*Editors*: D. Bao (San Francisco,
SFSU), D. Blecher (Houston), Bernhard 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 Editor*: K. Kaiser (Houston)

Houston Journal of Mathematics

Contents

**Fuchs, László,** Dept. of Mathematics, Tulane University, New Orleans, LA 70118, U.S.A.
(fuchs@tulane.edu),
**Salce, Luigi,** Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy
(salce@math.unipd.it), and
**Zanardo, Paolo**, Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy
(pzanardo@math.unipd.it).

Divisibility in cyclically presented modules over integral domains, pp. 663-680.

ABSTRACT. We consider divisibility properties of elements in cyclically presented modules over integral domains, with special focus on Bezout domains. We study gaps that signal a larger than expected increase in divisibility.

**Yingbo Han**, Xinyang Normal University, Xinyang ,46400, Henan, China
(yingbohan@163.com), and **Shuxiang Feng**, Xinyang Normal University, Xinyang ,46400, Henan, China
(shuxiangfeng78@163.com).

Montonicity formulas and the stability of **F**-stationary maps with potential,
pp. 681-713

ABSTRACT. In this paper, we introduce the notion of **F**-stationary map with potential with respect to the functional Φ_{F,H}. Then we use the stress-energy tensor to obtain the monotonicity formulas and vanishing theorems for these maps under some conditions on
**H**. We also obtain the first variation formula and the second variation formula for the functional
Φ_{F,H}. Then we study the stability of **F**-stationary map with potential form or into the standard sphere.

**Kamal Boussaf, Abdelbaki Boutabaa** and ** Alain Escassut,** Université Blaise Pascal, Clermont-Ferrand, France 63171
(Kamal.Boussaf@math.univ-bpclermont.fr) ,
(Abdelbaki.Boutabaa@math.univ-bpclermont.fr) ,
(Alain Escassut@math.univ-bpclermont.fr).

Growth of p-adic functions and applications, pp. 715-736.

ABSTRACT. Let K be an algebraically closed p-adic complete field of characteristic zero. We define the order of growth and the type of growth of an entire function f on K as done on the field C and show that they satisfy the same relations as in complex analysis, with regards to the coefficients of f. But here, another expression that we call cotype of f, depending on the number of zeros inside disks, is very important and we show under certain wide hypothesis, that this cotype is the product pf the growth by the type, a formula that has no equivalent in complex analysis and suggests that it might hold in the general case. We check that f and its derivative have the same growth order and the same growth type and present an asymptotic relation linking the numbers of zeros inside disks for two functions of same order. We show that the derivative of a transcenental entire function f has infinitely many zeros that are not zeros of f and particularly we show that f' cannot divide f when the p-adic absolute value of the number of zeros of f inside disks satisfies certain inequality and particularly when f is of finite order.

**M. Crampin,** Department of Mathematics, Ghent University,
Krijgslaan 281, B--9000 Gent, Belgium (m.crampin@btinternet.com.

On the construction of Riemannian metrics for Berwald spaces by averaging, pp. 737-750.

ABSTRACT. The construction of Riemannian metrics on the base manifold of any given Finsler space by averaging suitable objects over indicatrices, such that the Levi-Civita connection of the metric coincides with the canonical Berwald connection of the Finsler space when the Finsler space is a Berwald space, is discussed. Some examples of such metrics are already known, but several new ones, all in principle different, are defined and analysed.

Hyunjin Lee, The
Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang 790-784, Republic
of Korea
(lhjibis@hanmail.net), Young Jin Suh
and Changhwa Woo, Kyungpook National University, Taegu 702-701, Republic of Korea
(yjsuh@knu.ac.kr).

Real hypersurfaces in complex
two-plane Grassmannians with commuting Jacobi operators, pp. 751-766.

ABSTRACT.In this paper, we introduce a new commuting condition composed of the Jacobi operator R_{X},
the structure tensor φ and the shape operator A for Hopf hypersurfaces M in complex two-plane Grassmannians
G_{2}(**C**^{m+2}). By using such a commuting condition, we give a complete classification of Hopf hypersurfaces in
G_{2}(**C**^{m+2}).

Maps characterized by Lie products on nest algebras, pp. 767-777.

ABSTRACT. Let H be a complex Hilbert space and let T(N) be a nest algebra on H. We characterize linear maps f,g,h:T(N) →T(N) satisfying f([x,y])=[g(x),y]+[x,h(y)] for all x,y in T(N).

On quantum maps into quantum semigroups, pp. 779-790.

ABSTRACT. We analyze the recent examples of quantum semigroups defined by M.M. Sadr (M.M. Sadr: A kind of compact quantum semigroups. ArXiv:0808.2740v2 [math.OA]) who also brought up several open problems concerning these objects. These are defined as quantum families of maps from finite sets to a fixed compact quantum semigroup. We show that these are special cases of dual free products of quantum semigroups. This way we can answer all the questions stated by M.M. Sadr. Along the way we discuss the question whether restricting the comultiplication of a compact quantum group to a unital C*-subalgebra defines such a structure on the subalgebra. In the last section we show that the quantum family of all maps from a non-classical finite quantum space to a quantum group (even a finite classical group) might not admit any quantum group structure.

Profinite pro-C*-algebras and pro-C*-algebras of profinite groups, pp. 791-816.

ABSTRACT. We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the unitary representations of the group which factor through a finite group. We define natural homomorphisms from the C*-algebra of a locally compact group and its profinite completion to the pro-C*-algebra of the profinite completion of the group. We give some conditions for injectivity or surjectivity of these homomorphisms, but an important question remains open.

**Freeman, Daniel,** Department of Mathematics, University of Texas at Austin, Austin, TX 78712
(freeman@math.utexas.edu), **Poore, Daniel,** Department of Mathematics, Pomona College, Claremont, CA 92711
(dep02007@mymail.pomona.edu),** Wei, Ann Rebecca,** Department of Mathematics, Northwestern University, Evanston, IL 60208
(rwei@math.northwestern.edu), and
**Wyse, Madeline,** Department of Mathematics, Pomona College, Claremont, CA 92711 (mkw02007@mymail.pomona.edu).

Moving Parseval frames for vector bundles, pp. 817-832.

ABSTRACT. arseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold.

**S. H. Kulkarni, **Indian Institute of Technology Madras, Chennai-600036, India
(shk@iitm.ac.in).

The group of invertible elements of a real Banach algebra, pp. 833-836.

ABSTRACT.The following result is proved: Let A be a commutative real Banach algebra with unit 1. Let G denote the group of invertible elements of A and let G_{1} be the connected component of G containing 1. If the quotient group G/ G_{1} contains an element of finite order other than G_{1}, then the order of such an element must be 2. If the group G/ G_{1} is of finite order, then its order must be 2^{n} for some nonnegative integer n.

**Kucerovsky, Dan **University of New Brunswick at Fredericton, NB, Canada E3B 5A3
and **Sarraf, Aydin **University of New Brunswick at Fredericton, NB, Canada E3B 5A3
(dkucerov@unb.ca).

Schur multipliers and matrix products, pp. 837-850.

ABSTRACT. We give necessary and sufficient conditions for a Schur map to be a homomorphism with respect to ordinary matrix multiplication, with some generalizations to the infinite-dimensional case. In the finite-dimensional case, we find that a Schur multiplier distributes over matrix multiplication if and only if the coefficients of the Schur matrix are of the form a_{ij}=f(i)/f(j) for some f. In addition, it is shown that it is possible to enumerate all *-preserving multiplicative Schur maps on M_{n}(R). We also study the relation of Schur maps to the extreme points of certain sets.

A new interpolation inequality and its applications to a semiconductor model, pp. 851-874.

ABSTRACT. In this paper we study a drift-diffusion semiconductor model with variable electron mobility. We obtain conditions under which the electron density is bounded above and bounded away from 0 below. Part of the proof is based upon a new interpolation inequality which seems to be of interest on its own right.

Multiplication by monomials on BMOA, pp. 875-883.

ABSTRACT.In the recent and important context of Nevanlinna-Pick interpolation theory for classes of holomorphic functions with constraints, the starting point has been to describe the common invariant subspaces of the operators of multiplication by S

**Will Brian**, Mathematics Department, Tulane University, 6823
St. Charles Ave.**Jan van Mill**, Korteweg-de Vries Institute for Mathematics, University of Amsterdam (j.vanMill@va.nl), and **Rolf Suabedissen**, University of Oxford (suabedis@maths.ox.ac.uk).

Homogeneity and generalizations of 2-point sets, pp. 885-898.

ABSTRACT. We prove the existence of homogeneous *n*-point sets (i.e., subsets of the plane which meet every line in exactly *n* points) for every finite *n* ≥ 3. We also show that for every zero-dimensional subset *A* of the real line there is a subset *X* of the plane such that every line intersects *X* in a topological copy of *A*.

**Paul J. Szeptycki,** Department of Mathematics and Statistics, York University, Toronto, ON Canada M3J 1P3
(szeptyck@yorku.ca) and **Artur H. Tomita**, Departamento de Matemática, Instituto de Matemática e
Estatístíca, Universidade de São Paulo, Rua do Matão, 1010 – Cidade Universitária - CEP 05508-090, São Paulo, Brasil
(tomita@ime.usp.br).

Countable compactness of powers of HFD groups, pp. 899-916.

ABSTRACT. Let G be the product of continuum copies of 2. Hajnal and Juhász, in 1976, constructed under CH a topological
subgroup H of G that is an HFD with the following property
(P) the projection of H is onto G(I) for each countable subset I of the continuum, where G(I) is the product of I copies of 2.
Such examples are countably compact without non-trivial convergent se-
quences and were
first used by van Douwen to show that countable com-
pactness is not productive in the class of topological groups.
We show that the HFD constructed via Random reals have the countable
power countably compact.
We construct examples to show that the property HFD + (P) in a topological group does not decide the countable compactness of its powers. We
show under CH that there exists an HFD group satisfying (P) such that
its n-th power is countably compact but its (n + 1)^{st} power is not for every
positive integer n. We also show under CH that there is an HFD group that
is countably compact in all its powers. Furthermore, we show that there are
2 to the continuum many non-homeomorphic such HFD groups.

Remarks on star-Menger spaces, pp. 917-925.

ABSTRACT. In this paper, we show the following statements: (1) There exists a Tychonoff star-Menger space having a regular-closed subspace which is not star-Menger; (2) There exists a Hausdorff star-Menger space having a regular-closed G

Sequence-covering maps on generalized metric spaces, pp. 927-943.

ABSTRACT. Let f: X→ Y be a map. f is a sequence-covering map if whenever {y

** Hui Li,** Beijing University of Technology, Beijing 100124, China
(lihui86@emails.bjut.edu.cn) and
** Liang-Xue Peng (**Corresponding author)**,** Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn).

Some properties on monotonically meta-Lindelöf spaces and related conclusions,
pp. 945-956.

ABSTRACT. In this note we point out that there is a compact Hausdorff separable monotonically Lindelöf space which is not metrizable. If X is a monotonically meta-Lindelöf regular space and has caliber ω_{1}, then X is hereditarily Lindelöf. We prove that if X is a compact Hausdorff monotonically meta-Lindelöf space then the following are equivalent: X is hereditarily Lindelöf; X is perfect; the closure of D is perfect for any discrete subspace D of X; s(X)≤ω. We show that if X is a compact Hausdorff monotonically meta-Lindelöf space and has caliber ω_{1}, then there do not exist an uncountable collection {U_{α}:α∈ω_{1}} of nonempty open subsets of X and a subset {x_{α},y_{α}:α∈ω_{1}} of X such that: (1) x_{α}∈U_{α} and X\{y_{α}} contains the closure of U_{α} for each α∈ω_{1}; (2) For each α∈ω_{1}, the set {x_{α},y_{α}}⊂∩{U_{β}:β<α} or {x_{α},y_{α}}⊂X\∪{U_{β}:β<α}. We also get some conclusions of monotonically meta-Lindelöf spaces which are similar with some known conclusions of monotonically countably metacompact spaces. We show that if X is a regular monotonically countably metacompact space and {U_{α}:α∈ω_{1}} is an uncountable collection of open sets of X such that X\U_{α} is compact for each α∈ω_{1} then {U_{α}:α∈ω_{1}} is not point-countable. Some known conclusions on monotonically countably metacompact spaces can be gotten by this conclusion.

** Hui Li **and** Liang-Xue Peng*,** Beijing University of Technology, Beijing 100124, China (*Corresponding author) (pengliangxue@bjut.edu.cn)
(Peng), (lihui86@emails.bjut.edu.cn)
(Li).

On hereditarily normal rectifiable spaces and paratopological groups,
pp. 957-968.

ABSTRACT. In this note, we investigate hereditarily normal rectifiable
spaces and discuss some properties of paratopological groups. We
mainly show that every hereditarily normal rectifiable space with
a non-trivial convergent sequence has
a regular G_{δ}-diagonal. We prove that if G is a
hereditarily normal rectifiable space then every compact subspace
of G is metrizable.
We finally show that every hereditarily normal SIN
paratopological group of countable type is first countable.

**Valentin Gutev, **Department of Mathematics, Faculty of Science, University of
Malta, Msida MSD 2080, Malta
(valentin.gutev@um.edu.mt), and
**Takamitsu Yamauchi, **Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790-8577, Japan
(yamauchi.takamitsu.ts@ehime-u.ac.jp).

Factorising lower semi-continuous mappings,
pp. 969-986.

ABSTRACT. In this paper, we deal with factorisations of lower semi-continuous mappings
through metrizable spaces. Every metrizable space has a
σ-discrete base for its topology, i.e. for the family of its
open sets. One of our main results is that a lower semi-continuous mapping from a
space X to the nonempty closed subsets of a metrizable space Y
can be factorised through a metrizable domain if and only if the
pre-image of the topology of Y has a σ-discrete base of
cozero-sets. Several special cases are considered, also several
applications are presented.

**Krzyżanowska, Iwona,** Institute of Mathematics, University of Gdańsk, 80-952 Gdańsk, Wita Stwosza 57, Poland
(Iwona.Krzyzanowska@mat.ug.edu.pl), and
**Szafraniec, Zbigniew**, Institute of Mathematics, University of Gdańsk, 80-952 Gdańsk, Wita Stwosza 57, Poland
(Zbigniew.Szafraniec@mat.ug.edu.pl).

Polynomial mappings into a Stiefel manifold and immersions,
pp. 987-1006.

ABSTRACT. For a polynomial mapping from
S^{n-k} to the Stiefel manifold
V_{k}(R^{n}), where n-k is even,
there is presented an effective method of expressing the corresponding
element of the homotopy group π_{n-k}V_{k}(R^{n})≅ Z
in terms of signatures of quadratic forms.
There is also given a method of computing the intersection number
for a polynomial immersion S^{m}→R^{2m}.