Strongly meager and strong measure zero sets

 In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID="ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658 Received: 6 January 1999 / Revised version: 20 July 1999 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID=" ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658     

On linear independence of the values of generalized Heine series

 A linear independence measure is obtained for values of solutions of a certain system of functional equations. This result is then applied to a rather general class of q–hypergeometric series, for example to the values of q–analogues of exponential and Bessel functions at several algebraic points. Received: 18 October 2000 / Revised version: 2 August 2001 / Published online: 16 October 2002 RID="★" ID="★" The author is grateful to Alexander von Humboldt Foundation for support and to the Department of Mathematics of the University of Cologne for the kind hospitality. He also thanks Peter Bundschuh for many useful discussions.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

A new proof of regularity of weak solutions of the <Emphasis Type="Italic">H</Emphasis>-surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions of the H-surface system are locally H?lder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Hélein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in , this time assuming in addition that decays at infinity like . Received: 10 May 2001 / Accepted: 7 June 2001 / Published online: 18 January 2002 The author gratefully acknowledges the generous support of Alexander von Humboldt Foundation, and the hospitality of Mathematisches Institut der Universit?t Bonn, where this research has been carried out. In particular, many thanks are due to Professor Stefan Hildebrandt.     

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small . Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002 RID="*" ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China     

Higman’s question revisited

We construct a two-generated group with the co-recursively enumerable word problem that has no presentation by recursive permutations. This answers Higman’s question and exemplifies a group with the minimal possible number of generators. The previous article [1], in which that question was claimed settled, contains an incorrigible error. Dedicated to the 60th birthday of Academician Yu. L. Ershov Supported by the Alexander von Humboldt Foundation. This result was obtained during my work at the University of Heidelberg (Germany) as Alexander von Humboldt Research Fellow. The term a “II-group” was proposed by A. Nies. Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 134–144, March–April, 2000.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Distortion theorems for locally univalent Bloch functions

A holomorphic functionf defined on the unit disk d is called a Bloch function provided {fx73-02} For α ∃ (0,1] letB∞(α)denote the class of locally univalent Bloch functionsf normalized by ∥f∥_B ≤1f(0) = 0 andf’(0) = α. A type of subordination theorem is established for B∞(α). This subordination theorem is used to derive sharp growth, distortion, curvature and covering theorems for B∞(α). Supported as a Feodor Lynen Fellow of the Alexander von Humboldt Foundation. Research supported in part by a National Science Foundation grant.     

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.     

On crossing numbers of hypercubes and cube connected cycles

We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs.The research of both authors was supported by Alexander von Humboldt Foundation, Bonn, Germany.     

New central polynomials for the matrix algebra

Forn≥3 we find a central polynomial of degree (n−1)²+4 for then×n matrix algebra over a field of characteristic 0. Forn=3,4 our polynomial coincides with the known central polynomials of minimal degree and forn>4 the result gives new central polynomials. Until now, forn>4 the minimal degree of the known central polynomials wasn ². Partially supported by the Alexander von Humboldt Foundation in Germany and by Grant MM2/91 of the Ministry of Education and Science in Bulgaria.     

One- and two-term edgeworth expansions for a finite population sample mean. Exact results. I

We prove the validity of one- and two-term Edgeworth expansions under optimal conditions (a Cramer-type smoothness condition and the minimal moment conditions) and provide precise bounds for the remainders of expansions. The bounds depend explicitly on the ratiop=N/n, whereN andn denote the sample size and the population size, respectively. Supported by the Alexander von Humboldt Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 277–294, July–September, 2000.     

Lipschitz conditions in conformally invariant metrics

We consider two conformally invariant metrics in proper subdomains of euclideann-spaceR ⁿ. We show that Lipschitz mappings in these metrics include the class of quasiconformal mappings as a proper subclass, yet these Lipschitz mappings have many properties similar to those of quasiconformal mappings. Research supported in part by the U.S. National Science Foundation and the A.P. Sloan Foundation. Research supported in part by the Alexander von Humboldt Foundation.     

Numerical simulation of a modified KdV equation on the whole real axis

The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme.Supported by the National Natural Science Foundation of China under Grant No. 10401020, the Alexander von Humboldt Foundation, and the Key Project of China High Performance Scientific Computation Research.     

Approximation of p-multiplier Operators via their Spectral Projections

Conditions for a p-multiplier $\psi: {\mathbb{Z}} \to {\mathbb{C}}Conditions for a p-multiplier are presented which ensure that the corresponding operator T_ψ, acting in , can be approximated by linear combinations of p-multiplier projections coming from the uniform operator closed, unital algebra of operators generated by T_ψ. Functions of bounded variation on play an important role, as do certain Λ (p)-sets. Dedicated to the memory of H. H. Schaefer Werner J. Ricker: Former Alexander von Humboldt Fellow at the Universit?t Tübingen, hosted by Prof. H.H. Schaefer from Sept. 1987 – Feb. 1988.     

Dimensions of splines on 0-stars inR 3

In this note some recurrence relations for the dimensions of the spline spaces on so called 0-stars are presented. The work of this author is supported by the Alexander von Humboldt Foundation This paper is completed at Free Univ. Berlin, Inst. of Math., Arnimallee 2–6, 1000 Berlin 33, Germany     

On the existence of closed timelike geodesics in compact spacetimes. II

 The purpose of this paper is on the one hand to extend and generalize, in terms of Clifford translations, some results in a previous paper (Math. Z. 239 (2002), 277–291) concerning the existence of closed timelike geodesics in compact spacetimes, and on the other hand to prove that a compact flat spacetime (M, g) contains a closed timelike geodesic if and only if the fundamental group π₁(M) contains a non-trivial timelike translation. Received: 22 January 2002; in final form: 12 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 53C50, 53C22.     

A generalization of the Dual Ellentuck Theorem

 We prove versions of the Dual Ramsey Theorem and the Dual Ellentuck Theorem for families of partitions which are defined in terms of games. Received: 8 July 1999 Published online: 19 December 2002 RID="*" ID="*" The author wishes to thank the Swiss National Science Foundation for supporting him. The authors thank the referee for helpful comments. Mathematics Subject Classification (2000): 03E02, 05D10, 03E35 Key words or phrases: Dual Ramsey Theorem – Dual Ellentuck Theorem – Partitions – Games     

Quantization of curvature for compact surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n⋆</Emphasis></Superscript>

For minimal surfaces in spheres, there is a well known conjecture about the quantization of intrinsic curvature which has been solved only in special cases so far. We recall an intrinsic and an extrinsic version for the known results and extend them to compact non-minimal surfaces in spheres. In particular we discuss special classes like Willmore surfaces and surfaces with parallel mean curvature vector. Mathematics Subject Classification (2000):53C42, 53A10.H.Li is partially supported by a research fellowship of the Alexander von Humboldt Stiftung 2001/2002 and the Zhongdian grant of NSFC. U. Simon is partially supported by DFG 163/Si-7-2 and a Chinese–German research cooperation of NSFC and DFG.