Related papers: Box-Counting Dimension Using Triangles
The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in literature, we do not ask the symbol space to have finite box-counting…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we…
We compute the box dimension of a spiral trajectory around a hyperbolic saddle loop, as the simplest example of a hyperbolic saddle polycycle. In cases of weak foci and limit cycles, Zubrinic and Zupanovic show that the box dimension of a…
This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) $BV_{I}$ all fractal continuous functions with bounded variation is…
Shape is one of the most important visual attributes to characterize objects, playing a important role in pattern recognition. There are various approaches to extract relevant information of a shape. An approach widely used in shape…
We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an…
This article is an introductory work to a larger research project devoted to pure, applied and philosophical aspects of dimension theory. It concerns a novel approach toward an alternate dimension theory foundation: the point-dimension…
The numerical dimension is a numerical measure of the positivity of a pseudo-effective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama (2004) and Boucksom et al. (2004). We prove the equality…
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…
Quantitative analysis of the structure of star clusters is crucial for understanding their formation and evolution. In this article, we explore the application of fractal dimension analysis to study the evolution of star clusters, also…
In this paper, we discuss the Higuchi algorithm which serves as a widely used estimator for the box-counting dimension of the graph of a bounded function $f : [0,1] \to \R$. We formulate the method in a mathematically precise way and show…
Multiple resolution analysis of two dimensional structures composed of randomly adsorbed penetrable rods, for densities below the percolation threshold, has been carried out using box-counting functions. It is found that at relevant…
We examine Frostman-type characterisations and other extremal measure criteria for a range of fractal dimensions of sets. In particular we derive properties of the less familiar modified lower box dimension and upper correlation dimension.…
Mixed labyrinth fractals are dendrites in the unit square introduced by Cristea and Steinsky. They were studied recently by Cristea and Leobacher with respect to the lengths of arcs in the fractals. In this article we first give a…
Algorithmic fractal dimensions -- constructs of computability theory -- have recently been used to answer open questions in classical geometric measure theory, questions of mathematical analysis whose statements do not involve computability…
Several two-dimensional studies in spiral galaxies show that HII star formation regions have a fractal distribution, with a fractal dimension of approximately 2.3. In this work, the fractal dimension is calculated through the box-counting…
This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for…
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several…
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…