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Dot product
The dot product, also called the scalar product, is an operation on two vectors that returns a single number, the product of their lengths and the cosine of the angle between them.
Double factorial
The double factorial of a nonnegative integer n, written n!!, is the product of all positive integers up to n with the same parity as n.
Double-precision floating-point format
Double-precision floating-point format, known as binary64, is the 64-bit number format defined by IEEE 754 that offers about 15 to 17 significant decimal digits of precision.
Doubly stochastic matrix
A doubly stochastic matrix, also called a bistochastic matrix, is a square matrix of nonnegative numbers in which every row and every column sums to 1.
Drazin inverse
The Drazin inverse is a unique pseudoinverse of a square matrix that inverts its invertible part and annihilates its nilpotent part, introduced by M. P. Drazin in 1958.
Drinfeld center
The Drinfeld center of a monoidal category is a braided monoidal category whose objects are pairs with half-braidings, the categorical counterpart of Drinfeld's quantum double of a Hopf algebra.
Drinfeld double
The Drinfeld double, also called the quantum double, is a Hopf algebra built from a Hopf algebra and its dual, introduced by Vladimir Drinfeld in 1986.
Du Val singularity
A Du Val singularity, also called a rational double point or Kleinian singularity, is a surface singularity classified by the Dynkin diagrams A, D, and E, named after Patrick Du Val.
Dual basis
A dual basis is a basis of the dual space V matched to a basis of V: each functional gives 1 on one basis vector and 0 on the others.
Dual map
The dual map, also called the transpose, is the linear map between dual spaces that a linear map induces, running in the opposite direction; its matrix is the transpose.
Dual module
The dual module of a module M is Hom_R(M, R), the module of linear maps from M to the ring R, generalizing dual vector spaces.
Dual number
A dual number is an expression of the form a + εb, where ε is nonzero with ε² = 0, introduced by Clifford in 1873 and used in automatic differentiation.
Dual pair of topological vector spaces
A dual pair of topological vector spaces is two vector spaces over the same field with a non-degenerate bilinear form and topologies built from that pairing.
Dual quaternion
A dual quaternion is an eight-component number, formed from two quaternions and the nilpotent dual unit ε, that represents a rigid-body pose, combining rotation and translation compactly.
Dual representation
The dual representation, or contragredient representation, is the representation a group or Lie algebra induces on the dual space of its representation space, preserving the natural pairing.
Dual space
The dual space of a vector space V is the vector space V of all linear functionals on V, that is, linear maps from V to its scalar field.
Dualizing sheaf
The dualizing sheaf is a coherent sheaf on a Noetherian scheme representing duality on coherent sheaves, generalizing the canonical bundle to singular spaces and underlying Serre and Grothendieck duality.
Duffin–Schaeffer conjecture
The Duffin–Schaeffer conjecture, posed by R. J. Duffin and A. C. Schaeffer in 1941, gave a criterion for when almost every real number is approximated infinitely often by reduced fractions.
Duodecimal
Duodecimal, also called dozenal or base twelve, is a positional numeral system using twelve digits. Its base twelve divides evenly by 2, 3, 4, and 6, so fractions like 1/3 terminate.
Dustin Clausen
Dustin Clausen is an American mathematician known for algebraic K-theory and for developing condensed mathematics with Peter Scholze; he has been a permanent professor at IHES since 2023.
Dyadics
Dyadics, also called dyadic tensors or dyadic products, are second-order tensors built as sums of dyads, introduced by Josiah Willard Gibbs in the early 1880s.
Dynkin diagram
A Dynkin diagram is a graph encoding a root system, with one vertex per simple root, introduced by Evgeny Dynkin in 1947 to classify semisimple Lie algebras.
E (mathematical constant)
e, also known as Euler's number or Napier's constant, is the base of the natural logarithm, an irrational and transcendental number approximately equal to 2.71828, named for Leonhard Euler.
E7 (mathematics)
E7 is an exceptional simple Lie group and Lie algebra, one of five exceptional algebras, of dimension 133 and rank 7, distinguished by a 56-dimensional fundamental representation.
E8 (mathematics)
E8 (Lie group E8, E8 Lie algebra) is the largest of the five exceptional simple Lie algebras, a 248-dimensional structure of continuous symmetry with 240 roots.
Easton's theorem
Easton's theorem is a result in set theory, proved by William Easton, stating that the continuum function at regular cardinals is constrained only by monotonicity and König's theorem.
Ehrhart polynomial
The Ehrhart polynomial of a lattice polytope counts lattice points in its dilates, proved a polynomial by Eugène Ehrhart in 1962 and generalized to higher dimensions by Pick's theorem.
Eigendecomposition of a matrix
Eigendecomposition, or eigenvalue decomposition, is the factorization of a square matrix as A = XΛX⁻¹ with eigenvectors; it exists when the matrix has an eigenbasis and underlies PCA and quantum mechanics.
Eigenfunction
An eigenfunction is a nonzero function that a linear operator reproduces as a scalar multiple of itself, the function-space analog of an eigenvector of a matrix.
Eigenvalue
An eigenvalue is a scalar satisfying Az = λz for a nonzero eigenvector z of a matrix A, meaning the transformation stretches that vector without rotating it.