Related papers: Fractals corresponding to the metallic means seque…
Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several…
We prove the existence of fractal solutions to a class of linear ordinary differential equations.This reveals the possibility of chaos in the very short time limit of the evolution even of a linear one dimensional dynamical system.
The interstellar medium is structured as a hierachy of gas clouds, that looks self-similar over 6 orders of magnitude in scales and 9 in masses. This is one of the more extended fractal in the Universe. At even larger scales, the ensemble…
In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This…
Spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal…
Fractals are ubiquitous natural emergences that have gained increased attention in engineering applications, thanks to recent technological advancements enabling the fabrication of structures spanning across many spatial scales. We show how…
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic…
Fractal geometry deals mainly with irregularity and captures the complexity of a structure or phenomenon. In this article, we focus on the approximation of set-valued functions using modern machinery on the subject of fractal geometry. We…
We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a…
The article considers classification task of fractal time series by the meta algorithms based on decision trees. Binomial multiplicative stochastic cascades are used as input time series. Comparative analysis of the classification…
The small-angle scattering curves of deterministic mass fractals are studied and analyzed in the momentum space. In the fractal region, the curve I(q)q^D is found to be log-periodic with a good accuracy, and the period is equal to the…
We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.
This article introduces the novel notion of dimension preserving approximation for continuous functions defined on $[0,1]$ and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed…
In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension. This dimension stands out from the…
Novikov's problem of semiclassical orbits of quasi-electrons in a normal metal leads to a correspondance between 3-ply periodic functions in R and fractals in R P^2. These fractals are the complement of infinitely many open sets labeled by…
Some aspects of Cauchy integrals on sets with dimension larger than 1 are briefly discussed.
We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less…