Related papers: A class of random Cantor measures, with applicatio…
We show that the Fourier transform of Patterson-Sullivan measures associated to convex cocompact groups of isometries of real hyperbolic space decays polynomially quickly at infinity. The proof is based on the $L^2$-flattening theorem…
This paper presents a comprehensive introduction to the Hausdorff measure, a fundamental tool in fractal geometry and geometric measure theory. We begin by defining the Hausdorff outer measure on subsets of metric spaces, followed by a…
We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.…
We have built a new kind of manifolds which leads to an alternative new geometrical space. The study of the nowhere differentiable functions via a family of mean functions leads to a new characterization of this category of functions. A…
While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is…
Fractals are self-repeating patterns which have dimensions given by fractions rather than integers. While the dimension of a system unambiguously defines its properties, a fractional dimensional system can exhibit interesting properties.…
Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be…
The usual fluid equations describing the large-scale evolution of mass density in the universe can be written as local in the density, velocity divergence, and velocity potential fields. As a result, the perturbative expansion in small…
In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are…
The curves of scaling behavior is a significant concept in fractal dimension analysis of complex systems. However, the underlying rationale of this kind of curves for fractal cities is not yet clear. The aim of this paper is at researching…
We study the spectrum of quantized open maps, as a model for the resonance spectrum of quantum scattering systems. We are particularly interested in open maps admitting a fractal repeller. Using the ``open baker's map'' as an example, we…
The lattice Schwinger model (SM), the discrete version of QED in 1+1 dimensions, is a well-studied test bench for lattice gauge theories. Here we study the fractal properties of the SM. We reveal the self-similarity of the ground state,…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial…
Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes we obtain expressions similar to those of the…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the…
For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta…
We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We…
We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff…
We consider the entanglement properties of free fermions in one dimension and review an approach which relates the problem to the solution of a certain differential equation. The single-particle eigenfunctions of the entanglement…