HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 43, No. 4, 2017

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

Ulrich Albrecht,  316-A Parker Hall,  Department of Mathematics and Statistics,  Auburn University Auburn, AL 36849 (albreuf@auburn.edu).
Valued Baer groups, pp. 1031-1044
ABSTRACT. This paper investigates properties of a finite valuated p-group A which are determined by its endomorphism ring R. A pair of contra-variant functors between the categoy of finite valuated p-groups and the category of finite left modules is introduced to study the splitting of exact sequences of valuated p-groups.

Yongyi Gu, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China (gdguyongyi@163.com), Najva Aminakbari, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China (najvaaminakbary@yahoo.com), Wenjun Yuan(corresponding author), School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China (wjyuan1957@126.com), and Yonghong Wu, Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia (y.wu@curtin.edu.au).
Meromorphic solutions of a class of algebraic differential equations related to Painleve equation III", pp. 1045-1055.
ABSTRACT. In this paper, the complex method is employed to derive meromorphic solutions to a class of algebraic differential equations related to Painleve equation III, and then we illustrate our main results by some computer simulations. As an example, we obtain meromorphic solutions of a nonlinear evolution equation by the application of our results.


Jianjun Zhang, Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing, 210013, China (zhangjianjun1982@163.com).
On transcendental meromorphic solutions of certain types of nonlinear differential equations, pp. 1057-1070.
ABSTRACT. By utilizing Nevanlinna's value distribution theory of meromorphic function, we are able to give the existence condition and the form of a transcendental meromorphic solution of some certain types of nonlinear differential equations.

Kozin, Nikita, Rice University, Department of Mathematics, TX 77005 (nikita.kozin@rice.edu) and Majeti, Deepak, Rice University, Department of Computer Science, TX 77005 (deepak.majeti@rice.edu).
On the arithmetic of one Del Pezzo surface over the field with three elements, pp. 1071-1085.
ABSTRACT. We discuss the problem of existence of rational curves on a certain del Pezzo surface from a computational point of view and suggest a computer algorithm implementing search. In particular, our computations reveal that the surface contains 920 rational curves with parametrizations of degree 8 and does not contain rational curves for a smaller degree.

Ali Maalaoui, Department of Mathematics and Natural Sciences, American University of Ras Al Khaimah, UAE (ali.maalaoui@aurak.ae) and Vittorio Martino, Dipartimento di Matematica, Università di Bologna, Italy (vittorio.martino3@unibo.it).
Contact type hypersurfaces and Legendre duality, pp. 1087-1098.
ABSTRACT. In this paper we study contact type hypersurfaces embedded in four-dimensional Kähler manifolds. We are interested whether the so called Legendre duality can be performed and we will show that this can be related to some convexity assumptions, giving a sufficient condition. As an application, in the case of convex hypersurfaces in R4, we will explicitly complete this duality.

de Lima, Henrique, Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB 58429-970 (henrique@mat.ufcg.edu.br), dos Santos, Fábio, Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB 58429-970 (fabio@mat.ufcg.edu.br), Araújo, Jogli, Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB 58429-970 (jogli@mat.ufcg.edu.br), and Velásquez, Marco Antonio, Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB 58429-970 (marco.velasquez@mat.ufcg.edu.br).
Complete maximal spacelike submanifolds immersed in a locally symmetric semi-Riemannian space, pp. 1099-1110.
ABSTRACT. In this paper, we deal with complete maximal spacelike submanifolds immersed with flat normal bundle in a locally symmetric semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and, as application, we show that such a spacelike submanifold must be totally geodesic or the square norm of its second fundamental form must be bounded. We point out that our results extend classical theorems due to Ishihara and Nishikawa.

Katherine Heller, North Central College, 30 N. Brainard St. Naperville, IL 60540, (kheller@noctrl.edu)
Commutators of Linear Fractional Composition Operators on S2(D), pp. 1111-1131.
ABSTRACT. We study compactness of the linear fractionally induced commutator [Cφ*,Cψ] on the weighted Hardy space S2(D). We show that if φ and ψ are automorphisms of the disc, the commutator is compact if and only if both φ and ψ are rotations. When φ and ψ are linear fractional self-maps of the disc, the commutator is compact if and only if both are parabolic with the same boundary fixed point or both are hyperbolic with the same boundary fixed point and with non-boundary fixed points as conjugate reciprocals.

Ju Myung Kim, Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea (kjm21@kaist.ac.kr).
K1- and Ku1-approximation properties, pp. 1133-1145.
ABSTRACT. We prove that if the dual space X* of a Banach space X has the Ku1-approximation property (Ku1-AP), then X has the K1-approximation property (K1-AP). It is also shown that if X* has the K1-AP, then X has the Ku1-AP whenever X* has the Radon-Nikodym property. As a consequence, it follows that there exists a Banach space failing to have the Ku1-AP, which gives an affirmative answer to a question of the author.

M. Amini, Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran (mamini@modares.ac.ir) and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran (mamini@ipm.ir), A.R. Medghalchi, Department of Mathematics, Kharazmi University, 50 Taleghani Avenue, Tehran 15618, Iran (a_medghalchi@khu.ac.ir), and H. Nikpey, Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran (hamednikpey@gmail.com).
On tensor products of injective operator spaces, 1147-1163.
ABSTRACT. We show that an infinite dimensional injective operator space has an infinite dimensional operator subspace, completely isometric to the row or column Hilbert spaces or the operator space of all bounded sequences. When the third case does not happen, we characterize the injective operator space as a finite direct sum of spaces of bounded operators B(H,K) with H or K finite dimensional. We characterize injective operator spaces injective spacial tensor product. We give a similar characterization for injectivity of the Haagerup tensor product of injective operator spaces.

Yuan Li, School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P.R. China (liyuan0401@aliyun.com)
Complete order structures for completely bounded maps involving trace class operators, pp. 1165-1185.
ABSTRACT. We firstly establish the matrix orders for the *-vector spaces consisted respectively by all completely bounded linear maps on Banch *-algebra of all trace-class operators, by all bounded linear operator on  the tensor product of Hilbert spaces and by all bounded linear maps from Banch *-algebra of all trace-class operators into von Neumann algebra of all bounded linear operators. Then we show that the first and second matrix ordered *-vector spaces can be complete order embedded into the second and third matrix ordered *-vector spaces, respectively.

Moslehian, Mohammad Sal, Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran (moslehian@um.ac.ir), Zamani, Ali, Department of Mathematics, Farhangian University, Semnan, Iran (zamani.ali85@yahoo.com), and Dehghani, Mahdi, Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P. O. Box 87317-53153, Kashan, Iran (m.dehghani@kashanu.ac.ir).
Characterizations of smooth spaces by ρ*-orthogonality, pp.1187-1208.
ABSTRACT. The aim of this paper is to present some results concerning the ρ*-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if T is a nonzero linear (I, ρ*)-orthogonality preserving mapping between real normed spaces X and Y, then ||Tx|| is between ||T|| ||x||/3 and 3[T] ||x|| for all x in X, where [T]=inf{||Tx||: x is aunit vector in X}. We also show that the orthogonality defined by ρ* is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the ρ*-orthogonality.

Loga, Christopher Ryan, Southwestern Adventist University, Keene, TX 76059 (c.ryan.loga@gmail.com).
An extension theorem for matrix weighted Sobolev spaces on Lipschitz domains, pp. 1209-1233.
ABSTRACT. We recall the history of the extension problem for unweighted and scalar weighted Sobolev space. We then prove, with a few preliminaries, that a similar extension result holds for matrix weighted Sobolev space when considering a domain that is Lipschitz.

Helmut Maier, Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany (helmut.maier@uni-ulm.de) and Michael Th. Rassias, Institute of Mathematics, University of Zürich, CH-8057, Zürich, Switzerland (michail.rassias@math.uzh.ch).
Asymptotics for moments of certain cotangent sums for arbitrary exponents, pp. 1235-1249.
ABSTRACT. In this paper we extend a result on the asymptotics of moments of certain cotangent sums associated to the Estermann and Riemann zeta functions for arbitrary positive real exponents.

Dan-Virgil Voiculescu Department of Mathematics, University of California at Berkeley, Berkeley CA, 94720-3840 (dvv@math.berkeley.edu)
Lebesgue decomposition of functionals and unique preduals for commutants modulo normed ideals, pp.1251-1262.
ABSTRACT. We prove an analogue of the Lebesgue decomposition for continuous functionals on the commutant modulo a reflexive normed ideal of an n-tuple of hermitian operators for which there are quasicentral approximate units relative to the normed ideal. Using results of Godefroy-Talagrand and Pfitzner we derive from this strong uniqueness of the predual of such a commutant modulo a normed ideal. Lebesgue decomposition of functionals and unique preduals for commutants modulo normed ideals.

Medini, Andrea, Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria (andrea.medini@univie.ac.at) and  Zdomskyy, Lyubomyr, Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria (lzdomsky@gmail.com).
Productively Lindelöf spaces of countable tightness, pp. 1263-1272.
ABSTRACT. Michael asked whether every productively Lindelöf space is powerfully Lindelöf. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindelöf space of countable tightness is powerfully Lindelöf. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindelöf space X is powerfully Lindelöf if every open cover of Xω admits a point-continuum refinement consisting of basic open sets. This strengthens a result of Burton and Tall. Finally, we show that separation axioms are not relevant to Michael's question: if there exists a counterexample (possibly not even T0), then there exists a regular (actually, zero-dimensional) counterexample.

M.G. Tkachenko, Departamento de Matematicas, Universidad Autonoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, mexico City,  Mexico (mich@xanum.uam.mx). V.V. Tkachuk Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa 09340, Mexico City, Mexico  (vova@xanum.uam.mx).
More reflections in small continuous images, pp.1273-1289.
ABSTRACT. We show that it is independent of ZFC whether the Fréchet-Urysohn property in compact spaces reflects in continuous images of weight ≤ω1. It turns out that π-weight and π-character are reflectable in small continuous images but if there exists a measurable cardinal, then the functional tightness does not have this kind of reflection. It is established that, under CH, the property of having a Gδ-point reflects in continuous images of weight at most ω1. This fact implies that every compact continuous image of a Corson countably compact space has a dense set of Gδ-points, i.e., a problem of Kalenda has a positive answer under the Continuum Hypothesis. We also prove that it is independent of ZFC whether countable network weight reflects in continuous images of weight ≤ω1.

Marzougui, Habib, University of Carthage, Faculty of Sciences of Bizerte, Jarzouna, 7021, Tunisia (habib.marzougui@fsb.rnu.tn), and Naghmouchi Issam, University of Carthage, Faculty of Sciences of Bizerte, Jarzouna, 7021, Tunisia (issam.nagh@gmail.com).
On totally periodic ω-limit sets, pp. 1291-1303.
ABSTRACT. An ω-limit set of a continuous self-mapping of a com- pact metric space X is said to be totally periodic if all of its points are periodic. We say that X has the ω-FTP property, provided that for each continuous self-mapping f of X, every totally periodic ω-limit set is finite. First, we show that connected components of every totally periodic ω-limit set are finite. Second, we show on one hand that a zero-dimensional compact metric space has the ω-FTP property, and on the other hand for the wide class of one-dimensional continua, we prove that a hereditarily locally connected X has the ω-FTP property if and only if X is completely regular. This holds in particular when X is a local dendrite with a discrete set of branch points, and in a graph. As a consequence, for continuous maps on either a zero-dimensional compact metric space or a completely regular continuum, only two conditions are needed in the definition of ω-chaos (see Remark 2). For higher di- mensions, we show that any compact metric space X containing a topological n-ball (n ≥ 2) does not have the ω-FTP property. This holds for any topological compact manifold of dimension greater than 1.

I.  Sánchez, IMAC, Universitat Jaume I, Campus de Riu Sec, 12071 Castellon, Spain (isr.uami@gmail.com) and Mikhail G. Tkachenko, Departamento de Matematicas, Universidad Autonoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, D.F., Mexico (mich@xanum.uam.mx).
A quasitopological modification of paratopological groups, pp. 1305-1321.
ABSTRACT. For an arbitrary paratopological group H, we associate to H a quasitopological group Q2(H) which shares with H the underlying set and group structure. It turns out that Q2(H) is Tychonoff if and only if H is Hausdorff. This association is a covariant functor from the category of paratopological groups to the category of quasitopological groups which commutes with arbitrary products and respects dense subgroups. We study the influence of this functor in some cardinal invariants.

Delgado, Alberto L., Department of Mathematics, Illinois State University, Normal, IL 61790 (ald196883@gmail.com), and Timm, Mathew, Department of Mathematics, Bradley University, Peoria, IL 61625 (mtimm@bradley.edu).
Spaces with regular nonabelian self covers, pp. 1323-1336.
ABSTRACT. We construct two continua, one infinite dimensional and the other homeomorphic to the Sierpinski carpet, having the property that for any finite group G, the continua have finite sheeted regular self covers with G as their group of deck transformations. In particular, the groups in question can be nonabelian. The fundamental groups of the two continua are non-cohopfian and possess each finite group as a quotient.

Juan L.G. Guirao, Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena, Hospital de Marina, 30202–Cartagena, Región de Murcia, Spain (juan.garcia@upct.es) and Jaume Llibre, Departament de Matemàtiques. Universitat Autònoma de Barcelona, Bellaterra, 08193–Barcelona, Cata (jllibre@mat.uab.cat).
Topological entropy and periods of self-maps on compact manifolds, pp. 1337-1347.
ABSTRACT. Let (𝕄,ƒ) be a discrete dynamical system induced by a self-map ƒ defined on a smooth compact connected n-dimensional manifold 𝕄. We provide sufficient conditions in terms of the Lefschetz zeta function in order that: (1) ƒ has positive topological entropy when ƒ is 𝒞∞, and (2) ƒ has infinitely many periodic points when ƒ is 𝒞1 and ƒ(𝕄) ⊆ Int(𝕄). Moreover, for the particular manifolds 𝕊n, 𝕊n × 𝕊m,ℂPnand ℍPn we improve the previous sufficient conditions.