*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 r^{k}m 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 DC_{3}×
C_{n} and A_{4}× C_{n}, 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 DC_{3}×
C_{n } and A_{4}× C_{n }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(x_{1},...,x_{n}) is a multilinear polynomial over C with n non-commuting variables. If [μ(f(r_{1},...,r_{n})] is in Z(R) for all r_{1},...,r_{n} in I , then there exists an idempotent element e in Q such that RCe=IC and f(x_{1},...,x_{n}) is central valued on eRce.

Masses, discriminants, and Galois groups of tame quartic and quintic extensions of local fields, pp. 397-404.

ABSTRACT. Let

**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⊂R^{n} 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)
S_{r}(F):={x∈R^{n}: 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 R^{n}\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
l^{p}_{n}, 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 G_{2} C(^{ m+2})
that are tubes
over a totally geodesic G_{2} C(^{ m+1}) in G_{2} 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 G_{2} C(^{ m+2})

**Guoping, Zha**n, 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. .

Exponentially dichotomous generators of evolution bisemigroups on admissible function spaces, pp. 549-569.

ABSTRACT. To an evolution family

**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 T_{M}
,
pp. 571-581.

ABSTRACT.
We characterize surjective isometries of the unit spheres of the
Tsirelson space T and the modified Tsirelson space T_{M}. Applying
the results we give an affirmative answer for spaces
T and T_{M}
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 *n*th
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 *n*th 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
*a*_{1},…,*a _{n}* in

**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.