HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 36, No. 1, 2010

Editors: G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), H. Brezis (Paris), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (Radboud), 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

Chiang-Hsieh, Hung-Jen, Dept. of Math., National Chung Cheng University, Chia-Yi 621, Taiwan (hchiang@math.ccu.edu.tw), Smith, Neal O., Dept. of Math., Augusta State University, Augusta GA 30904 (nsmith12@aug.edu), and Wang, Hsin-Ju, Dept. of Math., National Chung Cheng University, Chia-Yi 621, Taiwan (hjwang@math.ccu.edu.tw).
Commutative rings with toroidal zero-divisor graphs, pp. 1-31.
ABSTRACT. We investigate the genus number of compact Riemann surface in which the zero-divisor graphs of commutative rings can be embedded and explicitly determine all finite commutative rings (up to isomorphism) such that their zero-divisor graphs are either planar or toroidal.

Dobbs, David E., University of Tennessee, Knoxville, TN 37996-1300 (dobbs@math.utk.edu).
When a minimal overring is a going-down domain , pp. 33-42.
ABSTRACT. Let R contained in T be a minimal ring extension of (commutative integral) domains. If R is integrally closed in T, then R is a going-down domain if and only if T is a going-down domain. The preceding assertion can be generalized to the context of weak Baer going-down rings. If R is integrally closed and T is a Prufer domain, then R is a Prufer domain. If T is integral over R and T is a going-down domain, then R is a going-down domain if and only if the extension R contained in T satisfies the going-down property. An example is given of an integral minimal overring extension R contained in T of two-dimensional domains such that T is a going-down (in fact, Prufer) domain and R is a treed domain which is not a going-down domain.

López-Ramos, Juan A., Department of Algebra and Analysis, University of Almería, 04120, Almería, Spain (jlopez@ual.es).
Relative spectral sequences with applications to Gorenstein dimensions. , pp. 43-53.
ABSTRACT. We prove that Grothendieck's classical results on spectral sequences can be extended to derived functors relative to complete cotorsion theories and use them to get applications on Gorenstein dimensions of modules.

Bin Chen, Department of Mathematics, Zhejiang University, Hangzhou,  P.R.China, 310028 (chenbinn@zju.edu.cn) and Lili Zhao, Department of Mathematics, Shanghai Jiaotong University, Minhang, Shanghai, P.R.China, 200240 (zhaolili@sjtu.edu.cn).
Randers metrics of sectional flag curvature, pp. 55-67.
ABSTRACT. A Finsler metric is of sectional flag curvature if its flag curvature depends only on the section. In this article, we characterize Randers metrics osectional flag curvature. It is proved that any non-Riemannian Randers metric of sectional flag curvature must have constant flag curvature if the dimension is greater than two.

Abbassi, Mohamed T.K.,  Département des Mathématiques, Université Sidi Mohamed Ben Abdallah, B.P. 1796, Fés, Morocco (mtk_abbassi@Yahoo.fr),  Calvaruso, Giovanni and Perrone, Domenico, Dipartimento di Matematica "E. De Giorgi", Universitá del Salento, Lecce, Italy (giovanni.calvaruso@unisalento.it), (domenico.perrone@unisalento.it).
Harmonic maps defined by the geodesic flow, pp. 69-90.
ABSTRACT.  Let (M,g) be a Riemannian manifold. We equip the unit tangent sphere bundle T₁ M of (M,g) and its unit tangent sphere bundle T_r T₁M of radius r>0 with arbitrary Riemannian g-natural metrics. When (M,g) is two-point homogeneous and both T₁ M and T₁T₁M are equipped with the Sasaki metrics, the geodesic flow vector field is harmonic and determines a harmonic map  [E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles,  Diff. Geom. Appl., 13  (2000), 77-93]. We prove that if arbitrary Riemannian g-natural metrics are considered, then the geodesic flow is still a harmonic vector field, and it also defines a harmonic map under some conditions on the g-natural metrics. This permits to exhibit large families of harmonic maps defined in a compact Riemannian manifold and having a target space with a highly nontrivial geometry. In particular, explicit examples are provided on the unit tangent sphere bundle of the sphere S ⁿ and the flat torus Tⁿ. Moreover, the geodesic flow being a Killing vector field is characterized in terms of harmonicity of the corresponding map and of properties of the base manifold.

Pedro L.Q. Pergher, Universidade Federal de São Carlos, Departamento de Matemática, Rodovia Washington Luiz, km. 235, Caixa Postal 676, CEP 13.565-905, São Carlos, SP, Brazil (pergher@dm.ufscar.br), Hemant K. Singh, and Tej B. Singh, Department of Mathematics, University of Delhi, Delhi 110 007, India (hksinghdu@gmail.com), (tbsingh@maths.du.ac.in) .
On Z₂ and S¹ free actions on spaces of cohomology type (a,b), pp. 137-146.
ABSTRACT. Let S¹ be the group of unitary complex numbers endowed with the complex multiplication, and Z₂ the cyclic group of order two. If X is a topological space, denote by Hⁱ(X,G) its i-th cohomology group with coefficients in G. For a natural number n, suppose X is a simply connected finite CW complex satisfying H^j (X,Z)=Z if j=0, n, 2n or 3n, and H^j (X,Z)=0 otherwise; here, Z is the group of integers. Let u₁, u₂ and u₃ generate Hⁿ(X,Z), H²ⁿ(X,Z) and H³ⁿ(X,Z), respectively. We say that X has type (a,b), for integers a and b, if u₁u₁=au₂ and u₁u₂=bu₃. In this paper, we show that Z₂ can not act freely on a space of type (a,b) if a is odd and b is even, and that S¹ can not act freely on a space of type (a,b) if a is different from zero. For the remaining pairs (a,b), we may have free actions, and thus it makes sense to ask for the possible cohomology rings of the corresponding orbit spaces. In this direction, we determine the possible Z₂-cohomology rings of orbit spaces of free actions of Z₂ on spaces of type (a,b), where a and b are even, and of free actions of S¹ on spaces of type (0,b). As a consequence of these cohomological calculations, we also obtain some results of the Borsuk-Ulam type, concerning the existence of equivariant maps from the m-dimensional sphere, equipped with standard G-actions (G = Z₂ or S¹), into X, where X is a space of type (a,b) equipped with arbitrary G-actions.

Rong, Feng,  Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, P.R. China  (frong@sjtu.edu.cn).
Robust parabolic curves in C^m (m≥3), pp. 147-155.
ABSTRACT. We show the existence of families of holomorphic self-maps of C^m, m≥3, tangent to the identity at an isolated fixed point, which do not have robust parabolic curves at that point. When m=3, this was shown by Abate and Tovena.

Montserrat Bruguera, Departamento de Matematica Aplicada I, Universidad Politecnica de Catalunya C/ Sicilia # 208, 2 1 (08013) Barcelona, Spain (m.montserrat.bruguera@upc.edu), Constancio Hernandez and  Mikhail Tkachenko, Departamento de Matematicas, Universidad Autonoma Metropolitana, Av. San Rafael Atlixco # 186, Col. Vicentina, C.P. 09340, Iztapalapa, Mexico, D.F. (chg@xanum.uam.mx), (mich@xanum.uam.mx).
Raikov completion and the Hartman-Mycielski construction, pp. 157-165.
ABSTRACT. Hartman and Mycielski proved that every topological group G is topologically isomorphic to a closed subgroup of a connected, locally connected topological group. Analyzing the Hartman--Mycielski construction, one can verify that the group and its Hartman--Mycielski completion share many properties such as metrizability, separability, and omega-narrowness. Further, if G is Abelian, divisible, torsion, or torsion-free, so is its Hartman--Mycielski completion.
It is easy to see that the Hartman--Mycielski completion of a group is not Raikov complete, except for the trivial case |G| = 1. However, for a metrizable group G, we describe the Raikov completion of a group in terms of known objects of Topological Algebra and Functional Analysis.

Christopher Mouron, Department of Mathematics and Computer Science, Rhodes College, Memphis, TN 38112 and Department of Mathematics, The University of Alabama at Birmingham, Birmingham, AL 39254 (mouronc@rhodes.edu).
Continua that admit expansive Zⁿ⁺¹ actions but not expansive Zⁿ actions, pp. 167-180.
ABSTRACT. For each set of positive integers k and n, a k-dimensional continuum is constructed that admits an expansive Zⁿ⁺¹ action but does not admit an expansive Zⁿ action.

S. Andima, Long Island University - C.W. Post Campus, Department of Mathematics, Brookville, NY 11548, USA (SAndima@liu.edu), R. Kopperman, City College of CUNY, Department of Mathematics, New York, NY 10031, USA (rdkcc@ccny.cuny.edu), P. Nickolas, University of Wollongong, School of Mathematics and Applied Statistics, NSW 2522, Australia (peter@uow.edu.au), and S. Popvassilev, City College of CUNY, Department of Mathematics, New York, NY 10031, USA,  (strash.pop@gmail.com).
A family of asymmetric Ellis-type theorems, pp. 181-198.
ABSTRACT. Bouziad in 1996 generalized theorems of Montgomery (1936) and Ellis (1957), by proving that each Čech-complete space with a separately continuous group operation must be a topological group. We generalize this result by dropping the requirement that the spaces be Hausdorff or even T₁. Our theorems then apply to groups with "asymmetric" topologies, such as the additive group of reals with the upper topology, whose open sets are the open upper rays. We use the fact that each topological space has an associated second topology, which we call the "k-dual", and we consider cases where the bitopological space consisting of the original topology and its k-dual is a "Hausdorff k-bispace", the latter being a bitopological parallel to the topological concept of a Hausdorff k-space, but in which neither topology need be Hausdorff. Suppose a group has a topology in which the group multiplication is separately continuous. Assume also that the bitopological space described above is a Hausdorff k-bitopological space. One of our results is that if the join of the two topologies is Čech-complete, then inversion is a homeomorphism between the original space and its k-dual, and the group operation is jointly continuous with respect to both topologies. The same conclusion holds more generally if the join is assumed to be a Baire p-space, p-σ-fragmentable by a complete sequence of covers.

Javier Camargo, Escuela de Matematicas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, COLOMBIA (jecamar@uis.edu.co), (jcamargo@matem.unam.mx).
Openness of induced maps and homeomorphisms, pp. 199-213.
ABSTRACT. We show a class of maps between continua such that if either of the induced maps C(f) or HS(f) is open, then f is a homeomorphism. Also, we give a characterization of openness of the induced map HS(f).

de Leo, Lorenzo, Universidad Complutense de Madrid, 28040 Madrid, Spain, and Tkachenko, Mikhail, Universidad Autonoma Metropolitana de Mexico - Iztapalapa, 09340 Mexico D.F. (mich@xanum.uam.mx).
The maximal omega-narrow group topology on Abelian groups, pp. 215-227.
ABSTRACT. For an Abelian group G, we consider the maximal omega-narrow group topology T on G induced by all homomorphisms of G to second-countable topological Abelian groups. We study the properties of the topological group (G,T) and we prove, among other results, that every uncountable Abelian group equipped with the maximal omega-narrow topology is a first category space which is neither a P-group nor R-factorizable. A comparison of the maximal omega-narrow group topology and the Bohr topology on Abelian groups is also presented.

Javier Camargo, Escuela de Matematicas, Facultad de Ciencias, Universidad Industrial de Santander, Ciudad Universitaria, Carrera 27 Calle 9, Bucaramanga, Santander, A.A. 678, COLOMBIA.  (jecamar@uis.edu.co)  and (jaencamargo@gmail.com).
On the openness of induced map C(f) for dendroids, pp. 229-235.
ABSTRACT. We show that if f is a map between dendroids such that the induced map C(f) is open, then f is a homeomorphism, giving a positive answer to the question posed in Openness of induced mappings by J. J. Charatonik, W. J. Charatonik and A. Illanes.

Choi, Yemon, School of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, England (Current address: Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada) (y.choi.97@cantab.net).
Simplicial homology of strong semilattices of Banach algebras, pp. 237-260.
ABSTRACT. Certain semigroups are known to admit a 'strong semilattice decomposition' into simpler pieces. We introduce a class of Banach algebras that generalise the l¹-convolution algebras of such semigroups, and obtain a disintegration theorem for their simplicial homology.
Using this we show that for any Clifford semigroup S of amenable groups, l¹(S) is simplicially trivial: this generalises previous results of the author (Glasgow Math. Journal, 2006). Some other applications are presented.

Scott Beaver, Mathematics Department, Western Oregon University, 345 N. Monmouth Ave., Monmouth, OR 97361 (beavers@wou.edu).
A weighted Wiener's lemma for integral operators with Schur-type or essential-supremum kernel decay conditions, pp. 261-273.
ABSTRACT. In communication theory, one may encounter integral operators whose kernels' off-diagonal decay is strictly weaker than exponential. In this paper it is shown that if such an operator is invertible, the kernel of the inverse operator exhibits the same type of decay, extending the Banach *-algebraic techniques of Gröchenig and Leinert to the setting of integral operators on the space of square-integrable functions. En route, symmetry of the algebras under consideration is established.

Kenneth Dykema, Department of Mathematics, Texas A&M University, College Station TX 77843-3368, USA (kdykema@math.tamu.edu) and Francesco Fidaleo, Dipartimento di Matematica, II Universita di Roma ``Tor Vergata'', Via della Ricerca Scientifica, 00133 Roma, Italy (fidaleo@mat.uniroma2.it).
Unique mixing of the shift on the C* algebras generated by the q-canonical commutation relations, pp.275-281.
ABSTRACT. The shift on the C*-algebras generated by the Fock representation of the q-commutation relations has the strong ergodic property of unique mixing, when |q|<1.

Gilles Godefroy, Université Paris 6 - Institut de Mathématiques de Jussieu 175, rue du Chevaleret, 75013 Paris (godefroy@math.jussieu.fr).
Remarks on non-linear embeddings between Banach spaces, pp.283-287.
ABSTRACT.If a separable Banach space X isometrically embeds into a Banach space Y through a non-linear embedding, then Y contains many well-complemented subspaces which are linearly isometric to X.

Mortini, Raymond, Université Paul Verlaine - Metz, Département de Mathématique, Ile du Saulcy, F-57045 Metz France (mortini@univ-metz.fr), Sasane, Amol, Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom (A.J.Sasane@lse.ac.uk), and Wick, Brett D., Department of Mathematics, University of South Carolina, LeConte College, 1523 Greene Street, Columbia, SC 29208, USA (wick@math.sc.edu).
The Corona Theorem and Stable Rank for the algebra C +BH^∞, pp.289-302.
ABSTRACT. Let B be a Blaschke product. We prove in several different ways the Corona theorem for the algebra H^∞_B:=C+BH^∞. That is, we show the equivalence of the classical Corona Condition  Σ |f_j | > δ> 0 on data f₁, …, f_n in  H^∞_B and the solvability of the Bezout equation  Σ g_jf_j =1 for g₁, …, g_n. Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of H^∞_B is 1. Analogous results are obtained also for A(D)_B.

Kichenassamy, Satyanad, Laboratoire de Mathématiques, Université de Reims Champagne-Ardenne, F-51687 Reims Cedex 2, France (satyanad.kichenassamy@univ-reims.fr).
Improving Hölder's inequality , pp.303-312.
ABSTRACT.We show that the remainder in Hölder's inequality (Rogers, 1888; Hölder, 1889) may be computed exactly. It satisfies functional equations, and possesses monotonicity and scaling properties. We obtain as a consequence improvements of recent sharpenings (Aldaz, 2008) of the classical inequality.

Sergiu Aizicovici, Department of Mathematics, Ohio University, Athens, OH 45701, USA (aizicovi@math.ohiou.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).
Multiple positive solutions for a p-Laplacian Dirichlet problem with superdiffusive reaction, pp.313-333.
ABSTRACT. A parametric Dirichlet elliptic problem, driven by the p-Laplacian, and with a superdiffusive reaction is studied. First, it is shown that there exists a λ^* > 0, such that for every parameter λ ≥λ^*, the problem has a strictly positive smooth solution. Subsequently, by strengthening the conditions on the Caratheodory nonlinearity f(z,x) that appears in the equation under consideration, it is shown that for all λ > λ^* the problem has two positive solutions.

Jaume Llibre, Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalunya, Spain (jllibre@mat.uab.cat) and Víctor F. Sirvent, Departamento de Matemáticas, Universidad Simón Bolívar, Caracas 1086-A, Venezuela (vsirvent@usb.ve).
Erratum: Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces, pp.335-336.
ABSTRACT. We had a mistake in our article ``Minimal sets of periods for Morse-Smale diffeomorphisms on orientable compact surfaces'', Houston J. of Math. 35 (2009), 835-855. The correction of it originates that the minimal set of periods cannot have even periods.