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1.
A semigroup S is called collapsing if there exists a positiveinteger n such that for every subset of n elements in S, atleast two distinct words of length n on these letters are equalin S. In particular, S is collapsing whenever it satisfies alaw. Let U(A) denote the group of units of a unitary associativealgebra A over a field k of characteristic zero. If A is generatedby its nilpotent elements, then the following conditions areequivalent: U(A) is collapsing; U(A) satisfies some semigrouplaw; U(A) satisfies the Engel condition; U(A) is nilpotent;A is nilpotent when considered as a Lie algebra.  相似文献   

2.
A norm on a group G is a function N mapping G into the set ofnon-negative real numbers such that for each x and y in G, N(xy–1) N(x)+N(y) and N(e) = 0, where e is the identity element ofG. It is shown here that if F(X) is the free topological groupon any completely regular Hausdorff space X and H is a subgroupof F(X) generated by a finite subset of X, then any norm onH can be extended to a continuous norm on F(X).  相似文献   

3.
Dualizing Complex and the Canonical Element Conjecture II   总被引:1,自引:0,他引:1  
In this paper we continue our study of the Canonical ElementConjecture (henceforth C.E.C.) via the dualizing complex. Throughoutthe work (A, m, k) will denote a noetherian complete local ringA of dimension n, m its maximal ideal and k=A/m. Since A iscomplete, we can find a complete local Gorenstein ring (R, mR,k) (complete intersection) such that dim R=dim A and A=R/I.Let denote the canonical module of A, that is, =HomR (A, R),which may be identified with the annihilator of I in R, an idealof R. When A is a domain, we change notation and denote I byP; in this case P is a height 0 prime ideal of R.  相似文献   

4.
The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue Sq(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power Tr(Mn)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to Sq(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in Sq(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gln) corresponding to the Liealgebra gln, the q-Schur algebra Sq(n, r), and the Hecke algebra,H(Ar–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gln). We shall denotethis ‘extended’ version of the q-Schur algebra bySv(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.  相似文献   

5.
A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.  相似文献   

6.
The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by adbc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.  相似文献   

7.
A graph G with maximum valency r is called critical if r + 1colours are needed for an edgecolouring, but every proper subgraphrequires at most r. In this note we consider the minimum orderf(r, g) of a critical graph of maximum valency r and girth g.We show that f(r, 3) = r+1 or r+2 according as r is even orodd, f(r, 4) = 2r+1,f(3, 5) = 9 and f(3, 6) = 15.  相似文献   

8.
Let p(n) denote the ordinary partition function. Subbarao conjecturedthat in every arithmetic progression r (mod t) there are infinitelymany integers N = r (mod t) for which p(N) is even, and infinitelymany integers M = r (mod t) for which p(M) is odd. We provethe conjecture for every arithmetic progression whose modulusis a power of 2. 2000 Mathematics Subject Classification 11P83.  相似文献   

9.
Let A be an order integral over a valuation ring V in a centralsimple F-algebra, where F is the fraction field of V. We showthat (a) if (Vh, Fh) is the Henselization of (V, F), then Ais a semihereditary maximal order if and only if AVVh is a semihereditarymaximal order, generalizing the result by Haile, Morandi andWadsworth, and (b) if J(V) is a principal ideal of V, then asemihereditary V-order is an intersection of finitely many conjugatesemihereditary maximal orders; if not, then there is only onemaximal order containing the V-order. 1991 Mathematics SubjectClassification 16H05.  相似文献   

10.
Shapiro's cyclic sum is defined by , If K is the cone in Rn of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T2, where T is the non-linear operatordefined by (Tx)i = xni+1/(xni+2 + xni+3)2for i = 1,2,...,n. It is conjectured that Tx = Skx, where Sis the shift operator in Rn, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, ai(x) = ani+1–k(x) fori = 1,2,...,n.  相似文献   

11.
The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation utt = (k(x)ux)x + F(x, t) andin the Neumann boundary condition k(0)ux(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=ut(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J1(w)= ||u(x, t;w) – µ(x)||02 and J2(w) = ||ut(x, T;w)– (x)||02 can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.  相似文献   

12.
This work, investigates the asymptotics for Euler’s q-exponentialEq(z), Ramanujan’s function Aq(z), Jackson’s q-Besselfunction Jv(2) (z; q), the Stieltjes–Wigert orthogonalpolynomials Sn(x; q) and q-Laguerre polynomials Ln() (x; q)as q approaches 1.  相似文献   

13.
A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and pe exactly divides |F|, then a ZpF-latticeM is determined up to isomorphism by its finite quotient M/pe+1M.If M is a free Zp-module of rank d, this is equivalent to sayingthat representations of F in GLd(Zp) are determined up to equivalenceby their images modulo pe+1. 1991 Mathematics Subject Classification20E18, 22E20.  相似文献   

14.
Let B and C be continuous nonlinear operators in a Banach space.Assume that B satisfies a dissipativeness condition, and thatthe range of C is relatively compact. Some sufficient conditionsare given for the existence of solutions of the equation u'(t)= B(t, u(t)) + C(t, u(t)).  相似文献   

15.
On a Topological Property of certain Calkin Algebras   总被引:1,自引:0,他引:1  
Let X = 1p, 1 p < , or X = c0, B(X) be the algebra of allbounded linear operators on X, H(X) be the ideal of compactoperators in B(X), and C(X) = B(X)/H(X) be the Calkin algebraon X. For TB(X), let ||T||c = dist(T, H(X)) be the essentialnorm of T that is the norm of T+H(X) in C(X). It is shown thatfor any operator TB(X) and any number 0 < t < 1, thereexists a closed infinite dimensional subspace Z Z X such that ||Tx|| t||T||c, for all x Z. As a consequence, it is shown that every (not necessarily complete)submultiplicative norm on the Calkin algebra C(X) is equivalentto the quotient norm || ||c on C(X).  相似文献   

16.
For n a positive integer, a group G is called core-n if H/HGhas order at most n for every subgroup H of G (where HG is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.  相似文献   

17.
Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.  相似文献   

18.
To a finitely generated profinite group G, a formal Dirichletseries PG(s)=nan/ns is associated, where an = |G:H|=n µG(H).It is proved that G is prosoluble if and only if the sequence{an}nN is multiplicative, that is, ars = aras for any pairof coprime positive integers r and s. This extends the analogousresult on the probabilistic zeta function of finite groups.  相似文献   

19.
For a smooth function g, the module R(X)+gR(X) is dense in C(X)if and only if R(Z) is dense in C(Z), where X is compact andnowhere dense and Z is the subset of X on which g vanishes.The "localness" of the module R(X)+gR(X) is also investigated.  相似文献   

20.
In this paper, we provide a new method to produce stable equivalencesof Morita type. Our main results can be stated as follows. LetA and B be two finite-dimensional k-algebras over a field k.Suppose that two bimodules AMB and BNA define a stable equivalenceof Morita type between A and B and that R is a generator forA-modules. Then there is a stable equivalence of Morita typedefined by X and Y between the endomorphism algebra EndA(R)of the module R and the endomorphism algebra EndB(NAR) of themodule NAR. If M and N satisfy the property that both (NA–,MB–) and (MB–, NA–) are adjoint pairs of functors,then so do the modules X and Y. Moreover, we show that the self-injectivedimension and the Gorenstein property are invariant under stableequivalences of Morita type with the above-mentioned adjointproperty.  相似文献   

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