Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 18
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. , an ],so that 1027 is denoted [0; 2, 1, 2, 3].finite intersection propertyThe propertyof a collection C of subsets of a set X that, forevery finite subcollection {C1 , . . . , Cn } of C, theintersection C1 ∩ · · · ∩ Cn is non-empty.finite ordinal A natural number, regarded asan ordinal number. See ordinal.finite setA set that contains only finitelymany elements. Equivalently, a set whose cardinality is a natural number.first category The class of topological spacethat is the countable union of nowhere densesubsets. Such a space is also called meager. Seenowhere dense subset, second category space.first countable space A topological space Xthat has a countable basis at each point x ∈ X.That is, for each x ∈ X there is a countablecollection Bx of neighborhoods of x such that ifU ⊂ X is an open set containing x, then thereis a set B ∈ Bx with x ∈ B ⊂ U .first fundamental form The quadratic formdefined on tangent vectors to a surface M inEuclidean space R3 by taking the square of thelength of the vector.
If a portion of the surface© 2001 by CRC Press LLCthen a tangent vector can be represented as alinear combination of the vectors Xu =dy dzdx dy dz( dxdu , du , du ) and Xv = ( dv , dv , dv ). Letfirst order languageA first order languageL for first order logic consists of the followingalphabet of symbols:(i.) (, ) (parentheses)(ii.) ¬, → (logical connectives)(iii.) an infinite list of variables v1 , v2 , .
. .(iv.) a symbol = for equality (which is optional)(v.) a quantifier ∀(vi.) predicate symbols: for each positiveinteger n, a particular, possibly empty, set ofsymbols, called n-place predicate symbols(vii.) constant symbols: a particular, possibly empty, set of symbols, called constant symbols(viii.) function symbols: for each positive integer n, a particular, possibly empty, set of symbols, called n-place function symbols (constantsymbols are sometimes called 0-place functionsymbols).In item (ii.), any complete set of logical connectives could be used. In item (v.), the universalquantifier ∀ could be replaced by the existentialquantifier ∃.Such a language is called a first order language because the quantifier ranges over variables only, as opposed to a second order language, where there are two types of quantifiers.Some examples of first order languages arethe language of set theory and the language ofelementary number theory.The language of set theory is a first order language that contains equality and one two-placefirst order logicpredicate ∈.
In this language, the variables areintended to represent sets, and ∈ is intended tobe interpreted as “is an element of”.The language of elementary number theoryis a first order language that contains equality,a single constant symbol 0, one two-place predicate <, one one-place function symbol S, andthree two-place function symbols +, ·, and E.In this language, the variables are intended torepresent natural numbers, S is intended to beinterpreted as the successor function, and 0, <,+, ·, and E are intended to be interpreted as 0,the usual ordering on the natural numbers, addition, multiplication, and exponentiation, respectively.first order logicA formal logic with symbols from a first order language, rules that tellwhich expressions from the language are wellformed formulas, a semantic notion of truth (seestructure, satisfy), and a syntactical notion ofprovability (see predicate calculus, proof).First order logic is also called predicate logic.fixed point Let f : X → X be continuous.
Apoint x0 ∈ X is a fixed point for f if f (x0 ) = x0 .See Brouwer Fixed-Point Theorem.focal property of a conicA property of aconic section with regard to its focus or foci.For an ellipse, this property is lines drawn fromthe foci to a fixed point on the ellipse make equalangles with the tangent at the point.
For a hyperbola, it is lines drawn from the foci to a fixedpoint on the hyperbola make an angle that is bisected by the tangent at the point. For a parabola,it is the line from the focus to a fixed point on theparabola makes an angle with the tangent equalto that made by the tangent with the line parallelto the axis of the parabola passing through thepoint.focusA point or points in the plane, corresponding to a given conic section, whose rolevaries depending upon the type of conic. Anellipse may be thought of as the locus of pointsin the plane whose distances from the foci havea constant sum. A hyperbola may be thoughtof as the locus of points in the plane whose distances from the foci have a constant difference.A parabola may be thought of as the locus of© 2001 by CRC Press LLCpoints in the plane whose distances from the focus and a given line (see also directrix) are equal.foliationA family {Nλ : λ ∈ } of arcwiseconnected pairwise disjoint subsets covering agiven manifold M such that every point in Mhas a local coordinate system (x 1 , .
. . , x n ) sothat each Nλ is given by x n−k+1 =constant, . . . ,x n = constant for some 0 ≤ k ≤ n.foot of perpendicularSuppose l is a line inthe Euclidean (or hyperbolic) plane and P is apoint not lying on l. Then the unique point Qlying on the line l such that the line through Pand Q is perpendicular to l is called the foot ofthe perpendicular from P to l.forgetful functorA functor F from a category C to Set that assigns to each objectA ∈Obj(C) its underlying set (also denoted byA) and to each morphism f : A → B in Cthe function f : A → B. Thus, the functor“forgets” any additional properties that the objects and morphisms in C have.
For example, ifC = Grp, then a group A ∈ C is mapped to theset A and a group homomorphism f : A → Bis mapped to the function f ; all group-theoreticproperties possessed by A and f are ignored.See functor.formal proofformal theorySee proof.See theory.four-spaceThe topological vector spaceformed by taking the Cartesian product of fourcopies of the real line, denoted E 4 , R4 or R4 . Apoint in four-space is uniquely determined by anordered quadruple (a, b, c, d) of real numbers.fractionIf a and b are integers with b =0, then the fraction a ÷ b denotes the rationalnumber resulting from the quotient a ÷ b.Fréchet filterSee filter.free variable Let L be a first order language.If x is a variable and α is a well-formed formula of L, then x occurs free in α (or x is afree variable in α) is defined by induction on thecomplexity of α, as follows:functor(i.) If α is an atomic formula, then x occursfree in α if x occurs in α.(ii.) If α = (¬β), then x occurs free in (¬β)if x occurs free in β.(iii.) If α = (β → γ ), then x occurs free in(β → γ ) if x occurs free in β or in γ .(iv.) If α = ∀vi β, then x occurs free in ∀vi βif x occurs free in β and x = vi .As an example, v1 and v3 occur free, whilev2 does not occur free, in∀v2 (v1 = v2 → ∀v1 (v1 = v3 )).Frenet frameAn orthonormal frame{T (t), N (t), B(s)} of vectors at the point C(s)on a given curve in R3 , giving a moving coordinate system along the curve.
Assume C hasthree continuous derivatives and that C (t) andC (t) are linearly independent. The first vector, T , is the unit vector tangent to the curve,C (t)given by ||C (t)|| . The second unit vector, N , isthe principal normal to the curve. It lies in theplane spanned by C (t) and C (t), is perpendicular to T , and is chosen so that it makes an acuteangle with C (t). The third vector, B, is thebinormal vector. It is defined by B = T × N .Frenet’s formulasEquations that relate thefundamental geometric invariants of a curve inEuclidean space or, more generally, in a Riemannian 3-manifold.Suppose C(s) is a curve possessing three continuous derivatives, parameterized by arc length.Assume that C (s) = 0.
Then the curve has aFrenet frame (T , N, B) satisfying the followinglinear system of differential equations:C (s) = T (s)T (s) = k(s)N(s)N (s) = −k(s)T (s) + τ (s)B(s)B (s) = −τ (s)N (s)The function k(s) is the geodesic curvature, andthe function τ (s) is the torsion of the curve. SeeFrenet frame.Frobenius integrability condition The condition that must be satisfied by a k-dimensionaldistribution in an n-dimensional manifold in order for the distribution to be tangent to the leaves© 2001 by CRC Press LLCof a k-dimensional foliation.
Given a manifoldM, a distribution assigns to each point Pin M a k-dimensional subspace of the tangentspace at P . It is integrable if the manifold isthe union of k-dimensional submanifolds, suchthat the k-plane (p) is the tangent plane of thek-manifold through p. The Frobenius conditionsays that if X and Y are vector fields defined ina neighborhood of P such that X(Q) and Y (Q)lie in (Q), then the Lie bracket [X, Y ](Q) alsolies in (Q). See foliation.Frobenius TheoremA theorem that givesnecessary and sufficient conditions for a distribution in a manifold to be tangent to the leavesof a foliation. Given a manifold M, a distribution assigns to each point P in M a kdimensional subspace of the tangent space at P .It is integrable if the manifold is the union of kdimensional submanifolds, such that the k-plane(p) is the tangent plane of the k-manifoldthrough p.
The Frobenius Theorem says that is integrable if and only if, whenever X andY are vector fields defined in a neighborhood ofP such that X(Q) and Y (Q) lie in (Q), thenthe Lie bracket [X, Y ](Q) also lies in (Q).frustrumThe portion of a cone lying between its base and a plane parallel to the base.full subcategorySee subcategory.function If X and Y are sets, then a functionfrom X to Y is a relation f ⊆ X × Y (oftenwritten f : X → Y ) with the property that(x, y), (x, z) ∈ f implies y = z.
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