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Displaying 201 – 220 of 344

On the 1/2 Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth.

Hany M. Farag (2002)

Revista Matemática Iberoamericana

In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...

On the Absolute Continuity of a Surface Representation

Hans Martin Reimann (1971)

Commentarii mathematici Helvetici

On the additivity of Gudder integral

Jozef Dravecký, Ján Šipoš (1980)

Mathematica Slovaca

On the analytic capacity and curvature of some Cantor sets with non-σ-finite length.

Pertti Mattila (1996)

Publicacions Matemàtiques

We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫01 r−3 h(r)2 dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.

On the asymptotics of counting functions for Ahlfors regular sets

Dušan Pokorný, Marc Rauch (2022)

Commentationes Mathematicae Universitatis Carolinae

We deal with the so-called Ahlfors regular sets (also known as $s$ -regular sets) in metric spaces. First we show that those sets correspond to a certain class of tree-like structures. Building on this observation we then study the following question: Under which conditions does the limit ${lim}_{ε \to 0 +} ε^{s} N (ε, K)$ exist, where $K$ is an $s$ -regular set and $N (ε, K)$ is for instance the $ε$ -packing number of $K$ ?

On the average of inner and outer measures

D. Fremlin (1991)

Fundamenta Mathematicae

On the Basic Extension Theorem in Measure Theory.

Heinz König (1985)

Mathematische Zeitschrift

On the Carathéodory method of the extension of measures and integrals

Beloslav Riečan (1977)

Mathematica Slovaca

On the classification of Markov chains via occupation measures

Onésimo Hernández-Lerma, Jean Lasserre (2000)

Applicationes Mathematicae

We consider a Markov chain on a locally compact separable metric space $X$ and with a unique invariant probability. We show that such a chain can be classified into two categories according to the type of convergence of the expected occupation measures. Several properties in each category are investigated.

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

On the complexification of the Weierstrass non-differentiable function.

Barański, Krzysztof (2002)

Annales Academiae Scientiarum Fennicae. Mathematica

On the complexity of some $σ$ -ideals of $σ$ -P-porous sets

Luděk Zajíček, Miroslav Zelený (2003)

Commentationes Mathematicae Universitatis Carolinae

Let $𝐏$ be a porosity-like relation on a separable locally compact metric space $E$ . We show that the $σ$ -ideal of compact $σ$ - $𝐏$ -porous subsets of $E$ (under some general conditions on $𝐏$ and $E$ ) forms a $Π_{1}^{1}$ -complete set in the hyperspace of all compact subsets of $E$ , in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. It is shown in the cases of the $σ$ -ideals of $σ$ -porous sets, $σ$ - $〈 g 〉$ -porous sets, $σ$ -strongly porous sets, $σ$ -symmetrically porous sets...

On the complexity of sums of Dirichlet measures

Sylvain Kahane (1993)

Annales de l'institut Fourier

Let $M$ be the set of all Dirichlet measures on the unit circle. We prove that $M + M$ is a non Borel analytic set for the weak* topology and that $M + M$ is not norm-closed. More precisely, we prove that there is no weak* Borel set which separates $M + M$ from $D^{⊥}$ (or even $L_{0}^{⊥})$ , the set of all measures singular with respect to every measure in $M$ . This extends results of Kaufman, Kechris and Lyons about $D^{⊥}$ and $H^{⊥}$ and gives many examples of non Borel analytic sets.

On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (2013)

Annales scientifiques de l'École Normale Supérieure

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X, d)$ , and its conformal dimension ${dim}_{A R} (X, d)$ . Using a sequence of finite coverings of $(X, d)$ , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute ${dim}_{A R} (X, d)$ using the critical exponent $Q_{N}$ associated to the combinatorial modulus.

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