On the divergence of spherical sums of double Fourier-Haar series

. : [0, +) [0, +) - , u+ (u) (u)=o(u lnu). [0, 1]² f , ¦f¦ L([0, 1]²), - [0, 1]².     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net of ideals of R such that, where E() is the subgroup generated by all transvections of the net subgroup G(). and is the normalizer of G(). The subgroup E() is normal in. To study the factor group we introduce an intermediate subgroup F(), E() F() G(). The group is finite and is connected with permutations in the symmetric group. The factor group G()/F() is Abelian — these are the values of a certain determinant. In the calculation of F()/E() appears the SK₁-functor. Results are stated without proof.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 13–20, 1979.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

The convergence rates of empirical Bayes estimation in a multiple linear regression model

Empirical Bayes (EB) estimation of the parameter vector =(,²) in a multiple linear regression modelY=X+ is considered, where is the vector of regression coefficient, N(0,² I) and ² is unknown. In this paper, we have constructed the EB estimators of by using the kernel estimation of multivariate density function and its partial derivatives. Under suitable conditions it is shown that the convergence rates of the EB estimators areO(n ^{-(k-1)(k-2)/k(2k+p+1)}), where the natural numberk3, 1/3<<1, andp is the dimension of vector .The project is supported by the National Natural Science Foundation of China.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

A Stability Estimate for a Solution to the Problem of Determination of Two Coefficients of a Hyperbolic Equation

We consider the problem of determination of two coefficients (x) and q(x) in a hyperbolic equation. Here (x) is the coefficient of the first derivative with respect to t and q(x) is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk D. Oscillations are excited by the impulse function (t)(x·) supported on the straight line t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside D is measured at the points of the boundary of D together with the normal derivative on some time interval of a fixed length T for two different values of the parameter . We prove that, for a sufficiently large T, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.     

Topological influence of the presence of fully-λ-bases in locally convex spaces

This article reveals the topological impact of fully--bases in locally convex spaces where carries either the traditional normal topology or the fairly generalized-topology of Ruckle. It has been established that the generalized nuclearity of plays a significant role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully--base ( being equipped with normal topology or-topology) has been investigated.We acknowledge with thanks the suggestions of the referee.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

Lacunary (0; 0, 1) interpolation on the roots of Jacobi polynomials and their derivatives,respectively. I (Existence,explicit formulae,unicity)

(0; 0, 1) , {x _k <x _k ^* <x _k+1} _k=1 ^n–1 {x _{k k=1} ⁿ }., I, , _n (x)=P _n ^{(, )} (x)–n- , =, n3 . , x ₀=+1 x _n+1= –1. II .

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To the memory of Paul Erds

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The research was supported by the Hungarian National Foundation for Scientific Research under Grant # T 914 244.     

Irregular but nonfractal drums andn-dimensional weyl conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set R ⁿ (n 1) with irregular but nonfractal boundary. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN _i ()(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ _i ()=() –N _i () for large, where() is the Weyl term. This implies that for the irregular but nonfractal drum , not only the volume || _n is spectral invariant but also the area of boundary || _n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.     

On the angular derivative and univalence

f . , , — , A f f(). , , f() 0 . , , ,A , f . , f() - f() . , , . (1976) ( ¦f(z)¦<1) . . (1969) ( ).     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Projektive G-Faserräume

Let > 0 be an integer. A projective -fibre space is formed by a covering of a projective geometry with -1 isomorphic geometries. The double elliptic space (Sphärischer Raum) is an example of a 2-fibre space. This note deals with projective -fibre spaces which are structured by multy valued orderfunctions (This notion was introduced by W. JUNKERS [2] for projective geometries) the range of which is a group G. If such an ordered -fibre space has the property ¦G¦=, it is called projective G-fibre space. It is proved that the desarguesian projective G-fibre spaces V are exactly those, which are induced by a vector space S over a field K (commutative or not) having a normal subgroup P K^*(·) with K^* P such that GK^*/P and SV^*/P. This theorem is a generalization of the well-known case P=K^*.