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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…
Given strong local Dirichlet forms and $\mathbb{R}^N$-valued functions on a metrizable space, we introduce the concepts of geodesic distance and intrinsic distance on the basis of these objects. They are defined in a geometric and an…
Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measures on the sets and may be used to carry well known flows and…
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…
Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for…
This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schr\"{o}dinger Equation and provides insights into its consequences. The study presents a General Solution for…
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.
In this paper, we construct new multifractal measures, on the Euclidean space $\mathbb{R}^n$, in a similar manner to Hewitt-Stomberg measures but using the class of all $n$-dimensional half-open binary cubes of covering sets in the…
We develop a new definition of fractals which can be considered as an abstraction of the fractals determined through self-similarity. The definition is formulated through imposing conditions which are governed the relation between the…
Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original abstract: An integral…
Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size.…
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…
There are many research available on the study of real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for vector-valued fractal interpolation…
The depth function of three numbers representing curvatures of three mutually tangent circles is introduced. Its 2D plot leads to a partition of the moduli space of the triples of mutually tangent circles/disks that is unexpectedly a…
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…
While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is…
We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on…
The term fractal describes a class of complex structures exhibiting self-similarity across different scales. Fractal patterns can be created by using various techniques such as finite subdivision rules and iterated function systems. In this…