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Displaying 321 – 340 of 343

Traces of Besov spaces on fractal h-sets and dichotomy results

António M. Caetano, Dorothee D. Haroske (2015)

Studia Mathematica

We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there occurs an...

Traces of Oscillating Functions.

Jean-Marie Aubry (1999)

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

Transformations preserving the Hausdorff-Besicovitch dimension

Sergio Albeverio, Mykola Pratsiovytyi, Grygoriy Torbin (2008)

Open Mathematics

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution...

Triangulations of closed sets and bases in function spaces.

Jonsson, Alf (2004)

Annales Academiae Scientiarum Fennicae. Mathematica

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on $ℝ^{d}$ . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

Unimodular Pisot substitutions and their associated tiles

Jörg M. Thuswaldner (2006)

Journal de Théorie des Nombres de Bordeaux

Let $σ$ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_{1}, ..., X_{d}$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_{i}$ ( $1 \leq i \leq d$ ) of these fractals. To this matter we define a certain graph, the so-called contact graph $𝒞$ of $σ$ . If $σ$ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_{1}, ..., \partial X_{d}$ . From this graph...

Uniqueness of Brownian motion on Sierpiński carpets

Martin Barlow, Richard F. Bass, Takashi Kumagai, Alexander Teplyaev (2010)

Journal of the European Mathematical Society

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi´nski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

Upper Estimate of Concentration and Thin Dimensions of Measures

H. Gacki, A. Lasota, J. Myjak (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

Gabriele Bonanno, Giovanni Molica Bisci, Vicenţiu Rădulescu (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

Variational fractals

Umberto Mosco (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Vector-valued fractal interpolation functions and their box dimension.

Peter R. Massopust (1991)

Aequationes mathematicae

Walk dimension and function spaces on self-similar fractals

Katarzyna Pietruska-Pałuba (2009)

Banach Center Publications

We outline the construction of Brownian motion on certain self-similar fractals and introduce the notion of walk dimension. We then show how the probabilistic approach relates to the theory of function spaces on fractals.

Wavelet Analysis and Covariance Structure of some Classes of Non-Stationary Processes.

Charles-Antoine Guérin (2000)

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

Wavelet analysis of the multivariate fractional brownian motion

Jean-François Coeurjolly, Pierre-Olivier Amblard, Sophie Achard (2013)

ESAIM: Probability and Statistics

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behaviour of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral...

Wavelet constructions in non-linear dynamics.

Dutkay, Dorin Ervin, Jørgensen, Palle E.T. (2005)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Wavelet techniques for pointwise regularity

Stéphane Jaffard (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$ , and denoted by $C_{E}^{α} (x_{0})$ . We show how properties of $E$ are transferred into properties of $C_{E}^{α} (x_{0})$ . Applications are given in multifractal analysis.

Wavelets on fractals.

Dorin E. Dutkay, Palle E.T. Jorgensen (2006)

Revista Matemática Iberoamericana

We show that there are Hilbert spaces constructed from the Hausdorff measures Hs on the real line R with 0 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C ⊂ [0,1], the Hilbert space is a separable subspace of L2(R, (dx)s) where s = log3(2). While we develop the general theory of multiresolutions in fractal Hilbert spaces, the emphasis is on the case of scale 3 which covers the traditional Cantor set C.

Wavelets on Fractals and Besov Spaces.

Alf Jonsson (1998)

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

Well-posedness and fractals via fixed point theory.

Chifu, Cristian, Petruşel, Gabriela (2008)

Fixed Point Theory and Applications [electronic only]

Where are typical $C^{1}$ functions one-to-one?

Zoltán Buczolich, András Máthé (2006)

Mathematica Bohemica

Suppose $F \subset [0, 1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^{1} [0, 1]$ is one-to-one on $F$ ? If ${\underset{̲}{dim}}_{B} F < 1 / 2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B} F = 1 / 2$ for which the answer is no. If $C_{α}$ is the middle- $α$ Cantor set we prove that the answer is yes if and only if $dim (C_{α}) \leq 1 / 2 .$ There are $F$ ’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.

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