For $\varepsilon = 0$, it is $\delta e^{K x}$.
A form of [[Grönwall's inequality|07EJHQYR9A]] uses it: let $f : [a, b] \to \mathbb{R}^n$ be continuous and differentiable on $(a, b)$, with $\|f(a)\| \le \delta$ and $\|f'(x)\| \le K \|f(x)\| + \varepsilon$ on $(a, b)$. Then $\|f(x)\| \le G_{\delta, K, \varepsilon}(x - a)$ on $[a, b]$.
The [[estimate for approximate solutions|0JTPMCCZT2]] is stated with this bound.

## Explanations

### L1 human

Suppose that a quantity starts at most at $\delta$ and grows at most at rate $K$ times itself plus $\varepsilon$. The Grönwall bound is the largest value that it can reach.
It is the solution of the extreme case, where both inequalities are equalities.
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# Total order

## Summary

A total order is a partial order in which any two elements are comparable.
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<document source="https://math2.tamnd.com/node/0FRYPZ6D0X.md" tag="0FRYPZ6D0X" kind="statement">
# Trace is a coefficient of the characteristic polynomial

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
Then the coefficient of $t^{n-1}$ in the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$ is $-\tr A$, where $\tr$ is the [[trace|0XC1RDBY7A]].
So $p_A(t) = t^n - (\tr A)\, t^{n-1} + \cdots$.

## Remarks

In the Leibniz expansion of $\det(t I_n - A)$, only the product of the diagonal entries $(t - a_{11}) \cdots (t - a_{nn})$ contributes to $t^{n-1}$.
For $n = 2$, $p_A(t) = t^2 - (\tr A) t + \det A$.
Together with [[the constant coefficient|0T0X8GTYYS]], it shows that $\tr$ and $\det$ are invariants of similarity.

## Explanations

### L1 human

The characteristic polynomial contains the trace as its second coefficient, with a minus sign.
For a $2 \times 2$ matrix, the full polynomial is $t^2 - (\text{trace})\,t + \text{determinant}$.
So the trace and the determinant are part of the information of the characteristic polynomial.
</document>

<document source="https://math2.tamnd.com/node/0FS3RHF6KE.md" tag="0FS3RHF6KE" kind="statement">
# Infinite pigeonhole principle

## Statement

Let $A$ be an infinite set, let $B$ be a finite set, and let $f : A \to B$ be a function.
Then there is an element $b \in B$ such that the fibre $f^{-1}(b)$ is infinite.

## Remarks

The proof is short. If each of the finitely many fibres were finite, then $A$, their union, would be finite.
Example: in any infinite sequence of integers, infinitely many terms have the same remainder modulo $10$.
The statement is the starting point of the infinite form of [[Ramsey's theorem|00TQV3F5S4]].

## Explanations

### L1 human

If you put infinitely many objects into finitely many boxes, at least one box gets infinitely many of them.
For example, colour each natural number red or blue. Then one of the two colours is used infinitely often.
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<document source="https://math2.tamnd.com/node/0FS6B46E5T.md" tag="0FS6B46E5T" kind="statement">
# Determinant of a matrix with a zero row

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
If one row of $A$ is zero, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This follows from linearity in that row: the zero row is $0$ times itself, so $\det A = 0 \cdot \det A = 0$.
The same holds for a zero column.

## Explanations

### L1 human

A zero row means that one edge of the box spanned by the rows has length zero.
Such a box has no volume.
So the determinant is zero.
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<document source="https://math2.tamnd.com/node/0FS972T0ZH.md" tag="0FS972T0ZH" kind="statement">
# Bertrand's ballot theorem

## Statement

Let $p$ and $q$ be natural numbers with $p > q$.
In an election, candidate $A$ gets $p$ votes and candidate $B$ gets $q$ votes.
The votes are counted one at a time, in an order chosen uniformly at random among all orders.
Then the probability that $A$ is strictly ahead of $B$ after each vote is counted is
$$\frac{p - q}{p + q}.$$

## Remarks

Equivalently, the number of such counting orders, as words with $p$ letters $A$ and $q$ letters $B$, is $\frac{p - q}{p + q} \binom{p + q}{p}$.
Example: for $p = 3$ and $q = 1$, the probability is $\frac{1}{2}$. Of the $4$ orders, $AAAB$ and $AABA$ work.
A short proof uses the reflection principle for [[lattice paths|040WY0TT2G]].

## History

William Allen Whitworth published the result in 1878.
Joseph Bertrand found it again in 1887, and the theorem has his name.

## Explanations

### L1 human

If the winner gets many more votes than the loser, the winner is likely to lead the whole time.
The formula makes this precise: the chance of an unbroken lead is the margin divided by the total number of votes.
</document>

<document source="https://math2.tamnd.com/node/0FSCXYG856.md" tag="0FSCXYG856" kind="statement">
# Reversing a path changes the sign of a line integral

## Statement

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a piecewise $C^1$ curve with image $C$, and let $\gamma^- \colon [a, b] \to \mathbb{R}^n$, $\gamma^-(t) = \gamma(a + b - t)$, be the reversed curve.
For each continuous vector field $F$ on $C$ and each continuous function $f$ on $C$,
$$\int_{\gamma^-} F \cdot d\mathbf{r} = -\int_\gamma F \cdot d\mathbf{r}, \qquad \int_{\gamma^-} f \, ds = \int_\gamma f \, ds.$$

## Remarks

Proof: by the [[parametric formula|0W3W1VYWBP]], $(\gamma^-)'(t) = -\gamma'(a + b - t)$. The substitution $u = a + b - t$ then gives both identities.
The [[line integral of a vector field|0YS8RX87F4]] depends on the orientation of the curve. The [[line integral of a scalar field|0M50E4WYWJ]] does not, because it uses only $\|\gamma'\|$.
More generally, under a $C^1$ change of parameter $\varphi$ with $\varphi' > 0$, the vector line integral does not change. With $\varphi' < 0$ it changes sign.

## Explanations

### L1 human

If a force helps you along a path, it works against you on the way back.
So the work, a line integral of a vector field, changes sign when you reverse the path. The length of the path and the mass of a wire do not.
</document>

<document source="https://math2.tamnd.com/node/0FSJA9M43W.md" tag="0FSJA9M43W" kind="statement">
# Multiplicativity of the totient function

## Statement

Let $m$ and $n$ be positive integers that are [[coprime|0GFKP8F8AG]].
Then $\varphi(m n) = \varphi(m) \varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof uses the [[Chinese remainder theorem|0FNN9B4W64]]. It gives a bijection between the residues modulo $m n$ and the pairs of residues modulo $m$ and modulo $n$. A residue is coprime to $m n$ if and only if both parts are coprime to their moduli.
The hypothesis is necessary: $\varphi(4) = 2$, but $\varphi(2) \varphi(2) = 1$.
So $\varphi$ is a [[multiplicative function|0AP80TT660]].
For example, $\varphi(36) = \varphi(4) \varphi(9) = 2 \cdot 6 = 12$.

## Explanations

### L1 human

When $m$ and $n$ have no common factor, a number is coprime to $m n$ exactly when it is coprime to $m$ and to $n$.
The two conditions are independent, so the counts multiply.
For example, $\varphi(15) = \varphi(3) \varphi(5) = 2 \cdot 4 = 8$.
</document>

<document source="https://math2.tamnd.com/node/0FSNQ0GNCC.md" tag="0FSNQ0GNCC" kind="definition">
# Identity function

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The identity function of $A$ is the [[function|0XHDTE5YMZ]] $\mathrm{id}_A \colon A \to A$ with $\mathrm{id}_A(a) = a$ for each $a \in A$.

## Remarks

For each function $f \colon A \to B$, $f \circ \mathrm{id}_A = f$ and $\mathrm{id}_B \circ f = f$. See [[function composition|07HF9YCWPP]].
The identity function is a [[bijection|0SNQEF7Z53]], and it is its own inverse.
Its graph is the diagonal $\{(a, a) \mid a \in A\}$.

## Explanations

### L1 human

The identity function does nothing: each input comes out unchanged.
It plays the role for composition that the number $1$ plays for multiplication.
</document>

<document source="https://math2.tamnd.com/node/0FTH2E9TAS.md" tag="0FTH2E9TAS" kind="concept">
# Complex conjugate

## Summary

The complex conjugate of a complex number has the same real part and the opposite imaginary part.
The conjugate of $a + b i$ is $a - b i$. In the [[complex plane|0CTE7A31XQ]], it is the reflection of the point in the real axis.
</document>

<document source="https://math2.tamnd.com/node/0FWBQ53MPJ.md" tag="0FWBQ53MPJ" kind="concept">
# Choice function

## Summary

A choice function on a collection of nonempty sets picks one element from each set.
</document>

<document source="https://math2.tamnd.com/node/0FWJ0VW561.md" tag="0FWJ0VW561" kind="definition">
# Orthonormal basis

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$.
An *orthonormal basis* of $V$ is a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$ of $V$ that is an [[orthonormal family|0RW9TNNT87]].

## Remarks

An orthonormal family of $\dim V$ vectors is an orthonormal basis.
In an orthonormal basis, coordinates are inner products: $v = \sum_i \langle v, e_i \rangle e_i$.
The vectors $\frac{1}{\sqrt 2}(1, 1)$ and $\frac{1}{\sqrt 2}(1, -1)$ form an orthonormal basis of $\R^2$.
In infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.

## Explanations

### L1 human

An orthonormal basis is a set of perpendicular unit vectors that spans the space.
Each coordinate of a vector is its inner product with one basis vector.
No linear system needs to be solved to find coordinates.
</document>

<document source="https://math2.tamnd.com/node/0FWVRHBRWM.md" tag="0FWVRHBRWM" kind="concept">
# Transposition

## Summary

A transposition is a permutation that swaps two elements and fixes all other elements.
In cycle notation it is a $2$-cycle $(a\ b)$.
</document>

<document source="https://math2.tamnd.com/node/0FXDW992DS.md" tag="0FXDW992DS" kind="concept">
# Initial value problem

## Summary

An initial value problem is an [[ordinary differential equation|0NT64JV96A]] together with a condition that fixes the value of the unknown function at one point.
A solution must satisfy both the equation and the condition. See the [[definition|075VBY4K4X]].
</document>

<document source="https://math2.tamnd.com/node/0FXJ7YFMMX.md" tag="0FXJ7YFMMX" kind="concept">
# Solvable group

## Summary

A solvable group is a group with a subnormal series whose factor groups are all abelian.
Equivalently, its derived series reaches the trivial subgroup.
</document>

<document source="https://math2.tamnd.com/node/0FY42N02DV.md" tag="0FY42N02DV" kind="concept">
# Algebraic multiplicity

## Summary

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.
</document>

<document source="https://math2.tamnd.com/node/0FYQMKFE77.md" tag="0FYQMKFE77" kind="concept">
# Order of quantifiers

## Summary

In a statement with several quantifiers, the order of different quantifiers changes the meaning.
For a [[predicate|0T3DPBG981]] $P(x, y)$, the statement $\exists y \, \forall x \, P(x, y)$ implies $\forall x \, \exists y \, P(x, y)$, but the converse is false in general.
Two quantifiers of the same type commute: $\forall x \, \forall y \, P(x, y)$ is equivalent to $\forall y \, \forall x \, P(x, y)$, and the same holds for $\exists$.

## Remarks

On the integers, $\forall x \, \exists y \, (y > x)$ is true, but $\exists y \, \forall x \, (y > x)$ is false.
In $\forall x \, \exists y$, the choice of $y$ can depend on $x$.
The definitions of continuity and of uniform continuity differ only in the order of quantifiers.

## Explanations

### L1 human

"Every person has a mother" is $\forall x \, \exists y$: each person has their own mother.
"Some person is the mother of every person" is $\exists y \, \forall x$: one mother for all.
The first sentence is true, and the second is false.
</document>

<document source="https://math2.tamnd.com/node/0FYX44GP92.md" tag="0FYX44GP92" kind="concept">
# Recurrence relation

## Summary

A recurrence relation defines each term of a sequence from earlier terms. Recurrences are a main tool to describe counting sequences, and solving one means finding a closed formula for its terms. The node [[recurrence relation|07TZV87Y0T]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0FZFHK97P7.md" tag="0FZFHK97P7" kind="statement">
# Second derivative test

## Statement

Let $f$ be a real function differentiable on an open interval around $c$, with $f'(c) = 0$.
Assume that the [[second derivative|0Y9D2FJRQ4]] $f''(c)$ exists.
If $f''(c) > 0$, then $f$ has a strict [[local minimum|0NS8BP96QX]] at $c$. If $f''(c) < 0$, then $f$ has a strict local maximum at $c$.

## Remarks

If $f''(c) > 0$, then $f'(x)/(x - c) \to f''(c)$. So $f'$ is negative just left of $c$ and positive just right of $c$, and the [[first derivative test|01A2EGXJEM]] applies.
If $f''(c) = 0$, the test gives no answer. The functions $x^4$, $-x^4$, and $x^3$ at $0$ show all three cases.
Mathlib states the minimum case as `isLocalMin_of_deriv_deriv_pos`.

## Explanations

### L1 human

At a stationary point where the graph bends upward, the point is the bottom of a valley.
Where it bends downward, it is a hilltop.
</document>

<document source="https://math2.tamnd.com/node/0G01TS1RRP.md" tag="0G01TS1RRP" kind="definition">
# Deficient number

## Definition

A positive integer $n$ is deficient if the sum of its positive divisors other than $n$ is less than $n$.
Equivalently, $\sigma(n) < 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

Each prime $p$ is deficient, since its only smaller divisor is $1$.
More generally, each prime power is deficient. For example, $1 + 2 + 4 = 7 < 8$.
Each divisor of a deficient number or of a [[perfect number|0SB7Q8SYSF]], other than the number itself, is deficient.

## Explanations

### L1 human

A deficient number has "too few" divisors: its smaller divisors add up to less than the number.
Most small numbers are deficient, for example $1, 2, 3, 4, 5, 7, 8, 9, 10$.
</document>

<document source="https://math2.tamnd.com/node/0G0ASKNA6Z.md" tag="0G0ASKNA6Z" kind="concept">
# Cardinality

## Summary

The cardinality of a set is a measure of its size, the number of its elements.
Two sets have the same cardinality exactly when they are equinumerous.
</document>

<document source="https://math2.tamnd.com/node/0G0B3DRG65.md" tag="0G0B3DRG65" kind="example">
# Cyclic group of order n

## Example

Let $n$ be a positive integer.
The integers modulo $n$, $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n - 1\}$, with addition modulo $n$, form a [[cyclic group|0KVGH0J5B3]] of [[order|0B85S2V2DR]] $n$.
The residue $1$ generates it, because each residue $k$ is the sum of $k$ copies of $1$.
Each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and this group is often written $C_n$.

## Remarks

A residue $k$ generates $\mathbb{Z}/n\mathbb{Z}$ exactly when $\gcd(k, n) = 1$.
The [[group of $n$th roots of unity|0WX11EA3F3]] is a cyclic group of order $n$ in multiplicative notation.
The rotations of a regular $n$-gon also form a cyclic group of order $n$.

## Explanations

### L1 human

Think of a clock with $n$ hours, where you add hours and wrap around after $n$.
This clock arithmetic gives the cyclic group of order $n$.
Every cyclic group of order $n$ looks the same as this clock, up to the names of the elements.
</document>

<document source="https://math2.tamnd.com/node/0G0F7FGCGP.md" tag="0G0F7FGCGP" kind="statement">
# Transcendence of pi

## Statement

The number $\pi$ is [[transcendental|00SW8B91E7]] over $\mathbb{Q}$.

## Remarks

The statement follows from the Lindemann–Weierstrass theorem: $e^\alpha$ is transcendental for every nonzero algebraic number $\alpha$.
Indeed, if $\pi$ were algebraic, then $i \pi$ would be algebraic and nonzero, but $e^{i \pi} = -1$ is algebraic.
So $\sqrt{\pi}$ is also transcendental, and it is not [[constructible|0FMN2MN7TM]].

## History

Charles Hermite proved the transcendence of $e$ in 1873.
Ferdinand von Lindemann proved in 1882 that $e^\alpha$ is transcendental for every nonzero algebraic $\alpha$, and so that $\pi$ is transcendental.
Karl Weierstrass proved a more general statement in 1885.

## Explanations

### L1 human

The number $\pi$ is not a root of any nonzero polynomial with rational coefficients.
This is much stronger than the irrationality of $\pi$, and the proof is much harder.
</document>

<document source="https://math2.tamnd.com/node/0G0RFPG8HR.md" tag="0G0RFPG8HR" kind="concept">
# Eigenspace

## Summary

The eigenspace of a linear map for a scalar $\lambda$ is the kernel of the map minus $\lambda$ times the identity.
It consists of the eigenvectors for $\lambda$ together with zero.
</document>

<document source="https://math2.tamnd.com/node/0G1022YXVQ.md" tag="0G1022YXVQ" kind="concept">
# Galton–Watson process

## Summary

A Galton–Watson process models a population. Each individual has a random number of children, independently of the others and with one common law.
The node [[Galton–Watson process|0H631CZP5K]] gives the precise definition.

## Remarks

The process comes from the study of the extinction of family names by Francis Galton.
</document>

<document source="https://math2.tamnd.com/node/0G24VQ1088.md" tag="0G24VQ1088" kind="statement">
# Sylvester's determinant theorem

## Statement

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$.
Then
$$\det(I_m + AB) = \det(I_n + BA),$$
where $I_m$ and $I_n$ are [[identity matrices|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The two sides are determinants of different sizes, $m$ and $n$.
For a column $u$ and a row $v^T$, it gives $\det(I_m + u v^T) = 1 + v^T u$.
The statement is also called the Weinstein–Aronszajn identity.

## Explanations

### L1 human

The products $AB$ and $BA$ can have very different sizes.
Still, adding the identity to them gives the same determinant.
This can turn a large determinant into a small one.
</document>

<document source="https://math2.tamnd.com/node/0G2YJN0H5S.md" tag="0G2YJN0H5S" kind="statement">
# The complex exponential is entire

## Statement

The [[complex exponential|054VXHMCA3]] $\exp : \mathbb{C} \to \mathbb{C}$ is an [[entire function|0419794R0T]]: it is complex differentiable at each $z \in \mathbb{C}$.

## Remarks

Proof: $\exp$ is given by a power series with infinite radius of convergence, and [[power series are holomorphic in their disc of convergence|0YMP521ZR5]].
Its [[derivative|019H1Z6419]] is $\exp$ itself.

## Explanations

### L1 human

The exponential function has a complex derivative at every point of the plane.
So it is a holomorphic function without any exceptional points.
</document>

<document source="https://math2.tamnd.com/node/0G31VJ5YS5.md" tag="0G31VJ5YS5" kind="definition">
# Tensor product of vector spaces

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.
Let $L$ be the vector space over $F$ with basis the set $V \times W$; write $[v, w]$ for the basis vector of $(v, w)$.
Let $N$ be the subspace of $L$ spanned by all elements of these forms, for $v, v' \in V$, $w, w' \in W$ and $a \in F$:
$$[v + v', w] - [v, w] - [v', w], \qquad [v, w + w'] - [v, w] - [v, w'],$$
$$[av, w] - a[v, w], \qquad [v, aw] - a[v, w].$$
The *tensor product* of $V$ and $W$ is the quotient space $V \otimes W = L / N$.
For $v \in V$ and $w \in W$, the class of $[v, w]$ is written $v \otimes w$.

## Remarks

The map $(v, w) \mapsto v \otimes w$ is bilinear, and the vectors $v \otimes w$ span $V \otimes W$.
Not every vector of $V \otimes W$ is of the form $v \otimes w$; for example, $e_1 \otimes e_1 + e_2 \otimes e_2$ in $F^2 \otimes F^2$ is not.
The tensor product is characterised by its [[universal property|0YDRV01R8F]].
For finite-dimensional spaces, $V^* \otimes W$ is isomorphic to the space of linear maps from $V$ to $W$.

## Explanations

### L1 human

The tensor product combines two vector spaces into one space of formal products $v \otimes w$.
These products follow the rules of bilinearity and nothing more.
It turns bilinear maps into linear maps, which are easier to study.
</document>

<document source="https://math2.tamnd.com/node/0G3DBXRRQJ.md" tag="0G3DBXRRQJ" kind="statement">
# Continuous dependence on initial data

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $K \ge 0$, and let $f : J \times U \to \mathbb{R}^n$ be continuous.
Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$.
Let $a < b$ with $[a, b] \subseteq J$, and let $\gamma_1, \gamma_2 : [a, b] \to U$ be [[solutions|04SW7WECS8]] of $x' = f(t, x)$.
Then
$$\|\gamma_1(t) - \gamma_2(t)\| \le \|\gamma_1(a) - \gamma_2(a)\|\, e^{K (t - a)} \quad \text{for each } t \in [a, b].$$

## Remarks

So the solution at time $t$ is a Lipschitz function of the initial value, with constant $e^{K(t - a)}$.
The proof applies [[Grönwall's inequality|07EJHQYR9A]] to $u(t) = \|\gamma_1(t) - \gamma_2(t)\|$.
Existence, uniqueness and continuous dependence together make the initial value problem well posed, in the sense of Jacques Hadamard.
For equal initial values, the bound gives [[uniqueness|025PKW8AMR]] again.

## Explanations

### L1 human

If two solutions start close together, they stay close for a while.
Their distance can grow, but at most exponentially, with rate given by the Lipschitz constant.
So a small error in the initial value gives a small error in the solution over a bounded time.
</document>

<document source="https://math2.tamnd.com/node/0G3RRM41X9.md" tag="0G3RRM41X9" kind="concept">
# Dedekind cut

## Summary

A Dedekind cut splits the rational numbers into a lower part and an upper part.
Each real number corresponds to exactly one cut, so the cuts give a construction of the [[real numbers|08J2M1JYAG]] from the [[rational numbers|0NCJ5WXN6Z]].
</document>

<document source="https://math2.tamnd.com/node/0G48PYM96Q.md" tag="0G48PYM96Q" kind="concept">
# Markov chain

## Summary

A Markov chain is a random process in which the next state depends only on the present state, and not on the earlier history.
The node [[Markov chain|033TJBZDG0]] gives the precise definition in discrete time with a countable state space.
</document>

<document source="https://math2.tamnd.com/node/0G4C8SJ6PV.md" tag="0G4C8SJ6PV" kind="statement">
# Automorphism group of a cyclic group

## Statement

Let $n$ be a positive integer and let $G$ be a [[cyclic group|0KVGH0J5B3]] of order $n$.
For each integer $a$ with $\gcd(a, n) = 1$, the map $\sigma_a \colon x \mapsto x^a$ is an automorphism of $G$.
The map $a \bmod n \mapsto \sigma_a$ is an [[isomorphism|08MG7H8JW5]] from the [[group of units|0FJ90Y9926]] $(\mathbb{Z}/n\mathbb{Z})^\times$ to $\operatorname{Aut}(G)$.

## Remarks

An automorphism of $G = \langle g \rangle$ is determined by the image of $g$, and this image must be a generator $g^a$ with $\gcd(a, n) = 1$.
Thus $|\operatorname{Aut}(G)| = \varphi(n)$, where $\varphi$ is Euler's totient function.
For the infinite cyclic group, $\operatorname{Aut}(\mathbb{Z}) \cong \{1, -1\}$.

## Explanations

### L1 human

An automorphism of a cyclic group must send a generator to a generator.
Each choice of a generator gives exactly one automorphism.
For $\mathbb{Z}/5\mathbb{Z}$, the generators are $1, 2, 3, 4$, so there are four automorphisms, given by multiplication by $1, 2, 3, 4$.
</document>

<document source="https://math2.tamnd.com/node/0G4D3ZKTP8.md" tag="0G4D3ZKTP8" kind="example">
# Sum of the reciprocals of the factorials

## Example

The series
$$\sum_{n=0}^{\infty} \frac{1}{n!} = 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \cdots$$
converges, and its sum is the number [[e|0MVCFMW7S4]] $\approx 2.71828$.

## Remarks

The series converges by the [[ratio test|0QNY6ZBXPN]], since the ratio of consecutive terms is $1/(n+1) \to 0$.
It is the value at $x = 1$ of the power series $\sum x^n / n!$ of the [[exponential function|0RMVS7RXZX]].
The series converges fast. The tail after the term $1/N!$ is less than $1/(N! \, N)$.

## Explanations

### L1 human

The terms shrink very fast, because $n!$ grows fast.
Only a few terms give a good value of $e$: the first five terms already give $2.708$.
</document>

<document source="https://math2.tamnd.com/node/0G4F0XB9X2.md" tag="0G4F0XB9X2" kind="concept">
# Entire function

## Summary

An entire function is a function that is holomorphic on the whole complex plane, see the [[definition|0419794R0T]].
Polynomials and the exponential function are entire.
</document>

<document source="https://math2.tamnd.com/node/0G4J7Z7WPJ.md" tag="0G4J7Z7WPJ" kind="definition">
# Completion of a metric space

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
A completion of $X$ is a pair $(\hat{X}, \iota)$ of a [[complete metric space|08EWT7J0SC]] $\hat{X}$ and an [[isometry|0878DN3WCS]] $\iota \colon X \to \hat{X}$ with [[dense|0D198GPMTJ]] image $\iota(X)$.

## Remarks

Each metric space has a completion. One construction takes the Cauchy sequences of $X$ and identifies two of them when the distance between their terms tends to $0$.
The completion is unique: for two completions $(\hat{X}, \iota)$ and $(\hat{X}', \iota')$, there is a unique isometric isomorphism $\varphi \colon \hat{X} \to \hat{X}'$ with $\varphi \circ \iota = \iota'$.
The completion of $\mathbb{Q}$ with the usual metric is $\mathbb{R}$. This is the [[construction of the reals from Cauchy sequences|0D8HE681SC]].
If $X$ is already complete, then $X$ with the identity map is a completion of $X$.

## Explanations

### L1 human

The completion fills the holes of a metric space with new points, and it adds no more points than it needs.
Each new point is a limit of a Cauchy sequence of old points.
</document>

<document source="https://math2.tamnd.com/node/0G4MFRQSGN.md" tag="0G4MFRQSGN" kind="definition">
# Cylindrical coordinates

## Definition

The cylindrical coordinate map is $C \colon [0, \infty) \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}^3$,
$$C(r, \theta, z) = (r \cos \theta, \ r \sin \theta, \ z).$$
A triple $(r, \theta, z)$ with $C(r, \theta, z) = (x, y, z)$ is a triple of cylindrical coordinates of $(x, y, z)$.
Here $r = \sqrt{x^2 + y^2}$ is the distance to the $z$-axis, $\theta$ is the azimuth, and $z$ is the height.

## Remarks

The pair $(r, \theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$. So, away from the $z$-axis, the coordinates are unique if $r > 0$ and $\theta \in [0, 2\pi)$.
The Jacobian determinant of $C$ is $r$, so the volume element is $dV = r \, dr \, d\theta \, dz$.
Example: the solid cylinder $x^2 + y^2 \le a^2$, $0 \le z \le h$ has volume $\int_0^h \int_0^{2\pi} \int_0^a r \, dr \, d\theta \, dz = \pi a^2 h$.
Some texts write $\rho$ or $s$ for $r$, and $\varphi$ for $\theta$.

## Explanations

### L1 human

Cylindrical coordinates are polar coordinates in the floor plan, together with the height.
Surfaces $r = $ constant are cylinders around the $z$-axis, which gives the coordinates their name.
</document>

<document source="https://math2.tamnd.com/node/0G4W89YJRK.md" tag="0G4W89YJRK" kind="definition">
# Oriented angle

## Definition

Let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ be nonzero vectors in the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Let $u \cdot v = u_1 v_1 + u_2 v_2$ and $\det(u, v) = u_1 v_2 - u_2 v_1$.
The oriented angle from $u$ to $v$ is the class modulo $2\pi$ of each real number $\theta$ with
$$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|} \quad \text{and} \quad \sin \theta = \frac{\det(u, v)}{\|u\| \, \|v\|}.$$
For points $A$, $B$ and $C$ of $\mathbb{R}^2$ with $A \neq B$ and $C \neq B$, the oriented angle $\measuredangle ABC$ is the oriented angle from $A - B$ to $C - B$.

## Remarks

Such a $\theta$ exists because $(u \cdot v)^2 + \det(u, v)^2 = \|u\|^2 \|v\|^2$. It is unique modulo $2\pi$.
Swapping the vectors changes the sign: the oriented angle from $v$ to $u$ is minus the oriented angle from $u$ to $v$.
The [[unoriented angle|0ZEHZY85E6]] between $u$ and $v$ is the absolute value of the representative of the oriented angle in $(-\pi, \pi]$.
Mathlib's `EuclideanGeometry.oangle` works in each oriented Euclidean plane, with values in $\mathbb{R} / 2\pi\mathbb{Z}$. It gives $0$ when a vector is zero.

## Explanations

### L1 human

The oriented angle remembers the direction of the turn. Turning from $(1, 0)$ to $(0, 1)$ is a quarter turn anticlockwise, so the oriented angle is $\pi / 2$. Turning back, from $(0, 1)$ to $(1, 0)$, gives $-\pi / 2$.
</document>

<document source="https://math2.tamnd.com/node/0G4X36AE0X.md" tag="0G4X36AE0X" kind="definition">
# Equilibrium point

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
A point $x^* \in U$ is an equilibrium point of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if $f(x^*) = 0$.

## Remarks

The point $x^*$ is an equilibrium point exactly when the constant function $x(t) = x^*$ is a [[solution|04SW7WECS8]].
For the [[logistic equation|0CC4VC7KR3]] $x' = r x (1 - x/K)$ with $r, K > 0$, the equilibrium points are $0$ and $K$.
The equilibrium points are the fixed points of the [[flow|02903SSA48]]: $\varphi_t(x^*) = x^*$ for each $t$.
The behaviour of solutions near an equilibrium point is the subject of [[Lyapunov stability|092S6VBECQ]].

## Explanations

### L1 human

An equilibrium point is a state where the velocity is zero.
A system that starts there stays there for all time.
</document>

<document source="https://math2.tamnd.com/node/0G52BPNPKG.md" tag="0G52BPNPKG" kind="definition">
# Clique

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A clique in $G$ is a set $S$ of vertices of $G$ such that any two distinct vertices of $S$ are adjacent.

## Remarks

A set $S$ is a clique if and only if the [[induced subgraph|0V05FC4GKQ]] on $S$ is a [[complete graph|03WKB2H474]].
A set is a clique in $G$ if and only if it is an [[independent set|0KZJ82WEPP]] in the [[complement graph|0RNMP6VCR2]].
To decide whether a graph has a clique of a given size is an NP-complete problem.

## Explanations

### L1 human

A clique is a group of vertices that are all joined to each other.
In a social network, it is a group of people who all know each other.
</document>

<document source="https://math2.tamnd.com/node/0G558GJQMW.md" tag="0G558GJQMW" kind="definition">
# Multivariate normal distribution

## Definition

Let $k \ge 1$ and let $X = (X_1, \dots, X_k)^{\mathsf{T}}$ be a random vector in $\mathbb{R}^k$.
Then $X$ is multivariate normal if there exist $\mu \in \mathbb{R}^k$, an integer $\ell \ge 1$ and a real $k \times \ell$ matrix $A$ with this property.
There is a vector $Z = (Z_1, \dots, Z_\ell)^{\mathsf{T}}$ of independent $\mathcal{N}(0, 1)$ random variables with
$$X = AZ + \mu.$$
Then the law of $X$ depends only on $\mu$ and on $\Sigma = AA^{\mathsf{T}}$. It is written $\mathcal{N}_k(\mu, \Sigma)$.

## Remarks

Here $\mu$ is the mean vector, with $\mu_i = E[X_i]$, and $\Sigma$ is the covariance matrix, with $\Sigma_{ij} = \operatorname{Cov}(X_i, X_j)$.
An equivalent condition: $X$ is multivariate normal if and only if $a^{\mathsf{T}} X$ has a [[normal distribution|07RXBVTVZB]] for each $a \in \mathbb{R}^k$. Here a constant counts as normal with variance $0$.
If $\Sigma$ is positive definite, $X$ has the density
$$f(x) = \frac{1}{\sqrt{(2\pi)^k \det \Sigma}} \exp\left(-\tfrac{1}{2} (x - \mu)^{\mathsf{T}} \Sigma^{-1} (x - \mu)\right), \quad x \in \mathbb{R}^k.$$
The components of a multivariate normal vector are independent if and only if they are uncorrelated, that is, if $\Sigma$ is diagonal.

## Explanations

### L1 human

A multivariate normal vector is a list of normal random variables that may be linked to each other.
The covariance matrix says how strongly each pair of components moves together. In two dimensions, the density looks like a hill with elliptical contour lines.
</document>

<document source="https://math2.tamnd.com/node/0G5AV9Q4RR.md" tag="0G5AV9Q4RR" kind="definition">
# Closed set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
A subset $F$ of $X$ is closed in $X$ when its [[complement|0B1FSS1VZW]] $X \setminus F$ is [[open|01ZQQDCERP]] in $X$.

## Remarks

The sets $\emptyset$ and $X$ are both open and closed.
A set can be neither open nor closed: $[0, 1)$ in $\mathbb{R}$ is an example.
In a metric space, $F$ is closed if and only if it [[contains the limits of its convergent sequences|09GQ3ZTQJ9]].

## Explanations

### L1 human

Closed is not the opposite of open. A closed set is a set whose outside is open.
The interval $[0, 1]$ is closed in $\mathbb{R}$, because its outside $(-\infty, 0) \cup (1, \infty)$ is open.
</document>

<document source="https://math2.tamnd.com/node/0G65751VSW.md" tag="0G65751VSW" kind="statement">
# Derivative of the inverse function

## Statement

Let $V, W \subseteq \mathbb{R}^n$ be open, and let $f \colon V \to W$ be a bijection.
Suppose that $f$ is differentiable at $a \in V$ and that $f^{-1}$ is differentiable at $b = f(a)$.
Then $Df(a)$ is invertible, and
$$D(f^{-1})(b) = \big( Df(a) \big)^{-1}.$$
In terms of [[Jacobian matrices|0YF0VXKXGB]], $J_{f^{-1}}(b) = J_f(a)^{-1}$.

## Remarks

Proof: apply the [[chain rule|0EAAS2EHDR]] to $f^{-1} \circ f = \mathrm{id}_V$ and $f \circ f^{-1} = \mathrm{id}_W$. This gives $D(f^{-1})(b) \, Df(a) = I$ and $Df(a) \, D(f^{-1})(b) = I$.
The hard part is the differentiability of $f^{-1}$. The [[inverse function theorem|0K9HSJFCMJ]] gives it when $f$ is of class $C^1$ and $Df(a)$ is invertible.
For polar coordinates $f(r, \theta) = (r \cos \theta, r \sin \theta)$ with $r > 0$, $J_f = \begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}$, and the Jacobian matrix of a local inverse is the inverse of this matrix.

## Explanations

### L1 human

Undoing a map undoes its linear approximation.
So the derivative of the inverse is the inverse matrix of the derivative, taken at the matching point.
</document>

<document source="https://math2.tamnd.com/node/0G6D8E7CTA.md" tag="0G6D8E7CTA" kind="statement">
# Cauchy sequences are bounded

## Statement

Let $(a_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] of real numbers.
Then $(a_n)$ is [[bounded|05FQ3VGAR9]].

## Remarks

Take $\varepsilon = 1$ to get $N$ with $|a_n - a_N| < 1$ for all $n \ge N$. Then $|a_n| \le \max(|a_0|, \dots, |a_{N-1}|, |a_N| + 1)$ for all $n$.
The statement holds in each metric space.
The proof of the [[Cauchy criterion|0ZA4AX4MW3]] uses this fact.

## Explanations

### L1 human

From some index on, all terms are within distance $1$ of one fixed term.
The finitely many terms before that index are bounded too, so the whole sequence is bounded.
</document>

<document source="https://math2.tamnd.com/node/0G77XFBV00.md" tag="0G77XFBV00" kind="definition">
# Lexicographic order

## Definition

Let $(A, \le_A)$ and $(B, \le_B)$ be [[partially ordered sets|011C29Q1S1]].
The lexicographic order on the [[Cartesian product|0C81YBVTWH]] $A \times B$ is the relation $\le$ with $(a, b) \le (a', b')$ when
$$a <_A a' \quad \text{or} \quad (a = a' \text{ and } b \le_B b').$$

## Remarks

It is a partial order. When $\le_A$ and $\le_B$ are [[total orders|0316AYH1SC]], it is a total order.
Example: in $\mathbb{N} \times \mathbb{N}$, $(1, 5) < (2, 0)$ and $(2, 0) < (2, 3)$.
A dictionary orders words in the same way, letter by letter.

## Explanations

### L1 human

The lexicographic order is the dictionary order.
"cat" comes before "cow", because the first letters agree and "a" comes before "o".
</document>

<document source="https://math2.tamnd.com/node/0G78GP5TF1.md" tag="0G78GP5TF1" kind="definition">
# Vector projection

## Definition

Let $n$ be a positive integer, let $a$ be a vector in $\mathbb{R}^n$, and let $b$ be a nonzero vector in $\mathbb{R}^n$. Let $a \cdot b = a_1 b_1 + \dots + a_n b_n$ be the dot product.
The vector projection of $a$ onto $b$ is the vector
$$\operatorname{proj}_b a = \frac{a \cdot b}{b \cdot b} \, b.$$
The scalar projection of $a$ onto $b$ is the real number $\dfrac{a \cdot b}{\|b\|}$, where $\|b\|$ is the [[length|0XKBKGTCPC]] of $b$.

## Remarks

The vector $a - \operatorname{proj}_b a$ is the vector rejection of $a$ from $b$. Its dot product with $b$ is $0$, so it is [[perpendicular to $b$|03NK6RSRQ1]].
Among all multiples $c b$ of $b$, the projection is the one that makes $\|a - c b\|$ smallest.
If $\theta$ is the angle between $a \neq 0$ and $b$, then the scalar projection is $\|a\| \cos \theta$.

## Explanations

### L1 human

The projection answers the question "how much of $a$ points along $b$?". For $a = (3, 4)$ and $b = (1, 0)$ the projection is $(3, 0)$. The rest, $(0, 4)$, is perpendicular to $b$.
</document>

<document source="https://math2.tamnd.com/node/0G7WM8DX4P.md" tag="0G7WM8DX4P" kind="concept">
# Parametrised curve

## Summary

A parametrised curve in $\mathbb{R}^n$ is a continuous [[vector-valued function|018GTRS7A4]] of one real variable, defined on an interval.
One can think of the parameter as time and of the curve as the trajectory of a moving point.
</document>

<document source="https://math2.tamnd.com/node/0G8G1TH5WK.md" tag="0G8G1TH5WK" kind="statement">
# Uniqueness of the complete ordered field

## Statement

Let $F$ and $K$ be [[complete ordered fields|0A9E6X0S3E]].
Then there is exactly one field isomorphism $\varphi \colon F \to K$ that preserves the order, that is, with $\varphi(a) \le \varphi(b)$ whenever $a \le b$.

## Remarks

The proof first maps the rational elements of $F$ to those of $K$. Each other element is the supremum of the rational elements below it, and $\varphi$ maps it to the supremum of their images.
In fact each field isomorphism between two complete ordered fields preserves the order, because the nonnegative elements are exactly the squares.

## Explanations

### L1 human

Two people may build the real numbers in different ways, for example with Dedekind cuts and with Cauchy sequences.
This theorem says that they get the same number system, with only different names for the numbers.
</document>

<document source="https://math2.tamnd.com/node/0G8VEZ3WTK.md" tag="0G8VEZ3WTK" kind="concept">
# Equivalence relation

## Summary

An equivalence relation is a relation that behaves like equality: it is reflexive, symmetric and transitive.
</document>

<document source="https://math2.tamnd.com/node/0G9EKM0E15.md" tag="0G9EKM0E15" kind="concept">
# Sequence

## Summary

A sequence is an ordered list of elements of a set, with one term for each natural number. Sequences are the basic objects of the study of limits in analysis. The node [[sequence|0QE8STGJH2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GA1V8NN8C.md" tag="0GA1V8NN8C" kind="statement">
# Inscribed angle theorem

## Statement

Let $A$, $B$ and $V$ be distinct points of a [[circle|060W5CBZP6]] with the centre $O$ in the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\mathbb{R} / 2\pi\mathbb{Z}$.
Then
$$\measuredangle AOB = 2 \, \measuredangle AVB.$$

## Remarks

For the unoriented [[angles|0GJAJC6TNZ]], with values in $[0, \pi]$: if $V$ and $O$ are not on opposite sides of the line $AB$, then $\angle AVB = \frac{1}{2} \angle AOB$. If they are on opposite sides, then $\angle AVB = \pi - \frac{1}{2} \angle AOB$.
Thus the inscribed angle does not change when $V$ moves on one arc from $A$ to $B$.
[[Thales's theorem|0GSDHYMA1T]] is the case where $AB$ is a diameter.
The oldest known version is in Propositions 20 and 21 of Book III of Euclid's Elements.

## Explanations

### L1 human

Fix two points $A$ and $B$ on a circle. The angle at the centre that looks at the arc $AB$ is twice the angle at any point of the circle on the other arc. The proof starts with the case where the line $VO$ passes through $B$. Then the triangle $VOA$ is isosceles, and its angle sum gives the result.
</document>

<document source="https://math2.tamnd.com/node/0GA6G43EA9.md" tag="0GA6G43EA9" kind="statement">
# Derivative of arcsine

## Statement

The arcsine is differentiable on $(-1, 1)$, and
$$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \quad \text{for } -1 < x < 1.$$

## Remarks

Let $y = \arcsin x$. The [[inverse function rule|0M0RZ3HSB0]] gives the derivative $\frac{1}{\cos y}$. By the [[Pythagorean identity|03H7WZ4T3K]], $\cos y = \sqrt{1 - x^2}$, since $\cos y > 0$ for $|y| < \pi/2$.
The arcsine is not differentiable at $\pm 1$. Its graph has vertical tangents there.
Similarly, $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$. Mathlib states the main claim as `Real.hasDerivAt_arcsin`.

## Explanations

### L1 human

The derivative of the arcsine is an algebraic function, although the arcsine itself is not.
It becomes infinite at the ends of the interval $[-1, 1]$.
</document>

<document source="https://math2.tamnd.com/node/0GA9K57A39.md" tag="0GA9K57A39" kind="example">
# Fibonacci sequence

## Example

The Fibonacci sequence $(F_n)_{n \ge 0}$ is the sequence of natural numbers defined by
$$F_0 = 0, \qquad F_1 = 1, \qquad F_{n+2} = F_{n+1} + F_n.$$
It is an example of a sequence defined by a [[recurrence relation|07TZV87Y0T]], and in particular of a [[linear recurrence with constant coefficients|081THG13JZ]].
Its first terms are
$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$$

## Remarks

The ratios $F_{n+1} / F_n$ tend to the [[golden ratio|0SC8T7548D]] $\varphi = \frac{1 + \sqrt{5}}{2}$.
[[Binet's formula|0404D8TN7Y]] gives $F_n$ in closed form.
$F_{n+1}$ is the number of ways to tile a strip of length $n$ with squares of length $1$ and dominoes of length $2$.

## Explanations

### L1 human

Each Fibonacci number is the sum of the two before it: $5 + 8 = 13$, $8 + 13 = 21$, and so on.
The numbers grow by a factor close to $1.618$ at each step.
</document>

<document source="https://math2.tamnd.com/node/0GBG8B49AA.md" tag="0GBG8B49AA" kind="concept">
# Product topology

## Summary

The product topology is the natural topology on a Cartesian product of topological spaces.
It is the coarsest topology for which all projections onto the factors are continuous.
</document>

<document source="https://math2.tamnd.com/node/0GBJACNAP6.md" tag="0GBJACNAP6" kind="concept">
# Independence of two random variables

## Summary

Two random variables are independent when each event about the first one is independent of each event about the second one.
The node [[independence of two random variables|0E9502NT92]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GBZDAFPX8.md" tag="0GBZDAFPX8" kind="statement">
# Epsilon-delta definition of a limit

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number.
Then $\lim_{x \to a} f(x) = L$ if and only if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ such that $|f(x) - L| < \varepsilon$ for all $x \in D$ with $0 < |x - a| < \delta$.

## Remarks

Each open interval around a point contains an interval $(L - \varepsilon, L + \varepsilon)$, and each such interval is open. This gives the equivalence with the [[definition|0TXTSEMHR5]].
Many texts take this condition as the definition of the limit.
Mathlib states a form for metric spaces as `Metric.tendsto_nhds_nhds`.

## Explanations

### L1 human

For each error tolerance $\varepsilon$, we can force $f(x)$ within $\varepsilon$ of $L$.
We do this by taking $x$ within a suitable distance $\delta$ of $a$.
</document>

<document source="https://math2.tamnd.com/node/0GBZM0MT0F.md" tag="0GBZM0MT0F" kind="concept">
# Binomial coefficient

## Summary

The binomial coefficient $\binom{n}{k}$ is the number of ways to choose $k$ elements from an $n$-element set. These numbers are the coefficients in the expansion of $(x + y)^n$. The node [[binomial coefficient|092GEF38YD]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GCM7B6PBK.md" tag="0GCM7B6PBK" kind="concept">
# Concave function

## Summary

A concave function on an interval is a function whose graph lies on or above each of its chords. A function is concave exactly when its negative is convex. The node [[concave function|0EF2VG8XKQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GCVEV6BJA.md" tag="0GCVEV6BJA" kind="definition">
# Mutual independence of events

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]], let $I$ be an index set, and let $A_i \in \mathcal{F}$ be an [[event|00SQGJNRTB]] for each $i \in I$.
The events $(A_i)_{i \in I}$ are mutually independent if for each finite nonempty subset $J \subseteq I$,
$$P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).$$

## Remarks

For two events this is the [[independence of two events|0P0MZ4X682]].
For three events $A, B, C$ the definition asks for four equations: one for each of the pairs $\{A, B\}$, $\{A, C\}$, $\{B, C\}$ and one for the triple $\{A, B, C\}$.
Mutual independence implies [[pairwise independence|0N57D1XY6T]]. The converse is false: see [[pairwise independent events that are not mutually independent|0AR4B3K5QD]].
If we replace some of the events $A_i$ by their complements, the family stays mutually independent.

## Explanations

### L1 human

For many events, independence must hold for every group of them, not only for every pair.
For three tosses of a fair coin, the events "toss $k$ gives heads" for $k = 1, 2, 3$ are mutually independent, and all three happen with probability $1/8$.
</document>

<document source="https://math2.tamnd.com/node/0GCWJS2EGB.md" tag="0GCWJS2EGB" kind="definition">
# Eccentricity

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$.
If $Q$ is a circle, its eccentricity is $e = 0$. Otherwise, let $F$ be a [[focus|0RBKKNH4GV]] of $Q$ and $\ell$ the [[directrix|0E5CXYNEPX]] for $F$. The eccentricity of $Q$ is the constant ratio
$$e = \frac{d(P, F)}{d(P, \ell)}, \qquad P \in Q.$$

## Remarks

An ellipse has $0 \leq e < 1$, a parabola has $e = 1$ and a hyperbola has $e > 1$.
For an ellipse or a hyperbola with the semi-major axis $a$ and the foci at the distance $c$ from the centre, $e = c / a$.
Two conic sections with the same eccentricity are similar.

## Explanations

### L1 human

The eccentricity tells how stretched a conic is. A circle has $e = 0$. The ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ has $c = 4$ and so $e = 4/5$. As $e$ gets near $1$, the ellipse becomes long and thin.
</document>

<document source="https://math2.tamnd.com/node/0GDNHS9J2X.md" tag="0GDNHS9J2X" kind="definition">
# Curl

## Definition

Let $U \subseteq \mathbb{R}^3$ be open, and let $F = (P, Q, R) \colon U \to \mathbb{R}^3$ be a [[vector field|0N67NBKQ5T]] whose first [[partial derivatives|05AXG3QRKR]] exist on $U$.
The curl of $F$ is the vector field $\operatorname{curl} F \colon U \to \mathbb{R}^3$,
$$\operatorname{curl} F = \nabla \times F = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right).$$
A field with $\operatorname{curl} F = 0$ on $U$ is irrotational, or curl-free.

## Remarks

As a mnemonic, $\nabla \times F$ is the formal determinant of the matrix with rows $(\mathbf{i}, \mathbf{j}, \mathbf{k})$, $(\partial_x, \partial_y, \partial_z)$ and $(P, Q, R)$.
For $F(x, y, z) = (-y, x, 0)$, $\operatorname{curl} F = (0, 0, 2)$. For $F(x, y, z) = (x, y, z)$, $\operatorname{curl} F = 0$.
For a plane field $(P, Q)$, one uses the scalar curl $\partial Q / \partial x - \partial P / \partial y$, the third component of the curl of $(P, Q, 0)$. It appears in [[Green's theorem|0HR3N02317]].
Take small loops orthogonal to a unit vector $n$. The component of $\operatorname{curl} F$ along $n$ is the limit of the circulation of $F$ around these loops, divided by the enclosed area. This follows from [[Stokes' theorem|0AKRKVHW94]].

## Explanations

### L1 human

Put a tiny paddle wheel into a fluid with velocity field $F$.
If it spins, the curl is not zero there. The curl points along the axis about which the wheel spins fastest.
</document>

<document source="https://math2.tamnd.com/node/0GDQRETR59.md" tag="0GDQRETR59" kind="concept">
# Prüfer sequence

## Summary

The Prüfer sequence encodes a labelled tree on $n$ vertices as a sequence of $n - 2$ labels. This encoding is a bijection, and it proves Cayley's formula. The node [[Prüfer sequence|09PV9JSY6E]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GE4DVQKR8.md" tag="0GE4DVQKR8" kind="example">
# The quaternions

## Example

The quaternions $\mathbb{H}$ are the real vector space with basis $1, i, j, k$.
Each quaternion is $a + b i + c j + d k$ with $a, b, c, d \in \mathbb{R}$.
The multiplication is $\mathbb{R}$-bilinear, $1$ is the identity, and the basis elements obey
$$i^2 = j^2 = k^2 = i j k = -1.$$
These rules give $i j = k = -j i$, $j k = i = -k j$ and $k i = j = -i k$.
With this multiplication, $\mathbb{H}$ is a [[division ring|0WZXK05ME7]] that is not commutative.

## Remarks

The conjugate of $q = a + b i + c j + d k$ is $\bar{q} = a - b i - c j - d k$, and $q \bar{q} = a^2 + b^2 + c^2 + d^2$.
So a nonzero $q$ has the inverse $q^{-1} = \bar{q} / (a^2 + b^2 + c^2 + d^2)$.
The quaternions are not a field, because $i j \neq j i$.

## Explanations

### L1 human

The quaternions extend the complex numbers with two more square roots of $-1$, called $j$ and $k$.
You can divide by every nonzero quaternion.
But the order of multiplication matters: $i j = k$ and $j i = -k$.
</document>

<document source="https://math2.tamnd.com/node/0GEQ42PTGB.md" tag="0GEQ42PTGB" kind="definition">
# Critical point

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
The point $a$ is a critical point of $f$ if either $f$ is not differentiable at $a$, or $f$ is differentiable at $a$ with $f'(a) = 0$.
In the second case, $a$ is also called a stationary point.

## Remarks

By [[Fermat's theorem|0PCVBRV9VJ]], each local extremum of $f$ in $I$ is a critical point.
The converse fails: $x^3$ has a critical point at $0$, but no local extremum.
For $|x|$, the point $0$ is a critical point where $f$ is not differentiable.

## Explanations

### L1 human

Critical points are the candidates for local maxima and minima.
There the tangent is horizontal, or there is no tangent at all.
</document>

<document source="https://math2.tamnd.com/node/0GETX79CJN.md" tag="0GETX79CJN" kind="concept">
# Normal core

## Summary

The normal core of a subgroup $H$ of a group $G$ is the largest normal subgroup of $G$ that is contained in $H$.
It is the intersection of all conjugates of $H$.
</document>

<document source="https://math2.tamnd.com/node/0GF6AJEBZ0.md" tag="0GF6AJEBZ0" kind="statement">
# Induction from an arbitrary base case

## Statement

Let $m$ be a natural number and let $P(n)$ be a property of natural numbers $n$ with $n \ge m$.
Suppose that $P(m)$ holds.
Suppose also that for each natural number $n \ge m$, $P(n)$ implies $P(n + 1)$.
Then $P(n)$ holds for every natural number $n \ge m$.

## Remarks

The proof applies ordinary [[induction|0ASJVEFA9J]] to the property $Q(k)$: "$P(m + k)$ holds".
For example, induction from $m = 4$ shows that $2^n \ge n^2$ for every natural number $n \ge 4$.

## Explanations

### L1 human

Sometimes a property is false for small numbers and true from some point on.
Then you start the induction at that point instead of at $0$.
The dominoes start falling at domino number $m$, and the dominoes before it do not matter.
</document>

<document source="https://math2.tamnd.com/node/0GFGHRT2FJ.md" tag="0GFGHRT2FJ" kind="concept">
# Subfield

## Summary

A subfield of a field is a subset that is a field with the same addition and multiplication.
The [[definition of a subfield|07QY5RPFBR]] lists the closure conditions.
</document>

<document source="https://math2.tamnd.com/node/0GFGK40582.md" tag="0GFGK40582" kind="concept">
# Pointwise convergence

## Summary

A sequence of functions converges pointwise when its values converge at every point of the domain. It is the weakest of the usual modes of convergence for functions. The node [[pointwise convergence|056FSKG32J]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GFKP8F8AG.md" tag="0GFKP8F8AG" kind="definition">
# Coprime integers

## Definition

Two integers $a$ and $b$ are coprime when their [[greatest common divisor|02WF878BH1]] is $1$, that is, when $\gcd(a, b) = 1$.

## Remarks

Equivalently, $a$ and $b$ are coprime when the only integers that divide both of them are $1$ and $-1$.
For example, $8$ and $15$ are coprime, but $8$ and $12$ are not, because $4$ divides both.
The integers $1$ and $-1$ are coprime to each integer. The integer $0$ is coprime only to $1$ and $-1$.
By [[Bézout's identity|0DQVHMYAGA]], $a$ and $b$ are coprime if and only if $a x + b y = 1$ for some integers $x$ and $y$.

## Explanations

### L1 human

Two numbers are coprime when they have no common factor except $1$.
The numbers do not need to be prime themselves: $9$ and $10$ are not prime, but they are coprime.
A fraction $a / b$ is in lowest terms exactly when $a$ and $b$ are coprime.
</document>

<document source="https://math2.tamnd.com/node/0GG9TVW90M.md" tag="0GG9TVW90M" kind="concept">
# Bijective function

## Summary

A function is bijective when it is both injective and surjective, so each element of the codomain is the value at exactly one input.
</document>

<document source="https://math2.tamnd.com/node/0GGE0Z0JQ1.md" tag="0GGE0Z0JQ1" kind="definition">
# Ring homomorphism

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
A ring homomorphism from $R$ to $S$ is a map $f \colon R \to S$ with these three properties.

1. $f(a + b) = f(a) + f(b)$ for all $a, b \in R$.
2. $f(a b) = f(a) f(b)$ for all $a, b \in R$.
3. $f(1) = 1$.

## Remarks

A ring homomorphism also satisfies $f(0) = 0$ and $f(-a) = -f(a)$.
For each integer $n \ge 1$, the map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ that sends $a$ to its class is a ring homomorphism.
The map $\mathbb{Z} \to \mathbb{Z}$, $a \mapsto 2a$, preserves sums but is not a ring homomorphism.
A bijective ring homomorphism is a [[ring isomorphism|0B6QC0C0G1]].

## Explanations

### L1 human

A ring homomorphism translates one ring into another and keeps the arithmetic.
It does not matter if you add or multiply first and then apply $f$, or apply $f$ first.
Reduction modulo $n$ is the main example.
</document>

<document source="https://math2.tamnd.com/node/0GGKJGZF6D.md" tag="0GGKJGZF6D" kind="concept">
# P-value

## Summary

The p-value of a test is the probability, under the null hypothesis, to get a result at least as extreme as the observed result. A small p-value is evidence against the null hypothesis.
The node [[p-value|0EHYZFZ4PK]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GGXWDNKBR.md" tag="0GGXWDNKBR" kind="concept">
# Forest

## Summary

A forest is a graph without cycles, that is, a disjoint union of trees. The spanning forests of a graph generalize spanning trees to graphs that are not connected. The node [[forest|09JP6G9FTA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GHDDVAH8P.md" tag="0GHDDVAH8P" kind="definition">
# Frobenius endomorphism

## Definition

Let $p$ be a prime number and let $R$ be a [[commutative ring|0GYGHKX834]] of [[characteristic|0P7CH1CND5]] $p$.
The Frobenius endomorphism of $R$ is the map $F \colon R \to R$, $F(x) = x^p$.
It is a [[ring homomorphism|0GGE0Z0JQ1]] from $R$ to $R$.

## Remarks

The map $F$ preserves products because $R$ is commutative. It preserves sums by the [[freshman's dream|0396VGFA78]].
If $R$ is an [[integral domain|09S3PFBEPM]], then $F$ is injective.
If $R$ is a [[finite field|08RTDHX1SK]], then $F$ is an automorphism of $R$.
On $\mathbb{F}_p$, the map $F$ is the identity, by Fermat's little theorem.

## Explanations

### L1 human

In characteristic $p$, raising to the $p$-th power respects addition and multiplication.
So it is a homomorphism from the ring to itself.
This simple map is a basic tool in the study of finite fields.
</document>

<document source="https://math2.tamnd.com/node/0GHS47PDAQ.md" tag="0GHS47PDAQ" kind="concept">
# Universal quantifier

## Summary

The universal quantifier $\forall$ expresses "for all".
A universally quantified statement says that a [[predicate|0T3DPBG981]] holds for every element of its domain.
</document>

<document source="https://math2.tamnd.com/node/0GHWEH6H15.md" tag="0GHWEH6H15" kind="statement">
# General solution of an inhomogeneous linear equation

## Statement

Let $I$ be an open interval, let $a_0, \dots, a_{n-1}, b : I \to \mathbb{R}$ be continuous, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y$.
Let $y_p$ be a solution of $L y = b$ on $I$, and let $(y_1, \dots, y_n)$ be a [[fundamental system|0H4KNNCHPY]] of $L y = 0$ on $I$.
Then a function $y : I \to \mathbb{R}$ is a solution of $L y = b$ on $I$ if and only if
$$y = y_p + c_1 y_1 + \dots + c_n y_n$$
for some real numbers $c_1, \dots, c_n$.

## Remarks

Proof: $y$ solves $L y = b$ if and only if $L(y - y_p) = 0$, by linearity. Then $y - y_p$ is a combination of the fundamental system.
The [[method of undetermined coefficients|0XRB469Z32]] and [[variation of parameters|0Y6TH5PQVS]] find a particular solution $y_p$.
Example: $y'' + y = 1$ has $y_p = 1$, so its solutions are $1 + c_1 \cos x + c_2 \sin x$.

## Explanations

### L1 human

To solve a linear equation with a right side, find one solution by any method.
Then add all the solutions of the equation without right side.
This gives every solution, and each only one time.
</document>

<document source="https://math2.tamnd.com/node/0GJ2HRG95T.md" tag="0GJ2HRG95T" kind="definition">
# Euclidean domain

## Definition

A Euclidean domain is an [[integral domain|09S3PFBEPM]] $R$ with a function $\delta \colon R \setminus \{0\} \to \mathbb{N}$ with this property.
For all $a, b \in R$ with $b \neq 0$, there are $q, r \in R$ such that $a = b q + r$, and $r = 0$ or $\delta(r) < \delta(b)$.
The function $\delta$ is a Euclidean function of $R$.

## Remarks

The absolute value is a Euclidean function on $\mathbb{Z}$. The degree is a Euclidean function on $K[x]$ for a field $K$.
Some authors also require $\delta(a) \le \delta(a b)$ for all nonzero $a, b$. This changes the class of Euclidean domains only in appearance, because a ring with a Euclidean function has one with this extra property.
Every Euclidean domain is a [[principal ideal domain|0RZYDVZ475]].

## Explanations

### L1 human

In a Euclidean domain, you can always divide with a remainder that is smaller than the divisor.
The function $\delta$ measures what smaller means.
This is all you need to run the Euclidean algorithm.
</document>

<document source="https://math2.tamnd.com/node/0GJ7T0NK49.md" tag="0GJ7T0NK49" kind="definition">
# Polynomial evaluation

## Definition

Let $R$ be a [[ring|0894J980RK]], let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$ and let $c \in R$.
The value of $f$ at $c$ is the element
$$f(c) = a_0 + a_1 c + a_2 c^2 + \dots + a_n c^n \in R.$$

## Remarks

The value $f(c)$ does not depend on the choice of $n$, because the extra coefficients are $0$.
For $f = x^2 - 2$ in $\mathbb{Z}[x]$, $f(3) = 7$.
If $R$ is commutative, then $(f g)(c) = f(c) g(c)$ and $(f + g)(c) = f(c) + g(c)$. This gives the [[evaluation homomorphism|0125QVNCFQ]].
Horner's rule computes $f(c) = a_0 + c(a_1 + c(a_2 + \dots + c a_n))$ with $n$ multiplications.

## Explanations

### L1 human

To evaluate $f$ at $c$, you put $c$ in place of $x$ and compute.
For $f = x^2 + 1$ and $c = 2$, you get $f(2) = 5$.
</document>

<document source="https://math2.tamnd.com/node/0GJ83E78A5.md" tag="0GJ83E78A5" kind="statement">
# Universal property of the quotient group

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $\pi \colon G \to G/N$ be the projection $\pi(g) = gN$ to the [[quotient group|0KYX2N69J0]].
Let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]] with $N \subseteq \ker \varphi$.
Then there is exactly one group homomorphism $\bar{\varphi} \colon G/N \to H$ with $\bar{\varphi} \circ \pi = \varphi$, and it is given by $\bar{\varphi}(gN) = \varphi(g)$.

## Remarks

The map $\bar{\varphi}$ is well defined because $\varphi$ is constant on each coset of $N$.
The image of $\bar{\varphi}$ is the image of $\varphi$, and the kernel of $\bar{\varphi}$ is $(\ker \varphi)/N$.
With $N = \ker \varphi$, this gives the [[first isomorphism theorem|0CXXJNG5B6]].

## Explanations

### L1 human

A homomorphism that sends all of $N$ to the identity cannot tell apart elements of the same coset of $N$.
So it already makes sense on the cosets, that is, on the quotient group.
This is the standard way to define a homomorphism out of a quotient group.
</document>

<document source="https://math2.tamnd.com/node/0GJAJC6TNZ.md" tag="0GJAJC6TNZ" kind="concept">
# Angle at a point

## Summary

The angle $\angle ABC$ at the point $B$ is the opening between the rays from $B$ through $A$ and from $B$ through $C$. It is a number from $0$ to $\pi$, and it does not depend on the order of the two rays.
</document>

<document source="https://math2.tamnd.com/node/0GJBJ19NXR.md" tag="0GJBJ19NXR" kind="definition">
# Mersenne number

## Definition

For a natural number $n$, the $n$-th Mersenne number is $M_n = 2^n - 1$.

## Remarks

The first values are $M_1 = 1$, $M_2 = 3$, $M_3 = 7$, $M_4 = 15$ and $M_5 = 31$.
In binary, $M_n$ is written with $n$ digits $1$.
Some texts call $M_n$ a Mersenne number only when $n$ is prime.
A Mersenne number that is prime is a [[Mersenne prime|06GPJF9V7M]].

## Explanations

### L1 human

A Mersenne number is one less than a power of $2$.
For example, $2^{10} - 1 = 1023$, which is $1111111111$ in binary.
</document>

<document source="https://math2.tamnd.com/node/0GJM24EXP9.md" tag="0GJM24EXP9" kind="statement">
# Cancellation in an integral domain

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b, c \in R$ with $a \neq 0$.
If $a b = a c$, then $b = c$.

## Remarks

The proof is short: $a b = a c$ gives $a (b - c) = 0$, and $a \neq 0$ gives $b - c = 0$.
The converse also holds: a nonzero commutative ring with this cancellation property is an integral domain.
Cancellation fails in $\mathbb{Z}/6\mathbb{Z}$: $2 \cdot 1 = 2 \cdot 4$, but $1 \neq 4$.

## Explanations

### L1 human

In the integers, you can cancel a nonzero factor from both sides of an equation.
From $3 b = 3 c$ you get $b = c$.
This works in every integral domain, even when you cannot divide by $a$.
</document>

<document source="https://math2.tamnd.com/node/0GJWQZCPZM.md" tag="0GJWQZCPZM" kind="statement">
# Trace is linear

## Statement

Let $R$ be a commutative ring, let $A$ and $B$ be $n \times n$ matrices over $R$, and let $c \in R$.
Then $\tr(A + B) = \tr A + \tr B$ and $\tr(cA) = c \tr A$, where $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

Thus the trace is a linear map from $M_n(R)$ to $R$.
Both identities follow at once from the definition, because the diagonal of $A + B$ is the sum of the diagonals.
Over a field, the trace is a [[linear form|07YQ1C78G0]] on the vector space of $n \times n$ matrices.

## Explanations

### L1 human

The trace of a sum is the sum of the traces.
The trace of a multiple is the same multiple of the trace.
This holds because we only add diagonal entries, and addition and scaling act entry by entry.
</document>

<document source="https://math2.tamnd.com/node/0GJZCE87WP.md" tag="0GJZCE87WP" kind="statement">
# Addition formulas for hyperbolic functions

## Statement

For all real numbers $x$ and $y$,
$$\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y,$$
$$\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y.$$

## Remarks

Expand the right sides with the definitions and use [[$e^{x+y} = e^x e^y$|0E8HNVBCSP]].
With $y = x$, they give $\sinh 2x = 2 \sinh x \cosh x$ and $\cosh 2x = \cosh^2 x + \sinh^2 x$.
Mathlib states the first formula as `Real.sinh_add`.

## Explanations

### L1 human

These formulas look like the trigonometric addition formulas.
The only change is the plus sign in the formula for $\cosh$.
</document>

<document source="https://math2.tamnd.com/node/0GJZE2E4RX.md" tag="0GJZE2E4RX" kind="counterexample">
# Function for which the iterated integrals differ

## Counterexample

One could guess that the two [[iterated integrals|0P4WX7Y502]] of a function over a square are always equal when both exist.
This is false.
Let $f(x, y) = \dfrac{x^2 - y^2}{(x^2 + y^2)^2}$ on $(0, 1] \times (0, 1]$. Then
$$\int_0^1 \int_0^1 f(x, y) \, dy \, dx = \frac{\pi}{4}, \qquad \int_0^1 \int_0^1 f(x, y) \, dx \, dy = -\frac{\pi}{4}.$$

## Remarks

For fixed $x > 0$, $\partial_y \big( y / (x^2 + y^2) \big) = f(x, y)$. So the inner integral is $1/(1 + x^2)$, and its integral over $[0, 1]$ is $\pi / 4$. Since $f(y, x) = -f(x, y)$, the other order gives $-\pi / 4$.
The function is not integrable on the square: $\int_0^1 \int_0^1 |f| \, dy \, dx = \infty$. So [[Fubini's theorem|0R0T1VWH5X]] does not apply.
A discrete example is $a_{xy} = 1$ if $x = y$, $a_{xy} = -1$ if $x = y + 1$, and $a_{xy} = 0$ otherwise, on the positive integers. The two iterated sums are $0$ and $1$.

## Explanations

### L1 human

Near the corner $(0, 0)$ the function has large positive values and large negative values.
The two orders of integration cancel these parts in different ways, so they give different answers.
</document>

<document source="https://math2.tamnd.com/node/0GJZW9JFZQ.md" tag="0GJZW9JFZQ" kind="concept">
# Transitive relation

## Summary

A relation is transitive when it passes along chains.
If a first element is related to a second, and the second to a third, then the first is related to the third.
</document>

<document source="https://math2.tamnd.com/node/0GK1WB9R98.md" tag="0GK1WB9R98" kind="definition">
# Minimal polynomial of an algebraic element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be [[algebraic|0NPW1DWECM]] over $K$.
The minimal polynomial of $a$ over $K$ is the [[monic polynomial|0BDX4MBKT4]] $m_a \in K[x]$ of smallest [[degree|0DPF6WRZW6]] such that $m_a(a) = 0$.

## Remarks

The minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at $a$, their difference has smaller degree and vanishes at $a$, so it is zero.
The minimal polynomial [[is irreducible|0NFEG48RDR]], and it [[divides each polynomial that vanishes at $a$|06K7SARZKR]].
The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$. Over $\mathbb{R}$, it is $x - \sqrt{2}$.
The minimal polynomial of $i$ over $\mathbb{R}$ is $x^2 + 1$.

## Explanations

### L1 human

Among all the polynomial equations that an algebraic element satisfies, the minimal polynomial is the simplest one.
It depends on the base field: a bigger base field can give a smaller minimal polynomial.
</document>

<document source="https://math2.tamnd.com/node/0GK68PZ6KH.md" tag="0GK68PZ6KH" kind="statement">
# Solution space of a homogeneous linear equation has dimension n

## Statement

Let $n \ge 1$, let $I$ be an open interval, and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous.
Let $S$ be the set of solutions $y : I \to \mathbb{R}$ of $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
Then $S$ is a real vector space of dimension $n$.
More precisely, for each $x_0 \in I$, the map $S \to \mathbb{R}^n$, $y \mapsto \left(y(x_0), y'(x_0), \dots, y^{(n-1)}(x_0)\right)$ is a linear bijection.

## Remarks

The [[superposition principle|0F2VBPXNP9]] gives that $S$ is a vector space. The map is injective by the uniqueness of solutions of the initial value problem, and surjective by their existence on all of $I$.
A basis of $S$ is a [[fundamental system of solutions|0H4KNNCHPY]].
Example: the solutions of $y'' + y = 0$ on $\mathbb{R}$ are $C_1 \cos x + C_2 \sin x$, a space of dimension $2$.

## Explanations

### L1 human

A solution is fixed by its starting data: the value and the first $n - 1$ derivatives at one point.
These are $n$ numbers that you can choose freely.
So there are exactly $n$ independent solutions, and each solution is a combination of them.
</document>

<document source="https://math2.tamnd.com/node/0GKJ78948V.md" tag="0GKJ78948V" kind="definition">
# Separable space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is separable when there is a [[countable|0ZG61TFBES]] subset $D \subseteq X$ that is [[dense|0D198GPMTJ]] in $X$.

## Remarks

The real line is separable, because $\mathbb{Q}$ is countable and dense. In the same way $\mathbb{R}^n$ is separable, with the dense set $\mathbb{Q}^n$.
Each [[second-countable space|00RE1R4W7N]] is separable. For metric spaces, the converse is also true.
An uncountable set with the [[discrete metric|0NA4FK7B8S]] is not separable, because each dense subset is the whole set.

## Explanations

### L1 human

A list of points can approximate a separable space. Each point of the space has points of the list as close as you want.
For $\mathbb{R}$, the list of all fractions does the job.
</document>

<document source="https://math2.tamnd.com/node/0GKQKF0FXJ.md" tag="0GKQKF0FXJ" kind="example">
# The alternating group A5 is simple

## Example

The group $A_5$ has order $60$.
Its [[conjugacy classes|0DXD4QSZ8A]] have sizes $1$, $15$, $20$, $12$ and $12$.
The class of size $1$ is $\{e\}$, and the class of size $20$ contains the $3$-cycles.
The class of size $15$ contains the products of two disjoint transpositions.
The $24$ five-cycles split into two classes of size $12$.
A normal subgroup $N$ of $A_5$ is a union of conjugacy classes, and it contains the class $\{e\}$.
By [[Lagrange's theorem|3C53TD]], $|N|$ divides $60$.
No sum of $1$ and some of the numbers $15, 20, 12, 12$ divides $60$, except $1$ and $60$.
Thus $N = \{e\}$ or $N = A_5$, and $A_5$ is simple.
It is the smallest simple group that is not abelian.
</document>

<document source="https://math2.tamnd.com/node/0GMPYM0NSR.md" tag="0GMPYM0NSR" kind="definition">
# Adjugate matrix

## Definition

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$ with [[cofactors|0XPKR18APA]] $C_{ij}$.
The *adjugate* of $A$ is the $n \times n$ matrix $\operatorname{adj}(A)$ with entries
$$\operatorname{adj}(A)_{ij} = C_{ji}.$$
So $\operatorname{adj}(A)$ is the [[transpose|09DXWYRD43]] of the matrix of cofactors $(C_{ij})$.

## Remarks

For $n = 2$, $\operatorname{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
The adjugate satisfies [[$A \operatorname{adj}(A) = \det(A) I_n$|0528HKWZ1Z]].
Older texts call it the *classical adjoint*; it is not the adjoint of an inner product space.

## Explanations

### L1 human

To build the adjugate, replace each entry by its cofactor, and then transpose.
The adjugate is almost the inverse: it is the inverse multiplied by the determinant.
For $2 \times 2$ matrices, swap the diagonal entries and negate the other two.
</document>

<document source="https://math2.tamnd.com/node/0GMV7GP8W0.md" tag="0GMV7GP8W0" kind="statement">
# Absolute value is multiplicative

## Statement

For all real numbers $a$ and $b$, $|a b| = |a| \, |b|$.
Here $|\cdot|$ is the [[absolute value|00S7C68D51]].

## Remarks

Proof: check the four cases of the signs of $a$ and $b$.
In particular $|a^n| = |a|^n$ for each natural number $n$, and $|a / b| = |a| / |b|$ for $b \neq 0$.
The [[modulus of complex numbers|0HYNZE940T]] is also multiplicative.

## Explanations

### L1 human

You can take the absolute value before or after a product, and you get the same result.
For example, $|(-2) \cdot 3| = |-6| = 6$ and $|-2| \cdot |3| = 2 \cdot 3 = 6$.
</document>

<document source="https://math2.tamnd.com/node/0GNCRPYXYV.md" tag="0GNCRPYXYV" kind="concept">
# Subring

## Summary

A subring is a subset of a ring that contains $0$ and $1$ and is closed under the ring operations.
It is a ring with the operations of the larger ring.
The [[definition of a subring|0859D58WPD]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0GNG2KVPZJ.md" tag="0GNG2KVPZJ" kind="statement">
# Nilpotent groups are solvable

## Statement

Let $G$ be a [[nilpotent group|0Q7MMQHC1P]].
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

A central series $\{e\} = N_0 \subseteq \cdots \subseteq N_n = G$ is a subnormal series, and each factor $N_{i+1}/N_i$ is central in $G/N_i$, so it is abelian.
The converse is false: $S_3$ is solvable, but it is not nilpotent.
So there are strict inclusions: abelian groups, then nilpotent groups, then solvable groups.

## Explanations

### L1 human

A central series has layers that commute with the whole group.
In particular, each layer is abelian.
So a nilpotent group is solvable.
</document>

<document source="https://math2.tamnd.com/node/0GNGPF7KS5.md" tag="0GNGPF7KS5" kind="definition">
# Antichain

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $A$ of $P$ is an antichain when, for all $a, b \in A$ with $a \neq b$, neither $a \le b$ nor $b \le a$.

## Remarks

In the power set of $\{1, 2, 3\}$, the set $\{\{1\}, \{2\}, \{3\}\}$ is an antichain.
A chain and an antichain have at most one common element.
The largest size of an antichain is the width of the poset.

## Explanations

### L1 human

An antichain is the opposite of a [[chain|0GP9ASVQ1X]]: its elements are side by side.
Under divisibility, the primes form an antichain, because no prime divides another prime.
</document>

<document source="https://math2.tamnd.com/node/0GNVN3CA2P.md" tag="0GNVN3CA2P" kind="statement">
# Dirichlet's theorem on arithmetic progressions

## Statement

Let $a$ and $d$ be positive integers that are [[coprime|0GFKP8F8AG]].
Then there are infinitely many [[prime numbers|0SN0MAEKYX]] of the form $a + n d$, with $n$ a natural number.
That is, there are infinitely many primes $p$ with $p \equiv a \pmod{d}$.

## Remarks

The hypothesis $\gcd(a, d) = 1$ is necessary. Each term of $a + n d$ is a multiple of $\gcd(a, d)$, so at most one term can be prime when $\gcd(a, d) > 1$.
For $d = 4$, the theorem gives infinitely many primes $p \equiv 1 \pmod 4$ and infinitely many primes $p \equiv 3 \pmod 4$.
A special case with an elementary proof is that there are [[infinitely many primes congruent to 1 modulo n|0T1BSD7H3M]].
The primes are in fact equally distributed among the $\varphi(d)$ possible classes, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## History

Dirichlet proved the theorem in 1837. His proof introduced Dirichlet characters and Dirichlet $L$-functions.

## Explanations

### L1 human

The sequence $3, 7, 11, 15, 19, 23, \dots$ adds $4$ at each step. It contains infinitely many primes.
In general, a sequence $a, a + d, a + 2d, \dots$ contains infinitely many primes, unless $a$ and $d$ have a common factor that blocks them all.
</document>

<document source="https://math2.tamnd.com/node/0GP9ASVQ1X.md" tag="0GP9ASVQ1X" kind="definition">
# Chain

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
A subset $C$ of $P$ is a chain when, for all $a, b \in C$, $a \le b$ or $b \le a$.

## Remarks

So a chain is a subset on which $\le$ is a [[total order|0316AYH1SC]].
In the power set of $\mathbb{N}$, the sets $\{0\} \subseteq \{0, 1\} \subseteq \{0, 1, 2\} \subseteq \dots$ form a chain.
Chains are the key notion in [[Zorn's lemma|04RXTQA5ZA]] and in the [[Hausdorff maximal principle|0KM4KVR9MV]].

## Explanations

### L1 human

A chain is a part of the order that lies on one line.
In the divisors of $12$, the set $\{1, 2, 4, 12\}$ is a chain, but $\{2, 3\}$ is not.
</document>

<document source="https://math2.tamnd.com/node/0GPBVKE31D.md" tag="0GPBVKE31D" kind="concept">
# Complete residue system

## Summary

A complete residue system modulo $n$ is a set of integers with exactly one element in each residue class modulo $n$.
The [[definition of a complete residue system|0E22JKS4W4]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0GPFKS96T7.md" tag="0GPFKS96T7" kind="statement">
# Cardinality of the power set

## Statement

Let $A$ be a [[set|0SJ3RX9TXA]].
Then the [[power set|0BC7KX51Z8]] of $A$ is equinumerous with the set $\{0, 1\}^A$ of functions from $A$ to $\{0, 1\}$, so
$$|\mathcal{P}(A)| = 2^{|A|}.$$

## Remarks

The bijection sends a subset $S$ to its [[indicator function|0VSB0XF4G7]] $\mathbf{1}_S$.
For a finite set with $n$ elements, the power set has $2^n$ elements.
For $A = \mathbb{N}$, $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].

## Explanations

### L1 human

A subset is a choice of "in" or "out" for each element.
With $n$ elements, there are $2 \cdot 2 \cdots 2 = 2^n$ such choices.
</document>

<document source="https://math2.tamnd.com/node/0GPG24Q3GR.md" tag="0GPG24Q3GR" kind="concept">
# Cauchy distribution

## Summary

The Cauchy distribution is a continuous distribution on $\mathbb{R}$ with a bell-shaped density and heavy tails. It has no mean and no variance.
The node [[Cauchy distribution|0NA6Q82YR7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GPQ67XV7N.md" tag="0GPQ67XV7N" kind="statement">
# The medians meet at the centroid

## Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$, and let $G = \frac{1}{3}(A + B + C)$ be its [[centroid|0X7EEWES5G]].
Then $G$ lies on each of the three [[medians|0XW8HX8VEV]] of $ABC$.
More precisely, let $M_A$ be the midpoint of $B$ and $C$. Then $G = A + \frac{2}{3}(M_A - A)$, and the same holds for the vertices $B$ and $C$.

## Remarks

The formula shows that $G$ divides each median in the ratio $2 : 1$, counted from the vertex.
The proof is one line: $A + \frac{2}{3}\left(\frac{B + C}{2} - A\right) = \frac{1}{3}(A + B + C)$.
The statement is true in each affine space over a field in which $2$ and $3$ are invertible. Mathlib states it for each simplex.

## Explanations

### L1 human

Draw the three medians of a paper triangle. They always cross at one point. This point is two thirds of the way from each vertex to the middle of the opposite side.
</document>

<document source="https://math2.tamnd.com/node/0GQ3H7VW3V.md" tag="0GQ3H7VW3V" kind="definition">
# Skewness

## Definition

Let $X$ be a real random variable with $E[|X|^3] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$.
The skewness of $X$ is
$$\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right] = \frac{\mu_3}{\sigma^3},$$
where $\mu_3$ is the third [[central moment|0ZT24EC90K]].

## Remarks

A distribution that is symmetric about its mean has skewness $0$. The converse is false.
Positive skewness usually means a longer tail on the right. Negative skewness usually means a longer tail on the left.
The skewness does not change under $X \mapsto aX + b$ with $a > 0$. For $a < 0$, it changes sign.
The exponential distribution has skewness $2$.

## Explanations

### L1 human

Skewness tells to which side a distribution leans.
Incomes have positive skewness: most values are moderate, and a few are very large.
</document>

<document source="https://math2.tamnd.com/node/0GQ938GDE0.md" tag="0GQ938GDE0" kind="concept">
# Independence of random variables

## Summary

A family of random variables is independent, or mutually independent, when events about different members of the family are always mutually independent.
The node [[independence of random variables|0XCE070V8B]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GR2APWTAC.md" tag="0GR2APWTAC" kind="statement">
# Quotient of continuous functions is continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f, g \colon D \to \mathbb{R}$ be [[continuous|0P27W9RW0G]] functions with $g(x) \ne 0$ for all $x \in D$.
Then the quotient $f/g$, given by $(f/g)(x) = f(x)/g(x)$, is continuous.

## Remarks

The proof uses [[sequential continuity|0PAQM3H57X]] and the [[quotient rule for limits|03S6GWWKEZ]] of sequences.
If $g$ has zeros, then $f/g$ is continuous on the set where $g$ is not zero.
Mathlib states it as `Continuous.div`.

## Explanations

### L1 human

Division by a continuous function causes no problem away from its zeros.
Near a point where $g$ is not zero, $g$ stays away from zero.
</document>

<document source="https://math2.tamnd.com/node/0GRAATMGP1.md" tag="0GRAATMGP1" kind="concept">
# Wedge product of forms

## Summary

The wedge product combines a $k$-form and an $l$-form into a $(k + l)$-form.
It is bilinear and alternating, and it encodes signed areas and volumes. It makes the [[differential forms|0E3G0GRPZ6]] on a set into an algebra.
</document>

<document source="https://math2.tamnd.com/node/0GRM3BS0DD.md" tag="0GRM3BS0DD" kind="counterexample">
# Curl-free field that is not conservative

## Counterexample

One could guess that each $C^1$ vector field with zero [[curl|0GDNHS9J2X]] on an open set is [[conservative|05R9NF2J49]].
This is false.
Let $U = \{(x, y, z) \in \mathbb{R}^3 : (x, y) \ne (0, 0)\}$, which is $\mathbb{R}^3$ minus the $z$-axis, and let
$$F(x, y, z) = \left( \frac{-y}{x^2 + y^2}, \ \frac{x}{x^2 + y^2}, \ 0 \right).$$
Then $\nabla \times F = 0$ on $U$, but $F$ is not conservative on $U$.

## Remarks

The third component of $\nabla \times F$ is $\partial_x \big( x / (x^2 + y^2) \big) - \partial_y \big( -y / (x^2 + y^2) \big) = \frac{y^2 - x^2}{(x^2 + y^2)^2} - \frac{y^2 - x^2}{(x^2 + y^2)^2} = 0$, and the other two components are zero.
On the unit circle $\gamma(t) = (\cos t, \sin t, 0)$, $t \in [0, 2\pi]$, we have $F(\gamma(t)) \cdot \gamma'(t) = 1$. So the circulation is $2\pi \ne 0$, and [[path independence|0BFN21TF5F]] fails.
The domain $U$ is not [[simply connected|075C629XTH]], so this does not contradict the theorem for [[convex|0B22P010D6]] or simply connected sets.
Locally, $F = \nabla \theta$, where $\theta$ is the polar angle of $(x, y)$. But no continuous choice of $\theta$ exists on all of $U$.

## Explanations

### L1 human

The field circles around the $z$-axis, and its strength falls off with distance in just the right way to have no local swirl.
But a loop around the axis picks up work $2\pi$, so no potential can exist. The hole in the domain causes this.
</document>

<document source="https://math2.tamnd.com/node/0GS0AW6JRY.md" tag="0GS0AW6JRY" kind="statement">
# Angles in the same segment are equal

## Statement

Let $A$, $B$, $C$ and $D$ be distinct [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$. Let $\measuredangle$ be the [[oriented angle|03YN70NK0Z]], with values in $\mathbb{R} / 2\pi\mathbb{Z}$.
Then
$$2 \, \measuredangle ACB = 2 \, \measuredangle ADB.$$
For the unoriented angles: if $C$ and $D$ are on the same side of the line $AB$, then $\angle ACB = \angle ADB$. If they are on opposite sides, then $\angle ACB + \angle ADB = \pi$.

## Remarks

The statement follows from the [[inscribed angle theorem|0GA1V8NN8C]]: both sides are equal to the oriented angle $\measuredangle AOB$ at the centre $O$.
The factor $2$ is necessary. It removes the difference of $\pi$ between points on the two arcs.
The case of opposite sides says that the opposite angles of a convex cyclic quadrilateral add up to $\pi$.

## Explanations

### L1 human

A segment of a circle is the region between a chord $AB$ and one of its arcs. From every point of that arc, the chord $AB$ is seen under the same angle. From the other arc, it is seen under the supplementary angle.
</document>

<document source="https://math2.tamnd.com/node/0GS27TKF1V.md" tag="0GS27TKF1V" kind="definition">
# Completely multiplicative function

## Definition

An [[arithmetic function|0H2EPRZC0D]] $f$ is completely multiplicative if $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.

## Remarks

Each completely multiplicative function is [[multiplicative|0AP80TT660]]. The converse fails: [[Euler's totient function|0VV04M41AN]] has $\varphi(4) = 2$ but $\varphi(2)^2 = 1$.
A completely multiplicative function is determined by its values at the primes.
Examples are $n \mapsto n^k$, the [[Liouville function|08RQAPXF1T]] and, for a fixed odd prime $p$, the [[Legendre symbol|01J8AMV9RS]] $n \mapsto \left(\frac{n}{p}\right)$.

## Explanations

### L1 human

A completely multiplicative function respects every product, not only products of coprime numbers.
So it is enough to know its values at the primes, as for the function $n \mapsto n^2$.
</document>

<document source="https://math2.tamnd.com/node/0GS74FGFEK.md" tag="0GS74FGFEK" kind="definition">
# Degree of a field extension

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
The field $L$ is a vector space over $K$, with the addition of $L$ and the multiplication $K \times L \to L$ of $L$.
The degree of $L/K$ is the dimension of this vector space, and it is written $[L : K]$.

## Remarks

The degree is a positive integer or an infinite cardinal.
One has $[L : K] = 1$ if and only if $L = K$.
Examples: $[\mathbb{C} : \mathbb{R}] = 2$, with the basis $1, i$. Also, $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$, with the basis $1, \sqrt{2}$.
The field $\mathbb{R}$ has infinite degree over $\mathbb{Q}$, because $\mathbb{Q}$ is countable and $\mathbb{R}$ is not.

## Explanations

### L1 human

The degree counts how many elements of the big field you need, as a basis, to write every element with coefficients in the small field.
Every complex number is $a + b i$ with $a, b$ real, so the degree of $\mathbb{C}$ over $\mathbb{R}$ is $2$.
</document>

<document source="https://math2.tamnd.com/node/0GSDHYMA1T.md" tag="0GSDHYMA1T" kind="statement">
# Thales's theorem

## Statement

Let $A$, $B$ and $C$ be distinct points of a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$. Suppose that $AC$ is a diameter, so that the centre is the [[midpoint|0S706JBZBG]] of $A$ and $C$.
Then the [[angle|0GJAJC6TNZ]] $\angle ABC$ is a right angle: $\angle ABC = \pi / 2$.
Conversely, if $A$, $B$, $C$ are distinct points with $\angle ABC = \pi / 2$, then $B$ is on the circle with the diameter $AC$.

## Remarks

Thales's theorem is the special case of the inscribed angle theorem for a central angle of $\pi$.
An equivalent form: the circumcentre of a right triangle is the midpoint of its hypotenuse.
A proof: let $O$ be the centre. The triangles $OAB$ and $OBC$ are isosceles, and the angle sum of $ABC$ is $\pi$.

## History

The theorem is part of Proposition 31 of Book III of Euclid's Elements. Tradition gives it to Thales of Miletus, and sometimes to Pythagoras.

## Explanations

### L1 human

Take a diameter of a circle and a third point on the circle. Join this point to the two ends of the diameter. The angle at the third point is always a right angle.
</document>

<document source="https://math2.tamnd.com/node/0GSW76Q14N.md" tag="0GSW76Q14N" kind="concept">
# Ideal generated by a subset

## Summary

The ideal generated by a subset of a ring is the smallest ideal that contains the subset.
In a commutative ring, it is the set of the finite linear combinations of elements of the subset.
The [[definition of the generated ideal|030TZ0DQWT]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0GSZ7B0CGR.md" tag="0GSZ7B0CGR" kind="object">
# Extended real number line

## Description

The extended real number line is the set $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$.
It has two new elements, $-\infty$ and $+\infty$, that are not [[real numbers|08J2M1JYAG]].
Its order extends the order of $\mathbb{R}$, with $-\infty < x < +\infty$ for each real number $x$.

## Remarks

In $\overline{\mathbb{R}}$, each subset has a [[supremum|0BTMQ04556]] and an infimum. For example, $\sup \mathbb{N} = +\infty$ and $\sup \emptyset = -\infty$.
Some operations extend: $x + \infty = +\infty$ for each real $x$. Other expressions, such as $\infty - \infty$, stay undefined.
The set $\overline{\mathbb{R}}$ is not a field.

## Explanations

### L1 human

The extended real line adds two end points to the number line, one at each end.
This lets you write $\lim_{x \to \infty}$ and $\sup$ of unbounded sets with symbols instead of words.
</document>

<document source="https://math2.tamnd.com/node/0GT4WWGWH2.md" tag="0GT4WWGWH2" kind="concept">
# System of linear equations

## Summary

A system of linear equations is a finite collection of linear equations in the same unknowns.
A solution of the system is a solution of all the equations at the same time.
</document>

<document source="https://math2.tamnd.com/node/0GT85FBPTS.md" tag="0GT85FBPTS" kind="example">
# Decimal expansion as a series

## Example

Let $(d_n)_{n \ge 1}$ be a sequence of digits, with $d_n \in \{0, 1, \dots, 9\}$.
The decimal $0.d_1 d_2 d_3 \dots$ is the sum of the series
$$\sum_{n=1}^{\infty} \frac{d_n}{10^n}.$$
This series converges, and its sum lies in $[0, 1]$.

## Remarks

The series converges by the [[comparison test|04Z5G0THN6]] with the [[geometric series|01X5CVV1Q5]] $\sum 9/10^n = 1$.
Two different digit sequences can give the same number. For example, $0.4999\ldots = 0.5$.
Conversely, each real number in $[0, 1)$ has a decimal expansion.

## Explanations

### L1 human

A decimal like $0.314$ means $3/10 + 1/100 + 4/1000$.
An infinite decimal is the same idea with infinitely many terms, so it is a series.
</document>

<document source="https://math2.tamnd.com/node/0GTBGMZXV8.md" tag="0GTBGMZXV8" kind="definition">
# Triple integral

## Definition

Let $B = [a_1, b_1] \times [a_2, b_2] \times [a_3, b_3]$ with $a_i < b_i$, and let $f \colon B \to \mathbb{R}$ be bounded.
Partition each interval $[a_i, b_i]$, and let $B_1, \dots, B_N$ be the resulting closed sub-boxes, with volumes $\Delta V_1, \dots, \Delta V_N$.
For points $p_k \in B_k$, the sum $S = \sum_{k=1}^{N} f(p_k) \, \Delta V_k$ is a Riemann sum of $f$.
The function $f$ is Riemann integrable on $B$ if there is $I \in \mathbb{R}$ with this property. For each $\varepsilon > 0$, there is $\delta > 0$ with $|S - I| < \varepsilon$ for all Riemann sums whose sub-boxes have diameter less than $\delta$.
Then $I$ is the triple integral of $f$ over $B$, written $\iiint_B f(x, y, z) \, dV$.

## Remarks

This is the case $n = 3$ of the multiple integral over a box. The case $n = 2$ is the [[double integral|0NKBMFC18R]].
For a bounded solid $E \subseteq B$, the integral over $E$ is the integral over $B$ of $f$ times the indicator function of $E$. The volume of $E$ is $\iiint_E 1 \, dV$, when this integral exists.
By [[Fubini's theorem|0R0T1VWH5X]], for continuous $f$ the triple integral equals an iterated integral. For example, $\iiint_{[0,1]^3} x y z \, dV = (1/2)^3 = 1/8$.

## Explanations

### L1 human

A triple integral adds up the values of $f$ over a solid, each weighted by the volume of a small piece.
If $f$ is a density, the triple integral is the total mass of the solid.
</document>

<document source="https://math2.tamnd.com/node/0GTEHJQRFK.md" tag="0GTEHJQRFK" kind="definition">
# Gaussian prime

## Definition

A Gaussian prime is a prime element of the ring $\mathbb{Z}[i]$ of [[Gaussian integers|0B268GVX95]].
That is, it is a nonzero Gaussian integer $\pi$ that is not a unit, with the following property.
For all Gaussian integers $\alpha$ and $\beta$, if $\pi \mid \alpha \beta$, then $\pi \mid \alpha$ or $\pi \mid \beta$.

## Remarks

Since $\mathbb{Z}[i]$ is a [[unique factorization domain|0SPYEFH7H4]], the Gaussian primes are also the irreducible elements: the non-units that are not a product of two non-units.
A Gaussian integer $\alpha$ is a Gaussian prime if and only if one of two conditions holds.
Either $N(\alpha)$ is a prime number, or $\alpha$ is a unit times a prime $p \equiv 3 \pmod 4$.
For example, $1 + i$, $2 + i$ and $3$ are Gaussian primes. But $5 = (2 + i)(2 - i)$ is not.

## Explanations

### L1 human

Gaussian primes play the role of prime numbers in the Gaussian integers.
Some ordinary primes, such as $5$, split into two Gaussian primes. Others, such as $3$, stay prime.
</document>

<document source="https://math2.tamnd.com/node/0GTZ2F3PQY.md" tag="0GTZ2F3PQY" kind="statement">
# Cavalieri's principle

## Statement

Let $E \subseteq \mathbb{R}^3$ be a Jordan measurable solid that lies between the planes $z = a$ and $z = b$.
For $z \in [a, b]$, let $E_z = \{(x, y) : (x, y, z) \in E\}$ be its cross-section at height $z$, and suppose that each $E_z$ is Jordan measurable with area $A(z)$.
Then
$$\operatorname{vol}(E) = \int_a^b A(z) \, dz.$$
In particular, if two such solids have cross-sections of equal area at each height $z$, then they have equal volume.

## Remarks

This follows from [[Fubini's theorem|0R0T1VWH5X]] applied to the indicator function of $E$.
Example: the ball of radius $r$ has cross-sections of area $\pi (r^2 - z^2)$. A cylinder of radius $r$ and height $r$ with a cone removed has the same cross-sections, which gives the volume $\tfrac{2}{3} \pi r^3$ for a half ball.
The plane version says that two regions with equal chord lengths on each line parallel to a fixed line have equal area.
Archimedes used a similar method in the 3rd century BC. Cavalieri developed it as the method of indivisibles in 1635.

## Explanations

### L1 human

Cut two solids into thin horizontal slices.
Suppose that each slice of one has the same area as the slice of the other at the same height. Then the two solids have the same volume, like two stacks of coins of the same size.
</document>

<document source="https://math2.tamnd.com/node/0GV2573FP4.md" tag="0GV2573FP4" kind="statement">
# Bijection principle

## Statement

Let $A$ and $B$ be finite sets.
If there is a bijection $f : A \to B$, then $|A| = |B|$.
Conversely, if $|A| = |B|$, then there is a bijection from $A$ to $B$.

## Remarks

A bijective proof shows that two sets have the same size by a bijection between them. The proof of the [[symmetry of binomial coefficients|0R7JS3DP3W]] is a standard example. It uses the map $S \mapsto X \setminus S$ on the subsets of a finite set $X$.
[[Dyck words|0R9JR0465V]] and [[binary trees|02A8EMC83C]] are both counted by [[Catalan numbers|092DRBFKMG]], and a bijection explains this.

## Explanations

### L1 human

To show that two collections have the same size, you can pair each element of one with exactly one element of the other.
For example, the subsets of $\{1, 2, 3\}$ pair with the words of length 3 in the letters 0 and 1. A subset gives the word with 1 in the positions of its elements.
</document>

<document source="https://math2.tamnd.com/node/0GVDFAY4KW.md" tag="0GVDFAY4KW" kind="concept">
# Disjunctive normal form

## Summary

A formula in disjunctive normal form is a disjunction of conjunctions of literals. Every Boolean function has such a form, which can be read off from its truth table. The node [[disjunctive normal form|06S4WGSEQA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0GVMR0C3S6.md" tag="0GVMR0C3S6" kind="definition">
# Moment of inertia

## Definition

Let $E \subseteq \mathbb{R}^3$ be a bounded Jordan measurable set with a continuous density $\delta \colon E \to [0, \infty)$, and let $L$ be a line in $\mathbb{R}^3$.
For $x \in E$, let $d_L(x)$ be the distance from $x$ to $L$.
The moment of inertia of $E$ about $L$ is
$$I_L = \iiint_E d_L(x)^2 \, \delta(x) \, dV.$$
In particular, $I_z = \iiint_E (x^2 + y^2) \, \delta(x, y, z) \, dV$ is the moment of inertia about the $z$-axis.

## Remarks

For a plane lamina $D \subseteq \mathbb{R}^2$, the moments of inertia about the coordinate axes are $I_x = \iint_D y^2 \delta \, dA$ and $I_y = \iint_D x^2 \delta \, dA$. These are [[double integrals|0NKBMFC18R]].
For a point mass $m$ at distance $r$ from the axis, the moment of inertia is $m r^2$.
Example: a solid cylinder of radius $a$, height $h$ and constant density has mass $M = \delta \pi a^2 h$ and $I_z = \delta \int_0^h \int_0^{2\pi} \int_0^a r^3 \, dr \, d\theta \, dz = \tfrac{1}{2} M a^2$.
For rotation about $L$ with angular velocity $\omega$, the kinetic energy is $\tfrac{1}{2} I_L \omega^2$.

## Explanations

### L1 human

Mass far from the axis is harder to set spinning than mass close to it.
The moment of inertia adds up all the mass, weighted by the square of its distance from the axis.
</document>

<document source="https://math2.tamnd.com/node/0GW1ZZH0R4.md" tag="0GW1ZZH0R4" kind="statement">
# Fundamental theorem of symmetric polynomials

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f \in R[x_1, \dots, x_n]$ be [[symmetric|06CTFY2DEZ]].
Then there is a unique polynomial $g \in R[y_1, \dots, y_n]$ with
$$f = g(e_1, \dots, e_n),$$
where $e_1, \dots, e_n$ are the [[elementary symmetric polynomials|0ZMY8441GW]].

## Remarks

Equivalently, the map $R[y_1, \dots, y_n] \to R[x_1, \dots, x_n]$, $y_k \mapsto e_k$, is injective, and its image is the ring of symmetric polynomials.
One proof is by induction on the leading monomial in the lexicographic order. Each step subtracts a product of the $e_k$ with the same leading monomial.
Example: $x_1^2 + x_2^2 = e_1^2 - 2 e_2$.
So a symmetric expression in the roots of a polynomial over a field $K$ is an element of $K$.

## Explanations

### L1 human

Every symmetric polynomial can be written in exactly one way using the elementary symmetric polynomials.
The coefficients stay in the same ring.
</document>

<document source="https://math2.tamnd.com/node/0GWD21EWRW.md" tag="0GWD21EWRW" kind="definition">
# Hyperbolic sine

## Definition

The hyperbolic sine is the function $\sinh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\sinh x = \frac{e^x - e^{-x}}{2}.$$

## Remarks

It is odd and strictly increasing, and it is a bijection of $\mathbb{R}$ onto $\mathbb{R}$.
Its power series is $\sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$, the sine series without the signs.
Mathlib defines it as `Real.sinh`.

## Explanations

### L1 human

The hyperbolic sine is the odd part of $e^x$.
For large $x$, it is close to $e^x / 2$.
</document>

<document source="https://math2.tamnd.com/node/0GWN3CMRK8.md" tag="0GWN3CMRK8" kind="concept">
# One-point compactification

## Summary

The one-point compactification of a topological space adds one point at infinity and makes the space compact.
For example, the plane with one added point is the sphere.
</document>

<document source="https://math2.tamnd.com/node/0GXA8Z205E.md" tag="0GXA8Z205E" kind="concept">
# Area under a curve

## Summary

For a nonnegative continuous function $f$ on $[a, b]$, the area under its graph is the [[integral|0QM4TXT3DR]] $\int_a^b f(x) \, dx$. The integral approximates the region by thin rectangles, through [[Riemann sums|0HBY48G8G1]]. For a function of any sign, the integral counts area below the $x$-axis as negative.

## Remarks

The area between the graphs of $f \le g$ over $[a, b]$ is $\int_a^b (g(x) - f(x)) \, dx$.
For example, the area under $x^2$ over $[0, 1]$ is $1/3$. In the 3rd century BC, Archimedes found the area under a parabola by the method of exhaustion.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes areas from antiderivatives.

## Explanations

### L1 human

The integral measures the area of the region under a graph.
We fill the region with thin rectangles and add up their areas.
</document>

<document source="https://math2.tamnd.com/node/0GXCXCX3QN.md" tag="0GXCXCX3QN" kind="statement">
# Integral of sine

## Statement

For all real numbers $a$ and $b$,
$$\int_a^b \sin x \, dx = \cos a - \cos b.$$

## Remarks

The [[derivative of cosine|0V2DV29EGT]] is $-\sin$, so $-\cos$ is an antiderivative of $\sin$.
For example, $\int_0^\pi \sin x \, dx = 2$.
Mathlib states it as `integral_sin`.

## Explanations

### L1 human

The area under one arch of the sine curve is exactly $2$.
This follows from the antiderivative $-\cos x$.
</document>

<document source="https://math2.tamnd.com/node/0GXKM1R7E4.md" tag="0GXKM1R7E4" kind="concept">
# Telescoping series

## Summary

A telescoping series is a series whose terms have the form $a_n = b_n - b_{n+1}$ for a sequence $(b_n)$. Its partial sums collapse: $\sum_{n=0}^{N} (b_n - b_{n+1}) = b_0 - b_{N+1}$. Thus the series converges if and only if $(b_n)$ converges, and then its sum is $b_0 - \lim b_n$.

## Remarks

For example, $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$, so $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1$.
The finite form $\sum_{k=0}^{n-1} (f(k+1) - f(k)) = f(n) - f(0)$ is a discrete analogue of the [[fundamental theorem of calculus|0NHGY0N0JB]]. Mathlib states it as `Finset.sum_range_sub`.

## Explanations

### L1 human

In a telescoping sum, each term cancels part of the next term, like the segments of a telescope that slide into each other.
Only the first piece and the last piece survive.
</document>

<document source="https://math2.tamnd.com/node/0GXMPDNYX3.md" tag="0GXMPDNYX3" kind="concept">
# Laurent series

## Summary

A Laurent series is a series $\sum_{n \in \mathbb{Z}} a_n (z - c)^n$ with powers of $z - c$ of all integer degrees, see the [[definition|0X3XD2ATB0]].
It represents holomorphic functions on annuli, also near isolated singularities where a Taylor series does not exist.
</document>

<document source="https://math2.tamnd.com/node/0GXNHEHQNA.md" tag="0GXNHEHQNA" kind="statement">
# Units of Z/nZ are the residues coprime to n

## Statement

Let $n \geq 1$ and $a$ be integers.
The residue class $[a]$ is a unit of the ring $\mathbb{Z}/n\mathbb{Z}$ if and only if $\gcd(a, n) = 1$.

## Remarks

If $\gcd(a, n) = 1$, then Bézout's lemma gives integers $x$ and $y$ with $a x + n y = 1$. So $[a][x] = [1]$.
If $[a][x] = [1]$, then $a x - 1$ is a multiple of $n$, so each common divisor of $a$ and $n$ divides $1$.
The condition does not depend on the representative, because $a \equiv b \pmod{n}$ implies $\gcd(a, n) = \gcd(b, n)$.
For $n = 12$, the units are $[1]$, $[5]$, $[7]$ and $[11]$.

## Explanations

### L1 human

A residue has a multiplicative inverse modulo $n$ exactly when it has no common factor with $n$.
Bézout's lemma gives the inverse.
So the [[group of units of Z/nZ|0F8X8ZRQ53]] is the set of residues coprime to $n$.
</document>

<document source="https://math2.tamnd.com/node/0GYDV9FE98.md" tag="0GYDV9FE98" kind="statement">
# Second derivative test in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$, and let $a \in U$ with $\nabla f(a) = 0$.
Let $H = H_f(a)$ be the [[Hessian matrix|0SCSQWXRA4]] of $f$ at $a$.

1. If $H$ is [[positive definite|01RBSSGMY5]], then $f$ has a strict [[local minimum|04CPG9K1C4]] at $a$.
2. If $H$ is negative definite, then $f$ has a strict local maximum at $a$.
3. If $H$ has a positive and a negative eigenvalue, then $a$ is a [[saddle point|00N3T18QNA]] of $f$.

In the other cases the test gives no conclusion.

## Remarks

For $n = 2$, let $D = f_{xx}(a) f_{yy}(a) - f_{xy}(a)^2$ be the determinant of $H$. If $D > 0$ and $f_{xx}(a) > 0$, then $a$ is a local minimum. If $D > 0$ and $f_{xx}(a) < 0$, then $a$ is a local maximum. If $D < 0$, then $a$ is a saddle point.
The proof uses the [[second-order Taylor expansion|0P30YYZN85]]. If $H$ is positive definite, then $h^{\mathsf{T}} H h \ge m \|h\|^2$ for some $m > 0$, and this term dominates the remainder.
The functions $x^4 + y^4$, $-(x^4 + y^4)$ and $x^4 - y^4$ all have zero Hessian at the origin. The origin is a minimum, a maximum and a saddle point, respectively, so the test cannot decide when $H$ is singular and semidefinite.
By Sylvester's criterion, $H$ is positive definite if and only if all its leading principal minors are positive.

## Explanations

### L1 human

At a critical point the tangent plane is flat, so the second derivatives decide the shape of the graph.
If the graph bends upward in every direction, the point is a minimum. If it bends down in every direction, the point is a maximum.
If it bends up in some directions and down in others, the point is a saddle.
</document>

<document source="https://math2.tamnd.com/node/0GYGHKX834.md" tag="0GYGHKX834" kind="definition">
# Commutative ring

## Definition

A commutative ring is a [[ring|0894J980RK]] $R$ such that $a \cdot b = b \cdot a$ for all $a, b \in R$.

## Remarks

The integers $\mathbb{Z}$, the rational numbers $\mathbb{Q}$ and the polynomials in one variable over $\mathbb{Q}$ are commutative rings.
The ring of $2 \times 2$ real matrices is not commutative.
Every [[field|0QA8CHH8MG]] is a commutative ring.

## Explanations

### L1 human

In a commutative ring, the order of the factors in a product does not matter.
The integers are the model: $3 \cdot 5 = 5 \cdot 3$.
</document>

<document source="https://math2.tamnd.com/node/0GYQK9NDJF.md" tag="0GYQK9NDJF" kind="concept">
# Cayley graph

## Summary

The Cayley graph of a group $G$ with a generating set $S$ has the elements of $G$ as vertices. It has an edge from $g$ to $g s$ for each $g \in G$ and $s \in S$.
It encodes the structure of the group as a graph.
</document>

<document source="https://math2.tamnd.com/node/0GZ3D8F921.md" tag="0GZ3D8F921" kind="statement">
# Divergence theorem on a rectangle in the plane

## Statement

Let $a < b$ and $c < d$, and let $R = [a, b] \times [c, d]$.
Let $F = (P, Q)$ be a vector field of class $C^1$ on an open set that contains $R$.
Then
$$\iint_R \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) dA = \int_c^d \big( P(b, y) - P(a, y) \big) \, dy + \int_a^b \big( Q(x, d) - Q(x, c) \big) \, dx.$$

## Remarks

The right side is the outward flux $\oint_{\partial R} F \cdot n \, ds$. Here the outward unit normal $n$ is $(1, 0)$, $(-1, 0)$, $(0, 1)$ and $(0, -1)$ on the right, left, top and bottom sides.
Proof: by [[Fubini's theorem|0R0T1VWH5X]], $\iint_R \partial P / \partial x \, dA = \int_c^d \int_a^b \partial P / \partial x \, dx \, dy$. The fundamental theorem of calculus turns the inner integral into $P(b, y) - P(a, y)$. The $Q$ term is similar.
This is the simplest case of the [[divergence theorem|0JX86PSSA9]]. In the plane, the divergence theorem is equivalent to [[Green's theorem|0HR3N02317]].

## Explanations

### L1 human

The total source strength inside a rectangle equals the net flow out through its four sides.
In each coordinate direction, this is the fundamental theorem of calculus, applied along each line through the rectangle.
</document>

<document source="https://math2.tamnd.com/node/0GZ6Y1H4RN.md" tag="0GZ6Y1H4RN" kind="concept">
# Ascending chain condition

## Summary

A partially ordered set satisfies the ascending chain condition when every increasing sequence of its elements is eventually constant.
The [[definition of the ascending chain condition|0ZCAXY9563]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/0GZB2SN0PH.md" tag="0GZB2SN0PH" kind="example">
# Function space

## Example

Let $F$ be a field, let $X$ be a set, and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
The set $V^X$ of all functions $f \colon X \to V$ is a vector space over $F$ with the pointwise operations
$$(f + g)(x) = f(x) + g(x), \qquad (af)(x) = a\, f(x)$$
for $f, g \in V^X$, $a \in F$ and $x \in X$.
The zero vector is the function with value $0$ at each point.

## Remarks

For $X = \{1, \dots, n\}$ and $V = F$, this is the [[coordinate space|0XM8BZD76G]] $F^n$.
The continuous functions $[0, 1] \to \R$ form a subspace of $\R^{[0, 1]}$.
If $X$ is infinite and $V \ne 0$, then $V^X$ is infinite-dimensional.

## Explanations

### L1 human

We add two functions by adding their values at each point.
We scale a function by scaling its values.
With these rules, functions behave like vectors, and the zero function is the zero vector.
</document>

<document source="https://math2.tamnd.com/node/0GZNSHMG4Q.md" tag="0GZNSHMG4Q" kind="example">
# Dirichlet integral

## Example

The improper integral of $\frac{\sin x}{x}$ over $(0, \infty)$ converges, and
$$\int_0^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}.$$
The integrand extends continuously to $x = 0$ with value $1$. So only the upper limit is improper.
[[Integration by parts|08Y1BBQTK9]] gives $\int_1^b \frac{\sin x}{x} \, dx = \cos 1 - \frac{\cos b}{b} - \int_1^b \frac{\cos x}{x^2} \, dx$. The last integral converges absolutely, so the limit exists.

## Remarks

The integral does not converge [[absolutely|03GP7KMBYM]]: $\int_0^\infty \frac{|\sin x|}{x} \, dx = \infty$.
So $\frac{\sin x}{x}$ is not Lebesgue integrable on $(0, \infty)$.
The value $\pi/2$ needs other methods, such as differentiation under the integral sign or contour integration.

## Explanations

### L1 human

The area under $\frac{\sin x}{x}$ alternates in sign, and the pieces shrink.
The positive and negative pieces cancel to give the value $\pi/2$.
</document>

<document source="https://math2.tamnd.com/node/0GZWXM2DKE.md" tag="0GZWXM2DKE" kind="concept">
# Characteristic equation

## Summary

The characteristic equation of a homogeneous linear differential equation with constant coefficients is the polynomial equation obtained from the trial solution $e^{r x}$.
Its roots determine the solutions of the differential equation. See the [[definition|0QDXMG46QW]].
</document>

<document source="https://math2.tamnd.com/node/0H11ZPX4ND.md" tag="0H11ZPX4ND" kind="concept">
# Conjugate subgroup

## Summary

A conjugate of a subgroup $H$ of a group $G$ is a subgroup $gHg^{-1}$ for some $g \in G$.
Conjugate subgroups are isomorphic.
</document>

<document source="https://math2.tamnd.com/node/0H1GN6P9R5.md" tag="0H1GN6P9R5" kind="concept">
# Fundamental matrix

## Summary

A fundamental matrix of a homogeneous linear system $x' = A(t) x$ is a matrix function whose columns are $n$ linearly independent solutions.
It is the matrix form of a [[fundamental system of solutions|0DS2QFYJHE]]. See the [[definition|0MQ24RBG93]].
</document>

<document source="https://math2.tamnd.com/node/0H1JFKE7H4.md" tag="0H1JFKE7H4" kind="concept">
# Hyperbolic sine

## Summary

The hyperbolic sine is half the difference of $e^x$ and $e^{-x}$. It is an odd, strictly increasing function on $\mathbb{R}$. The node [[hyperbolic sine|0GWD21EWRW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0H2EPRZC0D.md" tag="0H2EPRZC0D" kind="definition">
# Arithmetic function

## Definition

An arithmetic function is a function $f \colon \mathbb{Z}_{>0} \to \mathbb{C}$ from the positive integers to the complex numbers.

## Remarks

Examples are [[Euler's totient function|0VV04M41AN]] $\varphi$, the [[divisor function|0YEW47VJ0J]] $\sigma_k$ and the [[Möbius function|0C8QENHYZW]] $\mu$.
Some texts allow values in another commutative ring, or set $f(0) = 0$ and use all natural numbers as the domain.
The [[Dirichlet convolution|0KH1N66QWW]] makes the arithmetic functions into a commutative ring.

## Explanations

### L1 human

An arithmetic function is a sequence $f(1), f(2), f(3), \dots$ of numbers, seen as a function of $n$.
Number theory studies such functions when they describe arithmetic properties of $n$, such as its number of divisors.
</document>

<document source="https://math2.tamnd.com/node/0H3W614GEC.md" tag="0H3W614GEC" kind="definition">
# Pseudometric space

## Definition

Let $X$ be a set.
A pseudometric on $X$ is a function $d \colon X \times X \to \mathbb{R}$ such that, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. $d(x, y) = d(y, x)$.
3. $d(x, z) \le d(x, y) + d(y, z)$.

A pseudometric space is a pair $(X, d)$, where $X$ is a set and $d$ is a pseudometric on $X$.

## Remarks

Each [[metric|063W5JMY85]] is a pseudometric. A pseudometric is a metric if and only if $d(x, y) = 0$ implies $x = y$.
On the set of functions $f \colon [0, 1] \to \mathbb{R}$, the rule $d(f, g) = |f(0) - g(0)|$ is a pseudometric, but not a metric.
The relation $d(x, y) = 0$ is an equivalence relation, and $d$ gives a metric on the set of its classes.

## Explanations

### L1 human

A pseudometric measures distance, but it can fail to see the difference between two points.
For example, measure the distance between two people by the difference of their ages. Then two people of the same age have distance zero.
</document>

<document source="https://math2.tamnd.com/node/0H3WZXCZNF.md" tag="0H3WZXCZNF" kind="concept">
# Semidirect product

## Summary

A semidirect product builds a group from a group $N$ and a group $H$ that acts on $N$ by automorphisms.
It generalises the direct product, and $N$ becomes a normal subgroup of the result.
</document>

<document source="https://math2.tamnd.com/node/0H3YY8CPKH.md" tag="0H3YY8CPKH" kind="object">
# Euclidean group

## Description

The Euclidean group $\mathrm{E}(n)$ is the set of all [[Euclidean isometries|0J1N3WTKM4]] of $\mathbb{R}^n$, with the composition of maps as the operation. It is a [[group|G5S1M9]]: the identity map is the neutral element, and the inverse of an isometry is an isometry.
Each element has the form $x \mapsto A x + b$, with $A$ in the orthogonal group $\mathrm{O}(n)$ and $b \in \mathbb{R}^n$.
The direct isometries, with $\det A = 1$, form a subgroup $\mathrm{SE}(n)$ of index $2$. Its elements are the rigid motions.

## Remarks

The [[translations|087VJM5Q77]] form an abelian normal subgroup $T \cong \mathbb{R}^n$, and $\mathrm{E}(n) / T \cong \mathrm{O}(n)$.
The group $\mathrm{E}(n)$ has $n(n+1)/2$ degrees of freedom: $3$ for $n = 2$ and $6$ for $n = 3$.
The group is not abelian for $n \geq 2$. For example, a rotation about the origin and a translation do not commute in general.
The symmetry group of each figure in $\mathbb{R}^n$ is a subgroup of $\mathrm{E}(n)$.

## Explanations

### L1 human

The Euclidean group collects all the ways to move space without changing distances. Two such moves done one after the other give again such a move, and each move can be undone. In the plane, three numbers fix a direct move: two for a shift and one for a turn.
</document>

<document source="https://math2.tamnd.com/node/0H47DMFN3S.md" tag="0H47DMFN3S" kind="definition">
# Inner automorphism

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
The inner automorphism given by $g$ is the map $c_g \colon G \to G$, $c_g(x) = gxg^{-1}$.
The set of all inner automorphisms of $G$ is written $\operatorname{Inn}(G)$.

## Remarks

Each $c_g$ is an [[automorphism|0K1SCPBNKJ]] of $G$, with inverse $c_{g^{-1}}$.
Since $c_g \circ c_h = c_{gh}$, the map $g \mapsto c_g$ is a homomorphism from $G$ to $\operatorname{Aut}(G)$, and $\operatorname{Inn}(G)$ is its image.
The subgroup $\operatorname{Inn}(G)$ is normal in $\operatorname{Aut}(G)$, because $\alpha \circ c_g \circ \alpha^{-1} = c_{\alpha(g)}$ for each automorphism $\alpha$.
In an [[abelian group|082GHNPJ53]], each inner automorphism is the identity map.

## Explanations

### L1 human

Each element of a group gives a symmetry of the group: conjugate everything by this element.
These symmetries are called inner, because they come from inside the group.
In a group where all elements commute, conjugation changes nothing.
</document>

<document source="https://math2.tamnd.com/node/0H4KNNCHPY.md" tag="0H4KNNCHPY" kind="definition">
# Fundamental system of solutions

## Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous. Let $S$ be the vector space of solutions on $I$ of the [[homogeneous linear equation|0EQZSAB9EC]] $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
A fundamental system of solutions of the equation is a basis $(y_1, \dots, y_n)$ of $S$.

## Remarks

Since $S$ has [[dimension n|0GK68PZ6KH]], a family of $n$ solutions is a fundamental system if and only if it is linearly independent. The [[Wronskian test|0MHE4B1QCQ]] checks this.
Each solution is then $c_1 y_1 + \dots + c_n y_n$ for unique constants $c_1, \dots, c_n$.
Example: $(\cos x, \sin x)$ is a fundamental system of $y'' + y = 0$ on $\mathbb{R}$.

## Explanations

### L1 human

A fundamental system is a set of building blocks.
Each solution of the homogeneous equation is a unique combination of these $n$ solutions.
</document>

<document source="https://math2.tamnd.com/node/0H4R3RESGV.md" tag="0H4R3RESGV" kind="definition">
# Limit inferior

## Definition

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
For each $n \in \mathbb{N}$, let $i_n = \inf \{a_k : k \ge n\}$.
The sequence $(i_n)$ is increasing and bounded above.
The limit inferior of $(a_n)$ is
$$\liminf_{n \to \infty} a_n = \lim_{n \to \infty} i_n = \sup_{n \in \mathbb{N}} \inf_{k \ge n} a_k.$$

## Remarks

The limit exists by the [[monotone convergence theorem|0FM6N4R0VB]].
It satisfies $\liminf a_n = -\limsup (-a_n)$, where $\limsup$ is the [[limit superior|0P8BA7T7S5]].
Always $\inf_n a_n \le \liminf a_n \le \limsup a_n \le \sup_n a_n$.
For $a_n = (-1)^n$, the limit inferior is $-1$ and the limit superior is $1$.

## Explanations

### L1 human

The limit inferior is the lowest value that the sequence keeps coming back to, in the long run.
Each number below it is undercut by only finitely many terms.
</document>

<document source="https://math2.tamnd.com/node/0H5GKDTB01.md" tag="0H5GKDTB01" kind="definition">
# Disjunction

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The disjunction of $p$ and $q$ is the proposition $p \lor q$, read "$p$ or $q$".
The proposition $p \lor q$ is false when $p$ is false and $q$ is false. In all other cases, $p \lor q$ is true.

| $p$ | $q$ | $p \lor q$ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

## Remarks

This "or" is inclusive: $p \lor q$ is true when both $p$ and $q$ are true.
The exclusive "or" is true when exactly one of $p$ and $q$ is true.
The [[union|07KVB4JSC7]] of two sets uses disjunction: $x \in A \cup B$ means $x \in A \lor x \in B$.

## Explanations

### L1 human

The sentence "$4$ is even or $4$ is odd" is a disjunction.
It is true, because one part is true.
In mathematics, "or" also allows both parts to be true.
</document>

<document source="https://math2.tamnd.com/node/0H5J02P8S9.md" tag="0H5J02P8S9" kind="definition">
# Upper bound

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $u \in P$ is an upper bound of $S$ when $s \le u$ for each $s \in S$.
The set $S$ is bounded above when it has an upper bound.

## Remarks

An upper bound need not be in $S$. The number $1$ is an upper bound of $[0, 1)$ in $\mathbb{R}$, and so is $5$.
A lower bound is defined in the same way with $u \le s$.
The least upper bound of $S$, when it exists, is the supremum of $S$.

## Explanations

### L1 human

An upper bound is a ceiling for the set: no element of the set is above it.
The set $\{1, 1/2, 1/3, \dots\}$ has upper bounds $1$, $2$, $100$ and many more.
</document>

<document source="https://math2.tamnd.com/node/0H5MR9B6GT.md" tag="0H5MR9B6GT" kind="concept">
# Line integral of a vector field

## Summary

The line integral of a [[vector field|0299VCW1TH]] along an oriented curve adds up the component of the field along the direction of motion.
In physics it is the work that a force field does on a particle that moves along the curve.
</document>

<document source="https://math2.tamnd.com/node/0H5SYBSED9.md" tag="0H5SYBSED9" kind="concept">
# Gamma function

## Summary

The gamma function extends the factorial to real arguments. For $s > 0$ it is an improper integral, and it satisfies $\Gamma(n + 1) = n!$ for natural numbers $n$. The node [[gamma function|0V3FZMVB4K]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0H5WWC8RZX.md" tag="0H5WWC8RZX" kind="statement">
# The order of a finite field is a prime power

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]].
Then the [[characteristic|0P7CH1CND5]] of $F$ is a prime number $p$, and $|F| = p^n$ with $n = [F : \mathbb{F}_p]$.

## Remarks

The proof: a finite field has nonzero characteristic, and [[the characteristic of a domain is $0$ or prime|00758QM6JJ]]. The [[prime subfield|0VPFBAY7MW]] is $\mathbb{F}_p$, and $F$ is a vector space of dimension $n$ over it, so it has $p^n$ elements.
So there is no field with $6$ or $10$ elements.
The converse, that a field of order $p^n$ exists for each $n \ge 1$, is the [[existence theorem|0T8M3YC5RG]].

## Explanations

### L1 human

A finite field is a vector space over the field with $p$ elements.
A vector space of dimension $n$ over it has $p^n$ elements.
So the size of a finite field is always a power of a prime.
</document>

<document source="https://math2.tamnd.com/node/0H5ZAZCX2T.md" tag="0H5ZAZCX2T" kind="statement">
# Taylor's theorem with the Lagrange remainder

## Statement

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.
Then there is $\xi$ strictly between $a$ and $x$ with
$$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{(k + 1)!} (x - a)^{k+1},$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

For $k = 0$ this is the [[mean value theorem|0BARQY52N5]].
If $|f^{(k+1)}| \le M$ between $a$ and $x$, then $|f(x) - P_k(x)| \le M |x - a|^{k+1} / (k + 1)!$.
Mathlib states it as `taylor_mean_remainder_lagrange`.

## Explanations

### L1 human

The error of the Taylor polynomial looks like the next term of the polynomial.
Only the derivative is taken at an unknown point $\xi$ instead of at $a$.
</document>

<document source="https://math2.tamnd.com/node/0H60H1Q0SX.md" tag="0H60H1Q0SX" kind="statement">
# Changing the order of integration

## Statement

Let $D \subseteq \mathbb{R}^2$ be a region that has two descriptions,
$$D = \{(x, y) : a \le x \le b, \ \varphi_1(x) \le y \le \varphi_2(x)\} = \{(x, y) : c \le y \le d, \ \psi_1(y) \le x \le \psi_2(y)\},$$
where $\varphi_1 \le \varphi_2$ are continuous on $[a, b]$ and $\psi_1 \le \psi_2$ are continuous on $[c, d]$.
Let $f \colon D \to \mathbb{R}$ be continuous.
Then
$$\int_a^b \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) \, dy \, dx = \int_c^d \int_{\psi_1(y)}^{\psi_2(y)} f(x, y) \, dx \, dy,$$
and both equal the double integral of $f$ over $D$.

## Remarks

The proof applies [[Fubini's theorem|0R0T1VWH5X]] on a rectangle that contains $D$ to $f$ times the indicator function of $D$.
The main work is to describe $D$ again in the other order. One way is to sketch the region.
Example: $\int_0^1 \int_x^1 e^{y^2} \, dy \, dx$ has no elementary inner antiderivative. The region is $0 \le x \le y \le 1$, so the integral equals $\int_0^1 \int_0^y e^{y^2} \, dx \, dy = \int_0^1 y e^{y^2} \, dy = (e - 1)/2$.
For unbounded regions or discontinuous integrands, a change of order needs extra hypotheses.

## Explanations

### L1 human

An iterated integral over a region can sometimes be hard in one order and easy in the other.
Changing the order means describing the same region by slicing it in the other direction, and then integrating in the new order.
</document>

<document source="https://math2.tamnd.com/node/0H628RSGWX.md" tag="0H628RSGWX" kind="concept">
# Algebraic element

## Summary

Let $L/K$ be a field extension.
An element of $L$ is algebraic over $K$ if it is a root of a nonzero polynomial with coefficients in $K$.
The [[definition of an algebraic element|0NPW1DWECM]] states it.
</document>

<document source="https://math2.tamnd.com/node/0H631CZP5K.md" tag="0H631CZP5K" kind="definition">
# Galton–Watson process

## Definition

Let $\mu$ be a probability distribution on $\{0, 1, 2, \dots\}$, the offspring distribution. Let $(\xi_{n, i})_{n \ge 0, \, i \ge 1}$ be [[i.i.d.|0AJN7WHBJZ]] random variables with law $\mu$.
The Galton–Watson process with offspring distribution $\mu$ is the [[stochastic process|0QH5ENKZM1]] $(Z_n)_{n \ge 0}$ with
$$Z_0 = 1 \quad \text{and} \quad Z_{n+1} = \sum_{i=1}^{Z_n} \xi_{n, i} \quad \text{for } n \ge 0.$$
Here $Z_n$ is the size of generation $n$, and an empty sum is $0$.

## Remarks

$(Z_n)$ is a [[Markov chain|033TJBZDG0]] on $\{0, 1, 2, \dots\}$, and $0$ is an absorbing state. Extinction is the event that $Z_n = 0$ for some $n$.
Let $m$ be the mean of $\mu$ and let $G$ be its [[probability-generating function|0Z8PSSAVBA]]. Then $E[Z_n] = m^n$, and the generating function of $Z_n$ is the $n$-fold composition $G \circ \dots \circ G$.
The extinction probability is the smallest solution $q \in [0, 1]$ of $G(q) = q$.
Assume $\mu(\{1\}) < 1$. Then $q = 1$ if $m \le 1$, and $q < 1$ if $m > 1$.



## Explanations

### L1 human

The Galton–Watson process follows a family tree in which each person has a random number of children.
If people have on average at most one child, the family dies out with probability $1$. If they have more, it can survive forever.
</document>

<document source="https://math2.tamnd.com/node/0H6DP8CHND.md" tag="0H6DP8CHND" kind="concept">
# Del operator

## Summary

The del operator $\nabla = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right)$ is a vector of partial derivative operators on $\mathbb{R}^n$.
It gives a short notation for the [[gradient|0ABK9V9Y64]] $\nabla f$, the [[divergence|0MWV6ZTK6S]] $\nabla \cdot F$, the [[curl|0NF6GQNXMY]] $\nabla \times F$ in $\mathbb{R}^3$, and the [[Laplacian|0E0B6RA8K0]] $\nabla^2 f = \nabla \cdot \nabla f$.

## Remarks

The notation treats $\nabla$ like a vector, but it is an operator. For example, $\nabla \cdot F$ and $F \cdot \nabla$ are different: the first is a function, and the second is the operator $\sum_i F_i \, \partial / \partial x_i$.
The directional derivative of $f$ along $v$ is $(v \cdot \nabla) f = v \cdot \nabla f$.
The symbol $\nabla$ is called nabla.

## Explanations

### L1 human

Del is a shorthand that makes vector calculus formulas look like vector algebra.
Multiply it by a function to get the gradient. Dot it with a field to get the divergence, and cross it with a field to get the curl.
</document>

<document source="https://math2.tamnd.com/node/0H6KWAYS87.md" tag="0H6KWAYS87" kind="statement">
# Product of power series

## Statement

Let $R$ be a commutative ring, let $f, g \in R[\![X]\!]$ be [[formal power series|072W67KFYK]], and let $n$ be a natural number.
Then
$$[X^n](f g) = \sum_{i + j = n} [X^i] f \cdot [X^j] g = \sum_{i=0}^{n} [X^i] f \cdot [X^{n-i}] g.$$

## Remarks

In many texts this formula is the definition of the product of formal power series.
The sequence of coefficients of $fg$ is the convolution of the sequences of $f$ and $g$. This is the Cauchy product.
Example: $[X^n] \frac{1}{(1 - X)^2} = \sum_{i=0}^{n} 1 \cdot 1 = n + 1$.

## Explanations

### L1 human

To find the coefficient of $X^n$ in a product, collect all ways to get $X^n$ as $X^i$ times $X^j$.
Add the products of the matching coefficients.
</document>

<document source="https://math2.tamnd.com/node/0H7C3T1GJY.md" tag="0H7C3T1GJY" kind="concept">
# Scalar potential

## Summary

A scalar potential of a vector field is a scalar function whose [[gradient|0ABK9V9Y64]] is the field, or minus the field in the physics convention.
A field has a scalar potential exactly when it is [[conservative|0PKG97PX9Y]].
</document>

<document source="https://math2.tamnd.com/node/0H8H7ZTHW2.md" tag="0H8H7ZTHW2" kind="concept">
# Cumulative distribution function

## Summary

The cumulative distribution function of a real random variable $X$ gives, for each real $x$, the probability that $X$ is at most $x$.
It determines the law of $X$. The node [[cumulative distribution function|0MXMS3AE13]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0H8Q4MANBD.md" tag="0H8Q4MANBD" kind="statement">
# Every finite tree with at least two vertices has a leaf

## Statement

Let $T$ be a finite [[tree|0MJ0JS72DZ]] with at least two vertices.
Then $T$ has a vertex of [[degree|0T0F6CCS9W]] $1$.

## Remarks

In fact $T$ has at least two leaves: the two ends of a longest path are leaves.
Proof idea: take a longest path $v_0, \dots, v_k$. A neighbour of $v_k$ other than $v_{k-1}$ would extend the path or close a cycle, so $\deg(v_k) = 1$.
The lemma is the base of induction proofs on trees, as in the proof that [[a tree on n vertices has n − 1 edges|0VSNHTHK8T]].

## Explanations

### L1 human

A finite tree with at least one edge always has an end point.
Walk along the tree as far as you can without going back. You stop at a vertex with only one edge.
</document>

<document source="https://math2.tamnd.com/node/0H8SZFFDXJ.md" tag="0H8SZFFDXJ" kind="statement">
# Rule of Sarrus

## Statement

Let $K$ be a commutative ring and let $a, b, c, d, e, f, g, h, i \in K$.
Then
$$\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Copy the first two columns to the right of the matrix.
Then the three positive terms are the products along the diagonals that go down to the right.
The three negative terms are products along the diagonals that go up to the right in the same picture.
This rule is the [[Leibniz formula|016PE4VZ8Q]] for $n = 3$; it has no analogue with diagonals for $n \ge 4$.

## Explanations

### L1 human

The rule of Sarrus is a memory aid for $3 \times 3$ determinants.
Write the first two columns again to the right of the matrix.
Add the three products down to the right, and subtract the three products up to the right.
</document>

<document source="https://math2.tamnd.com/node/0H97Q4Z2SF.md" tag="0H97Q4Z2SF" kind="definition">
# Plane

## Definition

Let $n \geq 3$ be an integer. A plane in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ is a set of the form
$$P = \{\, p + s u + t v \;:\; s, t \in \mathbb{R} \,\},$$
where $p$ is a point of $\mathbb{R}^n$. Here $u$ and $v$ are linearly independent [[vectors|0QN3D80AWP]] of $\mathbb{R}^n$.

## Remarks

Three points that are not on one [[line|0MP9BCF6DP]] lie in exactly one plane.
In $\mathbb{R}^3$, the planes are also the solution sets of the linear equations $a x + b y + c z = d$ with $(a, b, c) \neq (0, 0, 0)$. See the [[equation of a plane|07SDJ5DNE4]].
In the language of affine geometry, a plane is an [[affine subspace|0PYW26VNT6]] of dimension 2.
The word plane also names the whole [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$.

## Explanations

### L1 human

A plane in space is like a sheet of glass that goes on forever. To describe it, pick one point on it and two directions in it that are not parallel. Each point of the plane is the start point plus some amount of the first direction plus some amount of the second.
</document>

<document source="https://math2.tamnd.com/node/0H9K5D3RDV.md" tag="0H9K5D3RDV" kind="definition">
# Abelianisation

## Definition

Let $G$ be a [[group|G5S1M9]] and let $[G, G]$ be its [[commutator subgroup|0ASGSTXSBW]].
The abelianisation of $G$ is the [[quotient group|0KYX2N69J0]] $G^{\mathrm{ab}} = G/[G, G]$.

## Remarks

The group $G^{\mathrm{ab}}$ is [[abelian|082GHNPJ53]], by [[this criterion|0MS8QGA6S5]].
If $G$ is abelian, then $G^{\mathrm{ab}} = G$.
The abelianisation of the symmetric group $S_n$ for $n \geq 2$ is cyclic of order $2$, and the sign of a permutation gives the isomorphism.

## Explanations

### L1 human

The abelianisation forces all elements of a group to commute, and changes nothing more than it must.
It keeps as much information about the group as an abelian group can keep.
For a group that is already abelian, it does nothing.
</document>

<document source="https://math2.tamnd.com/node/0HATW2VWAZ.md" tag="0HATW2VWAZ" kind="concept">
# Rectifiable curve

## Summary

A [[parametrised curve|0G7WM8DX4P]] is rectifiable when the lengths of the polygonal paths inscribed in it are bounded.
The least upper bound of these lengths is the length of the curve.
</document>

<document source="https://math2.tamnd.com/node/0HAZEFH137.md" tag="0HAZEFH137" kind="statement">
# Interior is the largest open subset

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $\operatorname{int} S \subseteq S$, the set $\operatorname{int} S$ is open, and each open set $U$ of $X$ with $U \subseteq S$ satisfies $U \subseteq \operatorname{int} S$.

## Remarks

Here $\operatorname{int} S$ is the [[interior|0C38S0KMJK]] of $S$. The openness is the statement [[interior is open|0PNWCM559P]].
A consequence: $S$ is open if and only if $\operatorname{int} S = S$.
If $S \subseteq T$, then $\operatorname{int} S \subseteq \operatorname{int} T$.

## Explanations

### L1 human

Among all open sets inside $S$, the interior is the biggest one.
Each open set inside $S$ is already inside the interior.
</document>

<document source="https://math2.tamnd.com/node/0HB0XKDXHH.md" tag="0HB0XKDXHH" kind="example">
# Binomial series

## Example

Let $\alpha$ be a real number.
For each natural number $k$, let
$$\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}.$$
The binomial series is the [[power series|0VP56B4EMX]] $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$.
For $|x| < 1$ its sum is $(1 + x)^{\alpha}$.

## Remarks

If $\alpha$ is a natural number, then $\binom{\alpha}{k} = 0$ for $k > \alpha$. The series is then the binomial formula for $(1 + x)^{\alpha}$.
Otherwise the [[radius of convergence is 1|0KX1MXJ5MR]].
For $\alpha = -1$ it is the geometric series $\sum (-1)^k x^k = 1/(1 + x)$.
For $\alpha = 1/2$ it gives $\sqrt{1 + x} = 1 + x/2 - x^2/8 + \cdots$.

## Explanations

### L1 human

The binomial series extends the binomial formula to exponents that are not natural numbers.
The expansion then has infinitely many terms, and it holds only for small $x$.
</document>

<document source="https://math2.tamnd.com/node/0HB8SXHPFA.md" tag="0HB8SXHPFA" kind="definition">
# Markov property

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $T \subseteq [0, \infty)$, and let $(\mathcal{F}_t)_{t \in T}$ be a filtration: σ-algebras with $\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}$ for $s \le t$.
Let $(X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in $(S, \mathcal{S})$ such that each $X_t$ is $\mathcal{F}_t$-measurable.
The process has the Markov property with respect to $(\mathcal{F}_t)$ if, for all $s \le t$ in $T$ and each $A \in \mathcal{S}$,
$$P(X_t \in A \mid \mathcal{F}_s) = P(X_t \in A \mid X_s) \quad \text{almost surely}.$$

## Remarks

Here $P(B \mid \mathcal{G}) = E[1_B \mid \mathcal{G}]$ is a [[conditional expectation|0YAMJRK04E]].
For a discrete-time process with a countable state space and the natural filtration, the property becomes $P(X_{n+1} = j \mid X_0, \dots, X_n) = P(X_{n+1} = j \mid X_n)$. This defines a Markov chain.
The property says that the future is [[conditionally independent|0ZJ87ZZ8PK]] of the past, given the present.
The strong Markov property is the same statement with a stopping time in place of the fixed time $s$.

## Explanations

### L1 human

A process with the Markov property forgets how it reached its present state.
To predict its future, the present state is all the information that you need.
</document>

<document source="https://math2.tamnd.com/node/0HBQC2GJXT.md" tag="0HBQC2GJXT" kind="definition">
# Limit point

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]], let $S \subseteq X$ and let $x \in X$.
The point $x$ is a limit point of $S$ when each [[open set|01ZQQDCERP]] $U$ with $x \in U$ contains a point of $S$ other than $x$.

## Remarks

The point $x$ need not be in $S$. In $\mathbb{R}$, the point $0$ is a limit point of $\{1/n : n \ge 1\}$.
In a metric space, $x$ is a limit point of $S$ if and only if each ball $B(x, \varepsilon)$ contains a point of $S \setminus \{x\}$.
The [[closure|0DRKT1QJR7]] of $S$ is the union of $S$ and the set of its limit points.
A point of $S$ that is not a limit point of $S$ is an [[isolated point|01PKQYTTMT]] of $S$.

## Explanations

### L1 human

A limit point is a point that other points of the set crowd around.
For the set $\{1, 1/2, 1/3, \ldots\}$, the points crowd around $0$, so $0$ is a limit point, but $1$ is not.
</document>

<document source="https://math2.tamnd.com/node/0HBRW4E6G1.md" tag="0HBRW4E6G1" kind="definition">
# Intersection

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The intersection of $A$ and $B$ is the set
$$A \cap B = \{x \in A \mid x \in B\}.$$
So an object $x$ is an [[element|0EZ758XE73]] of $A \cap B$ when $x \in A$ and $x \in B$.

## Remarks

Example: $\{1, 2\} \cap \{2, 3\} = \{2\}$.
Intersection is commutative and associative, and $A \cap B \subseteq A$.
When $A \cap B = \varnothing$, the sets are [[disjoint|0W4RPA8QH3]].
The intersection of a family of sets is in [[intersection of a family of sets|0839XNQXDK]].

## Explanations

### L1 human

The intersection keeps only what the two sets have in common.
The intersection of the even numbers and the multiples of $3$ is the set of multiples of $6$.
</document>

<document source="https://math2.tamnd.com/node/0HBS1F00DN.md" tag="0HBS1F00DN" kind="statement">
# Induction on the integers

## Statement

Let $P(z)$ be a property of [[integers|0C4JS9RF5M]] $z$.
Suppose that $P(0)$ holds.
Suppose that for each natural number $n$, $P(n)$ implies $P(n + 1)$.
Suppose that for each natural number $n$, $P(-n)$ implies $P(-n - 1)$.
Then $P(z)$ holds for every integer $z$.

## Remarks

The proof applies ordinary [[induction|0ASJVEFA9J]] two times: one time to $P(n)$ and one time to $P(-n)$, for natural numbers $n$.
Each integer is $n$ or $-n$ for some natural number $n$, thus the two inductions cover all of $\mathbb{Z}$.

## Explanations

### L1 human

You start at $0$ and you walk in both directions on the number line.
One induction goes up through $1, 2, 3, \dots$, and the other goes down through $-1, -2, -3, \dots$.
</document>

<document source="https://math2.tamnd.com/node/0HBY48G8G1.md" tag="0HBY48G8G1" kind="definition">
# Riemann sum

## Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$, and let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
Let $t = (t_1, \dots, t_n)$ be tags with $t_i \in [x_{i-1}, x_i]$ for each $i$.
The Riemann sum of $f$ for the tagged partition $(P, t)$ is
$$S(f, P, t) = \sum_{i=1}^{n} f(t_i) (x_i - x_{i-1}).$$

## Remarks

The left rule takes $t_i = x_{i-1}$, the right rule takes $t_i = x_i$, and the midpoint rule takes the midpoints.
For bounded $f$, each Riemann sum lies between the lower and the upper [[Darboux sum|0QRQVH0CRW]] for $P$.
Mathlib has a general form for boxes, `BoxIntegral.integralSum`.

## Explanations

### L1 human

A Riemann sum adds up areas of rectangles.
Each rectangle stands on a piece of the partition, and its height is the value of $f$ at the chosen tag.
</document>

<document source="https://math2.tamnd.com/node/0HE5WDQ9FJ.md" tag="0HE5WDQ9FJ" kind="concept">
# Zermelo–Fraenkel set theory

## Summary

Zermelo–Fraenkel set theory, ZF, is an axiomatic theory of [[sets|0SJ3RX9TXA]] in [[first-order logic|086QKPF0A6]] with one binary relation $\in$.
Its axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas.
ZF with the [[axiom of choice|0HTGB13FE2]] is ZFC, the standard foundation of most of mathematics.

## Remarks

The [[axiom of extensionality|0TB16T2C97]] says that a set is determined by its elements.
Separation forms only subsets $\{x \in A \mid P(x)\}$ of a given set $A$. This avoids [[Russell's paradox|0HESMMSYEG]].
The [[continuum hypothesis|0MN2MER9WN]] is independent of ZFC, if ZFC is consistent.

## Explanations

### L1 human

ZF is a short list of rules that says which sets exist.
For example, for two sets there is a set that contains both, and for each set there is the set of its subsets.
From these rules, one can build the numbers, functions and most other objects of mathematics.
</document>

<document source="https://math2.tamnd.com/node/0HEF46RPEK.md" tag="0HEF46RPEK" kind="statement">
# Union of open sets is open

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(U_i)_{i \in I}$ be a family of [[open subsets|01ZQQDCERP]] of $X$, indexed by any set $I$.
Then the [[union|05GK32H72H]] $\bigcup_{i \in I} U_i$ is open in $X$.

## Remarks

Proof: let $x$ be in the union. Then $x \in U_j$ for some $j \in I$, and there is $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U_j$. This ball is also in the union. Then use [[open sets of a metric space|0A26X79RTB]].
The family can be infinite, and even uncountable.
In a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.

## Explanations

### L1 human

Each point of the union lies in one of the open sets, and that set already gives a small ball about the point.
For example, the union of the intervals $(1/n, 1)$ for $n \ge 2$ is the open interval $(0, 1)$.
</document>

<document source="https://math2.tamnd.com/node/0HESMMSYEG.md" tag="0HESMMSYEG" kind="statement">
# Russell's paradox

## Statement

There is no [[set|0SJ3RX9TXA]] $R$ such that, for each set $x$, $x \in R$ if and only if $x \notin x$.

## Remarks

Proof: if such an $R$ exists, apply the condition to $x = R$. Then $R \in R$ if and only if $R \notin R$, which is a contradiction.
So the collection $\{x \mid x \notin x\}$ is not a set. Unrestricted comprehension, which makes a set from each property, is inconsistent.
It follows that there is no set of all sets. Otherwise, separation would give $R$ as a subset of it.

## History

Bertrand Russell found the paradox in May 1901.
Ernst Zermelo found it independently, by 1902 and possibly in 1899, but he did not publish it.
In a 1902 letter, Russell told Gottlob Frege that the paradox arises in the logical system of Frege.
Axiomatic set theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], avoid it with a restricted form of comprehension.

## Explanations

### L1 human

Try to form "the set of all sets that do not contain themselves".
Ask whether this set contains itself: both answers lead to a contradiction.
So some collections are too large to be sets.
</document>

<document source="https://math2.tamnd.com/node/0HF30P75G4.md" tag="0HF30P75G4" kind="concept">
# Hamiltonian cycle

## Summary

A Hamiltonian cycle is a cycle that passes through every vertex of a graph. The search for such cycles is a classical hard problem, related to the travelling salesman problem. The node [[Hamiltonian cycle|0HKXNG2RS1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0HFREYRSZP.md" tag="0HFREYRSZP" kind="concept">
# Rotation

## Summary

A rotation is a motion of a Euclidean space that keeps at least one point fixed and keeps the orientation. In the plane, a rotation turns each point about a centre through the same angle.
</document>

<document source="https://math2.tamnd.com/node/0HFSFVMWQB.md" tag="0HFSFVMWQB" kind="definition">
# Taylor polynomial

## Definition

Let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
The Taylor polynomial of degree $k$ of $f$ at $a$ is
$$P_k(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j = f(a) + f'(a)(x - a) + \cdots + \frac{f^{(k)}(a)}{k!} (x - a)^k.$$

## Remarks

It is the unique polynomial $P$ of degree at most $k$ with $P^{(j)}(a) = f^{(j)}(a)$ for $j = 0, \dots, k$, using [[higher derivatives|084NGSTWM7]].
For $k = 1$, its graph is the [[tangent line|01R1HW3Y42]].
The difference $R_k = f - P_k$ is the remainder. [[Taylor's theorem|03QHBVW4TT]] says that it is small near $a$.
Mathlib has `taylorWithinEval` for Taylor polynomials within a set.

## Explanations

### L1 human

The Taylor polynomial copies the value and the first $k$ derivatives of $f$ at $a$.
It is the polynomial that best fits $f$ near $a$.
</document>

<document source="https://math2.tamnd.com/node/0HG11FW8MJ.md" tag="0HG11FW8MJ" kind="definition">
# First-countable space

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
The space $X$ is first-countable when each point $x \in X$ has a sequence $(N_n)_{n \ge 1}$ of [[neighbourhoods|09CJ00H3NW]] with this property: each neighbourhood of $x$ contains some $N_n$.

## Remarks

Such a sequence is a [[countable|0ZG61TFBES]] neighbourhood basis at $x$. One can choose the sets $N_n$ open, and also decreasing.
Each metric space is first-countable: take $N_n = B(x, 1/n)$, as in [[open balls form a basis|02EEX96DFS]].
Each [[second-countable space|00RE1R4W7N]] is first-countable.
In a first-countable space, a point is in the closure of a set $A$ exactly when it is a limit of a sequence in $A$.

## Explanations

### L1 human

At each point, you can shrink to the point along a countable list of neighbourhoods.
In a metric space, the balls of radius $1/n$ do this.
</document>

<document source="https://math2.tamnd.com/node/0HGVCWQGY3.md" tag="0HGVCWQGY3" kind="statement">
# Pigeonhole principle

## Statement

Let $A$ and $B$ be finite sets with $|A| > |B|$, and let $f : A \to B$ be a function.
Then there are elements $a \neq a'$ of $A$ with $f(a) = f(a')$.
In other words, $f$ is not injective.

## Remarks

In the usual image, if $n + 1$ pigeons sit in $n$ holes, then some hole has at least two pigeons.
Example: among any $13$ people, two have their birthdays in the same month.
The [[generalized pigeonhole principle|0BY7GN3449]] gives a fibre with more than $n$ elements. The [[infinite pigeonhole principle|0FS3RHF6KE]] is a version for infinite sets.

## Explanations

### L1 human

If there are more objects than boxes, some box has two objects.
For example, in any group of 367 people, two share a birthday, because a year has at most 366 days.
</document>

<document source="https://math2.tamnd.com/node/0HH4Y7HSRQ.md" tag="0HH4Y7HSRQ" kind="concept">
# Sum of a series

## Summary

The sum of a convergent series is the limit of its partial sums. It gives a meaning to an infinite sum of real numbers. The node [[sum of a series|0K0H5E3Z5N]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0HH6JX5WZZ.md" tag="0HH6JX5WZZ" kind="definition">
# Generating function

## Definition

Let $R$ be a commutative ring and let $(a_n)_{n \ge 0}$ be a sequence in $R$.
The ordinary generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]
$$A(x) = \sum_{n \ge 0} a_n x^n \in R[\![x]\!].$$

## Remarks

Example: the sequence $1, 1, 1, \dots$ has the generating function $\sum_n x^n = \frac{1}{1 - x}$, the [[geometric series|0R8JR0ARJ4]].
Operations on sequences become operations on series. The [[product of power series|0H6KWAYS87]] corresponds to the convolution $c_n = \sum_{i+j=n} a_i b_j$.
The [[exponential generating function|0KE6NBTVY8]] is a variant with the weights $x^n / n!$.

## Explanations

### L1 human

A generating function stores a whole sequence as the coefficients of one power series.
Algebra with the series then gives facts about the sequence, such as recurrences or closed formulas.
</document>

<document source="https://math2.tamnd.com/node/0HHHDCXZV4.md" tag="0HHHDCXZV4" kind="concept">
# Differentiable function of several variables

## Summary

A function of several variables is differentiable at a point when, near that point, it is close to a linear function plus a constant. The error is small compared with the distance to the point.
The linear function is the [[total derivative|0FNKFT8H6E]] of the function at the point.
</document>

<document source="https://math2.tamnd.com/node/0HJ0HDA6W7.md" tag="0HJ0HDA6W7" kind="statement">
# Solutions for distinct real roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and suppose that the [[characteristic polynomial|0QDXMG46QW]] $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ has $n$ distinct real roots $r_1, \dots, r_n$.
Then $\left(e^{r_1 x}, \dots, e^{r_n x}\right)$ is a [[fundamental system|0H4KNNCHPY]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$ on $\mathbb{R}$.
So the solutions are exactly the functions $y = c_1 e^{r_1 x} + \dots + c_n e^{r_n x}$ with real constants $c_1, \dots, c_n$.

## Remarks

Each $e^{r_k x}$ is a solution because $p(r_k) = 0$. The Wronskian at $0$ is the Vandermonde determinant $\prod_{j < k} (r_k - r_j)$, which is not zero. So the [[Wronskian test|0MHE4B1QCQ]] gives independence.
Example: $y'' - 3 y' + 2 y = 0$ has the solutions $c_1 e^{x} + c_2 e^{2x}$.

## Explanations

### L1 human

Each root of the characteristic polynomial gives one exponential solution.
If the $n$ roots are different, these $n$ exponentials are independent, and their combinations are all the solutions.
</document>

<document source="https://math2.tamnd.com/node/0HJ28H9Y95.md" tag="0HJ28H9Y95" kind="example">
# Thomae's function

## Example

Define $f \colon \mathbb{R} \to \mathbb{R}$ as follows.
For irrational $x$, $f(x) = 0$. For rational $x = p/q$ in lowest terms with $q > 0$, $f(x) = 1/q$.
Then $f$ is [[continuous|0EEE2CZB2A]] at each irrational number and discontinuous at each rational number.

## Remarks

Let $a$ be irrational and let $\varepsilon > 0$. Only finitely many $p/q$ near $a$ have $1/q \ge \varepsilon$. So $f(x) < \varepsilon$ for all $x$ close enough to $a$.
At a rational $a$, irrational points near $a$ have value $0$, but $f(a) > 0$.
The function is periodic with period $1$.
No function on $\mathbb{R}$ is continuous exactly at the rationals.

## Explanations

### L1 human

The value $1/q$ is small when the denominator $q$ is large.
Near an irrational point, all rationals have large denominators, so the values there are small.
</document>

<document source="https://math2.tamnd.com/node/0HJ4DTW6FN.md" tag="0HJ4DTW6FN" kind="statement">
# Second isomorphism theorem for rings

## Statement

Let $R$ be a [[ring|0894J980RK]], let $S$ be a [[subring|0859D58WPD]] of $R$ and let $I$ be a two-sided ideal of $R$.
Then $S + I = \{s + x : s \in S, x \in I\}$ is a subring of $R$, and $I$ is a two-sided ideal of $S + I$.
The intersection $S \cap I$ is a two-sided ideal of $S$.
The map $S/(S \cap I) \to (S + I)/I$, $s + (S \cap I) \mapsto s + I$, is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

The proof applies the [[first isomorphism theorem|0ZMYX7GMH0]] to the map $S \to (S + I)/I$, $s \mapsto s + I$. It is surjective, and its kernel is $S \cap I$.
The theorem is also called the diamond theorem, after the picture of the four rings $S \cap I$, $S$, $I$ and $S + I$.

## Explanations

### L1 human

The theorem compares two quotients.
Divide the subring $S$ by its part inside $I$, or divide the larger ring $S + I$ by $I$.
The two quotients are the same ring.
</document>

<document source="https://math2.tamnd.com/node/0HJHYMGN3C.md" tag="0HJHYMGN3C" kind="concept">
# Discrete logarithm

## Summary

A discrete logarithm of an element $h$ of a cyclic group to a base $g$ is an exponent $k$ such that $g^k = h$.
The [[definition of the discrete logarithm|09B4X51J09]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0HKXNG2RS1.md" tag="0HKXNG2RS1" kind="definition">
# Hamiltonian cycle

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A Hamiltonian cycle of $G$ is a [[cycle|07CT9RDY7Q]] in $G$ that contains every vertex of $G$.

## Remarks

A Hamiltonian path is a [[path|0JDE56HWQ8]] that contains every vertex of $G$.
The names come from William Rowan Hamilton. His icosian game asks for a Hamiltonian cycle on the edges of a dodecahedron.
To decide whether a graph has a Hamiltonian cycle is an NP-complete problem.

## Explanations

### L1 human

A Hamiltonian cycle is a round trip that visits every vertex exactly once.
It is about visiting all vertices, while an Eulerian circuit is about using all edges.
</document>

<document source="https://math2.tamnd.com/node/0HM05N3FJD.md" tag="0HM05N3FJD" kind="concept">
# Random experiment

## Summary

A random experiment is a procedure that we can repeat as often as we like and that has a well-defined set of possible results.
The set of these results is the [[sample space|0W7NFW23RF]] of the experiment.
The experiment is random when it has more than one possible result, and deterministic when it has exactly one.

## Remarks

Each repetition of the experiment is a trial. Each trial gives exactly one [[outcome|0Y5R7TPNT1]].
A toss of a coin and a roll of a die are random experiments.

## Explanations

### L1 human

A random experiment is an action whose result you cannot predict, but whose possible results you know in advance.
When you roll a die, you do not know the number that you will get, but you know that it is one of $1, 2, 3, 4, 5, 6$.
</document>

<document source="https://math2.tamnd.com/node/0HM0HQCK6N.md" tag="0HM0HQCK6N" kind="concept">
# Distributive lattice

## Summary

A distributive lattice is a lattice in which meet and join distribute over each other. Lattices of sets and Boolean algebras are the main examples. The node [[distributive lattice|0T36P6TXQ6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0HM584C465.md" tag="0HM584C465" kind="example">
# RSA cryptosystem

## Example

The RSA cryptosystem is an example of [[public-key cryptography|01E9MNJYEX]].
To make the keys, choose two distinct primes $p$ and $q$, and put $n = pq$ and $\varphi(n) = (p - 1)(q - 1)$, the value of [[Euler's totient function|0VV04M41AN]].
Choose an integer $e$ with $1 < e < \varphi(n)$ and $\gcd(e, \varphi(n)) = 1$. Let $d$ be the [[inverse|00GXTQMY8Z]] of $e$ modulo $\varphi(n)$.
The public key is $(n, e)$, and the private key is $d$.
To encrypt a message $m$ with $0 \le m < n$, compute $c = m^e \bmod n$. To decrypt, compute $c^d \bmod n$, which is equal to $m$.
A small case: $p = 5$, $q = 11$, $n = 55$, $\varphi(n) = 40$, $e = 3$ and $d = 27$, since $3 \cdot 27 = 81 \equiv 1 \pmod{40}$.
The message $m = 2$ gives $c = 2^3 = 8$, and $8^{27} \equiv 2 \pmod{55}$.

## Remarks

Decryption works by [[Euler's theorem|0VC4DVE7DQ]]: $ed = 1 + k \varphi(n)$, so $m^{ed} \equiv m \pmod n$ when $\gcd(m, n) = 1$. The general case follows from the [[Chinese remainder theorem|0FNN9B4W64]].
The security rests on the belief that it is hard to factor $n$ when $p$ and $q$ are large.
Ron Rivest, Adi Shamir and Leonard Adleman described the system publicly in 1977.

## Explanations

### L1 human

Anyone can raise a message to the power $e$ modulo $n$. To undo this, you need $d$, and to find $d$ you need the factors of $n$.
Multiplying two large primes is easy, but recovering them from the product is believed to be very hard.
</document>

<document source="https://math2.tamnd.com/node/0HM732ZNVC.md" tag="0HM732ZNVC" kind="statement">
# A sequence converges iff its liminf and limsup agree

## Statement

Let $(a_n)$ be a bounded real sequence.
Then $(a_n)$ converges if and only if $\liminf_{n \to \infty} a_n = \limsup_{n \to \infty} a_n$.
In this case, $\lim_{n \to \infty} a_n$ is equal to this common value.

## Remarks

The proof of the direction from right to left uses $\inf_{k \ge n} a_k \le a_n \le \sup_{k \ge n} a_k$ and the [[squeeze theorem|0XMF00T4XQ]].
In the extended real line, the statement holds for every sequence. Then convergence to $+\infty$ or to $-\infty$ is allowed.
For $(-1)^n$, the two values are $-1$ and $1$. They differ, so the sequence diverges.

## Explanations

### L1 human

The limit inferior and the limit superior are the lowest and the highest points where the sequence keeps returning.
If these two points are the same, the sequence has nowhere else to go, so it converges to that point.
</document>

<document source="https://math2.tamnd.com/node/0HNATXFEV5.md" tag="0HNATXFEV5" kind="concept">
# Order on the natural numbers

## Summary

The order on the natural numbers is the relation $\le$ on $\mathbb{N}$.
Here $a \le b$ means that $a$ comes at or before $b$ when we count.
It is a total order, and $0$ is its least element.
</document>

<document source="https://math2.tamnd.com/node/0HNK6KT05K.md" tag="0HNK6KT05K" kind="concept">
# Centre of mass

## Summary

The centre of mass of a body is the average position of its mass, with each point weighted by the density there.
For a body with a continuous density it is a quotient of [[triple integrals|0698M7C7AX]], or of double integrals for a plane lamina.
</document>

<document source="https://math2.tamnd.com/node/0HNMFG8F8K.md" tag="0HNMFG8F8K" kind="statement">
# Eigenvectors with distinct eigenvalues are linearly independent

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $v_1, \dots, v_k$ be [[eigenvectors|0N6960W0BV]] of $T$ for eigenvalues $\lambda_1, \dots, \lambda_k$ that are pairwise distinct.
Then $v_1, \dots, v_k$ are [[linearly independent|0CRMGPAH83]].

## Remarks

The proof is by induction on $k$: apply $T - \lambda_k \id_V$ to a linear relation, and the term of $v_k$ disappears.
So the sum of the eigenspaces of $T$ for distinct eigenvalues is a direct sum.
In particular, an endomorphism of an $n$-dimensional space has at most $n$ distinct eigenvalues.

## Explanations

### L1 human

Eigenvectors for different eigenvalues point in genuinely different directions.
None of them is a combination of the others.
This is the key fact behind diagonalisation.
</document>

<document source="https://math2.tamnd.com/node/0HPHKCGEQ0.md" tag="0HPHKCGEQ0" kind="statement">
# Generalised Stokes theorem

## Statement

Let $\Omega$ be an oriented smooth manifold of dimension $n$ with boundary $\partial \Omega$, and give $\partial \Omega$ the induced orientation.
Let $\omega$ be a smooth [[differential form|0QHTX7WQEY]] of degree $n - 1$ on $\Omega$ with compact support, and let $d\omega$ be its [[exterior derivative|004FRCXH6M]]. Then
$$\int_\Omega d\omega = \int_{\partial \Omega} \omega.$$

## Remarks

For $\Omega = [a, b]$ and a function $f$, the theorem is the fundamental theorem of calculus, $\int_a^b f'(x) \, dx = f(b) - f(a)$. The boundary point $a$ gets the negative orientation.
For a curve and a 0-form, it is the [[gradient theorem|0SD237FEJD]]. For a plane region and a 1-form, it is [[Green's theorem|0HR3N02317]].
For a surface in $\mathbb{R}^3$ and a 1-form, it is the classical [[Stokes' theorem|0AKRKVHW94]]. For a solid in $\mathbb{R}^3$ and a 2-form, it is the [[divergence theorem|0JX86PSSA9]].
The orientation and the compact support are needed so that both integrals are defined.

## Explanations

### L1 human

The integral of a derivative over a region equals the integral of the original form over the boundary of the region.
One formula covers the fundamental theorem of calculus and the three integral theorems of vector calculus.
</document>

<document source="https://math2.tamnd.com/node/0HQARMKEV5.md" tag="0HQARMKEV5" kind="definition">
# Archimedean ordered field

## Definition

An [[ordered field|010JCV9J4E]] $F$ is Archimedean when, for all $x$ and $y$ in $F$ with $y > 0$, there is a natural number $n$ with $n y > x$.
Here $n y$ means $y + y + \dots + y$, with $n$ terms.

## Remarks

With $y = 1$, the definition gives the [[Archimedean property|0A74SX55QP]]: for each $x$, some natural number exceeds $x$. Conversely, this property implies the definition.
An equivalent condition: for each $\varepsilon > 0$ in $F$, there is a natural number $n \ge 1$ with $1/n < \varepsilon$.
The field of rational functions $\mathbb{R}(t)$, with the order in which $t$ is larger than every real number, is not Archimedean.

## Explanations

### L1 human

You can reach any distance with enough small steps of a fixed size.
In an Archimedean field there is no number so large that no multiple of a given positive number can pass it.
</document>

<document source="https://math2.tamnd.com/node/0HQFZAP3ND.md" tag="0HQFZAP3ND" kind="concept">
# Coefficient matrix

## Summary

The coefficient matrix of a system of linear equations is the matrix of the coefficients of the unknowns.
Its row $i$ holds the coefficients of equation $i$.
</document>

<document source="https://math2.tamnd.com/node/0HR3N02317.md" tag="0HR3N02317" kind="statement">
# Green's theorem

## Statement

Let $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\mathbb{R}^2$, and let $D$ be the bounded region enclosed by $C$.
Let $L$ and $M$ be real functions with continuous [[partial derivatives|05AXG3QRKR]] on an open set that contains $D \cup C$.
Then
$$\oint_C (L \, dx + M \, dy) = \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA.$$

## Remarks

Positive orientation means that $C$ goes counterclockwise, so $D$ is on the left.
Proof sketch: first let $D$ be a region between the graphs of two functions of $x$. Write the [[double integral|0NKBMFC18R]] of $\partial L / \partial y$ as an iterated integral, and use the fundamental theorem of calculus. This gives $\oint_C L \, dx = -\iint_D \partial L / \partial y \, dA$. The $M$ term is similar, and general regions are cut into such pieces.
Example: with $L = -y$ and $M = x$, the right side is $2$ times the [[area of D|09CBQB6V67]].
Green's theorem is the planar case of [[Stokes' theorem|0AKRKVHW94]]. George Green stated a similar result in 1828, and Riemann gave a proof in 1851.

## Explanations

### L1 human

The theorem says that the total circulation of a plane field around a closed curve equals the sum of the small circulations inside.
The small circulations of neighbouring squares cancel along shared edges, so only the outer boundary is left.
</document>

<document source="https://math2.tamnd.com/node/0HR573CHNM.md" tag="0HR573CHNM" kind="definition">
# Linear span

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $S \subseteq V$.
The *linear span* of $S$ is the set $\operatorname{span}(S)$ of all [[linear combinations|0V8F3E48E6]] of finitely many vectors of $S$:
$$\operatorname{span}(S) = \{a_1 v_1 + \cdots + a_k v_k : k \ge 0, \ a_i \in F, \ v_i \in S\}.$$

## Remarks

The span is a [[linear subspace|00YC1D652K]] of $V$, and it is the intersection of all subspaces of $V$ that contain $S$.
The span of the empty set is $\{0\}$.
In $\R^3$, the span of one nonzero vector is a line through the origin.
The span of two non-parallel vectors is a plane through the origin.

## Explanations

### L1 human

The span of some vectors is everything we can reach from them by scaling and adding.
Two arrows in space that do not point along the same line span a whole plane.
The span is always a subspace.
</document>

<document source="https://math2.tamnd.com/node/0HRJ2Y74T3.md" tag="0HRJ2Y74T3" kind="definition">
# Rearrangement of a series

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a real [[series|0JH4MN4K1C]].
Let $\sigma \colon \mathbb{N} \to \mathbb{N}$ be a bijection.
The series $\sum_{n=0}^{\infty} a_{\sigma(n)}$ is a rearrangement of $\sum a_n$.

## Remarks

A rearrangement uses each term of the original series exactly once, in a new order.
[[Rearrangements of an absolutely convergent series have the same sum|0J76TP9FGD]].
For a [[conditionally convergent|02YRQXZDCS]] series, rearrangements can change the sum, by the [[Riemann rearrangement theorem|0E7VBYGFF7]].

## Explanations

### L1 human

A rearrangement adds up the same terms in a different order.
For finite sums the order does not matter, but for infinite series it can matter.
</document>

<document source="https://math2.tamnd.com/node/0HRVY67CNV.md" tag="0HRVY67CNV" kind="statement">
# Solutions for repeated roots

## Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$.
Let $r$ be a real root of $p$ of multiplicity $m \ge 1$.
Then the $m$ functions $e^{r x}, x e^{r x}, \dots, x^{m-1} e^{r x}$ are linearly independent solutions of the equation on $\mathbb{R}$.
If all the roots of $p$ are real, the union of these families over the distinct roots of $p$ is a [[fundamental system|0H4KNNCHPY]].

## Remarks

For order $2$: if $r$ is a double root of $r^2 + a_1 r + a_0$, the solutions are $(c_1 + c_2 x) e^{r x}$.
Example: $y'' - 2 y' + y = 0$ has $p(r) = (r - 1)^2$, so its solutions are $(c_1 + c_2 x) e^{x}$.
The same result holds for complex roots, with complex solutions. For the real form, see [[complex conjugate roots|0BEZX8F2XW]].

## Explanations

### L1 human

A double root gives only one exponential, but we need two solutions.
The extra solution is the exponential times $x$.
A root of multiplicity $m$ gives the exponential times $1, x, \dots, x^{m-1}$.
</document>

<document source="https://math2.tamnd.com/node/0HTGB13FE2.md" tag="0HTGB13FE2" kind="statement">
# Axiom of choice

## Statement

Let $X$ be a [[set|0SJ3RX9TXA]] of nonempty sets.
Then there is a [[choice function|0YXEQWRGP0]] on $X$, that is, a function $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

## Remarks

An equivalent form: for each [[family|05Q6G4EH2G]] $(A_i)_{i \in I}$ of nonempty sets, the product $\prod_{i \in I} A_i$ is nonempty.
Over ZF, it is equivalent to [[Zorn's lemma|04RXTQA5ZA]], to the [[well-ordering theorem|0N109NG5ZX]] and to the [[comparability of cardinals|0YBHSJMN3J]].
It is independent of ZF, if ZF is consistent.
Kurt Gödel showed in 1938 that ZF does not refute it, and Paul Cohen showed in 1963 that ZF does not prove it.

## History

Ernst Zermelo formulated the axiom in 1904 for his proof of the [[well-ordering theorem|0N109NG5ZX]].

## Explanations

### L1 human

For finitely many nonempty boxes, you can pick one object from each box, one box at a time.
For infinitely many boxes without a rule to pick, the axiom of choice says that a selection still exists.
From infinitely many pairs of shoes, you can take the left shoe of each pair.
From infinitely many pairs of identical socks, there is no such rule, and you need the axiom.
</document>

<document source="https://math2.tamnd.com/node/0HV01NH20W.md" tag="0HV01NH20W" kind="concept">
# Reduced residue system

## Summary

A reduced residue system modulo $n$ is a set of integers with one element in each residue class of integers coprime to $n$.
The [[definition of a reduced residue system|0J0QEN1DZR]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0HV1SKWPMS.md" tag="0HV1SKWPMS" kind="statement">
# Rouché–Capelli theorem

## Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix over $F$, and let $b \in F^m$.
The [[system|0GT4WWGWH2]] $Ax = b$ has a solution if and only if $\rank A = \rank [A \mid b]$, where $[A \mid b]$ is the [[augmented matrix|0T6MWGZSXA]] and $\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].
If a solution exists, then the solution set is an affine subspace of $F^n$ of dimension $n - \rank A$.
In particular, the solution is unique if and only if $\rank A = n$.

## Remarks

The theorem is also called the Kronecker–Capelli theorem.
If $\rank [A \mid b] = \rank A + 1$, then the system is [[inconsistent|0X6V9B1V10]].
The dimension $n - \rank A$ is the number of free variables.

## Explanations

### L1 human

The theorem says that a system has a solution exactly when the right side adds no new independent information.
If adding the column $b$ raises the rank, then $b$ is not a combination of the columns of $A$, and no solution exists.
When solutions exist, each unknown without a pivot is free, and it adds one dimension to the solution set.
</document>

<document source="https://math2.tamnd.com/node/0HV9Y7WG5R.md" tag="0HV9Y7WG5R" kind="definition">
# Group homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
A group homomorphism from $G$ to $H$ is a map $\varphi \colon G \to H$ such that $\varphi(ab) = \varphi(a) \varphi(b)$ for all $a, b \in G$.
Here the product $ab$ is taken in $G$ and the product $\varphi(a) \varphi(b)$ is taken in $H$.

## Remarks

The map $n \mapsto 2^n$ is a homomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Q}^\times, \cdot)$.
The map $x \mapsto e^x$ is a homomorphism from $(\mathbb{R}, +)$ to the positive real numbers with multiplication.
For each pair of groups, the map that sends each element of $G$ to the identity of $H$ is a homomorphism, the trivial homomorphism.
The composite of two homomorphisms is a homomorphism.

## Explanations

### L1 human

A homomorphism translates one group into another so that products go to products.
It does not matter if you multiply first and then translate, or translate first and then multiply.
The logarithm turns multiplication of positive numbers into addition, and this is why it is a homomorphism.
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# Eccentricity

## Summary

The eccentricity of a conic section is a non-negative real number that measures how far the conic is from a circle. It is $0$ for a circle, between $0$ and $1$ for an ellipse, $1$ for a parabola and greater than $1$ for a hyperbola.
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<document source="https://math2.tamnd.com/node/0HW7BNHHGF.md" tag="0HW7BNHHGF" kind="statement">
# Limit comparison test

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences with $a_n \ge 0$ and $b_n > 0$ for all $n$.
Assume that $a_n / b_n \to c$ as $n \to \infty$, for a real number $c$ with $0 < c < \infty$.
Then $\sum a_n$ converges if and only if $\sum b_n$ converges.

## Remarks

For large $n$ we have $\frac{c}{2} b_n \le a_n \le 2c\, b_n$. The [[comparison test|04Z5G0THN6]] then gives both directions.
For example, $\frac{n}{n^3 + 1} / \frac{1}{n^2} \to 1$, so $\sum n/(n^3 + 1)$ converges, like $\sum 1/n^2$.
If $c = 0$, convergence of $\sum b_n$ still implies convergence of $\sum a_n$. The converse can fail.

## Explanations

### L1 human

If two series of positive terms have terms of the same size in the long run, they behave the same way.
So you can replace a complicated term by a simple one with the same leading behaviour.
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<document source="https://math2.tamnd.com/node/0HX8HZ1WTQ.md" tag="0HX8HZ1WTQ" kind="definition">
# Gram matrix

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$.
The *Gram matrix* of $v_1, \dots, v_k$ is the $k \times k$ matrix $G$ with entries
$$G_{ij} = \langle v_i, v_j \rangle \qquad (1 \le i, j \le k),$$
where $\langle \cdot, \cdot \rangle$ is the [[inner product|0NB0AWGJBW]].

## Remarks

The Gram matrix is Hermitian and [[positive semidefinite|01MMVW8CKC]].
It is invertible if and only if $v_1, \dots, v_k$ are linearly independent.
For real column vectors $v_i \in \R^n$ with the dot product, $G = A^T A$, where $A$ has the columns $v_1, \dots, v_k$.
Its determinant is the square of the $k$-dimensional volume of the parallelepiped spanned by the $v_i$.

## Explanations

### L1 human

The Gram matrix records all inner products of a list of vectors.
It contains all lengths and all angles between them.
Two lists with the same Gram matrix have the same shape, up to an isometry.
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<document source="https://math2.tamnd.com/node/0HXEX0DPRA.md" tag="0HXEX0DPRA" kind="definition">
# One-sided derivative

## Definition

Let $f$ be a real function defined on an interval $[a, a + r)$ with $r > 0$.
The right derivative of $f$ at $a$ is the [[one-sided limit|0909PWZR62]]
$$f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h},$$
when it exists.
The left derivative $f'_-(a)$ is defined in the same way, for $f$ on $(a - r, a]$ and $h \to 0^-$.

## Remarks

A function on an open interval is differentiable at $a$ if and only if both one-sided derivatives exist and are equal. Their common value is the [[derivative|0F1WR9C83K]].
One-sided derivatives give a meaning to derivatives at the endpoints of a closed interval.
For $|x|$ at $0$, the right derivative is $1$ and the left derivative is $-1$.
Mathlib expresses them with `HasDerivWithinAt` on a half line.

## Explanations

### L1 human

A one-sided derivative only looks at one side of the point.
At a corner of a graph, the slopes from the left and from the right exist but differ.
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<document source="https://math2.tamnd.com/node/0HY15D6WTX.md" tag="0HY15D6WTX" kind="definition">
# Finitely generated group

## Definition

A [[group|G5S1M9]] $G$ is finitely generated when there is a finite subset $S$ of $G$ with $\langle S \rangle = G$, where $\langle S \rangle$ is the [[subgroup generated by $S$|0RP7VWQ0YN]].

## Remarks

Each finite group is finitely generated, because it is generated by the finite set of all its elements.
The group $(\mathbb{Z}^n, +)$ is finitely generated by the $n$ standard basis vectors.
The group $(\mathbb{Q}, +)$ is not finitely generated: the subgroup generated by finitely many fractions has a common denominator, and it does not contain all fractions.

## Explanations

### L1 human

A group is finitely generated when a finite list of elements is enough to build all elements.
The group may still be infinite, as the integers show: one generator $1$ gives all integers.
The rational numbers need infinitely many generators.
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<document source="https://math2.tamnd.com/node/0HY1MXMEN9.md" tag="0HY1MXMEN9" kind="statement">
# Totient of a prime

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $\varphi(p) = p - 1$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

The proof: the only divisors of $p$ are $1$ and $p$. So each of $1, \dots, p - 1$ is coprime to $p$, and $p$ is not.
The converse holds: if $n > 1$ and $\varphi(n) = n - 1$, then $n$ is prime.
The result is the case $k = 1$ of the [[totient of a prime power|0AVY8HKBGD]].

## Explanations

### L1 human

For a prime $p$, each number from $1$ to $p - 1$ has no common factor with $p$.
For example, $\varphi(7) = 6$, because $1, 2, 3, 4, 5, 6$ are all coprime to $7$.
</document>

<document source="https://math2.tamnd.com/node/0HYC2RHD3M.md" tag="0HYC2RHD3M" kind="statement">
# Eigenvalues of a positive definite matrix are positive

## Statement

Let $A$ be a [[positive definite|0KX66CZMBB]] $n \times n$ complex Hermitian matrix.
Then each [[eigenvalue|0BQAA2BK3Q]] of $A$ is a positive real number.

## Remarks

If $Ax = \lambda x$ with $x \ne 0$, then $0 < x^* A x = \lambda\, x^* x$, and $x^* x > 0$.
The converse holds: a Hermitian matrix with only positive eigenvalues is positive definite, by the spectral theorem.
So $\det A > 0$ and $\tr A > 0$ for a positive definite $A$.

## Explanations

### L1 human

A positive definite matrix stretches each of its eigenvector directions by a positive factor.
It never flips or crushes a direction.
So its eigenvalues, its determinant and its trace are all positive.
</document>

<document source="https://math2.tamnd.com/node/0HYFEEZAEM.md" tag="0HYFEEZAEM" kind="statement">
# Injective iff it has a left inverse

## Statement

Let $A$ be a nonempty set, let $B$ be a set and let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Then $f$ is [[injective|0PVG1CRAJN]] if and only if $f$ has a [[left inverse|02486SD328]].

## Remarks

The hypothesis that $A$ is nonempty is necessary. The empty function from $\varnothing$ to a nonempty set is injective, but there is no function from a nonempty set to $\varnothing$.
To build a left inverse, fix $a_0 \in A$. Send each $b = f(a)$ to $a$, and send each $b \notin f(A)$ to $a_0$.

## Explanations

### L1 human

An injective function loses no information, so you can go back from each value to its unique input.
The values that $f$ never takes can go anywhere, and you need one point $a_0$ to send them to.
</document>

<document source="https://math2.tamnd.com/node/0HYFWKKY3C.md" tag="0HYFWKKY3C" kind="concept">
# Polar form of a quadratic form

## Summary

The polar form of a quadratic form $Q$ is the symmetric bilinear form $B(u, v) = Q(u + v) - Q(u) - Q(v)$.
</document>

<document source="https://math2.tamnd.com/node/0HYNZE940T.md" tag="0HYNZE940T" kind="statement">
# Modulus is multiplicative

## Statement

For all complex numbers $z$ and $w$,
$$|z w| = |z| \, |w|,$$
where $|\cdot|$ is the [[modulus|0KW2X0TFBX]].

## Remarks

Proof: $|z w|^2 = z w \overline{z w} = (z \bar{z})(w \bar{w}) = |z|^2 |w|^2$, and both sides are nonnegative.
For $z = a + b i$ and $w = c + d i$, this is the identity $(a^2 + b^2)(c^2 + d^2) = (a c - b d)^2 + (a d + b c)^2$.
On the real numbers, it gives that the [[absolute value is multiplicative|0GMV7GP8W0]].

## Explanations

### L1 human

The length of a product is the product of the lengths.
For example, $|(1 + i)(1 - i)| = |2| = 2$, and $|1 + i| \cdot |1 - i| = \sqrt{2} \cdot \sqrt{2} = 2$.
</document>

<document source="https://math2.tamnd.com/node/0HYP606RYF.md" tag="0HYP606RYF" kind="concept">
# Student's t-test

## Summary

A Student's t-test is a [[hypothesis test|087P03NEND]] whose test statistic has [[Student's t-distribution|0NQ5798SB1]] under the null hypothesis. It mostly tests hypotheses about the means of normal populations with unknown variance.
In the one-sample t-test, $X_1, \dots, X_n$ are i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with unknown $\sigma$, and $H_0 \colon \mu = \mu_0$. The statistic is
$$t = \frac{\bar{X}_n - \mu_0}{S_n / \sqrt{n}},$$
with the [[sample mean|0QNRBHWV35]] $\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$. Under $H_0$, $t \sim t_{n - 1}$.

## Remarks

The two-sample t-test compares the means of two independent normal samples with sizes $n_1$, $n_2$ and equal variances. It uses the pooled standard deviation $s_p$ and $t = (\bar{X}_1 - \bar{X}_2) / \big(s_p \sqrt{1/n_1 + 1/n_2}\big)$, with $n_1 + n_2 - 2$ degrees of freedom.
For two samples with unequal variances, Welch's t-test uses an approximate number of degrees of freedom.
The paired t-test is the one-sample t-test applied to the differences of paired observations.
For large samples, the t-test gives almost the same results as a test with the normal law, because $t_\nu$ approaches $\mathcal{N}(0, 1)$.

## Explanations

### L1 human

A t-test checks whether an average differs from a given value, or whether two averages differ, more than chance alone can explain.
It divides the difference by its estimated standard error, and compares the result with the t-distribution.
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<document source="https://math2.tamnd.com/node/0HYPR64JM2.md" tag="0HYPR64JM2" kind="statement">
# Mean value property

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, and identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + iy \leftrightarrow (x, y)$.
Let $u : U \to \mathbb{R}$ be harmonic: $u$ is twice continuously differentiable and $u_{xx} + u_{yy} = 0$ on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Then
$$u(c) = \frac{1}{2\pi} \int_0^{2\pi} u(c + r e^{i\theta}) \, d\theta.$$

## Remarks

The value of $u$ at the center of the circle is the average of $u$ on the circle.
For holomorphic functions, the same formula holds, and it follows from [[Cauchy's integral formula|0DG10TVYPN]] with $a = c$. On a disc, $u$ is the [[real part of a holomorphic function|031BG625Y3]], which gives the harmonic case.
The property also holds for averages over the solid disc. A continuous function with the mean value property on all small circles is harmonic.

## Explanations

### L1 human

If $u$ is harmonic, its value at a point is the average of its values on each circle around that point. This is true as long as the closed disc stays in the domain.
For example, $u = x^2 - y^2$ at $0$ is $0$, and on the unit circle it is $\cos 2\theta$, whose average is $0$.
</document>

<document source="https://math2.tamnd.com/node/0HYRJF4WNQ.md" tag="0HYRJF4WNQ" kind="concept">
# Norm of a Gaussian integer

## Summary

The norm of a Gaussian integer $a + bi$ is the integer $a^2 + b^2$. It is multiplicative.
The [[definition of the norm|00HPT88ZX0]] gives the precise form.
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<document source="https://math2.tamnd.com/node/0HYV2NME07.md" tag="0HYV2NME07" kind="concept">
# Hamiltonian graph

## Summary

A Hamiltonian graph is a graph that has a cycle through all of its vertices. There is no simple test for this property, but degree conditions give sufficient criteria. The node [[Hamiltonian graph|090WPKFABX]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0HZNE5QZDJ.md" tag="0HZNE5QZDJ" kind="definition">
# Quadratic form

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
A *quadratic form* on $V$ is a map $Q \colon V \to F$ such that:

1. $Q(av) = a^2 Q(v)$ for all $a \in F$ and $v \in V$.
2. The map $(u, v) \mapsto Q(u + v) - Q(u) - Q(v)$ is a [[bilinear form|0SQZMXB6GP]] on $V$.

## Remarks

On $F^n$, the quadratic forms are the maps $Q(x) = \sum_{i,j} a_{ij} x_i x_j$, that is, homogeneous polynomials of degree $2$.
In matrix notation, $Q(x) = x^T A x$ for an $n \times n$ matrix $A$.
If the characteristic of $F$ is not $2$, the matrix $A$ can be chosen symmetric, and then it is unique.
For example, $Q(x, y) = x^2 + 3xy + 2y^2$ is a quadratic form on $\R^2$.

## Explanations

### L1 human

A quadratic form is a polynomial in which every term has degree two, such as $x^2 - 4xy + y^2$.
It is the simplest kind of function after linear functions.
The squared length $x_1^2 + \cdots + x_n^2$ is the main example.
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<document source="https://math2.tamnd.com/node/0J0BBCMJ5W.md" tag="0J0BBCMJ5W" kind="concept">
# Pivot

## Summary

A pivot of a matrix in row echelon form is the leading entry of a nonzero row.
In elimination, a pivot is the entry that we use to clear the other entries of its column.
</document>

<document source="https://math2.tamnd.com/node/0J0FCWV32D.md" tag="0J0FCWV32D" kind="concept">
# Exponential function

## Summary

The exponential function $x \mapsto e^x$ is the function that equals its own derivative and takes the value $1$ at $0$. It turns sums into products, and it is the sum of the power series $\sum x^n / n!$. The node [[exponential function|0RMVS7RXZX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0J0QEN1DZR.md" tag="0J0QEN1DZR" kind="definition">
# Reduced residue system

## Definition

Let $n$ be a positive integer.
A reduced residue system modulo $n$ is a set $R$ of integers with these properties.
Each element of $R$ is [[coprime|0GFKP8F8AG]] to $n$.
No two distinct elements of $R$ are congruent modulo $n$.
Each integer coprime to $n$ is congruent modulo $n$ to some element of $R$.

## Remarks

Thus $R$ contains exactly one element of each [[residue class|0QRV8FPK5M]] of integers coprime to $n$. It has $\varphi(n)$ elements, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].
For example, $\{1, 5, 7, 11\}$ is a reduced residue system modulo $12$.
If $R$ is a reduced residue system modulo $n$ and $\gcd(a, n) = 1$, then $\{ a r : r \in R \}$ is also a reduced residue system modulo $n$. This fact gives a proof of [[Euler's theorem|0VC4DVE7DQ]].

## Explanations

### L1 human

Take a [[complete residue system|0E22JKS4W4]] modulo $n$, and keep only the numbers with no common factor with $n$.
For $n = 10$, the numbers $0, 1, \dots, 9$ give the reduced residue system $\{1, 3, 7, 9\}$.
</document>

<document source="https://math2.tamnd.com/node/0J0SK5FF4M.md" tag="0J0SK5FF4M" kind="definition">
# Quotient map

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a subspace of $V$.
The *quotient map* is the map
$$\pi \colon V \to V / W, \qquad \pi(v) = v + W,$$
from $V$ to the [[quotient space|0BR3FY5Q8M]] $V / W$.

## Remarks

The map $\pi$ is linear and surjective, and its kernel is $W$.
Each linear map $T \colon V \to U$ with $W \subseteq \ker T$ factors through $\pi$, see [[universal property of the quotient space|0ZERJX6PH6]].
In Mathlib, it is `Submodule.mkQ`.

## Explanations

### L1 human

The quotient map sends each vector to the class of all vectors that differ from it by an element of $W$.
It forgets exactly the information in $W$.
The vectors that it sends to zero are exactly the vectors of $W$.
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<document source="https://math2.tamnd.com/node/0J0XBKG64S.md" tag="0J0XBKG64S" kind="concept">
# Proof by contradiction

## Summary

A proof by contradiction proves a [[proposition|0ZGC7ZESYN]] $p$ in two steps.
It assumes $\neg p$, and it gets a contradiction from this assumption, that is, a statement $q$ together with $\neg q$.
Then $\neg p$ is false, so $p$ is true by the [[double negation law|0NFAGWH84K]].

## Remarks

Another name is reductio ad absurdum.
A classical example: $\sqrt{2}$ is irrational. Assume $\sqrt{2} = a/b$ in lowest terms. Then $a$ and $b$ are both even, which is a contradiction.
The last step uses classical logic. A proof of $\neg p$ from a contradiction with $p$ is valid also in intuitionistic logic.

## Explanations

### L1 human

To show that something is true, suppose that it is false and follow the consequences.
If you reach an impossible situation, the supposition was wrong.
So the statement must be true.
</document>

<document source="https://math2.tamnd.com/node/0J1E5ZRZRM.md" tag="0J1E5ZRZRM" kind="concept">
# Fixed field

## Summary

The fixed field of a group of automorphisms of a field is the subfield of the elements that every automorphism in the group leaves unchanged.
The [[definition of the fixed field|0KMJ6919WS]] states it.
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<document source="https://math2.tamnd.com/node/0J1N3WTKM4.md" tag="0J1N3WTKM4" kind="concept">
# Euclidean isometry

## Summary

A Euclidean isometry is a map of a Euclidean space to itself that keeps the distance between each two points. Translations, rotations and reflections are examples. The Euclidean isometries of $\mathbb{R}^n$ form the Euclidean group.
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<document source="https://math2.tamnd.com/node/0J1R5NP36S.md" tag="0J1R5NP36S" kind="definition">
# Liouville number

## Definition

A real number $x$ is a Liouville number if for each positive integer $n$ there are integers $p$ and $q$ with $q > 1$ and
$$0 < \left| x - \frac{p}{q} \right| < \frac{1}{q^n}.$$

## Remarks

The standard example is Liouville's constant $\sum_{k=1}^{\infty} 10^{-k!} = 0.110001000\ldots$.
Each Liouville number is irrational, and in fact [[transcendental|079EGSD32Y]].
The numbers $e$ and $\pi$ are transcendental, but they are not Liouville numbers.
The Liouville numbers are uncountable, but they form a set of Lebesgue measure zero.

## Explanations

### L1 human

A Liouville number has rational approximations that are extremely good, better than any fixed power of $1/q$.
Algebraic numbers cannot be approximated this well, so a Liouville number is not algebraic.
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<document source="https://math2.tamnd.com/node/0J1TT091TN.md" tag="0J1TT091TN" kind="statement">
# Solutions of Pell's equation are powers of the fundamental solution

## Statement

Let $d$ be a positive integer that is not a perfect square, and let $(x_1, y_1)$ be the [[fundamental solution|0KCYJSB9FW]] of $x^2 - d y^2 = 1$.
Let $x$ and $y$ be integers with $x \ge 1$, $y \ge 0$ and $x^2 - d y^2 = 1$.
Then there is a unique natural number $n$ with $x + y \sqrt{d} = (x_1 + y_1 \sqrt{d})^n$.

## Remarks

Conversely, each power $(x_1 + y_1 \sqrt{d})^n = x_n + y_n \sqrt{d}$ gives a solution $(x_n, y_n)$.
The integer solutions are exactly the pairs with $x + y \sqrt{d} = \pm (x_1 + y_1 \sqrt{d})^n$ for $n \in \mathbb{Z}$.
For $d = 2$: $(3 + 2\sqrt{2})^2 = 17 + 12 \sqrt{2}$, so $(17, 12)$ is the next solution after $(3, 2)$.
In algebraic terms, the units of norm $1$ in $\mathbb{Z}[\sqrt{d}]$ form the group $\{\pm 1\} \times \mathbb{Z}$.

## Explanations

### L1 human

Once you know the smallest solution, you get all the others by taking powers.
Expand $(x_1 + y_1 \sqrt{d})^n$ and read the integer part and the coefficient of $\sqrt{d}$.
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<document source="https://math2.tamnd.com/node/0J29GM30Z7.md" tag="0J29GM30Z7" kind="example">
# Existence of nth roots by the intermediate value theorem

## Example

Let $n \ge 1$ be a natural number and let $y \ge 0$ be real.
Then there is a unique real $x \ge 0$ with $x^n = y$.
For existence, let $b = \max(1, y)$. The function $p(x) = x^n$ is [[continuous|0C5TP6RYSV]], with $p(0) = 0 \le y$ and $p(b) \ge b \ge y$.
The [[intermediate value theorem|00KAFBR32E]] gives $x \in [0, b]$ with $x^n = y$.

## Remarks

Uniqueness holds because $x \mapsto x^n$ is strictly increasing on $[0, \infty)$.
The number $x$ is the $n$-th root $\sqrt[n]{y}$.
For example, it proves that $\sqrt{2}$ exists as a real number.

## Explanations

### L1 human

The graph of $x^n$ starts at $0$ and grows past $y$.
Since the graph has no gaps, it reaches the height $y$ at some point, which is the root.
</document>

<document source="https://math2.tamnd.com/node/0J2C7BDJQA.md" tag="0J2C7BDJQA" kind="statement">
# Polar decomposition

## Statement

Let $A$ be an $n \times n$ complex matrix.
Then there are a [[unitary matrix|0AR68JTMDT]] $U$ and a positive semidefinite Hermitian matrix $P$ with
$$A = UP.$$
The factor $P$ is unique: $P = (A^* A)^{1/2}$, the [[square root|0BJVDCF38Y]] of $A^* A$.
If $A$ is invertible, then $U$ is also unique.

## Remarks

It follows from the singular value decomposition $A = W \Sigma V^*$: take $U = W V^*$ and $P = V \Sigma V^*$.
For $n = 1$, it is the polar form $z = e^{i\theta} r$ of a complex number.
There is also a left polar decomposition $A = P' U$ with $P' = (A A^*)^{1/2}$.

## Explanations

### L1 human

Every square matrix is a stretch along perpendicular axes, followed by a rotation, possibly with a reflection.
This generalises writing a complex number as a positive length times a unit complex number.
The stretch part is always unique.
</document>

<document source="https://math2.tamnd.com/node/0J2WAR83WP.md" tag="0J2WAR83WP" kind="concept">
# Inverse function

## Summary

The inverse function of a bijection reverses it: it sends each value back to the input that gives it.
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<document source="https://math2.tamnd.com/node/0J3578562S.md" tag="0J3578562S" kind="definition">
# Equivalent norms

## Definition

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$.
A norm on $V$ is a function $\|\cdot\| \colon V \to [0, \infty)$ such that, for all $x, y \in V$ and all scalars $\lambda$: $\|x\| = 0$ only for $x = 0$, $\|\lambda x\| = |\lambda| \, \|x\|$, and $\|x + y\| \le \|x\| + \|y\|$.
Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $V$ are equivalent when there are real numbers $c > 0$ and $C > 0$ with
$$c \, \|x\|_a \le \|x\|_b \le C \, \|x\|_a \quad \text{for all } x \in V.$$

## Remarks

Each norm gives a metric $d(x, y) = \|x - y\|$. Two norms are equivalent if and only if their metrics are strongly [[equivalent|0TWQ5PRKWS]].
For norms, equivalence is the same as giving the same [[metric topology|0A6YRNZB2R]].
On a finite-dimensional space, [[all norms are equivalent|03MC70ZRNR]].
On $C([0, 1])$, the sup norm and the norm $\int_0^1 |f(t)| \, dt$ are not equivalent.

## Explanations

### L1 human

Equivalent norms measure the same vectors as small or large, up to a fixed factor.
So they agree on all questions of convergence and continuity.
</document>

<document source="https://math2.tamnd.com/node/0J3K4Y1KPD.md" tag="0J3K4Y1KPD" kind="concept">
# Cycle type

## Summary

The cycle type of a permutation of a finite set is the list of the lengths of the cycles in its disjoint cycle decomposition.
For a permutation in $S_n$ it is a partition of $n$.
</document>

<document source="https://math2.tamnd.com/node/0J438GAC8K.md" tag="0J438GAC8K" kind="statement">
# Conjugacy classes of the symmetric group correspond to cycle types

## Statement

Let $\sigma, \tau \in S_n$.
Then $\sigma$ and $\tau$ are [[conjugate|0DXD4QSZ8A]] in $S_n$ if and only if they have the same [[cycle type|0J3K4Y1KPD]].

## Remarks

The key formula is $\pi (a_1\ a_2\ \dots\ a_k) \pi^{-1} = (\pi(a_1)\ \pi(a_2)\ \dots\ \pi(a_k))$ for each $\pi \in S_n$.
So conjugation keeps the cycle type, and a suitable $\pi$ sends the cycles of $\sigma$ onto the cycles of $\tau$.
Thus the number of conjugacy classes of $S_n$ is the number of partitions of $n$.
For example, $S_4$ has $5$ conjugacy classes.

## Explanations

### L1 human

Conjugation by $\pi$ renames the points, but it keeps the shape of each cycle.
Conversely, two permutations of the same shape differ only by a renaming of the points.
So in $S_n$, conjugate means "same shape".
</document>

<document source="https://math2.tamnd.com/node/0J53NE674D.md" tag="0J53NE674D" kind="statement">
# The symmetric group on five or more letters is not solvable

## Statement

Let $n \geq 5$.
Then the [[symmetric group|0515A6D2Y1]] $S_n$ is not [[solvable|0FXJ7YFMMX]].

## Remarks

The group $A_n$ is [[simple|0ARPRJQWZ2]] and not abelian, so its commutator subgroup is $A_n$, and the derived series of $A_n$ is constant.
So $A_n$ is not solvable, and then $S_n$ is not solvable, because [[subgroups of solvable groups are solvable|05Y0F9M0TS]].
The groups $S_1, S_2, S_3$ and $S_4$ are solvable.
By Galois theory, this is the reason why the general polynomial equation of degree $n \geq 5$ is not solvable by radicals.

## Explanations

### L1 human

From five letters on, the even permutations form a simple group that is not abelian.
Such a group cannot be cut into abelian layers.
So the full symmetric group cannot be cut into abelian layers either.
</document>

<document source="https://math2.tamnd.com/node/0J5EMJRRFM.md" tag="0J5EMJRRFM" kind="concept">
# Characteristic polynomial of a linear map

## Summary

The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of its matrix in any basis.
</document>

<document source="https://math2.tamnd.com/node/0J5VGJYYC3.md" tag="0J5VGJYYC3" kind="statement">
# Congruence is compatible with powers

## Statement

Let $n$ be a positive integer, let $a$ and $b$ be integers, and let $k$ be a natural number.
If $a \equiv b \pmod n$, then $a^k \equiv b^k \pmod n$.

## Remarks

The proof is an induction on $k$ with [[compatibility with multiplication|072D5G4SK6]].
The exponent cannot be reduced modulo $n$ in general. For example, $2^5 = 32 \equiv 2 \pmod 3$, but $2^{5 \bmod 3} = 2^2 \equiv 1 \pmod 3$.
[[Euler's theorem|0VC4DVE7DQ]] tells when one can reduce the exponent, and by which modulus.

## Explanations

### L1 human

To find $10^{100} \bmod 9$, note that $10 \equiv 1 \pmod 9$.
So $10^{100} \equiv 1^{100} = 1 \pmod 9$, without a computation of the large number.
</document>

<document source="https://math2.tamnd.com/node/0J5XR7BW2S.md" tag="0J5XR7BW2S" kind="statement">
# Intersecting secants theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, with $A \neq D$ and $B \neq C$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
Suppose that the lines $AD$ and $BC$ meet at a point $P$ outside the circle.
Then
$$d(P, A)\, d(P, D) = d(P, B)\, d(P, C).$$

## Remarks

Both products are equal to the [[power|02E9VT1RVQ]] $d(P, O)^2 - r^2$ of $P$, where $O$ is the centre and $r$ the radius.
A proof: the triangles $PAC$ and $PBD$ are similar, because they have the same angle at $P$ and equal inscribed angles at $C$ and $D$.
If the second line becomes tangent, then $B = C$, and the statement becomes the [[tangent–secant theorem|0QX7E9KAAX]].

## Explanations

### L1 human

From a point outside a circle, draw two lines that cut the circle. On each line, multiply the distances from the point to the two crossing points. The two products are equal.
</document>

<document source="https://math2.tamnd.com/node/0J6A752DTZ.md" tag="0J6A752DTZ" kind="statement">
# Lyapunov's stability theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and let $x^* \in U$ be an equilibrium point of $x' = f(x)$.
Let $W \subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \to \mathbb{R}$ be a [[Lyapunov function|04P7B5XC22]] for $x^*$.
(1) Then $x^*$ is [[Lyapunov stable|092S6VBECQ]].
(2) If $V$ is a strict Lyapunov function, then $x^*$ is [[asymptotically stable|05Y1JAM3Y5]].
(3) If $W = U = \mathbb{R}^n$, $V$ is strict, and $V(x) \to \infty$ as $\|x\| \to \infty$, then $x^*$ is globally asymptotically stable.

## Remarks

Proof idea for (1): choose $\varepsilon$ so small that the sphere $S$ of radius $\varepsilon$ around $x^*$ lies in $W$. Let $m > 0$ be the minimum of $V$ on $S$. A solution that starts where $V < m$ keeps $V < m$, so it cannot reach $S$.
For the [[damped harmonic oscillator|0RNXVVTJTJ]] $x' = y$, $y' = -x - c y$ with $c > 0$, the energy $V = \frac{1}{2}(x^2 + y^2)$ has $\dot{V} = -c y^2 \le 0$. This gives stability, and [[LaSalle's invariance principle|0659J5X7B5]] gives asymptotic stability.
The method needs no solution formula. The difficulty is to find a suitable $V$.

## History

The method is named after Aleksandr Lyapunov, who introduced it in his study of the stability of motion.

## Explanations

### L1 human

If some energy-like quantity decreases along all motions near an equilibrium, the motions stay near it.
If the quantity decreases strictly, the motions must also run down to the equilibrium.
</document>

<document source="https://math2.tamnd.com/node/0J6AKYZQT8.md" tag="0J6AKYZQT8" kind="statement">
# Associativity of composition

## Statement

Let $f \colon A \to B$, $g \colon B \to C$ and $h \colon C \to D$ be [[functions|0XHDTE5YMZ]].
Then
$$h \circ (g \circ f) = (h \circ g) \circ f.$$

## Remarks

Both sides send each $a \in A$ to $h(g(f(a)))$, so they are equal.
Because of this law, one can write $h \circ g \circ f$ without brackets.
With the [[identity functions|0FSNQ0GNCC]], the functions from a set to itself form a monoid under composition.

## Explanations

### L1 human

When three machines are chained, it does not matter which two you connect first.
The input passes through $f$, then $g$, then $h$ in both cases.
</document>

<document source="https://math2.tamnd.com/node/0J6G5NJC5R.md" tag="0J6G5NJC5R" kind="statement">
# Kuratowski's theorem

## Statement

A subdivision of a [[graph|0JD5C82QMB]] $H$ is a graph that $H$ gives when some edges are replaced by paths with new inner vertices.
Let $G$ be a finite graph.
Then $G$ is [[planar|029JQEK15B]] if and only if no [[subgraph|0P11X5HKAX]] of $G$ is a subdivision of $K_5$ or of $K_{3,3}$.

## Remarks

One direction is easy: a subdivision of a nonplanar graph is nonplanar, and [[$K_5$ and $K_{3,3}$ are not planar|030A3FHN14]].
Wagner's theorem is a related result: a finite graph is planar if and only if it has neither $K_5$ nor $K_{3,3}$ as a minor.

## History

Kazimierz Kuratowski published the theorem in 1930.
Orrin Frink and Paul Smith proved it independently in the same year, but they did not publish their proof.
Lev Pontryagin reportedly proved it around 1927, and in the Soviet Union it was also known under his name.

## Explanations

### L1 human

There are exactly two kinds of obstacle to drawing a graph without crossings.
A graph is planar unless it contains a copy of $K_5$ or $K_{3,3}$ in which some edges are stretched into paths.
</document>

<document source="https://math2.tamnd.com/node/0J6M03WSS6.md" tag="0J6M03WSS6" kind="definition">
# Hyperbolic functions

## Definition

The hyperbolic functions are the functions $\sinh$, $\cosh$ and $\tanh$ on $\mathbb{R}$, together with their reciprocals:
$$\coth x = \frac{\cosh x}{\sinh x} \ (x \ne 0), \qquad \operatorname{sech} x = \frac{1}{\cosh x}, \qquad \operatorname{csch} x = \frac{1}{\sinh x} \ (x \ne 0).$$
Here [[$\sinh$|0GWD21EWRW]], [[$\cosh$|0EQFWKJ37J]] and [[$\tanh$|0MWYQTZ07Z]] are the hyperbolic sine, cosine and tangent.

## Remarks

They satisfy identities similar to the trigonometric identities, often with a change of sign.
The point $(\cosh t, \sinh t)$ lies on the hyperbola $x^2 - y^2 = 1$, as $(\cos t, \sin t)$ lies on the circle.
Vincenzo Riccati introduced the hyperbolic functions in 1757.

## Explanations

### L1 human

The hyperbolic functions are built from $e^x$ and $e^{-x}$.
They play the role for the hyperbola that sine and cosine play for the circle.
</document>

<document source="https://math2.tamnd.com/node/0J6Z6H5VQJ.md" tag="0J6Z6H5VQJ" kind="definition">
# Discrete random variable

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
$X$ is discrete if there is a [[countable set|0ZG61TFBES]] $S \subseteq \mathbb{R}$ with $P(X \in S) = 1$.
A distribution $\mu$ on $\mathbb{R}$ is discrete if $\mu(S) = 1$ for some countable set $S$.

## Remarks

Then $P(X \in B) = \sum_{x \in B \cap S} P(X = x)$ for each Borel set $B$. So the [[probability mass function|0WF4EJP91W]] $x \mapsto P(X = x)$ determines the law of $X$.
The cumulative distribution function of a discrete random variable is a step function when the set of values has no limit points. It jumps by $P(X = x)$ at each value $x$.
The Bernoulli, binomial, geometric and Poisson distributions are discrete.

## Explanations

### L1 human

A discrete random variable takes values that you can list, such as $0, 1, 2, \dots$.
The number of heads in ten coin tosses and the number of calls in an hour are discrete random variables.
</document>

<document source="https://math2.tamnd.com/node/0J768YF1ZE.md" tag="0J768YF1ZE" kind="definition">
# Nine-point circle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. Let $M_A$, $M_B$ and $M_C$ be the [[midpoints|0S706JBZBG]] of the sides $BC$, $CA$ and $AB$.
These three points are not collinear. The nine-point circle of $ABC$ is the [[circumcircle|03VZJE6C8C]] of the triangle $M_A M_B M_C$.

## Remarks

The circle also contains the feet of the three [[altitudes|0C9MZY2RNA]], and the midpoints of the segments $AH$, $BH$ and $CH$, where $H$ is the [[orthocentre|0WB619CA0V]]. This gives the nine points of the name.
Its centre is the midpoint of $H$ and the circumcentre $O$ of $ABC$. Its radius is half of the circumradius of $ABC$.
Brianchon and Poncelet proved in 1821 that all nine points lie on one circle. In 1822, Feuerbach proved that the circle touches the incircle and the three excircles.

## Explanations

### L1 human

Join the midpoints of the sides of a triangle to get a smaller triangle. The circle through its corners is the nine-point circle. It also passes through six other special points of the big triangle.
</document>

<document source="https://math2.tamnd.com/node/0J76TP9FGD.md" tag="0J76TP9FGD" kind="statement">
# Rearrangements of an absolutely convergent series have the same sum

## Statement

Let $\sum a_n$ be a real series that [[converges absolutely|0W3K3QPEWT]], with sum $A$.
Let $\sigma \colon \mathbb{N} \to \mathbb{N}$ be a bijection.
Then the [[rearrangement|0HRJ2Y74T3]] $\sum a_{\sigma(n)}$ converges absolutely, and its sum is $A$.

## Remarks

Given $\varepsilon > 0$, choose $N$ with $\sum_{n > N} |a_n| < \varepsilon$. For large $m$, the first $m$ terms of the rearrangement include $a_0, \dots, a_N$. So the two partial sums differ by less than $\varepsilon$.
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], the converse also holds for real series.
Mathlib states a general form for sums over a type, as `Equiv.tsum_eq`.

## Explanations

### L1 human

When the series converges absolutely, the far terms have a small total size, whatever their order.
So a new order changes the partial sums only by a small amount.
</document>

<document source="https://math2.tamnd.com/node/0J78KBPFZB.md" tag="0J78KBPFZB" kind="concept">
# Riemann sphere

## Summary

The Riemann sphere is the complex plane with one added point $\infty$, see the [[definition|0CDMXGPSES]].
Stereographic projection identifies it with the unit sphere in $\mathbb{R}^3$.
It is the natural domain and target of [[Möbius transformations|0SG74WBFA2]] and of meromorphic functions.
</document>

<document source="https://math2.tamnd.com/node/0J7H0YBFDJ.md" tag="0J7H0YBFDJ" kind="concept">
# Endomorphism

## Summary

An endomorphism of a vector space is a linear map from the space to itself.
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<document source="https://math2.tamnd.com/node/0J7HF2PF1H.md" tag="0J7HF2PF1H" kind="definition">
# Picard–Lindelöf hypotheses

## Definition

Let $t_{\min} \le t_0 \le t_{\max}$ be real numbers, let $J = [t_{\min}, t_{\max}]$, let $x_0 \in \mathbb{R}^n$, and let $a, L, K \ge 0$.
Let $B = \{x \in \mathbb{R}^n : \|x - x_0\| \le a\}$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function.
The function $f$ satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$ if these four conditions hold.
(1) For each $t \in J$, the map $x \mapsto f(t, x)$ is $K$-Lipschitz on $B$ ([[Lipschitz condition|055VVSN00V]]).
(2) For each $x \in B$, the map $t \mapsto f(t, x)$ is continuous on $J$.
(3) For each $t \in J$ and $x \in B$, $\|f(t, x)\| \le L$.
(4) $L \cdot \max(t_{\max} - t_0,\ t_0 - t_{\min}) \le a$.

## Remarks

Conditions (1) and (2) together imply that $f$ is continuous on $J \times B$.
Condition (4) makes the time interval short enough. A curve that starts at $x_0$ with speed at most $L$ cannot leave $B$ during the time interval $J$.
Under these hypotheses, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique solution of $x' = f(t, x)$, $x(t_0) = x_0$ on $J$.
A [[$C^1$ vector field satisfies these hypotheses|03Z9V0Y78A]] on a small time interval.

## Explanations

### L1 human


These are the conditions that make the proof of existence and uniqueness work.
The field must be continuous in time, Lipschitz in space and bounded near the start point.
The time interval must be so short that the solution stays in the region where we control the field.
</document>

<document source="https://math2.tamnd.com/node/0J7PHRBNNS.md" tag="0J7PHRBNNS" kind="statement">
# Logarithm of a quotient

## Statement

For all real $x > 0$ and $y > 0$,
$$\ln \frac{x}{y} = \ln x - \ln y.$$

## Remarks

Apply the [[product rule for logarithms|0K2FNY6DYV]] to $x = \frac{x}{y} \cdot y$.
With $x = 1$ it gives $\ln(1/y) = -\ln y$.
Mathlib states it as `Real.log_div`.

## Explanations

### L1 human

The logarithm of a quotient is the difference of the logarithms.
Division becomes subtraction.
</document>

<document source="https://math2.tamnd.com/node/0J817RTVG6.md" tag="0J817RTVG6" kind="definition">
# Primitive root of unity

## Definition

Let $n$ be a positive integer.
A complex number $\zeta$ is a primitive $n$th root of unity when $\zeta^n = 1$ and $\zeta^k \neq 1$ for each integer $k$ with $1 \le k < n$.

## Remarks

Equivalently, $\zeta$ has order exactly $n$ in the multiplicative group $\mathbb{C} \setminus \{0\}$.
If $\zeta$ is a primitive $n$th root of unity, then $1, \zeta, \zeta^2, \dots, \zeta^{n-1}$ are all the $n$th [[roots of unity|0S61N75H45]].
The number $e^{2 \pi i k / n}$ is primitive exactly when $\gcd(k, n) = 1$. Thus there are $\varphi(n)$ primitive $n$th roots of unity, where $\varphi$ is Euler's totient function.

## Explanations

### L1 human

The number $i$ is a primitive $4$th root of unity: $i, i^2 = -1, i^3 = -i$ are not $1$, and $i^4 = 1$.
The number $-1$ is a $4$th root of unity, but not a primitive one, because already $(-1)^2 = 1$.
</document>

<document source="https://math2.tamnd.com/node/0J87V9K1GE.md" tag="0J87V9K1GE" kind="statement">
# Sperner's theorem

## Statement

Let $E$ be a set with $n$ elements.
Let $\mathcal{S}$ be a family of subsets of $E$ in which no set is a proper subset of another.
Then $|\mathcal{S}| \le \binom{n}{\lfloor n/2 \rfloor}$.
Equality holds if and only if $\mathcal{S}$ is the family of all subsets of size $\lfloor n/2 \rfloor$, or of all subsets of size $\lceil n/2 \rceil$.

## Remarks

A family with this property is a Sperner family, or an antichain of sets.
For each $k$, the subsets of size $k$ form an antichain. The [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$ is largest for $k = \lfloor n/2 \rfloor$.
Lubell gave a short proof in 1966, with the Lubell–Yamamoto–Meshalkin inequality.

## History

Emanuel Sperner published the theorem in 1928.

## Explanations

### L1 human

You want many subsets of a set, but no one of them may contain another.
The best you can do is to take all subsets of half the size.
</document>

<document source="https://math2.tamnd.com/node/0J8ANSYKAC.md" tag="0J8ANSYKAC" kind="definition">
# Composition of an integer

## Definition

Let $n$ be a natural number.
A composition of $n$ is a finite sequence $(a_1, \dots, a_k)$ of positive integers with $a_1 + \dots + a_k = n$.
The number $k$ is the number of parts of the composition.

## Remarks

Unlike a [[partition|0404R9P9J3]], a composition records the order of its parts.
The compositions of $3$ are $(3)$, $(2, 1)$, $(1, 2)$ and $(1, 1, 1)$.
The number $0$ has exactly one composition, the empty sequence.

## Explanations

### L1 human

A composition of $n$ writes $n$ as an ordered sum of positive integers.
So $2 + 1$ and $1 + 2$ are two different compositions of $3$.
</document>

<document source="https://math2.tamnd.com/node/0J8T4K4YTE.md" tag="0J8T4K4YTE" kind="statement">
# Taylor's theorem in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$ be an integer, and let $f \colon U \to \mathbb{R}$ be of [[class C^k|0TCBZMEYGN]] on $U$.
Let $a \in U$.
For a [[multi-index|0PJXBR6QA3]] $\alpha = (\alpha_1, \dots, \alpha_n)$, write $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $h^\alpha = h_1^{\alpha_1} \cdots h_n^{\alpha_n}$, and $D^\alpha f = \partial^{|\alpha|} f / \partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}$.
Then
$$f(a + h) = \sum_{|\alpha| \le k} \frac{D^\alpha f(a)}{\alpha!} \, h^\alpha + R_k(h), \qquad \lim_{h \to 0} \frac{R_k(h)}{\|h\|^k} = 0.$$
If $f$ is of class $C^{k+1}$ and the segment from $a$ to $a + h$ lies in $U$, then
$$R_k(h) = \sum_{|\beta| = k + 1} \frac{k + 1}{\beta!} \, h^\beta \int_0^1 (1 - t)^k \, D^\beta f(a + t h) \, dt.$$

## Remarks

The sum over $|\alpha| \le k$ is the Taylor polynomial of order $k$ of $f$ at $a$.
For $k = 1$ the statement says that $f$ is differentiable at $a$, with $f(a + h) = f(a) + \nabla f(a) \cdot h + o(\|h\|)$.
One proof applies the one-variable Taylor theorem to $g(t) = f(a + t h)$ on $[0, 1]$, together with the chain rule.
The case $k = 2$ is the [[second-order Taylor expansion|0P30YYZN85]].

## Explanations

### L1 human

Near a point, a smooth function of several variables looks like a polynomial in the displacement $h$.
The coefficients of that polynomial are the partial derivatives at the point, divided by factorials.
The error is smaller than $\|h\|^k$, so the approximation gets better quickly as $h$ shrinks.
</document>

<document source="https://math2.tamnd.com/node/0J8XE7W68F.md" tag="0J8XE7W68F" kind="definition">
# Reachability

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
The vertex $v$ is reachable from $u$ if there is a [[walk|0VJZM0F9YM]] in $G$ from $u$ to $v$.

## Remarks

Reachability is an equivalence relation on the vertices. It is reflexive by the walk of length $0$, symmetric by reversal of a walk, and transitive by concatenation.
Its equivalence classes are the [[connected components|01VVP10MWJ]] of $G$.
For a [[directed graph|0R5HTV7REV]], reachability along arcs is reflexive and transitive, but in general it is not symmetric.

## Explanations

### L1 human

A vertex is reachable from another one when you can go from the first to the second along edges.
On a map of islands with bridges, two places are reachable from each other when they are on islands linked by bridges.
</document>

<document source="https://math2.tamnd.com/node/0J8Y1NCY0K.md" tag="0J8Y1NCY0K" kind="concept">
# Limit of a function of several variables

## Summary

The limit of a function of several variables at a point $a$ is the value that $f(x)$ approaches when $x$ approaches $a$ in [[Euclidean space|0XPR1E85W0]]. The approach can be from any direction and along any route.
The distance in $\mathbb{R}^n$ measures how close $x$ is to $a$.
</document>

<document source="https://math2.tamnd.com/node/0J97P5KZY0.md" tag="0J97P5KZY0" kind="definition">
# Bernoulli distribution

## Definition

Let $p \in [0, 1]$.
The Bernoulli distribution with parameter $p$ is the distribution on $\{0, 1\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(1) = p, \qquad p(0) = 1 - p.$$
Equivalently, $p(k) = p^k (1 - p)^{1-k}$ for $k \in \{0, 1\}$. One writes $X \sim \operatorname{Bernoulli}(p)$.

## Remarks

The mean is $p$ and the variance is $p(1 - p)$.
The indicator $\mathbf{1}_A$ of an event $A$ has the Bernoulli distribution with parameter $P(A)$.
It is the distribution of the outcome of one [[Bernoulli trial|0PHC9FE2V5]], with $1$ for success. It is the [[binomial distribution|0VRWZDWW6B]] with $n = 1$.

## Explanations

### L1 human

The Bernoulli distribution describes one yes-or-no experiment: $1$ means yes and $0$ means no.
A fair coin, with $1$ for heads, has the Bernoulli distribution with $p = 1/2$.
</document>

<document source="https://math2.tamnd.com/node/0J9TVTNB4K.md" tag="0J9TVTNB4K" kind="statement">
# Determinant of a block triangular matrix

## Statement

Let $K$ be a commutative ring, let $A$ be an $m \times m$ matrix, $B$ an $m \times n$ matrix and $D$ an $n \times n$ matrix over $K$.
Then the [[block matrix|0ASDFEQS6P]] satisfies
$$\det \begin{pmatrix} A & B \\ 0 & D \end{pmatrix} = \det A \cdot \det D,$$
where $0$ is the $n \times m$ zero matrix and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The same holds for the lower block triangular matrix $\begin{pmatrix} A & 0 \\ C & D \end{pmatrix}$.
It generalizes the rule for [[triangular matrices|0K1KKMFHYH]].
For a general block matrix, $\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} \ne \det A \det D - \det B \det C$ in general.

## Explanations

### L1 human

When a matrix has a block of zeros in its lower left corner, its determinant splits.
It is the product of the determinants of the two diagonal blocks.
The block $B$ does not matter, just like the entries above the diagonal of a triangular matrix.
</document>

<document source="https://math2.tamnd.com/node/0JAEHXTVV5.md" tag="0JAEHXTVV5" kind="concept">
# Riemann sum

## Summary

A Riemann sum approximates an integral by a finite sum of areas of rectangles. Each rectangle uses the value of the function at a chosen point of a subinterval. The node [[Riemann sum|0HBY48G8G1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JARTXX4XP.md" tag="0JARTXX4XP" kind="concept">
# Euler method

## Summary

The Euler method is a numerical method to approximate the solution of an [[initial value problem|075VBY4K4X]] $y' = f(t, y)$, $y(t_0) = y_0$.
With a step size $h > 0$, it sets $t_{k+1} = t_k + h$ and $y_{k+1} = y_k + h\, f(t_k, y_k)$ for $k = 0, 1, 2, \ldots$. Then $y_k$ approximates $y(t_k)$.
It follows the tangent line of the solution for one step at a time.

## Remarks

The error of one step, the local truncation error, is $\frac{h^2}{2} y''(\tau)$ for some $\tau$ between $t_k$ and $t_{k+1}$, when $y$ is twice continuously differentiable.
Let $f$ be $L$-[[Lipschitz|055VVSN00V]] in $y$ with $L > 0$, and let $|y''| \le M$ on the interval. Then the global error satisfies $|y(t_k) - y_k| \le \frac{h M}{2 L} \left( e^{L (t_k - t_0)} - 1 \right)$. The proof uses a [[discrete Grönwall inequality|0NTM3J5MFE]].
So the method is of first order: halving the step size roughly halves the error. It is the simplest Runge–Kutta method.
Leonhard Euler proposed it in his book Institutionum calculi integralis.

## Explanations

### L1 human

At each step, we compute the slope at the current point and walk a short distance $h$ along it.
The result is a broken line that stays close to the true solution when $h$ is small.
For $y' = y$, $y(0) = 1$, the method gives $y_k = (1 + h)^k$, and with $h = 1/k$ this tends to $e$ at time $1$.
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<document source="https://math2.tamnd.com/node/0JB228DCWB.md" tag="0JB228DCWB" kind="definition">
# Image of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *image* of $T$, also called the *range*, is
$$\im T = \{T(v) : v \in V\} \subseteq W.$$

## Remarks

The image is a [[linear subspace|00YC1D652K]] of $W$.
For the map $x \mapsto Ax$ of a matrix $A$, the image is the [[column space|03AJKXH1AV]] of $A$.
The map $T$ is surjective if and only if $\im T = W$.
The dimension of the image is the [[rank|0ZZXSBTR4J]] of $T$.

## Explanations

### L1 human

The image of a linear map is everything that the map can produce.
If a map sends ordinary space onto a plane, then its image is that plane.
The image is always a subspace.
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<document source="https://math2.tamnd.com/node/0JB3DP1BMF.md" tag="0JB3DP1BMF" kind="definition">
# Linear combination

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
Let $v_1, \dots, v_k \in V$.
A *linear combination* of $v_1, \dots, v_k$ is a vector of the form
$$a_1 v_1 + a_2 v_2 + \cdots + a_k v_k,$$
where $a_1, \dots, a_k \in F$ are the *coefficients*.
For an arbitrary family $(v_i)_{i \in I}$, a linear combination is a sum $\sum_{i \in I} a_i v_i$ in which only finitely many coefficients $a_i$ are nonzero.

## Remarks

The empty linear combination, with $k = 0$, is the zero vector.
The vector $(3, 5)$ is the linear combination $3 e_1 + 5 e_2$ of the standard basis vectors of $F^2$.
The set of all linear combinations of a family is its [[linear span|0DTY8B9HMS]].
The [[matrix–vector product|0PDVQEP0ZC]] $Ax$ is a linear combination of the columns of $A$, with the entries of $x$ as coefficients.

## Explanations

### L1 human

A linear combination mixes some vectors: scale each one by a number, then add the results.
With the vectors $(1, 0)$ and $(0, 1)$ we can make every vector of the plane in this way.
Infinite sums are not allowed: only finitely many coefficients can be nonzero.
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<document source="https://math2.tamnd.com/node/0JBJYT69ZV.md" tag="0JBJYT69ZV" kind="concept">
# Graph of a function

## Summary

The graph of a function is the set of all pairs of an input and its value.
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<document source="https://math2.tamnd.com/node/0JBMYAA9QH.md" tag="0JBMYAA9QH" kind="statement">
# Ratio formula for the radius of convergence

## Statement

Let $\sum a_n (x - c)^n$ be a real power series with $a_n \ne 0$ for all large $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for some $L$ in $[0, \infty]$.
Then the [[radius of convergence|0660QX1VPR]] of the series is $R = 1/L$, with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

The proof applies the [[ratio test|0QNY6ZBXPN]] to $\sum a_n (x - c)^n$. The ratio of consecutive terms tends to $L |x - c|$.
The formula needs the limit to exist. The [[Cauchy–Hadamard theorem|07E5MB1CA2]] works without this assumption.
For example, $\sum x^n / n!$ has $L = 0$ and $R = \infty$.
Mathlib states a form for nonzero $L$ as `FormalMultilinearSeries.ofScalars_radius_eq_inv_of_tendsto`.

## Explanations

### L1 human

When the ratio of consecutive coefficients settles to $L$, the coefficients behave like $L^n$.
So the series behaves like a geometric series with ratio $L |x - c|$.
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<document source="https://math2.tamnd.com/node/0JC5X4HWR2.md" tag="0JC5X4HWR2" kind="statement">
# Vector calculus identities

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, let $\phi, \psi \colon U \to \mathbb{R}$ be scalar fields, and let $A \colon U \to \mathbb{R}^3$ be a vector field.
If these fields are of class $C^1$, then
$$\nabla (\phi \psi) = \phi \nabla \psi + \psi \nabla \phi, \qquad \nabla \cdot (\psi A) = \psi \, \nabla \cdot A + \nabla \psi \cdot A, \qquad \nabla \times (\psi A) = \psi \, \nabla \times A + \nabla \psi \times A.$$
If these fields are of class $C^2$, then
$$\nabla \times (\nabla \phi) = 0, \qquad \nabla \cdot (\nabla \times A) = 0, \qquad \nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \Delta A,$$
where $\Delta A$ applies the Laplacian to each component of $A$.

## Remarks

Here $\nabla \phi$ is the [[gradient|0NS2QYVC4Y]], $\nabla \cdot A$ the [[divergence|02NSMQVQWK]], $\nabla \times A$ the [[curl|0GDNHS9J2X]], and $\Delta$ the [[Laplacian|0XQ2ZVN1PZ]].
The product rules follow from the product rule for partial derivatives, applied to each component.
The identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]] follow from the symmetry of second derivatives.
Also, $\nabla \cdot (\nabla \phi) = \Delta \phi$ and $\Delta (\phi \psi) = \phi \Delta \psi + 2 \nabla \phi \cdot \nabla \psi + \psi \Delta \phi$.

## Explanations

### L1 human

These identities are the algebra rules of vector calculus.
They let you expand derivatives of products and simplify repeated derivatives, the way the ordinary product rule does in one variable.
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<document source="https://math2.tamnd.com/node/0JCQGPY8Z4.md" tag="0JCQGPY8Z4" kind="concept">
# Stirling numbers of the second kind

## Summary

The Stirling numbers of the second kind $S(n, k)$ count the partitions of an $n$-element set into $k$ nonempty blocks. They connect ordinary powers with falling factorials and they sum to the Bell numbers. The node [[Stirling numbers of the second kind|0ZN07RQ3JS]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0JCSQNNP2M.md" tag="0JCSQNNP2M" kind="definition">
# Converse relation

## Definition

Let $R \subseteq A \times B$ be a [[binary relation|08EJC6Q25H]] from $A$ to $B$.
The converse of $R$ is the relation from $B$ to $A$
$$R^{-1} = \{(b, a) \in B \times A \mid (a, b) \in R\}.$$
So $b \mathrel{R^{-1}} a$ if and only if $a \mathrel{R} b$.

## Remarks

Other names are the inverse relation and the transpose. Another notation is $R^{T}$.
The converse of $<$ on $\mathbb{R}$ is $>$.
The converse of the converse of $R$ is $R$.
For a [[bijection|0SNQEF7Z53]] $f$, the converse of the graph of $f$ is the graph of the [[inverse function|0W766JAPSN]].

## Explanations

### L1 human

To get the converse, read each pair backwards.
The converse of "is a child of" is "is a parent of".
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<document source="https://math2.tamnd.com/node/0JD5C82QMB.md" tag="0JD5C82QMB" kind="definition">
# Graph

## Definition

A graph, or simple graph, is a pair $G = (V, E)$ of a set $V$ and a set $E$ of $2$-element subsets of $V$.
The elements of $V$ are the vertices of $G$, and the elements of $E$ are its edges.
Two vertices $u$ and $v$ are adjacent if $\{u, v\} \in E$. The notation for this edge is $uv$, and the notation for adjacency is $u \sim v$.

Equivalently, a graph is a set $V$ with a relation $\sim$ on $V$ that is symmetric and irreflexive.
A graph is finite if $V$ is finite.

## Remarks

In a simple graph, no edge joins a vertex to itself, and two vertices are joined by at most one edge. A [[multigraph|01M5T0KDG7]] drops the second condition, and some variants allow loops.
In a [[directed graph|0R5HTV7REV]] the edges are ordered pairs.
The set $E$ is the [[edge set|0WNGQBPPQX]] of $G$.

## Explanations

### L1 human

A graph is a set of points with lines between some pairs of points.
For example, the people at a party are vertices, and two people are joined by an edge when they know each other.
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<document source="https://math2.tamnd.com/node/0JD5ZNW0F4.md" tag="0JD5ZNW0F4" kind="definition">
# Analytic function

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$.
The function $f$ is analytic at $a \in I$ if there are $r > 0$ and a [[power series|0VP56B4EMX]] $\sum c_n (x - a)^n$ that converges to $f(x)$ for all $x$ with $|x - a| < r$.
It is analytic on $I$ if it is analytic at each point of $I$.

## Remarks

An analytic function is [[smooth|0BE3QDH73A]], and its power series at $a$ is its [[Taylor series|0225QPWRRE]] at $a$.
Polynomials, the exponential function, sine, and cosine are analytic on $\mathbb{R}$.
A smooth function need not be analytic. See the [[smooth non-analytic function|09RTYYHXKA]].
Mathlib defines analyticity at a point as `AnalyticAt`.

## Explanations

### L1 human

An analytic function equals a convergent power series near each point.
So it is determined near a point by its derivatives at that point.
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<document source="https://math2.tamnd.com/node/0JDAQHWA9N.md" tag="0JDAQHWA9N" kind="definition">
# Convergence in distribution

## Definition

Let $X, X_1, X_2, \dots$ be real random variables, possibly on different probability spaces, with [[cumulative distribution functions|0MXMS3AE13]] $F, F_1, F_2, \dots$.
The sequence $(X_n)$ converges in distribution to $X$ if
$$\lim_{n \to \infty} F_n(x) = F(x)$$
for each $x \in \mathbb{R}$ at which $F$ is continuous. One writes $X_n \xrightarrow{d} X$.

## Remarks

By the portmanteau theorem, an equivalent condition is $E[g(X_n)] \to E[g(X)]$ for each bounded continuous $g \colon \mathbb{R} \to \mathbb{R}$. This form extends to random variables in metric spaces.
The condition depends only on the laws. So it is the weak convergence of the laws of the $X_n$ to the law of $X$.
The restriction to continuity points is necessary. For $X_n = 1/n$ and $X = 0$, $F_n(0) = 0$ for all $n$, but $F(0) = 1$.
The central limit theorem is a result about convergence in distribution.

## Explanations

### L1 human

Convergence in distribution means that the probabilities for $X_n$ come close to those for $X$. The values of $X_n$ themselves need not come close to the values of $X$.
For example, the number of heads in $n$ tosses, after standardization, converges in distribution to a standard normal variable.
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<document source="https://math2.tamnd.com/node/0JDB8YJBSR.md" tag="0JDB8YJBSR" kind="definition">
# Triangle

## Definition

Let $K$ be a field and let $\mathcal{A}$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space.
A triangle in $\mathcal{A}$ is an ordered triple $(A, B, C)$ of points of $\mathcal{A}$ that are not [[collinear|0AFNNTRTJE]]. It is written $ABC$ or $\triangle ABC$.
The points $A$, $B$ and $C$ are its vertices. The segments $BC$, $CA$ and $AB$ are its sides.

## Remarks

Three points are not collinear if and only if they are affinely independent: the vectors $B - A$ and $C - A$ are linearly independent.
In a Euclidean space, the side lengths are often written $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$. The angles of the triangle are $\angle BAC$, $\angle CBA$ and $\angle ACB$.
A triangle is isosceles if two sides have the same length, equilateral if all three do, and scalene if no two do. It is a right triangle if one angle is $\pi / 2$.
Mathlib's `Affine.Triangle` is an affinely independent family of three points, indexed by $\{0, 1, 2\}$.

## Explanations

### L1 human

Pick three points that are not on one straight line and join each pair. The result is a triangle. If the three points were on one line, the "triangle" would be flat, and it would have no area.
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<document source="https://math2.tamnd.com/node/0JDE56HWQ8.md" tag="0JDE56HWQ8" kind="definition">
# Path

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A path in $G$ is a [[walk|0VJZM0F9YM]] $(w_0, w_1, \dots, w_k)$ whose vertices $w_0, w_1, \dots, w_k$ are pairwise distinct.

## Remarks

A path uses each edge at most once, so every path is a [[trail|062MQGP8M5]].
A path of length $k$ has $k + 1$ vertices. Its vertices and edges form a subgraph isomorphic to the [[path graph|075WR6W3Y3]] $P_{k+1}$.
Each walk from $u$ to $v$ contains a path from $u$ to $v$. The node [[every walk contains a path|09VZBHB33V]] states this.

## Explanations

### L1 human

A path is a walk that never visits the same vertex twice.
On a map, it is a route that never passes through the same crossing again.
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<document source="https://math2.tamnd.com/node/0JEJATBVD7.md" tag="0JEJATBVD7" kind="concept">
# Continuity in R^n

## Summary

Take a function between subsets of Euclidean spaces, and a point $a$. The function is continuous at this point when small changes of the input near $a$ give small changes of the value.
For a function of several variables, the input can change in all coordinates at once.
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# Sign is a homomorphism

## Statement

Let $n \geq 2$.
The map $\operatorname{sgn} \colon S_n \to \{1, -1\}$ is a surjective [[group homomorphism|0JYTG5K9Y4]], where $\{1, -1\}$ is a group under multiplication.
That is, $\operatorname{sgn}(\sigma \tau) = \operatorname{sgn}(\sigma) \operatorname{sgn}(\tau)$ for all $\sigma, \tau \in S_n$.

## Remarks

If $\sigma$ is a product of $m$ transpositions and $\tau$ is a product of $l$ transpositions, then $\sigma \tau$ is a product of $m + l$ transpositions.
The map is surjective because $\operatorname{sgn}((1\ 2)) = -1$.
Its [[kernel|0W88CEGQP1]] is the [[alternating group|0Z5905Q73D]] $A_n$.

## Explanations

### L1 human

When you do two permutations one after the other, you join their chains of swaps.
The number of swaps adds, so the signs multiply.
This is exactly the rule of a homomorphism.
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<document source="https://math2.tamnd.com/node/0JEVZAFDY6.md" tag="0JEVZAFDY6" kind="statement">
# The cumulative distribution function determines the distribution

## Statement

Let $\mu$ and $\nu$ be [[probability distributions|0QB913WM5E]] on $\mathbb{R}$ with its Borel σ-algebra.
If $\mu((-\infty, x]) = \nu((-\infty, x])$ for each $x \in \mathbb{R}$, then $\mu(B) = \nu(B)$ for each Borel set $B \subseteq \mathbb{R}$.

## Remarks

In other words, two real random variables with the same [[cumulative distribution function|0MXMS3AE13]] are identically distributed.
The half-lines $(-\infty, x]$ are closed under finite intersections and generate the [[Borel σ-algebra|0BVHZXWKCD]]. The proof uses Dynkin's π-λ theorem to pass from these half-lines to all Borel sets.
So to describe a real distribution, it is enough to give its cumulative distribution function.

## Explanations

### L1 human

If you know $P(X \le x)$ for every $x$, then you know the probability of every reasonable event about $X$.
For example, $P(a < X \le b)$ is a difference of two values of the function, and more complex sets come from such intervals.
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<document source="https://math2.tamnd.com/node/0JF0TVYGWT.md" tag="0JF0TVYGWT" kind="concept">
# Affine map

## Summary

An affine map is a map between [[affine spaces|09XNAPTJQ7]] that is a linear map followed by a translation. It sends lines to lines or points, and it keeps parallel lines parallel and ratios of lengths along a line.
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<document source="https://math2.tamnd.com/node/0JF1XXGWB4.md" tag="0JF1XXGWB4" kind="definition">
# Parametrised curve

## Definition

Let $I \subseteq \mathbb{R}$ be an interval with more than one point.
A parametrised curve in $\mathbb{R}^n$ is a [[continuous|0S9EK5MQPE]] map $\gamma \colon I \to \mathbb{R}^n$, $\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t))$.
Its image, or trace, is the set $\gamma(I) = \{\gamma(t) : t \in I\}$.
For $r \ge 1$, the curve is a $C^r$ curve if each component $\gamma_i$ is $r$ times continuously differentiable on $I$.
When $I = [a, b]$, the points $\gamma(a)$ and $\gamma(b)$ are the initial point and the end point of $\gamma$.

## Remarks

A curve is different from its image. The maps $t \mapsto (\cos t, \sin t)$ on $[0, 2\pi]$ and $t \mapsto (\cos 2t, \sin 2t)$ on $[0, 2\pi]$ have the same image, the unit circle, but the second goes around it twice.
The helix $\gamma(t) = (\cos t, \sin t, t)$, $t \in \mathbb{R}$, is a $C^\infty$ curve in $\mathbb{R}^3$.
The curve is simple if $\gamma$ is injective, and it is closed if $I = [a, b]$ and $\gamma(a) = \gamma(b)$.

## Explanations

### L1 human

A parametrised curve tells where a moving point is at each time $t$.
It records both the path and the way the point travels along it, including its speed and direction.
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# Regular space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is regular when the property that follows holds.
For each [[closed set|0G5AV9Q4RR]] $F \subseteq X$ and each point $x \in X \setminus F$, there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $x \in U$, $F \subseteq V$ and $U \cap V = \varnothing$.

## Remarks

Names vary. Some authors call a space T3 when it is regular and T0, and then it is also [[Hausdorff|0MXB4CKHFW]]. Other authors include this condition in the word regular.
Equivalently, $X$ is regular if and only if each neighbourhood of a point $x$ contains a closed neighbourhood of $x$.
Each metric space is regular. Each [[normal|0AJ646N339]] T1 space is regular.

## Explanations

### L1 human

In a regular space, you can separate a point from a closed set that does not contain it, by two disjoint open sets.
Hausdorff spaces only separate points from points.
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# Residue

## Definition

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < r$, so that $c$ is an [[isolated singularity|0TYB3HM9KE]] of $f$.
Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the [[Laurent expansion|06WHRARZKY]] of $f$ on this punctured disc.
The residue of $f$ at $c$ is the coefficient
$$\operatorname{Res}(f, c) = a_{-1}.$$

## Remarks

For each $\rho$ with $0 < \rho < r$, $\operatorname{Res}(f, c) = \frac{1}{2 \pi i} \oint_{|z - c| = \rho} f(z) \, dz$. This follows from the coefficient formula for $n = -1$.
The residue is $0$ at a [[removable singularity|0P18JC72RE]]. It can be $0$ at a pole too: $\operatorname{Res}(1/z^2, 0) = 0$.
For example, $\operatorname{Res}(1/z, 0) = 1$ and $\operatorname{Res}(e^{1/z}, 0) = 1$.
Residues at simple poles are easy to compute, see [[residue at a simple pole|01991NCP7H]].

## Explanations

### L1 human

When you integrate a Laurent series term by term around a small circle, every term gives $0$ except the term $a_{-1}/(z - c)$, which gives $2 \pi i a_{-1}$.
So the residue is the one coefficient that the integral remembers.
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<document source="https://math2.tamnd.com/node/0JG1KZSPFZ.md" tag="0JG1KZSPFZ" kind="statement">
# Turán's theorem

## Statement

Let $n \ge r \ge 1$ be integers, and let $G$ be a simple [[graph|0JD5C82QMB]] with $n$ vertices that has no [[clique|0G52BPNPKG]] with $r + 1$ vertices.
Then $G$ has at most as many edges as the [[Turán graph|0VTCABADBE]] $T(n, r)$.
In particular, $G$ has at most $\left(1 - \frac{1}{r}\right) \frac{n^2}{2}$ edges.

## Remarks

The bound is sharp, because $T(n, r)$ has no clique with $r + 1$ vertices.
The case $r = 2$ is [[Mantel's theorem|0MXEHSCYCS]] on graphs without triangles.
The theorem is a central result of extremal graph theory.

## History

Pál Turán described the theorem and the Turán graphs in 1941.

## Explanations

### L1 human

How many edges can a graph have if no $r + 1$ of its vertices are all joined to each other?
The answer is the Turán graph: split the vertices into $r$ balanced teams and join all pairs across teams.
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<document source="https://math2.tamnd.com/node/0JG9W3KA6B.md" tag="0JG9W3KA6B" kind="concept">
# Connected component

## Summary

A connected component of a graph is a maximal piece in which all vertices are reachable from each other. The components split each graph into connected parts. The node [[connected component|01VVP10MWJ]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0JGB7A6GTS.md" tag="0JGB7A6GTS" kind="definition">
# Rank of a linear map

## Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *rank* of $T$ is
$$\rank T = \dim \im T,$$
the [[dimension|03GSZ8RTZJ]] of the [[image|0WSFKKT6FF]] of $T$.

## Remarks

For the map $x \mapsto Ax$ of a matrix $A$, the rank is the [[rank of the matrix|0ZC9YTXZRY]].
The rank satisfies $\rank T \le \dim V$ and $\rank T \le \dim W$.
The [[rank–nullity theorem|0W4FSW1TVB]] relates the rank and the [[nullity|0APCFW7639]].
The rank of a composition satisfies $\rank(T \circ S) \le \min(\rank T, \rank S)$.

## Explanations

### L1 human

The rank of a linear map counts the dimensions of its output.
A map from space onto a plane has rank $2$.
The zero map has rank $0$.
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<document source="https://math2.tamnd.com/node/0JGEQXQ4AC.md" tag="0JGEQXQ4AC" kind="concept">
# Presentation of a group

## Summary

A presentation $\langle S \mid R \rangle$ describes a group by a set $S$ of generators and a set $R$ of relations.
The group is the quotient of the free group on $S$ by the normal subgroup generated by $R$.
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<document source="https://math2.tamnd.com/node/0JGZCDT5DN.md" tag="0JGZCDT5DN" kind="example">
# Space of polynomials of bounded degree

## Example

Let $F$ be a field and let $n \ge 0$ be an integer.
Let $F[x]_{<n}$ be the set of polynomials in $x$ with coefficients in $F$ and of degree less than $n$, with the zero polynomial included.
It is a [[vector space|000AYPA9AJ]] over $F$, with the usual addition of polynomials and multiplication by constants.
The monomials $1, x, x^2, \dots, x^{n-1}$ form a [[basis|0KCQ9Y3QKA]], so $\dim F[x]_{<n} = n$.

## Remarks

The set of polynomials of degree exactly $n$ is not a vector space.
It does not contain $0$, and it is not closed under addition.
The whole polynomial ring $F[x]$ is an infinite-dimensional vector space with basis $1, x, x^2, \dots$.
[[Differentiation|0SS770NS8B]] is a linear map from $F[x]_{<n}$ to $F[x]_{<n-1}$ for $n \ge 1$.

## Explanations

### L1 human

Polynomials of small degree can be added and scaled, and the result stays a polynomial of small degree.
So they form a vector space.
A polynomial $a_0 + a_1 x + a_2 x^2$ is the same data as the list $(a_0, a_1, a_2)$.
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# Series

## Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a [[real sequence|0CTV783HYT]].
The series with terms $a_n$ is the sequence $(s_N)_{N \in \mathbb{N}}$ given by
$$s_N = a_0 + a_1 + \cdots + a_N = \sum_{n=0}^{N} a_n.$$
The notation for the series is $\sum_{n=0}^{\infty} a_n$, or $\sum a_n$.

## Remarks

The number $s_N$ is a [[partial sum|0NB643K60Q]] of the series.
The symbol $\sum_{n=0}^{\infty} a_n$ also denotes the [[sum of the series|0K0H5E3Z5N]], when the series is convergent. The context shows which meaning is intended.
A series can start at another index, for example $\sum_{n=1}^{\infty} 1/n$.

## Explanations

### L1 human

A series is an infinite sum $a_0 + a_1 + a_2 + \cdots$.
We cannot add infinitely many numbers directly, so we add the first terms and watch what happens as we add more.
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<document source="https://math2.tamnd.com/node/0JHGAV9H25.md" tag="0JHGAV9H25" kind="statement">
# Eigenspaces of a self-adjoint operator are orthogonal

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be a [[self-adjoint|0NEGN5CRSC]] linear map.
Let $v$ and $w$ be eigenvectors of $T$ for distinct eigenvalues $\lambda \ne \mu$.
Then $v$ and $w$ are [[orthogonal|0Y2X7P2WB7]].
So the [[eigenspaces|0G0RFPG8HR]] of $T$ for distinct eigenvalues are orthogonal to each other.

## Remarks

The proof: $\lambda \langle v, w \rangle = \langle T(v), w \rangle = \langle v, T(w) \rangle = \mu \langle v, w \rangle$, since $\mu$ is real.
So $(\lambda - \mu) \langle v, w \rangle = 0$, and $\langle v, w \rangle = 0$.

## Explanations

### L1 human

For a symmetric matrix, eigenvectors for different eigenvalues are perpendicular.
So the eigenspaces meet at right angles.
This is the first step towards an orthonormal basis of eigenvectors.
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<document source="https://math2.tamnd.com/node/0JHM6JK7FD.md" tag="0JHM6JK7FD" kind="example">
# The ring of integers

## Example

Let $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ be the set of integers, with the usual addition and multiplication.
Then $\mathbb{Z}$ is a [[commutative ring|0GYGHKX834]].
Its zero element is $0$ and its identity element is $1$.

## Remarks

The only [[units|019NFGFPR5]] of $\mathbb{Z}$ are $1$ and $-1$.
The product of two nonzero integers is not zero, thus $\mathbb{Z}$ is an [[integral domain|09S3PFBEPM]].
The ring $\mathbb{Z}$ is not a [[field|0QA8CHH8MG]], because $2$ has no multiplicative inverse in $\mathbb{Z}$.

## Explanations

### L1 human

The integers are the first example of a ring.
You can add, subtract and multiply two integers, and the result is an integer.
You cannot always divide: $1 / 2$ is not an integer.
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<document source="https://math2.tamnd.com/node/0JHXPVFKT0.md" tag="0JHXPVFKT0" kind="concept">
# Exponential generating function

## Summary

An exponential generating function encodes a sequence $(a_n)$ as the series $\sum a_n x^n / n!$. It is the natural tool to count labelled structures, because products give binomial convolutions. The node [[exponential generating function|0KE6NBTVY8]] gives the precise definition.
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<document source="https://math2.tamnd.com/node/0JJ6GDDE7Z.md" tag="0JJ6GDDE7Z" kind="concept">
# Underdamped

## Summary

A [[damped harmonic oscillator|0RNXVVTJTJ]] $x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = 0$ with $\omega_0 > 0$ is underdamped when its damping ratio satisfies $0 < \zeta < 1$.
Then the characteristic roots are $-\zeta \omega_0 \pm i \omega_1$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$.
Each nonzero solution has the form $x(t) = A e^{-\zeta \omega_0 t} \cos(\omega_1 t + \varphi)$ with $A > 0$: an oscillation whose amplitude decays exponentially.

## Remarks

The other cases are critical damping, $\zeta = 1$, and overdamping, $\zeta > 1$. In these cases the solution changes sign at most one time.
The damped frequency $\omega_1$ is smaller than the undamped frequency $\omega_0$.
The zeros of an underdamped solution are spaced $\pi / \omega_1$ apart.

## Explanations

### L1 human

An underdamped system overshoots its rest position again and again.
Each swing is smaller than the previous one, by the same factor.
A guitar string or a car on soft springs behaves like this.
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<document source="https://math2.tamnd.com/node/0JJ9J1XZ8G.md" tag="0JJ9J1XZ8G" kind="definition">
# Boundary

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The boundary of $S$ is the [[set difference|0DT0TCV75N]]
$$\partial S = \overline{S} \setminus \operatorname{int} S,$$
where $\overline{S}$ is the [[closure|0DRKT1QJR7]] and $\operatorname{int} S$ is the [[interior|0C38S0KMJK]] of $S$.

## Remarks

A point $x$ is in $\partial S$ if and only if each open set that contains $x$ meets both $S$ and $X \setminus S$.
The boundary is closed, and $\partial S = \partial (X \setminus S)$.
In $\mathbb{R}$, the boundary of $(0, 1)$ is $\{0, 1\}$, and the boundary of $\mathbb{Q}$ is $\mathbb{R}$.
Another name for the boundary is the frontier.

## Explanations

### L1 human

The boundary is the edge of a set. It has the points where you can step into the set and also out of it with a very small step.
For a disc in the plane, the boundary is its circle.
</document>

<document source="https://math2.tamnd.com/node/0JJRENWMP6.md" tag="0JJRENWMP6" kind="statement">
# Cauchy's estimates

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$.
Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ with $|z - c| = r$.
Then for each integer $n \ge 0$,
$$|f^{(n)}(c)| \le \frac{n! \, M}{r^n}.$$

## Remarks

Proof: apply the [[estimation lemma|0DJMGPN8K9]] to the [[formula for the derivatives|0F0H5P5DED]] with $a = c$. The circle has length $2 \pi r$, and $|f(z) / (z - c)^{n+1}| \le M / r^{n+1}$ on it.
For $n = 1$, the estimate is $|f'(c)| \le M / r$.
If $f(z) = \sum a_n (z - c)^n$ is the Taylor series, the estimate is $|a_n| \le M / r^n$.

## Explanations

### L1 human

If a holomorphic function is small on a circle, then its derivatives at the center cannot be large.
A larger circle with the same bound $M$ gives a better bound for the derivatives. This is the idea of [[Liouville's theorem|0EHKVRF4J6]].
</document>

<document source="https://math2.tamnd.com/node/0JJV80RXHD.md" tag="0JJV80RXHD" kind="statement">
# Abelian simple groups have prime order

## Statement

Let $G$ be an [[abelian|082GHNPJ53]] [[group|G5S1M9]].
Then $G$ is [[simple|0YMYMNQ0QZ]] if and only if $G$ is finite of prime order.
In that case, $G$ is [[cyclic|0KVGH0J5B3]], and $G \cong \mathbb{Z}/p\mathbb{Z}$ for a prime number $p$.

## Remarks

In an abelian group, each subgroup is normal, so a simple abelian group has no subgroup except $\{e\}$ and itself.
Then $G = \langle g \rangle$ for each $g \neq e$, so $G$ is cyclic; $\mathbb{Z}$ is not simple, because $2\mathbb{Z}$ is a proper nontrivial subgroup.
A finite cyclic group of order $n$ has a subgroup of order $d$ for each divisor $d$ of $n$, so $n$ must be prime.

## Explanations

### L1 human

In an abelian group, all subgroups are normal.
So a simple abelian group has no subgroups at all except the trivial ones.
Only the groups of prime order have this property.
</document>

<document source="https://math2.tamnd.com/node/0JK6M18QZH.md" tag="0JK6M18QZH" kind="definition">
# Kronecker product

## Definition

Let $F$ be a field, let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $F$, and let $B$ be a $p \times q$ matrix over $F$.
The *Kronecker product* $A \otimes B$ is the $mp \times nq$ [[block matrix|0ASDFEQS6P]]
$$A \otimes B = \begin{pmatrix} a_{11} B & \cdots & a_{1n} B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{pmatrix}.$$

## Remarks

It satisfies $(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$ when the products $AC$ and $BD$ are defined.
If $A$ and $B$ are invertible, then $(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$.
It is not commutative in general, but $A \otimes B$ and $B \otimes A$ are permutation equivalent.
It is the matrix of the tensor product of two linear maps, in the tensor product bases ordered suitably.

## Explanations

### L1 human

The Kronecker product replaces each entry of $A$ with that entry times the whole matrix $B$.
For example, $I_2 \otimes B$ is the block diagonal matrix with two copies of $B$.
It is the matrix form of the tensor product.
</document>

<document source="https://math2.tamnd.com/node/0JKW7W7XFR.md" tag="0JKW7W7XFR" kind="concept">
# Combinatorics

## Summary

Combinatorics is the part of mathematics that studies finite or countable discrete structures.
It asks three main types of question about such structures.
Enumerative questions ask how many structures of a given type exist.
Existence questions ask if a structure with given properties exists.
Extremal questions ask how large or small a structure with given properties can be.

Its basic objects are finite sets, arrangements of elements, subsets, partitions and graphs.
Its basic tools are counting rules, bijections, recurrences, generating functions and the pigeonhole principle.

## Remarks

The [[rule of sum|0EE2V3CHFK]] and the [[rule of product|0VPK4X73Q0]] are the first counting rules.
[[Binomial coefficients|092GEF38YD]] count the subsets of a given size of a finite set.
Graph theory is a large subfield of combinatorics. Its basic notion is a [[graph|0JD5C82QMB]].

## Explanations

### L1 human

Combinatorics is the mathematics of counting and arranging.
A typical question is: in how many ways can 5 people sit in a row?
The answer is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$, and combinatorics gives general methods for questions of this type.
</document>

<document source="https://math2.tamnd.com/node/0JM7PA288B.md" tag="0JM7PA288B" kind="statement">
# Cauchy's theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number that divides $|G|$.
Then $G$ has an element of [[order|0WE2NHZJ7B]] $p$.

## Remarks

Equivalently, $G$ has a [[subgroup|0X1JXT47GA]] of order $p$, namely the cyclic subgroup generated by such an element.
It is a partial converse of [[Lagrange's theorem|3C53TD]]: a prime divisor of $|G|$ is always the order of a subgroup.
One proof lets $\mathbb{Z}/p\mathbb{Z}$ act by cyclic shifts on the set of tuples $(g_1, \dots, g_p)$ with $g_1 g_2 \cdots g_p = e$, and uses the [[fixed point lemma|098D2E7DEH]].
The [[first Sylow theorem|0V98WY94B8]] generalises the result to prime powers.

## History

Augustin-Louis Cauchy discovered the theorem in 1845.

## Explanations

### L1 human

Lagrange says that the order of an element divides the order of the group.
Cauchy says that, for prime numbers, each divisor really appears as the order of an element.
For example, a group of order $15$ has an element of order $3$ and an element of order $5$.
</document>

<document source="https://math2.tamnd.com/node/0JMR2HM2N2.md" tag="0JMR2HM2N2" kind="concept">
# Logical equivalence

## Summary

Two formulas are logically equivalent when they have the same truth value in every case.
</document>

<document source="https://math2.tamnd.com/node/0JMTY8MF54.md" tag="0JMTY8MF54" kind="concept">
# Sum of subspaces

## Summary

The sum of two subspaces is the set of all sums of a vector of the first subspace and a vector of the second subspace.
It is the smallest subspace that contains both.
</document>

<document source="https://math2.tamnd.com/node/0JN0SQG2JH.md" tag="0JN0SQG2JH" kind="concept">
# Kernel of a ring homomorphism

## Summary

The kernel of a ring homomorphism is the set of the elements that it sends to zero.
It is an ideal of the domain of the homomorphism.
The [[definition of the kernel|0AAHE7GEW6]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0JN9P4A944.md" tag="0JN9P4A944" kind="definition">
# Sample variance

## Definition

Let $n \ge 2$ and let $X_1, \dots, X_n$ be real random variables with [[sample mean|0QNRBHWV35]] $\bar{X}_n$.
The sample variance is the [[statistic|036756NFD6]]
$$S_n^2 = \frac{1}{n - 1} \sum_{i=1}^{n} \left(X_i - \bar{X}_n\right)^2.$$
The sample standard deviation is $S_n = \sqrt{S_n^2}$.

## Remarks

If the $X_i$ are i.i.d. with [[variance|007QED3E0Y]] $\sigma^2 < \infty$, then $E[S_n^2] = \sigma^2$. So $S_n^2$ is an unbiased estimator of $\sigma^2$.
The divisor $n - 1$ instead of $n$ is Bessel's correction. The version with divisor $n$ has expectation $\frac{n-1}{n} \sigma^2$, so it has the [[bias|01BD2TFFP4]] $-\sigma^2/n$.
The reason is that $\sum_i (X_i - \bar{X}_n)^2 \le \sum_i (X_i - \mu)^2$: the deviations from $\bar{X}_n$ are on average smaller than those from the true mean $\mu$.
For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $(n - 1) S_n^2 / \sigma^2$ has the chi-squared distribution with $n - 1$ degrees of freedom, and it is independent of $\bar{X}_n$.



## Explanations

### L1 human

The sample variance is the average squared distance of the data from their mean, but with the divisor $n - 1$.
For the data $4$, $7$, $10$, the mean is $7$, the squared deviations add up to $18$, and the sample variance is $18/2 = 9$.
</document>

<document source="https://math2.tamnd.com/node/0JNCBFMPXR.md" tag="0JNCBFMPXR" kind="statement">
# Lindemann–Weierstrass theorem

## Statement

Let $\alpha_1, \dots, \alpha_n$ be distinct [[algebraic numbers|0MB4TGMXG6]].
Then $e^{\alpha_1}, \dots, e^{\alpha_n}$ are linearly independent over the field of algebraic numbers.
That is, if $\beta_1, \dots, \beta_n$ are algebraic numbers with $\beta_1 e^{\alpha_1} + \dots + \beta_n e^{\alpha_n} = 0$, then $\beta_1 = \dots = \beta_n = 0$.

## Remarks

A consequence: $e^{\alpha}$ is [[transcendental|079EGSD32Y]] for each nonzero algebraic number $\alpha$. In particular, $e$ is transcendental.
Another consequence: $\pi$ is transcendental. Otherwise $i\pi$ would be algebraic, and $e^{i\pi} + 1 = 0$ would contradict the theorem.
So the circle cannot be squared with ruler and compass.

## History

Ferdinand von Lindemann proved in 1882 that $e^{\alpha}$ is transcendental for each nonzero algebraic $\alpha$.
Karl Weierstrass proved the general statement in 1885.

## Explanations

### L1 human

The exponential function sends nonzero algebraic numbers to transcendental numbers.
With $\alpha = 1$ this shows that $e$ is transcendental, and with Euler's formula it shows the same for $\pi$.
</document>

<document source="https://math2.tamnd.com/node/0JNMJEDD93.md" tag="0JNMJEDD93" kind="statement">
# Convolution theorem for the Laplace transform

## Statement

Let $f, g : [0, \infty) \to \mathbb{C}$ be piecewise continuous, and let $M, a$ be real constants with $|f(t)| \le M e^{a t}$ and $|g(t)| \le M e^{a t}$ for all $t \ge 0$.
Let the convolution $f * g : [0, \infty) \to \mathbb{C}$ be
$$(f * g)(t) = \int_0^t f(\tau)\, g(t - \tau)\, d\tau.$$
Let $F$ and $G$ be the [[Laplace transforms|05XT5AVJBK]] of $f$ and $g$.
Then for each $s \in \mathbb{C}$ with $\operatorname{Re} s > a$, the Laplace transform of $f * g$ converges at $s$, and $\mathcal{L}\{f * g\}(s) = F(s)\, G(s)$.

## Remarks

The proof writes $F(s) G(s)$ as a double integral over the quadrant $\tau, u \ge 0$, and changes variables to $t = \tau + u$. The absolute convergence for $\operatorname{Re} s > a$ justifies the change of the order of integration.
The convolution is commutative: $f * g = g * f$.
Example: $\mathcal{L}^{-1}\left\{\frac{1}{s^2 (s - 1)}\right\}$ is the convolution of $t$ and $e^t$, that is $e^t - 1 - t$.
In [[solving initial value problems|0K3HFZJK5Q]], the solution with zero initial values is the convolution of the forcing term with the impulse response.

## Explanations

### L1 human

Multiplying two transforms corresponds to a special kind of product of the functions, the convolution.
The convolution mixes the past values of one function with the other, as a memory of earlier inputs.
</document>

<document source="https://math2.tamnd.com/node/0JNXSS9TJ9.md" tag="0JNXSS9TJ9" kind="concept">
# Universal set

## Summary

A universal set is a set that contains all the objects under discussion.
In most set theories, there is no set of all sets, so a universal set is always relative to a context.
</document>

<document source="https://math2.tamnd.com/node/0JP0W07Y9A.md" tag="0JP0W07Y9A" kind="statement">
# Fundamental theorem of algebra

## Statement

Let $n \ge 1$ be an integer and let $a_0, a_1, \dots, a_n$ be [[complex numbers|09FTAET5Z4]] with $a_n \neq 0$.
Let $p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$.
Then there is a complex number $z_0$ with $p(z_0) = 0$.

## Remarks

Proof with [[Liouville's theorem|0EHKVRF4J6]]: if $p$ has no zero, then $1/p$ is entire. Since $|p(z)| \to \infty$ as $|z| \to \infty$, $1/p$ is bounded, so it is constant, which is false for $n \ge 1$.
By induction on $n$, it follows that $p(z) = a_n (z - z_1) \cdots (z - z_n)$ for some $z_1, \dots, z_n \in \mathbb{C}$. So $p$ has exactly $n$ roots, counted with multiplicity.
The theorem holds for polynomials with real coefficients too, but the roots can be non-real: $z^2 + 1$ has the roots $i$ and $-i$.

## Explanations

### L1 human

Each polynomial equation of degree at least $1$ has a solution in the complex numbers.
So we never need to add new numbers to $\mathbb{C}$ to solve polynomial equations, unlike the step from $\mathbb{R}$ to $\mathbb{C}$ for $x^2 = -1$.
</document>

<document source="https://math2.tamnd.com/node/0JP0ZF57PN.md" tag="0JP0ZF57PN" kind="statement">
# Kőnig's theorem

## Statement

Let $G$ be a finite [[bipartite graph|07DVHJ8N0A]].
Then the largest number of edges in a [[matching|0Z87BHAFTB]] of $G$ is equal to the smallest number of vertices in a [[vertex cover|0KDADE9RQ8]] of $G$.

## Remarks

In every graph, a vertex cover has at least as many vertices as a matching has edges. Kőnig's theorem says that equality holds for bipartite graphs.
The theorem fails for other graphs. The triangle $K_3$ has matchings with $1$ edge only, but each vertex cover has $2$ vertices.

## History

Dénes Kőnig published the theorem in 1931.
Jenő Egerváry found it independently in the same year, in a more general form with weights.

## Explanations

### L1 human

In a bipartite graph, the most pairs that you can match at once is the same as the fewest vertices that touch every edge.
Each of the two numbers gives a certificate that the other one is optimal.
</document>

<document source="https://math2.tamnd.com/node/0JPCHTXWG0.md" tag="0JPCHTXWG0" kind="concept">
# Function of several variables

## Summary

A function of several real variables is a [[function|0EPAT51DXQ]] $f \colon D \to \mathbb{R}$ whose domain $D$ is a subset of [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ for some $n \ge 2$.
Its value at a point $x = (x_1, \dots, x_n)$ of $D$ is written $f(x)$ or $f(x_1, \dots, x_n)$.
Examples are $f(x, y) = x^2 + y^2$ on $\mathbb{R}^2$ and $g(x, y, z) = xyz$ on $\mathbb{R}^3$.

## Remarks

When the values lie in $\mathbb{R}^m$ with $m \ge 2$, the function is a [[vector-valued function|018GTRS7A4]] of several variables.
The domain is often an open subset of $\mathbb{R}^n$, so that each point of the domain has a whole ball around it in the domain.

## Explanations

### L1 human

A function of two variables takes two numbers and gives one number. An example is the height of a hill above the point $(x, y)$ of a map.
Each input is now a point of the plane or of space, not a point of a line.
</document>

<document source="https://math2.tamnd.com/node/0JPDTEDCFD.md" tag="0JPDTEDCFD" kind="statement">
# Subgroups of abelian groups are normal

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $g \in G$ and $h \in H$, commutativity gives $ghg^{-1} = hgg^{-1} = h \in H$.
The converse is false: in the quaternion group $Q_8$ each subgroup is normal, but $Q_8$ is not abelian.
More generally, each subgroup of the center of any group is normal.

## Explanations

### L1 human

In an abelian group, conjugation does nothing, because $ghg^{-1} = h$.
So a subgroup can never be moved by conjugation.
Thus all subgroups are normal.
</document>

<document source="https://math2.tamnd.com/node/0JPH2Y4DFV.md" tag="0JPH2Y4DFV" kind="concept">
# Central binomial coefficient

## Summary

The central binomial coefficients $\binom{2n}{n}$ are the middle entries of the even rows of Pascal's triangle. They count balanced lattice paths and appear in the formula for Catalan numbers. The node [[central binomial coefficient|01WHD9T0V6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JPYSD5E0B.md" tag="0JPYSD5E0B" kind="definition">
# Order of an element

## Definition

Let $G$ be a [[group|G5S1M9]] with identity element $e$, and let $g \in G$.
When there is a positive integer $n$ with $g^n = e$, the order of $g$ is the smallest such $n$.
When there is no such $n$, the element $g$ has infinite order.
The notation for the order of $g$ is $\operatorname{ord}(g)$ or $|g|$.

## Remarks

The identity element is the only element of order $1$.
In $(\mathbb{Z}, +)$, each nonzero element has infinite order.
In the multiplicative group $\{1, i, -1, -i\}$, the element $-1$ has order $2$ and the element $i$ has order $4$.
In a finite group, each element has finite order, and the order of $g$ equals the [[order|0B85S2V2DR]] of the subgroup that $g$ generates.

## Explanations

### L1 human

Take an element and multiply it by itself again and again.
The order is the number of steps that you need to come back to the identity for the first time.
For a rotation by $90$ degrees, you need $4$ steps, so its order is $4$.
</document>

<document source="https://math2.tamnd.com/node/0JQ1KNQW2K.md" tag="0JQ1KNQW2K" kind="statement">
# Order of the general linear group over a finite field

## Statement

Let $q$ be a prime power, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $n \geq 1$.
The group $\mathrm{GL}_n(\mathbb{F}_q)$ of the invertible $n \times n$ matrices over $\mathbb{F}_q$ has [[order|0B85S2V2DR]] $\prod_{k=0}^{n-1} (q^n - q^k) = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$.

## Remarks

A matrix is invertible exactly when its columns are linearly independent.
There are $q^n - 1$ choices for the first column. Column $k+1$ has $q^n - q^k$ choices, because it must avoid the $q^k$ vectors in the span of the first $k$ columns.
For example, $|\mathrm{GL}_2(\mathbb{F}_2)| = 3 \cdot 2 = 6$, and $\mathrm{GL}_2(\mathbb{F}_2) \cong S_3$.
Also, $|\mathrm{GL}_3(\mathbb{F}_2)| = 7 \cdot 6 \cdot 4 = 168$.

## History

Évariste Galois computed the order of the general linear group over a prime field in 1832, in his last letter to Chevalier.

## Explanations

### L1 human

To build an invertible matrix, choose its columns one after the other.
Each new column must not lie in the span of the columns that you already chose.
Counting the choices at each step gives the product formula.
</document>

<document source="https://math2.tamnd.com/node/0JQ5Y3WWVE.md" tag="0JQ5Y3WWVE" kind="example">
# Alternating harmonic series

## Example

The alternating harmonic series is
$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$$
It converges by the [[alternating series test|0PK8QH1W1V]], since $1/n$ decreases to $0$.
Its sum is $\ln 2 \approx 0.6931$.

## Remarks

The series of absolute values is the [[harmonic series|0CXTMAY5PX]], which diverges. So the convergence is [[conditional|02YRQXZDCS]].
The value $\ln 2$ comes from the [[Mercator series|0YZXNE3HFR]] $\ln(1 + x) = x - x^2/2 + x^3/3 - \cdots$ at $x = 1$, with [[Abel's theorem|0S4EMDQS2F]].
By the [[Riemann rearrangement theorem|0E7VBYGFF7]], a reordering of the terms can give any real sum.

## Explanations

### L1 human

The partial sums are $1, 0.5, 0.833, 0.583, 0.783, \dots$.
They swing around $\ln 2 \approx 0.693$ and close in on it slowly.
</document>

<document source="https://math2.tamnd.com/node/0JQE130DHW.md" tag="0JQE130DHW" kind="statement">
# Weierstrass approximation theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then for every $\varepsilon > 0$ there is a polynomial $p$ with $|f(x) - p(x)| < \varepsilon$ for all $x \in [a, b]$.

## Remarks

Equivalently, there is a sequence of polynomials that converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.
One proof uses the [[Bernstein polynomials|0C4VWBFY73]] on $[0, 1]$, together with [[uniform continuity|0W0TZDRF9C]] of $f$.
The Stone–Weierstrass theorem generalizes the result to compact spaces. Mathlib states the theorem as `polynomialFunctions_closure_eq_top`.

## History

Karl Weierstrass proved the theorem in 1885.

## Explanations

### L1 human

Every continuous function on a closed bounded interval is uniformly close to a polynomial.
So polynomials can imitate any continuous graph as closely as we like.
</document>

<document source="https://math2.tamnd.com/node/0JQWWFCS9R.md" tag="0JQWWFCS9R" kind="definition">
# Ordered pair

## Definition

An ordered pair $(a, b)$ has a first entry $a$ and a second entry $b$, with this property:
$$(a, b) = (c, d) \iff a = c \text{ and } b = d.$$
In set theory, the Kuratowski definition is $(a, b) = \{\{a\}, \{a, b\}\}$. It has the property above.

## Remarks

In general $(a, b) \neq (b, a)$, but the sets $\{a, b\}$ and $\{b, a\}$ are equal.
Ordered pairs are the elements of a [[Cartesian product|0C81YBVTWH]].
An ordered triple can be defined as $(a, b, c) = ((a, b), c)$.

## Explanations

### L1 human

A point in the plane is an ordered pair of coordinates.
The points $(1, 2)$ and $(2, 1)$ are different, so the order matters.
The Kuratowski definition only shows that sets can model pairs.
</document>

<document source="https://math2.tamnd.com/node/0JRC32D6K2.md" tag="0JRC32D6K2" kind="definition">
# Multiplication of natural numbers

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]], let $S$ be the [[successor function|0832XVR512]] and let $+$ be the [[addition|0W5DSDFQ4G]] of $\mathbb{N}$.
Multiplication is the function $\cdot \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(a, b) \mapsto a \cdot b$, with these two properties for all $a$ and $b$ in $\mathbb{N}$:
$$a \cdot 0 = 0, \qquad a \cdot S(b) = a \cdot b + a.$$

## Remarks

For each $a$, these equations are a [[definition by recursion|0DSHAVH45F]] on $b$. Thus exactly one such function exists.
From the definition, $a \cdot 1 = a \cdot 0 + a = a$.
We often write $ab$ for $a \cdot b$.

## Explanations

### L1 human

Multiplication is repeated addition: $3 \cdot 4$ is $3 + 3 + 3 + 3 = 12$.
The definition says this in two rules: zero copies give $0$, and one more copy adds one more $a$.
</document>

<document source="https://math2.tamnd.com/node/0JRW4X1MDC.md" tag="0JRW4X1MDC" kind="definition">
# Inverse Laplace transform

## Definition

Let $a$ be a real number and let $F$ be a complex function on the half-plane $\{s \in \mathbb{C} : \operatorname{Re} s > a\}$.
An inverse Laplace transform of $F$ is a piecewise continuous function $f : [0, \infty) \to \mathbb{C}$ with two properties. First, there are constants $M, b$ with $|f(t)| \le M e^{b t}$ for $t \ge 0$. Second, the [[Laplace transform|05XT5AVJBK]] of $f$ equals $F(s)$ for each $s$ with $\operatorname{Re} s > \max(a, b)$.
The notation is $f = \mathcal{L}^{-1}\{F\}$.

## Remarks

By Lerch's theorem, two such functions with the same transform agree at each point where both are continuous. So $\mathcal{L}^{-1}\{F\}$ is unique in practice, and unique among continuous functions.
Let $\gamma$ be a real number larger than the real parts of all singularities of $F$. Under suitable conditions, the Bromwich integral recovers $f$: $f(t) = \frac{1}{2 \pi i} \lim_{T \to \infty} \int_{\gamma - iT}^{\gamma + iT} e^{s t} F(s)\, ds$.
In practice, one splits $F$ into partial fractions and reads the inverse of each term from a table. For example, $\mathcal{L}^{-1}\{1/(s - a)\} = e^{a t}$.

## Explanations

### L1 human

The inverse transform goes back from the $s$ side to the time side.
After we solve a problem in terms of $F(s)$, the inverse transform gives the solution as a function of time.
</document>

<document source="https://math2.tamnd.com/node/0JT92VK307.md" tag="0JT92VK307" kind="concept">
# Bernstein polynomial

## Summary

A Bernstein polynomial of a function on $[0, 1]$ is a weighted average of its values at the points $k/n$. For a continuous function, these polynomials converge uniformly to the function. The node [[Bernstein polynomial|0C4VWBFY73]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JTC045NNZ.md" tag="0JTC045NNZ" kind="concept">
# Legendre symbol

## Summary

The Legendre symbol $\left(\frac{a}{p}\right)$ takes the value $1$, $-1$ or $0$, and it tells whether $a$ is a square modulo an odd prime $p$.
The [[definition of the Legendre symbol|01J8AMV9RS]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0JTGRV4WT4.md" tag="0JTGRV4WT4" kind="definition">
# Cardinality

## Definition

Let $A$ be a [[set|0SJ3RX9TXA]].
The cardinality of $A$, written $|A|$, is an object assigned to $A$ such that, for all sets $A$ and $B$,
$$|A| = |B| \iff A \text{ and } B \text{ are } \text{[[equinumerous|08EJHHMB5R]]}.$$
For a finite set $A$, $|A|$ is the number $n$ of its elements, that is, the unique $n \in \mathbb{N}$ with $A \approx \{1, \dots, n\}$.
For an infinite set, $|A|$ is a [[cardinal number|0QKPNZKNEA]].

## Remarks

Other notations are $\#A$, $\operatorname{card}(A)$ and $\overline{\overline{A}}$.
With the [[axiom of choice|0HTGB13FE2]], one can take $|A|$ to be the least [[ordinal|0RDJK7EQYY]] equinumerous with $A$.
$|A| \le |B|$ means that there is an [[injection|0PVG1CRAJN]] from $A$ to $B$. See [[order of cardinals|01GTP8M5RG]].

## Explanations

### L1 human

The cardinality of $\{a, b, c\}$ is $3$.
For infinite sets, cardinality is a new kind of number that records which sets can be put in bijection.
For example, $|\mathbb{N}| = |\mathbb{Z}|$, but $|\mathbb{N}| < |\mathbb{R}|$.
</document>

<document source="https://math2.tamnd.com/node/0JTPMCCZT2.md" tag="0JTPMCCZT2" kind="statement">
# Estimate for approximate solutions

## Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$ and let $K \ge 0$. Let $f : J \times U \to \mathbb{R}^n$ satisfy $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$.
Let $a < b$ with $[a, b] \subseteq J$, and let $\delta, \varepsilon_1, \varepsilon_2 \ge 0$.
Let $\gamma_1, \gamma_2 : [a, b] \to U$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $\|\gamma_i'(t) - f(t, \gamma_i(t))\| \le \varepsilon_i$ for $i = 1, 2$ and $t \in (a, b)$.
If $\|\gamma_1(a) - \gamma_2(a)\| \le \delta$, then for each $t \in [a, b]$,
$$\|\gamma_1(t) - \gamma_2(t)\| \le G_{\delta, K, \varepsilon_1 + \varepsilon_2}(t - a),$$
where $G$ is the [[Grönwall bound|0FRX8JD7QH]].

## Remarks

A curve with $\|\gamma'(t) - f(t, \gamma(t))\| \le \varepsilon$ is an $\varepsilon$-approximate solution.
For $\varepsilon_1 = \varepsilon_2 = 0$, the result is [[continuous dependence on initial data|0G3DBXRRQJ]].
The proof applies the derivative form of [[Grönwall's inequality|07EJHQYR9A]] with the additional term $\varepsilon_1 + \varepsilon_2$ to $\|\gamma_1 - \gamma_2\|$.

## Explanations

### L1 human

A curve that almost solves the equation stays close to a true solution.
The error has two sources: the difference of initial values and the defects $\varepsilon_i$.
Both can grow at most exponentially in time.
</document>

<document source="https://math2.tamnd.com/node/0JTRMBQ1SA.md" tag="0JTRMBQ1SA" kind="definition">
# System of linear equations

## Definition

Let $F$ be a field and let $m, n \ge 1$ be integers.
Consider a *system of linear equations* over $F$, with $m$ equations in the unknowns $x_1, \dots, x_n$. Such a system is a list of [[linear equations|0Q3ZT8J05R]]
$$\begin{aligned} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n &= b_1, \\ &\ \ \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n &= b_m, \end{aligned}$$
where all $a_{ij}$ and $b_i$ are elements of $F$.
A *solution* of the system is a tuple $(s_1, \dots, s_n) \in F^n$ that is a solution of each of the $m$ equations.
The *solution set* of the system is the set of all its solutions.

## Remarks

With the [[coefficient matrix|0HQFZAP3ND]] $A = (a_{ij})$, the vector $b = (b_1, \dots, b_m)^T$ and $x = (x_1, \dots, x_n)^T$, the system is the single equation $Ax = b$.
Over a field, a system has no solution, exactly one solution, or more than one solution.
When $F$ is infinite, "more than one" means infinitely many.

## Explanations

### L1 human

A system of linear equations asks for values of the unknowns that make several linear equations true together.
For example, $x + y = 3$ and $x - y = 1$ together have the single solution $x = 2$, $y = 1$.
In the plane, each equation is a line, and a solution is a point on all the lines.
</document>

<document source="https://math2.tamnd.com/node/0JTRSJKCFB.md" tag="0JTRSJKCFB" kind="statement">
# Tychonoff's theorem

## Statement

Let $(X_i)_{i \in I}$ be a family of [[compact spaces|0QPHW703PV]], indexed by any set $I$.
Then the product $\prod_{i \in I} X_i$, with the [[product topology|04S9HX3GAP]], is compact.

## Remarks

For finitely many factors, the statement follows from [[product of compact sets is compact|09ZKE5XQ74]].
For arbitrary products, the theorem is equivalent to the axiom of choice.
The statement fails for the [[box topology|0CTM1EHG9P]]: the box topology on $\{0, 1\}^{\mathbb{N}}$ is discrete and infinite.
For example, $[0, 1]^I$ is compact for each set $I$.

## History

Andrey Tikhonov proved the theorem in 1930 for powers of the closed unit interval. He stated the general theorem in 1935, with the remark that the same proof works.

## Explanations

### L1 human

Any product of compact spaces is compact, even with infinitely many factors.
It works because a basic open set of the product restricts only finitely many coordinates.
</document>

<document source="https://math2.tamnd.com/node/0JV30XN73Z.md" tag="0JV30XN73Z" kind="concept">
# Integer partition

## Summary

An integer partition is a way to write a natural number as a sum of positive integers, without regard to order. Partitions are central objects of combinatorics, number theory and the representation theory of symmetric groups. The node [[integer partition|0404R9P9J3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JVJ2WCXZH.md" tag="0JVJ2WCXZH" kind="concept">
# Gaussian elimination

## Summary

Gaussian elimination is an algorithm that changes a matrix into a [[row echelon form|0VYCDGMJJJ]] by [[elementary row operations|04XWH28452]].
It works column by column, from left to right.
In each column, it chooses a nonzero entry as [[pivot|0J0BBCMJ5W]] and moves its row up.
Then it adds multiples of that row to the rows below, to make their entries in that column zero.
Applied to the [[augmented matrix|0T6MWGZSXA]] of a [[system of linear equations|0GT4WWGWH2]], it gives an equivalent system that [[back substitution|0ATFVE08MW]] solves.
For an $n \times n$ matrix, it uses on the order of $n^3$ arithmetic operations.

## Remarks

The algorithm also computes the [[rank|0ZC9YTXZRY]] of a matrix and, with a record of the row operations, its [[determinant|00KDZRW4C0]].
It also tests if a system is [[consistent|0X6V9B1V10]]: the system is inconsistent exactly when a row $(0, \dots, 0 \mid c)$ with $c \ne 0$ appears.
The method was known in China in the book *The Nine Chapters on the Mathematical Art*, long before Gauss.

## Explanations

### L1 human

Gaussian elimination is the method of solving equations by removing one unknown at a time.
We use the first equation to remove $x_1$ from all the other equations, then the second to remove $x_2$, and so on.
At the end, the last equation has the fewest unknowns, and we solve upwards.
</document>

<document source="https://math2.tamnd.com/node/0JVSNJCN68.md" tag="0JVSNJCN68" kind="concept">
# Split polynomial

## Summary

A polynomial splits over a field when it is a product of polynomials of degree one with coefficients in that field.
The [[definition of a split polynomial|0SHMC00RJV]] states it.
</document>

<document source="https://math2.tamnd.com/node/0JW13B48YP.md" tag="0JW13B48YP" kind="concept">
# Vector projection

## Summary

The vector projection of a vector $a$ onto a nonzero vector $b$ is the multiple of $b$ that is nearest to $a$. It is the shadow of $a$ on the line of $b$, under light that falls perpendicular to that line.
</document>

<document source="https://math2.tamnd.com/node/0JW2FHW7G3.md" tag="0JW2FHW7G3" kind="definition">
# Existential quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The existential quantification of $P$ over $X$ is the proposition $\exists x \in X, \, P(x)$, read "there exists $x$ in $X$ such that $P(x)$".
It is true when $P(a)$ is true for at least one $a \in X$, and false otherwise.

## Remarks

When $X$ is empty, $\exists x \in X, \, P(x)$ is false for each $P$.
For a finite set $X = \{a_1, \dots, a_n\}$, the statement is equivalent to $P(a_1) \lor \dots \lor P(a_n)$.
The [[uniqueness quantifier|0N3ADS4CHH]] $\exists!$ says that exactly one element exists.

## Explanations

### L1 human

"There is an even prime number" is $\exists n \in \mathbb{N}, \, (n \text{ is even} \land n \text{ is prime})$.
To prove it, one example is enough: $n = 2$.
</document>

<document source="https://math2.tamnd.com/node/0JW6MDHAZD.md" tag="0JW6MDHAZD" kind="concept">
# Saddle point

## Summary

A saddle point of a differentiable function of several variables is a critical point that is not a [[local extremum|0WFBH1Q3WC]].
Near a saddle point, the function takes values both larger and smaller than its value at the point.
</document>

<document source="https://math2.tamnd.com/node/0JW7KT6PQW.md" tag="0JW7KT6PQW" kind="concept">
# Minimum spanning tree

## Summary

A minimum spanning tree of a weighted connected graph is a spanning tree of least total weight. It is a classical problem of combinatorial optimization, solved by greedy algorithms. The node [[minimum spanning tree|0Z367CGNRM]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JWCX0TW2S.md" tag="0JWCX0TW2S" kind="definition">
# Concyclic points

## Definition

Let $S$ be a set of points of the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]].
The points of $S$ are concyclic if they lie on one [[circle|060W5CBZP6]]. That is, there are a [[plane|0V3XAMF3C1]] $P$ that contains $S$, a point $c \in P$ and a real number $r \geq 0$ with $d(x, c) = r$ for each $x \in S$.

## Remarks

Three points that are not collinear are always concyclic. Thus each triangle has a circumcircle.
Four or more points of the plane are not concyclic in general. Four concyclic points are the vertices of a cyclic quadrilateral.
The centre $c$ is on the perpendicular bisector of each two distinct points of $S$.
Mathlib's `EuclideanGeometry.Concyclic` asks for points that are cospherical and coplanar. This is the same condition.

## Explanations

### L1 human

Concyclic points are points on one common circle. The four corners of a rectangle are concyclic: the circle has its centre at the crossing of the diagonals. The four points $(0, 0)$, $(1, 0)$, $(2, 0)$ and $(0, 1)$ are not concyclic, because a circle meets a line in at most two points.
</document>

<document source="https://math2.tamnd.com/node/0JWM6GJNQW.md" tag="0JWM6GJNQW" kind="statement">
# An orthonormal family is linearly independent

## Statement

Let $V$ be an inner product space over $\R$ or $\C$.
Then each [[orthonormal family|0RW9TNNT87]] in $V$ is [[linearly independent|0CRMGPAH83]].

## Remarks

If $\sum_i a_i e_i = 0$, then the inner product with $e_j$ gives $a_j = 0$.
More generally, a family of nonzero pairwise orthogonal vectors is linearly independent.
So an orthonormal family in an $n$-dimensional space has at most $n$ vectors.

## Explanations

### L1 human

Vectors at right angles cannot be combinations of each other.
To find the coefficient of $e_j$ in a combination, take the inner product with $e_j$.
If the combination is zero, then every coefficient is zero.
</document>

<document source="https://math2.tamnd.com/node/0JWMJSTBSG.md" tag="0JWMJSTBSG" kind="concept">
# Divergent series

## Summary

A divergent series is a series whose partial sums do not have a finite limit. Summation methods give values to some divergent series. The node [[divergent series|0NBRP6093V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0JWRJX2RMP.md" tag="0JWRJX2RMP" kind="concept">
# Bayesian inference

## Summary

Bayesian inference is a method of statistical inference that treats an unknown parameter $\theta$ as a random variable. It uses [[Bayes' theorem|0NM6E9VW04]] to update a probability distribution for $\theta$ when data arrive.
One starts from a [[prior distribution|00NBBSXRQS]] with density $\pi(\theta)$ and a model with [[likelihood|05FSTXM92W]] $L(\theta \mid x) = f(x \mid \theta)$. The [[posterior distribution|0KJ7ZS8AJA]] has the density
$$\pi(\theta \mid x) = \frac{f(x \mid \theta)\, \pi(\theta)}{\int_\Theta f(x \mid \theta')\, \pi(\theta')\, d\theta'}.$$

## Remarks

In short, the posterior is proportional to the likelihood times the prior.
Example: with a $\operatorname{Beta}(a, b)$ prior for the success probability $p$, and $k$ successes in $n$ Bernoulli($p$) trials, the posterior is $\operatorname{Beta}(a + k, b + n - k)$ (see [[beta distribution|0PP5ZAJPP7]]). The prior is conjugate.
The posterior gives point estimates, such as the posterior mean, and credible intervals.
For new observations, the posterior predictive distribution is $f(\tilde{x} \mid x) = \int_\Theta f(\tilde{x} \mid \theta)\, \pi(\theta \mid x)\, d\theta$, if $\tilde{x}$ and $x$ are conditionally independent given $\theta$.

## Explanations

### L1 human

Bayesian inference starts with a belief about an unknown quantity, written as a probability distribution.
Each new piece of data updates this belief with Bayes' theorem. The result, the posterior, combines the prior belief with the evidence.
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<document source="https://math2.tamnd.com/node/0JWT4PPQ7B.md" tag="0JWT4PPQ7B" kind="statement">
# L'Hôpital's rule

## Statement

Let $I$ be an open interval and let $c \in I$.
Let $f, g$ be real functions differentiable on $I \setminus \{c\}$, with $g'(x) \ne 0$ for all $x \in I \setminus \{c\}$.
Assume that $f(x) \to 0$ and $g(x) \to 0$ as $x \to c$, and that $f'(x)/g'(x) \to L$ as $x \to c$.
Then $f(x)/g(x) \to L$ as $x \to c$.

## Remarks

The proof uses the [[Cauchy mean value theorem|02SYZAWM2K]] on intervals between $c$ and $x$.
Similar forms hold for the [[indeterminate form|0S7G92KKA7]] $\infty/\infty$, for one-sided limits, and for $x \to \pm\infty$.
For example, $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \frac{1}{2}$.
Mathlib states a form as `deriv.lhopital_zero_nhds`.

## History

Johann Bernoulli found the rule and sent it to Guillaume de l'Hôpital in a letter of 1694.
L'Hôpital published it in his textbook of 1696.

## Explanations

### L1 human

Near $c$, both $f$ and $g$ are close to their tangent lines through $(c, 0)$.
So their ratio is close to the ratio of their slopes.
</document>

<document source="https://math2.tamnd.com/node/0JWZX6FKSH.md" tag="0JWZX6FKSH" kind="definition">
# Sylow p-subgroup

## Definition

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Write $|G| = p^n m$ with $n \geq 0$ and $p \nmid m$.
A Sylow $p$-subgroup of $G$ is a [[subgroup|0X1JXT47GA]] of $G$ of order $p^n$.
The set of the Sylow $p$-subgroups of $G$ is written $\operatorname{Syl}_p(G)$, and its size is written $n_p$.

## Remarks

A Sylow $p$-subgroup is a maximal [[p-subgroup|06QYRTX9KQ]] of $G$.
If $p$ does not divide $|G|$, then the only Sylow $p$-subgroup is $\{e\}$.
In $S_3$, the Sylow $3$-subgroup is $A_3$, and there are three Sylow $2$-subgroups, each of order $2$.

## Explanations

### L1 human

A Sylow subgroup takes all the factors $p$ of the order of the group at once.
It is as large as a $p$-subgroup can be by Lagrange's theorem.
The Sylow theorems say that such subgroups exist and that they are all alike.
</document>

<document source="https://math2.tamnd.com/node/0JX86PSSA9.md" tag="0JX86PSSA9" kind="statement">
# Divergence theorem

## Statement

Let $V \subseteq \mathbb{R}^3$ be compact with a piecewise smooth boundary surface $S = \partial V$, oriented by the outward unit normal $n$.
If $F$ is a vector field of class $C^1$ on an open set that contains $V$, then
$$\iiint_V \nabla \cdot F \, dV = \iint_S F \cdot n \, dS.$$

## Remarks

The left side is the [[triple integral|0GTBGMZXV8]] of the [[divergence|02NSMQVQWK]], the total source strength in $V$. The right side is the outward [[flux|0YPRY7T4PK]] through the boundary.
Example: for $F(x, y, z) = (x, y, z)$ and the unit ball, $\nabla \cdot F = 3$. So both sides equal $3 \cdot \tfrac{4}{3} \pi = 4\pi$.
With $g F$ in place of $F$, for a $C^1$ function $g$, one gets $\iiint_V (F \cdot \nabla g + g \, \nabla \cdot F) \, dV = \iint_S g \, F \cdot n \, dS$. This is a form of integration by parts.
The same statement holds in $\mathbb{R}^n$. For a [[rectangle in the plane|0GZ3D8F921]], the proof is a direct use of the fundamental theorem of calculus.

## History

Lagrange discovered the theorem in 1762, and Gauss proved special cases from 1813.
Ostrogradsky gave the first proof of the general theorem in 1826, in his work on heat flow.

## Explanations

### L1 human

What flows out of a closed surface must come from sources inside it.
The divergence theorem makes this exact: the total divergence inside equals the net flux out through the boundary.
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<document source="https://math2.tamnd.com/node/0JXTQ392ZM.md" tag="0JXTQ392ZM" kind="statement">
# Legendre's theorem on continued fractions

## Statement

Let $\alpha$ be a real number, and let $p$ and $q$ be positive integers such that
$$\left| \alpha - \frac{p}{q} \right| < \frac{1}{2 q^2}.$$
Then $p / q$ is a [[convergent|0SRWNR42BQ]] of the [[continued fraction expansion|0SJ7Z6VKWK]] of $\alpha$.

## Remarks

So the very good rational approximations of $\alpha$ are all convergents.
The constant $\frac{1}{2}$ cannot be replaced by $1$: not every fraction with $\left| \alpha - \frac{p}{q} \right| < \frac{1}{q^2}$ is a convergent.
A use: it shows that the solutions of [[Pell's equation|0AF3DANJ76]] come from the convergents of $\sqrt{d}$.

## History

Adrien-Marie Legendre proved the theorem in his Essai sur la théorie des nombres of 1798.

## Explanations

### L1 human

If a fraction is extremely close to $\alpha$, then the continued fraction algorithm finds it.
So to look for the best approximations of a number, you only need to compute its convergents.
</document>

<document source="https://math2.tamnd.com/node/0JXZKX526R.md" tag="0JXZKX526R" kind="concept">
# Square matrix

## Summary

A square matrix is a matrix with the same number of rows and columns.
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<document source="https://math2.tamnd.com/node/0JYF6PTKEW.md" tag="0JYF6PTKEW" kind="statement">
# Lifting-the-exponent lemma

## Statement

Let $p$ be an odd prime, let $x$ and $y$ be integers with $p \nmid x$, $p \nmid y$ and $p \mid x - y$, and let $n$ be a positive integer.
Then $\nu_p(x^n - y^n) = \nu_p(x - y) + \nu_p(n)$, where $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For $p = 2$ the formula needs a stronger hypothesis. If $x$ and $y$ are odd and $4 \mid x - y$, then $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(n)$.
For $p = 2$, odd $x$ and $y$, and even $n$, the general formula is $\nu_2(x^n - y^n) = \nu_2(x - y) + \nu_2(x + y) + \nu_2(n) - 1$.
For example, $\nu_3(10^9 - 1) = \nu_3(9) + \nu_3(9) = 4$, and indeed $10^9 - 1 = 3^4 \cdot 37 \cdot 333667$.

## History

Édouard Lucas had some of the ideas by 1878. Mihai Manea published the modern form in 2006.

## Explanations

### L1 human

When $p$ divides $x - y$, the lemma tells you exactly how many factors $p$ are in $x^n - y^n$.
You add the factors $p$ of $x - y$ and the factors $p$ of $n$.
Olympiad problems often use the lemma.
</document>

<document source="https://math2.tamnd.com/node/0JYTG5K9Y4.md" tag="0JYTG5K9Y4" kind="concept">
# Group homomorphism

## Summary

A group homomorphism is a map between groups that preserves the group operation.
It is the basic way to compare two groups.
</document>

<document source="https://math2.tamnd.com/node/0JZ27DMJ5D.md" tag="0JZ27DMJ5D" kind="concept">
# Dihedral group

## Summary

The dihedral group $D_n$ is the group of the symmetries of a regular polygon with $n$ sides, for $n \geq 3$.
It has $n$ rotations and $n$ reflections, so it has order $2n$.
</document>

<document source="https://math2.tamnd.com/node/0JZGW75C5Z.md" tag="0JZGW75C5Z" kind="statement">
# Power rule

## Statement

Let $n \ge 1$ be a natural number.
Then the function $x \mapsto x^n$ is differentiable on $\mathbb{R}$, and its derivative at $x$ is $n x^{n-1}$.

## Remarks

Induction on $n$ with the [[product rule|0V9A43H7WV]] gives $(x^{n+1})' = (x^n \cdot x)' = n x^{n-1} \cdot x + x^n = (n + 1) x^n$.
The same formula holds for negative integers $n$ when $x \ne 0$, and for [[real exponents|0KY08TWDMQ]] when $x > 0$.
Mathlib states it as `hasDerivAt_pow`.

## Explanations

### L1 human

To differentiate a power, bring the exponent down as a factor and lower the exponent by one.
For example, the derivative of $x^5$ is $5 x^4$.
</document>

<document source="https://math2.tamnd.com/node/0JZSRWRBZE.md" tag="0JZSRWRBZE" kind="definition">
# Combination

## Definition

Let $X$ be a set and let $k$ be a natural number.
A $k$-combination of $X$ is a subset $S \subseteq X$ with $|S| = k$.

The set of all $k$-combinations of $X$ is written $\binom{X}{k}$.

## Remarks

A combination does not record an order of its elements. Two lists with the same elements in a different order give the same combination.
For $X = \{a, b, c\}$, the 2-combinations are $\{a, b\}$, $\{a, c\}$ and $\{b, c\}$.
The [[number of k-element subsets|0BFMR2FQ8B]] of an $n$-element set is the [[binomial coefficient|092GEF38YD]] $\binom{n}{k}$.

## Explanations

### L1 human

A combination is a selection of $k$ objects where the order does not matter.
Choosing a team of 2 from Ann, Bob and Cy gives 3 teams, because the team $\{$Ann, Bob$\}$ is the same as $\{$Bob, Ann$\}$.
</document>

<document source="https://math2.tamnd.com/node/0JZY3R908H.md" tag="0JZY3R908H" kind="definition">
# Galois group of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be nonzero.
Let $L$ be a [[splitting field|02NZZ1J0FH]] of $f$ over $K$.
The Galois group of $f$ over $K$ is the group $\operatorname{Aut}(L/K)$ of the [[automorphisms of $L/K$|0XVR72M51M]].

## Remarks

By the [[uniqueness of splitting fields|069A3BC3EP]], the group does not depend on the choice of $L$, up to isomorphism.
Each automorphism permutes the distinct roots $r_1, \dots, r_m$ of $f$ in $L$, and it is determined by this permutation, because the roots generate $L$. So the group is isomorphic to a subgroup of the symmetric group $S_m$.
The Galois group of $x^2 - 2$ over $\mathbb{Q}$ has order $2$. The Galois group of $x^3 - 2$ over $\mathbb{Q}$ is isomorphic to $S_3$.
If $f$ is separable, then $L/K$ is a [[Galois extension|0DS2F7GK05]].

## Explanations

### L1 human

The Galois group of a polynomial lists the ways to permute its roots that respect all the algebraic relations between them.
It measures how complicated the polynomial is.
For example, a polynomial over $\mathbb{Q}$ can be solved by radicals exactly when its Galois group is solvable.
</document>

<document source="https://math2.tamnd.com/node/0K06X9QXDX.md" tag="0K06X9QXDX" kind="definition">
# Convex set

## Definition

A subset $S$ of $\mathbb{R}^n$ is convex if for all $x, y \in S$ and each real number $t$ with $0 \leq t \leq 1$,
$$(1 - t)\, x + t\, y \in S.$$
That is, $S$ contains the segment from $x$ to $y$.

## Remarks

The same definition works in each real vector space or real affine space. Mathlib's `Convex` is stated for a module over an ordered semiring.
A convex set contains each convex combination $\sum_i w_i x_i$ of its points, an [[affine combination|0KJH6SRFZK]] with $w_i \geq 0$.
The empty set and the whole space are convex. Each intersection of convex sets is convex, but a union of convex sets need not be convex.
Balls, half-spaces, lines and triangles with their inside are convex.

## Explanations

### L1 human

A set is convex if you can see each of its points from each other point, with the line of sight inside the set. A disc is convex. A ring is not: the segment between two opposite points of the ring passes through the hole.
</document>

<document source="https://math2.tamnd.com/node/0K090GVHVN.md" tag="0K090GVHVN" kind="statement">
# Laplace transform of a derivative

## Statement

Let $f : [0, \infty) \to \mathbb{C}$ be continuously differentiable, and let $M, a$ be real constants with $|f(t)| \le M e^{a t}$ for all $t \ge 0$.
Let $F$ be the [[Laplace transform|05XT5AVJBK]] of $f$.
Then for each $s \in \mathbb{C}$ with $\operatorname{Re} s > a$, the Laplace transform of $f'$ converges at $s$ and
$$\mathcal{L}\{f'\}(s) = s F(s) - f(0).$$

## Remarks

Proof: integration by parts gives $\int_0^T f'(t) e^{-st}\, dt = f(T) e^{-sT} - f(0) + s \int_0^T f(t) e^{-st}\, dt$. The term $f(T) e^{-sT}$ tends to $0$ for $\operatorname{Re} s > a$.
Applied twice, under the same hypotheses on $f$ and $f'$: $\mathcal{L}\{f''\}(s) = s^2 F(s) - s f(0) - f'(0)$.
This rule is the basis of [[solving initial value problems with the Laplace transform|0K3HFZJK5Q]]. The initial values enter the transformed equation directly.

## Explanations

### L1 human

Differentiation in time becomes multiplication by $s$ after the transform, with a correction by the initial value.
So a differential equation becomes an algebraic equation for $F$.
</document>

<document source="https://math2.tamnd.com/node/0K0A06X2MQ.md" tag="0K0A06X2MQ" kind="concept">
# Upper and lower Darboux sums

## Summary

The upper and lower Darboux sums of a bounded function for a partition use the supremum and the infimum of the function on each subinterval. They bound the area under the graph from above and from below. The node [[upper and lower Darboux sums|0QRQVH0CRW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0K0GE6BCYW.md" tag="0K0GE6BCYW" kind="definition">
# Path-connected space

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
A path in $X$ from $x$ to $y$ is a [[continuous map|0SY9ECQ19Q]] $\gamma \colon [0, 1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$.
The space $X$ is path-connected when $X$ is nonempty and for all $x, y \in X$ there is a path in $X$ from $x$ to $y$.
A subset of $X$ is path-connected when it is path-connected in the [[subspace topology|0D5YHQGDJY]].

## Remarks

Some authors do not require $X$ to be nonempty.
Each convex subset of $\mathbb{R}^n$ is path-connected: use the segment $\gamma(t) = (1 - t)x + ty$.
[[Path-connected spaces are connected|0ZMT1GP4KM]]. The [[topologist's sine curve|0K3T4A659R]] is connected but not path-connected.

## Explanations

### L1 human

A space is path-connected when you can walk from each point to each other point along a continuous path inside the space.
This is a concrete way to show that a space is in one piece.
</document>

<document source="https://math2.tamnd.com/node/0K0H5E3Z5N.md" tag="0K0H5E3Z5N" kind="definition">
# Sum of a series

## Definition

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers with partial sums $s_N = \sum_{n=0}^{N} a_n$.
The sum of the series is the real number
$$\sum_{n=0}^{\infty} a_n = \lim_{N \to \infty} s_N.$$

## Remarks

The sum is well defined because a sequence has at most one limit.
The sum is linear: $\sum (a_n + b_n) = \sum a_n + \sum b_n$ and $\sum c\, a_n = c \sum a_n$ when $\sum a_n$ and $\sum b_n$ converge. This follows from the [[algebra of limits|07XBM3A9J5]].
For example, $\sum_{n=0}^{\infty} 1/2^n = 2$.

## Explanations

### L1 human

The sum of a series is the number that its running totals approach.
For $1 + 1/2 + 1/4 + \cdots$, the totals $1, 1.5, 1.75, 1.875, \dots$ approach $2$, so the sum is $2$.
</document>

<document source="https://math2.tamnd.com/node/0K1E2S6Z69.md" tag="0K1E2S6Z69" kind="statement">
# Distance from a point to a plane

## Statement

Let $a$, $b$, $c$ and $d$ be real numbers with $(a, b, c) \neq (0, 0, 0)$, and let $P = \{(x, y, z) \in \mathbb{R}^3 : a x + b y + c z = d\}$ be a [[plane|07SDJ5DNE4]].
Let $X = (x_0, y_0, z_0)$ be a point of $\mathbb{R}^3$. The distance from $X$ to $P$ is the smallest [[Euclidean distance|0NPPGHQNXQ]] $d(X, Q)$ for $Q \in P$. Then
$$\operatorname{dist}(X, P) = \frac{|a x_0 + b y_0 + c z_0 - d|}{\sqrt{a^2 + b^2 + c^2}}.$$

## Remarks

The smallest distance is reached at exactly one point of $P$, the orthogonal projection of $X$ on $P$. With $n = (a, b, c)$ this point is $X - \frac{a x_0 + b y_0 + c z_0 - d}{a^2 + b^2 + c^2} \, n$.
The number $(a x_0 + b y_0 + c z_0 - d) / \|n\|$ is the signed distance. Its sign tells on which side of the plane the point is.
The same formula holds for a hyperplane $\{x : a \cdot x = d\}$ with $a \neq 0$ in $\mathbb{R}^n$: the distance from $x_0$ is $|a \cdot x_0 - d| / \|a\|$.

## Explanations

### L1 human

The nearest point of a plane is straight below or above you, along the normal vector. For the floor plane $z = 0$, the formula gives $|z_0|$, your height above the floor. Tilted planes need the factor $\sqrt{a^2 + b^2 + c^2}$, because $(a, b, c)$ need not be a unit vector.
</document>

<document source="https://math2.tamnd.com/node/0K1EWAR6AF.md" tag="0K1EWAR6AF" kind="statement">
# Lipschitz functions are uniformly continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be [[Lipschitz|0QNVQ4VD9T]].
Then $f$ is [[uniformly continuous|0Q4DR37DVE]].

## Remarks

Let $K > 0$ be a Lipschitz constant. Given $\varepsilon > 0$, take $\delta = \varepsilon / K$.
The converse fails. The function $\sqrt{x}$ is uniformly continuous on $[0, 1]$ but not Lipschitz.
Mathlib states it as `LipschitzWith.uniformContinuous`.

## Explanations

### L1 human

A Lipschitz bound controls the change of $f$ in the same way at every point.
So one input tolerance works for the whole domain.
</document>

<document source="https://math2.tamnd.com/node/0K1KKMFHYH.md" tag="0K1KKMFHYH" kind="statement">
# Determinant of a triangular matrix

## Statement

Let $K$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ upper or lower [[triangular matrix|0N4XY7Z3NN]] over $K$.
Then
$$\det A = a_{11} a_{22} \cdots a_{nn},$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In the [[Leibniz formula|016PE4VZ8Q]], all terms except the term of the identity permutation contain an entry below, or above, the diagonal, so they are zero.
Row reduction to a triangular matrix, with this rule, is a practical way to compute determinants.
A triangular matrix over a field is invertible if and only if all its diagonal entries are nonzero.

## Explanations

### L1 human

For a triangular matrix, the determinant is the product of the diagonal entries.
The entries above the diagonal do not matter.
For example, $\det \begin{pmatrix} 2 & 7 \\ 0 & 3 \end{pmatrix} = 6$.
</document>

<document source="https://math2.tamnd.com/node/0K1N1Y4EGZ.md" tag="0K1N1Y4EGZ" kind="statement">
# Integral of 1/x

## Statement

Let $a, b$ be real numbers of the same sign, both nonzero.
Then
$$\int_a^b \frac{dx}{x} = \ln \frac{b}{a}.$$
In particular, $\int_1^x \frac{dt}{t} = \ln x$ for $x > 0$.

## Remarks

The function $\ln |x|$ is an antiderivative of $1/x$ on each of $(0, \infty)$ and $(-\infty, 0)$, by the [[derivative of the logarithm|07X1QA6BRW]].
Some texts use the second formula as the definition of the [[natural logarithm|0044CPHJBV]].
The integral over an interval that contains $0$ does not exist.
Mathlib states it as `integral_inv`, under the hypothesis $0 \notin [a, b]$.

## Explanations

### L1 human

The power rule fails for $x^{-1}$, since it would divide by $0$.
The logarithm fills this gap: its derivative is $1/x$.
</document>

<document source="https://math2.tamnd.com/node/0K1SCPBNKJ.md" tag="0K1SCPBNKJ" kind="concept">
# Automorphism group

## Summary

The automorphism group of a group $G$ is the group of all isomorphisms from $G$ to itself, with composition as the operation.
It describes the symmetries of the structure of $G$.
</document>

<document source="https://math2.tamnd.com/node/0K26YKNBSA.md" tag="0K26YKNBSA" kind="definition">
# Zero divisor

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $a \in R$ is a left zero divisor if there is a nonzero $x \in R$ with $a x = 0$.
An element $a \in R$ is a right zero divisor if there is a nonzero $y \in R$ with $y a = 0$.
An element is a zero divisor if it is a left or a right zero divisor.

## Remarks

In a [[commutative ring|0GYGHKX834]], left and right zero divisors are the same.
In $\mathbb{Z}/6\mathbb{Z}$, the classes of $2$ and $3$ are zero divisors, because $2 \cdot 3 = 0$.
Some authors do not count $0$ as a zero divisor. With the definition above, $0$ is a zero divisor of every nonzero ring.
A [[unit|019NFGFPR5]] is never a zero divisor.

## Explanations

### L1 human

In the integers, a product is zero only when one factor is zero.
In other rings this can fail.
In the integers modulo $6$, $2 \cdot 3 = 6 = 0$, but neither $2$ nor $3$ is zero.
</document>

<document source="https://math2.tamnd.com/node/0K28YXMS7Q.md" tag="0K28YXMS7Q" kind="definition">
# Scalar multiplication of matrices

## Definition

Let $R$ be a ring, let $c \in R$, and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The *scalar multiple* $cA$ is the $m \times n$ matrix with entries
$$(cA)_{ij} = c\, a_{ij}$$
for $1 \le i \le m$ and $1 \le j \le n$.

## Remarks

Over a field $F$, matrix addition and scalar multiplication make $M_{m \times n}(F)$ a [[vector space|000AYPA9AJ]] of dimension $mn$.
The rules $c(A + B) = cA + cB$, $(c + d)A = cA + dA$, $(cd)A = c(dA)$ and $1A = A$ hold.
For example, $2 \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ 0 & 6 \end{pmatrix}$.

## Explanations

### L1 human

Multiplying a matrix by a number means multiplying every entry by that number.
The size of the matrix does not change.
Doubling a matrix doubles every number in the table.
</document>

<document source="https://math2.tamnd.com/node/0K2DMACEZ4.md" tag="0K2DMACEZ4" kind="statement">
# Brahmagupta–Fibonacci identity

## Statement

Let $a$, $b$, $c$ and $d$ be elements of a commutative ring. Then
$$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 = (ac + bd)^2 + (ad - bc)^2.$$

## Remarks

So the product of two sums of two squares is again a sum of two squares.
For example, $5 \cdot 13 = 65$, and the identity gives $65 = 1^2 + 8^2 = 7^2 + 4^2$.
For complex numbers, the identity says $|z|^2 |w|^2 = |zw|^2$ with $z = a + bi$ and $w = c + di$.
It is a step in the proof of the [[sum of two squares theorem|04NPWJMA6H]].

## History

The identity appears in the Arithmetica of Diophantus, of the third century.
Brahmagupta found it again in the seventh century, and Fibonacci published it in Europe in 1225.

## Explanations

### L1 human

If two numbers are sums of two squares, then so is their product.
You can check the identity by expanding both sides: the cross terms $2abcd$ cancel.
</document>

<document source="https://math2.tamnd.com/node/0K2FNY6DYV.md" tag="0K2FNY6DYV" kind="statement">
# Logarithm of a product

## Statement

For all real $x > 0$ and $y > 0$,
$$\ln(x y) = \ln x + \ln y.$$

## Remarks

By [[the sum rule for exp|0E8HNVBCSP]], $e^{\ln x + \ln y} = e^{\ln x} e^{\ln y} = x y$. Now [[take logarithms|0RKW2X63YR]].
So $\ln$ is a group isomorphism from $((0, \infty), \cdot)$ to $(\mathbb{R}, +)$.
Mathlib states it as `Real.log_mul`, for nonzero $x$ and $y$.

## Explanations

### L1 human

The logarithm turns multiplication into addition.
This property made logarithm tables useful for hand computation.
</document>

<document source="https://math2.tamnd.com/node/0K2TX12X58.md" tag="0K2TX12X58" kind="definition">
# Singular value

## Definition

Let $A$ be an $m \times n$ complex matrix.
The matrix $A^* A$ is positive semidefinite, so its [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$ are real and nonnegative.
The *singular values* of $A$ are the numbers
$$\sigma_i = \sqrt{\lambda_i} \qquad (1 \le i \le n).$$

## Remarks

The number of nonzero singular values is the rank of $A$.
The largest singular value is the operator norm $\max_{\norm{x} = 1} \norm{Ax}$.
For a normal matrix, the singular values are the absolute values of the eigenvalues.
Some authors list only the first $\min(m, n)$ values.

## Explanations

### L1 human

The singular values of a matrix measure how much it stretches space in its main directions.
The unit sphere goes to an ellipsoid, and the singular values are the lengths of its semi-axes.
Unlike eigenvalues, they exist for every matrix, also rectangular ones.
</document>

<document source="https://math2.tamnd.com/node/0K387S2PKQ.md" tag="0K387S2PKQ" kind="statement">
# Minkowski's theorem

## Statement

Let $n$ be a positive integer, and let $L \subset \mathbb{R}^n$ be a lattice with determinant $d(L)$.
Let $S \subseteq \mathbb{R}^n$ be a convex set that is symmetric about the origin: $x \in S$ implies $-x \in S$.
If the volume of $S$ is greater than $2^n d(L)$, then $S$ contains a point of $L$ other than $0$.

## Remarks

The most common case is $L = \mathbb{Z}^n$, with $d(L) = 1$: a symmetric convex set of volume greater than $2^n$ contains a nonzero integer point.
The bound is sharp: the open cube $(-1, 1)^n$ has volume $2^n$ and contains no nonzero integer point.
Applications include proofs of [[Fermat's theorem on sums of two squares|043B3ENSTY]] and [[Lagrange's four-square theorem|0KYYSMEVHH]].

## History

Hermann Minkowski proved the theorem in 1889. It started the geometry of numbers.

## Explanations

### L1 human

A large, round, centered region in the plane must contain a lattice point other than the origin.
In the plane with $L = \mathbb{Z}^2$, the area must be more than $4$.
</document>

<document source="https://math2.tamnd.com/node/0K3HFZJK5Q.md" tag="0K3HFZJK5Q" kind="concept">
# Solving initial value problems with the Laplace transform

## Summary

To solve an [[initial value problem|075VBY4K4X]] for a [[linear equation with constant coefficients|0A6PGTTP2A]], one applies the [[Laplace transform|05XT5AVJBK]] to both sides of the equation.
The [[rule for derivatives|0K090GVHVN]] turns the equation and the initial values into one algebraic equation for the transform $Y(s)$ of the unknown $y(t)$.
One solves this equation for $Y(s)$ and then finds $y = \mathcal{L}^{-1}\{Y\}$ with the [[inverse Laplace transform|0JRW4X1MDC]].

## Remarks

Example: for $y'' + y = 0$, $y(0) = 0$, $y'(0) = 1$, the transform gives $s^2 Y(s) - 1 + Y(s) = 0$. So $Y(s) = 1/(s^2 + 1)$ and $y(t) = \sin t$.
For $a_n y^{(n)} + \cdots + a_0 y = g(t)$ with zero initial values, $Y(s) = G(s) / P(s)$, where $P(s) = a_n s^n + \cdots + a_0$ and $G$ is the transform of $g$. The [[convolution theorem|0JNMJEDD93]] gives $y = h * g$ with $h = \mathcal{L}^{-1}\{1/P\}$.
The method handles discontinuous forcing, such as a [[step function|09C4JP7Y0M]], and impulses, such as a [[Dirac delta|0KPHENZ5S7]].

## Explanations

### L1 human

The Laplace transform turns a differential equation into an algebra problem.
We solve the algebra problem, which is often easy, and then transform back to get the solution.
The initial values are built in from the start, so there are no constants to find at the end.
</document>

<document source="https://math2.tamnd.com/node/0K3MWA7DMC.md" tag="0K3MWA7DMC" kind="statement">
# Factor theorem

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $c \in R$.
Then $x - c$ divides $f$ in $R[x]$ if and only if $c$ is a [[root|00EHX6FWC0]] of $f$.

## Remarks

The proof uses the [[remainder theorem|0TEDQH56RW]]: $f = (x - c) q + f(c)$. So $x - c$ divides $f$ exactly when $f(c) = 0$.
For example, $1$ is a root of $x^3 - 1$, and $x^3 - 1 = (x - 1)(x^2 + x + 1)$.
The theorem gives the bound on the [[number of roots|0KJVQV3E0F]] of a polynomial over a field.

## Explanations

### L1 human

Each root gives a linear factor.
If $f(c) = 0$, then you can write $f$ as $(x - c)$ times another polynomial.
Conversely, a factor $x - c$ makes $f(c) = 0$.
</document>

<document source="https://math2.tamnd.com/node/0K3T4A659R.md" tag="0K3T4A659R" kind="counterexample">
# Topologist's sine curve

## Counterexample

Let
$$T = \left\{ \left( x, \sin \tfrac{1}{x} \right) : 0 < x \le 1 \right\} \cup \{ (0, 0) \} \subseteq \mathbb{R}^2,$$
with the [[subspace topology|0D5YHQGDJY]] of the plane.
Then $T$ is [[connected|0R37DFJKFW]] but not [[path-connected|0K0GE6BCYW]]. Thus connected spaces need not be path-connected.

## Remarks

The graph part $G$ is connected, as the continuous image of $(0, 1]$. The origin is in the closure of $G$, and a set between a connected set and its closure is connected. Thus $T$ is connected.
No path joins the origin to a point of $G$. Near $x = 0$, the curve goes up and down between $-1$ and $1$ infinitely often, and a path cannot follow it continuously.
Some texts use the closed version, which adds the whole segment $\{0\} \times [-1, 1]$. It is compact, connected and not path-connected.
The space $T$ is also not locally connected.

## Explanations

### L1 human

The curve $\sin(1/x)$ oscillates faster and faster as $x$ goes to $0$. The origin is a limit of the curve, so the space is in one piece.
But you cannot walk along the curve to the origin, because the curve never settles down.
</document>

<document source="https://math2.tamnd.com/node/0K4NMMJBG2.md" tag="0K4NMMJBG2" kind="definition">
# Confidence interval

## Definition

Let $X$ be a sample whose law $P_\theta$ depends on an unknown parameter $\theta \in \Theta$. Let $\tau \colon \Theta \to \mathbb{R}$ be a function and let $\gamma \in (0, 1)$.
A confidence interval for $\tau(\theta)$ with confidence level $\gamma$ is a random interval $[L(X), U(X)]$, where $L \le U$ are [[statistics|036756NFD6]], such that
$$P_\theta\big(L(X) \le \tau(\theta) \le U(X)\big) = \gamma \quad \text{for each } \theta \in \Theta.$$
If only "$\ge \gamma$" holds for each $\theta$, the interval is a conservative confidence interval.

## Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with $\sigma$ known. Then $\bar{X}_n \pm 1.96\, \sigma / \sqrt{n}$ is a confidence interval for $\mu$ with level approximately $0.95$.
If $\sigma$ is unknown, one replaces $\sigma$ with the sample standard deviation and $1.96$ with a quantile of Student's t-distribution with $n - 1$ degrees of freedom.
The probability refers to the random interval, not to $\theta$. After one computes the interval from data, it either contains $\tau(\theta)$ or it does not.
The confidence level is often written as $1 - \alpha$, for example $95\%$ for $\alpha = 0.05$.

## Explanations

### L1 human

A $95\%$ confidence interval comes from a procedure that captures the true value in $95\%$ of repeated experiments.
It does not mean that the true value has a $95\%$ probability to be in one particular computed interval.
</document>

<document source="https://math2.tamnd.com/node/0K4VQB7HWR.md" tag="0K4VQB7HWR" kind="statement">
# Determinant of a matrix with two equal rows

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
If two different rows of $A$ are equal, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 2 of the definition of the determinant.
The same holds for two equal columns.
Over a field, more generally, $\det A = 0$ when the rows of $A$ are linearly dependent.

## Explanations

### L1 human

Two equal rows mean that the rows span less than the full space.
Then the box spanned by the rows is flat, and its volume is zero.
So the determinant is zero.
</document>

<document source="https://math2.tamnd.com/node/0K4WCNTMHG.md" tag="0K4WCNTMHG" kind="definition">
# Rayleigh quotient

## Definition

Let $A$ be a [[Hermitian|0P17NGQQMQ]] $n \times n$ complex matrix, and let $x \in \C^n$ with $x \ne 0$.
The *Rayleigh quotient* of $A$ and $x$ is the real number
$$R(A, x) = \frac{x^* A x}{x^* x}.$$

## Remarks

For each $x \ne 0$, $\lambda_{\min} \le R(A, x) \le \lambda_{\max}$, where $\lambda_{\min}$ and $\lambda_{\max}$ are the least and the greatest [[eigenvalues|0BQAA2BK3Q]] of $A$.
If $x$ is an eigenvector for $\lambda$, then $R(A, x) = \lambda$.
So $\lambda_{\max} = \max_{x \ne 0} R(A, x)$ and $\lambda_{\min} = \min_{x \ne 0} R(A, x)$.

## Explanations

### L1 human

The Rayleigh quotient measures how much a symmetric matrix stretches a given direction, on average.
Its largest value over all directions is the largest eigenvalue.
This gives a way to estimate eigenvalues without solving the characteristic equation.
</document>

<document source="https://math2.tamnd.com/node/0K4ZWTFSV5.md" tag="0K4ZWTFSV5" kind="definition">
# Homogeneous first-order equation

## Definition

Let $J$ be an open interval and let $F : J \to \mathbb{R}$ be a continuous function.
A homogeneous first-order equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = F\left(\frac{y}{x}\right)$$
for an unknown function $y$ of $x$. Its domain is an interval that does not contain $0$, with $y(x)/x \in J$.

## Remarks

An equivalent form is $M(x, y)\, dx + N(x, y)\, dy = 0$, where $M$ and $N$ are homogeneous functions of the same degree $d$: $M(t x, t y) = t^d M(x, y)$ and $N(t x, t y) = t^d N(x, y)$ for $t \ne 0$.
The substitution $y = x v$ gives $x v' = F(v) - v$. This is a [[separable equation|0V4E623THY]] for $v$.
Example: $y' = (x + y)/x = 1 + y/x$ gives $x v' = 1$, so $v = \ln|x| + C$ and $y = x \ln|x| + C x$.
This use of the word homogeneous is different from its use for [[homogeneous linear equations|0EQZSAB9EC]].

## Explanations

### L1 human

In a homogeneous equation, the slope $y'$ depends only on the ratio $y/x$.
So the slope field is the same along each line through the origin.
The new unknown $v = y/x$ captures this, and the equation becomes separable.
</document>

<document source="https://math2.tamnd.com/node/0K57YG3BB8.md" tag="0K57YG3BB8" kind="concept">
# Differentiable function

## Summary

A differentiable function is a function that has a derivative at each point of its domain. Near each point, it is well approximated by a linear function. The node [[differentiable function|0A61B60B5R]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0K5TWJAGPD.md" tag="0K5TWJAGPD" kind="example">
# Action by left multiplication

## Example

Let $G$ be a [[group|G5S1M9]].
The map $G \times G \to G$, $(g, x) \mapsto gx$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.
The two axioms are $ex = x$ and $g(hx) = (gh)x$, and they follow from the group axioms.
This action is [[transitive|02GK6HZYBT]], because $(yx^{-1})x = y$ for all $x, y \in G$.
The [[stabiliser|0AYMP41THN]] of each point is $\{e\}$, because $gx = x$ implies $g = e$.
In particular, the action is [[faithful|0P7G9NGG2G]], and this gives [[Cayley's theorem|0151VYVX27]].
</document>

<document source="https://math2.tamnd.com/node/0K5W69AC0J.md" tag="0K5W69AC0J" kind="object">
# Gaussian integers

## Description

The ring of Gaussian integers is
$$\mathbb{Z}[i] = \{a + b i : a, b \in \mathbb{Z}\},$$
where $i \in \mathbb{C}$ satisfies $i^2 = -1$.
It is a [[subring|0859D58WPD]] of $\mathbb{C}$, with $(a + b i)(c + d i) = (a c - b d) + (a d + b c) i$.
As a subring of a field, it is an [[integral domain|09S3PFBEPM]].

## Remarks

The [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ is multiplicative.
The [[units of $\mathbb{Z}[i]$|06RH4XAGRY]] are $1, -1, i, -i$.
The ring $\mathbb{Z}[i]$ is a [[Euclidean domain|0PN13BB7BT]], so it has unique factorization.
A prime number $p$ is a sum of two squares if and only if $p$ is not prime in $\mathbb{Z}[i]$. This holds exactly for $p = 2$ and the primes $p \equiv 1 \pmod 4$.

## Explanations

### L1 human

The Gaussian integers are the points with integer coordinates in the complex plane.
You add and multiply them as complex numbers.
They form a ring with a theory of primes and factorization, like the integers.
</document>

<document source="https://math2.tamnd.com/node/0K665H5V1S.md" tag="0K665H5V1S" kind="definition">
# Matrix exponential

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$.
The *matrix exponential* of $A$ is the $n \times n$ matrix
$$e^A = \exp(A) = \sum_{k=0}^{\infty} \frac{1}{k!} A^k = I_n + A + \frac{1}{2} A^2 + \frac{1}{6} A^3 + \cdots,$$
with $A^0 = I_n$.

## Remarks

The series converges for each matrix $A$, entry by entry.
For a diagonal matrix $D$ with entries $d_i$, $e^D$ is diagonal with entries $e^{d_i}$.
If $A = P D P^{-1}$, then $e^A = P e^D P^{-1}$.
The function $x(t) = e^{tA} x_0$ solves the linear differential equation $x' = Ax$ with $x(0) = x_0$.

## Explanations

### L1 human

The matrix exponential uses the familiar series of $e^x$, with a matrix in place of the number $x$.
It is the main tool for solving systems of linear differential equations.
For $A = \begin{pmatrix} 0 & -t \\ t & 0 \end{pmatrix}$, $e^A$ is the rotation of the plane by the angle $t$.
</document>

<document source="https://math2.tamnd.com/node/0K67Q60Z8P.md" tag="0K67Q60Z8P" kind="statement">
# Complement of an open set is closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $U \subseteq X$.
Then $U$ is [[open|01ZQQDCERP]] in $X$ if and only if $X \setminus U$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: $X \setminus (X \setminus U) = U$, so the claim is the definition of a closed set applied to $X \setminus U$.
Thus the closed sets determine the open sets, and the open sets determine the closed sets.

## Explanations

### L1 human

Taking the complement turns open sets into closed sets and closed sets back into open sets.
For example, $(0, 1)$ is open in $\mathbb{R}$, so $(-\infty, 0] \cup [1, \infty)$ is closed.
</document>

<document source="https://math2.tamnd.com/node/0K6CH0MT0F.md" tag="0K6CH0MT0F" kind="concept">
# Alternating multilinear map

## Summary

An alternating multilinear map is a multilinear map that is zero whenever two of its arguments are equal.
</document>

<document source="https://math2.tamnd.com/node/0K78D81EF7.md" tag="0K78D81EF7" kind="definition">
# Young diagram

## Definition

A Young diagram is a finite set $D \subseteq \mathbb{N} \times \mathbb{N}$ with this property: if $(i, j) \in D$, $i' \le i$ and $j' \le j$, then $(i', j') \in D$.
The elements of $D$ are the cells, or boxes, of the diagram.
A cell $(i, j)$ is drawn as a unit square in row $i$ and column $j$, with rows numbered from the top.

## Remarks

The rows of a Young diagram have weakly decreasing lengths $\lambda_1 \ge \lambda_2 \ge \dots$. Thus a Young diagram with $n$ cells is the same as a [[partition|0404R9P9J3]] of $n$.
This drawing convention is the English one. The French convention draws row $0$ at the bottom.
Transposing a diagram, $(i, j) \mapsto (j, i)$, gives the diagram of the conjugate partition.

## Explanations

### L1 human

A Young diagram draws a partition as rows of boxes, left-aligned, with the longest row at the top.
The partition $4 + 2 + 1$ gives rows of $4$, $2$ and $1$ boxes.
</document>

<document source="https://math2.tamnd.com/node/0K7P6PR838.md" tag="0K7P6PR838" kind="concept">
# Maclaurin series

## Summary

The Maclaurin series of a smooth function is its Taylor series at the point $0$. It is a power series in $x$. The node [[Maclaurin series|043SJ5RF59]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0K7TD9BV2D.md" tag="0K7TD9BV2D" kind="definition">
# Idempotent matrix

## Definition

Let $R$ be a ring and let $P$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $P$ is *idempotent* when $P^2 = P$.

## Remarks

The matrices $0$, $I_n$ and $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ are idempotent.
Over a field, an idempotent matrix is the matrix of a [[projection|033DA1WE8Y]] onto its image along its kernel.
If $P$ is idempotent, then so is $I - P$.
Over a field, the eigenvalues of an idempotent matrix are in $\{0, 1\}$, and its rank equals its trace in characteristic $0$.

## Explanations

### L1 human

An idempotent matrix does nothing new when we apply it twice.
A projection onto a line is the main example: once a point is on the line, projecting again keeps it there.
So $P^2 = P$.
</document>

<document source="https://math2.tamnd.com/node/0K83KEJ43P.md" tag="0K83KEJ43P" kind="statement">
# Isolated zeros theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$, and let $z_0 \in U$.
Then exactly one of these two cases holds.

1. There is $r > 0$ such that $f(z) = 0$ for each $z$ with $|z - z_0| < r$.
2. There is $r > 0$ such that $f(z) \neq 0$ for each $z$ with $0 < |z - z_0| < r$.

## Remarks

Proof: if case 1 does not hold, then $f(z) = (z - z_0)^m g(z)$ near $z_0$ with $g(z_0) \neq 0$, where $m$ is the [[order|064N002TX4]] of $f$ at $z_0$. By continuity, $g \neq 0$ near $z_0$.
On a [[domain|097KNJFJ7V]], it follows that the zeros of a holomorphic function that is not identically zero have no accumulation point in the domain.
The real function $x \mapsto e^{-1/x^2}$, with value $0$ at $0$, is infinitely differentiable and has all derivatives $0$ at $0$. The theorem has no analogue for such real functions.

## Explanations

### L1 human

Near a point, a holomorphic function is either zero everywhere or zero only at that point.
So the zeros of a nonzero holomorphic function cannot pile up inside its domain, unlike the zeros of $\sin(1/x)$ near $0$.
</document>

<document source="https://math2.tamnd.com/node/0K8F7PVDMX.md" tag="0K8F7PVDMX" kind="concept">
# Associativity

## Summary

A binary operation is associative when the grouping of three elements does not change the result.
Then the parentheses in a product of several elements can be omitted.
</document>

<document source="https://math2.tamnd.com/node/0K8SGQSV7G.md" tag="0K8SGQSV7G" kind="statement">
# Compactness and sequential compactness agree in metric spaces

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[sequentially compact|0CARW59WD2]].

## Remarks

One proof of the hard direction uses the [[Lebesgue number lemma|040CFNJEAP]] and the fact that a sequentially compact metric space is [[totally bounded|0WE5BBHGM6]].
Both conditions are also equivalent to: $K$ is complete and totally bounded. This is the statement [[compact iff complete and totally bounded|054Q5PW5GX]].
Without a metric, the two notions differ. For example, the product $\{0, 1\}^{[0, 1]}$ is compact but not sequentially compact.

## Explanations

### L1 human

In a metric space, you can test compactness with sequences. A set is compact exactly when each sequence in it has a subsequence with a limit in the set.
This test is often easier than the test with open covers.
</document>

<document source="https://math2.tamnd.com/node/0K96Z2S699.md" tag="0K96Z2S699" kind="concept">
# Removable discontinuity

## Summary

A removable discontinuity is a point where the limit of a function exists but differs from the value of the function. Redefining the value at the point makes the function continuous there. The node [[removable discontinuity|0QG9C579SR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0K98R7A4VC.md" tag="0K98R7A4VC" kind="concept">
# Lower central series

## Summary

The lower central series of a group $G$ is the sequence $G = \gamma_1(G) \supseteq \gamma_2(G) \supseteq \cdots$ with $\gamma_{i+1}(G) = [\gamma_i(G), G]$.
The group is nilpotent if and only if this series reaches the trivial subgroup.
</document>

<document source="https://math2.tamnd.com/node/0K9AAJ8G51.md" tag="0K9AAJ8G51" kind="statement">
# Continuous image of a compact set is compact

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], let $f \colon X \to Y$ be a [[continuous map|0SY9ECQ19Q]] and let $K \subseteq X$ be [[compact|00R0X1WGPX]].
Then the [[image|0Q3F20XB0A]] $f(K)$ is compact in $Y$.

## Remarks

Proof: let $(V_i)$ be open sets of $Y$ that cover $f(K)$. The sets $f^{-1}(V_i)$ are open and cover $K$. A finite subfamily covers $K$, and the matching $V_i$ cover $f(K)$.
A consequence is that a [[continuous real function on a compact set attains its bounds|0V4DET59NW]].
Compactness is a [[topological property|02SMVGC690]].

## Explanations

### L1 human

Continuous maps keep compactness.
For example, the image of $[0, 1]$ under a continuous map into the plane is a compact curve, so it is closed and bounded.
</document>

<document source="https://math2.tamnd.com/node/0K9GBTZ1FX.md" tag="0K9GBTZ1FX" kind="definition">
# Parallelepiped

## Definition

Let $p$ be a point of $\mathbb{R}^3$, and let $a$, $b$ and $c$ be linearly independent [[vectors|0QN3D80AWP]] in $\mathbb{R}^3$.
The parallelepiped with the vertex $p$ and the edges $a$, $b$ and $c$ is the set
$$\{\, p + s a + t b + u c \;:\; s, t, u \in [0, 1] \,\}.$$

## Remarks

The parallelepiped has 8 vertices $p + s a + t b + u c$ with $s, t, u \in \{0, 1\}$, 12 edges and 6 faces. Opposite faces are congruent parallelograms in parallel planes.
Its volume is the absolute value of the [[scalar triple product|059Q2XWJDC]] $a \cdot (b \times c)$.
In Mathlib, the parallelepiped of a finite family of vectors $v_i$ is the set of the sums $\sum_i t_i v_i$ with each $t_i \in [0, 1]$. It does not ask for linear independence.
A cube and a rectangular cuboid are special parallelepipeds.

## Explanations

### L1 human

Take a cube and push it sideways so that its faces lean, but stay flat. The result is a parallelepiped. Each face is a parallelogram, and opposite faces are copies of each other.
</document>

<document source="https://math2.tamnd.com/node/0K9HSJFCMJ.md" tag="0K9HSJFCMJ" kind="statement">
# Inverse function theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$, and let $f \colon U \to \mathbb{R}^n$ be of [[class C^k|0TCBZMEYGN]] on $U$.
Let $a \in U$, and suppose that the derivative $Df(a)$ is invertible, that is, the [[Jacobian determinant|03W6Q1MFR5]] of $f$ at $a$ is not zero.
Then there are open sets $V \subseteq U$ with $a \in V$ and $W \subseteq \mathbb{R}^n$ with $f(a) \in W$ such that $f$ restricted to $V$ is a bijection from $V$ onto $W$.
Its inverse $f^{-1} \colon W \to V$ is of class $C^k$, and $D(f^{-1})(f(x)) = Df(x)^{-1}$ for each $x \in V$.

## Remarks

The theorem is local. The map $f(x, y) = (e^x \cos y, e^x \sin y)$ has invertible derivative at each point of $\mathbb{R}^2$, but it is not injective, since $f(x, y + 2\pi) = f(x, y)$.
Continuity of the derivative matters. The function $g(x) = x + 2 x^2 \sin(1/x)$, $g(0) = 0$, has $g'(0) = 1$, but $g$ is not injective on any interval around $0$.
A standard proof applies the contraction mapping principle to $x \mapsto x + Df(a)^{-1}(y - f(x))$ near $a$.
The formula for the derivative of the inverse is the [[derivative of the inverse function|0G65751VSW]].

## Explanations

### L1 human

If the linear approximation of $f$ at $a$ is invertible, then $f$ itself is invertible near $a$.
So one can solve the system $f(x) = y$ for $x$ when $y$ is close to $f(a)$, and the solution depends smoothly on $y$.
</document>

<document source="https://math2.tamnd.com/node/0K9QCK6WJC.md" tag="0K9QCK6WJC" kind="statement">
# General Leibniz rule

## Statement

Let $n$ be a natural number, and let $f$ and $g$ be $n$ times differentiable real functions on an open interval $I$.
Then $f g$ is $n$ times differentiable on $I$, and
$$(f g)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}.$$

## Remarks

The proof is an induction on $n$ with the [[product rule|0V9A43H7WV]], like the proof of the binomial theorem.
For $n = 1$ it is the product rule. For $n = 2$ it is $(f g)'' = f'' g + 2 f' g' + f g''$.
Mathlib states it as `iteratedDeriv_mul`.

## Explanations

### L1 human

The rule for the $n$-th derivative of a product looks like the binomial formula for $(a + b)^n$.
Each term distributes $n$ differentiations between the two factors.
</document>

<document source="https://math2.tamnd.com/node/0K9QVPJYJF.md" tag="0K9QVPJYJF" kind="concept">
# Ramsey number

## Summary

The Ramsey number $R(r, s)$ is the least $n$ such that every two-colouring of the edges of $K_n$ has a blue $K_r$ or a red $K_s$. Ramsey numbers measure how large a structure must be to contain order. The node [[Ramsey number|01DSVCRF6X]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KA4S8R39M.md" tag="0KA4S8R39M" kind="definition">
# Regular graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $k$ be a natural number.
The graph $G$ is $k$-regular, or regular of degree $k$, if each vertex of $G$ has [[degree|0T0F6CCS9W]] $k$.
A graph is regular if it is $k$-regular for some $k$.

## Remarks

A $0$-regular graph has no edges. A $1$-regular graph is a perfect matching. A finite $2$-regular graph is a disjoint union of cycles.
A $3$-regular graph is also called cubic. The [[Petersen graph|0NR95CTFCZ]] is cubic.
The complete graph $K_n$ is $(n-1)$-regular. By the [[handshaking lemma|0XAP8WHNES]], a finite $k$-regular graph on $n$ vertices has $kn/2$ edges.

## Explanations

### L1 human

A graph is regular when all vertices have the same number of neighbours.
The corners and edges of a cube form a $3$-regular graph.
</document>

<document source="https://math2.tamnd.com/node/0KA7A4ENVJ.md" tag="0KA7A4ENVJ" kind="statement">
# Determinant via the Schur complement

## Statement

Let $K$ be a commutative ring and let $A$ be an invertible $p \times p$ matrix over $K$.
Let $B$, $C$ and $D$ be matrices over $K$ of sizes $p \times q$, $q \times p$ and $q \times q$.
Then
$$\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det A \cdot \det(D - C A^{-1} B),$$
where $\det$ is the [[determinant|00KDZRW4C0]].
The matrix $D - C A^{-1} B$ is the *Schur complement* of $A$ in the [[block matrix|0ASDFEQS6P]].

## Remarks

The proof factors the block matrix as $\begin{pmatrix} I & 0 \\ C A^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ 0 & D - C A^{-1} B \end{pmatrix}$.
Then it uses the rule for [[block triangular matrices|0J9TVTNB4K]].
For $p = q = 1$ and $a \ne 0$, it gives $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a(d - cb/a) = ad - bc$.

## Explanations

### L1 human

Block elimination removes the block $C$, in the same way as Gaussian elimination removes an entry.
What remains in the lower right corner is the Schur complement.
The determinant is then the product of two smaller determinants.
</document>

<document source="https://math2.tamnd.com/node/0KABP2HWMQ.md" tag="0KABP2HWMQ" kind="concept">
# Upper and lower Darboux integrals

## Summary

The upper Darboux integral of a bounded function is the infimum of its upper Darboux sums. The lower one is the supremum of its lower Darboux sums. A function is integrable when the two agree. The node [[upper and lower Darboux integrals|09FQT057T7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KABPMRHM2.md" tag="0KABPMRHM2" kind="concept">
# Characteristic subgroup

## Summary

A characteristic subgroup of a group is a subgroup that each automorphism of the group maps onto itself.
Each characteristic subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0KAMF19NV8.md" tag="0KAMF19NV8" kind="statement">
# Sum of binomial coefficients

## Statement

Let $n$ be a natural number.
Then
$$\sum_{k=0}^{n} \binom{n}{k} = 2^n.$$

## Remarks

For $n = 4$: $1 + 4 + 6 + 4 + 1 = 16$.
The identity is the [[binomial theorem|0EHWPWGHB2]] with $x = y = 1$.
Combinatorially, both sides count the subsets of an $n$-element set. The [[number of subsets of a finite set|0C5RDJE73X]] is $2^n$, and $\binom{n}{k}$ of them have size $k$.

## Explanations

### L1 human

The numbers in row $n$ of [[Pascal's triangle|05DME4TZ3K]] add up to $2^n$.
This is because each subset of an $n$-element set has some size $k$ from $0$ to $n$, and there are $2^n$ subsets in total.
</document>

<document source="https://math2.tamnd.com/node/0KAN330PAH.md" tag="0KAN330PAH" kind="definition">
# Nondegenerate bilinear form

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *nondegenerate* when, for each $u \in V$,
$$B(u, v) = 0 \text{ for all } v \in V \implies u = 0.$$
A bilinear form that is not nondegenerate is *degenerate*.

## Remarks

$B$ is nondegenerate if and only if its matrix in a basis has a nonzero determinant.
So, in finite dimensions, the condition in the first argument is equivalent to the same condition in the second argument.
$B$ is nondegenerate if and only if $u \mapsto B(u, \cdot)$ is an isomorphism from $V$ to its dual space.
Each inner product is nondegenerate; the form $B(x, y) = x_1 y_1$ on $\R^2$ is degenerate.

## Explanations

### L1 human

A form is nondegenerate when it can detect every nonzero vector.
For each nonzero $u$, some $v$ gives $B(u, v) \ne 0$.
In matrix terms, the matrix of the form is invertible.
</document>

<document source="https://math2.tamnd.com/node/0KB1ND7XDS.md" tag="0KB1ND7XDS" kind="statement">
# Subgroups of cyclic groups are cyclic

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
Then $H$ is cyclic.

## Remarks

Let $g$ generate $G$.
If $H \neq \{e\}$, let $m$ be the smallest positive integer with $g^m \in H$; then division with remainder shows that $H = \langle g^m \rangle$.
For example, each subgroup of $(\mathbb{Z}, +)$ has the form $m\mathbb{Z}$, generated by $m$.

## Explanations

### L1 human

A subgroup of a cyclic group is generated by the smallest positive power of the generator that it contains.
Any other power in the subgroup is a multiple of this one, or the subgroup would contain a smaller positive power.
Thus the subgroup is again cyclic.
</document>

<document source="https://math2.tamnd.com/node/0KBEMAJAYA.md" tag="0KBEMAJAYA" kind="statement">
# Distributive law

## Statement

For all natural numbers $a$, $b$ and $c$, $a \cdot (b + c) = a \cdot b + a \cdot c$.
Here $+$ is the [[addition|0W5DSDFQ4G]] and $\cdot$ is the [[multiplication|0JRC32D6K2]] of natural numbers.

## Remarks

This is the left distributive law. With commutativity of multiplication, it gives the right law $(b + c) \cdot a = b \cdot a + c \cdot a$.
The proof from the recursive definitions uses [[induction|0ASJVEFA9J]] on $c$.
The same law holds in the integers, the rationals, the reals and the complex numbers.

## Explanations

### L1 human

To multiply a number by a sum, you can multiply it by each part and then add.
For example, $3 \cdot (10 + 2) = 30 + 6 = 36$, and this is how many people compute $3 \cdot 12$.
</document>

<document source="https://math2.tamnd.com/node/0KBFAWYRTE.md" tag="0KBFAWYRTE" kind="concept">
# Covariance

## Summary

The covariance of two random variables measures how they vary together. It is positive when large values of one go with large values of the other, and negative in the opposite case.
The node [[covariance|0DJ8W3HP2D]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KBTP8Q2FS.md" tag="0KBTP8Q2FS" kind="concept">
# Reduced word

## Summary

A reduced word on a set $S$ is a finite string of letters $s$ and $s^{-1}$ with $s \in S$, with no letter next to its inverse.
The reduced words on $S$ are the elements of the free group on $S$.
</document>

<document source="https://math2.tamnd.com/node/0KC00K7KY4.md" tag="0KC00K7KY4" kind="concept">
# Bernoulli number

## Summary

The Bernoulli numbers are a sequence of rational numbers $B_n$, the coefficients of $t^n / n!$ in the expansion of $t / (e^t - 1)$.
The [[definition of the Bernoulli numbers|0RAS24E1R5]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0KC0F51T6A.md" tag="0KC0F51T6A" kind="statement">
# Stirling's formula

## Statement

As $n \to \infty$,
$$n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n.$$
That is, the ratio $\frac{n!}{\sqrt{2 \pi n} \, (n/e)^n}$ tends to $1$.

## Remarks

The relation $\sim$ is [[asymptotic equivalence|00SB5EA3VW]]. The absolute error grows, but the relative error tends to $0$.
Comparing $\ln n! = \sum_{k \le n} \ln k$ with $\int_1^n \ln x \, dx$ gives the form $n! \sim C \sqrt{n} (n/e)^n$. The [[Wallis product|0XCYA4YERS]] gives $C = \sqrt{2\pi}$.
A sharper form is $n! = \sqrt{2 \pi n} (n/e)^n \big(1 + \frac{1}{12 n} + O(n^{-2})\big)$. Mathlib states the limit as `Stirling.tendsto_stirlingSeq_sqrt_pi`.

## History

Abraham de Moivre first stated a less precise form of the result.
In 1730, James Stirling showed that the constant is $\sqrt{2\pi}$.

## Explanations

### L1 human

Stirling's formula gives the size of $n!$ for large $n$.
For example, it gives $10! \approx 3.599 \times 10^6$, while $10! = 3628800$.
</document>

<document source="https://math2.tamnd.com/node/0KC1NMG90B.md" tag="0KC1NMG90B" kind="concept">
# Wronskian

## Summary

The Wronskian of $n$ functions is the determinant of the matrix of the functions and their derivatives up to order $n - 1$.
It is a test for the linear independence of solutions of a linear differential equation. See the [[definition|0N5NF1RGBP]].

## Remarks

It is named after Józef Hoene-Wroński.
</document>

<document source="https://math2.tamnd.com/node/0KCE8CG86P.md" tag="0KCE8CG86P" kind="concept">
# Idempotent matrix

## Summary

An idempotent matrix is a square matrix that is equal to its own square.
</document>

<document source="https://math2.tamnd.com/node/0KCQ9Y3QKA.md" tag="0KCQ9Y3QKA" kind="concept">
# Basis

## Summary

A basis of a vector space is a linearly independent family of vectors that spans the space.
Each vector of the space is then a linear combination of the basis vectors in exactly one way.
</document>

<document source="https://math2.tamnd.com/node/0KCT6DPXJK.md" tag="0KCT6DPXJK" kind="concept">
# Venn diagram

## Summary

A Venn diagram is a picture of the relations between a small number of sets.
Each set is a closed curve, often a circle, inside a rectangle for the [[universal set|0BPYRMHJ48]].
For $n$ sets, the curves cut the plane into $2^n$ regions, one for each combination of "in" and "out" of the sets.

## Remarks

John Venn made these diagrams popular in the 1880s.
The diagrams make laws such as [[De Morgan's laws for sets|0VK0W4D5Y5]] visible, but a picture is not a proof.
An Euler diagram is similar, but it shows only the regions that are not empty.

## Explanations

### L1 human

Draw two overlapping circles $A$ and $B$.
The overlap is $A \cap B$, both circles together are $A \cup B$, and the region outside both circles is the complement of $A \cup B$.
</document>

<document source="https://math2.tamnd.com/node/0KCYJSB9FW.md" tag="0KCYJSB9FW" kind="definition">
# Fundamental solution of Pell's equation

## Definition

Let $d$ be a positive integer that is not a perfect square.
The fundamental solution of [[Pell's equation|0AF3DANJ76]] $x^2 - d y^2 = 1$ is the solution $(x_1, y_1)$ with $x_1 > 1$ and $y_1 > 0$ such that $x_1$ is as small as possible.

## Remarks

The fundamental solution exists by the [[existence of nontrivial solutions|0EM6C9SZN5]].
Among the solutions with $x > 1$ and $y > 0$, it also has the smallest $y$ and the smallest value of $x + y \sqrt{d}$.
For example, the fundamental solution for $d = 2$ is $(3, 2)$, and for $d = 7$ it is $(8, 3)$.
It appears among the convergents of the continued fraction of $\sqrt{d}$.

## Explanations

### L1 human

The fundamental solution is the smallest solution with positive entries.
All other solutions come from it by taking [[powers|0J1TT091TN]] of $x_1 + y_1 \sqrt{d}$.
</document>

<document source="https://math2.tamnd.com/node/0KDADE9RQ8.md" tag="0KDADE9RQ8" kind="definition">
# Vertex cover

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A vertex cover of $G$ is a set $S$ of vertices of $G$ that contains at least one end of each edge.

## Remarks

A set $S$ is a vertex cover if and only if its complement is an [[independent set|0KZJ82WEPP]].
For the [[complete bipartite graph|0V0GC7BQAN]] $K_{m,n}$, the least size of a vertex cover is $\min\{m, n\}$.
Each vertex of a cover meets at most one edge of a [[matching|0Z87BHAFTB]]. Thus every vertex cover has at least as many vertices as any matching has edges.

## Explanations

### L1 human

A vertex cover is a set of vertices that touches every edge.
Think of guards at crossings: each street must have a guard at one of its two ends.
</document>

<document source="https://math2.tamnd.com/node/0KDJD249DT.md" tag="0KDJD249DT" kind="statement">
# Bolzano–Weierstrass theorem

## Statement

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
Then $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that converges to a real number.

## Remarks

The proof takes a [[monotone subsequence|0MHGJRZMTM]]. It is bounded, so it converges by the [[monotone convergence theorem|0FM6N4R0VB]].
The limit of the subsequence lies in each closed interval $[m, M]$ that contains all terms, by [[the preservation of non-strict inequalities|0N32V0E3XJ]].
The theorem holds in $\mathbb{R}^k$ for each $k$. It says that each closed and bounded subset of $\mathbb{R}^k$ is sequentially compact.
The theorem fails in $\mathbb{Q}$. The decimal approximations of $\sqrt{2}$ are bounded, and no subsequence has a limit in $\mathbb{Q}$.

## History

Bernard Bolzano proved the result in 1817, as a lemma for the intermediate value theorem. Karl Weierstrass proved it again about fifty years later and made it a central tool of analysis.

## Explanations

### L1 human

A bounded sequence cannot run away, so its terms must crowd together somewhere.
The theorem picks infinitely many terms that approach one point of crowding.
For $(-1)^n$, the even terms form a subsequence that converges to $1$.
</document>

<document source="https://math2.tamnd.com/node/0KE6NBTVY8.md" tag="0KE6NBTVY8" kind="definition">
# Exponential generating function

## Definition

Let $K$ be a field of characteristic $0$, for example $\mathbb{Q}$ or $\mathbb{R}$, and let $(a_n)_{n \ge 0}$ be a sequence in $K$.
The exponential generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]
$$\hat{A}(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!} \in K[\![x]\!].$$

## Remarks

The constant sequence $1$ has the exponential generating function $e^x = \sum_n x^n / n!$.
For exponential generating functions, the product corresponds to the binomial convolution $c_n = \sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$.
They suit labelled structures. For example, the [[derangement numbers|03WY4HHMT1]] have the exponential generating function $\frac{e^{-x}}{1 - x}$.

## Explanations

### L1 human

An exponential generating function divides the $n$-th term by $n!$ before it puts it in the series.
This makes the series of fast-growing sequences, such as $n!$, easy to handle: the sequence $n!$ gives $\frac{1}{1 - x}$.
</document>

<document source="https://math2.tamnd.com/node/0KF3J2KAER.md" tag="0KF3J2KAER" kind="statement">
# Classification of Pythagorean triples

## Statement

Let $m$ and $n$ be [[coprime|0GFKP8F8AG]] positive integers with $m > n$ and $m - n$ odd.
Then $(m^2 - n^2, \, 2mn, \, m^2 + n^2)$ is a primitive [[Pythagorean triple|01G1WY1GVX]].
Conversely, each primitive Pythagorean triple $(a, b, c)$ with $b$ even arises in this way from exactly one such pair $(m, n)$.
Each Pythagorean triple is $(ka, kb, kc)$ for a positive integer $k$ and a primitive triple $(a, b, c)$, up to the order of the first two entries.

## Remarks

The formula is called Euclid's formula.
For example, $(m, n) = (2, 1)$ gives $(3, 4, 5)$, and $(m, n) = (3, 2)$ gives $(5, 12, 13)$.
One proof writes $b^2 = (c - a)(c + a)$ and uses unique factorization. Another uses the rational points on the unit circle.

## Explanations

### L1 human

Euclid's formula produces every right triangle with whole-number sides, from two numbers $m$ and $n$.
The identity $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ shows that the formula always works.
</document>

<document source="https://math2.tamnd.com/node/0KFYGSHHF5.md" tag="0KFYGSHHF5" kind="concept">
# Directed graph

## Summary

A directed graph is a set of vertices with arrows between some ordered pairs of vertices. Directed graphs model one-way relations such as links, flows and dependencies. The node [[directed graph|0R5HTV7REV]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KFZ0QX83G.md" tag="0KFZ0QX83G" kind="statement">
# The alternating group is generated by three-cycles

## Statement

Let $n \geq 3$.
The [[alternating group|0Z5905Q73D]] $A_n$ is [[generated|0RP7VWQ0YN]] by the $3$-cycles in $S_n$.

## Remarks

Each $3$-cycle is even, so the $3$-cycles lie in $A_n$.
An even permutation is a product of an even number of transpositions, and each product of two transpositions is a product of $3$-cycles.
For distinct $a, b, c, d$, the formulas are $(a\ b)(a\ c) = (a\ c\ b)$ and $(a\ b)(c\ d) = (a\ c\ b)(a\ c\ d)$.

## Explanations

### L1 human

An even permutation is a chain of swaps, taken two at a time.
Each pair of swaps can be written with $3$-cycles.
So the $3$-cycles are enough to build all even permutations.
</document>

<document source="https://math2.tamnd.com/node/0KG4D8AP6G.md" tag="0KG4D8AP6G" kind="definition">
# Norm induced by an inner product

## Definition

Let $V$ be an [[inner product space|0QVNCJXRY3]] over $\R$ or $\C$, with inner product $\langle \cdot, \cdot \rangle$.
The *norm induced by the inner product* is the map
$$\norm{x} = \sqrt{\langle x, x \rangle} \qquad (x \in V).$$

## Remarks

It is a [[norm|0V2WP9B1MF]]; the triangle inequality follows from the [[Cauchy–Schwarz inequality|0RFWW56Y22]].
For the dot product on $\R^n$, it is the Euclidean length $\sqrt{x_1^2 + \cdots + x_n^2}$.
It satisfies the [[parallelogram law|026HPGYGS4]], and the [[polarisation identity|05FKHRT0B4]] recovers the inner product from it.

## Explanations

### L1 human

The length of a vector is the square root of its inner product with itself.
In the plane, this is the Pythagorean formula $\sqrt{x^2 + y^2}$.
So every inner product gives a way to measure lengths.
</document>

<document source="https://math2.tamnd.com/node/0KH1N66QWW.md" tag="0KH1N66QWW" kind="definition">
# Dirichlet convolution

## Definition

Let $f$ and $g$ be [[arithmetic functions|0H2EPRZC0D]].
Their Dirichlet convolution is the arithmetic function $f * g$ given by
$$(f * g)(n) = \sum_{d \mid n} f(d) \, g\!\left(\frac{n}{d}\right)$$
for each positive integer $n$, where $d$ runs over the [[positive divisors|0FCWA16FZE]] of $n$.

## Remarks

The convolution is commutative and associative, and it distributes over addition.
The identity element is $\varepsilon$, with $\varepsilon(1) = 1$ and $\varepsilon(n) = 0$ for $n > 1$.
An arithmetic function $f$ has an inverse for $*$ if and only if $f(1) \ne 0$.
For example, with the constant function $\mathbf{1}$, $(\mathbf{1} * \mathbf{1})(n)$ is the [[number of divisors|0BKJT6WG96]] of $n$.

## Explanations

### L1 human

The Dirichlet convolution combines two functions by a sum over the factorizations $n = d \cdot e$.
It is the natural product for functions defined by sums over divisors.
</document>

<document source="https://math2.tamnd.com/node/0KJ7ZS8AJA.md" tag="0KJ7ZS8AJA" kind="concept">
# Posterior probability

## Summary

The posterior probability of a hypothesis $H$ is its [[conditional probability|0FJB06AD2P]] $P(H \mid D)$ given the observed data $D$.
By [[Bayes' theorem|0NM6E9VW04]], $P(H \mid D) = P(D \mid H) \, P(H) / P(D)$. So the posterior is proportional to the likelihood $P(D \mid H)$ times the [[prior probability|00NBBSXRQS]] $P(H)$.
For an unknown parameter $\theta$ with prior density $\pi(\theta)$ and likelihood $L(\theta \mid x)$, the posterior density is $\pi(\theta \mid x) = L(\theta \mid x) \, \pi(\theta) / \int L(t \mid x) \, \pi(t) \, dt$.

## Remarks

The posterior of one experiment can be the prior of the next experiment. So Bayesian updates can be done one observation at a time.
A credible interval is an interval that has a given posterior probability, for example $95\%$.

## Explanations

### L1 human

The posterior is what you believe after you see the data.
It mixes what you believed before (the prior) with how well each hypothesis explains the data (the likelihood).
</document>

<document source="https://math2.tamnd.com/node/0KJH6SRFZK.md" tag="0KJH6SRFZK" kind="concept">
# Affine combination

## Summary

An affine combination of points is a weighted sum of the points whose weights add up to 1. It makes sense in an [[affine space|09XNAPTJQ7]], where a sum of points with arbitrary weights does not.
</document>

<document source="https://math2.tamnd.com/node/0KJVQV3E0F.md" tag="0KJVQV3E0F" kind="statement">
# A polynomial over a field has at most as many roots as its degree

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be a nonzero polynomial of [[degree|0DPF6WRZW6]] $n$.
Then $f$ has at most $n$ roots in $K$.
More precisely, the sum of the [[multiplicities|03XVP8Q4GY]] of the roots of $f$ in $K$ is at most $n$.

## Remarks

The proof is an induction on $n$ with the [[factor theorem|0K3MWA7DMC]]: a root $c$ gives $f = (x - c) g$ with $\deg g = n - 1$.
The same statement holds over every [[integral domain|09S3PFBEPM]].
It fails without this hypothesis: in $\mathbb{Z}/8\mathbb{Z}$, the polynomial $x^2 - 1$ has the four roots $1, 3, 5, 7$.

## Explanations

### L1 human

A polynomial of degree $2$ over the real numbers has at most two roots.
In general, each root takes away one linear factor, and you can do this at most $n$ times.
Strange rings like $\mathbb{Z}/8\mathbb{Z}$ can break this rule, because they have zero divisors.
</document>

<document source="https://math2.tamnd.com/node/0KJXHN0TW0.md" tag="0KJXHN0TW0" kind="statement">
# One-sided limits of a monotone function exist

## Statement

Let $a < c < b$ and let $f \colon (a, b) \to \mathbb{R}$ be increasing.
Then both [[one-sided limits|0909PWZR62]] of $f$ at $c$ exist, and
$$\lim_{x \to c^-} f(x) = \sup_{a < x < c} f(x) \le f(c) \le \inf_{c < x < b} f(x) = \lim_{x \to c^+} f(x).$$

## Remarks

The proof is the same as for the [[monotone convergence theorem|0FM6N4R0VB]]. The supremum exists because $f(c)$ is an upper bound.
A similar statement holds for decreasing functions, with the inequalities reversed.
So $f$ is continuous at $c$ exactly when the two one-sided limits are equal. Otherwise $c$ is a [[jump discontinuity|05B0FKFXQA]].
Mathlib states the left limit as `Monotone.tendsto_nhdsLT`.

## Explanations

### L1 human

An increasing function can only move up as $x$ approaches $c$ from the left.
Its values are bounded by $f(c)$, so they settle at a limit.
</document>

<document source="https://math2.tamnd.com/node/0KM2CJ80G7.md" tag="0KM2CJ80G7" kind="definition">
# Primitive root modulo n

## Definition

Let $n$ be a positive integer.
A primitive root modulo $n$ is an integer $g$ coprime to $n$ whose [[multiplicative order|0SBRX5CZ5R]] modulo $n$ is $\varphi(n)$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

Equivalently, $g$ is a primitive root modulo $n$ when the powers $g, g^2, \dots, g^{\varphi(n)}$ represent all the residue classes coprime to $n$.
So a primitive root exists exactly when the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic, and then $\overline{g}$ generates this group.
For example, $3$ is a primitive root modulo $7$: its powers are $3, 2, 6, 4, 5, 1$ modulo $7$.
There is no primitive root modulo $8$: each odd $a$ satisfies $a^2 \equiv 1 \pmod 8$, but $\varphi(8) = 4$.

## Explanations

### L1 human

A primitive root is a number whose powers run through all the residues coprime to $n$ before they come back to $1$.
Modulo $5$, the powers of $2$ are $2, 4, 3, 1$. So $2$ is a primitive root modulo $5$, but $4$ is not, since $4^2 \equiv 1$.
</document>

<document source="https://math2.tamnd.com/node/0KM4KVR9MV.md" tag="0KM4KVR9MV" kind="statement">
# Hausdorff maximal principle

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]].
Then each [[chain|0GP9ASVQ1X]] of $P$ is a subset of a maximal chain.
A maximal chain is a chain that is not a proper subset of another chain of $P$.

## Remarks

Over ZF, the principle is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].
With the empty chain, it says that each partially ordered set has a maximal chain.
Felix Hausdorff proved it in 1914.

## Explanations

### L1 human

Start with any chain in the order.
You can extend it until no element can be added without breaking the chain property.
</document>

<document source="https://math2.tamnd.com/node/0KMJ6919WS.md" tag="0KMJ6919WS" kind="definition">
# Fixed field

## Definition

Let $L$ be a [[field|0QA8CHH8MG]] and let $H$ be a group of ring automorphisms of $L$.
The fixed field of $H$ is
$$L^H = \{ a \in L : \sigma(a) = a \text{ for every } \sigma \in H \}.$$

## Remarks

The set $L^H$ is a [[subfield|07QY5RPFBR]] of $L$, because each $\sigma \in H$ respects the field operations.
For $L = \mathbb{C}$ and $H = \{\mathrm{id}, \sigma\}$, with $\sigma$ complex conjugation, $L^H = \mathbb{R}$.
If $H$ is finite, then $[L : L^H] = |H|$ and $L/L^H$ is a [[Galois extension|0DS2F7GK05]], by a theorem of Emil Artin.
More generally, a group acting on a ring has a fixed subring, the ring of invariants.

## Explanations

### L1 human

The fixed field is the part of the field that a group of symmetries does not move.
In the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], fixed fields turn subgroups into intermediate fields.
</document>

<document source="https://math2.tamnd.com/node/0KMN32XGPF.md" tag="0KMN32XGPF" kind="statement">
# Determinant of a linear map is the determinant of its matrix

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ with a basis $B$, and let $T \colon V \to V$ be a linear map.
Then the [[determinant of $T$|0X0HQYSZ0K]] equals the determinant of the [[matrix of $T$|0YQSGNYE3K]] in $B$:
$$\det T = \det [T]_B.$$

## Remarks

The key fact is $\det(P^{-1} A P) = \det A$, which follows from the product rule.
Over a commutative ring, the same statement holds for free modules of finite rank.

## Explanations

### L1 human

We can compute the determinant of a linear map in any basis.
We write its matrix in that basis, and we take the determinant of the matrix.
Every basis gives the same answer.
</document>

<document source="https://math2.tamnd.com/node/0KMVFSHFGZ.md" tag="0KMVFSHFGZ" kind="statement">
# Strong induction

## Statement

Let $P(n)$ be a property of natural numbers $n$.
Suppose that for each natural number $n$, if $P(k)$ holds for every natural number $k < n$, then $P(n)$ holds.
Then $P(n)$ holds for every natural number $n$.

## Remarks

The hypothesis for $n = 0$ asks for a proof of $P(0)$ with no assumption, because no natural number is less than $0$.
The proof applies ordinary [[induction|0ASJVEFA9J]] to the property "$P(k)$ holds for every $k < n$".
A standard use is the proof that each integer $n \ge 2$ is a product of primes.

## Explanations

### L1 human

In ordinary induction, you may use only the case just before $n$.
In strong induction, you may use all the cases before $n$.
This helps when $n$ breaks into smaller pieces that are not just $n - 1$, for example when $n = a b$ with $a$ and $b$ smaller than $n$.
</document>

<document source="https://math2.tamnd.com/node/0KN22AAW5R.md" tag="0KN22AAW5R" kind="definition">
# Monotone sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is increasing if $a_n \le a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is decreasing if $a_n \ge a_{n+1}$ for all $n \in \mathbb{N}$.
The sequence is monotone if it is increasing or decreasing.
The sequence is strictly increasing if $a_n < a_{n+1}$ for all $n$, and strictly decreasing if $a_n > a_{n+1}$ for all $n$.

## Remarks

Some texts say non-decreasing for increasing in this sense, and they keep the word increasing for strictly increasing.
By induction, $(a_n)$ is increasing if and only if $a_m \le a_n$ for all $m \le n$. This is the definition of a monotone function on $\mathbb{N}$.
The sequence $1/n$ is strictly decreasing. The sequence $(-1)^n$ is not monotone.

## Explanations

### L1 human

A monotone sequence always moves in one direction.
An increasing sequence never goes down, and a decreasing sequence never goes up.
</document>

<document source="https://math2.tamnd.com/node/0KN2S81K72.md" tag="0KN2S81K72" kind="concept">
# Ring isomorphism

## Summary

A ring isomorphism is a bijective ring homomorphism.
Two rings are isomorphic when there is a ring isomorphism between them.
The [[definition of a ring isomorphism|0B6QC0C0G1]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0KN2V7840E.md" tag="0KN2V7840E" kind="statement">
# The exponential function is strictly increasing

## Statement

The exponential function is strictly increasing on $\mathbb{R}$.
That is, $x < y$ implies $\exp(x) < \exp(y)$.

## Remarks

The derivative is [[$\exp$ itself|0EX9GFJ0B3]], and it is [[positive|0P9BNBW2W3]]. Then [[positive derivative|07J5X2T1ZX]] gives the claim.
Also $\exp(x) \to \infty$ as $x \to \infty$ and $\exp(x) \to 0$ as $x \to -\infty$. So $\exp$ is a bijection from $\mathbb{R}$ onto $(0, \infty)$.
Mathlib states it as `Real.exp_strictMono`.

## Explanations

### L1 human

The exponential function always grows as $x$ grows.
So it takes each positive value exactly once.
</document>

<document source="https://math2.tamnd.com/node/0KN7EPH7Z0.md" tag="0KN7EPH7Z0" kind="concept">
# Restriction of an endomorphism to an invariant subspace

## Summary

The restriction of an endomorphism to an invariant subspace is the endomorphism of the subspace that agrees with the original map on it.
</document>

<document source="https://math2.tamnd.com/node/0KN93VDSXA.md" tag="0KN93VDSXA" kind="statement">
# Ceva's theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \neq Y$, let the signed ratio $\frac{XZ}{ZY}$ be the real number $\lambda$ with $Z - X = \lambda\,(Y - Z)$.
Let $O$ be a point that is not on the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$. Suppose that the lines $AO$, $BO$ and $CO$ meet the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.
Then
$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.$$
Conversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $1$. Then the lines $AD$, $BE$ and $CF$ meet in one point or are all parallel.

## Remarks

The signed ratio $\frac{AF}{FB}$ is positive if $F$ is between $A$ and $B$, and negative otherwise.
The theorem uses only ratios of segments on one line. So it is true in each affine plane over each field.
For the three medians, each ratio is $1$. The converse then shows again that the medians meet in one point.

## History

Giovanni Ceva published the theorem in 1678, in his work De lineis rectis. Yusuf al-Mu'taman ibn Hud, an 11th century king of Zaragoza, had proved it much earlier.

## Explanations

### L1 human

Draw three lines from the corners of a triangle through one common inside point. Each line cuts the opposite side into two parts. Multiply the three ratios of the parts, going round the triangle in one direction: the product is always $1$.
</document>

<document source="https://math2.tamnd.com/node/0KNAN9F27P.md" tag="0KNAN9F27P" kind="definition">
# Maximum degree

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with a nonempty vertex set $V$.
The maximum degree of $G$ is
$$\Delta(G) = \max_{v \in V} \deg(v),$$
the largest [[degree|0T0F6CCS9W]] of a vertex of $G$.

## Remarks

The complete graph $K_n$ has $\Delta = n - 1$.
A [[greedy colouring|04AAKZNXHD]] uses at most $\Delta(G) + 1$ colours. [[Brooks' theorem|0Y4TV2D5RQ]] improves this to $\Delta(G)$ in most cases.

## Explanations

### L1 human

The maximum degree is the largest number of edges at any vertex.
In a star with one centre and $5$ leaves, the maximum degree is $5$.
</document>

<document source="https://math2.tamnd.com/node/0KNJWEVSMG.md" tag="0KNJWEVSMG" kind="example">
# p-norm on R^n

## Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number.
For $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, let
$$\|x\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p}, \qquad \|x\|_\infty = \max_{1 \le i \le n} |x_i|.$$
Then $\|\cdot\|_p$ and $\|\cdot\|_\infty$ are norms on $\mathbb{R}^n$.
They are $0$ only at $x = 0$, they satisfy $\|\lambda x\| = |\lambda| \, \|x\|$ for real $\lambda$, and they satisfy the triangle inequality $\|x + y\| \le \|x\| + \|y\|$.

## Remarks

The triangle inequality for $\|\cdot\|_p$ is the Minkowski inequality.
The metric $\|x - y\|_p$ is the [[p-metric|0TFDZN6AXP]]. For $p = 2$, the norm is the Euclidean norm.
For each $x$, $\|x\|_p \to \|x\|_\infty$ as $p \to \infty$.
All these norms are [[equivalent|03MC70ZRNR]], and $\mathbb{R}^n$ is a [[Banach space|0BDTTM24DP]] with each of them.

## Explanations

### L1 human

The p-norms give different ways to measure the length of a vector from its coordinates.
For $p = 1$, you add the absolute values. For $p = 2$, you get the usual length. For $p = \infty$, you take the largest coordinate.
</document>

<document source="https://math2.tamnd.com/node/0KP0QNXPRT.md" tag="0KP0QNXPRT" kind="concept">
# Homogeneous linear equation

## Summary

A homogeneous linear equation is a linear differential equation $L y = 0$ with right side zero.
Its solutions form a vector space whose dimension is the order of the equation. See the [[definition|0EQZSAB9EC]].
</document>

<document source="https://math2.tamnd.com/node/0KPHENZ5S7.md" tag="0KPHENZ5S7" kind="object">
# Dirac delta function

## Description

Let $C_c^\infty(\mathbb{R})$ be the space of infinitely differentiable functions $\varphi : \mathbb{R} \to \mathbb{C}$ that vanish outside a bounded set.
The Dirac delta is the distribution $\delta$ that maps each test function $\varphi \in C_c^\infty(\mathbb{R})$ to $\delta[\varphi] = \varphi(0)$.
Informally, one writes $\int_{-\infty}^{\infty} \delta(t)\, \varphi(t)\, dt = \varphi(0)$, as if $\delta$ were a function that is $0$ away from $0$ and has total integral $1$.
No actual function has this property. The delta is a generalized function.

## Remarks

As a measure, $\delta$ gives the mass $1$ to each set that contains $0$ and the mass $0$ to the other sets.
The shifted delta $\delta(t - a)$ maps $\varphi$ to $\varphi(a)$. For $a > 0$, its Laplace transform is $e^{-a s}$, and for $a = 0$ the usual convention gives $1$.
In the sense of distributions, $\delta$ is the derivative of the [[Heaviside step function|09C4JP7Y0M]].
In differential equations, $\delta(t - a)$ models an impulse: a large force that acts for a very short time around $t = a$.

## Explanations

### L1 human

The delta function is an idealized spike at $0$, infinitely tall and infinitely thin, with area $1$.
It is not a real function. But it makes sense when we integrate it against a smooth function, and the result is the value of that function at $0$.
</document>

<document source="https://math2.tamnd.com/node/0KQZYGEJTF.md" tag="0KQZYGEJTF" kind="definition">
# Signature of a quadratic form

## Definition

Let $V$ be a real vector space of finite dimension $n$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Choose a basis in which $Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + \cdots + a_n x_n^2$.
Let $p$, $q$ and $r$ be the numbers of indices $i$ with $a_i > 0$, $a_i < 0$ and $a_i = 0$.
The *signature* of $Q$ is the triple $(p, q, r)$.

## Remarks

By [[Sylvester's law of inertia|0WFED3K3DQ]], the triple does not depend on the choice of the basis.
For a symmetric matrix $A$ of $Q$, the numbers $p$, $q$ and $r$ count the positive, negative and zero eigenvalues of $A$, with multiplicity.
Conventions vary: some authors write only $(p, q)$, or the number $p - q$.
The Minkowski form $x_1^2 - x_2^2 - x_3^2 - x_4^2$ on $\R^4$ has $p = 1$, $q = 3$ and $r = 0$.

## Explanations

### L1 human

The signature counts the plus signs, the minus signs and the zeros in a real quadratic form, after diagonalisation.
It is the complete invariant of a real quadratic form.
Two real quadratic forms are equivalent exactly when they have the same signature.
</document>

<document source="https://math2.tamnd.com/node/0KRH5157WP.md" tag="0KRH5157WP" kind="concept">
# Stationary distribution

## Summary

A stationary distribution of a Markov chain is a distribution of the state that does not change in one step of the chain.
The node [[stationary distribution|02KYGSCTCN]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KRMJRDRZT.md" tag="0KRMJRDRZT" kind="concept">
# Continuity at a point

## Summary

A function is continuous at a point when points near the point are sent to points near its image.
In topological spaces, nearness is expressed with neighbourhoods.
</document>

<document source="https://math2.tamnd.com/node/0KRVRV5V80.md" tag="0KRVRV5V80" kind="concept">
# Symmetric bilinear form associated with a quadratic form

## Summary

Let $Q$ be a quadratic form over a field of characteristic not two.
The symmetric bilinear form associated with $Q$ is the unique symmetric bilinear form $B$ with $B(v, v) = Q(v)$.
</document>

<document source="https://math2.tamnd.com/node/0KS0KRDV4Y.md" tag="0KS0KRDV4Y" kind="concept">
# Degree of a vertex

## Summary

The degree of a vertex is the number of edges that meet it. Degrees are the simplest numerical invariants of a graph. The node [[degree of a vertex|0T0F6CCS9W]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KSM23SN3H.md" tag="0KSM23SN3H" kind="definition">
# Nowhere dense set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
The set $A$ is nowhere dense in $X$ when the [[interior|0C38S0KMJK]] of its [[closure|0DRKT1QJR7]] is empty, that is, $\operatorname{int} \overline{A} = \emptyset$.

## Remarks

The set $\mathbb{Z}$ is nowhere dense in $\mathbb{R}$. So is each finite subset of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is nowhere dense in $\mathbb{R}$, though it is uncountable.
The set $\mathbb{Q}$ is not nowhere dense in $\mathbb{R}$, because its closure is $\mathbb{R}$.
A closed set is nowhere dense if and only if its complement is open and [[dense|0D198GPMTJ]].

## Explanations

### L1 human

A nowhere dense set is thin everywhere: inside each open set, you can find a smaller open set that misses it completely.
The integers on the real line are an example, since there are gaps between them.
</document>

<document source="https://math2.tamnd.com/node/0KT8TGXD8K.md" tag="0KT8TGXD8K" kind="concept">
# Equivalence relation induced by a function

## Summary

Each function induces an equivalence relation on its domain: two elements are equivalent when the function has the same value at both.
</document>

<document source="https://math2.tamnd.com/node/0KTGJW1ZTH.md" tag="0KTGJW1ZTH" kind="concept">
# Jordan measure

## Summary

The Jordan measure, or Jordan content, extends the volume of boxes in [[R^n|0XPR1E85W0]] to some bounded sets. Finite unions of boxes approximate these sets well from inside and from outside.
A bounded set has Jordan measure exactly when its indicator function is Riemann integrable, and then the measure is the integral of that function.
</document>

<document source="https://math2.tamnd.com/node/0KTW26C80J.md" tag="0KTW26C80J" kind="concept">
# Free group

## Summary

The free group on a set $S$ is the group of reduced words on $S$, with concatenation followed by reduction as the operation.
It is the group generated by $S$ with no relations other than those that the group axioms force.
</document>

<document source="https://math2.tamnd.com/node/0KVGH0J5B3.md" tag="0KVGH0J5B3" kind="concept">
# Cyclic group

## Summary

A cyclic group is a group that one element generates.
Up to isomorphism, the cyclic groups are the integers and the integers modulo $n$ under addition.
</document>

<document source="https://math2.tamnd.com/node/0KVQ9D86BM.md" tag="0KVQ9D86BM" kind="statement">
# Variance under scaling

## Statement

Let $X$ be a real random variable with finite [[variance|007QED3E0Y]] and let $a \in \mathbb{R}$. Then
$$\operatorname{Var}(aX) = a^2 \operatorname{Var}(X).$$

## Remarks

Proof: $E[aX] = a E[X]$ by [[linearity of expectation|0P7PTC8DDE]]. So $(aX - E[aX])^2 = a^2 (X - E[X])^2$, and one takes expectations.
For the standard deviation, the formula becomes $\sigma_{aX} = |a| \, \sigma_X$.
In particular, $\operatorname{Var}(-X) = \operatorname{Var}(X)$.

## Explanations

### L1 human

If you measure in centimeters instead of meters, every value is multiplied by $100$. The variance is then multiplied by $100^2 = 10000$, because it is measured in squared units.
</document>

<document source="https://math2.tamnd.com/node/0KW2X0TFBX.md" tag="0KW2X0TFBX" kind="definition">
# Modulus of a complex number

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with real part $a = \operatorname{Re} z$ and imaginary part $b = \operatorname{Im} z$.
The modulus, or absolute value, of $z$ is the nonnegative real number
$$|z| = \sqrt{a^2 + b^2}.$$

## Remarks

For a real number $z = a$, the modulus is $\sqrt{a^2} = |a|$, the [[absolute value|00S7C68D51]] of $a$.
In the [[complex plane|0CTE7A31XQ]], $|z|$ is the distance from $z$ to $0$, by the Pythagorean theorem. The distance between $z$ and $w$ is $|z - w|$.
For example, $|3 + 4 i| = 5$.

## Explanations

### L1 human

The modulus of a complex number is its length, that is, its distance from $0$ in the plane.
The point $3 + 4 i$ is the corner of a right triangle with sides $3$ and $4$, so its distance from $0$ is $5$.
</document>

<document source="https://math2.tamnd.com/node/0KW3PN547K.md" tag="0KW3PN547K" kind="concept">
# Diameter of a set

## Summary

The diameter of a subset of a metric space is the supremum of the distances between pairs of its points.
It generalizes the diameter of a circle.
</document>

<document source="https://math2.tamnd.com/node/0KWFFMDF8T.md" tag="0KWFFMDF8T" kind="concept">
# Connected graph

## Summary

A connected graph is a graph in one piece, in which any two vertices are joined by a walk. Connectivity is a basic hypothesis of many results, such as the existence of spanning trees. The node [[connected graph|0FFAYNMAJZ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KWWEXW0RN.md" tag="0KWWEXW0RN" kind="statement">
# Reduction modulo p test for irreducibility

## Statement

Let $f \in \mathbb{Z}[x]$ be a [[monic polynomial|0BDX4MBKT4]] and let $p$ be a prime number.
Let $\bar{f} \in \mathbb{F}_p[x]$ be the polynomial whose coefficients are those of $f$ modulo $p$.
If $\bar{f}$ is [[irreducible|00X2A9G9W3]] in $\mathbb{F}_p[x]$, then $f$ is irreducible in $\mathbb{Z}[x]$ and in $\mathbb{Q}[x]$.

## Remarks

The proof: a factorisation $f = g h$ into monic factors of positive degree gives $\bar{f} = \bar{g} \bar{h}$, with $\deg \bar{g} = \deg g$ and $\deg \bar{h} = \deg h$.
Irreducibility in $\mathbb{Q}[x]$ then follows by [[Gauss's lemma|0QFV5Y1Z5X]].
The converse is false: $x^4 + 1$ is irreducible over $\mathbb{Q}$, but it is reducible modulo every prime.
Example: $x^3 + x + 1$ has no root in $\mathbb{F}_2$, so it is irreducible over $\mathbb{F}_2$, and so over $\mathbb{Q}$.

## Explanations

### L1 human

A factorisation over the integers stays a factorisation after reduction modulo $p$.
So if the reduced polynomial does not factor, the original one does not factor either.
Over a small finite field, you can check irreducibility by a finite search.
</document>

<document source="https://math2.tamnd.com/node/0KWXYZPJKN.md" tag="0KWXYZPJKN" kind="definition">
# Change of basis matrix

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with two [[bases|0KCQ9Y3QKA]] $B = (b_1, \dots, b_n)$ and $B' = (b'_1, \dots, b'_n)$.
The *change of basis matrix* from $B$ to $B'$ is the $n \times n$ matrix $P = (p_{ij})$ defined by
$$b'_j = \sum_{i=1}^{n} p_{ij} b_i \qquad (1 \le j \le n).$$
Its column $j$ is the [[coordinate vector|04CYQHN6MC]] $[b'_j]_B$.

## Remarks

For each $v \in V$, $[v]_B = P [v]_{B'}$.
The matrix $P$ is invertible, and $P^{-1}$ is the change of basis matrix from $B'$ to $B$.
$P$ is the matrix of the identity map of $V$ in the bases $B'$ (domain) and $B$ (codomain).
Conventions differ between authors; some call $P^{-1}$ the change of basis matrix.

## Explanations

### L1 human

The change of basis matrix lists the new basis vectors in the old coordinates.
It translates coordinates in the new basis into coordinates in the old basis.
Its inverse translates in the other direction.
</document>

<document source="https://math2.tamnd.com/node/0KX1MXJ5MR.md" tag="0KX1MXJ5MR" kind="statement">
# Radius of convergence of the binomial series

## Statement

Let $\alpha$ be a real number that is not a natural number.
Then the [[binomial series|0HB0XKDXHH]] $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$ has radius of convergence $1$.

## Remarks

The coefficients are nonzero, and $\binom{\alpha}{k+1} / \binom{\alpha}{k} = (\alpha - k)/(k + 1)$. Its absolute value tends to $1$, so the [[ratio formula|0JBMYAA9QH]] gives $R = 1$.
At $x = \pm 1$ the behavior depends on $\alpha$. For $\alpha > 0$ the series converges absolutely at both endpoints.
Mathlib states the result as `binomialSeries_radius_eq_one`.

## Explanations

### L1 human

Unless the series stops after finitely many terms, it converges exactly inside $(-1, 1)$, plus possibly some endpoints.
This matches the fact that $(1 + x)^{\alpha}$ has trouble at $x = -1$.
</document>

<document source="https://math2.tamnd.com/node/0KX66CZMBB.md" tag="0KX66CZMBB" kind="concept">
# Positive definite matrix

## Summary

A positive definite matrix is a Hermitian matrix $A$ with $x^* A x > 0$ for every nonzero vector $x$.
</document>

<document source="https://math2.tamnd.com/node/0KXJTZSW9B.md" tag="0KXJTZSW9B" kind="example">
# Usual metric on the real line

## Example

On the set $\mathbb{R}$ of [[real numbers|08J2M1JYAG]], let
$$d(x, y) = |x - y|,$$
where $|\cdot|$ is the [[absolute value|00S7C68D51]].
Then $d$ is a [[metric|063W5JMY85]] on $\mathbb{R}$.
The [[triangle inequality|03HF5PMY5T]] for real numbers gives the triangle inequality for $d$.

## Remarks

The number $|x - y|$ is the absolute difference of $x$ and $y$.
This metric is the case $n = 1$ of the [[Euclidean metric|0E1MMCAY5B]].
The same formula with the [[modulus|0KW2X0TFBX]] gives a metric on the complex numbers.

## Explanations

### L1 human

The distance between two numbers is the length of the segment between them on the number line.
For example, the distance between $-2$ and $3$ is $|-2 - 3| = 5$.
</document>

<document source="https://math2.tamnd.com/node/0KXQKES075.md" tag="0KXQKES075" kind="concept">
# Number of prime factors counted with multiplicity

## Summary

The function $\Omega(n)$ counts the prime factors of a positive integer $n$, with each prime counted as often as it divides $n$.
The [[definition of the number of prime factors with multiplicity|08420828EY]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0KXQKX9P63.md" tag="0KXQKX9P63" kind="concept">
# Splitting field

## Summary

A splitting field of a polynomial over a field is a smallest extension field over which the polynomial splits into linear factors.
It is generated by the roots of the polynomial.
The [[definition of a splitting field|02NZZ1J0FH]] states it.
</document>

<document source="https://math2.tamnd.com/node/0KXSK8TEEB.md" tag="0KXSK8TEEB" kind="statement">
# Continuous partial derivatives imply differentiability

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that each [[partial derivative|09EGC6F8PT]] $\partial f_j / \partial x_i$ exists at each point of $U$, for $1 \le i \le n$ and $1 \le j \le m$, and that each of them is continuous at $a$.
Then $f$ is [[differentiable|0135XZZQ8M]] at $a$.

## Remarks

If the partial derivatives exist on $U$ and are continuous on all of $U$, then $f$ is differentiable on $U$, and $Df$ is continuous. So $f$ is of class $C^1$ on $U$.
The converse is false. The function $f(x) = \|x\|^2 \sin(1/\|x\|)$ for $x \ne 0$, with $f(0) = 0$, is differentiable on $\mathbb{R}^n$, but its partial derivatives are not continuous at $0$.
Without continuity, the existence of the partial derivatives is not enough, see [[this counterexample|0TNZC55X3J]].

## Explanations

### L1 human

The partial derivatives give a candidate for the derivative: the matrix of all $\partial f_j / \partial x_i$ at $a$.
Continuity of the partial derivatives near $a$ is what makes this candidate a good linear approximation in all directions, and not only along the axes.
In practice, this theorem is how you check differentiability: compute the partial derivatives and see that they are continuous.
</document>

<document source="https://math2.tamnd.com/node/0KY08TWDMQ.md" tag="0KY08TWDMQ" kind="statement">
# Derivative of x^a

## Statement

Let $a$ be a real number. The function $x \mapsto x^a$ is differentiable on $(0, \infty)$, and
$$\frac{d}{dx} x^a = a x^{a-1} \quad \text{for } x > 0.$$

## Remarks

Write $x^a = e^{a \ln x}$. The [[chain rule|0RXX2PT1MQ]] with [[$\exp' = \exp$|0EX9GFJ0B3]] and [[$\ln' x = 1/x$|07X1QA6BRW]] gives $e^{a \ln x} \cdot \frac{a}{x} = a x^{a-1}$.
For example, $\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}$.
Similarly, $\frac{d}{dx} a^x = a^x \ln a$ for a fixed $a > 0$. Mathlib states the main claim as `Real.hasDerivAt_rpow_const`.

## Explanations

### L1 human

The power rule for whole exponents also works for real exponents, if $x > 0$.
Bring the exponent down and lower it by one.
</document>

<document source="https://math2.tamnd.com/node/0KY7GS695G.md" tag="0KY7GS695G" kind="concept">
# Stirling numbers of the first kind

## Summary

The Stirling numbers of the first kind count the permutations of $n$ elements with exactly $k$ cycles. They are the coefficients of the rising and falling factorials as polynomials. The node [[Stirling numbers of the first kind|0Z22VQH17V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0KYP951QWH.md" tag="0KYP951QWH" kind="definition">
# Discrete uniform distribution

## Definition

Let $S$ be a nonempty [[finite set|0BBTH3YSZA]] with $n$ elements.
The discrete uniform distribution on $S$ is the distribution with the [[probability mass function|0WF4EJP91W]]
$$p(s) = \frac{1}{n} \quad \text{for } s \in S.$$
A random variable $X$ is uniform on $S$ if $P(X = s) = 1/n$ for each $s \in S$.

## Remarks

Then $P(X \in A) = |A| / n$ for each $A \subseteq S$. This is the [[classical definition of probability|0AKNPYQ04R]].
For $S = \{a, a + 1, \dots, b\}$ with integers $a \le b$ and $n = b - a + 1$, the mean is $(a + b)/2$ and the variance is $(n^2 - 1)/12$.
The result of a fair die is uniform on $\{1, \dots, 6\}$, with mean $7/2$.

## Explanations

### L1 human

The discrete uniform distribution is "all outcomes equally likely".
A fair die, a fair coin and a well-shuffled deck all give uniform distributions.
</document>

<document source="https://math2.tamnd.com/node/0KYX2N69J0.md" tag="0KYX2N69J0" kind="concept">
# Quotient group

## Summary

The quotient group $G/N$ of a group $G$ by a normal subgroup $N$ is the set of cosets of $N$, with the product $(aN)(bN) = abN$.
It collapses $N$ to the identity and keeps the rest of the structure of $G$.
</document>

<document source="https://math2.tamnd.com/node/0KYYSMEVHH.md" tag="0KYYSMEVHH" kind="statement">
# Lagrange's four-square theorem

## Statement

Each natural number $n$ is a sum of four squares of integers: $n = a^2 + b^2 + c^2 + d^2$ for some integers $a$, $b$, $c$ and $d$.

## Remarks

For example, $7 = 4 + 1 + 1 + 1$ and $15 = 9 + 4 + 1 + 1$.
Three squares are not enough: $7$ is not a sum of three squares, by [[Legendre's three-square theorem|0B64Q7XNE3]].
By [[Euler's four-square identity|08WTN6SCQ6]], it is enough to prove the theorem for primes.
Other proofs use [[Minkowski's theorem|0K387S2PKQ]] or the Hurwitz quaternions.

## History

Diophantus knew examples, and Bachet stated the theorem in his Latin translation of the Arithmetica of 1621.
Lagrange gave the first proof in 1770.

## Explanations

### L1 human

Every whole number is a sum of at most four perfect squares, if you allow $0$ as a square.
Some numbers, such as $7$, $15$ and $23$, need all four.
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# Quadratic formula

## Statement

Let $a$, $b$ and $c$ be complex numbers with $a \neq 0$, and let $s$ be a complex number with $s^2 = b^2 - 4 a c$.
Then, for each complex number $x$,
$$a x^2 + b x + c = 0 \iff x = \frac{-b + s}{2 a} \text{ or } x = \frac{-b - s}{2 a}.$$

## Remarks

Such an $s$ exists, by the [[existence of nth roots|0YM6P6C206]] for $n = 2$.
The number $\Delta = b^2 - 4 a c$ is the discriminant. For real $a, b, c$, there are two real roots if $\Delta > 0$, one double real root if $\Delta = 0$, and two conjugate non-real roots if $\Delta < 0$.
The same formula holds in each field in which $2 \neq 0$.

## Explanations

### L1 human

The formula solves every quadratic equation.
For $x^2 + 2 x + 5 = 0$, the discriminant is $4 - 20 = -16$, so $s = 4 i$, and the roots are $-1 + 2 i$ and $-1 - 2 i$.
Completing the square proves the formula: the equation becomes $(2 a x + b)^2 = b^2 - 4 a c$.
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# Trace of a linear map

## Summary

The trace of an endomorphism of a finite-dimensional vector space is the trace of its matrix in any basis.
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# Independent set

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
An independent set in $G$ is a set $S$ of vertices of $G$ such that no two vertices of $S$ are adjacent.

## Remarks

A set is independent in $G$ if and only if it is a [[clique|0G52BPNPKG]] in the [[complement graph|0RNMP6VCR2]].
A set $S$ is independent if and only if its complement in the vertex set is a [[vertex cover|0KDADE9RQ8]].
The largest size of an independent set is the independence number $\alpha(G)$. Each colour class of a [[colouring|0874138YP6]] is an independent set.

## Explanations

### L1 human

An independent set is a set of vertices with no edge between any two of them.
In a conflict graph, it is a set of tasks that can all happen at the same time.
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# Diffeomorphism

## Summary

A diffeomorphism is a bijection between open sets, or between manifolds, such that both the map and its inverse are [[differentiable|0HHHDCXZV4]].
Diffeomorphisms are the changes of coordinates of calculus, and they preserve all differentiable structure.
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# Antisymmetric relation

## Summary

A relation is antisymmetric when two different elements are never related in both directions.
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# Derivative of an inverse function

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be continuous and strictly monotone.
Assume that $f$ is differentiable at $a \in I$ with $f'(a) \ne 0$, and let $b = f(a)$.
Then $f^{-1}$ is differentiable at $b$, and
$$(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.$$

## Remarks

The inverse is [[continuous|07BY47HG5S]], so $f^{-1}(b + k) \to a$ as $k \to 0$. The difference quotient of $f^{-1}$ at $b$ is the reciprocal of a difference quotient of $f$ at $a$.
If $f'(a) = 0$, then $f^{-1}$ is not differentiable at $b$. An example is $f(x) = x^3$ at $a = 0$.
For example, it gives the [[derivative of the logarithm|07X1QA6BRW]] from the derivative of the exponential function.

## Explanations

### L1 human

The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
The reflection turns a slope $m$ into the slope $1/m$.
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# Block diagonal matrix

## Definition

Let $R$ be a ring and let $A_1, \dots, A_k$ be square matrices over $R$, where $A_i$ has order $n_i$.
The *block diagonal matrix* $\operatorname{diag}(A_1, \dots, A_k)$ is the [[block matrix|0ASDFEQS6P]] of order $n_1 + \cdots + n_k$
$$\begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_k \end{pmatrix},$$
whose block in position $(i, i)$ is $A_i$. The other blocks are zero matrices.

## Remarks

This matrix is also called the *direct sum* $A_1 \oplus \cdots \oplus A_k$.
Products of block diagonal matrices with matching block sizes are computed block by block.
Over a commutative ring, $\det \operatorname{diag}(A_1, \dots, A_k) = \det A_1 \cdots \det A_k$.
The [[Jordan normal form|085HQ5G5TE]] is a block diagonal matrix of [[Jordan blocks|07PH4RX0X5]].

## Explanations

### L1 human

A block diagonal matrix has square blocks along the diagonal and zeros everywhere else.
Each block acts on its own group of coordinates and ignores the other coordinates.
So we can study each block separately.
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# Linearity of differentiation

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$.
Let $\alpha$ and $\beta$ be real numbers.
Then $\alpha f + \beta g$ is differentiable at $a$, and
$$(\alpha f + \beta g)'(a) = \alpha f'(a) + \beta g'(a).$$

## Remarks

The difference quotient of $\alpha f + \beta g$ is $\alpha$ times that of $f$ plus $\beta$ times that of $g$. The limit laws give the result.
The case $\alpha = \beta = 1$ is the sum rule $(f + g)' = f' + g'$. The case $\beta = 0$ is the [[constant multiple rule|0Z9JB71BS0]].
Mathlib states the sum rule as `deriv_add`.

## Explanations

### L1 human

Differentiation respects sums and constant factors.
So we can differentiate a sum term by term.
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# Intercept theorem

## Statement

Let $S$ be a point of the Euclidean plane, and let $\ell_1$ and $\ell_2$ be two distinct [[lines|0MP9BCF6DP]] through $S$. Let $A$ and $B$ be points of $\ell_1$, and let $C$ and $D$ be points of $\ell_2$, all different from $S$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.
Assume that the line $AC$ is parallel to the line $BD$. Then
$$\frac{|SA|}{|SB|} = \frac{|SC|}{|SD|} = \frac{|AC|}{|BD|}.$$

## Remarks

If also $A \neq B$ and $A$ and $B$ are on the same ray from $S$, then the theorem also gives $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$.
A converse holds for two rays from $S$: if $A, B$ are on one ray, $C, D$ on the other, and $\frac{|SA|}{|AB|} = \frac{|SC|}{|CD|}$, then $AC$ is parallel to $BD$.
The theorem says that the triangles $SAC$ and $SBD$ are [[similar|0YF6QXXWB5]]. The [[homothety|0ZG1PNEP35]] with the centre $S$ that sends $A$ to $B$ also sends $C$ to $D$.

## History

Tradition attributes the theorem to Thales of Miletus. He may have used it to find the heights of the pyramids of Egypt.

## Explanations

### L1 human

Draw two lines from one point, and cut them with two parallel lines. The parallel lines cut both lines in the same proportions. This is how you can find the height of a tree from the length of its shadow and the shadow of a stick.
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# Convex function on R^n

## Definition

A set $C \subseteq \mathbb{R}^n$ is convex if $t x + (1 - t) y \in C$ for all $x, y \in C$ and all $t \in [0, 1]$.
Let $C \subseteq \mathbb{R}^n$ be convex.
A function $f \colon C \to \mathbb{R}$ is convex if, for all $x, y \in C$ and all $t \in [0, 1]$,
$$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y).$$
The function $f$ is strictly convex if the inequality is strict for all $x \ne y$ and all $t \in (0, 1)$.

## Remarks

If $C$ is open and $f$ is differentiable on $C$, then $f$ is convex if and only if $f(y) \ge f(x) + \nabla f(x) \cdot (y - x)$ for all $x, y \in C$.
Let $C$ be open and let $f$ be of class $C^2$. Then $f$ is convex if and only if its [[Hessian matrix|0SCSQWXRA4]] is positive semidefinite at each point of $C$.
Each local minimum of a convex function is a global minimum, and a strictly convex function has at most one global minimum.
A function can be convex in each variable separately but not convex. The function $f(x, y) = x y$ is linear in each variable, but $f(t, -t) = -t^2$ is not convex.

## Explanations

### L1 human

A convex function is bowl-shaped: the straight segment between two points of its graph never goes below the graph.
Examples are $\|x\|^2$ and, on all of $\mathbb{R}^n$, each linear function.
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# Bernoulli's inequality

## Statement

Let $x$ be a real number with $x \ge -1$, and let $n$ be a natural number.
Then
$$(1 + x)^n \ge 1 + n x.$$

## Remarks

When $x \neq 0$ and $n \ge 2$, the inequality is strict.
For natural numbers $n$, the inequality also holds when $x \ge -2$.
A proof by [[induction|0ASJVEFA9J]] on $n$ multiplies $(1 + x)^n \ge 1 + n x$ by $1 + x \ge 0$ and uses $n x^2 \ge 0$.

## Explanations

### L1 human

A small growth rate $x$, applied $n$ times, gives at least $n$ times that growth.
For example, $1.1^{10}$ is at least $1 + 10 \cdot 0.1 = 2$; its true value is about $2.59$.
The difference comes from the growth on the growth, which the right side ignores.
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# Fermat's little theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer that $p$ does not divide.
Then $a^{p-1} \equiv 1 \pmod p$.

## Remarks

An equivalent form is [[Fermat's little theorem in the form a^p = a|0BMXWXDVJZ]]: $a^p \equiv a \pmod p$ for each integer $a$.
In the ring $\mathbb{Z}/p\mathbb{Z}$, the statement is $x^{p-1} = 1$ for each nonzero $x$. It follows from [[Lagrange's theorem|3C53TD]] for the multiplicative [[group|G5S1M9]] of $\mathbb{Z}/p\mathbb{Z}$, which has $p - 1$ elements.
[[Euler's theorem|0VC4DVE7DQ]] generalizes the result to each modulus $n$.
The converse is false. For example, $2^{340} \equiv 1 \pmod{341}$, but $341 = 11 \cdot 31$. See [[Fermat pseudoprime|02M2K4QH9P]].

## History

Fermat stated the theorem in a letter of 1640, without a proof. Euler published the first known proof in 1736. Leibniz had a proof earlier, in an unpublished manuscript.

## Explanations

### L1 human

Take $p = 5$ and $a = 2$: $2^4 = 16 = 3 \cdot 5 + 1$, so $2^4 \equiv 1 \pmod 5$.
The theorem says that this always happens for a prime $p$ and a number $a$ that $p$ does not divide.
It lets you reduce large powers: $2^{100} = (2^4)^{25} \equiv 1 \pmod 5$.
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# Conjunction

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The conjunction of $p$ and $q$ is the proposition $p \land q$, read "$p$ and $q$".
The proposition $p \land q$ is true when $p$ is true and $q$ is true. In all other cases, $p \land q$ is false.

| $p$ | $q$ | $p \land q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

## Remarks

Conjunction is commutative and associative: $p \land q$ has the same truth value as $q \land p$.
The [[intersection|0HBRW4E6G1]] of two sets uses conjunction: $x \in A \cap B$ means $x \in A \land x \in B$.

## Explanations

### L1 human

The sentence "$6$ is even and $6$ is divisible by $3$" is a conjunction.
It is true, because both parts are true.
If one part is false, the whole sentence is false.
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# Curl of a gradient is zero

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $\phi \colon U \to \mathbb{R}$ be of class $C^2$.
Then
$$\nabla \times (\nabla \phi) = 0 \quad \text{on } U.$$

## Remarks

Proof: the first component of the [[curl|0GDNHS9J2X]] of $\nabla \phi$ is $\partial_y \partial_z \phi - \partial_z \partial_y \phi$, and this is zero by [[Schwarz's theorem|0NNQY8WCW6]]. The other components are similar.
So a [[conservative|05R9NF2J49]] $C^1$ vector field is irrotational. This gives a quick test: if $\operatorname{curl} F \ne 0$ at some point, then $F$ has no scalar potential.
In $\mathbb{R}^n$, the same argument shows $\partial_i F_j = \partial_j F_i$ for all $i, j$ when $F = \nabla \phi$ with $\phi$ of class $C^2$.
The converse needs a condition on $U$. It holds on [[convex sets|0B22P010D6]], but [[not on every open set|0GRM3BS0DD]].

## Explanations

### L1 human

A gradient field has no swirl.
Going around a small loop in a landscape, you return to the same height. So the field $\nabla \phi$ has no circulation, and its curl is zero.
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# Cayley's formula

## Statement

Let $n \ge 2$ be an integer.
Then the number of [[trees|0MJ0JS72DZ]] with vertex set $\{1, 2, \dots, n\}$ is $n^{n-2}$.

## Remarks

The trees are labelled: two trees are the same only if they have the same edge set. Thus the formula counts the [[spanning trees|0FQ5B8S1ZC]] of $K_n$.
For $n = 3$ there are $3$ trees, and for $n = 4$ there are $16$.
A proof uses the [[Prüfer sequence|09PV9JSY6E]], a bijection between these trees and the sequences of length $n - 2$ over $\{1, \dots, n\}$.

## History

Carl Wilhelm Borchardt found the formula in 1860.
Arthur Cayley extended it in 1889, and the formula has his name.

## Explanations

### L1 human

Label $n$ points with the numbers $1$ to $n$ and count the ways to connect them by a tree.
For $4$ points there are $4^2 = 16$ ways: $12$ paths and $4$ stars.
</document>

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# The number of Sylow subgroups divides the index

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.
Then $n_p$ divides the [[index|0DTXBQPA2S]] $[G : P]$.

## Remarks

By [[this formula|09ZJ6CAK7S]], $n_p = [G : N_G(P)]$, and $[G : P] = [G : N_G(P)] \cdot [N_G(P) : P]$.
If $|G| = p^n m$ with $p \nmid m$, then $n_p$ divides $m$.
For example, in a group of order $15$, $n_3$ divides $5$ and $n_3 \equiv 1 \pmod 3$, so $n_3 = 1$.

## Explanations

### L1 human

The number of Sylow subgroups is the size of a conjugation orbit.
Orbit sizes divide the order of the group, and here even the part of the order without $p$.
This gives a strong limit on $n_p$.
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# Analytic continuation

## Summary

Let $U \subseteq V$ be open subsets of $\mathbb{C}$, with $U$ nonempty and $V$ a [[domain|097KNJFJ7V]], and let $f$ be [[holomorphic|0DG0Z72CQD]] on $U$.
An analytic continuation of $f$ to $V$ is a function $F$ that is holomorphic on $V$ with $F(z) = f(z)$ for each $z \in U$.
By the [[identity theorem|0YBVKMYX0W]], $f$ has at most one analytic continuation to $V$.

## Remarks

The geometric series $\sum_{n \ge 0} z^n$ is holomorphic on the unit disc, and $1/(1 - z)$ is its analytic continuation to $\mathbb{C} \setminus \{1\}$.
The Riemann zeta function $\sum_{n \ge 1} n^{-s}$ on $\operatorname{Re} s > 1$ has an analytic continuation to $\mathbb{C} \setminus \{1\}$.
Continuation along a path is a more general process. Its result can depend on the path: the [[logarithm|0X2FY491PF]] continued once around $0$ changes by $2\pi i$.

## Explanations

### L1 human

A formula such as a power series often defines a function only on a small region.
Analytic continuation extends the function to a larger region in the only possible holomorphic way.
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# Four colour theorem

## Statement

Let $G$ be a finite loopless [[planar graph|029JQEK15B]].
Then $\chi(G) \le 4$.

## Remarks

In terms of maps: four colours are enough for the regions of a map in the plane. Regions with a common boundary segment get different colours.
The bound is sharp, because $K_4$ is planar and $\chi(K_4) = 4$.

## History

Francis Guthrie stated the problem in 1852.
Kenneth Appel and Wolfgang Haken proved the theorem in 1976, with the help of a computer.
Robertson, Sanders, Seymour and Thomas gave a simpler proof in 1997. Georges Gonthier and Benjamin Werner checked a proof in the Coq proof assistant in 2005.

## Explanations

### L1 human

Four colours are enough to colour any map, so that neighbouring countries always have different colours.
It was the first major theorem whose proof needed a computer.
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# Binary operation

## Summary

A binary operation on a set combines two elements of the set and gives one element of the same set.
Addition and multiplication of numbers are the basic examples.
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# Uniform continuity

## Summary

A function is uniformly continuous when one input tolerance $\delta$ controls the output error $\varepsilon$ at all points of the domain at once. This is stronger than continuity at each point. The node [[uniform continuity|0Q4DR37DVE]] gives the precise definition.
</document>

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# Every linearly independent set extends to a basis

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $L \subseteq V$ be a [[linearly independent|0CRMGPAH83]] set.
Then there is a [[basis|0KCQ9Y3QKA]] $B$ of $V$ with $L \subseteq B$.

## Remarks

For finite-dimensional $V$, add to $L$, one at a time, a vector that is not in the span of the current set.
This stops after finitely many steps.
In general, the proof uses Zorn's lemma.
A stronger form: if $L \subseteq S$, with $L$ independent and $S$ spanning, then a basis $B$ exists with $L \subseteq B \subseteq S$.
For example, $\{(1, 1, 0)\}$ extends to the basis $\{(1, 1, 0), (0, 1, 0), (0, 0, 1)\}$ of $\R^3$.

## Explanations

### L1 human

We can always complete an independent set of vectors to a basis.
We keep adding vectors that point in new directions until the set reaches the whole space.
This shows that an independent set can never be larger than a basis.
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<document source="https://math2.tamnd.com/node/0M4TCS0RRG.md" tag="0M4TCS0RRG" kind="statement">
# Classification of the Platonic solids

## Statement

Let $P$ be a [[Platonic solid|017E5PRS1Q]] whose faces have $p$ sides each, with $q$ faces at each vertex. Then $\{p, q\}$ is one of
$$\{3, 3\}, \quad \{4, 3\}, \quad \{3, 4\}, \quad \{5, 3\}, \quad \{3, 5\}.$$
Each of these five cases occurs, by the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron and the regular icosahedron. Up to similarity, there is exactly one Platonic solid for each case.

## Remarks

A proof: let $V$, $E$ and $F$ be the numbers of vertices, edges and faces. Then $pF = 2E = qV$, and [[Euler's formula|09HT3P35YP]] $V - E + F = 2$ gives $\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{E} > \frac{1}{2}$.
With $p, q \geq 3$, this inequality has exactly the five solutions above.
A geometric proof uses the angles: at each vertex, the $q$ face angles $\frac{(p - 2)\pi}{p}$ must add up to less than $2\pi$.

## History

Theaetetus, a contemporary of Plato, described all five solids. He may have given the first proof that no other convex regular polyhedra exist. Euclid's Elements contains a geometric proof.

## Explanations

### L1 human

Only five perfectly regular convex solids exist. At each corner, at least three faces meet, and their angles must add up to less than a full turn. This leaves room only for three, four or five triangles, three squares or three pentagons.
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# Line integral of a scalar field

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $f \colon C \to \mathbb{R}$ be [[continuous|0S9EK5MQPE]].
For a partition $a = t_0 < t_1 < \dots < t_k = b$ and points $\tau_i \in [t_{i-1}, t_i]$, let $\Delta s_i$ be the length of the restriction of $\gamma$ to $[t_{i-1}, t_i]$.
The line integral of $f$ along $\gamma$ is the limit
$$\int_\gamma f \, ds = \lim \sum_{i=1}^{k} f(\gamma(\tau_i)) \, \Delta s_i,$$
taken as $\max_i (t_i - t_{i-1}) \to 0$.

## Remarks

The limit exists, and it does not depend on the choice of the points $\tau_i$.
For a $C^1$ curve, $\int_\gamma f \, ds = \int_a^b f(\gamma(t)) \, \|\gamma'(t)\| \, dt$. This is the [[parametric formula|0W3W1VYWBP]].
For $f = 1$ the line integral is the length of $\gamma$. If $\gamma$ is a wire with linear density $f$, the line integral is its mass.
The value does not change under a reparametrisation of $\gamma$, in either direction.

## Explanations

### L1 human

Walk along the curve, and at each small step multiply the length of the step by the value of $f$ there.
The sum of these products, in the limit of small steps, is the line integral with respect to arc length.
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# Existence of inverses modulo n

## Statement

Let $n$ be a positive integer and let $a$ be an integer.
Then $a$ has a [[modular multiplicative inverse|00GXTQMY8Z]] modulo $n$ if and only if $a$ and $n$ are [[coprime|0GFKP8F8AG]].

## Remarks

If $\gcd(a, n) = 1$, [[Bézout's identity|0DQVHMYAGA]] gives integers $x$ and $y$ with $a x + n y = 1$. Then $a x \equiv 1 \pmod n$.
Conversely, if $a x \equiv 1 \pmod n$, then $a x + n y = 1$ for some integer $y$, so each common divisor of $a$ and $n$ divides $1$.
The [[extended Euclidean algorithm|0964YQFSA8]] computes the inverse.
For example, $4$ has no inverse modulo $6$, because $\gcd(4, 6) = 2$.

## Explanations

### L1 human

You can "divide by $a$" modulo $n$ exactly when $a$ and $n$ have no common factor.
Modulo $10$, the numbers $1, 3, 7, 9$ have inverses, since $1 \cdot 1$, $3 \cdot 7$ and $9 \cdot 9$ all end with the digit $1$.
The numbers $2, 4, 5, 6, 8$ have no inverse modulo $10$.
</document>

<document source="https://math2.tamnd.com/node/0M6G75E95E.md" tag="0M6G75E95E" kind="statement">
# The only ideals of a field are zero and the whole field

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $I$ be an [[ideal|0MYSA6MYNA]] of $K$.
Then $I = \{0\}$ or $I = K$.

## Remarks

The proof: if $a \in I$ and $a \neq 0$, then $1 = a^{-1} a \in I$, so $r = r \cdot 1 \in I$ for each $r \in K$.
The converse holds: a nonzero commutative ring whose only ideals are $\{0\}$ and the whole ring is a field.
For each nonzero $a$, the ideal $(a)$ is the whole ring, so $1 = b a$ for some $b$.

## Explanations

### L1 human

If an ideal of a field contains a nonzero element, then it contains $1$, because you can divide.
Once an ideal contains $1$, it contains every element.
So a field has only the two trivial ideals.
</document>

<document source="https://math2.tamnd.com/node/0M6W4XNST6.md" tag="0M6W4XNST6" kind="statement">
# Universal property of the product topology

## Statement

Let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CNQXVBHCX]], let $X = \prod_{i \in I} X_i$ have the [[product topology|04S9HX3GAP]], and let $\pi_i \colon X \to X_i$ be the projections.
Let $Z$ be a topological space and let $f \colon Z \to X$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if the [[composition|07HF9YCWPP]] $\pi_i \circ f \colon Z \to X_i$ is continuous for each $i \in I$.

## Remarks

Proof of the converse: preimages of the subbasic sets $\pi_i^{-1}(U)$ under $f$ are $(\pi_i \circ f)^{-1}(U)$, which are open. Preimages respect unions and intersections, so all preimages of open sets are open.
Thus a map into a product is continuous if and only if all its coordinate functions are continuous. For example, $t \mapsto (\cos t, \sin t)$ is continuous from $\mathbb{R}$ to $\mathbb{R}^2$.
The statement fails for the [[box topology|0CTM1EHG9P]].

## Explanations

### L1 human

To check that a map into a product is continuous, check each coordinate separately.
This property characterises the product topology among all topologies on the product.
</document>

<document source="https://math2.tamnd.com/node/0M79HRJCRR.md" tag="0M79HRJCRR" kind="definition">
# Elementary matrix

## Definition

Let $F$ be a field and let $n \ge 1$ be an integer.
An $n \times n$ *elementary matrix* over $F$ is a matrix that results from the [[identity matrix|06DW9BAWB1]] $I_n$ by one [[elementary row operation|04XWH28452]].
There are three types:

1. $P_{ij}$, for $i \ne j$: exchange rows $i$ and $j$ of $I_n$.
2. $D_i(c)$, for $c \ne 0$: multiply row $i$ of $I_n$ by $c$.
3. $T_{ij}(c)$, for $i \ne j$ and $c \in F$: add $c$ times row $j$ of $I_n$ to row $i$.

## Remarks

For each $n \times p$ matrix $A$, the product $EA$ is the result of the same row operation on $A$.
The product $AE$ performs the matching column operation on $A$.
Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type.
Each invertible matrix over a field is a product of elementary matrices.

## Explanations

### L1 human

An elementary matrix stores one row operation as a matrix.
To do the operation on $A$, we multiply $A$ on the left by this matrix.
Then a whole elimination becomes a product of elementary matrices.
</document>

<document source="https://math2.tamnd.com/node/0M7ARED5Q7.md" tag="0M7ARED5Q7" kind="concept">
# Pythagorean triple

## Summary

A Pythagorean triple is a triple of positive integers $a$, $b$, $c$ with $a^2 + b^2 = c^2$, the side lengths of a right triangle.
The [[definition of a Pythagorean triple|01G1WY1GVX]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0M7DEDKSYB.md" tag="0M7DEDKSYB" kind="statement">
# Euler's criterion

## Statement

Let $p$ be an odd prime and let $a$ be an integer.
Then
$$a^{(p-1)/2} \equiv \left(\frac{a}{p}\right) \pmod p,$$
where $\left(\frac{a}{p}\right)$ is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

So for $p \nmid a$, the integer $a$ is a quadratic residue modulo $p$ if and only if $a^{(p-1)/2} \equiv 1 \pmod p$.
For example, $3^3 = 27 \equiv -1 \pmod 7$, so $3$ is a quadratic nonresidue modulo $7$.
The proof uses [[Fermat's little theorem|0M25PRN3WQ]]: $x = a^{(p-1)/2}$ satisfies $x^2 \equiv 1$, so $x \equiv \pm 1 \pmod p$.
A consequence is the [[multiplicativity of the Legendre symbol|0NW1JP7TBR]].

## History

Leonhard Euler found the criterion.

## Explanations

### L1 human

To test whether $a$ is a square modulo $p$, raise $a$ to the power $(p-1)/2$.
The result is $1$ for a square and $-1$ for a non-square, modulo $p$.
With [[fast modular exponentiation|04DR5PM25N]], this test is quick even for large $p$.
</document>

<document source="https://math2.tamnd.com/node/0M83V17KHX.md" tag="0M83V17KHX" kind="concept">
# Complementary subspaces

## Summary

Two subspaces of a vector space are complementary when the space is their direct sum.
</document>

<document source="https://math2.tamnd.com/node/0M856Q1AMZ.md" tag="0M856Q1AMZ" kind="definition">
# Dual space

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
The *dual space* of $V$ is the vector space
$$V^* = \Hom_F(V, F)$$
of all [[linear forms|07YQ1C78G0]] on $V$.
Its operations are pointwise: $(f + g)(v) = f(v) + g(v)$ and $(af)(v) = a f(v)$.

## Remarks

It is a special case of the [[space of linear maps|0X9MBPP2Z3]].
If $\dim V = n$ is finite, then $\dim V^* = n$, see [[the dimension of the dual space|00889YD15K]].
The elements of $F^n$ act on column vectors by the dot product, and this identifies $(F^n)^*$ with $F^n$.
For infinite-dimensional $V$, the space $V^*$ is not isomorphic to $V$.

## Explanations

### L1 human

The dual space collects all the ways to measure a vector by one linear number.
On $\R^3$, each linear form is a dot product with a fixed vector.
So, in finite dimensions, the dual space looks like the original space, but its elements play a different role.
</document>

<document source="https://math2.tamnd.com/node/0M85DXB5RM.md" tag="0M85DXB5RM" kind="concept">
# Equivalent norms

## Summary

Two norms on a vector space are equivalent when each is bounded by a constant multiple of the other.
Equivalent norms give the same open sets, the same convergent sequences and the same Cauchy sequences.
</document>

<document source="https://math2.tamnd.com/node/0M893X7YY7.md" tag="0M893X7YY7" kind="definition">
# Adjoining a root of a polynomial

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be [[irreducible|00X2A9G9W3]].
The field obtained by adjoining a root of $f$ to $K$ is the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$.
The map $K \to K[x]/(f)$, $c \mapsto c + (f)$, identifies $K$ with a subfield, and the class $\alpha = x + (f)$ is a root of $f$.

## Remarks

The quotient is a field, because [[$f$ is irreducible|0AWMPX35CW]].
Its degree over $K$ is $\deg f$, with the basis $1, \alpha, \dots, \alpha^{\deg f - 1}$.
For example, adjoining a root of $x^2 + 1$ to $\mathbb{R}$ gives a field isomorphic to $\mathbb{C}$.

## Explanations

### L1 human

You can create a root of a polynomial that has none.
Compute with polynomials in $x$, and replace $f(x)$ by zero each time it appears.
This is how $\mathbb{C}$ arises from $\mathbb{R}$ and $x^2 + 1$.
</document>

<document source="https://math2.tamnd.com/node/0M8EWQJ1YV.md" tag="0M8EWQJ1YV" kind="statement">
# Vieta's formulas

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in R[x]$.
Suppose that $f = a_n (x - r_1)(x - r_2) \cdots (x - r_n)$ with $r_1, \dots, r_n \in R$.
Then for $1 \le k \le n$,
$$a_n \, e_k(r_1, \dots, r_n) = (-1)^k a_{n-k},$$
where $e_k$ is the $k$-th [[elementary symmetric polynomial|0ZMY8441GW]].

## Remarks

If $a_n$ is a unit, this says $e_k(r_1, \dots, r_n) = (-1)^k a_{n-k}/a_n$.
For $x^2 + b x + c = (x - r_1)(x - r_2)$, it gives $r_1 + r_2 = -b$ and $r_1 r_2 = c$.
The proof expands the product and compares coefficients.
Over a field, the hypothesis says that $f$ [[splits|0SHMC00RJV]] with roots $r_1, \dots, r_n$, counted with multiplicity.

## History

The formulas are named after the French mathematician François Viète, who derived them for positive roots.

## Explanations

### L1 human

Vieta's formulas say that the sum of the roots, the sum of their products in pairs, and so on, are given by the coefficients.
You can know these sums without finding the roots.
</document>

<document source="https://math2.tamnd.com/node/0M8MHYERTY.md" tag="0M8MHYERTY" kind="definition">
# Nilpotent endomorphism

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *nilpotent* when $T^k = 0$ for some integer $k \ge 1$.
The least such $k$ is the *index* of $T$.

## Remarks

If $\dim V = n$ is finite and $T$ is nilpotent, then $T^n = 0$.
In finite dimensions, $T$ is nilpotent if and only if its characteristic polynomial is $t^n$.
Over an algebraically closed field, this means that $0$ is its only eigenvalue.
A matrix is nilpotent when its map is nilpotent, see [[nilpotent matrix|07BSPZPSMF]].
The derivative on the space of polynomials of degree less than $n$ is nilpotent.

## Explanations

### L1 human

A nilpotent map destroys everything after a few steps.
Each application pushes vectors further towards zero, until they vanish.
The shift $e_1 \mapsto 0$, $e_2 \mapsto e_1$, $e_3 \mapsto e_2$ in $F^3$ is nilpotent of index $3$.
</document>

<document source="https://math2.tamnd.com/node/0M8TGGP9JP.md" tag="0M8TGGP9JP" kind="statement">
# Approximation property of the supremum

## Statement

Let $S$ be a nonempty subset of the real numbers that is bounded above, and let $b = \sup S$ be its [[supremum|0BTMQ04556]].
For each real number $c < b$, there is an element $s$ of $S$ with $c < s \le b$.

## Remarks

Proof: if no such $s$ exists, then $c$ is an upper bound of $S$ that is less than $b$. This contradicts the definition of the supremum.
An equivalent form: for each $\varepsilon > 0$, there is $s$ in $S$ with $b - \varepsilon < s$.
For an infimum $a = \inf S$ and each $c > a$, there is $s$ in $S$ with $a \le s < c$.

## Explanations

### L1 human

You can find elements of a set as close to its supremum as you want.
If you lower the ceiling even a little, then some element of the set sticks out above it.
</document>

<document source="https://math2.tamnd.com/node/0M98XYVX6S.md" tag="0M98XYVX6S" kind="definition">
# Coefficient of a power series

## Definition

Let $R$ be a commutative ring and let $n$ be a natural number.
The coefficient of $X^n$ is the map $[X^n] : R[\![X]\!] \to R$ that sends a [[formal power series|072W67KFYK]] $f = \sum_{m \ge 0} a_m X^m$ to $a_n$.

## Remarks

The map $[X^n]$ is $R$-linear: $[X^n](f + g) = [X^n] f + [X^n] g$ and $[X^n](r f) = r\, [X^n] f$.
Two series are equal if and only if all their coefficients are equal.
The coefficient of a product is given by the node [[product of power series|0H6KWAYS87]].

## Explanations

### L1 human

The coefficient $[X^n] f$ is the number in front of $X^n$ in the series $f$.
For $f = 1 + 2X + 3X^2 + \dots$, the coefficient $[X^2] f$ is $3$.
</document>

<document source="https://math2.tamnd.com/node/0M9MX8YW0A.md" tag="0M9MX8YW0A" kind="concept">
# Incentre

## Summary

The incentre of a triangle is the point where the three internal angle bisectors meet. It is the point inside the triangle at the same distance from all three sides, and it is the centre of the incircle.
</document>

<document source="https://math2.tamnd.com/node/0M9XEDF1HJ.md" tag="0M9XEDF1HJ" kind="definition">
# Step function

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be a function.
The function $f$ is a step function if it is constant on each open subinterval of some [[partition|069P20KYQT]] $(x_0, \dots, x_n)$ of $[a, b]$.
That is, there are real numbers $c_1, \dots, c_n$ with $f(x) = c_i$ for all $x \in (x_{i-1}, x_i)$.

## Remarks

The values of $f$ at the points $x_i$ are arbitrary.
A step function is Riemann integrable, and $\int_a^b f = \sum_{i=1}^{n} c_i (x_i - x_{i-1})$.
The floor function $\lfloor x \rfloor$ restricted to $[a, b]$ is a step function.

## Explanations

### L1 human

The graph of a step function looks like a staircase.
It is flat on each piece of a partition, and it can jump at the cut points.
</document>

<document source="https://math2.tamnd.com/node/0MA5B744K3.md" tag="0MA5B744K3" kind="definition">
# Field generated by adjoining elements

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $S \subseteq L$.
The field $K(S)$ is the intersection of all the [[subfields|07QY5RPFBR]] of $L$ that contain $K$ and $S$.
It is the smallest [[intermediate field|02NBKVQFRQ]] of $L/K$ that contains $S$.

## Remarks

For $S = \{a_1, \dots, a_n\}$ one writes $K(a_1, \dots, a_n)$.
The elements of $K(S)$ are the quotients $f(s_1, \dots, s_n)/g(s_1, \dots, s_n)$ with $f, g$ polynomials over $K$, $s_i \in S$ and $g(s_1, \dots, s_n) \neq 0$.
One has $K(a, b) = K(a)(b)$.
For example, $\mathbb{Q}(\sqrt{2}) = \{ p + q \sqrt{2} : p, q \in \mathbb{Q} \}$ inside $\mathbb{R}$.

## Explanations

### L1 human

To adjoin elements to a field, you take the smallest field that contains the old field and the new elements.
It contains everything you can get from them by addition, subtraction, multiplication and division.
</document>

<document source="https://math2.tamnd.com/node/0MB48XDBY3.md" tag="0MB48XDBY3" kind="statement">
# Schwarz lemma

## Statement

Let $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ be the open unit disc.
Let $f : \mathbb{D} \to \mathbb{D}$ be [[holomorphic|0DG0Z72CQD]] on $\mathbb{D}$ with $f(0) = 0$.
Then $|f(z)| \le |z|$ for each $z \in \mathbb{D}$, and $|f'(0)| \le 1$.
If $|f(z_0)| = |z_0|$ for some $z_0 \neq 0$ in $\mathbb{D}$, or if $|f'(0)| = 1$, then there is $a \in \mathbb{C}$ with $|a| = 1$ such that $f(z) = a z$ for each $z \in \mathbb{D}$.

## Remarks

Proof idea: $g(z) = f(z)/z$, with $g(0) = f'(0)$, is holomorphic on $\mathbb{D}$. The [[maximum modulus principle|048F2P90WR]] on the discs $|z| \le r$ gives $|g| \le 1/r$ for each $r < 1$, so $|g| \le 1$.
It follows that the bijective holomorphic maps $\mathbb{D} \to \mathbb{D}$ with $f(0) = 0$ are the rotations $z \mapsto az$ with $|a| = 1$.
The Schwarz–Pick theorem is the version without the condition $f(0) = 0$.

## Explanations

### L1 human

A holomorphic map from the unit disc to itself that keeps $0$ fixed cannot move any point farther away from $0$.
The only maps that keep the distance to $0$ of some point are the rotations.
</document>

<document source="https://math2.tamnd.com/node/0MB4TGMXG6.md" tag="0MB4TGMXG6" kind="definition">
# Algebraic number

## Definition

A complex number $\alpha$ is algebraic if there is a nonzero polynomial $f$ with rational coefficients such that $f(\alpha) = 0$.
The degree of $\alpha$ is the smallest degree of such a polynomial.

## Remarks

One may use integer coefficients instead, by clearing denominators.
Examples are each rational number (degree $1$), $\sqrt{2}$ and $i$ (degree $2$), and $\sqrt[3]{2}$ (degree $3$).
The algebraic numbers form a field, and this field is countable.
A complex number that is not algebraic is [[transcendental|079EGSD32Y]].

## Explanations

### L1 human

An algebraic number is a solution of a polynomial equation with integer coefficients.
For example, $\sqrt{2}$ is algebraic because it solves $x^2 - 2 = 0$.
</document>

<document source="https://math2.tamnd.com/node/0MB8HQW495.md" tag="0MB8HQW495" kind="definition">
# Self-adjoint operator

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *self-adjoint* when $T^* = T$, where $T^*$ is the [[adjoint|0R9NMSM1ZP]] of $T$.
Equivalently, $\langle T(v), w \rangle = \langle v, T(w) \rangle$ for all $v, w \in V$.

## Remarks

In an orthonormal basis, $T$ is self-adjoint if and only if its matrix is [[symmetric|0VR671PHE5]] (over $\R$) or [[Hermitian|0P17NGQQMQ]] (over $\C$).
Orthogonal projections are self-adjoint.
The eigenvalues of a self-adjoint map are real, see [[the reality of the eigenvalues|0C4NT4GT3X]].
Self-adjoint maps are the subject of the [[spectral theorem|06J4PXRG22]].

## Explanations

### L1 human

A self-adjoint map can move freely from one side of the inner product to the other.
For real matrices, these are the symmetric matrices.
They model observable quantities in quantum mechanics, because their eigenvalues are real.
</document>

<document source="https://math2.tamnd.com/node/0MBJVDSK1R.md" tag="0MBJVDSK1R" kind="concept">
# Sum of ideals

## Summary

Let $R$ be a [[ring|0894J980RK]] and let $I$ and $J$ be left [[ideals|0MYSA6MYNA]] of $R$.
The sum of $I$ and $J$ is $I + J = \{a + b : a \in I, b \in J\}$.
It is a left ideal, and it is the smallest left ideal that contains $I$ and $J$.

## Remarks

The sum $I + J$ is the ideal [[generated by|030TZ0DQWT]] $I \cup J$.
In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = d\mathbb{Z}$, where $d$ is the greatest common divisor of $m$ and $n$.
The ideals $I$ and $J$ are [[coprime|0MDJP82TEH]] when $I + J = R$.

## Explanations

### L1 human

To add two ideals, take all the sums of one element from each.
In the integers, the sum of the multiples of $4$ and the multiples of $6$ is the set of the multiples of $2$.
</document>

<document source="https://math2.tamnd.com/node/0MBV2VHMXA.md" tag="0MBV2VHMXA" kind="concept">
# Generators of a group

## Summary

A subset $S$ of a [[group|G5S1M9]] $G$ generates $G$ when the [[subgroup generated by $S$|0RP7VWQ0YN]] is all of $G$.
Then each element of $G$ is a finite product of elements of $S$ and of their inverses. The elements of $S$ are called generators of $G$.

## Remarks

The set $\{1\}$ generates $(\mathbb{Z}, +)$, and so does the set $\{2, 3\}$.
Each group is generated by the set of all its elements.
A group generated by one element is a cyclic group, and a group generated by a finite set is a finitely generated group.

## Explanations

### L1 human

Generators are building blocks for a group: you can reach each element from them with products and inverses.
A small set of generators gives a short description of a big group.
For example, all symmetries of a square come from one rotation and one reflection.
</document>

<document source="https://math2.tamnd.com/node/0MBWQKM8QR.md" tag="0MBWQKM8QR" kind="statement">
# Monotonicity of the integral

## Statement

Let $f, g \colon [a, b] \to \mathbb{R}$ be Riemann integrable with $f(x) \le g(x)$ for all $x \in [a, b]$.
Then
$$\int_a^b f(x) \, dx \le \int_a^b g(x) \, dx.$$

## Remarks

Each [[Riemann sum|0HBY48G8G1]] of $f$ is at most the Riemann sum of $g$ for the same tagged partition.
In particular, if $m \le f \le M$ on $[a, b]$, then $m (b - a) \le \int_a^b f \le M (b - a)$.
Mathlib states it as `intervalIntegral.integral_mono_on`.

## Explanations

### L1 human

A larger function has a larger integral.
If one graph lies below another, the area under it is smaller.
</document>

<document source="https://math2.tamnd.com/node/0MC0AQZSHE.md" tag="0MC0AQZSHE" kind="statement">
# A finite abelian group is cyclic iff its exponent equals its order

## Statement

Let $G$ be a finite [[abelian group|082GHNPJ53]].
Then $G$ is [[cyclic|0KVGH0J5B3]] if and only if the [[exponent|0ZDW0H0VX4]] of $G$ equals $|G|$.

## Remarks

If $G$ is cyclic with generator $g$, then the order of $g$ is $|G|$, so the exponent is $|G|$.
For the converse, write $G \cong \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$, by the [[classification|07CQR9BW3R]]. The exponent is $k_u$ and the order is $k_1 \cdots k_u$, so they are equal only when $u \leq 1$.
The abelian condition is necessary: $S_3$ has exponent $6$ and order $6$, but it is not cyclic.
For example, $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $\lambda(n) = \varphi(n)$.

## Explanations

### L1 human

In a cyclic group, one element has order equal to the size of the group.
For a finite abelian group, it is enough that the exponent equals the size.
Then the group is cyclic.
</document>

<document source="https://math2.tamnd.com/node/0MCDDSCB9Z.md" tag="0MCDDSCB9Z" kind="statement">
# Orthogonal projection is the closest point

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, let $W$ be a finite-dimensional subspace of $V$, and let $v \in V$.
Let $P_W$ be the [[orthogonal projection|0B8KZRT28M]] onto $W$.
Then, for all $w \in W$,
$$\norm{v - P_W(v)} \le \norm{v - w},$$
with equality only for $w = P_W(v)$.

## Remarks

The proof: $v - w = (v - P_W(v)) + (P_W(v) - w)$ is a sum of orthogonal vectors, so the Pythagorean theorem applies.
The number $\norm{v - P_W(v)}$ is the distance from $v$ to $W$.
The Hilbert projection theorem gives the same result for closed convex sets in Hilbert spaces.
This is the basis of the method of [[least squares|0BYTQ4X1HC]].

## Explanations

### L1 human

The nearest point of a subspace to a given vector is the foot of the perpendicular.
Every other point of the subspace is farther away, by the Pythagorean theorem.
This turns problems of best approximation into problems of projection.
</document>

<document source="https://math2.tamnd.com/node/0MCK8K2D7Z.md" tag="0MCK8K2D7Z" kind="statement">
# Epsilon characterisation of Cauchy sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $(x_n)_{n \ge 1}$ be a sequence in $X$.
For each $N \ge 1$, let $T_N = \{x_n : n \ge N\}$.
Then the following two conditions are equivalent:

1. For each real $\varepsilon > 0$ there is an integer $N$ such that $d(x_m, x_n) < \varepsilon$ for all $m, n \ge N$.
2. The [[diameters|0692CGQF00]] $\operatorname{diam}(T_N)$ tend to $0$ as $N \to \infty$.

The sequence $(x_n)$ is a Cauchy sequence when it satisfies these conditions.

## Remarks

Condition 1 is the usual definition of a Cauchy sequence.
Each convergent sequence is a Cauchy sequence: if $x_n \to x$, then $d(x_m, x_n) \le d(x_m, x) + d(x, x_n)$.
In $\mathbb{Q}$ with the usual metric, the decimal approximations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt{2}$ form a Cauchy sequence that does not converge in $\mathbb{Q}$.
Each Cauchy sequence is [[bounded|0P953MG05E]].

## Explanations

### L1 human

The terms of a Cauchy sequence get closer and closer to each other.
You can check this without knowing a limit, and this is the point. A Cauchy sequence looks as if it converges. But the space can have no point for it to converge to.
</document>

<document source="https://math2.tamnd.com/node/0MCR8ZAXQD.md" tag="0MCR8ZAXQD" kind="definition">
# Minimal element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $m \in S$ is a minimal element of $S$ when no $s \in S$ satisfies $s < m$.
Equivalently, for each $s \in S$, $s \le m$ implies $s = m$.

## Remarks

It is the [[maximal element|0TZEEFTNCC]] for the reversed order $\ge$.
In $\{2, 3, 4, 6, 9\}$ under divisibility, the minimal elements are $2$ and $3$.
A [[well-founded relation|0230ECX1Q5]] is one in which each nonempty subset has a minimal element.

## Explanations

### L1 human

A minimal element has nothing below it.
Among the integers greater than $1$ under divisibility, the minimal elements are the primes.
</document>

<document source="https://math2.tamnd.com/node/0MCXX29HWY.md" tag="0MCXX29HWY" kind="definition">
# Algebraic closure

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
An algebraic closure of $K$ is a [[field extension|0WQ1A8SHTZ]] $\overline{K}/K$ such that $\overline{K}/K$ is [[algebraic|09EYMKMSB9]] and $\overline{K}$ is [[algebraically closed|02SJG1HC32]].

## Remarks

The field $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$, by the fundamental theorem of algebra.
The field $\overline{\mathbb{Q}}$ of [[algebraic numbers|0QFTETCHKB]] is an algebraic closure of $\mathbb{Q}$. The field $\mathbb{C}$ is not, because $\mathbb{C}/\mathbb{Q}$ is not algebraic.
If $K$ is infinite, then $\overline{K}$ has the same cardinality as $K$. If $K$ is finite, then $\overline{K}$ is countably infinite.
An algebraic closure is a largest algebraic extension of $K$, and a smallest algebraically closed field that contains $K$.

## Explanations

### L1 human

The algebraic closure adds to a field all the roots of all its polynomials, and nothing else.
For the rationals, it gives the algebraic numbers, not all the complex numbers.
</document>

<document source="https://math2.tamnd.com/node/0MCYJ72VGA.md" tag="0MCYJ72VGA" kind="concept">
# Complement graph

## Summary

The complement of a graph has the same vertices and exactly the edges that the graph does not have. It swaps cliques and independent sets. The node [[complement graph|0RNMP6VCR2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0MDJP82TEH.md" tag="0MDJP82TEH" kind="definition">
# Coprime ideals

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ and $J$ be ideals of $R$.
The ideals $I$ and $J$ are coprime if $I + J = R$, where $I + J$ is the [[sum of the ideals|0MBJVDSK1R]].
Equivalently, there are $a \in I$ and $b \in J$ with $a + b = 1$.

## Remarks

In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = \gcd(m, n)\mathbb{Z}$. So $m\mathbb{Z}$ and $n\mathbb{Z}$ are coprime if and only if $\gcd(m, n) = 1$.
If $I$ and $J$ are coprime, then $I \cap J = I J$.
Two distinct [[maximal ideals|09J6KGT65G]] are always coprime.

## Explanations

### L1 human

For integers, $3$ and $5$ are coprime, and $2 \cdot 3 - 1 \cdot 5 = 1$.
So $1$ is a multiple of $3$ plus a multiple of $5$.
Coprime ideals generalize this: $1$ is an element of $I$ plus an element of $J$.
</document>

<document source="https://math2.tamnd.com/node/0MDMZ965VE.md" tag="0MDMZ965VE" kind="concept">
# Exterior algebra

## Summary

The exterior algebra of a vector space is the associative algebra generated by the space in which the square of every vector is zero.
</document>

<document source="https://math2.tamnd.com/node/0ME21QN125.md" tag="0ME21QN125" kind="concept">
# Symmetric difference

## Summary

The symmetric difference of two sets is the set of the objects that are elements of exactly one of the two sets.
</document>

<document source="https://math2.tamnd.com/node/0MES3KS0ZC.md" tag="0MES3KS0ZC" kind="statement">
# Division algorithm for polynomials

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $g \in R[x]$ be a [[monic polynomial|0BDX4MBKT4]].
Then there are unique polynomials $q, r \in R[x]$ such that $f = q g + r$ and $\deg r < \deg g$.

## Remarks

The polynomials $q$ and $r$ are the quotient and the remainder. Long division of polynomials computes them.
Over a [[field|0QA8CHH8MG]] $K$, every nonzero $g$ is a unit times a monic polynomial. So the statement holds for every nonzero $g \in K[x]$.
For example, $x^3 + 1 = (x^2 - x + 1)(x + 1) + 0$ and $x^2 + 1 = (x - 1)(x + 1) + 2$.
The statement fails for a divisor that is not monic: in $\mathbb{Z}[x]$, no $q$ and $r$ with $\deg r < 1$ give $x = q \cdot 2x + r$.

## Explanations

### L1 human

This is the polynomial version of division with remainder for integers.
You can divide by a monic polynomial and get a remainder of smaller degree.
The degree plays the role that the size of a number plays for integers.
</document>

<document source="https://math2.tamnd.com/node/0MFJJV97RJ.md" tag="0MFJJV97RJ" kind="concept">
# Norm induced by an inner product

## Summary

The norm induced by an inner product is the square root of the inner product of a vector with itself.
</document>

<document source="https://math2.tamnd.com/node/0MH1WK89PP.md" tag="0MH1WK89PP" kind="statement">
# Derivative of a polynomial

## Statement

Let $p(x) = \sum_{k=0}^{n} a_k x^k$ with real coefficients $a_k$.
Then $p$ is differentiable on $\mathbb{R}$, and
$$p'(x) = \sum_{k=1}^{n} k a_k x^{k-1}.$$

## Remarks

It follows from the [[power rule|0JZGW75C5Z]] and the [[linearity of differentiation|0M0TKFRX2A]].
So the derivative of a polynomial of degree $n \ge 1$ is a polynomial of degree $n - 1$.
Mathlib states it as `Polynomial.hasDerivAt`.

## Explanations

### L1 human

We differentiate a polynomial term by term with the power rule.
For example, the derivative of $3 x^2 + 5 x + 7$ is $6 x + 5$.
</document>

<document source="https://math2.tamnd.com/node/0MHE4B1QCQ.md" tag="0MHE4B1QCQ" kind="statement">
# Wronskian test for linear independence of solutions

## Statement

Let $I$ be an open interval, let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous, and let $y_1, \dots, y_n$ be solutions on $I$ of the [[homogeneous linear equation|0EQZSAB9EC]] $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$.
Let $W = W(y_1, \dots, y_n)$ be their [[Wronskian|0N5NF1RGBP]].
Then these three conditions are equivalent.
(a) The functions $y_1, \dots, y_n$ are linearly independent on $I$.
(b) There is a point $x_0 \in I$ with $W(x_0) \ne 0$.
(c) For each $x \in I$, $W(x) \ne 0$.

## Remarks

So the Wronskian of $n$ solutions is either zero at each point of $I$ or zero at no point of $I$. [[Abel's identity|0VGWKX2JHB]] gives the same result for $n = 2$.
For arbitrary functions, $W(x_0) \ne 0$ at one point still implies linear independence. But the converse fails: $x^2$ and $x |x|$ are linearly independent on $\mathbb{R}$, and their Wronskian is $0$ at each point.
Example: $e^{x}$ and $e^{-x}$ solve $y'' - y = 0$, and their Wronskian is $-2$, so they are independent.

## Explanations

### L1 human

To check that $n$ solutions are independent, compute one determinant at one point.
If it is not zero, the solutions are independent and form a basis of all solutions.
For solutions of a linear equation, the value at one point already decides the question.
</document>

<document source="https://math2.tamnd.com/node/0MHGJRZMTM.md" tag="0MHGJRZMTM" kind="statement">
# Every sequence has a monotone subsequence

## Statement

Let $(a_n)$ be a real sequence.
Then $(a_n)$ has a [[subsequence|0R8J3ZQMJS]] that is [[monotone|0KN22AAW5R]].

## Remarks

The usual proof uses peaks. An index $n$ is a peak if $a_m \le a_n$ for all $m > n$.
If there are infinitely many peaks, the terms at the peaks form a decreasing subsequence.
If there are finitely many peaks, start after the last one. Each index then has a later index with a larger term, and this gives a strictly increasing subsequence.
With the [[monotone convergence theorem|0FM6N4R0VB]], the lemma gives the [[Bolzano–Weierstrass theorem|0KDJD249DT]].

## Explanations

### L1 human

Think of the terms as heights of hills seen from left to right.
Either infinitely many hills have no higher hill to their right, or from some point on you can always find a higher hill further right.
</document>

<document source="https://math2.tamnd.com/node/0MJ04A50QW.md" tag="0MJ04A50QW" kind="concept">
# Matching

## Summary

A matching in a graph is a set of edges without common ends. Matchings model pairing and assignment problems. The node [[matching|0Z87BHAFTB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0MJ0JS72DZ.md" tag="0MJ0JS72DZ" kind="definition">
# Tree

## Definition

A tree is a [[graph|0JD5C82QMB]] that is [[connected|0FFAYNMAJZ]] and [[acyclic|0XNN9Z8E5A]].

## Remarks

Let $T$ be a finite graph with $n \ge 1$ vertices. Then these conditions are equivalent: $T$ is a tree; $T$ is connected with $n - 1$ edges; $T$ is acyclic with $n - 1$ edges.
A graph is a tree if and only if any two vertices are joined by exactly one path. The node [[unique paths in trees|096B7Q5742]] states this.
The [[path graphs|075WR6W3Y3]] and the stars $K_{1,n}$ are trees.

## Explanations

### L1 human

A tree is a connected graph with no loops.
A family tree and the folders on a computer are pictures of trees.
</document>

<document source="https://math2.tamnd.com/node/0MKABWDRNC.md" tag="0MKABWDRNC" kind="definition">
# Upper triangular matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $U = (u_{ij})$ of order $n$ over $R$ is *upper triangular* when $u_{ij} = 0$ for all $i > j$.
It is *strictly upper triangular* when $u_{ij} = 0$ for all $i \ge j$.

## Remarks

Sums and products of upper triangular matrices of the same order are upper triangular.
The [[determinant of a triangular matrix|0K1KKMFHYH]] is the product of its diagonal entries.
The diagonal entries of an upper triangular matrix over a field are its [[eigenvalues|0BQAA2BK3Q]].
A system $Ux = b$ with $U$ upper triangular and invertible is solved by [[back substitution|0ATFVE08MW]].

## Explanations

### L1 human

An upper triangular matrix has zeros everywhere below its diagonal.
Its nonzero entries fill a triangle at the top right.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ is upper triangular.
</document>

<document source="https://math2.tamnd.com/node/0MKRHZB3WS.md" tag="0MKRHZB3WS" kind="concept">
# Change of basis matrix

## Summary

The change of basis matrix from one basis to another is a square matrix.
Its columns are the coordinate vectors of the new basis vectors in the old basis.
It converts new coordinates into old coordinates.
</document>

<document source="https://math2.tamnd.com/node/0MM085RX00.md" tag="0MM085RX00" kind="definition">
# Product of vector spaces

## Definition

Let $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.
The *product* $V \times W$ is the set of pairs $(v, w)$ with $v \in V$ and $w \in W$, with the operations
$$(v, w) + (v', w') = (v + v', w + w'), \qquad a(v, w) = (av, aw)$$
for $v, v' \in V$, $w, w' \in W$ and $a \in F$.
It is a vector space over $F$ with zero vector $(0, 0)$.

## Remarks

For finitely many spaces, the product is also the [[external direct sum|0S7W6EG32V]], written $V \oplus W$.
The [[dimension of a direct product|06HN47VPTH]] is $\dim V + \dim W$.
For example, $F^m \times F^n$ is isomorphic to $F^{m+n}$.
The projections $(v, w) \mapsto v$ and $(v, w) \mapsto w$ are linear maps.

## Explanations

### L1 human

The product of two vector spaces has vectors that are pairs: one vector from each space.
We add pairs part by part and scale both parts.
The plane $\R^2$ is the product of two copies of the line $\R$.
</document>

<document source="https://math2.tamnd.com/node/0MM45BT6HC.md" tag="0MM45BT6HC" kind="definition">
# One-point compactification

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $\infty$ be a point not in $X$. Let $X^* = X \cup \{\infty\}$.
The one-point compactification, or Alexandroff extension, of $X$ is the set $X^*$ with the following topology.
The open sets are the sets $U \in \tau$ and the sets $X^* \setminus C$, where $C \subseteq X$ is [[closed|0G5AV9Q4RR]] and [[compact|00R0X1WGPX]] in $X$.

## Remarks

The space $X^*$ is compact, and $X$ is an open subspace of $X^*$.
The space $X^*$ is Hausdorff if and only if $X$ is [[locally compact|0EQMWQ4WMJ]] and [[Hausdorff|0MXB4CKHFW]].
The one-point compactification of $\mathbb{R}^n$ is homeomorphic to the sphere $S^n$, by stereographic projection. For $n = 1$, it is the circle.

## Explanations

### L1 human

You make a non-compact space compact by adding one point, and you declare that the new point is near everything outside large compact sets.
The real line becomes a circle: both ends meet at the new point.
</document>

<document source="https://math2.tamnd.com/node/0MMCB8RXGA.md" tag="0MMCB8RXGA" kind="statement">
# A ring homomorphism is injective iff its kernel is zero

## Statement

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then $f$ is injective if and only if its [[kernel|0AAHE7GEW6]] is $\{0\}$.

## Remarks

The proof uses only the addition: $f(a) = f(b)$ holds if and only if $f(a - b) = 0$, that is, $a - b \in \ker f$.
The same criterion holds for group homomorphisms and for linear maps.

## Explanations

### L1 human

If two elements have the same image, then their difference goes to $0$.
So when only $0$ goes to $0$, no two different elements have the same image.
</document>

<document source="https://math2.tamnd.com/node/0MN2MER9WN.md" tag="0MN2MER9WN" kind="statement">
# Continuum hypothesis

## Statement

There is no [[set|0SJ3RX9TXA]] $S$ with $\aleph_0 < |S| < 2^{\aleph_0}$.
Equivalently, with the axiom of choice, $2^{\aleph_0} = \aleph_1$.

## Remarks

In words: each infinite subset of $\mathbb{R}$ is countable or equinumerous with $\mathbb{R}$.
The hypothesis is independent of ZFC: if ZFC is consistent, then ZFC proves neither the hypothesis nor its negation.
The generalized continuum hypothesis says that $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for each ordinal $\alpha$.

## History

Georg Cantor proposed the hypothesis in 1878.
It was the first of the problems that David Hilbert presented in 1900.
Kurt Gödel showed in 1940 that ZFC does not refute it, and Paul Cohen showed in 1963 that ZFC does not prove it.

## Explanations

### L1 human

Cantor showed that $\mathbb{R}$ is larger than $\mathbb{N}$.
The continuum hypothesis says that no size lies strictly between them.
The usual axioms of set theory cannot decide whether this is true.
</document>

<document source="https://math2.tamnd.com/node/0MN882VJ4P.md" tag="0MN882VJ4P" kind="statement">
# Derivative as a little-o approximation

## Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, and let $f \colon U \to W$. Let $x \in U$, and let $A \colon V \to W$ be a bounded linear map.
Then $A$ is the [[Fréchet derivative|08EVNYZF42]] of $f$ at $x$ if and only if
$$f(x + h) = f(x) + A h + r(h), \qquad \text{where } r(h) = o(\|h\|) \text{ as } h \to 0.$$
Here $r(h) = o(\|h\|)$ means that for each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|r(h)\| \le \varepsilon \|h\|$ for each $h$ with $\|h\| < \delta$ and $x + h \in U$.

## Remarks

The remainder $r(h)$ is defined by the equation, so the statement says that the error of the linear approximation tends to $0$ faster than $\|h\|$.
The little-o form includes $h = 0$, where $r(0) = 0$. This is why it uses $\le$ and does not divide by $\|h\|$.
This form is often the most convenient one for proofs, for example of the [[chain rule|0EAAS2EHDR]].

## Explanations

### L1 human

The equation splits $f(x + h)$ into three parts: the value at $x$, a linear part in $h$, and an error.
The derivative condition says only that the error is tiny compared with the step: halving the step more than halves the error.
</document>

<document source="https://math2.tamnd.com/node/0MNF0BYFQC.md" tag="0MNF0BYFQC" kind="concept">
# Algebraic number

## Summary

An algebraic number is a complex number that is a root of a nonzero polynomial with rational coefficients.
The [[definition of an algebraic number|0MB4TGMXG6]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0MNHXBSX3M.md" tag="0MNHXBSX3M" kind="definition">
# Prime element

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
An element $p \in R$ is prime if it has these three properties.

1. $p \neq 0$.
2. $p$ is not a [[unit|019NFGFPR5]].
3. For all $a, b \in R$, if $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Remarks

A nonzero $p$ is prime if and only if the principal ideal $(p)$ is a [[prime ideal|06HC6HK0Z8]].
The prime elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ a prime number.
In an integral domain, [[prime elements are irreducible|0F0K8DB9TD]].

## Explanations

### L1 human

Euclid's lemma says that a prime number dividing a product divides one of the factors.
A prime element of a ring is an element with this property.
In the integers, $6$ is not prime: $6$ divides $2 \cdot 3$, but not $2$ or $3$.
</document>

<document source="https://math2.tamnd.com/node/0MNM54WB99.md" tag="0MNM54WB99" kind="definition">
# Geometric sequence

## Definition

Let $a$ and $r$ be real numbers.
The geometric sequence with initial term $a$ and common ratio $r$ is the [[real sequence|0CTV783HYT]] $(a\,r^n)_{n \in \mathbb{N}}$.
Its terms are $a, a r, a r^2, a r^3, \dots$, and each term is $r$ times the term before it.

## Remarks

Many texts require $a \ne 0$ and $r \ne 0$, so that the quotient of two consecutive terms is always $r$.
The sequence $1, 2, 4, 8, \dots$ has $a = 1$ and $r = 2$. The sequence $1, 1/2, 1/4, \dots$ has $r = 1/2$.
For $|r| < 1$, the sequence [[converges to $0$|0Q7K4AXE53]]. For $r > 1$ and $a > 0$, it [[tends to infinity|0TZK91NQE3]].
The sum of the terms is the [[geometric series|01X5CVV1Q5]].

## Explanations

### L1 human

In a geometric sequence, you go from one term to the next by multiplication with the same number $r$.
If $r = 3$ and you start at $2$, you get $2, 6, 18, 54, \dots$.
</document>

<document source="https://math2.tamnd.com/node/0MNZCN4BQK.md" tag="0MNZCN4BQK" kind="definition">
# Product of rings

## Definition

Let $I$ be a set and let $(R_i)_{i \in I}$ be a family of [[rings|0894J980RK]].
The product $\prod_{i \in I} R_i$ is the set of families $(a_i)_{i \in I}$ with $a_i \in R_i$ for each $i \in I$.
Its addition is $(a_i)_{i \in I} + (b_i)_{i \in I} = (a_i + b_i)_{i \in I}$.
Its multiplication is $(a_i)_{i \in I} \cdot (b_i)_{i \in I} = (a_i \cdot b_i)_{i \in I}$.
With these operations, $\prod_{i \in I} R_i$ is a ring, with zero $(0)_{i \in I}$ and identity $(1)_{i \in I}$.

## Remarks

For two rings $R$ and $S$, the product is $R \times S = \{(r, s) : r \in R, s \in S\}$.
For each $j \in I$, the projection $\prod_{i \in I} R_i \to R_j$, $(a_i)_{i \in I} \mapsto a_j$, is a [[ring homomorphism|0GGE0Z0JQ1]].
The product of two nonzero rings is not an [[integral domain|09S3PFBEPM]]: in $R \times S$, $(1, 0) \cdot (0, 1) = (0, 0)$.

## Explanations

### L1 human

To multiply two pairs, you multiply the first entries and the second entries apart.
For example, in $\mathbb{Z} \times \mathbb{Z}$, $(2, 3) \cdot (4, 5) = (8, 15)$.
</document>

<document source="https://math2.tamnd.com/node/0MP9BCF6DP.md" tag="0MP9BCF6DP" kind="concept">
# Line

## Summary

A line is a straight figure of dimension one that extends without end in both directions. In a Euclidean space, through each two distinct points there is exactly one line.
</document>

<document source="https://math2.tamnd.com/node/0MPAJRB23T.md" tag="0MPAJRB23T" kind="statement">
# Diagonalisable iff the minimal polynomial has distinct roots

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$ with [[minimal polynomial|03NDVCSHSB]] $m_A$.
Then $A$ is [[diagonalisable|07SQCNV233]] over $F$ if and only if
$$m_A(t) = (t - \lambda_1)(t - \lambda_2) \cdots (t - \lambda_k)$$
for some pairwise distinct $\lambda_1, \dots, \lambda_k \in F$.

## Remarks

In words: $m_A$ splits over $F$ and has no repeated roots.
If $A^2 = A$, then $m_A$ divides $t^2 - t = t(t - 1)$, so each [[projection|033DA1WE8Y]] matrix is diagonalisable.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has $m_A = (t - 1)^2$, so it is not diagonalisable.

## Explanations

### L1 human

The minimal polynomial decides whether a matrix is diagonalisable.
The matrix is diagonalisable exactly when this polynomial is a product of distinct linear factors.
A repeated factor, like $(t - 1)^2$, signals a Jordan block of size at least $2$.
</document>

<document source="https://math2.tamnd.com/node/0MPZGAZ2GK.md" tag="0MPZGAZ2GK" kind="definition">
# Centre of mass

## Definition

Let $E \subseteq \mathbb{R}^n$ be a bounded Jordan measurable set, and let $\delta \colon E \to [0, \infty)$ be a continuous density with total mass $M = \int_E \delta(x) \, dx > 0$.
The centre of mass of $E$ with density $\delta$ is the point $\bar{x} = (\bar{x}_1, \dots, \bar{x}_n)$ with
$$\bar{x}_i = \frac{1}{M} \int_E x_i \, \delta(x) \, dx, \qquad i = 1, \dots, n.$$
If $\delta$ is constant, $\bar{x}$ is the centroid of $E$, and it does not depend on the value of the constant.

## Remarks

For $n = 3$ the integrals are [[triple integrals|0GTBGMZXV8]], and for a plane lamina, $n = 2$, they are [[double integrals|0NKBMFC18R]].
For point masses $m_1, \dots, m_k$ at $p_1, \dots, p_k$, the centre of mass is $\sum_j m_j p_j / \sum_j m_j$. The definition above replaces the sum by an integral.
Example: the half disc $\{x^2 + y^2 \le a^2, \ y \ge 0\}$ with constant density has centroid $(0, 4a / (3\pi))$.
If $E$ and $\delta$ are symmetric under a reflection, then $\bar{x}$ lies in the mirror of that reflection.

## Explanations

### L1 human

The centre of mass is the balance point of a body.
If the body is supported at that point, gravity does not turn it in any direction.
</document>

<document source="https://math2.tamnd.com/node/0MQ24RBG93.md" tag="0MQ24RBG93" kind="definition">
# Fundamental matrix

## Definition

Let $I$ be an open interval and let $A : I \to \mathbb{R}^{n \times n}$ be continuous.
A fundamental matrix of the homogeneous [[linear system|0A6QZ367GX]] $x' = A(t) x$ is a differentiable function $\Phi : I \to \mathbb{R}^{n \times n}$ with these two properties.
For each $t \in I$, $\Phi'(t) = A(t)\, \Phi(t)$.
For each $t \in I$, the matrix $\Phi(t)$ is invertible.

## Remarks

Equivalently, the columns of $\Phi$ are $n$ solutions that form a basis of the space of solutions.
Each solution is $x(t) = \Phi(t)\, c$ for a unique $c \in \mathbb{R}^n$, and the solution with $x(t_0) = x_0$ is $\Phi(t) \Phi(t_0)^{-1} x_0$.
If $\Phi'(t) = A(t) \Phi(t)$ and $\Phi(t_0)$ is invertible at one point $t_0$, then $\Phi(t)$ is invertible at each point, by [[Liouville's formula|09PS3S0EF6]].
For a constant matrix $A$, $\Phi(t) = e^{tA}$ is a fundamental matrix. See [[the matrix exponential as the solution|0YC8JT513J]].

## Explanations

### L1 human

A fundamental matrix puts $n$ independent solutions side by side as columns.
Then one matrix product gives each solution: choose the vector $c$ of coefficients.
</document>

<document source="https://math2.tamnd.com/node/0MQQQ5NA2S.md" tag="0MQQQ5NA2S" kind="statement">
# Compact Hausdorff spaces are normal

## Statement

Let $X$ be a [[compact space|0QPHW703PV]] that is [[Hausdorff|0MXB4CKHFW]].
Then $X$ is [[normal|0AJ646N339]].

## Remarks

Proof sketch: closed subsets of $X$ are compact, by [[closed subsets of a compact set are compact|066HW6D9A7]]. In a Hausdorff space, a point and a disjoint compact set have disjoint open neighbourhoods. A second use of compactness separates two disjoint compact sets.
With [[Urysohn's lemma|01EY3VR7ZD]], this shows that compact Hausdorff spaces have many continuous real functions.

## Explanations

### L1 human

In a compact Hausdorff space, you first separate points by open sets, and then compactness lets you combine finitely many of these choices.
The result separates any two disjoint closed sets.
</document>

<document source="https://math2.tamnd.com/node/0MQXVZEBMT.md" tag="0MQXVZEBMT" kind="statement">
# Change of basis formula

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Let $B, B'$ be bases of $V$, and let $C, C'$ be bases of $W$.
Let $P$ be the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$, and $Q$ the change of basis matrix from $C$ to $C'$.
Then, for each linear map $T \colon V \to W$,
$$[T]_{C',B'} = Q^{-1} [T]_{C,B} P,$$
where $[T]$ denotes the [[matrix of $T$|0YQSGNYE3K]] in the given bases.
In particular, for an endomorphism $T$ of $V$ and $C = B$, $C' = B'$, one gets $[T]_{B'} = P^{-1} [T]_B P$.

## Remarks

The formula follows from [[the matrix of a composition|05Q1B7MH0M]], applied to $T = \id_W \circ T \circ \id_V$.
For endomorphisms, the two matrices are [[similar|0E89HQATM8]].
For linear maps between two spaces, the two matrices are [[equivalent|0CMHSGTWXB]].

## Explanations

### L1 human

When we change bases, the matrix of a linear map changes in a fixed way.
We translate the input from the new to the old coordinates, apply the old matrix, and translate the result back.
This gives the product $Q^{-1} A P$.
</document>

<document source="https://math2.tamnd.com/node/0MR0BYF0K2.md" tag="0MR0BYF0K2" kind="concept">
# Normal matrix

## Summary

A normal matrix is a complex square matrix that commutes with its conjugate transpose.
</document>

<document source="https://math2.tamnd.com/node/0MRE6XRE5V.md" tag="0MRE6XRE5V" kind="definition">
# Series of functions

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$.
The series of functions $\sum_{n=0}^{\infty} f_n$ is the sequence of partial sums $s_N = \sum_{n=0}^{N} f_n$.
The series converges pointwise, or uniformly, on $S$ if the sequence $(s_N)$ converges [[pointwise|056FSKG32J]], or [[uniformly|08FPABDQ4C]], on $S$.
The limit is the sum $\sum_{n=0}^{\infty} f_n(x)$.

## Remarks

Power series and Fourier series are series of functions.
The [[Weierstrass M-test|0NX5NWAFC9]] is the main tool to prove uniform convergence.
By the [[uniform limit theorem|0X9XT05JVC]], a uniformly convergent series of continuous functions has a continuous sum.

## Explanations

### L1 human

A series of functions adds infinitely many functions, point by point.
At each $x$ it is an ordinary series of numbers.
</document>

<document source="https://math2.tamnd.com/node/0MS8QGA6S5.md" tag="0MS8QGA6S5" kind="statement">
# A quotient is abelian iff the subgroup contains the commutator subgroup

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the [[quotient group|0KYX2N69J0]] $G/N$ is [[abelian|082GHNPJ53]] if and only if $[G, G] \subseteq N$, where $[G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].

## Remarks

The cosets $gN$ and $hN$ commute exactly when $g^{-1} h^{-1} g h \in N$.
Thus $[G, G]$ is the smallest normal subgroup of $G$ with an abelian quotient.
A subgroup that contains $[G, G]$ is automatically normal.

## Explanations

### L1 human

To make a group abelian by a quotient, you must send each commutator to the identity.
So the subgroup that you divide by must contain all commutators.
If it does, the quotient is abelian.
</document>

<document source="https://math2.tamnd.com/node/0MSCQNFTM8.md" tag="0MSCQNFTM8" kind="statement">
# Test for exactness

## Statement

Let $R = (\alpha, \beta) \times (\gamma, \delta)$ be an open rectangle in $\mathbb{R}^2$.
Let $M, N : R \to \mathbb{R}$ be continuous functions such that the partial derivatives $\partial M / \partial y$ and $\partial N / \partial x$ exist and are continuous on $R$.
Then the equation $M\, dx + N\, dy = 0$ is [[exact|0EC7AGJ0PX]] on $R$ if and only if
$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \quad \text{on } R.$$

## Remarks

The condition is necessary on each open set: if $\psi_x = M$ and $\psi_y = N$, the mixed partial derivatives of $\psi$ are equal, so $M_y = N_x$.
For the converse, fix $(x_0, y_0) \in R$ and set $\psi(x, y) = \int_{x_0}^{x} M(s, y_0)\, ds + \int_{y_0}^{y} N(x, t)\, dt$.
The converse holds on each simply connected open set. It can fail on other sets: $(-y\, dx + x\, dy)/(x^2 + y^2) = 0$ satisfies the condition on $\mathbb{R}^2 \setminus \{0\}$, but it is not exact there.

## Explanations

### L1 human

If $\psi$ exists, then $M$ and $N$ are its two partial derivatives.
Their cross derivatives must agree, because the order of differentiation does not matter for nice functions.
On a rectangle, this easy test is also enough.
</document>

<document source="https://math2.tamnd.com/node/0MSRQHCNN9.md" tag="0MSRQHCNN9" kind="concept">
# Existential quantifier

## Summary

The existential quantifier $\exists$ expresses "there exists".
An existentially quantified statement says that a [[predicate|0T3DPBG981]] holds for at least one element of its domain.
</document>

<document source="https://math2.tamnd.com/node/0MSSD1VCW9.md" tag="0MSSD1VCW9" kind="concept">
# Order on the reals

## Summary

The order on the reals is the relation $\le$ on $\mathbb{R}$ that compares two real numbers by their position on the number line.
It extends the order of $\mathbb{Q}$ and makes $\mathbb{R}$ an [[ordered field|010JCV9J4E]].
</document>

<document source="https://math2.tamnd.com/node/0MSWYWHYF5.md" tag="0MSWYWHYF5" kind="definition">
# Alternating multilinear map

## Definition

Let $F$ be a field, let $n \ge 1$, and let $V$ and $W$ be vector spaces over $F$.
A [[multilinear map|0443XNARDV]] $f \colon V^n \to W$ is *alternating* when $f(v_1, \dots, v_n) = 0$ for each family with $v_i = v_j$ for some $i \ne j$.

## Remarks

An alternating map changes sign when two arguments are exchanged.
If the characteristic of $F$ is not $2$, then this sign rule implies that the map is alternating.
An alternating map is zero on each linearly dependent family.
The [[determinant|00KDZRW4C0]] is an alternating multilinear map of the rows of a square matrix.

## Explanations

### L1 human

An alternating map gives zero as soon as two inputs are the same.
It follows that swapping two inputs changes the sign of the value.
The signed area of the parallelogram of two plane vectors is an alternating bilinear map.
</document>

<document source="https://math2.tamnd.com/node/0MT2S0ZX4Y.md" tag="0MT2S0ZX4Y" kind="definition">
# Contraction mapping

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
A map $f \colon X \to X$ is a contraction when there is a real number $k$ with $0 \le k < 1$ such that
$$d\bigl(f(x), f(y)\bigr) \le k \, d(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

A contraction is a [[Lipschitz map|0TMZ21WKCN]] with a constant less than $1$. Thus it is uniformly continuous.
The map $x \mapsto x/2$ on $\mathbb{R}$ is a contraction with $k = 1/2$.
The condition $d(f(x), f(y)) < d(x, y)$ for all $x \neq y$ is weaker. The map $x \mapsto x + 1/x$ on $[1, \infty)$ satisfies it, but it is not a contraction and has no fixed point.

## Explanations

### L1 human

A contraction pulls the points of the space closer together, each time by at least the same factor.
If you apply it again and again, all points are pulled toward one point.
</document>

<document source="https://math2.tamnd.com/node/0MTBGVZ4XJ.md" tag="0MTBGVZ4XJ" kind="statement">
# First isomorphism theorem for linear maps

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map.
Then the map
$$\bar T \colon V / \ker T \to \im T, \qquad v + \ker T \mapsto T(v),$$
is well defined, and it is a linear isomorphism.
Here $V / \ker T$ is the [[quotient space|0BR3FY5Q8M]] by the [[kernel|0S23E6YE83]], and $\im T$ is the [[image|0WSFKKT6FF]] of $T$.

## Remarks

In finite dimensions, it gives the [[rank–nullity theorem|0W4FSW1TVB]]: $\dim V - \dim \ker T = \dim \im T$.
The same theorem holds for groups, rings and modules.
For a surjective map $T$, it says $V / \ker T \cong W$.

## Explanations

### L1 human

A linear map identifies vectors that differ by an element of the kernel.
After we glue these vectors together, the map becomes one to one.
So the quotient by the kernel is a copy of the image.
</document>

<document source="https://math2.tamnd.com/node/0MTENFBW3Q.md" tag="0MTENFBW3Q" kind="definition">
# One-form

## Definition

Let $U \subseteq \mathbb{R}^n$ be open.
A one-form on $U$ is a [[differential form|0QHTX7WQEY]] of degree $1$, that is, a sum $\omega = f_1 \, dx_1 + \dots + f_n \, dx_n$ with functions $f_i \colon U \to \mathbb{R}$.
At $x \in U$, it gives the linear map $\omega_x(v) = f_1(x) v_1 + \dots + f_n(x) v_n$ for $v \in \mathbb{R}^n$.

## Remarks

For a differentiable function $g$ on $U$, the differential of $g$ is the one-form $dg = \sum_i \frac{\partial g}{\partial x_i} \, dx_i$. So $dg_x(v)$ is the directional derivative of $g$ at $x$ in direction $v$.
For a continuous $\omega$ and a piecewise $C^1$ curve $\gamma \colon [a, b] \to U$, the integral is $\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}\big( \gamma'(t) \big) \, dt$. This is the [[line integral|0YS8RX87F4]] of the [[vector field|0N67NBKQ5T]] $(f_1, \dots, f_n)$.
Under this correspondence, $dg$ corresponds to $\nabla g$, and $\omega$ is of the form $dg$ exactly when the field is [[conservative|05R9NF2J49]].
On $\mathbb{R}^2 \setminus \{0\}$, the angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ is a one-form. Its integral around a closed curve is $2\pi$ times the winding number of the curve about the origin.

## Explanations

### L1 human

A one-form is a rule that, at each point, turns a direction vector into a number, in a linear way.
Integrating a one-form along a curve adds up these numbers for the velocity of the curve. It is the same thing as the work of a vector field.
</document>

<document source="https://math2.tamnd.com/node/0MTNR28K60.md" tag="0MTNR28K60" kind="concept">
# Poisson point process

## Summary

A Poisson point process is a random set of points in which the numbers of points in disjoint regions are independent Poisson random variables. On the time axis, it models events that occur at random times with a constant rate.
The node [[Poisson point process|05TG0SE4HT]] gives the precise definition of the homogeneous process on $[0, \infty)$.
</document>

<document source="https://math2.tamnd.com/node/0MVCFMW7S4.md" tag="0MVCFMW7S4" kind="object">
# e

## Description

The number $e$ is the value of the [[exponential function|0RMVS7RXZX]] at $1$:
$$e = \exp(1) = \sum_{n=0}^{\infty} \frac{1}{n!} \approx 2.718281828.$$
It is also the limit $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$.
It is the base of the natural logarithm, so $\ln e = 1$.

## Remarks

The series is the [[sum of the reciprocals of the factorials|0G4D3ZKTP8]].
The number $e$ is [[irrational|0W4GQSG7B6]]. Charles Hermite proved in 1873 that it is transcendental.
Jacob Bernoulli introduced the constant in 1683, in a problem of continuous compound interest. Leonhard Euler began to use the letter $e$ for it.

## Explanations

### L1 human

The number $e$ is about $2.718$. It is the natural base for exponential growth.
An amount that grows by $100\%$ per year, compounded continuously, becomes $e$ times larger in one year.
</document>

<document source="https://math2.tamnd.com/node/0MVDQEAB60.md" tag="0MVDQEAB60" kind="definition">
# Sequential continuity

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is sequentially continuous if the following holds.
For each sequence $(x_n)$ in $D$ that [[converges|0XKWSSY8P2]] to a point $a \in D$, the sequence $(f(x_n))$ converges to $f(a)$.

## Remarks

For real functions, [[sequential continuity is the same as continuity|0PAQM3H57X]].
The definition is useful to prove that a function is not continuous. It is enough to find one sequence $x_n \to a$ with $f(x_n) \not\to f(a)$.
Mathlib defines it for topological spaces, as `SeqContinuous`.

## Explanations

### L1 human

A sequentially continuous function preserves limits of sequences.
The function of the limit is the limit of the function values.
</document>

<document source="https://math2.tamnd.com/node/0MVH4Y1CRZ.md" tag="0MVH4Y1CRZ" kind="definition">
# Noetherian ring

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].
That is, for every chain $I_1 \subseteq I_2 \subseteq \dots$ of ideals of $R$, there is $N$ with $I_n = I_N$ for all $n \ge N$.

## Remarks

The ring $R$ is Noetherian if and only if every ideal of $R$ is [[finitely generated|030TZ0DQWT]].
Every field and every [[principal ideal domain|0RZYDVZ475]] is Noetherian.
By [[Hilbert's basis theorem|0YKA9J3XEE]], $R[x_1, \dots, x_n]$ is Noetherian when $R$ is.
The polynomial ring in infinitely many variables $x_1, x_2, \dots$ over a field is not Noetherian: $(x_1) \subset (x_1, x_2) \subset \dots$ never stops.
For a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.

## Explanations

### L1 human

In a Noetherian ring, each ideal is generated by a finite list of elements.
This makes many arguments work by induction, because a growing chain of ideals must stop.
Rings met in practice, like $\mathbb{Z}[x, y]$, are Noetherian.
</document>

<document source="https://math2.tamnd.com/node/0MVTCFW42K.md" tag="0MVTCFW42K" kind="statement">
# Exponentials dominate powers in sequences

## Statement

Let $k$ be a natural number and let $r$ be a real number with $r > 1$.
Then
$$\lim_{n \to \infty} \frac{n^k}{r^n} = 0.$$

## Remarks

For $k = 1$ and $r = 2$, it says $n / 2^n \to 0$. The terms are $1/2, 2/4, 3/8, 4/16, \dots$.
One proof uses the ratio of consecutive terms. The ratio $\frac{(n+1)^k}{n^k} \cdot \frac{1}{r}$ tends to $1/r < 1$, so the terms eventually shrink by a fixed factor.
In the language of [[little o notation|04TSHB9H8D]], it says $n^k = o(r^n)$.
The node [[exponential growth beats polynomial growth|0CT6CBGTYY]] gives the version for real variables.

## Explanations

### L1 human

A power such as $n^{10}$ grows fast, but an exponential such as $1.01^n$ grows faster in the end.
For small $n$ the power can be much larger. For large $n$ the exponential always wins.
</document>

<document source="https://math2.tamnd.com/node/0MW2WXTHMC.md" tag="0MW2WXTHMC" kind="definition">
# Equicontinuity

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $\mathcal{F}$ be a set of functions from $X$ to $Y$.
The family $\mathcal{F}$ is equicontinuous at a point $a \in X$ when the property that follows holds.
For each real $\varepsilon > 0$ there is a real $\delta > 0$ such that $d_Y(f(x), f(a)) < \varepsilon$ for all $f \in \mathcal{F}$ and all $x \in X$ with $d_X(x, a) < \delta$.
The family $\mathcal{F}$ is equicontinuous when it is equicontinuous at each point of $X$.

## Remarks

The number $\delta$ depends on $\varepsilon$ and $a$, but not on $f$.
Each finite set of continuous functions is equicontinuous.
The family of all functions $f \colon \mathbb{R} \to \mathbb{R}$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ is equicontinuous: take $\delta = \varepsilon$.
The family $f_n(x) = x^n$ on $[0, 1]$ is not equicontinuous at $x = 1$.

## Explanations

### L1 human

Each continuous function has its own $\delta$ for each $\varepsilon$. An equicontinuous family has one $\delta$ that works for all functions in the family.
The functions $x^n$ become steeper and steeper near $1$, so no common $\delta$ exists there.
</document>

<document source="https://math2.tamnd.com/node/0MWKJ44V5A.md" tag="0MWKJ44V5A" kind="object">
# Natural numbers

## Description

The natural numbers are the numbers $0, 1, 2, 3, \dots$ that count the elements of a finite set.
We write $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ for the set of natural numbers.
Each natural number $n$ has a next natural number $n + 1$, its successor.
The [[Peano axioms|0QFT7Z76N5]] describe $\mathbb{N}$ with the number $0$ and the [[successor function|0AK9TGSNXY]].
The [[addition|0S13V32CPB]], the [[multiplication|06J5DE801P]] and the [[order|0HNATXFEV5]] of $\mathbb{N}$ come from $0$ and the successor function.

## Remarks

Some authors start the natural numbers at $1$. They then write $\mathbb{N}_0$ or $\mathbb{Z}_{\ge 0}$ for the set $\{0, 1, 2, \dots\}$.
Every nonempty set of natural numbers has a least element. This is the [[well-ordering principle|01BX15HSRR]].
The [[integers|0C4JS9RF5M]] contain the natural numbers as the integers that are not negative.

## Explanations

### L1 human

The natural numbers are the numbers that you use to count: zero, one, two, three, and so on.
There is no largest natural number, because you can always add one more.
All the other number systems of this course, such as the integers and the real numbers, start from the natural numbers.
</document>

<document source="https://math2.tamnd.com/node/0MWV6ZTK6S.md" tag="0MWV6ZTK6S" kind="concept">
# Divergence

## Summary

The divergence of a differentiable [[vector field|0299VCW1TH]] is a scalar function. At each point, it measures the net rate of outward flow of the field per unit volume.
In Cartesian coordinates it is the sum of the [[partial derivatives|09EGC6F8PT]] $\partial F_i / \partial x_i$ of the components.
</document>

<document source="https://math2.tamnd.com/node/0MWYQTZ07Z.md" tag="0MWYQTZ07Z" kind="definition">
# Hyperbolic tangent

## Definition

The hyperbolic tangent is the function $\tanh \colon \mathbb{R} \to \mathbb{R}$ defined by
$$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}.$$

## Remarks

The quotient is defined everywhere, since $\cosh x \ge 1$.
It is odd and strictly increasing, with range $(-1, 1)$. Also $\tanh x \to \pm 1$ as $x \to \pm \infty$.
Mathlib defines it as `Real.tanh`.

## Explanations

### L1 human

The hyperbolic tangent is an S-shaped curve between $-1$ and $1$.
It is close to $x$ near $0$ and close to $\pm 1$ far away.
</document>

<document source="https://math2.tamnd.com/node/0MX1HZBBXJ.md" tag="0MX1HZBBXJ" kind="statement">
# A series of nonnegative terms converges iff its partial sums are bounded

## Statement

Let $(a_n)$ be a real sequence with $a_n \ge 0$ for all $n$.
Then the series $\sum a_n$ converges if and only if its [[partial sums|0NB643K60Q]] $s_N = \sum_{n=0}^{N} a_n$ are bounded above.
In this case, $\sum_{n=0}^{\infty} a_n = \sup_N s_N$.

## Remarks

The partial sums increase, since $s_{N+1} - s_N = a_{N+1} \ge 0$. The result is then the [[monotone convergence theorem|0FM6N4R0VB]].
If the partial sums are not bounded, they tend to $+\infty$, and one writes $\sum a_n = +\infty$.
The [[comparison test|04Z5G0THN6]], the [[integral test|019QZ6RRVH]] and the [[Cauchy condensation test|0B96KMDDD1]] all rest on this fact.

## Explanations

### L1 human

When all terms are nonnegative, the running total can only go up.
So it either stays below some ceiling and converges, or it grows past every bound.
</document>

<document source="https://math2.tamnd.com/node/0MXB4CKHFW.md" tag="0MXB4CKHFW" kind="definition">
# Hausdorff space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is Hausdorff when for all points $x \neq y$ of $X$ there are [[open sets|01ZQQDCERP]] $U$ and $V$ with $x \in U$, $y \in V$ and $U \cap V = \varnothing$.

## Remarks

Each [[metric space is Hausdorff|0P9DCDES73]]. Subspaces and products of Hausdorff spaces are Hausdorff.
In a Hausdorff space, [[limits of sequences are unique|0D8T984XEE]] and [[compact subsets are closed|0MYFJVMN7M]].
Each Hausdorff space is [[T1|05GQJXBP3F]]. An infinite set with the [[cofinite topology|0B906ZBTMZ]] is T1 but not Hausdorff.
The Hausdorff condition is also called T2.

## Explanations

### L1 human

In a Hausdorff space, you can put any two distinct points into two open sets that do not meet.
Most spaces of analysis, such as $\mathbb{R}^n$, are Hausdorff.
</document>

<document source="https://math2.tamnd.com/node/0MXCP7MXWJ.md" tag="0MXCP7MXWJ" kind="problem">
# Coupon collector's problem

## Problem

There are $n \ge 1$ different types of coupons. Each draw gives a coupon of a uniform random type, independently of the other draws.
Let $T$ be the number of draws needed to collect all $n$ types. Find $E[T]$.

## Remarks

Solution: write $T = t_1 + \dots + t_n$, where $t_i$ is the number of draws to get a new type when $i - 1$ types are known.
Each $t_i$ counts trials up to the first success, with success probability $p_i = (n - i + 1)/n$. So $E[t_i] = 1/p_i$ by the [[mean of the geometric distribution|0DB0R4N5XX]].
By [[linearity of expectation|0P7PTC8DDE]], $E[T] = n \left(\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}\right) = n H_n$, where $H_n$ is the $n$-th harmonic number. So $E[T] = n \ln n + \gamma n + \frac{1}{2} + o(1)$, with the Euler–Mascheroni constant $\gamma \approx 0.5772$.
The $t_i$ are independent, so $\operatorname{Var}(T) = \sum_i (1 - p_i)/p_i^2 < \pi^2 n^2 / 6$.

## Explanations

### L1 human

To collect all $n$ pictures of a series, you need about $n \ln n$ purchases, not $n$.
The last pictures are the slow ones: when only one type is missing, you wait on average $n$ draws for it. For $n = 50$, $E[T] = 50 H_{50} \approx 225$.
</document>

<document source="https://math2.tamnd.com/node/0MXEHSCYCS.md" tag="0MXEHSCYCS" kind="statement">
# Mantel's theorem

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n$ vertices and no cycle of length $3$.
Then $G$ has at most $\lfloor n^2/4 \rfloor$ edges.

## Remarks

The bound is sharp: the [[complete bipartite graph|0V0GC7BQAN]] $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$ has no triangle and has $\lfloor n^2/4 \rfloor$ edges.
The theorem is the case $r = 2$ of [[Turán's theorem|0JG1KZSPFZ]].

## History

Willem Mantel stated the theorem in 1907.

## Explanations

### L1 human

A graph on $n$ vertices without a triangle can have at most about a quarter of $n^2$ edges.
Split the vertices into two equal halves and join each pair across: this graph reaches the bound.
</document>

<document source="https://math2.tamnd.com/node/0MXFJ0J00R.md" tag="0MXFJ0J00R" kind="concept">
# Deficient number

## Summary

A deficient number is a positive integer whose positive divisors other than itself have a sum less than the number.
The [[definition of a deficient number|0G01TS1RRP]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0MXGYCH4SY.md" tag="0MXGYCH4SY" kind="concept">
# Perfect number

## Summary

A perfect number is a positive integer that is equal to the sum of its positive divisors other than itself.
The [[definition of a perfect number|0SB7Q8SYSF]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0MXMS3AE13.md" tag="0MXMS3AE13" kind="definition">
# Cumulative distribution function

## Definition

Let $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
The cumulative distribution function of $X$ is the function $F_X \colon \mathbb{R} \to [0, 1]$ with
$$F_X(x) = P(X \le x) \quad \text{for } x \in \mathbb{R}.$$
More generally, the cumulative distribution function of a [[probability distribution|0QB913WM5E]] $\mu$ on $\mathbb{R}$ is $F(x) = \mu((-\infty, x])$.

## Remarks

For $a < b$, $P(a < X \le b) = F_X(b) - F_X(a)$.
$F_X$ is nondecreasing and right-continuous, with limit $0$ at $-\infty$ and limit $1$ at $+\infty$. Conversely, each function with these properties is the cumulative distribution function of some distribution.
The jump of $F_X$ at $x$ is $P(X = x) = F_X(x) - \lim_{t \to x^-} F_X(t)$.
For a fair die, $F_X(x) = \lfloor x \rfloor / 6$ for $0 \le x < 6$, $F_X(x) = 0$ for $x < 1$, and $F_X(x) = 1$ for $x \ge 6$.

## Explanations

### L1 human

The cumulative distribution function adds up the probability from the left up to the point $x$.
Its graph starts at height $0$ on the far left, never goes down, and reaches height $1$ on the far right.
</document>

<document source="https://math2.tamnd.com/node/0MXN7AFBPE.md" tag="0MXN7AFBPE" kind="statement">
# Open connected subsets of R^n are path-connected

## Statement

Let $n \ge 1$ and let $U \subseteq \mathbb{R}^n$ be open and [[connected|0R37DFJKFW]], for the Euclidean topology.
Then $U$ is [[path-connected|0K0GE6BCYW]] or empty.

## Remarks

Proof: fix $x \in U$, and let $P$ be the set of points of $U$ that a path in $U$ joins to $x$. Each [[open ball|0DC3MPR7C9]] in $U$ is convex, so $P$ and $U \setminus P$ are both open. As $U$ is connected and $x \in P$, we get $P = U$.
The proof works in each locally path-connected space, for example in each open subset of a normed space.
In fact, any two points of $U$ can be joined by a path made of finitely many line segments.

## Explanations

### L1 human

For open sets in $\mathbb{R}^n$, being in one piece and being walkable are the same.
The bad behaviour of the topologist's sine curve cannot happen in an open set, because each point has a small ball around it.
</document>

<document source="https://math2.tamnd.com/node/0MXT6PRZZ6.md" tag="0MXT6PRZZ6" kind="concept">
# Marginal distribution

## Summary

A marginal distribution is the distribution of one variable, or of some of the variables, of a random vector, without regard to the other variables.
The node [[marginal distribution|0AXP2Z5N9H]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0MYDNRT7PP.md" tag="0MYDNRT7PP" kind="definition">
# Characteristic function

## Definition

Let $X$ be a real random variable with law $\mu_X$. Let $i$ be the imaginary unit in the [[complex numbers|09FTAET5Z4]].
The characteristic function of $X$ is the function $\varphi_X \colon \mathbb{R} \to \mathbb{C}$ with
$$\varphi_X(t) = E\left[e^{itX}\right] = E[\cos(tX)] + i \, E[\sin(tX)] = \int_{\mathbb{R}} e^{itx} \, \mu_X(dx).$$

## Remarks

The expectation exists for all $t$, because $|e^{itX}| = 1$. The second form uses [[Euler's formula|04R3CY41VJ]].
Always $\varphi_X(0) = 1$, $|\varphi_X(t)| \le 1$, and $\varphi_X$ is uniformly continuous on $\mathbb{R}$.
For $a, b \in \mathbb{R}$, $\varphi_{aX + b}(t) = e^{itb} \varphi_X(at)$.
If $E[|X|^k] < \infty$, then $\varphi_X$ is $k$ times continuously differentiable and $\varphi_X^{(k)}(0) = i^k E[X^k]$.

## Explanations

### L1 human

The characteristic function is like the moment-generating function, but with an imaginary argument.
This small change makes it finite for every distribution, so it is a tool that always works.
</document>

<document source="https://math2.tamnd.com/node/0MYFJVMN7M.md" tag="0MYFJVMN7M" kind="statement">
# Compact subsets of a Hausdorff space are closed

## Statement

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $K$ be a [[compact subset|00R0X1WGPX]] of $X$.
Then $K$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: fix $y \notin K$. For each $x \in K$, choose disjoint open sets $U_x \ni x$ and $V_x \ni y$. Finitely many $U_{x_1}, \ldots, U_{x_n}$ cover $K$. Then $V_{x_1} \cap \cdots \cap V_{x_n}$ is an open set that contains $y$ and misses $K$.
The Hausdorff condition is needed. With the [[indiscrete topology|0XJ02JBD6S]] on a set with two points, each one-point set is compact but not closed.
Each metric space is Hausdorff, so compact subsets of metric spaces are closed.

## Explanations

### L1 human

In a Hausdorff space, you can separate a point outside $K$ from each point of $K$ by open sets.
Compactness lets you do this with finitely many open sets at once, which gives an open set about the point that misses $K$.
</document>

<document source="https://math2.tamnd.com/node/0MYSA6MYNA.md" tag="0MYSA6MYNA" kind="definition">
# Ideal

## Definition

Let $R$ be a [[ring|0894J980RK]].
A left ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$.

If $R$ is a [[commutative ring|0GYGHKX834]], a left ideal is simply called an ideal.

## Remarks

In $\mathbb{Z}$, the set $n\mathbb{Z}$ of the multiples of $n$ is an ideal for every integer $n$.
Every ring $R$ has the ideals $\{0\}$ and $R$.
An ideal contains $1$ if and only if it is all of $R$.
In a noncommutative ring, one also uses right ideals and [[two-sided ideals|0DDA6CY7GJ]].

## Explanations

### L1 human

An ideal is a set that absorbs multiplication.
The even integers are an example: an even number times any integer is even.
Ideals play the role in rings that normal subgroups play in groups.
</document>

<document source="https://math2.tamnd.com/node/0MZB06ST2B.md" tag="0MZB06ST2B" kind="concept">
# Power series

## Summary

A power series is an infinite sum of terms $a_n (x - c)^n$, with constant coefficients $a_n$ and center $c$. Inside its radius of convergence, it defines a function. The node [[power series|0VP56B4EMX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0N019F6RE9.md" tag="0N019F6RE9" kind="definition">
# Separable extension

## Definition

Let $L/K$ be an [[algebraic extension|09EYMKMSB9]].
An element $a \in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ is a [[separable polynomial|03J8BS7FMV]].
The extension $L/K$ is separable if every element of $L$ is separable over $K$.

## Remarks

An irreducible polynomial over $K$ is separable if and only if its derivative is not the zero polynomial.
So every algebraic extension of a field of characteristic $0$ is separable.
Every algebraic extension of a finite field is separable.
For a prime $p$ and the field $K = \mathbb{F}_p(t)$ of rational functions, the extension $K(t^{1/p})/K$ is not separable: the minimal polynomial $x^p - t$ equals $(x - t^{1/p})^p$.

## Explanations

### L1 human

In a separable extension, the roots of each minimal polynomial are all different.
Problems arise only in characteristic $p$, where a derivative like that of $x^p - t$ can vanish.
</document>

<document source="https://math2.tamnd.com/node/0N0PXCTYA7.md" tag="0N0PXCTYA7" kind="definition">
# Subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] with identity element $e$.
A subgroup of $G$ is a subset $H \subseteq G$ with these three properties.
The identity $e$ is in $H$.
For all $a, b \in H$, the product $ab$ is in $H$.
For each $a \in H$, the inverse $a^{-1}$ is in $H$.
The notation $H \leq G$ means that $H$ is a subgroup of $G$.

## Remarks

A subgroup $H$ is a group with the operation of $G$ restricted to $H$, and its identity is the identity of $G$.
Each group $G$ has the subgroups $\{e\}$ and $G$.
The even integers $2\mathbb{Z}$ form a subgroup of $(\mathbb{Z}, +)$.
The natural numbers do not form a subgroup of $(\mathbb{Z}, +)$, because $-1$ is not a natural number.

## Explanations

### L1 human

A subgroup is a smaller group that lives inside a bigger group and uses the same operation.
To check that a subset is a subgroup, you check that it contains the identity and that it is closed under products and inverses.
For example, the rotations of a square form a subgroup of all the symmetries of the square.
</document>

<document source="https://math2.tamnd.com/node/0N109NG5ZX.md" tag="0N109NG5ZX" kind="statement">
# Well-ordering theorem

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Then each [[set|0SJ3RX9TXA]] $A$ has a [[well-order|0F1D6PK9RZ]].

## Remarks

Another name is Zermelo's theorem.
Over ZF, it is equivalent to the axiom of choice and to [[Zorn's lemma|04RXTQA5ZA]].
A well-order of $\mathbb{R}$ is hard or impossible to visualize, because its construction needs the axiom of choice.
With a well-order, each set allows proofs by [[transfinite induction|0ZHQMQ2C1N]].

## History

Ernst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.

## Explanations

### L1 human

The theorem says that each set has an order with a first element, a next element, and so on, possibly beyond infinity.
For $\mathbb{N}$ the usual order works.
For $\mathbb{R}$ such an order exists, but it is very hard to picture.
</document>

<document source="https://math2.tamnd.com/node/0N10YRNNWV.md" tag="0N10YRNNWV" kind="statement">
# Irrationality of square roots of non-squares

## Statement

Let $n$ be a natural number.
Then $\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.

## Remarks

If $n = m^2$, then $\sqrt{n} = m$ is rational. The content is the other direction.
A proof: if $\sqrt{n} = a/b$, then $a^2 = n b^2$. By [[unique factorization|084HH1NDCB]], each prime has an even exponent in $a^2$ and in $b^2$, so also in $n$. Then $n$ is a square.
For example, $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{12}$ are irrational.
The same argument shows that the $k$-th root of $n$ is irrational unless $n$ is a $k$-th power.

## Explanations

### L1 human

The square root of a whole number is either a whole number or an irrational number.
It is never a fraction such as $\frac{3}{2}$.
</document>

<document source="https://math2.tamnd.com/node/0N1JQXZXHC.md" tag="0N1JQXZXHC" kind="definition">
# Riemann integrable function

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be a function.
The function $f$ is Riemann integrable on $[a, b]$ if there is a real number $I$ with the following property.
For each $\varepsilon > 0$ there is $\delta > 0$ such that $|S(f, P, t) - I| < \varepsilon$ for each tagged partition $(P, t)$ of $[a, b]$ with [[mesh|04GM4NYXJB]] $\| P \| < \delta$.
Here $S(f, P, t)$ is the [[Riemann sum|0HBY48G8G1]] of $f$ for $(P, t)$.

## Remarks

The number $I$ is unique. It is the [[Riemann integral|0QM4TXT3DR]] of $f$.
A Riemann integrable function is bounded.
[[Continuous functions|06SESV7GQY]] and [[monotone functions|04YKAYDQPW]] on $[a, b]$ are Riemann integrable. The [[Dirichlet function|03Z4JC4YRJ]] is not.

## Explanations

### L1 human

A function is Riemann integrable if its Riemann sums settle down to one number as the partitions get finer.
It does not matter which tags we choose in the pieces.
</document>

<document source="https://math2.tamnd.com/node/0N1MY1T0W9.md" tag="0N1MY1T0W9" kind="definition">
# Matrix

## Definition

Let $R$ be a ring, for example a field, and let $m, n \ge 0$ be integers.
An $m \times n$ *matrix* over $R$ is a family $A = (a_{ij})$ of elements $a_{ij} \in R$, indexed by the pairs $(i, j)$ with $1 \le i \le m$ and $1 \le j \le n$.
The element $a_{ij}$ is the *entry* of $A$ in row $i$ and column $j$.
The notation for $A$ is a rectangular array with $m$ rows and $n$ columns:
$$A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}.$$
The set of all $m \times n$ matrices over $R$ is written $M_{m \times n}(R)$ or $R^{m \times n}$.

## Remarks

Two matrices are equal when they have the same size and the same entries in each position.
A $1 \times n$ matrix is a *row vector*, and an $m \times 1$ matrix is a *column vector*.
Mathlib uses arbitrary finite index types in place of $\{1, \dots, m\}$ and $\{1, \dots, n\}$.

## Explanations

### L1 human

A matrix is a table of numbers with a fixed number of rows and columns.
The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ has 2 rows and 3 columns, so it is a $2 \times 3$ matrix.
We name an entry by its row first and then its column.
</document>

<document source="https://math2.tamnd.com/node/0N1ZWFJDN5.md" tag="0N1ZWFJDN5" kind="statement">
# Completeness via Cauchy sequences

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
Then $X$ is [[complete|08EWT7J0SC]] if and only if each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ has a [[convergent|06JYPJ4WHH]] subsequence.

## Remarks

The key fact: if a Cauchy sequence $(x_n)$ has a subsequence that converges to $x$, then $x_n \to x$. For $\varepsilon > 0$, choose $N$ for $\varepsilon / 2$ in the Cauchy condition and a term $x_{n_k}$ with $n_k \ge N$ and $d(x_{n_k}, x) < \varepsilon / 2$.
A consequence: a metric space in which each sequence has a convergent subsequence is complete. Thus each [[sequentially compact|0CARW59WD2]] metric space is complete.

## Explanations

### L1 human

To show that a space is complete, you need not find the limit of a whole Cauchy sequence.
It is enough to find a limit of some subsequence, and then the whole sequence follows it.
</document>

<document source="https://math2.tamnd.com/node/0N288P2TB5.md" tag="0N288P2TB5" kind="concept">
# Radius of convergence

## Summary

Each power series with center $c$ has a unique $R$ in $[0, \infty]$ such that it converges for $|x - c| < R$ and diverges for $|x - c| > R$. This $R$ is its radius of convergence. The node [[radius of convergence|0660QX1VPR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0N28QN4CYA.md" tag="0N28QN4CYA" kind="definition">
# Symmetric matrix

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *symmetric* when $A^T = A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$.
Equivalently, $a_{ij} = a_{ji}$ for all $i$ and $j$.

## Remarks

The matrix $\begin{pmatrix} 1 & 7 \\ 7 & 2 \end{pmatrix}$ is symmetric.
For each $m \times n$ matrix $B$ over a commutative ring, $B^T B$ and $B B^T$ are symmetric.
A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, by the [[spectral theorem|06J4PXRG22]].
Symmetric matrices represent [[symmetric bilinear forms|0AXZNE5XNH]] in a basis.

## Explanations

### L1 human

A symmetric matrix looks the same after we flip it over its main diagonal.
The entry in row $i$, column $j$ equals the entry in row $j$, column $i$.
Distance tables between cities are symmetric, because the distance from $A$ to $B$ equals the distance from $B$ to $A$.
</document>

<document source="https://math2.tamnd.com/node/0N2P0XRTDF.md" tag="0N2P0XRTDF" kind="statement">
# A homomorphism is injective iff its kernel is trivial

## Statement

Let $G$ and $H$ be [[groups|G5S1M9]], let $e_G$ be the identity element of $G$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
Then $\varphi$ is injective if and only if $\ker \varphi = \{e_G\}$.

## Remarks

If $\varphi(a) = \varphi(b)$, then $\varphi(a^{-1}b) = e_H$, so $a^{-1}b \in \ker \varphi$.
More generally, $\varphi(a) = \varphi(b)$ exactly when $a$ and $b$ lie in the same coset of $\ker \varphi$.

## Explanations

### L1 human

To check that a homomorphism is injective, you only need to check one fibre: the elements that go to the identity.
If only the identity goes to the identity, then no two elements can go to the same place.
This is much easier than comparing all pairs of elements.
</document>

<document source="https://math2.tamnd.com/node/0N2SP6K9ZC.md" tag="0N2SP6K9ZC" kind="concept">
# Cumulant generating function

## Summary

The cumulant generating function of a real random variable is the logarithm of its moment-generating function. Its Taylor coefficients at $0$ are the cumulants.
The node [[cumulant generating function|0Q2QC81VTW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0N2WPCC00T.md" tag="0N2WPCC00T" kind="statement">
# The rationals form a field

## Statement

The [[rational numbers|0NCJ5WXN6Z]] $\mathbb{Q}$, with their addition and multiplication, form a field.
That is, $\mathbb{Q}$ is a commutative ring with $0 \neq 1$, and each rational number $q \neq 0$ has a multiplicative inverse $q^{-1}$ with $q \cdot q^{-1} = 1$.

## Remarks

The inverse of $a / b$ with $a \neq 0$ is $b / a$.
The field $\mathbb{Q}$ is the smallest subfield of the real numbers. Each field of characteristic $0$ contains a copy of $\mathbb{Q}$.
The [[integers|0C4JS9RF5M]] are not a field, and $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$.

## Explanations

### L1 human

In the rational numbers, you can add, subtract, multiply, and divide by any number that is not zero.
The result is always a rational number again.
</document>

<document source="https://math2.tamnd.com/node/0N32V0E3XJ.md" tag="0N32V0E3XJ" kind="statement">
# Limits preserve non-strict inequalities

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences that converge to real numbers $a$ and $b$.
Assume that there is a natural number $N$ such that $a_n \le b_n$ for all $n \ge N$.
Then $a \le b$.

## Remarks

Strict inequalities do not pass to the limit. For example, $0 < 1/n$ for all $n \ge 1$, but $\lim 1/n = 0$.
With a constant sequence $b_n = c$, it says: if $a_n \le c$ for large $n$, then $\lim a_n \le c$.
The proof is by contradiction. If $a > b$, take $\varepsilon = (a - b)/2$. Then for large $n$ we get $b_n < a_n$.

## Explanations

### L1 human

If one sequence is always below another, its limit cannot be above the limit of the other.
The limits can be equal, even when the terms are never equal.
</document>

<document source="https://math2.tamnd.com/node/0N3ADS4CHH.md" tag="0N3ADS4CHH" kind="definition">
# Uniqueness quantifier

## Definition

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
The proposition $\exists! x \in X, \, P(x)$ is true when exactly one $a \in X$ makes $P(a)$ true.
In terms of the other quantifiers, it is the proposition
$$\exists x \in X, \, \bigl( P(x) \land \forall y \in X, \, (P(y) \to y = x) \bigr).$$

## Remarks

A proof of $\exists! x, \, P(x)$ usually has two parts: existence, an $x$ with $P(x)$, and uniqueness, if $P(x)$ and $P(y)$ then $x = y$.
For example, $\exists! x \in \mathbb{R}, \, 2x + 1 = 7$ is true, with $x = 3$.

## Explanations

### L1 human

"There is exactly one even prime" says two things.
There is an even prime, namely $2$, and no other number is an even prime.
</document>

<document source="https://math2.tamnd.com/node/0N3ADXHNTQ.md" tag="0N3ADXHNTQ" kind="definition">
# Composite number

## Definition

A composite number is a natural number $n$ such that $n > 1$ and $n$ is not a [[prime number|0SN0MAEKYX]].

## Remarks

Equivalently, $n$ is composite when $n = a b$ for some natural numbers $a$ and $b$ with $1 < a < n$ and $1 < b < n$.
The first composite numbers are $4, 6, 8, 9, 10, 12$.
The number $1$ is neither prime nor composite.

## Explanations

### L1 human

A composite number is a number that you can write as a product of two smaller numbers, each greater than $1$.
For example, $15 = 3 \cdot 5$ is composite.
You can arrange a composite number of dots as a rectangle with at least two rows and at least two columns.
</document>

<document source="https://math2.tamnd.com/node/0N3WFZEFPM.md" tag="0N3WFZEFPM" kind="statement">
# Liouville's theorem on Diophantine approximation

## Statement

Let $\alpha$ be a real [[algebraic number|0MB4TGMXG6]] of degree $d \ge 2$.
Then there is a constant $c > 0$ such that $\left| \alpha - \frac{p}{q} \right| > \frac{c}{q^d}$ for all integers $p$ and $q$ with $q > 0$.
As a consequence, each [[Liouville number|0J1R5NP36S]] is [[transcendental|079EGSD32Y]].

## Remarks

The proof uses the minimal polynomial $f$ of $\alpha$: $q^d f(p/q)$ is a nonzero integer, and the mean value theorem bounds $f(p/q)$ by $|\alpha - p/q|$.
For a Liouville number $x$, the approximations are better than $c / q^d$ for every $d$, so $x$ is not algebraic.
Roth's theorem improves the exponent $d$ to $2 + \varepsilon$ for each $\varepsilon > 0$.

## History

Joseph Liouville proved the bound in 1844. With it, he gave the first proof that transcendental numbers exist.

## Explanations

### L1 human

Algebraic numbers cannot be approximated by fractions too well: the error is at least a constant times $1/q^d$.
A number with better approximations of every order cannot be algebraic.
This gave the first explicit transcendental numbers.
</document>

<document source="https://math2.tamnd.com/node/0N3Y8G340X.md" tag="0N3Y8G340X" kind="example">
# Hilbert's hotel

## Example

A hotel has rooms numbered $1, 2, 3, \dots$, one for each positive integer, and each room has a guest.
A new guest arrives. The manager moves the guest of room $n$ to room $n + 1$ for each $n$. Then room $1$ is free for the new guest.
Next, infinitely many new guests $g_1, g_2, \dots$ arrive. The manager moves the guest of room $n$ to room $2n$, and gives room $2k - 1$ to $g_k$.
In both cases, each guest has a room and no two guests share a room.

## Remarks

The maps $n \mapsto n + 1$ and $n \mapsto 2n$ are [[injections|0PVG1CRAJN]] of $\mathbb{N}_{>0}$ into a proper subset of itself. So the set of rooms is equinumerous with a proper subset.
This cannot happen for a [[finite set|0BBTH3YSZA]].
David Hilbert introduced the hotel in a lecture of 1924 to 1925, and George Gamow made it popular in a book of 1947.

## Explanations

### L1 human

A full finite hotel cannot take a new guest, but a full infinite hotel can.
The example shows that "as many as" for infinite sets does not behave like counting.
</document>

<document source="https://math2.tamnd.com/node/0N4F5FMQZW.md" tag="0N4F5FMQZW" kind="concept">
# Quotient map

## Summary

The quotient map of a vector space onto a quotient space sends each vector to its coset.
It is a surjective linear map whose kernel is the subspace.
</document>

<document source="https://math2.tamnd.com/node/0N4XY7Z3NN.md" tag="0N4XY7Z3NN" kind="concept">
# Upper triangular matrix

## Summary

An upper triangular matrix is a square matrix whose entries below the main diagonal are all zero.
</document>

<document source="https://math2.tamnd.com/node/0N57D1XY6T.md" tag="0N57D1XY6T" kind="definition">
# Pairwise independence

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(A_i)_{i \in I}$ be a family of events.
The family is pairwise independent if $A_i$ and $A_j$ are [[independent|0P0MZ4X682]] for all $i, j \in I$ with $i \ne j$, that is, $P(A_i \cap A_j) = P(A_i) \, P(A_j)$.

## Remarks

[[Mutual independence|0GCVEV6BJA]] implies pairwise independence, but the converse is false. The node [[pairwise independent events that are not mutually independent|0AR4B3K5QD]] gives an example.
The same definition applies to random variables: any two of them are independent.
Pairwise independence is enough for some results.
Examples are the formula for the variance of a sum and the weak law of large numbers with finite variance.

## Explanations

### L1 human

Pairwise independence checks the events two at a time.
It can happen that each pair of events is independent, but a group of three events is not.
</document>

<document source="https://math2.tamnd.com/node/0N5FCV2YV6.md" tag="0N5FCV2YV6" kind="definition">
# Boolean ring

## Definition

A Boolean ring is a ring $R$ with identity such that $x^2 = x$ for every $x \in R$.

## Remarks

In a Boolean ring, $x + x = 0$ for every $x$, and the ring is commutative.
The ring $\mathbb{Z}/2\mathbb{Z}$ is a Boolean ring. The power set of a set is a Boolean ring, with symmetric difference as addition and intersection as multiplication.
Each Boolean ring is a [[Boolean algebra|06GKBTAA5E]] with $x \wedge y = xy$, $x \vee y = x + y + xy$ and $\neg x = 1 + x$. This gives a correspondence between Boolean rings and Boolean algebras.

## Explanations

### L1 human

A Boolean ring is a ring in which squaring changes nothing.
It is the same thing as a Boolean algebra, written with plus and times instead of or and and.
</document>

<document source="https://math2.tamnd.com/node/0N5GP3YT8T.md" tag="0N5GP3YT8T" kind="definition">
# Square root of a positive semidefinite matrix

## Definition

Let $A$ be a [[positive semidefinite|01MMVW8CKC]] $n \times n$ complex Hermitian matrix.
There is exactly one positive semidefinite matrix $B$ with $B^2 = A$.
This matrix is the *square root* of $A$, written $A^{1/2}$ or $\sqrt{A}$.

## Remarks

If $A = U D U^*$ with $U$ unitary and $D$ diagonal with entries $d_i \ge 0$, then $A^{1/2} = U D^{1/2} U^*$, with entries $\sqrt{d_i}$ in $D^{1/2}$.
$A^{1/2}$ commutes with each matrix that commutes with $A$.
Without the condition $B \succeq 0$, a matrix can have many square roots; for example, $I_2$ has infinitely many.

## Explanations

### L1 human

A positive semidefinite matrix has exactly one positive semidefinite square root.
In an orthonormal basis of eigenvectors, we take the square root of each eigenvalue.
This is like the positive square root of a nonnegative number.
</document>

<document source="https://math2.tamnd.com/node/0N5M924N3P.md" tag="0N5M924N3P" kind="definition">
# Falling factorial

## Definition

Let $x$ be an element of a commutative ring, for example a real number, and let $k$ be a natural number.
The falling factorial of $x$ of order $k$ is
$$(x)_k = x (x - 1) (x - 2) \cdots (x - k + 1) = \prod_{i=0}^{k-1} (x - i).$$
For $k = 0$ the product is empty, and $(x)_0 = 1$.

## Remarks

Other notations are $x^{\underline{k}}$ and $P(n, k)$.
For natural numbers $n \ge k$, $(n)_k = n! / (n - k)!$. For $k > n$, $(n)_k = 0$.
For natural numbers $n$ and $k$, $(n)_k$ is the [[number of injections|0X0N833FF8]] from a $k$-element set to an $n$-element set.

## Explanations

### L1 human

The falling factorial starts like the [[factorial|0WFCGVMRRC]] but stops after $k$ factors.
For example, $(5)_3 = 5 \cdot 4 \cdot 3 = 60$ is the number of ways to award gold, silver and bronze among 5 runners.
</document>

<document source="https://math2.tamnd.com/node/0N5N15HWZ7.md" tag="0N5N15HWZ7" kind="concept">
# Cross product

## Summary

The cross product is a binary operation on the vectors of three-dimensional space. It takes two vectors and gives a vector that is perpendicular to both. The length of the result is the area of the parallelogram with the two vectors as sides.
</document>

<document source="https://math2.tamnd.com/node/0N5NF1RGBP.md" tag="0N5NF1RGBP" kind="definition">
# Wronskian

## Definition

Let $I$ be an interval and let $f_1, \dots, f_n : I \to \mathbb{R}$ be $n - 1$ times differentiable functions.
The Wronskian of $f_1, \dots, f_n$ is the function $W(f_1, \dots, f_n) : I \to \mathbb{R}$ given by
$$W(f_1, \dots, f_n)(x) = \det \begin{pmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{pmatrix}.$$

## Remarks

For two functions, $W(f, g) = f g' - f' g$.
Example: $W(\cos, \sin) = \cos^2 + \sin^2 = 1$.
The [[Wronskian test|0MHE4B1QCQ]] uses $W$ to decide whether solutions of a linear equation are linearly independent.

## Explanations

### L1 human

The rows of the matrix are the functions, then their first derivatives, and so on.
The determinant measures whether the vectors of starting data of the functions are independent at the point $x$.
</document>

<document source="https://math2.tamnd.com/node/0N5NJZKKN3.md" tag="0N5NJZKKN3" kind="concept">
# Idempotent element

## Summary

An idempotent element of a ring is an element that is equal to its own square.
The [[definition of an idempotent element|0X4JKZXJ86]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/0N5NR32NQG.md" tag="0N5NR32NQG" kind="statement">
# Classification of conics by the discriminant

## Statement

Let $Q = \{\, (x, y) \in \mathbb{R}^2 : A x^2 + B x y + C y^2 + D x + E y + F = 0 \,\}$ be a non-degenerate [[conic section|00HJCMC2CD]]. Let $\Delta = B^2 - 4AC$ be the discriminant.
Then $Q$ is
- an [[ellipse|0VKCCEXBJ1]] if and only if $\Delta < 0$,
- a [[parabola|04S2GMHX82]] if and only if $\Delta = 0$,
- a [[hyperbola|0WN5G117YC]] if and only if $\Delta > 0$.

Moreover, $Q$ is a [[circle|060W5CBZP6]] if and only if $A = C$ and $B = 0$.

## Remarks

The discriminant is $-4$ times the determinant of $\begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$, the matrix of the quadratic part.
A rotation or a translation of the coordinates does not change $\Delta$ or $A + C$. So the type of a conic does not depend on the choice of Cartesian coordinates.
A hyperbola with $A + C = 0$ is rectangular: its asymptotes are perpendicular.

## Explanations

### L1 human

The terms of degree $2$ decide the shape of a conic. For $x^2 + y^2 = 1$, $\Delta = -4 < 0$ and we get a circle, a special ellipse. For $y = x^2$, $\Delta = 0$ and we get a parabola. For $x y = 1$, $\Delta = 1 > 0$ and we get a hyperbola.
</document>

<document source="https://math2.tamnd.com/node/0N5QW9K7MH.md" tag="0N5QW9K7MH" kind="statement">
# Degree of a simple extension is the degree of the minimal polynomial

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$ of degree $n$.
Then $[K(a) : K] = n$, and $1, a, a^2, \dots, a^{n-1}$ is a basis of $K(a)$ over $K$.

## Remarks

The proof: by [[the structure theorem|0ZN39QDTP9]], $K(a) \cong K[x]/(m_a)$. By polynomial division, the classes of $1, x, \dots, x^{n-1}$ are a basis of $K[x]/(m_a)$.
For example, $[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3$, with the basis $1, \sqrt[3]{2}, \sqrt[3]{4}$.
With the [[tower law|0A7KJ68XHT]], the degree of every element of a finite extension $L/K$ divides $[L : K]$.

## Explanations

### L1 human

The degree of the field you get by adding one algebraic element is the degree of its minimal polynomial.
Each element of $K(a)$ is a polynomial in $a$ of degree less than $n$, in exactly one way.
</document>

<document source="https://math2.tamnd.com/node/0N5ZSZMYH6.md" tag="0N5ZSZMYH6" kind="concept">
# Fermat primality test

## Summary

The Fermat primality test is a probabilistic test of whether an integer $n > 1$ is prime.
It chooses a base $a$ with $1 < a < n - 1$ and computes $a^{n-1} \bmod n$ by [[modular exponentiation|04DR5PM25N]].
If the result is not $1$, then $n$ is composite, by [[Fermat's little theorem|0M25PRN3WQ]]. If the result is $1$, then $n$ is called a probable prime to the base $a$.

## Remarks

A probable prime can be composite. Such a number is a [[Fermat pseudoprime|02M2K4QH9P]].
The [[Carmichael numbers|0TA7WT68RK]] pass the test for each base coprime to them. So repeated tests with many bases cannot detect all composite numbers.
The [[Miller–Rabin primality test|0SZCSYHW60]] is a stronger test of the same kind, without this weakness.

## Explanations

### L1 human

To test $n = 15$ with $a = 2$, compute $2^{14} = 16384 \equiv 4 \pmod{15}$.
The result is not $1$, so $15$ is surely composite. The test does not find a factor; it only proves that one exists.
</document>

<document source="https://math2.tamnd.com/node/0N67NBKQ5T.md" tag="0N67NBKQ5T" kind="definition">
# Vector field

## Definition

Let $S \subseteq \mathbb{R}^n$.
A vector field on $S$ is a [[vector-valued function|018GTRS7A4]] $F \colon S \to \mathbb{R}^n$, $F(x) = (F_1(x), \dots, F_n(x))$.
The vector field is continuous, or of class $C^k$, if each component $F_i$ is [[continuous|0S9EK5MQPE]], or of [[class C^k|0TCBZMEYGN]], on $S$.

## Remarks

One pictures $F$ by an arrow $F(x)$ drawn with its tail at the point $x$.
For a differentiable function $f \colon U \to \mathbb{R}$ on an open set $U$, the gradient $\nabla f$ is a vector field on $U$. Such fields are gradient fields.
The field $F(x, y) = (-y, x)$ on $\mathbb{R}^2$ describes a counterclockwise rotation about the origin. The field $F(x) = x$ points away from the origin.
In $\mathbb{R}^3$ one often writes $F = P \, \mathbf{i} + Q \, \mathbf{j} + R \, \mathbf{k}$, with $P$, $Q$, $R$ the components of $F$.

## Explanations

### L1 human

A vector field attaches an arrow to each point.
Think of the wind on a weather map: at each place there is a wind direction and a wind speed.
</document>

<document source="https://math2.tamnd.com/node/0N681SSTHC.md" tag="0N681SSTHC" kind="concept">
# Simple extension

## Summary

A simple extension is a field extension that is generated by one element.
The [[definition of a simple extension|0P6SBR7YM2]] states it.
</document>

<document source="https://math2.tamnd.com/node/0N6960W0BV.md" tag="0N6960W0BV" kind="concept">
# Eigenvector

## Summary

An eigenvector of a linear map from a space to itself is a nonzero vector that the map sends to a scalar multiple of itself.
</document>

<document source="https://math2.tamnd.com/node/0N6R902FG1.md" tag="0N6R902FG1" kind="concept">
# Monoid

## Summary

A monoid is a semigroup that has an identity element.
Examples are the natural numbers with addition and the maps from a set to itself with composition.
</document>

<document source="https://math2.tamnd.com/node/0N7ATVF2X8.md" tag="0N7ATVF2X8" kind="concept">
# Classification of equilibria of planar linear systems

## Summary

Let $A$ be a real $2 \times 2$ matrix with $\det A \neq 0$, let $\tau = \operatorname{tr} A$ and $\Delta = \det A$, and consider $x' = A x$ in the [[phase plane|05VDGH129F]].
The origin is a saddle if $\Delta < 0$. It is a node if $\Delta > 0$ and $\tau^2 - 4 \Delta \ge 0$, and a spiral, or focus, if $\tau \neq 0$ and $\tau^2 - 4 \Delta < 0$. It is a centre if $\tau = 0$ and $\Delta > 0$.
A node or a spiral is stable for $\tau < 0$ and unstable for $\tau > 0$.

## Remarks

The eigenvalues are the roots of $\lambda^2 - \tau \lambda + \Delta = 0$. A saddle has real eigenvalues of opposite signs. A node has real eigenvalues of the same sign. A spiral or a centre has complex conjugate eigenvalues.
The case $\tau^2 = 4 \Delta$ gives a repeated eigenvalue. The origin is then a star node or a degenerate node.
The trace and the determinant do not change under a change of basis $A \mapsto C^{-1} A C$. So the classification depends only on the system, not on the coordinates.
The [[damped harmonic oscillator|0RNXVVTJTJ]] gives a stable spiral when it is underdamped and a stable node when it is overdamped.

## Explanations

### L1 human

For a linear system in the plane, two numbers, the trace and the determinant, tell the shape of the phase portrait near the origin.
For a node, trajectories run into or out of the origin along straight directions. They turn around it for a spiral, and they go around it in closed loops for a centre.
</document>

<document source="https://math2.tamnd.com/node/0N7ENQRCXN.md" tag="0N7ENQRCXN" kind="concept">
# Kronecker product

## Summary

The Kronecker product of two matrices is the block matrix whose blocks are the entries of the first matrix times the second matrix.
</document>

<document source="https://math2.tamnd.com/node/0N8NGKDZDZ.md" tag="0N8NGKDZDZ" kind="concept">
# Equilibrium point

## Summary

An equilibrium point of a differential equation is a state that gives a constant solution.
For an autonomous system $x' = f(x)$, these are the zeros of $f$. See the [[definition|0G4X36AE0X]].
</document>

<document source="https://math2.tamnd.com/node/0N91N98EG5.md" tag="0N91N98EG5" kind="statement">
# Mean value theorem for integrals

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then there is $c \in [a, b]$ with
$$\int_a^b f(x) \, dx = f(c) \, (b - a).$$

## Remarks

Let $m$ and $M$ be the minimum and the maximum of $f$, from the [[extreme value theorem|0WS1A8E2P9]]. By [[monotonicity|0MBWQKM8QR]], the average $\frac{1}{b - a} \int_a^b f$ lies in $[m, M]$. The [[intermediate value theorem|00KAFBR32E]] gives $c$.
The number $\frac{1}{b - a} \int_a^b f$ is the average value of $f$ on $[a, b]$.
Mathlib has a form for averages, `exists_eq_interval_average`.

## Explanations

### L1 human

A continuous function takes its average value somewhere.
The area under the graph equals the area of a rectangle whose height is a value of $f$.
</document>

<document source="https://math2.tamnd.com/node/0N9E4G2P97.md" tag="0N9E4G2P97" kind="definition">
# Fermat prime

## Definition

A Fermat prime is a [[Fermat number|0R03B54V4Z]] $F_n = 2^{2^n} + 1$ that is a [[prime number|0SN0MAEKYX]].

## Remarks

The numbers $F_0, \dots, F_4$, that is $3, 5, 17, 257, 65537$, are prime. No other Fermat prime is known.
Fermat conjectured that all Fermat numbers are prime. Euler refuted this in 1732 with $F_5 = 641 \cdot 6700417$.
A regular $n$-gon is constructible with ruler and compass if and only if $n$ is a power of $2$ times distinct Fermat primes.
This is the Gauss–Wantzel theorem.

## Explanations

### L1 human

A Fermat prime is a prime of the form $2^{2^n} + 1$.
Only five of them are known, and it is an open problem whether there are more.
</document>

<document source="https://math2.tamnd.com/node/0N9R5NN6TW.md" tag="0N9R5NN6TW" kind="definition">
# Nilpotent element

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $x \in R$ is nilpotent if there is a positive integer $n$ such that $x^n = 0$.

## Remarks

In $\mathbb{Z}/8\mathbb{Z}$, the class of $2$ is nilpotent, because $2^3 = 8 = 0$.
The matrix $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent: its square is the zero matrix.
In a nonzero ring, a nilpotent element is a [[zero divisor|0K26YKNBSA]]: if $n$ is the least integer with $x^n = 0$, then $x \cdot x^{n-1} = 0$ and $x^{n-1} \neq 0$.
An [[integral domain|09S3PFBEPM]] has no nilpotent element except $0$.

## Explanations

### L1 human

A nilpotent element is not zero, but some power of it is zero.
Numbers do not behave like this.
In the integers modulo $8$, the element $2$ does: $2 \cdot 2 \cdot 2 = 8$, and $8$ is $0$ modulo $8$.
</document>

<document source="https://math2.tamnd.com/node/0N9SA25Z3V.md" tag="0N9SA25Z3V" kind="definition">
# Symmetric relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is symmetric when, for all $a, b \in A$, $a \mathrel{R} b$ implies $b \mathrel{R} a$.

## Remarks

Equality is symmetric. The relation $\le$ on $\mathbb{R}$ is not symmetric, because $1 \le 2$ but not $2 \le 1$.
A relation $R$ is symmetric exactly when it is equal to its [[converse|0JCSQNNP2M]].

## Explanations

### L1 human

"Is a sibling of" is symmetric: if Ann is a sibling of Bob, then Bob is a sibling of Ann.
"Is a parent of" is not symmetric.
</document>

<document source="https://math2.tamnd.com/node/0NA0732YAG.md" tag="0NA0732YAG" kind="concept">
# Basis of a topology

## Summary

A basis of a topology is a family of open sets such that every open set is a union of members of the family.
A basis lets you describe a topology by a small set of simple open sets, such as open intervals or open balls.
</document>

<document source="https://math2.tamnd.com/node/0NA4FK7B8S.md" tag="0NA4FK7B8S" kind="example">
# Discrete metric

## Example

Let $X$ be a set.
For $x, y \in X$, let $d(x, y) = 0$ if $x = y$ and $d(x, y) = 1$ if $x \neq y$.
Then $d$ is a [[metric|063W5JMY85]] on $X$, the discrete metric.

## Remarks

For each $x \in X$, the open ball of radius $1/2$ about $x$ is $\{x\}$. Thus each subset of $X$ is open, and the metric gives the [[discrete topology|0S2TEHX4E1]].
A sequence converges in this metric if and only if it is eventually constant.

## Explanations

### L1 human

In the discrete metric, two points are either equal or at distance $1$.
No point is close to another point, so the space has no interesting limits.
</document>

<document source="https://math2.tamnd.com/node/0NA6Q82YR7.md" tag="0NA6Q82YR7" kind="definition">
# Cauchy distribution

## Definition

Let $x_0 \in \mathbb{R}$ and $\gamma > 0$.
The Cauchy distribution with location $x_0$ and scale $\gamma$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \frac{1}{\pi \gamma \left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)} = \frac{\gamma}{\pi \left((x - x_0)^2 + \gamma^2\right)}, \quad x \in \mathbb{R}.$$
The standard Cauchy distribution has $x_0 = 0$ and $\gamma = 1$, with density $\frac{1}{\pi(1 + x^2)}$.

## Remarks

The cumulative distribution function is $F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan\left(\frac{x - x_0}{\gamma}\right)$. The median is $x_0$.
The integral $\int |x| f(x) \, dx$ is infinite, so the distribution has no mean. It also has no variance.
If $X_1, \dots, X_n$ are independent standard Cauchy, then their average $(X_1 + \dots + X_n)/n$ is again standard Cauchy. So the law of large numbers fails without a finite mean.
The ratio $X/Y$ of two independent standard normal random variables is standard Cauchy.

## Explanations

### L1 human

The Cauchy distribution looks like a bell curve, but its tails are much heavier: very large values are not rare.
Shine a lamp at a uniform random angle onto a wall. The point where the light hits the wall has a Cauchy distribution.
</document>

<document source="https://math2.tamnd.com/node/0NACHJYPKW.md" tag="0NACHJYPKW" kind="statement">
# Ore's theorem

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \ge 3$ vertices.
Suppose that $\deg(u) + \deg(v) \ge n$ for any two distinct vertices $u$ and $v$ that are not adjacent.
Then $G$ is [[Hamiltonian|090WPKFABX]].

## Remarks

[[Dirac's theorem|0TC99BRSSB]] is a special case.
The Bondy–Chvátal theorem of 1976 extends it. It says that $G$ is Hamiltonian if and only if its closure is Hamiltonian.
The closure comes from $G$ when you add edges $uv$ with $\deg(u) + \deg(v) \ge n$ again and again.

## History

Øystein Ore proved the theorem in 1960.

## Explanations

### L1 human

Ore's theorem looks only at pairs of vertices that are not neighbours.
If each such pair has many edges together, the graph has a round trip through all vertices.
</document>

<document source="https://math2.tamnd.com/node/0NAHXR1FRH.md" tag="0NAHXR1FRH" kind="concept">
# Holomorphic function

## Summary

A holomorphic function is a complex function that is complex differentiable at each point of an open subset of $\mathbb{C}$, see the [[definition|0DG0Z72CQD]].
Holomorphic functions are the main objects of complex analysis.
They are infinitely differentiable and they are equal to their Taylor series near each point.
</document>

<document source="https://math2.tamnd.com/node/0NAKHQ9XYY.md" tag="0NAKHQ9XYY" kind="statement">
# Apollonius's theorem

## Statement

Let $A$, $B$ and $C$ be points of the Euclidean space $\mathbb{R}^n$, and let $D$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$. Let $|XY|$ denote the [[distance|0NPPGHQNXQ]] between the points $X$ and $Y$.
Then
$$|AB|^2 + |AC|^2 = 2 \left( |AD|^2 + |BD|^2 \right).$$

## Remarks

For a triangle $ABC$ with the side lengths $a = |BC|$, $b = |CA|$, $c = |AB|$, the [[median|0XW8HX8VEV]] from $A$ thus has the length $m_a = \frac{1}{2} \sqrt{2 b^2 + 2 c^2 - a^2}$.
In vector form, with $u = B - A$ and $v = C - A$, the statement is the parallelogram law $\|u + v\|^2 + \|u - v\|^2 = 2\|u\|^2 + 2\|v\|^2$.
For an isosceles triangle with $|AB| = |AC|$, the median $AD$ is perpendicular to $BC$, and the theorem reduces to the [[Pythagorean theorem|0VP3A1290K]].

## History

The theorem is Proposition 122 of Book VII of the Collection of Pappus of Alexandria. It was possibly in the lost work Plane Loci of Apollonius of Perga.

## Explanations

### L1 human

The theorem relates the two sides from $A$ to the median from $A$. If you know the three sides of a triangle, it gives the length of each median. For the right triangle with legs $3$, $4$ and hypotenuse $5$, the median to the hypotenuse has length $\frac{1}{2}\sqrt{18 + 32 - 25} = \frac{5}{2}$.
</document>

<document source="https://math2.tamnd.com/node/0NANDX7V02.md" tag="0NANDX7V02" kind="concept">
# Harmonic number

## Summary

The harmonic numbers are the sums $1 + 1/2 + \cdots + 1/n$ of the reciprocals of the first positive integers. They grow like the natural logarithm of $n$. The node [[harmonic number|0D2Y6G4C4S]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NB0AWGJBW.md" tag="0NB0AWGJBW" kind="concept">
# Inner product

## Summary

An inner product on a real or complex vector space is a function of two vectors.
It is linear in the first vector, conjugate symmetric, and positive on nonzero vectors.
</document>

<document source="https://math2.tamnd.com/node/0NB31CC4SN.md" tag="0NB31CC4SN" kind="example">
# exp(1/z) has an essential singularity at 0

## Example

Let $f(z) = e^{1/z}$ for $z \in \mathbb{C} \setminus \{0\}$, where $e^w$ is the [[complex exponential|054VXHMCA3]].
The function $f$ is holomorphic on $\mathbb{C} \setminus \{0\}$, so $0$ is an isolated singularity of $f$.
Its [[Laurent series|0X3XD2ATB0]] at $0$ is
$$e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n!} z^{-n} = 1 + \frac{1}{z} + \frac{1}{2! \, z^2} + \cdots \qquad (z \neq 0).$$
It has infinitely many nonzero terms with negative powers, so $0$ is an [[essential singularity|0YKDJBV26S]] of $f$.

## Remarks

A direct check: for real $x \to 0^+$, $e^{1/x} \to \infty$, and for real $x \to 0^-$, $e^{1/x} \to 0$. So $f$ has no limit at $0$ and $|f|$ does not tend to $\infty$.
In each punctured disc $0 < |z| < \delta$, $f$ takes each value $w \neq 0$ infinitely often: the solutions of $e^{1/z} = w$ are $z = 1 / (\operatorname{Log} w + 2\pi i k)$ with $k \in \mathbb{Z}$ and $\operatorname{Log} w + 2\pi i k \neq 0$, and these points tend to $0$ as $|k| \to \infty$.
The value $0$ is never taken. This agrees with [[Picard's great theorem|08QA8QTM71]], which allows one exception.
The residue of $f$ at $0$ is the coefficient of $z^{-1}$, which is $1$.

## Explanations

### L1 human

Near $0$, the number $1/z$ is huge and can point in any direction, so $e^{1/z}$ can be huge, tiny, or anything in between.
That is why no single value or infinity describes $e^{1/z}$ near $0$.
</document>

<document source="https://math2.tamnd.com/node/0NB643K60Q.md" tag="0NB643K60Q" kind="definition">
# Partial sum

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a [[series|0JH4MN4K1C]] of real numbers and let $N$ be a natural number.
The $N$-th partial sum of the series is the finite sum
$$s_N = \sum_{n=0}^{N} a_n.$$

## Remarks

The partial sums satisfy $s_0 = a_0$ and $s_N = s_{N-1} + a_N$ for $N \ge 1$.
The terms are the differences of the partial sums: $a_N = s_N - s_{N-1}$ for $N \ge 1$.
For the series $\sum 1/2^n$, the partial sums are $1, 3/2, 7/4, 15/8, \dots$, and $s_N = 2 - 1/2^N$.

## Explanations

### L1 human

The partial sums are the running totals of the series.
You add one more term at each step and record the total.
</document>

<document source="https://math2.tamnd.com/node/0NB689782W.md" tag="0NB689782W" kind="concept">
# Asymptotic stability

## Summary

An equilibrium point is asymptotically stable if it is Lyapunov stable and nearby solutions converge to it as time tends to infinity.
See the [[definition|05Y1JAM3Y5]].
</document>

<document source="https://math2.tamnd.com/node/0NB8VPB93V.md" tag="0NB8VPB93V" kind="definition">
# Composition series

## Definition

Let $G$ be a [[group|G5S1M9]].
A composition series of $G$ is a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is a [[simple group|0YMYMNQ0QZ]].

## Remarks

Equivalently, the inclusions are strict and each $G_{i-1}$ is a maximal proper normal subgroup of $G_i$.
Each finite group has a composition series, but the group $\mathbb{Z}$ has none.
For example, $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 3 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ and $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 2 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ are composition series.

## Explanations

### L1 human

A composition series cuts a group into layers that cannot be cut further.
Each layer is a simple group.
It is like the factorisation of an integer into prime numbers.
</document>

<document source="https://math2.tamnd.com/node/0NBRP6093V.md" tag="0NBRP6093V" kind="definition">
# Divergent series

## Definition

A [[series|0JH4MN4K1C]] $\sum a_n$ of real numbers is divergent if it is not a [[convergent series|0CVVKY7GY2]].
Thus its sequence of partial sums has no limit in $\mathbb{R}$.

## Remarks

The series $1 + 1 + 1 + \cdots$ diverges, and its partial sums tend to $+\infty$.
The series $1 - 1 + 1 - 1 + \cdots$ diverges, and its partial sums oscillate between $1$ and $0$.
The [[harmonic series|0CXTMAY5PX]] diverges, although its terms tend to $0$.
Summation methods, such as Cesàro summation, give values to some divergent series.

## Explanations

### L1 human

A divergent series has no sum in the usual sense.
Its running totals grow without bound or keep jumping around.
</document>

<document source="https://math2.tamnd.com/node/0NC2A3HWGT.md" tag="0NC2A3HWGT" kind="statement">
# The integral of a continuous function is an antiderivative

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be continuous, and let $a \in I$.
Then $F(x) = \int_a^x f(t) \, dt$ is an [[antiderivative|0BW9N5CDSB]] of $f$ on $I$.

## Remarks

The integral exists, since [[continuous functions are integrable|06SESV7GQY]]. The [[first fundamental theorem|063H5KJC69]] applies at each point of $I$.
So every continuous function on an open interval has an antiderivative.
Mathlib states the derivative as `Continuous.deriv_integral`.

## Explanations

### L1 human

Every continuous function has an antiderivative.
The area function from a fixed starting point is one.
</document>

<document source="https://math2.tamnd.com/node/0NC9R64TXP.md" tag="0NC9R64TXP" kind="concept">
# Planar graph

## Summary

A planar graph is a graph that can be drawn in the plane with no crossing edges. Planar graphs are the subject of Euler's formula and of the four colour theorem. The node [[planar graph|029JQEK15B]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NCJ5WXN6Z.md" tag="0NCJ5WXN6Z" kind="object">
# Rational numbers

## Description

A rational number is a number of the form $a / b$, where $a$ and $b$ are [[integers|0C4JS9RF5M]] and $b \neq 0$.
We write $\mathbb{Q}$ for the set of rational numbers.
Two fractions $a / b$ and $c / d$ are the same rational number when $a d = b c$.
With addition and multiplication, $\mathbb{Q}$ is a [[field|0N2WPCC00T]], and with its usual order it is an [[ordered field|09RFKMBQ63]].
The set $\mathbb{Q}$ contains $\mathbb{Z}$, because each integer $a$ is the fraction $a / 1$.

## Remarks

Each rational number has exactly one [[representation in lowest terms|0R64A2CVN4]] $a / b$ with $b > 0$.
The set $\mathbb{Q}$ is countable.
Not every real number is rational: $\sqrt{2}$ is [[irrational|0DZGSDHD92]].

## Explanations

### L1 human

The rational numbers are the fractions, such as $1/2$, $-3/4$ and $7 = 7/1$.
Many fractions name the same number: $1/2$, $2/4$ and $3/6$ are all equal.
In $\mathbb{Q}$, you can divide by any number except $0$.
</document>

<document source="https://math2.tamnd.com/node/0NCPXH194Y.md" tag="0NCPXH194Y" kind="concept">
# Graph colouring

## Summary

A colouring of a graph assigns colours to the vertices so that adjacent vertices get different colours. Colourings model scheduling and other conflict problems. The node [[graph colouring|0874138YP6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NDV5R6Z6X.md" tag="0NDV5R6Z6X" kind="definition">
# Principal part

## Definition

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $f$ be [[holomorphic|0DG0Z72CQD]] on the punctured disc $0 < |z - c| < R$.
Let $f(z) = \sum_{n \in \mathbb{Z}} a_n (z - c)^n$ be the [[Laurent expansion|06WHRARZKY]] of $f$ on this punctured disc.
The principal part of $f$ at $c$ is the series
$$\sum_{n = 1}^{\infty} a_{-n} (z - c)^{-n},$$
the part of the [[Laurent series|0X3XD2ATB0]] with the negative powers of $z - c$.

## Remarks

The principal part converges for each $z \neq c$. The difference $f(z) - \sum_{n \ge 1} a_{-n} (z - c)^{-n}$ extends to a holomorphic function on $|z - c| < R$.
The principal part is $0$ at a [[removable singularity|0P18JC72RE]], it has finitely many nonzero terms at a [[pole|0WPQYW8MTE]], and it has infinitely many at an [[essential singularity|0YKDJBV26S]].
For example, the principal part of $\frac{e^z}{z^2}$ at $0$ is $\frac{1}{z^2} + \frac{1}{z}$.

## Explanations

### L1 human

The principal part is the piece of the function that is responsible for its bad behavior at $c$.
If you subtract it, what is left is holomorphic at $c$.
</document>

<document source="https://math2.tamnd.com/node/0NDVHBVBXB.md" tag="0NDVHBVBXB" kind="concept">
# Linear recurrence with constant coefficients

## Summary

A linear recurrence with constant coefficients makes each term a fixed linear combination of the previous terms. Its solutions have closed forms in terms of the roots of its characteristic polynomial. The node [[linear recurrence with constant coefficients|081THG13JZ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NE1DK7H4T.md" tag="0NE1DK7H4T" kind="concept">
# Right coset

## Summary

A right coset of a subgroup $H$ of a group $G$ is a set $Hg$, with $g \in G$. You get it when you multiply each element of $H$ on the right by $g$.
The right cosets of $H$ partition $G$.
</document>

<document source="https://math2.tamnd.com/node/0NE233WGD1.md" tag="0NE233WGD1" kind="concept">
# Conditional convergence

## Summary

A series converges conditionally when it converges but the series of absolute values diverges. The sum of such a series depends on the order of its terms. The node [[conditional convergence|02YRQXZDCS]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NE2W2XPQJ.md" tag="0NE2W2XPQJ" kind="statement">
# Implicit function theorem for a surjective derivative

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $m \le n$, and let $F \colon U \to \mathbb{R}^m$ be of class $C^1$.
Let $p \in U$ with $F(p) = 0$, and suppose that $DF(p) \colon \mathbb{R}^n \to \mathbb{R}^m$ is surjective.
Then there are $m$ coordinates $x_{i_1}, \dots, x_{i_m}$ and an open neighbourhood $W$ of $p$ with this property. The set $\{x \in W : F(x) = 0\}$ is the graph of a $C^1$ function of the other $n - m$ coordinates.

## Remarks

The derivative is surjective if and only if the [[Jacobian matrix|0YF0VXKXGB]] $J_F(p)$ has rank $m$. Then some $m$ columns of $J_F(p)$ are linearly independent, and these columns choose the coordinates.
After a renumbering of coordinates, the statement is the [[implicit function theorem|0BXBADR94C]] with $y = (x_{i_1}, \dots, x_{i_m})$.
If $DF(x)$ is surjective at each point of the [[level set|0Y5Q8Q9WJ7]] $F^{-1}(0)$, this set is locally a graph near each of its points. So it is a $C^1$ submanifold of dimension $n - m$.
Example: for $F(x, y, z) = x^2 + y^2 + z^2 - 1$, $DF$ is surjective on the unit sphere. So the sphere is locally the graph of a function of two coordinates.

## Explanations

### L1 human

The condition that the derivative is onto means the $m$ equations are independent at the point.
Then $m$ of the variables are determined by the others, and the solution set has $n - m$ free directions.
</document>

<document source="https://math2.tamnd.com/node/0NEGN5CRSC.md" tag="0NEGN5CRSC" kind="concept">
# Self-adjoint operator

## Summary

A self-adjoint operator on a finite-dimensional inner product space is a linear map from the space to itself that equals its adjoint.
</document>

<document source="https://math2.tamnd.com/node/0NF1KGDXX3.md" tag="0NF1KGDXX3" kind="concept">
# Restriction of a function

## Summary

The restriction of a function to a subset of its domain is the function that has the same values but only on that subset.
</document>

<document source="https://math2.tamnd.com/node/0NF6GQNXMY.md" tag="0NF6GQNXMY" kind="concept">
# Curl

## Summary

The curl of a differentiable [[vector field|0299VCW1TH]] in $\mathbb{R}^3$ is a vector field that measures the rotation of the field at each point.
Its direction is the axis of the strongest rotation, by the right-hand rule, and its length measures the rate of rotation.
</document>

<document source="https://math2.tamnd.com/node/0NFAGWH84K.md" tag="0NFAGWH84K" kind="statement">
# Double negation law

## Statement

In classical logic, for each [[proposition|0ZGC7ZESYN]] $p$, the proposition $\neg \neg p$ is [[logically equivalent|06GYGS41JK]] to $p$.
$$\neg \neg p \equiv p$$

## Remarks

The implication $p \to \neg \neg p$ is also valid in intuitionistic logic.
The other direction, $\neg \neg p \to p$, needs classical logic.
This direction is the basis of [[proof by contradiction|0J0XBKG64S]].

## Explanations

### L1 human

"It is not true that $5$ is not prime" says the same as "$5$ is prime".
Two negations cancel each other.
</document>

<document source="https://math2.tamnd.com/node/0NFEG48RDR.md" tag="0NFEG48RDR" kind="statement">
# The minimal polynomial is irreducible

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$.
Then the [[minimal polynomial|0GK1WB9R98]] $m_a$ of $a$ over $K$ is [[irreducible|00X2A9G9W3]] over $K$.

## Remarks

The proof: if $m_a = g h$ with $g, h \in K[x]$ of smaller degree, then $g(a) h(a) = 0$ in the field $L$. So $g(a) = 0$ or $h(a) = 0$, and this contradicts the minimality of $\deg m_a$.
Conversely, a monic irreducible polynomial $f \in K[x]$ with $f(a) = 0$ is the minimal polynomial of $a$.
For example, $x^3 - 2$ is irreducible over $\mathbb{Q}$, so it is the minimal polynomial of $\sqrt[3]{2}$ over $\mathbb{Q}$.

## Explanations

### L1 human

The minimal polynomial cannot be factored over the base field.
If it could, one of the factors would already vanish at the element, and that factor would be a smaller polynomial.
</document>

<document source="https://math2.tamnd.com/node/0NFPJVJJHX.md" tag="0NFPJVJJHX" kind="concept">
# Probability space

## Summary

A probability space is the mathematical model of a random experiment: a sample space, a σ-algebra of events and a probability measure on the events.
The node [[probability space|0ZYBZ34RT2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NHGY0N0JB.md" tag="0NHGY0N0JB" kind="statement">
# Fundamental theorem of calculus

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be continuous.
Then the function $F(x) = \int_a^x f(t) \, dt$ is differentiable on $(a, b)$, with $F'(x) = f(x)$.
If $G$ is continuous on $[a, b]$ and $G' = f$ on $(a, b)$, then
$$\int_a^b f(x) \, dx = G(b) - G(a).$$

## Remarks

The first part is the [[first fundamental theorem|063H5KJC69]]. The second part is the [[second fundamental theorem|0XYAFCN0PT]].
Together they say that differentiation and integration are inverse operations.
The theorem reduces the computation of integrals to the search for [[antiderivatives|0BW9N5CDSB]].

## History

James Gregory published the first statement and proof of a geometric form of the theorem.
Isaac Barrow proved a more general version. Isaac Newton and Gottfried Leibniz then built the calculus around it.

## Explanations

### L1 human

The fundamental theorem connects the two main ideas of calculus.
The derivative of an area function is the height of the graph, and an integral is a change in an antiderivative.
</document>

<document source="https://math2.tamnd.com/node/0NHK9MFFW9.md" tag="0NHK9MFFW9" kind="concept">
# Perfect matching

## Summary

A perfect matching is a matching that covers every vertex of a graph. The theorems of Hall and Tutte say when a graph has one. The node [[perfect matching|05ZPH8KF9K]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NHXGY2C5J.md" tag="0NHXGY2C5J" kind="concept">
# Sample variance

## Summary

The sample variance of a sample measures the spread of its values around the sample mean. With the divisor $n - 1$, it is an unbiased estimator of the variance of the population.
The node [[sample variance|0JN9P4A944]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NK8JTE1NN.md" tag="0NK8JTE1NN" kind="concept">
# von Mangoldt function

## Summary

The von Mangoldt function $\Lambda$ takes the value $\log p$ at each power of a prime $p$, and $0$ at all other positive integers.
The [[definition of the von Mangoldt function|087EMZTHWN]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0NKBMFC18R.md" tag="0NKBMFC18R" kind="definition">
# Double integral

## Definition

Let $R = [a, b] \times [c, d]$ with $a < b$ and $c < d$, and let $f \colon R \to \mathbb{R}$ be bounded.
Choose partitions $a = x_0 < \dots < x_k = b$ and $c = y_0 < \dots < y_l = d$, and let $R_{ij} = [x_{i-1}, x_i] \times [y_{j-1}, y_j]$ with area $\Delta A_{ij} = (x_i - x_{i-1})(y_j - y_{j-1})$.
For points $p_{ij} \in R_{ij}$, the sum $S = \sum_{i, j} f(p_{ij}) \, \Delta A_{ij}$ is a Riemann sum of $f$.
The function $f$ is Riemann integrable on $R$ if there is $I \in \mathbb{R}$ with the following property. For each $\varepsilon > 0$ there is $\delta > 0$ such that $|S - I| < \varepsilon$ for all such sums whose rectangles have diameter less than $\delta$.
Then $I$ is the double integral of $f$ over $R$, written $\iint_R f(x, y) \, dA$ or $\iint_R f(x, y) \, dx \, dy$.

## Remarks

Each [[continuous|0S9EK5MQPE]] function on $R$ is Riemann integrable on $R$.
For a bounded region $D \subseteq R$, one integrates $f \mathbf{1}_D$ over $R$, where $\mathbf{1}_D$ is $1$ on $D$ and $0$ elsewhere. This gives [[integrals over non-rectangular regions|0TXCQ9CST1]].
In practice one computes double integrals as [[iterated integrals|0P4WX7Y502]], by [[Fubini's theorem|0R0T1VWH5X]]. For example, $\iint_{[0, 1]^2} x y \, dA = \int_0^1 \int_0^1 x y \, dy \, dx = 1/4$.
The integral is linear in $f$, and $\iint_R 1 \, dA = (b - a)(d - c)$ is the area of $R$.

## Explanations

### L1 human

Cut the rectangle into small pieces, multiply the area of each piece by a value of $f$ in it, and add.
As the pieces get smaller, these sums approach the double integral.
For $f \ge 0$ this is the volume under the graph of $f$.
</document>

<document source="https://math2.tamnd.com/node/0NM626FH6Q.md" tag="0NM626FH6Q" kind="concept">
# Series

## Summary

A series is the infinite sum of the terms of a sequence. It is studied through its partial sums, the sums of finitely many first terms. The node [[series|0JH4MN4K1C]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0NM6E9VW04.md" tag="0NM6E9VW04" kind="statement">
# Bayes' theorem

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ with $P(A) > 0$ and $P(B) > 0$.
Then
$$P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}.$$

## Remarks

Let $A_1, A_2, \dots$ be a finite or countable partition of $\Omega$ into events of positive probability. With the [[law of total probability|0E5G5FMT2R]], $P(A_j \mid B) = \dfrac{P(B \mid A_j) \, P(A_j)}{\sum_i P(B \mid A_i) \, P(A_i)}$.
In Bayesian language, $P(A)$ is the [[prior probability|00NBBSXRQS]], $P(B \mid A)$ is the likelihood and $P(A \mid B)$ is the [[posterior probability|0KJ7ZS8AJA]].
The proof: both $P(A \mid B) \, P(B)$ and $P(B \mid A) \, P(A)$ are equal to $P(A \cap B)$.

## History

The theorem is named after Thomas Bayes. Pierre-Simon Laplace found it independently and developed it further.

## Explanations

### L1 human

Bayes' theorem turns "the chance of the evidence given the cause" into "the chance of the cause given the evidence".
A medical test tells you how often sick people test positive. Bayes' theorem tells you how likely a person with a positive test is to be sick.
</document>

<document source="https://math2.tamnd.com/node/0NMF95YK8W.md" tag="0NMF95YK8W" kind="statement">
# Variance as the mean of the square minus the square of the mean

## Statement

Let $X$ be a real random variable with $E[X^2] < \infty$. Then
$$\operatorname{Var}(X) = E[X^2] - (E[X])^2.$$

## Remarks

Proof: with $\mu = E[X]$, $(X - \mu)^2 = X^2 - 2\mu X + \mu^2$. By [[linearity of expectation|0P7PTC8DDE]], $E[(X - \mu)^2] = E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2$.
As a corollary, $(E[X])^2 \le E[X^2]$.
For a fair die, $E[X^2] = 91/6$ and $E[X] = 7/2$, so $\operatorname{Var}(X) = 91/6 - 49/4 = 35/12$.
In floating-point arithmetic, this formula can lose much precision when $E[X^2]$ and $(E[X])^2$ are close.

## Explanations

### L1 human

This formula is often the fastest way to compute a variance: find the average of the squares and subtract the square of the average.
</document>

<document source="https://math2.tamnd.com/node/0NMG7W7Q11.md" tag="0NMG7W7Q11" kind="concept">
# Implication

## Summary

Implication is the logical operation "if ... then".
The implication from $p$ to $q$ is false only when $p$ is true and $q$ is false.
</document>

<document source="https://math2.tamnd.com/node/0NMGJJ19HD.md" tag="0NMGJJ19HD" kind="statement">
# Epsilon-net characterisation of total boundedness

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
Then $S$ is [[totally bounded|0WE5BBHGM6]] if and only if for each real $\varepsilon > 0$ there is a finite set $F \subseteq S$ with
$$S \subseteq \bigcup_{y \in F} B(y, \varepsilon).$$

## Remarks

Such a finite set $F$ is an $\varepsilon$-net of $S$. Here $B(y, \varepsilon)$ is an [[open ball|0DC3MPR7C9]].
Proof: an $\varepsilon$-net gives a cover by balls of diameter at most $2\varepsilon$. Conversely, from a cover by sets $A_k$ of diameter less than $\varepsilon$, choose one point of $S$ in each $A_k$ that meets $S$.
The centres can be taken in $S$ or in $X$. Both choices give the same notion.

## Explanations

### L1 human

For each $\varepsilon$, you can place finitely many points in $S$ such that each point of $S$ is within $\varepsilon$ of one of them.
For $[0, 1]$ and $\varepsilon = 0.1$, the points $0, 0.1, 0.2, \ldots, 1$ do the job.
</document>

<document source="https://math2.tamnd.com/node/0NMN00FNW1.md" tag="0NMN00FNW1" kind="definition">
# Transvection

## Definition

Let $R$ be a commutative ring, let $n \ge 2$, let $i \ne j$ be indices in $\{1, \dots, n\}$ and let $c \in R$.
Let $E_{ij}$ be the $n \times n$ matrix with entry $1$ in position $(i, j)$ and $0$ elsewhere.
The *transvection matrix* is
$$T_{ij}(c) = I_n + c E_{ij}.$$

## Remarks

The product $T_{ij}(c) A$ adds $c$ times row $j$ of $A$ to row $i$; this is an [[elementary matrix|01REXW3WWE]] of the row-addition type.
Its determinant is $1$, and its inverse is $T_{ij}(-c)$.
Over a field, the transvections generate the [[special linear group|0V9PMN0RBH]].
In the plane, $\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}$ is a horizontal shear.

## Explanations

### L1 human

A transvection slides each point parallel to one axis, by an amount that depends on another coordinate.
A deck of cards that we push sideways is a picture of a shear.
Areas and volumes do not change, which matches the determinant $1$.
</document>

<document source="https://math2.tamnd.com/node/0NMWG2PHJ0.md" tag="0NMWG2PHJ0" kind="example">
# Birthday problem

## Example

Assume that each person has a birthday on one of $365$ days, and that each day is equally likely.
Assume also that the birthdays of different people are independent.
Let $p(n)$ be the probability that among $n$ people, at least two have the same birthday.
By the [[rule for the complement|0PQEBTYDZM]], for $1 \le n \le 365$,
$$p(n) = 1 - \frac{365 \cdot 364 \cdots (365 - n + 1)}{365^n}.$$
Then $p(22) \approx 0.476$ and $p(23) \approx 0.507$. So $23$ people are enough for a probability above $1/2$.

## Remarks

The answer is surprising because there are many pairs: $23$ people form $\binom{23}{2} = 253$ pairs.
For $n$ much smaller than $365$, $p(n) \approx 1 - e^{-n(n-1)/730}$.
For $n = 70$, $p(n) > 0.999$. For $n \ge 366$, $p(n) = 1$ by the pigeonhole principle.

## Explanations

### L1 human

It is easier to compute the chance that all birthdays are different.
The second person must avoid $1$ day, the third person must avoid $2$ days, and so on. These small losses multiply quickly.
</document>

<document source="https://math2.tamnd.com/node/0NMXDCNNKR.md" tag="0NMXDCNNKR" kind="statement">
# Trace is the sum of the eigenvalues

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Assume that the characteristic polynomial of $A$ splits over $F$ as $p_A(t) = (t - \lambda_1) \cdots (t - \lambda_n)$.
Then
$$\tr A = \lambda_1 + \lambda_2 + \cdots + \lambda_n.$$

## Remarks

The $\lambda_i$ are the [[eigenvalues|0BQAA2BK3Q]] of $A$, each repeated according to its [[algebraic multiplicity|0FY42N02DV]].
The hypothesis always holds over $\C$.
The statement follows by comparing the coefficients of $t^{n-1}$, see [[the trace as a coefficient|0FRYPZ6D0X]].
For example, $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ has eigenvalues $1$ and $3$, and trace $4$.

## Explanations

### L1 human

The trace of a matrix is the sum of its eigenvalues, counted with multiplicity.
For a diagonal matrix this is clear, because the eigenvalues are the diagonal entries.
The statement says that it stays true for every complex matrix.
</document>

<document source="https://math2.tamnd.com/node/0NN5JEMTVE.md" tag="0NN5JEMTVE" kind="example">
# 0.999... equals 1

## Example

The [[decimal expansion|06XY4X9430]] $0.999\dots$, with the digit $9$ at every place after the point, represents the real number $1$.
By definition, $0.999\dots = \sum_{k=1}^{\infty} 9 \cdot 10^{-k}$.
This is a geometric series with first term $9/10$ and ratio $1/10$, thus its sum is $\frac{9/10}{1 - 1/10} = 1$.
Thus the number $1$ has two decimal expansions, $1.000\dots$ and $0.999\dots$.

## Remarks

Another proof: the $n$th truncation is $1 - 10^{-n}$. The supremum of these numbers is $1$, by the [[Archimedean property|0A3GWZ86AY]].
The same happens for each number with a finite expansion. For example, $0.25 = 0.24999\dots$.

## Explanations

### L1 human

Many people feel that $0.999\dots$ is a little less than $1$.
But the difference $1 - 0.999\dots$ is smaller than $10^{-n}$ for every $n$.
The only nonnegative real number with this property is $0$.
Another view: $1/3 = 0.333\dots$, and three times this gives $1 = 0.999\dots$.
</document>

<document source="https://math2.tamnd.com/node/0NNDQVJ54Q.md" tag="0NNDQVJ54Q" kind="statement">
# Closed formula for Catalan numbers

## Statement

Let $n$ be a natural number, and let $C_n$ be the $n$-th [[Catalan number|092DRBFKMG]].
Then
$$C_n = \frac{1}{n + 1} \binom{2n}{n} = \frac{(2n)!}{(n + 1)!\, n!}.$$

## Remarks

The fraction is an integer, because $C_n = \binom{2n}{n} - \binom{2n}{n+1}$.
The numerator $\binom{2n}{n}$ is the [[central binomial coefficient|01WHD9T0V6]].
Example: $C_4 = \frac{1}{5} \binom{8}{4} = \frac{70}{5} = 14$.

## Explanations

### L1 human

Among the $\binom{2n}{n}$ grid paths from $(0, 0)$ to $(n, n)$, exactly one in $n + 1$ stays below the diagonal.
So the Catalan number is the central binomial coefficient divided by $n + 1$.
</document>

<document source="https://math2.tamnd.com/node/0NNQ4KFEX7.md" tag="0NNQ4KFEX7" kind="definition">
# Associativity

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
The operation $\ast$ is associative when $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

## Remarks

Addition and multiplication of real numbers are associative.
Subtraction of integers is not associative: $(1 - 1) - 1 = -1$, but $1 - (1 - 1) = 1$.
Composition of maps from a set to itself is associative.
For an associative operation, each way to put parentheses in $a_1 \ast a_2 \ast \cdots \ast a_n$ gives the same result.

## Explanations

### L1 human

Associativity says that you can do the operations in a chain in any grouping.
For example, $(2 + 3) + 4$ and $2 + (3 + 4)$ are both $9$.
Thus we can write $2 + 3 + 4$ without parentheses.
</document>

<document source="https://math2.tamnd.com/node/0NNQY8WCW6.md" tag="0NNQY8WCW6" kind="statement">
# Schwarz's theorem for C^2 functions

## Statement

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$.
Then for all $i, j \in \{1, \dots, n\}$ and each $a \in U$,
$$\frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) = \frac{\partial^2 f}{\partial x_j \, \partial x_i}(a).$$

## Remarks

A function of class $C^2$ is twice differentiable at each point, so this is a case of the [[symmetry of second derivatives|01V9FPBBVK]].
The result is also called Clairaut's theorem. By induction, for a function of class $C^k$ the [[partial derivatives|02RSXVM1ZN]] of order at most $k$ do not depend on the order of differentiation.
So for $C^k$ functions one can use [[multi-index notation|0PJXBR6QA3]] $\partial^\alpha f$ without ambiguity.

## Explanations

### L1 human

If the second partial derivatives are continuous, then $f_{xy} = f_{yx}$.
This saves work: in a $C^2$ function of $n$ variables, only $n(n+1)/2$ of the $n^2$ second partial derivatives can differ.
</document>

<document source="https://math2.tamnd.com/node/0NNR68BFRY.md" tag="0NNR68BFRY" kind="definition">
# Reduced row echelon form

## Definition

Let $F$ be a field and let $A$ be a matrix over $F$.
The matrix $A$ is in *reduced row echelon form* when these three conditions hold:

1. $A$ is in [[row echelon form|0VYCDGMJJJ]].
2. The leading entry of each nonzero row is $1$.
3. In each column that has a leading entry, all the other entries are $0$.

## Remarks

The matrix $\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is in reduced row echelon form.
Each matrix is [[row equivalent|006BHFWMAP]] to exactly one matrix in reduced row echelon form.
[[Gauss–Jordan elimination|098D6FQN3N]] computes this matrix.
The number of nonzero rows of the reduced row echelon form of $A$ is the [[rank|0ZC9YTXZRY]] of $A$.

## Explanations

### L1 human

Reduced row echelon form is the simplest form that row operations can reach.
Each pivot is a $1$, and it has zeros above it and below it.
From this form we can read the solutions of the system directly.
</document>

<document source="https://math2.tamnd.com/node/0NNRDW10RZ.md" tag="0NNRDW10RZ" kind="concept">
# Zero divisor

## Summary

A zero divisor of a ring is an element that gives zero when you multiply it by some nonzero element.
The [[definition of a zero divisor|0K26YKNBSA]] gives the precise condition.
</document>

<document source="https://math2.tamnd.com/node/0NNRNGKMFM.md" tag="0NNRNGKMFM" kind="definition">
# Moment-generating function

## Definition

Let $X$ be a real random variable.
The moment-generating function of $X$ is the function
$$M_X(t) = E\left[e^{tX}\right], \quad t \in \mathbb{R},$$
with values in $(0, \infty]$.

## Remarks

Always $M_X(0) = 1$. The set of $t$ with $M_X(t) < \infty$ is an interval that contains $0$.
If $M_X$ is finite on an open interval around $0$, then all [[moments|0PH0VHTQS0]] of $X$ are finite and $M_X$ determines the law of $X$.
Some random variables have $M_X(t) = \infty$ for all $t \ne 0$. An example is a Cauchy random variable.
For $X \sim \operatorname{Exp}(\lambda)$, $M_X(t) = \lambda / (\lambda - t)$ for $t < \lambda$, and $M_X(t) = \infty$ for $t \ge \lambda$.

## Explanations

### L1 human

The moment-generating function is a tool that turns a distribution into a function of one real variable.
It is useful because sums of independent variables become products, and moments become derivatives.
</document>

<document source="https://math2.tamnd.com/node/0NNYDD1AVR.md" tag="0NNYDD1AVR" kind="definition">
# Eulerian trail

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] or a [[multigraph|01M5T0KDG7]].
An Eulerian trail of $G$ is a [[trail|062MQGP8M5]] in $G$ that uses every edge of $G$ exactly once.
An Eulerian circuit is an Eulerian trail that starts and ends at the same vertex.

## Remarks

A graph with an Eulerian circuit is called Eulerian.
[[Euler's theorem|020422MDQF]] characterizes the connected graphs with an Eulerian circuit by the parity of the degrees.
The [[Seven Bridges of Königsberg|0YCW781Q9D]] form a multigraph with no Eulerian trail.

## Explanations

### L1 human

An Eulerian trail draws the whole graph in one stroke, without lifting the pen and without going over any line twice.
An Eulerian circuit also ends where it starts.
</document>

<document source="https://math2.tamnd.com/node/0NP5N5PYW5.md" tag="0NP5N5PYW5" kind="example">
# Cantor pairing function

## Example

The Cantor pairing function is the function $\pi \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ with
$$\pi(x, y) = \frac{(x + y)(x + y + 1)}{2} + y.$$
It is a [[bijection|0SNQEF7Z53]].
It lists the pairs along the diagonals $x + y = 0, 1, 2, \dots$: $\pi(0, 0) = 0$, $\pi(1, 0) = 1$, $\pi(0, 1) = 2$, $\pi(2, 0) = 3$.

## Remarks

So $\mathbb{N} \times \mathbb{N}$ is [[countably infinite|0SEZ3JGZP7]]. See [[product of countable sets is countable|08G4XH86JC]].
The value $(x + y)(x + y + 1)/2$ is the number of pairs on the earlier diagonals, and $y$ is the position on the diagonal.

## Explanations

### L1 human

Write the pairs $(x, y)$ in an infinite grid.
Walk through the grid along the short diagonals, one after the other, and number the pairs as you go.
Each pair gets exactly one number, and each number gets exactly one pair.
</document>

<document source="https://math2.tamnd.com/node/0NPPGHQNXQ.md" tag="0NPPGHQNXQ" kind="concept">
# Euclidean distance

## Summary

The Euclidean distance between two points of a Euclidean space is the length of the line segment that joins them. In Cartesian coordinates, the Pythagorean theorem gives a formula for it.
</document>

<document source="https://math2.tamnd.com/node/0NPW1DWECM.md" tag="0NPW1DWECM" kind="definition">
# Algebraic element

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$.
The element $a$ is algebraic over $K$ if there is a nonzero polynomial $f \in K[x]$ with $f(a) = 0$.

## Remarks

The number $\sqrt{2}$ is algebraic over $\mathbb{Q}$, because it is a [[root|00EHX6FWC0]] of $x^2 - 2$.
Every element $a \in K$ is algebraic over $K$, as a root of $x - a$.
The element $a$ is algebraic over $K$ if and only if $[K(a) : K]$ is finite.
An element that is not algebraic is [[transcendental|00SW8B91E7]].

## Explanations

### L1 human

An algebraic element solves a polynomial equation with coefficients in the smaller field.
The algebraic numbers are the complex numbers that are algebraic over $\mathbb{Q}$.
</document>

<document source="https://math2.tamnd.com/node/0NQ1H2X495.md" tag="0NQ1H2X495" kind="concept">
# Quotient topology

## Summary

The quotient topology is the topology on a set of equivalence classes that comes from a topological space.
It formalises the idea of gluing points of a space together.
</document>

<document source="https://math2.tamnd.com/node/0NQ5798SB1.md" tag="0NQ5798SB1" kind="definition">
# Student's t-distribution

## Definition

Let $\nu > 0$ be a real number.
Student's t-distribution with $\nu$ degrees of freedom, written $t_\nu$, is the law on $\mathbb{R}$ with the density
$$f(t) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi}\, \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2}, \qquad t \in \mathbb{R}.$$

## Remarks

Let $Z$ be standard normal and $V \sim \chi^2_\nu$ (see [[chi-squared distribution|0RFMMBHFTV]]), [[independent|0XCE070V8B]]. Then $Z / \sqrt{V / \nu} \sim t_\nu$.
For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n \ge 2$, $\frac{\bar{X}_n - \mu}{S_n / \sqrt{n}} \sim t_{n - 1}$, with the [[sample mean|0QNRBHWV35]] $\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$.
The mean is $0$ for $\nu > 1$, and the variance is $\nu / (\nu - 2)$ for $\nu > 2$. For $\nu = 1$, $t_1$ is the standard [[Cauchy distribution|0NA6Q82YR7]].
As $\nu \to \infty$, the density of $t_\nu$ converges to the standard normal density.



## Explanations

### L1 human

The t-distribution looks like the bell curve, but it gives more probability to values far from $0$.
It takes into account the extra uncertainty when the standard deviation comes from a small sample. With many degrees of freedom, it is almost the normal law.
</document>

<document source="https://math2.tamnd.com/node/0NQMZ89X4A.md" tag="0NQMZ89X4A" kind="concept">
# Transcendental element

## Summary

An element of a field extension is transcendental over the smaller field if it is not algebraic over it.
The [[definition of a transcendental element|00SW8B91E7]] states it.
</document>

<document source="https://math2.tamnd.com/node/0NQSV2EWDR.md" tag="0NQSV2EWDR" kind="concept">
# Winding number

## Summary

The winding number of a closed curve around a point is the integer that counts the counterclockwise turns of the curve around the point. See the [[definition|0X7SBS8E0F]].
</document>

<document source="https://math2.tamnd.com/node/0NR82KMYKD.md" tag="0NR82KMYKD" kind="statement">
# Triangle inequality for integrals

## Statement

Let $f \colon [a, b] \to \mathbb{R}$ be Riemann integrable.
Then $|f|$ is Riemann integrable on $[a, b]$, and
$$\Big| \int_a^b f(x) \, dx \Big| \le \int_a^b |f(x)| \, dx.$$

## Remarks

The inequality follows from $-|f| \le f \le |f|$ and the [[monotonicity of the integral|0MBWQKM8QR]].
The converse of the first claim fails. On $[0, 1]$, let $f = 1$ on the rationals and $f = -1$ elsewhere. Then $|f| = 1$ is integrable, but $f$ is not.
Mathlib states the inequality as `intervalIntegral.norm_integral_le_integral_norm`.

## Explanations

### L1 human

Cancellation can only make an integral smaller in size.
The size of the integral is at most the integral of the size of the function.
</document>

<document source="https://math2.tamnd.com/node/0NR95CTFCZ.md" tag="0NR95CTFCZ" kind="example">
# Petersen graph

## Example

The Petersen graph is the [[graph|0JD5C82QMB]] whose vertices are the $2$-element subsets of $\{1, 2, 3, 4, 5\}$.
Two vertices are adjacent if and only if they are disjoint subsets.

It has $\binom{5}{2} = 10$ vertices and $15$ edges, and it is $3$-[[regular|0KA4S8R39M]].

## Remarks

Its shortest cycle has length $5$. It is not [[Hamiltonian|090WPKFABX]], and it is not [[planar|029JQEK15B]].
Its [[chromatic number|0C5M7AM6FH]] is $3$, and its edges cannot be coloured with $3$ colours.
A common drawing shows it as a pentagon outside, a pentagram inside, and five spokes.

## Explanations

### L1 human

The Petersen graph is a small graph with $10$ vertices and $15$ edges.
It is a standard counterexample: many statements that seem true for all cubic graphs fail for it.
</document>

<document source="https://math2.tamnd.com/node/0NRJGW864D.md" tag="0NRJGW864D" kind="object">
# Quaternion group

## Description

The quaternion group $Q_8$ is the set $\{1, -1, i, -i, j, -j, k, -k\}$ of quaternions, with quaternion multiplication as the operation.
The multiplication follows from the rules $i^2 = j^2 = k^2 = ijk = -1$, where $-1$ commutes with each element.
These rules give $ij = k$, $jk = i$, $ki = j$ and $ji = -k$, $kj = -i$, $ik = -j$.
The group has order $8$, and it is not [[abelian|082GHNPJ53]].

## Remarks

The element $-1$ is the only element of order $2$, and the six elements $\pm i, \pm j, \pm k$ have order $4$.
The [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of $Q_8$ are both $\{1, -1\}$, and $Q_8 / \{1, -1\}$ is the [[Klein four-group|0PMRQJVTME]].
Each subgroup of $Q_8$ is [[normal|0CB0DX5BVV]], although $Q_8$ is not abelian.
The groups $Q_8$ and $D_4$ are not isomorphic, because $D_4$ has five elements of order $2$.

## Explanations

### L1 human

The quaternion group is a small group of order $8$ in which the elements do not commute.
You can think of $i$, $j$, $k$ as three square roots of $-1$ whose products go around in a circle.
It is a good test case, because many simple guesses about groups fail for it.
</document>

<document source="https://math2.tamnd.com/node/0NRQPQX3VA.md" tag="0NRQPQX3VA" kind="statement">
# Composition of continuous functions is continuous

## Statement

Let $D, E \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function with $f(D) \subseteq E$, and let $g \colon E \to \mathbb{R}$.
If $f$ is continuous at $a \in D$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$.
In particular, if $f$ and $g$ are [[continuous|0P27W9RW0G]], then $g \circ f$ is continuous.

## Remarks

Given $\varepsilon > 0$, choose $\eta > 0$ for $g$ at $f(a)$, and then $\delta > 0$ for $f$ at $a$ and the tolerance $\eta$. See the [[epsilon-delta form|0W8YHHGCJG]].
For example, $x \mapsto \sin(x^2)$ is continuous.
Mathlib states it for topological spaces, as `Continuous.comp`.

## Explanations

### L1 human

A small change of $x$ gives a small change of $f(x)$.
This in turn gives a small change of $g(f(x))$.
</document>

<document source="https://math2.tamnd.com/node/0NRT0E2QT2.md" tag="0NRT0E2QT2" kind="statement">
# Connected components are closed

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $x \in X$.
The connected component of $x$ is the [[union|05GK32H72H]] $C(x)$ of all [[connected subsets|0R37DFJKFW]] of $X$ that contain $x$.
Then $C(x)$ is connected and [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: a union of connected sets with a common point is connected, so $C(x)$ is the largest connected set that contains $x$. The [[closure|0DRKT1QJR7]] of $C(x)$ is also connected, so it equals $C(x)$.
The components of $X$ form a partition of $X$.
Components need not be open. In $\mathbb{Q}$, the components are the single points, which are not open.
If $X$ has only finitely many components, then each of them is also open.

## Explanations

### L1 human

The component of a point is the largest piece of the space that contains the point and is in one piece.
Taking the closure cannot make the piece larger, so each component is closed.
</document>

<document source="https://math2.tamnd.com/node/0NRYHJPQG2.md" tag="0NRYHJPQG2" kind="concept">
# Linear map

## Summary

A linear map is a map between vector spaces over the same field that preserves addition and scalar multiplication.
</document>

<document source="https://math2.tamnd.com/node/0NRYPM9Y27.md" tag="0NRYPM9Y27" kind="definition">
# Finite extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is finite if its [[degree|0GS74FGFEK]] $[L : K]$ is finite.
Otherwise $L/K$ is an infinite extension.

## Remarks

A finite extension is not the same as an extension of finite fields. For example, $\mathbb{C}/\mathbb{R}$ is finite, and both fields are infinite.
The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is finite of degree $3$.
The field $K(x)$ of rational functions is an infinite extension of $K$, because $1, x, x^2, \dots$ are linearly independent over $K$.
[[Finite extensions are algebraic|0DFYTNZ6AM]].

## Explanations

### L1 human

A finite extension is one that is a finite-dimensional vector space over the smaller field.
Each element of it is a combination of finitely many fixed elements, with coefficients in the smaller field.
</document>

<document source="https://math2.tamnd.com/node/0NS0TDA4E2.md" tag="0NS0TDA4E2" kind="statement">
# Characteristic function of the normal distribution

## Statement

Let $\mu \in \mathbb{R}$, let $\sigma^2 > 0$ and let $X \sim \mathcal{N}(\mu, \sigma^2)$. Then for each $t \in \mathbb{R}$,
$$\varphi_X(t) = E\left[e^{itX}\right] = \exp\left(i \mu t - \tfrac{1}{2} \sigma^2 t^2\right).$$
In particular, the standard normal distribution has $\varphi(t) = e^{-t^2/2}$.

## Remarks

One proof: the [[moment-generating function|0B1G4E21G7]] $E[e^{zX}] = \exp(\mu z + \sigma^2 z^2/2)$ holds for real $z$. Both sides are entire functions of $z$, so the identity extends to all complex $z$, and one puts $z = it$.
Another proof: $\varphi(t) = e^{-t^2/2}$ for $\mathcal{N}(0, 1)$ is the solution of $\varphi'(t) = -t \varphi(t)$ with $\varphi(0) = 1$. Integration by parts gives this differential equation.
So the standard normal density is, up to the factor $\sqrt{2\pi}$, its own Fourier transform.

## Explanations

### L1 human

The characteristic function of a normal variable is again a bell-shaped function.
A narrow normal distribution, with small $\sigma$, has a wide characteristic function, and conversely.
</document>

<document source="https://math2.tamnd.com/node/0NS0X75RS1.md" tag="0NS0X75RS1" kind="statement">
# Derivative of a map into R^n by components

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, let $a \in U$, and let $L \colon \mathbb{R}^n \to \mathbb{R}^m$ be linear with components $L_1, \dots, L_m$.
Then $f$ is [[differentiable|0135XZZQ8M]] at $a$ with $Df(a) = L$ if and only if, for each $j \in \{1, \dots, m\}$, the component $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$.

## Remarks

So you can differentiate a vector-valued map one component at a time. The derivative of $f_j$ is row $j$ of the [[Jacobian matrix|047NDTEPCV]] of $f$.
The proof compares norms: each $|v_j| \le \|v\|$, and $\|v\| \le \sqrt{m} \max_j |v_j|$, for $v \in \mathbb{R}^m$.
This is the case of [[a map into a product|0QQMXFTX0A]] with $m$ factors equal to $\mathbb{R}$.

## Explanations

### L1 human

The map $f(t) = (\cos t, \sin t)$ has the derivative $h \mapsto (-h \sin t, \, h \cos t)$ at $t$.
You get it by differentiating $\cos t$ and $\sin t$ separately.
</document>

<document source="https://math2.tamnd.com/node/0NS1YTGM19.md" tag="0NS1YTGM19" kind="definition">
# Product of two subgroups

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[subgroups|0X1JXT47GA]] of $G$.
The product of $H$ and $K$ is the subset $HK = \{hk : h \in H, k \in K\}$ of $G$.

## Remarks

The set $HK$ is a subgroup of $G$ if and only if $HK = KH$.
It need not be a subgroup. In the symmetric group $S_3$, two distinct subgroups of order $2$ have a product with $4$ elements, and $4$ does not divide $6$.
When $HK$ is a subgroup, it is the [[subgroup generated by|0RP7VWQ0YN]] $H \cup K$.

## Explanations

### L1 human

To multiply two subgroups, multiply each element of the first by each element of the second.
The result contains both subgroups, but it may fail to be a subgroup, because a product $hk h'k'$ need not have the form $h''k''$.
In an abelian group, the product of two subgroups is always a subgroup.
</document>

<document source="https://math2.tamnd.com/node/0NS2QYVC4Y.md" tag="0NS2QYVC4Y" kind="definition">
# Gradient

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].
The gradient of $f$ at $a$ is the unique vector $\nabla f(a) \in \mathbb{R}^n$ such that
$$Df(a)(v) = \nabla f(a) \cdot v \quad \text{for each } v \in \mathbb{R}^n,$$
where $\cdot$ is the dot product of [[Euclidean space|0XPR1E85W0]].
In coordinates, it is the vector of [[partial derivatives|09EGC6F8PT]]
$$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \dots, \frac{\partial f}{\partial x_n}(a)\right).$$
Other notations are $\operatorname{grad} f(a)$ and $\nabla f(a)$, where $\nabla$ is the [[del operator|0H6DP8CHND]].

## Remarks

The vector exists and is unique because each linear map $\mathbb{R}^n \to \mathbb{R}$ is $v \mapsto w \cdot v$ for exactly one $w \in \mathbb{R}^n$.
For $f(x, y) = x^2 + 3 x y$, the gradient is $\nabla f(x, y) = (2x + 3y, \, 3x)$.
A gradient is a column vector, and the derivative $Df(a)$ is the row vector of the same entries. Some books do not make this distinction.
The map $a \mapsto \nabla f(a)$ is a [[vector field|0299VCW1TH]] on $U$, the gradient field of $f$.

## Explanations

### L1 human

The gradient collects all slopes of $f$ along the axes into one arrow.
On a map of a hill, the gradient at a point is an arrow that points straight uphill. The arrow is longer where the hill is steeper.
</document>

<document source="https://math2.tamnd.com/node/0NS58JP68E.md" tag="0NS58JP68E" kind="concept">
# Construction of the integers from the natural numbers

## Summary

The integers can be constructed as classes of pairs of [[natural numbers|0MWKJ44V5A]].
The idea is that the pair $(a, b)$ stands for the difference $a - b$.
On the set $\mathbb{N} \times \mathbb{N}$, define $(a, b) \sim (c, d)$ when $a + d = b + c$. This relation is an equivalence relation.
The set $\mathbb{Z}$ of [[integers|0C4JS9RF5M]] is the set of the equivalence classes $[(a, b)]$.
The operations on classes are $[(a, b)] + [(c, d)] = [(a + c, b + d)]$ and $[(a, b)] \cdot [(c, d)] = [(a c + b d, a d + b c)]$.
These operations do not depend on the choice of the pairs in the classes.

## Remarks

The negative of $[(a, b)]$ is $[(b, a)]$.
The map $n \mapsto [(n, 0)]$ embeds $\mathbb{N}$ in $\mathbb{Z}$, and it respects addition and multiplication.
Each class contains a pair of the form $(n, 0)$ or $(0, n)$. These classes are the integers $n$ and $-n$.

## Explanations

### L1 human

The pairs $(5, 2)$, $(4, 1)$ and $(3, 0)$ all stand for the integer $3$, because each first number is $3$ more than the second.
The pairs $(2, 5)$ and $(0, 3)$ stand for $-3$.
The construction makes new numbers from the natural numbers only, without the assumption that negative numbers already exist.
</document>

<document source="https://math2.tamnd.com/node/0NS8BP96QX.md" tag="0NS8BP96QX" kind="definition">
# Local minimum

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
The function $f$ has a local minimum at $a$ if there is $\delta > 0$ with $f(x) \ge f(a)$ for all $x \in D$ with $|x - a| < \delta$.
It is a strict local minimum if $f(x) > f(a)$ for all such $x \ne a$.

## Remarks

So $f$ has a local minimum at $a$ exactly when $-f$ has a [[local maximum|00999TYPXB]] at $a$.
For example, $x^2$ has a strict local minimum at $0$.
Mathlib defines it as `IsLocalMin`.

## Explanations

### L1 human

A local minimum is the bottom of a valley of the graph.
Nearby values are not lower.
</document>

<document source="https://math2.tamnd.com/node/0NSD1TPHR0.md" tag="0NSD1TPHR0" kind="definition">
# Meromorphic function

## Definition

Let $U \subseteq \mathbb{C}$ be an open set.
A function $f$ is meromorphic on $U$ if there is a set $P \subseteq U$ with these properties.

1. $P$ has no accumulation point in $U$.
2. $f$ is [[holomorphic|0DG0Z72CQD]] on $U \setminus P$.
3. Each point of $P$ is a [[pole|0WPQYW8MTE]] of $f$.

## Remarks

Rational functions $p/q$, with polynomials $p$ and $q \neq 0$, are meromorphic on $\mathbb{C}$. So are $\tan z$, $\frac{1}{\sin z}$ and the Gamma function.
If $U$ is a [[domain|097KNJFJ7V]], $g$ and $h$ are holomorphic on $U$, and $h$ is not identically zero, then $g/h$ is meromorphic on $U$. The meromorphic functions on a domain form a field.
The function $e^{1/z}$ is not meromorphic on $\mathbb{C}$, because its singularity at $0$ is [[essential|0YKDJBV26S]].
The set $P$ can be infinite. It is countable, and each compact subset of $U$ contains only finitely many points of $P$.

## Explanations

### L1 human

A meromorphic function is holomorphic except at some isolated points, where it has poles.
For example, $\frac{1}{z^2 + 1}$ is holomorphic on $\mathbb{C}$ except at $i$ and $-i$, where it goes to infinity.
</document>

<document source="https://math2.tamnd.com/node/0NSR9V8TED.md" tag="0NSR9V8TED" kind="concept">
# Normal space

## Summary

A normal space is a topological space in which any two disjoint closed sets can be put into disjoint open sets.
In normal spaces, there are many continuous real functions, by Urysohn's lemma.
</document>

<document source="https://math2.tamnd.com/node/0NT64JV96A.md" tag="0NT64JV96A" kind="concept">
# Ordinary differential equation

## Summary

An ordinary differential equation is a [[differential equation|003R0V5M7V]] for an unknown function of one variable.
It relates the function, its derivatives and the variable. See the [[definition|0P97V20RW7]].
</document>

<document source="https://math2.tamnd.com/node/0NTF6DMVAP.md" tag="0NTF6DMVAP" kind="concept">
# Direct proof

## Summary

A direct proof of an [[implication|0ZNX6DQ3EJ]] $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps.
The steps use axioms, definitions and theorems that are already proved.

## Remarks

Example: if $n$ is odd, then $n^2$ is odd. Write $n = 2k + 1$. Then $n^2 = 2(2k^2 + 2k) + 1$, which is odd.
When a direct proof is hard, a [[proof by contrapositive|0BB5TPGQ1N]] or a [[proof by contradiction|0J0XBKG64S]] can help.

## Explanations

### L1 human

In a direct proof you go straight from the hypothesis to the conclusion.
You write down what the hypothesis means, then you compute or reason until you reach the conclusion.
</document>

<document source="https://math2.tamnd.com/node/0NTM3J5MFE.md" tag="0NTM3J5MFE" kind="statement">
# Discrete Grönwall inequality

## Statement

Let $(u_k)_{k \ge 0}$ be a sequence of real numbers with $u_0 \ge 0$, and let $(b_k)_{k \ge 0}$ and $(c_k)_{k \ge 0}$ be sequences of non-negative real numbers.
Assume that $u_{k+1} \le (1 + c_k)\, u_k + b_k$ for each $k \ge 0$.
Then for each $n \ge 0$,
$$u_n \le \left( u_0 + \sum_{k=0}^{n-1} b_k \right) \exp\left( \sum_{k=0}^{n-1} c_k \right).$$

## Remarks

Proof idea: by induction, $u_n \le u_0 \prod_{k<n} (1 + c_k) + \sum_{k<n} b_k \prod_{k<j<n} (1 + c_j)$. Then use $1 + c \le e^{c}$.
It is the discrete analogue of [[Grönwall's inequality|07EJHQYR9A]], with sums in place of integrals.
It bounds the growth of errors in numerical methods such as the [[Euler method|0JARTXX4XP]].

## Explanations

### L1 human

At each step, the quantity grows at most by a factor $1 + c_k$ and an extra amount $b_k$.
The inequality says that the total growth is at most exponential in the sum of the rates.
</document>

<document source="https://math2.tamnd.com/node/0NTSE7EREA.md" tag="0NTSE7EREA" kind="statement">
# Exterior angle theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]]. Let $D$ be a point such that $C$ lies strictly between $B$ and $D$ on the line $BC$.
Then the exterior angle at $C$ is the sum of the two interior angles at the other vertices:
$$\angle ACD = \angle BAC + \angle CBA.$$

## Remarks

The angles $\angle ACD$ and $\angle ACB$ add up to $\pi$. Thus the statement follows from the [[angle sum of a triangle|05MDKQBEAF]].
A consequence is that the exterior angle is greater than each of the two remote interior angles. This weaker form does not need the parallel postulate.

## History

In Book I of Euclid's Elements, the weaker form is Proposition 16. The form with the sum is part of Proposition 32 of the same book. Many school books call the form with the sum the exterior angle theorem.

## Explanations

### L1 human

Extend one side of a triangle past a corner. The angle outside the triangle at that corner is the exterior angle. It is exactly as large as the two far angles of the triangle together.
</document>

<document source="https://math2.tamnd.com/node/0NTWN57GM3.md" tag="0NTWN57GM3" kind="concept">
# Multiplicative function

## Summary

A multiplicative function is an arithmetic function with $f(1) = 1$ that takes products of coprime integers to products of values.
The [[definition of a multiplicative function|0AP80TT660]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0NV6E43P4A.md" tag="0NV6E43P4A" kind="statement">
# Z/pZ is a field for p prime

## Statement

Let $n \ge 2$ be an integer.
The ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is a [[field|0QA8CHH8MG]] if and only if $n$ is a prime number.

## Remarks

If $n = a b$ with $1 < a, b < n$, then the classes of $a$ and $b$ are nonzero and their product is $0$. So $\mathbb{Z}/n\mathbb{Z}$ is not an integral domain.
If $n = p$ is prime and $p \nmid a$, then Bézout's identity gives $a x + p y = 1$ for some integers $x, y$. So $x$ is an inverse of $a$ modulo $p$.
The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$. It is the smallest [[finite field|08RTDHX1SK]] of characteristic $p$.

## Explanations

### L1 human

Modulo $5$, every nonzero class has an inverse: $1 \cdot 1 = 1$, $2 \cdot 3 = 6 \equiv 1$ and $4 \cdot 4 = 16 \equiv 1$.
Modulo $6$, the class of $2$ has no inverse, because $2 x$ is always even.
Division works modulo $n$ exactly when $n$ is prime.
</document>

<document source="https://math2.tamnd.com/node/0NVPNC12JY.md" tag="0NVPNC12JY" kind="definition">
# Open cover

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $A \subseteq X$.
An open cover of $A$ is a family $(U_i)_{i \in I}$ of [[open subsets|01ZQQDCERP]] of $X$ with $A \subseteq \bigcup_{i \in I} U_i$.
A subcover is a subfamily $(U_j)_{j \in J}$, with $J \subseteq I$, that is still an open cover of $A$.
A subcover is finite when $J$ is a finite set.

## Remarks

The intervals $(-n, n)$ for $n \ge 1$ form an open cover of $\mathbb{R}$. It has no finite subcover.
The intervals $(1/n, 1)$ for $n \ge 2$ form an open cover of $(0, 1)$ with no finite subcover.
A set $K$ is [[compact|00R0X1WGPX]] when each open cover of $K$ has a finite subcover.

## Explanations

### L1 human

An open cover is a collection of open sets that together cover the whole set.
You can think of them as patches, and a finite subcover uses only finitely many patches to cover the set.
</document>

<document source="https://math2.tamnd.com/node/0NW1JP7TBR.md" tag="0NW1JP7TBR" kind="statement">
# Multiplicativity of the Legendre symbol

## Statement

Let $p$ be an odd prime, and let $a$ and $b$ be integers.
Then
$$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right) \left(\frac{b}{p}\right),$$
where the symbols are [[Legendre symbols|01J8AMV9RS]].

## Remarks

The proof follows from [[Euler's criterion|0M7DEDKSYB]], because $(ab)^{(p-1)/2} = a^{(p-1)/2} b^{(p-1)/2}$.
In words: modulo $p$, the product of two nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue.
For example, $3$ and $5$ are nonresidues modulo $7$, and $15 \equiv 1 \pmod 7$ is a residue.
So the map $a \mapsto \left(\frac{a}{p}\right)$ is a completely multiplicative function of $a$.

## Explanations

### L1 human

To decide whether a product is a square modulo $p$, you can test each factor.
Two "no" answers give "yes", just as two negative signs give a positive sign.
</document>

<document source="https://math2.tamnd.com/node/0NW6CH38YR.md" tag="0NW6CH38YR" kind="concept">
# T1 space

## Summary

A T1 space is a topological space in which, for any two distinct points, each one has an open set that misses the other.
Equivalently, every one-point set is closed.
</document>

<document source="https://math2.tamnd.com/node/0NWXRAMEE7.md" tag="0NWXRAMEE7" kind="statement">
# Second Borel–Cantelli lemma

## Statement

Let $A_1, A_2, \dots$ be [[mutually independent|0GCVEV6BJA]] events in a probability space $(\Omega, \mathcal{F}, P)$ with $\sum_{n=1}^{\infty} P(A_n) = \infty$.
Then
$$P\left(\limsup_{n \to \infty} A_n\right) = 1.$$
So almost surely, infinitely many of the events $A_n$ occur.

## Remarks

Proof: the complement of $\limsup_n A_n$ is $\bigcup_n \bigcap_{m \ge n} A_m^c$. By independence and $1 - x \le e^{-x}$, $P\left(\bigcap_{m = n}^{N} A_m^c\right) = \prod_{m=n}^{N} (1 - P(A_m)) \le \exp\left(-\sum_{m=n}^{N} P(A_m)\right)$.
The right side tends to $0$ as $N \to \infty$. So each $\bigcap_{m \ge n} A_m^c$ has probability $0$, and so does their union.
Independence is necessary. For $A_n = A$ with $0 < P(A) < 1$ for all $n$, the sum diverges but $P(\limsup_n A_n) = P(A) < 1$.
With the [[first lemma|0T6WS7TT77]], it follows that for independent events, $P(\limsup_n A_n)$ is $0$ or $1$.

## Explanations

### L1 human

If independent events have chances that add up to infinity, then infinitely many of them happen, with probability $1$.
For example, in an infinite sequence of tosses of a fair coin, the pattern "ten heads in a row" appears infinitely often.
</document>

<document source="https://math2.tamnd.com/node/0NX5NWAFC9.md" tag="0NX5NWAFC9" kind="statement">
# Weierstrass M-test

## Statement

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$.
Assume that there are real numbers $M_n \ge 0$ with $|f_n(x)| \le M_n$ for all $n$ and all $x \in S$, and $\sum_{n=0}^{\infty} M_n < \infty$.
Then the [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges absolutely and uniformly on $S$.

## Remarks

Absolute convergence at each $x$ follows from the [[comparison test|04Z5G0THN6]]. For $m < n$, $|s_n(x) - s_m(x)| \le \sum_{k=m+1}^{n} M_k$. So the partial sums are [[uniformly Cauchy|01XAY17QA3]].
For example, $\sum \frac{\sin(n x)}{n^2}$ converges uniformly on $\mathbb{R}$, with $M_n = 1/n^2$.
Mathlib states it as `tendstoUniformlyOn_tsum`.

## Explanations

### L1 human

If each function is bounded by a number, and these numbers have a finite sum, then the series of functions converges uniformly.
The bounds do not depend on $x$, so the convergence is the same everywhere.
</document>

<document source="https://math2.tamnd.com/node/0NXGPJWNFN.md" tag="0NXGPJWNFN" kind="statement">
# Sum of independent normal random variables

## Statement

Let $X \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal{N}(\mu_2, \sigma_2^2)$ be [[independent|0E9502NT92]], with $\sigma_1^2, \sigma_2^2 > 0$. Then
$$X + Y \sim \mathcal{N}(\mu_1 + \mu_2, \, \sigma_1^2 + \sigma_2^2).$$

## Remarks

Proof with [[characteristic functions|0NS0TDA4E2]]: $\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t) = \exp(i(\mu_1 + \mu_2)t - (\sigma_1^2 + \sigma_2^2) t^2 / 2)$. By the [[uniqueness theorem|0T8BYAFEXX]], this fixes the law.
By induction, a sum of $n$ independent normal variables is normal. Its mean is the sum of the means and its variance is the sum of the variances.
With the rule for $aX + b$, every linear combination $a_1 X_1 + \dots + a_n X_n$ of independent normal variables is normal, if some $a_i \ne 0$.
The independence condition is necessary. Two normal variables that are not jointly normal can have a sum that is not normal.

## Explanations

### L1 human

Adding independent normal errors gives another normal error. The means add, and the variances add, but the standard deviations do not add.
</document>

<document source="https://math2.tamnd.com/node/0NXJM01QX1.md" tag="0NXJM01QX1" kind="definition">
# Euclidean isometry

## Definition

Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$.
A Euclidean isometry of $\mathbb{R}^n$ is a map $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with
$$d(f(p), f(q)) = d(p, q) \quad \text{for all } p, q \in \mathbb{R}^n.$$

## Remarks

Each Euclidean isometry is an [[affine map|0JF0TVYGWT]] $f(x) = A x + b$, with an orthogonal $n \times n$ matrix $A$ and a vector $b \in \mathbb{R}^n$. Thus $f$ is a bijection, and its inverse is also an isometry.
The isometry is direct if $\det A = 1$ and indirect if $\det A = -1$. The direct isometries keep the orientation of figures.
Mathlib's `AffineIsometryEquiv` is an affine bijection that keeps distances, between normed affine spaces.

## Explanations

### L1 human

An isometry moves the plane or the space without stretching, bending or tearing it. Lengths and angles do not change. In the plane, examples are a shift, a turn about a point and a mirror image in a line.
</document>

<document source="https://math2.tamnd.com/node/0NXWQK2Y56.md" tag="0NXWQK2Y56" kind="concept">
# Adjoining a root of a polynomial

## Summary

Let $f$ be an irreducible polynomial over a field $K$.
To adjoin a root of $f$ to $K$, one forms the field $K[x]/(f)$, in which the class of $x$ is a root of $f$.
The [[definition of this construction|0M893X7YY7]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0NYM0ZGAK9.md" tag="0NYM0ZGAK9" kind="statement">
# Theorem on friends and strangers

## Statement

Colour each edge of the [[complete graph|03WKB2H474]] $K_6$ red or blue.
Then there are three vertices such that the three edges between them have the same colour.

## Remarks

Proof: a vertex $v$ has $5$ edges, so at least $3$ of them have one colour, say blue, by the [[pigeonhole principle|0HGVCWQGY3]]. If an edge between their ends is blue, it closes a blue triangle with $v$. Otherwise these three ends form a red triangle.
The number $6$ is the least possible: $K_5$ has a colouring with no triangle of one colour.
Thus the [[Ramsey number|01DSVCRF6X]] $R(3, 3)$ is $6$.

## Explanations

### L1 human

At any party of six people, there are three people who all know each other, or three who are all strangers to each other.
With five people this can fail.
</document>

<document source="https://math2.tamnd.com/node/0NYZ2M45G9.md" tag="0NYZ2M45G9" kind="definition">
# Closed map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a closed map when for each [[closed set|0G5AV9Q4RR]] $F$ of $X$, the [[image|0Q3F20XB0A]] $f(F)$ is closed in $Y$.

## Remarks

A continuous map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is closed. This gives [[continuous bijections from compact to Hausdorff spaces are homeomorphisms|0430ETCMKD]].
The projection $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is open but not closed: it maps the closed set $\{(x, y) : xy = 1\}$ to $\mathbb{R} \setminus \{0\}$.
A map can be closed but not open. A constant map $\mathbb{R} \to \mathbb{R}$ is closed, but the image of $\mathbb{R}$ is a point, which is not open.

## Explanations

### L1 human

A closed map keeps closed sets closed when you push them forward.
Open maps and closed maps are different: a map can have one property without the other.
</document>

<document source="https://math2.tamnd.com/node/0NZ8JA25WK.md" tag="0NZ8JA25WK" kind="concept">
# Module over a ring

## Summary

A module over a ring is an abelian group with a multiplication by the elements of the ring.
This multiplication satisfies the same axioms as for a vector space.
</document>

<document source="https://math2.tamnd.com/node/0NZ8MDBN6Q.md" tag="0NZ8MDBN6Q" kind="definition">
# Adjacency matrix

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $\{v_1, \dots, v_n\}$.
The adjacency matrix of $G$ is the $n \times n$ matrix $A = (a_{ij})$ with
$$a_{ij} = \begin{cases} 1 & \text{if } v_i \sim v_j, \\ 0 & \text{otherwise.} \end{cases}$$

## Remarks

The matrix $A$ is symmetric and has zeros on the diagonal. The sum of row $i$ is the [[degree|0T0F6CCS9W]] of $v_i$.
The entry $(A^k)_{ij}$ is the number of [[walks|0VJZM0F9YM]] of length $k$ from $v_i$ to $v_j$.
Another order of the vertices gives the matrix $P^{T} A P$ for a permutation matrix $P$.

## Explanations

### L1 human

The adjacency matrix is a table of zeros and ones. A $1$ in row $i$ and column $j$ means that vertices $i$ and $j$ are joined.
For a triangle, every entry off the diagonal is $1$.
</document>

<document source="https://math2.tamnd.com/node/0NZVJYZVTZ.md" tag="0NZVJYZVTZ" kind="statement">
# Angle-angle similarity

## Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean space $\mathbb{R}^n$, and let $\angle$ denote the [[angle at a point|0GJAJC6TNZ]].
If $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$, then $ABC$ and $A'B'C'$ are [[similar|0YF6QXXWB5]].

## Remarks

By the [[angle sum of a triangle|05MDKQBEAF]], the third angles are then equal too. So the criterion is also called angle-angle-angle similarity.
Another criterion for similarity is side-angle-side: two equal ratios of sides, and equal angles between those sides.
In hyperbolic geometry, triangles with equal angles are congruent, so the criterion has a different meaning there.

## Explanations

### L1 human

The shape of a triangle depends only on its angles. If two angles match, the third one matches too, since the three always add up to $\pi$. So two triangles with matching angles are scaled copies of each other.
</document>

<document source="https://math2.tamnd.com/node/0P05RJC5XA.md" tag="0P05RJC5XA" kind="definition">
# Leading coefficient

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a nonzero polynomial in $R[x]$ of [[degree|0DPF6WRZW6]] $d$.
The leading coefficient of $f$ is $a_d$.
The leading coefficient of the zero polynomial is $0$.

## Remarks

The leading coefficient of $3x^4 - x + 2$ is $3$.
A polynomial with leading coefficient $1$ is [[monic|0BDX4MBKT4]].
In an [[integral domain|09S3PFBEPM]], the leading coefficient of $f g$ is the product of the leading coefficients of $f$ and $g$.

## Explanations

### L1 human

The leading coefficient is the number in front of the highest power of $x$.
For large values of $x$, the leading term $a_d x^d$ decides the size of a real polynomial.
</document>

<document source="https://math2.tamnd.com/node/0P0DJHBGBB.md" tag="0P0DJHBGBB" kind="definition">
# Cesàro mean

## Definition

Let $(a_n)_{n \ge 1}$ be a [[real sequence|0CTV783HYT]].
For each integer $n \ge 1$, the $n$-th Cesàro mean of $(a_n)$ is the arithmetic mean of its first $n$ terms,
$$\sigma_n = \frac{a_1 + a_2 + \cdots + a_n}{n}.$$

## Remarks

The sequence $(-1)^{n+1}$ diverges, but its Cesàro means are $1, 0, 1/3, 0, 1/5, \dots$, and they tend to $0$.
If $(a_n)$ converges, the [[Cesàro means converge to the same limit|0RDRPBMNAC]].
Cesàro summation applies the Cesàro means to the partial sums of a series. A series is Cesàro summable with sum $A$ when the means of its partial sums tend to $A$.

## Explanations

### L1 human

The Cesàro mean is the running average of the sequence.
Averaging smooths out oscillations. The sequence $1, 0, 1, 0, \dots$ has no limit, but its running averages tend to $1/2$.
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<document source="https://math2.tamnd.com/node/0P0MZ4X682.md" tag="0P0MZ4X682" kind="definition">
# Independence of two events

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \in \mathcal{F}$ be [[events|00SQGJNRTB]].
The events $A$ and $B$ are independent if
$$P(A \cap B) = P(A) \, P(B).$$

## Remarks

If $P(B) > 0$, then $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$, with the [[conditional probability|0FJB06AD2P]].
If $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$.
An event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \in \{0, 1\}$.
Independence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ are never independent.

## Explanations

### L1 human

Two events are independent when knowing that one happened tells you nothing about the other.
Roll a fair die two times. The events "the first roll is $6$" and "the second roll is $6$" are independent.
Both happen with probability $\frac16 \cdot \frac16 = \frac{1}{36}$.
</document>

<document source="https://math2.tamnd.com/node/0P0XDY9FZN.md" tag="0P0XDY9FZN" kind="statement">
# Classification of isometries of the plane

## Statement

Let $f$ be a [[Euclidean isometry|0J1N3WTKM4]] of the plane $\mathbb{R}^2$. Then $f$ is of exactly one of these five types:
- the identity,
- a [[translation|087VJM5Q77]] by a nonzero vector,
- a [[rotation|0HFREYRSZP]] about a point through an angle $\theta \notin 2\pi\mathbb{Z}$,
- a [[reflection|06BRDF3F0E]] in a line,
- a [[glide reflection|0167F9B344]] with a nonzero glide vector.

## Remarks

Write $f(x) = A x + b$ with $A$ orthogonal. If $\det A = 1$, then $f$ is the identity, a translation or a rotation. If $\det A = -1$, then $f$ is a reflection or a glide reflection.
The fixed points tell the types apart. A translation and a glide reflection have no fixed point. A rotation has one, and a reflection fixes a whole line.
Each isometry of the plane is the composite of at most three reflections in lines.

## Explanations

### L1 human

Each rigid motion of the plane is a shift, a turn, a mirror image, or a mirror image with a shift along the mirror. A turn and a shift keep the clockwise order of points. A mirror image and a glide reverse it.
</document>

<document source="https://math2.tamnd.com/node/0P11X5HKAX.md" tag="0P11X5HKAX" kind="definition">
# Subgraph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]].
A subgraph of $G$ is a graph $H = (V', E')$ with $V' \subseteq V$ and $E' \subseteq E$.
Each edge of $E'$ has both ends in $V'$, because $H$ is a graph.

## Remarks

A subgraph is spanning if $V' = V$.
An [[induced subgraph|0V05FC4GKQ]] keeps all edges of $G$ between the chosen vertices.
In the cycle $C_4$ with vertices $1, 2, 3, 4$, the path $1 - 2 - 3$ is a subgraph.

## Explanations

### L1 human

To get a subgraph, delete some vertices and some edges of a graph.
When you delete a vertex, you must also delete the edges at that vertex.
</document>

<document source="https://math2.tamnd.com/node/0P17NGQQMQ.md" tag="0P17NGQQMQ" kind="concept">
# Hermitian matrix

## Summary

A Hermitian matrix is a complex square matrix that is equal to its conjugate transpose.
</document>

<document source="https://math2.tamnd.com/node/0P18JC72RE.md" tag="0P18JC72RE" kind="definition">
# Removable singularity

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$.
The singularity $c$ is removable if there is a function $g$ that is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| < r$ with $g(z) = f(z)$ for each $z$ with $0 < |z - c| < r$.

## Remarks

The standard example is $f(z) = \frac{\sin z}{z}$ at $0$. With $g(0) = 1$, the function $g$ is the sum of $1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \cdots$, which is holomorphic.
The singularity is removable if and only if the [[principal part|0NDV5R6Z6X]] of $f$ at $c$ is $0$.
[[Riemann's theorem|0BSQDZWF34]] says that the singularity is removable if $f$ is bounded near $c$.

## Explanations

### L1 human

A removable singularity is a hole in the domain of the function that you can fill with one value.
After you fill it, the function is holomorphic at the point too.
</document>

<document source="https://math2.tamnd.com/node/0P21AVFFSP.md" tag="0P21AVFFSP" kind="definition">
# Equivalent matrices

## Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$.
The matrix $B$ is *equivalent* to $A$ when there are an [[invertible|0EB9N0S33C]] $n \times n$ matrix $P$ and an invertible $m \times m$ matrix $Q$ over $F$ with
$$B = Q^{-1} A P.$$

## Remarks

Two matrices are equivalent if and only if they represent the same linear map $F^n \to F^m$ in two pairs of bases.
Two $m \times n$ matrices over a field are equivalent if and only if they have the same [[rank|0ZC9YTXZRY]].
A matrix of rank $r$ is equivalent to the block matrix $\begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}$.
[[Similarity|0E89HQATM8]] is a finer relation: similar matrices are equivalent, but equivalent square matrices need not be similar.

## Explanations

### L1 human

Matrix equivalence allows independent changes of basis on the input side and on the output side.
With so much freedom, only the rank survives.
Every matrix of rank $r$ becomes the identity of size $r$, padded with zeros.
</document>

<document source="https://math2.tamnd.com/node/0P234N3M28.md" tag="0P234N3M28" kind="definition">
# Hyperbola

## Definition

Let $F_1$ and $F_2$ be distinct points of the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $a$ be a real number with $0 < 2a < d(F_1, F_2)$.
The hyperbola with the foci $F_1$, $F_2$ and the semi-major axis $a$ is the set
$$\{\, P \in \mathbb{R}^2 \;:\; |d(P, F_1) - d(P, F_2)| = 2a \,\}.$$

## Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{c^2 - a^2}$. For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$, the hyperbola is the set of the points with $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$.
This hyperbola has the two asymptotes $y = \pm \frac{b}{a} x$. Its [[eccentricity|0HVX863SAJ]] is $e = c / a > 1$.
For $a = b$, the asymptotes are perpendicular, and the hyperbola is rectangular. The graph of $y = 1/x$ is a rectangular hyperbola.

## Explanations

### L1 human

A hyperbola has two separate branches, like two opposite bows. On one branch, the points are nearer to one focus; on the other branch, they are nearer to the other focus. The difference of the two distances is always $2a$.
</document>

<document source="https://math2.tamnd.com/node/0P24YW565F.md" tag="0P24YW565F" kind="definition">
# Matrix of a bilinear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$.
Let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.
The *matrix* of $B$ in this basis is the $n \times n$ matrix $M$ with entries
$$M_{ij} = B(e_i, e_j) \qquad (1 \le i, j \le n).$$

## Remarks

If $x$ and $y$ are the coordinate vectors of $u$ and $v$, then $B(u, v) = x^T M y$.
The map $B \mapsto M$ is a linear bijection from the bilinear forms on $V$ to the $n \times n$ matrices over $F$.
For the dot product on $\R^n$ and the standard basis, $M$ is the identity matrix.

## Explanations

### L1 human

A bilinear form is fixed by its values on pairs of basis vectors.
We store these $n^2$ values in a matrix.
Then the form becomes a product of a row vector, the matrix and a column vector.
</document>

<document source="https://math2.tamnd.com/node/0P27W9RW0G.md" tag="0P27W9RW0G" kind="definition">
# Continuous function

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is continuous if it is [[continuous at|0EEE2CZB2A]] each point of $D$.
For a subset $S \subseteq D$, $f$ is continuous on $S$ if its restriction to $S$ is continuous.

## Remarks

Sums, [[products|0VGMHCF4SC]], [[quotients|0GR2APWTAC]], and [[compositions|0NRQPQX3VA]] of continuous functions are continuous.
[[Polynomial functions are continuous|0C5TP6RYSV]]. So are the exponential function, the sine, and the cosine.
On a closed bounded interval, a continuous function has the [[intermediate value property|00KAFBR32E]] and [[attains its extreme values|0WS1A8E2P9]].
Mathlib defines it for topological spaces, as `Continuous`.

## Explanations

### L1 human

A continuous function has no jumps or breaks at any point of its domain.
On an interval, we can draw its graph without lifting the pen.
</document>

<document source="https://math2.tamnd.com/node/0P30YYZN85.md" tag="0P30YYZN85" kind="statement">
# Second-order Taylor expansion

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be of [[class C^2|0TCBZMEYGN]] on $U$, and let $a \in U$.
Let $\nabla f(a)$ be the [[gradient|0NS2QYVC4Y]] of $f$ at $a$, and let $H_f(a)$ be the [[Hessian matrix|0SCSQWXRA4]] of $f$ at $a$.
Then
$$f(a + h) = f(a) + \nabla f(a) \cdot h + \tfrac{1}{2} \, h^{\mathsf{T}} H_f(a) \, h + R(h), \qquad \lim_{h \to 0} \frac{R(h)}{\|h\|^2} = 0.$$

## Remarks

This is the case $k = 2$ of [[Taylor's theorem in several variables|0J8T4K4YTE]], written with matrices. The quadratic term collects the terms with $|\alpha| = 2$.
For $n = 2$ and $h = (u, v)$, the quadratic term is $\tfrac{1}{2} \left( f_{xx}(a) u^2 + 2 f_{xy}(a) u v + f_{yy}(a) v^2 \right)$.
At a critical point the linear term is zero, so the sign of $h^{\mathsf{T}} H_f(a) h$ controls the behaviour of $f$ near $a$. This is the basis of the [[second derivative test|0GYDV9FE98]].

## Explanations

### L1 human

A function of class $C^2$ is close to a quadratic polynomial near each point.
The gradient gives the linear part and the Hessian gives the quadratic part.
The error goes to zero faster than $\|h\|^2$.
</document>

<document source="https://math2.tamnd.com/node/0P3A06QNDP.md" tag="0P3A06QNDP" kind="statement">
# Negation of a universal statement

## Statement

Let $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.
In classical logic, the negation of $\forall x \in X, \, P(x)$ is logically equivalent to $\exists x \in X, \, \neg P(x)$.
$$\neg \bigl( \forall x \in X, \, P(x) \bigr) \equiv \exists x \in X, \, \neg P(x)$$

## Remarks

The direction from $\exists x, \, \neg P(x)$ to $\neg \forall x, \, P(x)$ is valid in intuitionistic logic. The other direction needs classical logic.
This law explains why one [[counterexample|0ED72CTBNP]] refutes a universal statement.
For a finite $X$, it is a form of [[De Morgan's laws|09PKJKA6J0]].

## Explanations

### L1 human

"Not every student passed" means "some student did not pass".
To deny that all elements have a property, you show one element without it.
</document>

<document source="https://math2.tamnd.com/node/0P3KZK0FTV.md" tag="0P3KZK0FTV" kind="definition">
# Solvable group

## Definition

A [[group|G5S1M9]] $G$ is solvable when it has a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is [[abelian|082GHNPJ53]].

## Remarks

Equivalently, the [[derived series|0Q5G5DJ4P2]] of $G$ reaches $\{e\}$ after finitely many steps.
A finite group is solvable if and only if its [[composition factors|0PHTC93M58]] are all cyclic of prime order.
The group $S_4$ is solvable, by the series $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$.
The name comes from Galois theory: a polynomial equation over $\mathbb{Q}$ is solvable by radicals if and only if its Galois group is solvable.

## Explanations

### L1 human

A solvable group can be built from abelian groups in finitely many layers.
Abelian groups are easy to understand, so solvable groups are close to easy.
The smallest group that is not solvable is $A_5$, of order $60$.
</document>

<document source="https://math2.tamnd.com/node/0P3M4ZY9X8.md" tag="0P3M4ZY9X8" kind="concept">
# Divergent sequence

## Summary

A divergent sequence is a sequence that has no limit. Its terms can grow without bound or oscillate. The node [[divergent sequence|01HR2AH0BX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0P3VCBDEE2.md" tag="0P3VCBDEE2" kind="concept">
# Ordered field

## Summary

An ordered field is a field with a total order that is compatible with addition and with multiplication by positive elements.
The [[rational numbers|0NCJ5WXN6Z]] and the [[real numbers|08J2M1JYAG]] are ordered fields.
</document>

<document source="https://math2.tamnd.com/node/0P42JB4E3M.md" tag="0P42JB4E3M" kind="definition">
# Glide reflection

## Definition

Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be a hyperplane, with $a \neq 0$, and let $\sigma_H$ be the [[reflection|06BRDF3F0E]] in $H$. Let $w \in \mathbb{R}^n$ be a nonzero vector with $w \cdot a = 0$, and let $T_w$ be the [[translation|087VJM5Q77]] by $w$.
The glide reflection with the axis $H$ and the glide vector $w$ is the map $T_w \circ \sigma_H$.

## Remarks

Because $w$ is parallel to $H$, the two maps commute: $T_w \circ \sigma_H = \sigma_H \circ T_w$.
A glide reflection is an indirect isometry with no fixed point. Its square is the translation by $2w$.
If the translation vector is perpendicular to $H$ in place of parallel, then $T_w \circ \sigma_H$ is the reflection in a hyperplane parallel to $H$.
In the plane, the axis is the glide line. Glide reflections are symmetries of many frieze and wallpaper patterns.

## Explanations

### L1 human

First reflect the plane in a line, then slide it along that line. Neither step alone gives the result. The map $(x, y) \mapsto (x + 1, -y)$ is the glide reflection with the axis $y = 0$ and the glide vector $(1, 0)$.
</document>

<document source="https://math2.tamnd.com/node/0P4FTHHMYS.md" tag="0P4FTHHMYS" kind="concept">
# Standard deviation

## Summary

The standard deviation of a random variable is the square root of its variance. It measures the spread of the values in the same units as the variable.
The node [[standard deviation|0VTN5RKNC5]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0P4JRMSWNH.md" tag="0P4JRMSWNH" kind="concept">
# Degree of a polynomial

## Summary

The degree of a nonzero polynomial is the largest exponent of $x$ with a nonzero coefficient.
The [[definition of the degree|0DPF6WRZW6]] gives the details and the convention for the zero polynomial.
</document>

<document source="https://math2.tamnd.com/node/0P4NQD746K.md" tag="0P4NQD746K" kind="statement">
# Sine is 2π-periodic

## Statement

For every real $x$,
$$\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x.$$
The number $2\pi$ is the smallest positive period of $\sin$ and of $\cos$.

## Remarks

From [[$\cos \pi = -1$ and $\sin \pi = 0$|01A8JGR4JM]], the [[addition formulas|028XEQCE88]] give $\cos 2\pi = 1$ and $\sin 2\pi = 0$. Apply the addition formulas again with $y = 2\pi$.
So $\sin$ and $\cos$ are [[periodic|089KWVYD69]] functions.
Also $\sin(x + \pi) = -\sin x$ and $\sin(x + \pi/2) = \cos x$. Mathlib states the periodicity as `Real.sin_periodic`.

## Explanations

### L1 human

A full turn around the unit circle has length $2\pi$, and it brings the point back to where it started.
So the graphs of sine and cosine repeat after each interval of length $2\pi$.
</document>

<document source="https://math2.tamnd.com/node/0P4WX7Y502.md" tag="0P4WX7Y502" kind="definition">
# Iterated integral

## Definition

Let $R = [a, b] \times [c, d]$ and let $f \colon R \to \mathbb{R}$.
Suppose that, for each $x \in [a, b]$, the function $y \mapsto f(x, y)$ is integrable on $[c, d]$, and that $F(x) = \int_c^d f(x, y) \, dy$ is integrable on $[a, b]$.
Then the iterated integral of $f$, first in $y$ and then in $x$, is
$$\int_a^b \int_c^d f(x, y) \, dy \, dx = \int_a^b \left( \int_c^d f(x, y) \, dy \right) dx = \int_a^b F(x) \, dx.$$
The iterated integral in the other order, $\int_c^d \int_a^b f(x, y) \, dx \, dy$, is defined in the same way.

## Remarks

Iterated integrals in three or more variables are defined by repeating this step.
The inner bounds may depend on the outer variable, as in $\int_0^1 \int_0^x f(x, y) \, dy \, dx$. This integrates over the triangle $0 \le y \le x \le 1$.
An iterated integral is different in principle from the [[double integral|0NKBMFC18R]], but [[Fubini's theorem|0R0T1VWH5X]] says that they are equal under suitable hypotheses.
The two orders of integration can give [[different values|0GJZE2E4RX]] when the hypotheses fail.

## Explanations

### L1 human

To compute an iterated integral, work from the inside out.
First integrate in $y$ as if $x$ were a constant; the result is a function of $x$ only. Then integrate this function in $x$.
</document>

<document source="https://math2.tamnd.com/node/0P506SPZP2.md" tag="0P506SPZP2" kind="counterexample">
# Non-commutativity of matrix multiplication

## Counterexample

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, as real $2 \times 2$ matrices.
Then
$$AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.$$
Thus $AB \ne BA$.
This shows that [[matrix multiplication|032Q9FB8MT]] of $n \times n$ matrices is not commutative for $n = 2$.

## Remarks

The same example, extended with zero rows and zero columns, shows that $M_n(R)$ is not commutative for each $n \ge 2$ and each nonzero ring $R$.
Two matrices $A$ and $B$ with $AB = BA$ are said to *commute*.
Some pairs commute, for example a matrix and the [[identity matrix|06DW9BAWB1]], or two [[diagonal matrices|044JNC7ZAA]].

## Explanations

### L1 human

For numbers, $2 \cdot 3 = 3 \cdot 2$, but for matrices the order of the factors matters.
Here $AB$ and $BA$ are different matrices.
A reason is that $AB$ means "first apply $B$, then $A$", and two actions done in a different order can give different results.
</document>

<document source="https://math2.tamnd.com/node/0P64VE79X3.md" tag="0P64VE79X3" kind="statement">
# A function with bounded derivative is Lipschitz

## Statement

Let $I$ be an interval, let $f \colon I \to \mathbb{R}$ be continuous and differentiable at each interior point of $I$, and let $K \ge 0$.
If $|f'(x)| \le K$ for each interior point $x$ of $I$, then $f$ is [[Lipschitz|0QNVQ4VD9T]] with constant $K$.

## Remarks

The [[mean value theorem|0BARQY52N5]] gives $|f(y) - f(x)| = |f'(c)| \, |y - x| \le K |y - x|$.
For example, $\sin$ is Lipschitz with constant $1$ on $\mathbb{R}$.
Mathlib states a form for convex sets as `Convex.lipschitzOnWith_of_nnnorm_deriv_le`.

## Explanations

### L1 human

If the slope of the graph never exceeds $K$ in size, no chord can be steeper than $K$.
</document>

<document source="https://math2.tamnd.com/node/0P6KRD5B7A.md" tag="0P6KRD5B7A" kind="statement">
# Picard–Lindelöf theorem

## Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ with constants $a, L, K$.
Let $B$ be the closed ball of radius $a$ around $x_0$.
Then there is exactly one continuous curve $\alpha : J \to B$ with $\alpha(t_0) = x_0$ and with this property. For each $t \in J$, $\alpha$ has the derivative $f(t, \alpha(t))$ at $t$ within $J$.

## Remarks

The proof uses the [[integral equation form|015KK993C5]] and the fact that [[an iterate of the Picard operator is a contraction|08XDXP86XE]]. The limit of [[Picard iteration|0RJ5NHWYTP]] is the solution.
A common form: let $f$ be continuous on a neighbourhood of $(t_0, x_0)$ and locally Lipschitz in $x$. Then the [[initial value problem|075VBY4K4X]] has a unique solution on $[t_0 - \varepsilon, t_0 + \varepsilon]$ for some $\varepsilon > 0$.
The theorem is local. For $x' = 1 + x^2$, $x(0) = 0$, the solution $\tan t$ exists only for $|t| < \pi/2$.
Without the Lipschitz condition, solutions exist by the [[Peano existence theorem|098YP831QY]], but they need not be unique. See the [[non-uniqueness example|00K7ASZ1PR]].

## History

The theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.

## Explanations

### L1 human

If the vector field is regular enough, the present state determines the whole motion near the start time.
Regular enough means continuous in time and Lipschitz in the state.
Then there is one and only one solution through the initial point, at least for a short time.
</document>

<document source="https://math2.tamnd.com/node/0P6PMJB4MA.md" tag="0P6PMJB4MA" kind="statement">
# Compactness via open covers

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $K \subseteq X$.
Then $K$ is [[compact|00R0X1WGPX]] if and only if each family $(U_i)_{i \in I}$ of open subsets of $X$ with $K \subseteq \bigcup_{i \in I} U_i$ has a finite subfamily $U_{i_1}, \ldots, U_{i_n}$ with $K \subseteq U_{i_1} \cup \cdots \cup U_{i_n}$.

## Remarks

Proof: the open sets of the [[subspace topology|0D5YHQGDJY]] on $K$ are the sets $U \cap K$ with $U$ open in $X$. So covers of $K$ by open sets of $K$ and covers by open sets of $X$ match each other.
For $K = X$, the statement is the definition of a [[compact space|0QPHW703PV]].
Thus compactness of a subset does not depend on the space around it. If $K \subseteq Y \subseteq X$, then $K$ is compact in $Y$ if and only if it is compact in $X$.

## Explanations

### L1 human

To test if a subset is compact, you can use open sets of the big space.
Each time you cover the subset with open sets, finitely many of them must already be enough.
</document>

<document source="https://math2.tamnd.com/node/0P6SBR7YM2.md" tag="0P6SBR7YM2" kind="definition">
# Simple extension

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is simple if there is an element $\theta \in L$ with $L = K(\theta)$.
Such an element $\theta$ is a primitive element, or a generator, of $L/K$.

## Remarks

The extensions $\mathbb{C}/\mathbb{R}$, generated by $i$, and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$, generated by $\sqrt{2}$, are simple.
The extension $\mathbb{Q}(\sqrt{3}, \sqrt{7})/\mathbb{Q}$ is simple, with the generator $\sqrt{3} + \sqrt{7}$.
By the [[primitive element theorem|0SVX98HYV1]], every finite separable extension is simple.
For finite fields, the term "primitive element" usually means a [[generator of the multiplicative group|0YAA9CJQXF]], which is a stronger condition.

## Explanations

### L1 human

A simple extension is built by adding one new element to a field.
Simple extensions are well understood: there is one kind for an algebraic generator and one kind for a transcendental generator.
</document>

<document source="https://math2.tamnd.com/node/0P799D6AMN.md" tag="0P799D6AMN" kind="definition">
# Bilinear form

## Definition

Let $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.
A *bilinear form* on $V$ is a map $B \colon V \times V \to F$ such that, for all $u, v, w \in V$ and $a \in F$:

1. $B(u + v, w) = B(u, w) + B(v, w)$ and $B(au, w) = a B(u, w)$.
2. $B(u, v + w) = B(u, v) + B(u, w)$ and $B(u, aw) = a B(u, w)$.

## Remarks

The [[dot product|0S52TNNTH9]] on $\R^n$ is a bilinear form.
For each $n \times n$ matrix $M$ over $F$, the map $B(x, y) = x^T M y$ is a bilinear form on $F^n$.
The bilinear forms on $V$ form a vector space over $F$; if $\dim V = n$, it has dimension $n^2$.
More generally, a bilinear map $V \times W \to F$ uses two different spaces.

## Explanations

### L1 human

A bilinear form takes two vectors and gives a number.
It is linear in the first vector and also linear in the second vector.
The dot product is the main example, but bilinear forms need not be symmetric or positive.
</document>

<document source="https://math2.tamnd.com/node/0P7A2MJD24.md" tag="0P7A2MJD24" kind="definition">
# Coefficient matrix

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] over $F$ with $m$ equations in $n$ unknowns, where equation $i$ is $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$.
The *coefficient matrix* of the system is the $m \times n$ [[matrix|0VD9ZSWWJG]] $A = (a_{ij})$.
Its entry in row $i$ and column $j$ is the coefficient of $x_j$ in equation $i$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 1 & -3 \end{pmatrix}$.
With $x = (x_1, \dots, x_n)^T$ and $b = (b_1, \dots, b_m)^T$, the system is $Ax = b$.
If $A$ is square and [[invertible|0EB9N0S33C]], the system has exactly one solution, $x = A^{-1} b$.

## Explanations

### L1 human

The coefficient matrix keeps only the numbers in front of the unknowns, in a rectangular table.
Each row is one equation, and each column is one unknown.
This table lets us work with the whole system as one object.
</document>

<document source="https://math2.tamnd.com/node/0P7CH1CND5.md" tag="0P7CH1CND5" kind="definition">
# Characteristic of a ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
For a positive integer $n$, write $n \cdot 1 = 1 + 1 + \dots + 1$, with $n$ terms.
If there is a positive integer $n$ with $n \cdot 1 = 0$ in $R$, then the characteristic of $R$ is the least such $n$.
If there is no such $n$, then the characteristic of $R$ is $0$.

## Remarks

The rings $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ have characteristic $0$.
For $n \ge 1$, the ring $\mathbb{Z}/n\mathbb{Z}$ has characteristic $n$.
The characteristic of $R$ is the nonnegative generator of the kernel of the unique ring homomorphism $\mathbb{Z} \to R$.
The [[characteristic of an integral domain is zero or prime|00758QM6JJ]].

## Explanations

### L1 human

Start with $1$ and keep adding $1$.
In the integers you never come back to $0$, so the characteristic is $0$.
Modulo $5$, you get $0$ after five steps, so the characteristic is $5$.
</document>

<document source="https://math2.tamnd.com/node/0P7FZHXZ5S.md" tag="0P7FZHXZ5S" kind="concept">
# Composition of an integer

## Summary

A composition of an integer is a way to write it as an ordered sum of positive integers. Compositions are the ordered analogue of partitions, and they are easy to count. The node [[composition of an integer|0J8ANSYKAC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0P7G9NGG2G.md" tag="0P7G9NGG2G" kind="concept">
# Faithful action

## Summary

A group action is faithful when only the identity element of the group fixes each point.
Then different group elements move the points in different ways.
</document>

<document source="https://math2.tamnd.com/node/0P7PTC8DDE.md" tag="0P7PTC8DDE" kind="statement">
# Linearity of expectation

## Statement

Let $X$ and $Y$ be integrable real random variables on one probability space, and let $a, b \in \mathbb{R}$.
Then $aX + bY$ is integrable and
$$E[aX + bY] = a \, E[X] + b \, E[Y].$$

## Remarks

The statement is the linearity of the Lebesgue integral. It holds for all integrable $X$ and $Y$, also when they are not independent.
By induction, $E[X_1 + \dots + X_n] = E[X_1] + \dots + E[X_n]$.
A standard use is the method of indicators. If $N$ counts how many of the events $A_1, \dots, A_n$ occur, then $N = \sum_i 1_{A_i}$ and $E[N] = \sum_i P(A_i)$.
For nonnegative random variables, additivity $E[X + Y] = E[X] + E[Y]$ holds without integrability, in $[0, \infty]$.

## Explanations

### L1 human

The average of a sum is the sum of the averages, always.
If a random permutation of $n$ letters goes into $n$ envelopes, each letter is in its own envelope with probability $1/n$. So the expected number of correct letters is $n \cdot 1/n = 1$.
</document>

<document source="https://math2.tamnd.com/node/0P84B554AS.md" tag="0P84B554AS" kind="statement">
# Vandermonde determinant

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $x_1, \dots, x_n \in K$.
Let $V$ be the $n \times n$ *Vandermonde matrix* with entries $V_{ij} = x_i^{\,j-1}$.
Then
$$\det V = \prod_{1 \le i < j \le n} (x_j - x_i),$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Over a field, $\det V \ne 0$ if and only if the $x_i$ are pairwise distinct.
Then, for all $y_1, \dots, y_n$, there is exactly one polynomial $p$ of degree less than $n$ with $p(x_i) = y_i$ for all $i$.
For $n = 2$, the matrix is $\begin{pmatrix} 1 & x_1 \\ 1 & x_2 \end{pmatrix}$ with determinant $x_2 - x_1$.

## Explanations

### L1 human

A Vandermonde matrix lists the powers $1, x, x^2, \dots$ of some numbers, one row for each number.
Its determinant is the product of all differences of the numbers.
So it is nonzero exactly when the numbers are distinct.
</document>

<document source="https://math2.tamnd.com/node/0P8BA7T7S5.md" tag="0P8BA7T7S5" kind="definition">
# Limit superior

## Definition

Let $(a_n)$ be a [[bounded sequence|05FQ3VGAR9]] of real numbers.
For each $n \in \mathbb{N}$, let $s_n = \sup \{a_k : k \ge n\}$.
The sequence $(s_n)$ is decreasing and bounded below.
The limit superior of $(a_n)$ is
$$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} s_n = \inf_{n \in \mathbb{N}} \sup_{k \ge n} a_k.$$

## Remarks

The limit exists because a [[decreasing sequence bounded below converges|02V2GYAC8P]].
For a sequence that is not bounded above, one sets $\limsup a_n = +\infty$. For a sequence that is bounded above and not below, the same formula is used in the extended real line, and it can give $-\infty$.
For $a_n = (-1)^n (1 + 1/(n+1))$, the limit superior is $1$, although no term equals $1$.
For a bounded sequence, it is the largest [[cluster point|0RCKQN1E09]].

## Explanations

### L1 human

The limit superior is the highest value that the sequence keeps coming back to, in the long run.
Each number above it is exceeded by only finitely many terms.
</document>

<document source="https://math2.tamnd.com/node/0P8CY9V437.md" tag="0P8CY9V437" kind="example">
# Action by conjugation

## Example

Let $G$ be a [[group|G5S1M9]].
The map $G \times G \to G$, $(g, x) \mapsto gxg^{-1}$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.
Indeed, $exe^{-1} = x$ and $g(hxh^{-1})g^{-1} = (gh)x(gh)^{-1}$.
The [[orbits|0YQPZB5Q1G]] of this action are the [[conjugacy classes|0DEBXAJ5SA]] of $G$.
The [[stabiliser|0AYMP41THN]] of $x$ is the [[centraliser|02YYV36BFM]] $C_G(x)$.
The [[fixed points|0RRWNMYRXC]] are the elements of the [[centre|0DZ93493VN]] $Z(G)$.
For an [[abelian group|082GHNPJ53]], each element is a fixed point, and each orbit has one element.
</document>

<document source="https://math2.tamnd.com/node/0P953MG05E.md" tag="0P953MG05E" kind="definition">
# Bounded set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
The set $S$ is bounded when there is a real number $M$ such that $d(x, y) \le M$ for all $x, y \in S$.

## Remarks

The empty set and each finite set are bounded.
If $X$ is not empty, then $S$ is bounded if and only if $S$ lies in some [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$ with $x \in X$ and $r \ge 0$.
In $\mathbb{R}$ with the usual metric, $S$ is bounded in this sense if and only if $S$ is bounded above and bounded below.
Boundedness depends on the metric: $\mathbb{R}$ is not bounded for $|x - y|$, but it is bounded for $\min(|x - y|, 1)$.

## Explanations

### L1 human

A set is bounded when you can put all of it inside one large ball.
For example, the interval $(0, 1)$ is bounded, but the set $\mathbb{N}$ of natural numbers is not bounded in $\mathbb{R}$.
</document>

<document source="https://math2.tamnd.com/node/0P97V20RW7.md" tag="0P97V20RW7" kind="definition">
# Ordinary differential equation

## Definition

Let $n \ge 1$ be an integer, let $U$ be a subset of $\mathbb{R}^{n+2}$, and let $F : U \to \mathbb{R}$ be a function.
An ordinary differential equation of [[order|0XK0G19SK8]] $n$ is an equation
$$F\left(x, y, y', \dots, y^{(n)}\right) = 0$$
for an unknown real function $y$ of one real variable $x$. Here $y^{(k)}$ is the $k$-th derivative of $y$.
The equation must depend on $y^{(n)}$.
If $y$ takes values in $\mathbb{R}^m$ and $F$ takes values in $\mathbb{R}^m$, the equation is a system of $m$ ordinary differential equations.

## Remarks

The word ordinary means that the unknown function depends on one variable only. An equation with partial derivatives of a function of several variables is a partial differential equation.
The equation is in [[explicit form|08WWTN738R]] when it is solved for the highest derivative: $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$.
A [[solution|07VX34QXAV]] is a function on an interval that satisfies the equation at each point of the interval.

## Explanations

### L1 human

An ordinary differential equation is a rule that links a function of one variable with its derivatives.
For example, $y'' + y = 0$ links $y$ with its second derivative. The functions $\sin x$ and $\cos x$ satisfy it.
</document>

<document source="https://math2.tamnd.com/node/0P9BNBW2W3.md" tag="0P9BNBW2W3" kind="statement">
# The exponential function is positive

## Statement

For every real $x$, $\exp(x) > 0$.

## Remarks

By [[the sum rule|0E8HNVBCSP]], $\exp(x) = \exp(x/2)^2 \ge 0$. Also $\exp(x) \exp(-x) = 1$, so $\exp(x) \ne 0$.
For $x \ge 0$, it also follows from the series, since all terms are nonnegative and the first is $1$.
Mathlib states it as `Real.exp_pos`.

## Explanations

### L1 human

The graph of $e^x$ stays above the $x$-axis.
It comes close to $0$ as $x \to -\infty$, but it never reaches $0$.
</document>

<document source="https://math2.tamnd.com/node/0P9DCDES73.md" tag="0P9DCDES73" kind="statement">
# Metric spaces are Hausdorff

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], with the [[metric topology|0A6YRNZB2R]].
Then $X$ is a [[Hausdorff space|0MXB4CKHFW]].

## Remarks

Proof: for $x \neq y$, let $r = d(x, y) / 2 > 0$. The [[open balls|0DC3MPR7C9]] $B(x, r)$ and $B(y, r)$ are open. They are disjoint: a point $z$ in both would give $d(x, y) \le d(x, z) + d(z, y) < 2r$, by the [[triangle inequality|01M6CW9VM6]].
Thus each [[metrizable space|09KDHB82SG]] is Hausdorff. A non-Hausdorff space, such as an infinite set with the cofinite topology, is not metrizable.
For a pseudometric, the statement fails: points at distance $0$ cannot be separated.

## Explanations

### L1 human

Two distinct points of a metric space are at a positive distance. Balls of half that radius about them do not meet.
</document>

<document source="https://math2.tamnd.com/node/0P9T8WHNMH.md" tag="0P9T8WHNMH" kind="statement">
# Existence of a complementary subspace

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ be a subspace of $V$.
Then $U$ has a [[complement|0M83V17KHX]] in $V$: there is a subspace $W$ of $V$ with $V = U \oplus W$.

## Remarks

Proof: extend a basis $B_U$ of $U$ to a basis $B$ of $V$, and let $W$ be the span of $B \setminus B_U$.
For infinite-dimensional $V$, this uses the extension of independent sets to bases, and hence Zorn's lemma.
The statement fails for modules over rings: the submodule $2\Z$ of $\Z$ has no complement.
For closed subspaces of topological vector spaces, a closed complement need not exist.

## Explanations

### L1 human

Every subspace has a partner subspace that fills up the rest of the space.
We find it by completing a basis of the subspace and taking the span of the new vectors.
The partner is not unique, but its dimension is fixed.
</document>

<document source="https://math2.tamnd.com/node/0PAQM3H57X.md" tag="0PAQM3H57X" kind="statement">
# Continuity is equivalent to sequential continuity

## Statement

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
Then $f$ is [[continuous|0P27W9RW0G]] if and only if $f$ is [[sequentially continuous|0MVDQEAB60]].

## Remarks

The forward direction follows from the [[epsilon-delta form|0W8YHHGCJG]] of continuity.
For the converse, suppose that $f$ is not continuous at $a$. Then there is $\varepsilon > 0$ and for each $n$ a point $x_n$ with $|x_n - a| < 1/n$ and $|f(x_n) - f(a)| \ge \varepsilon$.
The equivalence holds in all metric spaces. Mathlib states it for sequential spaces, as `continuous_iff_seqContinuous`.

## Explanations

### L1 human

We can test continuity with sequences.
A function is continuous exactly when it sends every convergent sequence to a convergent sequence with the right limit.
</document>

<document source="https://math2.tamnd.com/node/0PARCKE8WP.md" tag="0PARCKE8WP" kind="example">
# Sup norm on bounded continuous functions

## Example

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $C_b(X)$ be the real vector space of bounded [[continuous maps|0SY9ECQ19Q]] $f \colon X \to \mathbb{R}$.
For $f \in C_b(X)$, let $\|f\|_\infty = \sup_{x \in X} |f(x)|$, with $\|f\|_\infty = 0$ when $X$ is empty.
Then $\|\cdot\|_\infty$ is a norm on $C_b(X)$, the sup norm, and $C_b(X)$ is a [[Banach space|0BDTTM24DP]] with this norm.

## Remarks

The [[supremum|0BTMQ04556]] is finite because $f$ is bounded.
The metric $\|f - g\|_\infty$ is the [[uniform metric|06HE9J7DXF]]. Its completeness is [[completeness of the space of bounded continuous functions|09GRCZVR5E]].
A sequence converges in the sup norm if and only if it converges uniformly.
For a compact space $X$, each continuous real function is bounded, so $C_b(X) = C(X)$.

## Explanations

### L1 human

The sup norm of a function is the height of the highest point of its graph, in absolute value.
Two functions are close in this norm when their graphs are close everywhere.
</document>

<document source="https://math2.tamnd.com/node/0PASQ34GHG.md" tag="0PASQ34GHG" kind="statement">
# Law of sines

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be its side lengths, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]]. Let $\alpha = \angle BAC$, $\beta = \angle CBA$ and $\gamma = \angle ACB$ be its [[angles|0GJAJC6TNZ]].
Then
$$\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}.$$

## Remarks

The common value of the three quotients is $2R$, where $R$ is the radius of the [[circumcircle|03VZJE6C8C]] of the triangle.
The angles of a triangle are in $(0, \pi)$, so their sines are positive and the quotients exist.
From two angles and one side, the law of sines gives the other sides. From two sides and an angle that is not between them, it can give two different triangles. This is the ambiguous case.

## History

Ptolemy knew an equivalent form, with chords in place of sines, in the 2nd century.

## Explanations

### L1 human

In a triangle, a larger angle sits opposite a longer side. The law of sines makes this exact: each side divided by the sine of the opposite angle gives the same number. For an equilateral triangle with side $1$, each quotient is $1 / \sin(\pi/3) = 2/\sqrt{3}$.
</document>

<document source="https://math2.tamnd.com/node/0PB7K8M7FG.md" tag="0PB7K8M7FG" kind="definition">
# Linear differential equation of order n

## Definition

Let $n \ge 1$, let $I$ be an open [[interval|0Q1D85B8G6]], and let $a_0, \dots, a_{n-1}, b : I \to \mathbb{R}$ be continuous functions.
The linear differential equation of order $n$ with these coefficients and right side $b$ is
$$y^{(n)} + a_{n-1}(x)\, y^{(n-1)} + \dots + a_1(x)\, y' + a_0(x)\, y = b(x)$$
for an unknown $n$ times differentiable function $y : I \to \mathbb{R}$.
With the operator $L y = y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y$, the equation is $L y = b$.

## Remarks

The map $L$ is linear: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2$ for constants $c_1, c_2$.
An equation $a_n(x) y^{(n)} + \dots + a_0(x) y = b(x)$ with continuous $a_n$ and no zero of $a_n$ on $I$ takes this form after division by $a_n$.
The equation is [[homogeneous|0EQZSAB9EC]] if $b = 0$. The case $n = 1$ is the [[linear first-order equation|0Z3YKRQZ48]].
For each $x_0 \in I$ and each $(c_0, \dots, c_{n-1}) \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$.

## Explanations

### L1 human

In a linear equation, the unknown $y$ and its derivatives appear only to the first power, each multiplied by a function of $x$.
For example, $y'' + x y' + y = \cos x$ is linear, but $y'' + y^2 = 0$ is not.
Linear equations have a rich theory, because sums and multiples of solutions behave well.
</document>

<document source="https://math2.tamnd.com/node/0PBFS92AJP.md" tag="0PBFS92AJP" kind="definition">
# Jordan block

## Definition

Let $F$ be a field, let $\lambda \in F$, and let $k \ge 1$.
The *Jordan block* $J_k(\lambda)$ is the $k \times k$ matrix with entries
$$(J_k(\lambda))_{ij} = \begin{cases} \lambda & \text{if } j = i, \\ 1 & \text{if } j = i + 1, \\ 0 & \text{otherwise}. \end{cases}$$

## Remarks

For example, $J_3(\lambda) = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}$.
$J_k(\lambda) - \lambda I_k$ is nilpotent of index $k$.
Its only eigenvalue is $\lambda$, with algebraic multiplicity $k$ and geometric multiplicity $1$.
Some authors put the $1$ entries below the diagonal.

## Explanations

### L1 human

A Jordan block is the simplest matrix that is not diagonalisable, when its size is at least $2$.
It has one eigenvalue and only one line of eigenvectors.
Every complex matrix is built from such blocks, by the Jordan normal form.
</document>

<document source="https://math2.tamnd.com/node/0PBN8VEKY8.md" tag="0PBN8VEKY8" kind="concept">
# Annihilating polynomial

## Summary

An annihilating polynomial of an endomorphism $T$ is a polynomial $p$ with $p(T) = 0$.
</document>

<document source="https://math2.tamnd.com/node/0PBWWQD9KY.md" tag="0PBWWQD9KY" kind="problem">
# Monty Hall problem

## Problem

A game show has three closed doors. A car is behind one door, chosen uniformly at random, and a goat is behind each of the two other doors.
The player chooses one door. Then the host, who knows where the car is, opens one of the two other doors and shows a goat.
If both other doors hide goats, the host chooses one of them with probability $1/2$ each. The host always opens a door and always offers a switch.
The player can keep the first door or switch to the other closed door. Which choice gives the larger probability of the car?

## Remarks

Under these assumptions, switching wins with probability $2/3$ and staying wins with probability $1/3$. The player's first door hides the car with probability $1/3$, and the host's action does not change this.
The conditional probability that switching wins, given the door that the host opens, is also $2/3$. [[Bayes' theorem|0NM6E9VW04]] gives this.
The answer depends on the rules of the host. If the host opens a random other door and it shows a goat by chance, both choices win with probability $1/2$.

## History

Steve Selvin posed the problem in a letter to The American Statistician in 1975. It became famous through the column of Marilyn vos Savant in Parade magazine.

## Explanations

### L1 human

Staying wins only when your first choice was right, which happens one time in three.
Switching wins every time your first choice was wrong, which happens two times in three.
</document>

<document source="https://math2.tamnd.com/node/0PBY4E9GQF.md" tag="0PBY4E9GQF" kind="concept">
# Skew-symmetric matrix

## Summary

A skew-symmetric matrix is a square matrix whose transpose is its negative.
</document>

<document source="https://math2.tamnd.com/node/0PCKBBMEXC.md" tag="0PCKBBMEXC" kind="statement">
# Order of the symmetric group

## Statement

Let $n$ be a positive integer.
The [[symmetric group|0515A6D2Y1]] $S_n$ has [[order|0B85S2V2DR]] $n!$.

## Remarks

A permutation $\sigma$ has $n$ choices for $\sigma(1)$, then $n - 1$ choices for $\sigma(2)$, and so on, down to $1$ choice for $\sigma(n)$.
For example, $|S_3| = 6$ and $|S_4| = 24$.

## Explanations

### L1 human

To make a permutation, you choose where each symbol goes, one symbol after the other.
Each choice removes one place from the next choices.
So the number of permutations is $n \cdot (n-1) \cdots 1 = n!$.
</document>

<document source="https://math2.tamnd.com/node/0PCV9EDJM2.md" tag="0PCV9EDJM2" kind="concept">
# Multinomial coefficient

## Summary

A multinomial coefficient counts the ways to distribute $n$ distinct objects into labelled boxes of given sizes. It extends the binomial coefficient from two boxes to any number of boxes. The node [[multinomial coefficient|00K0RF9BFY]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0PCVBRV9VJ.md" tag="0PCVBRV9VJ" kind="statement">
# Fermat's theorem on stationary points

## Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$.
Assume that $f$ has a [[local maximum|00999TYPXB]] or a [[local minimum|0NS8BP96QX]] at $a$, and that $f$ is differentiable at $a$.
Then $f'(a) = 0$.

## Remarks

At a local maximum, the difference quotient $(f(a + h) - f(a))/h$ is $\le 0$ for small $h > 0$ and $\ge 0$ for small $h < 0$. So the limit is both $\le 0$ and $\ge 0$.
The point must be interior. On $[0, 1]$, the function $x$ has a maximum at $1$, where its derivative is $1$.
Mathlib states it as `IsLocalExtr.deriv_eq_zero`.

## Explanations

### L1 human

At the top of a smooth hill, the tangent line is horizontal.
If the slope were positive or negative, moving a little would give a higher value.
</document>

<document source="https://math2.tamnd.com/node/0PD57PSHNR.md" tag="0PD57PSHNR" kind="definition">
# Circuit

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A circuit in $G$ is a closed [[trail|062MQGP8M5]] of positive length.
That is a walk $(w_0, w_1, \dots, w_k)$ with $k \ge 1$, $w_k = w_0$, and pairwise distinct edges.

## Remarks

A circuit may visit a vertex more than once. A [[cycle|07CT9RDY7Q]] is a circuit with no repeated vertex other than the start.
In a simple graph each circuit has length at least $3$.
Some authors use the word circuit for a cycle. Others use the word tour.

## Explanations

### L1 human

A circuit is a trail that ends where it starts.
It is a round trip that never uses the same edge twice.
</document>

<document source="https://math2.tamnd.com/node/0PDBVPP233.md" tag="0PDBVPP233" kind="counterexample">
# Converse of Lagrange's theorem fails for the alternating group A4

## Counterexample

The converse of [[Lagrange's theorem|3C53TD]] would say this. If $G$ is a finite group and $d$ is a positive divisor of $|G|$, then $G$ has a subgroup of order $d$.
This is false.
Let $A_4$ be the group of the even permutations of $\{1, 2, 3, 4\}$, under composition.
It has order $12$, and $6$ divides $12$, but $A_4$ has no subgroup of order $6$.

## Remarks

The elements of $A_4$ are the identity, the three products of two disjoint transpositions, and the eight $3$-cycles.
A subgroup $H$ of order $6$ would have index $2$, and thus it would contain the square of each element of $A_4$.
The square of each $3$-cycle is a $3$-cycle, and each $3$-cycle is such a square. So $H$ would contain all eight $3$-cycles, which is too many.
The group $A_4$ is the smallest group for which the converse fails.

## Explanations

### L1 human

Lagrange's theorem says which sizes a subgroup can have, but it does not promise that each allowed size occurs.
The group $A_4$ has $12$ elements, and $6$ is an allowed size, but no subgroup has $6$ elements.
For cyclic groups the converse is true, so a counterexample must be a group that is not cyclic.
</document>

<document source="https://math2.tamnd.com/node/0PDVQEP0ZC.md" tag="0PDVQEP0ZC" kind="concept">
# Matrix–vector product

## Summary

The product of an $m \times n$ matrix and a column vector with $n$ entries is a column vector with $m$ entries.
Its entry $i$ is the dot product of row $i$ of the matrix with the vector.
</document>

<document source="https://math2.tamnd.com/node/0PEJH7AC74.md" tag="0PEJH7AC74" kind="definition">
# Transitive closure

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The transitive closure of $R$ is the relation $R^{+}$ on $A$ such that $a \mathrel{R^{+}} b$ when there are $n \ge 1$ and $x_0, \dots, x_n \in A$ with
$$a = x_0 \mathrel{R} x_1 \mathrel{R} x_2 \cdots x_{n-1} \mathrel{R} x_n = b.$$

## Remarks

The relation $R^{+}$ is [[transitive|0CYRECG2YQ]], it contains $R$, and it is a subset of each transitive relation that contains $R$.
It is also the intersection of all transitive relations on $A$ that contain $R$.
Example: the transitive closure of "is a parent of" is "is an ancestor of".

## Explanations

### L1 human

The transitive closure adds all the shortcuts.
In a graph, $a \mathrel{R^{+}} b$ means that there is a path of one or more arrows from $a$ to $b$.
</document>

<document source="https://math2.tamnd.com/node/0PF6TM8TWT.md" tag="0PF6TM8TWT" kind="concept">
# Algebraically closed field

## Summary

An algebraically closed field is a field in which every nonconstant polynomial has a root.
The complex numbers are the main example.
The [[definition of an algebraically closed field|02SJG1HC32]] gives equivalent conditions.
</document>

<document source="https://math2.tamnd.com/node/0PFPWKBDTJ.md" tag="0PFPWKBDTJ" kind="definition">
# Restriction of an endomorphism to an invariant subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
Let $W$ be a $T$-[[invariant subspace|0ACX5QT8TE]] of $V$.
The *restriction* of $T$ to $W$ is the endomorphism
$$T|_W \colon W \to W, \qquad T|_W(w) = T(w).$$

## Remarks

The map is well defined because $T(W) \subseteq W$, and it is linear.
If $V$ is finite-dimensional, then the characteristic polynomial of $T|_W$ divides that of $T$.
Take a basis of $V$ that extends a basis of $W$.
In this basis, the matrix of $T$ is block upper triangular, and its first diagonal block is the matrix of $T|_W$.

## Explanations

### L1 human

If a map keeps a subspace inside itself, then it is also a map of that subspace.
This smaller map is the restriction.
Its eigenvalues are some of the eigenvalues of the original map.
</document>

<document source="https://math2.tamnd.com/node/0PG2H7WKGG.md" tag="0PG2H7WKGG" kind="concept">
# Divisibility

## Summary

Divisibility is the relation between two integers $a$ and $b$ that holds when $b$ is an integer multiple of $a$.
Then $a$ is a divisor of $b$, and we write $a \mid b$.
The [[definition of divisibility|096PW7Q014]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0PH0VHTQS0.md" tag="0PH0VHTQS0" kind="definition">
# Moment

## Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$.
The $k$-th moment of $X$ is
$$m_k = E[X^k].$$
The $k$-th absolute moment is $E[|X|^k]$.

## Remarks

By the [[law of the unconscious statistician|01ZWX875AT]], $m_k = \int x^k \, \mu_X(dx)$. So the moments depend only on the law $\mu_X$ of $X$.
The first moment is the mean. The variance is $m_2 - m_1^2$.
If $E[|X|^k] < \infty$, then $E[|X|^j] < \infty$ for each $j \le k$.
The [[Cauchy distribution|0NA6Q82YR7]] has no moments of order $k \ge 1$.

## Explanations

### L1 human

Moments are numbers that sum up the shape of a distribution.
The first gives the center, the second with the first gives the spread, and the higher ones describe asymmetry and the weight of the tails.
</document>

<document source="https://math2.tamnd.com/node/0PHAC833JF.md" tag="0PHAC833JF" kind="concept">
# Generating function

## Summary

A generating function encodes a sequence as the coefficients of a formal power series. It turns counting problems into algebra with series. The node [[generating function|0HH6JX5WZZ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0PHBS78HKT.md" tag="0PHBS78HKT" kind="statement">
# First supplement to quadratic reciprocity

## Statement

Let $p$ be an odd prime.
Then
$$\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2},$$
where the left side is a [[Legendre symbol|01J8AMV9RS]].

## Remarks

So $\left(\frac{-1}{p}\right) = 1$ if $p \equiv 1 \pmod 4$, and $\left(\frac{-1}{p}\right) = -1$ if $p \equiv 3 \pmod 4$.
The proof is [[Euler's criterion|0M7DEDKSYB]] with $a = -1$.
For example, $2^2 = 4 \equiv -1 \pmod 5$, but $-1$ is not a square modulo $7$.

## Explanations

### L1 human

The number $-1$ is a square modulo an odd prime $p$ exactly when $p$ leaves remainder $1$ on division by $4$.
So $-1$ is a square modulo $5$, $13$ and $17$, but not modulo $3$, $7$ and $11$.
</document>

<document source="https://math2.tamnd.com/node/0PHC9FE2V5.md" tag="0PHC9FE2V5" kind="concept">
# Bernoulli trial

## Summary

A Bernoulli trial is a [[random experiment|0HM05N3FJD]] with exactly two possible outcomes, called success and failure.
The probability $p$ of success is the same each time the trial is repeated, and the failure has probability $q = 1 - p$.
A sequence of Bernoulli trials usually means independent repetitions of one trial. The number of successes in $n$ such trials has the [[binomial distribution|0VRWZDWW6B]].

## Remarks

The outcome of one trial, coded as $1$ for success and $0$ for failure, has the [[Bernoulli distribution|0J97P5KZY0]].
The trials are named after Jacob Bernoulli, who studied them in his book Ars Conjectandi of 1713.
Examples are a coin toss, with heads as success, and a roll of a die, with a six as success and $p = 1/6$.

## Explanations

### L1 human

A Bernoulli trial is a yes-or-no experiment.
Many questions have this form: does the coin show heads, does the patient recover, does the part fail.
</document>

<document source="https://math2.tamnd.com/node/0PHDYNR24D.md" tag="0PHDYNR24D" kind="concept">
# Null space

## Summary

The null space of a matrix is the set of vectors that the matrix sends to zero.
It is the solution set of the homogeneous system with this matrix.
</document>

<document source="https://math2.tamnd.com/node/0PHN07KVQT.md" tag="0PHN07KVQT" kind="concept">
# Residue

## Summary

The residue of a function at an isolated singularity $c$ is the coefficient of $(z - c)^{-1}$ in its Laurent series, see the [[definition|0JFGKKWWFK]].
It is $\frac{1}{2\pi i}$ times the integral of the function around a small circle around $c$.
</document>

<document source="https://math2.tamnd.com/node/0PHT6NBYPB.md" tag="0PHT6NBYPB" kind="statement">
# Heine–Borel theorem in finite-dimensional normed spaces

## Statement

Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, with a norm $\|\cdot\|$ and the metric $d(x, y) = \|x - y\|$.
Then a subset $K \subseteq V$ is [[compact|00R0X1WGPX]] if and only if $K$ is [[closed|0G5AV9Q4RR]] and [[bounded|0P953MG05E]].

## Remarks

Proof: choose a basis. By [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the Euclidean norm of the coordinates. Equivalent norms have the same closed sets, bounded sets and compact sets. Then use the [[Heine–Borel theorem|07PFYV65MB]] in $\mathbb{R}^n$ or $\mathbb{C}^n = \mathbb{R}^{2n}$.
In other words, each finite-dimensional normed space is a [[proper metric space|0Z4V98VN5E]].
The converse also holds: if the closed unit ball is compact, the space is [[finite-dimensional|0VZVV8VME5]].

## Explanations

### L1 human

In each finite-dimensional normed space, the compact sets are the closed bounded sets, just as in $\mathbb{R}^n$.
</document>

<document source="https://math2.tamnd.com/node/0PHTC93M58.md" tag="0PHTC93M58" kind="concept">
# Composition factors

## Summary

The composition factors of a group are the factor groups of a composition series.
By the Jordan–Hölder theorem, they do not depend on the choice of the series, up to order and isomorphism.
</document>

<document source="https://math2.tamnd.com/node/0PJXBR6QA3.md" tag="0PJXBR6QA3" kind="concept">
# Multi-index notation

## Summary

A multi-index of dimension $n$ is an $n$-tuple $\alpha = (\alpha_1, \dots, \alpha_n)$ of non-negative integers.
With $|\alpha| = \alpha_1 + \dots + \alpha_n$, $\alpha! = \alpha_1! \cdots \alpha_n!$, $x^\alpha = x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ and $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$, formulas of several variables take the short form of their one-variable versions.

## Remarks

Here $\partial_i$ is the [[partial derivative|02RSXVM1ZN]] with respect to $x_i$, and $\partial^\alpha f$ is a partial derivative of order $|\alpha|$.
For $\alpha = (2, 0, 1)$, $|\alpha| = 3$, $\alpha! = 2$, $x^\alpha = x_1^2 x_3$ and $\partial^\alpha f = \partial^3 f / \partial x_1^2 \, \partial x_3$.
The notation $\partial^\alpha f$ does not show the order of differentiation. It is unambiguous when $f$ is of class $C^{|\alpha|}$, by [[Schwarz's theorem|0NNQY8WCW6]].
Examples of use are [[Taylor's theorem in several variables|0J8T4K4YTE]] and the multinomial theorem $(x_1 + \dots + x_n)^k = \sum_{|\alpha| = k} \frac{k!}{\alpha!} x^\alpha$.

## Explanations

### L1 human

Multi-index notation is a bookkeeping device.
One symbol $\alpha$ records how many times each variable appears, so one formula covers all mixed derivatives or monomials of a given order.
</document>

<document source="https://math2.tamnd.com/node/0PK8QH1W1V.md" tag="0PK8QH1W1V" kind="statement">
# Alternating series test

## Statement

Let $(b_n)$ be a decreasing real sequence with $b_n \to 0$ as $n \to \infty$.
Then the [[alternating series|017BR1M98J]] $\sum_{n=0}^{\infty} (-1)^n b_n$ converges.
Moreover, its sum $S$ and its partial sums $s_N = \sum_{n=0}^{N} (-1)^n b_n$ satisfy $|S - s_N| \le b_{N+1}$ for each $N$.

## Remarks

The even partial sums $s_0, s_2, s_4, \dots$ decrease, and the odd partial sums $s_1, s_3, \dots$ increase. Each odd partial sum is below each even one, so both sequences converge by the [[monotone convergence theorem|0FM6N4R0VB]]. Their limits agree, since $s_{2k} - s_{2k+1} = b_{2k+1} \to 0$.
The hypothesis that $(b_n)$ decreases is needed. Without it, an alternating series with terms tending to $0$ can diverge.
Mathlib states the convergence for an antitone sequence, in `Antitone.tendsto_alternating_series_of_tendsto_zero`.

## History

Leibniz discussed the criterion in an unpublished work of 1676. For this reason it is also called the Leibniz test.

## Explanations

### L1 human

Each step goes in the other direction and is shorter than the step before.
So the partial sums swing back and forth with smaller and smaller swings, and they settle down.
</document>

<document source="https://math2.tamnd.com/node/0PKG97PX9Y.md" tag="0PKG97PX9Y" kind="concept">
# Conservative vector field

## Summary

A conservative vector field is a [[vector field|0299VCW1TH]] that is the [[gradient|0ABK9V9Y64]] of a scalar function, called a [[scalar potential|0H7C3T1GJY]] of the field.
Its line integrals depend only on the end points of the path, and they vanish around each closed path.
</document>

<document source="https://math2.tamnd.com/node/0PKRYB8ATM.md" tag="0PKRYB8ATM" kind="concept">
# Limit of a function

## Summary

The limit of a function at a point is the value that the function approaches as its argument approaches the point. It does not depend on the value at the point itself. The node [[limit of a function|0TXTSEMHR5]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0PM88Z9W4M.md" tag="0PM88Z9W4M" kind="statement">
# Projections are continuous

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and give $X \times Y$ the [[product topology|0Z57EC6YX3]].
Then the projections $\pi_1 \colon X \times Y \to X$, $(x, y) \mapsto x$, and $\pi_2 \colon X \times Y \to Y$, $(x, y) \mapsto y$, are [[continuous|0SY9ECQ19Q]].

## Remarks

Proof: for $U$ open in $X$, we have $\pi_1^{-1}(U) = U \times Y$, which is a basic open set. The proof for $\pi_2$ is the same.
The projections are also [[open maps|0SQ2PC596Y]], but in general they are not closed maps.
The product topology is the coarsest topology on $X \times Y$ that makes both projections continuous.

## Explanations

### L1 human

Taking one coordinate of a point is a continuous operation.
For example, the map $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is continuous.
</document>

<document source="https://math2.tamnd.com/node/0PMRQJVTME.md" tag="0PMRQJVTME" kind="object">
# Klein four-group

## Description

The Klein four-group $V$ is the group with four elements $e, a, b, c$, in which $e$ is the identity and each element is its own inverse.
The product of two distinct elements of $\{a, b, c\}$ is the third one.
It is [[abelian|082GHNPJ53]], and it is [[isomorphic|08MG7H8JW5]] to the [[direct product|00HF4RJ5QN]] $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

## Remarks

The Klein four-group is the smallest group that is not [[cyclic|0KVGH0J5B3]], because it has no element of order $4$.
It is the symmetry group of a rectangle that is not a square.
In $S_4$, the set $\{e, (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}$ is a normal subgroup isomorphic to $V$.
Each group of order $4$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$ or to $V$.

## Explanations

### L1 human

The Klein four-group is like two light switches that you can flip separately.
Each switch flipped two times is back to the start.
Flipping both switches is the third element that is not the identity.
</document>

<document source="https://math2.tamnd.com/node/0PMT9YJZVW.md" tag="0PMT9YJZVW" kind="statement">
# Order of an inverse

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
Then $g$ and its inverse $g^{-1}$ have the same [[order|0WE2NHZJ7B]].
In particular, $g$ has infinite order if and only if $g^{-1}$ has infinite order.

## Remarks

For each positive integer $n$, $(g^{-1})^n = (g^n)^{-1}$, and $g^n = e$ if and only if $(g^n)^{-1} = e$.
For example, in $\{1, i, -1, -i\}$, both $i$ and $i^{-1} = -i$ have order $4$.

## Explanations

### L1 human

The inverse of an element walks the same path back to the identity, in the opposite direction.
It needs the same number of steps.
A rotation by $90$ degrees and a rotation by $-90$ degrees both have order $4$.
</document>

<document source="https://math2.tamnd.com/node/0PMY0YQYXX.md" tag="0PMY0YQYXX" kind="statement">
# The continued fraction of a rational number terminates

## Statement

Let $x$ be a real number.
The [[continued fraction expansion|0SJ7Z6VKWK]] of $x$ is finite if and only if $x$ is rational.

## Remarks

If $x = a/b$ with $b > 0$, the expansion follows the [[Euclidean algorithm|0TRNT4X7J0]] on $a$ and $b$. So it stops after finitely many steps.
Conversely, a finite simple continued fraction is a rational number.
For example, $\frac{43}{19}$: $43 = 2 \cdot 19 + 5$, $19 = 3 \cdot 5 + 4$, $5 = 1 \cdot 4 + 1$, $4 = 4 \cdot 1$. So $\frac{43}{19} = [2; 3, 1, 4]$.
A rational number has exactly two finite expansions, since $[a_0; \dots, a_n] = [a_0; \dots, a_n - 1, 1]$ when $a_n > 1$.

## Explanations

### L1 human

For a fraction, the continued fraction algorithm is the Euclidean algorithm in another form.
The remainders get smaller, so the process must end.
An irrational number gives an infinite continued fraction.
</document>

<document source="https://math2.tamnd.com/node/0PN13BB7BT.md" tag="0PN13BB7BT" kind="statement">
# The Gaussian integers are a Euclidean domain

## Statement

The ring of [[Gaussian integers|0K5W69AC0J]] $\mathbb{Z}[i]$ is a [[Euclidean domain|0GJ2HRG95T]], with the [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ as Euclidean function.
That is, for all $z, w \in \mathbb{Z}[i]$ with $w \neq 0$, there are $q, r \in \mathbb{Z}[i]$ with $z = q w + r$ and $N(r) < N(w)$.

## Remarks

The proof: write $z / w = x + y i$ in $\mathbb{C}$. Let $q = m + n i$, where $m$ and $n$ are integers nearest to $x$ and $y$.
Then $|z/w - q|^2 \le \tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2}$, so $r = z - q w$ has $N(r) \le \tfrac{1}{2} N(w) < N(w)$.
A consequence is that $\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.

## Explanations

### L1 human

To divide $z$ by $w$, compute the complex quotient and round both coordinates to the nearest integer.
The rounding error is small, so the remainder is smaller than $w$ in norm.
This makes the Euclidean algorithm work for Gaussian integers.
</document>

<document source="https://math2.tamnd.com/node/0PN2BXRS8G.md" tag="0PN2BXRS8G" kind="statement">
# Argument principle

## Statement

Let $c \in \mathbb{C}$ and $r > 0$.
Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on an open set that contains the closed disc $\{ z : |z - c| \le r \}$.
Suppose that $f$ has no zeros and no poles on the circle $|z - c| = r$.
Let $Z$ be the number of zeros of $f$ in the open disc $|z - c| < r$, each counted as many times as its [[order|064N002TX4]].
Let $P$ be the number of poles of $f$ in the open disc, each counted as many times as its [[order|0YK27TJ4WF]].
Then
$$\frac{1}{2\pi i} \oint_{|z - c| = r} \frac{f'(z)}{f(z)} \, dz = Z - P,$$
where the [[circle integral|0T7ABX4CWE]] goes once around the circle counterclockwise.

## Remarks

The function $f'/f$ is the logarithmic derivative of $f$.
At a zero of order $m$, it has a simple pole with residue $m$; at a pole of order $m$, it has a simple pole with residue $-m$.
So the theorem follows from the [[residue theorem|032HKYS329]].
The left side is also the [[winding number|0X7SBS8E0F]] around $0$ of the closed curve $\theta \mapsto f(c + r e^{i\theta})$, $\theta \in [0, 2\pi]$.

## Explanations

### L1 human

Walk once around the circle and watch the point $f(z)$.
Each zero inside the circle makes $f(z)$ turn once around $0$ in the positive direction, and each pole makes it turn once in the negative direction.
For example, $f(z) = z^3$ on the unit circle turns three times around $0$, and $f$ has a zero of order $3$ at $0$.
</document>

<document source="https://math2.tamnd.com/node/0PN4QJE969.md" tag="0PN4QJE969" kind="definition">
# Farey sequence

## Definition

Let $n$ be a positive integer.
The Farey sequence $F_n$ of order $n$ is the increasing sequence of all fractions $\frac{a}{b}$ with $0 \le a \le b \le n$ and $\gcd(a, b) = 1$.
Each fraction is in lowest terms, and $F_n$ starts with $\frac{0}{1}$ and ends with $\frac{1}{1}$.

## Remarks

For example, $F_5 = \left( \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1} \right)$.
If $\frac{a}{b} < \frac{c}{d}$ are neighbours in $F_n$, then $bc - ad = 1$.
The length of $F_n$ is $1 + \sum_{k=1}^{n} \varphi(k)$, with $\varphi$ [[Euler's totient function|0VV04M41AN]].
The sequences are named after John Farey, whose letter about them appeared in 1816.

## Explanations

### L1 human

Write all fractions between $0$ and $1$ with denominators up to $n$, in lowest terms and in increasing order.
Neighbouring fractions in this list are always very close in a precise sense: their cross difference is $1$.
</document>

<document source="https://math2.tamnd.com/node/0PN9WMRWT7.md" tag="0PN9WMRWT7" kind="statement">
# The constructible numbers form a field closed under square roots

## Statement

The set of the [[constructible numbers|0FMN2MN7TM]] is a [[subfield|07QY5RPFBR]] of $\mathbb{C}$.
If $z$ is constructible, then the square roots of $z$ are constructible.
Moreover, the field of constructible numbers is the smallest subfield of $\mathbb{C}$ that is closed under square roots.

## Remarks

The proof of closure uses classical constructions: parallels and similar triangles give products and quotients, and a circle on a diameter of length $1 + x$ gives $\sqrt{x}$.
Conversely, the intersection points of lines and circles have coordinates that come from the old ones by field operations and square roots.
So a number is constructible if and only if it is obtained from $\mathbb{Q}$ by finitely many field operations and square roots.
The constructible real numbers form an ordered field in which every positive element has a square root.

## Explanations

### L1 human

You can add, subtract, multiply and divide lengths with straightedge and compass, and you can take square roots.
Nothing else is possible, because lines and circles only lead to equations of degree one or two.
</document>

<document source="https://math2.tamnd.com/node/0PNPG2PP9H.md" tag="0PNPG2PP9H" kind="statement">
# Jordan's lemma

## Statement

Let $a > 0$ and $R > 0$ be real numbers.
Let $C_R$ be the semicircle $\{ R e^{i\theta} : \theta \in [0, \pi] \}$ in the upper half-plane, with the parametrization $\theta \mapsto R e^{i\theta}$.
Let $g$ be a continuous complex function on $C_R$, and let $f(z) = e^{iaz} g(z)$ for $z \in C_R$, where $e^w$ is the [[complex exponential|054VXHMCA3]].
Let $M_R = \max_{\theta \in [0, \pi]} |g(R e^{i\theta})|$.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $C_R$ satisfies
$$\left| \int_{C_R} f(z) \, dz \right| \le \frac{\pi}{a} M_R.$$

## Remarks

If $M_R \to 0$ as $R \to \infty$, then $\int_{C_R} f(z) \, dz \to 0$ as $R \to \infty$.
The bound does not contain the length $\pi R$ of $C_R$, so it is stronger than the [[estimation lemma|0DJMGPN8K9]] for this type of integrand.
A similar result holds for the semicircle in the lower half-plane when $a < 0$.
The lemma is a main step in the [[evaluation of real integrals|0476QH9846]] such as $\int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx$.

## Explanations

### L1 human

On the semicircle, $|e^{iaz}| = e^{-aR \sin \theta}$, and this number is very small except near the two ends of the semicircle.
So the factor $e^{iaz}$ kills most of the long path.
This is why the bound stays small even when the radius $R$ is large.
</document>

<document source="https://math2.tamnd.com/node/0PNRKMRC55.md" tag="0PNRKMRC55" kind="concept">
# Random variable

## Summary

A random variable is a quantity whose value depends on the outcome of a random experiment.
Formally it is a measurable function on the sample space of a probability space. The node [[random variable|0DKRR22AEQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0PNWCM559P.md" tag="0PNWCM559P" kind="statement">
# Interior is open

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then the [[interior|0C38S0KMJK]] $\operatorname{int} S$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: $\operatorname{int} S$ is a union of open sets, and each union of open sets is open by the axioms of a topology.
It follows that $\operatorname{int}(\operatorname{int} S) = \operatorname{int} S$.

## Explanations

### L1 human

The interior is made by putting together open sets, and open sets put together give an open set.
</document>

<document source="https://math2.tamnd.com/node/0PP5ZAJPP7.md" tag="0PP5ZAJPP7" kind="definition">
# Beta distribution

## Definition

Let $\alpha > 0$ and $\beta > 0$.
The beta distribution with parameters $\alpha$ and $\beta$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} & \text{if } 0 < x < 1, \\ 0 & \text{otherwise,} \end{cases}$$
where $B(\alpha, \beta) = \int_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} \, dt$ is the beta function.

## Remarks

The mean is $\alpha / (\alpha + \beta)$ and the variance is $\dfrac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$.
$\operatorname{Beta}(1, 1)$ is the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$.
The beta function satisfies $B(\alpha, \beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha + \beta)$.
In Bayesian statistics, the beta distribution is the conjugate prior of the success probability $p$ of binomial data. A $\operatorname{Beta}(\alpha, \beta)$ prior and $k$ successes in $n$ trials give a $\operatorname{Beta}(\alpha + k, \beta + n - k)$ posterior.

## Explanations

### L1 human

The beta distribution describes a random proportion, a number between $0$ and $1$.
Large $\alpha$ pushes the mass toward $1$, large $\beta$ pushes it toward $0$, and $\alpha = \beta$ gives a shape that is symmetric about $1/2$.
</document>

<document source="https://math2.tamnd.com/node/0PPG793642.md" tag="0PPG793642" kind="statement">
# Determinant is multilinear in the rows

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $1 \le i \le n$.
Let $A$, $B$, $C$ be $n \times n$ matrices over $K$ that agree outside row $i$.
Let $a$, $b$, $c$ be row $i$ of $A$, $B$, $C$, and assume $c = sa + tb$ with $s, t \in K$.
Then
$$\det C = s \det A + t \det B,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 1 of the definition of the determinant.
The determinant is not additive in the whole matrix: in general $\det(A + B) \ne \det A + \det B$.
By $\det A^T = \det A$, the determinant is also multilinear in the columns.

## Explanations

### L1 human

If we fix all rows but one, then the determinant is a linear function of the remaining row.
So a sum in one row splits the determinant into a sum of two determinants.
A factor in one row comes out of the determinant.
</document>

<document source="https://math2.tamnd.com/node/0PPP2J2DJV.md" tag="0PPP2J2DJV" kind="statement">
# Second isomorphism theorem

## Statement

Let $G$ be a [[group|G5S1M9]], let $S$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then $SN$ is a subgroup of $G$, $N$ is normal in $SN$, and $S \cap N$ is normal in $S$.
Moreover, the map $s(S \cap N) \mapsto sN$ is an [[isomorphism|08MG7H8JW5]] from $S/(S \cap N)$ to $SN/N$.

## Remarks

The proof applies the [[first isomorphism theorem|0CXXJNG5B6]] to the homomorphism $S \to SN/N$, $s \mapsto sN$, which is surjective with kernel $S \cap N$.
For finite groups, it gives $|SN| = |S| \, |N| / |S \cap N|$.
It is enough that $S$ normalizes $N$, that is, $sNs^{-1} = N$ for each $s \in S$.

## Explanations

### L1 human

Picture a diamond with $SN$ at the top, $S$ and $N$ in the middle, and $S \cap N$ at the bottom.
The theorem says that the two slanted sides opposite each other give the same quotient group.
For integers, with $S = 4\mathbb{Z}$ and $N = 6\mathbb{Z}$, it gives $4\mathbb{Z}/12\mathbb{Z} \cong 2\mathbb{Z}/6\mathbb{Z}$, both cyclic of order $3$.
</document>

<document source="https://math2.tamnd.com/node/0PQ2GGY1QV.md" tag="0PQ2GGY1QV" kind="definition">
# Null space

## Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
The *null space* of $A$ is
$$\operatorname{Null}(A) = \{x \in F^n : Ax = 0\}.$$
It is the [[kernel|0S23E6YE83]] of the linear map $x \mapsto Ax$.

## Remarks

The null space is a subspace of $F^n$, of dimension $n - \rank A$, the [[nullity|0APCFW7639]] of $A$.
Row equivalent matrices have the same null space.
The null space of $\begin{pmatrix} 1 & 1 \end{pmatrix}$ is the line spanned by $(1, -1)^T$.
Each vector of the null space is orthogonal to each row of $A$, for the dot product.

## Explanations

### L1 human

The null space of a matrix is the set of solutions of $Ax = 0$.
We find a basis of it by row reduction: each free variable gives one basis vector.
It is the kernel of the matrix, viewed as a linear map.
</document>

<document source="https://math2.tamnd.com/node/0PQEBTYDZM.md" tag="0PQEBTYDZM" kind="statement">
# Probability of the complement

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A \in \mathcal{F}$ be an [[event|00SQGJNRTB]].
Let $A^c = \Omega \setminus A$ be the [[complement|0B1FSS1VZW]] of $A$.
Then
$$P(A^c) = 1 - P(A).$$

## Remarks

The proof: $A$ and $A^c$ are disjoint and $A \cup A^c = \Omega$, so $P(A) + P(A^c) = P(\Omega) = 1$ by additivity.
With $A = \Omega$, the rule gives $P(\varnothing) = 0$.
The rule is useful when $A^c$ is simpler than $A$. For example, the probability of at least one six in four rolls of a fair die is $1 - (5/6)^4 \approx 0.518$.

## Explanations

### L1 human

Either $A$ happens or it does not, and the two chances add up to $1$.
If the chance of rain is $0.3$, then the chance of no rain is $0.7$.
</document>

<document source="https://math2.tamnd.com/node/0PQQGFAWSH.md" tag="0PQQGFAWSH" kind="statement">
# Alternating sum of binomial coefficients

## Statement

Let $n$ be a positive integer.
Then
$$\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.$$

## Remarks

For $n = 0$ the sum is $1$, so the hypothesis $n \ge 1$ is necessary.
For $n = 4$: $1 - 4 + 6 - 4 + 1 = 0$.
The identity is the [[binomial theorem|0EHWPWGHB2]] with $x = -1$ and $y = 1$. It says that a nonempty finite set has as many subsets of even size as of odd size.

## Explanations

### L1 human

Add and subtract the numbers of a row of [[Pascal's triangle|05DME4TZ3K]] in turn, and you get $0$.
For a nonempty set, add or remove one fixed element. This pairs the even subsets with the odd subsets.
</document>

<document source="https://math2.tamnd.com/node/0PR5P9Z4KM.md" tag="0PR5P9Z4KM" kind="statement">
# Principal axis theorem

## Statement

Let $A$ be a real symmetric $n \times n$ matrix, and let $Q(x) = x^T A x$ for $x \in \R^n$.
Then there are an [[orthogonal matrix|0W02QESPJQ]] $P$ and real numbers $\lambda_1, \dots, \lambda_n$ such that, for $x = P y$,
$$Q(x) = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2.$$
The numbers $\lambda_i$ are the eigenvalues of $A$, and the columns of $P$ are orthonormal eigenvectors of $A$.

## Remarks

It follows from the [[spectral theorem|0DGZBQWZWN]] $A = P D P^T$ with $D$ diagonal.
The lines spanned by the columns of $P$ are the *principal axes* of $Q$.
For example, $Q(x) = 5x_1^2 + 4x_1 x_2 + 2x_2^2$ has the eigenvalues $6$ and $1$, so it becomes $6y_1^2 + y_2^2$.
So the curve $Q(x) = 1$ is an ellipse, with semi-axes $1/\sqrt 6$ and $1$.

## Explanations

### L1 human

Each real quadratic form becomes a sum of squares after a rotation of the axes, possibly with a reflection.
The new axes are the principal axes of the form.
This is how we find the shape of an ellipse or a hyperbola from its equation.
</document>

<document source="https://math2.tamnd.com/node/0PR88XHDMB.md" tag="0PR88XHDMB" kind="statement">
# Dirichlet convolution of multiplicative functions is multiplicative

## Statement

Let $f$ and $g$ be [[multiplicative functions|0AP80TT660]].
Then their [[Dirichlet convolution|0KH1N66QWW]] $f * g$ is a multiplicative function.

## Remarks

Let $m$ and $n$ be coprime. The proof uses that each divisor of $mn$ is a unique product of a divisor of $m$ and a divisor of $n$.
With $f = \mathrm{id}_k \colon n \mapsto n^k$ and $g = \mathbf{1}$, the theorem gives the [[multiplicativity of the divisor function|04WC36XKV6]], since $\sigma_k = \mathrm{id}_k * \mathbf{1}$.
The Dirichlet inverse of a multiplicative function is also multiplicative.

## Explanations

### L1 human

Multiplicative functions stay multiplicative when you convolve them.
So a new function built from known ones by divisor sums keeps the property, and you can compute it from prime powers.
</document>

<document source="https://math2.tamnd.com/node/0PRAS55318.md" tag="0PRAS55318" kind="definition">
# Angle between two vectors

## Definition

Let $n$ be a positive integer, and let $u$ and $v$ be nonzero [[vectors|0QN3D80AWP]] in $\mathbb{R}^n$.
Let $u \cdot v = u_1 v_1 + \dots + u_n v_n$ be their dot product, and let $\|u\|$ and $\|v\|$ be their [[lengths|0XKBKGTCPC]].
The angle between $u$ and $v$ is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that
$$\cos \theta = \frac{u \cdot v}{\|u\| \, \|v\|}.$$
The Cauchy–Schwarz inequality $|u \cdot v| \leq \|u\| \, \|v\|$ makes sure that the quotient is in $[-1, 1]$. Thus $\theta = \arccos\left(\frac{u \cdot v}{\|u\| \, \|v\|}\right)$ exists.

## Remarks

The angle is $0$ if and only if $v = c u$ for a real number $c > 0$. It is $\pi$ if and only if $v = c u$ for a real number $c < 0$.
The angle does not depend on the order of the vectors. It does not change when you multiply $u$ or $v$ by a positive number.
Mathlib uses the same formula for all vectors. With the convention $0 / 0 = 0$, the angle between the zero vector and any vector is then $\pi / 2$.

## Explanations

### L1 human

Put the two arrows at the same start point. The angle between them is the opening between the arrows, measured in radians from $0$ to $\pi$. For example, $(1, 0)$ and $(0, 1)$ have dot product $0$, so the angle between them is $\arccos 0 = \pi / 2$, a right angle.
</document>

<document source="https://math2.tamnd.com/node/0PSKT5W817.md" tag="0PSKT5W817" kind="concept">
# Identity function

## Summary

The identity function of a set sends each element to itself.
</document>

<document source="https://math2.tamnd.com/node/0PSRCMCM5X.md" tag="0PSRCMCM5X" kind="definition">
# Moore–Penrose inverse

## Definition

Let $A$ be an $m \times n$ complex matrix.
There is exactly one $n \times m$ complex matrix $A^+$ with these four properties:

1. $A A^+ A = A$.
2. $A^+ A A^+ = A^+$.
3. $(A A^+)^* = A A^+$.
4. $(A^+ A)^* = A^+ A$.

This matrix is the *Moore–Penrose inverse*, or *pseudoinverse*, of $A$, where $^*$ is the [[conjugate transpose|0DXKE10PHF]].

## Remarks

If $A$ is invertible, then $A^+ = A^{-1}$.
If the columns of $A$ are linearly independent, then $A^+ = (A^* A)^{-1} A^*$.
For $b \in \C^m$, the vector $A^+ b$ is the least squares solution of $Ax = b$ with the least norm.
If $A = U \Sigma V^*$ is a singular value decomposition, then $A^+ = V \Sigma^+ U^*$, where $\Sigma^+$ inverts the nonzero singular values and transposes.

## Explanations

### L1 human

The pseudoinverse extends the inverse to all matrices, also singular and rectangular ones.
It gives the best possible solution of $Ax = b$ when no exact solution exists or when there are many.
For the zero matrix, the pseudoinverse is the zero matrix of the transposed size.
</document>

<document source="https://math2.tamnd.com/node/0PT52XQZD7.md" tag="0PT52XQZD7" kind="concept">
# Product of two subgroups

## Summary

The product of two subgroups $H$ and $K$ of a group is the set $HK$ of all products $hk$ with $h \in H$ and $k \in K$.
It is a subgroup exactly when $HK = KH$.
</document>

<document source="https://math2.tamnd.com/node/0PTBM7K0R2.md" tag="0PTBM7K0R2" kind="concept">
# Algebraic extension

## Summary

An algebraic extension is a field extension in which every element of the larger field is algebraic over the smaller field.
The [[definition of an algebraic extension|09EYMKMSB9]] states it.
</document>

<document source="https://math2.tamnd.com/node/0PTF1PTJK0.md" tag="0PTF1PTJK0" kind="definition">
# Endomorphism

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
An *endomorphism* of $V$ is a [[linear map|0NRYHJPQG2]] $T \colon V \to V$.
The set of all endomorphisms of $V$ is written $\End_F(V)$.

## Remarks

An endomorphism is also called a *linear operator* on $V$.
Endomorphisms can be composed with each other, so their powers $T^k$ and polynomials $p(T)$ are defined.
Under composition and addition, $\End_F(V)$ is the [[algebra of endomorphisms|0E3X2G1ZQB]].
Eigenvalues, the trace and the determinant are defined for endomorphisms of finite-dimensional spaces.

## Explanations

### L1 human

An endomorphism maps a space into itself.
So we can apply it again and again, and compare a vector with its image.
In a basis, an endomorphism of an $n$-dimensional space is a square $n \times n$ matrix.
</document>

<document source="https://math2.tamnd.com/node/0PTVNH4GPS.md" tag="0PTVNH4GPS" kind="concept">
# Linear congruence

## Summary

A linear congruence is a [[congruence|0TPW25CBN6]] $a x \equiv b \pmod n$, with given integers $a$ and $b$ and a positive integer $n$.
Its solutions are the integers $x$ that satisfy it.
Let $d = \gcd(a, n)$. The congruence has a solution if and only if $d \mid b$.
In this case, the solutions form exactly $d$ residue classes modulo $n$, and one residue class modulo $n / d$.

## Remarks

When $\gcd(a, n) = 1$, the unique solution modulo $n$ is $x \equiv a^{-1} b \pmod n$, where $a^{-1}$ is the [[modular multiplicative inverse|00GXTQMY8Z]] of $a$.
The congruence $a x \equiv b \pmod n$ is the same as the [[linear Diophantine equation|0R9GF3JD0K]] $a x - n y = b$.
For example, $6 x \equiv 4 \pmod{10}$ has $d = 2$. Its solutions are $x \equiv 4 \pmod 5$, that is, $x \equiv 4$ or $x \equiv 9 \pmod{10}$.

## Explanations

### L1 human

A linear congruence is a linear equation in modular arithmetic.
To solve $2 x \equiv 3 \pmod 5$, find the inverse $3$ of $2$ modulo $5$, and multiply: $x \equiv 9 \equiv 4 \pmod 5$.
Check: $2 \cdot 4 = 8 \equiv 3 \pmod 5$.
</document>

<document source="https://math2.tamnd.com/node/0PV74TR5FP.md" tag="0PV74TR5FP" kind="concept">
# Maximal ideal

## Summary

A maximal ideal of a ring is a proper ideal that is not contained in a larger proper ideal.
The maximal ideals of $\mathbb{Z}$ are the ideals $p\mathbb{Z}$ for $p$ prime.
The [[definition of a maximal ideal|09J6KGT65G]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0PVCTCKD3C.md" tag="0PVCTCKD3C" kind="statement">
# Expectation with respect to the binomial distribution

## Statement

Let $n \ge 1$ be an integer, let $p \in [0, 1]$, and let $X$ have the [[binomial distribution|0VRWZDWW6B]] $\operatorname{Bin}(n, p)$. Then
$$E[X] = \sum_{k=0}^{n} k \binom{n}{k} p^k (1 - p)^{n - k} = np.$$

## Remarks

A short proof: $X$ has the same law as $Y_1 + \dots + Y_n$ with independent [[Bernoulli|0J97P5KZY0]] $Y_i$ with $E[Y_i] = p$. Then [[linearity of expectation|0P7PTC8DDE]] gives $np$.
A direct proof uses $k \binom{n}{k} = n \binom{n-1}{k-1}$ and the binomial theorem.
The variance of $X$ is $np(1 - p)$.

## Explanations

### L1 human

In $n$ trials with success probability $p$, you expect $np$ successes.
For $100$ tosses of a fair coin, the expected number of heads is $50$.
</document>

<document source="https://math2.tamnd.com/node/0PVG1CRAJN.md" tag="0PVG1CRAJN" kind="definition">
# Injective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is injective when, for all $a, a' \in A$, $f(a) = f(a')$ implies $a = a'$.

## Remarks

Other names are one-to-one function and injection.
The function $x \mapsto 2x$ on $\mathbb{R}$ is injective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $f(1) = f(-1)$.
An injective function with a nonempty domain has a [[left inverse|02486SD328]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].

## Explanations

### L1 human

An injective function never sends two inputs to the same place.
To prove injectivity, assume $f(a) = f(a')$ and deduce $a = a'$.
</document>

<document source="https://math2.tamnd.com/node/0PVJRACB9H.md" tag="0PVJRACB9H" kind="statement">
# The exponential series has infinite radius of convergence

## Statement

The [[radius of convergence|0660QX1VPR]] of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $\infty$.
So the series converges absolutely for every real $x$.

## Remarks

The ratio of consecutive coefficients is $\frac{1/(n+1)!}{1/n!} = \frac{1}{n + 1} \to 0$. The [[ratio formula|0JBMYAA9QH]] then gives radius $\infty$.
The same holds for the series of sine, cosine, $\sinh$ and $\cosh$.
Mathlib states it as `NormedSpace.expSeries_radius_eq_top`.

## Explanations

### L1 human

The factorials in the denominators grow faster than every power $x^n$.
So the terms of the series go to $0$ fast, for each fixed $x$.
</document>

<document source="https://math2.tamnd.com/node/0PWE3BCHVJ.md" tag="0PWE3BCHVJ" kind="concept">
# Infinite set

## Summary

An infinite set is a set that is not finite.
</document>

<document source="https://math2.tamnd.com/node/0PWX18FM4G.md" tag="0PWX18FM4G" kind="concept">
# Cauchy sequence

## Summary

In a Cauchy sequence, any two terms with large enough indices are as near to each other as we wish. A metric space is complete when each Cauchy sequence in it converges, and the real numbers form a complete space. The node [[Cauchy sequence|0QZMTN5AXX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0PXN9G4MX1.md" tag="0PXN9G4MX1" kind="statement">
# Orthogonal complement of the orthogonal complement

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $W$ be a subspace of $V$.
Then
$$(W^\perp)^\perp = W,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.

## Remarks

Each vector of the subspace is orthogonal to each vector of its orthogonal complement, so $W \subseteq (W^\perp)^\perp$. Both have dimension $\dim V - \dim W^\perp = \dim W$.
In an infinite-dimensional Hilbert space, $(W^\perp)^\perp$ is the closure of $W$.

## Explanations

### L1 human

Taking the orthogonal complement twice gives back the original subspace.
The complement of the complement of a line is the same line.
</document>

<document source="https://math2.tamnd.com/node/0PXW0BN7DW.md" tag="0PXW0BN7DW" kind="concept">
# Commutator of two elements

## Summary

The commutator of two elements $g$ and $h$ of a group is $[g, h] = g^{-1} h^{-1} g h$.
It is the identity exactly when $g$ and $h$ commute.
</document>

<document source="https://math2.tamnd.com/node/0PYEZM6CN9.md" tag="0PYEZM6CN9" kind="concept">
# Liouville number

## Summary

A Liouville number is a real number that has rational approximations $p/q$ with error smaller than $1/q^n$ for each $n$.
The [[definition of a Liouville number|0J1R5NP36S]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0PYSR20HGF.md" tag="0PYSR20HGF" kind="concept">
# Composition of relations

## Summary

The composition of two relations relates $a$ to $c$ when there is an intermediate element $b$.
The first relation relates $a$ to $b$, and the second relation relates $b$ to $c$.
</document>

<document source="https://math2.tamnd.com/node/0PYW26VNT6.md" tag="0PYW26VNT6" kind="concept">
# Affine subspace

## Summary

An affine subspace of an [[affine space|09XNAPTJQ7]] is a subset that is itself an affine space, for the same action of the vectors. Points, lines and planes are the affine subspaces of dimension 0, 1 and 2.
</document>

<document source="https://math2.tamnd.com/node/0Q0H13G8ZB.md" tag="0Q0H13G8ZB" kind="concept">
# Closed ball

## Summary

A closed ball in a metric space is the set of all points at distance at most a given radius from a given centre.
</document>

<document source="https://math2.tamnd.com/node/0Q0Z1BEXTF.md" tag="0Q0Z1BEXTF" kind="definition">
# Centraliser

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The centraliser of $S$ in $G$ is the set $C_G(S) = \{g \in G : gs = sg \text{ for all } s \in S\}$.
For an element $a \in G$, the notation $C_G(a)$ means $C_G(\{a\})$.

## Remarks

The centraliser $C_G(S)$ is a [[subgroup|0X1JXT47GA]] of $G$.
The centraliser of $G$ itself is the [[centre|0DZ93493VN]] $Z(G)$.
The element $a$ lies in its own centraliser $C_G(a)$, and so does each power of $a$.

## Explanations

### L1 human

The centraliser of an element is the set of elements that commute with it.
If the element commutes with everything, its centraliser is the whole group.
The size of the centraliser tells you how many conjugates the element has.
</document>

<document source="https://math2.tamnd.com/node/0Q1D85B8G6.md" tag="0Q1D85B8G6" kind="concept">
# Interval

## Summary

An interval is a set of real numbers that contains each real number between two of its elements.
Examples are $[0, 1]$, $(0, 1)$, $[0, \infty)$ and $\mathbb{R}$ itself.
</document>

<document source="https://math2.tamnd.com/node/0Q1FKQKTKJ.md" tag="0Q1FKQKTKJ" kind="concept">
# Exact equation

## Summary

An exact equation is a first-order equation $M\, dx + N\, dy = 0$ whose left side is the total differential of a function $\psi(x, y)$.
Its solutions lie on the level curves $\psi(x, y) = C$. See the [[definition|0EC7AGJ0PX]].
</document>

<document source="https://math2.tamnd.com/node/0Q2MDD13G4.md" tag="0Q2MDD13G4" kind="concept">
# Singular solution

## Summary

A singular solution of a differential equation is a solution along which uniqueness of solutions fails.
It is often the envelope of a family of solutions, and the general solution does not contain it. See the [[definition|0CZA0NCFWP]].
</document>

<document source="https://math2.tamnd.com/node/0Q2MVQ7JMT.md" tag="0Q2MVQ7JMT" kind="definition">
# Internal direct product

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ and $K$ be [[normal subgroups|0CB0DX5BVV]] of $G$.
The group $G$ is the internal direct product of $H$ and $K$ when $H \cap K = \{e\}$ and $HK = G$.

## Remarks

In this case, $hk = kh$ for all $h \in H$ and $k \in K$, because $hkh^{-1}k^{-1}$ lies in $H \cap K$.
Then the map $(h, k) \mapsto hk$ is an [[isomorphism|08MG7H8JW5]] from the [[direct product|00HF4RJ5QN]] $H \times K$ to $G$.
Also, each element of $G$ is a product $hk$ with $h \in H$ and $k \in K$ in exactly one way.
For example, $\mathbb{Z}/6\mathbb{Z}$ is the internal direct product of $\{0, 3\}$ and $\{0, 2, 4\}$.

## Explanations

### L1 human

Sometimes a group already contains two subgroups that behave like the two factors of a direct product.
They meet only in the identity, they fill the group together, and they are both normal.
Then the group is the direct product of these subgroups, up to isomorphism.
</document>

<document source="https://math2.tamnd.com/node/0Q2QC81VTW.md" tag="0Q2QC81VTW" kind="definition">
# Cumulant generating function

## Definition

Let $X$ be a real random variable with [[moment-generating function|0NNRNGKMFM]] $M_X$.
The cumulant generating function of $X$ is
$$K_X(t) = \ln M_X(t) = \ln E\left[e^{tX}\right],$$
for each $t \in \mathbb{R}$ with $M_X(t) < \infty$.
If $M_X$ is finite on an open interval around $0$, the cumulants $\kappa_n$ of $X$ are the coefficients in $K_X(t) = \sum_{n=1}^{\infty} \kappa_n t^n / n!$, so $\kappa_n = K_X^{(n)}(0)$.

## Remarks

The first cumulants are $\kappa_1 = E[X]$, $\kappa_2 = \operatorname{Var}(X)$ and $\kappa_3 = E[(X - E[X])^3]$.
For independent $X$ and $Y$, $K_{X+Y} = K_X + K_Y$ by the [[product rule for moment-generating functions|0WRZ170YEH]]. So cumulants of independent sums add.
For $X \sim \mathcal{N}(\mu, \sigma^2)$, $K_X(t) = \mu t + \sigma^2 t^2 / 2$. For $X \sim \operatorname{Poisson}(\lambda)$, $K_X(t) = \lambda(e^t - 1)$, so all cumulants equal $\lambda$.
$K_X$ is convex on the interval where it is finite.

## Explanations

### L1 human

Taking the logarithm turns products into sums. So the cumulant generating function of a sum of independent variables is the sum of their cumulant generating functions.
This makes cumulants convenient for sums, for example in the central limit theorem.
</document>

<document source="https://math2.tamnd.com/node/0Q3AKZV74D.md" tag="0Q3AKZV74D" kind="statement">
# Stolz–Cesàro theorem

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences and let $L$ be a real number.
Assume that $(b_n)$ is strictly increasing and that $b_n \to +\infty$ as $n \to \infty$.
Assume that
$$\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L.$$
Then $a_n / b_n \to L$ as $n \to \infty$.

## Remarks

There is also a $0/0$ form. If $a_n \to 0$, $b_n \to 0$, $(b_n)$ is strictly decreasing and the quotient of the differences tends to $L$, then $a_n / b_n \to L$.
With $b_n = n$ it gives the [[convergence of the Cesàro means|0RDRPBMNAC]].
For example, with $a_n = 1 + 2 + \cdots + n$ and $b_n = n^2$, the quotient of differences is $(n+1)/(2n+1) \to 1/2$, so $a_n / n^2 \to 1/2$.
It is a discrete analogue of [[L'Hôpital's rule|0JWT4PPQ7B]].

## History

Otto Stolz proved the $\infty/\infty$ case in a book of 1885. Ernesto Cesàro gave it in an article of 1888.

## Explanations

### L1 human

To compare two sequences that grow without bound, it is enough to compare their steps.
If each step of $a_n$ is about $L$ times the step of $b_n$, then $a_n$ is about $L$ times $b_n$ in the long run.
</document>

<document source="https://math2.tamnd.com/node/0Q3F20XB0A.md" tag="0Q3F20XB0A" kind="definition">
# Image of a set

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $A$.
The image of $S$ under $f$ is the set
$$f(S) = \{f(s) \mid s \in S\} = \{b \in B \mid \exists s \in S, \, f(s) = b\}.$$

## Remarks

Another notation is $f[S]$. It avoids a confusion with $f(a)$ for an element $a$.
The image $f(A)$ of the whole domain is the [[range|0E9GTAV478]] of $f$.
Example: for $f(x) = x^2$ on $\mathbb{R}$, $f([-1, 2]) = [0, 4]$.
The image of a union is the union of the images, but the image of an intersection can be smaller. See [[image of an intersection|0RX7DDBYCK]].

## Explanations

### L1 human

To compute $f(S)$, apply $f$ to each element of $S$ and collect the results.
For $f(x) = x + 1$ and $S = \{1, 2, 3\}$, the image is $\{2, 3, 4\}$.
</document>

<document source="https://math2.tamnd.com/node/0Q3HYFT9RQ.md" tag="0Q3HYFT9RQ" kind="statement">
# Integral of a power

## Statement

Let $n \ge 0$ be an integer and let $a, b \in \mathbb{R}$.
Then
$$\int_a^b x^n \, dx = \frac{b^{n+1} - a^{n+1}}{n + 1}.$$

## Remarks

By the [[power rule|0JZGW75C5Z]], $x^{n+1}/(n + 1)$ is an antiderivative of $x^n$. Apply the [[second fundamental theorem|0XYAFCN0PT]].
The same formula holds for each real exponent $r \ne -1$, on intervals inside $(0, \infty)$. The case $r = -1$ gives the [[logarithm|0K1N1Y4EGZ]].
Mathlib states it as `integral_pow`.

## Explanations

### L1 human

To integrate $x^n$, raise the power by one and divide by the new power.
For example, $\int_0^1 x^2 \, dx = 1/3$.
</document>

<document source="https://math2.tamnd.com/node/0Q3ZT8J05R.md" tag="0Q3ZT8J05R" kind="concept">
# Linear equation

## Summary

A linear equation is an equation in which each unknown appears only to the first power and is multiplied only by a constant.
Linear equations are the basic objects of linear algebra.
</document>

<document source="https://math2.tamnd.com/node/0Q4CBD3HX0.md" tag="0Q4CBD3HX0" kind="concept">
# Domain

## Summary

The domain of a function is the set of its inputs.
</document>

<document source="https://math2.tamnd.com/node/0Q4CJ5NAT1.md" tag="0Q4CJ5NAT1" kind="concept">
# Arc length parametrisation

## Summary

An arc length parametrisation of a [[regular curve|0V7090SSPC]] uses the distance travelled along the curve as the parameter.
In this parametrisation the curve has speed $1$ at each point.
</document>

<document source="https://math2.tamnd.com/node/0Q4DR37DVE.md" tag="0Q4DR37DVE" kind="definition">
# Uniform continuity

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is uniformly continuous if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(y)| < \varepsilon$ for all $x, y \in D$ with $|x - y| < \delta$.

## Remarks

Each uniformly continuous function is continuous. In the [[epsilon-delta form|0W8YHHGCJG]] of continuity, $\delta$ may depend on the point. Here one $\delta$ works for all points.
The function $x^2$ is continuous on $\mathbb{R}$ but [[not uniformly continuous|0A8JXHYTJC]].
Mathlib defines it for uniform spaces, as `UniformContinuous`.
Eduard Heine published the first definition of uniform continuity, in 1870.

## Explanations

### L1 human

For a uniformly continuous function, one input tolerance works everywhere in the domain.
The graph has no places where it becomes steeper and steeper without bound.
</document>

<document source="https://math2.tamnd.com/node/0Q4QJWEHW3.md" tag="0Q4QJWEHW3" kind="concept">
# Second derivative

## Summary

The second derivative of a function is the derivative of its derivative. It measures how the slope changes, and so how the graph bends. The node [[second derivative|0Y9D2FJRQ4]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Q5A47J48D.md" tag="0Q5A47J48D" kind="statement">
# Darboux's theorem

## Statement

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be differentiable on $[a, b]$, with [[one-sided derivatives|0HXEX0DPRA]] at the endpoints.
Let $y$ be a real number strictly between $f'(a)$ and $f'(b)$.
Then there is $c \in (a, b)$ with $f'(c) = y$.

## Remarks

Assume $f'(a) < y < f'(b)$. The function $g(x) = f(x) - y x$ has $g'(a) < 0 < g'(b)$. So its minimum on $[a, b]$, given by the [[extreme value theorem|0WS1A8E2P9]], is at an interior point $c$. [[Fermat's theorem|0PCVBRV9VJ]] gives $g'(c) = 0$.
So a derivative has the intermediate value property, even when it is not continuous.
In particular, a derivative has no jump discontinuities. Mathlib states it as `exists_hasDerivWithinAt_eq_of_gt_of_lt`.

## Explanations

### L1 human

The derivative of a function cannot skip values, just as a continuous function cannot.
For example, no function has derivative $0$ on the left of a point and $1$ on the right.
</document>

<document source="https://math2.tamnd.com/node/0Q5CG2NEQ8.md" tag="0Q5CG2NEQ8" kind="concept">
# Statistic

## Summary

A statistic is a quantity computed from a sample, by a fixed rule that does not depend on unknown parameters. Examples are the sample mean and the sample maximum.
The node [[statistic|036756NFD6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Q5G5DJ4P2.md" tag="0Q5G5DJ4P2" kind="concept">
# Derived series

## Summary

The derived series of a group is the sequence of iterated commutator subgroups $G \supseteq G' \supseteq G'' \supseteq \cdots$.
A group is solvable if and only if this series reaches the trivial subgroup.
</document>

<document source="https://math2.tamnd.com/node/0Q5NA8Y362.md" tag="0Q5NA8Y362" kind="concept">
# Galois group of a polynomial

## Summary

The Galois group of a polynomial over a field is the Galois group of its splitting field.
It acts on the roots of the polynomial by permutations.
The [[definition of the Galois group of a polynomial|0JZY3R908H]] states it.
</document>

<document source="https://math2.tamnd.com/node/0Q5SM60HTH.md" tag="0Q5SM60HTH" kind="definition">
# Permutation matrix

## Definition

Let $n \ge 1$ and let $\sigma$ be a permutation of $\{1, \dots, n\}$.
The *permutation matrix* of $\sigma$ is the $n \times n$ matrix $P_\sigma$ with entries
$$(P_\sigma)_{ij} = \begin{cases} 1 & \text{if } i = \sigma(j), \\ 0 & \text{otherwise.} \end{cases}$$
A *permutation matrix* is a matrix of the form $P_\sigma$.

## Remarks

The matrix $P_\sigma$ sends the standard basis vector $e_j$ to $e_{\sigma(j)}$.
With this convention, $P_\sigma P_\tau = P_{\sigma \tau}$; some authors use the transposed convention.
Each permutation matrix is invertible with $P_\sigma^{-1} = P_\sigma^T$, and $\det P_\sigma = \sgn \sigma$.
The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is the permutation matrix that exchanges two coordinates.

## Explanations

### L1 human

A permutation matrix is the identity matrix with its columns shuffled.
Multiplying a vector by it shuffles the entries of the vector in the same way.
Each row and each column has exactly one $1$.
</document>

<document source="https://math2.tamnd.com/node/0Q70YFB6ZM.md" tag="0Q70YFB6ZM" kind="concept">
# Regular graph

## Summary

A regular graph is a graph in which all vertices have the same degree. Regular graphs are a basic family with strong symmetry and counting properties. The node [[regular graph|0KA4S8R39M]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Q7K4AXE53.md" tag="0Q7K4AXE53" kind="example">
# The sequence r^n tends to zero for |r| < 1

## Example

Let $r$ be a real number with $|r| < 1$.
Then $r^n \to 0$ as $n \to \infty$.
For $r = 0$ this is clear. For $0 < |r| < 1$, write $1/|r| = 1 + h$ with $h > 0$.
By Bernoulli's inequality, $(1 + h)^n \ge 1 + n h > n h$ for each $n \ge 1$.
Thus $0 \le |r^n| < 1/(n h)$, and the [[squeeze theorem|0XMF00T4XQ]] gives $|r^n| \to 0$, so $r^n \to 0$.

## Remarks

The [[geometric sequence|0MNM54WB99]] $a r^n$ then tends to $0$ for each real $a$.
For $r = 1$ the sequence is constant. For $r = -1$ it is $1, -1, 1, \dots$, which diverges.
For $|r| > 1$ the sequence $|r^n|$ [[tends to infinity|0TZK91NQE3]].

## Explanations

### L1 human

Multiply a number by $1/2$ again and again: $1, 1/2, 1/4, 1/8, \dots$.
The result gets closer and closer to $0$. The same happens for each factor between $-1$ and $1$.
</document>

<document source="https://math2.tamnd.com/node/0Q7M6K6VD7.md" tag="0Q7M6K6VD7" kind="statement">
# Holomorphic functions with nonzero derivative are conformal

## Statement

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]], and let $z_0 \in U$.
If $f'(z_0) \neq 0$, then $f$ is [[conformal|0VHNKP1N3K]] at $z_0$.
In particular, if $f'(z) \neq 0$ for each $z \in U$, then $f$ is conformal on $U$.

## Remarks

Write $f'(z_0) = r e^{i\varphi}$ with $r > 0$; then the real derivative of $f$ at $z_0$ is $h \mapsto r e^{i\varphi} h$, a rotation by $\varphi$ followed by a scaling by $r$.
The converse also holds: a map that is conformal at each point of an open set $U$ is holomorphic on $U$ with a nonzero derivative.
The condition $f'(z_0) \neq 0$ is necessary: $z \mapsto z^2$ doubles the angles between curves through $0$.
The exponential function is conformal at each point but not injective, so a conformal map need not be injective.

## Explanations

### L1 human

Near $z_0$, a [[complex differentiable|0D1C9Y1EXA]] function acts almost like $z \mapsto f(z_0) + f'(z_0)(z - z_0)$.
Multiplication by the complex number $f'(z_0)$ turns the plane and scales it.
Turning and scaling keep all angles, so $f$ keeps the angles between curves through $z_0$.
</document>

<document source="https://math2.tamnd.com/node/0Q7MMQHC1P.md" tag="0Q7MMQHC1P" kind="concept">
# Nilpotent group

## Summary

A nilpotent group is a group that has a central series of finite length.
Equivalently, its lower central series reaches the trivial subgroup, or its upper central series reaches the whole group.
</document>

<document source="https://math2.tamnd.com/node/0Q88KQD17A.md" tag="0Q88KQD17A" kind="definition">
# Conjugacy class

## Definition

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
The conjugacy class of $a$ is the set $\operatorname{Cl}(a) = \{gag^{-1} : g \in G\}$.

## Remarks

The conjugacy classes are the equivalence classes of the relation "is [[conjugate to|0DXD4QSZ8A]]", so they partition $G$.
They are the [[orbits|0YQPZB5Q1G]] of the [[action by conjugation|0P8CY9V437]].
The class of $a$ is $\{a\}$ if and only if $a$ lies in the [[centre|0DZ93493VN]] of $G$.
In the symmetric group $S_n$, the conjugacy classes are given by the [[cycle types|0J438GAC8K]].

## Explanations

### L1 human

The conjugacy class of an element collects all the elements that look like it after a renaming by conjugation.
In an abelian group, each element is alone in its class.
In $S_3$, the three transpositions form one class, and the two $3$-cycles form another class.
</document>

<document source="https://math2.tamnd.com/node/0Q8BZXVT1Z.md" tag="0Q8BZXVT1Z" kind="definition">
# Orientation of a real vector space

## Definition

Let $V$ be a real vector space of finite dimension $n \ge 1$.
Two ordered [[bases|0KCQ9Y3QKA]] $B$ and $B'$ of $V$ have the *same orientation* when the [[change of basis matrix|0MKRHZB3WS]] from $B$ to $B'$ has positive [[determinant|00KDZRW4C0]].
This is an equivalence relation with exactly two classes.
An *orientation* of $V$ is a choice of one of these two classes.

## Remarks

A vector space with a chosen orientation is *oriented*, and the bases in the chosen class are *positively oriented*.
The standard orientation of $\R^n$ is the class of the standard basis.
In $\R^2$, exchanging the two basis vectors reverses the orientation.
An invertible linear map keeps the orientation if and only if its determinant is positive.

## Explanations

### L1 human

An orientation fixes what counts as turning to the left, or as a right-handed system.
In the plane, a counterclockwise pair of basis vectors and a clockwise pair have opposite orientations.
A mirror reflection changes the orientation, and a rotation keeps it.
</document>

<document source="https://math2.tamnd.com/node/0Q8DGAZSJ4.md" tag="0Q8DGAZSJ4" kind="statement">
# Third Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.
Then the number $n_p$ of [[Sylow p-subgroups|0053CQE9Y2]] of $G$ satisfies $n_p \equiv 1 \pmod p$.

## Remarks

The proof lets a Sylow $p$-subgroup $P$ act by conjugation on $\operatorname{Syl}_p(G)$, and $P$ is the only fixed point.
Together with [[the divisibility condition|0M3PYJ3TSE]], this often forces $n_p = 1$.
In $S_4$, $n_2 = 3$ and $n_3 = 4$, and both numbers are $\equiv 1$ modulo the prime.

## Explanations

### L1 human

The number of Sylow $p$-subgroups is one more than a multiple of $p$.
This simple fact, together with a divisibility condition, leaves only a few possible values.
Often the only possible value is $1$, and then the Sylow subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0Q8HZTJC7G.md" tag="0Q8HZTJC7G" kind="definition">
# Orthogonal projection

## Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$.
By the [[orthogonal decomposition|0SP7P0G7KQ]], each $v \in V$ is $v = w + w'$ with unique $w \in W$ and $w' \in W^\perp$.
The *orthogonal projection* onto $W$ is the map $P_W \colon V \to V$, $P_W(v) = w$.

## Remarks

$P_W$ is a linear [[projection|033DA1WE8Y]] with image $W$ and kernel $W^\perp$, and it is self-adjoint.
For an orthonormal basis $e_1, \dots, e_k$ of $W$, $P_W(v) = \sum_{i=1}^{k} \langle v, e_i \rangle e_i$.
For the line spanned by $u \ne 0$, $P(v) = \frac{\langle v, u \rangle}{\langle u, u \rangle} u$.
$P_W(v)$ is the point of $W$ [[closest to $v$|0MCDDSCB9Z]].

## Explanations

### L1 human

The orthogonal projection drops a perpendicular from a vector onto a subspace.
The foot of the perpendicular is the projection.
In $\R^3$, projecting onto the $xy$-plane sets the $z$-coordinate to zero.
</document>

<document source="https://math2.tamnd.com/node/0Q8Y862DTZ.md" tag="0Q8Y862DTZ" kind="concept">
# Outer automorphism group

## Summary

The outer automorphism group of a group is the quotient of its automorphism group by the inner automorphisms.
It measures how many automorphisms do not come from conjugation.
</document>

<document source="https://math2.tamnd.com/node/0Q9EZS90GY.md" tag="0Q9EZS90GY" kind="definition">
# Polynomial ring

## Definition

Let $R$ be a [[ring|0894J980RK]].
A polynomial over $R$ is a sequence $(a_0, a_1, a_2, \dots)$ of elements of $R$ such that $a_n = 0$ for all but finitely many $n$.
The element $a_n$ is the coefficient of degree $n$.
The polynomial ring $R[x]$ is the set of these polynomials, with these operations:
$$(a_n)_n + (b_n)_n = (a_n + b_n)_n, \qquad (a_n)_n \cdot (b_n)_n = (c_n)_n, \quad c_n = \sum_{i + j = n} a_i b_j.$$
With these operations, $R[x]$ is a ring.

## Remarks

Let $x$ be the polynomial $(0, 1, 0, 0, \dots)$, and identify $r \in R$ with $(r, 0, 0, \dots)$.
Then each polynomial with $a_n = 0$ for $n > d$ is $a_0 + a_1 x + \dots + a_d x^d$, in exactly one way.
If $R$ is a [[commutative ring|0GYGHKX834]], then $R[x]$ is commutative.
A polynomial is a formal expression and not a function: over $\mathbb{F}_2$, $x^2 + x$ is a nonzero polynomial, but it is zero at both elements of $\mathbb{F}_2$.

## Explanations

### L1 human

A polynomial is a finite list of coefficients.
You add polynomials coefficient by coefficient.
You multiply them with the rule $x^i \cdot x^j = x^{i + j}$ and the distributive law.
</document>

<document source="https://math2.tamnd.com/node/0Q9PGXD9E4.md" tag="0Q9PGXD9E4" kind="definition">
# Diagonal matrix

## Definition

Let $R$ be a ring and let $n \ge 0$ be an integer.
A [[square matrix|0JXZKX526R]] $D = (d_{ij})$ of order $n$ over $R$ is *diagonal* when $d_{ij} = 0$ for all $i \ne j$.
For $d_1, \dots, d_n \in R$, the matrix $\operatorname{diag}(d_1, \dots, d_n)$ is the diagonal matrix with $d_{ii} = d_i$ for each $i$.

## Remarks

The product of two diagonal matrices is diagonal: $\operatorname{diag}(a_1, \dots, a_n) \operatorname{diag}(b_1, \dots, b_n) = \operatorname{diag}(a_1 b_1, \dots, a_n b_n)$.
Over a commutative ring, diagonal matrices commute with each other.
The [[determinant|00KDZRW4C0]] of a diagonal matrix is the product of its diagonal entries.
A matrix that is similar to a diagonal matrix is [[diagonalisable|07SQCNV233]].

## Explanations

### L1 human

A diagonal matrix has numbers only on its main diagonal, from the top left to the bottom right.
As a map, it stretches each coordinate axis by its own factor.
Diagonal matrices are easy to add, multiply and raise to powers.
</document>

<document source="https://math2.tamnd.com/node/0QA1HBD9Z4.md" tag="0QA1HBD9Z4" kind="concept">
# Convergence in probability

## Summary

A sequence of random variables converges in probability when, for each tolerance, the probability of a deviation larger than this tolerance tends to $0$.
The node [[convergence in probability|0EFDDXTVD6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0QA8CHH8MG.md" tag="0QA8CHH8MG" kind="definition">
# Field

## Definition

A field is a [[commutative ring|0GYGHKX834]] $K$ with these two properties.

1. $1 \neq 0$ in $K$.
2. Every nonzero element of $K$ is a [[unit|019NFGFPR5]]: for each $a \in K$ with $a \neq 0$, there is $b \in K$ with $a b = 1$.

## Remarks

The rings $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields. The ring $\mathbb{Z}$ is not a field.
For a prime number $p$, the ring [[$\mathbb{Z}/p\mathbb{Z}$ is a field|0NV6E43P4A]].
Every field is an [[integral domain|09S3PFBEPM]].
A [[division ring|0WZXK05ME7]] has the same axioms without the commutativity of the multiplication.

## Explanations

### L1 human

A field is a number system where the four operations of arithmetic work.
You can add, subtract, multiply, and divide by any number that is not zero.
The rational numbers are the simplest example.
</document>

<document source="https://math2.tamnd.com/node/0QAJHFGZZJ.md" tag="0QAJHFGZZJ" kind="concept">
# Uniqueness quantifier

## Summary

The uniqueness quantifier $\exists!$ expresses "there exists exactly one".
</document>

<document source="https://math2.tamnd.com/node/0QAR0XDZH4.md" tag="0QAR0XDZH4" kind="definition">
# Joint probability distribution

## Definition

Let $X_1, \dots, X_n$ be real [[random variables|0DKRR22AEQ]] on one probability space $(\Omega, \mathcal{F}, P)$.
Their joint distribution is the [[law|0TST4MHA3A]] of the random vector $(X_1, \dots, X_n) \colon \Omega \to \mathbb{R}^n$. It is the probability measure on the Borel sets of $\mathbb{R}^n$ with
$$\mu(B) = P((X_1, \dots, X_n) \in B).$$
Their joint cumulative distribution function is $F(x_1, \dots, x_n) = P(X_1 \le x_1, \dots, X_n \le x_n)$.

## Remarks

For discrete variables, the joint probability mass function $p(x_1, \dots, x_n) = P(X_1 = x_1, \dots, X_n = x_n)$ determines the joint distribution.
The variables have a joint density $f$ if $\mu(B) = \int_B f(x) \, dx$ for each Borel set $B \subseteq \mathbb{R}^n$.
The joint distribution determines the [[marginal distributions|0AXP2Z5N9H]], but the marginal distributions do not determine the joint distribution.
For two rolls of a fair die, the joint mass function of the two results is $1/36$ at each pair $(i, j)$ with $1 \le i, j \le 6$.

## Explanations

### L1 human

The joint distribution describes several random quantities together, not one by one.
For example, it says how likely it is that a person is tall and also heavy.
It gives more than the chance of each of the two alone.
</document>

<document source="https://math2.tamnd.com/node/0QB913WM5E.md" tag="0QB913WM5E" kind="definition">
# Probability distribution

## Definition

Let $E$ be a set with a σ-algebra $\mathcal{E}$.
A probability distribution on $(E, \mathcal{E})$ is a [[probability measure|0VGMEBVW1M]] $\mu$ on $(E, \mathcal{E})$.
A probability distribution on $\mathbb{R}$ is a probability measure on $\mathbb{R}$ with its [[Borel σ-algebra|0BVHZXWKCD]] $\mathcal{B}(\mathbb{R})$.

## Remarks

Each random variable $X$ with values in $E$ gives a distribution, its [[law|0TST4MHA3A]] $B \mapsto P(X \in B)$. Conversely, each distribution $\mu$ is the law of some random variable, for example the identity map on $(E, \mathcal{E}, \mu)$.
A distribution on $\mathbb{R}$ is determined by its [[cumulative distribution function|0MXMS3AE13]].
Important types are discrete distributions, given by a [[probability mass function|0WF4EJP91W]], and absolutely continuous distributions, given by a [[probability density function|04YG690N5V]].

## Explanations

### L1 human

A distribution tells you which values are possible and how likely each range of values is.
For a fair die, the distribution gives each of the values $1, \dots, 6$ the probability $1/6$.
</document>

<document source="https://math2.tamnd.com/node/0QBYFZV91P.md" tag="0QBYFZV91P" kind="concept">
# Subspace metric

## Summary

The subspace metric on a subset of a metric space is the restriction of the metric to pairs of points of the subset.
It makes each subset of a metric space a metric space.
</document>

<document source="https://math2.tamnd.com/node/0QD4E5JYPT.md" tag="0QD4E5JYPT" kind="statement">
# Reciprocal rule

## Statement

Let $g$ be a real function on an open interval $I$, differentiable at $a \in I$, with $g(a) \ne 0$.
Then $1/g$ is differentiable at $a$, and
$$\left( \frac{1}{g} \right)'(a) = -\frac{g'(a)}{g(a)^2}.$$

## Remarks

The difference quotient is $-\frac{g(a + h) - g(a)}{h} \cdot \frac{1}{g(a + h) g(a)}$. Continuity of $g$ at $a$ gives the limit.
For example, $(1/x)' = -1/x^2$ for $x \ne 0$.
Mathlib states it as `HasDerivAt.inv`.

## Explanations

### L1 human

When $g$ grows, $1/g$ shrinks, so the two derivatives have opposite signs.
The factor $1/g^2$ shows that $1/g$ changes fast where $g$ is small.
</document>

<document source="https://math2.tamnd.com/node/0QD83RAN8G.md" tag="0QD83RAN8G" kind="concept">
# Contour line

## Summary

A contour line of a function $f$ of two real variables is a curve in the plane along which $f$ has a constant value $c$.
It is a [[level set|0B2RZ0778N]] $\{(x, y) : f(x, y) = c\}$ of $f$, drawn in the $(x, y)$-plane.
A contour map draws the contour lines for several equally spaced values of $c$. It shows the shape of the [[graph|0JBJYT69ZV]] of $f$ in the plane.

## Remarks

On a topographic map, $f$ is the height above sea level, and the contour lines join points of equal height.
Where the contour lines of equally spaced values are close together, the function changes fast.

## Explanations

### L1 human

Imagine that you cut a hill with horizontal planes at heights $10$, $20$, $30$ and so on, and you look at the cuts from above.
Each cut gives a closed curve on the map, and these curves are the contour lines.
</document>

<document source="https://math2.tamnd.com/node/0QDXMG46QW.md" tag="0QDXMG46QW" kind="definition">
# Characteristic equation

## Definition

Let $a_0, \dots, a_{n-1}$ be real numbers, and consider the homogeneous [[linear equation with constant coefficients|0A6PGTTP2A]]
$$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = 0.$$
Its characteristic polynomial is $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0$.
Its characteristic equation is the polynomial equation $p(r) = 0$ for an unknown complex number $r$.

## Remarks

For each complex $r$, the function $y = e^{r x}$ satisfies $y^{(n)} + \dots + a_0 y = p(r)\, e^{r x}$. So $e^{rx}$ is a solution if and only if $p(r) = 0$.
Example: the equation $y'' - 3 y' + 2 y = 0$ has the characteristic equation $r^2 - 3 r + 2 = 0$, with roots $1$ and $2$.
The [[characteristic polynomial of a linear recurrence|0BWDJGQYST]] plays the same role for recurrences, with $r^k$ in place of $e^{r x}$.

## Explanations

### L1 human

Try a solution of the form $e^{r x}$.
Each derivative multiplies it by $r$, so the equation becomes $p(r) e^{rx} = 0$.
Since $e^{rx}$ is never zero, $r$ must be a root of $p$.
</document>

<document source="https://math2.tamnd.com/node/0QE5PFJY7V.md" tag="0QE5PFJY7V" kind="definition">
# Equivalence relation induced by a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The equivalence relation induced by $f$ is the relation $\sim_f$ on $A$ with
$$a \sim_f a' \iff f(a) = f(a')$$
for all $a, a' \in A$.

## Remarks

Another name is the kernel of $f$.
The relation $\sim_f$ is an [[equivalence relation|0SGTMBEWRF]], because equality is one.
Its classes are the nonempty preimages $f^{-1}(\{b\})$ of the points $b \in B$.
The function $f$ is [[injective|0PVG1CRAJN]] exactly when $\sim_f$ is equality.

## Explanations

### L1 human

Take the function that sends each person to their birthday.
Two people are equivalent under it when they have the same birthday.
The function forgets everything except its value, and the relation records what it forgets.
</document>

<document source="https://math2.tamnd.com/node/0QE8STGJH2.md" tag="0QE8STGJH2" kind="definition">
# Sequence

## Definition

Let $X$ be a set.
A sequence in $X$ is a function $a \colon \mathbb{N} \to X$, where $\mathbb{N} = \{0, 1, 2, \dots\}$.
The value $a(n)$ is the $n$-th term of the sequence, with the notation $a_n$.
The notation for the sequence is $(a_n)_{n \in \mathbb{N}}$, or $(a_n)$ when the index set is clear.

## Remarks

Many texts start the index at $1$, and they use a function on $\{1, 2, 3, \dots\}$. The theory does not change.
The terms of a sequence can repeat. For example, $(1, 0, 1, 0, \dots)$ is the sequence with $a_n = 1$ for even $n$ and $a_n = 0$ for odd $n$.
A sequence of real numbers is a [[real sequence|0CTV783HYT]].

## Explanations

### L1 human

A sequence is an infinite list $a_0, a_1, a_2, \dots$ of elements of a set.
The order matters, and the same element can occur more than once.
For example, the squares $0, 1, 4, 9, 16, \dots$ form the sequence $a_n = n^2$.
</document>

<document source="https://math2.tamnd.com/node/0QEAJB91GZ.md" tag="0QEAJB91GZ" kind="concept">
# Closure

## Summary

The closure of a subset of a topological space is the intersection of all closed sets that contain it.
It is the smallest closed set that contains the subset.
In a metric space, it is the set of limits of convergent sequences of points of the subset.
</document>

<document source="https://math2.tamnd.com/node/0QEBN8DGVE.md" tag="0QEBN8DGVE" kind="statement">
# Holomorphic functions are infinitely differentiable

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Then the complex derivative $f'$ is holomorphic on $U$.
Thus, for each $n \ge 1$, the $n$-th complex derivative $f^{(n)}$ exists and is holomorphic on $U$.
As a function of two real variables $(x, y)$ with $z = x + iy$, $f$ has continuous partial derivatives of all orders on $U$.

## Remarks

Proof: near each point, $f$ is the sum of a power series by [[holomorphic functions are analytic|0826SGV3R0]]. A [[power series can be differentiated term by term|0YMP521ZR5]], and the result is again a power series.
There is no real analogue: $x \mapsto x |x|$ is differentiable on $\mathbb{R}$, but its derivative $2|x|$ is not differentiable at $0$.
The [[formula for the derivatives|0F0H5P5DED]] gives each $f^{(n)}$ as a circle integral.

## Explanations

### L1 human

Once a complex function has one derivative on an open set, it has a second one, a third one, and so on forever.
So in complex analysis there are no functions that are differentiable only once.
</document>

<document source="https://math2.tamnd.com/node/0QED0GPYKY.md" tag="0QED0GPYKY" kind="concept">
# Alternating series

## Summary

An alternating series is a series whose terms alternate in sign. Such a series converges when the absolute values of its terms decrease to zero. The node [[alternating series|017BR1M98J]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0QEM68Y0DE.md" tag="0QEM68Y0DE" kind="statement">
# Cayley–Hamilton theorem for linear maps

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map with [[characteristic polynomial|0J5EMJRRFM]] $p_T$.
Then $p_T(T) = 0$, where $p_T(T)$ is the [[value of $p_T$ at $T$|0E3MV6E6MG]].

## Remarks

It follows from the [[Cayley–Hamilton theorem for matrices|0F765KFC2J]] applied to the matrix of $T$ in a basis.
The map $p \mapsto p([T]_B)$ corresponds to $p \mapsto p(T)$ under $T \mapsto [T]_B$.
The statement also holds for endomorphisms of finitely generated modules over a commutative ring, with a suitable characteristic polynomial.

## Explanations

### L1 human

The theorem does not depend on a basis.
A linear map satisfies its characteristic polynomial, because each of its matrices does.
</document>

<document source="https://math2.tamnd.com/node/0QF1MZ9FRG.md" tag="0QF1MZ9FRG" kind="concept">
# Infimum

## Summary

The infimum of a set is its greatest lower bound.
It is a lower bound of the set that is greater than or equal to every other lower bound.
It is the dual notion of the [[supremum|0R3H5PANRQ]].
</document>

<document source="https://math2.tamnd.com/node/0QF4ZEFG52.md" tag="0QF4ZEFG52" kind="statement">
# Eigenvalues are the roots of the minimal polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $\lambda \in F$.
Let $T \colon V \to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.
Then $\lambda$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$ if and only if $m_T(\lambda) = 0$.

## Remarks

If $T(v) = \lambda v$ with $v \ne 0$, then $0 = m_T(T) v = m_T(\lambda) v$, so $m_T(\lambda) = 0$.
Conversely, if $m_T = (x - \lambda) q$, then $q(T) \ne 0$ and $T - \lambda \id_V$ kills the image of $q(T)$.
So the minimal and the characteristic polynomial have the same roots, possibly with different multiplicities.

## Explanations

### L1 human

The minimal polynomial and the characteristic polynomial have the same roots: the eigenvalues.
Only the multiplicities of the roots can differ.
So the minimal polynomial is a second way to find the eigenvalues.
</document>

<document source="https://math2.tamnd.com/node/0QFT7Z76N5.md" tag="0QFT7Z76N5" kind="concept">
# Peano axioms

## Summary

The Peano axioms are axioms for the [[natural numbers|0MWKJ44V5A]].
They use a set $\mathbb{N}$, an element $0$ of $\mathbb{N}$ and a function $S \colon \mathbb{N} \to \mathbb{N}$, the [[successor function|0AK9TGSNXY]].
The axioms state three properties of these data.
First, $S(n) \neq 0$ for each $n$ in $\mathbb{N}$.
Second, $S$ is injective: if $S(m) = S(n)$, then $m = n$.
Third, the induction axiom holds: if a subset $A$ of $\mathbb{N}$ contains $0$ and contains $S(n)$ for each $n$ in $A$, then $A = \mathbb{N}$.

## Remarks

Giuseppe Peano published the axioms in 1889. Richard Dedekind gave a similar system in 1888.
The induction axiom speaks about all subsets of $\mathbb{N}$, thus it is a second-order axiom. First-order Peano arithmetic replaces it with an axiom schema.
The axioms are the base for the [[principle of mathematical induction|0ASJVEFA9J]] and for [[definition by recursion|0DSHAVH45F]].

## Explanations

### L1 human

The Peano axioms say what is special about counting: you start at $0$ and you always go to the next number.
Two different numbers never have the same next number, and you never come back to $0$.
The induction axiom says that you reach every natural number in this way.
</document>

<document source="https://math2.tamnd.com/node/0QFTETCHKB.md" tag="0QFTETCHKB" kind="statement">
# The algebraic numbers form a field

## Statement

Let $\overline{\mathbb{Q}}$ be the set of the complex numbers that are [[algebraic|0NPW1DWECM]] over $\mathbb{Q}$.
Then $\overline{\mathbb{Q}}$ is a [[subfield|07QY5RPFBR]] of $\mathbb{C}$.

## Remarks

The proof: if $a, b$ are algebraic over $\mathbb{Q}$, then $\mathbb{Q}(a, b)$ is a finite extension of $\mathbb{Q}$, by the [[tower law|0A7KJ68XHT]]. So $a + b$, $a - b$, $a b$ and $a/b$ for $b \neq 0$ are algebraic, because [[finite extensions are algebraic|0DFYTNZ6AM]].
More generally, for every extension $L/K$, the elements of $L$ that are algebraic over $K$ form a subfield of $L$.
The field $\overline{\mathbb{Q}}$ is countable and algebraically closed. It is an [[algebraic closure|0MCXX29HWY]] of $\mathbb{Q}$.

## Explanations

### L1 human

The sum and the product of two algebraic numbers are algebraic.
For example, $\sqrt{2} + \sqrt{3}$ is a root of $x^4 - 10x^2 + 1$.
The proof does not need to find such a polynomial: it uses only dimensions of vector spaces.
</document>

<document source="https://math2.tamnd.com/node/0QFV5Y1Z5X.md" tag="0QFV5Y1Z5X" kind="statement">
# Gauss's lemma

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] with [[field of fractions|0VF11ZTTMD]] $K$ and let $f \in R[x]$ be a [[primitive polynomial|0SP14DZN0G]].
Then $f$ is [[irreducible|00X2A9G9W3]] in $R[x]$ if and only if $f$ is irreducible in $K[x]$.

## Remarks

The key step: if $f = g h$ in $K[x]$, then one can scale $g$ and $h$ by elements of $K$ so that they are in $R[x]$ and $f = g h$ still holds. This uses that the [[product of primitive polynomials is primitive|08C1NWBNHA]].
For $R = \mathbb{Z}$: a primitive integer polynomial that factors over $\mathbb{Q}$ already factors over $\mathbb{Z}$, into factors of the same degrees.
The hypothesis that $f$ is primitive is necessary: $2x$ is irreducible in $\mathbb{Q}[x]$, but $2x = 2 \cdot x$ is a product of two nonunits in $\mathbb{Z}[x]$.

## Explanations

### L1 human

To show that an integer polynomial has no factorisation over the rationals, it is enough to check factorisations over the integers.
Fractions do not help to factor a primitive polynomial.
This is why criteria such as Eisenstein's work with integer coefficients.
</document>

<document source="https://math2.tamnd.com/node/0QG0Q3NDT9.md" tag="0QG0Q3NDT9" kind="concept">
# Fundamental solution of Pell's equation

## Summary

The fundamental solution of Pell's equation $x^2 - d y^2 = 1$ is its smallest solution in positive integers.
The [[definition of the fundamental solution|0KCYJSB9FW]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0QG9C579SR.md" tag="0QG9C579SR" kind="definition">
# Removable discontinuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$ be a limit point of $D$.
The point $a$ is a removable discontinuity of $f$ if the [[limit|0TXTSEMHR5]] $L = \lim_{x \to a} f(x)$ exists and $L \ne f(a)$.

## Remarks

Change the value of $f$ at $a$ to $L$. The new function is [[continuous at|0EEE2CZB2A]] $a$.
For example, let $f(x) = 1$ for $x \ne 0$ and $f(0) = 0$. Then $0$ is a removable discontinuity.
Some texts also call $a$ removable when $a \notin D$ and the limit exists. An example is $\sin(x)/x$ at $0$.

## Explanations

### L1 human

At a removable discontinuity, the graph has a single misplaced point.
Moving this one point repairs the continuity.
</document>

<document source="https://math2.tamnd.com/node/0QH5ENKZM1.md" tag="0QH5ENKZM1" kind="concept">
# Stochastic process

## Summary

A stochastic process is a family $(X_t)_{t \in T}$ of [[random variables|0DKRR22AEQ]] on one [[probability space|0ZYBZ34RT2]] $(\Omega, \mathcal{F}, P)$, all with values in the same measurable space $(S, \mathcal{S})$.
The set $T$ is the index set, often a set of times such as $\{0, 1, 2, \dots\}$ or $[0, \infty)$. The set $S$ is the state space.
For each outcome $\omega \in \Omega$, the function $t \mapsto X_t(\omega)$ is a sample path of the process.

## Remarks

The process is in discrete time if $T$ is countable, such as $\{0, 1, 2, \dots\}$, and in continuous time if $T$ is an interval.
The finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \dots, X_{t_n})$ for finite sets $\{t_1, \dots, t_n\} \subseteq T$. The Kolmogorov extension theorem builds a process from a consistent family of such laws.
Important examples are the Bernoulli process, random walks, Markov chains, the Poisson process and the Wiener process.

## Explanations

### L1 human

A stochastic process is a random quantity that changes with time.
Examples are the price of a share day by day and the number of customers in a queue.
At each time it is a random variable, and one run of the whole process gives a random path.
</document>

<document source="https://math2.tamnd.com/node/0QHEWQD30B.md" tag="0QHEWQD30B" kind="concept">
# Predicate

## Summary

A predicate is a statement with variables that becomes a [[proposition|0ZGC7ZESYN]] when each variable gets a value.
</document>

<document source="https://math2.tamnd.com/node/0QHTX7WQEY.md" tag="0QHTX7WQEY" kind="definition">
# Differential form

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $0 \le k \le n$.
Take a multi-index $I = (i_1, \dots, i_k)$ with $1 \le i_1 < \dots < i_k \le n$. Let $dx_I$ be the function of $k$ vectors $v_1, \dots, v_k \in \mathbb{R}^n$ given by
$$dx_I(v_1, \dots, v_k) = \det \big[ (v_j)_{i_l} \big]_{l, j = 1}^{k},$$
the determinant of a $k \times k$ matrix. The entry of this matrix in row $l$ and column $j$ is the $i_l$-th component of $v_j$.
A differential $k$-form on $U$ is a sum $\omega = \sum_I f_I \, dx_I$ over these multi-indices, where each $f_I \colon U \to \mathbb{R}$ is a function. At $x \in U$, it gives the function $\omega_x(v_1, \dots, v_k) = \sum_I f_I(x) \, dx_I(v_1, \dots, v_k)$.

## Remarks

The form is of class $C^r$ if all $f_I$ are of [[class C^r|0TCBZMEYGN]]. A $0$-form is a function, and a [[1-form|0MTENFBW3Q]] is $\sum_i f_i \, dx_i$.
Each $\omega_x$ is multilinear and alternating: it changes sign when two of the vectors are swapped. The forms $dx_I$ give a basis of such functions, so there are $\binom{n}{k}$ of them, and no nonzero forms of degree $k > n$.
In the [[wedge product|0S166D363M]] notation, $dx_I = dx_{i_1} \wedge \dots \wedge dx_{i_k}$. On $\mathbb{R}^3$, $dx \wedge dy \wedge dz$ gives the signed volume of the parallelepiped spanned by three vectors.
In $\mathbb{R}^3$, 1-forms and 2-forms correspond to vector fields, and 0-forms and 3-forms to functions. The [[exterior derivative|004FRCXH6M]] then corresponds to the gradient, curl and divergence.

## Explanations

### L1 human

A $k$-form measures small $k$-dimensional pieces: a 1-form measures small arrows, a 2-form small parallelograms, and so on.
Integrating a $k$-form over a $k$-dimensional surface adds up these measurements over the whole surface.
</document>

<document source="https://math2.tamnd.com/node/0QHX5GGABE.md" tag="0QHX5GGABE" kind="definition">
# Affine map

## Definition

Let $K$ be a field. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$, and let $B$ be an affine space over a $K$-vector space $W$.
A map $f : A \to B$ is an affine map if there is a linear map $L : V \to W$ such that
$$f(p + v) = f(p) + L(v) \quad \text{for each point } p \text{ of } A \text{ and each vector } v \text{ of } V.$$
The map $L$ is unique. It is the linear part of $f$, and it satisfies $L(q - p) = f(q) - f(p)$ for all points $p$ and $q$ of $A$.

## Remarks

The affine maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ are the maps $x \mapsto M x + b$, with $M$ a real $m \times n$ matrix and $b$ a vector of $\mathbb{R}^m$.
An affine map keeps [[affine combinations|0KJH6SRFZK]]. In particular it sends [[midpoints|0S706JBZBG]] to midpoints.
An affine transformation is a bijective affine map from $A$ to itself. The affine transformations of $A$ form a group under composition.
Mathlib's `AffineMap` is this notion, with the linear part as a field of the structure.

## Explanations

### L1 human

An affine map is a linear map that may also shift the origin. In the plane, examples are rotations, stretches, shears and translations, and the compositions of these. A grid of squares goes to a grid of parallelograms.
</document>

<document source="https://math2.tamnd.com/node/0QHZX1PK8V.md" tag="0QHZX1PK8V" kind="statement">
# Irrationality of pi

## Statement

The number [[$\pi$|0C1VYVNV2N]] is irrational.

## Remarks

Ivan Niven gave a short proof by calculus. Assume $\pi = a/b$, and let $f(x) = \frac{x^n (a - b x)^n}{n!}$.
Then $\int_0^\pi f(x) \sin x \, dx$ is a positive integer for every $n$. But it tends to $0$ as $n \to \infty$, a contradiction.
In 1882, Ferdinand von Lindemann proved the stronger fact that $\pi$ is transcendental. Mathlib states irrationality as `irrational_pi`.

## History

Johann Heinrich Lambert gave the first proof in the 1760s. He used a continued fraction for $\tan x$.

## Explanations

### L1 human

The number $\pi$ is not a fraction of two integers.
So its decimal expansion never ends and never repeats.
</document>

<document source="https://math2.tamnd.com/node/0QJJAQ6ZYZ.md" tag="0QJJAQ6ZYZ" kind="concept">
# Orientation of a real vector space

## Summary

An orientation of a finite-dimensional real vector space is a choice of one of the two classes of ordered bases.
Two bases are in the same class when the change of basis matrix has positive determinant.
</document>

<document source="https://math2.tamnd.com/node/0QK6K857HS.md" tag="0QK6K857HS" kind="concept">
# Matrix of a bilinear form

## Summary

The matrix of a bilinear form in a basis is the square matrix of the values of the form on all pairs of basis vectors.
</document>

<document source="https://math2.tamnd.com/node/0QK7FPE6VE.md" tag="0QK7FPE6VE" kind="statement">
# Dimension of the coordinate space

## Statement

Let $F$ be a field and let $n \ge 0$ be an integer.
The [[coordinate space|0XM8BZD76G]] $F^n$ has [[dimension|03GSZ8RTZJ]] $n$ over $F$.

## Remarks

The [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$ is a basis of $F^n$ with $n$ vectors.
A consequence: each $n$-dimensional space over $F$ is isomorphic to $F^n$.
In Mathlib, the index type is `Fin n`, and the space is the type of functions `Fin n → F`.

## Explanations

### L1 human

The space of lists of $n$ numbers has dimension $n$.
The $n$ standard unit vectors form a basis.
For example, $\R^3$ has dimension $3$.
</document>

<document source="https://math2.tamnd.com/node/0QKA345AW1.md" tag="0QKA345AW1" kind="concept">
# Implicit differentiation

## Summary

Implicit differentiation finds the derivative of a function that an equation $F(x, y) = 0$ defines, without solving the equation for $y$.
One differentiates both sides of $F(x, y(x)) = 0$ with the [[chain rule|0EAAS2EHDR]] and solves the result for $y'(x)$.

## Remarks

If $F$ is of class $C^1$ and $\partial F / \partial y \ne 0$ at the point, then $y'(x) = -\dfrac{\partial F / \partial x}{\partial F / \partial y}$, evaluated at $(x, y(x))$. The [[implicit function theorem|0BXBADR94C]] guarantees that $y(x)$ exists and is differentiable.
For the circle $x^2 + y^2 = 1$, differentiation gives $2 x + 2 y y' = 0$. So $y' = -x / y$ at each point with $y \ne 0$, on both the upper and the lower arc.
With several variables, $F(x, y, z) = 0$ gives $\partial z / \partial x = -F_x / F_z$ and $\partial z / \partial y = -F_y / F_z$ where $F_z \ne 0$. Here $F_x$, $F_y$, $F_z$ are the [[partial derivatives|05AXG3QRKR]] of $F$.

## Explanations

### L1 human

Sometimes a curve is given by an equation and not by a formula $y = f(x)$.
You can still find its slope: treat $y$ as a function of $x$, differentiate the equation term by term, and solve for $y'$.
</document>

<document source="https://math2.tamnd.com/node/0QKJ6NT0R9.md" tag="0QKJ6NT0R9" kind="concept">
# Derivative function

## Summary

The derivative function of a differentiable function $f$ is the function $f'$ that gives the derivative of $f$ at each point. Repeating the process gives higher derivatives. The node [[derivative function|027K9RMZ43]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0QKPNZKNEA.md" tag="0QKPNZKNEA" kind="definition">
# Cardinal number

## Definition

A cardinal number is the [[cardinality|0JTGRV4WT4]] $|A|$ of some set $A$.
In ZFC, the von Neumann definition takes $|A|$ to be the least [[ordinal|0RDJK7EQYY]] $\kappa$ such that $\kappa$ is [[equinumerous|08EJHHMB5R]] with $A$. A cardinal number is then an ordinal that is not equinumerous with any smaller ordinal.

## Remarks

The finite cardinals are the natural numbers $0, 1, 2, \dots$.
The least infinite cardinal is [[aleph-null|0V2H5PGJ9B]], the cardinality of $\mathbb{N}$.
Cardinals have an order, an addition, a multiplication and an exponentiation.
The sum $|A| + |B|$ is the cardinality of a disjoint union, and $|A| \cdot |B| = |A \times B|$.
The power $|B|^{|A|}$ is the cardinality of the set of functions from $A$ to $B$.

## Explanations

### L1 human

A cardinal number answers "how many?".
For finite sets, the answers are $0, 1, 2, \dots$.
For infinite sets, there are many different answers, such as $\aleph_0$ for $\mathbb{N}$ and $\mathfrak{c}$ for $\mathbb{R}$.
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# Euclidean domains are principal ideal domains

## Statement

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]].
Then $R$ is a [[principal ideal domain|0RZYDVZ475]].

## Remarks

The proof: let $I \neq 0$ be an ideal and let $b \in I$ be a nonzero element with $\delta(b)$ minimal. For $a \in I$, write $a = b q + r$. Then $r \in I$ and $\delta(r) < \delta(b)$ is impossible, so $r = 0$ and $I = (b)$.
The converse is false: $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not Euclidean.

## Explanations

### L1 human

Take a nonzero element of the ideal that is as small as possible.
Every other element of the ideal leaves remainder zero when you divide it by this one.
So the smallest element generates the ideal.
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# Element

## Summary

An element of a set is one of the objects that the set contains.
The relation "is an element of" is written with the symbol $\in$.
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# Product of rings

## Summary

The product of a family of rings is the set of tuples of their elements, with addition and multiplication in each coordinate.
The [[definition of the product of rings|0MNZCN4BQK]] gives the operations.
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# Riemann integral

## Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[Riemann integrable|0N1JQXZXHC]].
The Riemann integral of $f$ over $[a, b]$ is the unique number $I$ in the definition of Riemann integrability.
It is written
$$\int_a^b f(x) \, dx.$$

## Remarks

For $f \ge 0$, the integral equals the area under the graph of $f$ and above the $x$-axis.
By convention, $\int_a^a f = 0$ and $\int_b^a f = -\int_a^b f$.
The [[fundamental theorem of calculus|0NHGY0N0JB]] computes the integral from an antiderivative.
The integral is named after Bernhard Riemann.

## Explanations

### L1 human

The Riemann integral is the limit of sums of areas of thin rectangles under the graph.
It gives the signed area between the graph and the $x$-axis.
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# The Galois group has order at most the degree

## Statement

Let $L/K$ be a [[finite extension|0NRYPM9Y27]].
Then the group $\operatorname{Aut}(L/K)$ of [[automorphisms of $L/K$|0XVR72M51M]] is finite, and
$$|\operatorname{Aut}(L/K)| \le [L : K].$$

## Remarks

For a simple extension $L = K(a)$, an automorphism is determined by $\sigma(a)$, which is a root in $L$ of the minimal polynomial of $a$. There are at most $[L : K]$ such roots.
Equality holds if and only if $L/K$ is a [[Galois extension|0DS2F7GK05]].
For $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$, the inequality is strict: $1 < 3$.

## Explanations

### L1 human

A finite extension has only finitely many symmetries over the base field, and at most as many as its degree.
When there are exactly as many as the degree, the extension is Galois.
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<document source="https://math2.tamnd.com/node/0QMYQ1X03H.md" tag="0QMYQ1X03H" kind="definition">
# Orthocentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. The three [[lines|0MP9BCF6DP]] that contain the [[altitudes|0C9MZY2RNA]] of $ABC$ meet in exactly one point $H$.
This point $H$ is the orthocentre of $ABC$.

## Remarks

The orthocentre is inside the triangle if the triangle is acute. In a right triangle, it is the vertex at the right angle. In an obtuse triangle, it is outside.
In an equilateral triangle, the orthocentre is the same point as the centroid, the circumcentre and the incentre.
The Book of Lemmas, attributed to Archimedes, uses the fact that the three altitudes meet. Al-Nasawi stated and proved it explicitly in the 11th century.
Mathlib's `Affine.Triangle.orthocenter` defines it as the Monge point of the triangle.

## Explanations

### L1 human

Draw the three altitudes of a triangle, each from a corner at a right angle to the opposite side. The three lines always pass through one common point. For the right triangle with the vertices $(0, 0)$, $(3, 0)$ and $(0, 4)$, this point is the corner $(0, 0)$.
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# Galois extension

## Summary

A Galois extension is an algebraic field extension that is normal and separable.
Galois extensions are the extensions to which the fundamental theorem of Galois theory applies.
The [[definition of a Galois extension|0DS2F7GK05]] states it.
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<document source="https://math2.tamnd.com/node/0QN3D80AWP.md" tag="0QN3D80AWP" kind="concept">
# Euclidean vector

## Summary

A Euclidean vector is a geometric quantity with a length and a direction, often drawn as an arrow from one point to another.
Vectors can be added and multiplied by real numbers, and with these operations they form a real vector space.
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# Contraction mapping

## Summary

A contraction mapping is a map from a metric space to itself that shrinks all distances by at least a fixed factor less than one.
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# Constructible number

## Summary

A constructible number is a complex number that is a point of the plane reachable from $0$ and $1$ by a straightedge and compass construction.
The [[definition of a constructible number|0FMN2MN7TM]] states it.
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# Sample mean

## Definition

Let $X_1, \dots, X_n$ be real random variables, for example an [[i.i.d.|0AJN7WHBJZ]] sample.
The sample mean is the [[statistic|036756NFD6]]
$$\bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i.$$
For observed values $x_1, \dots, x_n$, the sample mean is the number $\bar{x} = (x_1 + \dots + x_n)/n$.

## Remarks

If the $X_i$ are i.i.d. with mean $\mu$, then $E[\bar{X}_n] = \mu$ by [[linearity of expectation|0P7PTC8DDE]]. So $\bar{X}_n$ is an unbiased estimator of $\mu$.
If also $\operatorname{Var}(X_i) = \sigma^2 < \infty$, then $\operatorname{Var}(\bar{X}_n) = \sigma^2 / n$.
For i.i.d. $\mathcal{N}(\mu, \sigma^2)$ variables, $\bar{X}_n \sim \mathcal{N}(\mu, \sigma^2/n)$.
By the law of large numbers, $\bar{X}_n \to \mu$ almost surely.

## Explanations

### L1 human

The sample mean is the ordinary average of the data.
For the measurements $4$, $7$ and $10$, the sample mean is $21/3 = 7$.
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<document source="https://math2.tamnd.com/node/0QNVQ4VD9T.md" tag="0QNVQ4VD9T" kind="definition">
# Lipschitz continuity

## Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $K \ge 0$ be real.
The function $f$ is Lipschitz with constant $K$ if
$$|f(x) - f(y)| \le K |x - y| \quad \text{for all } x, y \in D.$$
It is Lipschitz if it is Lipschitz with some constant $K$.

## Remarks

[[Lipschitz functions are uniformly continuous|0K1EWAR6AF]].
A differentiable function on an interval with $|f'| \le K$ is [[Lipschitz with constant $K$|0P64VE79X3]].
For example, $|x|$ and $\sin x$ are Lipschitz with constant $1$. The function $\sqrt{x}$ on $[0, 1]$ is not Lipschitz.
A Lipschitz map with constant $K < 1$ from a set to itself is a [[contraction|0CZRT49QNG]]. Mathlib defines it as `LipschitzWith`.

## Explanations

### L1 human

The slope of each chord of the graph of a Lipschitz function is at most $K$ in absolute value.
So the function never changes faster than a fixed rate.
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<document source="https://math2.tamnd.com/node/0QNY6ZBXPN.md" tag="0QNY6ZBXPN" kind="statement">
# Ratio test

## Statement

Let $(a_n)$ be a real sequence with $a_n \ne 0$ for all $n$.
Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for a real number $L$ with $L < 1$.
Then the series $\sum a_n$ converges [[absolutely|0W3K3QPEWT]].

## Remarks

Choose $q$ with $L < q < 1$. For $n \ge N$ we get $|a_{n+1}| \le q |a_n|$, so $|a_n| \le |a_N| q^{n-N}$. The [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] gives the result.
For $L > 1$ the series diverges, as the [[divergence part of the ratio test|08AM0KRCZ1]] states.
For $L = 1$ the test gives no answer. The series $\sum 1/n$ diverges and $\sum 1/n^2$ converges, and both have $L = 1$.
For example, $\sum x^n / n!$ converges for each real $x$, since the ratio is $|x|/(n+1) \to 0$.

## Explanations

### L1 human

If each term is eventually less than a fixed fraction $q < 1$ of the term before, the terms shrink at least as fast as $q^n$.
So the series behaves at least as well as a convergent geometric series.
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# Transitive action

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a nonempty set $X$.
The action is transitive when for all $x, y \in X$ there is a $g \in G$ with $g \cdot x = y$.

## Remarks

The action is transitive if and only if it has exactly one [[orbit|0YQPZB5Q1G]].
The action of $S_n$ on $\{1, \dots, n\}$ is transitive.
The action of $G$ on the left cosets of a subgroup $H$ is transitive. Each transitive action of $G$ is isomorphic to such an action.

## Explanations

### L1 human

A transitive action can move any point to any other point.
So the set does not split into separate pieces under the action.
The rotations of a square act transitively on its four corners.
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# Compact space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is compact when each [[open cover|0NVPNC12JY]] of $X$ has a finite subcover.

## Remarks

Some authors, mainly in the French tradition, also require a compact space to be Hausdorff, and they call the general notion quasi-compact.
Each finite topological space is compact. An infinite set with the [[discrete topology|0S2TEHX4E1]] is not compact.
For metric spaces, compactness is the same as [[sequential compactness|0CARW59WD2]], by [[this theorem|0K8SGQSV7G]].
The [[continuous image of a compact space is compact|0K9AAJ8G51]].

## Explanations

### L1 human

In a compact space, each way to cover the space by open sets can be reduced to finitely many of them.
This finiteness lets you turn local facts, which hold near each point, into global facts with one bound for the whole space.
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<document source="https://math2.tamnd.com/node/0QPYHHNRXC.md" tag="0QPYHHNRXC" kind="statement">
# Von Dyck's theorem

## Statement

Let $G = \langle S \mid R \rangle$, let $H$ be a [[group|G5S1M9]], and let $f \colon S \to H$ be a function.
Let $\varphi \colon F_S \to H$ be the homomorphism with $\varphi(s) = f(s)$ for each $s \in S$.
If $\varphi(r) = e$ for each $r \in R$, then there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\psi \colon G \to H$ with $\psi(s N) = f(s)$ for each $s \in S$.

## Remarks

Here $N$ is the normal closure of $R$ in $F_S$, so $G = F_S / N$.
The proof combines the [[universal property of the free group|0CDWKDYWN6]] with the [[universal property of the quotient group|0GJ83E78A5]], because $N \subseteq \ker \varphi$.
If the elements $f(s)$ generate $H$, then $\psi$ is surjective, so $H$ is a quotient of $G$.
The name refers to Walther von Dyck, who gave the first systematic study of group presentations in the early 1880s.

## Explanations

### L1 human

To map a presented group to another group, choose the images of the generators.
If these images obey all the relations, then the choice gives exactly one homomorphism.
You do not need to check anything else.
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<document source="https://math2.tamnd.com/node/0QQ0F12962.md" tag="0QQ0F12962" kind="example">
# Cycle graph

## Example

Let $n \ge 3$ be an integer.
The cycle graph $C_n$ is the [[graph|0JD5C82QMB]] with vertex set $\mathbb{Z}/n\mathbb{Z}$ in which $i$ and $j$ are adjacent if and only if $j - i = \pm 1$.
Its edges are $\{0, 1\}, \{1, 2\}, \dots, \{n-2, n-1\}$ and $\{n-1, 0\}$.

## Remarks

$C_n$ has $n$ vertices and $n$ edges, and it is $2$-[[regular|0KA4S8R39M]] and connected.
$C_3$ is the complete graph $K_3$.
The chromatic number of $C_n$ is $2$ for even $n$ and $3$ for odd $n$. The odd cycles are the obstruction in the theorem that [[a graph is bipartite iff it has no odd cycle|01R1HPEK6S]].

## Explanations

### L1 human

A cycle graph is a ring of $n$ vertices, each joined to the next one and the last joined to the first.
It looks like a polygon with $n$ corners.
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# Positive definite matrix

## Definition

Let $A$ be an $n \times n$ complex [[Hermitian matrix|0P17NGQQMQ]].
The matrix $A$ is *positive definite* when
$$x^* A x > 0 \qquad \text{for all } x \in \C^n \text{ with } x \ne 0.$$
A real [[symmetric matrix|0VR671PHE5]] $A$ is positive definite when $x^T A x > 0$ for all $x \in \R^n$ with $x \ne 0$.

## Remarks

For a Hermitian $A$, the number $x^* A x$ is always real.
The identity matrix and $\begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$ are positive definite.
$A$ is positive definite if and only if all its eigenvalues are positive.
$A$ is positive definite if and only if $\langle x, y \rangle = y^* A x$ is an inner product on $\C^n$.

## Explanations

### L1 human

A positive definite matrix gives a positive number $x^T A x$ for every nonzero vector $x$.
Such matrices behave like positive numbers in many ways: they have square roots and positive determinants.
They appear as Hessians at minima and as covariance matrices.
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# Derivative of a map into a product

## Statement

Let $V$, $W_1$ and $W_2$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, and let $x \in U$.
Let $f_1 \colon U \to W_1$ and $f_2 \colon U \to W_2$ be Fréchet differentiable at $x$.
Then the map $F \colon U \to W_1 \times W_2$, $F(y) = (f_1(y), f_2(y))$, is Fréchet differentiable at $x$, and
$$DF(x)(h) = \big(Df_1(x)(h), \, Df_2(x)(h)\big) \quad \text{for each } h \in V.$$
Here $W_1 \times W_2$ has the norm $\|(w_1, w_2)\| = \max(\|w_1\|, \|w_2\|)$.

## Remarks

The converse also holds: if $F$ is differentiable at $x$, then so are its two components, because each projection $W_1 \times W_2 \to W_i$ is a bounded linear map.
Any norm on $W_1 \times W_2$ that is equivalent to the max norm gives the same derivative.
Applied $m$ times with $W_i = \mathbb{R}$, this gives the [[derivative of a map into R^n by components|0NS0X75RS1]].

## Explanations

### L1 human

To differentiate a map with two outputs, differentiate each output and put the two results side by side.
The remainder of the pair is small exactly when both remainders are small.
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# Number of odd-degree vertices is even

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]].
Then the number of vertices of $G$ with odd [[degree|0T0F6CCS9W]] is even.

## Remarks

It follows from the [[handshaking lemma|0XAP8WHNES]]: the sum of all degrees is even, so it has an even number of odd terms.
Thus no graph has exactly one vertex of odd degree. For example, no graph has the degrees $3, 2, 2, 2$.

## Explanations

### L1 human

The total of all degrees is even, because it is twice the number of edges.
A sum of whole numbers is even only if it has an even number of odd terms.
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<document source="https://math2.tamnd.com/node/0QQRP17Z7S.md" tag="0QQRP17Z7S" kind="concept">
# Linear differential equation of order n

## Summary

A linear differential equation is an equation in which the unknown function and its derivatives occur linearly, with coefficients that are functions of the variable.
Its solutions are a particular solution plus the solutions of the associated homogeneous equation, which form a vector space. See the [[definition|0PB7K8M7FG]].
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# Asymptotic equivalence

## Summary

Two functions are asymptotically equivalent at a point when their ratio tends to $1$ there. The notation is $f \sim g$. The node [[asymptotic equivalence|00SB5EA3VW]] gives the precise definition.
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# Orthogonal matrix

## Definition

Let $n \ge 1$ and let $Q$ be a real $n \times n$ matrix.
The matrix $Q$ is *orthogonal* when
$$Q^T Q = I_n,$$
where $Q^T$ is the [[transpose|09DXWYRD43]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].

## Remarks

Equivalently, $Q$ is invertible with $Q^{-1} = Q^T$, and then also $Q Q^T = I_n$.
Equivalently, the columns of $Q$ form an orthonormal basis of $\R^n$; the same holds for the rows.
The map $x \mapsto Qx$ keeps the dot product, so it is a linear isometry of $\R^n$.
The determinant of an orthogonal matrix is $1$ or $-1$.

## Explanations

### L1 human

Orthogonal matrices describe the rigid motions of space that fix the origin: rotations and reflections.
They keep all lengths and angles.
To invert an orthogonal matrix, we only need to transpose it.
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# In a UFD irreducible elements are prime

## Statement

Let $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $p \in R$.
Then $p$ is [[irreducible|052SDEPZH7]] if and only if $p$ is [[prime|0MNHXBSX3M]].

## Remarks

[[Prime elements are irreducible|0F0K8DB9TD]] in every integral domain. The other direction needs unique factorisation.
The proof: suppose $p$ is irreducible and $p c = a b$. Factor $a$, $b$ and $c$ into irreducibles. By uniqueness, $p$ is an associate of an irreducible factor of $a$ or of $b$.
Conversely, an integral domain in which every nonzero nonunit is a product of irreducibles, and every irreducible is prime, is a unique factorisation domain.

## Explanations

### L1 human

In the integers, Euclid's lemma says that a prime that divides a product divides a factor.
In a unique factorisation domain, the same thing is true for every irreducible element.
In $\mathbb{Z}[\sqrt{-5}]$ this fails, and so factorisation is not unique there.
</document>

<document source="https://math2.tamnd.com/node/0QRQVH0CRW.md" tag="0QRQVH0CRW" kind="definition">
# Upper and lower Darboux sums

## Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$ be bounded, and let $P = (x_0, \dots, x_n)$ be a [[partition|069P20KYQT]] of $[a, b]$.
For each $i$, let $M_i = \sup f([x_{i-1}, x_i])$ and $m_i = \inf f([x_{i-1}, x_i])$.
The upper and lower Darboux sums of $f$ for $P$ are
$$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$

## Remarks

Always $(b - a) \inf f \le L(f, P) \le U(f, P) \le (b - a) \sup f$.
For a [[refinement|0EEKF7YFBW]] $Q$ of $P$, we have $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$.
So $L(f, P) \le U(f, Q)$ for any two partitions $P$ and $Q$.

## Explanations

### L1 human

The upper sum covers the graph with rectangles that reach its highest point on each piece.
The lower sum uses rectangles that reach its lowest point on each piece.
</document>

<document source="https://math2.tamnd.com/node/0QRRYXG7V7.md" tag="0QRRYXG7V7" kind="statement">
# Recurrence for derangements

## Statement

Let $n$ be a natural number, and let $D_m$ be the number of [[derangements|0TH61JR04G]] of an $m$-element set.
Then
$$D_{n+2} = (n + 1)\,(D_{n+1} + D_n).$$

## Remarks

Together with $D_0 = 1$ and $D_1 = 0$, the recurrence gives all values: $1, 0, 1, 2, 9, 44, 265, \dots$
A proof looks at a derangement $\sigma$ of $\{1, \dots, n + 2\}$ and the value $i = \sigma(n+2)$. There are $n + 1$ choices of $i$. The case $\sigma(i) = n + 2$ gives $D_n$ derangements, and the other case gives $D_{n+1}$.
Another recurrence is $D_{n+1} = (n + 1) D_n + (-1)^{n+1}$.

## Explanations

### L1 human

Look at where the last element goes, say to position $i$.
Either $i$ goes back to the last position, and the other $n$ elements form a derangement. Or it does not, and this acts like a derangement of $n + 1$ elements.
</document>

<document source="https://math2.tamnd.com/node/0QRV8FPK5M.md" tag="0QRV8FPK5M" kind="definition">
# Residue class

## Definition

Let $n$ be a positive integer and let $a$ be an integer.
The residue class of $a$ modulo $n$ is the set $\overline{a} = \{ b \in \mathbb{Z} : b \equiv a \pmod n \}$ of the integers [[congruent|0TPW25CBN6]] to $a$ modulo $n$.
Thus $\overline{a} = \{ a + k n : k \in \mathbb{Z} \} = a + n \mathbb{Z}$.

## Remarks

Two residue classes $\overline{a}$ and $\overline{b}$ are equal if $a \equiv b \pmod n$, and they have no common element otherwise.
There are exactly $n$ residue classes modulo $n$: $\overline{0}, \overline{1}, \dots, \overline{n - 1}$.
Other notations are $[a]$, $[a]_n$ and $a \bmod n$.
The residue classes are the elements of the ring of [[integers modulo n|01N3AYAA6N]].

## Explanations

### L1 human

Modulo $3$, each integer falls in one of three boxes: the multiples of $3$, the numbers $1$ more than a multiple of $3$, and the numbers $2$ more.
Each box is a residue class. For example, $\overline{1} = \{ \dots, -5, -2, 1, 4, 7, \dots \}$.
</document>

<document source="https://math2.tamnd.com/node/0QS8P74F2Z.md" tag="0QS8P74F2Z" kind="definition">
# Semisimple endomorphism

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
The map $T$ is *semisimple* when, for each $T$-[[invariant subspace|0ACX5QT8TE]] $W$ of $V$, there is a $T$-invariant subspace $W'$ with $V = W \oplus W'$.

## Remarks

The map $T$ is semisimple if and only if its minimal polynomial has no repeated irreducible factor.
Over an algebraically closed field, $T$ is semisimple if and only if it is [[diagonalisable|07SQCNV233]].
The rotation of $\R^2$ by $90$ degrees is semisimple, but it is not diagonalisable over $\R$.

## Explanations

### L1 human

Semisimple maps generalise diagonalisable maps to fields where eigenvalues can be missing.
They have no hidden nilpotent part.
Over $\C$, semisimple simply means diagonalisable.
</document>

<document source="https://math2.tamnd.com/node/0QSPNASE50.md" tag="0QSPNASE50" kind="statement">
# The reals form an ordered field

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition, multiplication and [[order|0337CYB0VD]] $\le$, form an [[ordered field|010JCV9J4E]].
That is, $\le$ is a total order with two properties for all real numbers $a$, $b$ and $c$.
If $a \le b$, then $a + c \le b + c$. If $0 \le a$ and $0 \le b$, then $0 \le a b$.

## Remarks

The ordered field $\mathbb{R}$ contains $\mathbb{Q}$ as an ordered subfield.
The order of $\mathbb{R}$ is the only order that makes $\mathbb{R}$ an ordered field. The nonnegative reals are exactly the squares.

## Explanations

### L1 human

All the usual rules for inequalities hold for the real numbers.
For example, from $a \le b$ you get $a + 5 \le b + 5$, and from $a \ge 0$ and $b \ge 0$ you get $a b \ge 0$.
</document>

<document source="https://math2.tamnd.com/node/0QT00PB5G2.md" tag="0QT00PB5G2" kind="concept">
# Homeomorphism

## Summary

A homeomorphism is a continuous bijection between topological spaces whose inverse is also continuous.
Homeomorphic spaces are the same from the point of view of topology.
</document>

<document source="https://math2.tamnd.com/node/0QTHB6PE52.md" tag="0QTHB6PE52" kind="statement">
# The group of units of Z/pZ is cyclic

## Statement

Let $p$ be a prime number.
Then the group $(\mathbb{Z}/p\mathbb{Z})^\times$ is [[cyclic|0KVGH0J5B3]] of order $p - 1$.

## Remarks

So $(\mathbb{Z}/p\mathbb{Z})^\times$ is isomorphic to the additive group $\mathbb{Z}/(p-1)\mathbb{Z}$, but no simple general formula for a generator is known.
A generator is a primitive root modulo $p$, and there are $\varphi(p - 1)$ of them.
For $p = 7$, the powers of $3$ modulo $7$ are $1, 3, 2, 6, 4, 5$, so $3$ is a primitive root modulo $7$.
More generally, $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$ for an odd prime $p$. Gauss first proved this.

## Explanations

### L1 human

For a prime $p$, one residue modulo $p$ has powers that give all the nonzero residues.
Such a residue is a primitive root.
So the multiplicative group modulo $p$ is cyclic.
</document>

<document source="https://math2.tamnd.com/node/0QTZDKJEDV.md" tag="0QTZDKJEDV" kind="statement">
# Frattini argument

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $H$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $H$.
Then $G = N_G(P) H$, where $N_G(P)$ is the [[normaliser|0YGN3VAA0D]] of $P$ in $G$.

## Remarks

For $g \in G$, the subgroup $gPg^{-1}$ lies in $H$, because $H$ is normal, and it is a Sylow $p$-subgroup of $H$.
By the [[second Sylow theorem|09GCQ5STVJ]] in $H$, there is $h \in H$ with $gPg^{-1} = hPh^{-1}$, so $h^{-1}g \in N_G(P)$.
The argument is used in the proof that a finite [[nilpotent group|0Q7MMQHC1P]] is the [[direct product of its Sylow subgroups|00Y9N80VSS]].

## History

Giovanni Frattini used the argument in a paper from 1885.
He took it, as he said himself, from a paper of Alfredo Capelli from 1884.

## Explanations

### L1 human

Conjugation by an element of $G$ moves $P$ to another Sylow subgroup of $H$.
An element of $H$ can move it back.
So each element of $G$ is an element of $H$ times an element that normalises $P$.
</document>

<document source="https://math2.tamnd.com/node/0QV2JS9RJ3.md" tag="0QV2JS9RJ3" kind="concept">
# Partial order

## Summary

A partial order is a relation that is reflexive, antisymmetric and transitive.
A set with a partial order is a partially ordered set, or poset.
</document>

<document source="https://math2.tamnd.com/node/0QV9VNNB4Z.md" tag="0QV9VNNB4Z" kind="concept">
# Distance in a graph

## Summary

The distance between two vertices of a graph is the length of a shortest walk between them. It makes a connected graph into a metric space. The node [[distance in a graph|0WPZ5YPAGR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0QVD7WBFGN.md" tag="0QVD7WBFGN" kind="concept">
# Integral domain

## Summary

An integral domain is a nonzero commutative ring in which the product of two nonzero elements is never zero.
The integers are the model example.
The [[definition of an integral domain|09S3PFBEPM]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0QVDWMKQ0T.md" tag="0QVDWMKQ0T" kind="definition">
# Homothety

## Definition

Let $K$ be a field, and let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space $V$. Let $c$ be a point of $A$ and let $k$ be a nonzero element of $K$.
The homothety with the centre $c$ and the ratio $k$ is the map $h : A \to A$ with
$$h(p) = c + k (p - c) \quad \text{for each point } p \text{ of } A.$$

## Remarks

A homothety is an [[affine map|0JF0TVYGWT]] with the linear part $v \mapsto k v$. It fixes its centre, and it sends each line to a parallel line.
The ratio $k = 1$ gives the identity. The ratio $k = -1$ gives the point reflection in $c$.
In a Euclidean space, a homothety with the ratio $k$ multiplies each distance by $|k|$, each area by $k^2$ and each volume by $|k|^3$.
Mathlib's `AffineMap.homothety` allows each $k$, also $k = 0$, which gives the constant map with the value $c$.

## Explanations

### L1 human

A homothety is a zoom centred at a point. With $k = 2$, each point moves to twice its distance from the centre, on the same ray. A triangle becomes a triangle with sides twice as long, and the same angles.
</document>

<document source="https://math2.tamnd.com/node/0QVNCJXRY3.md" tag="0QVNCJXRY3" kind="concept">
# Inner product space

## Summary

An inner product space is a real or complex vector space together with an inner product.
</document>

<document source="https://math2.tamnd.com/node/0QW084460E.md" tag="0QW084460E" kind="statement">
# Dimension of a subspace

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
Then $W$ is finite-dimensional, and $\dim W \le \dim V$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

A basis of $W$ is a linearly independent family in $V$, so it has at most $\dim V$ vectors.
Equality holds only when $W = V$, see [[a subspace of full dimension is the whole space|0WGAXEK99E]].
For infinite-dimensional $V$, the inequality $\dim W \le \dim V$ of cardinals still holds.

## Explanations

### L1 human

A subspace cannot have more independent directions than the space around it.
A plane in ordinary space has dimension $2$, which is at most $3$.
The only subspace with the same dimension is the whole space.
</document>

<document source="https://math2.tamnd.com/node/0QW64ARSHB.md" tag="0QW64ARSHB" kind="concept">
# Simply connected domain

## Summary

A simply connected domain is a path-connected set in which each closed loop can be shrunk continuously to a point without leaving the set.
Informally, the set is in one piece and has no hole that a loop can go around.
On such domains, curl-free vector fields are conservative.
</document>

<document source="https://math2.tamnd.com/node/0QWR8WTZEW.md" tag="0QWR8WTZEW" kind="statement">
# Every element of a finite field of order q satisfies x^q = x

## Statement

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
Then $a^q = a$ for every $a \in F$, and
$$x^q - x = \prod_{a \in F} (x - a) \quad \text{in } F[x].$$

## Remarks

The proof: the [[group of units|05THZKR9AN]] $F^\times$ has order $q - 1$, so $a^{q-1} = 1$ for $a \neq 0$, by [[Lagrange's theorem|3C53TD]]. Then $a^q = a$ for all $a$, also for $a = 0$.
So the $q$ elements of $F$ are $q$ distinct roots of $x^q - x$, which has degree $q$. This gives the factorisation.
For $q = p$ prime, $a^p = a$ in $\mathbb{Z}/p\mathbb{Z}$ is Fermat's little theorem.

## Explanations

### L1 human

In a field with $q$ elements, raising to the power $q$ changes nothing.
So the polynomial $x^q - x$ has every element of the field as a root.
</document>

<document source="https://math2.tamnd.com/node/0QX19C7K1C.md" tag="0QX19C7K1C" kind="concept">
# Order of a zero

## Summary

The order of a zero $z_0$ of a holomorphic function $f$ is the smallest $m$ with $f^{(m)}(z_0) \neq 0$, see the [[definition|064N002TX4]].
Near such a zero, $f(z) = (z - z_0)^m g(z)$ with $g(z_0) \neq 0$.
</document>

<document source="https://math2.tamnd.com/node/0QX7E9KAAX.md" tag="0QX7E9KAAX" kind="statement">
# Tangent–secant theorem

## Statement

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be a point outside $C$.
Let a [[tangent line|0T7DX8NY1X]] through $P$ touch $C$ at $T$. Let a second line through $P$ meet $C$ at the two points $G_1$ and $G_2$.
Then
$$d(P, T)^2 = d(P, G_1)\, d(P, G_2).$$

## Remarks

Both sides are equal to the [[power|02E9VT1RVQ]] of $P$ with respect to $C$.
It is the limit case of the [[intersecting secants theorem|0J5XR7BW2S]] where one secant becomes a tangent.
The theorem is Proposition 36 of Book III of Euclid's Elements.

## Explanations

### L1 human

From a point outside a circle, the square of the tangent length equals the product of the two distances along any secant. For the unit circle and $P = (3, 0)$, the secant along the $x$-axis meets the circle at distances $2$ and $4$. So the tangent length is $\sqrt{8}$.
</document>

<document source="https://math2.tamnd.com/node/0QXM0A9878.md" tag="0QXM0A9878" kind="concept">
# Picard–Lindelöf hypotheses

## Summary

The Picard–Lindelöf hypotheses on a vector field are continuity in time, a Lipschitz condition in space and a bound near the initial point. They also ask for a short enough time interval.
They are the conditions of the [[Picard–Lindelöf theorem|0P6KRD5B7A]]. See the [[definition|0J7HF2PF1H]].
</document>

<document source="https://math2.tamnd.com/node/0QY87R9CY3.md" tag="0QY87R9CY3" kind="statement">
# Convergent sequences are Cauchy

## Statement

Let $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.
Then $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].

## Remarks

Let $L$ be the limit and let $\varepsilon > 0$. Take $N$ with $|a_n - L| < \varepsilon/2$ for $n \ge N$. Then $|a_m - a_n| \le |a_m - L| + |L - a_n| < \varepsilon$ for $m, n \ge N$.
The statement holds in each metric space.
The converse holds in $\mathbb{R}$, but it fails in $\mathbb{Q}$ and in other spaces that are not complete.

## Explanations

### L1 human

If all late terms are close to one number $L$, then they are also close to each other.
Two terms that are each within $\varepsilon/2$ of $L$ are within $\varepsilon$ of each other.
</document>

<document source="https://math2.tamnd.com/node/0QZCY9PE3Q.md" tag="0QZCY9PE3Q" kind="definition">
# Presentation of a group

## Definition

Let $S$ be a set, let $F_S$ be the [[free group|0KTW26C80J]] on $S$, and let $R \subseteq F_S$.
Let $N$ be the smallest [[normal subgroup|0CB0DX5BVV]] of $F_S$ that contains $R$, the normal closure of $R$.
The group with presentation $\langle S \mid R \rangle$ is the [[quotient group|0KYX2N69J0]] $F_S / N$.
A group $G$ has the presentation $\langle S \mid R \rangle$ when $G$ is [[isomorphic|08MG7H8JW5]] to $F_S / N$.

## Remarks

The elements of $S$ are the generators and the elements of $R$ are the relators; a relation $x = y$ stands for the relator $y^{-1} x$.
The cyclic group of order $n$ has the presentation $\langle a \mid a^n \rangle$, and the dihedral group of order $16$ has the presentation $\langle r, f \mid r^8, f^2, (r f)^2 \rangle$.
Each group has a presentation, and each finite group has a presentation with $S$ and $R$ finite.
A group is finitely presented when it has a presentation with $S$ and $R$ finite.

## Explanations

### L1 human

A presentation lists generators and the rules that they obey.
The group is the largest group that these generators make, subject only to these rules.
Presentations are often the shortest way to describe a group.
</document>

<document source="https://math2.tamnd.com/node/0QZMTN5AXX.md" tag="0QZMTN5AXX" kind="definition">
# Cauchy sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]].
The sequence is a Cauchy sequence if the following holds for each real number $\varepsilon > 0$.
There is a natural number $N$ such that $|a_m - a_n| < \varepsilon$ for all natural numbers $m, n \ge N$.

## Remarks

In a metric space $(X, d)$, the definition is the same with $d(a_m, a_n)$ in place of $|a_m - a_n|$.
The definition does not mention a limit. In $\mathbb{R}$, the [[Cauchy criterion|0ZA4AX4MW3]] shows that the Cauchy sequences are exactly the convergent sequences.
It is not enough that $|a_{n+1} - a_n| \to 0$. The sequence $a_n = \sqrt{n}$ has this property, but it is not Cauchy.

## Explanations

### L1 human

In a Cauchy sequence, the terms become close to each other, not only to their neighbours.
From some index on, any two terms differ by less than the error that you allow.
</document>

<document source="https://math2.tamnd.com/node/0R03B54V4Z.md" tag="0R03B54V4Z" kind="definition">
# Fermat number

## Definition

For a natural number $n$, the $n$-th Fermat number is $F_n = 2^{2^n} + 1$.

## Remarks

The first values are $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ and $F_4 = 65537$.
The Fermat numbers satisfy $F_0 F_1 \cdots F_{n-1} = F_n - 2$ for $n \ge 1$.
A Fermat number that is prime is a [[Fermat prime|0N9E4G2P97]].

## Explanations

### L1 human

The Fermat numbers grow very fast, because the exponent itself is a power of $2$.
For example, $F_5 = 2^{32} + 1 = 4294967297$.
</document>

<document source="https://math2.tamnd.com/node/0R0D0XY4HT.md" tag="0R0D0XY4HT" kind="concept">
# Coefficient of a power series

## Summary

The coefficient of $X^n$ in a formal power series is the $n$-th term of its sequence. Coefficient extraction is how a generating function gives back its counting numbers. The node [[coefficient of a power series|0M98XYVX6S]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0R0MYT0BJD.md" tag="0R0MYT0BJD" kind="statement">
# Continuity of the sum of a uniformly convergent series of functions

## Statement

Let $S \subseteq \mathbb{R}$, and let $f_n \colon S \to \mathbb{R}$ be continuous for each $n \ge 0$.
If the [[series|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges uniformly on $S$, then its sum is continuous on $S$.

## Remarks

The partial sums are continuous, and they converge uniformly. Apply the [[uniform limit theorem|0X9XT05JVC]].
Uniform convergence often comes from the [[M-test|0NX5NWAFC9]]. Mathlib states the result in this form as `continuous_tsum`.
For example, $\sum_{n=1}^{\infty} \frac{\cos(n x)}{n^2}$ is continuous on $\mathbb{R}$.

## Explanations

### L1 human

Adding infinitely many continuous functions gives a continuous function, if the sum converges uniformly.
</document>

<document source="https://math2.tamnd.com/node/0R0T1VWH5X.md" tag="0R0T1VWH5X" kind="statement">
# Fubini's theorem

## Statement

Let $R = [a, b] \times [c, d]$, and let $f \colon R \to \mathbb{R}$ be Riemann integrable on $R$.
Suppose that $y \mapsto f(x, y)$ is integrable on $[c, d]$ for each $x \in [a, b]$, and that $x \mapsto f(x, y)$ is integrable on $[a, b]$ for each $y \in [c, d]$.
Then
$$\iint_R f \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy.$$
In particular, this holds for each [[continuous|0S9EK5MQPE]] function $f$ on $R$.

## Remarks

The left side is the [[double integral|0NKBMFC18R]], and the other two are [[iterated integrals|0P4WX7Y502]].
The Lebesgue version: let $f$ be Lebesgue measurable on $\mathbb{R}^n \times \mathbb{R}^m$ with $\int |f| < \infty$. Then the double integral equals both iterated integrals, where the inner integrals exist for almost all values of the outer variable.
Without integrability of $|f|$, the [[iterated integrals can differ|0GJZE2E4RX]]. For non-negative functions, [[Tonelli's theorem|067SYD2J4B]] removes the integrability hypothesis.
Euler knew the case of continuous functions. Lebesgue proved a version for bounded measurable functions in 1904, and Fubini proved the version for integrable functions in 1907.

## Explanations

### L1 human

To integrate over a rectangle, you can slice it into lines, integrate along each line, and then integrate the results.
Fubini's theorem says that both slicing directions give the same answer as the double integral, for well-behaved functions.
</document>

<document source="https://math2.tamnd.com/node/0R0X73XHAT.md" tag="0R0X73XHAT" kind="statement">
# Abelian groups are solvable

## Statement

Let $G$ be an [[abelian group|082GHNPJ53]].
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

The series $\{e\} \trianglelefteq G$ has one factor, $G$ itself, and this factor is abelian.
Equivalently, the commutator subgroup $G^{(1)}$ is already $\{e\}$.
The converse is false: $S_3$ is solvable, by the series $\{e\} \trianglelefteq A_3 \trianglelefteq S_3$, but it is not abelian.

## Explanations

### L1 human

An abelian group is already one abelian layer.
So it is solvable with a series of length one.
Solvable groups generalise abelian groups.
</document>

<document source="https://math2.tamnd.com/node/0R2X8K9QA7.md" tag="0R2X8K9QA7" kind="concept">
# Content of a polynomial

## Summary

The content of a polynomial over a unique factorisation domain is a greatest common divisor of its coefficients.
The [[definition of the content|08VV7YJRB7]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0R37DFJKFW.md" tag="0R37DFJKFW" kind="definition">
# Connected set

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The set $S$ is connected when $S$, with the [[subspace topology|0D5YHQGDJY]], is a [[connected space|0D939CMME7]].

## Remarks

Equivalently, there are no open sets $U, V$ of $X$ with $S \subseteq U \cup V$, $S \cap U \cap V = \varnothing$, $S \cap U \neq \varnothing$ and $S \cap V \neq \varnothing$.
The [[connected subsets of the real line are the intervals|0B9ZQFMQNV]].
The closure of a connected set is connected. A union of connected sets with a common point is connected.

## Explanations

### L1 human

A subset is connected when it is in one piece, as a space of its own.
For example, the set $\mathbb{Q}$ of rationals is not connected in $\mathbb{R}$: the sets $(-\infty, \sqrt{2})$ and $(\sqrt{2}, \infty)$ split it.
</document>

<document source="https://math2.tamnd.com/node/0R3B7DX55J.md" tag="0R3B7DX55J" kind="definition">
# T0 space

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is T0 when for all $x \neq y$ in $X$, some [[open set|01ZQQDCERP]] contains exactly one of $x$ and $y$.

## Remarks

Each [[T1 space|05GQJXBP3F]] is T0.
The Sierpiński space $\{0, 1\}$, with open sets $\varnothing$, $\{1\}$ and $\{0, 1\}$, is T0 but not T1.
An [[indiscrete space|0XJ02JBD6S]] with at least two points is not T0.

## Explanations

### L1 human

In a T0 space, no two distinct points have exactly the same open sets around them.
The open sets may still fail to separate the points in both directions.
</document>

<document source="https://math2.tamnd.com/node/0R3H35G96G.md" tag="0R3H35G96G" kind="definition">
# Derived series

## Definition

Let $G$ be a [[group|G5S1M9]].
The derived series of $G$ is the sequence of subgroups $G^{(0)} = G$ and $G^{(i+1)} = [G^{(i)}, G^{(i)}]$ for $i \geq 0$.
Here $[H, H]$ is the [[commutator subgroup|0ASGSTXSBW]] of $H$.

## Remarks

Each $G^{(i)}$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and $G^{(i)} / G^{(i+1)}$ is abelian.
The group $G$ is [[solvable|0FXJ7YFMMX]] if and only if $G^{(n)} = \{e\}$ for some $n$, and the least such $n$ is the derived length.
For $S_4$, the derived series is $S_4 \supseteq A_4 \supseteq V \supseteq \{e\}$.
For $n \geq 5$, the derived series of $S_n$ is $S_n \supseteq A_n \supseteq A_n \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

Each step of the derived series takes the commutator subgroup of the last step.
So each step removes an abelian layer from the group.
If you reach the identity, the group is solvable.
</document>

<document source="https://math2.tamnd.com/node/0R3H5PANRQ.md" tag="0R3H5PANRQ" kind="concept">
# Supremum

## Summary

The supremum of a set is its least upper bound.
It is an upper bound of the set that is less than or equal to every other upper bound.
A set can have a supremum that is not an element of the set.
</document>

<document source="https://math2.tamnd.com/node/0R41PVFA39.md" tag="0R41PVFA39" kind="definition">
# Type I and type II errors

## Definition

Consider a [[hypothesis test|087P03NEND]] of $H_0 \colon \theta \in \Theta_0$ against $H_1 \colon \theta \in \Theta_1$ that rejects $H_0$ when the data $X$ are in a rejection region $R$.
A type I error occurs when $H_0$ is true and the test rejects $H_0$. Its probability at $\theta \in \Theta_0$ is $P_\theta(X \in R)$.
A type II error occurs when $H_0$ is false and the test does not reject $H_0$. Its probability at $\theta \in \Theta_1$ is $\beta(\theta) = P_\theta(X \notin R)$.

## Remarks

The size of the test is $\alpha = \sup_{\theta \in \Theta_0} P_\theta(X \in R)$.
The power of the test at $\theta \in \Theta_1$ is $1 - \beta(\theta)$.
For a fixed sample size, a smaller rejection region lowers the probability of a type I error. It raises the probability of a type II error.
In the frequentist practice, one fixes a bound $\alpha$ on the type I error, and then chooses a test with a small type II error.

## Explanations

### L1 human

A type I error is a false alarm: the test says "effect" when there is none. A type II error is a miss: the test says "no effect" when there is one.
For a medical test, a positive result for a healthy person is a type I error. A negative result for an ill person is a type II error.
</document>

<document source="https://math2.tamnd.com/node/0R46A1XY60.md" tag="0R46A1XY60" kind="definition">
# Almost surely

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]].
An [[event|00SQGJNRTB]] $E \in \mathcal{F}$ happens almost surely if $P(E) = 1$.
More generally, a property of outcomes holds almost surely (a.s.) if an event of probability $0$ contains all outcomes where it fails.

## Remarks

The property need not hold for all outcomes. In this sense, "almost surely" is weaker than "surely".
Two random variables $X$ and $Y$ are equal almost surely if $P(X = Y) = 1$. They then have the same law.
A countable intersection of events that happen almost surely also happens almost surely.
It is the probability form of "almost everywhere" in measure theory.

## Explanations

### L1 human

If you choose a uniform random number in $[0, 1]$, it is almost surely irrational. Rational outcomes are possible, but together they have probability $0$.
If you toss a fair coin forever, you get heads at least once almost surely.
</document>

<document source="https://math2.tamnd.com/node/0R47MAD57S.md" tag="0R47MAD57S" kind="definition">
# Monoid

## Definition

A monoid is a triple $(M, \ast, e)$, where $(M, \ast)$ is a [[semigroup|0XMEM406JM]] and $e \in M$ is an [[identity element|07WYQPB8E3]] for $\ast$.
Thus $(a \ast b) \ast c = a \ast (b \ast c)$ and $e \ast a = a \ast e = a$ for all $a, b, c \in M$.

## Remarks

The natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ with addition form a monoid with identity $0$.
The integers with multiplication form a monoid with identity $1$.
The identity element of a monoid is unique, thus some authors write a monoid as the pair $(M, \ast)$.
Each [[group|G5S1M9]] is a monoid in which each element has an [[inverse|0ZNZH0EXFX]].

## Explanations

### L1 human

A monoid is a set with an associative operation and an element that does nothing.
The strings of letters with concatenation form a monoid, and the empty string is the identity.
In a monoid, an element need not have an inverse: you cannot undo the addition of a letter.
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<document source="https://math2.tamnd.com/node/0R5HTV7REV.md" tag="0R5HTV7REV" kind="definition">
# Directed graph

## Definition

A directed graph, or digraph, is a pair $D = (V, A)$ of a set $V$ and a relation $A \subseteq V \times V$.
The elements of $V$ are the vertices of $D$.
The elements of $A$ are the arcs, or directed edges, of $D$.
An arc $(u, v)$ goes from its tail $u$ to its head $v$.

## Remarks

With this definition a digraph can have loops $(v, v)$, and it has at most one arc from $u$ to $v$. Some authors exclude loops.
A [[graph|0JD5C82QMB]] is the same as a digraph whose relation is symmetric and irreflexive.
The out-degree of $v$ is the number of arcs with tail $v$, and the in-degree is the number of arcs with head $v$.

## Explanations

### L1 human

A directed graph is a graph whose lines are arrows.
One-way streets between crossings form a directed graph.
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<document source="https://math2.tamnd.com/node/0R5QPQ3AGJ.md" tag="0R5QPQ3AGJ" kind="definition">
# Translation

## Definition

Let $v$ be a [[vector|0QN3D80AWP]] of $\mathbb{R}^n$. The translation by $v$ is the map
$$T_v \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad T_v(p) = p + v.$$

## Remarks

Each translation is a [[Euclidean isometry|0J1N3WTKM4]], and it is direct. For $v \neq 0$, it has no fixed point.
$T_v \circ T_w = T_{v + w}$ and $T_v^{-1} = T_{-v}$. So the translations form an abelian group, isomorphic to $\mathbb{R}^n$.
In the plane, the composite of the reflections in two parallel lines is a translation.
Mathlib's `AffineEquiv.constVAdd` is the translation by a vector in an affine space.

## Explanations

### L1 human

A translation slides every point by the same arrow. Shapes keep their size, their form and their direction. The translation by $(2, 1)$ sends the point $(0, 0)$ to $(2, 1)$ and the point $(3, -1)$ to $(5, 0)$.
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<document source="https://math2.tamnd.com/node/0R64A2CVN4.md" tag="0R64A2CVN4" kind="statement">
# Fraction in lowest terms

## Statement

Let $q$ be a [[rational number|0NCJ5WXN6Z]].
Then there is exactly one pair of [[integers|0C4JS9RF5M]] $a$ and $b$ with $b > 0$, $\gcd(a, b) = 1$ and $q = a / b$.

## Remarks

The fraction $a / b$ is the irreducible fraction, or the fraction in lowest terms, of $q$.
To find it from any fraction $c / d$ with $d \neq 0$, divide $c$ and $d$ by $\pm \gcd(c, d)$, with the sign that makes the denominator positive.
For example, $-6 / -4$ has lowest terms $3 / 2$.

## Explanations

### L1 human

You can simplify each fraction until the top and the bottom have no common factor larger than $1$.
For example, $12 / 18$ becomes $2 / 3$ when you divide both numbers by $6$.
The result does not depend on the order of the simplification steps.
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<document source="https://math2.tamnd.com/node/0R6EWFKRCF.md" tag="0R6EWFKRCF" kind="concept">
# Compact set

## Summary

A compact subset of a topological space is a subset that is a compact space with the subspace topology.
Equivalently, each cover of the subset by open sets of the space has a finite subcover.
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<document source="https://math2.tamnd.com/node/0R6M158SQ9.md" tag="0R6M158SQ9" kind="definition">
# Annihilator of a subspace

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.
The *annihilator* of $W$ is the subset of the [[dual space|07JBSZE0V4]]
$$W^0 = \{f \in V^* : f(w) = 0 \text{ for all } w \in W\}.$$

## Remarks

The annihilator $W^0$ is a subspace of $V^*$.
If $W_1 \subseteq W_2$, then $W_2^0 \subseteq W_1^0$.
In finite dimensions, $\dim W + \dim W^0 = \dim V$, see [[the dimension of the annihilator|06Q1EFVH0Q]].
The annihilator of the line spanned by $(1, 1)$ in $F^2$ is spanned by the form $(x, y) \mapsto x - y$.

## Explanations

### L1 human

The annihilator of a subspace collects the linear forms that are zero on all of it.
In $\R^3$, the forms that vanish on a plane through the origin are the multiples of one form, whose kernel is that plane.
A larger subspace has a smaller annihilator.
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<document source="https://math2.tamnd.com/node/0R74MZYNRB.md" tag="0R74MZYNRB" kind="statement">
# Universal property of the quotient ring

## Statement

Let $R$ be a [[ring|0894J980RK]], let $I$ be a two-sided ideal of $R$ and let $\pi \colon R \to R/I$ be the quotient map.
Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]] with $I \subseteq \ker f$.
Then there is exactly one ring homomorphism $\bar{f} \colon R/I \to S$ such that $\bar{f} \circ \pi = f$.
It is given by $\bar{f}(a + I) = f(a)$.

## Remarks

The map $\bar{f}$ is well defined: if $a + I = b + I$, then $a - b \in I \subseteq \ker f$, so $f(a) = f(b)$.
The image of $\bar{f}$ is the image of $f$, and $\ker \bar{f} = (\ker f)/I$.
With $I = \ker f$, this gives the [[first isomorphism theorem|0ZMYX7GMH0]].

## Explanations

### L1 human

A homomorphism that sends all of $I$ to zero does not see the difference between $a$ and $a + x$ for $x \in I$.
So it already makes sense on the cosets of $I$.
This gives a homomorphism from the quotient ring.
</document>

<document source="https://math2.tamnd.com/node/0R7BTEWCJ2.md" tag="0R7BTEWCJ2" kind="statement">
# Compact subsets of a metric space are bounded

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $K$ be a [[compact subset|00R0X1WGPX]] of $X$.
Then $K$ is [[bounded|0P953MG05E]].

## Remarks

Proof: if $K$ is empty, it is bounded. Otherwise fix $x \in K$. The [[open balls|0DC3MPR7C9]] $B(x, n)$ for $n \ge 1$ cover $K$, so finitely many of them do. The largest of these balls contains $K$.
A compact subset of a metric space is also [[closed|0G5AV9Q4RR]], because metric spaces are Hausdorff.
The converse fails in general: a closed bounded set need not be compact. With the metric $\min(|x - y|, 1)$, the real line is closed and bounded, but not compact.

## Explanations

### L1 human

A compact set can be covered by finitely many balls about one point, and then it fits in the largest of them.
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<document source="https://math2.tamnd.com/node/0R7JS3DP3W.md" tag="0R7JS3DP3W" kind="statement">
# Symmetry of binomial coefficients

## Statement

Let $n$ and $k$ be natural numbers with $k \le n$.
Then
$$\binom{n}{k} = \binom{n}{n - k}.$$

## Remarks

For a set $X$ with $n$ elements, the map $S \mapsto X \setminus S$ is a bijection from the $k$-subsets of $X$ to the $(n-k)$-subsets. This is a [[bijective proof|0GV2573FP4]] of the identity.
The identity makes each row of [[Pascal's triangle|05DME4TZ3K]] read the same from left to right and from right to left.

## Explanations

### L1 human

To choose $k$ people for a team is the same as to choose the $n - k$ people who stay out.
For example, $\binom{5}{2} = \binom{5}{3} = 10$.
</document>

<document source="https://math2.tamnd.com/node/0R7SVNYTJS.md" tag="0R7SVNYTJS" kind="statement">
# pi is greater than 3

## Statement

$$\pi > 3.$$

## Remarks

The [[cosine series|08W5RN375X]] alternates. At $x = 3/2$, its terms decrease in size from the term $x^4/4!$ on. So $\cos(3/2) \ge 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}$ at $x = 3/2$. This is about $0.070 > 0$.
Since $\cos 2 < 0$, the zero of $\cos$ in $[0, 2]$ lies in $(3/2, 2)$. So $\pi/2 > 3/2$.
Archimedes proved the sharper bounds $\frac{223}{71} < \pi < \frac{22}{7}$. Mathlib states the claim as `Real.pi_gt_three`.

## Explanations

### L1 human

We show that $\cos x$ is still positive at $x = 3/2$.
So its first zero $\pi/2$ comes later, and $\pi > 3$.
</document>

<document source="https://math2.tamnd.com/node/0R7VXTYHYZ.md" tag="0R7VXTYHYZ" kind="statement">
# A linear isometry preserves the inner product

## Statement

Let $V$ and $W$ be inner product spaces over $\R$ or $\C$, with the induced norms, and let $T \colon V \to W$ be a [[linear isometry|07SJXTWVQW]].
Then, for all $u, v \in V$,
$$\langle T(u), T(v) \rangle = \langle u, v \rangle.$$

## Remarks

The proof applies the [[polarisation identity|05FKHRT0B4]], which writes inner products in terms of norms.
Conversely, a linear map that keeps inner products keeps norms.
So a linear isometry also keeps angles and orthogonality.

## Explanations

### L1 human

If a linear map keeps lengths, then it also keeps angles.
The reason is that lengths determine inner products, by polarisation.
So rigid linear motions keep the whole geometry.
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<document source="https://math2.tamnd.com/node/0R84PF4JV7.md" tag="0R84PF4JV7" kind="statement">
# Dimension of a tensor product

## Statement

Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then
$$\dim (V \otimes W) = \dim V \cdot \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]] over $F$.

## Remarks

It follows from the [[basis of a tensor product|0SH5VT8QYA]], which has $\dim V \cdot \dim W$ vectors.
For example, $F^m \otimes F^n$ has dimension $mn$, like the space of $m \times n$ matrices.
Compare this with the direct sum: $\dim (V \oplus W) = \dim V + \dim W$.

## Explanations

### L1 human

The dimension of a tensor product is the product of the dimensions.
The dimension of a direct sum is their sum instead.
So $\R^2 \otimes \R^3$ has dimension $6$.
</document>

<document source="https://math2.tamnd.com/node/0R8J3ZQMJS.md" tag="0R8J3ZQMJS" kind="definition">
# Subsequence

## Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a [[sequence|0QE8STGJH2]] in a set $X$.
Let $(n_k)_{k \in \mathbb{N}}$ be a strictly increasing sequence of natural numbers: $n_0 < n_1 < n_2 < \cdots$.
The sequence $(a_{n_k})_{k \in \mathbb{N}}$ is a subsequence of $(a_n)$.

## Remarks

Equivalently, a subsequence is the composite $a \circ \varphi$ of the sequence $a \colon \mathbb{N} \to X$ with a strictly increasing map $\varphi \colon \mathbb{N} \to \mathbb{N}$.
By induction, $n_k \ge k$ for each $k$.
The even terms $a_0, a_2, a_4, \dots$ form a subsequence, with $n_k = 2k$.

## Explanations

### L1 human

You get a subsequence when you delete some terms of a sequence and keep infinitely many terms in their old order.
From $1, 2, 3, 4, \dots$ you can keep the even numbers $2, 4, 6, \dots$.
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<document source="https://math2.tamnd.com/node/0R8JR0ARJ4.md" tag="0R8JR0ARJ4" kind="example">
# Geometric series as a power series

## Example

Let $R$ be a commutative ring and let $a \in R$.
In the ring $R[\![X]\!]$ of [[formal power series|072W67KFYK]],
$$(1 - aX) \sum_{n \ge 0} a^n X^n = 1.$$
Thus $1 - aX$ is a unit, and
$$\frac{1}{1 - aX} = \sum_{n \ge 0} a^n X^n.$$

## Remarks

The proof is a telescoping product. The coefficient of $X^n$ in the product is $a^n - a \cdot a^{n-1} = 0$ for $n \ge 1$, and it is $1$ for $n = 0$.
With $a = 1$, it gives $\frac{1}{1 - X} = 1 + X + X^2 + \dots$, the [[generating function|0HH6JX5WZZ]] of the constant sequence $1$.
Over $\mathbb{R}$ the same identity holds as a convergent series for $|aX| < 1$, but the formal identity needs no convergence.

## Explanations

### L1 human

Multiply $1 + aX + a^2 X^2 + \dots$ by $1 - aX$. Every term cancels with the next one, and only $1$ is left.
So the geometric series is the inverse of $1 - aX$.
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<document source="https://math2.tamnd.com/node/0R8X9XA83A.md" tag="0R8X9XA83A" kind="statement">
# A linear map is determined by its values on a basis

## Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $(b_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$.
If $S, T \colon V \to W$ are [[linear maps|0NRYHJPQG2]] with $S(b_i) = T(b_i)$ for all $i \in I$, then $S = T$.
Moreover, for each family $(w_i)_{i \in I}$ of vectors of $W$, there is exactly one linear map $T \colon V \to W$ with $T(b_i) = w_i$ for all $i \in I$.

## Remarks

The map is $T(\sum_i a_i b_i) = \sum_i a_i w_i$.
The values on a basis can be chosen freely; the values on a merely spanning family cannot.
This statement is the reason why a linear map between finite-dimensional spaces is described by a matrix.

## Explanations

### L1 human

A linear map is known as soon as we know where it sends the basis vectors.
Every other vector is a combination of basis vectors, and linearity tells us where it goes.
Conversely, we may send the basis vectors anywhere we like.
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<document source="https://math2.tamnd.com/node/0R8YXRC9AJ.md" tag="0R8YXRC9AJ" kind="statement">
# Integers modulo a prime form a field

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then the ring $\mathbb{Z}/p\mathbb{Z}$ of the [[integers modulo p|01N3AYAA6N]] is a field.
That is, each nonzero element of $\mathbb{Z}/p\mathbb{Z}$ has a multiplicative inverse.

## Remarks

The proof: if $\overline{a} \neq \overline{0}$, then $p \nmid a$, so $\gcd(a, p) = 1$. [[Bézout's identity|0DQVHMYAGA]] gives $a x + p y = 1$, and then $\overline{a} \cdot \overline{x} = \overline{1}$.
The converse holds: if $n > 1$ is composite, with $n = a b$ and $1 < a, b < n$, then $\overline{a} \cdot \overline{b} = \overline{0}$, so $\mathbb{Z}/n\mathbb{Z}$ is not a field.
The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$ or $\mathrm{GF}(p)$. It is the finite field with $p$ elements.

## Explanations

### L1 human

Modulo $7$, you can divide by each nonzero number. For example, $3 \cdot 5 = 15 \equiv 1 \pmod 7$, so $5$ plays the part of $1/3$.
Modulo $6$ this fails: no number $x$ gives $2 x \equiv 1 \pmod 6$, because $2 x$ is always even.
</document>

<document source="https://math2.tamnd.com/node/0R9B0B5458.md" tag="0R9B0B5458" kind="statement">
# Characterization of the floor

## Statement

Let $x$ be a real number and let $n$ be an integer.
Then $n = \lfloor x \rfloor$ if and only if $n \le x < n + 1$.
In particular, $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$.

## Remarks

Thus the [[floor|0YVRNEDSVG]] of $x$ is the unique integer $n$ with $n \le x < n + 1$.
The fractional part of $x$ is $x - \lfloor x \rfloor$. It lies in the interval $[0, 1)$.
An equivalent form: $x - 1 < \lfloor x \rfloor \le x$.

## Explanations

### L1 human

The intervals $[n, n + 1)$ for integers $n$ cover the number line without overlap.
Each real number lies in exactly one of them, and the floor of the number is the left end of that interval.
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<document source="https://math2.tamnd.com/node/0R9D4CEJEB.md" tag="0R9D4CEJEB" kind="statement">
# Divisibility by 3

## Statement

Let $n$ be a natural number with decimal digits $d_k d_{k-1} \dots d_0$, so that $n = \sum_{i=0}^{k} d_i 10^i$.
Then $3 \mid n$ if and only if $3 \mid d_0 + d_1 + \dots + d_k$.

## Remarks

More precisely, $n \equiv d_0 + d_1 + \dots + d_k \pmod 3$. This holds because $10 \equiv 1 \pmod 3$, so $10^i \equiv 1 \pmod 3$ by [[compatibility with powers|0J5VGJYYC3]].
For example, $3 \mid 7{,}341$, because $7 + 3 + 4 + 1 = 15$ and $3 \mid 15$.
The rule for [[divisibility by 9|0ESHD7ZQDX]] has the same proof.

## Explanations

### L1 human

Add the digits of the number. If the sum is a multiple of $3$, the number is a multiple of $3$, and if not, it is not.
You can repeat: for $987{,}654$ the sum is $39$, and $3 + 9 = 12$ is a multiple of $3$. So $3 \mid 987{,}654$.
</document>

<document source="https://math2.tamnd.com/node/0R9F850CTH.md" tag="0R9F850CTH" kind="statement">
# Gcd of Fibonacci numbers

## Statement

Let $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$\gcd(F_m, F_n) = F_{\gcd(m, n)}.$$

## Remarks

Example: $\gcd(F_{12}, F_{18}) = \gcd(144, 2584) = 8 = F_6$.
In particular, $F_m$ divides $F_n$ when $m$ divides $n$. Consecutive Fibonacci numbers are coprime, because $\gcd(n, n+1) = 1$ and $F_1 = 1$.
The proof follows the Euclidean algorithm and uses the [[addition formula|00CN2F997Q]].

## Explanations

### L1 human

The greatest common divisor of two Fibonacci numbers is again a Fibonacci number.
To find which one, take the greatest common divisor of the two indices.
</document>

<document source="https://math2.tamnd.com/node/0R9GF3JD0K.md" tag="0R9GF3JD0K" kind="concept">
# Linear Diophantine equation

## Summary

A linear Diophantine equation in two unknowns is an equation $a x + b y = c$ with given integers $a$, $b$ and $c$.
Its solutions are the pairs of integers $(x, y)$ that satisfy it.
When $a$ and $b$ are not both $0$, the equation has an integer solution if and only if $\gcd(a, b)$ divides $c$.
In this case, let $d = \gcd(a, b)$ and let $(x_0, y_0)$ be one solution. Then the solutions are the pairs $(x_0 + k b / d, \; y_0 - k a / d)$ with $k$ an integer.

## Remarks

[[Bézout's identity|0DQVHMYAGA]] gives the existence of a solution, and the [[extended Euclidean algorithm|0964YQFSA8]] computes one.
For example, $6 x + 10 y = 7$ has no integer solution, because $\gcd(6, 10) = 2$ does not divide $7$.
The equation is a special case of a [[Diophantine equation|08B1RTW7VZ]].

## Explanations

### L1 human

Ask whether you can pay $c$ with coins of value $a$ and $b$, when you can also get coins back as change.
Each amount that you can pay is a multiple of $\gcd(a, b)$, and each such multiple is possible.
For example, $3 x + 5 y = 1$ has the solution $x = 2$, $y = -1$.
</document>

<document source="https://math2.tamnd.com/node/0R9JR0465V.md" tag="0R9JR0465V" kind="definition">
# Dyck word

## Definition

A Dyck word is a finite word $w$ over the two-letter alphabet $\{U, D\}$ with these two properties.
The word $w$ has as many letters $U$ as letters $D$.
Each prefix of $w$ has at least as many letters $U$ as letters $D$.

The semilength of a Dyck word is its number of letters $U$, which is half its length.

## Remarks

Other common alphabets are $\{(, )\}$ and $\{[, ]\}$. Then the Dyck words are the balanced strings of brackets.
The Dyck words of semilength $2$ are $UUDD$ and $UDUD$.
Read $U$ as a step $(1, 1)$ and $D$ as a step $(1, -1)$. A Dyck word is then a [[lattice path|040WY0TT2G]] from $(0, 0)$ to $(2n, 0)$ that never goes below the horizontal axis.
The Dyck words of semilength $n$ are [[counted by the Catalan number|017SRZ7NE9]] $C_n$.

## Explanations

### L1 human

A Dyck word is a correct sequence of opening and closing brackets.
The sequence $(()())$ is a Dyck word, but $())($ is not, because after three symbols there are more closing than opening brackets.
</document>

<document source="https://math2.tamnd.com/node/0R9NMSM1ZP.md" tag="0R9NMSM1ZP" kind="concept">
# Adjoint of a linear map

## Summary

The adjoint of a linear map between finite-dimensional inner product spaces is a linear map in the opposite direction.
It moves the map from one side of the inner product to the other.
</document>

<document source="https://math2.tamnd.com/node/0R9PHTGVNW.md" tag="0R9PHTGVNW" kind="concept">
# One-sided derivative

## Summary

A one-sided derivative is the limit of difference quotients from one side of a point only. A function is differentiable at an interior point exactly when its left and right derivatives exist and agree. The node [[one-sided derivative|0HXEX0DPRA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0R9R0Y1HDK.md" tag="0R9R0Y1HDK" kind="object">
# Unitary group

## Description

Let $n \ge 1$.
The *unitary group* is the set
$$U(n) = \{U \in M_n(\C) : U^* U = I_n\}$$
of complex $n \times n$ [[unitary matrices|0AR68JTMDT]], with matrix multiplication.
It is a subgroup of the general linear group $GL_n(\C)$.

## Remarks

It is the group of linear isometries of $\C^n$ with the standard inner product.
$U(1)$ is the circle group $\{z \in \C : \abs{z} = 1\}$.
The unitary matrices with determinant $1$ form the special unitary group $SU(n)$.
$U(n)$ is a compact connected Lie group of real dimension $n^2$.

## Explanations

### L1 human

The unitary group collects all complex matrices that keep the length of every vector.
For $n = 1$, these are multiplications by complex numbers of absolute value $1$, that is, rotations of the complex plane.
The group appears throughout physics.
</document>

<document source="https://math2.tamnd.com/node/0RAM4E07NR.md" tag="0RAM4E07NR" kind="statement">
# Limit of a sum of sequences

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.
If $a_n \to a$ and $b_n \to b$ as $n \to \infty$, then $a_n + b_n \to a + b$ as $n \to \infty$.

## Remarks

Given $\varepsilon > 0$, choose $N$ so that $|a_n - a| < \varepsilon/2$ and $|b_n - b| < \varepsilon/2$ for all $n \ge N$. The triangle inequality then gives $|(a_n + b_n) - (a + b)| < \varepsilon$.
With the constant sequence $-1$ and the [[product rule for limits|0734RSVF2V]], it also gives $a_n - b_n \to a - b$.
It is one of the rules of the [[algebra of limits|07XBM3A9J5]].

## Explanations

### L1 human

If $a_n$ is close to $a$ and $b_n$ is close to $b$, then $a_n + b_n$ is close to $a + b$.
The two errors add, so we make each error smaller than half of the allowed error.
</document>

<document source="https://math2.tamnd.com/node/0RAS24E1R5.md" tag="0RAS24E1R5" kind="definition">
# Bernoulli number

## Definition

The Bernoulli numbers $B_0, B_1, B_2, \dots$ are the rational numbers given by the power series expansion
$$\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}.$$
Equivalently, $B_0 = 1$ and $\sum_{k=0}^{n} \binom{n+1}{k} B_k = 0$ for each $n \ge 1$.

## Remarks

The first values are $B_0 = 1$, $B_1 = -\frac{1}{2}$, $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$ and $B_6 = \frac{1}{42}$. Also $B_n = 0$ for each odd $n \ge 3$.
Some authors use the convention $B_1 = +\frac{1}{2}$, from the series $\frac{t e^t}{e^t - 1}$.
They appear in the formulas for the sums of powers $1^k + 2^k + \dots + n^k$, and in the values $\zeta(2n)$ of the Riemann zeta function.
They are named after Jacob Bernoulli. Seki Takakazu tabulated them in his Katsuyō Sanpō of 1712.

## Explanations

### L1 human

The Bernoulli numbers are a sequence of fractions that appear in many formulas, for example in the formula for $1^k + 2^k + \dots + n^k$.
You can compute them one after another from a simple recurrence.
</document>

<document source="https://math2.tamnd.com/node/0RBKKNH4GV.md" tag="0RBKKNH4GV" kind="concept">
# Focus of a conic

## Summary

A focus of a conic is a special point that is used to define the conic. An ellipse and a hyperbola have two foci, a parabola has one, and the focus of a circle is its centre.
</document>

<document source="https://math2.tamnd.com/node/0RBQS1JJCR.md" tag="0RBQS1JJCR" kind="concept">
# Quadratic form

## Summary

A quadratic form on a vector space is a map $Q$ to the scalars with $Q(av) = a^2 Q(v)$.
Also, $Q(u + v) - Q(u) - Q(v)$ must be bilinear in $u$ and $v$.
In coordinates, it is a homogeneous polynomial of degree two.
</document>

<document source="https://math2.tamnd.com/node/0RBT5PB74K.md" tag="0RBT5PB74K" kind="statement">
# Bessel's inequality

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Let $e_1, \dots, e_k$ be an [[orthonormal family|0RW9TNNT87]] in $V$.
Then, for each $x \in V$,
$$\sum_{i=1}^{k} \abs{\langle x, e_i \rangle}^2 \le \norm{x}^2.$$

## Remarks

The proof applies the Pythagorean theorem to $x = \sum_i \langle x, e_i \rangle e_i + r$, where $r$ is orthogonal to each $e_i$.
Equality holds if and only if $x$ lies in the span of $e_1, \dots, e_k$.
The inequality also holds for infinite orthonormal families, with a series on the left.

## Explanations

### L1 human

The squared lengths of the projections of a vector onto perpendicular directions add up to at most its squared length.
The missing part is the squared length of the component outside these directions.
If the directions span the whole space, then nothing is missing.
</document>

<document source="https://math2.tamnd.com/node/0RCKQN1E09.md" tag="0RCKQN1E09" kind="definition">
# Cluster point of a sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $c$ be a real number.
The number $c$ is a cluster point of $(a_n)$ if the following holds for each $\varepsilon > 0$ and each natural number $N$.
There is a natural number $n \ge N$ such that $|a_n - c| < \varepsilon$.

## Remarks

Equivalently, for each $\varepsilon > 0$ there are infinitely many indices $n$ with $|a_n - c| < \varepsilon$.
In $\mathbb{R}$, the number $c$ is a cluster point of $(a_n)$ if and only if some [[subsequence|0R8J3ZQMJS]] of $(a_n)$ converges to $c$.
The sequence $(-1)^n$ has the two cluster points $1$ and $-1$, and no limit.
For a bounded sequence, the [[limit superior|0P8BA7T7S5]] is the largest cluster point and the [[limit inferior|0H4R3RESGV]] is the smallest.

## Explanations

### L1 human

A cluster point is a number that the sequence visits again and again, as closely as you like.
A limit is a cluster point that the sequence approaches with all its late terms, not only with some of them.
</document>

<document source="https://math2.tamnd.com/node/0RCP929Y1M.md" tag="0RCP929Y1M" kind="concept">
# Cyclic quadrilateral

## Summary

A cyclic quadrilateral is a quadrilateral whose four vertices lie on one circle. Its sides are chords of that circle.
</document>

<document source="https://math2.tamnd.com/node/0RCQZ36MME.md" tag="0RCQZ36MME" kind="concept">
# Indicator function

## Summary

The indicator function of a subset takes the value $1$ on the subset and the value $0$ outside it.
</document>

<document source="https://math2.tamnd.com/node/0RCV1ZFP8G.md" tag="0RCV1ZFP8G" kind="concept">
# Boolean ring

## Summary

A Boolean ring is a ring in which every element is idempotent. Boolean rings are Boolean algebras written in the language of rings, with exclusive or as addition. The node [[Boolean ring|0N5FCV2YV6]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0RDJK7EQYY.md" tag="0RDJK7EQYY" kind="definition">
# Ordinal number

## Definition

A set $X$ is transitive when each element of an element of $X$ is an element of $X$.
An ordinal number, in the definition of von Neumann, is a transitive set $\alpha$ that is [[well-ordered|0F1D6PK9RZ]] by the relation $\in$, with $\beta < \gamma$ when $\beta \in \gamma$.

## Remarks

The first ordinals are $0 = \varnothing$, $1 = \{0\}$, $2 = \{0, 1\}$, and in general $n + 1 = n \cup \{n\}$.
The set $\omega = \{0, 1, 2, \dots\}$ of all finite ordinals is the first infinite ordinal.
Each well-ordered set is [[order isomorphic|0Y74KN99A5]] to exactly one ordinal, its order type.
Each ordinal $\alpha$ is the set of all ordinals smaller than $\alpha$.

## Explanations

### L1 human

Ordinals count positions: first, second, third, and then beyond all finite positions.
After $0, 1, 2, \dots$ comes $\omega$, then $\omega + 1$, and so on.
They let you continue a step-by-step construction past infinity.
</document>

<document source="https://math2.tamnd.com/node/0RDP2HEFEW.md" tag="0RDP2HEFEW" kind="definition">
# Arccosine

## Definition

The arccosine is the function $\arccos \colon [-1, 1] \to [0, \pi]$ such that, for each $x \in [-1, 1]$, $\arccos x$ is the unique $y \in [0, \pi]$ with $\cos y = x$.

## Remarks

The cosine is strictly decreasing and continuous on $[0, \pi]$, with range $[-1, 1]$. So $\arccos$ is well defined and [[continuous|07BY47HG5S]].
For all $x \in [-1, 1]$, $\arccos x = \frac{\pi}{2} - \arcsin x$.
For example, $\arccos 1 = 0$ and $\arccos(-1) = \pi$. Mathlib defines `Real.arccos` by the identity above.

## Explanations

### L1 human

The arccosine of $x$ is the angle between $0$ and $\pi$ whose cosine is $x$.
</document>

<document source="https://math2.tamnd.com/node/0RDRPBMNAC.md" tag="0RDRPBMNAC" kind="statement">
# Cesàro means of a convergent sequence converge

## Statement

Let $(a_n)_{n \ge 1}$ be a real sequence that converges to a real number $L$.
For $n \ge 1$, let $\sigma_n = (a_1 + \cdots + a_n)/n$ be its [[Cesàro mean|0P0DJHBGBB]].
Then $\sigma_n \to L$ as $n \to \infty$.

## Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_k - L| < \varepsilon$ for $k > N$. For $n > N$,
$$|\sigma_n - L| \le \frac{1}{n} \sum_{k=1}^{N} |a_k - L| + \frac{n - N}{n} \varepsilon.$$
The first term tends to $0$, so $|\sigma_n - L| < 2\varepsilon$ for large $n$.
The converse fails. The means of $(-1)^n$ tend to $0$, but $(-1)^n$ diverges.
The statement is a special case of the [[Stolz–Cesàro theorem|0Q3AKZV74D]] with $b_n = n$.

## Explanations

### L1 human

If the terms settle near $L$, then the running average also settles near $L$.
The early terms that are far from $L$ count less and less, because we divide by $n$.
</document>

<document source="https://math2.tamnd.com/node/0RE6X3BGPN.md" tag="0RE6X3BGPN" kind="example">
# Space of matrices

## Example

Let $F$ be a field and let $m, n \ge 1$.
The set $M_{m \times n}(F)$ of $m \times n$ [[matrices|0VD9ZSWWJG]] over $F$ is a [[vector space|000AYPA9AJ]] over $F$, with [[matrix addition|0Y7H3SAVA1]] and [[scalar multiplication|09V6DDACXR]].
Its zero vector is the [[zero matrix|0ZTKJJDGR7]].
The matrices $E_{ij}$, with entry $1$ in position $(i, j)$ and $0$ elsewhere, form a basis, so $\dim M_{m \times n}(F) = mn$.

## Remarks

The map that lists the entries of a matrix row by row is a linear isomorphism $M_{m \times n}(F) \to F^{mn}$.
The space $M_{m \times n}(F)$ is isomorphic to the [[space of linear maps|0X9MBPP2Z3]] from $F^n$ to $F^m$.
The symmetric $n \times n$ matrices form a subspace of dimension $n(n+1)/2$.

## Explanations

### L1 human

Matrices of one fixed size can be added and scaled, so they form a vector space.
A $2 \times 2$ matrix holds four numbers, so this space has dimension $4$.
It is $F^4$ written as a square instead of a row.
</document>

<document source="https://math2.tamnd.com/node/0REMT01N36.md" tag="0REMT01N36" kind="statement">
# Generating function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] random variables with values in $\{0, 1, 2, \dots\}$, and let $S = X_1 + \dots + X_n$.
Then for each complex $z$ with $|z| \le 1$, the [[probability-generating functions|0Z8PSSAVBA]] satisfy
$$G_S(z) = G_{X_1}(z) \, G_{X_2}(z) \cdots G_{X_n}(z).$$

## Remarks

Proof: $z^S = z^{X_1} \cdots z^{X_n}$. The factors are independent, so the [[product rule for expectations|05AW1G0WE1]] applies.
Example: a sum of $n$ independent Bernoulli($p$) variables has the generating function $(1 - p + pz)^n$. This is the generating function of $\operatorname{Bin}(n, p)$.
For a random number $N$ of i.i.d. terms $X_i$ with $N$ independent of the $X_i$, the sum $S_N = X_1 + \dots + X_N$ has $G_{S_N}(z) = G_N(G_{X_1}(z))$.

## Explanations

### L1 human

Adding independent counts corresponds to multiplying their generating functions.
This turns the hard computation of the law of a sum into a product of power series.
</document>

<document source="https://math2.tamnd.com/node/0RFMMBHFTV.md" tag="0RFMMBHFTV" kind="definition">
# Chi-squared distribution

## Definition

Let $k \ge 1$ be an integer and let $Z_1, \dots, Z_k$ be [[i.i.d.|0AJN7WHBJZ]] standard normal random variables.
The chi-squared distribution with $k$ degrees of freedom, written $\chi^2_k$, is the law of
$$Q = Z_1^2 + Z_2^2 + \dots + Z_k^2.$$

## Remarks

It has the [[density|04YG690N5V]] $f(x) = \frac{x^{k/2 - 1} e^{-x/2}}{2^{k/2}\, \Gamma(k/2)}$ for $x > 0$, and $f(x) = 0$ for $x \le 0$. So $\chi^2_k$ is the [[gamma distribution|06395GBNWD]] with shape $k/2$ and rate $1/2$.
If $Q \sim \chi^2_k$, then $E[Q] = k$ and $\operatorname{Var}(Q) = 2k$.
If $Q_1 \sim \chi^2_{k_1}$ and $Q_2 \sim \chi^2_{k_2}$ are independent, then $Q_1 + Q_2 \sim \chi^2_{k_1 + k_2}$.
For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n$ with [[sample variance|0JN9P4A944]] $S_n^2$, $(n - 1) S_n^2 / \sigma^2 \sim \chi^2_{n - 1}$.

## Explanations

### L1 human

Square $k$ independent standard normal numbers and add them: the result has the chi-squared distribution with $k$ degrees of freedom.
Its mean is $k$, and it has values in $[0, \infty)$ only.
</document>

<document source="https://math2.tamnd.com/node/0RFP7F3SEW.md" tag="0RFP7F3SEW" kind="definition">
# Composition factors

## Definition

Let $G$ be a [[group|G5S1M9]] with a [[composition series|0XV2A49X6Q]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$.
The composition factors of $G$ are the simple groups $G_i / G_{i-1}$ for $i = 1, \dots, n$, counted with multiplicity.
The number $n$ is the composition length of $G$.

## Remarks

The [[Jordan–Hölder theorem|0E088ZP533]] shows that the composition factors do not depend on the series, up to order and isomorphism.
The composition factors of $\mathbb{Z}/12\mathbb{Z}$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$.
The composition factors of $S_4$ are $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$.
Groups that are not isomorphic can have the same composition factors, for example $\mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

## Explanations

### L1 human

The composition factors are the simple pieces of a group.
They are like the prime factors of an integer, counted with multiplicity.
But unlike for integers, the pieces do not determine the group.
</document>

<document source="https://math2.tamnd.com/node/0RFWW56Y22.md" tag="0RFWW56Y22" kind="statement">
# Cauchy–Schwarz inequality

## Statement

Let $V$ be an [[inner product space|0QVNCJXRY3]] over $\R$ or $\C$, with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$.
Then, for all $x, y \in V$,
$$\abs{\langle x, y \rangle} \le \norm{x} \norm{y}.$$
Equality holds if and only if $x$ and $y$ are linearly dependent.

## Remarks

For the dot product on $\R^n$, it says $\left(\sum_i x_i y_i\right)^2 \le \left(\sum_i x_i^2\right)\left(\sum_i y_i^2\right)$.
In a real inner product space, it allows the definition of the angle $\theta \in [0, \pi]$ with $\cos\theta = \langle x, y \rangle / (\norm{x} \norm{y})$ for nonzero $x$ and $y$.
One proof uses that $\norm{x - ty}^2 \ge 0$ for all real $t$, so a quadratic has a nonpositive discriminant.
It implies the [[triangle inequality|0FJZMBX0ZZ]] for the induced norm.

## Explanations

### L1 human

The inner product of two vectors is at most the product of their lengths, in absolute value.
In the plane, this is the fact that $\abs{\cos\theta} \le 1$.
Equality happens only when the vectors lie on one line.
</document>

<document source="https://math2.tamnd.com/node/0RG9SPQSWM.md" tag="0RG9SPQSWM" kind="definition">
# Subgroup generated by a subset

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.
The subgroup generated by $S$, written $\langle S \rangle$, is the intersection of all [[subgroups|0X1JXT47GA]] of $G$ that contain $S$.

## Remarks

By [[the intersection property|0ZTGMDNVTB]], $\langle S \rangle$ is a subgroup, and it is contained in each subgroup that contains $S$.
Its elements are the products $s_1^{\varepsilon_1} s_2^{\varepsilon_2} \cdots s_k^{\varepsilon_k}$ with $k \geq 0$, $s_i \in S$ and $\varepsilon_i \in \{1, -1\}$, where the empty product ($k = 0$) is the identity.
In particular, $\langle \emptyset \rangle = \{e\}$.

## Explanations

### L1 human

Start with some elements of a group, and add everything that you can make from them with products and inverses.
The result is the subgroup generated by these elements.
For example, in $(\mathbb{Z}, +)$, the subgroup generated by $\{4, 6\}$ is the set $2\mathbb{Z}$ of even integers.
</document>

<document source="https://math2.tamnd.com/node/0RGGF8ZS6D.md" tag="0RGGF8ZS6D" kind="definition">
# Stabiliser

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
The stabiliser of $x$ is the set $G_x = \{g \in G : g \cdot x = x\}$.

## Remarks

The stabiliser $G_x$ is a [[subgroup|0X1JXT47GA]] of $G$, but in general it is not normal.
For the action of $S_n$ on $\{1, \dots, n\}$, the stabiliser of $n$ is a subgroup isomorphic to $S_{n-1}$.
For the action of $G$ on itself by conjugation, the stabiliser of $a$ is the [[centraliser|02YYV36BFM]] $C_G(a)$.

## Explanations

### L1 human

The stabiliser of a point collects the group elements that keep this point in place.
They may move other points.
For the rotations of a square about its centre, only the identity keeps a corner in place.
</document>

<document source="https://math2.tamnd.com/node/0RGWYWWBM5.md" tag="0RGWYWWBM5" kind="definition">
# Incircle

## Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with [[incentre|0M9MX8YW0A]] $I$ and inradius $r$. Let $d$ be the Euclidean distance.
The incircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $I$ and the radius $r$:
$$\{\, X \in \mathbb{R}^2 \;:\; d(X, I) = r \,\}.$$

## Remarks

The incircle touches each side of the triangle at exactly one point, the foot of the perpendicular from $I$ to that side.
Let $T$ be the area of $ABC$ and $s$ half of its perimeter. Then $T = r s$, so $r = T / s$.
An excircle is a circle outside the triangle that touches one side and the extensions of the other two sides. Each triangle has three excircles.

## Explanations

### L1 human

The incircle is the biggest circle that fits inside a triangle. It touches each side once. Cut the triangle into three pieces from $I$ to the corners: each piece has the height $r$, and this gives $T = r s$.
</document>

<document source="https://math2.tamnd.com/node/0RGZ10DFPG.md" tag="0RGZ10DFPG" kind="definition">
# Submodule

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A *submodule* of $M$ is a subset $N \subseteq M$ such that:

1. $N$ is a subgroup of the abelian group $(M, +)$.
2. $rx \in N$ for all $r \in R$ and $x \in N$.

## Remarks

A submodule is a module over $R$ with the restricted operations.
The submodules of $R$, as a left module over itself, are the left ideals of $R$.
The $\Z$-submodules of an abelian group are its subgroups, and over a field the submodules are the [[subspaces|00YC1D652K]].
Intersections and sums of submodules are submodules.

## Explanations

### L1 human

A submodule is a module inside a bigger module.
It contains the sums and the scalar multiples of its elements.
For example, the even integers are a submodule of $\Z$ over $\Z$.
</document>

<document source="https://math2.tamnd.com/node/0RH1TPDGCQ.md" tag="0RH1TPDGCQ" kind="definition">
# Torsion subgroup

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]].
The torsion subgroup of $A$ is $A_T = \{a \in A : a \text{ has finite order}\}$.
The group $A$ is a torsion group when $A_T = A$, and it is torsion-free when $A_T = \{e\}$.

## Remarks

The set $A_T$ is a [[subgroup|0X1JXT47GA]] because $A$ is commutative: if $a^m = e$ and $b^n = e$, then $(a b)^{m n} = e$.
In a nonabelian group, the elements of finite order need not form a subgroup. In $\langle x, y \mid x^2, y^2 \rangle$, the product $x y$ has infinite order.
The [[quotient group|0KYX2N69J0]] $A / A_T$ is torsion-free, and $A_T$ is a characteristic subgroup of $A$.
For $A = \mathbb{Q}/\mathbb{Z}$, each element has finite order, so $A_T = A$, but the orders have no upper bound.

## Explanations

### L1 human

The torsion subgroup collects the elements that return to the identity after finitely many steps.
In an abelian group, these elements form a subgroup.
Removing them leaves a group in which only the identity has finite order.
</document>

<document source="https://math2.tamnd.com/node/0RH4E3CAV9.md" tag="0RH4E3CAV9" kind="definition">
# Projection

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *projection* on $V$ is an [[endomorphism|0J7H0YBFDJ]] $P \colon V \to V$ with $P \circ P = P$.

## Remarks

If $P$ is a projection, then $V = \im P \oplus \ker P$, and $P$ acts as the identity on $\im P$.
Conversely, for each [[direct sum|0XYYT5KNTX]] $V = U \oplus W$, the map $u + w \mapsto u$ is a projection with image $U$ and kernel $W$.
If $P$ is a projection, then $I - P$ is the projection with image $\ker P$ and kernel $\im P$.
In an inner product space, an [[orthogonal projection|0B8KZRT28M]] is a projection whose kernel is orthogonal to its image.

## Explanations

### L1 human

A projection maps space onto a subspace and leaves the vectors of that subspace where they are.
Projecting a second time changes nothing.
The shadow of an object on the floor, in light from a fixed direction, is a picture of a projection.
</document>

<document source="https://math2.tamnd.com/node/0RJ5NHWYTP.md" tag="0RJ5NHWYTP" kind="concept">
# Picard iteration

## Summary

Picard iteration is the construction of a solution of $x' = f(t, x)$, $x(t_0) = x_0$ as a limit of successive approximations.
One starts with the constant curve $\varphi_0(t) = x_0$ and sets $\varphi_{k+1} = P \varphi_k$, where $P$ is the [[Picard operator|00WAVB68F9]]: $\varphi_{k+1}(t) = x_0 + \int_{t_0}^{t} f(s, \varphi_k(s))\, ds$.
Under the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]], the curves $\varphi_k$ converge uniformly on the time interval to the unique solution.

## Remarks

Example: for $x' = x$, $x(0) = 1$, the iterates are $\varphi_k(t) = \sum_{j=0}^{k} t^j / j!$. They converge to $e^t$.
Let $K$ be the Lipschitz constant and $T$ the length of the time interval. The error after $k$ steps is at most $C (K T)^{k} / k!$ for a constant $C$. See [[an iterate of the Picard operator is a contraction|08XDXP86XE]].

## Explanations

### L1 human

Start with a rough guess, the constant $x_0$.
Put the guess into the right side of the integral equation to get a better guess, and repeat.
Each step improves the approximation, and the guesses converge to the solution.
</document>

<document source="https://math2.tamnd.com/node/0RJ5TZH6G0.md" tag="0RJ5TZH6G0" kind="statement">
# Limit of the cumulative distribution function at infinity

## Statement

Let $X$ be a real random variable with [[cumulative distribution function|0MXMS3AE13]] $F_X$.
Then
$$\lim_{x \to +\infty} F_X(x) = 1.$$

## Remarks

The proof: the events $\{X \le n\}$ increase with $n$ and their union is $\Omega$. So [[continuity from below|0X8HBKCSYB]] gives $F_X(n) \to 1$, and $F_X$ is nondecreasing.
In the same way, $\lim_{x \to -\infty} F_X(x) = 0$, because the events $\{X \le -n\}$ decrease to $\varnothing$.
The proof uses that $X$ takes only real values. For a random variable with values in $[-\infty, +\infty]$, the limit is $1 - P(X = +\infty)$.

## Explanations

### L1 human

For a very large $x$, it is almost sure that $X \le x$, because $X$ is some real number.
So the probability $F_X(x)$ gets as close to $1$ as we like.
</document>

<document source="https://math2.tamnd.com/node/0RJF668P1T.md" tag="0RJF668P1T" kind="definition">
# Linear map

## Definition

Let $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.
A map $T \colon V \to W$ is *linear* when, for all $u, v \in V$ and $a \in F$,
$$T(u + v) = T(u) + T(v), \qquad T(av) = a\, T(v).$$
A linear map is also called a *linear transformation*.

## Remarks

A linear map sends $0$ to $0$ and linear combinations to linear combinations: $T(\sum a_i v_i) = \sum a_i T(v_i)$.
For an $m \times n$ matrix $A$, the map $F^n \to F^m$, $x \mapsto Ax$, is linear.
Rotations of $\R^2$ about the origin are linear; translations by a nonzero vector are not linear.
Each linear map between finite-dimensional spaces has a [[matrix|0YQSGNYE3K]] in a choice of bases.

## Explanations

### L1 human

A linear map respects the two operations of a vector space.
The image of a sum is the sum of the images, and the image of a scaled vector is the scaled image.
In the plane, linear maps send lines through the origin to lines through the origin, or to the origin, and keep parallel lines parallel.
</document>

<document source="https://math2.tamnd.com/node/0RJMNEN3BJ.md" tag="0RJMNEN3BJ" kind="definition">
# Symmetric group

## Definition

Let $X$ be a set.
The symmetric group $\operatorname{Sym}(X)$ is the set of all bijections $\sigma \colon X \to X$, with composition of maps as the operation.
For a positive integer $n$, the symmetric group $S_n$ is $\operatorname{Sym}(\{1, 2, \dots, n\})$.
An element of $S_n$ is a permutation of degree $n$.

## Remarks

The identity map is the identity element, and the inverse of $\sigma$ is the inverse map $\sigma^{-1}$.
In this node, the product $\sigma \tau$ means first $\tau$, then $\sigma$.
The group $S_n$ is not [[abelian|082GHNPJ53]] for $n \geq 3$, because $(1\ 2)(2\ 3) \neq (2\ 3)(1\ 2)$.
If $X$ and $Y$ have the same cardinality, then $\operatorname{Sym}(X)$ and $\operatorname{Sym}(Y)$ are [[isomorphic|08MG7H8JW5]].

## Explanations

### L1 human

The symmetric group collects all the ways to shuffle the elements of a set.
Two shuffles combine into one shuffle when you do them one after the other.
Each shuffle can be undone, so the shuffles form a group.
</document>

<document source="https://math2.tamnd.com/node/0RJSBR9GW1.md" tag="0RJSBR9GW1" kind="concept">
# Totally disconnected space

## Summary

A totally disconnected space is a topological space whose only connected subsets are the empty set and the single points.
</document>

<document source="https://math2.tamnd.com/node/0RJZV05MJA.md" tag="0RJZV05MJA" kind="definition">
# Polar coordinates

## Definition

The polar coordinate map is $P \colon [0, \infty) \times \mathbb{R} \to \mathbb{R}^2$, $P(r, \theta) = (r \cos \theta, r \sin \theta)$.
A pair $(r, \theta)$ with $P(r, \theta) = (x, y)$ is a pair of polar coordinates of the point $(x, y)$. Here $r = \sqrt{x^2 + y^2}$ is the radial coordinate and $\theta$ is the angular coordinate.

## Remarks

Each point other than the origin has exactly one pair of polar coordinates with $r > 0$ and $\theta \in [0, 2\pi)$. The origin has $r = 0$ and any $\theta$.
For $x > 0$, $\theta = \arctan(y / x)$ is one valid angle. In general one uses the two-argument function $\operatorname{atan2}(y, x)$, with values in $(-\pi, \pi]$.
The restriction of $P$ to $(0, \infty) \times (0, 2\pi)$ is a $C^\infty$ [[diffeomorphism|0Z1Y7JHGBR]] onto the plane without the half-line $\{(x, 0) : x \ge 0\}$. Its Jacobian determinant is $r$.
The circle of radius $a$ about the origin is $r = a$, and the ray at angle $\alpha$ is $\theta = \alpha$.

## Explanations

### L1 human

Instead of moving $x$ units right and $y$ units up, you turn by the angle $\theta$ and then walk the distance $r$.
Both descriptions name the same point.
</document>

<document source="https://math2.tamnd.com/node/0RKB5MKC05.md" tag="0RKB5MKC05" kind="definition">
# Circumcircle

## Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with [[circumcentre|0F98FZK14A]] $O$ and circumradius $R = d(O, A)$, where $d$ is the Euclidean distance.
The circumcircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $O$ and the radius $R$:
$$\{\, X \in \mathbb{R}^2 \;:\; d(X, O) = R \,\}.$$
It is the unique circle that contains $A$, $B$ and $C$.

## Remarks

For a triangle in $\mathbb{R}^n$, the circumcircle is the circle in the plane of the triangle with the centre $O$ and the radius $R$. Mathlib's `Affine.Simplex.circumsphere` is the sphere with this centre and radius in the ambient space.
The [[law of sines|0PASQ34GHG]] gives $2R = a / \sin \alpha$, with the side $a$ opposite the angle $\alpha$.
A polygon whose vertices all lie on one circle is a cyclic polygon. Each triangle is cyclic, but not each quadrilateral is.

## Explanations

### L1 human

Through any three points that are not on one line, there is exactly one circle. For a triangle, this is the circumcircle. For the right triangle with the vertices $(0, 0)$, $(4, 0)$ and $(0, 3)$, the circumcircle has the centre $(2, 1.5)$ and the radius $2.5$.
</document>

<document source="https://math2.tamnd.com/node/0RKF1ZT5JJ.md" tag="0RKF1ZT5JJ" kind="concept">
# Complement

## Summary

The complement of a set is the set of all objects of the universe that are not elements of the set.
</document>

<document source="https://math2.tamnd.com/node/0RKG5YW66Y.md" tag="0RKG5YW66Y" kind="concept">
# LU decomposition

## Summary

An *LU decomposition* of a square matrix $A$ is a factorization $A = LU$.
Here $L$ is a [[lower triangular matrix|00J3KZP8Y0]] and $U$ is an [[upper triangular matrix|0N4XY7Z3NN]].
[[Gaussian elimination|0JVJ2WCXZH]] without row exchanges produces such a factorization: $U$ is the row echelon form, and $L$ stores the multipliers of the row operations.
With row exchanges, one gets $PA = LU$ for a [[permutation matrix|08SP3DF8NB]] $P$, and each square matrix over a field has such a factorization.

## Remarks

An invertible matrix has an LU decomposition with $L$ unit lower triangular if and only if all its leading principal minors are nonzero.
This decomposition is then unique.
The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has no LU decomposition, but it has one after a row exchange.
To solve $Ax = b$, solve $Ly = b$ by forward substitution and $Ux = y$ by [[back substitution|0ATFVE08MW]].
The decomposition also gives $\det A = \det L \det U$.

## Explanations

### L1 human

An LU decomposition writes a matrix as a product of two triangular matrices.
Triangular systems are easy to solve, so we solve $Ax = b$ in two easy steps.
It is useful when we solve many systems with the same $A$ and different right sides.
</document>

<document source="https://math2.tamnd.com/node/0RKQQQACY7.md" tag="0RKQQQACY7" kind="definition">
# Slope field

## Definition

Let $\Omega$ be an open subset of $\mathbb{R}^2$ and let $f : \Omega \to \mathbb{R}$ be a function.
The slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \in \Omega$ the line through $(x, y)$ with slope $f(x, y)$.
In a picture, one draws a short segment of this line at each point of a grid in $\Omega$.

## Remarks

Let $u$ be a differentiable function on an interval. It is a [[solution|04SW7WECS8]] of $y' = f(x, y)$ if and only if its graph is tangent to the slope field at each of its points.
For $y' = y$, the segments are horizontal on the $x$-axis and get steeper as $|y|$ grows.
For a system $x' = F(x)$ in the plane, the same picture with arrows is a picture of the vector field $F$.

## Explanations

### L1 human

A slope field shows the direction that a solution must take at each point.
You can sketch a solution without a formula: start at a point and follow the small segments.
This gives a quick picture of how all the solutions behave.
</document>

<document source="https://math2.tamnd.com/node/0RKW2X63YR.md" tag="0RKW2X63YR" kind="statement">
# Logarithm is the inverse of the exponential

## Statement

For every real $x > 0$, $e^{\ln x} = x$. For every real $y$, $\ln(e^y) = y$.
The function $\ln$ is continuous and strictly increasing on $(0, \infty)$, and its range is $\mathbb{R}$.

## Remarks

The two identities say that $\ln$ and $\exp$ are inverse bijections between $(0, \infty)$ and $\mathbb{R}$.
Continuity follows from the [[continuity of inverse functions|07BY47HG5S]], since $\exp$ is [[strictly increasing|0KN2V7840E]] and continuous.
Also $\ln x \to \infty$ as $x \to \infty$, and $\ln x \to -\infty$ as $x \to 0^+$. Mathlib states the first identity as `Real.exp_log`.

## Explanations

### L1 human

Taking $\ln$ undoes taking $\exp$, and the other way round.
The graph of $\ln$ is the mirror image of the graph of $e^x$ in the line $y = x$.
</document>

<document source="https://math2.tamnd.com/node/0RMKABS2S5.md" tag="0RMKABS2S5" kind="definition">
# Sum of subspaces

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$.
The *sum* of $U$ and $W$ is
$$U + W = \{u + w : u \in U, \ w \in W\}.$$
For subspaces $U_1, \dots, U_k$, the sum is $U_1 + \cdots + U_k = \{u_1 + \cdots + u_k : u_i \in U_i\}$.

## Remarks

The sum $U + W$ is a subspace, and it is the smallest subspace of $V$ that contains $U \cup W$.
The sum is the [[span|0DTY8B9HMS]] of $U \cup W$.
When $U \cap W = \{0\}$, the sum is a [[direct sum|0XYYT5KNTX]].
For finite-dimensional subspaces, [[Grassmann's formula|06HTJNCMPA]] gives $\dim(U + W) = \dim U + \dim W - \dim(U \cap W)$.

## Explanations

### L1 human

The sum of two subspaces collects every vector that we can build from one piece in each subspace.
In space, the sum of two different lines through the origin is the plane that contains both lines.
It is the smallest subspace that holds both subspaces.
</document>

<document source="https://math2.tamnd.com/node/0RMVS7RXZX.md" tag="0RMVS7RXZX" kind="definition">
# Exponential function

## Definition

The exponential function $\exp \colon \mathbb{R} \to \mathbb{R}$ is defined by
$$\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots.$$

## Remarks

The [[power series|0VP56B4EMX]] converges absolutely for every real $x$, because [[its radius of convergence is infinite|0PVJRACB9H]].
One also writes $e^x$ for $\exp(x)$, where $e = \exp(1)$ is the number [[e|0MVCFMW7S4]].
Other common definitions use the limit $\lim_{n \to \infty} (1 + x/n)^n$, or the equation $f' = f$ with $f(0) = 1$. They give the same function.
Mathlib defines `Real.exp` as the real part of the complex exponential series.

## Explanations

### L1 human

The exponential function is an infinite polynomial with the coefficients $1/n!$.
It grows faster than every polynomial, and its derivative is itself.
</document>

<document source="https://math2.tamnd.com/node/0RN00NTM3S.md" tag="0RN00NTM3S" kind="concept">
# Inner automorphism

## Summary

An inner automorphism of a group is the automorphism given by conjugation by a fixed element.
The inner automorphisms form a normal subgroup of the automorphism group.
</document>

<document source="https://math2.tamnd.com/node/0RN765S7DH.md" tag="0RN765S7DH" kind="concept">
# Proper metric space

## Summary

A proper metric space is a metric space in which each closed ball is compact.
In such a space, a set is compact if and only if it is closed and bounded.
</document>

<document source="https://math2.tamnd.com/node/0RN88KZBJ3.md" tag="0RN88KZBJ3" kind="concept">
# Homogeneous first-order equation

## Summary

A homogeneous first-order equation is an equation $y' = F(y/x)$, in which the slope depends only on the ratio $y/x$.
The substitution $y = x v$ changes it into a separable equation. See the [[definition|0K4ZWTFSV5]].
</document>

<document source="https://math2.tamnd.com/node/0RNGDMHT7M.md" tag="0RNGDMHT7M" kind="concept">
# Complex power

## Summary

A complex power $z^w$ with $z \neq 0$ is a number $e^{w \ell}$ where $\ell$ is a logarithm of $z$, see the [[definition|0ZV9A7EV0F]].
The principal value is $e^{w \operatorname{Log} z}$.
</document>

<document source="https://math2.tamnd.com/node/0RNMP6VCR2.md" tag="0RNMP6VCR2" kind="definition">
# Complement graph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]].
The complement of $G$ is the graph $\overline{G}$ with vertex set $V$.
Two distinct vertices are adjacent in $\overline{G}$ if and only if they are not adjacent in $G$.

## Remarks

The complement of the complement of $G$ is $G$.
The complement of the complete graph $K_n$ is the graph on $n$ vertices with no edges.
A [[clique|0G52BPNPKG]] of $G$ is an [[independent set|0KZJ82WEPP]] of $\overline{G}$. This links Ramsey numbers to colourings of $K_n$.

## Explanations

### L1 human

To get the complement, remove all edges of the graph and draw all the missing ones.
If $G$ records which people know each other, then $\overline{G}$ records which people are strangers.
</document>

<document source="https://math2.tamnd.com/node/0RNXVVTJTJ.md" tag="0RNXVVTJTJ" kind="example">
# Damped harmonic oscillator

## Example

Let $m > 0$, $k > 0$ and $c > 0$ be constants. The damped harmonic oscillator is the equation
$$m\, x'' + c\, x' + k\, x = 0, \quad \text{that is} \quad x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = 0,$$
with $\omega_0 = \sqrt{k/m}$ and damping ratio $\zeta = c / \left(2 \sqrt{m k}\right)$.
The characteristic roots are $r = \omega_0 \left(-\zeta \pm \sqrt{\zeta^2 - 1}\right)$, and there are three cases.
If $\zeta > 1$ (overdamped), the roots are real, distinct and negative, and $x = c_1 e^{r_1 t} + c_2 e^{r_2 t}$.
If $\zeta = 1$ (critically damped), $r = -\omega_0$ is a double root, and $x = (c_1 + c_2 t) e^{-\omega_0 t}$.
If $0 < \zeta < 1$ ([[underdamped|0JJ6GDDE7Z]]), $x = e^{-\zeta \omega_0 t} \left(c_1 \cos \omega_1 t + c_2 \sin \omega_1 t\right)$ with $\omega_1 = \omega_0 \sqrt{1 - \zeta^2}$.

## Remarks

In each case, each solution tends to $0$ as $t \to \infty$. The origin is an asymptotically stable [[equilibrium point|0N8NGKDZDZ]] of the system for $(x, x')$.
The term $c x'$ models a friction force proportional to the velocity.
The energy $E = \frac{1}{2} m (x')^2 + \frac{1}{2} k x^2$ satisfies $E' = -c (x')^2 \le 0$. So the energy decreases along each solution.

## Explanations

### L1 human

Friction takes energy from the oscillating mass.
With little friction, the mass still oscillates, but the swings get smaller.
With much friction, the mass creeps back to rest without any swing.
</document>

<document source="https://math2.tamnd.com/node/0RP20HFYM3.md" tag="0RP20HFYM3" kind="concept">
# Positive definite quadratic form

## Summary

A positive definite quadratic form is a real quadratic form that takes a positive value at every nonzero vector.
</document>

<document source="https://math2.tamnd.com/node/0RP7VWQ0YN.md" tag="0RP7VWQ0YN" kind="concept">
# Subgroup generated by a subset

## Summary

The subgroup generated by a subset of a group is the smallest subgroup that contains the subset.
Its elements are the finite products of elements of the subset and their inverses.
</document>

<document source="https://math2.tamnd.com/node/0RPDTWY2JS.md" tag="0RPDTWY2JS" kind="definition">
# Trace of a linear map

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *trace* of $T$ is
$$\tr T = \tr [T]_B,$$
the [[trace|0XC1RDBY7A]] of the matrix of $T$ in a basis $B$ of $V$.

## Remarks

The value does not depend on $B$: by the [[change of basis formula|0MQXVZEBMT]], another basis gives $P^{-1} [T]_B P$, and $\tr(P^{-1} A P) = \tr A$.
The trace is a linear map $\End_F(V) \to F$, and $\tr(ST) = \tr(TS)$.
The trace of the identity map of $V$ is $\dim V \cdot 1$.
A [[projection|033DA1WE8Y]] on a real or complex space has trace equal to its rank.

## Explanations

### L1 human

We define the trace of a linear map by its matrix in some basis.
Another basis gives a different matrix, but the trace stays the same.
So the trace belongs to the map, not to the basis.
</document>

<document source="https://math2.tamnd.com/node/0RQ8DHHR8V.md" tag="0RQ8DHHR8V" kind="statement">
# Cantor set is homeomorphic to the space of binary sequences

## Statement

Let $\{0, 1\}$ have the [[discrete topology|0S2TEHX4E1]] and let $\{0, 1\}^{\mathbb{N}}$ have the [[product topology|04S9HX3GAP]].
Then the map
$$\varphi \colon \{0, 1\}^{\mathbb{N}} \to C, \qquad \varphi((a_n)_{n \ge 1}) = \sum_{n \ge 1} \frac{2 a_n}{3^n}$$
is a [[homeomorphism|05GZ11PWRY]] onto the [[Cantor set|0BXNDZFQME]] $C$.

## Remarks

The map is a bijection, because each point of $C$ has exactly one base-3 expansion with digits in $\{0, 2\}$.
It is continuous: if two sequences agree in their first $N$ terms, their images differ by at most $3^{-N}$.
The space $\{0, 1\}^{\mathbb{N}}$ is compact by [[Tychonoff's theorem|0JTRSJKCFB]], and $C$ is Hausdorff. Thus $\varphi$ is a homeomorphism, by [[this theorem|0430ETCMKD]].
A space homeomorphic to $C$ is a Cantor space. By a theorem of Brouwer, these are exactly the nonempty compact metrizable spaces that are totally disconnected and have no isolated points.

## Explanations

### L1 human

Each point of the Cantor set is described by an infinite list of choices: left or right third, at each step.
These lists are the binary sequences, and the topology of the Cantor set matches the product topology on them.
</document>

<document source="https://math2.tamnd.com/node/0RQPZZ44T4.md" tag="0RQPZZ44T4" kind="definition">
# Identity element

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] on $S$.
An element $e \in S$ is a left identity when $e \ast a = a$ for all $a \in S$.
An element $e \in S$ is a right identity when $a \ast e = a$ for all $a \in S$.
An element $e \in S$ is an identity element (or two-sided identity) when it is both a left identity and a right identity.

## Remarks

The number $0$ is the identity element for addition on $\mathbb{Z}$, and $1$ is the identity element for multiplication.
If $e$ is a left identity and $f$ is a right identity, then $e = e \ast f = f$.
Thus an operation has at most one two-sided identity element.

## Explanations

### L1 human

An identity element is an element that does nothing when you combine it with another element.
Adding $0$ to a number gives the same number, and multiplying a number by $1$ gives the same number.
Some operations have no identity element, for example addition on the positive integers.
</document>

<document source="https://math2.tamnd.com/node/0RQSNVTFVZ.md" tag="0RQSNVTFVZ" kind="concept">
# Prime-counting function

## Summary

The prime-counting function $\pi(x)$ gives the number of primes that are less than or equal to a real number $x$.
The [[definition of the prime-counting function|0XRJB5SSNE]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0RRNVX1VKA.md" tag="0RRNVX1VKA" kind="statement">
# Wedderburn's little theorem

## Statement

Let $D$ be a [[division ring|0WZXK05ME7]] with a finite number of elements.
Then the multiplication of $D$ is commutative, so $D$ is a [[field|0QA8CHH8MG]].

## Remarks

The hypothesis of finiteness is necessary: the [[quaternions|0GE4DVQKR8]] are an infinite division ring that is not commutative.
A consequence is that every finite domain, commutative or not, is a field.

## History

Joseph Wedderburn gave the first proof in 1905.
Leonard Eugene Dickson gave another proof soon after.
The first proof of Wedderburn had a gap, and for this reason some historians credit Dickson with the first correct proof.

## Explanations

### L1 human

In an infinite division ring, $a b$ and $b a$ can be different.
The theorem says that this cannot happen when the ring is finite.
So there are no finite analogues of the quaternions.
</document>

<document source="https://math2.tamnd.com/node/0RRWNMYRXC.md" tag="0RRWNMYRXC" kind="concept">
# Fixed points of an action

## Summary

A fixed point of a group action is a point that each element of the group keeps in place.
The fixed points are exactly the points whose orbit has one element.
</document>

<document source="https://math2.tamnd.com/node/0RT96J4D3H.md" tag="0RT96J4D3H" kind="definition">
# Linear form

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *linear form* on $V$ is a [[linear map|0NRYHJPQG2]] $f \colon V \to F$, where $F$ is a vector space over itself.
Thus $f(u + v) = f(u) + f(v)$ and $f(av) = a f(v)$ for all $u, v \in V$ and $a \in F$.

## Remarks

A linear form is also called a *linear functional* or a *covector*.
On $F^n$, each linear form is $x \mapsto c_1 x_1 + \cdots + c_n x_n$ for unique scalars $c_1, \dots, c_n$.
The [[trace|0XC1RDBY7A]] is a linear form on the space of $n \times n$ matrices.
The linear forms on $V$ form the [[dual space|07JBSZE0V4]] of $V$.

## Explanations

### L1 human

A linear form takes a vector and returns one number, in a linear way.
On the plane, the map $(x, y) \mapsto 2x - 3y$ is a linear form.
Its kernel is the line $2x = 3y$, unless the form is zero.
</document>

<document source="https://math2.tamnd.com/node/0RV3AEXPT6.md" tag="0RV3AEXPT6" kind="definition">
# Focus of a conic

## Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$.
Suppose that $Q$ is an [[ellipse|0VKCCEXBJ1]] or a [[hyperbola|0WN5G117YC]] with the foci $F_1$ and $F_2$, as in their definitions. Then $F_1$ and $F_2$ are the foci of $Q$. If $Q$ is a [[circle|060W5CBZP6]], its centre is its only focus.
If $Q$ is a [[parabola|04S2GMHX82]] with the focus $F$ and a directrix, as in its definition, then $F$ is the only focus of $Q$.

## Remarks

The foci are determined by the conic. For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the foci are $(\pm c, 0)$ with $c^2 = a^2 - b^2$.
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci are $(\pm c, 0)$ with $c^2 = a^2 + b^2$.
Let $Q$ be a conic that is not a circle, and let $F$ be a focus of $Q$. Then $Q$ is the set of the points whose distances to $F$ and to a line, the [[directrix|0E5CXYNEPX]], have a constant ratio $e > 0$.

## Explanations

### L1 human

The foci are the two pins in the string construction of an ellipse. A parabolic mirror sends the light of a lamp at its focus out in parallel rays. For the parabola $y = x^2/4$, the focus is $(0, 1)$.
</document>

<document source="https://math2.tamnd.com/node/0RVD9R2V99.md" tag="0RVD9R2V99" kind="concept">
# Cauchy product

## Summary

The Cauchy product of two series is the series whose $n$-th term is the sum of the products $a_k b_{n-k}$. It is the discrete convolution of the two sequences of terms. The node [[Cauchy product|0B1K9PVWE9]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0RVRNR7KA1.md" tag="0RVRNR7KA1" kind="concept">
# Gaussian prime

## Summary

A Gaussian prime is a prime element of the ring of Gaussian integers.
The [[definition of a Gaussian prime|0GTEHJQRFK]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0RVYE75E2M.md" tag="0RVYE75E2M" kind="definition">
# Converse

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The converse of the [[implication|0ZNX6DQ3EJ]] $p \to q$ is the implication $q \to p$.

## Remarks

An implication and its converse are not [[logically equivalent|06GYGS41JK]].
For example, "if $n = 2$, then $n$ is prime" is true, but its converse "if $n$ is prime, then $n = 2$" is false.
The inverse of $p \to q$ is $\neg p \to \neg q$. It is the [[contrapositive|04T1CT7212]] of the converse.

## Explanations

### L1 human

The converse reads the arrow backwards.
"If an animal is a dog, then it is a mammal" is true.
Its converse, "if an animal is a mammal, then it is a dog", is false.
</document>

<document source="https://math2.tamnd.com/node/0RW8KPMXF6.md" tag="0RW8KPMXF6" kind="definition">
# Spherical coordinates

## Definition

The spherical coordinate map is $S \colon [0, \infty) \times [0, \pi] \times \mathbb{R} \to \mathbb{R}^3$,
$$S(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \ \rho \sin \varphi \sin \theta, \ \rho \cos \varphi).$$
A triple $(\rho, \varphi, \theta)$ with $S(\rho, \varphi, \theta) = (x, y, z)$ is a triple of spherical coordinates of $(x, y, z)$.
Here $\rho = \sqrt{x^2 + y^2 + z^2}$ is the distance to the origin, $\varphi$ is the polar angle from the positive $z$-axis, and $\theta$ is the azimuth.

## Remarks

The pair $(\rho \sin \varphi, \theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$, and $\cos \varphi = z / \rho$ for $\rho > 0$.
The coordinates are unique for points off the $z$-axis if $\rho > 0$, $0 < \varphi < \pi$ and $\theta \in [0, 2\pi)$.
The sphere of radius $a$ is $\rho = a$, and $\varphi = \pi / 4$ is a cone around the positive $z$-axis.
This is a common convention in mathematics. Physics texts often write $(r, \theta, \varphi)$, with $\theta$ the polar angle and $\varphi$ the azimuth.

## Explanations

### L1 human

To locate a point in space, give its distance from the origin, how far it tilts away from the north pole, and its longitude.
This is like latitude and longitude on the Earth, with the polar angle measured from the north pole.
</document>

<document source="https://math2.tamnd.com/node/0RW9TNNT87.md" tag="0RW9TNNT87" kind="concept">
# Orthonormal family

## Summary

An orthonormal family is a family of unit vectors that are pairwise orthogonal.
</document>

<document source="https://math2.tamnd.com/node/0RWE7CPJ32.md" tag="0RWE7CPJ32" kind="concept">
# Probability theory

## Summary

Probability theory is the branch of mathematics that studies chance in a precise way.
It models a random experiment by a [[probability space|0ZYBZ34RT2]]: a sample space, a family of events and a probability measure that obeys the [[probability axioms|0DAX8PMAJ4]].
Its main objects are events, random variables, their distributions and their expected values.
Its main results include limit theorems, such as the law of large numbers and the central limit theorem.

## Remarks

The modern axioms are due to Andrey Kolmogorov, who based them on measure theory in 1933.
Statistics uses probability theory to draw conclusions from data.

## Explanations

### L1 human

Probability theory gives numbers to uncertain things, such as the chance of rain or the chance of a six on a die.
The rules for these numbers are few and simple, but they give strong results.
For example, the average of many fair coin tosses is close to one half.
</document>

<document source="https://math2.tamnd.com/node/0RWKYX8339.md" tag="0RWKYX8339" kind="concept">
# Cartesian coordinate system

## Summary

A Cartesian coordinate system in the [[Euclidean plane|0XW7V9ZZ05]] is a choice of a unit of length and of two perpendicular directed lines, the axes. The axes meet at a point $O$, the origin.
Each point $P$ then has a unique pair $(x, y)$ of real numbers, its coordinates. The numbers $x$ and $y$ are the signed distances from the origin to the perpendicular projections of $P$ on the two axes.
In a [[Euclidean space|0XPR1E85W0]] of dimension $n$, a Cartesian coordinate system has $n$ mutually perpendicular axes through one origin, and each point has $n$ coordinates.
The map from a point to its coordinates is a bijection onto $\mathbb{R}^n$ that keeps distances.

## Remarks

In three dimensions, the usual names of the coordinates are $x$, $y$ and $z$.
The two axes of the plane divide it into four quadrants. The three coordinate planes of space divide it into eight octants.
Descartes and Fermat developed the idea in the 17th century.

## Explanations

### L1 human

A Cartesian coordinate system is like the grid of a map. You choose a starting point and two perpendicular directions. Then you describe each point by how far you go along each direction. For example, "3 to the right and 2 up" is the point $(3, 2)$.
</document>

<document source="https://math2.tamnd.com/node/0RWMRTPC87.md" tag="0RWMRTPC87" kind="statement">
# Jensen's inequality

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be [[convex|0VV0HSW1ZS]].
Let $x_1, \dots, x_n \in I$, and let $w_1, \dots, w_n \ge 0$ with $w_1 + \cdots + w_n = 1$.
Then
$$f\Big( \sum_{i=1}^{n} w_i x_i \Big) \le \sum_{i=1}^{n} w_i f(x_i).$$

## Remarks

The case $n = 2$ is the definition of convexity. Induction on $n$ gives the general case.
For a concave function, the inequality is reversed. With $f = \ln$ this gives the [[AM–GM inequality|0DRF0F3XC9]].
Mathlib states it as `ConvexOn.map_sum_le`.

## History

Johan Jensen proved the inequality in 1906.

## Explanations

### L1 human

The value of a convex function at an average is at most the average of its values.
The weights $w_i$ say how much each point counts in the average.
</document>

<document source="https://math2.tamnd.com/node/0RWQ5VZ6VQ.md" tag="0RWQ5VZ6VQ" kind="statement">
# Every infinite set has a countably infinite subset

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $A$ be an [[infinite set|01MKW4R283]].
Then there is an [[injective function|0PVG1CRAJN]] $f \colon \mathbb{N} \to A$, so $A$ has a [[countably infinite|0SEZ3JGZP7]] subset $f(\mathbb{N})$.

## Remarks

The proof chooses $a_0 \in A$, then $a_1 \in A \setminus \{a_0\}$, and so on. Each set $A \setminus \{a_0, \dots, a_n\}$ is nonempty because $A$ is infinite.
The axiom of countable choice is enough, but some choice is needed: in ZF alone, the statement cannot be proved.
It follows that $\aleph_0$ is the least infinite cardinal.

## Explanations

### L1 human

In an infinite set you can always pick one more element.
Picking forever gives a sequence of different elements, which is a copy of $\mathbb{N}$ inside the set.
</document>

<document source="https://math2.tamnd.com/node/0RX7DDBYCK.md" tag="0RX7DDBYCK" kind="statement">
# Image of an intersection

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ and $T$ be [[subsets|0FE8Q5S42Y]] of $A$.
Then the [[image|0Q3F20XB0A]] of $S \cap T$ is a subset of the intersection of the images:
$$f(S \cap T) \subseteq f(S) \cap f(T).$$

## Remarks

The inclusion can be strict. For the constant function $f \colon \{1, 2\} \to \{0\}$, $S = \{1\}$ and $T = \{2\}$, the left side is $\varnothing$ and the right side is $\{0\}$.
Equality holds for all $S$ and $T$ if and only if $f$ is [[injective|0PVG1CRAJN]].
For unions, equality always holds: $f(S \cup T) = f(S) \cup f(T)$.

## Explanations

### L1 human

If $b = f(x)$ with $x$ in both $S$ and $T$, then $b$ is in both images.
But $b$ can be in both images for another reason: $b = f(s) = f(t)$ with $s \in S$, $t \in T$ and $s \neq t$.
</document>

<document source="https://math2.tamnd.com/node/0RXD6EZYBK.md" tag="0RXD6EZYBK" kind="statement">
# Distributivity of matrix multiplication

## Statement

Let $R$ be a ring.
Let $A$ be an $m \times n$ matrix over $R$, and let $B$ and $C$ be $n \times p$ matrices over $R$.
Then $A(B + C) = AB + AC$.
Let $D$ and $E$ be $m \times n$ matrices over $R$, and let $G$ be an $n \times p$ matrix over $R$.
Then $(D + E)G = DG + EG$.

## Remarks

Both identities follow entry by entry from the distributive law in $R$.
We need both identities, because [[matrix multiplication|032Q9FB8MT]] is not commutative.
Together with [[associativity|073E4XC016]], they make the square matrices a [[ring|0FHVPRAE60]].

## Explanations

### L1 human

Multiplication of matrices spreads over addition, as for numbers.
For example, $A(B + C)$ is the same as $AB + AC$.
We must keep the order: $A$ stays on the left in both terms.
</document>

<document source="https://math2.tamnd.com/node/0RXK2Z808Q.md" tag="0RXK2Z808Q" kind="counterexample">
# A continuous nowhere differentiable function

## Counterexample

Let $\varphi \colon \mathbb{R} \to \mathbb{R}$ be the zigzag function with $\varphi(x) = |x|$ for $-1 \le x \le 1$ and $\varphi(x + 2) = \varphi(x)$ for all $x$.
Define
$$f(x) = \sum_{n=0}^{\infty} \Big( \frac{3}{4} \Big)^n \varphi(4^n x).$$
The function $f$ is continuous on $\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.

## Remarks

Since $0 \le \varphi \le 1$, the [[M-test|0NX5NWAFC9]] with $M_n = (3/4)^n$ and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.
Fix $x$. For each $m$, choose $\delta_m = \pm \frac{1}{2} 4^{-m}$ so that no integer lies strictly between $4^m x$ and $4^m (x + \delta_m)$.
Then $\Big| \frac{f(x + \delta_m) - f(x)}{\delta_m} \Big| \ge \frac{3^m + 1}{2}$. This tends to infinity while $\delta_m \to 0$, so $f'(x)$ does not exist.
This function replaces the cosine in the [[Weierstrass function|01A0MK1730]] by a piecewise linear zigzag.

## Explanations

### L1 human

Each term adds smaller and steeper zigzags to the graph.
The sum is continuous, but its graph has a corner at every scale near every point.
</document>

<document source="https://math2.tamnd.com/node/0RXKW044R4.md" tag="0RXKW044R4" kind="concept">
# Completely multiplicative function

## Summary

A completely multiplicative function is an arithmetic function with $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n$.
The [[definition of a completely multiplicative function|0GS27TKF1V]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0RXS8R6K37.md" tag="0RXS8R6K37" kind="definition">
# Preorder

## Definition

Let $A$ be a set.
A preorder on $A$ is a [[binary relation|08EJC6Q25H]] $\le$ on $A$ that is [[reflexive|0XEGNW851S]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le c$ imply $a \le c$.

## Remarks

Divisibility on $\mathbb{Z}$ is a preorder. It is not antisymmetric, because $2 \mid -2$ and $-2 \mid 2$.
An antisymmetric preorder is a [[partial order|011C29Q1S1]].
For a preorder, $a \le b$ and $b \le a$ together define an [[equivalence relation|0SGTMBEWRF]]. The preorder gives a partial order on the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

Compare people by height with "is at most as tall as".
This is a preorder: two different people can have the same height.
</document>

<document source="https://math2.tamnd.com/node/0RXX2PT1MQ.md" tag="0RXX2PT1MQ" kind="statement">
# Chain rule

## Statement

Let $f$ be a real function on an open interval $I$ that is differentiable at $a \in I$.
Let $g$ be a real function on an open interval $J$ with $f(I) \subseteq J$, and assume that $g$ is differentiable at $f(a)$.
Then $g \circ f$ is differentiable at $a$, and
$$(g \circ f)'(a) = g'(f(a)) \, f'(a).$$

## Remarks

A naive proof multiplies and divides by $f(a + h) - f(a)$. This fails when $f(a + h) = f(a)$ for small $h$. A correct proof uses the linear approximation of $g$ at $f(a)$.
For example, $(\sin(x^2))' = 2 x \cos(x^2)$.
In Leibniz notation, $\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}$. Mathlib states the rule as `HasDerivAt.comp`.

## History

Gottfried Wilhelm Leibniz seems to have used the rule first, in a memoir of 1676.

## Explanations

### L1 human

Suppose $y$ changes $f'(a)$ times as fast as $x$, and $z$ changes $g'(y)$ times as fast as $y$.
Then $z$ changes $g'(y) f'(a)$ times as fast as $x$.
The rates multiply.
</document>

<document source="https://math2.tamnd.com/node/0RY0B72XAS.md" tag="0RY0B72XAS" kind="statement">
# Structure theorem for finitely generated abelian groups

## Statement

Let $G$ be a [[finitely generated|082DWWXMRJ]] [[abelian group|082GHNPJ53]].
Then $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$ for an integer $n \geq 0$ and prime powers $q_1, \ldots, q_t$.
Also $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$.
The rank $n$, the multiset of the $q_i$ and the sequence of the $k_i$ are unique.

## Remarks

The finite part is the [[torsion subgroup|0DBWXQRXQJ]] of $G$, and $\mathbb{Z}^n$ is a [[free abelian group|0WJX8MGG37]] of rank $n$.
So $G$ is finite if and only if $n = 0$, and then the theorem is the [[classification of finite abelian groups|07CQR9BW3R]].
Each finitely generated torsion-free abelian group is free abelian. The group $(\mathbb{Q}, +)$ shows that the condition "finitely generated" is necessary.
The theorem is a special case of the structure theorem for finitely generated modules over a principal ideal domain.

## History

Henry John Stephen Smith solved the case of finitely presented abelian groups in 1861, with the Smith normal form of integer matrices.
Henri Poincaré proved the theorem for finitely generated abelian groups in 1900, with a matrix proof.
Emmy Noether extended Kronecker's proof to finitely generated abelian groups in 1926.

## Explanations

### L1 human

A finitely generated abelian group is a free part $\mathbb{Z}^n$ plus a finite part.
The finite part is a sum of cyclic groups.
So a short list of integers describes the group completely.
</document>

<document source="https://math2.tamnd.com/node/0RY4GGMEQG.md" tag="0RY4GGMEQG" kind="statement">
# Lucas's theorem

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $m$ and $n$ be natural numbers with base-$p$ digits $m = \sum_{i=0}^{k} m_i p^i$ and $n = \sum_{i=0}^{k} n_i p^i$, where $0 \le m_i, n_i < p$.
Then
$$\binom{m}{n} \equiv \prod_{i=0}^{k} \binom{m_i}{n_i} \pmod p,$$
with the convention $\binom{a}{b} = 0$ when $b > a$.

## Remarks

A consequence: $p \mid \binom{m}{n}$ if and only if $n_i > m_i$ for some digit $i$.
For example, with $p = 3$: $10 = 101_3$ and $4 = 011_3$. The theorem gives $\binom{10}{4} \equiv \binom{1}{0} \binom{0}{1} \binom{1}{1} = 0 \pmod 3$, and indeed $\binom{10}{4} = 210$.
For $p = 2$, the binomial coefficient $\binom{m}{n}$ is odd if and only if $m$ has a binary digit $1$ at each place where $n$ has one.

## History

Édouard Lucas published the theorem in 1878.

## Explanations

### L1 human

To find a binomial coefficient modulo a prime $p$, write both numbers in base $p$.
Then compute small binomial coefficients digit by digit and multiply them.
This explains the pattern of odd numbers in Pascal's triangle, which looks like the Sierpiński triangle.
</document>

<document source="https://math2.tamnd.com/node/0RYTGBJAGN.md" tag="0RYTGBJAGN" kind="definition">
# Big O notation

## Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$.
The notation $f(x) = O(g(x))$ as $x \to \infty$ means that there are real numbers $C > 0$ and $x_1 \ge x_0$ with
$$|f(x)| \le C |g(x)| \quad \text{for all } x \ge x_1.$$

## Remarks

The same definition works as $x \to a$, with $|x - a| < \delta$ in place of $x \ge x_1$.
If $g(x) \ne 0$ for large $x$, then $f = O(g)$ means that $f/g$ is bounded for large $x$.
For example, $3x^2 + 5x = O(x^2)$ as $x \to \infty$.
Mathlib defines it for filters, as `Asymptotics.IsBigO`.

## Explanations

### L1 human

The statement $f = O(g)$ says that $f$ grows at most as fast as $g$, up to a constant factor.
It ignores constant factors and small values of $x$.
</document>

<document source="https://math2.tamnd.com/node/0RYYK8PG4S.md" tag="0RYYK8PG4S" kind="statement">
# Law of excluded middle

## Statement

In classical logic, for each [[proposition|0ZGC7ZESYN]] $p$, the proposition $p \lor \neg p$ is true.

## Remarks

The formula $p \lor \neg p$ is a [[tautology|0D1AHYPM8N]].
Some systems take the law as an axiom.
Intuitionistic logic does not accept the law in general.
The law is the basis of [[proof by cases|0WCP0PVET6]] on $p$ and $\neg p$.

## Explanations

### L1 human

The law says that there is no third option: a statement is true, or its negation is true.
For example, each natural number $n$ is even or not even.
</document>

<document source="https://math2.tamnd.com/node/0RZ89QWTSJ.md" tag="0RZ89QWTSJ" kind="concept">
# Separable polynomial

## Summary

A separable polynomial is a polynomial that is coprime to its formal derivative.
Over a field, these are the polynomials with no repeated root in an algebraic closure.
The [[definition of a separable polynomial|03J8BS7FMV]] gives the condition.
</document>

<document source="https://math2.tamnd.com/node/0RZCTT30H5.md" tag="0RZCTT30H5" kind="statement">
# A nontrivial finite p-group has nontrivial centre

## Statement

Let $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]] with $G \neq \{e\}$.
Then the [[centre|0DZ93493VN]] $Z(G)$ is not $\{e\}$.

## Remarks

In the [[class equation|08BPSC6YTE]] $|G| = |Z(G)| + \sum [G : C_G(a_i)]$, the left side and each term of the sum are divisible by $p$.
So $p$ divides $|Z(G)|$, and $|Z(G)| \geq p$.
Equivalently, apply the [[fixed point lemma|098D2E7DEH]] to the action of $G$ on itself by conjugation.
A consequence is that each finite $p$-group is [[nilpotent|0Q7MMQHC1P]].

## Explanations

### L1 human

The conjugacy classes of a $p$-group that have more than one element all have sizes divisible by $p$.
The order of the group is also divisible by $p$.
So the classes with one element, which form the centre, cannot be just the identity.
</document>

<document source="https://math2.tamnd.com/node/0RZH6W98HK.md" tag="0RZH6W98HK" kind="concept">
# Interior

## Summary

The interior of a subset of a topological space is the union of all open sets inside it.
It is the largest open set contained in the subset.
</document>

<document source="https://math2.tamnd.com/node/0RZYDVZ475.md" tag="0RZYDVZ475" kind="definition">
# Principal ideal domain

## Definition

A principal ideal domain is an [[integral domain|09S3PFBEPM]] $R$ in which every ideal is [[principal|00A2EDFT9Q]].
That is, for every ideal $I$ of $R$, there is $a \in R$ with $I = (a)$.

## Remarks

Every [[Euclidean domain is a principal ideal domain|0QKVE1GD9A]]. So $\mathbb{Z}$, $K[x]$ for a field $K$, and $\mathbb{Z}[i]$ are principal ideal domains.
The ring $\mathbb{Z}[x]$ is [[not a principal ideal domain|06E3AG9TM2]].
Every principal ideal domain is a [[unique factorization domain|06YMXJMXYS]].
The ring $\mathbb{Z}[\tfrac{1 + \sqrt{-19}}{2}]$ is a principal ideal domain that is not a Euclidean domain.

## Explanations

### L1 human

In a principal ideal domain, every ideal is the set of the multiples of one element.
For the integers, every ideal is $n\mathbb{Z}$ for some $n$.
This makes divisibility in a principal ideal domain behave much like divisibility of integers.
</document>

<document source="https://math2.tamnd.com/node/0S049HG1W5.md" tag="0S049HG1W5" kind="statement">
# Power-reduction formula for cosine

## Statement

For every real $x$,
$$\cos^2 x = \frac{1 + \cos 2x}{2}, \qquad \sin^2 x = \frac{1 - \cos 2x}{2}.$$

## Remarks

The [[addition formula|028XEQCE88]] gives $\cos 2x = \cos^2 x - \sin^2 x$. With the [[Pythagorean identity|03H7WZ4T3K]], $\cos 2x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$.
These formulas give $\int_0^{\pi} \cos^2 x \, dx = \frac{\pi}{2}$.
Mathlib states the first one as `Real.cos_sq`.

## Explanations

### L1 human

A square of a cosine is a cosine of the double angle, shifted and scaled.
This turns squares into first powers, which are easy to integrate.
</document>

<document source="https://math2.tamnd.com/node/0S13V32CPB.md" tag="0S13V32CPB" kind="concept">
# Addition of natural numbers

## Summary

Addition of natural numbers is the operation that sends two [[natural numbers|0MWKJ44V5A]] $a$ and $b$ to their sum $a + b$.
The sum $a + b$ is the number that you get when you start at $a$ and apply the [[successor function|0AK9TGSNXY]] $b$ times.
</document>

<document source="https://math2.tamnd.com/node/0S166D363M.md" tag="0S166D363M" kind="definition">
# Wedge product of forms

## Definition

Let $U \subseteq \mathbb{R}^n$ be open. For indices $j_1, \dots, j_m \in \{1, \dots, n\}$, let $dx_{j_1} \wedge \dots \wedge dx_{j_m}$ be $0$ if two indices are equal. Otherwise it is $\operatorname{sgn}(\sigma) \, dx_I$, where $I$ is the increasing rearrangement of the indices and $\sigma$ is the permutation that sorts them.
For [[differential forms|0QHTX7WQEY]] $\alpha = \sum_I f_I \, dx_I$ of degree $k$ and $\beta = \sum_J g_J \, dx_J$ of degree $l$ on $U$, the wedge product is the $(k + l)$-form
$$\alpha \wedge \beta = \sum_{I, J} f_I \, g_J \; dx_{i_1} \wedge \dots \wedge dx_{i_k} \wedge dx_{j_1} \wedge \dots \wedge dx_{j_l}.$$

## Remarks

The product is bilinear and associative. It is graded commutative: $\beta \wedge \alpha = (-1)^{k l} \, \alpha \wedge \beta$. In particular, $dx_i \wedge dx_j = -dx_j \wedge dx_i$ and $dx_i \wedge dx_i = 0$.
For a $0$-form $f$, the product $f \wedge \beta$ is the form $f \beta$.
In the plane, $(a \, dx + b \, dy) \wedge (c \, dx + d \, dy) = (a d - b c) \, dx \wedge dy$. The coefficient is the determinant, which is the signed area of the parallelogram spanned by $(a, b)$ and $(c, d)$.
For 1-forms $\alpha$ and $\beta$ and vectors $v, w$, $(\alpha \wedge \beta)(v, w) = \alpha(v) \beta(w) - \alpha(w) \beta(v)$.

## Explanations

### L1 human

The wedge product is a multiplication of forms in which order matters up to sign, and a form times itself is zero.
It is the algebra behind determinants, signed areas and signed volumes.
</document>

<document source="https://math2.tamnd.com/node/0S19CWFZM2.md" tag="0S19CWFZM2" kind="statement">
# Size of an orbit times the size of the stabiliser

## Statement

Let a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \in X$.
Then $|G \cdot x| \cdot |G_x| = |G|$.
In particular, the size of each [[orbit|0YQPZB5Q1G]] divides $|G|$.

## Remarks

This follows from the [[orbit-stabiliser theorem|0E2HSCKZ6C]] and from $[G : G_x] \cdot |G_x| = |G|$, which is [[Lagrange's theorem|3C53TD]].
For example, the rotation group of a cube acts transitively on the $6$ faces, and the stabiliser of a face has $4$ elements. So the group has order $24$.

## Explanations

### L1 human

Each point of the orbit is reached by exactly $|G_x|$ group elements.
So the group elements are counted by the orbit size times the stabiliser size.
This is a good way to compute the order of a symmetry group.
</document>

<document source="https://math2.tamnd.com/node/0S1FJ8VEX9.md" tag="0S1FJ8VEX9" kind="concept">
# Hausdorff space

## Summary

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighbourhoods.
It is the most used separation axiom, and it makes limits of sequences unique.
</document>

<document source="https://math2.tamnd.com/node/0S1KG0XE40.md" tag="0S1KG0XE40" kind="statement">
# Variation of constants formula for linear systems

## Statement

Let $I$ be an open interval, let $A : I \to \mathbb{R}^{n \times n}$ and $b : I \to \mathbb{R}^n$ be continuous, and let $\Phi$ be a [[fundamental matrix|0MQ24RBG93]] of $x' = A(t) x$ on $I$.
Let $t_0 \in I$ and $x_0 \in \mathbb{R}^n$.
Then the unique solution on $I$ of $x' = A(t) x + b(t)$, $x(t_0) = x_0$ is
$$x(t) = \Phi(t)\, \Phi(t_0)^{-1} x_0 + \Phi(t) \int_{t_0}^{t} \Phi(s)^{-1}\, b(s)\, ds, \qquad t \in I.$$

## Remarks

For a constant matrix $A$, take $\Phi(t) = e^{tA}$. The formula becomes $x(t) = e^{(t - t_0) A} x_0 + \int_{t_0}^{t} e^{(t - s) A}\, b(s)\, ds$.
The formula comes from the trial solution $x(t) = \Phi(t)\, c(t)$: the equation gives $\Phi(t)\, c'(t) = b(t)$. This is [[variation of parameters|0Y6TH5PQVS]] for systems.
For $n = 1$, it is the [[solution formula for linear first-order equations|06B9W45XV0]].

## Explanations

### L1 human


The first term is the free motion from the initial state.
The integral adds up the effect of the force $b(s)$ at each earlier time $s$, carried forward to time $t$ by the homogeneous system.
</document>

<document source="https://math2.tamnd.com/node/0S21GNW9ZB.md" tag="0S21GNW9ZB" kind="concept">
# Image of a homomorphism

## Summary

The image of a group homomorphism $\varphi \colon G \to H$ is the set of the values $\varphi(g)$ for $g \in G$.
It is a subgroup of $H$.
</document>

<document source="https://math2.tamnd.com/node/0S23E6YE83.md" tag="0S23E6YE83" kind="concept">
# Kernel of a linear map

## Summary

The kernel of a linear map is the set of vectors that the map sends to zero.
It is a subspace of the domain.
</document>

<document source="https://math2.tamnd.com/node/0S29Y4ZW0Y.md" tag="0S29Y4ZW0Y" kind="concept">
# Surface area

## Summary

The surface area of a surface is a measure of its two-dimensional size.
For a [[parametrised surface|0ZG9QF40HE]], it is the double integral of the length of the normal vector over the parameter domain.
</document>

<document source="https://math2.tamnd.com/node/0S2SYHTWPY.md" tag="0S2SYHTWPY" kind="definition">
# Smallest prime factor

## Definition

Let $n$ be a natural number with $n \ge 2$.
The smallest prime factor of $n$ is the smallest [[prime number|0SN0MAEKYX]] $p$ such that $p \mid n$.
It exists by the [[existence of a prime divisor|0SPBEDVNW1]].

## Remarks

The smallest prime factor of $n$ is also the smallest divisor $d$ of $n$ with $d > 1$.
The number $n$ is prime if and only if its smallest prime factor is $n$.
If $n$ is composite, its smallest prime factor $p$ satisfies $p^2 \le n$. See [[a composite number has a prime factor at most its square root|0EB6BBMQMC]].
For example, the smallest prime factor of $91 = 7 \cdot 13$ is $7$.

## Explanations

### L1 human

Divide $n$ by $2$, then by $3$, then by $4$, and so on. The first number that divides $n$ exactly is the smallest prime factor.
It is always prime, because a smaller factor of it would also divide $n$ and would come earlier.
</document>

<document source="https://math2.tamnd.com/node/0S2TEHX4E1.md" tag="0S2TEHX4E1" kind="example">
# Discrete topology

## Example

Let $X$ be a set and let $\tau = \mathcal{P}(X)$ be the set of all subsets of $X$.
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the discrete topology. The space $(X, \tau)$ is a discrete space.

## Remarks

Each subset of $X$ is both open and closed. Each point $x$ gives an open set $\{x\}$.
It is the finest topology on $X$. It is the [[metric topology|0A6YRNZB2R]] of the [[discrete metric|0NA4FK7B8S]].
Each function from a discrete space to any topological space is [[continuous|0SY9ECQ19Q]].
A discrete space is [[compact|0QPHW703PV]] if and only if it is finite.

## Explanations

### L1 human

In the discrete topology, every set is open, so each point is isolated from all other points.
The integers $\mathbb{Z}$, as a subspace of $\mathbb{R}$, carry the discrete topology.
</document>

<document source="https://math2.tamnd.com/node/0S3SMRH0Z5.md" tag="0S3SMRH0Z5" kind="definition">
# Continuous linear map

## Definition

Let $V$ and $W$ be vector spaces over $\mathbb{R}$ or $\mathbb{C}$, with [[norms|0J3578562S]] $\|\cdot\|_V$ and $\|\cdot\|_W$.
A map $T \colon V \to W$ is a continuous linear map when $T$ is linear and [[continuous|0SY9ECQ19Q]] for the metrics $\|x - y\|_V$ and $\|x - y\|_W$.

## Remarks

A linear map is continuous if and only if it is bounded, that is, $\|Tx\|_W \le C \|x\|_V$ for some constant $C$ and all $x$. This is [[bounded linear maps are continuous|0F3K4C7XYW]].
The smallest such $C$ is the [[operator norm|0ZTEH055YF]] of $T$.
On a finite-dimensional space, [[each linear map is continuous|0C13E623WD]].
The derivative $p \mapsto p'$ on polynomials, with the sup norm on $[0, 1]$, is linear but not continuous: $\|x^n\|_\infty = 1$ but $\|n x^{n-1}\|_\infty = n$.

## Explanations

### L1 human

A continuous linear map respects the vector space structure and also the notion of closeness.
In infinite dimensions, a linear map can fail to be continuous, so this is a real condition.
</document>

<document source="https://math2.tamnd.com/node/0S4EMDQS2F.md" tag="0S4EMDQS2F" kind="statement">
# Abel's theorem

## Statement

Let $(a_n)$ be a real sequence such that the series $\sum_{n=0}^{\infty} a_n$ [[converges|0CVVKY7GY2]].
Let $G(x) = \sum_{n=0}^{\infty} a_n x^n$ for $-1 < x < 1$.
Then $G(x) \to \sum_{n=0}^{\infty} a_n$ as $x \to 1$ from the left.

## Remarks

The [[power series|0VP56B4EMX]] converges on $(-1, 1)$, since its radius of convergence is at least $1$.
The theorem gives the sum of the [[alternating harmonic series|0JQ5Y3WWVE]]. Apply it to the [[Mercator series|0YZXNE3HFR]] for $\ln(1 + x)$.
The converse fails. For $a_n = (-1)^n$, $G(x) = 1/(1 + x) \to 1/2$, but $\sum (-1)^n$ diverges.
Mathlib states it as `Real.tendsto_tsum_powerSeries_nhdsWithin_lt`.

## Explanations

### L1 human

A power series that converges at the endpoint $x = 1$ is continuous there from the left.
So its value at $1$ is the limit of its values inside the interval.
</document>

<document source="https://math2.tamnd.com/node/0S4JF619V7.md" tag="0S4JF619V7" kind="statement">
# Convergent sequences are bounded

## Statement

Let $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.
Then $(a_n)$ is [[bounded|05FQ3VGAR9]]: there is a real number $K \ge 0$ such that $|a_n| \le K$ for all $n \in \mathbb{N}$.

## Remarks

Let $L$ be the limit. Take $\varepsilon = 1$ in the [[epsilon-N definition|09G6KHS6GY]] to get $N$. Then $K = \max(|a_0|, \dots, |a_{N-1}|, |L| + 1)$ works.
The converse is false. The sequence $(-1)^n$ is bounded and divergent.
The statement holds in each metric space, where bounded means contained in some ball.

## Explanations

### L1 human

After some index, all terms are close to the limit, so they lie in a small interval.
Only finitely many terms come before that index, and finitely many numbers are always bounded.
</document>

<document source="https://math2.tamnd.com/node/0S4WTRE9GG.md" tag="0S4WTRE9GG" kind="statement">
# Inverse of a product

## Statement

Let $R$ be a commutative ring and let $A$ and $B$ be [[invertible|0EB9N0S33C]] $n \times n$ matrices over $R$.
Then $AB$ is invertible, and
$$(AB)^{-1} = B^{-1} A^{-1}.$$

## Remarks

The proof is the computation $(AB)(B^{-1} A^{-1}) = A (B B^{-1}) A^{-1} = A A^{-1} = I_n$, and the same in the other order.
By induction, $(A_1 \cdots A_k)^{-1} = A_k^{-1} \cdots A_1^{-1}$.
Thus the invertible $n \times n$ matrices form a group, the [[general linear group|0FCHE902D7]].

## Explanations

### L1 human

To undo two steps, we undo the last step first.
If we first apply $B$ and then $A$, then we must first undo $A$ and then undo $B$.
So the inverse of $AB$ is $B^{-1} A^{-1}$, with the order reversed.
</document>

<document source="https://math2.tamnd.com/node/0S4ZEZM7KE.md" tag="0S4ZEZM7KE" kind="concept">
# Spanning tree

## Summary

A spanning tree of a graph is a tree inside the graph that reaches every vertex. Spanning trees are the minimal connected spanning subgraphs, with uses in network design and in counting. The node [[spanning tree|0FQ5B8S1ZC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0S52TNNTH9.md" tag="0S52TNNTH9" kind="concept">
# Dot product

## Summary

The dot product of two vectors of $\R^n$ is the sum of the products of their corresponding coordinates.
</document>

<document source="https://math2.tamnd.com/node/0S60DVMVPN.md" tag="0S60DVMVPN" kind="concept">
# Upper bound

## Summary

Let $S$ be a subset of a partially ordered set.
An upper bound of $S$ is an element that is at least as large as each element of $S$.
</document>

<document source="https://math2.tamnd.com/node/0S61N75H45.md" tag="0S61N75H45" kind="definition">
# Root of unity

## Definition

Let $n$ be a positive integer.
A complex number $z$ is an $n$th root of unity when $z^n = 1$.
A complex number $z$ is a root of unity when it is an $n$th root of unity for some positive integer $n$.

## Remarks

The same definition works in any field, or in any commutative ring.
The $n$th roots of unity in $\mathbb{C}$ form a group under multiplication, written $\mu_n$.
For example, the $4$th roots of unity are $1$, $i$, $-1$ and $-i$.
Each root of unity $z$ satisfies $|z| = 1$, because $|z|^n = |z^n| = 1$.

## Explanations

### L1 human

A root of unity is a number that gives $1$ when you multiply it by itself often enough.
The number $-1$ is one, because $(-1)^2 = 1$, and so is $i$, because $i^4 = 1$.
</document>

<document source="https://math2.tamnd.com/node/0S6GMNJ62Y.md" tag="0S6GMNJ62Y" kind="concept">
# Biconditional

## Summary

The biconditional is the logical operation "if and only if".
It joins two [[propositions|0ZGC7ZESYN]] into one proposition that is true when both parts have the same truth value.
</document>

<document source="https://math2.tamnd.com/node/0S6T9KQNC0.md" tag="0S6T9KQNC0" kind="concept">
# Confidence interval

## Summary

A confidence interval is an interval computed from the data. Over repetitions of the experiment, it contains an unknown parameter with a fixed probability, such as $95\%$.
The node [[confidence interval|0K4NMMJBG2]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0S706JBZBG.md" tag="0S706JBZBG" kind="concept">
# Midpoint

## Summary

The midpoint of a line segment is the point of the segment that is at the same distance from both endpoints. It divides the segment into two halves.
</document>

<document source="https://math2.tamnd.com/node/0S7495H4HM.md" tag="0S7495H4HM" kind="statement">
# Order of the group of units of Z/nZ is Euler's totient

## Statement

Let $n \geq 1$ be an integer.
Then the [[order|0B85S2V2DR]] of the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is $\varphi(n)$.
Here $\varphi(n)$ is the number of integers $a$ with $0 \leq a \leq n - 1$ and $\gcd(a, n) = 1$.

## Remarks

The proof is a count, because the [[units of Z/nZ are the residues coprime to n|0GXNHEHQNA]].
For a prime $p$, the order is $\varphi(p) = p - 1$.
For example, $(\mathbb{Z}/9\mathbb{Z})^\times = \{1, 2, 4, 5, 7, 8\}$ has order $\varphi(9) = 6$.
With [[Lagrange's theorem|3C53TD]], this gives Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod{n}$ for each $a$ coprime to $n$.

## Explanations

### L1 human

The size of the group of units modulo $n$ is the number of residues coprime to $n$.
Euler's totient function counts exactly these residues.
So the group has $\varphi(n)$ elements.
</document>

<document source="https://math2.tamnd.com/node/0S7A0EMPF4.md" tag="0S7A0EMPF4" kind="definition">
# Algebraic multiplicity

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Let $\lambda$ be an [[eigenvalue|0BQAA2BK3Q]] of $T$.
The *algebraic multiplicity* of $\lambda$ is the largest integer $m \ge 1$ such that $(t - \lambda)^m$ divides the [[characteristic polynomial|0J5EMJRRFM]] $p_T(t)$.

## Remarks

The sum of the algebraic multiplicities of all eigenvalues is at most $\dim V$, with equality when $p_T$ splits over $F$.
For $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the eigenvalue $1$ has algebraic multiplicity $2$.
The algebraic multiplicity is at least the [[geometric multiplicity|099TWVJ909]].

## Explanations

### L1 human

An eigenvalue can be a root of the characteristic polynomial several times.
The algebraic multiplicity counts how many times.
For the identity matrix of size $n$, the eigenvalue $1$ has algebraic multiplicity $n$.
</document>

<document source="https://math2.tamnd.com/node/0S7G92KKA7.md" tag="0S7G92KKA7" kind="concept">
# Indeterminate form

## Summary

An indeterminate form is a type of [[limit|0TXTSEMHR5]] where the limits of the parts do not determine the limit of the whole.
The seven common forms are $0/0$, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^{\infty}$, and $\infty^0$.

## Remarks

For example, $x \to 0$ and $x^2 \to 0$, but $x^2/x \to 0$, $x/x \to 1$, and $x/x^2$ has no finite limit. All three are of the form $0/0$.
[[L'Hôpital's rule|0JWT4PPQ7B]] treats the forms $0/0$ and $\infty/\infty$. Logarithms turn the forms $0^0$, $1^{\infty}$, and $\infty^0$ into the form $0 \cdot \infty$.
Forms such as $1/0^+$ and $0^{\infty}$ are not indeterminate.

## Explanations

### L1 human

Some limits cannot be found by plugging in the limits of the parts.
Then a finer analysis is needed, because many different answers are possible.
</document>

<document source="https://math2.tamnd.com/node/0S7W6EG32V.md" tag="0S7W6EG32V" kind="concept">
# External direct sum

## Summary

The external direct sum of a family of vector spaces is the space of families of vectors, one vector from each space.
Only finitely many entries of such a family are nonzero.
For finitely many spaces it is the product of the spaces.
</document>

<document source="https://math2.tamnd.com/node/0S7Y9Q990Q.md" tag="0S7Y9Q990Q" kind="definition">
# Centre of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The centre of $G$ is the set $Z(G) = \{z \in G : zg = gz \text{ for all } g \in G\}$.

## Remarks

The centre is an [[abelian|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] of $G$, and it is [[characteristic|0KABPMRHM2]].
The group $G$ is abelian if and only if $Z(G) = G$.
The centre of the symmetric group $S_n$ is trivial for $n \geq 3$.
The centre of $\mathrm{GL}_n(\mathbb{R})$ is the set of the nonzero scalar matrices $\lambda I$.

## Explanations

### L1 human

The centre collects the elements that commute with everybody.
The identity is always in the centre.
In an abelian group the centre is everything, and in many nonabelian groups it is only the identity.
</document>

<document source="https://math2.tamnd.com/node/0S8FE7ZMFM.md" tag="0S8FE7ZMFM" kind="concept">
# Chain

## Summary

A chain in a partially ordered set is a subset in which any two elements are comparable.
</document>

<document source="https://math2.tamnd.com/node/0S8WX8A4V9.md" tag="0S8WX8A4V9" kind="statement">
# Hartman–Grobman theorem

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be continuously differentiable. Let $\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$, and let $x^* \in U$ be a [[hyperbolic equilibrium point|0C5PREPZWG]].
Let $A = Df(x^*)$.
Then there are open neighbourhoods $V \subseteq U$ of $x^*$ and $W \subseteq \mathbb{R}^n$ of $0$, and a homeomorphism $h : V \to W$ with $h(x^*) = 0$, as follows.
For each $x \in V$ and each $t \in \mathbb{R}$ such that $\varphi_s(x) \in V$ for all $s$ between $0$ and $t$, $h(\varphi_t(x)) = e^{tA} h(x)$.

## Remarks

So near $x^*$ the flow is topologically conjugate to the flow of the [[linearisation|09ZWYRQ41J]] $y' = A y$, given by the [[matrix exponential|0YC8JT513J]]. The map $h$ sends trajectories to trajectories and keeps the direction of time.
The map $h$ is in general only continuous, not differentiable.
The hypothesis of hyperbolicity is necessary. For $x' = -x^3$ and $x' = x^3$ on $\mathbb{R}$, the linearisation at $0$ is the same, but the two phase portraits are not conjugate.

## History

The theorem is named after Philip Hartman and David M. Grobman. It is also called the linearization theorem.

## Explanations

### L1 human

Near a hyperbolic equilibrium, a nonlinear system looks like its linear approximation, after a continuous change of coordinates.
So the pattern of trajectories of the linear system, such as a saddle or a spiral, also appears in the nonlinear system.
</document>

<document source="https://math2.tamnd.com/node/0S9EK5MQPE.md" tag="0S9EK5MQPE" kind="definition">
# Continuity in R^n

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}^m$, and let $a \in D$.
The function $f$ is continuous at $a$ when, for each $\varepsilon > 0$, there is a $\delta > 0$ such that $\|f(x) - f(a)\| < \varepsilon$ for each $x \in D$ with $\|x - a\| < \delta$.
Here $\|\cdot\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].
The function $f$ is continuous on $D$ when it is continuous at each point of $D$.

## Remarks

Let $a$ be a limit point of $D$. Then $f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$, with the [[limit of a function of several variables|0J8Y1NCY0K]].
A function into $\mathbb{R}^m$ is continuous if and only if each component function is continuous.
Sums, products and compositions of continuous functions are continuous. So each polynomial in $x_1, \dots, x_n$ is continuous on $\mathbb{R}^n$.
Continuity in each variable separately is weaker: see the [[separately continuous function that is not continuous|0C0KPAKK9Y]].

## Explanations

### L1 human

A continuous function has no jumps: if you move the input a little, the output moves only a little.
In the plane, "a little" means inside a small disc, so the condition must hold for moves in every direction.
</document>

<document source="https://math2.tamnd.com/node/0S9GGTR0EX.md" tag="0S9GGTR0EX" kind="concept">
# Orthonormal basis

## Summary

An orthonormal basis of a finite-dimensional inner product space is a basis whose vectors have length one and are pairwise orthogonal.
</document>

<document source="https://math2.tamnd.com/node/0S9ZVZ34MD.md" tag="0S9ZVZ34MD" kind="statement">
# Correspondence theorem

## Statement

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the map $K \mapsto K/N$ is a bijection from the set of [[subgroups|0X1JXT47GA]] $K$ of $G$ with $N \subseteq K$ to the set of all subgroups of $G/N$. Here $G/N$ is the [[quotient group|0KYX2N69J0]].
Let $K$ and $L$ be subgroups of $G$ that contain $N$. Then $K \subseteq L$ if and only if $K/N \subseteq L/N$. Also, $K$ is normal in $G$ if and only if $K/N$ is normal in $G/N$.

## Remarks

The inverse map sends a subgroup $\bar{K}$ of $G/N$ to its preimage $\{g \in G : gN \in \bar{K}\}$.
The bijection is an isomorphism of [[lattices of subgroups|0B8BW4YWTF]], and it keeps indices: $[L : K] = [L/N : K/N]$ when $N \subseteq K \subseteq L$.
For example, the subgroups of $\mathbb{Z}/12\mathbb{Z}$ correspond to the subgroups $d\mathbb{Z}$ of $\mathbb{Z}$ with $d$ dividing $12$.

## Explanations

### L1 human

The subgroups of a quotient group $G/N$ are the same as the subgroups of $G$ that contain $N$.
Passing to the quotient does not create new subgroups or lose old ones above $N$.
So the picture of the subgroups above $N$ in $G$ is the full picture of the subgroups of $G/N$.
</document>

<document source="https://math2.tamnd.com/node/0SAAHC36J9.md" tag="0SAAHC36J9" kind="definition">
# Generalised eigenvector

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$.
A *generalised eigenvector* of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and
$$(T - \lambda \id_V)^k (v) = 0$$
for some integer $k \ge 1$.
The least such $k$ is the *rank* of the generalised eigenvector.

## Remarks

The generalised eigenvectors of rank $1$ are the [[eigenvectors|0N6960W0BV]].
If a generalised eigenvector for $\lambda$ exists, then $\lambda$ is an eigenvalue.
For $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the vector $e_2$ is a generalised eigenvector for $1$ of rank $2$, and it is not an eigenvector.

## Explanations

### L1 human

Some matrices do not have enough eigenvectors to make a basis.
Generalised eigenvectors fill the gap: they reach zero after several applications of $T - \lambda \id$ instead of one.
They are the building blocks of the Jordan normal form.
</document>

<document source="https://math2.tamnd.com/node/0SAMR34WXC.md" tag="0SAMR34WXC" kind="statement">
# Derivative of tangent

## Statement

The tangent function is differentiable at every $x$ with $\cos x \ne 0$, and
$$\tan'(x) = \frac{1}{\cos^2 x} = 1 + \tan^2 x.$$

## Remarks

The [[quotient rule|0BS0NJPKVE]] gives $\tan' = \frac{\cos \cdot \cos - \sin \cdot (-\sin)}{\cos^2}$. The [[Pythagorean identity|03H7WZ4T3K]] turns the numerator into $1$.
The derivative is positive. So $\tan$ is strictly increasing on $(-\pi/2, \pi/2)$.
Mathlib states it as `Real.hasDerivAt_tan`.

## Explanations

### L1 human

The tangent function always rises, and it rises faster and faster near its asymptotes.
Its slope at $0$ is $1$.
</document>

<document source="https://math2.tamnd.com/node/0SAR26CKPS.md" tag="0SAR26CKPS" kind="statement">
# Dirichlet convolution formula

## Statement

Let $f$ and $g$ be arithmetic functions, and let $n$ be a positive integer.
Then
$$(f * g)(n) = \sum_{\substack{a, b \ge 1 \\ ab = n}} f(a) \, g(b),$$
where the sum runs over the ordered pairs of positive integers $(a, b)$ with $ab = n$, and $*$ is the [[Dirichlet convolution|0KH1N66QWW]].

## Remarks

The map $d \mapsto (d, n/d)$ is a bijection from the divisors of $n$ to these pairs. This gives the formula.
The symmetric form shows at once that $f * g = g * f$.
For example, $(f * g)(6) = f(1)g(6) + f(2)g(3) + f(3)g(2) + f(6)g(1)$.

## Explanations

### L1 human

Each divisor $d$ of $n$ comes with its partner $n/d$.
So a sum over divisors is the same as a sum over the ways to write $n$ as an ordered product of two factors.
</document>

<document source="https://math2.tamnd.com/node/0SB7Q8SYSF.md" tag="0SB7Q8SYSF" kind="definition">
# Perfect number

## Definition

A positive integer $n$ is perfect if it is equal to the sum of its positive divisors other than $n$.
Equivalently, $\sigma(n) = 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The first perfect numbers are $6$, $28$, $496$ and $8128$. For example, $28 = 1 + 2 + 4 + 7 + 14$.
The [[Euclid–Euler theorem|0WETJCKNRG]] describes all even perfect numbers by [[Mersenne primes|06GPJF9V7M]].
No odd perfect number is known, and it is an open problem whether one exists.

## Explanations

### L1 human

A perfect number is the sum of its smaller divisors.
The divisors of $6$ other than $6$ are $1$, $2$ and $3$, and $1 + 2 + 3 = 6$.
</document>

<document source="https://math2.tamnd.com/node/0SB9FNZECG.md" tag="0SB9FNZECG" kind="definition">
# Path component

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]].
For $x, y \in X$, write $x \simeq y$ when there is a [[path|0K0GE6BCYW]] in $X$ from $x$ to $y$.
This is an [[equivalence relation|0SGTMBEWRF]] on $X$. Its classes are the path components of $X$.

## Remarks

The relation is reflexive by constant paths, symmetric by $t \mapsto \gamma(1 - t)$, and transitive by joining two paths one after the other.
The path component of $x$ is the largest path-connected subset of $X$ that contains $x$.
Each path component lies in a [[connected component|0NRT0E2QT2]]. The [[topologist's sine curve|0K3T4A659R]] is one connected component but has two path components.
Path components need not be closed.

## Explanations

### L1 human

The path component of a point is the set of all points that you can reach from it along a path.
</document>

<document source="https://math2.tamnd.com/node/0SBRX5CZ5R.md" tag="0SBRX5CZ5R" kind="definition">
# Multiplicative order

## Definition

Let $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.
The multiplicative order of $a$ modulo $n$ is the smallest positive integer $k$ such that $a^k \equiv 1 \pmod n$.
The notation for it is $\operatorname{ord}_n(a)$.

## Remarks

The order exists, because $a^{\varphi(n)} \equiv 1 \pmod n$ by [[Euler's theorem|0VC4DVE7DQ]].
For each integer $m \ge 0$, $a^m \equiv 1 \pmod n$ if and only if $\operatorname{ord}_n(a) \mid m$. In particular, $\operatorname{ord}_n(a) \mid \varphi(n)$.
The order of $a$ modulo $n$ is the order of $\overline{a}$ in the group $(\mathbb{Z}/n\mathbb{Z})^\times$.
For example, $\operatorname{ord}_7(2) = 3$, because $2^1 = 2$, $2^2 = 4$ and $2^3 = 8 \equiv 1 \pmod 7$.

## Explanations

### L1 human

Compute the powers $a, a^2, a^3, \dots$ modulo $n$, until you reach $1$. The number of steps is the order.
Modulo $10$, the powers of $3$ are $3, 9, 7, 1$, so the order of $3$ modulo $10$ is $4$.
</document>

<document source="https://math2.tamnd.com/node/0SC8T7548D.md" tag="0SC8T7548D" kind="object">
# Golden ratio

## Description

The golden ratio is the real number
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.$$
It is the positive root of the polynomial $x^2 - x - 1$, so $\varphi^2 = \varphi + 1$.
The other root is $\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = -\frac{1}{\varphi}$.

## Remarks

Two positive numbers $a > b$ are in the golden ratio when $\frac{a + b}{a} = \frac{a}{b}$. Then $\frac{a}{b} = \varphi$.
The number $\varphi$ is irrational, because $\sqrt{5}$ is irrational.
The ratios of consecutive [[Fibonacci numbers|0GA9K57A39]] tend to $\varphi$. [[Binet's formula|0404D8TN7Y]] uses $\varphi$ and $\psi$.

## Explanations

### L1 human

Cut a stick into a long part and a short part. Do it so that the whole stick relates to the long part as the long part relates to the short part.
The ratio of long to short is then $\varphi$, about $1.618$.
</document>

<document source="https://math2.tamnd.com/node/0SCSQWXRA4.md" tag="0SCSQWXRA4" kind="definition">
# Hessian matrix

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $a \in U$ be a point at which all second [[partial derivatives|02RSXVM1ZN]] of $f$ exist.
The Hessian matrix of $f$ at $a$ is the $n \times n$ matrix
$$H_f(a) = \left( \frac{\partial^2 f}{\partial x_i \, \partial x_j}(a) \right)_{1 \le i, j \le n},$$
whose entry in row $i$ and column $j$ is $\partial^2 f / \partial x_i \, \partial x_j$ at $a$.
It is also written $\nabla^2 f(a)$ or $D^2 f(a)$.

## Remarks

If the second partial derivatives of $f$ are continuous near $a$, then $H_f(a)$ is a symmetric matrix, by [[Schwarz's theorem|0NNQY8WCW6]].
For $f(x, y) = x^2 + 3 x y - y^2$, the Hessian matrix at each point is $\begin{pmatrix} 2 & 3 \\ 3 & -2 \end{pmatrix}$.
The Hessian matrix is the Jacobian matrix of the gradient map $x \mapsto \nabla f(x)$, when this map is differentiable.
At a critical point, the signs of the eigenvalues of $H_f(a)$ decide the [[second derivative test|0GYDV9FE98]].

## Explanations

### L1 human

The gradient tells how a function tilts; the Hessian tells how it bends.
For a bowl like $x^2 + y^2$ the Hessian is $2 I$, and both directions bend upward. For a saddle like $x^2 - y^2$ one direction bends up and the other bends down.
</document>

<document source="https://math2.tamnd.com/node/0SD237FEJD.md" tag="0SD237FEJD" kind="statement">
# Gradient theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $\phi \colon U \to \mathbb{R}$ be of class $C^1$, and let $\gamma \colon [a, b] \to U$ be a piecewise $C^1$ curve.
Then
$$\int_\gamma \nabla \phi \cdot d\mathbf{r} = \phi(\gamma(b)) - \phi(\gamma(a)).$$

## Remarks

Proof for a $C^1$ curve: by the [[chain rule along a curve|0A8TJ0VHC7]], $(\phi \circ \gamma)'(t) = \nabla \phi(\gamma(t)) \cdot \gamma'(t)$. Then the [[parametric formula|0W3W1VYWBP]] and the fundamental theorem of calculus give the result. For a piecewise $C^1$ curve, add the pieces.
So the line integral of a gradient field depends only on the end points. Around a closed curve it is $0$.
Example: for $\phi(x, y) = x^2 y$ and any path from $(0, 0)$ to $(1, 2)$, $\int_\gamma (2 x y, x^2) \cdot d\mathbf{r} = 2$.
This is the case of curves in the [[generalised Stokes theorem|0HPHKCGEQ0]]: the boundary of a curve consists of its two end points.

## Explanations

### L1 human

This is the fundamental theorem of calculus for curves.
To integrate the gradient of $\phi$ along a path, you only need the values of $\phi$ at the end and at the start.
</document>

<document source="https://math2.tamnd.com/node/0SD528P0EK.md" tag="0SD528P0EK" kind="definition">
# Minor

## Definition

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix over $K$, and let $1 \le k \le \min(m, n)$.
A *minor of order $k$* of $A$ is the [[determinant|00KDZRW4C0]] of a $k \times k$ submatrix of $A$.
Such a submatrix consists of the entries in $k$ chosen rows and $k$ chosen columns of $A$.
For an $n \times n$ matrix $A$, delete row $i$ and column $j$ to get an $(n-1) \times (n-1)$ matrix.
The minor $M_{ij}$ is the determinant of this matrix.

## Remarks

The minors $M_{ij}$ are called *first minors*.
Over a field, the [[rank|0ZC9YTXZRY]] of $A$ is the largest $k$ such that $A$ has a nonzero minor of order $k$.
The minors with the same chosen rows and columns are *principal minors*.

## Explanations

### L1 human

A minor is the determinant of a smaller square piece of a matrix.
For a $3 \times 3$ matrix, we delete one row and one column and take the $2 \times 2$ determinant of what is left.
Minors appear in the expansion of determinants and in the formula for the inverse.
</document>

<document source="https://math2.tamnd.com/node/0SDC934R52.md" tag="0SDC934R52" kind="concept">
# Polynomial ring

## Summary

The polynomial ring $R[x]$ over a ring $R$ is the ring of the polynomials in one variable $x$ with coefficients in $R$.
The [[definition of the polynomial ring|0Q9EZS90GY]] gives the construction.
</document>

<document source="https://math2.tamnd.com/node/0SDFXN1VDQ.md" tag="0SDFXN1VDQ" kind="definition">
# Dihedral angle

## Definition

Let $P_1$ and $P_2$ be two [[planes|0V3XAMF3C1]] in $\mathbb{R}^3$ that meet in a line. Let $n_1$ and $n_2$ be [[normal vectors|0WQJJCCTC8]] of $P_1$ and $P_2$, and let $\cdot$ be the dot product.
The dihedral angle between $P_1$ and $P_2$ is the unique real number $\varphi$ with $0 \leq \varphi \leq \pi / 2$ such that
$$\cos \varphi = \frac{|n_1 \cdot n_2|}{\|n_1\| \, \|n_2\|}.$$

## Remarks

The value does not depend on the choice of the normal vectors. The absolute value is necessary because $-n_1$ is also a normal vector of $P_1$.
For two half-planes with a common edge line, the dihedral angle is in $[0, \pi]$. Take a point $O$ on the edge. Let $u_1$ and $u_2$ be vectors that are perpendicular to the edge and point from $O$ into the two half-planes. The dihedral angle is the [[angle between $u_1$ and $u_2$|0ZEHZY85E6]].
For the planes $z = 0$ and $x = 0$, the normal vectors $(0, 0, 1)$ and $(1, 0, 0)$ have dot product $0$, so the dihedral angle is $\pi / 2$.
The dihedral angles of a polyhedron are the angles between the planes of two faces with a common edge, measured inside the solid.

## Explanations

### L1 human

Open a book partly. The two covers are half-planes that meet along the spine. The dihedral angle is how far the book is open: $0$ when it is closed and $\pi$ when it lies flat.
</document>

<document source="https://math2.tamnd.com/node/0SDMPMCN43.md" tag="0SDMPMCN43" kind="definition">
# Multilinear map

## Definition

Let $F$ be a field, let $n \ge 1$, and let $V_1, \dots, V_n$ and $W$ be vector spaces over $F$.
A map $f \colon V_1 \times \cdots \times V_n \to W$ is *multilinear* when it is linear in each variable separately.
That is, for each $i$ and each fixed choice of $v_j \in V_j$ for $j \ne i$, the map $v \mapsto f(v_1, \dots, v_{i-1}, v, v_{i+1}, \dots, v_n)$ is a [[linear map|0NRYHJPQG2]] $V_i \to W$.

## Remarks

For $n = 1$, a multilinear map is a linear map, and for $n = 2$ it is a *bilinear map*.
The dot product on $\R^n$ and the matrix product $M_n(F) \times M_n(F) \to M_n(F)$ are bilinear.
The [[determinant|00KDZRW4C0]] is multilinear in the $n$ columns of a matrix.
A multilinear map is in general not linear on the product space: $f(2v_1, 2v_2) = 4 f(v_1, v_2)$ for a bilinear $f$.

## Explanations

### L1 human

A multilinear map takes several vectors and is linear in each one, while the others stay fixed.
The product $xy$ of two real numbers is a simple example: it is linear in $x$ and linear in $y$.
It is not linear in the pair $(x, y)$.
</document>

<document source="https://math2.tamnd.com/node/0SEDGXG9S3.md" tag="0SEDGXG9S3" kind="definition">
# Cross product

## Definition

Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be [[vectors|0QN3D80AWP]] in $\mathbb{R}^3$.
The cross product of $a$ and $b$ is the vector
$$a \times b = (a_2 b_3 - a_3 b_2, \; a_3 b_1 - a_1 b_3, \; a_1 b_2 - a_2 b_1).$$

## Remarks

For the standard basis vectors $e_1$, $e_2$, $e_3$ of $\mathbb{R}^3$: $e_1 \times e_2 = e_3$, $e_2 \times e_3 = e_1$ and $e_3 \times e_1 = e_2$.
If $\theta$ is the angle between nonzero vectors $a$ and $b$, then $\|a \times b\| = \|a\| \, \|b\| \sin \theta$. This is the area of the parallelogram with the sides $a$ and $b$.
The cross product is [[perpendicular to its factors|0T4QHJXSD4]] and [[anticommutative|07ZS685JJC]]. It is bilinear, but it is not associative.
In the usual right-handed coordinates, $a$, $b$ and $a \times b$ follow the right-hand rule. The physics notation $a \wedge b$ is also used.

## Explanations

### L1 human

The cross product of two arrows in space gives a third arrow that sticks out at right angles to both. For example, $(1, 0, 0) \times (0, 1, 0) = (0, 0, 1)$. The longer and the more perpendicular the two arrows are, the longer their cross product.
</document>

<document source="https://math2.tamnd.com/node/0SEX9F9J5Z.md" tag="0SEX9F9J5Z" kind="definition">
# Order of a group

## Definition

Let $G$ be a [[group|G5S1M9]].
The order of $G$, written $|G|$, is the cardinality of the underlying set of $G$.
When $G$ is finite, $|G|$ is the number of elements of $G$.
When $G$ is infinite, $G$ has infinite order.

## Remarks

The [[trivial group|0BNPC9850Y]] has order $1$, and the [[group of $n$th roots of unity|0WX11EA3F3]] has order $n$.
The group $(\mathbb{Z}, +)$ has infinite order.
[[Lagrange's theorem|3C53TD]] says that the order of a subgroup of a finite group divides the order of the group.

## Explanations

### L1 human

The order of a group tells you how big the group is.
For example, the rotations of a square that map the square to itself form a group of order $4$.
The word "order" has a second meaning in group theory: the order of an element.
</document>

<document source="https://math2.tamnd.com/node/0SEZ3JGZP7.md" tag="0SEZ3JGZP7" kind="definition">
# Countably infinite set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countably infinite when there is a [[bijective function|0SNQEF7Z53]] $f \colon \mathbb{N} \to A$.

## Remarks

Another name is denumerable. Its cardinality is [[aleph-null|0V2H5PGJ9B]].
Examples are $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$, and also the set of primes.
A set is countably infinite exactly when it is [[countable|0ZG61TFBES]] and [[infinite|01MKW4R283]].

## Explanations

### L1 human

A countably infinite set can be listed as $a_0, a_1, a_2, \dots$ without end, and each element appears exactly one time.
The integers have such a list: $0, 1, -1, 2, -2, \dots$.
</document>

<document source="https://math2.tamnd.com/node/0SF85PY0QN.md" tag="0SF85PY0QN" kind="statement">
# Product rule in several variables

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f, g \colon U \to \mathbb{R}$, and let $a \in U$.
Suppose that $f$ and $g$ are [[differentiable|0135XZZQ8M]] at $a$.
Then the product $f g$ is differentiable at $a$, and its derivative is
$$D(f g)(a) = f(a) \, Dg(a) + g(a) \, Df(a).$$
That is, $D(f g)(a)(h) = f(a) \, Dg(a)(h) + g(a) \, Df(a)(h)$ for each $h \in \mathbb{R}^n$.

## Remarks

In terms of the [[gradient|0ABK9V9Y64]], the rule reads $\nabla (f g) = f \nabla g + g \nabla f$.
The same rule holds for each partial derivative: $\partial_i (f g) = f \, \partial_i g + g \, \partial_i f$.
More generally, let $B$ be a bounded bilinear map. If $f$ and $g$ are [[Fréchet differentiable|08EVNYZF42]] at $a$, then $D(B(f, g))(a)(h) = B(Df(a) h, g(a)) + B(f(a), Dg(a) h)$. This gives product rules for the dot product and the cross product.

## Explanations

### L1 human

This is the familiar rule $(f g)' = f' g + f g'$, applied to the change in each direction at once.
For $f = x$ and $g = y$ on $\mathbb{R}^2$, the product $x y$ has the derivative $h \mapsto y h_1 + x h_2$ at the point $(x, y)$.
</document>

<document source="https://math2.tamnd.com/node/0SG74WBFA2.md" tag="0SG74WBFA2" kind="concept">
# Möbius transformation

## Summary

A Möbius transformation is a map $z \mapsto \frac{az + b}{cz + d}$ of the Riemann sphere to itself, with $ad - bc \neq 0$, see the [[definition|0B346DDAM5]].
These maps are bijective and holomorphic, they preserve angles, and they [[map circles and lines to circles and lines|0SXDHQM2MH]].
</document>

<document source="https://math2.tamnd.com/node/0SGTMBEWRF.md" tag="0SGTMBEWRF" kind="definition">
# Equivalence relation

## Definition

Let $A$ be a set.
An equivalence relation on $A$ is a [[binary relation|08EJC6Q25H]] $\sim$ on $A$ that is [[reflexive|0XEGNW851S]], [[symmetric|0N9SA25Z3V]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \in A$:

1. $a \sim a$;
2. $a \sim b$ implies $b \sim a$;
3. $a \sim b$ and $b \sim c$ imply $a \sim c$.

## Remarks

For a fixed integer $n \ge 1$, congruence modulo $n$ on $\mathbb{Z}$ is an equivalence relation: $a \sim b$ when $n$ divides $a - b$.
An equivalence relation cuts the set into [[equivalence classes|082RMRSY26]], and these form a [[partition|09CCE86A6D]].
Each [[function|0XHDTE5YMZ]] gives an equivalence relation. See [[equivalence relation induced by a function|0QE5PFJY7V]].

## Explanations

### L1 human

An equivalence relation says when two objects are "the same for our purpose".
Two integers are congruent modulo $2$ when they have the same parity.
Then all even numbers count as one kind, and all odd numbers as another.
</document>

<document source="https://math2.tamnd.com/node/0SH5VT8QYA.md" tag="0SH5VT8QYA" kind="statement">
# Basis of a tensor product

## Statement

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.
Let $(e_i)_{i \in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$ and let $(f_j)_{j \in J}$ be a basis of $W$.
Then the family $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is a basis of the [[tensor product|0BF2DE2SYK]] $V \otimes W$.

## Remarks

So each vector of $V \otimes W$ is uniquely $\sum_{i,j} c_{ij}\, e_i \otimes f_j$ with only finitely many nonzero $c_{ij} \in F$.
For $v = \sum_i a_i e_i$ and $w = \sum_j b_j f_j$, $v \otimes w = \sum_{i,j} a_i b_j\, e_i \otimes f_j$.
Thus the coefficients of a vector $v \otimes w$ form a matrix of rank at most $1$.

## Explanations

### L1 human

From a basis of each factor, we get a basis of the tensor product.
Its vectors are all the products of a basis vector of $V$ and a basis vector of $W$.
A vector of $V \otimes W$ is like a matrix of coefficients.
</document>

<document source="https://math2.tamnd.com/node/0SHC6Z304Z.md" tag="0SHC6Z304Z" kind="statement">
# Closure is the smallest closed superset

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S \subseteq \overline{S}$, the set $\overline{S}$ is closed, and each closed set $F$ of $X$ with $S \subseteq F$ satisfies $\overline{S} \subseteq F$.

## Remarks

Here $\overline{S}$ is the [[closure|0DRKT1QJR7]] of $S$. That it is closed is the statement [[closure is closed|02TZDKG50Q]].
This is the dual of the statement that the [[interior is the largest open subset|0HAZEFH137]].
If $S \subseteq T$, then $\overline{S} \subseteq \overline{T}$.

## Explanations

### L1 human

Among all closed sets that contain $S$, the closure is the smallest one.
For example, each closed set that contains $(0, 1)$ in $\mathbb{R}$ also contains $[0, 1]$.
</document>

<document source="https://math2.tamnd.com/node/0SHMC00RJV.md" tag="0SHMC00RJV" kind="definition">
# Split polynomial

## Definition

Let $L$ be a [[field|0QA8CHH8MG]] and let $f \in L[x]$ be nonzero.
The polynomial $f$ splits over $L$ if there are $c, a_1, \dots, a_n \in L$ with
$$f = c \, (x - a_1)(x - a_2) \cdots (x - a_n).$$

## Remarks

Then $c$ is the leading coefficient of $f$, $n = \deg f$, and $a_1, \dots, a_n$ are the [[roots|00EHX6FWC0]] of $f$, not necessarily distinct.
The polynomial $x^2 - 2$ splits over $\mathbb{R}$, but not over $\mathbb{Q}$.
The polynomial $x^2 + 1$ splits over $\mathbb{C}$, but not over $\mathbb{R}$.
Over an [[algebraically closed field|02SJG1HC32]], every nonzero polynomial splits.

## Explanations

### L1 human

A polynomial splits when you can factor it completely into linear factors.
Then you can read all its roots from the factors.
</document>

<document source="https://math2.tamnd.com/node/0SHPXM2HWH.md" tag="0SHPXM2HWH" kind="example">
# Pendulum equation

## Example

Let $g > 0$ be the acceleration of gravity and $\ell > 0$ the length of a rigid massless rod, with a point mass at its end. Let $\theta(t)$ be the angle of the rod from the downward vertical at time $t$.
Without friction, the motion satisfies the pendulum equation
$$\theta''(t) + \frac{g}{\ell} \sin \theta(t) = 0.$$
As a first-order system, it is $\theta' = \omega$, $\omega' = -\frac{g}{\ell} \sin \theta$. The function $E(\theta, \omega) = \frac{1}{2} \omega^2 - \frac{g}{\ell} \cos \theta$ is a [[first integral|0SS1P1YC9A]].

## Remarks

For small angles, $\sin \theta \approx \theta$. This gives the [[harmonic oscillator|0YG0ZJPTWG]] $\theta'' + \frac{g}{\ell} \theta = 0$, with the solution $\theta(t) = \theta_0 \cos\left(\sqrt{g/\ell}\, t\right)$ for $\theta(0) = \theta_0$, $\theta'(0) = 0$, and the period $2 \pi \sqrt{\ell / g}$.
For the nonlinear equation, the period depends on the amplitude. It is given by an elliptic integral and it is longer than $2 \pi \sqrt{\ell / g}$.
The equilibrium $(0, 0)$, hanging down, is stable. The equilibrium $(\pi, 0)$, standing up, is a [[hyperbolic|0C5PREPZWG]] saddle, so it is unstable.

## Explanations

### L1 human

Gravity pulls the mass back toward the lowest point, with a force proportional to $\sin \theta$.
For small swings, this is nearly proportional to $\theta$, and the pendulum behaves like a spring.
For large swings, the motion is slower, and with enough energy the pendulum turns over the top.
</document>

<document source="https://math2.tamnd.com/node/0SJ3RX9TXA.md" tag="0SJ3RX9TXA" kind="definition">
# Set

## Definition

A set is a collection of objects, called the elements of the set.
For an object $x$ and a set $A$, either $x$ is an [[element|0EZ758XE73]] of $A$, written $x \in A$, or it is not, written $x \notin A$.
Two sets $A$ and $B$ are equal when they have the same elements: for each $x$, $x \in A$ if and only if $x \in B$.
A finite set can be written by a list of its elements in braces, for example $\{1, 2, 3\}$.

## Remarks

The sets $\{1, 2, 3\}$, $\{3, 2, 1\}$ and $\{1, 1, 2, 3\}$ are the same set.
In [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], "set" and "$\in$" are primitive notions, and the axioms say how they behave.
Not every collection can be a set. [[Russell's paradox|0HESMMSYEG]] shows this.

## Explanations

### L1 human

A set is like a bag of objects, where only membership counts.
The bag $\{1, 2, 3\}$ contains $1$, $2$ and $3$, and nothing else.
You cannot put the same object in twice, and the objects have no order.
</document>

<document source="https://math2.tamnd.com/node/0SJ7Z6VKWK.md" tag="0SJ7Z6VKWK" kind="definition">
# Continued fraction expansion of a real number

## Definition

Let $x$ be a real number.
Put $x_0 = x$ and $a_0 = \lfloor x_0 \rfloor$.
For $n \ge 0$, if $x_n \neq a_n$, put $x_{n+1} = \dfrac{1}{x_n - a_n}$ and $a_{n+1} = \lfloor x_{n+1} \rfloor$. If $x_n = a_n$, the process stops.
The continued fraction expansion of $x$ is the [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \dots]$, finite or infinite.

## Remarks

Here $\lfloor y \rfloor$ is the largest integer that is at most $y$. Since $0 < x_n - a_n < 1$, each $a_{n+1}$ with $n \ge 0$ is a positive integer.
For example, $\frac{43}{19} = [2; 3, 1, 4]$, and $\sqrt{2} = [1; 2, 2, 2, \dots]$.
The process stops if and only if $x$ is rational, by the [[termination theorem|0PMY0YQYXX]].
For a rational $x$, the steps are the steps of the [[Euclidean algorithm|0TRNT4X7J0]] on its numerator and denominator.

## Explanations

### L1 human

Take the integer part of $x$, then flip the remaining fractional part, and repeat.
The integer parts that you collect are the terms of the continued fraction of $x$.
</document>

<document source="https://math2.tamnd.com/node/0SJNCJCCQG.md" tag="0SJNCJCCQG" kind="concept">
# Falling factorial

## Summary

The falling factorial $(x)_k$ is the product of $k$ consecutive descending factors that start at $x$. It counts ordered selections without repetition, and it is a basis of the polynomials that is useful in finite difference calculus. The node [[falling factorial|0N5M924N3P]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SK1VHZ5R2.md" tag="0SK1VHZ5R2" kind="statement">
# Irreducibility of the cyclotomic polynomials over the rationals

## Statement

For every integer $n \ge 1$, the [[cyclotomic polynomial|0W0KVBM54S]] $\Phi_n$ is [[irreducible|00X2A9G9W3]] in $\mathbb{Q}[x]$.

## Remarks

For $n = p$ prime, [[Eisenstein's criterion|02BPKD7TMD]] applied to $\Phi_p(x + 1)$ with the prime $p$ proves the statement.
The general case is harder. One proof shows that if $\zeta$ is a root of an irreducible factor of $\Phi_n$, then so is $\zeta^p$ for each prime $p \nmid n$.
A consequence: $\Phi_n$ is the [[minimal polynomial|0GK1WB9R98]] over $\mathbb{Q}$ of each primitive $n$-th root of unity. So the degree of $\mathbb{Q}(e^{2\pi i/n})$ over $\mathbb{Q}$ is $\varphi(n)$.

## Explanations

### L1 human

The polynomial $\Phi_n$ cannot be split into smaller polynomials with rational coefficients.
So all the primitive $n$-th roots of unity are conjugate over the rationals: no rational polynomial equation separates them.
</document>

<document source="https://math2.tamnd.com/node/0SK90238BH.md" tag="0SK90238BH" kind="definition">
# Sign of a permutation

## Definition

Let $n \geq 2$ and let $\sigma \in S_n$.
Write $\sigma = \tau_1 \tau_2 \cdots \tau_m$ as a product of [[transpositions|0FWVRHBRWM]].
The sign of $\sigma$ is $\operatorname{sgn}(\sigma) = (-1)^m$.

## Remarks

The parity of $m$ does not depend on the decomposition, so the sign is well defined.
Equivalently, $\operatorname{sgn}(\sigma) = (-1)^{N(\sigma)}$, where $N(\sigma)$ is the number of pairs $i < j$ with $\sigma(i) > \sigma(j)$.
The sign of a permutation is the determinant of its permutation matrix.

## Explanations

### L1 human

Each permutation is a product of swaps, and the number of swaps is always even or always odd.
The sign records this: $+1$ for an even number and $-1$ for an odd number.
A single swap has sign $-1$.
</document>

<document source="https://math2.tamnd.com/node/0SKVEMXJWF.md" tag="0SKVEMXJWF" kind="statement">
# Stability of linear systems via eigenvalues

## Statement

Let $A$ be a real $n \times n$ matrix and consider the [[linear system with constant coefficients|0VJF99CZ0Z]] $x' = A x$ on $\mathbb{R}^n$, with the equilibrium point $0$.
(1) The point $0$ is [[asymptotically stable|05Y1JAM3Y5]] if and only if each eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda < 0$.
(2) If some eigenvalue $\lambda$ of $A$ has $\operatorname{Re} \lambda > 0$, then $0$ is unstable.
(3) The point $0$ is [[Lyapunov stable|092S6VBECQ]] if and only if two conditions hold. Each eigenvalue has $\operatorname{Re} \lambda \le 0$. Each eigenvalue with $\operatorname{Re} \lambda = 0$ has equal algebraic and geometric multiplicities.

## Remarks

The proof uses the [[matrix exponential|0YC8JT513J]]. The entries of $e^{tA}$ are sums of terms $t^k e^{\lambda t}$. Here $\lambda$ is an eigenvalue and $k$ is less than the size of its largest Jordan block.
In case (1), there are constants $C, \alpha > 0$ with $\|e^{tA}\| \le C e^{-\alpha t}$ for $t \ge 0$. So the origin is globally and exponentially stable.
For $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, both eigenvalues are $0$ but the solution $(t, 1)$ is unbounded. So $0$ is unstable, in accordance with (3).
The Routh–Hurwitz criterion tests the condition of (1) from the coefficients of the characteristic polynomial, without computing the roots.

## Explanations

### L1 human

For a linear system, the eigenvalues decide everything.
Eigenvalues with negative real part give decaying solutions, and an eigenvalue with positive real part gives a growing solution.
</document>

<document source="https://math2.tamnd.com/node/0SKYFQAFYD.md" tag="0SKYFQAFYD" kind="concept">
# General solution and particular solution

## Summary

The general solution of an [[ordinary differential equation|0P97V20RW7]] of order $n$ is a family of [[solutions|04SW7WECS8]] that depends on $n$ arbitrary constants $C_1, \dots, C_n$.
In the usual case, each solution of the equation is a member of the family, with the possible exception of [[singular solutions|0Q2MDD13G4]].
A particular solution is one solution of the equation. It is often the member of the general solution for one choice of the constants.

## Remarks

The general solution of $y'' + y = 0$ is $y = C_1 \cos x + C_2 \sin x$. The choice $C_1 = 1$, $C_2 = 0$ gives the particular solution $y = \cos x$.
An [[initial value problem|075VBY4K4X]] fixes the constants. For an equation of order $n$, one needs $n$ initial conditions.
For a linear equation, the [[general solution of the inhomogeneous equation|0GHWEH6H15]] is one particular solution plus the general solution of the homogeneous equation.

## Explanations

### L1 human

The general solution is the list of all answers, written with free constants.
A particular solution is one answer from the list.
For $y' = 2x$, the general solution is $y = x^2 + C$, and $y = x^2 + 5$ is a particular solution.
</document>

<document source="https://math2.tamnd.com/node/0SM1REAVYJ.md" tag="0SM1REAVYJ" kind="statement">
# Edge bound for planar graphs

## Statement

Let $G$ be a finite simple connected [[planar graph|029JQEK15B]] with $v \ge 3$ vertices and $e$ edges.
Then $e \le 3v - 6$.

## Remarks

Proof idea: each face is bounded by at least $3$ edges, and each edge touches at most $2$ faces. Thus $3f \le 2e$, and [[Euler's formula|0W1C598HMS]] gives the bound.
If $G$ also has no cycle of length $3$, then $e \le 2v - 4$.
A consequence: each finite simple planar graph has a vertex of [[degree|0T0F6CCS9W]] at most $5$.

## Explanations

### L1 human

A planar graph cannot have too many edges: at most about three times its number of vertices.
This simple count already shows that some graphs, such as $K_5$, are not planar.
</document>

<document source="https://math2.tamnd.com/node/0SN0MAEKYX.md" tag="0SN0MAEKYX" kind="definition">
# Prime number

## Definition

A prime number is a natural number $p$ such that $p > 1$ and the only positive divisors of $p$ are $1$ and $p$.

## Remarks

The first prime numbers are $2, 3, 5, 7, 11, 13$. The number $2$ is the only even prime.
The number $1$ is not prime by this definition. This choice makes the [[prime factorization|0AEKDFCYFA]] of each integer unique.
A natural number $n > 1$ that is not prime is a [[composite number|0N3ADXHNTQ]].
An equivalent property for $p > 1$ is [[Euclid's lemma|051VYKA1EX]]: if $p \mid a b$, then $p \mid a$ or $p \mid b$.

## Explanations

### L1 human

A prime number is a number greater than $1$ that you cannot split as a product of two smaller natural numbers.
For example, $7$ is prime, but $6 = 2 \cdot 3$ is not.
The primes are the building blocks of the natural numbers under multiplication.
</document>

<document source="https://math2.tamnd.com/node/0SNJM8HSD3.md" tag="0SNJM8HSD3" kind="statement">
# Erdős–Szekeres theorem

## Statement

Let $r$ and $s$ be positive integers.
Let $a_1, a_2, \dots, a_N$ be a sequence of $N = (r - 1)(s - 1) + 1$ distinct real numbers.
Then the sequence has an increasing subsequence of length $r$ or a decreasing subsequence of length $s$.

## Remarks

The bound is sharp: some sequence of $(r - 1)(s - 1)$ distinct numbers has neither type of subsequence.
For $r = s = 3$: each sequence of $5$ distinct numbers has a monotone subsequence of length $3$.
A short proof labels each term with the lengths of the longest increasing and decreasing subsequences that end at it. The labels are distinct, and the [[pigeonhole principle|0HGVCWQGY3]] finishes the proof.

## History

Paul Erdős and George Szekeres proved the theorem in 1935.

## Explanations

### L1 human

A long enough list of different numbers always contains a long subsequence that goes up or one that goes down. The terms need not be adjacent.
For example, $3, 1, 4, 5, 2$ has the increasing subsequence $3, 4, 5$.
</document>

<document source="https://math2.tamnd.com/node/0SNQEF7Z53.md" tag="0SNQEF7Z53" kind="definition">
# Bijective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is bijective when it is [[injective|0PVG1CRAJN]] and [[surjective|0XKH881BZ7]].
Equivalently, for each $b \in B$, there is exactly one $a \in A$ with $f(a) = b$.

## Remarks

Other names are bijection and one-to-one correspondence.
The function $x \mapsto 2x + 1$ on $\mathbb{R}$ is bijective.
A function is bijective if and only if it has an [[inverse function|0W766JAPSN]]. See [[bijective iff it has a two-sided inverse|01VG2A0BHW]].
Two finite sets have the same number of elements exactly when there is a bijection between them.

## Explanations

### L1 human

A bijection pairs the elements of $A$ and the elements of $B$ perfectly.
Each element of $A$ has one partner in $B$, and each element of $B$ has one partner in $A$.
</document>

<document source="https://math2.tamnd.com/node/0SP14DZN0G.md" tag="0SP14DZN0G" kind="definition">
# Primitive polynomial

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
A polynomial $f \in R[x]$ is primitive if every $r \in R$ that divides $f$ in $R[x]$ is a unit.
Here $r$ divides $f$ when $r$ divides each coefficient of $f$.

## Remarks

If $R$ is a [[unique factorisation domain|06YMXJMXYS]], then $f$ is primitive if and only if its [[content|08VV7YJRB7]] is a unit.
In $\mathbb{Z}[x]$, $3x^2 + 2x - 1$ is primitive and $6x^2 + 4x - 2$ is not.
Every [[monic polynomial|0BDX4MBKT4]] is primitive.
The word primitive polynomial has another meaning over a finite field: a polynomial whose root is a [[primitive element|0YAA9CJQXF]].

## Explanations

### L1 human

A primitive polynomial has coefficients with no common factor.
Over the integers, this means that the gcd of the coefficients is $1$.
</document>

<document source="https://math2.tamnd.com/node/0SP7P0G7KQ.md" tag="0SP7P0G7KQ" kind="statement">
# A space is the direct sum of a subspace and its orthogonal complement

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, and let $W$ be a finite-dimensional subspace of $V$.
Then
$$V = W \oplus W^\perp,$$
where $W^\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$ and $\oplus$ is the [[direct sum|0XYYT5KNTX]].

## Remarks

For an orthonormal basis $e_1, \dots, e_k$ of $W$, each $v \in V$ is $w + (v - w)$ with $w = \sum_i \langle v, e_i \rangle e_i \in W$ and $v - w \in W^\perp$.
The statement holds for each closed subspace of a Hilbert space.
It can fail for a subspace that is not closed in an infinite-dimensional space.

## Explanations

### L1 human

Each vector splits in exactly one way into a part in $W$ and a part perpendicular to $W$.
The part in $W$ is the shadow of the vector on $W$.
The part perpendicular to $W$ is what remains.
</document>

<document source="https://math2.tamnd.com/node/0SPBEDVNW1.md" tag="0SPBEDVNW1" kind="statement">
# Existence of a prime divisor

## Statement

Let $n$ be a natural number with $n \ge 2$.
Then there is a [[prime number|0SN0MAEKYX]] $p$ such that $p \mid n$.

## Remarks

One proof takes the smallest divisor $d$ of $n$ with $d > 1$. Each divisor $e$ of $d$ with $e > 1$ also divides $n$, so $e = d$. Thus $d$ is prime.
This smallest divisor is the [[smallest prime factor|0S2SYHTWPY]] of $n$.
The result is also true for $n = 0$, since each prime divides $0$. It is false for $n = 1$.

## Explanations

### L1 human

Split $n$ into two smaller factors when you can, and repeat with one of the factors.
The numbers get smaller each time, so the process stops at a number that you cannot split. This number is a prime, and it divides $n$.
For example, $60 = 2 \cdot 30$, so $2$ is a prime divisor of $60$.
</document>

<document source="https://math2.tamnd.com/node/0SPBPW3K61.md" tag="0SPBPW3K61" kind="definition">
# Hypergeometric distribution

## Definition

Let $N$, $K$ and $n$ be integers with $N \ge 1$, $0 \le K \le N$ and $0 \le n \le N$.
The hypergeometric distribution with parameters $N$, $K$ and $n$ has the [[probability mass function|0WF4EJP91W]]
$$p(k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}} \quad \text{for } \max(0, n + K - N) \le k \le \min(n, K).$$

## Remarks

A population has $N$ objects, and $K$ of them are successes. Draw $n$ objects without replacement, each subset of size $n$ equally likely. Then the number of successes in the sample has this distribution.
The mean is $nK/N$. The variance is $n \frac{K}{N} \frac{N - K}{N} \frac{N - n}{N - 1}$ for $N \ge 2$.
When $N$ is large compared with $n$, the distribution is close to the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $K/N$.
The probability of exactly $2$ aces in a hand of $5$ cards is $\binom{4}{2} \binom{48}{3} / \binom{52}{5}$.

## Explanations

### L1 human

An urn has red and blue balls. Draw a few balls without putting them back, and count the red ones.
The count has the hypergeometric distribution. Each draw changes the urn, so the draws are not independent.
</document>

<document source="https://math2.tamnd.com/node/0SPE8WNWFV.md" tag="0SPE8WNWFV" kind="concept">
# Connected set

## Summary

A connected set is a subset of a topological space that is connected in the subspace topology.
</document>

<document source="https://math2.tamnd.com/node/0SPYAGE7T0.md" tag="0SPYAGE7T0" kind="statement">
# Change of basis for a bilinear form

## Statement

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$, with two bases $\mathcal{B}$ and $\mathcal{B}'$.
Let $P$ be the [[change of basis matrix|0MKRHZB3WS]] whose columns are the coordinate vectors of the vectors of $\mathcal{B}'$ in $\mathcal{B}$.
Let $B$ be a bilinear form on $V$, with [[matrix|0QK6K857HS]] $M$ in $\mathcal{B}$ and $M'$ in $\mathcal{B}'$.
Then
$$M' = P^T M P.$$

## Remarks

Two square matrices $M$ and $M'$ are *congruent* when $M' = P^T M P$ for an invertible matrix $P$.
So two matrices represent the same bilinear form in two bases if and only if they are congruent.
Congruence keeps the rank and the symmetry of a matrix, but in general not its eigenvalues.
Compare this with the rule $P^{-1} A P$ for the matrix of a linear map.

## Explanations

### L1 human

When we change the basis, the matrix of a bilinear form changes by $P^T M P$.
Both vectors change coordinates, so $P$ appears twice.
For a linear map, the rule is $P^{-1} A P$ instead.
</document>

<document source="https://math2.tamnd.com/node/0SPYEFH7H4.md" tag="0SPYEFH7H4" kind="statement">
# The Gaussian integers form a Euclidean domain

## Statement

Let $\alpha$ and $\beta$ be [[Gaussian integers|0B268GVX95]] with $\beta \neq 0$.
Then there are Gaussian integers $q$ and $r$ with $\alpha = q \beta + r$ and $N(r) < N(\beta)$, where $N$ is the [[norm|00HPT88ZX0]].
So $\mathbb{Z}[i]$, with the norm as Euclidean function, is a Euclidean domain.

## Remarks

To find $q$, round the real part and the imaginary part of the complex number $\alpha / \beta$ to nearest integers. This gives even $N(r) \le \frac{1}{2} N(\beta)$.
As a consequence, $\mathbb{Z}[i]$ is a principal ideal domain and a unique factorization domain.
The quotient and the remainder are not always unique.
This is the analogue of the [[division algorithm|0CHQKZFK5S]] for integers.

## History

Carl Friedrich Gauss introduced the Gaussian integers in 1832, in his second paper on biquadratic reciprocity. In it, he proved unique factorization.

## Explanations

### L1 human

You can divide one Gaussian integer by another and get a remainder that is smaller in norm than the divisor.
So the Euclidean algorithm works in $\mathbb{Z}[i]$, and factorization into primes is unique, as for the integers.
</document>

<document source="https://math2.tamnd.com/node/0SPZ1H3N9V.md" tag="0SPZ1H3N9V" kind="concept">
# Refinement of a partition

## Summary

A refinement of a partition is a partition that contains all the points of the given one, and possibly more. Two partitions always have a common refinement. The node [[refinement of a partition|0EEKF7YFBW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SQ1F2BEDS.md" tag="0SQ1F2BEDS" kind="definition">
# Branch of the logarithm

## Definition

Let $U$ be an open subset of $\mathbb{C} \setminus \{0\}$.
A branch of the logarithm on $U$ is a continuous function $L : U \to \mathbb{C}$ such that $e^{L(z)} = z$ for each $z \in U$.
Thus $L(z)$ is a [[complex logarithm|0X2FY491PF]] of $z$ for each $z \in U$.

## Remarks

Each branch $L$ of the logarithm on $U$ is holomorphic on $U$, with $L'(z) = 1/z$.
If $U$ is a [[domain|097KNJFJ7V]] and $L_1, L_2$ are two branches on $U$, then $L_1 - L_2$ is the constant $2 \pi i k$ for one $k \in \mathbb{Z}$.
There is no branch of the logarithm on $\mathbb{C} \setminus \{0\}$, or on any open set that contains a circle with center $0$.
The [[principal branch|0YKY4M3KRQ]] $\operatorname{Log}$ is a branch on $\mathbb{C} \setminus (-\infty, 0]$. The functions $\operatorname{Log} z + 2 \pi i k$ with $k \in \mathbb{Z}$ are the other branches on this set.

## Explanations

### L1 human

The logarithm has many values at each point, so to get a function we must choose one value at each point, and the choice must change continuously.
Such a choice is a branch.
If you go once around $0$, the argument grows by $2\pi$, so no continuous choice is possible on a full circle around $0$.
</document>

<document source="https://math2.tamnd.com/node/0SQ2PC596Y.md" tag="0SQ2PC596Y" kind="definition">
# Open map

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is an open map when for each [[open set|01ZQQDCERP]] $U$ of $X$, the [[image|0Q3F20XB0A]] $f(U)$ is open in $Y$.

## Remarks

The [[projections of a product|0PM88Z9W4M]] $X \times Y \to X$ are open maps.
A continuous bijection is a [[homeomorphism|05GZ11PWRY]] if and only if it is an open map.
The map $\mathbb{R} \to \mathbb{R}$, $x \mapsto x^2$, is continuous but not open: the image of $(-1, 1)$ is $[0, 1)$.

## Explanations

### L1 human

An open map keeps open sets open when you push them forward.
Continuity is about pulling open sets back, so the two notions are different.
</document>

<document source="https://math2.tamnd.com/node/0SQDB87Z3T.md" tag="0SQDB87Z3T" kind="definition">
# Quadratic residue

## Definition

Let $n$ be a positive integer and let $q$ be an integer.
Then $q$ is a quadratic residue modulo $n$ if there is an integer $x$ with $x^2 \equiv q \pmod n$.
Otherwise $q$ is a quadratic nonresidue modulo $n$.

## Remarks

Many texts also require $\gcd(q, n) = 1$, and so they do not count $0$ as a quadratic residue.
Modulo $7$, the squares of $1, \dots, 6$ are $1, 4, 2, 2, 4, 1$. So the nonzero quadratic residues modulo $7$ are $1$, $2$ and $4$.
Modulo an odd prime $p$, exactly $(p - 1)/2$ of the classes $1, \dots, p - 1$ are quadratic residues.
The product of two quadratic residues is a quadratic residue.

## Explanations

### L1 human

A quadratic residue modulo $n$ is a number that is a perfect square in the arithmetic modulo $n$.
For example, $2$ is not a square in the integers, but $3^2 = 9 \equiv 2 \pmod 7$. So $2$ is a quadratic residue modulo $7$.
</document>

<document source="https://math2.tamnd.com/node/0SQSQP748R.md" tag="0SQSQP748R" kind="statement">
# Principal value of the argument

## Statement

Let $z$ be a complex number with $z \neq 0$.
Then $z$ has exactly one [[argument|0ZZRSX91QR]] $\theta$ with $-\pi < \theta \le \pi$.
This argument is the principal value of the argument, written $\operatorname{Arg} z$.

## Remarks

Proof: the arguments of $z$ form a set $\theta_0 + 2 \pi \mathbb{Z}$, and each interval $(-\pi, \pi]$ of length $2\pi$ contains exactly one element of such a set.
For example, $\operatorname{Arg}(-1) = \pi$, $\operatorname{Arg}(-i) = -\pi/2$ and $\operatorname{Arg} x = 0$ for real $x > 0$.
Some authors use the interval $[0, 2\pi)$ instead.

## Explanations

### L1 human

Many angles describe the same direction, because full turns do not change it.
The principal value picks one of them, the one between $-180$ and $180$ degrees, with $180$ allowed and $-180$ not.
</document>

<document source="https://math2.tamnd.com/node/0SQTMQTSRZ.md" tag="0SQTMQTSRZ" kind="concept">
# Even and odd permutations

## Summary

A permutation of a finite set is even when it is a product of an even number of transpositions, and odd otherwise.
The even permutations form the alternating group.
</document>

<document source="https://math2.tamnd.com/node/0SQV2VJTGY.md" tag="0SQV2VJTGY" kind="concept">
# Parallelepiped

## Summary

A parallelepiped is a solid in three-dimensional space with six faces, each of which is a parallelogram. It relates to a parallelogram as a cube relates to a square.
</document>

<document source="https://math2.tamnd.com/node/0SQZMXB6GP.md" tag="0SQZMXB6GP" kind="concept">
# Bilinear form

## Summary

A bilinear form on a vector space is a map of two vectors to the scalars.
It is linear in each vector when the other vector is fixed.
</document>

<document source="https://math2.tamnd.com/node/0SRATCFF1W.md" tag="0SRATCFF1W" kind="statement">
# Subgroups of the integers

## Statement

Let $H$ be a [[subgroup|0X1JXT47GA]] of the [[additive group of integers|0BSCJA5G53]] $(\mathbb{Z}, +)$.
Then there is exactly one integer $n \geq 0$ with $H = n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$.

## Remarks

When $H \neq \{0\}$, the integer $n$ is the smallest positive element of $H$.
For $m, n \geq 0$, $m\mathbb{Z} \subseteq n\mathbb{Z}$ holds exactly when $n$ divides $m$.
A consequence is that, for integers $a$ and $b$, the subgroup $a\mathbb{Z} + b\mathbb{Z}$ is $d\mathbb{Z}$ with $d = \gcd(a, b)$, which gives Bezout's identity.

## Explanations

### L1 human

Each subgroup of the integers is the set of multiples of one number.
For example, the set of all sums $4x + 6y$ with integers $x, y$ is the set of multiples of $2$.
The number $0$ gives the trivial subgroup $\{0\}$.
</document>

<document source="https://math2.tamnd.com/node/0SRE7K4E49.md" tag="0SRE7K4E49" kind="statement">
# Characteristic subgroups are normal

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[characteristic subgroup|0KABPMRHM2]] of $G$.
Then $H$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.

## Remarks

Each [[inner automorphism|0RN00NTM3S]] $c_g$ is an automorphism, so $gHg^{-1} = c_g(H) = H$ for each $g \in G$.
The converse is false: in the Klein four-group $V = \{e, a, b, ab\}$, each subgroup is normal, but an automorphism can swap $\{e, a\}$ and $\{e, b\}$.

## Explanations

### L1 human

Normal means fixed by the inner automorphisms, and characteristic means fixed by all automorphisms.
All automorphisms include the inner ones.
So a characteristic subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0SRWNR42BQ.md" tag="0SRWNR42BQ" kind="definition">
# Convergents of a continued fraction

## Definition

Let $[a_0; a_1, a_2, \dots]$ be a [[simple continued fraction|06XF7BCFFJ]].
For each $n$ such that $a_n$ exists, the $n$-th convergent is the rational number $\dfrac{p_n}{q_n} = [a_0; a_1, \dots, a_n]$.
Here $p_n$ and $q_n$ are given by $p_{-2} = 0$, $p_{-1} = 1$, $q_{-2} = 1$, $q_{-1} = 0$ and
$$p_n = a_n p_{n-1} + p_{n-2}, \qquad q_n = a_n q_{n-1} + q_{n-2} \quad (n \ge 0).$$

## Remarks

The recurrences give $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$. So $\gcd(p_n, q_n) = 1$, and the fraction $p_n / q_n$ is in lowest terms.
For $\sqrt{2} = [1; 2, 2, \dots]$, the convergents are $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \dots$.
For a general [[continued fraction|0BK6JBK58W]], the same recurrences hold with $p_n = b_n p_{n-1} + a_n p_{n-2}$ and $q_n = b_n q_{n-1} + a_n q_{n-2}$.

## Explanations

### L1 human

The convergents are the fractions that you get when you cut a continued fraction off after finitely many terms.
They approach the value of the continued fraction, alternately from below and from above.
</document>

<document source="https://math2.tamnd.com/node/0SRYJGXKR4.md" tag="0SRYJGXKR4" kind="concept">
# Convergence in distribution

## Summary

A sequence of random variables converges in distribution when their cumulative distribution functions converge to a limit at each point where the limit is continuous.
The node [[convergence in distribution|0JDAQHWA9N]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SS1P1YC9A.md" tag="0SS1P1YC9A" kind="definition">
# First integral

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function.
Let $H : U \to \mathbb{R}$ be a continuously differentiable function. It is a first integral of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if
$$\nabla H(x) \cdot f(x) = 0 \quad \text{for each } x \in U,$$
and if $H$ is not constant on any non-empty open subset of $U$.

## Remarks

The condition means that $\frac{d}{dt} H(\gamma(t)) = 0$ for each [[solution|04SW7WECS8]] $\gamma$. So $H$ is constant along each solution.
Each [[trajectory|0BVT6VSB3S]] lies in a level set $\{x : H(x) = c\}$. In the plane, the level curves of $H$ give the phase portrait.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, the function $H(x, y) = x^2 + y^2$ is a first integral.
The Hamiltonian of a time-independent [[Hamiltonian system|0C2AKPDCD9]] is a first integral.

## Explanations

### L1 human

A first integral is a quantity that does not change while the system moves, such as the energy of a system without friction.
Knowing it tells us that each motion stays on one level set of this quantity.
</document>

<document source="https://math2.tamnd.com/node/0SS770NS8B.md" tag="0SS770NS8B" kind="example">
# Differentiation as a linear map

## Example

Let $F$ be a field and let $F[x]$ be the vector space of polynomials over $F$.
The *formal derivative* is the map $D \colon F[x] \to F[x]$ with
$$D\left(\sum_{k=0}^{n} a_k x^k\right) = \sum_{k=1}^{n} k\, a_k x^{k-1}.$$
It is a [[linear map|0NRYHJPQG2]]: $D(p + q) = D(p) + D(q)$ and $D(cp) = c\, D(p)$ for all polynomials $p, q$ and all $c \in F$.

## Remarks

Over a field of characteristic $0$, the kernel of $D$ is the space of constant polynomials.
On the space $F[x]_{<n}$ of polynomials of degree less than $n$, $D$ is nilpotent: $D^n = 0$.
In the basis $1, x, x^2$ of $F[x]_{<3}$, the matrix of $D$ is $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}$.
Over $\R$, differentiation is also a linear map on the space of differentiable functions.

## Explanations

### L1 human

Differentiation is linear: the derivative of a sum is the sum of the derivatives, and constants come out.
So calculus gives important examples of linear maps.
On polynomials of bounded degree, we can even write differentiation as a matrix.
</document>

<document source="https://math2.tamnd.com/node/0SSKBQN58M.md" tag="0SSKBQN58M" kind="statement">
# Hurwitz's theorem

## Statement

Let $G$ be a [[domain|097KNJFJ7V]] in $\mathbb{C}$, that is, a connected open set.
Let $(f_k)$ be a sequence of [[holomorphic|0DG0Z72CQD]] functions on $G$ that converges uniformly on each compact subset of $G$ to a function $f$.
Suppose that $f$ is not identically zero on $G$, and that $f$ has a zero of [[order|064N002TX4]] $m$ at a point $z_0 \in G$.
Then for each sufficiently small $\rho > 0$ there is an integer $N$ with the following property.
For each $k \ge N$, the function $f_k$ has exactly $m$ zeros in the disc $|z - z_0| < \rho$, each counted as many times as its order.

## Remarks

The limit $f$ is holomorphic on $G$ by the [[theorem on uniform limits|0431RN0R1N]].
A consequence: if no $f_k$ has a zero in $G$, then $f$ either has no zero in $G$ or is identically zero.
A second consequence: if each $f_k$ is injective on $G$, then $f$ is injective or constant.
The radius $\rho$ must be small: $f_n(z) = z - 1 + \frac{1}{n}$ has one zero in the unit disc, but its limit $z - 1$ has none there.

## Explanations

### L1 human

Zeros of holomorphic functions do not suddenly appear or disappear under uniform limits.
Near a zero of the limit, each function far along in the sequence has the same number of zeros.
The proof compares $f_k$ with $f$ on a small circle, as in [[Rouché's theorem|0D430QR02W]].
</document>

<document source="https://math2.tamnd.com/node/0SSS6ERQ0S.md" tag="0SSS6ERQ0S" kind="concept">
# Estimator

## Summary

An estimator is a rule that computes a guess of an unknown parameter from the data. It is a statistic that one uses for this purpose.
The node [[estimator|0TTNRP5PFQ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SSSM1D475.md" tag="0SSSM1D475" kind="statement">
# The degree of a constructible number is a power of two

## Statement

Let $z \in \mathbb{C}$ be a [[constructible number|0FMN2MN7TM]].
Then $z$ is algebraic over $\mathbb{Q}$, and $[\mathbb{Q}(z) : \mathbb{Q}] = 2^r$ for some integer $r \ge 0$.

## Remarks

The proof: there is a tower $\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n$ with $z \in F_n$ and $[F_j : F_{j-1}] = 2$ for each $j$. By the [[tower law|0A7KJ68XHT]], $[F_n : \mathbb{Q}] = 2^n$, and $[\mathbb{Q}(z) : \mathbb{Q}]$ divides it.
So the [[minimal polynomial|0GK1WB9R98]] of $z$ over $\mathbb{Q}$ has degree a power of two.
The converse is false: some algebraic numbers whose degree is a power of two are not constructible.
A sufficient condition: the splitting field of the minimal polynomial of $z$ has degree a power of two over $\mathbb{Q}$.

## Explanations

### L1 human

Each construction step at most doubles the degree of the field.
So a number whose minimal polynomial has degree $3$, like $\sqrt[3]{2}$, can never be constructed.
This simple test settles several problems that were open for two thousand years.
</document>

<document source="https://math2.tamnd.com/node/0ST21PGDAP.md" tag="0ST21PGDAP" kind="concept">
# Almost surely

## Summary

An event happens almost surely when its probability is $1$. The event can still fail for some outcomes, but these outcomes together have probability $0$.
The node [[almost surely|0R46A1XY60]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ST8QF9T3T.md" tag="0ST8QF9T3T" kind="definition">
# Adjoint of a linear map

## Definition

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$, and let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *adjoint* of $T$ is the unique linear map $T^* \colon W \to V$ such that
$$\langle T(v), w \rangle = \langle v, T^*(w) \rangle \qquad \text{for all } v \in V \text{ and } w \in W.$$

## Remarks

The existence and uniqueness follow from the [[Riesz representation theorem|04GYPXDXNH]].
It satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$.
In orthonormal bases, the matrix of $T^*$ is the [[conjugate transpose|0DXKE10PHF]] of the matrix of $T$.
It satisfies $\ker T^* = (\im T)^\perp$.

## Explanations

### L1 human

The adjoint lets us move a map from one side of an inner product to the other.
For real matrices with the dot product, the adjoint is the transpose: $(Ax) \cdot y = x \cdot (A^T y)$.
For complex matrices, it is the conjugate transpose.
</document>

<document source="https://math2.tamnd.com/node/0STK8KEY9G.md" tag="0STK8KEY9G" kind="statement">
# Moments are derivatives of the moment-generating function at zero

## Statement

Let $X$ be a real random variable, and let $h > 0$ with $M_X(t) = E[e^{tX}] < \infty$ for all $t \in (-h, h)$.
Then $X$ has finite [[moments|0PH0VHTQS0]] of all orders, the [[moment-generating function|0NNRNGKMFM]] $M_X$ is infinitely differentiable on $(-h, h)$, and for each integer $k \ge 0$,
$$M_X^{(k)}(0) = E\left[X^k\right].$$
Moreover, $M_X(t) = \sum_{k=0}^{\infty} E[X^k] \, t^k / k!$ for $|t| < h$.

## Remarks

The proof differentiates under the expectation sign. The bound $|x|^k e^{t x} \le C(e^{a x} + e^{-a x})$ for $|t| < a < h$ justifies this by dominated convergence.
Example: for $X \sim \operatorname{Exp}(\lambda)$, $M_X(t) = \lambda/(\lambda - t)$, so $M_X'(0) = 1/\lambda$ and $M_X''(0) = 2/\lambda^2$. So $\operatorname{Var}(X) = 1/\lambda^2$.

## Explanations

### L1 human

This result is the reason for the name: the function "generates" the moments.
To find $E[X^k]$, you take the $k$-th derivative of $M_X$ and put $t = 0$.
</document>

<document source="https://math2.tamnd.com/node/0STPPDXGME.md" tag="0STPPDXGME" kind="statement">
# C^1 functions are strictly differentiable

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]] on $U$.
Then $f$ is [[strictly differentiable|0146MBQNNE]] at each point $a \in U$, with linear map $Df(a)$.

## Remarks

Proof: let $\varepsilon > 0$ and choose $r > 0$ with $\|Df(z) - Df(a)\| \le \varepsilon$ for $\|z - a\| < r$. Apply the [[mean value inequality|02CYB184RC]] to $g(z) = f(z) - Df(a) z$ on the ball of radius $r$. This gives $\|f(x) - f(y) - Df(a)(x - y)\| \le \varepsilon \|x - y\|$ for $x, y$ in that ball.
For a function of one variable on an open interval, strict differentiability at each point is equivalent to being of class $C^1$.

## Explanations

### L1 human

If the derivative changes continuously, then near $a$ it stays close to $Df(a)$.
So every chord of the graph near $a$ has a slope close to $Df(a)$, which is what strict differentiability asks.
</document>

<document source="https://math2.tamnd.com/node/0STX0WRDNG.md" tag="0STX0WRDNG" kind="statement">
# Density of the rationals

## Statement

Let $x$ and $y$ be real numbers with $x < y$.
Then there is a [[rational number|0NCJ5WXN6Z]] $q$ with $x < q < y$.

## Remarks

Proof idea: choose $n$ with $1/n < y - x$ by the [[Archimedean property|0WH916TJNY]]. Then some fraction $m/n$ with $m \in \mathbb{Z}$ lies between $x$ and $y$.
Thus between two real numbers there are infinitely many rational numbers.
The [[irrationals are also dense|04J6G4H78J]].

## Explanations

### L1 human

Between any two different real numbers, however close, there is a fraction.
For example, between $\sqrt{2} = 1.41421\dots$ and $1.4143$ lies the fraction $1.41425 = 28285/20000$.
</document>

<document source="https://math2.tamnd.com/node/0SV8K0TDGY.md" tag="0SV8K0TDGY" kind="concept">
# Sigma-algebra

## Summary

A σ-algebra on a set is a family of subsets that contains the whole set and is closed under complements and countable unions.
In probability theory, the σ-algebra is the family of events, the sets to which we give a probability. The node [[sigma-algebra|04WSW8T3SY]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SVN6X695K.md" tag="0SVN6X695K" kind="statement">
# Almost sure convergence implies convergence in probability

## Statement

Let $X, X_1, X_2, \dots$ be real random variables on one probability space.
If $X_n \to X$ [[almost surely|04P7D63W8P]], then $X_n \to X$ [[in probability|0EFDDXTVD6]].

## Remarks

Proof: fix $\varepsilon > 0$ and let $A_n = \bigcup_{m \ge n} \{|X_m - X| > \varepsilon\}$. The events $A_n$ decrease, and their intersection has probability $0$ by almost sure convergence.
By the [[continuity of probability|0X8HBKCSYB]], $P(A_n) \to 0$. Since $\{|X_n - X| > \varepsilon\} \subseteq A_n$, the claim follows.
The converse is false. Let $X_n$ be independent with $P(X_n = 1) = 1/n$ and $P(X_n = 0) = 1 - 1/n$. Then $X_n \to 0$ in probability.
But by the second Borel–Cantelli lemma, $X_n = 1$ for infinitely many $n$ almost surely, so $X_n$ does not converge to $0$ almost surely.

## Explanations

### L1 human

If almost every run of the experiment gives a sequence that settles down at $X$, then for large $n$ a large deviation becomes unlikely.
</document>

<document source="https://math2.tamnd.com/node/0SVR29SV1V.md" tag="0SVR29SV1V" kind="definition">
# Quotient space

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
For $v \in V$, the *coset* of $v$ is $v + W = \{v + w : w \in W\}$.
The *quotient space* $V / W$ is the set of all cosets $v + W$ with $v \in V$, with the operations
$$(u + W) + (v + W) = (u + v) + W, \qquad a(v + W) = av + W$$
for $u, v \in V$ and $a \in F$.

## Remarks

The operations are well defined, because $W$ is a subspace: the result does not depend on the choice of $u$ and $v$ in their cosets.
Two cosets $u + W$ and $v + W$ are equal if and only if $u - v \in W$.
The zero vector of $V / W$ is the coset $0 + W = W$.
In $\R^2$ with $W$ the $x$-axis, the cosets are the horizontal lines, and $V / W$ is isomorphic to $\R$.

## Explanations

### L1 human

In the quotient space $V / W$, we treat two vectors as the same when they differ by a vector of $W$.
Each element of $V / W$ is a copy of $W$ shifted by some vector.
Dividing the plane by a line gives the set of all lines parallel to it, which behaves like a single line.
</document>

<document source="https://math2.tamnd.com/node/0SVX98HYV1.md" tag="0SVX98HYV1" kind="statement">
# Primitive element theorem

## Statement

Let $L/K$ be a [[finite|0NRYPM9Y27]] [[separable extension|0N019F6RE9]].
Then $L/K$ is a [[simple extension|0P6SBR7YM2]]: there is $\theta \in L$ with $L = K(\theta)$.

## Remarks

Every finite extension of a field of characteristic $0$ is separable, so it is simple. For example, $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \mathbb{Q}(\sqrt{2} + \sqrt{3})$.
For an infinite field $K$, the proof shows that $\theta = a + c b$ works for all but finitely many $c \in K$.
For finite fields, a [[primitive element|0YAA9CJQXF]] of $L$ generates $L$ over $K$.
Without separability the statement can fail: there are finite extensions of degree $p^2$ in characteristic $p$ with no primitive element.

## History

Évariste Galois sketched a proof for splitting fields over the rationals in his First Memoir of 1831, published in 1846.
Ernst Steinitz proved the theorem in its modern form in 1910.
Emil Artin later developed Galois theory without primitive elements.

## Explanations

### L1 human

A finite separable extension, even one built from many elements, can be generated by one element.
So many statements about finite extensions reduce to the case of one algebraic element and its minimal polynomial.
</document>

<document source="https://math2.tamnd.com/node/0SVYSP8HCG.md" tag="0SVYSP8HCG" kind="concept">
# Principal ideal

## Summary

A principal ideal is an ideal generated by a single element.
In a commutative ring, it is the set of the multiples of that element.
The [[definition of a principal ideal|00A2EDFT9Q]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0SWTJ5VC07.md" tag="0SWTJ5VC07" kind="concept">
# Linear equation with constant coefficients

## Summary

A linear equation with constant coefficients is a linear differential equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = b(x)$ with real constants $a_0, \dots, a_{n-1}$.
The roots of its characteristic polynomial give all solutions of the homogeneous equation. See the [[definition|0A6PGTTP2A]].
</document>

<document source="https://math2.tamnd.com/node/0SWXJHKJ02.md" tag="0SWXJHKJ02" kind="concept">
# Sample space

## Summary

The sample space of a random experiment is the set of all its possible outcomes.
The node [[sample space|0W7NFW23RF]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SXDHQM2MH.md" tag="0SXDHQM2MH" kind="statement">
# Möbius transformations map circles and lines to circles and lines

## Statement

A generalized circle in the [[Riemann sphere|0CDMXGPSES]] $\widehat{\mathbb{C}}$ is either a circle in $\mathbb{C}$, or a set $L \cup \{\infty\}$ where $L$ is a line in $\mathbb{C}$.
Let $f$ be a [[Möbius transformation|0B346DDAM5]].
Then for each generalized circle $C$, the image $f(C)$ is a generalized circle.

## Remarks

A circle can go to a line: the map $z \mapsto \frac{1}{z}$ sends the circle $|z - 1| = 1$, which contains $0$, to the line $\operatorname{Re} w = \frac{1}{2}$ together with $\infty$.
The image of a generalized circle is a line together with $\infty$ exactly when the generalized circle contains the point that $f$ sends to $\infty$.
The proof checks the result for $z \mapsto z + b$, $z \mapsto az$ and $z \mapsto \frac{1}{z}$, because each Möbius transformation is a composition of such maps.

## Explanations

### L1 human

For a Möbius map, a line is only a circle through the point $\infty$.
So the theorem says that circles go to circles, if we count lines as circles.
For example, $z \mapsto 2z + 3$ sends each circle to a circle with twice the radius.
</document>

<document source="https://math2.tamnd.com/node/0SXDQ6TN00.md" tag="0SXDQ6TN00" kind="statement">
# Dimension of a quotient space

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a subspace of $V$.
Then the [[quotient space|0BR3FY5Q8M]] $V / W$ is finite-dimensional, and
$$\dim(V / W) + \dim W = \dim V,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

The number $\dim(V / W) = \dim V - \dim W$ is the *codimension* of $W$ in $V$.
The formula is the [[rank–nullity theorem|0W4FSW1TVB]] for the [[quotient map|0N4F5FMQZW]] $V \to V / W$.
For example, a plane $W$ in $\R^3$ gives $\dim(\R^3 / W) = 1$.

## Explanations

### L1 human

Dividing by a subspace removes its dimensions.
What remains has the dimension of the space minus the dimension of the subspace.
Ordinary space divided by a plane leaves one dimension.
</document>

<document source="https://math2.tamnd.com/node/0SXHBTNM9T.md" tag="0SXHBTNM9T" kind="concept">
# Acyclic graph

## Summary

An acyclic graph is a graph without cycles. Acyclic graphs are the forests, and their connected components are trees. The node [[acyclic graph|0XNN9Z8E5A]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0SXZ68HV6F.md" tag="0SXZ68HV6F" kind="example">
# Integral of 1/z around the unit circle

## Example

The [[circle integral|0T7ABX4CWE]] of $f(z) = 1/z$ over the unit circle $|z| = 1$ is
$$\oint_{|z| = 1} \frac{dz}{z} = 2 \pi i.$$
Computation: the path is $\gamma(\theta) = e^{i\theta}$ with $\theta \in [0, 2\pi]$, so $\gamma'(\theta) = i e^{i\theta}$. Then
$$\oint_{|z| = 1} \frac{dz}{z} = \int_0^{2\pi} \frac{i e^{i\theta}}{e^{i\theta}} \, d\theta = \int_0^{2\pi} i \, d\theta = 2 \pi i.$$
More generally, let $c, w \in \mathbb{C}$ and $R > 0$. If $|w - c| < R$, then $\oint_{|z - c| = R} \frac{dz}{z - w} = 2 \pi i$. If $|w - c| > R$, the integral is $0$.

## Remarks

For each integer $n \neq -1$, $\oint_{|z| = 1} z^n \, dz = 0$, because $z^n$ has a primitive on $\mathbb{C} \setminus \{0\}$.
The value is not $0$, so $1/z$ has no primitive on $\mathbb{C} \setminus \{0\}$, and [[Cauchy's integral theorem|01MPZXYVYF]] does not apply on $\mathbb{C} \setminus \{0\}$.
This integral is the base of [[Cauchy's integral formula|0DG10TVYPN]], of the [[residue|0JFGKKWWFK]] and of the [[winding number|0X7SBS8E0F]].

## Explanations

### L1 human

On the unit circle, $dz$ is $i z \, d\theta$, so $dz / z$ is just $i \, d\theta$.
Going once around adds up $i \, d\theta$ over a full turn of $2\pi$, which gives $2 \pi i$.
</document>

<document source="https://math2.tamnd.com/node/0SY9ECQ19Q.md" tag="0SY9ECQ19Q" kind="definition">
# Continuous map of topological spaces

## Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be [[topological spaces|0CNQXVBHCX]] and let $f \colon X \to Y$ be a function.
The function $f$ is continuous when for each [[open set|01ZQQDCERP]] $V \in \tau_Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is in $\tau_X$.

## Remarks

Equivalently, $f$ is continuous if and only if preimages of closed sets are closed, as in [[continuity via closed sets|0B8GPXFSP7]].
For metric spaces, this is the same as the epsilon-delta condition, by [[continuity via open sets|0YF82GBTND]].
A function is continuous if and only if it is [[continuous at each point|07N7W5C29X]].
Constant maps and identity maps are continuous, and the [[composition of continuous maps is continuous|05NK26G0DR]].

## Explanations

### L1 human

A continuous map does not tear the space apart: the points that go into an open set come from an open set.
This definition uses only open sets, so it works in each topological space.
</document>

<document source="https://math2.tamnd.com/node/0SYR10HD4R.md" tag="0SYR10HD4R" kind="concept">
# Cauchy principal value

## Summary

The Cauchy principal value gives a value to some improper integrals that do not converge in the usual sense, see the [[definition|0V3RDQV3ZX]].
It removes a symmetric neighborhood of the singular point, or integrates over a symmetric interval $[-R, R]$, and then takes the limit.
</document>

<document source="https://math2.tamnd.com/node/0SZ1KMNWVE.md" tag="0SZ1KMNWVE" kind="statement">
# Order of the alternating group

## Statement

Let $n \geq 2$.
The [[alternating group|0Z5905Q73D]] $A_n$ has [[order|0B85S2V2DR]] $n!/2$, and $[S_n : A_n] = 2$.

## Remarks

The sign map is a surjective homomorphism onto a group of order $2$ with kernel $A_n$, so $S_n / A_n$ has order $2$.
Another proof: the map $\sigma \mapsto (1\ 2) \sigma$ is a bijection from the even permutations to the odd permutations.
For example, $|A_4| = 12$ and $|A_5| = 60$.

## Explanations

### L1 human

Half of all permutations are even.
To see this, multiply by one fixed swap: this changes each even permutation into an odd one, and back.
So $S_n$ splits into two halves of the same size.
</document>

<document source="https://math2.tamnd.com/node/0SZCSYHW60.md" tag="0SZCSYHW60" kind="concept">
# Miller–Rabin primality test

## Summary

The Miller–Rabin test is a probabilistic [[primality test|0EGJZHZ02C]] for an odd integer $n > 2$.
Write $n - 1 = 2^s d$ with $d$ odd, and choose a base $a$ with $1 < a < n - 1$.
The number $n$ passes for base $a$ if $a^d \equiv 1 \pmod n$, or if $a^{2^r d} \equiv -1 \pmod n$ for some $r$ with $0 \le r < s$.
Each odd prime $n$ passes for each base. If $n$ fails for some base, then $n$ is composite.

## Remarks

The prime case follows from [[Fermat's little theorem|0M25PRN3WQ]] and the fact that $\pm 1$ are the only square roots of $1$ modulo a prime.
If $n$ is an odd composite number, then at most $\frac{1}{4}$ of the bases let it pass. So $k$ random bases give an error probability of at most $4^{-k}$.
Gary Miller found the test in 1976, in a deterministic form that depends on the extended Riemann hypothesis. Michael Rabin made it an unconditional probabilistic test in 1980.

## Explanations

### L1 human

The test checks a property that every prime has, for a few random bases.
A composite number fails for most bases, so a few rounds expose it with high probability.
The powers are fast to compute by [[modular exponentiation|04DR5PM25N]].
</document>

<document source="https://math2.tamnd.com/node/0T0EJDGHMB.md" tag="0T0EJDGHMB" kind="definition">
# Normal convergence

## Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ be bounded functions for $n \ge 0$.
The [[series of functions|0MRE6XRE5V]] $\sum_{n=0}^{\infty} f_n$ converges normally on $S$ if
$$\sum_{n=0}^{\infty} \sup_{x \in S} |f_n(x)| < \infty.$$

## Remarks

By the [[Weierstrass M-test|0NX5NWAFC9]], a normally convergent series converges absolutely and uniformly on $S$.
The converse fails. For constant functions $f_n = (-1)^n / (n + 1)$, the series converges uniformly but not normally.
René Baire introduced the concept in 1908.

## Explanations

### L1 human

Normal convergence means that the sizes of the functions, measured by their suprema, form a convergent series.
It is a simple condition that is stronger than uniform convergence.
</document>

<document source="https://math2.tamnd.com/node/0T0F6CCS9W.md" tag="0T0F6CCS9W" kind="definition">
# Degree of a vertex

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $v$ be a vertex of $G$ with a finite [[neighbourhood|01T3TMGTTE]] $N(v)$.
The degree of $v$ is the number
$$\deg(v) = |N(v)|$$
of vertices adjacent to $v$. It is also the number of edges that contain $v$.

## Remarks

A vertex of degree $0$ is isolated, and a vertex of degree $1$ is a leaf.
A graph is locally finite when each vertex has finite degree. Each finite graph is locally finite.
In a [[multigraph|01M5T0KDG7]], the degree counts edges with multiplicity, and a loop counts twice.

## Explanations

### L1 human

The degree of a vertex is the number of edges at that vertex.
In a friendship graph it is the number of friends of a person.
</document>

<document source="https://math2.tamnd.com/node/0T0JWJ3M47.md" tag="0T0JWJ3M47" kind="example">
# Bertrand paradox

## Example

Take a circle and an equilateral triangle inscribed in it. Choose a chord of the circle "at random". What is the probability that the chord is longer than a side of the triangle?
Three natural ways to choose the chord give three different answers.
Choose two independent uniform points on the circle and join them. The probability is $1/3$.
Choose a uniform radius, then a uniform point on it, and take the chord perpendicular to the radius through that point. The probability is $1/2$.
Choose a uniform point in the disc and take the chord with that point as midpoint. The probability is $1/4$.

## Remarks

The paradox shows that "at random" has no meaning until one fixes a probability measure. The three methods give three different distributions on the set of chords.
Joseph Bertrand gave the example in his book Calcul des probabilités of 1889, as a criticism of the principle of indifference.

## Explanations

### L1 human

The question sounds complete, but it is not: it does not say how to choose the chord.
Each method is fair in its own sense, and each one gives a different answer. In probability you must always say exactly how the random choice is made.
</document>

<document source="https://math2.tamnd.com/node/0T0KMZPYV9.md" tag="0T0KMZPYV9" kind="definition">
# Normal core

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The normal core of $H$ in $G$ is the subgroup $\operatorname{Core}_G(H) = \bigcap_{g \in G} gHg^{-1}$.

## Remarks

The core is a [[normal subgroup|0CB0DX5BVV]] of $G$, and it contains each normal subgroup of $G$ that lies in $H$.
The subgroup $H$ is normal exactly when $\operatorname{Core}_G(H) = H$.
The core is the kernel of the action of $G$ on the left cosets of $H$ by left multiplication.

## Explanations

### L1 human

Some subgroups are not normal, but they still contain normal subgroups.
The normal core is the biggest of these.
You find it by keeping only the elements that lie in every conjugate of the subgroup.
</document>

<document source="https://math2.tamnd.com/node/0T0X8GTYYS.md" tag="0T0X8GTYYS" kind="statement">
# Determinant is a coefficient of the characteristic polynomial

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $A$ be an $n \times n$ matrix over $K$.
Then the constant coefficient of the [[characteristic polynomial|0VSSPAFM2C]] $p_A(t) = \det(t I_n - A)$ is
$$p_A(0) = (-1)^n \det A,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The proof sets $t = 0$: $p_A(0) = \det(-A) = (-1)^n \det A$.
So, over a field, $0$ is an eigenvalue of $A$ if and only if $\det A = 0$.

## Explanations

### L1 human

The constant term of the characteristic polynomial is the determinant, up to a sign.
This follows when we put $t = 0$.
So a matrix with determinant zero has the eigenvalue $0$.
</document>

<document source="https://math2.tamnd.com/node/0T13HM377F.md" tag="0T13HM377F" kind="definition">
# Alternating group

## Definition

Let $n$ be a positive integer.
The alternating group $A_n$ is the set of the [[even permutations|0SQTMQTSRZ]] in $S_n$, with composition as the operation.

## Remarks

For $n \geq 2$, $A_n$ is the [[kernel|0W88CEGQP1]] of the [[sign homomorphism|0JEJJCB0EX]], so it is a [[normal subgroup|0CB0DX5BVV]] of $S_n$.
For $n \geq 2$, $A_n$ is also the [[commutator subgroup|0ASGSTXSBW]] of $S_n$.
The group $A_3$ is cyclic of order $3$, and $A_4$ has order $12$.
The group $A_n$ is abelian if and only if $n \leq 3$.

## Explanations

### L1 human

The alternating group keeps only the even permutations.
The product of two even permutations is even, so these form a group.
For $n \geq 5$ it is one of the most important examples of a simple group.
</document>

<document source="https://math2.tamnd.com/node/0T1BSD7H3M.md" tag="0T1BSD7H3M" kind="statement">
# Infinitely many primes congruent to 1 modulo n

## Statement

Let $n$ be a positive integer.
Then there are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p \equiv 1 \pmod{n}$.

## Remarks

This is the case $a = 1$ of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].
The case $a = 1$ has an elementary proof with cyclotomic polynomials. A prime factor $p$ of $\Phi_n(m)$ that does not divide $n$ satisfies $p \equiv 1 \pmod n$.
For example, the primes $p \equiv 1 \pmod{10}$ start with $11, 31, 41, 61, 71$.

## Explanations

### L1 human

Fix a number $n$, for example $n = 6$. Then infinitely many primes leave the remainder $1$ when you divide them by $6$.
The first ones are $7, 13, 19, 31, 37$.
</document>

<document source="https://math2.tamnd.com/node/0T1Q8D0FW8.md" tag="0T1Q8D0FW8" kind="statement">
# Fundamental theorem of Galois theory

## Statement

Let $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $G = \operatorname{Gal}(L/K)$.

1. The maps $H \mapsto L^H$ and $E \mapsto \operatorname{Gal}(L/E)$ are inverse bijections between the subgroups $H$ of $G$ and the [[intermediate fields|02NBKVQFRQ]] $E$ of $L/K$.
2. These bijections reverse inclusions: $H_1 \subseteq H_2$ if and only if $L^{H_1} \supseteq L^{H_2}$.
3. For every subgroup $H$, $[L : L^H] = |H|$ and $[L^H : K] = |G|/|H|$.
4. The extension $L^H/K$ is normal if and only if $H$ is a normal subgroup of $G$. Then $\operatorname{Gal}(L^H/K) \cong G/H$.

Here $L^H$ is the [[fixed field|0KMJ6919WS]] of $H$.

## Remarks

For $L = \mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $K = \mathbb{Q}$, the group $G$ has order $4$ and three subgroups of order $2$. They match the three intermediate fields $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$ and $\mathbb{Q}(\sqrt{6})$.
The top field $L$ matches the trivial subgroup, and the base field $K$ matches $G$.
A finite Galois extension has only finitely many intermediate fields, because a finite group has finitely many subgroups.
The theorem is used to prove that the general equation of degree $5$ is not solvable by radicals.

## History

Évariste Galois proved the theorem in his development of Galois theory.

## Explanations

### L1 human

The theorem turns questions about fields into questions about groups.
Each intermediate field matches one subgroup of the Galois group, and the bigger the field, the smaller the subgroup.
Finite groups are often easier to study than fields.
</document>

<document source="https://math2.tamnd.com/node/0T233AZ1GE.md" tag="0T233AZ1GE" kind="statement">
# Degree of a product

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $f, g \in R[x]$.
Then $\deg(f g) = \deg f + \deg g$, with the [[convention|0DPF6WRZW6]] $\deg 0 = -\infty$.

## Remarks

The proof: if $f$ and $g$ are nonzero with [[leading coefficients|0P05RJC5XA]] $a$ and $b$, then the coefficient of $x^{\deg f + \deg g}$ in $f g$ is $a b \neq 0$.
Over every ring, $\deg(f g) \le \deg f + \deg g$.
The equality fails in $(\mathbb{Z}/4\mathbb{Z})[x]$: $(2x + 1)(2x + 1) = 4x^2 + 4x + 1 = 1$.

## Explanations

### L1 human

When you multiply two polynomials, the highest terms multiply to give the highest term of the product.
For example, $(x^2 + 1)(x^3 - x)$ has degree $2 + 3 = 5$.
This needs the product of the leading coefficients to be nonzero.
</document>

<document source="https://math2.tamnd.com/node/0T2NTJCV83.md" tag="0T2NTJCV83" kind="definition">
# Least element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $l \in S$ is the least element of $S$ when $l \le s$ for each $s \in S$.

## Remarks

Another name is minimum. By antisymmetry, a set has at most one least element.
Each nonempty subset of $\mathbb{N}$ has a least element. This is the well-ordering principle of $\mathbb{N}$. See [[well-order|0F1D6PK9RZ]].
The interval $(0, 1]$ in $\mathbb{R}$ has no least element.

## Explanations

### L1 human

The least element is below all others.
In the power set of a set under $\subseteq$, the least element is the empty set.
</document>

<document source="https://math2.tamnd.com/node/0T2WYDTRP2.md" tag="0T2WYDTRP2" kind="definition">
# Subspace metric

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A$ be a [[subset|0FE8Q5S42Y]] of $X$.
The subspace metric on $A$ is the function $d_A \colon A \times A \to \mathbb{R}$ with $d_A(a, b) = d(a, b)$ for all $a, b \in A$.
It is the [[restriction|0D2MXBV21T]] of $d$ to $A \times A$.
The metric space $(A, d_A)$ is a subspace of $(X, d)$.

## Remarks

The function $d_A$ is a metric, because each axiom for $d_A$ is a special case of the same axiom for $d$.
For example, the interval $[0, 1]$ with $d(x, y) = |x - y|$ is a subspace of the [[real line|0KXJTZSW9B]].
The open sets of $(A, d_A)$ are the sets $U \cap A$ with $U$ open in $X$. Thus the topology of $d_A$ is the [[subspace topology|0D5YHQGDJY]].

## Explanations

### L1 human

A subspace keeps the distances of the big space and forgets the other points.
For example, the unit circle in the plane, with the Euclidean distance, is a subspace of $\mathbb{R}^2$. The distance between two points of the circle is the length of the straight chord, not of the arc.
</document>

<document source="https://math2.tamnd.com/node/0T32ME0A6H.md" tag="0T32ME0A6H" kind="statement">
# Power series of arctangent

## Statement

For every real $x$ with $|x| \le 1$,
$$\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n + 1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots.$$

## Remarks

For $|x| < 1$, integrate the [[geometric series|01X5CVV1Q5]] $\frac{1}{1 + t^2} = \sum (-1)^n t^{2n}$ term by term from $0$ to $x$. Its sum is the [[derivative of arctangent|05P4B4Q22V]].
At $x = \pm 1$, the series converges by the alternating series test, and [[Abel's theorem|0S4EMDQS2F]] gives the equality. At $x = 1$ this is the [[Leibniz formula|037FD5NBRT]] $\pi/4 = 1 - 1/3 + 1/5 - \cdots$.
The radius of convergence is $1$. Mathlib states the series for $|x| < 1$ as `Real.hasSum_arctan`.

## Explanations

### L1 human

The arctangent is a power series with odd powers and the simple coefficients $\pm 1/(2n+1)$.
At $x = 1$, it gives a famous series for $\pi/4$.
</document>

<document source="https://math2.tamnd.com/node/0T36P6TXQ6.md" tag="0T36P6TXQ6" kind="definition">
# Distributive lattice

## Definition

A lattice $(L, \vee, \wedge)$ is distributive if $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$ for all $x, y, z \in L$.

## Remarks

The condition is equivalent to its dual: $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$ for all $x, y, z$.
Every totally ordered set is a distributive lattice, with $\max$ as join and $\min$ as meet. Every [[Boolean algebra|06GKBTAA5E]] is a distributive lattice.
The diamond lattice $M_3$ and the pentagon lattice $N_5$ are the basic lattices that are not distributive.

## Explanations

### L1 human

A lattice is distributive when meet distributes over join, like intersection over union for sets.
Then join also distributes over meet.
</document>

<document source="https://math2.tamnd.com/node/0T3DPBG981.md" tag="0T3DPBG981" kind="definition">
# Predicate

## Definition

Let $X$ be a set.
A predicate on $X$ is a statement $P(x)$ with a variable $x$ that ranges over $X$, such that $P(a)$ is a [[proposition|0ZGC7ZESYN]] for each $a \in X$.
More generally, a predicate $P(x_1, \dots, x_n)$ has $n$ variables, and $n$ is its arity.

## Remarks

The predicate "$x > 3$" on the integers gives the true proposition "$5 > 3$" and the false proposition "$1 > 3$".
A predicate on $X$ determines the set $\{x \in X \mid P(x)\}$ of the values that make it true. See [[set-builder notation|0TMZ89Q5KA]].
A predicate with two variables is a [[binary relation|08EJC6Q25H]].

## Explanations

### L1 human

A predicate is a sentence with a blank, such as "___ is prime".
It is not true or false yet.
When you fill the blank with a number, you get a proposition.
</document>

<document source="https://math2.tamnd.com/node/0T3GK0XCBC.md" tag="0T3GK0XCBC" kind="concept">
# Phase portrait

## Summary

A phase portrait of an autonomous system is a picture of a representative set of its [[trajectories|0BVT6VSB3S]] in the phase space, usually the [[phase plane|05VDGH129F]].
Arrows on the curves show the direction of time.
It shows the [[equilibrium points|0G4X36AE0X]], the periodic orbits and the long-time behaviour of the solutions.

## Remarks

The [[phase line|081QSEAGEY]] is the phase portrait of a scalar autonomous equation.
For the undamped [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the phase portrait in the $(x, x')$ plane consists of the origin and the circles around it.

## Explanations

### L1 human

A phase portrait is a map of all the possible motions of a system.
One look shows where the system rests, where it cycles and where it goes in the long run.
</document>

<document source="https://math2.tamnd.com/node/0T3RZQXDYD.md" tag="0T3RZQXDYD" kind="concept">
# Limit superior

## Summary

The limit superior of a sequence is the limit of the suprema of its tails. It describes the largest value near which the sequence lies infinitely often. The node [[limit superior|0P8BA7T7S5]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0T3WVTD675.md" tag="0T3WVTD675" kind="concept">
# Frobenius endomorphism

## Summary

The Frobenius endomorphism of a commutative ring of prime characteristic $p$ is the map that sends each element to its $p$-th power.
It is a ring homomorphism.
The [[definition of the Frobenius endomorphism|0GHDDVAH8P]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0T4EJ8ZJM9.md" tag="0T4EJ8ZJM9" kind="counterexample">
# Dirichlet function

## Counterexample

Let $f \colon \mathbb{R} \to \mathbb{R}$ be given by $f(x) = 1$ for rational $x$ and $f(x) = 0$ for irrational $x$.
Then $f$ is not [[continuous at|0EEE2CZB2A]] any point.
Indeed, each real $a$ is the limit of a sequence of rationals and of a sequence of irrationals. Their images tend to $1$ and to $0$, so one of them does not tend to $f(a)$.

## Remarks

The function $f$ is the indicator function of $\mathbb{Q}$.
Every rational number is a period of $f$.
The function is [[not Riemann integrable|03Z4JC4YRJ]] on any interval with more than one point.

## Explanations

### L1 human

Rational and irrational numbers lie mixed together in every small interval.
So the function jumps between $0$ and $1$ near every point, and it has no limit anywhere.
</document>

<document source="https://math2.tamnd.com/node/0T4QHJXSD4.md" tag="0T4QHJXSD4" kind="statement">
# The cross product is perpendicular to its factors

## Statement

Let $a$ and $b$ be vectors in $\mathbb{R}^3$, and let $a \times b$ be their [[cross product|0N5N15HWZ7]]. Let $u \cdot v = u_1 v_1 + u_2 v_2 + u_3 v_3$ be the dot product.
Then
$$a \cdot (a \times b) = 0 \quad \text{and} \quad b \cdot (a \times b) = 0.$$

## Remarks

By the [[test with the dot product|03NK6RSRQ1]], $a \times b$ is perpendicular to $a$ and to $b$ when all three vectors are nonzero.
The proof is a direct computation. For example, $a \cdot (a \times b) = a_1(a_2 b_3 - a_3 b_2) + a_2(a_3 b_1 - a_1 b_3) + a_3(a_1 b_2 - a_2 b_1)$, and the six terms cancel in pairs.
If $a$ and $b$ are linearly independent, then $a \times b$ is a normal vector of the plane through $0$ that contains $a$ and $b$.

## Explanations

### L1 human

Lay two pencils flat on a table, starting at the same point. Their cross product points straight up from the table, or straight down. So it makes a right angle with each pencil.
</document>

<document source="https://math2.tamnd.com/node/0T6D4PE51C.md" tag="0T6D4PE51C" kind="example">
# Clairaut's equation

## Example

Let $f$ be a twice continuously differentiable function on an open interval $J$.
Clairaut's equation is the implicit first-order equation
$$y = x\, y' + f(y'),$$
for an unknown $y$ with $y' \in J$.
For each constant $C \in J$, the line $y = C x + f(C)$ is a solution. These lines form the [[general solution|0SKYFQAFYD]].
If $f'' \ne 0$ on $J$, the curve with the parametric form $x = -f'(p)$, $y = f(p) - p f'(p)$ for $p \in J$ is also a solution. It is the envelope of the lines, and it is a [[singular solution|0CZA0NCFWP]].

## Remarks

Differentiation of the equation gives $y''\left(x + f'(y')\right) = 0$. The case $y'' = 0$ gives the lines, and the case $x + f'(y') = 0$ gives the envelope.
Example: for $f(p) = p^2$, the lines are $y = C x + C^2$, and the singular solution is the parabola $y = -x^2/4$.
Each line of the general solution is tangent to the singular solution at one point.

## Explanations

### L1 human

Clairaut's equation has a family of straight lines as solutions, one for each slope $C$.
These lines wrap around a curve, their envelope, and this curve is also a solution.
No choice of $C$ gives the envelope, so it is a new, singular solution.
</document>

<document source="https://math2.tamnd.com/node/0T6MWGZSXA.md" tag="0T6MWGZSXA" kind="concept">
# Augmented matrix

## Summary

The augmented matrix of a system of linear equations is the coefficient matrix with the column of constant terms added on the right.
</document>

<document source="https://math2.tamnd.com/node/0T6WS7TT77.md" tag="0T6WS7TT77" kind="statement">
# First Borel–Cantelli lemma

## Statement

Let $A_1, A_2, \dots$ be [[events|00SQGJNRTB]] in a probability space $(\Omega, \mathcal{F}, P)$ with $\sum_{n=1}^{\infty} P(A_n) < \infty$.
Then
$$P\left(\limsup_{n \to \infty} A_n\right) = 0, \quad \text{where} \quad \limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{m=n}^{\infty} A_m.$$
So almost surely, only finitely many of the events $A_n$ occur.

## Remarks

The event $\limsup_n A_n$ is the set of outcomes that lie in infinitely many $A_n$. It is written $\{A_n \text{ i.o.}\}$, for "infinitely often".
Proof: for each $n$, $P(\limsup_m A_m) \le P\left(\bigcup_{m \ge n} A_m\right) \le \sum_{m \ge n} P(A_m)$ by [[Boole's inequality|0EGSEE9J9R]]. The right side tends to $0$ as $n \to \infty$.
The lemma needs no independence. The [[second Borel–Cantelli lemma|0NWXRAMEE7]] gives a partial converse for independent events.
The lemma holds in each measure space: if $\sum_n \mu(A_n) < \infty$, then $\mu(\limsup_n A_n) = 0$.

## History

The lemma is named after Émile Borel and Francesco Paolo Cantelli.

## Explanations

### L1 human

If the chances of the events add up to a finite number, then after some time none of them happens any more, with probability $1$.
For example, if $P(A_n) = 1/n^2$, then only finitely many $A_n$ occur.
</document>

<document source="https://math2.tamnd.com/node/0T71SSYZ39.md" tag="0T71SSYZ39" kind="statement">
# Cauchy–Riemann equations

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function.
Identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + i y \leftrightarrow (x, y)$.
Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$ be the [[real and imaginary parts|02ZE6JHJ7G]] of $f$, as real functions of $(x, y)$ on $U$.
Let $z_0 = x_0 + i y_0 \in U$.
Then $f$ is [[complex differentiable|0D1C9Y1EXA]] at $z_0$ if and only if the following two conditions hold. First, $u$ and $v$ are differentiable at $(x_0, y_0)$ as functions of two real variables. Second, they satisfy
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$
at $(x_0, y_0)$.
In this case $f'(z_0) = \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0)$.

## Remarks

For $f(z) = z^2$ we have $u = x^2 - y^2$ and $v = 2xy$. Then $u_x = 2x = v_y$ and $u_y = -2y = -v_x$.
The real differentiability of $u$ and $v$ is necessary. The function $u(x, y) = \sqrt{|xy|}$, $v = 0$ satisfies the two equations at $(0, 0)$, but $u + iv$ is not complex differentiable at $0$.
The two equations together are the one complex equation $\frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x}$.

## Explanations

### L1 human

A complex derivative must be the same in each direction. If you compare the derivative along the real axis with the derivative along the imaginary axis, you get the two equations.
So the equations are a test with partial derivatives for a complex derivative.
For example, $\bar z = x - i y$ has $u_x = 1$ and $v_y = -1$, so it fails the test.
</document>

<document source="https://math2.tamnd.com/node/0T7ABX4CWE.md" tag="0T7ABX4CWE" kind="definition">
# Circle integral

## Definition

Let $c \in \mathbb{C}$ and let $R > 0$ be a real number.
Let $f$ be a complex function that is continuous on the circle $\{z : |z - c| = R\}$.
The circle integral of $f$ over this circle is the [[contour integral|00FWNP9TXY]] along the path $\gamma(\theta) = c + R e^{i\theta}$, $\theta \in [0, 2\pi]$:
$$\oint_{|z - c| = R} f(z) \, dz = \int_0^{2\pi} f(c + R e^{i\theta}) \, i R e^{i\theta} \, d\theta.$$

## Remarks

The path goes once around the circle, in the positive (counterclockwise) direction.
The length of the path is $2 \pi R$.
The basic example is the [[integral of 1/z|0SXZ68HV6F]]: $\oint_{|z| = 1} \frac{dz}{z} = 2 \pi i$.

## Explanations

### L1 human

A circle integral is a contour integral where the path is a circle, traveled once counterclockwise.
Circles are the paths that appear most in complex analysis, for example in [[Cauchy's integral formula|0DG10TVYPN]].
</document>

<document source="https://math2.tamnd.com/node/0T7DX8NY1X.md" tag="0T7DX8NY1X" kind="concept">
# Tangent line to a circle

## Summary

A tangent line to a circle is a line that meets the circle in exactly one point, the point of tangency. It does not enter the inside of the circle, and it is perpendicular to the radius at the point of tangency.
</document>

<document source="https://math2.tamnd.com/node/0T89W1QZ9K.md" tag="0T89W1QZ9K" kind="statement">
# Integration in polar coordinates

## Statement

Let $0 \le r_1 < r_2$ and $\alpha < \beta \le \alpha + 2\pi$, and let $D = \{(r \cos \theta, r \sin \theta) : r_1 \le r \le r_2, \ \alpha \le \theta \le \beta\}$.
Let $f \colon D \to \mathbb{R}$ be continuous.
Then
$$\iint_D f(x, y) \, dx \, dy = \int_\alpha^\beta \int_{r_1}^{r_2} f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta.$$

## Remarks

This is the [[change of variables formula|00N5FHP1CV]] for the [[polar coordinate map|0RJZV05MJA]], whose Jacobian determinant is $r$. The boundary rays and the origin have area zero and do not change the integral.
Example: the area of the disc of radius $a$ is $\int_0^{2\pi} \int_0^a r \, dr \, d\theta = \pi a^2$.
For a region $\alpha \le \theta \le \beta$, $0 \le r \le g(\theta)$ with $g$ continuous, the formula gives the area $\tfrac{1}{2} \int_\alpha^\beta g(\theta)^2 \, d\theta$.
The Gaussian integral follows from $\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dx \, dy = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi$. Hence $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$.

## Explanations

### L1 human

In polar coordinates a small piece of the plane is almost a rectangle with sides $dr$ and $r \, d\theta$.
So its area is $r \, dr \, d\theta$, and the factor $r$ appears in the integral.
</document>

<document source="https://math2.tamnd.com/node/0T8BYAFEXX.md" tag="0T8BYAFEXX" kind="statement">
# The characteristic function determines the distribution

## Statement

Let $X$ and $Y$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\varphi_X$ and $\varphi_Y$.
If $\varphi_X(t) = \varphi_Y(t)$ for all $t \in \mathbb{R}$, then $X$ and $Y$ have the same [[law|0TST4MHA3A]].

## Remarks

The proof uses the Lévy inversion formula. For real $a < b$ that are continuity points of the [[cumulative distribution function|0MXMS3AE13]] $F_X$,
$$F_X(b) - F_X(a) = \lim_{T \to \infty} \frac{1}{2\pi} \int_{-T}^{T} \frac{e^{-ita} - e^{-itb}}{it} \, \varphi_X(t) \, dt.$$
If $\int |\varphi_X(t)| \, dt < \infty$, then $X$ has the continuous density $f_X(x) = \frac{1}{2\pi} \int e^{-itx} \varphi_X(t) \, dt$.
The same result holds for random vectors in $\mathbb{R}^n$, with $\varphi_X(t) = E[e^{i \langle t, X \rangle}]$.

## Explanations

### L1 human

Two different distributions cannot have the same characteristic function.
So, to identify a distribution, it is enough to compute its characteristic function and recognize it.
</document>

<document source="https://math2.tamnd.com/node/0T8M3YC5RG.md" tag="0T8M3YC5RG" kind="statement">
# Existence of finite fields of every prime power order

## Statement

Let $p$ be a prime number and let $n \ge 1$.
Then there is a [[finite field|08RTDHX1SK]] of order $q = p^n$.

## Remarks

The proof: let $L$ be a [[splitting field|01441KDGBY]] of $f = x^q - x$ over $\mathbb{F}_p$. The [[derivative|0AD4VS02H3]] $f' = q x^{q-1} - 1 = -1$ has no common root with $f$, so $f$ has $q$ distinct roots in $L$, by the [[criterion for repeated roots|067F8YVYMH]].
The roots of $f$ form a subfield of $L$, because $(a + b)^q = a^q + b^q$ and $(a b)^q = a^q b^q$. This subfield has $q$ elements.
So for each $n$ there is an irreducible polynomial of degree $n$ over $\mathbb{F}_p$.

## History

E. H. Moore first proved the classification of finite fields in 1893: there is a field of each prime power order, and all fields of a given order are isomorphic.

## Explanations

### L1 human

The field with $p^n$ elements is made of the roots of $x^{p^n} - x$.
The proof checks that these roots are distinct and closed under the field operations.
</document>

<document source="https://math2.tamnd.com/node/0T92VGGVWP.md" tag="0T92VGGVWP" kind="concept">
# Bernoulli equation

## Summary

A Bernoulli equation is a first-order equation $y' + P(x) y = Q(x) y^n$.
The substitution $v = y^{1-n}$ changes it into a linear equation. See the [[definition|0ZB84QCDKR]].

## Remarks

Jacob Bernoulli discussed the equation in 1695. Leibniz published a solution in the same year.
</document>

<document source="https://math2.tamnd.com/node/0TA7WT68RK.md" tag="0TA7WT68RK" kind="definition">
# Carmichael number

## Definition

A Carmichael number is a [[composite number|0N3ADXHNTQ]] $n$ such that $a^{n-1} \equiv 1 \pmod n$ for each integer $a$ [[coprime|0GFKP8F8AG]] to $n$.

## Remarks

Thus a Carmichael number is a [[Fermat pseudoprime|02M2K4QH9P]] to each base coprime to it.
Korselt's criterion: a composite $n$ is a Carmichael number if and only if $n$ is square-free and $p - 1 \mid n - 1$ for each prime $p \mid n$.
The smallest Carmichael number is $561 = 3 \cdot 11 \cdot 17$. Here $2$, $10$ and $16$ divide $560$.
Alford, Granville and Pomerance proved in 1994 that there are infinitely many Carmichael numbers.

## Explanations

### L1 human

A Carmichael number behaves like a prime for the Fermat test, for each base that has no common factor with it.
So the [[Fermat primality test|0N5ZSZMYH6]] cannot detect it, unless the base shares a factor with it.
</document>

<document source="https://math2.tamnd.com/node/0TB16T2C97.md" tag="0TB16T2C97" kind="statement">
# Axiom of extensionality

## Statement

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
If $A$ and $B$ have the same [[elements|0EZ758XE73]], then $A = B$:
$$\forall A \, \forall B \, \bigl( \forall x \, (x \in A \leftrightarrow x \in B) \to A = B \bigr).$$

## Remarks

It is one of the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
The converse is a rule of logic: equal sets have the same elements.
In practice, $A = B$ is often proved with $A \subseteq B$ and $B \subseteq A$. See [[subset|0FE8Q5S42Y]].

## Explanations

### L1 human

A set is nothing more than its elements.
If two sets contain exactly the same things, they are the same set, however you describe them.
For example, $\{x \in \mathbb{R} \mid x^2 = 1\}$ and $\{-1, 1\}$ are equal.
</document>

<document source="https://math2.tamnd.com/node/0TBYMWPWWB.md" tag="0TBYMWPWWB" kind="statement">
# The successor function is injective

## Statement

Let $S \colon \mathbb{N} \to \mathbb{N}$ be the [[successor function|0832XVR512]] of the natural numbers.
For all natural numbers $m$ and $n$, if $S(m) = S(n)$, then $m = n$.

## Remarks

This is one of the [[Peano axioms|0QFT7Z76N5]]. In a construction of $\mathbb{N}$, for example in set theory, it is a theorem.
An equivalent form is: if $m \neq n$, then $m + 1 \neq n + 1$.

## Explanations

### L1 human

Two different numbers never have the same next number.
If you know the number after $n$, then you know $n$.
</document>

<document source="https://math2.tamnd.com/node/0TC99BRSSB.md" tag="0TC99BRSSB" kind="statement">
# Dirac's theorem on Hamiltonian cycles

## Statement

Let $G$ be a finite simple [[graph|0JD5C82QMB]] with $n \ge 3$ vertices.
Suppose that every vertex of $G$ has [[degree|0T0F6CCS9W]] at least $n/2$.
Then $G$ is [[Hamiltonian|090WPKFABX]].

## Remarks

The hypothesis says that the [[minimum degree|0TKZCSGCCA]] is at least $n/2$.
The theorem follows from [[Ore's theorem|0NACHJYPKW]]: if all degrees are at least $n/2$, then each sum of two degrees is at least $n$.
The bound is sharp. The complete bipartite graph $K_{k, k+1}$ has minimum degree $k < (2k+1)/2$, and it is not Hamiltonian.

## History

G. A. Dirac published the theorem in 1952.

## Explanations

### L1 human

If each vertex is joined to at least half of the other vertices, there is a round trip through all vertices.
A graph with this many edges cannot avoid a Hamiltonian cycle.
</document>

<document source="https://math2.tamnd.com/node/0TCBZMEYGN.md" tag="0TCBZMEYGN" kind="statement">
# C^k function characterised by continuous derivatives

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $k \ge 1$ be an integer.
Say that $f$ is of class $C^k$ on $U$ when the following is true for each component $f_j$ and each $r \le k$. Each [[partial derivative|02RSXVM1ZN]] $\partial^r f_j / \partial x_{i_r} \cdots \partial x_{i_1}$ of order $r$ exists on $U$ and is continuous on $U$.
Then $f$ is of class $C^k$ on $U$ if and only if these two conditions hold.

1. For each $r$ with $0 \le r \le k$, the [[iterated derivative|0EY05Z9RPN]] $D^r f$ exists on $U$ and is continuous on $U$.
2. For each $r$ with $0 \le r < k$, the map $D^r f$ is differentiable on $U$.

## Remarks

So in finite dimensions, $C^k$ means $k$ times continuously differentiable, in either the sense of partial derivatives or the sense of Fréchet derivatives.
A function is of class $C^\infty$, or smooth, when it is of class $C^k$ for each $k$.
Each polynomial in $x_1, \dots, x_n$ is of class $C^\infty$ on $\mathbb{R}^n$. The function $x \mapsto \|x\|$ is of class $C^\infty$ on $\mathbb{R}^n \setminus \{0\}$, but it is not differentiable at $0$.

## Explanations

### L1 human

There are two ways to count derivatives: one variable at a time with partial derivatives, or all variables at once with total derivatives.
For functions on open subsets of $\mathbb{R}^n$, both ways give the same classes $C^1, C^2, \dots$, as long as you also ask for continuity.
</document>

<document source="https://math2.tamnd.com/node/0TCHAS04WK.md" tag="0TCHAS04WK" kind="definition">
# Convex hull

## Definition

Let $X$ be a subset of $\mathbb{R}^n$. The convex hull of $X$ is the intersection of all [[convex sets|02AS45T3H9]] of $\mathbb{R}^n$ that contain $X$:
$$\operatorname{conv}(X) = \bigcap \{\, S \subseteq \mathbb{R}^n \;:\; S \supseteq X \text{ and } S \text{ is convex} \,\}.$$

## Remarks

$\operatorname{conv}(X)$ is convex, and it is the smallest convex set that contains $X$.
It is also the set of all convex combinations $\sum_{i=1}^{k} w_i x_i$, with $k \geq 1$, $x_i \in X$, $w_i \geq 0$ and $\sum_i w_i = 1$. These are the [[affine combinations|0KJH6SRFZK]] with nonnegative weights.
The convex hull of a compact set is compact, and the convex hull of an open set is open.
The convex hull of a finite set is a convex polytope. For three points not on a line, it is the triangle with its inside.

## Explanations

### L1 human

For a finite set of points in the plane, put a rubber band round all the points and let it go. The band takes the shape of the boundary of the convex hull. The convex hull of the four corners of a square is the full square.
</document>

<document source="https://math2.tamnd.com/node/0TD904MG06.md" tag="0TD904MG06" kind="concept">
# Orientable surface

## Summary

A surface in space is orientable if one can choose a [[unit normal vector|073JKP99K2]] at each point in a continuous way. The choice must be continuous over the whole surface.
Such a choice is an orientation. The Möbius strip is the standard surface that has no orientation.
</document>

<document source="https://math2.tamnd.com/node/0TDY6JZHKS.md" tag="0TDY6JZHKS" kind="statement">
# Product of the gcd and the lcm

## Statement

Let $a$ and $b$ be natural numbers.
Then $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a b$, where $\gcd$ is the [[greatest common divisor|02WF878BH1]] and $\operatorname{lcm}$ is the [[least common multiple|09S5SN35DJ]].

## Remarks

For integers $a$ and $b$ of any sign, the same argument gives $\gcd(a, b) \cdot \operatorname{lcm}(a, b) = |a b|$.
Thus $\operatorname{lcm}(a, b) = a b / \gcd(a, b)$ when $a$ and $b$ are not both $0$. This is the usual way to compute the lcm, with the [[Euclidean algorithm|0TRNT4X7J0]].
For example, $\gcd(4, 6) = 2$, $\operatorname{lcm}(4, 6) = 12$ and $2 \cdot 12 = 24 = 4 \cdot 6$.

## Explanations

### L1 human

Think of the prime factors of $a$ and $b$. The gcd takes, for each prime, the smaller of the two exponents, and the lcm takes the larger one.
The smaller exponent plus the larger exponent is the sum of the two exponents.
So the product of the gcd and the lcm has the same prime factors as $a b$.
</document>

<document source="https://math2.tamnd.com/node/0TEDQH56RW.md" tag="0TEDQH56RW" kind="statement">
# Remainder theorem

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]], let $f \in R[x]$ and let $c \in R$.
Then the remainder of the [[division|0MES3KS0ZC]] of $f$ by $x - c$ is the constant polynomial $f(c)$.
That is, there is $q \in R[x]$ with $f = (x - c) q + f(c)$.

## Remarks

The proof: division by the monic polynomial $x - c$ gives $f = (x - c) q + r$ with $r$ a constant. [[Evaluation|0GJ7T0NK49]] at $c$ gives $r = f(c)$.
For example, the remainder of $x^3 - 2x + 5$ divided by $x - 2$ is $2^3 - 4 + 5 = 9$.

## Explanations

### L1 human

You do not need long division to find the remainder of $f$ divided by $x - c$.
You only compute $f(c)$.
</document>

<document source="https://math2.tamnd.com/node/0TEJ6QF7K9.md" tag="0TEJ6QF7K9" kind="definition">
# Affine space

## Definition

Let $K$ be a field and let $V$ be a vector space over $K$.
An affine space over $V$ is a nonempty set $A$ with a map $A \times V \to A$, written $(p, v) \mapsto p + v$, that has these properties.
For each point $p$ of $A$: $p + 0 = p$.
For each point $p$ of $A$ and all vectors $u$ and $v$ of $V$: $(p + u) + v = p + (u + v)$.
For all points $p$ and $q$ of $A$, there is exactly one vector $v$ of $V$ with $p + v = q$. This vector is written $q - p$ or $\overrightarrow{pq}$.
The elements of $A$ are the points, and $V$ is the space of translations of $A$. The dimension of $A$ is the dimension of $V$.

## Remarks

Each vector space $V$ is an affine space over itself, with the addition of $V$ as the action. In particular $\mathbb{R}^n$ is an affine space over $\mathbb{R}^n$.
For all points $p$, $q$ and $r$, Chasles's relation holds: $(q - p) + (r - q) = r - p$.
Mathlib calls this structure an additive torsor, `AddTorsor`. Its definition works for an additive group $V$ in place of a vector space.
A [[Euclidean space|0XPR1E85W0]] is a real affine space of finite dimension whose space of translations has an inner product.

## Explanations

### L1 human

In an affine space you can subtract two points to get the arrow from one to the other. You can also add an arrow to a point. But you cannot add two points: there is no special origin. A good picture is the plane without its coordinate axes.
</document>

<document source="https://math2.tamnd.com/node/0TFDZN6AXP.md" tag="0TFDZN6AXP" kind="example">
# p-metric on R^n

## Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_p(x, y) = \left( \sum_{i=1}^{n} |x_i - y_i|^p \right)^{1/p}.$$
Then $d_p$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$.

## Remarks

The triangle inequality for $d_p$ is the Minkowski inequality.
For $p = 1$ it is the [[taxicab metric|07CJNE6ZAQ]], and for $p = 2$ it is the [[Euclidean metric|0E1MMCAY5B]].
For $0 < p < 1$, the same formula does not give a metric when $n \ge 2$: the triangle inequality fails.
The metric $d_p$ comes from the [[p-norm|0KNJWEVSMG]]: $d_p(x, y) = \|x - y\|_p$.

## Explanations

### L1 human

The p-metrics form a family of ways to combine the differences of the coordinates into one distance.
A larger $p$ gives more weight to the largest difference, and for very large $p$ the distance is close to the [[sup metric|0VTAXQEWQD]].
</document>

<document source="https://math2.tamnd.com/node/0TGV80E6A4.md" tag="0TGV80E6A4" kind="concept">
# Big O notation

## Summary

Big O notation compares the growth of two functions. The statement $f = O(g)$ means that $|f|$ is at most a constant times $|g|$ near a given point or at infinity. The node [[big O notation|0RYTGBJAGN]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0TH61JR04G.md" tag="0TH61JR04G" kind="definition">
# Derangement

## Definition

Let $X$ be a set.
A derangement of $X$ is a [[permutation|0BQMWCNS8S]] $\sigma$ of $X$ such that $\sigma(x) \neq x$ for all $x \in X$.
In other words, it is a permutation with no fixed point.

## Remarks

The derangements of $\{1, 2, 3\}$ are $231$ and $312$ in one-line notation.
The [[number of derangements|03WY4HHMT1]] of an $n$-element set is written $D_n$ or $!n$.

## Explanations

### L1 human

A derangement is a shuffle in which nothing stays in its place.
Suppose that $n$ people leave their hats, and each person later takes a hat at random. A derangement is an outcome in which nobody gets their own hat.
</document>

<document source="https://math2.tamnd.com/node/0TH78S7GBZ.md" tag="0TH78S7GBZ" kind="concept">
# Arc length

## Summary

The arc length of a curve is its length, measured along the curve. It is the supremum of the lengths of inscribed polygonal paths, and an integral of the speed for smooth curves. The node [[arc length|0X7G61297X]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0TH9CPW3VS.md" tag="0TH9CPW3VS" kind="statement">
# Reduction of a higher-order equation to a first-order system

## Statement

Let $n \ge 1$, let $U$ be a subset of $\mathbb{R} \times \mathbb{R}^n$, and let $f : U \to \mathbb{R}$ be a function.
Consider the equation $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ and the first-order system for $z = (z_1, \dots, z_n)$:
$$z_1' = z_2, \quad z_2' = z_3, \quad \dots, \quad z_{n-1}' = z_n, \quad z_n' = f(x, z_1, \dots, z_n).$$
Let $I$ be an open interval.
If $y$ is a [[solution|04SW7WECS8]] of the equation on $I$, then $z = \left(y, y', \dots, y^{(n-1)}\right)$ is a solution of the system on $I$.
Conversely, let $z$ be a solution of the system on $I$. Then $y = z_1$ is a solution of the equation on $I$, and $z_k = y^{(k-1)}$ for $k = 1, \dots, n$.

## Remarks

The initial conditions $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$ become the one condition $z(x_0) = (c_0, \dots, c_{n-1})$. So results on first-order [[initial value problems|075VBY4K4X]] apply to equations of each order.
For $y'' = -y$, the system is $z_1' = z_2$, $z_2' = -z_1$.
The same method changes a system of equations of order $n$ into a larger first-order system.

## Explanations

### L1 human

The trick is to give new names to the derivatives of $y$.
Each new unknown is the derivative of the one before it, and the last one is given by the equation.
So we only need a theory of first-order systems.
</document>

<document source="https://math2.tamnd.com/node/0THBK9V825.md" tag="0THBK9V825" kind="concept">
# Function composition

## Summary

The composition of two functions applies the first function and then the second.
</document>

<document source="https://math2.tamnd.com/node/0THJ7416DX.md" tag="0THJ7416DX" kind="concept">
# Conditional independence

## Summary

Two events are conditionally independent given a third event when they are independent for the conditional probability given the third event.
The node [[conditional independence|0ZJ87ZZ8PK]] gives the precise definition for events.
</document>

<document source="https://math2.tamnd.com/node/0TJ1V9DDPC.md" tag="0TJ1V9DDPC" kind="statement">
# Third isomorphism theorem

## Statement

Let $G$ be a [[group|G5S1M9]], and let $N$ and $K$ be [[normal subgroups|0CB0DX5BVV]] of $G$ with $N \subseteq K$.
Then $K/N$ is a normal subgroup of $G/N$, and the map $(gN)(K/N) \mapsto gK$ is an [[isomorphism|08MG7H8JW5]] from $(G/N)/(K/N)$ to $G/K$.

## Remarks

The proof applies the [[first isomorphism theorem|0CXXJNG5B6]] to the surjective homomorphism $G/N \to G/K$, $gN \mapsto gK$, whose kernel is $K/N$.
For example, $(\mathbb{Z}/12\mathbb{Z})/(4\mathbb{Z}/12\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}$.
For finite groups, it gives $[G : K] = [G/N : K/N]$.

## Explanations

### L1 human

If you divide by $N$ and then by the image of $K$, you get the same result as dividing by $K$ in one step.
It looks like cancelling fractions: $(G/N)/(K/N) = G/K$.
The theorem makes this cancellation precise.
</document>

<document source="https://math2.tamnd.com/node/0TJ8QBJX5K.md" tag="0TJ8QBJX5K" kind="statement">
# Term-by-term differentiation

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be differentiable for each $n \ge 0$.
Assume that $\sum_{n} f_n(x_0)$ converges for some $x_0 \in [a, b]$, and that $\sum_{n} f_n'$ converges uniformly on $[a, b]$.
Then $\sum_{n} f_n$ converges uniformly on $[a, b]$, its sum is differentiable, and
$$\Big( \sum_{n=0}^{\infty} f_n \Big)' = \sum_{n=0}^{\infty} f_n'.$$

## Remarks

Apply [[the theorem on sequences|0CS42KEM5C]] to the partial sums.
The derivative series often converges uniformly by the M-test. Mathlib states that form as `hasDerivAt_tsum`.
Power series satisfy the hypotheses on each closed interval inside the interval of convergence.

## Explanations

### L1 human

We may differentiate a series term by term, if the series of derivatives converges uniformly.
Then the derivative of the sum is the sum of the derivatives.
</document>

<document source="https://math2.tamnd.com/node/0TK8Y3J5RW.md" tag="0TK8Y3J5RW" kind="definition">
# Interval

## Definition

A subset $I$ of $\mathbb{R}$ is an interval when, for all $x$ and $y$ in $I$ and each real $z$ with $x \le z \le y$, also $z \in I$.
For real numbers $a \le b$, the notation is:
$$[a, b] = \{x \in \mathbb{R} : a \le x \le b\}, \qquad (a, b) = \{x \in \mathbb{R} : a < x < b\},$$
$$[a, b) = \{x \in \mathbb{R} : a \le x < b\}, \qquad (a, b] = \{x \in \mathbb{R} : a < x \le b\}.$$
The set $[a, b]$ is a closed interval and $(a, b)$ is an open interval.
For each real $a$, the notation is $[a, \infty) = \{x : x \ge a\}$, $(a, \infty) = \{x : x > a\}$, $(-\infty, a] = \{x : x \le a\}$ and $(-\infty, a) = \{x : x < a\}$.

## Remarks

By the [[completeness axiom|0AXWET80YW]], each interval is one of the sets above, or $\mathbb{R}$, or $\emptyset$.
In some countries, the notation $]a, b[$ replaces $(a, b)$.
The symbols $\infty$ and $-\infty$ are not real numbers. They show only that the interval has no bound on that side.

## Explanations

### L1 human

An interval is a piece of the number line without holes.
A square bracket means that the end point belongs to the interval, and a round bracket means that it does not.
For example, $[0, 1)$ contains $0$ and $0.999$, but not $1$.
</document>

<document source="https://math2.tamnd.com/node/0TK9MCFHBW.md" tag="0TK9MCFHBW" kind="concept">
# Cylindrical coordinates

## Summary

Cylindrical coordinates describe a point of space by the [[polar coordinates|0BSW4T7THS]] of its projection to the $xy$-plane and by its height $z$ above that plane.
They simplify problems with symmetry about an axis, such as integrals over cylinders and cones.
</document>

<document source="https://math2.tamnd.com/node/0TKG7T3K0E.md" tag="0TKG7T3K0E" kind="statement">
# Bounded derivative on a convex set implies Lipschitz

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$, and let $C \subseteq U$ be a convex set.
Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in C$, with the operator norm.
Then $f$ is Lipschitz on $C$ with constant $M$: $\|f(y) - f(x)\| \le M \, \|y - x\|$ for all $x, y \in C$.

## Remarks

Because $C$ is convex, the segment between two points of $C$ lies in $C$. So this follows from the [[mean value inequality|02CYB184RC]].
Convexity matters. Take the annulus $1 < \|x\| < 2$ in $\mathbb{R}^2$ without the points on the negative $x$-axis, and the polar angle $\theta \in (-\pi, \pi)$ on it. Then $\|\nabla \theta\| = 1/\|x\| \le 1$, but two points just above and just below the slit are close, and their angles differ by almost $2\pi$.
A function of class $C^1$ on $U$ is Lipschitz on each closed ball in $U$, because $\|Df\|$ is bounded on the compact ball.

## Explanations

### L1 human

A Lipschitz function cannot change faster than a fixed rate $M$ per unit of distance.
A bound on the derivative gives such a rate. For this, you must be able to walk between any two points along a straight line inside the domain.
</document>

<document source="https://math2.tamnd.com/node/0TKP3TDD02.md" tag="0TKP3TDD02" kind="concept">
# Bell numbers

## Summary

The Bell numbers $B_n$ count all the partitions of an $n$-element set. They are the row sums of the Stirling numbers of the second kind. The node [[Bell numbers|0WW85SQVS3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0TKYAEJ8N7.md" tag="0TKYAEJ8N7" kind="object">
# Orthogonal group

## Description

Let $n \ge 1$.
The *orthogonal group* is the set
$$O(n) = \{Q \in M_n(\R) : Q^T Q = I_n\}$$
of real $n \times n$ [[orthogonal matrices|0W02QESPJQ]], with matrix multiplication.
It is a subgroup of the general linear group $GL_n(\R)$.

## Remarks

It is the group of linear isometries of $\R^n$ with the dot product.
Each element has determinant $1$ or $-1$, and $\det \colon O(n) \to \{1, -1\}$ is a surjective homomorphism.
Its kernel is the [[special orthogonal group|0CGYNDCWRR]].
$O(n)$ is a compact Lie group of dimension $n(n-1)/2$.

## Explanations

### L1 human

The orthogonal group collects all rotations and reflections of $n$-dimensional space that fix the origin.
Composing two of them gives another one, and each one can be undone.
In the plane, it consists of the rotations about the origin and the reflections in lines through the origin.
</document>

<document source="https://math2.tamnd.com/node/0TKZCSGCCA.md" tag="0TKZCSGCCA" kind="definition">
# Minimum degree

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]] with a nonempty vertex set $V$.
The minimum degree of $G$ is
$$\delta(G) = \min_{v \in V} \deg(v),$$
the smallest [[degree|0T0F6CCS9W]] of a vertex of $G$.

## Remarks

The [[maximum degree|0KNAN9F27P]] $\Delta(G)$ is the largest degree. Always $\delta(G) \le \Delta(G)$.
Each [[cycle graph|0QQ0F12962]] has $\delta = 2$, and each tree with at least two vertices has $\delta = 1$.
A condition on $\delta(G)$ gives a Hamiltonian cycle in [[Dirac's theorem|0TC99BRSSB]].

## Explanations

### L1 human

The minimum degree is the smallest number of edges at any vertex.
If $\delta(G) = 3$, then each vertex has at least three neighbours.
</document>

<document source="https://math2.tamnd.com/node/0TMZ21WKCN.md" tag="0TMZ21WKCN" kind="definition">
# Lipschitz map

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function and let $K \ge 0$ be a real number.
The map $f$ is Lipschitz with constant $K$ when
$$d_Y\bigl(f(x), f(y)\bigr) \le K \, d_X(x, y) \quad \text{for all } x, y \in X.$$
The map $f$ is Lipschitz when it is Lipschitz with some constant $K \ge 0$.

## Remarks

The smallest such $K$ is the Lipschitz constant of $f$.
The map $x \mapsto |x|$ on $\mathbb{R}$ is Lipschitz with constant $1$. The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is not Lipschitz, because $\sqrt{x} / x$ is not bounded near $0$.
A Lipschitz map with constant $K < 1$ is a [[contraction|0MT2S0ZX4Y]].
A differentiable $f \colon \mathbb{R} \to \mathbb{R}$ with $|f'(x)| \le K$ for all $x$ is Lipschitz with constant $K$, by the mean value theorem.

## Explanations

### L1 human

A Lipschitz map cannot stretch distances by more than the factor $K$.
For a real function, this means that the slopes of all its chords are between $-K$ and $K$.
</document>

<document source="https://math2.tamnd.com/node/0TMZ89Q5KA.md" tag="0TMZ89Q5KA" kind="concept">
# Set-builder notation

## Summary

Set-builder notation describes a [[set|0SJ3RX9TXA]] by a property of its [[elements|0EZ758XE73]].
For a set $X$ and a [[predicate|0T3DPBG981]] $P(x)$ on $X$, the set $\{x \in X \mid P(x)\}$ has as elements exactly the $a \in X$ such that $P(a)$ is true.

## Remarks

A colon is a common alternative to the bar: $\{x \in X : P(x)\}$.
Example: $\{n \in \mathbb{Z} \mid n \text{ is even}\}$ is the set of even integers.
The form $\{f(x) \mid x \in X\}$ denotes the set of the values $f(x)$. This is the [[image|0Q3F20XB0A]] of $X$ under $f$.
Without the bound $x \in X$, the notation $\{x \mid P(x)\}$ can lead to [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

Instead of a list of elements, you give a test.
$\{x \in \mathbb{R} \mid x^2 = 4\}$ means "all real numbers whose square is $4$".
This set is $\{-2, 2\}$.
</document>

<document source="https://math2.tamnd.com/node/0TNR9BQ871.md" tag="0TNR9BQ871" kind="concept">
# Powers of a diagonalisable matrix

## Summary

Let $F$ be a field, and let $A$ be an $n \times n$ matrix over $F$ with $A = P D P^{-1}$.
Here $P$ is invertible and $D$ is diagonal with entries $\lambda_1, \dots, \lambda_n$.
Then, for each integer $k \ge 0$,
$$A^k = P D^k P^{-1},$$
where $D^k$ is the diagonal matrix with entries $\lambda_1^k, \dots, \lambda_n^k$.
So the powers of a [[diagonalisable|07SQCNV233]] matrix are easy to compute.

## Remarks

The factors $P^{-1} P$ in $(P D P^{-1})^k$ cancel in pairs.
The same idea gives $p(A) = P\, p(D)\, P^{-1}$ for each polynomial $p$.
Over $\R$ or $\C$, it also gives the matrix exponential $e^A = P e^D P^{-1}$.
It is the standard method to solve linear recurrences, such as the Fibonacci recurrence.

## Explanations

### L1 human

To raise a diagonalisable matrix to a power, change to the basis of eigenvectors.
There the matrix is diagonal, and its power is the diagonal of the powers of the eigenvalues.
Then change back to the old basis.
</document>

<document source="https://math2.tamnd.com/node/0TNS65QB87.md" tag="0TNS65QB87" kind="concept">
# Prime subfield

## Summary

The prime subfield of a field is its smallest subfield.
It is $\mathbb{Q}$ or $\mathbb{F}_p$, up to isomorphism, by the characteristic.
The [[definition of the prime subfield|0VPFBAY7MW]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0TNZC55X3J.md" tag="0TNZC55X3J" kind="counterexample">
# Partial derivatives do not imply differentiability

## Counterexample

One could guess that a function of several variables is differentiable at a point when all its partial derivatives exist there.
This is false.
Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x, y) \ne (0, 0)$ and $f(0, 0) = 0$.
Both [[partial derivatives|09EGC6F8PT]] of $f$ exist at $(0, 0)$, and both are $0$.
But $f$ is not continuous at $(0, 0)$, so $f$ is not [[differentiable|0HHHDCXZV4]] at $(0, 0)$.

## Remarks

The function is $0$ on both axes, so $f(h, 0) = f(0, h) = 0$ for each $h$, and both difference quotients at the origin are $0$.
On the line $y = x$ the value is $1/2$, so $f$ is not continuous at the origin, as in the [[separately continuous function|0C0KPAKK9Y]]. A function that is differentiable at a point is continuous there.
Even all directional derivatives are not enough. For $g(x, y) = x^2 y / (x^4 + y^2)$ off the origin and $g(0, 0) = 0$, each directional derivative at the origin exists, but $g = 1/2$ on the parabola $y = x^2$ minus the origin.
If the partial derivatives exist near the point and are continuous at it, then the function is differentiable, by the [[criterion with continuous partial derivatives|0KXSK8TEEB]].

## Explanations

### L1 human

The partial derivatives look only along the two axes.
A function can be flat on both axes and still do something wild between them, here on the diagonal.
Differentiability asks for a good linear approximation in all directions at once, which is much more.
</document>

<document source="https://math2.tamnd.com/node/0TP2QKJ8DE.md" tag="0TP2QKJ8DE" kind="definition">
# Circumcentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be the plane through $A$, $B$ and $C$, their [[affine span|0YMB1676R1]].
The circumcentre of $ABC$ is the unique point $O$ of $P$ with
$$d(O, A) = d(O, B) = d(O, C).$$
This common distance $R$ is the circumradius of $ABC$.

## Remarks

The point $O$ exists and is unique. In the plane, it is the common point of the three [[perpendicular bisectors|055VNYT0PB]] of the sides. Two of them meet in one point because the sides are not parallel.
The circumcentre is inside the triangle if the triangle is acute, on the longest side if it is right, and outside if it is obtuse.
In a right triangle, the circumcentre is the midpoint of the hypotenuse, by [[Thales's theorem|0GSDHYMA1T]].
Mathlib defines the circumcentre of each simplex in the same way, inside the affine span of the vertices.

## Explanations

### L1 human

The circumcentre is the point that is equally far from the three corners of a triangle. To find it, draw the perpendicular bisectors of two sides; they cross at the circumcentre. A circle with this centre through one corner passes through all three.
</document>

<document source="https://math2.tamnd.com/node/0TPW25CBN6.md" tag="0TPW25CBN6" kind="definition">
# Congruence modulo n

## Definition

Let $n$ be a positive integer, and let $a$ and $b$ be integers.
The integer $a$ is congruent to $b$ modulo $n$ when $n$ [[divides|096PW7Q014]] $a - b$.
The notation is $a \equiv b \pmod{n}$. The integer $n$ is the modulus of the congruence.

## Remarks

Equivalently, $a \equiv b \pmod n$ if and only if $a$ and $b$ have the same remainder on division by $n$, that is, $a \bmod n = b \bmod n$.
Congruence modulo $n$ is an equivalence relation on the integers. Its classes are the [[residue classes|0QRV8FPK5M]] modulo $n$.
For example, $38 \equiv 14 \pmod{12}$, because $12 \mid 24$.
Some texts allow each integer $n$ as a modulus. Then congruence modulo $0$ is equality, and congruence modulo $-n$ is congruence modulo $n$.

## Explanations

### L1 human

Two integers are congruent modulo $n$ when they differ by a multiple of $n$.
On a $12$-hour clock, $14$ o'clock and $2$ o'clock show the same time, because $14 \equiv 2 \pmod{12}$.
</document>

<document source="https://math2.tamnd.com/node/0TPWS0K1BC.md" tag="0TPWS0K1BC" kind="statement">
# An algebraic extension of an algebraic extension is algebraic

## Statement

Let $K \subseteq L \subseteq M$ be fields.
If $M/L$ and $L/K$ are [[algebraic|09EYMKMSB9]], then $M/K$ is algebraic.

## Remarks

The proof: let $a \in M$ have minimal polynomial $x^n + c_{n-1} x^{n-1} + \dots + c_0$ over $L$. The field $E = K(c_0, \dots, c_{n-1})$ is finite over $K$, and $E(a)$ is finite over $E$. By the [[tower law|0A7KJ68XHT]], $E(a)/K$ is finite, so $a$ is algebraic over $K$.
Also, if $M/K$ is algebraic, then $M/L$ and $L/K$ are algebraic.

## Explanations

### L1 human

Algebraic extensions can be stacked.
If each step adds only algebraic elements, then the whole tower adds only algebraic elements.
</document>

<document source="https://math2.tamnd.com/node/0TR9SR9FTB.md" tag="0TR9SR9FTB" kind="definition">
# Strictly monotone function

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is strictly increasing if $f(x) < f(y)$ for all $x, y \in D$ with $x < y$.
It is strictly decreasing if $f(x) > f(y)$ for all $x, y \in D$ with $x < y$.
It is strictly monotone if it is strictly increasing or strictly decreasing.

## Remarks

With $\le$ and $\ge$ in place of $<$ and $>$, we get increasing, decreasing, and monotone functions.
A strictly monotone function is injective.
For example, $x^3$ is strictly increasing on $\mathbb{R}$, and $x^2$ is strictly increasing on $[0, \infty)$.
Mathlib defines strictly increasing maps as `StrictMono`.

## Explanations

### L1 human

A strictly increasing function always goes up: a larger input gives a larger output.
Its graph rises from left to right, with no flat parts.
</document>

<document source="https://math2.tamnd.com/node/0TRNT4X7J0.md" tag="0TRNT4X7J0" kind="concept">
# Euclidean algorithm

## Summary

The Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] of two natural numbers $a$ and $b$.
It uses two rules: $\gcd(a, 0) = a$, and $\gcd(a, b) = \gcd(b, a \bmod b)$ when $b \neq 0$.
It applies the second rule until the second number is $0$. Then the first number is the result.
The algorithm stops, because the second number gets smaller at each step.

## Remarks

The second rule is true because $a$ and $b$ have the same common divisors as $b$ and $a \bmod b$.
The number of steps is at most a constant times the number of digits of the smaller input. Lamé's theorem gives this bound.
The algorithm also works in each Euclidean domain, for example in the ring of polynomials over a field.

## Explanations

### L1 human

To find $\gcd(48, 18)$, divide again and again, and keep the divisor and the remainder.
$48 = 2 \cdot 18 + 12$, then $18 = 1 \cdot 12 + 6$, then $12 = 2 \cdot 6 + 0$.
The last nonzero remainder is $6$, so $\gcd(48, 18) = 6$.
</document>

<document source="https://math2.tamnd.com/node/0TS7VZJBWD.md" tag="0TS7VZJBWD" kind="definition">
# Orthogonal vectors

## Definition

Let $V$ be an inner product space over $\R$ or $\C$.
Two vectors $x, y \in V$ are *orthogonal*, written $x \perp y$, when $\langle x, y \rangle = 0$, where $\langle \cdot, \cdot \rangle$ is the [[inner product|0NB0AWGJBW]].

## Remarks

The relation is symmetric, because $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
The zero vector is orthogonal to every vector, and it is the only vector orthogonal to itself.
In $\R^2$ with the dot product, $(1, 2)$ and $(-2, 1)$ are orthogonal.

## Explanations

### L1 human

Orthogonal is the general word for perpendicular.
Two vectors are orthogonal when their inner product is zero.
In the plane and in space, this means that they meet at a right angle.
</document>

<document source="https://math2.tamnd.com/node/0TST4MHA3A.md" tag="0TST4MHA3A" kind="definition">
# Law of a random variable

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $(E, \mathcal{E})$ be a set with a σ-algebra, and let $X \colon \Omega \to E$ be a [[random variable|0DKRR22AEQ]].
The law of $X$, also called the distribution of $X$, is the function $P_X \colon \mathcal{E} \to [0, 1]$ with
$$P_X(B) = P(X^{-1}(B)) = P(X \in B) \quad \text{for } B \in \mathcal{E}.$$

## Remarks

$P_X$ is a [[probability distribution|0QB913WM5E]] on $(E, \mathcal{E})$. It is the image measure, or pushforward, of $P$ under $X$, and one also writes $P \circ X^{-1}$ or $X_* P$.
Notation: $X \sim \mu$ means that the law of $X$ is $\mu$.
Two random variables can have the same law but be different functions. For a fair coin toss, the indicator of heads and the indicator of tails both have the Bernoulli law with parameter $1/2$.

## Explanations

### L1 human

The law of $X$ forgets the sample space and keeps only the chances of the values of $X$.
For the sum of two dice, the law says that $7$ has probability $6/36$, $2$ has probability $1/36$, and so on.
</document>

<document source="https://math2.tamnd.com/node/0TTA57SEJG.md" tag="0TTA57SEJG" kind="definition">
# Convergence in mean

## Definition

Let $r \ge 1$ be real. Let $X, X_1, X_2, \dots$ be real random variables on one probability space with $E[|X|^r] < \infty$ and $E[|X_n|^r] < \infty$ for each $n$.
The sequence $(X_n)$ converges in $r$-th mean, or in $L^r$, to $X$ if
$$\lim_{n \to \infty} E\left[|X_n - X|^r\right] = 0.$$
For $r = 1$ one says convergence in mean, and for $r = 2$ convergence in mean square.

## Remarks

By [[Markov's inequality|05PWWHYBX5]], $P(|X_n - X| > \varepsilon) \le E[|X_n - X|^r] / \varepsilon^r$. So convergence in $r$-th mean implies convergence in probability.
For $s \ge r \ge 1$, convergence in $s$-th mean implies convergence in $r$-th mean.
Convergence in mean does not imply almost sure convergence, and almost sure convergence does not imply convergence in mean.
Example: on $[0, 1]$ with the uniform measure, $X_n = n \, 1_{[0, 1/n]}$ converges to $0$ almost surely but $E[X_n] = 1$ for all $n$.

## Explanations

### L1 human

Convergence in mean asks that the average size of the error $|X_n - X|^r$ goes to $0$.
Mean square convergence, with $r = 2$, is common in statistics, where it says that the mean squared error of an estimator goes to $0$.
</document>

<document source="https://math2.tamnd.com/node/0TTDGTEFMN.md" tag="0TTDGTEFMN" kind="concept">
# Conditional expectation

## Summary

The conditional expectation of a random variable given some information is its best estimate when one knows that information. It is itself a random variable.
The node [[conditional expectation|0YAMJRK04E]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0TTNRP5PFQ.md" tag="0TTNRP5PFQ" kind="definition">
# Estimator

## Definition

Let $\Theta$ be a set of parameters, and for each $\theta \in \Theta$ let $P_\theta$ be a probability distribution. Let $X_1, \dots, X_n$ be a sample whose joint law is determined by an unknown $\theta \in \Theta$. Let $\tau \colon \Theta \to \mathbb{R}^k$ be a function.
An estimator of $\tau(\theta)$ is a [[statistic|036756NFD6]] $\hat{\tau} = g(X_1, \dots, X_n)$ with values in $\mathbb{R}^k$, where $g$ does not depend on $\theta$.
The value of $\hat{\tau}$ for observed data is an estimate of $\tau(\theta)$.

## Remarks

The definition does not ask that $\hat{\tau}$ is close to $\tau(\theta)$. One measures the quality of an estimator with its [[bias|01BD2TFFP4]], its variance and its [[mean squared error|0V4VSCBXP3]].
For an i.i.d. sample, the sample mean is an estimator of the mean, and the sample variance is an estimator of the variance.
One often writes $\hat{\theta}$ for an estimator of $\theta$ itself.

## Explanations

### L1 human

An estimator is a recipe that turns data into a guess of an unknown quantity.
For example, the fraction of "yes" answers in a poll is an estimator of the fraction of "yes" voters in the whole population.
</document>

<document source="https://math2.tamnd.com/node/0TV07AQT62.md" tag="0TV07AQT62" kind="definition">
# Real exponentiation

## Definition

Let $a > 0$ and let $x$ be a real number.
The power $a^x$ is
$$a^x = e^{x \ln a}.$$

## Remarks

For natural numbers $x = n$, this agrees with the product of $n$ factors $a$. For $x = p/q$, it agrees with $\sqrt[q]{a^p}$.
The usual rules hold: $a^{x + y} = a^x a^y$, $(a^x)^y = a^{x y}$, and $(a b)^x = a^x b^x$.
For $a = e$, it gives $e^x = \exp(x)$, since $\ln e = 1$. Mathlib defines `Real.rpow` with extra conventions for $a \le 0$.

## Explanations

### L1 human

To raise $a$ to a real power $x$, multiply the logarithm of $a$ by $x$ and apply the exponential function.
This fills in all powers between the rational ones continuously.
</document>

<document source="https://math2.tamnd.com/node/0TVJZYB6SH.md" tag="0TVJZYB6SH" kind="concept">
# Compact space

## Summary

A compact space is a topological space in which each open cover has a finite subcover.
Closed bounded intervals of the real line are the basic examples.
</document>

<document source="https://math2.tamnd.com/node/0TWPTQ0AD7.md" tag="0TWPTQ0AD7" kind="definition">
# Incentre

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. For a point $X$ and a line $\ell$, let $d(X, \ell)$ be the [[distance from the point to the line|01AAYNSBN6]].
The incentre of $ABC$ is the unique point $I$ in the interior of the triangle with
$$d(I, BC) = d(I, CA) = d(I, AB).$$
This common distance $r$ is the inradius of $ABC$.

## Remarks

The point $I$ lies on each of the three internal angle bisectors of the triangle. Thus the three bisectors meet in one point.
Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be the side lengths, with $d$ the [[Euclidean distance|0NPPGHQNXQ]]. Then $I = \dfrac{aA + bB + cC}{a + b + c}$.
Outside the triangle there are three more points at the same distance from the three side lines. They are the excentres.
Mathlib's `Affine.Simplex.incenter` defines the incentre of each simplex in a Euclidean affine space.

## Explanations

### L1 human

The incentre is the point inside a triangle that is equally far from the three sides. You find it as the crossing point of two angle bisectors. For the right triangle with the vertices $(0, 0)$, $(3, 0)$ and $(0, 4)$, the incentre is $(1, 1)$.
</document>

<document source="https://math2.tamnd.com/node/0TWQ5PRKWS.md" tag="0TWQ5PRKWS" kind="definition">
# Equivalent metrics

## Definition

Let $X$ be a set and let $d_1$ and $d_2$ be metrics on $X$.
The metrics $d_1$ and $d_2$ are equivalent when they have the same [[open sets|01ZQQDCERP]], that is, when they give the same [[metric topology|0A6YRNZB2R]] on $X$.
The metrics are strongly equivalent when there are real numbers $a > 0$ and $b > 0$ with
$$a \, d_1(x, y) \le d_2(x, y) \le b \, d_1(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

Strongly equivalent metrics are equivalent: each ball for one metric contains a ball with the same centre for the other metric.
On $\mathbb{R}^n$, the [[taxicab metric|07CJNE6ZAQ]], the [[Euclidean metric|0E1MMCAY5B]] and the [[sup metric|0VTAXQEWQD]] are strongly equivalent.
On $\mathbb{R}$, the metrics $|x - y|$ and $\min(|x - y|, 1)$ are equivalent but not strongly equivalent, because only the second one is bounded.
Equivalent metrics have the same convergent sequences and the same continuous maps. Completeness can change: see the Remarks of [[complete metric space|08EWT7J0SC]].

## Explanations

### L1 human

Equivalent metrics can give different numbers for distances, but they agree on which points are near each other.
For example, on the plane, the taxicab distance is never more than $\sqrt{2}$ times the straight distance, and never less than it.
</document>

<document source="https://math2.tamnd.com/node/0TWTS88TWN.md" tag="0TWTS88TWN" kind="definition">
# Invertible matrix

## Definition

Let $R$ be a commutative ring and let $A$ be an $n \times n$ [[square matrix|0JXZKX526R]] over $R$.
The matrix $A$ is *invertible* when there is an $n \times n$ matrix $B$ over $R$ such that
$$AB = BA = I_n,$$
where $I_n$ is the [[identity matrix|06DW9BAWB1]].
The matrix $B$ is then unique; it is the *inverse* of $A$, written $A^{-1}$.

## Remarks

Over a field, other names are *nonsingular* and *nondegenerate*.
A matrix over $R$ is invertible if and only if its [[determinant is a unit|0783S9D0ZH]] of $R$.
Over a field, a [[one-sided inverse is two-sided|0753Y2X5K6]], and $A$ is invertible if and only if $Ax = 0$ has only the solution $x = 0$.
The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not invertible over $\R$, because its determinant is $0$.

## Explanations

### L1 human

A matrix is invertible when another matrix undoes it.
If $A$ sends $x$ to $y$, then $A^{-1}$ sends $y$ back to $x$.
For $2 \times 2$ matrices, $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ when $ad - bc \ne 0$.
</document>

<document source="https://math2.tamnd.com/node/0TX2YH2FQS.md" tag="0TX2YH2FQS" kind="definition">
# Symmetric bilinear form associated with a quadratic form

## Definition

Let $F$ be a field of characteristic not $2$, and let $V$ be a finite-dimensional vector space over $F$.
Let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
The *symmetric bilinear form associated with* $Q$ is
$$B(u, v) = \tfrac{1}{2} \bigl( Q(u + v) - Q(u) - Q(v) \bigr) \qquad (u, v \in V).$$

## Remarks

$B$ is a [[symmetric bilinear form|0AXZNE5XNH]] with $B(v, v) = Q(v)$, and it is the only one.
Conversely, each symmetric bilinear form $B$ gives the quadratic form $v \mapsto B(v, v)$; the two constructions are inverse to each other.
For $Q(x) = x^T A x$ with $A$ symmetric, $B(x, y) = x^T A y$.
It is half of the [[polar form|0HYFWKKY3C]] of $Q$.

## Explanations

### L1 human

Outside characteristic $2$, quadratic forms and symmetric bilinear forms are the same thing in two forms.
From $B$, we get $Q$ by putting the same vector twice.
From $Q$, we get $B$ by this formula.
</document>

<document source="https://math2.tamnd.com/node/0TXCQ9CST1.md" tag="0TXCQ9CST1" kind="concept">
# Integration over a non-rectangular region

## Summary

To integrate a function $f$ over a bounded region $D$ that is not a rectangle, take a rectangle $R \supseteq D$. Then integrate over $R$ the function equal to $f$ on $D$ and to $0$ on $R \setminus D$.
For a region between two graphs, $D = \{(x, y) : a \le x \le b, \ \varphi_1(x) \le y \le \varphi_2(x)\}$ with $\varphi_1 \le \varphi_2$ continuous, a continuous $f$ gives $\iint_D f \, dA = \int_a^b \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) \, dy \, dx$.

## Remarks

Such a region is normal with respect to the $x$-axis. A region of the form $\{c \le y \le d, \ \psi_1(y) \le x \le \psi_2(y)\}$ is normal with respect to the $y$-axis.
The definition does not depend on the choice of $R$. For continuous $f$, the integral exists when the boundary of $D$ has [[Jordan content|0KTGJW1ZTH]] zero.
Example: for $D = \{0 \le x \le 1, \ x^2 \le y \le 1\}$, $\iint_D x \, dA = \int_0^1 x (1 - x^2) \, dx = 1/4$.
The [[double integral|0WAR4F23B4]] over $D$ is computed as an [[iterated integral|0WZCHE1ZF9]] with variable bounds.

## Explanations

### L1 human

Most regions in practice are not rectangles, but many lie between two curves.
Then the inner integral runs from the lower curve to the upper curve, and the outer integral runs over the range of the other variable.
</document>

<document source="https://math2.tamnd.com/node/0TXNRQZY0H.md" tag="0TXNRQZY0H" kind="concept">
# Primitive polynomial

## Summary

A primitive polynomial is a polynomial whose coefficients have no common factor except units.
The [[definition of a primitive polynomial|0SP14DZN0G]] gives the condition.
</document>

<document source="https://math2.tamnd.com/node/0TXTSEMHR5.md" tag="0TXTSEMHR5" kind="definition">
# Limit of a function

## Definition

Let $D$ be a subset of $\mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function, and let $a$ be a [[limit point|07EGCNHAVW]] of $D$.
A real number $L$ is the limit of $f$ at $a$ if the following holds.
For each open interval $V$ containing $L$ there is an open interval $U$ containing $a$ with $f(x) \in V$ for all $x \in U \cap D$ with $x \ne a$.
The notation is $\lim_{x \to a} f(x) = L$, or $f(x) \to L$ as $x \to a$.

## Remarks

The value of $f$ at $a$ plays no role, and $a$ need not lie in $D$.
The [[epsilon-delta definition|0GBZDAFPX8]] states the same condition with numbers. The [[sequential characterization|000MXHEY22]] states it with sequences.
The limit is unique when it exists, since $a$ is a limit point of $D$.
Mathlib expresses limits with filters, as `Filter.Tendsto`.

## Explanations

### L1 human

The limit of $f$ at $a$ is the value that $f(x)$ approaches as $x$ comes close to $a$.
Only the values near $a$ matter, not the value at $a$ itself.
</document>

<document source="https://math2.tamnd.com/node/0TXWV1JGG1.md" tag="0TXWV1JGG1" kind="statement">
# Product of a subgroup and a normal subgroup is a subgroup

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
Then the [[product|0PT52XQZD7]] $HN$ is a subgroup of $G$, and $HN = NH$.

## Remarks

If $H$ is also normal, then $HN$ is normal in $G$.
The proof uses $hN = Nh$ for each $h \in H$, so that $HN = \bigcup_{h \in H} hN = \bigcup_{h \in H} Nh = NH$.
This product appears in the second isomorphism theorem.

## Explanations

### L1 human

The product of two subgroups can fail to be a subgroup, but not when one of them is normal.
A normal subgroup lets you move its elements past other elements, at the cost of changing them inside the subgroup.
So products of the form $hn$ stay of that form when you multiply them.
</document>

<document source="https://math2.tamnd.com/node/0TYB3HM9KE.md" tag="0TYB3HM9KE" kind="definition">
# Isolated singularity

## Definition

Let $c \in \mathbb{C}$ and let $f$ be a complex function.
The point $c$ is an isolated singularity of $f$ if there is $r > 0$ such that $f$ is defined and [[holomorphic|0DG0Z72CQD]] on the punctured disc $\{z : 0 < |z - c| < r\}$.
The value of $f$ at $c$, if there is one, plays no role.

## Remarks

Each isolated singularity is of exactly one of three types: [[removable|0P18JC72RE]], a [[pole|0WPQYW8MTE]], or [[essential|0YKDJBV26S]].
For example, $0$ is an isolated singularity of $1/z$, of $\frac{\sin z}{z}$ and of $e^{1/z}$.
The point $0$ is not an isolated singularity of $\frac{1}{\sin(1/z)}$, because the poles $\frac{1}{k\pi}$ accumulate at $0$. It is not an isolated singularity of $\operatorname{Log} z$ either, because $\operatorname{Log}$ is not holomorphic on any punctured disc around $0$.

## Explanations

### L1 human

An isolated singularity is a single bad point of a function, with only good points around it.
The function is holomorphic on a small disc around the point, with only the center removed.
</document>

<document source="https://math2.tamnd.com/node/0TYMASG33Y.md" tag="0TYMASG33Y" kind="statement">
# Characteristic polynomial does not depend on the basis

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $P$ be an invertible $n \times n$ matrix over $K$.
Then the [[characteristic polynomials|0VSSPAFM2C]] of $A$ and $P^{-1} A P$ are equal:
$$p_{P^{-1} A P}(t) = p_A(t).$$
So [[similar matrices|0E89HQATM8]] have the same characteristic polynomial.

## Remarks

The proof is $\det(t I - P^{-1} A P) = \det(P^{-1} (t I - A) P) = \det(t I - A)$.
Hence the characteristic polynomial of a linear map does not depend on the basis.
The converse fails: $I_2$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ both have $p(t) = (t - 1)^2$, but they are not similar.

## Explanations

### L1 human

A change of basis does not change the characteristic polynomial.
So the eigenvalues, the trace and the determinant, which we can read from it, are the same in every basis.
They are properties of the linear map, not of one matrix.
</document>

<document source="https://math2.tamnd.com/node/0TYQNHVB4C.md" tag="0TYQNHVB4C" kind="statement">
# Surjective iff it has a right inverse

## Statement

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
Assume the [[axiom of choice|0HTGB13FE2]].
Then $f$ is [[surjective|0XKH881BZ7]] if and only if $f$ has a [[right inverse|0DDPZAG5ER]].

## Remarks

If $f \circ h = \mathrm{id}_B$, then each $b$ equals $f(h(b))$, so $f$ is surjective. This direction does not use choice.
For the other direction, choose for each $b \in B$ an element $h(b)$ of the nonempty set $f^{-1}(\{b\})$.
The statement "each surjection has a right inverse" is equivalent to the axiom of choice.

## Explanations

### L1 human

For a surjection, each $b$ has at least one input that maps to it.
A right inverse picks one such input for each $b$.
For infinitely many $b$ with no rule to pick, you need the axiom of choice.
</document>

<document source="https://math2.tamnd.com/node/0TZ2DPT0JV.md" tag="0TZ2DPT0JV" kind="concept">
# Principal part

## Summary

The principal part of a function at an isolated singularity is the part of its Laurent series with the negative powers, see the [[definition|0NDV5R6Z6X]].
</document>

<document source="https://math2.tamnd.com/node/0TZAG60ZYT.md" tag="0TZAG60ZYT" kind="statement">
# Row rank equals column rank

## Statement

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$.
Then the dimension of the [[row space|069MCGX635]] of $A$ equals the dimension of the [[column space|03AJKXH1AV]] of $A$.
Equivalently, $\rank A^T = \rank A$, where $A^T$ is the [[transpose|09DXWYRD43]].

## Remarks

One proof uses row reduction: row operations keep the row space and the dimension of the column space.
In reduced row echelon form, both dimensions equal the number of pivots.
Another proof uses the [[annihilator|0ASHNYF7AG]]: the dual map has the transposed matrix and the same rank.
Over $\C$, also $\rank A^* = \rank A$.

## Explanations

### L1 human

The number of independent rows of a matrix always equals the number of independent columns.
This is surprising, because rows and columns live in different spaces.
For a $3 \times 5$ matrix, it shows at once that the rank is at most $3$.
</document>

<document source="https://math2.tamnd.com/node/0TZEEFTNCC.md" tag="0TZEEFTNCC" kind="definition">
# Maximal element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $m \in S$ is a maximal element of $S$ when no $s \in S$ satisfies $m < s$.
Equivalently, for each $s \in S$, $m \le s$ implies $s = m$.

## Remarks

A maximal element need not be the [[greatest element|0Z7HDDXX3X]], and a set can have several maximal elements.
In $\{\{1\}, \{2\}, \{1, 2, 3\}, \{4\}\}$ under $\subseteq$, the maximal elements are $\{1, 2, 3\}$ and $\{4\}$.
[[Zorn's lemma|04RXTQA5ZA]] gives maximal elements in many infinite posets.

## Explanations

### L1 human

A maximal element has nothing above it.
It may still be incomparable with other elements, so it is not necessarily above everything.
</document>

<document source="https://math2.tamnd.com/node/0TZK91NQE3.md" tag="0TZK91NQE3" kind="example">
# The sequence r^n tends to infinity for r > 1

## Example

Let $r$ be a real number with $r > 1$.
Then $r^n \to +\infty$ as $n \to \infty$, in the sense of a [[sequence tending to infinity|09AZCV6GMC]].
Write $r = 1 + h$ with $h > 0$. By Bernoulli's inequality, $r^n \ge 1 + n h$ for each natural number $n$.
Let $M$ be a real number, and choose a natural number $N$ with $N h > M$. Then $r^n > M$ for all $n \ge N$.

## Remarks

For $r < -1$ the sequence $r^n$ does not tend to $+\infty$ or to $-\infty$. Its terms change sign, but $|r^n| \to +\infty$.
Exponential growth is faster than polynomial growth. The node [[exponentials dominate powers in sequences|0MVTCFW42K]] makes this precise.

## Explanations

### L1 human

Doubling again and again gives $1, 2, 4, 8, 16, \dots$, and these numbers grow without bound.
Each factor $r > 1$ does the same, but a factor close to $1$ needs more steps.
</document>

<document source="https://math2.tamnd.com/node/0V032VWTHD.md" tag="0V032VWTHD" kind="concept">
# Connected space

## Summary

A topological space is connected when it is not the union of two disjoint nonempty open sets.
A connected space is in one piece.
</document>

<document source="https://math2.tamnd.com/node/0V05FC4GKQ.md" tag="0V05FC4GKQ" kind="definition">
# Induced subgraph

## Definition

Let $G = (V, E)$ be a [[graph|0JD5C82QMB]] and let $S \subseteq V$.
The subgraph induced by $S$ is the graph $G[S]$ with vertex set $S$ and edge set
$$\{ \{u, v\} \in E : u, v \in S \}.$$
A subgraph of this form is an induced subgraph.

## Remarks

An induced subgraph is a [[subgraph|0P11X5HKAX]] that keeps every edge of $G$ between its vertices.
In $K_4$, every subgraph induced by $3$ vertices is a triangle. A path on these $3$ vertices is a subgraph, but it is not induced.
A [[clique|0G52BPNPKG]] induces a complete graph, and an [[independent set|0KZJ82WEPP]] induces a graph with no edges.

## Explanations

### L1 human

Choose some vertices and keep all the edges between them.
The result is the induced subgraph on those vertices.
</document>

<document source="https://math2.tamnd.com/node/0V0778DCW9.md" tag="0V0778DCW9" kind="statement">
# Variance of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $E[X_i^2] < \infty$ for each $i$. Then
$$\operatorname{Var}(X_1 + \dots + X_n) = \operatorname{Var}(X_1) + \dots + \operatorname{Var}(X_n).$$

## Remarks

The proof uses the [[variance of a sum|0D86GJEZ39]]: independent variables are [[uncorrelated|077GN9HDR6]], so each covariance term is $0$.
So the formula holds more generally for pairwise uncorrelated $X_i$.
For i.i.d. $X_i$ with variance $\sigma^2$, the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$ has variance $\sigma^2 / n$.
For $X \sim \operatorname{Bin}(n, p)$, the formula gives $\operatorname{Var}(X) = np(1 - p)$, as $X$ is a sum of $n$ independent Bernoulli variables.

## History

The formula is also called the Bienaymé formula, after Irénée-Jules Bienaymé.

## Explanations

### L1 human

For independent variables, variances simply add.
This is why an average of many independent measurements is more precise than one measurement: its variance gets smaller like $1/n$.
</document>

<document source="https://math2.tamnd.com/node/0V0E04DWX2.md" tag="0V0E04DWX2" kind="statement">
# Sign of a cycle

## Statement

Let $k \geq 1$ and let $\sigma$ be a $k$-cycle in $S_n$.
Then $\operatorname{sgn}(\sigma) = (-1)^{k-1}$.

## Remarks

A $k$-cycle is a product of $k - 1$ transpositions, which gives the formula.
Thus a cycle is an [[even permutation|0SQTMQTSRZ]] exactly when its length is odd.
With the [[disjoint cycle decomposition|0YWDAZPRPQ]], the sign of $\sigma \in S_n$ is $(-1)^{n - c}$, where $c$ is the number of cycles, with the fixed points counted as $1$-cycles.

## Explanations

### L1 human

A cycle of length $k$ is a chain of $k - 1$ swaps.
So a $3$-cycle is even and a transposition is odd.
The length and the parity of a cycle are opposite.
</document>

<document source="https://math2.tamnd.com/node/0V0GC7BQAN.md" tag="0V0GC7BQAN" kind="example">
# Complete bipartite graph

## Example

Let $m$ and $n$ be natural numbers.
The complete bipartite graph $K_{m,n}$ is the [[graph|0JD5C82QMB]] whose vertex set is a disjoint union $A \sqcup B$ with $|A| = m$ and $|B| = n$.
Two vertices are adjacent if and only if one is in $A$ and the other is in $B$.

## Remarks

$K_{m,n}$ has $m + n$ vertices and $mn$ edges.
It is [[bipartite|07DVHJ8N0A]] with parts $A$ and $B$. For $m, n \ge 1$ its [[chromatic number|0FJBBCXYPS]] is $2$.
$K_{1,n}$ is a star. [[$K_{3,3}$ is not planar|030A3FHN14]].

## Explanations

### L1 human

Split the vertices into two groups and join every vertex of one group to every vertex of the other group.
There are no edges inside a group.
</document>

<document source="https://math2.tamnd.com/node/0V11FEXK08.md" tag="0V11FEXK08" kind="statement">
# Frobenius coin problem

## Statement

Let $a$ and $b$ be [[coprime|0GFKP8F8AG]] integers with $a > 1$ and $b > 1$.
Then $ab - a - b$ is not of the form $ax + by$ with natural numbers $x$ and $y$.
Every integer $n > ab - a - b$ is of this form.

## Remarks

The number $ab - a - b$ is the Frobenius number of $a$ and $b$: the largest amount that one cannot pay with coins of values $a$ and $b$.
For example, with coins of $3$ and $5$, the amount $7$ is impossible, but $8$, $9$, $10$ and all larger amounts are possible.
Exactly $(a - 1)(b - 1)/2$ positive integers are not of the form $ax + by$.
For three or more coin values, no such simple formula is known.

## History

James Joseph Sylvester found the formula for two coins in 1882.

## Explanations

### L1 human

With coins of values $a$ and $b$, you can pay all large amounts when $a$ and $b$ are coprime.
The theorem gives the largest amount that you cannot pay: $ab - a - b$.
</document>

<document source="https://math2.tamnd.com/node/0V1EJCT8G6.md" tag="0V1EJCT8G6" kind="concept">
# Derivative

## Summary

The derivative of a function at a point is the limit of the slopes of secant lines through that point. It measures the rate of change of the function, and it is the slope of the tangent line. The node [[derivative|0F1WR9C83K]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0V2DV29EGT.md" tag="0V2DV29EGT" kind="statement">
# Derivative of cosine

## Statement

The cosine function is differentiable on $\mathbb{R}$, and $\cos'(x) = -\sin x$ for all real $x$.

## Remarks

[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\frac{(-1)^n x^{2n}}{(2n)!}$ is $-\frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!}$ for $n \ge 1$.
So $\sin'' = -\sin$ and $\cos'' = -\cos$. Both functions solve $f'' + f = 0$.
Mathlib states it as `Real.hasDerivAt_cos`.

## Explanations

### L1 human

The slope of the cosine curve at $x$ is $-\sin x$.
The cosine curve falls where the sine is positive.
</document>

<document source="https://math2.tamnd.com/node/0V2H5PGJ9B.md" tag="0V2H5PGJ9B" kind="object">
# Aleph-null

## Description

Aleph-null, written $\aleph_0$, is the [[cardinality|0JTGRV4WT4]] of the set $\mathbb{N}$ of natural numbers.
It is the cardinality of each [[countably infinite set|0SEZ3JGZP7]].
It is the least infinite [[cardinal number|0QKPNZKNEA]]: each infinite set has cardinality at least $\aleph_0$, with the axiom of choice.

## Remarks

Other names are aleph-zero and aleph-nought.
$\aleph_0 + \aleph_0 = \aleph_0 \cdot \aleph_0 = \aleph_0$, but $2^{\aleph_0} > \aleph_0$ by [[Cantor's theorem|0WTEMGYV2C]].
As an ordinal, $\aleph_0$ is $\omega$, the set of the finite ordinals.

## Explanations

### L1 human

$\aleph_0$ is the size of the infinity of counting.
The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ all have this size.
</document>

<document source="https://math2.tamnd.com/node/0V2WP9B1MF.md" tag="0V2WP9B1MF" kind="concept">
# Norm

## Summary

A norm on a real or complex vector space is a function that gives each vector a length.
It satisfies the triangle inequality and absolute homogeneity, and only the zero vector has length zero.
</document>

<document source="https://math2.tamnd.com/node/0V33BNKZ9S.md" tag="0V33BNKZ9S" kind="concept">
# Smooth function

## Summary

A smooth function is a function with continuous derivatives of all orders. Smoothness is the top of the scale of differentiability classes $C^k$. The node [[smooth function|0BE3QDH73A]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0V33SPH7RR.md" tag="0V33SPH7RR" kind="statement">
# Intersection of subspaces

## Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$.
Then $U \cap W$ is a linear subspace of $V$.
It is the largest subspace of $V$ that is contained in both $U$ and $W$.
More generally, the intersection of each nonempty family of subspaces of $V$ is a subspace of $V$.

## Remarks

The union $U \cup W$ is a subspace only when $U \subseteq W$ or $W \subseteq U$.
The smallest subspace that contains $U \cup W$ is the [[sum|0JMTY8MF54]] $U + W$.
In $\R^3$, two different planes through the origin meet in a line through the origin.

## Explanations

### L1 human

A vector that lies in two subspaces stays in both of them when we add or scale.
So the common part of two subspaces is again a subspace.
Two planes through the origin in space cross in a line, which is a subspace.
</document>

<document source="https://math2.tamnd.com/node/0V3CFG1KK9.md" tag="0V3CFG1KK9" kind="definition">
# Semigroup

## Definition

A semigroup is a pair $(S, \ast)$, where $S$ is a set and $\ast$ is a [[binary operation|0M418FE7BB]] on $S$ that is [[associative|0K8F7PVDMX]].
Thus $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in S$.

## Remarks

The positive integers with addition form a semigroup that has no identity element.
Each [[monoid|0N6R902FG1]] and each [[group|G5S1M9]] is a semigroup.
Some authors allow the empty set as a semigroup, and others do not.

## Explanations

### L1 human

A semigroup is the simplest structure with an associative operation.
You can combine elements and the grouping does not matter, but maybe nothing acts as "do nothing" and no step can be undone.
An example is the set of nonempty words over an alphabet, with the operation that writes one word after the other.
</document>

<document source="https://math2.tamnd.com/node/0V3FZMVB4K.md" tag="0V3FZMVB4K" kind="definition">
# Gamma function

## Definition

For a real number $s > 0$, the gamma function is
$$\Gamma(s) = \int_0^\infty t^{s-1} e^{-t} \, dt.$$

## Remarks

The [[improper integral|0VCTBJMEJF]] converges at $0$ because $s > 0$, and at $\infty$ because $e^{-t}$ decays fast. Both claims follow from the [[comparison test|0FR1BDKK10]].
Integration by parts gives $\Gamma(s + 1) = s \, \Gamma(s)$. Also $\Gamma(1) = 1$, so [[$\Gamma$ extends the factorial|0FDCW77H44]].
The function extends to all real numbers except $0, -1, -2, \dots$, by the recursion. Mathlib defines this extension as `Real.Gamma`.
The notation $\Gamma$ is due to Adrien-Marie Legendre.

## Explanations

### L1 human

The gamma function is an integral that depends on a parameter $s$.
At positive integers it gives factorials, so it interpolates the factorial.
</document>

<document source="https://math2.tamnd.com/node/0V3RDQV3ZX.md" tag="0V3RDQV3ZX" kind="definition">
# Cauchy principal value

## Definition

Let $a < b < c$ be [[real numbers|08J2M1JYAG]].
Let $f$ be a function on $[a, c] \setminus \{b\}$ that is integrable on $[a, b - \varepsilon]$ and on $[b + \varepsilon, c]$ for each $\varepsilon$ with $0 < \varepsilon < \min(b - a, c - b)$.
If the limit
$$\operatorname{p.v.} \int_a^c f(x) \, dx = \lim_{\varepsilon \to 0^+} \left( \int_a^{b - \varepsilon} f(x) \, dx + \int_{b + \varepsilon}^{c} f(x) \, dx \right)$$
exists, it is the Cauchy principal value of the integral of $f$ over $[a, c]$.
Let $f$ be a function on $\mathbb{R}$ that is integrable on each interval $[-R, R]$.
If the limit $\operatorname{p.v.} \int_{-\infty}^{\infty} f(x) \, dx = \lim_{R \to \infty} \int_{-R}^{R} f(x) \, dx$ exists, it is the Cauchy principal value of the integral of $f$ over $\mathbb{R}$.

## Remarks

The integral $\int_{-1}^{1} \frac{dx}{x}$ does not exist as an improper integral, but $\operatorname{p.v.} \int_{-1}^{1} \frac{dx}{x} = 0$, because the two parts cancel.
Similarly, $\operatorname{p.v.} \int_{-\infty}^{\infty} x \, dx = 0$, because $\int_{-R}^{R} x \, dx = 0$ for each $R$.
When the improper integral exists, the principal value exists and has the same value.
Principal values occur in the [[evaluation of real integrals by residues|0476QH9846]] when the integrand has a pole on the real axis.

## Explanations

### L1 human

Sometimes an integral is infinite on the left of a bad point and infinite on the right, with opposite signs.
The principal value cuts out a small symmetric interval around the bad point, so that the two sides can cancel.
For $\frac{1}{x}$ on $[-1, 1]$, the areas on the two sides cancel exactly.
</document>

<document source="https://math2.tamnd.com/node/0V3XAMF3C1.md" tag="0V3XAMF3C1" kind="concept">
# Plane

## Summary

A plane is a flat surface of dimension two that extends without end. It is the two-dimensional analogue of a point, which has dimension zero, and of a line, which has dimension one.
</document>

<document source="https://math2.tamnd.com/node/0V49JEZW06.md" tag="0V49JEZW06" kind="concept">
# Laplace transform

## Summary

The Laplace transform is an integral transform. It sends a function $f$ of a real variable $t \ge 0$ to the function $F(s) = \int_0^\infty f(t) e^{-st}\, dt$ of a complex variable $s$.
It is named after Pierre-Simon Laplace. See the [[definition|05XT5AVJBK]].
</document>

<document source="https://math2.tamnd.com/node/0V4DET59NW.md" tag="0V4DET59NW" kind="statement">
# Continuous real function on a compact set attains its bounds

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]], let $K \subseteq X$ be a nonempty [[compact set|00R0X1WGPX]] and let $f \colon X \to \mathbb{R}$ be a [[continuous map|0SY9ECQ19Q]].
Then there are points $a, b \in K$ with $f(a) \le f(x) \le f(b)$ for all $x \in K$.

## Remarks

Proof: the set $f(K)$ is compact in $\mathbb{R}$, because the [[continuous image of a compact set is compact|0K9AAJ8G51]]. By the [[Heine–Borel theorem|07PFYV65MB]], it is closed and bounded. Thus it contains its [[supremum|0BTMQ04556]] and its infimum.
For $K = [a, b]$, this is the extreme value theorem of calculus.
Compactness is needed: $f(x) = x$ on $(0, 1)$ has no maximum, and $f(x) = x$ on $\mathbb{R}$ is not bounded.

## Explanations

### L1 human

A continuous function on a compact set has a largest value and a smallest value, and it reaches them at some points.
On an open interval, the function can come closer and closer to a value without reaching it.
</document>

<document source="https://math2.tamnd.com/node/0V4E623THY.md" tag="0V4E623THY" kind="definition">
# Separable equation

## Definition

Let $I$ and $J$ be open [[intervals|0Q1D85B8G6]] of $\mathbb{R}$, and let $g : I \to \mathbb{R}$ and $h : J \to \mathbb{R}$ be continuous functions.
A separable equation is a first-order [[ordinary differential equation|0P97V20RW7]] of the form
$$y' = g(x)\, h(y),$$
for an unknown function $y$ of $x$ with values in $J$.

## Remarks

Method: if $h(y) \ne 0$, write the equation as $y'/h(y) = g(x)$ and integrate. With antiderivatives $H$ of $1/h$ and $G$ of $g$, the solutions satisfy $H(y(x)) = G(x) + C$ for a constant $C$.
For each zero $c$ of $h$, the constant function $y = c$ is a solution. The method above does not find these solutions.
Example: $y' = x y$ has the solutions $y = C e^{x^2/2}$ with a real constant $C$.
[[Exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are separable.

## Explanations

### L1 human

In a separable equation, the right side is a product: one factor depends only on $x$ and the other only on $y$.
Put everything with $y$ on one side and everything with $x$ on the other side, then integrate both sides.
</document>

<document source="https://math2.tamnd.com/node/0V4QXPGJQK.md" tag="0V4QXPGJQK" kind="concept">
# Vacuous truth

## Summary

A vacuous truth is an [[implication|0ZNX6DQ3EJ]] or a universal statement that is true because its hypothesis is never true.
An implication $p \to q$ with a false $p$ is true, and a statement $\forall x \in X, \, P(x)$ with an empty $X$ is true.

## Remarks

Example: "each element of the [[empty set|0ECVK59Y03]] is an even number" is vacuously true.
Vacuous truth explains why the empty set is a [[subset|0FE8Q5S42Y]] of every set.

## Explanations

### L1 human

"All unicorns in this room are blue" is true when the room has no unicorns.
There is nothing that can break the statement.
This convention keeps the rules of logic simple and regular.
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<document source="https://math2.tamnd.com/node/0V4VB6MQDC.md" tag="0V4VB6MQDC" kind="statement">
# Fibonacci numbers as sums of binomial coefficients

## Statement

Let $n$ be a natural number, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].
Then
$$F_{n+1} = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n - k}{k}.$$

## Remarks

Example: $F_6 = \binom{5}{0} + \binom{4}{1} + \binom{3}{2} = 1 + 4 + 3 = 8$.
Combinatorial proof: $F_{n+1}$ counts the tilings of a strip of length $n$ by squares and dominoes. A tiling with $k$ dominoes has $n - k$ tiles, and $\binom{n-k}{k}$ choices fix the positions of the dominoes.
In [[Pascal's triangle|05DME4TZ3K]], the sum runs along a shallow diagonal.

## Explanations

### L1 human

Add the numbers of Pascal's triangle along the shallow diagonals, and you get the Fibonacci numbers.
Each term counts the ways to tile a strip with a fixed number of dominoes.
</document>

<document source="https://math2.tamnd.com/node/0V4VSCBXP3.md" tag="0V4VSCBXP3" kind="definition">
# Mean squared error

## Definition

Let $\hat{\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$, with $E_\theta[\hat{\theta}^2] < \infty$, where $E_\theta$ is the expectation when $\theta$ is the true parameter.
The mean squared error of $\hat{\theta}$ at $\theta$ is
$$\operatorname{MSE}_\theta(\hat{\theta}) = E_\theta\left[(\hat{\theta} - \theta)^2\right].$$

## Remarks

It decomposes as $\operatorname{MSE}_\theta(\hat{\theta}) = \operatorname{Var}_\theta(\hat{\theta}) + \operatorname{Bias}_\theta(\hat{\theta})^2$, with the [[bias|01BD2TFFP4]] $\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta$.
For an unbiased estimator, the mean squared error is the [[variance|007QED3E0Y]].
For the sample mean of $n$ i.i.d. variables with variance $\sigma^2$, the mean squared error is $\sigma^2/n$.
For $n$ predictions $\hat{y}_i$ of observed values $y_i$, one also calls $\frac{1}{n}\sum_i (y_i - \hat{y}_i)^2$ the mean squared error.

## Explanations

### L1 human

The mean squared error measures how far, on average, an estimator is from the truth.
It is large if the estimator spreads widely, or if it is systematically off, or both.
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<document source="https://math2.tamnd.com/node/0V547YQRMV.md" tag="0V547YQRMV" kind="statement">
# Subgroups of index two are normal

## Statement

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ of [[index|0DTXBQPA2S]] $2$.
Then $H$ is [[normal|0CB0DX5BVV]] in $G$.

## Remarks

For $g \notin H$, the left cosets are $H$ and $gH = G \setminus H$, and the right cosets are $H$ and $Hg = G \setminus H$.
Thus $gH = Hg$ for each $g \in G$, and [[the coset criterion|01E7SWYJX3]] applies.
For example, the alternating group $A_n$ has index $2$ in the symmetric group $S_n$, and thus it is normal.

## Explanations

### L1 human

When a subgroup has only two cosets, one coset is the subgroup and the other is everything else.
This is true for the left cosets and for the right cosets.
So the left and right cosets agree, and the subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0V58FPM3EQ.md" tag="0V58FPM3EQ" kind="definition">
# Minimal polynomial

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
The *minimal polynomial* of $T$ is the monic polynomial $m_T \in F[x]$ of least degree with $m_T(T) = 0$.
It is the monic generator of the ideal of [[annihilating polynomials|0PBN8VEKY8]] of $T$.
The minimal polynomial of a square matrix is defined in the same way.

## Remarks

The minimal polynomial exists and is unique.
It [[divides each annihilating polynomial|0VTAD25A3G]], in particular the [[characteristic polynomial|0VSSPAFM2C]].
For $I_2$, the minimal polynomial is $x - 1$; for $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, it is $(x - 1)^2$.
Similar matrices have the same minimal polynomial.

## Explanations

### L1 human

The minimal polynomial is the shortest polynomial relation that a matrix satisfies.
All other relations are multiples of it.
Its roots are exactly the eigenvalues of the matrix.
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# Characteristic function

## Summary

The characteristic function of a real random variable $X$ is the function $t \mapsto E[e^{itX}]$. It is the Fourier transform of the law of $X$, and it exists for every random variable.
The node [[characteristic function|0MYDNRT7PP]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0V5GYWR8R8.md" tag="0V5GYWR8R8" kind="concept">
# Leading coefficient

## Summary

The leading coefficient of a nonzero polynomial is its coefficient of highest degree.
The [[definition of the leading coefficient|0P05RJC5XA]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0V63D9V0FY.md" tag="0V63D9V0FY" kind="definition">
# Regular polygon

## Definition

Let $n \geq 3$, and let $P_1 P_2 \cdots P_n$ be a convex polygon in the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$. Write $P_0 = P_n$ and $P_{n+1} = P_1$.
The polygon is regular if all sides have the same length, $d(P_i, P_{i+1}) = d(P_1, P_2)$, and all [[angles|0GJAJC6TNZ]] are equal, $\angle P_{i-1} P_i P_{i+1} = \angle P_n P_1 P_2$, for $i = 1, \dots, n$.

## Remarks

Each interior angle of a regular convex $n$-gon is $\frac{(n - 2)\pi}{n}$. Each exterior angle is $\frac{2\pi}{n}$.
The vertices are [[concyclic|049DCJZ448]], and the polygon also has an incircle. The vertices of a regular $n$-gon with the circumradius $1$ can be $\left(\cos \frac{2\pi k}{n}, \sin \frac{2\pi k}{n}\right)$, for $k = 0, \dots, n - 1$.
With the side $s$, the area is $\frac{1}{4} n s^2 \cot \frac{\pi}{n}$.
Star polygons, such as the pentagram, are also called regular polygons. They are not convex.

## Explanations

### L1 human

A regular polygon looks the same from each corner. Equal sides alone are not enough: a rhombus has four equal sides but is regular only when it is a square. Equal angles alone are not enough either, as a rectangle shows.
</document>

<document source="https://math2.tamnd.com/node/0V6GAWTBYW.md" tag="0V6GAWTBYW" kind="definition">
# Biconditional

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The biconditional of $p$ and $q$ is the proposition $p \leftrightarrow q$, read "$p$ if and only if $q$".
The proposition $p \leftrightarrow q$ is true when $p$ and $q$ are both true or both false. Otherwise, it is false.

| $p$ | $q$ | $p \leftrightarrow q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

## Remarks

Other notations are $p \Leftrightarrow q$ and "$p$ iff $q$".
The proposition $p \leftrightarrow q$ has the same truth value as $(p \to q) \land (q \to p)$.

## Explanations

### L1 human

"A triangle has three equal sides if and only if it has three equal angles" is a biconditional.
It says two things: each property gives the other.
To prove a biconditional, you usually prove the two [[implications|0ZNX6DQ3EJ]] separately.
</document>

<document source="https://math2.tamnd.com/node/0V7090SSPC.md" tag="0V7090SSPC" kind="concept">
# Regular curve

## Summary

A regular curve is a continuously differentiable [[parametrised curve|0G7WM8DX4P]] whose [[tangent vector|0ZCZK72B45]] is never zero.
At each point a regular curve has a well-defined tangent line, and it can be reparametrised by arc length.
</document>

<document source="https://math2.tamnd.com/node/0V7E9KP6Q2.md" tag="0V7E9KP6Q2" kind="definition">
# Square root function

## Definition

For a real number $x \ge 0$, the square root $\sqrt{x}$ is the unique real number $y \ge 0$ with $y^2 = x$.
The square root function is the function $x \mapsto \sqrt{x}$ on $[0, \infty)$.

## Remarks

The number $y$ exists by the [[intermediate value theorem|00KAFBR32E]] for $t \mapsto t^2$. It is unique because $t^2$ is [[strictly increasing|0TR9SR9FTB]] on $[0, \infty)$.
The function is continuous and strictly increasing. It is differentiable on $(0, \infty)$, but not at $0$.
For $x > 0$, $\sqrt{x} = x^{1/2}$. Also $\sqrt{x^2} = |x|$ for every real $x$. Mathlib defines `Real.sqrt`, with $\sqrt{x} = 0$ for $x < 0$.

## Explanations

### L1 human

The square root of $x$ is the nonnegative number whose square is $x$.
For example, $\sqrt{9} = 3$, not $-3$.
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<document source="https://math2.tamnd.com/node/0V7G6WMFK0.md" tag="0V7G6WMFK0" kind="concept">
# Covariance matrix

## Summary

The covariance matrix of a random vector is the square matrix of the covariances of all pairs of its components.
The node [[covariance matrix|02CBHV4ZZQ]] gives the precise definition.
</document>

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# Standard basis

## Description

Let $F$ be a field and let $n \ge 1$ be an integer.
For $i = 1, \dots, n$, let $e_i \in F^n$ be the vector with entry $1$ in position $i$ and $0$ in the other positions.
The *standard basis* of the coordinate space $F^n$ is the family $(e_1, \dots, e_n)$.
It is a [[basis|0KCQ9Y3QKA]] of $F^n$, and each $x = (x_1, \dots, x_n)$ is equal to $x_1 e_1 + \cdots + x_n e_n$.

## Remarks

The coordinates of a vector in the standard basis are its entries.
In $\R^3$, the standard basis vectors are often written $\mathbf{i}, \mathbf{j}, \mathbf{k}$.
The column $j$ of an $m \times n$ matrix $A$ is $A e_j$.
The standard basis of $\R^n$ is an orthonormal basis for the dot product.

## Explanations

### L1 human

The standard basis of $F^n$ consists of the vectors with one entry $1$ and all other entries $0$.
In the plane, these are the unit arrows along the $x$-axis and the $y$-axis.
The coordinates of a vector in this basis are just its entries.
</document>

<document source="https://math2.tamnd.com/node/0V8046R0P9.md" tag="0V8046R0P9" kind="definition">
# Quotient topology

## Definition

Let $(X, \tau)$ be a [[topological space|0CNQXVBHCX]] and let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on $X$.
Let $q \colon X \to X / {\sim}$ be the map to the [[quotient set|0ZVHXDSZC7]] that sends $x$ to its class.
The quotient topology on $X / {\sim}$ is
$$\{ V \subseteq X / {\sim} : q^{-1}(V) \in \tau \}.$$

## Remarks

It is the finest topology on $X / {\sim}$ for which $q$ is continuous.
A [[preimage|0FGCFY1AZX]] $q^{-1}(V)$ is a union of classes. Thus $V$ is open if and only if the union of the classes in $V$ is open in $X$.
Quotients need not keep good properties: a quotient of a Hausdorff space need not be [[Hausdorff|0MXB4CKHFW]].
Examples are the [[circle|0WH1GVBJNH]], the [[torus|03NM7DZXXC]] and the [[Möbius strip|04AJE0M271]].

## Explanations

### L1 human

You make a quotient space by gluing the points of each class into one point.
For example, gluing the two ends of $[0, 1]$ gives a circle.
</document>

<document source="https://math2.tamnd.com/node/0V8F3E48E6.md" tag="0V8F3E48E6" kind="concept">
# Linear combination

## Summary

A linear combination of vectors is a sum of scalar multiples of the vectors.
</document>

<document source="https://math2.tamnd.com/node/0V948DE5KA.md" tag="0V948DE5KA" kind="statement">
# Logarithm of a power

## Statement

For every real $x > 0$ and every natural number $n$,
$$\ln(x^n) = n \ln x.$$

## Remarks

Induction on $n$ with the [[product rule for logarithms|0K2FNY6DYV]] gives the claim.
For [[real exponents|0TV07AQT62]], $\ln(x^a) = a \ln x$ holds by the definition $x^a = e^{a \ln x}$.
Mathlib states it as `Real.log_pow`.

## Explanations

### L1 human

The logarithm brings an exponent down as a factor.
For example, $\ln(x^3) = 3 \ln x$.
</document>

<document source="https://math2.tamnd.com/node/0V94H674MB.md" tag="0V94H674MB" kind="definition">
# Determinant of a family of vectors in a basis

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
For vectors $v_1, \dots, v_n \in V$, let $M$ be the $n \times n$ matrix whose column $j$ is the [[coordinate vector|04CYQHN6MC]] $[v_j]_B$.
The *determinant of $(v_1, \dots, v_n)$ in the basis $B$* is
$$\det_B(v_1, \dots, v_n) = \det M.$$

## Remarks

The map $\det_B \colon V^n \to F$ is the unique alternating multilinear form with $\det_B(b_1, \dots, b_n) = 1$.
The vectors $v_1, \dots, v_n$ form a basis of $V$ if and only if $\det_B(v_1, \dots, v_n) \ne 0$.
For another basis $B'$, $\det_{B'} = \det_{B'}(b_1, \dots, b_n) \cdot \det_B$.

## Explanations

### L1 human

Write $n$ vectors in the coordinates of a fixed basis, and put the coordinates side by side as columns.
The determinant of this square matrix measures the signed volume of the box that the vectors span.
The unit is the box of the basis vectors.
</document>

<document source="https://math2.tamnd.com/node/0V98WY94B8.md" tag="0V98WY94B8" kind="statement">
# First Sylow theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $k \geq 0$ with $p^k \mid |G|$.
Then $G$ has a [[subgroup|0X1JXT47GA]] of order $p^k$.
In particular, $G$ has a [[Sylow p-subgroup|0053CQE9Y2]].

## Remarks

For $k = 1$, this is [[Cauchy's theorem|0JM7PA288B]].
Let $p^n$ be the largest power of $p$ that divides $|G|$. A proof by Wielandt lets $G$ act by left multiplication on the subsets of $G$ of size $p^n$.
For example, a group of order $12$ has a subgroup of order $4$ and a subgroup of order $3$.

## History

Ludwig Sylow proved the Sylow theorems in 1872, and he published them in Mathematische Annalen.

## Explanations

### L1 human

The converse of Lagrange's theorem is false in general.
But it is true for prime powers: each prime power that divides the order of the group is the order of a subgroup.
The largest such subgroups are the Sylow subgroups.
</document>

<document source="https://math2.tamnd.com/node/0V9A43H7WV.md" tag="0V9A43H7WV" kind="statement">
# Product rule

## Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$.
Then $f g$ is differentiable at $a$, and
$$(f g)'(a) = f'(a) g(a) + f(a) g'(a).$$

## Remarks

Write $f(a + h) g(a + h) - f(a) g(a) = (f(a + h) - f(a)) g(a + h) + f(a) (g(a + h) - g(a))$. Divide by $h$ and use the [[continuity of $g$ at $a$|07ZMX4ANJS]].
For example, $(x \sin x)' = \sin x + x \cos x$.
Mathlib states it as `HasDerivAt.mul`.

## History

The discovery of the rule is credited to Gottfried Leibniz.

## Explanations

### L1 human

Think of $f g$ as the area of a rectangle with sides $f$ and $g$.
When both sides grow a little, the area grows by $f' g + f g'$, up to a tiny corner piece.
</document>

<document source="https://math2.tamnd.com/node/0V9ME8GQE7.md" tag="0V9ME8GQE7" kind="statement">
# Dirichlet's test

## Statement

Let $(a_n)$ be a [[monotone|0KN22AAW5R]] real sequence with $a_n \to 0$.
Let $(b_n)$ be a real sequence whose [[partial sums|0NB643K60Q]] are bounded. That is, there is $M$ with $\left| \sum_{k=1}^{n} b_k \right| \le M$ for all $n$.
Then the series $\sum_{n=1}^{\infty} a_n b_n$ converges.

## Remarks

The proof uses summation by parts and the [[Cauchy criterion for series|06PVXMY0TH]].
With $b_n = (-1)^n$ it gives the [[alternating series test|0PK8QH1W1V]].
It also shows that $\sum \sin(n x) / n$ converges for each real $x$.
Mathlib states a form with a Cauchy sequence of partial sums, as `Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded`.

## History

The test is named after Peter Gustav Lejeune Dirichlet. It was published after his death, in 1862.

## Explanations

### L1 human

The partial sums of $\sum b_n$ do not grow, and the weights $a_n$ shrink steadily to $0$.
Together these give enough cancellation for $\sum a_n b_n$ to converge.
</document>

<document source="https://math2.tamnd.com/node/0V9PMN0RBH.md" tag="0V9PMN0RBH" kind="object">
# Special linear group

## Description

Let $F$ be a field and let $n \ge 1$.
The *special linear group* is the set
$$SL_n(F) = \{A \in GL_n(F) : \det A = 1\}$$
of $n \times n$ matrices over $F$ with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.
It is a subgroup of the [[general linear group|0FCHE902D7]] $GL_n(F)$.

## Remarks

It is the kernel of the group homomorphism $\det \colon GL_n(F) \to F^\times$, so it is a normal subgroup.
The quotient $GL_n(F) / SL_n(F)$ is isomorphic to $F^\times$.
Over $\R$, its elements are the linear maps that keep volume and orientation.
$SL_2(\Z)$ is the analogous group of integer matrices with determinant $1$.

## Explanations

### L1 human

The special linear group collects the matrices with determinant $1$.
In the real case, these maps keep the volume of every region, and they do not flip orientation.
Shears, such as $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$, are typical elements.
</document>

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# Preorder

## Summary

A preorder is a relation that is reflexive and transitive.
It compares elements, but two different elements can be equivalent under it.
</document>

<document source="https://math2.tamnd.com/node/0VARVQ9ZW1.md" tag="0VARVQ9ZW1" kind="definition">
# Power series in the complex plane

## Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \ge 0}$ be a sequence of [[complex numbers|09FTAET5Z4]].
The power series with center $c$ and coefficients $a_n$ is the series
$$\sum_{n=0}^{\infty} a_n (z - c)^n$$
in the complex variable $z$.
The series converges at a point $z \in \mathbb{C}$ if the partial sums $\sum_{n=0}^{N} a_n (z - c)^n$ have a limit in $\mathbb{C}$ as $N \to \infty$. The sum of the series is the function that maps each such $z$ to this limit.

## Remarks

Each power series converges at $z = c$, with sum $a_0$.
The geometric series $\sum_{n \ge 0} z^n$ converges if and only if $|z| < 1$, and then its sum is $1/(1 - z)$.
The set where the series converges contains an open disc and is contained in its closure, see [[disc of convergence|045EWAXY9P]].
A [[formal power series|033WGN00BF]] is the same expression without any question of convergence.

## Explanations

### L1 human

A power series is a polynomial with infinitely many terms.
For some values of $z$ the terms get small fast enough and the series has a sum, and for other values it does not.
For example, $\sum z^n$ has a sum when $|z| < 1$, but not when $|z| \ge 1$, because then the terms do not go to $0$.
</document>

<document source="https://math2.tamnd.com/node/0VB8BFRABH.md" tag="0VB8BFRABH" kind="definition">
# Square root

## Definition

Let $x$ be a real number with $x \ge 0$.
The square root of $x$, written $\sqrt{x}$, is the unique real number $y$ with $y \ge 0$ and $y^2 = x$.

## Remarks

The [[existence of square roots|068BZBXE0C]] shows that exactly one such $y$ exists.
The function $x \mapsto \sqrt{x}$ on $[0, \infty)$ is increasing, and $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \ge 0$.
For each real $x$, $\sqrt{x^2} = |x|$, the [[absolute value|00S7C68D51]] of $x$.

## Explanations

### L1 human

The square root of $x$ is the side length of a square with area $x$.
The equation $y^2 = 25$ has two solutions, $5$ and $-5$. The symbol $\sqrt{25}$ means the positive one, $5$.
</document>

<document source="https://math2.tamnd.com/node/0VB9G5B034.md" tag="0VB9G5B034" kind="statement">
# Multiplicativity of the Möbius function

## Statement

The [[Möbius function|0C8QENHYZW]] $\mu$ is a [[multiplicative function|0AP80TT660]].
That is, $\mu(1) = 1$ and $\mu(mn) = \mu(m) \mu(n)$ for all coprime positive integers $m$ and $n$.

## Remarks

The function $\mu$ is not completely multiplicative: $\mu(4) = 0$, but $\mu(2)^2 = 1$.
The proof: for coprime $m$ and $n$, the product $mn$ is squarefree exactly when $m$ and $n$ are both squarefree. Then $\omega(mn) = \omega(m) + \omega(n)$.

## Explanations

### L1 human

For example, $\mu(5) = -1$ and $\mu(6) = 1$, and $\mu(30) = -1 = \mu(5) \mu(6)$.
When $m$ and $n$ have no common prime, their prime factors just combine, so the signs multiply.
</document>

<document source="https://math2.tamnd.com/node/0VBWK9XYDG.md" tag="0VBWK9XYDG" kind="statement">
# Kummer's theorem

## Statement

Let $p$ be a prime number, and let $m$ and $n$ be natural numbers.
Then $\nu_p \binom{m + n}{m}$ is the number of carries when one adds $m$ and $n$ in base $p$. Here $\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].

## Remarks

For example, in base $2$, $3 + 1 = 11_2 + 1_2 = 100_2$ has $2$ carries, and $\binom{4}{1} = 4 = 2^2$.
The theorem follows from [[Legendre's formula|01WPV9PHN7]] in the form with the digit sums $s_p$.
A consequence: $p \mid \binom{m+n}{m}$ if and only if there is at least one carry.

## History

Ernst Kummer proved the theorem in 1852.

## Explanations

### L1 human

Write $m$ and $n$ in base $p$ and add them by hand. Count how many times you carry a $1$.
That count is the exponent of $p$ in the binomial coefficient $\binom{m+n}{m}$.
For $p = 10$ the rule fails, because $10$ is not prime. The theorem needs a prime base.
</document>

<document source="https://math2.tamnd.com/node/0VBZH90R1A.md" tag="0VBZH90R1A" kind="concept">
# Internal direct product

## Summary

A group $G$ is the internal direct product of two normal subgroups $H$ and $K$ when $H \cap K = \{e\}$ and $HK = G$.
Then $G$ is isomorphic to the direct product $H \times K$.
</document>

<document source="https://math2.tamnd.com/node/0VC4DVE7DQ.md" tag="0VC4DVE7DQ" kind="statement">
# Euler's theorem

## Statement

Let $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.
Then $a^{\varphi(n)} \equiv 1 \pmod n$, where $\varphi$ is [[Euler's totient function|0VV04M41AN]].

## Remarks

When $n = p$ is prime, $\varphi(p) = p - 1$, and the result is [[Fermat's little theorem|0M25PRN3WQ]].
A proof uses [[Lagrange's theorem|3C53TD]]: the group $(\mathbb{Z}/n\mathbb{Z})^\times$ has order $\varphi(n)$, by the [[number of invertible residues modulo n|06RJWZCG95]].
The result lets us reduce exponents. If $\gcd(a, n) = 1$ and $k \equiv l \pmod{\varphi(n)}$, then $a^k \equiv a^l \pmod n$.
For example, $\varphi(10) = 4$, so $7^{4} \equiv 1 \pmod{10}$, and the last digit of $7^{222}$ is the last digit of $7^2 = 49$, that is, $9$.

## History

Euler proved the theorem in 1763.

## Explanations

### L1 human

Powers of $a$ modulo $n$ repeat in a cycle, and the cycle comes back to $1$.
Euler's theorem gives one exponent that always brings $a$ back to $1$, namely $\varphi(n)$.
For $n = 12$ and $a = 5$: $\varphi(12) = 4$ and $5^4 = 625 = 52 \cdot 12 + 1$.
</document>

<document source="https://math2.tamnd.com/node/0VCTBJMEJF.md" tag="0VCTBJMEJF" kind="definition">
# Improper integral

## Definition

Let $f \colon [a, \infty) \to \mathbb{R}$ be Riemann integrable on $[a, b]$ for each $b > a$.
The improper integral of $f$ over $[a, \infty)$ converges if the following [[limit|0Y12QYTJK3]] exists in $\mathbb{R}$:
$$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx.$$
Similarly, let $f \colon (a, b] \to \mathbb{R}$ be Riemann integrable on $[c, b]$ for each $c \in (a, b)$.
Then $\int_a^b f(x) \, dx = \lim_{c \to a^+} \int_c^b f(x) \, dx$, if this [[one-sided limit|0909PWZR62]] exists.

## Remarks

If the limit does not exist, the improper integral diverges.
For an integral over $(-\infty, \infty)$, or with several bad points, we split the interval. Each piece must converge separately.
For example, $\int_1^\infty \frac{dx}{x^2} = 1$, and $\int_0^1 \frac{dx}{\sqrt{x}} = 2$. But $\int_1^\infty \frac{dx}{x}$ diverges.

## Explanations

### L1 human

The Riemann integral needs a bounded interval and a bounded function.
An improper integral removes these limits by a limit of ordinary integrals.
</document>

<document source="https://math2.tamnd.com/node/0VD9ZSWWJG.md" tag="0VD9ZSWWJG" kind="concept">
# Matrix

## Summary

A matrix is a rectangular array of elements of a ring or a field, arranged in rows and columns.
Matrices represent linear maps and systems of linear equations.
</document>

<document source="https://math2.tamnd.com/node/0VDD7W0EKZ.md" tag="0VDD7W0EKZ" kind="statement">
# Morera's theorem

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be continuous.
Assume that $\oint_{\partial T} f(z) \, dz = 0$ for each closed triangle $T \subseteq U$, where $\partial T$ is the boundary of $T$ traveled once and the integral is a [[contour integral|00FWNP9TXY]].
Then $f$ is [[holomorphic|0DG0Z72CQD]] on $U$.

## Remarks

It is enough to know that $\oint_\gamma f(z) \, dz = 0$ for each closed piecewise continuously differentiable path $\gamma$ in $U$, because the boundary of a triangle is such a path.
Proof idea: on each disc in $U$, $F(z) = \int_{[c, z]} f(w) \, dw$ is a primitive of $f$. So $F$ is holomorphic, and $f = F'$ is holomorphic because [[derivatives of holomorphic functions are holomorphic|0QEBN8DGVE]].
Together with [[Goursat's theorem|0BRG85WXYC]], this shows: a continuous $f$ is holomorphic on $U$ if and only if its integrals over all triangles in $U$ are zero.
A main use is the theorem on [[uniform limits of holomorphic functions|0431RN0R1N]].

## Explanations

### L1 human

Morera's theorem is a converse of Cauchy's theorem.
If the integral of a continuous function around each small triangle is zero, then the function is holomorphic.
</document>

<document source="https://math2.tamnd.com/node/0VEFSSKX18.md" tag="0VEFSSKX18" kind="definition">
# Poisson distribution

## Definition

Let $\lambda > 0$.
The Poisson distribution with parameter $\lambda$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots$$
Here $k!$ is the [[factorial|0WFCGVMRRC]] of $k$. One writes $X \sim \operatorname{Poisson}(\lambda)$.

## Remarks

The values add up to $1$ because $\sum_{k \ge 0} \lambda^k / k! = e^{\lambda}$.
The mean and the variance are both $\lambda$.
It is the limit of the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $\lambda / n$ as $n \to \infty$. This is the [[Poisson limit theorem|0XKTZRWTXX]].
Siméon Denis Poisson introduced the distribution in 1837.

## Explanations

### L1 human

The Poisson distribution counts rare events that happen at random, such as calls to a help desk in one hour.
If $\lambda = 2$ calls come per hour on average, the chance of no call in an hour is $e^{-2} \approx 0.135$.
</document>

<document source="https://math2.tamnd.com/node/0VEKNG02W0.md" tag="0VEKNG02W0" kind="statement">
# Volume of a box

## Statement

Let $a_i < b_i$ for $i = 1, \dots, n$, and let $Q = [a_1, b_1] \times \dots \times [a_n, b_n] \subseteq \mathbb{R}^n$.
Then $Q$ is Jordan measurable, and its volume is
$$\operatorname{vol}(Q) = \int_Q 1 \, dV = (b_1 - a_1)(b_2 - a_2) \cdots (b_n - a_n).$$

## Remarks

For a box, the [[Jordan measure|0565DC8RMK]] and the Riemann integral of the constant function $1$ give the same value.
The integral can be computed with [[Fubini's theorem|0R0T1VWH5X]] as an iterated integral of $1$, one factor $b_i - a_i$ at a time.
Open and half-open boxes with the same edges have the same volume, because the faces of $Q$ have volume zero.
For $n = 3$ this is the formula length times width times height, a case of base area times height. The volume of a general solid $E$ is the [[triple integral|0GTBGMZXV8]] $\iiint_E 1 \, dV$.

## Explanations

### L1 human

The volume of a box is the product of its side lengths.
In the theory of multiple integrals this is the starting point: Riemann sums and Jordan measure are both built from volumes of small boxes.
</document>

<document source="https://math2.tamnd.com/node/0VF11ZTTMD.md" tag="0VF11ZTTMD" kind="definition">
# Field of fractions

## Definition

Let $R$ be an [[integral domain|09S3PFBEPM]].
On the set of pairs $(a, b)$ with $a, b \in R$ and $b \neq 0$, let $(a, b) \sim (c, d)$ if $a d = b c$.
This is an equivalence relation. Write $a/b$ for the class of $(a, b)$.
The field of fractions $\operatorname{Frac}(R)$ is the set of these classes, with the operations
$$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}, \qquad \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}.$$
It is a [[field|0QA8CHH8MG]], and the map $R \to \operatorname{Frac}(R)$, $a \mapsto a/1$, is an injective ring homomorphism.

## Remarks

The product $b d$ is not zero because $R$ is an integral domain. This makes the operations well defined.
The inverse of a nonzero $a/b$ is $b/a$.
Universal property: every injective ring homomorphism from $R$ to a field $K$ extends to exactly one ring homomorphism from $\operatorname{Frac}(R)$ to $K$.
The field of fractions of the polynomial ring $K[x]$ over a field $K$ is the field $K(x)$ of rational functions.

## Explanations

### L1 human

This is how the rational numbers come from the integers.
A fraction is a pair of a numerator and a nonzero denominator.
Two fractions are equal when cross-multiplication gives the same result, as with $1/2 = 2/4$.
</document>

<document source="https://math2.tamnd.com/node/0VFW4Y69BT.md" tag="0VFW4Y69BT" kind="definition">
# Quotient ring

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $I$ be a [[two-sided ideal|0DDA6CY7GJ]] of $R$.
For $a \in R$, the coset of $a$ is $a + I = \{a + x : x \in I\}$.
The quotient ring $R/I$ is the set of the cosets of $I$, with these operations:
$$(a + I) + (b + I) = (a + b) + I, \qquad (a + I)(b + I) = a b + I.$$
These operations do not depend on the choice of $a$ and $b$ in their cosets.
With them, $R/I$ is a ring with zero $0 + I = I$ and identity $1 + I$.

## Remarks

Two elements $a, b \in R$ have the same coset if and only if $a - b \in I$.
The map $\pi \colon R \to R/I$, $\pi(a) = a + I$, is a surjective [[ring homomorphism|0GGE0Z0JQ1]] with [[kernel|0AAHE7GEW6]] $I$.
For $n \ge 1$, the quotient $\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$.
The quotient $R/R$ is the [[zero ring|008TBCYDPX]].

## Explanations

### L1 human

To make the quotient ring, you declare all the elements of $I$ to be zero.
Two elements become equal when their difference is in $I$.
For $\mathbb{Z}/5\mathbb{Z}$, the numbers $2$ and $7$ become the same element.
</document>

<document source="https://math2.tamnd.com/node/0VG02M1FEZ.md" tag="0VG02M1FEZ" kind="statement">
# Every polynomial has a root in some extension field

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ with $\deg f \ge 1$.
Then there is a [[field extension|0WQ1A8SHTZ]] $L/K$ with $[L : K] \le \deg f$ such that $f$ has a [[root|00EHX6FWC0]] in $L$.

## Remarks

The proof: let $g$ be an irreducible factor of $f$ in $K[x]$. [[Adjoining a root of $g$|0M893X7YY7]] gives the field $L = K[x]/(g)$ of degree $\deg g$ over $K$.
The class of $x$ in $L$ is a root of $g$, and so of $f$.
The irreducible factor $g$ exists because $K[x]$ is a [[unique factorisation domain|0D6KH8Y59N]].
Repeating the step gives a [[splitting field|02NZZ1J0FH]] of $f$.

## Explanations

### L1 human

A polynomial with no root in a field gets a root in a bigger field.
The bigger field is built from the polynomial itself, without any reference to the complex numbers.
</document>

<document source="https://math2.tamnd.com/node/0VG7VNEWMG.md" tag="0VG7VNEWMG" kind="concept">
# Abundant number

## Summary

An abundant number is a positive integer whose positive divisors other than itself have a sum greater than the number.
The [[definition of an abundant number|0VS0MP1P5H]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0VGFHTM75Y.md" tag="0VGFHTM75Y" kind="concept">
# Continuous map

## Summary

A continuous map is a function that sends nearby points to nearby points.
Between topological spaces, this means that preimages of open sets are open. Between metric spaces, it is the same as the epsilon-delta condition.
</document>

<document source="https://math2.tamnd.com/node/0VGKB2NC4T.md" tag="0VGKB2NC4T" kind="example">
# Improper integral of x^p on an unbounded interval

## Example

Let $p$ be a real number. The [[improper integral|0VCTBJMEJF]] $\int_1^\infty x^p \, dx$ converges if and only if $p < -1$.
In this case,
$$\int_1^\infty x^p \, dx = -\frac{1}{p + 1}.$$
For $p \ne -1$, [[the power formula|0Q3HYFT9RQ]] gives $\int_1^b x^p \, dx = \frac{b^{p+1} - 1}{p + 1}$. This has a finite limit as $b \to \infty$ exactly when $p + 1 < 0$.
For $p = -1$, $\int_1^b \frac{dx}{x} = \ln b$, [[which tends to infinity|0K1N1Y4EGZ]].

## Remarks

This example parallels the [[$p$-series test|0ZNSSK8ZK1]] through the [[integral test|019QZ6RRVH]].
Mathlib states a form as `integrableOn_Ioi_rpow_iff`.

## Explanations

### L1 human

The function $x^p$ must decay faster than $1/x$ for the area over $[1, \infty)$ to be finite.
The function $1/x$ itself is just too slow.
</document>

<document source="https://math2.tamnd.com/node/0VGMEBVW1M.md" tag="0VGMEBVW1M" kind="definition">
# Probability measure

## Definition

Let $\Omega$ be a set and let $\mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]] on $\Omega$.
A probability measure on $(\Omega, \mathcal{F})$ is a function $P \colon \mathcal{F} \to [0, 1]$ with these two properties.
$P(\Omega) = 1$.
For each sequence $A_1, A_2, \dots$ of elements of $\mathcal{F}$ with $A_i \cap A_j = \varnothing$ for all $i \ne j$,
$$P\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} P(A_n).$$

## Remarks

These are the [[probability axioms|0DAX8PMAJ4]]. In other words, a probability measure is a measure of total mass $1$.
Countable additivity implies $P(\varnothing) = 0$ and finite additivity: $P(A \cup B) = P(A) + P(B)$ for disjoint $A$ and $B$.
On a finite set $\Omega$ with the power set as $\mathcal{F}$, each probability measure has the form $P(A) = \sum_{\omega \in A} p(\omega)$ with $p(\omega) \ge 0$ and $\sum_{\omega \in \Omega} p(\omega) = 1$.

## Explanations

### L1 human

A probability measure is a rule that gives each event a number between $0$ and $1$.
The whole sample space gets $1$, and the number of a union of separate pieces is the sum of the numbers of the pieces.
For a fair die, each face gets $1/6$, and the event "even" gets $3/6 = 1/2$.
</document>

<document source="https://math2.tamnd.com/node/0VGMHCF4SC.md" tag="0VGMHCF4SC" kind="statement">
# Product of continuous functions is continuous

## Statement

Let $D \subseteq \mathbb{R}$ and let $f, g \colon D \to \mathbb{R}$ be [[continuous|0P27W9RW0G]] functions.
Then the sum $f + g$ and the product $f g$, given by $(f g)(x) = f(x) g(x)$, are continuous.

## Remarks

The proof uses [[sequential continuity|0PAQM3H57X]] and the [[product rule for limits|0734RSVF2V]] of sequences.
In particular, $c f$ is continuous for each constant $c$.
Mathlib states it for topological monoids, as `Continuous.mul`.

## Explanations

### L1 human

If $f(x)$ and $g(x)$ change only a little when $x$ changes a little, so does $f(x) g(x)$.
</document>

<document source="https://math2.tamnd.com/node/0VGPKZ8MRK.md" tag="0VGPKZ8MRK" kind="statement">
# Continuity is equivalent to sequential continuity in metric spaces

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then $f$ is [[continuous|0SY9ECQ19Q]] if and only if $f(x_n) \to f(x)$ for each sequence $(x_n)$ in $X$ that [[converges|06JYPJ4WHH]] to a point $x$.

## Remarks

The condition on sequences is sequential continuity.
Proof of the converse: if $f$ is not continuous at $x$, then by [[epsilon-delta continuity|0EKGKY5TGH]] there are $\varepsilon > 0$ and points $x_n$ with $d(x_n, x) < 1/n$ and $d(f(x_n), f(x)) \ge \varepsilon$. Then $x_n \to x$, but $f(x_n) \not\to f(x)$.
The same equivalence holds at each single point.
In general topological spaces, continuity implies sequential continuity, but the converse can fail.

## Explanations

### L1 human

A continuous map sends convergent sequences to convergent sequences, with the right limit.
To show that a map is not continuous, it is often enough to find one sequence $x_n \to x$ with $f(x_n) \not\to f(x)$.
</document>

<document source="https://math2.tamnd.com/node/0VGPX25NXQ.md" tag="0VGPX25NXQ" kind="concept">
# Conditional variance

## Summary

The conditional variance of $X$ given some information measures the spread of $X$ around its conditional expectation. It is itself a random variable.
The node [[conditional variance|0F1SN93KXE]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VGSYBWHV2.md" tag="0VGSYBWHV2" kind="definition">
# Inner product

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a vector space over $\mathbb{K}$.
An *inner product* on $V$ is a map $\langle \cdot, \cdot \rangle \colon V \times V \to \mathbb{K}$ such that, for all $x, y, z \in V$ and $a, b \in \mathbb{K}$:

1. $\langle ax + by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle$.
2. $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
3. $\langle x, x \rangle > 0$ if $x \ne 0$.

## Remarks

Over $\R$, the conjugation does nothing, so an inner product is a symmetric positive definite bilinear form.
Over $\C$, conditions 1 and 2 give $\langle x, by \rangle = \bar b \langle x, y \rangle$, so the map is conjugate linear in the second argument.
Many physicists, and Mathlib, use the opposite convention: linear in the second argument.
The [[dot product|0S52TNNTH9]] on $\R^n$ and $\langle x, y \rangle = \sum_i x_i \overline{y_i}$ on $\C^n$ are inner products.

## Explanations

### L1 human

An inner product generalises the dot product to abstract vector spaces.
It lets us talk about lengths, angles and perpendicular vectors.
On the space of continuous functions on $[0, 1]$, the integral $\int_0^1 f(t) g(t)\, dt$ is an inner product.
</document>

<document source="https://math2.tamnd.com/node/0VGWKX2JHB.md" tag="0VGWKX2JHB" kind="statement">
# Abel's identity

## Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, and let $y_1, y_2$ be solutions on $I$ of
$$y'' + p(x)\, y' + q(x)\, y = 0.$$
Let $W = y_1 y_2' - y_1' y_2$ be their [[Wronskian|0N5NF1RGBP]], and let $x_0 \in I$.
Then for each $x \in I$,
$$W(x) = W(x_0) \exp\left( -\int_{x_0}^{x} p(t)\, dt \right).$$

## Remarks

Proof idea: a computation with the equation gives $W' = -p W$, a linear first-order equation for $W$.
So $W$ is zero at each point or at no point of $I$.
For an equation of order $n$ with coefficient $a_{n-1}$ of $y^{(n-1)}$, the same formula holds with $p$ replaced by $a_{n-1}$. [[Liouville's formula|09PS3S0EF6]] extends it to linear systems.
Niels Henrik Abel found the identity.

## Explanations

### L1 human

You can find the Wronskian of two solutions without knowing the solutions.
Only the coefficient $p$ of $y'$ matters.
For $y'' + y = 0$, we have $p = 0$, so the Wronskian of two solutions is constant.
</document>

<document source="https://math2.tamnd.com/node/0VH1TC1PWF.md" tag="0VH1TC1PWF" kind="definition">
# Companion matrix

## Definition

Let $F$ be a field, let $n \ge 1$, and let $p(t) = t^n + c_{n-1} t^{n-1} + \cdots + c_1 t + c_0$ be a monic polynomial over $F$.
The *companion matrix* of $p$ is the $n \times n$ matrix
$$C(p) = \begin{pmatrix} 0 & 0 & \cdots & 0 & -c_0 \\ 1 & 0 & \cdots & 0 & -c_1 \\ 0 & 1 & \cdots & 0 & -c_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \cdots & 1 & -c_{n-1} \end{pmatrix}.$$

## Remarks

The [[characteristic polynomial|0VSSPAFM2C]] and the [[minimal polynomial|03NDVCSHSB]] of $C(p)$ both equal $p$.
In the basis $e_1, \dots, e_n$, the map of $C(p)$ sends $e_i$ to $e_{i+1}$ for $i < n$.
Some authors use the transpose of this matrix as the companion matrix.
Companion matrices are the blocks of the [[rational canonical form|042JWAR3Q0]].

## Explanations

### L1 human

For each monic polynomial, the companion matrix is a matrix with exactly that characteristic polynomial.
It shifts each basis vector to the next one, and the polynomial fixes the image of the last one.
So questions about roots of polynomials become questions about eigenvalues of matrices.
</document>

<document source="https://math2.tamnd.com/node/0VHAVXW08V.md" tag="0VHAVXW08V" kind="statement">
# Subsequences of a convergent sequence converge to the same limit

## Statement

Let $(a_n)$ be a real sequence that converges to a real number $L$.
Let $(a_{n_k})_{k \in \mathbb{N}}$ be a [[subsequence|0R8J3ZQMJS]] of $(a_n)$.
Then $a_{n_k} \to L$ as $k \to \infty$.

## Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_n - L| < \varepsilon$ for $n \ge N$. Since $n_k \ge k$, we get $|a_{n_k} - L| < \varepsilon$ for all $k \ge N$.
It gives a test for divergence. If two subsequences have different limits, the sequence diverges. For $(-1)^n$ the even terms tend to $1$ and the odd terms tend to $-1$.

## Explanations

### L1 human

If all terms eventually stay close to $L$, then the terms that you keep also stay close to $L$.
Deleting terms cannot change where the sequence is going.
</document>

<document source="https://math2.tamnd.com/node/0VHJRXNAAS.md" tag="0VHJRXNAAS" kind="concept">
# Chromatic number

## Summary

The chromatic number of a graph is the least number of colours in a proper colouring. It is a central invariant of graph theory, and the four colour theorem bounds it for planar graphs. The node [[chromatic number|0C5M7AM6FH]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VHNKP1N3K.md" tag="0VHNKP1N3K" kind="definition">
# Conformal map

## Definition

Let $U$ be an open subset of the [[complex plane|0CTE7A31XQ]], identified with $\mathbb{R}^2$, and let $z_0 \in U$.
Let $f : U \to \mathbb{C}$ be differentiable at $z_0$ as a map from an open subset of $\mathbb{R}^2$ to $\mathbb{R}^2$, with Jacobian matrix $J$ at $z_0$.
The map $f$ is conformal at $z_0$ if $J = \lambda Q$ for a real number $\lambda > 0$ and a rotation matrix $Q$. A rotation matrix is an orthogonal $2 \times 2$ matrix with determinant $1$.
The map $f$ is conformal on $U$ if it is conformal at each point of $U$.

## Remarks

Equivalently, $f$ is conformal at $z_0$ when its real derivative at $z_0$ is the map $h \mapsto a h$ for a complex number $a \neq 0$.
A map that is conformal at $z_0$ preserves the angles between curves through $z_0$, and their orientation.
Some authors also allow orientation-reversing maps, such as $z \mapsto \bar{z}$, whose Jacobian is a positive number times an orthogonal matrix.
Other authors use "conformal map" for an injective holomorphic map on an open set.

## Explanations

### L1 human

A conformal map can stretch and turn a small figure, but it does not distort its shape.
If two curves cross at an angle of $30$ degrees, then their images cross at the same angle, in the same direction.
For example, $z \mapsto 2z$ and $z \mapsto iz$ are conformal at each point.
</document>

<document source="https://math2.tamnd.com/node/0VHSHB9B6J.md" tag="0VHSHB9B6J" kind="concept">
# Symmetric polynomial

## Summary

A symmetric polynomial is a polynomial in several variables that does not change when you permute the variables.
The [[definition of a symmetric polynomial|06CTFY2DEZ]] states it.
</document>

<document source="https://math2.tamnd.com/node/0VJBSK15SN.md" tag="0VJBSK15SN" kind="statement">
# Finite-dimensional spaces are isomorphic iff they have equal dimension

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then $V$ and $W$ are [[isomorphic|0B8T10CE7N]] if and only if $\dim V = \dim W$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

An isomorphism sends a basis to a basis, so it keeps the dimension.
Conversely, a bijection between bases of $V$ and $W$ extends to a linear isomorphism.
The statement holds for all vector spaces, with dimension as a cardinal number.
For example, $M_{2 \times 3}(F)$, $F^6$ and $F[x]_{<6}$ are pairwise isomorphic.

## Explanations

### L1 human

Up to isomorphism, a finite-dimensional vector space is determined by one number: its dimension.
All three-dimensional real spaces behave like $\R^3$.
Only the names of the vectors differ.
</document>

<document source="https://math2.tamnd.com/node/0VJF99CZ0Z.md" tag="0VJF99CZ0Z" kind="definition">
# Homogeneous linear system with constant coefficients

## Definition

Let $n \ge 1$ and let $A$ be a real $n \times n$ matrix.
The homogeneous linear system with constant coefficients given by $A$ is the [[system of linear differential equations|0A6QZ367GX]]
$$x'(t) = A\, x(t)$$
for an unknown differentiable function $x : \mathbb{R} \to \mathbb{R}^n$.

## Remarks

Its solutions are $x(t) = e^{t A} x(0)$. See [[the matrix exponential as the solution|0YC8JT513J]].
If $A v = \lambda v$, then $x(t) = e^{\lambda t} v$ is a solution. The [[eigenvalue method|08ZWFDJP3H]] builds all solutions from such solutions.
Example: for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, the solutions are rotations: $x(t) = \left(a \cos t + b \sin t, -a \sin t + b \cos t\right)$.

## Explanations

### L1 human

The rate of change of the vector $x$ is the matrix $A$ applied to $x$, and $A$ does not change with time.
This is the vector version of $y' = a y$, whose solutions are $e^{a t} y(0)$.
</document>

<document source="https://math2.tamnd.com/node/0VJZM0F9YM.md" tag="0VJZM0F9YM" kind="definition">
# Walk

## Definition

Let $G$ be a [[graph|0JD5C82QMB]], let $k$ be a natural number, and let $u, v$ be vertices of $G$.
A walk of length $k$ from $u$ to $v$ is a sequence of vertices $(w_0, w_1, \dots, w_k)$ with $w_0 = u$, $w_k = v$, and $w_{i-1} \sim w_i$ for $1 \le i \le k$.
Its edges are the edges $w_{i-1} w_i$.

## Remarks

A walk may repeat vertices and edges. A walk of length $0$ is a single vertex.
A [[trail|062MQGP8M5]] has no repeated edge, and a [[path|0JDE56HWQ8]] has no repeated vertex.
A walk is closed if $u = v$.

## Explanations

### L1 human

A walk is a route through a graph that goes from vertex to vertex along edges.
You may visit the same vertex or use the same edge more than once.
</document>

<document source="https://math2.tamnd.com/node/0VK0W4D5Y5.md" tag="0VK0W4D5Y5" kind="statement">
# De Morgan's laws for sets

## Statement

Let $U$ be a set and let $A$ and $B$ be [[subsets|0FE8Q5S42Y]] of $U$. Write $X^c = U \setminus X$ for the [[complement|0B1FSS1VZW]] of a subset $X$ in $U$.
Then
$$(A \cup B)^{c} = A^{c} \cap B^{c},$$
$$(A \cap B)^{c} = A^{c} \cup B^{c}.$$

## Remarks

The laws come from [[De Morgan's laws|09PKJKA6J0]] of logic, applied to $x \in A$ and $x \in B$.
They hold also for arbitrary families: the complement of $\bigcup_i A_i$ is $\bigcap_i A_i^c$, and the complement of $\bigcap_i A_i$ is $\bigcup_i A_i^c$.
With set differences, $C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B)$ for each set $C$.

## Explanations

### L1 human

An element is outside $A \cup B$ exactly when it is outside $A$ and outside $B$.
An element is outside $A \cap B$ exactly when it is outside $A$ or outside $B$.
A [[Venn diagram|0KCT6DPXJK]] with two circles shows both laws.
</document>

<document source="https://math2.tamnd.com/node/0VK5V9P8G3.md" tag="0VK5V9P8G3" kind="statement">
# Size of a conjugacy class is the index of the centraliser

## Statement

Let $G$ be a [[group|G5S1M9]] and let $a \in G$.
Then $|\operatorname{Cl}(a)| = [G : C_G(a)]$, where $\operatorname{Cl}(a)$ is the [[conjugacy class|0DEBXAJ5SA]] of $a$ and $C_G(a)$ is its [[centraliser|02YYV36BFM]].
In particular, if $G$ is finite, then $|\operatorname{Cl}(a)|$ divides $|G|$.

## Remarks

This is the [[orbit-stabiliser theorem|0E2HSCKZ6C]] for the [[action by conjugation|0P8CY9V437]].
Directly, $gag^{-1} = hah^{-1}$ if and only if $h^{-1}g \in C_G(a)$.
For example, in $S_3$ the centraliser of $(1\ 2\ 3)$ is $A_3$, of index $2$, and the class of $(1\ 2\ 3)$ has $2$ elements.

## Explanations

### L1 human

Conjugates of $a$ by two elements are equal exactly when the two elements differ by something that commutes with $a$.
So the conjugates of $a$ correspond to the cosets of the centraliser.
An element that commutes with many elements has few conjugates.
</document>

<document source="https://math2.tamnd.com/node/0VK6VEH345.md" tag="0VK6VEH345" kind="statement">
# Banach fixed point theorem

## Statement

Let $(X, d)$ be a nonempty [[complete metric space|08EWT7J0SC]] and let $f \colon X \to X$ be a [[contraction|0MT2S0ZX4Y]] with constant $k \in [0, 1)$.
Then $f$ has exactly one fixed point $x^*$, that is, a point with $f(x^*) = x^*$.
Moreover, for each $x_0 \in X$, the sequence $x_{n+1} = f(x_n)$ converges to $x^*$, and
$$d(x_n, x^*) \le \frac{k^n}{1 - k} \, d(x_1, x_0) \quad \text{for all } n \ge 0.$$

## Remarks

Proof idea: $d(x_{n+1}, x_n) \le k^n d(x_1, x_0)$, so the sequence $(x_n)$ is a [[Cauchy sequence|0MCK8K2D7Z]] by a geometric series bound. Its limit is a fixed point because $f$ is continuous. Two fixed points $x, y$ satisfy $d(x, y) \le k \, d(x, y)$, so $x = y$.
Completeness is needed: $x \mapsto x/2$ on $(0, 1]$ is a contraction without a fixed point.
The theorem gives the existence of solutions of ordinary differential equations in the Picard–Lindelöf theorem.

## History

Stefan Banach stated the theorem in 1922.

## Explanations

### L1 human

Apply the contraction again and again, starting from any point. The points get closer to each other, and in a complete space they converge.
The limit does not move under $f$, and it is the only point that does not move.
</document>

<document source="https://math2.tamnd.com/node/0VKCCEXBJ1.md" tag="0VKCCEXBJ1" kind="concept">
# Ellipse

## Summary

An ellipse is a closed plane curve with two foci. For each point of the curve, the sum of the distances to the two foci is the same constant. A circle is the special case of an ellipse where the two foci are the same point.
</document>

<document source="https://math2.tamnd.com/node/0VKD1TP7F3.md" tag="0VKD1TP7F3" kind="concept">
# Metric space

## Summary

A metric space is a set with a distance function between its points.
The distance is zero only from a point to itself, it is symmetric, and it satisfies the triangle inequality.
Metric spaces are the setting for limits, continuity and completeness beyond the real line.
</document>

<document source="https://math2.tamnd.com/node/0VKKP8X6Q6.md" tag="0VKKP8X6Q6" kind="statement">
# Lagrange multipliers

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $m < n$, and let $f \colon U \to \mathbb{R}$ and $g = (g_1, \dots, g_m) \colon U \to \mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]].
Let $M = \{x \in U : g(x) = 0\}$, and let $a \in M$ be a [[local extremum|04CPG9K1C4]] of the restriction of $f$ to $M$.
Suppose that the [[gradients|0NS2QYVC4Y]] $\nabla g_1(a), \dots, \nabla g_m(a)$ are linearly independent.
Then there is a unique $\lambda = (\lambda_1, \dots, \lambda_m) \in \mathbb{R}^m$ such that
$$\nabla f(a) = \lambda_1 \nabla g_1(a) + \dots + \lambda_m \nabla g_m(a).$$

## Remarks

The numbers $\lambda_i$ are the Lagrange multipliers. Equivalently, $(a, \lambda)$ is a critical point of the Lagrangian $L(x, \lambda) = f(x) - \lambda \cdot g(x)$.
The hypothesis on the gradients is the constraint qualification. It says that the [[Jacobian matrix|0YF0VXKXGB]] $Dg(a)$ has rank $m$.
The proof uses the [[implicit function theorem|0BXBADR94C]]. Near $a$, the set $M$ is the graph of a $C^1$ function of $n - m$ variables. The gradient of $f$ is orthogonal to the tangent space of $M$ at $a$.
The condition is necessary, not sufficient. Each candidate point needs a further check, for example a comparison of values on a compact set $M$.

## Explanations

### L1 human

At a constrained extremum, a small move along the constraint set does not change $f$ to first order.
So $\nabla f(a)$ has no component along the constraint set, and it is a combination of the normals $\nabla g_i(a)$.
This turns a constrained problem into a system of $n + m$ equations in $n + m$ unknowns.
</document>

<document source="https://math2.tamnd.com/node/0VM8B2SB4P.md" tag="0VM8B2SB4P" kind="concept">
# Decimal expansion

## Summary

A decimal expansion writes a nonnegative real number as an integer part and an infinite sequence of digits after the decimal point.
It represents the number as a sum of multiples of powers of $10$.
For example, $1/4 = 0.25000\dots$ and $1/3 = 0.333\dots$.
</document>

<document source="https://math2.tamnd.com/node/0VMG8TA8VF.md" tag="0VMG8TA8VF" kind="definition">
# Limit of a function of several variables

## Definition

Let $D \subseteq \mathbb{R}^n$ and let $f \colon D \to \mathbb{R}^m$.
Let $a \in \mathbb{R}^n$ be a limit point of $D$: each open ball about $a$ contains a point of $D$ other than $a$.
Let $L \in \mathbb{R}^m$.
When this holds, $f(x)$ tends to $L$ as $x$ tends to $a$, with the notation $\lim_{x \to a} f(x) = L$.
For each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|f(x) - L\| < \varepsilon$ for each $x \in D$ with $0 < \|x - a\| < \delta$.
Here $\|\cdot\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].

## Remarks

A limit is unique when it exists, because $a$ is a limit point of $D$.
The value $f(a)$, and even whether $a$ is in $D$, has no effect on the limit.
A function into $\mathbb{R}^m$ tends to $L$ if and only if each component $f_i$ tends to $L_i$.
To show that a limit does not exist, it is enough to find two [[paths with different limits|0BJ6H85DRQ]].

## Explanations

### L1 human

The definition says that you can make $f(x)$ as close to $L$ as you want. For this, take $x$ close enough to $a$ but not equal to $a$.
In the plane, "close to $a$" means inside a small disc about $a$, so $x$ can come from every direction at once.
</document>

<document source="https://math2.tamnd.com/node/0VMWGYY35Q.md" tag="0VMWGYY35Q" kind="statement">
# Cartan–Dieudonné theorem

## Statement

Let $V$ be a real inner product space of finite dimension $n$.
Then each linear isometry $T \colon V \to V$ is a composition of at most $n$ [[reflections|06BRDF3F0E]] in hyperplanes of $V$.

## Remarks

The identity map is the composition of zero reflections.
In $\R^2$, each rotation is the composition of two reflections in lines through the origin.
The theorem holds more generally for nondegenerate symmetric bilinear forms on spaces over fields of characteristic not $2$.

## Explanations

### L1 human

Every rigid linear motion of $n$-dimensional space is a sequence of at most $n$ mirror reflections.
So reflections are the building blocks of all isometries.
In the plane, a rotation by $\theta$ is the product of two reflections in lines at the angle $\theta/2$.
</document>

<document source="https://math2.tamnd.com/node/0VMWPGGG3C.md" tag="0VMWPGGG3C" kind="concept">
# Smallest prime factor

## Summary

The smallest prime factor of a natural number $n \ge 2$ is the least prime that divides $n$.
The [[definition of the smallest prime factor|0S2SYHTWPY]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0VN4H8M12X.md" tag="0VN4H8M12X" kind="statement">
# If 2^n + 1 is prime then n is a power of 2

## Statement

Let $n$ be a positive integer.
If $2^n + 1$ is a [[prime number|0SN0MAEKYX]], then $n = 2^k$ for a natural number $k$.
So every prime of the form $2^n + 1$ with $n \ge 1$ is a [[Fermat number|0R03B54V4Z]].

## Remarks

The proof: if $n = k m$ with $m > 1$ odd, then $x + 1$ divides $x^m + 1$ for $x = 2^k$. So $2^k + 1$ is a proper divisor of $2^n + 1$.
For example, $2^6 + 1 = 65 = 5 \cdot 13$, and $5 = 2^2 + 1$.
The converse is false, because $F_5 = 2^{32} + 1$ is composite.

## Explanations

### L1 human

An odd factor of the exponent gives a factorization of $2^n + 1$.
So the only exponents that can give primes are the powers of $2$, which have no odd factor except $1$.
</document>

<document source="https://math2.tamnd.com/node/0VNE161QCK.md" tag="0VNE161QCK" kind="statement">
# Polynomials of degree two or three are irreducible iff they have no roots

## Statement

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ with $\deg f = 2$ or $\deg f = 3$.
Then $f$ is [[irreducible|00X2A9G9W3]] over $K$ if and only if $f$ has no [[root|00EHX6FWC0]] in $K$.

## Remarks

The proof: if $f = g h$ with $\deg g, \deg h \ge 1$, then $\deg g + \deg h \le 3$, so one factor has degree $1$. A factor $a x + b$ of degree $1$ gives the root $-b/a$. The [[factor theorem|0K3MWA7DMC]] gives the converse.
For example, $x^3 - 2$ has no root in $\mathbb{Q}$, so it is irreducible over $\mathbb{Q}$.
The statement fails in degree $4$: $(x^2 + 1)^2$ has no real root, but it is not irreducible over $\mathbb{R}$.

## Explanations

### L1 human

A factorisation of a polynomial of degree $2$ or $3$ must have a factor of degree $1$.
A factor of degree $1$ is the same as a root.
So to test irreducibility in low degree, you only look for roots.
</document>

<document source="https://math2.tamnd.com/node/0VP3A1290K.md" tag="0VP3A1290K" kind="statement">
# Pythagorean theorem

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $d$ be the [[Euclidean distance|0NPPGHQNXQ]], and let $\angle ABC$ be the [[angle at $B$|0GJAJC6TNZ]].
Then
$$d(A, C)^2 = d(A, B)^2 + d(B, C)^2$$
if and only if $\angle ABC = \pi / 2$.

## Remarks

The direction "if" is the classical theorem: in a right triangle, the square of the hypotenuse is the sum of the squares of the legs, $a^2 + b^2 = c^2$. The direction "only if" is its converse.
In vector form: for vectors $u$ and $v$ with $u \cdot v = 0$, $\|u + v\|^2 = \|u\|^2 + \|v\|^2$.
The theorem is the special case $\gamma = \pi / 2$ of the [[law of cosines|060ZVZBV3E]].
Mathlib states the equivalence for all points $A$, $B$ and $C$, with its convention for the angle at a repeated point.

## History

The theorem carries the name of the Greek philosopher Pythagoras. Euclid's Elements proves it as Proposition 47 of Book I, and the converse as Proposition 48.

## Explanations

### L1 human

Build a square on each side of a right triangle. The two small squares together have the same area as the big square on the longest side. For the sides $3$, $4$ and $5$: $9 + 16 = 25$.
</document>

<document source="https://math2.tamnd.com/node/0VP4TBVJ19.md" tag="0VP4TBVJ19" kind="definition">
# Frobenius automorphism of a finite field

## Definition

Let $F$ be a [[finite field|08RTDHX1SK]] of characteristic $p$.
The Frobenius automorphism of $F$ is the map $\varphi \colon F \to F$, $\varphi(x) = x^p$.

## Remarks

The map $\varphi$ is the [[Frobenius endomorphism|0GHDDVAH8P]] of $F$. It is injective, as is every field homomorphism, so it is bijective, because $F$ is finite.
The map $\varphi$ fixes every element of the prime subfield $\mathbb{F}_p$, by Fermat's little theorem.
If $|F| = p^n$, then $\varphi^n$ is the identity, and the automorphisms of $F$ are exactly $\mathrm{id}, \varphi, \dots, \varphi^{n-1}$.
So $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by $\varphi$.

## Explanations

### L1 human

Raising to the power $p$ respects addition in characteristic $p$, by the freshman's dream.
In a finite field, this map permutes the elements, and every symmetry of the field is a power of it.
</document>

<document source="https://math2.tamnd.com/node/0VP56B4EMX.md" tag="0VP56B4EMX" kind="definition">
# Power series

## Definition

Let $c$ be a real number and let $(a_n)_{n \ge 0}$ be a real sequence.
The power series with center $c$ and coefficients $a_n$ is the [[series|0JH4MN4K1C]]
$$\sum_{n=0}^{\infty} a_n (x - c)^n,$$
taken as a function of the real variable $x$.

## Remarks

The series always converges at $x = c$, with sum $a_0$.
The set of $x$ where it converges is an interval around $c$. Its half length is the [[radius of convergence|0660QX1VPR]].
Polynomials are power series with only finitely many nonzero coefficients.
Complex power series use the same definition, with a complex variable $z$.

## Explanations

### L1 human

A power series is a polynomial with infinitely many terms.
For each $x$, we put $x$ into every term and add up the resulting series of numbers.
</document>

<document source="https://math2.tamnd.com/node/0VPCJWGMP9.md" tag="0VPCJWGMP9" kind="definition">
# Space of linear maps

## Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$.
The *space of linear maps* $\Hom_F(V, W)$, also written $L(V, W)$, is the set of all [[linear maps|0NRYHJPQG2]] $V \to W$. It has the operations
$$(S + T)(v) = S(v) + T(v), \qquad (aT)(v) = a\, T(v)$$
for $S, T \in \Hom_F(V, W)$, $a \in F$ and $v \in V$.
It is a vector space over $F$, with the zero map as zero vector.

## Remarks

For $V = F^n$ and $W = F^m$, the matrix map is an isomorphism $\Hom_F(F^n, F^m) \cong M_{m \times n}(F)$.
The [[dimension of the space of linear maps|0Z9N1ER0W0]] is $\dim V \cdot \dim W$ in finite dimensions.
For $W = F$, it is the [[dual space|07JBSZE0V4]] of $V$, and for $W = V$, it is the space of [[endomorphisms|0J7H0YBFDJ]] of $V$.

## Explanations

### L1 human

Linear maps can be added and scaled, value by value, and the result is again linear.
So the linear maps from $V$ to $W$ form a vector space themselves.
In coordinates, this is the vector space of matrices of a fixed size.
</document>

<document source="https://math2.tamnd.com/node/0VPFBAY7MW.md" tag="0VPFBAY7MW" kind="definition">
# Prime subfield

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
The prime subfield of $K$ is the intersection of all the [[subfields|07QY5RPFBR]] of $K$.
It is the smallest subfield of $K$.

## Remarks

If the [[characteristic|0P7CH1CND5]] of $K$ is a prime $p$, then the prime subfield is isomorphic to $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
If the characteristic of $K$ is $0$, then the prime subfield is isomorphic to $\mathbb{Q}$.
The prime subfield of $\mathbb{R}$ is $\mathbb{Q}$.

## Explanations

### L1 human

Every subfield contains $0$ and $1$, and so it contains all the sums $1 + 1 + \dots + 1$ and their quotients.
These elements form the prime subfield.
In the real numbers, they are exactly the rational numbers.
</document>

<document source="https://math2.tamnd.com/node/0VPK4X73Q0.md" tag="0VPK4X73Q0" kind="statement">
# Rule of product

## Statement

Let $A$ and $B$ be finite sets.
Then the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$ is finite and $|A \times B| = |A| \cdot |B|$.

More generally, for finite sets $A_1, \dots, A_k$, the product $A_1 \times \dots \times A_k$ has $|A_1| \cdots |A_k|$ elements.

## Remarks

A common form of the rule: a choice is made in $k$ steps, and step $i$ has $n_i$ options, whatever the earlier steps are. Then there are $n_1 n_2 \cdots n_k$ choices in total.
The [[number of functions between finite sets|08QQH9F7G0]] is a direct consequence.

## Explanations

### L1 human

If you choose one item from a first group and then one item from a second group, you multiply the sizes.
For example, with 3 shirts and 4 trousers there are $3 \cdot 4 = 12$ outfits.
</document>

<document source="https://math2.tamnd.com/node/0VQA7F720Z.md" tag="0VQA7F720Z" kind="concept">
# Sequence tending to infinity

## Summary

A real sequence tends to infinity when its terms eventually exceed each given bound. Such a sequence diverges, and we say that its limit is $+\infty$. The node [[sequence tending to infinity|09AZCV6GMC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VQQCVG8WA.md" tag="0VQQCVG8WA" kind="concept">
# Combinations with repetition

## Summary

A combination with repetition is a selection of $k$ elements from a set in which the order does not matter and elements can repeat. Such selections are multisets of size $k$, and stars and bars counts them. The node [[combinations with repetition|0YEQG3VBYC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VQYCCQVJ5.md" tag="0VQYCCQVJ5" kind="concept">
# Nowhere dense set

## Summary

A nowhere dense subset of a topological space is a set whose closure has empty interior.
Such a set is not dense in any nonempty open set.
</document>

<document source="https://math2.tamnd.com/node/0VR671PHE5.md" tag="0VR671PHE5" kind="concept">
# Symmetric matrix

## Summary

A symmetric matrix is a square matrix that is equal to its transpose.
</document>

<document source="https://math2.tamnd.com/node/0VRD8GC5BM.md" tag="0VRD8GC5BM" kind="definition">
# Isometric isomorphism

## Definition

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]].
An isometric isomorphism from $X$ to $Y$ is a [[bijective|0SNQEF7Z53]] [[isometry|0878DN3WCS]] $f \colon X \to Y$.
The spaces $X$ and $Y$ are isometric when there is an isometric isomorphism from $X$ to $Y$.

## Remarks

Because each isometry is injective, an isometry is an isometric isomorphism if and only if it is surjective.
The inverse of an isometric isomorphism is again an isometric isomorphism, and the composition of two of them is one.
Each isometric isomorphism is a [[homeomorphism|05GZ11PWRY]].
The map $x \mapsto x + 1$ is an isometric isomorphism from $[0, 1]$ to $[1, 2]$ with the usual metric.

## Explanations

### L1 human

Isometric spaces are copies of each other: a bijection matches their points, and it keeps all the distances.
Each fact about distances in one space is also true in the other space.
</document>

<document source="https://math2.tamnd.com/node/0VRWZDWW6B.md" tag="0VRWZDWW6B" kind="definition">
# Binomial distribution

## Definition

Let $n \ge 0$ be an integer and let $p \in [0, 1]$.
The binomial distribution with parameters $n$ and $p$ is the distribution on $\{0, 1, \dots, n\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \binom{n}{k} p^k (1 - p)^{n-k} \quad \text{for } k = 0, 1, \dots, n.$$
Here $\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]]. One writes $X \sim \operatorname{B}(n, p)$ or $X \sim \operatorname{Bin}(n, p)$.

## Remarks

The values add up to $1$ by the [[binomial theorem|0EHWPWGHB2]]: $\sum_{k} \binom{n}{k} p^k (1 - p)^{n-k} = (p + 1 - p)^n = 1$.
The number of successes in $n$ independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$ has this distribution. The factor $\binom{n}{k}$ counts the positions of the $k$ successes.
The mean is $np$ and the variance is $np(1 - p)$.
The probability of exactly $2$ heads in $4$ tosses of a fair coin is $\binom{4}{2} / 2^4 = 6/16$.

## Explanations

### L1 human

Repeat a yes-or-no experiment $n$ times, independently, and count the yes answers.
The count has the binomial distribution. Each sequence with $k$ yes answers has probability $p^k (1 - p)^{n-k}$, and there are $\binom{n}{k}$ such sequences.
</document>

<document source="https://math2.tamnd.com/node/0VS0MP1P5H.md" tag="0VS0MP1P5H" kind="definition">
# Abundant number

## Definition

A positive integer $n$ is abundant if the sum of its positive divisors other than $n$ is greater than $n$.
Equivalently, $\sigma(n) > 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The smallest abundant number is $12$, since $1 + 2 + 3 + 4 + 6 = 16 > 12$.
The smallest odd abundant number is $945$.
Each multiple $kn$ with $k \ge 2$ of a [[perfect number|0SB7Q8SYSF]] $n$ is abundant.

## Explanations

### L1 human

An abundant number has "too many" divisors: its smaller divisors add up to more than the number.
Compare with [[perfect numbers|0SB7Q8SYSF]], where the sum is equal, and [[deficient numbers|0G01TS1RRP]], where it is smaller.
</document>

<document source="https://math2.tamnd.com/node/0VSB0XF4G7.md" tag="0VSB0XF4G7" kind="definition">
# Indicator function

## Definition

Let $X$ be a set and let $A$ be a [[subset|0FE8Q5S42Y]] of $X$.
The indicator function of $A$ is the [[function|0XHDTE5YMZ]] $\mathbf{1}_A \colon X \to \{0, 1\}$ with
$$\mathbf{1}_A(x) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}$$

## Remarks

Other names are characteristic function and the notation $\chi_A$.
For subsets $A$ and $B$ of $X$, $\mathbf{1}_{A \cap B} = \mathbf{1}_A \mathbf{1}_B$ and $\mathbf{1}_{A^c} = 1 - \mathbf{1}_A$.
The map $A \mapsto \mathbf{1}_A$ is a [[bijection|0SNQEF7Z53]] from the [[power set|0BC7KX51Z8]] of $X$ to the set of functions from $X$ to $\{0, 1\}$.

## Explanations

### L1 human

The indicator function answers "is $x$ in $A$?" with $1$ for yes and $0$ for no.
For $A$ the even integers, $\mathbf{1}_A(4) = 1$ and $\mathbf{1}_A(5) = 0$.
</document>

<document source="https://math2.tamnd.com/node/0VSNHTHK8T.md" tag="0VSNHTHK8T" kind="statement">
# A tree on n vertices has n − 1 edges

## Statement

Let $T$ be a finite [[tree|0MJ0JS72DZ]] with $n$ vertices.
Then $T$ has exactly $n - 1$ edges.

## Remarks

A tree has at least one vertex, because it is connected. Thus $n \ge 1$.
Proof by induction on $n$: for $n \ge 2$ delete a leaf, which exists by the [[leaf lemma|0H8Q4MANBD]]. The rest is a tree with $n - 1$ vertices.
A consequence: the sum of the degrees in a tree is $2n - 2$.

## Explanations

### L1 human

Build a tree by adding one vertex at a time, each joined to the tree by one new edge.
You start with $1$ vertex and $0$ edges, so at the end there is always one edge fewer than vertices.
</document>

<document source="https://math2.tamnd.com/node/0VSNX4P2Q0.md" tag="0VSNX4P2Q0" kind="statement">
# Cancellation laws

## Statement

Let $G$ be a [[group|G5S1M9]] and let $a, b, c \in G$.
If $ab = ac$, then $b = c$.
If $ba = ca$, then $b = c$.

## Remarks

The first law is left cancellation and the second law is right cancellation.
To prove the first law, multiply $ab = ac$ on the left by $a^{-1}$.
A consequence is that in the [[Cayley table|0YB2NRP0T5]] of a finite group, no element occurs two times in one row or in one column.
The laws can fail in a [[monoid|0N6R902FG1]]: in $(\mathbb{Z}, \cdot)$, $0 \cdot 1 = 0 \cdot 2$ but $1 \neq 2$.

## Explanations

### L1 human

In a group, you can remove the same element from both sides of an equation, if it is on the same side of each product.
This works because each element has an inverse that undoes it.
Take care with the side: $ab = ca$ does not give $b = c$ in general.
</document>

<document source="https://math2.tamnd.com/node/0VSSPAFM2C.md" tag="0VSSPAFM2C" kind="concept">
# Characteristic polynomial

## Summary

The characteristic polynomial of a square matrix $A$ is the polynomial $\det(t I - A)$ in the variable $t$.
Its roots in the field are the eigenvalues of $A$.
</document>

<document source="https://math2.tamnd.com/node/0VT2QK6X6Q.md" tag="0VT2QK6X6Q" kind="concept">
# Riccati equation

## Summary

A Riccati equation is a first-order equation $y' = q_0(x) + q_1(x) y + q_2(x) y^2$, with a right side quadratic in $y$.
It can be changed into a second-order linear equation. See the [[definition|0E1ZZBHZ3Z]].
</document>

<document source="https://math2.tamnd.com/node/0VT6P0GNQR.md" tag="0VT6P0GNQR" kind="concept">
# Gram–Schmidt process

## Summary

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$ be [[linearly independent|0CRMGPAH83]].
The *Gram–Schmidt process* defines vectors $u_1, \dots, u_k$ by $u_1 = v_1$ and, for $2 \le j \le k$,
$$u_j = v_j - \sum_{i=1}^{j-1} \frac{\langle v_j, u_i \rangle}{\langle u_i, u_i \rangle} u_i.$$
Then the vectors $e_j = u_j / \norm{u_j}$ form an [[orthonormal family|0RW9TNNT87]] with the same [[span|0DTY8B9HMS]] as $v_1, \dots, v_j$ for each $j$.

## Remarks

Each step subtracts from $v_j$ its projections on the earlier vectors $u_i$.
For $v_1 = (1, 1)$ and $v_2 = (1, 0)$ in $\R^2$, it gives $u_1 = (1, 1)$ and $u_2 = (1/2, -1/2)$.
In floating point arithmetic, the modified Gram–Schmidt process is more stable.
In matrix form, the process gives the [[QR decomposition|0ERWG4NXQ9]].

## Explanations

### L1 human

The Gram–Schmidt process turns any independent list of vectors into a list of perpendicular unit vectors.
It treats the vectors one at a time, and it removes from each one its components along the earlier ones.
Then it scales each result to length one.
</document>

<document source="https://math2.tamnd.com/node/0VTAD25A3G.md" tag="0VTAD25A3G" kind="statement">
# Minimal polynomial divides every annihilating polynomial

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.
If $p \in F[x]$ and $p(T) = 0$, then $m_T$ divides $p$.

## Remarks

The proof divides with remainder: $p = q m_T + r$ with $\deg r < \deg m_T$.
Then $r(T) = 0$, so $r = 0$ by the minimality of $\deg m_T$.

## Explanations

### L1 human

Every polynomial that kills $T$ is a multiple of the minimal polynomial.
If a matrix satisfies $A^2 = A$, then its minimal polynomial divides $x^2 - x$.
So it is $x$, $x - 1$ or $x^2 - x$.
</document>

<document source="https://math2.tamnd.com/node/0VTAXQEWQD.md" tag="0VTAXQEWQD" kind="example">
# Sup metric on R^n

## Example

Let $n \ge 1$ be an integer.
For $x = (x_1, \ldots, x_n)$ and $y = (y_1, \ldots, y_n)$ in $\mathbb{R}^n$, let
$$d_\infty(x, y) = \max_{1 \le i \le n} |x_i - y_i|.$$
Then $d_\infty$ is a [[metric|063W5JMY85]] on $\mathbb{R}^n$, the sup metric.

## Remarks

It is also called the Chebyshev distance or the maximum metric.
For all $x, y$ in $\mathbb{R}^n$, $d_\infty(x, y) \le d_2(x, y) \le d_1(x, y) \le n \, d_\infty(x, y)$, where $d_2$ is the [[Euclidean metric|0E1MMCAY5B]] and $d_1$ is the [[taxicab metric|07CJNE6ZAQ]].
It is the limit of the [[p-metric|0TFDZN6AXP]] $d_p$ as $p \to \infty$.

## Explanations

### L1 human

The sup metric takes only the largest difference of the coordinates.
On a chessboard, it is the number of moves that a king needs to go from one square to another.
</document>

<document source="https://math2.tamnd.com/node/0VTCABADBE.md" tag="0VTCABADBE" kind="example">
# Turán graph

## Example

Let $n \ge r \ge 1$ be integers.
The Turán graph $T(n, r)$ has $n$ vertices, split into $r$ parts whose sizes differ by at most $1$.
Two vertices are adjacent if and only if they are in different parts.

## Remarks

Two vertices in the same part are not adjacent. Among any $r + 1$ vertices, two are in the same part, so $T(n, r)$ has no [[clique|0G52BPNPKG]] with $r + 1$ vertices.
The number of edges of $T(n, r)$ is $\left\lfloor \left(1 - \frac{1}{r}\right) \frac{n^2}{2} \right\rfloor$.
$T(n, 2)$ is the [[complete bipartite graph|0V0GC7BQAN]] $K_{\lfloor n/2 \rfloor, \lceil n/2 \rceil}$, and $T(n, n)$ is the [[complete graph|03WKB2H474]] $K_n$.

## Explanations

### L1 human

To build $T(n, r)$, split $n$ people into $r$ teams of almost equal size, and join two people when they are in different teams.
It is the graph with the most edges that has no clique of size $r + 1$.
</document>

<document source="https://math2.tamnd.com/node/0VTH8KPCG2.md" tag="0VTH8KPCG2" kind="concept">
# Ring homomorphism

## Summary

A ring homomorphism is a map between two rings that preserves addition, multiplication and the identity element.
The [[definition of a ring homomorphism|0GGE0Z0JQ1]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0VTN5RKNC5.md" tag="0VTN5RKNC5" kind="definition">
# Standard deviation

## Definition

Let $X$ be a real random variable with finite [[variance|007QED3E0Y]].
The standard deviation of $X$ is the [[square root|0VB8BFRABH]]
$$\sigma_X = \sqrt{\operatorname{Var}(X)} = \sqrt{E\left[(X - E[X])^2\right]}.$$

## Remarks

The standard deviation is in the same units as $X$, but the variance is in squared units.
For $a, b \in \mathbb{R}$, $\sigma_{aX + b} = |a| \, \sigma_X$.
For the [[normal distribution|07RXBVTVZB]] $\mathcal{N}(\mu, \sigma^2)$, the standard deviation is $\sigma$. About $68\%$ of the mass lies within one standard deviation of the mean.

## Explanations

### L1 human

The standard deviation is a typical size of the distance between a value and the mean.
A small standard deviation means that the values stay close to the mean. A large one means that they spread out.
</document>

<document source="https://math2.tamnd.com/node/0VTQTPQGAV.md" tag="0VTQTPQGAV" kind="definition">
# Basis

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A *basis* of $V$ is a family $(b_i)_{i \in I}$ of vectors of $V$ that is [[linearly independent|0CRMGPAH83]] and [[spans|088RYKAWK9]] $V$.

## Remarks

A family $(b_i)_{i \in I}$ is a basis if and only if each $v \in V$ is equal to $\sum_{i \in I} a_i b_i$ for exactly one finitely supported family of scalars $(a_i)_{i \in I}$.
The scalars $a_i$ are the [[coordinates|04CYQHN6MC]] of $v$ in the basis.
The [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$ is a basis of $F^n$; the vectors $(1, 1)$ and $(1, -1)$ form another basis of $\R^2$.
[[Every vector space has a basis|00MS9VMQ9Z]], and all bases of a space have the same cardinality.

## Explanations

### L1 human

A basis is a set of directions that is just right: enough to reach every vector, and none of them redundant.
Once we fix a basis, each vector gets a unique list of coordinates.
A vector space has many bases, but all of them have the same number of vectors.
</document>

<document source="https://math2.tamnd.com/node/0VTQXTCR2G.md" tag="0VTQXTCR2G" kind="concept">
# Transcendental number

## Summary

A transcendental number is a complex number that is not algebraic, such as $e$ or $\pi$.
The [[definition of a transcendental number|079EGSD32Y]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0VV04M41AN.md" tag="0VV04M41AN" kind="definition">
# Euler's totient function

## Definition

Let $n$ be a positive integer.
Euler's totient function $\varphi(n)$ is the number of integers $k$ with $1 \le k \le n$ such that $k$ and $n$ are [[coprime|0GFKP8F8AG]].

## Remarks

For example, $\varphi(1) = 1$, $\varphi(9) = 6$ and $\varphi(12) = 4$. The integers coprime to $12$ in $\{1, \dots, 12\}$ are $1, 5, 7, 11$.
The number $\varphi(n)$ is the number of elements of a [[reduced residue system|0J0QEN1DZR]] modulo $n$.
If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$, then $\varphi(n) = n \prod_{i=1}^{r} \left(1 - \frac{1}{p_i}\right)$.
The Lean library Mathlib sets $\varphi(0) = 0$.

## Explanations

### L1 human

Count the numbers from $1$ to $n$ that have no common factor with $n$.
For $n = 10$, these are $1, 3, 7, 9$, so $\varphi(10) = 4$.
</document>

<document source="https://math2.tamnd.com/node/0VV0HSW1ZS.md" tag="0VV0HSW1ZS" kind="definition">
# Convex function

## Definition

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be a function.
The function $f$ is convex if
$$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y)$$
for all $x, y \in I$ and all $t \in [0, 1]$.

## Remarks

Geometrically, each chord of the graph lies on or above the graph.
For example, $x^2$, $|x|$, and $e^x$ are convex on $\mathbb{R}$. Linear functions are convex.
A twice differentiable function with [[$f'' \ge 0$ is convex|0Z41C17TGP]].
Mathlib defines convexity on a set as `ConvexOn`.

## Explanations

### L1 human

A convex function bends upward, like a bowl.
The straight segment between two points of the graph never goes below the graph.
</document>

<document source="https://math2.tamnd.com/node/0VV3ZMRF88.md" tag="0VV3ZMRF88" kind="concept">
# Number of divisors

## Summary

The number of divisors $d(n)$ of a positive integer $n$ is the count of its positive divisors.
The [[definition of the number of divisors|0BKJT6WG96]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0VVK64Q4JK.md" tag="0VVK64Q4JK" kind="concept">
# Discriminant of a polynomial

## Summary

The discriminant of a polynomial is a quantity computed from its coefficients that is zero exactly when the polynomial has a repeated root.
The [[definition of the discriminant|0CCPN09EMJ]] gives the formula in terms of the roots.
</document>

<document source="https://math2.tamnd.com/node/0VVPQNVM8S.md" tag="0VVPQNVM8S" kind="statement">
# Order of a power

## Statement

Let $G$ be a [[group|G5S1M9]] and let $g \in G$ be an element of finite [[order|0WE2NHZJ7B]] $n$.
Let $k$ be an integer.
Then the order of $g^k$ is $n / \gcd(n, k)$.

## Remarks

Here $\gcd(n, 0) = n$, so that $g^0 = e$ has order $1$.
In particular, $g^k$ has order $n$ if and only if $\gcd(n, k) = 1$.
For example, if $g$ has order $12$, then $g^8$ has order $12 / 4 = 3$.

## Explanations

### L1 human

A power of an element returns to the identity faster than the element itself, or at the same time.
If $g$ has order $12$, then $g^8$ is at the identity after $k$ steps exactly when $12$ divides $8k$, that is, when $3$ divides $k$.
Thus $g^8$ has order $3$.
</document>

<document source="https://math2.tamnd.com/node/0VVY4H92G2.md" tag="0VVY4H92G2" kind="statement">
# Term test

## Statement

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers.
Then $a_n \to 0$ as $n \to \infty$.
Equivalently, if $(a_n)$ does not converge to $0$, then the series $\sum a_n$ diverges.

## Remarks

For the proof, let $s$ be the sum. Then $a_n = s_n - s_{n-1} \to s - s = 0$.
The converse is false. The terms of the [[harmonic series|0CXTMAY5PX]] tend to $0$, and the series diverges.
For example, $\sum (-1)^n$ and $\sum n/(n+1)$ diverge by this test.

## Explanations

### L1 human

If you keep adding terms that do not get small, the total cannot settle down.
So small terms are necessary for convergence, but they are not enough.
</document>

<document source="https://math2.tamnd.com/node/0VW90BBSJ4.md" tag="0VW90BBSJ4" kind="concept">
# Continuously differentiable function

## Summary

A continuously differentiable function is a differentiable function whose derivative is continuous. The classes $C^k$ ask for $k$ derivatives, the last one continuous. The node [[continuously differentiable function|06SKCHDQT5]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VW9E4QXR6.md" tag="0VW9E4QXR6" kind="concept">
# Likelihood function

## Summary

The likelihood function gives, for each value of an unknown parameter, the probability or the density of the observed data under that value. It is a function of the parameter, with the data fixed.
The node [[likelihood function|05FSTXM92W]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VX0CPRQTZ.md" tag="0VX0CPRQTZ" kind="statement">
# Euler line

## Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$. Let $O$ be its [[circumcentre|0F98FZK14A]], $G = \frac{1}{3}(A + B + C)$ its [[centroid|0X7EEWES5G]] and $H$ its [[orthocentre|0WB619CA0V]].
Then
$$H = O + 3\,(G - O).$$
In particular, the points $O$, $G$ and $H$ are [[collinear|0AFNNTRTJE]].
If the triangle is not equilateral, then $O \neq G$, and the line through $O$ and $G$ is the Euler line of the triangle.

## Remarks

The formula says that $G$ lies between $O$ and $H$, with $d(G, H) = 2\, d(O, G)$.
An equivalent form is $H - O = (A - O) + (B - O) + (C - O)$.
The centre of the nine-point circle is the midpoint of $O$ and $H$, so it is also on the Euler line.
In an equilateral triangle, the points $O$, $G$ and $H$ are the same point.

## History

Euler showed in 1765 that the orthocentre, the circumcentre and the centroid of a triangle are collinear.

## Explanations

### L1 human

Three famous centres of a triangle always lie on one straight line. The centroid is on this line between the other two. It is twice as far from the orthocentre as from the circumcentre.
</document>

<document source="https://math2.tamnd.com/node/0VX9EP2GS8.md" tag="0VX9EP2GS8" kind="concept">
# Minimal polynomial (field theory)

## Summary

Let $a$ be algebraic over a field $K$.
The minimal polynomial of $a$ over $K$ is the monic polynomial of smallest degree in $K[x]$ that has $a$ as a root.
The [[definition of the minimal polynomial|0GK1WB9R98]] states it.
</document>

<document source="https://math2.tamnd.com/node/0VXJHZ1G9J.md" tag="0VXJHZ1G9J" kind="concept">
# Standard score

## Summary

The standard score of a value is the number of standard deviations by which it lies above or below the mean.
The node [[standard score|05RZ4YZG4E]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0VXYV3NQQ1.md" tag="0VXYV3NQQ1" kind="statement">
# Gradient is orthogonal to level sets

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$, and let $c \in \mathbb{R}$.
Let $I \subseteq \mathbb{R}$ be an open interval and let $\gamma \colon I \to U$ be a curve that lies in the [[level set|0Y5Q8Q9WJ7]] $L_c(f)$, so that $f(\gamma(t)) = c$ for each $t \in I$.
Let $t_0 \in I$, and suppose that $\gamma$ is differentiable at $t_0$ and $f$ is differentiable at $\gamma(t_0)$.
Then $\nabla f(\gamma(t_0)) \cdot \gamma'(t_0) = 0$.

## Remarks

The proof is one step: $f \circ \gamma$ is constant, so its derivative is $0$, and by the [[chain rule for a curve|0A8TJ0VHC7]] that derivative is $\nabla f(\gamma(t_0)) \cdot \gamma'(t_0)$.
So the [[gradient|0NS2QYVC4Y]] is orthogonal to each tangent vector of the level set at the point. Let $\nabla f \ne 0$ and let $f$ be of class $C^1$. Then the level set near the point is a smooth hypersurface, and $\nabla f$ is a normal vector to it.
For $f(x, y, z) = x^2 + y^2 + z^2$, the level set at $c = 1$ is the unit sphere, and $\nabla f(p) = 2p$ is normal to the sphere at $p$.

## Explanations

### L1 human

Along a contour line the height does not change, so the slope along the line is $0$.
The gradient gives the slope in each direction by a dot product, so it must be at right angles to the contour line.
</document>

<document source="https://math2.tamnd.com/node/0VY3WPP9Y3.md" tag="0VY3WPP9Y3" kind="statement">
# Bézout's identity in a Euclidean domain

## Statement

Let $R$ be a [[Euclidean domain|0GJ2HRG95T]] and let $a, b \in R$.
Then $a$ and $b$ have a [[greatest common divisor|0F7KNJ1ZYN]] $d$, and there are $s, t \in R$ with
$$d = s a + t b.$$

## Remarks

The [[Euclidean algorithm|023VY493K0]] computes $d$. The extended Euclidean algorithm also computes $s$ and $t$, by back substitution.
In $\mathbb{Z}$: $\gcd(252, 198) = 18 = 4 \cdot 252 - 5 \cdot 198$.
The same identity holds in every [[principal ideal domain|0RZYDVZ475]], because the ideal $(a, b)$ is principal.
If $d \neq 0$, the coefficients are not unique: $s + k b/d$ and $t - k a/d$ also work for each $k \in R$.

## Explanations

### L1 human

The gcd of two elements is a combination of them with coefficients in the ring.
For example, $\gcd(4, 6) = 2 = 2 \cdot 4 - 1 \cdot 6$.
This is the main tool to solve linear equations $s a + t b = c$ in a Euclidean domain.
</document>

<document source="https://math2.tamnd.com/node/0VYCDGMJJJ.md" tag="0VYCDGMJJJ" kind="concept">
# Row echelon form

## Summary

A matrix is in row echelon form when its zero rows are at the bottom.
Also, the first nonzero entry of each row is to the right of the first nonzero entry of the row above.
</document>

<document source="https://math2.tamnd.com/node/0VZ48N083F.md" tag="0VZ48N083F" kind="concept">
# Coprime integers

## Summary

Two integers are coprime when no integer other than $1$ and $-1$ divides both of them.
The [[definition of coprime integers|0GFKP8F8AG]] states this with the greatest common divisor.
</document>

<document source="https://math2.tamnd.com/node/0VZVV8VME5.md" tag="0VZVV8VME5" kind="statement">
# Compact closed unit ball implies finite dimension

## Statement

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$.
If the closed unit ball $\{ x \in X : \|x\| \le 1 \}$ is [[compact|00R0X1WGPX]], then $X$ is finite-dimensional.

## Remarks

Proof: if $X$ is infinite-dimensional, build unit vectors $x_1, x_2, \ldots$ by [[Riesz's lemma|097C7KCAT5]], with $x_{n+1}$ at distance at least $\tfrac{1}{2}$ from the span of $x_1, \ldots, x_n$. That span is finite-dimensional, so it is [[complete|0ABP2F2ZP7]] and closed. Then $\|x_m - x_n\| \ge \tfrac{1}{2}$ for $m \neq n$, so no subsequence converges, and the ball is not [[sequentially compact|0K8SGQSV7G]].
With the [[Heine–Borel theorem in finite dimension|0PHT6NBYPB]], we get: a normed space is finite-dimensional if and only if its closed unit ball is compact.
Equivalently, a normed space is locally compact if and only if it is finite-dimensional.

## Explanations

### L1 human

In infinite dimensions, the closed unit ball is never compact: there is always room for infinitely many unit vectors that stay far apart.
This is one of the main differences between finite and infinite dimensions.
</document>

<document source="https://math2.tamnd.com/node/0W02QESPJQ.md" tag="0W02QESPJQ" kind="concept">
# Orthogonal matrix

## Summary

An orthogonal matrix is a real square matrix whose transpose is its inverse, so that its columns form an orthonormal basis.
</document>

<document source="https://math2.tamnd.com/node/0W0KVBM54S.md" tag="0W0KVBM54S" kind="definition">
# Cyclotomic polynomial

## Definition

Let $n \ge 1$ be an integer.
The $n$-th cyclotomic polynomial is
$$\Phi_n(x) = \prod_{\substack{1 \le k \le n \\ \gcd(k, n) = 1}} \left(x - e^{2 \pi i k / n}\right) \in \mathbb{C}[x].$$
Its roots are the primitive $n$-th roots of unity: the complex numbers $\zeta$ with $\zeta^n = 1$ and $\zeta^m \neq 1$ for $0 < m < n$.

## Remarks

The polynomial $\Phi_n$ is [[monic|0BDX4MBKT4]] of degree $\varphi(n)$, where $\varphi$ is Euler's totient function.
For each $n \ge 1$, $x^n - 1 = \prod_{d \mid n} \Phi_d(x)$. By induction on $n$, this shows that $\Phi_n$ has integer coefficients.
The first values are $\Phi_1 = x - 1$, $\Phi_2 = x + 1$, $\Phi_3 = x^2 + x + 1$, $\Phi_4 = x^2 + 1$ and $\Phi_6 = x^2 - x + 1$.
For a prime $p$, $\Phi_p = x^{p-1} + x^{p-2} + \dots + x + 1$.

## Explanations

### L1 human

The $n$-th roots of unity are the corners of a regular $n$-gon on the unit circle.
The cyclotomic polynomial $\Phi_n$ collects only the corners that are not roots of unity of a smaller order.
The name means "circle dividing".
</document>

<document source="https://math2.tamnd.com/node/0W0TZDRF9C.md" tag="0W0TZDRF9C" kind="statement">
# Continuous functions on a closed bounded interval are uniformly continuous

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then $f$ is [[uniformly continuous|0A3CK6DVVB]] on $[a, b]$.

## Remarks

Suppose not. Then there is $\varepsilon > 0$ and points $x_n, y_n$ with $|x_n - y_n| < 1/n$ and $|f(x_n) - f(y_n)| \ge \varepsilon$. A [[convergent subsequence|0KDJD249DT]] of $(x_n)$ leads to a contradiction with continuity.
The interval must be closed and bounded. The function $1/x$ is continuous on $(0, 1]$ but not uniformly continuous there.
The name honors Eduard Heine and Georg Cantor. Mathlib states it for compact sets, as `IsCompact.uniformContinuousOn_of_continuous`.

## Explanations

### L1 human

On a closed bounded interval, continuity at each point upgrades to uniform continuity.
There is no room for the function to become steeper and steeper.
</document>

<document source="https://math2.tamnd.com/node/0W1C598HMS.md" tag="0W1C598HMS" kind="statement">
# Euler's formula for planar graphs

## Statement

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[planar graph|029JQEK15B]], drawn in the plane with no crossing edges.
Let $v$, $e$ and $f$ be the numbers of vertices, edges and faces of the drawing, where the unbounded face counts.
Then $v - e + f = 2$.

## Remarks

Proof idea: if the graph has a cycle, delete an edge of the cycle. This removes one edge and one face.
At the end you have a [[spanning tree|0FQ5B8S1ZC]], with $e = v - 1$ and $f = 1$.
The same formula holds for the vertices, edges and faces of a convex polyhedron.

## Explanations

### L1 human

Draw a connected graph without crossings and count its vertices, edges and regions.
Vertices minus edges plus regions is always $2$. For a square, this is $4 - 4 + 2$.
</document>

<document source="https://math2.tamnd.com/node/0W1K48V5SG.md" tag="0W1K48V5SG" kind="concept">
# Rearrangement of a series

## Summary

A rearrangement of a series is the series of the same terms taken in a new order, given by a bijection of the indices. Rearrangements keep the sum of an absolutely convergent series. The node [[rearrangement of a series|0HRJ2Y74T3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0W1QMKJ9W8.md" tag="0W1QMKJ9W8" kind="concept">
# Uniform convergence on a set

## Summary

Uniform convergence on a subset of the domain asks for uniform closeness only at the points of that subset. A sequence can converge uniformly on some subsets and not on others. The node [[uniform convergence on a set|08HJRTG5N7]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0W276SSSCG.md" tag="0W276SSSCG" kind="concept">
# Linear approximation

## Summary

A linear approximation of a function $f$ near a point $a$ replaces $f$ by an affine function. This function has the same value and the same derivative at $a$.
For a differentiable map $f$ from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$, it is $f(x) \approx f(a) + Df(a)(x - a)$, where $Df(a)$ is the [[total derivative|0FNKFT8H6E]].
The error is small compared with $\|x - a\|$ as $x \to a$.

## Remarks

For a function of one variable, the graph of the linear approximation is the tangent line. For a function of two variables, it is the [[tangent plane|035V802J58]].
For $f(x) = \sqrt{x}$ near $a = 4$, the approximation $\sqrt{x} \approx 2 + (x - 4)/4$ gives $\sqrt{4.1} \approx 2.025$, and the true value is $2.0248\ldots$.

## Explanations

### L1 human

Near a point, a smooth function looks like a straight line, or a flat plane in more variables.
So you can estimate values of the function near the point with simple linear arithmetic.
</document>

<document source="https://math2.tamnd.com/node/0W29TBTY12.md" tag="0W29TBTY12" kind="concept">
# Riemann integral

## Summary

The Riemann integral of a function on a closed bounded interval is the limit of its Riemann sums as the partitions get finer. For a nonnegative function, it is the area under the graph. The node [[Riemann integral|0QM4TXT3DR]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0W2QAVJQ1W.md" tag="0W2QAVJQ1W" kind="definition">
# Cofactor

## Definition

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$, with $n \ge 2$.
The *cofactor* of $A$ in position $(i, j)$ is
$$C_{ij} = (-1)^{i+j} M_{ij},$$
where $M_{ij}$ is a first [[minor|0BT9BYWVRY]] of $A$. It is the minor that results when row $i$ and column $j$ are deleted.

## Remarks

The signs $(-1)^{i+j}$ form a checkerboard pattern, with $+$ in the top left corner.
The cofactors give the [[Laplace expansion|0C08BQ3M05]] of the determinant.
The transpose of the matrix of cofactors is the [[adjugate matrix|07W9H8QF38]].

## Explanations

### L1 human

A cofactor is a minor with a sign.
The sign is $+$ or $-$, according to a checkerboard pattern.
For the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the cofactors are $C_{11} = d$, $C_{12} = -c$, $C_{21} = -b$ and $C_{22} = a$.
</document>

<document source="https://math2.tamnd.com/node/0W3K3QPEWT.md" tag="0W3K3QPEWT" kind="definition">
# Absolute convergence

## Definition

Let $(a_n)$ be a real sequence.
The series $\sum a_n$ converges absolutely if the series $\sum |a_n|$ of absolute values [[converges|0CVVKY7GY2]].

## Remarks

The series $\sum |a_n|$ has nonnegative terms. So it converges exactly when its partial sums are bounded.
[[Absolutely convergent series converge|0Y123X48GA]], but the converse fails. The [[alternating harmonic series|0JQ5Y3WWVE]] converges, but not absolutely.
A convergent series that does not converge absolutely is [[conditionally convergent|02YRQXZDCS]].
The same definition works for complex terms, with the modulus in place of the absolute value.

## Explanations

### L1 human

A series converges absolutely if it still converges after we make all its terms nonnegative.
Then cancellation between positive and negative terms is not needed for convergence.
</document>

<document source="https://math2.tamnd.com/node/0W3W1VYWBP.md" tag="0W3W1VYWBP" kind="statement">
# Line integral as an integral over the parameter interval

## Statement

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a $C^1$ curve with image $C$.
Let $f \colon C \to \mathbb{R}$ be continuous, and let $F \colon C \to \mathbb{R}^n$ be a continuous vector field.
Then
$$\int_\gamma f \, ds = \int_a^b f(\gamma(t)) \, \|\gamma'(t)\| \, dt, \qquad \int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt.$$

## Remarks

The left sides are the [[line integral of a scalar field|0M50E4WYWJ]] and the [[line integral of a vector field|0YS8RX87F4]]. The right sides are ordinary integrals in one variable.
The formula follows from $\Delta s_i = \int_{t_{i-1}}^{t_i} \|\gamma'(t)\| \, dt$, the mean value theorem for each component of $\gamma$, and the uniform continuity of $f \circ \gamma$ and $\gamma'$.
The formulas extend to piecewise $C^1$ curves, by summing over the pieces.
Example: for $F(x, y) = (-y, x)$ and $\gamma(t) = (\cos t, \sin t)$, $t \in [0, 2\pi]$, the integrand is $\sin^2 t + \cos^2 t = 1$, so $\int_\gamma F \cdot d\mathbf{r} = 2\pi$.

## Explanations

### L1 human

To compute a line integral, write the curve with a parameter $t$.
Then $ds$ becomes $\|\gamma'(t)\| \, dt$ and $d\mathbf{r}$ becomes $\gamma'(t) \, dt$, and the line integral is an ordinary integral in $t$.
</document>

<document source="https://math2.tamnd.com/node/0W41XG4W60.md" tag="0W41XG4W60" kind="statement">
# Complete subspaces are closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $A \subseteq X$.
If $A$ with the [[subspace metric|0T2WYDTRP2]] is [[complete|08EWT7J0SC]], then $A$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: let $x \in \overline{A}$. By [[closure is the set of limits of sequences|003QP7A8HG]], there is a sequence in $A$ with limit $x$. It is a Cauchy sequence in $A$, so it has a limit in $A$. Limits are unique in $X$, so $x \in A$.
The space $X$ need not be complete here.
For example, $\mathbb{Q}$ is not closed in $\mathbb{R}$, so $\mathbb{Q}$ is not complete.
In a complete space, a subset is complete if and only if it is closed, by [[closed subsets of complete spaces are complete|0FBXJZX3MK]].

## Explanations

### L1 human

A complete subset contains the limits of its Cauchy sequences, and thus of its convergent sequences.
A set that keeps all its limits is closed.
</document>

<document source="https://math2.tamnd.com/node/0W44K2VNBD.md" tag="0W44K2VNBD" kind="definition">
# Sesquilinear form

## Definition

Let $V$ be a vector space over $\C$.
A *sesquilinear form* on $V$ is a map $\varphi \colon V \times V \to \C$ such that, for all $u, v, w \in V$ and $a \in \C$:

1. $\varphi(u + v, w) = \varphi(u, w) + \varphi(v, w)$ and $\varphi(au, w) = a\, \varphi(u, w)$.
2. $\varphi(u, v + w) = \varphi(u, v) + \varphi(u, w)$ and $\varphi(u, aw) = \bar a\, \varphi(u, w)$.

## Remarks

Here the form is linear in the first argument; many authors, especially in physics, use linearity in the second argument instead.
A sesquilinear form is *Hermitian* when $\varphi(v, u) = \overline{\varphi(u, v)}$ for all $u, v$.
On $\C^n$, each sesquilinear form is $\varphi(x, y) = y^* M x$ for one $n \times n$ complex matrix $M$.
A complex [[inner product|0NB0AWGJBW]] is a positive definite Hermitian sesquilinear form.

## Explanations

### L1 human

Over the complex numbers, bilinear forms cannot be positive, because $\varphi(iv, iv) = -\varphi(v, v)$.
Sesquilinear forms fix this by conjugating the scalar in one argument.
The standard inner product $\sum_i x_i \overline{y_i}$ on $\C^n$ is the main example.
</document>

<document source="https://math2.tamnd.com/node/0W4CN9E2W5.md" tag="0W4CN9E2W5" kind="concept">
# Convergent sequence

## Summary

A convergent sequence is a sequence that has a limit. Convergent sequences are bounded, and their limits obey the algebra of limits. The node [[convergent sequence|0AZ3R52593]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0W4FSW1TVB.md" tag="0W4FSW1TVB" kind="statement">
# Rank–nullity theorem

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a vector space over $F$.
Let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
Then
$$\dim \im T + \dim \ker T = \dim V,$$
that is, $\rank T + \operatorname{nullity} T = \dim V$, with the [[rank|0ZZXSBTR4J]] and the [[nullity|0APCFW7639]] of $T$.

## Remarks

For an $m \times n$ matrix $A$, this says $\rank A + \dim \{x : Ax = 0\} = n$.
A proof extends a basis of $\ker T$ to a basis of $V$; the images of the new vectors form a basis of $\im T$.
The theorem also follows from the [[first isomorphism theorem|0MTBGVZ4XJ]] $V / \ker T \cong \im T$.

## Explanations

### L1 human

Each dimension of the domain either survives in the image or is crushed into the kernel.
So the dimensions of image and kernel add up to the dimension of the domain.
A map from $\R^5$ to $\R^3$ has a kernel of dimension at least $2$.
</document>

<document source="https://math2.tamnd.com/node/0W4GQSG7B6.md" tag="0W4GQSG7B6" kind="statement">
# Irrationality of e

## Statement

The number [[$e$|0MVCFMW7S4]] is irrational.

## Remarks

Assume $e = a/b$ with positive integers $a, b$. Then $x = b! \big( e - \sum_{n=0}^{b} \frac{1}{n!} \big)$ is an integer.
But the [[series for $e$|0G4D3ZKTP8]] gives $0 < x < \sum_{k=1}^{\infty} (b + 1)^{-k} = \frac{1}{b} \le 1$. No integer lies strictly between $0$ and $1$.
This short proof is due to Joseph Fourier.

## History

Leonhard Euler wrote the first proof in 1737. It was published seven years later.

## Explanations

### L1 human

The series for $e$ converges so fast that $e$ cannot be a fraction.
If $e = a/b$, a suitable multiple of the tail of the series would be an integer between $0$ and $1$.
</document>

<document source="https://math2.tamnd.com/node/0W4RPA8QH3.md" tag="0W4RPA8QH3" kind="definition">
# Disjoint sets

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The sets $A$ and $B$ are disjoint when $A \cap B = \varnothing$.
A family of sets $(A_i)_{i \in I}$ is pairwise disjoint when $A_i \cap A_j = \varnothing$ for all $i, j \in I$ with $i \neq j$.

## Remarks

Example: $\{1, 2\}$ and $\{3, 4\}$ are disjoint, but $\{1, 2\}$ and $\{2, 3\}$ are not.
The blocks of a [[partition|09CCE86A6D]] are pairwise disjoint.
For finite disjoint sets, the number of elements of $A \cup B$ is the sum of the numbers of elements of $A$ and $B$.

## Explanations

### L1 human

Disjoint sets do not overlap.
The even integers and the odd integers are disjoint, because no integer is both.
</document>

<document source="https://math2.tamnd.com/node/0W4X9G6263.md" tag="0W4X9G6263" kind="concept">
# Unique factorisation domain

## Summary

A unique factorisation domain is an integral domain in which every nonzero nonunit is a product of irreducible elements, in an essentially unique way.
The fundamental theorem of arithmetic says that $\mathbb{Z}$ is one.
The [[definition of a unique factorisation domain|06YMXJMXYS]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0W50MT1C6X.md" tag="0W50MT1C6X" kind="concept">
# Trail

## Summary

A trail is a walk in a graph that uses each edge at most once. Trails are the setting of Eulerian problems. The node [[trail|062MQGP8M5]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0W5DSDFQ4G.md" tag="0W5DSDFQ4G" kind="definition">
# Addition of natural numbers

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]] and let $S$ be the [[successor function|0832XVR512]].
Addition is the function $+ \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, $(a, b) \mapsto a + b$, with these two properties for all $a$ and $b$ in $\mathbb{N}$:
$$a + 0 = a, \qquad a + S(b) = S(a + b).$$

## Remarks

For each $a$, these equations are a [[definition by recursion|0DSHAVH45F]] on $b$. Thus exactly one such function exists.
With $1 = S(0)$, the second equation gives $a + 1 = S(a)$.
For example, $2 + 2 = S(2 + 1) = S(S(2 + 0)) = S(S(2)) = 4$.

## Explanations

### L1 human

To add $b$ to $a$, you start at $a$ and count up $b$ steps.
The definition says this in two rules: adding $0$ changes nothing, and adding one more step gives the next number.
</document>

<document source="https://math2.tamnd.com/node/0W5NJQ8MKY.md" tag="0W5NJQ8MKY" kind="definition">
# Harmonic function

## Definition

Let $U \subseteq \mathbb{R}^n$ be open.
A function $u \colon U \to \mathbb{R}$ is harmonic on $U$ if $u$ is of [[class C^2|0TCBZMEYGN]] on $U$ and its [[Laplacian|0XQ2ZVN1PZ]] vanishes on $U$:
$$\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \dots + \frac{\partial^2 u}{\partial x_n^2} = 0.$$

## Remarks

Each affine function is harmonic. In the plane, $x^2 - y^2$, $x y$ and $e^x \sin y$ are harmonic, and so is $\ln(x^2 + y^2)$ on $\mathbb{R}^2 \setminus \{0\}$.
On $\mathbb{R}^3 \setminus \{0\}$, $1 / \|x\|$ is harmonic. It is the potential of a unit point charge at the origin.
The real and imaginary parts of a holomorphic function of $z = x + i y$ are harmonic functions of $(x, y)$.
Mean value property: let the closed ball of centre $a$ and radius $r$ lie in $U$. Then $u(a)$ is the average of $u$ on the sphere of radius $r$ about $a$, and also on the ball.

## Explanations

### L1 human

A harmonic function has no local bumps or dips: its value at each point is the average of its values around the point.
A stretched rubber sheet held at fixed heights on its rim has a harmonic height function inside.
</document>

<document source="https://math2.tamnd.com/node/0W62TKMW7V.md" tag="0W62TKMW7V" kind="statement">
# Law of total probability for an event and its complement

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B \in \mathcal{F}$ with $0 < P(B) < 1$. Let $B^c = \Omega \setminus B$.
Then
$$P(A) = P(A \mid B) \, P(B) + P(A \mid B^c) \, (1 - P(B)).$$

## Remarks

This is the [[law of total probability|0E5G5FMT2R]] for the partition $\{B, B^c\}$, together with $P(B^c) = 1 - P(B)$.
The formula is the denominator of [[Bayes' theorem|0NM6E9VW04]] in the case of two hypotheses.

## Explanations

### L1 human

Either $B$ happens or it does not.
So the chance of $A$ is its chance when $B$ happens times $P(B)$, plus its chance when $B$ does not happen times $1 - P(B)$.
</document>

<document source="https://math2.tamnd.com/node/0W6M3PSD08.md" tag="0W6M3PSD08" kind="definition">
# Dual basis

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
Let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
The *dual basis* of $B$ is the family $(b_1^*, \dots, b_n^*)$ of elements of the [[dual space|07JBSZE0V4]] $V^*$. These elements are defined by
$$b_i^*(b_j) = \delta_{ij} \qquad (1 \le i, j \le n),$$
where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

## Remarks

The dual basis is a basis of $V^*$.
The form $b_i^*$ returns coordinate $i$: $b_i^*(a_1 b_1 + \cdots + a_n b_n) = a_i$.
Each $f \in V^*$ is $f = \sum_i f(b_i)\, b_i^*$.
For the standard basis of $F^n$, $e_i^*$ is the map $x \mapsto x_i$.

## Explanations

### L1 human

Each vector of the dual basis reads off one coordinate.
The form $b_1^*$ gives the first coordinate in the basis $B$, the form $b_2^*$ gives the second, and so on.
The condition $b_i^*(b_j) = \delta_{ij}$ says exactly this.
</document>

<document source="https://math2.tamnd.com/node/0W766JAPSN.md" tag="0W766JAPSN" kind="definition">
# Inverse function

## Definition

Let $f \colon A \to B$ be a [[bijective function|0SNQEF7Z53]].
The inverse function of $f$ is the function $f^{-1} \colon B \to A$ such that, for each $b \in B$, $f^{-1}(b)$ is the unique $a \in A$ with $f(a) = b$.

## Remarks

Then $f^{-1} \circ f = \mathrm{id}_A$ and $f \circ f^{-1} = \mathrm{id}_B$. See [[bijective iff it has a two-sided inverse|01VG2A0BHW]].
The inverse $f^{-1}$ is a bijection, and its inverse is $f$.
Example: the inverse of $x \mapsto 2x + 1$ on $\mathbb{R}$ is $y \mapsto (y - 1)/2$.
Do not confuse $f^{-1}(b)$ with the [[preimage|0FGCFY1AZX]] $f^{-1}(\{b\})$, which exists for every function.

## Explanations

### L1 human

To find the inverse of $y = 2x + 1$, solve for $x$: $x = (y - 1)/2$.
The inverse runs the function backwards.
Its graph is the graph of $f$ reflected in the line $y = x$.
</document>

<document source="https://math2.tamnd.com/node/0W77CVBCHQ.md" tag="0W77CVBCHQ" kind="definition">
# Normal subgroup

## Definition

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[subgroup|0X1JXT47GA]] of $G$.
The subgroup $N$ is normal in $G$ when $gng^{-1} \in N$ for all $g \in G$ and all $n \in N$.
The notation is $N \trianglelefteq G$.

## Remarks

Equivalently, $gNg^{-1} = N$ for each $g \in G$, that is, $N$ equals each of its [[conjugates|0H11ZPX4ND]].
Each group $G$ has the normal subgroups $\{e\}$ and $G$.
In the symmetric group $S_3$, the subgroup $\{e, (1\,2\,3), (1\,3\,2)\}$ is normal, but the subgroup $\{e, (1\,2)\}$ is not.

## Explanations

### L1 human

A normal subgroup does not move when you conjugate it by any element of the group.
This is the condition that lets you divide the group by the subgroup and get a new group.
In an abelian group, every subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0W7FBYS1F1.md" tag="0W7FBYS1F1" kind="concept">
# Uncountable set

## Summary

An uncountable set is a set that is not countable.
It is too large to be listed in a sequence.
</document>

<document source="https://math2.tamnd.com/node/0W7NFW23RF.md" tag="0W7NFW23RF" kind="definition">
# Sample space

## Definition

Let $E$ be a [[random experiment|0HM05N3FJD]].
A sample space of $E$ is a nonempty [[set|0SJ3RX9TXA]] $\Omega$ with two properties.
Each performance of $E$ gives exactly one element of $\Omega$.
Each element of $\Omega$ is a possible result of $E$.
The usual notation for a sample space is $\Omega$ or $S$.

## Remarks

The elements of $\Omega$ are mutually exclusive: two different elements cannot both be the result of one trial. They are also collectively exhaustive: some element is always the result.
For one toss of a coin, $\Omega = \{H, T\}$. For two rolls of a die, $\Omega = \{1, \dots, 6\}^2$ has $36$ elements.
One experiment can have more than one sample space. For a card drawn from a deck, $\Omega$ can be the set of $52$ cards or the set of the $4$ suits.
A sample space can be finite, countably infinite or uncountable. The time until a lamp fails has the sample space $[0, \infty)$.

## Explanations

### L1 human

The sample space is the list of everything that can happen in the experiment.
For a die, the list is $1, 2, 3, 4, 5, 6$. Every question about the die is then a question about this list.
</document>

<document source="https://math2.tamnd.com/node/0W809K6AW0.md" tag="0W809K6AW0" kind="definition">
# Transpose

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The *transpose* of $A$ is the $n \times m$ matrix $A^T$ with entries
$$(A^T)_{ij} = a_{ji}$$
for $1 \le i \le n$ and $1 \le j \le m$.

## Remarks

The transpose of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ is $\begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.
The rules $(A^T)^T = A$, $(A + B)^T = A^T + B^T$ and $(cA)^T = c A^T$ hold.
Over a commutative ring, $(AB)^T = B^T A^T$, see [[transpose of a product|09GK5PS824]].
The [[matrix of the dual map is the transpose|0ZPB3180AY]] of the matrix of the map.

## Explanations

### L1 human

To transpose a matrix, flip it over its main diagonal.
Row $1$ becomes column $1$, row $2$ becomes column $2$, and so on.
A $2 \times 3$ matrix becomes a $3 \times 2$ matrix.
</document>

<document source="https://math2.tamnd.com/node/0W88CEGQP1.md" tag="0W88CEGQP1" kind="concept">
# Kernel of a homomorphism

## Summary

The kernel of a group homomorphism is the set of elements that it sends to the identity.
It is a normal subgroup, and it measures how far the homomorphism is from being injective.
</document>

<document source="https://math2.tamnd.com/node/0W8ETY2AZ6.md" tag="0W8ETY2AZ6" kind="concept">
# Tangent half-angle substitution

## Summary

The tangent half-angle substitution sets $t = \tan(x/2)$ in an integral. It turns each rational expression in $\sin x$ and $\cos x$ into a rational function of $t$. Partial fractions then finish the computation.

## Remarks

The substitution gives $\sin x = \frac{2t}{1 + t^2}$, $\cos x = \frac{1 - t^2}{1 + t^2}$ and $dx = \frac{2 \, dt}{1 + t^2}$.
It works on intervals where $x/2$ stays inside $(-\pi/2, \pi/2)$.
For example, it gives $\int \frac{dx}{\sin x} = \ln |\tan(x/2)| + C$.

## Explanations

### L1 human

This substitution turns trigonometric integrals into integrals of fractions of polynomials.
Those integrals are always solvable by partial fractions.
</document>

<document source="https://math2.tamnd.com/node/0W8QHVSN95.md" tag="0W8QHVSN95" kind="concept">
# Branch cut

## Summary

A branch cut is a curve that is removed from the plane so that a multivalued function has a continuous single-valued branch on the rest. See the [[definition|06JYC06JHY]].
The usual branch cut for the logarithm is the negative real axis.
</document>

<document source="https://math2.tamnd.com/node/0W8YHHGCJG.md" tag="0W8YHHGCJG" kind="statement">
# Epsilon-delta definition of continuity

## Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$.
Then $f$ is [[continuous at|0EEE2CZB2A]] $a$ if and only if the following holds.
For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(a)| < \varepsilon$ for all $x \in D$ with $|x - a| < \delta$.

## Remarks

Each open interval around a point contains an interval of the form $(c - r, c + r)$ with $r > 0$. This gives the equivalence.
Many texts take this condition as the definition of continuity.
Mathlib states a form for metric spaces as `Metric.continuousAt_iff`.

## Explanations

### L1 human

For each allowed error $\varepsilon$ in the output, there is a tolerance $\delta$ in the input.
Inputs within $\delta$ of $a$ give outputs within $\varepsilon$ of $f(a)$.
</document>

<document source="https://math2.tamnd.com/node/0W93FJAC8F.md" tag="0W93FJAC8F" kind="concept">
# Fermat pseudoprime

## Summary

A Fermat pseudoprime to a base $a$ is a composite number $n$ that satisfies the congruence $a^{n-1} \equiv 1 \pmod n$ of Fermat's little theorem.
The [[definition of a Fermat pseudoprime|02M2K4QH9P]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0W95W33CMP.md" tag="0W95W33CMP" kind="concept">
# Equivalence class

## Summary

The equivalence class of an element is the set of all elements that are equivalent to it.
</document>

<document source="https://math2.tamnd.com/node/0W97J8QED7.md" tag="0W97J8QED7" kind="statement">
# Chinese remainder theorem for rings

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $I_1, \dots, I_n$ be ideals of $R$, with $n \ge 1$.
Suppose that $I_j$ and $I_k$ are [[coprime|0MDJP82TEH]] for all $j \neq k$.
Then the map
$$R/(I_1 \cap \dots \cap I_n) \to R/I_1 \times \dots \times R/I_n, \qquad a + (I_1 \cap \dots \cap I_n) \mapsto (a + I_1, \dots, a + I_n)$$
is a [[ring isomorphism|0B6QC0C0G1]].

## Remarks

Under the same hypothesis, $I_1 \cap \dots \cap I_n = I_1 I_2 \cdots I_n$.
For $R = \mathbb{Z}$ and pairwise coprime integers $m_1, \dots, m_n$, the theorem gives $\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z} \cong \mathbb{Z}/m_1\mathbb{Z} \times \dots \times \mathbb{Z}/m_n\mathbb{Z}$.
For example, $\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$.

## Explanations

### L1 human

For integers, the theorem says that a number modulo $15$ is the same information as the number modulo $3$ and modulo $5$.
For each pair of remainders, exactly one class modulo $15$ has them.
The ring version says the same for any family of pairwise coprime ideals.
</document>

<document source="https://math2.tamnd.com/node/0WA7ZR95V4.md" tag="0WA7ZR95V4" kind="statement">
# Time translation of solutions of an autonomous system

## Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be a function. Let $\gamma : I \to U$ be a solution of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ on an interval $I$.
Let $c \in \mathbb{R}$ and let $I - c = \{t - c : t \in I\}$.
Then the function $\delta : I - c \to U$ given by $\delta(t) = \gamma(t + c)$ is also a solution of $x' = f(x)$.

## Remarks

Proof: by the chain rule, $\delta'(t) = \gamma'(t + c) = f(\gamma(t + c)) = f(\delta(t))$.
So it is enough to solve an initial value problem with initial time $0$. The solution with $x(t_0) = x_0$ is $t \mapsto \varphi(t - t_0)$, where $\varphi$ solves $x(0) = x_0$.
The result fails for non-autonomous equations. For $x' = t$, the solution $t^2/2$ is a solution, but $(t+1)^2/2$ is not.

## Explanations

### L1 human

If the law of motion does not change with time, a motion that starts later is the same motion, only delayed.
So shifting a solution in time gives another solution.
</document>

<document source="https://math2.tamnd.com/node/0WAR4F23B4.md" tag="0WAR4F23B4" kind="concept">
# Double integral

## Summary

The double integral of a function of two variables over a region of the plane is a limit of Riemann sums. The partitions of the region into small rectangles become finer and finer.
For a non-negative function it is the volume of the solid between the region and the graph.
</document>

<document source="https://math2.tamnd.com/node/0WAY5WX8NY.md" tag="0WAY5WX8NY" kind="concept">
# Congruence modulo n

## Summary

Two integers are congruent modulo a positive integer $n$ when their difference is a multiple of $n$.
Modular arithmetic is the arithmetic of integers up to this relation.
The [[definition of congruence modulo n|0TPW25CBN6]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0WB57D9488.md" tag="0WB57D9488" kind="definition">
# Sine and cosine

## Definition

The sine and cosine functions $\sin, \cos \colon \mathbb{R} \to \mathbb{R}$ are defined by the power series
$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.$$

## Remarks

Both [[series|02KTRHQMHN]] [[converge|08W5RN375X]] absolutely for every real $x$.
Sine is odd and cosine is even. Also $\sin 0 = 0$ and $\cos 0 = 1$.
Geometrically, $(\cos x, \sin x)$ is the point of the unit circle at arc length $x$ from $(1, 0)$, counterclockwise.
Mathlib defines `Real.sin` and `Real.cos` from the complex exponential, with $\cos x + i \sin x = e^{i x}$.

## Explanations

### L1 human

Sine and cosine describe motion around the unit circle.
In analysis, we define them by power series, and we then prove their geometric properties.
</document>

<document source="https://math2.tamnd.com/node/0WB619CA0V.md" tag="0WB619CA0V" kind="concept">
# Orthocentre

## Summary

The orthocentre of a triangle is the point where the three lines that contain its altitudes meet. It is inside the triangle exactly when all angles are acute.
</document>

<document source="https://math2.tamnd.com/node/0WB6DH499T.md" tag="0WB6DH499T" kind="concept">
# Path-connected space

## Summary

A topological space is path-connected when any two of its points can be joined by a continuous path.
Path-connected spaces are connected, but the converse is false.
</document>

<document source="https://math2.tamnd.com/node/0WBFFC8XZ9.md" tag="0WBFFC8XZ9" kind="concept">
# Subgraph

## Summary

A subgraph of a graph is a graph formed by some of its vertices and some of its edges. Subgraphs are how one speaks of a structure, such as a cycle or a tree, inside a graph. The node [[subgraph|0P11X5HKAX]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WBRJXXDPZ.md" tag="0WBRJXXDPZ" kind="concept">
# Norm on the Gaussian integers

## Summary

The norm of a Gaussian integer $a + b i$ is $a^2 + b^2$, the square of its absolute value.
The [[definition of the norm|03AN1V4ZFJ]] gives its properties.
</document>

<document source="https://math2.tamnd.com/node/0WBTWFXJQG.md" tag="0WBTWFXJQG" kind="concept">
# Integration by partial fractions

## Summary

Integration by partial fractions is a method to integrate [[rational functions|0WCZH4AFMT]]. We write the rational function as a polynomial plus a sum of simple fractions. Each simple fraction has a known antiderivative.

## Remarks

Over $\mathbb{R}$, the simple fractions have the forms $\frac{A}{(x - r)^k}$ and $\frac{B x + C}{(x^2 + p x + q)^k}$, with $p^2 < 4 q$.
Their antiderivatives use powers, the [[logarithm|0K1N1Y4EGZ]], and the [[arctangent|065XW2V6C3]].
For example, $\frac{1}{x^2 - 1} = \frac{1}{2} \Big( \frac{1}{x - 1} - \frac{1}{x + 1} \Big)$.
So every rational function has an elementary antiderivative.

## Explanations

### L1 human

A complicated fraction of polynomials splits into a sum of simple fractions.
We can integrate each simple fraction by a standard formula.
</document>

<document source="https://math2.tamnd.com/node/0WC28W9DNM.md" tag="0WC28W9DNM" kind="object">
# Cardinality of the continuum

## Description

The cardinality of the continuum, written $\mathfrak{c}$, is the [[cardinal number|0QKPNZKNEA]] $2^{\aleph_0}$.
It is the cardinality of the [[power set|0BC7KX51Z8]] of $\mathbb{N}$, and of the set of all sequences of $0$ and $1$.
It is also the cardinality of $\mathbb{R}$. See [[the reals have the cardinality of the continuum|075PM9FH7A]].

## Remarks

By [[Cantor's theorem|0WTEMGYV2C]], $\mathfrak{c} > \aleph_0$.
Other sets of cardinality $\mathfrak{c}$ are each nonempty open interval of $\mathbb{R}$, the plane $\mathbb{R}^2$ and the set of irrational numbers.
The [[continuum hypothesis|0MN2MER9WN]] says that $\mathfrak{c} = \aleph_1$.

## Explanations

### L1 human

$\mathfrak{c}$ is the size of the real line.
It is a larger infinity than $\aleph_0$.
Surprisingly, a short interval, a whole line and a whole plane all have this same size.
</document>

<document source="https://math2.tamnd.com/node/0WCKRBHFFD.md" tag="0WCKRBHFFD" kind="definition">
# Clique number

## Definition

Let $G$ be a finite [[graph|0JD5C82QMB]].
The clique number $\omega(G)$ is the largest number of vertices of a [[clique|0G52BPNPKG]] in $G$.

## Remarks

The vertices of a clique need different colours. Thus $\omega(G) \le \chi(G)$, where $\chi(G)$ is the [[chromatic number|0C5M7AM6FH]].
For the [[complete graph|03WKB2H474]], $\omega(K_n) = n$. For the [[cycle graph|0QQ0F12962]] $C_n$ with $n \ge 4$, $\omega(C_n) = 2$.
The inequality $\omega(G) \le \chi(G)$ can be strict: for $C_5$, $\omega = 2$ and $\chi = 3$.

## Explanations

### L1 human

The clique number is the size of the largest group of vertices that are all joined to each other.
It gives a lower bound for the number of colours that a graph needs.
</document>

<document source="https://math2.tamnd.com/node/0WCP0PVET6.md" tag="0WCP0PVET6" kind="concept">
# Proof by cases

## Summary

A proof by cases splits the situation into a finite number of cases that cover all possibilities.
Then it proves the statement in each case.
The basic form is this rule: from $p \lor q$, $p \to r$ and $q \to r$, conclude $r$.

## Remarks

Another name is proof by exhaustion, mainly when the cases are many.
Example: $n^2 + n$ is even for each integer $n$. If $n$ is even, $n^2$ and $n$ are even. If $n$ is odd, $n^2$ and $n$ are odd. In both cases, the sum is even.
A split into $p$ and $\neg p$ uses the [[law of excluded middle|0RYYK8PG4S]].

## Explanations

### L1 human

When one argument does not work for all cases, split the cases.
Integers are even or odd, so a proof for even integers and a proof for odd integers together cover all integers.
Make sure that no case is missing.
</document>

<document source="https://math2.tamnd.com/node/0WCPH5Z7FA.md" tag="0WCPH5Z7FA" kind="statement">
# Trace of a linear map is the trace of its matrix

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ with a basis $B$, and let $T \colon V \to V$ be a linear map.
Then the [[trace of $T$|0KZDNGA561]] equals the trace of the [[matrix of $T$|0YQSGNYE3K]] in $B$:
$$\tr T = \tr [T]_B.$$

## Remarks

The key fact is that similar matrices have the same trace, by $\tr(AB) = \tr(BA)$.
With a basis-free definition, the trace is the map $\End_F(V) \cong V^* \otimes V \to F$ induced by evaluation $f \otimes v \mapsto f(v)$.

## Explanations

### L1 human

We can compute the trace of a linear map in any basis that we like.
We write the matrix in that basis and add its diagonal entries.
Every basis gives the same number.
</document>

<document source="https://math2.tamnd.com/node/0WCWA1KT3M.md" tag="0WCWA1KT3M" kind="concept">
# Binomial distribution

## Summary

The binomial distribution with parameters $n$ and $p$ is the distribution of the number of successes in $n$ independent Bernoulli trials.
Each trial has the success probability $p$.
The node [[binomial distribution|0VRWZDWW6B]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WCX7NB4N7.md" tag="0WCX7NB4N7" kind="concept">
# Separable extension

## Summary

A separable extension is an algebraic extension in which the minimal polynomial of every element has no repeated roots.
The [[definition of a separable extension|0N019F6RE9]] states it.
</document>

<document source="https://math2.tamnd.com/node/0WCY42EV9V.md" tag="0WCY42EV9V" kind="concept">
# Group action

## Summary

An action of a group $G$ on a set $X$ lets each element of $G$ move the points of $X$. The moves must respect the group operation.
Equivalently, it is a group homomorphism from $G$ to the symmetric group of $X$.
</document>

<document source="https://math2.tamnd.com/node/0WCZH4AFMT.md" tag="0WCZH4AFMT" kind="definition">
# Rational function

## Definition

Let $p$ and $q$ be real polynomials, with $q$ not the zero polynomial.
Let $D = \{ x \in \mathbb{R} : q(x) \ne 0 \}$.
The function $f \colon D \to \mathbb{R}$ given by $f(x) = p(x)/q(x)$ is a rational function.

## Remarks

A nonzero polynomial has only finitely many roots, so $D$ is $\mathbb{R}$ minus a finite set.
Each rational function is continuous on $D$, since [[polynomials are continuous|0C5TP6RYSV]] and [[quotients of continuous functions are continuous|0GR2APWTAC]].
For example, $1/x$ and $(x^2 + 1)/(x - 2)$ are rational functions.

## Explanations

### L1 human

A rational function is a quotient of two polynomials.
It is defined everywhere except at the zeros of the denominator.
</document>

<document source="https://math2.tamnd.com/node/0WD3SE7S1X.md" tag="0WD3SE7S1X" kind="statement">
# R/M is a field iff M is maximal

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an ideal of $R$.
Then the [[quotient ring|0VFW4Y69BT]] $R/M$ is a [[field|0QA8CHH8MG]] if and only if $M$ is a [[maximal ideal|09J6KGT65G]].

## Remarks

The proof uses the [[correspondence theorem|0D1R4SVMBQ]]: the ideals of $R/M$ are the images of the ideals of $R$ that contain $M$.
So $M$ is maximal if and only if $R/M$ is nonzero and has only the two ideals $\{0\}$ and $R/M$.
A nonzero commutative ring with only these two ideals is a field, by the [[converse for fields|0M6G75E95E]].
For example, $\mathbb{Z}/p\mathbb{Z}$ is a field because $p\mathbb{Z}$ is maximal for $p$ prime.

## Explanations

### L1 human

The quotient by an ideal is a field exactly when the ideal is as large as possible.
In the integers, $(5)$ is maximal, and $\mathbb{Z}/5\mathbb{Z}$ is a field.
The ideal $(6)$ is not maximal, and $\mathbb{Z}/6\mathbb{Z}$ is not a field.
</document>

<document source="https://math2.tamnd.com/node/0WDAKKN4NC.md" tag="0WDAKKN4NC" kind="counterexample">
# x^2 sin(1/x) is differentiable with discontinuous derivative

## Counterexample

Let $f(x) = x^2 \sin(1/x)$ for $x \ne 0$, and let $f(0) = 0$.
Then $f$ is [[differentiable|0A61B60B5R]] on $\mathbb{R}$, but $f'$ is not continuous at $0$.
At $0$, the difference quotient is $h \sin(1/h)$, which tends to $0$. So $f'(0) = 0$.
For $x \ne 0$, $f'(x) = 2 x \sin(1/x) - \cos(1/x)$. This has no limit as $x \to 0$, because $\cos(1/x)$ has none.

## Remarks

So $f$ is differentiable but not [[of class $C^1$|06SKCHDQT5]].
The discontinuity of $f'$ at $0$ is essential. By [[Darboux's theorem|0Q5A47J48D]], a derivative can have no jump discontinuities.

## Explanations

### L1 human

The factor $x^2$ flattens the oscillation of $\sin(1/x)$ enough to give a slope $0$ at the origin.
But near the origin the slopes keep swinging between about $-1$ and $1$.
</document>

<document source="https://math2.tamnd.com/node/0WDNRBXX7K.md" tag="0WDNRBXX7K" kind="definition">
# Cauchy–Euler equation

## Definition

Let $n \ge 1$ and let $a_0, \dots, a_n$ be real numbers with $a_n \ne 0$.
The Cauchy–Euler equation with these coefficients is the [[linear differential equation|0PB7K8M7FG]]
$$a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \dots + a_1 x\, y' + a_0\, y = 0$$
for an unknown function $y$ on the interval $x > 0$.

## Remarks

The substitution $x = e^{t}$, $u(t) = y(e^t)$ changes it into a [[linear equation with constant coefficients|0A6PGTTP2A]] for $u$.
For $n = 2$, the trial solution $y = x^m$ gives the indicial equation $a_2 m (m - 1) + a_1 m + a_0 = 0$.
For order $2$: a double root $m$ gives the solutions $x^m$ and $x^m \ln x$. Roots $\alpha \pm i \beta$ with $\beta \ne 0$ give $x^\alpha \cos(\beta \ln x)$ and $x^\alpha \sin(\beta \ln x)$.
Example: $x^2 y'' - 2 y = 0$ gives $m(m-1) - 2 = 0$, so $m = 2$ or $m = -1$, and the solutions are $c_1 x^2 + c_2 x^{-1}$.

## Explanations

### L1 human

In this equation, each derivative $y^{(k)}$ is multiplied by $x^k$.
So powers $x^m$ behave like exponentials do for constant coefficients: the equation becomes a polynomial equation in $m$.
</document>

<document source="https://math2.tamnd.com/node/0WE2NHZJ7B.md" tag="0WE2NHZJ7B" kind="concept">
# Order of an element

## Summary

The order of an element of a group is the smallest positive integer $n$ such that the $n$th power of the element is the identity.
When no such $n$ exists, the element has infinite order.
</document>

<document source="https://math2.tamnd.com/node/0WE5BBHGM6.md" tag="0WE5BBHGM6" kind="definition">
# Totally bounded set

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \subseteq X$.
The set $S$ is totally bounded when for each real $\varepsilon > 0$ there are finitely many sets $A_1, \ldots, A_n \subseteq X$ with $S \subseteq A_1 \cup \cdots \cup A_n$ and $\operatorname{diam}(A_k) < \varepsilon$ for each $k$.
The space $X$ is totally bounded when $X$ itself is a totally bounded set.

## Remarks

Here $\operatorname{diam}$ is the [[diameter|0692CGQF00]].
Each totally bounded set is [[bounded|0P953MG05E]]. The converse is false: an infinite set with the [[discrete metric|0NA4FK7B8S]] is bounded but not totally bounded.
In $\mathbb{R}^n$ with the Euclidean metric, a set is totally bounded if and only if it is bounded.
The [[epsilon-net characterisation|0NMGJJ19HD]] gives an equivalent form with balls.

## Explanations

### L1 human

A totally bounded set can be cut into finitely many small pieces, for each size of piece that you choose.
A bounded set only fits in one big ball, and this is not enough in infinite dimensions.
</document>

<document source="https://math2.tamnd.com/node/0WETJCKNRG.md" tag="0WETJCKNRG" kind="statement">
# Euclid–Euler theorem

## Statement

Let $n$ be an even positive integer.
Then $n$ is a [[perfect number|0SB7Q8SYSF]] if and only if $n = 2^{p-1} (2^p - 1)$, where $2^p - 1$ is a [[Mersenne prime|06GPJF9V7M]].

## Remarks

The primes $2^p - 1$ for $p = 2, 3, 5, 7$ give the perfect numbers $6$, $28$, $496$ and $8128$.
So the even perfect numbers correspond one to one to the Mersenne primes.
The proof uses the multiplicativity of the [[sum of divisors|009QCP5T3Z]]: $\sigma(2^{p-1} q) = (2^p - 1)(q + 1)$ for an odd prime $q$.

## History

Euclid proved in the Elements that $2^{p-1}(2^p - 1)$ is perfect when $2^p - 1$ is prime.
Ibn al-Haytham conjectured the converse around 1000 CE.
Euler proved the converse in the 18th century, in a paper published after his death.

## Explanations

### L1 human

Euclid found a recipe for perfect numbers: take a prime of the form $2^p - 1$ and multiply it by $2^{p-1}$.
Euler showed that the recipe gives every even perfect number.
Whether an odd perfect number exists is still unknown.
</document>

<document source="https://math2.tamnd.com/node/0WF4EJP91W.md" tag="0WF4EJP91W" kind="definition">
# Probability mass function

## Definition

Let $X$ be a [[discrete random variable|0J6Z6H5VQJ]] on a probability space $(\Omega, \mathcal{F}, P)$.
The probability mass function of $X$ is the function $p_X \colon \mathbb{R} \to [0, 1]$ with
$$p_X(x) = P(X = x) \quad \text{for } x \in \mathbb{R}.$$
More generally, a probability mass function on a countable set $S$ is a function $p \colon S \to [0, 1]$ with $\sum_{s \in S} p(s) = 1$.

## Remarks

The set $\{x : p_X(x) > 0\}$ is countable, and $\sum_x p_X(x) = 1$.
For each Borel set $B$, $P(X \in B) = \sum_{x \in B} p_X(x)$.
For a fair die, $p_X(k) = 1/6$ for $k = 1, \dots, 6$ and $p_X(x) = 0$ for other $x$.

## Explanations

### L1 human

The probability mass function is a table: each possible value with its probability.
The entries are never negative and they add up to $1$.
</document>

<document source="https://math2.tamnd.com/node/0WF5AQYTW6.md" tag="0WF5AQYTW6" kind="concept">
# Reduced row echelon form

## Summary

A matrix is in reduced row echelon form when it is in row echelon form and each leading entry is $1$.
Also, each leading entry is the only nonzero entry in its column.
</document>

<document source="https://math2.tamnd.com/node/0WFBH1Q3WC.md" tag="0WFBH1Q3WC" kind="concept">
# Local maximum and minimum

## Summary

A [[function of several real variables|0JPCHTXWG0]] has a local maximum at a point when no nearby point of the domain gives a larger value.
It has a local minimum when no nearby point gives a smaller value, and a local extremum is either of the two.
</document>

<document source="https://math2.tamnd.com/node/0WFCGVMRRC.md" tag="0WFCGVMRRC" kind="definition">
# Factorial

## Definition

Let $n$ be a natural number.
The factorial of $n$ is the natural number $n!$ defined by the recursion
$$0! = 1, \qquad (n+1)! = (n+1) \cdot n!.$$
For $n \ge 1$ this gives $n! = 1 \cdot 2 \cdots n$, the product of the positive integers up to $n$.

## Remarks

The first values are $0! = 1$, $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$ and $5! = 120$.
The value $n!$ is the [[number of permutations|0CEBCS0E0E]] of an $n$-element set.
The convention $0! = 1$ agrees with the empty product, and it makes the [[factorial formula for binomial coefficients|033KTKKRV6]] true for $k = 0$ and $k = n$.

## Explanations

### L1 human

The factorial $n!$ multiplies all whole numbers from $1$ to $n$.
It counts the ways to put $n$ different books in a row: $n$ choices for the first place, $n - 1$ for the second, and so on.
</document>

<document source="https://math2.tamnd.com/node/0WFCNG0MZC.md" tag="0WFCNG0MZC" kind="concept">
# Ordered pair

## Summary

An ordered pair is a pair of objects in which the order matters.
Two ordered pairs are equal exactly when their first entries are equal and their second entries are equal.
</document>

<document source="https://math2.tamnd.com/node/0WFED3K3DQ.md" tag="0WFED3K3DQ" kind="statement">
# Sylvester's law of inertia

## Statement

Let $A$ be a real symmetric $n \times n$ matrix.
Let $P$ and $P'$ be invertible real $n \times n$ matrices such that $D = P^T A P$ and $D' = P'^T A P'$ are diagonal.
Then $D$ and $D'$ have the same number of positive diagonal entries and the same number of negative diagonal entries.
They also have the same number of zero diagonal entries.

## Remarks

So these three numbers are invariants of the quadratic form $x \mapsto x^T A x$; they give its [[signature|07EZA80NEH]].
They are also the numbers of positive, negative and zero eigenvalues of $A$, counted with multiplicity.
Equivalently, two real symmetric matrices are congruent if and only if they have the same numbers of positive, negative and zero eigenvalues.

## History

James Joseph Sylvester published the theorem in 1852.

## Explanations

### L1 human

A real quadratic form can be written as a sum of squares in many ways.
The numbers of plus signs and minus signs are always the same.
For example, $x^2 + 2xy$ always has one plus sign and one minus sign.
</document>

<document source="https://math2.tamnd.com/node/0WFF6AZPBP.md" tag="0WFF6AZPBP" kind="definition">
# Monotone function

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[preorders|0RXS8R6K37]] and let $f \colon P \to Q$ be a [[function|0XHDTE5YMZ]].
The function $f$ is monotone when, for all $a, b \in P$, $a \le_P b$ implies $f(a) \le_Q f(b)$.

## Remarks

Other names are order-preserving and, for real functions, monotonically increasing or non-decreasing.
The function $f$ is antitone when $a \le b$ implies $f(b) \le f(a)$.
The map $S \mapsto f(S)$ on [[power sets|0BC7KX51Z8]] is monotone for $\subseteq$.

## Explanations

### L1 human

A monotone function never reverses the order of two inputs.
On $\mathbb{R}$, $x \mapsto x^3$ is monotone, and $x \mapsto x^2$ is not, because $-2 \le 1$ but $4 > 1$.
</document>

<document source="https://math2.tamnd.com/node/0WFMKK7DFZ.md" tag="0WFMKK7DFZ" kind="problem">
# Gambler's ruin

## Problem

A gambler starts with $k$ units of money, where $0 \le k \le N$ for an integer $N \ge 1$. In each round, the gambler wins one unit with probability $p \in (0, 1)$ and loses one unit with probability $q = 1 - p$. The rounds are independent.
The game stops when the fortune reaches $0$ (ruin) or $N$ (the target).
Find the probability $P_k$ that the gambler reaches $N$ before $0$.

## Remarks

The fortune is a [[random walk|0C08YMV95H]] on $\{0, 1, \dots, N\}$ that stops at $0$ and $N$. By the [[law of total probability|0E5G5FMT2R]] for the first round, $P_k = p P_{k+1} + q P_{k-1}$ for $0 < k < N$, with $P_0 = 0$ and $P_N = 1$.
For a fair game, $p = q = 1/2$, the solution is $P_k = k/N$.
For $p \ne q$, the solution is $P_k = \dfrac{1 - (q/p)^k}{1 - (q/p)^N}$.
In a fair game against an opponent with unlimited money ($N \to \infty$), the gambler is ruined with probability $1$.

## History

An early form of the problem appears in a letter of Blaise Pascal to Pierre de Fermat in 1656.

## Explanations

### L1 human

A gambler who plays a fair game against a much richer opponent will almost surely lose everything in the end.
With $10$ units and a target of $100$ in a fair game, the chance to reach the target is only $10/100 = 0.1$.
</document>

<document source="https://math2.tamnd.com/node/0WFZX06698.md" tag="0WFZX06698" kind="concept">
# Inverse hyperbolic functions

## Summary

The inverse hyperbolic functions are the inverses of the hyperbolic sine, cosine and tangent, with a restricted domain for the cosine. They have explicit formulas in terms of the logarithm and the square root. The node [[inverse hyperbolic functions|04D7CHT5V1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WG2A3HE0S.md" tag="0WG2A3HE0S" kind="statement">
# Newton's identities

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
Let $e_k$ be the [[elementary symmetric polynomials|0ZMY8441GW]] and $p_k$ the [[power sums|0AS1FZT6BM]] in $x_1, \dots, x_n$, with $e_0 = 1$ and $e_k = 0$ for $k > n$.
Then for every $k \ge 1$,
$$k \, e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i} \, p_i.$$

## Remarks

The first cases are $e_1 = p_1$, $2 e_2 = e_1 p_1 - p_2$ and $3 e_3 = e_2 p_1 - e_1 p_2 + p_3$.
For $k > n$, the left side is $0$, and the identity expresses $p_k$ through $p_{k-1}, \dots, p_{k-n}$.
Applied to the roots of a polynomial, the identities compute the power sums of the roots from the coefficients.
Applied to the eigenvalues of a matrix $A$, they relate the traces of the powers of $A$ to the coefficients of its characteristic polynomial.

## History

Isaac Newton found the identities around 1666.
Albert Girard had already found the first four formulas in 1629.

## Explanations

### L1 human

Newton's identities convert between two ways of describing the roots of a polynomial: the sums of their powers, and the coefficients.
Each identity gives the next quantity from the earlier ones.
</document>

<document source="https://math2.tamnd.com/node/0WG43AC0EV.md" tag="0WG43AC0EV" kind="example">
# Two-element Boolean algebra

## Example

The two-element Boolean algebra is the set $\{0, 1\}$ with these operations.
$a \wedge b = 1$ if and only if $a = b = 1$.
$a \vee b = 0$ if and only if $a = b = 0$.
$\neg 0 = 1$ and $\neg 1 = 0$.
It satisfies the axioms of a [[Boolean algebra|06GKBTAA5E]], with $0$ as least element and $1$ as greatest element.

## Remarks

If $1$ means true and $0$ means false, then $\wedge$, $\vee$ and $\neg$ are the connectives and, or and not.
An equation in the Boolean operations holds in all Boolean algebras if and only if it holds in the two-element Boolean algebra.
It is isomorphic to the power set of a set with one element.

## Explanations

### L1 human

The two-element Boolean algebra is the algebra of true and false.
It is also the algebra of one bit in a digital circuit.
</document>

<document source="https://math2.tamnd.com/node/0WGAXEK99E.md" tag="0WGAXEK99E" kind="statement">
# A subspace of full dimension is the whole space

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
If $\dim W = \dim V$, then $W = V$.

## Remarks

A basis of $W$ is a linearly independent family of $\dim V$ vectors in $V$.
So it is a basis of $V$, see [[linearly independent set of the right size is a basis|096V7H13H9]].
The statement fails for infinite-dimensional spaces.
In $F[x]$, the subspace of polynomials with $p(0) = 0$ has the same dimension as $F[x]$, but it is not equal to $F[x]$.

## Explanations

### L1 human

If a subspace is as large as the whole space, measured by dimension, then it is the whole space.
For example, a two-dimensional subspace of the plane is the plane.
This lets us prove that two subspaces are equal by an inclusion and a count.
</document>

<document source="https://math2.tamnd.com/node/0WGDPYR30Q.md" tag="0WGDPYR30Q" kind="definition">
# Continuous uniform distribution

## Definition

Let $a < b$ be real numbers.
The continuous uniform distribution on the [[interval|0TK8Y3J5RW]] $[a, b]$ is the distribution on $\mathbb{R}$ with the [[probability density function|04YG690N5V]]
$$f(x) = \begin{cases} \dfrac{1}{b - a} & \text{if } a \le x \le b, \\ 0 & \text{otherwise.} \end{cases}$$
One writes $X \sim \operatorname{U}(a, b)$.

## Remarks

For $a \le c \le d \le b$, $P(c \le X \le d) = (d - c)/(b - a)$.
The cumulative distribution function is $F(x) = (x - a)/(b - a)$ for $a \le x \le b$, with $F(x) = 0$ for $x < a$ and $F(x) = 1$ for $x > b$.
The mean is $(a + b)/2$ and the variance is $(b - a)^2 / 12$.
If $U \sim \operatorname{U}(0, 1)$ and $F$ is a cumulative distribution function with [[quantile function|0B3Q3XDM5X]] $Q$, then $Q(U)$ has the cumulative distribution function $F$.

## Explanations

### L1 human

The uniform distribution is "a point chosen at random in an interval, with no preference".
For $\operatorname{U}(0, 10)$, the chance to land between $2$ and $5$ is $3/10$, the length of the piece over the length of the interval.
</document>

<document source="https://math2.tamnd.com/node/0WH1GVBJNH.md" tag="0WH1GVBJNH" kind="example">
# Circle as a quotient of the real line

## Example

On $\mathbb{R}$, let $s \sim t$ when $s - t \in \mathbb{Z}$. Let $\mathbb{R} / \mathbb{Z}$ be the set of classes, with the [[quotient topology|0V8046R0P9]].
Let $S^1 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ be the unit circle, with the [[subspace topology|0D5YHQGDJY]] of $\mathbb{R}^2$.
Then the map $[t] \mapsto (\cos 2\pi t, \sin 2\pi t)$ is a [[homeomorphism|05GZ11PWRY]] from $\mathbb{R} / \mathbb{Z}$ to $S^1$.

## Remarks

The map is well defined and continuous by the [[universal property of the quotient topology|0FF2FJ8CH2]], and it is bijective.
The space $\mathbb{R} / \mathbb{Z}$ is compact, because it is the image of $[0, 1]$ under the quotient map. The circle is Hausdorff. Thus the map is a homeomorphism, by [[this theorem|0430ETCMKD]].
Equivalently, $S^1$ is homeomorphic to $[0, 1]$ with the two endpoints $0$ and $1$ glued together.
Addition of classes makes $\mathbb{R} / \mathbb{Z}$ a group, the circle group.

## Explanations

### L1 human

Wrap the real line around a circle of length $1$. Points that differ by an integer land at the same place.
The quotient $\mathbb{R} / \mathbb{Z}$ records exactly this wrapping.
</document>

<document source="https://math2.tamnd.com/node/0WH916TJNY.md" tag="0WH916TJNY" kind="statement">
# There is a natural number n with 1/n below any positive epsilon

## Statement

Let $\varepsilon$ be a real number with $\varepsilon > 0$.
Then there is a natural number $n \ge 1$ with $\dfrac{1}{n} < \varepsilon$.

## Remarks

Proof: since the [[reals are Archimedean|0A3GWZ86AY]], there is a natural number $n > 1/\varepsilon$. Then $n \ge 1$ and $1/n < \varepsilon$.
Thus $\inf \{1/n : n \ge 1\} = 0$. There are no positive infinitely small real numbers.
The same holds in each Archimedean ordered field.

## Explanations

### L1 human

The fractions $1, 1/2, 1/3, \dots$ get as small as you want.
For $\varepsilon = 0.001$, the number $n = 1001$ works, because $1/1001 < 0.001$.
</document>

<document source="https://math2.tamnd.com/node/0WHCA4SVP2.md" tag="0WHCA4SVP2" kind="concept">
# Invariant factors and elementary divisors

## Summary

Let $G$ be a finitely generated [[abelian group|082GHNPJ53]], and write $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/k_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/k_u\mathbb{Z}$ with $1 < k_1 \mid k_2 \mid \cdots \mid k_u$.
The integers $k_1, \ldots, k_u$ are the invariant factors of $G$.
In the other form $G \cong \mathbb{Z}^n \oplus \mathbb{Z}/q_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/q_t\mathbb{Z}$, the prime powers $q_1, \ldots, q_t$ are the elementary divisors of $G$.

## Remarks

The [[structure theorem|0RY0B72XAS]] says that the rank $n$, the invariant factors and the elementary divisors are unique, and that they determine $G$ up to isomorphism.
The elementary divisors are the prime power factors of the invariant factors. For example, the invariant factors $2, 12$ give the elementary divisors $2, 4, 3$.
For a finite group $G$, the product of the invariant factors is $|G|$, and the largest invariant factor $k_u$ is the [[exponent|0ZDW0H0VX4]] of $G$.
The same terms are used for finitely generated modules over a principal ideal domain.

## Explanations

### L1 human

There are two standard ways to write a finitely generated abelian group as a sum of cyclic groups.
The invariant factors are orders that divide each other.
The elementary divisors are prime power orders.
</document>

<document source="https://math2.tamnd.com/node/0WJGR81HBD.md" tag="0WJGR81HBD" kind="statement">
# Taylor's theorem with the integral remainder

## Statement

Let $k \ge 0$, let $a \ne x$ be real numbers, and let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function on $J$ such that $f^{(k+1)}$ exists and is continuous on $J$.
Then
$$f(x) = P_k(x) + \int_a^x \frac{f^{(k+1)}(t)}{k!} (x - t)^k \, dt,$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

For $k = 0$ it is the [[fundamental theorem of calculus|0NHGY0N0JB]], $f(x) = f(a) + \int_a^x f'(t)\, dt$. Repeated [[integration by parts|08Y1BBQTK9]] gives the general case.
The hypotheses can be weakened to absolute continuity of $f^{(k)}$.
Mathlib states it as `taylor_integral_remainder_of_absolutelyContinuous`.

## Explanations

### L1 human

The integral form gives the error exactly, as an average of the next derivative with weights.
It needs no unknown point $\xi$.
</document>

<document source="https://math2.tamnd.com/node/0WJX8MGG37.md" tag="0WJX8MGG37" kind="concept">
# Free abelian group

## Summary

A free abelian group is an abelian group with a basis. Each element is an integer combination of finitely many basis elements in exactly one way.
The free abelian group of finite rank $n$ is isomorphic to $\mathbb{Z}^n$.
</document>

<document source="https://math2.tamnd.com/node/0WK0Y8RH1M.md" tag="0WK0Y8RH1M" kind="statement">
# Dimension theorem for vector spaces

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Any two [[bases|0KCQ9Y3QKA]] of $V$ have the same cardinality.

## Remarks

For finite bases, this follows from the [[Steinitz exchange lemma|08ZDRN6H41]].
For infinite bases, the proof uses cardinal arithmetic: each vector of one basis is a combination of finitely many vectors of the other basis.
The common cardinality is the [[dimension|03GSZ8RTZJ]] of $V$.
The theorem holds more generally for free modules over a commutative ring that is not the zero ring.

## Explanations

### L1 human

However we choose a basis of a space, it always has the same number of vectors.
So "the number of vectors in a basis" is a property of the space itself.
We call this number the dimension.
</document>

<document source="https://math2.tamnd.com/node/0WKEF78PMV.md" tag="0WKEF78PMV" kind="statement">
# Sylvester's criterion

## Statement

Let $A$ be an $n \times n$ complex Hermitian matrix.
For $1 \le k \le n$, let $\Delta_k$ be the determinant of the upper left $k \times k$ submatrix of $A$.
Then $A$ is [[positive definite|0KX66CZMBB]] if and only if $\Delta_k > 0$ for all $k = 1, \dots, n$.

## Remarks

The numbers $\Delta_k$ are the *leading principal [[minors|0BT9BYWVRY]]* of $A$.
For $\begin{pmatrix} a & b \\ b & c \end{pmatrix}$ real, the test is $a > 0$ and $ac - b^2 > 0$.
For positive semidefinite matrices, the analogous test needs all principal minors, not only the leading ones, to be nonnegative.

## Explanations

### L1 human

Sylvester's criterion tests positive definiteness with $n$ determinants.
We check the top left $1 \times 1$, $2 \times 2$, and so on, up to the full matrix.
All of them must be positive.
</document>

<document source="https://math2.tamnd.com/node/0WKG15TV3G.md" tag="0WKG15TV3G" kind="concept">
# Fermat number

## Summary

A Fermat number is an integer of the form $2^{2^n} + 1$ for a natural number $n$.
The [[definition of a Fermat number|0R03B54V4Z]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0WKG8CYY6A.md" tag="0WKG8CYY6A" kind="statement">
# Cholesky decomposition

## Statement

Let $A$ be a [[positive definite|0KX66CZMBB]] $n \times n$ complex Hermitian matrix.
Then there is exactly one [[lower triangular matrix|00J3KZP8Y0]] $L$ with positive real diagonal entries such that
$$A = L L^*.$$
If $A$ is real, then $L$ is real.

## Remarks

The factor $L$ is the *Cholesky factor* of $A$.
For example, $\begin{pmatrix} 4 & 2 \\ 2 & 5 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$.
It needs about half the work of an LU decomposition, and it solves systems $Ax = b$ with positive definite $A$.
A positive semidefinite matrix also has a decomposition $A = L L^*$, but $L$ need not be unique.

## Explanations

### L1 human

A positive definite matrix is the product of a lower triangular matrix and its conjugate transpose.
This is like writing a positive number as the square of its positive square root.
It is the fastest standard way to solve linear systems with such a matrix.
</document>

<document source="https://math2.tamnd.com/node/0WM7DJCM6G.md" tag="0WM7DJCM6G" kind="statement">
# Product of exponentials

## Statement

For all complex numbers $z$ and $w$,
$$e^{z + w} = e^{z} e^{w},$$
where $e^{z} = \sum_{n=0}^{\infty} z^n / n!$ is the complex exponential function.

## Remarks

The proof multiplies the two series and uses the binomial theorem.
It follows that $e^{z} \neq 0$ and $e^{-z} = 1 / e^{z}$ for each $z$.
With [[Euler's formula|04R3CY41VJ]], it gives the addition formulas for $\cos$ and $\sin$.

## Explanations

### L1 human

The exponential function turns sums into products, for complex numbers as for real numbers.
For example, $e^{i \theta} e^{i \varphi} = e^{i (\theta + \varphi)}$: turning by $\theta$ and then by $\varphi$ is turning by $\theta + \varphi$.
</document>

<document source="https://math2.tamnd.com/node/0WMDV48VSN.md" tag="0WMDV48VSN" kind="statement">
# Volume of a ball in R^3

## Statement

Let $R > 0$, and let $B = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le R^2\}$.
Then
$$\operatorname{vol}(B) = \frac{4}{3} \pi R^3.$$

## Remarks

In [[spherical coordinates|05WKRC7HWT]], $\operatorname{vol}(B) = \int_0^{2\pi} \int_0^\pi \int_0^R \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = 2\pi \cdot 2 \cdot R^3 / 3$.
By slices, the cross-section at height $z$ is a disc of area $\pi (R^2 - z^2)$, and $\int_{-R}^{R} \pi (R^2 - z^2) \, dz = \tfrac{4}{3} \pi R^3$. This is [[Cavalieri's principle|0GTZ2F3PQY]].
This is the case $n = 3$ of the [[volume of a ball in R^n|00YH28R457]], since $\Gamma(5/2) = \tfrac{3}{4} \sqrt{\pi}$.
Archimedes found this volume, by comparison with a cone and a cylinder.

## Explanations

### L1 human

The ball of radius $R$ fills two thirds of the cylinder of radius $R$ and height $2R$ around it, since that cylinder has volume $2 \pi R^3$.
Any of the standard methods of multiple integration gives the same value.
</document>

<document source="https://math2.tamnd.com/node/0WMJSEFYFE.md" tag="0WMJSEFYFE" kind="statement">
# A prime is a Gaussian prime if and only if it is 3 modulo 4

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]].
Then $p$ is a [[Gaussian prime|0GTEHJQRFK]] if and only if $p \equiv 3 \pmod 4$.

## Remarks

If $p = 2$, then $2 = (1 + i)(1 - i)$, and $1 + i$ is a Gaussian prime.
If $p \equiv 1 \pmod 4$, then $p = a^2 + b^2 = (a + bi)(a - bi)$ by [[Fermat's theorem on sums of two squares|043B3ENSTY]], and $a \pm bi$ are Gaussian primes.
If $p \equiv 3 \pmod 4$, then $p$ is not a sum of two squares. So no Gaussian integer has norm $p$, and $p$ cannot split.
For example, $3$ and $7$ are Gaussian primes, but $13 = (3 + 2i)(3 - 2i)$.

## Explanations

### L1 human

An ordinary prime stays prime among the Gaussian integers exactly when it leaves remainder $3$ on division by $4$.
The other primes factor, because they are sums of two squares.
</document>

<document source="https://math2.tamnd.com/node/0WN1QXR7R3.md" tag="0WN1QXR7R3" kind="concept">
# Indefinite integral

## Summary

The indefinite integral of a function is the family of all its antiderivatives on an interval. It is written as one antiderivative plus an arbitrary constant. The node [[indefinite integral|03VHANA6ZK]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WN5G117YC.md" tag="0WN5G117YC" kind="concept">
# Hyperbola

## Summary

A hyperbola is a plane curve with two foci. For each point of the curve, the difference of the distances to the two foci has the same absolute value. It has two branches that are mirror images of each other.
</document>

<document source="https://math2.tamnd.com/node/0WNGQBPPQX.md" tag="0WNGQBPPQX" kind="definition">
# Edge set

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] with vertex set $V$ and adjacency relation $\sim$.
The edge set of $G$ is the set
$$E(G) = \{ \{u, v\} : u, v \in V,\ u \sim v \}$$
of unordered pairs of adjacent vertices.

## Remarks

Each edge is a $2$-element subset of $V$, because $\sim$ is irreflexive.
The number of edges $|E(G)|$ is the size of $G$, and the number of vertices $|V|$ is its order.
The [[handshaking lemma|0XAP8WHNES]] relates $|E(G)|$ to the degrees of the vertices.

## Explanations

### L1 human

The edge set is the list of all lines of the graph, with each line given by its two end points.
A triangle on the vertices $a, b, c$ has the edge set $\{\{a, b\}, \{b, c\}, \{a, c\}\}$.
</document>

<document source="https://math2.tamnd.com/node/0WPDNXGSSN.md" tag="0WPDNXGSSN" kind="statement">
# Hoeffding's inequality

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $a_i \le X_i \le b_i$ almost surely, where $a_i < b_i$ are reals. Let $S_n = X_1 + \dots + X_n$.
Then for each $t > 0$,
$$P\left(S_n - E[S_n] \ge t\right) \le \exp\left(-\frac{2t^2}{\sum_{i=1}^n (b_i - a_i)^2}\right).$$

## Remarks

The same bound holds for $P(S_n - E[S_n] \le -t)$. So $P(|S_n - E[S_n]| \ge t) \le 2 \exp\left(-2t^2 / \sum_i (b_i - a_i)^2\right)$.
The proof uses the [[Chernoff bound|014PEBGDYF]] together with Hoeffding's lemma: $E[e^{s(X - E[X])}] \le e^{s^2 (b - a)^2 / 8}$ if $a \le X \le b$.
For $n$ independent tosses of a coin with $P(\text{heads}) = p$, the fraction $H$ of heads satisfies $P(|H - p| \ge \varepsilon) \le 2 e^{-2n\varepsilon^2}$.

## History

The inequality is named after Wassily Hoeffding.

## Explanations

### L1 human

A sum of many independent bounded terms is very unlikely to be far from its mean.
The probability of a large deviation decreases like $e^{-c t^2}$. This is why polls with a few thousand people give reliable results.
</document>

<document source="https://math2.tamnd.com/node/0WPQYW8MTE.md" tag="0WPQYW8MTE" kind="definition">
# Pole

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$, so that $f$ is holomorphic on $0 < |z - c| < r$ for some $r > 0$.
The singularity $c$ is a pole of $f$ if $|f(z)| \to \infty$ as $z \to c$. This means that for each $K > 0$ there is $\delta > 0$ such that $|f(z)| > K$ for each $z$ with $0 < |z - c| < \delta$.

## Remarks

The point $c$ is a pole of $f$ if and only if $f(z) = \frac{g(z)}{(z - c)^m}$ near $c$, for an integer $m \ge 1$ and a function $g$ that is holomorphic near $c$ with $g(c) \neq 0$. The integer $m$ is the [[order of the pole|0YK27TJ4WF]].
Equivalently, $1/f$ extends to a holomorphic function near $c$ with a [[zero|0CEGEASY49]] at $c$.
For example, $\frac{1}{z^2 (z - 1)}$ has a pole of order $2$ at $0$ and a pole of order $1$, a simple pole, at $1$.

## Explanations

### L1 human

A pole is a point where the function goes to infinity, in the way $1/z$ or $1/z^3$ does at $0$.
Near a pole, $1/f$ is a holomorphic function with a zero there.
</document>

<document source="https://math2.tamnd.com/node/0WPZ5YPAGR.md" tag="0WPZ5YPAGR" kind="definition">
# Distance in a graph

## Definition

Let $G$ be a [[graph|0JD5C82QMB]] and let $u, v$ be vertices of $G$.
If $v$ is [[reachable|0J8XE7W68F]] from $u$, the distance $d(u, v)$ is the smallest length of a [[walk|0VJZM0F9YM]] from $u$ to $v$.
If $v$ is not reachable from $u$, then $d(u, v) = \infty$.

## Remarks

A shortest walk is a [[path|0JDE56HWQ8]], so $d(u, v)$ is also the smallest length of a path. Such a path is a geodesic.
On a [[connected|0FFAYNMAJZ]] graph, $d$ is a metric: $d(u, v) = 0$ only for $u = v$, it is symmetric, and it satisfies the triangle inequality.

## Explanations

### L1 human

The distance between two vertices is the smallest number of edges that you must cross to go from one to the other.
In the cycle $C_6$, opposite vertices have distance $3$.
</document>

<document source="https://math2.tamnd.com/node/0WQ19TJV24.md" tag="0WQ19TJV24" kind="concept">
# Distance from a point to a set

## Summary

In a metric space, the distance from a point $x$ to a nonempty set $A$ is the infimum of the distances $d(x, a)$ with $a \in A$.
</document>

<document source="https://math2.tamnd.com/node/0WQ1A8SHTZ.md" tag="0WQ1A8SHTZ" kind="definition">
# Field extension

## Definition

Let $K$ and $L$ be [[fields|0QA8CHH8MG]].
The field $L$ is an extension of $K$ if $K$ is a [[subfield|07QY5RPFBR]] of $L$.
One writes $L/K$ for this field extension.

## Remarks

Often one also calls $L$ an extension of $K$ when there is an injective [[ring homomorphism|0GGE0Z0JQ1]] $K \to L$. Then one identifies $K$ with its image.
An extension $L/K$ makes $L$ a vector space over $K$. Its dimension is the [[degree of the extension|0GS74FGFEK]].
Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{R}/\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$.
The notation $L/K$ does not mean a quotient.

## Explanations

### L1 human

A field extension is a bigger field that contains a given field.
For example, the complex numbers extend the real numbers by an element $i$ with $i^2 = -1$.
Field theory studies how the bigger field is built from the smaller one.
</document>

<document source="https://math2.tamnd.com/node/0WQE3G7B88.md" tag="0WQE3G7B88" kind="definition">
# Surface area

## Definition

Let $D \subseteq \mathbb{R}^2$ be a bounded [[Jordan measurable|0565DC8RMK]] closed set, and let $r \colon D \to \mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is injective on the interior of $D$.
The surface area of $S = r(D)$ is
$$\operatorname{area}(S) = \iint_D \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.$$

## Remarks

The integrand is the length of the [[normal vector|0BJDQ2HBH1]]. It is the area of the parallelogram spanned by $r_u$ and $r_v$, which approximates the image of a small parameter rectangle.
For the graph $z = f(x, y)$ over $D$, the area is $\iint_D \sqrt{1 + f_x^2 + f_y^2} \, dx \, dy$.
For the unit sphere, $\| r_\varphi \times r_\theta \| = \sin \varphi$, and the area is $\int_0^{2\pi} \int_0^\pi \sin \varphi \, d\varphi \, d\theta = 4\pi$.
The value does not depend on the parametrisation. A piecewise smooth surface gets the sum of the areas of its pieces. Inscribed polyhedra do not give a good definition: the Schwarz lantern is a counterexample.

## Explanations

### L1 human

To find the area of a curved surface, cut the parameter domain into small rectangles.
Each one maps to a small curved patch, which is nearly a parallelogram, and you add up the areas of these parallelograms.
</document>

<document source="https://math2.tamnd.com/node/0WQJJCCTC8.md" tag="0WQJJCCTC8" kind="concept">
# Normal vector

## Summary

A normal vector of a line, a plane or a surface is a nonzero vector that is perpendicular to it. For a curved surface, a normal vector at a point is perpendicular to the tangent plane at that point.
</document>

<document source="https://math2.tamnd.com/node/0WQNRCNY3M.md" tag="0WQNRCNY3M" kind="concept">
# Expected value

## Summary

The expected value of a random variable is its average value, where each value has a weight equal to its probability.
The node [[expected value|0DCT0QZVQ1]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WQVXP7PDB.md" tag="0WQVXP7PDB" kind="concept">
# Moment-generating function

## Summary

The moment-generating function of a real random variable $X$ is the function $t \mapsto E[e^{tX}]$. When it is finite near $0$, its derivatives at $0$ are the moments of $X$.
The node [[moment-generating function|0NNRNGKMFM]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WQXY32Q8D.md" tag="0WQXY32Q8D" kind="concept">
# Borel sigma-algebra

## Summary

The Borel σ-algebra of a topological space is the smallest σ-algebra that contains all open sets. Its elements are the Borel sets.
On the real line it is the usual family of events for real random variables. The node [[Borel sigma-algebra|0BVHZXWKCD]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0WRZ170YEH.md" tag="0WRZ170YEH" kind="statement">
# Moment-generating function of a sum of independent random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \dots + X_n$.
Then for each $t \in \mathbb{R}$, the [[moment-generating functions|0NNRNGKMFM]] satisfy
$$M_S(t) = M_{X_1}(t) \, M_{X_2}(t) \cdots M_{X_n}(t),$$
in $(0, \infty]$.

## Remarks

Proof: $e^{tS} = e^{tX_1} \cdots e^{tX_n}$. The factors are nonnegative and independent, so the [[product rule for expectations|05AW1G0WE1]] applies.
For i.i.d. $X_i$, $M_S(t) = M_{X_1}(t)^n$.
Example: if $X_i \sim \operatorname{Exp}(\lambda)$ are independent, then $M_S(t) = (\lambda/(\lambda - t))^n$ for $t < \lambda$. This is the moment-generating function of the gamma distribution $\operatorname{Gamma}(n, \lambda)$.

## Explanations

### L1 human

When you add independent random variables, their moment-generating functions multiply.
This makes it easy to identify the law of a sum, if you know which law has the product as its moment-generating function.
</document>

<document source="https://math2.tamnd.com/node/0WS1A8E2P9.md" tag="0WS1A8E2P9" kind="statement">
# Extreme value theorem

## Statement

Let $a \le b$ and let $f \colon [a, b] \to \mathbb{R}$ be [[continuous|0P27W9RW0G]].
Then there are $c, d \in [a, b]$ with $f(c) \le f(x) \le f(d)$ for all $x \in [a, b]$.

## Remarks

By the [[boundedness theorem|0ATVXC3NCC]], $M = \sup f([a, b])$ is finite. If $f(x) < M$ for all $x$, then $1/(M - f(x))$ is continuous and unbounded, which is a contradiction.
Each hypothesis is needed. The function $x$ on $[0, 1)$ has no maximum, and $x$ on $[0, \infty)$ is not bounded above.
Mathlib states it for compact sets, as `IsCompact.exists_isMinOn` and `IsCompact.exists_isMaxOn`.

## History

Bernard Bolzano proved the theorem in the 1830s. His proof stayed unpublished until 1930.

## Explanations

### L1 human

On a closed bounded interval, a continuous function has a highest value and a lowest value.
Its graph has a top point and a bottom point.
</document>

<document source="https://math2.tamnd.com/node/0WS9P12FQR.md" tag="0WS9P12FQR" kind="concept">
# Subspace topology

## Summary

The subspace topology is the topology that a subset of a topological space gets from the whole space.
Its open sets are the intersections of the subset with the open sets of the whole space.
</document>

<document source="https://math2.tamnd.com/node/0WSFKKT6FF.md" tag="0WSFKKT6FF" kind="concept">
# Image of a linear map

## Summary

The image of a linear map is the set of all values of the map.
It is a subspace of the codomain.
</document>

<document source="https://math2.tamnd.com/node/0WTEMGYV2C.md" tag="0WTEMGYV2C" kind="statement">
# Cantor's theorem

## Statement

Let $A$ be a [[set|0SJ3RX9TXA]] and let $f \colon A \to \mathcal{P}(A)$ be a [[function|0XHDTE5YMZ]] to the [[power set|0BC7KX51Z8]] of $A$.
Then $f$ is not [[surjective|0XKH881BZ7]].
So $|A| < |\mathcal{P}(A)|$.

## Remarks

Proof: let $D = \{a \in A \mid a \notin f(a)\}$. If $D = f(d)$ for some $d \in A$, then $d \in D$ if and only if $d \notin D$, a contradiction.
The map $a \mapsto \{a\}$ is an injection, so $|A| \le |\mathcal{P}(A)|$, and the inequality is strict.
It follows that there is no largest cardinal. The proof is a form of the [[diagonal argument|0XCH8QVDCQ]], and it is close to [[Russell's paradox|0HESMMSYEG]].

## Explanations

### L1 human

For each set, its set of subsets is strictly larger.
For a set with $3$ elements, this is $3 < 8$.
For infinite sets, it gives an endless tower of larger and larger infinities.
</document>

<document source="https://math2.tamnd.com/node/0WW060JR6P.md" tag="0WW060JR6P" kind="definition">
# Nullcline

## Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f = (f_1, \ldots, f_n) : U \to \mathbb{R}^n$ be a function, and consider the [[autonomous system|07BNRFQ0PJ]] $x_j' = f_j(x_1, \ldots, x_n)$ for $j = 1, \ldots, n$.
For $j \in \{1, \ldots, n\}$, the $j$-th nullcline of the system is the set
$$N_j = \{x \in U : f_j(x) = 0\}.$$

## Remarks

On the $j$-th nullcline, the velocity has no component in the direction $x_j$. In the plane, trajectories cross the $x$-nullcline vertically and the $y$-nullcline horizontally.
The [[equilibrium points|0G4X36AE0X]] are the points that lie on all the nullclines.
The nullclines divide the [[phase plane|05VDGH129F]] into regions where each component of the velocity has a constant sign.
For a planar linear system, the nullclines are lines through the origin, or the whole plane when a row of the matrix is zero.

## Explanations

### L1 human

A nullcline is the set of states where one of the variables stops changing for an instant.
Drawing the nullclines is a quick first step to sketch a phase portrait.
</document>

<document source="https://math2.tamnd.com/node/0WW85SQVS3.md" tag="0WW85SQVS3" kind="definition">
# Bell numbers

## Definition

Let $n$ be a natural number.
The Bell number $B_n$ is the number of partitions of an $n$-element set into nonempty blocks.
Equivalently, it is the sum of [[Stirling numbers of the second kind|0ZN07RQ3JS]]:
$$B_n = \sum_{k=0}^{n} S(n, k).$$

## Remarks

The first values, for $n = 0, 1, \dots, 6$, are $1, 1, 2, 5, 15, 52, 203$.
$B_0 = 1$, because the empty set has exactly one partition, the partition with no blocks.
The values satisfy a [[recurrence|0110GBHPPB]] with binomial coefficients.

## Explanations

### L1 human

The Bell number $B_n$ counts all ways to split $n$ labelled objects into groups, with any number of groups.
For $\{1, 2, 3\}$ there are $5$ ways: $123$, $1|23$, $2|13$, $3|12$ and $1|2|3$.
</document>

<document source="https://math2.tamnd.com/node/0WWRD82HD2.md" tag="0WWRD82HD2" kind="concept">
# Right inverse

## Summary

A right inverse of a function chooses, for each target value, an input that the function sends to it.
</document>

<document source="https://math2.tamnd.com/node/0WX11EA3F3.md" tag="0WX11EA3F3" kind="object">
# Group of nth roots of unity

## Description

Let $n$ be a positive integer.
The group of $n$th roots of unity is the set $\mu_n = \{z \in \mathbb{C} : z^n = 1\}$ with complex multiplication.
It is a finite [[abelian group|082GHNPJ53]] of order $n$, and its elements are $e^{2\pi i k / n}$ for $k = 0, 1, \ldots, n - 1$.
It is cyclic, and the element $e^{2\pi i / n}$ generates it.

## Remarks

The group $\mu_n$ is a subgroup of the [[circle group|04EAFVFTF7]].
For $n = 4$, the group is $\{1, i, -1, -i\}$.
A generator of $\mu_n$ is called a primitive $n$th root of unity, and there are $\varphi(n)$ of them, where $\varphi$ is Euler's totient function.

## Explanations

### L1 human

The $n$th roots of unity are $n$ points on the unit circle that cut it into $n$ equal arcs.
Multiplying two of them adds their angles, so the product is again one of these points.
They behave like the rotations of a regular polygon with $n$ sides.
</document>

<document source="https://math2.tamnd.com/node/0WXBGBTA3D.md" tag="0WXBGBTA3D" kind="statement">
# Gram–Schmidt vectors are orthogonal

## Statement

Let $V$ be an inner product space over $\R$ or $\C$, and let $v_1, \dots, v_k \in V$ be linearly independent.
Let $u_1, \dots, u_k$ be the vectors of the [[Gram–Schmidt process|0VT6P0GNQR]].
Then $u_1, \dots, u_k$ are nonzero and pairwise [[orthogonal|0Y2X7P2WB7]], and
$$\operatorname{span}(u_1, \dots, u_j) = \operatorname{span}(v_1, \dots, v_j) \qquad (1 \le j \le k).$$

## Remarks

The proof is by induction on $j$: by construction, $\langle u_j, u_i \rangle = 0$ for $i < j$.
The vector $u_j$ is nonzero, because $v_j$ is not in $\operatorname{span}(v_1, \dots, v_{j-1})$.
If the $v_i$ are only spanning, then the process produces some zero vectors, and dropping them still gives an orthogonal family with the same span.

## Explanations

### L1 human

The vectors made by Gram–Schmidt are really perpendicular to each other.
Also, the first $j$ new vectors span the same space as the first $j$ old vectors.
So no information is lost at any step.
</document>

<document source="https://math2.tamnd.com/node/0WYTM4C9QG.md" tag="0WYTM4C9QG" kind="statement">
# A monotone function has countably many discontinuities

## Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be monotone.
Then the set of points of $I$ where $f$ is not continuous is countable.

## Remarks

Assume $f$ increasing. By [[the existence of one-sided limits|0KJXHN0TW0]], each discontinuity $c$ inside $I$ gives an open interval $(f(c^-), f(c^+))$.
These intervals are disjoint. Each contains a rational number, so there are only countably many of them.
All these discontinuities are [[jumps|05B0FKFXQA]].
Mathlib states it as `Monotone.countable_not_continuousAt`.

## Explanations

### L1 human

Each discontinuity of an increasing function is a jump up.
The jumps take disjoint pieces of the vertical axis, so there can be only countably many.
</document>

<document source="https://math2.tamnd.com/node/0WZCHE1ZF9.md" tag="0WZCHE1ZF9" kind="concept">
# Iterated integral

## Summary

An iterated integral integrates a function of several variables with respect to one variable at a time. At each step, it keeps the other variables fixed.
It reduces a multiple integral to a sequence of integrals in one variable.
</document>

<document source="https://math2.tamnd.com/node/0WZTM0EFR6.md" tag="0WZTM0EFR6" kind="definition">
# Hölder continuity

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \colon X \to Y$ be a function, and let $C \ge 0$ and $\alpha > 0$ be real numbers.
The map $f$ is Hölder continuous with constant $C$ and exponent $\alpha$ when
$$d_Y\bigl(f(x), f(y)\bigr) \le C \, d_X(x, y)^{\alpha} \quad \text{for all } x, y \in X.$$

## Remarks

For $\alpha = 1$, this is the condition for a [[Lipschitz map|0TMZ21WKCN]] with constant $C$.
The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is Hölder continuous with $C = 1$ and $\alpha = 1/2$.
A Hölder continuous map is uniformly continuous: for $\varepsilon > 0$ and $C > 0$, take $\delta = (\varepsilon / C)^{1/\alpha}$.
For a real function on an interval, Hölder continuity with an exponent $\alpha > 1$ forces the function to be constant.

## Explanations

### L1 human

The Hölder condition is a weaker form of the Lipschitz condition.
It allows a map to be steep, as $\sqrt{x}$ is near $0$, but it still controls how fast the values can change.
</document>

<document source="https://math2.tamnd.com/node/0WZXK05ME7.md" tag="0WZXK05ME7" kind="definition">
# Division ring

## Definition

A division ring is a [[ring|0894J980RK]] $D$ with these two properties.

1. $1 \neq 0$ in $D$.
2. Every nonzero element of $D$ is a [[unit|019NFGFPR5]]: for each $a \neq 0$ in $D$, there is $b \in D$ with $a b = b a = 1$.

## Remarks

A commutative division ring is a [[field|0QA8CHH8MG]].
The [[quaternions|0GE4DVQKR8]] are a division ring that is not commutative.
By [[Wedderburn's little theorem|0RRNVX1VKA]], every finite division ring is a field.

## Explanations

### L1 human

In a division ring, you can divide by every nonzero element, as in a field.
But $a b$ and $b a$ can be different.
The quaternions of Hamilton are the standard example.
</document>

<document source="https://math2.tamnd.com/node/0X05GS525E.md" tag="0X05GS525E" kind="concept">
# Boolean function

## Summary

A Boolean function is a function from $n$-tuples of bits to bits. Boolean functions are the objects that logic formulas and digital circuits compute. The node [[Boolean function|0A58THH5XV]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X0HQYSZ0K.md" tag="0X0HQYSZ0K" kind="concept">
# Determinant of a linear map

## Summary

The determinant of an endomorphism of a finite-dimensional vector space is the determinant of its matrix in any basis.
</document>

<document source="https://math2.tamnd.com/node/0X0N833FF8.md" tag="0X0N833FF8" kind="statement">
# Number of injections

## Statement

Let $A$ and $B$ be finite sets with $|A| = k$ and $|B| = n$.
Then the number of injective functions from $A$ to $B$ is the [[falling factorial|0N5M924N3P]]
$$(n)_k = n (n - 1) \cdots (n - k + 1).$$

## Remarks

If $k > n$, then $(n)_k = 0$, and there is no injection. This is the [[pigeonhole principle|0HGVCWQGY3]].
An injection from $\{1, \dots, k\}$ to $B$ is a $k$-permutation of $B$: an ordered list of $k$ distinct elements of $B$.
For $k = n$ the count is $n!$, the [[number of permutations|0CEBCS0E0E]].

## Explanations

### L1 human

List the elements of $A$ and give each one an image in $B$ that is not used yet.
The first has $n$ choices, the second $n - 1$, and so on, for $k$ steps.
</document>

<document source="https://math2.tamnd.com/node/0X1GSWNBN1.md" tag="0X1GSWNBN1" kind="concept">
# Union

## Summary

The union of sets is the set of all objects that are elements of at least one of the sets.
</document>

<document source="https://math2.tamnd.com/node/0X1JXT47GA.md" tag="0X1JXT47GA" kind="concept">
# Subgroup

## Summary

A subgroup of a group is a subset that is itself a group under the same operation.
</document>

<document source="https://math2.tamnd.com/node/0X2FY491PF.md" tag="0X2FY491PF" kind="definition">
# Complex logarithm

## Definition

Let $z$ be a complex number with $z \neq 0$.
A complex logarithm of $z$ is a complex number $w$ with $e^w = z$, where $e^w$ is the [[complex exponential|054VXHMCA3]] of $w$.
The notation $\log z$ means the set of all complex logarithms of $z$, or one chosen element of this set.

## Remarks

Let $|z|$ be the [[modulus|0KW2X0TFBX]] of $z$, let $\ln$ be the real natural logarithm, and let $\theta$ be an [[argument|0ZZRSX91QR]] of $z$. Then the logarithms of $z$ are exactly the numbers $\ln |z| + i (\theta + 2 \pi k)$ with $k \in \mathbb{Z}$.
The number $0$ has no logarithm, because $e^w \neq 0$ for each $w$.
So $\log$ is a [[multivalued function|03CPD1HGF3]] on $\mathbb{C} \setminus \{0\}$. Its [[principal branch|0YKY4M3KRQ]] is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, with $-\pi < \operatorname{Arg} z \le \pi$.
For example, the logarithms of $-1$ are $i \pi (2k + 1)$ with $k \in \mathbb{Z}$, and $\operatorname{Log}(-1) = i \pi$.

## Explanations

### L1 human

A logarithm of $z$ is an answer to the question: $e$ to which power gives $z$?
Because $e^{w + 2\pi i} = e^w$, each answer gives infinitely many other answers, spaced by $2\pi i$ on a vertical line.
The real part of each answer is $\ln |z|$, and the imaginary part is an angle of $z$.
</document>

<document source="https://math2.tamnd.com/node/0X2GJPNPFN.md" tag="0X2GJPNPFN" kind="statement">
# Hall's theorem for bipartite graphs

## Statement

Let $G$ be a finite [[bipartite graph|07DVHJ8N0A]] with parts $X$ and $Y$.
For $W \subseteq X$, let $N(W)$ be the set of vertices of $Y$ that are adjacent to some vertex of $W$.
Then $G$ has a [[matching|0Z87BHAFTB]] that covers every vertex of $X$ if and only if $|N(W)| \ge |W|$ for every $W \subseteq X$.

## Remarks

If $|X| = |Y|$, such a matching is a [[perfect matching|05ZPH8KF9K]]. Thus the condition then characterizes the perfect matchings.
A consequence: a finite regular bipartite graph with at least one edge has a perfect matching.
The theorem is the graph form of [[Hall's marriage theorem|08WCB90WND]]: let $X$ index the sets and let $Y$ be their union.

## Explanations

### L1 human

Think of $X$ as workers and $Y$ as jobs, with an edge when a worker can do a job.
Every worker can get a different job exactly when this holds.
Each group of workers can do, together, at least as many jobs as there are workers in the group.
</document>

<document source="https://math2.tamnd.com/node/0X2J0VY6DN.md" tag="0X2J0VY6DN" kind="concept">
# Discrete random variable

## Summary

A discrete random variable is a random variable whose values lie, with probability one, in a countable set.
Its law is described by the probabilities of the single values. The node [[discrete random variable|0J6Z6H5VQJ]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X2T63GFQQ.md" tag="0X2T63GFQQ" kind="statement">
# Countable intersection of dense open sets is dense

## Statement

Let $X$ be a [[Baire space|0D5KR33H7X]] and let $(U_n)_{n \in \mathbb{N}}$ be a sequence of open subsets of $X$, each [[dense|0D198GPMTJ]] in $X$.
Then $\bigcap_{n} U_n$ is dense in $X$.

## Remarks

Proof: let $F_n = X \setminus U_n$. Each $F_n$ is closed, and $\operatorname{int} F_n = X \setminus \overline{U_n} = \emptyset$. Thus $\operatorname{int} \bigcup_n F_n = \emptyset$. Its complement $\bigcap_n U_n$ therefore has closure $X$.
The converse also holds, so this property characterises Baire spaces.
In $\mathbb{R}$, for an enumeration $q_1, q_2, \ldots$ of $\mathbb{Q}$, the sets $U_n = \mathbb{R} \setminus \{q_n\}$ are open and dense. Their intersection is the set of irrational numbers, which is dense.
The intersection need not be open.

## Explanations

### L1 human

Each dense open set leaves out only a thin closed set.
In a Baire space, removing countably many thin sets still leaves a set that is everywhere.
</document>

<document source="https://math2.tamnd.com/node/0X308R2DBE.md" tag="0X308R2DBE" kind="statement">
# The exterior derivative squares to zero

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $\omega$ be a differential $k$-form on $U$ with coefficients of class $C^2$.
Then
$$d(d\omega) = 0.$$

## Remarks

Proof: for $\omega = f \, dx_I$, the [[exterior derivative|004FRCXH6M]] gives $d(d\omega) = \sum_{j, l} \partial_l \partial_j f \, dx_l \wedge dx_j \wedge dx_I$.
By [[Schwarz's theorem|0NNQY8WCW6]], $\partial_l \partial_j f = \partial_j \partial_l f$, while $dx_l \wedge dx_j = -dx_j \wedge dx_l$. So the terms cancel in pairs, and the terms with $j = l$ are zero.
In $\mathbb{R}^3$, the cases $k = 0$ and $k = 1$ are the identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]].
It follows that each [[exact form|0F1AVGX26N]] with $C^2$ coefficients is closed.

## Explanations

### L1 human

Taking the exterior derivative twice always gives zero, because mixed second partial derivatives are equal while the wedge product changes sign.
This one rule contains the facts that gradients have no curl and curls have no divergence.
</document>

<document source="https://math2.tamnd.com/node/0X3EG7Y65T.md" tag="0X3EG7Y65T" kind="concept">
# Path component

## Summary

A path component of a topological space is a maximal path-connected subset.
The path components form a partition of the space.
</document>

<document source="https://math2.tamnd.com/node/0X3TEM69AA.md" tag="0X3TEM69AA" kind="definition">
# Congruent triangles

## Definition

Let $n$ be a positive integer, and let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$.
The triangles are congruent, written $ABC \cong A'B'C'$, if there is an isometry $f : \mathbb{R}^n \to \mathbb{R}^n$ with $f(A) = A'$, $f(B) = B'$ and $f(C) = C'$.
An isometry is a map $f$ with $d(f(x), f(y)) = d(x, y)$ for all points $x$ and $y$, where $d$ is the [[Euclidean distance|0NPPGHQNXQ]].

## Remarks

Congruence is an equivalence relation on triangles.
Congruent triangles have equal corresponding sides and equal corresponding angles. The [[side-side-side criterion|0BRBYRK6PA]] gives the converse.
The definition depends on the order of the vertices. For example, $ABC \cong ACB$ holds only when $d(A, B) = d(A, C)$.
Mathlib's `Congruent` asks for equal distances between all pairs of corresponding points of two indexed families. In a Euclidean space this is equivalent, by the side-side-side criterion.

## Explanations

### L1 human

Two triangles are congruent if you can cut one out of paper and lay it exactly on the other. You may slide it, turn it, and flip it over. Then corresponding sides and corresponding angles match.
</document>

<document source="https://math2.tamnd.com/node/0X3WGX95FE.md" tag="0X3WGX95FE" kind="concept">
# Continuous function

## Summary

A continuous function is a function that is continuous at each point of its domain. Small changes of the argument then give small changes of the value. The node [[continuous function|0P27W9RW0G]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X3XD2ATB0.md" tag="0X3XD2ATB0" kind="definition">
# Laurent series

## Definition

Let $c \in \mathbb{C}$ and let $(a_n)_{n \in \mathbb{Z}}$ be complex numbers.
The Laurent series with center $c$ and coefficients $a_n$ is the series
$$\sum_{n = -\infty}^{\infty} a_n (z - c)^n$$
in the complex variable $z$.
It converges at a point $z \neq c$ if both series $\sum_{n \ge 0} a_n (z - c)^n$ and $\sum_{n \ge 1} a_{-n} (z - c)^{-n}$ converge. Its sum at $z$ is then the sum of the two limits.

## Remarks

A [[power series|0VARVQ9ZW1]] is a Laurent series with $a_n = 0$ for each $n < 0$.
If the series converges on a nonempty open set, then it converges absolutely on an [[annulus|0AJ0YVK743]] $r < |z - c| < R$, and uniformly on each smaller closed annulus.
For example, $e^{1/z} = \sum_{n \ge 0} \frac{1}{n!} z^{-n}$ for each $z \neq 0$.
The part with negative powers is the [[principal part|0NDV5R6Z6X]].

## Explanations

### L1 human

A Laurent series is like a power series, but it can also have negative powers of $z - c$.
The negative powers allow it to describe functions that blow up at $c$, such as $1/z$ or $e^{1/z}$.
</document>

<document source="https://math2.tamnd.com/node/0X3Y0ZSKNF.md" tag="0X3Y0ZSKNF" kind="concept">
# Hypergeometric distribution

## Summary

The hypergeometric distribution describes the number of successes in a sample drawn without replacement from a finite population.
The node [[hypergeometric distribution|0SPBPW3K61]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X45ZM731G.md" tag="0X45ZM731G" kind="example">
# Cayley transform

## Example

Let $\mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im} z > 0 \}$ be the upper half-plane and $\mathbb{D} = \{ w \in \mathbb{C} : |w| < 1 \}$ the open unit disc.
The Cayley transform is
$$W(z) = \frac{z - i}{z + i}.$$
It is a [[Möbius transformation|0B346DDAM5]] with $a = 1$, $b = -i$, $c = 1$, $d = i$, and $ad - bc = 2i \neq 0$.
For $z \in \mathbb{C}$, $|W(z)| < 1$ holds exactly when $z$ is nearer to $i$ than to $-i$, that is, when $\operatorname{Im} z > 0$.
The map $W$ is a bijection from $\mathbb{H}$ onto $\mathbb{D}$, with inverse $w \mapsto i \frac{1 + w}{1 - w}$.

## Remarks

Some values are $W(i) = 0$, $W(0) = -1$, $W(1) = -i$, $W(-1) = i$ and $W(\infty) = 1$.
The map $W$ sends the real line together with $\infty$ onto the unit circle.
It is an explicit case of the [[Riemann mapping theorem|02WHMEG5ZR]] for $\mathbb{H}$.
In hyperbolic geometry, it connects the half-plane model and the disc model.

## Explanations

### L1 human

The Cayley transform turns the upper half-plane into the unit disc.
A point of $\mathbb{H}$ is nearer to $i$ than to $-i$, so the quotient of the two distances is less than $1$.
It lets you move a problem on the half-plane to the disc, where it can be easier.
</document>

<document source="https://math2.tamnd.com/node/0X4JKZXJ86.md" tag="0X4JKZXJ86" kind="definition">
# Idempotent element

## Definition

Let $R$ be a [[ring|0894J980RK]].
An element $e \in R$ is idempotent if $e^2 = e$.

## Remarks

In every ring, $0$ and $1$ are idempotent.
In $\mathbb{Z}/6\mathbb{Z}$, the classes of $3$ and $4$ are idempotent, because $9 \equiv 3$ and $16 \equiv 4 \pmod 6$.
If $e$ is idempotent, then $1 - e$ is idempotent, and $e (1 - e) = 0$.
In the [[product|0MNZCN4BQK]] $R \times S$, the elements $(1, 0)$ and $(0, 1)$ are idempotent.

## Explanations

### L1 human

An idempotent does not change when you square it.
Among the integers, only $0$ and $1$ have this property.
Other rings can have more idempotents, for example the projection matrices.
</document>

<document source="https://math2.tamnd.com/node/0X4NT20S3G.md" tag="0X4NT20S3G" kind="example">
# Forced oscillation and resonance

## Example

Let $\omega_0 > 0$, $\zeta \ge 0$, $F > 0$ and $\omega > 0$ be constants. The forced oscillator is
$$x'' + 2 \zeta \omega_0\, x' + \omega_0^2\, x = F \cos(\omega t).$$
If $\zeta > 0$, it has the particular solution $x_p(t) = a(\omega) \cos(\omega t - \delta)$ for a phase $\delta$, with amplitude
$$a(\omega) = \frac{F}{\sqrt{\left(\omega_0^2 - \omega^2\right)^2 + \left(2 \zeta \omega_0 \omega\right)^2}}.$$
Each solution is $x_p$ plus a solution of the [[damped equation|0RNXVVTJTJ]], which tends to $0$. So $x_p$ is the steady state.
If $\zeta = 0$ and $\omega = \omega_0$, a particular solution is $x_p(t) = \frac{F}{2 \omega_0}\, t \sin(\omega_0 t)$, which is unbounded. This is resonance.

## Remarks

For $0 < \zeta < 1/\sqrt{2}$, the amplitude $a(\omega)$ is largest at $\omega = \omega_0 \sqrt{1 - 2 \zeta^2}$. For small $\zeta$, the maximum is close to $F / (2 \zeta \omega_0^2)$, which is large.
For $\zeta = 0$ and $\omega \ne \omega_0$, all solutions are bounded: $x_p = F \cos(\omega t) / (\omega_0^2 - \omega^2)$.
The factor $t$ in the resonant solution comes from the rule of the [[method of undetermined coefficients|0XRB469Z32]]: $i \omega_0$ is a root of the characteristic polynomial.

## Explanations

### L1 human

If you push a swing at the right rhythm, each push adds to the motion, and the swing goes higher and higher.
This is resonance: the driving frequency matches the natural frequency $\omega_0$.
Friction limits the growth, but the response is still very large near $\omega_0$.
</document>

<document source="https://math2.tamnd.com/node/0X52TNMTCJ.md" tag="0X52TNMTCJ" kind="statement">
# Tietze extension theorem

## Statement

Let $X$ be a [[normal space|0AJ646N339]], let $A \subseteq X$ be [[closed|0G5AV9Q4RR]], and let $f \colon A \to \mathbb{R}$ be [[continuous|0SY9ECQ19Q]] for the subspace topology on $A$.
Then there is a continuous $F \colon X \to \mathbb{R}$ whose [[restriction|0D2MXBV21T]] to $A$ is $f$.
If also $|f(a)| \le M$ for all $a \in A$, then one can choose $F$ with $|F(x)| \le M$ for all $x \in X$.

## Remarks

The proof applies [[Urysohn's lemma|01EY3VR7ZD]] many times and sums a uniformly convergent series of functions.
The converse holds: if each continuous function from each closed subset to $\mathbb{R}$ extends, then $X$ is normal.
The closedness of $A$ is needed: the function $\sin(1/x)$ on $(0, 1]$ has no continuous extension to $[0, 1]$.

## Explanations

### L1 human

On a normal space, each continuous real function on a closed set can be extended to the whole space, without losing continuity.
You can also keep the same bound on the function.
</document>

<document source="https://math2.tamnd.com/node/0X55M9XET4.md" tag="0X55M9XET4" kind="definition">
# Uncountable set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is uncountable when it is not [[countable|0ZG61TFBES]], that is, when there is no [[injective function|0PVG1CRAJN]] from $A$ to $\mathbb{N}$.

## Remarks

The real numbers and the [[power set|0BC7KX51Z8]] of $\mathbb{N}$ are uncountable.
With the axiom of choice, a set is uncountable exactly when its cardinality is at least $\aleph_1$, the least cardinal larger than $\aleph_0$.
A superset of an uncountable set is uncountable.

## Explanations

### L1 human

For an uncountable set, every attempt to list its elements misses some of them.
[[Cantor's diagonal argument|0XCH8QVDCQ]] shows how to find a missing element.
</document>

<document source="https://math2.tamnd.com/node/0X5QVCF06N.md" tag="0X5QVCF06N" kind="concept">
# Elementary symmetric polynomial

## Summary

The elementary symmetric polynomials in $n$ variables are the sums of all products of $k$ distinct variables, for $k = 0, \dots, n$.
They generate all symmetric polynomials.
The [[definition of the elementary symmetric polynomials|0ZMY8441GW]] gives the formula.
</document>

<document source="https://math2.tamnd.com/node/0X60VE98HN.md" tag="0X60VE98HN" kind="concept">
# Quantile function

## Summary

The quantile function of a real distribution is a generalized inverse of its cumulative distribution function.
For a level $p$ between $0$ and $1$, it gives the smallest value $x$ with probability at least $p$ of not being exceeded. The node [[quantile function|0B3Q3XDM5X]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X6AB5P3G1.md" tag="0X6AB5P3G1" kind="concept">
# Algebraic closure

## Summary

An algebraic closure of a field $K$ is an algebraic extension of $K$ that is algebraically closed.
It exists and is unique up to an isomorphism that fixes $K$.
The [[definition of an algebraic closure|0MCXX29HWY]] states it.
</document>

<document source="https://math2.tamnd.com/node/0X6C0MAF84.md" tag="0X6C0MAF84" kind="concept">
# Hyperbolic tangent

## Summary

The hyperbolic tangent is the quotient of the hyperbolic sine by the hyperbolic cosine. It is a bounded, odd, strictly increasing function with range $(-1, 1)$. The node [[hyperbolic tangent|0MWYQTZ07Z]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0X6V9B1V10.md" tag="0X6V9B1V10" kind="concept">
# Consistent and inconsistent systems

## Summary

A system of equations is consistent when it has at least one solution, and inconsistent when it has no solution.
</document>

<document source="https://math2.tamnd.com/node/0X7CVDE3BG.md" tag="0X7CVDE3BG" kind="statement">
# Impossibility of squaring the circle

## Statement

The number $\sqrt{\pi}$ is not a [[constructible number|0FMN2MN7TM]].
So the circle cannot be [[squared|0BSXRER81B]] with straightedge and compass.

## Remarks

The proof: every constructible number is [[algebraic over $\mathbb{Q}$|0SSSM1D475]]. If $\sqrt{\pi}$ were constructible, then $\pi = (\sqrt{\pi})^2$ would be constructible and so algebraic.
This contradicts the [[transcendence of $\pi$|0G0F7FGCGP]].

## History

The impossibility was proved in 1882, as a consequence of Lindemann's proof that $\pi$ is transcendental.
It had been known for decades that the transcendence of $\pi$ would imply the impossibility.

## Explanations

### L1 human

A square with the area of the unit circle has side $\sqrt{\pi}$.
Constructible numbers are algebraic, and $\pi$ is not, so this side cannot be constructed.
</document>

<document source="https://math2.tamnd.com/node/0X7EEWES5G.md" tag="0X7EEWES5G" kind="concept">
# Centroid

## Summary

The centroid of a finite set of points is their average, the arithmetic mean of their positions. The centroid of a triangle is the centroid of its three vertices, and it is the point where the three medians meet.
</document>

<document source="https://math2.tamnd.com/node/0X7G61297X.md" tag="0X7G61297X" kind="definition">
# Arc length

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be continuous.
The arc length of $\gamma$ is
$$L(\gamma) = \sup_{P} \sum_{i=1}^{k} \| \gamma(t_i) - \gamma(t_{i-1}) \|,$$
where the supremum runs over all [[partitions|069P20KYQT]] $P = (t_0, \dots, t_k)$ of $[a, b]$.
The curve $\gamma$ is rectifiable if $L(\gamma) < \infty$.

## Remarks

Each sum is the length of a polygonal path through points of the curve. Adding points does not decrease this length.
If $\gamma$ is continuously differentiable, then $L(\gamma) = \int_a^b \| \gamma'(t) \| \, dt$.
For the graph of a $C^1$ function $f$ on $[a, b]$, this gives $L = \int_a^b \sqrt{1 + f'(x)^2} \, dx$.
Some continuous curves are not rectifiable, for example the graph of the [[Weierstrass function|01A0MK1730]].

## Explanations

### L1 human

The arc length of a curve is the length of a string laid along it.
We approximate the curve by straight segments and take the best approximation.
</document>

<document source="https://math2.tamnd.com/node/0X7SBS8E0F.md" tag="0X7SBS8E0F" kind="definition">
# Winding number

## Definition

Let $\gamma : [a, b] \to \mathbb{C}$ be a closed path that is continuous and piecewise continuously differentiable. Let $w \in \mathbb{C}$ be a point that is not on the image of $\gamma$.
The winding number of $\gamma$ around $w$ is
$$n(\gamma, w) = \frac{1}{2 \pi i} \oint_\gamma \frac{dz}{z - w},$$
where the integral is a [[contour integral|00FWNP9TXY]].

## Remarks

The number $n(\gamma, w)$ is an integer. It counts how many times $\gamma$ goes around $w$ counterclockwise, minus the times it goes around $w$ clockwise.
As a function of $w$, it is constant on each connected component of the complement of the image of $\gamma$. It is $0$ on the unbounded component.
For the circle $|z - c| = R$, traveled once counterclockwise, $n = 1$ for $|w - c| < R$ and $n = 0$ for $|w - c| > R$, by the [[integral of 1/(z - w)|0SXZ68HV6F]].

## Explanations

### L1 human

Stand at the point $w$ and watch a point that moves along the closed curve. The winding number is the number of full turns that you make.
A circle around you gives $1$, the same circle traveled twice gives $2$, and a circle that does not go around you gives $0$.
</document>

<document source="https://math2.tamnd.com/node/0X89GC65N1.md" tag="0X89GC65N1" kind="definition">
# Altitude of a triangle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$, and let $\cdot$ be the dot product. Let $H_A$ be the orthogonal projection of $A$ onto the [[line|0MP9BCF6DP]] $BC$:
$$H_A = B + \frac{(A - B) \cdot (C - B)}{(C - B) \cdot (C - B)} \, (C - B).$$
The altitude of $ABC$ from $A$ is the segment from $A$ to $H_A$. The point $H_A$ is the foot of the altitude, and $h_a = \|A - H_A\|$ is its length.
The altitudes from $B$ and from $C$ are defined in the same way.

## Remarks

The formula uses the [[vector projection|0JW13B48YP]] of $A - B$ onto $C - B$. The length $h_a$ is the [[distance from $A$ to the line $BC$|01AAYNSBN6]].
The word altitude also names the whole line through $A$ and $H_A$, or the length $h_a$. Mathlib's `Affine.Simplex.altitude` is the line.
With $a = \|C - B\|$, the area of the triangle is $\frac{1}{2} a h_a$.
Let the right angle of a right triangle be at $C$. Then the leg $AC$ is the altitude from $A$, and the leg $BC$ is the altitude from $B$.

## Explanations

### L1 human

An altitude is the height of a triangle, measured from one corner. Drop a plumb line from the corner to the opposite side, or to its extension. The foot of the altitude can be outside the triangle when the triangle has an obtuse angle.
</document>

<document source="https://math2.tamnd.com/node/0X8HBKCSYB.md" tag="0X8HBKCSYB" kind="statement">
# Continuity of probability measures

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]].
If $A_1 \subseteq A_2 \subseteq \cdots$ are events and $A = \bigcup_{n=1}^{\infty} A_n$, then $\lim_{n \to \infty} P(A_n) = P(A)$.
If $B_1 \supseteq B_2 \supseteq \cdots$ are events and $B = \bigcap_{n=1}^{\infty} B_n$, then $\lim_{n \to \infty} P(B_n) = P(B)$.

## Remarks

The first part is continuity from below. Its proof writes $A$ as the disjoint union of $A_1$ and the sets $A_{n+1} \setminus A_n$, and uses countable additivity.
The second part, continuity from above, follows from the first part with the complements $B_n^c$.
For a general measure $\mu$, continuity from above needs $\mu(B_1) < \infty$. A [[probability measure|0VGMEBVW1M]] is finite, so this holds.
For example, $P(X \le x + 1/n) \to P(X \le x)$ for a real random variable $X$. This gives the right continuity of the cumulative distribution function.

## Explanations

### L1 human

If events grow step by step, their probabilities grow toward the probability of the limit event.
For example, the chance that a fair coin shows heads at least once in the first $n$ tosses is $1 - 2^{-n}$. It tends to the chance $1$ of heads at some toss.
</document>

<document source="https://math2.tamnd.com/node/0X8NMST0J2.md" tag="0X8NMST0J2" kind="concept">
# Reflexive relation

## Summary

A relation on a set is reflexive when each element is related to itself.
</document>

<document source="https://math2.tamnd.com/node/0X9MBPP2Z3.md" tag="0X9MBPP2Z3" kind="concept">
# Space of linear maps

## Summary

The space of linear maps between two vector spaces is the set of all linear maps from the first space to the second.
Its operations are pointwise addition and pointwise scalar multiplication.
</document>

<document source="https://math2.tamnd.com/node/0X9XT05JVC.md" tag="0X9XT05JVC" kind="statement">
# Uniform limit theorem

## Statement

Let $S \subseteq \mathbb{R}$, and let $f_n \colon S \to \mathbb{R}$ be continuous for each $n$.
If $f_n \to f$ [[uniformly on $S$|08HJRTG5N7]], then $f$ is continuous on $S$.

## Remarks

Fix $c \in S$ and $\varepsilon > 0$. Choose $n$ with $|f_n - f| < \varepsilon/3$ on $S$, and $\delta$ for the [[continuity|0EEE2CZB2A]] of $f_n$ at $c$.
Then $|f(x) - f(c)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(c)| + |f_n(c) - f(c)| < \varepsilon$ for $|x - c| < \delta$.
The [[sequence $x^n$ on $[0, 1]$|0YSRR1JTKZ]] shows that pointwise convergence is not enough. Mathlib states the theorem as `TendstoUniformly.continuous`.

## History

Augustin-Louis Cauchy claimed in 1821 that a series of continuous functions has a continuous sum, whenever it converges.
In 1826, Niels Henrik Abel found counterexamples among Fourier series. In modern terms, Cauchy's argument needs uniform convergence.

## Explanations

### L1 human

If continuous graphs approach a limit graph uniformly, the limit graph has no jumps.
A jump in the limit would force some $f_n$ to jump as well.
</document>

<document source="https://math2.tamnd.com/node/0XADJKM699.md" tag="0XADJKM699" kind="concept">
# Boolean algebra

## Summary

A Boolean algebra is a structure with meet, join and complement that satisfies the laws of classical logic and of set algebra. It is the algebra of propositional logic and of digital circuits. The node [[Boolean algebra|06GKBTAA5E]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XAP8WHNES.md" tag="0XAP8WHNES" kind="statement">
# Handshaking lemma

## Statement

Let $G$ be a finite [[graph|0JD5C82QMB]] with vertex set $V$ and [[edge set|0WNGQBPPQX]] $E$.
Then
$$\sum_{v \in V} \deg(v) = 2\, |E|.$$

## Remarks

Proof: count the pairs $(v, e)$ with $v \in e$. Each edge has two ends, and each vertex $v$ lies in $\deg(v)$ edges. This is [[double counting|0F8YBHF2W3]].
A direct consequence is that the [[number of odd-degree vertices is even|0QQQQCPZA8]].
For a $3$-regular graph on $n$ vertices, it gives $3n = 2 |E|$, so $n$ is even.

## Explanations

### L1 human

At a party, add up the number of hands each person shakes.
Each handshake is counted twice, once for each of the two people, so the total is twice the number of handshakes.
</document>

<document source="https://math2.tamnd.com/node/0XB0J0P64D.md" tag="0XB0J0P64D" kind="concept">
# Simple continued fraction

## Summary

A simple continued fraction is a continued fraction with all partial numerators equal to $1$ and positive integer partial denominators after the first term.
The [[definition of a simple continued fraction|06XF7BCFFJ]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0XB9GQTN9G.md" tag="0XB9GQTN9G" kind="concept">
# Poisson distribution

## Summary

The Poisson distribution with parameter $\lambda$ describes the number of events in a fixed interval of time or space.
It applies when the events occur independently at a constant mean rate.
The node [[Poisson distribution|0VEFSSKX18]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XBRG3965F.md" tag="0XBRG3965F" kind="concept">
# Negative binomial distribution

## Summary

The negative binomial distribution describes the number of failures before the $r$-th success in a sequence of independent Bernoulli trials.
The node [[negative binomial distribution|0XN7KEVSSS]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XC1RDBY7A.md" tag="0XC1RDBY7A" kind="concept">
# Trace of a matrix

## Summary

The trace of a square matrix is the sum of the entries on its main diagonal.
</document>

<document source="https://math2.tamnd.com/node/0XCE070V8B.md" tag="0XCE070V8B" kind="definition">
# Independence of random variables

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $I$ be an index set. For each $i \in I$, let $X_i$ be a [[random variable|0DKRR22AEQ]] with values in $(E_i, \mathcal{E}_i)$.
The family $(X_i)_{i \in I}$ is independent if for each finite nonempty set $J \subseteq I$ and each choice of $B_j \in \mathcal{E}_j$ for $j \in J$,
$$P\left(\bigcap_{j \in J} \{X_j \in B_j\}\right) = \prod_{j \in J} P(X_j \in B_j).$$

## Remarks

Equivalently, the σ-algebras $\sigma(X_i) = \{X_i^{-1}(B) : B \in \mathcal{E}_i\}$ are [[independent|09T8ZHST1V]].
For two random variables this is the [[independence of two random variables|0E9502NT92]].
Mutual independence implies pairwise independence, but the converse is false, as for [[events|0AR4B3K5QD]].
For real random variables $X_1, \dots, X_n$, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.

## Explanations

### L1 human

Independent random variables come from separate parts of an experiment that do not influence each other.
For $n$ tosses of a coin, the results of the tosses are independent random variables.
</document>

<document source="https://math2.tamnd.com/node/0XCH8QVDCQ.md" tag="0XCH8QVDCQ" kind="concept">
# Cantor's diagonal argument

## Summary

Cantor's diagonal argument shows that some infinite sets are [[uncountable|0X55M9XET4]].
Given a list $s_0, s_1, s_2, \dots$ of infinite sequences of $0$ and $1$, define a sequence $t$ by $t(n) = 1 - s_n(n)$.
Then $t$ differs from each $s_n$ at position $n$, so $t$ is not in the list.
So the set of all such sequences is not countable.

## Remarks

Georg Cantor published the argument in 1891. His first proof that $\mathbb{R}$ is uncountable, from 1874, used another method.
The same idea proves [[Cantor's theorem|0WTEMGYV2C]] for every set.
Diagonal arguments also appear in the proofs of the halting problem and of Gödel's incompleteness theorems.

## Explanations

### L1 human

Write the list of sequences as the rows of a table.
Go down the diagonal and change each entry.
The new row differs from row $n$ at column $n$, so it cannot be any row of the table.
</document>

<document source="https://math2.tamnd.com/node/0XCVV5Q500.md" tag="0XCVV5Q500" kind="definition">
# Geometric distribution

## Definition

Let $p \in (0, 1]$.
The geometric distribution with parameter $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = (1 - p)^k \, p \quad \text{for } k = 0, 1, 2, \dots$$

## Remarks

It is the distribution of the number of failures before the first success in independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$.
The other common convention counts the trials up to and including the first success. That number $Y = X + 1$ has $P(Y = k) = (1 - p)^{k-1} p$ for $k \ge 1$.
For $X$ as in the definition, $P(X \ge k) = (1 - p)^k$, the mean is $(1 - p)/p$ and the variance is $(1 - p)/p^2$.
The geometric distribution is the only distribution on $\{0, 1, 2, \dots\}$ with the [[memoryless property|0ZAZZ34WQS]].

## Explanations

### L1 human

Toss a coin until the first head. The number of tails before it has the geometric distribution.
Each extra tail multiplies the probability by $1 - p$, so the probabilities form a geometric sequence.
</document>

<document source="https://math2.tamnd.com/node/0XCYA4YERS.md" tag="0XCYA4YERS" kind="statement">
# Wallis product

## Statement

The partial products of the following infinite product converge to $\pi/2$:
$$\frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4 n^2}{4 n^2 - 1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots.$$

## Remarks

Let $I_n = \int_0^{\pi/2} \sin^n x \, dx$. The [[reduction formula|012RYET5E2]] gives $I_{2n}$ and $I_{2n+1}$ as explicit products.
The sequence $I_n$ is decreasing, and $I_{2n+1} / I_{2n} \to 1$ by the [[squeeze theorem|0XMF00T4XQ]]. This ratio is a partial product times $2/\pi$.
Mathlib states it as `Real.tendsto_prod_pi_div_two`.

## History

John Wallis published the product in 1656.

## Explanations

### L1 human

The Wallis product writes $\pi/2$ as an infinite product of simple fractions.
The partial products converge slowly to $\pi/2$.
</document>

<document source="https://math2.tamnd.com/node/0XCYGCCX8V.md" tag="0XCYGCCX8V" kind="statement">
# Number of elements of each order in a cyclic group

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] of finite order $n$, and let $d$ be a positive divisor of $n$.
Then $G$ has exactly $\varphi(d)$ elements of [[order|0WE2NHZJ7B]] $d$, where $\varphi$ is Euler's totient function.

## Remarks

Here $\varphi(d)$ is the number of integers $k$ with $1 \leq k \leq d$ and $\gcd(k, d) = 1$.
The elements of order $d$ are the generators of the unique subgroup of order $d$.
Counting all elements by their orders gives the identity $\sum_{d \mid n} \varphi(d) = n$.
In particular, $G$ has exactly $\varphi(n)$ generators.

## Explanations

### L1 human

In $\mathbb{Z}/6\mathbb{Z}$, the element $0$ has order $1$ and the element $3$ has order $2$. The elements $2$ and $4$ have order $3$, and $1$ and $5$ have order $6$.
These counts are $1, 1, 2, 2$, which are $\varphi(1), \varphi(2), \varphi(3), \varphi(6)$.
The totient function counts the elements of each order.
</document>

<document source="https://math2.tamnd.com/node/0XD22PBMWA.md" tag="0XD22PBMWA" kind="statement">
# Monotonicity of the cumulative distribution function

## Statement

Let $X$ be a real random variable with [[cumulative distribution function|0MXMS3AE13]] $F_X$.
Then $F_X$ is nondecreasing: if $x \le y$, then $F_X(x) \le F_X(y)$.

## Remarks

The proof: $\{X \le x\} \subseteq \{X \le y\}$, so the claim follows from the [[monotonicity of probability|0464Z0JP2E]].
More precisely, $F_X(y) - F_X(x) = P(x < X \le y) \ge 0$.
A nondecreasing function has at most countably many points of discontinuity. So $P(X = x) > 0$ for at most countably many $x$.

## Explanations

### L1 human

As $x$ moves to the right, the event "$X \le x$" contains more outcomes, so its probability cannot go down.
</document>

<document source="https://math2.tamnd.com/node/0XDB5AW7M1.md" tag="0XDB5AW7M1" kind="definition">
# Jacobi symbol

## Definition

Let $n$ be a positive odd integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \cdots p_k^{e_k}$, and let $a$ be an integer.
The Jacobi symbol is
$$\left(\frac{a}{n}\right) = \left(\frac{a}{p_1}\right)^{e_1} \cdots \left(\frac{a}{p_k}\right)^{e_k},$$
where each factor on the right is a [[Legendre symbol|01J8AMV9RS]].
For $n = 1$, the empty product gives $\left(\frac{a}{1}\right) = 1$.

## Remarks

For a prime $n$, the Jacobi symbol equals the Legendre symbol.
If $\left(\frac{a}{n}\right) = -1$, then $a$ is not a quadratic residue modulo $n$.
The converse fails: $\left(\frac{2}{15}\right) = \left(\frac{2}{3}\right) \left(\frac{2}{5}\right) = 1$, but $2$ is not a square modulo $15$.
The symbol is $0$ exactly when $\gcd(a, n) > 1$.

## Explanations

### L1 human

The Jacobi symbol extends the Legendre symbol from odd primes to all odd positive integers.
It multiplies the Legendre symbols for the prime factors of $n$.
Its main use is fast computation, because it obeys [[quadratic reciprocity|0YY06KP3ZQ]] without the need to factor numbers.
</document>

<document source="https://math2.tamnd.com/node/0XDK9CGYKJ.md" tag="0XDK9CGYKJ" kind="concept">
# Rational function

## Summary

A rational function is a function given by a quotient of two polynomials. It is defined and continuous wherever the denominator is not zero. The node [[rational function|0WCZH4AFMT]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XE690QF65.md" tag="0XE690QF65" kind="definition">
# Even and odd functions

## Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function.
The function $f$ is even if $f(-x) = f(x)$ for all $x \in \mathbb{R}$.
It is odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$.

## Remarks

The graph of an even function is symmetric about the $y$-axis. The graph of an odd function is symmetric about the origin.
For example, $x^2$ and $\cos x$ are even, and $x^3$ and $\sin x$ are odd.
Each function splits in a unique way into an even part and an odd part: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$.
Leonhard Euler introduced the two notions in 1727. Mathlib defines even functions as `Function.Even`.

## Explanations

### L1 human

An even function takes the same value at $x$ and at $-x$.
An odd function takes opposite values at $x$ and at $-x$.
</document>

<document source="https://math2.tamnd.com/node/0XEGNW851S.md" tag="0XEGNW851S" kind="definition">
# Reflexive relation

## Definition

Let $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.
The relation $R$ is reflexive when $a \mathrel{R} a$ for each $a \in A$.

## Remarks

On $\mathbb{R}$, the relation $\le$ is reflexive and the relation $<$ is not.
A relation is irreflexive when no element is related to itself.
Reflexivity is one of the conditions for an [[equivalence relation|0SGTMBEWRF]] and for a [[partial order|011C29Q1S1]].

## Explanations

### L1 human

In the graph of a reflexive relation, each point has a loop to itself.
"Has the same birthday as" is reflexive, because each person has the same birthday as themselves.
</document>

<document source="https://math2.tamnd.com/node/0XFC1VQYEK.md" tag="0XFC1VQYEK" kind="statement">
# Multinomial theorem

## Statement

Let $R$ be a commutative ring, let $x_1, \dots, x_m \in R$, and let $n$ be a natural number.
Then
$$(x_1 + \dots + x_m)^n = \sum_{k_1 + \dots + k_m = n} \binom{n}{k_1, \dots, k_m} x_1^{k_1} \cdots x_m^{k_m}.$$
The sum runs over all $m$-tuples $(k_1, \dots, k_m)$ of natural numbers with sum $n$, and the coefficients are [[multinomial coefficients|00K0RF9BFY]].

## Remarks

For $m = 2$ it is the [[binomial theorem|0EHWPWGHB2]].
Example: $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2xz$.
With all $x_i = 1$ it gives $\sum \binom{n}{k_1, \dots, k_m} = m^n$.

## Explanations

### L1 human

To expand a power of a sum of $m$ terms, pick one term from each of the $n$ factors.
The monomial $x_1^{k_1} \cdots x_m^{k_m}$ appears once for each way to decide which factors give which term.
</document>

<document source="https://math2.tamnd.com/node/0XFZ1BMF5Q.md" tag="0XFZ1BMF5Q" kind="statement">
# Euler's partition theorem

## Statement

Let $n$ be a natural number.
Then the number of [[partitions|0404R9P9J3]] of $n$ into odd parts equals the number of partitions of $n$ into distinct parts.

## Remarks

For $n = 5$, the partitions into odd parts are $5$, $3 + 1 + 1$ and $1 + 1 + 1 + 1 + 1$. The partitions into distinct parts are $5$, $4 + 1$ and $3 + 2$.
A proof with generating functions: $\prod_{k \ge 1} (1 + x^k) = \prod_{k \ge 1} \frac{1 - x^{2k}}{1 - x^k} = \prod_{k \text{ odd}} \frac{1}{1 - x^k}$.
Glaisher gave a bijection. It merges equal parts in pairs, again and again, until all parts are distinct.

## Explanations

### L1 human

Count the ways to write $n$ with only odd summands, and the ways to write it with summands that are all different.
The two counts are always equal, although the partitions themselves are not the same.
</document>

<document source="https://math2.tamnd.com/node/0XGCZ4GDZ6.md" tag="0XGCZ4GDZ6" kind="concept">
# Meagre set

## Summary

A meagre subset of a topological space is a countable union of nowhere dense sets.
Meagre sets are small in the sense of Baire category.
</document>

<document source="https://math2.tamnd.com/node/0XH79S06J3.md" tag="0XH79S06J3" kind="concept">
# Archimedean ordered field

## Summary

An Archimedean ordered field is an [[ordered field|0P3VCBDEE2]] that has no infinitely large and no infinitely small positive elements.
In such a field, the natural numbers are not bounded above.
The fields $\mathbb{Q}$ and $\mathbb{R}$ are Archimedean.
</document>

<document source="https://math2.tamnd.com/node/0XHDTE5YMZ.md" tag="0XHDTE5YMZ" kind="definition">
# Function

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
A function from $A$ to $B$ is a [[binary relation|08EJC6Q25H]] $f \subseteq A \times B$ such that, for each $a \in A$, there is exactly one $b \in B$ with $(a, b) \in f$.
This $b$ is the value of $f$ at $a$, written $f(a)$.
The notation $f \colon A \to B$ says that $f$ is a function from $A$ to $B$.

## Remarks

The set $A$ is the [[domain|093CM69AGS]] and the set $B$ is the [[codomain|0F9JHP8AJN]] of $f$.
Many authors take the triple $(A, B, f)$ as the function. Then two functions with the same pairs but different codomains are different.
Two functions $f, g \colon A \to B$ are equal when $f(a) = g(a)$ for each $a \in A$.
Other names are map and mapping.

## Explanations

### L1 human

A function is a machine: you put in an element $a$ of $A$, and it gives back one element $f(a)$ of $B$.
Each input gives exactly one output, but two inputs can give the same output.
For example, $f(x) = x^2$ on $\mathbb{R}$ sends both $2$ and $-2$ to $4$.
</document>

<document source="https://math2.tamnd.com/node/0XHM30WBVB.md" tag="0XHM30WBVB" kind="definition">
# Finitely generated module

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
The module $M$ is *finitely generated* when there are $x_1, \dots, x_n \in M$ such that each $x \in M$ is
$$x = r_1 x_1 + \cdots + r_n x_n \qquad \text{for some } r_1, \dots, r_n \in R.$$

## Remarks

Over a field, the finitely generated modules are the finite-dimensional vector spaces.
$\Z^n$ and $\Z / n\Z$ are finitely generated over $\Z$; the group $\Q$ is not.
Over a Noetherian ring, each submodule of a finitely generated module is finitely generated.
The finitely generated modules over a principal ideal domain are classified by the structure theorem.

## Explanations

### L1 human

A module is finitely generated when finitely many elements produce all others by linear combinations.
This is the module analogue of a finite-dimensional vector space.
Unlike in vector spaces, the generators need not be independent.
</document>

<document source="https://math2.tamnd.com/node/0XHWTD3MPV.md" tag="0XHWTD3MPV" kind="concept">
# Derangement

## Summary

A derangement is a permutation that moves every element. Derangements answer the classical hat-check problem, and their number is a standard application of inclusion and exclusion. The node [[derangement|0TH61JR04G]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XHZW0VCPR.md" tag="0XHZW0VCPR" kind="concept">
# K-colourable graph

## Summary

A graph is $k$-colourable when it has a proper colouring with $k$ colours. The least such $k$ is the chromatic number. The node [[k-colourable graph|0Y7VDAX8Q3]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XJ02JBD6S.md" tag="0XJ02JBD6S" kind="example">
# Indiscrete topology

## Example

Let $X$ be a set and let $\tau = \{\varnothing, X\}$.
Then $\tau$ is a [[topology|0CNQXVBHCX]] on $X$, the indiscrete topology, also called the trivial topology.

## Remarks

It is the coarsest topology on $X$.
Each function from any topological space to an indiscrete space is [[continuous|0SY9ECQ19Q]].
In an indiscrete space, each sequence converges to each point. Thus, if $X$ has at least two points, the space is not [[Hausdorff|0MXB4CKHFW]] and not [[metrizable|09KDHB82SG]].
Each indiscrete space is [[compact|0QPHW703PV]] and [[connected|0D939CMME7]].

## Explanations

### L1 human

In the indiscrete topology, the only open sets are the empty set and the whole space.
The topology cannot tell any two points apart.
</document>

<document source="https://math2.tamnd.com/node/0XJVF81QF6.md" tag="0XJVF81QF6" kind="definition">
# Consistent estimator

## Definition

For each $n \ge 1$, let $\hat{\theta}_n$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$ that uses the first $n$ observations of a sample.
The sequence $(\hat{\theta}_n)$ is consistent if, for each $\theta \in \Theta$, $\hat{\theta}_n \to \theta$ [[in probability|0EFDDXTVD6]] when $\theta$ is the true parameter:
$$\lim_{n \to \infty} P_\theta\left(|\hat{\theta}_n - \theta| > \varepsilon\right) = 0 \quad \text{for each } \varepsilon > 0.$$

## Remarks

If the convergence is almost sure, the sequence is strongly consistent.
By the weak law of large numbers, the sample mean of an i.i.d. sample is a consistent estimator of the mean.
If the mean squared error of $\hat{\theta}_n$ tends to $0$, then $(\hat{\theta}_n)$ is consistent. This follows from Markov's inequality.
Consistent estimators can be biased, and unbiased estimators can fail to be consistent. For example, the estimator $X_1$ of the mean uses only one observation: it is unbiased but not consistent, unless $X_1$ is almost surely constant.

## Explanations

### L1 human

A consistent estimator gets closer and closer to the truth when you collect more and more data.
This is a minimal requirement for a reasonable estimator.
</document>

<document source="https://math2.tamnd.com/node/0XJXTBCB2T.md" tag="0XJXTBCB2T" kind="concept">
# Orthogonal complement

## Summary

The orthogonal complement of a subspace of an inner product space is the set of all vectors orthogonal to every vector of the subspace.
</document>

<document source="https://math2.tamnd.com/node/0XK0G19SK8.md" tag="0XK0G19SK8" kind="concept">
# Order of a differential equation

## Summary

The order of a [[differential equation|003R0V5M7V]] is the highest order of a derivative of the unknown function in it.
See the [[definition|09M2RDB61G]].
</document>

<document source="https://math2.tamnd.com/node/0XK6FECMW5.md" tag="0XK6FECMW5" kind="statement">
# Decomposition of a permutation into transpositions

## Statement

Let $n \geq 2$.
Each permutation in $S_n$ is a product of [[transpositions|0FWVRHBRWM]].

## Remarks

A $k$-cycle is a product of $k - 1$ transpositions, for example $(a_1\ a_2\ \dots\ a_k) = (a_1\ a_k)(a_1\ a_{k-1}) \cdots (a_1\ a_2)$.
With the [[disjoint cycle decomposition|0YWDAZPRPQ]], this gives the result, and the identity is $(1\ 2)(1\ 2)$.
The decomposition is not unique, but the parity of the number of factors is unique.

## Explanations

### L1 human

You can sort any arrangement of objects by swapping two objects at a time.
So each permutation is a chain of swaps.
Different chains can give the same permutation, but they are all of even length or all of odd length.
</document>

<document source="https://math2.tamnd.com/node/0XKBKGTCPC.md" tag="0XKBKGTCPC" kind="concept">
# Length of a vector

## Summary

The length of a [[Euclidean vector|0QN3D80AWP]] is the nonnegative real number that measures its size. It is the distance between the start point and the end point of an arrow that represents the vector.
</document>

<document source="https://math2.tamnd.com/node/0XKH881BZ7.md" tag="0XKH881BZ7" kind="definition">
# Surjective function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The function $f$ is surjective when, for each $b \in B$, there is an $a \in A$ with $f(a) = b$.

## Remarks

Other names are onto function and surjection.
Equivalently, the [[range|0E9GTAV478]] of $f$ is the whole [[codomain|0F9JHP8AJN]] $B$.
The function $x \mapsto x^3$ on $\mathbb{R}$ is surjective, and $x \mapsto x^2$ on $\mathbb{R}$ is not, because $-1$ is not a value.

## Explanations

### L1 human

A surjective function hits every element of the codomain at least once.
To prove surjectivity, take an arbitrary $b \in B$ and find an input $a$ with $f(a) = b$.
</document>

<document source="https://math2.tamnd.com/node/0XKJPFS2W0.md" tag="0XKJPFS2W0" kind="definition">
# Periodic continued fraction

## Definition

An infinite [[simple continued fraction|06XF7BCFFJ]] $[a_0; a_1, a_2, \dots]$ is periodic if there are integers $N \ge 0$ and $k \ge 1$ with $a_{n+k} = a_n$ for all $n \ge N$.
It is purely periodic if this holds with $N = 0$.

## Remarks

One writes the repeating block with a bar, for example $\sqrt{2} = [1; \overline{2}]$ and $\sqrt{3} = [1; \overline{1, 2}]$.
By theorems of Euler and Lagrange, a real number has a periodic continued fraction if and only if it is a quadratic irrational. That is, it is an irrational root of a quadratic equation with integer coefficients.
The period of $\sqrt{d}$ gives the [[fundamental solution of Pell's equation|0KCYJSB9FW]].

## Explanations

### L1 human

A periodic continued fraction repeats the same block of terms forever.
The golden ratio $[1; 1, 1, 1, \dots]$ is the simplest example.
</document>

<document source="https://math2.tamnd.com/node/0XKTZRWTXX.md" tag="0XKTZRWTXX" kind="statement">
# Poisson limit theorem

## Statement

Let $\lambda > 0$, and let $p_n \in [0, 1]$ satisfy $n p_n \to \lambda$ as $n \to \infty$. Let $S_n \sim \operatorname{Bin}(n, p_n)$.
Then for each integer $k \ge 0$,
$$\lim_{n \to \infty} P(S_n = k) = \lim_{n \to \infty} \binom{n}{k} p_n^k (1 - p_n)^{n - k} = e^{-\lambda} \frac{\lambda^k}{k!}.$$
So $S_n$ converges [[in distribution|0JDAQHWA9N]] to the [[Poisson distribution|0VEFSSKX18]] with parameter $\lambda$.

## Remarks

Proof: write $\binom{n}{k} p_n^k = \frac{n(n-1)\cdots(n-k+1)}{n^k} \cdot \frac{(n p_n)^k}{k!}$. The first factor tends to $1$, and $(1 - p_n)^{n-k} \to e^{-\lambda}$.
In practice, $\operatorname{Bin}(n, p)$ is close to $\operatorname{Poisson}(np)$ when $n$ is large and $p$ is small.
Le Cam's inequality gives an explicit bound on the error of this approximation.

## Explanations

### L1 human

The number of rare events among many independent chances is approximately Poisson.
Examples are the number of misprints on a page, or the number of large insurance claims in a year.
</document>

<document source="https://math2.tamnd.com/node/0XKWSSY8P2.md" tag="0XKWSSY8P2" kind="definition">
# Limit of a sequence

## Definition

Let $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $L$ be a real number.
The sequence $(a_n)$ converges to $L$ if the following holds for each open interval $U$ that contains $L$.
There is a natural number $N$ such that $a_n \in U$ for all $n \ge N$.
Then $L$ is a limit of $(a_n)$, with the notation $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$.

## Remarks

The [[epsilon-N definition of the limit of a sequence|09G6KHS6GY]] is an equivalent form, with intervals $(L - \varepsilon, L + \varepsilon)$.
A sequence has at most one limit, by the [[uniqueness of limits|0ZXJ2455NJ]]. So the notation $\lim_{n \to \infty} a_n$ is well defined.
The same definition works in each topological space, with open neighbourhoods of $L$ in place of open intervals.
In Mathlib, the limit is the statement `Filter.Tendsto a Filter.atTop (nhds L)`.

## Explanations

### L1 human

The sequence $(a_n)$ tends to $L$ when its terms come close to $L$ and stay close.
However small a window around $L$ you choose, from some index on, all terms are inside the window.
For example, $1/(n+1)$ tends to $0$.
</document>

<document source="https://math2.tamnd.com/node/0XM8BZD76G.md" tag="0XM8BZD76G" kind="object">
# Coordinate space

## Description

Let $F$ be a field and let $n \ge 0$ be an integer.
The *coordinate space* $F^n$ is the set of $n$-tuples $(x_1, \dots, x_n)$ with $x_i \in F$.
It is a [[vector space|000AYPA9AJ]] over $F$ with the operations
$$(x_1, \dots, x_n) + (y_1, \dots, y_n) = (x_1 + y_1, \dots, x_n + y_n), \qquad a(x_1, \dots, x_n) = (ax_1, \dots, ax_n).$$
Its zero vector is $(0, \dots, 0)$.

## Remarks

The space $F^n$ has [[dimension|03GSZ8RTZJ]] $n$, with the [[standard basis|0V7ZD6P7RK]] $e_1, \dots, e_n$.
More generally, for a set $I$, the set $F^I$ of all functions $I \to F$ is a vector space with pointwise operations.
Each $n$-dimensional vector space over $F$ is isomorphic to $F^n$.
The real spaces $\R^2$ and $\R^3$ model the plane and ordinary space.

## Explanations

### L1 human

The coordinate space $F^n$ is the space of lists of $n$ numbers.
We add two lists position by position, and we scale a list by scaling each entry.
Every finite-dimensional vector space looks like one of these spaces after we choose a basis.
</document>

<document source="https://math2.tamnd.com/node/0XM8ZA9RNM.md" tag="0XM8ZA9RNM" kind="concept">
# Sieve of Eratosthenes

## Summary

The sieve of Eratosthenes is an algorithm that finds all the [[prime numbers|0SN0MAEKYX]] up to a given bound $N$.
It writes the integers from $2$ to $N$. It takes the smallest number that is not crossed out, which is a prime $p$, and crosses out the multiples $p^2, p^2 + p, p^2 + 2p, \dots$ up to $N$.
It repeats this step while $p^2 \le N$. The numbers that are not crossed out at the end are the primes up to $N$.

## Remarks

It is enough to start at $p^2$, because a smaller multiple of $p$ has a smaller prime factor, and an earlier step crossed it out.
The algorithm needs about $N \log \log N$ operations.
The idea of the sieve is the basis of sieve methods in analytic number theory.

## Explanations

### L1 human

For $N = 30$, cross out the multiples of $2$ from $4$, then the multiples of $3$ from $9$, then the multiples of $5$ from $25$.
Then $7^2 = 49 > 30$, so you stop.
The numbers left are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
</document>

<document source="https://math2.tamnd.com/node/0XMEM406JM.md" tag="0XMEM406JM" kind="concept">
# Semigroup

## Summary

A semigroup is a set with an associative binary operation.
It need not have an identity element or inverses.
</document>

<document source="https://math2.tamnd.com/node/0XMF00T4XQ.md" tag="0XMF00T4XQ" kind="statement">
# Squeeze theorem

## Statement

Let $(a_n)$, $(b_n)$ and $(c_n)$ be real sequences and let $L$ be a real number.
Assume that there is a natural number $N$ such that $a_n \le c_n \le b_n$ for all $n \ge N$.
Assume that $a_n \to L$ and $b_n \to L$ as $n \to \infty$.
Then $c_n \to L$ as $n \to \infty$.

## Remarks

The theorem gives the existence of the limit of $(c_n)$. It does not assume it.
For example, $-1/n \le \sin(n)/n \le 1/n$, so $\sin(n)/n \to 0$.
The same statement holds for the [[limit of a function|0TXTSEMHR5]] at a point.
Other names are the sandwich theorem and, in some countries, the two policemen theorem.

## Explanations

### L1 human

If a sequence is trapped between two sequences that go to the same limit, it must go there too.
For large $n$, the upper and lower sequences are both close to $L$, and $c_n$ lies between them.
</document>

<document source="https://math2.tamnd.com/node/0XN3K7RSZY.md" tag="0XN3K7RSZY" kind="concept">
# Multivariate normal distribution

## Summary

The multivariate normal distribution is the extension of the normal distribution to random vectors in $\mathbb{R}^k$. A mean vector and a covariance matrix determine it.
The node [[multivariate normal distribution|0G558GJQMW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XN66MZN0Y.md" tag="0XN66MZN0Y" kind="concept">
# Equicontinuity

## Summary

A family of functions is equicontinuous when all its members are continuous in the same way. For each tolerance, one neighbourhood works for all of them at the same time.
</document>

<document source="https://math2.tamnd.com/node/0XN7KEVSSS.md" tag="0XN7KEVSSS" kind="definition">
# Negative binomial distribution

## Definition

Let $r \ge 1$ be an integer and let $p \in (0, 1]$.
The negative binomial distribution with parameters $r$ and $p$ is the distribution on $\{0, 1, 2, \dots\}$ with the [[probability mass function|0WF4EJP91W]]
$$p(k) = \binom{k + r - 1}{k} p^r (1 - p)^k \quad \text{for } k = 0, 1, 2, \dots$$

## Remarks

It is the distribution of the number of failures before the $r$-th success in independent Bernoulli trials with success probability $p$. The last trial is a success, and the $k$ failures are among the other $k + r - 1$ trials.
For $r = 1$ it is the [[geometric distribution|0XCVV5Q500]]. The sum of $r$ independent geometric random variables with parameter $p$ has the negative binomial distribution with parameters $r$ and $p$.
The mean is $r(1 - p)/p$ and the variance is $r(1 - p)/p^2$.
Some authors count the trials, or the successes before a fixed number of failures. Check the convention of each source.

## Explanations

### L1 human

Toss a coin until you have seen $r$ heads, and count the tails on the way.
This count has the negative binomial distribution. For $r = 1$ you wait only for the first head.
</document>

<document source="https://math2.tamnd.com/node/0XNN9Z8E5A.md" tag="0XNN9Z8E5A" kind="definition">
# Acyclic graph

## Definition

A [[graph|0JD5C82QMB]] is acyclic if it has no [[cycle|07CT9RDY7Q]].

## Remarks

An acyclic graph is also called a [[forest|09JP6G9FTA]]. A connected acyclic graph is a [[tree|0MJ0JS72DZ]].
The [[path graphs|075WR6W3Y3]] and the graphs with no edges are acyclic. The [[cycle graphs|0QQ0F12962]] are not.
A finite acyclic graph with $n$ vertices and $c$ components has $n - c$ edges.

## Explanations

### L1 human

A graph is acyclic when you cannot go around any loop in it.
Each edge then splits its piece of the graph in two when you delete it.
</document>

<document source="https://math2.tamnd.com/node/0XNZWJ96HK.md" tag="0XNZWJ96HK" kind="definition">
# Rotation

## Definition

Let $c$ be a point of $\mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ matrix with $\det A = 1$. The rotation about $c$ with the matrix $A$ is the map
$$R \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad R(p) = c + A\,(p - c).$$
In the [[Euclidean plane|0XW7V9ZZ05]] $\mathbb{R}^2$, the rotation about $c$ through the angle $\theta \in \mathbb{R}$ is the case
$$A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$$

## Remarks

Each rotation is a direct [[Euclidean isometry|0J1N3WTKM4]] that fixes $c$. In the plane, each orthogonal matrix with determinant $1$ has this form, so each plane rotation has an angle.
In the plane, rotations about one centre commute, and their angles add modulo $2\pi$. Rotations about different centres do not commute in general.
In the plane, the composite of the reflections in two lines that meet at $c$ is a rotation about $c$.
Mathlib's `Orientation.rotation` is the rotation through an angle of an oriented Euclidean plane, as a linear map.

## Explanations

### L1 human

A plane rotation turns the whole plane about one fixed point through one angle. With $\theta > 0$, the turn is anticlockwise. The rotation about $(0, 0)$ through $\pi/2$ sends $(1, 0)$ to $(0, 1)$ and $(0, 1)$ to $(-1, 0)$.
</document>

<document source="https://math2.tamnd.com/node/0XPKR18APA.md" tag="0XPKR18APA" kind="concept">
# Cofactor

## Summary

The cofactor in position $(i, j)$ of a square matrix is the first minor $M_{ij}$ with the sign $(-1)^{i+j}$.
</document>

<document source="https://math2.tamnd.com/node/0XPR1E85W0.md" tag="0XPR1E85W0" kind="object">
# Euclidean space

## Description

Let $n$ be a positive integer.
The Euclidean $n$-space is the set $\mathbb{R}^n$ of the $n$-tuples $x = (x_1, \dots, x_n)$ of real numbers, with the dot product
$$x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.$$
The elements of $\mathbb{R}^n$ are its points, and also its vectors.
The dot product gives the length $\|x\| = \sqrt{x \cdot x}$ of a vector $x$.
It also gives the distance $\|x - y\|$ between two points $x$ and $y$, and the angle between two nonzero vectors.
For $n = 2$ this is the [[Euclidean plane|0XW7V9ZZ05]], and for $n = 3$ it is the space of elementary solid geometry.

## Remarks

In a more abstract form, a Euclidean space is a real [[affine space|09XNAPTJQ7]] of finite dimension whose space of translations has an inner product. Each such space of dimension $n$ is isometric to $\mathbb{R}^n$ with the dot product.
The notation $\mathbb{E}^n$ is also common for the Euclidean $n$-space.

## Explanations

### L1 human

The Euclidean $n$-space is the space of ordinary geometry in $n$ dimensions. A point is a list of $n$ coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.
</document>

<document source="https://math2.tamnd.com/node/0XPX5NC67E.md" tag="0XPX5NC67E" kind="concept">
# Spectrum of a matrix

## Summary

The spectrum of a square matrix is the set of its eigenvalues.
</document>

<document source="https://math2.tamnd.com/node/0XQ2ZVN1PZ.md" tag="0XQ2ZVN1PZ" kind="definition">
# Laplacian

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$.
The Laplacian of $f$ is the function $\Delta f \colon U \to \mathbb{R}$,
$$\Delta f = \operatorname{div}(\nabla f) = \nabla \cdot \nabla f,$$
the [[divergence|02NSMQVQWK]] of the [[gradient|0NS2QYVC4Y]] of $f$. It is also written $\nabla^2 f$.

## Remarks

In Cartesian coordinates, $\Delta f$ is the [[sum of the unmixed second partial derivatives|05RCT1522R]] of $f$.
The operator $\Delta$ is linear, and it maps functions of [[class C^k|0TCBZMEYGN]] to functions of class $C^{k-2}$ for $k \ge 2$.
Let $f$ be of class $C^2$, and let $\bar{f}_r$ be the average of $f$ on the sphere of radius $r$ about $x$. Then $\Delta f(x)$ is $2n$ times the limit of $(\bar{f}_r - f(x)) / r^2$ as $r \to 0$. So $\Delta f(x) > 0$ means that $f(x)$ is below its nearby averages.
The Laplacian appears in the heat equation $\partial_t u = \Delta u$, the wave equation and Laplace's equation $\Delta u = 0$.

## Explanations

### L1 human

The Laplacian compares the value of a function at a point with its values nearby.
If $f$ at the point is lower than the average around it, the Laplacian is positive. If it is higher, the Laplacian is negative.
</document>

<document source="https://math2.tamnd.com/node/0XR2QK49V6.md" tag="0XR2QK49V6" kind="concept">
# Greatest element

## Summary

The greatest element of a subset of a partially ordered set is an element of the subset that is above every element of the subset.
</document>

<document source="https://math2.tamnd.com/node/0XR7A7RSXB.md" tag="0XR7A7RSXB" kind="definition">
# Negation

## Definition

Let $p$ be a [[proposition|0ZGC7ZESYN]].
The negation of $p$ is the proposition $\neg p$, read "not $p$".
The proposition $\neg p$ is true when $p$ is false, and $\neg p$ is false when $p$ is true.

| $p$ | $\neg p$ |
|---|---|
| T | F |
| F | T |

## Remarks

Other notations for $\neg p$ are $\sim p$ and $\overline{p}$.
In classical logic, $\neg \neg p$ has the same truth value as $p$. This is the [[double negation law|0NFAGWH84K]].

## Explanations

### L1 human

To negate a sentence, say that it is not the case.
The negation of "$5$ is prime" is "$5$ is not prime".
The first sentence is true, so its negation is false.
</document>

<document source="https://math2.tamnd.com/node/0XRB469Z32.md" tag="0XRB469Z32" kind="concept">
# Method of undetermined coefficients

## Summary

The method of undetermined coefficients finds a particular solution of a [[linear equation with constant coefficients|0A6PGTTP2A]] $L y = b(x)$ when $b$ has a special form.
The right side must be a sum of terms $x^k e^{\lambda x}$, $x^k e^{\lambda x} \cos(\omega x)$ and $x^k e^{\lambda x} \sin(\omega x)$.
For such a term, one tries a function of the same type, with a general polynomial of degree $k$ and unknown coefficients. If $\lambda$ (or $\lambda + i\omega$) is a root of multiplicity $s$ of the [[characteristic polynomial|0QDXMG46QW]], one multiplies the trial function by $x^s$.
Substitution into the equation gives linear equations for the unknown coefficients.

## Remarks

Example: for $y'' + y = x$, try $y_p = A x + B$. Then $A x + B = x$, so $y_p = x$.
Example with resonance: for $y'' + y = \cos x$, the number $i$ is a simple root of $r^2 + 1$. So try $y_p = x (A \cos x + B \sin x)$, which gives $y_p = \frac{1}{2} x \sin x$.
The method works because the derivatives of these functions stay in a finite-dimensional space. For other right sides, use [[variation of parameters|0Y6TH5PQVS]].

## Explanations

### L1 human

Guess that the answer looks like the right side, with unknown numbers in it.
Put the guess into the equation and match the coefficients.
If the guess already solves the homogeneous equation, multiply it by $x$.
</document>

<document source="https://math2.tamnd.com/node/0XRDYJQES6.md" tag="0XRDYJQES6" kind="concept">
# Farey sequence

## Summary

The Farey sequence of order $n$ is the increasing list of reduced fractions between $0$ and $1$ with denominators at most $n$.
The [[definition of a Farey sequence|0PN4QJE969]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0XRJB5SSNE.md" tag="0XRJB5SSNE" kind="definition">
# Prime-counting function

## Definition

For a real number $x$, the prime-counting function $\pi(x)$ is the number of [[prime numbers|0SN0MAEKYX]] $p$ with $p \le x$.

## Remarks

For example, $\pi(10) = 4$, because the primes up to $10$ are $2, 3, 5, 7$. Also $\pi(100) = 25$.
The function is a step function. It increases by $1$ at each prime.
The [[prime number theorem|0EHBY5Y84N]] says that $\pi(x)$ grows like $x / \ln x$.
The symbol $\pi$ here has no relation to the number $3.14159\dots$.

## Explanations

### L1 human

The function $\pi(x)$ answers the question: how many primes are there up to $x$?
The answer grows without bound, by [[Euclid's theorem|0292EPPPT0]], but the primes become rarer as $x$ grows.
</document>

<document source="https://math2.tamnd.com/node/0XRRYF1HPE.md" tag="0XRRYF1HPE" kind="statement">
# Dini's theorem

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be continuous functions with $f_n(x) \le f_{n+1}(x)$ for all $n$ and all $x$.
Assume that $(f_n)$ converges [[pointwise|056FSKG32J]] to a continuous function $f$ on $[a, b]$.
Then $(f_n)$ converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.

## Remarks

The same holds for decreasing sequences, and for any compact space in place of $[a, b]$.
One proof uses the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. Another uses the open sets $\{x : f(x) - f_n(x) < \varepsilon\}$, which increase and cover $[a, b]$.
All hypotheses are needed. The sequence $x^n$ on $[0, 1]$ is monotone, but its limit is not continuous. Mathlib states a version as `Monotone.tendstoLocallyUniformly_of_forall_tendsto`.

## History

The theorem is named after Ulisse Dini. He gave the first version in an 1878 book on real functions, printed in Pisa.

## Explanations

### L1 human

If continuous functions increase to a continuous limit on a closed bounded interval, they get close to it everywhere at once.
Monotonicity and compactness rule out the bad behavior of examples such as $x^n$.
</document>

<document source="https://math2.tamnd.com/node/0XSDBCP4RZ.md" tag="0XSDBCP4RZ" kind="statement">
# Eisenstein's lemma

## Statement

Let $p$ be an odd prime and let $a$ be a positive integer with $p \nmid a$.
Then
$$\left(\frac{a}{p}\right) = (-1)^{\sum_{u} \lfloor a u / p \rfloor},$$
where the sum runs over the even integers $u = 2, 4, \dots, p - 1$, and $\left(\frac{a}{p}\right)$ is the [[Legendre symbol|01J8AMV9RS]].

## Remarks

For example, take $p = 7$ and $a = 3$. The sum is $\lfloor 6/7 \rfloor + \lfloor 12/7 \rfloor + \lfloor 18/7 \rfloor = 0 + 1 + 2 = 3$, so $\left(\frac{3}{7}\right) = -1$.
A common variant holds for odd $a$: then $\left(\frac{a}{p}\right) = (-1)^{\sum_{k=1}^{(p-1)/2} \lfloor k a / p \rfloor}$.
The proof is similar to the proof of [[Gauss's lemma|037ACYXB5Z]].
Eisenstein used the lemma in a proof of [[quadratic reciprocity|085MJS6907]] that counts lattice points in a triangle.

## Explanations

### L1 human

The lemma turns the question "is $a$ a square modulo $p$?" into a count of integer parts of fractions.
The parity of the count gives the answer.
</document>

<document source="https://math2.tamnd.com/node/0XSJ25Z9M4.md" tag="0XSJ25Z9M4" kind="definition">
# Linear isomorphism

## Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$.
A *linear isomorphism* from $V$ to $W$ is a bijective [[linear map|0NRYHJPQG2]] $T \colon V \to W$.
The spaces $V$ and $W$ are *isomorphic*, written $V \cong W$, when a linear isomorphism $V \to W$ exists.

## Remarks

The inverse map $T^{-1} \colon W \to V$ of a linear isomorphism is linear.
A linear map is an isomorphism if and only if it sends some basis, or each basis, of $V$ to a basis of $W$.
For an $n \times n$ matrix $A$ over $F$, the map $x \mapsto Ax$ is an isomorphism of $F^n$ if and only if $A$ is [[invertible|0EB9N0S33C]].
[[Finite-dimensional spaces are isomorphic if and only if they have equal dimension|0VJBSK15SN]].

## Explanations

### L1 human

A linear isomorphism is a perfect translation between two vector spaces.
It matches the vectors one to one and respects addition and scaling.
Isomorphic spaces have the same linear algebra, even when their vectors look different.
</document>

<document source="https://math2.tamnd.com/node/0XTNJ6SVAC.md" tag="0XTNJ6SVAC" kind="concept">
# Cluster point of a sequence

## Summary

A cluster point of a sequence is a point near which infinitely many terms of the sequence lie. In the real numbers, the cluster points are exactly the limits of the convergent subsequences. The node [[cluster point of a sequence|0RCKQN1E09]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XTY86BZKY.md" tag="0XTY86BZKY" kind="concept">
# Galois group

## Summary

The Galois group of a field extension $L/K$ is the group of the automorphisms of $L$ that fix every element of $K$.
The [[definition of the Galois group|0XVR72M51M]] gives the details.
</document>

<document source="https://math2.tamnd.com/node/0XV2A49X6Q.md" tag="0XV2A49X6Q" kind="concept">
# Composition series

## Summary

A composition series of a group is a subnormal series in which each factor group is simple.
It cannot be refined further, and its factors are unique up to order by the Jordan–Hölder theorem.
</document>

<document source="https://math2.tamnd.com/node/0XVEZDE9J8.md" tag="0XVEZDE9J8" kind="concept">
# Unit of a ring

## Summary

A unit of a ring is an element that has a multiplicative inverse in the ring.
The [[definition of a unit|019NFGFPR5]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/0XVGJE7Q04.md" tag="0XVGJE7Q04" kind="statement">
# The principal logarithm is holomorphic off the branch cut

## Statement

Let $S = \mathbb{C} \setminus (-\infty, 0]$ be the slit plane, the complement of the standard [[branch cut|06JYC06JHY]].
Let $\operatorname{Log}$ be the [[principal branch|0YKY4M3KRQ]] of the logarithm.
Then $\operatorname{Log}$ is [[holomorphic|0DG0Z72CQD]] on $S$, and
$$\operatorname{Log}'(z) = \frac{1}{z}$$
for each $z \in S$.

## Remarks

Proof idea: on $S$, $\operatorname{Log}$ is continuous and it is the inverse of the [[complex exponential|054VXHMCA3]] on the strip $\{w : -\pi < \operatorname{Im} w < \pi\}$. The inverse function rule gives $\operatorname{Log}'(z) = 1 / e^{\operatorname{Log} z} = 1/z$.
$\operatorname{Log}$ is not continuous at the points of $(-\infty, 0)$. For $x < 0$, its values tend to $\ln|x| + i\pi$ from above the axis and to $\ln|x| - i\pi$ from below.

## Explanations

### L1 human

Away from the cut along the negative real axis, the principal logarithm behaves like the real logarithm: its derivative is $1/z$.
On the cut itself, the angle jumps from about $\pi$ to about $-\pi$, so the function is not even continuous there.
</document>

<document source="https://math2.tamnd.com/node/0XVR72M51M.md" tag="0XVR72M51M" kind="definition">
# Galois group

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
An automorphism of $L/K$ is a ring isomorphism $\sigma \colon L \to L$ with $\sigma(c) = c$ for every $c \in K$.
These automorphisms form a [[group|G5S1M9]] $\operatorname{Aut}(L/K)$ under composition.
If $L/K$ is a [[Galois extension|0DS2F7GK05]], this group is the Galois group of $L/K$, written $\operatorname{Gal}(L/K)$.

## Remarks

Some authors call $\operatorname{Aut}(L/K)$ the Galois group for every extension $L/K$.
The group $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ has two elements: the identity and complex conjugation.
The group $\operatorname{Aut}(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q})$ is trivial, because an automorphism must send $\sqrt[3]{2}$ to a real cube root of $2$.
For a finite extension, $|\operatorname{Aut}(L/K)| \le [L : K]$, by the [[bound on the order|0QM98DNWVW]].

## Explanations

### L1 human

The Galois group collects the symmetries of a bigger field that leave the smaller field unchanged.
Each symmetry permutes the roots of every polynomial over the smaller field.
</document>

<document source="https://math2.tamnd.com/node/0XW54DH28V.md" tag="0XW54DH28V" kind="concept">
# Upper central series

## Summary

The upper central series of a group $G$ is the sequence $\{e\} = Z_0 \subseteq Z_1 \subseteq Z_2 \subseteq \cdots$, where $Z_1$ is the centre and $Z_{i+1}/Z_i$ is the centre of $G/Z_i$.
The group is nilpotent if and only if this series reaches $G$.
</document>

<document source="https://math2.tamnd.com/node/0XW7V9ZZ05.md" tag="0XW7V9ZZ05" kind="object">
# Euclidean plane

## Description

The Euclidean plane is the set $\mathbb{R}^2$ of the ordered pairs $(x, y)$ of real numbers, with the dot product $(x_1, y_1) \cdot (x_2, y_2) = x_1 x_2 + y_1 y_2$.
The elements of $\mathbb{R}^2$ are the points of the plane.
The dot product gives the length of a vector, the distance between two points and the angle between two vectors.
The distance between the points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ is $\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}$.
The Euclidean plane is the [[Euclidean space|0XPR1E85W0]] of dimension 2.

## Remarks

Each Euclidean space of dimension 2 is isometric to $\mathbb{R}^2$. A choice of a [[Cartesian coordinate system|0RWKYX8339]] gives such an isometry.
The plane with a fixed Cartesian coordinate system is also called the Cartesian plane.
Books I to IV and VI of Euclid's Elements study the geometry of the plane.

## Explanations

### L1 human

Think of the Euclidean plane as an infinite flat sheet of paper. Two numbers, the coordinates $x$ and $y$, fix the position of each point. For example, the points $(0, 0)$ and $(3, 4)$ are at distance $\sqrt{9 + 16} = 5$.
</document>

<document source="https://math2.tamnd.com/node/0XW8HX8VEV.md" tag="0XW8HX8VEV" kind="concept">
# Median of a triangle

## Summary

A median of a triangle is the segment from a vertex to the midpoint of the opposite side. Each triangle has three medians, and they meet at one point, the centroid.
</document>

<document source="https://math2.tamnd.com/node/0XWCKJZRYH.md" tag="0XWCKJZRYH" kind="statement">
# A root of an integer is an integer or irrational

## Statement

Let $x$ be a real number, let $n$ be a positive integer and let $m$ be an [[integer|0C4JS9RF5M]] with $x^n = m$.
If $x$ is not an integer, then $x$ is [[irrational|080T2TXVQV]].

## Remarks

Thus, a rational number $x$ with $x^n \in \mathbb{Z}$ is an integer.
For example, $\sqrt[3]{2}$ is irrational, because $1^3 < 2 < 2^3$ shows that it is not an integer.
The statement follows from the rational root theorem for the polynomial $X^n - m$.

## Explanations

### L1 human

Take a root of a whole number, for example $\sqrt{10}$ or $\sqrt[3]{20}$.
Either the root is a whole number, as for $\sqrt{9} = 3$, or it is not a fraction at all.
A root of a whole number is never a fraction like $5/2$.
</document>

<document source="https://math2.tamnd.com/node/0XWG0JGDFW.md" tag="0XWG0JGDFW" kind="concept">
# Factorial

## Summary

The factorial of a natural number $n$ is the product of the positive integers up to $n$. It counts the orderings of $n$ distinct objects and appears in most counting formulas. The node [[factorial|0WFCGVMRRC]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0XYAFCN0PT.md" tag="0XYAFCN0PT" kind="statement">
# Second fundamental theorem of calculus

## Statement

Let $a < b$, and let $F \colon [a, b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.
Assume that $F'$ extends to a [[Riemann integrable|0N1JQXZXHC]] function $f$ on $[a, b]$.
Then
$$\int_a^b f(x) \, dx = F(b) - F(a).$$

## Remarks

For a partition $P$, the [[mean value theorem|0BARQY52N5]] gives tags $t_i$ with $F(x_i) - F(x_{i-1}) = f(t_i)(x_i - x_{i-1})$. So $F(b) - F(a)$ is a [[Riemann sum|0HBY48G8G1]] of $f$ for each partition.
One writes $F(b) - F(a) = \big[ F(x) \big]_a^b$.
The hypothesis that $f$ is integrable is needed. Some differentiable functions have derivatives that are not Riemann integrable.
Mathlib states it as `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le`.

## Explanations

### L1 human

To integrate $f$, find a function $F$ with derivative $f$.
The integral is the change of $F$ from $a$ to $b$.
</document>

<document source="https://math2.tamnd.com/node/0XYKE8BE7X.md" tag="0XYKE8BE7X" kind="definition">
# Normal matrix

## Definition

Let $A$ be an $n \times n$ complex matrix.
The matrix $A$ is *normal* when
$$A A^* = A^* A,$$
where $A^*$ is the [[conjugate transpose|0DXKE10PHF]] of $A$.

## Remarks

Hermitian, skew-Hermitian and unitary matrices are normal.
A real matrix is normal when $A A^T = A^T A$; real symmetric and orthogonal matrices are examples.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not normal.
A matrix is normal if and only if its map on $\C^n$ is a [[normal operator|0C8BR8XRTR]] for the standard inner product.

## Explanations

### L1 human

A matrix is normal when it commutes with its conjugate transpose.
Normal matrices are exactly the matrices that a unitary change of basis makes diagonal.
They include the symmetric, the Hermitian and the unitary matrices.
</document>

<document source="https://math2.tamnd.com/node/0XYP2KKSPC.md" tag="0XYP2KKSPC" kind="definition">
# Normal vector

## Definition

Let $H$ be a [[line|0MP9BCF6DP]] in $\mathbb{R}^2$ or a [[plane|0V3XAMF3C1]] in $\mathbb{R}^3$. Let $\cdot$ be the dot product.
A normal vector of $H$ is a nonzero vector $n$ such that $n \cdot (x - y) = 0$ for all points $x$ and $y$ of $H$.

## Remarks

If $n$ is a normal vector of $H$, then the normal vectors of $H$ are exactly the vectors $c n$ with $c \neq 0$.
The plane $\{(x, y, z) : a x + b y + c z = d\}$ with $(a, b, c) \neq (0, 0, 0)$ has the normal vector $(a, b, c)$. The line $\{(x, y) : a x + b y = c\}$ with $(a, b) \neq (0, 0)$ has the normal vector $(a, b)$.
For the plane $\{p + s u + t v : s, t \in \mathbb{R}\}$, the [[cross product|0N5N15HWZ7]] $u \times v$ is a normal vector.
A unit normal vector is a normal vector of length 1.

## Explanations

### L1 human

A normal vector sticks straight out of a flat object. Think of a pencil standing upright on a table: it is normal to the table. Each arrow that lies in the table makes a right angle with the pencil, so its dot product with the pencil is $0$.
</document>

<document source="https://math2.tamnd.com/node/0XYVDWGQX2.md" tag="0XYVDWGQX2" kind="statement">
# De Morgan's laws in a Boolean algebra

## Statement

Let $A$ be a [[Boolean algebra|06GKBTAA5E]].
Then $\neg(a \vee b) = \neg a \wedge \neg b$ and $\neg(a \wedge b) = \neg a \vee \neg b$ for all $a, b \in A$.

## Remarks

Proof idea: in a Boolean algebra each element has a unique complement. Check that $\neg a \wedge \neg b$ satisfies the two complement equations for $a \vee b$.
In the [[power set Boolean algebra|004J7ZEFAT]], the laws become $X \setminus (A \cup B) = (X \setminus A) \cap (X \setminus B)$ and its dual.
In the [[two-element Boolean algebra|0WG43AC0EV]], they are the laws of logic for not, and, and or.

## History

The laws are named after Augustus De Morgan. His book Formal Logic of 1847 states them.

## Explanations

### L1 human

The complement of an or is the and of the complements, and the complement of an and is the or of the complements.
Not (rain or snow) means no rain and no snow.
</document>

<document source="https://math2.tamnd.com/node/0XYYT5KNTX.md" tag="0XYYT5KNTX" kind="concept">
# Direct sum of subspaces

## Summary

A vector space is the direct sum of a family of subspaces when each vector is a sum of vectors from the subspaces.
This sum must be unique.
</document>

<document source="https://math2.tamnd.com/node/0XZ37KAB1Z.md" tag="0XZ37KAB1Z" kind="concept">
# Disjoint sets

## Summary

Two sets are disjoint when they have no element in common.
</document>

<document source="https://math2.tamnd.com/node/0Y0JGC2K9T.md" tag="0Y0JGC2K9T" kind="concept">
# Rooted tree

## Summary

A rooted tree is a tree with a chosen vertex, the root, which gives each edge a direction away from it. Rooted trees model hierarchies and recursive data. The node [[rooted tree|0C9J6W2A8A]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Y0KFHF47G.md" tag="0Y0KFHF47G" kind="definition">
# Graph of a function

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The graph of $f$ is the set
$$\Gamma(f) = \{(a, f(a)) \mid a \in A\} \subseteq A \times B.$$

## Remarks

For a real function $f \colon \mathbb{R} \to \mathbb{R}$, the graph is a curve in the plane $\mathbb{R}^2$.
Let $G \subseteq A \times B$.
$G$ is the graph of a function from $A$ to $B$ if and only if each $a \in A$ is the first entry of exactly one pair.
In the set-theoretic [[definition of a function|0XHDTE5YMZ]], the function and its graph are the same set.

## Explanations

### L1 human

The graph of $f(x) = x^2$ is the parabola $\{(x, x^2) \mid x \in \mathbb{R}\}$.
Each vertical line meets it in exactly one point. This is the vertical line test.
</document>

<document source="https://math2.tamnd.com/node/0Y0MQYST0M.md" tag="0Y0MQYST0M" kind="concept">
# Modus ponens

## Summary

Modus ponens is the rule of inference that gets $q$ from the two premises $p$ and $p \to q$.
$$\frac{p \qquad p \to q}{q}$$
It is valid: when the two premises are true, the conclusion $q$ is true.

## Remarks

Most proof systems for [[propositional logic|09K9MWD2QQ]] use modus ponens as a rule.
Its partner modus tollens gets $\neg p$ from $\neg q$ and $p \to q$.

## Explanations

### L1 human

Modus ponens is the most common step in a proof.
If you know that $n$ is divisible by $4$, and that each number divisible by $4$ is even, then you know that $n$ is even.
</document>

<document source="https://math2.tamnd.com/node/0Y0WS5BCJ7.md" tag="0Y0WS5BCJ7" kind="definition">
# Inflection point

## Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be continuous, and let $c \in I$.
The point $c$ is an inflection point of $f$ if there is $\delta > 0$ with one of the following properties.
The function $f$ is [[convex|0VV0HSW1ZS]] on $(c - \delta, c]$ and [[concave|0EF2VG8XKQ]] on $[c, c + \delta)$.
Or $f$ is concave on $(c - \delta, c]$ and convex on $[c, c + \delta)$.

## Remarks

If $f$ is twice differentiable at an inflection point $c$, then $f''(c) = 0$.
The converse fails: $f(x) = x^4$ has $f''(0) = 0$, but $0$ is not an inflection point.
The function $x^3$ has an inflection point at $0$. So does $\sin x$, at each multiple of $\pi$.
Some texts also ask the graph to have a tangent line at $c$.

## Explanations

### L1 human

At an inflection point, the graph changes from bending up to bending down, or the reverse.
The graph crosses its tangent line there.
</document>

<document source="https://math2.tamnd.com/node/0Y123X48GA.md" tag="0Y123X48GA" kind="statement">
# Absolutely convergent series converge

## Statement

Let $(a_n)$ be a real sequence.
If the series $\sum a_n$ [[converges absolutely|0W3K3QPEWT]], then $\sum a_n$ converges.
Moreover, $\left| \sum_{n=0}^{\infty} a_n \right| \le \sum_{n=0}^{\infty} |a_n|$.

## Remarks

The proof uses the [[Cauchy criterion for series|06PVXMY0TH]]. For $n > m$ the triangle inequality gives $\left| \sum_{k=m+1}^{n} a_k \right| \le \sum_{k=m+1}^{n} |a_k|$.
The converse fails: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but $\sum 1/n$ diverges.
The result holds in every complete normed space. Mathlib states it as `Summable.of_norm`.

## Explanations

### L1 human

Cancellation between terms of opposite signs can only make partial sums smaller in absolute value.
So if the series converges without any cancellation, it also converges with it.
</document>

<document source="https://math2.tamnd.com/node/0Y12QYTJK3.md" tag="0Y12QYTJK3" kind="definition">
# Limit at infinity

## Definition

Let $D \subseteq \mathbb{R}$ be a set that is not bounded above, and let $f \colon D \to \mathbb{R}$ be a function.
A real number $L$ is the limit of $f$ at $+\infty$ if for each $\varepsilon > 0$ there is $M$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $x > M$.
The notation is $\lim_{x \to \infty} f(x) = L$.
The limit at $-\infty$ is defined in the same way, with $x < M$.

## Remarks

A sequence is a function on $\mathbb{N}$. For sequences this definition gives the [[limit of a sequence|0XKWSSY8P2]].
For example, $\lim_{x \to \infty} 1/x = 0$.
Mathlib expresses $x \to \infty$ by the filter `Filter.atTop`.

## Explanations

### L1 human

The limit at infinity is the value that $f(x)$ approaches as $x$ grows without bound.
The graph then gets close to a horizontal line $y = L$.
</document>

<document source="https://math2.tamnd.com/node/0Y17PEKF24.md" tag="0Y17PEKF24" kind="definition">
# Diagonalisable matrix

## Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
The matrix $A$ is *diagonalisable* over $F$ when there is an invertible $n \times n$ matrix $P$ over $F$ such that $P^{-1} A P$ is a [[diagonal matrix|044JNC7ZAA]].
An endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.

## Remarks

So $A$ is diagonalisable if and only if it is [[similar|0E89HQATM8]] to a diagonal matrix.
The diagonal entries of $P^{-1} A P$ are the eigenvalues of $A$, and the columns of $P$ are eigenvectors.
The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not diagonalisable over any field.
Over $\R$, the rotation matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is not diagonalisable; over $\C$, it is.

## Explanations

### L1 human

A diagonalisable matrix becomes a diagonal matrix in a suitable basis.
In that basis, the map just stretches each coordinate axis by a factor.
This makes powers, exponentials and other functions of the matrix easy to compute.
</document>

<document source="https://math2.tamnd.com/node/0Y1MG9R5VN.md" tag="0Y1MG9R5VN" kind="statement">
# Catalan recurrence

## Statement

Let $n$ be a natural number, and let $C_m$ be the [[Catalan numbers|092DRBFKMG]].
Then
$$C_{n+1} = \sum_{i=0}^{n} C_i\, C_{n-i}.$$

## Remarks

With the definition of the node [[Catalan numbers|092DRBFKMG]], the statement holds by definition. Its content is in the combinatorial families that satisfy it.
Example: $C_4 = C_0 C_3 + C_1 C_2 + C_2 C_1 + C_3 C_0 = 5 + 2 + 2 + 5 = 14$.
For [[Dyck words|0R9JR0465V]], the recurrence splits a word at the first return to level $0$.

## Explanations

### L1 human

A structure of size $n + 1$ often splits into a left part of size $i$ and a right part of size $n - i$.
Add over all $i$ the number of ways to build the two parts, and you get the next Catalan number.
</document>

<document source="https://math2.tamnd.com/node/0Y2X7P2WB7.md" tag="0Y2X7P2WB7" kind="concept">
# Orthogonal vectors

## Summary

Two vectors of an inner product space are orthogonal when their inner product is zero.
</document>

<document source="https://math2.tamnd.com/node/0Y34JMJNE9.md" tag="0Y34JMJNE9" kind="statement">
# Gauss–Wantzel theorem

## Statement

Let $n \ge 3$.
The regular $n$-gon can be constructed with straightedge and compass if and only if $n = 2^k p_1 p_2 \cdots p_t$, with $k, t \ge 0$ and $p_1, \dots, p_t$ distinct Fermat primes.

## Remarks

A Fermat prime is a prime number of the form $2^{2^m} + 1$. The known Fermat primes are $3, 5, 17, 257$ and $65537$.
The regular $n$-gon is constructible if and only if $e^{2\pi i/n}$ is a [[constructible number|0FMN2MN7TM]]. Its degree over $\mathbb{Q}$ is $\varphi(n)$, by the [[irreducibility of $\Phi_n$|0SK1VHZ5R2]].
The condition on $n$ holds if and only if $\varphi(n)$ is a power of $2$.
So the $17$-gon and the $15$-gon are constructible, but the $7$-gon and the $9$-gon are not.

## History

Carl Friedrich Gauss proved in 1796 that the regular $17$-gon is constructible.
In his 1801 book Disquisitiones Arithmeticae, he gave the sufficient condition.
Pierre Wantzel gave a full proof of the necessity in 1837.

## Explanations

### L1 human

The theorem tells exactly which regular polygons you can draw with straightedge and compass.
The answer depends on the prime factors of the number of sides.
Only Fermat primes, each used at most once, and powers of $2$ are allowed.
</document>

<document source="https://math2.tamnd.com/node/0Y3KSN2E9X.md" tag="0Y3KSN2E9X" kind="concept">
# Series of functions

## Summary

A series of functions is an infinite sum of functions defined on a common set. Its convergence is the convergence of its partial sums, pointwise or uniformly. The node [[series of functions|0MRE6XRE5V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Y4B7P0GS8.md" tag="0Y4B7P0GS8" kind="statement">
# The order modulo p divides p − 1

## Statement

Let $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ be an integer with $p \nmid a$.
Then the [[multiplicative order|0SBRX5CZ5R]] of $a$ modulo $p$ divides $p - 1$.

## Remarks

The proof: $a^{p-1} \equiv 1 \pmod p$ by [[Fermat's little theorem|0M25PRN3WQ]], and each exponent $m$ with $a^m \equiv 1$ is a multiple of the order.
The order is equal to $p - 1$ exactly when $a$ is a [[primitive root|0KM2CJ80G7]] modulo $p$.
For example, modulo $13$, the order of $3$ is $3$ and the order of $2$ is $12$. Both divide $12$.

## Explanations

### L1 human

The powers of $a$ modulo $p$ go round a cycle. The length of the cycle always divides $p - 1$.
Modulo $11$, the powers of $5$ are $5, 3, 4, 9, 1$. The cycle has length $5$, and $5 \mid 10$.
</document>

<document source="https://math2.tamnd.com/node/0Y4NF1DRKD.md" tag="0Y4NF1DRKD" kind="statement">
# Trace of a product is symmetric

## Statement

Let $R$ be a commutative ring.
Let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $R$.
Then $\tr(AB) = \tr(BA)$, where $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

Both sides equal $\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} b_{ji}$.
The matrices $AB$ and $BA$ can have different sizes, and the identity still holds.
A consequence is $\tr(P^{-1} A P) = \tr A$, so similar matrices have the same trace.
For three matrices, only cyclic permutations are allowed in general: $\tr(ABC) = \tr(BCA)$, but $\tr(ABC)$ can differ from $\tr(BAC)$.

## Explanations

### L1 human

Although $AB$ and $BA$ are usually different matrices, they have the same trace.
Each diagonal sum adds the same products $a_{ij} b_{ji}$, only in another order.
This fact makes the trace a property of a linear map, not of the basis that we choose.
</document>

<document source="https://math2.tamnd.com/node/0Y4TV2D5RQ.md" tag="0Y4TV2D5RQ" kind="statement">
# Brooks' theorem

## Statement

Let $G$ be a finite connected simple [[graph|0JD5C82QMB]] with [[maximum degree|0KNAN9F27P]] $\Delta$.
Suppose that $G$ is not a [[complete graph|03WKB2H474]] and not a [[cycle graph|0QQ0F12962]] of odd length.
Then $\chi(G) \le \Delta$.

## Remarks

A [[greedy colouring|04AAKZNXHD]] gives $\chi(G) \le \Delta + 1$ for every graph. Brooks' theorem improves this bound by $1$ in all other cases.
The two exceptions need $\Delta + 1$ colours: $\chi(K_n) = n$ with $\Delta = n - 1$, and an odd cycle has $\chi = 3$ with $\Delta = 2$.

## History

R. Leonard Brooks published the theorem in 1941.

## Explanations

### L1 human

If each vertex has at most $\Delta$ neighbours, then $\Delta + 1$ colours are always enough.
Brooks' theorem says that $\Delta$ colours are enough, except for complete graphs and odd cycles.
</document>

<document source="https://math2.tamnd.com/node/0Y5Q8Q9WJ7.md" tag="0Y5Q8Q9WJ7" kind="definition">
# Level set

## Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$ be a [[function of several variables|0JPCHTXWG0]], and let $c \in \mathbb{R}$.
The level set of $f$ at the value $c$ is the set
$$L_c(f) = \{x \in D : f(x) = c\}.$$
It is the [[preimage|01AHG4G9RT]] $f^{-1}(\{c\})$ of the one-point set $\{c\}$.
For $n = 2$ a level set is also called a level curve, and for $n = 3$ it is also called a level surface.

## Remarks

For $f(x, y) = x^2 + y^2$, the level set at $c > 0$ is the circle of radius $\sqrt{c}$ about the origin. The level set at $c = 0$ is the point $(0, 0)$, and the level set at $c < 0$ is empty.
The level sets of $f$ for different values of $c$ are disjoint, and their union is the domain $D$.
At a point where $f$ is differentiable with a nonzero [[gradient|0ABK9V9Y64]], the gradient is [[orthogonal to the level set|0VXYV3NQQ1]].
A level set of a function of two variables is a [[contour line|0QD83RAN8G]] of its graph.

## Explanations

### L1 human

On a map of a hill, a level set is the set of all places at one fixed height.
For the bowl $f(x, y) = x^2 + y^2$, each level set at a positive height is a circle, and the circles grow as the height grows.
</document>

<document source="https://math2.tamnd.com/node/0Y5R7TPNT1.md" tag="0Y5R7TPNT1" kind="definition">
# Outcome

## Definition

Let $\Omega$ be a [[sample space|0W7NFW23RF]] of a random experiment.
An outcome of the experiment is an [[element|0EZ758XE73]] $\omega$ of $\Omega$.

## Remarks

Two different outcomes cannot both happen in one trial.
An outcome $\omega$ gives the event $\{\omega\}$, an elementary event. An outcome and an event are different things: an [[event|00SQGJNRTB]] is a set of outcomes.
For a roll of a die, the outcomes are $1, 2, 3, 4, 5, 6$, and "the result is even" is the event $\{2, 4, 6\}$, not an outcome.

## Explanations

### L1 human

An outcome is one thing that can happen.
When you toss two coins, "heads then tails" is one outcome, and there are four outcomes in total.
</document>

<document source="https://math2.tamnd.com/node/0Y5RHNRKJZ.md" tag="0Y5RHNRKJZ" kind="definition">
# Centroid

## Definition

Let $k$ be a positive integer, and let $K$ be a field in which $k \neq 0$. Let $A$ be an [[affine space|09XNAPTJQ7]] over a $K$-vector space, and let $p_1, \dots, p_k$ be points of $A$.
The centroid of $p_1, \dots, p_k$ is the [[affine combination|0KJH6SRFZK]] with all weights equal to $\frac{1}{k}$:
$$G = \frac{1}{k} p_1 + \frac{1}{k} p_2 + \dots + \frac{1}{k} p_k.$$
The centroid of a triangle $ABC$ is the centroid $G = \frac{1}{3}(A + B + C)$ of its vertices.

## Remarks

In $\mathbb{R}^n$, the coordinates of the centroid are the means of the coordinates of the points.
In $\mathbb{R}^n$, the centroid is the point $x$ that makes the sum $\sum_i \|x - p_i\|^2$ smallest.
The centroid of two points is their [[midpoint|0S706JBZBG]].
The centroid of a triangle is also the centre of mass of a uniform triangular plate.

## Explanations

### L1 human

The centroid is the average of the points. For the triangle with the vertices $(0, 0)$, $(6, 0)$ and $(0, 3)$, the centroid is $\left(\frac{0 + 6 + 0}{3}, \frac{0 + 0 + 3}{3}\right) = (2, 1)$. A cardboard triangle balances on a pin at this point.
</document>

<document source="https://math2.tamnd.com/node/0Y5X3SDBCZ.md" tag="0Y5X3SDBCZ" kind="statement">
# Fermat's Last Theorem

## Statement

Let $n$ be an integer with $n > 2$.
Then there are no positive integers $a$, $b$ and $c$ with $a^n + b^n = c^n$.

## Remarks

For $n = 2$ there are infinitely many solutions, the [[Pythagorean triples|01G1WY1GVX]].
It is enough to prove the theorem for $n = 4$ and for each odd prime $n$, because each $n > 2$ has one of these as a divisor.
The cases [[n = 4|0E741HSMJ1]] and [[n = 3|07NKG38VEG]] have elementary proofs.

## History

Pierre de Fermat stated the theorem around 1637 and wrote that he had a proof.
Andrew Wiles gave the first proof in 1994, with a step completed jointly with Richard Taylor. The proof was published in 1995.
The proof uses the modularity of semistable elliptic curves.

## Explanations

### L1 human

The equation $a^2 + b^2 = c^2$ has many whole-number solutions, such as $3^2 + 4^2 = 5^2$.
For cubes, fourth powers and all higher powers, the theorem says that there are none.
The statement is simple, but the proof took more than 350 years.
</document>

<document source="https://math2.tamnd.com/node/0Y6QNF1J49.md" tag="0Y6QNF1J49" kind="statement">
# Tail sum formula for expectation

## Statement

Let $X$ be a random variable with values in $[0, \infty)$. Then
$$E[X] = \int_0^\infty P(X > t) \, dt,$$
with both sides in $[0, \infty]$.
If $X$ takes values in $\{0, 1, 2, \dots\}$, then
$$E[X] = \sum_{n=1}^{\infty} P(X \ge n).$$

## Remarks

The proof writes $X = \int_0^\infty 1_{\{X > t\}} \, dt$ and exchanges expectation and integral by Tonelli's theorem.
In terms of the [[cumulative distribution function|0MXMS3AE13]] $F$, the first formula reads $E[X] = \int_0^\infty (1 - F(t)) \, dt$.
For $X \sim \operatorname{Exp}(\lambda)$, the formula gives $E[X] = \int_0^\infty e^{-\lambda t} \, dt = 1/\lambda$.

## Explanations

### L1 human

You can compute an average by adding the chances that $X$ passes each level.
For a count $X$, each $n \le X$ adds $1$ to $X$, and it does so with probability $P(X \ge n)$.
</document>

<document source="https://math2.tamnd.com/node/0Y6TH5PQVS.md" tag="0Y6TH5PQVS" kind="concept">
# Variation of parameters

## Summary

Variation of parameters is a method to find a particular solution of an inhomogeneous linear equation from a [[fundamental system|0H4KNNCHPY]] of the homogeneous equation.
Let $p, q, f$ be continuous on an open interval $I$. Let $(y_1, y_2)$ be a fundamental system of $y'' + p y' + q y = 0$ on $I$, with [[Wronskian|0N5NF1RGBP]] $W$.
One looks for $y = u_1 y_1 + u_2 y_2$ with the extra condition $u_1' y_1 + u_2' y_2 = 0$. Then the equation $y'' + p y' + q y = f$ becomes $u_1' y_1' + u_2' y_2' = f$.
This linear system gives $u_1' = -y_2 f / W$ and $u_2' = y_1 f / W$, so a particular solution is
$$y_p = -y_1 \int \frac{y_2 f}{W}\, dx + y_2 \int \frac{y_1 f}{W}\, dx.$$

## Remarks

The method works for each continuous right side, unlike the [[method of undetermined coefficients|0XRB469Z32]].
Example: for $y'' + y = \tan x$ on $(-\pi/2, \pi/2)$, with $y_1 = \cos x$, $y_2 = \sin x$ and $W = 1$, one gets $y_p = -\cos x \ln\left(\sec x + \tan x\right)$.
For first-order systems, the same idea gives the [[variation of constants formula|0S1KG0XE40]].

## Explanations

### L1 human

The general solution of the homogeneous equation is $c_1 y_1 + c_2 y_2$ with constants.
The idea is to let the constants vary with $x$.
A smart extra condition makes the equations for $u_1'$ and $u_2'$ simple.
</document>

<document source="https://math2.tamnd.com/node/0Y74KN99A5.md" tag="0Y74KN99A5" kind="definition">
# Order isomorphism

## Definition

Let $(P, \le_P)$ and $(Q, \le_Q)$ be [[partially ordered sets|011C29Q1S1]].
An order isomorphism from $P$ to $Q$ is a [[bijective function|0SNQEF7Z53]] $f \colon P \to Q$ such that, for all $a, b \in P$,
$$a \le_P b \iff f(a) \le_Q f(b).$$

## Remarks

The inverse of an order isomorphism is an order isomorphism.
The map $x \mapsto 2x$ is an order isomorphism from $\mathbb{Z}$ to the even integers.
The ordered sets $\mathbb{N}$ and $\mathbb{Z}$ are not order isomorphic, because $\mathbb{N}$ has a least element and $\mathbb{Z}$ has none.
A [[monotone|0WFF6AZPBP]] bijection need not be an order isomorphism, when the orders are not total.

## Explanations

### L1 human

Order isomorphic sets have the same shape as ordered sets.
Only the names of the elements differ.
</document>

<document source="https://math2.tamnd.com/node/0Y7H3SAVA1.md" tag="0Y7H3SAVA1" kind="concept">
# Matrix addition

## Summary

Matrix addition is the operation that adds two matrices of the same size entry by entry.
</document>

<document source="https://math2.tamnd.com/node/0Y7PVW4Q44.md" tag="0Y7PVW4Q44" kind="definition">
# Directional derivative

## Definition

Let $U \subseteq \mathbb{R}^n$ be an open set, let $f \colon U \to \mathbb{R}^m$, let $a \in U$, and let $v \in \mathbb{R}^n$.
The directional derivative of $f$ at $a$ in the direction $v$ is the limit
$$D_v f(a) = \lim_{t \to 0} \frac{f(a + t v) - f(a)}{t},$$
when this limit exists in $\mathbb{R}^m$.
Other notations for it are $\partial_v f(a)$ and $\nabla_v f(a)$.

## Remarks

For $v = e_i$, the directional derivative is the [[partial derivative|09EGC6F8PT]] $\partial f / \partial x_i$ at $a$.
Some books require that $v$ is a unit vector, so that $D_v f(a)$ is a rate of change per unit of distance.
If $D_v f(a)$ exists and $c \in \mathbb{R}$, then $D_{cv} f(a)$ exists and equals $c \, D_v f(a)$.
When $f$ is [[differentiable|0HHHDCXZV4]] at $a$, the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$, see [[this statement|0A6PMR20N4]].

## Explanations

### L1 human

Stand at the point $a$ on the graph of $f$ and walk in the direction $v$.
The directional derivative is the slope that you feel at the first step.
For $f(x, y) = x^2 + y^2$ at $a = (1, 0)$ and $v = (0, 1)$, you get $f(1, t) = 1 + t^2$, so the slope is $0$. You start along the tangent of the circle $x^2 + y^2 = 1$, where $f$ is constant.
</document>

<document source="https://math2.tamnd.com/node/0Y7VDAX8Q3.md" tag="0Y7VDAX8Q3" kind="definition">
# K-colourable graph

## Definition

Let $k$ be a natural number.
A [[graph|0JD5C82QMB]] $G$ is $k$-colourable if it has a [[colouring|0874138YP6]] with the set of colours $\{1, 2, \dots, k\}$.

## Remarks

A $k$-colourable graph is also $(k+1)$-colourable.
A graph is $2$-colourable if and only if it is [[bipartite|07DVHJ8N0A]].
The [[complete graph|03WKB2H474]] $K_n$ is $k$-colourable if and only if $k \ge n$.

## Explanations

### L1 human

A graph is $k$-colourable when $k$ colours are enough to colour it properly.
A triangle needs three colours, so it is $3$-colourable but not $2$-colourable.
</document>

<document source="https://math2.tamnd.com/node/0Y85GYEGSF.md" tag="0Y85GYEGSF" kind="statement">
# Existence of an orthonormal basis

## Statement

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$.
Then $V$ has an [[orthonormal basis|0S9GGTR0EX]].
Moreover, each orthonormal family in $V$ extends to an orthonormal basis of $V$.

## Remarks

The proof applies the [[Gram–Schmidt process|0VT6P0GNQR]] to any basis of $V$.
For the extension, first extend the orthonormal family to a basis, and then apply Gram–Schmidt; it keeps the first vectors.
Hence each finite-dimensional real inner product space is isometric to $\R^n$ with the dot product.

## Explanations

### L1 human

Every finite-dimensional inner product space has a basis of perpendicular unit vectors.
We construct one with the Gram–Schmidt process.
So every such space looks like $\R^n$ or $\C^n$ with the standard inner product.
</document>

<document source="https://math2.tamnd.com/node/0Y8FRYX8Q2.md" tag="0Y8FRYX8Q2" kind="concept">
# Sequence of functions

## Summary

A sequence of functions is a sequence $(f_n)$ of functions $f_n \colon S \to \mathbb{R}$ on a common set $S$. For each fixed $x \in S$, the values $f_n(x)$ form a [[sequence of real numbers|0QE8STGJH2]]. Such sequences can converge in several ways, mainly [[pointwise|056FSKG32J]] or [[uniformly|08FPABDQ4C]].

## Remarks

The two modes give different properties of the limit. A uniform limit of continuous functions is continuous, but a pointwise limit need not be.
Examples are the Taylor polynomials of a function, and the partial sums of a power series.

## Explanations

### L1 human

A sequence of functions is a list of graphs, one for each $n$.
We ask whether the graphs approach the graph of a limit function, and in what sense.
</document>

<document source="https://math2.tamnd.com/node/0Y95Z7MT02.md" tag="0Y95Z7MT02" kind="statement">
# Distinct eigenvalues imply diagonalisable

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
If $A$ has $n$ pairwise distinct eigenvalues in $F$, then $A$ is [[diagonalisable|07SQCNV233]] over $F$.

## Remarks

Eigenvectors for the $n$ distinct eigenvalues are [[linearly independent|0HNMFG8F8K]], so they form a basis of $F^n$.
The converse fails: $I_n$ is diagonal, but it has only one eigenvalue.
For example, $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ has eigenvalues $1$ and $3$, so it is diagonalisable.

## Explanations

### L1 human

If an $n \times n$ matrix has $n$ different eigenvalues, then it has $n$ independent eigenvectors.
So it is diagonalisable.
This sufficient condition is easy to check with the characteristic polynomial.
</document>

<document source="https://math2.tamnd.com/node/0Y97E2DMB7.md" tag="0Y97E2DMB7" kind="statement">
# Binary trees are counted by Catalan numbers

## Statement

Let $n$ be a natural number.
Then the number of [[binary trees|02A8EMC83C]] with exactly $n$ nodes is the [[Catalan number|092DRBFKMG]] $C_n$.

## Remarks

Here a binary tree is either empty or a root node with an ordered pair of binary trees as its left and right subtrees.
For $n = 3$ there are $C_3 = 5$ binary trees.
Proof idea: a nonempty tree with $n + 1$ nodes has a left subtree with $i$ nodes and a right subtree with $n - i$ nodes. This gives the [[Catalan recurrence|0Y1MG9R5VN]].

## Explanations

### L1 human

Remove the root of a binary tree, and you get a left tree and a right tree.
The sizes of the two trees add up to one less than the size of the original tree, and this gives the Catalan recurrence.
</document>

<document source="https://math2.tamnd.com/node/0Y9D2FJRQ4.md" tag="0Y9D2FJRQ4" kind="definition">
# Second derivative

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable on $I$.
If $f'$ is differentiable at $a \in I$, the second derivative of $f$ at $a$ is $f''(a) = (f')'(a)$.

## Remarks

It is the [[higher derivative|084NGSTWM7]] $f^{(2)}$. Another notation is $\frac{d^2 f}{dx^2}$.
If $f'' > 0$ on $I$, then $f$ is [[convex|0VV0HSW1ZS]] on $I$.
In mechanics, if $f(t)$ is the position at time $t$, then $f''(t)$ is the acceleration.

## Explanations

### L1 human

The second derivative is the rate of change of the slope.
It tells how the graph bends: upward where $f'' > 0$ and downward where $f'' < 0$.
</document>

<document source="https://math2.tamnd.com/node/0Y9H52BT03.md" tag="0Y9H52BT03" kind="definition">
# Order on the natural numbers

## Definition

Let $a$ and $b$ be [[natural numbers|0MWKJ44V5A]].
The notation $a \le b$ means that there is a natural number $c$ with $a + c = b$, where $+$ is the [[addition|0W5DSDFQ4G]] of $\mathbb{N}$.
The notation $a < b$ means that $a \le b$ and $a \neq b$.

## Remarks

The relation $\le$ is a total order on $\mathbb{N}$: it is reflexive, antisymmetric and transitive, and $a \le b$ or $b \le a$ for all $a$ and $b$.
The order agrees with the operations: if $a \le b$, then $a + c \le b + c$ and $a c \le b c$ for each natural number $c$.
Each nonempty set of natural numbers has a least element for this order. This is the [[well-ordering principle|01BX15HSRR]].

## Explanations

### L1 human

The number $a$ is at most $b$ when you can get from $a$ to $b$ by adding something.
For example, $3 \le 7$ because $3 + 4 = 7$, but $7 \le 3$ is false because no natural number added to $7$ gives $3$.
</document>

<document source="https://math2.tamnd.com/node/0YA7X86K05.md" tag="0YA7X86K05" kind="concept">
# Characteristic polynomial of a linear recurrence

## Summary

The characteristic polynomial of a linear recurrence encodes its coefficients as a polynomial. Its roots give the geometric solutions, and they lead to closed formulas for all solutions. The node [[characteristic polynomial of a linear recurrence|0BWDJGQYST]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YAA9CJQXF.md" tag="0YAA9CJQXF" kind="definition">
# Primitive element of a finite field

## Definition

Let $F$ be a [[finite field|08RTDHX1SK]] of order $q$.
An element $\alpha \in F$ is a primitive element of $F$ if $\alpha$ generates the multiplicative group $F^\times$.
That is, $F^\times = \{ \alpha, \alpha^2, \dots, \alpha^{q-1} = 1 \}$.

## Remarks

Primitive elements exist, because [[$F^\times$ is cyclic|0EBZ3G455G]]. There are exactly $\varphi(q - 1)$ of them, where $\varphi$ is Euler's totient function.
The element $2$ is a primitive element of $\mathbb{F}_3$ and of $\mathbb{F}_5$, but not of $\mathbb{F}_7$. The element $3$ is a primitive element of $\mathbb{F}_7$.
A primitive element $\alpha$ of $F$ generates $F$ as a [[simple extension|0P6SBR7YM2]] of its prime subfield. The converse is false in general.

## Explanations

### L1 human

A primitive element is one whose powers give every nonzero element of the field.
It works like a base for logarithms in the finite field.
</document>

<document source="https://math2.tamnd.com/node/0YAG43TCXQ.md" tag="0YAG43TCXQ" kind="concept">
# Mersenne number

## Summary

A Mersenne number is an integer of the form $2^n - 1$, one less than a power of two.
The [[definition of a Mersenne number|0GJBJ19NXR]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0YAMJRK04E.md" tag="0YAMJRK04E" kind="definition">
# Conditional expectation

## Definition

Let $X$ be an integrable real random variable on a probability space $(\Omega, \mathcal{F}, P)$, and let $\mathcal{G} \subseteq \mathcal{F}$ be a [[σ-algebra|04WSW8T3SY]].
A conditional expectation of $X$ given $\mathcal{G}$ is an integrable random variable $Z$ with these two properties:
1. $Z$ is $\mathcal{G}$-measurable.
2. $E[Z \, 1_G] = E[X \, 1_G]$ for each $G \in \mathcal{G}$.
Such a $Z$ exists, and two such random variables are equal almost surely. One writes $Z = E[X \mid \mathcal{G}]$.
For a random variable $Y$, $E[X \mid Y] = E[X \mid \sigma(Y)]$, where $\sigma(Y)$ is the σ-algebra generated by $Y$.

## Remarks

The existence follows from the Radon–Nikodym theorem. For $E[X^2] < \infty$, $E[X \mid \mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\mathcal{G})$.
For a discrete $Y$, $E[X \mid Y] = g(Y)$ with $g(y) = E[X \, 1_{\{Y = y\}}] / P(Y = y)$ when $P(Y = y) > 0$. This $g(y)$ is the expectation of $X$ under the [[conditional probability|0FJB06AD2P]] $P(\cdot \mid Y = y)$.
The conditional expectation is linear and monotone. If $W$ is $\mathcal{G}$-measurable and $WX$ is integrable, then $E[WX \mid \mathcal{G}] = W \, E[X \mid \mathcal{G}]$.
If $X$ is independent of $\mathcal{G}$, then $E[X \mid \mathcal{G}] = E[X]$.

## Explanations

### L1 human

$E[X \mid Y]$ is the average value of $X$ when you know $Y$. It changes with $Y$, so it is a random variable.
For two dice with results $D_1$ and $D_2$, $E[D_1 + D_2 \mid D_1] = D_1 + 3.5$.
</document>

<document source="https://math2.tamnd.com/node/0YB2NRP0T5.md" tag="0YB2NRP0T5" kind="concept">
# Cayley table

## Summary

A Cayley table of a finite set with a binary operation is a square table. It shows the result of the operation for each pair of elements.
For a finite group, it shows the full multiplication of the group.
</document>

<document source="https://math2.tamnd.com/node/0YB93GWTKK.md" tag="0YB93GWTKK" kind="definition">
# Absorbing Markov chain

## Definition

Let $(X_n)$ be a time-homogeneous [[Markov chain|033TJBZDG0]] with [[transition matrix|0F1KBVY161]] $P = (p_{ij})$ on a state space $S$.
A state $i \in S$ is absorbing if $p_{ii} = 1$. A state that is not absorbing is transient.
The chain is an absorbing Markov chain if it has at least one absorbing state and each state can reach an absorbing state.
That is, for each $i \in S$ there are an absorbing state $j$ and an $n \ge 0$ with $p^{(n)}_{ij} > 0$.

## Remarks

For a finite absorbing chain, write $P$ in the canonical form with the transient states first: $P = \begin{pmatrix} Q & R \\ 0 & I \end{pmatrix}$.
The fundamental matrix is $N = (I - Q)^{-1} = \sum_{n=0}^{\infty} Q^n$. The entry $N_{ij}$ is the expected number of visits to the transient state $j$, starting from the transient state $i$.
The expected number of steps before absorption from $i$ is the $i$-th entry of $N \mathbf{1}$, where $\mathbf{1}$ is the column vector of ones. The absorption probabilities are the entries of $NR$.
A finite absorbing chain reaches an absorbing state with probability $1$.

## Explanations

### L1 human

An absorbing chain wanders for a while and then gets stuck in a trap state forever.
The fortune in the gambler's ruin problem is an example: the states $0$ and $N$ are traps.
</document>

<document source="https://math2.tamnd.com/node/0YBDSJARWN.md" tag="0YBDSJARWN" kind="concept">
# Absolute convergence

## Summary

A series converges absolutely when the series of the absolute values of its terms converges. Absolutely convergent series converge, and their sums do not depend on the order of the terms. The node [[absolute convergence|0W3K3QPEWT]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YBHSJMN3J.md" tag="0YBHSJMN3J" kind="statement">
# Comparability of cardinals

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
Then there is an [[injective function|0PVG1CRAJN]] from $A$ to $B$ or an injective function from $B$ to $A$.
So $|A| \le |B|$ or $|B| \le |A|$.

## Remarks

With the [[Schröder–Bernstein theorem|0A9V6RKQG6]], it follows that the [[order of cardinals|01GTP8M5RG]] is a total order.
The statement is equivalent to the axiom of choice over ZF.
One proof uses [[Zorn's lemma|04RXTQA5ZA]] on the set of injective partial functions from $A$ to $B$.

## Explanations

### L1 human

Any two sets can be compared in size: one of them fits into the other.
Without the axiom of choice, two sets can exist such that neither fits into the other.
</document>

<document source="https://math2.tamnd.com/node/0YBR6QPY4Y.md" tag="0YBR6QPY4Y" kind="example">
# Groups of order pq

## Example

Let $p < q$ be prime numbers such that $p$ does not divide $q - 1$, and let $G$ be a group of order $pq$.
The number $n_q$ divides $p$ and satisfies $n_q \equiv 1 \pmod q$, so $n_q = 1$, because $p < q$.
The number $n_p$ divides $q$ and satisfies $n_p \equiv 1 \pmod p$; since $p \nmid q - 1$, $n_p = 1$.
So $G$ has a [[normal|0CB0DX5BVV]] subgroup $P$ of order $p$ and a normal subgroup $Q$ of order $q$.
They meet only in $e$, so $G$ is the [[internal direct product|0VBZH90R1A]] of $P$ and $Q$.
Thus $G \cong \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z} \cong \mathbb{Z}/pq\mathbb{Z}$, and $G$ is cyclic.
For example, each group of order $15$ is cyclic.
If $p$ divides $q - 1$, there is also a group of order $pq$ that is not cyclic, for example the dihedral group $D_3$ of order $6$.
</document>

<document source="https://math2.tamnd.com/node/0YBVKMYX0W.md" tag="0YBVKMYX0W" kind="statement">
# Identity theorem

## Statement

Let $D \subseteq \mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f, g : D \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.
Let $S = \{z \in D : f(z) = g(z)\}$.
Assume that $S$ has an accumulation point in $D$. This means that there is $z_0 \in D$ such that each disc with center $z_0$ contains a point of $S$ other than $z_0$.
Then $f(z) = g(z)$ for each $z \in D$.

## Remarks

Proof idea: apply the [[isolated zeros theorem|0K83KEJ43P]] to $f - g$. The set of points near which $f - g = 0$ is open, closed in $D$ and not empty, so it is $D$ because $D$ is connected.
For example, if $f$ is entire and $f(1/n) = 0$ for each $n \ge 1$, then $f = 0$.
The connectedness of $D$ is necessary: on the union of two disjoint discs, a function can be $0$ on one disc and $1$ on the other.
The theorem makes [[analytic continuation|0M3YQJ81NB]] unique.

## Explanations

### L1 human

Two holomorphic functions on a domain that agree on a small piece, even on a convergent sequence of points with its limit, agree everywhere.
So a holomorphic function is fixed everywhere by its values on a small disc.
</document>

<document source="https://math2.tamnd.com/node/0YBY3VYX0K.md" tag="0YBY3VYX0K" kind="statement">
# Term-by-term integration

## Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be Riemann integrable for each $n \ge 0$.
If $\sum_{n} f_n$ converges uniformly on $[a, b]$, then its sum is Riemann integrable, and
$$\int_a^b \sum_{n=0}^{\infty} f_n(x) \, dx = \sum_{n=0}^{\infty} \int_a^b f_n(x) \, dx.$$

## Remarks

Apply [[the theorem on sequences|0BWGW68B66]] to the partial sums.
For example, integrating $\frac{1}{1 + t} = \sum (-t)^n$ term by term over $[0, x]$, with $|x| < 1$, gives the Mercator series for $\ln(1 + x)$.
Mathlib states a version with summable sup norms as `intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm`.

## Explanations

### L1 human

We may integrate a uniformly convergent series term by term.
The integral of the sum is the sum of the integrals.
</document>

<document source="https://math2.tamnd.com/node/0YC8JT513J.md" tag="0YC8JT513J" kind="statement">
# Matrix exponential as the solution of a linear system

## Statement

Let $A$ be a real $n \times n$ matrix, and for $t \in \mathbb{R}$ let $e^{t A} = \sum_{k=0}^{\infty} \frac{t^k}{k!} A^k$, a series that converges for each $t$.
Then the map $t \mapsto e^{t A}$ is differentiable on $\mathbb{R}$, and
$$\frac{d}{dt}\, e^{t A} = A\, e^{t A} = e^{t A} A.$$
Consequently, for each $x_0 \in \mathbb{R}^n$, the function $x(t) = e^{t A} x_0$ is the unique solution on $\mathbb{R}$ of the [[initial value problem|075VBY4K4X]] $x' = A x$, $x(0) = x_0$.

## Remarks

Uniqueness: if $x$ is a solution, then $\frac{d}{dt}\left(e^{-tA} x(t)\right) = 0$, so $e^{-tA} x(t) = x_0$.
If $A = P D P^{-1}$ with $D$ diagonal, then $e^{tA} = P e^{tD} P^{-1}$, and $e^{tD}$ is diagonal with the entries $e^{t \lambda_k}$.
The matrix $e^{tA}$ is the [[fundamental matrix|0MQ24RBG93]] of $x' = A x$ with value $I$ at $t = 0$.

## Explanations

### L1 human

For one equation $y' = a y$, the solution is $e^{a t} y_0$.
The matrix exponential is the same formula for a system, with the matrix $A$ in place of the number $a$.
The series is the same as the series of $e^x$.
</document>

<document source="https://math2.tamnd.com/node/0YCDZ7N8VD.md" tag="0YCDZ7N8VD" kind="definition">
# Augmented matrix

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] $Ax = b$ over $F$, with [[coefficient matrix|0HQFZAP3ND]] $A$ of size $m \times n$ and constant vector $b \in F^m$.
The *augmented matrix* of the system is the $m \times (n+1)$ matrix $[A \mid b]$.
Its first $n$ columns are the columns of $A$, and its last column is $b$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the augmented matrix $\left(\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -3 & 0 \end{array}\right)$.
[[Elementary row operations|04XWH28452]] on the augmented matrix do not change the solution set of the system.
More generally, $[A \mid B]$ denotes the matrix $A$ with the columns of a matrix $B$ added on the right.

## Explanations

### L1 human

The augmented matrix is the full system written as a table of numbers.
The vertical bar reminds us where the equals signs were.
We do row reduction on this table instead of rewriting the equations each time.
</document>

<document source="https://math2.tamnd.com/node/0YCG648KDD.md" tag="0YCG648KDD" kind="concept">
# Open map

## Summary

An open map is a function between topological spaces that sends open sets to open sets.
Open maps need not be continuous, and continuous maps need not be open.
</document>

<document source="https://math2.tamnd.com/node/0YCW781Q9D.md" tag="0YCW781Q9D" kind="example">
# Seven Bridges of Königsberg

## Example

The city of Königsberg had four land areas, joined by seven bridges over the river Pregel.
The problem asked for a walk that crosses each bridge exactly once.

Model each land area as a vertex and each bridge as an edge. This gives a connected [[multigraph|01M5T0KDG7]] with $4$ vertices and $7$ edges.
The degrees of the vertices are $5$, $3$, $3$ and $3$.
An [[Eulerian trail|0NNYDD1AVR]] can have at most two vertices of odd degree, by the theorem on [[degrees along an Eulerian trail|0CDVDGWGEG]].
Here all four degrees are odd, so the walk does not exist.

## Remarks

Leonhard Euler solved the problem in 1736. His analysis is often seen as the start of graph theory.
The problem has no solution even if the walk does not have to return to its start.

## Explanations

### L1 human

Each time a walk passes through a land area, it uses two bridges there, one to enter and one to leave.
So a land area in the middle of the walk needs an even number of bridges.
Königsberg had four land areas with an odd number of bridges.
</document>

<document source="https://math2.tamnd.com/node/0YDRV01R8F.md" tag="0YDRV01R8F" kind="statement">
# Universal property of the tensor product

## Statement

Let $F$ be a field, and let $V$, $W$ and $U$ be vector spaces over $F$.
Let $h \colon V \times W \to U$ be a bilinear map.
Then there is exactly one linear map $\tilde h \colon V \otimes W \to U$ such that
$$\tilde h(v \otimes w) = h(v, w) \qquad \text{for all } v \in V \text{ and } w \in W,$$
where $V \otimes W$ is the [[tensor product|0BF2DE2SYK]].

## Remarks

So the bilinear maps $V \times W \to U$ correspond one to one with the [[linear maps|0NRYHJPQG2]] $V \otimes W \to U$.
This property determines $V \otimes W$ up to a unique isomorphism that fixes the vectors $v \otimes w$.
Uniqueness holds because the vectors $v \otimes w$ span $V \otimes W$.

## Explanations

### L1 human

Every bilinear map out of $V \times W$ factors through the tensor product in exactly one way.
So the study of bilinear maps reduces to the study of linear maps.
This property is the usual definition of the tensor product in more abstract settings.
</document>

<document source="https://math2.tamnd.com/node/0YE5868HZB.md" tag="0YE5868HZB" kind="concept">
# Open ball

## Summary

An open ball in a metric space is the set of all points at distance strictly less than a given radius from a given centre.
Open balls are the basic open sets of a metric space.
</document>

<document source="https://math2.tamnd.com/node/0YE597NHGB.md" tag="0YE597NHGB" kind="definition">
# Cayley graph

## Definition

Let $G$ be a [[group|G5S1M9]] and let $S$ be a [[generating set|0MBV2VHMXA]] of $G$.
The Cayley graph $\Gamma(G, S)$ is the directed graph with vertex set $G$.
For each $g \in G$ and $s \in S$, it has an edge with colour $s$ from $g$ to $g s$.

## Remarks

For $G = \mathbb{Z}$ and $S = \{1, -1\}$, the Cayley graph is an infinite path. For $\mathbb{Z}/n\mathbb{Z}$ with a generator and its inverse, it is the cycle $C_n$.
The Cayley graph of a free group with respect to a basis is a tree.
The [[left multiplication action|0K5TWJAGPD]] of $G$ on itself acts on $\Gamma(G, S)$, so each Cayley graph is vertex-transitive.
Arthur Cayley first considered these graphs for finite groups in 1878.

## Explanations

### L1 human

A Cayley graph draws a group as a network.
Each element is a point, and each generator gives an arrow from $g$ to $g s$.
Paths in the graph correspond to words in the generators.
</document>

<document source="https://math2.tamnd.com/node/0YEQG3VBYC.md" tag="0YEQG3VBYC" kind="definition">
# Combinations with repetition

## Definition

Let $X$ be a set and let $k$ be a natural number.
A $k$-combination with repetition of $X$ is a multiset of size $k$ with elements in $X$.
That is a function $m : X \to \mathbb{N}$ with $\sum_{x \in X} m(x) = k$, where $m(x)$ is the multiplicity of $x$.

For natural numbers $n$ and $k$, the multiset coefficient $\left(\!\binom{n}{k}\!\right)$ is the number of $k$-combinations with repetition of an $n$-element set.

## Remarks

The value is $\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}$. This is [[stars and bars|009TEMTQKH]].
For $X = \{a, b\}$ and $k = 2$ there are $3$ multisets: $\{a, a\}$, $\{a, b\}$ and $\{b, b\}$.
Unlike a [[combination|0JZSRWRBZE]], a combination with repetition can use an element more than once.

## Explanations

### L1 human

A combination with repetition is a choice of $k$ items from $n$ kinds. You can take the same kind more than once, and the order does not matter.
An example is a bag of 3 fruits from apples, pears and plums.
</document>

<document source="https://math2.tamnd.com/node/0YEW47VJ0J.md" tag="0YEW47VJ0J" kind="definition">
# Divisor function

## Definition

Let $k$ be a natural number.
The divisor function $\sigma_k$ is the [[arithmetic function|0H2EPRZC0D]] given by
$$\sigma_k(n) = \sum_{d \mid n} d^k,$$
where the sum runs over the [[positive divisors|0FCWA16FZE]] $d$ of $n$.

## Remarks

The case $k = 0$ is the [[number of divisors|0BKJT6WG96]], and the case $k = 1$ is the [[sum of divisors|009QCP5T3Z]].
For example, $\sigma_2(6) = 1 + 4 + 9 + 36 = 50$.
The same formula defines $\sigma_k$ for each real or complex number $k$.
The function $\sigma_k$ is [[multiplicative|04WC36XKV6]].

## Explanations

### L1 human

To compute $\sigma_k(n)$, list the divisors of $n$, raise each one to the power $k$, and add the results.
</document>

<document source="https://math2.tamnd.com/node/0YF0VXKXGB.md" tag="0YF0VXKXGB" kind="definition">
# Jacobian matrix

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, and let $a \in U$.
Suppose that each [[partial derivative|09EGC6F8PT]] $\partial f_i / \partial x_j$ exists at $a$, for $1 \le i \le m$ and $1 \le j \le n$.
The Jacobian matrix of $f$ at $a$ is the $m \times n$ matrix
$$J_f(a) = \begin{pmatrix} \dfrac{\partial f_1}{\partial x_1}(a) & \cdots & \dfrac{\partial f_1}{\partial x_n}(a) \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m}{\partial x_1}(a) & \cdots & \dfrac{\partial f_m}{\partial x_n}(a) \end{pmatrix},$$
whose entry in row $i$ and column $j$ is $\partial f_i / \partial x_j (a)$.
Other notations are $Df(a)$, $f'(a)$ and $\dfrac{\partial (f_1, \dots, f_m)}{\partial (x_1, \dots, x_n)}$.

## Remarks

If $f$ is [[differentiable|0135XZZQ8M]] at $a$, then $Df(a)(h) = J_f(a) \, h$ for each column vector $h \in \mathbb{R}^n$.
Row $i$ of $J_f(a)$ is the transpose of the [[gradient|0ABK9V9Y64]] of $f_i$ at $a$. For $m = 1$, the Jacobian matrix is the gradient written as a row.
For the polar map $f(r, \theta) = (r \cos \theta, r \sin \theta)$, the Jacobian matrix is $\begin{pmatrix} \cos \theta & -r \sin \theta \\ \sin \theta & r \cos \theta \end{pmatrix}$.
For $m = n$, its determinant is the [[Jacobian determinant|0ZCSMYHTBP]].

## Explanations

### L1 human

The Jacobian matrix lists how each output changes with each input: one row for each output, one column for each input.
It is the multivariable form of the derivative. It turns a small step $h$ in the input into the approximate step $J_f(a) h$ in the output.
</document>

<document source="https://math2.tamnd.com/node/0YF6QXXWB5.md" tag="0YF6QXXWB5" kind="concept">
# Similar triangles

## Summary

Two triangles are similar when they have the same shape, but possibly different sizes. Then one is a scaled copy of the other, after a rigid motion and possibly a reflection.
</document>

<document source="https://math2.tamnd.com/node/0YF82GBTND.md" tag="0YF82GBTND" kind="statement">
# Continuity via open sets

## Statement

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be a function.
Then the following two conditions are equivalent:

1. For each $a \in X$ and each $\varepsilon > 0$ there is $\delta > 0$ such that $d_X(x, a) < \delta$ implies $d_Y(f(x), f(a)) < \varepsilon$ for all $x \in X$.
2. For each [[open set|01ZQQDCERP]] $V$ of $Y$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ is open in $X$.

## Remarks

Condition 2 is the definition of a [[continuous map|0SY9ECQ19Q]] between topological spaces. Thus, for metric spaces, the epsilon-delta definition and the topological definition agree.
Proof that 1 implies 2: let $a \in f^{-1}(V)$. Choose $\varepsilon$ with $B(f(a), \varepsilon) \subseteq V$, then $\delta$ from condition 1. Then $B(a, \delta) \subseteq f^{-1}(V)$.
Images of open sets need not be open: the constant map $\mathbb{R} \to \mathbb{R}$, $x \mapsto 0$, is continuous, but the image of $\mathbb{R}$ is $\{0\}$.

## Explanations

### L1 human

The topological form of continuity uses only open sets and does not mention distances.
This form lets us define continuity in spaces that have no metric at all.
</document>

<document source="https://math2.tamnd.com/node/0YG0ZJPTWG.md" tag="0YG0ZJPTWG" kind="example">
# Harmonic oscillator

## Example

Let $m > 0$ and $k > 0$ be constants, and let $\omega_0 = \sqrt{k/m}$.
The equation of the harmonic oscillator is
$$m\, x''(t) + k\, x(t) = 0, \quad \text{that is} \quad x'' + \omega_0^2\, x = 0.$$
It models a mass $m$ on a spring with stiffness $k$, where $x$ is the displacement from the rest position.
Its characteristic roots are $\pm i \omega_0$, so its solutions are $x(t) = c_1 \cos(\omega_0 t) + c_2 \sin(\omega_0 t) = A \cos(\omega_0 t + \varphi)$ with constants $A \ge 0$ and $\varphi$.

## Remarks

Each nonzero solution is periodic with period $2 \pi / \omega_0$. The period does not depend on the amplitude $A$.
The energy $E = \frac{1}{2} m (x')^2 + \frac{1}{2} k x^2$ is constant along each solution. It is a [[first integral|0SS1P1YC9A]].
With friction, the equation becomes the [[damped harmonic oscillator|0RNXVVTJTJ]].

## Explanations

### L1 human

The spring pulls the mass back with a force proportional to the displacement.
The mass overshoots the rest position, and the motion repeats forever as a sine wave.
A stiffer spring or a lighter mass gives a faster oscillation.
</document>

<document source="https://math2.tamnd.com/node/0YGN3VAA0D.md" tag="0YGN3VAA0D" kind="concept">
# Normaliser

## Summary

The normaliser of a subset $S$ of a group $G$ is the set of elements $g$ with $gSg^{-1} = S$.
It is the largest subgroup in which a given subgroup is normal.
</document>

<document source="https://math2.tamnd.com/node/0YH1M56SBN.md" tag="0YH1M56SBN" kind="statement">
# Sum of independent Poisson random variables

## Statement

Let $X_1, \dots, X_n$ be [[independent|0XCE070V8B]] random variables with $X_i \sim \operatorname{Poisson}(\lambda_i)$ and $\lambda_i > 0$. Then
$$X_1 + \dots + X_n \sim \operatorname{Poisson}(\lambda_1 + \dots + \lambda_n).$$

## Remarks

Proof with [[generating functions|0REMT01N36]]: $G_{X_i}(z) = e^{\lambda_i (z - 1)}$, so the sum has $G(z) = e^{(\lambda_1 + \dots + \lambda_n)(z - 1)}$. This is the generating function of $\operatorname{Poisson}(\lambda_1 + \dots + \lambda_n)$.
A direct proof for $n = 2$ uses the binomial theorem: $\sum_{k=0}^{m} e^{-\lambda_1} \frac{\lambda_1^k}{k!} e^{-\lambda_2} \frac{\lambda_2^{m-k}}{(m-k)!} = e^{-(\lambda_1 + \lambda_2)} \frac{(\lambda_1 + \lambda_2)^m}{m!}$.
Conversely, given $X_1 + X_2 = m$, the conditional law of $X_1$ is $\operatorname{Bin}(m, \lambda_1 / (\lambda_1 + \lambda_2))$.

## Explanations

### L1 human

If calls come to two help desks as independent Poisson counts, the total number of calls is again a Poisson count. Its mean is the sum of the two means.
</document>

<document source="https://math2.tamnd.com/node/0YHFMXJJAZ.md" tag="0YHFMXJJAZ" kind="statement">
# Irrationality of the square root of a prime

## Statement

Let $p$ be a prime number.
Then $\sqrt{p}$ is an [[irrational number|080T2TXVQV]].

## Remarks

This generalizes the [[irrationality of the square root of 2|0DZGSDHD92]].
More generally, the square root of a natural number $n$ is irrational unless $n$ is a perfect square. See [[the roots of integers|0XWCKJZRYH]].

## Explanations

### L1 human

The numbers $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{7}$ are all irrational.
The proof for $\sqrt{2}$ works for each prime: if $a^2 = p b^2$, then $p$ divides $a$, and then $p$ also divides $b$.
</document>

<document source="https://math2.tamnd.com/node/0YJ3WRR9RY.md" tag="0YJ3WRR9RY" kind="example">
# Integers modulo n as a quotient group

## Example

Let $n$ be a positive integer.
The subgroup $n\mathbb{Z}$ of the [[additive group of integers|0BSCJA5G53]] is normal, because $\mathbb{Z}$ is abelian.
The [[quotient group|0KYX2N69J0]] $\mathbb{Z}/n\mathbb{Z}$ has the $n$ cosets $0 + n\mathbb{Z}, 1 + n\mathbb{Z}, \ldots, (n-1) + n\mathbb{Z}$.
The operation is $(a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z}$, which is addition modulo $n$.

## Remarks

The coset $a + n\mathbb{Z}$ is the set of integers with the same remainder as $a$ on division by $n$.
Thus $\mathbb{Z}/n\mathbb{Z}$ is the cyclic group of order $n$ of clock arithmetic.
For $n = 0$, the quotient $\mathbb{Z}/0\mathbb{Z}$ is isomorphic to $\mathbb{Z}$.

## Explanations

### L1 human

Two integers are in the same coset of $n\mathbb{Z}$ when their difference is a multiple of $n$.
So the quotient group keeps only the remainder modulo $n$.
Adding cosets is the same as adding remainders and reducing modulo $n$.
</document>

<document source="https://math2.tamnd.com/node/0YK27TJ4WF.md" tag="0YK27TJ4WF" kind="definition">
# Order of a pole

## Definition

Let $c$ be a [[pole|0WPQYW8MTE]] of $f$.
The order of the pole $c$ is the unique integer $m \ge 1$ with the following property. On some punctured disc $0 < |z - c| < \delta$,
$$f(z) = \frac{g(z)}{(z - c)^m},$$
where $g$ is [[holomorphic|0DG0Z72CQD]] on the disc $|z - c| < \delta$ and $g(c) \neq 0$.
A pole of order $1$ is a simple pole.

## Remarks

The order of the pole of $f$ at $c$ is the [[order of the zero|064N002TX4]] of $1/f$ at $c$.
In the Laurent series $\sum a_n (z - c)^n$ of $f$ at $c$, the order is $m$ if and only if $a_{-m} \neq 0$ and $a_n = 0$ for each $n < -m$.
Many authors give a pole of order $m$ the signed order $-m$. Then zeros have positive order and poles have negative order.
For example, $\frac{1}{\sin^2 z}$ has a pole of order $2$ at each point $k\pi$ with $k \in \mathbb{Z}$.

## Explanations

### L1 human

The order of a pole says how fast the function goes to infinity at the point.
Near a pole of order $m$, the function behaves like a constant times $1/(z - c)^m$.
</document>

<document source="https://math2.tamnd.com/node/0YKA9J3XEE.md" tag="0YKA9J3XEE" kind="statement">
# Hilbert's basis theorem

## Statement

Let $R$ be a [[Noetherian|0MVH4Y1CRZ]] [[commutative ring|0GYGHKX834]].
Then the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is Noetherian.

## Remarks

By induction, $R[x_1, \dots, x_n]$ is Noetherian for every $n$. In particular, every ideal of $K[x_1, \dots, x_n]$ for a field $K$ is finitely generated.
The proof studies the ideals of the leading coefficients of the polynomials of each degree in an ideal of $R[x]$.
In algebraic geometry, the theorem says that every set of polynomial equations has the same solutions as a finite subset of it.

## History

David Hilbert proved the theorem in 1890, for polynomials in several variables over a field, in an article on invariant theory.
His proof was not constructive. It did not give a method to find the finite set of generators.

## Explanations

### L1 human

An ideal of polynomials can contain infinitely many polynomials.
The theorem says that a finite list of them always generates the whole ideal.
So every system of polynomial equations can be replaced by a finite one.
</document>

<document source="https://math2.tamnd.com/node/0YKDJBV26S.md" tag="0YKDJBV26S" kind="definition">
# Essential singularity

## Definition

Let $c$ be an [[isolated singularity|0TYB3HM9KE]] of $f$.
The singularity $c$ is essential if it is neither a [[removable singularity|0P18JC72RE]] nor a [[pole|0WPQYW8MTE]] of $f$.

## Remarks

The singularity $c$ is essential if and only if the [[principal part|0NDV5R6Z6X]] of $f$ at $c$ has infinitely many nonzero terms.
Equivalently, $f(z)$ has no limit in $\mathbb{C}$ as $z \to c$, and $|f(z)|$ does not tend to $\infty$.
The standard example is [[exp(1/z) at 0|0NB31CC4SN]]. The function $\sin(1/z)$ also has an essential singularity at $0$.
Near an essential singularity, $f$ comes close to each complex value, by the [[Casorati–Weierstrass theorem|02HHNPC8ED]].

## Explanations

### L1 human

An essential singularity is a bad point that is neither a removable hole nor a point where the function simply goes to infinity.
Near such a point, the function behaves wildly: along different paths it can go to $0$, to infinity, or to any other value.
</document>

<document source="https://math2.tamnd.com/node/0YKY4M3KRQ.md" tag="0YKY4M3KRQ" kind="definition">
# Principal branch

## Definition

Let $F$ be a [[multivalued function|03CPD1HGF3]] on a set $X \subseteq \mathbb{C}$, so that $F(z)$ is a set of complex numbers for each $z \in X$.
A principal branch of $F$ is a function $f$ on $X$ with $f(z) \in F(z)$ for each $z \in X$, chosen by a standard convention. Its value $f(z)$ is the principal value of $F$ at $z$.
For the main functions of complex analysis, the conventions are these. Let $z \neq 0$, and let $\operatorname{Arg} z \in (-\pi, \pi]$ be the [[principal value of the argument|0SQSQP748R]].

- The principal [[logarithm|0X2FY491PF]] is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$, where $\ln$ is the real natural logarithm.
- The principal square root is $\sqrt{z} = e^{\frac{1}{2} \operatorname{Log} z}$.
- For $w \in \mathbb{C}$, the principal [[power|0ZV9A7EV0F]] is $z^w = e^{w \operatorname{Log} z}$.

## Remarks

The imaginary part of $\operatorname{Log} z$ is in $(-\pi, \pi]$. For example, $\operatorname{Log} i = i\pi/2$ and $\operatorname{Log}(-1) = i\pi$.
These principal branches are continuous and holomorphic on $\mathbb{C} \setminus (-\infty, 0]$, see [[the principal logarithm is holomorphic off the branch cut|0XVGJE7Q04]]. They are not continuous at the points of $(-\infty, 0)$.
Some authors use $\operatorname{Arg} z \in [0, 2\pi)$ instead. This gives a different principal branch.

## Explanations

### L1 human

A multivalued function gives many values at a point. The principal branch is the rule that everybody agrees on for picking one value.
For the logarithm, the rule is to take the angle between $-180$ and $180$ degrees.
</document>

<document source="https://math2.tamnd.com/node/0YM6P6C206.md" tag="0YM6P6C206" kind="statement">
# Existence of nth roots of complex numbers

## Statement

Let $z$ be a complex number and let $n$ be a positive integer.
Then there is a complex number $w$ with $w^n = z$.

## Remarks

For $z = r e^{i \theta}$ with $r > 0$, one solution is $w = r^{1/n} e^{i \theta / n}$, by the [[polar form|0BRXFNQ5JV]].
For $z \neq 0$, there are exactly $n$ solutions. They are $w$ times the $n$th [[roots of unity|0E021GF2B8]].
This fact is a special case of the fundamental theorem of algebra.

## Explanations

### L1 human

Every complex number has a square root, a cube root, and so on.
For example, $-1$ has the square roots $i$ and $-i$, and $i$ has the square roots $\pm \frac{\sqrt{2}}{2}(1 + i)$.
To take an $n$th root, take the $n$th root of the length and divide the angle by $n$.
</document>

<document source="https://math2.tamnd.com/node/0YMB1676R1.md" tag="0YMB1676R1" kind="concept">
# Affine span

## Summary

The affine span of a set of points is the smallest [[affine subspace|0PYW26VNT6]] that contains them. For two distinct points it is the line through them.
</document>

<document source="https://math2.tamnd.com/node/0YMP521ZR5.md" tag="0YMP521ZR5" kind="statement">
# Power series are holomorphic in the disc of convergence

## Statement

Let $\sum_{n \ge 0} a_n (z - c)^n$ be a [[power series|0VARVQ9ZW1]] with radius of convergence $R > 0$, and let $D = \{z : |z - c| < R\}$ be its [[disc of convergence|045EWAXY9P]].
Let $f(z) = \sum_{n=0}^{\infty} a_n (z - c)^n$ for $z \in D$.
Then $f$ is [[holomorphic|0DG0Z72CQD]] on $D$, and for each $z \in D$,
$$f'(z) = \sum_{n=1}^{\infty} n a_n (z - c)^{n-1}.$$
The series for $f'$ has the same radius of convergence $R$.

## Remarks

By induction, $f$ has complex derivatives of all orders on $D$, and $a_n = f^{(n)}(c) / n!$. So the power series is the Taylor series of $f$ at $c$.
For example, the [[complex exponential|054VXHMCA3]] has $R = \infty$, so it is [[entire|0G2YJN0H5S]].
The converse is also true: [[holomorphic functions are analytic|0826SGV3R0]].

## Explanations

### L1 human

Inside its disc of convergence, a power series can be differentiated term by term, like a polynomial.
The result is again a power series with the same disc, so you can do it again and again.
</document>

<document source="https://math2.tamnd.com/node/0YMSEZE1VH.md" tag="0YMSEZE1VH" kind="definition">
# Free abelian group

## Definition

Let $A$ be an [[abelian group|082GHNPJ53]], written additively.
A basis of $A$ is a subset $B \subseteq A$ such that each $a \in A$ has exactly one expression $a = k_1 b_1 + \cdots + k_n b_n$ with distinct $b_i \in B$ and nonzero integers $k_i$.
The group $A$ is a free abelian group when it has a basis.

## Remarks

The additive group $\mathbb{Z}$ is free abelian with basis $\{1\}$, and $\mathbb{Z}^2$ is free abelian with basis $\{(1, 0), (0, 1)\}$.
All bases of $A$ have the same cardinality, the rank, and a free abelian group of finite rank $n$ is [[isomorphic|08MG7H8JW5]] to $\mathbb{Z}^n$.
The free abelian group on a set $S$ is the [[abelianisation|0AQZV0905N]] of the [[free group|0KTW26C80J]] $F_S$.
Each subgroup of a free abelian group is free abelian.

## Explanations

### L1 human

A free abelian group is like a vector space over the integers with a basis.
Each element is an integer combination of basis elements in exactly one way.
The standard examples are $\mathbb{Z}^n$ for $n \geq 0$.
</document>

<document source="https://math2.tamnd.com/node/0YMT0D2442.md" tag="0YMT0D2442" kind="concept">
# Irreducible element

## Summary

An irreducible element of an integral domain is a nonzero nonunit that is not a product of two nonunits.
The [[definition of an irreducible element|052SDEPZH7]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0YMYMNQ0QZ.md" tag="0YMYMNQ0QZ" kind="concept">
# Simple group

## Summary

A simple group is a nontrivial group whose only normal subgroups are the trivial subgroup and the group itself.
The finite simple groups are the building blocks of all finite groups.
</document>

<document source="https://math2.tamnd.com/node/0YP7WYXF9V.md" tag="0YP7WYXF9V" kind="definition">
# Direct product of groups

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]].
The direct product $G \times H$ is the set of ordered pairs $(g, h)$ with $g \in G$ and $h \in H$, with the operation $(g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2)$.

## Remarks

The identity is $(e_G, e_H)$, and the inverse of $(g, h)$ is $(g^{-1}, h^{-1})$.
If $G$ and $H$ are finite, then $|G \times H| = |G| \cdot |H|$.
The [[order|0WE2NHZJ7B]] of $(g, h)$ is the least common multiple of the orders of $g$ and $h$, when both are finite.
For example, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ is cyclic of order $6$, but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is not cyclic.

## Explanations

### L1 human

The direct product runs two groups side by side.
An element is a pair, and you multiply the first parts in $G$ and the second parts in $H$.
The two parts never interact.
</document>

<document source="https://math2.tamnd.com/node/0YPRY7T4PK.md" tag="0YPRY7T4PK" kind="definition">
# Flux through a surface

## Definition

Let $S \subseteq \mathbb{R}^3$ be an [[oriented surface|0EKZW08FQ5]] with unit normal field $n$, and let $F$ be a continuous [[vector field|0N67NBKQ5T]] on $S$.
The flux of $F$ through $S$ is the [[surface integral|0DQ8ERJZNS]]
$$\iint_S F \cdot n \, dS.$$

## Remarks

If $r \colon D \to \mathbb{R}^3$ is a regular [[parametrisation|05788J51WZ]] of $S$ with $r_u \times r_v$ pointing in the direction of $n$, then the flux is $\iint_D F\big( r(u, v) \big) \cdot \big( r_u \times r_v \big) \, du \, dv$.
Reversing the orientation of $S$ changes the sign of the flux. The notation $\iint_S F \cdot d\mathbf{S}$, with $d\mathbf{S} = n \, dS$, is common.
Example: for $F(x, y, z) = (x, y, z)$ and the unit sphere with the outward normal $n(p) = p$, we get $F \cdot n = 1$, so the flux is the area $4\pi$.
If $F$ is the velocity field of a fluid, the flux is the volume of fluid that crosses $S$ per unit time. Flow in the direction of $n$ counts as positive.

## Explanations

### L1 human

Flux measures how much of a field goes through a surface.
Only the part of the field perpendicular to the surface counts. Flow along the surface does not cross it.
</document>

<document source="https://math2.tamnd.com/node/0YQ458D7PF.md" tag="0YQ458D7PF" kind="statement">
# Fundamental theorem of contour integration

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be continuous.
Let $F : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] with $F'(z) = f(z)$ for each $z \in U$.
Let $\gamma : [a, b] \to U$ be a continuous and piecewise continuously differentiable path.
Then the [[contour integral|00FWNP9TXY]] of $f$ along $\gamma$ is
$$\int_\gamma f(z) \, dz = F(\gamma(b)) - F(\gamma(a)).$$
In particular, $\int_\gamma f(z) \, dz = 0$ if $\gamma$ is closed.

## Remarks

The function $F$ is a primitive of $f$ on $U$. The integral depends only on the end points of $\gamma$.
For each integer $n \neq -1$, $z^n$ has the primitive $z^{n+1}/(n+1)$ on $\mathbb{C} \setminus \{0\}$. So $\oint_\gamma z^n \, dz = 0$ for each closed path $\gamma$ that does not pass through $0$.
The function $1/z$ has no primitive on $\mathbb{C} \setminus \{0\}$, because its [[integral over the unit circle|0SXZ68HV6F]] is $2 \pi i \neq 0$.

## Explanations

### L1 human

This is the complex version of $\int_a^b F'(t) \, dt = F(b) - F(a)$.
If you know a function whose derivative is $f$, you get the integral along any path from its values at the two ends.
</document>

<document source="https://math2.tamnd.com/node/0YQPZB5Q1G.md" tag="0YQPZB5Q1G" kind="concept">
# Orbit

## Summary

The orbit of a point $x$ under a group action is the set of all points to which elements of the group move $x$.
</document>

<document source="https://math2.tamnd.com/node/0YQQYET8YQ.md" tag="0YQQYET8YQ" kind="statement">
# Intersecting chords theorem

## Statement

Let $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the chords $AC$ and $BD$ meet at a point $S$ that is inside both segments.
Then
$$d(A, S)\, d(S, C) = d(B, S)\, d(S, D).$$

## Remarks

Both products are equal to $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius. This is minus the [[power|02E9VT1RVQ]] of $S$.
A proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem.
The theorem is Proposition 35 of Book III of Euclid's Elements.

## Explanations

### L1 human

Draw two chords of a circle that cross each other. Each chord is cut into two pieces. The product of the two pieces is the same for both chords.
</document>

<document source="https://math2.tamnd.com/node/0YQSGNYE3K.md" tag="0YQSGNYE3K" kind="concept">
# Matrix of a linear map

## Summary

The matrix of a linear map in two bases is a matrix with one column for each basis vector of the domain.
Its column $j$ is the coordinate vector of the image of basis vector $j$.
</document>

<document source="https://math2.tamnd.com/node/0YRG241HQQ.md" tag="0YRG241HQQ" kind="statement">
# Pascal's rule

## Statement

Let $n$ and $k$ be natural numbers.
Then
$$\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

## Remarks

For $n = 4$ and $k = 1$ it gives $\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10$.
Combinatorial proof: fix one element $x$ of an $(n+1)$-element set. A $(k+1)$-subset contains $x$, and then it has $k$ other elements, or it omits $x$.
The rule builds [[Pascal's triangle|05DME4TZ3K]] row by row.

## Explanations

### L1 human

Each number in Pascal's triangle is the sum of the two numbers above it.
This rule is the same fact: a team of $k+1$ from $n+1$ people either includes a fixed person or does not.
</document>

<document source="https://math2.tamnd.com/node/0YS40B38JD.md" tag="0YS40B38JD" kind="statement">
# Closed balls are closed

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r \ge 0$.
Then the [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: let $y \notin \bar{B}(x, r)$ and let $\varepsilon = d(x, y) - r > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \ge d(x, y) - d(y, z) > r$ by the [[triangle inequality|01M6CW9VM6]]. Thus the complement of $\bar{B}(x, r)$ is open.
For $r = 0$, this shows that each one-point set $\{x\}$ is closed in a metric space.

## Explanations

### L1 human

A point outside the closed ball is at some positive distance beyond its edge.
A small ball about that point stays outside, so the outside is open.
</document>

<document source="https://math2.tamnd.com/node/0YS8RX87F4.md" tag="0YS8RX87F4" kind="definition">
# Line integral of a vector field

## Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $F \colon C \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
For a partition $a = t_0 < t_1 < \dots < t_k = b$ and points $\tau_i \in [t_{i-1}, t_i]$, consider the sum $\sum_{i=1}^{k} F(\gamma(\tau_i)) \cdot \big( \gamma(t_i) - \gamma(t_{i-1}) \big)$.
The line integral of $F$ along $\gamma$ is the limit of these sums as $\max_i (t_i - t_{i-1}) \to 0$. It is written
$$\int_\gamma F \cdot d\mathbf{r} \quad \text{or} \quad \int_\gamma F_1 \, dx_1 + \dots + F_n \, dx_n.$$

## Remarks

The limit exists, and it does not depend on the choice of the points $\tau_i$.
For a $C^1$ curve, $\int_\gamma F \cdot d\mathbf{r} = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, dt$, by the [[parametric formula|0W3W1VYWBP]].
For a regular curve with unit tangent vector $T$, $\int_\gamma F \cdot d\mathbf{r} = \int_\gamma (F \cdot T) \, ds$. So it is the line integral of the scalar field $F \cdot T$.
The value depends on the direction of travel. [[Reversing the path|0FSCXYG856]] changes its sign.

## Explanations

### L1 human

At each small step along the curve, take the dot product of the field with the step.
This counts only the part of the field that pushes along the curve, and the line integral is the total of these contributions.
</document>

<document source="https://math2.tamnd.com/node/0YSCSX5YZC.md" tag="0YSCSX5YZC" kind="concept">
# Ordinal number

## Summary

Ordinal numbers extend the natural numbers to measure the length of well-ordered sets, including infinite ones.
Each well-ordered set is order isomorphic to exactly one ordinal.
</document>

<document source="https://math2.tamnd.com/node/0YSE802YCQ.md" tag="0YSE802YCQ" kind="definition">
# Group action

## Definition

Let $G$ be a [[group|G5S1M9]] with identity $e$, and let $X$ be a set.
A (left) action of $G$ on $X$ is a map $G \times X \to X$, $(g, x) \mapsto g \cdot x$, with these two properties:

1. $e \cdot x = x$ for each $x \in X$.
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.

## Remarks

For each $g \in G$, the map $x \mapsto g \cdot x$ is a bijection of $X$, with inverse $x \mapsto g^{-1} \cdot x$.
The map that sends $g$ to this bijection is a [[group homomorphism|0JYTG5K9Y4]] from $G$ to the [[symmetric group|0515A6D2Y1]] $\operatorname{Sym}(X)$.
Conversely, each homomorphism $G \to \operatorname{Sym}(X)$ gives an action of $G$ on $X$.
A right action is a map $X \times G \to X$ with $x \cdot e = x$ and $(x \cdot g) \cdot h = x \cdot (gh)$.

## Explanations

### L1 human

A group action lets the elements of a group move the points of a set.
The identity does not move any point, and moving by $h$ and then by $g$ is the same as moving by $gh$.
For example, the rotations of a square move the four corners of the square.
</document>

<document source="https://math2.tamnd.com/node/0YSRR1JTKZ.md" tag="0YSRR1JTKZ" kind="counterexample">
# x^n converges pointwise but not uniformly on [0, 1]

## Counterexample

Let $f_n(x) = x^n$ on $[0, 1]$. Let $f(x) = 0$ for $0 \le x < 1$ and $f(1) = 1$.
For $0 \le x < 1$, $x^n \to 0$ by the [[geometric sequence limit|0MNM54WB99]]. So $f_n \to f$ [[pointwise|056FSKG32J]] on $[0, 1]$.
But $\sup_{0 \le x < 1} |x^n - 0| = 1$ for every $n$. So the convergence is not [[uniform|08FPABDQ4C]].

## Remarks

Each $f_n$ is continuous, but $f$ is not. This also shows the failure, by the [[uniform limit theorem|0X9XT05JVC]].
On $[0, r]$ with $0 < r < 1$, the convergence is uniform, since $\sup_{[0, r]} x^n = r^n \to 0$.

## Explanations

### L1 human

Near $x = 1$, the values $x^n$ stay close to $1$ for a long time before they drop to $0$.
So no single $n$ makes all the values small at once.
</document>

<document source="https://math2.tamnd.com/node/0YT586TKXF.md" tag="0YT586TKXF" kind="definition">
# Cyclic subgroup generated by an element

## Definition

Let $G$ be a [[group|G5S1M9]] and let $g \in G$.
The cyclic subgroup generated by $g$ is the set $\langle g \rangle = \{g^k : k \in \mathbb{Z}\}$ of all integer powers of $g$.

## Remarks

By [[the laws of exponents|00DRK90Z6K]], $g^j (g^k)^{-1} = g^{j-k}$, thus $\langle g \rangle$ is a [[subgroup|0X1JXT47GA]] of $G$.
It is the [[subgroup generated by|0RP7VWQ0YN]] the set $\{g\}$.
In additive notation, $\langle g \rangle = \{kg : k \in \mathbb{Z}\}$, for example $\langle 3 \rangle = 3\mathbb{Z}$ in $(\mathbb{Z}, +)$.

## Explanations

### L1 human

Take one element and all its powers, positive and negative, and the identity.
These powers form a subgroup, the smallest one that contains the element.
In the integers with addition, the cyclic subgroup generated by $3$ is the set of multiples of $3$.
</document>

<document source="https://math2.tamnd.com/node/0YTCMFB4HC.md" tag="0YTCMFB4HC" kind="concept">
# Critical point

## Summary

A critical point of a real function is a point where the derivative is zero or does not exist. All interior local extrema of the function occur at critical points. The node [[critical point|0GEQ42PTGB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YTJ3RN3GE.md" tag="0YTJ3RN3GE" kind="concept">
# Dyck word

## Summary

A Dyck word is a balanced word of opening and closing brackets. Dyck words form the Dyck language of formal language theory, and those of semilength $n$ are counted by the Catalan number $C_n$. The node [[Dyck word|0R9JR0465V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YV2CKP2AV.md" tag="0YV2CKP2AV" kind="concept">
# Root of unity

## Summary

A root of unity is a number whose power with some positive integer exponent is $1$.
In $\mathbb{C}$, the $n$th roots of unity are the $n$ points $e^{2 \pi i k / n}$ on the unit circle. They are the vertices of a regular $n$-gon.
</document>

<document source="https://math2.tamnd.com/node/0YVDC5PYDW.md" tag="0YVDC5PYDW" kind="definition">
# Conditional probability distribution

## Definition

Let $X$ and $Y$ be random variables on one probability space.
Discrete case: let $X$ and $Y$ be discrete and let $x$ satisfy $P(X = x) > 0$. The conditional distribution of $Y$ given $X = x$ has the [[probability mass function|0WF4EJP91W]]
$$p_{Y \mid X}(y \mid x) = P(Y = y \mid X = x) = \frac{P(X = x, \, Y = y)}{P(X = x)}.$$
Continuous case: let $(X, Y)$ have a joint density $f_{X, Y}$, and let $f_X$ be the [[marginal|0AXP2Z5N9H]] density of $X$. For each $x$ with $f_X(x) > 0$, the conditional distribution of $Y$ given $X = x$ has the [[density|04YG690N5V]]
$$f_{Y \mid X}(y \mid x) = \frac{f_{X, Y}(x, y)}{f_X(x)}.$$

## Remarks

In the discrete case, $p_{Y \mid X}(\cdot \mid x)$ is the law of $Y$ under the [[conditional probability|0FJB06AD2P]] $P(\cdot \mid X = x)$.
In both cases, the joint law is the marginal times the conditional: $f_{X, Y}(x, y) = f_X(x) \, f_{Y \mid X}(y \mid x)$.
If $X$ and $Y$ are independent, the conditional distribution of $Y$ given $X = x$ is the law of $Y$, for each such $x$.
In general measure theory, one defines it as a regular conditional distribution. This is a probability kernel $\kappa$ with $P(X \in A, Y \in B) = \int_A \kappa(x, B) \, \mu_X(dx)$.

## Explanations

### L1 human

The conditional distribution answers the question: "If I know $X$, how is $Y$ distributed?"
For a person of height $180$ cm, it describes the possible weights of that person and their chances.
</document>

<document source="https://math2.tamnd.com/node/0YVKNB7V5F.md" tag="0YVKNB7V5F" kind="concept">
# Metric topology

## Summary

In the metric topology of a metric space, the open sets are the sets that contain a small ball about each of their points.
It turns each metric space into a topological space.
</document>

<document source="https://math2.tamnd.com/node/0YVQKBNV2R.md" tag="0YVQKBNV2R" kind="concept">
# Skewness

## Summary

The skewness of a distribution measures its asymmetry about the mean.
The node [[skewness|0GQ3H7VW3V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YVRNEDSVG.md" tag="0YVRNEDSVG" kind="definition">
# Floor function

## Definition

Let $x$ be a real number.
The floor of $x$, written $\lfloor x \rfloor$, is the greatest [[integer|0C4JS9RF5M]] $n$ with $n \le x$.
The floor function is the function $\mathbb{R} \to \mathbb{Z}$, $x \mapsto \lfloor x \rfloor$.

## Remarks

The greatest such integer exists because the [[reals are Archimedean|0A3GWZ86AY]] and because of the [[well-ordering principle|01BX15HSRR]].
For each integer $n$, $\lfloor n \rfloor = n$. For example, $\lfloor \pi \rfloor = 3$ and $\lfloor -0.5 \rfloor = -1$.
The [[characterization of the floor|0R9B0B5458]] gives an equivalent definition.

## Explanations

### L1 human

The floor of a number rounds it down to a whole number.
On the number line, it is the nearest integer at the number or to its left.
For negative numbers, take care: the floor of $-1.2$ is $-2$, not $-1$.
</document>

<document source="https://math2.tamnd.com/node/0YVVN6R715.md" tag="0YVVN6R715" kind="statement">
# Homotopy invariance of line integrals of closed forms

## Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $F \colon U \to \mathbb{R}^3$ be a vector field of class $C^1$ with $\nabla \times F = 0$ on $U$.
Let $c_0, c_1 \colon [0, 1] \to U$ be piecewise smooth [[closed curves|0D032NSZKY]].
Suppose that there is a piecewise smooth map $H \colon [0, 1] \times [0, 1] \to U$ with $H(t, 0) = c_0(t)$, $H(t, 1) = c_1(t)$ and $H(0, s) = H(1, s)$ for all $s, t \in [0, 1]$.
Then
$$\oint_{c_0} F \cdot d\mathbf{r} = \oint_{c_1} F \cdot d\mathbf{r}.$$

## Remarks

Proof: apply [[Stokes' theorem|0AKRKVHW94]] to the surface $H([0, 1]^2)$. The curl term is zero, and the integrals along the curves $s \mapsto H(0, s)$ and $s \mapsto H(1, s)$ cancel.
If $c_1$ is a constant loop, then $\oint_{c_0} F \cdot d\mathbf{r} = 0$.
On a [[simply connected|075C629XTH]] domain, each loop deforms to a point, and a smoothing argument gives a piecewise smooth $H$. So curl-free $C^1$ fields on such domains are [[conservative|05R9NF2J49]].
The condition $\nabla \times F = 0$ means that the 1-form $F_1 \, dx + F_2 \, dy + F_3 \, dz$ is closed. This explains the name of the theorem.

## Explanations

### L1 human

For a field with no swirl, the work around a loop does not change when you slide and bend the loop continuously inside the domain.
A loop that can shrink to a point therefore has zero work.
</document>

<document source="https://math2.tamnd.com/node/0YVVXT3A0B.md" tag="0YVVXT3A0B" kind="definition">
# Number of distinct prime factors

## Definition

For a positive integer $n$, $\omega(n)$ is the number of distinct [[prime numbers|0SN0MAEKYX]] that divide $n$.
So if $n = p_1^{e_1} \cdots p_k^{e_k}$ with distinct primes $p_i$ and $e_i \ge 1$, then $\omega(n) = k$.

## Remarks

For example, $\omega(1) = 0$, $\omega(12) = 2$ and $\omega(30) = 3$.
The function $\omega$ is additive: $\omega(mn) = \omega(m) + \omega(n)$ for coprime $m$ and $n$.
Compare with the [[number of prime factors with multiplicity|08420828EY]] $\Omega(n)$. We have $\omega(n) \le \Omega(n)$, with equality exactly for squarefree $n$.

## Explanations

### L1 human

The function $\omega(n)$ counts the different primes in $n$ and ignores repeats.
For $360 = 2^3 \cdot 3^2 \cdot 5$, the different primes are $2$, $3$ and $5$, so $\omega(360) = 3$.
</document>

<document source="https://math2.tamnd.com/node/0YWDAZPRPQ.md" tag="0YWDAZPRPQ" kind="statement">
# Decomposition of a permutation into disjoint cycles

## Statement

Let $X$ be a finite set and let $\sigma \in \operatorname{Sym}(X)$.
Then $\sigma$ is a product of cycles of length at least $2$ whose sets of moved points are pairwise disjoint.
This product is unique up to the order of the factors.

## Remarks

The cycles come from the orbits of $\langle \sigma \rangle$ on $X$: each orbit with more than one point gives one cycle.
The order of the factors does not matter, because [[disjoint cycles commute|07E6D8R4D6]].
For example, the permutation $1 \mapsto 2$, $2 \mapsto 1$, $3 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 3$ in $S_5$ is $(1\ 2)(3\ 4\ 5)$.

## Explanations

### L1 human

Start at a point and follow the permutation until you return.
This gives one cycle; then start again at a point that you did not visit yet.
When all points are visited, the permutation is the product of the cycles that you found.
</document>

<document source="https://math2.tamnd.com/node/0YWZJEABB2.md" tag="0YWZJEABB2" kind="concept">
# Catalan numbers

## Summary

The Catalan numbers $C_n$ form a sequence that counts many recursive structures, such as Dyck words, binary trees and bracketings. They satisfy a convolution recurrence and the formula $C_n = \binom{2n}{n}/(n+1)$. The node [[Catalan numbers|092DRBFKMG]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0YXEHVA93R.md" tag="0YXEHVA93R" kind="definition">
# Tangent plane to a graph

## Definition

Let $U \subseteq \mathbb{R}^2$ be open, let $f \colon U \to \mathbb{R}$, and let $(a, b) \in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].
The tangent plane to the [[graph|0JBJYT69ZV]] $z = f(x, y)$ at the point $(a, b, f(a, b))$ is the plane in $\mathbb{R}^3$ with the equation
$$z = f(a, b) + \frac{\partial f}{\partial x}(a, b)\,(x - a) + \frac{\partial f}{\partial y}(a, b)\,(y - b).$$
More generally, let $U \subseteq \mathbb{R}^n$ be open and let $f$ be differentiable at $a \in U$. The tangent hyperplane to the graph at $(a, f(a))$ is the set of points $(x, z) \in \mathbb{R}^n \times \mathbb{R}$ with $z = f(a) + \nabla f(a) \cdot (x - a)$.

## Remarks

The vector $\big(\partial f / \partial x(a, b), \, \partial f / \partial y(a, b), \, -1\big)$ is normal to the tangent plane.
For $f(x, y) = x^2 + y^2$ at $(1, 1)$, the tangent plane is $z = 2 + 2(x - 1) + 2(y - 1)$, that is, $z = 2x + 2y - 2$.
The tangent plane exists only where $f$ is differentiable. The existence of the [[partial derivatives|09EGC6F8PT]] alone gives a plane, but this plane need not touch the graph well.

## Explanations

### L1 human

Put a flat sheet on top of a smooth hill so that it touches the hill at one point and does not tilt against it.
That sheet is the tangent plane. Near the point of contact, the hill and the sheet are almost the same.
</document>

<document source="https://math2.tamnd.com/node/0YXEQWRGP0.md" tag="0YXEQWRGP0" kind="definition">
# Choice function

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]] of nonempty sets.
A choice function on $X$ is a [[function|0XHDTE5YMZ]] $c \colon X \to \bigcup X$ such that $c(S) \in S$ for each $S \in X$.

## Remarks

For a finite set $X$, a choice function exists without any axiom of choice.
For the set of nonempty subsets of $\mathbb{N}$, the function that sends each subset to its least element is a choice function.
The [[axiom of choice|0HTGB13FE2]] says that each set of nonempty sets has a choice function.

## Explanations

### L1 human

Think of $X$ as a collection of boxes, none empty.
A choice function names one object in each box.
Sometimes a rule gives it, such as "take the smallest number".
</document>

<document source="https://math2.tamnd.com/node/0YY06KP3ZQ.md" tag="0YY06KP3ZQ" kind="statement">
# Quadratic reciprocity for the Jacobi symbol

## Statement

Let $m$ and $n$ be coprime positive odd integers.
Then
$$\left(\frac{m}{n}\right) \left(\frac{n}{m}\right) = (-1)^{\frac{m-1}{2} \cdot \frac{n-1}{2}},$$
where the symbols are [[Jacobi symbols|0XDB5AW7M1]].

## Remarks

The supplements also extend: for a positive odd $n$, $\left(\frac{-1}{n}\right) = (-1)^{(n-1)/2}$ and $\left(\frac{2}{n}\right) = (-1)^{(n^2-1)/8}$.
The proof uses the [[law of quadratic reciprocity|085MJS6907]] for each pair of prime factors.
With this law, one computes $\left(\frac{a}{n}\right)$ by an algorithm like the [[Euclidean algorithm|0TRNT4X7J0]], with no factorization.

## Explanations

### L1 human

The Jacobi symbol obeys the same reciprocity law as the Legendre symbol.
So you can flip the symbol, reduce the top number modulo the bottom number, and repeat.
This gives a fast way to compute Legendre symbols for large primes.
</document>

<document source="https://math2.tamnd.com/node/0YY8VX28JR.md" tag="0YY8VX28JR" kind="concept">
# Moore–Penrose inverse

## Summary

The Moore–Penrose inverse of a matrix is the unique matrix that satisfies four equations of Penrose.
It gives the least squares solution of least norm of a linear system.
</document>

<document source="https://math2.tamnd.com/node/0YYMY95FWS.md" tag="0YYMY95FWS" kind="definition">
# Matrix of a linear map

## Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$ with [[bases|0KCQ9Y3QKA]] $B = (b_1, \dots, b_n)$ and $C = (c_1, \dots, c_m)$.
Let $T \colon V \to W$ be a [[linear map|0NRYHJPQG2]].
The *matrix of $T$* in the bases $B$ and $C$ is the $m \times n$ matrix $[T]_{C,B} = (a_{ij})$ defined by
$$T(b_j) = \sum_{i=1}^{m} a_{ij} c_i \qquad (1 \le j \le n).$$
Equivalently, column $j$ of $[T]_{C,B}$ is the [[coordinate vector|04CYQHN6MC]] $[T(b_j)]_C$.

## Remarks

For each $v \in V$, $[T(v)]_C = [T]_{C,B} [v]_B$.
The map $T \mapsto [T]_{C,B}$ is a linear isomorphism $\Hom_F(V, W) \to M_{m \times n}(F)$.
For the rotation of $\R^2$ by an angle $\theta$, in the standard basis, the matrix is $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$.
Mathlib writes this map as `LinearMap.toMatrix`.

## Explanations

### L1 human

To write a linear map as a matrix, apply the map to each basis vector.
Write the coordinates of each image as a column.
These columns, side by side, are the matrix.
</document>

<document source="https://math2.tamnd.com/node/0YYT36R4YD.md" tag="0YYT36R4YD" kind="concept">
# Conformal map

## Summary

A conformal map is a map that preserves angles and their orientation at each point, but not necessarily lengths, see the [[definition|0VHNKP1N3K]].
In the plane, the conformal maps on an open set are the [[holomorphic|0DG0Z72CQD]] functions with a derivative that is never zero.
</document>

<document source="https://math2.tamnd.com/node/0YYWCNTPWR.md" tag="0YYWCNTPWR" kind="definition">
# Quotient group

## Definition

Let $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.
The quotient group $G/N$ is the set of [[left cosets|03DEH43F4H]] $\{gN : g \in G\}$ with the operation $(aN)(bN) = (ab)N$.

## Remarks

The operation is well defined because $N$ is normal: if $aN = a'N$ and $bN = b'N$, then $abN = a'b'N$.
The identity of $G/N$ is $N = eN$, and the inverse of $aN$ is $a^{-1}N$.
When $G$ is finite, $|G/N| = [G : N] = |G| / |N|$.
The map $g \mapsto gN$ is a surjective homomorphism from $G$ to $G/N$ with kernel $N$.

## Explanations

### L1 human

A quotient group treats all elements of one coset as one element.
For example, the integers modulo $2$ come from the integers when you treat all even numbers as one element and all odd numbers as another.
The normality of $N$ makes sure that the product of two cosets does not depend on the chosen elements.
</document>

<document source="https://math2.tamnd.com/node/0YZ3T5Q7XN.md" tag="0YZ3T5Q7XN" kind="example">
# Determinant as a group homomorphism

## Example

Let $n$ be a positive integer and let $\mathrm{GL}_n(\mathbb{R})$ be the group of invertible $n \times n$ real matrices under multiplication.
The determinant is a map $\det \colon \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times$ to the group of nonzero real numbers under multiplication.
It is a [[group homomorphism|0JYTG5K9Y4]], because $\det(AB) = \det(A) \det(B)$ for all $A, B$.

## Remarks

The [[kernel|0W88CEGQP1]] of $\det$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$ of matrices of determinant $1$, so this group is normal in $\mathrm{GL}_n(\mathbb{R})$.
The determinant is surjective onto $\mathbb{R}^\times$, so $\mathrm{GL}_n(\mathbb{R}) / \mathrm{SL}_n(\mathbb{R}) \cong \mathbb{R}^\times$.
The same holds over each field $F$, with $\mathrm{GL}_n(F)$, $\mathrm{SL}_n(F)$ and $F^\times$.

## Explanations

### L1 human

The determinant turns a product of matrices into a product of numbers.
That is exactly what a homomorphism does.
The matrices that it sends to $1$ form an important subgroup, the special linear group.
</document>

<document source="https://math2.tamnd.com/node/0YZXNE3HFR.md" tag="0YZXNE3HFR" kind="statement">
# Mercator series

## Statement

For every real $x$ with $-1 < x \le 1$,
$$\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots.$$

## Remarks

For $|x| < 1$, integrate the [[geometric series|01X5CVV1Q5]] $\frac{1}{1 + t} = \sum (-t)^n$ term by term from $0$ to $x$.
At $x = 1$ the series converges by the alternating series test. [[Abel's theorem|0S4EMDQS2F]] gives $\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \cdots$.
Mathlib states the form $\sum_{n \ge 0} \frac{x^{n+1}}{n+1} = -\ln(1 - x)$ as `Real.hasSum_pow_div_log_of_abs_lt_one`.

## History

Johannes Hudde found the series in 1656 and Isaac Newton in 1665, but neither of them published it.
Nicholas Mercator found it independently and used it in his book Logarithmotechnia of 1668.

## Explanations

### L1 human

The logarithm of $1 + x$ is a power series in $x$, for $x$ in $(-1, 1]$.
At $x = 1$ it gives the alternating harmonic series, with sum $\ln 2$.
</document>

<document source="https://math2.tamnd.com/node/0Z05VCBQ7Y.md" tag="0Z05VCBQ7Y" kind="concept">
# Cycle

## Summary

A cycle in a graph is a closed walk that visits each of its vertices once and returns to its start. Cycles are central in graph theory, for example in the definitions of trees and of bipartite graphs. The node [[cycle|07CT9RDY7Q]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Z0NE3J6JX.md" tag="0Z0NE3J6JX" kind="concept">
# Pell's equation

## Summary

Pell's equation is the Diophantine equation $x^2 - d y^2 = 1$, where $d$ is a positive integer that is not a perfect square.
The [[definition of Pell's equation|0AF3DANJ76]] gives the precise form.
</document>

<document source="https://math2.tamnd.com/node/0Z1G2F8DGY.md" tag="0Z1G2F8DGY" kind="statement">
# Power series of complex cosine

## Statement

For each $z \in \mathbb{C}$, the series $\sum_{n \ge 0} (-1)^n z^{2n} / (2n)!$ converges absolutely, and
$$\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots,$$
where $\cos$ is the [[complex cosine|0A7YDKRNR6]].

## Remarks

Proof: put $iz$ and $-iz$ in the series of the [[complex exponential|054VXHMCA3]] and add. The terms with odd powers cancel, and $i^{2n} = (-1)^n$.
In the same way, $\sin z = \sum_{n \ge 0} (-1)^n z^{2n+1} / (2n+1)!$ for each $z \in \mathbb{C}$.
The series shows that $\cos$ is an even function: $\cos(-z) = \cos z$.

## Explanations

### L1 human

The series of $\cos$ has only the even powers of $e^z$'s series, with signs that alternate.
For example, $\cos(0.1) \approx 1 - 0.005 = 0.995$ from the first two terms.
</document>

<document source="https://math2.tamnd.com/node/0Z1Y7JHGBR.md" tag="0Z1Y7JHGBR" kind="definition">
# Diffeomorphism

## Definition

Let $U, V \subseteq \mathbb{R}^n$ be open, and let $k \ge 1$ be an integer or $k = \infty$.
A map $f \colon U \to V$ is a $C^k$ diffeomorphism if $f$ is a bijection of [[class C^k|0TCBZMEYGN]] on $U$. In addition, $f^{-1} \colon V \to U$ must be of class $C^k$ on $V$.
The sets $U$ and $V$ are $C^k$ diffeomorphic if there is a $C^k$ diffeomorphism from $U$ to $V$.

## Remarks

If $f$ is a diffeomorphism, the chain rule applied to $f^{-1} \circ f = \mathrm{id}$ shows that $Df(x)$ is invertible for each $x \in U$.
Conversely, a $C^k$ bijection $f \colon U \to V$ with $Df(x)$ invertible at each point is a $C^k$ diffeomorphism, by the [[inverse function theorem|0K9HSJFCMJ]].
The map $x \mapsto x^3$ is a $C^\infty$ bijection of $\mathbb{R}$, but it is not a diffeomorphism, because its inverse $x \mapsto x^{1/3}$ is not differentiable at $0$.
Polar coordinates $(r, \theta) \mapsto (r \cos \theta, r \sin \theta)$ give a $C^\infty$ diffeomorphism from $(0, \infty) \times (0, 2\pi)$ onto the plane without the closed half-line $\{(x, 0) : x \ge 0\}$.

## Explanations

### L1 human

A diffeomorphism is a smooth change of coordinates that can be undone smoothly.
It may stretch and bend space, but it does not tear, fold or crush it.
</document>

<document source="https://math2.tamnd.com/node/0Z22VQH17V.md" tag="0Z22VQH17V" kind="definition">
# Stirling numbers of the first kind

## Definition

Let $n$ and $k$ be natural numbers.
The unsigned Stirling number of the first kind $c(n, k)$ is the number of [[permutations|0BQMWCNS8S]] of an $n$-element set that have exactly $k$ disjoint cycles.
Fixed points count as cycles of length $1$.

## Remarks

Other notations are $\left[ {n \atop k} \right]$ and $|s(n, k)|$. The signed numbers are $s(n, k) = (-1)^{n-k} c(n, k)$.
Example: $c(3, 2) = 3$. The permutations of $\{1, 2, 3\}$ with two cycles are the three transpositions.
The numbers satisfy $\sum_k c(n, k) = n!$ and $x (x + 1) \cdots (x + n - 1) = \sum_k c(n, k)\, x^k$.
The values satisfy a [[recurrence|0C5BFX94J9]] in $n$ and $k$.

## Explanations

### L1 human

Write a permutation as a set of cycles, for example $(1\,2)(3)$.
$c(n, k)$ counts the permutations of $n$ elements that split into exactly $k$ cycles.
</document>

<document source="https://math2.tamnd.com/node/0Z2P41SVDP.md" tag="0Z2P41SVDP" kind="concept">
# Separable space

## Summary

A separable space is a topological space with a countable dense subset.
</document>

<document source="https://math2.tamnd.com/node/0Z367CGNRM.md" tag="0Z367CGNRM" kind="definition">
# Minimum spanning tree

## Definition

Let $G$ be a finite [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]] with edge set $E$, and let $w : E \to \mathbb{R}$ be a weight function.
The weight of a subgraph is the sum of the weights of its edges.
A minimum spanning tree of $(G, w)$ is a [[spanning tree|0FQ5B8S1ZC]] of $G$ whose weight is the smallest among all spanning trees of $G$.

## Remarks

A minimum spanning tree exists, because a finite graph has finitely many spanning trees. It is unique when all weights are distinct.
Kruskal's algorithm and Prim's algorithm find a minimum spanning tree with greedy choices.

## Explanations

### L1 human

Suppose that each road between towns has a cost to build.
A minimum spanning tree is the cheapest set of roads that links all towns.
</document>

<document source="https://math2.tamnd.com/node/0Z3FEWZ8HA.md" tag="0Z3FEWZ8HA" kind="statement">
# Diagonalisation of a quadratic form

## Statement

Let $F$ be a field of characteristic not $2$, and let $V$ be a vector space over $F$ of finite dimension $n$.
Let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Then $V$ has a basis $(e_1, \dots, e_n)$ and there are $a_1, \dots, a_n \in F$ such that, for all $x_1, \dots, x_n \in F$,
$$Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.$$

## Remarks

The basis is [[orthogonal|01TXN7ETH2]] for the associated symmetric bilinear form, and $a_i = Q(e_i)$.
In matrix terms, each symmetric matrix $A$ over $F$ has an invertible $P$ such that $P^T A P$ is diagonal.
The numbers $a_i$ are not unique; for example, each $a_i$ can be multiplied by a nonzero square.
Over $\R$, the orthogonal diagonalisation of the [[principal axis theorem|0PR5P9Z4KM]] is a special case.

## Explanations

### L1 human

Every quadratic form becomes a sum of squares with coefficients after a good change of variables.
For example, $x^2 + 2xy = (x + y)^2 - y^2$.
With $u = x + y$ and $v = y$, the form becomes $u^2 - v^2$.
</document>

<document source="https://math2.tamnd.com/node/0Z3GDY89T5.md" tag="0Z3GDY89T5" kind="statement">
# Decomposition into generalised eigenspaces

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map.
Assume that the characteristic polynomial of $T$ splits over $F$ as $p_T(t) = (t - \lambda_1)^{m_1} \cdots (t - \lambda_k)^{m_k}$ with pairwise distinct $\lambda_i$.
Then
$$V = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k},$$
where $V_{\lambda_i}$ is the [[generalised eigenspace|0AJ035ZTZE]] of $\lambda_i$, and $\dim V_{\lambda_i} = m_i$.

## Remarks

The hypothesis holds for every $T$ when $F$ is algebraically closed, for example $F = \C$.
Each $V_{\lambda_i}$ is $T$-invariant, and $T - \lambda_i \id$ is nilpotent on it.
A proof uses the coprime polynomials $(t - \lambda_i)^{m_i}$ and the [[Cayley–Hamilton theorem|0F765KFC2J]].
This [[direct sum|0XYYT5KNTX]] decomposition is the first step towards the [[Jordan normal form|085HQ5G5TE]].

## Explanations

### L1 human

Over $\C$, the space splits into pieces, one for each eigenvalue.
On each piece, the map is $\lambda$ times the identity plus a nilpotent part.
So a hard problem breaks into simpler problems, one for each eigenvalue.
</document>

<document source="https://math2.tamnd.com/node/0Z3YKRQZ48.md" tag="0Z3YKRQZ48" kind="definition">
# Linear first-order equation

## Definition

Let $I$ be an open [[interval|0Q1D85B8G6]] and let $p, q : I \to \mathbb{R}$ be continuous functions.
The linear first-order equation with coefficient $p$ and right side $q$ is the equation
$$y' + p(x)\, y = q(x)$$
for an unknown differentiable function $y : I \to \mathbb{R}$.
It is homogeneous if $q = 0$ on $I$, and inhomogeneous otherwise.

## Remarks

The [[solution formula|06B9W45XV0]] gives each solution explicitly, with an [[integrating factor|01THH3Y3YP]].
The homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$, where $P$ is an antiderivative of $p$ and $C$ is a constant.
Many authors write the equation as $y' = a(x) y + b(x)$, with $a = -p$ and $b = q$.

## Explanations

### L1 human

The equation is linear because $y$ and $y'$ appear only to the first power, with coefficients that depend on $x$.
There is no product such as $y\, y'$ and no function such as $\sin y$.
This structure is the reason why there is a formula for all the solutions.
</document>

<document source="https://math2.tamnd.com/node/0Z409N8RDG.md" tag="0Z409N8RDG" kind="concept">
# Intermediate field

## Summary

An intermediate field of an extension $L/K$ is a subfield of $L$ that contains $K$.
The [[definition of an intermediate field|02NBKVQFRQ]] states it.
</document>

<document source="https://math2.tamnd.com/node/0Z41C17TGP.md" tag="0Z41C17TGP" kind="statement">
# A function with nonnegative second derivative is convex

## Statement

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be twice differentiable.
If $f''(x) \ge 0$ for all $x \in I$, then $f$ is [[convex|0VV0HSW1ZS]].

## Remarks

The derivative $f'$ has a nonnegative derivative, so $f'$ is [[increasing|026BJBQNGJ]]. Then [[convexity follows|005JT76HCK]].
If $f'' > 0$ on $I$, then $f$ is strictly convex. The converse fails: $x^4$ is strictly convex, but its second derivative at $0$ is $0$.
Mathlib states it as `convexOn_of_deriv2_nonneg`.

## Explanations

### L1 human

A nonnegative [[second derivative|0Y9D2FJRQ4]] means that the slope never decreases.
So the graph bends upward everywhere.
</document>

<document source="https://math2.tamnd.com/node/0Z424HK8CQ.md" tag="0Z424HK8CQ" kind="definition">
# Kernel of a homomorphism

## Definition

Let $G$ and $H$ be [[groups|G5S1M9]], let $e_H$ be the identity element of $H$, and let $\varphi \colon G \to H$ be a [[group homomorphism|0JYTG5K9Y4]].
The kernel of $\varphi$ is the set $\ker \varphi = \{g \in G : \varphi(g) = e_H\}$.

## Remarks

The kernel is a [[normal subgroup|0CB0DX5BVV]] of $G$.
The kernel of the map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, $k \mapsto k + n\mathbb{Z}$, is $n\mathbb{Z}$.
The kernel of the determinant map from $\mathrm{GL}_n(\mathbb{R})$ to $\mathbb{R}^\times$ is the special linear group $\mathrm{SL}_n(\mathbb{R})$.

## Explanations

### L1 human

The kernel is the part of the group that the homomorphism forgets: everything in it goes to the identity.
If the kernel is only the identity, the homomorphism forgets nothing, and it is injective.
For the map that sends an integer to its remainder modulo $3$, the kernel is the set of multiples of $3$.
</document>

<document source="https://math2.tamnd.com/node/0Z47M1B2VS.md" tag="0Z47M1B2VS" kind="statement">
# Burnside's lemma

## Statement

Let a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a finite set $X$.
For $g \in G$, let $X^g = \{x \in X : g \cdot x = x\}$.
Then the number of [[orbits|0YQPZB5Q1G]] is $|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$.

## Remarks

The proof counts the pairs $(g, x)$ with $g \cdot x = x$ in two ways, and uses $|G_x| = |G| / |G \cdot x|$.
For example, the rotation group $\mathbb{Z}/4\mathbb{Z}$ of a square acts on the $2^4 = 16$ colourings of the corners with two colours.
The fixed colourings number $16$, $2$, $4$ and $2$, so there are $(16 + 2 + 4 + 2)/4 = 6$ colourings up to rotation.

## History

William Burnside stated and proved the lemma in his 1897 book on finite groups, and he attributed it to Frobenius, 1887.
Cauchy already knew the formula in 1845.
For this reason, the lemma is sometimes called "the lemma that is not Burnside's".

## Explanations

### L1 human

The number of orbits is the average number of points that a group element fixes.
This lets you count objects up to symmetry without listing them.
You count the fixed objects for each symmetry, add, and divide by the number of symmetries.
</document>

<document source="https://math2.tamnd.com/node/0Z49NTR1DZ.md" tag="0Z49NTR1DZ" kind="definition">
# Symmetric difference

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The symmetric difference of $A$ and $B$ is the set
$$A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A).$$
An object $x$ is an element of $A \mathbin{\triangle} B$ when $x$ is an element of exactly one of $A$ and $B$.

## Remarks

Another notation is $A \ominus B$.
Example: $\{1, 2, 3\} \mathbin{\triangle} \{2, 3, 4\} = \{1, 4\}$.
Also $A \mathbin{\triangle} B = (A \cup B) \setminus (A \cap B)$.
The [[power set|0BC7KX51Z8]] of a set is a [[group|G5S1M9]] under symmetric difference, with identity $\varnothing$.

## Explanations

### L1 human

The symmetric difference is the set version of "exclusive or".
It keeps what is in one set or the other, but not what is in both.
</document>

<document source="https://math2.tamnd.com/node/0Z49YH6ZWY.md" tag="0Z49YH6ZWY" kind="concept">
# Taylor series

## Summary

The Taylor series of a smooth function at a point is the power series built from all the derivatives of the function at that point. For analytic functions, it converges to the function near the point. The node [[Taylor series|0225QPWRRE]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0Z4BWARC38.md" tag="0Z4BWARC38" kind="statement">
# Determinant of the identity matrix

## Statement

Let $K$ be a commutative ring and let $n \ge 1$.
Then $\det I_n = 1$, where $I_n$ is the [[identity matrix|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 3 of the definition of the determinant.
It fixes the scale: the unit cube has volume $1$.
More generally, the determinant of a diagonal matrix is the product of its diagonal entries.

## Explanations

### L1 human

The identity matrix leaves every vector where it is.
It does not change any volume.
So its determinant is $1$.
</document>

<document source="https://math2.tamnd.com/node/0Z4V98VN5E.md" tag="0Z4V98VN5E" kind="definition">
# Proper metric space

## Definition

A [[metric space|063W5JMY85]] $(X, d)$ is proper when each [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$, with $x \in X$ and $r \ge 0$, is [[compact|00R0X1WGPX]].

## Remarks

In a proper metric space, a subset is compact if and only if it is closed and bounded. Thus the [[Heine–Borel theorem|07PFYV65MB]] says that $\mathbb{R}^n$ is proper.
Each proper metric space is complete and [[locally compact|0EQMWQ4WMJ]].
An infinite set with the [[discrete metric|0NA4FK7B8S]] is not proper: the closed ball of radius $1$ is the whole set, which is not compact.

## Explanations

### L1 human

A proper metric space is a space where the Heine–Borel test for compactness works: closed and bounded is enough.
Euclidean spaces are proper, but many infinite-dimensional spaces are not.
</document>

<document source="https://math2.tamnd.com/node/0Z57EC6YX3.md" tag="0Z57EC6YX3" kind="definition">
# Product topology

## Definition

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be [[topological spaces|0CNQXVBHCX]].
The product topology on the [[Cartesian product|0C81YBVTWH]] $X \times Y$ is the topology with [[basis|06EX0FHBTF]]
$$\{ U \times V : U \in \tau_X, \ V \in \tau_Y \}.$$

## Remarks

This family is a basis of a topology, because $(U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2)$.
A set $W \subseteq X \times Y$ is open if and only if for each $(x, y) \in W$ there are open $U \ni x$ and $V \ni y$ with $U \times V \subseteq W$.
For metric spaces, it is the metric topology of the [[product metric|044D4AQV8V]]. The product topology on $\mathbb{R} \times \mathbb{R}$ is the usual topology of $\mathbb{R}^2$.
The [[projections are continuous|0PM88Z9W4M]]. For products of any family of spaces, see the [[product topology on an arbitrary product|04S9HX3GAP]].

## Explanations

### L1 human

The open sets of $X \times Y$ are built from open rectangles $U \times V$.
In the plane, open rectangles and open discs give the same topology.
</document>

<document source="https://math2.tamnd.com/node/0Z5905Q73D.md" tag="0Z5905Q73D" kind="concept">
# Alternating group

## Summary

The alternating group $A_n$ is the group of the even permutations of $n$ symbols.
It is a normal subgroup of index $2$ in $S_n$ for $n \geq 2$.
</document>

<document source="https://math2.tamnd.com/node/0Z5R9TVWQH.md" tag="0Z5R9TVWQH" kind="definition">
# Interval of convergence

## Definition

Let $\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].
Its interval of convergence is the set of real numbers $x$ for which the series converges.

## Remarks

Let $R$ be the [[radius of convergence|0660QX1VPR]]. The interval of convergence contains $(c - R, c + R)$ and lies in $[c - R, c + R]$.
Each endpoint must be checked on its own. For $\sum x^n / n$, the interval is $[-1, 1)$.
For $\sum x^n$ it is $(-1, 1)$, and for $\sum x^n / n^2$ it is $[-1, 1]$.

## Explanations

### L1 human

The radius of convergence fixes the interval up to its two endpoints.
At each endpoint the series may converge or diverge, so the interval may be open, closed, or half open.
</document>

<document source="https://math2.tamnd.com/node/0Z7HDDXX3X.md" tag="0Z7HDDXX3X" kind="definition">
# Greatest element

## Definition

Let $(P, \le)$ be a [[partially ordered set|011C29Q1S1]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $P$.
An element $g \in S$ is the greatest element of $S$ when $s \le g$ for each $s \in S$.

## Remarks

Another name is maximum. By antisymmetry, a set has at most one greatest element.
A greatest element is a [[maximal element|0TZEEFTNCC]] and an [[upper bound|0H5J02P8S9]] of $S$ that lies in $S$.
The interval $[0, 1)$ in $\mathbb{R}$ has no greatest element.

## Explanations

### L1 human

The greatest element beats every other element of the set.
In $\{1, 2, 3, 6\}$ under divisibility, $6$ is the greatest element, because each element divides $6$.
</document>

<document source="https://math2.tamnd.com/node/0Z83DHFMYM.md" tag="0Z83DHFMYM" kind="concept">
# Computing the inverse by row reduction

## Summary

Row reduction computes the inverse of a square matrix, or shows that no inverse exists.
Let $A$ be an $n \times n$ matrix over a field $F$, and form the $n \times 2n$ [[augmented matrix|0T6MWGZSXA]] $[A \mid I_n]$.
Apply [[Gauss–Jordan elimination|098D6FQN3N]] to it.
If the left block becomes $I_n$, then the result is $[I_n \mid A^{-1}]$, and the right block is the inverse.
If the left block gets a zero row, then $A$ is not [[invertible|0EB9N0S33C]].

## Remarks

The method works because each row operation is left multiplication by an [[elementary matrix|01REXW3WWE]].
If $E_k \cdots E_1 A = I_n$, then $E_k \cdots E_1 = A^{-1}$, and the same product changes $I_n$ into $A^{-1}$.
For $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, the method gives $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$.

## Explanations

### L1 human

We write the identity matrix next to $A$ and row reduce the whole wide matrix.
When the left half becomes the identity, the right half has become $A^{-1}$.
The same row operations that turn $A$ into $I$ turn $I$ into $A^{-1}$.
</document>

<document source="https://math2.tamnd.com/node/0Z87BHAFTB.md" tag="0Z87BHAFTB" kind="definition">
# Matching

## Definition

Let $G$ be a [[graph|0JD5C82QMB]].
A matching in $G$ is a set $M$ of edges of $G$ such that no two distinct edges of $M$ have a common end.

## Remarks

A vertex is matched if it is an end of an edge of $M$. Otherwise it is unmatched.
A maximal matching is not contained in a larger matching. A maximum matching has the largest number of edges, the matching number $\nu(G)$.
Every maximum matching is maximal, but the converse is false: in the [[path graph|075WR6W3Y3]] $P_4$, the middle edge alone is a maximal matching.

## Explanations

### L1 human

A matching pairs some vertices along edges, and no vertex is in two pairs.
For example, it can pair workers with jobs that they can do, one job per worker.
</document>

<document source="https://math2.tamnd.com/node/0Z8ATBPM5W.md" tag="0Z8ATBPM5W" kind="counterexample">
# The complex conjugate is not holomorphic

## Counterexample

Let $f : \mathbb{C} \to \mathbb{C}$ be the [[complex conjugate|0BJV24C4SY]] map, $f(z) = \bar z$.
The function $f$ is continuous, but it is not [[complex differentiable|0D1C9Y1EXA]] at any point $z_0 \in \mathbb{C}$.
Thus $f$ is not [[holomorphic|0DG0Z72CQD]] on any nonempty open set.
Proof: the difference quotient at $z_0$ is
$$\frac{f(z_0 + h) - f(z_0)}{h} = \frac{\bar h}{h}.$$
It is $1$ for each real $h \neq 0$ and $-1$ for each purely imaginary $h \neq 0$. So it has no limit as $h \to 0$.

## Remarks

The real and imaginary parts $u = x$ and $v = -y$ are differentiable, but $u_x = 1$ and $v_y = -1$. So the [[Cauchy–Riemann equations|0T71SSYZ39]] fail at each point.
As a map of $\mathbb{R}^2$, $f$ is the reflection in the real axis. It is real linear but not complex linear.
For the same reason, $\operatorname{Re} z$, $\operatorname{Im} z$ and $|z|$ are not holomorphic on any nonempty open set.

## Explanations

### L1 human

As a map of the plane, conjugation is a mirror, and it is very smooth.
But a complex derivative must be the same in each direction, and conjugation fixes the real direction and reverses the imaginary direction.
So the quotient gives $1$ in one direction and $-1$ in the other, and there is no complex derivative.
</document>

<document source="https://math2.tamnd.com/node/0Z8CSYVHDJ.md" tag="0Z8CSYVHDJ" kind="statement">
# Local homeomorphism from the inverse function theorem

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}^n$ be of class $C^1$ with $Df(x)$ invertible for each $x \in U$.
Then each $a \in U$ has an open neighbourhood $V \subseteq U$ such that $f(V)$ is open and $f$ restricted to $V$ is a homeomorphism onto $f(V)$.
In particular, $f$ is an open map: $f(O)$ is open for each open set $O \subseteq U$.

## Remarks

This follows from the [[inverse function theorem|0K9HSJFCMJ]], since a $C^1$ inverse is [[continuous|0S9EK5MQPE]].
For each $a$, the hypothesis is needed only at $a$: if $Df(a)$ is invertible, $f$ is a homeomorphism from a neighbourhood of $a$ onto an open set.
A local homeomorphism need not be injective. The map $(x, y) \mapsto (e^x \cos y, e^x \sin y)$ is a local homeomorphism from $\mathbb{R}^2$ onto $\mathbb{R}^2 \setminus \{0\}$, but it is not injective.

## Explanations

### L1 human

Where the derivative is invertible, the map neither folds nor crushes space in a small region.
So small open sets go to open sets, and the map can be undone continuously on each small region.
</document>

<document source="https://math2.tamnd.com/node/0Z8FM54P86.md" tag="0Z8FM54P86" kind="statement">
# Countable union of countable sets is countable

## Statement

Assume the [[axiom of choice|0HTGB13FE2]].
Let $I$ be a [[countable|0ZG61TFBES]] set and let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]] such that each $A_i$ is countable.
Then the [[union|05GK32H72H]] $\bigcup_{i \in I} A_i$ is countable.

## Remarks

Proof: choose for each $i$ a surjection from $\mathbb{N}$ onto $A_i$, when $A_i$ is nonempty. Together they give a surjection from a subset of $\mathbb{N} \times \mathbb{N}$ onto the union, and $\mathbb{N} \times \mathbb{N}$ is countable.
The proof uses the axiom of countable choice to choose the surjections. Without any choice, the statement cannot be proved in ZF.
When the enumerations of the $A_i$ are given, no choice is needed.

## Explanations

### L1 human

List each set $A_i$ as a row of a grid.
Then list the grid diagonal by diagonal.
Each element of the union appears in the list.
</document>

<document source="https://math2.tamnd.com/node/0Z8PSSAVBA.md" tag="0Z8PSSAVBA" kind="definition">
# Probability-generating function

## Definition

Let $X$ be a random variable with values in $\{0, 1, 2, \dots\}$ and with [[mass function|0WF4EJP91W]] $p(k) = P(X = k)$.
The probability-generating function of $X$ is
$$G_X(z) = E\left[z^X\right] = \sum_{k=0}^{\infty} p(k) \, z^k,$$
for each complex $z$ with $|z| \le 1$.

## Remarks

$G_X$ is the [[generating function|0HH6JX5WZZ]] of the sequence $p(0), p(1), \dots$. The series converges absolutely for $|z| \le 1$, and $G_X(1) = 1$.
$G_X$ determines the law of $X$: $P(X = k) = G_X^{(k)}(0)/k!$.
If $E[X] < \infty$, then $E[X] = G_X'(1)$, with the left derivative at $1$. Also $E[X(X - 1)] = G_X''(1)$ when it is finite.
For $X \sim \operatorname{Poisson}(\lambda)$, $G_X(z) = e^{\lambda(z - 1)}$. For $X \sim \operatorname{Bin}(n, p)$, $G_X(z) = (1 - p + pz)^n$.

## Explanations

### L1 human

The probability-generating function packs all the probabilities of a count into one function.
Derivatives of this function at $0$ give back the probabilities, and derivatives at $1$ give the moments.
</document>

<document source="https://math2.tamnd.com/node/0Z962ECCXR.md" tag="0Z962ECCXR" kind="concept">
# Trial division

## Summary

Trial division is the simplest method to factor an integer $n > 1$ or to test whether it is prime.
It divides $n$ by the integers $d = 2, 3, 4, \dots$ in turn, and it stops at the first $d$ that divides $n$, or when $d^2 > n$.
If no $d$ with $d^2 \le n$ divides $n$, then $n$ is prime.

## Remarks

The method is correct because [[a composite number has a prime factor at most its square root|0EB6BBMQMC]].
It is enough to test the primes $d$, but a test of all integers $d$ is also correct.
For a number $n$ with $k$ digits, the method can need about $\sqrt{n} \approx 10^{k/2}$ divisions. Thus it is slow for large $n$.

## Explanations

### L1 human

To test whether $91$ is prime, try to divide it by $2$, $3$, $5$ and $7$. You can stop after $9$, because $10^2 > 91$.
The division by $7$ works: $91 = 7 \cdot 13$. So $91$ is not prime.
</document>

<document source="https://math2.tamnd.com/node/0Z9JB71BS0.md" tag="0Z9JB71BS0" kind="statement">
# Constant multiple rule

## Statement

Let $f$ be a real function on an open interval $I$, differentiable at $a \in I$, and let $c$ be a real number.
Then $c f$ is differentiable at $a$, and $(c f)'(a) = c f'(a)$.

## Remarks

It is the case $\beta = 0$ of the [[linearity of differentiation|0M0TKFRX2A]].
For example, $(5 x^2)' = 10 x$.
Mathlib states it as `deriv_const_mul`.

## Explanations

### L1 human

Stretching a graph vertically by a factor $c$ multiplies all its slopes by $c$.
</document>

<document source="https://math2.tamnd.com/node/0Z9N1ER0W0.md" tag="0Z9N1ER0W0" kind="statement">
# Dimension of the space of linear maps

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then the [[space of linear maps|0X9MBPP2Z3]] $\Hom_F(V, W)$ is finite-dimensional, and
$$\dim \Hom_F(V, W) = \dim V \cdot \dim W.$$

## Remarks

With bases of $V$ and $W$, the map that sends $T$ to its matrix is an isomorphism $\Hom_F(V, W) \cong M_{m \times n}(F)$, with $n = \dim V$ and $m = \dim W$.
The matrices $E_{ij}$ form a basis of $M_{m \times n}(F)$, so its dimension is $mn$.
In particular, the [[dual space|07JBSZE0V4]] of $V$ has dimension $\dim V$.

## Explanations

### L1 human

A linear map from an $n$-dimensional space to an $m$-dimensional space is an $m \times n$ matrix.
A matrix of this size has $mn$ free entries.
So the space of these maps has dimension $mn$.
</document>

<document source="https://math2.tamnd.com/node/0ZA4AX4MW3.md" tag="0ZA4AX4MW3" kind="statement">
# Cauchy criterion for sequences

## Statement

Let $(a_n)$ be a real sequence.
Then $(a_n)$ converges if and only if $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].

## Remarks

One direction is [[convergent sequences are Cauchy|0QY87R9CY3]].
For the other direction, a Cauchy sequence is [[bounded|0G6D8E7CTA]], so it has a convergent subsequence by the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. A Cauchy sequence with a convergent subsequence converges to the same limit.
The criterion says that $\mathbb{R}$ is a complete metric space. It fails in $\mathbb{Q}$.
Its value is that it proves convergence without a candidate for the limit.

## Explanations

### L1 human

To show that a sequence has a limit, you often need to know the limit first.
The Cauchy criterion avoids this. It is enough to show that the terms become close to each other.
</document>

<document source="https://math2.tamnd.com/node/0ZAF9KHDDF.md" tag="0ZAF9KHDDF" kind="concept">
# Greatest common divisor in an integral domain

## Summary

A greatest common divisor of two elements of an integral domain is a common divisor that every common divisor divides.
It is unique up to associates when it exists.
The [[definition of a greatest common divisor|0F7KNJ1ZYN]] gives the conditions.
</document>

<document source="https://math2.tamnd.com/node/0ZAZZ34WQS.md" tag="0ZAZZ34WQS" kind="concept">
# Memorylessness

## Summary

A random waiting time $X \ge 0$ is memoryless if, for all $s, t \ge 0$ with $P(X > s) > 0$,
$$P(X > s + t \mid X > s) = P(X > t).$$
So the time already waited does not change the law of the remaining waiting time.
Among the distributions on $[0, \infty)$ with $P(X > t) > 0$ for each $t$, only the [[exponential distributions|028Z8WC5Y9]] are memoryless. Among the distributions on $\{0, 1, 2, \dots\}$, with $s$ and $t$ integers, only the [[geometric distributions|0XCVV5Q500]] are memoryless.

## Remarks

For $X \sim \operatorname{Exp}(\lambda)$, $P(X > s + t) = e^{-\lambda(s + t)} = P(X > s) \, P(X > t)$. This is the memoryless property.
The proof of the characterization uses that $G(t) = P(X > t)$ satisfies $G(s + t) = G(s) \, G(t)$ and is right-continuous. The only such functions with values in $(0, 1]$ are $G(t) = e^{-\lambda t}$ with $\lambda \ge 0$.

## Explanations

### L1 human

A memoryless waiting time does not "age".
If a bus comes at exponential times and you have already waited ten minutes, your expected further wait is the same as when you arrived.
</document>

<document source="https://math2.tamnd.com/node/0ZB2JG1M9C.md" tag="0ZB2JG1M9C" kind="concept">
# Nilpotent element

## Summary

A nilpotent element of a ring is an element that has a power equal to zero.
The [[definition of a nilpotent element|0N9R5NN6TW]] states this precisely.
</document>

<document source="https://math2.tamnd.com/node/0ZB3M68NYQ.md" tag="0ZB3M68NYQ" kind="concept">
# Pairwise independence

## Summary

A family of events, or of random variables, is pairwise independent when any two of its members are independent.
It is weaker than mutual independence. The node [[pairwise independence|0N57D1XY6T]] gives the precise definition for events.
</document>

<document source="https://math2.tamnd.com/node/0ZB7J071WQ.md" tag="0ZB7J071WQ" kind="statement">
# A complex number times its conjugate

## Statement

Let $z$ be a complex number, let $\bar{z}$ be its [[conjugate|0BJV24C4SY]] and let $|z|$ be its [[modulus|0KW2X0TFBX]].
Then
$$z \bar{z} = |z|^2.$$
In particular, $z \bar{z}$ is a nonnegative real number.

## Remarks

With $z = a + b i$: $(a + b i)(a - b i) = a^2 - (b i)^2 = a^2 + b^2$.
This identity gives the formula for the [[inverse of a complex number|0FQRAV5SKH]].
It also gives a quick proof that the [[modulus is multiplicative|0HYNZE940T]].

## Explanations

### L1 human

When you multiply a complex number by its conjugate, the imaginary parts cancel.
What remains is the square of the length of the number, for example $(3 + 4 i)(3 - 4 i) = 9 + 16 = 25$.
</document>

<document source="https://math2.tamnd.com/node/0ZB84QCDKR.md" tag="0ZB84QCDKR" kind="definition">
# Bernoulli equation

## Definition

Let $I$ be an open interval, let $P, Q : I \to \mathbb{R}$ be continuous functions, and let $n$ be a real number.
A Bernoulli equation is an [[ordinary differential equation|0P97V20RW7]] of the form
$$y' + P(x)\, y = Q(x)\, y^{n}$$
for an unknown function $y$ on a subinterval of $I$. If $n$ is not an integer, the function must satisfy $y > 0$.

## Remarks

For $n = 0$ and $n = 1$ the equation is [[linear|0Z3YKRQZ48]]. Some authors exclude these two values.
For $n \ne 1$, the substitution $v = y^{1 - n}$ changes the equation, on the set where $y \ne 0$, into the linear equation $v' + (1 - n) P(x)\, v = (1 - n) Q(x)$.
For $n > 0$, the function $y = 0$ is also a solution.
The [[logistic equation|0CC4VC7KR3]] is a Bernoulli equation with $n = 2$.

## Explanations

### L1 human

A Bernoulli equation is a linear equation with an extra power of $y$ on the right side.
It is not linear, but a change of the unknown makes it linear.
Then the [[solution formula|06B9W45XV0]] for linear equations solves it.
</document>

<document source="https://math2.tamnd.com/node/0ZC78099PA.md" tag="0ZC78099PA" kind="statement">
# Gradient points in the direction of steepest ascent

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}$ be differentiable at $a \in U$, and suppose that $\nabla f(a) \ne 0$.
Then for each unit vector $u \in \mathbb{R}^n$, the [[directional derivative|0Y7PVW4Q44]] satisfies $-\|\nabla f(a)\| \le D_u f(a) \le \|\nabla f(a)\|$.
The upper bound is attained if and only if $u = \nabla f(a) / \|\nabla f(a)\|$, and the lower bound if and only if $u = -\nabla f(a) / \|\nabla f(a)\|$.

## Remarks

The proof combines $D_u f(a) = \nabla f(a) \cdot u$, from the statement that [[the derivative gives the directional derivatives|0A6PMR20N4]], with the Cauchy–Schwarz inequality $|w \cdot u| \le \|w\| \, \|u\|$ and its case of equality.
So the [[gradient|0NS2QYVC4Y]] gives the direction of fastest increase, and its norm is the fastest rate of increase per unit of distance.
The method of gradient descent uses the opposite direction $-\nabla f(a)$ to look for a minimum of $f$.

## Explanations

### L1 human

If you stand on a hillside and want to climb as fast as possible, walk in the direction of the gradient.
If you walk at right angles to the gradient, your height does not change at first: you follow a contour line.
</document>

<document source="https://math2.tamnd.com/node/0ZC7FYE0NZ.md" tag="0ZC7FYE0NZ" kind="statement">
# Orbits partition the set

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.
The relation $x \sim y$ if and only if $y = g \cdot x$ for some $g \in G$ is an equivalence relation on $X$.
Its equivalence classes are the [[orbits|0YQPZB5Q1G]], so each point of $X$ lies in exactly one orbit.

## Remarks

Reflexivity uses $e \cdot x = x$, symmetry uses $x = g^{-1} \cdot (g \cdot x)$, and transitivity uses $h \cdot (g \cdot x) = (hg) \cdot x$.
If $X$ is finite, then $|X|$ is the sum of the sizes of the distinct orbits.
The [[cosets|03DEH43F4H]] of a subgroup and the [[conjugacy classes|0DEBXAJ5SA]] of a group are examples of orbits.

## Explanations

### L1 human

The group action cuts the set into separate pieces, the orbits.
Inside one piece, each point can reach each other point.
No point can reach a point of another piece.
</document>

<document source="https://math2.tamnd.com/node/0ZC9YTXZRY.md" tag="0ZC9YTXZRY" kind="concept">
# Rank of a matrix

## Summary

The rank of a matrix is the dimension of the span of its columns, which equals the dimension of the span of its rows.
</document>

<document source="https://math2.tamnd.com/node/0ZCAXY9563.md" tag="0ZCAXY9563" kind="definition">
# Ascending chain condition

## Definition

Let $(P, \le)$ be a partially ordered set.
The set $P$ satisfies the ascending chain condition if every sequence $a_1 \le a_2 \le a_3 \le \dots$ of elements of $P$ is eventually constant.
That is, there is an integer $N$ with $a_n = a_N$ for all $n \ge N$.

## Remarks

Equivalently, every nonempty subset of $P$ has a maximal element. This equivalence uses the axiom of dependent choice.
The set $\mathbb{N}$ with its usual order does not satisfy the condition. With the reverse order, it does.
For the set of the ideals of a ring, ordered by inclusion, the condition defines [[Noetherian rings|0MVH4Y1CRZ]].
The ideals of $\mathbb{Z}$ satisfy the condition: a chain $(a_1) \subseteq (a_2) \subseteq \dots$ means $a_2 \mid a_1$, $a_3 \mid a_2$, and so on.

## Explanations

### L1 human

The condition says that you cannot climb forever.
Every increasing chain stops growing after a finite number of steps.
For the ideals of the integers, a strictly increasing chain like $(12) \subset (6) \subset (3) \subset (1)$ must end.
</document>

<document source="https://math2.tamnd.com/node/0ZCK8DQYN1.md" tag="0ZCK8DQYN1" kind="concept">
# Closed set

## Summary

A closed set in a topological space is a set whose complement is open.
In a metric space, the closed sets are the sets that contain the limits of all their convergent sequences.
</document>

<document source="https://math2.tamnd.com/node/0ZCSMYHTBP.md" tag="0ZCSMYHTBP" kind="concept">
# Jacobian determinant

## Summary

The Jacobian determinant of a map from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^n$ at a point is the determinant of its [[Jacobian matrix|047NDTEPCV]] there.
Its absolute value is the factor by which the map scales small volumes near the point. Its sign tells whether the map keeps or reverses orientation.
</document>

<document source="https://math2.tamnd.com/node/0ZCWPNZTQ3.md" tag="0ZCWPNZTQ3" kind="concept">
# Companion matrix

## Summary

The companion matrix of a monic polynomial is a square matrix with ones just below the diagonal.
Its last column holds the negated coefficients, and all other entries are zero.
Its characteristic polynomial is the given polynomial.
</document>

<document source="https://math2.tamnd.com/node/0ZCY0Z80B6.md" tag="0ZCY0Z80B6" kind="concept">
# Neighbourhood

## Summary

The neighbourhood of a vertex in a graph is the set of vertices adjacent to it. Many local properties of graphs, such as degrees, are defined through neighbourhoods. The node [[neighbourhood|01T3TMGTTE]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZCZK72B45.md" tag="0ZCZK72B45" kind="concept">
# Tangent vector of a curve

## Summary

The tangent vector of a differentiable [[parametrised curve|0G7WM8DX4P]] at a parameter value is the derivative of the curve there.
If the curve is the path of a moving point, the tangent vector is its velocity, and its length is the speed.
</document>

<document source="https://math2.tamnd.com/node/0ZD2CF6ZS0.md" tag="0ZD2CF6ZS0" kind="concept">
# Dual basis

## Summary

The dual basis of a basis $b_1, \dots, b_n$ is the family of linear forms $f_1, \dots, f_n$ with $f_i(b_j) = 1$ when $i = j$ and $0$ otherwise.
</document>

<document source="https://math2.tamnd.com/node/0ZDDNBQSC4.md" tag="0ZDDNBQSC4" kind="definition">
# Angle at a point

## Definition

Let $A$, $B$ and $C$ be points of a [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ with $A \neq B$ and $C \neq B$.
The angle $\angle ABC$ is the [[angle between the vectors|0ZEHZY85E6]] $A - B$ and $C - B$. Thus it is the unique real number $\theta$ with $0 \leq \theta \leq \pi$ such that
$$\cos \theta = \frac{(A - B) \cdot (C - B)}{\|A - B\| \, \|C - B\|}.$$
The point $B$ is the vertex of the angle.

## Remarks

The angle is symmetric: $\angle ABC = \angle CBA$.
The angle is $\pi$ if and only if $B$ lies strictly between $A$ and $C$ on a line. It is $0$ if and only if $A$ and $C$ are on the same ray from $B$.
Mathlib's `EuclideanGeometry.angle` uses the same formula for all points. It gives $\pi / 2$ when $A = B$ or $C = B$.
The angle is a right angle when $\theta = \pi / 2$, acute when $\theta < \pi / 2$ and obtuse when $\theta > \pi / 2$.

## Explanations

### L1 human

To measure the angle at $B$, stand at $B$ and look at $A$, then turn to look at $C$. The angle is how far you turn, using the short way round. The order of $A$ and $C$ does not matter.
</document>

<document source="https://math2.tamnd.com/node/0ZDW0H0VX4.md" tag="0ZDW0H0VX4" kind="concept">
# Exponent of a group

## Summary

The exponent of a group is the least positive integer $n$ with $g^n = e$ for each element $g$, when such an integer exists.
It is the least common multiple of the orders of the elements.
</document>

<document source="https://math2.tamnd.com/node/0ZEHZY85E6.md" tag="0ZEHZY85E6" kind="concept">
# Angle between two vectors

## Summary

The angle between two nonzero [[vectors|0QN3D80AWP]] measures the turn from the direction of one to the direction of the other. It is a number from $0$ to $\pi$. The dot product of the two vectors and their lengths give its cosine.
</document>

<document source="https://math2.tamnd.com/node/0ZEPNS9BXK.md" tag="0ZEPNS9BXK" kind="statement">
# g to the order of the group is the identity

## Statement

Let $G$ be a finite [[group|G5S1M9]] with identity element $e$, and let $g \in G$.
Then $g^{|G|} = e$.

## Remarks

The [[order|0WE2NHZJ7B]] of $g$ divides $|G|$ by [[this corollary|08V7JHS3QZ]], and [[a power is $e$ when the order divides the exponent|0F5QC101Q7]].
For the group $(\mathbb{Z}/p\mathbb{Z})^\times$ of order $p - 1$, with $p$ prime, this gives Fermat's little theorem $a^{p-1} \equiv 1 \pmod p$ for $p \nmid a$.
For the group $(\mathbb{Z}/n\mathbb{Z})^\times$ of order $\varphi(n)$, it gives Euler's theorem.

## Explanations

### L1 human

If a group has $n$ elements, then the $n$th power of each element is the identity.
For example, in a group of order $6$, each element $g$ satisfies $g^6 = e$.
This simple fact proves Fermat's little theorem in number theory.
</document>

<document source="https://math2.tamnd.com/node/0ZERJX6PH6.md" tag="0ZERJX6PH6" kind="statement">
# Universal property of the quotient space

## Statement

Let $F$ be a field, let $V$ and $U$ be vector spaces over $F$, and let $W$ be a subspace of $V$.
Let $\pi \colon V \to V / W$ be the [[quotient map|0N4F5FMQZW]].
Let $T \colon V \to U$ be a [[linear map|0NRYHJPQG2]] with $W \subseteq \ker T$.
Then there is exactly one linear map $\bar T \colon V / W \to U$ with $\bar T \circ \pi = T$.
It is given by $\bar T(v + W) = T(v)$.

## Remarks

The condition $W \subseteq \ker T$ makes $\bar T$ well defined.
The kernel of $\bar T$ is $\ker T / W$, and its image is $\im T$.
For $W = \ker T$, the map $\bar T$ is injective, which gives the [[first isomorphism theorem|0MTBGVZ4XJ]].

## Explanations

### L1 human

A linear map that kills every vector of $W$ cannot tell apart two vectors that differ by an element of $W$.
So it makes sense on the cosets, and it becomes a map from the quotient space.
This property describes the quotient space completely.
</document>

<document source="https://math2.tamnd.com/node/0ZET0ZH16W.md" tag="0ZET0ZH16W" kind="definition">
# Median of a triangle

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\mathbb{R}^n$. Let $M_A$ be the [[midpoint|0S706JBZBG]] of $B$ and $C$.
The median from $A$ is the segment $\{\, A + t (M_A - A) : t \in [0, 1] \,\}$ from $A$ to $M_A$. The medians from $B$ and from $C$ are defined in the same way, with the midpoints of $CA$ and of $AB$.

## Remarks

Some texts call the whole line through $A$ and $M_A$ the median.
The three medians meet at the [[centroid|0X7EEWES5G]]. See [[the medians meet at the centroid|0GPQ67XV7N]].
Each median divides the triangle into two triangles of equal area.
[[Apollonius's theorem|0NAKHQ9XYY]] gives the length of a median from the side lengths.

## Explanations

### L1 human

Join a corner of a triangle to the middle of the opposite side. The segment is a median. If the triangle is a thin plate of uniform material, it balances on each median.
</document>

<document source="https://math2.tamnd.com/node/0ZFZ8ZG3EN.md" tag="0ZFZ8ZG3EN" kind="concept">
# Minimum degree

## Summary

The minimum degree of a finite graph is the smallest degree of its vertices. Many theorems guarantee a structure, such as a long cycle, from a lower bound on the minimum degree. The node [[minimum degree|0TKZCSGCCA]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZG1PNEP35.md" tag="0ZG1PNEP35" kind="concept">
# Homothety

## Summary

A homothety is a transformation that scales the space by a fixed factor from a fixed point, its centre. It sends each figure to a figure of the same shape.
</document>

<document source="https://math2.tamnd.com/node/0ZG3Z5JZ85.md" tag="0ZG3Z5JZ85" kind="concept">
# Symmetry group

## Summary

The symmetry group of an object is the group of all transformations of the ambient space that map the object onto itself. These transformations must keep the relevant structure of the object.
The operation is composition of transformations.
It is often written $\operatorname{Sym}(X)$ for an object $X$.

## Remarks

The symmetry group of a regular $n$-gon in the plane is the [[dihedral group|0JZ27DMJ5D]] of order $2n$.
The rotations of a regular tetrahedron form a group [[isomorphic|08MG7H8JW5]] to the [[alternating group|0Z5905Q73D]] $A_4$.
The symmetry group of an object is a [[subgroup|0X1JXT47GA]] of the group of all transformations of the ambient space.

## Explanations

### L1 human

A symmetry of an object is a motion that leaves the object looking the same.
Two symmetries done one after the other give another symmetry, and each symmetry can be undone.
So the symmetries of an object form a group, and this group measures how symmetric the object is.
</document>

<document source="https://math2.tamnd.com/node/0ZG61TFBES.md" tag="0ZG61TFBES" kind="definition">
# Countable set

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countable when there is an [[injective function|0PVG1CRAJN]] $f \colon A \to \mathbb{N}$.

## Remarks

Equivalently, $A$ is [[finite|0BBTH3YSZA]] or [[countably infinite|0SEZ3JGZP7]].
A nonempty set is countable exactly when there is a [[surjection|0XKH881BZ7]] from $\mathbb{N}$ to it.
Some authors use "countable" only for countably infinite sets, and "at most countable" for this notion.
Subsets of countable sets are countable.

## Explanations

### L1 human

A countable set is one that you can list: first element, second element, and so on.
The list may stop, or it may go on forever, but each element appears at some finite position.
</document>

<document source="https://math2.tamnd.com/node/0ZG9QF40HE.md" tag="0ZG9QF40HE" kind="concept">
# Parametrised surface

## Summary

A parametrised surface is a surface in $\mathbb{R}^3$ given by a map of two parameters into space.
Surfaces in Stokes' theorem and the divergence theorem are often given in this form.
From a parametrisation one can compute the surface area, the normal vectors and the curvature of the surface.
</document>

<document source="https://math2.tamnd.com/node/0ZGC7ZESYN.md" tag="0ZGC7ZESYN" kind="concept">
# Proposition

## Summary

A proposition is a statement that is either true or false.
In classical logic, each proposition has exactly one of the two truth values, true or false.
For example, "$2 + 2 = 4$" is a true proposition, and "$7$ is even" is a false proposition.
A question or a command is not a proposition, because it has no truth value.

## Remarks

Letters such as $p$, $q$ and $r$ usually denote propositions.
An open sentence such as "$x > 3$" is not a proposition until $x$ has a value. It is a [[predicate|0T3DPBG981]].
The connectives [[negation|0XR7A7RSXB]], [[conjunction|0M28ASWG8T]] and [[disjunction|0H5GKDTB01]] make new propositions from given propositions.

## Explanations

### L1 human

A proposition is a sentence that you can mark with "true" or "false".
"It rains in Paris now" is a proposition, because it is true or false at a given time.
"Close the door" is not a proposition, because it is neither true nor false.
</document>

<document source="https://math2.tamnd.com/node/0ZH5AJ2ED7.md" tag="0ZH5AJ2ED7" kind="concept">
# Baire space

## Summary

A Baire space is a topological space in which countable unions of closed sets with empty interior have empty interior.
Equivalently, countable intersections of dense open sets are dense.
</document>

<document source="https://math2.tamnd.com/node/0ZHQMQ2C1N.md" tag="0ZHQMQ2C1N" kind="statement">
# Transfinite induction

## Statement

Let $R$ be a [[well-founded relation|0230ECX1Q5]] on a set $A$ and let $P(x)$ be a [[predicate|0T3DPBG981]] on $A$.
Assume that, for each $x \in A$, if $P(y)$ holds for each $y \in A$ with $y \mathrel{R} x$, then $P(x)$ holds.
Then $P(x)$ holds for each $x \in A$.

## Remarks

Proof: if the set of $x$ with $\neg P(x)$ is nonempty, it has an $R$-minimal element $m$. Then $P(y)$ holds for each $y \mathrel{R} m$, so $P(m)$ holds, a contradiction.
For $<$ on $\mathbb{N}$, this is strong induction.
For [[ordinal numbers|0RDJK7EQYY]], one often splits the step into three cases: zero, successor ordinals and limit ordinals.

## Explanations

### L1 human

Transfinite induction extends induction on $\mathbb{N}$ to every well-founded order.
You show: if the statement holds for everything below $x$, then it holds for $x$.
Since you cannot go down forever, the statement holds everywhere.
</document>

<document source="https://math2.tamnd.com/node/0ZHSTAG8ZY.md" tag="0ZHSTAG8ZY" kind="concept">
# Exterior derivative

## Summary

The exterior derivative extends the differential of a function to [[differential forms|0E3G0GRPZ6]] of higher degree. It maps $k$-forms to $(k + 1)$-forms.
In $\mathbb{R}^3$ it unifies the gradient, curl and divergence, and it gives a common form of the theorems of Green, Stokes and Gauss. Élie Cartan described it in its current form in 1899.
</document>

<document source="https://math2.tamnd.com/node/0ZHWKQMVX5.md" tag="0ZHWKQMVX5" kind="concept">
# Finite-dimensional vector space

## Summary

A finite-dimensional vector space is a vector space that has a finite basis, or equivalently a finite spanning set.
</document>

<document source="https://math2.tamnd.com/node/0ZJ87ZZ8PK.md" tag="0ZJ87ZZ8PK" kind="definition">
# Conditional independence

## Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B, C \in \mathcal{F}$ with $P(C) > 0$.
The events $A$ and $B$ are conditionally independent given $C$ if
$$P(A \cap B \mid C) = P(A \mid C) \, P(B \mid C),$$
where $P(\cdot \mid C)$ is the [[conditional probability|0FJB06AD2P]] given $C$.

## Remarks

If also $P(B \cap C) > 0$, the condition is equivalent to $P(A \mid B \cap C) = P(A \mid C)$.
Conditional independence does not imply independence, and independence does not imply conditional independence.
Example: choose one of two coins at random, a fair coin or a coin with two heads, and toss it two times. Given the coin, the two tosses are independent. Without this knowledge, they are dependent.
For random variables $X$ and $Y$ given $Z$, the definition uses the conditional distributions given $Z$.

## Explanations

### L1 human

Two results can be linked only through a hidden common cause.
When you know the cause, the link disappears. This is conditional independence given the cause.
</document>

<document source="https://math2.tamnd.com/node/0ZJM9D7NPS.md" tag="0ZJM9D7NPS" kind="statement">
# Canonical isomorphism with the double dual

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $V^{**} = (V^*)^*$ be the dual of the [[dual space|07JBSZE0V4]].
The *evaluation map* $\operatorname{ev} \colon V \to V^{**}$ is defined by $\operatorname{ev}(v)(f) = f(v)$ for $v \in V$ and $f \in V^*$.
This map is linear and injective.
If $V$ is finite-dimensional, then $\operatorname{ev}$ is an isomorphism.

## Remarks

The evaluation map does not depend on a choice of basis, so it is called *canonical* or *natural*.
In finite dimensions, it is an isomorphism because it is injective and $\dim V^{**} = \dim V^* = \dim V$.
For infinite-dimensional $V$, the evaluation map is not surjective.

## Explanations

### L1 human

A vector $v$ can act on linear forms: it sends each form $f$ to the number $f(v)$.
This turns each vector into an element of the double dual.
In finite dimensions, every element of the double dual comes from exactly one vector in this way.
</document>

<document source="https://math2.tamnd.com/node/0ZKPH2BD1T.md" tag="0ZKPH2BD1T" kind="concept">
# Cycle notation

## Summary

Cycle notation writes a permutation as a list of cycles.
The cycle $(a_1\ a_2\ \dots\ a_k)$ sends $a_i$ to $a_{i+1}$ for $i < k$, sends $a_k$ to $a_1$, and fixes all other points.
A cycle with $k$ entries is a $k$-cycle, and a $1$-cycle is the identity, so it is usually not written.

## Remarks

The same cycle has $k$ different notations, because you can start the list at any entry: $(1\ 2\ 3) = (2\ 3\ 1) = (3\ 1\ 2)$.
The inverse of $(a_1\ a_2\ \dots\ a_k)$ is $(a_k\ \dots\ a_2\ a_1)$.
Each permutation of a finite set is a product of [[disjoint cycles|0YWDAZPRPQ]], so cycle notation can show each permutation.
For example, in $S_5$ the permutation with $1 \mapsto 3$, $3 \mapsto 1$, $2 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 2$ is $(1\ 3)(2\ 4\ 5)$.

## Explanations

### L1 human

Cycle notation follows each symbol to where it goes, until you come back to the start.
Each round trip is a cycle, and you write it in parentheses.
This is shorter than a table, and it shows the structure of the permutation directly.
</document>

<document source="https://math2.tamnd.com/node/0ZKZBJPG2J.md" tag="0ZKZBJPG2J" kind="statement">
# Absorption law

## Statement

Let $L$ be a lattice with join $\vee$ and meet $\wedge$.
Then $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$ for all $a, b \in L$.

## Remarks

In a lattice defined as a partially ordered set, the proof is short. Since $a \wedge b \le a$, the join of $a$ and $a \wedge b$ is $a$. The second law is dual.
In the algebraic definition of a lattice, the two absorption laws are axioms, together with associativity and commutativity.
The absorption laws imply idempotence: $a \vee a = a \vee (a \wedge (a \vee a)) = a$.

## Explanations

### L1 human

Absorption says that combining $a$ with something made from $a$ gives back $a$.
For sets: $A \cup (A \cap B) = A$, because $A \cap B$ is already inside $A$.
</document>

<document source="https://math2.tamnd.com/node/0ZMT1GP4KM.md" tag="0ZMT1GP4KM" kind="statement">
# Path-connected spaces are connected

## Statement

Let $X$ be a [[path-connected space|0K0GE6BCYW]].
Then $X$ is [[connected|0D939CMME7]].

## Remarks

Proof: suppose open sets $U$ and $V$ split $X$, with $x \in U$ and $y \in V$. Take a path $\gamma$ from $x$ to $y$. Then $\gamma([0, 1])$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]] and [[connectedness of intervals|0B9ZQFMQNV]]. But $U$ and $V$ split it, a contradiction.
The converse fails, as the [[topologist's sine curve|0K3T4A659R]] shows. It holds for [[open subsets of R^n|0MXN7AFBPE]].

## Explanations

### L1 human

If you can walk between any two points, the space cannot be split into two separate open parts.
</document>

<document source="https://math2.tamnd.com/node/0ZMY8441GW.md" tag="0ZMY8441GW" kind="definition">
# Elementary symmetric polynomials

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]] and let $n \ge 1$.
For $0 \le k \le n$, the $k$-th elementary symmetric polynomial in $x_1, \dots, x_n$ is
$$e_k = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} x_{i_1} x_{i_2} \cdots x_{i_k} \in R[x_1, \dots, x_n].$$

## Remarks

So $e_0 = 1$, $e_1 = x_1 + \dots + x_n$ and $e_n = x_1 x_2 \cdots x_n$. One sets $e_k = 0$ for $k > n$.
In three variables, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$.
The generating identity is $\prod_{i=1}^n (t + x_i) = \sum_{k=0}^n e_k t^{n-k}$.
Each $e_k$ is [[symmetric|06CTFY2DEZ]] and homogeneous of degree $k$.

## Explanations

### L1 human

The elementary symmetric polynomials are the simplest symmetric polynomials: the sum of the variables, the sum of their products in pairs, and so on.
Up to sign, they are the coefficients of the polynomial whose roots are the variables.
</document>

<document source="https://math2.tamnd.com/node/0ZMYX7GMH0.md" tag="0ZMYX7GMH0" kind="statement">
# First isomorphism theorem for rings

## Statement

Let $f \colon R \to S$ be a [[ring homomorphism|0GGE0Z0JQ1]].
Then the [[kernel|0AAHE7GEW6]] $\ker f$ is a two-sided ideal of $R$, and the image $f(R)$ is a subring of $S$.
The map $R/\ker f \to f(R)$, $a + \ker f \mapsto f(a)$, is a [[ring isomorphism|0B6QC0C0G1]].
In particular, if $f$ is surjective, then $R/\ker f \cong S$.

## Remarks

The reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ has kernel $n\mathbb{Z}$.
The evaluation map $\mathbb{R}[x] \to \mathbb{C}$, $p \mapsto p(i)$, is surjective with kernel $(x^2 + 1)$. So $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$.
The theorem has the same form for groups and for modules.

## Explanations

### L1 human

A homomorphism forgets exactly the information in its kernel.
If you divide out the kernel, the homomorphism becomes a bijection onto its image.
So every image of a ring is a quotient of that ring.
</document>

<document source="https://math2.tamnd.com/node/0ZN07RQ3JS.md" tag="0ZN07RQ3JS" kind="definition">
# Stirling numbers of the second kind

## Definition

Let $n$ and $k$ be natural numbers.
The Stirling number of the second kind $S(n, k)$ is the number of partitions of an $n$-element set into $k$ nonempty blocks.
A partition into blocks is a set of pairwise disjoint nonempty subsets whose union is the whole set.

## Remarks

Other notations are $\left\{ {n \atop k} \right\}$ and $S_n^{(k)}$.
Boundary values: $S(0, 0) = 1$, $S(n, 0) = 0$ for $n \ge 1$, and $S(n, k) = 0$ for $k > n$.
Example: $S(4, 2) = 7$. The partitions of $\{1, 2, 3, 4\}$ into two blocks are $1|234$, $2|134$, $3|124$, $4|123$, $12|34$, $13|24$ and $14|23$.
The values satisfy a [[recurrence|0C019SDTER]], and the sum $\sum_k S(n, k)$ is the [[Bell number|0WW85SQVS3]] $B_n$.

## Explanations

### L1 human

$S(n, k)$ counts the ways to split $n$ labelled objects into $k$ groups, when the groups have no labels and none is empty.
For example, there are $S(3, 2) = 3$ ways to split three people into two teams.
</document>

<document source="https://math2.tamnd.com/node/0ZN2J42SK7.md" tag="0ZN2J42SK7" kind="statement">
# Determinant of a 2 by 2 matrix

## Statement

Let $K$ be a commutative ring and let $a, b, c, d \in K$.
Then
$$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is the [[Leibniz formula|016PE4VZ8Q]] for $n = 2$, with the identity and one transposition.
Over $\R$, $\abs{ad - bc}$ is the area of the parallelogram spanned by the columns $(a, c)$ and $(b, d)$.
For example, $\det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = 4 - 6 = -2$.

## Explanations

### L1 human

The determinant of a $2 \times 2$ matrix is the product of the main diagonal minus the product of the other diagonal.
It is zero exactly when one column is a multiple of the other, over a field.
Then the matrix squeezes the plane onto a line or a point.
</document>

<document source="https://math2.tamnd.com/node/0ZN39QDTP9.md" tag="0ZN39QDTP9" kind="statement">
# K(a) is isomorphic to K[x] modulo the minimal polynomial

## Statement

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$.
Then the [[evaluation map|0125QVNCFQ]] $K[x] \to L$, $f \mapsto f(a)$, induces a ring isomorphism
$$K[x]/(m_a) \cong K(a).$$
In particular, $K(a) = K[a] = \{ f(a) : f \in K[x] \}$.

## Remarks

The proof: the kernel of the evaluation map is $(m_a)$, by [[the divisibility property|06K7SARZKR]]. The [[first isomorphism theorem|0ZMYX7GMH0]] gives $K[x]/(m_a) \cong K[a]$.
Since $m_a$ is irreducible, $K[x]/(m_a)$ is a [[field|0AWMPX35CW]]. So $K[a]$ is a field, and $K[a] = K(a)$.
If $a$ is transcendental over $K$, then the evaluation map is injective, and it extends to an isomorphism $K(x) \cong K(a)$.

## Explanations

### L1 human

An algebraic element adds nothing to the field beyond the relation given by its minimal polynomial.
So $K(a)$ is the same as polynomials in a variable, computed modulo the minimal polynomial.
For example, $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}[x]/(x^2 - 2)$.
</document>

<document source="https://math2.tamnd.com/node/0ZN9EZTEM6.md" tag="0ZN9EZTEM6" kind="concept">
# Conditional probability distribution

## Summary

The conditional distribution of $Y$ given $X = x$ is the distribution of $Y$ when one knows that $X$ has the value $x$.
The node [[conditional probability distribution|0YVDC5PYDW]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZND7NQ970.md" tag="0ZND7NQ970" kind="definition">
# Eigenvalue

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
A scalar $\lambda \in F$ is an *eigenvalue* of $T$ when there is a vector $v \in V$ with $v \ne 0$ and
$$T(v) = \lambda v.$$
An eigenvalue of an $n \times n$ matrix $A$ over $F$ is an eigenvalue of the map $F^n \to F^n$, $x \mapsto Ax$.

## Remarks

So $\lambda$ is an eigenvalue of $T$ if and only if $T - \lambda \id_V$ is not injective.
In finite dimensions, this holds if and only if $\det(T - \lambda \id_V) = 0$.
The eigenvalues of the diagonal matrix $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ are $2$ and $3$.
The rotation of $\R^2$ by $90$ degrees has no real eigenvalue.

## Explanations

### L1 human

Most vectors change direction when a linear map acts on them.
An eigenvalue is a scaling factor $\lambda$ such that some nonzero vector only gets stretched by $\lambda$, without a change of its line.
These special directions often reveal the structure of the map.
</document>

<document source="https://math2.tamnd.com/node/0ZNMMA9TF5.md" tag="0ZNMMA9TF5" kind="concept">
# Intersection

## Summary

The intersection of sets is the set of all objects that are elements of each of the sets.
</document>

<document source="https://math2.tamnd.com/node/0ZNSSK8ZK1.md" tag="0ZNSSK8ZK1" kind="statement">
# p-series test

## Statement

Let $p$ be a real number.
The series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if and only if $p > 1$.

## Remarks

For $p = 1$ it is the [[harmonic series|0CXTMAY5PX]], which diverges. For $p \le 0$ the terms do not tend to $0$.
One proof uses the [[Cauchy condensation test|0B96KMDDD1]]. The condensed series is $\sum 2^n \cdot 2^{-np} = \sum (2^{1-p})^n$, a geometric series with ratio $2^{1-p}$.
Another proof uses the [[integral test|019QZ6RRVH]] with $f(x) = x^{-p}$.
For $p > 1$ the sum is the value $\zeta(p)$ of the Riemann zeta function. For example, $\zeta(2) = \pi^2/6$, which is the [[Basel problem|0DMY75D6FV]].

## Explanations

### L1 human

The terms $1/n^p$ shrink faster when $p$ is larger.
The value $p = 1$ is the border. Above it the sum is finite, and at it or below it the sum is infinite.
</document>

<document source="https://math2.tamnd.com/node/0ZNX6DQ3EJ.md" tag="0ZNX6DQ3EJ" kind="definition">
# Implication

## Definition

Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]].
The implication from $p$ to $q$ is the proposition $p \to q$, read "if $p$, then $q$".
The proposition $p$ is the hypothesis, and the proposition $q$ is the conclusion.
The proposition $p \to q$ is false when $p$ is true and $q$ is false. In all other cases, $p \to q$ is true.

| $p$ | $q$ | $p \to q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

## Remarks

Another notation for $p \to q$ is $p \Rightarrow q$.
The proposition $p \to q$ has the same truth value as $\neg p \lor q$.
When $p$ is false, $p \to q$ is true. This is [[vacuous truth|0V4QXPGJQK]].
The [[converse|0RVYE75E2M]] of $p \to q$ is $q \to p$. It can have a different truth value.

## Explanations

### L1 human

Think of $p \to q$ as a promise: "if it rains, I take an umbrella".
You break the promise only when it rains and you take no umbrella.
When it does not rain, you keep the promise, whatever you do.
</document>

<document source="https://math2.tamnd.com/node/0ZNXZB4HNB.md" tag="0ZNXZB4HNB" kind="concept">
# Power sum symmetric polynomial

## Summary

The $k$-th power sum in $n$ variables is the sum of the $k$-th powers of the variables.
The [[definition of the power sums|0AS1FZT6BM]] states it.
</document>

<document source="https://math2.tamnd.com/node/0ZNZH0EXFX.md" tag="0ZNZH0EXFX" kind="concept">
# Inverse element

## Summary

An inverse of an element undoes the element: their combination gives the identity element.
For example, $-a$ is the inverse of $a$ for addition.
</document>

<document source="https://math2.tamnd.com/node/0ZPB3180AY.md" tag="0ZPB3180AY" kind="statement">
# Matrix of the dual map is the transpose

## Statement

Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Let $B$ be a basis of $V$ and $C$ a basis of $W$, with [[dual bases|0ZD2CF6ZS0]] $B^*$ and $C^*$.
Let $T \colon V \to W$ be a linear map with [[dual map|0AZ5ARP287]] $T^*$.
Then
$$[T^*]_{B^*, C^*} = \left([T]_{C, B}\right)^T,$$
where $[\,\cdot\,]$ is the [[matrix of a linear map|0YQSGNYE3K]] and the exponent $T$ is the [[transpose|09DXWYRD43]].

## Remarks

The proof computes entry $(j, i)$ of $[T^*]$ as $T^*(c_i^*)(b_j) = c_i^*(T(b_j))$, which is entry $(i, j)$ of $[T]$.
With the [[annihilator|0ASHNYF7AG]], this gives a proof that [[row rank equals column rank|0TZAG60ZYT]].

## Explanations

### L1 human

The dual map is the linear algebra form of the transpose of a matrix.
In dual bases, its matrix is the transpose of the matrix of the original map.
This explains why rows and columns play symmetric roles.
</document>

<document source="https://math2.tamnd.com/node/0ZPQHH37PF.md" tag="0ZPQHH37PF" kind="concept">
# Grönwall bound

## Summary

The Grönwall bound is the solution $x \mapsto \delta e^{Kx} + \frac{\varepsilon}{K}(e^{Kx} - 1)$ of $v' = K v + \varepsilon$, $v(0) = \delta$, with the value $\delta + \varepsilon x$ when $K = 0$.
It is the upper bound in a form of [[Grönwall's inequality|07EJHQYR9A]]. See the [[definition|0FRX8JD7QH]].
</document>

<document source="https://math2.tamnd.com/node/0ZPV68RN1X.md" tag="0ZPV68RN1X" kind="concept">
# Recursively defined sequence

## Summary

A recursively defined sequence is a [[sequence|0QE8STGJH2]] given by its first terms and a rule that computes each later term from the terms before it. A typical form is $a_0 = c$ and $a_{n+1} = f(n, a_n)$ for a given value $c$ and a given function $f$. The recursion theorem shows that exactly one sequence satisfies such a rule.

## Remarks

The factorials satisfy $0! = 1$ and $(n+1)! = (n+1) \cdot n!$.
The Fibonacci numbers satisfy $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. Here each term needs the two terms before it.
In analysis, the sequence $a_0 = 1$, $a_{n+1} = \frac{1}{2}(a_n + 2/a_n)$ converges to $\sqrt{2}$. It is [[Newton's method|07EXVNF9VP]] for $x^2 - 2$.
The rule that links the terms is a recurrence relation.

## Explanations

### L1 human

A recursive rule tells you how to get the next term from the terms you already have.
Start with $a_0 = 1$ and the rule $a_{n+1} = 2 a_n$, and you get $1, 2, 4, 8, \dots$.
</document>

<document source="https://math2.tamnd.com/node/0ZQJ7KEFQ5.md" tag="0ZQJ7KEFQ5" kind="statement">
# Tower law for indices

## Statement

Let $G$ be a [[group|G5S1M9]], let $H$ be a [[subgroup|0X1JXT47GA]] of $G$, and let $K$ be a subgroup of $H$.
Then $[G : K] = [G : H] \cdot [H : K]$, where $[\cdot : \cdot]$ is the [[index|0DTXBQPA2S]].

## Remarks

The equation holds as an equation of cardinal numbers; in particular $[G : K]$ is finite if and only if $[G : H]$ and $[H : K]$ are both finite.
For example, in $(\mathbb{Z}, +)$ with $H = 2\mathbb{Z}$ and $K = 6\mathbb{Z}$, the indices are $[\mathbb{Z} : 6\mathbb{Z}] = 6 = 2 \cdot 3$.

## Explanations

### L1 human

Each coset of $H$ in $G$ splits into $[H : K]$ cosets of $K$.
So the number of cosets of $K$ is the number of cosets of $H$ times $[H : K]$.
It is like a box with several drawers, each with the same number of compartments.
</document>

<document source="https://math2.tamnd.com/node/0ZS54B9GJF.md" tag="0ZS54B9GJF" kind="statement">
# The logarithm is strictly concave

## Statement

The [[natural logarithm|0044CPHJBV]] is strictly [[concave|0EF2VG8XKQ]] on $(0, \infty)$.
That is, $\ln(t x + (1 - t) y) > t \ln x + (1 - t) \ln y$ for all $x \ne y$ in $(0, \infty)$ and $t \in (0, 1)$.

## Remarks

The [[derivative of $\ln x$|07X1QA6BRW]] is $1/x$, and its second derivative is $-1/x^2 < 0$.
The concavity gives the [[AM–GM inequality|0DRF0F3XC9]] through Jensen's inequality.
Mathlib states it as `strictConcaveOn_log_Ioi`.

## Explanations

### L1 human

The graph of $\ln x$ bends downward everywhere.
Every chord of the graph lies below the graph.
</document>

<document source="https://math2.tamnd.com/node/0ZT24EC90K.md" tag="0ZT24EC90K" kind="definition">
# Central moment

## Definition

Let $X$ be a real random variable and let $k \ge 1$ be an integer with $E[|X|^k] < \infty$. Let $\mu = E[X]$.
The $k$-th central moment of $X$ is
$$\mu_k = E\left[(X - \mu)^k\right].$$

## Remarks

Always $\mu_1 = 0$, and $\mu_2 = \operatorname{Var}(X)$ is the [[variance|007QED3E0Y]].
Central moments do not change when one adds a constant to $X$. For $a \in \mathbb{R}$, the central moments of $aX$ are $a^k \mu_k$.
The binomial theorem gives $\mu_k = \sum_{j=0}^{k} \binom{k}{j} (-\mu)^{k - j} m_j$ in terms of the [[moments|0PH0VHTQS0]] $m_j = E[X^j]$, with $m_0 = 1$.
For a distribution that is symmetric about its mean, all odd central moments are $0$.

## Explanations

### L1 human

Central moments measure the shape of a distribution around its center, and not its position.
The third gives asymmetry and the fourth gives the weight of the tails.
</document>

<document source="https://math2.tamnd.com/node/0ZTEH055YF.md" tag="0ZTEH055YF" kind="definition">
# Operator norm

## Definition

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be a [[continuous linear map|0S3SMRH0Z5]].
The operator norm of $T$ is
$$\|T\| = \sup \{ \|Tx\|_W : x \in V, \ \|x\|_V \le 1 \}.$$

## Remarks

The [[supremum|0BTMQ04556]] is finite because $T$ is [[bounded|0F3K4C7XYW]]. If $V \neq \{0\}$, then $\|T\| = \sup_{x \neq 0} \|Tx\|_W / \|x\|_V$.
It is the smallest $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$.
The operator norm is a norm on the space $L(V, W)$ of continuous linear maps. It satisfies $\|ST\| \le \|S\| \, \|T\|$. If $W$ is a [[Banach space|0BDTTM24DP]], so is $L(V, W)$.
For a diagonal matrix on $\mathbb{R}^n$ with the Euclidean norm, the operator norm is the largest absolute value of a diagonal entry.

## Explanations

### L1 human

The operator norm tells you how much $T$ can stretch a vector at most.
For example, the map $x \mapsto 3x$ on $\mathbb{R}$ has operator norm $3$.
</document>

<document source="https://math2.tamnd.com/node/0ZTGMDNVTB.md" tag="0ZTGMDNVTB" kind="statement">
# Intersection of subgroups is a subgroup

## Statement

Let $G$ be a [[group|G5S1M9]] and let $(H_i)_{i \in I}$ be a family of [[subgroups|0X1JXT47GA]] of $G$, indexed by a nonempty set $I$.
Then the intersection $\bigcap_{i \in I} H_i$ is a subgroup of $G$.

## Remarks

The identity is in each $H_i$, and if $a, b$ are in each $H_i$, then $ab^{-1}$ is in each $H_i$.
The union of two subgroups is in general not a subgroup: in $(\mathbb{Z}, +)$, $2 \in 2\mathbb{Z}$ and $3 \in 3\mathbb{Z}$, but $5$ is in neither.
The union of two subgroups $H$ and $K$ is a subgroup only when $H \subseteq K$ or $K \subseteq H$.

## Explanations

### L1 human

If several subgroups each pass the subgroup test, then their common part also passes it.
For example, $2\mathbb{Z} \cap 3\mathbb{Z} = 6\mathbb{Z}$ in the integers.
This fact lets us define the smallest subgroup that contains a given subset.
</document>

<document source="https://math2.tamnd.com/node/0ZTKJJDGR7.md" tag="0ZTKJJDGR7" kind="object">
# Zero matrix

## Description

Let $R$ be a ring and let $m, n \ge 0$ be integers.
The $m \times n$ *zero matrix* over $R$ is the [[matrix|0VD9ZSWWJG]] $0_{m \times n}$ whose entries are all $0$.
It is the identity element for [[matrix addition|0Y7H3SAVA1]]: $A + 0 = 0 + A = A$ for each $m \times n$ matrix $A$.
For each matrix $B$ with $n$ rows, the product $0_{m \times n} B$ is a zero matrix.

## Remarks

The $2 \times 2$ zero matrix is $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.
The zero matrix is the matrix of the zero linear map in any pair of bases.
A product of two nonzero matrices can be a zero matrix, for example $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2 = 0$.

## Explanations

### L1 human

The zero matrix is the table with $0$ in every position.
It plays the role of the number $0$: adding it to a matrix changes nothing.
There is one zero matrix for each size.
</document>

<document source="https://math2.tamnd.com/node/0ZV14XRNF0.md" tag="0ZV14XRNF0" kind="concept">
# Closed and exact forms

## Summary

A [[differential form|0E3G0GRPZ6]] is closed if its [[exterior derivative|0ZHSTAG8ZY]] is zero, and exact if it is the exterior derivative of another form.
Each exact form is closed. Whether each closed form is exact depends on the shape of the domain, so these notions detect holes in the domain.
</document>

<document source="https://math2.tamnd.com/node/0ZV2YKKZ58.md" tag="0ZV2YKKZ58" kind="definition">
# Solution of a linear recurrence

## Definition

Let $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \dots, c_{d-1}$.
A solution of $E$ is a sequence $u : \mathbb{N} \to R$ such that
$$u_{n+d} = \sum_{i=0}^{d-1} c_i\, u_{n+i} \quad \text{for all } n \in \mathbb{N}.$$

## Remarks

Each choice of the initial values $u_0, \dots, u_{d-1}$ gives exactly one solution.
The sum of two solutions is a solution, and a multiple of a solution is a solution. So the solutions form the [[space of solutions|05BDYVQQ0T]].
For $u_{n+2} = u_{n+1} + u_n$, the Lucas numbers $2, 1, 3, 4, 7, 11, \dots$ form a solution.

## Explanations

### L1 human

A solution is a sequence that obeys the rule of the recurrence at every step.
The rule alone does not fix the sequence: you also choose the first $d$ terms.
</document>

<document source="https://math2.tamnd.com/node/0ZV9A7EV0F.md" tag="0ZV9A7EV0F" kind="definition">
# Complex power

## Definition

Let $z, w \in \mathbb{C}$ with $z \neq 0$.
The complex powers of $z$ with exponent $w$ are the numbers $e^{w \ell}$, where $\ell$ runs through the [[complex logarithms|0X2FY491PF]] of $z$.
The principal value of $z^w$ is
$$z^w = e^{w \operatorname{Log} z},$$
where $\operatorname{Log}$ is the [[principal branch|0YKY4M3KRQ]] of the logarithm and $e^u$ is the [[complex exponential|054VXHMCA3]] of $u$.

## Remarks

The powers of $z$ with exponent $w$ are the numbers $e^{w \operatorname{Log} z} e^{2 \pi i k w}$ with $k \in \mathbb{Z}$. There is one value if $w$ is an integer, there are $q$ values if $w = p/q$ in lowest terms with $q \ge 2$, and there are infinitely many values otherwise.
For an integer $w = n$, $z^n$ is the usual product of $n$ factors $z$, or of $-n$ factors $1/z$.
For example, $i^i = e^{i \cdot i \pi/2} = e^{-\pi/2}$, a positive real number.
The usual rules can fail: with principal values, $(z^a)^b \neq z^{ab}$ for some $z, a, b$. A common convention for $z = 0$ is $0^0 = 1$ and $0^w = 0$ for $w \neq 0$.

## Explanations

### L1 human

For real numbers, $a^b = e^{b \ln a}$ when $a > 0$. The complex power uses the same formula with a complex logarithm.
Because the logarithm has many values, the power can have many values too, and the principal value picks one of them.
</document>

<document source="https://math2.tamnd.com/node/0ZVEAYVRC3.md" tag="0ZVEAYVRC3" kind="concept">
# Law of large numbers

## Summary

The law of large numbers says that the average of many independent repetitions of a random experiment tends to the expected value.
For an [[i.i.d.|0AJN7WHBJZ]] sequence $X_1, X_2, \dots$ with finite mean $\mu$, the sample mean $\bar{X}_n = (X_1 + \dots + X_n)/n$ tends to $\mu$.
The [[weak law|0BFDHF2H7N]] gives convergence in probability, and the [[strong law|0DFWHQ839T]] gives almost sure convergence.

## Remarks

The law fails without a finite mean. For i.i.d. standard Cauchy variables, $\bar{X}_n$ is standard Cauchy for each $n$ and does not converge.
For indicator variables of an event $A$, the law says that the relative frequency of $A$ tends to $P(A)$. This links probability with long-run frequency.

## Explanations

### L1 human

If you toss a fair coin many times, the fraction of heads comes close to $1/2$.
The law of large numbers is the precise form of this experience. It is the reason why averages of many measurements are reliable.
</document>

<document source="https://math2.tamnd.com/node/0ZVHXDSZC7.md" tag="0ZVHXDSZC7" kind="definition">
# Quotient set

## Definition

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$.
The quotient set of $A$ by $\sim$ is the set
$$A/{\sim} = \{[a] \mid a \in A\}$$
of all [[equivalence classes|082RMRSY26]].
The canonical projection is the [[function|0XHDTE5YMZ]] $\pi \colon A \to A/{\sim}$ with $\pi(a) = [a]$.

## Remarks

For congruence modulo $n$ on $\mathbb{Z}$, the quotient set $\mathbb{Z}/n\mathbb{Z}$ has the $n$ elements $[0], [1], \dots, [n-1]$.
The canonical projection is [[surjective|0XKH881BZ7]].
A function on $A/{\sim}$ is often defined by a formula on representatives. The formula must give the same value for equivalent representatives.

## Explanations

### L1 human

In the quotient set, each class becomes a single point.
From $\mathbb{Z}$ with congruence modulo $2$ you get a set with two elements: "even" and "odd".
</document>

<document source="https://math2.tamnd.com/node/0ZVQKTRKVQ.md" tag="0ZVQKTRKVQ" kind="concept">
# Bipartite graph

## Summary

A bipartite graph is a graph whose vertices split into two parts with every edge between the parts. Bipartite graphs model relations between two kinds of object, and they are the setting of matching theory. The node [[bipartite graph|07DVHJ8N0A]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZXJ2455NJ.md" tag="0ZXJ2455NJ" kind="statement">
# Uniqueness of limits

## Statement

Let $(a_n)$ be a real sequence and let $L$ and $L'$ be real numbers.
If $(a_n)$ converges to $L$ and $(a_n)$ converges to $L'$, then $L = L'$.

## Remarks

The proof takes $\varepsilon = |L - L'|/2$ in the [[epsilon-N definition|09G6KHS6GY]]. If $L \ne L'$, then for large $n$ the term $a_n$ would be closer than $\varepsilon$ to both numbers, which is impossible.
The statement holds in each metric space and, more generally, in each Hausdorff topological space.
The statement fails in some spaces that are not Hausdorff. In a set with the trivial topology, each sequence converges to each point.

## Explanations

### L1 human

A sequence cannot approach two different numbers at the same time.
If the two numbers are a distance $d$ apart, the terms cannot be within $d/2$ of both of them.
</document>

<document source="https://math2.tamnd.com/node/0ZXSF2YQV2.md" tag="0ZXSF2YQV2" kind="concept">
# Isometric isomorphism

## Summary

An isometric isomorphism is a bijective isometry between metric spaces.
Two metric spaces with an isometric isomorphism between them have the same metric properties.
</document>

<document source="https://math2.tamnd.com/node/0ZXW95S44M.md" tag="0ZXW95S44M" kind="statement">
# Second supplement to quadratic reciprocity

## Statement

Let $p$ be an odd prime.
Then
$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8},$$
where the left side is a [[Legendre symbol|01J8AMV9RS]].

## Remarks

So $2$ is a quadratic residue modulo $p$ if $p \equiv \pm 1 \pmod 8$, and a nonresidue if $p \equiv \pm 3 \pmod 8$.
For example, $3^2 \equiv 2 \pmod 7$, but $2$ is not a square modulo $5$.
One proof uses [[Gauss's lemma|037ACYXB5Z]] with $a = 2$.

## History

Euler proved the second supplement.

## Explanations

### L1 human

Whether $2$ is a square modulo $p$ depends only on the remainder of $p$ on division by $8$.
The remainders $1$ and $7$ give a square, and the remainders $3$ and $5$ do not.
</document>

<document source="https://math2.tamnd.com/node/0ZXZ82FTX2.md" tag="0ZXZ82FTX2" kind="concept">
# Finite field

## Summary

A finite field is a field with a finite number of elements.
For each prime power $q$ there is exactly one finite field of order $q$, up to isomorphism, and there are no other finite fields.
The [[definition of a finite field|08RTDHX1SK]] states it.
</document>

<document source="https://math2.tamnd.com/node/0ZY9DX8FBF.md" tag="0ZY9DX8FBF" kind="concept">
# Probability density function

## Summary

A probability density function of a real random variable $X$ is a nonnegative function $f$ with this property.
For each set $B$ of values, the integral of $f$ over $B$ is the probability that $X$ lies in $B$.
The node [[probability density function|04YG690N5V]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZYBZ34RT2.md" tag="0ZYBZ34RT2" kind="definition">
# Probability space

## Definition

A probability space is a triple $(\Omega, \mathcal{F}, P)$ with these parts.
$\Omega$ is a nonempty set, the [[sample space|0W7NFW23RF]].
$\mathcal{F}$ is a [[σ-algebra|04WSW8T3SY]] on $\Omega$. Its elements are the events.
$P$ is a [[probability measure|0VGMEBVW1M]] on $(\Omega, \mathcal{F})$.

## Remarks

For one roll of a fair die: $\Omega = \{1, \dots, 6\}$, $\mathcal{F}$ is the power set of $\Omega$, and $P(A) = |A| / 6$.
A discrete probability space has a countable $\Omega$, the power set as $\mathcal{F}$, and $P(A) = \sum_{\omega \in A} p(\omega)$ for a function $p \ge 0$ with total sum $1$.
The interval $[0, 1]$ with its Borel σ-algebra and the length of intervals as $P$ is a probability space. It models a point chosen uniformly at random in $[0, 1]$.

## Explanations

### L1 human

A probability space puts three things together: what can happen, which questions we can ask about it, and how likely each answer is.
Every problem in probability starts, at least in the background, with one probability space.
</document>

<document source="https://math2.tamnd.com/node/0ZZ4RHTR9X.md" tag="0ZZ4RHTR9X" kind="concept">
# Taylor polynomial

## Summary

The Taylor polynomial of degree $k$ of a function at a point has the same value and the same first $k$ derivatives there. It approximates the function near the point. The node [[Taylor polynomial|0HFSFVMWQB]] gives the precise definition.
</document>

<document source="https://math2.tamnd.com/node/0ZZ7ZYDR8W.md" tag="0ZZ7ZYDR8W" kind="concept">
# Meromorphic function

## Summary

A meromorphic function on an open set is a function that is holomorphic on the set except at isolated points, which are poles. See the [[definition|0NSD1TPHR0]].
Quotients of holomorphic functions are meromorphic.
</document>

<document source="https://math2.tamnd.com/node/0ZZQ2E1MTE.md" tag="0ZZQ2E1MTE" kind="statement">
# Determinant of a transpose

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
Then $\det A^T = \det A$, where $A^T$ is the [[transpose|09DXWYRD43]] of $A$ and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In the [[Leibniz formula|016PE4VZ8Q]], the term of $\sigma$ for $A^T$ equals the term of $\sigma^{-1}$ for $A$, and $\sgn(\sigma^{-1}) = \sgn(\sigma)$.
So each property of the determinant for rows also holds for columns.

## Explanations

### L1 human

Exchanging rows and columns does not change the determinant.
So we can expand, reduce or test a determinant by columns as well as by rows.
Every rule for rows has a twin rule for columns.
</document>

<document source="https://math2.tamnd.com/node/0ZZR8D7Y1Z.md" tag="0ZZR8D7Y1Z" kind="concept">
# Subnormal series

## Summary

A subnormal series of a group is a finite chain of subgroups from the trivial subgroup to the whole group.
Each subgroup in the chain is normal in the next one.
Its factor groups are the quotients of consecutive terms.
</document>

<document source="https://math2.tamnd.com/node/0ZZRSX91QR.md" tag="0ZZRSX91QR" kind="definition">
# Argument of a complex number

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with $z \neq 0$, and let $|z|$ be its [[modulus|0KW2X0TFBX]].
An argument of $z$ is a real number $\theta$ with
$$z = |z| (\cos \theta + i \sin \theta).$$
The notation $\arg z$ means the set of all arguments of $z$, or one chosen argument.

## Remarks

If $\theta$ is an argument of $z$, then the arguments of $z$ are exactly the numbers $\theta + 2 \pi k$ with $k \in \mathbb{Z}$.
The number $0$ has no argument in this sense. Some conventions set $\arg 0 = 0$.
For example, the arguments of $i$ are $\pi/2 + 2 \pi k$, and those of $-1$ are $\pi + 2 \pi k$.

## Explanations

### L1 human

The argument says in which direction a complex number points from $0$.
The number $1 + i$ points at $45$ degrees, so $\pi/4$ is an argument of it.
A full turn brings you back to the same direction, so adding $2\pi$ gives another argument.
</document>

<document source="https://math2.tamnd.com/node/0ZZXSBTR4J.md" tag="0ZZXSBTR4J" kind="concept">
# Rank of a linear map

## Summary

The rank of a linear map is the dimension of its image.
</document>

<document source="https://math2.tamnd.com/node/3C53TD.md" tag="3C53TD" kind="statement">
# Lagrange's theorem

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $H$ be a subgroup of $G$.
Then the order of $H$ divides the order of $G$.

## Remarks

The quotient $|G| / |H|$ is the number of left cosets of $H$ in $G$.
</document>

<document source="https://math2.tamnd.com/node/AAYEQE.md" tag="AAYEQE" kind="computation">
# The subgroups of the symmetric group S4

## Description

The program in GAP lists each subgroup of $S_4$, the [[group|G5S1M9]] of the permutations of $\{1, 2, 3, 4\}$. It writes each subgroup with the sorted list of its elements to the certificate, and it prints the number of subgroups of each order.

The checker is a short Python program that does not use GAP. It finds the subgroups of $S_4$ again with its own search. It starts with the trivial subgroup and adds the subgroup that a known subgroup and one more element generate, until it finds no new subgroup. It accepts the certificate only when the lines of the certificate are these subgroups, each one time, and the order of each one divides 24.

## Remarks

The orders divide 24, as [[Lagrange's theorem|3C53TD]] says. The converse of the theorem is not true for each finite group, but it is true for $S_4$. For each divisor $d$ of 24, $S_4$ has a subgroup of order $d$. The alternating group $A_4$ of order 12 has no subgroup of order 6, so the converse is false for $A_4$.
</document>

<document source="https://math2.tamnd.com/node/G5S1M9.md" tag="G5S1M9" kind="concept">
# Group

## Summary

A group is a set $G$ with a binary operation $\cdot$ that has these three properties.
The operation is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c$ in $G$.
There is an identity element $e$ in $G$ such that $e \cdot a = a \cdot e = a$ for all $a$ in $G$.
Each element $a$ of $G$ has an inverse $a^{-1}$ in $G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

The order of a finite group is its number of elements.

## Remarks

[[Lagrange's theorem|3C53TD]] gives a property of the order of a finite group.
</document>