Related papers: Fractal Intersections and Products via Algorithmic…
IFS fractals - the attractors of Iterated Function Systems - have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design…
We provide the first known upper bounds for the packing dimension of weighted singular and weighted $\omega$-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the…
Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence…
We compute the Hausdorff dimension of sets defined by the growth of weighted products of multiple digits at arbitrary positions in $d$-decaying Gauss-like iterated function systems. We provide the complete Hausdorff dimensional result for…
Developing a robust generalization measure for the performance of machine learning models is an important and challenging task. A lot of recent research in the area focuses on the model decision boundary when predicting generalization. In…
We determine the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity by renormalizing appropriate composite operators. We use these results to deduce the fractal…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
The concept of fractal index is introduced in connection with the idea of universal class $h$ of particles or quasiparticles, termed fractons, which obey fractal statistics. We show the relation between fractons and conformal field…
Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the…
A simple multifractal coarsening model is suggested that can explain the observed dynamical behavior of the fractal dimension in a wide range of coarsening fractal systems. It is assumed that the minority phase (an ensemble of droplets) at…
The interstellar medium seems to have an underlying fractal structure which can be characterized through its fractal dimension. However, interstellar clouds are observed as projected two-dimensional images, and the projection of a…
We present some work relating to fractal transformations on masked iterated function systems and demonstrate how well known algorithms for generating fractal transformations can be modifed for these systems. We also demonstrate that these…
By a "happy fractal" we mean a metric space with bounded geometry in the sense of a doubling condition and a lot of paths of finite length, so that any pair of points can be connected by a path whose length is less than or equal to a…
This paper investigates a class of deterministic fractals whose construction is governed by arithmetic sequences. We introduce the essential fractal prime set P_{ess} , a variant of the Cantor set constructed using the sequence of prime…
Measuring the complexity of high-dimensional data in physical systems becomes a critical factor in determining the information and quality of the systems. However, traditional metrics, such as Lyapunov exponent, fractal dimension, and…
Intricate comparison between two given tertiary structures of proteins is as important as the comparison of their functions. Several algorithms have been devised to compute the similarity and dissimilarity among protein structures. But,…
Perfect fractals are mathematical objects that, because they are generated by recursive processes, have self-similarity and infinite complexity. In particular, they also have a fractional dimension. Although several proposals for the study…
The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical…
Fractal geometry, defined by self-similar patterns across scales, is crucial for understanding natural structures. This work addresses the fractal inverse problem, which involves extracting fractal codes from images to explain these…
The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space…