Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 19
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It is standard to write f (x) = y when (x, y) ∈ f . Seerelation.function space (1) Let X and Y be topological spaces. The function space Y X is the setof all continuous maps from X into Y . Thisspace can be given several topologies, the mostcommon being the compact-open topology. Seecompact-open topology.(2) Any topological space whose elementsare functions on some common domain.functorEither a covariant functor or a contravariant functor.
If no description is specified,fundamental cyclethen the functor is assumed to be covariant. Seecovariant functor.fundamental cycleIf M is a compact, orientable manifold of dimension n, then the ndimensional homology of M is an infinite cyclicgroup. A generator of Hn (M) is a fundamentalcycle. If M is a polyhedral manifold, then thefundamental cycle in simplicial homology canbe given by the n-chain which is the sum of then-simplices. See homology group.Fundamental Theorem of ArithmeticIf nis an integer greater than 1, then n is either aprime number or can be expressed as a productof prime numbers, uniquely, except for order.For example, 24 = 2 × 2 × 2 × 3 and 30 =2 × 3 × 5.Fundamental Theorem of the Theory ofCurvesA curve in Euclidean space R3 isuniquely determined up to rigid motion by itsgeodesic curvature κ and torsion τ , as functions of its arc length parameter s. More precisely, given two continuous functions κ(s) andτ (s) of one real variable such that κ > 0, andgiven initial values X(0) and X (0) in R3 with© 2001 by CRC Press LLC|X (0)| = 1, there is a unique curve X(s) whosecurvature is κ and torsion is τ .
κ is usually takento be continuously differentiable and X then hasthree continuous derivatives.Fundamental Theorem of the Theory of SurfacesLet S be a surface in Euclidean threespace parameterized byX(u, v) = (x(u, v), y(u, v), z(u, v)) ,where x, y, and z are assumed to have continuous third-order partial derivatives. Then Spossesses a first fundamental form g and a second fundamental form L satisfying the Gaussequations and the Codazzi-Mainardi equations.The fundamental theorem of the theory of surfaces states the converse, namely, if g(u, v) isa positive definite symmetric tensor (i.e., an inner product), and L(u, v) is a symmetric tensor,with g having continuous second derivatives andL having continuous first derivatives, and if gand L satisfy the Gauss and Codazzi-Mainardiequations, then they (locally) determine a surface S uniquely up to rigid motion. See firstfundamental form, Gauss equations, CodazziMainardi equations.geodesic correspondenceasL(s, χ ) =Gn=1Gδ set A countable intersection of open sets.See Fσ set.Gauss equationsA system of partial differential equations arising in the theory of surfaces.
If M is a surface in R3 with local coordinates (u1 , u2 ), its geometric invariants can bedescribed by its first fundamental formgij (u1 , u2 ), and second fundamental formLij (u1 , u2 ). The Christoffel symbols ijk aredetermined by the first fundamental form. Inorder for functions gij and Lij , i, j = 1, 2 tobe the first and second fundamental forms ofa surface, certain integrability conditions (arising from equality of mixed partial derivatives)must be satisfied.
One set of conditions, theGauss equations, relate the determinant of thesecond fundamental form to an expression involving only the first fundamental form (and itsfirst and second partial derivatives). See alsoChristoffel symbols, first fundamental form.general Cantor setA Cantor set in whichintervals of length 3αn are removed at stage n for0 < α < 1. The resulting set, Cα = ∩∞n=1 In ,is a closed set with length 1 − α which forms atotally disconnected compact topological spacein which every element is a limit point of the set.See Cantor set.generalized continuum hypothesisstatementThe2ℵα = ℵα+1for all ordinals α. (Both this statement andits negation are independent of the axioms ofZermelo-Fraenkel set theory with the Axiom ofChoice.) See continuum hypothesis.generalized Riemann hypothesis An assertion concerning the zeros of functions that aresimilar to the Riemann zeta function.
Thesefunctions are called L-functions and are defined© 2001 by CRC Press LLC∞χ (n)nswhere χ is a Dirichlet character, that is, a real orcomplex valued function defined on the positiveintegers so that(i.) χ (mn) = χ (m)χ (n) for all m and n;(ii.) there is a positive integer k so that χ (n+k) = χ (n) for all n;(iii.) χ (n) = 0 if gcd(n, k) > 1.The generalized Riemann hypothesis conjecturesthat all zeros of the function L(s, χ ) on the “critical strip” (those complex numbers s = x + iysuch that 0 < x < 1) must lie on the “criticalline” (x = 21 ). Notice that L(s, 1) = ζ (s), so thegeneralized Riemann hypothesis “agrees with”the Riemann hypothesis.
See also Riemann zetafunction, Riemann Hypothesis.General Position TheoremA name givento a number of theorems asserting, for variousclasses of maps of spaces X into spaces Y , thatsuch maps may be approximated by maps having simpler structure. These “generic” mapsusually have minimum possible dimensionalityof intersections or self-intersections. A simpleexample is the statement that two affine subspaces of dimensions k and l in n-dimensionalspace can be perturbed so that they intersect ina subspace of dimension n − (k + l).generating curve A surface of revolution inEuclidean space can be parameterized byX(t, θ ) = (r(t) cos(θ ), r(t) sin(θ ), z(t)) ,where r(t) > 0 and z(t) are continuous functions. The curve σ (t) = (r(t), z(t)) is the generating curve of the surface.geodesicIn a manifold with metric (nondegenerate smoothly varying quadratic form oneach tangent space), a curve of minimal lengthbetween two points. An example of geodesicsare arcs of great circles on a sphere.geodesic correspondenceA smooth mapφ : S −→ S between two surfaces which takesgeodesics of S to geodesics of S .
An isometricmapping is automatically a geodesic correspondence, but the converse is not true.geodesic curvatureAn example of a geodesic correspondence isgivenby central projection of the hemispherez = 1 − x 2 − y 2 to the plane z = 1. Great circle arcs in the hemisphere correspond to straightlines in the plane under this projection.geodesic curvatureIf C is a curve lying ona surface S in Euclidean space R3 , parameterized by unit speed, then its second derivative oracceleration vector C is perpendicular to thevelocity vector C . The length of the component of C tangent to the surface is the geodesiccurvature of the curve. It measures the amountof bending the curve undergoes within the surface, as opposed to the amount of bending dueto the bending of the surface itself.geometric realizationLet S be an abstractsimplicial complex.
If S is isomorphic to thevertex scheme of a simplicial complex K, thenK is the geometric realization of S. The geometric realization of an abstract simplicial complex is unique up to isomorphism.geometry on a surface The measurement oflengths of curves, angles between curves, andareas of figures lying on a surface. This is alsocalled intrinsic geometry, to distinguish it fromproperties of a surface that depend on how thesurface sits in space.Gödel numberA Gödel numbering (arithmetization) is an effective method of coding nonnumerical objects by natural numbers.
For example, there is a computable bijection from theset T of all Turing machine programs to the setof natural numbers. The inverse of this bijection is also computable, so that, given a Turingmachine program, one can effectively find thenatural number assigned to it, which is calledthe Gödel number of the program, and given anatural number, one can effectively “decode” itin order to find the Turing machine program thatcorresponds to it. The notation ϕe is used to denote the Turing computable partial function withGödel number e; i.e., ϕe is Turing computablevia the Turing program with Gödel number e.Gödel was the first to use Gödel numbers in hisproof of his Incompleteness Theorem.© 2001 by CRC Press LLCGödel set theory The same as Bernays-Gödelset theory.
See Bernays-Gödel set theory.great circleA circle on a sphere formed byintersecting the sphere with a plane that passesthrough the center of the sphere. If P and Q aretwo points on the sphere, then a curve of leastlength joining P to Q is an arc of a great circle.See also geodesic.greatest common divisor(1) If a and b arenonzero integers, then the greatest common divisor of a and b, denoted gcd(a, b) is the largestinteger that is a divisor of both a and b.
Forexample, the greatest common divisor of 28 and36 is 4 (the common divisors of 28 and 36 are±1, ±2, and ±4).(2) Alternatively, in a Euclidean ring,R, gcd(a, b) is an element (not necessarilyunique) d of R satisfying(i.) d is a divisor of both a and b;(ii.) if x is a divisor of a and b, then x is alsoa divisor of d.greatest elementGiven a set A with an ordering ≤ on A, an element u ∈ A is said to be agreatest element of A if, for all x ∈ A, x ≤ u.Note that if A has a greatest element, then it isunique. Compare with least element.greatest lower boundLet A be an orderedset and let B ⊆ A.
An element l ∈ A is saidto be a greatest lower bound (or infimum) for Bif it is a lower bound for B (i.e., for all x ∈ B,l ≤ x) and if it is the greatest element in the setof all lower bounds for B (i.e., for all y ∈ A, iffor all x ∈ B, y ≤ x, then y ≤ l). Note that if aset has a greatest lower bound, then it is unique.Compare with least upper bound.groupA non-empty set G with a productmap G × G −→ G ((g, h) is taken to gh), aninverse map G −→ G (g is taken to g −1 ), and adistinguished element called the identity (oftendenoted 0, 1, or e) satisfying ge = eg = g,for g ∈ G.
These satisfy the relationships thatg1 (g2 g3 ) = (g1 g2 )g3 and gg −1 = e = g −1 g.Common examples include topologicalgroups and Lie groups.group of symmetriesgroup of motionsA group of lengthpreserving transformations, or rigid motions, inEuclidean n-space. A group is a non-empty setwith a binary associative operation that containsan identity and an inverse of each one of its elements. The rigid motions of the Euclidean planeare translations, rotations, reflections, and glidereflections.© 2001 by CRC Press LLCgroup of symmetriesOf a figure, a groupof motions that transform a figure into itself.See group of motions. For example, the groupof symmetries of an equilateral triangle is thegroup of six elements that can be identified withthe permutations of the vertices.hexahedronHhalf line A connected unbounded and propersubset of a line in Euclidean space.
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