Mathematics

The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root, and therefore factors completely into linear factors.

The fundamental theorem of arithmetic states that every integer greater than one has a unique prime factorization, apart from the order of its factors.

The fundamental theorem of calculus connects differentiation with integration, explaining how accumulation produces antiderivatives and how antiderivatives evaluate definite integrals.

Galois theory relates field extensions to groups of symmetries, explaining when polynomial equations can be solved by radicals and how intermediate fields are structured.

Game theory mathematically analyzes strategic interactions in which each participant’s outcomes depend on the choices of others.

The gamma function extends factorials to noninteger arguments and is fundamental in analysis, probability, and mathematical physics.

Gaussian elimination solves linear systems by reversible row operations and underlies matrix factorization, rank computation, and many numerical solvers.

A Gaussian process is a stochastic process whose every finite collection of values has a joint Gaussian distribution, specified by a mean function and a covariance kernel.

A generalized linear model relates a response variable’s conditional mean to a linear predictor through a link function, accommodating several probability distributions.

A generating function encodes a sequence as coefficients of a series, enabling algebraic and analytic methods to solve counting problems, recurrences, and probability calculations.

A geodesic generalizes a straight line to curved spaces, locally minimizing distance in Riemannian geometry and describing free motion in general relativity.

A geometric series sums terms related by a constant multiplier; its infinite form converges when the multiplier’s absolute value is less than one.

The branch of mathematics that studies shape, size, relative position, and the properties of space, including Euclidean and non-Euclidean geometries.

A foundational theorem establishing that every semantic consequence in classical first-order logic has a formal proof.

Two mathematical theorems establishing limits on completeness and internal consistency proofs in effectively axiomatized theories of arithmetic.

The gradient is a vector describing the direction and rate of greatest local increase of a differentiable scalar-valued function.

Gradient descent is an iterative optimization method that reduces a differentiable objective by moving opposite to its gradient.

A Gram matrix records pairwise inner products of vectors, encoding their lengths, angles, linear dependence, and associated geometric volumes.

The Gram–Schmidt process converts an ordered set of linearly independent vectors into an orthonormal set spanning the same subspace.

The graph of a function is the set of ordered pairs associating every input in its domain with its corresponding output.

Graph theory studies vertices and their connections, providing mathematical foundations for analyzing networks, discrete structures, and computational problems.

The greatest common divisor is the largest positive integer that divides each of a given collection of integers without a remainder.

A group action describes how a group transforms a set or mathematical structure in a way compatible with its multiplication.

Group theory studies algebraic structures that describe symmetry, reversible operations, and the relationships between transformations.

The Hahn–Banach theorem extends linear functionals while preserving bounds and provides fundamental separation and duality principles in functional analysis.

A half-space is a region on one side of a hyperplane, with the boundary either included or excluded.

Harmonic analysis studies functions and operators through frequency decomposition, oscillation, symmetry, and quantitative estimates.

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighborhoods.

A parabolic partial differential equation describing heat conduction and diffusion through the evolution of a spatially distributed quantity.

The Heine–Borel theorem states that a subset of finite-dimensional Euclidean space is compact if and only if it is closed and bounded.

The Hessian matrix collects a function’s second-order partial derivatives, describing its local curvature and supporting optimization and critical-point classification.

A probabilistic model of sequential observations generated by unobserved states that evolve according to a Markov chain.

A Hilbert space is a complete inner product space that extends Euclidean geometry to settings including infinite-dimensional spaces of sequences and functions.

Hilbert's Program sought to secure classical mathematics through formal axiomatization and finitary consistency proofs, establishing a research agenda whose original aims were limited by Gödel's incompleteness theorems.

A decimal positional system of numeration, developed in India and transmitted through the Islamic world, that represents numbers using ten digits, including zero.

A homeomorphism is a continuous bijection with a continuous inverse, expressing equivalence between topological spaces.

Homogeneous coordinates represent points by nonzero tuples defined up to a common scale, enabling a unified treatment of affine geometry, infinity, and projection.

A homomorphism is a map between mathematical structures that preserves their specified operations or relations.

A Householder transformation is a reflection across a hyperplane, used to eliminate vector components and construct stable matrix factorizations.

A hyperparameter governs a model’s structure, learning procedure, or higher-level probability distribution, rather than serving as an ordinary fitted parameter.

A hyperplane is a flat subspace of codimension one, generalizing a line in a plane and a plane in three-dimensional space.

An ideal is an additive subgroup of a ring that absorbs multiplication by ring elements, enabling quotient constructions and generalizing divisibility.

An identity matrix is a square matrix with ones on its main diagonal and zeros elsewhere, acting as the neutral element for matrix multiplication.

The image of a linear map is the subspace of its codomain consisting of all outputs attained by the map.

The image of a function is the set of outputs it actually attains, distinguished from its specified codomain.

An indicator function represents membership in a set by assigning one to its elements and zero to all other elements.

An infinite series is an ordered sum of infinitely many terms, assigned an ordinary sum when its finite partial sums approach a limit.

Information theory mathematically quantifies uncertainty and establishes fundamental limits on data compression and reliable communication.

An injective function maps distinct inputs to distinct outputs, so every element of its codomain has at most one preimage.

An inner product pairs vectors with scalars, extending the dot product and defining lengths, angles, orthogonality, and projections in real or complex vector spaces.

Integer programming optimizes decisions subject to constraints requiring some or all variables to take integer values.

An integral measures accumulation by combining continuously varying quantities and, through the fundamental theorem of calculus, connects accumulation with differentiation.

A system of constructive reasoning in which proofs supply evidence, and excluded middle and double-negation elimination are not generally valid.

An inverse function reverses a bijective function, assigning each output its unique original input.

An inverse matrix reverses the action of an invertible square matrix, yielding the identity matrix when multiplied by the original matrix in either order.

An inverse problem infers unknown parameters, structures, or states from observations using a model of how those observations are produced.

An isometry is a distance-preserving map between metric spaces, expressing exact geometric equivalence when it is also surjective.

An isomorphism is an invertible structure-preserving map that identifies mathematical objects as equivalent in their specified structure.

The Jacobian matrix collects a function’s first-order partial derivatives and, when the function is differentiable, represents its local linear approximation.

Jensen’s inequality states that a convex function evaluated at an average cannot exceed the average of its values.