Related papers: Dynamical Systems, Fractal Geometry and Diophantin…
The symmetry reduction of dynamical systems that are invariant under changes of global scale is well-understood for classical theories of particles, and fields. The excision of the superfluous degree of freedom generating such rescalings…
In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents…
We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large…
Symmetry properties of stochastic dynamical systems described by stochastic differential equation of Stratonovich type and related conserved quantities are discussed, extending previous results by Misawa. New conserved quantities are given…
For a spatial characteristic, there exist commonly fat-tail frequency distributions of fragment-size and -mass of glass, areas enclosed by city roads, and pore size/volume in random packings. In order to give a new analytical approach for…
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with…
An elementary system leading to the notions of fractional integrals and derivatives is considered. Various physical situations whose description is associated with fractional differential equations of motion are discussed.
A number of dynamical system formulations have been proposed over the last few years to analyse cosmological solutions in $f(R)$ gravity. The aim of this article is to provide a brief introduction to the different approaches, presenting…
We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the…
The Hausdorff dimensions of certain sets of real numbers described in terms of the \alpha-L\"uroth expansion are given.
We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian…
We briefly review our recent results on the geometry of nonholonomic manifolds and Lagrange--Finsler spaces and fractional calculus with Caputo derivatives. Such constructions are used for elaborating analogous models of fractional gravity…
We propose a new and general method for deriving exact density functionals in one dimension for lattice gases with finite-range pairwise interactions. Corresponding continuum functionals are derived by applying a proper limiting procedure.…
By appropriate scaling of coupling constants a one-parameter family of ensembles of two-dimensional geometries is obtained, which interpolates between the ensembles of (generalized) causal dynamical triangulations and ordinary dynamical…
We investigate numerically and experimentally dynamical systems having three interacting frequencies: a discrete mapping (a circle map), an exactly solvable model (a system of coupled ordinary differential equations), and an experimental…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
In this article we study the energy level spectrum of fractals which have block-hierarchical structures. We develop a method to study the spectral properties in terms of linearization of spectral decimation procedure and verify it…
Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed…
In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…
This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.