*Editors*: D. Bao (San Francisco, SFSU), D. Blecher
(Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers),
B. Dacorogna (Lausanne), M. Dugas (Baylor), M.
Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), S. Harvey (Rice), A. Haynes (Houston), Y. Hattori
(Matsue, Shimane), W. B. Johnson (College Station), M. Rojas (College Station),
Min Ru (Houston), S.W. Semmes (Rice), D. Werner (FU Berlin).

*Managing Editors*: B. G. Bodmann and K. Kaiser (Houston)

Houston Journal of Mathematics

*Contents*

R. Raveendra Prathap Department of Mathematics, Manonmaniam Sundaranar
University, Tirunelveli 627012, Tamil Nadu, India (rasuraveendraprathap@gmail.com)
and T. Tamizh Chelvam Department of Mathematics, Manonmaniam Sundaranar
University, Tirunelveli 627012, Tamil Nadu, India. ORCID:0000-0002-1878-7847
(tamche59@gmail.com).

Complement graph of the square graph of ﬁnite abelian groups, pp. 845–857

ABSTRACT. Let G be a ﬁnite abelian group. The square graph Γ_{sq}(G) of G is the
simple undirected graph with vertex set G in which two distinct vertices x and y are
adjacent if and only if x + y = 2t for some t ∈ G and 2t≠0 where 0 is the identity
of the group G. In this paper, we discuss the diameter and the girth of the
complement graph Γ_{sq}(G). We give a necessary and suﬃcient condition for
Γ_{sq}(G) to be self-centered. Also, we obtain the independence number, the clique
number and the chromatic number of Γ_{sq}(G) and hence we prove that Γ_{sq}(G) is
weakly perfect. Also, we discuss when the complement graph Γ_{sq}(G) is perfect.
Further, we give a necessary and suﬃcient condition for Γ_{sq}(G) to be vertex
pancyclic.

J. A. Nido Valencia Maestría en Ciencias de la Complejidad, Universidad Autónoma
de la Ciudad de México, San Lorenzo 290, Col. Del Valle, C.P. 03100, Mexico City,
Mexico (antonio.nido@uacm.edu.mx), H. G. Salazar Pedroza Departamento de
Ingeniería en Minas, Metalurgia y Geología, Universidad de Guanajuato, Ex Hacienda
de San Matías S/N, Col. San Javier, C.P. 36020, Guanajuato, Gto., Mexico
(hg.salazar@ugto.mx), and L. M. Villegas Silva Departamento de Matemáticas,
Universidad Autónoma Metropolitana Iztapalapa, Avenida San Rafael Atlixco 186, C.P.
09340, Mexico City, Mexico (villegas63@gmail.com).

Compactness phenomena around the class of locally projective modules, pp.
859–895

ABSTRACT. In this paper we examine compactness phenomena in some classes
of modules. We ﬁrst demonstrate that a κ-projective module M is projective
when the size κ of M is weakly compact. We introduce the class of κ-locally
projective modules and prove that they have the compactness property for
R when κ is a singular or a weakly compact cardinal and R is a PID. In the
case when κ is singular, we show that Shelah’s Singular Compactness Theorem
holds for these modules and also for torsionless modules. We show that for
some non weakly compact cardinal κ, compactness does not hold for locally
projective modules. Finally, we prove some compactness properties for U-torsionless
modules, where U is a bimodule, for certain special classes of large cardinals
κ.

Qianyun Wang School of Mathematical Sciences, Shanghai Jiao Tong University,
Shanghai, 200240, China (wangqy1226@gmail.com).

The closed range property for the , pp. 897–914 -operator

ABSTRACT. In this paper, we study the closed range property for the -operator on
the unit disc, the punctured disc, the annulus, and a special Stein domain. The necessary
conditions for the closed range property of -operator are also given on these domains
endowed with the Kähler metrics.

Xiao-Min Li Department of Mathematics, Ocean University of China, Qingdao,
Shandong 266100, People’s Republic of China (lixiaomin@ouc.edu.cn), Cong-Cong Wu
Department of Mathematics, Ocean University of China, Qingdao, Shandong 266100,
People’s Republic of China (wucongcong1233@163.com)., and Hong-Xun Yi
Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s
Republic of China (hxyi@sdu.edu.cn).

Dirichlet series satisfying a Riemann type functional equation and sharing a set, pp.
915-933

ABSTRACT. In 2011, Li proved that if two L-functions L_{1} and L_{2} satisfy the same
functional equation with a(1) = 1 and L_{1}^{−1}(c_{j}) = L_{2}^{−1}(c_{j}) for two ﬁnite distinct
complex numbers c_{1} and c_{2}, then L_{1} = L_{2}. We prove that if two L-functions L_{1} and L_{2}
satisfy the same functional equation with a(1) = 1 and E_{L1}(S) = E_{L2}(S) for a ﬁnite
set S = {c_{1},c_{2}}, where c_{1} and c_{2} are two ﬁnite distinct complex values, then
L_{1} = L_{2}.

Xiu Ji Department of Mathematics, Beijing Institute of Technology, Beijing, 100081,
China (jixiu1106@163.com) and Tongzhu Li Department of Mathematics, Beijing
Institute of Technology, Beijing, 100081, China (litz@bit.edu.cn).

Conformal homogeneous spacelike hypersurfaces with two distinct principal curvatures in
Lorentzian space forms, pp. 935–951

ABSTRACT. Let M_{1}^{n+1}(c) be an (n + 1)-dimensional Lorentzian space form and
𝒞(M_{1}^{n+1}(c)) denote the conformal transformation group of M_{1}^{n+1}(c). A spacelike
hypersurface f : M^{n} → M_{1}^{n+1}(c) is called a conformal homogeneous spacelike
hypersurface, If there exists a subgroup G ⊂𝒞(M_{1}^{n+1}(c)) such that the orbit
G(p) = f(M^{n}),p ∈ f(M^{n}). In this paper, we classify completely all conformal
homogeneous spacelike hypersurfaces with two distinct principal curvatures
under the conformal transformation group of M_{1}^{n+1}(c) when the dimension
n ≥ 3.

Abolfazl Mohajer Universität Mainz, Fachbereich 08, Institut für Mathematik, 55099
Mainz, Germany (mohajer@uni-mainz.de).

Shimura varieties and abelian covers of the line, pp. 953-972

ABSTRACT. We prove that under some conditions on the monodromy, families of
abelian covers of the projective line do not give rise to (higher dimensional) Shimura
subvarieties in A_{g}. This is achieved by a reduction to p argument. We also use another
method based on monodromy computations to show that two dimensional subvarieties in
the above locus are not special. In particular it is shown that such families have usually
large monodromy groups. Together with our earlier results, the above mentioned results
contribute to classifying the special families in the moduli space of abelian varieties and
partially completes the work of several authors including the author’s previous
work.

Víctor Cañulef-Aguilar Facultad de Matemáticas, Pontiﬁcia Universidad Católica de
Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile (vacanulef@uc.cl) and
Duvan Henao Facultad de Matemáticas, Pontiﬁcia Universidad Católica de
Chile

Vicuña Mackenna 4860, Macul, Santiago, Chile (dhenao@mat.puc.cl).

Hölder estimates for the Neumann problem in a domain with holes and a relation
formula between the Dirichlet and Neumann problems, pp. 973–1004

ABSTRACT. In this paper we study the dependence of the Hölder estimates on the
geometry of a domain with holes for the Neumann problem. For this, we study the
Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk
(and in the exterior of the disk), from which we get a relation between harmonic
extensions and harmonic functions with prescribed Neumann condition on the
boundary of the disk (for both interior and exterior problems). A novelty of
this work is that we deal directly with the Hölder regularity of the single layer
potentials of the Dirichlet and Neumann problems for the Poisson equation,
something that most of the times seems to be avoided by studying the Newtonian
potential.

Mehmet Çelk Texas A&M University-Commerce, Department of Mathematics,
Commerce, TX 75429, USA (mehmet.celik@tamuc.edu), Sönmez Şahutoğlu
University of Toledo, Department of Mathematics & Statistics, Toledo, OH
43606, USA (sonmez.sahutoglu@utoledo.edu), and Emil J. Straube Texas
A&M University, Department of Mathematics, College Station, TX 77843, USA
(straube@math.tamu.edu).

Compactness of Hankel operators with continuous symbols on convex domains, pp.
1005-1016

ABSTRACT. Let Ω be a bounded convex domain in ℂ^{n}, n ≥ 2, 1 ≤ q ≤ (n − 1), and
ϕ ∈ C(Ω). If the Hankel operator H_{ϕ}^{q−1} on (0,q − 1)–forms with symbol ϕ is compact,
then ϕ is holomorphic along q–dimensional analytic (actually, aﬃne) varieties in the
boundary. We also prove a partial converse: if the boundary contains only ‘ﬁnitely many’
varieties, 1 ≤ q ≤ n, and ϕ ∈ C(Ω) is analytic along the ones of dimension q (or higher),
then H_{ϕ}^{q−1} is compact.

Lorenzo Galeotti Amsterdam University College, Science Park 113, 1098 XG
Amsterdam, The Netherlands (l.galeotti@uva.nl), Aymane Hanaﬁ Trinity College,
University of Cambridge, Cambridge CB2 1TQ, England (ah859@alumni.cam.ac.uk), and
Benedikt Löwe Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55,
20146 Hamburg, Germany (loewe@math.uni-hamburg.de).

Relations between notions of gaplessness for non-Archimedean ﬁelds, pp. 1017–1031

ABSTRACT. In this paper, we are studying the relationship between notions of
gaplessness for non-Archimedean ordered ﬁelds, in particular three generalised
Bolzano-Weierstraß properties: the Sikorski property (which fails for saturated ﬁelds),
the Keisler-Schmerl property, and the weak Bolzano-Weierstraß property introduced by
Carl, Galeotti, and Löwe. We show that the weak Bolzano-Weierstraß property is
“Bolzano-Weierstraß minus Cauchy completeness” and is equivalent the tree property of
the base number of the ﬁeld. We furthermore improve on a number of results by Carl,
Galeotti, and Löwe.

Daciberg L. Gonçalves Department of Mathematics - IME, University of São Paulo,
Rua do Matão 1010, CEP: 05508-090 - São Paulo - SP - Brazil (dlgoncal@ime.usp.br),
Thaís F. M. Monis São Paulo State University (Unesp) , IGCE - Department
of Mathematics, Av. 24 A 1515, CEP: 13506-900 - Rio Claro - SP - Brazil
(thais.monis@unesp.br), and Stanisław Spież Institute of Mathematics - Polish
Academy of Sciences, IMPAN, ul. Śniadeckich 8, 00-656 - Warsaw - Poland
(spiez@impan.pl).

Deﬁcient and multiple points of maps into a manifold, pp. 1033–1052

ABSTRACT. The notion of Hopf’s absolute degree is classically deﬁned for
mappings between manifolds of the same dimension, even when they are not
necessarily orientable. In this paper, we extend such notion for mappings from more
general spaces into manifolds. Once we have stablished a version of Hopf’s
absolute degree for certain maps f : X → M, where M is a manifold but X
need not be, we study the sets of deﬁcient and multiple points of f. In case of
the set of deﬁcient points, we estimate its dimension. For multiple points, we
study its density in X, and we also provide examples where its complement is
dense.

Dawei Wang Department of Mathematics, University of Utah, 155 S 1400 E, JWB 233,
Salt Lake City, UT 84112 (Dawei.Wang@utah.edu).

On the capacity dimension of the boundary of CAT(0) spaces, pp. 1053–1074

ABSTRACT. In this paper, we study the capacity dimension of the boundary of
CAT(0) spaces. We ﬁrst compare the two metrics on the boundary of a hyperbolic
CAT(0) space, i.e., the visual metric and the conical metric, and prove that they give the
same capacity dimension of the boundary. Then we study the capacity dimension of the
boundary of buildings, which is an important class of CAT(0) spaces. Finally, we give a
possible method to prove the ﬁniteness of the asymptotic dimension of CAT(0)
spaces.