*Editors*: G. Auchmuty (Houston), D. Bao (San Francisco,
SFSU), D. Blecher (Houston), Bernhard G. Bodmann, H. Brezis (Paris and Rutgers),
B. Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M.
Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori
(Matsue, Shimane), W. B. Johnson (College Station), M. Rojas (College Station),
Min Ru (Houston), S.W. Semmes (Rice).

*Managing Editor*: K. Kaiser (Houston)

Houston Journal of Mathematics

*Contents*

**Estrada, Sergio,** University of Murcia, Murcia, Spain
(sestrada@um.es), and **Özdemir, Salahattin,** Dokuz Eylül University, Izmir, Turkey
(salahattin.ozdemir@deu.edu.tr).

Relative homological algebra in categories of representations of infinite quivers,
pp. 343-362.

ABSTRACT. In the first part of this paper, we
prove the existence of torsion free covers in the category of representations of
quivers for a wide class of quivers included in the class of the so-called
source injective representation quivers, provided that any direct sum of torsion
free and injective modules is injective. In the second part, we prove the
existence of Ƒ_{cw}-covers for any quiver Q and any ring R with unity, where Ƒ_{cw} is the class of all “componentwise” flat representations of Q.

**Gyu Whan Chang,** Department of Mathematics, University of Incheon,
Incheon 406-772, Korea,
(whan@incheon.ac.kr).

Prüfer v-multiplication domains and valuation ideals, pp. 363-371.

ABSTRACT. Let D be an integral domain, and let t be the (so-called) t-operation on D .
A nonzero ideal I of D is called a w-ideal if I = {x ∈ D| xJ ⊆ I
for some nonzero finitely generated ideal J of D with J_{t} = D}.
In this paper, we introduce the notion of t -valuation ideals, and we then prove that D is a
Prüfer v-multiplication domain if and only if each w -ideal of
D is an intersection of t-valuation ideals. We also prove that
D is a ring of Krull type if and only if each w-ideal of
D is a finite intersection of t-valuation ideals. As a corollary, we have
that a Noetherian domain D is a Krull domain if and only if each primary
w-ideal of D is a t-valuation ideal.

Mendes, Carla,
Centro de Matemática, Universidade do Minho,
4710-057 Braga, Portugal
(cmendes@math.uminho.pt).

Fixed points in MS_{n}-algebras, pp. 373-386.

ABSTRACT.
A MS_{n}-algebra, n ∈ N, is an Ockham algebra (L,∧,∨,f,0,1)
satisfying x ≤ f^{2}(x). The class of all MS_{n}-algebras is
a variety denoted by **MS _{n}**. For each
subvariety

**B. Goldsmith,** School of Mathematical Sciences, Dublin Institute of Technology, Aungier
Street, Dublin 2, Ireland, (brendan.goldsmith@dit.ie)) and
**P. Zanardo,** Dipartimento di Matematica Pura e Applicata, University
of Padova, Via Trieste 63, 35121 Padova, Italy (pzanardo@math.unipd.it).

On Maximal relatively divisible submodules, pp. 387-494.

ABSTRACT.
A torsion-free module M over an integral domain R has relatively divisible (RD-) submodules which are maximal with respect to inclusion. There are situations in which the number of non-isomorphic maximal RD-submodules is small; Goebel and Goldsmith have investigated this and related questions in the context of Abelian groups. We address corresponding problems for modules over arbitrary domains. We obtain results relating to the level of coherency of a ring R, and establish connections between the level of coherency and the minimum number of generators of RD-submodules of a given R-module. Under some natural restrictions, we prove that an R-module G, all of whose maximal RD-submodules are isomorphic to a fixed free module X of infinite rank, is itself free. We investigate R-modules G all of whose maximal RD-submodules are isomorphic to a direct product, over an infinite set I, of copies of R. We first show that, for any slender integral domain R, that such a direct product has infinitely many non-isomorphic maximal RD-submodules. Moreover, when R is a slender valuation domain, whose field of fractions has projective dimension 1 and G is an R-module with all maximal RD-submodules isomorphic to a direct product, over an infinite set I, of copies of R , we prove that G itself is isomorphic to this direct product. Consequently, in a wide range of situations no such module can exist, for instance if R is either a maximal Pruefer domain or a discrete valuation domain.

**Manjra, Said,** Department of Mathematics, Imam University.
P.O.Box: 240337. Riyadh 11322. Saudi Arabia.
(smanjra@uottawa.ca).

An arithmetic study of the formal Laplace transform in several variables,
pp. 405-426.

ABSTRACT.
Let K be a number field, and let
K(x_{1},...,x_{d}) be the field of rational fractions in the variables x_{1},...,x_{d}. In this paper, we introduce two kinds of Laplace transform adapted to solutions of the differential K(x_{1},...,x_{d})-modules with regular singularities, and give some of their basic differential and arithmetic properties.
The purpose of this article is to provide some tools which might be useful, in particular, for the arithmetic study of the differential K(x_{1},...,x_{d})-modules associated to E-functions in several variables.

On trees with double domination number equal to 2-domination number plus one, pp. 427-440.

ABSTRACT. .A vertex of a graph is said to dominate itself and all of its neighbors. A subset D ⊆ V(G) is a 2 -dominating set of G if every vertex of V(G)\ D is dominated by at least two vertices of D , while it is a double dominating set of G if every vertex of G is dominated by at least two vertices of D . The 2 -domination (double domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (double dominating, respectively) set of G . We characterize all trees with the double domination number equal to the 2 -domination number plus one.

**Ayman Badawi,** American Univ of Sharjah, Department of Mathematics, Box 26666, Sharjah, UAE ( abadawi@aus.edu
)and **Ahmad Yousefian Darani,** Department of Mathematics, University of Mohaghegh Ardabili, P. O. Box 179, Ardabil, Iran (yousefian@uma.ac.ir
).

On weakly 2-absorbing ideal of commutative rings,
pp. 441-452.

ABSTRACT. Let R be a commutative ring with identity 1 not equal to 0. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper
ideal I of R is called a weakly 2-absorbing ideal of R if whenever abc is not equal to 0 for some a, b, c in R and abc
is in I, then
ab is in I or ac is in I or bc is in I. For example, every proper ideal of a quasi-local ring
(R,M) with M^{3} equals {0} is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal
I of R with I^{3} not equal to {0} is a 2-absorbing ideal of R. We show that every proper ideal
of a commutative ring R is a weakly 2-absorbing ideal if and only if
either R is a quasi-local ring with maximal ideal M such that M^{3} equals {0} or
R is ring-isomorphic to (R_{1} × F) where R_{1} is a quasi-local ring with maximal ideal M such that M^{2}
equals {0} and F is a field or R is ring-isomorphic to (F_{1} × F_{2} × F_{3}) for some fields F_{1}, F_{2}, F_{3}.

**Dube, Themba,** Department of Mathematical Sciences, University of South
Africa, PO Box 392, 0003 Unisa, South Africa (dubeta@unisa.ac.za),
and **Naidoo, Inderasan,** Department of Mathematical Sciences, University of
South Africa, PO Box 392, 0003 Unisa, South Africa
(naidoi@unisa.ac.za).

Round squares in the category of frames, pp. 453-473.

ABSTRACT. A commutative square in the category Frm of frames and their homomorphisms will
be called round if there are parallel morphisms which when replaced with their
right adjoints yield a commutative diagram, albeit not necessarily in Frm. We
consider several cases of round squares and characterize compact, Lindelöf,
realcompact and paracompact frames in terms of round squares. We also show that
a strong nearness frame is complete if and only if any completable uniform
homomorphism into it gives rise to a round square when lifted to completions.

**Yanchang Chen**, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, P. R. China
and College of Mathematics and Information Science, Henan Normal University,
Xinxiang 453007, P. R. China**Yanying Wang**, College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, P. R. China
(wyanying2003@yahoo.com.cn).

Ring structures of rational equivariant cohomology rings and ring homomorphisms between them, pp. 475-485.

ABSTRACT. We give an explicit description of rational equivariant cohomology of G-equivariantly formal manifolds in terms of algebra, where G is the circle group. This makes it possible to determine the number of equivariant cohomology rings (up to isomorphism) of such 2- and 4-dimensional G-manifolds. Moreover, we obtain an analytic description of the ring homomorphism between equivariant cohomology rings of such two G-manifolds induced by a G-equivariant map.

Effective

ABSTRACT.Given a finite subset σ of the unit disc

**M. Essmaili, **Faculty of Mathematical and Computer Science, Tarbiat Moallem University, 50 Taleghani Avenue, 15618 Tehran,
Iran (m.essmaili@tmu.ac.ir) and **M. Filali,** Department of Mathematical Sciences,
University of Oulu, P.O. Box 3000, Oulu 90014, Finland (mahmoud.filali@oulu.fi).

Φ-amenability and character amenability of some classes of Banach algebras, pp. 515-529.

ABSTRACT.We study the notions of -amenability and character amenability of the semigroup algebra
1^{1}(S), where S is a semilattice. We also consider the character amenability of semigroup algebras of
uniformly locally finite inverse semigroups. As a consequence, we characterize the character amenability
of Brandt semigroup algebras. Moreover, we study these notions for module extension of Banach algebras
with application to semigroup algebras.

**Forrest, Brian E.,** University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (beforres@math.uwaterloo.ca),
**Lee, Hun Hee**, Department of Mathematical Sciences, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea
(hunheelee@snu.ac.kr),
and **Samei, Ebrahim, **
University of Saskatchewan, Saskatoon, SK S7N 5E6, Canada (samei@math.usask.ca).

Projectivity of modules over Segal algebras, pp. 531-560.

ABSTRACT. In this paper we will study the projectivity of various natural modules associated to operator Segal algebras of the Fourier algebra of a locally compact group G. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces.
We show that projectivity often implies that the underlying group is discrete or even finite.
We will also look at the projectivity for modules of A_{cb}(G), the closure of A(G) in the space of its completely bounded multipliers. Here we give some evidence to show that weak amenability of G plays an important role.

**Kunal Mukherjee,** Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai - 600113, India (kunal@imsc.res.in).

Singular masas and measure-multiplicity invariant, pp. 561-598.

ABSTRACT. This paper studies singular masas in finite von Neumann algebras from a bimodule point of view.
The structure of the standard Hilbert space as the natural bimodule over a masa is characterized by a measure class
(left-right measure) and a multiplicity function. This paper focuses on relations between properties of a
singular masa and its left-right measure. Part of the analysis is mainly concerned with masas for which
the left-right measure is the class of product measure. An example of a Tauer masa whose left-right measure
is the class of product measure is presented. It is shown that for each nonempty subset S of the extended
natural numbers which contains infinity, there exist uncountably many pairwise non conjugate singular masas in
the free group factors whose Pukánszky invariant is S.

**Chunfang Shen,** Department of Mathematics, Hefei Normal University, Hefei, Anhui Province, PR China, 230061 (xjiangfeng@163.com).

Three positive solutions for non-linear multi-point boundary value problem with nonlinearity depending on all order derivatives, pp. 599-609.

ABSTRACT. In this paper, by using the Avery and Peterson fixed point theorem, existence of at least three positive solutions for the third-order multi-point boundary value problem with nonlinearity depending on all order derivative are established. An example is given to illustrate the main results.

**Chunnuan Cheng** and **Yan Xu**, Institute of Mathematics, School of Mathematics, Nanjing Normal University,
Nanjing 210046, P.R.China (chengchunnuan@126.com),
(xuyan@njnu.edu.cn).

Normal families of meromorphic functions concerning differential polynomials , pp. 611-623.

ABSTRACT. Huang and Gu proved that a family F of meromorphic functions in a domain D is
normal, if, for two analytic functions a(z) (not equivalent to 0), b(z) in D and all f in F, (1) f(z) ≠ ∞ when a(z)=0 ; (2) f'(z)-a(z)f^{2}(z) ≠ b(z); (3) all poles of f(z) are of multiplicity at least 4. In this paper, we first give an example to show that condition (3) is sharp, and prove that our
counterexample is unique in some sense. Also, two normality criteria are given, which extend the result of Huang and Gu.
.

**Jie Zhang, **College of Science, China University of
Mining and Technology, Xuzhou 221116, PR China
(zhangjie1981@cumt.edu.cn).

Existence of entire solution of some certain type difference equation, pp. 625-635.

ABSTRACT. In this paper, we investigate the existence and growth of the solutions of
some difference or differential equations by utilizing Nevanlinna theory and
Wiman-Valiron theorem.

**Das, Pratulananda,** Jadavpur University, Kolkata - 700032, West Bengal, India
(pratulananda@yahoo.co.in).

Certain types of open covers and selection principles using ideals, pp. 637-650.

ABSTRACT. In this note we make a new and very general approach to the
study of open covers and selection principles by using the very general notion
of ideals and investigate some of its consequences. Our results present a more
general form of some very recent statistical variants of open covers and related
selection principles introduced by Di Maio and Kocinac (G. Di Maio, Lj.D.R.
Kocinak,
Statistical convergence in topology, Topology Appl., 156 (2008),

28 - 45.).

** Charatonik, Wlodzimierz J.,** Missouri University of Science
and Technology, Department of Mathematics and Statistics, 65409
Rolla, MO, USA (wjcharat@mst.edu),
**Wright, Evan P.,** Department of Mathematics, State University
of New York at Stony Brook, Stony Brook, NY 11794, USA
(evanpw@math.sunysb.edu), and
**Zafiridou, Sophia S., ** Department of Mathematics,
University of Patras, 26500 Patras, Greece
(zafeirid@math.upatras.gr).

Dendrites with a countable set of end points and
universality, pp. 651-666.

ABSTRACT. We introduce a notion of ramification degree for dendrites and we
use it to show that in the family of all dendrites with a countable
set of end points, there is no universal element. Moreover, we
characterize a dendrite with a countable
set of end points using this
ramification degree. Finally, we investigate the problem of
existence of minimal dendrite in some families of dendrites with a
countable set of end points.

**Brodskiy, Nikolay**, Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996
(brodskiy@math.utk.edu), **Dydak, Jerzy**, Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996 (dydak@@math.utk.edu),** Labuz, Brendon,** Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996
(labuz@math.utk.edu), **Mitra, Atish,** Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996
(ajmitra@math.utk.edu).

Rips complexes and covers in the uniform category, pp. 667-699.

ABSTRACT. James introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut introduced a theory of covers for uniform spaces generalizing their results for topological groups. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section 7, the paper of Berestovskii and Plaut generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions.
In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc. One way to do it is by embedding X in a space with good local properties and this is done in Section 6. Another way is by systematic use of Rips complexes. In the topological category one uses paths in X originating from a base point to construct the universal covering space X̃. We use paths in Rips complexes and their homotopy classes possess a natural uniform structure, a generalization of the basic topology on X̃. Applying Rips complexes leads to a natural class of uniform spaces for which our theory of covering maps works as well as the classical one, namely the class of locally uniform joinable spaces. In the case of metric continua (compact and connected metric spaces) that class is identical with pointed 1-movable spaces, a well-understood class of spaces introduced by shape theorists. The class of pointed 1-movable continua contains all planar subcontinua (examples: Hawaiian Earring and the suspension of the Cantor set) and is preserved by continuous maps. The most notable continuum not being pointed 1-movable is the dyadic solenoid. As an application of our results we present an exposition of Prajs' homogeneous curve that is path-connected but not locally connected.

**Sergey Antonyan,** Univesrsidad Nacional Autonoma de Mexico, Mexico City, 04510 Mexico
(antonyan@unam.mx).

Locally compact subgroup actions on topological groups, pp. 701-716.

ABSTRACT. Let X be a Hausdorff topological group
and G a locally compact subgroup of X.

We show that X admits a locally finite
sigma-discrete G-functionally open cover each member of which is G-homeomorphic
to a twisted product.

If, in addition, the space of connected components of G
is compact and X is normal, then X itself is G-homeomorphic to a twisted
product. This implies that X is homeomorphic to the product of a Euclidean space
and a normal space.

Using these results we prove the inequality dim X ≤
dim X/G + dim G for every Hausdorff topological group X and a locally compact
subgroup G of X.

**Michalik, Daria,** Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszynski University, Woycickiego 1/3, 01-938 Warsaw, Poland
(d.michalik@uksw.edu.pl).

Embedding and factorization properties of the product of generalized Sierpiński curves, pp. 717-732.

ABSTRACT. We prove that there is the universal space for the class of n-dimensional metric spaces in the Cartesian
product Σ(τ)^{n+1} of generalized Sierpiński curves. It is also shown that the set of
embeddings of any n-dimensional
metric space X into this universal space is a residual set in C(X,Σ(τ)^{n+1}).
Other properties of product of generalized Sierpinski curves are also proved.