Electronic Edition Vol. 32, No. 2, 2006

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), W. B. Johnson (College Station), J. Nagata (Osaka), V. I. Paulsen (Houston), , S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


Jack Maney, Department of Mathematical Sciences, The University of South Dakota, 414 E. Clark St., Vermillion, SD 57069 (jmaney@usd.edu).
On the boundary map and overrings, pp. 325-335.
ABSTRACT. This paper uses a generalization of the length function, due to Coykendall, to study the overrrings of an half factorial domain.

David E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300 (dobbs@math.utk.edu) and Jay Shapiro, Department of Mathematics, George Mason University, Fairfax, Virginia 22030-4444 (jshapiro@gmu.edu).
Descent of Divisibility Properties of Integral Domains to Fixed Rings, pp. 337-353.
ABSTRACT. Let G be a finite group acting via ring automorphisms on an integral domain R. The operation R goes to RG is shown to commute with the formations of integral closure and complete integral closure, as well as with the formations of certain rings of fractions and certain conductor overrings. The extension RG a subring of R satisfies GD (the going-down property). As a result, the following are among the classes of integral domains that are stable under the operation R goes to RG: integrally closed domains, seminormal domains, completely integrally closed domains, Krull domains, Prufer domains, going-down domains, locally divided domains, divided domains, locally pseudo-valuation domains, pseudo-valuation domains, and almost pseudo-valuation domains.

S. Hedayat and R. Nekooei, Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran (s_hedayat@mail.uk.ac.ir), (rnekooei@mail.uk.ac.ir).
Prime and Radical Submodules of Free Modules over a PID, pp. 355-367.
ABSTRACT. In this paper the notion of prime matrix is introduced. It is shown that if R is a PID then every full rank prime submodule of a free module of finite rank is the row space of a prime matrix. Hence the notion of a prime matrix may be regarded as a generalization of the notion of a prime element. Finally, using prime matrices, we
obtain the radical of submodules of a free module of finite rank, as well as the radical submodules.

S. Hedayat and R. Nekooei, Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran (s_hedayat@mail.uk.ac.ir), (rnekooei@mail.uk.ac.ir).
Primary Decomposition of Submodules of a Finitely Generated Module over a PID, pp. 369-377.
ABSTRACT. In this paper we present a method for calculating a reduced primary decomposition for submodules of a finitely generated module over a PID.

Ayman Badawi, Department of Mathematics and Statistics, The American Universit of Sharjah, P.O. Box 26666 Sharjah, United Arab Emirates (abadawi@aus.ac.ae) and David E. Dobbs, Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996-1300, U. S. A. (dobbs@math.utk.edu).
Strong Ring Extensions and phi-pseudo-valuation rings, pp. 379-398.
ABSTRACT. In this paper, we extend the concept of strong extensions of domains to the context of (commutative) rings with zero-divisors. We show that the theory of strong extensions of rings resembles that of strong extensions of domains.

P. Malcolmson, Department of Mathematics, Wayne State University, Detroit MI 48202 (petem@math.wayne.edu) and Frank Okoh, Department of Mathematics, Wayne State University, Detroit MI 48202 (okoh@math.wayne.edu).
A class of integral domains between factorial domains and idf-domains, pp. 399-421.
ABSTRACT. For a non-zero element a in an integral domain R, consider the set D(a) of non-associate irreducible divisors of all of the powers of that element. The domain R is idpf (irreducible divisors of powers finite) if for every non-zero element a in R, the set D(a) is finite. The domain is idf (irreducible divisors finite) if the set of non-associate irreducible divisors of a is finite for every non-zero element a. We locate these concepts amongst other generalizations of factorial domains. A modification of a construction due to Samuel leads to a domain that is idf but not idpf. We determine the idpf-subrings of the ring of Gaussian integers. A canonical problem is the determination of the irreducible affine varieties that are idpf. We show that every Krull domain, in particular every Dedekind domain, is idpf; hence the coordinate ring of a nonsingular curve is idpf. The coordinate rings of some familiar singular affine plane curves are shown to be idpf in positive characteristic but fail to be idpf in zero characteristic.

Guido, Daniele and Isola, Tommaso, Dipartimento di Matematica, Univ. Roma "Tor Vergata", Via della Ricerca Scientifica 1, I-00133 Roma - Italy (guido@mat.uniroma2.it), (isola@mat.uniroma2.it).
Tangential dimensions II. Measures, pp. 423-444.
ABSTRACT. Notions of (pointwise) tangential dimension are considered, for measures of Rn. Under regularity conditions (volume doubling), the upper resp. lower tangential dimension at a point x of a measure can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to the given measure at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in D. Guido, T. Isola, "Tangential dimensions I. Metric spaces", Houston J. Mathematics 31(4), 2005, 1023-1045, and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in [1] and [2] in the framework of Alain Connes' noncommutative geometry.
[1] D. Guido, T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, Journal of Functional Analysis, Vol. 203(2), , 2003, pp. 362-400.
[2] D. Guido, T. Isola, Dimensions and spectral triples for fractals in RN , Advances in Operator Algebras and Mathematical Physics; Proceedings of the Conference held in Sinaia, Romania, June 2003, F. Boca, O. Bratteli, R. Longo H. Siedentop Eds., Theta Series in Advanced Mathematics, Bucharest 2005. Paper available through ArXiv.

Labouriau, Isabel S., Centro de Matemática da Universidade do Porto, R. do Campo Alegre, 687, 4 169 Porto, Portugal (islabour@fc.up.pt) and Ruas, Maria A. S.,Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil (maasruas@icmc.usp.br).
Invariants for Bifurcations, pp. 445-458.
ABSTRACT. Bifurcation problems with one parameter are studied here. We develop a method for computing a topological invariant, the number of fold points in a stable one-parameter unfolding for any given bifurcation of finite codimension.
We introduce another topological invariant, the algebraic number of folds. The invariant gives the number of complex solutions to the equations of fold points in a stabilization, an upper bound for the number of fold points in any unfolding. It can be computed by algebraic methods, we show that it is finite for germs of finite codimension. An open question is whether this value is always attained as the maximum number of fold points in a stable unfolding.
We compute these two invariants for simple bifurcations in one dimension, answering the question above in the affirmative. We discuss other invariants in the literature and verify that the algebraic number of folds and the Milnor number form a complete set of invariants for simple bifurcations in one dimension.

Popvassilev, Strashimir G., Department of Mathematics, University of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, Louisiana 70504-1010, U.S.A. (pgs2889@louisiana.edu), (www.louisiana.edu/~pgs2889).
Base-family paracompactness, pp. 459-469.
ABSTRACT. Call a topological space base-family paracompact if it has an open base every subfamily of which has a subfamily with the same union, such that the latter subfamily is locally finite at each point of its union. Proto-metrizable spaces are base-family paracompact. A T1 space is metrizable if and only if its product with a converging sequence is base-family paracompact.

Francis Jordan,  Department of Mathematical Sciences, Georgia Southern University, Statesboro, Ga. 30458 (fjordan@georgiasouthern.edu).
The S4 continua in the sense of Michael are precisely the dendrites, pp. 471-487.
ABSTRACT. A continuum X is said be S4 if for every partition P of X into compacta there is a continuous selector for P.  It is known that every S4 continuum is a dendrite.  In this paper we prove the opposite implication.  The result will follow from a general theorem on the existence of continuous selectors and a structure theorem on partitions of dendrites.

Jan van Mill, Faculty of Sciences, Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands (vanmill@cs.vu.nl).
Not all homogeneous Polish spaces are products, pp. 489-492.
ABSTRACT. We prove that not every homogeneous Polish space is the product of one of its quasi-components and a totally disconnected space. This answers a question of Aarts and Oversteegen.

Ofelia T. Alas, Instituto de Matematica e Estatıstica, Universidade de Sao Paulo, Caixa Postal 66281, 05311-970 Sao Paulo, Brasil (alas@ime.usp.br) and R. G. Wilson, Departamento de Matematicas, Universidad Autonoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340, Mexico, D.F., Mexico (rgw@xanum.uam.mx).
Minimal properties between T1 and T2, pp. 493-504.
ABSTRACT. A space is a US-space if every convergent sequence has a unique limit; it is an SC-space if each convergent sequence together with its limit is closed and is a KC-space if every compact subset is closed. We study the existence of spaces which are minimal with respect to these properties. We develop a number of results regarding minimal SC-spaces, we show that the class of infinite minimal US-spaces is empty and we give a consistent example of a Tychonoff topology which contains no minimal KC-topology

I. G. Todorov, Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, N. Ireland} (i.todorov@qub.ac.uk).
Synthetic properties of ternary masa-bimodules, pp. 505-519.
ABSTRACT. We provide a detailed description of the support of the masa-bimodules, which are ternary rings of operators (we call such masa-bimodules ternary). We show that the reflexive masa-bimodule generated by any synthetic masa-bimodule and a finite number of ternary masa-bimodules in an appropriate mutual position, is always synthetic and characterise the hereditarily synthetic masa-bimodules of this type. We give some corollaries of these results

Pengtong Li, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (pengtonglee@vip.sina.com), Jipu Ma, Department of Mathematics, Nanjing University, Nanjing 210093, China, and Jing Wu, Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA (ucfjing@yahoo.com).
Additive Derivations of Certain Reflexive Algebras, pp. 521-530.
ABSTRACT. Let L be a J-subspace lattice on a Banach space, AlgL be the associated reflexive algebra and A be a subalgebra of AlgL containing all finite rank operators in AlgL. Under an assumption, we prove that every additive derivation from A into AlgL is linear and quasi-spatial. This result can apply to those reflexive algebras with atomic Boolean subspace lattices and pentagon subspace lattices, respectively.

Kucerovsky, D. and Ng,P.W., University of New Brunswick at Fredericton, Fredericton, NB E3B 5A3 Canada (dan@math.unb.ca), (pwn@math.unb.ca).
The corona factorization property and approximate unitary equivalence, pp. 531-550.
ABSTRACT. We study Rordam's group, KL(A,B), and a corona factorization condition. Our key technical result is a lemma showing that approximate unitary equivalence preserves the purely large property of Elliott and Kucerovsky. Using this, we characterize KL(A,B) as a group of purely large extensions under approximate unitary equivalence, generalizing a theorem of Kasparov's. Then we prove the following:
Let B be a stable and separable C*-algebra. Then the following are equivalent (for weakly nuclear extensions):
i. The corona algebra of B has a certain quasi-invertibility property, which we here call the corona factorization property.
ii. Rordam's group KL1(A,B) is isomorphic to the group of full essential extensions of A by B.
iii. Every strongly full and positive element of the corona algebra of B is properly infinite.
iv. Every norm-full extension of B is absorbing with respect to approximate unitary equivalence.
v. Every norm-full extension of B is absorbing with respect to ordinary unitary equivalence.
vi. Every norm-full extension of B is absorbing with respect to weak equivalence.
vii. Every norm-full trivial extension of B is absorbing with respect to unitary equivalence.
viii. A K-theoretical uniqueness result for maps into M(B)/B.
We show that if X is the infinite Cartesian product of spheres, then the stabilization of C(X) does not have the corona factorization property.

Timur Oikhberg, University of California - Irvine, Irvine CA 92697 (toikhber@math.uci.edu).
Operator spaces with complete bases, lacking completely unconditional bases, pp. 551-561.
ABSTRACT. We construct a Hilbertian operator space X such that the set of completely bounded operators on X consists of Hilbert-Schmidt perturbations of a certain representation of the second dual to the James space. This space possesses an orthonormal basis (ei) such that all basis projections are completely contractive, yet any n-dimensional block subspace has complete unconditionality constant of at least c n1/2 (c is a constant).

Prüss, Jan, University of Halle, 06099 Halle, Germany (pruess@mathematik.uni-halle.de), Rhandi, Abdelaziz, University of Marrakesh, 40000 Marrakesh, Morocco (rhandi@ucam.ac.ma), and Schnaubelt, Roland, University of Halle, 06099 Halle, Germany (schnaubelt@mathematik.uni-halle.de).
The domain of elliptic operators on Lp(Rd) with unbounded drift coefficients, pp. 563-576.
ABSTRACT. We consider elliptic operators on Rd of second order where the diffusion coefficients are uniformly elliptic and the drift coefficients can grow as |x| log|x|. We show that the domain in Lp(Rd) is the intersection of the Sobolev space W2,p(Rd) and the domain of the drift term, and that A generates a strongly continuous semigroup on Lp(Rd). Our approach relies on a Dore-Venni type theorem on sums of non commuting operators in Lp(Rd). The description of the domain implies global regularity of the density of the invariant measure of the corresponding transition probabilities (if the measure exists), i.e., the density belongs to W2,q(Rd) for all finite q.

Anders Olofsson, Falugatan 22 1tr, SE-113 32 Stockholm, Sweden (anderso@math.su.se).
An inequality for sums of subharmonic and superharmonic functions, pp. 577-588.
ABSTRACT. We prove a general inequality for the distributional Laplacian of a sum of a subharmonic and a superharmonic function postcomposed with a convex function of linear growth. We use this inequality to show that convex functions of linear growth operate by means of postcomposition on the class of sums of subharmonic and superharmonic functions.

Zhihua Chen, Department of Mathematics, Tongji University, Shanghai, P.R. China (zzzhhc@tongji.edu.cn) and Min Ru, Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA (minru@math.uh.edu).
A uniqueness theorem for moving targets with truncated multiplicities, pp. 589-601.
ABSTRACT. After Nevanlinna's result that any two non-constant meromorphic functions f and g sharing the same inverse images (regardless of multiplicities) for five distinct values must be equal, H. Cartan declared that there are at most two meromorphic functions on C which have the same inverse images regardless of multiplicities. Although Cartan's proof contained a gap, Cartan's declaration is true if one assumes that meromorphic functions share four distinct values counted with multiplicities truncated by 2. In 1998, H. Fujimoto extended such a restricted Cartan's declaration to the case of holomorphic curves sharing hyperplanes in a n-dimensional projective space. This paper generalizes Fujimoto's result to the moving targets.

Robert J. McCann, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 (mccann@math.toronto.edu).
Stable rotating binary stars and fluid in a tube, pp. 603-631.
ABSTRACT. This paper considers compressible fluid models for a Newtonian rotating star. For fixed mass and large angular momentum, stable solutions to the associated Navier-Stokes-Poisson system are constructed in the form of slow, uniformly rotating binary stars with specified mass ratio. The variational method employed was suggested by Elliott Lieb; it predicts uniform rotation as a consequence rather than an assumption. The density profiles of the solutions are local energy minimizers in the Wasserstein L-infinity metric; no global energy minimum can be achieved. A one-dimensional toy model admitting explicit solution is also introduced which caricatures the situation: to any specified number of components and their masses corresponds a single family of solutions, parameterized by angular velocity up to the point of equatorial break-up; here the equilibrium model breaks down as the atmosphere of the lightest star in the system begins to drift into space.