User:Kompik/bookmarks/Math

General

 * WikiProject Mathematics

Disciplines

 * Template:Areas of mathematics

Notation

 * Glossary of mathematical symbols
 * List of logic symbols
 * List of mathematical symbols by subject

Theorems

 * List of lemmas
 * List of theorems

Calculus

 * Category:Theorems in calculus

Continuity

 * Absolute continuity
 * Continuous function
 * Intermediate value theorem
 * Uniform continuity

Limits

 * Stolz–Cesàro theorem

Series

 * Telescoping series

Convergence tests

 * Cauchy condensation test
 * Cauchy's convergence test
 * Ratio test

Derivatives

 * Differentiation rules
 * Differentiation of trigonometric functions
 * Leibniz integral rule for differentiation under the integral sign

Theorems

 * Darboux's theorem (analysis)
 * Mean value theorem

Single variable

 * Derivative test
 * Fermat's theorem (stationary points)

Multivariable

 * Second partial derivative test (Hessian matrix)

Integration

 * Integration by partial fractions
 * Integration by parts
 * Trigonometric substitution

Integration by substitution

 * Euler substitution
 * Tangent half-angle substitution

Combinatorics

 * Bell number
 * Catalan number
 * Derangement
 * Enumerative combinatorics
 * Inclusion–exclusion principle
 * Pigeonhole principle
 * Ramsey's theorem
 * Stirling numbers of the first kind
 * Stirling numbers of the second kind

Analytic geometry

 * Line (geometry)
 * Normal (geometry)

Geometric inequalities

 * Category:Geometric inequalities
 * Category:Triangle inequalities
 * List of triangle inequalities

Circle

 * Inscribed angle (Inscribed angle theorem)
 * Power of a point

Right triangle

 * Geometric mean theorem (right triangle altitude theorem)
 * Pythagorean theorem

Triangle

 * Category:Theorems about triangles
 * Centroid (Barycenter)
 * Cevian (Stewart's theorem)
 * Circumradius and inradius
 * Circumscribed circle (Incircle and excircles of a triangle)
 * Euler's theorem in geometry: $$ d^2=R (R-2r) $$ or equivalently $$ \frac{1}{R-d} + \frac{1}{R+d} = \frac{1}{r}.$$
 * Geometric mean theorem
 * Median (geometry)
 * Menelaus's theorem
 * Morley's trisector theorem
 * Napoleon's theorem
 * Thales's theorem
 * Van Aubel's theorem

Solid geometry

 * Volume

Polyhedra

 * Frustum

Real analysis

 * Semi-continuity
 * Stone–Weierstrass theorem
 * Universal chord theorem

Trigonometry

 * List of trigonometric identities
 * Exact trigonometric constants

Algebraic geometry

 * Hilbert's Nullstellensatz
 * Scheme (mathematics)
 * Zariski topology

Algebraic topology

 * Category:Theorems in algebraic topology

Fundamental group

 * Seifert–Van Kampen theorem

Homology theory

 * Homology (mathematics)
 * Category:Homology theory
 * Mayer–Vietoris sequence

Amenability

 * Amenable group
 * Følner sequence

Calculus of variations

 * Category:Calculus of variations
 * Euler–Lagrange equation
 * Fundamental lemma of calculus of variations (du Bois-Reymond lemma)

Category theory

 * Template:Category theory
 * Category:Category theory

Basic definitions

 * Category (mathematics) and Morphisms
 * Small category

Additive categories

 * Category:Additive categories
 * Abelian categories (Kernel (category theory), Cokernel)
 * Mitchell's embedding theorem (also known as the Freyd–Mitchell theorem)

Categories

 * Enriched category
 * Comma category
 * Filtered category

Limits and colimits

 * Category:Limits (category theory)
 * Complete category
 * Direct limit
 * Equaliser (mathematics)
 * Inverse limit
 * Product (category theory)

Functors

 * Adjoint functors
 * Yoneda lemma

Monads

 * Codensity monad

Monoidal categories

 * Braided monoidal category
 * Enriched category
 * Cartesian closed category
 * Symmetric monoidal category

Specific categories

 * Category:Categories in category theory
 * Category of finite-dimensional Hilbert spaces
 * Category of finite sets
 * Category of relations
 * Category of sets
 * Category of topological spaces

Commutative algebra

 * Spectrum of a ring
 * Zariski topology

Complex analysis

 * Analytic continuation
 * Cauchy's integral formula $$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,dz.\,$$
 * Cauchy's integral theorem (also known as the Cauchy–Goursat theorem) $$\int_C f(z)\,dz = 0. $$
 * Cauchy–Riemann equations (Looman–Menchoff theorem)
 * Dirichlet series
 * Holomorphic function (Analyticity of holomorphic functions)

Specific functions

 * Euler function (Pentagonal number theorem)
 * Riemann zeta function

Continued fractions

 * Category:Continued fractions
 * Padé approximant

Differential geometry

 * Category:Differential geometry
 * Riemannian manifold

Differential topology

 * Category:Differential topology
 * Exotic sphere
 * Hairy ball theorem
 * Poincaré–Hopf theorem
 * Whitney embedding theorem

Algebraic elements and algebraically closed fields

 * Algebraic element (Minimal polynomial (field theory))
 * Algebraically closed field (Algebraic closure)

Field extensions

 * Degree of a field extension (Finite extension)
 * Galois extension
 * Normal extension
 * Separable extension
 * Simple extension
 * Splitting field

Finite fields

 * Category:Finite fields
 * Frobenius endomorphism
 * Wedderburn's little theorem (Artin–Zorn theorem)

Galois theory

 * Category:Galois theory
 * Galois group
 * Inverse Galois problem

Fixed-point theorems

 * Category:Fixed-point theorems
 * Banach fixed-point theorem
 * Brouwer fixed-point theorem
 * Markov–Kakutani fixed-point theorem
 * Schauder fixed-point theorem

Fourier analysis

 * Category:Fourier analysis
 * Category:Theorems in Fourier analysis

Fourier series

 * Parseval's identity

Fourier transform

 * Template:Fourier transforms
 * Parseval's theorem
 * Sinc function

Functional analysis

 * Category:Banach spaces
 * Category:Normed spaces
 * Category:Theorems in functional analysis
 * Template:Functional analysis
 * Template:Banach spaces

Basic examples of normed spaces

 * Lp space
 * L-infinity
 * Sequence space

Basic results

 * Closed graph theorem
 * Hahn–Banach theorem
 * Open mapping theorem (functional analysis) (Banach–Schauder theorem)
 * Uniform boundedness principle (Banach–Steinhaus theorem)

Dual spaces

 * Banach–Alaoglu theorem
 * Eberlein–Šmulian theorem
 * Krein–Smulian theorem
 * Weak topology

Hilbert spaces

 * Orthonormal basis (Bessel's inequality, Parseval's identity)

Locally convex topological vector spaces

 * Krein–Milman theorem

Riesz spaces

 * Riesz space

Various (classes of) spaces

 * Besov space
 * Orlicz space

Functional equations

 * Cauchy's functional equation

General topology

 * Category:General topology
 * Category:Theorems in topology

Basic notions

 * Topological space
 * Base and subbase
 * Closure (topology) - Kuratowski's closure-complement problem
 * Neighbourhood basis (local basis)

Examples and counterexamples, specific spaces

 * Category:Topological spaces
 * Counterexamples in Topology
 * List of topologies
 * Baire space (set theory)
 * Cantor cube, Cantor space
 * Lower limit topology (Sorgenfrey line)
 * Moore plane
 * Order topology
 * Sorgenfrey plane
 * Topologist's sine curve

Baire category theorem

 * Baire category theorem (Meagre set, Nowhere dense set)
 * Baire space

Compactifications

 * Category:Compactification (mathematics)
 * Alexandroff extension
 * Stone–Čech compactification
 * Wallman compactification

Compactness

 * Category:Compactness (mathematics)
 * Compact space

Results

 * Alexander subbase theorem
 * Tube lemma
 * Tychonoff's theorem

Generalizations

 * Compactly generated space
 * Countably compact space
 * H-closed space
 * Limit point compact
 * Locally compact space
 * Lindelöf space
 * Metacompact space
 * Pseudocompact space
 * Relatively compact subspace
 * σ-compact space
 * Sequentially compact space

Connectedness

 * Connected space
 * Arcwise connected
 * Extremally disconnected space
 * Locally connected space
 * Path connected (Path (topology))
 * Simply connected space
 * Totally disconnected space
 * Zero-dimensional space

Continuum theory

 * Continuum (topology)
 * Category:Continuum theory
 * Indecomposable continuum
 * Pseudo-arc

Filters

 * Filter (mathematics)
 * Filters in topology
 * Cauchy filter

Nets

 * Net (mathematics)
 * Subnet (mathematics)

Sequences

 * Sequential space

Dimension theory

 * Category:Dimension theory
 * Hausdorff dimension (Hausdorff measure)
 * Inductive dimension

Generalizations

 * Convergence space
 * Pretopological space
 * Proximity space

Maps

 * Open and closed maps
 * Quotient map
 * Retraction (topology)

Metric spaces

 * Complete metric space
 * Pseudometric space
 * Totally bounded space

Metrizable spaces

 * Completely metrizable space
 * Uniformizable space and Completely uniformizable space

Metrization theorems

 * Bing metrization theorem
 * Nagata–Smirnov metrization theorem
 * Urysohn's metrization theorem

Paracompact spaces

 * Partition of unity
 * Refinement (topology), Star refinement

Selection principles

 * Menger space
 * Hurewicz space
 * Rothberger space

Separation axioms

 * Hausdorff space - Collectionwise Hausdorff space
 * Normal space (Tietze extension theorem, Urysohn's lemma) - Collectionwise normal space, Monotonically normal space
 * Tychonoff space = completely regular T2 space
 * Urysohn and completely Hausdorff spaces (completely Hausdorff space = functionally Hausdorff space)

Topological constructions

 * Disjoint union (topology)
 * Final topology
 * Initial topology
 * Product topology
 * Quotient space (topology)
 * Subspace (topology)

Topological games

 * Banach–Mazur game

Uniform spaces

 * Complete uniform space
 * Totally bounded space

Various classes of spaces

 * Glossary of topology
 * Category:Properties of topological spaces
 * Alexandrov topology (Alexandrov-discrete space or finitely generated space)
 * Developable space
 * Door space
 * P-space
 * Shrinking space

Graph theory

 * Cayley graph
 * Extremal graph theory
 * Moore graph

Algebraic graph theory

 * Category:Algebraic graph theory
 * Laplacian matrix
 * Spectral graph theory

Graph algorithms

 * Floyd–Warshall algorithm is an algorithm for finding shortest paths.
 * Kruskal's algorithm

Graph coloring

 * Four color theorem

Graph invariants

 * Category:Graph invariants
 * Girth (graph theory)

Graph polynomials

 * Tutte polynomial

Group theory

 * List of group theory topics
 * Template:Group theory sidebar
 * Template:Group navbox
 * Category:Theorems in group theory
 * Solvable group and derived series

Elementary results

 * Cauchy's theorem (group theory)
 * Center (group theory)
 * Characteristic subgroup
 * Inner automorphism

Cosets, normal subgroups, quotient groups

 * Congruence relation
 * Isomorphism theorems
 * Lagrange's theorem (group theory) (Index of a subgroup)

Homomorphisms

 * Inner automorphism

Subgroups

 * Characteristic subgroup
 * Commutator, Commutator subgroup

Finite groups

 * Category:Finite groups
 * Category:Theorems about finite groups
 * Sylow theorems

Group actions

 * Category:Group actions (mathematics)
 * Burnside's lemma
 * Orbit-stabilizer theorem
 * Pólya enumeration theorem

Infinite group theory

 * Category:Infinite group theory
 * Profinite group
 * Residually finite group
 * Sofic group

Specific groups

 * Dihedral group $Dn$
 * Klein four-group (Vierergruppe)
 * Symmetric group $Sn$

Harmonic analysis

 * Category:Harmonic analysis
 * Category:Theorems in harmonic analysis
 * Plancherel theorem
 * Pontryagin duality

Homotopy theory

 * Category:Homotopy theory
 * Homotopy group

History of mathematics

 * History of mathematical analysis
 * History of mathematical notation
 * History of mathematics education
 * History of number theory

Timelines

 * Category:Mathematics timelines
 * Timeline of mathematics
 * Timeline of number theory

Inequalities

 * Inequality of arithmetic and geometric means (Generalized mean)

Integration

 * Template:Integrals
 * Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral, Luzin integral or Perron integral,
 * Riemann integral
 * Riemann–Stieltjes integral

Lebesgue integration

 * Monotone convergence theorem
 * Dominated convergence theorem

Basic notions

 * Basis (linear algebra) (Hamel basis)
 * Dimension (vector space) (Dimension theorem for vector spaces)

Basic results

 * Category:Theorems in linear algebra
 * Rouché–Capelli theorem

Affine spaces

 * Affine transformation
 * Barycentric coordinate system

Determinants

 * Category:Determinants
 * Permanent (mathematics)

Eigenvalues and eigenvectors

 * Companion matrix
 * Gershgorin circle theorem

Inner product spaces

 * Frame (linear algebra)

First-order logic

 * List of rules of inference
 * Existential generalization (existential introduction)
 * Existential instantiation (existential elimination)

Model theory

 * Category:Model theory
 * Categorical theory (Morley's categoricity theorem)
 * Compactness theorem
 * Löwenheim–Skolem theorem
 * Ultraproduct (Łoś's theorem)

Proof theory

 * Ordinal analysis - proof-theoretic ordinal

Matrix theory

 * Category:Matrix theory
 * Category:Matrices
 * List of named matrices
 * Template:Matrix classes
 * Kronecker product
 * Moore–Penrose inverse
 * Perron–Frobenius theorem
 * Schur complement
 * Singular value decomposition

Hermitian matrices

 * Schur–Horn theorem

Matrix calculus

 * Jacobi's formula
 * Matrix exponential

Matrix decompositions

 * Cholesky decomposition
 * QR decomposition
 * LU decomposition

Types of matrices

 * Doubly stochastic matrix (Muirhead's Inequality)

Measure theory

 * Category:Measure theory
 * Category:Measures (measure theory)
 * Category:Theorems in measure theory
 * Borel–Cantelli lemma
 * Fatou's lemma
 * Lebesgue measure
 * Peano–Jordan measure
 * Outer measure
 * Zero–one law (Kolmogorov's zero–one law)

Real numbers

 * Template:Real numbers
 * Construction of the real numbers
 * Real closed field

Elementary number theory

 * Binomial coefficients: Lucas's theorem, Kummer's theorem
 * Factorial: Legendre's formula

Congruences and modular arithmetic

 * Chinese remainder theorem
 * Lucas's theorem
 * Wilson's theorem

Additive number theory

 * Category:Additive number theory
 * Goldbach's conjecture
 * Waring's problem

Algebraic number theory

 * Category:Algebraic number theory
 * Algebraic number field (Ring of integers)
 * Cyclotomic field
 * Ideal class group, Class number (number theory)

Analytic number theory

 * Category:Analytic number theory
 * Category:Theorems in analytic number theory
 * Dirichlet character
 * Mertens' theorems (Meissel–Mertens constant)
 * Ramanujan's master theorem

Arithmetic functions

 * Category:Arithmetic functions
 * Category:Additive functions
 * Category:Multiplicative functions
 * Landau's function
 * Möbius function
 * Prime omega function
 * Von Mangoldt function

Diophantine equations

 * Integer triangle, Heronian triangle - sides and the area are integers
 * Mordell curve - Mordell's equation y2 = x3 + n
 * Pell's equation
 * Pythagorean triple (Formulas for generating Pythagorean triples)
 * Ramanujan–Nagell equation

Geometry of numbers

 * Category:Geometry of numbers
 * Minkowski's theorem

Primality tests

 * AKS primality test
 * Lucas–Lehmer primality test

Prime numbers

 * Category:Prime numbers
 * Category:Theorems about prime numbers
 * List of prime numbers
 * Bertrand's postulate (Proof of Bertrand's postulate)
 * Prime number theorem
 * Primes in arithmetic progression (Dirichlet's theorem on arithmetic progressions, Green–Tao theorem)

Special primes

 * Category:Classes of prime numbers
 * Factorial prime
 * Mersenne prime (Mersenne conjectures)
 * Primorial prime

Quadratic residues and quadratic reciprocity

 * Category:Quadratic residue
 * Euler's criterion
 * Gauss's lemma (number theory)
 * Legendre symbol (Jacobi symbol, Kronecker symbol)
 * Proofs of quadratic reciprocity
 * Zolotarev's lemma

Sums of squares

 * Sum of two squares theorem
 * Lagrange's four-square theorem (Euler's four-square identity, Hurwitz quaternion)
 * Legendre's three-square theorem
 * Sum of squares function

Transcendental numbers

 * Category:Transcendental numbers
 * Lindemann–Weierstrass theorem
 * Gelfond–Schneider theorem (Gelfond–Schneider constant)

Various

 * Normal number
 * Irrationality measure

Open problems

 * List of unsolved problems in mathematics
 * Category:Unsolved problems in mathematics
 * List of conjectures
 * Collatz conjecture
 * Lonely runner conjecture
 * Moore graph
 * Moving sofa problem
 * Singmaster's conjecture

Order theory

 * Szpilrajn extension theorem states that every partial order is contained in a total order.

Pointless topology

 * Complete Heyting algebra (Frames and locales)

Polynomials

 * Category:Polynomials
 * Algebraic numbers and transcendental numbers
 * Gauss's lemma (polynomials)
 * Geometrical properties of polynomial roots

Basic definitions and results

 * Factor theorem
 * Horner's method

Cubic, quartic, quintic polynomials

 * Bring radical or ultraradical of a real number a is the unique real root of the polynomial $$x^5 + x + a.$$

Symmetric polynomials

 * Elementary symmetric polynomial
 * Newton's identities

Probability and statistics

 * Simpson's paradox

Inequalities

 * Chebyshev's inequality
 * Markov's inequality

Representation theory of finite groups

 * Maschke's theorem
 * Representation theory of the symmetric group
 * Schur polynomial (Littlewood–Richardson rule)

Basic definitions

 * Characteristic (algebra)

Euclidean domains

 * Eisenstein integers and Gaussian integers
 * Unique factorization domain

Set theory

 * Category:Set theory

Books

 * Equivalents of the Axiom of Choice
 * Naive Set Theory (book)
 * Set Theory: An Introduction to Independence Proofs
 * The Higher Infinite

Axiom of choice

 * Category:Axiom of choice
 * Axiom of countable choice
 * Axiom of dependent choice
 * Tarski's theorem about choice
 * Teichmüller–Tukey lemma
 * Ultrafilter lemma
 * Well-ordering theorem
 * Zorn's lemma

Basics

 * Pairing function (Fueter–Pólya theorem - there is only one pairing function which is a quadratic polynomial.)

Cardinals

 * Cardinal number
 * Category:Cardinal numbers
 * Cantor's theorem
 * Cardinality of the continuum
 * Cofinality
 * Hartogs number
 * Schröder–Bernstein theorem
 * Scott's trick
 * Von Neumann cardinal assignment

Cardinal arithmetic

 * Cardinal arithmetic
 * Easton's theorem
 * König's theorem (set theory)

Descriptive set theory

 * Baire space (set theory)
 * Cantor space
 * Polish space
 * Suslin scheme
 * Tree (descriptive set theory)

Finite/infinite sets

 * Amorphous set
 * Dedekind-infinite set
 * Finite set

Forcing

 * Category:Forcing (mathematics)
 * Countable chain condition
 * Cohen forcing
 * Easton forcing
 * Solovay model

Independence results

 * Category:Independence results
 * List of statements independent of ZFC
 * Axiom of constructibility V=L (Constructible universe)
 * Continuum hypothesis
 * Martin's axiom
 * Martin's maximum
 * Proper forcing axiom
 * Shoenfield's absoluteness theorem

Infinitary combinatorics

 * Stationary set

Infinite games

 * Axiom of determinacy
 * Banach–Mazur game
 * Borel determinacy theorem
 * Choquet game

Small uncountable cardinals

 * Cardinal characteristic of the continuum
 * Cichoń's diagram
 * Set theory of the real line

Infinite trees

 * Tree (set theory)
 * Category:Trees (set theory)
 * Kőnig's lemma

Ordinals

 * Ordinal number
 * Category:Ordinal numbers
 * Computable ordinal (Recursive ordinal)
 * Ordinal arithmetic

Ultrafilters

 * Ultrafilter
 * Ultrafilter (set theory)
 * Ramsey ultrafilter
 * Rudin–Keisler ordering

Wellfoundedness

 * Category:Wellfoundedness
 * Well-founded relation (Hereditarily well-founded set)

ZFC

 * Zermelo–Fraenkel set theory
 * Axiom of regularity (Von Neumann universe)

Special functions

 * Category:Special hypergeometric functions
 * Lambert W function

Beta function

 * Euler integral

Dirichlet beta function

 * Catalan's constant $$G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} - \cdots$$

Gamma function

 * Category:Gamma and related functions
 * Incomplete gamma function

Zeta and L-functions

 * L-function
 * List of zeta functions
 * Category:Zeta and L-functions
 * Dirichlet eta function
 * Dirichlet L-function

Stochastic differential equations

 * Category:Stochastic differential equations
 * Stochastic partial differential equation