Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 16
Текст из файла (страница 16)
. . an ∈ A,(a1 , . . . , an ) ∈ P A⇔ (h(a1 ), . . . , h(an )) ∈ P B ,(ii.) for each constant symbol c,h(cA ) = cB ,and© 2001 by CRC Press LLC(iii.) for each n-ary function symbol f andevery a1 . . . , an ∈ A,h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )).If there is an embedding of A into B, then Ais isomorphic to a substructure of B.(2) An injective map f of a space X into aspace Y such that if Z = f (X), then the mapf : X → Z, obtained by restricting the codomain of f , is a homeomorphism.empty setements.A set denoted ∅ which has no el-enumeration An enumeration of a set A is asurjection f : N → A; i.e., a function f whichhas domain N and range A. Such a functionis called an enumeration because it “lists” theelements of A.
An enumeration need not be aninjection (i.e., one-to-one), nor list the elementsof A in any particular order.equal geometric figuresTwo figures suchthat one can be moved coincident with the othervia a transformation.equal sets Two sets A and B which have thesame elements; that is, if for all x, x ∈ A ifand only if x ∈ B. In formal ZF (ZermeloFraenkel) set theory, this is called the Axiom ofExtensionality.equiangular polygonA polygon whose interior angles all have the same measure.equiangular spiral A spiral given by the polar equation r = ekθ , where k is a constant. Alsoknown as the logarithmic or exponential spiral,or the spiral of Bernoulli.equidistantA set of objects such that anypair of objects in the set is the same distanceapart as any other pair of objects in the set.equilateral A figure with sides, all of whichhave the same length.equilateral trianglethree sides congruent.A triangle having allequinumerous setsTwo sets A and B suchthat there is a bijection, or one-to-one correspon-equipollent setsdence, between them; i.e., there is a functionf : A → B such that f is both injective andsurjective.
For example, if N denotes the set ofnatural numbers, Q denotes the set of rationalnumbers, and R denotes the set of real numbers,then N and Q are equinumerous, while Q and Rare not equinumerous.equipollent setsTwo sets A and B whichhave a bijection f : A → B between them.equivalent bases for a topological space LetX be a topological space. The bases B and B areequivalent if they generate the same topology onX. That is, for all B ∈ B, if x ∈ B then thereexists B ∈ B so that x ∈ B ⊂ B.
Converselyfor all B ∈ B , if x ∈ B then there exists B ∈ Bso that x ∈ B ⊂ B .equivalent setsTwo sets A and B such thatthere exists a bijection f : A → B.Erlangen ProgramThe name given to amethod for studying the geometry of a spaceX.Initiated by Felix Klein, the program proposed a study of the geometric properties of aspace X that remain invariant under a specifiedgroup of one-to-one continuous transformationsof the space.For example, the geometry of the Euclideanplane can be described by the group of rigid motions of R2 that take congruent figures to oneanother.escribed circle of a triangleA circle tangent to one side of the triangle as well as to theextensions of the other two sides.EuclideanSatisfying the postulates of Euclid’s Elements.Euclidean algorithmA method for determining the greatest common divisor of twononzero integers using repeated application ofthe division algorithm.
See division algorithm.To find gcd(10, 46), begin by using the division algorithm to determine the remainder obtained when 44 is divided by 12:46 = 4(10) + 6 .© 2001 by CRC Press LLCNext, repeat the division algorithm with 10 and6 (the dividend and remainder from above):10 = 1(6) + 4 .This procedure (repeating the division algorithmwith the previous dividend and remainder) is repeated until a 0 remainder is obtained (note thatthis is guaranteed to occur eventually, since theremainders are necessarily decreasing). To continue the illustration, repeat the division algorithm with 6 and 4 to obtain,6 = 1(4) + 2 ,then apply the division algorithm one more timeto get4 = 2(2) + 0 .The last nonzero remainder will always be thegreatest common divisor of the two original integers.To illustrate the algorithm more succinctly,461064= 4(10) + 6= 1(6) + 4= 1(4) + 2= 2(2) + 0As 2 is the last nonzero remainder, we concludethat gcd(10, 46) = 2.Also known as Euclid’s algorithm.Euclidean geometryOrdinary plane orthree-dimensional geometry.
More generally,it can refer to any geometry in which eachpoint is uniquely described by an ordered setof n numbers, the coordinates of the point, andwhere the distance d(x, y) between two pointsxn ) and y = (y1 , . . . , yn ) is givenx = (x1 , . . . , n2by d(x, y) =i=1 (yi − xi ) .Euclidean plane Two-dimensional Euclideanspace, in which each point is uniquely describedby an ordered pair of real numbers (x, y), anddistance between points P1 = (x1 , y1 ) and P2 =(x2 , y2 ) is given byd(P1 , P2 ) = (x2 − x1 )2 + (y2 − y1 )2 .Euclidean polyhedronIn R3 , a solidbounded by polygons. More generally, the set ofEuler’s summation formulapoints belonging to the simplices of a Euclideansimplicial complex in Rn .Euclidean space A space that has a Euclideangeometry. See Euclidean geometry.Euler characteristicLet K be a simplicialcomplex of dimension n and let αm be the number of simplices of dimension m.
Then theEuler-Poincaré characteristic, χ (K), of K isdefined by:χ (K) =n(−1)m αm .m=0The most common version of the Euler-Poincarécharacteristic occurs in the case where K hasdimension two. If we let V be the number ofvertices, E be the number of edges, and F be thenumber of faces of K, then χ (K) = V −E +F .The Euler-Poincaré characteristic is an invariant of the complex; that is, it is independentof the triangulation of the complex K.If βp is the pth Betti number of K, that isβp = rankHp (K)/Tp (K) where Tp (K) is thetorsion subgroup of Hp (K), thenχ (K) =n(−1)p βp .p=0Euler phi function The arithmetic function,denoted ϕ, which, for any positive integer n, returns the number of positive integers less than orequal to n which are relatively prime to n.
(Seearithmetic function.) That is, φ(n) = #{i : 1 ≤i ≤ n and (i, n) = 1}. For example, φ(6) =2, φ(13) = 12. The value of φ(n) is even for alln > 1. It is multiplicative; its value at a primepower is given by φ(p i ) = p i−1 (p − 1). It isalso called the totient function.Euler-Poincaré classGiven an orientablevector bundle ξ , with base space B, on Rn , theprimary obstruction in Hn (B; Z) for constructing a cross-section of the associated (n − 1)sphere bundle.
The Euler-Poincare class of amanifold is that of its tangent bundle.Euler-Poincaré formulateristic.© 2001 by CRC Press LLCSee Euler charac-Euler productIf f is a multiplicative function (a real or complex valued function definedon the positive integers with the property that ifgcd(m, n) = 1, then f (mn) = f (m)f (n)) and∞f (n) converges absolutely, thenthe seriesn=1∞f (n) =1 + f (p) + f (p 2 ) + · · ·,pn=1where the product is taken over all primes. Thisproduct is called the Euler product of the series.If f is completely multiplicative (f (mn) =f (m)f (n) for all m, n), in which case f (pk ) =f (p)k for each k and p, then the Euler productabove can be simplified using our knowledge ofgeometric series and we have∞f (n) =pn=11.1 − f (p)Euler product formulaThe Euler productfor certain Dirichlet series. See Euler product.For example, using the Euler product (withf (n) = 1 for all n), we can express the Riemannzeta function as a product.
Namely,ζ (s) =p11− sp−1for all real numbers s > 1.Euler’s criterionLet p be an odd prime. Ifp is not a divisor of the integer a, then a is ap−1quadratic residue of p if and only if a 2 is onep−1more than a multiple of p (note that a 2 is always either one more or one less than a multipleof p by Fermat’s Little Theorem).Euler’s summation formula A formula thatspecifies the error involved when a partial sumof an arithmetic function is approximated by anintegral. Specifically, the formula states thatif a and b are real numbers with a < b andf is continuously differentiable on the intervalEuler’s Theorem of Polyhedrons[a, b], thenorf (k) b=f (x) dxaa<k≤b+ baf (x)(x − [x]) dx+ (b − [b])f (b)− (a − [a])f (a) .Here, [x] denotes the greatest integer less thanor equal to x (the so-called greatest integer function).The Euler-Maclaurin summation formula isa special case of Euler’s formula when a and bare integers.
Namely,bf (k)=k=af (x) dxa b1f (x)(x − [x] − ) dx2a11+ f (a) − f (b) .22Euler’s Theorem of PolyhedronsFor Euclidean space, this theorem states that V − E +F = 2 for any simple polyhedron, where V =number of vertices, E = number of edges, andF = number of faces in the polyhedron. Thistheorem may be generalized to state that, for anyfinite CW complex, α0 − α1 + α2 = 2, whereαi = number of i-cells of the CW complex.A diagramfg0 −→ A −→ B −→ C −→ 0in the category of modules is exact if f is injective, g is surjective, and the kernel of g is equal tothe image of f . An exact functor is an additivefunctor F : C → D between categories of modules satisfying the property that the exactness ofthe diagramfg0 −→ A −→ B −→ C −→ 0implies the exactness of eitherF (f )F (g)0 −→ F (A) −→ F (B) −→ F (C) −→ 0© 2001 by CRC Press LLCF (f )depending on whether F is covariant or contravariant, respectively. See diagram, additivefunctor, covariant functor.exact sequence of groupsof groupsA0A finite sequencef0f1f3fnfn−1A1 A2...An−1 An→ → →→→is exact if Im(fi−1 ) = Ker(fi ) for i = 1, 2,.
. . , n − 1.An infinite sequence of groups b+exact functorF (g)0 −→ F (C) −→ F (B) −→ F (A) −→ 0 ,...fi−2fi−1 fifi+1Ai−1Ai Ai+1...→→→→is exact if Im(fi−1 ) = Ker(fi ) for all i ∈ Z.A short exact sequence of groups is an exactsequence 1 → A → B → C → 1, where 1denotes the trivial group, i is injective, and πis surjective. In this case we say that B is anextension of A by C.existential quantifierSee quantifier.existential sentenceLet L be a first orderlanguage and let σ be a sentence of L.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
