Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 22
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For instance, a set S is finite if there is a nonnegative integer n and abijection f : S → {1, 2, 3, . . . , n} (that is, ifS has n elements) and S is infinite, otherwise.Alternatively, a set is infinite if there is a bijection from the set to a proper subset of itself. Inhis theory of transfinite numbers, Georg Cantor dealt with various “sizes” of infinity, dis-© 2001 by CRC Press LLCinitial segmentA subset of a well-orderedset W which has the form {x ∈ W : x < w},where w ∈ W .(2) The function i : X → Y between twosets X and Y , with X ⊆ Y , defined by i(x) = x.See also function.integer An element of the set {.
. . , −4, −3,−2, −1, 0, 1, 2, 3, 4, . . . } consisting of allwhole numbers. The set of all integers is usuallydenoted Z or Z.integral curvatureIf S is a surface in Euclidean space and A is a measurable subset of S,then the integral curvature of A is the area of itsimage under the Gauss map to the unit sphere.That is, it is the area of the set of unit vectorsthat are outer normals to support planes of thesurface at points of A.
For smooth surfaces,it can be computed intrinsically by integratingthe Gaussian curvature over the region A. Fora polyhedron, the integral curvature is concentrated at the vertices.interior angle If P is a simple closed polygonenclosing a region R, then the interior angle ata vertex V is measured by the magnitude of therotation that carries one edge of P adjacent to Vto the other edge, the rotation performed withinR.interior of closed curveinterior of closed curveThe bounded component of the complement of a simple closedcurve.
By the Jordan Curve Theorem, the complement of the curve consists of exactly two connected components.interior of polygon The bounded componentof the complement of a polygon P , which, as acurve, is closed and has no self-intersections.Any two points in the interior can be joined bya continuous curve that does not intersect P ,while any two points in the other component ofthe complement of P (the exterior of the polygon) can be joined by a continuous curve thatdoes not intersect P . But no point in the interiorcan be joined to any point in the exterior by acontinuous curve that does not intersect P .
Thisfact is the content of the Jordan Curve Theoremfor polygons.interior of polyhedronA closed connectedpolyhedral surface in Euclidean space R3 has acomplement consisting of two path-connectedcomponents. One of these two, the boundedcomponent, is called the interior of the polyhedron.intersection of sets If X and Y are sets, thenthe intersection of X and Y , denoted X∩Y , is theset consisting of all elements that are common toboth X and Y . Symbolically, X ∩ Y = {z : z ∈X and z ∈ Y }. More generally, if {Xα }α∈ is afamily of sets, then the intersection α∈ Xα isthe set consisting of all elements that are common to all Xα .
See also Boolean algebra, lattice.inverse correspondencetion.See inverse func-inverse functionLet X, Y be sets and suppose that f ⊆ X × Y is a function. If f isone-to-one (f (x1 ) = f (x2 ) implies x1 = x2 ),then the inverse function f −1 is the unique function obtained by interchanging the coordinatesin the ordered pairs belonging to f . Symbolically, f −1 = {(y, x) : (x, y) ∈ f }. It can beverified that f −1 f (x) = x and ff −1 (y) = yfor all x ∈ X and all y in the range of f . Thefunction f −1 is also one-to-one and its inversesatisfies (f −1 )−1 = f . If f is not one-to-one,then f −1 is not a function. See function.© 2001 by CRC Press LLCinverse morphismSuppose that C is a category. If f ∈ HomC (A, B) is an invertiblefunction, then the inverse morphism f −1 is theunique morphism in HomC (B, A) satisfying(i.) ff −1 = 1B ,(ii.)f −1 f = 1A .inverse relationLet X, Y be sets and suppose that R ⊆ X × Y is a relation.
The inverserelation R −1 is the relation obtained by interchanging the coordinates of all ordered pairs inR. Symbolically, R −1 = {(y, x) ∈ Y × X :(x, y) ∈ R}. See relation.involuteA curve associated with a givencurve C as follows: the tangent lines to the curveform a surface, an involute is a curve on this surface which is orthogonal to the tangent lines.
IfC is parameterized by arc length s, then the involutes are given by Ic (s) = C(s)+(c −s)C (s), ca constant.involutionA transformation that is its owninverse. In geometry, the reflection across astraight line is an involution of the plane.irrational number A real number that is notrational. That is, a real number that cannot beexpressed as a quotient of integers. Examples ofirrational numbers√ are π ,√e (the base of the natural logarithm), 2, and 6 (in fact, the squareroot of any integer, other than a perfect square,will be irrational).irreducible quadratic polynomial The polynomial ax 2 + bx + c with real coefficients isirreducible (over the field of real numbers) ifit cannot be expressed as the product of twonon-constant polynomials with real coefficients.This occurs if and only if b2 − 4ac < 0.Any polynomial with real coefficients can befactored (using only real coefficients) into theproduct of linear factors and irreducible quadratic polynomials.irreflexive relationA relation R ⊆ X × Xon a set X such that there is no x ∈ X with(x, x) ∈ R.
For example, if R consists of allordered pairs of real numbers (a, b) such thata < b, then R is irreflexive.isosceles triangleisolated pointA point x in a topologicalspace X such that the singleton set {x} is openin X. Equivalently, x ∈/ X \ {x}. Thus, x is isolated in X if and only if it is not an accumulationpoint in X.More generally, x is an isolated point of asubset A ⊆ X if x ∈ A and there is an openU ⊆ X with U ∩ A = {x}. That is, x ∈/ A \ {x},and so x is an isolated point of A if and only ifit is in A but is not an accumulation point of A.isometric surfacesTwo surfaces S and S ,for which there is a bijection from S to S , whichtakes every curve in S to a curve in S of thesame length. Assuming the surfaces are differentiable, they are isometric if there is a diffeomorphism from S to S which pulls the firstfundamental form of S back to the first fundamental form of S.
See first fundamental form.isomorphic orderings Two orderings (X, ≤)and (X , ≤ ) such that there exists a bijectionfrom X to X which is order-preserving. Moreprecisely, the orderings are isomorphic if there isa bijection f : X → X such that if x1 , x2 ∈ Xand x1 ≤ x2 , then f (x1 ) ≤ f (x2 ).isomorphism Let L be a first order language,and let A and B be structures for L, where A andB are the universes of A and B, respectively.A function h : A → B is an isomorphism ofstructures if h is injective and surjective, and© 2001 by CRC Press LLC(i.) for each n-ary predicate symbol P andevery a1 , .
. . an ∈ A,(a1 , . . . , an ) ∈ P A⇔ (h(a1 ), . . . , h(an )) ∈ P B ,(ii.) for each constant symbol c,h(cA ) = cB ,and(iii.) for each n-ary function symbol f andevery a1 . . . , an ∈ A,h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )).If there is an isomorphism of A onto B, thenA and B are isomorphic structures (notation:A∼= B).isoperimetric For two curves, C and C , theproperty of having the same length. The isoperimetric inequality in the plane states that, amongall curves isoperimetric to a given simple closedcurve C, the circle encloses the maximum area.isoscelesAn isosceles polygon is a polygonpossessing two sides of the same length.
Theterm is usually applied to triangles or trapezoids.In the case of a trapezoid, the sides are generallytaken to be opposite sides.isosceles triangle A triangle possessing twosides of equal length. See also isosceles.jumpJjump Let A be a set of natural numbers. Thejump of A (also called the Turing jump of A) isthe halting set, relativized to A; i.e., the jumpof A is the set {e : ϕeA (e) is defined}, where ϕeAdenotes the partial A–computable (A–recursive)function with Gödel number e. The jump of Ais denoted by A .© 2001 by CRC Press LLCKurepa treeKK3 surfaceAn algebraic surface that issmooth, has a global holomorphic 2-form, andfirst homology group of rank 0. Part of an important class of surfaces in algebraic geometry,named after three mathematicians: Kummer,Kähler, and Kodaira.
An example is the intersection of three generic quadric hypersurfacesin P5 . See hypersurface.Kirby calculusA method of specifyingsurgery operations on a manifold in terms of theidentifications to be performed on the meridianof a solid torus embedded in the manifold.Kleene’s hierarchyAlternate (rarely used)terminology for the arithmetical hierarchy. Seearithmetical hierarchy.Kleinian groupA subgroup G of the groupof Möbius transformations, with the propertythat there exists some point z of the extendedplane C ∪ {∞} at which G acts discontinuously,i.e., the stabilizer Gz is finite, and there exists aneighborhood U of z which is fixed by all theelements of Gz , but whose only fixed point under any element of Gz is z.
See linear fractionalfunction. Examples are given by the first homotopy group of Riemann surfaces.© 2001 by CRC Press LLCKodaira dimensionA rational invariant ofa smooth projective variety V , named after theJapanese mathematician Kunihiko Kodaira. Itis the maximum of the dimensions of φn (V ),where φn is the rational map associated to thenth power of the canonical bundle, over all positive integers n for which this power has globalsections. If no such n exists, the Kodaira dimension is defined to be −∞.k-perfect number A positive integer n having the property that the sum of its positive divisors is kn, i.e., σ (n) = kn.
Thus, a 2-perfectnumber is the same as a perfect number. Thesmallest 3-perfect number is 120. The smallest4-perfect number is 30,240.Kurepa treeA tree of height ω1 with nouncountable levels but at least ω2 uncountablebranches. Thus, for each α < ω1 , the α-level ofT , Levα (T ), given byt ∈ T : ordertype({s ∈ T : s < t}) = αis countable, Levω1 (T ) is the first empty level ofT , and there are at least ω2 different sets B ⊆ Ttotally ordered by < (branches) that are uncountable. Kurepa’s Hypothesis (KH) is that there exists a Kurepa tree, but KH is independent of theaxioms of set theory. In fact, KH is independentof ZFC+GCH.For any regular cardinal κ, a κ-Kurepa treeis a tree of height κ in which all levels have sizeless than κ and there are at least κ + brancheswith length κ.
See also Aronszajn tree, Suslintree.Lie groupmultiple of two integers is the product of the twointegers (i.e., ab = gcd(a, b) · LCM(a, b).) Seegreatest common divisor.LLatin square An n×n array of numbers suchthat each row and column of the array containsthe same numbers and each number appears exactly once in every row and column. For example,123231312and13221332 .1lattice A non-empty set X, together with twobinary operations ∪, ∩ on X (called union andintersection, respectively), which satisfy the following conditions for all A, B, C ∈ X:(i.) (A ∪ B) ∪ C = A ∪ (B ∪ C);(ii.) (A ∩ B) ∩ C = A ∩ (B ∩ C);(iii.) A ∪ B = B ∪ A;(iv.) A ∩ B = B ∩ A;(v.) (A ∪ B) ∩ A = A;(vi.) (A ∩ B) ∪ A = A.leafA manifold that is a maximal integralsubmanifold of an integrable distribution.
Given a manifold M, a distribution assigns toeach point P in M a k-dimensional subspace ofthe tangent space at P . It is integrable if the manifold is the union of k-dimensional immersedsubmanifolds, such that the k-plane (p) is thetangent plane of the k-manifold through p. Aleaf is a maximal connected integral submanifold of the distribution.least common multipleFor two nonzerointegers a and b, the smallest positive integerL that is a multiple of both a and b, is denoted LCM(a, b). Equivalently, LCM(a, b) isthe unique positive integer that is a multipleof both a and b and is a divisor of all othercommon multiples of a and b.
For example,LCM(14, 8) = 56 and LCM(3, 5) = 15. Notethat the least common multiple of two nonzerointegers will always be a divisor of the productof the two integers. In fact, the product of thegreatest common divisor and the least common© 2001 by CRC Press LLCleast element Given a set A with an ordering≤ on A, an element l ∈ A is said to be a leastelement of A if, for all x ∈ A, l ≤ x. Notethat if A has a least element, then it is unique.Compare with greatest element.least upper boundLet A be an ordered setand let B ⊆ A. An element u ∈ A is said tobe a least upper bound (or supremum) for B ifit is an upper bound for B (i.e., for all x ∈ B,x ≤ u) and it is the least element in the set ofall upper bounds for B (i.e., for all y ∈ A, if forall x ∈ B, x ≤ y, then u ≤ y).
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