Related papers: Spectral Asymptotics for $V$-variable Sierpinski G…
In view of promising applications of fractal nanostructures, we analyze the spectra of quantum particles in the Sierpinski carpet and study the non-correlated electron gas in this geometry. We show that the spectrum exhibits scale…
We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a…
The diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly. The fractals considered are of the class generated iteratively by successive dilations and translations, and include generalizations of the…
We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" : at each…
The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using…
Deterministic and random fractals, within the framework of Iterated Function Systems, have been used to model and study a wide range of phenomena across many areas of science and technology. However, for many applications deterministic…
Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure:…
A recently developed wavelet based approach is employed to characterize the scaling behavior of spectral fluctuations of random matrix ensembles, as well as complex atomic systems. Our study clearly reveals anti-persistent behavior and…
Fractal geometry is the study of sets which exhibit the same pattern at multiple scales. Developing tools to study these sets is of great interest. One step towards developing some of these tools is recognizing the duality between…
One of the ways that analysis on fractals is more complicated than analysis on manifolds is that the asymptotic behavior of the spectral counting function $N(t)$ has a power law modulated by a nonconstant multiplicatively periodic function.…
Fractals are a basic tool to phenomenologically describe natural objects having a high degree of temporal or spatial variability. From a physical point of view the fractal properties of natural systems can also be interpreted by using the…
Following \cite{Visintin}, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we…
V-variable fractals, where $V$ is a positive integer, are intuitively fractals with at most $V$ different "forms" or "shapes" at all levels of magnification. In this paper we describe how V-variable fractals can be used for the purpose of…
We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpi\'{n}ski gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any…
We develop the foundation of the spectral analysis on Barlow-Evans projective limit fractals, or vermiculated spaces, which corresponds to symmetric Markov processes on these spaces. For some new examples, such as the generalized Laakso…
In this work, we examine the relationship between geometry and spectrum of regions with fractal boundary. The relationship is well-understood for fractal harps in one dimension, but largely open for fractal drums in larger dimensions. To…
The Sierpinski gasket is known to support an exotic stochastic process called the asymptotically one-dimensional diffusion. This process displays local anisotropy, as there is a preferred direction of motion which dominates at the…
The fluctuations in the elastic light scattering spectra of normal and dysplastic human cervical tissues analyzed through wavelet transform based techniques reveal clear signatures of self-similar behavior in the spectral fluctuations.…
We investigate the sandpile model on the two--dimensional Sierpinski gasket fractal. We find that the model displays novel critical behavior, and we analyze the distribution functions of avalanche sizes, lifetimes and topplings and…
By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated…