Electronic Edition Vol. 35, No. 1, 2009

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), H. Brezis (Paris), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. Hausen (Houston), 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


Oman, Greg, Otterbein College, Mathematical Sciences Department, One Otterbein College, Westerville OH, 43081-2006 (ggoman@gmail.com).
Some Results on Jonsson Modules over a Commutative Ring, pp. 1-12.
Let M be an infinite unitary module over a commutative ring R with identity. M is called Jonsson over R provided every proper submodule of M has smaller cardinality than M; M is large if M has cardinality larger than R. Extending results of Gilmer and Heinzer, we prove that if M is Jonsson over R, then either M is isomorphic to R and R is a field, or M is a torsion module. We show that there are no large Jonsson modules of regular or singular strong limit cardinality. In particular, the Generalized Continuum Hypothesis (GCH) implies there are no large Jonsson modules. Necessary and sufficient conditions are given for an infinitely generated Jonsson module to be countable. As applications, we prove there are no large uniserial or Artinian modules. Under the GCH, we derive a new characterization of the quasi-cyclic groups.

Chapman, S. T., Sam Houston State University, Department of Mathematics and Statistics, Box2206 Huntsville, Texas 77341-2206 (scott.chapman@shsu.edu) ,  Schmid, W. A., Institut fuer Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universitaet Graz, Heinrichstrasse 36, 8010 Graz, Austria (wolfgang.schmid@uni-graz.at), and Smith, W. W., The University of North Carolina at Chapel Hill, Department of Mathematics, Phillips Hall, Chapel Hill, NC 27599, USA (wwsmith@email.unc.edu).
Quasi-half-factorial subsets of abelian torsion groups, pp. 13-22.
ABSTRACT. If G is an abelian torsion group with generating subset G0, then by a classical result in the theory of non-unique factorizations, the block monoid B(G0) is a half-factorial monoid if each of its atoms has cross number 1. In this case, G0 is called a half-factorial set. In this note, we introduce the notion of a k-quasi-half-factorial set and show for many abelian torsion groups that G0 k-quasi-half-factorial implies that G0 is half-factorial. We moreover show in general that G0 k-quasi-half-factorial implies that G0 is weakly half factorial, a condition which has been of interest in the recent literature.

May, Coy L., Department of Mathematics, Towson University, Towson, MD 21252 (cmay@towson.edu).
The Real Genus of Direct Products Zn× G, pp. 23-37.
ABSTRACT. Let G be a finite group. The real genus of G is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider the problem of finding the real genus of the direct product Zn× G , in which one factor is cyclic and the other factor G is a familiar group with well-known properties or, alternately, a group for which the real genus has already been determined. We focus on groups that are generated by two elements, one of which is an involution. Let G be a finite non-abelian group of this type, and assume the order n of the cyclic factor is relatively prime to |G|. Our main result is the determination of the real genus of this direct product in terms of n, |G|, and two parameters associated with the group G. We then give a range of applications of this result. Each genus formula yields a sequence of values for the real genus g such that there exists a group of genus g.

Meng Wu, Chern Institute of Mathematics, Nankai University, Tianjin, P. R. China 300071 (wumengzy@yahoo.com.cn, and Yunhui Wu, Department of Mathematics, #151 Thayer Street, Brown University, Providence, Rhode Island 02912 (yunhuiwuzy@yahoo.com.cn).
On the geometry of spheres with positive curvature, pp. 39-48.
ABSTRACT. Let M be an n-dimensional complete connected Riemannian manifold with sectional curvature bigger than or equal to 1 and diameter bigger than π/2, and N be a closed connected totally geodesic submanifold. In this paper we show that N is homeomorphic to a sphere if there exist a point x in N with rad(x) in M bigger than π/2. If we further assume that the radius of M is bigger than π/2, we give a two-dimensional example to show that the antipodal map A of M restricted to a complete totally geodesic submanifold may not agree with that of M.

Cordero, Luis A., Universidade de Santiago, 15782 Santiago de Compostela, Spain (cordero@zmat.usc.es), and Parker, Phillip E., Wichita State University, Wichita KS 67260-0033, USA.(phil@math.wichita.edu).
Isometry Groups of Pseudoriemannian 2-step Nilpotent Lie Groups, pp. 49-72.
ABSTRACT.  While still a semidirect product, the isometry group can be strictly larger than the obvious Riemannian analogue I aut. In fact, there are three relevant groups of isometries, I splI autI, and I spl < I aut < is possible when the center is degenerate. When the center is nondegenerate, I spl = I.

Lev Birbrair, Departamento de Matematia, Universidade Federa do Ceara, Av. Humberto Monte, s/n Campus do Pici - Bloco 914, CEP: 60.455-760 Fortaleca-CE, Brazil (birb@ufc.br) and Dirk Siersma, Department of Mathematics, Utrecht University P.O.Box 80.010, 3508 TA Utrecht, The Netherlands (siersma@math.uu.nl).
Metric properties of conflict sets, pp. 73-80.
ABSTRACT. In this paper In this paper we show that the tangent cone of a conflict set in Euclidean space is a linear affine cone over a conflict set of a smaller dimension. We prove that the conflict sets in the Euclidean plane have no cuspidal singularities. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone.

Guoxin Wei, Department of Mathematics, Saga University, Saga, 840-8502, Japan (weigx03@mails.tsinghua.edu.cn).
New examples of Willmore hypersurfaces in a sphere, pp. 81-92.
ABSTRACT.  It is well known that any totally geodesic hypersurface is a Willmore hypersurface. Li and Vrancken (2003) got some new examples of Willmore surfaces in a sphere. Guo, Li and Wang (2001) obtained Willmore tori. In this paper, we find some new examples of Willmore hypersurfaces in a sphere.

Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea (yjsuh@knu.ac.kr) and Guoxin Wei, Department of Mathematics, Saga University, Saga, 840-8502, Japan (weigx03@mails.tsinghua.edu.cn).
Complete spacelike hypersurfaces in anti-De Sitter space H1n+1(-1), pp. 93-102.
ABSTRACT. In this paper, by investigating complete k-maximal spacelike hypersurfaces Mn in anti-de Sitter space  H1n+1(-1) with two distinct principal curvatures, we give a characterization of a hyperbolic cylinder H1(- n\ k})× Hn-1(- n\ (n-k)).

Kraszewski, Jan, Mathematical Institute, Wroclaw University, pl. Grunwaldzki 2/4, 50-156 Wroclaw, Poland (kraszew@math.uni.wroc.pl)
Everywhere meagre and everywhere null sets , pp. 103-111.
ABSTRACT. We introduce new classes of small subsets of the reals (connected with Mycielski ideals), having natural combinatorial definitions, namely everywhere meagre and everywhere null sets. We investigate properties of these sets, in particular we show that these classes are closed under taking products and projections. We also prove several relations between these classes and other well-known classes of small subsets of the reals (e.g. universally meagre sets or strongly meagre sets).

Oleksandr V. Maslyuchenko, 58012, vul. Kotsyubyns'koho, 2, Chernivtsi National University, Department of Mathematical Analysis, Chernivtsi, Ukraine (omasl@ukr.net).
The oscillation of quasi-continuous functions on pairwise attainable spaces, pp. 113-130.
ABSTRACT. We introduce a class of attainable and pairwise attainable spaces which includes metrizable spaces. On this classes we develop a new technique of constructing of functions with given oscillations. We characterize the oscillations of quasi-continuous functions. In particular, we prove that the function g defined on a pairwise attainable space is the oscillation of some quasi-continuous function f if and only if g is nonnegative upper semi-continuous function with a meager support.

Hiroshi, Hosokawa, Tokyo Gakugei University, Nukuikita-machi, Koganei-shi,Tokyo, 184, Japan (hosokawa@u-gakugei.ac.jp).
Mutual aposyndesis of n-fold hyperspaces, pp. 131-137.
ABSTRACT. A continuum is a compact connected metric space. The n-fold hyperspace of a continuum X is the space of all nonempty closed subsets of X with at most n components. We show that if n>1, then for any continuum X, the n-fold hyperspace of X is mutually aposyndetic.

Shou Lin, Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, P. R. China; Institute of Mathematics, Ningde Teachers College, Ningde 352100, P. R. China (linshou@public.ndptt.fj.cn), Zhangyong Cai, Department of Mathematics and computer science, Guangxi Teachers College, Nanning 530023, P. R. China (zycaigxu2002@126.com) and Chuan Liu, Department of Mathematics, Ohio University-Zanesville, Ohio, OH 43701, USA (liuc1@ohio.edu)
The closed mappings on k-semistratifiable spaces pp. 139-147.
ABSTRACT. In 1999, Y. Tanaka and C. Liu posed the following question: Let f be a closed mapping from a topological space X onto a topological space Y. Under what conditions on X or Y the boundary of each fiber of the mapping f has some nice properties? In this paper it was shown that if X is a k-, and k-semistratifiable space and Y is some special spaces then the boundary of each fiber of the mapping f is a Lindelöf subset of a compact subset. This improves some results about closed mappings on generalized metric spaces.

Alas, Ofelia T., Universidade de São Paulo, 05311-970 São Paulo, Brasil (alas@ime.usp.br), Tkachenko, Michael G., Universidad Autónoma Metropolitana, 09340, México D. F., México (mich@xanum.uam.mx), and Wilson, Richard G., Universidad Autónoma Metropolitana, 09340, México D. F., México (rgw@xanum.uam.mx).
Which topologies have immediate predecessors in the poset of Hausdorff topologies? , pp. 149-158.
ABSTRACT. It is known that each topology on a set X which is not H-closed has an immediate predecessor in the poset of Hausdorff topologies on X. In this paper, we show that all submaximal H-closed topologies which are not minimal Hausdorff, as well as certain classes of dispersed H-closed topologies, also have such predecessors. We give examples of second countable H-closed topologies which are not upper in this poset, answering in the negative a question of N. Carlson (2007).

Michael Lacey, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 USA (lacey@math.gatech.edu) and Erin Terwilleger, Department of Mathematics, U-3009, University of Connecticut, Storrs, CT 06269 USA (terwilleger@math.uconn.edu).
Hankel operators in several complex variables and product BMO, pp. 159-183.
ABSTRACT. We consider the Hilbertian Hardy space on the product of tori, and the "little" Hankel operator which maps these spaces into themselves. These operators are indexed by their symbol b, a function that can be taken to be jointly analytic in all the variables present. We show that these operators are bounded iff the symbol b is in Chang-Fefferman BMO, genearlizing the classical result of Nehari, and a recent result of S. Ferguson and one of the authors on the bi-disk. This result is proved by an argument which inducts on the number of complex variables present. This is is carried out with a particular form of a Journe Lemma, which occurs implicitly in the work of Pipher.

Jia-feng, Lü, College of Mathematics and Physics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China (jiafenglv@126.com)
On modules with piecewise-Koszul towers, pp. 185-207.
ABSTRACT. The concept of (strongly) weakly piecewise-Koszul module is introduced. Let A be a piecewise-Koszul algebra and M a finitely generated graded A-module. Then M is weakly piecewise-Koszul if and only if M admits a tower of piecewise-Koszul modules. As applications of the approximation chain, we show that the finitistic dimension conjecture is true in the category of weakly piecewise-Koszul modules, and if M is (strongly) weakly piecewise-Koszul, then the Koszul dual of M is not only finitely generated, but also generated in degree zero. In particular, if M is perfect, then M is strongly piecewise-Koszul if and only if the Koszul dual of M is finitely generated and generated in degree zero. Furthermore, we show that M is weakly piecewise-Koszul if and only if the Koszul dual of G(M) is finitely generated and generated in degree zero.

Alexey I. Popov, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G,2G1. Canada (apopov@math.ualberta.ca)
Schreier singular operators, pp. 209-222.
ABSTRACT. In this paper we further investigate recently introduced Schreier singular operators. We show that the class of Schreier singular operators is stable under left and right multiplication by bounded operators. We also present a characterization of Schreier singular operators in terms of spreading models.
Androulakis, Dodos, Sirotkin, and Troitsky (Israel J. Math, to appear) showed that under cerain conditions on the space the product of sufficiently many Schreier singular operators is compact. We extend this result to a broader class of spaces. Finally, we show that this cannot be extended to arbitrary Banach spaces by presenting an example of a finitely strictly singular operator which is not even polynomially compact.

Goblet, Jordan, Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium (goblet@math.ucl.ac.be).
Lipschitz extension of multiple Banach-valued functions in the sense of Almgren, pp. 223-231.
ABSTRACT. For any positive integer Q, a Q(Y)-valued function f on X is essentially a rule assigning Q unordered and not necessarily distinct elements of Y to each element of X. Equivalently f maps X into the space Q(Y) of Q unordered points in Y. We study the Lipschitz extension problem in this context by using two general Lipschitz extension theorems recently proved by U. Lang and T. Schlichenmaier. We prove that the pair (X,Q(Y)) has the Lipschitz extension property if Y is a Banach space and X is a metric space with a finite Nagata dimension. We also show that Q(Y) is an absolute Lipschitz retract if Y is a finite algebraic dimensional Banach space.

Yan, Qiming, Department of Mathematics, Fudan University, Shanghai 200433, P.R. China, (yan_qiming@hotmail.com), and Chen, Zhihua, Department of Mathmetics, Tongji University, Shanghai 200092, P.R. China, (zzzhhc@tongji.edu.cn).
Some results on the uniqueness theorem of meromorphic mappings pp. 233-246.
ABSTRACT. In this article, some results about the uniqueness theorem of meromorphic mappings in several complex variables are proved.

Park, Young Ja, Department of Mathematics, Hoseo University, Asan 336-795, South Korea  (ypark@hoseo.edu).
An Extremal function for Sobolev trace inequality, pp. 247-251
ABSTRACT. We explain how an extremal function for the Sobolev trace inequality might be conjectured. We prove that this function is indeed an actual minimizer when we consider the space of functions which share values on the boundary with the conjectured function.

Daws, Matthew, School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom (matt.daws@cantab.net),  Le Pham, Hung, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2E1, Canada (hlpham@math.ualberta.ca) and White, Stuart, Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, United Kingdom (s.white@maths.gla.ac.uk).
Conditions implying the uniqueness of the weak *-topology on certain group algebras, pp. 253-276.
ABSTRACT. We investigate possible preduals of the measure algebra M(G) of a locally compact group and the Fourier algebra A(G) of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak*-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals C0(G) of M(G) and C*(G) of A(G) are uniquely determined. In both cases we consider a natural comultiplication and show that the canonical predual gives rise to the unique weak*-topology making both the multiplication separately weak*-continuous and the comultiplication weak*-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure.

Qi Han, Department of Mathematics, the University of Houston, Houston, TX 77204 (k.l.han@hotmail.com).
On complex analytic solutions of the partial differential equation (uz1)m+(uz2)m = um , pp. 277-289.
ABSTRACT. In this paper, we study the characterization and classification of the left-factors (in the sense of composition of meromorphic functions of several complex variables in the factorization theory) of the entire solutions of certain homogenous first-order partial differential equations, which is complemented by several examples to show the accuracy. Further, we give the complex analytic, i.e., entire or meromorphic, solutions of those equations for the case n=2.

Long Wei, The School of Science, Hangzhou Dianzi University,Xiasha Hangzhou Zhejiang 310018, China (alongwei@163.com).
On the number of nodal bubbling solutions to a sinh-Poisson equation, pp. 291-326.
ABSTRACT. We show that for ε>0 small, there exist arbitrarily many nodal solutions for the semi-linear equation ∆u + 2ε2sinh u=0 posed on a bounded smooth domain Ω in R2 with homogeneous Neumann boundary condition. More precisely, for ε sufficiently small and any given positive integers L ≥ 1, there exists a family of nodal solution uε that develops 2L boundary singularities and which with the property 2ε2   sinh u → 4 π Σ2Lj=1 (-1) j-1δξ ,where ( ξ1,···,ξ 2L) are critical points of some functional defined explicitly in terms of the associated Green's function. This solution has at least L+1 nodal domains. No assumption on the geometry, nor the topology of the domain is needed.

Songxiao Li, Department of Mathematics, JiaYing University, 514015, Meizhou, GuangDong, China (jyulsx@163.com), and Stevo Stevic Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia (sstevic@ptt.rs).
Composition followed by differentiation between H and α-Bloch spaces, pp.327-340.
ABSTRACT. The boundedness and compactness of the product of differentiation and composition operators between the space of bounded analytic functions on the unit disk and the Bloch-type space are discussed in this paper.