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
Papiya Bhattacharjee, Department of Mathematical Sciences, Charles E. Schmidt
College of Science, Florida Atlantic University, Boca Raton, FL 33431, USA
(pbhattacharjee@fau.edu), Lee Klingler, Department of Mathematics, Oregon State
University, Corvallis, OR 97331-4605, USA (klinglel@oregonstate.edu), and Warren
Wm. McGovern, Wilkes Honors College, Florida Atlantic University, Jupiter, FL
33458, USA (warren.mcgovern@fau.edu).
Martínez frames, pp. 463–487.
ABSTRACT. We call a frame Martínez if it is an algebraic frame with FIP in
which every element is a d-element. The study of Martinez frames and the
d-operator in this article has led to a better understanding of the q-nucleus defined
in [15]. We generalize the construction of the q-nucleus to arbitrary sets of
primes and investigate this operator from a topological perspective on the prime
spectrum.
Patrick W. Keef, Department of Mathematics and Statistics, Whitman College, Walla
Walla, WA 99362 (keef@whitman.edu).
Generalizing a result of Hausen and Johnson on Jacobson radicals of endomorphism
rings, pp. 489–495.
ABSTRACT. A results of Hausen and Johnson (1978) describing the Jacobson
radicals of the endomorphism rings of primary Abelian groups that are sufficiently
projective is extended to the class of countably totally projective groups. In so
doing, the arguments used to prove this earlier characterization are simplified
considerably.
Gregor Dolinar, University of Ljubljana, Faculty of Electrical Engineering, Trža-ška
cesta 25, SI-1000 Ljubljana, Slovenia, and IMFM, Jadranska 19, SI-1000 Ljubljana,
Slovenia (gregor.dolinar@fe.uni-lj.si), Bojan Kuzma, University of Primorska,
Glagoljaška 8, SI-6000 Koper, Slovenia, and IMFM, Jadranska 19, SI-1000 Ljubljana,
Slovenia (bojan.kuzma@upr.si), Janko Marovt, University of Maribor, Faculty of
Economics and Business, Razlagova 14, SI-2000 Maribor, Slovenia, and IMFM,
Jadranska 19, SI-1000 Ljubljana, Slovenia (janko.marovt@um.si), and Edward Poon,
Department of Mathematics, Embry-Riddle Aeronautical University, 3700 Willow Creek
Road, Prescott, AZ, USA (edward.poon@erau.edu).
Spectrum preservers on densely defined unbounded operators, pp. 497–507.
ABSTRACT. We classify additive bijections on the set of possibly unbounded
self-adjoint operators which preserve the spectrum. Similar problem is considered also on
the set of infinitesimal generators of 𝒞0-semigroups.
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).
On the conjecture of Chen and Yi, pp. 509–530.
ABSTRACT. In the paper, we discuss the uniqueness problem of meromorphic function
f that shares a1,a2 and ∞ CM with Δcf, where a1 and a2 are two distinct entire
functions such that ρ(ai) < 1, i = 1,2 and c ∈ ℂ ∖{0}. The obtained results confirm the
conjecture posed by Chen and Yi [2]. Also in the paper, we improve the recent result of
Huang and Zhang [8]. Moreover, we exhibit some examples to show that our results are
best possible.
Wei Chen, School of Sciences, Chongqing University of Posts and Telecommunications,
Chongqing, 400065, P.R. China (weichensdu@126.com), Qiong Wang, Chongqing Key
Laboratory of Intelligent Analysis and Decision on Complex Systems, Chongqing
University of Posts and Telecommunications, Chongqing, 400065, P.R. China
(qiongwangsdu@126.com), and Liu Yang, School of Mathematics and Physics, Anhui
University of Technology, Maanshan, 243032, P.R. China (yangliu6@ahut.edu.cn).
On generalized Fermat Diophantine functional equations in ℂn and Picard type
theorems, pp. 531–549.
ABSTRACT. This paper concerns entire and meromorphic solutions of the generalized
Fermat Diophantine functional equations hfp + kgq = 1 in ℂn, where h,k are
meromorphic coefficients in several complex variables and p,q ≥ 1 are integers
with (p,q)≠(1,1). As applications, we determine when entire solutions of the
simple-looking functional equation fp + gq = 1 in ℂ reduce to constant and
then apply the result to show two well-known Picard type theorems in a direct
manner.
Hui Li, School of Science, China University of Mining and Technology-Beijing, Beijing
100083, P. R. China (lihui2021@amss.ac.cn), Mingliang Fang, School of Sciences,
Hangzhou Dianzi University, Hangzhou 310012, P. R. China (mlfang@hdu.edu.cn), and
Xiao Yao, School of Mathematical Sciences and LPMC, Nankai University, Tianjin
300071, P. R. China (yaoxiao@nankai.edu.cn).
A difference analogue of Hayman-Clunie’s theorem, pp. 551–566.
ABSTRACT. In 1959, Hayman proved a landmark result concerning the meromorphic
function f with its second derivative f′′ [Ann. Math. 70 (1959), 9-42]. Shortly after
Hayman’s paper, Clunie completely settled the case for nth derivative f(n) (n ≥ 3) in
1962. The whole picture is still not clear for the case n = 1. Over the past few decades, a
lot of work has been made in exploring the difference analogue of the value distribution
theory of meromorphic functions. This progress is largely attributed to the development
of the difference logarithmic derivative lemma, which is considered to be the
most important tool in this field. However, the strength of this technique is still
insufficient to prove the difference type of the celebrated Hayman-Clunie’s theorem.
In this paper, we develop a new technique to establish the difference type of
Hayman-Clunie’s theorem for all n ≥ 1. More precisely, we prove that if the
transcendental meromorphic function f and its forward difference Δcnf have
only finite number of zeros and poles in the whole complex plane, we obtain
f(z) = R(z)exp(h(z) + C1z), where R(z) is a rational function, h(z) is an entire function
of period c or 2c, and C1 is a constant. Moreover, we have a complete classification when
c and 2c happen, which depends on the total number of zeros and poles of
f.
Bogdan D. Suceavă, Department of Mathematics, California State University,
Fullerton, McCarthy Hall 154, Fullerton, CA 92831-6850 (bsuceava@fullerton.edu).
The spread of the shape operator as a curvature invariant for a smooth hypersurface, pp.
567–577.
ABSTRACT. The spread of a matrix was originally introduced in linear algebra by L.
Mirsky, in 1956. The first question we investigate in our present study is the behavior of
a limiting process involving the spread of the shape operator in the neighborhood of an
umbilical point; the problem is interesting because it naturally yields a “zero over zero”
limiting situation. Secondly, by regarding the spread of the shape operator as a curvature
invariant we state and prove a fundamental inequality of B.-Y. Chen type between
the intrinsic quantities and the extrinsic quantities at a point on a smooth
hypersurface.
Hongmei Zhu, College of Mathematics and Information Science, Henan Normal
University, Xinxiang, 453007, P.R. China (zhm403@163.com), and Lumin Song, College
of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007,
P.R. China (15837318225@163.com).
On the scalar curvature of Kropina metrics I, pp. 579–602.
ABSTRACT. In this paper, we study the scalar curvature defined by H. Akbar-Zadeh in
Finsler geometry. We prove that a Kropina metric is of weakly isotropic scalar curvature
if and only if it is an Einstein metric. Further, we give a negative answer to Yamabe
problem on Kropina metrics with isotropic S-curvature.
Sajad Salami, Institute of Mathematics and Statistics & State University of Rio de
Janeiro & Rio de Janeiro, Brazil (sajad.salami@ime.uerj.br), and Tony Shaska,
Department of Mathematics & Statistics, Oakland University, Roch-ester Hills, MI
(tanush@umich.edu).
Local and global heights on weighted projective varieties, pp. 603–636.
ABSTRACT. We investigate local and global weighted heights a-la Weil for weighted
projective spaces via Cartier and Weil divisors and extend the definition of weighted
heights on weighted projective spaces from [5] to weighted varieties and closed
subvarieties. We prove that any line bundle on a weighted variety admits a locally
bounded weighted M-metric. Using this fact, we define local and global weighted heights
for weighted varieties in weighted projective spaces and their closed subschemes, and
show their fundamental properties.
Andrea Ammerlaan, Nipissing University, Department of Computer Science &
Mathematics, 100 College Drive, Box 5002, North Bay, Ontario, Canada, P1B 8L7
(ajammerlaan879@my.nipissingu.ca), Ana Anušić, Nipissing University, Department
of Computer Science & Mathematics, 100 College Drive, Box 5002, North Bay, Ontario,
Canada, P1B 8L7 (anaa@nipissingu.ca), and Logan C. Hoehn, Nipissing University,
Department of Computer Science & Mathematics, 100 College Drive, Box 5002, North
Bay, Ontario, Canada, P1B 8L7 (loganh@nipissingu.ca).
Radial departures and plane embeddings of arc-like continua, pp. 637–666.
ABSTRACT. We study the problem of Nadler and Quinn from 1972, which asks
whether, given an arc-like continuum X and a point x ∈ X, there exists an embedding of
X in ℝ2 for which x is an accessible point. We develop the notion of a radial
departure of a map f : [−1,1] → [−1,1], and establish a simple criterion in terms of
the bonding maps in an inverse system on intervals to show that there is an
embedding of the inverse limit for which a given point is accessible. Using this
criterion, we give a partial affirmative answer to the problem of Nadler and
Quinn, under some technical assumptions on the bonding maps of the inverse
system.
Lei Mou, School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China (moulei@cnu.edu.cn), and Yanhui Huang, School of Mathematical Sciences,
Capital Normal University, Beijing 100048, China (yanhuihuang94@163.com).
Star countability of products of subspaces of ordinals, pp. 667–676.
ABSTRACT. For an infinite cardinal κ, a topological space X is called κ-compact if
every F ⊆ X with |F|≥ κ has an accumulation point. A space X is said to be star
countable (respectively star Lindelof) if for every open cover 𝒰 of X, there exists a
countable subset (respectively a Lindelof subspace) F of X such that St(F,𝒰) = X. In
this paper, we give a characterization when A × B is κ-compact for subspaces A and
B of an ordinal ł, where κ > ø is a regular cardinal. We also show that for
subspaces A and B of an ordinal, A × B is star countable if and only if it is star
Lindelof.
Meng Bao, School of Sciences and Arts, Suqian University, Suqian, 263800, P. R. China
(mengbao95213@163.com), and Xiaoquan Xu, School of mathematics and statistics,
Minnan Normal University, Zhangzhou 363000, P. R. China (xiqxu2002@163.com).
On some kinds of factorizable topological groups, pp. 677–691.
ABSTRACT. Based on ℝ-factorizable topological groups and ℳ-factorizable topological
groups, we introduce four classes of factorizabilities on topological groups, named
Pℳ-factorizability, Pm-factorizability, Sℳ-factorizabil-ity and PSℳ-factorizability,
respectively. Then it is shown that a topological group G is Pm-factorizable iff G is
Pℝ-factorizable, and G is PSℳ-factorizable iff G is Pℳ-factorizable. Some properties
of these classes of groups are investigated.
J. A. Martínez-Cadena, Departamento de Matemáticas, Facultad de
Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior s/n,
Ciudad Universitaria, Apartado Postal 04510, Ciudad de México, México
(martinezcadenajuan@gmail.com), and Á. Tamariz-Mascarúa, Departamento de
Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México,
Circuito Exterior s/n, Ciudad Universitaria, Apartado Postal 04510, Ciudad de México,
México (atamariz@unam.mx).
Some properties involving feeble compactness, I: Compact-boundedness and selective
feeble compactness, pp. 693–714.
ABSTRACT. We study some subclasses of feebly compact Hausdorff spaces such as the
(weakly) compact-bounded spaces and selectively (sequentially) feebly compact spaces,
paying special attention to topological groups.