Related papers: Fractional perimeter from a fractal perspective
The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational…
This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
The Koch snowflake is a classical example of a planar curve with infinite perimeter enclosing a finite, positive area. Although such examples are well known individually, classical treatments typically analyze each construction in isolation…
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern…
We consider the geometry of a class of fractal sets in $\mathbb{R}^{2}$ that generalise the famous Koch curve and Koch snowflake. While the classical Koch curve is defined by an iterative process that divides a line segment into three parts…
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…
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
We discuss the definition and measurability questions of random fractals and find under certain conditions a formula for upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.
The main goal of this paper has a double purpose. On the one hand, we propose a new definition in order to compute the fractal dimension of a subset respect to any fractal structure, which completes the theory of classical box-counting…
Since the recent dissertation by Steffen Winter, for certain self-similar sets $F$ the growth behaviour of the Minkowski functionals of the parallel sets $F_\varepsilon := \{x\in \mathbb R^d : d(x,F)\leq \varepsilon\}$ as $\varepsilon…
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with…
In previous papers by A. Kameyama and by J. Kigami distances on fractals have been discussed having two different but similar properties. One property is that the maps defining the fractal are Lipschitz of prescribed constants less than 1,…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering…
Using ultraproduct techniques we define a nonstandard Minkowski dimension which exists for all bounded sets and which has the property that $\dim(A\times B)=\dim(A)+\dim(B).$ That is, our new dimension is product-summable. To illustrate our…
In the present paper we define statistically self-similar sets, and, using a modification of method described K.J.Falconer find a Hausdorff dimension of a statistically self-similar set.
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…
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…
Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures…