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The Hausdorff lower semicontinuous envelope of the length in the plane

Raphaël Cerf (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

Przemysław Liszka (2014)

Open Mathematics

Very recently bounds for the L q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities...

The M-components of level sets of continuous functions in WBV.

Coloma Ballester, Vicent Caselles (2001)

Publicacions Matemàtiques

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and...

The mean curvature measure

Quiyi Dai, Neil S. Trudinger, Xu-Jia Wang (2012)

Journal of the European Mathematical Society

We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...

The measure algebra does not always embed

Alan Dow, Klaas Hart (2000)

Fundamenta Mathematicae

The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.

The measure extension problem for vector lattices

J. D. Maitland Wright (1971)

Annales de l'institut Fourier

Let V be a boundedly σ -complete vector lattice. If each V -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a σ -additive measure on the generated σ -field then V is said to have the measure extension property. Various sufficient conditions on V which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: V has the...

The multifractal box dimensions of typical measures

Frédéric Bayart (2012)

Fundamenta Mathematicae

We compute the typical (in the sense of Baire’s category theorem) multifractal box dimensions of measures on a compact subset of ℝ d . Our results are new even in the context of box dimensions of measures.

Currently displaying 61 – 80 of 150