HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 37, No. 4 , 2011

Editors: G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), D. Blecher (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (Radboud), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. A. Johnson (Houston), W. B. Johnson (College Station),  V. I. Paulsen (Houston), M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

 Houston Journal of Mathematics


Contents

A. Benobaid, Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia, (g200502650@kfupm.edu.sa) and A. Mimouni, Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, P. O. Box 278, Dhahran 31261, Saudi Arabia (amimouni@kfupm.edu.sa).
Compact and coprime packedness with respect to star operations, pp. 1043-1061.
ABSTRACT  In this paper, we will study the notions of compactly and coprimely packed rings with respect to a star operation of finite type. We extend well-known results and investigate more properties of these notions in different settings such as Prufer-like settings, Noetherian-like settings and pullbacks. Particular attention is paid to the t-operation as the often usual star operation of finite type. Examples to illustrate the scopes and limits of the results are constructed.

Krzywkowski, Marcin, Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland (marcin.krzywkowski@gmail.com).
Hat problem on odd cycles, pp. 1063-1069.
ABSTRACT. The topic is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of a win. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The hat problem on a graph was solved for trees and for the cycle on four vertices. Then Uriel Feige conjectured that for any graph the maximum chance of success in the hat problem is equal to the maximum chance of success for the hat problem on the maximum clique in the graph. He provided several results that support this conjecture, and solved the hat problem for bipartite graphs and planar graphs containing a triangle. We make a step towards proving the conjecture of Feige. We solve the hat problem on all cycles of odd length. Of course, the maximum chance of success for the hat problem on the cycle on three vertices is three fourths. We prove that the hat number of every odd cycle of length at least five is one half, which is consistent with the conjecture of Feige.
Mathematica Code

Huang, Libing, School of Mathematics Sciences, Nankai University, Tianjin 300071, P. R. China (huanglb@nankai.edu.cn).
Einstein Finsler metrics on S3 with nonconstant flag curvature, pp. 1071-1086.
ABSTRACT.  It is a well-known result in Riemannian geometry that a three-manifold with constant Ricci curvature must be of constant sectional curvature. But in Finsler geometry, this fact may not be true. In this paper, we constructed a two parameter family of almost regular Finsler metrics on S3. They have constant Ricci curvature +1, but their flag curvatures are nonconstant.

Marian Ioan Munteanu, Faculty of Mathematics, University Alexandru Ioan Cuza of Iasi, Bd. Carol I, n. 11, 700506 - Iaşi, Romania (munteanu@uaic.ro), Ana-Irina Nistor, Faculty of Mathematics, University Alexandru Ioan Cuza of Iasi, Bd. Carol I, n. 11, 700506 - Iaşi, Romania (ana.irina.nistor@gmail.com).
On the geometry of the second fundamental form of translation surfaces in E3, pp. 1087-1102.
ABSTRACT. In this paper we study the second fundamental form of translation surfaces in the 3-dimensional Euclidean space.A non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII is given. A classification of translation surfaces for which KII and the mean curvature H are proportional is obtained.

M. Crampin, Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B--9000 Gent, Belgium  (crampin@btinternet.com).
On Landsberg spaces and the Landsberg-Berwald problem, pp. 1103-1124.
ABSTRACT.  This paper is concerned with the geometry of a class of Finsler spaces called Landsberg spaces. A Landsberg space may be characterized by the fact that its fundamental tensor is covariant constant along horizontal curves with respect to its Berwald connection. A Finsler space whose Berwald connection is affine is called a Berwald space. Berwald spaces are necessarily Landsbergian, but whether there are y-global Landsberg spaces which are not of Berwald type is not known. Resolving this question is the Landsberg-Berwald problem of the title. The paper deals with several topics in Landsberg geometry which are related mainly by the possibility that the results obtained may throw light on the Landsberg-Berwald problem. It is assumed throughout that the dimension of the base manifold is at least 3. It is shown that a Landsberg space over a compact base, which is R-quadratic, is necessarily Berwaldian. A model for the holonomy algebra of a Landsberg space is proposed. Finally, the technique of averaging the fundamental tensor over the indicatrix is discussed, and it is shown that for a Landsberg space, with the correct interpretations, the averaged Berwald connection is the Levi-Civita connection of the averaged metric.

Cho, Jong-Taek, Department of Mathematics, Chonnam National University, CNU, The Institute of Basic Science, Kwangju, 500-757, Korea (jtcho@chonnam.ac.kr), and Inoguchi, Jun-ichi, Department of Mathematics Education, Utsunomiya University, Utsunomiya, 321-8505, Japan (inoguchi@sci.kj.yamagata-u.ac.jp).
Curvatures and symmetries of tangent sphere bundles, pp. 1125-1142.
ABSTRACT. This paper has two purposes. (1) Holomorphic sectional curvature and ξ-sectional curvature of tangent sphere bundles are investigated. In particular, tangent sphere bundles of constant holomorphic sectional curvature or of constant ξ-sectional curvature are classified. (2) Hypersurface geometry of tangent sphere bundles is developed. Tangent sphere bundles with pseudo-parallel shape operator or η-parallel shape operator are classified.

Robert L. Moore and T.T. Trent, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350 (rmoore@gp.as.ua.edu).
Co-rank one interpolation in CSL algebras, pp. 1143-1156.
ABSTRACT. The interpolation problem for CSL algebras is this: Given a CSL algebra AlgL, and given two operators X and Y, does there exist an operator A in AlgL such that AX=Y? We discuss this problem in the case where X has co-rank one and give necessary and sufficient conditions for the existence of A.

Choi, Changsun, KAIST, Daejeon, Daejeon 305-701, Korea (cschoi@kaist.ac.kr), and Kim, Ju Myung, Pohang Mathematics Institute, San 31 Hyoja Dong, Nam-Gu, Pohang, Gyeong-Buk, 790-784, Korea (kjm21@postech.ac.kr).
Hahn-Banach theorem for the compact convergence topology and applications to approximation properties, pp. 1157-1164.
ABSTRACT. Let X and Y be Banach spaces. We show that if X*  or Y* has the Radon-Nikodym property and S is a space of weak* to weak continuous compact operators from X* into Y, then given ε>0 every norm continuous linear functional φ on S can be extended to a linear functional ψ on B(X*,Y), without increasing the norm more than ε so that ψ is continuous with respect to the compact convergence topology. From this result we obtain various applications.

Ricard, Éric,Université de Franche-Comté, 25030 Besançon Cedex, France, and Xu, Quanhua,Université de Franche-Comté, 25030 Besançon Cedex, France (quanhua.xu@univ-fcomte.fr).
Complex interpolation of weighted noncommutative Lp-spaces, pp. 1165-1179.
ABSTRACT. Let M be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace τ. Let d be an injective positive measurable operator with respect to (M,τ) such that d-1 is also measurable. Define
Lp(d)={x in L0(M) | dx+xd in  Lp(d)(M)} and ||x||Lp(d)=||dx+xd||p.
We show that for 1≤p0<p1≤∞, 0<θ<1 and α0≥0, α1≥0, the interpolation equality (Lp0(dα0),Lp1(dα1))θ =Lp(dα) holds with equivalent norms, where 1/p=(1-θ)/p0+θ/p1 and α=(1-θ)α0+θα1.

Shinji Yamashita, Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, JAPAN (s-yamashita@math.kyushu-u.ac.jp).
Circle correspondence C^-algebras , pp. 1181-1202.
ABSTRACT.  We investigate Cuntz-Pimsner C*-algebras associated with certain correspondences of the unit circle. We analyze these C*-algebras by analogy with irrational rotation algebras and Cuntz algebras. We study the fixed point algebras of certain actions of finite groups, and calculate the entropy of a certain endomorphism. We also study the induced map of the dual action of the gauge action on K-groups.

Elke Wolf, Mathematical Institute University of Paderborn, Warbuger Str. 100, 33098 Paderborn Germany (lichte@math.uni-paderborn.de).
Compact differences of weighted composition operators between weighted Bergman spaces of infinite order, pp. 1203-1209.
ABSTRACT. We investigate the relation between the compactness of differences of weighted composition operators and the compactness of the involved single weighted composition operators in the setting of weighted Bergman spaces of infinite order.

Ryle, Julie, The University of Alabama, Tuscaloosa AL 35487 (julieryle@gmail.com), and Trent, Tavan, The University of Alabama, Tuscaloosa AL 35487 (ttrent@as.ua.edu). 
A Corona Theorem for certain subalgebras of H∞(D), pp. 1211-1226.
ABSTRACT. Consider algebras of the form C+BH∞(D), where B is a Blaschke product, H(D) denotes the bounded analytic functions on the unit disk, and C denotes the complex numbers. We answer a conjecture of Mortini, Sasane, and Wick by showing that the Corona theorem holds for infinitely many functions in these algebras.

Gournay, Antoine, Institut de Mathématiques, Université de Neuchâtel, Rue É.-Argand 11, CH-2000 Neuchâtel, Switzerland (antoine.gournay@unine.ch).
Widths of lp-balls, pp. 1227-1248.
ABSTRACT. We say a map ƒ: X→ Y is an ε-embedding if it is continuous and the diameter of the fibers is less than ε. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), an(X). It is the smallest real number such that there exists an ε-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. Following Gromov, we take the slightly different point of view by looking at the smallest dimension n for which there exists a ε-embedding to a polyhedron of dimension n. Our bounds are obtained using Hadamard matrices, the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of (n+1) points not contained in a hemisphere (obtained by methods very close to those of Ivanov and Pichugov). We are also able to give a complete description in dimension 3 for 1≤p≤2.

Qingzhai Fan, Department of Mathematics, Shanghai Maritime University, 1550 Haigang Ave. in New Harbor City, Shanghai 201306, China (qzfan@shmtu.edu.cn), and Xiaochun Fang, Department of mathematics, Tongji University, Shanghai, 200092, China  (xfang@tongji.edu.cn).
Non-simple tracial approximation, pp. 1249-1263.
ABSTRACT. We show that the following properties of C*-algebras in the class Ω are inherited by C*-algebras in the class TAΩ:  (1) K1-surjective property, (2) K1-injective property, (3)Stable weak cancelation property, (4) Stable finite property, (5) Having at least one tracial state.

Ding, Juntang, School of Mathematical Sciences, Shanxi University, Taiyuan 030006, P.R. China, and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa  (djuntang@sxu.edu.cn), and Guo, Bao-Zhu, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100190, P.R. China, and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa (bzguo@iss.ac.cn) .
Global and blow-up solutions for nonlinear parabolic equations with a gradient term, pp. 1265-1277.
ABSTRACT.  In this paper, we are concerned with the a kind of nonlinear parabolic equations with a gradient term and Neumann boundary condition in a bounded domain in N-dimensional Euclidean space with smooth boundary. The upper and lower solution technique is adopted in investigations. The sufficient conditions for the existence of global positive solution and an upper estimate of global solution are given. Moreover, under some appropriate assumptions on the coefficients functions, we prove the existence of blow-up positive solution. An upper bound of “blow-up time” is also presented

Ehsani, Dariush, (dehsani.math@gmail.com)
Boundary value problems on product domains, pp. 1279-1295.
ABSTRACT. We consider the inhomogeneous Dirichlet problem on product domains. The main result is the asymptotic expansion of the solution in terms of increasing smoothness up to the boundary. In particular, we show the exact nature of the singularities of the solution at singularities of the boundary by constructing singular functions which make up an asymptotic expansion of the solution.

Sergio Macias, Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Ciudad Universitaria, Mexico D. F., C. P. 04510, Mexico  (macias@servidor.unam.mx).
On the idempotency of the set function T, pp. 1297-1305.
ABSTRACT. In 1980, Bellamy asked: If X and Y are indecomposable continua, is T idempotent on their product? Even for only the closed sets of their product? (see Problem 164 of the Houston Problem Book). We present a negative answer to the first question by showing that the set function T is not idempotent on the product of two continua one of which is indecomposable. In particular, T is not idempotent on the product of two indecomposable continua. The second question remains open.

Cencelj, Matija, University of Ljubljana, Faculty of Education, Kardeljeva pl. 16, SI-1000 Ljubljana, Slovenia (matija.cencelj@guest.arnes.si), Dydak, Jerzy, University of Tennessee, Knoxville, TN 37996, USA, (dydak@math.utk.edu), Smrekar, Jaka, University of Ljubljana, FMF, Jadranska 19, SI-1000, Ljubljana, Slovenia, (jaka.smrekar@fmf.uni-lj.si), and Vavpetič, Aleš, University of Ljubljana, FMF, Jadranska 19, SI-1000, Ljubljana, Slovenia (ales.vavpetic@fmf.uni-lj.si).
Sublinear Higson corona and Lipschitz extensions, pp. 1307-1322.
ABSTRACT. We show that the dimension of the sublinear Higson corona of a metric space X is the smallest non-negative integer m with the following property: Any norm-preserving asymptotically Lipschitz function from a closed subset A of X to the Euclidean space of dimension m+1 extends to a norm-preserving asymptotically Lipschitz function from X to the Euclidean space of dimension m+1. As an application we obtain another proof of the following result of Dranishnikov and Smith: Let X be a cocompact proper metric space, which is M-connected for some M, and has the asymptotic Assouad-Nagata dimension finite. Then this dimension equals the dimension of the sublinear Higson corona of X.

Van Nall, Dept. of Mathematics, University of Richmond, Richmond, VA 23173 (vnall@richmond.edu).
Inverse limits with set valued functions, pp. 1323-1332.
ABSTRACT. We begin to answer the question of which continua can be homeomorphic to an inverse limit with a single upper semi-continuous bonding map from [0,1] to the set of closed subsets of [0,1]. Several continua including [0,1]×[0,1] and all compact manifolds with dimension greater than one cannot be homeomorphic to such an inverse limit. It is also shown that if the upper semi-continuous bonding maps have only zero dimensional pointvalues, then the dimension of the inverse limit does not exceed the dimension of the factor spaces.

Alejandro Illanes, Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Cuidad Universitaria, México D.F., 04510, México (illanes@matem.unam.mx) and  Martínez-Montejano, Jorge M., Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Cuidad Universitaria, México D.F., 04510, México (jorge@matematicas.unam.mx).
Zero-dimensional closed set aposyndesis and symmetric Products, pp.1333-1346.
ABSTRACT. A continuum is a compact, connected, metric space. It is said that a continuum X is zero-dimensional closed set aposyndetic provided that for each zero-dimensional closed subset A of X and for each p in X minus A, there exists a subcontinuum M of X such that p is in the interior of M and M does not intersect A. In this paper we show that if X is a continuum and n is greater or equal to two, then the n-fold symmetric product of X is zero-dimensional closed set aposyndetic.

Gerald Beer, Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, California 90032, USA (gbeer@cslanet.calstatela.edu), Camillo Costantini, Dipartimento di Matematica dell' Univerita di Torino, Via Carlo Alberto 10, 10123 Torino, Italy (camillo.costantini@unito.it), and Sandro Levi, Dipartimento di Matematica e Applicazioni, Universita' di Milano-Bicocca, ViaCozzi 53, 20125 Milano, Italy (sandro.levi@unimib.it)
Total boundedness in metrizable spaces, pp. 1347-1362.
ABSTRACT. We show that a metric space (X,d ) is separable if and only if the bornology of its d-bounded subsets agrees with the bornology of ρ-totally bounded subsets with respect to some equivalent remetrization ρ. We also show that the bornology of d-totally bounded subsets agrees with the bornology of ρ-bounded subsets with respect to some equivalent remetrization if and only if the former bornology has a countable cofinal subfamily. Finally, we characterize those bornologies on a metrizable space that are bornologies of totally bounded sets as determined by some metric compatible with the topology.

Er-Guang Yang, School of Mathematical Sciences, Nanjing Normal University,Nanjing 210046, P.R. China; School of Mathematics & Physics, Anhui University of Technology, Maanshan 243002, P.R. China (egyang@126.com), and Wei-Xue Shi Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China (wxshi@nju.edu.cn)
Strong developments and metrization theorems, pp. 1363-1372.
ABSTRACT. We present characterizations of a strongly developable space interms of g-functions from which we deduce some criteria for the metrizability of a topological space, and with these criteria we give short or direct proofs of some metrization theorems that appeared in the literature.

Alas, Ofélia. T., Instituto de Matemática e Estatística (IME-USP), Universidade de São Paulo, São Paulo, SP - CEP 05508-090 - Brazil (alas@ime.usp.br),  Aurichi, Leandro F., Instituto de Ciências Matemáticas e de Computação (ICMC-USP), Universidade de São Paulo, São Carlos, SP - CEP 13566-590 - Brazil (aurichi@icmc.usp.br),  Junqueira, Lúcia R. Instituto de Matemática e Estatística (IME-USP), Universidade de São Paulo, São Paulo, SP - CEP 05508-090 - Brazil. (lucia@ime.usp.br), and Tall, Franklin D. Department of Mathematics, University of Toronto, Toronto, Ontario - M5S 2E4 - Canada  (f.tall@utoronto.ca).  
Non-productively Lindelöf spaces and small cardinals, pp.1373-1381.
ABSTRACT. In this paper we concentrate on finding conditions for a Lindelöf space to not be productively Lindelöf , using small cardinals.