HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 40, No. 3 , 2014

Editors:  D. Bao (San Francisco, SFSU), D. Blecher (Houston), Bernhard G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station),  M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice).
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics



Contents

Fuchs, László, Dept. of Mathematics, Tulane University, New Orleans, LA 70118, U.S.A. (fuchs@tulane.edu), Salce, Luigi, Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy (salce@math.unipd.it), and Zanardo, Paolo, Dipartimento di Matematica, Università di Padova, 35121 Padova, Italy (pzanardo@math.unipd.it).
Divisibility in cyclically presented modules over integral domains, pp. 663-680.
ABSTRACT. We consider divisibility properties of elements in cyclically presented modules over integral domains, with special focus on Bezout domains. We study gaps that signal a larger than expected increase in divisibility.

Yingbo Han, Xinyang Normal University, Xinyang ,46400, Henan, China (yingbohan@163.com), and Shuxiang Feng, Xinyang Normal University, Xinyang ,46400, Henan, China (shuxiangfeng78@163.com).
Montonicity formulas and the stability of F-stationary maps with potential, pp. 681-713
ABSTRACT. In this paper, we introduce the notion of F-stationary map with potential with respect to the functional ΦF,H. Then we use the stress-energy tensor to obtain the monotonicity formulas and vanishing theorems for these maps under some conditions on H. We also obtain the first variation formula and the second variation formula for the functional ΦF,H. Then we study the stability of F-stationary map with potential form or into the standard sphere.

Kamal Boussaf, Abdelbaki Boutabaa and Alain Escassut, Université Blaise Pascal, Clermont-Ferrand, France 63171 (Kamal.Boussaf@math.univ-bpclermont.fr) , (Abdelbaki.Boutabaa@math.univ-bpclermont.fr) , (Alain Escassut@math.univ-bpclermont.fr).
 Growth of p-adic functions and applications, pp. 715-736.
ABSTRACT. Let K be an algebraically closed p-adic complete field of characteristic zero. We define the order of growth and the type of growth of an entire function f on K as done on the field C and show that they satisfy the same relations as in complex analysis, with regards to the coefficients of f. But here, another expression that we call cotype of f, depending on the number of zeros inside disks, is very important and we show under certain wide hypothesis, that this cotype is the product pf the growth by the type, a formula that has no equivalent in complex analysis and suggests that it might hold in the general case. We check that f and its derivative have the same growth order and the same growth type and present an asymptotic relation linking the numbers of zeros inside disks for two functions of same order. We show that the derivative of a transcenental entire function f has infinitely many zeros that are not zeros of f and particularly we show that f' cannot divide f when the p-adic absolute value of the number of zeros of f inside disks satisfies certain inequality and particularly when f is of finite order.

M. Crampin, Department of Mathematics, Ghent University, Krijgslaan 281, B--9000 Gent, Belgium (m.crampin@btinternet.com.
On the construction of Riemannian metrics for Berwald spaces by averaging, pp. 737-750.
ABSTRACT. The construction of Riemannian metrics on the base manifold of any given Finsler space by averaging suitable objects over indicatrices, such that the Levi-Civita connection of the metric coincides with the canonical Berwald connection of the Finsler space when the Finsler space is a Berwald space, is discussed. Some examples of such metrics are already known, but several new ones, all in principle different, are defined and analysed.

Hyunjin Lee, The Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea (lhjibis@hanmail.net), Young Jin Suh and Changhwa Woo, Kyungpook National University, Taegu 702-701, Republic of Korea (yjsuh@knu.ac.kr).
Real hypersurfaces in complex two-plane Grassmannians with commuting Jacobi operators, pp. 751-766.
ABSTRACT.In this paper, we introduce a new commuting condition composed of the Jacobi operator RX, the structure tensor φ and the shape operator A for Hopf hypersurfaces M in complex two-plane Grassmannians G2(Cm+2). By using such a commuting condition, we give a complete classification of Hopf hypersurfaces in G2(Cm+2).  

Cheng-Kai Liu, Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan (R.O.C.) (ckliu@cc.ncue.edu.tw).
Maps characterized by Lie products on nest algebras,  pp. 767-777.
ABSTRACT. Let H be a complex Hilbert space and let T(N) be a nest algebra on H. We characterize linear maps f,g,h:T(N) →T(N) satisfying f([x,y])=[g(x),y]+[x,h(y)] for all x,y in T(N).
 

Sołtan, Piotr M., Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland, and Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland (piotr.soltan@fuw.edu.pl).
On quantum maps into quantum semigroups, pp. 779-790.
ABSTRACT. We analyze the recent examples of quantum semigroups defined by M.M. Sadr (M.M. Sadr: A kind of compact quantum semigroups. ArXiv:0808.2740v2 [math.OA]) who also brought up several open problems concerning these objects. These are defined as quantum families of maps from finite sets to a fixed compact quantum semigroup. We show that these are special cases of dual free products of quantum semigroups. This way we can answer all the questions stated by M.M. Sadr. Along the way we discuss the question whether restricting the comultiplication of a compact quantum group to a unital C*-subalgebra defines such a structure on the subalgebra. In the last section we show that the quantum family of all maps from a non-classical finite quantum space to a quantum group (even a finite classical group) might not admit any quantum group structure.

El Harti, Rachid, Dept. of Mathematics and Computer Sciences, Faculty of Sciences and Techniques, Univ. Hassan I, BP 577, Settat, Morocco (relharti@gmail.com), Phillips, N. Christopher, Dept. of Mathematics, Univ. of Oregon, Eugene OR 97403-1222, USA, and Pinto, Paulo R., Dept. of Mathematics, Inst. Superior Técnico, Technical Univ. of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal (ppinto@math.ist.utl.pt).

Profinite pro-C*-algebras and pro-C*-algebras of profinite groups, pp. 791-816.
ABSTRACT. We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the unitary representations of the group which factor through a finite group. We define natural homomorphisms from the C*-algebra of a locally compact group and its profinite completion to the pro-C*-algebra of the profinite completion of the group. We give some conditions for injectivity or surjectivity of these homomorphisms, but an important question remains open.

Freeman, Daniel, Department of Mathematics, University of Texas at Austin, Austin, TX 78712 (freeman@math.utexas.edu), Poore, Daniel, Department of Mathematics, Pomona College, Claremont, CA 92711 (dep02007@mymail.pomona.edu), Wei, Ann Rebecca, Department of Mathematics, Northwestern University, Evanston, IL 60208 (rwei@math.northwestern.edu), and Wyse, Madeline, Department of Mathematics, Pomona College, Claremont, CA 92711 (mkw02007@mymail.pomona.edu).
Moving Parseval frames for vector bundles, pp. 817-832.
ABSTRACT. arseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold.

S. H. Kulkarni,  Indian Institute of Technology Madras, Chennai-600036,  India (shk@iitm.ac.in).
The group of invertible elements of a real Banach algebra, pp. 833-836.
ABSTRACT.The following result is proved: Let A be a commutative real Banach algebra with unit 1. Let G denote the group of invertible elements of A and let G1 be the connected component of G containing 1. If the quotient group G/ G1 contains an element of finite order other than G1, then the order of such an element must be 2. If the group G/ G1 is of finite order, then its order must be 2n for some nonnegative integer n.

Kucerovsky, Dan University of New Brunswick at Fredericton, NB, Canada E3B 5A3 and Sarraf, Aydin University of New Brunswick at Fredericton, NB, Canada E3B 5A3 (dkucerov@unb.ca).
Schur multipliers and matrix products, pp. 837-850.
ABSTRACT. We give necessary and sufficient conditions for a Schur map to be a homomorphism with respect to ordinary matrix multiplication, with some generalizations to the infinite-dimensional case. In the finite-dimensional case, we find that a Schur multiplier distributes over matrix multiplication if and only if the coefficients of the Schur matrix are of the form aij=f(i)/f(j) for some f. In addition, it is shown that it is possible to enumerate all *-preserving multiplicative Schur maps on Mn(R). We also study the relation of Schur maps to the extreme points of certain sets.

Xu, Xiangsheng, Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762 (xxu@math.msstate.edu)
A new interpolation inequality and its applications to a semiconductor model, pp. 851-874.
ABSTRACT. In this paper we study a drift-diffusion semiconductor model with variable electron mobility. We obtain conditions under which the electron density is bounded above and bounded away from 0 below. Part of the proof is based upon a new interpolation inequality which seems to be of interest on its own right.
 

Niteesh Sahni and Dinesh Singh, Department of Mathematics, University of Delhi, Delhi 110007, India (niteeshsahni@gmail.com), (dineshsingh1@gmail.com).
Multiplication by monomials on BMOA, pp. 875-883.
ABSTRACT.In the recent and important context of Nevanlinna-Pick interpolation theory for classes of holomorphic functions with constraints, the starting point has been to describe the common invariant subspaces of the operators of multiplication by S2 and S3 on the Hardy classes. In the same spirit this paper offers as a new result the charcterization of the common invariant subspaces of S2 and S3 on the space BMOA. In addition we present a new, elementary and short proof of the invariant subspace characterization of the operator S on BMOA and use this to give a new proof of the invariant subspace characterization of the backward shift on the Hardy space H1 that is also short and elementary.

Will Brian, Mathematics Department, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118 (wbrian.math@gmail.com),   Jan van Mill, Korteweg-de Vries Institute for Mathematics, University of Amsterdam (j.vanMill@va.nl), and Rolf Suabedissen, University of Oxford (suabedis@maths.ox.ac.uk).
Homogeneity and generalizations of 2-point sets, pp. 885-898.
ABSTRACT. We prove the existence of homogeneous n-point sets (i.e., subsets of the plane which meet every line in exactly n points) for every finite n ≥ 3. We also show that for every zero-dimensional subset A of the real line there is a subset X of the plane such that every line intersects X in a topological copy of A.  

Paul J. Szeptycki, Department of Mathematics and Statistics, York University, Toronto, ON Canada M3J 1P3 (szeptyck@yorku.ca) and Artur H. Tomita, Departamento de Matemática, Instituto de Matemática e Estatístíca, Universidade de São Paulo, Rua do Matão, 1010 – Cidade Universitária - CEP 05508-090, São Paulo, Brasil (tomita@ime.usp.br).
Countable  compactness  of  powers  of  HFD groups, pp. 899-916.
ABSTRACT. Let G be the product of continuum copies of 2. Hajnal and Juhász, in 1976, constructed under CH a topological subgroup H of G that is an HFD with the following property (P) the projection of H is onto G(I) for each countable subset I of the continuum, where G(I) is the product of I copies of 2. Such examples are countably compact without non-trivial convergent se- quences and were first used by van Douwen to show that countable com- pactness is not productive in the class of topological groups. We show that the HFD constructed via Random reals have the countable power countably compact. We construct examples to show that the property HFD + (P) in a topological group does not decide the countable compactness of its powers. We show under CH that there exists an HFD group satisfying (P) such that its n-th power is countably compact but its (n + 1)st power is not for every positive integer n. We also show under CH that there is an HFD group that is countably compact in all its powers. Furthermore, we show that there are 2 to the continuum many non-homeomorphic such HFD groups.

Yan-Kui Song, Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210046, China (songyankui@njnu.edu.cn).  
Remarks on star-Menger spaces, pp. 917-925.

ABSTRACT. In this paper, we show the following statements: (1) There exists a Tychonoff star-Menger space having a regular-closed subspace which is not star-Menger; (2) There exists a Hausdorff star-Menger space having a regular-closed Gδ-subspace which is not star-Menger; (3) Assuming some cardinal assumptions, there exists a Tychonoff strongly star-Menger space having a regular-closed subspace which is not strongly star-Menger; (4) An open Fσ-subset of a strongly star-Menger space is strongly star-Menger. The above statements give an answer to a question of Kočinac.

Fucai Lin, Department of Mathematics and Information Science, Zhangzhou Normal University,Fujian, Zhangzhou, 363000, P.R.China (linfucai2008@yahoo.com.cn) and Shou Lin, Institute of Mathematics, Ningde Teachers' College, Ningde, Fujian 352100, P. R. China (linshou@public.ndptt.fj.cn).
Sequence-covering maps on generalized metric spaces, pp. 927-943.

ABSTRACT. Let f: X→ Y be a map. f is a sequence-covering map if whenever {yn} is a convergent sequence in Y there is a convergent sequence {xn} in X with each xn ∈ f-1(yn); f is a 1-sequence-covering map if for each y ∈ Y there is x ∈ f-1(y) such that whenever {yn} is a sequence converging to y in Y there is a sequence {xn} converging to x in X with each xn ∈f-1(yn). In this paper, we mainly discuss the sequence-covering maps on generalized metric spaces, and give an affirmative answer to a question raised in a 2010 paper by the authors,  and some related questions. Moreover, we also prove that open and closed maps preserve strongly monotonically monolithity, and closed sequence-covering maps preserve spaces with a σ-point-discrete k-network. Some questions about sequence-covering maps on generalized metric spaces are posed.
 

Hui Li,  Beijing University of Technology, Beijing 100124, China (lihui86@emails.bjut.edu.cn) and Liang-Xue Peng (Corresponding author), Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn).
Some properties on monotonically meta-Lindelöf spaces and related conclusions, pp. 945-956.
ABSTRACT. In this note we point out that there is a compact Hausdorff separable monotonically Lindelöf space which is not metrizable. If X is a monotonically meta-Lindelöf regular space and has caliber ω1, then X is hereditarily Lindelöf. We prove that if X is a compact Hausdorff monotonically meta-Lindelöf space then the following are equivalent: X is hereditarily Lindelöf; X is perfect; the closure of D is perfect for any discrete subspace D of X; s(X)≤ω. We show that if X is a compact Hausdorff monotonically meta-Lindelöf space and has caliber ω1, then there do not exist an uncountable collection {Uα:α∈ω1} of nonempty open subsets of X and a subset {xα,yα:α∈ω1} of X such that: (1) xα∈Uα and X\{yα} contains the closure of Uα for each α∈ω1; (2) For each α∈ω1, the set {xα,yα}⊂∩{Uβ:β<α} or {xα,yα}⊂X\∪{Uβ:β<α}. We also get some conclusions of monotonically meta-Lindelöf spaces which are similar with some known conclusions of monotonically countably metacompact spaces. We show that if X is a regular monotonically countably metacompact space and {Uα:α∈ω1} is an uncountable collection of open sets of X such that X\Uα is compact for each α∈ω1 then {Uα:α∈ω1} is not point-countable. Some known conclusions on monotonically countably metacompact spaces can be gotten by this conclusion.

Hui Li and Liang-Xue Peng*, Beijing University of Technology, Beijing 100124, China (*Corresponding author)  (pengliangxue@bjut.edu.cn) (Peng),  (lihui86@emails.bjut.edu.cn) (Li).
On hereditarily normal rectifiable spaces and paratopological groups, pp. 957-968.
ABSTRACT. In this note, we investigate hereditarily normal rectifiable spaces and discuss some properties of paratopological groups. We mainly show that every hereditarily normal rectifiable space with a non-trivial convergent sequence has a regular Gδ-diagonal. We prove that if G is a hereditarily normal rectifiable space then every compact subspace of G is metrizable. We finally show that every hereditarily normal SIN paratopological group of countable type is first countable.

Valentin Gutev, Department of Mathematics, Faculty of Science, University of Malta, Msida MSD 2080, Malta (valentin.gutev@um.edu.mt), and Takamitsu Yamauchi, Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790-8577, Japan (yamauchi.takamitsu.ts@ehime-u.ac.jp).
Factorising lower semi-continuous mappings, pp. 969-986.
ABSTRACT. In this paper, we deal with factorisations of lower semi-continuous mappings through metrizable spaces. Every metrizable space has a σ-discrete base for its topology, i.e. for the family of its open sets. One of our main results is that a lower semi-continuous mapping from a space X to the nonempty closed subsets of a metrizable space Y can be factorised through a metrizable domain if and only if the pre-image of the topology of Y has a σ-discrete base of cozero-sets. Several special cases are considered, also several applications are presented.

Krzyżanowska, Iwona, Institute of Mathematics, University of Gdańsk, 80-952 Gdańsk, Wita Stwosza 57, Poland (Iwona.Krzyzanowska@mat.ug.edu.pl), and Szafraniec, Zbigniew, Institute of Mathematics, University of Gdańsk, 80-952 Gdańsk, Wita Stwosza 57, Poland (Zbigniew.Szafraniec@mat.ug.edu.pl).
Polynomial mappings into a Stiefel manifold and immersions, pp. 987-1006.
ABSTRACT. For a polynomial mapping from Sn-k to the Stiefel manifold Vk(Rn), where n-k is even, there is presented an effective method of expressing the corresponding element of the homotopy group πn-kVk(Rn)≅ Z in terms of signatures of quadratic forms. There is also given a method of computing the intersection number for a polynomial immersion Sm→R2m.