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In the class of self-affine sets on ℝⁿ we study a subclass for which the geometry is rather tractable. A type is a standardized position of two intersecting pieces. For a self-affine tiling, this can be identified with an edge or vertex type. We assume that the number of types is finite. We study the topology of such fractals and their boundary sets, and we show how new finite type fractals can be constructed. For finite type self-affine tiles in the plane we give an algorithm which decides whether...

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

Self-decomposable probability measures on Banach spaces

A. Kumar, B. Schreiber (1975)

Studia Mathematica

Self-similar centralizers of circle maps.

Cowen, Robert (2001)

Southwest Journal of Pure and Applied Mathematics [electronic only]

Self-similar Jordan arcs and the graph directed systems of similarities.

Tetenov, A.V. (2006)

Sibirskij Matematicheskij Zhurnal

Self-Similar Lattice Tillings.

Karlheinz Gröchenig, Andrew Haas (1994/1995)

The journal of Fourier analysis and applications [[Elektronische Ressource]]

Self-similar random fractal measures using contraction method in probabilistic metric spaces.

Kolumbán, József, Soós, Anna, Varga, Ibolya (2003)

International Journal of Mathematics and Mathematical Sciences

Self-similar simplices.

Hertel, Eike (2000)

Beiträge zur Algebra und Geometrie

Semicontinuity of dimension and measure for locally scaling fractals

L. B. Jonker, J. J. P. Veerman (2002)

Fundamenta Mathematicae

The basic question of this paper is: If you consider two iterated function systems close to each other in an appropriate topology, are the dimensions of their respective invariant sets close to each other? It is well known that the Hausdorff dimension (and Lebesgue measure) of the invariant set does not depend continuously on the iterated function system. Our main result is that (with a restriction on the "non-conformality" of the transformations) the Hausdorff dimension is a lower semicontinuous...

Semicontinuity of the Face-Function of a Covex Set

Victor Klee, Michael Martin (1971)

Commentarii mathematici Helvetici

Semicontinuous integrands as jointly measurable maps

Oriol Carbonell-Nicolau (2014)

Commentationes Mathematicae Universitatis Carolinae

Suppose that $(X, 𝒜)$ is a measurable space and $Y$ is a metrizable, Souslin space. Let $𝒜^{u}$ denote the universal completion of $𝒜$ . For $x \in X$ , let $\underset{̲}{f} (x, \cdot)$ be the lower semicontinuous hull of $f (x, \cdot)$ . If $f : X \times Y \to \bar{ℝ}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable, then $\underset{̲}{f}$ is $(𝒜^{u} \otimes ℬ (Y), ℬ (\bar{ℝ}))$ -measurable.

Semiring of Sets

Roland Coghetto (2014)

Formalized Mathematics

Schmets [22] has developed a measure theory from a generalized notion of a semiring of sets. Goguadze [15] has introduced another generalized notion of semiring of sets and proved that all known properties that semiring have according to the old definitions are preserved. We show that this two notions are almost equivalent. We note that Patriota [20] has defined this quasi-semiring. We propose the formalization of some properties developed by the authors.

Semiring of Sets: Examples

Roland Coghetto (2014)

Formalized Mathematics

This article proposes the formalization of some examples of semiring of sets proposed by Goguadze [8] and Schmets [13].

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