*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*

**László Szalay** J. Selye University, 94501 Komárno, Bratislavská cesta 3322, Slovakia (szalayl@ujs.sk)

Computational algorithm for solving the diophantine equations
2^{n} ± *α* ⋅ 2^{m} + *α*^{2} = *x*^{2}, pp. 295-306.

ABSTRACT. In this paper, we construct an algorithm for solving the diophantine equations 2^{n} ± *α* ⋅ 2^{m} + *α*^{2} = *x*^{2}, where *α* is a given odd prime such that 2 is a non-quadratic residue modulo *α*. Applying the implementation of the procedure in Maple, apart from the plus case with the condition *n* < *m* we solve completely the problem for *α* < 3 ⋅ 10^{6}. The theoretical background relies on the treatment worked out to solve the equation 2^{n} + 2^{m} + 1 = *x*^{2}.

**Huaifu Liu** College of Applied Sciences, Beijing University of Technology, Beijing 100124, China (liuhf@bjut.edu.cn)
and **Xiaohuan Mo** Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China (moxh@pku.edu.cn)

The explicit construction on Finsler warped product metrics of scalar flag curvature, pp. 307-321.

ABSTRACT. By refining Chen-Shen-Zhao and Liu-Mo-Zhang equations characterizing Finsler warped product metrics of scalar flag curvature, we construct infinitely many non-spherically symmetric warped product Finsler metrics of scalar flag curvature.

**Ke Yan** School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
(1120160015@mail.nankai.edu.cn),
**Hui Wang** College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, P.R. China
(wanghui0801@njupt.edu.cn), and **Shaoqiang Deng**
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
(dengsq@nankai.edu.cn)

Killing Frames on Cohomogeneity One Finsler Spaces, pp. 323-341.

ABSTRACT. In this article, we study Killing frames on cohomogeneity one Finsler spaces. We first prove the existence of normal geodesics. Then we give a coordinate-free formula for the geodesic spray and S-curvature. Finally, we apply the formula to calculate the S-curvature of cohomogeneity one (*α*, *β*)-metrics, and give a condition for a Kropina space or a Randers space to have almost isotropic S-curvature.

**Yong Ji** School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, Jiangsu, P.R.China (imjiyong@126.com)
, **Yunping Wang** School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, Jiangsu, P.R.China
(yunpingwangj@126.com), and **Cao Zhao** School of Mathematical and Physics, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, P.R. China (izhaocao@126.com)

Mean dimension for free semigroup actions, pp. 343-356.

ABSTRACT. We consider finitely generated free semigroup actions on a compact metric space and generalize mean dimension, Lindenstrauss’ metric mean dimension for free semigroup actions. We give a Bufetov formula for metric mean dimension of free semigroup action. We also show the relationship between mean topological dimension and metric mean dimension.

**Elie Alhajjar** Army Cyber Institute, United States Military Academy, 252 Thayer Hall, West Point, NY 10996, U.S.A.
(elie.alhajjar@westpoint.edu) and **Travis B. Russell** Army Cyber Institute, United States Military Academy, 216 Thayer Hall, West Point, NY 10996, U.S.A.
(travis.russell@westpoint.edu)

Maximally entangled correlation sets, pp. 357-376.

ABSTRACT. We study the set of quantum correlations generated by actions on maximally entangled states. We show that such correlations are dense in their own convex hull. As a consequence, we show that these correlations are dense in the set of synchronous quantum correlations. We introduce the concept of corners of correlation sets and show that every local or non-signalling correlation can be realized as the corner of a synchronous local or non-signalling correlation. We provide partial results for other correlation sets.

**Menassie Ephrem** Department of Mathematics and Statistics, Coastal Carolina University, Conway, SC 29528–6054
(menassie@coastal.edu)

Primitive Ideals of Labelled Graph *C*^{*}-algebras, pp. 377-387.

ABSTRACT. Given a directed graph and a labeling, one forms the labelled graph *C*^{*}-algebra by taking a weakly left–resolving labelled space and considering a universal generating family of partial isometries and projections. In this paper we provide characterization for primitive ideals of labelled graph *C*^{*}-algebras.

**Mrinal Kanti Roychowdhury** School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive,
Edinburg, TX 78539-2999, USA. (mrinal.roychowdhury@utrgv.edu)

The quantization of the standard triadic Cantor distribution, pp. 389-407.

ABSTRACT. The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. For a given *k* ≥ 2, let {*S*_{j} : 1 ≤ *j* ≤ *k*} be a set of *k* contractive similarity mappings such that $S_j(x)=\frac 1 {2k-1} x +\frac{2 (j-1)} {2k-1}$ for all *x* ∈ ℝ, and let $P= \frac 1 k \sum_{j=1}^kP\circ S_j^{-1}$. Then, *P* is a unique Borel probability measure on ℝ such that *P* has support the Cantor set generated by the similarity mappings *S*_{j} for 1 ≤ *j* ≤ *k*. In this paper, for the probability measure *P*, when *k* = 3, we investigate the optimal sets of *n*-means and the *n*th quantization errors for all *n* ≥ 2. We further show that the quantization coefficient does not exist though the quantization dimension exists.

**Costel Peligrad** Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA
(costel.peligrad@uc.edu)

Invariant subspaces of generalized Hardy algebras associated with compact abelian group actions on W*-algebras, pp. 409-433.

ABSTRACT. We consider an action of a compact abelian group whose dual is any subgroup of the additive group of real numbers (so, an archimedean linearly ordered group) or a direct product (or sum) of such groups on a W*-algebra, *M*. We define the generalized Hardy subspace of the Hilbert space of a standard representation the algebra, and the Hardy subalgebra of analytic elements of *M* with respect to the action. We prove that in many relevant situations, the Hardy algebra is hereditarily reflexive, that is, every unital subalgebra is completely determined by its lattice of invariant subspaces. Our results contain and improve on several classical and more recent studies in the field. In particular if every non zero spectral subspace, contains a unitary operator, the condition is satisfied and therefore the Hardy algebra is hereditarily reflexive. This is the case if the action is the dual action on a crossed product, or an ergodic action, or, if, in some situations, the fixed point algebra is a factor.

**R. Radha** Department of Mathematics, Indian Institute of Technology Madras, Chennai–600 036, India,
(radharam@iitm.ac.in)
**Saswata Adhikari** Department of Mathematics, Indian Institute of Technology Madras, Chennai–600 036, India,
(saswata.adhikari@gmail.com)

Shift-invariant spaces with countably many mutually orthogonal generators on the Heisenberg group, pp. 435-463.

ABSTRACT. Let *E*(𝒜) denote the shift-invariant space associated with a countable family 𝒜 of functions in *L*^{2}(ℍ^{n}) with mutually orthogonal generators, where ℍ^{n} denotes the Heisenberg group. The characterizations for the collection *E*(𝒜) to be orthonormal, Bessel sequence, Parseval frame and so on are obtained in terms of the group Fourier transform of the Heisenberg group. These results are derived using such type of results which were proved for twisted shift-invariant spaces and characterized in terms of Weyl transform.

**L. C. Hoehn** Nipissing University, Department of Computer Science & Mathematics, 100 College Drive, Box 5002, North Bay, Ontario, Canada, P1B 8L7 (loganh@nipissingu.ca)
**L. G. Oversteegen** University of Alabama at Birmingham, Department of Mathematics, Birmingham, AL 35294, USA
(overstee@uab.edu), **E. D. Tymchatyn** University of Saskatchewan, Department of Mathematics and Statistics, 106 Wiggins road, Saskatoon, Canada, S7N 5E6 (tymchat@math.usask.ca)

A canonical parameterization of paths in ℝ^{n}, pp. 465-489.

ABSTRACT. For sufficiently tame paths in ℝ^{n}, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an alternative notion of path length, which we call len. Like Euclidean path length, len is invariant under isometries of ℝ^{n}, is monotone with respect to sub-paths, and for any two points in ℝ^{n} the straight line segment between them has minimal len length. Unlike Euclidean path length, the len length of *any* path is defined (i.e., finite) and len is continuous relative to the uniform distance between paths. We use this notion to obtain characterizations of those families of paths which can be reparameterized to be equicontinuous or compact. Finally, we use this parametrization to obtain a canonical homeomorphism between certain families of arcs.

**Chong Shen** School of Mathematics and Statistics, Beijing Institute of Technology, Fangshan District， Beijing 100081, China
(shenchong0520@163.com), **Hadrian Andradi** National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616,
and Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Indonesia 55281 (hadrian.andradi@gmail.com
(Corresponding author)), **Dongsheng Zhao** National Institute of Education, Nanyang Technological University, 1 Nanyang Walk,
Singapore 637616 (dongsheng.zhao@nie.edu.sg),
and **Fugui Shi ** School of Mathematics and Statistics, Beijing Institute of Technology, Fangshan District， Beijing 100081,
China (fuguishi@bit.edu.cn)

*S**I*_{2}-topology
on *T*_{0} spaces, pp. 491-505.

ABSTRACT. This paper provides a new approach to defining a topology by using irreducible sets from a given *T*_{0} space. This derived topology, called *S**I*_{2}-topology, leads to a weak sobriety and continuity, called *δ*-sobriety and *S**I*_{2}-continuity, respectively. It is proved that the *δ*-sober *C*-spaces are exactly the *s*_{2}-topological spaces on *s*_{2}-continuous posets. Moreover, it turns out that the *S**I*_{2}-topology on a *T*_{0} space can be described completely in terms of convergence, and this convergence structure is topological whenever the given space is *S**I*_{2}-continuous.

**Rongxin Shen** Department of Mathematics, Taizhou University, Taizhou 225300, P. R. China
(srx20212021@163.com) and **Ziqin Feng** Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
(zzf0006@auburn.edu)

On *ω*^{ω}-bases and *ω*^{ω}-weak bases, pp. 507-518.

ABSTRACT. We discuss the spaces with an *ω*^{ω}-base and spaces with an *ω*^{ω}-weak base in this paper. The main results are: (1) Every quasi-first-countable (weakly quasi-first-countable) space has an *ω*^{ω}-base (*ω*^{ω}-weak base); (2) Every space with an *ω*^{ω}-weak base has countable cs^{*}-character; (3) For a topological group *G*, the following are equivalent: (i) *G* is an ℳ_{ω}-space; (ii) *G* is a weakly quasi-first-countable space; (iii) *G* is a sequential space with an *ω*^{ω}-base; (iv) *G* is a sequential space with an *ω*^{ω}-weak base; (v) *G* is a sequential space with countable cs^{*}-character. Several examples relating to *ω*^{ω}-bases and *ω*^{ω}-weak bases are given.

**Pratulananda Das**
Department of Mathematics, Jadavpur University, Kolkata-32, West Bengal, India (pratulananda@yahoo.co.in),
**Upasana Samanta** Department of Mathematics, Jadavpur University, Kolkata-32, West Bengal, India
(samantaupasana@gmail.com), and **Debraj Chandra**
Department of Mathematics, University of Gour Banga, Malda-732103, West Bengal, India (debrajchandra1986@gmail.com)

A more balanced approach to ideal variation of *γ*-covers, pp. 519-535.

ABSTRACT. In this note, we introduce the notion of *G*-ℐ-*γ* cover as a generalized and more balanced version of ℐ-*γ*-cover and study some of its basic selection properties as also certain Ramsey like properties and splittability properties. Certain characterizations of *γ*-sets are obtained in terms of *G*-ℐ-*γ* covers. This new approach helps to establish some of the results proved for *γ*-covers, which in turn help us to produce a version of Scheepers’ diagram in terms of *G*-ℐ-*γ* covers.

**Włodzimierz J. Charatonik** Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W 12th St, Rolla MO 65409-0020 (wjcharat@mst.edu
), **Robert P. Roe** Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W 12th St, Rolla MO 65409-0020 (rroe@mst.edu), and **Ismail Uğur Tiryaki**
Abant Izzet Baysal University, Faculty of Science and Letters, Department of Mathematics, 14280, Bolu, Turkey (ismail@ibu.edu.tr)

Open diameters on graphs, pp. 537-559.

ASTRACT. We prove that every connected finite graph admits a metric for which the diameter mapping is open.