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We study, both with numerical simulations and theoretical methods, a cellular automata model for continuum equations describing growth processes in the presence of an external flux of particles. As a result of local instabilities we find a…
City size distributions are known to be well approximated by power laws across a wide range of countries. But such distributions are also meaningful at other spatial scales, such as within certain regions of a country. Using data from…
In the interstellar medium, as well as in the Universe, large density fluctuations are observed, that obey power-law density distributions and correlation functions. These structures are hierarchical, chaotic, turbulent, but are also…
We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…
We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what…
We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what…
Galaxies and clusters distributions show two major properties: (i) the positions of galaxies and clusters are characterized by a power law distribution indicating properties with respect to their positions. (ii) The distribution of masses…
We consider a plate made from an isotropic but brittle elastic material, which is used to span a rigid aperture, across which a small pressure difference is applied. The problem we address is to find the structure which uses the least…
Second order perturbation theory predicts a specific dependence of the bispectrum, or three-point correlation function in the Fourier transform domain, on the shape of the configuration of its three wave vector arguments, which can be taken…
We investigate the geometry of the critical fluctuations for a general system undergoing a thermal second order phase transition. Adopting a generalized effective action for the local description of the fluctuations of the order parameter…
We suggest a new method for estimating the fractal dimension of the spatial distribution of galaxies, the method of selected cylinders. We show the capabilities of this method by constructing a two-point conditional column density for…
Problems of calculus of variations with variable endpoints cannot be solved without transversality conditions. Here, we establish such type of conditions for fractional variational problems with the Caputo derivative. We consider: the…
This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a…
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial…
We have built a new kind of manifolds which leads to an alternative new geometrical space. The study of the nowhere differentiable functions via a family of mean functions leads to a new characterization of this category of functions. A…
We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes,…
We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie's modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free…
A companion paper presents a worked model for evolution through inflation to initial conditions for an isocurvature model for structure formation. It is shown here that the model is consistent with the available observational constraints…
Viscoelasticity and related phenomena are of great importance in the study of mechanical properties of material especially, biological materials. Certain materials show some complex effects in mechanical tests, which cannot be described by…