HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 36, No. 2, 2010

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)

Contents

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+xm-1w ) 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.

Chen, Bang-Yen, Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, U.S.A. (bychen@math.msu.edu), and Van der Veken, Joeri, Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200 B, B-3001 Leuven, Belgium (joeri.vanderveken@wis.kuleuven.be).
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 Gδ-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 Lp projection bodies.

Spiros, A. Argyros, National Technical University of Athens, 15780 Athens, Greece (sargyros@math.ntua.gr), Irene Deliyanni, 18 Neapoleos st., 15341 Athens, Greece, (ideliyanni@yahoo.gr), Andreas G. Tolias, Department of Mathematics, University of the Aegean, 83200 Karlovasi, Greece (atolias@math.aegean.gr).
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 (en ) on which there exist striclty singular non-compact diagonal operators. We also prove that the space of diagonal operators of the space X, with respect to the basis (en ) contains an isomorphic copy of l(N).

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 Rn, pp. 637-652.
ABSTRACT. In R2 (or R3) 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 Rn(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.