Electronic Edition Vol. 38, No. 2 , 2012

Editors: G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), D. Blecher (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), J. A. Johnson (Houston), W. B. Johnson (College Station),  V. I. Paulsen (Houston), M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


Azizi, A., Shiraz University, 71457-44776, Shiraz, Iran (aazizi@shirazu.ac.ir) and Nikseresht, A., Shiraz University, 71457-44776, Shiraz, Iran (ashkan_nikseresht@yahoo.com).
Simplified radical formula in modules, pp. 333-344.
ABSTRACT  Let R be a commutative ring with identity and B a submodule of an R-module M. The intersection of all prime submodules of M containing B is denoted by rad(B). We say that a module M satisfies the simplified radical formula, when for every submodule B of M and each x in rad(B), x=rm+b, where r, m and b are elements of R, M and B, respectively and rkm is in B, for some natural number k. Also it is said that a ring R satisfies the simplified radical formula, if every R-module satisfies the simplified radical formula. It is shown that a Noetherian ring satisfies the simplified radical formula if and only if it is a ZPI-ring, and we prove that every one dimensional valuation domain satisfies the simplified radical formula. Furthermore we will characterize zero dimensional local rings which satisfy the simplified radical formula. Moreover it is proved that every serial module satisfies the simplified radical formula.

Etayo, J. J., Departamento de Álgebra, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain (jetayo@mat.ucm.es) and Martínez, E., Departamento de Matemáticas Fundamentales, UNED, Paseo Senda del Rey 9, 280-40- Madrid, Spain (emartinez@mat.uned.es).
The symmetric crosscap number of the families of groups DC3× Cn and A4× Cn, pp. 345-358.
ABSTRACT. Every finite group G acts as an automorphism group of some non-orientable Klein surfaces without boundary. The minimal genus of these surfaces is called the symmetric crosscap number and denoted by σ˜(G). The systematic study about the symmetric crosscap number was begun by C. L. May who also calculated it for certain finite groups. It is known that 3 cannot be the symmetric crosscap number of a group. Conversely, all integers non-congruent with 3 or 7 modulo 12 are the symmetric crosscap number of some group. Here we obtain the symmetric crosscap number for the families of groups DC3× Cn and A4× Cn and we prove that their values cover a quarter of the numbers congruent with 3 modulo 12 and three quarters of the numbers congruent with 7 modulo 12. As a consequence there are only five integers lower than 100 which are not known if they are the symmetric crosscap number of some group.

Kim, Hwankoo, Hoseo University, Asan 336-795, Korea (hkkim@hoseo.edu) and Wang, Fanggui, Sichuan Normal University, Chengdu 610068, China.
On φ-strong Mori rings, pp. 359-371.
ABSTRACT.We introduce a new class of rings which is closely related to the class of strong Mori domains which was introduced by the second author. Let H = {R | R is a commutative ring with 1 ≠ 0 and the nilradical n(R) is a divided prime ideal of R}. Let R∈ H and let T(R) be the total quotient ring of R and define φ from T(R) to R localized at n(R) by φ( a/b ) = a/b for every a ∈ R and regular element b of R. A nonnil ideal I is a φ-w-ideal if φ(I) is a w-ideal of φ(R) and a φ-ring R is called a φ-SM ring if it satisfies the ascending chain condition on φ-w-ideals. We show that the theory of φ-SM rings resembles that of strong Mori domains.

Vincenzo De Filippis, DI.S.I.A., Faculty of Engineering University of Messina, Contrada Di Dio, 98166, Messina, Italy (defilippis@unime.it) and Feng Wei, School of Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China (daoshuo@bit.edu.cn), (daoshuo@hotmail.com).
Posner's second theorem for skew derivations on multilinear polynomials on left ideals, pp. 373-395.
ABSTRACT. Let R be a prime ring of characteristic different from 2 with symmetric Martindale quotient ring Q and extended centroid C and let I be a nonzero left ideal of R. Suppose that μ is a nonzero skew derivation of R with associated automorphism α and that f(x1,...,xn) is a multilinear polynomial over C with n non-commuting variables. If [μ(f(r1,...,rn)] is in Z(R) for all r1,...,rn in I , then there exists an idempotent element e in Q such that RCe=IC and f(x1,...,xn) is central valued on eRce.

Awtrey, Chad Elon University, Elon, NC 27244 (cawtrey@elon.edu).
Masses, discriminants, and Galois groups of tame quartic and quintic extensions of local fields, pp. 397-404.
ABSTRACT.  Let K be a finite extension of the p-adic numbers with p not equal to 2 or 5, and let LK be a ramified extension. We count the number of nonisomorphic extensions where the Galois group of the splitting field of L is equal to one of the ten transitive subgroups of S4 and S5.

Constales, Denis, Department of Mathematical Analysis, Ghent University, and Laboratory for Chemical Technology, Ghent University, Belgium (Denis.Constales@UGent.be), Krausshar, Soeren, Fachbereich Mathematik, Technische Universität Darmstadt, Germany (Krausshar@mathematik.tu-darmstadt.de) and Ryan, John, Department of Mathematics, University of Arkansas, AR 72701, USA (jryan@uark.edu).
Hyperbolic Dirac and Laplace operators on examples of hyperbolic spin manifolds, pp. 405-420.
ABSTRACT. Fundamental solutions of hyperbolic Dirac operators and hyperbolic versions of the Laplace operator are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of Rn by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein and Poincaré type series. As main goal we develop Cauchy and Green type integral formulas and describe Hardy space decompositions for spinor sections of the associated spinor bundles on these manifolds.

Peter Hästö, Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu, Finland (peter.hasto@helsinki.fi), Zair Ibragimov,  Department of Mathematics, California State University, Fullerton, CA 92831, USA (zibragimov@fullerton.edu), and David Minda, Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221-0025, USA (minda@ucmail.uc.ed).
Convex sets of constant width and 3-diameter, pp. 421-443.
ABSTRACT. In this article we introduce the notion of a 3-diameter of planar sets of constant width. We obtain analogues of the isodiametric inequality and the Blaschke--Lebesgue Theorem for 3-diameter of constant width sets. Namely, we prove that among all the sets of given constant width, disks have the smallest 3-diameter and Reuleaux triangles have the largest 3-diameter. We also discuss sets (called d_3 complete) where addition of another point must increase the 3-diameter. The notion of a constant 3-diameter set is introduced, and we prove that infinitely many noncircular examples exist, and one is given explicitly.

Rataj, Jan and Zajíĺček, Luděk, Charles University, Faculty of Mathematics and Physics, Sokolovska 83, 186 75 Praha 8, Czech Republic (rataj@karlin.mff.cuni.cz), (zajicek@karlin.mff.cuni.cz).
Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces, pp. 445-467.
ABSTRACT. Let F⊂Rn be a closed set and n=2 or n=3. S. Ferry (1975) proved that then, for almost all r>0, the level set (distance sphere, r-boundary) Sr(F):={x∈Rn:  dist(x,F)=r} is a topological (n-1)-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's result is an easy consequence of the only fact that the distance function d(x)=dist(x,F) is locally DC and has no stationary point in Rn\F. Using this observation, we show that Ferry's (and even Fu's) result extends to sufficiently smooth normed linear spaces X with dim X∈{2,3} (e.g., to lpn, n=2,3, p≥2), which improves and generalizes a result of R. Gariepy and W.D. Pepe (1972). By the same method we also generalize Fu's result to Riemannian manifolds and improve a result of K. Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.

Juan de Dios Pérez, Department of Geometry and Topology, University of Granada, 18071 Granada, Spain (jdperez@ugr.es), Jung Taek Oh and Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea (yjsuh@knu.ac.kr).
Compact real hypersurfaces in complex two-plane Grassmannians, pp. 469-492.
ABSTRACT. In this paper we give some characterizations of real hypersurfaces of type A in complex two-plane Grassmannians G2 C( m+2) that are tubes over a totally geodesic G2 C( m+1) in G2 C( m+2) in terms of the squared norm of the covariant derivatives of the shape operator A by estimating the inequality of the Laplacian defined on compact real hypersurfaces in G2 C( m+2)

Guoping, Zhan, Department of Mathematics, Nanjing University, Nanjing , 210093 P. R. China (dg0921014@smail.nju.edu.cn), Liangwen, Liao, Department of Mathematics, Nanjing University ,Nanjing, 210093 P. R. China (maliao@nju.edu.cn).
 Area of non-escaping parameters of the sine family, pp. 493-524.
ABSTRACT.  We investigate the dynamics of the sine family on the parameter plane. We consider those non-escaping parameters, whose iteration do not go to infinity. We prove that the area of the set of all non-escaping parameters is finite in any vertical strip of finite width. .

Veech, William A., Mathematics Department, MS 136, Rice University, 6100 S. Main St., Houston, TX, 77005 (veech@rice.edu).
Martin boundary for the similarity walk in a planar triangle , pp. 525-548
ABSTRACT. Computation of the limiting distribution of a natural random walk on a given planar triangle is employed along with the Choquet and (sub)martingale theorems to achieve a unique "Poisson" representation for each element of the cone of nonnegative harmonic functions for this walk. .

Nguyen Thieu Huy, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology 1-Dai Co Viet, Hanoi, Vietnam. (huynguyen@mail.hut.edu.vn) and Rainer Nagel, Universität Tübingen, Mathematisches Institut, auf der Morgenstelle 10, 72076 Tübingen, Germany (rana@fa.uni-tuebingen.de).
Exponentially dichotomous generators of evolution bisemigroups on admissible function spaces,  pp. 549-569.
ABSTRACT. To an evolution family U=(U(t,s))t≥ s≥ 0 of bounded operators on a Banach space X and through the integral equation u(t)=U(t,s)u(s)+∫st U(t,ξ)f(ξ)dξ, we associate an operator GZ acting on Banach spaces of X-valued functions corresponding to admissible Banach function spaces. These spaces contain the Lp spaces (1≤ p<∞), the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We will show that the exponential dichotomy of U is equivalent to the exponential dichotomy of the operator GZ generating a bisemigroup (T(t))t ∈R. We also prove that the exponential dichotomy of GZ is robust under small perturbations by bounded operators. This leads to applications to vector-valued Wiener-Hopf and to Riccati equations.

Dong-Ni Tan, Department of Mathematics, Tianjin University of Technology, Tianjin 300384, and School of Mathematical Science, Nankai University, Tianjin 300071, China (0110127@mail.nankai.edu.cn)
Isometries of the unit spheres of the Tsirelson space T and the modified Tsirelson space TM , pp. 571-581.
ABSTRACT. We characterize surjective isometries of the unit spheres of the Tsirelson space T and the modified Tsirelson space TM. Applying the results we give an affirmative answer for spaces T and TM to Tingley's problem whether every onto isometry between the unit spheres of two real normed spaces is necessarily the restriction of a linear or affine map on the whole space.

Karassev, Alexandre, Dept. of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada (alexandk@nipissingu.ca), Krupski, Paweł, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (krupski@math.uni.wroc.pl), Todorov, Vladimir, Dept. of Mathematics, UACG, 1 H. Smirnenski blvd., 1046 Sofia, Bulgaria (vtt-fte@uacg.bg), and Valov, Vesko, Dept. of Computer Science and Mathematics, Nipissing University, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7, Canada (veskov@nipissingu.ca).
Generalized Cantor manifolds and homogeneity pp. 583-609.
ABSTRACT. A classical theorem of Alexandroff states that every n-dimensional compactum X contains an n-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and Vn-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class C is a strong Cantor manifold (or at least a Cantor manifold) with respect to C. Here, the class C is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered.

Akira Koyama, Department of Mathematics, Faculty of Science, Shizuoka University, Suruga, Shizuoka, 422-8529, Japan, sakoyam@ipc.shizuoka.ac.jp, Józef Krasinkiewicz, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland, jokra@impan.pl, and Stanisław Spież, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warsaw, Poland, spiez@impan.pl.
Embeddings into products and symmetric products – an algebraic approach, pp. 611-641.
ABSTRACT. In this article we mostly study algebraic properties of n-dimensional “cyclic” compacta lying either in products of n curves or in the nth symmetric product of a curve. The basic results have been obtained for compacta admitting essential maps into the n-sphere. One of the main results asserts that if a compactum X admits such a mapping and X embeds in a product of n curves then there exists an algebraically essential map from X into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of an algebraically essential mapping from X into the n-torus is equivalent to the existence of some elements a1,…,an in H¹(X) whose cup product is not zero, and implies that rank of H¹(X)≥n and catX>n. In particular, it follows that the n-sphere, n>1, is not embeddable in the nth symmetric product of any curve. The case n=2 answers in the negative a question of Illanes and Nadler.

Mańka, Roman, Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O.Box 21, 00-956 Warszawa, Poland (r.manka@impan.pl).
Locally connected curves admit small retractions onto graphs, pp. 643-651.
ABSTRACT. We prove that for every locally connected 1-dimensional metric continuum and for every ε>0 there exists a graph which is an ε-retract of the continuum. For the proof we give, among other things, an introductory exposition of uniformly locally arcwise connected sets in an arbitrary metric space.