# kb

id: list-of-publications-in-mathematics-20260505-8
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-8.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
==== "Полиномиальный алгоритм в линейном программировании" ==== Khachiyan, Leonid Genrikhovich (1979). Полиномиальный алгоритм в линейном программировании [A polynomial algorithm for linear programming]. Doklady Akademii Nauk SSSR (in Russian). 244: 1093–1096.. Khachiyan's work on the ellipsoid method. This was the first polynomial time algorithm for linear programming. == Early manuscripts == These are publications that are not necessarily relevant to a mathematician nowadays, but are nonetheless important publications in the history of mathematics. === Moscow Mathematical Papyrus === This is one of the earliest mathematical treatises that still survives today. The Papyrus contains 25 problems involving arithmetic, geometry, and algebra, each with a solution given. Written in Ancient Egypt at approximately 1850 BC. === Rhind Mathematical Papyrus === Ahmes (scribe) One of the oldest mathematical texts, dating to the Second Intermediate Period of ancient Egypt. It was copied by the scribe Ahmes (properly Ahmose) from an older Middle Kingdom papyrus. It laid the foundations of Egyptian mathematics and in turn, later influenced Greek and Hellenistic mathematics. Besides describing how to obtain an approximation of π only missing the mark by less than one per cent, Ahmes make use of a kind of an analogue of the cotangent. === Archimedes Palimpsest ===


id: list-of-publications-in-mathematics-20260505-9
file: data/en.wikipedia.org/wiki/List_of_publications_in_mathematics-9.md
category: reference
title: List of publications in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_mathematics
date_saved: 2026-05-05T08:16:32.440767+00:00
instance: kb-cron
parent: list-of-publications-in-mathematics-20260505-0
summary:
=== Naive Set Theory === Paul Halmos An undergraduate introduction to not-very-naive set theory which has lasted for decades. It is still considered by many to be the best introduction to set theory for beginners. While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo–Fraenkel set theory and gives correct and rigorous definitions for basic objects. Where it differs from a "true" axiomatic set theory book is its character: There are no long-winded discussions of axiomatic minutiae, and there is next to nothing about topics like large cardinals. Instead it aims, and succeeds, in being intelligible to someone who has never thought about set theory before. === Cardinal and Ordinal Numbers === Wacław Sierpiński The nec plus ultra reference for basic facts about cardinal and ordinal numbers. If you have a question about the cardinality of sets occurring in everyday mathematics, the first place to look is this book, first published in the early 1950s but based on the author's lectures on the subject over the preceding 40 years. === Set Theory: An Introduction to Independence Proofs === Kenneth Kunen This book is not really


id: list-of-q-analogs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_q-analogs-0.md
category: reference
title: List of q-analogs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_q-analogs
date_saved: 2026-05-05T08:16:33.618494+00:00
instance: kb-cron
parent: list-of-q-analogs-20260505-0
summary:
This is a list of q-analogs in mathematics and related fields. == Algebra == Iwahori–Hecke algebra Quantum affine algebra Quantum enveloping algebra Quantum group == Analysis == Jackson integral q-derivative q-difference polynomial Quantum calculus == Combinatorics == LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity == Orthogonal polynomials == q-Bessel polynomials q-Charlier polynomials q-Hahn polynomials q-Jacobi polynomials: Big q-Jacobi polynomials Continuous q-Jacobi polynomials Little q-Jacobi polynomials q-Krawtchouk polynomials q-Laguerre polynomials Continuous q-Legendre polynomials q-Meixner polynomials q-Meixner–Pollaczek polynomials q-Racah polynomials == Probability and statistics == Gaussian q-distribution q-exponential distribution q-Weibull distribution Tsallis q-Gaussian Tsallis entropy == Special functions == Basic hypergeometric series Elliptic gamma function Hahn–Exton q-Bessel function Jackson q-Bessel function q-exponential q-gamma function q-theta function == See also == Lists of mathematics topics


id: list-of-mathematical-topics-in-quantum-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory-0.md
category: reference
title: List of mathematical topics in quantum theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_quantum_theory
date_saved: 2026-05-05T08:16:34.798624+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-quantum-theory-20260505-0
summary:
This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions. == Mathematical formulation of quantum mechanics == bra–ket notation canonical commutation relation complete set of commuting observables Heisenberg picture Hilbert space Interaction picture Measurement in quantum mechanics quantum field theory quantum logic quantum operation Schrödinger picture semiclassical statistical ensemble wavefunction wave–particle duality Wightman axioms WKB approximation == Schrödinger equation == quantum mechanics, matrix mechanics, Hamiltonian (quantum mechanics) particle in a box particle in a ring particle in a spherically symmetric potential quantum harmonic oscillator hydrogen atom ring wave guide particle in a one-dimensional lattice (periodic potential) Fock symmetry in theory of hydrogen == Symmetry == identical particles angular momentum angular momentum operator rotational invariance rotational symmetry rotation operator translational symmetry Lorentz symmetry Parity transformation Noether's theorem Noether charge Spin (physics) isospin Aman matrices scale invariance spontaneous symmetry breaking supersymmetry breaking == Quantum states == quantum number Pauli exclusion principle quantum indeterminacy uncertainty principle wavefunction collapse zero-point energy bound state coherent state squeezed coherent state density state Fock state, Fock space vacuum state quasinormal mode no-cloning theorem quantum entanglement


id: list-of-random-number-generators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_random_number_generators-0.md
category: reference
title: List of random number generators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_random_number_generators
date_saved: 2026-05-05T08:16:35.976218+00:00
instance: kb-cron
parent: list-of-random-number-generators-20260505-0
summary:
Random number generators are important in many kinds of technical applications, including physics, engineering or mathematical computer studies (e.g., Monte Carlo simulations), cryptography and gambling (on game servers). This list includes many common types, regardless of quality or applicability to a given use case. == Pseudorandom number generators (PRNGs) == The following algorithms are pseudorandom number generators. == Cryptographic algorithms == Cipher algorithms and cryptographic hashes can be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators. These include: Stream ciphers. Popular choices are Salsa20 or ChaCha (often with the number of rounds reduced to 8 for speed), ISAAC, HC-128 and RC4. Block ciphers in counter mode. Common choices are AES (which is very fast on systems supporting it in hardware), TwoFish, Serpent and Camellia. Cryptographic hash functions A few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a `true' random stream to a computationally difficult problem. These approaches are theoretically important but are too slow to be practical in most applications. They include: Blum–Micali algorithm (1984) Blum Blum Shub


id: list-of-real-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-0.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
This is a list of articles that are considered real analysis topics. See also: glossary of real and complex analysis. == General topics == === Limits === Limit of a sequence Subsequential limit – the limit of some subsequence Limit of a function (see List of limits for a list of limits of common functions) One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions === Sequences and series === (see also list of mathematical series) Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number Harmonic progression – a


id: list-of-real-analysis-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_real_analysis_topics-1.md
category: reference
title: List of real analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_real_analysis_topics
date_saved: 2026-05-05T08:16:37.250027+00:00
instance: kb-cron
parent: list-of-real-analysis-topics-20260505-0
summary:
== Fundamental theorems == Monotone convergence theorem – relates monotonicity with convergence Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc Taylor's theorem – gives an approximation of a k {\displaystyle k} times differentiable function around a given point by a k {\displaystyle k} -th order Taylor-polynomial. L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms Abel's theorem – relates the limit of a power series to the sum of its coefficients Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function Darboux's theorem – states that all functions that result from the


id: list-of-properties-of-sets-of-reals-20260505-0
file: data/en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals-0.md
category: reference
title: List of properties of sets of reals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_properties_of_sets_of_reals
date_saved: 2026-05-05T08:16:38.430648+00:00
instance: kb-cron
parent: list-of-properties-of-sets-of-reals-20260505-0
summary:
This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology. == Definability properties == Borel set Analytic set C-measurable set Projective set Inductive set Infinity-Borel set Suslin set Homogeneously Suslin set Weakly homogeneously Suslin set Set of uniqueness == Regularity properties == Property of Baire Lebesgue measurable Universally measurable set Perfect set property Universally Baire set == Largeness and smallness properties == Meager set Comeager set - A comeager set is one whose complement is meager. Null set Conull set Dense set Nowhere dense set


id: list-of-recreational-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_recreational_number_theory_topics-0.md
category: reference
title: List of recreational number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_recreational_number_theory_topics
date_saved: 2026-05-05T08:16:39.664179+00:00
instance: kb-cron
parent: list-of-recreational-number-theory-topics-20260505-0
summary:
This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging problems posed purely for their own sake. See list of number theory topics for pages dealing with aspects of number theory with more consolidated theories. == Number sequences == Integer sequence Fibonacci sequence Golden mean base Fibonacci coding Lucas sequence Padovan sequence Figurate numbers Polygonal number Triangular number Square number Pentagonal number Hexagonal number Heptagonal number Octagonal number Nonagonal number Decagonal number Centered polygonal number Centered square number Centered pentagonal number Centered hexagonal number Tetrahedral number Pyramidal number Triangular pyramidal number Square pyramidal number Pentagonal pyramidal number Hexagonal pyramidal number Heptagonal pyramidal number Octahedral number Star number Perfect number Quasiperfect number Almost perfect number Multiply perfect number Hyperperfect number Semiperfect number Primitive semiperfect number Unitary perfect number Weird number Untouchable number Amicable number Sociable number Abundant number Deficient number Amenable number Aliquot sequence Super-Poulet number Lucky number Powerful number Primeval number Palindromic number Telephone number Triangular square number Harmonic divisor number Sphenic number Smith number Double Mersenne number Zeisel number Heteromecic number Niven numbers Superparticular number Highly composite number Highly totient


id: list-of-mathematics-reference-tables-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_reference_tables-0.md
category: reference
title: List of mathematics reference tables
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_reference_tables
date_saved: 2026-05-05T08:16:41.067246+00:00
instance: kb-cron
parent: list-of-mathematics-reference-tables-20260505-0
summary:
See also: List of reference tables == Mathematics == List of mathematical topics List of statistical topics List of mathematical functions List of mathematical theorems List of mathematical proofs List of matrices List of numbers List of relativistic equations List of small groups Mathematical constants Sporadic group Table of Clebsch-Gordan coefficients Table of derivatives Table of divisors Table of integrals Table of mathematical symbols Table of prime factors Taylor series Timeline of mathematics Trigonometric identities Truth table


id: list-of-mathematical-topics-in-relativity-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity-0.md
category: reference
title: List of mathematical topics in relativity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_relativity
date_saved: 2026-05-05T08:16:42.212375+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-relativity-20260505-0
summary:
This is a list of mathematical topics in relativity, by Wikipedia page. == Special relativity == Foundational issues principle of relativity speed of light faster-than-light biquaternion conjugate diameters four-vector four-acceleration four-force four-gradient four-momentum four-velocity hyperbolic orthogonality hyperboloid model light-like Lorentz covariance Lorentz group Lorentz transformation Lorentz–FitzGerald contraction hypothesis Minkowski diagram Minkowski space Poincaré group proper length proper time rapidity relativistic wave equations relativistic mass split-complex number unit hyperbola world line == General relativity == black holes no-hair theorem Hawking radiation Hawking temperature Black hole entropy charged black hole rotating black hole micro black hole Schwarzschild black hole Schwarzschild metric Schwarzschild radius Reissner–Nordström black hole Immirzi parameter closed timelike curve cosmic censorship hypothesis chronology protection conjecture Einstein–Cartan theory Einstein's field equation geodesic gravitational redshift Penrose–Hawking singularity theorems Pseudo-Riemannian manifold stress–energy tensor worm hole == Cosmology == anti-de Sitter space Ashtekar variables Batalin–Vilkovisky formalism Big Bang Cauchy horizon cosmic inflation cosmic microwave background cosmic variance cosmological constant dark energy dark matter de Sitter space Friedmann–Lemaître–Robertson–Walker metric horizon problem large-scale structure of the cosmos Randall–Sundrum model warped geometry Weyl curvature hypothesis


id: list-of-representation-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_representation_theory_topics-0.md
category: reference
title: List of representation theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_representation_theory_topics
date_saved: 2026-05-05T08:16:43.462653+00:00
instance: kb-cron
parent: list-of-representation-theory-topics-20260505-0
summary:
This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. See also: Glossary of representation theory == General representation theory == Linear representation Unitary representation Trivial representation Irreducible representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation Schur's lemma Restricted representation == Representation theory of groups == Group representation Group ring Maschke's theorem Regular representation Character (mathematics) Character theory Class function Representation theory of finite groups Modular representation theory Frobenius reciprocity Restricted representation Induced representation Peter–Weyl theorem Young tableau Spherical harmonic Hecke operator Representation theory of the symmetric group Representation theory of diffeomorphism groups Permutation representation Affine representation Projective representation Central extension == Representation theory of Lie groups and Lie algebras == Representation of a Lie group Lie algebra representation, Representation of a Lie superalgebra Universal enveloping algebra Casimir element Infinitesimal character Harish-Chandra homomorphism Fundamental representation Antifundamental representation Bifundamental representation Adjoint representation Weight (representation theory) Cartan's theorem Spinor Wigner's classification, Representation theory of the Poincaré group Wigner–Eckart theorem Stone–von Neumann theorem Orbit method Kirillov character formula Weyl character formula Discrete series representation Principal series representation Borel–Weil–Bott theorem


id: list-of-formulas-in-riemannian-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-0.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. == Christoffel symbols, covariant derivative == In a smooth coordinate chart, the Christoffel symbols of the first kind are given by Γ k i j = 1 2 ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 ( g k i , j + g k j , i − g i j , k ) , {\displaystyle \Gamma _{kij}={\frac {1}{2}}\left({\frac {\partial }{\partial x^{j}}}g_{ki}+{\frac {\partial }{\partial x^{i}}}g_{kj}-{\frac {\partial }{\partial x^{k}}}g_{ij}\right)={\frac {1}{2}}\left(g_{ki,j}+g_{kj,i}-g_{ij,k}\right)\,,} and the Christoffel symbols of the second kind by Γ m i j = g m k Γ k i j = 1 2 g m k ( ∂ ∂ x j g k i + ∂ ∂ x i g k j − ∂ ∂ x k g i j ) = 1 2 g m k ( g k i , j + g k j , i − g i j , k ) . {\displaystyle {\begin{aligned}\Gamma


id: list-of-formulas-in-riemannian-geometry-20260505-1
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-1.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i j k l = ∂ Γ j k l ∂ x i − ∂ Γ i k l ∂ x j + ( Γ j k p Γ i p l − Γ i k p Γ j p l ) {\displaystyle {R_{ijk}}^{l}={\frac {\partial \Gamma _{jk}^{l}}{\partial x^{i}}}-{\frac {\partial \Gamma _{ik}^{l}}{\partial x^{j}}}+{\big (}\Gamma _{jk}^{p}\Gamma _{ip}^{l}-\Gamma _{ik}^{p}\Gamma _{jp}^{l}{\big )}} R ( u , v ) w = ∇ u ∇ v w − ∇ v ∇ u w − ∇ [ u , v ] w {\displaystyle R(u,v)w=\nabla _{u}\nabla _{v}w-\nabla _{v}\nabla _{u}w-\nabla _{[u,v]}w} ==== (3,1) Riemann curvature tensor ==== R j k l i = ∂ Γ l j i ∂ x k − ∂ Γ k j i ∂ x l + ( Γ k p i Γ l j p − Γ l p i Γ k j p ) {\displaystyle {R_{jkl}^{i}}={\frac {\partial \Gamma _{lj}^{i}}{\partial x^{k}}}-{\frac {\partial \Gamma _{kj}^{i}}{\partial x^{l}}}+{\big (}\Gamma _{kp}^{i}\Gamma _{lj}^{p}-\Gamma _{lp}^{i}\Gamma _{kj}^{p}{\big )}} ==== Ricci curvature ==== R i k = R i j k j {\displaystyle R_{ik}={R_{ijk}}^{j}} Ric ⁡ ( v , w ) = tr ⁡ ( u ↦ R ( u , v ) w ) {\displaystyle \operatorname {Ric} (v,w)=\operatorname {tr} (u\mapsto


id: list-of-formulas-in-riemannian-geometry-20260505-2
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-2.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
div g ⁡ Ric = 1 2 d R {\displaystyle \operatorname {div} _{g}\operatorname {Ric} ={\frac {1}{2}}dR} Equivalently: g p q ∇ p G q k = 0 {\displaystyle g^{pq}\nabla _{p}G_{qk}=0} div g ⁡ G = 0 {\displaystyle \operatorname {div} _{g}G=0} ==== Ricci identity ==== If X {\displaystyle X} is a vector field then ∇ i ∇ j X k − ∇ j ∇ i X k = − R i j p k X p , {\displaystyle \nabla _{i}\nabla _{j}X^{k}-\nabla _{j}\nabla _{i}X^{k}=-{R_{ijp}}^{k}X^{p},} which is just the definition of the Riemann tensor. If ω {\displaystyle \omega } is a one-form then ∇ i ∇ j ω k − ∇ j ∇ i ω k = R i j k p ω p . {\displaystyle \nabla _{i}\nabla _{j}\omega _{k}-\nabla _{j}\nabla _{i}\omega _{k}={R_{ijk}}^{p}\omega _{p}.} More generally, if T {\displaystyle T} is a (0,k)-tensor field then ∇ i ∇ j T l 1 ⋯ l k − ∇ j ∇ i T l 1 ⋯ l k = R i j l 1 p T p l 2 ⋯ l k + ⋯ + R i j l k p T l 1 ⋯ l k − 1 p . {\displaystyle \nabla _{i}\nabla _{j}T_{l_{1}\cdots


id: list-of-formulas-in-riemannian-geometry-20260505-3
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-3.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
R i k ℓ m = 1 2 ( ∂ 2 g i m ∂ x k ∂ x ℓ + ∂ 2 g k ℓ ∂ x i ∂ x m − ∂ 2 g i ℓ ∂ x k ∂ x m − ∂ 2 g k m ∂ x i ∂ x ℓ ) {\displaystyle R_{ik\ell m}={\frac {1}{2}}\left({\frac {\partial ^{2}g_{im}}{\partial x^{k}\partial x^{\ell }}}+{\frac {\partial ^{2}g_{k\ell }}{\partial x^{i}\partial x^{m}}}-{\frac {\partial ^{2}g_{i\ell }}{\partial x^{k}\partial x^{m}}}-{\frac {\partial ^{2}g_{km}}{\partial x^{i}\partial x^{\ell }}}\right)} R ℓ i j k = ∂ ∂ x j Γ ℓ i k − ∂ ∂ x k Γ ℓ i j {\displaystyle R^{\ell }{}_{ijk}={\frac {\partial }{\partial x^{j}}}\Gamma ^{\ell }{}_{ik}-{\frac {\partial }{\partial x^{k}}}\Gamma ^{\ell }{}_{ij}} == Conformal change == Let g {\displaystyle g} be a Riemannian or pseudo-Riemannian metric on a smooth manifold M {\displaystyle M} , and φ {\displaystyle \varphi } a smooth real-valued function on M {\displaystyle M} . Then g ~ = e 2 φ g {\displaystyle {\tilde {g}}=e^{2\varphi }g} is also a Riemannian metric on M {\displaystyle M} . We say that g ~ {\displaystyle {\tilde {g}}} is (pointwise) conformal to g {\displaystyle g} . Evidently, conformality of metrics is an equivalence relation. Here


id: list-of-formulas-in-riemannian-geometry-20260505-4
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-4.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
=== Hodge operator on p-forms === ∗ ~ i 1 ⋯ i n − p j 1 ⋯ j p = e ( n − 2 p ) φ ∗ i 1 ⋯ i n − p j 1 ⋯ j p {\displaystyle {\widetilde {\ast }}_{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}=e^{(n-2p)\varphi }\ast _{i_{1}\cdots i_{n-p}}^{j_{1}\cdots j_{p}}} ∗ ~ = e ( n − 2 p ) φ ∗ {\displaystyle {\widetilde {\ast }}=e^{(n-2p)\varphi }\ast } === Codifferential on p-forms === d ∗ ~ j 1 ⋯ j p − 1 i 1 ⋯ i p = e − 2 φ ( d ∗ ) j 1 ⋯ j p − 1 i 1 ⋯ i p − ( n − 2 p ) e − 2 φ ∇ i 1 φ δ j 1 i 2 ⋯ δ j p − 1 i p {\displaystyle {\widetilde {d^{\ast }}}_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}=e^{-2\varphi }(d^{\ast })_{j_{1}\cdots j_{p-1}}^{i_{1}\cdots i_{p}}-(n-2p)e^{-2\varphi }\nabla ^{i_{1}}\varphi \delta _{j_{1}}^{i_{2}}\cdots \delta _{j_{p-1}}^{i_{p}}} d ∗ ~ = e − 2 φ d ∗ − ( n − 2 p ) e − 2 φ ι ∇ φ {\displaystyle {\widetilde {d^{\ast }}}=e^{-2\varphi }d^{\ast }-(n-2p)e^{-2\varphi }\iota _{\nabla \varphi }} === Laplacian on functions === Δ ~ Φ = e −


id: list-of-formulas-in-riemannian-geometry-20260505-5
file: data/en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry-5.md
category: reference
title: List of formulas in Riemannian geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
date_saved: 2026-05-05T08:16:45.030637+00:00
instance: kb-cron
parent: list-of-formulas-in-riemannian-geometry-20260505-0
summary:
∂ ∂ t R i k = 1 2 ( ∇ p ∇ k v i p + ∇ i ( div ⁡ v ) k − ∇ i ∇ k ( tr g ⁡ v ) − Δ v i k ) + 1 2 R i p v p k − 1 2 R i p k q v p q {\displaystyle {\frac {\partial }{\partial t}}R_{ik}={\frac {1}{2}}{\Big (}\nabla ^{p}\nabla _{k}v_{ip}+\nabla _{i}(\operatorname {div} v)_{k}-\nabla _{i}\nabla _{k}(\operatorname {tr} _{g}v)-\Delta v_{ik}{\Big )}+{\frac {1}{2}}R_{i}^{p}v_{pk}-{\frac {1}{2}}R_{i}{}^{p}{}_{k}{}^{q}v_{pq}} ∂ ∂ t R = div g ⁡ div g ⁡ v − Δ ( tr g ⁡ v ) − ⟨ v , Ric ⟩ g {\displaystyle {\frac {\partial }{\partial t}}R=\operatorname {div} _{g}\operatorname {div} _{g}v-\Delta (\operatorname {tr} _{g}v)-\langle v,\operatorname {Ric} \rangle _{g}} ∂ ∂ t d μ g = 1 2 g p q v p q d μ g {\displaystyle {\frac {\partial }{\partial t}}d\mu _{g}={\frac {1}{2}}g^{pq}v_{pq}\,d\mu _{g}} ∂ ∂ t ∇ i ∇ j Φ = ∇ i ∇ j ∂ Φ ∂ t − 1 2 g k p ( ∇ i v j p + ∇ j v i p − ∇ p v i j ) ∂ Φ ∂ x k {\displaystyle


id: list-of-rules-of-inference-20260505-0
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-0.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
This is a list of rules of inference, logical laws that relate to mathematical formulae. == Introduction == Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation φ ⊢ ψ {\displaystyle \varphi \vdash \psi } indicates such a subderivation from the temporary assumption φ {\displaystyle \varphi } to ψ {\displaystyle \psi } . == Rules for propositional calculus == === Rules for negations === Reductio ad absurdum (or Negation Introduction) φ ⊢ ψ {\displaystyle \varphi \vdash \psi } φ ⊢ ¬ ψ _ {\displaystyle {\underline {\varphi \vdash \lnot \psi }}} ¬ φ {\displaystyle \lnot \varphi } Reductio ad absurdum (related to the law of excluded middle) ¬ φ ⊢ ψ {\displaystyle \lnot \varphi \vdash \psi


id: list-of-rules-of-inference-20260505-1
file: data/en.wikipedia.org/wiki/List_of_rules_of_inference-1.md
category: reference
title: List of rules of inference
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rules_of_inference
date_saved: 2026-05-05T08:16:46.303073+00:00
instance: kb-cron
parent: list-of-rules-of-inference-20260505-0
summary:
ψ {\displaystyle \psi } Restriction 1: β {\displaystyle \beta } is a variable which does not occur in φ {\displaystyle \varphi } . Restriction 2: There is no occurrence, free or bound, of β {\displaystyle \beta } in ψ {\displaystyle \psi } . Restriction 3: β {\displaystyle \beta } is not mentioned in any hypothesis or undischarged assumptions. == Rules of substructural logic == The following are special cases of universal generalization and existential elimination; these occur in substructural logics, such as linear logic. Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem) α ⊢ β {\displaystyle \alpha \vdash \beta } α , α ⊢ β ¯ {\displaystyle {\overline {\alpha ,\alpha \vdash \beta }}} Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) α , α , γ ⊢ β _ {\displaystyle {\underline {\alpha ,\alpha ,\gamma \vdash \beta }}} α , γ ⊢ β {\displaystyle \alpha ,\gamma \vdash \beta } == Table: Rules of Inference == The rules above can be summed up in the following table. The "Tautology" column shows how to interpret the notation of a given rule. All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth


id: list-of-runge-kutta-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-0.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation d y d t = f ( t , y ) . {\displaystyle {\frac {dy}{dt}}=f(t,y).} Explicit Runge–Kutta methods take the form y n + 1 = y n + h ∑ i = 1 s b i k i k 1 = f ( t n , y n ) , k 2 = f ( t n + c 2 h , y n + h ( a 21 k 1 ) ) , k 3 = f ( t n + c 3 h , y n + h ( a 31 k 1 + a 32 k 2 ) ) , ⋮ k i = f ( t n + c i h , y n + h ∑ j = 1 i − 1 a i j k j ) . {\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}\\k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\\&\;\;\vdots \\k_{i}&=f\left(t_{n}+c_{i}h,y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}\right).\end{aligned}}} Stages for implicit methods of s stages take the more general form, with the solution to be found over all s k i = f ( t n + c i h , y n + h ∑ j = 1 s a i j k j


id: list-of-runge-kutta-methods-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-1.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
0 0 0 0 0 1 / 3 1 / 3 0 0 0 2 / 3 − 1 / 3 1 0 0 1 1 − 1 1 0 1 / 8 3 / 8 3 / 8 1 / 8 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\1/3&1/3&0&0&0\\2/3&-1/3&1&0&0\\1&1&-1&1&0\\\hline &1/8&3/8&3/8&1/8\\\end{array}}} ==== Ralston's fourth-order method ==== This fourth order method has minimum truncation error. 0 0 0 0 0 2 5 2 5 0 0 0 14 − 3 5 16 − 2 889 + 1 428 5 1 024 3 785 − 1 620 5 1 024 0 0 1 − 3 365 + 2 094 5 6 040 − 975 − 3 046 5 2 552 467 040 + 203 968 5 240 845 0 263 + 24 5 1 812 125 − 1000 5 3 828 3 426 304 + 1 661 952 5 5 924 787 30 − 4 5 123 {\displaystyle {\begin{array}{c|cccc}0&0&0&0&0\\{\frac {2}{5}}&{\frac {2}{5}}&0&0&0\\{\frac {14-3{\sqrt {5}}}{16}}&{\frac {-2\,889+1\,428{\sqrt {5}}}{1\,024}}&{\frac {3\,785-1\,620{\sqrt {5}}}{1\,024}}&0&0\\1&{\frac {-3\,365+2\,094{\sqrt {5}}}{6\,040}}&{\frac {-975-3\,046{\sqrt {5}}}{2\,552}}&{\frac {467\,040+203\,968{\sqrt {5}}}{240\,845}}&0\\\hline &{\frac {263+24{\sqrt {5}}}{1\,812}}&{\frac {125-1000{\sqrt {5}}}{3\,828}}&{\frac {3\,426\,304+1\,661\,952{\sqrt {5}}}{5\,924\,787}}&{\frac {30-4{\sqrt {5}}}{123}}\\\end{array}}} === Fifth-order methods === ==== Nyström's fifth-order method ==== This fifth-order method was a correction of the one proposed originally by Kutta's work. 0 0 0


id: list-of-runge-kutta-methods-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-2.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
=== Gauss–Legendre methods === These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau: 1 2 − 3 6 1 4 1 4 − 3 6 1 2 + 3 6 1 4 + 3 6 1 4 1 2 1 2 1 2 + 3 2 1 2 − 3 2 {\displaystyle {\begin{array}{c|cc}{\frac {1}{2}}-{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {\sqrt {3}}{6}}\\{\frac {1}{2}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}+{\frac {\sqrt {3}}{6}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}\\\end{array}}} The Gauss–Legendre method of order six has Butcher tableau: 1 2 − 15 10 5 36 2 9 − 15 15 5 36 − 15 30 1 2 5 36 + 15 24 2 9 5 36 − 15 24 1 2 + 15 10 5 36 + 15 30 2 9 + 15 15 5 36 5 18 4 9 5 18 − 5 6 8 3 − 5 6 {\displaystyle {\begin{array}{c|ccc}{\frac {1}{2}}-{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}&{\frac {2}{9}}-{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{30}}\\{\frac {1}{2}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{24}}&{\frac {2}{9}}&{\frac {5}{36}}-{\frac {\sqrt {15}}{24}}\\{\frac {1}{2}}+{\frac {\sqrt {15}}{10}}&{\frac {5}{36}}+{\frac {\sqrt {15}}{30}}&{\frac {2}{9}}+{\frac {\sqrt {15}}{15}}&{\frac {5}{36}}\\\hline &{\frac {5}{18}}&{\frac {4}{9}}&{\frac {5}{18}}\\&-{\frac {5}{6}}&{\frac {8}{3}}&-{\frac {5}{6}}\end{array}}} === Diagonally Implicit Runge–Kutta methods ===


id: list-of-runge-kutta-methods-20260505-3
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-3.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
1 / 2 1 / 2 0 0 0 2 / 3 1 / 6 1 / 2 0 0 1 / 2 − 1 / 2 1 / 2 1 / 2 0 1 3 / 2 − 3 / 2 1 / 2 1 / 2 3 / 2 − 3 / 2 1 / 2 1 / 2 {\displaystyle {\begin{array}{c|cccc}1/2&1/2&0&0&0\\2/3&1/6&1/2&0&0\\1/2&-1/2&1/2&1/2&0\\1&3/2&-3/2&1/2&1/2\\\hline &3/2&-3/2&1/2&1/2\\\end{array}}} === Lobatto methods === There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta. ==== Lobatto IIIA methods ==== The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule: 0 0 0 1 1


id: list-of-runge-kutta-methods-20260505-4
file: data/en.wikipedia.org/wiki/List_of_Runge–Kutta_methods-4.md
category: reference
title: List of Runge–Kutta methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods
date_saved: 2026-05-05T08:16:47.604001+00:00
instance: kb-cron
parent: list-of-runge-kutta-methods-20260505-0
summary:
The fifth-order method is given by 0 1 9 − 1 − 6 18 − 1 + 6 18 3 5 − 6 10 1 9 11 45 + 7 6 360 11 45 − 43 6 360 3 5 + 6 10 1 9 11 45 + 43 6 360 11 45 − 7 6 360 1 9 4 9 + 6 36 4 9 − 6 36 {\displaystyle {\begin{array}{c|ccc}0&{\frac {1}{9}}&{\frac {-1-{\sqrt {6}}}{18}}&{\frac {-1+{\sqrt {6}}}{18}}\\{\frac {3}{5}}-{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {7{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {43{\sqrt {6}}}{360}}\\{\frac {3}{5}}+{\frac {\sqrt {6}}{10}}&{\frac {1}{9}}&{\frac {11}{45}}+{\frac {43{\sqrt {6}}}{360}}&{\frac {11}{45}}-{\frac {7{\sqrt {6}}}{360}}\\\hline &{\frac {1}{9}}&{\frac {4}{9}}+{\frac {\sqrt {6}}{36}}&{\frac {4}{9}}-{\frac {\sqrt {6}}{36}}\\\end{array}}} ==== Radau IIA methods ==== The ci of this method are zeros of d s − 1 d x s − 1 ( x s − 1 ( x − 1 ) s ) {\displaystyle {\frac {d^{s-1}}{dx^{s-1}}}(x^{s-1}(x-1)^{s})} . The first-order method is equivalent to the backward Euler method. The third-order method is given by 1 / 3 5 / 12 − 1 / 12 1 3 / 4 1 / 4 3 / 4 1 / 4 {\displaystyle {\begin{array}{c|cc}1/3&5/12&-1/12\\1&3/4&1/4\\\hline &3/4&1/4\\\end{array}}} The fifth-order method is given by 2 5 − 6 10 11 45 − 7 6


id: list-of-self-intersecting-polygons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_self-intersecting_polygons-0.md
category: reference
title: List of self-intersecting polygons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_self-intersecting_polygons
date_saved: 2026-05-05T08:16:48.847957+00:00
instance: kb-cron
parent: list-of-self-intersecting-polygons-20260505-0
summary:
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: the crossed quadrilateral, with four edges the antiparallelogram, a crossed quadrilateral with alternate edges of equal length the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel Star polygons pentagram, with five edges hexagram, with six edges heptagram, with seven edges octagram, with eight edges enneagram or nonagram, with nine edges decagram, with ten edges hendecagram, with eleven edges dodecagram, with twelve edges icositetragram, with twenty four edges 257-gram, with two hundred and fifty seven edges == See also == List of regular polytopes and compounds § Stars Complex polygon


id: list-of-mathematical-series-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-0.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, 0 0 {\displaystyle 0^{0}} is taken to have the value 1 {\displaystyle 1} { x } {\displaystyle \{x\}} denotes the fractional part of x {\displaystyle x} B n ( x ) {\displaystyle B_{n}(x)} is a Bernoulli polynomial. B n {\displaystyle B_{n}} is a Bernoulli number, and here, B 1 = − 1 2 . {\displaystyle B_{1}=-{\frac {1}{2}}.} E n {\displaystyle E_{n}} is an Euler number. ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. ψ n ( z ) {\displaystyle \psi _{n}(z)} is a polygamma function. Li s ⁡ ( z ) {\displaystyle \operatorname {Li} _{s}(z)} is a polylogarithm. ( n k ) {\displaystyle n \choose k} is binomial coefficient exp ⁡ ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle x} == Sums of powers == See Faulhaber's formula. ∑ k = 0 m k n − 1 = B n ( m + 1 ) − B n n {\displaystyle \sum _{k=0}^{m}k^{n-1}={\frac {B_{n}(m+1)-B_{n}}{n}}} The first few values are:


id: list-of-mathematical-series-20260505-1
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-1.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ k 2 z k k ! = ( z + z 2 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{2}{\frac {z^{k}}{k!}}=(z+z^{2})e^{z}} (cf. second moment of Poisson distribution) ∑ k = 0 ∞ k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{3}{\frac {z^{k}}{k!}}=(z+3z^{2}+z^{3})e^{z}} ∑ k = 0 ∞ k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z {\displaystyle \sum _{k=0}^{\infty }k^{4}{\frac {z^{k}}{k!}}=(z+7z^{2}+6z^{3}+z^{4})e^{z}} ∑ k = 0 ∞ k n z k k ! = z d d z ∑ k = 0 ∞ k n − 1 z k k ! = e z T n ( z ) {\displaystyle \sum _{k=0}^{\infty }k^{n}{\frac {z^{k}}{k!}}=z{\frac {d}{dz}}\sum _{k=0}^{\infty }k^{n-1}{\frac {z^{k}}{k!}}\,\!=e^{z}T_{n}(z)} where T n ( z ) {\displaystyle T_{n}(z)} is the Touchard polynomials. === Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship === ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin ⁡ z {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}=\sin z} ∑ k = 0 ∞ z


id: list-of-mathematical-series-20260505-2
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-2.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 0 ∞ ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 − 1 − z z , | z | < 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {(4k)!}{2^{4k}{\sqrt {2}}(2k)!(2k+1)!}}z^{k}={\sqrt {\frac {1-{\sqrt {1-z}}}{z}}},|z|<1} ∑ k = 0 ∞ 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin ⁡ z ) 2 , | z | ≤ 1 {\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k}(k!)^{2}}{(k+1)(2k+1)!}}z^{2k+2}=\left(\arcsin {z}\right)^{2},|z|\leq 1} ∑ n = 0 ∞ ∏ k = 0 n − 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + ∑ n = 0 ∞ α ∏ k = 0 n − 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin ⁡ z , | z | ≤ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(4k^{2}+\alpha ^{2})}{(2n)!}}z^{2n}+\sum _{n=0}^{\infty }{\frac {\alpha \prod _{k=0}^{n-1}[(2k+1)^{2}+\alpha ^{2}]}{(2n+1)!}}z^{2n+1}=e^{\alpha \arcsin {z}},|z|\leq 1} === Binomial coefficients === ( 1


id: list-of-mathematical-series-20260505-3
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-3.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ A ∈ P ( E ) c a r d ( A ) = n 2 n − 1 , where E is a finite set, and card( E ) = n {\displaystyle \sum _{A\ \in \ {\mathcal {P}}(E)}card(A)=n2^{n-1}{\text{, where }}E{\text{ is a finite set, and card(}}E{\text{) = n}}} == Trigonometric functions == Sums of sines and cosines arise in Fourier series. ∑ k = 1 ∞ cos ⁡ ( k θ ) k = − 1 2 ln ⁡ ( 2 − 2 cos ⁡ θ ) = − ln ⁡ ( 2 sin ⁡ θ 2 ) , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\cos(k\theta )}{k}}=-{\frac {1}{2}}\ln(2-2\cos \theta )=-\ln \left(2\sin {\frac {\theta }{2}}\right),0<\theta <2\pi } ∑ k = 1 ∞ sin ⁡ ( k θ ) k = π − θ 2 , 0 < θ < 2 π {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin(k\theta )}{k}}={\frac {\pi -\theta }{2}},0<\theta <2\pi } ∑ k = 1 ∞ ( − 1 ) k − 1 k cos ⁡ ( k θ ) = 1 2 ln ⁡ ( 2 + 2 cos ⁡ θ ) = ln ⁡ ( 2 cos ⁡ θ 2 ) ,


id: list-of-mathematical-series-20260505-4
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-4.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ m ≠ 0 n − 1 1 1 − z m = n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{1-z_{m}}}={\frac {n-1}{2}}} ∑ m ≠ 0 n − 1 1 ( 1 − z m ) 2 = ( n − 1 ) ( 5 − n ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {1}{(1-z_{m})^{2}}}={\frac {(n-1)(5-n)}{12}}} Let k {\displaystyle k} be an integer 0 < k < n {\displaystyle 0<k<n} then we also got: ∑ m = 0 n − 1 z m k = 0 {\displaystyle \sum _{m=0}^{n-1}z_{m}^{k}=0} ∑ m = 0 n − 1 ( x − z m ) k = n ⋅ x k {\displaystyle \sum _{m=0}^{n-1}(x-z_{m})^{k}=n\cdot x^{k}} ∑ m ≠ 0 n − 1 ( z m ) k 1 − z m = 2 k − n − 1 2 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{1-z_{m}}}={\frac {2k-n-1}{2}}} ∑ m ≠ 0 n − 1 ( z m ) k ( 1 − z m ) 2 = 6 k ( n + 2 − k ) − ( n + 1 ) ( n + 5 ) 12 {\displaystyle \sum _{m\neq 0}^{n-1}{\frac {(z_{m})^{k}}{(1-z_{m})^{2}}}={\frac {6k(n+2-k)-(n+1)(n+5)}{12}}} == Rational functions == ∑ n = a + 1 ∞


id: list-of-mathematical-series-20260505-5
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-5.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
S ( a , b ) = ( − 1 ) c ( ∑ k = 0 ∞ ( − 1 ) k a k + d − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) = ( − 1 ) c ( S ( a , d ) − ∑ k = 0 c − 1 ( − 1 ) k a k + d ) {\displaystyle S(a,b)=(-1)^{c}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{ak+d}}-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)=(-1)^{c}\left(S(a,d)-\sum _{k=0}^{c-1}{\frac {(-1)^{k}}{ak+d}}\right)} what that means is that all the infinite choices of a {\displaystyle a} and b {\displaystyle b} can essentially be boiled down to the cases where g c d ( a , b ) = 1 {\displaystyle gcd(a,b)=1} and a > b > 0 {\displaystyle a>b>0} . If we assume those two things we can then write: S ( a , b ) = 1 a ( π 2 sin ⁡ ( π b a ) − 2 ∑ m = 0 ⌊ a 2 ⌋ cos ⁡ ( π ( 2 m + 1 ) b a ) ln ⁡ ( sin ⁡ ( π 2 m + 1 2 a ) ) )


id: list-of-mathematical-series-20260505-6
file: data/en.wikipedia.org/wiki/List_of_mathematical_series-6.md
category: reference
title: List of mathematical series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_series
date_saved: 2026-05-05T08:16:50.313770+00:00
instance: kb-cron
parent: list-of-mathematical-series-20260505-0
summary:
∑ k = 1 ∞ 1 T e k = 1 1 + 1 4 + 1 10 + 1 20 + 1 35 + ⋯ = 3 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{Te_{k}}}={\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{20}}+{\frac {1}{35}}+\cdots ={\frac {3}{2}}} Where T e n = ∑ k = 1 n T k {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}} === Exponential and logarithms === ∑ k = 0 ∞ 1 ( 2 k + 1 ) ( 2 k + 2 ) = 1 1 × 2 + 1 3 × 4 + 1 5 × 6 + 1 7 × 8 + 1 9 × 10 + ⋯ = ln ⁡ 2 {\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}={\frac {1}{1\times 2}}+{\frac {1}{3\times 4}}+{\frac {1}{5\times 6}}+{\frac {1}{7\times 8}}+{\frac {1}{9\times 10}}+\cdots =\ln 2} ∑ k = 1 ∞ 1 2 k k = 1 2 + 1 8 + 1 24 + 1 64 + 1 160 + ⋯ = ln ⁡ 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2^{k}k}}={\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{24}}+{\frac {1}{64}}+{\frac {1}{160}}+\cdots =\ln 2} ∑ k = 1 ∞ ( − 1 ) k + 1 2 k k + ∑ k = 1 ∞ ( − 1 ) k + 1 3 k k


id: list-of-set-identities-and-relations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-0.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The binary operations of set union ( ∪ {\displaystyle \cup } ) and intersection ( ∩ {\displaystyle \cap } ) satisfy many identities. Several of these identities or "laws" have well established names. == Notation == Throughout this article, capital letters (such as A , B , C , L , M , R , S , {\displaystyle A,B,C,L,M,R,S,} and X {\displaystyle X} ) will denote sets. On the left hand side of an identity, typically, L {\displaystyle L} will be the leftmost set, M {\displaystyle M} will be the middle set, and R {\displaystyle R} will be the rightmost set. This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity. For example, the identity ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ ( M ∖ R ) {\displaystyle (L\,\setminus \,M)\,\setminus \,R~=~(L\,\setminus \,R)\,\setminus \,(M\,\setminus \,R)} may be read as: ( Left set


id: list-of-set-identities-and-relations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-1.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ L = L (Idempotence) L ∩ L = L (Idempotence) L △ L = ∅ (Nilpotence of index 2) L ∖ L = ∅ (Nilpotence of index 2) {\displaystyle {\begin{alignedat}{10}L\cup L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\cap L&\;=\;&&L&&\quad {\text{ (Idempotence)}}\\[1.4ex]L\,\triangle \,L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]L\setminus L&\;=\;&&\varnothing &&\quad {\text{ (Nilpotence of index 2)}}\\[1.4ex]\end{alignedat}}} Domination/Absorbing element: Definition: z {\displaystyle z} is called a left absorbing element of a binary operator ∗ {\displaystyle \,\ast \,} if z ∗ R = z {\displaystyle z\,\ast \,R=z} for all R {\displaystyle R} and it is called a right absorbing element of ∗ {\displaystyle \,\ast \,} if L ∗ z = z {\displaystyle L\,\ast \,z=z} for all L . {\displaystyle L.} A left absorbing element that is also a right absorbing element if called an absorbing element. Absorbing elements are also sometime called annihilating elements or zero elements. A universe set is an absorbing element of binary union ∪ . {\displaystyle \cup .} The empty set ∅ {\displaystyle \varnothing } is an absorbing element of binary intersection ∩ {\displaystyle \cap } and binary Cartesian product × , {\displaystyle \times ,} and it is also a left absorbing element of set subtraction ∖ : {\displaystyle \,\setminus :} X


id: list-of-set-identities-and-relations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-2.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
X ∖ ( L ∩ R ) = ( X ∖ L ) ∪ ( X ∖ R ) Also written ( L ∩ R ) ∁ = L ∁ ∪ R ∁ (De Morgan's law) X ∖ ( L ∪ R ) = ( X ∖ L ) ∩ ( X ∖ R ) Also written ( L ∪ R ) ∁ = L ∁ ∩ R ∁ (De Morgan's law) {\displaystyle {\begin{alignedat}{10}X\setminus (L\cap R)&=(X\setminus L)\cup (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cap R)^{\complement }=L^{\complement }\cup R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]X\setminus (L\cup R)&=(X\setminus L)\cap (X\setminus R)&&\qquad {\text{ Also written }}\quad &&(L\cup R)^{\complement }=L^{\complement }\cap R^{\complement }&&\quad &&{\text{ (De Morgan's law)}}\\[1.4ex]\end{alignedat}}} === Commutativity === Unions, intersection, and symmetric difference are commutative operations: L ∪ R = R ∪ L (Commutativity) L ∩ R = R ∩ L (Commutativity) L △ R = R △ L (Commutativity) {\displaystyle {\begin{alignedat}{10}L\cup R&\;=\;&&R\cup L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\cap R&\;=\;&&R\cap L&&\quad {\text{ (Commutativity)}}\\[1.4ex]L\,\triangle R&\;=\;&&R\,\triangle L&&\quad {\text{ (Commutativity)}}\\[1.4ex]\end{alignedat}}} Set subtraction is not commutative. However, the commutativity of set subtraction can be characterized: from ( L ∖ R ) ∩ ( R ∖ L ) = ∅ {\displaystyle (L\,\setminus \,R)\cap (R\,\setminus \,L)=\varnothing } it follows that: L ∖


id: list-of-set-identities-and-relations-20260505-3
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-3.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
===== Empty set ===== A set L {\displaystyle L} is empty if the sentence ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true, where the notation x ∉ L {\displaystyle x\not \in L} is shorthand for ¬ ( x ∈ L ) . {\displaystyle \lnot (x\in L).} If L {\displaystyle L} is any set then the following are equivalent: L {\displaystyle L} is not empty, meaning that the sentence ¬ [ ∀ x ( x ∉ L ) ] {\displaystyle \lnot [\forall x(x\not \in L)]} is true (literally, the logical negation of " L {\displaystyle L} is empty" holds true). (In classical mathematics) L {\displaystyle L} is inhabited, meaning: ∃ x ( x ∈ L ) {\displaystyle \exists x(x\in L)} In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set L {\displaystyle L} that is not empty (where by definition, " L {\displaystyle L} is empty" means that the statement ∀ x ( x ∉ L ) {\displaystyle \forall x(x\not \in L)} is true) might not have an inhabitant (which is an x {\displaystyle x}


id: list-of-set-identities-and-relations-20260505-4
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-4.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
=== Precedence rules === There is no universal agreement on the order of precedence of the basic set operators. Nevertheless, many authors use precedence rules for set operators, although these rules vary with the author. One common convention is to associate intersection L ∩ R = { x : ( x ∈ L ) ∧ ( x ∈ R ) } {\displaystyle L\cap R=\{x:(x\in L)\land (x\in R)\}} with logical conjunction (and) L ∧ R {\displaystyle L\land R} and associate union L ∪ R = { x : ( x ∈ L ) ∨ ( x ∈ R ) } {\displaystyle L\cup R=\{x:(x\in L)\lor (x\in R)\}} with logical disjunction (or) L ∨ R , {\displaystyle L\lor R,} and then transfer the precedence of these logical operators (where ∧ {\displaystyle \,\land \,} has precedence over ∨ {\displaystyle \,\lor \,} ) to these set operators, thereby giving ∩ {\displaystyle \,\cap \,} precedence over ∪ . {\displaystyle \,\cup .\,} So for example, L ∪ M ∩ R {\displaystyle L\cup M\cap R} would mean L ∪ ( M ∩ R ) {\displaystyle L\cup (M\cap R)} since it would be associated with the logical statement L ∨ M ∧ R = L ∨ ( M ∧


id: list-of-set-identities-and-relations-20260505-5
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-5.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∖ R = L ∖ ( M ∖ R ) {\textstyle \;(L\setminus M)\setminus R=L\setminus (M\setminus R)\;} if and only if ( R ∖ M ) ∖ L = R ∖ ( M ∖ L ) , {\displaystyle \;(R\setminus M)\setminus L=R\setminus (M\setminus L),\;} where the only difference between the left and right hand side set equalities is that the locations of L and R {\displaystyle L{\text{ and }}R} have been swapped. === Distributivity === Definition: If ∗ and ∙ {\displaystyle \ast {\text{ and }}\bullet } are binary operators then ∗ {\displaystyle \,\ast \,} left distributes over ∙ {\displaystyle \,\bullet \,} if L ∗ ( M ∙ R ) = ( L ∗ M ) ∙ ( L ∗ R ) for all L , M , R {\displaystyle L\,\ast \,(M\,\bullet \,R)~=~(L\,\ast \,M)\,\bullet \,(L\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R} while ∗ {\displaystyle \,\ast \,} right distributes over ∙ {\displaystyle \,\bullet \,} if ( L ∙ M ) ∗ R = ( L ∗ R ) ∙ ( M ∗ R ) for all L , M , R . {\displaystyle (L\,\bullet \,M)\,\ast \,R~=~(L\,\ast \,R)\,\bullet \,(M\,\ast \,R)\qquad \qquad {\text{ for all }}L,M,R.} The operator ∗ {\displaystyle \,\ast


id: list-of-set-identities-and-relations-20260505-6
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-6.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∪ ( M ∩ R ) = ( L ∪ M ) ∩ ( L ∪ R ) (Left-distributivity of ∪ over ∩ ) L ∪ ( M ∪ R ) = ( L ∪ M ) ∪ ( L ∪ R ) (Left-distributivity of ∪ over ∪ ) L ∩ ( M ∪ R ) = ( L ∩ M ) ∪ ( L ∩ R ) (Left-distributivity of ∩ over ∪ ) L ∩ ( M ∩ R ) = ( L ∩ M ) ∩ ( L ∩ R ) (Left-distributivity of ∩ over ∩ ) L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) (Left-distributivity of ∩ over △ ) L × ( M ∩ R ) = ( L × M ) ∩ ( L × R ) (Left-distributivity of × over ∩ ) L × ( M ∪ R ) = ( L × M ) ∪ ( L × R ) (Left-distributivity of × over ∪ ) L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) (Left-distributivity of × over ∖ )


id: list-of-set-identities-and-relations-20260505-7
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-7.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∖ ( M ∪ R ) ⊆ ( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cup \,R)~&~~{\color {red}{\subseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cup \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cap \,R)\\[1.4ex]\end{alignedat}}} L ∖ ( M ∩ R ) ⊇ ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) {\displaystyle {\begin{alignedat}{5}L\,\setminus \,(M\,\cap \,R)~&~~{\color {red}{\supseteq }}~~&&\color {black}{\,}(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R)~~=~~L\,\setminus \,(M\,\cup \,R)\\[1.4ex]\end{alignedat}}} where equality holds for one (or equivalently, for both) of the above two inclusion formulas if and only if L ∩ M = L ∩ R . {\displaystyle L\,\cap \,M=L\,\cap \,R.} The following statements are equivalent: L ∩ M = L ∩ R {\displaystyle L\cap M\,=\,L\cap R} L ∖ M = L ∖ R {\displaystyle L\,\setminus \,M\,=\,L\,\setminus \,R} L ∖ ( M ∩ R ) = ( L ∖ M ) ∩ ( L ∖ R ) ; {\displaystyle L\,\setminus \,(M\,\cap \,R)=(L\,\setminus \,M)\,\cap \,(L\,\setminus \,R);} that is, ∖ {\displaystyle \,\setminus \,} left distributes over ∩ {\displaystyle \,\cap \,} for these three particular sets L ∖ ( M ∪ R ) = ( L ∖ M ) ∪ ( L ∖ R )


id: list-of-set-identities-and-relations-20260505-8
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-8.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) △ R = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ L ) ∪ ( L ∩ M ∩ R ) (the three outermost sets are pairwise disjoint) {\displaystyle {\begin{alignedat}{4}(L\setminus M)~\triangle ~R&=(L\setminus (M\cup R))\cup (R\setminus L)\cup (L\cap M\cap R)~~~{\text{ (the three outermost sets are pairwise disjoint) }}\\\end{alignedat}}} ( L ∖ M ) × R = ( L × R ) ∖ ( M × R ) (Distributivity) {\displaystyle (L\,\setminus M)\times R=(L\times R)\,\setminus (M\times R)~~~~~{\text{ (Distributivity)}}} ==== L\(M ⁎ R) ==== Set subtraction on the left, and parentheses on the right L ∖ ( M ∪ R ) = ( L ∖ M ) ∩ ( L ∖ R ) (De Morgan's law) = ( L ∖ M ) ∖ R = ( L ∖ R ) ∖ M {\displaystyle {\begin{alignedat}{3}L\setminus (M\cup R)&=(L\setminus M)&&\,\cap \,(&&L\setminus R)~~~~{\text{ (De Morgan's law) }}\\&=(L\setminus M)&&\,\,\setminus &&R\\&=(L\setminus R)&&\,\,\setminus &&M\\\end{alignedat}}} L ∖ ( M ∩ R ) = ( L ∖ M ) ∪ ( L ∖ R ) (De Morgan's law) {\displaystyle {\begin{alignedat}{4}L\setminus (M\cap R)&=(L\setminus M)\cup (L\setminus R)~~~~{\text{ (De Morgan's law) }}\\\end{alignedat}}} where the above two sets that are the subjects of De Morgan's laws always


id: list-of-set-identities-and-relations-20260505-9
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-9.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∪ M ) ∪ ( M ∪ R ) = L ∪ M ∪ R ( L ∪ M ) ∩ ( M ∪ R ) = M ∪ ( L ∩ R ) ( L ∪ M ) ∖ ( M ∪ R ) = L ∖ ( M ∪ R ) ( L ∪ M ) △ ( M ∪ R ) = ( L ∖ ( M ∪ R ) ) ∪ ( R ∖ ( L ∪ M ) ) = ( L △ R ) ∖ M ( L ∩ M ) ∪ ( M ∩ R ) = M ∩ ( L ∪ R ) ( L ∩ M ) ∩ ( M ∩ R ) = L ∩ M ∩ R ( L ∩ M ) ∖ ( M ∩ R ) = ( L ∩ M ) ∖ R ( L ∩ M ) △ ( M ∩ R ) = [ ( L ∩ M ) ∪ ( M ∩ R ) ] ∖ ( L ∩ M ∩ R ) ( L ∖ M ) ∪ ( M ∖ R ) = ( L ∪ M ) ∖


id: list-of-set-identities-and-relations-20260505-10
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-10.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L ∖ M ) ∪ ( L ∖ R ) = L ∖ ( M ∩ R ) ( L ∖ M ) ∩ ( L ∖ R ) = L ∖ ( M ∪ R ) ( L ∖ M ) ∖ ( L ∖ R ) = ( L ∩ R ) ∖ M ( L ∖ M ) △ ( L ∖ R ) = L ∩ ( M △ R ) = ( L ∩ M ) △ ( L ∩ R ) {\displaystyle {\begin{alignedat}{9}(L\,\setminus M)&\,\cup \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cap \,R)\\[1.4ex](L\,\setminus M)&\,\cap \,&&(&&L\,\setminus R)&&\;=\;&&L\,\setminus \,(M\,\cup \,R)\\[1.4ex](L\,\setminus M)&\,\setminus \,&&(&&L\,\setminus R)&&\;=\;&&(L\,\cap \,R)\,\setminus \,M\\[1.4ex](L\,\setminus M)&\,\triangle \,&&(&&L\,\setminus R)&&\;=\;&&L\,\cap \,(M\,\triangle \,R)\\[1.4ex]&\,&&\,&&\,&&\;=\;&&(L\cap M)\,\triangle \,(L\cap R)\\[1.4ex]\end{alignedat}}} === Other simplifications === Other properties: L ∩ M = R and L ∩ R = M if and only if M = R ⊆ L . {\displaystyle L\cap M=R\;{\text{ and }}\;L\cap R=M\qquad {\text{ if and only if }}\qquad M=R\subseteq L.} If L ⊆ M {\displaystyle L\subseteq M} then L ∖ R = L ∩ ( M ∖ R ) . {\displaystyle L\setminus R=L\cap (M\setminus R).} L × ( M ∖ R ) = ( L × M ) ∖ ( L × R ) {\displaystyle L\times


id: list-of-set-identities-and-relations-20260505-11
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-11.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( L × M × R ) ∩ ( L 2 × M 2 × R 2 ) = ( L ∩ L 2 ) × ( M ∩ M 2 ) × ( R ∩ R 2 ) {\displaystyle (L\times M\times R)\cap \left(L_{2}\times M_{2}\times R_{2}\right)~=~\left(L\cap L_{2}\right)\times \left(M\cap M_{2}\right)\times \left(R\cap R_{2}\right)} === Binary ⋃ of finite ⨯ === ( L × R ) ∪ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ ( L 2 ∖ L ) × R 2 ] ∪ [ ( L ∩ L 2 ) × ( R ∪ R 2 ) ] = [ L × ( R ∖ R 2 ) ] ∪ [ L 2 × ( R 2 ∖ R ) ] ∪ [ ( L ∪ L 2 ) × ( R ∩ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\cup ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\setminus L_{2}\right)\times R\right]~\cup ~\left[\left(L_{2}\setminus L\right)\times R_{2}\right]~\cup ~\left[\left(L\cap L_{2}\right)\times \left(R\cup R_{2}\right)\right]\\[0.5ex]~&=~\left[L\times \left(R\setminus R_{2}\right)\right]~\cup ~\left[L_{2}\times \left(R_{2}\setminus R\right)\right]~\cup ~\left[\left(L\cup L_{2}\right)\times \left(R\cap R_{2}\right)\right]\\\end{alignedat}}} === Difference \ of finite ⨯ === ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 )


id: list-of-set-identities-and-relations-20260505-12
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-12.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== Arbitrary families of sets == Let ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} ( R j ) j ∈ J , {\displaystyle \left(R_{j}\right)_{j\in J},} and ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} be indexed families of sets. Whenever the assumption is needed, then all indexing sets, such as I {\displaystyle I} and J , {\displaystyle J,} are assumed to be non-empty. === Definitions === A family of sets or (more briefly) a family refers to a set whose elements are sets. An indexed family of sets is a function from some set, called its indexing set, into some family of sets. An indexed family of sets will be denoted by ( L i ) i ∈ I , {\displaystyle \left(L_{i}\right)_{i\in I},} where this notation assigns the symbol I {\displaystyle I} for the indexing set and for every index i ∈ I , {\displaystyle i\in I,} assigns the symbol L i {\displaystyle L_{i}} to the value of the function at i . {\displaystyle i.} The function itself may then be denoted by the symbol L ∙ , {\displaystyle L_{\bullet },} which is obtained from


id: list-of-set-identities-and-relations-20260505-13
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-13.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
otherwise, if you are working in a model in which "the class of all things x {\displaystyle x} " is not a set (by far the most common situation) then ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} is undefined because ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} consists of everything, which makes ⋂ i ∈ ∅ L i {\displaystyle {\textstyle \bigcap \limits _{i\in \varnothing }}L_{i}} a proper class and not a set. Assumption: Henceforth, whenever a formula requires some indexing set to be non-empty in order for an arbitrary intersection to be well-defined, then this will automatically be assumed without mention. A consequence of this is the following assumption/definition: A finite intersection of sets or an intersection of finitely many sets refers to the intersection of a finite collection of one or more sets. Some authors adopt the so called nullary intersection convention, which is the convention that an empty intersection of sets is equal to some canonical set. In particular, if all sets are subsets of some set X {\displaystyle X} then some author may declare that the empty intersection of these sets be equal to X


id: list-of-set-identities-and-relations-20260505-14
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-14.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
⋂ i ∈ I ( L i ∪ R i ) {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~\color {Red}{\neq }\color {Black}{}~~\bigcap _{i\in I}\left(L_{i}\cup R_{i}\right)~} (an example of this is given above). The single intersection on the right hand side must be over all pairs ( i , j ) ∈ I × I : {\displaystyle (i,j)\in I\times I:} ( ⋂ i ∈ I L i ) ∪ ( ⋂ i ∈ I R i ) = ⋂ j ∈ I i ∈ I , ( L i ∪ R j ) . {\displaystyle ~\left(\bigcap _{i\in I}L_{i}\right)\cup \left(\bigcap _{i\in I}R_{i}\right)~~=~~\bigcap _{\stackrel {i\in I,}{j\in I}}\left(L_{i}\cup R_{j}\right).~} ==== Arbitrary ⋂'s and arbitrary ⋃'s ==== ===== Incorrectly distributing by swapping ⋂ and ⋃ ===== Naively swapping ⋃ i ∈ I {\displaystyle \;{\textstyle \bigcup \limits _{i\in I}}\;} and ⋂ j ∈ J {\displaystyle \;{\textstyle \bigcap \limits _{j\in J}}\;} may produce a different set The following inclusion always holds: In general, equality need not hold and moreover, the right hand side depends on how for each fixed i ∈ I , {\displaystyle i\in I,} the sets ( S i , j ) j ∈ J {\displaystyle \left(S_{i,j}\right)_{j\in J}} are labelled; and analogously, the left hand side


id: list-of-set-identities-and-relations-20260505-15
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-15.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Equality in Inclusion 1 ∪∩ is a subset of ∩∪ can hold under certain circumstances, such as in 7e, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( i , j ) ∈ I × J {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(i,j)\in I\times J}} (that is, S i , j : = L i ∖ R j {\displaystyle S_{i,j}\colon =L_{i}\setminus R_{j}} with the same indexing sets I {\displaystyle I} and J {\displaystyle J} ), or such as in 7f, which is the special case where ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is ( L i ∖ R j ) ( j , i ) ∈ J × I {\displaystyle \left(L_{i}\setminus R_{j}\right)_{(j,i)\in J\times I}} (that is, S ^ j , i : = L i ∖ R j {\displaystyle {\hat {S}}_{j,i}\colon =L_{i}\setminus R_{j}} with the indexing sets I {\displaystyle I} and J {\displaystyle J} swapped). For a correct formula that extends the distributive laws, an approach other than just switching ∪ {\displaystyle \cup } and ∩


id: list-of-set-identities-and-relations-20260505-16
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-16.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Example application: To apply the general formula to the case of ( C k ) k ∈ K {\displaystyle \left(C_{k}\right)_{k\in K}} and ( D l ) l ∈ L , {\displaystyle \left(D_{l}\right)_{l\in L},} use I : = { 1 , 2 } , {\displaystyle I\colon =\{1,2\},} J 1 : = K , {\displaystyle J_{1}\colon =K,} J 2 : = L , {\displaystyle J_{2}\colon =L,} and let T 1 , k : = C k {\displaystyle T_{1,k}\colon =C_{k}} for all k ∈ J 1 {\displaystyle k\in J_{1}} and let T 2 , l : = D l {\displaystyle T_{2,l}\colon =D_{l}} for all l ∈ J 2 . {\displaystyle l\in J_{2}.} Every map f ∈ ∏ J ∙ = def ∏ i ∈ I J i = J 1 × J 2 = K × L {\displaystyle f\in {\textstyle \prod }J_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \prod \limits _{i\in I}}J_{i}=J_{1}\times J_{2}=K\times L} can be bijectively identified with the pair ( f ( 1 ) , f ( 2 ) ) ∈ K × L {\displaystyle \left(f(1),f(2)\right)\in K\times L} (the inverse sends ( k , l ) ∈ K × L {\displaystyle (k,l)\in K\times L} to the map f ( k , l ) ∈


id: list-of-set-identities-and-relations-20260505-17
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-17.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
and if I = J {\displaystyle I=J} then also: === Cartesian products Π of arbitrarily many sets === ==== Intersections ⋂ of Π ==== If ( S i , j ) ( i , j ) ∈ I × J {\displaystyle \left(S_{i,j}\right)_{(i,j)\in I\times J}} is a family of sets then Moreover, a tuple ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} belongs to the set in Eq. 8 above if and only if x i ∈ S i , j {\displaystyle x_{i}\in S_{i,j}} for all i ∈ I {\displaystyle i\in I} and all j ∈ J . {\displaystyle j\in J.} In particular, if ( L i ) i ∈ I {\displaystyle \left(L_{i}\right)_{i\in I}} and ( R i ) i ∈ I {\displaystyle \left(R_{i}\right)_{i\in I}} are two families indexed by the same set then ( ∏ i ∈ I L i ) ∩ ∏ i ∈ I R i = ∏ i ∈ I ( L i ∩ R i ) {\displaystyle \left(\prod _{i\in I}L_{i}\right)\cap \prod _{i\in I}R_{i}~=~\prod _{i\in I}\left(L_{i}\cap R_{i}\right)} So for instance, ( L × R ) ∩ ( L 2 × R 2 ) = ( L ∩ L 2 ) × ( R ∩ R


id: list-of-set-identities-and-relations-20260505-18
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-18.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L × ( ⋃ i ∈ I R i ) = ⋃ i ∈ I ( L × R i ) (Left-distributivity of × over ∪ ) L × ( ⋂ i ∈ I R i ) = ⋂ i ∈ I ( L × R i ) (Left-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) ( ⋃ i ∈ I L i ) × R = ⋃ i ∈ I ( L i × R ) (Right-distributivity of × over ∪ ) ( ⋂ i ∈ I L i ) × R = ⋂ i ∈ I ( L i × R ) (Right-distributivity of × over ⋂ i ∈ I when I ≠ ∅ ) {\displaystyle {\begin{alignedat}{5}L\times \left(\bigcup _{i\in I}R_{i}\right)&\;=\;\;&&\bigcup _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]L\times \left(\bigcap _{i\in I}R_{i}\right)&\;=\;\;&&\bigcap _{i\in I}(L\times R_{i})\qquad &&{\text{ (Left-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\left(\bigcup _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcup _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\cup \,{\text{)}}\\[1.4ex]\left(\bigcap _{i\in I}L_{i}\right)\times R&\;=\;\;&&\bigcap _{i\in I}(L_{i}\times R)\qquad &&{\text{ (Right-distributivity of }}\,\times \,{\text{ over }}\,\bigcap _{i\in I}\,{\text{ when }}I\neq \varnothing \,{\text{)}}\\[1.4ex]\end{alignedat}}} ==== Distributing arbitrary Π over arbitrary ⋃ ==== Suppose that


id: list-of-set-identities-and-relations-20260505-19
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-19.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
( ∏ i ∈ I L i ) ∖ ∏ i ∈ I R i = ⋃ j ∈ I ∏ i ∈ I { L j ∖ R j if i = j L i if i ≠ j = ⋃ j ∈ I [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] = ⋃ L j ⊈ R j j ∈ I , [ ( L j ∖ R j ) × ∏ j ≠ i i ∈ I , L i ] {\displaystyle {\begin{alignedat}{9}\left(\prod _{i\in I}L_{i}\right)~\setminus ~\prod _{i\in I}R_{i}~&=~\;~\bigcup _{j\in I}\;~\prod _{i\in I}{\begin{cases}L_{j}\,\setminus \,R_{j}&{\text{ if }}i=j\\L_{i}&{\text{ if }}i\neq j\\\end{cases}}\\[0.5ex]~&=~\;~\bigcup _{j\in I}\;~{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.5ex]~&=~\bigcup _{\stackrel {j\in I,}{L_{j}\not \subseteq R_{j}}}{\Big [}\left(L_{j}\,\setminus \,R_{j}\right)~\times ~\prod _{\stackrel {i\in I,}{j\neq i}}L_{i}{\Big ]}\\[0.3ex]\end{alignedat}}} so for instance, ( L × R ) ∖ ( L 2 × R 2 ) = [ ( L ∖ L 2 ) × R ] ∪ [ L × ( R ∖ R 2 ) ] {\displaystyle {\begin{alignedat}{9}\left(L\times R\right)~\setminus ~\left(L_{2}\times R_{2}\right)~&=~\left[\left(L\,\setminus \,L_{2}\right)\times R\right]~\cup ~\left[L\times \left(R\,\setminus \,R_{2}\right)\right]\\\end{alignedat}}} and ( L × M × R ) ∖ ( L 2 × M 2 ×


id: list-of-set-identities-and-relations-20260505-20
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-20.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Any such set R {\displaystyle R} necessarily contains f ( A ) {\displaystyle f(A)} as a subset. Any set not entirely contained in the domain of f {\displaystyle f} cannot be f {\displaystyle f} -saturated. A = f − 1 ( f ( A ) ) . {\displaystyle A=f^{-1}(f(A)).} A ⊇ f − 1 ( f ( A ) ) {\displaystyle A\supseteq f^{-1}(f(A))} and A ⊆ domain ⁡ f . {\displaystyle A\subseteq \operatorname {domain} f.} The inclusion L ∩ domain ⁡ f ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap \operatorname {domain} f\subseteq f^{-1}(f(L))} always holds, where if A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} then this becomes A ⊆ f − 1 ( f ( A ) ) . {\displaystyle A\subseteq f^{-1}(f(A)).} A ⊆ domain ⁡ f {\displaystyle A\subseteq \operatorname {domain} f} and if a ∈ A {\displaystyle a\in A} and x ∈ domain ⁡ f {\displaystyle x\in \operatorname {domain} f} satisfy f ( x ) = f ( a ) , {\displaystyle f(x)=f(a),} then x ∈ A . {\displaystyle x\in A.} Whenever a fiber of f {\displaystyle f} intersects A , {\displaystyle A,} then A {\displaystyle A} contains the entire fiber.


id: list-of-set-identities-and-relations-20260505-21
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-21.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
If ( A i ) i ∈ I {\displaystyle \left(A_{i}\right)_{i\in I}} is a family of arbitrary subsets of X = domain ⁡ f , {\displaystyle X=\operatorname {domain} f,} which means that A i ⊆ X {\displaystyle A_{i}\subseteq X} for all i , {\displaystyle i,} then Conditional Equality 10a becomes: === (Pre)Images of binary set operations === Throughout, let L {\displaystyle L} and R {\displaystyle R} be any sets and let f : X → Y {\displaystyle f:X\to Y} be any function. Summary As the table below shows, set equality is not guaranteed only for images of: intersections, set subtractions, and symmetric differences. Preimages preserve set operations Preimages of sets are well-behaved with respect to all basic set operations: f − 1 ( L ∪ R ) = f − 1 ( L ) ∪ f − 1 ( R ) f − 1 ( L ∩ R ) = f − 1 ( L ) ∩ f − 1 ( R ) f − 1 ( L ∖ R ) = f − 1 ( L ) ∖ f − 1 ( R ) f − 1 ( L △ R ) = f − 1 ( L ) △


id: list-of-set-identities-and-relations-20260505-22
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-22.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The containment f ( L △ R ) ⊋ f ( L ) △ f ( R ) {\displaystyle ~f(L~\triangle ~R)~\supsetneq ~f(L)~\triangle ~f(R)~} is strict: { y } = f ( L △ R ) ≠ f ( L ) △ f ( R ) = { y } △ { y } = ∅ {\displaystyle \{y\}~=~f\left(L~\triangle ~R\right)~\neq ~f(L)~\triangle ~f(R)~=~\{y\}\triangle \{y\}~=~\varnothing } In words: functions might not distribute over symmetric difference △ {\displaystyle \,\triangle \,} (which can be defined as the set subtraction of two sets: L △ R = ( L ∪ R ) ∖ ( L ∩ R ) {\displaystyle L\triangle R=(L\cup R)\setminus (L\cap R)} ). The containment f ( L ∩ R ) ⊊ f ( L ) ∩ f ( R ) {\displaystyle ~f(L\cap R)~\subsetneq ~f(L)\cap f(R)~} is strict: ∅ = f ( ∅ ) = f ( L ∩ R ) ≠ f ( L ) ∩ f ( R ) = { y } ∩ { y } = { y } {\displaystyle \varnothing ~=~f(\varnothing )~=~f(L\cap R)~\neq ~f(L)\cap f(R)~=~\{y\}\cap \{y\}~=~\{y\}} In words: functions might not distribute over set intersection ∩ {\displaystyle \,\cap \,} (which can be defined as the set subtraction of two sets:


id: list-of-set-identities-and-relations-20260505-23
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-23.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( L △ R ) = f ( L ) △ f ( R ) for all L , R ⊆ X . {\displaystyle f(L\,\triangle \,R)=f(L)\,\triangle \,f(R)\;{\text{ for all }}\,L,R\subseteq X.} (The equals sign = {\displaystyle \,=\,} can be replaced with ⊆ {\displaystyle \,\subseteq \,} ). Any one of the four statements (b) - (e) but with the words "for all" replaced with any one of the following: "for all singleton subsets" In particular, the statement that results from (d) gives a characterization of injectivity that explicitly involves only one point (rather than two): f {\displaystyle f} is injective if and only if f ( x ) ∉ f ( X ∖ { x } ) for every x ∈ X . {\displaystyle f(x)\not \in f(X\setminus \{x\})\;{\text{ for every }}\,x\in X.} "for all disjoint singleton subsets" For statement (d), this is the same as: "for all singleton subsets" (because the definition of "pairwise disjoint" is satisfies vacuously by any family that consists of exactly 1 set). "for all disjoint subsets" In particular, if a map is not known to be injective then barring additional information, there is no guarantee that any of the equalities in statements (b) - (e)


id: list-of-set-identities-and-relations-20260505-24
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-24.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ domain ⁡ f ⊆ R {\displaystyle L\cap \operatorname {domain} f\,\subseteq R} R ⊆ L {\displaystyle R\subseteq L} L ⊆ R {\displaystyle L\subseteq R} In addition, the following always hold: f ( f − 1 ( L ) ∩ R ) = L ∩ f ( R ) {\displaystyle f\left(f^{-1}(L)\cap R\right)~=~L\cap f(R)} f ( f − 1 ( L ) ∪ R ) = ( L ∩ Im ⁡ f ) ∪ f ( R ) {\displaystyle f\left(f^{-1}(L)\cup R\right)~=~(L\cap \operatorname {Im} f)\cup f(R)} ===== Conditions for f(L\R) = f(L)\f(R) ===== f ( L ∖ R ) ⊇ f ( L ) ∖ f ( R ) always holds {\displaystyle f(L\setminus R)~\supseteq ~f(L)\setminus f(R)\qquad \qquad {\text{ always holds}}} Characterizations of equality: The following statements are equivalent: f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~=~f(L)\setminus f(R)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\setminus R)~\subseteq ~f(L)\setminus f(R)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} L ∩ f − 1 ( f ( R ) ) = L ∩ R ∩


id: list-of-set-identities-and-relations-20260505-25
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-25.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
Characterizations of equality: The following statements are equivalent: f ( L △ R ) = f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)=f(L)~\triangle ~f(R)} f ( L △ R ) ⊆ f ( L ) △ f ( R ) {\displaystyle f\left(L~\triangle ~R\right)\subseteq f(L)~\triangle ~f(R)} f ( L ∖ R ) = f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)=f(L)\,\setminus \,f(R)} and f ( R ∖ L ) = f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)=f(R)\,\setminus \,f(L)} f ( L ∖ R ) ⊆ f ( L ) ∖ f ( R ) {\displaystyle f(L\,\setminus \,R)\subseteq f(L)\,\setminus \,f(R)} and f ( R ∖ L ) ⊆ f ( R ) ∖ f ( L ) {\displaystyle f(R\,\setminus \,L)\subseteq f(R)\,\setminus \,f(L)} L ∩ f − 1 ( f ( R ) ) ⊆ R {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~R} and R ∩ f − 1 ( f ( L ) ) ⊆ L {\displaystyle R\cap f^{-1}(f(L))~\subseteq ~L} The inclusions L ∩ f − 1 ( f ( R ) ) ⊆ f − 1 ( f ( L ) ) {\displaystyle L\cap f^{-1}(f(R))~\subseteq ~f^{-1}(f(L))} and R ∩ f −


id: list-of-set-identities-and-relations-20260505-26
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-26.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
f ( X ∖ R ) = f ( X ) ∖ { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } . {\displaystyle f(X\setminus R)\,=\,f(X)\setminus \left\{y\in f(R):f^{-1}(y)\subseteq R\right\}.} From this it follows that: f ( X ∖ R ) = f ( X ) ∖ f ( R ) if and only if f ( R ) = { y ∈ f ( R ) : f − 1 ( y ) ⊆ R } if and only if f − 1 ( f ( R ) ) ⊆ R . {\displaystyle f(X\setminus R)=f(X)\setminus f(R)\quad {\text{ if and only if }}\quad f(R)=\left\{y\in f(R):f^{-1}(y)\subseteq R\right\}\quad {\text{ if and only if }}\quad f^{-1}(f(R))\subseteq R.} If f R := { y ∈ f ( X ) : f − 1 ( y ) ⊆ R } {\displaystyle f_{R}:=\left\{y\in f(X):f^{-1}(y)\subseteq R\right\}} then f ( X ∖ R ) = f ( X ) ∖ f R , {\displaystyle f(X\setminus R)=f(X)\setminus f_{R},} which can be written more symmetrically as f ( X ∖ R ) = f X ∖ f R {\displaystyle f(X\setminus R)=f_{X}\setminus f_{R}} (since f X = f ( X ) {\displaystyle f_{X}=f(X)} ). =====


id: list-of-set-identities-and-relations-20260505-27
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-27.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ∩ f − 1 ( y ) ⊆ L ∖ R {\displaystyle L\cap f^{-1}(y)\subseteq L\setminus R~} if and only if L ∩ R ∩ f − 1 ( y ) = ∅ {\displaystyle ~L\cap R\cap f^{-1}(y)=\varnothing ~} if and only if R ∩ f − 1 ( y ) ⊆ R ∖ L {\displaystyle ~R\cap f^{-1}(y)\subseteq R\setminus L~} if and only if y ∉ f ( L ∩ R ) . {\displaystyle ~y\not \in f(L\cap R).} The sets U {\displaystyle U} and V {\displaystyle V} mentioned above could, in particular, be any of the sets f ( L ∪ R ) , Im ⁡ f , {\displaystyle f(L\cup R),\;\operatorname {Im} f,} or Y , {\displaystyle Y,} for example. === (Pre)Images of set operations on (pre)images === Let L {\displaystyle L} and R {\displaystyle R} be arbitrary sets, f : X → Y {\displaystyle f:X\to Y} be any map, and let A ⊆ X {\displaystyle A\subseteq X} and C ⊆ Y . {\displaystyle C\subseteq Y.} (Pre)Images of operations on images Since f ( L ) ∖ f ( L ∖ R ) = { y ∈ f ( L ∩ R ) : L ∩ f − 1 ( y


id: list-of-set-identities-and-relations-20260505-28
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-28.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
The map F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}} should not be confused with the Cartesian product ∏ j ∈ J F j {\displaystyle \prod _{j\in J}F_{j}} of these maps, which is by definition is the map ∏ j ∈ J F j : ∏ j ∈ J X → ∏ j ∈ J Y j ( x j ) j ∈ J ↦ ( F j ( x j ) ) j ∈ J {\displaystyle {\begin{alignedat}{4}\prod _{j\in J}F_{j}:\;&&\prod _{j\in J}X&&~\;\to \;~&\prod _{j\in J}Y_{j}\\[0.3ex]&&\left(x_{j}\right)_{j\in J}&&~\;\mapsto \;~&\left(F_{j}\left(x_{j}\right)\right)_{j\in J}\\\end{alignedat}}} with domain ∏ j ∈ J X = X J {\displaystyle \prod _{j\in J}X=X^{J}} rather than X . {\displaystyle X.} Preimage and images of a Cartesian product Suppose F ∙ = ( F j ) j ∈ J : X → ∏ j ∈ J Y j . {\displaystyle F_{\bullet }=\left(F_{j}\right)_{j\in J}~:~X~\to ~\prod _{j\in J}Y_{j}.} If A ⊆ X {\displaystyle A~\subseteq ~X} then F ∙ ( A ) ⊆ ∏ j ∈ J F j ( A ) . {\displaystyle F_{\bullet }(A)~~\;\color {Red}{\subseteq }\color {Black}{}\;~~\prod _{j\in J}F_{j}(A).} If B ⊆ ∏ j ∈ J


id: list-of-set-identities-and-relations-20260505-29
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-29.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
where N {\displaystyle \mathbb {N} } denotes the natural numbers. A notation S ∙ = ( S i ) i ∈ I {\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}} indicates that S ∙ {\displaystyle S_{\bullet }} is a net directed by ( I , ≤ ) , {\displaystyle (I,\leq ),} which (by definition) is a sequence if the set I , {\displaystyle I,} which is called the net's indexing set, is the natural numbers (that is, if I = N {\displaystyle I=\mathbb {N} } ) and ≤ {\displaystyle \,\leq \,} is the natural order on N . {\displaystyle \mathbb {N} .} Disjoint and monotone sequences of sets If S i ∩ S j = ∅ {\displaystyle S_{i}\cap S_{j}=\varnothing } for all distinct indices i ≠ j {\displaystyle i\neq j} then S ∙ {\displaystyle S_{\bullet }} is called a pairwise disjoint or simply a disjoint. A sequence or net S ∙ {\displaystyle S_{\bullet }} of set is called increasing or non-decreasing if (resp. decreasing or non-increasing) if for all indices i ≤ j , {\displaystyle i\leq j,} S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} (resp. S i ⊇ S j {\displaystyle S_{i}\supseteq S_{j}} ). A sequence or net S ∙ {\displaystyle S_{\bullet }}


id: list-of-set-identities-and-relations-20260505-30
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-30.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
closed under finite intersections (resp. closed under finite unions) if whenever L , R ∈ L {\displaystyle L,R\in {\mathcal {L}}} then L ∩ R ∈ L {\displaystyle L\cap R\in {\mathcal {L}}} (respectively, L ∪ R ∈ L {\displaystyle L\cup R\in {\mathcal {L}}} ). closed under countable intersections (resp. closed under countable unions) if whenever L 1 , L 2 , L 3 , … {\displaystyle L_{1},L_{2},L_{3},\ldots } are elements of L {\displaystyle {\mathcal {L}}} then so is their intersections ⋂ i = 1 ∞ L i := L 1 ∩ L 2 ∩ L 3 ∩ ⋯ {\displaystyle \bigcap _{i=1}^{\infty }L_{i}:=L_{1}\cap L_{2}\cap L_{3}\cap \cdots } (resp. so is their union ⋃ i = 1 ∞ L i := L 1 ∪ L 2 ∪ L 3 ∪ ⋯ {\displaystyle \bigcup _{i=1}^{\infty }L_{i}:=L_{1}\cup L_{2}\cup L_{3}\cup \cdots } ). closed under complementation in (or with respect to) X {\displaystyle X} if whenever L ∈ L {\displaystyle L\in {\mathcal {L}}} then X ∖ L ∈ L . {\displaystyle X\setminus L\in {\mathcal {L}}.} A family L {\displaystyle {\mathcal {L}}} of sets is called a/an: π−system if L ≠ ∅ {\displaystyle {\mathcal {L}}\neq \varnothing } and L {\displaystyle {\mathcal {L}}} is closed under finite-intersections. Every


id: list-of-set-identities-and-relations-20260505-31
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-31.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
L ( ∪ ) { X } = { X } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\{X\}=\{X\}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∩ ) { ∅ } = { ∅ } if L ≠ ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\{\varnothing \}=\{\varnothing \}~~~~{\text{ if }}{\mathcal {L}}\neq \varnothing } L ( ∪ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cup )\;\varnothing =\varnothing } L ( ∩ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\cap )\;\varnothing =\varnothing } L ( ∖ ) ∅ = ∅ {\displaystyle {\mathcal {L}}\;(\setminus )\;\varnothing =\varnothing } ∅ ( ∖ ) R = ∅ {\displaystyle \varnothing \;(\setminus )\;{\mathcal {R}}=\varnothing } === Power set === P ( L ∩ R ) = P ( L ) ∩ P ( R ) {\displaystyle {\mathcal {P}}(L\cap R)~=~{\mathcal {P}}(L)\cap {\mathcal {P}}(R)} P ( L ∪ R ) = P ( L ) ( ∪ ) P ( R ) ⊇ P ( L ) ∪ P ( R ) . {\displaystyle {\mathcal {P}}(L\cup R)~=~{\mathcal {P}}(L)\ (\cup )\ {\mathcal {P}}(R)~\supseteq ~{\mathcal {P}}(L)\cup {\mathcal {P}}(R).} If L {\displaystyle L} and R {\displaystyle R} are subsets of a vector space X {\displaystyle X} and if s {\displaystyle s} is a scalar then P ( s


id: list-of-set-identities-and-relations-20260505-32
file: data/en.wikipedia.org/wiki/List_of_set_identities_and_relations-32.md
category: reference
title: List of set identities and relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_identities_and_relations
date_saved: 2026-05-05T08:16:53.246995+00:00
instance: kb-cron
parent: list-of-set-identities-and-relations-20260505-0
summary:
== References == Artin, Michael (1991). Algebra. Prentice Hall. ISBN 81-203-0871-9. Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Springer. ISBN 1-85233-905-5.. Bylinski, Czeslaw (2004). "Some Basic Properties of Sets". Journal of Formalized Mathematics. 1. Retrieved 5 October 2021. Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. "SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS". Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011. Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303. Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. ISBN 9780442030643. Zbl 0087.04403. {{cite book}}: ISBN / Date incompatibility


id: list-of-alternative-set-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_alternative_set_theories-0.md
category: reference
title: List of alternative set theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_alternative_set_theories
date_saved: 2026-05-05T08:16:54.419564+00:00
instance: kb-cron
parent: list-of-alternative-set-theories-20260505-0
summary:
In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory. == Alternative set theories == Alternative set theories include: Vopěnka's alternative set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory Pocket set theory Naive set theory S (set theory) Double extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements Scott–Potter set theory Constructive set theory Zermelo set theory General set theory Mac Lane set theory == See also == Non-well-founded set theory List of first-order theories § Set theories == Notes ==


id: list-of-set-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_set_theory_topics-0.md
category: reference
title: List of set theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_set_theory_topics
date_saved: 2026-05-05T08:16:55.603082+00:00
instance: kb-cron
parent: list-of-set-theory-topics-20260505-0
summary:
This page is a list of articles related to set theory. == Articles on individual set theory topics == == Lists related to set theory == Glossary of set theory List of large cardinal properties List of properties of sets of reals List of set identities and relations == Set theorists == == Societies and organizations == Association for Symbolic Logic The Cabal


id: list-of-types-of-sets-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_sets-0.md
category: reference
title: List of types of sets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_sets
date_saved: 2026-05-05T08:16:56.774657+00:00
instance: kb-cron
parent: list-of-types-of-sets-20260505-0
summary:
Sets can be classified according to the properties they have. == Relative to set theory == Empty set Finite set, Infinite set Countable set, Uncountable set Power set == Relative to a topology == Closed set Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere dense set == Relative to a metric == Bounded set Totally bounded set == Relative to measurability == Borel set Baire set Measurable set, Non-measurable set Universally measurable set == Relative to a measure == Negligible set Null set Haar null set == In a linear space == Convex set Balanced set, Absolutely convex set == Relative to the real/complex numbers == Fractal set == Ways of defining sets/Relation to descriptive set theory == Recursive set Recursively enumerable set Arithmetical set Diophantine set Hyperarithmetical set Analytical set Analytic set, Coanalytic set Suslin set Projective set Inhabited set == More general objects still called sets == Multiset == See also == List of set identities and relations – Equalities for combinations of sets List of types of functions


id: list-of-shapes-with-known-packing-constant-20260505-0
file: data/en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant-0.md
category: reference
title: List of shapes with known packing constant
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_shapes_with_known_packing_constant
date_saved: 2026-05-05T08:16:58.046518+00:00
instance: kb-cron
parent: list-of-shapes-with-known-packing-constant-20260505-0
summary:
The packing constant of a geometric body is the largest average density achieved by packing arrangements of congruent copies of the body. For most bodies the value of the packing constant is unknown. The following is a list of bodies in Euclidean spaces whose packing constant is known. Fejes Tóth proved that in the plane, a point symmetric body has a packing constant that is equal to its translative packing constant and its lattice packing constant. Therefore, any such body for which the lattice packing constant was previously known, such as any ellipse, consequently has a known packing constant. In addition to these bodies, the packing constants of hyperspheres in 8 and 24 dimensions are almost exactly known. == References ==


id: list-of-solids-derived-from-the-sphere-20260505-0
file: data/en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere-0.md
category: reference
title: List of solids derived from the sphere
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_solids_derived_from_the_sphere
date_saved: 2026-05-05T08:16:59.236190+00:00
instance: kb-cron
parent: list-of-solids-derived-from-the-sphere-20260505-0
summary:
This page lists solids derived from a sphere. == Solids from cutting a sphere with one or more planes == Dome Spherical cap Spherical sector Spherical segment Spherical shell Spherical wedge == Solids from deforming a sphere == Ellipsoid Spheroid Solid bounded by Morin surface Any Genus 0 surface == Solids from intersecting a sphere with other solids or curved planes == Reuleaux tetrahedron Spherical lens == Notes ==


id: list-of-spherical-symmetry-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_spherical_symmetry_groups-0.md
category: reference
title: List of spherical symmetry groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups
date_saved: 2026-05-05T08:17:01.639486+00:00
instance: kb-cron
parent: list-of-spherical-symmetry-groups-20260505-0
summary:
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. == Involutional symmetry == There are four involutional groups: no symmetry (C1), reflection symmetry (Cs), 2-fold rotational symmetry (C2), and central point symmetry (Ci). == Cyclic symmetry == There are four infinite cyclic symmetry families, with n = 2 or higher. (n may be 1 as a special case as no symmetry) == Dihedral symmetry == There are three infinite dihedral symmetry families, with n = 2 or higher (n may be 1 as a special case).


id: list-of-textbooks-in-thermodynamics-and-statistical-mechanics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics-0.md
category: reference
title: List of textbooks in thermodynamics and statistical mechanics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_textbooks_in_thermodynamics_and_statistical_mechanics
date_saved: 2026-05-05T08:17:02.992953+00:00
instance: kb-cron
parent: list-of-textbooks-in-thermodynamics-and-statistical-mechanics-20260505-0
summary:
A list of notable textbooks in thermodynamics and statistical mechanics, arranged by category and date. == Only or mainly thermodynamics == Fermi, Enrico (1956). Thermodynamics (New ed.). Dover Publications. ISBN 978-0486603612. {{cite book}}: ISBN / Date incompatibility (help) Van Ness, H. C. (1983). Understanding Thermodynamics. Dover Publications. ISBN 978-0486632773. Callen, Herbert (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8. Finn, C. B. P. (1993). Thermal Physics (2nd ed.). United States of America: CRC Press. ISBN 0-7487-4379-0. Zemansky, Mark W.; Dittman, Richard (1997). Heat and Thermodynamics: An Intermediate Textbook (7th ed.). McGraw-Hill. ISBN 978-0-070-17059-9. Hanson, Robert M.; Green, Susan (2008). Introduction to Molecular Thermodynamics. University Science Books. ISBN 978-1891389498, 978-1-938787-63-8. Pokrovskii, Vladimir (2020). Thermodynamics of Complex Systems: Principles and applications. IOP Publishing, Bristol, UK. ISBN 978-0-7503-3451-8. == Both thermodynamics and statistical mechanics == Reif, Frederick (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill. ISBN 0-07-051800-9. Sears, Francis W. (1975). Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Addison Wesley. ISBN 020106894X. Kittel, Charles (1969). Thermal Physics. Chichester: Wiley. ISBN 0-471-49030-X. 2e Kittel, Charles; and Kroemer, Herbert (1980) New York: W.H. Freeman ISBN 0-7167-1088-9 Mandl, Franz (1971). Statistical physics. Chichester: Wiley. ISBN 0-471-56658-6. 2e (1988)


id: list-of-statistics-articles-20260505-0
file: data/en.wikipedia.org/wiki/List_of_statistics_articles-0.md
category: reference
title: List of statistics articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_statistics_articles
date_saved: 2026-05-05T08:17:16.414516+00:00
instance: kb-cron
parent: list-of-statistics-articles-20260505-0
summary:
== 0–9 == 1.96 2SLS (two-stage least squares) – redirects to instrumental variable 3SLS – see three-stage least squares 68–95–99.7 rule 100-year flood == A == == B == == C == == D == == E == == F == == G == == H == == I == == J == == K == == L == == M == == N == == O == == P == == Q == == R == == S == == T == == U == == V == == W == == Y == Yamartino method Yates analysis Yates's correction for continuity Youden's J statistic Yule–Simon distribution == Z == == See also == Supplementary lists These lists include items which are somehow related to statistics however are not included in this index: List of statisticians List of important publications in statistics List of scientific journals in statistics Topic lists Outline of statistics List of probability topics Glossary of probability and statistics Glossary of experimental design Notation in probability and statistics List of probability distributions List of graphical methods List of fields of application of statistics List of stochastic processes topics Lists of statistics topics List of statistical packages


id: list-of-publications-in-statistics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_publications_in_statistics-0.md
category: reference
title: List of publications in statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_statistics
date_saved: 2026-05-05T08:17:17.689706+00:00
instance: kb-cron
parent: list-of-publications-in-statistics-20260505-0
summary:
This is a list of publications in statistics, organized by field. Some reasons why a particular publication might be regarded as important: Topic creator – A publication that created a new topic Breakthrough – A publication that changed scientific knowledge significantly Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of statistics. == Probability == Théorie analytique des probabilités Author: Pierre-Simon Laplace Publication data: 1820 (3rd ed.) Online version: Internet Archive; CNRS, with more accurate character recognition; Gallica-Math, complete PDF and PDFs by section Description: Introduced the Laplace transform, exponential families, and conjugate priors in Bayesian statistics. Pioneering asymptotic statistics, proved an early version of the Bernstein–von Mises theorem on the irrelevance of the (regular) prior distribution on the limiting posterior distribution, highlighting the asymptotic role of the Fisher information. Studies the influence of median and skewness in regression analysis. Inspired the field of robust regression, proposed the Laplace distribution and was the first to provide alternatives to Carl Friedrich Gauss's work on statistics. Importance: Topic creator, Breakthrough, Influence == Mathematical statistics == Mathematical Methods of Statistics Author: Harald Cramér Publication data: Princeton Mathematical Series, vol. 9. Princeton University Press,


id: list-of-publications-in-statistics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_publications_in_statistics-1.md
category: reference
title: List of publications in statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_statistics
date_saved: 2026-05-05T08:17:17.689706+00:00
instance: kb-cron
parent: list-of-publications-in-statistics-20260505-0
summary:
Authors: George E.P. Box and Gwilym M. Jenkins Publication data: Holden-Day, 1970 Description: Systematic approach to ARIMA and ARMAX modelling Importance: This book introduces ARIMA and associated input-output models, studies how to fit them and develops a methodology for time series forecasting and control. It has changed econometrics, process control and forecasting. == Applied statistics == Statistical Methods for Research Workers Author: R.A. Fisher Publication data: Edinburgh: Oliver & Boyd, 1925 (1st edition); London: Macmillan, 1970 (15th edition) Online version: http://psychclassics.yorku.ca/Fisher/Methods/ Description: The original manual for researchers, especially biologists, on how to statistically evaluate numerical data. Importance: Hugely influential text by the father of modern statistics that remained in print for more than 50 years. Responsible for the widespread use of tests of statistical significance. Statistical Methods Author: George W. Snedecor Publication data: 1937, Collegiate Press Description: One of the first comprehensive texts on statistical methods. Reissued as Statistical Methods Applied to Experiments in Agriculture and Biology in 1940 and then again as Statistical Methods with Cochran, WG in 1967. A classic text. Importance: Influence Principles and Procedures of Statistics with Special Reference to the Biological Sciences. Authors: Steel, R.G.D, and Torrie, J. H. Publication data: McGraw Hill (1960) 481


id: list-of-publications-in-statistics-20260505-2
file: data/en.wikipedia.org/wiki/List_of_publications_in_statistics-2.md
category: reference
title: List of publications in statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_publications_in_statistics
date_saved: 2026-05-05T08:17:17.689706+00:00
instance: kb-cron
parent: list-of-publications-in-statistics-20260505-0
summary:
Author: Charles Sanders Peirce and Joseph Jastrow Publication data: Peirce, Charles Sanders; Jastrow, Joseph (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences. 3: 73–83. Online version: http://psychclassics.yorku.ca/Peirce/small-diffs.htm Description: Peirce and Jastrow use logistic regression to estimate subjective probabilities of subjects's judgments of the heavier of two measurements, following a randomized controlled repeated measures design. Importance: The first randomized experiment, which also used blinding; it seems also to have been the first experiment for estimating subjective probabilities. The Design of Experiments Author: Fisher, RA Publication data: 1935, Oliver and Boyd, Edinburgh Description: The first textbook on experimental design Importance: Influence The Design and Analysis of Experiments Author: Oscar Kempthorne Publication data: 1950, John Wiley & Sons, New York (Reprinted with corrections in 1979 by Robert E. Krieger) Description: Early exposition of the general linear model using matrix algebra (following lecture notes of George W. Brown). Bases inference on the randomization distribution objectively defined by the experimental protocol, rather than a so-called "statistical model" expressing the subjective beliefs of a statistician: The normal model is regarded as a convenient approximation to the randomization-distribution, whose quality is assessed by theorems about moments and simulation experiments. Importance: The first


id: outline-of-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_statistics-0.md
category: reference
title: Outline of statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_statistics
date_saved: 2026-05-05T08:17:18.916400+00:00
instance: kb-cron
parent: outline-of-statistics-20260505-0
summary:
The following outline is provided as an overview of and topical guide to statistics: Statistics is a field of inquiry that studies the collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities; it is also used and misused for making informed decisions in all areas of business and government. == Nature of statistics == Statistics can be described as all of the following: An academic discipline: one with academic departments, curricula and degrees; national and international societies; and specialized journals. A scientific field (a branch of science) – widely recognized category of specialized expertise within science, and typically embodies its own terminology and nomenclature. Such a field will usually be represented by one or more scientific journals, where peer reviewed research is published. A formal science – branch of knowledge concerned with formal systems. A mathematical science – field of science that is primarily mathematical in nature but may not be universally considered subfields of mathematics proper. Statistics, for example, is mathematical in its methods but grew out of political arithmetic which merged with inverse probability and grew through applications in the social


id: list-of-string-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_string_theory_topics-0.md
category: reference
title: List of string theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_string_theory_topics
date_saved: 2026-05-05T08:17:20.266586+00:00
instance: kb-cron
parent: list-of-string-theory-topics-20260505-0
summary:
This is a list of string theory topics. == String theory == Strings Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string theory Type IIB string theory Heterotic string N=2 superstring M-theory Matrix theory Introduction to M-theory F-theory String field theory Matrix string theory Nonlinear sigma model Tachyon condensation RNS formalism String theory landscape History of string theory First superstring revolution Second superstring revolution === String duality === T-duality S-duality U-duality Montonen–Olive duality Mysterious duality === Particles and fields === Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole === Branes === D-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition == Supersymmetry == Supergravity Superspace Lie superalgebra Lie supergroup == Conformal field theory == Two-dimensional conformal field theory Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model Monstrous moonshine == Geometry == Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7) manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory == Holography == Holographic principle AdS/CFT correspondence ==


id: sum-of-squares-20260505-0
file: data/en.wikipedia.org/wiki/Sum_of_squares-0.md
category: reference
title: Sum of squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sum_of_squares
date_saved: 2026-05-05T08:17:21.476492+00:00
instance: kb-cron
parent: sum-of-squares-20260505-0
summary:
In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts: == Statistics == For partitioning of variance, see Partition of sums of squares For the "sum of squared deviations", see Least squares For the "sum of squared differences", see Mean squared error For the "sum of squared error", see Residual sum of squares For the "sum of squares due to lack of fit", see Lack-of-fit sum of squares For sums of squares relating to model predictions, see Explained sum of squares For sums of squares relating to observations, see Total sum of squares For sums of squared deviations, see Squared deviations from the mean For modelling involving sums of squares, see Analysis of variance For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance == Number theory == Fermat's theorem on sums of two squares says which primes are sums of two squares. The sum of two squares theorem generalizes Fermat's theorem to specify which composite numbers are the sums of two squares. Legendre's three-square theorem states which numbers can be expressed as the sum of three squares For representing an integer as a sum of squares of 4 integers, see


id: sums-of-powers-20260505-0
file: data/en.wikipedia.org/wiki/Sums_of_powers-0.md
category: reference
title: Sums of powers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Sums_of_powers
date_saved: 2026-05-05T08:17:23.640704+00:00
instance: kb-cron
parent: sums-of-powers-20260505-0
summary:
In mathematics and statistics, sums of powers occur in a number of contexts: Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities. There are only finitely many positive integers that are not sums of distinct squares. The largest one is 128. The same applies for sums of distinct cubes (largest one is 12,758), distinct fourth powers (largest is 5,134,240), etc. See for a generalization to sums of polynomials. Faulhaber's formula expresses 1 k + 2 k + 3 k + ⋯ + n k {\displaystyle 1^{k}+2^{k}+3^{k}+\cdots +n^{k}} as a polynomial in n, or alternatively in terms of a Bernoulli polynomial. Fermat's right triangle theorem states that there is no solution in positive integers for a 2 = b 4 + c 4 {\displaystyle a^{2}=b^{4}+c^{4}} and a 4 = b 4 + c 2 {\displaystyle a^{4}=b^{4}+c^{2}} . Fermat's Last Theorem states that x k + y k = z k {\displaystyle x^{k}+y^{k}=z^{k}} is impossible in positive integers with k > 2. The equation of a


id: list-of-sums-of-reciprocals-20260505-0
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-0.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
In mathematics and number theory, the sum of reciprocals (or sum of inverses) is defined as the sum of reciprocals of some series of positive integers (counting numbers). It is a sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given as a sequence whose nth term is defined as the sum of the first n reciprocals. For sums of reciprocals over a series of finitely many numbers, key questions include whether there is a simple expression for the value of the sum, whether the sum must be less than a certain value, and whether the sum is ever an integer. When the sum of reciprocals is over an infinite series, questions include: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to be large if the sum of its reciprocals diverges, and small if it converges.) Second, if it converges, what is a simple expression for the value it converges to? Is that value rational or irrational, and is that value


id: list-of-sums-of-reciprocals-20260505-1
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-1.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
A sum-free sequence of increasing positive integers is one for which no number is the sum of any subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570 . The sum of the reciprocals of the heptagonal numbers converges to a known value that is not only irrational but also transcendental, and for which there exists a complicated formula. The sum of the reciprocals of the twin primes, of which there may be finitely many or infinitely many, is known to be finite and is called Brun's constant, approximately 1.90216 . (By convention, the reciprocal of 5 {\displaystyle 5} appears twice in the sum.) The sum of the reciprocals of the Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479 . The prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706 . The sum of the reciprocals of the perfect powers (including duplicates) is 1 . The sum of the reciprocals of the perfect powers (excluding duplicates) is approximately 0.8745


id: list-of-sums-of-reciprocals-20260505-2
file: data/en.wikipedia.org/wiki/List_of_sums_of_reciprocals-2.md
category: reference
title: List of sums of reciprocals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals
date_saved: 2026-05-05T08:17:27.040927+00:00
instance: kb-cron
parent: list-of-sums-of-reciprocals-20260505-0
summary:
=== Divergent series === The n-th partial sum of the harmonic series, which is the sum of the reciprocals of the first n positive integers, diverges as n goes to infinity, albeit extremely slowly: The sum of the first 1043 terms is less than 100 . The difference between the cumulative sum and the natural logarithm of n converges to the Euler–Mascheroni constant, commonly denoted as γ , {\displaystyle \gamma ,} which is approximately 0.5772. The sum of the reciprocals of the primes diverges. Given coprime positive integers a and b, the sum of the reciprocal of the primes of the form an + b diverges. This result is used to prove the Dirichlet's theorem, which states that there are infinitely many primes p {\displaystyle p} satisfying p ≡ a ( m o d b ) {\displaystyle p\equiv a{\text{ }}(\mathrm {mod} {\text{ }}b)} . The sum of the reciprocals of the primes of the form 4n + 1 is divergent. By Fermat's theorem on sums of two squares, it follows that the sum of reciprocals of numbers of the form a 2 + b 2 , {\displaystyle a^{2}+b^{2},} where a and b are non-negative integers, not both equal to 0,


id: table-of-mathematical-symbols-by-introduction-date-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date-0.md
category: reference
title: Table of mathematical symbols by introduction date
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date
date_saved: 2026-05-05T08:17:28.247985+00:00
instance: kb-cron
parent: table-of-mathematical-symbols-by-introduction-date-20260505-0
summary:
The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. == See also == History of mathematical notation History of the Hindu–Arabic numeral system Glossary of mathematical symbols List of mathematical symbols by subject Mathematical notation Mathematical operators and symbols in Unicode == Sources == == External links == RapidTables: Math Symbols List Jeff Miller: Earliest Uses of Various Mathematical Symbols


id: list-of-types-of-systems-theory-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_systems_theory-0.md
category: reference
title: List of types of systems theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_systems_theory
date_saved: 2026-05-05T08:17:29.461574+00:00
instance: kb-cron
parent: list-of-types-of-systems-theory-20260505-0
summary:
This list of types of systems theory gives an overview of different types of systems theory, which are mentioned in scientific book titles or articles. The following more than 40 types of systems theory are all explicitly named systems theory and represent a unique conceptual framework in a specific field of science. Systems theory has been formalized since the 1950s, and a long set of specialized systems theories and cybernetics exist. In the beginnings, general systems theory was developed by Ludwig von Bertalanffy to overcome the over-specialisation of the modern times and as a worldview using holism. The systems theories nowadays are closer to the traditional specialisation than to holism, by interdependencies and mutual division by mutually-different specialists. == A == Abstract systems theory (also see: formal system) Action Theory Adaptive systems theory (also see: complex adaptive system) Applied general systems theory (also see: general systems theory) Applied multidimensional systems theory Archaeological systems theory (also see: Systems theory in archaeology) Systems theory in anthropology Associated systems theory == B == Behavioral systems theory Biochemical systems theory Biomatrix systems theory Body system == C == Complex adaptive systems theory (also see: complex adaptive system) Complex systems theory (also see: complex systems)


id: tarjan-s-algorithm-20260505-0
file: data/en.wikipedia.org/wiki/Tarjan's_algorithm-0.md
category: reference
title: Tarjan's algorithm
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Tarjan's_algorithm
date_saved: 2026-05-05T08:17:30.674857+00:00
instance: kb-cron
parent: tarjan-s-algorithm-20260505-0
summary:
Tarjan's algorithm may refer to one of several algorithms attributed to Robert Tarjan, including: Tarjan's strongly connected components algorithm Tarjan's off-line lowest common ancestors algorithm Tarjan's algorithm for finding bridges in an undirected graph Tarjan's algorithm for finding simple circuits in a directed graph == See also == List of algorithms == References ==


id: list-of-tessellations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_tessellations-0.md
category: reference
title: List of tessellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_tessellations
date_saved: 2026-05-05T08:17:31.940951+00:00
instance: kb-cron
parent: list-of-tessellations-20260505-0
summary:
This is a list of tessellations. == Spherical == == Planar == == Hyperbolic == == See also == Tessellation Uniform tiling Convex uniform honeycombs List of k-uniform tilings List of Euclidean uniform tilings Uniform tilings in hyperbolic plane == References ==


id: list-of-theorems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_theorems-0.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. == Logics and foundations == Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic) Cut-elimination theorem (proof theory) Deduction theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set theory) Feferman–Vaught theorem (model theory) Friedberg–Muchnik theorem (mathematical logic) Fundamental theorem of equivalence relations


id: list-of-theorems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_theorems-1.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Number theory == 15 and 290 theorems (number theory) Albert–Brauer–Hasse–Noether theorem (algebras) Ankeny–Artin–Chowla theorem (number theory) Apéry's theorem (number theory) Artin–Verdier duality theorem (number theory) ATS theorem (number theory) Auxiliary polynomial theorem (Diophantine approximation) Ax–Kochen theorem (number theory) Baker's theorem (number theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation) Behrend's theorem (number theory) Bertrand's postulate (number theory) Birch's theorem (algebraic number theory) Bombieri's theorem (number theory) Bombieri–Friedlander–Iwaniec theorem (number theory) Brauer–Siegel theorem (number theory) Brun's theorem (number theory) Brun–Titchmarsh theorem (number theory) Carmichael's theorem (Fibonacci numbers) Chebotarev's density theorem (number theory) Chen's theorem (number theory) Chowla–Mordell theorem (number theory) Cohn's irreducibility criterion (polynomials) Critical line theorem (number theory) Davenport–Schmidt theorem (number theory, Diophantine approximations) Dirichlet's approximation theorem (Diophantine approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic theory) Erdős–Kac theorem (number theory) Euclid's theorem (number theory) Euclid–Euler theorem (number theory) Euler's theorem (number theory) Fermat's Last Theorem (number theory) Fermat's little theorem (number theory) Fermat's theorem on sums of two squares (number theory) Ferrero–Washington theorem (algebraic number theory) Ford's theorem (number theory) Franel–Landau theorem (number theory) Gelfond–Schneider theorem (transcendental number theory) Glaisher's theorem (number theory)


id: list-of-theorems-20260505-2
file: data/en.wikipedia.org/wiki/List_of_theorems-2.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Linear and multilinear algebra; matrix theory == Amitsur–Levitzki theorem (linear algebra) Binomial inverse theorem (linear algebra) Birkhoff–Von Neumann theorem (linear algebra) Bregman–Minc inequality (discrete mathematics) Cauchy-Binet formula (linear algebra) Cayley–Hamilton theorem (Linear algebra) Dimension theorem for vector spaces (vector spaces, linear algebra) Euler's rotation theorem (geometry) Exchange theorem (linear algebra) Gamas's Theorem (multilinear algebra) Gershgorin circle theorem (matrix theory) Inverse eigenvalues theorem (linear algebra) Perron–Frobenius theorem (matrix theory) Principal axis theorem (linear algebra) Rank–nullity theorem (linear algebra) Rouché–Capelli theorem (Linear algebra) Sinkhorn's theorem (matrix theory) Specht's theorem (matrix theory) Spectral theorem (linear algebra, functional analysis) Sylvester's determinant theorem (determinants) Sylvester's law of inertia (quadratic forms) Witt's theorem (quadratic forms) == Associative rings and algebras == Artin–Wedderburn theorem (abstract algebra) Artin–Zorn theorem (algebra) Brauer–Cartan–Hua theorem (ring theory) Frobenius theorem (abstract algebras) Goldie's theorem (ring theory) Jacobson density theorem (ring theory) Jacobson–Bourbaki theorem (algebra) Levitzky's theorem (ring theory) Regev's theorem (ring theory) Skolem–Noether theorem (simple algebras) Wedderburn's little theorem (ring theory) Wedderburn's theorem (abstract algebra) == Nonassociative rings and algebras == Ado's theorem (Lie algebra) Goddard–Thorn theorem (vertex algebras) Hurwitz's theorem (normed division algebras) Jacobson–Morozov theorem (Lie algebra) Levi's theorem (Lie groups) Lie's theorem (Lie algebra) Poincaré–Birkhoff–Witt theorem (universal enveloping algebras)


id: list-of-theorems-20260505-3
file: data/en.wikipedia.org/wiki/List_of_theorems-3.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Functions of a complex variable == Akhiezer's theorem (complex analysis) Arakelyan's theorem (complex analysis) Area theorem (conformal mapping) (complex analysis) Beurling–Lax theorem (Hardy spaces) Bloch's theorem (complex analysis) Bôcher's theorem (complex analysis) Borel–Carathéodory theorem (complex analysis) Branching theorem (complex manifold) Carathéodory's theorem (complex analysis) Carleson–Jacobs theorem (complex analysis) Carlson's theorem (complex analysis) Cauchy integral theorem (complex analysis) Cauchy–Hadamard theorem (complex analysis) Clifford's theorem on special divisors (algebraic curves) Corona theorem (complex analysis) de Branges's theorem (complex analysis) De Franchis theorem (Riemann surfaces) Edge-of-the-wedge theorem (complex analysis) Farrell–Markushevich theorem (complex analysis) Fatou's theorem (complex analysis) Fundamental theorem of algebra (complex analysis) Gauss–Lucas theorem (complex analysis) Grunsky's theorem (complex analysis) Hadamard three-circle theorem (complex analysis) Hadamard three-lines theorem (complex analysis) Hardy's theorem (complex analysis) Hartogs–Rosenthal theorem (complex analysis) Harnack's theorem (complex analysis) Hurwitz's automorphisms theorem (algebraic curves) Hurwitz's theorem (complex analysis) Identity theorem (complex analysis) Identity theorem for Riemann surfaces (Riemann surfaces) Koebe 1/4 theorem (complex analysis) Lagrange inversion theorem (mathematical analysis, combinatorics) Lagrange reversion theorem (mathematical analysis, combinatorics) Laurent expansion theorem (complex analysis) Lindelöf's theorem (complex analysis) Liouville's theorem (complex analysis, entire functions) Liouville's theorem (conformal mappings) Looman–Menchoff theorem (complex analysis) Marden's theorem (polynomials) Mergelyan's theorem (complex analysis) Measurable Riemann


id: list-of-theorems-20260505-4
file: data/en.wikipedia.org/wiki/List_of_theorems-4.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Operator theory == Aronszajn–Smith theorem (functional analysis) Atiyah–Singer index theorem (elliptic differential operators, harmonic analysis) Atkinson's theorem (operator theory) Babuška–Lax–Milgram theorem (partial differential equations) Banach–Stone theorem (operator theory) Bauer–Fike theorem (spectral theory) Bounded inverse theorem (operator theory) Browder–Minty theorem (operator theory) Choi's theorem on completely positive maps (operator theory) Commutation theorem (von Neumann algebra) Fuglede's theorem (functional analysis) Gelfand–Mazur theorem (Banach algebra) Gelfand–Naimark theorem (functional analysis) Hardy–Littlewood maximal theorem (real analysis) Hellinger–Toeplitz theorem (functional analysis) Hilbert–Schmidt theorem (functional analysis) Hille–Yosida theorem (functional analysis) Kaplansky density theorem (von Neumann algebra) Kuiper's theorem (operator theory, topology) Lax–Milgram theorem (partial differential equations) Lions–Lax–Milgram theorem (partial differential equations) Lumer–Phillips theorem (semigroup theory) Marcinkiewicz theorem (functional analysis) Mazur–Ulam theorem (normed spaces) Mercer's theorem (functional analysis) Min-max theorem (functional analysis) Moreau's theorem (convex analysis) Nash–Moser theorem (mathematical analysis) Open mapping theorem (functional analysis) Peetre theorem (functional analysis) Riesz–Thorin theorem (functional analysis) Ryll-Nardzewski fixed-point theorem (functional analysis) Sazonov's theorem (functional analysis) Schröder–Bernstein theorems for operator algebras (operator algebras) Stinespring factorization theorem (operator theory) Stone's theorem on one-parameter unitary groups (functional analysis) Sz.-Nagy's dilation theorem (operator theory) Tomita's theorem (operator algebras) Von Neumann bicommutant theorem (functional analysis) Von Neumann's theorem (operator theory) == Calculus of variations and


id: list-of-theorems-20260505-5
file: data/en.wikipedia.org/wiki/List_of_theorems-5.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== General topology == Arzelà–Ascoli theorem (functional analysis) Baire category theorem (topology, metric spaces) Bing metrization theorem (general topology) Bolzano–Weierstrass theorem (real analysis, calculus) Borsuk–Ulam theorem (topology) Brouwer fixed-point theorem (topology) Cantor's intersection theorem (real analysis) Closed graph theorem (functional analysis) Extreme value theorem (calculus) Fixed-point theorems in infinite-dimensional spaces Hairy ball theorem (algebraic topology) Hahn–Mazurkiewicz theorem (continuum theory) Heine–Borel theorem (real analysis) Heine–Cantor theorem (metric geometry) Jordan curve theorem (topology) Kuratowski's closure-complement problem (topology) Lebesgue covering dimension (dimension theory) Metrization theorems (topological spaces) Nagata–Smirnov metrization theorem(general topology) Netto's theorem (topology) Parovicenko's theorem (topology) Tietze extension theorem (general topology) Tychonoff's theorem (general topology) == Algebraic topology == Acyclic models theorem (algebraic topology) Blakers–Massey theorem (homotopy theory) Bott periodicity theorem (homotopy theory) Brown's representability theorem (homotopy theory) Cellular approximation theorem (algebraic topology) Dold–Thom theorem (algebraic topology) Eilenberg–Ganea theorem (homological algebra, algebraic topology) Eilenberg–Zilber theorem (algebraic topology) Euler's polyhedron theorem (polyhedra) Excision theorem (homology theory) Freudenthal suspension theorem (homotopy theory) Hilton–Milnor theorem (algebraic topology) Homotopy excision theorem (algebraic topology) Hopf theorem (differential topology) Hurewicz theorem (algebraic topology) Künneth theorem (algebraic topology) Invariance of domain (algebraic topology) Landweber exact functor theorem (algebraic topology) Lefschetz fixed-point theorem (fixed points, algebraic topology) Lefschetz–Hopf theorem (topology)


id: list-of-theorems-20260505-6
file: data/en.wikipedia.org/wiki/List_of_theorems-6.md
category: reference
title: List of theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems
date_saved: 2026-05-05T08:17:33.142978+00:00
instance: kb-cron
parent: list-of-theorems-20260505-0
summary:
== Mechanics of deformable solids == Betti's theorem (physics) Castigliano's first and second theorems (structural analysis) Clapeyron's theorem (physics) Saint-Venant's theorem (physics) Theorem of three moments (physics) == Fluid mechanics == Buckingham π theorem (dimensional analysis) Helmholtz's theorems (physics) Kelvin's circulation theorem (physics) Kutta–Joukowski theorem (physics) Reynolds transport theorem (fluid dynamics) Taylor–Proudman theorem (physics) == Optics, electromagnetic theory == Blondel's theorem (electric power) (physics) Earnshaw's theorem (electrostatics) Maximum power theorem (electrical circuits) Norton's theorem (electrical networks) Optical theorem (physics) Poynting's theorem (physics) Thévenin's theorem (electrical circuits) == Classical thermodynamics, heat transfer == Carnot's theorem (thermodynamics) Clausius theorem (physics) == Quantum theory == Adiabatic theorem (physics) Bell's theorem (quantum mechanics) Bogoliubov–Parasyuk theorem (quantum field theory) Byers–Yang theorem (quantum mechanics) C-theorem (physics) Cluster decomposition theorem (quantum field theory) Coleman–Mandula theorem (quantum field theory) Elitzur's theorem (quantum field theory, statistical field theory) Furry's theorem (quantum field theory) Gell-Mann and Low theorem (quantum field theory) Gleason's theorem (Hilbert space) Goldstone's theorem (physics) Haag's theorem (quantum field theory) Haag–Łopuszański–Sohnius theorem (physics) Hellmann–Feynman theorem (physics) Kinoshita–Lee–Nauenberg theorem (quantum field theory) Kochen–Specker theorem (physics) Kramers' theorem (physics) Nielsen–Ninomiya theorem (quantum field theory) No-broadcasting theorem (quantum information theory) No-cloning theorem (quantum computation) No-communication theorem (quantum information theory) No-deleting


id: list-of-theorems-called-fundamental-20260505-0
file: data/en.wikipedia.org/wiki/List_of_theorems_called_fundamental-0.md
category: reference
title: List of theorems called fundamental
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_theorems_called_fundamental
date_saved: 2026-05-05T08:17:34.373025+00:00
instance: kb-cron
parent: list-of-theorems-called-fundamental-20260505-0
summary:
In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describes classification of regular curves in space up to translation and rotation. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself. == Fundamental theorems of mathematical topics == Fundamental theorem of algebra Fundamental theorem of algebraic K-theory Fundamental theorem of arithmetic Fundamental theorem of Boolean algebra Fundamental theorem of calculus Fundamental theorem of calculus for line integrals Fundamental theorem of curves Fundamental theorem of cyclic groups Fundamental theorem of dynamical systems Fundamental theorem of equivalence relations Fundamental theorem of exterior calculus Fundamental theorem


id: list-of-misnamed-theorems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_misnamed_theorems-0.md
category: reference
title: List of misnamed theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_misnamed_theorems
date_saved: 2026-05-05T08:17:35.596362+00:00
instance: kb-cron
parent: list-of-misnamed-theorems-20260505-0
summary:
This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, the items on this list illustrate Stigler's law of eponymy (which is not, of course, due to Stephen Stigler, who credits Robert K Merton). == Applied mathematics == Benford's law. This was first stated in 1881 by Simon Newcomb, and rediscovered in 1938 by Frank Benford. The first rigorous formulation and proof seems to be due to Ted Hill in 1988.; see also the contribution by Persi Diaconis. Bertrand's ballot theorem. This result concerning the probability that the winner of an election was ahead at each step of ballot counting was first published by W. A. Whitworth in 1878, but named after Joseph Louis François Bertrand who rediscovered it in 1887. A common proof uses André's reflection method, though the proof by Désiré André did not use any reflections. == Algebra == Burnside's lemma. This was stated and proved without attribution in Burnside's 1897 textbook, but it had previously been discussed by Augustin Cauchy, in 1845, and by Georg Frobenius in


id: list-of-misnamed-theorems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_misnamed_theorems-1.md
category: reference
title: List of misnamed theorems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_misnamed_theorems
date_saved: 2026-05-05T08:17:35.596362+00:00
instance: kb-cron
parent: list-of-misnamed-theorems-20260505-0
summary:
== Number theory == Bézout's theorem. The statement may have been made first by Isaac Newton in 1665. The matter of a proof was taken up by Colin MacLaurin (c. 1720) and Leonhard Euler as well as Étienne Bézout (c. 1750). However, Bézout's "proof" was incorrect. The first correct proof seems to be due mostly to Georges-Henri Halphen in the 1870s. Cramer's rule. It is named after Gabriel Cramer (1704–1752), who published the rule in his 1750 Introduction à l'analyse des lignes courbes algébriques, although Colin Maclaurin also published the method in his 1748 Treatise of Algebra (and probably knew of the method as early as 1729). Pell's equation. The solution of the equation x2 − dy2 = 1, where x and y are unknown positive integers and where d is a known positive integer which is not a perfect square, is ascribed to John Pell. It seems to have been discovered by Fermat, who set it as a challenge problem in 1657. The first European solution is found in a joint work in 1658 by John Wallis and Lord Brouncker; in 1668, a shorter solution was given in an edition of a third mathematician's work by Pell; see ref.


id: list-of-topologies-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topologies-0.md
category: reference
title: List of topologies
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topologies
date_saved: 2026-05-05T08:17:38.117299+00:00
instance: kb-cron
parent: list-of-topologies-20260505-0
summary:
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. == Discrete and indiscrete == Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open. == Cardinality and ordinals == Cocountable topology Given a topological space ( X , τ ) , {\displaystyle (X,\tau ),} the cocountable extension topology on X {\displaystyle X} is the topology having as a subbasis the union of τ and the family of all subsets of X {\displaystyle X} whose complements in X {\displaystyle X} are countable. Cofinite topology Double-pointed cofinite topology Ordinal number topology Pseudo-arc Ran space Tychonoff plank === Finite spaces === Discrete two-point space − The simplest example of a totally disconnected discrete space. Finite topological space Pseudocircle − A finite topological space on 4 elements that fails to satisfy any separation axiom besides T0. However, from the viewpoint of algebraic topology, it has the remarkable property that it


id: list-of-topologies-20260505-1
file: data/en.wikipedia.org/wiki/List_of_topologies-1.md
category: reference
title: List of topologies
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topologies
date_saved: 2026-05-05T08:17:38.117299+00:00
instance: kb-cron
parent: list-of-topologies-20260505-0
summary:
Alexandroff plank Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact. Arens square Bullet-riddled square - The space [ 0 , 1 ] 2 ∖ Q 2 , {\displaystyle [0,1]^{2}\setminus \mathbb {Q} ^{2},} where [ 0 , 1 ] 2 ∩ Q 2 {\displaystyle [0,1]^{2}\cap \mathbb {Q} ^{2}} is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable. Cantor tree Comb space Dieudonné plank Double origin topology Dunce hat (topology) Either–or topology Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point. Fort space Half-disk topology Hilbert cube − [ 0 , 1 / 1 ] × [ 0 , 1 / 2 ] × [ 0 , 1 / 3 ] × ⋯ {\displaystyle [0,1/1]\times [0,1/2]\times [0,1/3]\times \cdots } with the product topology. Infinite broom Integer broom topology K-topology Knaster–Kuratowski fan Long line (topology) Moore plane, also called the Niemytzki plane − A first countable, separable, completely regular, Hausdorff, Moore space that is not normal, Lindelöf, metrizable, second countable, nor locally compact. It also an uncountable closed


id: list-of-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topology_topics-0.md
category: reference
title: List of topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topology_topics
date_saved: 2026-05-05T08:17:39.346041+00:00
instance: kb-cron
parent: list-of-topology-topics-20260505-0
summary:
In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it


id: list-of-algebraic-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebraic_topology_topics-0.md
category: reference
title: List of algebraic topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebraic_topology_topics
date_saved: 2026-05-05T08:17:40.558645+00:00
instance: kb-cron
parent: list-of-algebraic-topology-topics-20260505-0
summary:
This is a list of algebraic topology topics. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. == Homology (mathematics) == Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal coefficient theorem Cohomology List of cohomology theories Cocycle class Cup product Cohomology ring De Rham cohomology Čech cohomology Alexander–Spanier cohomology Intersection cohomology Lusternik–Schnirelmann category Poincaré duality Fundamental class Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous mapping Borsuk–Ulam theorem Ham sandwich theorem Homology sphere == Homotopy theory == Homotopy Path (topology) Fundamental group Homotopy group Seifert–van


id: list-of-general-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_general_topology_topics-0.md
category: reference
title: List of general topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_general_topology_topics
date_saved: 2026-05-05T08:17:41.721700+00:00
instance: kb-cron
parent: list-of-general-topology-topics-20260505-0
summary:
This is a list of general topology topics. == Basic concepts == Topological space Topological property Open set, closed set Clopen set Closure Boundary Interior Density G-delta set, F-sigma set Closeness Neighborhood Continuity (topology) Homeomorphism Local homeomorphism Open and closed maps Embedding Germ Basis Subbasis Open cover Locally finite space Covering space Atlas == Limits == Limit point Net Filter Ultrafilter == Topological properties == Baire category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set === Compactness and countability === Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf space Sigma-compact space === Connectedness === Connected space Simply connected space Path connected space === Separation axioms === T0 space T1 space Hausdorff space Completely Hausdorff space Regular space Tychonoff space Normal space Urysohn's lemma Tietze extension theorem Paracompact Separated sets == Topological constructions == Direct sum and the dual construction product Subspace and the dual construction quotient Topological tensor product == Examples == Discrete space Locally constant function Trivial topology Cofinite topology Cocountable topology Finer topology Product topology Restricted product


id: list-of-geometric-topology-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_geometric_topology_topics-0.md
category: reference
title: List of geometric topology topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_geometric_topology_topics
date_saved: 2026-05-05T08:17:42.922521+00:00
instance: kb-cron
parent: list-of-geometric-topology-topics-20260505-0
summary:
This is a list of geometric topology topics. == Low-dimensional topology == === Knot theory === Knot (mathematics) Link (knot theory) Wild knots Examples of knots (and links) Unknot Trefoil knot Figure-eight knot (mathematics) Borromean rings Types of knots (and links) Torus knot Prime knot Alternating knot Hyperbolic link Knot invariants Crossing number Linking number Skein relation Knot polynomials Alexander polynomial Jones polynomial Knot group Writhe Quandle Seifert surface Braids Braid theory Braid group Kirby calculus === Surfaces === Genus (mathematics) Examples Positive Euler characteristic 2-disk Sphere Real projective plane Zero Euler characteristic Annulus Möbius strip Torus Klein bottle Negative Euler characteristic The boundary of the pretzel is a genus three surface Embedded/Immersed in Euclidean space Cross-cap Boy's surface Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification === Three-manifolds === Moise's Theorem (see also Hauptvermutung) Poincaré conjecture Thurston elliptization conjecture Thurston's geometrization conjecture Hyperbolic 3-manifolds Spherical 3-manifolds Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups Seifert fiber space Heegaard splitting Waldhausen conjecture Compression body Handlebody Incompressible surface Dehn's lemma Loop theorem (aka the Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination Examples 3-sphere Torus bundles Surface bundles over the


id: list-of-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_transforms-0.md
category: reference
title: List of transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_transforms
date_saved: 2026-05-05T08:17:44.168043+00:00
instance: kb-cron
parent: list-of-transforms-20260505-0
summary:
This is a list of transforms in mathematics. == Integral transforms == Abel transform Aboodh transform Bateman transform Fourier transform Fourier cosine transform Fourier sine transform Fractional Fourier transform Short-time Fourier transform Rectangular mask short-time Fourier transform Fourier–Bros–Iagolnitzer transform Gabor transform Hankel transform Hartley transform Hermite transform Hilbert transform Hilbert–Schmidt integral operator Jacobi transform Laguerre transform Laplace transform Inverse Laplace transform Two-sided Laplace transform Inverse two-sided Laplace transform Laplace–Carson transform Laplace–Stieltjes transform Legendre transform Associated Legendre transform Linear canonical transform Mellin transform Inverse Mellin transform Poisson–Mellin–Newton cycle N-transform Radon transform X-ray transform Shehu transform Stieltjes transformation Sumudu transform Wavelet transform (integral) Weierstrass transform == Discrete transforms == Binomial transform Discrete Fourier transform, DFT Fast Fourier transform, a popular implementation of the DFT Discrete cosine transform Modified discrete cosine transform Discrete Hartley transform Discrete sine transform Discrete wavelet transform Hadamard transform (or, Walsh–Hadamard transform) Fast wavelet transform Hankel transform, the determinant of the Hankel matrix Discrete Chebyshev transform Equivalent, up to a diagonal scaling, to a discrete cosine transform Finite Legendre transform Irrational base discrete weighted transform Mojette transform Number-theoretic transform Spherical Harmonic transform Stirling transform == Discrete-time transforms == These transforms have a continuous frequency domain: Discrete-time Fourier transform Z-transform


id: list-of-triangle-inequalities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-0.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the form "less than", "less than or equal to", "greater than", or "greater than or equal to". The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane. == Main parameters and notation == The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c; the semiperimeter s = ⁠a + b + c/2⁠ (half the perimeter p); the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the


id: list-of-triangle-inequalities-20260505-1
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-1.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
each holding with equality only when a = b = c. This says that in the non-equilateral case the harmonic mean of the sides is less than their geometric mean, which in turn is less than their arithmetic mean, and which in turn is less than their quadratic mean. == Angles == cos ⁡ A + cos ⁡ B + cos ⁡ C ≤ 3 2 . {\displaystyle \cos A+\cos B+\cos C\leq {\frac {3}{2}}.} ( 1 − cos ⁡ A ) ( 1 − cos ⁡ B ) ( 1 − cos ⁡ C ) ≥ cos ⁡ A ⋅ cos ⁡ B ⋅ cos ⁡ C . {\displaystyle (1-\cos A)(1-\cos B)(1-\cos C)\geq \cos A\cdot \cos B\cdot \cos C.} cos 4 ⁡ A 2 + cos 4 ⁡ B 2 + cos 4 ⁡ C 2 ≤ s 3 2 a b c {\displaystyle \cos ^{4}{\frac {A}{2}}+\cos ^{4}{\frac {B}{2}}+\cos ^{4}{\frac {C}{2}}\leq {\frac {s^{3}}{2abc}}} for semi-perimeter s, with equality only in the equilateral case. a + b + c ≥ 2 b c cos ⁡ A + 2 c a cos ⁡ B + 2 a b cos ⁡ C . {\displaystyle a+b+c\geq 2{\sqrt {bc}}\cos A+2{\sqrt {ca}}\cos B+2{\sqrt {ab}}\cos C.} sin


id: list-of-triangle-inequalities-20260505-2
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-2.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
a b + b c + c a ≥ 4 3 ⋅ T {\displaystyle ab+bc+ca\geq 4{\sqrt {3}}\cdot T} and T ≤ a b c 2 a + b + c a 3 + b 3 + c 3 + a b c ≤ 1 4 3 ( a + b + c ) 3 ( a b c ) 4 a 3 + b 3 + c 3 6 ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {abc}{2}}{\sqrt {\frac {a+b+c}{a^{3}+b^{3}+c^{3}+abc}}}\leq {\frac {1}{4}}{\sqrt[{6}]{\frac {3(a+b+c)^{3}(abc)^{4}}{a^{3}+b^{3}+c^{3}}}}\leq {\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}.} From the rightmost upper bound on T, using the arithmetic-geometric mean inequality, is obtained the isoperimetric inequality for triangles: T ≤ 3 36 ( a + b + c ) 2 = 3 9 s 2 {\displaystyle T\leq {\frac {\sqrt {3}}{36}}(a+b+c)^{2}={\frac {\sqrt {3}}{9}}s^{2}} for semiperimeter s. This is sometimes stated in terms of perimeter p as p 2 ≥ 12 3 ⋅ T , {\displaystyle p^{2}\geq 12{\sqrt {3}}\cdot T,} with equality for the equilateral triangle. This is strengthened by T ≤ 3 4 ( a b c ) 2 3 . {\displaystyle T\leq {\frac {\sqrt {3}}{4}}\left(abc\right)^{\frac {2}{3}}.} Bonnesen's inequality also strengthens the isoperimetric inequality: π 2 (


id: list-of-triangle-inequalities-20260505-3
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-3.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
== Altitudes == The altitudes ha, etc. each connect a vertex to the opposite side and are perpendicular to that side. They satisfy both h a + h b + h c ≤ 3 2 ( a + b + c ) {\displaystyle h_{a}+h_{b}+h_{c}\leq {\frac {\sqrt {3}}{2}}(a+b+c)} and h a 2 + h b 2 + h c 2 ≤ 3 4 ( a 2 + b 2 + c 2 ) . {\displaystyle h_{a}^{2}+h_{b}^{2}+h_{c}^{2}\leq {\frac {3}{4}}(a^{2}+b^{2}+c^{2}).} In addition, if a ≥ b ≥ c , {\displaystyle a\geq b\geq c,} then a + h a ≥ b + h b ≥ c + h c . {\displaystyle a+h_{a}\geq b+h_{b}\geq c+h_{c}.} We also have h a 2 ( b 2 + c 2 ) ⋅ h b 2 ( c 2 + a 2 ) ⋅ h c 2 ( a 2 + b 2 ) ≤ ( 3 8 ) 3 . {\displaystyle {\frac {h_{a}^{2}}{(b^{2}+c^{2})}}\cdot {\frac {h_{b}^{2}}{(c^{2}+a^{2})}}\cdot {\frac {h_{c}^{2}}{(a^{2}+b^{2})}}\leq \left({\frac {3}{8}}\right)^{3}.} For internal angle bisectors ta, tb, tc from vertices A, B, C and circumcenter R and incenter r, we have h a t a + h b t b + h c t c ≥ R + 4 r


id: list-of-triangle-inequalities-20260505-4
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-4.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
2 ( P A + P B + P C ) > A B + B C + C A > P A + P B + P C , {\displaystyle 2(PA+PB+PC)>AB+BC+CA>PA+PB+PC,} and more strongly than the second of these inequalities is: If A B {\displaystyle AB} is the shortest side of the triangle, then P A + P B + P C ≤ A C + B C . {\displaystyle PA+PB+PC\leq AC+BC.} We also have Ptolemy's inequality P A ⋅ B C + P B ⋅ C A > P C ⋅ A B {\displaystyle PA\cdot BC+PB\cdot CA>PC\cdot AB} for interior point P and likewise for cyclic permutations of the vertices. If we draw perpendiculars from interior point P to the sides of the triangle, intersecting the sides at D, E, and F, we have P A ⋅ P B ⋅ P C ≥ ( P D + P E ) ( P E + P F ) ( P F + P D ) . {\displaystyle PA\cdot PB\cdot PC\geq (PD+PE)(PE+PF)(PF+PD).} Further, the Erdős–Mordell inequality states that P A + P B + P C P D + P E + P F ≥ 2 {\displaystyle {\frac {PA+PB+PC}{PD+PE+PF}}\geq 2}


id: list-of-triangle-inequalities-20260505-5
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-5.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
P A 4 + P B 4 + P C 4 + k ( P A ⋅ P B ⋅ P C ) 4 3 ≥ 16 ( k + 3 ) r 4 {\displaystyle PA^{4}+PB^{4}+PC^{4}+k(PA\cdot PB\cdot PC)^{\frac {4}{3}}\geq 16(k+3)r^{4}} for k = 0, 1, ..., 9. Furthermore, for circumradius R, ( P A ⋅ P B ) 3 2 + ( P B ⋅ P C ) 3 2 + ( P C ⋅ P A ) 3 2 ≥ 12 R r 2 ; {\displaystyle \left(PA\cdot PB\right)^{\frac {3}{2}}+\left(PB\cdot PC\right)^{\frac {3}{2}}+\left(PC\cdot PA\right)^{\frac {3}{2}}\geq 12Rr^{2};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A ) 2 ≥ 8 ( R + r ) R r 2 ; {\displaystyle (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 8(R+r)Rr^{2};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A ) 2 ≥ 48 r 4 ; {\displaystyle (PA\cdot PB)^{2}+(PB\cdot PC)^{2}+(PC\cdot PA)^{2}\geq 48r^{4};} ( P A ⋅ P B ) 2 + ( P B ⋅ P C ) 2 + ( P C ⋅ P A )


id: list-of-triangle-inequalities-20260505-6
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-6.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
s ≤ ( 3 3 − 4 ) r + 2 R . {\displaystyle s\leq (3{\sqrt {3}}-4)r+2R.} We also have, for all acute triangles, s > 2 R + r . {\displaystyle s>2R+r.} For incircle center I, let AI, BI, and CI extend beyond I to intersect the circumcircle at D, E, and F respectively. Then A I I D + B I I E + C I I F ≥ 3. {\displaystyle {\frac {AI}{ID}}+{\frac {BI}{IE}}+{\frac {CI}{IF}}\geq 3.} In terms of the vertex angles we have cos ⁡ A ⋅ cos ⁡ B ⋅ cos ⁡ C ≤ ( r R 2 ) 2 . {\displaystyle \cos A\cdot \cos B\cdot \cos C\leq \left({\frac {r}{R{\sqrt {2}}}}\right)^{2}.} Denote as R A , R B , R C {\displaystyle R_{A},R_{B},R_{C}} the tanradii of the triangle. Then 4 R ≤ 1 R A + 1 R B + 1 R C ≤ 2 r {\displaystyle {\frac {4}{R}}\leq {\frac {1}{R_{A}}}+{\frac {1}{R_{B}}}+{\frac {1}{R_{C}}}\leq {\frac {2}{r}}} with equality only in the equilateral case, and 9 2 r ≤ R A + R B + R C ≤ 2 R + 1 2 r {\displaystyle {\frac {9}{2}}r\leq R_{A}+R_{B}+R_{C}\leq 2R+{\frac {1}{2}}r} with equality only in the equilateral case. ===


id: list-of-triangle-inequalities-20260505-7
file: data/en.wikipedia.org/wiki/List_of_triangle_inequalities-7.md
category: reference
title: List of triangle inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_triangle_inequalities
date_saved: 2026-05-05T08:17:45.554884+00:00
instance: kb-cron
parent: list-of-triangle-inequalities-20260505-0
summary:
=== Inscribed triangle === If three points D, E, F on the respective sides AB, BC, and CA of a reference triangle ABC are the vertices of an inscribed triangle, which thereby partitions the reference triangle into four triangles, then the area of the inscribed triangle is greater than the area of at least one of the other interior triangles, unless the vertices of the inscribed triangle are at the midpoints of the sides of the reference triangle (in which case the inscribed triangle is the medial triangle and all four interior triangles have equal areas): Area(DEF) ≥ min ( Area(BED), Area(CFE), Area(ADF) ) . {\displaystyle {\text{Area(DEF)}}\geq \min({\text{Area(BED), Area(CFE), Area(ADF)}}).} === Inscribed squares === An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle. (A right triangle has only two distinct inscribed squares.) If one of these squares has side length xa and another has side length xb with xa < xb, then 1 ≥ x a x b ≥ 2 2 3 ≈ 0.94. {\displaystyle 1\geq {\frac {x_{a}}{x_{b}}}\geq {\frac {2{\sqrt {2}}}{3}}\approx 0.94.} Moreover, for


id: list-of-trigonometric-identities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-0.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. == Pythagorean identities == The basic relationship between the sine and cosine is given by the Pythagorean identity: sin 2 ⁡ θ + cos 2 ⁡ θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,} where sin 2 ⁡ θ {\displaystyle \sin ^{2}\theta } means ( sin ⁡ θ ) 2 {\displaystyle {(\sin \theta )}^{2}} and cos 2 ⁡ θ {\displaystyle \cos ^{2}\theta } means ( cos ⁡ θ ) 2 . {\displaystyle {(\cos \theta )}^{2}.} This can be viewed as a version of the Pythagorean theorem, and follows from


id: list-of-trigonometric-identities-20260505-1
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-1.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
== Angle sum and difference identities == These are also known as the angle addition and subtraction theorems (or formulae). sin ⁡ ( α + β ) = sin ⁡ α cos ⁡ β + cos ⁡ α sin ⁡ β sin ⁡ ( α − β ) = sin ⁡ α cos ⁡ β − cos ⁡ α sin ⁡ β cos ⁡ ( α + β ) = cos ⁡ α cos ⁡ β − sin ⁡ α sin ⁡ β cos ⁡ ( α − β ) = cos ⁡ α cos ⁡ β + sin ⁡ α sin ⁡ β {\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}} The angle difference identities for sin ⁡ ( α − β ) {\displaystyle \sin(\alpha -\beta )} and cos ⁡ ( α − β ) {\displaystyle \cos(\alpha -\beta )} can be derived from the angle sum versions (and vice versa) by substituting − β {\displaystyle -\beta } for β {\displaystyle \beta } and using the


id: list-of-trigonometric-identities-20260505-2
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-2.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
tan ⁡ ( ∑ i θ i ) = sin ( ∑ i θ i ) / ∏ i cos ⁡ θ i cos ( ∑ i θ i ) / ∏ i cos ⁡ θ i = ∑ odd k ≥ 1 ( − 1 ) k − 1 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ∏ i ∈ A tan ⁡ θ i ∑ even k ≥ 0 ( − 1 ) k 2 ∑ A ⊆ { 1 , 2 , 3 , … } | A | = k ∏ i ∈ A tan ⁡ θ i = e 1 − e 3 + e 5 − ⋯ e 0 − e 2 + e 4 − ⋯ cot ⁡ ( ∑ i θ i ) = e 0 − e 2 + e 4 − ⋯ e 1 − e 3 + e 5 − ⋯ {\displaystyle {\begin{aligned}\tan {\Bigl (}\sum _{i}\theta _{i}{\Bigr )}&={\frac {{\sin }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}{{\cos }{\bigl (}\sum _{i}\theta _{i}{\bigr )}/\prod _{i}\cos \theta _{i}}}\\[10pt]&={\frac {\displaystyle \sum _{{\text{odd}}\ k\geq 1}(-1)^{\frac {k-1}{2}}\sum _{\begin{smallmatrix}A\subseteq \{1,2,3,\dots \}\\\left|A\right|=k\end{smallmatrix}}\prod _{i\in A}\tan \theta _{i}}{\displaystyle \sum


id: list-of-trigonometric-identities-20260505-3
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-3.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sec ⁡ ( α + β + γ ) = sec ⁡ α sec ⁡ β sec ⁡ γ 1 − tan ⁡ α tan ⁡ β − tan ⁡ α tan ⁡ γ − tan ⁡ β tan ⁡ γ csc ⁡ ( α + β + γ ) = sec ⁡ α sec ⁡ β sec ⁡ γ tan ⁡ α + tan ⁡ β + tan ⁡ γ − tan ⁡ α tan ⁡ β tan ⁡ γ . {\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\tan \alpha \tan \beta -\tan \alpha \tan \gamma -\tan \beta \tan \gamma }}\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\tan \alpha +\tan \beta +\tan \gamma -\tan \alpha \tan \beta \tan \gamma }}.\end{aligned}}} === Ptolemy's theorem === Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral A B C D {\displaystyle ABCD} , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths


id: list-of-trigonometric-identities-20260505-4
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-4.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ ( 3 θ ) = 3 sin ⁡ θ − 4 sin 3 ⁡ θ = 4 sin ⁡ θ sin ⁡ ( π 3 − θ ) sin ⁡ ( π 3 + θ ) cos ⁡ ( 3 θ ) = 4 cos 3 ⁡ θ − 3 cos ⁡ θ = 4 cos ⁡ θ cos ⁡ ( π 3 − θ ) cos ⁡ ( π 3 + θ ) tan ⁡ ( 3 θ ) = 3 tan ⁡ θ − tan 3 ⁡ θ 1 − 3 tan 2 ⁡ θ = tan ⁡ θ tan ⁡ ( π 3 − θ ) tan ⁡ ( π 3 + θ ) cot ⁡ ( 3 θ ) = 3 cot ⁡ θ − cot 3 ⁡ θ 1 − 3 cot 2 ⁡ θ sec ⁡ ( 3 θ ) = sec 3 ⁡ θ 4 − 3 sec 2 ⁡ θ csc ⁡ ( 3 θ ) = csc 3 ⁡ θ 3 csc 2 ⁡ θ − 4 {\displaystyle {\begin{aligned}\sin(3\theta )&=3\sin \theta -4\sin ^{3}\theta &&=4\sin \theta \sin \left({\tfrac {\pi }{3}}-\theta \right)\sin \left({\tfrac {\pi }{3}}+\theta \right)\\\cos(3\theta )&=4\cos ^{3}\theta -3\cos \theta &&=4\cos


id: list-of-trigonometric-identities-20260505-5
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-5.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
cos ⁡ ( ( n − 1 ) x + x ) = cos ⁡ ( ( n − 1 ) x ) cos ⁡ x − sin ⁡ ( ( n − 1 ) x ) sin ⁡ x cos ⁡ ( ( n − 1 ) x − x ) = cos ⁡ ( ( n − 1 ) x ) cos ⁡ x + sin ⁡ ( ( n − 1 ) x ) sin ⁡ x {\displaystyle {\begin{aligned}\cos {\bigl (}(n-1)x+x{\bigr )}&=\cos {\bigl (}(n-1)x{\bigr )}\cos x-\sin {\bigl (}(n-1)x{\bigr )}\sin x\\\cos {\bigl (}(n-1)x-x{\bigr )}&=\cos {\bigl (}(n-1)x{\bigr )}\cos x+\sin {\bigl (}(n-1)x{\bigr )}\sin x\end{aligned}}} It follows by induction that cos ⁡ ( n x ) {\displaystyle \cos(nx)} is a polynomial of cos ⁡ x , {\displaystyle \cos x,} the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition. Similarly, sin ⁡ ( n x ) {\displaystyle \sin(nx)} can be computed from sin ⁡ ( ( n − 1 ) x ) , {\displaystyle \sin((n-1)x),} sin ⁡ ( ( n − 2 ) x ) , {\displaystyle \sin((n-2)x),} and cos ⁡ x {\displaystyle \cos x} with sin ⁡ ( n x ) = 2 cos ⁡ x sin ⁡


id: list-of-trigonometric-identities-20260505-6
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-6.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae. === Product-to-sum identities === The product of two sines or cosines of different angles can be converted to a sum of trigonometric functions of a sum and difference of those angles: cos ⁡ θ cos ⁡ φ = 1 2 ( cos ⁡ ( θ − φ ) + cos ⁡ ( θ + φ ) ) , sin ⁡ θ sin ⁡ φ = 1 2 ( cos ⁡ ( θ − φ ) − cos ⁡ ( θ + φ ) ) , sin ⁡ θ cos ⁡ φ = 1 2 ( sin ⁡ ( θ + φ ) + sin ⁡ ( θ − φ ) ) , cos ⁡ θ sin ⁡ φ = 1 2 ( sin ⁡ ( θ + φ ) − sin ⁡ ( θ − φ ) ) .


id: list-of-trigonometric-identities-20260505-7
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-7.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
cot ⁡ ( z − a 1 ) cot ⁡ ( z − a 2 ) = − 1 + cot ⁡ ( a 1 − a 2 ) cot ⁡ ( z − a 1 ) + cot ⁡ ( a 2 − a 1 ) cot ⁡ ( z − a 2 ) . {\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).} === Finite products of trigonometric functions === For coprime integers n, m ∏ k = 1 n ( 2 a + 2 cos ⁡ ( 2 π k m n + x ) ) = 2 ( T n ( a ) + ( − 1 ) n + m cos ⁡ ( n x ) ) {\displaystyle \prod _{k=1}^{n}\left(2a+2\cos \left({\frac {2\pi km}{n}}+x\right)\right)=2\left(T_{n}(a)+{(-1)}^{n+m}\cos(nx)\right)} where Tn is the Chebyshev polynomial. The following relationship holds for the sine function ∏ k = 1 n − 1 sin ⁡ ( k π n ) = n 2 n − 1 . {\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}.} More generally for an integer n > 0 sin ⁡ ( n x ) = 2 n − 1 ∏ k = 0 n − 1 sin ⁡ ( k π n + x ) = 2


id: list-of-trigonometric-identities-20260505-8
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-8.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ ( A + B ) − sin ⁡ ( A − B ) = 2 cos ⁡ A sin ⁡ B . {\displaystyle \sin(A+B)-\sin(A-B)=2\cos A\sin B.} Then let's examine the following formula, 2 sin ⁡ α ∑ k = 1 n cos ⁡ ( 2 k − 1 ) α = 2 sin ⁡ α cos ⁡ α + 2 sin ⁡ α cos ⁡ 3 α + 2 sin ⁡ α cos ⁡ 5 α + ⋯ + 2 sin ⁡ α cos ⁡ ( 2 n − 1 ) α {\displaystyle 2\sin \alpha \sum _{k=1}^{n}\cos(2k-1)\alpha =2\sin \alpha \cos \alpha +2\sin \alpha \cos 3\alpha +2\sin \alpha \cos 5\alpha +\cdots +2\sin \alpha \cos(2n-1)\alpha } and this formula can be written by using the above identity, 2 sin ⁡ α ∑ k = 1 n cos ⁡ ( 2 k − 1 ) α = ∑ k = 1 n ( sin ⁡ ( 2 k α ) − sin ⁡ ( 2 ( k − 1 ) α ) ) = ( sin ⁡ 2 α − sin ⁡ 0 ) + ( sin ⁡ 4 α − sin ⁡ 2 α ) + ( sin ⁡ 6


id: list-of-trigonometric-identities-20260505-9
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-9.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
sin ⁡ x = x ∏ n = 1 ∞ ( 1 − x 2 π 2 n 2 ) , cos ⁡ x = ∏ n = 1 ∞ ( 1 − x 2 π 2 ( n − 1 2 ) ) 2 ) , sinh ⁡ x = x ∏ n = 1 ∞ ( 1 + x 2 π 2 n 2 ) , cosh ⁡ x = ∏ n = 1 ∞ ( 1 + x 2 π 2 ( n − 1 2 ) ) 2 ) . {\displaystyle {\begin{aligned}\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right),\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}} == Inverse trigonometric functions == The following identities give the result of composing a trigonometric function with an inverse trigonometric function. sin ⁡ ( arcsin ⁡ x ) = x cos ⁡ ( arcsin ⁡ x ) = 1 − x 2 tan ⁡ ( arcsin ⁡ x ) = x 1 − x 2 sin ⁡ ( arccos ⁡ x ) = 1 − x 2 cos ⁡ ( arccos ⁡ x ) = x tan ⁡ ( arccos


id: list-of-trigonometric-identities-20260505-10
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-10.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
arctan ⁡ 1 p = arctan ⁡ 1 p + q + arctan ⁡ q p 2 + p q + 1 . {\displaystyle \arctan {\frac {1}{p}}=\arctan {\frac {1}{p+q}}+\arctan {\frac {q}{p^{2}+pq+1}}.} The curious identity known as Morrie's law, cos ⁡ 20 ∘ ⋅ cos ⁡ 40 ∘ ⋅ cos ⁡ 80 ∘ = 1 8 , {\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},} is a special case of an identity that contains one variable: ∏ j = 0 k − 1 cos ⁡ ( 2 j x ) = sin ⁡ ( 2 k x ) 2 k sin ⁡ x . {\displaystyle \prod _{j=0}^{k-1}\cos \left(2^{j}x\right)={\frac {\sin \left(2^{k}x\right)}{2^{k}\sin x}}.} Similarly, sin ⁡ 20 ∘ ⋅ sin ⁡ 40 ∘ ⋅ sin ⁡ 80 ∘ = 3 8 {\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}}} is a special case of an identity with x = 20 ∘ {\displaystyle x=20^{\circ }} : sin ⁡ x ⋅ sin ⁡ ( 60 ∘ − x ) ⋅ sin ⁡ ( 60 ∘ + x ) = sin ⁡ 3 x 4 . {\displaystyle \sin x\cdot \sin \left(60^{\circ }-x\right)\cdot \sin \left(60^{\circ }+x\right)={\frac {\sin 3x}{4}}.} For the


id: list-of-trigonometric-identities-20260505-11
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-11.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
If n is an odd number ( n = 2 m + 1 {\displaystyle n=2m+1} ) we can make use of the symmetries to get ∏ k = 1 m tan ⁡ k π 2 m + 1 = 2 m + 1 {\displaystyle \prod _{k=1}^{m}\tan {\frac {k\pi }{2m+1}}={\sqrt {2m+1}}} The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved: ∏ k = 1 n sin ⁡ ( 2 k − 1 ) π 4 n = ∏ k = 1 n cos ⁡ ( 2 k − 1 ) π 4 n = 2 2 n {\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi }{4n}}=\prod _{k=1}^{n}\cos {\frac {\left(2k-1\right)\pi }{4n}}={\frac {\sqrt {2}}{2^{n}}}} === Computing π === An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula: π 4 = 4 arctan ⁡ 1 5 − arctan ⁡ 1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}} or, alternatively, by using an identity of Leonhard Euler: π 4 = 5 arctan


id: list-of-trigonometric-identities-20260505-12
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-12.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
== Further "conditional" identities for the case α + β + γ = 180° == A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied. The following formulae apply to arbitrary plane triangles and follow from α + β + γ = 180 ∘ , {\displaystyle \alpha +\beta +\gamma =180^{\circ },} as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur). tan ⁡ α + tan ⁡ β + tan ⁡ γ = tan ⁡ α tan ⁡ β tan ⁡ γ 1 = cot ⁡ β cot ⁡ γ + cot ⁡ γ cot ⁡ α + cot ⁡ α cot ⁡ β cot ⁡ ( α 2 ) + cot ⁡ ( β 2 ) + cot ⁡ ( γ 2 ) = cot ⁡ ( α 2 ) cot ⁡ ( β 2 ) cot ⁡ ( γ 2 ) 1 = tan ⁡ ( β 2 ) tan ⁡ ( γ 2 ) + tan ⁡ ( γ 2 ) tan ⁡ ( α 2 ) + tan ⁡ (


id: list-of-trigonometric-identities-20260505-13
file: data/en.wikipedia.org/wiki/List_of_trigonometric_identities-13.md
category: reference
title: List of trigonometric identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_trigonometric_identities
date_saved: 2026-05-05T08:17:46.795425+00:00
instance: kb-cron
parent: list-of-trigonometric-identities-20260505-0
summary:
=== Dirichlet kernel === The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity: 1 + 2 cos ⁡ x + 2 cos ⁡ ( 2 x ) + 2 cos ⁡ ( 3 x ) + ⋯ + 2 cos ⁡ ( n x ) = sin ⁡ ( ( n + 1 2 ) x ) sin ⁡ ( 1 2 x ) . {\displaystyle 1+2\cos x+2\cos(2x)+2\cos(3x)+\cdots +2\cos(nx)={\frac {\sin \left(\left(n+{\frac {1}{2}}\right)x\right)}{\sin \left({\frac {1}{2}}x\right)}}.} The convolution of any integrable function of period 2 π {\displaystyle 2\pi } with the Dirichlet kernel coincides with the function's n {\displaystyle n} th-degree Fourier approximation. The same holds for any measure or generalized function. === Tangent half-angle substitution === If we set t = tan ⁡ x 2 , {\displaystyle t=\tan {\frac {x}{2}},} then sin ⁡ x = 2 t 1 + t 2 ; cos ⁡ x = 1 − t 2 1 + t 2 ; e i x = 1 + i t 1 − i t ; d x = 2 d t 1 + t 2 , {\displaystyle \sin x={\frac {2t}{1+t^{2}}};\qquad \cos x={\frac {1-t^{2}}{1+t^{2}}};\qquad e^{ix}={\frac {1+it}{1-it}};\qquad dx={\frac {2\,dt}{1+t^{2}}},} where e i x =


id: outline-of-trigonometry-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_trigonometry-0.md
category: reference
title: Outline of trigonometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_trigonometry
date_saved: 2026-05-05T08:17:48.718303+00:00
instance: kb-cron
parent: outline-of-trigonometry-20260505-0
summary:
The following outline is provided as an overview of and topical guide to trigonometry: Trigonometry – branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. == Basics == Geometry – mathematics concerned with questions of shape, size, the relative position of figures, and the properties of space. Geometry is used extensively in trigonometry. Angle – the angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Ratio – a ratio indicates how many times one number contains another == Content of trigonometry == Trigonometry Trigonometric functions Trigonometric identities Euler's formula == Scholars == Archimedes Aristarchus Aryabhata Bhaskara I Claudius Ptolemy Euclid Hipparchus Madhava of Sangamagrama Ptolemy Pythagoras Regiomontanus == History == Aristarchus's inequality Bhaskara I's sine approximation formula Greek astronomy Indian astronomy Jyā, koti-jyā and utkrama-jyā Madhava's sine table Ptolemy's table of chords Rule of marteloio Āryabhaṭa's sine table == Fields ==


id: list-of-types-of-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_numbers-0.md
category: reference
title: List of types of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_numbers
date_saved: 2026-05-05T08:17:50.610585+00:00
instance: kb-cron
parent: list-of-types-of-numbers-20260505-0
summary:
Numbers can be classified according to how they are represented or according to the properties that they have. == Main types == Natural numbers ( N {\displaystyle \mathbb {N} } ): The counting numbers {1, 2, 3, ...} are commonly called natural numbers; however, other definitions include 0, so that the non-negative integers {0, 1, 2, 3, ...} are also called natural numbers. Natural numbers including 0 are also sometimes called whole numbers. Alternatively natural numbers not including 0 are also sometimes called whole numbers instead. Integers ( Z {\displaystyle \mathbb {Z} } ): Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. Rational numbers ( Q {\displaystyle \mathbb {Q} } ): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers ( R {\displaystyle \mathbb {R} } ): Numbers that correspond to points along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true. Irrational numbers ( R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} }


id: list-of-types-of-numbers-20260505-1
file: data/en.wikipedia.org/wiki/List_of_types_of_numbers-1.md
category: reference
title: List of types of numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_numbers
date_saved: 2026-05-05T08:17:50.610585+00:00
instance: kb-cron
parent: list-of-types-of-numbers-20260505-0
summary:
== Algebraic numbers == Algebraic number: Any number that is the root of a non-zero polynomial with rational coefficients. Transcendental number: Any real or complex number that is not algebraic. Examples include e and π. Trigonometric number: Any number that is the sine or cosine of a rational multiple of π. Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. == Non-standard numbers == Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets. Infinitesimals: These are smaller than any positive real number, but are nonetheless greater than zero. These were used in the initial development of calculus, and are used in synthetic


id: list-of-undecidable-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_undecidable_problems-0.md
category: reference
title: List of undecidable problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_undecidable_problems
date_saved: 2026-05-05T08:17:51.782326+00:00
instance: kb-cron
parent: list-of-undecidable-problems-20260505-0
summary:
In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not exist. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not. For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC. == Problems about abstract machines == The halting problem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it halts for every starting configuration). Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states and symbols). Rice's theorem states that for all nontrivial properties of partial functions, it is undecidable whether a given machine


id: list-of-undecidable-problems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_undecidable_problems-1.md
category: reference
title: List of undecidable problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_undecidable_problems
date_saved: 2026-05-05T08:17:51.782326+00:00
instance: kb-cron
parent: list-of-undecidable-problems-20260505-0
summary:
== Other problems == The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining if a given set of polyominoes can tile the plane. The problem of determining the Kolmogorov complexity of a string. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions, or in a piecewise-linear flow in three dimensions. Finding the capacity of an information-stable finite state machine channel. In network coding, determining whether a network is solvable. Determining whether a player has a winning strategy in a game of Magic: The Gathering. Planning in a partially observable Markov decision process. Planning air travel from one destination to another, when fares are taken into account. == See also == Lists of problems List of unsolved problems Reduction (complexity) Unknowability == Notes == == Bibliography == Brookshear, J. Glenn (1989). Theory of Computation: Formal Languages, Automata, and Complexity. Redwood City, California: Benjamin/Cummings Publishing Company, Inc. Appendix C includes impossibility of algorithms deciding if a grammar contains ambiguities, and impossibility of verifying program correctness by an algorithm


id: list-of-statements-independent-of-zfc-20260505-0
file: data/en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC-0.md
category: reference
title: List of statements independent of ZFC
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
date_saved: 2026-05-05T08:17:53.109673+00:00
instance: kb-cron
parent: list-of-statements-independent-of-zfc-20260505-0
summary:
The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC. == Axiomatic set theory == In 1931, Kurt Gödel proved his incompleteness theorems, establishing that many mathematical theories, including ZFC, cannot prove their own consistency. Assuming ω-consistency of such a theory, the consistency statement can also not be disproven, meaning it is independent. A few years later, other arithmetic statements were defined that are independent of any such theory, see for example Rosser's trick. The following set theoretic statements are independent of ZFC, among others: the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.); the generalized continuum hypothesis (GCH); a related independent


id: list-of-statements-independent-of-zfc-20260505-1
file: data/en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC-1.md
category: reference
title: List of statements independent of ZFC
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
date_saved: 2026-05-05T08:17:53.109673+00:00
instance: kb-cron
parent: list-of-statements-independent-of-zfc-20260505-0
summary:
One can write down a concrete polynomial p ∈ Z[x1, ..., x9] such that the statement "there are integers m1, ..., m9 with p(m1, ..., m9) = 0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). This follows from Yuri Matiyasevich's resolution of Hilbert's tenth problem; the polynomial is constructed so that it has an integer root if and only if ZFC is inconsistent. == Measure theory == A stronger version of Fubini's theorem for positive functions, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, is independent of ZFC. On the one hand, CH implies that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of [0, 1] equivalent to a well ordering of the cardinal ω1. A similar example can be constructed using MA. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman. It can also be deduced from a variant of Freiling's axiom of symmetry. == Topology == The Normal Moore Space conjecture, namely that every normal


id: list-of-euclidean-uniform-tilings-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Euclidean_uniform_tilings-0.md
category: reference
title: List of Euclidean uniform tilings
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Euclidean_uniform_tilings
date_saved: 2026-05-05T08:17:54.404656+00:00
instance: kb-cron
parent: list-of-euclidean-uniform-tilings-20260505-0
summary:
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex. These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color-uniform.) In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it


id: list-of-uniform-polyhedra-20260505-0
file: data/en.wikipedia.org/wiki/List_of_uniform_polyhedra-0.md
category: reference
title: List of uniform polyhedra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_uniform_polyhedra
date_saved: 2026-05-05T08:17:55.737868+00:00
instance: kb-cron
parent: list-of-uniform-polyhedra-20260505-0
summary:
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both. This list includes these: all 75 nonprismatic uniform polyhedra; a few representatives of the infinite sets of prisms and antiprisms; one degenerate polyhedron, Skilling's figure with overlapping edges. It was proven in Sopov (1970) that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. Not included are: The uniform polyhedron compounds. 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra) 11 Euclidean convex uniform


id: uniform-tilings-in-hyperbolic-plane-20260505-0
file: data/en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane-0.md
category: reference
title: Uniform tilings in hyperbolic plane
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane
date_saved: 2026-05-05T08:17:57.549662+00:00
instance: kb-cron
parent: uniform-tilings-in-hyperbolic-plane-20260505-0
summary:
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol {7,3}. Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}. == Wythoff construction == There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where ⁠1/p⁠ + ⁠1/q⁠ + ⁠1/r⁠ < 1, where p, q, r are


id: uniform-tilings-in-hyperbolic-plane-20260505-1
file: data/en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane-1.md
category: reference
title: Uniform tilings in hyperbolic plane
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane
date_saved: 2026-05-05T08:17:57.549662+00:00
instance: kb-cron
parent: uniform-tilings-in-hyperbolic-plane-20260505-0
summary:
=== (4 3 3) === The (4 3 3) triangle group, Coxeter group [(4,3,3)], orbifold (*433) contains these uniform tilings. Without right angles in the fundamental triangle, the Wythoff constructions are slightly different. For instance in the (4,3,3) triangle family, the snub form has six polygons around a vertex and its dual has hexagons rather than pentagons. In general the vertex figure of a snub tiling in a triangle (p,q,r) is p. 3.q.3.r.3, being 4.3.3.3.3.3 in this case below. === (4 4 3) === The (4 4 3) triangle group, Coxeter group [(4,4,3)], orbifold (*443) contains these uniform tilings. === (4 4 4) === The (4 4 4) triangle group, Coxeter group [(4,4,4)], orbifold (*444) contains these uniform tilings. === (5 3 3) === The (5 3 3) triangle group, Coxeter group [(5,3,3)], orbifold (*533) contains these uniform tilings. === (5 4 3) === The (5 4 3) triangle group, Coxeter group [(5,4,3)], orbifold (*543) contains these uniform tilings. === (5 4 4) === The (5 4 4) triangle group, Coxeter group [(5,4,4)], orbifold (*544) contains these uniform tilings. === (6 3 3) === The (6 3 3) triangle group, Coxeter group [(6,3,3)], orbifold (*633) contains these uniform tilings. ===


id: list-of-united-states-regional-mathematics-competitions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_United_States_regional_mathematics_competitions-0.md
category: reference
title: List of United States regional mathematics competitions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_United_States_regional_mathematics_competitions
date_saved: 2026-05-05T08:17:58.773121+00:00
instance: kb-cron
parent: list-of-united-states-regional-mathematics-competitions-20260505-0
summary:
Many math competitions in the United States have regional restrictions. Of these, most are statewide. For a more complete list, please visit here . The contests include: == Alabama == Alabama Statewide High School Mathematics Contest Virgil Grissom High School Math Tournament Vestavia Hills High School Math Tournament == Arizona == Great Plains Math League AATM State High School Contest == California == Bay Area Math Olympiad Berkeley Math Tournament Lawrence Livermore National Laboratories Annual High School Math Challenge Cal Poly Math Contest and Trimathlon [1] Polya Competition Bay Area Math Meet College of Creative Studies Math Competition LA Math Cup Math Day at the Beach hosted by CSULB[2] Math Field Day for San Diego Middle Schools Mesa Day Math Contest at UC Berkeley Santa Barbara County Math Superbowl Pomona College Mathematical Talent Search Redwood Empire Mathematics Tournament hosted by Humboldt State (middle and high school) San Diego Math League and San Diego Math Olympiad hosted by the San Diego Math Circle Santa Clara University High School Mathematics Contest Southern California Mathematics Competition, hosted at USC Stanford Mathematics Tournament UCSD/GSDMC High School Honors Mathematics Contest == Colorado == Colorado Mathematics Olympiad == District of Columbia == Moody's Mega Math ==


id: list-of-variational-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_variational_topics-0.md
category: reference
title: List of variational topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_variational_topics
date_saved: 2026-05-05T08:17:59.915350+00:00
instance: kb-cron
parent: list-of-variational-topics-20260505-0
summary:
This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction. Action (physics) Averaged Lagrangian Brachistochrone curve Calculus of variations Catenoid Cycloid Dirichlet principle Euler–Lagrange equation cf. Action (physics) Fermat's principle Functional (mathematics) Functional derivative Functional integral Geodesic Isoperimetry Lagrangian Lagrangian mechanics Legendre transformation Luke's variational principle Minimal surface Morse theory Noether's theorem Path integral formulation Plateau's problem Prime geodesic Principle of least action Soap bubble Soap film Tautochrone curve


id: vector-algebra-relations-20260505-0
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-0.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
The following are important identities in vector algebra. Identities that only involve the magnitude of a vector ‖ A ‖ {\displaystyle \|\mathbf {A} \|} and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier. == Magnitudes == The magnitude of a vector A can be expressed using the dot product: ‖ A ‖ 2 = A ⋅ A {\displaystyle \|\mathbf {A} \|^{2}=\mathbf {A\cdot A} } In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem: ‖ A ‖ 2 = A 1 2 + A 2 2 + A 3 2 {\displaystyle \|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}} == Inequalities == The Cauchy–Schwarz inequality: A ⋅ B ≤ ‖ A ‖ ‖ B ‖ {\displaystyle \mathbf {A} \cdot \mathbf {B} \leq \left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|} The triangle inequality: ‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A}


id: vector-algebra-relations-20260505-1
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-1.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
== Addition and multiplication of vectors == Commutativity of addition: A + B = B + A {\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} } . Commutativity of scalar product: A ⋅ B = B ⋅ A {\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} } . Anticommutativity of cross product: A × B = − ( B × A ) {\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-} (\mathbf {B} \times \mathbf {A} )} . Distributivity of multiplication by a scalar over addition: c ( A + B ) = c A + c B {\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} } . Distributivity of scalar product over addition: ( A + B ) ⋅ C = A ⋅ C + B ⋅ C {\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} } . Distributivity of vector product over addition: ( A + B ) × C = A × C + B × C {\displaystyle (\mathbf {A} +\mathbf {B} )\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} } . Scalar triple product: A ⋅ (


id: vector-algebra-relations-20260505-2
file: data/en.wikipedia.org/wiki/Vector_algebra_relations-2.md
category: reference
title: Vector algebra relations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_algebra_relations
date_saved: 2026-05-05T08:18:01.266487+00:00
instance: kb-cron
parent: vector-algebra-relations-20260505-0
summary:
A consequence of the previous equation: | A B C | D = ( A ⋅ D ) ( B × C ) + ( B ⋅ D ) ( C × A ) + ( C ⋅ D ) ( A × B ) . {\displaystyle |\mathbf {A} \,\mathbf {B} \,\mathbf {C} |\,\mathbf {D} =(\mathbf {A} \cdot \mathbf {D} )\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right).} In 3 dimensions, a vector D can be expressed in terms of basis vectors {A,B,C} as: D = D ⋅ ( B × C ) | A B C | A + D ⋅ ( C × A ) | A B C | B + D ⋅ ( A × B ) | A B C | C . {\displaystyle \mathbf {D} \ =\ {\frac {\mathbf {D} \cdot (\mathbf {B} \times \mathbf {C} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {A} +{\frac {\mathbf {D} \cdot (\mathbf {C} \times \mathbf {A} )}{|\mathbf {A} \,\mathbf {B} \,\mathbf {C} |}}\ \mathbf {B} +{\frac {\mathbf {D} \cdot (\mathbf {A} \times \mathbf {B} )}{|\mathbf {A} \,\mathbf {B}


id: vector-calculus-identities-20260505-0
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-0.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
The following are important identities involving derivatives and integrals in vector calculus. == Operator notation == === Gradient === For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad ⁡ ( f ) = ∇ f = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k {\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} } where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ψ ( x 1 , … , x n ) {\displaystyle \psi (x_{1},\ldots ,x_{n})} , also called a scalar field, the gradient is the vector field: ∇ ψ = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) ψ = ∂ ψ ∂ x 1 e 1 + ⋯ + ∂ ψ ∂ x n e n {\displaystyle


id: vector-calculus-identities-20260505-1
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-1.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
curl ⁡ F = ∇ × F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) × ( F x , F y , F z ) = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | = ( ∂ F z ∂ y − ∂ F y ∂ z ) i + ( ∂ F x ∂ z − ∂ F z ∂ x ) j + ( ∂ F y ∂ x − ∂ F x ∂ y ) k {\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}\\[1em]&={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}\\[1em]&=\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} \end{aligned}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. As the name implies the curl is a measure of how much nearby vectors tend in a circular direction. In Einstein notation,


id: vector-calculus-identities-20260505-2
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-2.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
∇ ⋅ ( A × B ) = ∇ A ⋅ ( A × B ) + ∇ B ⋅ ( A × B ) = ( ∇ A × A ) ⋅ B + ( ∇ B × A ) ⋅ B = ( ∇ A × A ) ⋅ B − ( A × ∇ B ) ⋅ B = ( ∇ A × A ) ⋅ B − A ⋅ ( ∇ B × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) {\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\nabla _{\mathbf {A} }\cdot (\mathbf {A} \times \mathbf {B} )+\nabla _{\mathbf {B} }\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} +(\nabla _{\mathbf {B} }\times \mathbf {A} )\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \nabla _{\mathbf {B} })\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla _{\mathbf {B} }\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}} An alternative method is to use the Cartesian components of


id: vector-calculus-identities-20260505-3
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-3.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
For the remainder of this article, Feynman subscript notation will be used where appropriate. == First derivative identities == For scalar fields ψ {\displaystyle \psi } , ϕ {\displaystyle \phi } and vector fields A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } , we have the following derivative identities. === Distributive properties === === First derivative associative properties === === Product rule for multiplication by a scalar === We have the following generalizations of the product rule in single-variable calculus. === Quotient rule for division by a scalar === === Chain rule === Let f ( x ) {\displaystyle f(x)} be a one-variable function from scalars to scalars, r ( t ) = ( x 1 ( t ) , … , x n ( t ) ) {\displaystyle \mathbf {r} (t)=(x_{1}(t),\ldots ,x_{n}(t))} a parametrized curve, ϕ : R n → R {\displaystyle \phi \!:\mathbb {R} ^{n}\to \mathbb {R} } a function from vectors to scalars, and A : R n → R n {\displaystyle \mathbf {A} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} a vector field. We have the following special cases of the multi-variable chain rule. For a vector transformation u : R n → R


id: vector-calculus-identities-20260505-4
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-4.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
=== Curl of divergence is not defined === The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore, ∇ × ( ∇ ⋅ A ) is undefined. {\displaystyle \nabla \times (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}} === Second derivative associative properties === === A mnemonic === The figure to the right is a mnemonic for some of these identities. The abbreviations used are: D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist. == Summary of important identities == === Differentiation === ==== Gradient ==== ==== Divergence ==== ==== Curl ==== ==== Vector-dot-Del Operator ==== ==== Second derivatives ==== ==== Third derivatives ==== === Integration === Below, the curly symbol ∂ means "boundary of" a surface or solid. ==== Surface–volume integrals ==== In the following surface–volume integral theorems, V denotes a three-dimensional volume with


id: vector-calculus-identities-20260505-5
file: data/en.wikipedia.org/wiki/Vector_calculus_identities-5.md
category: reference
title: Vector calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Vector_calculus_identities
date_saved: 2026-05-05T08:18:02.786492+00:00
instance: kb-cron
parent: vector-calculus-identities-20260505-0
summary:
∮ ∂ S A × ( B × d ℓ ) = ∬ S ( ∇ × ( A B T ) ) ⋅ d S + ∬ S ( ∇ ⋅ ( B A T ) ) × d S {\displaystyle \oint _{\partial S}\mathbf {A} \times (\mathbf {B} \times d{\boldsymbol {\ell }})\ =\ \iint _{S}\left(\nabla \times \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\right)\cdot d\mathbf {S} +\iint _{S}\left(\nabla \cdot \left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\right)\right)\times d\mathbf {S} } ∮ ∂ S ( B ⋅ d ℓ ) A = ∬ S ( d S ⋅ [ ∇ × B − B × ∇ ] ) A {\displaystyle \oint _{\partial S}(\mathbf {B} \cdot d{\boldsymbol {\ell }})\mathbf {A} =\iint _{S}(d\mathbf {S} \cdot \left[\nabla \times \mathbf {B} -\mathbf {B} \times \nabla \right])\mathbf {A} } Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral): ==== Endpoint-curve integrals ==== In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points q − p = ∂ P {\displaystyle \mathbf {q} -\mathbf {p} =\partial P} and integration along P is from


id: list-of-vector-spaces-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_vector_spaces_in_mathematics-0.md
category: reference
title: List of vector spaces in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_vector_spaces_in_mathematics
date_saved: 2026-05-05T08:18:04.096795+00:00
instance: kb-cron
parent: list-of-vector-spaces-in-mathematics-20260505-0
summary:
This is a list of vector spaces in abstract mathematics, by Wikipedia page. Banach space Besov space Bochner space Dual space Euclidean space Fock space Fréchet space Hardy space Hilbert space Hölder space LF-space Lp space Minkowski space Montel space Morrey–Campanato space Orlicz space Riesz space Schwartz space Sobolev space Tsirelson space


id: list-of-wavelet-related-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_wavelet-related_transforms-0.md
category: reference
title: List of wavelet-related transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_wavelet-related_transforms
date_saved: 2026-05-05T08:18:06.741131+00:00
instance: kb-cron
parent: list-of-wavelet-related-transforms-20260505-0
summary:
A list of wavelet related transforms: Continuous wavelet transform (CWT) Discrete wavelet transform (DWT) Multiresolution analysis (MRA) Lifting scheme Binomial QMF (BQMF) Fast wavelet transform (FWT) Complex wavelet transform Non or undecimated wavelet transform, the downsampling is omitted Newland transform, an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters in frequency space Wavelet packet decomposition (WPD), detail coefficients are decomposed and a variable tree can be formed Stationary wavelet transform (SWT), no downsampling and the filters at each level are different e-decimated discrete wavelet transform, depends on if the even or odd coefficients are selected in the downsampling Second generation wavelet transform (SGWT), filters and wavelets are not created in the frequency domain Dual-tree complex wavelet transform (DTCWT), two trees are used for decomposion to produce the real and complex coefficients WITS: Where Is The Starlet, a collection of a hundredth of wavelet names in -let and associated multiscale, directional, geometric, representations, from activelets to x-lets through bandelets, chirplets, contourlets, curvelets, noiselets, wedgelets ...


id: list-of-wenninger-polyhedron-models-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_models-0.md
category: reference
title: List of Wenninger polyhedron models
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_models
date_saved: 2026-05-05T08:18:08.055382+00:00
instance: kb-cron
parent: list-of-wenninger-polyhedron-models-20260505-0
summary:
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex regular and quasiregular polyhedra. Models listed here can be cited as "Wenninger Model Number N", or WN for brevity. The polyhedra are grouped in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings. == Platonic solids (regular convex polyhedra) W1 to W5 == == Archimedean solids (Semiregular) W6 to W18 == == Kepler–Poinsot polyhedra (Regular star polyhedra) W20, W21, W22 and W41 == == Stellations: models W19 to W66 == === Stellations of octahedron === === Stellations of dodecahedron === === Stellations of icosahedron === === Stellations of cuboctahedron === === Stellations of icosidodecahedron === == Uniform nonconvex solids


id: list-of-zeta-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_zeta_functions-0.md
category: reference
title: List of zeta functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_zeta_functions
date_saved: 2026-05-05T08:18:09.230852+00:00
instance: kb-cron
parent: list-of-zeta-functions-20260505-0
summary:
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur zeta function of a dynamical system Barnes zeta function or double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function of a number field Duursma zeta function of error-correcting codes Epstein zeta function of a quadratic form Goss zeta function of a function field Hasse–Weil zeta function of a variety Height zeta function of a variety Hurwitz zeta function, a generalization of the Riemann zeta function Igusa zeta function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization of the Riemann zeta function Local zeta function of a characteristic-p variety Matsumoto zeta function Minakshisundaram–Pleijel zeta function of a Laplacian Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic number Prime zeta function,


id: index-of-meteorology-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-0.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
This is a list of meteorology topics. The terms relate to meteorology, the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. (see also: List of meteorological phenomena) == A == advection aeroacoustics aerobiology aerology air parcel (in meteorology) air quality index (AQI) airshed (in meteorology) American Geophysical Union (AGU) American Meteorological Society (AMS) anabatic wind anemometer annular hurricane anticyclone (in meteorology) apparent wind Atlantic Oceanographic and Meteorological Laboratory (AOML) Atlantic hurricane season atmometer atmosphere Atmospheric Model Intercomparison Project (AMIP) Atmospheric Radiation Measurement (ARM) (atmospheric boundary layer [ABL]) planetary boundary layer (PBL) atmospheric chemistry atmospheric circulation atmospheric convection atmospheric dispersion modeling atmospheric electricity atmospheric icing atmospheric physics atmospheric pressure atmospheric sciences atmospheric stratification atmospheric thermodynamics atmospheric window (see under Threats) == B == ball lightning balloon (aircraft) baroclinity barotropicity barometer ("to measure atmospheric pressure") berg wind biometeorology blizzard bomb (meteorology) buoyancy Bureau of Meteorology (in Australia) == C == Canada Weather Extremes Canadian Hurricane Centre (CHC) Cape Verde-type hurricane capping inversion (in meteorology) (see "severe thunderstorms" in paragraph 5) carbon cycle carbon fixation carbon flux carbon monoxide (see under Atmospheric presence) ceiling balloon ("to determine the height of the base of clouds above ground level") ceilometer


id: index-of-meteorology-articles-20260505-1
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-1.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
== E == Earth's atmosphere Earth's magnetic field Earth System Research Laboratories (ESRL) economics of global warming Emagram effect of Hurricane Katrina on New Orleans (see also Hurricane Katrina effects by region) effect of sun angle on climate Energy meteorology Enhanced Fujita scale (EF scale) eolian processes equator (see under Equatorial seasons and climate) equilibrium level (EL) equivalent potential temperature equivalent temperature European Centre for Medium-Range Weather Forecasts (ECMWF) European Climate Change Programme (ECCP) European emission standards (for motor vehicles) European Severe Storms Laboratory (ESSL) European windstorm evaporation evaporative cooler evaporative cooling evaporite (a mineral sediment resulting from evaporation of saline water) evapotranspiration (ET) (sum of evaporation and plant transpiration) exhaust gas recirculation (EGR) (exhaust gas recycling) exosphere (layer of atmosphere) extratropical cyclone (mid-latitude cyclone) extreme weather extremes on Earth == F == fire whirl firestorm fog forensic meteorology free convective layer (FCL) freezing rain (front [meteorology]: see) surface weather analysis frontogenesis frontolysis frost frost creep (frost heave) frost flowers (frost castles) (ice castles) (ice ribbons) (ice blossoms) frost heaving (frost heave) frost law frost line (frost point: see) dew point (dewpoint) frostbite Fujita scale (F scale) (for measuring tornadoes) fulgurite (full lunar eclipse: see) lunar eclipse full-spectrum light funnel


id: index-of-meteorology-articles-20260505-2
file: data/en.wikipedia.org/wiki/Index_of_meteorology_articles-2.md
category: reference
title: Index of meteorology articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_meteorology_articles
date_saved: 2026-05-05T08:18:10.971529+00:00
instance: kb-cron
parent: index-of-meteorology-articles-20260505-0
summary:
== N == NASA Clean Air Study NASA Earth Observatory NASA World Wind (virtual globe) National Ambient Air Quality Standards (NAAQS) (in the US) National Center for Atmospheric Research (NCAR) (in the US) National Centers for Environmental Prediction (NCEP) (in the US) National Climatic Data Center (NCDC) (in the US) National Emissions Standards for Hazardous Air Pollutants (NESHAPS) (in the US) (National Environmental Satellite, Data and Information Service: see) National Oceanic and Atmospheric Administration (NOAA) (in the US) National Geomagnetism Program (in the US) National Hurricane Center (NHC) (in the US) National Map (in the US) National Oceanic and Atmospheric Administration (NOAA) (in the US) (National Severe Storms Forecast Center [NSSFC]: renamed) Storm Prediction Center (SPC) (in the US) National Severe Storms Laboratory (NSSL) (in the US) National Snow and Ice Data Center (NSIDC) (in the US) National Solar Observatory (in the US) National Weather Association (NWA) (in the US) National Weather Center (NWC) (in the US) National Weather Service bulletin for New Orleans region (at 10:11 a.m., August 28, 2005) National Weather Service (NWS) nautical almanac nephology nephoscope night sky nimbus cloud nitrogen cycle (nitrogen pollution: see) eutrophication (see under Atmospheric deposition) NOAA Weather Radio All Hazards (NWR) (of


id: outline-of-meteorology-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_meteorology-0.md
category: reference
title: Outline of meteorology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_meteorology
date_saved: 2026-05-05T08:18:12.231405+00:00
instance: kb-cron
parent: outline-of-meteorology-20260505-0
summary:
The following outline is provided as an overview of and topical guide to the field of Meteorology. Meteorology The interdisciplinary, scientific study of the Earth's atmosphere with the primary focus being to understand, explain, and forecast weather events. Meteorology, is applied to and employed by a wide variety of diverse fields, including the military, energy production, transport, agriculture, and construction. == Essence of meteorology == Meteorology Climate – the average and variations of weather in a region over long periods of time. Meteorology – the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting (in contrast with climatology). Weather – the set of all the phenomena in a given atmosphere at a given time. == Branches of meteorology == Microscale meteorology – the study of atmospheric phenomena about 1 km or less, smaller than mesoscale, including small and generally fleeting cloud "puffs" and other small cloud features Mesoscale meteorology – the study of weather systems about 5 kilometers to several hundred kilometers, smaller than synoptic scale systems but larger than microscale and storm-scale cumulus systems, such as sea breezes, squall lines, and mesoscale convective complexes Synoptic scale meteorology – is a horizontal length scale of the


id: outline-of-meteorology-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_meteorology-1.md
category: reference
title: Outline of meteorology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_meteorology
date_saved: 2026-05-05T08:18:12.231405+00:00
instance: kb-cron
parent: outline-of-meteorology-20260505-0
summary:
History of weather forecasting – prior to the invention of meteorological instruments, weather analysis and prediction relied on pattern recognition, which was not always reliable History of surface weather analysis – initially used to study storm behavior, now used to explain current weather and as an aid in short term weather forecasting == Meteorological phenomena == Atmospheric pressure – the pressure at any given point in the Earth's atmosphere Cloud – a visible mass of droplets or frozen crystals floating in the atmosphere above the surface of a planet Rain – precipitation in which separate drops of water fall to the Earth from clouds, a product of the condensation of atmospheric water vapor Snow – precipitation in the form of crystalline water ice, consisting of a multitude of snowflakes that fall from clouds Freezing rain – precipitation that falls from a cloud as snow, melts completely on its way down, then passes through a layer of below-freezing air becoming supercooled, at which point it will freeze upon impact with any object encountered Sleet – term used in the United States and Canada for precipitation consisting of small, translucent ice balls, usually smaller than hailstones Tropical cyclone – a storm system


id: list-of-atlantic-hurricane-records-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-0.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
As of November 2025, there have been 1,753 tropical cyclones of at least tropical storm intensity, 972 at hurricane intensity, and 342 at major hurricane intensity within the Atlantic Ocean since 1851, the first Atlantic hurricane season to be included in the official Atlantic tropical cyclone record. Though a majority of these cyclones have fallen within climatological averages, prevailing atmospheric conditions occasionally lead to anomalous tropical systems which at times reach extremes in statistical record-keeping including in duration and intensity. The scope of this list is limited to tropical cyclone records solely within the North Atlantic Ocean and is subdivided by their reason for notability. == Tropical cyclogenesis == === Most active / least active Atlantic hurricane seasons === Most Atlantic hurricane seasons prior to the weather satellite era include seven or fewer recorded tropical storms or hurricanes. As the usage of satellite data was not available until the mid-1960s, early storm counts are less reliable. Before the advent of the airplane or means of tracking storms, the ones recorded were storms that affected mainly populated areas. An undercount bias of zero to six tropical cyclones per year between 1851 and 1885 and zero to four per year between 1886


id: list-of-atlantic-hurricane-records-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-1.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
Generally speaking, the intensity of a tropical cyclone is determined by either the storm's maximum sustained winds or lowest barometric pressure. The following table lists the most intense Atlantic hurricanes in terms of their lowest barometric pressure. In terms of wind speed, Allen from 1980 and Melissa from 2025 are tied for the strongest Atlantic tropical cyclones on record, with maximum sustained winds of 190 mph (305 km/h). For many years, it was thought that Hurricane Camille also attained this intensity, but this conclusion was changed in 2014. The original measurements of Camille are suspect since wind speed instrumentation used at the time would likely be damaged by winds of such intensity. Nonetheless, their central pressures are low enough to rank them among the strongest recorded Atlantic hurricanes. This is also why despite there being 2 hurricanes in the month of January, neither of them are ranked here as both of them had a higher pressure than Unnamed Subtropical Storm (2023). Owing to their intensity, the strongest Atlantic hurricanes have all attained Category 5 classification. Hurricane Opal, the most intense Category 4 hurricane recorded, intensified to reach a minimum pressure of 916 mbar (hPa; 27.05 inHg), a pressure typical of


id: list-of-atlantic-hurricane-records-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records-2.md
category: reference
title: List of Atlantic hurricane records
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Atlantic_hurricane_records
date_saved: 2026-05-05T08:18:13.570754+00:00
instance: kb-cron
parent: list-of-atlantic-hurricane-records-20260505-0
summary:
=== Fastest intensification === Fastest intensification from a tropical depression to a hurricane (1-minute sustained surface winds) – 12 hoursHarvey 1981 – 35 mph (55 km/h) to 80 mph (130 km/h) – from 1200 UTC September 12 to 0000 UTC September 13 Fastest intensification from a tropical depression to a Category 5 hurricane (1-minute sustained surface winds) – 54 hoursMilton 2024 – 35 mph (55 km/h) to 175 mph (280 km/h) – from 1200 UTC October 5 to 1800 UTC October 7 Fastest intensification from a tropical storm to a Category 5 hurricane (1-minute sustained surface winds) – 24 hoursWilma 2005 – 70 mph (110 km/h) to 175 mph (275 km/h) – from 0600 UTC October 18 to 0600 UTC October 19 Maximum pressure drop in 12 hours – 83 mbarWilma 2005 – 975 millibars (28.8 inHg) to 892 millibars (26.3 inHg) – from 1800 UTC October 18 to 0600 UTC October 19 Maximum pressure drop in 24 hours – 97 mbarWilma 2005 – 979 millibars (28.9 inHg) to 882 millibars (26.0 inHg) – from 1200 UTC October 18 to 1200 UTC October 19 == Landfall == === Earliest and latest Atlantic landfalls === === Southern and northernmost Atlantic landfalls


id: list-of-countries-by-average-yearly-temperature-20260505-0
file: data/en.wikipedia.org/wiki/List_of_countries_by_average_yearly_temperature-0.md
category: reference
title: List of countries by average yearly temperature
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_countries_by_average_yearly_temperature
date_saved: 2026-05-05T08:18:14.880921+00:00
instance: kb-cron
parent: list-of-countries-by-average-yearly-temperature-20260505-0
summary:
This is a list of countries and sovereign states by temperature. Average yearly temperature is calculated by averaging the minimum and maximum daily temperatures in the country, averaged for the years 1991 – 2020, from World Bank Group, derived from raw gridded climatologies from the Climatic Research Unit. == See also == List of countries by average annual precipitation == References ==


id: list-of-meteorology-institutions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_meteorology_institutions-0.md
category: reference
title: List of meteorology institutions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_meteorology_institutions
date_saved: 2026-05-05T08:18:16.194620+00:00
instance: kb-cron
parent: list-of-meteorology-institutions-20260505-0
summary:
The following is a list of meteorology institutions around the world. == Governmental == === Africa === Egyptian Meteorological Authority Ghana Meteorological Agency Mozambique National Institute of Meteorology National Meteorological Office ONM (Algeria) South African Weather Service SAWS === Americas === Canada Environment and Climate Change Canada Meteorological Service of Canada Canadian Ice Service Canadian Meteorological Centre United States: National Center for Atmospheric Research NCAR National Oceanic and Atmospheric Administration NOAA National Severe Storms Laboratory National Climatic Data Center National Weather Service National Centers for Environmental Prediction National Hurricane Center Storm Prediction Center Naval Maritime Forecast Center/Joint Typhoon Warning Center Bermuda Weather Service Instituto Nacional de Meteorologia (Brazil) Institute of Hydrology, Meteorology and Environmental Studies (Colombia) Instituto Meteorológico Nacional (Costa Rica) Institute of Meteorology (Cuba) Instituto Nacional Meteorologia e Hidrologia (Ecuador) Instituto Nacional de Sismología, Vulcanología, Meteorología e Hidrología (Guatemala) Instituto Nacional De Meteorologia E Hidrologia (Venezuela) National Service of Meteorology and Hydrology of Peru Servicio Meteorológico Nacional (Argentina) Servicio Meteorológico Nacional (Mexico) Uruguayan Institute of Meteorology === Asia / Oceania === Afghanistan Meteorological Department Bureau of Meteorology (Australia) Bangladesh Meteorological Department BMD Central Weather Administration (Taiwan) China Meteorological Administration CMA Hong Kong Observatory HKO Macao Meteorological and Geophysical Bureau


id: list-of-rogue-waves-incidents-20260505-0
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-0.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
This list of rogue waves compiles incidents of known and likely rogue waves – also known as freak waves, monster waves, killer waves, and extreme waves. These are dangerous and rare ocean surface waves that unexpectedly reach at least twice the height of the tallest waves around them, and are often described by witnesses as "walls of water". They occur in deep water, usually far out at sea, and are a threat even to capital ships, ocean liners and land structures such as lighthouses. == Background == Anecdotal evidence from mariners' testimonies and incidents of wave damage to ships has long suggested the existence of rogue waves; however, their scientific measurement was positively confirmed only following measurements of the Draupner wave, a rogue wave at the Draupner platform, in the North Sea on 1 January 1995. In this event, minor damage was inflicted on the platform, confirming that the reading was valid. In modern oceanography, rogue waves are defined not as the biggest possible waves at sea, but instead as extreme sized waves for a given sea state. Many of these encounters are only reported in the media, and are not examples of open ocean rogue waves. Often a huge


id: list-of-rogue-waves-incidents-20260505-1
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-1.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
On 4 August 1498, during the third voyage of Christopher Columbus, Columbus's ships were in the Gulf of Paria off the southern tip of Trinidad when Columbus heard a tremendous roar from behind his flagship. He observed a wave approaching from astern that was as tall as his ships' masts. Columbus described the wave as lifting his ships higher than anything he ever had experienced before, after which the ships dropped into a huge trough. On 28 September 1853 a huge wave struck the emigrant ship Annie Jane while she was off Barra in the Outer Hebrides. The wave caused the poop deck to collapse, instantly crushing 200 people to death. Annie Jane soon sank, and her 102 survivors were washed ashore on a piece of her deck. On 11 March 1861 at midday the lighthouse on Eagle Island off the west coast of Ireland was struck by a large wave that smashed 23 panes, washing some of the lamps down the stairs, and broken glass damaged the reflectors beyond repair. In order to damage the uppermost portion of the lighthouse, water had to surmount a seaside cliff measuring 40 m (130 ft) in height and a further 26 m


id: list-of-rogue-waves-incidents-20260505-2
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-2.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
Water rose as high as the bottom of her lower foretop, filling it with water, sweeping her forward deck clear, smashing her forebridge – much of which was found in pieces on her upper deck – wrecking her chart house, shifting the roof of her conning tower, and flooding her forward main gun turret, mess decks, and flats. Five of her crew died, and 17 others suffered serious injuries. At midnight on 5–6 May 1916 the British polar explorer Ernest Shackleton was at the tiller of the small sailboat James Caird in the Southern Ocean during a storm when he thought he saw the bad weather clearing in the west, astern. He then realized that what he thought was a line of white clouds above a clear dark sky was actually the crest of a single enormous wave that struck and nearly swamped the boat. Shackleton reported that the wave was larger than any he had ever seen before in his 26 years of seafaring. On 29 August 1916 at about 4:40 p.m., the United States Navy armored cruiser USS Memphis was wrecked in Santo Domingo harbor in the Dominican Republic when struck in rapid succession by three waves of


id: list-of-rogue-waves-incidents-20260505-3
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-3.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
=== Second half of the 20th century === On 5 February 1963, the French Navy light cruiser Jeanne d'Arc encountered a rogue wave while serving as the training ship of the French Naval Academy. On 12 April 1966, the Italian liner Michelangelo was in the Atlantic Ocean steaming toward New York City in a storm, frequently encountering 20-metre (66 ft) waves, when a giant wave struck her, breaking 18 metres (59 ft) over her bow. It traveled 70 metres (230 ft) aft along her deck and struck her superstructure, smashing 2-centimetre (0.8 in) thick windows on her bridge over 25 metres (82 ft) above the waterline. It tore a hole in the superstructure, killed two passengers instantly, fatally injured a crewman who died a few hours later, and injured over 50 other people, 10 of them gravely. In 1968 the sailing yacht Suhaili was in the Southern Ocean in high winds and a 40-to-50-foot (12 to 15 m) swell when her captain came on deck and sighted a breaking wave approaching from astern which stretched from horizon to horizon and was twice the height of the swell, indicating a height of at least 80 feet (24 m). He climbed into


id: list-of-rogue-waves-incidents-20260505-4
file: data/en.wikipedia.org/wiki/List_of_rogue_waves_incidents-4.md
category: reference
title: List of rogue waves incidents
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_rogue_waves_incidents
date_saved: 2026-05-05T08:18:17.503231+00:00
instance: kb-cron
parent: list-of-rogue-waves-incidents-20260505-0
summary:
=== 21st century === The Bahamian-registered cruise ships MS Bremen and MS Caledonian Star encountered 30-meter (98 ft) freak waves in the South Atlantic in 2001. Bridge windows on both ships were smashed, and all power and instrumentation lost. Naval Research Laboratory ocean-floor pressure sensors detected a freak wave caused by Hurricane Ivan in the Gulf of Mexico in 2004. The wave was around 27.7 meters (91 ft) high from peak to trough, and around 200 meters (660 ft) long. Norwegian Dawn, (three waves in succession, off the coast of Georgia, 16 April 2005): "The sea had actually calmed down when the 21-metre (69 ft) wave seemed to come out of thin air… Our captain, who has 20 years on the job, said he never saw anything like it." "The water exerted enough force to shear off the welds for the aluminum rail supports on the [ninth- and tenth-level] balconies of two cabins, allowing the teak balcony rails to break loose and crash into the cabin windows. The broken glass filling the drains compounded the water damage by allowing a large amount of water to enter the two cabins and damage the carpets in 61 other cabins. The ship's operating


id: list-of-meteorologists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_meteorologists-0.md
category: reference
title: List of meteorologists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_meteorologists
date_saved: 2026-05-05T08:18:18.835134+00:00
instance: kb-cron
parent: list-of-meteorologists-20260505-0
summary:
== A–B == Cleveland Abbe Aristotle June Bacon-Bercey David Bates Francis Beaufort Tor Bergeron Jacob Bjerknes Vilhelm Bjerknes Howard B. Bluestein Bert Bolin Léon Teisserenc de Bort Harold E. Brooks Keith Browning David Brunt C.H.D. Buys Ballot Horace R. Byers == C–D == Matthew Cappucci DaNa Carlis Jule Gregory Charney Coching Chu Isaac Cline Balfour Currie John Dalton Robert Davies-Jones William Morris Davis Igor Delijanić Jean-André Deluc Charles A. Doswell III Heinrich Wilhelm Dove == E–F == V. Walfrid Ekman Kerry Emanuel John Farrah William Ferrel Rich Fields John Park Finley Michael Fish Robert FitzRoy Bert Foord Benjamin Franklin Tetsuya Theodore Fujita == G–H == Francis Galton Audronė Galvonaitė Allison Göhler Ernest Gold William M. Gray Thomas P. Grazulis Willis Ray Gregg George Hadley Edmond Halley Julius von Hann Kenneth Hare Ingrid Holford Brian Hoskins Luke Howard == J–K == Robert Johns Jan-Erik Johnsen John Knox Wladimir Köppen Shen Kuo == L–O == Hubert Lamb Christopher Landsea Leslie R. Lemon Johannes Letzmann Richard Lindzen Mònica López Edward Norton Lorenz David M. Ludlum Paul Markowski Charles F. Marvin Cliff Mass Patrick McTaggart-Cowan Pomponius Mela Milutin Milanković James T. Moore Joel Myers Aili Nurminen Isidoro Orlanski == P–R == Erik Palmén Allen


id: list-of-meteorological-photos-and-videos-20260505-0
file: data/en.wikipedia.org/wiki/List_of_meteorological_photos_and_videos-0.md
category: reference
title: List of meteorological photos and videos
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_meteorological_photos_and_videos
date_saved: 2026-05-05T08:18:20.153817+00:00
instance: kb-cron
parent: list-of-meteorological-photos-and-videos-20260505-0
summary:
There are several notable photographs or videos in the field of meteorology, the study of the Earth's atmosphere, climate, and weather. These images or videos may be referred to as historic, most important, most iconic, or most influential weather-related photographs or weather-related videos. == List == === 19th century === === 20th century === === 21st century === == See also == List of photographs considered the most important == References ==


id: list-of-atmospheric-optical-phenomena-20260505-0
file: data/en.wikipedia.org/wiki/List_of_atmospheric_optical_phenomena-0.md
category: reference
title: List of atmospheric optical phenomena
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_atmospheric_optical_phenomena
date_saved: 2026-05-05T08:18:21.455086+00:00
instance: kb-cron
parent: list-of-atmospheric-optical-phenomena-20260505-0
summary:
Atmospheric optical phenomena include: Afterglow Airglow Alexander's band, the dark region between the two bows of a double rainbow. Alpenglow Anthelion Anticrepuscular rays Aurora (northern and southern lights, aurora borealis and aurora australis) Belt of Venus Brocken Spectre Circumhorizontal arc Circumzenithal arc Cloud iridescence Crepuscular rays Earth's shadow Earthquake lights Glories Green flash Halos, of Sun or Moon, including sun dogs Haze Heiligenschein or halo effect, partly caused by the opposition effect Ice blink Light pillar Lightning Mirages (including Fata Morgana) Monochrome Rainbow Moon dog Moonbow Nacreous cloud/Polar stratospheric cloud Rainbow Sprite (lightning) Subsun Sun dog Tangent arc Tyndall effect Upper-atmospheric lightning, including red sprites, Blue jets, and ELVES Water sky == See also == == References ==


