Electronic Edition Vol. 33, No. 3, 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


D.D. Anderson and John S. Kintzinger, Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242 (dan-anderson@uiowa.edu), (JohnSKintzinger@netscape.net.
General ZPI-rings without Identity, pp. 631-634.
Abstract - General ZPI-rings without IdentityLet R be a commutative ring not necessarily having an identity. Then R is a general ZPI-ring if every ideal of R is a product of prime ideals. S. Mori showed that a general ZPI-ring without identity is either (1) an integral domain, (2) a ring R where every ideal of R including 0 is a power of R, (3) K times R where K is a field and R is a ring as in (2), or (4) K times D where K is a field and D is a domain with every nonzero ideal of D a power of D. The purpose of this paper is to prove that if R is a ring as in (2), then there is an SPIR S with S=R[1] having R as its maximal ideal. Moreover, there is a complete DVR (D,(p)) with D=(p)[1] so that S and R are homomorphic images of D and (p), respectively.

Chatham, R. Douglas, Department of Mathematics and Computer Science, Morehead State University, Morehead, KY 40351 (d.chatham@moreheadstate.edu) and Dobbs, David E., University of Tennessee, Knoxville, TN 37996 (dobbs@math.utk.edu).
Pairs of commutative rings in which all intermediate rings have the same dimension, pp. 635-647.
ABSTRACT. If n is a nonnegative integer or infinity, and R is a (commutative unital) ring contained in a (commutative unital) ring T, then (R,T) is said to be an n-dimensional pair if every ring S that both contains R and is contained in T has Krull dimension n. For n greater than zero, examples are given of n-dimensional pairs that are not integral extensions, including an infinite family of examples that are neither LO-pairs nor INC-pairs. The n-dimensional pair property transfers well in constructions involving pullbacks or passage to the associated reduced rings, but this property is not stable under passage to factor domains. Special attention is paid to the n-dimensional pairs whose first coordinate is a Jaffard domain or a residually Jaffard ring. Also, examples are given of infinity-dimensional pairs whose intermediate rings have prime ideal chains of arbitrarily large cardinality; and of a family of n-dimensional pairs arising from minimal overrings.

Ko, Seokku, Konkuk University, Chungjusi Chungbuk Korea 380-701 (seokko@kku.ac.kr).
Embedding Bordered Riemann Surfaces in 4-dimensional Riemannian Manifolds, pp. 649-661.
ABSTRACT. Any bordered Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we prove that, given any bordered Riemann surface S0, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. A model can be constructed by deforming a compactification surface of the given topologically equivalent complete Riemann surface S in the normal direction. This result along with previous Ko Embedding theorem(see "Embedding bordered Riemann surfaces in Riemannian Manifolds", Journal of Korean Mathematical Society, Vol. 30. no. 2, 1993) now shows that a bordered Riemann surface admits conformal models in any Riemannian manifold of dimension greater than or equal to 3.

Houston, Kevin, School of Mathematics, University of Leeds, Leeds, LS2 9JT, U.K. (k.houston@leeds.ac.uk).
Disentanglements and Whitney equisingularity, pp. 663-681.
ABSTRACT. A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if, and only if, the mu*-sequence for a hypersurface is constant in the family. This paper shows that similar results are true for families of finitely A-determined map-germs from Cn to C3, where n=2 or 3. Rather than the Milnor fibre we use the disentanglement of a map, and since a disentanglement can be viewed as a section of a stable discriminant we can apply Damon's theory which defines an analogue of the mu*-sequence. The constancy of this sequence is equivalent to Whitney equisingularity of the family in the n=2 case. For the other case it is shown, using extra information, that the image of the family is Whitney equisingular.

Qun He, Department of Applied Mathematics, Tongji University, Shanghai 200092, China (hequn@mail.tongji.edu.cn) and Yi-Bing Shen, Department of Mathematics, Zhejiang University, Hangzhou 310028, China (yibingshen@zju.edu.cn).
Some properties of harmonic maps for Finsler manifolds, pp. 683-699.
ABSTRACT.  This paper is to study further some properties of harmonic maps between Finsler manifolds. Some rigidity theorems for harmonic maps between Finsler manifolds are given. By introducing the stress-energy tensor of maps between Finsler manifolds, some integral formulas are obtained. Moreover, it is proved that any conformal strongly harmonic map from an n(>2)-dimensional Finsler manifold to a Finsler manifold must be homothetic.

Rezsõ L. Lovas, Institute of Mathematics, University of Debrecen H--4010 Debrecen, P.O.B. 12, Hungary (lovasr@math.klte.hu).
A note on Finsler-Minkowski norms, pp. 701-707.
ABSTRACT. By a Finsler--Minkowski norm, we mean a function on a real vector space which is positively homogeneous and positive on the non-zero vectors. We suppose that its metric tensor, i.e., the second derivative of its square is non-degenerate. Then we show that it is automatically positive definite.
The main idea of the proof is as follows. We suppose, without loss of generality, that our vector space is a Euclidean n-space. The unit sphere is a compact hypersurface, and therefore it has a point, the furthermost one from the origin, in which all the n-1 principal curvatures have the same sign. Finally, we establish a relation between the signs of these principal curvatures and the signature of the metric tensor.

Wladimir G. Boskoff, Department of Mathematics and Computer Science, Ovidius University, Constanţa, 900527, Romania (boskoff@univ-ovidius.ro) Marian G. Ciucă, Department of Mathematics and Computer Science, Ovidius University, Constanţa, Romania (mgciuca@univ-ovidius.ro) and Bogdan D. Suceavă, Department of Mathematics, California State University, Fullerton, CA 92834-6850, U.S.A. (bsuceava@fullerton.edu).
Distances induced by Barbilian's metrization procedure, pp. 709-717.
ABSTRACT.  Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Riemannian and generalized Lagrangian metrics. Then we prove an extension of Barbilian's metrization procedure.

Csikós, Balázs, Department of Geometry, Eötvös University, 1117 Budapest, Hungary (csikos@cs.elte.hu), Németh, Brigitta, Ybl Miklós Faculty of Engineering, Szent István University, 1146 Budapest, Hungary (Nemeth.Brigitta@ymmfk.szie.hu), and Verhóczki, László, Department of Geometry, Eötvös University, 1117 Budapest, Hungary (verhol@cs.elte.hu).
Volumes of principal orbits of isotropy subgroups in compact symmetric spaces, pp. 719-734.
ABSTRACT. Let (G,K) be a Riemannian symmetric pair of compact type such that G is simply connected. Take the compact Riemannian symmetric space G/K and the natural isometric action of the isotropy subgroup K on G/K. In this paper, we discuss the problem how to compute the volumes of the principal orbits in explicit form and how to find the unique principal orbit of maximal volume. Moreover, we study this problem in detail in some rank two irreducible symmetric spaces with different restricted root systems.

Takamitsu Yamauchi, Department of Mathematics, Shimane University, Matsue, 690-8504, Japan (t_yamauchi@riko.shimane-u.ac.jp).
A characterization of metrizable finitistic spaces and its applications to upper semicontinuous set-valued selections, pp. 735-751.
ABSTRACT. In this paper, we give two types of characterizations of finitistic spaces. One is in terms of perfect mappings from zero-dimensional metric spaces, which is analogous to K. Morita's characterization of finite-dimensional metrizable spaces. Another is by means of upper semicontinuous set-valued selections, which is an analogue of M. M. Čoban's characterization of finite-dimensional paracompact spaces. Characterizations of some classes of infinite-dimensional spaces and a generalization of finitistic spaces will also be given.

Gerardo Acosta, Instituto de Matematicas, Circuito Exterior, Ciudad Universitaria, Area de la Investigacion Cientifica, Mexico, D.F. 04510, Mexico (gacosta@matem.unam.mx) and Peyman Eslami, Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294 (peslami@math.uab.edu), and Lex G. Oversteegen Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294 (overstee@math.uab.edu).
On open maps between dendrites, pp. 753-770.
ABSTRACT. In this paper we mainly present two results, one dynamical and one topological, about open mappings between dendrites. The dynamical result states that if f is a homeomorphism from a dendrite X onto itself, then the omega limit set of any point of X is either a periodic orbit or a Cantor set. In the latter case, the restriction of f to this omega limit set is an adding machine. The topological result states that if f is an open map from a dendrite X onto a dendrite Y, then there exists n subcontinua X1, X2, ..., Xn of X such that X is the union of them, the intersection of any two of those subcontinua contains at most one element which is a critical point of f and the restriction of f to any set Xi is an open map from Xi onto Y that can be lifted, in a natural way, to a product space Zi x Y, for some compact and zero-dimensional space Zi

Guo-Fang Zhang, Department of Mathematics, Nanjing University, Nanjing 210093, China (tuopmath@nju.edu.cn) and Wei-Xue Shi, Department of Mathematics, Nanjing University, Nanjing 210093, China (wxshi@nju.edu.cn).
Characterizations of relative paracompactness by relative normality of product spaces, pp. 771-779.
ABSTRACT. In this paper, we mainly study the relative version of Tamano Theorem and give some characterizations of relative paracompactness in terms of relative normality of products of paracompact spaces and compact spaces, which gives an answer to a problem posed by Arhangel'skii in 2002.

A. V. Arhangel'skii, Ohio University, Athens, Ohio, 45701 (arhangel@math.ohiou.edu). J. van Mill, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands (vanmill@few.vu.nl). G. J. Ridderbos, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands (gfridder@few.vu.nl).
A new bound on the cardinality of power homogeneous compacta, pp. 781-793.
ABSTRACT. It was recently proved by R. de la Vega that if X is a homogeneous compactum then the cardinality of X is bounded by 2t(X), where t(X) denotes the tightness of X. We extend de la Vega's argument to show that the same inequality holds for power homogeneous compacta.

David Herrera Carrasco, Facultad de Ciencias Físico-Matemáticas de la Benemérita Universidad Autónoma de Puebla, rio verde y San Claudio, Ciudad Universitaria, Puebla Pue., México (dherrera@fcfm.buap.mx).
Dendrites with unique hyperspace, pp. 795-805.
ABSTRACT. For a metric continuum X let C(X) denote the hyperspace of subcontinua of X. The continuum X is said to have unique hyperspace of subcontinua provided that if Y is a continuum and C(X) is homeomorphic to C(Y), then X is homeomorphic to Y. We show in this paper the following: A dendrite which is not an arc has unique hyperspace of subcontinua if its set of end points is closed.

Granda, Larry M., Department of Mathematics, St. Louis University, St. Louis, MO 63103 (grandalm@slu.edu).
Representing homology classes of a surface by disjoint curves, pp. 807-813.
ABSTRACT. Conditions are given for when a collection of homology classes of a closed oriented surface of genus g may be represented by a collection of pairwise disjoint simple closed curves.

Park, Chun-Gil, Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea (cgpark@cnu.ac.kr).
Automorphisms on a C*-algebra and isomorphisms between Lie JC*-algebras associated with a generalized additive mapping, pp. 815-837.
It is shown that if an odd mapping satisfies a generalized additive functional equation, then the odd mapping is Cauchy additive, and we prove the Cauchy-Rassias stability of linear mappings in Banach modules over a unital C*-algebra for the generalized additive functional equation. As an application, we show that every almost linear bijective mapping on a unital C*-algebra is an automorphism under some conditions, and that every almost linear bijective mapping of a unital Lie JC*-algebra onto a unital Lie JC*-algebra is a Lie JC*-algebra isomorphism under some conditions.

Defant, Andreas, University of Oldenburg, D-26111, Oldenburg, Germany (defant@mathematik.uni-oldenburg.de),  García, Domingo,  Universidad
de Valencia, 46100 Burjasot (Valencia), Spain (domingo.garcia@uv.es), Maestre, Manuel,  Universidad de Valencia, 46100 Burjasot (Valencia),  Spain (manuel.maestre@uv.es), and Pérez-García, David, Universidad Rey Juan Carlos, 28933 Móstoles (Madrid), Spain (david.perez.garcia@urjc.es).
Extension of multilinear forms and polynomials from subspaces of L1-spaces, pp. 839-860.
Let X be a Banach space which has an unconditional basis and is a subspace of some  L1-space    Y. We show that X=l1 if and only if  every m-linear form S on X, has an m-linear extension T to Y satisfying that ||T||  is less than  or equal to Cm ||S||, where C > 0 is a constant independent of m. If we replace m-linear forms by m-homogeneous polynomials, then we can only show that X is ``close'' to l1.

Hopenwasser, Alan, University of Alabama, Tuscaloosa, AL 35487 (ahopenwa@bama.ua.edu).
Partial crossed product presentations for On and Mk(On) using amenable groups, pp. 861-876.
The Cuntz algebra  On is presented as a partial crossed product in which an amenable group partially acts on an abelian C*-algebra. The partial action is related to the Cuntz groupoid for On and connections are made with non-self-adjoint subalgebras of On, particularly the Volterra nest subalgebra. These ideas are also extended to the context of matrix algebras Mk(On) over the Cuntz algebra.

Dutkay, Dorin E., University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd, PO Box 161364, Orlando, FL 32816-1364, USA (ddutkay@mail.ucf.edu) and Jorgensen, Palle E. T., The University of Iowa, Department of Mathematics, Iowa City, IA 52242-1419, USA (palle-jorgensen@uiowa.edu).
Harmonic analysis and dynamics for affine iterated function systems, pp. 877-905.
ABSTRACT. We introduce a harmonic analysis for a class of affine iteration models in finite dimensions. Our results use Hilbert-space geometry, and we develop a new duality notion for affine and contractive iterated function systems (IFSs). Our applications include some new identities for the Fourier transform of the measures arising from infinite Bernoulli convolutions.

Amir Khosravi, Faculty of Mathematical Sciences and Computer Engineering, University For Teacher Education, 599 Taleghani Ave., Tehran 15614, Iran, (khosravi_amir@yahoo.com)and Mohammad Sadegh Asgari, Dept. of Math., Science and Research Branch, Islamic Azad University, Tehran, Iran (msasgari@yahoo.com).
Frames of subspaces and approximation of the inverse frame operator, pp. 907-920.
ABSTRACT. A frame of subspaces in a Hilbert space H allows that identity operator on H to be written as a sum of some bounded operators on H. This family of bounded operators on H is called an atomic resolution of the identity on H. We show the atomic resolution of the identity associated to a frame of subspaces have a certain minimum property relative to its associated norm. We further show that under extra condition every atomic resolution of the identity provides a frame of subspaces for H. We consider direct sum of frames of subspaces with respect to the same family of weights which is a frame of subspaces for their direct sum space. Frame theory of subspaces describes how one can choose the corresponding atomic resolution of the identity, which is interesting from mathematical point of view, but for applications it is a problem that requires to know the inverse frame operator S-1W,v on H. If the underlying Hilbert space is infinite dimensional it is hard to invert the frame operator SW,v. We show how the inverse of SW,v can be approximated by using the methods of linear algebra.

Gal, Nadia J., The University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152 (nadiagal@memphis.edu).
Isometric Equivalence of Differentiated Composition Operators between Spaces of Analytic Functions, pp. 921-926.
The differentiated composition operator on the Hardy space is defined as the composition with an analytic self-map of the disk, followed by differentiation. We consider the isometric equivalence problem of the differentiated composition operator on Hardy and Bergman spaces. Using Forrelli's form for the isometric isomorphism on the Hardy space, we obtain a result similar to the result of R. C. Wright for the isometric equivalence problem of composition operators on the Hardy space.

A. C. Ponce, Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083), Fédération Denis Poisson, Université François Rabelais, 37200 Tours France (ponce@lmpt.univ-tours.fr) and J. Van Schaftingen, Département de Mathématique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
The continuity of functions with N-th derivative measure, pp. 927-939.
ABSTRACT. We study the continuity of functions u whose mixed derivative ∂1…∂Nu is a measure. If u ∈ W1,1(RN), then we prove that u is continuous. The same conclusion holds for u∈ Wk,p(Q), with kp > N-1, where Q denotes a cube in RN. The key step in the proof consists in showing that the measure ∂1…∂Nu does not charge hyperplanes orthogonal to the coordinate axes.