Mathematics and statistics

Monotone convergence theorem

The monotone convergence theorem, or Beppo Levi's theorem, is a result of Lebesgue integration proved in 1906 stating that the integral of the limit of increasing nonnegative functions equals the limit of their integrals.

Monotonic function

A monotonic function, also called a monotone function, is a real function that preserves or reverses order, so its outputs move in one fixed direction.

Monster group

The Monster group, also called the friendly giant, is the largest of the 26 sporadic finite simple groups, with order about 8 × 10^53, predicted by Fischer and Griess in 1973.

Monstrous moonshine

Monstrous moonshine, or the Conway–Norton conjecture, is the unexpected connection between the monster group and modular functions that Richard Borcherds proved in 1992, earning a Fields Medal.

Monte Carlo dropout

Monte Carlo dropout is a technique for estimating a neural network's prediction uncertainty by keeping dropout active at inference and averaging multiple stochastic forward passes.

Monte Carlo estimation

Monte Carlo estimation estimates integrals, expectations, and probabilities by averaging a function over many random draws, with error shrinking as one over the square root of the sample count.

Monte Carlo localization

Monte Carlo localization (MCL) is a probabilistic algorithm, introduced in 1999 by Fox, Burgard, Dellaert, and Thrun, that estimates a robot's position from sensor data and a map.

Monte Carlo method

The Monte Carlo method, also known as Monte Carlo simulation, is a class of computational algorithms that rely on repeated random sampling to estimate quantities impractical to compute exactly.

Monte Carlo sampling

Monte Carlo sampling is a statistical method that estimates hard-to-compute sums, integrals, and probabilities by averaging many random samples from a probability model, with error shrinking as 1/√N.

Montgomery modular multiplication

Montgomery modular multiplication, or Montgomery reduction, is an algorithm introduced by Peter L. Montgomery in 1985 that computes products modulo an integer without division, enabling fast modular arithmetic in public-key cryptography.

Monty Hall problem

The Monty Hall problem, also known as the Monty Hall paradox, is a probability puzzle in which switching doors after the host reveals a goat wins with probability 2/3.

Moonshine module

The moonshine module is an infinite-dimensional vertex operator algebra constructed by Frenkel, Lepowsky, and Meurman, on which the Monster group acts as its full automorphism group.

MOORA method

The MOORA method, a multi-criteria decision making method introduced by Willem K. Brauers and Edmundas Kazimieras Zavadskas in 2006, ranks alternatives by normalizing criteria values and combining them into a single appraisal score.

Moore graph

A Moore graph is a regular graph of given degree and diameter containing the maximum possible number of vertices, attaining the Moore bound; examples include the Petersen graph.

Moore–Penrose inverse

The Moore–Penrose inverse, also called the Moore–Penrose pseudoinverse, is a generalized matrix inverse that exists uniquely for any matrix, and reduces to the ordinary inverse for nonsingular matrices.

Moran process

The Moran process, or Moran model, is a stochastic model of genetic drift in a finite population, introduced by P. A. P. Moran in 1958.

Moran's I

Moran's I is a statistic measuring spatial autocorrelation, whether similar values occur near each other, introduced by P. A. P. Moran in Biometrika in 1950.

Mordell–Weil theorem

The Mordell–Weil theorem states that the group of rational points of an abelian variety over a number field is finitely generated; Mordell proved the elliptic curve case in 1922.

Morita equivalence

Morita equivalence is an equivalence relation on rings, introduced by Kiiti Morita in 1958, holding when two rings have equivalent module categories, so module properties transfer; a ring and its matrix rings are equivalent.

Morley rank

Morley rank is an ordinal-valued dimension assigned to definable sets in a model, introduced by Michael Morley in 1965; in algebraically closed fields it equals Zariski dimension.

Morphism

A morphism is an abstract arrow from one object of a category to another, with composition and identities, defined without reference to functions or sets.

Morse homology

Morse homology is a homology theory for a smooth manifold built from a Morse function's critical points and gradient flow lines between them, canonically isomorphic to singular homology.

Morse theory

Morse theory is a mathematical method that recovers the topology of a smooth manifold from the critical points of a smooth real-valued function on it.

Morse–Kelley set theory

Morse–Kelley set theory (MK), also called Kelley–Morse set theory, is an axiomatic set theory with sets and classes whose class comprehension allows quantification over proper classes.

Mortality rate

A mortality rate measures how often death occurs in a population, calculated as deaths divided by population size over a set period, expressed per 1,000 or 100,000.

Moser spindle

The Moser spindle, or Moser graph, is a 7-vertex, 11-edge unit-distance graph requiring four colors, introduced by Leo and William Moser in 1961 as the first 4-chromatic unit-distance graph.

Motion estimation

Motion estimation is the computer vision task of computing the displacement of image content between video frames, producing motion vectors or dense optical flow fields.

Motive (algebraic geometry)

In algebraic geometry, a motive is a conjectural elementary building block of algebraic varieties, proposed by Alexander Grothendieck in 1964 so that all cohomology theories factor through the category of motives.

Motivic cohomology

Motivic cohomology is a cohomology theory for algebraic varieties, with bidegrees H^{p,q}, recovering Chow groups, the Picard group, and Milnor K-theory; Voevodsky realized it in the late 1990s.