HJM, Vol. 44, No. 3, 2018

HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 44, No. 3, 2018

Editors: D. Bao (San Francisco, SFSU), D. Blecher (Houston), B. 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 Editors: B. G. Bodmann and K. Kaiser (Houston)

Houston Journal of Mathematics

Contents

Ulrich Albrecht, Department of Mathematics, Auburn University, Auburn, AL 36849, U.S.A. (albreuf@mail.auburn.edu) and Daniel James, Department of Mathematics, Auburn University, Auburn, AL 36849, U.S.A. (jamesdw@mail.auburn.edu).
Dedekind Domains and the P-Rank of Ext, pp. 741-752.
ABSTRACT. This paper discusses to which extent the modules Ext¹_R(A,-) and Ext¹_R(-,A) determine the structure of A in case R is a Dedekind domain. We show that R has to be a Dedekind domain for the obtained results to hold.

Nganou, Jean B., University of Houston-Downtown, Houston TX 77002 (nganouj@uhd.edu).
Profinite completions of MV-algebras, pp. 753-767.
ABSTRACT. We provide a concrete description of the profinite completion of arbitrary MV-algebras in terms of their finite simple quotients, a description that generalizes the well known profinite completion of Boolean algebras as the power sets of their Stone spaces. We use the description found to prove the functoriality of the profinite completion and investigate classes of profinite MV-algebras that are profinite completions of some MV-algebras.

Nguyen Thi Nhung, Thang Long University, Vietnam (hoangnhung227@gmail.com) and Le Ngoc Quynh, An Giang University, Vietnam (nquynh1511@gmail.com).
Unicity of meromorphic mappings from complete Kähler manifolds into projective spaces, pp. 769-785.
ABSTRACT. Let M be a complete Kähler Manifold, whose universal covering is biholomorphic to a ball in C^m. In this article, we prove a uniqueness theorem for meromorphic mappings from M into projective space Pⁿ(C) sharing hyperplanes in general position under a general condition that the intersections of inverse images of any k+1 hyperplanes are of codimension at least two.

Jie Zhang, College of Mathematics, China University of Mining and Technology, Xuzhou 221116, PR China, (zhangjie1981@cumt.edu.cn), Liang Wen Liao, Department of Mathematics, Nanjing University, Nanjing 210093, PR China, (maliao@nju.edu.cn), and Qiang Xu, School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China, (xuqiangwang@jsnu.edu.cn).
On entire solutions of one type of differential and difference equations related to Brück's conjecture, pp. 787-802.
ABSTRACT. In this paper, we mainly consider one special type of differential equation F’-P=R.exp{a(z)}.(F-Q), and we also consider one certain difference equation f(z+c)- f(z)-1=exp{a(z)}.(f-1) under some conditions.

Weihong Yao, Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China (whyao@sjtu.edu.cn).
Zeroes of random polynomials with non-identically distributed coefficients, pp. 803-816.
ABSTRACT. This paper is concerned with the distribution of normalized zero-sets of random polynomials when the coefficients of the polynomials are random variables associated to independent (but not necessarily identical) distributions satisfying additional certain mild conditions. Our main result describes the expectation of such normalized zero-set when the degree of the random polynomial of one complex variable tends to infinity. This generalizes well-known results in the case of the Gaussian distribution. We also apply our main result to study certain random polynomials of several complex variables using the slicing method of value distribution theory.

Dimitrios Poulakis, Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (poulakis@math.auth.gr).
Integral points of algebraic curves with a totally imaginary point at infinity, pp. 817-829.
ABSTRACT. In this paper, we consider plane algebraic curves defined over a totally real number field, having a totally imaginary point at infinity, and we determine an upper bound of polynomial type for the size of their integral points.

Krzyżanowska, Iwona, Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland (Iwona.Krzyzanowska@mat.ug.edu.pl), and Nowel, Aleksandra, Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland (Aleksandra.Nowel@mat.ug.edu.pl).
Mappings into the Stiefel manifold and cross-cap singularities, pp. 831-846.
ABSTRACT. Take n>k>1 such that n−k is odd. In this paper we consider a mapping a from (n−k+1)-dimensional closed ball into the space of (n×k)-matrices such that its restriction to the sphere S^n−k goes into the Stiefel manifold Ṽ_k(ℝⁿ). We construct a homotopy invariant Λ of a|S^n−k which defines an isomorphism between (n−k)-th homotopy group Π_n−kṼ_k(ℝⁿ) and ℤ₂. It can be used to calculate in an effective way the class of a|S^n−k in Π_n−kṼ_k(ℝⁿ) for a polynomial mapping a and to find the number mod 2 of cross-cap singularities of a mapping from a closed m-dimensional ball into ℝ^2m−1, m even.

Glück, Jochen, Ulm University, 89069 Ulm, Germany (jochen.glueck@alumni.uni-ulm.de).
Growth rates and the peripheral spectrum of positive operators, pp. 847-872.
ABSTRACT. Let T be a positive operator on a complex Banach lattice. It is a long open problem whether the peripheral spectrum of T is always cyclic. We consider several growth conditions on T, involving its eigenvectors or its resolvent, and show that these conditions provide new sufficient criteria for the cyclicity of the peripheral spectrum of T. Moreover, we give an alternative proof of the recent result that every (WS)-bounded positive operator has cyclic peripheral spectrum. We also consider irreducible operators T. If such an operator is Abel bounded, then it is known that every peripheral eigenvalue of T is algebraically simple. We show that the same is true if T only fulfils the weaker condition of being (WS)-bounded.

Towsner, Henry, University of Pennsylvania, Pennsylvania, PA 19104 (htowsner@math.upenn.edu).
An inverse Ackermannian lower bound on the local unconditionality constant of the James space, pp. 873-885.
ABSTRACT. The proof that the James space is not locally unconditional appears to be non-constructive since it makes use of an ultraproduct construction. We use this as an example of the way new proof mining techniques can extract explicit bounds from ultraproduct arguments. As a result, we give an explicit combinatorial proof that the James space is not locally unconditional, which gives an inverse Ackermannian lower bound on the rate of growth of the local unconditionality constants of finitely generated subspaces of the James space.

Deepshikha, University of Delhi, Delhi-110007, India (dpmmehra@gmail.com) and Vashisht, Lalit K., University of Delhi, Delhi-110007, India (lalitkvashisht@gmail.com).
On weaving frames, pp. 887-915.
ABSTRACT. Bemrose, Casazza, Grochenig, Lammers, and Lynch introduced a new concept of weaving frames in separable Hilbert spaces. In this paper, we consider an infinite family of frames for separable Hilbert spaces and propose infinitely woven frames. We extend some results about woven frames to infinitely woven frames. It is observed that there exists an infinite family of frames for which finite weaving is possible but the family itself is not a frame under infinite weaving. Two different necessary conditions for infinitely woven frames involving frame bounds have been obtained. We prove that if an infinite family of frames woven into Riesz sequences, then it can weave into Riesz bases. Further a result has been obtained to weave Riesz sequences into a Riesz sequence by taking the diameter of Riesz sequences sufficiently small. Some perturbation results for infinitely woven frames are given.

Yu Zhou, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China (roczhou_fly@126.com), Zihou Zhang, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China (zhz@sues.edu.cn), and Chunyan Liu, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China (cyl@sues.edu.cn).
On representation of isometric embeddings between Hausdorff metric spaces of compact convex subsets, pp. 917-925.
ABSTRACT. We prove the following representation theorem: Let X (resp. Y) be a real Banach space satisfying that the set of all weak star exposed points of the dual unit ball of X (resp. Y) is weak star dense in the dual unit sphere of X (resp. Y), cc(X) (resp. cc(Y)) be the metric space of all compact convex subsets of X (resp. Y) endowed with the Hausdoff distance, and f be a surjective standard isometric embedding from cc(X) onto cc(Y). Then, (1) the restriction of f to X is a surjective linear isometric embedding from X onto Y; (2) for each compact convex subset A of X, f(A) is consisted of f(a) for all a in A.

Lancien, Gilles, Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France (gilles.lancien@univ-fcomte.fr), and Raja, Matias, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain (matias@um.es).
Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces, pp. 927-940.
ABSTRACT. In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for q< p, a q-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a p-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.

Pengcheng Niu, Department of Applied Mathematics, Northwestern Polytechnical University. Shaanxi, 710129, China (pengchengniu@nwpu.edu.cn) and Huiju Wang, Department of Applied Mathematics, Northwestern Polytechnical University. Shaanxi, 710129, China (huijuwang@mail.nwpu.edu.cn).
Gehring's lemma for Orlicz functions in metric measure spaces and higher integrability for convex integral functionals, pp. 941-974.
ABSTRACT. We derive Gehring's lemma in the Orlicz setting by proving boundedness of Hardy-Littlewood maximal functions in a doubling metric measure space. As an application, the higher integrability of minimizers for a convex integral functional is established by providing a general Caccioppoli type inequality.

Ng, P. W., University of Louisiana at Lafayette, Lafayette, LA, 70504 (png@louisiana.edu).
Nonstable absorption, pp. 975-1017.
ABSTRACT. Let X be a finite CW complex and B be a nounital separable simple C*-algebra with continuous scale. Let Ext(C(X),B) be the set of unitary equivalence classes of nonunital extensions of B by C(X). We show that Ext(C(X), B) is a group, and we also characterize the neutral element of Ext(C(X), B), with further information when B has additional regularity properties. We have similar results for a unital version of the functor. In the process, we prove some results involving nonstable absorption, some of which work for general (not necessarily simple) purely infinite corona algebras.

Pol, Elżbieta, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland (E.Pol@mimuw.edu.pl), and Pol, Roman, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland, (R.Pol@mimuw.edu.pl).
Splitting cubes into two Borel punctiform sets, pp. 1019-1027.
ABSTRACT. We present some new observations about splitting cubes into two punctiform sets (i.e., containing no non-trivial continuum) with possibly simple Borel structure and we discuss classical results of Mazurkiewicz, Kuratowski and Sierpiński concerning this topic.

Themba Dube, Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Pretoria, SOUTH AFRICA (dubeta@unisa.ac.za) and Oghenetega Ighedo, Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Pretoria, SOUTH AFRICA (ighedo@unisa.ac.za).
Concerning the summand intersection property in function rings, pp. 1029-1049.
ABSTRACT. A ring has the summand intersection property (SIP, for short) if the intersection of any collection of direct summands is a direct summand. The ring of continuous real-valued functions on a completely regular locale L is denoted by RL. We characterize, in terms of elements and sublocales, the locales L for which RL has SIP. We show that if L is covered by its connected components, and the connected components are open, then RL has SIP. The internal characterization of when RL has SIP in terms of elements of L leads, in a natural way, to a property that resembles the De Morgan property; but is strictly weaker. We thus term it the ``weak De Morgan property". We prove that if a commutative ring with identity has SIP, then the locale of its radical ideals is weakly De Morgan, and conversely if the ring is reduced.

K. S. Singh, Department of Mathematics, University of Delhi, Delhi-110007, India (ksomorjitmaths@gmail.com), H. K. Singh, Department of Mathematics, University of Delhi, Delhi-110007, India (hemantksingh@maths.du.ac.in), and T. B. Singh, Department of Mathematics, University of Delhi, Delhi-110007, India (tej_b_simgh@yahoo.co.in).
The set of balanced points of maps on cohomology lens spaces, pp. 1051-1061.
ABSTRACT. Jaworowski (2002) has determined the index of a lens space with a free action of the group G = S¹ and applied it to prove a Borsuk-Ulam-Yang type theorem for equivariant mappings from L_p^2m+1X→C^k. In this paper, we generalize the above results to a cohomology lens space. Also, we obtain suffcient conditions for nonexistence of equivariant maps X→ S^2k+1 and S^2k+1→ X, where X is a cohomology lens space.

Alas, Ofelia T., Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05311-970 São Paulo, Brasil (alas@ime.usp.br), Martínez-Cadena, Juan A., Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco 186, Apartado Postal 55-532, 09340, México, CDMX, México (jamc88@xanum.uam.mx), and Wilson, Richard G., Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco 186, Apartado Postal 55-532, 09340, México, CDMX, México (rgw@xanum.uam.mx).
Sequential and selective feeble compactness, pp.1063-1079.
ABSTRACT. We study two subclasses of the class of feebly compact spaces, that of the sequentially feebly compact spaces and that of the selectively feebly compact spaces. We show that conditions known to be equivalent to selective pseudocompactness in the class of Tychonoff spaces are also equivalent in the class of Hausdorff spaces. We characterize both maximal selective feeble compactness and maximal sequential feeble compactness and consider the problem of when a Tychonoff space has a sequentially pseudocompact compactification.