Electronic Edition Vol. 31, No. 4, 2005

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), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


J.S. Okon, Department of Mathematics, California State University San Bernardino, CA 92407 (jokon@csusb.edu), D.E. Rush, Department of Mathematics, University of California, Riverside, CA 92521 (rush@math.ucr.edu) and L.J. Wallace, Department of Mathematics, California State University, San Bernardino, CA 92407 (wallace@csusb.edu).
A Mori-Nagata Theorem for Lattices and Graded Rings, pp. 973-997.
ABSTRACT. We extend the global transform and the Mori-Nagata theorem, from integral domains to multiplicative lattices.

R.B.J.T. Allenby, Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, England (pmt6ra@leeds.ac.uk).
On the Upper near Frattini Subgroup of a Generalized Free Product, pp. 999-1005.
ABSTRACT. Azarian has shown that, for many types of generalized free product, the lower near Frattini subgroup is a subgroup of the amalgamated subgroup. Together with earlier work the present results show that, in all cases for which the lower near Frattini subgroup has been proved to be a subgroup of the amalgamated subgroup, so, too, is the (potentially larger) upper near Frattini subgroup contained in the amalgamated subgroup. This answers a question raised by Azarian.

David F. Anderson, Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, U. S. (anderson@math.utk.edu) and Ayman Badawi, Department of Mathematics & Statistics, American University Of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates (abadawi@ausharjah.edu).
On phi-Dedekind Rings and phi-Krull Rings, pp. 1007-1022.
ABSTRACT. The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Dedekind domains and Krull domains. Let H = {R | R is a commutative ring with 1 and Nil(R) is a divided prime ideal of R}. Let R in H, T(R) be the total quotient ring of R, and let phi be the map from R into RNil(R) (the localization of R at Nil(R)) such that phi(a/b) = a/b for every a in R and b in R\ Z(R). Then phi is a ring homomorphism from T(R) into RNil(R), and phi restricted to R is also a ring homomorphism from R into RNil(R) given by phi(x) = x /1 for every x in R. A nonnil ideal I of R is said to be phi-invertible if phi(I) is an invertible ideal of phi(R). If every nonnil ideal of R is phi-invertible, then we say that R is a phi-Dedekind ring. Also, we say that R is a phi-Krull ring if phi(R) is the intersection of {Vi}, where each Vi is a discrete phi-chained overring of phi(R), and for every nonnilpotent element x in R , phi(x) is a unit in all but finitely many Vi. We show that the theories of phi-Dedekind and phi-Krull rings resemble those of Dedekind and Krull domains.

Daniele Guido and Tommaso Isola, Dipartimento di Matematica, Universita di Roma ``Tor Vergata'', I--00133 Roma, Italy (guido@mat.uniroma2.it), (isola@mat.uniroma2.it).
Tangential dimensions I. Metric spaces, pp. 1023-1045.
ABSTRACT. Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a set, namely which is able to detect the "oscillations" of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in [D.Guido, T.Isola Journ. Funct. Analysis 203, (2003) 362-400], in the framework of Alain Connes' noncommutative geometry.

Paul Bankston, Marquette University, Milwaukee, WI 53201 (paulb@mscs.mu.edu).
Mapping Properties of Co-existentially Closed Continua, pp. , 1047-1063.
ABSTRACT. A continuous surjection between compacta is called co-existential if it is the second of two maps whose composition is a standard ultracopower projection. A continuum is called co-existentially closed if it is only a co-existential image of other continua. This notion is not only an exact dual of Abraham Robinson's existentially closed structures in model theory, it also parallels the definition of other classes of continua defined by what kinds of continuous images they can be. In this paper we continue our study of co-existentially closed continua, especially how they (and related continua) behave in certain mapping situations.

Stephen Lipscomb, 8809 Robert E. Lee Drive, Spotsylvania, Virginia 22553, USA (slipscom@umw.edu).
A Minimal Extension of the Iterated Function System for Sierpinski's Gasket to One Whose Attractor is the 2-simplex, pp. 1065-1083.
ABSTRACT. As an iterated function system, a pair of contractions (scalings by 1/2) yield an attractor, 1-simplex; a code space, Cantor's set; and an address map, quotient map from Cantor's set onto the 1-simplex known as (classical) adjacent-endpoint identification. In 1972, the author extended adjacent-endpoint identification to arbitrary code spaces. And for systems with (n+1) greater than 2 such contractions, (general) adjacent-endpoint identification yields an attractor that is a fractal known as the n-web. Each such n-web is a proper subspace of an n-simplex. Indeed, Sierpi\'nski's classical construction of his gasket begins with a 2-simplex (manifold) and ends with the (fractal) 2-web subspace. It is therefore natural (inverse of moving from manifolds to fractals) to seek a minimal (code space and address map) extension of the n-web system to an n-simplex system. Here, we extend the 2-web system (Sierpi\'nski-gasket system) to one whose attractor is the 2-simplex. For n > 2, however, it is an open problem to find such minimal extensions

Yasushi Hirata, Graduate School of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan (yhira@jb3.so-net.ne.jp).
Subnormal finite products of subspaces of ω 1, pp. 1085-1095.
ABSTRACT. The author and Kemoto characterized mild normality of finite products of subspaces of ω 1 in terms of stationarity. In this paper, we will show that the subshrinking property, subnormality, mild subnormality, and mild normality coincide for every finite product of subspaces of ω 1.

Yan-Kui Song, Department of Mathematics, Nanjing Normal University, Nanjing 210097, China (songyankui@njnu.edu.cn).
On relatively absolutely star-Lindelöf spaces, pp. 1097-1102.
ABSTRACT. The author defines and studies the spaces mentioned in the title.

Taras Banakh, Department of Mathematics and Mechanics, Lviv University, Universytetska 1, Lviv, 79000, Ukraine (tbanakh@franko.lviv.ua) and Dusan Repovs, Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana, Slovenia 1001 (dusan.repovs@fmf.uni-lj.si).
On linear realizations and local self-similarity of the universal Zarichnyi map, pp. 1103- 1114.
ABSTRACT. Answering a question of M.Zarichnyi we show that the universal Zarichnyi map is not locally self-similar. We also characterize linear operators homeomorphic to this map and on this base give a simple construction of it.

T. Mizokami, Department of Mathematics, Joetsu University of Education, Joetsu, Niigata 943-8512 Japan (mizokami@juen.ac.jp} and F. Suwada, Joint graduate school(PhD Program) Hyougo University of Teacher Education, Yashiro, Hyougo 673-1494 Japan.
On General Resolutions due to Networks, pp. 1115-1126.
ABSTRACT. We study the general resolutions of spaces such as M3-spaces, metrizable spaces and developable spaces under the condition that families of subsets of the initial spaces on which the resolutions are defined are σ-discrete closed networks.

Kaori Yamazaki, Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (kaori@math.tsukuba.ac.jp).
Controlled extensions of products of continuous functions, pp. 1127-1133.
ABSTRACT. For a topological space X and a subspace A of X, we prove that: A is C-embedded in X if and only if for any real-valued continuous function f on A, any non-negative real-valued function g on A, and any real-valued function H on X satisfying that the product of f and g equals the restriction of H to A and the zero-set of g is contained in that of f, there exist continuous extensions F and G of f and g, respectively, over X such that the product of F and G equals H and the zero-set of G is contained in that of F. The result essentially extends an original theorem of M. Frantz on compact metrizable domains (1995) and a subsequent theorem of S. Barov and J. J. Dijkstra on normal domains (2002). A version of complex-valued functions is also given.

Rajeev Kumar and Romesh Kumar, Department of Mathematics, University of Jammu, Jammu--180 006, (raj1k2@yahoo.co.in) ( omesh_jammu@yahoo.com).
On Finite Dimensional Algebras Generated by Composition Operators on Orlicz Sequence Spaces with Weight, pp. 1135-1152.
ABSTRACT. In this paper, we characterise the algebraic composition operators and the essentailly algebraicity of the Calkin image of composition operators on Orlicz sequence spaces with weight when the Orlicz function is having atmost polynomial growth. We also classify the essentially characteristic polynomials.

Bernhard Haak, Mathematisches Institut I, Universität Karlsruhe, Englerstrasse 2, 76128 Karlsruhe, Germany (Bernhard.Haak@math.uni-karlsruhe.de) and Christian Le Merdy, Laboratoire de Mathematiques,Universite de Franche-Comte, 25030 Besancon Cedex, France (lemerdy@math.univ-fcomte.fr).
α-Admissibility of Observation and Control Operators, pp. 1153-1167.
ABSTRACT. Given a strongly continuous semigroup T(t) on some Banach space, we introduce a variant of the admissibility of an operator with respect to this semigroup, called alpha-admissibility. Let -A be the generator of T(t). In the case when that semigroup is a bounded analytic one, the second named author had showed that the validity of the so-called Weiss conjecture is equivalent to A having a square function estimate. In this paper, we extend that characterisation to our new setting. Indeed we show that if A has a square function estimate, then alpha-admissibility is equivalent to an appropriate resolvent estimate.

Dan Kucerovsky, University of New Brunwick-- Fredericton, Fredericton, N.B., Canada E3B 5A3 (dkucerov @ unb.ca).
Properties of Strictly Positive Elements in C*- algebras, pp. 1169-1177.
ABSTRACT. We study the relationship of spectral properties of strictly positive elements of a C*-algebra to other properties of the algebra, in particular establishing the CS property for stable sigma-unital algebras that are not of real rank zero.

Benton L. Duncan, University of Nebraska-Lincoln, Lincoln, NE 68588-0323 (bduncan@math.unl.edu).
Universal Operator Algebras of Directed Graphs, pp. 1179-1198.
ABSTRACT. Given a directed graph, there exists a universal operator algebra and universal C*-algebra associated to the directed graph. For finite graphs this algebra decomposes as the universal free product of some building block operator algebras. For countable directed graphs, the universal operator algebras arise as direct limits of operator algebras of finite subgraphs. Finally, a method for computing the K-groups for universal operator algebras of directed graphs is given.

Zhijian Qiu, Research Institute of Mathematics, HanShan Normal University, ChaoZhou, GaungDong 521041, P. R. China (qiu@hstc.edu.cn}.
Carleson Measures On Circular Domains, pp. 1199-1206.
ABSTRACT. In this article, we study Carleson measures on circular domains. We characterize the Carleson measures on circular domains and extend the celebrated theorem of L. Carleson that gives an equivalence between a Carleson measure and the Carleson measure inequality from the unit disk to circular domains.

Ravi P. Agarwal, Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA (agarwal@fit.edu), Haishen Lü, Department of Applied Mathematics, Hohai University, Nanjing, 210098, China and Donal O'Regan, Department of Mathematics, National University of Ireland, Galway, Ireland.
Positive Solutions for the Singular p-Laplace Equation, pp. 1207-1220.
ABSTRACT. An existence theorem concerning positive solutions for a certain singular equation is established. The result is obtained using a fixed point theorem in cones.

Thomas Kühn, Mathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany (kuehn@mathematik.uni-leipzig.de) and Tomas Schonbek, Florida Atlantic University, Department of Mathematical Sciences, Boca Raton, FL 33431 (schonbek@fau.edu).
Compact Embeddings of Besov spaces into Orlicz and Lorentz-Zygmund Spaces, pp. 1221-1243.
ABSTRACT. Let 1 ≤  p,q < ∞, and let Ω be a bounded open subset of Rn. Then the Besov space Bpqn/p(Ω) embeds into the exponential Orlicz space Eν (Ω) generated by the function Φν(t) = t exp(tν) if and only if 0 < ν  ≤ q'. The embedding is even compact if ν < q'. We give two-sided estimates for the entropy numbers of these compact embeddings, thus generalizing previous results of H. Triebel (Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc., 66 (1993), 589-618) and the first author (Compact embeddings of Besov spaces in exponential Orlicz spaces, J. London Math. Soc. (2), 67 (2003), 235-244) in the case p=q.
Alternatively, the Orlicz spaces Eν(Ω) can also be seen as special Lorentz-Zygmund spaces Λ(w,Ω). We introduce a scale of weights w such that the corresponding Lorentz-Zygmund spaces are only minimally larger than the limiting Orlicz space Eq'(Ω), and such that Bpqn/p(Ω) still embeds compactly into Λ(w,Ω). We prove sharp estimates for approximation and entropy numbers of these embeddings.

Plamen Simeonov, Department of Mathematics, University of Houston-Downtown, One Main Street, Houston, Texas 77002 (simeonovp@uhd.edu).
A Weighted Energy Problem for a Class of Admissible Weights, pp. 1245-1260.
ABSTRACT. We study the minimization problem for weighted logarithmic energy integrals over the set of probability Borel measures supported on a closed subset of the extended complex plane. The weight is a nonnegative upper semi-continuous function that behaves like 1/|z| at infinity. We show that there exists a unique measure that minimizes the energy integral and we give a characterization of this measure in terms of a weighted logarithmic potential.