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Markovian diffusion processes yield a system of conservation laws which couple various conditional expectation values (local moments). Solutions of that closed system of deterministic partial differential equations stand for a regular alternative to erratic (irregular) sample paths that are associated with weak solutions of the original stochastic differential equations. We investigate an issue of local characteristics of motion in the non-Gaussian context, when moments of the probability measure may not exist. A particular emphasis is put on jump-type stochastic processes with the Ornstein– –Uhlenbeck–Cauchy process as a fully computable exemplary case.

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We derive explicit forms of Markovian transition probability densities for the velocity space and phase-space Brownian motion of a charged particle in a constant magnetic field.

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The incessantly growing area of applications of Clifford algebras and naturalness of their use in formulating problems for direct calculation entitles one to call them Clifford numbers. The generalized “universal” Clifford numbers are here introduced via k-ubic form \(Q_k\) replacing quadratic one in familiar construction of an appropriate ideal of tensor algebra. One of the epimorphic images of universal algebras \(k-C_n\cong T(V)\)/\(I(Q_k)\) is the algebra \(Cl_n^{(k)}\) with n generators and these are the algebras to be used here. Because generalized Clifford algebras \(Cl_n^{(k)}\) possess inherent \(Z_k\oplus Z_k\oplus {\mit \Lambda }\oplus Z_k \) grading — this makes them an efficient apparatus to deal with spin lattice systems. This efficiency is illustrated here by derivation of two major observations. Namely — partition functions for vector and planar Potts models and other model with \(Z_n\) invariant Hamiltonian are polynomials in generalized hyperbolic functions of the n-th order. Secondly, the problem of algorithmic calculation of the partition function for any vector Potts model as treated here is reduced to the calculation of Tr\((\gamma _{i_1}...\gamma _{i_s})\), where \(\gamma \)’s are the generators of the generalized Clifford algebra. Finally the expression for Tr\((\gamma _{i_1}...\gamma _{i_s})\), for arbitrary collection of such \(\gamma \) matrices is derived.

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Let \(({\cal A}, \{T_t\},\omega )\) be a dynamical system. Assume the detailed balance condition for \((\{T_t\}, \omega )\). We prove, under the new form of the spectral condition, the property of return to equilibrium for the considered dynamical system.

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Stochastic resonance is investigated in a chain of unidirectionally coupled threshold elements driven by independent noises and a plane travelling wave. Both stochastic resonance in an individual element embedded in the chain, characterized by a maximum of the signal-to-noise ratio for nonzero noise intensity, and stochastic resonance with spatiotemporal signal, characterized by a maximum of a spatiotemporal input–output correlation function, are observed. Both kinds of stochastic resonance can be enhanced due to proper coupling, although this effect is weaker than for bidirectional coupling and occurs for a smaller range of wavelengths of the plane wave. The enhancement is related to a maximum spatiotemporal synchronization among elements with the same phase of the periodic signal at input.

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A time series that represents daily values of the WIG index (the main index of Warsaw Stock Exchange) over last 5 years is examined. Non-Gaussian features of distributions of fluctuations, namely returns, over a time scale are considered. Some general properties like exponents of the long range correlation estimated by averaged volatility and detrended fluctuations analysis (DFA) as well as exponents describing a decay of tails of the cumulative distributions are found. Closing, the Zipf analysis for the WIG index time series translated into three letter text is presented.

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We present analytical and numerical studies of a chaotic model of a kicked magnetic moment (spin) in the presence of anisotropy and damping. There is an influence of the fractal structure of attractors and basins of attraction on mean transient lifetimes near chaotic crises and on noise-free stochastic resonance in this system. The observed oscillations of average transient times emerging on the background of the well-known power scaling law can be explained by simple geometric models of overlapping fractal sets. Using as the control parameter the amplitude of magnetic field pulses one finds that such measures of stochastic resonance as the input–output correlation function or the signal-to-noise ratio show multiple maxima characteristic of stochastic multiresonance. A simple adiabatic theory which takes into account the fractal structures of this model well explains numerical simulations.

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An analytical approach to the \(d\)-dimensional grain growth, which is a kind of the heterogeneous nucleation-and-growth phase transformation, is offered. The system is assumed to be driven by capillary forces. Another important operative assumption is that the system evolves under preservation of its hypervolume, which results in considering the process as a random walk in the space of grain sizes. A role of the initial condition imposed on the system behaviour, and how does the system behave upon a prescribed initial state, have been examined. A general conclusion appears, which states that this prescription does not affect the asymptotic system behavior, but may be of importance when inspecting the early-time domain more carefully, cf. the Weibull-type initial distribution. This study is addressed to some analogous theoretical descriptions concerning polycrystals as well as bubbles-containing systems. Some comparison to another modelling, in which a crucial role of local material gradients (fluxes) was emphasized, has been attached.

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Intermittency in time series of the time intervals between heart beats (RR intervals) extracted from 24 hour (portable) ECG is found for some cases of humans with arrhythmia. Laminar phases are found by sweeping a short (5 intervals) time window through the time series and calculating the standard deviation of the series in each window. 8 of the 18 arrhythmia cases studied had a bimodal distribution of the standard deviation values indicating some kind of intermittency. The distribution of lengths of the laminar phases identifies the intermittency obtained in human heart rate variability as Type I in the Pomeau and Manneville classification. Although the arrhythmia cases studied were medically very different — in those instances that intermittency did occur the probability distributions of laminar phase lengths were strikingly similar.

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A Random Walk (RW) realization in the square lattice, upon which a percolation cluster of sites, visited one by one by random walkers is built up (by direct Monte Carlo method), has been carried out towards its basic tendencies. It turns out that if the RW is realized near the site-percolation threshold, the process, as expected, decelerates. If, in turn, one systematically goes above the percolation threshold, being roughly about \(0.6\), towards the isotropic site-cluster regime, the process accelerates. Some drift superimposed on the RW realization as well as boundary conditions of certain types change the system behavior in a quite predictive way. Both new and interesting examples, emphasizing a possible applications of the phenomenon under study, are carefully mentioned. A finite-size effect always incorporated in the realized MC-algorithm is going to make the process apparently closer to reality. The notion of continuous phase (sub)transition has been discussed in the presented context.

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The family of model (prototype) structures (Sierpiński carpet, Sierpiński gasket, dendrites, etc.) have been chosen to test \(D_q\) and \(f(\alpha )\) as tools for structure-morphology analysis. It turns out that both are very sensitive to deviations from global regularity as well as, to some extent, also to local changes. Based on monotonic increasing property of log function and diffeomorphism of \(D_q\)/\(f(\alpha )\) the problems of uniqueness and inversibility are formulated and discussed.

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We have investigated the intermediate scaling regime in the phase ordering/separating kinetics of the three-dimensional system of the non-conserved scalar order parameter. It is demonstrated that the observed scaling behavior can be described in terms of two length scales \(L_H(t)\sim t^{2/5}\) and \(L_K(t)\sim t^{3/10}\). The quantity \(L_H(t)\) is related to the geometrical properties of the phase interface and describes time evolution of the characteristic domain size, surface area, and the mean curvature. The second length scale, \(L_K(t)\), determining the Gaussian curvature and the Euler characteristic, can be regarded as the topological measure of the phase interface. Also, we have shown that the existence of the two length scales has a simple physical interpretation and is related to the domains-necks decoupling process observed in the intermediate regime.

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It is known that nonlinear chemical systems may be used for direct information processing. In this paper we study properties of the perpendicular junction of two channels filled with an excitable medium as a function of a time difference of arriving impulses. It is shown that depending on this difference the cross junction works as a coincidence detector or as a signal switch.

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Transport properties of a single pore in a track-etched poly(ethylene terephthalate) membrane are characterized using statistical analysis. Probability density function, autocorrelation function, power spectrum, Hurst and detrended fluctuation analysis, as well as Orey’s index were the tools used to characterize the ion current behavior. The examined pore is conical in shape and has been obtained by one-sided electric field stopped etching. The pore has a highly nonlinear diode-like current-voltage characteristic, with preferential flow of ions in one direction. We show that the examined current fluctuations at –2V and +2V, however looking very similar, reflect differences in action of the system at the two polarities. The existence of longer memory for the weaker signal, recorded at –2V, has been found.

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The correspondence of the fractional Brownian motion to the statistically self-similar on different time scales kinetics of a single locust potassium channel is discussed. The parameters of the non-Markovian long-memory process, modelling the ionic transport, are derived in terms of the main statistical characteristics of the recorded current signal.