*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.

ABSTRACT.
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 G_{0},
then by a classical result in the theory of non-unique factorizations,
the block monoid B(G_{0}) is a half-factorial monoid if each of
its atoms has cross number 1. In this case, G_{0}
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 G_{0} k-quasi-half-factorial
implies that G_{0} is half-factorial. We moreover show
in general that G_{0} k-quasi-half-factorial implies
that G_{0} 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 Z_{n}× 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 Z_{n}× 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,

**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.

Complete spacelike hypersurfaces in anti-De Sitter space H

ABSTRACT. In this paper, by investigating complete k-maximal spacelike hypersurfaces M

**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
C_{0}(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 (u_{z1})^{m}+(u_{z2})^{m} =
u_{m }, 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 + 2ε^{2}sinh
u=0 posed on a bounded smooth domain Ω in R^{2}
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 u_{ε} that
develops 2L boundary singularities and which with the property 2ε^{2 }∫_{Ω} sinh u → 4 π Σ^{2L}_{j=1}_{ }(-1) ^{j-1}δ_{ξ
}_{j },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.