Electronic Edition Vol. 35, No. 4, 2009

Editors: H. Amann (Zürich), G. Auchmuty (San Francisco, SFSU), D. Bao (Houston), H. Brezis (Paris), K. Davidson (Waterloo), M. Gehrke (Radboud), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. Hausen (Houston), 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


Malcolmson, Peter, Wayne State University, Detroit MI 48202 (petem@math.wayne.edu) and Okoh, Frank, Wayne State University, Detroit MI 48202 (okoh@math.wayne.edu).
Factorization in subalgebras of the polynomial algebra, pp. 991-1012.
We investigate the factorization properties of subalgebras of the form K+fK[T] of the polynomial ring K[T], where f is in K[T], and the stability of these factorization properties under the standard operations. These subrings K+ fK[T] satisfy the known generalizations of factoriality except possibly HFD and IDPF. A domain is HFD if each non-zero element that is not a unit is a product of a unique number of irreducible elements. We prove that R= K+ fK[T] is HFD if and only if R = K[T]. A domain is IDPF if for every non-zero element a in the domain, the ascending sequence of sets of non-associate irreducible divisors of an stabilizes on a finite set. We prove that if the characteristic of K is zero, then R = K + fK[T] is IDPF if and only if R = K[T] When K has positive characteristic we prove that R is IDPF if and only if R = K[T] or f has only one root or K is algebraic over its prime subfield and f is a power of an irreducible polynomial. We compare the factorization properties of K + fK[T] with those of the subrings of the Gaussian integers.

Amir Mafi, Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran and School of Mathematics, Institute for Research in Fundamental Science (IPM), P. O. Box 19395-5746, Tehran, Iran.(a-mafi@araku.ac.ir) and Hero Saremi, Islamic Azad University of Arak, Arak, Iran  (h-saremi@iau-arak.ac.ir).
Cofinite modules and generalized local cohomology, pp. 1013-1019.
ABSTRACT. Let R be a commutative Noetherian ring, a an ideal of R, and M, N two  finitely generated R-modules. We prove that the generalized local cohomology modules Hta (M,N) are a-cofinite; that is, ExtiR (R/a, Hta(M,N)) is finitely generated for all i, t 0, in the following cases: (i) cd(a) =1, where cd is the cohomological dimension of a in R.(ii) dimR 2. Additionally, we show that if cd(a) = 1 then is ExtiR (M, Hta (N)) a-cofinite for all i, t ≥ 0.

Driss Bennis and Najib Mahdou, Department of Mathematics, Faculty of Sciences and Technology, University S. M. Ben Abdellah, Fez 30000, Morocco (driss_bennis@hotmail.com), (mahdou@hotmail.com).
Global Gorenstein dimensions of polynomial rings and of direct products of rings, pp. 1019-1028.
ABSTRACT. In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.

Musso, Emilio, Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy (emilio.musso@polito.it), and Nicolodi, Lorenzo, Dipartimento di Matematica, Universita' di Parma, 43100 Parma, Italy (lorenzo.nicolodi@unipr.it).
Conformal deformation of spacelike surfaces in Minkowski space, pp. 1029-1049.
ABSTRACT. We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and singular solutions. In particular, we show that isothermic surfaces are singular solutions of the system, which implies that a generic second order deformable surface is not isothermic. This differs from the situation in 3-dimensional conformal geometry, where isothermic surfaces coincide with deformable surfaces.

Yamamoto, Minoru, Department of Mathematics Aichi University of Education 1 Hirosawa, Igaya-cho, Kariya, Aichi 448-8542, Japan (minomoto@auecc.aichi-edu.ac.jp ).
The number of singular set components of fold maps between oriented surfaces, pp. 1051-1069.
ABSTRACT.  We study fold maps between closed oriented surfaces. A fold map is a smooth map with only fold singularities. We determine the number of singular set components of fold maps between closed oriented surfaces.

Jørgensen, Peter, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom (peter.jorgensen@ncl.ac.uk).
A new recollement for schemes, pp. 1071-1077.
ABSTRACT. Let X be a scheme, U an open subscheme, and Z the closed complement of U. Under some weak assumptions, a new recollement is established which expresses the derived category D(X) in terms of the derived category D(U) and DZ(X), the full subcategory of D(X) of complexes with support in Z.

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).
A chainable continuum that admits a homeomorphism with entropy of arbitrary value, pp. 1079-1090.
ABSTRACT. A chainable continuum is constructed with the following properties:
(1) for every epsilon from zero to infinity (inclusive) there exists a homeomorphism with entropy the value epsilon and (2) The continuum does not contain a pseudo-arc. This answers a question by W. Lewis.

Paul Bankston, Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI 53201-1881 (paulb@mscs.mu.edu).
Dendrites, topological graphs, and 2-dominance, pp. 1091-1102.
ABSTRACT. For each positive ordinal α, the reflexive and transitive binary relation of α-dominance between compacta was first defined in our paper [Mapping properties of co-existentially closed continua, Houston J. Math., 31 (2005), 1047-1063] using the ultracopower construction. Here we consider the important special case α =2, and show that any Peano compactum 2-dominated by a dendrite is itself a dendrite (with the same being true for topological graphs and trees). We also characterize the topological graphs that 2-dominate arcs (resp., simple closed curves) as those that have cut points of order 2 (resp., those that are not trees).

Lei Mou, Capital Normal University, Beijing, 100048 China (moulei@mail.cnu.edu.cn).
Base-cover metacompactness of Q×(ω1+1) and subspaces of ordinals, pp. 1103-1110.
ABSTRACT. In this paper, we prove: (1) The product of Q and ω1+1 is not base-cover metacompact, where Q is the subspace of all rationals in the real line with the usual topology and ω1+1 has the order topology. (2) Let X be a subspace of an ordinal. Then X is base-cover paracompact if and only if X is base-cover metacompact. The result (1) answers a question asked by Popvassilev.

Fang Xiaochun, Department of Mathematics, Tongji University, Shanghai 200092, China (xfang@mail.tongji.edu.cn), and Yang Xinbing, Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China (yangxinbing@zjnu.cn).
Real rank and stable Rank of C*-algebras with tracial rank zero, pp. 1111-1129.
ABSTRACT. It is known that a unital separable simple C*-algebra with tracial rank zero has stable rank one and real rank zero. In this note, we use the generalized inductive limits to construct a non-simple C*-algebra. Then we show that the tracial rank of this algebra is zero, but the stable rank of it is not one and the real rank of it is not zero.

David Ralston, Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174  (ralston@math.ohio-state.edu), (spetzo@hotmail.com).
Heaviness: An extension of a lemma of Y. Peres, pp. 1131-1141.
ABSTRACT. We provide an elementary proof of Y. Peres' lemma on the existence in certain dynamical systems of what we term heavy points, points whose ergodic averages consistently dominate the expected value of the ergodic averages. We also derive several generalizations of Peres' lemma by employing techniques from the simplified proof.

Vernicos, Constantin, Institut de Mathématiques et de Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier II, Case Courrier 051, Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France, (vernicos@math.univ-montp2.fr).
Spectral radius and amenability in Hilbert geometries, pp. 1143-1169.
ABSTRACT. We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a Fölner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the spectrum of the Hyperbolic geometry of the same dimension. Horoballs, from a purely metric point of view, and their relation with the bottom of the spectrum in Hilbert geometries are briefly studied.

Lee, Hun Hee, Department of Mathematics, Chungbuk National University, 410 Seongbong-ro, Heungduk-gu, Cheongju 361-763, Korea (hhlee@chungbuk.ac.kr)
Weak type (2,H) and weak cotype (2,H) of operator spaces, pp. 1171-1201.
ABSTRACT. Recently an operator space version of type and cotype, namely type (p,H) and cotype (q,H) of operator spaces for 1 &le p &le 2 &le q &infin and a subquadratic and homogeneous Hilbetian operator space H were introduced and investigated by the author. In this paper we define weak type (2,H) (resp. weak cotype (2,H)) of operator spaces, which lies strictly between type (2,H) (resp. cotype (2,H)) and type (p,H) for all p < 2 (resp. cotype(q,H) for all q > 2). This is an analogue of weak type 2 and weak cotype 2 in the Banach space case, so we develop analogous equivalent formulations. We also consider weak-H space, spaces with weak type (2,H) and weak cotype (2,H^*) simultaneously and establish corresponding equivalent formulations.

P.W. Ng, Department of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P. O. Box 41010, Lafayette, LA 70504-1010 (png@louisiana.edu) and E. Ruiz, Department of Mathematics, University of Hawaii Hilo, 200 W. Kawili St., Hilo, Hawaii 96720 (ruize@hawaii.edu)
The structure of the unitary groups of certain simple C*-algebras, pp. 1203-1232.
ABSTRACT. Let A be a simple unital separable C*-algebra such that either [i.] A is an AH-algebra with bounded dimension growth; or [ii.]A is an approximately splitting interval algebra.  We determine the structure of the topological group of unitaries (of A). We use this to study the structure of the automorphism group of A.

Costea, Serban, Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada (secostea@math.mcmaster.ca)
Strong A-infinity weights and Sobolev capacities in metric measure spaces, pp. 1233-1249.
ABSTRACT. This article studies strong A-infinity weights in Ahlfors Q-regular unbounded and geodesic metric measure spaces satisfying a weak (1,s)-Poincaré inequality for some s in (1,Q]. For a fixed s in (Q-1,Q], it is shown that a function u yields a strong A-infinity weight of the form w=exp(Qu) whenever the minimal s-weak upper gradient of u has sufficiently small Morrey s norm.

Dilworth, Steve J., University of South Carolina, Columbia, SC 29208 (dilworth@math.sc.edu), Odell, Edward, University of Texas at Austin, Austin, TX 78712-0257  (odell@math.utexas.edu), Schlumprecht, Thomas, Texas A&M University, College Station, TX 77843-3368  (schlump@math.tamu.edu), Zsák, András,  Peterhouse, Cambridge, CB2 1RD, UK and Lancaster University, Lancaster, LA1 4YF, UK (a.zsak@dpmms.cam.ac.uk).
Partial Unconditionality, pp. 1251-1311.
ABSTRACT. J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional. His proof gives an upper bound on the constant K(d) of near-unconditionality for d>0, which tends to infinity as d tends to 0. It is unknown if K(d) is uniformly bounded independently of d. This problem turns out to be closely related to the question whether every infinite-dimensional Banach space contains a quasi-greedy basic sequence. The notion of a quasi-greedy basic sequence was introduced by S. V. Konyagin and V. N. Temlyakov. We present an extension of Elton's result which includes Schreier unconditionality.
The proof involves a basic framework which we show can be also employed to prove other partial unconditionality results including that of convex unconditionality due to Argyros, Mercourakis and Tsarpalias. Various constants of partial unconditionality are defined and we investigate the relationships between them. We also explore the combinatorial problem underlying the problem of uniform boundedness of K(d), and show that K(d)>5/4 for all d.

M. Obradovic, Department of Mathematics, Faculty of Civil Engineering, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia  (obrad@grf.bg.ac.yu) and S. Ponnusamy, Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India  (samy@iitm.ac.in).
Univalency and convolution results associated with confluent hypergeometric functions, pp.1313-1328.
ABSTRACT. Given the confluent hypergeometric functions φ ( a;c;z ) , we place conditions on a and c to guarantee that zφ ( a;c;z ) will be in two subclasses of univalent functions. In addition, we obtain conditions to obtain some convolution results.