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Although geographic features, such as mountains and coastlines, are fractal, some studies have claimed that the fractal property is not universal. This claim, which is false, is mainly attributed to the strict definition of fractal…
The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using…
We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We…
The present work shows a novel fractal dimension method for shape analysis. The proposed technique extracts descriptors from the shape by applying a multiscale approach to the calculus of the fractal dimension of that shape. The fractal…
We use the self-similar tilings constructed by the second author in "Canonical self-affine tilings by iterated function systems" to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta…
The fractal dimension is a central quantity in nonlinear dynamics and can be estimated via several different numerical techniques. In this review paper we present a self-contained and comprehensive introduction to the fractal dimension. We…
In this paper, we define new fractal dimensions of a metric space in order to calculate the roughness of a set on a large scale. These fractal dimensions are called upper zeta dimension and lower zeta dimension. The upper zeta dimension is…
We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring…
In this work, we demonstrate how differentiable stochastic sampling techniques developed in the context of deep Reinforcement Learning can be used to perform efficient parameter inference over stochastic, simulation-based, forward models.…
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.…
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…
We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…
In Koeller \cite{koerprops} the twelve variants of the Reifenberg properties known to be instrumental in the theory of minimal surfaces were classified with respect to various Hausdorff measure based measure theoretic properties. The…
We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(\mu)$ for each homological dimension $i\ge 0$, assigned to a probability measure $\mu$ on a metric space. The case of…
The fractal dimension of large-scale galaxy clustering has been demonstrated to be roughly $D_F \sim 2$ from a wide range of redshift surveys. If correct, this statistic is of interest for two main reasons: fractal scaling is an implicit…
The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the…
Fractal behavior and long-range dependence have been observed in an astonishing number of physical systems. Either phenomenon has been modeled by self-similar random functions, thereby implying a linear relationship between fractal…
The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal…
Fractal dimension is an effective scaling exponent of characterizing scale-free phenomena such as cities. Urban growth can be described with time series of fractal dimension of urban form. However, how to explain the factors behind fractal…