## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 41, No. 3 , 2015

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)

Contents

Gollakota V.V. Hemasundar, Department of Mathematics, S.I.W.S. College,Wadala,Mumbai-400031, INDIA (gvvhemasundar@yahoo.co.in).
Families of compact RIEMANN surfaces, pp. 727-731.
ABSTRACT.The aim of this article is to indicate how Koebe’s General Uniformisation Theorem for planar Riemann surfaces may be used to construct families of compact Riemann surfaces of every genus (explicitly in case of g=1) in a very concrete way.

Calvaruso, Giovanni, Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy (giovanni.calvaruso@unisalento.it).

ABSTRACT. Let g denote an arbitrary 2n-dimensional Lie algebra. A symplectic structure on g is a closed two-form ω on g of maximal rank, that is, such that ωn ≠ 0. A paraKähler Lie algebra is an even-dimensional Lie algebra \g, equipped with a pair (J,g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental two-form Ω(X,Y)=g(X,JY) is symplectic. We completely classify left-invariant paraKähler structures on four-dimensional Lie algebras and then study the geometry of their paraKähler metrics, emphasizing different behaviors with respect to their Kähler analogues.

Leobardo Rosales, Keimyung University, Department of Mathematics, 1095 Dalgubeol-daero, Daegu, Republic of Korea, 704-701 (lrosales@gw.kmu.ac.kr).
The q-Valued minimal surface equation, pp. 749-765.
ABSTRACT. The q-valued minimal surface equation is a partial differential equation over the unit disk in the plane, but degenerate at the origin, producing solutions which if can be continuously extended at the origin, then have corresponding q-valued graphs which are stable immersed minimal surfaces in the unit cylinder. Here, we give a complete geometric description of the q-valued graph corresponding to a solution to the q-valued minimal surface equation in case we cannot extend the solution continuously across the origin.

Eunmi Pak, Young Jin Suh and Changhwa Woo, Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea, (legalgwch@knu.ac.kr).
ABSTRACT. In this paper, we have considered new restricted commuting conditions between the restricted Jacobi operator and the Ricci tensor for real hypersurfaces in complex two-plane Grassmannians G2(Cm+2). By using a new method of simultaneous diagonalzation for commuting symmetric operators, we verified a complete classification for real hypersurfaces in G2(Cm+2) with above conditions that become equivalent condition for hypersurfaces of Type~A.

Sievewright, Daniel, Western Michigan University, Kalamazoo, MI 49008, (daniel.s.sievewright@wmich.edu).
Deddens algebras for weighted shifts, pp. 785-814.
ABSTRACT. We describe the possible structure of the Deddens algebra associated with a weighted shift, focusing on injective weighted shifts of finite multiplicity. We give necessary and sufficient conditions for such an algebra to have a nontrivial invariant subspace. Then, several examples are given to show that we cannot strengthen the results about the structure of the Deddens algebra.

More symmetric polynomials related to p-norms, pp. 815-830.
ABSTRACT. Extending a previous result of the author, we define two additional new families of symmetric polynomials which can be used to state sufficient conditions for p-norm inequalities between two real n-tuples x and y, in the range p>2. When x consists of the eigenvalues of an n by n matrix A, we give a formula for the polynomials in terms of the entries of A, generalizing sums of principal subdeterminants of A.

Johnston, Nathaniel, Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario N1G~2W1, Canada and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L~3G1, Canada (nathaniel.johnston@uwaterloo.ca) and Kribs, David W., Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario N1G~2W1, Canada and Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L~3G1, Canada (dkribs@uoguelph.ca).
Duality of entanglement norms, pp. 831-847.
ABSTRACT. We consider four norms on tensor product spaces that have appeared in quantum information theory and demonstrate duality relationships between them. We show that the product numerical radius is dual to the robustness of entanglement, and we similarly show that the S(k)-norm is dual to the projective tensor norm. We show that, analogous to how the product numerical radius and the S(k)-norm characterize k-block positivity of operators, there is a natural version of the projective tensor norm that characterizes Schmidt number. In this way we obtain an elementary new proof of the cross norm criterion for separability, and we also generalize both the cross norm and realignment criteria to the case of arbitrary Schmidt number.

Ismail Kombe, Department of Mathematics, Faculty of Science and Letters, Istanbul Commerce University, Uskudar, Istanbul, Turkey (ikombe@ticaret.edu.tr)
Hardy and Rellich-type inequalities with remainders for Baouendi-Grushin vector fields, pp. 849-874.
ABSTRACT. In this paper we study Hardy and Rellich-type inequalities and their improved versions for the Baouendi-Grushin vector fields.

Hotovy, Ryan, Texas A&M University, College Station, TX 77843 (rhotovy@math.tamu.edu), Larson, David, Texas A&M University, College Station, TX 77843 (larson@math.tamu.edu), and Scholze, Sam, Texas A&M University, College Station, TX 77843 (scholzes@math.tamu.edu).
Binary frames, pp. 875-899.
ABSTRACT. We investigate frame theory over the binary field Z2. We consider general finite dimensional vector spaces over Z2. equipped with an (indefinite) inner product. We characterize precisely when two such spaces are unitarily equivalent in terms of a computable invariant we term the matricial spectrum. We investigate the existence of dual frames and Parseval frames for such spaces.

Kunszenti-Kovács, Dávid, ELTE TTK, Institute of Mathematics and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, 1117 Budapest, Pázmány P. sétány 1/C, Hungary (daku@fa.uni-tuebingen.de).
Almost weak polynomial stability of operators, pp. 901-913.
ABSTRACT. We investigate whether almost weak stability of an operator T on a Banach space X implies its almost weak polynomial stability. We show, using a modified version of the van der Corput Lemma, that if X is a Hilbert space and T a contraction, then the implication holds. On the other hand, based on a TDS arising from a two dimensional ODE, we give an explicit example of a contraction on a C0 space that is almost weakly stable, but its appropriate polynomial powers fail to converge weakly to zero along a subsequence of density 1. Finally we provide an application to convergence of polynomial multiple ergodic averages.

Banjade, Debendra P., Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, AL 35487 (dpbanjade@crimson.ua.edu) and Trent, Tavan T., Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, AL 35487 (ttrent@as.ua.edu).
Wolff's problem of ideals in the multiplier algebra on weighted Dirichlet space, pp. 915-932.
ABSTRACT. Wolff proved a theorem on ideal membership in the algebra of bounded analytic functions on the unit disk. We establish an analogous version for the multiplier algebra on weighted Dirichlet spaces.

McCann, Shawn, University of Regina, Regina, SK, S4S 0A2 (mccann1s@uregina.ca).
C*-algebras associated with topological group quivers II: K-groups , pp. 933-964.
ABSTRACT. Topological quivers generalize the notion of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to a topological quiver Q is a C*-correspondence, and in turn, a Cuntz-Pimsner algebra C*(Q). Given Γ a locally compact group and endomorphisms α and β of Γ, one may construct a topological quiver Qα, β(Γ) with vertex set Γ, and edge set Eα, β(Γ)={(x, y) in Γ x Γ where α(y)=β(x)}. In a previous paper, the author examined the Cuntz-Pimsner algebra Oα, β(Γ)= C*(Qα, β(Γ)) and found generators (and their relations) of Oα, β(Γ). In this paper, the author uses this information to create a six term exact sequence in order to calculate the K-groups of Oα, β(Γ) and in particular, those K-groups resulting from choosing Γ to be the d-torus with endomorphisms α and β as integral matrices with non-zero determinant.

Changguo Wei, School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, P.R. China (weicgqd@163.com).
On the classification of certain unital extensions of C*-algebras, pp. 965-991.
ABSTRACT. We pursue the classification of unital essential extensions up to unitary equivalence or isomorphism. We characterize the six-term exact sequences with base points of the unital extensions which are weakly unitarily equivalent and give a necessary and sufficient condition for their six-term exact sequences with base points being congruent. Using these results, we prove several UCTs for unital extensions which generalizes the UCT of Brown and Dadarlat. As an application, we give a generalization of the classic BDF-theory and classify certain classes of unital full extensions up to unitary.

Hou, Chunjuan, Huashang College, Guangdong University of Business Studies, Guangzhou,  511300,  China (houchunjuanhao@163.com), Hu, Shouchuan, Department of Mathematics, Missouri State University, Springfield, MO 65804 (shu@missouristate.edu), and Papageorgiou, Nikolas S., Department of  Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr).
Positive solutions for parametric nonlinear neumann problems with competing nonlinearities, pp. 993-1019.
ABSTRACT. We consider a parametric nonlinear Neumann problem driven by the p-Laplacian and with a reaction exhibiting the competing effects of a concave (p-sublinear) and of a convex (p-superlinear) term.  Using critical point theory together with truncation and comparison techniques, we prove a bifurcation type theorem describing the set of positive solutions as the parameter varies.

Banič, Iztok, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia (iztok.banic@uni-mb.si), Črepnjak, Matevž, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia (matevz.crepnjak@um.si), Erceg, Goran, Faculty of Natural Sciences and Mathematics, University of Split, 21000 Split, Croatia (gorerc@pmfst.hr), Merhar, Matej, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia (matej.merhar@uni-mb.si), and Milutinović, Uroš, Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia (uros.milutinovic@uni-mb.si).
Inducing functions between inverse limits with upper semicontinuous bonding functions, pp. 1021-1037.
ABSTRACT. In this paper we introduce the category CU in which the compact metric spaces are objects and upper semicontinuous functions from X to 2Y are morphisms from X to Y. We also introduce the category ICU of inverse sequences in CU. Then we investigate the induced functions between inverse limits of compact metric spaces with upper semicontinuous bonding functions. We provide criteria for their existence and prove that under suitable assumptions they have surjective graphs. We also show that taking such inverse limits is very close to being a functor (but is not a functor) from ICU to CU, if morphisms are mapped to induced functions. At the end of the paper we give a useful application of the mentioned results.

Van Nall, Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173 (vnall@richmond.edu).
More continua which are not the inverse limit with a closed subset of a unit square, pp. 1039-1050.
ABSTRACT. There are nice characterizations of continua that are homeomorphic to an inverse limit with a continuous function from [0,1] to [0,1]. Continua that are homeomorphic to an inverse limits with a single upper semi-continuous set valued function from [0,1] into the closed subsets of [0,1] have not been characterized, and so, in order to learn more about them, we add to the list of continua which cannot be obtained as such an inverse limit.

Dirk Hofmann, CIDMA -- Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal (dirk@ua.pt) and Gavin J. Seal, Mathematics Institute for Geometry and Applications, Ecole Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland (gavin.seal@fastmail.fm).
Exponentiable approach spaces, pp. 1051-1062.
ABSTRACT. In this note we present a characterisation of exponentiable approach spaces in terms of ultrafilter convergence.

Angelo Bella, Department of Mathematics and Computer Science, University of Catania, Città universitaria viale A. Doria 695125, Catania Italy (bella@dmi.unict.it) and Santi Spadaro, Departamento de Matematica, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Rua do Matao, 1010, Cidade Universitaria, 05508-090, Sao Paulo, SP Brazil (santidspadaro@gmail.com).
Infinite games and cardinal properties of topological spaces, pp. 1063-1077.
ABSTRACT. Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindelof first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic variant of the countable chain condition, introduced by Daniels, Kunen and Zhou in 1994. We prove that it is equivalent to countable pi-weight for spaces having countable local pi-weight at every point, and exploit it to give a new proof of Shapirovskii's classical bound on the number of regular open sets of a regular topological space.

Michalik, Daria, Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszynski University, Woycickiego 1/3, 01-938 Warsaw, Poland (d.michalik@uksw.edu.pl).
Embeddings into the product of generalized Sierpiński curves admitting extensions of maps, pp.1079-1086.
ABSTRACT. For each cardinal number τ we can construct a 1-dimensional space Σ(τ) - called generalization of Sierpiński curve of weight τ. We prove in this paper that for a metrizable space X and a family {fi}i ∈ N of mappings of X into itself the set of all embeddings h : X → Σ(τ)ω such that dim Cl(h(X)) ≤ dim X and all functions hfih-1 are extendable to mappings fi* : Cl(h(X)) → Cl(h(X)) is residual in C(X,Σ(τ)ω).

Daniel, D,. Lamar University, Beaumont, Texas 77710 (dale.daniel@lamar.edu) and Tuncali, M., Nipissing University, North Bay, Ontario P1B 8L7 (muratt@nipissingu.ca).
Embeddings of non-metric products in images of ordered compacta, pp. 1087-1096.
ABSTRACT. We generalize a construction of A. J. Ward to build continuous images of ordered compacta that contain a non-metrizable product. Such spaces have a relatively restricted structure. As a result, one may express in somewhat explicit form a mapping of an ordered compactum onto an appropriate subspace of a compactum containing such a non-metric product.

Er-Guang Yang, School of Mathematics & Physics, Anhui University of Technology, Maanshan 243002, P.R. China (egyang@126.com).
Properties defined with semi-continuous functions and some related spaces, pp. 1097-1106.
ABSTRACT. The notions of properties (UL)mwl, (UL)mK and (UL)m are introduced to generalize the notion of property (USC)m introduced by Ohta and Sakai. It is shown that spaces having property (UL)mwl (resp. (UL)mK, (UL)m) coincide with countably paracompact (resp. countably mesocompact, countably metacompact) spaces. As applications, some insertion theorems of countably paracompact spaces and countably mesocompact spaces are obtained.