Related papers: Fractal analysis of Pi normality
Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy…
The statistical precision of a chord method for estimating fractal dimension from a correlation integral is derived. The optimal chord length is determined, and a comparison is made to other estimators. These calculations use the…
We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian…
In this report we present experimental results using \emph{Haussdorf-Besicovich} fractal dimension for performing morphological galaxy classification. The fractal dimension is a topological, structural and spatial property that give us…
Using a large acceptance calorimeter and a stopped pion beam we have made a precise measurement of the rare Pi+ -> Pi0 e+ Nu,(pi_beta) decay branching ratio. We have evaluated the branching ratio by normalizing the number of observed…
A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely…
Quantitative analysis of the structure of star clusters is crucial for understanding their formation and evolution. In this article, we explore the application of fractal dimension analysis to study the evolution of star clusters, also…
Estimates of the fractal dimension $D$ of the set of galaxies in the universe, based on ever improving data sets, tend to settle on $D\approx 2$. This result raised a raging debate due to its glaring contradiction with astrophysical models…
The geometric characteristics of dust clouds provide important information on the physical processes that structure such clouds. One of such characteristics is the $2D$ fractal dimension $D$ of a cloud projected onto the sky plane. In…
Counts of galaxies as a function of apparent magnitude are among the most time-honored observations in cosmology. In this Letter, we focus on some statistical properties of these counts which are fundamental in order to characterize the…
Numerical characteristics of identities of finite-dimensional nonassociative algebras are studied. The main result is the construction of a four-dimensional simple unitary algebra with fractional PI-exponent strictly less than its…
Using a sample of 133 991 galaxies distributed in the sky region $100^{\circ} <\alpha<270^{\circ}$ and $7^{\circ}<\delta<65^{\circ}$, extracted from the SDSS NASA/AMES Value Added Galaxy Catalog (AMES-VAGC), we estimate the fractal…
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals.…
In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are…
A standard scientific study comprises two processes: one is to describe a thing, and the other is to understand how the thing works. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe…
Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(r\pi)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce…
An exact analytical analysis of anomalous diffusion on a fractal mesh is presented. The fractal mesh structure is a direct product of two fractal sets which belong to a main branch of backbones and side branch of fingers. The fractal sets…
Fractal AI is a theory for general artificial intelligence. It allows deriving new mathematical tools that constitute the foundations for a new kind of stochastic calculus, by modelling information using cellular automaton-like structures…
Some recent papers have argued that frequency should have the dimensions of angle/time, with the consequences that 1 Hz = $2\pi$ rad/s instead of 1 s$^{-1}$, and also that $\nu = \omega$ and $h = \hbar$. This letter puts the case that this…
We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and…