相关论文: On the Structure of the Littlewood Polynomials and…
The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper we present a self-contained and comprehensive introduction to the fractal dimension. We…
We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y=…
These are the lecture notes for a course at the Summer School on "Applied Analysis" at the Technical University Chemnitz in September 2011. We start with the definition of a fractal algebra and show that the fractal property is enormously…
The original notion of dimension for posets was introduced by Dushnik and Miller in 1941 and has been studied extensively in the literature. In 1992, Brightwell and Scheinerman developed the notion of fractional dimension as the natural…
We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically…
We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded…
We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…
Fractal surfaces are ubiquitous in nature as well as in the sciences. The examples range from the cloud boundaries to the corroded surfaces. Fractal dimension gives a measure of the irregularity in the object under study. We present a…
Recent advances in deep learning have transformed many fields by introducing generic embedding spaces, capable of achieving great predictive performance with minimal labeling effort. The geology field has not yet met such success. In this…
This work presents a new Visual Basic 6.0 application for estimating the fractal dimension of images, based on an optimized version of the box-counting algorithm. Following the attempt to separate the real information from noise, we…
We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this…
Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with…
We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(\mu)$ for each homological dimension $i\ge 0$, assigned to a probability measure $\mu$ on a metric space. The case of…
We consider the problem of localizing relevant subsets of non-rigid geometric shapes given only a partial 3D query as the input. Such problems arise in several challenging tasks in 3D vision and graphics, including partial shape similarity,…
A type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between…
Using the concept of self similarity in the structure of the proton at small $x$, we comment on possibility of a single positive fractal dimension of proton in analogy with classical monofractals. Plausible dynamics and physical…
This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its…
In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.
We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…