Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 25
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Withthe usual topology, the Möbius band is a nonorientable manifold.where µ is the Möbius function. In other words,F is the Dirichlet convolution of f and u (theunit function) if and only if f is the Dirichletconvolution of F and µ. See arithmetic function.Möbius transformationfunction.See linear fractionalMöbius transformation groupThe projective linear group PL(2, C) of all Möbius transformations. See linear fractional function.Named after the German mathematician AugustFerdinand Möbius.modelLet L be a first order language, σ bea sentence of L, and A be a structure for L.If A satisfies σ with some (and hence every)s : V → A, then A is a model of σ , and σ istrue in A. (Here, V is the set of variables ofL and A is the universe of A.) If is a set ofsentences of L, then A is a model of if A is amodel of every sentence in .The term model is sometimes synonymouswith the term structure.
See also structure.model completeA theory T of a first orderlanguage L such that, for all structures A and Bwhich are models of T , if A is a substructure ofB, then A is an elementary substructure of B.As an example, let L be the first order language with equality, whose only predicate symbol is <, and let R be the structure for L whoseMöbius band.© 2001 by CRC Press LLCmodus ponensuniverse is the set R of real numbers and where< is interpreted in the usual way.
Then thetheory of R, the set of all sentences true in R,is model complete.multipleThe integer c is a multiple of theinteger a if there exists an integer b so that ab =c. That is, c is a multiple of a if a is a divisor ofc. See also divisor.modus ponens The logical rule of inference“from A and (A → B), infer B.” Here, A andB can be any well-formed propositional (sentential) or first order formulas. Literally, modusponens means “the positing method”, where “toposit” means “to present as a fact” or “to postulate”.multiplicative function An arithmetic function f having the property that f (mn) = f (m)·f (n) whenever m and n are relatively prime.(See arithmetic function.) Many important functions, including the Euler phi function and theMöbius function µ, are multiplicative. The values of a multiplicative function depend only onits values at powers of primes: if n = p1i1 · · · pkikand f is multiplicative, thenmorphismA category has objects and morphisms.
Though a morphism is a primitive notion in category theory, it can be understood asan abstraction of the notion of function. Thefollowing categories are standard examples (theobjects are listed first, the morphisms second):topological spaces and continuous functions;Abelian groups and group homomorphisms;rings and ring homomorphisms; partiallyordered sets and monotone functions; complexBanach spaces and bounded linear transformations; sets and injective functions; sets and surjective functions.motion An element of the group of motions.See group of motions.© 2001 by CRC Press LLCf (n) = f (p1i1 ) · · · f (pkik ) .See also completely multiplicative function,strongly multiplicative function.mutually relatively prime set of integersA set of integers such that there is no integer dgreater than 1 which is a divisor of all membersof the set.
For instance, the set {2, 3, 4} is mutually relatively prime since the only commonpositive divisor of 2, 3, and 4 is 1. Note thatthe set is not pairwise relatively prime since thegreatest common divisor of 2 and 4 is 2. Seealso pairwise relatively prime numbers.normal spacenon-Euclidean Not satisfying the postulatesfrom Euclid’s Elements.Nnatural equivalencetion.See natural transforma-natural isomorphismmation.See natural transfor-natural numberA positive integer. The setof natural numbers is denoted N or N.natural transformationLet C, D be categories, and let F, G : C → D be functors.A natural transformation is a correspondence φthat sends every object A of C to a morphism φAof D such that, for every morphism f : A → Bof C, the diagramF (A)φA ↓G(A)F (f )−→−→G(f )F (B)↓ φBG(B)commutes.
The correspondence φ is a natural equivalence (or natural isomorphism) if, inaddition, φA is an isomorphism in D, for eachobject A of C.negative numberthan 0.A real number that is lessnegative of a number If n is a number, thennegative n (also referred to as the opposite of n),is the number −n = (−1) × n (i.e., the productof −1 and n). Alternatively, −n is the additiveinverse of the integer n (the unique integer k sothat n + k = 0).neighborhood A neighborhood of a point xin a topological space X is a set U such that Ucontains an open subset V of X with x ∈ V .neutral geometryThe portion of geometrythat can be derived without the use of Euclid’sparallel postulate. This is also referred to as“absolute geometry”, a term coined by JanosBolyai.© 2001 by CRC Press LLCnon-Euclidean geometry A class of geometrical systems not satisfying the postulates fromEuclid’s Elements.
Includes elliptic geometry,hyperbolic geometry, projective geometry, andspherical geometry.non-Euclidean space A space satisfying axioms that contradict the postulates from Euclid’sElements.non-Euclidean surfaceA surface that is asubset of a non-Euclidean space. See nonEuclidean space.nonprincipal ultrafilterAn ultrafilter Uover a Boolean algebra B with no b ∈ B suchthat U = {x ∈ B : b ≤ x}.normal bundleWhen a manifold is contained in Rn , the directions perpendicular to thetangent directions are normal. Forming a vectorspace at each point of the manifold, these directions yield a normal bundle over the manifold.Example: Let M denote a Möbius band inR3 . The normal bundle of M in R3 can be visualized by looking at the part of the bundle overthe middle circle of M: this part of the bundleis again a Möbius band.
One can see this bytaking a normal direction (perpendicular to thesurface of the Möbius band) at any point andwalking around the band along the middle circle. The normal vector will be pointing in theopposite direction. This is not true for the cylinder (S 1 × R1 ). See Möbius band.normal curvature At a point P on a surface,the curvature (with proper choice of sign) of thecurve formed by the intersection of the surfacewith the plane through the normal vector at Pand a unit vector in the tangent plane.normalized vectorA vector made to be oflength one by multiplying the vector by the reciprocal of its length.normal space A topological space X satisfying the following: given any two disjoint closedsubsets C and D of X, there exist disjoint opennormal to planesubsets U and V of X such that U ⊃ C andV ⊃ D.surface form the total space of a 1-sphere bundle.normal to planeA vector or line, passingthrough a given point in the plane, perpendicularto all lines in the plane passing through the point.If the plane in space is given by the equationnull objectnull setSee zero object.See empty set.number fielda(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 ,then the vector (a, b, c) is normal to the planeat the point (x0 , y0 , z0 ).normal topological spaceA topologicalspace X in which one-point sets are closed and,given any two closed, disjoint subsets A1 , A2 ofX, there exist disjoint open subsets U1 and U2of X such that A1 ⊂ U1 and A2 ⊂ U2 .
Examples of normal spaces include metric spaces andcompact Hausdorff spaces.normal to surface At a point on the surface,the vector orthogonal to the tangent vector spaceat the point.normal vector A vector at a point of a manifold (contained in Rn ) that is perpendicular to alltangent vectors at that point. For example, thenorth pole is normal to the surface of the earth.nowhere dense subset A subset A of a topological space X such that the closure of A contains no nonempty open subsets of X. Any discrete set is nowhere dense in a Hausdorff space.A more interesting example is the Cantor Set,which is not discrete and yet is a nowhere densesubset of the unit interval [0, 1].
See Cantor set.n-sphere bundle A fiber bundle whose fiberis the n-dimensional sphere, and whose structure group is a subgroup of the orthogonal groupO(n + 1). It consists of a base space B, a totalspace E, and a projection map π : E −→ B.There is a covering of B by open sets Ui andhomeomorphisms φi : Ui × S n −→ π −1 (Ui )such that π ◦ φi (x, q) = x. This identifiesπ −1 (x) with the n-sphere. When two sets Uiand Uj overlap, the two identifications are related by orthogonal transformations gij (x) ofS n . For example, if M is a surface in R3 , thenthe space of vectors of length one tangent to the© 2001 by CRC Press LLCSee algebraic number field.number of distinct prime divisors functionThe arithmetic function, denoted ω, which, forany positive integer n, returns the number of distinct prime divisors of n. (See arithmetic function.) For example, ω(12) = ω(24) = 2 ({2, 3}is the set of distinct prime divisors in both cases).It is additive.number of divisors function The arithmeticfunction, usually denoted τ or d, which, for anypositive integer n, returns the number of positive divisors of n, i.e., τ (n) = #{a : 1 ≤ a ≤n and a|n }.
(See arithmetic function.) For example, τ (12) = 6 ({1, 2, 3, 4, 6, 12} is the setof divisors). It is multiplicative; its value at aprime power is given byτ (pi ) = i + 1 .See also sum of kth powers of divisors function.number system A logically organized method for expressing numbers which may be visual(using writing or hand signs), oral (spoken), ortactile (e.g., the Braille system). A variety ofnumber systems have been used throughout history. The number system used by most culturestoday is a positional base-10 system. See baseof number system.number theoretic functionfunction.See arithmeticnumber theoryThat branch of mathematics involving the study of the integers and theirgeneralizations.numeral A physical representation of a number, often in written form.numeratorThe number a in the fraction ab .numerical Of or relating to numbers or computations involving numbers.one-to-one correspondenceOobjectA category has objects and morphisms.
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