Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 17
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Thesentence σ is an existential sentence if it has theform ∃v1 . . . ∃vn α, where α is quantifier-free,for some n ≥ 0.expansion of a languageLet L1 and L2be first order languages. The language L2 isan expansion of L1 if L1 ⊆ L2 ; i.e., L2 hasall the symbols of L1 , together with additionalpredicate symbols, constant symbols, or function symbols.Let L be a first order language, A a structurefor L, and let X ⊆ A, where A is the universe ofA. The expansion LX is the expansion obtainedfrom L by adding a new and distinct constantsymbol ca for each a ∈ X.expansion of a structureLet L1 and L2 befirst order languages such that L2 is an expansion of L1 , and let A be a structure for L1 .
Anexpansion of A to L2 gives interpretations in A,extreme pointthe universe of A, of the additional predicate,constant, and function symbols in L2 , while theinterpretations in A of the symbols in L1 remainthe same.Let L be a first order language, A a structurefor L, and let X ⊆ A, where A is the universeof A. The expansion AX is the expansion of Ato LX by interpreting each new constant symbolca of LX , for each a ∈ X, by a; i.e., caAX = a.This expansion is often denoted by (A, a)a∈X .See also expansion of a language.© 2001 by CRC Press LLCextensionSee substructure.extension of a mapping Suppose that A ⊂ Xand that f : A → Y is a map. Then F : X → Yis an extension of f if the restriction of F to A isequal to f ; that is, F (a) = f (a) for all a ∈ A.extreme pointA point of a convex set inEuclidean space which is not the midpoint ofa straight line joining two distinct points of theset.fiber productof order n is called the Farey dissection (of ordern) of the circle.
See also Farey dissection.FFσ setGδ set.A countable union of closed sets. SeefaceA boundary polygon of a Euclideanpolyhedron. More generally, an (n − 1)dimensional subspace F of a convex cell C inn-dimensional affine space An such that F isthe intersection of the boundary of C with an(n − 1)-dimensional subspace of An .face angle An angle between two edges of apolyhedron that share a vertex.factor of integer The integer b is a factor ofthe integer a if there exists an integer c so thata = bc. For example, 8 is a factor of 24 since24 = 8 × 3, but 8 is not a factor of 36 since thereis no integer c where 8c = 36.
See also divisor.family A family of sets is a function from anindex set to the set of subsets of a set X whosevalue at α ∈ is denoted by Xα . Although thefunction can be denoted in the usual way as aset of ordered pairs {(α, Xα ) : α ∈ }, it iscompletely specified by {Xα : α ∈ }.Farey arc For a given positive integer n, construct the Farey series Fn of order n, that is, theascending sequence of rational numbers ab between 0 and 1 with the property that 0 ≤ a ≤b ≤ n and GCD(a, b) = 1. Next, determinethe mediants of all consecutive elements of Fn .(See mediant.) A Farey arc is an interval of theform (p, q), where p and q are consecutive mediants. These intervals are usually visualized asarcs lying on the circle of circumference 1 onwhich the number x is represented by the pointPx lying x units counterclockwise from 0 (the“bottom” of the circle). Under the identification0 = 1 on this circle, we also include the arc1n( n+1, n+1) which runs from the “last” mediantto the first.
Each of these arcs contains exactlyone member of Fn The collection of Farey arcs© 2001 by CRC Press LLCFarey dissection The Farey dissection of order n is the collection of Farey arcs of order n.See also Farey arc.Farey sequenceIf n is a positive integer,the Farey sequence of order n (denoted by Fn )is the sequence of rational numbers, listed inincreasing order, whose denominator is no largerthan n. For example,F1 = (. . . ,−2 −1 0 1 2,, , , ,...)11 1 1 1−3 −1 0 1 1 3 2,, ,, , , , ,...)21 1 2 1 2 1−2 −1 −1 0 1 1 2 1 4F3 = (.
. . ,,,, , , , , , ,...)323 1 3 2 3 1 3F2 = (. . . ,The Farey sequences can be used to make rational approximations.Fermat number A number of the form Fn =n22 + 1, where n is a nonnegative integer. Forexample, F3 = 257. The Fermat numbersare named after Pierre de Fermat, a 17th century lawyer and amateur mathematician, whoconjectured that all Fn are prime. The integers Fn , n ≤ 4, are in fact prime, but no otherprime Fermat numbers are known. All Fn with5 ≤ n ≤ 27 (except n = 24) are known to becomposite.fiber Any f −1 (y) for y ∈ Y , where f : X →Y (that is, (X, Y, f )) is a fiber space. Sometimesspelled fibre.
See fiber space.fiber bundle A fiber bundle (over a space X)is a fibration f : F → X, namely a continuoussurjective map such that X can be covered byopen sets Uα over which the fibration is equivalent to the trivial one, the second projection ofthe Cartesian product Y × Uα → Uα , for Y asuitable space.fiber productIn complete generality, thecategory-theoretical product X ×S Y of X withY , where X, Y , and S are objects in a category C,and X and Y (with given morphisms) are thoughtof as objects in the category C/S. Here the category C/S has as its objects all morphisms toS.
A morphism in C/S from f : X → S tofiber spaceg : Y → S is any morphism h in C from X toY such that f = g ◦ h. Sometimes spelled fibreproduct.fiber spaceA triple consisting of two topological spaces X and Y , and a continuous mapf : X → Y , such that, for any cube I n ={(x1 , . . . xn ) : 0 ≤ xi ≤ 1}, any mapping φ :I n → X, and any homotopy ht : I n → Y withf ◦ φ = h0 , there is a homotopy φt : I n → Xwith φ0 = φ and f ◦ φt = ht for all t. Sometimes spelled fibre space.fiber sumGiven two objects X and Y in acategory C, the fiber product in the dual categoryC ◦ of these two objects over another object S.Sometimes spelled fibre sum.
See fiber product.Fibonacci sequence The recursive sequence{fn } = {1, 1, 2, 3, 5, 8, . . . } defined by the initial conditions f0 = f1 = 1 and the recursiveequation fn+1 = fn +fn−1 . The mathematicianLeonardo de Fibonacci originally developed thissequence to model the so-called “Rabbit Problem”: Suppose that rabbits mature in one month,that the gestation period for rabbits is also onemonth, that a female rabbit always gives birthto a breeding pair of rabbits, and that rabbitsnever die. If a male and female rabbit are left onan uninhabited island at birth, how many pairsof rabbits will there be after a given number ofmonths?Initially, there is one pair of rabbits (we letf0 denote the number (1) of pairs of immaturerabbits at the end of the “0th” month).
At theend of the first month, the rabbits have maturedand are able to reproduce (after a gestation period of one month, the female will give birth) sothat f1 = 1, as well. At the end of the secondmonth, the female rabbit has given birth to a pairof rabbits, so f2 = 2. At the end of the thirdmonth, the new pair of rabbits has just maturedand the first pair gives birth to another pair ofrabbits.
Thus, f3 = 3. At the end of the fourthmonth, both the original pair and the first pairof offspring give birth to pairs of rabbits, so thatf4 = 5. It is left to the reader to show that thenth term of the Fibonacci sequence provides thenumber of pairs of rabbits which are alive at theend of the nth month.© 2001 by CRC Press LLCfilter(1) A filter on a set S (or in P(S)) is acollection F of subsets of S such that (i.) S ∈ F,(ii.) A, B ∈ F implies A ∩ B ∈ F, for all A,B and (iii.) A ∈ F and A ⊆ B implies B ∈ F,for all A, B.
A proper filter does not contain theempty set. For example, if S is any infinite set,the Fréchet filter on S is the set of all cofinitesubsets of S. The Fréchet filter is a proper filter. Another important example of a proper filteris the club filter on some uncountable cardinalnumber κ. The club filter on κ is the set of allclub subsets of κ.(2) If (B, ∨, ∧, ∼, 1, 0) is a Boolean algebra, F ⊆ B is a filter in B if (i) 1 ∈ F, (ii)a, b ∈ F implies a ∧ b ∈ F, for all a, b, and(iii) a ∈ F and a ∧ b = a implies b ∈ F, forall a, b. For example, (P(N), ∪, ∩, ∼, N, Ø)is a Boolean algebra, and the set of all cofinitesubsets of N is a filter in this Boolean algebra.(3) If (P , ≤) is a partially ordered set, F ⊆ Pis a filter in P if (i.) for all a, b ∈ F, there existsc ∈ F such that c ≤ a and c ≤ b, and (ii.) forall a ∈ F and all x ∈ P , x ≤ a implies x ∈ F.filter in a partial orderA nonempty subsetG of a partial order (P , ≤) such that any twoelements of G are compatible in G and G isclosed upwards.
That is, for any p and q inG, there is an r ∈ G with r ≤ p and r ≤ q(compatibility), and for any p ∈ G, if q ∈ Pwith p ≤ q, then q ∈ G (closed up). Filterscapture large elements in a partial order in thesame way that ideals capture small elements.final object An object F in a category C withthe property that, for any object X in C, thereexists a unique morphism g ∈ HomC (X, F ).finite cardinalA natural number, regardedas a cardinal number.finite characterA set A (of sets) is of finitecharacter if A = ∅ and for all sets X, X is in Aif and only if every finite subset of X is in A.For example, let A be the collection of allsets that contain pairwise disjoint subsets of N.Then A has finite character.first order languagefinite continued fractionthe forma0 +A real number ofis parameterized byX(u, v) = (x(u, v), y(u, v), z(u, v)) ,1a1 +1a2 +1a3 +1...1an−1 + a1nwhere each ai is a real number.
If each ai isan integer, the fraction is said to be a simplefinite continued fraction. It can be shown thata real number is rational if and only if it can beexpressed as a finite (simple) continued fraction.For example,101=272+1+1E(u, v) = Xu · Xu , F (u, v) = Xu · XvandG(u, v) = Xv · Xv .Then any√tangent vector aXu + bXv has lengthgiven by Ea 2 + 2F ab + Gb2 . The first fundamental form is classically given in the form:ds 2 = Edu2 + 2F dudv + gdv 2 .12+ 31As an alternative to this cumbersome notation,the finite continued fraction decomposition of anumber is often abbreviated [a0 ; a1 , a2 , . .
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