## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 34, No. 2, 2008

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. Hausen (Houston), 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

May, Coy L., Towson University, 8000 York Road, Towson, Maryland, 21252, USA, (cmay@towson.edu) and Zimmerman, Jay, Towson University, 8000 York Road, Towson, Maryland, 21252, USA ( jzimmerman@towson.edu).
The symmetric genus of groups of odd order, pp. 319-338.
ABSTRACT.
Let G be a finite group of odd order. The symmetric genus G is the minimum genus of any Riemann surface on which G acts faithfully. Suppose G acts on a Riemann surface X of genus g ³ 2. If |G| > 8(g - 1), then |G| = K(g-1), where K is 15, 21/2, 9 or 33/4. We call these four types of groups LO1-groups through LO4-groups, respectively. We determine the characteristics of LO1-groups and LO2-groups and show that there are infinite families of each type. We also show that there are exactly ten odd order groups with genus between 2 and 26 inclusive. Finally, if G is an odd order group with symmetric genus of the form p+1 for an odd prime p or 2k + 1, for some positive integer k, then G is a metacyclic group with certain properties. We determine that, in the range between 26 and 200, most of the numbers of either of these forms are not the genus of an odd order group.

Families of numerical semigroups closed under finite intersections and for the Frobenius number, pp. 339-348.
ABSTRACT. In this paper, we introduce the concept of Frobenius variety. This notion generalizes several classes of numerical semigroups including Arf semigroups, sets of semigroups admitting certain patterns, saturated semigroups, and system proportionally modular numerical semigroups. We show that many of the properties that hold for these more specific classes of semigroups are satisfied by Frobenius varieties as well.

Kim, Hwankoo, Hoseo University, Asan 336-795, Korea (hkkim@office.hoseo.ac.kr), Kim, Eun Sup, Kyungpook National University, Taegu 702-701, Korea, and Park, Young Soo, Kyungpook National University, Taegu 702-701, Korea.
Injective Modules over Strong Mori Domains, pp. 349-360.
ABSTRACT. We show that an integral domain R is a strong Mori domain if and only if any direct sum of co-semi-divisorial injective R-modules is injective. We also show that if M is a nonzero co-semi-divisorial torsion module over a strong Mori domain R with w-dim(R)=1, then the semi-divisorial envelope of M is isomorphic to the direct sum of M localized at P, where P runs over the set of w-maximal ideals of R.

David F. Anderson,  Mathematics Department, The University of Tennessee, Knoxville, TN 37996-1300 (anderson@math.utk.edu).
On the diameter and girth of a zero-divisor graph, II, pp. 361-371.
ABSTRACT. The zero-divisor graph of a commutative ring R has the set of nonzero zero-divisors of R as its set of vertices, and two distinct vertices x and y are adjacent if and only if xy = 0. Let A be a subring of a commutative ring B. In this paper, we study the relationship between the diameters (resp., girths) of the zero-divisor graphs of A and B.

H. Pat Goeters, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310 (goetehp@auburn.edu), and Bruce Olberding, Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003-8001 (olberdin@nmsu.edu).
Pure submodules of completely decomposable modules, pp. 373-382.
ABSTRACT.  We examine the integral domain R with a submodule X of the quotient field Q of R such that the endomorphism ring of X is R and every rank one pure (in the sense of Matlis) submodule of a direct sum F of copies of X is isomorphic to X. Necessarily R is a GCD domain. If R is noetherian or atomic, then this property is characterized by R being a UFD.

Muge Kanuni, Department of Mathematics, Bogazici University, 34342 Istanbul, Turkey (muge.kanuni@boun.edu.tr).
Module properties of incidence rings, pp. 383-396.
ABSTRACT. In this survey, we consider I(X,R) both as a module over itself and as an R-module and investigate the necessary and sufficient conditions for the incidence ring to be divisible, injective, principally injective, prime, and multiplication. Moreover, we study some results about prime, multiplication, dense and essential modules, and give results about dense submodules in prime and multiplication modules.

Ayman Badawi, American University Of Sharjah, P.O. Box 26666 Sharjah, United Arab Emirates (abadawi@aus.edu), and Ali Jaballah, University Of Sharjah, P.O. Box 27272 Sharjah,United Arab Emirates (ajaballah@sharjah.ac.ae ).
Some finiteness conditions on the set of overrings of a phi-ring, pp. 397-408.
ABSTRACT.  Let H = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. For a ring R in H with total quotient ring T(R), Let phi be the natural ring homomorphism from T(R) into R_Nil(R). An integral domain R is said to be an FC-domain (in the sense of Gilmer) if each chain of distinct overrings of R is finite, and R is called an FO-domain if R has finitely many overrings. A ring R is called an FC-ring if each chain of distinct overrings of R is finite, and R is said to be an FO-ring if R has finitely many overrings. A ring R in H is said to be a phi-FC-ring if phi(R) is an FC-ring, and R is called a phi-FO-ring if phi(R) is an FO-ring. In this paper, we show that the theory of phi-FC-rings and phi-FO-rings resembles that of FC-domains and FO-domains.

Ulrich Albrecht, Department of Mathematics,  Auburn University,  Auburn, AL 36849,  U.S.A. (albreuf@mail.auburn.edu), Simion Breaz, Babes-Bolyai University, Faculty of Mathematics and Computer Science, Str. Mihail Kogalniceanu 1, 400084 Cluj-Napoca, Romania (bodo@math.ubbcluj.ro), and William Wickless, Department of Mathematics,
University of Connecticut, Storrs, CT 06269 U.S.A.(wickless@math.uconn.edu).
Finitely A-cogenerated abelian groups, pp. 409-421.
ABSTRACT. We give structure theorems for self-small abelian groups A of finite torsion-free rank such that every endomorphic image of A is a quasi-summand of A. Several examples and related results are given.

Abrams, G., Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, U.S.A. (abrams@math.uccs.edu), and Aranda Pino, G., Departamento de Algebra, Universidad Complutense de Madrid, 28040 Madrid, Spain (gonzaloa@mat.ucm.es).
The Leavitt path algebras of arbitrary graphs, pp. 423-442.
ABSTRACT. We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of row-finite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and semiprime. In our final result, we show that all Leavitt path algebras have zero Jacobson radical.

Weidong Wang, Department of Mathematics, China Three Gorges University, Hubei Yichang, 443002 P.R.China (wdwxh722@163.com), and Gangsong Leng, Department of Mathematics, Shanghai University, Shanghai, 200436, P.R.China (lenggangsong@163.com).
Some affine isoperimetric inequalities associated with  Lp- affine surface area, pp. 443-453.
ABSTRACT. In this paper, combining the Blaschke-Santalo inequality with the Lp-centro-affine inequality and the Lp-Petty projection inequality, respectively, some affine isoperimetric inequalities for Lp-affine surface area are shown.

Crasmareanu, Mircea, Faculty of Mathematics, University Al. I. Cuza, Iasi, 700506, Romania (mcrasm@uaic.ro).
Last multipliers as autonomous solutions of the Liouville equation of transport, pp. 455-466.
ABSTRACT. Using the characterization of last multipliers as solutions of the Liouville's transport equation, new results are given in this approach of ODE by providing several new characterizations, e.g. in terms of Witten and Marsden differentials or adjoint vector field. Applications to Hamiltonian vector fields on Poisson manifolds and vector fields on Riemannian manifolds are presented.
In Poisson case, the unimodular bracket considerably simplifies computations while, in the Riemannian framework, a Helmholtz type decomposition yields remarkable examples: one is the quadratic porous medium equation, the second (the autonomous version of the
previous) produces harmonic square functions, while the third refers to the gradient of the distance function with respect to a two dimensional rotationally symmetric metric. A final example relates the solutions of Helmholtz (particularly Laplace) equation to provide a last multiplier for a gradient vector field. A connection of our subject with gas dynamics in Riemannian setting is pointed at the end.

Li, Yao Wen, Department of Mathematics, NanJing University, NanJing, PR China, (lieyauvn@263.net) and Zou, Xiao Rong, Department of Mathematics, NanJing University, NanJing, PR China (xrzou@nju.edu.cn).
On the bi-Ricci curvature and some applications, pp. 467-481.
ABSTRACT. In this paper, we investigate the properties of the bi-Ricci curvature of a Riemannian manifold and use this geometrical quantity to study submanifolds in two ways. First, we shall prove a sharp lower bound of the bi-Ricci curvature of an immersed submanifold in a general Riemannian manifold and use the estimation to characterize the Clifford hypersurface S2((1-c²)1/2)× Sn-2(c) in the standard sphere Sn+1(1). Secondly, we shall prove that there are no nontrivial L2 harmonic 1-forms on a strongly stable hypersurface M of a general Riemannian manifold N when the bi-Ricci curvature of N is no less than certain lower bound, which gives a topological obstruction for the stability of M. The result about the nonexistence of nontrivial L2 harmonic forms on the strongly stable hypersurface M is also used to study the number of its ends.

Kamiyama, Yasuhiko and Tsukuda, Shuichi, Department of Mathematics, University of the Ryukyus, Okinawa 903-0213, Japan (kamiyama@sci.u-ryukyu.ac.jp), (tsukuda@math.u-ryukyu.ac.jp).
On the homology of configuration spaces of arachnoid mechanisms, pp. 483-499.
ABSTRACT. We determine the integral homology groups of the configuration space of a certain arachnoid mechanism -- that is, a parallel robot in 3-dimensional Euclidean space having n two-joined legs, with all joints of a fixed length a/2, joined together at a central point q, with the other end of the i-th leg at the i-th vertex of a regular polyhedron P in d-dimensional Euclidean space where d=2,3.

Yuji Akaike, Kure National College of Technology, 2-2-11 Aga-Minami Kure-shi Hiroshima 737-8506, Japan (akaike@kure-nct.ac.jp), Naotsugu Chinen, Okinawa National College of Technology, 905 Henoko Nago-shi Okinawa 905-2192, Japan (chinen@okinawa-ct.ac.jp), and Kazuo Tomoyasu, Miyakonojo National College of Technology, 473-1 Yoshio-cho Miyakonojo-shi Miyazaki 885-8567, Japan (tomoyasu@cc.miyakonojo-nct.ac.jp).
Large inductive dimension of the Smirnov remainder, pp. 501-510.
ABSTRACT. The purpose of this paper is to investigate the large inductive dimension of the remainder of the Smirnov compactification of n-dimensional Euclidean space with the usual metric, and give an application of it.

Wisloski, Gregory, Indiana University of Pennsylvania, Indiana, PA 15705 (wisloski@iup.edu), and Heath, Robert W., University of Pittsburgh, Pittsburgh, PA 15213
Diagonal characterizations of generalized metric spaces , pp. 511-518.
ABSTRACT. Using a well-known result of Ceder, we show that a variety of generalized metric spaces can be characterized using the diagonal of the space. Such characterizations will help illustrate the connections between these types of spaces.

Ambrozie, Calin-Grigore, Mathematical Institute, Zitna 25 11567 Prague, Czech Republic (ambrozie@math.cas.cz).
Functional commutant lifting and interpolation on generalized analytic polyhedra, pp. 519-543.
ABSTRACT We prove that for every generalized analytic polyhedron there exists a functional Hilbert space enabling the use of the commutant lifting technique for interpolation problems. This applies in particular to Caratheodory-Fejer type interpolation problems for bounded anaytic functions of several variables. The existence of the fractional transform - type solutions is then characterized in terms of positivity conditions. We show by a concrete example how to obtain such solutions.

Jasiczak, M., A. Mickiewicz University, Poznan, Poland (mjk@amu.edu.pl), (mjask@umich.edu).
Generators of the algebra of holomorphic functions with log-type growth, pp. 545-563.
ABSTRACT In this paper we address the problem of solvability of the Bézout equation in the algebra of holomorphic functions, which grow not faster than some power of logarithm of the distance to the boundary. It is proved that if the domain is smoothly bounded and strongly pseudoconvex then two such functions generate the algebra, if they are jointly invertible in the corresponding algebra of continuous functions. The method relies on explicit formulae for the Neumann operator given by Lieb and Range. As a result, we also obtain some regularity results for the Neumann operator.

Heil, Christopher, Georgia Institute of Technology, Atlanta, GA 30332-0160 (heil@math.gatech.edu), and Kutyniok, Gitta, Justus-Liebig-University Giessen,
35392 Giessen, Germany (gitta.kutyniok@math.uni-giessen.de).
Density of frames and Schauder bases of windowed exponentials, pp. 565-600.
ABSTRACT. This paper proves that every frame of windowed exponentials satisfies a Strong Homogeneous Approximation Property with respect to its canonical dual frame, and a Weak Homogeneous Approximation Property with respect to an arbitrary dual frame. As a consequence, a simple proof of the Nyquist density phenomenon satisfied by frames of windowed exponentials with one or finitely many generators is obtained. The more delicate cases of Schauder bases and exact systems of windowed exponentials are also studied. New results on the relationship between density and frame bounds for frames of windowed exponentials are obtained. In particular, it is shown that a tight frame of windowed exponentials must have uniform Beurling density.

Zhao, Chunyi, East China Normal University, Shanghai, China 200062 (agacyz@gmail.com).
Singular limits in a Liouville-type equation with singular sources, pp. 601-621.
ABSTRACT. This paper considers the existence of multiple-peak solutions for a Liouville-type equation with singular sources under homogeneous Dirichlet conditions. In the paper, by using the so-called "localized energy mathod"-a combination of Liapunov-Schmidt reduction method and variational techniques, we construct a solution which blows up at singular sources and serveral other points in the domain..

Ranjbar-Motlagh, Alireza, Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran (ranjbarm@sharif.edu).
Generalized Stepanov type theorem with applications over metric-measure spaces, pp. 623-635.
ABSTRACT. The main purpose of this article is to extend an Lp-type generalization of Stepanov's differentiability theorem in metric-measure space. This generalized Stepanov type theorem is applied to the Sobolev and bounded variation functions in order to show the Lp-type generalized differentiability for such functions. The proof of this generalized differentiability theorem is a combination of the proofs of Campanato and Stepanov theorems which is an extension of author's work to abstract spaces. Moreover, we give a positive answer to a question of Balogh-Rogovin-Zurcher about p-type generalized differentiability of BV functions over metric-measure spaces..