*Editors*: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao
(San Francisco, SFSU), H. Brezis (Paris), K. Davidson (Waterloo), M. Gehrke (Radboud), 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

** Buckner, Joshua **and ** Dugas, Manfred, ** Department of Mathematics, Baylor University, Waco, TX 76798
(manfred_dugas@baylor.edu).

Group algebras with Zassenhaus families of right
ideals, pp. 691-703.

ABSTRACT.
Let K be a field and A a K-algebra.
For a family F of right ideals of A we define
End_{S}(A, F) = {φ in End_{S}(A) : φ(X) in X
for all X in F}. We call F a Zassenhaus family
if A = End_{S}(A, F). We investigate the following question: For
which fields K and groups G does the group algebra K[G] have a
Zassenhaus family?

**Rüdiger Göbel,** Universität Duisburg-Essen, FB Mathematik, 45117 Essen, Germany
(ruediger.goebel@uni-due.de) and **Agnes T. Paras,** Institute of Mathematics, University of the Philippines at Diliman, 1101 Quezon City, Philippines
(agnes@math.upd.edu.ph).

Decompositions of reflexive groups and Martin's Axiom
, pp. 705-718.

ABSTRACT.
We begin with the answer to a problem of Gabriel Sabbagh given by Eklof-Shelah in 1987:

For any natural number m not 1, there is a locally free abelian group G (of cardinality the continuum) such that for all positive integers n, G is isomorphic to the direct sum of G and a free abelian group of rank n if and only if n is a multiple of m.
It is the aim of this paper to replace locally free by reflexive. Recall that a group G is reflexive if G is isomorphic by the evaluation map to its double-dual G** and note that G* = Hom(G,Z) where Z is the ring of integers. The existence of such group G (a pure subgroup of the Baer-Specker group) follows with the help of Martin's axiom (MA); in particular it holds under CH. The proof uses ideas from the Eklof-Shelah paper as well as methods from a more recent paper by Göbel-Shelah (2001) which establishes under MA the existence of a reflexive group G not isomorphic to the direct sum of G and Z. In all such situations it is
unknown if the result holds in ordinary ZFC.

**Ben Abdallah, Mohamed Jaouhar**,
Faculty of Sciences of Gafsa, Sidi Ahmed Zarroug, Gafsa 2112, Tunisia (mouhammad.jaouhar@gmail.com)
and **Jarboui, Noômen**, Faculty of Sciences of Sfax, Route
Soukra, Sfax 3018, Tunisia (noomenjarboui@yahoo.fr).

A note on the ring D[tu^{n}, n≥0], pp. 719-724.

ABSTRACT. Let D be a Noetherian domain. The main purpose of this paper is to prove that D[tu^{n}, n≥0] is a universally catenarian domain if and only if D is universally catenarian.

On modules which are self-slender, pp. 725-736.

ABSTRACT. This paper is an examination of the dual of the fundamental isomorphism relating homomorphism groups involving direct sums and direct products over arbitrary index sets. Recall that a module G is said to be self-slender if every homomorphism from a countable product of copies of G into G, vanishes on all but finitely many of the components of the product. Modules of this type are investigated. The simplest version of the results obtained is that under weak cardinality restrictions, there exist non-slender but self-slender Abelian groups.

**H. Ahmad, **(deceased), and **N. Colwell,
**Department of Mathematics, Saginaw Valley State University
University Center, MI 48710 (nccolwel@svsu.edu).

Factorization patterns of polynomials, pp. 737-744.

ABSTRACT. For a UFD, an algorithm is introduced that will efficiently produce the factorization pattern of a given polynomial of content 1. We also show that for some UFDs, like the rational integers, the factorization pattern can be
obtained from the polynomial values.

**Bahman Khosravi, **Department of Mathematics, Shahid Beheshti University, Tehran, Iran,
(khosravibahman@yahoo.com).

On Cayley graphs of left groups, pp. 745-755.

ABSTRACT. A. V. Kelarev and C. E. Praeger in [A. V. Kelarev and C. E. Praeger, On transitive Cayley graphs
of groups and semigroups, European J. Combin., 24 (2003),
59-72] gave necessary
and sufficient conditions for Cayley graphs of semigroups to be
vertex-transitive. Also S. Fan and Y. Zeng in [S. Fan and Y. Zeng, On Cayley graphs of Bands,
Semigroup Forum, 74 (2007), 99-105] gave a
description of all vertex-transitive Cayley graphs of finite
bands. In this paper we give similar descriptions for all
vertex-transitive Cayley graphs of left groups. Also we extend
some of the results to every direct product of a group and a band.

**Pitkethly, Jane,** La Trobe University, Victoria 3086, Australia
(J.Pitkethly@latrobe.edu.au).

A full duality that cannot be upgraded to a strong duality,
pp. 757-774.

ABSTRACT.
We solve a variant of the Full Versus Strong Problem of natural duality theory, by giving an example of a three-element algebra that is fully dualisable but not strongly dualisable. This example also shows that there is no general "Strong Duality Compactness Theorem".

**Ta Thi Hoai An**, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi, VietNam (tthan@math.ac.vn)
and **Ha Tran Phuong, **Department of Mathematics, Thai Nguyen University
of Education (hatranphuong@yahoo.com).

An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space, pp.
775-786.

ABSTRACT. Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation.

**Hu, Shengda,** Department de Mathematiques et de Statistique, Universite de Montreal, CP 6128 succ
Centre-Ville, Montreal, QC H3C 3J7, Canada
(shengda@dms.umontreal.ca).

Hamiltonian symmetries and reduction in generalized geometry , pp. 787-811.

ABSTRACT. A closed 3-form H in Ω^{3}_{0}(M) defines an
extension
of
Γ(TM) by Ω^{2}_{0}(M). This fact leads to
the
definition
of
the group of H-twisted Hamiltonian symmetries Ham(M, J; H) as
well as Hamiltonian action of Lie group and moment map in the category
of (twisted) generalized complex manifold. The Hamiltonian reduction in
the category of generalized complex geometry is then constructed. The
definitions and constructions are natural extensions of the
corresponding ones in the symplectic geometry. We describe cutting in
generalized complex geometry to show that it's a general phenomenon in
generalized geometry that topology change is often accompanied by
twisting (class) change.

**Vajiac, Mihaela**, Department of Mathematics and Computer Science, Chapman University, Orange, CA, 92866
(mbvajiac@chapman.edu).

Quantum-type products in symplectic geometry,
pp. 813-827.

ABSTRACT.
In this note gauge theory techniques and the theory of flat connections are used to show that the small quantum product is a deformation of the cup product on a symplectic manifold in a gauge theoretical sense. We also construct and analyze a space of products on the even cohomology space of the symplectic manifold which have the same properties as the quantum product (associative, commutative, Frobenius, and have unit).

**Masatomo Takahashi, **Muroran Institute of Tecnology, Muroran 050-8585,
JAPAN
(masatomo@mmm.muroran-it.ac.jp)

A sufficient condition that contact equivalence implies right equivalence for smooth function germs,
pp. 829-833.

ABSTRACT. If two smooth function germs are right equivalent, they are also contact equivalent.
In this note, we shall show as an application of singularity theory that if they are quasihomogeneous, then the converse also holds.

**Jaume Llibre,** Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalunya, Spain
(jllibre@mat.uab.cat) and **Víctor F. Sirvent,** Departamento de Matemáticas, Universidad Simón Bolívar, Caracas 1086-A, Venezuela (vsirvent@usb.ve).

Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces
pp. 835-855.

ABSTRACT.
We study the minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces; in fact our study extends to the diffeomorphisms
on these surfaces having all the periodic points hyperbolic and the same action
on the homology as the Morse-Smale diffeomorphisms.
We provide explicit information about these minimal sets of periods for the Morse--Smale diffeomorphisms on orientable
compact surfaces of genus 0, 1, 2 and 3. Additionally we give an algorithm which
allows the study of these minimal sets of periods for the Morse-Smale diffeomorphisms on orientable compact surfaces of arbitrary genus.

*Addendum: *A correction was received December 2, 2009. See
Vol36-1.html

**Garcia-Ferreira,** Salvador, Instituto de Matematicas (UNAM), Apartado
Postal 61-3, Xangari 58089, Morelia, Michoacan, Mexico
(sgarcia@matmor.unam.mx) and **
Gutev, Valentin,** School of Mathematical Sciences, University of
KwaZulu-Natal, Westville Campus, Private Bag X54001, Durban 4000, South Africa
(gutev@ukzn.ac.za).

Baire property and web-adjacent spaces, pp. 857-875.

ABSTRACT.
We introduce the concept of web-adjacent spaces which is motivated by
a construction of Krom. The purpose of the paper is to generalize a
result of Krom's by showing that if two spaces are web-adjacent, then
the one is a Baire space if and only if the another is a Baire space.
Further, we demonstrate that for two web-adjacent spaces the
corresponding Vietoris hyperspaces are also web-adjacent, consequently
the one Vietoris hyperspace is Baire if and only if the another is
Baire. This refines the answer to a question of McCoy.

**Yun Ziqiu,** Department of Mathematics, Suzhou University, 215006 P. R.China (yunziqiu@public1.sz.js.cn).

On sequential order of product spaces, pp. 877-890.

ABSTRACT.
In this paper, we give a canonical example of sequential spaces whose sequential order is greater than α for each α< ω1. As applications, some results about sequential order of product spaces are obtained and a question raised by T. Nagura and A. Shibakov is answered.

**Hirata, Yasushi, **Graduate School of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan
(yhira@jb3.so-net.ne.jp).

The collectionwise Hausdorff property of products of two or three subspaces of ordinals, pp. 891-901.

ABSTRACT.
It is known that every finite power of ω_{1} is hereditarily collectionwise Hausdorff. And it is easy to see that there is a subspace of (ω +1)×(ω_{1} +1) which is not collectionwise Hausdorff. We will prove that every product space of two subspaces of ordinals is collectionwise Hausdorff, but there is a product space of three subspaces of ordinals which is not collectionwise Hausdorff.

**Cynthia Farthing,** Department of Mathematics, Creighton University, 2500 California Plaza, Omaha, NE 68178, USA
(CynthiaFarthing@creighton.edu),
**David Pask,** School of Mathematics and Applied Statistics,
Austin Keane Building (15),
University of Wollongong,
NSW 2522, AUSTRALIA (dpask@uow.edu.au), and
**Aidan Sims,** School of Mathematics and Applied Statistics
Austin Keane Building (15), University of Wollongong
NSW 2522 AUSTRALIA (asims@uow.edu.au) .

Crossed products of k-graph C*-algebras by Z^{1}, pp.
903-933.

ABSTRACT. An action of Z^{l} by automorphisms of a k-graph induces an action of Z^{l} by automorphisms of the corresponding k-graph C^{*}-algebra. We show how to construct a (k+l)-graph whose C^{*}-algebra coincides with the crossed product of the original k-graph C^{*}-algebra by Z^{l}. We then investigate the structure of the crossed-product C^{*}-algebra.

Antonio
Aviles, University of Paris 7, Equipe de
Logique Mathématique, UFR de Mathematiques, 2 Place Jussieu, 75251 Paris,
FRANCE (avileslo@um.es),
(aviles@logique.jussieu.fr).

Automatic norm continuity of weak* homeomorphisms, pp. 935-943.

ABSTRACT.
We prove that in a certain class of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak* topology. Therefore, whithin this class, every weak* homeomorphism of dual balls is automatically norm continuous. The nonseparable versions of some of the classical sequence spaces belong to this class.

**Pradipta Bandyopadhyay**, Stat-Math Division, Indian Statistical
Institute, 202, B. T. Road, Kolkata 700 108, India
(pradipta@isical.ac.in) and **S. Dutta,**
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur
2008016, India (sudipta@iitk.ac.in).

Almost Constrained Subspaces of Banach spaces-II, pp. 945-957.

ABSTRACT.
A subspace Y of a Banach space X is an almost constrained (AC) subspace if any family of closed balls centred at points of Y that intersects in X also intersects in Y. In this paper, we show that a subspace H of finite codimension in C(K), the space of continuous functions on a compact Hausdorff space K, is an AC-subspace if and only if H is the range of a norm one projection in C(K). We also give a simple proof that the implication “AC → 1-complemented” holds for any subspace of c_{0}(Γ) and c.

**Astashkin, Sergei**, Department of Mathematics and Mechanics, Samara
State University, Acad. Pavlov,1
Samara 443011, Russia (astashkn@ssu.samara.ru),
** Semenov, Evgenii, ** Department of Mathematics, Voronezh State
University, Universitetskaya
pl.1, Voronezh 394006, Russia (semenov@func.vsu.ru)
and **Sukochev, Fedor**, School of Mathematics and Statistics, University of New South Wales, Kensington NSW 2052, Australia
(f.sukochev@unsw.edu.au).

Banach-Saks type properties in rearrangement- invariant spaces with the Kruglov property,
pp. 959-973

ABSTRACT.
We study Banach-Saks index sets for rearrangement-invariant (r.i.) function spaces. The main focus is on the class of r.i. spaces whose (classical) Banach-Saks index set is trivial, but for which the counterpart computed for weakly null sequences of independent random variables is not trivial. We discover and exploit an interesting connection with the Kruglov property in r.i. spaces.

Bianca Stroffolini
and Anna Verde, Dipartimento di
Matematica, Università di Napoli, Federico II, Via Cintia, I-80126 Napoli
(Italy) (bstroffo@unina.it),
(anverde@unina.it).

X-quasiconvexity in Carnot Groups and lower semicontinuity results, pp. 975-990.

ABSTRACT.
We characterize lower semicontinuity of integral functionals of the calculus of variations in the
setting of Carnot Groups. Accordingly, we introduce the notion of
X-quasiconvexity, that is referred to the family of Hörmander vector fields,
associated to the Group.