Glossary - D

All terms beginning with the letter 'D' are shown below:

decoupled system: A difference equation ykC1 D Ayk , or a differential equation y0 .t/ D Ay.t/, in which A is a diagonal matrix. The discrete evolution of each entry in yk (as a function of k ), or the continuous evolution of each entry in the vector-valued function y.t /, is unaffected by what happens to the other entries as k ! 1 or t ! 1. design matrix: The matrix X in the linear model y D Xˇ C !, where the columns of X are determined in some way by the observed values of some independent variables. determinant (of a square matrix A): The number det A defined inductively by a cofactor expansion along the first row of A. Also, ."1/r times the product of the diagonal entries in any echelon form U obtained from A by row replacements and r row interchanges (but no scaling operations). diagonal entries (in a matrix): Entries having equal row and column indices. diagonalizable (matrix): A matrix that can be written in factored form as PDP !1 , where D is a diagonal matrix and P is an invertible matrix. diagonal matrix: A square matrix whose entries not on the main diagonal are all zero. difference equation (or linear recurrence relation): An equation of the form xkC1 D Axk (k D 0; 1; 2; : : :) whose solution is a sequence of vectors, x0 ; x1 ; : : : : dilation: A mapping x 7! r x for some scalar r , with 1 < r . dimension: of a flat S : The dimension of the corresponding parallel subspace. of a set S : The dimension of the smallest flat containing S . of a subspace S : The number of vectors in a basis for S , written as dim S . of a vector space V : The number of vectors in a basis for V , written as dim V . The dimension of the zero space is 0. discrete linear dynamical system: A difference equation of the form xkC1 D Axk that describes the changes in a system (usually a physical system) as time passes. The physical system is measured at discrete times, when k D 0; 1; 2; : : : ; and the state of the system at time k is a vector xk whose entries provide certain facts of interest about the system. distance between u and v: The length of the vector u " v, denoted by dist .u; v/. distance to a subspace: The distance from a given point (vector) v to the nearest point in the subspace. distributive laws: (left) A.B C C / D AB C AC , and (right) .B C C /A D BA C CA, for all A, B , C . domain (of a transformation T ): The set of all vectors x for which T .x/ is defined. dot product: See inner product. dynamical system: See discrete linear dynamical system.

de Casteljau algorithm:	De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljau's algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value.
de Casteljau, Paul:	Paul de Casteljau (1930---2022) was a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, leading to the curves widely known as Bézier curves.
Divergence:	The divergence of a vector field \[ \nabla \bullet {\bf F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \] is a scalar field that describes the strength of local sources and sinks. If ∇ • F = 0, the field has no sources or sinks and is said to be ‘incompressible.'
Affine dependence relation:	An equation of the form c1 v1 C ! ! ! C cp vp D 0, where the weights c1 ; : : : ; cp are not all zero, and c1 C ! ! ! C cp D 0.
Affine hull (or affine span) of a set S:	The set of all affine combinations of points in S , denoted by aff S.
Affinely dependent set:	A set fv1 ; : : : ; vp g in Rn such that there are real numbers c1 ; : : : ; cp , not all zero, such that c1 C ! ! ! C cp D 0 and c1 v1 C ! ! ! C cp vp D 0.
Affinely independent set:	A set fv1 ; : : : ; vp g in Rn that is not affinely dependent.
Affine set (or affine subset):	A set S of points such that if p and q are in S , then .1 " t/p C t q 2 S for each real number t.
Affine transformation:	A mapping T W Rn ! Rm of the form T .x/ D Ax C b, with A an m # n matrix and b in Rm.
Algebraic multiplicity:	The multiplicity of an eigenvalue as a root of the characteristic equation.
Angle (between nonzero vectors u and v in R2 or R3/:	The angle # between the two directed line segments from the origin to the points u and v. Related to the scalar product by u ! v D kuk kvk cos #
Associative law of multiplication:	A.BC/ D .AB/C , for all A, B, C.
attractor (of a dynamical system in R2):	The origin when all trajectories tend toward 0.
Augmented matrix:	A matrix made up of a coefficient matrix for a linear system and one or more columns to the right. Each extra column contains the constants from the right side of a system with the given coefficient matrix.
Auxiliary equation:	A polynomial equation in a variable r, created from the coefficients of a homogeneous difference equation.