.
Finally, we give the characterization of associativity and commutativity of the elements
of the bicomplex Möbius gyrogroup. 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
Abhijit Banerjee, Department of Mathematics, University of Kalyani, West Bengal
741235, India (abanerjee_kal@yahoo.co.in), Sujoy Majumder, Department
of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India
(sm05math@gmail.com, sjm@raiganjuniversity.ac.in), and Debabrata Pramanik,
Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India
(debumath07@gmail.com).
Power of a meromorphic function sharing value with its k-th order directional derivative
in ℂm, pp. 225–257.
ABSTRACT. In the context of several complex variables, we investigate the uniqueness
problem for a power of a meromorphic function that shares a value with its k-th order
directional derivative in ℂm. Our results extend previous uniqueness theorems from the
one-variable case to higher dimensions. Furthermore, we provide numerous
examples to demonstrate that our results are, in certain senses, best possible.
Zekun Li, School of Mathematics, Sun Yat-sen University, Guangzhou 510275,
Guangdong, People’s Republic of China (lizk25@mail2.sysu.edu.cn), and Binlin Dai,
School of Mathematics, Shanghai University of Finance and Economics, Shanghai
200433, People’s Republic of China (bldai@mail.shufe.edu.cn).
Bicomplex Möbius gyrogroup and associated properties, pp. 259–278.
ABSTRACT. In this paper, we first extend the Möbius addition in the complex disk to
the bicomplex ball and study the properties of the Möbius addition ⊕ under the action
of the gyration operator gyr[A,Z], such as the gyrocommutative law and the nested
gyroautomorphism identity. Then, by using the properties of Möbius addition, we
demonstrate that all Möbius addition form a gyrogroup in bicomplex ball 𝔹.
Finally, we give the characterization of associativity and commutativity of the elements
of the bicomplex Möbius gyrogroup.
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).
Average shadowing property of set-valued mappings, pp. 279–292.
ABSTRACT. Shadowing, average shadowing properties and their connections to
transitivity, weakly mixing, specification property, topological ergodicity of set-valued
mappings are studied. Shadowing plus average shadowing implies transitivity, and
furthermore, it also implies weakly mixing. Shadowing plus specification implies average
shadowing. And Lyapunov stability plus average shadowing implies topological
ergodicity.
Lan-fang Zhang, College of Sciences, Inner Mongolia University of Technology, Hohhot
010051, China (1641048157@qq.com), and Ji-jun Ao, College of Sciences, Inner
Mongolia University of Technology, Hohhot 010051, China (george_ao78@sohu.com).
A fourth-order boundary value problem with transmission conditions dependent on the
eigenparameter, pp. 293–317.
ABSTRACT. This paper examines a fourth-order boundary value problem with
eigenparameter-dependent transmission conditions. The main purpose is to show the
continuity and differentiability of eigenvalues for the problem under consideration. To
this end, firstly, the boundary value problem is transformed into an operator problem
and the self-adjointness of the operator is proved. Then the expression of the
characteristic function is given by constructing the basic solutions. Finally, the continuity
and differentiability of eigenvalues with respect to each parameter are proved, especially
the differential expressions of eigenvalues with respect to each parameter are given.
Mykola Yaremenko, National Technical University of Ukraine, “Igor Sikorsky Kyiv
Polytechnic Institute”, Kyiv (Math.kiev@gmail.com).
Hölder regularity for quasilinear elliptic equations with generalized variable exponent
growth: A De Giorgi-Nash-Moser approach, pp. 319–338.
ABSTRACT. This paper establishes the Hölder continuity of bounded weak solutions to
a class of quasilinear elliptic equations in divergence form with nonstandard
growth governed by a variable exponent p(x). We consider equations of the
form div(A(x,u,∇u)) + b(x,u,∇u) = 0 under minimal structural conditions on
the coefficients A and b. The exponent p(x) is assumed only to be log-Hölder
continuous, which is the optimal regularity condition for the variable exponent
setting. By adapting the classical De Giorgi-Nash-Moser iteration to the variable
exponent Sobolev space framework, we prove that any bounded weak solution
u ∈ W1p(⋅)(Ω) belongs to C0,α(Ω) for some α ∈ (0,1). The Hölder exponent α and the
constant depend explicitly on the bound of the solution, the structural constants,
and the log-Hölder modulus of p(x). Our results extend and sharpen previous
regularity theorems that required stronger conditions on p(x), and they provide a
complete generalization of the classical theory to the variable exponent case.
Xu Xuecuibao, School of Mathematics, Shandong University, Jinan, China
(xuxuecuibao@gmail.com).
A correlation estimate for representation functions of binary quadratic forms, pp.
339–346.
ABSTRACT. Let f and g be positive definite integral binary quadratic forms, and let
be the corresponding representation function. Motivated by the work of Blomer and Granville on moments of representation functions of binary quadratic forms, we study the correlation
We prove that, if f and g have fundamental discriminants −D and −D′, then
whereas in general
The logarithm is unavoidable when the two discriminants agree, as is already shown by the classical form X2 + Y 2. The proof passes from the correlation sum to integral points on a nonsingular quaternary quadratic cone and applies the theorem of Browning and Heath-Brown on rational points on quadric surfaces.
Guodong Hua, School of Mathematics and Statistics, Weinan Normal University,
Weinan, Shaaxi Province, China 714099; Research Institute of Qindong
Mathematics, Weinan Normal University, Weinan, Shaaxi Province, China 714099
(gdhuasdu@163.com).
On the asymptotic distribution of a hybrid arithmetic function associated to coefficients
of general L-functions over arithmetic progressions, pp. 347–375.
ABSTRACT. Let a(n) be the n-th coefficient of the general L-functions L(s,𝒜) satisfying certain natural axioms, and let σ(n) and φ(n) be the sum-of-divisors function and the Euler totient function, respectively. In this paper, for b,c ∈ ℝ, we are interested in the asymptotic behaviour of the summatory function of the form
for sufficiently large x > 0, without the assumption of the Ramanujan conjecture. As a direct application, we also consider a hybrid arithmetic function associated to holomorphic newforms or Hecke-Maaß cusp forms for Γ = SL(2, ℤ) over arithmetic progressions, which constitutes a generalization of the previous results in this direction.
Qin Yang, School of Statistics and Data Science, Taizhou University, Taizhou 225300,
P.R. China (2032094290@qq.com), and Qianqian Sun, School of Statistics and Data
Science, Taizhou University, Taizhou 225300, P.R. China (909950213@qq.com).
On feathered quotient spaces with strong subgyrogroups, pp. 377–392.
ABSTRACT. The concept of strongly topological gyrogroups was introduced by Bao
and Lin in 2019. In this paper, we investigate some generalized metric properties of the
quotient space G∕H, where G is a strongly topological gyrogroup and H is a closed
strong subgyrogroup of G. It is shown that if G is a strongly topological gyrogroup, H is
a compact strong subgyrogroup of G such that the quotient space G∕H is feathered, then
there exists a compact strong subgyrogroup K of G with H ⊆ K such that G∕K is
metrizable. Then some equivalent characterizations of metrizability in the quotient space
G∕H are investigated.
Piyu Li, School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou
221018, China (lpy91132006@aliyun.com), Rongxin Shen, School of Mathematics and
Physics, Taizhou University, Taizhou 225300, China (srx20212021@163.com), and
Hanfeng Wang, Department of Mathematics, Shandong Agricultural University, Taian
271018, China (whfeng@sdau.edu.cn).
Answers to questions on pre-topological groups, pp. 393–403.
ABSTRACT. This paper presents the following main results: (1) We construct a
symmetrically pre-topological group that is not a strongly pre-topological group; (2)We
give an example of an almost topological group G with an open set U and an
element x ∈ U for which there exists no open neighborhood V of the identity
satisfying V xV ⊆ U; (3) We show the existence of a non-first-countable almost
topological group G and a closed pre-topological subgroup H of G such that
both the subspace H and the quotient space G∕H are first-countable. These
results provide answers to the corresponding questions raised in [F. Lin et al.,
Some properties of pre-topological groups, Topol. Appl., 362 (2025): 109204.]
Liang-Xue Peng, Department of Mathematics, School of Mathematics, Statistics
and Mechanics, Beijing University of Technology, Beijing 100124, China
(pengliangxue@bjut.edu.cn), and Mu-Wei Huang, Department of Mathematics, School
of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing
100124, China (2384890151@qq.com).
On some topological properties related to dense open subsets of a space, pp.
405–426.
ABSTRACT. In this article, we introduce some notions that can be characterized by dense open subsets of a space. These spaces have characterizations similar to semi-stratifiable, k-semi-stratifiable and stratifiable spaces, respectively. Accordingly, we call them by d-semi-stratifiable, k-d-semi-stratifiable and d-stratifiable, respectively. Several characterizations of these classes of spaces are established in this article.
We also get the following results. If X is a d-semi-stratifiable space, then X is subparacompact. If a space X satisfies that the set of all non-isolated points is a semi-stratifiable Gδ-subspace of X, then X is a semi-stratifiable space. If X is a d-stratifiable space, then X is monotonically CND-normal. Every monotonically CND-normal T1-space is collectionwise normal. Every d-stratifiable T0-space is paracompact. If f : X → Y is a pseudo-open closed surjective continuous map and X is a d-semi-stratifiable space, then Y is d-semi-stratifiable. If f : X → Y is a pseudo-open perfect map and X is a d-stratifiable (k-d-semi-stratifiable) space, then Y is d-stratifiable (k-d-semi-stratifiable). If X is a k-d-semi-stratifiable Fréchet-Urysohn space, then X is monotonically CND-normal. If X is a d-semi-stratifiable T1-space, then X is dually σ-closed discrete. We show that a Hausdorff space is k-semi-stratifiable (semi-stratifiable) if and only if it is k-d-semi-stratifiable (d-semi-stratifiable).