## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 44, No. 1, 2018

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)

Contents

May, Coy L., Towson University, Towson, MD., USA, (cmay@towson.edu) and Zimmerman, Jay, Towson University, Towson, MD., USA. (jzimmerman@towson.edu).
The symmetric genus of large odd order groups, pp. 1-19.
ABSTRACT. Let L be the set of integers n for which there is a solvable group G of order n generated by elements of prime order p. Then the set L has density 0 in the set of positive integers. Finite quotients of the Fuchsian triangle groups of type (3, 3, n), for n = 5, 7, 9 and 11, are generated by two elements of order 3. We call odd order groups of these four types LO-1 groups through LO-4 groups. The symmetric genus s(G) is the minimum genus of any Riemann surface on which G acts faithfully. If G acts on a Riemann surface X of genus g > 1, we say that G is a large odd order group since |G| > 8(g - 1). We obtain restrictions on the powers of the primes dividing the orders of such groups. In addition, we study the metabelian LO-3 groups, and classify the integers that are the orders of metabelian LO-3 groups.

Calugareanu, Grigore, Babes-Bolyai University, Cluj-Napoca, Romania (calu@math.ubbcluj.ro).
A new class of semiprime rings, pp. 21-30.
ABSTRACT. A ring R is called unit-semiprime if for any element a in R, a=0 whenever aua=0 for all units u in U(R). This turns out to be a proper class of semiprime rings which, among others, includes the reduced rings and the unit-regular rings, is closed to matrix extensions but is not Morita invariant. The usual Ring theoretic constructions are investigated and connections with some other known classes of rings are established.

Hesam Safa
, Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran (h.safa@ub.ac.ir), M. Farrokhi D. G., Institute for Advanced Studies in Basic Sciences (IASBS), and the Center for Research in Basic Sciences and Contemporary Technologies, IASBS, P.O.Box 45195-1159, Zanjan 66731-45137, Iran (m.farrokhi.d.g@gmail.com), (farrokhi@iasbs.ac.ir), and Mohammad Reza R. Moghaddam, Department of Mathematics, Khayyam University, International Campus and Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran (rezam@ferdowsi.um.ac.ir) .
Some properties of 2-auto-Engel groups, pp. 31-48.
ABSTRACT. For a given group G, with an element x in G and automorphism α in Aut(G), the nth autocommutator [x,nα] is defined recursively by [x,α]=x-1xα and [x,nα]=[[x,n- 1α],α] for all n>1. The group G is said to be n-auto-Engel if [x,nα]=[α,nx]=1, for all x∈G and all α∈Aut(G), where [α,x]=[x,α]-1. We study the structure of 2-auto-Engel groups and show that 2-auto-Engel groups indeed satisfy the equation α(x)α-1(x)=x2, for all x∈G and α∈Aut (G). Also, we give a precise description of all abelian 2-auto-Engel groups of finite 2-rank as well as 2-auto-Engel 2-groups, which are not purely non-abelian, and construct an infinite family of purely non-abelian 2-auto-Engel 2-groups.

Luck Darniére, Département de mathématiques, Facult\é des sciences, Université d'Angers, France (Luck.Darniere@univ-angers.fr) and Markus Junker, Mathematisches Institut, Abteilung für mathematische Logik, Albert-Ludwigs-Universität Freiburg, Deutschland (markus.junker@math.uni-freiburg.de)
Model completion of varieties of co-Heyting algebras, pp. 49-82.
ABSTRACT. It is known that exactly eight varieties of Heyting algebras have a model-completion. However no concrete axiomatization of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms satisfy a certain remarkable embedding theorem. For four of these six varieties (those which are locally finite) these two results provide a new proof of the existence of a model-completion with, in addition, an explicit and finite axiomatization.

Janusz Sokół, Faculty of Mathematics and Natural Sciences, University of Rzeszow, ul. Prof. Pigonia 1, 35-310 Rzeszow, Poland (jsokol@ur.edu.pl) and Derek K. Thomas, Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK (d.k.thomas@swansea.ac.uk).
Further results on a class of starlike functions related to the Bernoulli lemniscate, pp. 83-95.
ABSTRACT. The purpose of this paper is to provide further results for the class of starlike functions S L consisting of analytic functions f normalized by f(0) = f’(0) - 1 = 0 in the open unit disk |z|<1 satisfying the subordination condition zf’(z)/f(z) is subordinate to square root of 1+z. Various new results for the coe cients of f(z) are obtained, together with sharp inequalities for the Fekete-Szegö functional, the Hankel determinant, the coefficients of the inverse function, and the coefficients of log f(z)/z.

Mishev, Ilia, Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, CO 80309-0395, U.S.A. (ilia.mishev@colorado.edu).
On a class of transformations of sequences of complex numbers, pp. 97-119.
ABSTRACT. In this paper we consider a transformation La of sequences of complex numbers. We find the inverse transformation of La as well as the inverse of a related transformation. We explore a connection to the binomial transform and significantly generalize a previously known result. We also obtain new relations among many classical hypergeometric orthogonal polynomials as well as new formulas for sums involving terminating hypergeometric series.

Memarian Yashar,
Department of Mathematics, University of Notre-Dame, IN 46556, U.S.A. (y.memarian@gmail.com)
A note on the geometry of positively-curved Riemannian manifolds, pp. 121-145.
ABSTRACT. In this paper I present a comparison theorem for the waist of Riemannian manifolds with positive sectional curvature. The main theorem of this paper gives a partial positive answer to a conjecture formulated by M.Gromov in 2003. The content of this paper combines two aspects: classical volume comparison theorems of Riemannian geometry, and geometric measure theoretic ideas stemming from Almgren-Pitts Min-Max theory. I present the background which is needed in Riemannian Geometry as well as in geometric measure theory in different sections. One of my wishes was to give the reader a broad idea on the subject: which is estimation of metric invariants on concrete geometric spaces.

Deaconu, Valentin, University of Nevada, Reno, NV 89557-0084 (vdeaconu@unr.edu).
Group actions on graphs and C*-correspondences, pp. 147-168.
ABSTRACT. If G acts on a C*-correspondence H over the C*-algebra A (see Definition 3), then by the universal property G acts on the Cuntz-Pimsner algebra O(H) and we study the crossed product O(H)xG and the fixed point algebra. Using intertwiners, we define the Doplicher-Roberts algebra  of a representation  of a compact group G on H and prove that under certain conditions the fixed point algebra is isomorphic to the Doplicher-Roberts algebra. The action of G commutes with the gauge action on O(H), therefore G acts also on the core algebras. We give applications for the action of a group G on the C*-correspondence  associated to a topological graph E. If G is finite and E is discrete and locally finite, we prove that the crossed product C*(E)xG is isomorphic to the C*-algebra of a graph of C*-correspondences and stably isomorphic to a locally finite graph algebra. If C*(E) is simple and purely infinite and the action of G is outer, then the fixed point algebra and the crossed product are also simple and purely infinite with the same K-theory groups. We illustrate with several examples.

Linghui Kong, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China (konglinghui@mail.dlut.edu.cn) and Yufeng Lu, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China (lyfdlut@dlut.edu.cn).
Some algebraic properties of dual Toeplitz operators, pp. 169-185.
ABSTRACT. In this paper, we study some algebraic properties of dual Toeplitz operators, and prove that the dual Toeplitz operator commuting with an analytic dual Toeplitz operator must be also an analytic dual Toeplitz operator on the orthogonal complement of the Bergman space on the unit ball. Furthermore, we characterize when the sum of products of two dual Toeplitz operators is equal to a dual Toeplitz operator on the orthogonal complement of the Bergman space on the unit ball or the Hardy space on the sphere.

Turdebek N. Bekjan,, L. N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan} (bekjant@yahoo.com).
On dominated convergence in noncommutative integration, pp. 187-200.
ABSTRACT. Let M be a von Neumann algebra and let T: L0(M) → L0(M) be a linear bounded positive map. If (xn)n≥1 is a sequence in L0(M) converging to x almost uniformly in Lance's sense and (xn)n≥1 satisfies some kind of domination condition, then (T(xn))n≥1 converges to T(x) almost uniformly.

Kevin Beanland, Department of Mathematics, Washington and Lee University, Lexington, VA 24450 (beanlandk@wlu.edu and  Ryan M. Causey, Department of Mathematics, University of South Carolina, Columbia, SC 29208  (causeyrm@mailbox.sc.edu).
On a generalization of Bourgain's tree index, pp. 201-208.
ABSTRACT. For a Banach space X, a sequence of Banach spaces (Yn) and a Banach space Z with an unconditional basis, D. Alspach and B. Sari introduced a generalization of a Bourgain tree called a (⊕nYn)Z-tree in X. These authors also prove that any separable Banach space admitting a (⊕nYn)Z-tree with order ω1 admits a subspace isomorphic to (⊕nYn)Z. In this paper we give two new proofs of this result.

Boedihardjo, March, Texas A&M University, College Station, TX 77843-3368 (march@math.tamu.edu) and Dykema, Ken, Texas A&M University, College Station, TX 77843-3368 (kdykema@math.tamu.edu).
On algebra-valued R-diagonal elements, pp. 209-252.
ABSTRACT. For an element in an algebra-valued ∗-noncommutative probability space, equivalent conditions for algebra-valued R-diagonality (a notion introduced by Sniady and Speicher) are proved. Formal power series relations involving the moments and cumulants of such R-diagonal elements are proved. Decompositions of algebra-valued R-diagonal elements into products of the form unitary times self-adjoint are investigated; sufficient conditions, in terms of cumulants, for ∗-freeness of the unitary and the self-adjoint part are proved, and a tracial example is given where ∗-freeness fails. The particular case of algebra-valued circular elements is considered; as an application, the polar decompostion of the quasinilpotent DT-operator is described.

Benoît Collins,
Department of Mathematics, Kyoto University (collins@math.kyoto-u.ac.jp).
Haagerup's inequality and additivity violation of the Minimum Output Entropy, pp. 253-261.
ABSTRACT.We give a simple and conceptual proof of the fact that random unitary channels yield violation of the Minimum Output Entropy additivity. The proof relies on strong convergence of random unitary matrices and Haagerup's inequality.

Gehér, György Pál, MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1., Hungary; and MTA-DE "Lendület" Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary (gehergy@math.u-szeged.hu) or (gehergyuri@gmail.com).
Surjective Kuiper isometries, pp. 263-281.
ABSTRACT. Characterisations of surjective isometries with respect to the Kuiper distance on three classes of Borel probability measures on the real line (or equivalently, probability distribution functions) are presented here. These classes are the set of continuous, absolute continuous and general measures.

Thomas R. Hoffman and James P. Solazzo, Wall Building 124 F, Mathematics Department, Coastal Carolina University, Conway, SC 29528, (thoffman@coastal.edu), (jsolazzo@coastal.edu).
Complex two-graphs, pp. 283-300.
ABSTRACT. In A survey of two-graphs, J.J. Seidel lays out the connections between simple graphs, two-graphs, equiangular lines in Rk and strongly regular graphs. It is well known that there is a one-to-one correspondence between two-graphs and sets of equiangular lines in Rk. This article gives a generalization of two-graphs by allowing the entries of the Seidel matrix to be roots of unity beyond ± 1. Many of the results regarding \emph{real} two-graphs have a natural generalization in the complex setting including equiangular lines in Ck.

Sneh Lata, Department of Mathematics, Shiv Nadar University, School of Natural Sciences, Gautam Budh Nagar - 203207, Uttar Pradesh, India (sneh.lata@snu.edu.in) and Dinesh Singh, Department of Mathematics, University of Delhi, Delhi 110007, India (dineshsingh1@gmail.com)
A class of sub-Hardy Hilbert spaces associated with weighted shifts, pp. 301-308.
ABSTRACT. In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function $z$ is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries.

Kilbane, James, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom (jk511@cam.ac.uk) and Ostrovskii, M. I., Department of Mathematics and Computer Science, St John's University, NY 11439 (ostrovsm@stjohns.edu).
There is no finitely isometric Krivine's theorem, pp. 309-317.
ABSTRACT. We prove that for every real number p>1 except p=2 there exist a Banach space X isomorphic to lp and a finite subset U in lp, such that U is not isometric to a subset of X. This result shows that the finite isometric version of the Krivine theorem (which would be a strengthening of the Krivine theorem (1976)) does not hold.

Gokieli, Maria (m.gokieli@icm.edu.pl), Kenmochi, Nobuyuki (n.kenmochi@icm.edu.pl), and Niezgódka, Marek (m.niezgodka@icm.edu.pl), Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Tyniecka 15/17, 02-630 Warsaw, Poland.
A new compactness theorem for variational inequalities of parabolic type, pp. 319-350.
ABSTRACT. This paper deals with the weak solvability of fully nonlinear parabolic variational inequalities with time dependent convex constraints. As possible approaches to such problems, one has for instance the time-discretization method and the fixed point method of Schauder type with appropriate compactness theorems. In this paper, our attention is paid to the latter approach. However, no appropriate compactness theorem,  that would enable a direct application of the fixed point method to variational inequalities of parabolic type, has been established up to now. We thus start with setting up a new compactness theorem,  and then apply it to prove existence of solutions for a wide class of parabolic variational inequalities.

Zhangyong Cai, Department of Mathematics, Guangxi Teachers Education University, Nanning 530023, P.R. China, School of Mathematics, Sichuan University, Chengdu 610065, P.R. China (zycaigxu2002@126.com), Shou Lin, Institute of Mathematics, Ningde Normal University, Ningde 352100, P. R. China, Department of Mathematics, Minnan Normal University, Zhangzhou 363000, P.R. China (shoulin60@163.com), and Chuan Liu, Department of Mathematics, Ohio University Zanesville Campus, Zanesville, OH 43701, USA (liuc1@ohio.edu).
Copies of special spaces in free (abelian) paratopological groups, pp. 351-362.
ABSTRACT. Let FP(X) (AP(X)) denote the free paratopological group (free Abelian paratopological group) over a topological space X. In this paper, a homeomorphism theorem for the free Abelian paratopological group over a topological space X is established, which extends a result of A. Arhangel'skii. As an application, it is shown that if X is a Tychonoff space and P is a densely self-embeddable prime space with a q-point, then AP(X) contains a copy of P if and only if FP(X) contains a copy of P if and only if X contains a copy of P, which generalizes a theorem of K. Eda, H. Ohta and K. Yamada. At last, it is shown that if the free paratopological group FP(X) (the free Abelian paratopological group AP(X)) over a Tychonoff space X contains a non-trivial convergent sequence, then FP(X) (AP(X)) contains a closed copy of Arens' space, further, which gives an affirmative answer to a question in literature.

F. Azarpanah, Department of mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran(azarpanah@ipm.ir ) and M. Ghirati., Department of Mathematics, Yasouj University, Yasouj, Iran(ghirati@yu.ac.ir ) and A. Taherifar, Department of Mathematics, Yasouj University, Yasouj, Iran(ataherifar@yu.ac.ir and ataher54@gmail.com).
, pp. 363-383.
ABSTRACT. In [M. Ghirati and A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167(2014) 62-68], the authors have given a new representation for some closed ideals of C(X). For instance, they have shown that the intersection of all free maximal ideals of C(X), can be represented by the set of all continuous functions f such that Z(1-fg) is compact for each continuous function g. In this paper we generalize this representation for all closed ideals in C(X) and the intersections of all maximal ideals in C*(X) . For a completely regular Hausdorff space X, we construct a space λX in βX containing νX and show that X is Lindelöf if and only if X coincides with λX.

Lin, Fucai, Minnan Normal University, Zhangzhou 363000, PR China (linfucai2008@aliyun.com), Lin, Shou, Ningde Normal University, Ningde 352100, PR China(shoulin60@163.com), and Sakai, Masami, Kanagawa University, Hiratsuka 259-1293, Japan (sakaim01@kanagawa-u.ac.jp)
Point-countable covers and sequence-covering maps, pp. 385-397.
ABSTRACT. We answer some questions on the theory of generalized metric spaces posed in the book: S. Lin, Point-Countable Covers and Sequence-Covering Mappings (second edition), Beijing, China Science Press, 2015.