# kb

id: list-of-algorithms-20260505-2
file: data/en.wikipedia.org/wiki/List_of_algorithms-2.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Chien search: a recursive algorithm for determining roots of polynomials defined over a finite field Schreier–Sims algorithm: computing a base and strong generating set (BSGS) of a permutation group Todd–Coxeter algorithm: Procedure for generating cosets. === Computer algebra === Buchberger's algorithm: finds a Gröbner basis Cantor–Zassenhaus algorithm: factor polynomials over finite fields Faugère F4 algorithm: finds a Gröbner basis (also mentions the F5 algorithm) Gosper's algorithm: find sums of hypergeometric terms that are themselves hypergeometric terms Knuth–Bendix completion algorithm: for rewriting rule systems Multivariate division algorithm: for polynomials in several indeterminates Pollard's kangaroo algorithm (also known as Pollard's lambda algorithm): an algorithm for solving the discrete logarithm problem Polynomial long division: an algorithm for dividing a polynomial by another polynomial of the same or lower degree Risch algorithm: an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives) === Geometry === Closest pair problem: find the pair of points (from a set of points) with the smallest distance between them Collision detection algorithms: check for the collision or intersection of two given solids Cone algorithm: identify surface points Convex hull algorithms: determining the convex hull of a set of points Chan's algorithm Gift wrapping algorithm or Jarvis march


id: list-of-algorithms-20260505-3
file: data/en.wikipedia.org/wiki/List_of_algorithms-3.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Computation of π: Bailey–Borwein–Plouffe formula: (BBP formula) a spigot algorithm for the computation of the nth binary digit of π Borwein's algorithm: an algorithm to calculate the value of 1/π Chudnovsky algorithm: a fast method for calculating the digits of π Gauss–Legendre algorithm: computes the digits of pi Division algorithms: for computing quotient and/or remainder of two numbers Goldschmidt division Long division Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q. Non-restoring division Restoring division SRT division Exponentiation: Addition-chain exponentiation: exponentiation by positive integer powers that requires a minimal number of multiplications Exponentiating by squaring: an algorithm used for the fast computation of large integer powers of a number Hyperbolic and Trigonometric Functions: BKM algorithm: computes elementary functions using a table of logarithms CORDIC: computes hyperbolic and trigonometric functions using a table of arctangents Montgomery reduction: an algorithm that allows modular arithmetic to be performed efficiently when the modulus is large Multiplication algorithms: fast multiplication of two numbers Booth's multiplication algorithm: a multiplication algorithm that multiplies two signed binary numbers in two's complement notation Fürer's algorithm: an integer multiplication algorithm for very large numbers possessing a


id: list-of-algorithms-20260505-4
file: data/en.wikipedia.org/wiki/List_of_algorithms-4.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
=== Optimization algorithms === Hybrid Algorithms Alpha–beta pruning: search to reduce number of nodes in minimax algorithm A hybrid BFGS-Like method (see more https://doi.org/10.1016/j.cam.2024.115857) Branch and bound Bruss algorithm: see odds algorithm Chain matrix multiplication Combinatorial optimization: optimization problems where the set of feasible solutions is discrete Greedy randomized adaptive search procedure (GRASP): successive constructions of a greedy randomized solution and subsequent iterative improvements of it through a local search Hungarian method: a combinatorial optimization algorithm which solves the assignment problem in polynomial time Conjugate gradient methods (see more https://doi.org/10.1016/j.jksus.2022.101923) Constraint satisfaction AC-3 algorithm general algorithms for the constraint satisfaction Chaff algorithm: an algorithm for solving instances of the Boolean satisfiability problem Davis–Putnam algorithm: check the validity of a first-order logic formula Difference map algorithm general algorithms for the constraint satisfaction Davis–Putnam–Logemann–Loveland algorithm (DPLL): an algorithm for deciding the satisfiability of propositional logic formula in conjunctive normal form, i.e. for solving the CNF-SAT problem Exact cover problem Min conflicts algorithm general algorithms for the constraint satisfaction Algorithm X: a nondeterministic algorithm Dancing Links: an efficient implementation of Algorithm X Cross-entropy method: a general Monte Carlo approach to combinatorial and continuous multi-extremal optimization and importance sampling Differential evolution Dynamic Programming: problems


id: list-of-algorithms-20260505-5
file: data/en.wikipedia.org/wiki/List_of_algorithms-5.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Algorithms for calculating variance: avoiding instability and numerical overflow Approximate counting algorithm: allows counting large number of events in a small register Bayesian statistics Nested sampling algorithm: a computational approach to the problem of comparing models in Bayesian statistics Clustering algorithms Average-linkage clustering: a simple agglomerative clustering algorithm Canopy clustering algorithm: an unsupervised pre-clustering algorithm related to the K-means algorithm Chinese whispers Complete-linkage clustering: a simple agglomerative clustering algorithm DBSCAN: a density based clustering algorithm Expectation-maximization algorithm Fuzzy clustering: a class of clustering algorithms where each point has a degree of belonging to clusters FLAME clustering (Fuzzy clustering by Local Approximation of MEmberships): define clusters in the dense parts of a dataset and perform cluster assignment solely based on the neighborhood relationships among objects Fuzzy c-means k-means clustering: cluster objects based on attributes into partitions k-means++: a variation of this, using modified random seeds k-medoids: similar to k-means, but chooses datapoints or medoids as centers KHOPCA clustering algorithm: a local clustering algorithm, which produces hierarchical multi-hop clusters in static and mobile environments. Linde–Buzo–Gray algorithm: a vector quantization algorithm to derive a good codebook Lloyd's algorithm (Voronoi iteration or relaxation): group data points into a given number of categories, a popular


id: list-of-algorithms-20260505-6
file: data/en.wikipedia.org/wiki/List_of_algorithms-6.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Asymmetric (public key) encryption: ElGamal Elliptic curve cryptography MAE1 NTRUEncrypt RSA Digital signatures (asymmetric authentication): DSA, and its variants: ECDSA and Deterministic ECDSA EdDSA (Ed25519) RSA Cryptographic hash functions (see also the section on message authentication codes): BLAKE MD5 – Note that there is now a method of generating collisions for MD5 RIPEMD-160 SHA-1 – Note that there is now a method of generating collisions for SHA-1 SHA-2 (SHA-224, SHA-256, SHA-384, SHA-512) SHA-3 (SHA3-224, SHA3-256, SHA3-384, SHA3-512, SHAKE128, SHAKE256) Tiger (TTH), usually used in Tiger tree hashes WHIRLPOOL Cryptographically secure pseudo-random number generators Blum Blum Shub – based on the hardness of factorization Fortuna, intended as an improvement on Yarrow algorithm Linear-feedback shift register (note: many LFSR-based algorithms are weak or have been broken) Yarrow algorithm Key exchange Diffie–Hellman key exchange Elliptic-curve Diffie–Hellman (ECDH) Key derivation functions, often used for password hashing and key stretching Argon2 bcrypt PBKDF2 scrypt Message authentication codes (symmetric authentication algorithms, which take a key as a parameter): HMAC: keyed-hash message authentication Poly1305 SipHash Secret sharing, secret splitting, key splitting, M of N algorithms Blakey's scheme Shamir's secret sharing Symmetric (secret key) encryption: Advanced Encryption Standard (AES), winner of NIST competition, also known as Rijndael Blowfish


id: list-of-algorithms-20260505-7
file: data/en.wikipedia.org/wiki/List_of_algorithms-7.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Hopcroft's algorithm, Moore's algorithm, and Brzozowski's algorithm: algorithms for minimizing the number of states in a deterministic finite automaton Powerset construction: algorithm to convert nondeterministic automaton to deterministic automaton. Tarski–Kuratowski algorithm: a non-deterministic algorithm which provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy == Information theory and signal processing == === Coding theory === ==== Error detection and correction ==== BCH Codes Berlekamp–Massey algorithm Peterson–Gorenstein–Zierler algorithm Reed–Solomon error correction BCJR algorithm: decoding of error correcting codes defined on trellises (principally convolutional codes) Forward error correction Gray code Hamming codes Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits Hamming distance: sum number of positions which are different Hamming weight (population count): find the number of 1 bits in a binary word Redundancy checks Adler-32 Cyclic redundancy check Damm algorithm Fletcher's checksum Longitudinal redundancy check (LRC) Luhn algorithm: a method of validating identification numbers Luhn mod N algorithm: extension of Luhn to non-numeric characters Parity: simple/fast error detection technique Verhoeff algorithm ==== Lossless compression algorithms ==== Burrows–Wheeler transform: preprocessing useful for improving lossless compression Context tree weighting Delta encoding: aid to compression of data


id: list-of-algorithms-20260505-8
file: data/en.wikipedia.org/wiki/List_of_algorithms-8.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Cache algorithms CHS conversion: converting between disk addressing systems Double dabble: convert binary numbers to BCD Hash function: convert a large, possibly variable-sized amount of data into a small datum, usually a single integer that may serve as an index into an array Fowler–Noll–Vo hash function: fast with low collision rate Pearson hashing: computes 8-bit value only, optimized for 8-bit computers Zobrist hashing: used in the implementation of transposition tables Unicode collation algorithm Xor swap algorithm: swaps the values of two variables without using a buffer == Database algorithms == Algorithms for Recovery and Isolation Exploiting Semantics (ARIES): transaction recovery Join algorithms Block nested loop Hash join Nested loop join Sort-Merge Join The Chase == Distributed systems algorithms == Clock synchronization Berkeley algorithm Cristian's algorithm Intersection algorithm Marzullo's algorithm Consensus (computer science): agreeing on a single value or history among unreliable processors Chandra–Toueg consensus algorithm Paxos algorithm Raft (computer science) Detection of Process Termination Dijkstra-Scholten algorithm Huang's algorithm Lamport ordering: a partial ordering of events based on the happened-before relation Leader election: a method for dynamically selecting a coordinator Bully algorithm Mutual exclusion Lamport's Distributed Mutual Exclusion Algorithm Naimi-Trehel's log(n) Algorithm Maekawa's Algorithm Raymond's Algorithm Ricart–Agrawala Algorithm Snapshot algorithm: record


id: list-of-aperiodic-sets-of-tiles-20260505-0
file: data/en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles-0.md
category: reference
title: List of aperiodic sets of tiles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles
date_saved: 2026-05-05T08:12:50.348730+00:00
instance: kb-cron
parent: list-of-aperiodic-sets-of-tiles-20260505-0
summary:
In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles). A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions. An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".) The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete. == Explanations == == List == ==


id: outline-of-arithmetic-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_arithmetic-0.md
category: reference
title: Outline of arithmetic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_arithmetic
date_saved: 2026-05-05T08:12:51.611767+00:00
instance: kb-cron
parent: outline-of-arithmetic-20260505-0
summary:
The following outline is provided as an overview of and topical guide to arithmetic: Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms, and the use of such operations to perform more complicated calculations. Arithmetic is fundamental for both pure mathematics and its practical application to science and daily life. In pure mathematics, arithmetic is defined formally for various types of numbers, such as integers, fractions of integers (rational numbers), and real numbers. In practical application, arithmetic is usually approximate, operating on decimal or binary numbers, with rounding. == Essence of arithmetic == Elementary arithmetic Decimal arithmetic Decimal point Numeral Place value == History of arithmetic == == Arithmetic operations and related concepts == Order of operations Addition Summation – Answer after adding a sequence of numbers Additive inverse Subtraction – Taking away numbers Multiplication – Repeated addition Multiple – Product of multiplication Least common multiple Multiplicative inverse Division – Repeated subtraction Modulo – The remainder of division Quotient – Result of division Quotition and partition – How many parts are there, and what is the size


id: list-of-mathematical-artists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_artists-0.md
category: reference
title: List of mathematical artists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_artists
date_saved: 2026-05-05T08:12:52.917714+00:00
instance: kb-cron
parent: list-of-mathematical-artists-20260505-0
summary:
This is a list of artists who actively explored mathematics in their artworks. Art forms practised by these artists include painting, sculpture, architecture, textiles and origami. Some artists such as Piero della Francesca and Luca Pacioli went so far as to write books on mathematics in art. Della Francesca wrote books on solid geometry and the emerging field of perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De corporibus regularibus (Regular Solids), while Pacioli wrote De divina proportione (On Divine Proportion), with illustrations by Leonardo da Vinci, at the end of the fifteenth century. Merely making accepted use of some aspect of mathematics such as perspective does not qualify an artist for admission to this list. The term "fine art" is used conventionally to cover the output of artists who produce a combination of paintings, drawings and sculptures. == List == == References == == External links == Saint Louis University: List of mathematical artists, by field


id: list-of-axioms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_axioms-0.md
category: reference
title: List of axioms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_axioms
date_saved: 2026-05-05T08:12:54.221468+00:00
instance: kb-cron
parent: list-of-axioms-20260505-0
summary:
This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system. == ZF (the Zermelo–Fraenkel axioms without the axiom of choice) == Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity Axiom schema of specification See also Zermelo set theory. == Axiom of choice == With the Zermelo–Fraenkel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable. === Equivalents of AC === Hausdorff maximality theorem Well-ordering theorem Zorn's lemma === Stronger than AC === Axiom of global choice === Weaker than AC === Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of uniformization === Alternates incompatible with AC === Axiom of real determinacy == Other axioms of mathematical logic == Von Neumann–Bernays–Gödel axioms Continuum


id: list-of-banach-spaces-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Banach_spaces-0.md
category: reference
title: List of Banach spaces
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Banach_spaces
date_saved: 2026-05-05T08:12:55.569651+00:00
instance: kb-cron
parent: list-of-banach-spaces-20260505-0
summary:
In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well. == Classical Banach spaces == According to Diestel (1984, Chapter VII), the classical Banach spaces are those defined by Dunford & Schwartz (1958), which is the source for the following table. Glossary of symbols for the table below: F {\displaystyle \mathbb {F} } denotes the field of real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} K {\displaystyle K} is a compact Hausdorff space. p , q ∈ R {\displaystyle p,q\in \mathbb {R} } are real numbers with 1 < p , q < ∞ {\displaystyle 1<p,q<\infty } that are Hölder conjugates, meaning that they satisfy 1 q + 1 p = 1 {\displaystyle {\frac {1}{q}}+{\frac {1}{p}}=1} and thus also q = p p − 1 . {\displaystyle q={\frac {p}{p-1}}.} Σ {\displaystyle \Sigma } is a σ {\displaystyle \sigma } -algebra of sets. Ξ {\displaystyle \Xi } is an algebra of sets (for spaces only requiring finite additivity, such as the ba space). μ {\displaystyle \mu


id: list-of-works-by-nikolay-bogolyubov-20260505-0
file: data/en.wikipedia.org/wiki/List_of_works_by_Nikolay_Bogolyubov-0.md
category: reference
title: List of works by Nikolay Bogolyubov
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nikolay_Bogolyubov
date_saved: 2026-05-05T08:12:56.878536+00:00
instance: kb-cron
parent: list-of-works-by-nikolay-bogolyubov-20260505-0
summary:
List of some published works of Nikolay Bogolyubov in chronological order: == 1924 == N. N. Bogolyubov (1924). On the behavior of solutions of linear differential equations at infinity (Russian: О поведении решений линейных дифференциальных уравнений на бесконечности). == 1934 == N. M. Krylov and N. N. Bogolyubov (1934). On various formal expansions of non-linear mechanics (in Ukrainian). Kiev: Izdat. Zagal'noukr. Akad. Nauk. == 1937 == N. N. Bogoliubov and N. M. Krylov (1937). "La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire" (in French). Ann. Math. II 38: 65–113. Zbl. 16.86. == 1945 == Боголюбов Н. Н. (1945). О некоторых статистических методах в математической физике (in Russian). Киев: Изд-во АН УССР. N. N. Bogoliubov (1945). On Some Statistical Methods in Mathematical Physics (in Russian). Kyiv: Izdatelstvo AN USSR. == 1946 == N. N. Bogoliubov (1946). "Kinetic Equations" (in Russian). Journal of Experimental and Theoretical Physics 16 (8): 691–702. N. N. Bogoliubov (1946). "Kinetic Equations" (in English). Journal of Physics 10 (3): 265–274. == 1947 == N. N. Bogoliubov, K. P. Gurov (1947). "Kinetic Equations in Quantum Mechanics" (in Russian). Journal of Experimental and Theoretical Physics 17 (7): 614–628. N. N.


id: list-of-boolean-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Boolean_algebra_topics-0.md
category: reference
title: List of Boolean algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Boolean_algebra_topics
date_saved: 2026-05-05T08:12:58.151075+00:00
instance: kb-cron
parent: list-of-boolean-algebra-topics-20260505-0
summary:
This is a list of topics around Boolean algebra and propositional logic. == Articles with a wide scope and introductions == Algebra of sets Boolean algebra (structure) Boolean algebra Field of sets Logical connective Propositional calculus == Boolean functions and connectives == Ampheck Analysis of Boolean functions Balanced Boolean function Bent function Boolean algebras canonically defined Boolean function Boolean matrix Boolean-valued function Conditioned disjunction Evasive Boolean function Exclusive or Functional completeness Logical biconditional Logical conjunction Logical disjunction Logical equality Logical implication Logical negation Logical NOR Majority function Material conditional Minimal axioms for Boolean algebra Peirce arrow Read-once function Sheffer stroke Sole sufficient operator Symmetric Boolean function Symmetric difference Zhegalkin polynomial == Examples of Boolean algebras == Boolean domain Complete Boolean algebra Interior algebra Two-element Boolean algebra == Extensions of Boolean algebras == Derivative algebra (abstract algebra) Free Boolean algebra Monadic Boolean algebra == Generalizations of Boolean algebras == De Morgan algebra First-order logic Heyting algebra Lindenbaum–Tarski algebra Skew Boolean algebra == Syntax == Algebraic normal form Boolean conjunctive query Canonical form (Boolean algebra) Conjunctive normal form Disjunctive normal form Formal system == Technical applications == And-inverter graph Logic gate Boolean analysis == Theorems and specific laws == Boolean prime ideal


id: list-of-calculus-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_calculus_topics-0.md
category: reference
title: List of calculus topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_calculus_topics
date_saved: 2026-05-05T08:12:59.314130+00:00
instance: kb-cron
parent: list-of-calculus-topics-20260505-0
summary:
This is a list of calculus topics. == Limits == Limit (mathematics) Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function == Differential calculus == Derivative Notation Newton's notation for differentiation Leibniz's notation for differentiation Simplest rules Derivative of a constant Sum rule in differentiation Constant factor rule in differentiation Linearity of differentiation Power rule Chain rule Local linearization Product rule Quotient rule Inverse functions and differentiation Implicit differentiation Stationary point Maxima and minima First derivative test Second derivative test Extreme value theorem Differential equation Differential operator Newton's method Taylor's theorem L'Hôpital's rule General Leibniz rule Mean value theorem Logarithmic derivative Differential (calculus) Related rates Regiomontanus' angle maximization problem Rolle's theorem == Integral calculus == Antiderivative/Indefinite integral Simplest rules Sum rule in integration Constant factor rule in integration Linearity of integration Arbitrary constant of integration Cavalieri's quadrature formula Fundamental theorem of calculus Integration by parts Inverse chain rule method Integration by substitution Tangent half-angle substitution Differentiation under the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the secant function Integral of secant cubed Arclength Solid of revolution


id: outline-of-calculus-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_calculus-0.md
category: reference
title: Outline of calculus
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_calculus
date_saved: 2026-05-05T08:13:00.609028+00:00
instance: kb-cron
parent: outline-of-calculus-20260505-0
summary:
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of contemporary mathematics education. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. == Branches of calculus == Differential calculus Integral calculus Multivariable calculus Fractional calculus Differential Geometry == History of calculus == History of calculus Important publications in calculus == General calculus concepts == Continuous function Derivative Fundamental theorem of calculus Integral Limit Non-standard analysis Partial derivative Infinite Series == Calculus scholars == Sir Isaac Newton Gottfried Leibniz Leonhard Euler == Calculus lists == List of calculus topics == See also == Glossary of calculus Table of mathematical symbols == References == == External links == Weisstein, Eric W. "Calculus". MathWorld. Topics on Calculus at PlanetMath. Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF Calculus.org: The Calculus page at University of California, Davis – contains resources and links to other sites COW: Calculus on the Web Archived 2007-10-14 at the Wayback Machine at Temple University - contains resources ranging from pre-calculus and associated algebra Online Integrator (WebMathematica) from Wolfram Research The Role of


id: outline-of-category-theory-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_category_theory-0.md
category: reference
title: Outline of category theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_category_theory
date_saved: 2026-05-05T08:13:01.794317+00:00
instance: kb-cron
parent: outline-of-category-theory-20260505-0
summary:
The following outline is provided as an overview of and guide to category theory: Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics. == Essence of category theory == Category Functor Natural transformation == Branches of category theory == Homological algebra Diagram chasing Topos theory Enriched category theory Higher category theory Categorical logic Applied category theory == Specific categories == Category of sets Concrete category Category of small categories Category of vector spaces Category of graded vector spaces Category of chain complexes Category of finite dimensional Hilbert spaces Category of sets and relations Category of topological spaces Category of metric spaces Category of preordered sets Category of groups Category of abelian groups Category of rings


id: list-of-chaotic-maps-20260505-0
file: data/en.wikipedia.org/wiki/List_of_chaotic_maps-0.md
category: reference
title: List of chaotic maps
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_chaotic_maps
date_saved: 2026-05-05T08:13:03.039151+00:00
instance: kb-cron
parent: list-of-chaotic-maps-20260505-0
summary:
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems). == List of chaotic maps == == List of fractals == Cantor set de Rham curve Gravity set, or Mitchell-Green gravity set Julia set - derived from complex quadratic map Koch snowflake - special case of de Rham curve Lyapunov fractal Mandelbrot set - derived from complex quadratic map Menger sponge Newton fractal Nova fractal - derived from Newton fractal Quaternionic fractal - three dimensional complex quadratic map Sierpinski carpet Sierpinski triangle == References ==


id: list-of-mathematical-topics-in-classical-mechanics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_classical_mechanics-0.md
category: reference
title: List of mathematical topics in classical mechanics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_classical_mechanics
date_saved: 2026-05-05T08:13:04.549140+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-classical-mechanics-20260505-0
summary:
This is a list of mathematical topics in classical mechanics, by Wikipedia page. See also list of variational topics, correspondence principle. == Newtonian physics == Newton's laws of motion Inertia, Kinematics, rigid body Momentum, kinetic energy Parallelogram of force Circular motion Rotational speed Angular speed Angular momentum torque angular acceleration moment of inertia parallel axes rule perpendicular axes rule stretch rule centripetal force, centrifugal force, Reactive centrifugal force Laplace–Runge–Lenz vector Euler's disk elastic potential energy Mechanical equilibrium D'Alembert's principle Degrees of freedom (physics and chemistry) Frame of reference Inertial frame of reference Galilean transformation Principle of relativity == Conservation laws == Conservation of momentum Conservation of linear momentum Conservation of angular momentum Conservation of energy Potential energy Conservative force Conservation of mass == Law of universal gravitation == Projectile motion Kepler's laws of planetary motion Escape velocity Potential well Weightlessness Lagrangian point N-body problem Kolmogorov–Arnold–Moser theorem Virial theorem Gravitational binding energy Speed of gravity Newtonian limit Hill sphere Roche lobe Roche limit == Hamiltonian mechanics == Phase space Symplectic manifold Liouville's theorem (Hamiltonian) Poisson bracket Poisson algebra Poisson manifold Antibracket algebra Hamiltonian constraint Moment map Contact geometry Analysis of flows Nambu mechanics == Lagrangian mechanics == Action (physics) Lagrangian Euler–Lagrange


id: list-of-textbooks-on-classical-mechanics-and-quantum-mechanics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics-0.md
category: reference
title: List of textbooks on classical mechanics and quantum mechanics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_textbooks_on_classical_mechanics_and_quantum_mechanics
date_saved: 2026-05-05T08:13:06.136598+00:00
instance: kb-cron
parent: list-of-textbooks-on-classical-mechanics-and-quantum-mechanics-20260505-0
summary:
This is a list of notable textbooks on classical mechanics and quantum mechanics arranged according to level and surnames of the authors in alphabetical order. == Undergraduate == === Classical mechanics === Feynman, Richard P. (2005). The Feynman Lectures on Physics. Vol. 1 (2nd ed.). Addison-Wesley. ISBN 978-0-8053-9065-0. Halliday, David; Resnick, Robert (1970). Fundamentals of Physics. John Wiley & Sons. Chapters 1–21. Numerous subsequent editions. Hamill, Patrick (2014). A Student's Guide to Lagrangians and Hamiltonians. Cambridge University Press. ISBN 978-1107617520. Hand, Louis; Finch, Janet (1998). Analytical Mechanics. Cambridge University Press. ISBN 0521573270. Kibble, T. W.; Berkshire, F. H. (2004). Classical Mechanics. Imperial College Press. ISBN 1860944248. Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics. McGraw-Hill. ISBN 0-07-035048-5. Marion, Jerry; Thornton, Stephen (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole. ISBN 0534408966. Morin, David (2005). Introduction to Classical Mechanics: With Problems and Solutions. Cambridge University Press. ISBN 9780521876223. Müller-Kirsten, Harald J.W. (2024). Classical Mechanics and Relativity (2nd ed.). World Scientific. ISBN 9789811287114. Taylor, John (2005). Classical Mechanics. University Science Books. ISBN 978-981-12-8711-4. Young, Hugh D.; Freedman, Roger A. (2019). University Physics with Modern Physics (15th ed.). Pearson. ISBN 978-0135159552. === Quantum mechanics === Eisberg, Robert; Resnick, Robert


id: classification-of-low-dimensional-real-lie-algebras-20260505-0
file: data/en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras-0.md
category: reference
title: Classification of low-dimensional real Lie algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
date_saved: 2026-05-05T08:13:07.603621+00:00
instance: kb-cron
parent: classification-of-low-dimensional-real-lie-algebras-20260505-0
summary:
This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra. An English version and review of this classification was published by Popovych et al. in 2003. == Mubarakzyanov's Classification == Let g n {\displaystyle {\mathfrak {g}}_{n}} be n {\displaystyle n} -dimensional Lie algebra over the field of real numbers with generators e 1 , … , e n {\displaystyle e_{1},\dots ,e_{n}} , n ≤ 4 {\displaystyle n\leq 4} . For each algebra g {\displaystyle {\mathfrak {g}}} we adduce only non-zero commutators between basis elements. === One-dimensional === g 1 {\displaystyle {\mathfrak {g}}_{1}} , abelian. === Two-dimensional === 2 g 1 {\displaystyle 2{\mathfrak {g}}_{1}} , abelian R 2 {\displaystyle \mathbb {R} ^{2}} ; g 2.1 {\displaystyle {\mathfrak {g}}_{2.1}} , solvable a f f ( 1 ) = { ( a b 0 0 ) : a , b ∈ R } {\displaystyle {\mathfrak {aff}}(1)=\left\{{\begin{pmatrix}a&b\\0&0\end{pmatrix}}\,:\,a,b\in \mathbb {R} \right\}} , [ e 1 , e 2 ] = e 1 . {\displaystyle [e_{1},e_{2}]=e_{1}.} === Three-dimensional === 3 g 1 {\displaystyle 3{\mathfrak {g}}_{1}} , abelian, Bianchi I; g 2.1 ⊕ g 1 {\displaystyle {\mathfrak {g}}_{2.1}\oplus


id: classification-of-low-dimensional-real-lie-algebras-20260505-1
file: data/en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras-1.md
category: reference
title: Classification of low-dimensional real Lie algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
date_saved: 2026-05-05T08:13:07.603621+00:00
instance: kb-cron
parent: classification-of-low-dimensional-real-lie-algebras-20260505-0
summary:
g 4.1 {\displaystyle {\mathfrak {g}}_{4.1}} , indecomposable nilpotent, [ e 2 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ; {\displaystyle [e_{2},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};} g 4.2 {\displaystyle {\mathfrak {g}}_{4.2}} , indecomposable solvable, [ e 1 , e 4 ] = β e 1 , [ e 2 , e 4 ] = e 2 , [ e 3 , e 4 ] = e 2 + e 3 , β ≠ 0 ; {\displaystyle [e_{1},e_{4}]=\beta e_{1},\quad [e_{2},e_{4}]=e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3},\quad \beta \neq 0;} g 4.3 {\displaystyle {\mathfrak {g}}_{4.3}} , indecomposable solvable, [ e 1 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ; {\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{3},e_{4}]=e_{2};} g 4.4 {\displaystyle {\mathfrak {g}}_{4.4}} , indecomposable solvable, [ e 1 , e 4 ] = e 1 , [ e 2 , e 4 ] = e 1 + e 2 , [ e 3 , e 4 ] = e 2 + e 3 ; {\displaystyle [e_{1},e_{4}]=e_{1},\quad [e_{2},e_{4}]=e_{1}+e_{2},\quad [e_{3},e_{4}]=e_{2}+e_{3};} g 4.5 {\displaystyle {\mathfrak {g}}_{4.5}} , indecomposable solvable, [ e 1 , e 4 ] = α e 1 , [ e 2 , e 4


id: list-of-cohomology-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_cohomology_theories-0.md
category: reference
title: List of cohomology theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cohomology_theories
date_saved: 2026-05-05T08:13:25.127193+00:00
instance: kb-cron
parent: list-of-cohomology-theories-20260505-0
summary:
This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at the end of this article. == Notation == S = π = S 0 {\displaystyle S=\pi =S^{0}} is the sphere spectrum. S n {\displaystyle S^{n}} is the spectrum of the n {\displaystyle n} -dimensional sphere S n Y = S n ∧ Y {\displaystyle S^{n}Y=S^{n}\land Y} is the n {\displaystyle n} th suspension of a spectrum Y {\displaystyle Y} . [ X , Y ] {\displaystyle [X,Y]} is the abelian group of morphisms from the spectrum X {\displaystyle X} to the spectrum Y {\displaystyle Y} , given (roughly) as homotopy classes of maps. [ X , Y ] n = [ S n X , Y ] {\displaystyle [X,Y]_{n}=[S^{n}X,Y]} [ X , Y ] ∗ {\displaystyle [X,Y]_{*}} is the graded abelian group given as the sum of the groups [ X , Y ] n {\displaystyle [X,Y]_{n}} . π n ( X ) = [ S n , X ] = [ S , X ] n {\displaystyle \pi


id: list-of-cohomology-theories-20260505-1
file: data/en.wikipedia.org/wiki/List_of_cohomology_theories-1.md
category: reference
title: List of cohomology theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cohomology_theories
date_saved: 2026-05-05T08:13:25.127193+00:00
instance: kb-cron
parent: list-of-cohomology-theories-20260505-0
summary:
=== K theory with coefficients === Spectrum: KG G is some abelian group; for example the localization Z(p) at the prime p. Other K-theories can also be given coefficients. === Self conjugate K-theory === Spectrum: KSC Coefficient ring: to be written... The coefficient groups π i {\displaystyle \pi _{i}} (KSC) have period 4 in i, given by the sequence Z, Z2, 0, Z, repeated. Introduced by Donald W. Anderson in his unpublished 1964 University of California, Berkeley Ph.D. dissertation, "A new cohomology theory". === Connective K-theories === Spectrum: ku for connective K-theory, ko for connective real K-theory. Coefficient ring: For ku, the coefficient ring is the ring of polynomials over Z on a single class v1 in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 in dimension 8, the periodicity generator, modulo the relations that 2η = 0, x42 = 4v14, η3 = 0, and ηx = 0. Roughly speaking, this is K-theory with the negative dimensional parts killed off. === KR-theory === This is a cohomology theory defined for spaces with involution, from which many of the other K-theories can


id: list-of-combinatorial-computational-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_combinatorial_computational_geometry_topics-0.md
category: reference
title: List of combinatorial computational geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_combinatorial_computational_geometry_topics
date_saved: 2026-05-05T08:13:26.374691+00:00
instance: kb-cron
parent: list-of-combinatorial-computational-geometry-topics-20260505-0
summary:
List of combinatorial computational geometry topics enumerates the topics 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. See List of numerical computational geometry topics for another flavor of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. == Construction/representation == Boolean operations on polygons Convex hull Hyperplane arrangement Polygon decomposition Polygon triangulation Minimal convex decomposition Minimal convex cover problem (NP-hard) Minimal rectangular decomposition Tessellation problems Shape dissection problems Straight skeleton Stabbing line problem Triangulation Delaunay triangulation Point-set triangulation Polygon triangulation Voronoi diagram == Extremal shapes == Minimum bounding box (Smallest enclosing box, Smallest bounding box) 2-D case: Smallest bounding rectangle (Smallest enclosing rectangle) There are two common variants of this problem. In many areas of computer graphics, the bounding box (often abbreviated to bbox) is understood to be the smallest box delimited by sides parallel to coordinate axes which encloses the objects in question. In other applications, such as packaging, the problem is to find the smallest box the object (or objects) may fit in ("packaged"). Here


id: index-of-combinatorics-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_combinatorics_articles-0.md
category: reference
title: Index of combinatorics articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_combinatorics_articles
date_saved: 2026-05-05T08:13:27.553086+00:00
instance: kb-cron
parent: index-of-combinatorics-articles-20260505-0
summary:
== A == Abstract simplicial complex Addition chain Scholz conjecture Algebraic combinatorics Alternating sign matrix Almost disjoint sets Antichain Arrangement of hyperplanes Assignment problem Quadratic assignment problem Audioactive decay == B == Barcode Matrix code QR Code Universal Product Code Bell polynomials Bertrand's ballot theorem Binary matrix Binomial theorem Block design Balanced incomplete block design(BIBD) Symmetric balanced incomplete block design (SBIBD) Partially balanced incomplete block designs (PBIBDs) Block walking Boolean satisfiability problem 2-satisfiability 3-satisfiability Bracelet (combinatorics) Bruck–Chowla–Ryser theorem == C == Catalan number Cellular automaton Collatz conjecture Combination Combinatorial design Combinatorial number system Combinatorial optimization Combinatorial search Constraint satisfaction problem Conway's Game of Life Cycles and fixed points Cyclic order Cyclic permutation Cyclotomic identity == D == Data integrity Alternating bit protocol Checksum Cyclic redundancy check Luhn formula Error detection Error-detecting code Error-detecting system Message digest Redundancy check Summation check De Bruijn sequence Deadlock Delannoy number Dining philosophers problem Mutual exclusion Rendezvous problem Derangement Dickson's lemma Dinitz conjecture Discrete optimization Dobinski's formula == E == Eight queens puzzle Entropy coding Enumeration Algebraic enumeration Combinatorial enumeration Burnside's lemma Erdős–Ko–Rado theorem Euler number == F == Faà di Bruno's formula Factorial number system Family of sets Faulhaber's formula Fifteen puzzle Finite geometry


id: outline-of-combinatorics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_combinatorics-0.md
category: reference
title: Outline of combinatorics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_combinatorics
date_saved: 2026-05-05T08:13:28.713632+00:00
instance: kb-cron
parent: outline-of-combinatorics-20260505-0
summary:
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. == Essence of combinatorics == Matroid Greedoid Ramsey theory Van der Waerden's theorem Hales–Jewett theorem Umbral calculus, binomial type polynomial sequences Combinatorial species == Branches of combinatorics == Algebraic combinatorics Analytic combinatorics Arithmetic combinatorics Combinatorics on words Combinatorial design theory Enumerative combinatorics Extremal combinatorics Geometric combinatorics Graph theory Infinitary combinatorics Matroid theory Order theory Partition theory Probabilistic combinatorics Topological combinatorics === Multi-disciplinary fields that include combinatorics === Coding theory Combinatorial optimization Combinatorics and dynamical systems Combinatorics and physics Discrete geometry Finite geometry Phylogenetics == History of combinatorics == History of combinatorics == General combinatorial principles and methods == Combinatorial principles Trial and error, brute-force search, bogosort, British Museum algorithm Pigeonhole principle Method of distinguished element Mathematical induction Recurrence relation, telescoping series Generating functions as an application of formal power series Cyclic sieving Schrödinger method Exponential generating function Stanley's reciprocity theorem Binomial coefficients and their properties Combinatorial proof Double counting (proof technique) Bijective proof Inclusion–exclusion principle Möbius inversion formula Parity, even and odd permutations Combinatorial Nullstellensatz Incidence algebra Greedy algorithm Divide and conquer algorithm Akra–Bazzi method Dynamic programming Branch and bound Birthday attack, birthday paradox Floyd's


id: list-of-commutative-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_commutative_algebra_topics-0.md
category: reference
title: List of commutative algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_commutative_algebra_topics
date_saved: 2026-05-05T08:13:29.965551+00:00
instance: kb-cron
parent: list-of-commutative-algebra-topics-20260505-0
summary:
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. == Research fields == Combinatorial commutative algebra Invariant theory === Active research areas === Serre's multiplicity conjectures Homological conjectures == Basic notions == Commutative ring Module (mathematics) Ring ideal, maximal ideal, prime ideal Ring homomorphism Ring monomorphism Ring epimorphism Ring isomorphism Zero divisor Chinese remainder theorem == Classes of rings == Field (mathematics) Algebraic number field Polynomial ring Integral domain Boolean algebra (structure) Principal ideal domain Euclidean domain Unique factorization domain Dedekind domain Nilpotent elements and reduced rings Dual numbers Tensor product of fields Tensor product of R-algebras == Constructions with commutative rings == Quotient ring Field of fractions Product of rings Annihilator (ring theory) Integral closure == Localization and completion == Completion (ring theory) Formal power


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-0.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations. Vector algebra uses all dimensions and signatures, as does geometric algebra, notably 3+1 spacetime as well as 2 dimensions. == Basic concepts and operations == Geometric algebra (GA) is an extension or completion of vector algebra (VA). The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in G 3 {\displaystyle {\mathcal {G}}_{3}} the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally written boldface as the meaning is usually clear from the context. The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated I {\displaystyle I} . The ungeneralized 3D vector form of the geometric product is: a b = a ⋅ b + a ∧ b {\displaystyle ab=a\cdot b+a\wedge b} that is the sum


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-1
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-1.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
The pseudovector/bivector subalgebra of the geometric algebra of Euclidean 3-dimensional space form a 3-dimensional vector space themselves. Let the standard unit pseudovectors/bivectors of the subalgebra be i = e 2 e 3 {\displaystyle \mathbf {i} =\mathbf {e_{2}} \mathbf {e_{3}} } , j = e 1 e 3 {\displaystyle \mathbf {j} =\mathbf {e_{1}} \mathbf {e_{3}} } , and k = e 1 e 2 {\displaystyle \mathbf {k} =\mathbf {e_{1}} \mathbf {e_{2}} } , and the anti-commutative commutator product be defined as A × B = 1 2 ( A B − B A ) {\displaystyle A\times B={\tfrac {1}{2}}(AB-BA)} , where A B {\displaystyle AB} is the geometric product. The commutator product is distributive over addition and linear, as the geometric product is distributive over addition and linear. From the definition of the commutator product, i {\displaystyle \mathbf {i} } , j {\displaystyle \mathbf {j} } and k {\displaystyle \mathbf {k} } satisfy the following equalities: i × j = 1 2 ( i j − j i ) = 1 2 ( ( e 2 e 3 e 1 e 3 − e 1 e 3 e 2 e 3 ) = 1 2 ( − e 2 e 3 e


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-2
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-2.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
A × B = ( A 1 i + A 2 j + A 3 k ) × ( B 1 i + B 2 j + B 3 k ) = A 1 B 1 i × i + A 1 B 2 i × j + A 1 B 3 i × k + A 2 B 1 j × i + A 2 B 2 j × j + A 2 B 3 j × k + A 3 B 1 k × i + A 3 B 2 k × j + A 3 B 3 k × k = A 1 B 2 k − A 1 B 3 j − A 2 B 1 k + A 2 B 3 i + A 3 B 1 j − A 3 B 2 i = ( A 2 B 3 − A 3 B 2 ) i + ( A 3 B 1 − A 1 B 3 ) j + ( A 1 B 2 − A 2 B 1 ) k {\displaystyle {\begin{aligned}\mathbf {A} \times \mathbf {B} &=(A_{1}\mathbf {i} +A_{2}\mathbf {j} +A_{3}\mathbf {k} )\times (B_{1}\mathbf {i} +B_{2}\mathbf {j} +B_{3}\mathbf {k} )\\&=A_{1}B_{1}\mathbf {i} \times


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-3
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-3.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
This is a useful theoretic result. For numerical problems row reduction with pivots and other methods are more stable and efficient. When the wedge product is coupled with the Clifford product and put into a natural geometric context, the fact that the determinants are used in the expression of R N {\displaystyle {\mathbb {R} }^{N}} parallelogram area and parallelepiped volumes (and higher-dimensional generalizations thereof) also comes as a nice side-effect. As is also shown below, results such as Cramer's rule also follow directly from the wedge product's selection of non-identical elements. The result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule. Two variables example [ a b ] [ x y ] = a x + b y = c . {\displaystyle {\begin{bmatrix}a&b\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=ax+by=c.} Pre- and post-multiplying by a {\displaystyle a} and b {\displaystyle b} , ( a x + b y ) ∧ b = ( a ∧ b ) x = c ∧ b {\displaystyle (ax+by)\wedge b=(a\wedge b)x=c\wedge b} a ∧ ( a x + b y ) = ( a ∧ b ) y = a ∧ c {\displaystyle a\wedge (ax+by)=(a\wedge b)y=a\wedge c} Provided


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-4
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-4.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
v − u ^ ( u ^ ⋅ v ) = 1 ‖ u ‖ 2 ( ‖ u ‖ 2 v − u ( u ⋅ v ) ) {\displaystyle v-{\hat {u}}({\hat {u}}\cdot v)={\frac {1}{{\Vert u\Vert }^{2}}}({\Vert u\Vert }^{2}v-u(u\cdot v))} The rejection can be expressed as a single geometric algebraic product in a few different ways u u 2 ( u v − u ⋅ v ) = 1 u ( u ∧ v ) = u ^ ( u ^ ∧ v ) = ( v ∧ u ^ ) u ^ {\displaystyle {\frac {u}{{u}^{2}}}(uv-u\cdot v)={\frac {1}{u}}(u\wedge v)={\hat {u}}({\hat {u}}\wedge v)=(v\wedge {\hat {u}}){\hat {u}}} The similarity in form between the projection and the rejection is notable. The sum of these recovers the original vector v = u ^ ( u ^ ⋅ v ) + u ^ ( u ^ ∧ v ) {\displaystyle v={\hat {u}}({\hat {u}}\cdot v)+{\hat {u}}({\hat {u}}\wedge v)} Here the projection is in its customary vector form. An alternate formulation is possible that puts the projection in a form that differs from the usual vector formulation v = 1 u ( u ⋅ v ) + 1 u ( u ∧ v ) = (


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-5
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-5.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
Notice the similarities between this planar rejection result and the vector rejection result. To calculate the component of a vector outside of a plane we take the volume spanned by three vectors (trivector) and "divide out" the plane. Independent of any use of the geometric product it can be shown that this rejection in terms of the standard basis is x = 1 ( A u , v ) 2 ∑ i < j < k | w i w j w k u i u j u k v i v j v k | | u i u j u k v i v j v k e i e j e k | {\displaystyle x={\frac {1}{(A_{u,v})^{2}}}\sum _{i<j<k}{\begin{vmatrix}w_{i}&w_{j}&w_{k}\\u_{i}&u_{j}&u_{k}\\v_{i}&v_{j}&v_{k}\\\end{vmatrix}}{\begin{vmatrix}u_{i}&u_{j}&u_{k}\\v_{i}&v_{j}&v_{k}\\{e}_{i}&{e}_{j}&{e}_{k}\\\end{vmatrix}}} where ( A u , v ) 2 = ∑ i < j | u i u j v i v j | = − ( u ∧ v ) 2 {\displaystyle (A_{u,v})^{2}=\sum _{i<j}{\begin{vmatrix}u_{i}&u_{j}\\v_{i}&v_{j}\end{vmatrix}}=-(u\wedge v)^{2}} is the squared area of the parallelogram formed by u {\displaystyle u} , and v {\displaystyle v} . The (squared) magnitude of x {\displaystyle x} is ‖ x ‖ 2 = x ⋅ w = 1 ( A u , v ) 2 ∑ i


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-6
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-6.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
A 2 = − ( u ∧ v ) 2 = ∑ i < j | u i u j v i v j | 2 . {\displaystyle A^{2}=-(\mathbf {u} \wedge \mathbf {v} )^{2}=\sum _{i<j}{\begin{vmatrix}u_{i}&u_{j}\\v_{i}&v_{j}\end{vmatrix}}^{2}.} Note that this squared bivector is a geometric multiplication; this computation can alternatively be stated as the Gram determinant of the two vectors. === Angle between two vectors === ( sin ⁡ θ ) 2 = ‖ u × v ‖ 2 ‖ u ‖ 2 ‖ v ‖ 2 {\displaystyle ({\sin \theta })^{2}={\frac {{\Vert \mathbf {u} \times \mathbf {v} \Vert }^{2}}{{\Vert \mathbf {u} \Vert }^{2}{\Vert \mathbf {v} \Vert }^{2}}}} ( sin ⁡ θ ) 2 = − ( u ∧ v ) 2 u 2 v 2 {\displaystyle ({\sin \theta })^{2}=-{\frac {(\mathbf {u} \wedge \mathbf {v} )^{2}}{{\mathbf {u} }^{2}{\mathbf {v} }^{2}}}} === Volume of the parallelopiped formed by three vectors === In vector algebra, the volume of a parallelopiped is given by the square root of the squared norm of the scalar triple product: V 2 = ‖ ( u × v ) ⋅ w ‖ 2 = | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-7
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-7.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
d d t ( r ‖ r ‖ ) = 1 ‖ r ‖ 3 ( r × d r d t ) × r = ( r ^ × 1 ‖ r ‖ d r d t ) × r ^ {\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {r} }{\Vert \mathbf {r} \Vert }}\right)={\frac {1}{{\Vert \mathbf {r} \Vert }^{3}}}\left(\mathbf {r} \times {\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {r} =\left({\hat {\mathbf {r} }}\times {\frac {1}{\Vert \mathbf {r} \Vert }}{\frac {d\mathbf {r} }{dt}}\right)\times {\hat {\mathbf {r} }}} The equivalent geometric product generalization is d d t ( r ‖ r ‖ ) = 1 ‖ r ‖ 3 r ( r ∧ d r d t ) = 1 r ( r ^ ∧ d r d t ) {\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {r} }{\Vert \mathbf {r} \Vert }}\right)={\frac {1}{{\Vert \mathbf {r} \Vert }^{3}}}\mathbf {r} \left(\mathbf {r} \wedge {\frac {d\mathbf {r} }{dt}}\right)={\frac {1}{\mathbf {r} }}\left({\hat {\mathbf {r} }}\wedge {\frac {d\mathbf {r} }{dt}}\right)} Thus this derivative is the component of 1 ‖ r ‖ d r d t {\displaystyle {\frac {1}{\Vert \mathbf {r} \Vert }}{\frac {d\mathbf {r} }{dt}}} in the direction perpendicular to r {\displaystyle \mathbf {r} } . In other words, this is 1 ‖


id: list-of-mathematics-competitions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_competitions-0.md
category: reference
title: List of mathematics competitions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_competitions
date_saved: 2026-05-05T08:13:32.568377+00:00
instance: kb-cron
parent: list-of-mathematics-competitions-20260505-0
summary:
Mathematics competitions or mathematical olympiads are competitive events where participants complete a math test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof. == International mathematics competitions == World Mathematics Invitational — for all grade levels K-12, Taiwanese based international mathematics competition, with English and Chinese as the official languages of the papers. Championnat International de Jeux Mathématiques et Logiques — for all ages, mainly for French-speaking countries, but participation is not limited by language. China Girls Mathematical Olympiad (CGMO) — held annually for teams of girls representing different regions within China and a few other countries. European Girls' Mathematical Olympiad (EGMO) — since April 2012 Integration Bee — competition in integral calculus held in various institutions of higher learning in the United States and some other countries International Mathematical Modeling Challenge — team contest for high school students International Mathematical Olympiad (IMO) — the oldest international Olympiad, occurring annually since 1959. International Mathematics Competition for University Students (IMC) — international competition for undergraduate students. Mathematical Contest in Modeling (MCM) — team contest for undergraduates Mathematical Kangaroo — worldwide competition. Mental Calculation World Cup — contest for the best mental calculators Primary Mathematics


id: list-of-complex-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_complex_analysis_topics-0.md
category: reference
title: List of complex analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_complex_analysis_topics
date_saved: 2026-05-05T08:13:33.858629+00:00
instance: kb-cron
parent: list-of-complex-analysis-topics-20260505-0
summary:
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering. See also: glossary of real and complex analysis. == Overview == Complex numbers Complex plane Complex functions Complex derivative Holomorphic functions Harmonic functions Elementary functions Polynomial functions Exponential functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions Residue theory Isometries in the complex plane == Related fields == Number theory Hydrodynamics Thermodynamics Electrical engineering == Local theory == Holomorphic function Antiholomorphic function Cauchy–Riemann equations Conformal mapping Conformal welding Power series Radius of convergence Laurent series Meromorphic function Entire function Pole (complex analysis) Zero (complex analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati theorem Landau's constants Holomorphic functions are analytic Schwarzian derivative Analytic capacity Disk algebra Univalent function == Growth and distribution of values == Ahlfors theory Bieberbach conjecture Borel–Carathéodory theorem Corona theorem Hadamard three-circle theorem Hardy space Hardy's theorem Maximum modulus principle Nevanlinna theory Paley–Wiener theorem Phragmén-Lindelöf principle Progressive function


id: list-of-complexity-classes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_complexity_classes-0.md
category: reference
title: List of complexity classes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_complexity_classes
date_saved: 2026-05-05T08:13:35.107427+00:00
instance: kb-cron
parent: list-of-complexity-classes-20260505-0
summary:
This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics. Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.) "The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it. == References == == External links == Complexity Zoo - list of over 500 complexity classes and their properties


id: list-of-computability-and-complexity-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computability_and_complexity_topics-0.md
category: reference
title: List of computability and complexity topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computability_and_complexity_topics
date_saved: 2026-05-05T08:13:36.334916+00:00
instance: kb-cron
parent: list-of-computability-and-complexity-topics-20260505-0
summary:
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast). == Calculation == Lookup table Mathematical table Multiplication table Generating trigonometric tables History of computers Multiplication algorithm Peasant multiplication Division by two Exponentiating by squaring Addition chain Scholz conjecture Presburger arithmetic == Computability theory: models of computation == Arithmetic circuits Algorithm Procedure, recursion Finite-state automaton Mealy machine Minsky register machine Moore machine State diagram State transition system Deterministic finite automaton Nondeterministic finite automaton Generalized nondeterministic finite automaton Regular language Pumping lemma Myhill–Nerode theorem Regular expression Regular grammar Prefix grammar Tree automaton Pushdown automaton Context-free grammar Büchi automaton Chomsky hierarchy Context-sensitive language, context-sensitive grammar Recursively enumerable language Register machine Stack machine Petri net Post machine Rewriting Markov algorithm Term rewriting String rewriting system L-system Knuth–Bendix completion algorithm Star height Star height problem Generalized


id: computational-complexity-of-mathematical-operations-20260505-0
file: data/en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations-0.md
category: reference
title: Computational complexity of mathematical operations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations
date_saved: 2026-05-05T08:13:37.584907+00:00
instance: kb-cron
parent: computational-complexity-of-mathematical-operations-20260505-0
summary:
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm. == Arithmetic functions == This table lists the complexity of mathematical operations on integers. On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n-bit numbers in time O(n). == Algebraic functions == Here we consider operations over polynomials and n denotes their degree; for the coefficients we use a unit-cost model, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers. For this section M ( n ) {\displaystyle M(n)} indicates the time needed for multiplying two polynomials of degree at most n {\displaystyle n} . == Special functions == Many of the methods in this section are given in Borwein & Borwein. === Elementary functions === The elementary functions are constructed by


id: list-of-computer-algebra-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computer_algebra_systems-0.md
category: reference
title: List of computer algebra systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computer_algebra_systems
date_saved: 2026-05-05T08:13:38.747454+00:00
instance: kb-cron
parent: list-of-computer-algebra-systems-20260505-0
summary:
The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language. A CAS may include a user interface and graphics capability, and to be effective may require a large library of algorithms, efficient data structures, and a fast kernel. == General == These computer algebra systems are sometimes combined with "front end" programs that provide a better user interface, such as the general-purpose GNU TeXmacs. === Functionality === Below is a summary of significantly developed symbolic functionality in each of the systems. ^ via SymPy ^ via qepcad optional package Those which do not "edit equations" may have a GUI, plotting, ASCII graphic formulae and math font printing. The ability to generate plaintext files is also a sought-after feature because it allows a work to be understood by people who do not have a computer algebra system installed. === Operating system support === The software can run under their respective operating systems natively without emulation. Some systems must be compiled first using an appropriate compiler for the source language


id: list-of-computer-graphics-and-descriptive-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computer_graphics_and_descriptive_geometry_topics-0.md
category: reference
title: List of computer graphics and descriptive geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computer_graphics_and_descriptive_geometry_topics
date_saved: 2026-05-05T08:13:40.076467+00:00
instance: kb-cron
parent: list-of-computer-graphics-and-descriptive-geometry-topics-20260505-0
summary:
This is a list of computer graphics and descriptive geometry topics, by article name. == !-9 == 2D computer graphics 2D geometric model 3D computer graphics 3D modeling 3D projection 3D rendering == A == A-buffer Algorithmic art Aliasing Alpha compositing Alpha mapping Alpha to coverage Ambient occlusion Anamorphosis Anisotropic filtering Anti-aliasing Asymptotic decider Augmented reality Axis-aligned bounding box Axonometric projection == B == B-spline Back-face culling Barycentric coordinate system Beam tracing Bézier curve Bézier surface Bicubic interpolation Bidirectional reflectance distribution function Bidirectional scattering distribution function Bidirectional texture function Bilateral filter Bilinear interpolation Bin (computational geometry) Binary space partitioning Bit blit Bit plane Bitmap Bitmap textures Blend modes Blinn–Phong reflection model Bloom (shader effect) Bounding interval hierarchy Bounding sphere Bounding volume Bounding volume hierarchy Bresenham's line algorithm Bump mapping == C == Calligraphic projection Cel shading Channel (digital image) Checkerboard rendering Circular thresholding Clip coordinates Clipmap Clipping (computer graphics) Clipping path Collision detection Color depth Color gradient Color space Colour banding Color bleeding (computer graphics) Color cycling Composite Bézier curve Compositing Computational geometry Compute kernel Computer animation Computer art Computer graphics Computer graphics (computer science) Computer graphics lighting Computer-generated imagery Cone tracing Constructive solid geometry Control point (mathematics) Convex hull


id: list-of-conjectures-by-paul-erd-s-20260505-0
file: data/en.wikipedia.org/wiki/List_of_conjectures_by_Paul_Erdős-0.md
category: reference
title: List of conjectures by Paul Erdős
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_conjectures_by_Paul_Erdős
date_saved: 2026-05-05T08:13:41.334180+00:00
instance: kb-cron
parent: list-of-conjectures-by-paul-erd-s-20260505-0
summary:
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them. The unsolved problems are commonly known as Erdős problems. == Unsolved == The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3. The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers. The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus. The Erdős–Straus conjecture on the Diophantine equation 4/n = 1/x + 1/y + 1/z. The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals. The Erdős–Szekeres conjecture on the number of points needed to ensure that a point set contains a large convex polygon. The Erdős–Turán conjecture on additive bases of natural numbers. A conjecture on quickly growing integer sequences with rational reciprocal series. A conjecture with Norman Oler on circle packing in an equilateral triangle with a number of circles one less


id: list-of-conjectures-20260505-0
file: data/en.wikipedia.org/wiki/List_of_conjectures-0.md
category: reference
title: List of conjectures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_conjectures
date_saved: 2026-05-05T08:13:42.537850+00:00
instance: kb-cron
parent: list-of-conjectures-20260505-0
summary:
This is a list of notable mathematical conjectures. == Open problems == The following conjectures remain open. The (incomplete) column "cites" lists the number of results for a Google Scholar search for the term, in double quotes as of September 2022. == Conjectures now proved (theorems) == The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names. Deligne's conjecture on 1-motives == Disproved (no longer conjectures) == The conjectures in following list were not necessarily generally accepted as true before being disproved. Aharoni-Korman conjecture also known as the fishbone conjecture Atiyah conjecture (not a conjecture to start with) Borsuk's conjecture Chinese hypothesis (not a conjecture to start with) Doomsday conjecture Euler's sum of powers conjecture Ganea conjecture Generalized Smith conjecture Hauptvermutung Hedetniemi's conjecture, counterexample announced 2019 Hirsch conjecture (disproved 2010) Kaplansky unit conjecture Intersection graph conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies conjecture (disproved 1910) Tait's conjecture Von Neumann conjecture Weyl–Berry conjecture Williamson conjecture == Refutations of conventional wisdom == In contemporary mathematics, ideas are supposedly not accepted absent rigorous proof. Historically, some statements that were generally accepted were then shown to be false.


id: list-of-continuity-related-mathematical-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics-0.md
category: reference
title: List of continuity-related mathematical topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics
date_saved: 2026-05-05T08:13:43.792858+00:00
instance: kb-cron
parent: list-of-continuity-related-mathematical-topics-20260505-0
summary:
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. == Continuity of functions and measures == Continuous function Absolutely continuous function Absolute continuity of a measure with respect to another measure Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to the condition that every set whose Lebesgue measure is 0 has probability 0. Geometric continuity Parametric continuity Smoothness == Continuum == Continuum (set theory), the real line or the corresponding cardinal number Linear continuum, any ordered set that shares certain properties of the real line Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set. Cardinality of the continuum, a cardinal number that represents the


id: list-of-convexity-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_convexity_topics-0.md
category: reference
title: List of convexity topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_convexity_topics
date_saved: 2026-05-05T08:13:45.110385+00:00
instance: kb-cron
parent: list-of-convexity-topics-20260505-0
summary:
This is a list of convexity topics, by Wikipedia page. Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This is a convex combination of two colors allowing for transparency effects in computer graphics. Barycentric coordinates - a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of masses placed at its vertices. The coordinates are non-negative for points in the convex hull. Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger diameter. Solved by Hadwiger for the case of smooth convex bodies. Bond convexity - a measure of the non-linear relationship between price and yield duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. A basic form of convexity in finance. Carathéodory's theorem (convex hull) - If a point x of Rd lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its


id: list-of-coordinate-charts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_coordinate_charts-0.md
category: reference
title: List of coordinate charts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_coordinate_charts
date_saved: 2026-05-05T08:13:46.208272+00:00
instance: kb-cron
parent: list-of-coordinate-charts-20260505-0
summary:
This article contains a non-exhaustive list of coordinate charts for Riemannian manifolds and pseudo-Riemannian manifolds. Coordinate charts are mathematical objects of topological manifolds, and they have multiple applications in theoretical and applied mathematics. When a differentiable structure and a metric are defined, greater structure exists, and this allows the definition of constructs such as integration and geodesics. == Charts for Riemannian and pseudo-Riemannian surfaces == The following charts (with appropriate metric tensors) can be used in the stated classes of Riemannian and pseudo-Riemannian surfaces: == See also == Coordinate chart Coordinate system Metric tensor List of mathematics lists, particularly: List of multivariable calculus topics List of Fourier analysis topics List of differential geometry topics == References ==


id: list-of-common-coordinate-transformations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-0.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
This is a list of some of the most commonly used coordinate transformations. == 2-dimensional == Let ( x , y ) {\displaystyle (x,y)} be the standard Cartesian coordinates, and ( r , θ ) {\displaystyle (r,\theta )} the standard polar coordinates. === To Cartesian coordinates === ==== From polar coordinates ==== x = r cos ⁡ θ y = r sin ⁡ θ ∂ ( x , y ) ∂ ( r , θ ) = [ cos ⁡ θ − r sin ⁡ θ sin ⁡ θ − r cos ⁡ θ ] Jacobian = det ∂ ( x , y ) ∂ ( r , θ ) = r {\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\[5pt]{\frac {\partial (x,y)}{\partial (r,\theta )}}&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &{\phantom {-}}r\cos \theta \end{bmatrix}}\\[5pt]{\text{Jacobian}}=\det {\frac {\partial (x,y)}{\partial (r,\theta )}}&=r\end{aligned}}} ==== From log-polar coordinates ==== x = e ρ cos ⁡ θ , y = e ρ sin ⁡ θ . {\displaystyle {\begin{aligned}x&=e^{\rho }\cos \theta ,\\y&=e^{\rho }\sin \theta .\end{aligned}}} By using complex numbers ( x , y ) = x + i y ′ {\displaystyle (x,y)=x+iy'} , the transformation can be written as x + i y = e ρ + i θ {\displaystyle


id: list-of-common-coordinate-transformations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-1.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as illustrated). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent. If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged. All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation. === To Cartesian coordinates === ==== From spherical coordinates ==== x = ρ sin ⁡ θ cos ⁡ φ y = ρ sin ⁡ θ sin ⁡ φ z = ρ cos ⁡ θ ∂


id: list-of-common-coordinate-transformations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-2.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
∂ ( r , θ , h ) ∂ ( x , y , z ) = ( x x 2 + y 2 y x 2 + y 2 0 − y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) {\displaystyle {\frac {\partial (r,\theta ,h)}{\partial (x,y,z)}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}} ==== From spherical coordinates ==== r = ρ sin ⁡ φ h = ρ cos ⁡ φ θ = θ ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = ( sin ⁡ φ ρ cos ⁡ φ 0 cos ⁡ φ − ρ sin ⁡ φ 0 0 0 1 ) det ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = − ρ {\displaystyle {\begin{aligned}r&=\rho \sin \varphi \\h&=\rho \cos \varphi \\\theta &=\theta \\{\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&={\begin{pmatrix}\sin \varphi &\rho \cos \varphi &0\\\cos \varphi &-\rho \sin \varphi &0\\0&0&1\\\end{pmatrix}}\\\det {\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&=-\rho \end{aligned}}} === Arc-length, curvature and torsion from Cartesian coordinates === s = ∫ 0 t x ′ 2 + y ′ 2 + z ′


id: index-of-cryptography-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-0.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
Articles related to cryptography include: == A == A5/1 • A5/2 • ABA digital signature guidelines • ABC (stream cipher) • Abraham Sinkov • Acoustic cryptanalysis • Adaptive chosen-ciphertext attack • Adaptive chosen plaintext and chosen ciphertext attack • Advantage (cryptography) • ADFGVX cipher • Adi Shamir • Advanced Access Content System • Advanced Encryption Standard • Advanced Encryption Standard process • Adversary • AEAD block cipher modes of operation • Affine cipher • Agnes Meyer Driscoll • AKA (security) • Akelarre (cipher) • Alan Turing • Alastair Denniston • Al Bhed language • Alex Biryukov • Alfred Menezes • Algebraic Eraser • Algorithmically random sequence • Alice and Bob • All-or-nothing transform • Alphabetum Kaldeorum • Alternating step generator • American Cryptogram Association • AN/CYZ-10 • Anonymous publication • Anonymous remailer • Antoni Palluth • Anubis (cipher) • Argon2 • ARIA (cipher) • Arlington Hall • Arne Beurling • Arnold Cipher • Array controller based encryption • Arthur Scherbius • Arvid Gerhard Damm • Asiacrypt • Atbash • Attribute-based encryption • Attack model • Auguste Kerckhoffs • Authenticated encryption • Authentication • Authorization certificate • Autokey cipher • Avalanche effect == B == B-Dienst • Babington Plot • Baby-step


id: index-of-cryptography-articles-20260505-1
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-1.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== D == D'Agapeyeff cipher • Daniel J. Bernstein • Data Authentication Algorithm • Data Encryption Standard • Datagram Transport Layer Security • David Chaum • David Kahn • David Naccache • David Wagner • David Wheeler (computer scientist) • Davies attack • Davies–Meyer hash • DEAL • Decipherment • Decisional Diffie–Hellman assumption • Decorrelation theory • Decrypt • DeCSS • Defence Signals Directorate • Degree of anonymity • Delegated Path Discovery • Delegated Path Validation • Deniable encryption • Derek Taunt • Derived unique key per transaction • DES Challenges • DES supplementary material • DES-X • Deterministic encryption • DFC (cipher) • Dictionary attack • Differential cryptanalysis • Differential-linear attack • Differential power analysis • Diffie–Hellman key exchange • Diffie–Hellman problem • DigiCipher 2 • Digital Fortress • Digital rights management • Digital signature • Digital Signature Algorithm • Digital signature forgery • Digital timestamping • Digital watermarking • Dilly Knox • Dining cryptographers problem • Diplomatic bag • Direct Anonymous Attestation • Discrete logarithm • Disk encryption • Disk encryption hardware • Disk encryption software • Distance-bounding protocol • Distinguishing attack • Distributed.net • DMA attack • dm-crypt • Dmitry Sklyarov • DomainKeys • Don Coppersmith •


id: index-of-cryptography-articles-20260505-2
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-2.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== K == Kaisa Nyberg • Kalyna (cipher) • Kasiski examination • KASUMI • KCDSA • KeePass • Kerberos (protocol) • Kerckhoffs's principle • Kevin McCurley (cryptographer) • Key-agreement protocol • Key authentication • Key clustering • Key (cryptography) • Key derivation function • Key distribution center • Key escrow • Key exchange • Keyfile • Key generation • Key generator • Key management • Key-recovery attack • Key schedule • Key server (cryptographic) • Key signature (cryptography) • Keysigning • Key signing party • Key size • Key space (cryptography) • Keystream • Key stretching • Key whitening • KG-84 • KHAZAD • Khufu and Khafre • Kiss (cryptanalysis) • KL-43 • KL-51 • KL-7 • Kleptography • KN-Cipher • Knapsack problem • Known-key distinguishing attack • Known-plaintext attack • KOI-18 • KOV-14 • Kryha • Kryptos • KSD-64 • Kupyna • Kuznyechik • KW-26 • KW-37 • KY-3 • KY-58 • KY-68 • KYK-13 == L == Lacida • Ladder-DES • Lamport signature • Lars Knudsen • Lattice-based cryptography • Laurance Safford • Lawrie Brown • LCS35 • Leo Marks • Leonard Adleman • Leon Battista Alberti • Leo Rosen • Leslie Yoxall • LEVIATHAN (cipher) • LEX


id: index-of-cryptography-articles-20260505-3
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-3.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== R == Rabbit (cipher) • Rabin cryptosystem • Rabin signature • RadioGatún • Rail fence cipher • Rainbow table • Ralph Merkle • Rambutan (cryptography) • Random function • Randomness tests • Random number generator attack • Random oracle • RC2 • RC4 • RC5 • RC6 • Rebound attack • Reciprocal cipher • Red/black concept • REDOC • Red Pike (cipher) • Reflector (cipher machine) • Regulation of Investigatory Powers Act 2000 • Reihenschieber • Rekeying (cryptography) • Related-key attack • Replay attack • Reservehandverfahren • Residual block termination • Rijndael key schedule • Rijndael S-box • Ring signature • RIPEMD • Rip van Winkle cipher • Robert Morris (cryptographer) • Rockex • Rolf Noskwith • Ron Rivest • Room 40 • Root certificate • Ross J. Anderson • Rossignols • ROT13 • Rotor machine • RSA RSA • RSA-100 • RSA-1024 • RSA-110 • RSA-120 • RSA-129 • RSA-130 • RSA-140 • RSA-150 • RSA-1536 • RSA-155 • RSA-160 • RSA-170 • RSA-180 • RSA-190 • RSA-200 • RSA-2048 • RSA-210 • RSA-220 • RSA-230 • RSA-232 • RSA-240 • RSA-250 • RSA-260 • RSA-270 • RSA-280 • RSA-290 • RSA-300 • RSA-309 • RSA-310 • RSA-320


id: list-of-curves-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_curves_topics-0.md
category: reference
title: List of curves topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_curves_topics
date_saved: 2026-05-05T08:13:49.986081+00:00
instance: kb-cron
parent: list-of-curves-topics-20260505-0
summary:
This is an alphabetical index of articles related to curves used in mathematics. Acnode Algebraic curve Arc Asymptote Asymptotic curve Barbier's theorem Bézier curve Bézout's theorem Birch and Swinnerton-Dyer conjecture Bitangent Bitangents of a quartic Cartesian coordinate system Caustic Cesàro equation Chord (geometry) Cissoid Circumference Closed timelike curve concavity Conchoid (mathematics) Confocal Contact (mathematics) Contour line Crunode Cubic Hermite curve Curvature Curve orientation Curve fitting Curve-fitting compaction Curve of constant width Curve of pursuit Curves in differential geometry Cusp Cyclogon De Boor algorithm Differential geometry of curves Eccentricity (mathematics) Elliptic curve cryptography Envelope (mathematics) Fenchel's theorem Genus (mathematics) Geodesic Geometric genus Great-circle distance Harmonograph Hedgehog (curve) [1] Hilbert's sixteenth problem Hyperelliptic curve cryptography Inflection point Inscribed square problem intercept, y-intercept, x-intercept Intersection number Intrinsic equation Isoperimetric inequality Jordan curve Jordan curve theorem Knot Limit cycle Linking coefficient List of circle topics Loop (knot) M-curve Mannheim curve[2] Meander (mathematics) Mordell conjecture Natural representation Opisometer Orbital elements Osculating circle Osculating plane Osgood curve Parallel (curve) Parallel transport Parametric curve Bézier curve Spline (mathematics) Hermite spline Beta spline B-spline Higher-order spline NURBS Perimeter Pi Plane curve Pochhammer contour Polar coordinate system Prime geodesic Projective line Ray Regular parametric representation Reuleaux triangle Ribaucour curve[3][4]


id: differentiation-rules-20260505-0
file: data/en.wikipedia.org/wiki/Differentiation_rules-0.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. == Elementary rules of differentiation == Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb {R} } ) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers ( C {\textstyle \mathbb {C} } ). === Constant term rule === For any value of c {\textstyle c} , where c ∈ R {\textstyle c\in \mathbb {R} } , if f ( x ) {\textstyle f(x)} is the constant function given by f ( x ) = c {\textstyle f(x)=c} , then d f d x = 0 {\textstyle {\frac {df}{dx}}=0} . ==== Proof ==== Let c ∈ R {\textstyle c\in \mathbb {R} } and f ( x ) = c {\textstyle f(x)=c} . By the definition of the derivative: f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 ( c ) − ( c ) h = lim h → 0 0 h = lim


id: differentiation-rules-20260505-1
file: data/en.wikipedia.org/wiki/Differentiation_rules-1.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 , {\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},} wherever f {\textstyle f} is nonzero. In Leibniz's notation, this formula is written: d ( 1 f ) d x = − 1 f 2 d f d x . {\displaystyle {\frac {d\left({\frac {1}{f}}\right)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.} The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule. === Quotient rule === If f {\textstyle f} and g {\textstyle g} are functions, then: ( f g ) ′ = f ′ g − g ′ f g 2 , {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}},} wherever g {\textstyle g} is nonzero. This can be derived from the product rule and the reciprocal rule. === Generalized power rule === The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f {\textstyle f} and g {\textstyle g} , ( f g ) ′ = ( e g ln ⁡ f ) ′ = f g ( f ′ g f + g ′ ln ⁡ f ) , {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln


id: differentiation-rules-20260505-2
file: data/en.wikipedia.org/wiki/Differentiation_rules-2.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
Γ ′ ( x ) = ∫ 0 ∞ t x − 1 e − t ln ⁡ t d t = Γ ( x ) ( ∑ n = 1 ∞ ( ln ⁡ ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) , {\displaystyle {\begin{aligned}\Gamma '(x)&=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt\\&=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x),\end{aligned}}} with ψ ( x ) {\textstyle \psi (x)} being the digamma function, expressed by the parenthesized expression to the right of Γ ( x ) {\textstyle \Gamma (x)} in the line above. === Riemann zeta function === ζ ( x ) = ∑ n = 1 ∞ 1 n x {\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}} ζ ′ ( x ) = − ∑ n = 1 ∞ ln ⁡ n n x = − ln ⁡ 2 2 x − ln ⁡ 3 3 x − ln ⁡ 4 4 x − ⋯ = − ∑ p prime p − x ln ⁡ p ( 1 − p − x ) 2 ∏ q prime , q ≠ p 1 1 − q


id: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
file: data/en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi-0.md
category: reference
title: List of derivatives and integrals in alternative calculi
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi
date_saved: 2026-05-05T08:13:52.505671+00:00
instance: kb-cron
parent: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
summary:
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). == Table == In the following table; ψ ( x ) = Γ ′ ( x ) Γ ( x ) {\displaystyle \psi (x)={\frac {\Gamma '(x)}{\Gamma (x)}}} is the digamma function, K ⁡ ( x ) = e ζ ′ ( − 1 , x ) − ζ ′ ( − 1 ) = e z − z 2 2 + z 2 ln ⁡ ( 2 π ) − ψ ( − 2 ) ( z ) {\displaystyle \operatorname {K} (x)=e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=e^{{\frac {z-z^{2}}{2}}+{\frac {z}{2}}\ln(2\pi )-\psi ^{(-2)}(z)}} is the K-function, ( ! x ) = Γ ( x + 1 , − 1 ) e {\displaystyle (!x)={\frac {\Gamma (x+1,-1)}{e}}} is subfactorial, B a ( x ) = − a ζ ( − a + 1 , x ) {\displaystyle B_{a}(x)=-a\zeta (-a+1,x)} are the generalized to real


id: list-of-differential-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_differential_geometry_topics-0.md
category: reference
title: List of differential geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
date_saved: 2026-05-05T08:13:53.815515+00:00
instance: kb-cron
parent: list-of-differential-geometry-topics-20260505-0
summary:
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. == Differential geometry of curves and surfaces == === Differential geometry of curves === List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius of curvature Osculating circle Curve Fenchel's theorem === Differential geometry of surfaces === Theorema egregium Gauss–Bonnet theorem First fundamental form Second fundamental form Gauss–Codazzi–Mainardi equations Dupin indicatrix Asymptotic curve Curvature Principal curvatures Mean curvature Gauss curvature Elliptic point Types of surfaces Minimal surface Ruled surface Conical surface Developable surface Nadirashvili surface == Foundations == === Calculus on manifolds === See also multivariable calculus, list of multivariable calculus topics Manifold Differentiable manifold Smooth manifold Banach manifold Fréchet manifold Tensor analysis Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius theorem (differential topology) Integral curve === Differential topology === Diffeomorphism Large diffeomorphism Orientability characteristic class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding Whitney embedding theorem Critical value Sard's theorem Saddle point Morse


id: outline-of-discrete-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-0.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis. Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered. Logic – Study of correct reasoning Modal logic – Type of formal logic Set theory – Branch of mathematics that studies sets Number theory – Branch of pure mathematics Combinatorics – Branch of discrete mathematics Finite mathematics – Syllabus in college and university mathematics Graph theory – Area of discrete mathematics Digital geometry – Deals with digitized models or images of objects of the 2D or 3D Euclidean space Digital topology – Properties of 2D or 3D digital images that correspond to classic topological properties Algorithmics – Sequence of operations for


id: outline-of-discrete-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-1.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Left-hand side and right-hand side of an equation – Mathematical nomenclaturePages displaying short descriptions of redirect targets Linear equation – Equation that does not involve powers or products of variables Quadratic equation – Polynomial equation of degree two Solution point – Mathematical formula expressing equalityPages displaying short descriptions of redirect targets Arithmetic progression – Sequence of equally spaced numbers Recurrence relation – Pattern defining an infinite sequence of numbers Finite difference – Discrete analog of a derivative Difference operator – Pattern defining an infinite sequence of numbersPages displaying short descriptions of redirect targets Groups – Set with associative invertible operation Group isomorphism – Bijective group homomorphism Subgroups – Subset of a group that forms a group itselfPages displaying short descriptions of redirect targets Fermat's little theorem – A prime p divides a^p–a for any integer a Cryptography – Practice and study of secure communication techniques Faulhaber's formula – Expression for sums of powers === Mathematical relations === Binary relation – Relationship between elements of two sets Heterogeneous relation – Relationship between elements of two setsPages displaying short descriptions of redirect targets Reflexive relation – Binary relation that relates every element to itself Reflexive property of equality – Basic notion of


id: divisibility-rule-20260505-0
file: data/en.wikipedia.org/wiki/Divisibility_rule-0.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American. == Divisibility rules for numbers 1−30 == The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means. For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits. == Step-by-step examples == === Divisibility by 2 === First, take any number (for this example it will be 376) and note


id: divisibility-rule-20260505-1
file: data/en.wikipedia.org/wiki/Divisibility_rule-1.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Divisibility by 5 === Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5. If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8). If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25). Example. If the last digit is 0 110 (The original number) 11 0 (Take the last digit of the number, and check if it is 0 or


id: divisibility-rule-20260505-2
file: data/en.wikipedia.org/wiki/Divisibility_rule-2.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Take for instance the number 371 Change all occurrences of 7, 8 or 9 into 0, 1 and 2, respectively. In this example, we get: 301. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on. Now convert the first digit (3) into the following digit in the sequence 13264513... In our example, 3 becomes 2. Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So 301 becomes 21. Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = 7. If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should


id: divisibility-rule-20260505-3
file: data/en.wikipedia.org/wiki/Divisibility_rule-3.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern. This phenomenon forms the basis for Steps B and C. Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example: 341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES 67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and


id: divisibility-rule-20260505-4
file: data/en.wikipedia.org/wiki/Divisibility_rule-4.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Finding remainder of a number when divided by 7 7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, −1, −3, −2 Minimum magnitude sequence (1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, 6, 4, 5 Positive sequence Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7. Example: What is the remainder when 1036125837 is divided by 7? Multiplication of the rightmost digit = 1 × 7 = 7 Multiplication of the second rightmost digit = 3 × 3 = 9 Third rightmost digit = 8 × 2 = 16 Fourth rightmost digit = 5 × −1 = −5 Fifth rightmost digit = 2 × −3 = −6 Sixth rightmost digit = 1 × −2 = −2 Seventh rightmost digit = 6 × 1 = 6 Eighth rightmost digit = 3 × 3 =


id: divisibility-rule-20260505-5
file: data/en.wikipedia.org/wiki/Divisibility_rule-5.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Another digit grouping method of divisibility by 7 Method This method determines whether an integer is divisible by 7 by grouping digits in sets of three: Starting from the right (units place), divide the number into groups of three digits. If necessary, prepend leading zeros to complete the leftmost group. Starting from the rightmost group, alternately add and subtract the groups moving left. That is, subtract the second group from the first, add the third, subtract the fourth, and so on. Evaluate the resulting sum. If the result is divisible by 7, then the original number is divisible by 7; otherwise, it is not. Example To test the number 10,517,815: Group the digits: 010 | 517 | 815 Apply alternating operations: 815 − 517 + 010 = 308 Since 308 is divisible by 7, the original number is also divisible by 7. Proof of correctness This method is based on modular arithmetic. Observe that: 1000 ≡ −1 (mod 7) since 1000 leaves a remainder of 6 when divided by 7, and 6 ≡ −1 (mod 7). Any integer can be expressed as a sum of multiples of powers of 1000: N = a₀ + a₁·1000 + a₂·1000² + a₃·1000³ +


id: divisibility-rule-20260505-6
file: data/en.wikipedia.org/wiki/Divisibility_rule-6.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Composite divisors === A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful. A composite divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors. To test the divisibility of a number by a power of 2 or a power of 5 (2n or 5n, in which n is a positive integer), one only need to look at the last n digits of that number. To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}} , we can separately test for


id: divisibility-rule-20260505-7
file: data/en.wikipedia.org/wiki/Divisibility_rule-7.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Proof using basic algebra === Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them. By writing a number as the sum of each digit times a power of 10 each digit's power can be manipulated individually. ==== Case where all digits are summed ==== This method works for divisors that are factors of 10 − 1 = 9. Using 3 as an example, 3 divides 9 = 10 − 1. That means 10 ≡ 1 ( mod 3 ) {\displaystyle 10\equiv 1{\pmod {3}}} (see modular arithmetic). The same for all the higher powers of 10: 10 n ≡ 1 n ≡ 1 ( mod 3 ) . {\displaystyle 10^{n}\equiv 1^{n}\equiv 1{\pmod {3}}.} They are all congruent to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1: 100 ⋅ a + 10 ⋅ b + 1 ⋅ c ≡ ( 1 ) a + ( 1 ) b + ( 1 ) c (


id: table-of-divisors-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_divisors-0.md
category: reference
title: Table of divisors
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_divisors
date_saved: 2026-05-05T08:13:57.922976+00:00
instance: kb-cron
parent: table-of-divisors-20260505-0
summary:
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which ⁠n/m⁠ is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since ⁠21/7⁠ = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only list positive divisors. == Key to the tables == d(n) is the number of the positive divisors of n, including 1 and n itself σ(n) is the sum of the positive divisors of n, including 1 and n itself s(n) is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n a deficient number is greater than the sum of its proper divisors; that is, s(n) < n a perfect number equals the sum of its proper divisors; that is, s(n) = n an abundant number is lesser than the sum of its proper divisors; that is, s(n) > n a highly abundant number has a sum of positive divisors that is greater than


id: list-of-dualities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_dualities-0.md
category: reference
title: List of dualities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_dualities
date_saved: 2026-05-05T08:13:59.079468+00:00
instance: kb-cron
parent: list-of-dualities-20260505-0
summary:
== Mathematics == In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Alexander duality Alvis–Curtis duality Artin–Verdier duality Beta-dual space Coherent duality Conjugate hyperbola De Groot dual Dual abelian variety Dual basis in a field extension Dual bundle Dual curve Dual (category theory) Dual graph Dual group Dual object Dual pair Dual polygon Dual polyhedron Dual problem Dual representation Dual q-Hahn polynomials Dual q-Krawtchouk polynomials Dual space Dual topology Dual wavelet Duality (optimization) Duality (order theory) Duality of stereotype spaces Duality (projective geometry) Duality theory for distributive lattices Dualizing complex Dualizing sheaf Eckmann–Hilton duality Esakia duality Fenchel's duality theorem Grothendieck local duality Hodge dual Isbell duality Jónsson–Tarski duality Lagrange duality Langlands dual Lefschetz duality Local Tate duality Opposite category Poincaré duality Twisted Poincaré duality Poitou–Tate duality Pontryagin duality S-duality (homotopy theory) Schur–Weyl duality Series-parallel duality Serre duality Spanier–Whitehead duality Stone's duality Tannaka–Krein duality Verdier duality == Philosophy and religion == Dualism (philosophy of mind) Dualistic cosmology Epistemological dualism Soul dualism Yin and yang


id: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics-0.md
category: reference
title: List of dynamical systems and differential equations topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics
date_saved: 2026-05-05T08:14:00.321935+00:00
instance: kb-cron
parent: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
summary:
This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. == Dynamical systems, in general == Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and chaos theory Chaos theory Chaos argument Butterfly effect 0-1 test for chaos Bifurcation diagram Feigenbaum constant Sharkovskii's theorem Attractor Strange nonchaotic attractor Stability theory Mechanical equilibrium Astable Monostable Bistability Metastability Feedback Negative feedback Positive feedback Homeostasis Damping ratio Dissipative system Spontaneous symmetry breaking Turbulence Perturbation theory Control theory Non-linear control Adaptive control Hierarchical control Intelligent control Optimal control Dynamic programming Robust control Stochastic control System dynamics, system analysis Takens' theorem Exponential dichotomy Liénard's theorem Krylov–Bogolyubov theorem Krylov-Bogoliubov averaging method == Abstract dynamical systems == Measure-preserving dynamical system Ergodic theory Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmetic dynamics Sequential dynamical system Graph dynamical system Topological dynamical system == Dynamical systems, examples == List of chaotic maps Logistic map Lorenz attractor Lorenz-96 Iterated function system Tetration Ackermann function Horseshoe map Hénon map Arnold's cat map Population dynamics == Complex dynamics == Fatou set Julia set Mandelbrot set == Difference equations == Recurrence relation Matrix difference equation


id: list-of-eponyms-of-special-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_eponyms_of_special_functions-0.md
category: reference
title: List of eponyms of special functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_eponyms_of_special_functions
date_saved: 2026-05-05T08:14:02.125990+00:00
instance: kb-cron
parent: list-of-eponyms-of-special-functions-20260505-0
summary:
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym). Named symmetric functions, and other special polynomials, are included. == A == Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial Sir George Biddell Airy: Airy function Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara polynomial C. T. Anger: Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell polynomials Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson) == B == Ernest William Barnes: Barnes G-function E. T. Bell: Bell polynomials Bender–Dunne polynomial Jacob Bernoulli: Bernoulli polynomial Friedrich Bessel: Bessel function, Bessel–Clifford function H. Blasius: Blasius functions R. P. Boas, R. C. Buck: Boas–Buck polynomial Böhmer integral Erland Samuel Bring: Bring radical de Bruijn function Buchstab function Burchnall, Chaundy: Burchnall–Chaundy polynomial == C == Leonard Carlitz: Carlitz polynomial Arthur Cayley, Capelli: Cayley–Capelli operator Celine's polynomial Charlier polynomial Pafnuty Chebyshev: Chebyshev polynomials Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation Cyclotomic polynomials == D == H. G. Dawson:


id: list-of-mathematical-examples-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_examples-0.md
category: reference
title: List of mathematical examples
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_examples
date_saved: 2026-05-05T08:14:04.638441+00:00
instance: kb-cron
parent: list-of-mathematical-examples-20260505-0
summary:
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a general theorem, could perhaps be considered an "example"). The discussion page for list of mathematical topics has some comments on this. Eventually this page may have its own discussion page. This page links to itself in order that edits to this page will be included among related changes when the user clicks on that button. The concrete example within the article titled Rao–Blackwell theorem is perhaps one of the best ways for a probabilist ignorant of statistical inference to get a quick impression of the flavor of that subject. == Uncategorized examples, alphabetized == Alexander horned sphere All horses are the same color Cantor function Cantor set Checking if a coin is biased Concrete illustration of the central limit theorem Differential equations of mathematical physics Dirichlet function Discontinuous linear map Efron's non-transitive


id: list-of-exceptional-set-concepts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exceptional_set_concepts-0.md
category: reference
title: List of exceptional set concepts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exceptional_set_concepts
date_saved: 2026-05-05T08:14:06.643357+00:00
instance: kb-cron
parent: list-of-exceptional-set-concepts-20260505-0
summary:
This is a list of exceptional set concepts. In mathematics, and in particular in mathematical analysis, it is very useful to be able to characterise subsets of a given set X as 'small', in some definite sense, or 'large' if their complement in X is small. There are numerous concepts that have been introduced to study 'small' or 'exceptional' subsets. In the case of sets of natural numbers, it is possible to define more than one concept of 'density', for example. See also list of properties of sets of reals. Almost all Almost always Almost everywhere Almost never Almost surely Analytic capacity Closed unbounded set Cofinal (mathematics) Cofinite Dense set IP set 2-large Large set (Ramsey theory) Meagre set Measure zero Natural density Negligible set Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick set Thin set (Serre)


id: list-of-exponential-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exponential_topics-0.md
category: reference
title: List of exponential topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exponential_topics
date_saved: 2026-05-05T08:14:08.848554+00:00
instance: kb-cron
parent: list-of-exponential-topics-20260505-0
summary:
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. Accelerating change Approximating natural exponents (log base e) Artin–Hasse exponential Bacterial growth Baker–Campbell–Hausdorff formula Carlitz exponential Cell growth Barometric formula Beer–Lambert law Characterizations of the exponential function Catenary Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double exponential function Doubling time e-folding Elimination half-life Error exponent Euler's formula Euler's identity e (mathematical constant) Exponent Exponent bias Exponential (disambiguation) Exponential backoff Exponential decay Exponential dichotomy Exponential discounting Exponential diophantine equation Exponential dispersion model Exponential distribution Exponential error Exponential factorial Exponential family Exponential field Ordered exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy Exponential integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation Exponential object (category theory) Exponential polynomials—see also Touchard polynomials (combinatorics) Exponential response formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent measures Exponentiating by squaring Exponentiation Forgetting curve Four exponentials conjecture Fourier analysis List of Fourier analysis topics Gaussian function Gudermannian function Half-exponential function Half-life Hyperbolic function Inflation, inflation rate Interest


id: exterior-calculus-identities-20260505-0
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-0.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry. == Notation == The following notation is used in this article. === Manifold === M {\displaystyle M} , N {\displaystyle N} are n {\displaystyle n} -dimensional smooth manifolds, where n ∈ N {\displaystyle n\in \mathbb {N} } . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page. p ∈ M {\displaystyle p\in M} , q ∈ N {\displaystyle q\in N} denote one point on each of the manifolds. The boundary of a manifold M {\displaystyle M} is a manifold ∂ M {\displaystyle \partial M} , which has dimension n − 1 {\displaystyle n-1} . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} . === Tangent and cotangent bundles === T M {\displaystyle TM} , T ∗ M {\displaystyle T^{*}M} denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold M {\displaystyle M} . T p M {\displaystyle T_{p}M} , T q N {\displaystyle T_{q}N} denote the tangent spaces of M {\displaystyle M} , N {\displaystyle


id: exterior-calculus-identities-20260505-1
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-1.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Exterior derivative === The exterior derivative d k : Ω k ( M ) → Ω k + 1 ( M ) {\displaystyle d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)} is defined for all 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . We generally omit the subscript when it is clear from the context. For a 0 {\displaystyle 0} -form f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} we have d 0 f ∈ Ω 1 ( M ) {\displaystyle d_{0}f\in \Omega ^{1}(M)} as the 1 {\displaystyle 1} -form that gives the directional derivative, i.e., for the section X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} we have ( d 0 f ) ( X ) = ∂ X f {\displaystyle (d_{0}f)(X)=\partial _{X}f} , the directional derivative of f {\displaystyle f} along X {\displaystyle X} . For 0 < k ≤ n {\displaystyle 0<k\leq n} , ( d k ω ) ( X 0 , … , X k ) = ∑ 0 ≤ j ≤ k ( − 1 ) j d 0 ( ω ( X 0 , … , X ^ j , … , X k ) ) ( X


id: exterior-calculus-identities-20260505-2
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-2.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
A one-form α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} corresponds to the unique vector field α ♯ ∈ Γ ( T M ) {\displaystyle \alpha ^{\sharp }\in \Gamma (TM)} such that for all X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} , we have: α ( X ) = g ( α ♯ , X ) . {\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).} These mappings extend via multilinearity to mappings from k {\displaystyle k} -vector fields to k {\displaystyle k} -forms and k {\displaystyle k} -forms to k {\displaystyle k} -vector fields through ( A 1 ∧ A 2 ∧ ⋯ ∧ A k ) ♭ = A 1 ♭ ∧ A 2 ♭ ∧ ⋯ ∧ A k ♭ {\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }} ( α 1 ∧ α 2 ∧ ⋯ ∧ α k ) ♯ = α 1 ♯ ∧ α 2 ♯ ∧ ⋯ ∧ α k ♯ . {\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.} === Hodge star === For an n-manifold M, the Hodge star


id: exterior-calculus-identities-20260505-3
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-3.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Cohomology === The k {\displaystyle k} -th cohomology of a manifold M {\displaystyle M} and its exterior derivative operators d 0 , … , d n − 1 {\displaystyle d_{0},\ldots ,d_{n-1}} is given by H k ( M ) := ker ( d k ) im ( d k − 1 ) {\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}} Two closed k {\displaystyle k} -forms α , β ∈ Ω k ( M ) {\displaystyle \alpha ,\beta \in \Omega ^{k}(M)} are in the same cohomology class if their difference is an exact form i.e. [ α ] = [ β ] ⟺ α − β = d η for some η ∈ Ω k − 1 ( M ) {\displaystyle [\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)} A closed surface of genus g {\displaystyle g} will have 2 g {\displaystyle 2g} generators which are harmonic. === Dirichlet energy === Given α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , its Dirichlet energy is E D ( α ) := 1 2 ⟨ ⟨ d α , d α ⟩ ⟩ + 1 2 ⟨ ⟨ δ α ,


id: exterior-calculus-identities-20260505-4
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-4.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
⋆ ( λ 1 α + λ 2 β ) = λ 1 ( ⋆ α ) + λ 2 ( ⋆ β ) {\displaystyle {\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )} for λ 1 , λ 2 ∈ R {\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} } ( linearity ) ⋆ ⋆ α = s ( − 1 ) k ( n − k ) α {\displaystyle {\star }{\star }\alpha =s(-1)^{k(n-k)}\alpha } for α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , n = dim ⁡ ( M ) {\displaystyle n=\dim(M)} , and s = sign ⁡ ( g ) {\displaystyle s=\operatorname {sign} (g)} the sign of the metric ⋆ ( − 1 ) = s ( − 1 ) k ( n − k ) ⋆ {\displaystyle {\star }^{(-1)}=s(-1)^{k(n-k)}{\star }} ( inversion ) ⋆ ( f α ) = f ( ⋆ α ) {\displaystyle {\star }(f\alpha )=f({\star }\alpha )} for f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} ( commutative with 0 {\displaystyle 0} -forms ) ⟨ ⟨ α , α ⟩ ⟩ = ⟨ ⟨ ⋆ α , ⋆ α ⟩ ⟩ {\displaystyle \langle \!\langle


id: exterior-calculus-identities-20260505-5
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-5.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
( α ∧ β ∧ γ ) ( X , Y , Z ) = det [ α ( X ) α ( Y ) α ( Z ) β ( X ) β ( Y ) β ( Z ) γ ( X ) γ ( Y ) γ ( Z ) ] {\displaystyle (\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)&\alpha (Z)\\\beta (X)&\beta (Y)&\beta (Z)\\\gamma (X)&\gamma (Y)&\gamma (Z)\end{bmatrix}}} ( α 1 ∧ … ∧ α l ) ( X 1 , … , X l ) = det [ α 1 ( X 1 ) α 1 ( X 2 ) … α 1 ( X l ) α 2 ( X 1 ) α 2 ( X 2 ) … α 2 ( X l ) ⋮ ⋮ ⋱ ⋮ α l ( X 1 ) α l ( X 2 ) … α l ( X l ) ] {\displaystyle (\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}} === Projection and rejection === ( − 1 ) k ι X ⋆ α = ⋆ ( X ♭ ∧ α


id: exterior-calculus-identities-20260505-6
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-6.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Identities in Euclidean 3-space === Let Euclidean metric g ( X , Y ) := ⟨ X , Y ⟩ = X ⋅ Y {\displaystyle g(X,Y):=\langle X,Y\rangle =X\cdot Y} . We use ∇ = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) {\displaystyle \nabla =\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)} differential operator R 3 {\displaystyle \mathbb {R} ^{3}} ι X α = g ( X , α ♯ ) = X ⋅ α ♯ {\displaystyle \iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }} for α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} . d e t ( X , Y , Z ) = ⟨ X , Y × Z ⟩ = ⟨ X × Y , Z ⟩ {\displaystyle \mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle } ( scalar triple product ) X × Y = ( ⋆ ( X ♭ ∧ Y ♭ ) ) ♯ {\displaystyle X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }} ( cross product ) ι X α = − ( X × A ) ♭ {\displaystyle \iota _{X}\alpha =-(X\times A)^{\flat }} if α ∈ Ω 2 ( M )


id: list-of-factorial-and-binomial-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics-0.md
category: reference
title: List of factorial and binomial topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics
date_saved: 2026-05-05T08:14:15.007034+00:00
instance: kb-cron
parent: list-of-factorial-and-binomial-topics-20260505-0
summary:
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain Beta function Bhargava factorial Binomial coefficient Pascal's triangle Binomial distribution Binomial proportion confidence interval Binomial-QMF (Daubechies wavelet filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient Combination Combinatorial number system De Polignac's formula Difference operator Difference polynomials Digamma function Egorychev method Erdős–Ko–Rado theorem Euler–Mascheroni constant Faà di Bruno's formula Factorial Factorial moment Factorial number system Factorial prime Factoriangular number Gamma distribution Gamma function Gaussian binomial coefficient Gould's sequence Hyperfactorial Hypergeometric distribution Hypergeometric function identities Hypergeometric series Incomplete beta function Incomplete gamma function Jordan–Pólya number Kempner function Lah number Lanczos approximation Lozanić's triangle Macaulay representation of an integer Mahler's theorem Multinomial distribution Multinomial coefficient, Multinomial formula, Multinomial theorem Multiplicities of entries in Pascal's triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution Nörlund–Rice integral Pascal matrix Pascal's pyramid Pascal's simplex Pascal's triangle Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof of Bertrand's postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian


id: list-of-finite-simple-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-0.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. == Summary == The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = A3(2) and A2(4) both have order 20160, and that the group Bn(q) has the same order as Cn(q) for q odd, n > 2. The smallest of the latter pairs of groups are B3(3) and C3(3) which both have order 4585351680.) There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie


id: list-of-finite-simple-groups-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-1.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
=== Ree groups, 2G2(32n+1) === Simplicity: Simple for n ≥ 1. The group 2G2(3) is not simple, but its derived group 2G2(3)′ is a simple subgroup of index 3. Order: q3 (q3 + 1) (q − 1), where q = 32n+1 Schur multiplier: Trivial for n ≥ 1 and for 2G2(3)′. Outer automorphism group: 1⋅f⋅1, where f = 2n + 1. Other names: Ree(32n+1), R(32n+1), E2∗(32n+1) . Isomorphisms: The derived group 2G2(3)′ is isomorphic to A1(8). Remarks: 2G2(32n+1) has a doubly transitive permutation representation on 33(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements. == Sporadic groups == === Mathieu groups, M11, M12, M22, M23, M24 === === Janko groups, J1, J2, J3, J4 === === Conway groups, Co1, Co2, Co3 === === Fischer groups, Fi22, Fi23, Fi24′ === === Higman–Sims group, HS === Order: 29 ⋅ 32 ⋅ 53 ⋅ 7 ⋅ 11 = 44352000 Schur multiplier: Order 2. Outer automorphism group: Order 2. Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co2 and in Co3. === McLaughlin group, McL === Order: 27 ⋅ 36 ⋅ 53 ⋅


id: list-of-finite-dimensional-nichols-algebras-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-0.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts U q ( g ) + {\displaystyle U_{q}({\mathfrak {g}})^{+}} of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u q ( g ) + {\displaystyle u_{q}({\mathfrak {g}})^{+}} of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity. The following article lists all known finite-dimensional Nichols algebras B ( V ) {\displaystyle {\mathfrak {B}}(V)} where V {\displaystyle V} is a Yetter–Drinfel'd module over a finite group G {\displaystyle G} , where the group is generated by the support of V {\displaystyle V} . For more details on Nichols algebras see Nichols algebra. There are two major cases: G {\displaystyle G} abelian, which implies V {\displaystyle V} is


id: list-of-finite-dimensional-nichols-algebras-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-1.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
=== Nichols algebras from Coxeter groups === For every finite coxeter system ( W , S ) {\displaystyle (W,S)} the Nichols algebra over the conjugacy class(es) of reflections was studied in (reflections on roots of different length are not conjugate, see fourth example fellow). They discovered in this way the following first Nichols algebras over nonabelian groups : The case S 2 ≅ Z 2 {\displaystyle \mathbb {S} _{2}\cong \mathbb {Z} _{2}} is the rank 1 diagonal Nichols algebra u i ( A 1 ) + {\displaystyle u_{i}(A_{1})^{+}} of dimension 2. === Other Nichols algebras of rank 1 === === Nichols algebras of rank 2, type Gamma-3 === These Nichols algebras were discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2 type Gamma-4 === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2, type T === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 3 involving Gamma-3 === This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin. === Nichols algebras from diagram folding === The following families


id: list-of-first-order-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-0.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. == Preliminaries == For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: List or describe a set of sentences in the language Lσ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in Lσ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An Lσ theory may: be consistent: no proof of contradiction exists; be satisfiable: there exists a σ-structure for which the sentences of the theory are all true (by the completeness theorem, satisfiability is equivalent to consistency); be complete:


id: list-of-first-order-theories-20260505-1
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-1.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
~ has an infinite number of equivalence classes; ~ has exactly n equivalence classes (for any fixed positive integer n); All equivalence classes are infinite; All equivalence classes have size exactly n (for any fixed positive integer n). The theory of an equivalence relation with exactly 2 infinite equivalence classes is an easy example of a theory which is ω-categorical but not categorical for any larger cardinal. The equivalence relation ~ should not be confused with the identity symbol '=': if x=y then x~y, but the converse is not necessarily true. Theories of equivalence relations are not all that difficult or interesting, but often give easy examples or counterexamples for various statements. The following constructions are sometimes used to produce examples of theories with certain spectra; in fact by applying them to a small number of explicit theories T one gets examples of complete countable theories with all possible uncountable spectra. If T is a theory in some language, we define a new theory 2T by adding a new binary relation to the language, and adding axioms stating that it is an equivalence relation, such that there are an infinite number of equivalence classes all of which are models of


id: list-of-first-order-theories-20260505-2
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-2.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Boolean algebras == There are several different signatures and conventions used for Boolean algebras: The signature has two constants, 0 and 1, and two binary functions ∧ and ∨ ("and" and "or"), and one unary function ¬ ("not"). This can be confusing as the functions use the same symbols as the propositional functions of first-order logic. In set theory, a common convention is that the language has two constants, 0 and 1, and two binary functions · and +, and one unary function −. The three functions have the same interpretation as the functions in the first convention. Unfortunately, this convention clashes badly with the next convention: In algebra, the usual convention is that the language has two constants, 0 and 1, and two binary functions · and +. The function · has the same meaning as ∧, but a+b means a∨b∧¬(a∧b). The reason for this is that the axioms for a Boolean algebra are then just the axioms for a ring with 1 plus ∀x x2 = x. Unfortunately this clashes with the standard convention in set theory given above. The axioms are: The axioms for a distributive lattice (see above) ∀a a∧¬a = 0, ∀a a∨¬a =


id: list-of-first-order-theories-20260505-3
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-3.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Rings and fields == The signature of (unital) rings has two constants 0 and 1, two binary functions + and ×, and, optionally, one unary negation function −. Rings Axioms: Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive. Commutative rings The axioms for rings plus ∀x ∀y xy = yx. Fields The axioms for commutative rings plus ∀x (¬ x = 0 → ∃y xy = 1) and ¬ 1 = 0. Many of the examples given here have only universal, or algebraic axioms. The class of structures satisfying such a theory has the property of being closed under substructure. For example, a subset of a group closed under the group actions of multiplication and inverse is again a group. Since the signature of fields does not usually include multiplicative and additive inverse, the axioms for inverses are not universal, and therefore a substructure of a field closed under addition and multiplication is not always a field. This can be remedied by adding unary inverse functions to the language. For any positive integer n the property that all equations of degree n have a


id: list-of-first-order-theories-20260505-4
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-4.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀ A ∃ b ∃ c ∃ d b A ∧ c A ∧ d A ∧ ¬ b = c ∧ ¬ b = d ∧ ¬ c = d {\displaystyle \forall A\exists b\exists c\exists d\;bA\land cA\land dA\land \lnot b=c\land \lnot b=d\land \lnot c=d} (Every line has at least 3 points) Euclid did not state all the axioms for Euclidean geometry explicitly, and the first complete list was given by Hilbert in Hilbert's axioms. This is not a first-order axiomatization as one of Hilbert's axioms is a second order completeness axiom. Tarski's axioms are a first-order axiomatization of Euclidean geometry. Tarski showed this axiom system is complete and decidable by relating it to the complete and decidable theory of real closed fields. == Differential algebra == The theory DF of differential fields. The signature is that of fields (0, 1, +, −, ×) together with a unary function ∂, the derivation. The axioms are those for fields together with ∀ u ∀ v ∂ ( u v ) = u ∂ v + v ∂ u {\displaystyle \forall u\forall v\,\partial (uv)=u\,\partial v+v\,\partial u} ∀ u ∀ v ∂ ( u + v ) = ∂ u + ∂ v


id: list-of-first-order-theories-20260505-5
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-5.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀x ¬ Sx = 0 ∀x ¬ x = 0 → ∃y Sy = x ∀x∀y Sx = Sy → x = y ∀x x + 0 = x ∀x∀y x + Sy = S(x + y) ∀x x × 0 = 0 ∀x∀y x × Sy = (x × y) + x. IΣn is first-order Peano arithmetic with induction restricted to Σn formulas (for n = 0, 1, 2, ...). The theory IΣ0 is often denoted by IΔ0. This is a series of more and more powerful fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0 with an axiom stating that xy exists for all x and y (with the usual properties). First-order Peano arithmetic, PA. The "standard" theory of arithmetic. The axioms are the axioms of Robinson arithmetic above, together with the axiom scheme of induction: ϕ ( 0 ) ∧ ( ∀ x ϕ ( x ) → ϕ ( S x ) ) → ( ∀ x ϕ ( x ) ) {\displaystyle \phi (0)\wedge (\forall x\phi (x)\rightarrow \phi (Sx))\rightarrow (\forall x\phi (x))} for any formula φ in the language


id: list-of-formal-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formal_systems-0.md
category: reference
title: List of formal systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formal_systems
date_saved: 2026-05-05T08:14:21.897478+00:00
instance: kb-cron
parent: list-of-formal-systems-20260505-0
summary:
This is a list of formal systems, also known as logical calculi. == Mathematical == Functional calculus, a way to apply various types of functions to operators Matrix calculus, a specialized notation for multivariable calculus over spaces of matrices Umbral calculus, the combinatorics of certain operations on polynomials Vector calculus (also called vector analysis), comprising specialized notations for multivariable analysis of vectors in an inner-product space === Logical === Predicate calculus, specifies the rules of inference governing the logic of predicates Propositional calculus, specifies the rules of inference governing the logic of propositions === In theoretical computer science (Formal language) === Modal μ-calculus, a common temporal logic used by formal verification methods such as model checking Lambda calculus, a formulation of the theory of reflexive functions that has deep connections to computational theory Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus Rho calculus, introduced as a general means to uniformly integrate rewriting into lambda calculus Process calculus, a set of approaches to formulating formal models of concurrent systems Ambient calculus, a family of models for concurrent systems based on the concept of agent mobility Join calculus, a theoretical model for the design of distributed programming languages


id: list-of-formulas-in-elementary-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry-0.md
category: reference
title: List of formulas in elementary geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry
date_saved: 2026-05-05T08:14:23.334016+00:00
instance: kb-cron
parent: list-of-formulas-in-elementary-geometry-20260505-0
summary:
This is a short list of some common mathematical shapes and figures and the formulas that describe them. == Two-dimensional shapes == Sources: == Three-dimensional shapes == This is a list of volume formulas of basic shapes: Cone – 1 3 π r 2 h {\textstyle {\frac {1}{3}}\pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cone's height; Cube – a 3 {\textstyle a^{3}} , where a {\textstyle a} is the side's length; Cuboid – a b c {\textstyle abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the sides' length; Cylinder – π r 2 h {\textstyle \pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cylinder's height; Ellipsoid – 4 3 π a b c {\textstyle {\frac {4}{3}}\pi abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the semi-major and semi-minor axes' length; Sphere – 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} , where r {\textstyle r} is the radius; Parallelepiped – a b c K {\textstyle abc{\sqrt {K}}} , where a {\textstyle a} , b


id: list-of-fourier-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier_analysis_topics-0.md
category: reference
title: List of Fourier analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier_analysis_topics
date_saved: 2026-05-05T08:14:24.703918+00:00
instance: kb-cron
parent: list-of-fourier-analysis-topics-20260505-0
summary:
This is a list of Fourier analysis topics. == Fourier analysis == Multiplier (Fourier analysis) Fourier shell correlation Pinsky phenomenon === Fourier series === Generalized Fourier series Regressive discrete Fourier series Gibbs phenomenon Sigma approximation Dini test Poisson summation formula Spectrum continuation analysis Convergence of Fourier series Half-range Fourier series === Fourier transforms === List of Fourier-related transforms Fourier transform on finite groups Fractional Fourier transform Continuous Fourier transform Fourier operator Fourier inversion theorem Sine and cosine transforms Parseval's theorem Paley–Wiener theorem Projection-slice theorem Frequency spectrum === Discrete Fourier transforms === Discrete Fourier series Non-uniform discrete Fourier transform DFT matrix Entropy influence conjecture Fast cosine transform Fast Fourier transform Cyclotomic fast Fourier transform Sparse Fourier transform === Applications of Fourier analysis === Fourier amplitude sensitivity testing Fourier optics Quantum mechanics, for Wavefunctions Uncertainty principle Quantum Fourier transform == Algebra and analysis == Periodic function Almost periodic function ATS theorem Modulus of continuity Banach algebra Compact group Haar measure Hardy space Sobolev space Topological group === Harmonic analysis === Set of uniqueness Pontryagin duality Plancherel theorem Peter–Weyl theorem Fourier integral operator Oscillatory integral operator === Harmonic functions === Laplace operator Laplace equation Dirichlet problem Unit circle Unit disc Spherical harmonic ===


id: list-of-fourier-related-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier-related_transforms-0.md
category: reference
title: List of Fourier-related transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier-related_transforms
date_saved: 2026-05-05T08:14:26.129378+00:00
instance: kb-cron
parent: list-of-fourier-related-transforms-20260505-0
summary:
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. == Continuous transforms == Applied to functions of continuous arguments, Fourier-related transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform: the Fourier transform may be considered a special case of the imaginary axis of the bilateral Laplace transform Fourier transform, with special cases: Fourier series When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values


id: index-of-fractal-related-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_fractal-related_articles-0.md
category: reference
title: Index of fractal-related articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_fractal-related_articles
date_saved: 2026-05-05T08:14:27.328439+00:00
instance: kb-cron
parent: index-of-fractal-related-articles-20260505-0
summary:
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics. 1/f noise Apollonian gasket Attractor Box-counting dimension Cantor distribution Cantor dust Cantor function Cantor set Cantor space Chaos theory Coastline Constructal theory Dimension Dimension theory Dragon curve Fatou set Fractal Fractal antenna Fractal art Fractal compression Fractal flame Fractal landscape Fractal transform Fractint Graftal Iterated function system Horseshoe map How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Julia set Koch snowflake L-system Lebesgue covering dimension Lévy C curve Lévy flight List of fractals by Hausdorff dimension Lorenz attractor Lyapunov fractal Mandelbrot set Menger sponge Minkowski–Bouligand dimension Multifractal analysis Olbers' paradox Perlin noise Power law Rectifiable curve Scale-free network Self-similarity Sierpinski carpet Sierpiński curve Sierpinski triangle Space-filling curve T-square (fractal) Topological dimension


id: list-of-fractals-by-hausdorff-dimension-20260505-0
file: data/en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension-0.md
category: reference
title: List of fractals by Hausdorff dimension
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
date_saved: 2026-05-05T08:14:28.578190+00:00
instance: kb-cron
parent: list-of-fractals-by-hausdorff-dimension-20260505-0
summary:
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. == Deterministic fractals == == Random and natural fractals == == See also == Fractal dimension Hausdorff dimension Scale invariance == Notes and references == == Further reading == Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. W.H. Freeman. ISBN 0-7167-1186-9. Peitgen, Heinz-Otto (1988). Saupe, Dietmar (ed.). The Science of Fractal Images. Springer Verlag. ISBN 0-387-96608-0. Barnsley, Michael F. (1 January 1993). Fractals Everywhere. Morgan Kaufmann. ISBN 0-12-079061-0. Sapoval, Bernard; Mandelbrot, Benoît B. (2001). Universalités et fractales: jeux d'enfant ou délits d'initié?. Flammarion-Champs. ISBN 2-08-081466-4. == External links == The fractals on Mathworld Other fractals on Paul Bourke's website Soler's Gallery Fractals on mathcurve.com 1000fractales.free.fr - Project gathering fractals created with various software Fractals unleashed IFStile - software that computes the dimension of the boundary of self-affine tiles


id: list-of-functional-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_functional_analysis_topics-0.md
category: reference
title: List of functional analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_functional_analysis_topics
date_saved: 2026-05-05T08:14:29.761801+00:00
instance: kb-cron
parent: list-of-functional-analysis-topics-20260505-0
summary:
This is a list of functional analysis topics. See also: Glossary of functional analysis. == Hilbert space == == Functional analysis, classic results == == Operator theory == == Banach space examples == Lp space Hardy space Sobolev space Tsirelson space ba space == Real and complex algebras == == Topological vector spaces == == Amenability == Amenable group Von Neumann conjecture == Wavelets == == Quantum theory == See also list of mathematical topics in quantum theory == Probability == Free probability Bernstein's theorem == Non-linear == Fixed-point theorems in infinite-dimensional spaces == History == Stefan Banach (1892–1945) Hugo Steinhaus (1887–1972) John von Neumann (1903-1957) Alain Connes (born 1947) Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra


id: list-of-mathematical-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_functions-0.md
category: reference
title: List of mathematical functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_functions
date_saved: 2026-05-05T08:14:30.991878+00:00
instance: kb-cron
parent: list-of-mathematical-functions-20260505-0
summary:
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are "anonymous", with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions == Elementary functions == Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) === Algebraic functions === Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer (or equivalently rational) coefficients. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial. Rational functions: A ratio


id: list-of-types-of-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-0.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
In mathematics, functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. == Relative to set theory == These properties concern the domain, the codomain and the image of functions. Injective function: has a distinct value for each distinct input. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain. Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image. Also called a surjection or onto function. Bijective function: is both an injection and a surjection, and thus invertible. Identity function: maps any given element to itself. Constant function: has a fixed value regardless of its input. Empty function: whose domain equals the empty set. Set function: whose input is a set. Set-valued function: whose values are sets. Choice function called also selector or uniformizing function: assigns to each set one of its elements. == Relative to an operator (c.q. a group or other structure) == These properties concern how the function is affected by arithmetic operations on its argument. The following are special examples


id: list-of-types-of-functions-20260505-1
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-1.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
== Ways of defining functions/relation to type theory == Polynomial function: defined by evaluating a polynomial. Linear function; also affine function. Quadratic function Cubic function Quartic function Quintic function Rational function: ratio of two polynomial functions. In particular, Möbius transformation called also linear fractional function. Algebraic function: defined as the root of a polynomial equation. Transcendental function: analytic but not algebraic. Also hypertranscendental function. Composite function: is formed by the composition of two functions f and g, by mapping x to f (g(x)). Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value. Piecewise function: is defined by different expressions on different intervals. Computable function: an algorithm can do the job of the function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating what it should map to. For this purpose, the ↦ {\displaystyle \mapsto } symbol or Church's λ {\displaystyle \lambda } is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. f


id: list-of-examples-in-general-topology-20260505-0
file: data/en.wikipedia.org/wiki/List_of_examples_in_general_topology-0.md
category: reference
title: List of examples in general topology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_examples_in_general_topology
date_saved: 2026-05-05T08:14:33.479770+00:00
instance: kb-cron
parent: list-of-examples-in-general-topology-20260505-0
summary:
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete topology Double-pointed cofinite topology Extended real number line Finite topological space Hawaiian earring Hilbert cube Irrational cable on a torus Lakes of Wada Long line Order topology Lexicographical/dictionary order Ordinal number topology Real line Split interval Overlapping interval topology Moore plane Sierpiński space Sorgenfrey line Sorgenfrey plane Space-filling curve Topologist's sine curve Trivial topology Unit interval Zariski topology == See also == Counterexamples in Topology π-Base: An Interactive Encyclopedia of Topological Spaces


id: list-of-genetic-algorithm-applications-20260505-0
file: data/en.wikipedia.org/wiki/List_of_genetic_algorithm_applications-0.md
category: reference
title: List of genetic algorithm applications
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_genetic_algorithm_applications
date_saved: 2026-05-05T08:14:34.797128+00:00
instance: kb-cron
parent: list-of-genetic-algorithm-applications-20260505-0
summary:
This is a list of genetic algorithm (GA) applications. == Natural Sciences, Mathematics and Computer Science == Bayesian inference links to particle methods in Bayesian statistics and hidden Markov chain models Artificial creativity Chemical kinetics (gas and solid phases) Calculation of bound states and local-density approximations Code-breaking, using the GA to search large solution spaces of ciphers for the one correct decryption. Computer architecture: using GA to find out weak links in approximate computing such as lookahead. Configuration applications, particularly physics applications of optimal molecule configurations for particular systems like C60 (buckyballs) Construction of facial composites of suspects by eyewitnesses in forensic science. Data Center/Server Farm. Distributed computer network topologies Electronic circuit design, known as evolvable hardware Evolutionary image processing Feature selection for Machine Learning Feynman-Kac models File allocation for a distributed system Filtering and signal processing Finding hardware bugs. Game theory equilibrium resolution Genetic Algorithm for Rule Set Production Scheduling applications, including job-shop scheduling and scheduling in printed circuit board assembly. The objective being to schedule jobs in a sequence-dependent or non-sequence-dependent setup environment in order to maximize the volume of production while minimizing penalties such as tardiness. Satellite communication scheduling for the NASA Deep Space Network was shown


id: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
file: data/en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra-0.md
category: reference
title: List of geodesic polyhedra and Goldberg polyhedra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra
date_saved: 2026-05-05T08:14:36.120154+00:00
instance: kb-cron
parent: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
summary:
This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with m > 0 {\displaystyle m>0} and n ≥ 0 {\displaystyle n\geq 0} . T is the triangulation number, which is equal to T = m 2 + m n + n 2 {\displaystyle T=m^{2}+mn+n^{2}} . == Icosahedral == == Octahedral == == Tetrahedral == == References == Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4


id: outline-of-geometry-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_geometry-0.md
category: reference
title: Outline of geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_geometry
date_saved: 2026-05-05T08:14:37.296179+00:00
instance: kb-cron
parent: outline-of-geometry-20260505-0
summary:
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Modern geometry also extends into non-Euclidean spaces, topology, and fractal dimensions, bridging pure mathematics with applications in physics, computer science, and data visualization. == Types, methodologies, and terminologies of geometry. == Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic geometry Enumerative geometry Epipolar geometry Euclidean geometry Finite geometry Fractal geometry Geometry of numbers Hyperbolic geometry Incidence geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian geometry Ruppeiner geometry Solid geometry Spherical geometry Symplectic geometry Synthetic geometry Systolic geometry Taxicab geometry Toric geometry Transformation geometry Tropical geometry == General geometry concepts == === General concepts === Chirality Handedness Relative direction Mirror image Coordinate-free treatment Four-dimensional space Infinitesimal transformation Geometric progression Geometric shape Pi Angular velocity Linear velocity De Moivre's theorem Similar triangles


id: graduate-studies-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics-0.md
category: reference
title: Graduate Studies in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics
date_saved: 2026-05-05T08:14:38.533221+00:00
instance: kb-cron
parent: graduate-studies-in-mathematics-20260505-0
summary:
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published in hardcover and e-book formats. == List of books == == See also == Graduate Texts in Mathematics == Notes == == External links == Graduate Studies in Mathematics on AMS Bookstore ISSN 1065-7339


id: graduate-texts-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-0.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.


id: graduate-texts-in-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-1.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
== List of books == Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ISBN 978-1-4613-8170-9) Measure and Category – A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., ISBN 978-0-387-90508-2) Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ISBN 978-0-387-98726-2) A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., ISBN 978-0-387-94823-2) Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., ISBN 978-0-387-98403-2) Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, ISBN 978-3-540-90043-6) A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7) Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring, (1973, ISBN 978-3-540-90050-4) Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7) A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, ISBN 0-387-90056-X) Functions of One Complex Variable I, John B. Conway (1978, 2nd ed., ISBN 978-0-387-90328-6) Advanced Mathematical Analysis, Richard Beals (1973, ISBN 978-0-387-90065-0) Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, 2nd ed., ISBN 978-0-387-97845-1) Stable Mappings and Their Singularities, Martin Golubitsky, Victor Guillemin, (1974, ISBN 978-0-387-90072-8) Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian, (1974, ISBN


id: graduate-texts-in-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-2.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Koblitz (1993, 2nd ed., ISBN 978-0-387-97966-3) Representations of Compact Lie Groups, Theodor Bröcker, Tammo tom Dieck (1985, ISBN 978-3-540-13678-1) Finite Reflection Groups, L.C. Grove, C.T. Benson (1985, 2nd ed., ISBN 978-0-387-96082-1) Harmonic Analysis on Semigroups – Theory of Positive Definite and Related Functions, Christian Berg, Jens Peter Reus Christensen, Paul Ressel (1984, ISBN 978-0-387-90925-7) Galois Theory, Harold M. Edwards (1984, ISBN 978-0-387-90980-6) Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan (1984, ISBN 978-0-387-90969-1) Complex Analysis, Serge Lang (1999, 4th ed., ISBN 978-0-387-98592-3) Modern Geometry — Methods and Applications Part II: The Geometry and Topology of Manifolds, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1985, ISBN 978-0-387-96162-0) SL2(R), Serge Lang (1985, ISBN 978-0-387-96198-9) The Arithmetic of Elliptic Curves, Joseph H. Silverman (2009, 2nd ed., ISBN 978-0-387-09493-9) Applications of Lie Groups to Differential Equations, Peter J. Olver (2nd ed., 1993, ISBN 978-1-4684-0276-6) Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range (1986, ISBN 978-0-387-96259-7) Univalent Functions and Teichmüller Spaces, O. Lehto (1987, ISBN 978-1-4613-8654-4) Algebraic Number Theory, Serge Lang (1994, 2nd ed., ISBN 978-0-387-94225-4) Elliptic Curves, Dale Husemöller (2004, 2nd ed., ISBN 978-0-387-95490-5) Elliptic Functions, Serge Lang (1987, 2nd ed., ISBN 978-0-387-96508-6) Brownian Motion and Stochastic


id: graduate-texts-in-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-3.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Robert (2000, ISBN 978-0-387-98669-2) Theory of Bergman Spaces, Hakan Hedenmalm, Boris Korenblum, Kehe Zhu (2000, ISBN 978-0-387-98791-0) An Introduction to Riemann–Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen (2000, ISBN 978-1-4612-7070-6) Diophantine Geometry, Marc Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4) Introduction to Topological Manifolds, John M. Lee (2011, 2nd ed., ISBN 978-1-4419-7939-1) The Symmetric Group — Representations, Combinatorial Algorithms, and Symmetric Functions, Bruce E. Sagan (2001, 2nd ed., ISBN 978-0-387-95067-9) Galois Theory, Jean-Pierre Escofier (2001, ISBN 978-0-387-98765-1) Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-6) Problems in Analytic Number Theory, M. Ram Murty (2007, 2nd ed., ISBN 978-0-387-95143-0) Algebraic Graph Theory, Chris Godsil, Gordon Royle (2001, ISBN 978-0-387-95241-3) Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6) A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7) Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9) Algebra, Serge Lang (2002, Revised 3rd ed., ISBN 978-0-387-95385-4) Lectures on Discrete Geometry, Jiří Matoušek (2002, ISBN 978-0-387-95374-8) From Holomorphic Functions to Complex Manifolds, Klaus Fritzsche, Hans Grauert (2002, ISBN 978-0-387-95395-3) Partial Differential Equations, Jürgen Jost, (2013, 3rd ed., ISBN 978-1-4614-4808-2) Algebraic Functions and Projective Curves, David M. Goldschmidt, (2003, ISBN 978-0-387-95432-5) Matrices — Theory and Applications,


id: graduate-texts-in-mathematics-20260505-4
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-4.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Getz, Heekyoung Hahn (2024, ISBN 978-3-031-41151-9) An Invitation to Mathematical Logic, David Marker (2024, ISBN 978-3-031-55367-7) Fundamentals of Fourier Analysis, Loukas Grafakos (2024, ISBN 978-3-031-56499-4) A Course in Real Algebraic Geometry, Claus Scheiderer (2024, ISBN 978-3-031-69212-3) An Introduction to Microlocal Analysis, Peter Hintz (2025, ISBN 978-3-031-90705-0) Complex Analysis in One Variable and Riemann Surfaces, Mei-Chi Shaw, Charles M. Stanton (2025, ISBN 978-3-031-93641-8) Metric Lie Groups: Carnot-Carathéodory Spaces from the Homogeneous Viewpoint, Enrico Le Donne (2025, ISBN 978-3-031-98831-8) From Classical L-Functions to Modern Reciprocity Laws, Claus Sorensen (2025, ISBN 978-3-032-03034-4) Unitary Representations and Unitary Duals, Manfred Einsiedler, Thomas Ward (2025, ISBN 978-3-032-03898-2) Commutative Algebra, Shiro Goto, Kei-ichi Watanabe (2025, ISBN 978-981-95-4226-0) == See also == Graduate Studies in Mathematics == Notes == == External links == Springer-Verlag's Summary of Graduate Texts in Mathematics


id: list-of-graph-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graph_theory_topics-0.md
category: reference
title: List of graph theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graph_theory_topics
date_saved: 2026-05-05T08:14:41.041813+00:00
instance: kb-cron
parent: list-of-graph-theory-topics-20260505-0
summary:
This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory for basic terminology. == Examples and types of graphs == == Graph coloring == == Paths and cycles == == Trees == === Terminology === Node Child node Parent node Leaf node Root node Root (graph theory) === Operations === Tree structure Tree data structure Cayley's formula Kőnig's lemma Tree (set theory) (need not be a tree in the graph-theory sense, because there may not be a unique path between two vertices) Tree (descriptive set theory) Euler tour technique == Graph limits == Graphon == Graphs in logic == Conceptual graph Entitative graph Existential graph Laws of Form Logical graph == Mazes and labyrinths == Labyrinth Maze Maze generation algorithm == Algorithms == Ant colony algorithm Breadth-first search Depth-first search Depth-limited search FKT algorithm Flood fill Graph exploration algorithm Matching (graph theory) Max flow min cut theorem Maximum-cardinality search Shortest path Dijkstra's algorithm Bellman–Ford algorithm A* algorithm Floyd–Warshall algorithm Topological sorting Pre-topological order == Other topics == == Networks, network theory == See list of network theory topics == Hypergraphs == Helly family Intersection (Line) Graphs of hypergraphs


id: list-of-graphical-methods-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graphical_methods-0.md
category: reference
title: List of graphical methods
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graphical_methods
date_saved: 2026-05-05T08:14:42.275606+00:00
instance: kb-cron
parent: list-of-graphical-methods-20260505-0
summary:
This is a list of graphical methods with a mathematical basis. Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization. There is also a list of computer graphics and descriptive geometry topics. == Simple displays == Area chart Bar chart Histogram Variable-width bar chart Box plot Dispersion fan diagram Graph of a function Logarithmic graph paper Heatmap Line chart Pie chart Variable-radius pie chart Plotting Radar chart Scatterplot Sparkline Spiral graphic Stemplot Stripe graphic Tape diagram == Set theory == Venn diagram Karnaugh diagram == Descriptive geometry == Isometric projection Orthographic projection Perspective (graphical) == Engineering drawing == Technical drawing Graphical projection Mohr's circle Pantograph Circuit diagram Smith chart Sankey diagram == Systems analysis == Binary decision diagram Control-flow graph Functional flow block diagram Information flow diagram IDEF N2 chart Sankey diagram State diagram System context diagram Data-flow diagram == Cartography == Map projection Orthographic projection (cartography) Robinson projection Stereographic projection Dymaxion map Topographic map Craig retroazimuthal projection Hammer retroazimuthal projection == Biological sciences == Cladogram Punnett square Systems Biology Graphical Notation == Physical sciences == Free body diagram Greninger chart Phase diagram Wavenumber-frequency diagram Bode plot Nyquist plot Dalitz plot Feynman diagram Carnot Plot ==


id: list-of-graphs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graphs-0.md
category: reference
title: List of graphs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graphs
date_saved: 2026-05-05T08:14:43.462282+00:00
instance: kb-cron
parent: list-of-graphs-20260505-0
summary:
This partial list of graphs contains definitions of graphs and graph families. For collected definitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs. Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts. == Individual graphs == == Highly symmetric graphs == === Strongly regular graphs === The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ). === Symmetric graphs === A symmetric graph is one in which there is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric 3-regular graphs. Every strongly regular graph is symmetric, but not vice versa. === Semi-symmetric graphs === == Graph families == === Complete graphs === The complete graph on n {\displaystyle n} vertices is often called the n {\displaystyle n} -clique


id: list-of-group-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_group_theory_topics-0.md
category: reference
title: List of group theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_group_theory_topics
date_saved: 2026-05-05T08:14:44.910624+00:00
instance: kb-cron
parent: list-of-group-theory-topics-20260505-0
summary:
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. == Structures and operations == == Basic properties of groups == === Group homomorphisms === == Basic types of groups == === Simple groups and their classification === Classification of finite simple groups === Permutation and symmetry groups === == Concepts groups share with other mathematics == == Mathematical objects making use of a group operation == == Mathematical


id: list-of-small-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_small_groups-0.md
category: reference
title: List of small groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_small_groups
date_saved: 2026-05-05T08:14:46.116505+00:00
instance: kb-cron
parent: list-of-small-groups-20260505-0
summary:
The following list in mathematics contains the finite groups of small order up to group isomorphism. == Counts == For n = 1, 2, … the number of nonisomorphic groups of order n is 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence A000001 in the OEIS) For labeled groups, see (sequence A034383 in the OEIS). == Glossary == Each group is named by Small Groups Library as Goi, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of Z/nZ) Dihn: the dihedral group of order 2n (often the notation Dn or D2n is used) Dn: the dihedral group of order 2n, the same as Dihn (notation used in section List of small non-abelian groups) K4: the Klein four-group of order 4, isomorphic to Z2 × Z2 and Dih2 Sn: the symmetric group of degree n, containing the n! permutations of n elements An: the alternating group of degree n, containing the


id: list-of-harmonic-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_harmonic_analysis_topics-0.md
category: reference
title: List of harmonic analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_harmonic_analysis_topics
date_saved: 2026-05-05T08:14:47.331318+00:00
instance: kb-cron
parent: list-of-harmonic-analysis-topics-20260505-0
summary:
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematical physics and engineering. == Fourier analysis == === Fourier series === Periodic function Trigonometric function Trigonometric polynomial Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem Weyl differintegral Generalized Fourier series Orthogonal functions Orthogonal polynomials Empirical orthogonal functions Set of uniqueness === Fourier transform === Continuous Fourier transform Fourier inversion theorem Plancherel's theorem Convolution Convolution theorem Positive-definite function Poisson summation formula Paley–Wiener theorem Sobolev space Time–frequency representation Quantum Fourier transform == Topological groups == Topological abelian group Haar measure Discrete Fourier transform Dirichlet character Amenable group Von Neumann's conjecture Pontryagin duality Kronecker's theorem on diophantine approximation Almost periodic function Bohr compactification Wiener's tauberian theorem == Representation theory == Representation of a Lie group Unitary representation Irreducible representation Harish-Chandra character Restricted representation Induced representation Peter–Weyl theorem Spherical harmonic Casimir operator Hecke operator Stone–von Neumann theorem Discrete series representation Tempered representation Langlands program == Fast Fourier transform == Bluestein's FFT algorithm Cooley–Tukey FFT algorithm Rader's FFT algorithm Number-theoretic transform Irrational base discrete weighted


