## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 46, No. 1, 2020

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)

Contents

Zepeng Li, School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, P.R. China (lizp@lzu.edu.cn), Zehui Shao, Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, P.R. China (zshao@gzhu.edu.cn), and Enqiang Zhu, Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, P.R. China (zhuenqiang@gzhu.edu.cn).
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 χi(G) of a graph G is the smallest number k such that G admits an injective coloring with k colors. Hahn et al.(2002) proved that Δ <= χi(G) <= Δ2-Δ+1 for any graph G, where Δ is the maximum degree of G. For a constant c>=0, determining the injective chromatic number of which graphs is at most Δ+c is an interesting problem. In this paper, we investigate the injective colorings of generalized Petersen graphs P(n,k). We prove that χi(P(n,k))<= 5 for any generalized Petersen graph P(n,k) and χi(P(n,k))=3 if n is congruent to 0 mod 3 and k is not congruent to 0 mod 3. Furthermore, we determine the precise injective chromatic numbers of P(n,1) and P(n,2).

Xianjing Dong, Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, 210093, P.R.China (xjdong05@126.com).
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.

Jing-Jing Huang, Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV 89557 (jingjingh@unr.edu), and Huixi Li, Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV 89557 (huixil@urn.edu)
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.

Young Jin Suh, Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea (yjsuh@knu.ac.kr) and Doo Hyun Hwang, Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu 41566, Republic of Korea, (engus0322@knu.ac.kr).
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 Qm*=Q2k*. Next we give a complete classification of real hypersurfaces in Qm*=Q2k* whose structure Jacobi operator is of Codazzi type.

Jeffrey L. Boersema, Seattle University, Department of Mathematics, Seattle, WA 98133, USA (boersema@seattleu.edu).
K-theory for real C*-algebras via unitary elements with symmetries, Part II -- Natural transformations and KO*(R)-module operations, pp. 71-111.
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* and KU* natural transformations in terms of this picture. In addition we study several examples, including suspension algebras and dimension drop algebras, finding unitary elements to represent all non-trivial elements of KOi(A).

Alireza Ranjbar-Motlagh, Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11365-9415, Tehran, Iran (ranjbarm@sharif.edu).
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.

Robert J. Archbold, Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, United Kingdom (r.archbold@abdn.ac.uk) and Douglas W.B. Somerset, Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, United Kingdom (douglassomerset@yahoo.com).
Minimal primal ideals in the inner corona algebra of a C0(X)-algebra, pp. 117-150.
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.

LaLonde, Scott M., The University of Texas at Tyler, 3900 University Boulevard, Tyler, TX 75799 (slalonde@uttyler.edu)
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.

Mrinmay Biswas, Department of Mathematics and Statistics, IISER Kolkata, Mohanpur-741246, India (mb13rs044@iiserkol.ac.in).
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.

Darian McLaren, Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada, Sarah Plosker, Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada, and Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada (ploskers@brandonu.ca), and Christopher Ramsey, Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada, and Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada.
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.

Jean Goubault-Larrecq, LSV, CNRS & ENS de Cachan, 61 avenue du Président Wilson, 94235 CACHAN Cedex, France (goubault@lsv.fr) and Frédéric Mynard, NJCU, department of Mathematics, 2039 Kennedy Blvd, Jersey City, NJ 07305, USA (fmynard@njcu.edu) .
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 limL from the lattice of filters on L to L, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spaces and the opposite of so-called pretopological convergence coframes, between adherence spaces and the opposite of so-called adherence coframes, between topological spaces and the opposite of so-called topological coframes. All of our pointfree categories are complete and cocomplete, and topological over the category of coframes. Our final pointfree category, that of topological coframes, shares with the category of frames the property of being in a dual adjunction with the category of topological spaces. We show that the latter arises as a retract of the former, and that this retraction restricts to a reflection between frames and so-called strong topological coframes.

Lawson, Jimmie D., Louisiana State University, Baton Rouge, LA 70810 USA (lawson@math.lsu.edu), Wu, Guohua, Nanyang Technological University, Singapore (guohua.wu@ntu.edu.sg), and Xi, Xiaoyong, Nanyang Technological University, Singapore (xiaoyong.xi@ntu.edu.sg).
Well-filtered spaces, compactness, and the lower topology, pp. 283-294.
ABSTRACT. It has long been known that a locally compact T0-space is sober. The question has been asked whether locally compact could be weakened to core compact, and we give a positive answer to this question in this paper. We next turn to a detailed study of the lower topology of a partially ordered set, where the partial order may be the order of specialization of a T0-space. After deriving some of its elementary properties related to compactness, we consider T0-spaces with the distinguished property that every set closed in the lower topology is compact in the lower topology. A key result is that a dcpo equipped with the Scott-topology satisfying this distinguished property is well-filtered, and the result generalizes to a T0-space with its topology determined by its directed subsets. We also derive a converse of this result.