6,052 articles
Algebraic number theory
Algebraic number theory studies algebraic integers in finite-degree number fields over Q, using rings of integers, ideal class groups, Dirichlet's unit theorem, class field theory, and zeta and L-functions; it arose from Fermat's last theorem and differs from elementary number theory by restoring unique factorization through ideals.
Algebraic quantum field theory
Algebraic quantum field theory, also called local quantum physics, is a mathematical framework for relativistic quantum physics proposed by Rudolf Haag and Daniel Kastler in 1964.
Algebraic Riccati equation
The algebraic Riccati equation, also called the matrix Riccati equation, is a nonlinear matrix equation central to optimal control, including the linear quadratic regulator and Kalman filtering.
Algebraic stack
An algebraic stack, also called an Artin stack, is a stack in groupoids over schemes that locally looks like a scheme, used for moduli problems.
Algebraic structure
An algebraic structure, or algebraic system, is a set equipped with operations satisfying specified axioms; examples include groups, rings, fields, lattices, and vector spaces, studied collectively by universal algebra.
Algebraic topology
Algebraic topology is the branch of mathematics that studies topological spaces by assigning them algebraic objects like groups; its tools range from the fundamental group to persistent homology in data analysis.
Algebraic variety
An algebraic variety, also called an algebraic set, is the set of solutions of polynomial equations over a field and the central object of algebraic geometry.
Algebraically closed field
An algebraically closed field is a field in which every non-constant polynomial with coefficients in the field has a root in it, like the complex numbers.
Algorithmic Combinatorics on Partial Words
Algorithmic Combinatorics on Partial Words is a 2008 research monograph by Francine Blanchet-Sadri, published by Chapman & Hall/CRC and the first devoted to partial words.
Algorithmic probability
Algorithmic probability, also called Solomonoff or universal probability, is a mathematical prior introduced by Ray Solomonoff in 1960 that assigns shorter strings higher probability and underlies Solomonoff induction.
Algorithms for calculating variance
Algorithms for calculating variance are the numerical procedures used to compute variance, whose naive one-pass formula loses precision while Welford's and pairwise methods stay stable.
All models are wrong
"All models are wrong, but some are useful" is an aphorism attributed to statistician George E. P. Box, holding that models are judged by usefulness, not truth.
Allen–Cahn equation
The Allen–Cahn equation, also known as the time-dependent Ginzburg–Landau equation, is a second-order parabolic partial differential equation describing antiphase boundary motion, introduced by Samuel Allen and John Cahn in 1979.
Almost complex structure
An almost complex structure on a smooth manifold is a smooth field of linear maps J with J² = −id; it is integrable exactly when its Nijenhuis tensor vanishes.
Almost ideal demand system
The Almost Ideal Demand System (AIDS) is an econometric model that relates budget shares to prices and expenditure, introduced by Deaton and Muellbauer in 1980.
Almost integer
An almost integer is a real number very close to, but not equal to, an integer. The flagship example, e^(π√163), differs from 262537412640768744 by less than 10^-12.
Almost sure convergence
Almost sure convergence, also known as convergence with probability 1, means a sequence of random variables converges pointwise for every outcome except a set of probability zero.
Almost surely
Almost surely, also written with probability 1 or a.s., is a probability theory phrase meaning a property holds except on a set of probability zero; the strong law of large numbers is its central example.
Alpha recursion theory
Alpha recursion theory (Α-recursion theory), also called admissible recursion theory, is a branch of mathematical logic generalizing computability theory from the natural numbers to admissible ordinals.
Alternating direction method of multipliers
The alternating direction method of multipliers (ADMM) is an algorithm for convex optimization that splits large problems into smaller subproblems, with applications to the lasso, basis pursuit, and support vector machines.
Alternating group
The alternating group A_n is the group of even permutations of n objects, of order n!/2, and is simple, with no proper nontrivial normal subgroups, for every n at least 5.
Alternating optimization
Alternating optimization is a numerical method that minimizes a multivariable function by optimizing blocks of parameters in turn, underpinning the EM algorithm and alternating least squares.
Alternating-time temporal logic
Alternating-time temporal logic (ATL) is a modal temporal logic stating what groups of agents can enforce by strategy, introduced by Alur, Henzinger, and Kupferman and published in full in 2002.
Alternative algebra
Alternative algebra, also known as alternative ring, is an algebra in which every two elements generate an associative subalgebra; the octonions are the standard non-associative example.
Altitude (triangle)
An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side; the three altitudes meet at a point called the orthocenter.
AM–GM inequality
The AM–GM inequality (inequality of arithmetic and geometric means) states that the arithmetic mean of positive numbers is never below their geometric mean, with equality only when all numbers are equal.
Amenable Banach algebra
An amenable Banach algebra is a Banach algebra in which every bounded derivation into a dual bimodule is inner; the notion was introduced by B. E. Johnson in 1972.
Amenable group
An amenable group is a group admitting a left-invariant mean on bounded functions, introduced by John von Neumann in 1929 while studying the Banach–Tarski paradox.
American Community Survey
The American Community Survey (ACS) is a monthly U.S. Census Bureau survey covering more than 40 topics, including income, education, and housing, for communities nationwide since 2005.
American Mathematics Competitions
The American Mathematics Competitions (AMC) are secondary-school math contests administered by the Mathematical Association of America since 1950, leading through the AIME to the USA Mathematical Olympiad and the International Mathematical Olympiad team.