HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 47, No. 2, 2021

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), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and K. Kaiser (Houston)

 Houston Journal of Mathematics


Contents

Mostafa Abedi, Esfarayen University of Technology, Esfarayen, Iran (ms_abedi@yahoo.com).
Rings of quotients of the ring ℛL, pp. 271–293.

ABSTRACT. Let Q(ℛL) (resp. Qcl(ℛL)) denote the maximal ring of quotients (resp. classical ring of quotients) of the ring ℛL of real-valued continuous functions on a completely regular frame L. Then Q(ℛL) (Qcl(ℛL)) may be realized as the direct limit of the subrings ℛ(↓c), where c is a dense element (dense cozero element) of L. These representations of Q(ℛL) and Qcl(ℛL) are applied to describe equalities among various rings of quotients of ℛL. For example, we show that every dense open quotient of L is pseudocompact (that is, Q(ℛL) = Q∗(ℛL)) if and only if L is a pseudocompact P-frame.  

Claudia Mureşan, University of Cagliari, University of Bucharest (cmuresan@fmi.unibuc.ro).
Some properties of lattice congruences preserving involutions and their largest numbers in the finite case, pp. 295–320.

ABSTRACT. In this paper, we characterize the congruences of an arbitrary i–lattice, investigate the structure of the lattice they form and how it relates to the structure of the lattice of lattice congruences, then, for an arbitrary nonzero natural number n, we determine the largest possible number of congruences of an n–element i–lattice, along with the structures of the n–element i–lattices with this number of congruences. Our characterizations for the congruences of i–lattices have useful corollaries: a description of the atoms of the congruence lattices of i–lattices, congruence extensibility and characterization of subdirect irreducibility results. In terms of the relation between the above–mentioned problem on numbers of congruences of finite i–lattices and its analogue for lattices, while the n–element i–lattices with the largest number of congruences turn out to be exactly the n–element lattices whose number of congruences is either the largest or the second largest possible, we provide examples of pairs of n–element i–lattices and even pseudo–Kleene algebras such that one of them has strictly more congruences, but strictly less lattice congruences than the other.  

Gary G. Gundersen, University of New Orleans, Department of Mathematics, New Orleans, LA 70148, USA (ggunders@uno.edu), Janne M. Heittokangas, University of Eastern Finland, Department of Physics and Mathematics, P.O. Box 111, 80101 Joensuu, Finland; Taiyuan University of Technology, Department of Mathematics, Yingze West Street, No. 79, Taiyuan 030024, China (janne.heittokangas@uef.fi), and Zhi-Tao Wen, Shantou University, Department of Mathematics, Daxue Road, No. 243, Shantou 515063, China; Taiyuan University of Technology, Department of Mathematics, Yingze West Street, No. 79, Taiyuan 030024, China (zhtwen@stu.edu.cn).
Contour integral solutions of linear differential equations which include a generalization of the Airy integral, pp. 321–351.

ABSTRACT. The Airy integral is a well-known contour integral solution of Airy’s equation which has several applications and which has been used for mathematical illustrations due to its interesting properties. We present and derive properties of two families of contour integral solutions of linear differential equations. The first family includes the Airy integral and Airy’s equation, such that the family generalizes known properties of the Airy integral which include exponential decay growth in a certain sector. The second family includes a known example and contains a subfamily with interesting properties where a separate analysis of three pairwise linearly independent contour integral solutions of a particular equation is given.  

Jane Hawkins, Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina 27599-3250 (jmh@math.unc.edu), and Mónica Moreno Rocha, Centro de Investigación en Matemáticas, A.P. 402, C.P. 36023, Guanajuato, Gto., México (mmoreno@cimat.mx).
Unbounded Fatou components for elliptic functions over square lattices, pp. 353–374.

ABSTRACT. We show the existence of elliptic functions on the complex plane ℂ with a square period lattice Λ with unbounded Fatou components under iteration. For these functions, the imaginary axis lies in a single Fatou component, and projects onto a band on the torus ℂ∕Λ. Unlike all previous examples, the resulting Julia set is not Cantor because there are two attracting fixed points. There is an open set of parameter values within a family of functions for which these Fatou components exist. The first example of a toral band for a map with a parabolic fixed point is also given. This shows that unlike the situation for iterations of the Weierstrass elliptic ℘ function, some Fatou components that are unbounded in only one direction can occur for square lattices.  

Shuangshuang Yang, Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, P.R. China (ssyang1997@126.com), and Xianjing Dong, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, P.R. China (xjdong@amss.ac.cn).
A defect relation of meromorphic functions on a product of Riemann surfaces, pp. 375–392.

ABSTRACT. We consider the value distribution of nonconstant meromorphic functions on a product of several Riemann surfaces. When a certain growth condition is satisfied, we obtain a defect relation.  

Kit C. Chan, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403, USA (kchan@bgsu.edu), and Serge P. Phanzu, Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97201, USA (sphanzu@pdx.edu).
Every pure quasinormal operator has a supercyclic adjoint, pp. 393–407.

ABSTRACT. We prove that every pure quasinormal operator T on a separable, infinite dimensional Hilbert space H has a supercyclic adjoint. It follows that an operator having a pure quasinormal extension on H must have a supercyclic adjoint. Our result improves a result of Wogen who proved that every pure quasinormal operator on H has a cyclic adjoint. Motivated by our result, we further show that the pure quasinormal operator T has a hypercyclic adjoint if and only if ∥Tf∥ > ∥f∥ for every nonzero vector f in H.  

Ofelia T. Alas, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970 São Paulo, Brasil (alas@ime.usp.br), J. A. Martínez-Cadena, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340, México, D.F., México (martinezcadenajuan@gmail.com), and Richard G. Wilson, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340, México, D.F., México (rgw@xanum.uam.mx).
Product theorems in classes of feebly compact spaces, pp. 409–425.

ABSTRACT. We study two subclasses of the class of feebly compact spaces, namely weakly cellular-compact spaces and almost cellular-compact spaces. The first of these was introduced and studied by Pichardo-Mendoza, Tamariz-Mascarúa and Villegas-Rodríguez in the class of Tychonoff spaces, while the second is a subclass of the first which contains all cellular-compact spaces. We show that many of the results obtained by the above mentioned authors are valid in the class of Hausdorff spaces and we give conditions under which the product of an almost cellular-compact space and a compact space is almost cellular-compact. The main consistency results concern the preservation of this property under products with compact spaces.  

Meng Bao, School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China; College of Mathematics, Sichuan University, Chengdu 610064, P. R. China (mengbao95213@163.com), and Fucai Lin, School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P. R. China; Fujian Key Laboratory of Granular Computing and Application, Minnan Normal University, Zhangzhou 363000, P. R. China (linfucai2008@aliyun.com; linfucai@mnnu.edu.cn).
Submetrizability of strongly topological gyrogroups, pp. 427–443.

ABSTRACT. Topological gyrogroups, with a weaker algebraic structure without associative law, have been investigated recently. We prove that each T0-strongly topological gyrogroup is completely regular. We also prove that every T0-strongly topological gyrogroup with a countable pseudocharacter admits a weaker metrizable topology. Finally, we prove that the left coset space G∕H admits a weaker metrizable topology if H is an admissible L-subgyrogroup of a T0-strongly topological gyrogroup G.  

Meng Bao, College of Mathematics, Sichuan University, Chengdu 610064, P.R. China (mengbao95213@163.com), Xuewei Ling, Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210046, P.R. China (781736783@qq.com), and Xiaoquan Xu, School of mathematics and statistics, Minnan Normal University, Zhangzhou 363000, P.R. China (xiqxu2002@163.com).
Strongly topological gyrogroups and quotient with respect to L-subgyrogroups, pp. 445–466.

ABSTRACT. In this paper, we obtain the precise form of meromorphic functions u(z1,z2) in ℂ2 when the linearly independent operators auz1 + buz2 and cuz1 + duz2 have a common right factor with constants a,b,c,d. We also describe entire solutions of the partial differential equation
F (uz ,uz ,...,uz ) = 1 1 2 n
in ℂn under a more general definition in the sense of the prime functions. In addition, our results generalize the recent results in a work by Li.  

H. K. Elsayied, Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt (hkelsayied1989@yahoo.com), A. M. Tawfiq, Mathematics Department, Faculty of education, Ain-Shams University, Cairo, Egypt (abdelrhmanmagdi@edu.asu.edu.eg), and A. Elsharkawy, Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt (ayman_ramadan@science.tanta.edu.eg).
Special Smarandach curves according to the quasi frame in 4-dimensional Euclidean space E4, pp. 467–482.

ABSTRACT. The first purpose of this paper is to investigate the quasi frame, the quasi equations and show the relations between the Frenet and the quasi curvatures in E4. The second purpose is to study Smarandach curves in E4. We obtain the Frenet and the quasi invariants for Smarandach curves in E4 and by obtaining the Frenet’s curvatures we deduce the quasi’s curvatures.  

Olivier Olela Otafudu, Department of Mathematics and Applied Mathematics North-West University, Potchefstroom Campus, Potchefstroom 2520, South Africa (olmaolela@gmail.com).
On superample functions on extended quasi-pseudometric spaces, pp. 483–497.

ABSTRACT. In this article we consider the concept of superample function pairs in an extended quasi-metric space. We show, for instance, that there exists a covariant functor from the category of extended quasi-metric spaces and nonexpansive maps into the category of sets of all superample function pairs on an extended quasi-metric space and nonexpansive maps.  

Liang–Xue Peng (corresp. author), School of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn), and Ying Liu, School of Mathematics, Faculty of Science, Beijing University of Technology, Beijing 100124, China (15733268276@163.com).
Topological groups with a (strong) q-point, pp. 499–516.

ABSTRACT. We show that a topological group G has a q-point if and only if G is an open quasi-perfect preimage of a metric topological space (i.e. G is an M-space). We introduce a notion of strong q-spaces. We show that the product of countably many regular strong q-spaces is a strong q-space. The product of a strong q-space and a q-space is a q-space. We finally give a characterization of topological groups with a strong q-point. As a corollary, we have that the product of countably many sequential topological groups with a q-point is a strong q-space.  

Suqian Zhao, College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, P.R. China (suqianzhao@126.com), and Pedro L. Q. Pergher, Universidade Federal de São Carlos, Departamento de Matemática, São Carlos, SP, Brazil (pergher@dm.ufscar.br).
(Z2)k-actions fixing the disjoint union of odd-dimensional quaternionic projective spaces, pp. 517–533.

ABSTRACT. If M is a smooth, closed manifold and T : M → M is a smooth involution defined on M, then it is well known that the fixed point set of T, F, is a finite and disjoint union of closed submanifolds of M; the same is valid if M is equipped with a smooth (Z2)k-action Φ : (Z2)k × M → M, where (Z2)k is understood as the group generated by k commuting smooth involutions. In this setting, for a given F, a natural question is the classification, up to equivariant cobordism, of the pairs (M,Φ) for which the fixed point set is F. In this paper we contribute to this problem in the case k = 1, showing that if F consists of an union of odd-dimensional quaternionic projective spaces, then (M,T) bounds equivariantly; this result was already known for real and complex projective spaces. We also extend the result for k > 1, but imposing that the dimensions of the involved projective spaces (real, complex or quaternionic) are different.