Search results for "Hausdorff"

Efficient spatial designs using Hausdorff distances and Bayesian optimization

2021

An iterative Bayesian optimisation technique is presented to find spatial designs of data that carry much information. We use the decision theoretic notion of value of information as the design criterion. Gaussian process surrogate models enable fast calculations of expected improvement for a large number of designs, while the full-scale value of information evaluations are only done for the most promising designs. The Hausdorff distance is used to model the similarity between designs in the surrogate Gaussian process covariance representation, and this allows the suggested algorithm to learn across different designs. We study properties of the Bayesian optimisation design algorithm in a sy…

Genericity of dimension drop on self-affine sets

2017

We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.

2021

Abstract We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY ). We say that a metric space (Y, dY ) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY ) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasi…

On a Continuous Sárközy-Type Problem

2022

Abstract We prove that there exists a constant $\epsilon> 0$ with the following property: if $K \subset {\mathbb {R}}^2$ is a compact set that contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\dim _{\textrm {H}} K \leq 2 - \epsilon $.

Invariants of transverse foliations

2012

Abstract We construct two invariants for a pair of transverse one-dimensional foliations on the plane. If the set of separatrices is Hausdorff in the space of leaves, the invariant is a distinguished graph. In case there are a finite number of separatrices the invariant is an indexed link.

The Daugavet equation for polynomials

2007

In this paper we study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ‖Id + P‖ = 1 + ‖P‖ is satisfied for all weakly compact polynomials P : X −→ X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation max |ω|=1 ‖Id + ω P‖ = 1 + ‖P‖ for polynomials P : X −→ X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. The result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-h…

Relatively weakly open convex combinations of slices

2018

We show that c 0 c_0 and, in fact, C ( K ) C(K) for any scattered compact Hausdorff space K K have the property that finite convex combinations of slices of the unit ball are relatively weakly open.

Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open

2018

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has…

Boundary blow-up under Sobolev mappings

2014

We prove that for mappings $W^{1,n}(B^n, \R^n),$ continuous up to the boundary, with modulus of continuity satisfying certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For H\"older continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate.

Free vs. Locally Free Kleinian Groups

2015

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < < 1 are free. On the other hand we construct for any ε > > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < < 1 + + ε.