Dedication to Prof. Dr. Otto H. Kegel

Dedication

Dedication to Prof. Dr. Otto H. Kegel     

Preface: An Appreciation of Professor Jose B. Cruz,Jr.

Preface

Preface: An Appreciation of Professor Jose B. Cruz, Jr.     

The Entropy in Learning Theory. Error Estimates

We continue the investigation of some problems in learning theory in the setting formulated by F. Cucker and S. Smale. The goal is to find an estimator on the base of given data that approximates well the regression function of an unknown Borel probability measure defined on We assume that belongs to a function class It is known from previous works that the behavior of the entropy numbers of in the uniform norm plays an important role in the above problem. The standard way of measuring the error between a target function and an estimator is to use the norm ( is the marginal probability measure on X generated by ). This method has been used in previous papers. We continue to use this method in this paper. The use of the norm in measuring the error has motivated us to study the case when we make an assumption on the entropy numbers of in the norm. This is the main new ingredient of thispaper. We construct good estimators in different settings: (1) we know both and ; (2) we know but we do not know and (3) we only know that is from a known collection of classes but we do not know An estimator from the third setting is called a universal estimator.     

Quasilinear parabolic P.D.E.s with discontinuous hysteresis

This work deals with an initial- and boundary-value problem for a quasilinear parabolic equation that includes a possibly discontinuous hysteresis operator, :

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

This issue is dedicated to Professor Paul L. Butzer on the occasion of his 70th Birthday

Dedication

This issue is dedicated to Professor Paul L. Butzer on the occasion of his 70th Birthday     

Sensitivity vs. block sensitivity of Boolean functions

Senstivity and block sensitivity are important measures of complexity of Boolean functions. In this note we exhibit a Boolean function ofn variables that has sensitivity and block sensitivity (n). This demonstrates a quadratic separation of the two measures.c/o László Babai     

Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

Metric Geometry in Homogeneous Spaces of the Unitary Group of a C*-Algebra. Part II. Geodesics Joining Fixed Endpoints

This article focuses on the study of the metric geometry of homogeneous spaces (the unitary group of a C*-algebra modulo the unitary group of a C*-subalgebra ) where the invariant Finsler metric in is induced by the quotient norm of Under the assumption that is of compact type, i.e. when the unitary group is relatively compact in the strong operator topology, this work presents local and global versions of Hopf-Rinow-like theorems: given points there exists a minimal uniparametric group curve joining ρ₀ and ρ₁.     

Congruence properties in congruence permutable and in ideal determined varieties, with applications.

We define a weak version of EDPC (equationally definable principal congruences), called EDPC*, that is shown to be preserved under varietal closure in congruence permutable varieties. We show that if is a congruence permutable variety generated by a class then has EDPC iff has EDPC* iff has EDPC*. An equational condition is given which, if satisfied by implies that has the CEP (congruence extension property). Similar results are proved for ideal determined varieties. These results are applied to the variety of residuated lattices, with examples.Received January 15, 2004; accepted in final form October 8, 2004.     

Semicontinuous Mappings into T.V.S. with Applications to Mixed Vector Variational-Like Inequalities

Let (X, Z) be the space of continuous linear mappings between topological vector spaces, where Z is Hausdorff and preordered by a closed convex cone C. In this paper, we introduce a notion of semicontinuity to any function from a topological space into X. A notion of semicontinuity is also introduced to any function from a topological space into (X, Z). These two notions of semicontinuity are related by the embedding of X into (X, Z). Their basic properties are given. As an application, we derive some existence results for the mixed vector variational-like inequality. This work was partially supported by grants from the National Science Council of the Republic of China.     

D. G. near rings on certain groups

All the distributive near rings in which the index ofR inR is a prime have been determined. Certain d. g. near rings in whichR is the only normal subgroup ofR are also considered. It is also shown that if (R,+) is finite and not perfect then there is a distributive near ring in whichR ²0.     

Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces

Book Review

Approximation Theory in the Central Limit Theorem. Exact Results in Banach SpacesV. Paulauskas and A. Rackauskas: (MIA Series), Kluwer Academic Publishers, Dordrecht, Boston, London, 1989, 176 pp., ISBN 90-277-2875-9, Dfl154.00/US$79.00.     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.     

Weighted sums of I.I.D. Random variables attracted to integrals of stable processes

Summary Under the assumption of the convergence of partial sum processesZ _n (t) of i.i.d. random variables to an -stable Lévy processZ()(t), 0<2, the convergence of weighted sums f _n (u)dZ _n (u) to f(u)dZ ₍₎(u) is studied. The general convergence result is then applied to examine the domain of attraction of the fractional stable process.Supported in part by Grant-in-Aid for Scientific Research, No. 60740103, Ministry of Education, Science and Culture     

On a problem of L. Leindler concerning strong approximation by Fourier series and Lipschitz classes

Пусть? — возрастающа я непрерывная фцнкци я на [0,π],?(0)=0 и $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ Положим $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ Доказывается следую щая теорема.Пусть f∈ С[?π, π], ω(f, δ)=О(?(δ))) и $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ для x∈E?[?π, π], ¦E¦>0. Тогда д ля сопряженной функц ии f почти всюду на E выполн яется соотношение $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\psi \left( {\left| h \right|} \right)}}\left| {\tilde f\left( {x + h} \right) - \tilde f\left( x \right)} \right| = 0.$$ Из этой теоремы вытек ает положительное ре шение одной задачи Л. Лейндлера.     

Bivariate distributions with Weibull conditionals

, . , - . , , , .     

Differential operators on homogeneous spaces. I

Summary In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation of the enveloping algebra of by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI _x of _X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI _x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map of the cotangent-bundle ofX onto a nilpotent variety in , as studied e.g. in [BM1]. We prove, for instance, that grI _x is prime if and only if _X is birational with normal image. In general, we show that is prime. Equivalently, the associated variety ofI _x in is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI _y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.     

Some results on geodesic mappings of Riemannian manifolds satisfying the conditionR.R.=Q(S,R)

In this paper geodesically corresponding metricsg and on a manifoldM, dim 5, under the assumption that the tensorsR andS of the metricg satisfyR.R=Q(S, R), are considered. It is stated that the corresponding tensors and of not necessarily must satisfy . Certain relations between the curvatures ofg and are obtained.Supported by a post-doctoral fellowship of the researchcouncil of the KU Leuven; Bitnet FGBDA3O at BLEKUL11     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .