HOUSTON JOURNAL OF MATHEMATICS

 Electronic Edition Vol. 50, No. 4, 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

Yuxu Chen, Department of Mathematics, Sichuan University, Chengdu 610064, China (yuxuchen1210@.sina.com), Hui Kou, Department of Mathematics, Sichuan University, Chengdu 610064, China (kouhui@scu.edu.cn), and Zhenchao Lyu, Department of Mathematics, Sichuan University, Chengdu 610064, China (zhenchaolyu@scu.edu.cn).
Free algebras over continuous spaces, pp. 751–776.

ABSTRACT. Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. A c-space (resp., b-space) can be characterized as a continuous (resp., algebraic) directed space. We show that continuous spaces are just all retracts of algebraic spaces through topological ideals, which are topological generalization of rounded ideals. By means of this result, we prove the conjecture proposed by Keimel and Lawson in 2012 that the free algebras over c-spaces are still c-spaces. Besides, we show that the category of continuous spaces with way-below preserving maps as morphisms is cartesian closed.  

Mohammad Ali Parsa, Department of Mathematics, University of Birjand, Birjand, Iran 97175-615 (mohammadali.parsa@birjand.ac.ir), and Hosein Fazaeli Moghimi, Department of Mathematics, University of Birjand, Birjand, Iran 97175-615 (hfazaeli@birjand.ac.ir).
Sobrification of modules with I-adic topology, pp. 777–798.

ABSTRACT. Let R be a commutative ring with identity and M be an R-module. For any ideal I of R, the I-adic sobrification of M denoted s_IM, consists of the closure of elements of M for the I-adic topology on M. This article presents an algebraic theory for I-adic sobrification of modules. For this purpose, we show that s_IR admits naturally a topological ring structure which can be embedded in the I-adic completion ^I of R. Moreover, s_IM admits naturally a s_IR-module structure, and in particular, s_I can be viewed as an additive covariant functor from the category of R-modules to the category of s_IR-modules. Considering I a sequential ideal (as a new type of ideal) of a Noetherian ring R, it is shown that s_I is naturally isomorphic to s_IR ⊗− on finitely generated R-modules. We also study the left derived functors {L_ns_I}_n∈ℕ of s_I, and show that if R is an Artinian ring, then L_ns_I(M) is isomorphic to H_n(Mic(Xⁱ ⊗ M)) as the n-th homology of the microscope complex Mic(Xⁱ ⊗ M) with Xⁱ a free resolution of R∕Iⁱ.  

Grigore Călugăreanu, Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania (calu@math.ubbcluj.ro), and Horia F. Pop, Department of Computer Science, Babeş-Bolyai University, Cluj-Napoca, 400084, Romania (horia.pop@ubbcluj.ro).
On square stable range one matrices over commutative rings, pp. 799–824.

ABSTRACT. Khurana and Lam introduced the concept of left square stable range one (ssr1) for an element of a unital ring. In this paper, over commutative rings, we examine 2 × 2 matrices that satisfy the ssr1 condition. Our findings indicate significant differences from the stable range one (sr1) condition, necessitating the development of specialized techniques.

Among our results, we provide characterizations of 2 × 2 matrices that possess ssr1 in several cases: implicitly over commutative rings, nilpotent matrices over commutative reduced rings, and explicitly over elementary divisor domains.

As applications, we demonstrate that over commutative Bézout domains, ring multiples of idempotent 2 × 2 matrices have ssr1. Additionally, we characterize ssr1 matrices with a zero row (or zero column) and offer an explicit description of ssr1 integral matrices.

Building on these results, we further show that the Jacobson Lemma for ssr1 holds for 2 × 2 integral matrices, contingent on a conjecture regarding the greatest common divisors of their entries.

M.M. Czerwińska, Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224 (m.czerwinska@unf.edu).
Monotone properties in symmetric spaces of measurable operators, pp. 825–840.

ABSTRACT. Let ℳ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ. We study connections between the monotonicity properties of an operator x in a noncommutative symmetric space E(ℳ,τ) and its singular value function μ(x) in a symmetric Banach function space E on [0,τ(1)). Consequently, we reveal that the monotonic characteristics of symmetric operator spaces strongly correlate with their convex properties. Moreover, a new connection is uncovered between the k-extreme points and the upper monotone points within the framework of symmetric function and operator spaces.  

Timothy Rainone, Occidental College, Department of Mathematics, 1600 Campus Rd, Los Angeles, California, 90041 (trainone@oxy.edu).
C*-norms on cross-sectional algebras and conditional expectations, pp. 841–859.

ABSTRACT. C^∗-algebras that arise from groups, dynamical systems, topological groupoids, and Fell bundles are all constructed as norm completions of convolution, or rather, cross-sectional algebras. In each of these cases there is a C^∗-norm obtained from the spatial left-regular representation which allows for a natural evaluation map to be continuous and faithful. The continuous extension of this evaluation map to the C^∗-completion is known as a conditional expectation. In this piece we prove a fundamental factor lemma and apply it in each of these constructions to show that the left-regular norm is the smallest C^∗-norm making the expectation continuous.  

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).
Kato’s chaos of a multiple mapping and its continuous self-map, pp. 861–872.

ABSTRACT. In 2016, Hou and Wang introduced the concept of multiple mappings based on iterated function system, which is an important branch of fractal theory. In this paper, we introduce the definitions of sensitivity, accessibility, and Kato’s chaos of multiple mappings from a set-valued perspective. We show that a multiple mapping and its continuous self-maps do not imply each other in terms of sensitivity or accessibility. While a sufficient condition for multiple mappings to be sensitive, accessible and Kato’s chaotic is provided, respectively. And the sensitivity, accessibility, and Kato’s chaos of multiple mappings are preserved under topological conjugation.  

Runa Shimada, Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan (231s010s@stu.kobe-u.ac.jp).
Geometry on deformations of S₁ singularities, pp. 873–890.

ABSTRACT. To study a one parameter deformation of an S₁ singularity taking into consideration its differential geometric properties, we give a form representing the deformation using only diffeomorphisms on the source and isometries of the target. Using this form, we study differential geometric properties of S₁ singularities and the Whitney umbrellas appearing in the deformation.  

Tongzhu Li, Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China (litz@bit.edu.cn), and Bingxin Xie, Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China (xiebingxinchn@163.com).
Möbius semi-parallel hypersurfaces with three distinct principal curvatures in 𝕊ⁿ⁺¹, pp. 891–903.

ABSTRACT. Let x : Mⁿ → 𝕊ⁿ⁺¹ be an immersed hypersurface without umbilical points in an (n + 1)-dimensional sphere 𝕊ⁿ⁺¹, one can define two fundamental Möbius invariants on Mⁿ, the Möbius metric g and the Möbius second fundamental form B. Let R_m denote the Riemannian curvature operator of the Möbius metric g, the hypersurface x is called a Möbius semi-parallel hypersurface if R_m ∘B = 0. In this paper, we classify completely the Möbius semi-parallel hypersurfaces with three distinct principal curvatures up to a Möbius transformation.  

Y. Wang, School of Mathematics and Statistics, Henan Normal University, Xinxiang 453007, Henan, P. R. China (wyn051@163.com).
A classification of real hypersurfaces in nonflat complex space forms, pp. 905–920.

ABSTRACT. In this paper we present a classification theorem of real hypersurfaces in a nonflat complex space form in terms of the classification of the induced almost contact metric structures on the hypersurfaces.  

Dimitrios Poulakis. Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (poulakis@math.auth.gr).
On the number of integral points on curves of genus zero, pp. 921–931.

ABSTRACT. Let F(X,Y ) be an absolutely irreducible polynomial in ℤ[X,Y ] such that the algebraic curve defined by the equation F(X,Y ) = 0 is of genus zero with at least three infinite valuations. In this note, we compute an improved upper bound for the number of solutions (x,y) ∈ ℤ² to the equation F(X,Y ) = 0.  

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 higher power moments of coefficients of Dedekind zeta function twisted by Fourier coefficients of cusp forms, pp. 933–958.

ABSTRACT. Let K₃ be a non-normal cubic extension over ℚ, and let a_K3(n) denotes the n-th coefficient of the Dedekind zeta function associated to K₃∕ℚ. Let λ_g(n) be the n-th normalized Fourier coefficient attached to a primitive holomorphic cusp form of even integral weight k ≥ 2 for the full modular group Γ = SL(2, ℤ). In this paper, we investigate the asymptotic behaviour of higher power moments of a_K3(n) twisted by Fourier coefficients of cusp forms of the type

where ℓ₁ ≥ 2,ℓ₂ ≥ 1 are any given integers. Furthermore, as an application, we also establish the corresponding asymptotic formula of the variance of a_K3^ℓ₁(n)λ_g^2ℓ₂(n).

Jiewen Chen, School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi 710119, P.R. China (cje1993@163.com), and Bin Zhao, School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi 710119, P.R. China (zhaobin@snnu.edu.cn).
Countable type properties in quotient spaces of paratopological groups, pp. 959–975.

ABSTRACT. In this paper, we investigate countable type properties in quotient spaces of paratopological groups. It is mainly show that if H is a compact subgroup of a paratopological group G, then (1) G∕H has countable type ⇔ G∕H is feathered ⇔ G contains a compact subgroup K with H ⊂ K such that χ(K∕H,G∕H) ≤ ω and G∕K is quasi-metrizable; (2) G∕H is quasi-metrizable ⇔ G∕H is feathered and csf-countable ⇔ G∕H is feathered and has countable tightness ⇔ G∕H is a feathered K-space.  

Zheng Ping, Department of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (63113015@qq.com), Shou Lin, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P.R. China (shoulin60@163.com), and Rongxin Shen, Department of Mathematics, Taizhou University, Taizhou, Jiangsu 225300, P.R. China (srx20212021@163.com).
α_i-Fréchet spaces as the images of metric spaces, pp. 977–992.

ABSTRACT. In this paper we discuss how to represent α_i-Fréchet spaces for i ∈{1,2,3,4} as certain images of metric spaces, and how to characterize a space so that every quotient mapping onto the space is a pseudo-open-like mapping. It is known that a space X is an α₂-Fréchet (resp., α₄-Fréchet) space if and only if it is a strictly countably bi-quotient (resp., countably bi-quotient) image of a metric space. It is proved that (1) X is an α₁-Fréchet space if and only if it is an α₁-sequence-covering and pseudo-open image of a metric space; (2) X is an α₃-Fréchet space if and only if it is an α₃-pseudo-open image of a metric space. We also provide a few examples to show that some implications do not hold among α_i-spaces and α_i-Fréchet spaces, and to illustrate some irreversible relationships among the mappings discussed in this paper.