HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 50, No. 3, 2024

Editors:  D. Bao (San Francisco, SFSU), S. Berhanu (Temple), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), M. Gehrke (LIAFA, Paris7), R. M. Hardt (Rice), S. Harvey (Rice), A. Haynes (Houston), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station), H. Koivusalo (Bristol), M. Marsh (Sacramento), M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and A. Haynes (Houston)

 Houston Journal of Mathematics


Contents

Guo-Shuai Mao, Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China (maogsmath@163.com), and Lei Shi, Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China (1960991228@qq.com).
Congruence properties for the quadrinomial coefficients, pp. 497–508.

ABSTRACT. In this paper, we mainly obtain some congruence properties for the quadrinomial coefficients, one of which is analogous to the Wolstenholme’s theorem. The quadrinomial coefficient is defined by the coefficient of xk in the polynomial expansion of (1 + x + x2 + x3)n, where n and k are nonnegative integers.  

Spencer Chapman, Mathematics Department, Trinity University, San Antonio, TX 78212 (schapma3@trinity.edu), Eli B. Dugan, Mathematics and Statistics Department, Williams College, Williamstown, MA 01267 (ebd3@williams.edu), Shadi Gaskari, Mathematics Department, San Diego State University, San Diego, CA 92182 (shgaskari@gmail.com), Emi Lycan, Mathematics Department, San Diego State University, San Diego, CA 92182 (ronlycan1250@gmail.com), Sarah Mendoza De La Cruz, Mathematics Department, University of Texas at Austin, Austin, TX 78712 (sarahpaolam11@gmail.com), Christopher O’Neill, Mathematics Department, San Diego State University, San Diego, CA 92182 (cdoneill@sdsu.edu), and Vadim Ponomarenko, Mathematics Department, San Diego State University, San Diego, CA 92182 (vponomarenko@sdsu.edu).
Some asymptotic results on p-lengths of factorizations for numerical semigroups and arithmetical congruence monoids, pp. 509–523.

ABSTRACT. A factorization of an element x in a monoid (M,⋅) is an expression of x of the form x = u1z1⋅⋅⋅ukzk for irreducible elements u1,…,uk ∈ M, and the length of such a factorization is z1 + ⋅⋅⋅ + zk. We introduce the notion of p-length, a generalized notion of factorization length obtained from the ℓp-norm of the sequence (z1,…,zk), and present asymptotic results on extremal p-lengths of factorizations for large elements of numerical semigroups (additive submonoids of ℤ≥0) and arithmetical congruence monoids (certain multiplicative submonoids of ℤ≥1). Our results, inspired by analogous results for classical factorization length, demonstrate the types of combinatorial statements one may hope to obtain for sufficiently nice monoids, as well as the subtlety such asymptotic questions can have for general monoids.  

Nanhui Hu, Department of Mathematics, Shantou University, Shantou 515063, Guangdong, P. R. China; Department of Mathematics, Jiaying University, Meizhou 514015, Guangdong, P. R. China (hunhjyu@163.com), and Songxiao Li, Department of Mathematics, Shantou University, Shantou 515063, Guangdong, P. R. China (jyulsx@163.com).
Closure in the logarithmic Bloch-type norm of Dirichlet-Morrey spaces, pp. 525–542.

ABSTRACT. Two characterizations for the closure of Dirichlet-Morrey spaces in the logarithmic Bloch-type spaces are given in this paper. In particular, we get a new characterization for the spaces 𝒞ℬ(H2 ∩ℬ) and 𝒞ℬ(BMOA), respectively. Furthermore, we obtain a criterion for an interpolating Blaschke product to be in the closures.  

Guo-Shuai Mao, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, People’s Republic of China (maogsmath@163.com), and Hao Pan, School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, People’s Republic of China (haopan79@zoho.com).
On the Atkin-Swinnerton-Dyer type congruences for some sequences arising from Calabi-Yau equations, pp. 543–558.

ABSTRACT. In this paper, we show some supercongruences for the coefficients {An}n=0,1,2,… of the analytic solution y0(z) = ∑ n=0∞Anzn to the Calabi-Yau type equations Dy = 0 normalized by the condition y0(0) = A0 = 1, where D is a 4th-order linear differential operator.  

Mabud Ali Sarkar, Department of Mathematics, The University of Burdwan, Burdwan 713104, India (mabudji@gmail.com), and Absos Ali Shaikh, Department of Mathematics, The University of Burdwan, Burdwan 713104, India (aashaikh@math.buruniv.ac.in).
On the image of p-adic logarithm on principal units, pp. 559–591.

ABSTRACT. The p-adic logarithm appears in many places in number theory. Therefore, a comprehensive description of the image of the p-adic logarithm would be beneficial. In particular, it is important to figure out the image of 1 + 𝔪K, where K denotes an algebraic extension of ℚp and 𝔪K represents its maximal ideal. If the ramification index of K is strictly less than p − 1, then it is known that the p-adic logarithm serves as a bijection from 1 + 𝔪K to 𝔪K. If the ramification index is equal to or greater than p − 1, then the p-adic logarithm is no longer a bijection, and the situation is more complicated. Our main result is the computation of log p(1 + 𝔪K) in two distinct cases: first, when K = ℚp(ζp), a totally ramified p-cyclotomic extension of ℚp with a ramification index of p − 1; and second, when K is a quadratic extension of ℚ2, characterised by a ramification index of either 1 or 2. As an application, we compute the normalised p-adic regulator of the number field ℚ(ζp) by utilising the image of the p-adic logarithm on the units.  

Lin Hu, School of Mathematical Sciences, Chongqing Normal University, Chong-qing, 401331, P. R. China (2840788320@qq.com), Rebecca Smith, The University of Newcastle, Callaghan 2308, NSW, Australia (Rebecca.Smith@newcastle.edu.au), and Zhenliang Zhang, School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, P. R. China (zhliang_zhang@163.com).
Geometric mean of Lüroth digits in arithmetic progressions, pp. 593–626.

ABSTRACT. This paper explores the Hausdorff dimension analysis of the sets of real numbers for which the products of partial quotients in Lüroth expansion satisfy certain growth conditions. That is, for any x ∈ (0,1], let [d1(x),d2(x),…] denote its Lüroth expansion. Let Ψ : ℕ → (1,∞) be any function, we establish the Hausdorff dimension of the set:

Ef(Ψ) := {x ∈ (0,1] : ∘ ------------------------- n df(n)(x )d2f(n)(x)...dnf(n)(x) ≥ Ψ(n)
for infinitely many n ∈ ℕ },

where f(n) := cn + t be a linear function, c ∈ ℕ and t ∈ ℤ≥0. It is to be noted that when f(n) is a constant, the Hausdorff dimension stays the same irrespective of the value of the constant. The Hausdorff dimension increases as c increases, whereas, t does not impact the dimension.

 

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).
Weighted Erdős-Kac type theorems for higher moments of cusp form coefficients in short intervals, pp. 627–660.

ABSTRACT. Let f be a normalized primitive holomorphic Hecke eigenform of even integral weight κ for the full modular group Γ = SL(2, ℤ), and denote by λf(n) its n-th Fourier coefficient. In this paper, we establish weighted Erdős-Kac type theorems for the higher power moments of λf(n) in short intervals, by using the generalized Selberg-Delange method developed in recent decades. We also establish the asymptotic formulae for the same higher moments of λf(n) with prescribed number of prime divisors in short intervals. By analogy, we also obtain the similar results for the Fourier coefficients associated to two distinct cuspidal Hecke eigenforms.  

Zhuo Chen, Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, BN1 9QH Brighton, United Kingdom, and Michael Melgaard, Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, BN1 9QH Brighton, United Kingdom (m.melgaard@sussex.ac.uk).
Energy-localized Egorov theorem for matrix-valued Schrödinger operator without Cordes’s hyperbolicity condition, pp. 661–697.

ABSTRACT. We establish a semi-classical Egorov type theorem for a matrix-valued Schrödinger operator, frequently arising in important applications, which does not satisfy typically imposed hypotheses, e.g., Cordes’s hyperbolicity condition. The proof combines a number of new ideas and the techniques involve h-pseudodifferential operator calculus, Riesz projection formula, energy localization, and detailed study of the Heisenberg equations on symbol level using the time-dependent Schrödinger equation to justify an auxiliary localized initial condition which allows for a detailed analysis of the Hamiltonian flow and the derivation of key estimates for the solutions to new auxiliary Heisenberg equations.  

Qing Zhang, Anhui International Studies University, Anhui, 231201, China (zhangqing20241212@163.com), and Akbar Jahanbani, Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran (akbar.jahanbani92@gmail.com).
Modified Sombor index of graphs with a given girth, pp. 699–707.

ABSTRACT. The modified Sombor index of a graph G is defined as mSO(G) = ∑ uv∈E(G)√--1--- d2u+d2v, where du is the degree of the vertex u in G. In this paper, the minimum and maximum values of the modified Sombor index for connected graphs with girth l(l ≥ 3) are determined. Moreover, we obtain lower and upper bounds for the modified Sombor index which depend on the girth of a graph.  

Ofelia T. Alas, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970 São Paulo, Brasil (alas@ime.usp.br), L. Enrique Gutiérrez-Domínguez, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Pos-tal 55-532, 09340, México, D.F., México (luenriquegudo@gmail.com), Lucia R. Junqueira, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970 São Paulo, Brasil (lucia@ime.usp.br), 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).
Maximality and productivity of the weak Lindelöf property, pp. 709–723.

ABSTRACT. Necessary conditions are given in order that a topology be maximal with respect to being weakly Lindelöf and preservation of the weak Lindelöf, the almost Lindelöf and the weak linearly Lindelöf properties under products is studied.  

Zhangyong Cai, Center for Applied Mathematics of Guangxi (Nanning Normal University), College of Mathematics and Statistics, Nanning Normal University, Nanning 530100, P.R. China (zycaigxu2002@126.com), and Shou Lin, Institute of Mathematics, Ningde Normal University, Ningde 352100, P.R. China (shoulin60@163.com).
A question on countable tightness of free Abelian paratopological groups, pp. 725–732.

ABSTRACT. This paper continues to study the countable tightness of free Abelian paratopological groups. An open problem is whether the free Abelian paratopological group AP(X) on a metrizable space X has countable tightness if the subspace X′ consisting of all non-isolated points in X is separable. In this paper, around the above problem, it is established that if X is a metrizable space with only finitely many non-isolated points, then AP(X) has countable tightness.  

Fang Liu, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (58451372@qq.com), Shou Lin, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (shoulin60@163.com), Xin Liu, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (liuxintp@126.com), and Xiangeng Zhou, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (56667400@qq.com).
The symmetrizability in weak topological groups with ideal continuity, pp. 733–750.

ABSTRACT. A mapping which preserves ideal convergence of sequences between topological spaces is precisely an ℐsn-continuous mapping. Based on ℐsn-continuity, in this paper we consider some classes of weak topological groups, discuss the symmetrizability of weak ℐsn-topological groups, and define and study quasi-ℐ-sn-symmetrizable spaces as a common generalization of symmetrizable spaces and quasi-developable spaces.

Let ℐ be an admissible ideal on the set ℕ of natural numbers. The following main results are obtained.

(1) Every ℐ-sn-symmetrizable para-ℐsn-topological group is an ℐ-sn-develop-able space.

(2) Every ℐsn-Baire, quasi-ℐ-sn-symmetrizable para-ℐsn-topological group is an ℐsn-topological group.

These not only generalize some results related to the symmetrizability in weak topological groups, but also present a version in topological algebra using the notion of ideal convergence.