Related papers: On the Structure of the Littlewood Polynomials and…
In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modelling of phenomena exhibiting a complex self-referential…
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are…
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.
Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will…
In recent years there has been much interest -and progress- in understanding projections of many concrete fractals sets and measures. The general goal is to be able to go beyond general results such as Marstrand's Theorem, and quantify the…
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost…
A way to add an extra dimension is briefly discussed.
The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis.
We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal…
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…
The fractal dimension of a surface allows its degree of roughness to be characterized quantitatively. However, limited effort is attempted to calculate the fractal dimension of surfaces computed from precisely known atomic coordinates from…
Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions…
The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored…
We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close…
Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and…
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 offer the ability to generate fascinating geometric shapes with all sorts of unique characteristics (for instance, fractal geometry provides a basis for modelling infinite detail found in nature). While fractals are non-euclidean…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model…