Mathematics

A Sobolev space is a complete function space that measures both integrability and generalized differentiability, providing a principal framework for partial differential equations and variational problems.

The softmax function converts real-valued scores into a normalized probability distribution by exponentiating each score and dividing by the sum of all exponentials.

The source coding theorem identifies entropy as the fundamental limit on the rate of lossless compression of a discrete information source.

A matrix with sufficiently many zero entries that specialized storage and algorithms can reduce memory use and computational work.

A family of theorems expressing suitable linear operators through orthogonal eigenvectors or spectral measures.

Standard deviation measures dispersion around a mean as the square root of variance, expressed in the same units as the observations.

Standard error measures the sampling variability of a statistic, indicating how much its value varies across repeated samples.

A stationary distribution is a probability distribution preserved by the transition dynamics of a Markov process.

A stationary process is a stochastic process whose distributions, or specified statistical moments, are invariant under shifts of the time index.

Statistical hypothesis testing evaluates claims about populations or models using sample data and procedures with specified error properties.

Statistical independence is the property that joint probabilities factor into the product of the corresponding individual probabilities.

Statistics is the discipline of collecting, analyzing, and interpreting data to describe variation, draw inferences, and quantify uncertainty.

Stochastic gradient descent is an iterative optimization method that updates parameters using gradients estimated from randomly sampled observations.

A stochastic process is a family of random variables indexed by time or another parameter, used to describe uncertain systems and their dependence structure.

Stokes’ theorem equates integration of a derivative over an oriented region with integration of the original quantity over its boundary.

A family of symmetric, heavy-tailed probability distributions central to inference about normally distributed populations with unknown variance.

A subset is a set whose every element also belongs to another specified set, with equality permitted unless the subset is required to be proper.

The subspace topology equips a subset of a topological space with open sets obtained by intersecting it with the ambient space’s open sets.

A sufficient statistic preserves all information in a sample about an unknown parameter within a specified statistical model.

A surjective function maps its domain onto its entire codomain, so every element of the codomain has at least one preimage.

The symmetric group is the group of all permutations of a set, with composition as its operation, and is fundamental to the study of algebraic symmetry.

A system of linear equations is a collection of linear equations in shared unknowns, whose simultaneous solutions are characterized by matrix rank and computed through elimination or numerical methods.

A tangent space is the vector space of first-order directions at a point of a differentiable manifold.

A Taylor series is a power series whose coefficients are determined by a function’s derivatives at a chosen point, providing local representations and polynomial approximations.

A tensor is a multilinear mathematical object whose components transform consistently under changes of basis; in computing, the term also denotes a multidimensional array.

Tensor contraction pairs compatible tensor indices and sums over them, generalizing evaluation, matrix trace, and matrix multiplication.

A tensor field assigns a tensor to each point of a space, describing geometric or physical quantities independently of coordinates.

A tensor product combines vector spaces or modules into an object that represents multilinear relationships through linear maps.

A test statistic is a numerical function of sample data used to assess a statistical hypothesis against a reference distribution.

A theorem is a mathematical statement established by proof from specified assumptions and accepted rules of inference.

A time series is a sequence of observations indexed by time, studied to describe temporal patterns, model dependence, and forecast future values.

A topological space is a set equipped with open subsets satisfying axioms that define continuity, convergence, and other spatial properties without requiring a distance.

Topology studies properties of spaces preserved by homeomorphisms, using concepts such as continuity, connectedness, compactness, and algebraic invariants.

A metric or uniform space is totally bounded if it admits a finite covering at every prescribed scale of closeness.

A matrix encoding probabilities of movement between states, or, in linear algebra, conversion between coordinate bases.

The triangle inequality states that a direct distance or the size of a sum cannot exceed the corresponding sum of distances or sizes.

Trigonometry studies relationships among angles, triangle sides, and circular functions, providing tools for geometry, measurement, and periodic motion.

A truth table lists the truth values of a logical expression for every possible assignment of values to its variables.

Type I and Type II errors are the two ways a statistical hypothesis test can reach an incorrect decision about a null hypothesis.

Type theory studies formal systems that classify expressions by types, connecting mathematical foundations, logic, computation, and machine-checked proofs.

Uniform continuity requires a single input-distance bound to control output differences everywhere in a function’s domain.

Uniform convergence is convergence of functions in which one error bound eventually holds simultaneously at every point of the domain.

A probability distribution assigning equal probabilities to outcomes in a finite set, or constant density over a region of finite positive measure.

A unitary matrix is a complex square matrix whose conjugate transpose is its inverse, so it preserves inner products and vector lengths.

A universal property characterizes a mathematical construction through uniquely determined maps, specifying it up to a unique structure-preserving isomorphism.

A value function assigns an optimal objective value or an expected cumulative reward to a parameter, state, or state–action pair.

Value iteration is a dynamic programming algorithm that computes optimal value functions and policies by repeatedly applying Bellman optimality updates.

Variance measures dispersion by averaging squared deviations from a mean, providing a foundation for probability theory, statistical inference, and predictive modeling.

A vector field assigns a vector to each point of a space, describing quantities such as velocity, force, or local directions of motion.

A vector space is a set equipped with vector addition and scalar multiplication satisfying axioms that generalize the arithmetic of geometric vectors.

A weak solution satisfies a differential equation through integral identities or distributional derivatives, allowing less regularity than a classical solution.

A Wiener process is a continuous-time stochastic process with continuous paths and independent Gaussian increments, providing the standard mathematical model of Brownian motion.

Zermelo–Fraenkel set theory is a first-order axiomatic framework for sets that provides a foundation for much of mathematics.

Zero (0) is the number for an empty quantity. It is also the digit that marks an empty place in positional notation, and it is the additive identity in arithmetic.

A game in which players’ payoffs sum to zero at every outcome, so gains to some players are exactly balanced by losses to others.