HJM, Vol. 45, No. 4, 2019

HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 45, No. 4, 2019

Editors: D. Bao (San Francisco, SFSU), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B. Dacorogna (Lausanne), M. Dugas (Baylor), M. Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), S. Harvey (Rice), A. Haynes (Houston), 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

Shimon Garti, Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel and Ben-Gurion University of the Negev, Beer-Sheva, Israel (shimon.garty@mail.huji.ac.il), and Saharon Shelah, Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA(shelah@math.huji.ac.il).
Depth+ and Length+ of Boolean algebras, pp. 953-963.
ABSTRACT. We investigate the cardinal invariants Length+ and Depth+ in Boolean algebras. Given a sequence of Boolean algebras and an ultrafilter, we compare the value of the invariant of the product algebra with the ultrapower of the invariants of each algebra. We show that strict inequalities are provable in ZFC.

Ivanka Gradeva, University of Plovdiv ‘P. Hilendarski’ – 24 Tzar Asen Str., Plovdiv, Bulgaria (gradeva@uni-plovdiv.bg), and Yordan Epitropov, University of Plovdiv ‘P. Hilendarski’ – 24 Tzar Asen Str., Plovdiv, Bulgaria (epitropov@uni-plovdiv.bg).
Minimal central orthogonal idempotents of semisimple group algebras of metacyclic group, pp. 965-977.
ABSTRACT. Let K be an arbitrary field, whose characteristic does not divide the order of the metacyclic group G. In this paper we make the first step towards our examining the structure of the semisimple group algebra KG, namely - finding a complete system of its minimal central orthogonal idempotents. This is important, because every simple component of the Wedderburn decomposition of KG is a minimal two-sided ideal, which is generated by such an idempotent. For this purpose, in Section 2 we first determine the conjugacy classes in G and we calculate their number. With the restriction, that K is a field of decomposition of G, in Section 3 we find the central orthogonal idempotents of KG and we prove that they are minimal. We finish the examination with a theorem in which without the restriction on the field K we find one complete system of minimal central orthogonal idempotents of KG. In Section 4 we give examples in which for a specific group G and field K we compute explicit expressions for the minimal central orthogonal idempotents of KG.

Laura Cossu, Dipartimento di Matematica ‘Tullio Levi-Civita’ Via Trieste 63, 35121 Padova, Italy(lcossu@math.unipd.it), Paolo Zanardo, Dipartimento di Matematica ‘Tullio Levi-Civita’ Via Trieste 63, 35121 Padova, Italy (pzanardo@math.unipd.it).
Minimal Prüfer-Dress rings and products of idempotent matrices, pp. 979-994.
ABSTRACT. We investigate a special class of Pruefer domains, firstly introduced by A. Dress in 1965: the minimal Dress rings of a field K. We show that, for convenient choices of K, Dress rings may be valuation domains, or, more generally, Bézout domains admitting a weak algorithm. Then we focus on the minimal Dress ring D of the field of real rational functions: we describe its elements, we prove that it is a Dedekind domain and we characterize its non-principal ideals. Moreover, we study the products of 2x2 idempotent matrices over D, a subject of particular interest for Pruefer non-Bézout domains.

Han, Song-Chol, Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea (sc.han@ryongnamsan.edu.kp), (cioc1@ryongnamsan.edu.kp) and Pae, Won-Sok, Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea.
Note on ''Some properties of the Zariski topology of multiplication modules'': Proof of compactness of basic open sets, pp. 995-998.
ABSTRACT. This paper shows by a counter-example that the proof of Theorem 3.7 in [Houston J. Math. 36 (2010) 337] about the compactness of basic open sets in the prime spectrum of a multiplication module over a commutative ring with identity is incorrect, and then gives a new proof.

Mohamed Taoufiq Damir, Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland (mohamed.damir@aalto.fi), and Lenny Fukshansky, Department of Mathematics, 850 Columbia Avenue, Claremont McKenna College, Claremont, CA 91711, USA (lenny@cmc.edu).
Canonical basis twists of ideal lattices from real quadratic number fields, pp. 999-1019.
ABSTRACT. Ideal lattices in the plane coming from real quadratic number fields have been investigated by several authors in the recent years. In particular, it has been proved that every such ideal has a basis that can be twisted by the action of the diagonal group into a Minkowski reduced basis for a well-rounded lattice. We explicitly study such twists on the canonical bases of ideals, which are especially important in arithmetic theory of quadratic number fields and binary quadratic forms. Specifically, we prove that every fixed real quadratic field has only finitely many ideals whose canonical basis can be twisted into a well-rounded or a stable lattice in the plane. We demonstrate some explicit examples of such twists. We also briefly discuss the relation between stable and well-rounded twists of arbitrary ideal bases.

Xianjing Dong, Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, 210093, P.R.China (xjdong05@126.com).
Structure of entire solutions to general linear differential-difference equations with constant coefficients, pp. 1021-1036.
ABSTRACT. In this paper, we first give the structure of entire solutions to the general linear differential-difference equations with constant coefficients which has been studied by many authors. In addition, we also generalize a Varilon's theorem on the structure of entire solutions to a linear differential equation of infinite order.

Yecheng Zhu, Department of Applied Mathematics, Anhui University of Technology, Maanshan 243000, Anhui Province, People's Republic of China and Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui Province, People's Republic of China (zhuyc929@mail.ustc.edu.cn), and Qing Chen, Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui Province, People's Republic of China (qchen@ustc.edu.cn).
Classification and rigidity of λ-hypersurfaces in the weighted volume-preserving mean curvature flow, pp. 1037-1053.
ABSTRACT. In this paper, we study the classification and rigidity of λ-hypersurfaces without the assumption on polynomial volume growth in the weighted volume-preserving mean curvature flow and generalize many meaningful results of self-shrinkers to λ-hypersurfaces.

Deguang Zhong, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China (huachengzhon@163.com) and Wenjun Yuan (corresponding author), School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China (wjyuan1957@126.com).
On estimate of hyperbolically partial derivatives of K-quasiconformal mappings satisfying Poisson's equation and its applications, pp. 1055-1070.
ABSTRACT. Let ||g||_∞=sup_z∈D|g(z)|, where D is the open unit disk and g is continuous on the closed unit disk. The main aim of this paper is to give explicit estimates of hyperbolically partial derivative of K-quasiconformal mapping ω from open unit disk onto itself satisfying Poisson's equation Δ ω = g with Dirichlet boundary conditions when the norm of g is small enough. Moreover, the estimates we obtained are asymptotically sharp when the norm tends to zero. As applications, hyperbolic area distortion and Euclidean length distortion are obtained.

Shaoqiang Deng, School of Mathematical Sceinces and LPMC, Nankai University, Tianjin 300071, P.R. China. (dengsq@nankai.edu.cn).
Masoumeh Hosseini, Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan, 81746-73441-Iran. (hoseini_masomeh@ymail.com).
Huaifu Liu, College of Applied Sciences, Beijing University of Technology, Beijing 100124, P.R. China. (liuhf@bjut.edu.cn).
Hamid Reza Salimi Moghaddam, Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan, 81746-73441-Iran. (hr.salimi@sci.ui.ac.ir).
On left invariant (α,β)-metrics on some Lie groups, pp. 1071-1088.
ABSTRACT. We give the explicit formulas of the flag curvature of (α,β)-metrics of Berwald type, and correct an error of the first and third authors in a previous article. Then we prove that at any point of a connected non-commutative nilpotent Lie group, the flag curvature of any left invariant (α,β)-metric of Douglas type admits zero, positive and negative values, generalizing a theorem of Wolf. Moreover, we study left invariant (α,β)-metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, including Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for (α,β)-metrics to be of Berwald type, as well as some necessary and sufficient conditions for Randers metrics to be of Douglas type. We show that every left invariant (α,β)-metric of Douglas type, on Lie groups defined by Milnor, is a locally projectively flat Randers metric. We also give the explicit formulas of the flag curvature of left invariant Randers metrics of Douglas type on these spaces and show that, under a condition, the flag curvature is negative.

Bin Shen, Southeast University, Nanjing, 211189, PR China (shenbin@seu.edu.cn).
Strong maximum principle for a Finsler-Laplacian, pp. 1089-1117.
ABSTRACT. We study the Q-Laplacian operator on the nonreversible Finsler manifold. We prove the weak maximum principle in pointwise sense, the Hopf's Lemma and the strong maximum principle for Q-subharmonic functions. Referring to weak solutions, the maximum principle for weak solutions of equation -Qu=f is obtained by presenting the Sobolev's inequality. Also the strong maximum principle and the Harnack's inequality for weak solutions are obtained through the Local maximum principle and the weak Harnack's inequality.

Zhigang Wang (corresponding author), School of Mathematical Sciences, Harbin Normal University, Heilongjiang, China, 150025 (wangzg2003205@163.com), Meiling He, School of Mathematical Sciences, Harbin Normal University, Heilongjiang, China, 150025 (hemeilingd@163.com), and Yang Jiang, College of Math and Systematic Science, Shenyang Normal University, Liaoning, China, 110034 (xjiangyang@126.com).
Dual surfaces along spacelike curves in light cone and their singularity, pp. 1119-1151.
ABSTRACT. In this paper, we consider spacelike curves in the light-cone 2-space that is canonically embedded in the light-cone 3-space and the de Sitter 3-space in Minkowski space-time. To study the differential geometry of spacelike curves in the light cone, we propose a new type of frame called a light-cone frame, moving along a spacelike curve. Concerning the framework of the theory of the Legendrian dualities between pseudo-spheres, the dual relationships between these spacelike curves and the light-cone dual surface, the de Sitter dual surface, and the sphere-cone dual surface are revealed. Using the classical unfolding theory, the singularities of the hyperbolic evolute of the original curve and a classification of the singularities of these surfaces is found using several equivalent conditions. It is also revealed that the projections of the critical value sets of both the light-cone dual surface and the sphere-cone dual surface along a spacelike curve are the hyperbolic evolute of the spacelike curve. Finally, some relevant counterexamples are indicated.

Haller, Rainis, University of Tartu, Estonia (rainis.haller@ut.ee), Kuuseok, Paavo, University of Tartu, Estonia (paavo.kuuseok@gmail.com), and Põldvere, Märt, University of Tartu, Estonia (mart.poldvere@ut.ee).
On convex combinations of slices of the unit ball in Banach spaces, pp. 1153-1168.
ABSTRACT. Recently, Abrahamsen and Lima proved that, for scattered compact Hausdorff spaces K, every convex combination of slices of the unit ball of C(K) is open in the relative weak topology. They considered the following three properties for a Banach space: every finite convex combination of slices of the unit ball (P1) is relatively weakly open, (P2) has nonempty interior in the relative weak topology of the unit ball, (P3) intersects the unit sphere. We show that these properties are all different from each other. In particular, the 1-sum of two Banach spaces does not have property (P1), but it has property (P2) if both the spaces have property (P1); the space C₀(K) with K non-scattered has never property (P2), although it has property (P3).

Marius Mantoiu, Departamento de Matemáticas, Facultad de Ciencias Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile (mantoiu@uchile.cl).
Modulation spaces and representations for Rieffel's quantization, pp. 1169-1186.
ABSTRACT. We define localized modulation maps and modulation spaces of symbols suited to the study of Rieffel's deformation quantization pseudodifferential calculus. They are used to generate Hilbert space representations for the quantized C*-algebras, starting from covariant representations of the corresponding twisted C*-dynamical system. In the case of an Abelian undeformed algebra, orthogonal relations and extra information about the representations are obtained.

Alejandro Illanes, Veronica Martinez-de-la-Vega, Instituto de Matematicas, Universidad Nacional Autonoma de Mexico, Circuito Exterior, Cd. Universitaria, Mexico 04510, Ciudad de Mexico (illanes@matem.unam.mx, vmvm@matem.unam.mx), and Daria Michalik, Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszynski University, Woycickiego 1/3, 01-938 Warszawa, Poland (d.michalik@uksw.edu.pl).
Symmetric products as cones, Peano curves, pp. 1187-1195.
ABSTRACT. For a continuum X, let Fn(X) be the hyperspace of all nonempty subsets of X with at most n points. In this paper we prove that if X is a locally connected curve, then the following conditions are equivalent: (a) X is a cone, (b) Fn(X) is a cone for some n>1, and (c) Fn(X) is a cone for each n>1.

M. Farhadi, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (m.farhadi@math.iut.ac.ir) and M.R. Koushesh, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran (koushesh@cc.iut.ac.ir).
On closed subalgebras of C_B(X), pp. 1197-1207.
ABSTRACT. For a completely regular space X and a non-vanishing self-adjoint closed subalgebra H of C_B(X) which separates points from closed sets in X we construct the Gelfand spectrum sp(H) of H as an open subspace of the compactification of X generated by H. The simple construction of sp(H) enables easier examination of its properties. We illustrate this by an example showing that the space sp(H) is separable metrizable if and only if H is countably generated.

Angelo Bella, Department of Mathematics and Computer Science, University of Catania, viale A. Doria 6, 95125 Catania, Italy (bella@dmi.unict.it).
Discrete sets and the cardinality of a linearly Lindelöf space, pp. 1209-1214.
ABSTRACT. The depth g(X) of a space X is the supremum of the cardinalities of closures of discrete sets of X. Recently, Spadaro proved a cardinality bound for a Hausdorff space X involving g(X) and the Lindelof degree L(X). We try to extend this result by replacing the Lindelof degree with the linear Lindelof degree. We will do it for Tychonoff spaces and consistently in the general case.

Matevž Črepnjak , Faculty of Natural Sciences and Mathematics (Faculty of Chemistry and Chemical Engineering) University of Maribor, Koroška 160 (Smetanova 17), SI-2000 Maribor, Slovenia (matevz.crepnjak@um.si).
On classification of tent maps inverse limits: A counterexample, pp. 1215-1225.
ABSTRACT. Generalized tent functions are functions from [0,1] to [0,1] whose graphs are unions of two straight line segments, one from (0,0) to (a,b), and the other one from (a,b) to (1,0), where (a,b) is any point in [0,1]×[0,1]. The point (a,b) is called the top point of the graph of such function. I. Banič, M. Črepnjak, M. Merhar and U. Milutinović recently described a family F = { C_t | t ∈ [1,∞)} of curves in (0,1) × [0,1] and showed that for each positive integer n, the following holds true. If (a,b),(c,d)∈ C_n are top points of two generalized tent functions, then the corresponding inverse limits are homeomorphic. They also discuss if the same holds true for any t ∈ [1,∞). More precisely, let (a,b) and (c,d) be top points of two generalized tent functions. Are then the corresponding inverse limits homeomorphic, if (a,b),(c,d) ∈ C_t? They pose this question as an open problem. In this paper we construct a curve D in (0,1) × [0,1] satisfying the following properties: (1) D∉ F and (2) if (a,b),(c,d) ∈ D, then the corresponding inverse limits are homeomorphic. Using this curve and the Ingram conjecture we answer the above question in negative.

Jaume Llibre, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain, (jllibre@mat.uab.cat), and Marcio Gouveia, Universidade Estadual Paulista, 15054-000 Sao Jose Rio Preto, Sao Paulo, Brazil, (mra.gouveia@unesp.br).
Periods of periodic homeomorphisms of pinched surfaces with one or two branching points, pp. 1227-1243.
ABSTRACT. In this paper we characterize all the possible sets of periods of a periodic homeomorphism defined on compact connected pinched surfaces with one or two branching points.