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Mariusz Bialecki

Publications and source records attributed to Mariusz Bialecki.

8 recordsLinked to original sources
arXiv2012

An explanation of the shape of the universal curve of the Scaling Law for the Earthquake Recurrence Time Distributions

This paper presents an explanation of a possible mechanism underlying the shape of the universal curve of Scaling Law for Earthquake Recurrence Time Distributions. The presented simple stochastic cellular automaton model is reproducing the gamma distribution fit with the proper value of the parameter $γ$ characterizing Earth's seismicity and also imitates a deviation from the fit at the short interevent times, as observed in real data. Thus the model suggests an explanation of the universal pattern of rescaled Earthquake Recurrence Time Distributions in terms of combinatorial rules for accumulation and abrupt release of seismic energy.

physics.geo-ph↗
arXiv2012

Finite Random Domino Automaton

Finite version of Random Domino Automaton (FRDA) - recently proposed a toy model of earthquakes - is investigated. Respective set of equations describing stationary state of the FRDA is derived and compared with infinite case. It is shown that for the system of big size, these equations are coincident with RDA equations. We demonstrate a non-existence of exact equations for size N bigger then 4 and propose appropriate approximations, the quality of which is studied in examples obtained within Markov chains framework. We derive several exact formulas describing properties of the automaton, including time aspects. In particular, a way to achieve a quasi-periodic like behaviour of RDA is presented. Thus, based on the same microscopic rule - which produces exponential and inverse-power like distributions - we extend applicability of the model to quasi-periodic phenomena.

nlin.CG↗
arXiv2011

Analytic approach to stochastic cellular automata: exponential and inverse power distributions out of Random Domino Automaton

Inspired by extremely simplified view of the earthquakes we propose the stochastic domino cellular automaton model exhibiting avalanches. From elementary combinatorial arguments we derive a set of nonlinear equations describing the automaton. Exact relations between the average parameters of the model are presented. Depending on imposed triggering, the model reproduces both exponential and inverse power statistics of clusters.

nlin.CG↗
arXiv2006

On pattern structures of the N-soliton solution of the discrete KP equation over a finite field

The existence and properties of coherent pattern in the multisoliton solutions of the dKP equation over a finite field is investigated. To that end, starting with an algebro-geometric construction over a finite field, we derive a "travelling wave" formula for $N$-soliton solutions in a finite field. However, despite it having a form similar to its analogue in the complex field case, the finite field solutions produce patterns essentially different from those of classical interacting solitons.

nlin.SI↗
arXiv2005

Integrable 1D Toda cellular automata

First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation and next we perform a reduction of the dKP equation to the discrete 1D Toda equation. This gives a method of construction of solutions of the discrete 1D Toda equation taking values in a finite field.

nlin.SI↗
arXiv2005

Complex Relativity: Gravity and Electromagnetic Fields

We present new aspects of the electromagnetic field by introducting the natural potentials. These natural potentials are suitable for constructing the first order distortions of the metric tensor of Complex Relativity - the theory combining the General Relativity with the electromagnetic equations. A transition from antisymmetric tensors to the symmetric ones helps to define the natural potentials; their form fits a system of the Dirac matrices and this representation leads to distortion of the metric tensor. Our considerations have originated from the recent progresses in the asymmetric continuum theories. One version of such theories assumes an existence of the antisymmetric strain and stress fields; these fields originate due to some kind of internal friction in a continuum medium which have elastic bonds related to rotations of the particles.

physics.class-ph↗
arXiv2002

The Hirota equation over finite fields. Algebro-geometric approach and multisoliton solutions

We consider the Hirota equation (the discrete analog of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogs of the multisoliton solutions for which the wave functions and the $τ$-function can be found using rational functions. Within the class of multisoliton solutions we isolate generalized breather-type solutions which have no direct counterparts in the complex field case.

nlin.SI↗