Editors: D. Bao (San Francisco, SFSU), S. Berhanu (Temple), D. Blecher
(Houston), B. G. Bodmann (Houston),
M. Gehrke (CNRS),
Y. Hattori (Matsue, Shimane), A. Haynes (Houston), W. B. Johnson (College Station), H. Koivusalo (Bristol),
T. H. Lê (Mississippi),
M. Marsh (Sacramento),
M. Ru (Houston), S. W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and A. Haynes (Houston)
Houston Journal of Mathematics
Contents
Chetana Visave, Department of Mathematics, University of Mumbai
(visavechetana94@gmail.com), and Rajendra Deore, Department of Mathematics,
University of Mumbai (rpdeore@gmail.com).
Power graph of some finite and infinite groups, pp. 425–451.
ABSTRACT. The power graph of a group G is a graph 𝒫(G) with vertex set G and two
vertices x and y, x≠y are adjacent if there exists some integer k such that x = yk
or y = xk. We denote the complement of 𝒫(G) by 𝒫(G) and a proper power
graph of G by 𝒫∗(G). This paper investigates the Hamiltonian and Eulerian
nature of the power graphs of some finite Abelian and non-Abelian groups. We
also discuss properties like planarity, outerplanarity, diameter, girth including
neighbourhood and separating sets for a power graph, a proper power graph and
the complement of a proper power graph of some finite and infinite groups.
Sujoy Majumder, Department of Mathematics, Raiganj University, Raiganj,
West Bengal-733134, India (sm05math@gmail.com, sjm@raiganjuniversity.ac.in),
Nabadwip Sarkar, Department of Mathematics, Raiganj University, Raiganj, West
Bengal-733134, India (naba.iitbmath@gmail.com), and Debabrata Pramanik,
Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India
(debumath07@gmail.com).
Solutions of complex Fermat-type difference equations in several variables, pp.
453–482.
ABSTRACT. It is known that the functional equation fn(z) + gn(z) = 1 can be regarded as the Fermat-type equations over function fields, where n is a positive integer. The purpose of the paper is to find out the precise form of the finite order entire solutions of the following Fermat-type difference equations in ℂm such as
and
where n1 and n2 are two positive integers such that n1 + n2 > 2, aj ∈ ℂ for j = 1,2,3,4 such that a3a2 − a1a4≠0, c ∈ ℂm∖{0}, P1(z), P2(z) and Q(z) are non-zero polynomials in ℂm. The results of our paper, significantly generalize and improve the results due to Xu and Cao, Haldar and Ahamed and Allu. Also one of the main results of the paper is related to the following Fermat-type difference equations in ℂm
for finding finite order meromorphic solutions, where n1 and n2 are two positive integers such that n1 + n2 > 2, P1(z) and P2(z) are two non-zero polynomials in ℂm such that P1(z + c) ≡ P2(z). Moreover plenty of examples are provided to illustrate our findings.
Dongmei Wei, School of Mathematical Sciences, Suzhou University of Science and
Technology, Suzhou 215009, P.R.China; Institute of Mathematics, School of Mathematics,
Nanjing Normal University, Nanjing 210023, P.R.China (weidmei@yeah.net), and Fei Li,
Institute of Mathematics, School of Mathematics, Nanjing Normal University, Nanjing
210023, P.R.China (lf19980517@126.com), and Yan Xu, Institute of Mathematics,
School of Mathematics, Nanjing Normal University, Nanjing 210023, P.R.China
(xuyan@njnu.edu.cn).
Normality and the sequence of exceptional functions, pp. 483–500.
ABSTRACT. In this paper, we obtain some normality criteria for sequence
meromorphic functions that concerning the sequence of exceptional functions of
derivatives, which generalize related results due to Deng-Yang-Fang, Chang, Yang, and
Schwick.
Haoming Wang, School of Mathematics, Sun Yat-sen University, Xingang Road No.
135, Guangzhou, 510275, China (wanghm37@mail2.sysu.edu.cn).
On minimal predictable intensity of point processes, pp. 501–515.
ABSTRACT. An adapted, right-continuous, non-decreasing, integer-valued process with
unit jumps and starting at zero has a minimal predictable intensity if and only if it is a
standard Poisson process under an absolutely continuous transformation of measures.
Yingcui Zhao, School of Computer Science and Technology, Dongguan University of
Technology, No.1 Daxue Road, Dongguan City, 523808, Guangdong Province, China
(zycchaos@126.com).
Shadowing, average shadowing and transitive properties of multiple mappings, pp.
517–530.
ABSTRACT. Introduced by Hou and Wang in 2016, the concept of multiple
mappings—rooted in iterated function systems—constitutes an important branch of
fractal theory. In the present work, we propose the definitions of shadowing, average
shadowing, transitive, weakly mixing, mixing, chain transitive and chain mixing
properties of multiple mappings from a set-valued perspective. We show that both two
continuous self-maps have shadowing (respectively, average shadowing, transitive, weakly
mixing, mixing) property may not imply the multiple mappings they form has the
corresponding property. While a sufficient condition for multiple mappings to be
transitive (respective, weakly mixing, mixing, chain transitive, chain mixing) is given.
Both shadowing and average shadowing properties are invariant under the iterative
action of multiple mappings. Also, we study that for multiple mappings shadowing
property plus chain mixing implies mixing and average shadowing properties implies
chain transitivity.
John McCuan, School of Mathematics, 686 Cherry Street, Atlanta, GA 30332
(mccuan@pm.me).
Parametric mollification of C1 planar curves and constant curvature invariance, pp.
531–563.
ABSTRACT. Mollification is a remarkable procedure for approximating real valued functions by smooth C∞ functions. We describe geometric adaptations of this procedure for approximating certain curves, especially certain planar curves. For C1 curves with a weak curvature vector we give a mollification procedure leaving curves of constant curvature invariant in analogy to mollification of functions leaving solutions of Laplace’s equation (harmonic functions) invariant.
Generalization of the basic constructions are also given for curves in higher dimensional Euclidean spaces and curves of lower regularity.
Jackson S. Morrow, Department of Mathematics, University of North Texas, Denton,
TX 76203 (jackson.morrow@unt.edu), and Yueqiao Wu, Department of Mathematics,
Johns Hopkins University, Baltimore, MD 21218 (ywu347@jhu.edu).
Connections between K-stability and Vojta’s conjecture, pp. 565–579.
ABSTRACT. In this note, we use recent advances concerning the K-stability of ℚ-Fano
varieties to provide settings for which Vojta’s conjecture holds.
Yunru Bai, School of Science, Guangxi University of Science and Technology, Liuzhou
545006, Guangxi, China (yunrubai@163.com), Nikolaos S. Papageorgiou, Department
of Mathamatics, University of Craiova, 200585, Craiova, Romania, and Department of
Mathematics, National Technical University, Zograrou Compus, Athens 15780, Greece
(npapg@math.ntua.gr), and Shengda Zeng, National Center for Applied Mathematics
in Chongqing, and School of Mathematical Sciences, Chongqing Normal University,
Chongqing 401331, China (zengshengda@163.com).
Singular perturbation of a double phase eigenvalue problem, pp. 581–611.
ABSTRACT. We consider a Dirichlet problem driven by the double phase operator and
with a reaction having the combined effects of a singular term and of a parametric
superlinear term. Using variational tools together with truncation and comparison
techniques and critical groups, we prove an existence and multiplicity theorem which is
global in the parameter λ > 0.
Gülcan Kekeç, Istanbul University, Faculty of Science, Department of Mathematics,
34134, Vezne-ciler, Fatih, Istanbul, Türkiye (gulkekec@istanbul.edu.tr).
Schneider p−adic continued fractions, pp. 613–624.
ABSTRACT. We construct transcendental Schneider p−adic continued fractions by
applying Schneider p−adic continued fraction expansions of algebraic irrational p−adic
numbers.
Zhangyong Cai, [Zhangyong Cai]College of Mathematics and statistics, Center for
Applied Mathematics of Guangxi (Nanning Normal University), Nanning Normal
University, Nanning 530100, P.R. China (zycaigxu2002@126.com), and Yingqiao
Wang, College of Mathematics and statistics, Center for Applied Mathematics of
Guangxi (Nanning Normal University), Nanning Normal University, Nanning 530100,
P.R. China (248436725@qq.com).
A note on weak first-countability and first-countability in hyperspaces, pp.
625–634.
ABSTRACT. For a regular space X, let CL(X) (𝒦(X)) be the set of all nonempty closed (compact) subsets of X. Denote the set CL(X) with the Vietoris topology by (CL(X),V), with the locally finite topology by (CL(X),LF) and with the Fell topology by (CL(X),F), respectively. Weak first-countability and first-countability in hyperspaces are studied and the following results are obtained. (1) For a topological space X, (CL(X),LF) is weakly first-countable if and only if it is first-countable. (2) Let X be a topological space. (𝒦(X),F) is first-countable if and only if X is first-countable, every compact subset of X is (hereditarily) separable and the complement of every nonempty compact subset of X is hemicompact. (3) It is pointed out that for a topological space X, if (CL(X),V) is a Fréchet-Urysohn space with an ωω-base, then it is first-countable, which answers a question in literature. A few related questions are posed.