HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 37, No. 2 , 2011

Editors: G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), D. Blecher (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (Radboud), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), 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)

Houston Journal of Mathematics


Contents

Dobbs, David E., University of Tennessee, Knoxville, TN 37996 (dobbs@math.utk.edu), and Shapiro, Jay, George Mason University, Fairfax, VA 22030 (jshapiro@gmu.edu).
Universal quasi-lying-over rings, pp. 351-359.
ABSTRACT  All rings considered here are commutative with 1, and all subrings are unital. A ring R is called a UQLO-ring (resp., ULO-ring) if every extension with R contained in T satisfies the quasi-lying-over property (resp., lying-over property) for each ring extension T of R. Examples of ULO-rings are known in every (Krull) dimension. All rings of dimension at most 1 are UQLO-rings. The converse is proved for integral domains (more generally, for rings R such that each non-minimal prime ideal of R is semiregular).
Many examples and non-examples of UQLO-rings arise via the A+B construction, thanks to the following result. IfR is a total quotient ring with Property A, then R is a UQLO-ring (if and) only if R is a ULO-ring. For each positive integer n, there exists an n-dimensional ring R which is a UQLO-ring but not a ULO-ring.

Amir Mafi, Department of Mathematics, University of Kurdistan, Pasdaran ST., P.O. Box: 416, Sanandaj, Iran  and Institute for Research in Fundamental Science (IPM), P. O. Box 19395-5746, Tehran, Iran; (a_mafi@ipm.ir).
Finiteness of local cohomology modules over rings of small dimension , pp. 361-368.
ABSTRACT. Let R be a commutative Noetherian local  ring of dimension d, a an ideal of R, and M, N two finitely generated R-modules. We prove that if d≤2, then ExtpR (M, Hqa (N)) is a-cofinite for all p,q ≥ 0. Also, if  d≤3 then the set of associated primes of any quotient  of  ExtpR (R/a, Hqa(M,N)) and  ExttR (R/a, ExtpR (M, Hqa (N)))  are finite for all p,q,t ≥ 0.

Crampin, M., Department of Mathematical Physics and Astronomy, Ghent University, B--9000 Gent, Belgium (Crampin@btinternet.com).
Some remarks on the Finslerian version of Hilbert's fourth problem, pp. 369-391.
ABSTRACT. The Finslerian version of Hilbert's fourth problem is the problem of finding projective Finsler functions. Álvarez Paiva (J. Diff. Geom. 69 (2005) 353-378) has shown that projective absolutely homogeneous Finsler functions correspond to symplectic structures on the space of oriented straight lines, with certain properties. I give new and direct proofs of his main results, and show how they are related to the more classical formulations of the problem due to Hamel and Rapcsák.

Munteanu, Ovidiu, Transylvania University,  I. Maniu, No 50, 500091, Brasov, Romania (gh.munteanu@info.unitbv.ro)
Eigenvalue estimates for the Laplacian on Finsler spaces of Randers type, pp. 393-404.
ABSTRACT. In this paper we investigate upper and lower bounds for the eigenvalues of the Bao-Lackey Laplacian on Finsler manifolds of Randers type in terms of the eigenvalues of the Laplacian defined by the underlying Riemannian metric.

Ezequiel Barbosa and Marcos Montenegro, ICEx - Departamento de Matemática Universidade Federal de Minas Gerais. Av. Antônio Carlos, 6627 - Caixa Postal 702 - CEP 30161-970, Belo Horizonte, MG- Brazil  (ezequiel@mat.ufmg.br), (montene@mat.ufmg.br.
Multiplicity for the prescribed scalar curvature problem, pp. 405-413.
ABSTRACT.  In this work we construct multiple metrics on product manifolds conformal to the usual product metric having a same nonconstant positive scalar curvature. On such manifolds, R. Schoen exhibited multiple solutions for the Yamabe problem.

Pit-Mann, Wong (deceased) and Bing-Ye Wu, Minjiang University, Fuzhou 350108 China (bingyewu@yahoo.cn).
On the holomorphic sectional curvature of complex Finsler manifolds, pp. 415-433.
ABSTRACT. In this note we extend the result, expressing holomorphic sectional curvature in terms of Gaussian curvature of immersed complex curves, of H. Wu from the setting of Hermitian geometry to that of Finsler geometry, complex manifolds with smooth and strictly pseudoconvex Finsler metrics. Applications, concerning the negativity of holomorphic sectional curvature, are also given extending results in Hermitian geometry to Finsler geometry. In the last section we extend the results to the case where the Finsler metric is not smooth. This is useful because intrinsic metrics on complex manifolds and in general not smooth. The curvature property of smooth intrinsic metrics is due to B. Wong.

Menassie Ephrem, Department of Mathematics and Statistics, Coastal Carolina University, Conway, SC 29528-6054} (menassie@coastal.edu) and Jack Spielberg, Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804 (jack.spielberg@asu.edu).
K-theory of C*-algebras of directed graphs,pp. 435-447.
ABSTRACT. For a directed graph E, we compute the K-theory of the C*-algebra C*(E) from the Cuntz-Krieger generators and relations. First we compute the K-theory of the crossed product C*(E) × T, and then using duality and the Pimsner-Voiculescu exact sequence we compute the K-theory of C*(E) × T × Z. The method relies on the decomposition of C*(E) as an inductive limit of Toeplitz graph C*-algebras, indexed by the finite subgraphs of E. The proof and result require no special assumptions about the graph, and is given in graph-theoretic terms. This can be helpful if the graph is described by pictures rather than by a matrix.

Bellido, José C., ETSI Industriales, Universidad de Castilla-La Mancha. 13071 Ciudad Real. Spain (JoseCarlos.Bellido@uclm.es), and Mora-Corral, Carlos, BCAM – Basque Center for Applied Mathematics. Bizkaia Technology Park, building 500. 48160 Derio (Vizcaya). Spain (mora@bcamath.org).
Approximation of Hölder continuous homeomorphisms by piecewise affine homeomorphisms, pp. 449-500.
ABSTRACT. This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R² that is globally Hölder continuous of exponent α for some α in (0,1], and whose inverse is also globally Hölder continuous of exponent α can be approximated in the Hölder norm of exponent β by piecewise affine homeomorphisms, for some β in (0,α) that only depends on α. The proof is constructive. We adapt the proof of simplicial approximation in the supremum norm, and measure the side lengths and angles of the triangulation over which the approximating homeomorphism is piecewise affine. The approximation in the supremum norm, and a control on the minimum angle and on the ratio between the maximum and minimum side lengths of the triangulation suffice to obtain approximation in the Hölder norm.

Anoussis, M., University of the Aegean, GR-83 200, Karlovassi, Samos, Greece (mano@aegean.gr), Katavolos, A., University of Athens, Panepistimioupolis, GR-157 84, Athens, Greece (akatavol@math.uoa.gr), and Todorov, I.G., Queen's University Belfast, Belfast BT7 1NN, U.K. (i.todorov@qub.ac.uk).
Angles in C*-algebras, pp. 501-517.
ABSTRACT. In this work we characterise the C*-algebras A generated by projections with the property that every pair of projections in A has positive angle, as certain extensions of abelian algebras by algebras of compact operators. We show that this property is equivalent to a lattice theoretic property of projections and also to the property that the set of finite-dimensional *-subalgebras of A is directed.

Oliveira, Lina, Instituto Superior Técnico, Portugal (linaoliv@math.ist.utl.pt). 
Finite rank operators in Lie ideals of nest algebras, pp. 519-536.
ABSTRACT.  The finite rank operators lying in a norm closed Lie ideal of a continuous nest algebra are characterised as those finite rank operators satisfying a condition determined by a left order continuous homomorphism on the nest such that they form themselves an associative ideal. As a consequence, it is possible to identify the norm closure of the set of finite rank operators lying in the norm closed Lie ideal as consisting of the compact operators lying in a certain bimodule of the nest algebra. The decomposability of the finite rank operators, which is proved in this work, and the continuity of the nest are essential to obtain these results.

Ohno, Shûichi, Nippon Institute of Technology, 4-1-1 Gakuendai, Miyashiro, Minami-Saitama 345-8501, Japan (ohno@nit.ac.jp) and Stroethoff, Karel, Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812--0864, USA (karel.stroethoff@umontana.edu).
Weighted Composition Operators from Reproducing Hilbert Spaces to Bloch Spaces, pp. 537-558.
ABSTRACT. We define weighted composition operators induced by a fixed analytic function and an analytic self-map of the open unit disk. We will consider the questions of when the weighted composition operator maps boundedly or compactly one analytic function space to another. The function spaces which we deal with are the reproducing Hilbert spaces of analytic functions on the open unit disk, the Bloch space and the little Bloch space.

Hu, Shouchuan, Shandong Normal University, Jinan, Shandong, China, and Missouri State University, Springfield, MO 65804, USA (shu@missouristate.edu) and Papageorgiou, Nikolas S., National Technical University, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr).
Three nontrivial solutions for noncoercive asymptotically linear elliptic problems, pp. 559-576.
ABSTRACT. We consider a semilinear elliptic equation, assuming that the right and side nonlinearity is asymptotically linear and our conditions on it imply that the Euler functional of the problem is noncoercive. Using variational methods coupled with Morse theory and in particular the use of critical groups, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign.

Markus Niess, Katholische Universität Eichstätt-Ingolstadt MGF, D-85071 Eichstätt, Germany (markus.niess@ku-eichstaett.de).
On the sharpness of Jentzsch's Theorem - Generic properties, pp. 577-589.
ABSTRACT. Answers of questions posed by Luh are given.

S.Jain, V.K.Jain, Department of Mathematics, Bareilly College, Bareilly,U.P., India  and A.Verma, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, INDIA (arunreeta.verma@gmail.com).
Some general expansions of RR type, pp. 591-625.
ABSTRACT. Multiple series expansions for products similar to the ones in Rogers-Ramanujan identities are derived. These expansions for particular values yield full quota of expansions, for the products in which successive terms advance by the prime numbers 3,5,7,11,13,29,31,37, as infinite/double/triple series. Some multiple series expansions for multiples of these primes are also obtained.

Jishan Fan, Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, P.R.China (anjishan@njfu.com.cn) and  Tohru Ozawa, Department of Applied Physics, Waseda University, Tokyo, 169-8555, Japan (txozawa@waseda.jp).
Regularity criterion for the incompressible viscoelastic fluid system, pp. 627-636.
ABSTRACT.  The incompressible viscoelastic fluid system of the Oldroyd-B model is studied.

Klaas Pieter Hart, TU Delft, 2600 GA Delft, the Netherlands (k.p.hart@tudelft.nl) and Elżbieta Pol, University of Warsaw, 02-197 Warszawa, Poland (pol@mimuw.edu.pl).
On hereditarily indecomposable compacta and factorization of maps, pp. 637-644.
ABSTRACT. We employ a dual version of the Löwenheim-Skolem theorem to obtain a factorization theorem for maps with hereditarily indecomposable fibers. This enables us to obtain universal hereditarily indecomposable compact spaces as well as hereditarily indecomposable compactifications of any prescribed weight and dimension. We also reprove theorem of Mackowiak on the existence of universal hereditarily indecomposable continua.

Rongxin Shen, Department of Mathematics, Taizhou Teachers' College, Taizhou 225300, P. R. China (rxshen@yahoo.cn) and Shou Lin, Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, P. R. China (linshou@public.ndptt.fj.cn).
On discrete spaces and AP spaces, pp. 645-651.
ABSTRACT. In this paper, it is proved that a space Y is discrete if and only if every sequentially quotient mapping onto Y is bi-quotient (weak-open). Also, we discuss AP-spaces which are important generalizations of Frechet-Urysohn spaces. We give a new characterization of AP-spaces and prove that every space is an almost-open image of some AP-space.

Krupski, Pawel, Mathematical Institute, University of Wroclaw, Pl. Grunwaldzki 2/4, 50—384, Wroclaw, Poland, (krupski@math.uni.wroc.pl), and Tuncali, Murat, Department of Computer Science and Mathematics, Nipissing University, North Bay, Ontario, P1B 8L7, Canada ( muratt@nipissingu.ca) .
Maps of rank m , pp. 653-675
ABSTRACT. A function F from a space X to a space Y is said to be of rank m, if the cardinality of the set of all non-degenerate fibers of F does not exceed m. In this paper, we study general properties and find some invariants of  rank m maps on topological spaces. Some close relationships between them and monotone maps are revealed. Monotone maps on surfaces are approximated by countable rank monotone maps if the set of local separating points of the range space has a countable closure. Projective classes of countable rank maps and of fully closed maps are evaluated.

Fucai Lin, Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, P. R. China; Department of Mathematics, Sichuan University, Chengdou, 610064, P. R. China (linfucai2008@yahoo.com.cn) and Shou Lin, Department of Mathematics, Zhangzhou Normal University, Zhangzhou 363000, P. R. China; Institute of Mathematics, Ningde Teachers' College, Ningde, Fujian 352100, P. R. China (linshou@public.ndptt.fj.cn).
Uniform bases at non-isolated points and maps, pp. 677-688.
ABSTRACT. In this paper, the authors mainly discuss the images of spaces with an uniform base at non-isolated points, and obtain the following main results: (1) Perfect maps preserve spaces with an uniform base at non-isolated points; (2) Open and closed maps preserve regular spaces with an uniform base at non-isolated points; (3) Spaces with an uniform base at non-isolated points don't satisfy the decomposition theorem.

Dow, Alan, UNC Charlotte, Department of Mathematics and Statistics, 9201 University City Blvd., Charlotte, NC, U.S.A., 28223, mailto:adow@uncc.edu  and Tkachuk, Vladimir V., Universidad Autónoma Metropolitana, Mexico City, Mexico, 09340, mailto:vova@xanum.uam.mx.
.Connected compactifications of countable spaces , pp. 689-698.
ABSTRACT. Call a space X Tychonoff connectifiable if X has a connected Tychonoff extension or, equivalently, a connected Hausdorff compactification. We prove that for any countable dense-in-itself Tychonoff space X there exists a Tychonoff space Y such that X ⊂ Y and X is dense in Y while Y\ X is a singleton and Y is not Tychonoff connectifiable. In particular, the space Q of the rationals has a one-point Tychonoff extension which is not Tychonoff connectifiable. We also show that it is consistent with ZFC that the space Q∪ {p} has a connected compactification for any p ∈ βQ\ Q.