D
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d'Alembertian
.
A vector (PL) amd tensor operator. Writing a partial derivative with respect to x as ƒx, the d'Alembertian often appears in the literature as: ¤2 2 - 1/c2ƒt2, where c is the vacuum speed of light. In tensor notation, ¤2 f (glk f;l );l = glk ƒ xlxkf - G ƒxl f.  (The d'Alembertlian can more directly, and more generally, be defined in multivector theory [31] (by variation on the multiproduct. PL) as 2 - 1/u2 ƒt2, where u has the dimension of velocity.)
d'Alembert's Principle from Newton's Law (Lanczos)
.
On p. 88 of his book, The Variational Principles of Mechanics, first published in 1949, Cornelius says: "With a stroke of genius the eminent French nathematician and philosopher d'Alembert (1717-1783) succeeded in extending the applicability of the principle of virtual work from statics to dynamics. The simple but far-reaching idea of d'Alembert can be approached as follows. We start with the fundamental law of motion: 'mass times acceleration equal moving force' [vectors underlined]:

			mA = F (1)
and rewrite this equation in the form

			F _ mA = 0. (2)
We now define a vector I by the equation

			I = -mA. (3)
This vector I can be considered as a force, created by the motion. We call it the 'force of inertia'. With this concept the equation of Newton can be formulated as follows:

			F + I = 0. (4)
Apparently nothing is gained, since the intermediate step (3) gives merely a new name to the negative product of mass times acceleration. It is exactly this triviality which makes d'Alembert's principle such an ingenious invention and at the same time so open to distortion and misunderstanding." This Pricinciple, reducing dynamics to statics is actually antitonic, as showm elsewhere.
d'Alembert's Principle and Newton's Law from The Antitonic Principle (Hays)
.
Consider the antitone: J * K = CONSTANT. (1)

Take the logarithm of (1): log J + log K = 0. (2)

Relabel: F = log J, I = log K -> F + I = 0, d'Alembert's Principle. (3)
Relabel: I = -A. (4)

From (3), (4), F - mA = 0, or F = mA (5), Newton's Law (implicitly antitonic). PL Lagrange's Principle as Antitonic, Hamilton's Principle as Antitonic.

Darboux's theorem
.
Provides formula for expanding functions as infinite series, with a Taylor series (PL) as a special case. Let f(x) be analytic at all points along a line and let f(t) be any polynomial (PL) of degree n in t. For 0 t 1, differentiation (PL) yields: DtS m-1 (-1)m(z - a)m f(n - m)(t)f m(a + t(z - a)), which can be integrated (PL), (A Taylor series evolves by setting f(t - 1)n and letting n -> .)
decagon
.
A constructible regular (PL) 10-sided polygon (PL).
decidable
.
A theory is decidable iff an algorithm (PL) exists for determining if a given sentence (PL) is an element of the theory.
.
decimal
.
A word derived from Latin meaning "ten".
decimal numeration system
.
A positional (PL) numeration system Based upon ten elements, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . Thus, the symbol, 27458, denotes 2 X 10 3 + 7 X 103 + 4 X 102 + 5 X 10 + 8. The position of a digit (from right-most position) denotes its power of ten (starting in zero power on right). This system is believed to derive from the normal ten fingers of human hands, hence the use of the word, "digit", for both finger and numeral.
Dedekind cut
.
A partitioning ("cut") of the rational numbers into nonempty sets, S1 , S2, s. t. all numbers of S1 are less than the numbers of S2 and S 1 has no greatest number -- allowing for an irrational number in any "cut".
Dedekind ring
.
A commutative ring (PL) which is a Noetherian ring (PL) and integral domain (PL) containing the set of algebraic integers (PL) in its field of rationals (PL) s.t. its prime ideal (PL) is also a maximal ideal.
deficiency (binomial)
.
Given binomial coefficient, C(n, k), let n - k + 1 = ai bi, with 0 i k, and with bi containing only those prime factors greater than k. Then the number i for which bi > 1 is the deficiency of C(n, k).
del operator
.
PL nabla operator.
delta operator
.
A shift-invariant operator (PL), Q s. t. Qx is a nonzero constant: Qa = 0, for each constant a; for degree n-polynomial p(x), Qp(x) is of degree n - 1; each delta sequence has a unique basic polynomial sequence (PL).
De Morgan's duality law
.
For every statement (a.k.a. proposition) (PL) involving logical operators "and", "or", there is a corresponding logical statement interchanging these operators.
Denjoy integral
.
Extension of the TRiemann integral (PL) and the Lebesque integral (PL).
denominator
.
Given a rational number ("fraction") -- PL -- of the number n/d, the number d is the denominator of this mathematical structure, whereas n is its numerator (PL). (Also PL least common denominator.)
dense
.
A set S is dense in set T if T = S R where R is the limit of sequences of elements in S. Thus, the set of rational numbers is dense in the set of real numbers.
dependent variable
.
The functand (PL) in a function which is determined by another (independent) functand of the function.
de Rham cohomology
.
A formal arrangement for analytic problems, the kth deRham cohomology vector space is the space of all differential k-forms with exterior derivative (PL) equal to zero, modulo all boundaries of (k - 1) -forms. If so, this is the setup of the problem, it is the ordinary vector space.
derangement
.
A permutation (PL) s. t. no element retains its original position.
derivation
.
From "derive", a sequence of logical or computational steps from one structure to another. (PL derivation algebra.)
derivation algebra
.
Given an algebra (PL), A, over a field (PL), a derivation is a linear operator (PL), D, s. t. D(xy) = (Dx)y + x(Dy), for all x, y e A. The set D(A) of all derivatives of A in the subspace (PL) of an associative algebra (PL) of all the linear operators (PL) on A is a Lie algebra (PL), labeled a derivation algebra.
derivative
.
PL derivative operator /
derivative operator
For the standard notation, a prototype is d/dx, resembling a "fraction". This misleading notation gives rise to Menger's "trivial" interpretation of calculus due to its "atrocious languge", alluded to on the FRONTPAGE of this Website. It actually is a profound theorem that (dy/dx)*(dx/dy) = 1, although the notation resembles the "cancellation of fractions": (2/3)*(3/2) = 1. The "nonfractional" Leibnitz derivative notation avoids this confusion: (Dxy)*(Dyx) = 1.
Desargues' theorem
.
Given two triangles, ABC, A'B'C', with three straight lines joining the vertices of opposite sides. If these lines meet in a point (perspectival point,), then the three intersectiions of pairs of correspomding sides lie on a straight line (perspectival axis ). That is, two triangles which are perpective from a point are also [erspective from a line. This theorem is self-dual under the duality principle (PL) of projective geometry (PL).
Descartes' rule of signs
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Determines maximum number of positive and negative real roots of a polynomial equation/ Proceeding from lower power to highest, the sign change number is the number of positive real roots. Changing the power carrier to its negative yields a test equation for counting the negative real roots. Thus, 5x7 + 3x6 - 2x5 + 8x3 - 2x2 + 9x - 4 = 0 has 5 sign changes, hence, a maximum of 5 positive real roots. Then 5 (-x7) + 3(-x6) - 2(-x5) + 8(-x3) - 2(-x2) + 9(-x) - 4 = -5x7 + 3x6 + 2x5 - 8x3 + 2x2 - 9x - 4 = 0 has 4 sign changes, hence, the original equation has a maximum of 4 negative real roots. It can have, at most, 7 real roots, encompassed within this sum of 5 + 4 = 9.
descending chain condition
.
The dual of ascending chain condition (PL).
determinant
.
A "square" array of elements (numbers, etc.). usually written as:

              æ e11 e12 ... e1nö
              ç e21 e22 ... e2n÷  
              ç .............  ÷
              è en1 en2 ... 3nnø
A determinant of numbers equals a single number, which is computed by the Laplacian rule (PL). A matrix has an associated determinint iff it is square.
devil's curve
.
Studied by G. Cramer (1704-17520 in 1750, its equation is: y4 - a2y 2 = x4 - b2x2
.
diagonal
.
In a polygon, the diagonal connects nonintersecting vertices. In a diagonal matrix, the only non-null elements are those with equal row and column indices.
diagonalizable matrix
.
A square matrix , M, is diagonalizablie iff it can be written as the "conjugate" product, M = UDU-1, where D is a diagonal matrix (PL) whose entries are the eigenvalues of M and the entries of U are the associated eigenvectors.
diagonal matrix
.
PL diagonal.
diagram
.
PL standard dictionary entry.
diameter
.
A chord (PL) passing through and bisected by the center of a circle, conic, or solid of revolution generated by a conic. For example, a diameter of a parabola (having its center at the point at infinity on its axis) is a ray parallel to the axis of the parabola; it bisects chords parallel to the line tangent to the parabola at the endpoint of the ray.
Dido's problem
.
Find the geometric figure bounded by a line which has the maximum perimater. Solution: semicircle. (Problem appears in The Aeniad, written by the Roman poet, Vergil (70-19 BC)).
diffeomorphism
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A map betweeem manifolds s. t. the map is differentiable (PL) with differentiable inverse .
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difference
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Result of a subtraction operation.
difference equation
.
PL finite difference.
difference operator
.
PL finite difference.
differentiable function
.
PL analytic function.
differentiable manifold
.
PL smooth manifold.
differential
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PL differential form .
differential calculus
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The arithmetic of sets of functions of numbers (or the algebra) obtained by adjoining (to the finitary operations of arithmetic) the transfinitary (PL) operation of limit (PL).
differential equation
.
An equation containing one or more derivatives (PL) or differentials (PL).
differential (k-)form
.
A completely antisymmetric homogeneous r-tensor on a differentiable n-manifold, where 0 r n. These form a Grassmannian algebra (subsystem of a Clifford algebra, PL) labled the algebra of differential forms.
differential geometry
.
Study of metrical (PL) structures on Riemann manifolds. (PL differential topology.)
differential topology
.
Study of nometrical structures on Riemann manifolds. (PL differential geometry.)
differentiation
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Operational formation of limits (PL) of difference functions (PL)
digit
.
A numeral (PL) in a number declaration.
dihedral
.
PL standard dictionary.
dihedral angle
.
The angle between two planes.
dihedral group
.
The group (PL) of symmetries (PL) of an n-sided regular polygon; denoted Dn.
dilation
.
A similarity (PL) transformation (PL) which transforms a line into a parallel line whose length is a fixed multiple of the original line. A trivial dilation is a translation (PL). Any other is a central dilation. Two triangles related by a central dilation are labeled perspective triangles since the lines joining their vertices are concurrent. Thus, a dilation is a translation concatenated with an expansion.
dimension
.
One of the units (gnomons, PL) associated with a physical quantity, physically expressable in terms of the fundamental quantities of mass, length, time, and charge. It has been noted that the plane, for example, is 2-D only because we choose to make the point the 0-D space-gnomon, making the line segment 1-D and the plane 2-D, because it can be determined by two independent numbers. However, we are free to make the circle the space-gnomon, determined by three number (2 for center, 1 for radius), making the plane 3-D. (PL activithm.) For other concerns about dimension,
dimensional analysis
.
The techniques for judging the correctness of equations by considering the units of measurement of the variables in the equations:
  • the units on the left side of the equation must be the same as on the right side;
  • by changing measure functands, replace physical constants by dimensionless numbers (PL) that remain invariant regardless of the units of measurement chosen.
(Note that Gerhard A. Blass, in his book, Theoretical Physics, 1962, pp. 236-43, shows that Ludwig Boltzmann (1844-1906) -- in deriving the most probable distribution of molecules according to Maxwell-Boltzmann statistics -- just missed discovering the quantum constant of action, h, 20 years before Max Planck (1858-1938), by failing to write a constant of integration and consider its dimensional form of ACTION).
dimensional analysis algebra
.
The FORMS of dimensional analysis (PL) can be used an an algebra for deriving other FORMS from basic FORMS. Thus, the Planck-Einstein Law and the de Broglie Law of QUANTICS -- mixing terms such as momentum, energy from MECHANICS with terms such as frequency, wavelengthfrom OPTICS-ELECTROMAGNETICS -- can both be derived from the photonic law: ln = c. The intermediary is the concept of ACTION, h, from Lagrange's Principle, applicable both in MECHANICS and OPTICS-ELECTROMAGNETICS. To derive deB, we start with the Photonic Law (PL), which dimensionally is [ln] = [c] = [L T-1]. Compare with dimensional FORM of ACTION: [h] = [M L2 T-1]. Now, a dimensional part of the latter is [M L T-1], equivalent to that for momentum: [p] = [M LT-1]; and this becomes dimensionally equivalent to ACTION by bring in length: [h] = [p L]. And we can obtain a term of length via the wavelength of the photonic law: [l] = [L]. Hence, we have h = pl , which is de Broglie's Law. To derive PE, we start with the knowledge that light involves kinetic energy, which we wish to introduce into some part of PL. Now, dimensionally [E] = [M L2 T-2]. Using Planck's CONSTANT as intermediary: [h] = [(M L2)T-1]. Thus: [E] = [hT-1]. Taking WAVELENGTH from PL, we have: [El] = [hLT-1]. On the right, we can use the PL VACCUM LIGHT SPEED, c: [El] = [hc], for the ANTITONIC PE formula, El = hc. Converting to the standard form of PE, first take E = h(c/l), and the parenthetical expression is Photonically c/l = n, yielding the standard PE form, E = hn. So, we find, by DIMENSIONAL ALGEBRA, that DB and PE are IMPLICIT in PL, mixing MECHANICS with OPTICS and ELECTROMAGNETICS. (What else is possible by DIMENSIONAL ALGEBRA?)
dimensionless number
.
A number with no associated unit of measure (PL) (such as square feet or pounds). In the ratio of two numbers with the same unit of measure, the common unit "divides out".
dimension theory
.
The branch of topology that deals with various definitions of dimension. And PL measurement scales.
Diophantine analysis
.
PL diophantine equations.
diophantine equations
.
A vast and often difficult field of MATHEMATICS, named for Greek mathematician, Diophantus (c. 250 AD). Let f(x1, x2, ..., xn) be a polynomial in x1, x2, ..., xn with integer coefficients. It is Diophantine if solutions must be integral. Linear Example: 4x1 + 6x2= 24. A solution is x1 = 3, x2 = 2. (For a linear equation, a1x1 + a2x2 + ... + anxn = b, to be INTEGRALLY SOLVABLE, b must BE DIVISIBLE BY gcd(a1, a2, ..., an) -- as you find in the above example, wherein gcd(4, 6) = 2 divides 12. (Linear Diophantine equations have been useful in modeling chemical crystals.) Conditions have been found for SOLVABILITY of higher degree Diophantine equations. In 1912, the great German mathematician, David Hilbert (1862-1943), gave a List of Problems to be Solved -- one of which was a general solution for Diophantine equations. Eventually, it was proven that no such general solution can exist.
diophantus problem
.
The Greek mathematician, Diophantus (200?-284? A.D.), has been called "The Father of Algebra". But, as some one said about another subject, "Algebra has many fathers". al-Khwarizmi could be called "The Islamic Father of Algebra", so call Diophantus "The Greek Father of Algebra". Little is known about the life of Diophantus, except for an Algebraic Riddle quoted in The Greek Anthology, also called Palatine Anthology, a collection of Greek epigrams, songs, 3700 short poems, epitaphs, and rhetorical exercises. Some of the passages in The Greek Anthology resemble epitaphs on grave stones. The passage about Diophantus presents (in translation) a riddle about the phases of his life:
God granted him to be a boy for a sixth part of his life, and adding a twelfth part to this, He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! late-borne wretched child; after attaining the measure of half his father's life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life. Solution: Let
  • x DENOTE the age at death of Diophantus;
  • 1/6 x DENOTE boyhood;
  • 1/12 x DENOTE youth;
  • 1/7 xDENOTE batchelorhood ending in marriage;
  • 5 years after marriage, a son was born;
  • let 1/2 x + 4 denote the period from first fatherhood to Diophantus' death.
  • Then we have: 1/6 x + 1/12 x + 1/7 x + 5 + 1/2 x + 4 = x.

In solving, the least common denominator of these numbers (6, 12, 7, 2) is 12 x 7 = 84. Then,

  • 1/6 (84) = 14 years (boyhood);
  • 1/12 (84) = 7 years (youth);
  • 1/7 (84) = 12 years (batchelorhood to marriage);
  • 5 years after marriage a son; son lived half of father's life, 1/2(84) = 42;
  • 4 years later, death of Diophantus (apparently by suicide).

CHECKING (back to ARITHMETIC!): 14 + 7 + 12 + 5 + 42 + 4 = 21 + 17 + 46 = 38 + 46 = 84. Amswer: 84 years of life.

directed angle
.
An angle measured from one side (the initial side) to the other (the terminal side); often designating rotation in a positively or negatively defined direction: for example, -45° or +20°.
directed distance
.
For points A, B lying on a directed line, the directed distance AB is plus or minus the distance AB, according as the direction from A to B coincides with or is opposite that of the directed line.
directed graph (digraph)
.
A graph (PL) s. t. each edge is directed ("arrowed"); simple, if no loop (PL); complete, if edges bidrected ("doubly-arrowed"); oriented, if no bidirection; tournament, if totally unidireted.
directed line
.
A line on which the distance is indicated.
directed number
.
Label for a signed number.
directed set
.
A partially ordered set with ordering , such that every pair of elements in the set has an upper bound. Also, labeled directed system.
directional derivative
.
To know in which of the i, j, k directions a scalar function may most rapidly change, we use a directional derivative which allows choice of direction and calculation of the derivative in that direction, d(p0, p1), say, from p0(x 0, y0, z0) to p1(x1 , y1, z1). We can write a unit vector in terms of direction cosines (PL): v = icosa + jcosb + kcosg. Using point coordinates, define x1 - x0 x, y1 - y0 y, z1 - z0 z. The direction is d(p0, p1) = ix + jy + kz; and distance between points: s = ((x)2 + (y)2 + (z)2)) ½. Since d(p0, p1) has the same direction as vector v, and length equal to s, then d(p0, p1) /s = v, that is, i (x/ s) + y (y/s) + k (z/s) = icosa + j cosb + kcosg, or x/s = cosa, y/s = cosb, z/ s = cosg. These direction cosines remain constant as the distance between the two points decreases, hence, in the limit, we have the derivatives: Dsx = cosa, Dsy = cosb, Dsz = cosg. Thus, for the function, f(x, y, z) at the point, p(x0, y0, z0), we formally have the derivative as the sum of partial derivatives: Dsf = f x(x0, y0, z0)Dxf + fy(x0 , y0, z0)Dyf + fz(x0, y0, z0)Dzf, or Dsf = fx(x0 , y0, z0)cosa + fy(x0, y0, z0)cosb + fz(x0, y0 , z0)cosg. Part of this is found in the above vector written in terms of unit vectors and directional cosines: v = icosa + jcosb + kcosg. The rest can be obtained in another vector involving our function: w = i fx(x0, y0, z0) + jfy(x0, y0, z0) + kfz(x0, y0, z0 )jcosb + kcosg. Given vectors, v, w, and the result that i · i = j · j = k · k = 1, we obtain our functional derivative as the inner product (PL) of these two vectors: Dsf = fx(x 0, y0, z0)cosa + fy(x0 , y0, z0)cosb + fz(x0, y0, z0)cosg = v · w = Dsf . But what is this particular inner product? It is the gradient (PL) of the function, f(x0, y0, z0), at point, p(x0, y0, z0), derived via a directional derivative, an extension of orthogonality (PL).
direction angles
.
The angles between a line in space and Cartesian coordinate axes.
direction cosine
.
Cosine of angle between a vector (PL) and a coordinate axis, say, v = icos a + jcosb + kcosg
direct product
.
PL Cartesian product .
direct proof
.
Proving an assertion (PL) by deducing the desired conclusion as a logical consequence from the hypothesis and other known or given statements.
direct proportion
.
The relation between two functands (PL) with contant ratio (PL antitone). Example: y = kx, for proportionality constant k. Also labeled Also, labeled direct variation.
directrix
.
The fixed line that, together with a fixed point (the focus), defines a given conic; also, the (constant) ratio of the distance from a point of the conic to the focus and the distance from that point to the directrix is the eccentricity of the conic.
direct search factorization
.
Simplest prime factorization algorithm, by systematically testing trial divisors in increasing sequence. Inefficient; used for small numbers. For a number n, only divisors up to |_n_| (floor function PL) need to be tested, since n/(|_n_| + 1 < n.
direct sum
.
Given sets A, B, then = {a + b: a e A, b e B}. Can extend to arbitrary number of summands.
Dirichlet partial differential equation conditions
.
Elliptic (PL) partial differential equations (PL) are boundary value problems requiring data to be prescribed on the boundary of the solution domain. Dirichlet conditions prescribe the solution along the boundary domain, in contrast to Neumann conditions (PL).
.
Dirichlet Fourier series conditions
.
A piecewise regular (PL) function with a finite number of finite discontinuities and extrema can be expanded in a Fourier series (PL) converging to the function at continuous points and and the mean of the positive-negative limits at discontinuous points.
Dirichlet function
.
For two distinct real numbers, c, d (say, c = 1, d = 0, define, D(x) = {c, for x rational; d, for x irrational}.
discontinuity
.
A point at which a mathematical object is discontinuous, as in a jump (PL).
discrete set
.
A set with discrete topology (PL), which may be induced naturally, as in the integers embedded in the real numbers.
discrete topology
.
The topology on a set of points such that every subset of the set is open (PL). This set and the topology constitutes a discrete topological space.
discrete variable
.
A functand (PL) assuming natural number (PL) or integral (PL) values.
.
discriminant
.
The general quadratic equation: ax2 + bx + c = 0 has the solution: x = (b ±(b2 - 4ac))/2a. Clearly, the term b2 - 4a discriminates the type of roots of the equation -- real or complex, equal or distinct -- hence, is labeled the discriminant of a quadratic equation. The discriminant is an invariant of the equation: if the equation is transformed into one with other another functand -- x -> x', the discriminant of the new equation is simply a multiple of the former discriminat.
disjoint sets
.
Their union (PL) is null.
disjunction
.
The "or" operation in statement logic (PL).
disk (disc)
.
The collection of points within a radial distance from a fixed point in Euclidean n-space.
dissection
.
Any two rectilinear (PL) figures can be dissected into a finite number of pieces to form each other. (Wallace- Bolyai-gerwein theorem.)
distance
.
The length of a minimal path between two given sets or geometric configurations: the distance function on two distinct two points. For more advanced concepts PL http://www.harcourt.com/dictionary /browse/19/
distance function
.
PL distance.
.
distribution (generalized function)
.
The impulse integrand, being "instaneous", is not a function, but can be represented by a generalized function or distribution, defined as a continuous linear functional (PL) over a space of infinitely differentiable functions s. t. each continuous function has a g. f. derivative. (Example: Dirac's delta function, PL.) The distribution corresponding to a function g is T = W fg.The distribution corresponding to a measure M is Tm = Wfd m. Distributions differ from functions in being covariant (PL): the push firward. Thus, given a smooth function (PL) a : W1 W 2, a distribution, W on W1 pushes forward to a distribution on W2. On the other hand, a real function , f, on W2 pulls back to a function, f(a(x)) , on W1. The distribution topology (PL) is defined on the family of seminorms, NK, a(f) = supK||Daf|| . This agrees with the C-infinity topology on copmpact subsets, wherein a sequence converges, fn f, iff there exists a compact set, K, s. t. all fn are supported in K and every derivative, Dafn converges uniformaly to Daf in K.
distribution function
.
PL statistical distribution.
distributive lattice
.
A lattice in which every pair of elements has an join and a meet. PL complemented distributive lattice.
distributive law
.
Misleading label, dating back to Aristotle's Logic. It has persisted because of the medial operator which became popular with the advent of printing. The label commurator law better describes it. The concept and label operation induces the concepts and labels, operator and operand. Thus, in the standard medial notation 2 + 4, the operator is +, the operands are 2, 4. The prefix (a.k.a. Polish) notation encapsulates this: +2,4. (The ordered pair notation of set theory also encapsulates this, along with its sum: <<2,4>,6>.) In medial notation, we have (for naturals a,b,c): a*(b + c) = a*b + a*c. This (passing from medial to prefix notation) becomes: a*(b+c) -> *a+b,c and a*b + a*c -> +*a,b,*a,c, in general, a*(b+c) = a*b + a*c -> *a+b,c = +*a,b,*a,c, which displays operator commuting in *+ -> +*. Hence, commurator law is a better name than distributive law.
divergence
.
Failure of convergence (PL).
divergent sequence
.
Antonomy of convergent sequence (PL).
divergent series
.
Antonomy of convergent series (PL).
divide
.
PL division.
dividend
.
The operand of a division operation (PL).
division
.
Arithmetical inverse (PL) of multiplication, defined thus (for numbers a, b, c, with b ~= 0: a b = c (dividend divided by divisor equals quotient) iff a = b*c. Division is partial on naturals, integers, defined iff the dividend is a natural or integral multiple of the divisor, that is, a defined quotient. Rendering division total results in rationals, as vectors of integers with rules derived from conserving defined quotients.
division algrbra
.
PL division ring .
division algorithm
.
Given dividend D, divisor d ~= 0, quotient Q, and remainder R, then: D = d * Q + R, where 0 </= R < d. The proviso on remainder makes for an antitonic process (PL), which ends when remainder is less than divisor, otherwise the division process would not end. The antitonicity of the division algorithm makes it prototypical for every algorithm (PL).
division ring (d. algebra, skew field)
.
A ring (PL) in which every nonzero members has a multiplicative inverse, although multiplication is not necessarily commutative (PL). Explicitly, a set S(+, *) with additive associativity and commutativity and identity and inverse; multiplicative associativity and identity and inverse; let and right ditributivity (PL all these). In 1878 and 1880, Foebenius and Peirce proved that the only associative division algebras are the real numbers, the complex numbers, and the quaternions.
divisor
.
The operator of a division operation. PL dividend.
Dixon's factorization algorithm
.
To find integers x, y s. t. x2 y2 (modification of Fermat's factorization). If can, there is a "half=half" possibility thatGCD(n, x - y) is a factor of n. Choose randomly ri to compute g(ri) r2i mod n), and try to factor g(ri). If not easily factorable (up to a small divisor d), try another ri, say, of form è(n) ½ø + k, k = 1, 2, ..., allowing use of the quadratic sieve algorithm (PL). Continue finding and factoring the g(ri) until finding N d p, where p is the prime counting function (PL). Proceeding, for each g(ri), write g(r i) = p1ia1ip2ia2i...pNiaNi, and form the exponent vector:

                   æ a1i ö
                   ç a2i ÷
           v(ri) = ç     ÷
                   ç ·   ÷          
                   ç ·   ÷
                   è aNi ø.
If aki is even for any k, then g(ri) is square, solving x2 y2. Otherwise, search for a linear combination (of the above matrix), Sic iv(ri) of all even elements which "go to zero mod two", which is the only way the problem can be solved. So, replace the aij by bij = 0, for aij even; = 1, for a ij odd. And Gaussian elimination then solves bc = z for c; z "going to zero mod two". Knowing c, we have: Pkg(rk) Pkrk2 (mod 2) , the products being taken over all k for which ck = 1. With both sides square, a "half-half chance" of yielding a nontrivial factor of n. Otherwise, choose different z, and repeat procedure. In practice, this algorithm works faster than any using trial divisors; and is amenable to parallel processing, with each processor working on a different value of r.
dodecagon
.
A twelve-sided polygon /
dodecahedron
.
A polyhedron whose sections are dodeagons.
domain
.
The set of values for which a function is defined. Also the label for a connected set.
double cusp
.
A double point (PL).
double integral
.
Special case of a multiple integral (PL).
double point
.
A point which is traced out twice as a curve is traversed.
double root
.
Special case of multiple roots (PL).
dual (duality)
.
Describes a simple operation (PL) with two states. Only an odd number of applications of the operation changes the state; an even number leaves the state unchanged. (Example: complementation, PL, on sets or lattice elements.) PL quaternality .
dual graph
.
Whitney showed that the combinatoric dual graph and geometric dual graph are equivalent, hence, the label "dual graph". Give a graph, G, its geometric dual, G* is constructed by placing a vertex in each region of G (including its exterior region) s.t., if two regions have an edge, E, in common, the corresponding vertices are joined by an edge, E*, which crosses only E.
duality principle
.
All theorems of projective geometry (PL) occur in pairs in which the labels "point" and "line" interchange.
dual linear programming
.
PL linear programming.
dual space
.
A linear vector space induces a dual space of associated linear functions.
dual vector space
.
PL dual space.
duodecimal number system
.
A number system with base (PL) twqelve.
Durfee square
.
The largest square (of dots) in a Ferrers diagram (PL) of a partition (PL).
dyad
.
The vector version of a second rank tensor.
dynamical system
.
The description of one state evolving into another state over the course of time.
dynamic programming
.
A operations research (PL) procedure developed by Richard Bellman by proceeding backwards from the goal to the intial stage while arranging optimality at each stage.
Dynkin diagram
.
A graph whose connections represent irredudible subalgebras (PL) of a (PL).