Tootfinder

On Complex Dimensions and Heat Content of Self-Similar Fractals
William E. Hoffer, Michel L. Lapidus
https://arxiv.org/abs/2508.09512 https://arxiv.org/pdf…

On Complex Dimensions and Heat Content of Self-Similar Fractals
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 the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allow for non-disjoint over…

Hausdorff measure of cartesian product of Cantor sets
Siyuan Guo, Taylor Jones
https://arxiv.org/abs/2510.09701 https://arxiv.org/pdf/2510.09701

Hausdorff measure of cartesian product of Cantor sets
Hausdorff measure and Hausdorff dimension are useful tools to describe fractals. This paper investigates the bounds on the $d\log_32$-dimensional Hausdorff measure of the $d$-fold Cartesian product of the $1/3$ Cantor set, $\mathcal C^d$. By applying known theorems on the Hausdorff measure of fractals satisfying the strong open set condition and generalizing what has been done on $\mathcal C^2$, we compute stricter upper and lower bounds for the Hausdorff measure of $\mathcal C^d$ for several s…

Quantum-Circuit-Based Visual Fractal Image Generation in Qiskit and Analytics
Hillol Biswas
https://arxiv.org/abs/2508.18835 https://arxiv.org/pdf/2508.188…

Quantum-Circuit-Based Visual Fractal Image Generation in Qiskit and Analytics
As nature is ascribed as quantum, the fractals also pose some intriguing appearance which is found in many micro and macro observable entities or phenomena. Fractals show self-similarity across sizes; structures that resemble the entire are revealed when zoomed in. In Quantum systems, the probability density or wavefunction may exhibit recurring interference patterns at various energy or length scales. Fractals are produced by basic iterative rules (such as Mandelbrot or Julia sets), and they p…

Non-existence of several random fractals in the Brownian motion and the Brownian loop soup
Yifan Gao, Xinyi Li, Runsheng Liu, Wei Qian
https://arxiv.org/abs/2507.20324 https://

Non-existence of several random fractals in the Brownian motion and the Brownian loop soup
We develop a unified approach to establish the non-existence of three types of random fractals: (1) the pioneer triple points of the planar Brownian motion, answering an open question in [7], (2) the pioneer double cut points of the planar and three-dimensional Brownian motions, and (3) the double points on the boundaries of the clusters of the planar Brownian loop soup at the critical intensity, answering an open question in [39]. These fractals have the common feature that they are associated…

The dimension of well approximable numbers
Victor Beresnevich, Sanju Velani
https://arxiv.org/abs/2509.05687 https://arxiv.org/pdf/2509.05687

The dimension of well approximable numbers
In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming from Besicovitch's result, with a focus on the Mass Transference Principle, Ubiquity and Diophantine approximation on manifolds and fractals. We highlight the subtle yet profound connections between number theory and fractal geometry, and discuss several open p…

Scattering of Antiplane Shear Waves by Fractals in Strain Gradient Elasticity
E. Alevras, Th. Zisis, P. A. Gourgiotis
https://arxiv.org/abs/2507.18768 https://

Scattering of Antiplane Shear Waves by Fractals in Strain Gradient Elasticity
Wave manipulation is essential in various applications, including seismic wave protection, sound isolation, and acoustic device design. This study examines the scattering and trapping of antiplane SH waves in a microstructured solid embedded with rigid pins. The material's response is governed by the theory of strain gradient elasticity. A method is proposed for identifying the wavenumbers that induce resonance in the elastic body, based on specific material parameters and pin configurations. T…

Max-Min and 1-Bounded Space Algorithms for the Bin Packing Problem
Hiroshi Fujiwara, Rina Atsumi, Hiroaki Yamamoto
https://arxiv.org/abs/2508.18718 https://

Max-Min and 1-Bounded Space Algorithms for the Bin Packing Problem
In the (1-dimensional) bin packing problem, we are asked to pack all the given items into bins, each of capacity one, so that the number of non-empty bins is minimized. Zhu~[Chaos, Solitons \& Fractals 2016] proposed an approximation algorithm $MM$ that sorts the item sequence in a non-increasing order by size at the beginning, and then repeatedly packs, into the current single open bin, first as many of the largest items in the remaining sequence as possible and then as many of the smallest it…

Replaced article(s) found for cond-mat.dis-nn. https://arxiv.org/list/cond-mat.dis-nn/new
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- Emergence of non-ergodic multifractal quantum states in geometrical fractals
Fabio Salvati, Mikhail I. Katsnelson, Andrey A. Bagrov

What does the tree of life look like as it grows? Evolution and the multifractality of time
Kevin Hudnall, Raissa D'Souza
https://arxiv.org/abs/2509.25615 https://

What does the tree of life look like as it grows? Evolution and the multifractality of time
By unifying three foundational principles of modern biology, we develop a mathematical framework to analyze the growing tree of life. Contrary to the static case, where the analogy between phylogenetic trees and the tree that grows in soil is drawn, our framework shows that the living tree of life is analogous to a Cantor dust where each branch is a distinct fractal curve. The system as a whole is therefore multifractal in the sense that it consists of many unique fractals. The three foundation…