*Editors*: G. Auchmuty (Houston), D. Bao
(San Francisco, SFSU), H. Brezis (Paris), B. Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (Radboud), C. Hagopian (Sacramento),
R. M. Hardt (Rice), Y. Hattori (Matsue,
Shimane), 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

**Reza Ameri,** Department of Mathematics, Faculty of Basic
Sciences, University of Mazandaran Babolsar, Iran
(ameri@umz.ac.ir).

Some properties of the Zariski topology of multiplication modules, pp. 337-344.

ABSTRACT.
Let
R be a commutative ring with identity and M be a multiplication
R-module. We investigate the properties of Zariski topology on
Spec(M), the collection of all prime submodule of M. In particular,
we will prove that Spec(M) and Spec( R/ann(M)) are homeomorphic and
obtain some results, which are already, known for Spec(R). Finally,
we investigate the irreducible subsets of Spec(M).

**Adan-Bante,** Edith, Northern Illinois University,Watson Hall 320, DeKalb,
IL 60115-2888, USA
(EdithAdan@illinoisalumni.org).

On nilpotent groups and conjugacy classes, pp. 345-356.

ABSTRACT.
Fix a prime p. Let G be a finite nilpotent group, C and D be conjugacy classes of G of size p. Then either the product CD={cd| c in C, d in D} is a conjugacy class or is the union of at least (p+1)/2 distinct conjugacy classes of G. As an application of the previous result, given any nilpotent group G and any conjugacy class C of size p, we describe the square CC of C in terms of conjugacy classes of G.

**Rosales, J. C.**, Departamento de Álgebra, Universidad de
Granada, E-18071 Granada, Spain (jrosales@ugr.es), and
**Vasco, P.,** Departamento de Matemática, Universidade de
Trás-os-Montes e Alto Douro, 5001-801 Vila Real,
Portugal (pvasco@utad.pt).

The Frobenius variety of the saturated numerical semigroups, pp. 357-365.

ABSTRACT. A Frobenius variety is a nonempty family of numerical semigroups closed under finite intersections and under the adjoin of the Frobenius number. In this work we see that the variety of the saturated numerical semigroups, is the least Frobenius variety satisfying that for any integers m and r with m greater than or equal to 2 and r greater than m and not multiple of m, there exists an element of the variety with multiplicity m and smallest generator greater than the multiplicity equal to r. As a consequence we obtain that every saturated numerical semigroup admits a Toms decomposition. Finally, we give a characterization of the saturated numerical semigroups in terms of a certain type of Diophantine inequalities.

**Winfield, Christopher, J., **University of Wisconsin -
Madison, 1150 University Av., Madison, WI 53706 (cjwinfield2005@yahoo.com).

Solvability and non-solvability of some partial differential operators with polynomial coefficients,
pp. 367-392.

ABSTRACT. We
examine the local and semi-global solvability of partial differential
operators which in operator notation take the form L = P(∂_{x},∂_{y}+x^{m-1}∂_{w}
) for certain homogeneous polynomials P of degree two or greater and
for integers m ≥ 3. Using partial Fourier transforms we find a
condition that is equivalent to semi-global and, in turn, local
solvability of these operators. This condition is formulated in terms
of asymptotic behavior of transition matrices for certain canonical
bases arising from a Fourier representation of operators L.

**Mohammed Benalili** and **Hichem Boughazi,** Department of
Mathematics, Faculty of Sciences BP119 University Abou-Bekr Belkaïd. Tlemcen
Algeria (m_benalili@mail.univ-tlemcen.dz),
(h_boughazi@mail.univ-tlemcen.dz).

On the second Paneitz-Branson invariant, pp. 393-420.

ABSTRACT. We define the second Paneitz-Branson operator on a compact Einsteinian manifold of dimension n≥5 and we give sufficient conditions that make it attained.

Classification of marginally trapped surfaces with parallel mean curvature vector in Lorentzian space forms, pp. 421-449.

ABSTRACT. A space-like surface in a four-dimensional Lorentzian manifold is called marginally trapped if its mean curvature vector is light-like at each point. In this article, we prove that if a marginally trapped surface in a four- dimensional Minkowski space-time lying in a light cone, then it has parallel mean curvature vector. The main purpose of this article is to classify marginally trapped surfaces with parallel mean curvature vector in four-dimensional Lorentzian space forms. Our main results state that there are six families of such surfaces in the Minkowski space-time, eight families in the de Sitter space-time and eight families in the anti-de Sitter space-time. Conversely, marginally trapped surfaces with parallel mean curvature vector in four-dimensional Lorentzian space forms are obtained from these families. Explicit examples of such surfaces are presented. In addition we give a simple relation between marginally trapped surfaces with parallel mean curvature vector and biharmonic surfaces in four-dimensional Minkowski space-time.

**Bing-Ye Wu,** Minjiang University, Fuzhou 350108 China (bingyewu@yahoo.cn).

On hypersurfaces with two distinct principal curvatures in Euclidean space, pp. 451-467.

ABSTRACT. We investigate hypersurfaces in Euclidean space with two distinct principal curvatures and constant m-th mean curvature. By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in Euclidean space of constant m-th mean curvature and two distinct principal curvatures of multiplicities n-1,1 ( we assume that the m-th mean curvature is nonzero when m is greater than 1). As the result, we prove that any local hypersurface in Euclidean space of constant mean curvature and two distinct principal curvatures is an open part of a complete hypersurface of the same curvature properties. The corresponding result does not hold for m-th mean curvature when m is greater than 1.

**Alexander Blokh **and **Lex Oversteegen, **University of Alabama at
Birmingham Birmingham, AL 35294-1170** **(ablokh@math.uab.edu),
(overstee@math.uab.edu)

Monotone images of Cremer Julia sets, pp. 469-476.

ABSTRACT.
We show that if P is a quadratic polynomial with a fixed Cremer point and Julia
set J, then for any monotone map φ: J →
A from J onto a locally connected continuum A, A is a single
point.

**Mahmoud Filali,** and **Tero Vedenjuoksu, ** Department of
Mathematical Sciences, University of Oulu, P.O.Box 3000, 90014 Oulu, Finland (mahmoud.filali@oulu.fi),
(tero.vedenjuoksu@oulu.fi).

The
Stone-Cech compactification of a topological group and the
β-extension property, pp. 477-488.

ABSTRACT. Let G be a topological group which is not a P-group. Then the
Stone-Cech compactification βG of G is a semigroup with an
operation extending that of G such that G is contained in the
topological centre of βG if and only if G is pseudocompact. This generalizes a known result due to Baker and
Butcher for locally compact groups.

We see that Lindelöf P-groups have this extension property. A
non-discrete P-group without the extension property is also given.

**Eiichi, Matsuhashi,** Department of Mathematics, Faculty of Engineering,
Shimane University ,Matsue, Shimane 690-8504, Japan
(matsuhashi@riko.shimane-u.ac.jp).

Parametric Krasinkiewicz maps, cones and polyhedra,
pp. 489-498.

ABSTRACT.Let X,Y and Z be metrizable spaces with Y being a C-space
and let f : X → Y
be a perfect map.
If Z is a polyhedron or the cone over a compactum,
then the set {g in C(X,Z)|g|f^{-1}(y) : f^{-1}(y) → Z is a Krasinkiewicz map for each y in Y} is a dense
G_{δ}-subset
of the mapping space C(X,Z) with
the source limitation topology.

**Yuan Jun,** Department of Mathematics, Nanjing Xiaozhuang University, Nanjing, 211171, P.R.China,(yuanjun@graduate.shu.edu.cn,
**Leng Gangsong,** Shanghai University, Shanghai, China, and **Cheung Wing-Sum,** University of Hong Kong, China.

Convex bodies with minimal p-mean width,
pp. 499-511.

ABSTRACT.
In this paper, we generalize the minimal mean width to the Brunn-Minkowski-Firey theory. We characterize the minimal position of convex bodies in terms of isotropicity of a suitable measure and obtain a stability result for L_{p} projection bodies.

Strictly singular non-compact diagonal operators on HI spaces, pp. 513-566.

ABSTRACT. A Banach space is Hereditarily Indecomposable (HI) provided that none of its closed subspaces is the direct sum of two infinite dimensional further subspaces. We present the construction of an HI space X with a Schauder basis (e

**Goehle, Geoff,** Western Carolina University, Cullowhee, NC 28723
(grgoehle@email.wcu.edu).

The Mackey machine for crossed products by regular groupoids. I,
pp. 567-590.

ABSTRACT. We first describe a Rieffel induction system for groupoid crossed
products. We then use this induction system to show that, given a regular groupoid G and an action of G on an upper-semicontinuous bundle A, every irreducible representation of the crossed product C*(A,G) is induced from a representation of the group crossed product C*(A(u),S(u)) where u is a unit, A(u) is a fibre of A, and S(u) is a stabilizer subgroup of G.

**Popovych, Stanislav,** Kyiv Schevchenko University, Glushkova 2, Kyiv 03022, Ukraine (popovych@univ.kiev.ua).

On O*-representability and C*-representability of *-algebras, pp. 591-617.

ABSTRACT. Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. Applications to the *-doubles, the monomial *-algebras and several other classes of *-algebras are presented.

**Rivera-Noriega, Jorge, ** Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor CP62209, México (rnoriega@buzon.uaem.mx)

Two results over sets with big pieces of parabolic Lipschitz graphs,
pp. 619-635.

ABSTRACT. For a set E in (n+1)-Euclidean space with uniform big pieces of parabolic Lipschitz graphs (defined in the bulk of the paper) we first observe that certain parabolic singular integrals are bounded on E. If E is the boundary of certain type of non-cylindrical domain, and E is regular for the heat equation, we prove a weak reverse Hölder inequality for caloric measure over E.

**Zhuoran Du**,
College of Mathematics and Econometrics, Hunan University, 410082, Changsha,China.
(Zhuorandu@yahoo.com.cn).

Limit behaviour of solutions to linear hyperbolic equations with an
equivalued boundary on a small "hole" in R^{n}, pp.
637-652.

ABSTRACT. In R^{2} (or R^{3}) the limit behaviour of solutions to linear hyperbolic equations with an equivalued boundary on a small "hole" has
been discussed by A.Damlamian and Li Ta Tsien. In this paper
, we discuss the corresponding limit behaviour in R^{n}(n≥4) by using the transposition method.

**Yan, Qiming,** Department of Mathematics, Tongji University, Shanghai
200092, P.R. China (yan_qiming@hotmail.com).

Uniqueness theorem of meromorphic mappings with few moving hyperplanes, pp. 653-664.

ABSTRACT.
We prove a truncated uniqueness theorem of meromorphic
mappings with few moving hyperplanes.

**Jie Zhang** and **Liang-Wen Liao**, Department of Mathematics Nanjing
University Nanjing 210093 P. R. China (zhangjie@smail.nju.edu.cn),
(maliao@nju.edu.cn).

On Brück's conjecture on entire functions sharing
one value with their derivatives, pp. 665-674.

ABSTRACT.
In this paper, we study the question that an entire function and its
derivative share a small function by utilizing Nevanlinna theory and Wiman-Valiron
theory. We obtain some uniqueness theorems, which improve Chang-Zhu's result and
give a partial answer to Brueck's conjecture in this direction.