Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 6
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See also Axiom of Choice.chordA line segment with endpoints on acurve (usually a circle).Christoffel symbolsThe coefficients in local coordinates for a connection on a manifold.If (u1 , . . . , un ) is a local coordinate system ina manifold M and ∇ is a covariant derivativeoperator, then the derivatives of the coordinatefields ∂u∂ j can be written as linear combinationsof the coordinate fields:n∂∂∇ ∂=ijk k .j∂u∂ui ∂uk=1The functions ijk (u1 , . . .
, un ) are the Christoffel symbols. For the standard connection onEuclidean space Rn the Christoffel symbols areidentically zero in rectilinear coordinates, but ingeneral coordinate systems they do not vanisheven in Rn .Church-Turing ThesisIf a partial functionϕ on the natural numbers is computable by analgorithm in the intuitive sense, then ϕ is computable, in the formal, mathematical sense. (Afunction ϕ on the natural numbers is partial if itsdomain is some subset of the natural numbers.)See computable.This statement of the Church-Turing Thesis is a modern day rephrasing of independentstatements by Alonzo Church and Alan Turing. Church’s Thesis, published by Church in1936, states that the intuitively computable partial functions are exactly the general recursivefunctions, where the notion of general recursivefunction is a formalization of computable defined by Gödel.
Turing’s Thesis, published byTuring in 1936, states that the intuitively computable partial functions are exactly the partialfunctions which are Turing computable.© 2001 by CRC Press LLCThe Church-Turing Thesis is a statement thatcannot be proved; rather it must be accepted orrejected. The Church-Turing Thesis is, in general, accepted by mathematicians; evidence infavor of accepting the thesis is that all knownmethods of formalizing the notion of computability (see computable) have resulted in the sameclass of functions; i.e., a partial function ϕ ispartial recursive if and only if it is Turing computable, etc.The most important use of the Church-TuringThesis is to define formally the notion of noncomputability.
To show the lack of any algorithm to compute a function, it suffices by thethesis to show that the function is not partial recursive (or Turing computable, etc.). The converse of the Church-Turing Thesis is clearly true.circleThe curve consisting of all points in aplane which are a fixed distance (the radius ofthe circle) from a fixed point (the center of thecircle) in the plane.circle of curvature For a plane curve, a circleof curvature is the circle defined at a point on thecurve that is both tangent to the curve and hasthe same curvature as the curve at that point.
Fora space curve, the osculating circle is the circleof curvature.circle on sphere The intersection of the surface of the sphere with a plane.circular arccircular coneA segment of a circle.A cone whose base is a circle.circular cylindercircles.A cylinder whose bases arecircular helix A curve lying on the surface ofa circular cylinder that cuts the surface at a constant angle. It is parameterized by the equationsx = a sin t, y = a cos t, and z = bt, where aand b are real constants.circumcenter of triangle The center of a circle circumscribed about a given triangle.
Thecircumcenter coincides with the point commonto the three perpendicular bisectors of the triangle. See circumscribe.closed and unboundedcircumference of a circlelength, of a circle.The perimeter, orcircumference of a sphereThe circumference of a great circle of the sphere. See circumference of a circle, great circle.circumscribeGenerally a plane (or solid)figure F circumscribes a polygon (or polyhedron) P if the region bounded by F containsthe region bounded by P and if every vertex ofP is incident with F .
In such a case P is saidto be inscribed in F . See circumscribed circle,for example. In specific circumstances, figuresother than polygons and polyhedra may also becircumscribed.circumscribed prism A prism that containsthe interior of a cylinder in its interior, in such away that both bases of the prism circumscribe abase of the cylinder (and so each lateral face ofthe prism is tangent to the cylindrical surface);i.e., the cylinder is inscribed in the prism. Seecircumscribe.circumscribed pyramid A pyramid that contains, in its interior, the interior of a cone, in sucha way that the base of the pyramid circumscribesthe base of the cone and the vertex of the pyramid coincides with the vertex of the cone; i.e.,the cone is inscribed in the pyramid. See circumscribe.circumscribed circle A circle containing theinterior of a polygon in its interior, in such a waythat every vertex of the polygon is on the circle;i.e., the polygon is inscribed in the circle.circumscribed sphereA sphere that contains, in its interior, the region bounded by apolyhedron, in such a way that every vertex ofthe polyhedron is on the sphere; i.e., the polyhedron is inscribed in the sphere.
See circumscribe.circumscribed coneA cone that circumscribes a pyramid in such a way that the baseof the cone circumscribes the base of the pyramid and the vertex of the cone coincides withthe vertex of the pyramid; i.e., the pyramid isinscribed in the cone. See circumscribe.class The collection of all objects that satisfya given property. Every set is a class, but theconverse is not true. A class that is not a setis called a proper class; such a class is much“larger” than a set because it cannot be assigneda cardinality. See Bernays-Gödel set theory.circumscribed cylinder A cylinder that circumscribes a prism in such a way that both basesof the cylinder circumscribe a base of the prism;i.e., the prism is inscribed in the cylinder.
Seecircumscribe.circumscribed polygon A polygon that contains the region bounded by a closed curve (usually a circle) in the region it bounds, in such away that every side of the polygon is tangentto the closed curve; i.e., the closed curve is inscribed in the polygon.circumscribed polyhedronA polyhedronthat bounds a volume containing the volumebounded by a closed surface (usually a sphere)in such a way that every face of the polyhedronis tangent to the closed surface; i.e., the closedsurface is inscribed in the polyhedron. See circumscribe.© 2001 by CRC Press LLCclassifying spaceThe classifying space ofa topological group G is a space BG with theproperty that the set of equivalence classes ofvector bundles p : E −→ B with G-action is inbijective correspondence with the set [B, BG]of homotopy classes of maps from the space Bto BG.The space BG is unique up to homotopy, thatis, any two spaces satisfying the above propertyfor a fixed group G are homotopy equivalent.For G = Z/2, BZ/2 is an infinite projectivespace RP∞ , the union of all projective spacesRPn .
Since O(1) = Z/2, all line bundles overa space X are classified up to bundle homotopyequivalence by homotopy classes of maps fromX into RP∞ .closed and unboundedIflimit ordinal (in practice κ iscardinal), and C ⊆ κ, C isbounded if it satisfies (i.) forκ is a non-zeroan uncountableclosed and unevery sequenceclosed convex curveα0 < α1 < · · · < αβ . . .
of elements of C(where β < γ , for some γ < κ), the supremum of the sequence, β<γ αβ , is in C, and(ii.) for every α < κ, there exists β ∈ C suchthat β > α. A closed and unbounded subset ofκ is often called a club subset of κ.closed convex curve A curve C in the planewhich is a closed curve and is the boundary ofa convex figure A. That is, the line segmentjoining any two points in C lies entirely withinA.
Equivalently, if A is a closed bounded convex figure in the plane, then its boundary C is aclosed convex curve.closed convex surfaceThe boundary S ofa closed convex body in three-dimensional Euclidean space. S is topologically equivalent toa sphere and the line segment joining any twopoints in S lies in the bounded region boundedby S.closed formula A well-formed formula ϕ ofa first-order language such that ϕ has no freevariables.closed half line A set in R of the form [a, ∞)or (−∞, a] for some a ∈ R.closed half planeA subset of R2 consistingof a straight line L and exactly one of the two halfplanes which L determines. Thus, any closedhalf plane is either of the form {(x, y) : ax +by ≥ c} or {(x, y) : ax + by ≤ c}. The setsx ≥ c and x ≤ c are vertical closed half planes;y ≥ c and y ≤ c are horizontal half planes.closed map A function f : X → Y betweentopological spaces X and Y such that, for anyclosed set C ⊆ X, the image set f (C) is closedin Y .closed set(1) A subset A of a topologicalspace, such that the complement of A is open.See open set.
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