SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Some Remarks on p-Adic Analytic Groups

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

A Matrix Setting for the q-Schur Algebra

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 S_q(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 T^r(M_n)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 S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(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(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–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(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(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.     

Shapiro's Cyclic Sum

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 = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, 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, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Dualizing Complex and the Canonical Element Conjecture II

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, m_R,k) (complete intersection) such that dim R=dim A and A=R/I.Let denote the canonical module of A, that is, =Hom_R (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.     

Norms on Free Topological Groups

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).     

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(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) = ||u_t(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.     

Algebras with Collapsing Monomials

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.     

Profinite Groups with Multiplicative Probabilistic Zeta Function

To a finitely generated profinite group G, a formal Dirichletseries P_G(s)=_na_n/n^s is associated, where a_n = _|G:H|=n µ_G(H).It is proved that G is prosoluble if and only if the sequence{a_n}_nN is multiplicative, that is, a_rs = a_ra_s for any pairof coprime positive integers r and s. This extends the analogousresult on the probabilistic zeta function of finite groups.     

Connectedness and Stability of Julia Sets of the Composition of Polynomials of the Form z2+cn

For a sequence (c_n) of complex numbers, the quadratic polynomialsf_cn(z) := z² + c_n and the sequence (F_n) of iterates F_n := f_cn...f_c1areconsidered. The Fatou set F_(cn) is by definition the set ofall z C^ such that (F_n) is normal in some neighbourhood ofz, while the complement of F_(cn) is called the Julia set J_(cn).The aim of this article is to study the connectedness and stabilityof the Julia set J_(cn) provided that the sequence (c_n) is bounded.     

On Semihereditary Maximal Orders

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 (V^h, F^h) is the Henselization of (V, F), then Ais a semihereditary maximal order if and only if A_VV^h 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.     

Structural Stability of Constrained Polynomial Systems

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 ad–bc=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.     

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

Constructions of stable equivalences of Morita type for finite-dimensional algebras III

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 _AM_B and _BN_A 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 End_A(R)of the module R and the endomorphism algebra End_B(N_AR) of themodule N_AR. If M and N satisfy the property that both (N_A–,M_B–) and (M_B–, N_A–) 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.     

Parity of the Partition Function in Arithmetic Progressions, II

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.     

The Product Separability of the Generalized Free Product of Cyclic Groups

Let G be a group endowed with its profinite topology, then Gis called product separable if the profinite topology of G isHausdorff and, whenever H₁, H₂, ..., H_n are finitely generatedsubgroups of G, then the product subset H₁ H₂ ... H_n is closedin G. In this paper, we prove that if G=FxZ is the direct productof a free group and an infinite cyclic group, then G is productseparable. As a consequence, we obtain the result that if Gis a generalized free product of two cyclic groups amalgamatinga common subgroup, then G is also product separable. These resultsgeneralize the theorems of M. Hall Jr. (who proved the conclusionin the case of n=1, [3]), and L. Ribes and P. Zalesskii (whoproved the conclusion in the case of that G is a finite extensionof a free group, [6]).     

On the Girth of Graphs Critical with Respect to Edge-Colourings

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.     

Two Diophantine Birds with One Stone

Integer Solutions are found to the equations t²–3(a²,b², (a + b)², (a–b)²) = p², q², r², s². These lead surprisinglyto solutions to the equations u² + (c², d², (c + d)², (c –d)²) = p², q², v², w², with the same values of p and q.     

Structural Projections on JBW*-Triples

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.     

The Markov Oscillation Problem in Discrete Time

Consider a discrete-time regenerative phenomenon with associatedrenewal sequence u_n. General results for the supremum of u_m+1,..., u_m+n are developed for those renewal sequences {u_n} forwhich the first m + 1 elements match those of a fixed renewalsequence {v_n}, that is, u₀ = v₀, ..., u_m = v_m. A series of associatedlemmas are developed in the process.