HJM, Vol. 45, No. 2, 2019

HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 45, No. 2, 2019

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

Juan Climent Vidal, Universitat de València, Departament de Lògica i Filosofia de la Ciència, Av. de Blasco Ibáñez, 30-7a, 46010, València, Spain. (Juan.B.Climent@uv.es) and Enric Cosme Llópez, Université de Lyon, CNRS, ENS de Lyon, UCB Lyon 1, Laboratoire de L'Informatique du Parallélisme, 46 allée d'Italie, 69364, Lyon, France. (Enric.Cosme-Llopez@ens-lyon.fr).
Eilenberg theorems for many-sorted formations, pp. 321-369.
ABSTRACT. A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts S and a fixed S-sorted signature Σ, the concepts of formation of congruences with respect to Σ and of formation of Σ-algebras, we prove that the algebraic lattices of all Σ-congruence formations and of all Σ-algebra formations are isomorphic, which is an Eilenberg’s type theorem. Moreover, under a suitable condition on the free Σ-algebras and after defining the concepts of formation of congruences of finite index with respect to Σ, of formation of finite Σ-algebras, and of formation of regular languages with respect to Σ, we prove that the algebraic lattices of all Σ-finite index congruence formations, of all Σ-finite algebra formations, and of all Σ-regular language formations are isomorphic, which is also an Eilenberg’s type theorem.

Asir, T., Department of Mathematics, DDE, Madurai Kamaraj University, Madurai 625 021, Tamil Nadu, India (asirjacob75@gmail.com), Maimani, H. R., Mathematics Section, Department of Basic Sciences, Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran (maimani@ipm.ir), Pournaki, M. R., Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran (pournaki@ipm.ir), and Tamizh Chelvam, T., Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India (tamche59@gmail.com).
Some bounds for the genus of a class of graphs arising from rings, pp. 371-384.
ABSTRACT. Let R be a commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is R \ J(R), and two distinct vertices x and y are adjacent if and only if 1-xy is not a unit in R. In this paper, some bounds for the genus of Jacobson graphs are obtained. As an application, all commutative Artinian rings with nonzero identity whose Jacobson graphs are toroidal is classified up to isomorphism by a similar result for finite case. Finally, we obtain an isomorphism relation between two Jacobson graphs.

Nunokawa, Mamoru, University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan (mamoru_nuno@doctor.nifty.jp), Sokół, Janusz, University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland (jsokol@ur.edu.pl), and Trąbka-Więcław, Katarzyna, Lublin University of Technology, Mechanical Engineering Faculty, ul. Nadbystrzycka 36, 20-618 Lublin, Poland (k.trabka@pollub.pl).
New sufficient conditions for strong starlikeness, pp. 385-393.
ABSTRACT. This paper determines new sufficient conditions for strong starlikeness and some related properties. The proof rests on several genetralizations and corollaries from Nunokawa's lemma, On the Order of Strongly Starlikeness of Strongly Convex Functions, Proc. Japan Acad. 69, Ser. A (1993) 234-237.

Nan Wu, Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing,100083, People's Republic of China (wunan2007@163.com).
Deviations and spreads of holomorphic curves of finite lower order, pp. 395-411.
ABSTRACT. In this paper, we consider the relation between the number of maximum modulus points, spread and growth of a holomorphic curve. We use the method of I. I. Marchenko and E. Ciechanowicz to generalize their results of meromorphic functions to holomorphic curves.

Taylor, Michael, University of North Carolina, Chapel Hill NC 27599 (met@math.unc.edu).
The Weierstrass ℘-function as a distribution on a complex torus, and its Fourier Series, pp. 413-429.
ABSTRACT. We treat the Weierstrass ℘-function associated to a lattice in the complex plane as a principal value distribution on the quotient torus and compute its Fourier coefficients. The computation of these coefficients for nonzero frequencies is straightforward, but quite pretty. The "constant term" is more mysterious. It leads to a non-absolutely convergent doubly infinite series. This can be regarded as a version of an Eisenstein series, though as we discuss in Section 4 it differs from the "Eisenstein summation" of the series, as treated in Weil's monograph on elliptic functions. Material from Section 3 on the Fourier series of elliptic functions arising from the Weierstrass zeta function leads to a formula connecting our sum with the Eisenstein series treated in Weil's text, and thereby yields a rapidly convergent approximation to the constant term.

Ping Li, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China (pli@ustc.edu.cn), Wei-Ran Lü, College of Science, China University of Petroleum, Qingdao, Shandong, 266580, P. R. China (luwr@upc.edu.cn), and Chung-Chun Yang, Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, P. R. China (maccyang@163.com).
Entire solutions of certain types of nonlinear differential equations, pp. 431-437.
ABSTRACT. By utilizing classical Laguerre's theorem, we can resolve the entire solutions of nonlinear differential equations of the form: f(z)f''(z)=(p₁(z) exp(α₁z)+p₂(z) exp(α₂z))², where p₁(z) and p₂(z) are nonzero polynomials, and α₁, α₂ are distinct nonzero constants. The results of this paper extend or generalize a result of Titchmarsh. An example is provided to show that the results in this paper, in a sense, are best possible.

Jianjun Zhang, Mathematics and Information Technology School, Jiangsu Second Normal University, Beijing West Road 77, Nanjing, 210013, P.R.China, (zhangjianjun1982@163.com), Xiaoqing Lu (corresponding author), Mathematics and Information Technology School, Jiangsu Second Normal University, Beijing West Road 77, Nanjing, 210013, P.R.China (luxiaoqing1984@126.com), and Liangwen Liao, Department of Mathematics, Nanjing University, Hankou Road 22, Nanjing, 210093, P.R.China (maliao@nju.edu.cn).
On exact transcendental meromorphic solutions of nonlinear complex differential equations, pp. 439-453.
ABSTRACT. In this paper, we will deal with the existence and the form of transcendental meromorphic solutions of nonlinear differential equation fⁿ+Q_d(z,f)=p₁(z)e^α₁(z)+p₂(z)e^α₂(z),where n≥ 4 is an integer, Q_d(z,f) is a special differential polynomial in f of degree d = n-1 with rational functions as its coefficients, p₁, p₂ are non-vanishing rational functions and α₁,α₂ are nonconstant polynomials.In particular, we can show that Conjecture 1 (P. Li and C. C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math. Anal. Appl. 320 (2006), 827-835.) is true when Q_d(z,f) has a special form.

Brendan Guilfoyle, School of Science, Technology, Engineering and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland (brendan.guilfoyle@ittralee.ie) and Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom (wilhelm.klingenberg@durham.ac.uk).
A global version of a classical result of Joachimsthal, pp. 455-467.
ABSTRACT. A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove the following global analogue of this result. Suppose that two closed convex surfaces intersect with constant angle along a curve that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both non-orientable. We prove this by characterizing the constant angle intersection of two surfaces in Euclidean 3-space as the intersection of a Lagrangian surface and a foliated hypersurface in the space of oriented lines, endowed with its canonical neutral Kähler structure. This establishes a relationship between the principal directions of the two surfaces along the intersection curve in Euclidean space. A winding number argument yields the result. The method of proof is motivated by topology and, in particular, the slice problem for curves in the boundary of a 4-manifold.

Sun, Zonghan, Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China (sun-zh13@mails.tsinghua.edu.cn) , (bipfiic2008xj@163.com), and Zhang, Guangyuan, Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China (gyzhang@mail.tsinghua.edu.cn).
The properties of extremal surfaces in Ahlfors' theory of covering surfaces, pp. 469-495.
ABSTRACT. Ahlfors' second fundamental theorem in Ahlfors' theory of covering surfaces claims that for each fixed set E_q of q (q>2) extended complex numbers (called the special points), the spherical area of a covering surface is dominated by the number of times when this surface assumes the special points, and a constant h multiple of the spherical perimeter of this surface. The optimal value of the constant h is called Ahlfors' constant H₀(E_q), which depends on E_q in a rather complicated way. One may only consider the surfaces which don't assume special points. In this case, the optimal value of the constant h is denoted by h₀(E_q), which also depends on E_q in a rather complicated way. Few properties of H₀(E_q) and h₀(E_q) are known yet, and Zhang (2012) have determined h₀({-1,0,1})=4.034.... In order to determine H₀(E_q) and h₀(E_q) relatively conveniently, among all surfaces with a fixed bound of perimeters, we should consider the surfaces with the maximal area, which are called the extremal surfaces. In this paper, we prove that extremal surfaces have many good properties. For example, the boundary of an extremal surface could always be partitioned into finitely many convex circular arcs, with a common geodesic curvature (simply called the boundary geodesic curvature), and the end points of each of these circular arcs must be in E_q. The main result of this paper is an equality that H₀(E_q) or h₀(E_q) is exactly (q-2) divided by the limit of the boundary geodesic curvatures of extremal surfaces. This relation also gives a fast numerical algorithm to compute h₀({-1,0,1})=4.034...

Hamed Najafi, Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran (hamednajafi20@gmail.com).
A converse of the characterization of operator geometric means, pp. 497-508.
ABSTRACT. In this paper, we used the closed convex hull of unitary orbit of positive operators and completely positive linear maps to investigate reverse of the characterization of operator geometric means by positive block matrices.

Rim Abid, University of Tunis-El Manar, 2092-El Manar, Tunisia, and Karim Boulabiar, University of Tunis-El Manar, 2092-El Manar, Tunisia (karim.boulabiar@ipest.rnu.tn).
Algebras of disjointness preserving operatcors on Banach lattices, pp. 509-524.
ABSTRACT. Let A be an algebra of disjointness preserving operators on a Banach lattice X. We shall characterize the set of all nilpotent operators in A and we will deduce that if A is semiprime (i.e., with no nonzero nilpotent elements) then A is automatically commutative. This will lead us to show that if A is semiprime then A has an isometrically-isomorphic copy in the center of some Banach lattice.

Stammeier, Nicolai, Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0216 Oslo, Norway (nicolsta@math.uio.no).
Topological freeness for *-commuting covering maps, pp. 525-551.
ABSTRACT. We prove a close connection between *-commutativity and independence of group endomorphisms as considered by Cuntz-Vershik. This motivates the study of a family of *-commuting surjective local homeomorphisms of a compact Hausdorff space. Inspired by Ledrappier's shift, we describe interesting new examples related to cellular automata. To every such family we associate a universal C*-algebra that we then identify as the Cuntz-Nica-Pimsner algebra of a product system of Hilbert bimodules. This allows us to extend a result of Meier-Carlsen and Silvestrov which yields an application for irreversible algebraic dynamical systems.

Beanland, Kevin, Washington and Lee University, Lexington, VA 23220 (beanlandk@wlu.edu) and Kania, Tomasz, University of Warwick, Coventry, UK (tomasz.marcin.kania@gmail.com) and Laustsen, Niels Jakob, Fylde College, Lancaster University, Lancaster, UK (n.laustsen@lancaster.ac.uk).
The algebras of bounded operators on the Tsirelson and Baernstein spaces are not Grothendieck spaces, pp. 553-566.
ABSTRACT. We present two new examples of reflexive Banach spaces X for which the associated Banach algebra B(X) of bounded operators on X is not a Grothendieck space, namely X = T (the Tsirelson space) and X = B_p (the pth Baernstein space) for 1<p< infinity.

Timothy Ferguson, Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487 (tjferguson1@ua.edu).
Bounds on integral means of Bergman projections and their derivatives, pp. 567-588.
ABSTRACT. We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted L^p norms of certain derivatives of the original function in the &theta direction. These results easily imply the well known result that the Bergman projection is bounded from the Sobolev space W^k,p into itself for 1 < p < ∞. We also apply our results to derive certain regularity results involving extremal problems in Bergman spaces. Lastly, we construct a function that approaches 0 uniformly at the boundary of the unit disc but whose Bergman projection is not in H².

Mendoza, José M., Universidad Federal de São Carlos, São Carlos, Brazil (josearanda@dm.ufscar.br).
Existence of solutions for a nonhomogeneous semilinear fractional Laplacian problems, pp. 589-599.
ABSTRACT. In this paper we give existence results for a nonhomogeneous semilinear fractional laplacian problems with local coercivity in euclidean bounded domains using variational methods.

Włodzimierz J. Charatonik and Sahika Sahan, Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 West 12th St., Rolla, MO, 65409-0020 (wjcharat@mst.edu}, (ssxx4@mst.edu)
Zero-dimensional spaces homeomorphic to their Cartesian squares, pp. 601-608.
ABSTRACT. We show that there exists uncountably many zero-dimensional compact metric spaces homeomorphic to their cartesian squares as well as their n-fold symmetric products.

Roshan Adikari and Wayne Lewis, Department of Mathematics and Statistics; Texas Tech University, Lubbock, Texas 79409 (roshan.adikari@ttu.edu), (wayne.lewis@ttu.edu).
Endpoints of nondegenerate hereditarily decomposable chainable continua, pp. 609-624.
ABSTRACT. We show that a nondegenerate hereditarily decomposable chainable continuum must have a pair of opposite endpoints and use this result to investigate more on endpoints of such continua.

Liang-Xue Peng (Corresponding author), Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn) and Pei Zhang, Beijing University of Technology, Beijing 100124, China (zp19872016@163.com).
Some properties of bounded sets in certain topological spaces, pp. 625-646.
ABSTRACT. In the first part of this article we give some sufficient conditions under which a bounded set in a topological space (paratopological group) X is strongly bounded in X (p-bounded in X for every p∈ω^∗). We show that if X is a first-countable topological space, then every bounded subset of X is strongly bounded in X. If G is a paratopological group and for every U∈Ν(e) there exists a continuous homomorphism p_U:G→H_U onto a first-countable Hausdorff paratopological group H_U such that p_U^-1(Cl(V_U))⊂ Cl(U) for some neighborhood V_U of the neutral element of H_U, then any bounded set in G is strongly bounded in G. We also give some sufficient conditions under which a semitopological group G satisfies that for any U∈Ν(e) there exists a continuous homomorphism p_U:G→H_U onto a first-countable Hausdorff semitopological group H_U such that p_U^-1(Cl(V_U))⊂Cl(U) for some neighborhood V_U of the neutral element of H_U. In the last part of this note, we give some equivalent conditions for a bounded set in a Tychonoff topological space. We finally show that every pseudocompact submetacompact space is weakly Lindelöf.