HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 42, No. 4, 2016

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

John Harding, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM USA 88003 (jharding@nmsu.edu) and Taewon Yang, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM USA 88003 (yangtjong@gmail.com).
Sections in orthomodular structures of decompositions, pp. 1079-1092.
ABSTRACT. There is a family of constructions to produce orthomodular structures from modular lattices, lattices that are M and M*-symmetric, relation algebras, the idempotents of a ring, the direct product decompositions of a set or group or topological space, and from the binary direct product decompositions of an object in a suitable type of category. We show that an interval [0,a] of such an orthomodular structure constructed from A is again an orthomodular structure constructed from some B built from A. When A is a modular lattice, this B is an interval of A, and when A is a set, group, topological space, or more generally an object in a suitable category, this B is a factor of A.

Goddard, Bart, University of Texas Austin, Austin, TX 78712-0257, and Luca, Florian, Wits University, Wits 2050, South Africa (florian.luca@wits.ac.za).
Goddard twins, pp. 1093-1099.
ABSTRACT. A number is abundant (deficient) if the ratio σ(n)/n is > 2 (< 2, respectively). A Goddard number is a positive integer n such that D(n) is abundant, where D(n) stands for the number of all deficient numbers smaller than or equal to n. In this paper, we prove that there are infinitely many positive integers n such that n and n+1 are both Goddard.

Hakami, Ali H., Department of Mathematics, Jazan University, P.O. Box 277, Jazan 45142, Saudi Arabia (aalhakami@jazanu.edu.sa).
Counting zeros of quadratic forms with integer coefficients over Zp, pp. 1101-1110.
ABSTRACT. Let Q(x) = (x1, x2, ..., xn) be a quadratic form in n variables with integer coefficients, p an odd prime and Zp the integers (mod p). We obtain bounds on the number of solutions over Zp to the congruence Q(x) ≡ 0 (mod p) in a general rectangular box. We use Fourier series and exponential sums to obtain our results.

Nguyễn Thảo Nguyên Bùi, Department of Pedagogy, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Vietnam, (thaonguyen0802@gmail.com) and Tiến Sơn Phạm, Department of Mathematics, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Vietnam (sonpt@dlu.edu.vn).
On the subanalytically topological types of function germs, pp. 1111-1126. 
ABSTRACT. We present relationships between topological/bi-Lipschitz equivalence types of subanalytic function germs. For subanalytic C1-function germs with isolated singularities, the definitions of subanalytically topological equivalence types are coincide. We show that the Łojasiewicz exponent and the multiplicity of analytic function germs are invariants of the bi-Lipschitz K-equivalence.

Kalafat, Mustafa, Tunceli Üniversitesi, Aktuluk mh, 62000, Tunceli, Türkiye (mkalafat@tunceli.edu.tr) and Koca, Caner, Vanderbilt University, Department of Mathematics, Nashville, TN 37240, USA (caner.koca@vanderbilt.edu).
Compact complex surfaces of locally conformally flat type, pp. 1127-1139.
ABSTRACT. We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Moreover, we give a list of possibilities of such surfaces.

Costa, Ezio, Universidade Federal da Bahia - UFBA, 40170-110-Salvador/BA, Brazil (ezio@ufba.br), Ribeiro Jr, Ernani, Universidade Federal do Ceará - UFC, 60455-760-Fortaleza/CE, Brazil (ernani@mat.ufc.br), and Santos, Alex, Universidade Federal do Ceará - UFC, 60455-760-Fortaleza/CE, Brazil (alexsls@hotmail.com).
Modified Yamabe problem on four-dimensional compact manifolds., pp. 1141-1156.
ABSTRACT. The aim of this paper is to investigate the modified Yamabe problem on four-dimensional manifolds whose the modifiers invariants depending on the eigenvalues of the Weyl curvature tensor and they are described in terms of maximum and minimum of the bi-orthogonal (sectional) curvature. We provide some geometrical and topological properties on four-dimensional manifolds in terms of these invariants.

Sorin V. Sabau, Tokai University, Sapporo, 005-8601 Japan (sorin@tokai-u.ac.jp) and Minoru Tanaka, Tokai University, Hiratsuka, Kanagawa, 259-1292 Japan (tanaka@tokai-u.jp).
The cut locus and distance function from a closed subset of a Finsler manifold pp. 1157-1197.
ABSTRACT. We characterize the differentiable points of the distance function from a closed subset N of an arbitrary dimensional Finsler manifold in terms of the number of N-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset N, namely that it is a local tree comprised of countably many rectifiable Jordan arcs except for the endpoints of the cut locus, and that an intrinsic metric can be introduced in the cut locus and the intrinsic and induced topologies coincide. We should point out that some of these are new results even for Riemannian manifolds.

Steven M. Seubert and J. Gordon Wade , Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403-0221(sseuber@bgsu.edu), (gwade@bgsu.edu).
Dense sets of common cyclic vectors for complete operators on a Frechet space, pp. 1199-1216.
ABSTRACT. Various collections of linear maps on a Frechet space having a common collection of root spaces which span the entire space are shown to have dense sets of common cyclic vectors.

Rion, Kevin, Bridgewater State University, Bridgewater, MA 02325 (krion@bridgew.edu).
Convergence Properties of the Aluthge Sequence of Weighted Shifts , pp. 1217-1226.
ABSTRACT. In this paper, we show for any weighted shift operator with a weight sequence that is eventually bounded away from zero, the Aluthge sequence of that shift can only have quasinormal subsequential limits and that the sequence either converges in the strong operator topology or diverges to an interval of quasinormal shift operators.

Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, P7B 5E1, Canada (andrew.j.dean@lakeheadu.ca).
Classification of actions of compact groups on real approximately finite dimensional C*-algebras, pp. 1227-1243.
ABSTRACT. A K-theoretic classification is given of actions of compact groups on real C*-algebras arising as inductive limits of actions on finite dimensional real C*-algebras. The invariant consists of the K0 groups of the crossed products of the real algebra, its complexification, and its tensor product with the quaternions, by the action, and the maps between them induced by the natural inclusions of the real algebra into the complexification and the complexification to the tensor product with the quaternions. Here, the K0 groups are viewed as ordered groups with distinguished elements, and the K0 of the complexification is given the structure of a module over the K0 of the group C*-algebra.

Eleftherakis, G.K., University of Patras, Faculty of Sciences, Department of Mathematics, 26500, PATRAS, GREECE (gelefth@math.upatras.gr).
Stable isomorphism and strong Morita equivalence of operator algebras, pp. 1245-1266.
ABSTRACT. We introduce a Morita type equivalence: two operator algebras A and B are called strongly Delta-equivalent if they have completely isometric representations f and g respectively and there exists a ternary ring of operators M such that f (A) (resp. g(B) ) is equal to the norm closure of the linear span of the set M*g(B)M, (resp. Mf(A)M*). We study the properties of this equivalence. We prove that if two operator algebras A and B possessing countable approximate identities, are strongly Delta-equivalent, then their tensor products with the set of compact operators acting on an infinite dimensional Hilbert space are isomorphic. Conversely, if the above tensor products are isomorphic then A and B are strongly Delta-equivalent.

LaLonde, Scott M., The University of Texas at Tyler, 3900 University Boulevard, Tyler, TX 75799 (slalonde@uttyler.edu).
Equivalence and exact groupoids, pp. 1267-1290.
ABSTRACT.Given two locally compact Hausdorff groupoids G and H and a (G,H)-equivalence Z, one can construct the associated linking groupoid L. This is reminiscent of the linking algebra for Morita equivalent C*-algebras. Indeed, Sims and Williams reestablished Renault's equivalence theorem by realizing C*(L) as the linking algebra for C*(G) and C*(H). Since the proof that Morita equivalence preserves exactness for C*-algebras depends on the linking algebra, the linking groupoid should serve the same purpose for groupoid exactness and equivalence. We exhibit such a proof here.
 
Vernik, Alexander, Ben-Gurion University of the Negev, Be'er Sheva 8410501, Israel (vernika@math.bgu.ac.il).
Dilations of CP-maps commuting according to a graph, pp.1291-1329.
ABSTRACT. We study dilations of finite tuples of normal, completely positive and completely contractive maps (which we call CP-maps) acting on a von Neumann algebra, and commuting according to a graph G. We show that if G is acyclic, then a tuple commuting according to it has a simultaneous *-endomorphic dilation, which also commutes according to G. Conversely, if G has a cycle, we exhibit an example of a tuple of CP-maps commuting according to G, which does not have an *-endomorphic dilation to K, that also commutes according to G. To obtain these results we use dilation theory of representations of subproduct systems, as introduced and studied by Shalit and Solel. In the course of our investigations we also prove some results about those kinds of subproduct systems which arise from CP-maps commuting according to to a graph.

Costel Peligrad, Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA (peligrc@ucmail.uc.edu).
Invariant subspaces of algebras of analytic elements, pp. 1331-1344.
ABSTRACT. We consider a von Neumann algebra with separable predual together with a periodic group of automorphisms. We define a certain standard representation of the algebra on a Hilbert space and a subspace of that Hilbert space that is the analog of the classical Hardy space. We prove that, in certain conditions, the subalgebra consisting of the analytic elements of the algebra is reflexive in the sense that it is completely determined by its invariant subspaces. These results generalize several reflexivity results in the literature.

Dorantes-Aldama, Alejandro and Tamariz-Mascarúa, Ángel. Departamento de Matemáticas, Facultad de Ciencias, Circuito exterior s/n, Ciudad Universitaria, CP 04510, México D. F., Mexico. (alejandro_dorantes@hotmail.com), (atamariz@unam.com).
Some results on weakly pseudocompact spaces, pp. 1345-1366.
ABSTRACT. A space X is weakly pseudocompact if it is Gδ-dense in at least one of its compactifications. A compact space K in which X is densely embedded is called a simple compactification of X if K is the quotient space obtained by identifying a compact subset of βX\ X to one point. A space X is simple pseudocompact if it is a Gδ-dense subspace in one of its simple compactifications. In this article we improve several results about weakly pseudocompact spaces obtained by F. W. Eckertson and by S. García-Ferreira and A. García-Máynez. The main theorem characterizes weakly pseudocompact spaces as those spaces whose ring C*(X) contains a regular subring Q such that if f∈ Q and Z(f)=∅, then 1/f ∈ Q. Furthermore, we prove: (1) if X is locally pseudocompact, then X is weakly pseudocompact if and only if X is either compact or is not Lindelöf; (2) the simple pseudocompact spaces is the class constituted by compact and locally pseudocompact non-Lindelöf spaces; (3) the property of being a simple pseudocompact space is preserved by the images of perfect functions and the inverse images of open perfect functions; (4) if the product of a space X with a compact space is weakly pseudocompact then X is weakly pseudocompact.

Averbeck, N., Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA, and Raines, B. E., Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA (nathan_averbeck@baylor.edu), (brian_raines@baylor.edu).
Chaotic pairs for shift maps, pp. 1367-1372.
ABSTRACT.  In this short, simple paper we answer a question of Fu and You by considering the properties of chaotic pairs of points in the sense of Li and Yorke for shift maps on symbolic dynamical systems. We show that a pair of points (x, y), in a symbolic dynamical system is chaotic if, and only if, it has a thick set of agreements and infinitely many disagreements. We show that the Banach density of the pairs set of agreements does not indicate whether the pair is chaotic or not, unless that density is exactly one.

Manisha Aggarwal, Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016, India (manishaaggarwal.iitd@gmail.com) and S. Kundu, Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016, India (skundu@maths.iitd.ac.in).
More about the cofinally complete spaces and the Atsuji spaces, pp. 1373-1395.
ABSTRACT. Metric spaces satisfying properties stronger than completeness and weaker than compactness have been the subject of study for a number of articles over the years. Two such significant families of metric spaces are those of cofinally complete and Atsuji spaces. In the literature, one can find various equivalent characterizations of such spaces. The major goal of this paper is to give seven new characterizations of cofinally complete metric spaces and three for Atsuji spaces. Since all such spaces are complete, we also give various new equivalent conditions for the metric spaces to have an Atsuji completion or a cofinal completion.

Liang-Xue Peng (Corresponding author), Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn) and Ming-Yue Guo,  Beijing University of Technology, Beijing 100124, China (guomingyue@emails.bjut.edu.cn).
On spaces of step functions over GO-spaces and Menger property, pp. 1397-1416.
ABSTRACT. Given a GO-space (generalized ordered topological space) L, the Dedekind completion of L is denoted by cL. An element x∈cL is in T(L) if and only if x∈cL\L, or x=∞, or x∈L and x has the immediate successor in L. Points of T(L) that are in L are declared isolated. The other points inherit base neighborhoods from the Dedekind completion cL. We show that if L is a GO-space then T(L)n is covered by finitely many closed homeomorphic copies of a closed subspace of Cp(L, n+1). We also show that if L is a GO-space and T(L)n is Lindelöf (Menger) for each n then Sp(L, n) is Lindelöf (Menger) for each n, where Sp(L, n) is the subspace of Cp(L, n), which consists of all step functions with finitely many steps and constant functions. We show that if L is a countably compact GO-space then Cp(L, n) is Menger for each n if and only if T(L) is Lindelöf. If L is a first countable GO-space such that L' is countably compact and Y=ClL(L\L')∩L' is scattered with rank(Y)<ω1, then Cp(L, m) is a Menger space if and only if Cp(L, m) is a Lindelöf space, where m∈N. We finally show that if L is a GO-space then Sp(L, n) is dense in Cp(L, n) for each n∈N.