Sharif Digital Repository

Statistical properties of interbeat intervals cascade in human hearts are evaluated by considering the joint probability distribution P(Δx 2, τ 2 Δx1, τ 1) for two interbeat increments Δx1 and Δx2 of different time scales τ 1 and τ 2. We present evidence that the conditional probability distribution P(Δx2, τ2|Δx1, τ 1 ) may be described by a Chapman-Kolmogorov equation. The corresponding Kramers-Moyal (KM) coefficients are evaluated. The analysis indicates that while the first and second KM coefficients take on well-defined and significant values, the higher-order coefficients in the KM expansion are small. As a result, the joint probability distributions of the increments in the interbeat... 

Stochastic behavior of viscoelastic panels in supersonic flow under random aerodynamic pressure and inplane forces is investigated. The governing equations of motion are based on the Von Karman's large deflection equation and are considered with Kelvin's model of viscoelastic structural damping. The panel under study is two dimensional and simply supported for which the first order piston theory is used to account for the unsteady aerodynamic loading. Transformation of the governing partial differential equation to a set of ordinary differential equations is performed through the Galerkin averaging technique. The statistical response moment equations are generated for two modes using the... 

Stochastic behavior of viscoelastic panels in supersonic flow under random aerodynamic pressure and in-plane forces is investigated. The governing equations of motion are based on the Von Karman's large deflection equation and are considered with Kelvin's model of viscoelastic structural damping. The panel under study is two dimensional and simply supported for which the first order piston theory is used to account for the unsteady aerodynamic loading. Transformation of the governing partial differential equation to a set of ordinary differential equations is performed through the Galerkin averaging technique. The statistical response moment equations are generated for two modes using the... 

In this chapter, we present the details of Kramers–Moyal (KM) expansion and prove the Pawula theorem. The Fokker–Planck equation is then introduced and its short-term propagator is presented. Finally, we derive the master equation from the Chapman–Kolmogorov equation. © 2019, Springer Nature Switzerland AG  

In this chapter, we present the details of Kramers–Moyal (KM) expansion and prove the Pawula theorem. The Fokker–Planck equation is then introduced and its short-term propagator is presented. Finally, we derive the master equation from the Chapman–Kolmogorov equation. © 2019, Springer Nature Switzerland AG  

In this chapter we show the equivalence between the Langevin approach and the Fokker–Planck equation, and derive the equation for the statistical moments of the process whose dynamics is described by the Langevin equation. © 2019, Springer Nature Switzerland AG  

This chapter describes and discusses Langevin dynamics and the Fokker–Planck equation in higher dimension, and discrete-time evolution and discrete-time approximation of stochastic evolution equations. We close the chapter with calculations of short-time propagators of d-dimensional Fokker–Planck equation. © 2019, Springer Nature Switzerland AG  

In this chapter we show the equivalence between the Langevin approach and the Fokker–Planck equation, and derive the equation for the statistical moments of the process whose dynamics is described by the Langevin equation. © 2019, Springer Nature Switzerland AG  

This chapter describes and discusses Langevin dynamics and the Fokker–Planck equation in higher dimension, and discrete-time evolution and discrete-time approximation of stochastic evolution equations. We close the chapter with calculations of short-time propagators of d-dimensional Fokker–Planck equation. © 2019, Springer Nature Switzerland AG  

In this chapter we define notions of stochastic continuity and differentiability and describe Lindeberg’s condition for continuity of stochastic Markovian trajectories. We also show that the Fokker–Planck equation describes a continuous stochastic process. Finally, we derive the stationary solutions of the Fokker–Planck equation and define potential function of dynamics. © 2019, Springer Nature Switzerland AG  

In this chapter we define notions of stochastic continuity and differentiability and describe Lindeberg’s condition for continuity of stochastic Markovian trajectories. We also show that the Fokker–Planck equation describes a continuous stochastic process. Finally, we derive the stationary solutions of the Fokker–Planck equation and define potential function of dynamics. © 2019, Springer Nature Switzerland AG  

We describe a general method for analyzing a nonstationary stochastic process X (t) which, unlike many of the previous analysis methods, does not require X (t) to have any scaling feature. The method is used to study the fluctuations in the daily price of oil. It is shown that the returns time series, y (t) =ln [X (t+1) X (t)], is a stationary and Markov process, characterized by a Markov time scale tM. The coefficients of the Kramers-Moyal expansion for the probability density function P (y,t y0, t0) are computed. P (y,t, y0, t0) satisfies a Fokker-Planck equation, which is equivalent to a Langevin equation for y (t) that provides quantitative predictions for the oil price over times that... 

Stochastic behavior of panels in supersonic flow is investigated to assess the significance of including the damping caused by the strains resulting from axial extension of the panel. The governing equations of motion are based on the Von Karman's large deflection equation and are considered with Kelvin's model of structural damping. The panel under study is two dimensional and simply supported for which the first order piston theory is used to account for the unsteady aerodynamic loading. Transformation of the governing partial differential equation to a set of ordinary differential equations is performed through the Galerkin averaging technique. The statistical response moment equations... 

When the complete understanding of a complex system is not available, as, e.g., for systems considered in the real world, we need a top-down approach to complexity. In this approach, one may desire to understand general multipoint statistics. Here, such a general approach is presented and discussed based on examples from turbulence and sea waves. Our main idea is based on the cascade picture of turbulence, entangling fluctuations from large to small scales. Inspired by this cascade picture, we express the general multipoint statistics by the statistics of scale-dependent fluctuations of variables and relate it to a scale-dependent process, which finally is a stochastic cascade process. We... 

Several methods have been developed in the past for analyzing the porosity and other types of well logs for large-scale porous media, such as oil reservoirs, as well as their permeability distributions. We developed a method for analyzing the porosity logs φ(h) (where h is the depth) and similar data that are often nonstationary stochastic series. In this method one first generates a new stationary series based on the original data, and then analyzes the resulting series. It is shown that the series based on the successive increments of the log y(h)=φ(h+δh)-φ(h) is a stationary and Markov process, characterized by a Markov length scale hM. The coefficients of the Kramers-Moyal expansion for... 

Recent phase noise analysis techniques of oscillators mainly rely on solving a stochastic differential equation governing the phase noise process. This equation has been solved in the literature using a number of mathematical tools from probability theory like deriving the Fokker-Planck equation governing the phase noise probability density function. Here, a completely different approach for solving this equation in presence of white noise sources is introduced that is based on the Ito calculus for stochastic differential equations. Time-domain analytical expressions for the correlation of the noisy variables of the oscillator are derived that in asymptotically large times give the... 

We provide a simple interpretation of non-Gaussian nature of the light scattering-intensity fluctuations from an aging colloidal suspension of Laponite using the multiplicative cascade model, Markovian method, and volatility correlations. The cascade model and Markovian method enable us to reproduce most of recent empirical findings: long-range volatility correlations and non-Gaussian statistics of intensity fluctuations. We provide evidence that the intensity increments Δx (τ) =I (t+τ) -I (t), upon different delay time scales τ, can be described as a Markovian process evolving in τ. Thus, the τ dependence of the probability density function p (Δx,τ) on the delay time scale τ can be...