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Differentially closed field
A differentially closed field is a field with a derivation containing enough solutions of differential equations, the differential analog of an algebraically closed field, in a precise first-order sense.
Differentiation of trigonometric functions
Differentiation of trigonometric functions gives the derivatives of sine, cosine, tangent, cotangent, secant, and cosecant: in radians, differentiating sine gives cosine, and the derivatives cycle back after four steps.
Differentiation rules
Differentiation rules, also called differentiation formulas, are calculus identities that give the derivative of a function built from simpler parts, such as the product, quotient, and chain rules.
Diffusion equation
The diffusion equation is a partial differential equation describing how heat or concentration spreads through a medium over time, first derived by Joseph Fourier in 1822 and identical to the heat equation.
Diffusion process (Markov process)
A diffusion process is a continuous-time Markov process with continuous sample paths, described locally by drift and diffusion coefficients; Brownian motion is the canonical example.
Digamma function
The digamma function, written ψ(z) and also called the psi function, is the logarithmic derivative of the gamma function, first of the polygamma functions, with ψ(n+1) = Hn − γ.
Digital root
The digital root, also called the repeated digital sum, is the single digit left after repeatedly summing an integer's digits; it equals the remainder modulo 9.
Dihedral angle
A dihedral angle is the angle between two intersecting planes, measured on a plane normal to their shared edge; it describes faces of polyhedra and molecular torsion.
Dihedral group
The dihedral group is the group of symmetries of a regular n-sided polygon, consisting of n rotations and n reflections for 2n elements, and is a standard first example of a finite non-abelian group.
Dihedral group of order 8
The dihedral group of order 8, also called the symmetry group of a square, is the group of the eight symmetries of a square: four rotations and four reflections.
Dilemma
A dilemma is a logical argument form built from two conditional statements and a disjunctive premise, so a stated conclusion follows from either alternative; in ethics it names conflicting obligations.
Dilworth's theorem
Dilworth's theorem is a 1950 result in order theory stating that a finite poset can be partitioned into chains, whose minimum number equals its largest antichain.
Dimension
Dimension in topology is a numerical invariant, usually the Lebesgue covering dimension, counting a space's independent directions; homeomorphic spaces share it, and Euclidean space Rn has dimension n.
Dimension (vector space)
In linear algebra, the dimension of a vector space, also called its Hamel dimension, is the number of vectors in any basis of the space.
Dimension theory (algebra)
Dimension theory in commutative algebra studies the Krull dimension of a ring, the supremum of lengths of chains of prime ideals, introduced by Wolfgang Krull in 1928 and matching the geometric dimension of varieties.
Diophantine approximation
Diophantine approximation is the branch of number theory studying how closely a real number can be approximated by rationals with bounded denominators; key results include Dirichlet's and Roth's theorems.
Diophantine condition
A Diophantine condition is an arithmetic lower bound keeping frequency vectors from being too closely approximated by rationals, used in KAM theory to control small divisors.
Diophantine equation
A Diophantine equation, or Diophantine analysis, is a polynomial equation with rational coefficients whose integer or rational solutions are sought, as in Fermat's Last Theorem, proved in 1994.
Diophantine geometry
Diophantine geometry studies integral and rational points on systems of polynomial equations through algebraic geometry, named for Diophantus of Alexandria; Serge Lang coined the term in 1962.
Diophantine set
A Diophantine set is a set of tuples of positive integers that forms part of the solution set of a polynomial equation with integer coefficients, central to Hilbert's tenth problem.
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.
Dirac delta function
The Dirac delta function, or unit impulse, is a mathematical object that is zero everywhere except at one point and integrates to one, rigorously defined as a distribution.
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 proof
A direct proof establishes a statement by assuming its hypotheses and deriving the conclusion through an explicit chain of deductions, proving the proposition in its originally stated form rather than a transformed equivalent.
Direct sampling (Monte Carlo)
Direct sampling is a Monte Carlo technique that generates independent, unweighted draws exactly from a target probability distribution by transforming uniform random numbers, without rejecting candidates.
Direct search (optimization)
Direct search is a derivative-free optimization method that improves an objective function by sampling trial points and using function values alone, with no gradients or models.
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.
Directed acyclic graph
A directed acyclic graph (DAG), also called an acyclic digraph, is a directed graph containing no directed cycles, so it admits a topological ordering of its vertices.