Mathematics

Abstract algebra studies algebraic structures and the maps between them, emphasizing general properties of operations rather than particular calculations.

A function that transforms a neural network unit’s input into its output, typically introducing nonlinearity or constraining output values.

Adam is a gradient-based optimization algorithm that combines momentum with adaptive, coordinate-wise scaling based on estimates of gradient moments.

An affine map preserves affine combinations and can be expressed, after choosing coordinates, as a linear map followed by a translation.

An affine space is a space of points modeled on a vector space, with well-defined displacements but no distinguished origin.

Al-Khwarizmi was a ninth-century scholar whose works shaped algebra, numerical calculation, astronomy, and mathematical geography.

Algebra is the branch of mathematics that uses symbols and rules to study equations and operations, and the abstract structures that these operations form.

Algebraic geometry studies polynomial equations through the geometric spaces they define and the algebraic structures associated with those spaces.

An algorithm is a precisely specified procedure for performing a computation or solving a class of problems.

A property holds almost everywhere when it fails only within a set of measure zero relative to a specified measure.

In probability theory, a property holds almost surely when the set of outcomes on which it holds has probability one, although exceptions may exist.

A statistical proposition specifying parameter values or distributions against which a null hypothesis is tested.

Analysis of variance partitions observed variation into components to test differences among group means and effects of explanatory factors.

Analytic geometry studies geometric objects through coordinates and equations, translating relationships between shapes into algebraic problems.

Andrey Markov was a Russian mathematician whose work on dependent random variables established the foundations of Markov chains.

Archimedes was an ancient Greek mathematician and inventor whose surviving works established major results in geometry, mechanics, and hydrostatics.

Arithmetic is the branch of mathematics that studies numbers and the basic operations on them: addition, subtraction, multiplication, and division.

An axiom is a statement accepted as a starting assumption within a theory, from which other statements are derived using rules of inference.

The axiom of choice asserts that every set-indexed family of nonempty sets admits a function selecting one element from each member.

An axiom schema is a formal template specifying a family of axioms through permitted substitutions, often providing a finite description of infinitely many statements.

A Banach space is a normed vector space in which every Cauchy sequence converges to an element of the space.

A basis is a linearly independent spanning set of a vector space, giving each vector a unique representation as a finite linear combination.

A basis is a collection of open sets whose arbitrary unions constitute every open set of a topological space.

Bayes’ theorem relates conditional probabilities and provides the mathematical basis for updating probabilities in light of evidence.

Bayesian inference updates probability distributions for unknown quantities by combining prior information with observed data through Bayes’ theorem.

A Bayesian network represents a joint probability distribution through a directed acyclic graph and local conditional distributions, enabling reasoning under uncertainty.

A recursive equation relating a decision problem’s value to immediate rewards or costs and the value of subsequent states.

A discrete probability distribution describing a single binary outcome, with probability p of taking the value 1 and probability 1 − p of taking the value 0.

Bessel’s correction replaces n with n−1 in sample variance calculations to obtain an unbiased estimator of population variance when the mean is estimated from the same sample.

A flexible family of continuous probability distributions on the unit interval, widely used to model proportions and uncertainty about probabilities.

The bias of an estimator is the difference between its expected value under repeated sampling and the parameter it estimates.

The bias–variance tradeoff describes how systematic prediction error and sensitivity to training data jointly affect a model’s expected performance on unseen observations.

Big-O notation expresses an asymptotic upper bound on a function, widely used to describe algorithmic resource requirements and mathematical approximation errors.

A bijective function pairs every element of its domain with exactly one element of its codomain, and has a unique inverse function.

A binary number expresses a numerical value using base-two positional notation, with the digits 0 and 1.

A binary relation is a set of ordered pairs specifying which elements of one set are related to elements of another.

The binomial distribution describes the number of successes in a fixed number of independent trials with the same success probability.

A multiple-testing procedure that controls the probability of at least one false rejection by dividing the overall significance level among tests.

Boolean algebra studies operations on truth values and complemented distributive structures, providing a mathematical foundation for classical logic and digital circuits.

Bootstrap sampling estimates statistical uncertainty by repeatedly resampling observed data or simulating from a fitted model.

The Borel sigma-algebra is the smallest sigma-algebra containing every open set of a topological space, connecting topology with measurable structures.

A boundary value problem seeks a solution of a differential equation that also satisfies prescribed conditions at the boundary of its domain.

Calculus is the branch of mathematics that studies continuous change through limits, derivatives, integrals and infinite series.

Cantor’s diagonal argument constructs an object missing from any proposed enumeration, proving that infinite sets can have different cardinalities.

Cantor’s theorem states that every set has strictly smaller cardinality than its power set, establishing that there is no largest size of infinity.

Cardinality measures the size of a set through one-to-one correspondence, extending ordinary counting to distinguish different sizes of infinity.

The Cartesian product forms a set of ordered combinations by taking one element from each of several sets.

A Cauchy sequence has terms that become arbitrarily close to one another; in a complete metric space, every such sequence converges.

The Cauchy–Schwarz inequality bounds the absolute inner product of two vectors by the product of their norms, with equality precisely when the vectors are linearly dependent.

A family of probability theorems describing when suitably standardized sums of random variables converge in distribution to a normal distribution.

The chain rule expresses the derivative of a composite function in terms of the derivatives of its component functions.

Channel capacity is the supremum of information rates achievable over a specified communication channel with decoding error probability tending to zero.

A polynomial associated with a square matrix or finite-dimensional linear operator whose roots are its eigenvalues, counted with algebraic multiplicity.

Chebyshev’s inequality bounds the probability that a random variable deviates from its mean using only its variance.

Cholesky decomposition factors a positive-definite matrix into a triangular matrix and its conjugate transpose, enabling efficient linear-system solutions and statistical computations.

Claude Shannon was an American mathematician and engineer who founded information theory and established key principles of digital circuit design.

A closed set is a subset of a topological space whose complement is open; in metric spaces, it contains every limit of its convergent sequences.

The closure of a subset of a topological space is the smallest closed set containing it, equivalently the set of points whose every neighborhood meets that subset.

Cluster analysis groups observations according to similarity or statistical structure, revealing patterns without requiring predefined class labels.

The codomain is the specified target set of a function, containing every output but not necessarily consisting only of outputs actually attained.