Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary Given an integern 1, and the orthogonal polynomials _n (·; d) of degreen relative to some positive measured, the polynomial system induced by _n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of _n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

A Covering Space with No Compact Core

This note describes an example of a compact aspherical 2-complex X with a covering space , such that is finitely presented but has no compact core.I thank John Stallings for being my 'Senior Scientist' in 1996–1997, and for many superimaginative e-mails that led to this relationship.     

On the embedding of core-free projective images of normal subgroups

Sunto Siano G e due gruppi, : GG una proiettività e N un sottogruppo normale di G. Vengono provati alcuni risultati sulle proprietà di immersions in e G rispettivamente, del gruppi , la chiusura normale di N in modulo il nocciolo di N in , e .

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Supported by a C.N.R. grant.     

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Weakly Periodic Sequences of Bounded Linear Transformations: a Spectral Characterization

Let X and Y be two Hilbert spaces, and the space of bounded linear transformations from X into Y. Let {A } be a weakly periodic sequence of period T. Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to characterize {A _n} by a positive measure and a number T of -valued functions a ₀, . . . ,a _T–1; in the spectral form , where and is an -valued Borel set function on [0, 2) such that      

Hermite polynomials on the plane

The space of bivariate generalised Hermite polynomials of degree n is invariant under rotations. We exploit this symmetry to construct an orthonormal basis for which consists of the rotations of a single polynomial through the angles , ℓ=0,...n. Thus we obtain an orthogonal expansion which retains as much of the symmetry of as is possible. Indeed we show that a continuous version of this orthogonal expansion exists.      

An iterative method for the least squares symmetric solution of matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: with unknown symmetric matrix . By the iterative method, for any initial symmetric matrix , a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the solution with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution to a given matrix in Frobenius norm can be expressed as , where is the least norm symmetric solution of the new minimum residual problem: with . Given numerical examples are show that the iterative method is quite efficient.Research supported by Scientific Research Fund of Hunan Provincial Education Department of China (05C797), by China Postdoctoral Science Foundation (2004035645) and by National Natural Science Foundation of China (10571047).     

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

Regular Unitarily Invariant Spaces on the Complex Sphere

Let K be a compact space, let X be a closed subspace of C(K), and let be a positive measure on K. The triple is said to be regular if, for any positive function and for any , there exists a function such that on K and . The case where K is the unit sphere in and the subspace X is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for the regularity of a triple are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented. Bibliography: 16 titles.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .     

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number the Kronecker sequence m (m = 1,...,M) is equidistributed modulo one in the limit , and (ii) closed horocycles of length become equidistributed in the unit tangent bundle of a hyperbolic surface of finite area, as . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant the Kronecker sequence embedded in along a long closed horocycle becomes equidistributed in for almost all , provided that . This equidistribution result holds in fact under explicit diophantine conditions on (e.g. for = 2) provided that , with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n² modulo one.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. II

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups and construct -invariant generalizations of differentiable connections. In both regular and special cases of the representations of the relevant groups , we found all the affine nonholonomic -, -, and -connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations and also described the respective -invariant planar connections.     

Reducibility of lacunary polynomials,VII

Letk>1 and let be non-zero algebraic numbers contained in the field . It is shown that for almost all, in the sense of density integer vectorsn ₁,...,n _k the polynomial becomes irreducible over on dividing by the product of all factorsx–, where is a root of unity.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Gauss polynomials and the rotation algebra

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=ab with commuting witha andb, then the (generalized) binomial coefficient arising in the expansion

Spherical elastic-plastic waves

Zusammenfassung Es wird die Fortpflanzung elastisch-plastischer Spannungswellen in einem unendlichen Medium betrachtet, welches einer idealen Spannungs-Verformungs-Kurve folgt, Trescas Fliesskriterium unterworfen ist und einen sphärischen Hohlraum enthält, wobei an der Fläche des Hohlraumes ein Stoss angenommen wird. Ein rechnerisches Verfahren, basiert auf endliche Differenzen, wird entwickelt and ein Beispiel gegeben.

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Notation radial stress - tangential stress - K yield stress - _rr non-dimensional radial stress ( /K) - non-dimensional tangential stress ( /K) - , Lame's constants - K _b Bulk constant (=(3+2)/3) - v Poisson's constant - Material density - C Elastic wave velocity (=((+2)/)^1/2) - C _p Plastic wave velocity (=(K _b /)^1/2) - distance from center of cavity - r ₀ cavity radius - v non-dimensional radial co-ordinate (= /r ₀) - time - t non-dimensional time (=C /r ₀) - radial displacement - u non-dimensional radial displacement (=/r ₀) - particle velocity - v non-dimensional particle velocity (= /C) - pressure - P(t) non-dimensional pressure (= /K)