HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 33, No. 1, 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

Davey, Brian A., La Trobe University, Victoria 3086, Australia (B.Davey@latrobe.edu.au), Haviar, Miroslav, Matej Bel University, PdF, 974 01 Banska Bystrica, Slovak Republic (haviar@pdf.umb.sk), and Niven, Todd, La Trobe University, Victoria 3086, Australia (T.Niven@latrobe.edu.au).
When is a full duality strong?, pp. 1-22.
ABSTRACT. The relationship between full and strong dualities in the theory of natural dualities is not yet understood. Our aim in this paper is to present partial solutions to the Full versus Strong Problem, which asks if every full duality is necessarily strong. We introduce local versions of this problem and prove that they have affirmative solutions for four well-known classes of algebras: abelian groups, semilattices, relative Stone Heyting algebras and bounded distributive lattices. Along the way we provide some useful additions to the general theory.

Li, Shirong, Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, P. R. China (shirong@gxu.edu.cn) and Ballester-Bolinches, A., Departament d'Àlgebra, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, València, Spain (Adolfo.Ballester@uv.es)
On a class of finite groups, pp. 23-31.
ABSTRACT. Let L be the class of all finite groups G such that every supersolvable subgroup of G is nilpotent. It is proved in the paper that L is composed of solvable groups and is the largest saturated Fitting formation such that the intersection of L and U is contained in N, where U is the class of all supersolvable groups and N is the class of all nilpotent groups. Some properties of L-groups are also exhibited.

I. N. Cangül, Uludag Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, 16059 Bursa, Turkey (cangul@uludag.edu.tr), R. Sahin, S. İkikardes and Ö. Koruoğlu, Balıkesir Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, 10100 Balıkesir, Turkey (rsahin@balikesir.edu.tr), (skardes@balikesir.edu.tr) and (ozdenk@balikesir.edu.tr).
Power Subgroups of Some Hecke Groups II, pp. 33-42.
ABSTRACT. Let q ³ 3 be an odd integer and let H(lq) be the Hecke group associated to q. Let m be a positive integer and Hm(lq) be the m-th power subgroup of H(lq). In this paper, authors consider the power subgroups of some Hecke groups. They determine the abstract group structure and generators of these subgroups and generalise the results regarding the relations between power subgroups and commutator subgroup, given for the modular group to Hecke groups.

A. Abdollahi, S. M. Jafarian Amiri and A. Mohammadi Hassanabadi, Department of Mathematics, University of Isfahan, Isfahan 81746-73441 and Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, IRAN (a.abdollahi@math.ui.ac.ir), (sm.jafarian@sci.ui.ac.ir), (aamohaha@yahoo.com)
Groups with specific number of centralizers, pp. 43-57.
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.

Dobbs, David, University of Tennessee,Knoxville, Tennessee 37996 (dobbs@math.utk.edu) and Shapiro, Jay George Mason University, Fairfax, Virginia 22030 (jshapiro@gmu.edu).
Descent of Minimal Overrings of Integrally Closed Domains to Fixed Rings, pp. 59-82.
ABSTRACT.  Let G be a group acting via ring automorphisms on a commutative unital ring R. If G is finite, then the embedding  RG ® R is universally going-down, with generalizations to certain classes of locally finite actions by infinite groups. If R is an integrally closed integral domain with a minimal overring and G is finite such that the order of G is a unit of R, then RG has a minimal overring which is the G-fixed ring of the Kaplansky transform of some radical ideal of R.

David E. Rush, Department of Mathematics, University of California, Riverside, California 92521 (rush@math.ucr.edu), and  James S. Okon, and Laura J. Wallace,
Department of Mathematics, California State University, San Bernardino, California 92407 (jokon@csusb.edu), (wallace@csusb.edu).
A Mori-Nagata type theorem for seminormal Mori lattices, pp. 83-102.
ABSTRACT. It is shown that the complete integral closure of a seminormal Mori lattice is a Krull lattice. This extends a result of V. Barucci to multiplicative lattices. New results on rings are obtained by specializing to the case that the lattice is the set of homogeneous ideals of an M-graded ring where M is an abelian cancellative torsion-free monoid.

Albrecht, Ulrich, Auburn University, Auburn, AL 36849 (albreuf@auburn.edu).
Two-Sided Essential Submodules of Q(R) , pp. 103-123.
ABSTRACT.  The focus of this paper are essential submodules A of the maximal right ring of quotients, Q(R), of a right non-singular ring R. Since Q(R) is a R-R-bimodule, particular attention is given to submodules of the right R-module Q(R) which are also submodules of the left R-module Q(R). In this discussion, properties of R which are inherited by intermediate rings S contained between R and Q(R) are investigated. The results obtained are used to discuss homological properties of essential submodules A of Q(R). In particular, the paper addresses the question when S-closed submodules of finite direct sums of copies of A are direct summands.

Lu, Chin-Pi, University of Colorado at Denver, Denver, CO 80217-3364 (Sylvia.lu@cudenver.edu).
A Module whose prime Spectrum has the Surjective Natural Map, pp. 125-143.
ABSTRACT. Let R be a commutative ring with identity. The purpose of this paper is to introduce a new class of modules over R called primeful R-modules. Every nonzero primeful module possesses the non-empty prime spectrum with the surjective natural map. This class contains the family of finitely generated R-modules properly. We show that the theory of prime submodules of primeful modules resembles to that of finitely generated modules.

Arwini, Khadiga, University of Manchester, Manchester M60 1QD, UK (arwini2001@yahoo.com), Del Riego, L., Universidad Autonoma de San Luis Potosi, San Luis Potosi, SLP, 78900 Mexico (lilia@fciencias.uaslp.mx) and Dodson, C.T.J., University of Manchester, Manchester M60 1QD, UK (ctdodson@manchester.ac.uk).
Universal connection and curvature for statistical manifold geometry, pp. 145-161.
ABSTRACT. Statistical manifolds are representations of smooth families of probability density functions (ie subadditive measures of unit weight) that allow differential geometric methods to be applied to problems in stochastic processes, mathematical statistics and information theory. It is common to have to consider a number of linear connections on a given statistical manifold and so it is important to know the corresponding universal connection and curvature; then all linear connections and their curvatures are pullbacks. An important class of statistical manifolds is that arising from the exponential families and one particular family is that of gamma distributions, which we showed recently to have important uniqueness properties in stochastic processes. Here we provide formulae for universal connections and curvatures on exponential families and give an explicit example for the manifold of gamma distributions.

Fanai, Hamid-Reza, Department of Mathematical Sciences, Sharif University of Technology, P.O.Box 11365-9415, Tehran, Iran (fanai@sharif.ac.ir).
Invariant Measures under Geodesic Flow, pp. 163-167.
ABSTRACT. For a compact Riemannian manifold with negative curvature, the Liouville measure, the Bowen-Margulis measure and the Harmonic measure are three natural invariant measures under the geodesic flow. We show that if any two of the above three measure classes coincide then the space is locally symmetric, provided the function with respect to which the equilibrium state is the Harmonic measure, depends only on the foot points.

Stephen Leon Lipscomb, 8809 Robert E. Lee Drive, Spotsylvania, FA 22553 (slipscom@umw.edu).
The Sierpinski-cheese iterated function system extended to a 3-simplex system, pp. 169-207.
ABSTRACT. The n-web fractal -the attractor of the IFS of (n+1) contractions by 1/2 toward the vertices of an n-simplex - emerges from a manifold (n-simplex). The classical example is Sierpinski's gasket (2-web) which emerges from a 2-simplex. It is therefore natural (inverse of moving from manifolds to fractals) to seek, for each n greater than 1, an extension of the n-web system to an n-simplex system. For n = 2, the author recently provided such a (minimal) extension. Here, we extend the 3-web system to one whose attractor is the 3-simplex. For n greater than 3, the extension problem remains open.

Tkachuk, Vladimir V., Universidad Autónoma Metropolitana, Mexico City, Mexico, 09340, (vova@xanum.uam.mx).
Condensing function spaces into Σ-products of real lines , pp. 209-228.
ABSTRACT. We prove that if X is a Lindelöf Σ-space such that t(X)≤κ and πχ(X)≤κ then X has a π-base of order at most κ. This generalizes a theorem of Shapirovsky on existence of a π-base of order ≤κ in any compact space of tightness ≤κ. A famous theorem of Gul'ko says that if X is compact and Cp(X) is Lindelöf Σ then X is Corson compact, i.e., there exists an embedding of X into a Σ-product of real lines. We establish a general fact about Lindelöf Σ-spaces which implies that if both spaces X and Cp(X) are Lindelöf Σ then there is a rich family of retractions on the space X. As a consequence, for any Tychonoff space X, if Cp(X) is Lindelöf Σ then X can be condensed into a Σ-product of real lines. This gives an essential strengthening of the Gul'ko's theorem. As an application of this result, we show that if Cp(X) is a Lindelöf Σ-space and p(X)=ω then nw(X)≤2ω

Nadler, Sam B., Jr., West Virginia University, P.O. Box 6310, Morgantown, WV 26506-6310 (nadler@math.wvu.edu), and Pellicer-Covarrubias, Patricia, Departamento de Matemáticas, Facultad de Ciencias, Circuito ext s/n, Ciudad Universitaria 04510, México, D.F. (paty@ciencias.unam.mx).
Cones That Are ½-Homogeneous, pp. 229-247.
ABSTRACT. A space is ½-homogeneous provided that there are exactly two orbits for the action of the group of homeomorphisms of the space onto itself. Let X be a nonempty compact metric space such that the cone over X is ½-homogeneous. It is shown that if X is finite-dimensional, then X is an absolute neighborhood retract. A general theorem is proved which shows that finite dimensionality is necessary. It is shown that if X is a 1-dimensional continuum or if X does not contain certain types of triods in some nonempty open set, then X is an arc or a simple closed curve (assuming Cone(X) is ½-homogeneous). A number of corollaries are derived. Some unanswered questions are stated.

Chuan Liu, Department of Mathematics, Ohio University-Zanesville Campus, Zanesville, OH 43701 (liuc1@ohio.edu).
Notes on closed mappings, pp. 249-259.
ABSTRACT. In this paper, we characterize closed sigma-compact images of metric spaces as Fr\'echet, weakly quasi-first-countable spaces with a sigma-locally finite k-network; and discuss the fibers of closed mappings on some generalized metric spaces and improve some Tanaka's results and answered a Yun's question. It is also shown that g-metrizable spaces and spaces with a point-countable base are preserved by closed sequence-covering mappings, which gives an affirmative answer to a Lin's question.

Felix Capulin Perez,  Departamento de Matematicas, Facultad de Ciencias UNAM, Circuito exterior, C.P 04510, Mexico, D.F.(fcapulin@gmail.com) and Wlodzimierz  J. Charatonik,  Departament of Mathematics and Statistics, University of Missouri-Rolla, M.O. 65-409-002, U.S.A (wjcharat@umr.edu).
Retractions from C(X) onto X and continua of type N, pp. 261-272.
ABSTRACT.

If a metric continuum X is of type N, then there is no a retraction from the hyperspace of subcontinua C(X) onto F1(X), and X admits no mean. We also give an example which answers a question posed by T. J. Lee. The question is the following: Is there a fan X without the bend intersection property such that X is not of type N? The answer is affirmative, we show a fan.
Let A be a closed subset of X. A retraction is a mapping r from X onto A such that r restricted to A is the identity in A . A mean is a mapping m from X× X onto X such that
a) m((x,x))=x for each x in X, b) m((x,y))=m((y,x)) for each x, y in X.
X is a continuum of Type N if there exist in X an arc A=[p,q], two sequences of arcs An=[pn,p'n] and Bn=[qn,q'n] and points p"n in Bn--{qn,q 'n} and q"n in An--{pn,p'n} such that: 1. the sequences of arcs {An} and {Bn} converge to the arc A; 2. the sequences {pn}, {p'n} and {p"n} converge to the point p  3. the sequences {qn}, {q'n} and {q"n} converge to the point q; 4. each arc in X joining pn and p'n contains q"n;  5. each arc in X joining qn and q'n contains p"n.

Kehe Zhu,  Department of Mathematics, SUNY, Albany, NY 12222 (kzhu@math.albany.edu)
Compact composition operators on Bergman spaces of the unit ball, pp. 273-283.
ABSTRACT. Under a mild condition we show that a composition operator on the Bergman space of the unit ball is compact if and only if the inducing map has no finite angular derivative at any point on the unit sphere.

Constantinescu, Tiberiu, University of Texas at Dallas, Richardson, TX 75083 and El-Sissi, Nermine, Department of Mathematics, Trinity  University, 125 Michigan Avenue NE
Washington, DC 20017 (elsissin@trinitydc.edu).
Positive Definite Kernels and Lattice Paths, pp. 285-300.
ABSTRACT. In this paper, we explore the structure of positive definite kernels on the set of non-negative integers in terms of operator models. Particularly, we introduce two models, one of a Hessenberg type and another that we call ‘near tridiagonal.’ These models produce two distinct parametrizations of the kernels. We also describe the combinatorial nature of these parametrizations in terms of lattice paths of Dyck and Lukasiewicz types.

Stewart, David E., University of Iowa, IA 52242 (david-e-stewart@uiowa.edu).
Differentiating complementarity problems and fractional index convolution complementarity problems, pp. 301-322.
ABSTRACT. Two functions a and b are said to be complementary if a has values in a closed convex cone K (such as the non-negative orthant) while b has values in its dual cone K* (which can also be the non-negative orthant), yet the inner product of a(t) and b(t) is zero for (almost) all t. In this paper we consider implications of the form: ``If a and b are complementary functions, then the inner product of a(t) with the derivative b'(t) is zero for (almost) all t''. This is proved, for example, where a is in Lp and b' is in Lq, 1/p+1/q=1, where a is continuous and b has bounded variation, and where a and b' lie in dual Sobolev spaces. Consequences for more than one derivative are also shown: the inner product of a'(t) with b'(t) being non-negative and the inner product of a(t) with b''(t) non-positive for almost all t provided a and b satisfy mild regularity conditions. These implications can be used to prove conservation of energy in impact systems as well as existence and regularity results for dynamic complementarity problems of various kinds. In particular, it is shown that solutions exist for a convolution complementarity problem where b = k*a + q in Rn with k(t) ~ k0 t c, 0 < c < 1, for small t and k0 positive definite. Such problems arise in connection with the impact of a viscoelastic rod.