Related papers: Geometric functionals of fractal percolation
The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the $q$-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative…
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the…
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 type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between…
We give a geometrically exact treatment of percolation through voids around assemblies of randomly placed impermeable barrier particles, introducing a computationally inexpensive approach to finding critical barrier density thresholds…
Real networks are vulnerable to random failures and malicious attacks. However, when a node is harmed or damaged, it may remain partially functional, which helps to maintain the overall network structure and functionality. In this paper, we…
A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal…
The basic notion of percolation in physics assumes the emergence of a giant connected (percolation) cluster in a large disordered system when the density of connections exceeds some critical value. Until recently, the percolation phase…
We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the…
In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modelling of phenomena exhibiting a complex self-referential…
In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several…
The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot…
We estimate a Box-counting dimension of fractal surfaces which are generated by iterated function systems with a vertical contraction factor function on an arbitrary data set over rectangular grids and can express well a lot of natural…
Fractals have been at the heart of geophysical and geospatial studies in the recent past. We examine the emergent fractal character of water vapor distributions above the surface of the Earth as a function of both image resolution (number…
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 review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions.…
Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied…
In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the…
Cohesive particles form agglomerates that are usually very porous. Their geometry, particularly their fractal dimension, depends on the agglomeration process (diffusion-limited or ballistic growth by adding single particles or…
Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of…