Mathematics

Differential entropy measures the spread of a continuous probability distribution relative to a specified coordinate scale and reference measure.

A differential equation relates an unknown function to its derivatives, providing a mathematical framework for describing change and spatial variation.

A differential form is a smooth field of alternating multilinear maps used to define coordinate-independent differentiation and integration on manifolds.

Differential geometry studies smooth spaces and their geometric structures using calculus, including curvature, distances, and connections between local and global properties.

The dimension of a vector space is the number of vectors in a basis, measuring its independent linear directions over a specified field.

Dimensionality reduction represents data with fewer variables while seeking to preserve information relevant to analysis, visualization, or prediction.

A construction that combines vector spaces or modules as independent components, with unique decomposition and finite support in the algebraic infinite case.

A directed acyclic graph is a graph with directed edges and no directed cycles, used to represent ordered dependencies and probabilistic relationships.

A graph whose edges have a direction, representing ordered connections between vertices.

A directional derivative measures a function’s instantaneous rate of change at a point along a specified direction.

The domain of a function is the set of inputs for which the function assigns a uniquely determined output.

The dominated convergence theorem permits interchange of limits and integrals when a convergent sequence of measurable functions is bounded by a single integrable function.

Double-negation elimination is the classical logical principle that a proposition follows from the negation of its negation.

A dual space consists of scalar-valued linear functionals on a vector space, with continuity required in the topological setting.

A mathematical and computational method that solves structured problems by combining and reusing solutions to subproblems.

An effect size quantifies the magnitude of a difference or association, complementing statistical tests with information about scale and practical importance.

An eigenspace is the linear subspace consisting of the zero vector and all eigenvectors associated with a particular eigenvalue of a linear operator.

Eigenvalues and eigenvectors describe how a linear transformation scales particular vectors, providing tools for matrix analysis, dynamical systems, and statistics.

A notation in which repeated indices imply summation, simplifying expressions involving vectors, matrices, and tensors.

Empirical risk minimization selects a predictive model by minimizing its average loss on observed data, using that loss as a proxy for population risk.

The empty set is the unique set containing no elements, fundamental to set theory, logic, and mathematical constructions.

Information entropy measures the average uncertainty of a probability distribution and establishes fundamental limits on lossless data compression.

Entropy rate measures the asymptotic average information generated per symbol by a stochastic process, accounting for dependence between successive observations.

An equation states that two mathematical expressions are equal and may specify conditions that unknown quantities must satisfy.

An equivalence class is the set of all elements equivalent to a given element under a specified equivalence relation.

An equivalence relation is a reflexive, symmetric, and transitive relation that partitions a set into classes of mutually equivalent elements.

An estimator is a rule that uses observed data to infer an unknown population parameter or other statistical quantity.

The Euclidean algorithm computes the greatest common divisor by repeatedly replacing a pair of numbers with a divisor and its remainder.

Euclidean distance measures the straight-line separation between two points, calculated as the square root of the sum of squared coordinate differences.

A finite-dimensional space whose inner product determines the familiar geometry of distances, angles, and perpendicularity.

An event is a measurable set of outcomes in a probability model, representing a condition that may or may not occur.

The evidence lower bound is a variational objective that bounds log marginal likelihood from below and supports approximate inference and probabilistic model learning.

Exclusive OR is a logical operation that is true when exactly one of two inputs is true, equivalent to addition modulo two.

Expected value is the probability-weighted mean of a random variable, describing its theoretical average rather than a guaranteed outcome.

Experimental design is the statistical planning of experiments to estimate effects, control unwanted variation, and obtain interpretable evidence.

A function with a fixed positive base and a variable exponent, fundamental to calculus and models of proportional growth and decay.

The extreme value theorem states that a continuous real-valued function on a nonempty compact domain attains both a maximum and a minimum.

The factorial of a nonnegative integer is the product of all positive integers up to it, with zero factorial defined as one.

A factorial design investigates multiple factors together by combining their levels, allowing estimation of individual effects and interactions.

A family of algorithms that efficiently compute discrete Fourier transforms and their inverses, typically using O(N log N) arithmetic operations.

A feasible set is the collection of all candidate solutions that satisfy an optimization problem’s constraints and domain restrictions.

Fibonacci was a medieval mathematician from Pisa whose writings helped spread Hindu–Arabic arithmetic in Europe and included the sequence now named after him.

An integer sequence in which each term is the sum of the preceding two, linking recurrence relations, counting problems, and the golden ratio.

A field is an algebraic structure supporting addition, subtraction, multiplication, and division by every nonzero element.

A filtration is an increasing family of sigma-algebras representing the information available over time in a probabilistic model.

A finite field is a field with finitely many elements, existing precisely at prime-power sizes and supporting exact algebraic computation.

A numerical method that approximates derivatives by weighted differences of function values, converting differential equations into discrete algebraic equations.

The finite-element method approximates differential equations by dividing a domain into elements and solving for a finite number of coefficients in locally defined functions.

A first-countable space is a topological space in which every point has a countable neighborhood base, allowing closure and continuity to be characterized using sequences.

A formal system for expressing properties, relations, and quantified statements about objects, with precise rules for interpretation and deduction.

A fixed point is an element that a function maps to itself, central to existence theorems, iterative computation, dynamics, and recursive definitions.

A formal proof is a derivation whose statements and inference steps obey explicitly specified rules within a formal system.

A formal system specifies a symbolic language and explicit rules for deriving conclusions from axioms or assumptions.

A Fourier series represents a periodic function as a sum of sinusoidal components whose frequencies are integer multiples of a fundamental frequency.

A mathematical transform that represents a function in terms of its frequency components, connecting time or spatial descriptions with spectral descriptions.

A fraction expresses a quotient and, with integer terms and a nonzero denominator, represents a rational number.

A function is a mathematical correspondence assigning exactly one output to each element of a specified domain.

Function approximation represents a target function using a simpler or computationally tractable function, with accuracy measured by a specified error criterion.

Function composition combines functions by using the output of one as the input of another, with order and domain compatibility determining the result.

Functional analysis studies vector spaces equipped with notions of convergence and the operators and functionals acting on them, especially in infinite dimensions.