Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108)
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DICTIONARY OFClassicalANDTheoreticalmathematics© 2001 by CRC Press LLCa Volume in theComprehensive Dictionaryof MathematicsDICTIONARY OFClassicalANDTheoreticalmathematicsEdited byCatherine CavagnaroWilliam T. Haight, IICRC PressBoca Raton London New York Washington, D.C.© 2001 by CRC Press LLCPrefaceThe 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 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 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 mistakenly overlooked, and an included term as one “not quite yet cooked” by years of use by a broadmathematical community.
Such differences in taste cannot be circumnavigated, even by our wellintentioned and diligent authors. Mathematics is a living and breathing entity, changing daily, so alist of included terms may be regarded only as a snapshot 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, teachers,researchers, and professionals.Catherine CavagnaroWilliam T.
Haight, II© 2001 by CRC Press LLC© 2001 by CRC Press LLCCONTRIBUTORSCurtis BennettKrystyna KuperbergBowling Green State UniversityBowling Green, OhioAuburn UniversitySteve BensonThomas LaFramboiseUniversity of New HampshireDurham, New HampshireMarietta CollegeCatherine CavagnaroUniversity of the SouthSewanee, TennesseeAuburn, AlabamaMarietta, OhioAdam LewenbergUniversity of AkronAkron, OhioMinevra CorderoTexas Tech UniversityLubbock, TexasElena MarchisottoCalifornia State UniversityNorthridge, CaliforniaDouglas E. EnsleyShippensburg UniversityShippensburg, PennsylvaniaWilliam T.
Haight, IIRick MirandaColorado State UniversityFort Collins, ColoradoUniversity of the SouthSewanee, TennesseeEmma PreviatoWilliam HarrisBoston, MassachusettsGeorgetown CollegeGeorgetown, KentuckyV.V. RamanBoston UniversityRochester Institute of TechnologyPhil HotchkissPittsford, New YorkUniversity of St. ThomasSt. Paul, MinnesotaDavid A.
SingerCase Western Reserve UniversityMatthew G. HudelsonCleveland, OhioWashington State UniversityPullman, WashingtonDavid SmeadTamara HummelAllegheny CollegeMeadville, PennsylvaniaFurman UniversityGreenville, South CarolinaSam SmithMark J. JohnsonSt. Joseph’s UniversityCentral CollegePella, IowaPhiladelphia, PennsylvaniaPaul KapitzaAllegheny CollegeIllinois Wesleyan UniversityBloomington, IllinoisMeadville, Pennsylvania© 2001 by CRC Press LLCVonn WalterJerome WolbertOlga YiparakiUniversity of MichiganAnn Arbor, MichiganUniversity of ArizonaTucson, Arizona© 2001 by CRC Press LLCabsolute valueabscissa of convergenceFor the Dirichlet∞f (n)seriesns , the real number σc , if it exists,An=1Abelian categoryAn additive category C,which satisfies the following conditions, for anymorphism f ∈ HomC (X, Y ):(i.) f has a kernel (a morphism i ∈ HomC(X , X) such that f i = 0) and a co-kernel (amorphism p ∈ HomC (Y, Y ) such that pf = 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 morphisms;(iii.) if f is a monic, then it is a kernel; if fis an epic, then it is a co-kernel.See additive category.Abel’s summation identityIf a(n) is anarithmetical function (a real or complex valuedfunction defined on the natural numbers), defineA(x) =0a(n)n≤xif x < 1 ,if x ≥ 1 .If the function f is continuously differentiableon the interval [w, x], thena(n)f (n)=A(x)f (x)w<n≤x− A(w)f (w) x−A(t)f (t) dt .wabscissa of absolute convergenceFor the∞f (n)Dirichlet seriesns , the real number σa , if itn=1exists, such that the series converges absolutelyfor all complex numbers s = x +iy with x > σabut not for any s so that x < σa .
If the seriesconverges absolutely for all s, then σa = −∞and if the series fails to converge absolutely forany s, then σa = ∞. The set {x + iy : x > σa }is called the half plane of absolute convergencefor the series. See also abscissa of convergence.© 2001 by CRC Press LLCsuch that the series converges for all complexnumbers s = x + iy with x > σc but not forany s so that x < σc . If the series convergesabsolutely for all s, then σc = −∞ and if theseries fails to converge absolutely for any s, thenσc = ∞. The abscissa of convergence of theseries is always less than or equal to the abscissaof absolute convergence (σc ≤ σa ).
The set{x + iy : x > σc } is called the half plane ofconvergence for the series. See also abscissa ofabsolute convergence.absolute neighborhood retractA topological space W such that, whenever (X, A) is apair consisting of a (Hausdorff) normal spaceX and a closed subspace A, then any continuous function f : A −→ W can be extendedto a continuous function F : U −→ W , forU some open subset of X containing A.
Anyabsolute retract is an absolute neighborhood retract (ANR). Another example of an ANR is then-dimensional sphere, which is not an absoluteretract.absolute retract A topological space W suchthat, whenever (X, A) is a pair consisting of a(Hausdorff) normal space X and a closed subspace A, then any continuous function f : A −→W can be extended to a continuous functionF : X −→ W . For example, the unit intervalis an absolute retract; this is the content of theTietze Extension Theorem. See also absoluteneighborhood retract.absolute valuequantity(1) If r is a real number, therif r ≥ 0 ,−rif r < 0 .√Equivalently, |r| = r 2 . For example, | − 7|= |7| = 7 and | − 1.237| = 1.237. Also calledmagnitude of r.(2) If z = x + iy is a complex number, then|z|, also referredto as the norm or modulus of2 + y 2 . For example, |1 − 2i| =z,equalsx√√12 + 22 = 5.(3) In Rn (Euclidean n space), the absolutevalue of an element is its (Euclidean) distance|r| =abundant numberto the origin.
That is,|(a1 , a2 , . . . , an )| =a12 + a22 + · · · + an2 .In particular, if a is a real or complex number,then |a| is the distance from a to 0.abundant number A positive integer n having the property that the sum of its positive divisors is greater than 2n, i.e., σ (n) > 2n. Forexample, 24 is abundant, since1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 > 48 .additive functorAn additive functor F :C → D, between two additive categories, suchthat F (f + g) = F (f ) + F (g) for any f, g ∈HomC (A, B). See additive category, functor.Adem relations The relations in the Steenrodalgebra which describe a product of pth poweror square operations as a linear combination ofproducts of these operations.
For the square operations (p = 2), when 0 < i < 2j ,Sq i Sq j =0≤k≤[i/2]j −k−1i − 2kSq i+j −k Sq k ,The smallest odd abundant number is 945. Compare with deficient number, perfect number.accumulation point A point x in a topological space X such that every neighborhood of xcontains a point of X other than x. That is, for allopen U ⊆ X with x ∈ U , there is a y ∈ U whichis different from x. Equivalently, x ∈ X \ {x}.More generally, x is an accumulation pointof a subset A ⊆ X if every neighborhood of xcontains a point of A other than x.
That is, forall open U ⊆ X with x ∈ U , there is a y ∈U ∩ A which is different from x. Equivalently,x ∈ A \ {x}.additive category A category C with the following properties:(i.) the Cartesian product of any two elements of Obj(C) is again in Obj(C);(ii.) HomC (A, B) is an additive Abelian groupwith identity element 0, for any A, B ∈Obj(C);(iii.) the distributive laws f (g1 + g2 ) =f g1 + f g1 and (f1 + f2 )g = f1 g + f2 g hold formorphisms when the compositions are defined.See category.additive functionAn arithmetic function fhaving the property that f (mn) = f (m) + f (n)whenever m and n are relatively prime.
(Seearithmetic function). For example, ω, the number of distinct prime divisors function, is additive. The values of an additive function depend only on its values at powers of primes: ifn = p1i1 · · · pkik and f is additive, then f (n) =f (p1i1 ) + . . . + f (pkik ). See also completely additive function.© 2001 by CRC Press LLCwhere [i/2] is the greatest integer less than orequal to i/2 and the binomial coefficients in thesum are taken mod 2, since the square operationsare a Z/2-algebra.As a consequence of the values of the binomial coefficients, Sq 2n−1 Sq n = 0 for all valuesof n.The relations for Steenrod algebra of pthpower operations are similar.adjoint functorIf X is a fixed object in acategory X , the covariant functor Hom∗ : X →Sets maps A ∈Obj (X ) to HomX (X, A); f ∈HomX (A, A ) is mapped to f∗ : HomX (X, A)→ HomX (X, A ) by g → f g.
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