Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.

     

Weak partition properties on trees

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ ^<ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals.     

Billingsley's packing dimension

For a stochastic process on a finite state space, we define the notion of a packing measure based on the naturally defined cylinder sets. For any two measures , , corresponding to the same stochastic process, if

then we prove that

     

Frames for Weighted Shift-invariant Spaces

In this paper we prove the equivalence of the frame property and the closedness for a weighted shift-invariant space $$ V^p_\mu(\Phi) = \left\{\sum \limits^{r}_{i=1} \sum \limits_{j \in \mathbb{Z}^d} c_{i}(j)\phi_{i}(\cdot-j)\left \vert {\{c_{i}(j)\}}_{j \in \mathbb{Z}^{d}} \in {\ell_{\mu}^{p}}\right.\right\}, \quad p \in [1, \infty], $$ which corresponds to ${{\Phi = \Phi^r = (\phi_1, \phi_2, . . . , \phi_r)^T \in (W^{1}_\omega)^r}}$ . We, also, construct a sequence Φ^2k+1 and the sequence of spaces ${{V^{p}_{\mu} (\Phi^{2k+1})}}$ , ${k \in {\mathbb N}}$ , on ${\mathbb R}$ , with the useful properties in sampling, approximations and stability.     

Countably Complementable Linear Orderings

We say that a countable linear ordering is countably complementable if there exists a linear ordering , possibly uncountable, such that for any countable linear ordering , does not embed into if and only if embeds into . We characterize the linear orderings which are countably complementable. We also show that this property is equivalent to the countable version of the finitely faithful extension property introduced by Hagendorf. Using similar methods and introducing the notion of weakly countably complementable linear orderings, we answer a question posed by Rosenstein and prove the countable case of a conjecture of Hagendorf, namely, that every countable linear ordering satisfies the countable version of the totally faithful extension property. This research was partially supported by NSF grant DMS-0600824.     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

Some remarks on totally imperfect sets

We prove the following two theorems.

Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .

     

Uniformly bounded limit of fractional homomorphisms

We show that a bounded homomorphism is equivalent to a uniformly bounded family of fractional homomorphisms for any 0$">. We add this characterization to the Widder-Arendt-Kisynski theorem and relate it to -times integrated semigroups.

     

Some remarks on spreading models and mixed Tsirelson spaces

We prove that if a Banach space with a bimonotone shrinking basis does not contain spreading models but every block sequence of the basis contains a further block sequence which is a spreading model for every , then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space , such that , does not contain spreading models.

     

Spherical classes and the Lambda algebra

Let be Singer's invariant-theoretic model of the dual of the lambda algebra with , where denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, , into is a chain-level representation of the Lannes-Zarati dual homomorphism

The Lannes-Zarati homomorphisms themselves, , correspond to an associated graded of the Hurewicz map

Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in , of positive degree represents the homology class in for 2$">.

We also show that factors through , where denotes the differential of . Therefore, the problem of determining should be of interest.

     

Line Transversals to Translates of Unit Discs

Let be a family of convex figures in the plane. We say that has property T if there exists a line intersecting every member of . Also, the family has property T(k) if every k-membered subfamily of has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family of translates of B has property T(4) then the family , where , has property T. We also give some results concerning families of translates of the unit disc which has either property T(3) or property T(5).     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Sebestyén moment problem: The multi-dimensional case

Given a family of vectors in a Hilbert space we characterize the existence of a family of commuting contractions on having regular dilation and such that

The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .

     

Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

Stabilization of oscillators subject to dry friction: Finite time convergence versus exponential decay results

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion:

where is a smooth potential and is a convex function. The friction is modelized by the subdifferential term . When (dry friction condition), it was shown by Adly, Attouch, and Cabot (2006) that the unique solution to converges in a finite time toward an equilibrium state provided that . In this paper, we study the delicate case where the vector belongs to the boundary of the set . We prove that either the solution converges in a finite time or the speed of convergence is exponential. When , , , we obtain the existence of a critical coefficient below which every solution stabilizes in a finite time. It is also shown that the geometry of the set plays a central role in the analysis.

     

Normalizers of nest algebras

For a nest with associated nest algebra , we define , the normalizer of . We develop a characterization of elements of based on certain order homomorphisms of into itself. This characterization enables us to prove several structure theorems.

     

-actions whose fixed data has a section

Given a collection of real vector bundles over a closed manifold , suppose that, for some is of the form , where is the trivial one-dimensional bundle. In this paper we prove that if is the fixed data of a -action, then the same is true for the Whitney sum obtained from by replacing by . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.     

Combinatorial Aspects of the Splitting Number

This paper deals with the splitting number ${\mathfrak{s}}$ and polarized partition relations. In the first section we define the notion of strong splitting families, and prove that its existence is equivalent to the failure of the polarized relation $$\left(\begin{array}{lll}\mathfrak{s} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}\mathfrak{s} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ . We show that the existence of a strong splitting family is consistent with ZFC, and that the strong splitting number equals the splitting number, when it exists. Consequently, we can put some restriction on the possibility that s is singular. In the second section we deal with the polarized relation under the weak diamond, and we prove that the strong polarized relation $$\left(\begin{array}{lll}2^{\omega} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}2^{\omega} \\ \omega \end{array} \right)}^{1, 1}_{2}$$ is consistent with ZFC, even when cf ${(2^{\omega}) = \aleph_{1}}$ (hence the weak diamond holds).     

Generalized Cuntz-Krieger algebras

To a special embedding of circle algebras having the same spectrum, we associate an r-discrete, locally compact groupoid, similar to the Cuntz-Krieger groupoid. Its -algebra, denoted , is a continuous version of the Cuntz-Krieger algebras . The algebra is generated by an AT-algebra and a nonunitary isometry. We compute its K-theory under the assumption that the AT-algebra is simple.