HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 49, No. 1, 2023

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), 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

Jean Goubault-Larrecq, Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France (goubault@lsv.fr), and Bastien Laboureix, Université de Lorraine, LORIA, 54000 Nancy, France (bastien.laboureix@hotmail.fr).
Statures and sobrification ranks of Noetherian spaces, pp. 1–76.

ABSTRACT. There is a rich theory of maximal order types of well-partial-orders (wpos), pioneered by de Jongh and Parikh (1977) and Schmidt (1981). Every wpo is Noetherian in its Alexandroff topology, and there are more; this prompts us to investigate an analogue of that theory in the wider context of Noetherian spaces.

The notion of maximal order type does not seem to have a direct analogue in Noetherian spaces per se, but the equivalent notion of stature, investigated by Blass and Gurevich (2008) does: we define the stature ||X|| of a Noetherian space X as the ordinal rank of its poset of proper closed subsets. We obtain formulas for statures of sums, of products, of the space of words on a space X, of the space of finite multisets on X, in particular. They confirm previously known formulas on wpos, and extend them to Noetherian spaces.

The proofs are, by necessity, rather different from their wpo counterparts, and rely on explicit characterizations of the sobrifications of the corresponding spaces, as obtained by Finkel and the first author (2020).

We also give formulas for the statures of some natural Noetherian spaces that do not arise from wpos: spaces with the cofinite topology, Hoare powerspaces, powersets, and spaces of words on X with the so-called prefix topology.

Finally, because our proofs require it, and also because of its independent interest, we give formulas for the ordinal ranks of the sobrifications of each of those spaces, which we call their sobrification ranks.

 

Toni Harrison, Kent State University, Department of Mathematics, Kent, OH, USA (aharri60@kent.edu), Thomas Michael Keller, Texas State University, Department of Mathematics, San Marcos, TX, USA (tk04@txstate.edu), and Joshua Andrew Rice, Iowa State University, Department of Mathematics, Ames, IA, USA (jar238@iastate.edu).
Groups with a large conjugacy class relative to a normal subgroup, pp. 77–86.

ABSTRACT. In 2013, the first author studied a parameter e defined by the equation

√--(√-- ) |G | = k k + e ,

where k is the largest conjugacy class size of a finite group G. They then obtained a general bound for |G| and studied when equality occurs. Their parameter was motivated by earlier work of Snyder on character degrees; Snyder introduced a new parameter e′ and bounded |G| in terms of e′. Later, C. Durfee defined a relative version of Snyder’s parameter for groups with respect to a fixed normal subgroup N and studied upper bounds for |G∕N| in terms of the relative parameter. Following the lead on the character theoretic side again, we define the conjugacy class analogue eN of the relative parameter; in particular, we obtain the tight upper bound

|G∕N | ≤---p---(eN)2 (p− 1)2

and study when equality occurs. In addition, we establish some interesting properties for eN.

 

Hualin Miao, School of Mathematics, Hunan University, Changsha, Hunan, 410082, China (miaohualinmiao@163.com), Wei Luan, School of Mathematics, Hunan University, Changsha, Hunan, 410082, China (luanwei@hnu.edu.cn), Dexian Liu, College of science, Hunan University of Technology, Zhuzhou, Hunan , 412007, China (liudexian2010@126.com), and Qingguo Li, School of Mathematics, Hunan University, Changsha, Hunan, 410082, China (liqingguoli@aliyun.com).
Spaces of functions from RB-domains to quasicontinuous domains, pp. 87–103.

ABSTRACT. This paper mainly investigates function spaces from RB-domains to quasicontinuous domains. It is proved that the function space from an RB-domain to a quasicontinuous domain with property M∗ is quasicontinuous, and simultaneously, the Isbell-topology and the Scott-topology coincide. Without using step functions, we show that the function space from an RB-domain to a domain is continuous. In this case, the Isbell-topology equals the Scott-topology.  

Linke Ma, Department of Mathematics, Nanjing University, Hankou Road 22, Nanjing, 210093, P.R.China (cmlk912@163.com), Liangwen Liao, Department of Mathematics, Nanjing University, Hankou Road 22, Nanjing, 210093, P.R.China (maliao@nju.edu.cn), and Xujie Shi, School of Science, Nanjing University of Posts and Telecommunications, Wenyuan Road 9, Nanjing, 210093, P.R.China (shixujie163@163.com).
Uniqueness of entire functions concerning sharing the value with their derivatives and difference operator, pp. 105–116.

ABSTRACT. In this paper, we study the uniqueness of entire functions that share the value with their derivatives and difference operator simultaneously. Suppose f(z) is a nonconstant entire function of finite order, k is a positive integer, a and c are two nonzero finite constants. If f(z), f(k)(z) and Δcf(z) share a CM, then f(z) = Ceλz −a- A + a, where λk = A = (ln(A + 1))k, A and C are nonzero constants. Moreover, suppose f(z) is a nonconstant entire function. If f(z), f′′(z) and Δcf(z) share 0 CM, then f(z) = Ceλz, where λ2 = A.  

Doo Hyun Hwang, Reserach Institute of real and Complex Manifolds, Kyungpook National University, Daegu 41566, Republic of Korea (engus0322@naver.com), Imsoon Jeong, Department of Mathematics Education, Cheongju University, Chung-cheongbuk-do 28503, Republic of Korea (isjeong@cju.ac.kr), and Young Jin Suh, Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea (yjsuh@knu.ac.kr).
Yamabe and gradient Yamabe solitons on real hypersurfaces in the complex hyperbolic quadric, pp. 117–142.

ABSTRACT. In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex hyperbolic quadric Qm∗ = SO2,m0∕SO2SOm. Next as an application we can assert a classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex hyperbolic quadric Qm∗.  

Maryam Akbari, Faculty of mathematics, statistics and computer science, Semnan University, P. O. Box 35131-19111, Semnan, Iran (maryamakbari@semnan.ac.ir), and Fereidoun Habibian, Faculty of mathematics, statistics and computer science, Semnan University, P. O. Box 35131-19111, Semnan, Iran (fhabibian@semnan.ac.ir).
Higher dimensional amenability for a generalized matrix Banach algebra, pp. 143–156.

ABSTRACT. Let G := 𝔊(𝔄,M,N,𝔅) be a generalized matrix Banach algebra. Under certain conditions, we establish a relationship between the n-th (co)homology group of G and the n-th (co)homology group of 𝔄. The results are applied to show a connection between the higher dimensional amenability of G and 𝔄.  

March Boedihardjo, Department of Mathematics, UCLA, Los Angeles, CA 90045-1438 (march@math.tamu.edu), and Ken Dykema, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (kdykema@math.tamu.edu).
Erratum to “On algebra-valued R-diagonal elements”, pp. 157–158. Original paper available here.

Ayreena Bakhtawar, School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (a.bakhtawar@unsw.edu.au), Mumtaz Hussain, Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, Australia (m.hussain@latrobe.edu.au), Dmitry Kleinbock, Brandeis University, Waltham MA 02454-9110 (kleinboc@brandeis.edu), and Bao-Wei Wang, School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China (bwei_wang@hust.edu.cn).
Metrical properties for the weighted products of multiple partial quotients in continued fractions, pp. 159–194.

ABSTRACT. The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet’s theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grow at a certain rate. Recently it was observed that the growth of the product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet’s theorem. In this paper we consider the products of several consecutive partial quotients raised to different powers. Namely, we find the Lebesgue measure and the Hausdorff dimension of the following set:

{ } m∏−1 t ℰt(ψ ) := x ∈ [0,1) : ain+i(x) ≥ Ψ (n) for infinitely many n ∈ ℕ , i=0

where ti ∈ ℝ+ for all 0 ≤ i ≤ m − 1, and Ψ : ℕ → ℝ≥1 is a positive function.

 

G. Hoepfner, Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brasil (ghoepfner@ufscar.br), A. Raich, Department of Mathematical Sciences, SCEN 327, 1 University of Arkansas, Fayetteville, AR 72701 (araich@uark.edu), and P. Rampazo, Departamento de Ciências Exatas, Biológicas e da Terra, Universidade Federal Fluminense, RJ, 24220-900, Brasil (prampazo@id.uff.br).
Weighted Hardy spaces and ultradistributions, pp. 195–229.

ABSTRACT. The goal of this work is to identify certain classes of global ultradistributions as boundary values of generalized Hardy spaces defined on cones. The ultradistributions arise as elements of dual spaces of classes of globally Lq-integrable ultradifferentiable functions defined in terms of weight functions. We also demonstrate that global Lq-Gevrey functions are an example.  

Yutian Lei, Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China (leiyutian@njnu.edu.cn), and Xin Xu, Chern Institute Mathematics and LPMC, Nankai University, Tianjin, 300 071, China (9820210107@nankai.edu.cn).
A Liouville theorem for an integral equation of the Ginzburg-Landau type, pp. 231–245.

ABSTRACT. In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation

−→ ∫ u(1 − |u|2) u(x) = l + C ∗ n |x-−-y|n−α-dy. ℝ

Here u : ℝn → ℝk is a bounded, uniformly continuous and differentiable function with k ≥ 1 and 1 < α < n, −→l ∈ ℝk is a constant vector, and C∗ is a real constant. If u is the finite energy solution, we prove that |−→ l|∈{0,1}. Furthermore, we also give a Liouville type theorem (i.e., u ≡−→ l). This is consistent with the Cazenave-type Liouville theorem in PDE case.