User:Kompik/bookmarks/Math

General

• Wikipedia: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

Differentiation and derivatives

Derivatives

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

Theorems

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

Extrema

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

Geometry

Analytic geometry

• Line (geometry)
• Normal (geometry)

Plane 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

Field theory

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

• L^p 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

Convergence

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 D_n
• Klein four-group (Vierergruppe)
• Symmetric group S_n

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

Linear algebra

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)

Logic and model theory

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)

Number systems

Real numbers

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

Number theory

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

Representation theory of finite groups

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

Ring theory

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 calculus

Stochastic differential equations

• Category:Stochastic differential equations
• Stochastic partial differential equation