Related papers: Dimension and measure for generic continuous image…
We study the properties of "generic", in the sense of the Haar measure on the corresponding Grassmann manifold, subspaces of l^N_infinity of given dimension. We prove that every "well bounded" operator on such a subspace, say E, is a…
We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that…
A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function…
Let $\mu$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions…
We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…
We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following…
We prove the following result: For each closed $n$-dimensional manifold $M$ in a (finite or infinite-dimensional) Banach space $B$, and each positive real $m\leq n$ there exists a pseudomanifold $W^{n+1}\subset B$ such that $\partial…
Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where $X$ is a definable set and $R$ is the universe of the…
For a compact Hausdorff space $K$, we give descriptions of the dual of $C(K)^\delta$, the Dedekind completion of the Banach lattice $C(K)$ of continuous, real-valued functions on $K$. We characterize those functionals which are…
We introduce the generalized upper box dimension which is defined for any set, whether the set is bounded or unbounded. We study basic properties of the generalized upper box dimension. We prove that the generalized upper box and upper box…
In this paper we collect several examples of convergence of functions of random processes to generalized functionals of those processes. We remark that the limit is always finitely absolutely continuous with respect to Wiener measure. We…
This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…
We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the…
We introduce the notion of dynamical metric order of a continuous map on a compact metric space, study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with…
In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus,…
We introduce a family of dimensions, which we call the $\Phi$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by…
The classical Hausdorff dimension of finite or countable metric spaces is zero. Recently, we defined a variant, called \emph{finite Hausdorff dimension}, which is not necessarily trivial on finite metric spaces. In this paper we apply this…
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a…
The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension…
Let $K \subset \mathbb{R}^{2}$ be a rotation and reflection free self-similar set satisfying the strong separation condition, with dimension $\dim K = s > 1$. Intersecting $K$ with translates of a fixed line, one can study the $(s -…