*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*

Injective coloring of generalized Petersen graphs, pp. 1-12.

ABSTRACT. An injective coloring of a graph is a vertex coloring where two vertices have distinct c olors if a path of length two exists between them. The injective chromatic number χ

Existence and uniqueness of entire solutions to a linear differential-difference equation of infinite order, pp. 13-26.

ABSTRACT. In this paper, we investigate existence and uniqueness problem on the entire solutions to a linear differential-difference equation of infinite order in a certain linear space provided some necessary and sufficient conditions of growth are imposed.

On a generalization of a theorem of Popov, pp. 27-38.

ABSTRACT. In this paper, we obtain sharp estimates for the number of lattice points under and near the dilation of a general parabola, the former generalizing an old result of Popov. We apply Vaaler's lemma and the Erdős-Turán inequality to reduce the two underlying counting problems to mean values of a certain quadratic exponential sums, whose treatment is subject to classical analytic techniques.

Real hypersurfaces with structure Jacobi operator of Codazzi type in the complex hyperbolic quadric, pp. 39-70.

ABSTRACT. First we introduce the notion of structure Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric Q

K-theory for real C*-algebras via unitary elements with symmetries, Part II -- Natural transformations and KO

ABSTRACT. We extend the unitary picture of K-theory for real C*-algebras developed in an earlier paper by Terry Loring and the author, finding formulas for the KO

A remark on the Bourgain-Brezis-Mironescu characterization of constant functions, pp. 113-115.

ABSTRACT. The purpose of this paper is to describe a simple proof for a result originally presented by H. Brezis, with roots in a paper by J. Bourgain, H. Brezis and P. Mironescu.

Minimal primal ideals in the inner corona algebra of a C

ABSTRACT. This paper is concerned with the inner corona algebra of an algebra obtained by tensoring C(X) with K(H), where X is an infinite compact Hausdorff space and H a separable infinite-dimensional Hilbert space. Using ultrapowers, we exhibit a faithful family of irreducible representations of this algebra and study consequences.

On some permanence properties of exact groupoids, pp. 151-187.

ABSTRACT. A locally compact groupoid is said to be exact if its associated reduced crossed product functor is exact. In this paper, we establish some permanence properties of exactness, including generalizations of some known results for exact groups. Our primary goal is to show that exactness descends to certain types of closed subgroupoids, which in turn gives conditions under which the isotropy groups of an exact groupoid are guaranteed to be exact. As an initial step toward these results, we establish the exactness of any transformation groupoid associated to an action of an exact groupoid on a locally compact Hausdorff space. We also obtain a partial converse to this result, which generalizes a theorem of Kirchberg and Wassermann. We end with some comments on the weak form of exactness known as inner exactness.

On the best constant of the one-dimensional Bliss inequality., pp. 189-200.

ABSTRACT. In this article, we have established the domain invariance property of the best constant of the Bliss Inequality, which is a generalization of one dimensional Hardy’s Inequality. We have also proved that the best constant is never achieved except for a particular case.

On operator-valued measures, pp. 201-226.

ABSTRACT. We consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikodym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesgue decomposition theorem and Johnson’s atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the “cleanness” relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.

Convergence without points, pp. 227-282.

ABSTRACT. We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice L with a monotonic map lim

Well-filtered spaces, compactness, and the lower topology, pp. 283-294.

ABSTRACT. It has long been known that a locally compact T