Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 14
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Given any X ⊆ ω1 ,the set of α where X ∩ α = Sα is stationary inω1 . In other words, there are a large number ofα where Sα is the same as X up to α.To see why this implies the Continuum Hypothesis, notice that if X ⊆ ω, thenX∩α =Xfor any α ≥ ω. But C = [ω, ω1 ) is closed andunbounded, so there is an α ∈ C ∩S where Sα =X ∩ α = X. That is, each subset of the naturalnumbers appears in the diamond sequence, sothe number of subsets is at most ω1 .Other consequences of ♦ include the negation of Suslin’s Hypothesis; i.e., ♦ implies thereis an ω1 -Suslin tree.difference of sets For two sets X and Y , theset X\Y consisting of all elements of X whichare not elements of Y .
More precisely,X\Y = {x ∈ X : x ∈/ Y} .differentiable structureA compatible wayof assigning, to each point in a space, a homeomorphism from a neighborhood of that point toan open subset of n-dimensional real space Rn ,or of n-dimensional complex space Cn .diagram in the category In the category C, adiagram in which the vertices represent objectsof C and the arrows represent morphisms of C.See diagram, category.differential geometry The body of geometrythat investigates curves and surfaces in the immediate neighborhood of one of their points, using calculus, and analyzes what is implied aboutthe curve or surface as a whole on the basis ofthis local behavior. A more advanced aspect ofdifferential geometry is the possibility of constructing geometrical systems determined solelyby concepts and postulates that affect only theimmediate neighborhood of each point of thesystem.diameter The greatest distance between anytwo points of the body in question.differential invariant An expression that consists of certain functions, partial derivatives, andab·→ ·→ · · ·consists of vertices a, b, .
. . and arrows. Thevertices represent sets A, B, . . . and the arrowsrepresent functions between the sets.© 2001 by CRC Press LLCdihedral angledifferentials, which is invariant with respect tocertain transformations.dihedral angle The angle between two planes,measured as the angle in a plane perpendicularto the line of intersection of the two planes. Ifthe planes do not intersect, or coincide, the dihedral angle is zero.dilatationAn affinity that possesses a fixedpoint and maps every line onto a parallel of itself.dimensionAny one of many possible different topologically invariant measures of thesize of a topological space. Different definitions of dimension include the Lebesgue dimension, the homological dimension, the cohomological dimension, and the large and small inductive dimensions.
The large inductive dimension, which agrees with the Lebesgue and smallinductive dimensions when the space is separable and metrizable, is defined inductively as follows. We say that the empty set has dimension−1. Assuming that we have defined all spacesof dimension ≤ n, we say that a space X hasdimension ≤ n + 1 if, for any disjoint closedsubsets C and D of X, there is a closed subsetT of X with dimension T ≤ n such that X\Tis the union of two disjoint open subsets, onecontaining C and the other containing D.
Wethen say that a topological space has dimensionn if it has dimension ≤ n but it does not havedimension ≤ n − 1.dimension functionA function d : L →Z from a lattice L to the nonnegative integerssatisfying the conditions (i.) d(x+y)+d(xy) =d(x) + d(y) for all x, y ∈ L, and (ii.) if [x, y] isan elementary interval in L, then d(y) = d(x)+1.dimension of a complexLet X be a CWcomplex and let E be the set of cells of X. Thedimension of X is given by:dimX = sup{dim(e) : e ∈ E} .We say that X is finite dimensional if dimX isfinite and infinite dimensional otherwise.© 2001 by CRC Press LLCdimensions of a rectangleThe dimensionsthat fully describe a rectangle, namely lengthand width.Dimension Theorem of Affine GeometryGiven affine space An , with nondisjoint subspaces Ar and As of dimension r and s, respectively, then r + s = dim(Ar ∪ As ) + dim(Ar ∩As ).dimension theoryThe branch of topologydevoted to the definition and study of the notionof dimension in various classes of topologicalspaces.dimension typeTwo topological spaces Xand Y have the same dimension type if X ishomeomorphic to a subspace of Y and Y is homeomorphic to a subspace of X.dimension zeroA topological space X hasdimension zero if it has the discrete topology.See topological dimension.Dini surfaceA helicoidal surface in threedimensional Euclidean space, which is the surface of revolution of a tractrix.directed setA set D with a partial order ,such that for all a, b ∈ D there exists an elementc ∈ D so that a c and b c.directrix The polar, with respect to the reciprocating circle of a conic section, of the centerof the reciprocal circle of the conic section; thisapplies when the conic section is regarded asthe reciprocal of a circle.
Alternatively, if theconic section is regarded as a curve generatedby a point moving in the plane such that the ratio of its distance from a fixed point to a fixedline remains constant, then the directrix of theconic section is that fixed line.directrix of WilczynskiTwo straight linesassociated with a normal frame in projective differential geometry.Dirichlet convolutionThe arithmeticfunction f ∗g defined by (f ∗g)(n) = d f (d)g( dn ),where f and g are arithmetic functions, and dranges over the divisors of n.
(See arithmeticdistance functionfunction.) For example, if f = φ, the Eulerphi function, and g = τ , the number of divisorsfunction,disjoint setsTwo sets X and Y which haveno common elements. Symbolically, X and Yare disjoint ifX∩Y =∅.(φ ∗ τ )(10)= φ(1)τ (10)+φ(2)τ (5)+φ(5)τ (2)+φ(10)τ (1)= 18 .In fact, φ ∗ τ = σ , the sum of divisors function. The Dirichlet convolution is also calledthe Dirichlet product.Dirichlet inverse The Dirichlet inverse of anarithmetic function f is a function f −1 such thatthe Dirichlet convolution f ∗f −1 = I , the identity function.
(See arithmetic function, Dirichletconvolution.) A function f has a Dirichlet inverse if and only if f (1) = 0. When it exists, theinverse is unique. For example, µ, the Möbiusfunction, and u, the unit function, are Dirichletinverses of one another.Dirichlet multiplicationThe operation under which the Dirichlet convolution of two arithmetic functions is computed. It is commutativeand associative.
In fact, the set of arithmeticfunctions f such that f (1) = 0 forms a groupunder this operation. See also Dirichlet convolution.Dirichlet productSee Dirichlet convolution.discrete linear orderingA linear ordering≤ on a set A such that(i.) every element x ∈ A that has a successor(i.e., an element y ∈ A such that x < y) hasan immediate successor (i.e., there exists z ∈ Asuch that z is a successor of x and there does notexist y ∈ A with x < y < z), and(ii.) every element x ∈ A that has a predecessor (i.e., an element y ∈ A such that y < x)has an immediate predecessor (i.e., there existsz ∈ A such that z is a predecessor of x and theredoes not exist y ∈ A with z < y < x).The usual ordering ≤ on the set N of naturalnumbers is discrete.discrete topologyThe topology on a set X,consisting of all subsets of X.
That is, everysubset is open in X.© 2001 by CRC Press LLCdisjunctive normal formA propositional(sentential) formula of the formminAij ;i=1j =1i.e.,(A11 ∧ · · · ∧ A1m1 ) ∨ · · · ∨(An1 ∧ · · · ∧ Anmn ) ,where each Aij , 1 ≤ i ≤ n, 1 ≤ j ≤ mi , iseither a sentence symbol or the negation of asentence symbol. Every well-formed propositional formula is logically equivalent to one indisjunctive normal form.For example, if A, B, and C are sentencesymbols, then the well-formed formula ((A →B) → C) is logically equivalent to (A ∧ B ∧C) ∨ (A ∧ ¬B ∧ C) ∨ (A ∧ ¬B ∧ ¬C) ∨ (¬A ∧B ∧ C) ∨ (¬A ∧ ¬B ∧ C), which is a formulain disjunctive normal form.distancePart of the definition of a metricspace M. The distance function on M, d :M × M → R, must be nonnegative-valuedand satisfy (i.) d(P1 , P2 ) = 0 if and only ifP1 = P2 , (ii.) d(P1 , P2 ) = d(P2 , P1 ), and(iii.) d(P1 , P2 ) + d(P2 , P3 ) ≥ d(P1 , P3 ) forall P1 , P2 , P3 ∈ M.
Then the distance betweentwo points P1 and P2 is d(P1 , P2 ). We may thenalso define the distance between any two subsetsS and T of M to be the greatest lower bound ofthe set {d(s, t) : s ∈ S, t ∈ T }.In three-dimensional Euclidean space, the distance function is given byd((x1 , y1 , z1 ), (x2 , y2 , z2 ))= (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .distance function A function d : X × X →R, where X is a topological space and R is thedivision algorithmreal numbers, which satisfies the following threeconditions:(i.) d(x, y) ≥ 0, and d(x, y) = 0 if and onlyif x = y;(ii.) d(x, y) = d(y, x); and(iii.) d(x, y) + d(y, z) ≥ d(x, z).This last condition, known as the triangle inequality, generalizes the principle of plane geometry that the length of any side of a triangleis not longer than the sum of the lengths of theother two sides.double angle formulasThe trigonometricidentities sin 2θ = 2 sin θ cos θ and cos 2θ =1 − 2 sin2 θ.division algorithmIf a and b = 0 are in Z,there exist unique integers r so that a = bq + rand 0 ≤ r < b.
A clever repeated application ofthe division algorithm, known as Euclid’s algorithm, leads to the computation of the greatestcommon divisor of the integers a and b. SeeEuclidean algorithm.Division algorithms hold in other rings, suchas the polynomials with real coefficients.the dual bundle ξ ∗ of ξ has the projectiondivisor If a and b are elements of a ring andthere exists an element c in that ring satisfyingbc = a, then b (similarly, c) is a divisor ofa. For example, in the ring of integers, 6 is adivisor of 24 since 6 × 4 = 24 (and 4, 6, and24 are all integers), while 5 is not a divisor of24 since there is no integer c so that 5c = 24.However, in the ring Z36 (the integers mod 36),5 is a divisor of 24 since 5 × 12 = 24 in Z36(alternatively, 5 × 12 ≡ 24 (mod 36)).dodecagonA polygon having 12 sides.dodecahedronA polyhedron with 12 faces.The regular dodecahedron, the regular convex polyhedron having 12 pentagonal faces, 30edges, and 20 vertices, is one of the five platonicsolids.domainFor a binary relation R on two setsX and Y , the setdom(R) = {x : (x, y) ∈ R for some y ∈ Y } .Commonly, the relation R is a function f , R ={(x, y) : y = f (x)}, and the domain of thefunction f is the domain of the relation R, inthis case.
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