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Diophantus (Διόφαντος)
Diophantus (Διόφαντος), also known as Diophantus of Alexandria, was a Greek mathematician active about 250 CE, sometimes called the father of algebra, best known for his Arithmetica.
Direct integral
A direct integral assembles a family of Hilbert spaces, von Neumann algebras, or operators indexed by a measure space, generalizing the direct sum of Hilbert spaces.
Direct product
The direct product is a construction in algebra that combines structures of the same kind, such as groups or rings, into a single structure with coordinatewise operations.
Direct solver
A direct solver is a numerical linear algebra method that solves linear systems by factorizing the matrix, giving the solution in a fixed number of steps.
Direct sum
The direct sum is an algebraic construction combining modules, vector spaces, or abelian groups into one larger object, whose elements are tuples with only finitely many nonzero entries.
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 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 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 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 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'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.
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 valuation ring
A discrete valuation ring (DVR) is a principal ideal domain with exactly one nonzero prime ideal; examples include the power series ring k[[t]] and the integers localized at a prime.
Discriminant
The discriminant is a quantity computed from the coefficients of a polynomial or quadratic form that vanishes exactly when the object degenerates, as when two roots coincide.
Distance matrix
A distance matrix is a square matrix recording the pairwise distances between elements of a finite set, with a symmetric, zero-diagonal form when the distance is a metric.
Distance-regular graph
A distance-regular graph is a connected graph where, for any two vertices at a given distance, the counts of neighbors at each distance depend only on that distance.
Distributive property
The distributive property, also called the distributive law, is the arithmetic rule that a factor times a sum equals the sum of the factor times each addend, written a(b + c) = ab + ac.
Divisibility (ring theory)
Divisibility in ring theory studies when one element of a ring divides another, generalizing integer divisibility; properties like unique factorization into primes can fail in general rings.
Divisibility rule
A divisibility rule, also called a divisibility test, is a shortcut that checks whether an integer is divisible by another by examining its digits instead of dividing.
Division (mathematics)
Division is the arithmetic operation of taking the ratio of two numbers, the inverse of multiplication, where x is the dividend, y the divisor, and the result the quotient.
Division algorithm
The division algorithm is a theorem of elementary number theory: any integer divided by a nonzero divisor yields a unique quotient and remainder, with 0 ≤ r < divisor.
Division by zero
Division by zero, also called dividing by zero or a zero denominator, is dividing a number by zero, undefined for real numbers, with 0/0 an indeterminate form.
Division ring
A division ring, also called a skew field, is a nonzero ring in which every nonzero element has a multiplicative inverse, while multiplication need not commute.
Division sign
The division sign (÷), also called the obelus, is a mathematical symbol used to indicate division. Johann Rahn first used it for division in his 1659 book Teutsche Algebra.
Divisor
A divisor (also called a factor) of an integer n is an integer that divides n with no remainder; its study is the starting point of number theory.
Divisor (algebraic geometry)
In algebraic geometry, a divisor is a formal object on an algebraic variety or scheme recording codimension-1 subvarieties with integer multiplicities, tied to line bundles through the Picard group.
Divisor function
The divisor function, also called the sum-of-divisors function, is the number-theoretic function σk(n) summing the k-th powers of n's divisors; σ0 counts them and σ1 adds them up.
Dominating number
The dominating number, written 𝔡, is a cardinal characteristic of set theory: the smallest size of a family of natural-number functions that eventually dominates every such function.