*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*

A sufficient condition for the absence of irredundant bases, pp. 399-411.

ABSTRACT. A basis of identities for an algebra is

Keef, Patrick W.,

Endomorphism rings of mixed modules and a theorem of W. May, pp. 413-435.

ABSTRACT. A theorem of W. May (1990) on endomorphism rings of mixed modules over a complete discrete valuation domain is shown to be incorrect. A precise description is given of the class of modules for which the theorem remains valid, as well as a complete description of all possible counter-examples.

Degeneracy theorems for three meromorphic mappings sharing few hyperplanes, pp. 437-454.

ABSTRACT. In the all mentioned results of degeneracy problem for three meromorphic, we would like to emphasize that the number of hyperplanes is always assumed to be at least 2n+2. By improving the lemma on Cartan's auxiliary function and introducing some new techniques, in this paper, we prove some degeneracy theorems for three meromorphic mappings sharing less than 2n+2 hyperplanes in general position regardless of multiplicities. These results are improvements of some recent results.

Jianfei Wang,

Compositions and multiplications on holomorphic Campanato spaces, pp. 455-480.

ABSTRACT. The purpose of this paper is twofold. The first is to characterize holomorphic Campanato spaces defined in the unit ball of C

Zaili Yan,

Existence of homogeneous geodesics on homogeneous Randers space, pp. 481-493.

ABSTRACT. A geodesic of a Finsler space (M, F) is called homogeneous if it is the orbit of a one-parameter transformation group of isometries of (M, F). In this paper we show that every homogeneous Randers space admits at least a homogeneous geodesic through any point.

Vincze, Csaba,

Analytic properites and the asymptotic behavior of the area function of a Funk metric, pp. 495-520.

ABSTRACT. The classical version of the problem we are going to discuss here is to minimize the so-called volume product. The Blaschke-Santaló point as the solution of the optimization problem is well-known. As a kind of generalization of the classical problem we use the Funk metric induced by a compact convex body K. The Funk metric is constructed by moving the origin in the interior of the body. This gives a smoothly varying family F of Minkowski functionals to measure the length of vectors depending on the position too. At any given interior point, the volume of the translated body can be measured by the volume form coming from the Hessian of the energy function E=F^2/2. Since the volume and the area are proportional because of the usual homogenity properties it is enough to investigate one of them. The area as the function of the base point varying in the interior of K is called the area function of the Funk metric. We prove that it is a a strictly convex, locally analytic function and its value can be arbitrarily large near to the boundary of K. Therefore the (global) minima is attained at a uniquely determined interior point. One of the possible applications is to generalize the so-called Brickell theorem for convex bodies with the uniquely determined minimizer of the area function at the origin. It is based on Schneider's trick to prove the classical Brickell theorem. Since the Funk metrics are special cases of the Finslerian metric functions we present some more general investigations too. We prove that the minimizers of the area functions of Funk metrics induced by the indicatrices of a Finsler manifold constitute a smooth vector field, i.e. each indicatrix body can be translated in such a way that the minima is attained at the origin. The associated metric structure is called a Finsler manifold with balanced indicatrices. Some explicite examples are also presented.

**Gauss, Falko,** Universität Mannheim, 68131 Mannheim,
Germany (gauss@math.uni-mannheim.de).

Planar
tropical gravitational descendants with one tangency condition, pp. 521-543

ABSTRACT.
We prove a formula which allows us to recursively compute planar tropical
gravitational descendants which involve psi-classes of arbitrary power at marked
ends fixed by points and additionally a psi-class of power one at exactly one
marked end fixed by a line. Up to now almost exclusively tropical gravitational
descendants with psi-classes at marked ends restricted by points had been
considered.

Some geometrical properties of minimal graph on space forms with nonpositive curvature, pp. 545-570

ABSTRACT. For the minimal graph defined on two dimensional space forms with nonpositive curvature, we derive some geometrical properties, including the regularity and the strict convexity of the level sets, the curvature description of its steepest descents.

**Gardella, Eusebio,** Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62,
48149 Muenster, Germany (gardella@uni-muenster.de),
(eusegardella@gmail.com).

Circle actions on UHF-absorbing C^{*}-algebras,
pp. 571-601.

ABSTRACT. We study circle actions with the Rokhlin property, in relation to their restrictions to finite subgroups. We construct examples showing the following: the restriction of a circle action with the Rokhlin property (even on a real rank zero
C^{*}-algebra), need not have the Rokhlin property; and even if every restriction of a given circle action has the Rokhlin property, the circle action itself need not have it. As a positive result, we show that the restriction of a circle action with the Rokhlin property to the subgroup Z_{n }has the Rokhlin property if the underlying algebra absorbs the UHF-algebra of type n^{∞}. The condition on the algebra is also necessary in most cases of interest.
Despite the fact that there are no circle actions with the Rokhlin property on UHF-algebras, we construct many such actions on certain UHF-absorbing simple AT-algebras. Additionally, we show that circle actions with the Rokhlin property on O_{2}-absorbing C^{*}-algebras are generic, in a suitable sense.

Piecewise conjugacy for multivariable dynamics over the Jacobson spectrum of a C*-algebra, pp. 603-606.

ABSTRACT. We show that if (A,α ) and (B, β) are automorphic multivariable C*-dynamical systems with isometrically isomorphic tensor algebras (or semi crossed products), then the systems are piecewise conjugate over their Jacobson spectrum. This answers a question of Kakariadis and the author.

Lin, Minghua,

Power majorization between the roots of two polynomials, pp. 607-612.

ABSTRACT. It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Klemeš is confirmed.

Nathanial Brown,

Rokhlin dimension for C*-correspondences, pp. 613-643.

ABSTRACT. We extend the notion of Rokhlin dimension from topological dynamical systems to C*-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz-Pimsner and (hence) Cuntz-Pimsner algebras. As a consequence we provide new examples of classifiable C*-algebras: if A is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective H with finite Rokhlin dimension, the associated Cuntz-Pimsner algebra O, H is classifiable in the sense of Elliott's Program.

**S. Walters,** Department of Mathematics
and Statistics, University of Northern British Columbia, Prince George, B.C. V2N
4Z9, Canada (Samuel.Walters@unbc.ca).

Semiflat orbifold projections,
pp. 645-663.

ABSTRACT. We study K-theoretic properties of certain
projections in orbifolds of the irrational rotation algebra A (generated by the
usual canonical unitaries U,V) under the noncommutative Fourier transform F, the
order 4 automorphism that takes V to U, and U to V-inverse, and its square, the
flip automorphism. We call a projection semiflat when it has the form h + F(h)
where h is a flip-invariant projection such that hF(h) = 0. We compute the
semiflat positive cone in the K0 group of the fixed point subalgebra of A under
F (associated to classes of semiflat projections) and show that it is determined
by classes of positive trace and the vanishing of two topological invariants.
The semiflat projections are 3-dimensional and come in three basic topological
genera: (2,0,0), (1,1,2), (0,0,2). We also show that the traces of semiflat
projections cover all the expected trace values (namely 2 times the usual trace
range on the K0 group of the rotation algebra).

The specification property on a set-valued map and its inverse limit, pp. 665-677.

ABSTRACT. In this paper we consider dynamical properties of set-valued mappings and their implications on the assiciated inverse limit space. Specifically, we define the specification property and topological entropy for set-valued functions and prove some elementary results of these properties. We end with a few results regarding invariant measures for set-valued functions and their associated inverse limits.

Javier Casas-de la Rosa,

Star–β spaces and related properties, pp. 679-693.

ABSTRACT. Given a topological property P, a topological space X is called star–P if for any given open cover of X, it is possible to find a subset Y of X with the property P such that the star of Y related to the cover coincides with X. In this paper, we study two special classes of star-P spaces determined, respectively, by certain kind of sequences and relatively countably compact subsets.

Banič, Iztok

Markov pairs, quasi Markov functions and inverse limits, pp. 695-707.

ABSTRACT. We show that two inverse limits of inverse sequences of closed intervals and quasi Markov bonding functions are homeomorphic, if the inverse sequences follow the same pattern. This significantly improves Holte's result about when two inverse limits of inverse sequences with Markov interval maps as bonding functions are homeomorphic. Our result improves Holte's result in several directions: (1) it generalizes finite Markov partitions to Markov pairs of sets that may even be uncountable, (2) it generalizes Markov interval maps from I onto I to quasi Markov functions from I to J (so the domains and the codomains of the bonding functions are not necessarily the same interval), (3) we no longer require the bonding functions to be surjective, and (4) we no longer require the bonding functions to be continuous.

**Iván Martínez Ruiz,** and **Alejandro Ramírez Páramo,** and
**Armando Romero Morales,** Facultad de Ciencias Físico Matemáticas, Benémerita Universidad Autónoma de Puebla, Av. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue. C.P. 72570, México
(armando@mixteco.utm.mx).

Some remarks on the κ–closure operator,
pp. 709-720.

ABSTRACT. In this paper we analyze properties related with the κ-closure operator and elementary submodels satisfying certain properties. In particular, we study the κ–closure of a space Y wich is obtained by intersecting a space X with a elementary submodel M; for instance, we prove that Y is a normal space when X is normal and M is closed under t(X)-sequences. This theorem generalizes a result obtained by de la Vega R. (A new bound on the cardinality of homogeneous compacta, Top. Appl. 2008). Once more, we provide a generalization of the well known Arhangel'skii’s inequality, which gives a upper bound of Hausdorff spaces.

On the rings of functions which are discontinuous on a finite set, pp. 721-739.

ABSTRACT. Our aim is to introduce a ring of some discontinuous real-valued functions defined on a topological space X. By C(X)