Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 8
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The set {∧, →} is not a complete setof logical connectives.completely multiplicative function An arithmetic function f having the property thatf (mn) = f (m) · f (n) for all positive integersm and n. (See arithmetic function.) For example, λ, Liouville’s function, is completely multiplicative. The values of a completely multiplicative function depend only on its values at primes,since f (p i ) = (f (p))i . See also multiplicativefunction, strongly multiplicative function.complete theoryLet L be a first order language and let T be a (closed) theory of L. Thetheory T is complete if, for all sentences σ , either σ ∈ T or (¬σ ) ∈ T .If T is simply a collection of sentences, thenT is complete if for all sentences σ , either σ isa logical consequence of T or (¬σ ) is a logicalconsequence of T .
Equivalently, T is completeif, for all sentences σ , either σ is provable fromT or (¬σ ) is provable from T .Let A be a structure for L. The theory of A(denoted T h(A)), the set of all sentences of Lwhich are true in A, is a complete theory.completely normal topological spaceAtopological space X such that every subspaceof X is normal. In particular, X itself mustbe normal, and since normality is not generallypreserved in subspaces, complete normality isstronger than normality.
Complete normality isequivalent to requiring that, for all subsets A andB of X, if Ā ∩ B = A ∩ B̄ = ∅, then there aredisjoint open sets U and V with A ⊆ U andB ⊆ V.completely regular topological space A topological space X such that points are closed andpoints and closed sets can be separated by continuous functions. That is, for each x ∈ X, thesingleton {x} is closed, and for all closed C ⊆ Xwith x ∈/ C, there is a continuous f : X →[0, 1] such that f (x) = 0 and f (c) = 1 for allc ∈ C.complete metric spaceA topological spaceX with metric d such that any Cauchy sequencein X converges. That is, if {xn : n ∈ N} ⊆ Xis such that for any > 0 there is an N withd(xn , xm ) < for any n, m ≥ N , then there isan x ∈ X with xn → x.
For example, each Euclidean n-space Rn is a complete metric space.complete set of logical connectivesA setC of logical connectives such that, given anywell-formed propositional (sentential) formulaϕ, whose logical connectives are from among© 2001 by CRC Press LLCcomplex A collection of cells with the properties: (i.) if C is a cell in the complex, thenevery face of C is in the complex; and (ii.) everytwo cells in the complex have disjoint interiors.complex analytic fiber bundle A fiber bundle f : F → X where F and X are complexmanifolds and f is an analytic map. See fiberbundle.complex analytic structure On a real differential manifold M an integrable complex structure on the tangent bundle T M; namely, the dataof an invertible linear map Jp : Tp M → Tp Mon each tangent space at p ∈ M, such thatJp2 = − Identity, which varies smoothly withp and is integrable, i.e., admits an atlas withconstant transition functions.
Without the integrability condition, the data define an “almostcomplex structure” on M.complex conjugate bundleFor a complexvector bundle f : V → M, the conjugate bundle V is defined by taking the complex conjugatef α of each local map fα : Cn × Uα ∼= V |Uα =f −1 (Uα ) that defines the bundle restricted toUα , for a suitable covering Uα of M.complex dimensioncomplex dimension (1) For a vector space X,the dimension of X, considered as a vector spaceover the field C of complex numbers, as opposedto the real dimension, which is the dimension ofX as a vector space over the real numbers R.(2) (For a complex manifold M). The complex dimension of the tangent space Tp M at eachpoint p.(3) The dimension of a complex; i.e., thehighest of the dimensions of the cells that formthe complex.complex line bundle A complex vector bundle whose fibers have dimension 1.
See complexvector bundle.complex manifold A set of points M whichcan be covered with a family of subsets {Uα }α∈Ai.e., M = α∈A Uα , each of which is isomorphic to an open ball in complex n-space:{(z1 , . . . , zn ) ∈ Cn : |z1 |2 + . . . + |zn |2 = 1},for a fixed non-negative integer n.represents the extended complex plane. See complex plane.complex torusThe n-dimensional compactcomplex analytic manifold Cn /, where n isa positive integer and a complete lattice inCn . In dimension 1, the complex tori C/(Zω1 +Zω2 ), where ω1 and ω2 are complex numbersindependent over R, are all algebraic varieties,also called elliptic curves.complex vector bundleA complex vectorbundle (of dimension n) on a differentiable manifold M is a manifold E, given by a family ofcomplex vector spaces {Ep }p∈M , with a trivialization over an open covering {Uα }α∈A of M,namely diffeomorphisms φα : Cn × Uα →{Ep }p∈Uα .
If M and E are complex analyticmanifolds and a trivialization exists with φα biholomorphic maps, the bundle is said to be complex analytic.compositecomplex of linesIn projective geometry, aline complex is a subvariety of the Grassmannian Gr(2, 4) of all lines in (complex) projective3-space CP3 , which is the set of 2-dimensionalsubspaces of a 4-dimensional complex vectorspace. Gr(2, 4) is a quadric hypersurface inCP5 , thus an example of line complex is a “linear line complex”, the intersection of Gr(2, 4)with a hyperplane, e.g., all the lines in P3 thatmeet a given plane.complex planeThe topological space, denoted C or C, consisting of the set of complexnumbers, i.e., numbers of the form a +bi, wherea and b are real numbers and i 2 = −1.
C isusually visualized as the set of pairs (a, b) andhence the terminology plane.The term extended complex plane refers toC, together with a point at infinity and neighborhoods of the form {z : z > r} for realnumbers r.complex sphere (1) A sphere {z : |z − z0 | =r}, in the complex plane.(2) A unit sphere whose points are identifiedwith points in the complex plane by a stereographic projection, with the “north pole” identified with the point ∞. Such a sphere, therefore,© 2001 by CRC Press LLCSee composite number.composite numberAn integer, other than−1, 0, and 1, that is not a prime number. Thatis, a nonzero integer is composite if it has morethan two positive divisors. For example, 6 iscomposite since the positive divisors of 6 are1, 2, 3, and 6.
Just as prime numbers are usually assumed to be positive integers, a composite number is usually assumed to be positive aswell.composition of functionsSuppose that f :X → Y and g : Y → Z are functions. Thecomposition gf : X → Z is the function consisting of all ordered pairs (x, z) such that thereexists an element y ∈ Y with (x, y) ∈ f and(y, z) ∈ g.
See function.computable Let N be the set of natural numbers. Intuitively, a function f : N → N iscomputable if there is an algorithm, or effective procedure, which, given n ∈ N as input,produces f (n) as output in finitely many steps.There are no limitations on the amount of timeor “memory” (i.e., “scratch paper”) necessaryto compute f (n), except that they be finite.
Iff : Nk → N, then f is computable is definedanalogously.computableA function ϕ on N is partial if its domainis some subset of N; i.e., ϕ may not be definedon all inputs. A partial function ϕ on N is intuitively computable if there is an algorithm, oreffective procedure, which given n ∈ N as input,produces ϕ(n) as output in finitely many stepsif n ∈ dom(ϕ), and runs forever otherwise.For example, the function f (n, m) = n + mis intuitively computable, as is the function fwhich, on input n ∈ N, produces as output thenth prime number.
The function ϕ which, oninput n ∈ N, produces the output 1 if there existsa consecutive run of exactly n 5s in the decimalexpansion of π, and is undefined otherwise, isan intuitively computable partial function.The notion of computability has a formalmathematical definition; in order to say that afunction is not computable, one must have a formal mathematical definition.
There have beenseveral formalizations of the intuitive notion ofcomputability, all of which generate the sameclass of functions. Given here is the formalization of Turing computable. A second formalization is given in the definition of a partial recursive function. See partial recursive function.Other formalizations include that of register machine computability (Shepherdson–Sturgis,1963), general recursive functions (Gödel,1934), and λ-definable functions (Church, 1930).It has been proved that, for any partial functionϕ, ϕ is Turing computable if and only if ϕ ispartial recursive, if and only if ϕ is register machine computable, etc. See also Church-TuringThesis. Thus, the term computable can (mathematically) mean computable in any such formalization.A set A of natural numbers is computable ifits characteristic function is computable; i.e., thefunction1 if n ∈ AχA (n) =0 if n ∈ Ais recursive.A partial function ϕ on N is Turing computable if there is some Turing machine thatcomputes it.
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