Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 15
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See relation.© 2001 by CRC Press LLCdualThe dual of a concept represented bya diagram is the diagram in which the verticesare the same, but all arrows are reversed. Seediagram.dual bundletionGiven a bundle ξ , with projecπ :E→B,π : E → B ,with E = ∪p∈B [π −1 (p)]∗ , where [π −1 (p)]∗denotes the dual space of π −1 (p), and π takeseach [π −1 (p)]∗ to p.dual category The dual of a category C (alsoknown as the opposite of C) is the category C opwhich satisfies the following properties:(i.) Obj(C op ) = Obj(C);(ii.) HomC op (A, B) = HomC (B, A).Composition of morphisms in C op is defined byg op f op = (f g)op .
See category.dual complexThe set of dual cells of simplices of a complex. More specifically, considera simplicial complex C. Let C be the barycentric subdivision of C and, for any q-simplex σof C, let C(σ ) denote the union of all (n − q)simplices (n being the dimension of the manifold) of C . Then the set C ∗ = {C(σ ) : σ ∈ C}is the dual complex of C.dual convex cone Given a convex cone C ⊆Rn , the set {x ∈ Rn : (x, y) ≤ 0 for all y ∈ C}.Here (x, y) denotes the inner product of x andy.Dupin indicatrixIf M is a surface in R3 ,and P is a point in M, then a plane parallel tothe tangent plane to M at P and very close tothe tangent plane will intersect M in a curve thatis approximately a quadratic curve. The Dupinindicatrix is a quadratic curve that is similar tothis curve of intersection. If the principal curvatures κ1 and κ2 of the surface at P are bothpositive, then the Dupin indicatrix is given bydyadic compactumthe ellipse κ1 x 2 + κ2 y 2 = 1.
If κ1 > 0 > κ2 ,then the Dupin indicatrix is the pair of hyperbolas κ1 x 2 + κ2 y 2 = ±1. When one of theprincipal curvatures is 0, the indicatrix is a pairof parallel straight lines.duplication of cubeOne of the “Three Famous Greek Problems” from the classical Greekgeometers. In this problem, a cube is to be constructed with double the volume of a given cube.© 2001 by CRC Press LLCIt can be proved that this construction is impossible using a straight edge and compass alone.dyadic compactumLet X be the discretespace with two points. The infinite product ofcopies of X, with the product topology, is thedyadic compactum.
It is a compact, uncountable, totally disconnected, Hausdorff space,homeomorphic to the Cantor set. See also Cantor set.Eilenberg-Steenrod AxiomsEeccentric angleFor an ellipse, the angle θ ,where the ellipse is described parametrically bythe equations x = a cos θ , y = b sin θ . Similarly, the eccentric angle at (x, y) for a hyperbola described parametrically by the equationsx = a sec φ, y = b tan φ is φ.eccentric circles(1) For an ellipse, the circles centered at the center of the ellipse withdiameters equal to the lengths of the major andminor axes of the ellipse.(2) The two eccentric circles of a hyperbolaare those with center at the origin of the hyperbola, and with diameters equal to the lengths ofthe transverse and conjugate axes of the hyperbola.eccentricityFor a conic section, the ratioOA,whentheconicsection in question is rergarded as the reciprocal of a circle with radiusr and center A with respect to the circle havingcenter O.
Alternatively, if the conic section isregarded as a curve generated by a point moving in the plane such that the ratio of its distancefrom a fixed point to a fixed line remains constant, then the eccentricity of the conic is thatdistance ratio.effective Informally, the term effective is often used as in the definition of effective procedure, as a synonym for “algorithmic”.
Formally, the term effective is synonymous withcomputable, or recursive. See effective procedure, computable, recursive.effectively enumerableA set A of naturalnumbers such that there is an effective procedurewhich, when given a natural number n, will output 1 after finitely many steps if n ∈ A and willrun forever otherwise. Alternatively, A is effectively enumerable if there is an effective procedure that lists the elements of A.
In other words,for an effectively enumerable set A, if n ∈ A,© 2001 by CRC Press LLCone will find out eventually, but if n ∈ A, thenone will never know for sure.For example, the set of all natural numbers nsuch that there exists a consecutive run of exactlyn 5s in the decimal expansion of π is effectivelyenumerable.This notion is intuitive; the correspondingformal notion is recursively enumerable, alsoknown as computably enumerable. See recursively enumerable.effective procedureAn effective procedure(or algorithm) is a finite, precisely given list ofinstructions which is deterministic; i.e., at anystep during the execution of the instructions,there must be at most one instruction that canbe applied.
This notion is intuitive for a corresponding formal mathematical notion. See computable, recursive.Eilenberg-Steenrod Axioms Let T be a category of pairs of topological spaces and continuous maps and let A denote the category of Abelian groups. Suppose we have the following:(i.) A functor Hp : T → A for each integerp ≥ 0, whose value is denoted Hp (X, A).
Iff : (X, A) → (Y, B) is a continuous map, let(f∗ )p denote the induced map from Hp (X, A)to Hp (Y, B).(ii.) A natural transformation∂p : Hp (X, A) → Hp−1 (A)for each integer p ≥ 0, where A denotes the pair(A, ∅).These functors and natural transformationsmust satisfy the following three axioms fromcategory theory. All pairs are in T .Axiom 1. If i is the identity, then (i∗ )p is theidentity for each p.Axiom 2.
((k ◦ h)∗ )p = (k∗ )p ◦ (h∗ )p .Axiom 3. If f : (X, A) → (Y, B), then thefollowing diagram is commutative:Hp (X, A)∂p ↓Hp−1 (A)(f∗ )p→((f |A )∗ )p→Hp (Y, B)∂p ↓Hp−1 (B) .The Eilenberg-Steenrod axioms are the following five axioms:Exactness Axiom. The sequence· · · → Hp (A)(i∗ )p(π∗ )pHp (X)Hp (X, A)→→elementarily equivalent structures∂pHp−1 (A) → .
. .→is exact, where i : A → X and π : X → (X, A)are the inclusion maps.Homotopy Axiom. If h and k are homotopic,then (h∗ )p = (k∗ )p for each p.Excision Axiom. Given (X, A), let U bean open subset of X such that Ū ⊂ IntA. If(X \ U, A \ U ) is in A, then the inclusion (X \U, A \ U ) → (X, A) induces an isomorphismHp (X \ U, A \ U ) Hp (X, A) .Dimension Axiom.
If P is a one-point space,the Hp (P ) = {0} for p = 0 and H0 (P ) Z.Axiom of Compact Support. If z ∈Hp (X, A), there is a pair (X0 , A0 ) in T with X0and A0 compact, such that z is in the image ofthe homomorphism Hp (X0 , A0 ) → Hp (X, a)induced by the inclusion (X0 , A0 ) → (X, A).Any theory that satisfies these axioms is calleda homology theory on T . (See homology theory.) The first homology theory was defined forthe category of compact polyhedra. Later several other homology theories, such as singularhomology, were defined. Eilenberg and Steenrod then showed that the above axioms completely classified the homology groups on theclass of polyhedra.
There are also similar axioms for cohomology theory, except for the Axiom of Compact Support.elementarily equivalent structuresTwostructures A and B in the language L such that,for every sentence φ of L, A |= φ if and onlyif B |= φ; that is, φ is true in A if and only ifit is true in B. Elementary equivalence (writtenA ≡ B) expresses the property that L cannotdistinguish between the structures A and B.elementary diagramThe theory of all sentences which hold in a model A, using an extraconstant symbol for each element of A.
Moreprecisely, let A be a model in the language L, andlet LA be the expansion of L which adds a newconstant symbol ca for each a ∈ A. Then theelementary diagram of A is the set of all LA sentences which are true in the model A witheach ca interpreted by a.elementary embeddingLet L be a first order language, and let A and B be structures© 2001 by CRC Press LLCfor L, where A and B are the universes of Aand B, respectively. An elementary embeddingof A into B is an embedding h of A into Bwith the property that for every well-formed formula ϕ with free variables v1 , .
. . , vn and every n-tuple a1 , . . . , an of elements of A, if |=Aϕ[a1 , . . . an ], then |=B ϕ[h(a1 ), . . . , h(an )].See embedding, satisfy (for the definition of thenotation used here).elementary substructureLet L be a firstorder language, and let A and B be structures forL, where A is the universe of A. The structureA is an elementary substructure of B if(i.) A is a substructure of B, and(ii.) for all well-formed formulas ϕ withfree variables from among v1 , . . . , vn and all ntuples a1 , .
. . , an of elements of A, if|=A ϕ[a1 , . . . , an ], then |=B ϕ[a1 , . . . , an ].See satisfy (for the definition of the notationused here).If A is an elementary substructure of B, thenB is an elementary extension of A. The termelementary submodel is sometimes synonymouswith elementary substructure.As an example, let L be the first order language with equality whose only predicate symbol is <.
Let Q be the structure whose universeis the set Q of rational numbers and where <is interpreted in the usual way, and let R bethe structure whose universe is the set R of realnumbers and where < is interpreted in the usualway. Then Q is an elementary substructure ofR.element of a setOne of the objects x thatmakes up the set X, written x ∈ X. See set.element of coneAny line that lies on thesurface of a given cone and contains its vertex.element of cylinder The generator of a givencylinder in any fixed position, where the cylinderis thought of as being generated by a straightline moving along a given curve while remainingparallel to a fixed line.ellipse A proper conic section formed by theintersection of a plane with one nappe of thecone.
Alternatively, a conic section with eccentricity less than one.equinumerous setsellipsoidA surface whose intersection withany plane is either a point, a circle, or an ellipse.elliptic coneThe set of points consisting ofall the lines passing through a fixed ellipse anda fixed point not in the plane of the ellipse.elliptic cylinder The set of points consistingof all the lines passing through a fixed ellipseand parallel to a fixed line not parallel to theplane of the ellipse.elliptic pointA point on a surface at whichthe centers of curvature are all on the same sideof the surface normal; the normal sections areall concave or all convex.elliptic surface Any type of Riemann surfacethat can be mapped conformally on the closedcomplex plane.
More generally, a nonsingularsurface E having a surjective morphismπ :E→Sonto a nonsingular curve S whose generic fiberis a nonsingular elliptic curve.elliptic transformationtransformationz →A linear fractionalaz + bcz + don the complex numbers C, where a + d is real,and discriminant (a + d)2 − 4 is negative.embedding(1) Let L be a first order language, and let A and B be structures for L, withuniverses A and B for A and B, respectively. Afunction h : A → B is an embedding if h isinjective and(i.) for each n-ary predicate symbol P andevery a1 , .
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