HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 45, No. 3, 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), A. Haynes (Houston), 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

Zhengxin Chen (corresponding author), College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, P.R.China (chenzxmath@sina.com), and Chundan Zhu, College of Mathematics and Informatics, Fujian Normal University, Fuzhou 350117, P.R.China (375797432@qq.com).
Commuting automorphisms of the group consisting of the unit upper triangular matrices, pp. 647-658.
ABSTRACT. Let G be a group, an automorphism h of G is called a commuting automorphism, if for all x in G, h(x) commutes with x. Let Un(K) be the group consisting of all unit upper triangular matrices of order n over a field K with characteristic char(K) not equal to two. In this paper, we prove that if n is larger than 3, a map h on Un(K) is a commuting automorphism of Un(K) if and only if it is a central automorphism of Un(K). Moreover, the set of the commuting automorphisms of Un(K) is a normal subgroup of the automorphism group of Un(K).

Jinhua Yang, School of Mathematical Sciences, Xinjiang Normal University, Urumqi, 830054, P.R. China (1511337295@qq.com), Zhangcai Hu, Department of Mathematics, East China Normal University, Shanghai, 200062, P.R. China (1297760510@qq.com), and Xuecheng Pang, Department of Mathematics, East China Normal University, Shanghai, 200062, P.R. China ((xcpang@math.ecnu.edu.cn).
Quasinormal families of meromorphic functions, pp. 659-668.
ABSTRACT. We discuss the quasinormal family of meromorphic functions and get a result that the derivatives of meromorphic functions converge to the omitted functions in the spherical metric. In addition, an example is also provided to show that the quasinormal order can be positive infinity.

Jianren Long (corresponding author), School of Mathematical Science, Guizhou Normal University, Guiyang, 550001, P.R. China and School of Computer Science and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, P. R. China, (longjianren2004@163.com), XingqunWu, School of Mathematical Science, Guizhou Normal University, Guiyang, 550001, P.R. China  (jrlong@gznu.edu.cn.
A note on Fatou sets of solutions of complex linear differential equations, pp. 669-683.
ABSTRACT. We study the dynamical property and the growth of solutions of f''+A(z)f=F(z), where A(z) is an entire function of finite order, and F(z) is an entire function of order of growth less than that of A(z). On the one hand, some conditions on A(z) guaranteeing every non-trivial solution of the equation is of infinite lower order are obtained. On the other hand, we prove that the entire solutions of the equation have no Baker wandering domain.

Xiu Ji, Tongzhu Li, and Huafei Sun, Beijing Institute of Technology, Beijing, 100081, China (jixiu1106@163.com), (litz@bit.edu.cn), (huafeisun@bit.edu.cn).
Para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms, pp. 685-706.
ABSTRACT. Let Mn be an umbilic-free spacelike hypersurface in the (n+1)-dimensional Lorentzian space form M1 n+1(c). Three basic conformal invariants of Mn are the conformal 1-form C, the conformal second fundamental form B, and the Blaschke tensor A. The para-Blaschke tensor Dt = A + tB, which is a linear combination of A and B for some constant t, is a symmetric (0, 2)-tensor. A spacelike hypersurface is called a para-Blaschke isoparametric spacelike hypersurface if the conformal 1-form vanishes and the eigenvalues of the para-Blaschke tensor are constant. In this paper, we classify the para-Blaschke isoparametric spacelike hypersurfaces under the conformal group of M1 n+1(c).

Mustafa Dede, Department of Mathematics, University of Kilis 7 Aralik, Kilis, Turkey 79000 (mustafadede@kilis.edu.tr).
A new representation of tubular surfaces, pp. 707-720.
ABSTRACT. Assume we are given a polynomial curve in 3D Euclidean space. The basic idea behind the proposed frame is that the new binormal vector is computed by using the cross product of the first and the highest order derivatives of the curve. We call this new frame as Frenet-like curve frame or for short, Flc-frame. This new frame is proposed to decrease the number of singular points where we cannot define the Frenet frame. Surprisingly, it has been noticed that the proposed frame also decreases the undesirable rotation around the tangent vector of the curve. In addition, we introduce three new curvatures of a polynomial space curve. Then, we give the geometric interpretation of the new curvatures. Finally, we derive the parametric equation of some tube surfaces generated by the Flc-frame.

Min Su, School of Science, China University of Mining and Technology, Beijing 100083, P. R. China; School of Mathematics,Yunnan Normal University, Kunming 650500, P. R. China (mathesumin@yahoo.com) and Yuhua Li (Corresponding author) , School of Mathematics,Yunnan Normal University, Kunming 650500, P. R. China (liyuhua@ynnu.edu.cn).
Meromorphic functions and the Gauss map of complete minimal surfaces, pp. 721-729.
ABSTRACT. In this paper, we obtain a condition which guarantees the meromorphic function on the complex plane is the Gauss map of some complete minimal surface. In fact, we prove that if g1(z) and g2(z) are entire functions which have no common zeros at least one of their Taylor expansions at the origin has 2-order Fejér gaps, then the meromorphic function g1(z)⁄g2(z)(g2(z) is not identically zero) can be viewed as the Gauss map of some complete minimal surfaces.

Stephen Avsec, Department of Mathematics, Mailstop 3368, Texas A\&M University, College Station, TX 77843-3368 (stephen.avsec@gmail.com) and Isaac Goldbring, Department of Mathematics, University of California, Irvine, Irvine, CA, 92697-3875  (isaac@math.uci.edu).
Boundary amenability of groups via ultrapowers, pp. 731-741.
ABSTRACT. We use C*-algebra ultrapowers to give a new construction of the Stone-Cech compactification of a separable, locally compact space. We use this construction to give a new proof of the fact that groups that act isometrically, properly, and transitively on trees are boundary amenable.

Silvano Delladio, Department of Mathematics, University of Trento, via Sommarive 14 (Povo), 38123 Trento, Italy (silvano.delladio@unitn.it).
Density-degree function for subsets of Rn , pp. 743-762.
ABSTRACT. For all subsets E of Rn , we define a function dE measuring the density-degree of E at the points of Rn. We provide some results which involve dE. In particular we prove an approximation property stating that, given a bounded open set Ω, the following facts hold: (i) For all C<Ln(Ω) there exists a closed subset F of Ω such that Ln(F)>C and dF=n almost everywhere in F; (ii) For all C<Ln(Ω) and for every proper subinterval I of (n,+∞), there exists a closed subset F of Ω and an open subset U of Ω such that F⊃ Ω∖ U, Ln(U)< Ln(Ω;)− C. (hence Ln(F)>C) and dF(x)∈I for all x∈ Ω∖ U.

David Pask, Adam Sierakowski, and Aidan Sims, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, AUSTRALIA (dpask@uow.edu.au), (asierako@uow.edu.au), and (asims@uow.edu.au).
Unbounded quasitraces, stable finiteness and pure infiniteness, pp. 763-814.
ABSTRACT. We prove that if A is a sigma-unital exact C*-algebra of real rank zero, then every state on the zeroth K-group of A extends to a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for a twisted C*-algebra of a cofinal k-graph with no sources, stable finiteness is independent of the twisting cocycle. We also study pure infiniteness of twisted higher rank graph C*-algebras.

Nasiri, Leila, Department of Mathematics and computer science, Faculty of Science, Lorestan University, Khorramabad, Iran (leilanasiri468@gmail.com) and Bakherad, Mojtaba, Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran (bakherad@member.ams.org).
Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant, pp. 815-830.
ABSTRACT. In this paper, we present some operator inequalities involving positive linear maps that generalize and improve the derived results in some recent years. For instance, we prove some refinements of a reverse AM-GM operator inequality.

Raj, Kuldip, School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J & K, India (kuldipraj68@gmail.com ), Anand, Renu, School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J & K, India (renuanand71@gmail.com), Sharma, Charu, School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J & K, India (charu145.cs@gmail.com).
Matrix transformations on Lacunary Orlicz Sequence spaces and their Toeplitz duals, pp. 831-851.
ABSTRACT. In this paper we study some lacunary sequence spaces originated with infinite matrices and a sequence of Orlicz functions. We make an effort to study some algebraic and topological properties of these sequence spaces. Some inclusion relations between these spaces are also established. Furthermore, the β-,γ-duals and matrix transformation of these spaces are determined.

David Cruz-Uribe, OFS, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 (dcruzuribe@ua.edu), Kabe Moen, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 (kabe.moen@ua.edu), and  Hanh Van Nguyen, Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 (nguyenvanhanh@dhsphue.edu.vn).
Multilinear fractional Calderon-Zygmund operators on weighted Hardy spaces, pp. 853-871.
ABSTRACT. We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals (D.Cruz-Uribe, K.Moen, and H.V. Nguyen, The boundedness of multilinear Calderon-Zygmund operators on weighted and variable Hardy spaces, to appear in Publ. Mat.) For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.

Chun-Yu Lei, School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China (leichygzu@sina.cn).
On a Schrödinger-Poisson system with singular term and critical growth, pp. 873-892.
ABSTRACT. In this work, we study the Schrödinger-Poisson system with singular and critical growth terms in a bounded domain. By using the variational method, the Brézis-Lieb(Proc. Amer. Math. Soc. 88 (1983)) and Brézis-Nirenberg (Comm. Pure Appl. Math. 36 (1983)) classical techniques, the existence and multiplicity of positive solutions are established.

Paul Gartside, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (gartside@math.pitt.edu) and Jeremiah Morgan, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA (j.morgan@pitt.edu) .
Local networks for function spaces, pp. 893-923.
ABSTRACT. Let y be a point in a space Y, and suppose there is a countable family F of subsets of Y satisfying: whenever U is an open neighborhood of y, and y is in cl(A)\A for some subset A of Y, there is a D in F such that y ∈ D ⊆ U and D intersects A (respectively, D intersects A in an infinite set). Then we say F is a countable network (respectively, countable Pytkeev network) at y and that Y is (cn)y (respectively, (cp)y). Y is called (cn) (respectively, (cp), or strongly Pytkeev) if it is (cn)y (respectively, (cp)y) at every point y. Let Ck(X) (respectively, Cp(X)) denote the space of all continuous real-valued functions on X with the compact-open topology (respectively, pointwise topology). We show Ck(X) is (cp) if and only if it is (cn), and we characterize this both in terms of X and in terms of the order structure of the neighborhood filter of Ck(X). On the other hand, Cp(X) is known to be (cp) only when X is countable, but we characterize when Cp(X) is (cn) and demonstrate that Cp(X) can be (cn) for a wide range of spaces including the Double Arrow space and Alexandrov duplicate of the unit interval.

Er-Guang Yang, School of Mathematics & Physics, Anhui University of Technology, Maanshan 243002, P.R. China (egyang@126.com).
Characterizations of MCP and mcb  spaces with maps, pp.  925-933.
ABSTRACT. This paper is a continuation of [Characterizations of some spaces with maps to ordered topological vector spaces, Houston J. Math., 43 (2017), 678-689] in which some characterizations of countably paracompact spaces and cb-spaces in terms of maps to ordered topological vector spaces were presented. In this paper, we shall generalize the corresponding results to MCP-spaces and mcb-spaces.

Xiaodong Jia, School of Computer Science, University of Birmingham, Birmingham, B15 2TT, United Kingdom (jia.xiaodong@yahoo.com), Achim Jung, School of Computer Science, University of Birmingham, Birmingham, B15 2TT, United Kingdom (A.Jung@cs.bham.ac.uk), and Qingguo Li, College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, China (liqingguoli@aliyun.com)
A dichotomy result for locally compact sober dcpos, pp. 935-951.
ABSTRACT. The second author proved in 1989 that each cartesian closed category of pointed domains and Scott-continuous functions is contained in either the category of Lawson-compact domains or that of L-domains, and this result eventually led to a classification of continuous domains with respect to cartesian closedness. In this paper, we generalise this result to the category LcS of pointed locally compact sober dcpos and Scott-continuous functions, and show that any cartesian closed full subcategory of LcS is contained in either the category of stably compact dcpos or that of L-dcpos. (Note that for domains Lawson-compactness and stable compactness are equivalent.) As we will show, this entails that any candidate for solving the Jung-Tix problem in LcS must be stably compact. To prove our dichotomy result, we first show that any dcpo with a core-compact function space must be meet-continuous; then we prove that a function space in LcS is meet-continuous only if either its input dcpo is coherent or its output dcpo has complete principal ideals.