Mathematics and statistics

Möbius transformation

A Möbius transformation, also called a Möbius map, is a function (az+b)/(cz+d) with ad − bc ≠ 0 that maps the extended complex plane conformally to itself.

Modal algebra

A modal algebra, also called a Magari algebra, is a Boolean algebra equipped with a normal operator representing a modal connective, providing the algebraic semantics of propositional modal logic.

Modal logic

Modal logic is a family of formal logics extending classical logic with operators for necessity and possibility, plus logics of obligation, time, and belief, originating with C.I. Lewis in 1912.

Modal logic D

Modal logic D, also written KD, adds to K the axiom that what is necessary is possible, making it the modal basis of Standard Deontic Logic.

Modal logic of agency

The modal logic of agency formalizes what agents do, can do, and are responsible for, chiefly through Belnap and Perloff's "seeing to it that" (stit) operator.

Modal realism

Modal realism is the metaphysical thesis that possible worlds are concrete entities like our own world, associated above all with David Lewis's 1986 book On the Plurality of Worlds.

Modal μ-calculus

The modal μ-calculus is a temporal logic of programs extending modal logic with least and greatest fixed-point operators, used to model check recursive correctness properties.

Mode (statistics)

The mode in statistics is the value that occurs most often in a data set, one of the three classical measures of central tendency alongside the mean and the median.

Mode effect

A mode effect is a systematic difference in survey results arising from how data are collected, covering selection effects on who responds and measurement effects on how.

Model category

A model category is a category with three classes of morphisms, weak equivalences, fibrations, and cofibrations, introduced by Daniel Quillen in 1967 as an axiomatic framework for homotopy theory.

Model checking

Model checking is a formal verification technique that automatically checks whether a finite model of a system satisfies a temporal logic property, producing counterexamples on failure.

Model companion

A model companion of a first-order theory T is a model complete theory whose models each embed in models of T and vice versa, introduced by Abraham Robinson.

Model of computation

A model of computation is a formal mathematical description of what a computing device can do, the most famous being the Turing machine, introduced by Alan Turing in 1936.

Model theory

Model theory, also called the theory of models, is the branch of mathematical logic studying how formal languages are interpreted, especially first-order languages and their structures.

Model-based clustering is a statistical method that groups data by fitting a finite mixture model, usually of multivariate normal distributions, fitted with the EM algorithm and choosing the number of clusters by BIC.

Moderate deviations

Moderate deviations are asymptotic results in probability theory describing tail probabilities at intermediate scales, between the central limit theorem and large deviations, where Gaussian approximations still hold.

Moderated mediation

Moderated mediation, also known as moderated mediation analysis, is a statistical method testing whether the indirect effect of a variable through a mediator varies with a moderator, coined by James and Brett in 1984.

Moderation analysis

Moderation analysis is a regression-based statistical method for testing whether the relationship between an independent and dependent variable changes as a function of a third variable, the moderator.

Modern portfolio theory

Modern portfolio theory, also known as Markowitz portfolio theory, is a framework for choosing investments by balancing expected return against risk, introduced by Harry Markowitz in 1952.

Modest set

A modest set is a set whose elements have computational realizers, each realizer determining at most one element, serving as a notion of data type in realizability.

Modular arithmetic

Modular arithmetic, also called clock arithmetic, treats integers as congruent when they leave the same remainder; Gauss formalized it in 1801, and it underlies RSA cryptography.

Modular curve

A modular curve is an algebraic curve whose points parameterize elliptic curves with extra level structure, such as a point of order N, central to the arithmetic of elliptic curves.

Modular decomposition

Modular decomposition, also known as substitution decomposition, is a graph algorithm that recursively partitions a graph into modules, vertex sets sharing the same outside neighbors, and returns a tree.

Modular exponentiation

Modular exponentiation, also called Modpow, computes the remainder when a base raised to an exponent is divided by a modulus, and is the core operation of RSA and Diffie-Hellman.

Modular form

A modular form is a holomorphic function on the complex upper half-plane that transforms in a prescribed way under a subgroup of the modular group, holomorphic at the cusps.

Modular function

A modular function is a meromorphic function on the upper half-plane, invariant under a subgroup of the modular group and meromorphic at the cusps; the j-invariant is the principal example.

Modular group

The modular group is the group PSL(2, ℤ) of 2 × 2 integer matrices with determinant 1, acting on the upper half-plane and central to number theory.

Modular lattice

A modular lattice is a lattice satisfying the modular law, a weakening of distributivity formulated by Richard Dedekind around 1900; examples include subspaces of a vector space and normal subgroups of a group.

Modular multiplicative inverse

The modular multiplicative inverse, also called the modular inverse, is the analog of division in modular arithmetic: an integer b⁻¹ with b·b⁻¹ ≡ 1 (mod m). It exists when b and m are relatively prime.