Related papers: Scale-dependent Hausdorff dimensions in 2d gravity
We devise a geometric description of bounded systems at criticality in any dimension $d$. This is achieved by altering the flat metric with a space dependent scale factor $\gamma(x)$, $x$ belonging to a general bounded domain $\Omega$.…
We review the theoretical framework that establishes a crucial bridge between the general Steiner-type formula of Hug, Last, and Weil and the theory of complex (fractal) dimensions of Lapidus et all. Two novel families of geometric…
Universal scaling laws of fluctuations (the $\Delta$-scaling laws) can be derived for equilibrium and off-equilibrium systems when combined with the finite-size scaling analysis. In any system in which the second-order critical behavior can…
We formulate matter-coupled scaling-Carroll gravity as a gauge theory and analyze its associated gravity multiplet. After fixing the scaling symmetry, the theory is governed by the trace of the extrinsic curvature, the Carroll boost…
The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its…
Three-dimensional scalar electrodynamics, with a local U(1) gauge symmetry, is believed to be dual to a scalar theory with a global U(1) symmetry, near the phase transition point. The conjectured duality leads to definite predictions for…
Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical…
Many fluctuating systems consist of macroscopic structures in addition to noisy signals. Thus, for this class of fluctuating systems, the scaling behaviors are very complicated. Such phenomena are quite commonly observed in Nature, ranging…
As for the remarkable study on the estimate of the Hausdorff dimension of a self-similar set due to weak contractions (Kitada A. et al. Chaos, Solitons & Fractals 13 (2002) 363-366), we present a mathematically simplified form which will be…
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…
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…
We consider a dynamical triangulation model of euclidean quantum gravity where the topology is not fixed. This model is equivalent to a tensor generalization of the matrix model of two dimensional quantum gravity. A set of moves is given…
The optical spectra of fractal multilayer dielectric structures have been shown to possess spectral scalability, which has been found to be directly related to the structure's spatial (geometrical) self-similarity. Phase and amplitude…
We investigate a model of two-dimensional gravity with arbitrary scalar potential obtained by gauging a deformation of de Sitter or more general algebras, which accounts for the existence of an invariant energy scale. We obtain explicit…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
We study the dissipative bi-stable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff…
Within the causal dynamical triangulations approach to the quantization of gravity, striking evidence has emerged for the dynamical reduction of spacetime dimension on sufficiently small scales. Specifically, the spectral dimension…
We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain…
To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is…
Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In…