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In 1996 Y. Kifer obtained a variational formula for the Hausdorff dimension of the set of points for which the frequencies of the digits in the Cantor series expansion is given. In this note we present a slightly different approach to this…
This work provides the foundation for the finite element analysis of an elliptic problem which is the rotational analogue of the $p$-Laplacian and which appears as a model of the magnetic induction in a high-temperature superconductor…
In this article, we survey recent progress on self-similar $p$-energy forms on self-similar fractals, where $p\in(1,\infty)$. While for $p=2$ the notion of such forms coincides with that of self-similar Dirichlet forms and there have been…
The Feichtinger conjecture for exponentials asserts that the following property holds for every fat Cantor subset B of the circle group: the set of restrictions to B of exponential functions can be covered by Riesz sets. In their seminal…
We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set…
The Cantor set complementary to the Devil's Staircase associated with the Circle Map has a fractal dimension d approximately equal to 0.87, a value that is universal for a wide range of maps, such results being of a numerical character. In…
We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of…
For $\lambda>0$, let $E_{\lambda}$ be the self-similar set generated by the iterated function system (IFS) $\left \{ \frac{x}{3}, \frac{x+\lambda}{3} \right \}$. In this paper we study the structure of parameters $\lambda$ in which…
We establish asymptotics of growing one dimensional self-similar fractal graphs, they are networks that allow multiple weighted edges between nodes, in terms of quantum central limit theorems for algebraic probability spaces in pure state.…
In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this…
We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are…
The stochastic--quantum correspondence reinterprets quantum dynamics as arising from an underlying stochastic process on a configuration space. We generalize the correspondence by lifting an arbitrary stochastic kernel $\Gamma$ in finite…
Fibonacci word fractals are a class of fractals that have been studied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which possesses…
We introduce two fractals, in Euclidean spaces of dimension two and three respectively, such the $2$-conductive homogeneity holds but there is some $\eps \in (0, 1)$ so that the $p$-conductive homogeneity fails for every $p\in (1, 1+\eps)$.…
We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…
Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have…
We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kova\v{c}'s maximal restriction principle to show…
Self-destructive percolation with parameters $p,\delta$ is obtained by taking a site percolation configuration with parameter $p$, closing all sites belonging to infinite clusters, then opening every closed site with probability $\delta$,…
This work investigates the origin of Kardar-Parisi-Zhang (KPZ) scaling in the phase dynamics of one-dimensional and two-dimensional polariton condensates. We demonstrate that the key mechanism leading to the observed power laws for the…
The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. A textural fractal can be employed to explain the scale-free property of regional boundaries such as border lines,…