HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 44, No. 4, 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), A. Haynes (Houston), 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

Merlin Carl, Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany (merlin.carl@uni-konstanz.de), Lorenzo Galeotti, Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany (lorenzo.galeotti@gmail.com), and Benedikt Löwe, Institute for Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands (bloewe@science.uva.nl).
The Bolzano-Weierstrass theorem in generalised analysis, pp. 1081-1109.
ABSTRACT. Let κ be an uncountable regular cardinal with κ<κ= κ. We consider two totally ordered fields κ–ℝ and ℝκ, due to Sikorski and the second author, respectively, that serve as the κ-analogues of the real line and consider generalisations of the Bolzano-Weierstrass theorem for them, showing that for ℝκ, the weak κ-Bolzano-Weierstrass theorem is closely related to the tree property of κ.

Hammonds, Trajan, Dept. of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15289 (thammond@andrew.cmu.edu), Johnson, Jeremy,   Dept. of Mathematics, Humboldt State University, Arcata, CA 95521 (jsj132@humboldt.edu), Patini, Angela, Dept. of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 (apatini@sas.upenn.edu), and Walker, Robert M., Dept. of Mathematics, University of Michigan, Ann Arbor, MI, 48109  (robmarsw@umich.edu).
Counting roots of polynomials over Z/p2Z, pp. 1111-1119.
ABSTRACT. Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime integer p and f in (Z/pnZ)[x] any nonzero polynomial of degree d whose coefficients are not all divisible by p. For the case n=2, we prove a new efficient algorithm to count the roots of f in Z/p2Z within time (d+size(f)+log p),2+o(1), based on a formula conjectured by Cheng, Gao, Rojas, and Wan.

HongYan Xu, Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China (xhyhhh@126.com) and YinYing Kong (Corresponding author), School of Accounting, Guangdong University of Finance and Economics, Guangzhou, Guangdong 510320,China (kongcoco@hotmail.com).
The Nevanlinna and Valiron deficiencies of some q-difference-differential polynomials, pp. 1121-1134.
ABSTRACT. For a zero order meromorphic function f(z), the main purpose of this paper is to investigate the value distribution on f, f' and some types of q-difference-differential polynomials. We obtain some interesting results, which reveal the relation of Nevanlinna deficiencies among such functions concerning Valiron deficiency. Moreover, we give some examples to explain our conclusions.

Xiaoliang Cheng, Department of Mathematics, Jilin Normal University, Siping, Jilin, 136000, China (chengxiaoliang92@163.com), Qianyun Wang, School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China (wangqy1226@gmail.com), and Xu Zhang, Department of Mathematics, Jilin Normal University, Siping, Jilin, 136000, China (xzhang.2018math@yahoo.com)
On local holomorphic conformal embeddings, pp. 1135-1145.
ABSTRACT. We first develop some general properties for holomorphic conformal maps between Kähler manifolds, such as extension and algebraicity. Applying these properties, some rigidity results for holomorphic conformal maps from the unit disk to the product of unit balls are obtained.

Miguel Angel Javaloyes, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain (majava@um.es) and Henrique Vitório, Departamento de Matemática, Universidade Federal de Pernambuco, Cidade Universitária, Recife, Pernambuco, Brazil (henriquevitorio@dmat.ufpe.br).
Some properties of Zermelo navigation in pseudo-Finsler metrics under an arbitrary wind, pp. 1147-1179.
ABSTRACT. We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric F is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of F by a vector field W at each point, where W is an arbitrary vector field, without the classical restriction of W being mild (i.e. we allow the opposite of W to have F-norm bigger or equal to 1). Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when W is homothetic, we give a description of the geodesic flow of the translation and we prove that the flag curvature of the translation coincides with the one of the original one up to the addition of a non-positive constant. We give a proof of the latter by exploiting the fanning curve approach to the flag curvature. These results allow us to extend all the Randers spaceforms classified by D. Bao, C. Robles and Z. Shen (in J. Differential Geom., 66 (2004)) to geodesically complete conic Finsler manifolds with constant flag curvature.

Grieve, Nathan, Michigan State University, East Lansing, MI 48824 (grievena@msu.edu).
On arithmetic general theorems for polarized varieties, pp.  1181-1203.
ABSTRACT. We apply Schmidt's Subspace Theorem to establish Arithmetic General Theorems for projective varieties over number and function fields. Our first result extends an analogous result of M. Ru and P. Vojta. One aspect to its proof makes use of a filtration construction which appears in work of Autissier. Further, we consider work of M. Ru and J. T.-Y. Wang which pertains to an extension of K. F. Roth's theorem for projective varieties in the sense of D. McKinnon and M. Roth. Motivated by these works, we establish our second Arithmetic General Theorem, namely a form of Roth's theorem for exceptional divisors. Finally, we observe that our results give, within the context of Fano varieties, a sufficient condition for validity of the main inequalities predicted by Vojta.

Rajeev Gupta and Md. Ramiz Reza, Indian Institute of Science, Bangalore 560012 (rajgupta56@gmail.com),(ramiz.md@gmail.com).
Operator space structures on ℓ1 (n), pp. 1205-1212.
ABSTRACT. We show that the complex normed linear space ℓ¹(n) for n > 1 has no isometric embedding into complex matrices of size k for any natural number k and discuss a class of infinite-dimensional operator space structures on it.

Soltani Renani, Sima, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran (simasoltani@cc.iut.ac.ir).
Homological properties of some Banach modules related to completely continuous convolution operators, pp. 1213-1220.
ABSTRACT. For a locally compact group G, the notion of a Dunford-Pettis operator and the convolution product were used to introduce the function space DP(G). Closely related to this space is the space of all left uniformly measurable functions on G that denoted by LUM(G). Here, we investigate projectivity, injectivity and flatness of LUM(G) and DP(G) as Banach right modules over the group algebra.

T.S.S.R.K. Rao, Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore-560059, India (tss@isibang.ac.in).
Points of  strong  subdifferentiability  in  dual  spaces, pp.  1221-1226.
ABSTRACT. In this paper, motivated by a classical results of Franchetti and Paya on points of strong subdifferentiability, we exhibit several naturally occurring situations when the property of being strongly subdifferentiable can be lifted from a subspace to the entire space. Our methods rely on the deep analysis of strongly proximinal subspaces of finite codimension done by Godefroy and Indumathi and techniques from the theory of M-ideals.

Bernhard Burgstaller, Doppler Institute for Mathematical Physics, Trojanova 13, 12000 Praha, Czech Republic (bernhardburgstaller@yahoo.de).
A Green-Julg isomorphism for inverse semigroups, pp. 1227-1240.
ABSTRACT. For every finite unital inverse semigroup and equivariant C*-algebra we establish a Green-Julg isomorphism between equivariant K-theory of the C*-algebra and the non-equivariant K-theory of the crossed of the C*-algebra by the inverse semigroup.

Caleb Eckhardt, Department of Mathematics, Miami University, Oxford, OH, 45056 (eckharc@miamioh.edu).
Free groups and quasidiagonality, pp. 1241-1267.
ABSTRACT. We use free groups to settle a couple questions about the values of the Pimsner-Popa-Voiculescu modulus of quasidiagonality for a set of operators Ω, denoted by qd(Ω). Along the way we deduce information about the operator space structure of finite dimensional subspaces of ℂ[𝔽d] ⊂ C*lp(𝔽d) where C*lp(𝔽d) is the so-called lp-completion of ℂ[𝔽d]. Roughly speaking, we use free groups and qd(Ω) to put a quantitative face on the two known qualitative obstructions to quasidiagonality; absence of an amenable trace or the presence of a proper isometry. The modulus of quasidiagonality for a proper isometry is equal to 1. We show that qd(λa,λb) ∈ [1/2,√ 3/2] where a and b are free group generators and λ is the left regular representation. In another direction, we use certain lp representations of free groups constructed by Pytlik and Szwarc and a recent result of Ruan and Wiersma to show that qd(Ω) may be positive, yet arbitrarily close to zero when Ω is a set of unitaries.

Alan Bertl, Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849 (acb0039@auburn.edu).
Category theoretic characterizations of generalized inverse limits, pp. 1269-1291.
ABSTRACT. We construct a category GLim of generalized inverse sequences which admits a functor that maps sequences to their limits (in both the topological and category theoretical sense) in the standard category Top of topological spaces. This functor is shown to be right adjoint to a functor mapping each topological space U to a generalized inverse sequence the limit of which is homeomorphic to U. These constructions give rise to two characterizations of generalized inverse limits in the language of category theory.

Roger B. Eggleton, University of Newcastle, Callaghan, NSW 2308, Australia (roger@ilstu.edu), Michal Morayne, Robert Rałowski, Szymon Żeberski, Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, Wybrzeze Wyspianskiego 27, 50-370 Wrocław, Poland (michal.morayne@pwr.edu.pl), (robert.ralowski@pwr.edu.pl), (szymon.zeberski@pwr.edu.pl).
On midpoint-free subsets of some topological groups, pp. 1293-1311.
ABSTRACT. A subset of an abelian group is midpoint-free if it contains no three distinct elements a, b, c such that a+b=2c. We study midpoint-free sets in various classical topological groups. For every infinite cardinal k not greater than the cardinality of the continuum, we show that the real line can be partitioned into k-many maximal midpoint-free sets. Examples of closed maximal midpoint-free subsets are given for topological groups such as the real line, the complex plane, the one-dimensional sphere and the torus. Finally, among sets that are not regular, such as non-measurable sets, Bernstein sets, and Luzin sets, we study instances which are midpoint-free.

Alan Bertl and Michel Smith, Mathematics and Statistics, Auburn University, Auburn, Alabama 36849 (acb0039@auburn.edu) , (smith01@auburn.edu).
Generalized set-valued inverse limits with finite coordinate spaces, pp. 1313-1334.
ABSTRACT. Generalized set-valued inverse limits whose first coordinate space is a compact metric space and all of whose subsequent spaces are finite non-Hausdorff spaces are considered. A condition is stated that causes such inverse limit spaces to be compact metric spaces. Several examples are provided and the algebraic structure is examined in the case where the resultant metric space is a topological group. The example of the solenoid as a topological group is considered in detail.

David Herrera-Carrasco, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, 72570, Puebla, Mexico (dherrera@fcfm.buap.mx), María de J. López, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, 72570, Puebla, Mexico (mjlopez@fcfm.buap.mx), and Fernando Macías-Romero, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, 72570, Puebla, Mexico (fmacias@fcfm.buap.mx).
Almost meshed locally connected continua without unique n-fold hyperspace suspension, pp. 1335-1365.
ABSTRACT. For a metric continuum X and a positive integer n, we consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). Let HSn(X) be the quotient space which is obtained from Cn(X) by identifying Fn(X) to a one-point set, with the quotient topology. In this paper we prove that: (1) If X and Y are almost meshed locally connected continua and HSn(X) is homeomorphic to HSn(Y ), then X is homeomorphic to Y , for each n greater than or equal to 3; (2) There are almost meshed locally connected continua without unique n-fold hyperspace suspension, for each positive integer n.

Tingmei Gao, School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong, Shaanxi 723000, P.R. China (gtmgtmgtm@snut.edu.cn), and Shou Lin, Institute of Mathematics, Ningde Normal University, Ningde, Fujian 352100, P. R. China (shoulin60@163.com).
On perfect images of mu-spaces, pp. 1367-1376.
ABSTRACT. A space is called a μ-space if it can be embedded in the product of countably many paracompact Fσ-metrizable spaces. K. Nagami posed the following problem: is the perfect image of a μ-space a μ-space? By the saturated sets-topology of submetrizable spaces, in this paper, we gives a partial answer to Nagami's problem.

Jing, Zhang School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, P. R. China (zhangjing86@126.com).
A quasitopological modification of semitopological groups, pp. 1377-1388.
ABSTRACT. In this paper, for a given semitopological group H and each positive integer n , we define a quasitopological group Q2n(H). Some basic properties of the quasitopological group Q2n(H) are obtained. We also discuss relations between H and Q2n(H), some cardinal invariants are considered.

Jesús Díaz Reyes, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y Río Verde, Ciudad Universitaria, San Manuel Puebla, Pue., México C.P. 72570 (jdeisauzs@gmail.com), and Armando Romero Morales, Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca, carretera a Acatlima Km. 2.5 Huajuapan de León Oax., México C.P. 69000 (armando@mixteco.utm.mx).
The weak Urysohn number and upper bounds for cardinality of Hausdorff spaces, pp. 1389-1398.
ABSTRACT. In this article, following the line of research of Bonanzinga (On the Hausdorff number of a topological space, Houston J. Math. 2013) we introduce a new cardinal function, the weak Urysohn number of a space X, U*(X), to extend some famous inequalities due to Bella and Cammaroto (On the cardinality of Urysohn spaces, Canad. Math. Bull. 1988) and Alas (More topological cardinal inequalities) among others, from the class of Urysohn spaces to the class of Hausdorff spaces with U*(X)≤ω.