## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 41, No. 2 , 2015

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)

Contents

Rush, David E., University of California at Riverside, CA 92521 (rush@math.ucr.edu).
Modules, lattice modules and the set of congruences on a commutative monoid, pp. 367-382.
ABSTRACT.Let S be a commutative multiplicative monoid with a zero element. It is shown that the set of congruences on S is a lattice module over the multiplicative lattice of ideals of S. This gives a method of obtaining results on the set of congruences on a monoid by extending results on modules to lattice modules, and shows the relevance of examples and results on the set set of congruences on a monoid to the development of results on lattice modules. We illustrate these two aspects by extending specific results on modules to lattice modules, to get corresponding results on the set of congruences on a monoid, and by extending a well-known result of Drbohlav on the set of congruences on a monoid, to obtain a more general primary decomposition theorem on lattice modules than those in the literature.

May, Coy L., Towson University, Baltimore, MD 21252 (cmay@towson.edu).
M*-simple groups, pp. 383-396.
ABSTRACT. Let X be compact bordered Klein surface of algebraic genus g at least 2. Then X has at most 12(g - 1) automorphisms. A bordered surface for which the bound is attained is said to have maximal symmetry, and its full automorphism group is called an M*-group. In general, a quotient of an M*-group is again an M*-group, and it is natural to consider the notion of an M*-simple group, that is, an M*-group with no proper M*-quotient. Here we classify the non-solvable M*-simple groups with dihedral Sylow 2-subgroups. This in turn yields the classification of the M*-simple groups of even real genus. Finally, we apply our results to classify the M*-simple groups of real genus at most 600.

Cahen, Paul-Jean, Université Paul Cézanne, 13397 Marseille Cedex 20, France (pauljean.cahen@gmail.com), Dobbs, David E., University of Tennessee, Knoxville, TN 37996-1320, USA (dobbs@math.utk.edu), dobbs@math.utk.edu, and Lucas, Thomas G., University of North Carolina Charlotte, Charlotte, NC 28223, USA (tglucas@uncc.edu).
Valuative pairs of commutative rings, pp. 397-431.
ABSTRACT. The concept of a valuative domain is generalized in several ways to the context of a ring extension R ⊆ T. For instance, R is said to be weakly T-valuative (resp., a mock valuation ring of T) if, whenever xy ∈R with x,y ∈T, then at least one of the extensions R ⊆ R[x] and R ⊆ R[y] has no proper intermediate rings (resp., then either x ∈ R or y ∈ R). R is a mock valuation ring of T if and only if R is weakly T-valuative, R is integrally closed in T and at least one prime ideal P of R is such that the large quotient ring R(T,P) equals R. If R is weakly T-valuative, then R has at most three maximal ideals M such that M is the radical of (R:R x) for some x ∈ T. If R is a mock valuation ring of T, then R has at most one such M. If R is weakly T-valuative and has three such M, then R is integrally closed in T and there are (exactly) three minimal ring extensions of R inside T. If R is weakly T-valuative but not a mock valuation ring of T, then R has at least one M of the above form such that R ⊆ R[x] is a minimal ring extension. If R is weakly T-valuative but not integrally closed in T, then R has at most two such M such that R ⊆ R[x] is a minimal ring extension, and R has a unique such M such that R ⊆ R[x] is an integral minimal ring extension. If R is weakly T-valuative, then R(T,P) is a mock valuation ring of T for all but at most one prime ideal P of R; and such an exceptional P exists if and only if R is not integrally closed in T.

Shun Maeta, Division of Mathematics, Shimane University, Nishikawatsu 1060 Matsue, 690-8504 (shun.maeta@gmail.com).
Construction of triharmonic maps, pp. 433-444 .
ABSTRACT. Theory of harmonic maps has been applied into various fields in differential geometry. By extending the notion of harmonic maps, J. Eells and L. Lemaire introduced polyharmonic maps of order k. In 1989, S. B. Wang showed the Euler Lagrange equation of polyharmonic maps of order 3 (triharmonic maps). In this paper, we study triharmonic immersion into a sphere. We show the necessary and sufficient condition of triharmonic isometric immersion, and give some non trivial examples. Moreover, we also construct non harmonic triharmonic maps.

Ding-Gong Yang, School of Mathematical Science, Soochow University, Suzhou 215006, P.R. China, and Jin-Lin, Liu, Department of Mathematics, Yangzhou University, Yangzhou 225002, P.R. China (jlliu@yzu.edu.cn).

ABSTRACT. We introduce two new subclasses Hp,k(λ,A,B) and Kp,k(λ,A,B) of analytic and p-valent functions which are starlike with respect to k-symmetric points. Distortion bounds, inclusion relations, integral transforms and convolution properties for these classes are studied.

Yasheng Ye, Dept. of Math., University of Shanghai for Science and Technology, Shanghai 200093, P. R. China (yashengye@aliyun.com)Lei Shi, Dept. of Math., East China Normal University, Shanghai 200062, P. R. China (sishimath@163.com), and  Xuecheng Pang, Dept. of Math., East China Normal University, Shanghai 200062, P. R. China (xcpang@math.ecnu.edu.cn).
Normal families of holomorphic curves into Pn(C) for moving targets,  pp. 471-490.
ABSTRACT. In this paper, we obtain a Montel-type criterion for normal families of holomorphic curves into complex projective space for moving hyperplanes in general position. This result improves a previous result for moving hyperplanes in pointwise general position.. Peiyan

Peiyan Niu, Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R.China and Department of Mathematics, Anhui Science and Technology University, Chuzhou 233100, P.R.China (niupeiyan1983@126.com) and Yan Xu (Corresponding author), Institute of mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R.China (xuyan@njnu.edu.cn)/a>.
Normality concerning shared functions, pp. 481-490.
ABSTRACT. In this paper, we characterize non-normal sequences in a family of meromorphic functions, and give a normality criterion concerning shared functions, which extends a result of Chang.

Ricardo M. Martins and Marco A. Teixeira, Instituto de Matemática, Estatistica e Computação científica, Universidade Estadual de Campinas, 13083-859, Campinas/SP, Brasil. (rmiranda@ime.unicamp.br),
Minimal sets in double-perturbed differential equations, pp. 491-512.
ABSTRACT. In this article we study minimal sets of coupled systems of second order differential equations. We use methods in Averaging Theory, Lyapunov-Schmidt and Normal Forms Theory to analyze the birth of cylinders and tori in such systems. We provide applications of the developed methodology to study the existence of minimal sets in perturbations of some normal forms, and in a class of weakly coupled oscillators.

Ricardo Gallego Torromé and Paolo Piccione, Departamento de Matemática, Instituto de Matemática e Estatística Rua do Matão, 1010 São Paulo - SP - Brazil (rigato39@gmail.com), (piccione.p@gmail.com).
On the Lie group structure of pseudo-Finsler isometries, pp. 513-521.
ABSTRACT. Using an extension to isometries of the associated Sasaki structure, we establish a Lie transformation group structure for the set of isometries of a pseudo-Finsler conical metric.

Chimenton, Alessandro G., Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brazil (matgaio@gmail.com), Gomes, José Barbosa, Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil (barbosa.gomes@ufjf.edu.br), and Ruggiero, Rafael O., Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, Brazil (rorr@mat.puc-rio.br).
Transitivity of Finsler geodesic flows of compact surfaces without conjugate points and higher genus, and applications to Finsler rigidity problems, pp. 523-551.
ABSTRACT.  We show that if (M,F) is a C k-basic Finsler compact surface without conjugate points and genus greater than one whose Green bundles are continuous then (M,F) is Riemannian. The proof combines Riemann-Finsler geometry, the theory of Lagrangian bundles and the transitivity of Finsler geodesic flows of compact surfaces of genus greater than one. The transitivity of geodesic flows of compact Finsler surfaces without conjugate points and higher genus generalizes the work of P. Eberlein for compact Riemannian surfaces without conjugate points in the context of visibility manifolds.

Don Hadwin, University of New Hampshire (www.unh.edu), (don@unh.edu) Arkady Kitover, Community College of Philadelphia (www.ccp.edu)(akitover@ccp.edu), and  Mehmet Orhon, University of New Hampshire (www.unh.edu), (mo@unh.edu).
ABSTRACT. The classic result of Perron and Frobenius states that if  A and B  are matrices with nonnegative elements, such that A ≤ B,   A is irreducible, and the spectral radii of A  and B are equal then A = B . We extend this result to a large class of band irreducible positive operators on a large class of Banach lattices and provide examples to show that the conditions we put on the operators and the Banach lattices cannot be weakened.

Niteesh Sahni, University of Delhi, Delhi, 110007 and Shiv Nadar University, Dadri, Uttar Pradesh, India (niteeshsahni@gmail.com) and Dinesh Singh, University of Delhi, Delhi, 110007, India (dineshsingh1@gmail.com).
Lax-Halmos type theorems on Hp spaces, pp. 571-587.
ABSTRACT. In this paper we characterize for 0p that are invariant under multiplication by all powers of a finite Blaschke factor B, except the first power. Our result clearly generalizes the invariant subspace theorem obtained by Paulsen and Singh ( Modules over subalgebras of the disk algebra, Indiana Univ. Math. Jour. 55 (2006), 1751-1766) which has proved to be the starting point of important work on constrained Nevanlinna-Pick interpolation. Our method of proof can also be readily adapted to the case where the subspace is invariant under all positive powers of B(z) .The two results are in the mould of the classical Lax-Halmos Theorem and can be said to be Lax-Halmos type results in the finite multiplicity case for two commuting shifts and for a single shift respectively.

T. S. S. R. K. Rao, Stat-Math Unit, Indian Statistical Institute, R. V. College P.O. Bangalore 560059, India,
(tss@isibang.ac.in), (srin@fulbrightmail.org).
On intersections of ideals in Banach spaces, pp. 589-594.
ABSTRACT. The notion of an ideal in a Banach space was introduced by Godefroy, Kalton and Saphar. In this short note we are interested in studying finite intersections of ideals in Banach spaces. We show that for a Banach space X, if in the bidual X**, every ideal of finite codimension is the intersection of ideals of codimension one, then the same property holds in X.

Dosev, Detelin, Oklahoma State University, Department of Mathematics, Stillwater, OK 74075 (dosev@okstate.edu).
On  a class  of  operators  on  C(K), pp. 595-610.
ABSTRACT. We consider certain classes of operators on C(K) (including C(K)-strictly singular operators and sums of order homomorphism) and show that all operators in these classes are commutators. We use a theorem of Kalton for representation of a Borel measure on a compact space to prove our results. This work is part of the ongoing effort to classify the commutators on the Banach spaces which have Pelczynski decomposition.

Yixin Yang, Department of Mathematics and Statistics, SUNY At Albany, Albany, NewYork 12222 (yangyx314272@gmail.com).
Commutator on Hardy submodule over bidisc, pp. 611-620.
ABSTRACT. The commutator plays an important role in the study of submodule of the Hardy space over bidisc. In this paper, we will define some numerical invariants through the (self)cross-commutator and reveal their relationships. An index formula for the two variables Jordan block is obtained. We also give a matrix representation for the core operator and the representation of rank of the core operator. Some examples are given in the last section.

Richard C. Bradley, Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA (bradleyr@indiana.edu) and Zbigniew J. Jurek, Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-386 Wroclaw, Poland (zjjurek@math.uni.wroc.pl).
On a central limit theorem for shrunken weakly dependent random variables, pp. 621-638.
ABSTRACT. A central limit theorem is proved for some strictly stationary sequences of random variables that satisfy certain mixing conditions and are subjected to certain "shrinking operators''.  For independent, identically distributed random variables, this result was proved earlier by Housworth and Shao.

Müller, Paul F.X., Institute of Analysis, Johannes Kepler University, Linz, Austria (pfxm@bayou.uni-linz.ac.at) and Penteker, Johanna, Institute of Analysis, Johannes Kepler University, Linz, Austria (johanna.penteker@jku.at).
p-summing multiplication operators, dyadic Hardy spaces and atomic decomposition, pp. 639-668.
ABSTRACT. We constructively determine the Pietsch measure of a 2-summing multiplication operator acting on the Haar system in the dyadic Hardy spaces. The input for our construction is the atomic decomposition of an element in the dyadic Hardy spaces.

Banič, Iztok, Department of Mathematics, University of Maribor, 2000 Maribor, Slovenia (iztok.banic@uni-mb.si), and Martínez-de-la-Vega, Verónica, Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico (vmvm@matem.unam.mx).
Universal dendrite D3 as a generalized inverse limit, pp. 669-682.
ABSTRACT. We construct an upper semi-continuous function f from [0,1] to 2[0,1] such that the inverse limit of inverse sequence of intervals [0,1] and f as the only bonding function is homeomorphic to the Universal Dendrite D3.

Eiichi, Matsuhashi, Shimane University, Matsue, Shimane 690-8504, Japan (matsuhashi@riko.shimane-u.ac.jp) .
Whitney preserving maps onto certain Peano continua,  pp. 683-699.
ABSTRACT. Espinoza (2002) proved that a map f  from a continuum containing a dense arc component onto the unit interval I is a homeomorphism if and only if f  is a Whitney preserving map. The author (2012) showed that this result is also true when I is changed to a dendrite with a closed set of branch points. However it is known that  Espinoza's result is not true when I is changed to a dendrite with a dense set of branch points or a graph which admits an Eulerian path. In this paper we prove that if Y is a nondegenerate connected finite polyhedron without local cut-points (or a compact connected n-dimensional manifold (n > 1), or a 1-dimensional Peano continuum without free arcs), then there exist a continuum Z containing a dense arc component and a Whitney preserving map f from Z onto Y such that f is not a homeomorphism. Also, we prove some results which yield a lot of examples of Whitney preserving maps.

Ito, Noboru, Waseda Institute for Advanced Study, Tokyo, 169-8050, Japan (noboru@moegi.waseda.jp) and Takimura, Yusuke, Gakushuin Boys' Junior High School, Tokyo, 171-0031, Japan (Yusuke.Takimura@gakushuin.ac.jp).
Sub-chord diagrams of knot projections, pp. 701-725.
ABSTRACT. A chord diagram is a circle with paired points with each pair of points connected by a chord.  Every generic immersed spherical curve provides a chord diagram by associating each chord with two preimages of a double point.  Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves.  This study counts the difference in the numbers of sub-chord diagrams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1.1).  This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 1.2, Table): there has been no such invariant before.  The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence consisting of RI and strong RIII (Theorem 1.3).  Moreover, invariants of spherical curves under flypes are provided by counting sub-chord diagrams (Theorem 1.4).