Mathematics

A joint probability distribution describes the simultaneous behavior of two or more random variables, including their individual distributions and dependence relationships.

K-means clustering partitions numerical observations into a specified number of groups by minimizing their squared distances from cluster means.

Karl Weierstrass was a German mathematician whose work on rigorous foundations, complex functions, and approximation helped shape modern mathematical analysis.

First-order conditions characterizing constrained optima under suitable regularity assumptions, and certifying global optimality in convex problems.

The kernel of a linear map is the subspace of its domain consisting of all vectors mapped to the zero vector.

A family of mathematical learning techniques that use kernel functions to perform computations in implicit feature spaces.

Kripke semantics interprets logical formulas at worlds or states connected by accessibility relations, providing models for modal and intuitionistic logics.

Kullback–Leibler divergence measures the discrepancy between probability distributions through an expected logarithmic probability ratio.

Austrian-born American logician whose completeness, incompleteness, and set-theoretic results transformed the foundations of mathematics.

Lagrangian duality associates a constrained optimization problem with a multiplier-based problem that provides bounds and, under suitable conditions, optimality certificates.

A second-order differential operator that measures local spatial variation and underlies equations of potential theory, diffusion, waves, and quantum mechanics.

Laplace’s equation is a linear partial differential equation describing harmonic functions and source-free equilibrium fields.

The law of excluded middle asserts that every proposition satisfies the disjunction “P or not P,” a defining principle of classical logic.

A family of probability theorems describing when averages of many random observations converge to their expected value.

The Lebesgue integral defines integration through measure, extending classical integration and providing powerful theorems for convergence and probability.

Lebesgue measure extends length, area, and volume to a broad class of sets, providing a foundation for modern integration and probability.

The Levi-Civita connection is the unique torsion-free connection that preserves the metric on a Riemannian or pseudo-Riemannian manifold.

A likelihood function evaluates how observed data support different parameter values within a specified statistical model.

A limit describes the value approached by a function or sequence under a precisely specified process of convergence.

A line integral accumulates a scalar quantity or the tangential component of a vector field along a curve.

A numerical optimization procedure that selects a step size along a prescribed search direction.

Linear algebra studies vector spaces, linear maps, and systems of linear equations, providing tools for geometry, scientific computation, and data analysis.

A linear combination is a finite sum of vectors or other vector-space elements multiplied by scalar coefficients.

A linear functional is a scalar-valued linear map on a vector space, fundamental to duality, geometry, and functional analysis.

Linear independence is the property that a family of vectors has no nontrivial linear combination equal to zero.

A linear map is a function between vector spaces that preserves vector addition and scalar multiplication.

Linear programming optimizes a linear objective subject to linear constraints on continuous decision variables.

A statistical method that models a response variable as a linear combination of predictors and estimates relationships or predicts numerical outcomes.

Linear separability is the property that two classes of points can be placed on opposite sides of a single hyperplane.

The linear span of a set of vectors consists of all their finite linear combinations and is the smallest linear subspace containing them.

A linear subspace is a subset of a vector space that is itself a vector space under the inherited operations.

A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication.

Lipschitz continuity bounds changes in a function’s output by a fixed multiple of changes in its input.

The logistic function is an S-shaped mathematical function used to describe bounded growth, transform log-odds into probabilities, and provide nonlinear activation in neural networks.

Logistic regression models categorical outcomes by relating their probabilities to linear combinations of explanatory variables through the logistic function.

The logit is the natural logarithm of the odds of an event, mapping probabilities between zero and one to the real line.

A loss function assigns a numerical penalty to a decision or prediction, defining the errors that statistical and machine-learning procedures seek to minimize.

A fundamental result of first-order logic establishing the existence of models of different infinite cardinalities and elementary substructures of controlled size.

An Lp space is a space of measurable functions, identified up to equality almost everywhere, whose magnitude satisfies an integrability or essential boundedness condition.

LU decomposition expresses a matrix as triangular factors, usually with permutations, enabling efficient solution of linear systems and related computations.

A manifold is a space that locally resembles Euclidean space, providing a framework for studying shapes, coordinates, and geometric structures.

Manifold learning identifies low-dimensional structure in high-dimensional data through nonlinear representations that preserve selected geometric or neighborhood relationships.

Marginal likelihood is the probability or density of observed data under a model, obtained by averaging its likelihood over a prior distribution.

A Markov chain is a stochastic process whose future evolution, conditional on its present state, does not depend on its past states.

A family of sampling methods that uses Markov chains to approximate probability distributions and expectations.

A Markov decision process models sequential choices under uncertainty, using states, actions, transition probabilities, and rewards to define and optimize decision-making policies.

The Markov property states that, conditional on a process’s present state, its future evolution does not depend on its past history.

A martingale is an integrable stochastic process whose conditional expected future value equals its present value, relative to a specified information flow.

Mathematical analysis studies limits, continuity, differentiation, integration, and the behavior of functions and infinite processes.

A deductive proof method that establishes a statement for every natural number by proving an initial case and a general step to the next case.

Mathematical optimization studies how to find the best feasible solution to a problem defined by an objective function and constraints.

A mathematical proof is a deductive argument that shows a statement follows necessarily from accepted axioms, definitions and previously proven theorems.

Mathematics studies numbers, structures, space, and change through abstraction, logical reasoning, and proof, providing foundations for quantitative inquiry and computation.

A matrix is a rectangular array of mathematical entries used to represent linear transformations, systems of equations, and structured data.

Matrix diagonalization expresses a square matrix in an eigenvector basis, reducing its action to independent scalar multiplications along coordinate directions.

Matrix factorization expresses a matrix as a product of structured matrices, supporting numerical computation, low-rank approximation, and data modeling.

Matrix rank is the dimension of a matrix’s column space, equivalently its row space, and measures the number of independent directions represented by the matrix.

Matrix similarity relates square matrices that represent the same linear operator in different bases, preserving its algebraic structure and spectral properties.

The matrix trace is the sum of a square matrix’s diagonal entries, equal to the sum of its eigenvalues and invariant under changes of basis.

The transpose of a matrix exchanges its rows and columns, connecting matrix operations with duality, inner products, symmetry, and least-squares methods.