HOUSTON JOURNAL OF
MATHEMATICS

Electronic Edition Vol. 42, No. 1 , 2016

Editors:  D. Bao (San Francisco, SFSU), D. Blecher (Houston), Bernhard 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 Editor: K. Kaiser (Houston)

 Houston Journal of Mathematics


Contents

J.D.H. Smith, Department of Mathematics, Iowa State University Ames, Iowa 50011 (jdhsmith@iastate.edu).
Directional algebras, pp. 1-22.
ABSTRACT. Directional algebras are generalizations of dimonoids, which may themselves be regarded as directional semigroups. Given a constant-free type, a directional type is obtained by pointing to each of the arguments of the original, undirected type. For each axiomatization of a variety of algebras of constant-free type, a corresponding directional variety is determined. Dimonoids and digroups are shown to arise from the general procedure. For quasigroups, various choices of equational bases lead to various varieties of directional quasigroups. Under one natural axiomatization, the variety of quasigroups is shown to be directionally complete, in the sense that the corresponding directional variety is again the variety of quasigroups. Another axiomatization yields (4+2)-quasigroups. Digroups are equivalent to a certain class of (4+2)-quasigroups.

Mooney, Christopher Park, Westminster College, Fulton, MO 65251 (christopher.mooney@westminster-mo.edu).
τ-Complete factorization in commutative rings with zero-divisors, pp. 23-44.
ABSTRACT. Lots of progress has been made on generalized factorization techniques in integral domains, namely τ-factorization. There has also been substantial research done on investigating factorization in commutative rings with zero-divisors. There are many ways authors have decided to study factorization when zero-divisors present. This article focuses on the method τ-complete factorizations developed by D.D. Anderson and A. Frazier. τ-complete factorization is a natural way to think of refining factorizations into smaller pieces until one simply cannot refine the factorization any further. We see that this notion translates well into the case of commutative rings with zero-divisors and there is an interesting relationship between the τ-complete finite factorization properties and the original τ-finite factorization properties in rings with zero-divisors developed by the author in 2012.

Lele, Celestin, Dept. of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon (celestinlele@yahoo.com), and Nganou, Jean B, Dept. of Mathematics, University of Oregon, Eugene OR 97403 (nganou@uoregon.edu).
The Chang-Mundici's l-group of a BL-algebra, pp. 45-72.
ABSTRACT. For any BL-algebra L, we construct an abelian lattice-ordered group GL that coincides with the Chang-Mundici l-group of an MV-algebra when the BL-algebra is an MV-algebra. We prove that the Chang-Mundici group of the MV-center of any BL-algebra L is a direct summand in GL and that its complement S(L) is an ideal of GL. We find an alternate description of the group S(L) using the filter of dense elements of L, and generalize the group of PL-algebras as treated by Cignoli and Torrens(2000).

Kaiyun Wang and Bin Zhao (corresponding author), Department of Mathematics, Shaanxi Normal University, Xi'an 710062, China (wangkaiyun@snnu.edu.cn) (K. Wang), (zhaobin@snnu.edu.cn) (B. Zhao).
On Embeddings of Q-algebras, pp. 73-90 
 ABSTRACT. Firstly, we prove that every Q-algebra can be embedded into a unital Q-algebra. On the basis of this result, we also show that the category of unital Q-algebras is a reflective subcategory of the category of Q-algebras. Secondly, we prove that every Q-algebra is embeddable into the endomorphism Q-algebra of a Q-module. Finally, we introduce the concept of a Girard Q-algebra and show that every Q-algebra can be embedded into a Girard Q-algebra.

Jianming Qi,, Department of Mathematics and Physics Shanghai Dianji University, Shanghai, 201306, P. R. China. (qijianmingsdju@163.com), Fanning Meng (corresponding author), School of Mathematics and Information Science Guangzhou University, Guangzhou, 510006, P. R. China (mfnfdbx@163.com), and Wenjun Yuan, School of Mathematics and Information Science Guangzhou University, Guangzhou, 510006, P. R. China (wjyuan1957126.com).
Further results about the value distribution of meromorphic function with its k-th derivative, pp. 91-100.
ABSTRACT. Let k be an integer which is not less than 2. Let f(z) be a transcendental meromorphic function on the complex plane, all of whose zeros have multiplicity at least k+1, except possibly finitely many; and all of whose poles are multiple except possibly finite many. Let a(z) be the multiplication of R(z) and exp(t(z)), where R(z) is a rational function and t(z) is a transcendental entire function, and the hyper-order of f(z) is bigger than the hyper-order of a(z). Then the difference between the k-th derivative of f(z) and a(z) has infinitely many zeros.

Fan-Ning Meng, School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China  (jpmfnfdbx@yahoo.co.jp), Seiki Mori (corresponding author), Department of Mathematical Sciences, Yamagata University, Yamagata 8560, Japan (sp993gc9@shirt.ocn.ne.jp), and Wenjun Yuan,  School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China (wjyuan1957@126.com).
Schottky-Landau type theorem for holomorphic mappings, pp. 101-121.
ABSTRACT. P. Griffiths (1971) or K. Kodaira (1971) showed Schottky-Landau type theorems for holomorphic mappings f from a ball B(r) of radius r in complex n-space into a smooth complete canonical algebraic variety V or a smooth projective algebraic variety V of dimension n of general type. On the other hand, P. Griffiths et al.(1972 or 1973) showed the case of holomorphic mappings from B(r) into V minus D, where the sum of Chern classes of K and D is positive, and K is the canonical bundle over V. Their results are existence theorems of the upper bound R such that f is defined on B(r) for every r <R under some condition. In this note, we give Carlson-Griffiths-King's theorem with a little bit detail bound than Griffiths' one by using Kodaira's method and Griffiths' singular volume form. Especially, our main purpose is to give a concrete bound for the case of complex projective n-space V minus a collection of hyperplanes in general position.

N. Anghel, Department of Mathematics, University of North Texas, Denton, TX 76203 (anghel@unt.edu).
Curvature bundle morphisms for hypersurface generalized Dirac operators, pp. 123-142.
ABSTRACT. The datum of a generalized Dirac operator (in the sense of Gromov and Lawson) on a Riemannian manifold can be restricted to a hypersurface, and then suitably altered via a bundle morphism-valued one-form commuting with Clifford multiplication, to yield a generalized Dirac operator on the hypersurface. How do the associated curvature bundle morphisms, on the hypersurface and on the ambient manifold, relate? The Gauss-like relationship we are going to establish in this paper involves the shape operator of the hypersurface, a normal part of the curvature associated to the connection giving the Dirac operator on the ambient manifold, and the curvature of the one-form. Applications are then given to operators of Spin and Clifford type.

Taylor, Michael, University of North Carolina, Chapel Hill, NC 27599 (met@math.unc.edu).
Traveling wave solutions to NLS and NLKG equations in non-Euclidean settings, pp. 143-165.
ABSTRACT. We study traveling wave solutions to nonlinear Schrodinger (NLS) and nonlinear Klein-Gordon (NLKG) equations on a compact Riemannian manifold M, with a Killing field X, generating a group of isometries. The emphasis is on NLKG. Then if X has length less than 1 everywhere, one gets a semilinear elliptic PDE on M, to which standard variational techniques apply (for a natural class of nonlinearities and suitable Sobolev spaces), as reviewed in Section 1, though there remains the question of whether the associated waves are really (or just apparently) traveling, a point taken up in Section 2. In Sections 3 and 4 we consider sonic speed waves, in some situations that lead to subelliptic nonlinear PDE, and in Section 5 we consider some supersonic traveling waves.

Nikolaev, Igor, The Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, CANADA (igor.v.nikolaev@gmail.com).
Hyperbolic geometry, continued fractions and classification of the finitely generated totally ordered simple dimension groups, pp. 167-178.
ABSTRACT. We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic surface of genus greater or equal two. The main theorem says that isomorphism classes of the polycyclic totally ordered dimension groups are bijective with a generic subset of reals modulo the action of group GL(2, Z). The result is an extension of the Effros-Shen classification of the dicyclic dimension groups.

Deng, Guotai, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China (hilltower@163.com),  Liu, Chuntai, School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, P. R. China (lct984@163.com),  and  Ngai, Sze-Man, College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China, and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA (smngai@georgiasouthern.edu).
Dimensions of the boundary of a graph-directed self-similar set with overlaps, pp. 179-210.
ABSTRACT. We set up a framework for computing the Hausdorff and box dimensions of the boundary of each component of a graph self-similar family defined by a graph-directed iterated function system satisfying the so-called graph finite boundary type condition.  We show that the boundary of each component has the same Hausdorff and box dimension, with the corresponding Hausdorff measure being positive and σ-finite. These results are natural extensions of existing ones concerning the dimensions of the boundary of a self-similar tile.

Virdol, Cristian, Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Korea (cristian.virdol@gmail.com).
Drinfeld modules and subfields of division fields, pp. 211-221.
ABSTRACT. n this paper we shall generalize some results of Cojocaru and Duke who obtained some asymptotic formulas for the splitting of the rational primes p in some divisions fields associated to an elliptic curve E defined over Q, to the case of Drinfeld A-modules of arbitrary rank r>2. A similar asymptotic formula as in this paper was obtained by Hooley in his proof, under the Generalized Riemann Hypothesis (GRH), of the classical Artin's primitive roots conjecture. While Artin's conjecture is still unproved, the results in this paper are unconditional, as GRH for the function field case is known.

Duvenhage, Rocco, University of Pretoria, Pretoria 0002, South Africa (rocco.duvenhage@up.ac.za) and Ströh, Anton, University of Pretoria, Pretoria 0002, South Africa (anton.stroh@up.ac.za).
Disjointness of C*-dynamical systems, pp. 223-247.
ABSTRACT. We study ergodic theorems for disjoint C*- and W*-dynamical systems, where disjointness here is a noncommutative version of the concept introduced by Furstenberg for classical dynamical systems. We also consider specific examples of disjoint W*-dynamical systems. Lastly we use unique ergodicity and unique ergodicity relative to the fixed point algebra to give examples of disjoint C*-dynamical systems.

Bickel, Kelly, Bucknell University, Lewisburg, PA 17837 (kelly.bickel@bucknell.edu) and Wick, Brett D, Washington University in St. Louis, St. Louis, MO 63130 (wick@math.wustl.edu).
Well-localized operators on matrix weighted L2 spaces, pp. 249-283.
ABSTRACT. Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for almost-diagonal operators that played a key role in the proof of the A2 conjecture for dyadic shifts and related operators. In this paper, we obtain a generalization of their T1 theorem to the setting of matrix weights. Our theorem does differ slightly from the scalar results, a fact attributable almost completely to differences between the scalar and matrix Carleson Embedding Theorems. The main tools include a reduction to the study of well-localized operators, a new system of Haar functions adapted to matrix weights, and a matrix Carleson Embedding Theorem.

Castro, Hernán, Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile (hcastro@inst-mat.utalca.cl).
Uniqueness results for a singular non-linear Sturm-Liouville equation, pp. 285-306.
ABSTRACT. In this work we study the uniqueness of solutions to a non-linear Sturm-Liouville equation that is singular at the origin. We prove that uniqueness of solutions is guaranteed to hold when one imposes some appropriate behavior at the origin.

Sergiu Aizicovici, Department of Mathematics, Ohio University, Athens, OH 45701, USA (aizicovs@ohio.edu),  Nikolaos S. Papageorgiou, Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (npapg@math.ntua.gr), and  Vasile Staicu, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal (vasile@ua.pt).
Semilinear Neumann equations with indefinite and unbounded potential, pp. 307-340.
ABSTRACT. We consider a semilinear Neumann problem with an indefinite and unbounded potential, and a Carathéodory reaction term. Under asymptotic conditions on the reaction which make the energy functional coercive, we prove multiplicity theorems producing three or four solutions with sign information on them. Our approach combines variational methods based on the critical point theory with suitable perturbation and truncation techniques, and with Morse theory.

Janusz R. Prajs, California State University Sacramento, Department of Mathematics and Statistics, 6000 J Street, Sacramento, CA 95819, USA (prajs@csus.edu).
Semi-terminal continua in homogeneous spaces, pp. 341-371.
ABSTRACT. A semi-terminal continuum Y in a space X is defined by the condition that no two disjoint subcontinua of X intersect both Y and X-Y. Though numerous obvious examples of such continua can be found in arcs, trees and tree-like continua, these examples are related to the non-homogeneity of the space, and having semi-terminal continua in a homogeneous continuum is counter-intuitive. Recently, a large collection of homogeneous spaces with semi-terminal, non-terminal subcontinua has been found. This paper is devoted to studying these spaces and the general structure of homogeneous continua related to the presence of semi-terminal subcontinua.

Lipparini, Paolo, II Università di Roma (Tor Vergata) I-00133 Rome, Italy (lipparin@axp.mat.uniroma2.it)
For Hausdorff spaces, H-closed = D-pseudocompact for every ultrafilter D

ABSTRACT. We prove that, for an arbitrary topological space X, the following conditions are equivalent: (a) Every open cover of X has a finite subfamily with dense union; (b) X is D-pseudocompact, for every ultrafilter D. Locally, our result asserts that if X is weakly initially λ-compact, and 2μ ≤ λ, then X is D-pseudocompact, for every ultrafilter D over any set of cardinality ≤ μ. As a consequence, if 2μ ≤ λ, then the product of any family of weakly initially λ-compact spaces is weakly initially μ-compact.