Search results for "Hausdorff space"
showing 10 items of 64 documents
POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES
2007
We study in this paper the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K,X), and L∞(μ,X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every σ-finite measure μ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k > 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X∗∗ is the least possible, namely k k 1−k . We also give new examples of Banach spaces with the polynomial Daugavet property, namely L∞(μ,X…
Space-filling vs. Luzin's condition (N)
2013
Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not ne…
Conical upper density theorems and porosity of measures
2008
Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.
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…
Regular Minimality and Thurstonian-type modeling
2009
Abstract A Thurstonian-type model for pairwise comparisons is any model in which the response (e.g., “they are the same” or “they are different”) to two stimuli being compared depends, deterministically or probabilistically, on the realizations of two randomly varying representations (perceptual images) of these stimuli. The two perceptual images in such a model may be stochastically interdependent but each has to be selectively dependent on its stimulus. It has been previously shown that all possible discrimination probability functions for same–different comparisons can be generated by Thurstonian-type models of the simplest variety, with independent percepts and deterministic decision ru…
Weak regularity and consecutive topologizations and regularizations of pretopologies
2009
Abstract L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ -compact pretopology. On the other hand, it is proved that for each n ω there is a (regular) pretopology ρ (on a set of cardinality c ) such that ( RT ) k ρ > ( RT ) n ρ for each k n and ( RT ) n ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT -order ⩾ ω 0 . Moreover, all these pretopologies have the property…
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.
Visible parts and dimensions
2003
We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of n, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n−1, we have the almost sure lower bound n−1 for the Hausdorff dimensions of visible parts. We al…
Assouad Type Dimensions in Geometric Analysis
2021
We consider applications of the dual pair of the (upper) Assouad dimension and the lower (Assouad) dimension in analysis. We relate these notions to other dimensional conditions such as a Hausdorff content density condition and an integrability condition for the distance function. The latter condition leads to a characterization of the Muckenhoupt Ap properties of distance functions in terms of the (upper) Assouad dimension. It is also possible to give natural formulations for the validity of Hardy–Sobolev inequalities using these dual Assouad dimensions, and this helps to understand the previously observed dual nature of certain cases of these inequalities. peerReviewed
Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings
2010
Abstract We investigate how the integrability of the derivatives of Orlicz–Sobolev mappings defined on open subsets of R n affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.