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
Wladimir G. Boskoff, Department of Mathematics and Computer Science, Ovidius
University, Bd. Mamaia, nr. 124, Constan
a, Romania (boskoff@univ-ovidius.ro), and
Bogdan D. Suceavă, Department of Mathematics, California State University,
Fullerton, McCarthy Hall 154, Fullerton, CA 92831-6850 (bsuceava@fullerton.edu).
A natural representation of volumes yields a remarkable affine consequence, pp.
635–644.
ABSTRACT. At the beginning of the 20th Century there was a growing interest for the
investigation of the action of linear groups on the geometry of surfaces. In that context of
ideas, the quest for a connection between curvature and the behaviour of linear groups
rose naturally. Pursuing the original thought, we investigate how the geometric meaning
of this idea is intimately related to the concept of volume of parallelepiped boxes. We
show how the ratio of the Gaussian curvature divided by the fourth power of a
certain distance of interest in the geometry of surfaces can be represented as a
function of volumes. This geometric description explores the profound meaning of
a quantity considered by Ţiţeica in 1907, in a work that sparked a growing
interest in affine differential geometry, as an illustration of Felix Klein’s Erlangen
Program, in which the quest for geometric invariants was the main point of inquiry.
Kamel Touahri, Université de Tunis, Institut Préparatoire aux Études d’Ingénieurs de
Tunis, 2, Rue Jawaher Lel Nehru - 1089 Montfleury - Tunisie et Université de Tunis El
Manar, LR11ES12, Laboratoire Algèbre, Topologie, Arithmétique et Ordre, 2092 Tunis,
Tunisie (ktouahri@yahoo.fr).
On the average of some arithmetical functions under Ostrowski’s sum of digits, pp.
645–673.
ABSTRACT. Given two distinct integers m1,m2 ≥ 2, we set α1 = [0;1,m1] and
α2 = [0;1,m2] to be two Ostrowski’s representations, we denote by Sα1(n) and Sα2(n)
respectively the sum of digits functions in the Ostrowski α1 and α2−representations of a
given positive integer n. In this paper we are concerned with the estimate of the average
of some multiplicative arithmetical functions under digital constraints on the Ostrowski
sum of their digits.
James Jing Yu Zhao, School of Accounting, Guangzhou College of Technology and
Business, Foshan 528138, P. R. China (zhao@gzgs.edu.cn).
The extended reverse ultra log-concavity of transposed Boros-Moll sequences, pp.
675–692.
ABSTRACT. The Boros-Moll sequences {dℓ(m)}ℓ=0m arise in the study of
evaluation of a quartic integral. After the infinite log-concavity conjecture of the
sequences {dℓ(m)}ℓ=0m was proposed by Boros and Moll, a lot of interesting
inequalities on dℓ(m) were obtained, although the conjecture is still open. Since dℓ(m)
has two parameters, it is natural to consider the properties for the sequences
{dℓ(m)}m≥ℓ, which are called the transposed Boros-Moll sequences here. In this paper,
we mainly prove the extended reverse ultra log-concavity of the transposed
Boros-Moll sequences {dℓ(m)}m≥ℓ, and hence give an upper bound for the ratio
dℓ2(m)∕(dℓ(m − 1)dℓ(m + 1)). A lower bound for this ratio is also established which
implies a result stronger than the log-concavity of the sequences {dℓ(m)}m≥ℓ. As a
consequence, we also show that the transposed Boros-Moll sequences possess a
stronger log-concave property than the Boros-Moll sequences do. At last, we
propose some conjectures on the Boros-Moll sequences and their transposes.
Guodong Hua, School of Mathematics and Statistics, Weinan Normal University, Weinan
714099, Shaaxi Province, China; Research Institute of Qindong Mathematics, Weinan
Normal University, Weinan 714099, Shaaxi Province, China (gdhuasdu@163.com),
and Jiafan Zhang, School of Mathematics and Statistics, Weinan Normal
University, Weinan 714099, Shaaxi Province, China; Research Institute of Qindong
Mathematics, Weinan Normal University, Weinan 714099, Shaaxi Province, China
(Zhangjiafan_math@163.com).
The general divisor problem of Dirichlet coefficients of symmetric square L-functions over
certain sequences of positive integers, pp. 693–733.
ABSTRACT. Let f be a normalized primitive cuspidal Hecke eigenform of even integral
weight for the full modular group Γ = SL(2, ℤ), and denote by λsym2f(n) the n-th
normalized coefficient of the Dirichlet expansion of the symmetric square L-function
attached to f. In this paper, we investigate the asymptotic behaviour of the general
divisor problem associated to λsym2f(nν),(ν ≥ 2) over the set of certain polynomials of
arithmetic interests.
Kyung Seung Lee, RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju,
52828, Republic of Korea (kslhg33@gmail.com).
Rational period functions for Γ0+(3) with irrational poles, pp. 735–748.
ABSTRACT. This paper investigates rational period functions for the group Γ0+(3)
with irrational poles. Our attention is directed towards rational period functions with
non-zero poles. Using quadratic forms, we construct explicitly rational period functions
for Γ0+(3) with irrational poles.
Mabud Ali Sarkar, Department of Mathematics, The University of Burdwan,
Burdwan-713104, India; Department of Mathematics, Darjeeling Hills University,
Darjeeling-734313, India (mabudji@gmail.com), and Absos Ali Shaikh,
Department of Mathematics, The University of Burdwan, Burdwan-713104, India
(aashaikh@math.buruniv.ac.in).
Erratum to “On the image of p-adic logarithm on principal units”, pp. 749–750.
ABSTRACT. In this erratum, we omit Theorem 1.1 and its Corollary 1.2 from our
paper “On the Image of p-adic Logarithm on Principal Units,” Houston Journal of
Mathematics, Volume 50, Number 3, pp. 559–591, due to an error in the proof of
Theorem 1.1. There is no loss of the main result of the paper, as Theorem 1.5 covers
both Theorem 1.1 and its Corollary 1.2.
Doudou Zhu, School of Mathematics and Statistics, Zhengzhou University, Zheng-zhou
450001, PR China (zhudoudou0379@163.com), Jingli Ren, School of Mathematics and
Statistics, Zhengzhou University, Zhengzhou 450001, PR China (renjl@zzu.edu.cn),
and Tianze Wang, Institute of Mathematics, Henan Academy of Science,
Zhengzhou 450046, PR China; School of Mathematics and Statistics, North China
University of Water Resources and Electric Power, Zhengzhou 450046, PR China
(wtz@ncwu.edu.cn).
On the number of powers of 2 in pairs of linear Goldbach-Linnik equations, pp. 751–765.
ABSTRACT. In this paper, it is proved that k=19 is sufficient for the solvability of the
system of a pair of equations both with two primes and k powers of 2. This improves the
best known result of k=34.
Dongxin Zhang, Department of Mathematics and Statistics, North China University of
Water Resources and Electric Power, Jinshui E Road, Zhengzhou, Henan, 450046, P.
R. China; Institute of Mathematics, Henan Academy of Sciences, Zhengzhou,
Henan 450046, P. R. China (17634605691@163.com), Feng Zhao, Department of
Mathematics and Statistics, North China University of Water Resources and
Electric Power, Jinshui E Road, Zhengzhou, Henan, 450046, P. R. China; Institute
of Mathematics, Henan Academy of Sciences, Zhengzhou, Henan 450046, P.
R. China (zhaofeng@ncwu.edu.cn), and Shichuang Wang, Department of
Mathematics and Statistics, North China University of Water Resources and Electric
Power, Jinshui E Road, Zhengzhou, Henan, 450046, P. R. China; Institute of
Mathematics, Henan Academy of Sciences, Zhengzhou, Henan 450046, P. R. China
(x20231080769@stu.ncwu.edu.cn).
Mean value estimates for Fourier coefficients of cusp forms on special sets, pp.
767–784.
ABSTRACT. Let f be a normalized primitive holomorphic cusp form of even integral
weight k for the full modular group Γ = SL(2, ℤ). In this paper, we investigate the upper
bounds of the error terms associated with the mean value estimates for coefficients
λf⊗f⊗
⊗lf(n)of l-fold product L-functions, where f ⊗ f ⊗
⊗lf denotes the l-fold
product of l.
Xinyan Li, School of Mathematics and Statistics, North China University of
Water Resources and Electric Power, Zhengzhou, Henan 450046, P.R. China
(16692171030@163.com), and Xiaodong Zhao, Institute of Mathematics, Henan
Academy of Sciences, Zhengzhou, Henan 450046, P.R. China (zhaoxiaodong@hnas.ac.cn).
A pair of Goldbach-Linnik equations on unlike powers of primes, pp. 785–797.
ABSTRACT. It was proved that, for k = 128, every pair of large positive odd integers
satisfying certain necessary conditions can be expressed as the sum of a prime, four cubes
of primes, and k powers of 2. In this paper, we improve the value of k to 86.
Ramesh Kumar, Department of Mathematics, Faculty of Mathematical Sciences,
University of Delhi, Delhi, 110007, India (rameshkhichar1458@gmail.com), Abdul
Gaffar Khan, Department of Mathematics, Kirori Mal College, University of Delhi,
Delhi, 110007, India (gaffarkhan18@gmail.com), and Tarun Das, Department of
Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India
(tarukd@gmail.com).
Expansive and topologically stable measures for set-valued maps, pp. 799–815.
ABSTRACT. In this paper, we define positive expansive, positive shadowable and
topologically stable Borel probability measures with respect to an upper-semicontinuous
closed-valued map on a compact metric space X. We prove that the set of all
positive expansive measures is a Gδσ subset of the set of all the Borel probability
measures on X. We further prove that every positive expansive Borel probability
measure with positive pseudo-orbit tracing property is topologically stable.
Liang-Xue Peng, Department of Mathematics, School of Mathematics, Statistics
and Mechanics, Beijing University of Technology, Beijing 100124, China
(pengliangxue@bjut.edu.cn).
On semi-stratifiable spaces with open (G) and related results, pp. 817–834.
ABSTRACT. In this article, we show that if a semi-stratifiable space X satisfies open
(G), then X is developable and has a point-countable base. No separation axioms are
assumed in this result. If a space X is subparacompact M-scattered with respect to a
sequence {𝒰n : n ∈ ω} of open covers of X and satisfies open (G), then X is developable
and has a point-countable base. We show that if X is a perfect regular space, then X
is a σ-space if and only if X is a union of countably many σ-spaces. By this
result and a known result we get that if X is a perfect regular space and is a
union of countably many semi-stratifiable subspaces satisfying (G), then X is a
σ-space. If a space X has a point-finite open cover 𝒰 such that every U ∈𝒰 is a
semi-stratifiable subspace of X and satisfies open (G) (resp., (G)) then X is developable
(resp., has a σ-discrete network). If each space Xn satisfies (G) (resp., open (G))
for every n ∈ ω, then the product space X = ∏
n∈ωXn satisfies (G) (resp.,
open (G)). Hence a countable product of T1-spaces satisfying (G) is a D-space.
Yu Xue, School of Mathematics, Hunan University, Changsha, Hunan, 410082, China
(xuey@hnu.edu.cn), and Qingguo Li, School of Mathematics, Hunan University,
Changsha, Hunan, 410082, China (liqingguoli@aliyun.com).
Ordinal sum of T0 spaces, pp. 835–852.
ABSTRACT. The ordinal sums of a finite number of T0 spaces are introduced via the ordinal sums of posets given by Birkhoff. We then investigate whether the properties of sobriety, well-filteredness and being a d-space are preserved under this construction. Furthermore, we explore the relationship between the soberification of T0 spaces and the ordinal sum, and prove that for a finite family of T0 spaces, the ordinal sum of their sobrification coincides with the sobrification of their ordinal sum. This conclusion also holds for the well-filterification and D-completion of T0 spaces.