Mathematics

Maximum a posteriori estimation selects a parameter value that maximizes its posterior probability or density, combining observed data with a prior distribution.

Maximum likelihood estimation selects model parameters that maximize the likelihood of observed data under a specified statistical model.

Mean squared error measures estimation or prediction error by averaging squared differences between estimated and reference values.

The mean value theorem states that a differentiable real function attains its average rate of change at some interior point of an interval.

A measurable function is a map whose inverse images of measurable sets are measurable, providing a foundation for integration and probability.

A measurable space is a set equipped with a sigma-algebra specifying which subsets are measurable, without assigning them a measure.

Measure theory studies generalized notions of size for sets and provides a rigorous foundation for integration and probability.

The method of exhaustion establishes exact areas and volumes through finite geometric approximations whose errors can be made arbitrarily small.

A metric completion embeds a metric space densely and isometrically into a complete metric space, supplying precisely its missing Cauchy limits.

A metric space is a set equipped with a distance function that satisfies positivity, symmetry, and the triangle inequality.

A metric tensor is a smoothly varying symmetric bilinear form that defines local geometric measurements on a manifold, including lengths, angles, and spacetime intervals.

A mixed strategy is a probability distribution over a player’s pure strategies, allowing randomized choice in a game.

Model theory studies mathematical structures through formal languages, examining truth, definability, and the classification of models satisfying specified axioms.

Modular arithmetic studies integer calculations in which numbers differing by a multiple of a fixed modulus are treated as equivalent.

A module is an abelian group equipped with scalar multiplication by a ring, generalizing vector spaces and unifying many structures in algebra.

A function that consistently preserves or reverses the ordering of its inputs, with strict and non-strict variants.

A multilinear map is a function of several vector arguments that is linear in each argument separately when all the others are fixed.

Multiple testing concerns inference on several hypotheses and methods for controlling false-positive errors across the resulting collection of decisions.

A probability distribution for random vectors whose every linear combination is normally distributed, characterized by a mean vector and covariance matrix.

Mutual information measures statistical dependence by quantifying how much observing one random variable reduces uncertainty about another.

A strategy profile in a game in which no player can improve their payoff by changing their strategy alone.

A neighborhood of a point is a set containing an open set that contains the point, expressing proximity without requiring a distance.

Nested cross-validation separates model selection from performance estimation by placing an inner validation procedure inside an outer evaluation loop.

A net is a family indexed by a directed set that generalizes sequences and characterizes convergence in arbitrary topological spaces.

Newton’s method is an iterative numerical technique for solving nonlinear equations through successive local linear approximations.

The noisy-channel coding theorem identifies channel capacity as the limiting rate for communication with arbitrarily small decoding error.

Non-Euclidean geometry studies spaces that depart from Euclidean geometry, especially hyperbolic and elliptic spaces with different rules for parallel lines.

Norbert Wiener was an American mathematician who helped establish cybernetics and made foundational contributions to probability, mathematical analysis, and statistical prediction.

A norm measures the size of a vector, generalizing length through definiteness, absolute homogeneity, and the triangle inequality.

A continuous probability distribution defined by its mean and variance, with a symmetric bell-shaped density and a central role in statistical inference.

A normal subgroup is a subgroup invariant under conjugation, precisely the condition that allows its cosets to form a quotient group.

A normed vector space is a real or complex vector space equipped with a norm that measures vector size and induces a compatible metric.

A null hypothesis is a statistical claim used as a reference model for evaluating whether observed data warrant its rejection.

A null set is a measurable set assigned measure zero, expressing negligible size relative to a specified measure.

The null space of a linear map is the vector subspace of inputs mapped to zero, describing homogeneous solutions and nonuniqueness.

A number is a mathematical object used to count, measure and label. Number systems range from the natural numbers to the complex numbers and beyond.

Number theory studies integers, prime numbers, divisibility, and arithmetic equations, using elementary, algebraic, analytic, geometric, and computational methods.

Numerical analysis studies algorithms for approximating mathematical solutions and evaluates their accuracy, stability, convergence, and computational efficiency.

Numerical linear algebra studies accurate and efficient computational methods for solving linear systems, least-squares problems, and matrix eigenvalue problems.

Numerical stability describes how a computational method controls the propagation and amplification of errors, particularly those introduced by finite-precision arithmetic.

A mathematical function whose value is minimized or maximized to select the best feasible solution to an optimization problem.

An odds ratio compares the odds of an event between two groups and measures association between binary variables.

An open ball is the set of points whose distance from a specified center is strictly less than a positive radius.

An open set is a subset belonging to a topology, generalizing the idea that every point has a surrounding region contained within the set.

An ordered pair is a mathematical object with two components whose positions distinguish them and determine equality.

An ordinal number represents the order type of a well-ordered set, extending finite positions and counting into the transfinite.

Ordinary least squares estimates linear regression coefficients by minimizing the sum of squared differences between observed and fitted responses.

The orthogonal complement of a subset of an inner product space consists of all vectors perpendicular to every vector in that subset.

A real square matrix whose transpose equals its inverse, representing a linear transformation that preserves Euclidean lengths and angles.

Orthogonal projection maps a vector to its nearest point in a subspace, leaving a residual perpendicular to that subspace.

An orthonormal basis is a complete collection of mutually orthogonal unit vectors that provides coordinates through inner products.

A p-value measures how extreme an observed test statistic is under a specified null hypothesis and statistical model.

A partial derivative measures how a multivariable function changes with one input while its other inputs remain fixed.

A partial differential equation relates a function of several independent variables to its partial derivatives, describing spatial variation, evolution, and other multivariable relationships.

A partial order is a reflexive, antisymmetric, transitive relation that permits some pairs of elements to be incomparable.

The Peano axioms characterize natural numbers through zero, succession, and induction, providing a foundation for formal arithmetic.

Pierre-Simon Laplace was a French mathematician and astronomer whose work transformed celestial mechanics, probability theory, and mathematical physics.

Pointwise convergence is convergence of a sequence of functions at each fixed point of their common domain, without requiring uniform control across that domain.

The Poisson distribution is a discrete probability distribution for event counts, characterized by a single parameter equal to both its mean and variance.

Poisson's equation relates the Laplacian of an unknown function to a prescribed source, describing potentials and steady-state fields.