Mathematics and statistics

Consistent estimator

A consistent estimator, a concept also called statistical consistency, is a sequence of estimators that converges in probability to the true parameter value as the sample size grows.

Constrained sampling (statistics)

Constrained sampling is the task of drawing random samples from a probability distribution while restricting them to a feasible set, typically a convex polytope, using Markov chain Monte Carlo.

Constructible number

A constructible number, also called a Euclidean number, is a real number that can be built as a length using only an unmarked straightedge and compass.

Constructible universe

The constructible universe, or Gödel's constructible universe, is a class L of sets defined by Kurt Gödel in 1938 to prove the consistency of the axiom of choice.

Construction and structure of Kac–Moody algebras

Kac–Moody algebras are generally infinite-dimensional Lie algebras built from a generalized Cartan matrix by generators and relations, introduced independently by Victor Kac and Robert Moody in the late 1960s.

Construction of the complex numbers

The construction of the complex numbers builds C rigorously from R, as Hamilton's ordered pairs, 2×2 matrices, or the quotient R[x]/(x²+1), all isomorphic fields containing a square root of −1.

Construction of the real numbers

The construction of the real numbers, or construction of the reals, builds the real numbers from the rationals, chiefly through Cantor's Cauchy sequences and Dedekind's cuts, both published in 1872.

Constructive modal logic

Constructive modal logic is modal logic built on an intuitionistic propositional base, adding necessity and possibility operators; its basic system is Constructive K, introduced in 2001.

Constructive proof

A constructive proof establishes that a mathematical object exists by giving an explicit construction of it, rejecting proof by contradiction and the law of excluded middle in general.

Constructive set theory

Constructive set theory is a family of axiomatic set theories, the best known being CZF, that provide foundations for mathematics based on intuitionistic logic without the law of excluded middle.

Constructive solid geometry

Constructive solid geometry (CSG) is a solid-modeling technique that builds complex 3D objects from primitives such as boxes, spheres, and cylinders, using the Boolean operations union, intersection, and difference.

Constructivism (philosophy of mathematics)

Constructivism, also called constructive mathematics, is the philosophy of mathematics holding that existence requires a construction or proof, a view founded by L.E.J. Brouwer in 1907 that rejects the law of excluded middle.

Contact form

A contact form is a differential 1-form η on a smooth odd-dimensional manifold satisfying η ∧ (dη)^n ≠ 0, and its kernel is a contact structure, the subject of contact geometry.

Contact structure

A contact structure is a maximally nonintegrable field of tangent hyperplanes on an odd-dimensional manifold, arising from Sophus Lie's 1872 work and applied in mechanics, optics, and thermodynamics.

Container method

The container method, also called the method of hypergraph containers, is a technique in extremal combinatorics, developed independently around 2012 by Saxton and Thomason and by Balogh, Morris, and Samotij.

Content (polynomial)

In algebra, the content of a polynomial is the greatest common divisor of its coefficients; dividing by it leaves the primitive part of the polynomial.

Context-free grammar

A context-free grammar (CFG) is a formal grammar whose rules replace a single nonterminal symbol regardless of context, generating the context-free languages at level 2 of the Chomsky hierarchy.

Context-free language

A context-free language is a formal language generated by a context-free grammar and recognizable by a pushdown automaton, the standard formalism for programming-language syntax, introduced by Chomsky.

Contiguity (probability theory)

Contiguity, introduced by Lucien Le Cam around 1956, describes when two sequences of probability measures share their limiting behavior, letting asymptotic distributions transfer between them; it underlies local power of tests.

Contingency table

A contingency table, also called a cross tabulation or crosstab, is a matrix of counts for testing whether two categorical variables are associated, analyzed chiefly with the chi-square test.

Continual reassessment method

The continual reassessment method (CRM) is a Bayesian model-based design for phase I dose-finding trials, introduced in 1990, that refits a dose-toxicity model after each patient cohort.

Continued fraction

A continued fraction is an expression built from nested denominators representing any real number, finite exactly for rationals; its convergents give best rational approximations, underlying results like the 355/113 approximation of π.

Continuity correction

Continuity correction, also known as Yates's correction for continuity, is a statistical adjustment adding or subtracting 0.5 when a discrete distribution such as the binomial is approximated by the normal.

Continuous function

A continuous function, also called a continuous map, is a function whose output changes by arbitrarily small amounts when the input changes by a sufficiently small amount.

Continuous mapping theorem

The continuous mapping theorem, or Mann–Wald theorem, states that convergence of random variables is preserved under continuous functions, and underlies Slutsky's theorem and the delta method.

Continuous or discrete variable

In probability and statistics, a discrete variable takes countable values such as die rolls, while a continuous variable takes any value in an interval, like height or temperature.

Continuous spectrum (functional analysis)

The continuous spectrum of a linear operator, in functional analysis, is the part of its spectrum where injectivity holds and the range is dense but not surjective.

Continuous uniform distribution

The continuous uniform distribution, also called the rectangular distribution, gives every value in a fixed interval equal probability, with constant density across the interval and zero outside it.

Continuous wavelet transform

The continuous wavelet transform (CWT) is a signal analysis method that decomposes signals into wavelets of every scale and position, tracking frequency content over time.