Search results for "Open set"
showing 10 items of 40 documents
Covering by discrete and closed discrete sets.
2008
Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $\sigma$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.
Quasi-linear diffusion equations with gradient terms and L1 data
2004
Abstract In this article we study the following quasi-linear parabolic problem: u t − Δ u+|u| β−2 u| ∇ u| q =|u| α−2 u| ∇ u| p in Ω×]0,T[, u(x,t)=0 on ∂Ω×]0,T[, u(x,0)=u 0 (x) in Ω, where Ω is a bounded open set of R N and T>0. We prove that if α,β>1, 0⩽p u 0 ∈L 1 (Ω) .
A Nonlocal Mean Curvature Flow
2019
Consider a family { Γt}t≥0 of hypersurfaces embedded in \(\mathbb {R}^N\) parametrized by time t. Assume that each Γt = ∂Et, the boundary of a bounded open set Et in \(\mathbb {R}^N\).
Envelopes of open sets and extending holomorphic functions on dual Banach spaces
2010
We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.
Measures with predetermined regularity and inhomogeneous self-similar sets
2016
We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding self-similar set $E$ and the condensation set $C$. If the Assouad dimension of $C$ is strictly smaller than the Assouad dimension of $E$, then the upper regularity dimens…
Rectifiability and analytic capacity in the complex plane
1995
Analytic capacity and removable sets In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C . The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense? (19.1) If U is an open set in C containing E and f : U\E → C is a bounded analytic function, then f has an analytic extension to U . This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete so…
Weakly controlled Moran constructions and iterated functions systems in metric spaces
2011
We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.
SURVEY Towards a global view of dynamical systems, for the C1-topology
2011
AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.
In between the inequalities of Sobolev and Hardy
2015
We establish both sufficient and necessary conditions for the validity of the so-called Hardy-Sobolev inequalities on open sets of the Euclidean space. These inequalities form a natural interpolating scale between the (weighted) Sobolev inequalities and the (weighted) Hardy inequalities. The Assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions.
On the existence of attractors
2009
On every compact 3-manifold, we build a non-empty open set $\cU$ of $\Diff^1(M)$ such that, for every $r\geq 1$, every $C^r$-generic diffeomorphism $f\in\cU\cap \Diff^r(M)$ has no topological attractors. On higher dimensional manifolds, one may require that $f$ has neither topological attractors nor topological repellers. Our examples have finitely many quasi attractors. For flows, we may require that these quasi attractors contain singular points. Finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics.