1、DICTIONARY OFClassicalANDTheoreticalmathematics 2001 by CRC Press LLCa Volume in theComprehensive Dictionaryof MathematicsDICTIONARY OFClassicalANDTheoreticalmathematicsEdited byCatherine CavagnaroWilliam T. Haight, IIBoca Raton London New York Washington, D.C.CRC Press 2001 by CRC Press LLCPrefaceT
2、he Dictionary of Classical and Theoretical Mathematics, one volume of the ComprehensiveDictionary of Mathematics, includes entries from the fields of geometry, logic, number theory,set theory, and topology. The authors who contributed their work to this volume are professionalmathematicians, active
3、in both teaching and research.The goal in writing this dictionary has been to define each term rigorously, not to author alarge and comprehensive survey text in mathematics. Though it has remained our purpose to makeeach definition self-contained, some definitions unavoidably depend on others, and a
4、 modicum of“definition chasing” is necessitated. We hope this is minimal.The authors have attempted to extend the scope of this dictionary to the fringes of commonlyaccepted higher mathematics. Surely, some readers will regard an excluded term as being mistak-enly overlooked, and an included term as
5、 one “not quite yet cooked” by years of use by a broadmathematical community. Such differences in taste cannot be circumnavigated, even by our well-intentioned and diligent authors. Mathematics is a living and breathing entity, changing daily, so alist of included terms may be regarded only as a sna
6、pshot in time.We thank the authors who spent countless hours composing original definitions. In particular, thehelp of Dr. Steve Benson, Dr. William Harris, and Dr. Tamara Hummel was key in organizing thecollection of terms. Our hope is that this dictionary becomes a valuable source for students, te
7、achers,researchers, and professionals.Catherine CavagnaroWilliam T. Haight, II 2001 by CRC Press LLC 2001 by CRC Press LLCCONTRIBUTORSCurtis BennettBowling Green State UniversityBowling Green, OhioSteve BensonUniversity of New HampshireDurham, New HampshireCatherine CavagnaroUniversity of the SouthS
8、ewanee, TennesseeMinevra CorderoTexas Tech UniversityLubbock, TexasDouglas E. EnsleyShippensburg UniversityShippensburg, PennsylvaniaWilliam T. Haight, IIUniversity of the SouthSewanee, TennesseeWilliam HarrisGeorgetown CollegeGeorgetown, KentuckyPhil HotchkissUniversity of St. ThomasSt. Paul, Minne
9、sotaMatthew G. HudelsonWashington State UniversityPullman, WashingtonTamara HummelAllegheny CollegeMeadville, PennsylvaniaMark J. JohnsonCentral CollegePella, IowaPaul KapitzaIllinois Wesleyan UniversityBloomington, IllinoisKrystyna KuperbergAuburn UniversityAuburn, AlabamaThomas LaFramboiseMarietta
10、 CollegeMarietta, OhioAdam LewenbergUniversity of AkronAkron, OhioElena MarchisottoCalifornia State UniversityNorthridge, CaliforniaRick MirandaColorado State UniversityFort Collins, ColoradoEmma PreviatoBoston UniversityBoston, MassachusettsV.V. RamanRochester Institute of TechnologyPittsford, New
11、YorkDavid A. SingerCase Western Reserve UniversityCleveland, OhioDavid SmeadFurman UniversityGreenville, South CarolinaSam SmithSt. Josephs UniversityPhiladelphia, PennsylvaniaVonn WalterAllegheny CollegeMeadville, Pennsylvania 2001 by CRC Press LLCJerome WolbertUniversity of MichiganAnn Arbor, Mich
12、iganOlga YiparakiUniversity of ArizonaTucson, Arizona 2001 by CRC Press LLCabsolute valueAAbeliancategory An additive category C,which satisfies the following conditions, for anymorphism f HomC(X,Y):(i.) f has a kernel (a morphism i HomC(Xprime,X) such that fi= 0) and a co-kernel (amorphismp HomC(Y,
13、Yprime) such thatpf= 0);(ii.) f may be factored as the composition ofan epic (onto morphism) followed by a monic(one-to-one morphism) and this factorization isunique up to equivalent choices for these mor-phisms;(iii.) if f is a monic, then it is a kernel; if fis an epic, then it is a co-kernel.See
14、additive category.Abelssummationidentity If a(n) is anarithmetical function (a real or complex valuedfunction defined on the natural numbers), defineA(x)=braceleftBigg0ifxabut not for any s so that xais called the half plane of absolute convergencefor the series. See also abscissa of convergence.abs
15、cissaofconvergence For the Dirichletseriessummationtextn=1f(n)ns , the real number c, if it exists,such that the series converges for all complexnumbers s=x+iy with xc but not forany s so that xc is called the half plane ofconvergence for the series. See also abscissa ofabsolute convergence.absolute
16、neighborhoodretract A topolog-ical space W such that, whenever (X,A) is apair consisting of a (Hausdorff) normal spaceX and a closed subspace A, then any continu-ous function f:AW can be extendedto a continuous function F:UW, forU some open subset of X containing A.Anyabsolute retract is an absolute
17、 neighborhood re-tract (ANR). Another example of an ANR is then-dimensional sphere, which is not an absoluteretract.absoluteretract A topological spaceW suchthat, whenever (X,A) is a pair consisting of a(Hausdorff) normal space X and a closed sub-spaceA, thenanycontinuousfunctionf:AW can be extended
18、 to a continuous functionF:XW. For example, the unit intervalis an absolute retract; this is the content of theTietze Extension Theorem. See also absoluteneighborhood retract.absolute value (1)Ifr is a real number, thequantity|r|=braceleftbiggr if r 0 ,r if r 2n.Forexample, 24 is abundant, since1 +
19、2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 48 .Thesmallestoddabundantnumber is945. Com-pare with deficient number, perfect number.accumulationpoint A point x in a topolog-ical space X such that every neighborhood of xcontains a point ofX other thanx. That is, for allopenUX withxU, there is ayU whichis differe
20、nt from x. Equivalently, x Xx.More generally, x is an accumulation pointof a subset AX if every neighborhood of xcontains a point of A other than x. That is, forall open UX with xU, there is a yUA which is different from x. Equivalently,x Ax.additivecategory A category C with the fol-lowing properti
21、es:(i.) the Cartesian product of any two ele-ments of Obj(C) is again in Obj(C);(ii.) HomC(A,B)isanadditiveAbeliangroupwith identity element 0, for any A,BObj(C);(iii.) the distributive laws f(g1 +g2)=fg1+fg1 and(f1+f2)g=f1g+f2g hold formorphisms when the compositions are defined.See category.additi
22、vefunction An arithmetic function fhaving the property thatf(mn)=f(m)+f(n)whenever m and n are relatively prime. (Seearithmetic function). For example, , the num-ber of distinct prime divisors function, is ad-ditive. The values of an additive function de-pend only on its values at powers of primes:
23、ifn=pi11pikkand f is additive, then f(n)=f(pi11 )+.+f(pikk). See also completely ad-ditive function.additivefunctor An additive functor F:CD, between two additive categories, suchthat F(f+g)=F(f)+F(g)for any f,gHomC(A,B). See additive category, functor.Ademrelations The relations in the Steenrodalge
24、bra which describe a product of pth poweror square operations as a linear combination ofproducts of these operations. For the square op-erations (p= 2), when 0 0, f is in Baireclass if there is a sequence of functions fnconverging pointwise to f , with fnBnandn. A closed and unbounded subset of is o
25、ften called a club subset of .closedconvexcurve 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 con-vex figure in the plane, then its boundary C i
26、s aclosed convex curve.closedconvexsurface The boundary S ofa closed convex body in three-dimensional Eu-clidean space. S is topologically equivalent toa sphere and the line segment joining any twopoints in S lies in the bounded region boundedby S.closedformula A well-formed formula ofa first-order
27、language such that has no freevariables.closedhalfline A set in R of the forma,)or (,a for some aR.closedhalfplane A subset of R2 consistingofastraightlineLandexactlyoneofthetwohalfplanes which L determines. Thus, any closedhalf plane is either of the form(x,y):ax+byc or(x,y):ax+byc. The setsxc and
28、xc are vertical closed half planes;yc and yc are horizontal half planes.closedmap A function f:XY betweentopological spaces X and Y such that, for anyclosed set CX, the image set f(C)is closedin Y.closedset (1) A subset A of a topologicalspace, such that the complement of A is open.See open set. For
29、 example, the setsa,b anda are closed in the usual topology of the realline.(2)Aclosed set of ordinals is one that isclosed in the order topology. That is, Cis closed if, for any limit ordinal 0 there is an N withd(xn,xm)r for realnumbers plexsphere (1) A spherez:|zz0|=r, in the complex plane.(2) A
30、unit sphere whose points are identifiedwith points in the complex plane by a stereo-graphic projection, with the “north pole” iden-tifiedwiththepoint. Suchasphere, therefore,representstheextendedcomplexplane. Seecom-plex plextorus The n-dimensional compactcomplex analytic manifold Cn/Lambda1, where
31、n isa positive integer and Lambda1 a complete lattice inCn. In dimension 1, the complex tori C/(Z1+Z2), where 1 and 2 are complex numbersindependent over R, are all algebraic varieties,also called elliptic plexvectorbundle A complex vectorbundle (of dimensionn) on a differentiable man-ifold M is a m
32、anifold E, given by a family ofcomplex vector spacesEppM, with a trivial-ization over an open coveringUA of M,namely diffeomorphisms :CnUEppU .IfM and E are complex analyticmanifolds and a trivialization exists with bi-holomorphic maps, the bundle is said to be com-plex posite See composite positenu
33、mber An integer, other than1, 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 usu-ally assumed to be positive integers, a co
34、mpos-ite number is usually assumed to be positive positionoffunctions Suppose that f:XY and g:YZ are functions. Thecomposition gf:XZ is the function con-sisting of all ordered pairs (x, z) such that thereexists an element y Y with (x, y) f and(y,z)g. See putable Let N be the set of natural num-bers.
35、 Intuitively, a function f : N N iscomputable if there is an algorithm, or effec-tive 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
36、finite. Iff : Nk N, then f is computable is definedanalogously. 2001 by CRC Press LLCcomputableA function on N is partial if its domainis some subset of N; i.e., may not be definedon all inputs. A partial function on N is in-tuitively computable if there is an algorithm, oreffective procedure, which
37、 givennN as input,produces (n) as output in finitely many stepsif n dom(), and runs forever otherwise.For example, the function f(n,m)=n+mis intuitively computable, as is the function fwhich, on input nN, produces as output thenth prime number. The function which, oninputnN, produces the output 1 if
38、 there existsa consecutive run of exactly n 5s in the decimalexpansion of , 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 for-mal mathematical
39、 definition. There have beenseveral formalizations of the intuitive notion ofcomputability, all of which generate the sameclass of functions. Given here is the formaliza-tion of Turing computable. A second formal-ization is given in the definition of a partial re-cursive function. See partial recurs
40、ive function.Other formalizations include that of register ma-chine computability (ShepherdsonSturgis,1963), general recursive functions (Gdel,1934), and-definablefunctions(Church, 1930).It has been proved that, for any partial function, is Turing computable if and only if ispartial recursive, if an
41、d only if is register ma-chine computable, etc. See also Church-TuringThesis. Thus, the term computable can (math-ematically) mean computable in any such for-malization.A set A of natural numbers is computable ifits characteristic function is computable; i.e., thefunctionA(n)=braceleftbigg1ifnA0ifnn
42、egationslashAis recursive.A partial function on N is Turing com-putable if there is some Turing machine thatcomputes it. The notion of Turing machine wasformalized by Alan Turing in his 1936 Proceed-ings of the London Mathematical Society paper.A Turing machine consists of a bi-infinitetape, which i
43、s divided into cells, a reading headwhich can scan one cell of the tape at a time, afinite tape alphabet S =s0,s1,.,snof sym-bols which can be written on the tape, and a finitesetQ=q0,q1,.,qmof possible states. Thesets S and Q have the properties that SQ=,1,BS (where B stands for “blank”), andq0 Q i
44、s the designated initial state. A Turingmachine which is in state qjreading symbol sion its tape may perform one of three possibleactions: it may write over the symbol it is scan-ning, move the read head right (R), and go intoanother (possibly the same) state; it may writeover the symbol it is scann
45、ing, move the readhead left (L), and go into another (possibly thesame) state; or it may halt.The action of the Turing machine is governedby a Turing program, given by a transition func-tion, whose domain is some subset ofQS andwhose range is a subset of the set QSR,L.If (q,a)=(q,a,m), then the acti
46、on of the ma-chine is as follows. If the machine is in state q,reading symbol a on the tape, then it replaces aby a on the tape, moves the read head one cellto the right if m = R, moves the read head onecell to the left if m = L, and goes into state q.The Turing program halts if the machine is in as
47、tate q, reading a symbol a, and the transitionfunction is undefined on (q, a).A Turing machine computes a partial func-tion as follows: given input x1,.,xn, the tapeis initially set to.B1x1+1B1x2+1B.B1xn+1B.,where 1kindicates a string of k 1s, one symbol1 per cell, .B1x1+1indicates that all cells to
48、the left of the initial 1 on the tape are blank, and1xn+1B. indicates that all cells to the rightof the last 1 on the tape are blank. The read-ing head is positioned on the leftmost 1 on thetape, and the machine is set to the initial state q0.The output of the function (if any) is the numberof 1s on
49、 the tape when the machine halts, afterexecuting the program, if it ever halts.The following is a Turing machine programwhich computes the function f(x1,x2)=x1+x2, thus showing that f is Turing computable.The idea is that, given input.B1x1+1B1x2+1B.,the machine replaces the middle blank B by a1 (instructions 1 2), moves to the leftmost 1
