Mathematics and statistics

Exponential function

The exponential function, also called the natural exponential function, is the mathematical function exp(x) = e^x, with e ≈ 2.718; it is its own derivative and models growth and decay.

Exponential generating functions

An exponential generating function (EGF) attaches to a counting sequence the power series sum a_n x^n/n!, the standard tool of enumerative combinatorics for labeled structures like permutations and set partitions.

Exponential growth

Exponential growth, also called geometric growth, is increase at a rate proportional to the current value, giving a constant doubling time; it models populations, epidemics, and compound interest.

Exponential integral

The exponential integral, or Ei function, is a special function defined by the integral of e−t/t, introduced by Legendre in 1811 and arising throughout physics and engineering.

Exponential integrator

An exponential integrator is a time-stepping method for stiff or oscillatory semilinear differential equations that solves the linear part exactly through a matrix exponential, dating to the 1960s.

Exponential map (Riemannian geometry)

The exponential map of a Riemannian manifold sends a tangent vector at a point to the endpoint of the geodesic starting there, and underlies normal coordinates and geodesic completeness.

Exponential smoothing

Exponential smoothing, also known as Holt-Winters, is a family of time series forecasting methods that weight recent observations more heavily, proposed in the late 1950s.

Exponential stability

Exponential stability is a strong form of stability of an equilibrium point of a dynamical system: nearby solutions converge to it at least as fast as a decaying exponential.

Exponential sum

An exponential sum, also called a trigonometric sum, is a finite sum of complex roots of unity, central to analytic number theory since Gauss's 1801 work.

Exponential time differencing

Exponential time differencing (ETD) is a family of numerical methods for stiff semilinear differential equations that integrates the linear part exactly with matrix exponentials, dating to the 1960s.

Exponential time hypothesis

The exponential time hypothesis (ETH) is an unproven conjecture in computational complexity theory, proposed by Impagliazzo and Paturi in 2001, stating that 3-SAT cannot be decided in subexponential time.

Exponentiation

Exponentiation is the arithmetic operation that raises a base to an exponent, written as a superscript; for positive integer exponents it means repeated multiplication, extending to zero, negative, rational, real, and complex exponents.

Exponentiation by squaring

Exponentiation by squaring, also called binary exponentiation or the square-and-multiply algorithm, computes large powers using multiplications proportional to the logarithm of the exponent, making modular exponentiation in RSA practical.

Expression (mathematics)

In mathematics, an expression is a finite combination of numbers, variables, and operation symbols standing for a calculation, and an equation joins two expressions with an equal sign.

Ext functor

The Ext functor, or Ext group, is a family of functors in homological algebra defined as the right derived functors of the Hom functor, named for extensions.

Extended Euclidean algorithm

The extended Euclidean algorithm, also called extended GCD, computes the greatest common divisor of two integers plus Bézout coefficients, and is used to find modular inverses in cryptography.

Extended finite element method

The extended finite element method (XFEM) is a numerical technique of computational mechanics, reported by Moës, Dolbow, and Belytschko in 1999, that models cracks and interfaces without remeshing.

Extended Kalman filter

The extended Kalman filter (EKF) is a state-estimation algorithm applying the Kalman filter to nonlinear systems by linearizing about the current estimate, developed at NASA Ames in the early 1960s for Apollo lunar navigation.

Extended precision

Extended precision is an IEEE 754 floating-point format exceeding its associated basic format in precision and range, in practice usually Intel's x87 80-bit format with 64 significant bits.

Extendible cardinal

An extendible cardinal is a cardinal κ in set theory such that every rank segment starting at κ can be elementarily copied into a higher one.

Exterior algebra

Exterior algebra, also called Grassmann algebra, is the algebra of a vector space in which vectors anticommute under the wedge product. It packages oriented volumes and determinants.

Exterior derivative

The exterior derivative, written d, is an operator of differential geometry that takes a k-form to a (k+1)-form, unifying the gradient, curl, and divergence into one coordinate-free operation.

Exterior power

The k-th exterior power ΛᵏV of a vector space or module V is built from the k-fold tensor power by forcing tensors with a repeated factor to vanish.

Extinction probability of branching processes

The extinction probability of a branching process is the probability that a population dies out, equal to the smallest fixed point of the offspring generating function.

Extragradient method

The extragradient method is an iterative algorithm for solving variational inequalities and saddle points, introduced by G. M. Korpelevich in 1976 for bilinear saddle-point problems.

Extrapolation

Extrapolation is a numerical and statistical technique for estimating a function or dataset at points outside the range of observed data, the complement of interpolation.

Extremal combinatorics

Extremal combinatorics is the branch of combinatorics studying the maximum or minimum size of structures like graphs under prescribed properties, with classical results by Mantel and Turán.

Extremal graph theory

Extremal graph theory is the branch of combinatorics that studies how large a graph can be while avoiding a forbidden subgraph, with Turán's 1941 theorem as its prototype.

Extreme value index

The extreme value index (EVI), written γ or ξ, is the shape parameter governing the tails of the generalized extreme value distribution, used to estimate high quantiles in finance, hydrology, and insurance.