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
William Paulsen, Department of Mathematics and Statistics, 2105 East, Aggie Rd,
Jonesboro, AR 72401 (wpaulsen@astate.edu), and Graham West.
On perfect cuboids and CN-elliptic curves, pp. 227–240.
ABSTRACT. In this paper we show the connection between two famous unsolved
problems, the perfect cuboid problem, and the congruent number problem. In particular,
we show that if a perfect cuboid exists, then it would have a corresponding congruent
number, such that the triangle formed by the derived cuboid will have all three of its
Heron angles having the same corresponding congruent number. This gives a new way to
search for perfect cuboids, even if the perfect cuboid contains hundreds of digits. First we
show that no perfect cuboid exists for a congruent number which has rank 1.
Then we test all higher ranking congruent numbers via elliptic curves up to
7500.
Li-Yuan Wang, School of Physical and Mathematical Sciences, Nanjing Tech
University, Nanjing 211816, People’s Republic of China (wly@smail.nju.edu.cn), and
Zhi-Wei Sun, Department of Mathematics, Nanjing University, Nanjing, 210093,
People’s Republic of China (zwsun@nju.edu.cn).
On practical numbers of some special forms, pp. 241–247.
ABSTRACT. In this paper we study practical numbers of some special forms. For any
integers b ≥ 0 and c > 0, we show that if n2 + bn + c is practical for some integer n > 1,
then there are infinitely many nonnegative integers n with n2 + bn + c practical. We also
prove that there are infinitely many practical numbers of the form q4 + 2 with q
practical, and that there are infinitely many practical Pythagorean triples (a,b,c) with
gcd(a,b,c) = 6 (or gcd(a,b,c) = 4).
Lan Nguyen, Department of Mathematics, University of Wisconsin-Parkside
(nguyenl@uwp.edu).
Quantum functional equations and extension of non-prime supports for solutions with
rational field of coefficients, pp. 249–280.
ABSTRACT. In this paper, we provide a solution to a problem raised by Melvyn
Nathanson, concerning the extensions of supports for solutions of functional equations
arising from quantum arithmetic, in the case where the supports of these solutions are
not necessarily prime semigroups and their field of coefficients is ℚ. Contrary to the
prime semigroup support case, a solution in this case requires a different approach due to
the lack of prime indexed elements.
Narek Hovsepyan, Department of Mathematics, Rutgers University, NJ, USA
(narek.hovsepyan@rutgers.edu).
On the optimal analytic continuation from discrete data, pp. 281–293.
ABSTRACT. We consider analytic functions from a reproducing kernel Hilbert space.
Given that such a function is of order 𝜖 on a set of discrete data points, relative to its
global size, we ask how large can it be at a fixed point outside of the data set. We obtain
optimal bounds on this error of analytic continuation and describe its asymptotic
behavior in 𝜖. We also describe the maximizer function attaining the optimal error in
terms of the resolvent of a positive semidefinite, self-adjoint and finite rank
operator.
Masoumeh Hosseini, Department of Pure Mathematics, Faculty of Mathematics and
Statistics, University of Isfahan, Isfahan, 81746-73441-Iran (hoseini_masomeh@ymail.com),
and Hamid Reza Salimi Moghaddam, Department of Pure Mathematics, Faculty of
Mathematics and Statistics, University of Isfahan, Isfahan, 81746-73441-Iran
(hr.salimi@sci.ui.ac.ir, salimi.moghaddam@gmail.com).
Invariant Einstein Kropina metrics on Lie groups and homogeneous spaces, pp.
295–304.
ABSTRACT. In this article, we study Einstein Kropina metrics on Lie groups and
homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie
groups. As an example of this method, a family of non-Riemannian Einstein Kropina
metrics on the special orthogonal group SO(n) is given. Then, we classify all left
invariant Einstein Kropina metrics on simply connected 3-dimensional real Lie groups.
We provide a procedure to build Einstein Kropina metrics on homogeneous spaces. Using
this technique, we study invariant Einstein Kropina metrics on the spheres. Finally, we
show that projective spaces do not admit any homogeneous non-Riemannian Einstein
Kropina metric.
Noureddine Karim, Chouaib Doukkali University, Department of Mathematics,
Faculty of science, El Jadida, Morocco (noureddinekarim1894@gmail.com), Otmane
Benchiheb, Chouaib Doukkali University, Department of Mathematics, Faculty of
science, El Jadida, Morocco (otmane.benchiheb@gmail.com, benchiheb.o@ucd.ac.ma),
and Mohamed Amouch, Chouaib Doukkali University, Department of Mathematics,
Faculty of science, El Jadida, Morocco (amouch.m@ucd.ac.ma).
On compositional dynamics on spaces of analytic functions, pp. 305–317.
ABSTRACT. In this work, we investigate super-recurrence, super-rigidity, and
uniformly super-rigidity of composition operators acting on H(Ω) the space
of holomorphic functions on Ω, where Ω is either the complex plane ℂ or the
punctured plane ℂ∖{0}. We deduce the form of the symbol ϕ that generates a
super-recurrent, super-rigid, uniformly super-rigid composition operator Cϕ acting on
H(Ω).
Yong Chen, Department of Mathematics, Hangzhou Normal University, Hangzhou,
311121, P.R.China (ychen@hznu.edu.cn), Hansong Huang, Department of
Mathematics, East China University of Science and Technology, Shanghai, 200237,
P.R.China (hshuang@ecust.edu.cn), and Shan Li, Department of Mathematics, Jiangsu
University of Technology, Chang-zhou, 213001, P.R.China (lishan_math@163.com).
On compactness of composition operators on Bergman spaces over planar domains, pp.
319–336.
ABSTRACT. In this paper we consider compact composition operators acting between
Bergman spaces over planar domains. When those two domains equal the open unit disc,
it is well known that any analytic selfmap of that disc induces a bounded composition
operator, and that operator is compact if and only if the inducing selfmap has no finite
angular derivative on the unit circle. We reformulate that condition in geometric terms
and prove that it characterizes compact composition operators between Bergman spaces
over planar domains if the boundaries of those domains are C2-smooth. We
give examples showing that the smoothness condition cannot be dropped. We
also discuss some interesting phenomena related to domains with cusps on the
boundary.
Ved Prakash Gupta, School of Physical Sciences, Jawaharlal Nehru University, New
Delhi 110067 (vedgupta@mail.jnu.ac.in, ved.math@gmail.com), and Lav Kumar Singh,
School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067
(lavksingh@hotmail.com).
On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space, pp.
337–351.
ABSTRACT. It was shown in a previous paper that the Banach algebra
A := S2(ℓ2) ⊗γS2(ℓ2) is not Arens regular, where S2(ℓ2) denotes the Banach algebra of
the Hilbert-Schmidt operators on ℓ2. In this article, employing the notion of
limits along ultrafilters, we prove that the irregularity of S2(ℓ2) ⊗γS2(ℓ2) is not
strong. Along the way, we provide a class of functionals in A∗∗ which lie in the
topological center but are not in A; and, as a consequence, we deduce that
A∗∗ is not an annihilator Banach algebra with respect to any of the two Arens
products.
Basma Ammous, Department of Mathematics, Faculty of Sciences, University of Sfax,
BP 1171, Sfax 3000, Tunisia (ammous.basma@hotmail.fr), Lamia Dammak,
Department of Mathematics, Faculty of Sciences, University of Sfax, BP 1171, Sfax 3000,
Tunisia (dammaklamia@yahoo.fr), and Mohamed Hbaib, Department of
Mathematics, Faculty of Sciences, University of Sfax, BP 1171, Sfax 3000, Tunisia
(mmmhbaib@gmail.com).
Palindromic and quasi-palindromic Ruban continued fractions in ℚp, pp. 353–368.
ABSTRACT. The aim of this paper is to establish a new transcendence criterion of
p-adic continued fractions. We prove that a p-adic number whose the sequence of its
partial quotients is bounded in ℚp and begins with arbitrarily long palindromes is either
quadratic or transcendental.
Abdul Hassen, Department of Mathematics, Rowan University, 201 Mullica Hill Rd,
Glassboro, NJ 08028, USA (hassen@rowan.edu), Driss Drissi, Department of
Mathematics, Rowan University, 201 Mullica Hill Rd, Glassboro, NJ 08028, USA
(drissi@rowan.edu), Emily Hammett, Department of Mathematics, Rowan University,
201 Mullica Hill Rd, Glassboro, NJ 08028, USA (hammet22@students.rowan.edu), and
Quyn Ulrich, Department of Mathematics, Rowan University, 201 Mullica Hill Rd,
Glassboro, NJ 08028, USA (ulrich57@students.rowan.edu).
Hypergeometric polynomials and numbers of higher order, pp. 369–394.
ABSTRACT. In this paper, we shall consider what we refer to as Hypergeometric Bernoulli numbers and polynomials of higher order, which are given by
where TN−1(x) = ∑ n=0N−1xn∕n!. When N = 1, and t = 1, we have the Bernoulli polynomials and when z = 0, we have the Bernoulli numbers. In our case, Bkt(N,0) = Bkt(N) are the Hypergeometric Bernoulli numbers of order t.
Among other things, we shall prove that
J. A. Martínez-Cadena, Departamento de Matemáticas, Facultad de Ciencias,
Universidad Nacional Autónoma de México, Circuito exterior s/n, Ciudad Universitaria,
CP 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, CP 04510, Ciudad de México, México (atamariz@unam.mx).
The C-bounded-open topology on C(X), pp. 395–414.
ABSTRACT. For the collection of C-bounded subsets of X (usually called bounded),
Cb, and the collection of closed C-bounded subsets of X, Cb, we study the spaces
(C(X),𝒯Cb,u) and (C(X),𝒯Cb), where C(X) is the set of continuous real-valued
functions defined on the Tychonoff space X, 𝒯Cb,u is the topology of the uniform
convergence on the C-bounded subsets of X, and 𝒯Cb is the C-bounded-open
topology. The spaces (C(X),𝒯Cb,u) and (C(X),𝒯Cb) are also studied, where
now what the symbols 𝒯Cb,u and 𝒯Cb mean should be clear. We obtain: (1)
For countably compact infinite zero-dimensional spaces 𝒯Cb ⊊ 𝒯Cb; (2) The
space (C(X),𝒯Cb) is a topological group if and only if 𝒯Cb is contained in
the topology of uniform convergence on X, 𝒯u, if and only if, each element in
Cb is a C-compact subset of X; (3) If X is pseudocompact and each element
in Cb is a C-compact subset of X, then 𝒯Cb = 𝒯u ⊆𝒯Cb, and, moreover, if
there is an element in Cb which is not C-compact, then 𝒯u ⊊ 𝒯Cb; (4) For a
normal pseudocompact space X, (C(X),𝒯Cb) is metrizable if and only if it is a
q-space.
Iztok Banič, Faculty of Natural Sciences and Mathematics, University of Maribor,
Koroška 160, SI-2000 Maribor, Slovenia, Institute of Mathematics, Physics and
Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia, Andrej Marušič Institute,
University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia (iztok.banic@um.si),
Judy Kennedy, Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont,
TX 77710, USA (kennedy9905@gmail.com), and Piotr Minc, Department of
Mathematics, 218 Parker Hall, Auburn University, Auburn, AL 36849-5310, USA
(mincpio@auburn.edu).
Characterizations of 𝒫-like continua that do not have the fixed point property, pp.
415–450.
ABSTRACT. We give two characterizations of 𝒫-like continua X that do not have the fixed point property. Both characterizations are stated in terms of sequences of open covers of X that follow fixed-point-free patterns. We use these to characterize planar tree-like continua that do not have the fixed point property in terms of infinite sequences of tree-chains in the plane that follow fixed-point-free patterns. We also establish a useful relationship between these tree-chains and commutative simplicial diagrams that we use later to construct a finite sequence (of any given length) of tree-chains in the plane that follows a fixed-point-free pattern.
An earlier characterization of 𝒫-like continua with the fixed point property was given in 1994 by Feuerbacher based on a 1963 result by Mioduszewski. The Mioduszewski-Feuerbacher characterization is expressed in terms of almost commutative inverse diagrams. In contrast, our approach is more geometric, and it may potentially lead to new methods in the elusive search for a planar tree-like continuum without the fixed-point property.