HOUSTON JOURNAL OF MATHEMATICS

Electronic Edition Vol. 36, No. 3, 2010

Editors: G. Auchmuty (San Francisco, SFSU, D. Bao (Houston), H. Brezis (Paris), B.  Dacorogna (Lausanne), 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

Voloch, José Felipe, Dept. of Mathematics, University of Texas, Austin TX 78712 (voloch@math.utexas.edu)
Conics over function fields and the Artin-Tate conjecture, pp. 675-679.
ABSTRACT. We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields.

Nicola Garofalo, Purdue University, West Lafayette, IN 47907, USA, (garofalo@math.purdue.edu), and Niko Marola, Department of Mathematics and Systems Analysis, Helsinki University of Technology, P.O. Box 1100, FI-02014 TKK, Finland (niko.marola@tkk.fi).
Sharp capacitary estimates for rings in metric spaces, pp. 681-695.
ABSTRACT. We establish sharp estimates for the p-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. These estimates play an essential role in the study of the local behavior of p-harmonic Green's functions.

Kong Lingling, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China (KongLL111@nenu.edu.cn),  Gao Ruimei, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China (gaorm135@nenu.edu.cn), Pei Donghe, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China (peidh340@nenu.edu.cn), and Zhang Jianhua, Department of Mathematics, Yili Teacher's College, Yili 835000, P.R.China.
Singularities of generic lightcone Gauss maps and lightcone pedal surfaces of spacelike curves in Minkowski 4-space , pp. 697-710.
ABSTRACT. The main goal of this paper is to study singularities of generic lightcone Gauss maps and lightcone pedal surfaces of spacelike curves in Minkowski 4-space. To do this, we first construct lightcone height functions and extended lightcone height functions, and then show the relations between singularities of generic lightcone Gauss maps(resp. lightcone pedal surfaces) and that of lightcone height functions (resp. extended lightcone height functions). In addition some geometric properties of the spacelike curves are studied from geometrical point of view.
Editorial Addendum concerning an error in this paper.

Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea (yjsuh@knu.ac.kr), Juan De Dios Pérez, Departamento De Geometría Y Topología, Facultad De Ciencias, Universidad De Granada, 18071-Granada, Spain (jdperez@ugr.es), Hyung Jun Jin, Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea and Hae Houng Yang, Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea.
Real hypersurfaces in complex two–plane Grassmannians with D^⊥–parallel Lie derivatives, pp. 711-726.
ABSTRACT. We In this paper we give some non-existence properties of real hypersurfaces in complex two–plane Grassmannians G₂(C^m+2) in terms of D^⊥–parallel Lie derivatives for the structure tensor φ_i, i=1, 2, 3, the shape operator A, and the induced Riemannian metric tensor g along the distribition D^⊥=Span{ξ₁, ξ₂, ξ₃}.

Jeffrey Rauch, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (rauch@umich.edu) and Michael Taylor, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA (met@math.unc.edu).
Quadrature estimates for multidimensional integrals, pp. 727-749.
ABSTRACT.  We prove estimates for the error in the most straightforward discrete approximation to the integral of a compactly supported function of  n variables. The methods use Fourier analysis and interpolation theory, and also make contact with classical lattice point estimates. We also prove error estimates for the approximation of the integral over an interval by the trapezoidal rule and the midpoint rule..

Pandelis Dodos, National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece (pdodos@math.ntua.gr).
Definability under duality, pp. 781-792.
ABSTRACT. It is shown that if A is an analytic class of Banach spaces with separable dual, then the class of spaces isomorphic to a dual of a member of A is also analytic. The corresponding result for pre-duals is false.

Farid Behrouzi, Department of Mathematics, Alzahra University,Vanak Ave.,Postcode 1993891176 Iran (behrouzif@yahoo.com).
Riesz and quasi-compact endomorphisms of Lipschitz algebras, pp. 793-802.
ABSTRACT. In this paper, we study Riesz and quasi-compact endomorphisms of Lipschitz algebras. For a unital endomorphism of a little Lipschitz algebra, we establish a lower bound for its essential spectral radius. Also, we show that this lower bound can be attained by imposing some extra assumption.

Lin, Bor-Luh, University of Iowa, Iowa City, IA 52242 (bllin@math.uiowa.edu), and Fonf, Vladimir, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (fonf@math.bgu.ac.il).
Unbounded nested sequences of balls in Banach spaces, pp. 803-821.
ABSTRACT. It is not so difficult to prove that any closed convex solid cone in a Banach space is the union of an unbounded nested sequence of balls in some equivalent norm. In this paper the converse question is considered: namely, under what conditions is the union of an unbounded nested sequence of balls a cone? It was proved in joint work of the present authors, P. Bandyopadhyay, and M. Martin [Houston J. Math. 29 (2003), 173-193] that in finite dimensions such a union is always a cone. In this paper we reveal the following infinite-dimensional phenomena: it is possible that the union is a cone but no vertex of it belongs to the (closed) subspace generated by the centers of the balls. Thus an answer to the question above in general is "no". By using the concept of a norming cone, we establish necessary and sufficient conditions for the union of an unbounded nested sequence of balls to be a cone. In this context we get necessary and sufficient conditions for a Lindenstrauss space to be polyhedral, and treat also the space of affine continuous functions on a compact metrizable simplex. Some other related problems are considered.

Mathes, Ben, Colby College, Waterville, ME 04901 (dbmathes@colby.edu).
Strictly cyclic algebras with arbitrary prescribed Gelfand spectrum, pp. 823-828.
ABSTRACTGiven any compact subset C of the plane, G. Kalisch shows that there is an operator whose spectrum consists entirely of point spectrum and equals C. We use Banach algebra techniques and the theory of strictly cyclic algebras to give a new proof of this result. In the process, we construct many new examples of strictly cyclic algebras. In particular, we construct a semisimple commutative strictly cyclic algebra whose Gelfand spectrum is homeomorphic to an arbitrary compact subset of Euclidean space.

Zhe, Dong, Department of Mathematics, Zhejiang University, Hangzhou 310027, China (dongzhe@zju.edu.cn) and Ying Huang, Department of ISEE, Zhejiang University, Hangzhou 310027, China (listana@zju.edu.cn). 
Finite-representability and U-local reflexivity of operator spaces, pp. 829-842.
ABSTRACT. In this paper, we introduce the new concept 'U-local reflexivity' of operator spaces, and study the relationship between finite representability and U-local reflexivity.

Claudianor O. Alves, Unidade Academica de Matematica e Estatistica, Universidade Federal de Campina Grande, CEP: 58109-970, Campina Grande (PB), Brazil, (coalves@dme.ufcg.edu.br) , Paulo Cesar Carriao, Departamento de Matematica, Universidade Federal de Minas, Gerais, CEP: 31270-010, Belo Horizonte (MG), Brazil (carrion@mat.ufmg.br), and Olimpio Hiroshi Miyagaki, Departamento de Matematica, Universidade Federal de Vicosa, CEP: 36571-000, Vicosa (MG), Brazil (olimpio@ufv.br), (ohmiyagaki@gmail.com).
Multi-bump homoclinic orbits for a class of Hamiltonian systems with superquadratic potential, pp. 859-877.
ABSTRACT. This paper is concerned with the existence of homoclinic orbits of multi-bump type in the second order Hamiltonian system with superquadratic potential, without neither periodicity nor coercivity condition on the potentials. The homoclinic orbits of multi-bump type is obtained by variational methods combined with some arguments used by Ding and Tanaka.

Changming Ding, Department of Mathematics, Xiamen University, Xiamen, Fujian 361005, P.R.China (cding@xmu.edu.cn).
Chain stability of closed sets, pp. 879-885.
ABSTRACT. This article deals with chain stability of closed sets. It is shown that chain stability is strictly stronger than absolute stability, but it may lack asymptotic stability. A characterization of chain stability is given in terms of neighborhoods.

Mardešić, Sibe, University of Zagreb, Zagreb, Croatia (smardes@math.hr)
The standard resolution of the product of a compactum and a polyhedron consists of ANEs for metric spaces, pp. 887-904.
ABSTRACT. In order to study (strong) shape properties of the Cartesian product of a compactum X and a (non-compact) polyhedron P, the author has introduced in 2003 a particular resolution of such a product. It is an inverse limit of the product to an inverse system Y having some additional properties. The system Y consists of paracompact spaces Y_μ, μ in M, having the homotopy type of polyhedra. In the present paper, it is proved that the spaces Y_μ are stratifiable k-spaces and absolute neighborhood extensors for metric spaces. This makes it possible to use the homotopy extension property, for pairs of metric spaces, a result needed in various arguments concerning the shape of Cartesian products of a compactum X and a polyhedron P.

Lewis, Wayne, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 (wayne.lewis@ttu.edu) and Minc, Piotr, Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849 (mincpio@auburn.edu).
Drawing the pseudo-arc, pp. 905-934.
ABSTRACT. It is very likely that the pseudo-arc may occur as an attractor of some natural dynamical system. How would a picture of such a strange attractor look? Would it be recognized as the pseudo-arc, a hereditarily indecomposable continuum? This paper shows that it could be difficult. We notice that no black and white image can look hereditarily indecomposable on any raster device (like a computer screen or a printed page). We also try to generate the best computer picture of the pseudo-arc as it is possible under the circumstances. With that purpose in mind, we expand the pseudo-arc into an inverse limit with relatively simple, deterministically defined and easy to handle numerically n-crooked bonding maps. We use this expansion to assess numerical complexity of drawing the pseudo-arc with help from the Anderson-Choquet embedding theorem. We also generate graphs of n-crooked maps with large n's, and we prove that a rasterized image of such a graph does not look very crooked at all because it must contain a long straight linear vertical segment.

Eiichi, Matsuhashi, Department of Mathematics, Faculty of Engineering, Shimane University ,Matsue, Shimane 690-8504, Japan (matsuhashi@riko.shimane-u.ac.jp).
Some remarks on Whitney preserving maps, pp. 935-943.
ABSTRACT. Espinoza (2002) proved that every Whitney preserving map from a continuum containing a dense arc component onto the unit interval is a homeomorphism. In this paper we generalize this result. Also we deal with other topics related to Whitney preserving maps.

Marsh, M. M., California State University, Sacramento, CA 95819 (mmarsh@saclink.csus.edu).
Inverse limits of projective spaces and the fixed point property, pp. 945-957.
ABSTRACT. We consider inverse limits of even dimensional projective spaces with essential bonding mappings. J. Segal and T. Watanabe have shown that such inverse limits on complex projective space will have the fixed point property. We obtain some positive fixed point results for both the real and quaternionic projective spaces and we correct an error related to the real projective case in the author's paper Covering Spaces, Inverse Limits, and Induced Coincidence Producing Mappings.

Corona-Vázquez, Florencio., Licenciaturas en Física y Matemáticas,UNACH, 4a Ote-Nte 1428, La Pimineta, Tuxtla Gutiérrez, Chiapas, 29000, MÉXICO, (florencio.corona@unach.mx) and Escobedo, Raúl, Facultad de Ciencias Físico Matemáticas, BUAP, Ave. San Claudio y Río Verde, Ciudad Universitaria, San Manuel, Puebla, Pue., 72570, MÉXICO, (escobedo@fcfm.buap.mx).
Hyperspace suspension and the fixed point property, pp. 959-965.
ABSTRACT. We present fixed point theorems for hyperspace suspension of certain continua. We also give some examples, namely: two circle-like continua whose hyperspace suspension has the fixed point property, one of them is the continuum called the Warsaw circle and the other one is a compactification of the real line having an arc as its remainder; in addition, we see that a disk with a spiral is a continuum with the fixed point property such that its hyperspace suspension does not have this property.

D. Basile Università degli Studi di Catania, Dipartimento di Matematica e Informatica, Viale Andrea Doria 6, 95125 Catania, Italy. (basile@dmi.unict.it). G. J. Ridderbos Faculty of Electrical Engineering Mathematics and Computer Science, TU Delft, Postbus 5031, 2600 GA Delft, The Netherlands. (G.F.Ridderbos@tudelft.nl).
On the Cech number of Sigma-products, pp. 967-983.
ABSTRACT. We study the Cech-number of uncountable Sigma-products of the unit interval. We characterize this number in terms of dominating families in a suitably chosen poset. We also show that the Cech-number of the Sigma-product of ω₁ copies of the unit interval may be strictly bigger and strictly less than the compact covering number of the irrationals.