HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 33, No. 2, 2007

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), W. B. Johnson (College Station), J. Nagata (Osaka), V. I. Paulsen (Houston), Min Ru (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)


Houston Journal of Mathematics


Contents

Uçkun, Mustafa, University of İnönü , 44069 Malatya, Turkey (muckun@inonu.edu.tr) and Öztürk, Mehmet Ali, University of Cumhuriyet, 58140 Sivas, Turkey (maozturk@cumhuriyet.edu.tr).
On Trace of Symmetric Bi-Gamma-Derivations in Gamma-Near-Rings, pp. 323-339.
ABSTRACT. Let M be a 2-torsion free 3-prime left gamma-near-ring with multiplicative center C. Let x be an element of M and C(x) the centralizer of x in M. The aim of this paper is to study the trace of symmetric bi-gamma-derivations (also symmetric bi-generalized gamma-derivations) on M. Main results are the following theorems: Let D(.,.) be a non-zero symmetric bi-gamma-derivation of M and F(.,.) a symmetric bi-additive mapping of M. Let d and f be traces of D(.,.) and F(.,.), respectively. In this case (1) If d(M) is a subset of C, then M is a commutative ring. (2) If d(y), d(y) + d(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring. (3) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(M) is a subset of C, then M is a commutative ring. (4) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M associated with D(.,.) and f(y), f(y) + f(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a commutative ring.

Weixing Chen, Mathematics and Information Science School, Shandong Institute of Business and Technology, Yantai 264005, P. R. China (wxchen5888@163.com) and Wenting Tong, Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China (wttong@nju.edu.cn).
 On skew Armendariz rings and rigid rings, pp. 341-353.
ABSTRACT.  In this paper we study skew Armendariz rings and rigid rings, extending and improving some results of Hong et al. (2003) and some known results on Armendariz rings. New families of skew Armendariz rings are presented including the one in which the endomorphism is not monomorphic and the ring is not reduced.

Bezhanishvili, Guram  and Harding, John, New Mexico State University, Las Cruces, NM, USA 88003 (gbezhani@nmsu.edu), (jharding@nmsu.edu).
MacNeille completions of modal algebras, pp. 355-384.
ABSTRACT. For a modal algebra (B,f), there are two natural ways to extend f to an operation on the MacNeille completion of B. The resulting structures are called the lower and upper MacNeille completions of (B,f). In this paper we consider lower and upper MacNeille completions for various varieties of modal algebras. In particular, we characterize the varieties of closure algebras and diagonalizable algebras that are closed under lower and upper MacNeille completions. We also introduce the variety of Sierpinski algebras, and show that although this variety is not closed under lower or upper MacNeille completions, it follows from the axiom of choice that each Sierpinski algebra has a MacNeille completion that is also a Sierpinski algebra, and that this result implies the Boolean ultrafilter theorem.

Chang, Gyu Whan, Department of Mathematics, University of Incheon, Incheon 402-749, Korea (whan@incheon.ac.kr).
Quasi-invertible prime t-ideals, pp. 385-389.
ABSTRACT. Let G be a group and let cent(G) denote the set of centralizers of single elements of G. A group G is called n-centralizer if |cent(G)|=n. In this paper, for a finite group G, we give some interesting relations between |cent(G)| and the maximum number of the pairwise non-commuting elements in G. Also we characterize all n-centralizer finite groups for n=7 and 8. Using these results we prove that there is no finite group G with the property that |cent(G)|=|cent(G/Z(G))|=8, where Z(G) denotes the centre of G. This latter result answers positively a conjecture posed by A. R. Ashrafi.

Padmanabhan, R., Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (padman@cc.umanitoba.ca), McCune, W., Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439-4844, U.S.A. (mccune@mcs.anl.gov) and Veroff, R., Department of Computer Science, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A. (veroff@cs.unm.edu).
Lattice laws forcing distributivity under unique complementation., pp. 391-401.
ABSTRACT.  We give several new lattice identities valid in nonmodular lattices such that a uniquely complemented lattice satisfying any of these identities is necessarily Boolean. Since some of these identities are consequences of modularity as well, these results generalize the classical result of Birkhoff and von Neumann that every uniquely complemented modular lattice is Boolean. In particular, every uniquely complemented lattice in M∨V(N5), the least nonmodular variety, is Boolean.

Banks, William D., Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (bbanks@math.missouri.edu) and Luca, Florian, Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, México (fluca@matmor.unam.mx).
Sums of prime divisors and Mersenne numbers, pp. 403-413.
ABSTRACT. In this note, we study those positive integers n with the property that the sum of the distinct prime factors of n divides the n-th Mersenne number.

Picozza, Giampaolo, Dipartimento di Matematica, Università degli Studi "Roma Tre", Roma, Italy (picozza@mat.uniroma3.it).
A note on semistar Noetherian domains , pp. 415-432.
ABSTRACT.  We study semistar Noetherian domains, that is, domains having the ascending chain condition on "quasi semistar ideals''.We generalize several of the classical results that hold in Noetherian domains to the case of semistar operations stable and of finite type, for instance, Cohen Theorem, primary decomposition, principal ideal Theorem, Krull intersection Theorem, etc. We do this mainly by using a method that allows one to transfer properties already proved for star operations to the context of semistar operations. Furthermore, an analogue of the Hilbert basis Theorem for semistar Noetherian domain (with respect to stable semistar operations) is proved.

Ling Jia, Ludong University, 264025 Yantai, Shandong, China (jialing471@126.com) and Fang Li, Zhejiang University, 310027 Hangzhou, China.
On weak group entwining structures, pp. 433-450.
ABSTRACT. Group coalgebras and Hopf group coalgebras appeared in the work of Turaev [1] on homotopy quantum field theories. A purely algebraic study of Hopf group coalgebras was initiated by Virelizer [2], and then continued by Scaenepeel and Wang [3]. Virelizer laid the algebraic foundations and gave a generalized version of the Fundamental Theorem for Hopf group coalgebras, Wang introduced the notions of a group entwining structure and of a group coalgebra extension, and Scaenepeel proposed an alternative approach to Hopf group coalgebras and showed that Hopf group coalgebras are essentially Hopf algebras in a symmetric monoidal category. We asked what the weak bialgebras in this category would be. We have found  an answer to this question by introducing weak Hopf group coalgebras. This paper is devoted to studying the generalizations of entwining structure and coalgebra Galois extension in the setting of weak semi-Hopf group coalgebras, and have obtained  a relation between them, that is, a weak group coalgebra Galois extension can induce a unique compatible weak group entwining structure.
References:
1. V.G. Turaev, Homopoty Field Theory In Dimension 3 And Crossed Group Categories, Preprint GT/0005291
2. Alexis Virelizer, Hopf Group Coalgebras, Journal of Pure And Applied Algebra, 171(2002):75-122
3. S.Wang, Group Entwining Structures And Group Coalgebra Galois Extensions, Comm. Algebra,32(9)(2004):3437-3457

M. Crampin, Department of Mathematical Physics and Astronomy, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium and Department of Mathematics, King's College,
Strand, London WC2R 2LS, UK (Crampin@btinternet.com).
Isotropic and R-flat sprays, pp. 451-459.
ABSTRACT. It is shown that in dimension greater than 2 a spray is isotropic if and only if it is locally projectively R-flat.

Bang-Yen Chen, Department of Mathematics, Michigan State University, East Lansing, MI 48824-1029, U.S.A. (bychen@math.msu.edu.
Tension field, iterated Laplacian, type number and Gauss maps, pp. 461-481.
ABSTRACT. Let M be a Riemannian manifold. By applying the finite type theory we study maps from M into a Euclidean space whose tension field is an eigenmap of a p-iterated Laplacian for some natural number p. First, we prove that such maps are either of 1-type, of null 2-type, or of infinite type. Several examples are then given to illustrate that this result is sharp. Some applications of this result are also presented. The simplest examples of maps whose tension field is an eigenmap of an iterated Laplacian are those which have constant tension field. Next, we study hypersurfaces whose (classical or spherical) Gauss map has constant tension field. Finally, we prove that every spherical hypersurface with 2-type spherical Gauss map must have non-constant mean curvature.

Ralph D. Kopperman, Department of Mathematics, City College of New York, CUNY, New York, NY 10031, U. S. A. (rdkcc@ccny.cuny.edu) and Richard G. Wilson, Departamento de Matematicas, Universidad Autonoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340, Mexico, D.F., Mexico (rgw@xanum.uam.mx).
Separation and connectedness in spectral compactifications, pp. 483-497.
ABSTRACT. We continue the study of the relationship between properties of an inverse spectrum and those of the inverse limit and selected subspaces of its minimal points. It is shown that limits of inverse spectra of joincompact spaces with pairwise continuous bonding maps are connected if and only if the spaces are connected. Since finite T1-spaces are discrete, there are not enough finite spaces with higher separation properties to obtain the infinite spaces with these properties as limits of inverse systems of such finite spaces.
We show that many higher separation properties of the space of minimal points of the inverse limit result from conditions imposed on the bonding maps. This relationship is studied for the separation properties T1, regularity, complete regularity, normality and hereditary normality.

Alejandro Illanes, Instituto de Matematicas, UNAM, Circuito Exterior, Ciudad Universitaria, Mexico, 04510, D.F. (illanes@matem.unam.mx).
A tree-like continuum whose cone admits a fixed-point-free map, pp. 499-518.
ABSTRACT. In this paper we prove that the cone over the continuum which is the union of a simple triod and a spiral surrounding it does not have the fixed point property.

Antonyan, Natella, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Ciudad de México, 14380 México D.F., México (nantonya@itesm.mx).
An intrinsic characterization of G-pseudocompact spaces, pp. 519-530.
ABSTRACT. For any locally compact Hausdorff group G we give, in the realm of G-normal spaces, an intrinsic characterization of G-pseudocompact spaces. On this way we prove also an equivariant quantitative version of the well known Urysohn's Lemma.

Yajima, Yukinobu, Kanagawa University, Yokohama 221-8686, Japan (yajimy01@kanagawa-u.ac.jp).
Strong beta-spaces and their countable products, pp. 531-540.
ABSTRACT. We introduce a new concept called "strong beta-spaces". Strong beta-spaces are beta-spaces. Semi-stratifiable spaces, strong Sigma-spaces and strict p-spaces are all strong beta-space. Thus, the class of strong beta-spaces is well located in the classes of generalized metric spaces. This class does not coincide with the class of beta-spaces, but they coincide under the condition of paracompactness. As a merit of the class of strong beta-spaces, we show that it is countably productive. Moreover, it is shown that the class of normal strong beta-spaces (or paracompact beta-spaces) is countably productive if it is finitely productive.

Henrik Petersson, Chalmers/Göteborg University, School of Mathematical Sciences SE-412 96 Göteborg, Sweden (henripet@math.chalmers.se).
Complemented hypercyclic subspaces, pp. 541-553.
ABSTRACT. A sequence T=(Tn) of continuous linear operators Tn acting on a space X, is said to be hypercyclic if there is a vector x, called hypercyclic for T, such that (Tnx) forms a dense set. A hypercyclic subspace for T is an infinite dimensional closed subspace of X formed by, except for zero, hypercyclic vectors (for T). We establish a criterion for a sequence T of operators, acting on a separable Frechet space with a continuous norm, to have a complemented hypercyclic subspace. Our result complements previous results by several authors.  

Bennett, G.,  Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. (bennettg@indiana.edu).
Meaningful sequences, pp. 555-580.
ABSTRACT. The fundamental Theorem on Means suggests many new elementary inequalities, yet it offers no hint at all for proving them. Our aim here is to explore this gap.

Bernal-Gonzalez, Luis, Dept. Analisis Matematico, Apdo.41080 Sevilla, Spain (lbernal@us.es), Calderon-Moreno, Maria del Carmen, Dept. Analisis Matematico, Apdo.41080 Sevilla, Spain (mccm@us.es) and Bonilla, Antonio, Dept. Analisis Matematico, Apdo.38271 La Laguna, Tenerife, Spain (abonilla@ull.es).
 Compositional hypercyclicity equals supercyclicity, pp. 581-591.
ABSTRACT. In this note it is proved that the sequence of composition operators generated by automorphisms of a simply connected domain strictly contained in the complex plane is hypercyclic (that is, possesses some dense orbit) if and only if it is supercyclic (i.e., possesses some dense projective orbit). When the domain is the full complex plane, a result in this direction is also obtained. In addition, a number of statements about the corresponding cyclicity properties of single composition

Selby, Christina, Department of Mathematics, Purdue University, West Lafayette, IN 47907 (cselby@math.purdue.edu).
An extension and trace theorem for functions of H-bounded variation in Carnot Groups of step 2, pp. 593-616.
ABSTRACT. This paper provides an extension of a function u in BVH (Ω) to a function u0 in BVH(G), when Ω is H-admissible and G is a Carnot group of step 2. It is shown that H-admissible domains include non-characteristic domains and domains in groups of Heisenberg type which have a partial symmetry about characteristic points. An example is given of a domain that is C1,α , α < 1, that is not H-admissible. Further, when Ω is H-admissible a trace theorem is proved for u in BVH(Ω).

Brandolini, Barbara, Chiacchio, Francesco and Trombetti, Cristina, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 – Napoli, Italy (brandolini@unina.it), (francesco.chiacchio@unina.it), (cristina@unina.it).
Some remarks on nonlinear elliptic problems involving Hardy potentials, pp. 617-630.
ABSTRACT. In this note we prove an Hardy type inequality with a remainder term, where the potential depends only on a group of variables. Such a result allows us to show the existence of entropy solutions to a class of elliptic P.D.E.’s.