Showing papers in "Houston Journal of Mathematics in 2010"

Commutative rings with toroidal zero-divisor graphs

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.

Infinitesimal affine geometry of metric spaces endowed with a dilation structure

Abstract: We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group operations are not fundamental, but derived objects, and the generalization of affine geometry is not based on incidence relations.

On the second Paneitz-Branson invariant

Abstract: We define the second Paneitz-Branson operator on a compact Einsteinian manifold of dimension $n\geq 5$ and we give sufficient conditions that make it attained.

Sharp capacitary estimates for rings in metric spaces

Abstract: We establish sharp estimates for the p-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling mea- sure and supporting a Poincare inequality. These estimates play an essential role in the study of the local behavior of p-harmonic Green's functions. In this paper we establish sharp capacitary estimates for the metric rings with unrelated radii in a locally doubling metric measure space supporting a local (1,p)-Poincare inequality. A motivation for pursuing these estimates comes from the study of the asymptotic behavior of p- harmonic Green's functions in this geometric setting. Similarly to the classical case (for the latter the reader should see (30), (36) and (37)), capacitary estimates play a crucial role in studying the local behavior of such singular functions. For this aspect we refer the reader to the forthcoming paper by Danielli and the authors (11). Perhaps the most important model of a metric space with a rich non-Euclidean geometry is the Heisenberg group H n , whose underlying manifold is C n × R with the group law (z,t) ◦ (z ' ,t ' ) = (z + z ' ,t + t ' − 1 Im(zz ' )). Koranyi and Reimann (29) first computed explicitly the Q-capacity of a metric ring in H n . Here Q = 2n + 2 indicates the homogeneous dimension of H n attached to the non-isotropic group dilations ��(z,t) = (�z,� 2 t). Their method makes use of a suitable choice of "polar" coordinates in the group. The Heisenberg group is the prototype of a general class of nilpotent stratified Lie groups, nowadays known as Carnot groups. In this more general context, Heinonen and Holopainen (18) proved sharp estimates for the Q-capacity of a ring. Again, here Q indicates the homogeneous dimension attached to the non-isotropic dilations associated with the grading of the Lie algebra. In the paper (6) Capogna, Danielli and the first named author es- tablished sharp p-capacitary estimates, for the range 1 < p < ∞, for 2000 Mathematics Subject Classification. Primary: 31B15, 31C45; Secondary: 31C15.

Monotone images of cremer julia sets

Abstract: We show that if P is quadratic polynomial with a flxed Cremer point and Julia set J, then for any monotone map ' : J ! A from J onto a locally connected continuum A, A is a single point.

On Z_2 and S^1 free actions on spaces of cohomology type (a,b)

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 Hj (X,Z)=Z if j=0, n, 2n or 3n, and Hj (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

On O*-representability and C*-representability of *-algebras

Abstract: Characterization of the *-algebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications to *- doubles, monomial *-algebras and several other classes of *-algebras are presented.

Strictly singular non-compact diagonal operators on HI spaces

Abstract: We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$.

The Corona Theorem and stable rank for the algebra C plus BH infinity

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 ? |fj | > ?> 0 on data f1, �, fn in H?B and the solvability of the Bezout equation ? gjfj =1 for g1, �, gn. 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.

Ultrametric root counting

Abstract: Let $K$ be a complete non-archimedean field with a discrete valuation, $f\in K[X]$ a polynomial with non-vanishing discriminant, $A$ the valuation ring of $K$, and $\M$ the maximal ideal of $A$. The first main result of this paper is a reformulation of Hensel's lemma that connects the number of roots of $f$ with the number of roots of its reduction modulo a power of $\M$. We then define a condition --- {\em regularity} --- that yields a simple method to compute the exact number of roots of $f$ in $K$. In particular, we show that regularity implies that the number of roots of $f$ equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.

Model completion of varieties of co-Heyting algebras

Abstract: It is known that exactly eight varieties of Heyting algebras have a model-completion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms have a certain embedding property. For four of these six varieties (those which are locally finite) this actually provides a new proof of the existence of a model-completion, this time with an explicit and finite axiomatisation.

Harmonic maps defined by the geodesic flow

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 Tr 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.

Drawing the pseudo-arc

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 AndersonChoquet 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

A Unified Theory of Function Spaces and Hyperspaces: Local Properties

Abstract: Every convergence (in particular, every topology) τ on the hyperspace C (X, $) preimage-wise determines a convergence τ on C (X,Z), where X,Z are topological spaces and $ is the Sierpinski topology, so that f ∈ limτ⇑ F if and only if f −1(U) ∈ limτ F−1(U) for every open subset U of Z. Classical instances are the pointwise, compact-open and Isbell topologies, which are preimage-wise with respect to the topologies, whose open sets are the collections of, respectively, all (openly isotone) finitely generated, compactly generated and compact families of open subsets of X (compact families are precisely the open sets of the Scott topology); the natural (that is, continuous) convergence is preimage-wise with respect to the natural hyperspace convergence. It is shown that several fundamental local properties hold for a hyperspace convergence τ (at the whole space) if and only if they hold for τ on C (X,R) at the origin, provided that the underlying topology of X have some R-separation properties. This concerns character, tightness, fan tightness, strong fan tightness, and various Frechet properties (from the simple through the strong to that for finite sets) and corresponds to various covering properties (like Lindelof, Rothberger, Hurewicz) of the underlying space X. This way, many classical results are unified, extended and improved. Among new surprising results: the tightness and the character of the natural convergence coincide and are equal to the Lindelof number of the underlying space; The Frechet property coincides with the Frechet property for finite sets for the hyperspace topologies generated by compact networks.

Robust parabolic curves in C^m (m>=3)

Abstract: We show the existence of families of holomorphic self-maps of Cm, 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.

Definability under duality

Abstract: ) and endow-ing SB with the relative Effros-Borel structure, the set SB becomes the standardBorel space of all separable Banach spaces (see [AD], [AGR], [Bos] and [Ke]). Byidentifying any class of separable Banach spaces with a subset of SB, the spaceSB provides the appropriate frame for studying structural properties of classes ofBanach spaces. This identification is ultimately related to universality problems inBanach Space Theory. This is justified by a number of results ([AD], [DF], [D] and[DLo]) of which the following one, taken from [DF], is a sample.If A is an analytic subset of SB such that every X ∈ A is reflexive, then there existsa reflexive Banach space Y, with a Schauder basis, that contains isomorphic copiesof every X ∈ A.To see how such a result is used, let us consider the set UC consisting of all X ∈ SBwhich are uniformly convex. It is a classical fact (see [LT]) that UC contains onlyreflexive spaces. Moreover, it is easily checked that UC is a Borel subset of SB. Ap-plying the above result, we recover a recent result of E. Odell and Th. Schlumprecht[OS] asserting the existence of a separable reflexive space R containing an isomor-phic copy of every separable uniformly convex Banach space. The problem of theexistence of such a space was posed by Jean Bourgain [Bou2].(B) As we have already indicated, in applications one has to decide whether a givenclass of separable Banach spaces is analytic or not. Sometimes this is straight-forward to check invoking, simply, the definition of the class. There are classes,however, which are defined implicitly using a certain Banach space operation. Inthese cases, usually, deeper arguments are involved.