# kb

id: list-of-lie-groups-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md
category: reference
title: List of Lie groups topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Lie_groups_topics
date_saved: 2026-05-05T08:15:25.912263+00:00
instance: kb-cron
parent: list-of-lie-groups-topics-20260505-0
summary:
This is a list of Lie group topics, by Wikipedia page. == Examples == See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary group, special unitary group SU(2) SU(3) Orthogonal group, special orthogonal group Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane. Lorentz group Spinor group Symplectic group Exceptional groups G2 F4 E6 E7 E8 Affine group Euclidean group Poincaré group Heisenberg group == Lie algebras == Commutator Jacobi identity Universal enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra == Foundational results == One-parameter group, One-parameter subgroup Matrix exponential Infinitesimal transformation Lie's third theorem Maurer–Cartan form Cartan's theorem Cartan's criterion Local Lie group Formal group law Hilbert's fifth problem Hilbert–Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification (Lie group) == Semisimple theory == Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus


id: table-of-lie-groups-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md
category: reference
title: Table of Lie groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_Lie_groups
date_saved: 2026-05-05T08:15:27.102675+00:00
instance: kb-cron
parent: table-of-lie-groups-20260505-0
summary:
This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. == Real Lie groups and their algebras == Column legend Cpt: Is this group G compact? (Yes or No) π 0 {\displaystyle \pi _{0}} : Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). π 1 {\displaystyle \pi _{1}} : Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). UC: If G is not simply connected, gives the


id: list-of-limits-20260505-0
file: data/en.wikipedia.org/wiki/List_of_limits-0.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x. == Limits for general functions == === Definitions of limits and related concepts === lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} if and only if ∀ ε > 0 ∃ δ > 0 : 0 < | x − c | < δ ⟹ | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon } . This is the (ε, δ)-definition of limit. The limit superior and limit inferior of a sequence are defined as lim sup n → ∞ x n = lim n → ∞ ( sup m ≥ n x m ) {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)} and lim inf n → ∞ x n = lim n → ∞ ( inf m ≥ n x m ) {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)} . A function, f ( x ) {\displaystyle f(x)} , is said to be continuous at a point, c, if


id: list-of-limits-20260505-1
file: data/en.wikipedia.org/wiki/List_of_limits-1.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
If lim x → c f ( x ) = lim x → c h ( x ) = L {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} and f ( x ) ≤ g ( x ) ≤ h ( x ) {\displaystyle f(x)\leq g(x)\leq h(x)} for all x in an open interval that contains c, except possibly c itself, lim x → c g ( x ) = L . {\displaystyle \lim _{x\to c}g(x)=L.} This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c. == Polynomials and functions of the form xa == lim x → c a = a {\displaystyle \lim _{x\to c}a=a} === Polynomials in x === lim x → c x = c {\displaystyle \lim _{x\to c}x=c} lim x → c ( a x + b ) = a c + b {\displaystyle \lim _{x\to c}(ax+b)=ac+b} lim x → c x n = c n {\displaystyle \lim _{x\to c}x^{n}=c^{n}} if n is a positive integer lim x → ∞ x / a = { ∞ , a > 0 does not exist , a = 0 − ∞ ,


id: list-of-limits-20260505-2
file: data/en.wikipedia.org/wiki/List_of_limits-2.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
lim x → 1 ln ⁡ ( x ) x − 1 = 1 {\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1} lim x → 0 ln ⁡ ( x + 1 ) x = 1 {\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1} lim x → 0 − ln ⁡ ( 1 + a ( e − x − 1 ) ) x = a {\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a} . This limit follows from L'Hôpital's rule. lim x → 0 x ln ⁡ x = 0 {\displaystyle \lim _{x\to 0}x\ln x=0} , hence lim x → 0 x x = 1 {\displaystyle \lim _{x\to 0}x^{x}=1} lim x → ∞ ln ⁡ x x = 0 {\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0} === Logarithms to arbitrary bases === For b > 1, lim x → 0 + log b ⁡ x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty } lim x → ∞ log b ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=\infty } For b < 1, lim x → 0 + log b ⁡ x = ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty } lim x → ∞ log b ⁡ x = − ∞ {\displaystyle \lim _{x\to \infty


id: list-of-books-on-history-of-number-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md
category: reference
title: List of books on history of number systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems
date_saved: 2026-05-05T08:15:29.597547+00:00
instance: kb-cron
parent: list-of-books-on-history-of-number-systems-20260505-0
summary:
This list compiles notable works that explore the history and development of number systems across various civilizations and time periods. These works cover topics ranging from ancient numeral systems and arithmetic methods to the evolution of mathematical notations and the impact of numerals on science, trade, and culture. == Overview == Number systems have been central to the development of human civilization, enabling record-keeping, commerce, astronomy, and scientific advancement. Early systems such as tally marks and Roman numerals gradually gave way to more abstract and efficient representations like the Babylonian base-60 system and the Hindu–Arabic numerals, now standard worldwide. The invention of zero, positional notation, and symbolic mathematics has had profound philosophical and technological implications. == Notable works on the history of number systems == == Works on the history of zero == == Children's books on the history of numbers == == Historical texts == == References ==


id: list-of-mathematics-books-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md
category: reference
title: List of mathematics books
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_books
date_saved: 2026-05-05T08:15:30.833032+00:00
instance: kb-cron
parent: list-of-mathematics-books-20260505-0
summary:
This is a list of mathematics books including textbooks, expository works, popular mathematics fields, and historically significant treatises. == General works == Concrete Mathematics — Ronald Graham, Donald Knuth, and Oren Patashnik Concepts of Modern Mathematics — Ian Stewart Mathematics and the Imagination — Edward Kasner and James Newman Mathematics and Plausible Reasoning — George Pólya Mathematics: The Loss of Certainty — Morris Kline The Princeton Companion to Mathematics — Timothy Gowers What Is Mathematics? — Richard Courant and Herbert Robbins == Popular mathematics and biographies == A Mathematician's Apology — G. H. Hardy The Annotated Turing — Charles Petzold The Beauty of Fractals — Heinz-Otto Peitgen and Peter Richter The Emperor's New Mind — Roger Penrose Fermat's Enigma — Simon Singh God Created the Integers — Stephen Hawking Gödel, Escher, Bach — Douglas Hofstadter How Not to Be Wrong — Jordan Ellenberg The Man Who Loved Only Numbers — Paul Hoffman Prime Obsession — John Derbyshire Where Mathematics Comes From — George Lakoff and Rafael E. Núñez == Algebra == Algebra — Serge Lang Algebra: Chapter 0 — Paolo Aluffi == Calculus and analysis == Calculus on Manifolds — Michael Spivak Principles of Mathematical Analysis — Walter Rudin Introduction


id: list-of-regular-polytope-compounds-20260505-0
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces. == Two dimensional compounds == For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon. In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m. A further


id: list-of-regular-polytope-compounds-20260505-1
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive. === Euclidean 3-space compounds === The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}. === Hyperbolic 3-space compounds === C. W. L. Garner described two dual pairs of regular hyperbolic compound honeycombs in 1970: the compact pair 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5}, and the paracompact pair {6,3,3}[5{6,3,4}] and [5{4,3,6}]{3,3,6}. He did not consider vertex-regular compounds where the vertices are at infinity, or (reciprocally) cell-regular compounds where the cells are centred at infinity. In 2019, Peter McMullen (who focused only on the compact case) pointed out and filled a gap in Garner's proof of completeness, so that it is now proven that 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5} are the only compact regular


id: list-of-things-named-after-felix-hausdorff-20260505-0
file: data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md
category: reference
title: List of things named after Felix Hausdorff
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff
date_saved: 2026-05-05T08:15:33.394906+00:00
instance: kb-cron
parent: list-of-things-named-after-felix-hausdorff-20260505-0
summary:
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician. He made significant contributions to mathematics, specifically in the fields of set theory and topology. == Mathematics == Baker–Campbell–Hausdorff formula Gromov–Hausdorff convergence Hausdorff–Young inequality Hausdorff completion Hausdorff dimension Hausdorff distance Hausdorff gap Hausdorff maximal principle Hausdorff measure Hausdorff Medal Hausdorff moment problem Hausdorff paradox Hausdorff space == Places == Felix Hausdorff International Meeting Centre at the University of Greifswald Hausdorff Center for Mathematics == Space == 24947 Hausdorff == References ==


id: index-of-logarithm-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_logarithm_articles-0.md
category: reference
title: Index of logarithm articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_logarithm_articles
date_saved: 2026-05-05T08:15:34.685156+00:00
instance: kb-cron
parent: index-of-logarithm-articles-20260505-0
summary:
This is a list of logarithm topics, by Wikipedia page. See also the list of exponential topics. Acoustic power Amoeba (mathematics) Antilogarithm Apparent magnitude Baker's theorem Bel Benford's law Binary logarithm Bode plot Henry Briggs Bygrave slide rule Cologarithm Common logarithm Complex logarithm Discrete logarithm Discrete logarithm records e Representations of e El Gamal discrete log cryptosystem Harmonic series History of logarithms Hyperbolic sector Iterated logarithm Otis King Law of the iterated logarithm Linear form in logarithms Linearithmic List of integrals of logarithmic functions Log canonical singularity Log-likelihood ratio Log-log graph Log-normal distribution Log-periodic antenna Log semiring Log structure Log-Weibull distribution Logarithmic algorithm Logarithmic convolution Logarithmic decrement Logarithmic derivative Logarithmic differential Logarithmic differentiation Logarithmic distribution Logarithmic form Logarithmic graph paper Logarithmic growth Logarithmic identities Logarithmic mean Logarithmic number system Logarithmic scale Logarithmic spiral Logit LogSumExp Mantissa is a disambiguation page; see common logarithm for the traditional concept of mantissa; see significand for the modern concept used in computing. Matrix logarithm Mel scale Mercator projection Mercator series Moment magnitude scale John Napier Napierian logarithm Natural logarithm Natural logarithm of 2 Neper Offset logarithmic integral pH Plethystic logarithm Pollard's kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem


id: list-of-mathematical-logic-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md
category: reference
title: List of mathematical logic topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_logic_topics
date_saved: 2026-05-05T08:15:35.913586+00:00
instance: kb-cron
parent: list-of-mathematical-logic-topics-20260505-0
summary:
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms. == Working foundations == Peano axioms Giuseppe Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics) Ur-element Singleton (mathematics) Simple theorems in the algebra of sets Algebra of sets Power set Empty set Non-empty set Empty function Universe (mathematics) Axiomatization Axiomatic system Axiom schema Axiomatic method Formal system Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology Consistency proof Arithmetization of analysis Foundations of mathematics Formal language Principia Mathematica Hilbert's program Impredicative Definable real number Algebraic logic Boolean algebra (logic) Dialectica space categorical logic == Model theory == Finite model theory Descriptive complexity theory Model checking Trakhtenbrot's theorem Computable model theory Tarski's exponential function problem Undecidable problem Institutional model theory Institution (computer science) Non-standard analysis Non-standard calculus Hyperinteger Hyperreal number Transfer principle Overspill Elementary Calculus: An Infinitesimal Approach Criticism of non-standard analysis Standard part function Set theory Forcing (mathematics) Boolean-valued model Kripke semantics General frame Predicate logic First-order logic Infinitary logic Many-sorted logic Higher-order logic Lindström quantifier Second-order


id: outline-of-logic-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_logic-0.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid) and incorrect (or fallacious) inferences. Logicians study the criteria for the evaluation of arguments. == Foundations of logic == Philosophy of logic Analytic-synthetic distinction Antinomy A priori and a posteriori Definition Description Entailment Identity (philosophy) Inference Logical form Logical implication Logical truth Logical consequence Name Necessity Material conditional Meaning (linguistic) Meaning (non-linguistic) Paradox (list) Possible world Presupposition Probability Quantification Reason Reasoning Reference Semantics Strict conditional Syntax (logic) Truth Truth value Validity == Branches of logic == Affine logic Alethic logic Aristotelian logic Boolean logic Buddhist logic Bunched logic Categorical logic Classical logic Computability logic Deontic logic Dependence logic Description logic


id: outline-of-logic-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_logic-1.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
Finitary relation Antisymmetric relation Asymmetric relation Bijection Bijection, injection and surjection Binary relation Composition of relations Congruence relation Connected relation Converse relation Coreflexive relation Covering relation Cyclic order Dense relation Dependence relation Dependency relation Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence relation Partial function Partially ordered set Preorder Prewellordering Propositional function Quasitransitive relation Reflexive relation Serial relation Surjective function Symmetric relation Ternary relation Transitive relation Trichotomy (mathematics) Well-founded relation == Mathematical logic == Mathematical logic === Set theory === Set theory list Aleph null Bijection, injection and surjection Binary set Cantor's diagonal argument Cantor's first uncountability proof Cantor's theorem Cardinality of the continuum Cardinal number Codomain Complement (set theory) Constructible universe Continuum hypothesis Countable set Decidable set Denumerable set Disjoint sets Disjoint union Domain of a function Effective enumeration Element (mathematics) Empty function Empty set Enumeration Extensionality Finite set Forcing (mathematics) Function (set theory) Function composition Generalized continuum hypothesis Index set Infinite set Intension Intersection (set theory) Inverse function Large cardinal Löwenheim–Skolem theorem Map (mathematics) Multiset Morse–Kelley set theory Naïve set theory One-to-one correspondence Ordered pair Partition of a set Pointed set Power set Projection (set theory) Proper subset Proper superset Range of


id: outline-of-logic-20260505-2
file: data/en.wikipedia.org/wiki/Outline_of_logic-2.md
category: reference
title: Outline of logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_logic
date_saved: 2026-05-05T08:15:37.128800+00:00
instance: kb-cron
parent: outline-of-logic-20260505-0
summary:
=== Journals === Journal of Logic, Language and Information Journal of Philosophical Logic Linguistics and Philosophy === Books === A System of Logic Attacking Faulty Reasoning Begriffsschrift Categories (Aristotle) Charles Sanders Peirce bibliography De Interpretatione Gödel, Escher, Bach Introduction to Mathematical Philosophy Language, Truth, and Logic Laws of Form Novum Organum On Formally Undecidable Propositions of Principia Mathematica and Related Systems Organon Philosophy of Arithmetic Polish Logic Port-Royal Logic Posterior Analytics Principia Mathematica Principles of Mathematical Logic Prior Analytics Rhetoric (Aristotle) Sophistical Refutations Sum of Logic The Art of Being Right The Foundations of Arithmetic Topics (Aristotle) Tractatus Logico-Philosophicus == Logic organizations == Association for Symbolic Logic == Logicians == List of logicians List of philosophers of language == See also == Glossary of logic Index of logic articles Mathematics List of basic mathematics topics List of mathematics articles Philosophy List of basic philosophy topics List of philosophy topics Outline of philosophy Outline of discrete mathematics – for introductory set theory and other supporting material == References == == External links == Taxonomy of Logical Fallacies forall x: an introduction to formal logic, by P.D. Magnus, covers sentential and quantified logic Translation Tips, by Peter Suber, for translating from English


id: list-of-manifolds-20260505-0
file: data/en.wikipedia.org/wiki/List_of_manifolds-0.md
category: reference
title: List of manifolds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_manifolds
date_saved: 2026-05-05T08:15:38.253557+00:00
instance: kb-cron
parent: list-of-manifolds-20260505-0
summary:
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see Category:Manifolds and its subcategories. == Generic families of manifolds == Euclidean space, Rn n-sphere, Sn n-torus, Tn Real projective space, RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold Grassmann manifold Stiefel manifold Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics. == Manifolds of a specific dimension == === 1-manifolds === Circle, S1 Long line Real line, R Real projective line, RP1 ≅ S1 === 2-manifolds === Cylinder, S1 × R Klein bottle, RP2 # RP2 Klein quartic (a genus 3 surface) Möbius strip Real projective plane, RP2 Sphere, S2 Surface of genus g Torus Double torus === 3-manifolds === 3-sphere, S3 3-torus, T3 Poincaré homology sphere SO(3) ≅ RP3 Solid Klein bottle Solid torus Whitehead manifold Meyerhoff manifold Weeks manifold For more examples see 3-manifold. === 4-manifolds === Complex projective plane Del Pezzo surface E8 manifold Enriques surface Exotic R4 Hirzebruch surface K3 surface For more examples see 4-manifold. == Special types of manifolds == === Manifolds related


id: list-of-martin-gardner-mathematical-games-columns-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns-0.md
category: reference
title: List of Martin Gardner Mathematical Games columns
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Martin_Gardner_Mathematical_Games_columns
date_saved: 2026-05-05T08:15:39.569085+00:00
instance: kb-cron
parent: list-of-martin-gardner-mathematical-games-columns-20260505-0
summary:
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for Scientific American magazine. During the next 5.5 years, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "[cover]" in the table with a hyperlink to the cover. == Other articles by Gardner == Gardner wrote 5 other articles for Scientific American. His flexagon article in December 1956 was in all but name the first article in the series of Mathematical Games columns and led directly to the series which began the following month. These five articles are listed below. == References == == External links == A Quarter Century of Recreational Mathematics, by Martin Gardner preserved at the Internet Archive A subject index for the fifteen books of Martin Gardner's Mathematical Games columns The Top 10 Martin Gardner Scientific American Articles


id: list-of-named-matrices-20260505-0
file: data/en.wikipedia.org/wiki/List_of_named_matrices-0.md
category: reference
title: List of named matrices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_named_matrices
date_saved: 2026-05-05T08:15:45.006476+00:00
instance: kb-cron
parent: list-of-named-matrices-20260505-0
summary:
This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] . {\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.} and the zero matrix of dimension m × n {\displaystyle m\times n} . For example: O 2 × 3 = ( 0 0 0 0 0 0 ) {\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}} . Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in these areas. == Constant matrices == The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix


id: list-of-named-matrices-20260505-1
file: data/en.wikipedia.org/wiki/List_of_named_matrices-1.md
category: reference
title: List of named matrices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_named_matrices
date_saved: 2026-05-05T08:15:45.006476+00:00
instance: kb-cron
parent: list-of-named-matrices-20260505-0
summary:
== Matrices used in science and engineering == Cabibbo–Kobayashi–Maskawa matrix — a unitary matrix used in particle physics to describe the strength of flavour-changing weak decays. Density matrix — a matrix describing the statistical state of a quantum system. Hermitian, non-negative and with trace 1. Fundamental matrix (computer vision) — a 3 × 3 matrix in computer vision that relates corresponding points in stereo images. Fuzzy associative matrix — a matrix in artificial intelligence, used in machine learning processes. Gamma matrices — 4 × 4 matrices in quantum field theory. Gell-Mann matrices — a generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3). Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics and linear-quadratic regulator (LQR) systems. Irregular matrix — a matrix used in computer science which has a varying number of elements in each row. Overlap matrix — a type of Gramian matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. S matrix — a matrix in quantum mechanics that connects asymptotic (infinite past and future) particle states. Scattering matrix - a


id: list-of-mathematics-based-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics-based_methods-0.md
category: reference
title: List of mathematics-based methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics-based_methods
date_saved: 2026-05-05T08:15:46.364138+00:00
instance: kb-cron
parent: list-of-mathematics-based-methods-20260505-0
summary:
This is a list of mathematics-based methods. Adams' method (differential equations) Akra–Bazzi method (asymptotic analysis) Bisection method (root finding) Brent's method (root finding) Condorcet method (voting systems) Coombs' method (voting systems) Copeland's method (voting systems) Crank–Nicolson method (numerical analysis) D'Hondt method (voting systems) D21 – Janeček method (voting system) Discrete element method (numerical analysis) Domain decomposition method (numerical analysis) Epidemiological methods Euler's forward method Explicit and implicit methods (numerical analysis) Finite difference method (numerical analysis) Finite element method (numerical analysis) Finite volume method (numerical analysis) Highest averages method (voting systems) Method of exhaustion Method of infinite descent (number theory) Information bottleneck method Inverse chain rule method (calculus) Inverse transform sampling method (probability) Iterative method (numerical analysis) Jacobi method (linear algebra) Largest remainder method (voting systems) Level-set method Linear combination of atomic orbitals molecular orbital method (molecular orbitals) Method of characteristics Least squares method (optimization, statistics) Maximum likelihood method (statistics) Method of complements (arithmetic) Method of moving frames (differential geometry) Method of successive substitution (number theory) Monte Carlo method (computational physics, simulation) Newton's method (numerical analysis) Pemdas method (order of operation) Perturbation methods (functional analysis, quantum theory) Probabilistic method (combinatorics) Romberg's method (numerical analysis) Runge–Kutta method (numerical analysis) Sainte-Laguë method (voting


id: list-of-works-by-nicolas-minorsky-20260505-0
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-0.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
List of works by Nicolas Minorsky. == Books == Minorsky, N. (1947). Introduction to non-linear mechanics: Topological methods, analytical methods, non-linear resonance, relaxation oscillations. J.W. Edwards. ASIN B0007DXRVY. Minorsky, Nicolai (1955). Influence d'Henri Poincaré sur l'évolution moderne de la théorie des oscillations non linéaires [Influence of Henri Poincaré on the modern development of the theory of nonlinear oscillations]. Le livre du centenaire de la naissance de Henri Poincaré, 1854-1954 (in French). Paris, Gauthier-Villars. pp. 120–126. OCLC 10571426. Minorsky, N. (1958). Dynamics and Nonlinear Mechanics: The Theory of Oscillations. Surveys in Applied Mathematics. Wiley. pp. 110–206. ASIN B0006AVKQW. Minorsky, N. (1962). G. Szego; et al. (eds.). On some aspects of non-linear oscillations. Studies in Mathematical Analysis and Related Topics. Stanford Univ Pr (April 1962). ISBN 978-0804701402. {{cite book}}: ISBN / Date incompatibility (help) Minorsky, N. (1967). Théorie des Oscillations [Theory of Oscillations]. Memorial des Sciences Mathematiques, Fascicule CLXIII (in French). ASIN B000LQD7KI. Minorsky, Nicolai; Hetnarski, Richard B.; Mróz, Zenon (1967). Drgania nieliniowe [Nonlinear oscillations] (in Polish). Warszawa : Państwowe Wydawnictwo Naukowe. OCLC 749191659.{{cite book}}: CS1 maint: publisher location (link)


id: list-of-works-by-nicolas-minorsky-20260505-1
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-1.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
== Papers == Minorsky, N. (May 1922). "Directional stability of automatically steered bodies". Journal of the American Society of Naval Engineers. 34 (2): 280–309. doi:10.1111/j.1559-3584.1922.tb04958.x. Minorsky, N. (February 1927). "Phenomenon of direct-current self excitation in vacuum tubes circuits and in applications". J. Franklin Inst. 203 (2): 181–209. doi:10.1016/S0016-0032(27)92437-5. Minorsky, N. (April 1928). "La rotation de l'arc électrique dans champ magnétique radial" [The rotation of an arc in a radial magnetic field] (PDF). J. Phys. Radium (in French). 9 (4): 127–136. doi:10.1051/jphysrad:0192800904012700. Minorsky, N. (1928). "Mesure de la vitesse d'un aéronef par rapport au sol en l'absence supposée de tout repère extérieur" [Measuring the speed of an aircraft relative to the ground in the absence of any assumed approximate]. L'Aéronautique (in French). 113. Minorsky, N. (May 1930). "Automatic steering test". Journal of the American Society for Naval Engineers. 42 (2): 285–310. doi:10.1111/j.1559-3584.1930.tb05036.x. Minorsky, N. (June 1930). "Electronic conduction and ionization in crossed electric and magnetic fields". Journal of the Franklin Institute. 209 (6): 757–775. doi:10.1016/S0016-0032(30)91472-X. ISSN 0016-0032. Minorsky, N. (August 1934). "Ship stabilization by activated tank: An experimental investigation". The Engineer. 158: 154. Minorsky, N. (27 January 1936). "Une méthode d'intégration de quelques équations différentielles par un procédé électrique" [A method


id: list-of-works-by-nicolas-minorsky-20260505-2
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-2.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
Acad. Sci. 259. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les phénomènes paramétriques". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur ll'interaction des oscillations non linéaires". C. R. Acad. Sci. 261. Minorsky, N. (1965). "Sur les oscillations quasi discontinues". C. R. Acad. Sci. Mécanique. 261. Minorsky, N. (1966). "Sur la méthode stroboscopique". C. R. Acad. Sci. 263. Minorsky, N. (1967). "Comments "On asynchronous quenching"". IEEE Transactions on Automatic Control. 12 (2): 225–227. doi:10.1109/TAC.1967.1098559. Minorsky, N. (1968). "Sur les équations différentielles aux différences". C. R. Acad. Sci. 266.


id: list-of-works-by-nicolas-minorsky-20260505-3
file: data/en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky-3.md
category: reference
title: List of works by Nicolas Minorsky
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nicolas_Minorsky
date_saved: 2026-05-05T08:15:47.573957+00:00
instance: kb-cron
parent: list-of-works-by-nicolas-minorsky-20260505-0
summary:
== Conferences == Minorsky, N. (1948). Self-excited oscillations in systems possessing retarded actions. 7th Int. Congr. Appl. Math, London, England. Minorsky, N. (1951). Sur le phénomène Béthenod. Actes Colloq. Int. Vibrations linéaires, Porqurolles, France. Minorsky, N. (1952). Non-linear control systems. Conf. Automat. Contr., London:Butterworth. Minorsky, N. (1952). Sur la méthode stroboscopique. Mem. lett. Accad. Sci. Ist., Bologna. Minorsky, N. (1952). Sur quelques oscillatoires contenant les paramètres à inertie. Ann. Fac. Sci. Univ. d'Aix, Marseilles, France. Minorsky, N. (1957). Nouvelles méthodes de la théorie des oscillations. Conf. Semin. Mat. Univ. Bari. Minorsky, N. (1961). On synchronization. Int. Union Theor. Appl. Mech. Symp. Nonlinear Vibrations, Kiev, U.S.S.R. Minorsky, N. (1963). On synchronization. Tr. Mizh. Symp. po Nelin. Kolebelina, vol. 1, Kiev, U.S.S.R. Minorsky, N. (1964). Les vibrations forcées dans les systèmes non linéaires. Coll. Int. Centre Nat. Rech. Sci., no. 148, Marseilles, France. Minorsky, N. (1965). Sur les systèmes non autonomes. Colloques Inl. Centre Nat. Rech. Sci., vol 148. == Patents == US patent 1306552, N. Minorsky, "Gyrometer", published 1919-06-10, assigned to The Sperry Gyroscope company US patent 1372184, N. Minorsky, "Angular-velocity-indicating apparatus", published 1921-03-22, assigned to N. Minorsky US patent 1436280, N. Minorsky, "Automatic steering device", published 1922-11-21, assigned to N.


id: list-of-multivariable-calculus-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_multivariable_calculus_topics-0.md
category: reference
title: List of multivariable calculus topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multivariable_calculus_topics
date_saved: 2026-05-05T08:15:48.804210+00:00
instance: kb-cron
parent: list-of-multivariable-calculus-topics-20260505-0
summary:
This is a list of multivariable calculus topics. See also multivariable calculus, vector calculus, list of real analysis topics, list of calculus topics. Closed and exact differential forms Contact (mathematics) Contour integral Contour line Critical point (mathematics) Curl (mathematics) Current (mathematics) Curvature Curvilinear coordinates Del Differential form Differential operator Directional derivative Divergence Divergence theorem Double integral Equipotential surface Euler's theorem on homogeneous functions Exterior derivative Flux Frenet–Serret formulas Gauss's law Gradient Green's theorem Green's identities Harmonic function Helmholtz decomposition Hessian matrix Hodge star operator Inverse function theorem Irrotational vector field Isoperimetry Jacobian matrix Lagrange multiplier Lamellar vector field Laplacian Laplacian vector field Level set Line integral Matrix calculus Mixed derivatives Monkey saddle Multiple integral Newtonian potential Parametric equation Parametric surface Partial derivative Partial differential equation Potential Real coordinate space Saddle point Scalar field Solenoidal vector field Stokes' theorem Submersion Surface integral Symmetry of second derivatives Taylor's theorem Total derivative Vector field Vector operator Vector potential


id: table-of-newtonian-series-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_Newtonian_series-0.md
category: reference
title: Table of Newtonian series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_Newtonian_series
date_saved: 2026-05-05T08:15:50.124582+00:00
instance: kb-cron
parent: table-of-newtonian-series-20260505-0
summary:
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a n {\displaystyle a_{n}} written in the form f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n {\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}} where ( s n ) {\displaystyle {s \choose n}} is the binomial coefficient and ( s ) n {\displaystyle (s)_{n}} is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus. == List == The generalized binomial theorem gives ( 1 + z ) s = ∑ n = 0 ∞ ( s n ) z n = 1 + ( s 1 ) z + ( s 2 ) z 2 + ⋯ . {\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .} A proof for this identity can be obtained by showing that it satisfies the differential equation ( 1 + z ) d ( 1 + z ) s d z = s ( 1 + z ) s . {\displaystyle


id: list-of-np-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_NP-complete_problems-0.md
category: reference
title: List of NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_NP-complete_problems
date_saved: 2026-05-05T08:15:51.446008+00:00
instance: kb-cron
parent: list-of-np-complete-problems-20260505-0
summary:
This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems. As there are thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found in Garey & Johnson (1979). == Graphs and hypergraphs == Graphs occur frequently in everyday applications. Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g. Facebook or LinkedIn). 1-planarity 3-dimensional matching Bandwidth problem Bipartite dimension Capacitated minimum spanning tree Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem. Clique cover problem Clique problem Complete coloring, a.k.a. achromatic number Cycle rank Degree-constrained spanning tree Domatic number Dominating set, a.k.a. domination number NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. Feedback vertex set Feedback arc set Graph coloring Graph homomorphism problem Graph partition into


id: list-of-np-complete-problems-20260505-1
file: data/en.wikipedia.org/wiki/List_of_NP-complete_problems-1.md
category: reference
title: List of NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_NP-complete_problems
date_saved: 2026-05-05T08:15:51.446008+00:00
instance: kb-cron
parent: list-of-np-complete-problems-20260505-0
summary:
By the same article existence of bounded modular square roots with arbitrarily composite modulus. Given positive integers A , B , C ≥ 0 {\displaystyle \textstyle A,B,C\geq 0} , decide existence of an integer x ∈ [ 0 , C ] {\displaystyle x\in [0,C]} such that x 2 ≡ A mod B {\displaystyle x^{2}\equiv A{\bmod {B}}} . The problem remains NP-complete even if a prime factorization of B {\displaystyle B} is provided. Serializability of database histories Set cover (also called "minimum cover" problem). This is equivalent, by transposing the incidence matrix, to the hitting set problem. Set packing Set splitting problem Scheduling to minimize weighted completion time Block Sorting (Sorting by Block Moves) Sparse approximation Variations of the Steiner tree problem. Specifically, with the discretized Euclidean metric, rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. Three-dimensional Ising model == See also == Existential theory of the reals § Complete problems Karp's 21 NP-complete problems List of PSPACE-complete problems Reduction (complexity) == Notes == == References == General Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New


id: list-of-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_number_theory_topics-0.md
category: reference
title: List of number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_number_theory_topics
date_saved: 2026-05-05T08:15:52.778467+00:00
instance: kb-cron
parent: list-of-number-theory-topics-20260505-0
summary:
This is a list of topics in number theory. See also: List of recreational number theory topics Topics in cryptography == Divisibility rule == Composite number Highly composite number Even and odd numbers Parity Divisor, aliquot part Greatest common divisor Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number, prime power Bonse's inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial == Fractions == Rational number Unit fraction Irreducible fraction = in lowest terms Dyadic fraction Recurring decimal Cyclic number Farey sequence Ford circle Stern–Brocot tree Dedekind sum Egyptian fraction == Modular arithmetic == Montgomery reduction Modular exponentiation Linear congruence theorem Successive over-relaxation Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory) Congruence of squares Luhn formula Mod n cryptanalysis == Arithmetic functions == Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor


id: list-of-numeral-system-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numeral_system_topics-0.md
category: reference
title: List of numeral system topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numeral_system_topics
date_saved: 2026-05-05T08:15:54.075389+00:00
instance: kb-cron
parent: list-of-numeral-system-topics-20260505-0
summary:
This is a list of Wikipedia articles on topics of numeral system and "numeric representations" See also: computer numbering formats and number names. == Arranged by base == Radix, radix point, mixed radix, exponentiation Unary numeral system (base 1) Tally marks – Numeral form used for counting Binary numeral system (base 2) Negative base numeral system (base −2) Ternary numeral system numeral system (base 3) Balanced ternary numeral system (base 3) Negative base numeral system (base −3) Quaternary numeral system (base 4) Quater-imaginary base (base 2i) Quinary numeral system (base 5) Pentadic numerals – Scandinavian numeral system Senary numeral system (base 6) Septenary numeral system (base 7) Octal numeral system (base 8) Nonary (novenary) numeral system (base 9) Decimal (denary) numeral system (base 10) Bi-quinary coded decimal – Numeral encoding scheme Negative base numeral system (base −10) Duodecimal (dozenal) numeral system (base 12) Hexadecimal numeral system (base 16) Vigesimal numeral system (base 20) Sexagesimal numeral system (base 60) == Arranged by culture == Aegean numbers – Numeral system used by the Minoans and MycenaeansPages displaying short descriptions of redirect targets Australian Aboriginal enumeration – Counting system used by Australian Aboriginals Armenian numerals Babylonian numerals – Numeral systemPages displaying short descriptions


id: list-of-numerical-libraries-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_libraries-0.md
category: reference
title: List of numerical libraries
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_libraries
date_saved: 2026-05-05T08:15:55.235107+00:00
instance: kb-cron
parent: list-of-numerical-libraries-20260505-0
summary:
This is a list of numerical libraries, which are libraries used in software development for performing numerical calculations. It is not a complete listing but is instead a list of numerical libraries with articles on Wikipedia, with few exceptions. The choice of a typical library depends on a range of requirements such as: desired features (e.g. large dimensional linear algebra, parallel computation, partial differential equations), licensing, readability of API, portability or platform/compiler dependence (e.g. Linux, Windows, Visual C++, GCC), performance, ease-of-use, continued support from developers, standard compliance, specialized optimization in code for specific application scenarios or even the size of the code-base to be installed. == Multi-language == == C == == C++ == == Delphi == ALGLIB - an open source numerical analysis library. == .NET Framework languages: C#, F#, VB.NET and PowerShell == == Fortran == == Java == == OCaml == OCaml programming language has support for array programming in the standard library, also with a specific module named bigarrays for multi-dimensional, numerical arrays, with both C and Fortran layout options. A comprehensive support of numerical computations is provided by the library Owl Scientific Computing which provides methods for statistics, linear algebra (using OpenBLAS), differential equations, algorithmic


id: list-of-numerical-analysis-software-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-0.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
Listed here are notable end-user computer applications intended for use with numerical or data analysis: == Numerical-software packages == Analytica is a widely used proprietary software tool for building and analyzing numerical models. It is a declarative and visual programming language based on influence diagrams. FlexPro is a program to analyze and present measurement data. It has a rich Excel-like user interface and a built-in vector programming language FPScript has a syntax similar to MATLAB. FreeMat, an open-source MATLAB-like environment with a GNU General Public License (GPL). GNU Octave is a high-level programming language, intended for mainly numerical computing. It has a convenient command-line interface to solve linear and nonlinear problems numerically, and to perform other numerical experiments using a language that is compatible mostly with MATLAB. The 4.0 and newer releases of Octave include a GUI. Several independently developed Linux programs (Cantor, KAlgebra) also offer GUI front-ends to Octave. An active community provides technical support to users. GroovyLab (formerly jLab), a research platform to build an open-source MATLAB-like environment in pure Java and Groovy. Supports interpreted j-Scripts (MATLAB-like) and compiled GroovySci (extension to Groovy) scripts that give direct interfacing to Java code and scripting access to many popular Java


id: list-of-numerical-analysis-software-20260505-1
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-1.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
== Interface-oriented == Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. COMSOL Multiphysics is a finite-element analysis, solver and simulation software / FEA Software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. Dataplot is provided by NIST. DADiSP is a commercial program focused on digital signal processing (DSP) that combines the numerical ability of MATLAB with a spreadsheet-like interface. Easy Java Simulations (EJS) is an open-source software tool, written in Java, for generating simulations. Euler Mathematical Toolbox is a powerful numerical laboratory with a programming language that can handle real, complex and interval numbers, vectors and matrices. It can produce 2D/3D plots. FEATool Multiphysics is a MATLAB GUI toolbox for finite element FEM and PDE multiphysics simulations. FEniCS Project is a collection of project for automated solutions to partial differential equations (PDEs). Hermes is a C++ library of advanced adaptive finite element algorithms to solve PDEs and multiphysics coupled problems. Fityk is a curve fitting and data-analysis program. Primarily used for peak fitting and analyzing peak data. FlexPro is a commercial program for interactive and automated analysis and presentation of mainly measurement data. It supports many binary instrument data formats and has


id: list-of-numerical-analysis-software-20260505-2
file: data/en.wikipedia.org/wiki/List_of_numerical-analysis_software-2.md
category: reference
title: List of numerical-analysis software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical-analysis_software
date_saved: 2026-05-05T08:15:56.369537+00:00
instance: kb-cron
parent: list-of-numerical-analysis-software-20260505-0
summary:
acslX is a software application for modeling and evaluating the performance of continuous systems described by time-dependent, nonlinear differential equations. ADMB is a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation. AMPL is a mathematical modeling language for describing and solving high complexity problems for large-scale optimization. Calcpad is an open-source numerical software for engineering calculations. Supports numerical differentiation and integration, root and extrema finding, vector and matrix calculations. Ch, a commercial C/C++-based interpreted language with computational array for scientific numerical computation and visualization. APMonitor: APMonitor is a mathematical modeling language for describing and solving representations of physical systems in the form of differential and algebraic equations. Armadillo is C++ template library for linear algebra; includes various decompositions, factorisations, and statistics functions; its syntax (application programming interface (API) is similar to MATLAB. Clojure with numeric libraries Neanderthal, ClojureCUDA, and ClojureCL to call optimized matrix and linear algebra functions on CPU and GPU. Julia is designed for cloud parallel scientific computing in mind on LLVM-based just-in-time compilation (JIT) as a backend. Lightweight green threading (coroutines). Direct calls of C functions from code (no wrappers or special APIs needed), support for Unicode. Powerful shell-like abilities to manage


id: list-of-numerical-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-0.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
This is a list of numerical analysis topics. == General == Validated numerics Iterative method Rate of convergence — the speed at which a convergent sequence approaches its limit Order of accuracy — rate at which numerical solution of differential equation converges to exact solution Series acceleration — methods to accelerate the speed of convergence of a series Aitken's delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation — similar to Aitken's delta-squared process, but applied to the partial sums Van Wijngaarden transformation — for accelerating the convergence of an alternating series Abramowitz and Stegun — book containing formulas and tables of many special functions Digital Library of Mathematical Functions — successor of book by Abramowitz and Stegun Curse of dimensionality Local convergence and global convergence — whether you need a good initial guess to get convergence Superconvergence Discretization Difference quotient Complexity: Computational complexity of mathematical operations Smoothed analysis — measuring the expected performance of algorithms under slight random perturbations of worst-case inputs Symbolic-numeric computation — combination of symbolic and numeric methods Cultural and historical aspects: History of numerical solution of differential equations using computers Hundred-dollar, Hundred-digit Challenge problems


id: list-of-numerical-analysis-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-1.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Basic concepts === Types of matrices appearing in numerical analysis: Sparse matrix Band matrix Bidiagonal matrix Tridiagonal matrix Pentadiagonal matrix Skyline matrix Circulant matrix Triangular matrix Diagonally dominant matrix Block matrix — matrix composed of smaller matrices Stieltjes matrix — symmetric positive definite with non-positive off-diagonal entries Hilbert matrix — example of a matrix which is extremely ill-conditioned (and thus difficult to handle) Wilkinson matrix — example of a symmetric tridiagonal matrix with pairs of nearly, but not exactly, equal eigenvalues Convergent matrix — square matrix whose successive powers approach the zero matrix Algorithms for matrix multiplication: Strassen algorithm Coppersmith–Winograd algorithm Cannon's algorithm — a distributed algorithm, especially suitable for processors laid out in a 2d grid Freivalds' algorithm — a randomized algorithm for checking the result of a multiplication Matrix decompositions: LU decomposition — lower triangular times upper triangular QR decomposition — orthogonal matrix times triangular matrix RRQR factorization — rank-revealing QR factorization, can be used to compute rank of a matrix Polar decomposition — unitary matrix times positive-semidefinite Hermitian matrix Decompositions by similarity: Eigendecomposition — decomposition in terms of eigenvectors and eigenvalues Jordan normal form — bidiagonal matrix of a certain form; generalizes the eigendecomposition Weyr canonical


id: list-of-numerical-analysis-topics-20260505-2
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-2.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Spline (mathematics) — the piecewise polynomials used as interpolants Perfect spline — polynomial spline of degree m whose mth derivate is ±1 Cubic Hermite spline Centripetal Catmull–Rom spline — special case of cubic Hermite splines without self-intersections or cusps Monotone cubic interpolation Hermite spline Bézier curve De Casteljau's algorithm composite Bézier curve Generalizations to more dimensions: Bézier triangle — maps a triangle to R3 Bézier surface — maps a square to R3 B-spline Box spline — multivariate generalization of B-splines Truncated power function De Boor's algorithm — generalizes De Casteljau's algorithm Non-uniform rational B-spline (NURBS) T-spline — can be thought of as a NURBS surface for which a row of control points is allowed to terminate Kochanek–Bartels spline Coons patch — type of manifold parametrization used to smoothly join other surfaces together M-spline — a non-negative spline I-spline — a monotone spline, defined in terms of M-splines Smoothing spline — a spline fitted smoothly to noisy data Blossom (functional) — a unique, affine, symmetric map associated to a polynomial or spline See also: List of numerical computational geometry topics === Trigonometric interpolation === Trigonometric interpolation — interpolation by trigonometric polynomials Discrete Fourier transform — can be viewed as trigonometric interpolation


id: list-of-numerical-analysis-topics-20260505-3
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-3.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Orders of approximation Lebesgue's lemma Curve fitting Vector field reconstruction Modulus of continuity — measures smoothness of a function Least squares (function approximation) — minimizes the error in the L2-norm Minimax approximation algorithm — minimizes the maximum error over an interval (the L∞-norm) Equioscillation theorem — characterizes the best approximation in the L∞-norm Unisolvent point set — function from given function space is determined uniquely by values on such a set of points Stone–Weierstrass theorem — continuous functions can be approximated uniformly by polynomials, or certain other function spaces Approximation by polynomials: Linear approximation Bernstein polynomial — basis of polynomials useful for approximating a function Bernstein's constant — error when approximating |x| by a polynomial Remez algorithm — for constructing the best polynomial approximation in the L∞-norm Bernstein's inequality (mathematical analysis) — bound on maximum of derivative of polynomial in unit disk Mergelyan's theorem — generalization of Stone–Weierstrass theorem for polynomials Müntz–Szász theorem — variant of Stone–Weierstrass theorem for polynomials if some coefficients have to be zero Bramble–Hilbert lemma — upper bound on Lp error of polynomial approximation in multiple dimensions Discrete Chebyshev polynomials — polynomials orthogonal with respect to a discrete measure Favard's theorem — polynomials satisfying suitable 3-term


id: list-of-numerical-analysis-topics-20260505-4
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-4.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Algorithms for linear programming: Simplex algorithm Bland's rule — rule to avoid cycling in the simplex method Klee–Minty cube — perturbed (hyper)cube; simplex method has exponential complexity on such a domain Criss-cross algorithm — similar to the simplex algorithm Big M method — variation of simplex algorithm for problems with both "less than" and "greater than" constraints Interior point method Ellipsoid method Karmarkar's algorithm Mehrotra predictor–corrector method Column generation k-approximation of k-hitting set — algorithm for specific LP problems (to find a weighted hitting set) Linear complementarity problem Decompositions: Benders' decomposition Dantzig–Wolfe decomposition Theory of two-level planning Variable splitting Basic solution (linear programming) — solution at vertex of feasible region Fourier–Motzkin elimination Hilbert basis (linear programming) — set of integer vectors in a convex cone which generate all integer vectors in the cone LP-type problem Linear inequality Vertex enumeration problem — list all vertices of the feasible set === Convex optimization === Convex optimization Quadratic programming Linear least squares (mathematics) Total least squares Frank–Wolfe algorithm Sequential minimal optimization — breaks up large QP problems into a series of smallest possible QP problems Bilinear program Basis pursuit — minimize L1-norm of vector subject to linear constraints Basis pursuit denoising (BPDN) —


id: list-of-numerical-analysis-topics-20260505-5
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-5.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Semi-infinite programming — infinite number of variables and finite number of constraints, or other way around Shape optimization, Topology optimization — optimization over a set of regions Topological derivative — derivative with respect to changing in the shape Generalized semi-infinite programming — finite number of variables, infinite number of constraints === Uncertainty and randomness === Approaches to deal with uncertainty: Markov decision process Partially observable Markov decision process Robust optimization Wald's maximin model Scenario optimization — constraints are uncertain Stochastic approximation Stochastic optimization Stochastic programming Stochastic gradient descent Random optimization algorithms: Random search — choose a point randomly in ball around current iterate Simulated annealing Adaptive simulated annealing — variant in which the algorithm parameters are adjusted during the computation. Great Deluge algorithm Mean field annealing — deterministic variant of simulated annealing Bayesian optimization — treats objective function as a random function and places a prior over it Evolutionary algorithm Differential evolution Evolutionary programming Genetic algorithm, Genetic programming Genetic algorithms in economics MCACEA (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) — uses an evolutionary algorithm for every agent Simultaneous perturbation stochastic approximation (SPSA) Luus–Jaakola Particle swarm optimization Stochastic tunneling Harmony search — mimicks the improvisation process of musicians see also the


id: list-of-numerical-analysis-topics-20260505-6
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-6.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Rectangle method — first-order method, based on (piecewise) constant approximation Trapezoidal rule — second-order method, based on (piecewise) linear approximation Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation Adaptive Simpson's method Boole's rule — sixth-order method, based on the values at five equidistant points Newton–Cotes formulas — generalizes the above methods Romberg's method — Richardson extrapolation applied to trapezium rule Gaussian quadrature — highest possible degree with given number of points Chebyshev–Gauss quadrature — extension of Gaussian quadrature for integrals with weight (1 − x2)±1/2 on [−1, 1] Gauss–Hermite quadrature — extension of Gaussian quadrature for integrals with weight exp(−x2) on [−∞, ∞] Gauss–Jacobi quadrature — extension of Gaussian quadrature for integrals with weight (1 − x)α (1 + x)β on [−1, 1] Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature Gauss–Kronrod rules Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points Clenshaw–Curtis quadrature — based on expanding the integrand in terms of Chebyshev polynomials Adaptive quadrature — adapting the subintervals in which the integration interval is divided depending on the integrand Monte Carlo


id: list-of-numerical-analysis-topics-20260505-7
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-7.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
Finite difference — the discrete analogue of a differential operator Finite difference coefficient — table of coefficients of finite-difference approximations to derivatives Discrete Laplace operator — finite-difference approximation of the Laplace operator Eigenvalues and eigenvectors of the second derivative — includes eigenvalues of discrete Laplace operator Kronecker sum of discrete Laplacians — used for Laplace operator in multiple dimensions Discrete Poisson equation — discrete analogue of the Poisson equation using the discrete Laplace operator Stencil (numerical analysis) — the geometric arrangements of grid points affected by a basic step of the algorithm Compact stencil — stencil which only uses a few grid points, usually only the immediate and diagonal neighbours Higher-order compact finite difference scheme Non-compact stencil — any stencil that is not compact Five-point stencil — two-dimensional stencil consisting of a point and its four immediate neighbours on a rectangular grid Finite difference methods for heat equation and related PDEs: FTCS scheme (forward-time central-space) — first-order explicit Crank–Nicolson method — second-order implicit Finite difference methods for hyperbolic PDEs like the wave equation: Lax–Friedrichs method — first-order explicit Lax–Wendroff method — second-order explicit MacCormack method — second-order explicit Upwind scheme Upwind differencing scheme for convection — first-order scheme for convection–diffusion


id: list-of-numerical-analysis-topics-20260505-8
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-8.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Other methods === Spectral method — based on the Fourier transformation Pseudo-spectral method Method of lines — reduces the PDE to a large system of ordinary differential equations Boundary element method (BEM) — based on transforming the PDE to an integral equation on the boundary of the domain Interval boundary element method — a version using interval arithmetics Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics Godunov's scheme — first-order conservative scheme for fluid flow, based on piecewise constant approximation MUSCL scheme — second-order variant of Godunov's scheme AUSM — advection upstream splitting method Flux limiter — limits spatial derivatives (fluxes) in order to avoid spurious oscillations Riemann solver — a solver for Riemann problems (a conservation law with piecewise constant data) Properties of discretization schemes — finite volume methods can be conservative, bounded, etc. Discrete element method — a method in which the elements can move freely relative to each other Extended discrete element method — adds properties such as strain to each particle Movable cellular automaton — combination of cellular


id: list-of-numerical-analysis-topics-20260505-9
file: data/en.wikipedia.org/wiki/List_of_numerical_analysis_topics-9.md
category: reference
title: List of numerical analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics
date_saved: 2026-05-05T08:15:57.787804+00:00
instance: kb-cron
parent: list-of-numerical-analysis-topics-20260505-0
summary:
=== Grids and meshes === Grid classification / Types of mesh: Polygon mesh — consists of polygons in 2D or 3D Triangle mesh — consists of triangles in 2D or 3D Triangulation (geometry) — subdivision of given region in triangles, or higher-dimensional analogue Nonobtuse mesh — mesh in which all angles are less than or equal to 90° Point-set triangulation — triangle mesh such that given set of point are all a vertex of a triangle Polygon triangulation — triangle mesh inside a polygon Delaunay triangulation — triangulation such that no vertex is inside the circumcentre of a triangle Constrained Delaunay triangulation — generalization of the Delaunay triangulation that forces certain required segments into the triangulation Pitteway triangulation — for any point, triangle containing it has nearest neighbour of the point as a vertex Minimum-weight triangulation — triangulation of minimum total edge length Kinetic triangulation — a triangulation that moves over time Triangulated irregular network Quasi-triangulation — subdivision into simplices, where vertices are not points but arbitrary sloped line segments Volume mesh — consists of three-dimensional shapes Regular grid — consists of congruent parallelograms, or higher-dimensional analogue Unstructured grid Geodesic grid — isotropic grid on a sphere Mesh generation Image-based


id: list-of-numerical-computational-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics-0.md
category: reference
title: List of numerical computational geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_numerical_computational_geometry_topics
date_saved: 2026-05-05T08:15:58.970321+00:00
instance: kb-cron
parent: list-of-numerical-computational-geometry-topics-20260505-0
summary:
List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer-aided geometric design, and geometric modelling. See List of combinatorial computational geometry topics for another flavor of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character. == Curves == In the list of curves topics, the following ones are fundamental to geometric modelling. Parametric curve Bézier curve Spline Hermite spline Beta spline B-spline Higher-order spline NURBS Contour line == Surfaces == Bézier surface Isosurface Parametric surface == Other == Level-set method Computational topology


id: list-of-operator-splitting-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_operator_splitting_topics-0.md
category: reference
title: List of operator splitting topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_operator_splitting_topics
date_saved: 2026-05-05T08:16:00.110845+00:00
instance: kb-cron
parent: list-of-operator-splitting-topics-20260505-0
summary:
This is a list of operator splitting topics. == General == Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations GRADELA — simple gradient elasticity model Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices Paul Tseng — resolved question on convergence of matrix splitting algorithms PISO algorithm — pressure-velocity calculation for Navier-Stokes equations Projection method (fluid dynamics) — computational fluid dynamics method Reactive transport modeling in porous media — modeling of chemical reactions and fluid flow through the Earth's crust Richard S. Varga — developed matrix splitting Strang splitting — specific numerical method for solving differential equations using operator splitting


id: list-of-mathematic-operators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematic_operators-0.md
category: reference
title: List of mathematic operators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematic_operators
date_saved: 2026-05-05T08:16:01.392792+00:00
instance: kb-cron
parent: list-of-mathematic-operators-20260505-0
summary:
In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators. In the following L is an operator L : F → G {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}} which takes a function y ∈ F {\displaystyle y\in {\mathcal {F}}} to another function L [ y ] ∈ G {\displaystyle L[y]\in {\mathcal {G}}} . Here, F {\displaystyle {\mathcal {F}}} and G {\displaystyle {\mathcal {G}}} are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. == See also == List of transforms List of Fourier-related transforms Transfer operator Fredholm operator Borel transform Glossary of mathematical symbols


id: list-of-order-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_order_theory_topics-0.md
category: reference
title: List of order theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_order_theory_topics
date_saved: 2026-05-05T08:16:02.632494+00:00
instance: kb-cron
parent: list-of-order-theory-topics-20260505-0
summary:
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value and mathematical optimization. == Overview == Partially ordered set Preorder Totally ordered set Total preorder Chain Trichotomy Extended real number line Antichain Strict order Hasse diagram Directed acyclic graph Duality (order theory) Product order == Distinguished elements of partial orders == Greatest element (maximum, top, unit), Least element (minimum, bottom, zero) Maximal element, minimal element Upper bound Least upper bound (supremum, join) Greatest lower bound (infimum, meet) Limit superior and limit inferior Irreducible element Prime element Compact element == Subsets of partial orders == Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense set Linked set (upwards and downwards) Directed set (upwards and downwards) centered and σ-centered set Net (mathematics) Upper set and lower set Ideal and filter Ultrafilter == Special types of partial orders == Completeness (order theory) Dense order Distributivity (order theory) Modular lattice Distributive lattice


id: list-of-order-structures-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_order_structures_in_mathematics-0.md
category: reference
title: List of order structures in mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_order_structures_in_mathematics
date_saved: 2026-05-05T08:16:03.849108+00:00
instance: kb-cron
parent: list-of-order-structures-in-mathematics-20260505-0
summary:
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities) Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions Total orders, orderings that specify, for every two distinct elements, which one is less than the other Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) Well-orders, total orders in which every non-empty subset has a least element Well-quasi-orderings, a class of preorders generalizing the well-orders == See also == Glossary of order theory List of order theory topics


id: list-of-partial-differential-equation-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_partial_differential_equation_topics-0.md
category: reference
title: List of partial differential equation topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_partial_differential_equation_topics
date_saved: 2026-05-05T08:16:05.112454+00:00
instance: kb-cron
parent: list-of-partial-differential-equation-topics-20260505-0
summary:
This is a list of partial differential equation topics. == General topics == Partial differential equation Nonlinear partial differential equation list of nonlinear partial differential equations Boundary condition Boundary value problem Dirichlet problem, Dirichlet boundary condition Neumann boundary condition Stefan problem Wiener–Hopf problem Separation of variables Green's function Elliptic partial differential equation Singular perturbation Cauchy–Kovalevskaya theorem H-principle Atiyah–Singer index theorem Bäcklund transform Viscosity solution Weak solution Loewy decomposition of linear differential equations == Specific partial differential equations == Broer–Kaup equations Burgers' equation Euler equations Fokker–Planck equation Hamilton–Jacobi equation, Hamilton–Jacobi–Bellman equation Heat equation Laplace's equation Laplace operator Harmonic function Spherical harmonic Poisson integral formula Klein–Gordon equation Korteweg–de Vries equation Modified KdV–Burgers equation Maxwell's equations Navier–Stokes equations Poisson's equation Primitive equations (hydrodynamics) Schrödinger equation Wave equation == Numerical methods for PDEs == Finite difference Finite element method Finite volume method Boundary element method Multigrid Spectral method Computational fluid dynamics Alternating direction implicit == Related areas of mathematics == Calculus of variations Harmonic analysis Ordinary differential equation Sobolev space


id: list-of-partition-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_partition_topics-0.md
category: reference
title: List of partition topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_partition_topics
date_saved: 2026-05-05T08:16:06.339911+00:00
instance: kb-cron
parent: list-of-partition-topics-20260505-0
summary:
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set, partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see block matrix, and partition of the sum of squares in statistics problems, especially in the analysis of variance, quotition and partition, two ways of viewing the operation of division of integers. == Integer partitions == Composition (combinatorics) Ewens's sampling formula Ferrers graph Glaisher's theorem Landau's function Partition function (number theory) Pentagonal number theorem Plane partition Quotition and partition Rank of a partition Crank of a partition Solid partition Young tableau Young's lattice == Set partitions == Bell number Bell polynomials Dobinski's formula Cumulant Data clustering Equivalence relation Exact cover Knuth's Algorithm X Dancing Links Exponential formula Faà di Bruno's formula Feshbach–Fano partitioning Foliation Frequency partition Graph partition Kernel of a function Lamination (topology) Matroid partitioning Multipartition Multiplicative partition Noncrossing partition Ordered partition of a set Partition calculus Partition function (quantum field theory) Partition function (statistical mechanics) Derivation of the partition function Partition of an interval Partition of a


id: list-of-periodic-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_periodic_functions-0.md
category: reference
title: List of periodic functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_periodic_functions
date_saved: 2026-05-05T08:16:07.555333+00:00
instance: kb-cron
parent: list-of-periodic-functions-20260505-0
summary:
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions. == Smooth functions == All trigonometric functions listed have period 2 π {\displaystyle 2\pi } , unless otherwise stated. For the following trigonometric functions: Un is the nth up/down number, Bn is the nth Bernoulli number in Jacobi elliptic functions, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}} == Non-smooth functions == The following functions have period p {\displaystyle p} and take x {\displaystyle x} as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the floor function of n {\displaystyle n} and sgn {\displaystyle \operatorname {sgn} } is the sign function. K means Elliptic integral K(m) == Vector-valued functions == Epitrochoid Epicycloid (special case of the epitrochoid) Limaçon (special case of the epitrochoid) Hypotrochoid Hypocycloid (special case of the hypotrochoid) Spirograph (special case of the hypotrochoid) == Doubly periodic functions == Jacobi's elliptic functions Weierstrass's


id: list-of-permutation-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_permutation_topics-0.md
category: reference
title: List of permutation topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_permutation_topics
date_saved: 2026-05-05T08:16:08.751910+00:00
instance: kb-cron
parent: list-of-permutation-topics-20260505-0
summary:
This is a list of topics on mathematical permutations. == Particular kinds of permutations == Alternating permutation Circular shift Cyclic permutation Derangement Even and odd permutations—see Parity of a permutation Josephus permutation Parity of a permutation Separable permutation Stirling permutation Superpattern Transposition (mathematics) Unpredictable permutation == Combinatorics of permutations == Bijection Combination Costas array Cycle index Cycle notation Cycles and fixed points Cyclic order Direct sum of permutations Enumerations of specific permutation classes Factorial Falling factorial Permutation matrix Generalized permutation matrix Inversion (discrete mathematics) Major index Ménage problem Permutation graph Permutation pattern Permutation polynomial Permutohedron Rencontres numbers Robinson–Schensted correspondence Sum of permutations: Direct sum of permutations Skew sum of permutations Stanley–Wilf conjecture Symmetric function Szymanski's conjecture Twelvefold way == Permutation groups and other algebraic structures == === Groups === Alternating group Automorphisms of the symmetric and alternating groups Block (permutation group theory) Cayley's theorem Cycle index Frobenius group Galois group of a polynomial Jucys–Murphy element Landau's function Oligomorphic group O'Nan–Scott theorem Parker vector Permutation group Place-permutation action Primitive permutation group Rank 3 permutation group Representation theory of the symmetric group Schreier vector Strong generating set Symmetric group Symmetric inverse semigroup Weak order of permutations Wreath product Young symmetrizer Zassenhaus group


id: list-of-formulae-involving-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-0.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π. == Euclidean geometry == π = C d = C 2 r {\displaystyle \pi ={\frac {C}{d}}={\frac {C}{2r}}} where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, π = L w {\displaystyle \pi ={\frac {L}{w}}} where L and w are, respectively, the perimeter and the width of any curve of constant width. A = π r 2 {\displaystyle A=\pi r^{2}} where A is the area of a circle. More generally, A = π a b {\displaystyle A=\pi ab} where A is the area enclosed by an ellipse with semi-major axis a and semi-minor axis b. C = 2 π agm ⁡ ( a , b ) ( a 1 2 − ∑ n = 2 ∞ 2 n − 1 ( a n 2 − b n 2 ) ) {\displaystyle C={\frac {2\pi }{\operatorname {agm} (a,b)}}\left(a_{1}^{2}-\sum _{n=2}^{\infty }2^{n-1}(a_{n}^{2}-b_{n}^{2})\right)} where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and a n , b n {\displaystyle


id: list-of-formulae-involving-20260505-1
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-1.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.) == Formulae yielding π == === Integrals === 2 ∫ − 1 1 1 − x 2 d x = π {\displaystyle 2\int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx=\pi } (integrating two halves y ( x ) = 1 − x 2 {\displaystyle y(x)={\sqrt {1-x^{2}}}} to obtain the area of the unit circle) ∫ 0 2 4 − x 2 d x = π {\displaystyle \int _{0}^{2}{\sqrt {4-x^{2}}}\,dx=\pi } (integrating a quarter of a circle with a radius of two x 2 + y 2 = 4 {\displaystyle x^{2}+y^{2}=4} to obtain 4 π / 4 {\displaystyle {4\pi }/4} ) ∫ − ∞ ∞ sech ⁡ x d x = π {\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} x\,dx=\pi } ∫ − ∞ ∞ ∫ t ∞ e − 1 / 2 t 2 − x 2 + x t d x d t = ∫ − ∞ ∞ ∫


id: list-of-formulae-involving-20260505-2
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-2.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
∑ k = 0 ∞ ( − 1 ) k 2 10 k ( − 2 5 4 k + 1 − 1 4 k + 3 + 2 8 10 k + 1 − 2 6 10 k + 3 − 2 2 10 k + 5 − 2 2 10 k + 7 + 1 10 k + 9 ) = 2 6 π {\displaystyle \sum _{k=0}^{\infty }{\frac {{(-1)}^{k}}{2^{10k}}}\left(-{\frac {2^{5}}{4k+1}}-{\frac {1}{4k+3}}+{\frac {2^{8}}{10k+1}}-{\frac {2^{6}}{10k+3}}-{\frac {2^{2}}{10k+5}}-{\frac {2^{2}}{10k+7}}+{\frac {1}{10k+9}}\right)=2^{6}\pi } Plouffe's series for calculating arbitrary decimal digits of π: ∑ k = 1 ∞ k 2 k k ! 2 ( 2 k ) ! = π + 3 {\displaystyle \sum _{k=1}^{\infty }k{\frac {2^{k}k!^{2}}{(2k)!}}=\pi +3} === Other infinite series === ζ ( 2 ) = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ = π 2 6 {\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} (see also Basel problem and Riemann zeta function) ζ ( 4 ) = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + ⋯ = π 4 90 {\displaystyle \zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}} ζ


id: list-of-formulae-involving-20260505-3
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-3.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 k + 1 = ( − 1 ) k E 2 k 2 ( 2 k ) ! ( π 2 ) 2 k + 1 , k ∈ N 0 {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2k+1}}}=(-1)^{k}{\frac {E_{2k}}{2(2k)!}}\left({\frac {\pi }{2}}\right)^{2k+1},\quad k\in \mathbb {N} _{0}} where E 2 k {\displaystyle E_{2k}} is the 2 k {\displaystyle 2k} th Euler number. ∑ n = 0 ∞ ( 1 2 n ) ( − 1 ) n 2 n + 1 = 1 − 1 6 − 1 40 − ⋯ = π 4 {\displaystyle \sum _{n=0}^{\infty }{\binom {\frac {1}{2}}{n}}{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{6}}-{\frac {1}{40}}-\cdots ={\frac {\pi }{4}}} ∑ n = 0 ∞ 1 ( 4 n + 1 ) ( 4 n + 3 ) = 1 1 ⋅ 3 + 1 5 ⋅ 7 + 1 9 ⋅ 11 + ⋯ = π 8 {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(4n+1)(4n+3)}}={\frac {1}{1\cdot 3}}+{\frac {1}{5\cdot 7}}+{\frac {1}{9\cdot 11}}+\cdots ={\frac {\pi }{8}}} ∑ n = 1 ∞ ( − 1 ) ( n 2 + n ) / 2 + 1 | G ( ( − 1 ) n + 1


id: list-of-formulae-involving-20260505-4
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-4.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
π 4 = 5 arctan ⁡ 1 7 + 2 arctan ⁡ 3 79 {\displaystyle {\frac {\pi }{4}}=5\arctan {\frac {1}{7}}+2\arctan {\frac {3}{79}}} π 4 = 6 arctan ⁡ 1 8 + 2 arctan ⁡ 1 57 + arctan ⁡ 1 239 {\displaystyle {\frac {\pi }{4}}=6\arctan {\frac {1}{8}}+2\arctan {\frac {1}{57}}+\arctan {\frac {1}{239}}} π 4 = 12 arctan ⁡ 1 49 + 32 arctan ⁡ 1 57 − 5 arctan ⁡ 1 239 + 12 arctan ⁡ 1 110443 {\displaystyle {\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}} π 4 = 44 arctan ⁡ 1 57 + 7 arctan ⁡ 1 239 − 12 arctan ⁡ 1 682 + 24 arctan ⁡ 1 12943 {\displaystyle {\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}} === Infinite products === π 4 = ( ∏ p ≡ 1 ( mod 4 ) p p − 1 ) ⋅ ( ∏ p ≡ 3 ( mod 4 ) p p + 1 ) = 3 4 ⋅ 5 4 ⋅ 7 8 ⋅ 11 12 ⋅ 13 12 ⋯ , {\displaystyle {\frac {\pi }{4}}=\left(\prod _{p\equiv 1{\pmod {4}}}{\frac {p}{p-1}}\right)\cdot \left(\prod _{p\equiv 3{\pmod {4}}}{\frac {p}{p+1}}\right)={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot


id: list-of-formulae-involving-20260505-5
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-5.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
whenever a + b + c = a b c {\displaystyle a+b+c=abc} and a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are positive real numbers (see List of trigonometric identities). A special case is π = arctan ⁡ 1 + arctan ⁡ 2 + arctan ⁡ 3. {\displaystyle \pi =\arctan 1+\arctan 2+\arctan 3.} === Complex functions === e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} (Euler's identity) The following equivalences are true for any complex z {\displaystyle z} : e z ∈ R ↔ ℑ z ∈ π Z {\displaystyle e^{z}\in \mathbb {R} \leftrightarrow \Im z\in \pi \mathbb {Z} } e z = 1 ↔ z ∈ 2 π i Z {\displaystyle e^{z}=1\leftrightarrow z\in 2\pi i\mathbb {Z} } Also 1 e z − 1 = lim N → ∞ ∑ n = − N N 1 z − 2 π i n − 1 2 , z ∈ C . {\displaystyle {\frac {1}{e^{z}-1}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z-2\pi in}}-{\frac {1}{2}},\quad z\in \mathbb {C} .} Suppose a lattice Ω {\displaystyle \Omega } is generated by two periods ω 1 , ω 2 {\displaystyle \omega _{1},\omega _{2}} . We define the quasi-periods of this lattice by η


id: list-of-formulae-involving-20260505-6
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-6.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
( 2 n n ) ∼ 4 n π n {\displaystyle {\binom {2n}{n}}\sim {\frac {4^{n}}{\sqrt {\pi n}}}} (asymptotic growth rate of the central binomial coefficients) C n ∼ 4 n π n 3 {\displaystyle C_{n}\sim {\frac {4^{n}}{\sqrt {\pi n^{3}}}}} (asymptotic growth rate of the Catalan numbers) n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} (Stirling's approximation) log ⁡ n ! ≃ ( n + 1 2 ) log ⁡ n − n + log ⁡ 2 π 2 {\displaystyle \log n!\simeq \left(n+{\frac {1}{2}}\right)\log n-n+{\frac {\log 2\pi }{2}}} ∑ k = 1 n φ ( k ) ∼ 3 n 2 π 2 {\displaystyle \sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}} (where φ {\displaystyle \varphi } is Euler's totient function) ∑ k = 1 n φ ( k ) k ∼ 6 n π 2 {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}} The symbol ∼ {\displaystyle \sim } means that the ratio of the left-hand side and the right-hand side tends to one as n → ∞ {\displaystyle n\to \infty } . The symbol ≃ {\displaystyle \simeq } means that the difference between the left-hand side and the right-hand side tends to


id: list-of-formulae-involving-20260505-7
file: data/en.wikipedia.org/wiki/List_of_formulae_involving_π-7.md
category: reference
title: List of formulae involving π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulae_involving_π
date_saved: 2026-05-05T08:16:10.037546+00:00
instance: kb-cron
parent: list-of-formulae-involving-20260505-0
summary:
π = Γ ( 3 / 4 ) 4 agm ⁡ ( 1 , 1 / 2 ) 2 = Γ ( 1 / 4 ) 4 / 3 agm ⁡ ( 1 , 2 ) 2 / 3 2 {\displaystyle \pi ={\frac {\Gamma (3/4)^{4}}{\operatorname {agm} (1,1/{\sqrt {2}})^{2}}}={\frac {\Gamma \left({1/4}\right)^{4/3}\operatorname {agm} (1,{\sqrt {2}})^{2/3}}{2}}} (where agm is the arithmetic–geometric mean) π = agm ⁡ ( θ 2 2 ( 1 / e ) , θ 3 2 ( 1 / e ) ) {\displaystyle \pi =\operatorname {agm} \left(\theta _{2}^{2}(1/e),\theta _{3}^{2}(1/e)\right)} (where θ 2 {\displaystyle \theta _{2}} and θ 3 {\displaystyle \theta _{3}} are the Jacobi theta functions) agm ⁡ ( 1 , 2 ) = π ϖ {\displaystyle \operatorname {agm} (1,{\sqrt {2}})={\frac {\pi }{\varpi }}} (due to Gauss, ϖ {\displaystyle \varpi } is the lemniscate constant) N ⁡ ( 2 ϖ ) = e 2 π , N ⁡ ( ϖ ) = e π / 2 {\displaystyle \operatorname {N} (2\varpi )=e^{2\pi },\quad \operatorname {N} (\varpi )=e^{\pi /2}} (where N {\displaystyle \operatorname {N} } is the Gauss N-function) i π = Log ⁡ ( − 1 ) = lim n → ∞ n ( ( − 1 ) 1 / n


id: list-of-topics-related-to-20260505-0
file: data/en.wikipedia.org/wiki/List_of_topics_related_to_π-0.md
category: reference
title: List of topics related to π
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_topics_related_to_π
date_saved: 2026-05-05T08:16:11.270763+00:00
instance: kb-cron
parent: list-of-topics-related-to-20260505-0
summary:
This is a list of topics related to pi (π), the fundamental mathematical constant. 2π theorem Approximations of π Arithmetic–geometric mean Bailey–Borwein–Plouffe formula Basel problem Borwein's algorithm Buffon's needle Cadaeic Cadenza Chronology of computation of π Circle Euler's identity Six nines in pi Gauss–Legendre algorithm Gaussian function History of π A History of Pi Indiana Pi Bill Leibniz formula for pi Lindemann–Weierstrass theorem (Proof that π is transcendental) List of circle topics List of formulae involving π Liu Hui's π algorithm Mathematical constant (sorted by continued fraction representation) Mathematical constants and functions Method of exhaustion Milü Pi Pi (art project) Pi (letter) Pi Day PiFast PiHex Pi in the Sky Pilish Pimania Piphilology Proof that π is irrational Proof that 22/7 exceeds π Proof of Wallis product Rabbi Nehemiah Radian Ramanujan–Sato series Rhind Mathematical Papyrus Salamin–Brent algorithm Software for calculating π Squaring the circle Turn (geometry) Viète's formula


id: list-of-planar-symmetry-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_planar_symmetry_groups-0.md
category: reference
title: List of planar symmetry groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_planar_symmetry_groups
date_saved: 2026-05-05T08:16:12.595642+00:00
instance: kb-cron
parent: list-of-planar-symmetry-groups-20260505-0
summary:
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups 7 frieze groups – 2D line groups 17 wallpaper groups – 2D space groups. == Rosette groups == There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group. == Frieze groups == The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. == Wallpaper groups == The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular). The p1 and p2 groups, with no reflectional symmetry, are repeated in all classes. The


id: list-of-mathematical-properties-of-points-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_properties_of_points-0.md
category: reference
title: List of mathematical properties of points
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_properties_of_points
date_saved: 2026-05-05T08:16:13.832116+00:00
instance: kb-cron
parent: list-of-mathematical-properties-of-points-20260505-0
summary:
In mathematics, the following appear: Algebraic point Associated point Base point Closed point Divisor point Embedded point Extreme point Fermat point Fixed point Focal point Geometric point Hyperbolic equilibrium point Ideal point Inflection point Integral point Isolated point Generic point Heegner point Lattice hole, Lattice point Lebesgue point Midpoint Napoleon points Non-singular point Normal point Parshin point Periodic point Pinch point Point (geometry) Point source Rational point Recurrent point Regular point, Regular singular point Saddle point Semistable point Separable point Simple point Singular point of a curve Singular point of an algebraic variety Smooth point Special point Stable point Torsion point Vertex (curve) Weierstrass point == Calculus == Critical point (aka stationary point), any value v in the domain of a differentiable function of any real or complex variable, such that the derivative of v is 0 or undefined == Geometry == Antipodal point, the point diametrically opposite to another point on a sphere, such that a line drawn between them passes through the centre of the sphere and forms a true diameter Conjugate point, any point that can almost be joined to another by a 1-parameter family of geodesics (e.g., the antipodes of a sphere, which are linkable by


id: list-of-polygons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polygons-0.md
category: reference
title: List of polygons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polygons
date_saved: 2026-05-05T08:16:15.076521+00:00
instance: kb-cron
parent: list-of-polygons-20260505-0
summary:
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners. The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well. == Greek numbers == Polygons are primarily named by prefixes from Ancient Greek numbers. == Systematic polygon names == To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" connector is not included by some authors. Extending the system up to 999 is expressed with these prefixes. == List of n-gons by Greek numerical prefixes == == See also == Platonic solid


id: list-of-polyhedral-stellations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-0.md
category: reference
title: List of polyhedral stellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyhedral_stellations
date_saved: 2026-05-05T08:16:16.272155+00:00
instance: kb-cron
parent: list-of-polyhedral-stellations-20260505-0
summary:
In three-dimensional space, applying the operation of stellation to a polyhedron extends its faces (or edges and planes) until they generate new vertices that bound a newly formed figure. Stellation represents the dual action to faceting a polyhedron. Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices Mundi. Progress later ensued on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, with developments on different stellation methods occurring in the 1900s, principally from Coxeter et al. (1938) and soon afterward, Pawley (1975). A short generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the Platonic solids (including the fifty-nine icosahedral stellations), as well as for select Catalan solids (e.g., the rhombic dodecahedron and the rhombic triacontahedron). Stellations of non-convex uniform polyhedra of structure with facial planes passing through their centers (i.e., hemipolyhedral), which render unbounded vertices, are also included; these are stellations to infinity — per Wenninger (1983) — conforming to extensions on traditional definitions of polyhedra. == Background == === Star polytopes === Experimentation


id: list-of-polyhedral-stellations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_polyhedral_stellations-1.md
category: reference
title: List of polyhedral stellations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyhedral_stellations
date_saved: 2026-05-05T08:16:16.272155+00:00
instance: kb-cron
parent: list-of-polyhedral-stellations-20260505-0
summary:
== Stellations of Platonic solids == Only three of the five Platonic solids produce stellations: the regular octahedron, regular dodecahedron, and regular icosahedron. The regular tetrahedron and cube are unable to generate stellations when extending their faces, since extending their vertices only form one possible convex hull. === Stellations of the octahedron === The stella octangula (or stellated octahedron) is the only stellation of the regular octahedron. This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound: === Stellations of the dodecahedron === All stellations of the regular dodecahedron are Kepler-Poinsot polyhedra: === Stellations of the icosahedron === Coxeter et al. (1938) detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in Coxeter et al. (1999), the following table lists all stellations of the icosahedron per the Crennells' indexing (in it, the regular icosahedron (or snub octahedron) stellation core is indexed as "1"): "Cells" (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron. A subset of these are illustrated in Wenninger (1989), alongside constructions for physical models (W19–W66). == Stellations of Catalan solids == === Stellations of the rhombic dodecahedron ===


id: list-of-polynomial-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polynomial_topics-0.md
category: reference
title: List of polynomial topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polynomial_topics
date_saved: 2026-05-05T08:16:17.484027+00:00
instance: kb-cron
parent: list-of-polynomial-topics-20260505-0
summary:
This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. == Terminology == Degree: The maximum exponents among the monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. Graphing End behaviour – Concavity – Orientation – Tangency point – Inflection point – Point where concavity changes. == Basics == Polynomial Coefficient Monomial Polynomial long division Synthetic division Polynomial factorization Rational function Partial fraction Partial fraction decomposition over R Vieta's formulas Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem === Elementary abstract algebra === See also Theory of equations below. Polynomial ring Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) == Theory of equations == Polynomial family Quadratic function Cubic function Quartic function Quintic function Sextic function Septic function Octic function Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root (radical) Surd Square root


id: list-of-regular-polytopes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-0.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. == Overview == This table shows a summary of regular polytope counts by rank. == 1-polytopes == There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a dion and gives it the Schläfli symbol { }. Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. == 2-polytopes (polygons) == The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol {p}. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than


id: list-of-regular-polytopes-20260505-1
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-1.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
==== Spherical ==== In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations. The first few cases (n from 2 to 6) are listed below. Star-dihedra and hosohedra {p/q, 2} and {2, p/q} also exist for any star polygon {p/q}. === Stars === The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow. There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}. === Skew polyhedra === Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures. For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n}


id: list-of-regular-polytopes-20260505-2
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-2.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
A vertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex. An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge. A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face. A regular 5-polytope { p , q , r , s } {\displaystyle \{p,q,r,s\}} exists only if { p , q , r } {\displaystyle \{p,q,r\}} and { q , r , s } {\displaystyle \{q,r,s\}} are regular 4-polytopes. The space it fits in is based on the expression: cos 2 ⁡ ( π q ) sin 2 ⁡ ( π p ) + cos 2 ⁡ ( π r ) sin 2 ⁡ ( π s ) {\displaystyle {\frac {\cos ^{2}\left({\frac {\pi }{q}}\right)}{\sin ^{2}\left({\frac {\pi }{p}}\right)}}+{\frac {\cos ^{2}\left({\frac {\pi }{r}}\right)}{\sin ^{2}\left({\frac {\pi }{s}}\right)}}} < 1 {\displaystyle <1} : Spherical 4-space tessellation or 5-space polytope = 1 {\displaystyle =1} : Euclidean 4-space tessellation > 1 {\displaystyle >1} : hyperbolic 4-space tessellation Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The


id: list-of-regular-polytopes-20260505-3
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-3.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
==== Hyperbolic tilings ==== Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H2. As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1. There are a number of different ways to display the hyperbolic plane, including the Poincaré disk model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens. There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. {3,7}, {3,8}, {3,9} ... {3,∞} {4,5}, {4,6}, {4,7} ... {4,∞} {5,4}, {5,5}, {5,6} ... {5,∞} {6,4}, {6,5}, {6,6} ... {6,∞} {7,3}, {7,4}, {7,5} ... {7,∞} {8,3}, {8,4}, {8,5} ... {8,∞} {9,3}, {9,4}, {9,5} ... {9,∞} ... {∞,3}, {∞,4}, {∞,5} ... {∞,∞} A sampling: The tilings {p, ∞} have ideal vertices, on the


id: list-of-regular-polytopes-20260505-4
file: data/en.wikipedia.org/wiki/List_of_regular_polytopes-4.md
category: reference
title: List of regular polytopes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytopes
date_saved: 2026-05-05T08:16:18.735925+00:00
instance: kb-cron
parent: list-of-regular-polytopes-20260505-0
summary:
There are no regular compact or paracompact hyperbolic star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical. Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point. === 5-apeirotopes === ==== Tessellations of Euclidean 4-space ==== There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space: There are also the two improper cases {4,3,4,2}


id: list-of-ppad-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_PPAD-complete_problems-0.md
category: reference
title: List of PPAD-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_PPAD-complete_problems
date_saved: 2026-05-05T08:16:19.979284+00:00
instance: kb-cron
parent: list-of-ppad-complete-problems-20260505-0
summary:
This is a list of PPAD-complete problems. == Fixed-point theorems == Sperner's lemma Brouwer fixed-point theorem Kakutani fixed-point theorem == Game theory == Nash equilibrium Core of Balanced Games == Equilibria in game theory and economics == Fisher market equilibria Arrow-Debreu equilibria Approximate Competitive Equilibrium from Equal Incomes Finding clearing payments in financial networks == Graph theory == Fractional stable paths problems Fractional hypergraph matching (see also the NP-complete Hypergraph matching) Fractional strong kernel == Miscellaneous == Scarf's lemma Fractional bounded budget connection games == References == Papadimitriou, Christos (1994). "On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence". Journal of Computer and System Sciences. 48 (3): 498–532. CiteSeerX 10.1.1.321.7008. doi:10.1016/S0022-0000(05)80063-7. Paper available online at Papadimitriou's Homepage. C. Daskalakis, P. W. Goldberg and C.H. Papadimitriou (2009). "The Complexity of Computing a Nash Equilibrium". SIAM Journal on Computing. 39 (3): 195–259. CiteSeerX 10.1.1.68.6111. doi:10.1137/070699652. Xi Chen; Xiaotie Deng (2006). "Settling the complexity of two-player Nash equilibrium". Proc. 47th FOCS. pp. 261–272.


id: list-of-prime-knots-20260505-0
file: data/en.wikipedia.org/wiki/List_of_prime_knots-0.md
category: reference
title: List of prime knots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_prime_knots
date_saved: 2026-05-05T08:16:22.748133+00:00
instance: kb-cron
parent: list-of-prime-knots-20260505-0
summary:
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes. == Table of prime knots == === Six or fewer crossings === === Seven crossings === === Eight crossings === === Nine crossings === === Ten crossings === === Higher === Conway knot 11n34 Kinoshita–Terasaka knot 11n42 == Table of prime links == === Eight or fewer crossings === === Higher === == See also == List of knots List of mathematical knots and links Knot tabulation (−2,3,7) pretzel knot == Notes == == External links == "The Rolfsen Knot Table", The Knot Atlas. "KnotInfo", Indiana.edu.


id: list-of-mersenne-primes-and-perfect-numbers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers-0.md
category: reference
title: List of Mersenne primes and perfect numbers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers
date_saved: 2026-05-05T08:16:25.190446+00:00
instance: kb-cron
parent: list-of-mersenne-primes-and-perfect-numbers-20260505-0
summary:
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Euclid proved c. 300 BCE that every prime expressed as Mp = 2p − 1 has a corresponding perfect number Mp × (Mp+1)/2 = 2p − 1 × (2p − 1). For example, the Mersenne prime 22 − 1 = 3 leads to the corresponding perfect number 22 − 1 × (22 − 1) = 2 × 3 = 6.


id: list-of-repunit-primes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_repunit_primes-0.md
category: reference
title: List of repunit primes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_repunit_primes
date_saved: 2026-05-05T08:16:26.460363+00:00
instance: kb-cron
parent: list-of-repunit-primes-20260505-0
summary:
This is a list of repunit primes in various bases. == Base 2 repunit primes == Base-2 repunit primes are called Mersenne primes. == Base 3 repunit primes == The first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence A076481 in the OEIS), corresponding to n {\displaystyle n} of 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117, ... (sequence A028491 in the OEIS). == Base 4 repunit primes == The only base-4 repunit prime is 5 ( 11 4 {\displaystyle 11_{4}} ). 4 n − 1 = ( 2 n + 1 ) ( 2 n − 1 ) {\displaystyle 4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)} , and 3 always divides 2 n + 1 {\displaystyle 2^{n}+1} when n is odd and 2 n − 1 {\displaystyle 2^{n}-1} when n is even. For n greater than 2, both 2 n + 1 {\displaystyle 2^{n}+1} and 2 n − 1 {\displaystyle 2^{n}-1} are greater than 3, so removing the factor of 3 still leaves two factors greater than 1. Therefore, the number cannot be prime. == Base 5 repunit primes == The first few base-5 repunit primes are


id: list-of-repunit-primes-20260505-1
file: data/en.wikipedia.org/wiki/List_of_repunit_primes-1.md
category: reference
title: List of repunit primes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_repunit_primes
date_saved: 2026-05-05T08:16:26.460363+00:00
instance: kb-cron
parent: list-of-repunit-primes-20260505-0
summary:
2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39, 12, 22, 67, 120, 195, 48, 54, 463, 38, 41, 17, 808, 404, 46, 76, 793, 38, 28, 215, 37, 236, 59, 15, 514, 260, 498, 6, 2, 95, 3, ... (sequence A066180 in the OEIS) Smallest base b {\displaystyle b} such that R p ( − b ) {\displaystyle R_{p}(-b)} is prime (where p {\displaystyle p} is the n {\displaystyle n} th prime) are 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2,


id: probability-and-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Probability_and_statistics-0.md
category: reference
title: Probability and statistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Probability_and_statistics
date_saved: 2026-05-05T08:16:27.641847+00:00
instance: kb-cron
parent: probability-and-statistics-20260505-0
summary:
Probability and statistics are two closely related fields in mathematics that are sometimes combined for academic purposes. They are covered in multiple articles and lists: Probability Statistics Glossary of probability and statistics Notation in probability and statistics Timeline of probability and statistics Publications named for both fields include the following: Brazilian Journal of Probability and Statistics Counterexamples in Probability and Statistics Probability and Mathematical Statistics Theory of Probability and Mathematical Statistics == References ==


id: list-of-mathematical-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_proofs-0.md
category: reference
title: List of mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_proofs
date_saved: 2026-05-05T08:16:28.793352+00:00
instance: kb-cron
parent: list-of-mathematical-proofs-20260505-0
summary:
A list of articles with mathematical proofs: == Theorems of which articles are primarily devoted to proving them == Bertrand's postulate and a proof Estimation of covariance matrices Fermat's little theorem and some proofs Gödel's completeness theorem and its original proof Mathematical induction and a proof Proof that 0.999... equals 1 Proof that 22/7 exceeds π Proof that e is irrational Proof that π is irrational Proof that the sum of the reciprocals of the primes diverges == Articles devoted to theorems of which a (sketch of a) proof is given == Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π theorem (proof in progress) Burnside's lemma Cantor's theorem Cantor–Bernstein–Schroeder theorem Cayley's formula Cayley's theorem Clique problem (to do) Compactness theorem (very compact proof) Erdős–Ko–Rado theorem Euler's formula Euler's four-square identity Euler's theorem Five color theorem Five lemma Fundamental theorem of arithmetic Gauss–Markov theorem (brief pointer to proof) Gödel's incompleteness theorem Gödel's first incompleteness theorem Gödel's second incompleteness theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's lemma Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's theorem (group theory) Lagrange's


id: list-of-long-mathematical-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-0.md
category: reference
title: List of long mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs
date_saved: 2026-05-05T08:16:29.984359+00:00
instance: kb-cron
parent: list-of-long-mathematical-proofs-20260505-0
summary:
This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10,000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full. == Long proofs == The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. 1799 – The Abel–Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. 1890 – Killing's classification of simple complex Lie algebras, including his discovery of the exceptional Lie algebras, took 180 pages in 4 papers. 1894 – The ruler-and-compass construction of a polygon of 65537 sides by Johann Gustav Hermes took over 200 pages. 1905 – Emanuel Lasker's original proof of the Lasker–Noether theorem took


id: list-of-long-mathematical-proofs-20260505-1
file: data/en.wikipedia.org/wiki/List_of_long_mathematical_proofs-1.md
category: reference
title: List of long mathematical proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long_mathematical_proofs
date_saved: 2026-05-05T08:16:29.984359+00:00
instance: kb-cron
parent: list-of-long-mathematical-proofs-20260505-0
summary:
Several proofs of the existence of sporadic simple groups, such as the Lyons group, originally used computer calculations with large matrices or with permutations on billions of symbols. In most cases, such as the baby monster group, the computer proofs were later replaced by shorter proofs avoiding computer calculations. Similarly, the calculation of the maximal subgroups of the larger sporadic groups uses a lot of computer calculations. Proving that a particular number is prime 2004 – Verification of the Riemann hypothesis for the first 1013 zeros of the Riemann zeta function. 2007 – Verification that checkers is a draw. Calculations of large numbers of digits of π. 2010 – Showing that the Rubik's Cube can be solved in 20 moves. 2012 – Showing that Sudoku needs at least 17 clues. 2013 – Ternary Goldbach conjecture: Every odd number greater than 5 can be expressed as the sum of three primes. 2014 – Proof of Erdős discrepancy conjecture for the particular case C=2: every ±1-sequence of the length 1161 has a discrepancy at least 3; the original proof, generated by a SAT solver, had a size of 13 gigabytes and was later reduced to 850 megabytes. 2016 – Solving the Boolean


id: list-of-pspace-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_PSPACE-complete_problems-0.md
category: reference
title: List of PSPACE-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_PSPACE-complete_problems
date_saved: 2026-05-05T08:16:31.189075+00:00
instance: kb-cron
parent: list-of-pspace-complete-problems-20260505-0
summary:
Here are some of the more commonly known problems that are PSPACE-complete when expressed as decision problems. This list is in no way comprehensive. == Games and puzzles == Generalized versions of: == Logic == == Lambda calculus == Type inhabitation problem for simply typed lambda calculus == Automata and language theory == === Circuit theory === Integer circuit evaluation === Automata theory === === Formal languages === == Graph theory == == Others == == See also == List of NP-complete problems == Notes == == References == Garey, M.R.; Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 978-0-7167-1045-5. Eppstein's page on computational complexity of games The Complexity of Approximating PSPACE-complete problems for hierarchical specifications


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: 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-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-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-cat-documentaries-television-series-and-cartoons-20260505-0
file: data/en.wikipedia.org/wiki/List_of_cat_documentaries,_television_series_and_cartoons-0.md
category: reference
title: List of cat documentaries, television series and cartoons
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cat_documentaries,_television_series_and_cartoons
date_saved: 2026-05-05T08:18:34.681061+00:00
instance: kb-cron
parent: list-of-cat-documentaries-television-series-and-cartoons-20260505-0
summary:
List of cat documentaries, television series and cartoons includes serious documentaries, television series and cartoons, in alphabetical order, related to cats . == Documentaries == Cat Empire: Docuseries (2021), Ubique Film Cats: Caressing the Tiger (1991), National Geographic Cats: Choosing, Caring and Training (2008), Revolution LLC Kedi (2016), Termite Films The Lion in Your Living Room (2015), Canadian Broadcasting Company Science of Cats (2014), National Geographic Secret Life of Cats (2014), National Geographic The Standard of Perfection: Show Cats (2009), Public Broadcasting Service Understanding Cats (2009), Public Broadcasting Service The World of Cats (2008), Columbia River Entertainment === Big cats documentaries === Big Cats (2018, 3 episodes), BBC Lions of Darkness (1993) National Geographic Lions of the African Night (1987) National Geographic Tiger King: Murder, Mayhem and Madness (2020, 7 episodes), Netflix === Documentary and reality series === Cats 101 (2009–2012), Animal Planet My Cat from Hell (2011–2016), Animal Planet Must Love Cats (2011–2012), Animal Planet == Cartoons == Garfield (1978–present; TV series 1988–1994, 2009–2016, 2019–2020), Universal Press Syndicate Heathcliff (1980–1981, 1984–1985), Ruby-Spears Productions, et al. The Itchy & Scratchy Show (1988–1997, as an element of The Simpsons), 20th Television Krazy Kat (1913–1944), King Features Syndicate The Ren and Stimpy


id: list-of-nature-conservation-organizations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_nature_conservation_organizations-0.md
category: reference
title: List of nature conservation organizations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_nature_conservation_organizations
date_saved: 2026-05-05T08:18:36.001086+00:00
instance: kb-cron
parent: list-of-nature-conservation-organizations-20260505-0
summary:
This is a list of nature conservation organisations that primarily aim to protect species, their habitats, and ecosystems. == List of international conservation organisations == African Wildlife Foundation - ensuring that wildlife and wild lands thrive in modern Africa Bat Conservation International - working to conserve the world's bats and their habitats BirdLife International - a global partnership of organisations that strive to conserve birds Center for Biological Diversity - protecting endangered species through legal action, petitions, media and activism Conservation International - secure the critical benefits that nature provides to humanity Durrell Wildlife Conservation Trust - mission to save species from extinction EcoHealth Alliance - protecting the health of people, animals, and the environment from emerging infectious diseases Fauna and Flora International - safeguard the future of southern Africa's large mammal populations Greenpeace - raising environmental issues to public knowledge International Anti-Poaching Foundation - created a structured military-like approach to conservation in Africa International Union for Conservation of Nature - a partnership of 1400 organisations working in the field of nature conservation MarAlliance - non-profit that supports the effective management and conservation of large marine wildlife and their critical habitats with the support of dependent communities. National Audubon Society


id: index-of-conservation-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_conservation_articles-0.md
category: reference
title: Index of conservation articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_conservation_articles
date_saved: 2026-05-05T08:18:37.189423+00:00
instance: kb-cron
parent: index-of-conservation-articles-20260505-0
summary:
This is an index of conservation topics. It is an alphabetical index of articles relating to conservation biology and conservation of the natural environment. == A == Abiotic stress - Adaptive management - Adventive plant - Aerial-seeding - Agreed Measures for the Conservation of Antarctic Fauna and Flora - Agroecology - American Prairie Foundation - Anti-whaling - Assisted migration - Assisted migration of forests in North America == B == Biodegradation - Biodiversity - Biodiversity action plan - Biodiversity hotspot - Biogenic - Biodiversity Outcomes Framework - Biogeographic realm - Biogeography - Bioinformatics - Biological integrity - Biomagnification - Biomaterial - Biome - Biomimicry - Biomonitoring - Biophilia hypothesis - Biophilic design - Biopiracy - Bioregion - Bioregionalism - Biosafety - Biosalinity - Biosecurity - Biosphere - Biosphere reserve - Biostatistics - Biosurvey - Biotechnology - Bioterrorism - Biotransference - Bird conservation - Blue-listed - Bottom trawling - Buffer zone == C == Captive breeding - Cave conservation - Charismatic megafauna - CITES - Common species - Community-based conservation - Compassionate conservation - Conference of Governors - Conservation agriculture - Conservation area - Conservation authority - Conservation biology - Conservation Commons - Conservation community - Conservation dependent - Conservation designation


id: natural-regions-of-germany-20260505-0
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-0.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
This division of Germany into major natural regions takes account primarily of geomorphological, geological, hydrological, and pedological criteria in order to divide the country into large, physical units with a common geographical basis. Political boundaries play no part in this, apart from defining the national border. In addition to a division of Germany by natural regions, the federal authorities have also produced a division by so-called landscape areas (Landschaftsräume) that is based more on human utilisation of various regions and so has clearly different boundaries. == Groundwork by the Federal Institute of Regional Studies (BfL) == The natural region classification of Germany, as used today by the Federal Office for Nature Conservation (Bundesamt für Naturschutz or BfN) and by most state institutions, is largely based on the work in producing the Handbook of Natural Region Divisions of Germany between the years 1953 to 1962. This divided the present federal territory (then West and East Germany) into 86 so-called major landscape unit groups (Haupteinheitgruppen) each with a two-digit number between 01 and 90. These, in turn, were subdivided into up to ten, in some cases more, major landscape units (Haupteinheiten), each with a three-digit number. The handbook was accompanied by 1:100,000


id: natural-regions-of-germany-20260505-1
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-1.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
The first version by Heinrich Müller-Miny appeared in the updated handbook map, the overview maps of the individual map sheets from 1960 and in the ninth edition of the handbook (1962) In the map sheets from publication year 1964 this version was slightly changed: The Upper Palatine-Upper Main Hills (Oberpfälzisch-Obermainische Hügelland) was raised from the third to the second level. The secondary landscape region of Lössbörden was expanded, especially in the west. There was another change in the map sheets from 1979 (Sheet 182 Burghausen) until their last publication (Sheet München and Tegernsee 1994): The primary landscape region, Central Uplands (Mittelgebirge), was divided in two into Mittelgebirgsschwelle and Schichtstufenland The classification of the Natural regions in Saxony, published after that of the Institute for Regional Studies and continually revised, allocates the larger northern part of the major unit group of Upper Lusatia (Oberlausitz) to the Lössbörden, so that only the Lusatian Mountains (Lausitzer Bergland) in the south remains within the Central Uplands. For clarity the first and second levels of the major landscape regions are organised from north to south and from west to east. Within a secondary or tertiary landscape region the list follows the numerical order in the


id: natural-regions-of-germany-20260505-2
file: data/en.wikipedia.org/wiki/Natural_regions_of_Germany-2.md
category: reference
title: Natural regions of Germany
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Natural_regions_of_Germany
date_saved: 2026-05-05T08:18:38.455696+00:00
instance: kb-cron
parent: natural-regions-of-germany-20260505-0
summary:
Upper Rhine Plain (D53) 23 Rhine-Main Basin 22 Northern Upper Rhine Plain 21 Middle Upper Rhine Plain 20 Southern Upper Rhine Plain ==== South German Scarplands ==== Source: (arranged by escarpments Grundgebirge/Buntsandstein, Muschelkalk, Keuper-Lias and Malm from north to south, internally from west to east. 14 Odenwald-Spessart-Rhön (D55) 15 Black Forest (D54) Swabian-Franconian Gäue 12 Neckar and Tauber Gäu Plateaus (D57) 13 Main Franconia Plateau (D56) Swabian-Franconian Keuper-Lias Lands 10 Swabian Keuper-Lias Lands (D58) 11 Franconian Keuper-Lias Lands (D59) Swabian-Franconian Jura 09 Swabian Jura (D60) 08 Franconian Jura (D61) 16 Upper Rhine Region (Dinkelberg and Upper Rhine Valley) (D69) ==== Upper Palatine-Upper Main Hills ==== 07 Upper Palatine-Upper Main Hills (D62) === Alpine Foreland === Source: arranged from north to south and from west to east) ==== Northern Alpine Foreland ==== 04 Iller-Lech Plateau (D64) Lower Bavarian Upland and Isar-Inn Gravel Plateau (D65) 06 Lower Bavarian Upland 05 Isar-Inn Gravel Plateau ==== Southern Alpine Foreland ==== Pre-Alpine Hills and Moorland (= Southern Alpine Foreland; D66) 03 Subalpine New Moraine Land 02* Nagelfluh Hills and Basins between Lake Constance and Wertach === Alps === The following 3 groups were still counted as part of (ex-)group 02 in the Handbook; the German


id: list-of-glaciers-20260505-0
file: data/en.wikipedia.org/wiki/List_of_glaciers-0.md
category: reference
title: List of glaciers
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_glaciers
date_saved: 2026-05-05T08:18:39.696555+00:00
instance: kb-cron
parent: list-of-glaciers-20260505-0
summary:
A glacier (US: GLAY-shər) or (UK: ) is a persistent body of dense ice that is constantly moving under its own weight; it forms where the accumulation of snow exceeds its ablation (melting and sublimation) over many years, often centuries. Glaciers slowly deform and flow due to stresses induced by their weight, creating crevasses, seracs, and other distinguishing features. Because glacial mass is affected by long-term climate changes, e.g., precipitation, mean temperature, and cloud cover, glacial mass changes are considered among the most sensitive indicators of climate change. There are about 198,000 to 200,000 glaciers in the world. Catalogs of glaciers include: World Glacier Inventory World Glacier Monitoring Service Global Land Ice Measurements from Space (GLIMS) Glacier Database Randolph Glacier Inventory (RGI) == Glaciers by continent == === Africa === Africa, specifically East Africa, has contained glacial regions, possibly as far back as the last glacier maximum 10 to 15 thousand years ago. Seasonal snow does exist on the highest peaks of East Africa as well as in the Drakensberg Range of South Africa, the Stormberg Mountains, and the Atlas Mountains in Morocco. Currently, the only remaining glaciers on the continent exist on Mount Kilimanjaro, Mount Kenya, and the Rwenzori.


id: list-of-habitats-of-principal-importance-in-wales-20260505-0
file: data/en.wikipedia.org/wiki/List_of_habitats_of_principal_importance_in_Wales-0.md
category: reference
title: List of habitats of principal importance in Wales
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_habitats_of_principal_importance_in_Wales
date_saved: 2026-05-05T08:18:40.968477+00:00
instance: kb-cron
parent: list-of-habitats-of-principal-importance-in-wales-20260505-0
summary:
Wales is obliged by law to maintain lists of species and habitats of principal importance for biodiversity conservation; the other countries within the UK: Scotland, England and Northern Ireland, have their own laws for this purpose. Public bodies, including local authorities now have a legal duty to have regard to conserving biodiversity in the exercise of their normal functions. In Wales, that obligation originally derived from section 42 of the Natural Environment and Rural Communities (NERC) Act 2006. However, this requirement for Wales has since been superseded by an almost identical requirement enshrined within the Environment (Wales) Act 2016. == Selection == The habitats that have been designated to be of "principal importance for the purpose of conserving biodiversity" derive from lists originally drawn up for the UK Biodiversity Action Plan (UK BAP). These lists were reviewed in 2007, and the total number of UK BAP habitats increased from 45 to 65, and the number of UK BAP species increased from under 600 to 1,150. From these, the formal list just for Wales (and laid out below) now contains 53 of those 65 habitats. === Legal obligations === Section 6 of the Environment (Wales) Act 2016 places a legal obligation


id: list-of-herbaria-in-europe-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_Europe-0.md
category: reference
title: List of herbaria in Europe
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_Europe
date_saved: 2026-05-05T08:18:42.184594+00:00
instance: kb-cron
parent: list-of-herbaria-in-europe-20260505-0
summary:
This is a list of herbaria in Europe, organized first by region where the herbarium is located (using the United Nations geoscheme for Europe), then within each region by size of the collection. For other continents, see List of herbaria. == Eastern Europe == The tables below list herbaria located in Eastern Europe as defined by the United Nations geoscheme for Europe. === Black Sea === The following table includes herbaria located in countries on the Black Sea, including Bulgaria, Moldova, Romania, and Ukraine. === Czech Republic === The following table includes herbaria located in the Czech Republic. === Hungary === The following table includes herbaria located in Hungary. === Poland === The following table includes herbaria located in Poland. === Russia === The following table includes herbaria located in European Russia. === Ukraine === National Herbarium of Ukraine (KW) == Northern Europe == The tables below list herbaria located in Northern Europe as defined by the United Nations geoscheme for Europe: Ireland, the British Isles, Baltic states, Scandinavia, and Iceland. === Baltic states === The following table includes herbaria located in the Baltic states: Estonia, Latvia, and Lithuania. === Britain & Ireland === The following table includes herbaria located


id: list-of-herbaria-in-north-america-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_North_America-0.md
category: reference
title: List of herbaria in North America
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_North_America
date_saved: 2026-05-05T08:18:43.376729+00:00
instance: kb-cron
parent: list-of-herbaria-in-north-america-20260505-0
summary:
This is a list of herbaria in North America, organized first by country or region where the herbarium is located, then within each region by size of the collection. All herbarium codes follow the Index Herbariorum. For other continents, see List of herbaria. == Canada == == Central America and the Caribbean == The table below lists herbaria located in Central America and the Caribbean. == Mexico == == United States == == References ==


id: list-of-herbaria-in-turkey-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria_in_Turkey-0.md
category: reference
title: List of herbaria in Turkey
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria_in_Turkey
date_saved: 2026-05-05T08:18:44.619963+00:00
instance: kb-cron
parent: list-of-herbaria-in-turkey-20260505-0
summary:
The following is a list of herbaria in Turkey. Herbaria are established within faculties and institutes of credible universities. Those created in departments that are concerned with natural sciences such as botany, ecology, biogeography and climatology are mostly used to do research on the genetics of the plants, to examine their distribution on specific geographical locations and to protect them for future generations, while those affiliated with more practice-based departments such as pharmacy are used as a resource for drug production techniques. The first herbarium in Turkey was opened in 1933 within the Faculty of Science at Ankara University. Founded by Kurt Krause and Hikmet Birand, this herbarium is the oldest among its counterparts in the country and has the most number of species within its 200,000-piece collection. Therefore, it is also called Herbarium Turcicum (Turkish Herbarium). The Sugar Institute Herbarium, Atatürk University Faculty of Science Herbarium and Ege University Faculty of Pharmacy Herbarium are closed to visitors and scientific researchers. However, thanks to a specially created website, it is possible to access all the data related to the Ege University Faculty of Pharmacy Herbarium remotely. Altınbaş University Faculty of Pharmacy Herbarium, which opened in 2018, is the first herbarium


id: list-of-herbaria-20260505-0
file: data/en.wikipedia.org/wiki/List_of_herbaria-0.md
category: reference
title: List of herbaria
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_herbaria
date_saved: 2026-05-05T08:18:45.907753+00:00
instance: kb-cron
parent: list-of-herbaria-20260505-0
summary:
This is a list of active herbaria, organized first by continent where the herbarium is located, then within each continent by size of the collection. The list is based on the Index Herbariorum, a global directory of herbaria and their associated staff. == Background == A herbarium (plural "herbaria") is a collection of preserved plant specimens. These specimens may be whole plants or plant parts: these will usually be in a dried form, mounted on a sheet, but depending upon the material may also be kept in alcohol or other preservative. The same term is often used in mycology to describe an equivalent collection of preserved fungi and in phycology to describe a collection of algae. To preserve their form and color, plants collected in the field are spread flat on sheets of newsprint and dried, usually in a plant press, between blotters or absorbent paper. The specimens, which are then mounted on sheets of stiff white paper, are labeled with all essential data, such as collector, date and place found, description of the plant, elevation, and special habitat conditions. The sheet is then placed in a protective case. As a precaution against insect damage, the pressed plant is frozen


id: iucn-protected-area-categories-20260505-0
file: data/en.wikipedia.org/wiki/IUCN_protected_area_categories-0.md
category: reference
title: IUCN protected area categories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/IUCN_protected_area_categories
date_saved: 2026-05-05T08:18:47.158801+00:00
instance: kb-cron
parent: iucn-protected-area-categories-20260505-0
summary:
IUCN protected area categories, or IUCN protected area management categories, are categories used to classify protected areas in a system developed by the International Union for Conservation of Nature (IUCN). The enlisting of such areas is part of a strategy being used toward the conservation of the world's natural environment and biodiversity. The IUCN has developed the protected area management categories system to define, record and classify the wide variety of specific aims and concerns when categorising protected areas and their objectives. Further supplementary guidelines have been developed specific to marine protected areas (MPAs). This categorisation method is recognised on a global scale by national governments and international bodies such as the United Nations and the Convention on Biological Diversity. == Categories == === Category I === ==== Category Ia – strict nature reserve ==== A strict nature reserve (IUCN Category Ia) is an area which is protected from all but light human use in order to protect its biodiversity and also possibly its geological/geomorphological features. These areas are often home to dense native ecosystems where all human disturbance except scientific study, environmental monitoring and education is prohibited. Because these areas are so strictly protected, they provide ideal pristine environments


id: iucn-protected-area-categories-20260505-1
file: data/en.wikipedia.org/wiki/IUCN_protected_area_categories-1.md
category: reference
title: IUCN protected area categories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/IUCN_protected_area_categories
date_saved: 2026-05-05T08:18:47.158801+00:00
instance: kb-cron
parent: iucn-protected-area-categories-20260505-0
summary:
=== Category V – protected landscape or seascape === A protected landscape or protected seascape (IUCN Category V) covers a body of land or ocean with an explicit natural conservation plan, but usually also accommodates a range of for-profit activities. The main objective is to safeguard regions that have built up a distinct and valuable ecological, biological, cultural, or scenic character. In contrast with previous categories, Category V permits surrounding communities to interact more with the area, contributing to the area's sustainable management and engaging with its natural and cultural heritage. Landscapes and seascapes that fall into this category should represent an integral balance between people and nature and can sustain activities such as traditional agricultural and forestry systems on conditions that ensure the continued protection or ecological restoration of the area. Category V is one of the more flexible classifications of protected areas. As a result, protected landscapes and seascapes may be able to accommodate contemporary developments, such as ecotourism, at the same time as maintaining the historical management practices that may procure the sustainability of agrobiodiversity and aquatic biodiversity. === Category VI – protected area with sustainable use of natural resources === Though human involvement is a large


id: list-of-animals-by-number-of-legs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_animals_by_number_of_legs-0.md
category: reference
title: List of animals by number of legs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_animals_by_number_of_legs
date_saved: 2026-05-05T08:18:48.428358+00:00
instance: kb-cron
parent: list-of-animals-by-number-of-legs-20260505-0
summary:
The following is a list of selected animals in order of increasing number of legs, from 0 legs to 653 pairs of legs, the maximum recorded in the animal kingdom. Each entry provides the relevant taxa up to the rank of phylum. Each entry also provides the common name of the animal. If the relevant taxon includes different animals with different common names, then the entry provides the common name of a familiar example. If juveniles have fewer legs than adults, then the animal is listed by the number of legs recorded in mature adults. If this number varies among adults within the taxon, then this variation is noted in a comment. In counting legs, this list follows conventions adopted in the relevant literature. For example, millipedes with gonopods are listed by numbers that exclude leg pairs that become gonopods, but numbers for millipedes with telopods include leg pairs that become telopods. Animals have been selected so that each number from 0 to 55 leg pairs has one example listed. Each of these examples is listed by a number closely associated with the relevant taxon, either because that number is the one most commonly observed in that taxon or because


id: list-of-long-distance-footpaths-20260505-0
file: data/en.wikipedia.org/wiki/List_of_long-distance_footpaths-0.md
category: reference
title: List of long-distance footpaths
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_long-distance_footpaths
date_saved: 2026-05-05T08:18:49.672015+00:00
instance: kb-cron
parent: list-of-long-distance-footpaths-20260505-0
summary:
This is a list of some long-distance footpaths used for walking and hiking. == Africa == === Lesotho === 3 Ranges Trail: 288.3 km (179.1 mi) a hiking trail that passes three mountain ranges, namely Thaba Putsoa Range, Central Range and ends at Drakensburg Range. It starts at Lesotho lowest point, at the confluence of Makhaleng and Senqu and ends at the country's highest point, Thabana Ntlenyane (3482m). === Namibia === Fish River Canyon: 90 km (56 mi) route in the ǀAi-ǀAis/Richtersveld Transfrontier Park === South Africa === Otter Trail: 44 km (27 mi) section of the Garden Route along the Cape coast Drakensberg Grand Traverse: rugged 205 km (127 mi) trek in KwaZulu-Natal === Egypt === Sinai Trail: 250 km (160 mi) thru-hike from Nuweiba to Mount Catherine === Uganda === Sir Samuel and Lady Florence Baker Trail: 575 km (357 mi) trek through northern Uganda == Asia == === Bangladesh === Jhiri Path: ascent of Keokradong, one of the highest mountains in Bangladesh. === Bhutan === Snowman Trek: 328 km (204 mi) trek through high passes near the border with Tibet [1] === Hong Kong === Hong Kong Trail: 50 km (31 mi) across Hong Kong Island Lantau


