Mathematics and statistics

Abc conjecture

The abc conjecture, also known as the Oesterlé–Masser conjecture, is a 1985 conjecture in number theory about coprime integers satisfying a + b = c.

Abductive reasoning

Abductive reasoning, or inference to the best explanation, is a form of inference that reasons from an observation to the hypothesis that best explains it.

Abel Prize

The Abel Prize is an annual international prize in mathematics, established by Norway's parliament in 2002 to honor Niels Henrik Abel, and first awarded in 2003.

Abel–Jacobi map

The Abel–Jacobi map is the canonical homomorphism sending the degree-zero divisor classes of an algebraic curve to its Jacobian variety, defined by integrating holomorphic differentials along paths.

Abel–Ruffini theorem

The Abel–Ruffini theorem, also known as Abel's impossibility theorem, states that no formula in radicals can solve general polynomial equations of degree five or higher.

Abelian category

An abelian category is a mathematical category, axiomatized by Alexander Grothendieck in 1957, in which kernels, images, and exact sequences behave as they do for modules over a ring.

Abelian extension

An abelian extension is a Galois field extension whose Galois group is abelian; class field theory classifies such extensions of number fields by ray class groups.

Abelian group

An abelian group, or commutative group, is a group in which the group operation is commutative: xy = yx for all elements, named for Niels Henrik Abel.

Abelian variety

An abelian variety is a complete algebraic variety equipped with a commutative group structure, a higher-dimensional generalization of elliptic curves central to algebraic geometry and cryptography.

Abelianization

Abelianization is a construction in group theory that turns a group into its maximal abelian quotient by dividing out the subgroup generated by all commutators.

Abraham Fraenkel (אברהם הלוי פרנקל)

Abraham Fraenkel, born Adolf Abraham Halevi Fraenkel (1891–1965), was a German-born Israeli mathematician whose axiom of replacement produced the standard Zermelo–Fraenkel axioms of set theory.

Abraham Wald (Wald Ábrahám; 1902–1950) was a Hungarian mathematician and statistician who founded sequential analysis and whose analysis of returning bomber damage became the classic example of survivorship bias.

Abscissa and ordinate

In the Cartesian plane, the abscissa is a point's x-coordinate, its signed distance from the y-axis, and the ordinate is the y-coordinate, its signed distance from the x-axis.

Absolute continuity

Absolute continuity is a strengthening of continuity in analysis: a function is absolutely continuous on an interval, and a measure μ is absolutely continuous with respect to ν when ν(A) = 0 implies μ(A) = 0.

Absolute convergence

Absolute convergence is a property of an infinite series in which the series of its terms' absolute values converges; every absolutely convergent series can be rearranged without changing its sum.

Absolute Galois group

The absolute Galois group of a field K is the Galois group of its separable closure, a profinite topological group that packages the Galois theory of all algebraic extensions.

Absolute nodal coordinate formulation

The absolute nodal coordinate formulation (ANCF) is a nonlinear finite element method for flexible multibody dynamics, initiated in 1996 by Ahmed A. Shabana at the University of Illinois Chicago.

Absolute value

Absolute value, also called modulus, is the magnitude of a real or complex number without regard to its sign, written |x|, the distance from zero on the number line.

Absoluteness (mathematical logic)

In mathematical logic, absoluteness is the property of a formula whose truth value stays the same when evaluated in different models or inner structures, such as forcing extensions.

Abstract algebra

Abstract algebra, also called modern algebra, is the branch of mathematics that studies algebraic structures like groups, rings, and fields, rather than computing with particular number systems.

Abstract Wiener space

An abstract Wiener space is a Banach space with a Hilbert space and Gaussian measure, introduced by Leonard Gross in the 1960s as the framework for infinite-dimensional Gaussian analysis.

Abundant number

An abundant number, or excessive number, is a positive integer whose proper divisors add up to more than the number itself, the smallest being 12.

Accelerated aging

Accelerated aging is the testing of products at elevated stress, such as higher temperature or humidity, so that years of normal-use failures appear within days or weeks.

Accelerated failure time model

The accelerated failure time model is a survival analysis regression model in which covariates act on time to an event rather than on the hazard, via an acceleration factor.

Acceptable quality limit

The acceptable quality limit, also called the acceptable quality level, is the worst tolerable process average in acceptance sampling, indexed by standards like ISO 2859-1.

Acceptance sampling

Acceptance sampling is a statistical quality control method that inspects a random sample from a product lot to decide whether to accept or reject the whole lot.

Ackermann function

The Ackermann function, also called the Ackermann-Peter function, is a computable function that grows faster than every primitive recursive function, defined by Wilhelm Ackermann in 1928.

Action semantics

Action semantics is a framework for the formal semantics of programming languages, introduced by Peter D. Mosses in 1992, describing program behavior with composable actions.

Activation likelihood estimation

Activation likelihood estimation (ALE) is a coordinate-based meta-analytic method combining peak activation coordinates from many fMRI and PET experiments into a single statistical map of consistent activation.