*Editors*: D. Bao (San Francisco, SFSU), D. Blecher
(Houston), B. G. Bodmann (Houston), 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 Editors*: B. G. Bodmann and K. Kaiser (Houston)

Houston Journal of Mathematics

*Contents*

**Ulrich Albrecht,** Department of Mathematics, Auburn University, Auburn, AL
36849, U.S.A. (albreuf@mail.auburn.edu) and
**Gregg Scible,** Department of Mathematics, Valencia College, Orlando, FL
32811, U.S.A. (gscible@mail.valenciacollege.edu).

On the number of generators of submodules of Q^{r}(R), pp. 1-13.

ABSTRACT. This paper investigates maximal S-closed submodules of non-singular modules over a right non-singular ring R. In particular, it addresses the question when the projectivity of all such submodules forces the module itself to be projective. Related question concerning non-commutative chain domains are discussed.

**Anderson, D.D.,** Department of Mathematics, The University of Iowa,
Iowa City, IA 52242 (dan-anderson@uiowa.edu), and **Chun, Sangmin,**
Department of Mathematics, Seoul National University, Seoul 151-747, Republic of
Korea (schun@snu.ac.kr).

Ideals that are an irredundant
union of principal ideals, II, pp. 15-29

ABSTRACT. We investigate ideals of a commutative ring that are an irredundant
union of principal ideals. We show that in an atomic ring every prime ideal is
an irredundant union of principal ideals, but give an example of a non-atomic
ring in which every prime ideal is an irredundant union of principal ideals.
Special attention is paid to prime ideals that are a finite union of principal
ideals.

**Greg Oman,** Department of Mathematics, University of
Colorado, Colorado Springs, Colorado Springs, CO 80918
(goman@uccs.edu) and **Adam Salminen,
**Department of Mathematics, University of Evansville, Evansville IN
47722 (as341@evansville.edu).

Polynomial and power series rings with finite quotients, pp. 31-38

ABSTRACT. We determine the rings R with the property that the quotient ring R[X]/I
(respectively, R[[X]]/I) is finite for every nonzero ideal I of the polynomial
ring R[X] (respectively, of the power series ring R[[X]]). We also classify the
rings R such that R[X]/I (R[[X]]/I) is a finite left R[X]-module (R[[X]]-module)
for every nonzero left ideal I of R[X] (of R[[X]]).

**Maloney, Gregory R.**, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK
(gregory.maloney@ncl.ac.uk).

The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one, pp. 39-60.

ABSTRACT. Let G be an ordered group that is a direct sum of a rank-one torsion-free abelian group and a finite-rank torsion-free abelian group, with order structure arising from the natural order on the first summand. A necessary condition and a sufficient condition are given for G to have an ordered-group inductive limit representation using injective maps.

**Coykendall, Jim,**
Clemson University, Clemson, SC, 29634-0975
(jcoyken@clemson.edu), **Malcolmson, Peter,** Dept. of Mathematics, Wayne State University, Detroit, MI 48202-3622
(petem@math.wayne.edu), and **Okoh, Frank**, Dept. of Mathematics, Wayne State University, Detroit, MI 48202-3622
(okoh@math.wayne.edu).

Inert primes and factorization in extensions of quadratic orders, pp. 61-77.

ABSTRACT.
An integral domain D is half-factorial if every non-zero non-unit element has the property that each irreducible factorization of that element has the same length. Some Mersenne primes give rise to half-factorial subrings of quadratic number rings. If R is a half-factorial subring of a factorial quadratic number ring, then R[[T]] is also half-factorial. We compare half-factoriality with factoriality and other factorization properties in the context of quadratic orders and their ring extensions.

**Connor, Peter,**
Indiana University South Bend, South Bend, IN 46634 (pconnor@iusb.edu).

A note on special polynomials and minimal
surfaces, pp. 79-88.

ABSTRACT.When using Traizet's regeneration technique to construct minimal surfaces, the simplest nontrivial configurations are given as the roots of polynomials that satisfy a hypergeometric differential equation. We exhibit examples of simple minimal surfaces exhibiting the same behavior.

**Toru Sasahara,**
Division of Mathematics, Hachinohe Institute
of Technology, Hachinohe, Aomori 031-8501, Japan
(sasahara@hi-tech.ac.jp).

Real hypersurfaces in the complex projective plane attaining equality in a basic
inequality, pp. 89-94.

ABSTRACT. We determine non-Hopf hypersurfaces with constant mean curvature in
the complex projective plane which attain equality in a basic inequality between
the maximum Ricci curvature and the squared mean curvature.

**Perälä, Antti, **
University of Eastern Finland, 80101 Joensuu, Finland
(antti.perala@uef.fi),
**Taskinen, Jari, **University of Helsinki, 00014 Helsinki, Finland
(jari.taskinen@helsinki.fi), and
**Virtanen, Jani A., **University of Reading, Reading RG6 6AX, England
(j.a.virtanen@reading.ac.uk).

Toeplitz operators on Dirichlet-Besov spaces, 95-110.

ABSTRACT. We study Toeplitz operators on the Besov spaces Bp of the open unit disk when p>1. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator on Bp. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of
the existence of nontrivial compact Toeplitz operators and also consider Fredholm properties and
prove an index formula.

**Lacey, Michael T., **Georgia Institute of Technology, Atlanta Georgia
(lacey@math.gatech.edu)
and
** Dario Menas Arias, **Georgia Institute of Technology, Atlanta
Georgia (dario.mena@gatech.edu).

The sparse T1 theorem, pp. 111-127.

ABSTRACT.
We impose standard T1-type assumptions on a Calderon-Zygmund operator T, and deduce that
for bounded compactly supported functions f and g there is a sparse bilinear form B so
that |< Tf, g >|; < B(f,g).
The proof is short and elementary. The sparse bound quickly implies all the standard mapping
properties of a Calderon-Zygmund on a (weighted) L ^{p} space.

**J. Suárez de la Fuente**, Escuela Politécnica Avda. de la universidad s/n 10071 Cáceres Spain
(jesus@unex.es).

A Schur space that is not a uniform retract of its bidual, pp. 129-138.

ABSTRACT.
We give an example of a non-separable Banach space Z with the Schur property such that there is no uniformly continuous retraction of the bidual of Z onto Z.

**Ping Wong Ng, **Department of Mathematics, University of
Louisiana at Lafayette, 104 E University Ave, Lafayette, LA 70504
(png@louisiana.edu) and Leonel Robert,
Department of Mathematics, University of Louisiana at Lafayette, 104 E
University Ave, LA 70504
(lrobert@louisiana.edu).

The kernel of the determinant map on pure C*-algebras, pp. 139-168.

In a simple C*-algebra with suitable regularity properties, any unitary or
invertible element with de la Harpe--Skandalis determinant zero is a finite
product of commutators.

**Soroushmehr, Maedeh**, Mosaheb Institue of Mathematical Research, Kharazmi
University, 50 Taleghani Avenue, 64518,Tehran, Iran
(std_soroushmehr@khu.ac.ir).

On the character space of ultraproducts of Banach algebras and its applications, pp. 169-182.

ABSTRACT. We study the character space of ultraproducts of Banach algebra.
Then as an application, character amenability of ultrapowers
of Banach algebras are investigated.

**Hänninen, Timo S.**, Department of Mathematics and Statistics, University of Helsinki, Finland
(timo.s.hanninen@helsinki.fi).

Remark on median oscillation decomposition and dyadic pointwise domination, pp. 183-197.

ABSTRACT. In this note, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures. In the light of the analogy between median and mean oscillation, our extension can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO.
As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John-Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of zero complexity.
Furthermore, by using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity.

**Stavropoulos, Theodoros, ** National and Kapodistrian
University of Athens, Department of Mathematics
(tstavrop@math.uoa.gr).

L^{p}-bounds for maximal operators associated
to two dyadic structures, pp. 199-206.

ABSTRACT. We find an L^{p}-upper bound for the dyadic maximal
function Mφ with respect to two trees *T*
and *T* ' on a non-atomic probability space (X,μ)
given the L^{1} and L^{p} norm of φ.

**Helmut Maier**, Department of Mathematics, University of Ulm,
Helmholtzstrasse 18, 89081 Ulm, Germany
(helmut.maier@uni-ulm.de), and **Michael Th. Rassias**, Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland
and Institute for Advanced Study, Program in Interdisciplinary Studies,

1
Einstein Dr, Princeton, NJ 08540
(michail.rassias@math.uzh.ch), (michailrassias@math.princeton.edu).

Asymptotics for moments of certain cotangent sums, pp.
207-222.

In this paper we improve a result on the order of magnitude of certain
cotangent sums associated to the Estermann and the Riemann zeta functions.

**Gotchev, Ivan S., **Central Connecticut State University, New Britain, CT 06050
(gotchevi@ccsu.edu).

The non-Urysohn number of a topological space, pp. 223-235.

ABSTRACT.
We call a non-empty subset A of a topological space X
* finitely non-Urysohn* if for every non-empty finite subset F of A and
every family {U_{x }: x in F} of open neighborhoods U_{x} of x,
∩{cl(U_{x}): x in F} is not empty and we define the
* non-Urysohn number of X *
as follows: nu(X) = 1 + sup{|A| : A is a finitely non-Urysohn subset of X}. This
cardinal function is related to and inspired by the cardinal function
* Urysohn number of X*, denoted by U(X), recently introduced by
Bonanzinga, Cammaroto and Matveev.
Using this new cardinal function, for any topological space X and any subset A
of X, we prove the following inequalities:
(1) |cl_{θ}(A)| ≤ |A|^{κ(X)}nu(X),
(2) |[A]_{θ}| ≤ (|A|nu(X))^{κ(X)},
(3) |X| ≤ nu(X)^{κ(X)sLθ(X)}, and
(4) |X| ≤ nu(X)^{κ(X)aL(X)}.

In 1979, Arhangel'skii asked if the inequality |X| ≤ 2^{χ(X)wLc(X)} was true
for every Hausdorff space X. It follows from (3) that the
answer of this question is in the affirmative for all spaces with nu(X) not
greater than the cardinality of the continuum.
We also give a simple example
of a Hausdorff space X such that |cl_{θ}(A)| > |A|^{χ(X)}U(X)
and |cl_{θ}(A)| > (|A|U(X))^{χ(X)}. This example shows that
in (1) and (2) above nu(X) cannot be replaced by U(X) and answers some
questions posed by Bella and Cammaroto (1988), Bonanzinga, Cammaroto and Matveev
(2011), and Bonanzinga and Pansera (2012).

**Bataineh, Khaled,** Dept. of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, JORDAN
(khaledb@just.edu.jo) and
**Belkhirat, Abdelhadi**, Dept. of Mathematics, University of Bahrain, BAHRAIN
(abelkhirat@uob.edu.bh)

The derivatives of the Hoste and Przytycki polynomial for oriented links in the solid torus, 237-253.

ABSTRACT. In 1989 Hoste and Przytycki introduced a two-variable Laurent polynomial invariant for 1-trivial dichromatic links with oriented 2-sublink, which we view as an invariant Y(A; t) of oriented links in the solid torus. We show that the nth derivatives of Y with respect to A at A = 1 is a Vassiliev link invariant of order less than or equal to n. Then we give explicit formulas for Y(1; t) and for the first derivative of Y at A = 1. We explore the topological information involved in these two invariants for knots, and we prove the universality for the first derivative of Y at A = 1, as a type-one knot invariant using properties of the well-known Chebyshev Polynomials of Approximation Theory, which appear in the explicit formulas of these invariants.

**D. Basile**, Dipartimento di matematica e informatica, viale Andrea Doria 6, 95125 Catania, Italia (de.basile@gmail.com) and
**U. B. Darji**, Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
(ubdarj01@louisville.edu).

P-domination and Borel sets, pp. 255-262.

ABSTRACT. In recent years much attention has been enjoyed by topological spaces which are dominated by
second countable spaces. The origin of the concept dates back to the 1979 paper of Talagrand in
which it was shown that for a compact space X, C_{p}(X) is dominated by P, the set of irrationals, if and only if
C_{p}(X) is K-analytic. Cascales extended this result to spaces X which are angelic and finally
in 2005 Tkachuk proved that the Talagrand result is true for all Tychnoff spaces. In recent years,
the notion of P-domination has enjoyed attention independent of C_{p}(X). In particular, Cascales,
Orihuela and Tkachuk proved that a Dieudonné complete space is K-analytic if and only if it is dominated
by P. A notion related to P-domination is that of strong P-domination. Christensen had earlier
shown that a second countable space is strongly P-dominated if and only if it is completely metrizable. We show
that a very small modification of the definition of P-domination characterizes Borel subsets of Polish spaces.

**Kelly, James P.,** Department of Mathematics, Christopher Newport University, Newport News, VA 23606
(james.kelly@cnu.edu) and **Tennant, Tim,** Department of Mathematics, Baylor University, Waco, TX 76798
(timothy_tennant@baylor.edu).

Topological entropy of set-valued functions, pp. 263-282.

ABSTRACT.
Topological entropy is a widely studied indicator of chaos in topological dynamics. Here we give a generalized definition of topological entropy which may be applied to set-valued functions. We demonstrate that some of the well-known results concerning topological entropy of continuous (single-valued) functions extend naturally to set-valued functions while others must be altered. We also present sufficient conditions for a set-valued function to have positive or infinite topological entropy.

**D. N. Georgiou,** University of Patras, Department of Mathematics, 265 00 Patras, Greece
(georgiou@math.upatras.gr),
**A. C. Megaritis,** Technological Educational Institute of Western Greece, Department of Accounting and Finance,
302 00 Messolonghi, Greece (thanasismeg13@gmail.com), and
**F. Sereti**, University of Patras, Department of Mathematics,
265 00 Patras, Greece (seretifot@gmail.com).

A topological dimension greater than or equal to the classical covering dimension, pp. 283-298.

ABSTRACT. We introduce a dimension function for topological spaces what it is
called quasi covering dimension and it is denoted by dim_{q}.
We prove that this dimension function is always greater than or
equal to the classical covering dimension dim. We construct for
every n=1,2,… a hereditarily T_{4}-space (a compact
T_{1}-space) X such that dim_{q}(X)=n and
dim(X)=0 (dim_{q}(X)=n and dim(X)=1). Moreover, we
prove that there exists a compact Hausdorff space (a
Lindelöf hereditarily T_{4}-space) such that
dim(X)=0 and dim_{q}(X)≥1 (dim(X)=0
and dim_{q}(X)=∞). Finally, basic properties of the quasi
covering dimension, examples, and questions are given.