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Diagonal lemma
The diagonal lemma, also called the diagonalization or fixed-point lemma, is a mathematical logic theorem giving each arithmetic formula a sentence describing itself, key to Gödel's incompleteness theorems.
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.
Dialectica interpretation
The Dialectica interpretation, also called Gödel's functional interpretation or D-interpretation, is a proof interpretation reducing Heyting arithmetic to Gödel's quantifier-free System T, published by Kurt Gödel in 1958.
Dialetheism
Dialetheism is the view that some contradictions, called dialetheias, are both true and false, developed by Graham Priest and Richard Routley in the late 1970s.
Diameter
Diameter is the greatest distance between any two points of a set; in a circle it is a chord through the center and equals twice the radius.
Dice coefficient
The Dice coefficient, also called the Dice similarity coefficient or Sørensen–Dice coefficient, is a statistic that measures the overlap between two sets, ranging from 0 to 1.
Dichotomy
A dichotomy is a division of a whole into exactly two mutually exclusive, jointly exhaustive parts. In logic it divides a genus into two species by a single attribute.
Dickson nearfield
The Dickson nearfield, also called a regular nearfield, is a nearfield made by twisting the multiplication of a division ring, first constructed by Leonard Dickson in 1905.
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.
Die Grundlagen der Arithmetik
Die Grundlagen der Arithmetik (The Foundations of Arithmetic) is an 1884 book by the German philosopher and logician Gottlob Frege that argues arithmetic can be derived from logic alone.
Diffeomorphism
A diffeomorphism is a smooth bijection between differentiable manifolds whose inverse is also smooth. Diffeomorphic manifolds share all differential-topological properties, and classification up to diffeomorphism is finer than up to homeomorphism.
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.
Difference set
A difference set is a subset of a finite group whose pairwise quotients represent every nonidentity element exactly λ times, used to build symmetric block designs.
Differentiable function
A differentiable function, also called a derivable function or differentiable mapping, is a function with a derivative at each point of interest, the central hypothesis of differential calculus.
Differentiable manifold
A differentiable manifold, also called a smooth manifold, is a topological space locally resembling Euclidean space, equipped with a smooth structure allowing calculus, central to differential geometry.
Differential (mathematics)
In mathematics, a differential, also called a differential element, is the main linear part of a function's increment, denoted dy, introduced by Leibniz and published in 1684, and used today for approximation and error estimation.
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 algebraic equation
A differential algebraic equation (DAE) is an equation combining differential equations with algebraic constraints, imposing instantaneous relations among variables; it is applied in circuit simulation, mechanics, and control.
Differential calculus
Differential calculus is the branch of mathematics that studies how functions change through the derivative, which measures instantaneous rates of change and the slope of tangent lines.
Differential equation
A differential equation relates an unknown function to its derivatives, and solving it means finding a function that satisfies the equation; it dates to the late 17th century.
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.
Differential game
A differential game is a mathematical model in which players with conflicting objectives control a dynamical system evolving continuously in time, extending optimal control theory to multiple agents.
Differential geometry
Differential geometry, also known as differential geometry and topology, is the branch of geometry that studies curves and surfaces by analysis, centered on differentiable manifolds and supplying the language of general relativity.
Differential inclusion
A differential inclusion is a continuous-time evolution law x'(t) ∈ F(t, x(t)), generalizing an ordinary differential equation by allowing multiple velocities at each state, with origins in control theory.
Differential of a function
The differential of a function is the main, linear part of its change when the input changes, dy = f′(x) dx; the idea goes back to Leibniz in 1684.
Differential operator
A differential operator, also called a derivative operator, is an operator that maps a function to a function built from its derivatives. The basic example is D = d/dx, whose inverse denotes integration.
Differential quadrature method
The differential quadrature method (DQM) is a numerical technique that approximates derivatives as weighted sums of function values at grid points, introduced by Bellman and colleagues in 1972.
Differential topology
Differential topology is the branch of topology studying differentiable manifolds and mappings, using tools like transversality, Morse theory, and bordism, with landmark results including Milnor's 1956 exotic 7-spheres.