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
Mohammed Bachir, Laboratoire SAMM 4543, Université Paris 1
Panthéon-Sorbonne, Centre PMF, 90 rue Tolbiac, F-75634 Paris cedex 13, France
(Mohammed.Bachir@univ-paris1.fr), and Aris Daniilidis, Institut für Stochastik und
Wirtschaftsmathematik, VADOR E105-04, TU Wien, Wiedner Hauptstraße 8, A-1040
Wien (aris.daniilidis@tuwien.ac.at).
Trace convexity and Choquet theory, pp. 247–282.
ABSTRACT. We study the notion of trace-convexity for functions and respectively, for
subsets of a compact topological space. This notion generalizes both classical
convexity of vector spaces, as well as Choquet convexity for compact metric
spaces and provides an alternative description for the convexification for sets and
functions. We show that the class of upper semicontinuous convex-trace functions
attaining their maximum at exactly one Choquet-boundary point is residual and
we obtain several enhanced versions of the maximum principle, including a
multi-maximum principle for families of convex-trace functions, which generalize both
the classical Bauer’s theorem as well as its abstract version in the Choquet theory.
We illustrate our notions and results with concrete examples of three different
types.
Ming Xu, School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.
R. China (mgmgmgxu@163.com), Shaoqiang Deng, School of Mathematical Sciences
and LPMC, Nankai University, Tianjin 300071, P. R. China (dengsq@nankai.edu.cn), and
Zaili Yan, Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P.
R. China (yanzaili@nbu.edu.cn).
Geodesic orbit Finsler metrics on Euclidean spaces, pp. 283–303.
ABSTRACT. A Finsler space (M,F) is called a geodesic orbit space if any geodesic of
constant speed is the orbit of a one-parameter subgroup of isometries of (M,F). In this
paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We
will show that, in this case (M,F) is a fiber bundle over a symmetric Finsler space M1 of
non-compact type such that each fiber M2 is a totally geodesic nilmanifold
with a step-size at most 2, and the projection π : M → M1 is a Finslerian
submersion. Furthermore, when M1 has no Hermitian symmetric factors, the
fiber bundle description for M can be strengthened to M = M1 × M2 as coset
spaces, such that each product factor is totally geodesic in (M,F) and is a
geodesic orbit Finsler space itself. Finally, we use the techniques in this paper
to discuss the interaction between the geodesic orbit spaces and the negative
(non-positive) curved conditions, and provide new proofs for some of our previous
results.
Limiao Lin, School of Mathematics and statistics, FJKLMAA, Fujian Normal
University, Fuzhou, 350117, China (83343055@163.com), and Tongzhu Li,
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China
(litz@bit.edu.cn).
Hypersurfaces with harmonic conformal Gauss map in 𝕊4, pp. 305–323.
ABSTRACT. Let x : Mn → 𝕊n+1 be an oriented immersed hypersurface in
(n + 1)-dimensional sphere 𝕊n+1 with a global unit normal vector field ξ, then the
conformal Gauss map Φ = (H,Hx + ξ) : Mn → 𝕊1n+2 is invariant under the Möbius
transformations of 𝕊n+1, where H is the mean curvature of x and 𝕊1n+2 is the
(n + 2)-dimensional de Sitter space. In this paper, we classify completely the
hypersurfaces with harmonic conformal Gauss map in 𝕊4 up to a Möbius transformation
of 𝕊4.
Mikaël Pichot, McGill University, 805 Sherbrooke St. W., Montréal, Québec,
Canada, H3A 0B9 (mikael.pichot@mcgill.ca), and Erik Séguin, University
of Waterloo, 200 University Ave. W., Waterloo, Ontario, Canada, N2L 3G1
(e2seguin@uwaterloo.ca).
Positive definite maps on amenable groups, II, pp. 325–338.
ABSTRACT. We introduce an Ulam-type stability condition for positive definite
maps defined on a countable group and prove that this condition characterizes
amenability.
Khalil Ayadi, Sfax University, Higher Institute of Industrial Management
of Sfax, Faculty of Sciences, Lab AGTS, Tunisia (khalil.ayadi@isgis.usf.tn),
Chiheb Ben Bechir, Sfax University, Faculty of Sciences, Lab AGTS, Tunisia
(chihebbenbechir@gmail.com), and Maher Saadaoui, (mahersaadaoui716@gmail.com).
Specializable series resulting from folded continued fractions and transcendence, pp.
339–351.
ABSTRACT. In [A. J. Van Der Poorten and J. Shallit: A Specialised Continued
Fraction, Canad. J. Math., 5(1993), 1067–1079 ],the authors examine the continued
fraction expansion of the series ∑
k=2∞T−Fk, where (F
h) stands for the well-known
Fibonacci number sequence. This series is intriguing because it features a continued
fraction that can be specialized, allowing us to determine the continued fraction of a real
number ∑
k=2∞x−Fk by replacing the variable T by an integer x ≥ 2. Their proofs rely
on the Folding lemma and the Roth’s Theorem, respectively, to compute all partial
quotients of this series and to draw the conclusion that it is transcendental. We applied
their methodology to examine a number of additional fascinating classes of series
delivered in different formats.
Zhi-Qiang Lu, College of Science, Tianjin University of Technology, Tianjin 300384, P.
R. China (lzq@stud.tjut.edu.cn), and Yan-Ping Mu, College of Science, Tianjin
University of Technology, Tianjin 300384, P. R. China (yanping.mu@gmail.com).
Inversion relations and the q-Racah polynomials, pp. 353–368.
ABSTRACT. By the inversion relations given by Andrews, we expand the q-shifted
factorial (x,γδq∕x;q)n in terms of the q-Racah polynomials Rm(μ(x)). As one
application of this expansion, we express a 4ϕ3 series as a double sum, from which we
derive several Hecke-type identities. As another application, we evaluate the sum
∑
i=0Nωi ⋅ Rm(μ(q−i)) ⋅ (q−i,γδqi+1;q)n by orthogonality, where ωi is the weight
function for Rm(μ(x)). This allows us to derive some summation and transformation
formulas.
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).
Persistence and cw-topological stability for set-valued maps, pp. 369–380.
ABSTRACT. In this paper, we introduce the α-persistent property and cw-topological
stability for upper-semicontinuous closed-valued set-valued maps on a compact metric
space. We prove that the α-persistent property and the positive pseudo-orbit tracing
property are equivalent notions irrespective of the map considered. We further establish
that every upper-semicontinuous closed-valued map with the positive pseudo-orbit
tracing property is cw-topologically stable provided the induced shift map on the inverse
limit space is positive continuum-wise expansive and we use this to give sufficient
condition under which the positive pseudo-orbit tracing property, the α-persistent
property and cw-topological stability become equivalent notions. We give necessary
examples to support the results.
Huijun Hou, School of Mathematics, Hunan University, Changsha, Hunan, 410082, P.
R. China (houhuijun2021@163.com), Qingguo Li, School of Mathematics, Hunan
University, Changsha, Hunan, 410082, P. R. China (liqingguoli@aliyun.com), Hualin
Miao, School of Mathematics, Hunan University, Changsha, Hunan, 410082, P. R. China
(miaohualinmiao@163.com), and Dongsheng Zhao, Mathematics and Mathematics
Education, National Institute of Education, Nanyang Technological University, 1
Nanyang Walk, Singapore, 637616 (dongsheng.zhao@nie.edu.sg).
The reflective hulls of some subcategories in the category of all T0 spaces, pp.
381–395.
ABSTRACT. In this paper, we use some results obtained recently by Shen et al., to
answer two problems concerning the reflective hulls of subcategories of the
category of all T0 spaces with continuous mappings. One is about the category of
all k-bounded sober spaces and the other is about the category of all open
well-filtered spaces which was introduced to prove that every core-compact
well-filtered space is sober. We also consider the corresponding problem on the
category of all Γ-determined spaces. We obtain the result that the product of two
Γ-determined spaces is still Γ-determined and further show that the reflective
hull of the category of all Γ-determined spaces equals the category of all sober
spaces.
Chengyuan Wu, Department of Mathematics, National University of Singapore,
Singapore 119076 (wuchengyuan@u.nus.edu), Shiquan Ren, School of Mathematics and
Statistics, Henan University, Kaifeng 475004, China (renshiquan@henu.edu.cn), Jie Wu,
Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408,
China (wujie@bimsa.cn), and Kelin Xia, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 637371(xiakelin@ntu.edu.sg).
Weighted (co)homology and weighted Laplacian, pp. 397–429.
ABSTRACT. In this paper, we generalize the combinatorial Laplace operator of Horak
and Jost by introducing the ϕ-weighted coboundary operator induced by a weight
function ϕ. Our weight function ϕ is a generalization of Dawson’s weighted boundary
map. We show that our above-mentioned generalizations include new cases that are
not covered by previous literature. Our definition of weighted Laplacian for
weighted simplicial complexes is also applicable to weighted/unweighted graphs and
digraphs.
Taotao Cao, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an
710049, P.R. China (cttmath@163.com).
Some properties on compactness involving lower topology, pp. 431–443.
ABSTRACT. Well-filteredness is one of the key properties of T0 spaces in domain
theory. The Ω∗-compactness involving the lower topology has close connections with
well-filteredness. It is well-known that every Ω∗-compact and monotone determined space
is well-filtered. In this paper, we further explore the properties on Ω∗-compactness.
The main results include: (1) a monotone determined and well-filtered space
need not be Ω∗-compact; (2) if there exists an infinite chain with respect to
the specialization order of a T0 space, the space is not Ω∗-compact; (3) the
subcategory of all Ω∗-compact spaces is not reflective in the category of all T0
spaces.
Kodai Wada, Department of Mathematics, Kobe University, 1-1 Rokkodai-cho,
Nada-ku, Kobe 657-8501, Japan (wada@math.kobe-u.ac.jp).
CF-moves for virtual links, pp. 445–462.
ABSTRACT. Oikawa defined an unknotting operation for virtual knots called a
CF-move, and gave a classification of 2-component virtual links up to CF-moves by the
virtual linking number and his n-invariant. In particular, it was proved that a CF-move
characterizes the virtual linking numbers for 2-component odd virtual links. In this
paper, we extend this result by classifying odd virtual links and almost odd virtual links
with arbitrary number of components up to CF-moves, using virtual linking numbers.
Moreover, we extend Oikawa’s n-invariant and introduce two invariants for
3-component even virtual links. Using these invariants together with virtual
linking numbers, we classify 3-component even virtual links up to CF-moves.
As a result, a classification of 3-component virtual links up to CF-moves is
provided.