6,052 articles
Epistemic modal logic
Epistemic modal logic, also called epistemic logic, is the branch of modal logic that formalizes knowledge and belief with operators like "agent a knows that". Jaakko Hintikka's 1962 book founded the field.
Epsilon number
An epsilon number is a transfinite ordinal that is a fixed point of omega exponentiation, satisfying ω^α = α, with ε₀ the first and best known.
Equals sign
The equals sign, also called the equality sign, is a mathematical symbol stating that two expressions have the same value, introduced by Robert Recorde in 1557.
Equation
An equation is a mathematical statement that two expressions are equal, written with an equals sign; identities hold for all values, conditional equations for some, and contradictions for none.
Equation solving
Equation solving, also called solving an equation, is the task of finding values of unknowns that make equations true, either exactly in symbols or approximately by numerical methods.
Equational logic
Equational logic is the formal logic whose only formulas are identities s = t between terms, underlying universal algebra and term rewriting; Birkhoff proved it sound and complete in 1935.
Equiconsistency
Equiconsistency is a relation between formal theories in which each is consistent if and only if the other is; it underlies the consistency strength hierarchy comparing theories like ZFC.
Equilateral triangle
An equilateral triangle, also known as an equiangular triangle, is a triangle with all three sides of equal length and three interior angles of 60 degrees, the simplest regular polygon.
Equivalence class
An equivalence class is the subset collecting all elements related to a given element by an equivalence relation; the classes form a partition of the set.
Equivalence of categories
Equivalence of categories is the standard category-theoretic notion of sameness between two categories: a pair of functors going in opposite directions, inverse up to natural isomorphism.
Equivalence relation
An equivalence relation is a binary relation that is reflexive, symmetric, and transitive, so it behaves like equality; its equivalence classes form a partition of the set.
Equivalence test
An equivalence test is a statistical hypothesis test concluding two treatments or measurements differ by no more than a prespecified equivalence margin, typically using two one-sided tests.
Equivalents of the axiom of choice
The equivalents of the axiom of choice are statements provably equivalent to it over ZF, including Tychonoff's theorem, Zorn's lemma, and every vector space having a basis.
Erdős cardinal
An Erdős cardinal is a large cardinal defined by a partition property, named for Paul Erdős; the most-used case is κ(ω₁), and a Ramsey cardinal is the κ-Erdős cardinal.
Erdős discrepancy problem
The Erdős discrepancy problem, posed by Paul Erdős in the 1930s, asks whether every infinite sequence of 1s and −1s must have arbitrarily large discrepancy along arithmetic progressions.
Erdős–Bacon number
An Erdős–Bacon number is the sum of a person's Erdős number and Bacon number, their collaboration distances from mathematician Paul Erdős and actor Kevin Bacon through coauthored papers and shared film appearances.
Erdős–Faber–Lovász conjecture
A combinatorics conjecture, posed in 1972 by Paul Erdős, Vance Faber, and László Lovász, that edges of any linear hypergraph on n vertices can be colored with n colors.
Erdős–Kac theorem
The Erdős–Kac theorem, proved by Paul Erdős and Mark Kac in 1940, states that the number of distinct prime factors of an integer is normally distributed.
Erdős–Ko–Rado theorem
A theorem in extremal set theory stating that an intersecting family of k-element subsets of an n-element set has at most n−1 choose k−1 members, proved in 1938.
Erdős–Rényi model
The Erdős–Rényi model is a random graph model in which each possible edge appears independently with fixed probability, introduced by Paul Erdős and Alfréd Rényi in 1959.
Erdős–Straus conjecture
The Erdős–Straus conjecture is an unsolved problem in number theory, posed by Paul Erdős and Ernst Straus in 1948, stating that 4/n is always a sum of three unit fractions.
Erdős–Szemerédi theorem
A 1983 theorem in additive combinatorics stating that every finite set of integers has a large sum set or product set, of size at least |A|^{1+δ}.
Erdős–Tetali theorem
The Erdős–Tetali theorem is a 1990 result in additive number theory, proved by Paul Erdős and Prasad Tetali, that economical asymptotic bases of order k exist for every fixed k.
Erdős–Turán conjecture on additive bases
The Erdős–Turán conjecture on additive bases, posed by Paul Erdős and Pál Turán in 1941, is an open problem stating that representations cannot be bounded.
Erez Lieberman Aiden
Erez Lieberman Aiden, also known as Erez Aiden, is an American scientist who invented the Hi-C genome mapping method, co-created the Google Ngram Viewer, and introduced evolutionary graph theory.
Ergodic process
An ergodic process is a stochastic process in which time averages along one realization converge, with probability one, to ensemble averages, so one trajectory is statistically representative.
Ergodic theorem
The ergodic theorem states that for a measure-preserving transformation, the time average of an integrable function converges along almost every trajectory, equaling the space average when the system is ergodic.
Ergodic theory
Ergodic theory is the branch of dynamical systems theory that studies measure-preserving transformations, with roots in Boltzmann's ergodic hypothesis and applications to statistical mechanics, number theory, and combinatorics.
Ergodicity
Ergodicity is the property of a measure-preserving dynamical system where a single trajectory's time average equals its average over the whole space, central to ergodic theory.
Ergodicity and convergence to equilibrium of continuous-time Markov processes
Ergodicity and convergence to equilibrium of continuous-time Markov processes covers recurrence, stationary distributions, the ergodic theorem, and convergence rates from exponential spectral gaps to subgeometric ones.