Cavagnaro, Haight - Dictionary of Classical and Theoretical Mathematics (523108), страница 33
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A long exact sequenceof the form· · · → Hn (A) →→ Hn (X) → Hn (X, A)→ Hn−1 (A) → · · ·Riemann zeta functionrelates the relative to the ordinary homologygroups.relative homotopy groupGiven a topological space X and a subspace A, the nth relativehomotopy group πn (X, A) is the set of homotopy equivalence classes of maps f from the ndimensional ball B n to X such that f (S n−1 ) ⊆A where S n−1 , the n−1-sphere, is the boundaryof B n . A homotopy between two such maps fand g is required to carry S n−1 into A for all t. IfA ⊆ B ⊆ X, then there is a long exact sequenceof the form· · · → πn (X, B) →→ πn (X, A) → πn (B, A)→ πn−1 (X, B) → · · · .relatively open set A subset U of a topological space X such that U is a proper subset of asubspace A of X and U is an open subset of A.relative topology The topology on a subset Aof a topological space X that is the collection ofall intersections of A with open sets in X.
Thatis, U ⊆ A is open in the relative topology on Aif there is an open set V ⊆ X with U = V ∩ A.The relative topology makes any set A ⊆ X intoa subspace of X.repeating decimalA decimal representation. . . a4 a3 a2 a1 a0 .a−1 a−2 a−3 . . .of a real number for which there exist positive integers P and N, so that for every n ≥N, a−n = a−n−P , where a−n (resp. a−n−P )is the digit in the 10−n (resp. 10−n−P ) placein the decimal expansion of the number.
Forexample, the decimal expansion of the number1517990 is 1.532323 . . . , where the sequence 32 repeats forever (this is often written 1517990 = 1.532,where the bar over 32 means that this sequencerepeats forever). It can be shown that a real number has either a repeating or terminating decimalrepresentation if and only if that number is a rational number. Note that a terminating decimalrepresentation could be said to have “repeatingzeros” and therefore be a repeating decimal representation as well.representative of an equivalence classelement of an equivalence class.© 2001 by CRC Press LLCAnyrestriction of a functionIf f : A → B is afunction, the restriction of f to a set S (usuallyS is a subset of the domain), denoted by f |S, is{(x, y) : y = f (x) and x ∈ S}.retract A subspace A of a topological spaceX such that there exists a retraction r : X →A.
(See retraction.) Retracts are important inalgebraic topology because they cause the longexact homology and cohomology sequences tosplit as short exact sequences.retraction A continuous function r : X → Afrom a topological space X to a subspace A isa retraction of A if r(a) = a for all a ∈ A.Equivalently, r is a left inverse to the inclusionA → X.Riemann Hypothesis The conjecture that allof the nontrivial zeros of the Riemann zeta function have real part 21 (i.e., are of the form x + iywhere x = 21 ).
Bernhard Riemann stated in hismemoirs that it seemed likely to be true and ifproved could likely be used to prove that thereare infinitely many twin prime pairs. DavidHilbert listed it as one of the most important outstanding problems facing mathematicians at thedawn of the 20th century. Although it is knownthat there are infinitely many zeros of the zetafunction with real part 21 , it is still an open problem as this book is printed at the dawn of the21st century.
See also Riemann zeta function,generalized Riemann hypothesis.Riemannian geometry The study of the geometric properties of locally Euclidean manifolds. A Riemannian manifold is a manifoldwhose tangent space at each point p possessesa positive definite inner product g(p)(X, Y ),which varies continuously (usually smoothly)with the point p. This structure allows oneto define lengths, angles, and other geometricquantities.
The term Riemannian geometry issometimes used to refer specifically to ellipticgeometry, which is a non-Euclidean geometryin which the parallel postulate is replaced by thepostulate that straight lines always intersect.Riemann zeta function The Dirichlet series∞1ζ (s) =ns defined for (extendible to) alln=1right adjoint of a functorcomplex numbers s = 1. It can be shown thatζ (−2n) = 0 for all positive integers n. It isconjectured that the only other zeros are of theform s = 21 + iy (that is, have real part equalto 21 ). This conjecture is known as the RiemannHypothesis.right adjoint of a functorLet C and D becategories, and let F : C → D, G : D → C befunctors. G is the right adjoint of F (and Fis the left adjoint of G) if there is a bijection θbetween the collection of morphisms from F (A)to B and the collection of morphisms from Ato G(B) that is natural for objects A of C andobjects B of D.
Hence, this bijection θ sendsevery morphism f : F (A) → B to a morphismθ (f ) : A → G(B) so that both conditions (i.)θ (f ◦ F (g)) = (θ (f )) ◦ g, and (ii.) θ (h ◦ f ) =(G(h)) ◦ (θ (f )) are satisfied, for every pair ofmorphisms g : A → A and f : B → B . Forexample, if C is the category of groups and grouphomomorphisms, D is the category of sets andfunctions, the forgetful functor G : C → D is theright adjoint to the free group functor F : D →C.rigid motion An even transformation of (Euclidean) space which preserves lengths and angles. A rigid motion takes any geometric figureto one which is congruent to itself.
Because asymmetry of the plane reverses orientation, it isnot considered a rigid motion of the plane, although it is so viewed when the plane is thoughtof as a subset of three-dimensional space. Therigid motions form a group under composition;it is the component of the identity in the groupof isometries.root of a number (1) If n is a positive integerand a is a complex number, an nth root of a is acomplex number r such that r n = a.(2) If n is a positive integer and √r is a realnumber, the nth root of a, denoted n a, is theunique real number r so that√r n = a, if sucha number exists. If n is odd, n a always exists(and is positive when a is positive and negativewhen a is negative),while if n is even and a√is negative, n a does not exist, within the realnumbers.© 2001 by CRC Press LLCroot of equationA number that, when substituted in a given equation, makes the equation√2 andvalid.√ For example, the real numbers2− 2 are√ roots of the equation x = 2, sincex = ± 2 are solutions to the equation.
A rootof a polynomial p(x) is a root of the equationp(x) = 0.rotationA rigid motion of the plane whichfixes exactly one point, or a rigid motion ofthree-dimensional space which fixes the pointson exactly one line, which is the axis of rotation.ruled surfaceAn algebraic surface, birational to C × P1 , where C is a smooth projectivecurve. For example, a smooth quadric in P3 ,which is isomorphic to P1 × P1 .Russell’s ParadoxA paradox of naive (informal) or non-axiomatic set theory.
In naiveset theory, it is possible to form the set A = {x :x ∈ x}. Note that if A ∈ A, then A ∈ A, andif A ∈ A, then A ∈ A. This contradiction iscalled Russell’s Paradox.Russell’s Paradox was discovered by Bertrand Russell in 1901 and published by him in1903. The discovery of this and other paradoxesrevealed that set theory could not be used as alanguage to formalize mathematics in a naivefashion, so that an axiomatic approach, givingrules for which sets could exist, needed to be developed in order to avoid contradictions.
Thereare several axiomatizations of set theory, including ZF (Zermelo-Fraenkel set theory) andN BG (von Neumann-Bernays-Gödel set theory).Prior to the discovery of Russell’s Paradox, itwas believed that any definable collection; i.e.,any collection {x : P (x)} of objects x satisfyinga property P (x), is a set. The difficulty with this(and with Russell’s Paradox) is that some collections are, in some sense, “too big” to be sets.In ZF set theory, only definable collections thatare already subsets of existing sets can be sets(this is the Axiom of Comprehension, or Axiomof Subsets).
In this set theory, the collectionA above cannot be a set, since assuming it is aset leads to a contradiction. The N BG set theory differentiates between classes and sets. Inthis set theory, the collection A above is a classwhich is not a set.s-cobordismSsatisfiableLet L be a first order language,and let be a set of well-formed formulas of L.The set is satisfiable if there exists a structureA for L and a mapping s : V → A such thatfor each formula γ ∈ , A satisfies γ with s.(Here, V is the set of variables of L and A is theuniverse of A.)satisfy Let L be a first order language, α be awell-formed formula of L, A be a structure forL, V be the set of variables of L, and s : V →A (i.e., s assigns each variable in the languageto some element of the universe of A).
Thefunction s can be extended to a function s : T →A from the set T of all terms of L into A, byinduction, as follows.(i.) If x is a variable of L, thens(x) = s(x) .(ii.) If c is a constant symbol of L, thens(c) = cA ,where cA is the element of A assigned to c byA.(iii.) If t1 , . . . , tn are terms of L and f is ann-ary function symbol of L, thens(f (t1 , . . . , tn )) = f A (s(t1 ), . .
. , s(tn )) ,where f A is the n-ary function on A assignedto f by A.The structure A satisfies α with s (notation:|=A α[s]) and is defined by induction on thecomplexity of α as follows.(i.) If α = (t1 = t2 ), where t1 and t2 areterms of L, then A satisfies (t1 = t2 ) with s ifs(t1 ) = s(t2 ).(ii.) If α = P (t1 , . . . , tn ), where t1 , .
. . , tnare terms of L and P is an n-ary predicate symbol of L, then A satisfies P (t1 , . . . , tn ) with s if(s(t1 ), . . . , s(tn )) ∈ P A , where P A is the n-aryrelation on A assigned to P by A.(iii.) If α = (¬β), then A satisfies (¬β) withs if A does not satisfy β with s.© 2001 by CRC Press LLC(iv.) If α = (β → γ ), then A satisfies (β →γ ) with s if A satisfies (¬β) with s or A satisfiesγ with s.(v.) If α = ∀vβ, then A satisfies ∀vβ with sif for all a ∈ A, A satisfies β with the followingmodified version sa : V → A of s:s(x) if x = vsa (x) =a if x = v .Let a1 , . .
. , an ∈ A and let ϕ be a wellformed formula with free variables from amongv1 , . . . , vn . The notation |=A ϕ[a1 , . . . , an ]means that there is an s : V → A with s(vi ) =ai , for 1 ≤ i ≤ n, and A satisfies ϕ with s.saturated model A model A that realizes asmany types as possible. More precisely, if A isa model in the language L and X is any subsetof A, let LX be the expansion of L which addsa constant symbol cx for each x ∈ X.
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