1,733 articles
Density evolution
Density evolution is an analytical technique in coding theory that tracks the probability densities of messages exchanged by iterative decoders, predicting code ensemble performance and yielding the decoding threshold.
Depth (ring theory)
Depth, also called the depth of a module or of a local ring, is an invariant of commutative algebra measuring the length of a maximal regular sequence on a module.
Derivation (differential algebra)
In differential algebra, a derivation is a linear map on an algebra obeying the Leibniz product rule, the algebraic abstraction of differentiation begun by Joseph Fels Ritt in the 1930s.
Derived algebraic geometry
Derived algebraic geometry, or DAG, is a branch of mathematics studying derived schemes and stacks, classical schemes enhanced with homotopical, higher categorical data, with foundations laid in the 2000s by Toën, Vezzosi, and Lurie.
Derived category
A derived category is a mathematical construction that replaces an abelian category by a category of chain complexes, invented by Grothendieck and Verdier in the early 1960s.
Derived functor
A derived functor, in left and right variants, is a homological algebra construction measuring how far a functor fails to be exact, computed by applying it to a projective or injective resolution and taking homology.
Descartes' rule of signs
Descartes' rule of signs, also called Descartes' sign rule, is an algebra theorem stating that a polynomial's positive real roots number at most its coefficient sign changes.
Descent theory
Descent theory is a set of techniques in algebraic geometry and category theory for recovering an object from its pullback along a covering morphism, presented in general form by Alexander Grothendieck in 1960.
Dessin d'enfant
A dessin d'enfant, French for "child's drawing," is a finite bicolored graph on a surface that determines a Riemann surface; the absolute Galois group acts on these objects.
Determinacy and large cardinals
Determinacy and large cardinals refers to results linking determinacy axioms, which say certain infinite games have winning strategies, with large cardinal axioms, beginning with work from the 1960s onward.
Determinant
The determinant of a square matrix is a scalar that measures how the matrix scales oriented volume and is zero exactly when the matrix is not invertible.
DFT matrix
The DFT matrix is a complex square matrix whose entries are powers of an N-th root of unity, so multiplying it by a vector computes the discrete Fourier transform.
Diagonal matrix
A diagonal matrix is a square matrix in which every entry off the main diagonal is zero, so arithmetic, powers, determinants, and inverses reduce to operations on the diagonal entries.
Diagonalizable matrix
A diagonalizable matrix, or nondefective matrix, is a square matrix similar to a diagonal matrix, meaning it has n linearly independent eigenvectors and acts as simple scaling in the eigenbasis.
Diagonalization (linear algebra)
Diagonalization is a linear algebra method rewriting a square matrix A as A = PDP⁻¹, with D diagonal of eigenvalues and P's columns eigenvectors, making powers and functions of A easy to compute.
Dicyclic group
The dicyclic group Dic_n, also called the binary dihedral group, is a finite non-abelian group of order 4n defined by a presentation with generators a and b.
Difference of two squares
The difference of two squares is the algebraic identity a² − b² = (a − b)(a + b), one of the most recognizable factoring patterns in elementary algebra.
Differential algebra
Differential algebra is the branch of algebra studying rings and fields equipped with derivations, operations satisfying the formal rules of differentiation, founded by Joseph Fels Ritt.
Differential Galois theory
Differential Galois theory, known for linear equations as Picard–Vessiot theory, attaches to each linear differential equation a Galois group capturing algebraic dependencies among solutions and characterizing closed-form solvability.
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 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.
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.