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

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

Houston Journal of Mathematics

*Contents*

**Azarang, Alborz,** Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz-Iran
(a_azarang@scu.ac.ir).

On Fields with only finitely many maximal subrings, pp. 299-324.

ABSTRACT. Fields with only finitely many maximal subrings are completely
determined. We show that such fields are certain absolutely algebraic fields and
give some characterization of them. In particular, we observed that a field E
has only finitely many maximal subrings if and only if every saturated
descending chains of subring, beginning from E, is stationary; and have the same
length. Moreover, we prove that the last term of such chains is a unique subring
of E. We also determine when certain affine rings have only finitely many
maximal subrings. In particular, we prove that an affine integral domain R over
a field F has only finitely many maximal subrings if and only if F has only
finitely many maximal subrings and each generator of R over F is algebraic over
F, which is similar to the celebrated Zariski's Lemma.

**D.D. Anderson, **
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
(dan-anderson@uiowa.edu) and **Muhammad Zafrullah,** Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA
(mzafrullah@usa.net).

Cohen type theorems for a commutative ring, pp. 325-331

ABSTRACT.
Let R be a commutative ring with 1 not equal to 0. We show that if every prime
ideal containing a proper ideal is principal (resp., invertible, finitely
generated locally principal), then I is a finite product of principal
(resp., invertible, finitely generated locally principal) prime ideals. Let
R be an integral domain and * a finite character star operation on R.
We show that if every prime *-ideal containing a proper
*-ideal I is *-invertible, then I is a finite *-product of *-invertible prime
*-ideals and hence is *-invertible.

**Mamoru, Nunokawa,** University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan (mamoru_nuno@doctor.nifty.jp), and
**Janusz, Sokol,** Rzeszow University of Technology, Poland
(jsokol@prz.edu.pl).

New conditions for starlikeness and strongly starlikeness of order
alpha, pp. 333-344

ABSTRACT. Using a modified Nunokawa's lemma new conditions for starlikeness and
strongly starlikeness of order alpha are proved in this work.

**Xujie, Shi**, Nanjing University, Nanjing (yywn2006@sina.com),
**Liangwen Liao**, Nanjing University, Nanjing
(maliao@nju.edu.cn), and **Jie Zhang**, China University of Mining and Technology, Xuzhou
(zhangjie1981@cumt.edu.cn).

On a polynomial P such that P(Δ_{η}^{n}f) and P(f) sharing a small function, pp. 345-361.

ABSTRACT. In this paper, we investigate the difference counterpart of Brucke's
conjecture. We obtain that for entire function f of finite order who has a small
Borel exceptional entire function, if p is a polynomial such that P(Δ_{η}^{n}f) and P(f) sharing a small function CM, then we obtain the form of f, and we give the necessary and sufficient condition when f equal to Δ_{η}^{n}f.

**Hamid-Reza Fanaï** and **Atefeh Hasan-Zadeh, ** Department of Mathematical Sciences, Sharif University of
Technology, P.O.Box 11155-9415, Tehran, Iran (fanai@sharif.ac.ir),
(a-hasanzadeh@mehr.sharif.ac.ir).

A symplectic rigidity problem for 2-step nilmanifolds, pp. 363-374.

ABSTRACT.
We study a result of Gordon, Mao and Schueth about compact 2-step
nilmanifolds with symplectically conjugate flows, and consider this result as
a special case of a problem in Poisson and symplectic structures. In this
setting, via Poisson cohomology and other respective notions, we present a
proof of their result which extends not only symplectic concepts to Poisson
geometry,

but also 2-step nilmanifolds to manifolds with extensible
momentum maps.

Adeyemo, H. Praise, University of Ibadan, Ibadan, Oyo State, Nigeria
(ph.adeyemo@ui.edu.ng),
and
Sottile, Frank, Texas A&M University, College Station, Texas, USA,
(sottile@math.tamu.edu).

Equivariant cohomology theories and the pattern map, pp. 375-393.

ABSTRACT.
Billey and Braden defined a geometric pattern map on flag
manifolds which extends the generalized pattern map of Billey and
Postnikov on Weyl groups. The interaction of this torus
equivariant map with the Bruhat order and its action on line
bundles lead to formulas for its pullback on the equivariant
cohomology ring and on equivariant K-theory. These
formulas are in terms of the Borel presentation, the basis of
Schubert classes, and localization at torus fixed points.

Ghawadrah, Ghadeer,
Université Paris VI, Boîte 186, 4 Place
Jussieu, 75252 paris cedex 05, France (ghawadrah@math.jussieu.fr),
(g.ghawadrah@najah.edu).

The descriptive complexity of the family of Banach spaces with the bounded approximation property, pp.
395-401.

ABSTRACT. We show that the set of all separable Banach spaces that have the bounded approximation property (BAP) is a Borel subset of the set of all closed subspaces of C(Δ), where Δ is the Cantor set, equipped with the standard Effros-Borel structure.

**Søren Eilers**, Department of Mathematical Sciences,
University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
(eilers@math.ku.dk), **Gunnar Restorff**, Department of Science and Technology, University of the Faroe Islands,
Nóatún 3, FO-100 Tórshavn, Faroe Islands
(gunnarr@setur.fo), and **Efren Ruiz**, Department of Mathematics, University of Hawaii, Hilo,
200 W. Kawili St., Hawaii, 96720-4091, USA 9
(ruize@hawaii.edu).

Ideal related K-theory with coefficients,
403-458.

ABSTRACT.
In this paper, we define an invariant, which we believe should be the
substitute for total K-theory in the case when there is one distinguished
ideal. Moreover, some diagrams relating the new groups to the ordinary
K-groups with coefficients are constructed. These diagrams will in most cases
help to determine the new groups, and will in a companion paper be used to
prove a universal multi-coefficient theorem for the one distinguished ideal
case for a large class of algebras.

**Hui Li,** Research Center for Operator Algebras, Department of
Mathematics, East China Normal University, 500 Dongchuan Road, Shanghai 200241,
China (hli@math.ecnu.edu.cn).

Twisted Topological Graph Algebras, pp. 459-494.

ABSTRACT.
We define the notion of a twisted topological graph algebra associated to a
topological graph and a 1-cocycle on its edge set. We prove a stronger version
of a Vasselli's result. We expand Katsura's results to study twisted topological
graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem,
describe the gauge-invariant ideal structure. We find that a twisted topological
graph algebra is simple if and only if the corresponding untwisted one is
simple.

Pons, Matthew A., North Central College, Naperville, IL 60540, USA (mapons@noctrl.edu ).

The Adjoint of a linear fractional composition operator on the Dirichlet space, pp.
495-508.

ABSTRACT. Here we revisit the investigation of the adjoint of a composition
operator with linear fractional symbol acting on the Dirichlet space of the unit
disk. Earlier work shows that the adjoint can be expressed as another
composition operator plus a rank two operator. We work on the Dirichlet space
equipped with an equivalent norm, and hence a different inner product, to
determine how the adjoint representation differs in this setting.

**Bartosz K. Kwasniewski,** Department of Mathematics and Computer Science, The
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
(bartoszk@math.uwb.edu.pl).

Exel's crossed product and crossed products by
completely positive maps, pp. 509-567

ABSTRACT. We introduce crossed products
of a C*-algebra A by a completely positive map relative to an ideal in A. When
the map is multiplicative, they generalize various crossed products by
endomorphisms. When A is commutative they include C*-algebras associated to
Markov operators by Ionescu, Muhly, Vega, and to topological relations by
Brenken, but in general they are not modeled by topological quivers popularized
by Muhly and Tomforde.

We show that Exel's crossed product, generalized to
the case where A is not necessarily unital, is a relative crossed product of A
by the transfer operator L. We give natural conditions under which this crossed
product depends only on L. Moreover, the C*-algebra associated to an Exel system
by Exel and Royer always coincides with our unrelative crossed product by L.

As another non-trivial application of our construction we extend a result of
Brownlowe, Raeburn and Vittadello, by showing that the C*-algebra C*(E) of an
arbitrary infinite graph E can be realized as a crossed product of the diagonal
algebra D by a `Perron-Frobenious' operator L. The important difference to the
previous result is that in general there is no endomorphism α of D making (D, α,
L) an Exel system.

**Li, Yongjin,** Department of Mathematics, Sun Yat-sen University,
Guangzhou, 510275, P. R. China
(stslyj@mail.sysu.edu.cn), and **Bu,
Qingying,** Department of Mathematics,
University of Mississippi, Oxford, MS 38677, USA
(qbu@olemiss.edu).

New examples of non-reflexive Grothendieck spaces, pp. 569-575.

ABSTRACT. In this short paper, we characterize the Fremlin projective tensor
product X⊗Y that is a Grothendieck space, where X is a Banach sequence lattice
and Y is a Banach lattice. Then by using this characterization we provide new
examples of non-reflexive Grothendieck spaces.

**Benjamin Espinoza, **
Department of Mathematics, University of Pittsburgh
at Greensburg, 236 Frank A. Cassell Hall, 150 Finoli Drive, Greensburg, PA
15601, USA
(bee1@pitt.edu), **
Paul Gartside**, Department of Mathematics, University of Pittsburgh, 406
Thackeray Hall, Pittsburgh, PA 15260, USA
(gartside@math.pitt.edu), **
Merve Kovan-Bakan**, Department of Mathematics, University of Pittsburgh, 301
Thackeray Hall, Pittsburgh, PA 15260, USA
(mervekovan@gmail.com), and **Ana**
**Mamatelashvili**, Department of Mathematics, University of Pittsburgh, 301
Thackeray Hall, Pittsburgh, PA 15260, USA
(anm137@math.pitt.edu).

Strong arcwise connectedness, pp. 577-610.

ABSTRACT. A space is n-strong arc connected (n-sac) if for any n points in
the space there is an arc in the space visiting them in order. A space is
ω-strong arc connected (ω-sac) if it is n-sac for all n. We study these
properties in finite graphs, regular continua, and rational continua. There are
no 4-sac graphs, but there are 3-sac graphs and graphs which are 2-sac but not
3-sac. For every n there is an n-sac regular continuum, but no regular continuum
is ω-sac. There is an ω-sac rational continuum. For graphs we give a simple
characterization of those graphs which are 3-sac. It is shown, using ideas from
descriptive set theory, that there is no simple characterization of n-sac, or
ω-sac, rational continua.

**Li-Hong Xie,** School of Mathematics and Computational Science, Wuyi
University, Jiangmen 529020, P.R. China
(xielihong2011@aliyun.com) and
**Peng-Fei Yan,** School of Mathematics and Computational Science,
Wuyi University, Jiangmen 529020, P.R. China
(ypengfei@sina.com).

Expansions of set-valued mappings on stratifiable
spaces, pp. 611-624.

ABSTRACT. In this paper, we give some characterizations of stratifiable and
semistratifiable spaces by expansions of set-valued mappings.

Alejandro Illanes, Instituto de Matemáticas, Universidad Nacional Autónoma de
México, 04510, México, D.F.
(illanes@matem.unam.mx) and **Jorge M. Martínez-Montejano, **Facultad de
Ciencías, Universidad Nacional Autónoma de México, 04510, México, D.F.
(jorge@matematicas.unam.mx).

Z-sets in symmetric products, pp.
625-647.

ABSTRACT. A closed subset A of a continuum X (with metric D) is a Z-set in X,
provided that for each ε>0 there exists a map f_{ε}:X→X\A such that
d(x,f_{ε}(x))<ε for each x∈X. Given a continuum X and a positive integer
n, we consider the hyperspace F_{n}(X) of the nonempty closed subsets of
X with at most n points. We prove that F_{1}([0,1]) ) is a Z-set in F_{n}([0,1]))
if and only if n is even. Also, we consider the problem of determining the
continua X for which F_{1}(X) is a Z-set in F_{2}(X). We solve
this problem for finite graphs and compactifications of the ray with locally
connected remainder different of a simple closed curve. Lastly, we study several
significant examples.

**Liang-Xue Peng (Corresponding author),** Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn) and **
Hui Li,** Beijing University of Technology, Beijing 100124, China (lihui86@emails.bjut.edu.cn).

On monotone n-star
covering properties, pp. 649-667.

ABSTRACT.
In this note, we discuss
some properties of monotonically n-star Ρ spaces. We give an example to show
that there is a space X which has a monotonically star closed-and-discrete dense
subspace but X is not monotonically star closed-and-discrete. This gives a
partial answer to a question posed by S.G. Popvassilev and J.E. Porter
([Question 35(b), in: Topology Appl. 169 (2014) 87-98]). We point out that there
is a monotonically 2-star finite space which is monotonically star
closed-and-discrete but it is not star finite. We show that if X is a
monotonically star closed-and-discrete GO-space and g∈X, then (←,g) is
monotonically 2-star closed-and-discrete. Every open paracompact subspace of a
regular monotonically n-star closed-and-discrete space is monotonically n-star
closed-and-discrete, where n∈N. Every monotonically D-space is monotonically
star closed-and-discrete.

**Er-Guang Yang,** School of Mathematics & Physics, Anhui University of
Technology, Maanshan 243002, P.R. China
(egyang@126.com).

On MCP-spaces and mcb-spaces, 669-677.

ABSTRACT. In [Properties defined with semi-continuous functions and some related
spaces, Houston J. Math., 41 (2015), 1097-1106], the relationships between
properties defined with real-valued functions and some covering properties were
studied. In this paper, we shall continue with this study. Some other
properties, such as (UL)_{m}^{wl} and (UC)_{m }are
introduced and we show that spaces with these properties coincide with
MCP-spaces and mcb-spaces respectively. As an application, an insertion theorem
of MCP-spaces is obtained which corrects a mistake in [Monotone countable
paracompactness, Top. Appl., 101 (2000), 281-298].

**Er-Guang Yang,** School of Mathematics & Physics, Anhui University of
Technology, Maanshan 243002, P.R. China (egyang@126.com).

Characterizations of some spaces with maps to ordered topological vector spaces, pp.
679-689.

ABSTRACT. In this paper, we generalize real-valued functions in some earlier
results to maps with values into ordered topological vector spaces. Some
characterizations of countably paracompact spaces and cb-spaces in terms of maps
to ordered topological vector spaces are obtained which extend some known
results in the literature.