Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 13
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First, start with one point; attach the cellD n by identifying all points in its boundary withthat one point. Alternately, start with two points.Attach two 1-cells (intervals) by sending oneendpoint to each of the two points that formedthe previous stage. This constructs a circle, S 1 .Given S n−1 by iteration, form S n by identifying the boundaries of each of two n-cells withS n−1 . This latter construction is convenient forstudying real projective space.Any manifold can be constructed as a CWcomplex and such constructions are useful forcalculating the homology or cohomology of themanifold.cycle An element c in Cn such that ∂n (c) = 0,where Cn is a member of a chain complex overa ring R. See chain complex.cycle groupIf {Cn , ∂n } is a chain complex(of Abelian groups), then the kth cycle group Zkis the subgroup of Ck consisting of all elementsc such that ∂k (c) = 0.
The boundary groupcylinderBn = ∂n+1 /Cn+1 is a subgroup of the cyclegroup, since ∂n +1◦∂n = 0. See chain complex.cylinder A flat ruled surface in R3 , which isthe locus of points on a straight line (the generator) moving parallel to itself, and intersecting a given plane curve (the directrix). Thus,as the generator, which is perpendicular to theplane of the directrix, moves along the directrix,it sweeps out a cylinder.cylinder of revolution A surface formed bythe set of all lines passing through a given circleand perpendicular to the plane of the circle. Alsoknown as a right circular cylinder.© 2001 by CRC Press LLCcylindrical helix A helix whose path lies ona cylinder, forming a constant angle with the elements of the cylinder.
A cylindrical helix isdescribed, for example, by the parametric equationsx = cos ty = sin tz=t(−∞ < t < ∞).cylindrical surfaceA surface generated bya line in space moving along a curve, alwaysstaying parallel to a fixed line.Dedekind cutDDarboux’s frame If γ (s) is a curve on a surface M in Euclidean space R3 , parameterized byarc length, then there is an orthonormal frame(T , U, V ) defined along the curve called Darboux’s frame.
The first vector, T = γ (s), isthe unit tangent vector to the curve. The vectorV = ν(γ (s)) is the unit normal to the surfaceat the point. (This assumes that a normal vectorfield ν(P ) is specified.) The vector U = V × Tis the normal vector to the curve in the surface,whose direction is determined by the choice ofsurface normal. One can then define the analogof the Frenet equations for the curve:TUV=κg U + κ n V= −κg T+ τg V= −κn T − τg UThe functions κg and κn are the geodesic curvature and the normal curvature, respectively,of the curve in the surface. τg is the geodesictorsion.decagonA ten-sided polygon.decidable A set of objects of some sort is decidable if there is an effective procedure (algorithm) which, given an arbitrary object, decideswhether or not the object is in the set.
This notion, as defined, is an intuitive notion. To definethis notion formally requires the formal notionof computability and the ability to code effectively (Gödel number) the objects in question.See computable.For example, the set of tautologies of propositional logic is decidable; i.e., there is an effective procedure which, given a string from thelanguage of the propositional logic, can determine whether or not the string is a well-formedformula, and if it is, can determine whether ornot the well-formed formula is a tautology.
Asanother example, a theory in first order logic isdecidable if it is both complete and axiomatizable by a decidable set of axioms.© 2001 by CRC Press LLCdecision problemA decision problem asksif there exists an effective procedure (algorithm)for deciding the truth or falsity of an entire classof statements. If such an effective procedure exists, the problem is said to be solvable; otherwisethe problem is said to be unsolvable.This notion, as defined, is intuitive. To defineit formally requires the formal notion of computability, the ability to effectively code (Gödelnumber) the statements, and the Church–TuringThesis.For example, given an n-ary predicate R, thedecision problem associated with R asks if thereis an effective procedure for deciding, given arbitrary natural numbers a1 , .
. . , an , whetherR(a1 , . . . , an ) is true or false. This decisionproblem is solvable if R is recursive (computable) as a relation; otherwise it is unsolvable.There are many famous decision problems.Hilbert’s Tenth Problem asks if there is an effective procedure which, given a Diophantine equation p(x1 , .
. . , xn ) = 0, where p is a polynomialwith integer coefficients, determines whether ornot there is an integer solution. It was provedby results of Matijasevič (1970) and Davis, Putnam, and Robinson (1961) that Hilbert’s TenthProblem is not solvable. The decision problem for propositional logic, which asks if thereis an effective procedure which, given a wellformed formula of propositional logic, determines whether or not that formula is provable,was proved by Post (1921) to be solvable.
Onthe other hand, the decision problem (Entscheidungsproblem) for first-order logic is not solvable (Church, 1936; Turing, 1936); i.e., if L isa first-order language with a k-ary function orpredicate symbol, for k ≥ 2, then there is no effective procedure that can determine whether ornot an arbitrary sentence of L is logically valid.Dedekind cutSuppose that X is a subsetof R and A, B are two subsets of X with thefollowing properties:(i.) A ∩ B = ∅;(ii.) A ∪ B = X;(iii.) a < b for any a ∈ A, b ∈ B.The sets A and B form a Dedekind cut if A hasa last element and B has no first element, or ifA has no last element and B has a first element.X is an interval of R if and only if any choice ofdeductionA, B satisfying conditions (i.) through (iii.) isa Dedekind cut.If there exists a choice of A and B which isnot a Dedekind cut, then X can be extended to aninterval of R by including additional real numbers.
For example, if X consists of the rationalnumbers, then no choice of A, B is a Dedekindcut. However, X may be extended by includingthe irrational numbers. This technique can beused if X is a subset of any complete linearlyordered set.deductionSee proof.degree of unsolvabilitynumbers of A, the setFor a set of naturaldeg(A) = {B : B ≡T A};i.e., the class of all sets B of natural numberswhich are Turing equivalent to A.
A degree ofunsolvability is often called a Turing degree, orsimply a degree.As an example, if A is a recursive (computable) set, then deg(A) consists of all recursive(computable) sets. However, if A is computably(recursively) enumerable, then deg(A) does notconsist of all computably enumerable sets.deficient number A positive integer n havingthe property that the sum of its positive divisorsis less than 2n, i.e., σ (n) < 2n. For example,16 is deficient, sincede Morgan’s laws Let B be a Boolean algebra with binary operations ∪, ∩ and unary operation . If X, Y ∈ B, then:1 + 2 + 4 + 8 + 16 = 31 < 32 .(X ∪ Y ) = X ∩ Y .Compare with abundant number, perfect number.definition by recursionSee recursion.(X ∩ Y ) = X ∪ Y See Boolean algebra.denominatorab.The number b in the fractiondegenerate conic A conic section formed bya plane intersecting a conical surface either onlyin the vertex of the surface (resulting in a point),only at an element of the surface (resulting ina line), or only at two elements of the surface(resulting in two lines).dense linear ordering Given a set A with atleast two distinct elements and a linear ordering≤ on A, ≤ is dense if, for all x, y ∈ A withx < y, there exists z ∈ A with x < z < y.The usual ordering ≤ on Q, the set of rationalnumbers, is dense.degenerate simplex An n-simplex (map fromthe n-dimensional analog of a triangle to a space)whose image is less than n-dimensional.dense subsetA subset A of a topologicalspace X such that Ā = X, where Ā denotes theclosure of A in X.degree of an arc The degree measure of thecentral angle determining the arc, provided thearc is minor.
If the arc is major, its degree is 360minus the degree measure of the central angle.denumerable setA set A, equinumerouswith the set N of natural numbers; i.e., suchthat there is a bijection from A onto N. Forexample, the set Q of rational numbers is denumerable, while the set R of real numbers is notdenumerable.degree of mappingLet f : (S n , x0 ) →n(S , x0 ) be a continuous map where n ≥ 1 andlet f∗ : Hn (S n , x0 ) → Hn (S n , x0 ) be the induced map. If z0 is the generator of Hn (S n , x0 ),then f∗ (z0 ) = d · z0 for some integer d.
In thiscase d is the degree of the map f and is denoteddeg(f ).An important consequence of this is theBrouwer Fixed-Point Theorem: Every continuous map f : B n → B n has a fixed point.© 2001 by CRC Press LLCderived setThe set of accumulation pointsof a subset A of a topological space X. Thederived set of A is written A . The closure ofA is then A ∪ A , and A is closed if and only ifA ⊆ A.determined Let X be a subset of ωω . Associated with X is the game GX which is played bydifferential invarianttwo players A and B. The two players alternatechoosing natural numbers a1 , b1 , a2 , b2 , .
. . . Ifthe sequence (a1 , b1 , a2 , b2 , . . . ) ∈ X, thenplayer A wins. If not, player B wins. A strategy is a rule that tells the player (either A or B)which number to choose, based on the previouschoices of both players. A winning strategy isa strategy in which the player always wins. Thegame GX is determined if one of the two playershas a winning strategy.developable surface A ruled surface is a surface swept out by a straight line (called the generator) moving through space. It can be given aparameterization of the form X(u, v) = C(u) +vD(u) where C is a curve and C (u)×D(u) doesnot vanish. If the tangent plane to the surfaceat any point is tangent along the entire generator through the point, then the surface is calleda developable surface.
For example, a cylinderis a developable surface, as is a cone. A hyperboloid of one sheet is an example of a ruledsurface which is not a developable surface.diagonalFor any set X, the subset ={(x, x) : x ∈ X} of X × X. This is also commonly referred to as the relation of equality onX. See relation.diagonal intersectionIf κ is a regular uncountable cardinal and (Sα , α < κ) is a sequence of subsets of κ, the diagonal intersection of this sequence, denoted {Sα : α < κ},is {β : β ∈ α<β Sα }.diagramA diagramdiametral plane A plane containing all midpoints of a set of parallel chords of a surface.diamond A strengthening of the ContinuumHypothesis (denoted ♦) which asserts that thereis a sequence of sets Sα ⊆ α for α < ω1 , calledthe diamond sequence, which captures all subsets of ω1 in a certain way.
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