Related papers: Counting-Based Effective Dimension and Discrete Re…
With the decline of the Copenhagen interpretation of quantum mechanics and the recent experiments indicating that quantum mechanics does actually embody 'objective reality', one might ask if a 'mechanical', conceptual model for quantum…
Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several…
Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the…
The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical…
We show that the dimension of the geometric shape formed by the phenomenologically valid points inside a multi-dimensional parameter space can be used to characterise different new physics models and to define a quantitative measure for the…
Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time, when travelling…
High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a…
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…
Multiple resolution analysis of two dimensional structures composed of randomly adsorbed penetrable rods, for densities below the percolation threshold, has been carried out using box-counting functions. It is found that at relevant…
We propose a novel method to measure the fractal dimension of a submonolayer metal adatom system grown under conditions of limited diffusivity on a surface. The method is based on measuring the specular peak attenuation of He atoms…
Developing a robust generalization measure for the performance of machine learning models is an important and challenging task. A lot of recent research in the area focuses on the model decision boundary when predicting generalization. In…
This work joins aspects of reservoir optimization, information-theoretic optimal encoding, and at its center fractal analysis. We build on the observation that, due to the recursive nature of recurrent neural networks, input sequences…
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,…
While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is…
Decomposing predictive uncertainty into epistemic (model ignorance) and aleatoric (data ambiguity) components is central to reliable decision making, yet most methods estimate both from the same predictive distribution. Recent empirical and…
Small-angle scattering (SAS) of X-rays, neutrons or light from ensembles of randomly oriented and placed deterministic fractal structures are studied theoretically. In the standard analysis, a very few parameters can be determined from SAS…
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…
In the report submitted here, an elementary formula is presented that allows calculation of the fine structure constant, derived on the basis of the unified theory of the electromagnetic and gravitational field elaborated by the author.…
Atomic-resolution imaging with scanning transmission electron microscopy is a powerful tool for characterizing the nanoscale structure of materials, in particular features such as defects, local strains, and symmetry-breaking distortions.…
We investigate the extrinsic geometry of causal sets in $(1+1)$-dimensional Minkowski spacetime. The properties of boundaries in an embedding space can be used not only to measure observables, but also to supplement the discrete action in…