6,052 articles
Directed graph
A directed graph, or digraph, is a graph whose edges are ordered pairs of vertices, drawn as arrows followed in one direction, modeling one-way streets, social networks, and dependencies.
Direction (geometry)
Direction, also called spatial direction or the direction of a line, is which way a line, ray, or displacement points, independent of position and often of length.
Direction cosines
Direction cosines are the cosines of the angles a directed line or vector in 3D space makes with the positive coordinate axes, equal to the components of the unit vector along it.
Directional derivative
The directional derivative of a multivariable function is its rate of change at a point along a chosen direction, generalizing the ordinary derivative of single-variable calculus.
Dirichlet algebra
A Dirichlet algebra is a uniform algebra of continuous functions on a compact space whose real parts are dense in all real continuous functions, defined by Gleason in 1958.
Dirichlet boundary condition
The Dirichlet boundary condition, also known as the first-type or fixed boundary condition, requires a partial differential equation's solution to equal given values on the domain's boundary.
Dirichlet character
A Dirichlet character modulo q is a completely multiplicative, periodic function vanishing on integers not coprime to q, central to Dirichlet's 1837 theorem on primes in arithmetic progressions.
Dirichlet conditions
The Dirichlet conditions are sufficient conditions on a periodic function, including absolute integrability over one period, guaranteeing pointwise convergence of its Fourier series, first proved by Dirichlet in 1829.
Dirichlet convolution
Dirichlet convolution is a binary operation on arithmetic functions, combining two functions by summing f(d)g(n/d) over divisor pairs of n, named after the German mathematician Dirichlet.
Dirichlet distribution
The Dirichlet distribution is a continuous multivariate probability distribution over vectors of proportions summing to one, generalizing the beta distribution and serving as the conjugate prior to the multinomial distribution.
Dirichlet divisor problem
The Dirichlet divisor problem in number theory seeks the smallest exponent bounding the error term in the asymptotic formula for the summatory divisor function, conjectured to be 1/4.
Dirichlet eta function
The Dirichlet eta function, also called the alternating zeta function, is defined by the series 1 − 2−s + 3−s − ⋯, convergent for Re(s) > 0.
Dirichlet form
A Dirichlet form is a closed, symmetric, Markovian quadratic form on an L²-space, introduced by Beurling and Deny, linking potential theory to symmetric Markov processes.
Dirichlet integral
The Dirichlet integral is the improper integral of sin x over x from 0 to infinity, equal to π/2, named after Peter Gustav Lejeune Dirichlet.
Dirichlet L-function
A Dirichlet L-function (Dirichlet L-series) is a meromorphic function L(s, χ) defined by a Dirichlet character, introduced in 1837 to prove Dirichlet's theorem on primes in arithmetic progressions.
Dirichlet problem
The Dirichlet problem is the potential-theory problem of finding a function harmonic in a domain, satisfying the Laplace equation, that takes prescribed values on the domain's boundary.
Dirichlet process
The Dirichlet process is a stochastic process whose realizations are random probability distributions, used as a prior over unknown distributions, introduced by Thomas Ferguson in 1973.
Dirichlet process mixture model
A Dirichlet process mixture model is a Bayesian mixture model whose mixing distribution is random, allowing the number of components to be inferred from data.
Dirichlet series
A Dirichlet series is an infinite series of the form of a sum of coefficients divided by n to the power s, with s a complex variable.
Dirichlet space
The Dirichlet space is the Hilbert space of functions analytic on the open unit disk whose Dirichlet integral is finite, one of the three classical Hilbert spaces of holomorphic functions.
Dirichlet-multinomial distribution
The Dirichlet-multinomial distribution, also known as the multivariate Pólya distribution, is a probability distribution for count vectors whose category probabilities are drawn from a Dirichlet distribution.
Dirichlet's theorem on arithmetic progressions
Dirichlet's theorem on arithmetic progressions, or Dirichlet's theorem on primes, states that for coprime integers a and q, infinitely many primes satisfy p ≡ a (mod q), proved in 1837.
Discontinuous Galerkin method
The discontinuous Galerkin (DG) method is a finite element method for numerically solving partial differential equations, using discontinuous polynomials coupled through numerical fluxes at element boundaries.
Discrepancy theory
Discrepancy theory, also known as the theory of irregularities of distribution, is a branch of mathematics studying how evenly discrete objects can approximate balanced distributions, with roots in Hermann Weyl's work.
Discrete exterior calculus
Discrete exterior calculus, introduced in Anil Hirani's 2003 Caltech dissertation, translates differential forms and the exterior derivative onto simplicial meshes, with applications in computer graphics and electromagnetics.
Discrete Fourier transform
The discrete Fourier transform (DFT) is the form of Fourier analysis computed by computers, converting N samples into complex numbers giving the amplitude and phase of component sinusoids.
Discrete geometry
Discrete geometry, also called combinatorial geometry, is the branch of geometry studying discrete arrangements of points, lines, polygons, and polyhedra, with applications in robotics, computer graphics, and crystallography.
Discrete Laplace operator
The discrete Laplace operator is a linear operator, usually a matrix, that plays the role of the Laplacian on graphs and grids, encoding connectivity and approximating partial differential equations.
Discrete logarithm
The discrete logarithm is the exponent x satisfying g^x = h in a finite group; recovering it from g and h secures the 1976 Diffie-Hellman key exchange.
Discrete mathematics
Discrete mathematics is the branch of mathematics studying objects with distinct, separated values rather than smoothly varying ones, centered on combinatorics, graph theory, logic, and the mathematical foundations of computing.