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Sample records for linear fokker-planck coefficients

  1. Generalized multivariate Fokker-Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamics

    Frank, T.D.

    2002-01-01

    We study many particle systems in the context of mean field forces, concentration-dependent diffusion coefficients, generalized equilibrium distributions, and quantum statistics. Using kinetic transport theory and linear nonequilibrium thermodynamics we derive for these systems a generalized multivariate Fokker-Planck equation. It is shown that this Fokker-Planck equation describes relaxation processes, has stationary maximum entropy distributions, can have multiple stationary solutions and stationary solutions that differ from Boltzmann distributions

  2. A Lie-admissible method of integration of Fokker-Planck equations with non-linear coefficients (exact and numerical solutions)

    Fronteau, J.; Combis, P.

    1984-08-01

    A Lagrangian method is introduced for the integration of non-linear Fokker-Planck equations. Examples of exact solutions obtained in this way are given, and also the explicit scheme used for the computation of numerical solutions. The method is, in addition, shown to be of a Lie-admissible type

  3. Linear analysis of the momentum cooling Fokker-Planck equation

    Rosenzweig, J.B.

    1989-01-01

    In order to optimize the extraction scheme used to take antiprotons out of the accumulator, it is necessary to understand the basic processes involved. At present, six antiproton bunches per Tevatron store are removed sequentially by RF unstacking from the accumulator. The phase space dynamics of this process, with its accompanying phase displacement deceleration and phase space dilution of portions of the stack, can be modelled by numerical solution of the longitudinal equations of motion for a large number of particles. We have employed the tracking code ESME for this purpose. In between RF extractions, however, the stochastic cooling system is turned on for a short time, and we must take into account the effect of momentum stochastic cooling on the antiproton energy spectrum. This process is described by the Fokker-Planck equation, which models the evolution of the antiproton stack energy distribution by accounting for the cooling through an applied coherent drag force and the competing heating of the stack due to diffusion, which can arise from intra-beam scattering, amplifier noise and coherent (Schottky) effects. In this note we examine the aspects of the Fokker-Planck in the regime where the nonlinear terms due to Schottky effects are small. This discussion ultimately leads to solution of the equation in terms of an orthonormal set of functions which are closely related to the quantum simple-harmonic oscillator wave-functions. 5 refs

  4. Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma

    Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.

    2015-11-01

    In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.

  5. On the linear discrepancy model and risky shifts in group behavior: a nonlinear Fokker-Planck perspective

    Frank, T D

    2009-01-01

    Using a nonlinear Fokker-Planck perspective we re-formulate the linear discrepancy model proposed by Boster and colleagues that describes the emergence of risky shifts during group decision making. Analytical expressions for the stationary case are derived and risky shifts are obtained by Monte Carlo simulations. Striking similarities with the Kuramoto model for group synchronization are pointed out

  6. Expansion of the relativistic Fokker-Planck equation including non-linear terms and a non-Maxwellian background

    Shkarofsky, I.P.

    1997-01-01

    The relativistic Fokker-Planck collision term in Braams and Karney [Phys. Fluids B 1, 1355 (1989)] is expanded using Cartesian tensors (equivalent to associated Legendre spherical harmonics) retaining all non-linear terms and an arbitrary zeroth order distribution background. Expressions are given for collision terms between all harmonics and the background distribution in terms of the j and y functions in Braams and Karney. The results reduce to Braams and Karney for the first order harmonic term with a Maxwellian background and to those given by Shkarofsky [Can. J. Phys. 41, 1753 (1963)] in the non-relativistic limit. Expressions for the energy and momentum transfer associated with relativistic Coulomb collisions are given. The fast two dimensional Fokker-Planck solver in Shoucri and Shkarofsky [Comput. Phys. Commun. 82, 287 (1994)] has been extended to include the second order harmonic term. copyright 1997 American Institute of Physics

  7. Stationary solution of the Fokker-Planck equation for linearly coupled motion in an electron storage ring

    Chao, A.W.; Lee, M.J.

    1975-09-01

    Effects upon longitudinal bunch shape in a storage ring due to linear and nonlinear potential can be calculated by finding the stationary solution to the Fokker-Planck equation for the particle distribution. Effects upon transverse bunch shape of a stored electron beam due to photon emissions and damping can be calculated by this method. It has been found that this method can also be used for a case in which the transverse modes of oscillation are coupled to the energy deviation δ. Examples of lattice elements which produce linear coupling between these oscillations are skew quadrupole magnets and solenoid magnets. For the linearly coupled case the stationary solution has been found to be given by exp (ΣΣA/sub ij/ x/sub i/x/sub j/) with x/sub i/ the canonical variables (x,p/sub x/, y, p/sub y/, δ, p/sub δ/) and A /sub ij/ some constants. The solution for the values of A /sub ij/'s will be described in this report. It will be shown that this solution can be expressed in a compact form. For simple cases, this form of solution leads directly to analytic expressions for the values of A /sub ij/'s and the bunch shape can be calculated by integrating the distribution function over some of the coordinates; for the more complex cases, it can be conveniently adapted as an algorithm for numerical evaluation. 16 refs

  8. Exact solutions of the Fokker-Planck equation from an nth order supersymmetric quantum mechanics approach

    Schulze-Halberg, Axel [Escuela Superior de Fisica y Matematicas, IPN, Unidad Profesional Adolfo Lopez Mateos, Col. San Pedro Zacatenco, Edificio 9, 07738 Mexico D.F. (Mexico)], E-mail: xbataxel@gmail.com; Rivas, Jesus Morales [Universidad Autonoma Metropolitana - Azcapotzalco, CBI - Area de Fisica Atomica Molecular Aplicada, Av. San Pablo 180, Reynosa Azcapotzalco, 02200 Mexico D.F. (Mexico)], E-mail: jmr@correo.azc.uam.mx; Pena Gil, Jose Juan [Universidad Autonoma Metropolitana - Azcapotzalco, CBI - Area de Fisica Atomica Molecular Aplicada, Av. San Pablo 180, Reynosa Azcapotzalco, 02200 Mexico D.F. (Mexico)], E-mail: jjpg@correo.azc.uam.mx; Garcia-Ravelo, Jesus [Escuela Superior de Fisica y Matematicas, IPN, Unidad Profesional Adolfo Lopez Mateos, Col. San Pedro Zacatenco, Edificio 9, 07738 Mexico D.F. (Mexico)], E-mail: ravelo@esfm.ipn.mx; Roy, Pinaki [Physics and Applied Mathematics Unit, Indian Statistical Institute, Calcutta-700108 (India)], E-mail: pinaki@isical.ac.in

    2009-04-20

    We generalize the formalism of nth order Supersymmetric Quantum Mechanics (n-SUSY) to the Fokker-Planck equation for constant diffusion coefficient and stationary drift potential. The SUSY partner drift potentials and the corresponding solutions of the Fokker-Planck equation are given explicitly. As an application, we generate new solutions of the Fokker-Planck equation by means of our first- and second-order transformation.

  9. Exact solutions of the Fokker-Planck equation from an nth order supersymmetric quantum mechanics approach

    Schulze-Halberg, Axel; Rivas, Jesus Morales; Pena Gil, Jose Juan; Garcia-Ravelo, Jesus; Roy, Pinaki

    2009-01-01

    We generalize the formalism of nth order Supersymmetric Quantum Mechanics (n-SUSY) to the Fokker-Planck equation for constant diffusion coefficient and stationary drift potential. The SUSY partner drift potentials and the corresponding solutions of the Fokker-Planck equation are given explicitly. As an application, we generate new solutions of the Fokker-Planck equation by means of our first- and second-order transformation.

  10. Generalized Fokker-Planck equations for coloured, multiplicative Gaussian noise

    Cetto, A.M.; Pena, L. de la; Velasco, R.M.

    1984-01-01

    With the help of Novikov's theorem, it is possible to derive a master equation for a coloured, multiplicative, Gaussian random process; the coefficients of this master equation satisfy a complicated auxiliary integro-differential equation. For small values of the Kubo number, the master equation reduces to an approximate generalized Fokker-Planck equation. The diffusion coefficient is explicitly written in terms of correlation functions. Finally, a straightforward and elementary second order perturbative treatment is proposed to derive the same approximate Fokker-Planck equation. (author)

  11. Solving the Fokker-Planck equation on a massively parallel computer

    Mirin, A.A.

    1990-01-01

    The Fokker-Planck package FPPAC had been converted to the Connection Machine 2 (CM2). For fine mesh cases the CM2 outperforms the Cray-2 when it comes to time-integrating the difference equations. For long Legendre expansions the CM2 is also faster at computing the Fokker-Planck coefficients. 3 refs

  12. Nonlinear Fokker-Planck Equations Fundamentals and Applications

    Frank, Till Daniel

    2005-01-01

    Providing an introduction to the theory of nonlinear Fokker-Planck equations, this book discusses fundamental properties of transient and stationary solutions, emphasizing the stability analysis of stationary solutions by means of self-consistency equations, linear stability analysis, and Lyapunov's direct method. Also treated are Langevin equations and correlation functions. Nonlinear Fokker-Planck Equations addresses various phenomena such as phase transitions, multistability of systems, synchronization, anomalous diffusion, cut-off solutions, travelling-wave solutions and the emergence of power law solutions. A nonlinear Fokker-Planck perspective to quantum statistics, generalized thermodynamics, and linear nonequilibrium thermodynamics is given. Theoretical concepts are illustrated where possible by simple examples. The book also reviews several applications in the fields of condensed matter physics, the physics of porous media and liquid crystals, accelerator physics, neurophysics, social sciences, popul...

  13. Fully non-linear multi-species Fokker-Planck-Landau collisions for gyrokinetic particle-in-cell simulations of fusion plasma

    Hager, Robert; Yoon, E. S.; Ku, S.; D'Azevedo, E. F.; Worley, P. H.; Chang, C. S.

    2015-11-01

    We describe the implementation, and application of a time-dependent, fully nonlinear multi-species Fokker-Planck-Landau collision operator based on the single-species work of Yoon and Chang [Phys. Plasmas 21, 032503 (2014)] in the full-function gyrokinetic particle-in-cell codes XGC1 [Ku et al., Nucl. Fusion 49, 115021 (2009)] and XGCa. XGC simulations include the pedestal and scrape-off layer, where significant deviations of the particle distribution function from a Maxwellian can occur. Thus, in order to describe collisional effects on neoclassical and turbulence physics accurately, the use of a non-linear collision operator is a necessity. Our collision operator is based on a finite volume method using the velocity-space distribution functions sampled from the marker particles. Since the same fine configuration space mesh is used for collisions and the Poisson solver, the workload due to collisions can be comparable to or larger than the workload due to particle motion. We demonstrate that computing time spent on collisions can be kept affordable by applying advanced parallelization strategies while conserving mass, momentum, and energy to reasonable accuracy. We also show results of production scale XGCa simulations in the H-mode pedestal and compare to conventional theory. Work supported by US DOE OFES and OASCR.

  14. A Fokker-Planck treatment of stochastic particle motion within the framework of a fully coupled 6-dimensional formalism for electron-positron storage rings including classical spin motion in linear approximation

    Barber, D.P.; Heinemann, K.; Mais, H.; Ripken, G.

    1991-12-01

    In the following report we investigate stochastic particle motion in electron-positron storage ring in the framework of a Fokker-Planck treatment. The motion is described by using the canonical variables χ, p χ , z, p z , σ = s - cxt, p σ = ΔE/E 0 of the fully six-dimensional formalism. Thus synchrotron- and betatron-oscillations are treated simultaneously taking into account all kinds of coupling (synchro-betatron coupling and the coupling of the betatron oscillations by skew quadrupoles and solenoids). In order to set up the Fokker-Planck equation, action-angle variables of the linear coupled motion are introduced. The averaged dimensions of the bunch, resulting from radiation damping of the synchro-betatron oscillations and from an excitation of these oscillations by quantum fluctuations, are calculated by solving the Fokker-Planck equation. The surfaces of constant density in the six-dimensional phase space, given by six-dimensional ellipsoids, are determined. It is shown that the motion of such an ellipsoid under the influence of external fields can be described by six generating orbit vectors which may be combined into a six-dimenional matrix B(s). This 'bunch-shape matrix', B(s), contains complete information about the configuration of the bunch. Classical spin diffusion in linear approximation has also been included so that the dependence of the polarization vector on the orbital phase space coordinates can be studied and another derivation of the linearized depolarization time obtained. (orig.)

  15. Kappa and other nonequilibrium distributions from the Fokker-Planck equation and the relationship to Tsallis entropy

    Shizgal, Bernie D.

    2018-05-01

    This paper considers two nonequilibrium model systems described by linear Fokker-Planck equations for the time-dependent velocity distribution functions that yield steady state Kappa distributions for specific system parameters. The first system describes the time evolution of a charged test particle in a constant temperature heat bath of a second charged particle. The time dependence of the distribution function of the test particle is given by a Fokker-Planck equation with drift and diffusion coefficients for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. A second system involves the Fokker-Planck equation for electrons dilutely dispersed in a constant temperature heat bath of atoms or ions and subject to an external time-independent uniform electric field. The momentum transfer cross section for collisions between the two components is assumed to be a power law in reduced speed. The time-dependent Fokker-Planck equations for both model systems are solved with a numerical finite difference method and the approach to equilibrium is rationalized with the Kullback-Leibler relative entropy. For particular choices of the system parameters for both models, the steady distribution is found to be a Kappa distribution. Kappa distributions were introduced as an empirical fitting function that well describe the nonequilibrium features of the distribution functions of electrons and ions in space science as measured by satellite instruments. The calculation of the Kappa distribution from the Fokker-Planck equations provides a direct physically based dynamical approach in contrast to the nonextensive entropy formalism by Tsallis [J. Stat. Phys. 53, 479 (1988), 10.1007/BF01016429].

  16. Fokker-Planck code for the quasi-linear absorption of electron cyclotron waves in a tokamak plasma

    Meyer, R.L.; Giruzzi, G.; Krivenski, V.

    1986-01-01

    We present the solution of the kinetic equation describing the quasi-linear evolution of the electron momentum distribution function under the influence of the electron cyclotron wave absorption. Coulomb collisions and the dc electric field in a tokamak plasma. The solution of the quasi-linear equation is obtained numerically using a two-dimensional initial value code following an ADI scheme. Most emphasis is given to the full non-linear and self-consistent problem, namely, the wave amplitude is evaluated at any instant and any point in space according to the actual damping. This is necessary since wave damping is a very sensitive function of the slope of the local momentum distribution function because the resonance condition relates the electron momentum to the location of wave energy deposition. (orig.)

  17. Fokker-Planck and quasilinear codes

    Karney, C.F.F.

    1985-11-01

    The interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. Methods for solving this equation on a computer are discussed. 40 refs., 12 figs., 3 tabs

  18. Vectorized Fokker-Planck package for the CRAY-1

    McCoy, M.G.; Mirin, A.A.; Killeen, J.

    1979-08-01

    A program for the solution of the time-dependent, two dimensional, nonlinear, multi-species Fokker-Planck equation is described. The programming is written such that the loop structure is highly vectorizable on the CRAY FORTRAN Compiler. A brief discussion of the Fokker-Planck equation itself is followed by a description of the procedure developed to solve the equation efficiently. The Fokker-Planck equation is a second order partial differential equation whose coefficients depend upon moments of the distribution functions. Both the procedure for the calculation of these coefficients and the procedure for the time advancement of the equation itself must be done efficiently if significant overall time saving is to result. The coefficients are calculated in a series of nested loops, while time advancement is accomplished by a choice of either a splitting or an ADI technique. Overall, timing tests show that the vectorized CRAY program realizes up to a factor of 12 advantage over an optimized CDC-7600 program and up to a factor of 365 over a non-vectorized version of the same program on the CRAY

  19. Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

    Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

    2009-01-01

    The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

  20. Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

    Densmore, Jeffery D [Los Alamos National Laboratory; Warsa, James S [Los Alamos National Laboratory; Lowrie, Robert B [Los Alamos National Laboratory; Morel, Jim E [TEXAS A& M UNIV

    2008-01-01

    The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

  1. Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

    Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

    2009-09-01

    The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

  2. Stochastic reliability analysis using Fokker Planck equations

    Hari Prasad, M.; Rami Reddy, G.; Srividya, A.; Verma, A.K.

    2011-01-01

    The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well. It is also known as the Kolmogorov forward equation (diffusion). Hence, for any process, which evolves with time, the probability density function as a function of time can be represented with Fokker-Planck equation. In stochastic reliability analysis one is more interested in finding out the reliability or failure probability of the components or structures as a function of time rather than instantaneous failure probabilities. In this analysis the variables are represented with random processes instead of random variables. A random processes can be either stationary or non stationary. If the random process is stationary then the failure probability doesn't change with time where as in the case of non stationary processes the failure probability changes with time. In the present paper Fokker Planck equations have been used to find out the probability density function of the non stationary random processes. In this paper a flow chart has been provided which describes step by step process for carrying out stochastic reliability analysis using Fokker-Planck equations. As a first step one has to identify the failure function as a function of random processes. Then one has to solve the Fokker-Planck equation for each random process. In this paper the Fokker-Planck equation has been solved by using Finite difference method. As a result one gets the probability density values of the random process in the sample space as well as time space. Later at each time step appropriate probability distribution has to be identified based on the available probability density values. For checking the better fitness of the data Kolmogorov-Smirnov Goodness of fit test has been performed. In this way one can find out the distribution of the random process at each time step. Once one has the probability distribution

  3. Integral solution for the spherically symmetric Fokker-Planck equation

    Donoso, J.M.; Soler, M.

    1993-01-01

    We propose an integral method to deal with the spherically symmetric non-linear Fokker-Planck equation appearing in plasma physics. A probability transition expression is obtained, which takes into account the proper domain for the radial velocity component. The analytical and computational results are new, and the time evolution is completely satisfactory. The main achievement of the method is conservation of both the initial norm and energy for unlimited times, which has not been attained in the differential approach to the problem. (orig.)

  4. Fokker-Planck equation in the presence of a uniform magnetic field

    Dong, Chao, E-mail: chaodong@iphy.ac.cn [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Department of Nuclear Engineering, Seoul National University, Seoul 151-744 (Korea, Republic of); Zhang, Wenlu [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Li, Ding, E-mail: dli@ustc.edu.cn [Center for Plasma Theory and Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Department of Modern Physics, University of Science and Technology of China, Anhui Hefei 230026 (China)

    2016-08-15

    The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.

  5. Fokker-Planck equation in the presence of a uniform magnetic field

    Dong, Chao; Zhang, Wenlu; Li, Ding

    2016-01-01

    The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.

  6. Fokker-Planck modeling of pitting corrosion in underground pipelines

    Camacho, Eliana Nogueira [Risco Ambiental Engenharia, Rio de Janeiro, RJ (Brazil); Melo, Paulo F. Frutuoso e [Coordenacao dos Programas de Pos-Graduacao de Engenharia (COPPE/UFRJ), Rio de Janeiro, RJ (Brazil). Programa de Engenharia Nuclear; Saldanha, Pedro Luiz C. [Comissao Nacional de Energia Nuclear (CGRC/CNEN), Rio de Janeiro, RJ (Brazil). Coordenacao Geral de Reatores e Ciclo do Combustivel; Silva, Edson de Pinho da [Universidade Federal Rural do Rio de Janeiro (UFRRJ), Seropedica, RJ (Brazil). Dept. of Physics

    2011-07-01

    Full text: The stochastic nature of pitting corrosion has been recognized since the 1930s. It has been learned that this damage retains no memory of its past. Instead, the future state is determined only by the knowledge of its present state. This Markovian property that underlies the stochastic process governing pitting corrosion has been explored as a discrete Markovian process by many authors since the beginning of the 1990s for underground pipelines of the oil and gas industries and nuclear power plants. Corrosion is a genuine continuous time and space state Markovian process, so to model it as a discrete time and/or state space is an approximation to the problem. Markovian chains approaches, with an increasing number of states, could involve a large number of parameters, the transition rates between states, to be experimentally determined. Besides, such an increase in the number of states produces matrices with huge dimensions leading to time-consuming computational solutions. Recent approaches involving Markovian discrete process have overcome those difficulties but, on the other hand, a large number of soil and pipe stochastic variables have to be known. In this work we propose a continuous time and space state approach to the evolution of pit corrosion depths in underground pipelines. In order to illustrate the application of the model for defect depth growth a combination of real life data and Monte Carlo simulation was used. The process is described by a Fokker-Planck equation. The Fokker-Planck equation is completely determined by the knowledge of two functions known as the drift and diffusion coefficients. In this work we also show that those functions can be estimated from corrosion depth data from in-line inspections. Some particular forms of drift and diffusion coefficients lead to particular Fokker-Planck equations for which analytical solutions are known, as is the case for the Wiener process, the Ornstein-Uhlenbeck process and the Brownian motion

  7. Massively parallel Fokker-Planck calculations

    Mirin, A.A.

    1990-01-01

    This paper reports that the Fokker-Planck package FPPAC, which solves the complete nonlinear multispecies Fokker-Planck collision operator for a plasma in two-dimensional velocity space, has been rewritten for the Connection Machine 2. This has involved allocation of variables either to the front end or the CM2, minimization of data flow, and replacement of Cray-optimized algorithms with ones suitable for a massively parallel architecture. Calculations have been carried out on various Connection Machines throughout the country. Results and timings on these machines have been compared to each other and to those on the static memory Cray-2. For large problem size, the Connection Machine 2 is found to be cost-efficient

  8. A nonlinear Fokker-Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

    Marin, D.; Ribeiro, M. A.; Ribeiro, H. V.; Lenzi, E. K.

    2018-07-01

    We investigate the solutions for a set of coupled nonlinear Fokker-Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.

  9. Fokker-Planck equation in mirror research

    Post, R.F.

    1983-01-01

    Open confinement systems based on the magnetic mirror principle depend on the maintenance of particle distributions that may deviate substantially from Maxwellian distributions. Mirror research has therefore from the beginning relied on theoretical predictions of non-equilibrium rate processes obtained from solutions to the Fokker-Planck equation. The F-P equation plays three roles: Design of experiments, creation of classical standards against which to compare experiment, and predictions concerning mirror based fusion power systems. Analytical and computational approaches to solving the F-P equation for mirror systems will be reviewed, together with results and examples that apply to specific mirror systems, such as the tandem mirror

  10. Maximum Path Information and Fokker Planck Equation

    Li, Wei; Wang A., Q.; LeMehaute, A.

    2008-04-01

    We present a rigorous method to derive the nonlinear Fokker-Planck (FP) equation of anomalous diffusion directly from a generalization of the principle of least action of Maupertuis proposed by Wang [Chaos, Solitons & Fractals 23 (2005) 1253] for smooth or quasi-smooth irregular dynamics evolving in Markovian process. The FP equation obtained may take two different but equivalent forms. It was also found that the diffusion constant may depend on both q (the index of Tsallis entropy [J. Stat. Phys. 52 (1988) 479] and the time t.

  11. Massively parallel Fokker-Planck code ALLAp

    Batishcheva, A.A.; Krasheninnikov, S.I.; Craddock, G.G.; Djordjevic, V.

    1996-01-01

    The recently developed for workstations Fokker-Planck code ALLA simulates the temporal evolution of 1V, 2V and 1D2V collisional edge plasmas. In this work we present the results of code parallelization on the CRI T3D massively parallel platform (ALLAp version). Simultaneously we benchmark the 1D2V parallel vesion against an analytic self-similar solution of the collisional kinetic equation. This test is not trivial as it demands a very strong spatial temperature and density variation within the simulation domain. (orig.)

  12. On the Fokker-Planck theory of electron three-body recombination

    Sayasov, Yu. S.

    1977-01-01

    The Fokker-Planck theory of electron three-body recombination based on the concept of electron diffusion along the energy scale in the excited hydrogen-like atoms formed in the recombining plasmas, is extended in several respects. 1) An universal formula for population distribution of the excited atoms in strongly ionized plasmas was found under a sole assumption, that the cross-sections for the inelastic atom-electron collisions are governed by the classical impulse approximation. 2) A general Fokker-Planck theory of the recombination in a slightly ionized, two-temperature plasmas was formulated. The recombination coefficients for such plasmas were shown to possess some peculiar properties in case the electronic temperature differs appreciable from the atomic one. A few limitations of the existing schemas for calculation of the recombination kinetics are briefly discussed. (orig.) [de

  13. An Adjoint-based Numerical Method for a class of nonlinear Fokker-Planck Equations

    Festa, Adriano; Gomes, Diogo A.; Machado Velho, Roberto

    2017-01-01

    Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

  14. An Adjoint-based Numerical Method for a class of nonlinear Fokker-Planck Equations

    Festa, Adriano

    2017-03-22

    Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

  15. FOKKER-PLANCK ANALYSIS OF TRANSVERSE COLLECTIVE INSTABILITIES IN ELECTRON STORAGE RINGS

    Lindberg, R. R.

    2017-06-25

    We analyze single bunch transverse instabilities due to wakefields using a Fokker-Planck model. We expand on the work of Suzuki [1], writing out the linear matrix equation including chromaticity, both dipolar and quadrupolar transverse wakefields, and the effects of damping and diffusion due to the synchrotron radiation. The eigenvalues and eigenvectors determine the collective stability of the beam, and we show that the predicted threshold current for transverse instability and the profile of the unstable agree well with tracking simulations. In particular, we find that predicting collective stability for high energy electron beams at moderate to large values of chromaticity requires the full Fokker-Planck analysis to properly account for the effects of damping and diffusion due to synchrotron radiation.

  16. Fokker-Planck transport in solid state accelerator concepts

    Newberger, B.; Tajima, T.

    1989-01-01

    Particle transport in a crystalline solid under channeling conditions is considered by means of a Fokker-Planck description. The model includes electron multiple scattering, radiation damping and an accelerating electric field. Analytic solutions have been obtained using a harmonic potential model to describe the channeling forces. These solutions will be described

  17. Reduced Fokker-Planck models for fast particle distribution across a transition layer of disparate plasma temperatures

    Tang, Xian-Zhu; Berk, H. L.; Guo, Zehua; McDevitt, C. J.

    2014-03-01

    Across a transition layer of disparate plasma temperatures, the high energy tail of the plasma distribution can have appreciable deviations from the local Maxwellian distribution due to the Knudson layer effect. The Fokker-Planck equation for the tail particle population can be simplified in a series of practically useful limiting cases. The first is the approximation of background Maxwellian distribution for linearizing the collision operator. The second is the supra-thermal particle speed ordering of vTi ≪ v ≪ vTe for the tail ions and vTi ≪ vTe ≪ v for the tail electrons. Keeping both the collisional drag and energy scattering is essential for the collision operator to produce a Maxwellian tail distribution. The Fokker-Planck model for following the tail ion distribution for a given background plasma profile is explicitly worked out for systems of one spatial dimension, in both slab and spherical geometry. A third simplification is an expansion of the tail particle distribution using the spherical harmonics, which are eigenfunctions of the pitch angle scattering operator. This produces a set of coupled Fokker-Planck equations that contain energy-dependent spatial diffusion terms in two coordinates (position and energy), which originate from pitch angle scattering in the original Fokker-Planck equation. It is shown that the well-known diffusive Fokker-Planck model is a poor approximation of the two-mode truncation model, which itself has fundamental deficiency compared with the three-mode truncation model. The cause is the lack of even-symmetry representation in pitch dependence in the two-mode truncation model.

  18. Fokker--Planck/transport analyses of fusion plasmas in contemporary beam-driven tokamaks

    Mirin, A.A.; McCoy, M.G.; Killeen, J.; Rensink, M.E.; Shumaker, D.E.; Jassby, D.L.; Post, D.E.

    1978-04-01

    The properties of deuterium plasmas in experimental tokamaks heated and fueled by intense neutral-beam injection are evaluated with a Fokker-Planck/radial transport code coupled with a Monte Carlo neutrals treatment. Illustrative results are presented for the Poloidal Divertor Experiment at PPPL as a function of beam power and plasma recycling coefficient, R/sub c/. When P/sub beam/ = 8 MW at E/sub b/ = 60 keV, and R/sub c/ = 0.2, then approximately 0.5, [ 2 / 3 ] = 22 keV approximately 6 , and the D-D neutron intensity is 10 16 n/sec

  19. Steady state solution of the Fokker-Planck equation combined with unidirectional quasilinear diffusion under detailed balance conditions

    Hizanidis, K.

    1984-04-01

    The relativistic collisional Fokker-Planck equation combined with an externally imposed unidirectional quasilinear (rf) diffusion is solved for arbitrary values of rf diffusion coefficient under conditions of detailed balance of the staionary joint distribution involved. The detailed balance condition imposes a restriction on the functional form of the quasilinear diffusion coefficient which might be associated with the existence of a saturated spectrum of fluctuation in a quasilinearly rf-driven plasma

  20. Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation

    Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.

    2017-01-01

    The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.

  1. Fokker-Planck equation resolution for N variables. Application examples

    Munoz, A.; Garcia-Olivares, A.

    1994-01-01

    A set of problems which are reducible to Fokker-Planck equations are presented. Those problems have been solved by using the CHAPKOL library. This library of programs solves stochastic Fokker-Plank equations in one or several dimensions by using the Chapman- Kolmogorov integral. This method calculates the probability distribution at a time t + dt from a distribution given at time t through a convolution integral in which the integration is the product of the distribution function at time t and the Green function of the Fokker-Planck equation. The method have some numerical advantages when compared with finite differences algorithms. The accuracy of the method is analysed in several specific cases. (Author) 9 refs

  2. FIFPC, a fast ion Fokker--Planck code

    Fowler, R.H.; Callen, J.D.; Rome, J.A.; Smith, J.

    1976-07-01

    A computer code is described which solves the Fokker--Planck equation for the velocity space distribution of fast ions injected into a tokamak plasma. The numerical techniques are described and use of the code is outlined. The program is written in FORTRAN IV and is modularized in order to provide greater flexibility to the user. A program listing is provided and the results of sample cases are presented

  3. Chaotic universe dynamics using a Fokker-Planck equation

    Coule, D.H.; Olynyk, K.O.

    1987-07-01

    A Fokker-Planck equation that accounts for fluctuations in field and its conjugate momentum is solved numerically for the case of a λ phi 4 potential. Although the amount of inflation agrees closely with that expected classically, in certain cases (large initial fields or large dispersions),the ''slow rolling'' approximation appears invalid. In such cases inflation would stop prematurely before possibly restarting. 18 refs., 2 figs

  4. Derivation of a Fokker-Planck equation for bunched beams

    Ruggiero, A.G.

    1993-01-01

    This report investigates the derivation of the Fokker-Planck equation which is commonly used to evaluate the evolution with time of an ensemble of particles under the effect of external rf forces, cooling and forces of stochastic nature like intrabeam scattering. The conventional approach based on the classical work by Chandrasekhar is first exposed, where the phase delay and the momentum error of the particle are used. The method is then extended to the case the distribution function is expressed in terms of the amplitude of motion instead of the original rectilinear variables. The new Fokker-Planck equation is obtained with an averaging process over the phase distribution instead of the time-averaging as it was usually performed earlier, to avoid the appearance of a singularity behavior. The solution of the Fokker-Planck equation is chosen in the proper form which makes easier the evaluation of the beam lifetime in the presence of the separatrix of the rf buckets. Finally the numerical applications apply the Relativistic Heavy Ion Collider (RHIC)

  5. Numerical method for the nonlinear Fokker-Planck equation

    Zhang, D.S.; Wei, G.W.; Kouri, D.J.; Hoffman, D.K.

    1997-01-01

    A practical method based on distributed approximating functionals (DAFs) is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo [Phys. Rev. E 54, 931 (1996)] to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schroedinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem. copyright 1997 The American Physical Society

  6. Numerical Resolution of N-dimensional Fokker-Planck stochastic equations; Resolucion Numerica de Ecuaciones Estocasticas de tipo Fokker-Planck en Varias Dimensiones

    Garcia-Olivares, R A; Munoz Roldan, A

    1992-07-01

    This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs.

  7. A transformed path integral approach for solution of the Fokker-Planck equation

    Subramaniam, Gnana M.; Vedula, Prakash

    2017-10-01

    A novel path integral (PI) based method for solution of the Fokker-Planck equation is presented. The proposed method, termed the transformed path integral (TPI) method, utilizes a new formulation for the underlying short-time propagator to perform the evolution of the probability density function (PDF) in a transformed computational domain where a more accurate representation of the PDF can be ensured. The new formulation, based on a dynamic transformation of the original state space with the statistics of the PDF as parameters, preserves the non-negativity of the PDF and incorporates short-time properties of the underlying stochastic process. New update equations for the state PDF in a transformed space and the parameters of the transformation (including mean and covariance) that better accommodate nonlinearities in drift and non-Gaussian behavior in distributions are proposed (based on properties of the SDE). Owing to the choice of transformation considered, the proposed method maps a fixed grid in transformed space to a dynamically adaptive grid in the original state space. The TPI method, in contrast to conventional methods such as Monte Carlo simulations and fixed grid approaches, is able to better represent the distributions (especially the tail information) and better address challenges in processes with large diffusion, large drift and large concentration of PDF. Additionally, in the proposed TPI method, error bounds on the probability in the computational domain can be obtained using the Chebyshev's inequality. The benefits of the TPI method over conventional methods are illustrated through simulations of linear and nonlinear drift processes in one-dimensional and multidimensional state spaces. The effects of spatial and temporal grid resolutions as well as that of the diffusion coefficient on the error in the PDF are also characterized.

  8. Fokker-Planck-Rosenbluth-type equations for self-gravitating systems in the 1PN approximation

    Ramos-Caro, Javier; Gonzalez, Guillermo A

    2008-01-01

    We present two formulations of Fokker-Planck-Rosenbluth-type (FPR) equations for many-particle self-gravitating systems, with first-order relativistic corrections in the post-Newtonian approach (1PN). The first starts from a covariant Fokker-Planck equation for a simple gas, introduced recently by Chacon-Acosta and Kremer (2007 Phys. Rev. E 76 021201). The second derivation is based on the establishment of an 1PN-BBGKY hierarchy, developed systematically from the 1PN microscopic law of force and using the Klimontovich-Dupree (KD) method. We close the hierarchy by the introduction of a two-point correlation function that describes adequately the relaxation process. This picture reveals an aspect that is not considered in the first formulation: the contribution of ternary correlation patterns to the diffusion coefficients, as a consequence of the nature of 1PN interaction. Both formulations can be considered as a generalization of the equation derived by Rezania and Sobouti (2000 Astron. Astrophys. 354 1110), to stellar systems where the relativistic effects of gravitation play a significant role

  9. A fractional Fokker-Planck model for anomalous diffusion

    Anderson, Johan, E-mail: anderson.johan@gmail.com [Department of Earth and Space Sciences, Chalmers University of Technology, SE-412 96 Göteborg (Sweden); Kim, Eun-jin [Department of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH (United Kingdom); Moradi, Sara [Ecole Polytechnique, CNRS UMR7648, LPP, F-91128 Palaiseau (France)

    2014-12-15

    In this paper, we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.

  10. Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix

    Schulze-Halberg, Axel

    2012-01-01

    We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schrödinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.

  11. Integral propagator solvers for Vlasov-Fokker-Planck equations

    Donoso, J M; Rio, E del

    2007-01-01

    We briefly discuss the use of short-time integral propagators on solving the so-called Vlasov-Fokker-Planck equation for the dynamics of a distribution function. For this equation, the diffusion tensor is singular and the usual Gaussian representation of the short-time propagator is no longer valid. However, we prove that the path-integral approach on solving the equation is, in fact, reliable by means of our generalized propagator, which is obtained through the construction of an auxiliary solvable Fokker-Planck equation. The new representation of the grid-free advancing scheme describes the inherent cross- and self-diffusion processes, in both velocity and configuration spaces, in a natural manner, although these processes are not explicitly depicted in the differential equation. We also show that some splitting methods, as well as some finite-difference schemes, could fail in describing the aforementioned diffusion processes, governed in the whole phase space only by the velocity diffusion tensor. The short-time transition probability offers a stable and robust numerical algorithm that preserves the distribution positiveness and its norm, ensuring the smoothness of the evolving solution at any time step. (fast track communication)

  12. Numerical Resolution of N-dimensional Fokker-Planck stochastic equations

    Garcia-Olivares, R. A.; Munoz Roldan, A.

    1992-01-01

    This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs

  13. Fokker-Planck theory of electron cyclotron assisted startup and breakdown in Tokamaks

    Fidone, I.; Granata, G.

    1993-04-01

    The kinetic theory of plasma startup in a tokamak in the presence of electron cyclotron resonance heating is discussed. The linear theory of the X-mode and the upper-hybrid converted mode damping in low density and temperature plasmas are first reviewed. Then, the kinetic equation for the electron velocity distribution is considered, which is determined by the perpendicular electron cyclotron quasilinear diffusion operator, the parallel electric field, elastic and inelastic electron-neutral collisions and various losses. Two different time scales, namely the elastic electron-neutral collision time and the much longer ionization time, are identified. Thus a two time scale ordering procedure is legitimated for which the velocity distribution is determined by the quasilinear diffusion and the electron-neutral collision frequency; the ionization rate is computed using the Fokker-Planck solution for the electron velocity distribution

  14. Fokker-Planck simulation study of Alfven eigenmode burst

    Todo, Y.; Watanabe, T.; Park, Hyoung-Bin; Sato, T.

    2001-01-01

    Recurrent bursts of toroidicity-induced Alfven eigenmodes (TAEs) are reproduced with a Fokker-Planck-magnetohydrodynamic simulation where a fast-ion source and slowing down are incorporated self-consistently. The bursts take place at regular time intervals and the behaviors of all the TAEs are synchronized. The fast-ion transport due to TAE activity spatially broadens the classical fast-ion distribution and significantly reduces its peak value. Only a small change of the distribution takes place with each burst, leading to loss of a small fraction of the fast ions. The system stays close to the marginal stability state established through the interplay of the fast-ion source, slowing down, and TAE activity. (author)

  15. Computing generalized Langevin equations and generalized Fokker-Planck equations.

    Darve, Eric; Solomon, Jose; Kia, Amirali

    2009-07-07

    The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.

  16. Hypersonic expansion of the Fokker--Planck equation

    Fernandez-Feria, R.

    1989-01-01

    A systematic study of the hypersonic limit of a heavy species diluted in a much lighter gas is made via the Fokker--Planck equation governing its velocity distribution function. In particular, two different hypersonic expansions of the Fokker--Planck equation are considered, differing from each other in the momentum equation of the heavy gas used as the basis of the expansion: in the first of them, the pressure tensor is neglected in that equation while, in the second expansion, the pressure tensor term is retained. The expansions are valid when the light gas Mach number is O(1) or larger and the difference between the mean velocities of light and heavy components is small compared to the light gas thermal speed. They can be applied away from regions where the spatial gradient of the distribution function is very large, but it is not restricted with respect to the temporal derivative of the distribution function. The hydrodynamic equations corresponding to the lowest order of both expansions constitute two different hypersonic closures of the moment equations. For the subsequent orders in the expansions, closed sets of moment equations (hydrodynamic equations) are given. Special emphasis is made on the order of magnitude of the errors of the lowest-order hydrodynamic quantities. It is shown that if the heat flux vanishes initially, these errors are smaller than one might have expected from the ordinary scaling of the hypersonic closure. Also it is found that the normal solution of both expansions is a Gaussian distribution at the lowest order

  17. On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks

    Annunziato, Mario; Borzì , Alfio; Nobile, Fabio; Tempone, Raul

    2014-01-01

    appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the FokkerPlanck formalism allows considering a larger class of objectives. To illustrate the connection between the two control

  18. A multigroup flux-limited asymptotic diffusion Fokker-Planck equation

    Liu Chengan

    1987-01-01

    A more perfrect flux-limited method is applied to combine with asymptotic diffusion theory of the radiation transpore, and the high peaked component in the scattering angle is treated with Fokker-Planck methods, thus the flux-limited asymptotic diffusion Fokker-Planck equation has been founded. Since the equation is of diffusion form, it retains the simplity and the convenience of the classical diffusion theory, and improves precision in describing radiation transport problems

  19. Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries

    Nier, Francis

    2018-01-01

    This article is concerned with the maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic laplacian.

  20. Vlasov-Fokker-Planck modeling of magnetized plasma

    Thomas, Alexander

    2016-01-01

    Understanding the magnetic fields that can develop in high-power-laser interactions with solid-density plasma is important because such fields significantly modify both the magnitude and direction of electron heat fluxes. The dynamics of such fields evidently have consequences for inertial fusion energy applications, as the coupling of the laser beams with the walls or pellet and the development of temperature inhomogeneities are critical to the uniformity of the implosion and potentially the success of, for example, the National Ignition Facility. To study these effects, we used the code Impacta, a two-dimensional, fully implicit, Vlasov-Fokker-Planck code with self-consistent magnetic fields and a hydrodynamic ion model, designed for nanosecond time-scale laser-plasma interactions. Heat-flux effects in Ohm's law under non-local conditions was investigated; physics that is not well captured by standard numerical models but is nevertheless important in fusion-related scenarios. Under such conditions there are numerous interesting physical effects, such as collisional magnetic instabilities, amplification of magnetic fields, re-emergence of non-locality through magnetic convection, and reconnection of magnetic field lines and redistribution of thermal energy. In this project highlights included the first full-scale kinetic simulations of a magnetized hohlraum and the discovery of a new magnetic reconnection mechanism, as well as a completed PhD thesis and the production of a new code for Inertial Fusion research.

  1. Vlasov-Fokker-Planck modeling of magnetized plasma

    Thomas, Alexander [Univ. of Michigan, Ann Arbor, MI (United States)

    2016-08-01

    Understanding the magnetic fields that can develop in high-power-laser interactions with solid-density plasma is important because such fields significantly modify both the magnitude and direction of electron heat fluxes. The dynamics of such fields evidently have consequences for inertial fusion energy applications, as the coupling of the laser beams with the walls or pellet and the development of temperature inhomogeneities are critical to the uniformity of the implosion and potentially the success of, for example, the National Ignition Facility. To study these effects, we used the code Impacta, a two-dimensional, fully implicit, Vlasov-Fokker-Planck code with self-consistent magnetic fields and a hydrodynamic ion model, designed for nanosecond time-scale laser-plasma interactions. Heat-flux effects in Ohm’s law under non-local conditions was investigated; physics that is not well captured by standard numerical models but is nevertheless important in fusion-related scenarios. Under such conditions there are numerous interesting physical effects, such as collisional magnetic instabilities, amplification of magnetic fields, re-emergence of non-locality through magnetic convection, and reconnection of magnetic field lines and redistribution of thermal energy. In this project highlights included the first full-scale kinetic simulations of a magnetized hohlraum and the discovery of a new magnetic reconnection mechanism, as well as a completed PhD thesis and the production of a new code for Inertial Fusion research.

  2. Development of parallel Fokker-Planck code ALLAp

    Batishcheva, A.A.; Sigmar, D.J.; Koniges, A.E.

    1996-01-01

    We report on our ongoing development of the 3D Fokker-Planck code ALLA for a highly collisional scrape-off-layer (SOL) plasma. A SOL with strong gradients of density and temperature in the spatial dimension is modeled. Our method is based on a 3-D adaptive grid (in space, magnitude of the velocity, and cosine of the pitch angle) and a second order conservative scheme. Note that the grid size is typically 100 x 257 x 65 nodes. It was shown in our previous work that only these capabilities make it possible to benchmark a 3D code against a spatially-dependent self-similar solution of a kinetic equation with the Landau collision term. In the present work we show results of a more precise benchmarking against the exact solutions of the kinetic equation using a new parallel code ALLAp with an improved method of parallelization and a modified boundary condition at the plasma edge. We also report first results from the code parallelization using Message Passing Interface for a Massively Parallel CRI T3D platform. We evaluate the ALLAp code performance versus the number of T3D processors used and compare its efficiency against a Work/Data Sharing parallelization scheme and a workstation version

  3. A Fokker-Planck-Landau collision equation solver on two-dimensional velocity grid and its application to particle-in-cell simulation

    Yoon, E. S.; Chang, C. S., E-mail: cschang@pppl.gov [Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Korea Advanced Institute of Science and Technology, Yuseong-gu, DaeJeon 305-701 (Korea, Republic of)

    2014-03-15

    An approximate two-dimensional solver of the nonlinear Fokker-Planck-Landau collision operator has been developed using the assumption that the particle probability distribution function is independent of gyroangle in the limit of strong magnetic field. The isotropic one-dimensional scheme developed for nonlinear Fokker-Planck-Landau equation by Buet and Cordier [J. Comput. Phys. 179, 43 (2002)] and for linear Fokker-Planck-Landau equation by Chang and Cooper [J. Comput. Phys. 6, 1 (1970)] have been modified and extended to two-dimensional nonlinear equation. In addition, a method is suggested to apply the new velocity-grid based collision solver to Lagrangian particle-in-cell simulation by adjusting the weights of marker particles and is applied to a five dimensional particle-in-cell code to calculate the neoclassical ion thermal conductivity in a tokamak plasma. Error verifications show practical aspects of the present scheme for both grid-based and particle-based kinetic codes.

  4. Nonparametric estimates of drift and diffusion profiles via Fokker-Planck algebra.

    Lund, Steven P; Hubbard, Joseph B; Halter, Michael

    2014-11-06

    Diffusion processes superimposed upon deterministic motion play a key role in understanding and controlling the transport of matter, energy, momentum, and even information in physics, chemistry, material science, biology, and communications technology. Given functions defining these random and deterministic components, the Fokker-Planck (FP) equation is often used to model these diffusive systems. Many methods exist for estimating the drift and diffusion profiles from one or more identifiable diffusive trajectories; however, when many identical entities diffuse simultaneously, it may not be possible to identify individual trajectories. Here we present a method capable of simultaneously providing nonparametric estimates for both drift and diffusion profiles from evolving density profiles, requiring only the validity of Langevin/FP dynamics. This algebraic FP manipulation provides a flexible and robust framework for estimating stationary drift and diffusion coefficient profiles, is not based on fluctuation theory or solved diffusion equations, and may facilitate predictions for many experimental systems. We illustrate this approach on experimental data obtained from a model lipid bilayer system exhibiting free diffusion and electric field induced drift. The wide range over which this approach provides accurate estimates for drift and diffusion profiles is demonstrated through simulation.

  5. Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes

    Morel, J.E.

    1987-01-01

    The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs

  6. Fokker-Planck equation resolution for N variables-Application examples

    Munoz Roldan, A.; Garcia-Olivares, A.

    1994-01-01

    A set of problems which are reducible to Fokker-Planck equations are presented. Those problems have been solved by using the CHAPKOL library. This library of programs solves stochastic ''Fokker-Planck'' equations in one or several dimensions by using the Chapman-Kolmogorov integral. This method calculates the probability distribution at a time t+dt from a distribution given at time t through a convolution integral in which the integrant is the product of the distribution function at time t and the Green function of the Fokker-Planck equation. The method have some numerical advantages when compared with finite differences algorithms. The accuracy of the method is analysed in several specific cases

  7. Automatic mesh refinement and parallel load balancing for Fokker-Planck-DSMC algorithm

    Küchlin, Stephan; Jenny, Patrick

    2018-06-01

    Recently, a parallel Fokker-Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers was developed by the authors. Fokker-Planck-DSMC (FP-DSMC) is an augmentation of the classical DSMC algorithm, which mitigates the near-continuum deficiencies in terms of computational cost of pure DSMC. At each time step, based on a local Knudsen number criterion, the discrete DSMC collision operator is dynamically switched to the Fokker-Planck operator, which is based on the integration of continuous stochastic processes in time, and has fixed computational cost per particle, rather than per collision. In this contribution, we present an extension of the previous implementation with automatic local mesh refinement and parallel load-balancing. In particular, we show how the properties of discrete approximations to space-filling curves enable an efficient implementation. Exemplary numerical studies highlight the capabilities of the new code.

  8. Application of Fokker-Planck equation in positron diffusion trapping model

    Bartosova, I.; Ballo, P.

    2015-01-01

    This paper is a theoretical prelude to future work involving positron diffusion in solids for the purpose of positron annihilation lifetime spectroscopy (PALS). PALS is a powerful tool used to study defects present in materials. However, the behavior of positrons in solids is a process hard to describe. Various models have been established to undertake this task. Our preliminary model is based on the Diffusion Trapping Model (DTM) described by partial differential Fokker-Planck equation and is solved via time discretization. Fokker-Planck equation describes the time evolution of the probability density function of velocity of a particle under the influence of various forces. (authors)

  9. On the Connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck Control Frameworks

    Annunziato, Mario

    2014-09-01

    In the framework of stochastic processes, the connection between the dynamic programming scheme given by the Hamilton-Jacobi-Bellman equation and a recently proposed control approach based on the Fokker-Planck equation is discussed. Under appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the FokkerPlanck formalism allows considering a larger class of objectives. To illustrate the connection between the two control strategies, the cases of an Itō stochastic process and of a piecewise-deterministic process are considered.

  10. Fokker-Planck equation resolution for N variables. Application examples; Aplicaciones del programa CHAPKOL para la resolucion de ecuaciones Fokker-Planck en N variables

    Munoz, A; Garcia-Olivares, A

    1993-07-01

    A set of problems which are reducible to Fokker-Planck equations are presented. Those problems have been solved by using the CHAPKOL library. This library of programs solves stochastic Fokker-Plank equations in one or several dimensions by using the Chapman- Kolmogorov integral. This method calculates the probability distribution at a time t + dt from a distribution given at time t through a convolution integral in which the integration is the product of the distribution function at time t and the Green function of the Fokker-Planck equation. The method have some numerical advantages when compared with finite differences algorithms. The accuracy of the method is analysed in several specific cases. (Author) 9 refs.

  11. Contribution to the study of the Fokker-planck equation; Contribution a l'etude de l'equation de Fokker-planck

    Blaquiere, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

    1963-07-01

    In the first paragraphs of this report, the Fokker-Planck equation is presented using the presentation method due to S. Chandrasekhar. Certain conventional resolution methods are given, and then a consideration of the physical interpretation of its various terms leads to a new study method based on the use of Campbell's theorems. This gives a solution to the equation in an integral form. The integral kernel of the solution is a normal centred distribution. Finally, the use of the Laplace transformation leads to a simple determination of the parameters of this integral kernel and connects the present theory to the characteristic function method used in particular in the field of nuclear reactors. The method also makes it possible to calculate the moments of the different orders of the probability distribution without the necessity of solving the Fokker-Planck equation. (author) [French] Dans les premiers paragraphes de ce rapport, l'equation de FOKKER-PLANCK est introduite en utilisant le mode d'expose de S. CHANDRASEKHAR. Puis, apres avoir rappele certaines methodes classiques de resolution, l'interpretation physique de ses differents termes nous conduit a une nouvelle methode d'etude qui repose sur l'utilisation des theoremes de CAMPBELL. On est ainsi conduit a la solution de l'equation sous forme integrale. Le noyau integral de la solution est une distribution normale centree. Enfin l'emploi de la transformation de LAPLACE conduit a une determination simple des parametres de ce noyau integral, et relie la theorie actuelle a la methode de la fonction caracteristique associee, utilisee en particulier dans le domaine des reacteurs nucleaires. Finalement cette methode permet le calcul des moments des differents ordres de la distribution de probabilites, sans passer par la resolution souvent laborieuse de l'equation de FOKKER-PLANCK. (auteur)

  12. Electron acceleration by an obliquely propagating electromagnetic wave in the regime of validity of the Fokker-Planck-Kolmogorov approach

    Hizanidis, Kyriakos; Vlahos, L.; Polymilis, C.

    1989-01-01

    The relativistic motion of an ensemble of electrons in an intense monochromatic electromagnetic wave propagating obliquely in a uniform external magnetic field is studied. The problem is formulated from the viewpoint of Hamiltonian theory and the Fokker-Planck-Kolmogorov approach analyzed by Hizanidis (1989), leading to a one-dimensional diffusive acceleration along paths of constant zeroth-order generalized Hamiltonian. For values of the wave amplitude and the propagating angle inside the analytically predicted stochastic region, the numerical results suggest that the diffusion probes proceeds in stages. In the first stage, the electrons are accelerated to relatively high energies by sampling the first few overlapping resonances one by one. During that stage, the ensemble-average square deviation of the variable involved scales quadratically with time. During the second stage, they scale linearly with time. For much longer times, deviation from linear scaling slowly sets in.

  13. Numerical solution of modified fokker-planck equation with poissonian input

    Náprstek, Jiří; Král, Radomil

    2010-01-01

    Roč. 17, 3/4 (2010), s. 251-268 ISSN 1802-1484 R&D Projects: GA AV ČR(CZ) IAA200710805; GA ČR(CZ) GA103/09/0094 Institutional research plan: CEZ:AV0Z20710524 Keywords : Fokker-Planck equation * poisson ian exciation * numerical solution * transition effects Subject RIV: JN - Civil Engineering

  14. Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

    Shao-Hong Yan

    2014-01-01

    Full Text Available The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

  15. Single particle dynamics of many-body systems described by Vlasov-Fokker-Planck equations

    Frank, T.D.

    2003-01-01

    Using Langevin equations we describe the random walk of single particles that belong to particle systems satisfying Vlasov-Fokker-Planck equations. In doing so, we show that Haissinski distributions of bunched particles in electron storage rings can be derived from a particle dynamics model

  16. Invalidity of the spectral Fokker-Planck equation forCauchy noise driven Langevin equation

    Ditlevsen, Ove Dalager

    2004-01-01

    -called alpha-stable noise (or Levy noise) the Fokker-Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. In stead it has been attempted to formulate an equation for the characteristic function (the Fourier transform...

  17. Modelling of thermal transport using Fokker-Planck equations in laser produced plasma

    Nakarmi, J.J.; Jha, L.N.

    1996-12-01

    The kinetic equation with Fokker-Planck collision term has been presented to obtain the distribution function in the corona of inertial confinement fusion, in the presence of the self generated magnetic field. The resulting distribution has non-local form with the convolution in Maxwellian. An expression for thermal flux with self generated magnetic field is obtained. (author). 22 refs

  18. A purely Lagrangian method for the numerical integration of Fokker-Planck equations

    Combis, P.; Fronteau, J.

    1986-01-01

    A new numerical approach to Fokker-Planck equations is presented, in which the integration grid moves according to the solution of a differential system. The method is purely Lagrangian, the mean effect of the diffusion being inserted into the differential system itself

  19. The Fokker-Planck equation for ray dispersion in gyrotropic stratified media

    Golynski, S.M.

    1984-01-01

    The Hamilton equations of geometrical optics determine the rays of the relevant wave field in the short wavelength. We give a systematic derivation of the Fokker-Planck equation for the joint probability density of the position and unit direction vector of rays propagating in a gyrotropic stratified

  20. Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field

    Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

    2017-10-01

    In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

  1. Long-Time Dynamic Response and Stochastic Resonance of Subdiffusive Overdamped Bistable Fractional Fokker-Planck Systems

    Yan-Mei, Kang; Yao-Lin, Jiang

    2008-01-01

    To explore the influence of anomalous diffusion on stochastic resonance (SR) more deeply and effectively, the method of moments is extended to subdiffusive overdamped bistable fractional Fokker-Planck systems for calculating the long-time linear dynamic response. It is found that the method of moments attains high accuracy with the truncation order N = 10, and in normal diffusion such obtained spectral amplification factor (SAF) of the first-order harmonic is also confirmed by stochastic simulation. Observing the SAF of the odd-order harmonics we find some interesting results, i.e. for smaller driving frequency the decrease of sub diffusion exponent inhibits the stochastic resonance (SR), while for larger driving frequency the decrease of sub diffusion exponent enhances the second SR peak, but the first one vanishes and a double SR is induced in the third-order harmonic at the same time. These observations suggest that the anomalous diffusion has important influence on the bistable dynamics

  2. SU-E-T-22: A Deterministic Solver of the Boltzmann-Fokker-Planck Equation for Dose Calculation

    Hong, X; Gao, H [Shanghai Jiao Tong University, Shanghai, Shanghai (China); Paganetti, H [Massachusetts General Hospital, Boston, MA (United States)

    2015-06-15

    Purpose: The Boltzmann-Fokker-Planck equation (BFPE) accurately models the migration of photons/charged particles in tissues. While the Monte Carlo (MC) method is popular for solving BFPE in a statistical manner, we aim to develop a deterministic BFPE solver based on various state-of-art numerical acceleration techniques for rapid and accurate dose calculation. Methods: Our BFPE solver is based on the structured grid that is maximally parallelizable, with the discretization in energy, angle and space, and its cross section coefficients are derived or directly imported from the Geant4 database. The physical processes that are taken into account are Compton scattering, photoelectric effect, pair production for photons, and elastic scattering, ionization and bremsstrahlung for charged particles.While the spatial discretization is based on the diamond scheme, the angular discretization synergizes finite element method (FEM) and spherical harmonics (SH). Thus, SH is used to globally expand the scattering kernel and FFM is used to locally discretize the angular sphere. As a Result, this hybrid method (FEM-SH) is both accurate in dealing with forward-peaking scattering via FEM, and efficient for multi-energy-group computation via SH. In addition, FEM-SH enables the analytical integration in energy variable of delta scattering kernel for elastic scattering with reduced truncation error from the numerical integration based on the classic SH-based multi-energy-group method. Results: The accuracy of the proposed BFPE solver was benchmarked against Geant4 for photon dose calculation. In particular, FEM-SH had improved accuracy compared to FEM, while both were within 2% of the results obtained with Geant4. Conclusion: A deterministic solver of the Boltzmann-Fokker-Planck equation is developed for dose calculation, and benchmarked against Geant4. Xiang Hong and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang

  3. Analytic solution of the two-dimensional Fokker-Planck equation governing stochastic ion heating by a lower hybrid wave

    Malescio, G.

    1981-04-01

    The two-dimensional Fokker-Planck equation describing the ion motion in a coherent lower hybrid wave above the stochasticity threshold is analytically solved. An expression is given for the steady state power dissipation

  4. Simulating transient dynamics of the time-dependent time fractional Fokker-Planck systems

    Kang, Yan-Mei

    2016-09-01

    For a physically realistic type of time-dependent time fractional Fokker-Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker-Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.

  5. Space distribution and energy straggling of charged particles via Fokker-Planck equation

    Manservisi, S.; Molinari, V.; Nespoli, A.

    1996-01-01

    The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-know Bohr of Bethe formula is found as a first-order approximation of the Fokker-Planck equation

  6. Generalized Fokker-Planck theory for electron and photon transport in biological tissues: application to radiotherapy.

    Olbrant, Edgar; Frank, Martin

    2010-12-01

    In this paper, we study a deterministic method for particle transport in biological tissues. The method is specifically developed for dose calculations in cancer therapy and for radiological imaging. Generalized Fokker-Planck (GFP) theory [Leakeas and Larsen, Nucl. Sci. Eng. 137 (2001), pp. 236-250] has been developed to improve the Fokker-Planck (FP) equation in cases where scattering is forward-peaked and where there is a sufficient amount of large-angle scattering. We compare grid-based numerical solutions to FP and GFP in realistic medical applications. First, electron dose calculations in heterogeneous parts of the human body are performed. Therefore, accurate electron scattering cross sections are included and their incorporation into our model is extensively described. Second, we solve GFP approximations of the radiative transport equation to investigate reflectance and transmittance of light in biological tissues. All results are compared with either Monte Carlo or discrete-ordinates transport solutions.

  7. New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators

    Hassan Kamil Jassim

    2015-01-01

    Full Text Available We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.

  8. A Simple Map Between Fokker-Planck Equation and its Fractional form

    Zahran, M.A.; El-Shewy, E.K.

    2008-01-01

    A simple map between Fokker-Planck Equation (FPE) and its fractional form (FFPE), which recently formulates to describe sub diffusive processes, has been suggested. This connection based on a relation between k-orders for moments of ordinary time domain of FPE and the moments associated with fractional time domain of FFPE . Two classes of special interest of FFPE has been considered to outline this map

  9. Fokker-Planck modeling of current penetration during electron cyclotron current drive

    Merkulov, A.; Westerhof, E.; Schueller, F. C.

    2007-01-01

    The current penetration during electron cyclotron current drive (ECCD) on the resistive time scale is studied with a Fokker-Planck simulation, which includes a model for the magnetic diffusion that determines the parallel electric field evolution. The existence of the synergy between the inductive electric field and EC driven current complicates the process of the current penetration and invalidates the standard method of calculation in which Ohm's law is simply approximated by j-j cd =σE. Here it is proposed to obtain at every time step a self-consistent approximation to the plasma resistivity from the Fokker-Planck code, which is then used in a concurrent calculation of the magnetic diffusion equation in order to obtain the inductive electric field at the next time step. A series of Fokker-Planck calculations including a self-consistent evolution of the inductive electric field has been performed. Both the ECCD power and the electron density have been varied, thus varying the well known nonlinearity parameter for ECCD P rf [MW/m -3 ]/n e 2 [10 19 m -3 ] [R. W. Harvey et al., Phys. Rev. Lett 62, 426 (1989)]. This parameter turns out also to be a good predictor of the synergetic effects. The results are then compared with the standard method of calculations of the current penetration using a transport code. At low values of the Harvey parameter, the standard method is in quantitative agreement with Fokker-Planck calculations. However, at high values of the Harvey parameter, synergy between ECCD and E parallel is found. In the case of cocurrent drive, this synergy leads to the generation of large amounts of nonthermal electrons and a concomitant increase of the electrical conductivity and current penetration time. In the case of countercurrent drive, the ECCD efficiency is suppressed by the synergy with E parallel while only a small amount of nonthermal electrons is produced

  10. Fermi-Dirac-Fokker-Planck equation : well-posedness & long-time asymptotics

    Carrillo , José A.; Laurençot , Philippe; Rosado , Jesús

    2009-01-01

    International audience; A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a con...

  11. Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics

    Carrillo, José A.; Laurençot, Philippe; Rosado, Jesús

    2008-01-01

    A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asym...

  12. Quasilinear simulation of auroral kilometric radiation by a relativistic Fokker-Planck code

    Matsuda, Y.

    1991-01-01

    An intense terrestrial radiation called the auroral kilometric radiation (AKR) is believed to be generated by cyclotron maser instability. We study a quasilinear evolution of this instability by means of a two-dimensional relativistic Fokker-Planck code which treats waves and distributions self-consistently, including radiation loss and electron source and sink. We compare the distributions and wave amplitude with spacecraft observations to elucidate physical processes involved. 3 refs., 1 fig

  13. Collective motions and non-polynomial lagrangians in Fokker-Planck dynamics

    Spina, A.; Vucetich, H.

    1986-04-01

    A method based on the ideas of collective motion is applied to Fokker-Planck dynamics. The usual diagramatic techniques used in stochastic dynamics are enlarged in order to deal with the non-polynomial Lagrangians that appear in the theory. The technique is tested and applied to the case of a self-sustained sinusoidal oscillator whose statistical behaviour is well understood: the Poincare oscillator. (author)

  14. Solution of the Fokker-Planck equation for axially-channeled relativistic electrons

    Muralev, V.A.; Telegin, V.I.

    1981-01-01

    A method of the two dimensional kinetic equation of the Fokker-Planck type for axially-channeled electrons is proposed. This equation has been obtained recently by Beloshitsky and Kumakhov to describe the diffusion of channeling negative particles over the transverse energy and angular momentum. The results of computation of the dechanneling function of 1 GeV electrons in tungsten are presented. (author)

  15. Time dependent solutions of the Fokker-Planck equation for fast fusion ions

    Gnavi, G.; Gratton, F.T.; Heyn, M.

    1990-01-01

    Approximate time dependent solutions for the Fokker-Planck equation for fast fusion ions from an isotropic, monoenergetic source are presented, for the problem of D - T - He 3 reactions. The equations include the effect of diffusion, which is particularly noticeable in the distribution of particles of lower energy and in the formation of a tail of particles with energy higher than that of the source. (Author)

  16. An application of the ideas of collective motion to Fokker-Planck dynamics

    Spina, A.

    1985-08-01

    The implementation of ideas and techniques of field theory to statistical physics have proved invaluable both in deepening our understanding in this second area and as a powerful computational tool. In this paper we analyze some aspects of the application to Fokker-Planck dynamics for the case of self-sustained oscillators driven by white noise of a concept that has been found fruitful in quantum field theory, namely the collective coordinate method. (author)

  17. Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium.

    Ceccato, Alessandro; Frezzato, Diego

    2018-02-14

    The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.

  18. The Fokker-Planck equation for coupled Brown-Néel-rotation

    Weizenecker, Jürgen

    2018-02-01

    Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

  19. The Fokker-Planck equation for coupled Brown-Néel-rotation.

    Weizenecker, Jürgen

    2018-01-22

    Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

  20. Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes; Solucion de la ecuacion de transporte de Boltzmann-Fokker-Planck usando esquemas nodales exponenciales

    Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx

    2003-07-01

    There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

  1. Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

    Náprstek, Jiří; Král, Radomil

    2016-01-01

    Roč. 744, č. 1 (2016), č. článku 012177. ISSN 1742-6588. [International Conference on Motion and Vibration Control (MOVIC 2016) /13./ and International Conference on Recent Advances in Structural Dynamics (RASD 2016) /12./. Southampton, 04.07.2016-06.07.2016] R&D Projects: GA ČR(CZ) GP14-34467P; GA ČR(CZ) GA15-01035S Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * finite element method * single degree of freedom systems (SDOF) Subject RIV: JM - Building Engineering http://iopscience.iop.org/article/10.1088/1742-6596/744/1/012177

  2. FOKN: a relativistic Fokker-Planck code with large angle scattering and radiation losses

    Zimmerman, G.; Scharlemann, T.; Wood, L.; Weaver, T.; Chu, T.; Lee, G.

    1976-07-01

    FOKN is a computer code which employs a relativistic Fokker-Planck algorithm to evolve the distribution functions of the various mutually interacting components of a multi-species plasma forward in time, with the optional addition of high angle, large energy and momentum transfer interactions between the various charged species of the plasma. As a computational expediency, the latter processes are handled by transfer matrices which are generated separately by another code, RNUX, so that once specific transfer matrices are generated, they can be used over and over by FOKN provided the group structures are compatible

  3. Fokker-Planck Modelling of Delayed Loss of Charged Fusion Products in TFTR

    Edenstrasser, J.W.; Goloborod'ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A.; Zweben, S.

    1998-01-01

    The results of a Fokker-Planck simulation of the ripple-induced loss of charged fusion products in the Tokamak Fusion Test Reactor (TFTR) are presented. It is shown that the main features of the measured ''delayed loss'' of partially thermalized fusion products, such as the differences between deuterium-deuterium and deuterium-tritium discharges, the plasma current and major radius dependencies, etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR

  4. One-Dimensional Fokker-Planck Equation with Quadratically Nonlinear Quasilocal Drift

    Shapovalov, A. V.

    2018-04-01

    The Fokker-Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are derived and the Lie symmetries are found. A group invariant solution having the form of a traveling wave is found. Within the framework of Adomian's iterative method, the first iterations of an approximate solution of the Cauchy problem are obtained. Two illustrative examples of exact solutions are found.

  5. Some exact solutions for a unidimensional fokker-planck equation by using lie symmetries

    Hugo Hernán Ortíz-Álvarez

    2015-01-01

    Full Text Available The Fokker Planck equation appears in the study of diffusion phenomena, stochastics processes and quantum and classical mechanics. A particular case fromthis equation, ut − uxx − xux − u=0, is examined by the Lie group method approach. From the invariant condition it was possible to obtain the infinitesimal generators or vectors associated to this equation, identifying the corresponding symmetry groups. Exact solution were found for each one of this generators and new solution were constructed by using symmetry properties.

  6. 3-D Ray-tracing and 2-D Fokker-Planck Simulations of Radiofrequency Application to Tokamak Plasmas

    Cardinali, A.; Paoletti, F.; Bernabei, S.

    1999-01-01

    A state of the art numerical tool has been developed to simulate the propagation and the absorption of coexisting different types of waves in a tokamak geometry. The code includes a numerical solution of the three-dimensional (R, Z, Φ) toroidal wave equation for the electric field of the different waves in the WKBJ approximation. At each step of integration, the two-dimensional (v parallel, v perpendicular) Fokker-Planck equation is solved in the presence of quasilinear diffusion coefficients. The electron Landau damping of the waves is modeled taking into account the interaction of the wave electric fields with the quasilinearly modified distribution function. Consistently, the code calculates the radial profiles of non-inductively generated current densities, the transmitted power traces and the total power damping curves. Synergistic effects among the different type of waves (e.g., lower hybrid and ion Bernstein waves) are studied through the separation of the contributions of the single wave from the effects due to their coexistence

  7. High energy ion range and deposited energy calculation using the Boltzmann-Fokker-Planck splitting of the Boltzmann transport equation

    Mozolevski, I.E.

    2001-01-01

    We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann-Fokker-Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann-Fokker-Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann-Fokker-Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses

  8. A New Fokker-Planck Approach for the Relaxation-driven Evolution of Galactic Nuclei

    Vasiliev, Eugene

    2017-10-01

    We present an approach for simulating the collisional evolution of spherical isotropic stellar systems based on the one-dimensional Fokker-Planck equation. A novel aspect is that we use the phase volume as the argument of the distribution function instead of the traditionally used energy, which facilitates the solution. The publicly available code PhaseFlow implements a high-accuracy finite-element method for the Fokker-Planck equation, and can handle multiple-component systems, optionally with the central black hole and taking into account loss-cone effects and star formation. We discuss the energy balance in the general setting, and in application to the Bahcall-Wolf cusp around a central black hole, for which we derive a perturbative solution. We stress that the cusp is not a steady-state structure, but rather evolves in amplitude while retaining an approximately ρ \\propto {r}-7/4 density profile. Finally, we apply the method to the nuclear star cluster of the milky Way, and illustrate a possible evolutionary scenario in which a two-component system of lighter main-sequence stars and stellar-mass black holes develops a Bahcall-Wolf cusp in the heavier component and a weaker ρ \\propto {r}-3/2 cusp in the lighter, visible component, over the period of several Gyr. The present-day density profile is consistent with the recently detected mild cusp inside the central parsec, and is weakly sensitive to initial conditions.

  9. Cusping, transport and variance of solutions to generalized Fokker-Planck equations

    Carnaffan, Sean; Kawai, Reiichiro

    2017-06-01

    We study properties of solutions to generalized Fokker-Planck equations through the lens of the probability density functions of anomalous diffusion processes. In particular, we examine solutions in terms of their cusping, travelling wave behaviours, and variance, within the framework of stochastic representations of generalized Fokker-Planck equations. We give our analysis in the cases of anomalous diffusion driven by the inverses of the stable, tempered stable and gamma subordinators, demonstrating the impact of changing the distribution of waiting times in the underlying anomalous diffusion model. We also analyse the cases where the underlying anomalous diffusion contains a Lévy jump component in the parent process, and when a diffusion process is time changed by an uninverted Lévy subordinator. On the whole, we present a combination of four criteria which serve as a theoretical basis for model selection, statistical inference and predictions for physical experiments on anomalously diffusing systems. We discuss possible applications in physical experiments, including, with reference to specific examples, the potential for model misclassification and how combinations of our four criteria may be used to overcome this issue.

  10. Fokker-Planck description for the queue dynamics of large tick stocks

    Garèche, A.; Disdier, G.; Kockelkoren, J.; Bouchaud, J.-P.

    2013-09-01

    Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. “Jump” events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

  11. Fractional Fokker-Planck equation and oscillatory behavior of cumulant moments

    Suzuki, N.; Biyajima, M.

    2002-01-01

    The Fokker-Planck equation is considered, which is connected to the birth and death process with immigration by the Poisson transform. The fractional derivative in time variable is introduced into the Fokker-Planck equation in order to investigate an origin of oscillatory behavior of cumulant moments. From its solution (the probability density function), the generating function (GF) for the corresponding probability distribution is derived. We consider the case when the GF reduces to that of the negative binomial distribution (NBD), if the fractional derivative is replaced to the ordinary one. The H j moment derived from the GF of the NBD decreases monotonically as the rank j increases. However, the H j moment derived in our approach oscillates, which is contrasted with the case of the NBD. Calculated H j moments are compared with those of charged multiplicities observed in pp-bar, e + e - , and e + p collisions. A phenomenological meaning of introducing the fractional derivative in time variable is discussed

  12. Stationarity-conservation laws for fractional differential equations with variable coefficients

    Klimek, Malgorzata [Institute of Mathematics and Computer Science, Technical University of Czestochowa, Czestochowa (Poland)

    2002-08-09

    In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)

  13. Stationarity-conservation laws for fractional differential equations with variable coefficients

    Klimek, Malgorzata

    2002-01-01

    In this paper, we study linear fractional differential equations with variable coefficients. It is shown that, by assuming some conditions for the coefficients, the stationarity-conservation laws can be derived. The area where these are valid is restricted by the asymptotic properties of solutions of the respective equation. Applications of the proposed procedure include the fractional Fokker-Planck equation in (1+1)- and (d+1)-dimensional space and the fractional Klein-Kramers equation. (author)

  14. Efficient positive, conservative, Maxwellian preserving and implicit difference schemes for the 1-D isotropic Fokker-Planck-Landau equation; Schemas positifs, implicites, conservant l'energie et les etats d'equilibre pour l'equation de Fokker-Planck-Landau isotrope

    Buet, Ch. [CEA Bruyeres-le-Chatel, Dept. Sciences de la Simulation et de l' Information, Service Numerique Environnement et Constantes, 91 (France); Le Thanh, K.C. [CEA Bruyeres-le-Chatel, Service Physique des Plasmas et Electromagnetisme, 91 (France). Dept. de Physique Theorique et Appliquee

    2008-07-01

    The aim of this paper is to describe the discretization of the Fokker-Planck-Landau (FPL) collision term in the isotropic case, which models the self-collision for the electrons when they are totally isotropized by heavy particles background such as ions. The discussion focuses on schemes, which could preserve positivity, mass, energy and Maxwellian equilibrium. The Chang and Cooper method is widely used by plasma's physicists for the FPL equation (and for Fokker-Planck type equations). We present a new variant that is both positive and conservative contrary to the existing one's. We propose also a non Chang and Cooper 'type scheme on non-uniform grid, which is also both positive, conservative and equilibrium state preserving contrary to existing one's. The case of Coulombian potential is emphasized. We address also the problem of the time discretization. In particular we show how to recast some implicit methods to get band diagonal system and to solve it by direct method with a linear cost. (authors)

  15. Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations

    Kleinert, H.; Zatloukal, V.

    2013-11-01

    The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

  16. A closed-form solution for the two-dimensional Fokker-Planck equation for electron transport in the range of Compton Effect

    Rodriguez, B.D.A.; Vilhena, M.T.; Borges, V.; Hoff, G.

    2008-01-01

    In this paper we solve the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation for charged particles in a rectangular domain. To construct the solution we begin applying the P N approximation in the angular variable and the Laplace Transform in the x-variable, thus obtaining a first order linear differential equation in y-variable, which the solution is straightforward. The angular flux of electrons and the parameters of the medium are used for the calculation of the energy deposited by the secondary electrons generated by Compton Effect. The remaining effects will not be taken into account. The results will be presented under absorbed energy form in several points of interested. We present numerical simulations and comparisons with results obtained by using Geant4 (version 8) program which applies the Monte Carlo's technique to low energy libraries for a two-dimensional problem assuming the screened Rutherford differential scattering cross-section

  17. Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

    Colmenares, Pedro J.

    2018-05-01

    This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.

  18. New Fokker-Planck derivation of heavy gas models for neutron thermalization

    Larsen, E.W.; Williams, M.M.R.

    1990-01-01

    This paper is concerned with the derivation of new generalized heavy gas models for the infinite medium neutron energy spectrum equation. Our approach is general and can be used to derive improved Fokker-Planck approximations for other types of kinetic equations. In this paper we obtain two distinct heavy gas models, together with estimates for the corresponding errors. The models are shown in a special case to reduce to modified heavy gas models proposed earlier by Corngold (1962). The error estimates show that both of the new models should be more accurate than Corngold's modified heavy gas model, and that the first of the two new models should generally be more accurate than the second. (author)

  19. Studies of parallel algorithms for the solution of a Fokker-Planck equation

    Deck, D.; Samba, G.

    1995-11-01

    The study of laser-created plasmas often requires the use of a kinetic model rather than a hydrodynamic one. This model change occurs, for example, in the hot spot formation in an ICF experiment or during the relaxation of colliding plasmas. When the gradients scalelengths or the size of a given system are not small compared to the characteristic mean-free-path, we have to deal with non-equilibrium situations, which can be described by the distribution functions of every species in the system. We present here a numerical method in plane or spherical 1-D geometry, for the solution of a Fokker-Planck equation that describes the evolution of stich functions in the phase space. The size and the time scale of kinetic simulations require the use of Massively Parallel Computers (MPP). We have adopted a message-passing strategy using Parallel Virtual Machine (PVM)

  20. Quantal Brownian Motion from RPA dynamics: The master and Fokker-Planck equations

    Yannouleas, C.

    1984-05-01

    From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian Motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted. The RPA master equation agrees with the master equation derived by the time-dependent perturbative approaches which utilize Tamm-Dancoff Hilbert spaces and invoke the rotating wave approximation. Since the RPA yields a full, as well as a contracted description, it can account for both the kinetic and the unperturbed oscillator momenta. The RPA description of the quantal Brownian Motion contrasts with the descriptions provided by the time perturbative approaches whether they invoke or not the rotating wave approximation. The RPA description also contrasts with the phenomenological phase space quantization. (orig.)

  1. Fokker-Planck simulations of knock-on electron runaway avalanche and bursts in tokamaks

    Chiu, S.C.; Rosenbluth, M.N.; Harvey, R.W.; Chan, V.S.

    1998-01-01

    The avalanche of runaway electrons in an ohmic tokamak plasma triggered by knock-on collisions of traces of energetic electrons with the bulk electrons is simulated by the bounce averaged Fokker-Planck code, CQL3D. It is shown that even when the electric field is small for the production of Dreicer runaways, the knock-on collisions can produce significant runaway electrons in a fraction of a second at typical reactor parameters. The energy spectrum of these knock-on runaways has a characteristic temperature. The growth rate and temperature of the runaway distribution are determined and compared with theory. In simulations of pellet injection into high temperature plasmas, it is shown that a burst of Dreicer runaways may also occur depending on the cooling rate due to the pellet injection. Implications of these phenomena on disruption control in reactor plasmas are discussed. (author)

  2. Solution of the relativistic 2-D Fokker-Planck equation for LH current drive

    Hizanidis, K.; Hewett, D.W.; Bers, A.

    1984-03-01

    We solve numerically the steady-state two-dimensional relativistic Fokker-Planck equation with strong rf diffusion using spectra relevant to recent experiments in ALCATOR-C. The results (current generated, power dissipated, and the distribution of energetic electrons) are sensitive to the location of the spectrum in momentum space. Relativistic effects play an important role, especially for wide spectra. The dependence on the ionic charge number Z/sub i/ is also investigated. Particular attention is paid to the perpendicular temperature inside the resonant region and beyond, as well as to the angular energetic particle-temperature distribution, T/sub μ/, a function of the pitch angle parameter μ. The dependence of the perpendicular temperature on the location of the spectrum is also investigated analytically with a model based on the method of moments and the results compared with those found numerically

  3. Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

    Guarnieri, F.; Moon, W.; Wettlaufer, J. S.

    2017-09-01

    Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with a negative constant drift, described by a Fokker-Planck equation with a potential V (x ) =-[b ln(x ) +a x ] , for b >0 and a finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schrödinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a constant negative drift. We conclude with a comparison to other analytical methods and with numerical solutions.

  4. A Fokker-Planck based kinetic model for diatomic rarefied gas flows

    Gorji, M. Hossein; Jenny, Patrick

    2013-06-01

    A Fokker-Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker-Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau-Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

  5. Efficient statistically accurate algorithms for the Fokker-Planck equation in large dimensions

    Chen, Nan; Majda, Andrew J.

    2018-02-01

    Solving the Fokker-Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O (100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6

  6. Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes

    Ortega J, R.; Valle G, E. del

    2003-01-01

    There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S 4 with expansions of the dispersion cross sections until P 3 order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

  7. Dynamic least-squares kernel density modeling of Fokker-Planck equations with application to neural population.

    Shotorban, Babak

    2010-04-01

    The dynamic least-squares kernel density (LSQKD) model [C. Pantano and B. Shotorban, Phys. Rev. E 76, 066705 (2007)] is used to solve the Fokker-Planck equations. In this model the probability density function (PDF) is approximated by a linear combination of basis functions with unknown parameters whose governing equations are determined by a global least-squares approximation of the PDF in the phase space. In this work basis functions are set to be Gaussian for which the mean, variance, and covariances are governed by a set of partial differential equations (PDEs) or ordinary differential equations (ODEs) depending on what phase-space variables are approximated by Gaussian functions. Three sample problems of univariate double-well potential, bivariate bistable neurodynamical system [G. Deco and D. Martí, Phys. Rev. E 75, 031913 (2007)], and bivariate Brownian particles in a nonuniform gas are studied. The LSQKD is verified for these problems as its results are compared against the results of the method of characteristics in nondiffusive cases and the stochastic particle method in diffusive cases. For the double-well potential problem it is observed that for low to moderate diffusivity the dynamic LSQKD well predicts the stationary PDF for which there is an exact solution. A similar observation is made for the bistable neurodynamical system. In both these problems least-squares approximation is made on all phase-space variables resulting in a set of ODEs with time as the independent variable for the Gaussian function parameters. In the problem of Brownian particles in a nonuniform gas, this approximation is made only for the particle velocity variable leading to a set of PDEs with time and particle position as independent variables. Solving these PDEs, a very good performance by LSQKD is observed for a wide range of diffusivities.

  8. LLE-LLNL progress report on studies in nonlocal heat transport in spherical plasmas using the Fokker-Planck code SPARK

    Epperlein, E.M.

    1992-01-01

    Preliminary 1-D studies of nonlocal heat transport in spherical plasmas based on the Fokker-Planck code SPARK indicate significant levels of electron preheat and radial heat flux across a spherical heat sink surface kept at fixed temperature. However, the diffusive approximation to the Fokker-Planck equation is shown to be particularly sensitive to the nature of the inner surface boundary condition chosen. A suggested remedy is the inclusion of a target capsule in future simulations studies with SPARK

  9. From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag

    Plastino, A. R.; Curado, E. M. F.; Nobre, F. D.; Tsallis, C.

    2018-02-01

    Nonlinear Fokker-Planck equations endowed with power-law diffusion terms have proven to be valuable tools for the study of diverse complex systems in physics, biology, and other fields. The nonlinearity appearing in these evolution equations can be interpreted as providing an effective description of a system of particles interacting via short-range forces while performing overdamped motion under the effect of an external confining potential. This point of view has been recently applied to the study of thermodynamical features of interacting vortices in type II superconductors. In the present work we explore an embedding of the nonlinear Fokker-Planck equation within a Vlasov equation, thus incorporating inertial effects to the concomitant particle dynamics. Exact time-dependent solutions of the q -Gaussian form (with compact support) are obtained for the Vlasov equation in the case of quadratic confining potentials.

  10. Fokker-Planck description of the scattering of radio frequency waves at the plasma edge

    Hizanidis, Kyriakos; Kominis, Yannis; Tsironis, Christos; Ram, Abhay K.

    2010-01-01

    In magnetic fusion devices, radio frequency (rf) waves in the electron cyclotron (EC) and lower hybrid (LH) range of frequencies are being commonly used to modify the plasma current profile. In ITER, EC waves are expected to stabilize the neoclassical tearing mode (NTM) by providing current in the island region [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)]. The appearance of NTMs severely limits the plasma pressure and leads to the degradation of plasma confinement. LH waves could be used in ITER to modify the current profile closer to the edge of the plasma. These rf waves propagate from the excitation structures to the core of the plasma through an edge region, which is characterized by turbulence--in particular, density fluctuations. These fluctuations, in the form of blobs, can modify the propagation properties of the waves by refraction. In this paper, the effect on rf due to randomly distributed blobs in the edge region is studied. The waves are represented as geometric optics rays and the refractive scattering from a distribution of blobs is formulated as a Fokker-Planck equation. The scattering can have two diffusive effects--one in real space and the other in wave vector space. The scattering can modify the trajectory of rays into the plasma and it can affect the wave vector spectrum. The refraction of EC waves, for example, could make them miss the intended target region where the NTMs occur. The broadening of the wave vector spectrum could broaden the wave generated current profile. The Fokker-Planck formalism for diffusion in real space and wave vector space is used to study the effect of density blobs on EC and LH waves in an ITER type of plasma environment. For EC waves the refractive effects become important since the distance of propagation from the edge to the core in ITER is of the order of a meter. The diffusion in wave vector space is small. For LH waves the refractive effects are insignificant but the diffusion in wave vector space is

  11. Proof of the path integral representation of the nonlinear Fokker-Planck equation by means of Fourier series

    Dekker, H.

    1978-01-01

    The lagrangian for the action occurring in the path integral solution of the nonlinear Fokker-Planck equation with constant diffusion function is derived by means of a straightforward Fourier series analysis. In this manner the path between the prepoint and the postpoint in the short time propagator is not restricted a priori to the usually considered straight line. Earlier results by Graham, Stratonovich, Horsthemke and Back, and the author's are recovered and thus put on much safer ground. (Auth.)

  12. Quantum Stochastic Trajectories: The Fokker-Planck-Bohm Equation Driven by the Reduced Density Matrix.

    Avanzini, Francesco; Moro, Giorgio J

    2018-03-15

    The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.

  13. Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment

    Ge, Wenchao; Rodenburg, Brandon; Bhattacharya, M.

    2016-08-01

    Optically levitated nanoparticles have recently emerged as versatile platforms for investigating macroscopic quantum mechanics and enabling ultrasensitive metrology. In this paper we theoretically consider two damping regimes of an optically levitated nanoparticle cooled by cavityless parametric feedback. Our treatment is based on a generalized Fokker-Planck equation derived from the quantum master equation presented recently and shown to agree very well with experiment [B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, Quantum model of cooling and force sensing with an optically trapped nanoparticle, Optica 3, 318 (2016), 10.1364/OPTICA.3.000318]. For low damping, we find that the resulting Wigner function yields the single-peaked oscillator position distribution and recovers the appropriate energy distribution derived earlier using a classical theory and verified experimentally [J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state, Nat. Nano. 9, 358 (2014), 10.1038/nnano.2014.40]. For high damping, in contrast, we predict a double-peaked position distribution, which we trace to an underlying bistability induced by feedback. Unlike in cavity-based optomechanics, stochastic processes play a major role in determining the bistable behavior. To support our conclusions, we present analytical expressions as well as numerical simulations using the truncated Wigner function approach. Our work opens up the prospect of developing bistability-based devices, characterization of phase-space dynamics, and investigation of the quantum-classical transition using levitated nanoparticles.

  14. Fokker-Planck simulation of runaway electron generation in disruptions with the hot-tail effect

    Nuga, H., E-mail: nuga@p-grp.nucleng.kyoto-u.ac.jp; Fukuyama, A. [Department of Engineering, Kyoto University, Kyoto 615-8540 (Japan); Yagi, M. [National Institutes for Quantum and Radiological Science and Technology, Aomori 039-3212 (Japan)

    2016-06-15

    To study runaway electron generation in disruptions, we have extended the three-dimensional (two-dimensional in momentum space; one-dimensional in the radial direction) Fokker-Planck code, which describes the evolution of the relativistic momentum distribution function of electrons and the induced toroidal electric field in a self-consistent manner. A particular focus is placed on the hot-tail effect in two-dimensional momentum space. The effect appears if the drop of the background plasma temperature is sufficiently rapid compared with the electron-electron slowing down time for a few times of the pre-quench thermal velocity. It contributes to not only the enhancement of the primary runaway electron generation but also the broadening of the runaway electron distribution in the pitch angle direction. If the thermal energy loss during the major disruption is assumed to be isotropic, there are hot-tail electrons that have sufficiently large perpendicular momentum, and the runaway electron distribution becomes broader in the pitch angle direction. In addition, the pitch angle scattering also yields the broadening. Since the electric field is reduced due to the burst of runaway electron generation, the time required for accelerating electrons to the runaway region becomes longer. The longer acceleration period makes the pitch-angle scattering more effective.

  15. Loading, absorption, and Fokker-Planck calculations for upcoming ICRF experiments on ATF

    Shepard, T.D.; Carter, M.D.; Goulding, R.H.; Kwon, M.

    1989-01-01

    ICRF experiments on ATF at the 100-kW level are planned for the current 1989 operating period. These plans include the 2ω/sub cH/ regime at f/sub RF/ = 28.88 MHz, D(H) at 14.44 MHz, and 4 He( 3 He) and D( 3 He) at 9.63 MHz. ECH target plasmas have n/sub eO/ /approxreverse arrowlt/ 0.15 /times/ 10 20 m/sup /minus/3/ and B = 0.95 T. The density and temperature profiles obtained are broader than those from 1988, owing to recent field error corrections. The values used for target-plasma parameters in the calculations were taken from initial 1989 ATF data. Loading and absorption calculations have been performed using the 3D RF heating code ORION with a helically symmetric equilibrium, and Fokker-Planck calculations were performed using the steady-state code RFTRANS with two velocity dimensions and one spatial dimension. 6 refs., 3 figs

  16. Supersymmetric quantum mechanics method for the Fokker-Planck equation with applications to protein folding dynamics

    Polotto, Franciele; Drigo Filho, Elso; Chahine, Jorge; Oliveira, Ronaldo Junio de

    2018-03-01

    This work developed analytical methods to explore the kinetics of the time-dependent probability distributions over thermodynamic free energy profiles of protein folding and compared the results with simulation. The Fokker-Planck equation is mapped onto a Schrödinger-type equation due to the well-known solutions of the latter. Through a semi-analytical description, the supersymmetric quantum mechanics formalism is invoked and the time-dependent probability distributions are obtained with numerical calculations by using the variational method. A coarse-grained structure-based model of the two-state protein Tm CSP was simulated at a Cα level of resolution and the thermodynamics and kinetics were fully characterized. Analytical solutions from non-equilibrium conditions were obtained with the simulated double-well free energy potential and kinetic folding times were calculated. It was found that analytical folding time as a function of temperature agrees, quantitatively, with simulations and experiments from the literature of Tm CSP having the well-known 'U' shape of the Chevron Plots. The simple analytical model developed in this study has a potential to be used by theoreticians and experimentalists willing to explore, quantitatively, rates and the kinetic behavior of their system by informing the thermally activated barrier. The theory developed describes a stochastic process and, therefore, can be applied to a variety of biological as well as condensed-phase two-state systems.

  17. A combination between Laplace transformation technique and numerical approximations to the Fokker-Planck equation solutions

    Monticelli, Cintia O.; Wortmann, Sergio; Segatto, Cynthia F.

    2005-01-01

    In this work is obtained a hybrid solution to the Fokker-Planck equation with energy dependency, very used in ion implantation problems. The main idea relies on the application of Laplace transform in the energy variable, and finite-difference in the spatial variable and in the angular variable. This procedure leads to a symbolic matrix problem for the transformed energy. To solve this system, is needed to do the Laplace inverse of the (sI+A) matrix, where s is a complex parameter, I is the identity matrix and A is a square matrix that was proceeded from the finite-difference in the spatial variable and in the angular variable. The matrix A is not defective, then is taken decomposition of A in a sum of two others matrices, where one is defective. It leads a iterative inversion method, similar the source fixed method combined with the diagonalization method, then is obtained the values to the angular flux. Hereafter we can to determine the energy deposited into the electronic system and in the nuclear system of the target. To comprove the results obtained, the simulation of implantation of B into Si at energies ranging from 1 KeV to 50 MeV was carried out and compared with the results by software SRIM2003. (author)

  18. On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

    Balazs, N.L.

    1978-01-01

    In the classical theory of Brownian motion the Langevin equation can be considered as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, the Langevin equation can be reinterpreted as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta one can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. One may associate, according to Weyl, functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions the streaming can be expressed as a continuity equation of the Wigner function. It is found that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation. (Auth.)

  19. Application of the Fokker-Planck molecular mixing model to turbulent scalar mixing using moment methods

    Madadi-Kandjani, E.; Fox, R. O.; Passalacqua, A.

    2017-06-01

    An extended quadrature method of moments using the β kernel density function (β -EQMOM) is used to approximate solutions to the evolution equation for univariate and bivariate composition probability distribution functions (PDFs) of a passive scalar for binary and ternary mixing. The key element of interest is the molecular mixing term, which is described using the Fokker-Planck (FP) molecular mixing model. The direct numerical simulations (DNSs) of Eswaran and Pope ["Direct numerical simulations of the turbulent mixing of a passive scalar," Phys. Fluids 31, 506 (1988)] and the amplitude mapping closure (AMC) of Pope ["Mapping closures for turbulent mixing and reaction," Theor. Comput. Fluid Dyn. 2, 255 (1991)] are taken as reference solutions to establish the accuracy of the FP model in the case of binary mixing. The DNSs of Juneja and Pope ["A DNS study of turbulent mixing of two passive scalars," Phys. Fluids 8, 2161 (1996)] are used to validate the results obtained for ternary mixing. Simulations are performed with both the conditional scalar dissipation rate (CSDR) proposed by Fox [Computational Methods for Turbulent Reacting Flows (Cambridge University Press, 2003)] and the CSDR from AMC, with the scalar dissipation rate provided as input and obtained from the DNS. Using scalar moments up to fourth order, the ability of the FP model to capture the evolution of the shape of the PDF, important in turbulent mixing problems, is demonstrated. Compared to the widely used assumed β -PDF model [S. S. Girimaji, "Assumed β-pdf model for turbulent mixing: Validation and extension to multiple scalar mixing," Combust. Sci. Technol. 78, 177 (1991)], the β -EQMOM solution to the FP model more accurately describes the initial mixing process with a relatively small increase in computational cost.

  20. Theoretical background and implementation of the finite element method for multi-dimensional Fokker-Planck equation analysis

    Král, Radomil; Náprstek, Jiří

    2017-01-01

    Roč. 113, November (2017), s. 54-75 ISSN 0965-9978 R&D Projects: GA ČR(CZ) GP14-34467P; GA ČR(CZ) GA15-01035S Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * finite element method * simplex element * multi-dimensional problem * non-symmetric operator Subject RIV: JM - Building Engineering OBOR OECD: Mechanical engineering Impact factor: 3.000, year: 2016 https://www.sciencedirect.com/science/ article /pii/S0965997817301904

  1. A Finite-Orbit-Width Fokker-Planck solver for modeling of energetic particle interactions with waves, with application to Helicons in ITER

    Petrov Yuri V.

    2017-01-01

    Full Text Available The bounce-average (BA finite-difference Fokker-Planck (FP code CQL3D [1,2] now includes the essential physics to describe the RF heating of Finite-Orbit-Width (FOW ions in tokamaks. The FP equation is reformulated in terms of Constants-Of-Motion coordinates, which we select to be particle speed, pitch angle, and major radius on the equatorial plane thus obtaining the distribution function directly at this location. Full-orbit, low collisionality neoclassical radial transport emerges from averaging the local friction and diffusion coefficients along guiding center orbits. Similarly, the BA of local quasilinear RF diffusion terms gives rise to additional radial transport. The local RF electric field components needed for the BA operator are usually obtained by a ray-tracing code, such as GENRAY, or in conjunction with full-wave codes. As a new, practical application, the CQL3D-FOW version is used for simulation of alpha-particle heating by high-harmonic waves in ITER. Coupling of high harmonic or helicon fast waves power to electrons is a promising current drive (CD scenario for high beta plasmas. However, the efficiency of current drive can be diminished by parasitic channeling of RF power into fast ions, such as alphas, through finite Larmor-radius effects. We investigate possibilities to reduce the fast ion heating in CD scenarios.

  2. A closed-form solution for the two-dimensional Fokker-Planck equation for electron transport in the range of Compton Effect

    Rodriguez, B.D.A. [Universidade Federal Rio Grande do Sul, Programa de Pos-Graduacao em Engenharia Mecanica, Rua Portuguesa 218/304, 90650-12 Porto Alegre, RS (Brazil)], E-mail: barbara.arodriguez@gmail.com; Vilhena, M.T. [Universidade Federal Rio Grande do Sul, Departamento de Matematica Pura e Aplicada, Porto Alegre, RS (Brazil)], E-mail: vilhena@mat.ufrgs.br; Borges, V. [Universidade Federal Rio Grande do Sul, Programa de Pos-Graduacao em Engenharia Mecanica, Rua Portuguesa 218/304, 90650-12 Porto Alegre, RS (Brazil)], E-mail: borges@ufrgs.br; Hoff, G. [Pontificia Universidade Catolica do Rio Grande do Sul, Faculdade de Fisica, Porto Alegre, RS (Brazil)], E-mail: hoff@pucrs.br

    2008-05-15

    In this paper we solve the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation for charged particles in a rectangular domain. To construct the solution we begin applying the P{sub N} approximation in the angular variable and the Laplace Transform in the x-variable, thus obtaining a first order linear differential equation in y-variable, which the solution is straightforward. The angular flux of electrons and the parameters of the medium are used for the calculation of the energy deposited by the secondary electrons generated by Compton Effect. The remaining effects will not be taken into account. The results will be presented under absorbed energy form in several points of interested. We present numerical simulations and comparisons with results obtained by using Geant4 (version 8) program which applies the Monte Carlo's technique to low energy libraries for a two-dimensional problem assuming the screened Rutherford differential scattering cross-section.

  3. On self-consistent ray-tracing and Fokker-Planck modeling of the hard X-ray emission during lower-hybrid current driven in Tokamaks

    Bizarro, J.P.; Peysson, Y.; Bonoli, P.T.; Carrasco, J.; Dudok de Wit, T.; Fuchs, V.; Hoang, G.T.; Litaudon, X.; Moreau, D.; Pocheau, C.; Shkarofsky, I.P.

    1993-04-01

    A detailed investigation is presented on the ability of combined ray-tracing and Fokker-Planck calculations to predict the hard x-ray (HXR) emission during lower-hybrid (LH) current drive in tokamaks when toroidally induced-ray-stochasticity is important. A large number of rays is used and the electron distribution function is obtained by self-consistently iterating the appropriate LH power deposition and Fokker-Planck calculations. Most of the experimentally observed features of the HXR emission are correctly predicted. It is found that corrections due to radial diffusion of suprathermal electrons and to radiation scattering by the inner wall can be significant

  4. Improved method for calculating neoclassical transport coefficients in the banana regime

    Taguchi, M., E-mail: taguchi.masayoshi@nihon-u.ac.jp [College of Industrial Technology, Nihon University, Narashino 275-8576 (Japan)

    2014-05-15

    The conventional neoclassical moment method in the banana regime is improved by increasing the accuracy of approximation to the linearized Fokker-Planck collision operator. This improved method is formulated for a multiple ion plasma in general tokamak equilibria. The explicit computation in a model magnetic field shows that the neoclassical transport coefficients can be accurately calculated in the full range of aspect ratio by the improved method. The some neoclassical transport coefficients for the intermediate aspect ratio are found to appreciably deviate from those obtained by the conventional moment method. The differences between the transport coefficients with these two methods are up to about 20%.

  5. The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models

    Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.

    1997-05-01

    We report a new measurement of the velocity dispersion profile within 1' (3 pc) of the center of the globular cluster M15 (NGC 7078), using long-slit spectra from the 4.2 m William Herschel Telescope at La Palma Observatory. We obtained spatially resolved spectra for a total of 23 slit positions during two observing runs. During each run, a set of parallel slit positions was used to map out the central region of the cluster; the position angle used during the second run was orthogonal to that used for the first. The spectra are centered in wavelength near the Ca II infrared triplet at 8650 Å, with a spectral range of about 450 Å. We determined radial velocities by cross-correlation techniques for 131 cluster members. A total of 32 stars were observed more than once. Internal and external comparisons indicate a velocity accuracy of about 4 km s-1. The velocity dispersion profile rises from about σ = 7.2 +/- 1.4 km s-1 near 1' from the center of the cluster to σ = 13.9 +/- 1.8 km s-1 at 20". Inside of 20", the dispersion remains approximately constant at about 10.2 +/- 1.4 km s-1 with no evidence for a sharp rise near the center. This last result stands in contrast with that of Peterson, Seitzer, & Cudworth who found a central velocity dispersion of 25 +/- 7 km s-1, based on a line-broadening measurement. Our velocity dispersion profile is in good agreement with those determined in the recent studies of Gebhardt et al. and Dubath & Meylan. We have developed a new set of Fokker-Planck models and have fitted these to the surface brightness and velocity dispersion profiles of M15. We also use the two measured millisecond pulsar accelerations as constraints. The best-fitting model has a mass function slope of x = 0.9 (where 1.35 is the slope of the Salpeter mass function) and a total mass of 4.9 × 105 M⊙. This model contains approximately 104 neutron stars (3% of the total mass), the majority of which lie within 6" (0.2 pc) of the cluster center. Since the

  6. Design of foam-buffered high gain target with Fokker-Planck implosion simulation for thermal insulation and imprint mitigation

    Takeda, T.; Mima, K.; Norimatsu, T.; Nagatomo, H.; Nishiguchi, A.

    2003-01-01

    It is proposed that a thick foam layer on a plastic capsule of fusion pellet is effective not only for reducing the initial imprint, but also for solving the melting problem of cryogenic deuterium-tritium layer, in a reactor chamber. Investigated are the dependences of gain, thermal insulation for preventing the melting, and imprint mitigation of a foam-buffered target on the foam layer thickness. The imprint mitigation, the Rayleigh-Taylor growth factor and the fusion gain of a foam-buffered target are evaluated by the hydrodynamic implosion code HIMICO [A. Nishiguchi et al., Phys. Fluids B 4, 417 (1992)], which includes a Fokker-Planck transport code. As the result, it is found that high gain can be achieved by the foam-buffered target together with thermal insulation and imprint mitigation

  7. Using some results about the Lie evolution of differential operators to obtain the Fokker-Planck equation for non-Hamiltonian dynamical systems of interest

    Bianucci, Marco

    2018-05-01

    Finding the generalized Fokker-Planck Equation (FPE) for the reduced probability density function of a subpart of a given complex system is a classical issue of statistical mechanics. Zwanzig projection perturbation approach to this issue leads to the trouble of resumming a series of commutators of differential operators that we show to correspond to solving the Lie evolution of first order differential operators along the unperturbed Liouvillian of the dynamical system of interest. In this paper, we develop in a systematic way the procedure to formally solve this problem. In particular, here we show which the basic assumptions are, concerning the dynamical system of interest, necessary for the Lie evolution to be a group on the space of first order differential operators, and we obtain the coefficients of the so-evolved operators. It is thus demonstrated that if the Liouvillian of the system of interest is not a first order differential operator, in general, the FPE structure breaks down and the master equation contains all the power of the partial derivatives, up to infinity. Therefore, this work shed some light on the trouble of the ubiquitous emergence of both thermodynamics from microscopic systems and regular regression laws at macroscopic scales. However these results are very general and can be applied also in other contexts that are non-Hamiltonian as, for example, geophysical fluid dynamics, where important events, like El Niño, can be considered as large time scale phenomena emerging from the observation of few ocean degrees of freedom of a more complex system, including the interaction with the atmosphere.

  8. About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation; Sur les schemas positifs, conservant l'energie et les etats d'equilibre pour l'equation de Fokker-Planck-Landau isotrope

    Buet, Ch. [CEA Bruyeres-le-Chatel, Dept. des Sciences de la Simulation et de l' Information, 91 (France); Le Thanh, K.C. [CEA Bruyeres-le-Chatel, Dept. de Physique Theorique et Appliquee, 91 (France)

    2006-07-01

    The aim of this paper is to describe the discretization of the Fokker-Planck-Landau (FPL) collision term in the isotropic case which models the self collision for the electrons when they are totally isotropized by heavy particles background such as ions. The discussion focus on schemes which could preserve positivity, mass, energy and Maxwellian equilibrium. First, we analyze in detail the popular Chang and Cooper method for this non-linear collision term: derivation, conservation and positivity properties. We show that some variants of this method, based on the drift-diffusion form of the FPL operator, could not be positive or could not conserve the energy. We present a new variant of the Chang and Cooper method derived from the Landau form that is both positive and conservative. We also propose two new alternatives and simpler schemes for the FPL operator which show that the Chang and Cooper method is not the only way to construct positive, energy conservative and equilibrium state preserving schemes for this operator. For all these schemes, we explain clearly the properties of conservation of the density and the energy, the positivity of the solution and the conservation of the equilibrium states, or their lack. The case of Maxwellian and Coulombian potentials are emphasized. (authors)

  9. An adaptive, implicit, conservative, 1D-2V multi-species Vlasov-Fokker-Planck multi-scale solver in planar geometry

    Taitano, W. T.; Chacón, L.; Simakov, A. N.

    2018-07-01

    We consider a 1D-2V Vlasov-Fokker-Planck multi-species ionic description coupled to fluid electrons. We address temporal stiffness with implicit time stepping, suitably preconditioned. To address temperature disparity in time and space, we extend the conservative adaptive velocity-space discretization scheme proposed in [Taitano et al., J. Comput. Phys., 318, 391-420, (2016)] to a spatially inhomogeneous system. In this approach, we normalize the velocity-space coordinate to a temporally and spatially varying local characteristic speed per species. We explicitly consider the resulting inertial terms in the Vlasov equation, and derive a discrete formulation that conserves mass, momentum, and energy up to a prescribed nonlinear tolerance upon convergence. Our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the Vlasov operator and the Fokker-Planck collision operator. Numerical examples of varying degrees of complexity, including shock-wave propagation, demonstrate the favorable efficiency and accuracy properties of the scheme.

  10. Full-wave and Fokker Planck analysis of ICRF heating experiments in the Alcator C-Mod tokamak

    Bonoli, P.T.; Golovato, S.; Porkolab, M.; Takase, Y.

    1996-01-01

    The Alcator C-Mod device is a high field, high density, shaped tokamak with parameters a = 0.22 m, R 0 = 0.67 m, B 0 ≤ 9.0 T, κ ≤ 1.8, δ ≤ 0.8, and 1.0 x 10 20 m -3 n e (0) ≤ 1.0 x 10 21 m -3 . Four megawatt of ICRF power is available at 80 MHz. The wide operating range in magnetic field makes several heating schemes possible: (i) Second harmonic heating of hydrogen (f 0 = 2f CH ) at 2.6 T in (D-H); (ii) Fundamental heating of (H) (f 0 = f CH ) at 5.3T in a D-(H) plasma; and (iii) Fundamental heating of ( 3 He) (f 0 = f C 3 He ) at 7.9 T in a D-( 3 He) plasma. The most successful heating regime to date has been (H)-minority heating at 5.3 T. Pellet enhanced performance (PEP) modes have also been achieved in C-Mod in D-(H) at 5.3 T and in D-( 3 He) at 7.9 T, with a combination of intense ICRF heating and Li-pellet injection. A variety of numerical models are used to analyze these heating schemes. A 1-D full-wave code (FELICE) is used to study open-quotes single passclose quotes damping of the ICRF wavefront and damping of mode-converted ion Bernstein waves. A toroidal full-wave code (FISIC) is used to study interference and focussing effects of the ICRF waves as well as damping of the ICRF power upon multiple passes of the ICRF wavefront. A combined bounce averaged Fokker Planck and toroidal full-wave code (FPPRF) is used to study the ion tail formation, orbit losses, and the power partition of the ICRF tail to the background electrons and ions. Full-wave and Fokker Planck analyses confirm the strong single pass absorption of the ICRF power in D-(H) at 5.3 T. Analysis of PEP-mode plasmas in D-( 3 He) indicates improved wave focussing and 3 He-cyclotron absorption of the ICRF waves relative to L-mode. A dramatic increase in the transfer of 3 He tail power to the background deuterium is also found for PEP-mode plasmas

  11. Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

    Herda, Maxime; Rodrigues, L. Miguel

    2018-03-01

    The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L^2 space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ → ∞ to the strongly collisional regime τ → 0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ -dependent constraint on δ ensuring exponential decay with explicit τ -dependent rates towards the stationary solution. In the strongly collisional limit τ → 0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L^2 space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

  12. Numerical resolution of the Fokker-Planck equation for non-lineal cases

    Mastropiero, Daniel

    1997-01-01

    The author resolves the Fokker-Plank equation of the statistical thermodynamics of irreversible processes for the flow of particles in a medium, considering two cases: a) The diffusion coefficient of the medium depends on concentration; and b) The medium is unhomogeneous, i.e. the diffusion coefficient varies with the position. Different cases are analyzed and compared

  13. Generalized study of the return to equilibrium of a particle in a plasma (Fokker-Planck formalism) (1961); Etude generale du retour a l'equilibre d'une particule au sein d'un plasma (formalisme de Fokker-Planck) (1961)

    Salmon, J [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires; Faculte des Sciences de Caen, 14 (France)

    1961-07-01

    The author examines the problem of the return to equilibrium of a particle in a plasma and completely explains Fokker-Planck equation. After that, he studies the possibility of interpreting the return of the test particle to Maxwellian distribution, using the development - which is obtained. He discusses the validity limits of the Rosenbluth, MacDonald and Judd approximation. (author) [French] Examinant le probleme du retour a l'equilibre d'une particule test au sein d'un plasma en equilibre, l'auteur cherche a expliciter completement l'expression de l'operateur de Fokker-Planck. Il etudie ensuite les conditions de coherence, c'est-a-dire la possibilite pour le developpement obtenu de traduire le retour de la particule test a l'etat maxwellien et discute des limites de validite de la formule de 'Rosenbluth, Mac Donald et Judd'. (auteur)

  14. Representation of the Fokker-Planck collision term for Coulomb interaction as series of Legendre polynomials

    Almeida Ferreira, A.C. de.

    1984-01-01

    For problems with azimuthal symmetry in velocity space, the distribution function depends only on the speed and on the pitch angle. The angular dependence of the distribution function is expanded in Legendre polynomials, and the expansions of the collision integrals describing two-body Coulomb interactions in a plasma are determined through the use of the Rosenbluth potentials. The electron distribution function is written as a Maxwellian plus a deviation, and the representation in Legendre polynomials of the electron-electron collision term is given for both its linear and nonlinear part. To determine the representation of the electron-ion collision term it is assumed that the ion distribution is much narrower in velocity space than the electron distribution, and shifted from the origin by a flow velocity. The equations are presented in a form that is suitable for their use in a computer. (Author) [pt

  15. An approximate factorization procedure for solving nine-point elliptic difference equations. Application for a fast 2-D relativistic Fokker-Planck solver

    Peysson, Y.

    1997-09-01

    A full implicit numerical procedure based on the use of a nine-point difference operator is presented to solve the two dimensional (2 D ) relativistic Fokker-Planck equation for the current drive problem and synergetic effects between the lower hybrid and the electron cyclotron waves in tokamaks. As compared to the standard approach based on the use of a five-point difference operator [M. Shoucri, I. Shkarofsky, Comput. Phys. Comm. 82 (1994) 287], the convergence rate towards the steady state solution may be significantly enhanced with no loss of accuracy on the distribution function. Moreover, it is shown that the numerical stability may be strongly improved without a large degradation of the CPU time consumption as in the five-point scheme, making this approach very attractive for a fast solution of the 2-D Fokker-Planck equation on a fine grid in conjunction with other numerical codes for realistic plasma simulations. This new algorithm, based on an approximate matrix factorization technique, may be applied to all numerical problems with large sets of equations which involve nine-point difference operators. (author)

  16. An approximate factorization procedure for solving nine-point elliptic difference equations. Application for a fast 2-D relativistic Fokker-Planck solver

    Peysson, Y. [Association Euratom-CEA, CEA Grenoble, 38 (France). Dept. de Recherches sur la Fusion Controlee; Choucri, M. [Centre Canadien de Fusion Magnetique, Varennes, PQ (Canada)

    1997-09-01

    A full implicit numerical procedure based on the use of a nine-point difference operator is presented to solve the two dimensional (2{sub D}) relativistic Fokker-Planck equation for the current drive problem and synergetic effects between the lower hybrid and the electron cyclotron waves in tokamaks. As compared to the standard approach based on the use of a five-point difference operator [M. Shoucri, I. Shkarofsky, Comput. Phys. Comm. 82 (1994) 287], the convergence rate towards the steady state solution may be significantly enhanced with no loss of accuracy on the distribution function. Moreover, it is shown that the numerical stability may be strongly improved without a large degradation of the CPU time consumption as in the five-point scheme, making this approach very attractive for a fast solution of the 2-D Fokker-Planck equation on a fine grid in conjunction with other numerical codes for realistic plasma simulations. This new algorithm, based on an approximate matrix factorization technique, may be applied to all numerical problems with large sets of equations which involve nine-point difference operators. (author) 21 refs.

  17. Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations

    Jumarie, Guy

    2004-01-01

    There are presently two different models of fractional Brownian motions available in the literature: the Riemann-Liouville fractional derivative of white noise on the one hand, and the complex-valued Brownian motion of order n defined by using a random walk in the complex plane, on the other hand. The paper provides a comparison between these two approaches, and in addition, takes this opportunity to contribute some complements. These two models are more or less equivalent on the theoretical standpoint for fractional order between 0 and 1/2, but their practical significances are quite different. Otherwise, for order larger than 1/2, the fractional derivative model has no counterpart in the complex plane. These differences are illustrated by an example drawn from mathematical finance. Taylor expansion of fractional order provides the expression of fractional difference in terms of finite difference, and this allows us to improve the derivation of Fokker-Planck equation and Kramers-Moyal expansion, and to get more insight in their relation with stochastic differential equations of fractional order. In the case of multi-fractal systems, the Fokker-Planck equation can be solved by using path integrals, and the fractional dynamic equations of the state moments of the stochastic system can be easily obtained. By combining fractional derivative and complex white noise of order n, one obtains a family of complex-valued fractional Brownian motions which exhibits long-range dependence. The conclusion outlines suggestions for further research, mainly regarding Lorentz transformation of fractional noises

  18. Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions

    Jumarie, Guy

    2009-01-01

    A probability distribution of fractional (or fractal) order is defined by the measure μ{dx} = p(x)(dx) α , 0 α (D x α h α )f(x) provided by the modified Riemann Liouville definition, one can expand a probability calculus parallel to the standard one. A Fourier's transform of fractional order using the Mittag-Leffler function is introduced, together with its inversion formula; and it provides a suitable generalization of the characteristic function of fractal random variables. It appears that the state moments of fractional order are more especially relevant. The main properties of this fractional probability calculus are outlined, it is shown that it provides a sound approach to Fokker-Planck equation which are fractional in both space and time, and it provides new results in the information theory of non-random functions.

  19. Addendum: ``The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models'' (ApJ, 481, 267 [1997])

    Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.

    2003-03-01

    It has recently come to our attention that there are axis scale errors in three of the figures presented in Dull et al. (1997, hereafter D97). This paper presented Fokker-Planck models for the collapsed-core globular cluster M15 that include a dense, centrally concentrated population of neutron stars and massive white dwarfs. These models do not include a central black hole. Figure 12 of D97, which presents the predicted mass-to-light profile, is of particular interest, since it was used by Gerssen et al. (2002) as an input to their Jeans equation analysis of the Hubble Space Telescope (HST) STIS velocity measurements reported by van der Marel et al. (2002). On the basis of the original, incorrect version of Figure 12, Gerssen et al. (2002) concluded that the D97 models can fit the new data only with the addition of an intermediate-mass black hole. However, this is counter to our previous finding, shown in Figure 6 of D97, that the Fokker-Planck models predict the sort of moderately rising velocity dispersion profile that Gerssen et al. (2002) infer from the new data. Baumgardt et al. (2003) have independently noted this apparent inconsistency. We appreciate the thoughtful cooperation of Roeland van der Marel in resolving this issue. Using our corrected version of Figure 12 (see below), Gerssen et al. (2003) now find that the velocity dispersion profile that they infer from the D97 mass-to-light ratio profile is entirely consistent with the velocity dispersion profile presented in Figure 6 of D97. Gerssen et al. (2003) further find that there is no statistically significant difference between the fit to the van der Marel et al. (2002) velocity measurements provided by the D97 intermediate-phase model and that provided by their model, which supplements this D97 model with a 1.7+2.7-1.7×103Msolar black hole. Thus, the choice between models with and without black holes will require additional model predictions and observational tests. We present corrected versions of

  20. Exact statistical analysis of nonlinear dynamic power reactor models by the Fokker--Planck method. Part II. Reactor with on-off control

    Debosscher, A.F.; Dutre, W.L.

    1979-01-01

    The paper deals with the exact stochastic analysis of the low-frequency neutron density fluctuations in an on-off controlled nuclear power reactor without delayed neutrons and perturbed by Gaussian white reactivity noise. The stochastic process, being Markovian, is completely characterized by its first-order probability density function (pdf) and the transition pdf. The first-order pdf is the normalized solution to the time-independent Fokker--Planck equation (FPE). Using this pdf, a general expression for the moments is obtained. The conditions for stochastic stability in probability, in the mean, and in the mean-square are derived. The time-dependent FPE is solved using the Laplace transform technique, which results in four distinct expressions for the transition pdf, according to the relative magnitude of initial and final reactor power with respect to the regulator level. After Laplace inversion, a physical interpretation of the controller's effect on the stochastic process becomes possible. Finally, making use of the obtained pdf's, the spectral density of the reactor power fluctuations is calculated

  1. High-field transport of electrons and radiative effects using coupled force-balance and Fokker-Planck equations beyond the relaxation-time approximation

    Huang, Danhong; Apostolova, T.; Alsing, P.M.; Cardimona, D.A.

    2004-01-01

    The dynamics of a many-electron system under both dc and infrared fields is separated into a center-of-mass and a relative motion. The first-order force-balance equation is employed for the slow center-of-mass motion of electrons, and the Fokker-Planck equation is used for the ultrafast relative scattering motion of degenerate electrons. This approach allows us to include the anisotropic energy-relaxation process which has been neglected in the energy-balance equation in the past. It also leads us to include the anisotropic coupling to the incident infrared field with different polarizations. Based on this model, the transport of electrons is explored under strong dc and infrared fields by going beyond the relaxation-time approximation. The anisotropic dependence of the electron distribution function on the parallel and perpendicular kinetic energies of electrons is displayed with respect to the dc field direction, and the effect of anisotropic coupling to an incident infrared field with polarizations parallel and perpendicular to the applied dc electric field is shown. The heating of electrons is more accurately described beyond the energy-balance equation with the inclusion of an anisotropic coupling to the infrared field. The drift velocity of electrons is found to increase with the amplitude of the infrared field due to a suppressed momentum-relaxation process (or frictional force) under parallel polarization but decreases with the amplitude due to an enhanced momentum-relaxation process under perpendicular polarization

  2. Comparing linear probability model coefficients across groups

    Holm, Anders; Ejrnæs, Mette; Karlson, Kristian Bernt

    2015-01-01

    of the following three components: outcome truncation, scale parameters and distributional shape of the predictor variable. These results point to limitations in using linear probability model coefficients for group comparisons. We also provide Monte Carlo simulations and real examples to illustrate......This article offers a formal identification analysis of the problem in comparing coefficients from linear probability models between groups. We show that differences in coefficients from these models can result not only from genuine differences in effects, but also from differences in one or more...... these limitations, and we suggest a restricted approach to using linear probability model coefficients in group comparisons....

  3. Oligarchy as a phase transition: The effect of wealth-attained advantage in a Fokker-Planck description of asset exchange

    Boghosian, Bruce M.; Devitt-Lee, Adrian; Johnson, Merek; Li, Jie; Marcq, Jeremy A.; Wang, Hongyan

    2017-06-01

    The ;Yard-Sale Model; of asset exchange is known to result in complete inequality-all of the wealth in the hands of a single agent. It is also known that, when this model is modified by introducing a simple model of redistribution based on the Ornstein-Uhlenbeck process, it admits a steady state exhibiting some features similar to the celebrated Pareto Law of wealth distribution. In the present work, we analyze the form of this steady-state distribution in much greater detail, using a combination of analytic and numerical techniques. We find that, while Pareto's Law is approximately valid for low redistribution, it gives way to something more similar to Gibrat's Law when redistribution is higher. Additionally, we prove in this work that, while this Pareto or Gibrat behavior may persist over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Also in this work, we introduce a bias in favor of the wealthier agent-what we call Wealth-Attained Advantage (WAA)-and show that this leads to the phenomenon of ;wealth condensation; when the bias exceeds a certain critical value. In the wealth-condensed state, a finite fraction of the total wealth of the population ;condenses; to the wealthiest agent. We examine this phenomenon in some detail, and derive the corresponding modification to the Fokker-Planck equation. We observe a second-order phase transition to a state of coexistence between an oligarch and a distribution of non-oligarchs. Finally, by studying the asymptotic behavior of the distribution in some detail, we show that the onset of wealth condensation has an abrupt reciprocal effect on the character of the non-oligarchical part of the distribution. Specifically, we show that the above-mentioned gaussian decay at extremely large wealth is valid both above and below criticality, but degenerates to exponential decay precisely at criticality.

  4. Stochastic quantum inflation for a canonical scalar field with linear self-interaction potential

    Panotopoulos, Grigoris [CENTRA, Instituto Superior Tecnico, Universidade de Lisboa, Lisboa (Portugal)

    2017-10-15

    We apply Starobinsky's formalism of stochastic inflation to the case of a massless minimally coupled scalar field with linear self-interaction potential. We solve the corresponding Fokker-Planck equation exactly, and we obtain analytical expressions for the stochastic expectation values. (orig.)

  5. Hypocoercivity for linear kinetic equations conserving mass

    Dolbeault, Jean; Mouhot, Clé ment; Schmeiser, Christian

    2015-01-01

    We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

  6. Hypocoercivity for linear kinetic equations conserving mass

    Dolbeault, Jean

    2015-02-03

    We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

  7. Thermal linear expansion coefficient of structural graphites

    Virgil'ev, Yu.S.

    1995-01-01

    The data now available on radiation induced changes of linear thermal expansion coefficients (CTE) for native structural carbon materials (SCM) irradiated with high fluences are summarized. For different types of native and foreign SCM dose dependences of CTE changes in the temperature range of 300...1600 K and at fluences up to (2...3)x10 22 n/cm 2 (E>0.18 meV) are compared. On the base of this comparison factors defined the CTE changes under neutron irradiation are revealed and the explanation of observed phenomena is offered. Large number of the factors revealed does not allowed to calculate CTE radiation induced changes. 39 refs.; 16 figs.; 5 tabs

  8. New and precise construction of the local interstellar electron spectrum from the radio background and an application to the solar modulation of cosmic rays showing an incompatability of the electron and nuclei modulation using the spherically symmetric Fokker-Planck equation

    Rockstroh, J.M.

    1977-01-01

    Cosmic-ray electrons generate the observed radio-frequency background. Previous attempts in the literature to reconcile quantitatively the measured radio-frequency intensity with the intensity deduced from the electron spectrum measured at earth have culminated in the problem that to get the respective emissivities to agree, an unacceptably high interstellar B field must be chosen. In the light of new experimental data on the emissivity as deduced from H II region studies and on the functional dependence of the diffusion coefficient with solar radius and particle rigidity, the assumptions under which the electron emissivity comparison has been made have been reexamined closely. The paradox between predicted and measured emissivity was resolved by ascribing to the magnetic fields of the galaxy a distribution of magnetic field strengths. From modified synchrotron formulas, the interstellar electron spectrum has been constructed from the radio frequency emission data with greatly improved precision. The interstellar electron spectrum has been determined independently of the solar modulation and provides, therefore, an estimate of the absolute depth of the electron modulation. Then the measured electron, proton, and helium-nuclei fluxes were systematically compared to the predictions of the spherically symmetric Fokker-Planck equation using the electron modulation as a base. A previously unnoticed non-tracking of the modulation parameters was observed during the recent recovery that did not occur during the 1965 to 1969 period. Although the argument could be presented just as well by attributing the anomaly to the nuclei, the discussion here arbitrarily tailored it to the electrons, and this new phenomenon was named, the modulation reluctance of the cosmic-ray electrons

  9. When is quasi-linear theory exact. [particle acceleration

    Jones, F. C.; Birmingham, T. J.

    1975-01-01

    We use the cumulant expansion technique of Kubo (1962, 1963) to derive an integrodifferential equation for the average one-particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the equation for this function degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory only for this limited class of fluctuations.

  10. Estimations hypoelliptiques globales et compacit\\'e de la r\\'esolvante Estimations hypoelliptiques globales et compacit\\'e de la r\\'esolvante pour des op\\'erateurs de Fokker-Planck ou des laplaciens de Witten

    Helffer, B

    2004-01-01

    Ces derni\\`eres ann\\'ees, les estimations hypoelliptiques ont connu une nouvelle jeunesse en liaison avec des questions provenant de la th\\'eorie cin\\'etique des gaz. Dans cette direction de nombreux auteurs ont en effet eu besoin de d\\'emontrer des estimations maximales pour en d\\'eduire la compacit\\'e de l'op\\'erateur de Fokker-Planck et avoir des estimations sur la r\\'esolvante permettant d'aborder la question du retour \\`a l'\\'equilibre. Dans un article tr\\`es r\\'ecent, F.~H\\'erau et F.~Nier (inspir\\'es par des calculs explicites du livre de Risken) ont mis en \\'evidence les liens \\'etroits entre ces questions et des questions analogues pour un laplacien de Witten. L'\\'etude de ces liens est poursuivie et syst\\'ematis\\'ee dans un livre en pr\\'eparation \\'ecrit en collaboration avec F.~Nier dont nous allons pr\\'esenter quelques aspects ici en pr\\'esentant parfois un \\'eclairage diff\\'erent sur un probl\\`eme qui laisse encore beaucoup de conjectures non r\\'esolues.

  11. Stationary solutions of linear stochastic delay differential equations: applications to biological systems.

    Frank, T D; Beek, P J

    2001-08-01

    Recently, Küchler and Mensch [Stochastics Stochastics Rep. 40, 23 (1992)] derived exact stationary probability densities for linear stochastic delay differential equations. This paper presents an alternative derivation of these solutions by means of the Fokker-Planck approach introduced by Guillouzic [Phys. Rev. E 59, 3970 (1999); 61, 4906 (2000)]. Applications of this approach, which is argued to have greater generality, are discussed in the context of stochastic models for population growth and tracking movements.

  12. Linearized gyro-kinetic equation

    Catto, P.J.; Tsang, K.T.

    1976-01-01

    An ordering of the linearized Fokker-Planck equation is performed in which gyroradius corrections are retained to lowest order and the radial dependence appropriate for sheared magnetic fields is treated without resorting to a WKB technique. This description is shown to be necessary to obtain the proper radial dependence when the product of the poloidal wavenumber and the gyroradius is large (k rho much greater than 1). A like particle collision operator valid for arbitrary k rho also has been derived. In addition, neoclassical, drift, finite β (plasma pressure/magnetic pressure), and unperturbed toroidal electric field modifications are treated

  13. Correlated Levy Noise in Linear Dynamical Systems

    Srokowski, T.

    2011-01-01

    Linear dynamical systems, driven by a non-white noise which has the Levy distribution, are analysed. Noise is modelled by a specific stochastic process which is defined by the Langevin equation with a linear force and the Levy distributed symmetric white noise. Correlation properties of the process are discussed. The Fokker-Planck equation driven by that noise is solved. Distributions have the Levy shape and their width, for a given time, is smaller than for processes in the white noise limit. Applicability of the adiabatic approximation in the case of the linear force is discussed. (author)

  14. Measurement of linear attenuation coefficient of different materials

    Ali, M. M.

    2013-07-01

    In this research we study the linear attenuation coefficient from the materials concrete, brick, mixture concrete and iron. In the secondary standard dosimetry laboratory in Atomic Energy from different distance by use Cs-137 sours, chamber farmer 2675 A-600 cc-S/N 0511, and electrometer 2670 A-S/N 114. Found the value of linear attenuation coefficient of concert in the range 0.167 cm -1 , the brick in the range 0.063 -1 and mixture concrete and iron in the range 0.253cm -1 .(Author)

  15. Non-linear Bayesian update of PCE coefficients

    Litvinenko, Alexander

    2014-01-06

    Given: a physical system modeled by a PDE or ODE with uncertain coefficient q(?), a measurement operator Y (u(q), q), where u(q, ?) uncertain solution. Aim: to identify q(?). The mapping from parameters to observations is usually not invertible, hence this inverse identification problem is generally ill-posed. To identify q(!) we derived non-linear Bayesian update from the variational problem associated with conditional expectation. To reduce cost of the Bayesian update we offer a unctional approximation, e.g. polynomial chaos expansion (PCE). New: We apply Bayesian update to the PCE coefficients of the random coefficient q(?) (not to the probability density function of q).

  16. Non-linear Bayesian update of PCE coefficients

    Litvinenko, Alexander; Matthies, Hermann G.; Pojonk, Oliver; Rosic, Bojana V.; Zander, Elmar

    2014-01-01

    Given: a physical system modeled by a PDE or ODE with uncertain coefficient q(?), a measurement operator Y (u(q), q), where u(q, ?) uncertain solution. Aim: to identify q(?). The mapping from parameters to observations is usually not invertible, hence this inverse identification problem is generally ill-posed. To identify q(!) we derived non-linear Bayesian update from the variational problem associated with conditional expectation. To reduce cost of the Bayesian update we offer a unctional approximation, e.g. polynomial chaos expansion (PCE). New: We apply Bayesian update to the PCE coefficients of the random coefficient q(?) (not to the probability density function of q).

  17. Overall mass-transfer coefficients in non-linear chromatography

    Mollerup, Jørgen; Hansen, Ernst

    1998-01-01

    In case of mass transfer where concentration differences in both phases must be taken into account, one may define an over-all mass-transfer coefficient basd on the apparent over-all concentration difference. If the equilibrium relationship is linear, i.e. in cases where a Henry´s law relationshi...

  18. General guidelines solution for linear programming with fuzzy coefficients

    Sergio Gerardo de los Cobos Silva

    2013-08-01

    Full Text Available This work introduce to the Possibilistic Programming and the Fuzzy Programming as paradigms that allow to resolve problems of linear programming when the coefficients of the model or the restrictions on the same are presented as fuzzy numbers, rather than exact numbers (crisp. This work presents some examples based on [1].

  19. Distributing Correlation Coefficients of Linear Structure-Activity/Property Models

    Sorana D. BOLBOACA

    2011-12-01

    Full Text Available Quantitative structure-activity/property relationships are mathematical relationships linking chemical structure and activity/property in a quantitative manner. These in silico approaches are frequently used to reduce animal testing and risk-assessment, as well as to increase time- and cost-effectiveness in characterization and identification of active compounds. The aim of our study was to investigate the pattern of correlation coefficients distribution associated to simple linear relationships linking the compounds structure with their activities. A set of the most common ordnance compounds found at naval facilities with a limited data set with a range of toxicities on aquatic ecosystem and a set of seven properties was studied. Statistically significant models were selected and investigated. The probability density function of the correlation coefficients was investigated using a series of possible continuous distribution laws. Almost 48% of the correlation coefficients proved fit Beta distribution, 40% fit Generalized Pareto distribution, and 12% fit Pert distribution.

  20. A parameterization scheme for the x-ray linear attenuation coefficient and energy absorption coefficient.

    Midgley, S M

    2004-01-21

    A novel parameterization of x-ray interaction cross-sections is developed, and employed to describe the x-ray linear attenuation coefficient and mass energy absorption coefficient for both elements and mixtures. The new parameterization scheme addresses the Z-dependence of elemental cross-sections (per electron) using a simple function of atomic number, Z. This obviates the need for a complicated mathematical formalism. Energy dependent coefficients describe the Z-direction curvature of the cross-sections. The composition dependent quantities are the electron density and statistical moments describing the elemental distribution. We show that it is possible to describe elemental cross-sections for the entire periodic table and at energies above the K-edge (from 6 keV to 125 MeV), with an accuracy of better than 2% using a parameterization containing not more than five coefficients. For the biologically important elements 1 coefficients. At higher energies, the parameterization uses fewer coefficients with only two coefficients needed at megavoltage energies.

  1. Final Scientific/Technical Report, USDOE Award DE-FG-02ER54684, Recipient: CompX, Project Title: Fokker-Planck/Ray Tracing for Electron Bernstein and Fast Wave Modeling in Support of NSTX

    Harvey, R.W.

    2009-01-01

    This DOE grant supported fusion energy research, a potential long-term solution to the world's energy needs. Magnetic fusion, exemplified by confinement of very hot ionized gases, i.e., plasmas, in donut-shaped tokamak vessels is a leading approach for this energy source. Thus far, a mixture of hydrogen isotopes has produced 10's of megawatts of fusion power for seconds in a tokamak reactor at Princeton Plasma Physics Laboratory in New Jersey. The research grant under consideration, ER54684, uses computer models to aid in understanding and projecting efficacy of heating and current drive sources in the National Spherical Torus Experiment, a tokamak variant, at PPPL. The NSTX experiment explores the physics of very tight aspect ratio, almost spherical tokamaks, aiming at producing steady-state fusion plasmas. The current drive is an integral part of the steady-state concept, maintaining the magnetic geometry in the steady-state tokamak. CompX further developed and applied models for radiofrequency (rf) heating and current drive for applications to NSTX. These models build on a 30 year development of rf ray tracing (the all-frequencies GENRAY code) and higher dimensional Fokker-Planck rf-collisional modeling (the 3D collisional-quasilinear CQL3D code) at CompX. Two mainline current-drive rf modes are proposed for injection into NSTX: (1) electron Bernstein wave (EBW), and (2) high harmonic fast wave (HHFW) modes. Both these current drive systems provide a means for the rf to access the especially high density plasma--termed high beta plasma--compared to the strength of the required magnetic fields. The CompX studies entailed detailed modeling of the EBW to calculate the efficiency of the current drive system, and to determine its range of flexibility for driving current at spatial locations in the plasma cross-section. The ray tracing showed penetration into NSTX bulk plasma, relatively efficient current drive, but a limited ability to produce current over the whole

  2. Linear thermal expansion coefficient measurement technology in hot cell

    Park, Dae Gyu; Choo, Yong Sun; Ahn, Sang Bok; Hong, Kwon Pyo; Lee, K. S.

    1998-06-01

    To establish linear thermal expansion coefficient measurement technology in hot cell, we reviewed and evaluated various measuring technology by paper and these were compared with the data produced with pre-installed dilatometer in hot cell. Detailed contents are as follows; - The theory of test. - Review of characteristics for various measurement technology and compatibility with hot cell. - Review of standard testing regulations(ASTM). - System calibration of pre-installed dilatometer. - Performance test of pre-installed dilatometer. (author). 12 refs., 15 tabs., 8 figs

  3. Local energy decay for linear wave equations with variable coefficients

    Ikehata, Ryo

    2005-06-01

    A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

  4. Linear and Non-Linear Piezoresistance Coefficients in Cubic Semiconductors. I. Theoretical Formulations

    Durand, S.; Tellier, C. R.

    1996-02-01

    This paper constitutes the first part of a work devoted to applications of piezoresistance effects in germanium and silicon semiconductors. In this part, emphasis is placed on a formal explanation of non-linear effects. We propose a brief phenomenological description based on the multi-valleys model of semiconductors before to adopt a macroscopic tensorial model from which general analytical expressions for primed non-linear piezoresistance coefficients are derived. Graphical representations of linear and non-linear piezoresistance coefficients allows us to characterize the influence of the two angles of cut and of directions of alignment. The second part will primarily deal with specific applications for piezoresistive sensors. Cette publication constitue la première partie d'un travail consacré aux applications des effets piézorésistifs dans les semiconducteurs germanium et silicium. Cette partie traite essentiellement de la modélisation des effets non-linéaires. Après une description phénoménologique à partir du modèle de bande des semiconducteurs nous développons un modèle tensoriel macroscopique et nous proposons des équations générales analytiques exprimant les coefficients piézorésistifs non-linéaires dans des repères tournés. Des représentations graphiques des variations des coefficients piézorésistifs linéaires et non-linéaires permettent une pré-caractérisation de l'influence des angles de coupes et des directions d'alignement avant l'étude d'applications spécifiques qui feront l'objet de la deuxième partie.

  5. Partially linear varying coefficient models stratified by a functional covariate

    Maity, Arnab; Huang, Jianhua Z.

    2012-01-01

    We consider the problem of estimation in semiparametric varying coefficient models where the covariate modifying the varying coefficients is functional and is modeled nonparametrically. We develop a kernel-based estimator of the nonparametric

  6. Partially linear varying coefficient models stratified by a functional covariate

    Maity, Arnab

    2012-10-01

    We consider the problem of estimation in semiparametric varying coefficient models where the covariate modifying the varying coefficients is functional and is modeled nonparametrically. We develop a kernel-based estimator of the nonparametric component and a profiling estimator of the parametric component of the model and derive their asymptotic properties. Specifically, we show the consistency of the nonparametric functional estimates and derive the asymptotic expansion of the estimates of the parametric component. We illustrate the performance of our methodology using a simulation study and a real data application.

  7. Simultaneous interferometric measurement of linear coefficient of thermal expansion and temperature-dependent refractive index coefficient of optical materials.

    Corsetti, James A; Green, William E; Ellis, Jonathan D; Schmidt, Greg R; Moore, Duncan T

    2016-10-10

    Characterizing the thermal properties of optical materials is necessary for understanding how to design an optical system for changing environmental conditions. A method is presented for simultaneously measuring both the linear coefficient of thermal expansion and the temperature-dependent refractive index coefficient of a sample interferometrically in air. Both the design and fabrication of the interferometer is presented as well as a discussion of the results of measuring both a steel and a CaF2 sample.

  8. Heat conduction coefficient and coefficient of linear thermal expansion of electric insulation materials for superconducting magnetic system

    Deev, V.I.; Sobolev, V.P.; Kruglov, A.B.; Pridantsev, A.I.

    1984-01-01

    Results of experimental investigation of heat conduction coefficient and coefficient of linear thermal expansion and thermal shrinkages of the STEF-1 textolite-glass widely used in superconducting magnetic systems as electric insulating and structural material are presented. Samples of two types have been died: sample axisa is perpendicular to a plae of fiberglass layers ad sample axis is parallel to a plane of fiberglass layers. Heat conduction coefficient was decreased almost a five times with temperature decrease from 300 up to 5K and was slightly dependent on a sample type. Temperature variation of linear dimensions in a sample of the first type occurs in twice as fast as compared to the sample of the second type

  9. New Inference Procedures for Semiparametric Varying-Coefficient Partially Linear Cox Models

    Yunbei Ma

    2014-01-01

    Full Text Available In biomedical research, one major objective is to identify risk factors and study their risk impacts, as this identification can help clinicians to both properly make a decision and increase efficiency of treatments and resource allocation. A two-step penalized-based procedure is proposed to select linear regression coefficients for linear components and to identify significant nonparametric varying-coefficient functions for semiparametric varying-coefficient partially linear Cox models. It is shown that the penalized-based resulting estimators of the linear regression coefficients are asymptotically normal and have oracle properties, and the resulting estimators of the varying-coefficient functions have optimal convergence rates. A simulation study and an empirical example are presented for illustration.

  10. Transport coefficients from SU(3) Polyakov linear-σ model

    Tawfik, A.; Diab, A.

    2015-01-01

    In the mean field approximation, the grand potential of SU(3) Polyakov linear-σ model (PLSM) is analyzed for the order parameter of the light and strange chiral phase-transitions, σ l and σ s , respectively, and for the deconfinement order parameters φ and φ*. Furthermore, the subtracted condensate Δ l,s and the chiral order-parameters M b are compared with lattice QCD calculations. By using the dynamical quasiparticle model (DQPM), which can be considered as a system of noninteracting massive quasiparticles, we have evaluated the decay width and the relaxation time of quarks and gluons. In the framework of LSM and with Polyakov loop corrections included, the interaction measure Δ/T 4 , the specific heat c v and speed of sound squared c s 2 have been determined, as well as the temperature dependence of the normalized quark number density n q /T 3 and the quark number susceptibilities χ q /T 2 at various values of the baryon chemical potential. The electric and heat conductivity, σ e and κ, and the bulk and shear viscosities normalized to the thermal entropy, ζ/s and η/s, are compared with available results of lattice QCD calculations.

  11. Internal friction and linear expansion coefficient in zirconium and cobalt within the range of phase transitions

    Boyarskij, S.V.

    1986-01-01

    Experimental results are presented for internal friction and linear expansion coefficient at zirconium and cobalt in the temperature range from 440 K to the point of the phase transition of the first kind (1138 K for Zr and 706 for Co). Anomalous changes of the internal friction and linear expansion coefficient in the phase transition region are found. Theoretical considerations are given to explain the sharp decrease of the internal friction as temperature approaches the phase transition point

  12. Fuzzy solution of the linear programming problem with interval coefficients in the constraints

    Dorota Kuchta

    2005-01-01

    A fuzzy concept of solving the linear programming problem with interval coefficients is proposed. For each optimism level of the decision maker (where the optimism concerns the certainty that no errors have been committed in the estimation of the interval coefficients and the belief that optimistic realisations of the interval coefficients will occur) another interval solution of the problem will be generated and the decision maker will be able to choose the final solution having a complete v...

  13. Comparing Regression Coefficients between Nested Linear Models for Clustered Data with Generalized Estimating Equations

    Yan, Jun; Aseltine, Robert H., Jr.; Harel, Ofer

    2013-01-01

    Comparing regression coefficients between models when one model is nested within another is of great practical interest when two explanations of a given phenomenon are specified as linear models. The statistical problem is whether the coefficients associated with a given set of covariates change significantly when other covariates are added into…

  14. An Analytical-empirical Calculation of Linear Attenuation Coefficient of Megavoltage Photon Beams.

    Seif, F; Tahmasebi-Birgani, M J; Bayatiani, M R

    2017-09-01

    In this study, a method for linear attenuation coefficient calculation was introduced. Linear attenuation coefficient was calculated with a new method that base on the physics of interaction of photon with matter, mathematical calculation and x-ray spectrum consideration. The calculation was done for Cerrobend as a common radiotherapy modifier and Mercury. The values of calculated linear attenuation coefficient with this new method are in acceptable range. Also, the linear attenuation coefficient decreases slightly as the thickness of attenuating filter (Cerrobend or mercury) increased, so the procedure of linear attenuation coefficient variation is in agreement with other documents. The results showed that the attenuation ability of mercury was about 1.44 times more than Cerrobend. The method that was introduced in this study for linear attenuation coefficient calculation is general enough to treat beam modifiers with any shape or material by using the same formalism; however, calculating was made only for mercury and Cerrobend attenuator. On the other hand, it seems that this method is suitable for high energy shields or protector designing.

  15. Spin fluctuations and low temperature features of thermal coefficient of linear expansion of iron monosilicide

    Volkov, A.G.; Kortov, S.V.; Povzner, A.A.

    1996-01-01

    The low temperature measurements of thermal coefficient of linear expansion of strong paramagnet FeSi are carried out. The results obtained are discussed with in the framework of spin-fluctuation theory. It is shown that electronic part of the thermal coefficient of linear expansion is negative in the range of temperatures lower that of the semiconductor-metal phase transition. In metal phase it becomes positive. This specific features of the thermal coefficient is explained by the spin-fluctuation renormalization of d-electronic states density

  16. Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

    Camporesi, Roberto

    2011-01-01

    We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

  17. Linear Parametric Sensitivity Analysis of the Constraint Coefficient Matrix in Linear Programs

    Zuidwijk, Rob

    2005-01-01

    textabstractSensitivity analysis is used to quantify the impact of changes in the initial data of linear programs on the optimal value. In particular, parametric sensitivity analysis involves a perturbation analysis in which the effects of small changes of some or all of the initial data on an optimal solution are investigated, and the optimal solution is studied on a so-called critical range of the initial data, in which certain properties such as the optimal basis in linear programming are ...

  18. Invar hardening under keeping of low values of temperature coefficient of linear expansion

    Bashnin, Yu.A.; Shiryaeva, A.N.; Omel'chenko, A.V.

    1982-01-01

    Complex invar alloying with chromium, zirconium and nitrogen is conducted for increasing hardness and assuring low values of the temperature coefficient of linear expansion. It is shown that alloying with nitride-forming elements-chromium, zirconium and the following high-temperature saturation under high pressure with nitrogen provides the invar hardening at assuring a low temperature coefficient of linear expansion. Saturation with nitrogen under 100 MPa pressure at 1050 deg C during 3 hours permits to prepare an invar containing up to 0.2% N 2 uniformly distributed over the whole cross section of samples with 4 mm diameter. Nitrogen in invar alloys alloyed with chromium and zirconium affects the Curie point similarly to carbon and nickel shifting it towards higher temperatures, it slightly changes the value of the temperature coefficient of linear expansion and provides linear character of thermal expansion dependence on temperature in the +100 deg C - -180 deg C range

  19. Coefficient of linear attenuation of beer for γ rays of 662 keV

    Ortiz A, M. D.; Cano S, D.; Vega C, H. R.

    2017-10-01

    The coefficient of linear attenuation of the beer was determined by means of a transmission experiment with a source of Cs 137 and a gamma ray spectrometer with a NaI(Tl) detector of 7.62 cm in diameter and 7.62 cm in height, using narrow geometry. The pulse height spectrum was accumulated for 1 minute of live time, 7 beer thicknesses (0.6 cm) were used. By means of linear regression by weighted squares we determined the linear attenuation coefficient whose value was μ = 0.0843 ± 0.0007 cm -1 . The coefficient of linear attenuation of water is 2.2% times greater than that of beer and to the geometry of the experimental arrangement. (Author)

  20. Derivation of linear attenuation coefficients from CT numbers for low-energy photons

    Watanabe, Y.

    1999-01-01

    One can estimate photon attenuation properties from the CT number. In a standard method one assumes that the linear attenuation coefficient is proportional to electron density and ignores its nonlinear dependence on atomic number. When the photon energy is lower than about 50 keV, such as for brachytherapy applications, however, photoelectric absorption and Rayleigh scattering become important. Hence the atomic number must be explicitly considered in estimating the linear attenuation coefficient. In this study we propose a method to more accurately estimate the linear attenuation coefficient of low-energy photons from CT numbers. We formulate an equation that relates the CT number to the electron density and the effective atomic number. We use a CT calibration phantom to determine unknown coefficients in the equation. The equation with a given CT number is then solved for the effective atomic number, which in turn is used to calculate the linear attenuation coefficient for low-energy photons. We use the CT phantom to test the new method. The method significantly improves the standard method in estimating the attenuation coefficient at low photon energies (20keV≤E≤40keV) for materials with high atomic numbers. (author)

  1. Two media method for linear attenuation coefficient determination of irregular soil samples

    Vici, Carlos Henrique Georges

    2004-01-01

    In several situations of nuclear applications, the knowledge of gamma-ray linear attenuation coefficient for irregular samples is necessary, such as in soil physics and geology. This work presents the validation of a methodology for the determination of the linear attenuation coefficient (μ) of irregular shape samples, in such a way that it is not necessary to know the thickness of the considered sample. With this methodology irregular soil samples (undeformed field samples) from Londrina region, north of Parana were studied. It was employed the two media method for the μ determination. It consists of the μ determination through the measurement of a gamma-ray beam attenuation by the sample sequentially immersed in two different media, with known and appropriately chosen attenuation coefficients. For comparison, the theoretical value of μ was calculated by the product of the mass attenuation coefficient, obtained by the WinXcom code, and the measured value of the density sample. This software employs the chemical composition of the samples and supplies a table of the mass attenuation coefficients versus the photon energy. To verify the validity of the two media method, compared with the simple gamma ray transmission method, regular pome stone samples were used. With these results for the attenuation coefficients and their respective deviations, it was possible to compare the two methods. In this way we concluded that the two media method is a good tool for the determination of the linear attenuation coefficient of irregular materials, particularly in the study of soils samples. (author)

  2. Linear Parametric Sensitivity Analysis of the Constraint Coefficient Matrix in Linear Programs

    R.A. Zuidwijk (Rob)

    2005-01-01

    textabstractSensitivity analysis is used to quantify the impact of changes in the initial data of linear programs on the optimal value. In particular, parametric sensitivity analysis involves a perturbation analysis in which the effects of small changes of some or all of the initial data on an

  3. Feature Genes Selection Using Supervised Locally Linear Embedding and Correlation Coefficient for Microarray Classification.

    Xu, Jiucheng; Mu, Huiyu; Wang, Yun; Huang, Fangzhou

    2018-01-01

    The selection of feature genes with high recognition ability from the gene expression profiles has gained great significance in biology. However, most of the existing methods have a high time complexity and poor classification performance. Motivated by this, an effective feature selection method, called supervised locally linear embedding and Spearman's rank correlation coefficient (SLLE-SC 2 ), is proposed which is based on the concept of locally linear embedding and correlation coefficient algorithms. Supervised locally linear embedding takes into account class label information and improves the classification performance. Furthermore, Spearman's rank correlation coefficient is used to remove the coexpression genes. The experiment results obtained on four public tumor microarray datasets illustrate that our method is valid and feasible.

  4. Measurements of linear attenuation coefficients of irregular shaped samples by two media method

    Singh, Sukhpal; Kumar, Ashok; Thind, Kulwant Singh; Mudahar, Gurmel S.

    2008-01-01

    The linear attenuation coefficient values of regular and irregular shaped flyash materials have been measured without knowing the thickness of a sample using a new technique namely 'two media method'. These values have also been measured with a standard gamma ray transmission method and obtained theoretically with winXCOM computer code. From the comparison it is reported that the two media method has given accurate results of attenuation coefficients of flyash materials

  5. Integral linear momentum balance in combining flows for calculating the pressure drop coefficients

    Bollmann, A.

    1983-01-01

    Equations for calculating the loss coefficient in combining flows in tee functions are obtained by an integral linear momentum balance. It is a practice, when solving this type of problem, to neglect the pressure difference in the upstream location as well as the wall-fluid interaction in the lateral branch of the junction. In this work it is demonstrated the influence of the above parameters on the loss coefficient based on experimental values and by apropriate algebraic manipulation of the loss coefficient values published by previous investigators. (Author) [pt

  6. Densities, isobaric thermal expansion coefficients and isothermal compressibilities of linear alkylbenzene

    Zhou, X; Zhang, Z Y; Zhang, Q M; Liu, Q; Ding, Y Y; Zhou, L; Cao, J

    2015-01-01

    We report the measurements of the densities of linear alkylbenzene at three temperatures over 4 to 23 °C with pressures up to 10 MPa. The measurements have been analysed to yield the isobaric thermal expansion coefficients and, so far for the first time, isothermal compressibilities of linear alkylbenzene. Relevance of results for current generation (i.e., Daya Bay) and next generation (i.e. JUNO) large liquid scintillator neutrino detectors are discussed. (paper)

  7. Materials analysis using x-ray linear attenuation coefficient measurements at four photon energies

    Midgley, S M

    2005-01-01

    The analytical properties of an accurate parameterization scheme for the x-ray linear attenuation coefficient are examined. The parameterization utilizes an additive combination of N compositional- and energy-dependent coefficients. The former were derived from a parameterization of elemental cross-sections using a polynomial in atomic number. The compositional-dependent coefficients are referred to as the mixture parameters, representing the electron density and higher order statistical moments describing elemental distribution. Additivity is an important property of the parameterization, allowing measured x-ray linear attenuation coefficients to be written as linear simultaneous equations, and then solved for the unknown coefficients. The energy-dependent coefficients can be determined by calibration from measurements with materials of known composition. The inverse problem may be utilized for materials analysis, whereby the simultaneous equations represent multi-energy linear attenuation coefficient measurements, and are solved for the mixture parameters. For in vivo studies, the choice of measurement energies is restricted to the diagnostic region (approximately 20 keV to 150 keV), where the parameterization requires N ≥ 4 energies. We identify a mathematical pathology that must be overcome in order to solve the inverse problem in this energy regime. An iterative inversion strategy is presented for materials analysis using four or more measurements, and then tested against real data obtained at energies 32 keV to 66 keV. The results demonstrate that it is possible to recover the electron density to within ±4% and fourth mixture parameter. It is also a key finding that the second and third mixture parameters cannot be recovered, as they are of minor importance in the parameterization at diagnostic x-ray energies

  8. Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices

    Freund, Roland

    1989-01-01

    We consider conjugate gradient type methods for the solution of large sparse linear system Ax equals b with complex symmetric coefficient matrices A equals A(T). Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. We investigate conjugate gradient type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices. We propose a new approach with iterates defined by a quasi-minimal residual property. The resulting algorithm presents several advantages over the standard biconjugate gradient method. We also include some remarks on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

  9. Analysis of unexpected exits using the Fokker - Planck equation

    Herwaarden, van O.A.

    1996-01-01


    In this thesis exit problems are considered for stochastic dynamical systems with small random fluctuations. We study exit from a domain in the state space through a boundary, or a specified part of the boundary, that is unattainable in the underlying deterministic system. We analyze

  10. Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

    Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

    2013-01-01

    Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

  11. Exact Finite-Difference Schemes for d-Dimensional Linear Stochastic Systems with Constant Coefficients

    Peng Jiang

    2013-01-01

    Full Text Available The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

  12. System of coefficients for charged-particle beam linear transformation by a magnetic dipole element

    Tarantin, N.I.

    1979-01-01

    A new technique for consideration of dipole magnet ion-optical effect has been developed to study the problems of commutation and monochromatization of a charged particle beam. In a new form obtained are systematized coefficients of linear transformation (CLT) of the charged particle beam for radial and axial motions in a magnetic dipole element (MDE) including a dipole magnet and two gaps without magnetic field. Given is a method of graphic determination of MDE parameters and main CLT. The new form of coefficients and conditions of the transformations feasibility considerably facilitates the choice and calculation of dipole elements

  13. Estimating a graphical intra-class correlation coefficient (GICC) using multivariate probit-linear mixed models.

    Yue, Chen; Chen, Shaojie; Sair, Haris I; Airan, Raag; Caffo, Brian S

    2015-09-01

    Data reproducibility is a critical issue in all scientific experiments. In this manuscript, the problem of quantifying the reproducibility of graphical measurements is considered. The image intra-class correlation coefficient (I2C2) is generalized and the graphical intra-class correlation coefficient (GICC) is proposed for such purpose. The concept for GICC is based on multivariate probit-linear mixed effect models. A Markov Chain Monte Carlo EM (mcm-cEM) algorithm is used for estimating the GICC. Simulation results with varied settings are demonstrated and our method is applied to the KIRBY21 test-retest dataset.

  14. Path integral solutions of the master equation. [Lagrangian function, Ehrenfest-type theorem, Cauchy method, inverse functions

    Etim, E; Basili, C [Rome Univ. (Italy). Ist. di Matematica

    1978-08-21

    The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation the authors reproduce the result obtained by integrating over Fourier series coefficients and by other methods.

  15. Linear thermal expansion coefficient of MgAl2O4(s)

    Dash, A.; Samui, P.; Naik, Y.P.; Chaudhary, Z.S.

    2011-01-01

    The coefficient of linear thermal expansion (α av ) of MgAl 2 O 4 (s) has been determined using a Netzsch 402 PC dilatometer with Al 2 O 3 (s) as the push-rod. The change in length per unit length was recorded as a function of temperature between room temperature to 1273 K at a heating rate of 8 K.min /1 , in argon flowing atmosphere. The average of three measurements was quoted as the α av for MgAl 2 O 4 (s). The linear thermal expansion was measured to an accuracy of ±3%. (author)

  16. A Compensatory Approach to Multiobjective Linear Transportation Problem with Fuzzy Cost Coefficients

    Hale Gonce Kocken

    2011-01-01

    Full Text Available This paper deals with the Multiobjective Linear Transportation Problem that has fuzzy cost coefficients. In the solution procedure, many objectives may conflict with each other; therefore decision-making process becomes complicated. And also due to the fuzziness in the costs, this problem has a nonlinear structure. In this paper, fuzziness in the objective functions is handled with a fuzzy programming technique in the sense of multiobjective approach. And then we present a compensatory approach to solve Multiobjective Linear Transportation Problem with fuzzy cost coefficients by using Werner's and operator. Our approach generates compromise solutions which are both compensatory and Pareto optimal. A numerical example has been provided to illustrate the problem.

  17. The linear attenuation coefficients as features of multiple energy CT image classification

    Homem, M.R.P.; Mascarenhas, N.D.A.; Cruvinel, P.E.

    2000-01-01

    We present in this paper an analysis of the linear attenuation coefficients as useful features of single and multiple energy CT images with the use of statistical pattern classification tools. We analyzed four CT images through two pointwise classifiers (the first classifier is based on the maximum-likelihood criterion and the second classifier is based on the k-means clustering algorithm) and one contextual Bayesian classifier (ICM algorithm - Iterated Conditional Modes) using an a priori Potts-Strauss model. A feature extraction procedure using the Jeffries-Matusita (J-M) distance and the Karhunen-Loeve transformation was also performed. Both the classification and the feature selection procedures were found to be in agreement with the predicted discrimination given by the separation of the linear attenuation coefficient curves for different materials

  18. Linear Coefficient of Thermal Expansion of Porous Anodic Alumina Thin Films from Atomic Force Microscopy

    Zhang, Richard X; Fisher, Timothy; Raman, Arvind; Sands, Timothy D

    2009-01-01

    In this article, a precise and convenient technique based on the atomic force microscope (AFM) is developed to measure the linear coefficient of thermal expansion of a porous anodic alumina thin film. A stage was used to heat the sample from room temperature up to 450 K. Thermal effects on AFM probes and different operation modes at elevated temperatures were also studied, and a silicon AFM probe in the tapping mode was chosen for the subsequent measurements due to its temperature insensitivi...

  19. Self-Consistency Method to Evaluate a Linear Expansion Thermal Coefficient of Composite with Dispersed Inclusions

    V. S. Zarubin

    2015-01-01

    Full Text Available The rational use of composites as structural materials, while perceiving the thermal and mechanical loads, to a large extent determined by their thermoelastic properties. From the presented review of works devoted to the analysis of thermoelastic characteristics of composites, it follows that the problem of estimating these characteristics is important. Among the thermoelastic properties of composites occupies an important place its temperature coefficient of linear expansion.Along with fiber composites are widely used in the technique of dispersion hardening composites, in which the role of inclusions carry particles of high-strength and high-modulus materials, including nanostructured elements. Typically, the dispersed particles have similar dimensions in all directions, which allows the shape of the particles in the first approximation the ball.In an article for the composite with isotropic spherical inclusions of a plurality of different materials by the self-produced design formulas relating the temperature coefficient of linear expansion with volume concentration of inclusions and their thermoelastic characteristics, as well as the thermoelastic properties of the matrix of the composite. Feature of the method is the self-accountability thermomechanical interaction of a single inclusion or matrix particles with a homogeneous isotropic medium having the desired temperature coefficient of linear expansion. Averaging over the volume of the composite arising from such interaction perturbation strain and stress in the inclusions and the matrix particles and makes it possible to obtain such calculation formulas.For the validation of the results of calculations of the temperature coefficient of linear expansion of the composite of this type used two-sided estimates that are based on the dual variational formulation of linear thermoelasticity problem in an inhomogeneous solid containing two alternative functional (such as Lagrange and Castigliano

  20. Recovery coefficients as a test of system linearity of response in PET

    Geworski, L.; Munz, D.L.; Knoop, B.; Hofmann, M.; Knapp, W.H.

    2002-01-01

    Aim: New imaging protocols have created an increasing demand for quantitation in dedicated PET. Besides attenuation and scatter correction the recovery correction, accounting for the instrument's limited spatial resolution, has gained importance. For clinical practicability these corrections should work independent from the object, i.e. from the actual distribution of emitter and absorber. Aim of the study was to test this object independency, i.e. system linearity of response, by comparing recovery coefficients (RC) determined for different object geometries. In fact, this comparison may serve as a final test on system linearity of response, as measured on the quantitative accuracy by which the activity concentration in small lesions can be recovered. Method: For hot and cold spot imaging situations spatial distribution of activity is different. Therefore, scatter correction algorithm has to deal with different scatter distributions. If all factors disturbing system linearity, specifically scatter and attenuation, are corrected to a sufficient degree of accuracy, the system behaves linearly resulting in the theoretical relationship. CSRC = (1-HSRC). Thus, this equation, applied hot and cold spot measurements, will serve as a test on the effectiveness of the corrections and, hence, as a test of system linearity of response. Following IEC standard procedures (IEC 61675-1) measurements were done with and without interplane septa (2D/3D) on an ECAT EXACT 922 using a cylindrical phantom containing six spheres of different diameters (10 mm - 40 mm). All data were corrected for attenuation (transmission scan) and scatter (2D: deconvolution, 3D: scatter model), as implemented in the scanner's standard software. Recovery coefficients were determined for cold (CSRC) and hot (HSRC) lesions using both 2D and 3D acquisition mode. Results: CSRC directly measured versus CSRC calculated according to eq. (1) from HSRC resulted in an excellent agreement for both 2D and 3D data

  1. Apparent diffusion coefficient measurement in a moving phantom simulating linear respiratory motion.

    Kwee, Thomas C; Takahara, Taro; Muro, Isao; Van Cauteren, Marc; Imai, Yutaka; Nievelstein, Rutger A J; Mali, Willem P T M; Luijten, Peter R

    2010-10-01

    The aim of this study was to examine the effect of simulated linear respiratory motion on apparent diffusion coefficient (ADC) measurements. Six rectangular test tubes (14 × 92 mm) filled with either water, tomato ketchup, or mayonnaise were positioned in a box containing agarose gel. This box was connected to a double-acting pneumatic cylinder, capable of inducing periodic linear motion in the long-axis direction of the magnetic bore (23-mm stroke). Diffusion-weighted magnetic resonance imaging was performed for both the static and moving phantoms, and ADC measurements were made in the six test tubes in both situations. In the three test tubes whose long axes were parallel to the direction of motion, ADCs agreed well between the moving and static phantom situations. However, in two test tubes that were filled with fluids that had a considerably lower diffusion coefficient than the surrounding agarose gel, and whose long axes were perpendicular to the direction of motion, the ADCs agreed poorly between the moving and static phantom situations. ADC measurements of large homogeneous structures are not affected by linear respiratory motion. However, ADC measurements of inhomogeneous or small structures are affected by linear respiratory motion due to partial volume effects.

  2. Apparent diffusion coefficient measurement in a moving phantom simulating linear respiratory motion

    Kwee, T.C.; Takahara, Taro; Nievelstein, R.A.J.; Mali, W.P.T.M.; Luijten, P.R.; Muro, Isao; Imai, Yutaka; Cauteren, M. Van

    2010-01-01

    The aim of this study was to examine the effect of simulated linear respiratory motion on apparent diffusion coefficient (ADC) measurements. Six rectangular test tubes (14 x 92 mm) filled with either water, tomato ketchup, or mayonnaise were positioned in a box containing agarose gel. This box was connected to a double-acting pneumatic cylinder, capable of inducing periodic linear motion in the long-axis direction of the magnetic bore (23-mm stroke). Diffusion-weighted magnetic resonance imaging was performed for both the static and moving phantoms, and ADC measurements were made in the six test tubes in both situations. In the three test tubes whose long axes were parallel to the direction of motion, ADCs agreed well between the moving and static phantom situations. However, in two test tubes that were filled with fluids that had a considerably lower diffusion coefficient than the surrounding agarose gel, and whose long axes were perpendicular to the direction of motion, the ADCs agreed poorly between the moving and static phantom situations. ADC measurements of large homogeneous structures are not affected by linear respiratory motion. However, ADC measurements of inhomogeneous or small structures are affected by linear respiratory motion due to partial volume effects. (author)

  3. On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

    Olha P. Kupenko

    2013-01-01

    Full Text Available We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.

  4. Fast computation of the Maslov index for hyperbolic linear systems with periodic coefficients

    Chardard, F; Dias, F; Bridges, T J

    2006-01-01

    The Maslov index is a topological property of periodic orbits of finite-dimensional Hamiltonian systems that is widely used in semiclassical quantization, quantum chaology, stability of waves and classical mechanics. The Maslov index is determined from the analysis of a linear Hamiltonian system with periodic coefficients. In this paper, a numerical scheme is devised to compute the Maslov index for hyperbolic linear systems when the phase space has a low dimension. The idea is to compute on the exterior algebra of the ambient vector space, where the Lagrangian subspace representing the unstable subspace is reduced to a line. When the exterior algebra is projectified the Lagrangian subspace always forms a closed loop. The idea is illustrated by application to Hamiltonian systems on a phase space of dimension 4. The theory is used to compute the Maslov index for the spectral problem associated with periodic solutions of the fifth-order Korteweg de Vries equation

  5. Linear attenuation coefficients of tissues from 1 keV to 150 keV

    Böke, Aysun

    2014-09-01

    The linear attenuation coefficients and three interaction processes have been computed for liver, kidney, muscle, fat and for a range of x-ray energies from 1 keV to 150 keV. Molecular photoelectric absorption cross sections were calculated from atomic cross section data. Total coherent (Rayleigh) and incoherent (Compton) scattering cross sections were obtained by numerical integration over combinations of F2m(x) with the Thomson formula and Sm(x) with the Klein-Nishina formula, respectively. For the coherent (Rayleigh) scattering cross section calculations, molecular form factors were obtained from recent experimental data in the literature for values of xelements involved in tissue composition is 5 for liver, 47 for kidney, 44 for muscle and 3 for fat. The results are compared with previously published experimental and theoretical linear attenuation coefficients. In general, good agreement is obtained. The molecular form factors and scattering functions and cross sections are incorporated into a Monte Carlo program. The energy distributions of x-ray photons scattered from tissues have been simulated and the results are presented.

  6. Correlation coefficient based supervised locally linear embedding for pulmonary nodule recognition.

    Wu, Panpan; Xia, Kewen; Yu, Hengyong

    2016-11-01

    Dimensionality reduction techniques are developed to suppress the negative effects of high dimensional feature space of lung CT images on classification performance in computer aided detection (CAD) systems for pulmonary nodule detection. An improved supervised locally linear embedding (SLLE) algorithm is proposed based on the concept of correlation coefficient. The Spearman's rank correlation coefficient is introduced to adjust the distance metric in the SLLE algorithm to ensure that more suitable neighborhood points could be identified, and thus to enhance the discriminating power of embedded data. The proposed Spearman's rank correlation coefficient based SLLE (SC(2)SLLE) is implemented and validated in our pilot CAD system using a clinical dataset collected from the publicly available lung image database consortium and image database resource initiative (LICD-IDRI). Particularly, a representative CAD system for solitary pulmonary nodule detection is designed and implemented. After a sequential medical image processing steps, 64 nodules and 140 non-nodules are extracted, and 34 representative features are calculated. The SC(2)SLLE, as well as SLLE and LLE algorithm, are applied to reduce the dimensionality. Several quantitative measurements are also used to evaluate and compare the performances. Using a 5-fold cross-validation methodology, the proposed algorithm achieves 87.65% accuracy, 79.23% sensitivity, 91.43% specificity, and 8.57% false positive rate, on average. Experimental results indicate that the proposed algorithm outperforms the original locally linear embedding and SLLE coupled with the support vector machine (SVM) classifier. Based on the preliminary results from a limited number of nodules in our dataset, this study demonstrates the great potential to improve the performance of a CAD system for nodule detection using the proposed SC(2)SLLE. Copyright © 2016 Elsevier Ireland Ltd. All rights reserved.

  7. EP-based wavelet coefficient quantization for linear distortion ECG data compression.

    Hung, King-Chu; Wu, Tsung-Ching; Lee, Hsieh-Wei; Liu, Tung-Kuan

    2014-07-01

    Reconstruction quality maintenance is of the essence for ECG data compression due to the desire for diagnosis use. Quantization schemes with non-linear distortion characteristics usually result in time-consuming quality control that blocks real-time application. In this paper, a new wavelet coefficient quantization scheme based on an evolution program (EP) is proposed for wavelet-based ECG data compression. The EP search can create a stationary relationship among the quantization scales of multi-resolution levels. The stationary property implies that multi-level quantization scales can be controlled with a single variable. This hypothesis can lead to a simple design of linear distortion control with 3-D curve fitting technology. In addition, a competitive strategy is applied for alleviating data dependency effect. By using the ECG signals saved in MIT and PTB databases, many experiments were undertaken for the evaluation of compression performance, quality control efficiency, data dependency influence. The experimental results show that the new EP-based quantization scheme can obtain high compression performance and keep linear distortion behavior efficiency. This characteristic guarantees fast quality control even for the prediction model mismatching practical distortion curve. Copyright © 2014 IPEM. Published by Elsevier Ltd. All rights reserved.

  8. Effect of Different Concrete Grade on Radiation Linear Attenuation Coefficient (μ)

    Noor Azreen Masenwat; Mohammad Shahrizan Samsu; Mohamad Pauzi Ismail; Suhairy Sani; Ismail Mustapha; Nasharuddin Isa; Mohamad Haniza Mahmud

    2014-01-01

    In calculating the quantity of absorption of radiation and its relationship with the thickness of a material, linear attenuation coefficient (μ) of the material is one of the parameters to be taken into account. For normal concrete, the (μ) varies depending on the type of radiation used, 0.105 cm -1 for Co-60 and 0.123 cm -1 for Cs-137. Value (μ) is used in the calculation of the radiation absorption for concrete material does not take into account factors such concrete grades. In this research, concrete with different grades (Grade 15, Grade 20, Grade 25, Grade 30, Grade 35, Grade 40) are designed and manufactured with reference to the mixing method described in British Standard. Then, the linear attenuation (μ) for each grade are measured using the radiation from the source Co-60 and Cs-137 sources. This paper describes and discusses the impact of differences in concrete grade of linear attenuation (μ) for Co-60 source/ source Cs-137 and its relationship with the compressive strength. (author)

  9. A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

    Camporesi, Roberto

    2016-01-01

    We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

  10. Using the Coefficient of Determination "R"[superscript 2] to Test the Significance of Multiple Linear Regression

    Quinino, Roberto C.; Reis, Edna A.; Bessegato, Lupercio F.

    2013-01-01

    This article proposes the use of the coefficient of determination as a statistic for hypothesis testing in multiple linear regression based on distributions acquired by beta sampling. (Contains 3 figures.)

  11. Thermal Coefficient of Linear Expansion Modified by Dendritic Segregation in Nickel-Iron Alloys

    Ogorodnikova, O. M.; Maksimova, E. V.

    2018-05-01

    The paper presents investigations of thermal properties of Fe-Ni and Fe-Ni-Co casting alloys affected by the heterogeneous distribution of their chemical elements. It is shown that nickel dendritic segregation has a negative effect on properties of studied invars. A mathematical model is proposed to explore the influence of nickel dendritic segregation on the thermal coefficient of linear expansion (TCLE) of the alloy. A computer simulation of TCLE of Fe-Ni-Co superinvars is performed with regard to a heterogeneous distribution of their chemical elements over the whole volume. The ProLigSol computer software application is developed for processing the data array and results of computer simulation.

  12. Region specific optimization of continuous linear attenuation coefficients based on UTE (RESOLUTE)

    Ladefoged, Claes N; Benoit, Didier; Law, Ian

    2015-01-01

    The reconstruction of PET brain data in a PET/MR hybrid scanner is challenging in the absence of transmission sources, where MR images are used for MR-based attenuation correction (MR-AC). The main challenge of MR-AC is to separate bone and air, as neither have a signal in traditional MR images......-valued linear attenuation coefficients in bone that provides accurate reconstructed PET image data. A total of 164 [(18)F]FDG PET/MR patients were included in this study, of which 10 were used for training. MR-AC was based on either standard CT (reference), UTE or our method (RESOLUTE). The reconstructed PET...... on the reconstructed PET images, as well as limiting the number and extent of the outliers....

  13. Linear-fitting-based similarity coefficient map for tissue dissimilarity analysis in -w magnetic resonance imaging

    Yu Shao-De; Wu Shi-Bin; Xie Yao-Qin; Wang Hao-Yu; Wei Xin-Hua; Chen Xin; Pan Wan-Long; Hu Jiani

    2015-01-01

    Similarity coefficient mapping (SCM) aims to improve the morphological evaluation of weighted magnetic resonance imaging However, how to interpret the generated SCM map is still pending. Moreover, is it probable to extract tissue dissimilarity messages based on the theory behind SCM? The primary purpose of this paper is to address these two questions. First, the theory of SCM was interpreted from the perspective of linear fitting. Then, a term was embedded for tissue dissimilarity information. Finally, our method was validated with sixteen human brain image series from multi-echo . Generated maps were investigated from signal-to-noise ratio (SNR) and perceived visual quality, and then interpreted from intra- and inter-tissue intensity. Experimental results show that both perceptibility of anatomical structures and tissue contrast are improved. More importantly, tissue similarity or dissimilarity can be quantified and cross-validated from pixel intensity analysis. This method benefits image enhancement, tissue classification, malformation detection and morphological evaluation. (paper)

  14. Analysis of infant cry through weighted linear prediction cepstral coefficients and Probabilistic Neural Network.

    Hariharan, M; Chee, Lim Sin; Yaacob, Sazali

    2012-06-01

    Acoustic analysis of infant cry signals has been proven to be an excellent tool in the area of automatic detection of pathological status of an infant. This paper investigates the application of parameter weighting for linear prediction cepstral coefficients (LPCCs) to provide the robust representation of infant cry signals. Three classes of infant cry signals were considered such as normal cry signals, cry signals from deaf babies and babies with asphyxia. A Probabilistic Neural Network (PNN) is suggested to classify the infant cry signals into normal and pathological cries. PNN is trained with different spread factor or smoothing parameter to obtain better classification accuracy. The experimental results demonstrate that the suggested features and classification algorithms give very promising classification accuracy of above 98% and it expounds that the suggested method can be used to help medical professionals for diagnosing pathological status of an infant from cry signals.

  15. Synthesis of linear regression coefficients by recovering the within-study covariance matrix from summary statistics.

    Yoneoka, Daisuke; Henmi, Masayuki

    2017-06-01

    Recently, the number of regression models has dramatically increased in several academic fields. However, within the context of meta-analysis, synthesis methods for such models have not been developed in a commensurate trend. One of the difficulties hindering the development is the disparity in sets of covariates among literature models. If the sets of covariates differ across models, interpretation of coefficients will differ, thereby making it difficult to synthesize them. Moreover, previous synthesis methods for regression models, such as multivariate meta-analysis, often have problems because covariance matrix of coefficients (i.e. within-study correlations) or individual patient data are not necessarily available. This study, therefore, proposes a brief explanation regarding a method to synthesize linear regression models under different covariate sets by using a generalized least squares method involving bias correction terms. Especially, we also propose an approach to recover (at most) threecorrelations of covariates, which is required for the calculation of the bias term without individual patient data. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

  16. Predicting sediment sorption coefficients for linear alkylbenzenesulfonate congeners from polyacrylate-water partition coefficients at different salinities.

    Rico Rico, A.; Droge, S.T.J.|info:eu-repo/dai/nl/304834017; Hermens, J.L.M.|info:eu-repo/dai/nl/069681384

    2010-01-01

    The effect of the molecular structure and the salinity on the sorption of the anionic surfactant linear alkylbenzenesulfonate (LAS) to marine sediment has been studied. The analysis of several individual LAS congeners in seawater and of one specific LAS congener at different dilutions of seawater

  17. A Fresh Look at Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

    Camporesi, Roberto

    2016-01-01

    We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as…

  18. Non-linear dynamo waves in an incompressible medium when the turbulence dissipative coefficients depend on temperature

    A. D. Pataraya

    1997-01-01

    Full Text Available Non-linear α-ω; dynamo waves existing in an incompressible medium with the turbulence dissipative coefficients depending on temperature are studied in this paper. We investigate of α-ω solar non-linear dynamo waves when only the first harmonics of magnetic induction components are included. If we ignore the second harmonics in the non-linear equation, the turbulent magnetic diffusion coefficient increases together with the temperature, the coefficient of turbulent viscosity decreases, and for an interval of time the value of dynamo number is greater than 1. In these conditions a stationary solution of the non-linear equation for the dynamo wave's amplitude exists; meaning that the magnetic field is sufficiently excited. The amplitude of the dynamo waves oscillates and becomes stationary. Using these results we can explain the existence of Maunder's minimum.

  19. A new modified conjugate gradient coefficient for solving system of linear equations

    Hajar, N.; ‘Aini, N.; Shapiee, N.; Abidin, Z. Z.; Khadijah, W.; Rivaie, M.; Mamat, M.

    2017-09-01

    Conjugate gradient (CG) method is an evolution of computational method in solving unconstrained optimization problems. This approach is easy to implement due to its simplicity and has been proven to be effective in solving real-life application. Although this field has received copious amount of attentions in recent years, some of the new approaches of CG algorithm cannot surpass the efficiency of the previous versions. Therefore, in this paper, a new CG coefficient which retains the sufficient descent and global convergence properties of the original CG methods is proposed. This new CG is tested on a set of test functions under exact line search. Its performance is then compared to that of some of the well-known previous CG methods based on number of iterations and CPU time. The results show that the new CG algorithm has the best efficiency amongst all the methods tested. This paper also includes an application of the new CG algorithm for solving large system of linear equations

  20. Linear electro-optic coefficient in multilayer self-organized InAs quantum dot structures

    Akca, I.B.; Dana, A.; Aydinli, A.; Rossetti, M.; Li, L.; Dagli, N.; Fiore, A.

    2007-01-01

    The electro-optic coefficients of self-organized InAs quantum dot layers in molecular beam epitaxy grown laser structures in reverse bias have been investigated. Enhanced electrooptic coefficients compared to bulk GaAs were observed.

  1. Correlation between metal-ceramic bond strength and coefficient of linear thermal expansion difference

    Stella Crosara Lopes

    2009-04-01

    Full Text Available The purpose of this study was to evaluate the metal-ceramic bond strength (MCBS of 6 metal-ceramic pairs (2 Ni-Cr alloys and 1 Pd-Ag alloy with 2 dental ceramics and correlate the MCBS values with the differences between the coefficients of linear thermal expansion (CTEs of the metals and ceramics. Verabond (VB Ni-Cr-Be alloy, Verabond II (VB2, Ni-Cr alloy, Pors-on 4 (P, Pd-Ag alloy, and IPS (I and Duceram (D ceramics were used for the MCBS test and dilatometric test. Forty-eight ceramic rings were built around metallic rods (3.0 mm in diameter and 70.0 mm in length made from the evaluated alloys. The rods were subsequently embedded in gypsum cast in order to perform a tensile load test, which enabled calculating the CMBS. Five specimens (2.0 mm in diameter and 12.0 mm in length of each material were made for the dilatometric test. The chromel-alumel thermocouple required for the test was welded into the metal test specimens and inserted into the ceramics. ANOVA and Tukey's test revealed significant differences (p=0.01 for the MCBS test results (MPa, with PI showing higher MCBS (67.72 than the other pairs, which did not present any significant differences. The CTE (10-6 oC-1 differences were: VBI (0.54, VBD (1.33, VB2I (-0.14, VB2D (0.63, PI (1.84 and PD (2.62. Pearson's correlation test (r=0.17 was performed to evaluate of correlation between MCBS and CTE differences. Within the limitations of this study and based on the obtained results, there was no correlation between MCBS and CTE differences for the evaluated metal-ceramic pairs.

  2. Correlation between metal-ceramic bond strength and coefficient of linear thermal expansion difference.

    Lopes, Stella Crosara; Pagnano, Valéria Oliveira; Rollo, João Manuel Domingos de Almeida; Leal, Mônica Barbosa; Bezzon, Osvaldo Luiz

    2009-01-01

    The purpose of this study was to evaluate the metal-ceramic bond strength (MCBS) of 6 metal-ceramic pairs (2 Ni-Cr alloys and 1 Pd-Ag alloy with 2 dental ceramics) and correlate the MCBS values with the differences between the coefficients of linear thermal expansion (CTEs) of the metals and ceramics. Verabond (VB) Ni-Cr-Be alloy, Verabond II (VB2), Ni-Cr alloy, Pors-on 4 (P), Pd-Ag alloy, and IPS (I) and Duceram (D) ceramics were used for the MCBS test and dilatometric test. Forty-eight ceramic rings were built around metallic rods (3.0 mm in diameter and 70.0 mm in length) made from the evaluated alloys. The rods were subsequently embedded in gypsum cast in order to perform a tensile load test, which enabled calculating the CMBS. Five specimens (2.0 mm in diameter and 12.0 mm in length) of each material were made for the dilatometric test. The chromel-alumel thermocouple required for the test was welded into the metal test specimens and inserted into the ceramics. ANOVA and Tukey's test revealed significant differences (p=0.01) for the MCBS test results (MPa), with PI showing higher MCBS (67.72) than the other pairs, which did not present any significant differences. The CTE (10(-6) oC(-1)) differences were: VBI (0.54), VBD (1.33), VB2I (-0.14), VB2D (0.63), PI (1.84) and PD (2.62). Pearson's correlation test (r=0.17) was performed to evaluate of correlation between MCBS and CTE differences. Within the limitations of this study and based on the obtained results, there was no correlation between MCBS and CTE differences for the evaluated metal-ceramic pairs.

  3. Correlated linear response calculations of the C6 dispersion coefficients of hydrogen halides

    Sauer, S. P. A.; Paidarová, Ivana

    2007-01-01

    Roč. 3, 2-4 (2007), s. 399-421 ISSN 1574-0404 R&D Projects: GA AV ČR IAA401870702 Institutional research plan: CEZ:AV0Z40400503 Keywords : hydrogen halides * C6 dospersion coefficients * van der Waals coefficients * polarizability at imaginary frequences * SOPPA Subject RIV: CF - Physical ; Theoretical Chemistry

  4. With respect to coefficient of linear thermal expansion, bacterial vegetative cells and spores resemble plastics and metals, respectively.

    Nakanishi, Koichi; Kogure, Akinori; Fujii, Takenao; Kokawa, Ryohei; Deuchi, Keiji; Kuwana, Ritsuko; Takamatsu, Hiromu

    2013-10-09

    If a fixed stress is applied to the three-dimensional z-axis of a solid material, followed by heating, the amount of thermal expansion increases according to a fixed coefficient of thermal expansion. When expansion is plotted against temperature, the transition temperature at which the physical properties of the material change is at the apex of the curve. The composition of a microbial cell depends on the species and condition of the cell; consequently, the rate of thermal expansion and the transition temperature also depend on the species and condition of the cell. We have developed a method for measuring the coefficient of thermal expansion and the transition temperature of cells using a nano thermal analysis system in order to study the physical nature of the cells. The tendency was seen that among vegetative cells, the Gram-negative Escherichia coli and Pseudomonas aeruginosa have higher coefficients of linear expansion and lower transition temperatures than the Gram-positive Staphylococcus aureus and Bacillus subtilis. On the other hand, spores, which have low water content, overall showed lower coefficients of linear expansion and higher transition temperatures than vegetative cells. Comparing these trends to non-microbial materials, vegetative cells showed phenomenon similar to plastics and spores showed behaviour similar to metals with regards to the coefficient of liner thermal expansion. We show that vegetative cells occur phenomenon of similar to plastics and spores to metals with regard to the coefficient of liner thermal expansion. Cells may be characterized by the coefficient of linear expansion as a physical index; the coefficient of linear expansion may also characterize cells structurally since it relates to volumetric changes, surface area changes, the degree of expansion of water contained within the cell, and the intensity of the internal stress on the cellular membrane. The coefficient of linear expansion holds promise as a new index for

  5. Measurement of Linear Coefficient of Thermal Expansion and Temperature-Dependent Refractive Index Using Interferometric System

    Corsetti, James A.; Green, William E.; Ellis, Jonathan D.; Schmidt, Greg R.; Moore, Duncan T.

    2017-01-01

    A system combining an interferometer with an environmental chamber for measuring both coefficient of thermal expansion (CTE) and temperature-dependent refractive index (dn/dT) simultaneously is presented. The operation and measurement results of this instrument are discussed.

  6. Linear thermal expansion coefficient (at temperatures from 130 to 800 K) of borosilicate glasses applicable for coupling with silicon in microelectronics

    Sinev, Leonid S.; Petrov, Ivan D.

    2017-01-01

    Processing results of measurements of linear thermal expansion coefficients and linear thermal expansion of two brands of borosilicate glasses --- LK5 and Borofloat 33 --- are presented. The linear thermal expansion of glass samples have been determined in the temperature range 130 to 800 K (minus 143 to 526 $\\deg$C) using thermomechanical analyzer TMA7100. Relative imprecision of indirectly measured linear thermal expansion coefficients and linear thermal expansion of both glass brands is le...

  7. Experimental Study on the Tensile Strength and Linear Expansion Coefficient of Air Tunnel Terrazzo Surface

    Boping Li

    2015-01-01

    Full Text Available At present, studies on the surface tension of air tunnel terrazzo under wind load and how regularly it is affected by temperature are relatively less, and the measured results of the thermal expansion coefficient of terrazzo have not yet been given. In this paper, based on the top terrazzo surface structure of the inner wall of the wind tunnel, the tensile performance tests of terrazzo surface layer are conducted, while the thermal expansion coefficient of the six terrazzo test blocks were tested. The tests and analysis show that the construction of terrazzo surface, based on the proposed construction process, can effectively guarantee the reliable cement performance for the binding layer between mortar and concrete base layer, terrazzo surface layer and the cement mortar layer. And the thermal expansion coefficient of terrazzo can be valued at 1.06e-5/ºC.

  8. The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.

    Nakagawa, Shinichi; Johnson, Paul C D; Schielzeth, Holger

    2017-09-01

    The coefficient of determination R 2 quantifies the proportion of variance explained by a statistical model and is an important summary statistic of biological interest. However, estimating R 2 for generalized linear mixed models (GLMMs) remains challenging. We have previously introduced a version of R 2 that we called [Formula: see text] for Poisson and binomial GLMMs, but not for other distributional families. Similarly, we earlier discussed how to estimate intra-class correlation coefficients (ICCs) using Poisson and binomial GLMMs. In this paper, we generalize our methods to all other non-Gaussian distributions, in particular to negative binomial and gamma distributions that are commonly used for modelling biological data. While expanding our approach, we highlight two useful concepts for biologists, Jensen's inequality and the delta method, both of which help us in understanding the properties of GLMMs. Jensen's inequality has important implications for biologically meaningful interpretation of GLMMs, whereas the delta method allows a general derivation of variance associated with non-Gaussian distributions. We also discuss some special considerations for binomial GLMMs with binary or proportion data. We illustrate the implementation of our extension by worked examples from the field of ecology and evolution in the R environment. However, our method can be used across disciplines and regardless of statistical environments. © 2017 The Author(s).

  9. Microscopic model for the non-linear fluctuating hydrodynamic of 4 He superfluid helium deduced by maximum entropy method

    Alvarez R, J.T.

    1998-01-01

    This thesis presents a microscopic model for the non-linear fluctuating hydrodynamic of superfluid helium ( 4 He), model developed by means of the Maximum Entropy Method (Maxent). In the chapter 1, it is demonstrated the necessity to developing a microscopic model for the fluctuating hydrodynamic of the superfluid helium, starting from to show a brief overview of the theories and experiments developed in order to explain the behavior of the superfluid helium. On the other hand, it is presented the Morozov heuristic method for the construction of the non-linear hydrodynamic fluctuating of simple fluid. Method that will be generalized for the construction of the non-linear fluctuating hydrodynamic of the superfluid helium. Besides, it is presented a brief summary of the content of the thesis. In the chapter 2, it is reproduced the construction of a Generalized Fokker-Planck equation, (GFP), for a distribution function associated with the coarse grained variables. Function defined with aid of a nonequilibrium statistical operator ρhut FP that is evaluated as Wigneris function through ρ CG obtained by Maxent. Later this equation of GFP is reduced to a non-linear local FP equation from considering a slow and Markov process in the coarse grained variables. In this equation appears a matrix D mn defined with a nonequilibrium coarse grained statistical operator ρhut CG , matrix elements are used in the construction of the non-linear fluctuating hydrodynamics equations of the superfluid helium. In the chapter 3, the Lagrange multipliers are evaluated for to determine ρhut CG by means of the local equilibrium statistical operator ρhut l -tilde with the hypothesis that the system presents small fluctuations. Also are determined the currents associated with the coarse grained variables and furthermore are evaluated the matrix elements D mn but with aid of a quasi equilibrium statistical operator ρhut qe instead of the local equilibrium operator ρhut l -tilde. Matrix

  10. A Lie-Deprit perturbation algorithm for linear differential equations with periodic coefficients

    Casas Pérez, Fernando; Chiralt Monleon, Cristina

    2014-01-01

    A perturbative procedure based on the Lie-Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear di erential equations with periodic coe cients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous sys- tem and also to its characteristic ...

  11. Polaron effects on the linear and the nonlinear optical absorption coefficients and refractive index changes in cylindrical quantum dots with applied magnetic field

    Wu Qingjie; Guo Kangxian; Liu Guanghui; Wu Jinghe

    2013-01-01

    Polaron effects on the linear and the nonlinear optical absorption coefficients and refractive index changes in cylindrical quantum dots with the radial parabolic potential and the z-direction linear potential with applied magnetic field are theoretically investigated. The optical absorption coefficients and refractive index changes are presented by using the compact-density-matrix approach and iterative method. Numerical calculations are presented for GaAs/AlGaAs. It is found that taking into account the electron-LO-phonon interaction, not only are the linear, the nonlinear and the total optical absorption coefficients and refractive index changes enhanced, but also the total optical absorption coefficients are more sensitive to the incident optical intensity. It is also found that no matter whether the electron-LO-phonon interaction is considered or not, the absorption coefficients and refractive index changes above are strongly dependent on the radial frequency, the magnetic field and the linear potential coefficient.

  12. Using the Ridge Regression Procedures to Estimate the Multiple Linear Regression Coefficients

    Gorgees, HazimMansoor; Mahdi, FatimahAssim

    2018-05-01

    This article concerns with comparing the performance of different types of ordinary ridge regression estimators that have been already proposed to estimate the regression parameters when the near exact linear relationships among the explanatory variables is presented. For this situations we employ the data obtained from tagi gas filling company during the period (2008-2010). The main result we reached is that the method based on the condition number performs better than other methods since it has smaller mean square error (MSE) than the other stated methods.

  13. Linear thermal expansion coefficient of cast Fe-Ni invar and Fe-Ni-Co superinvar alloys

    Ogorodnikova, O.M.; Chermenskaya, E.V.; Rabinovich, S.V.; Grachev, S.V.

    1999-01-01

    Cast invar alloys Fe-Ni (28-35 wt. % Ni) are investigated using metallography, dilatometry and X-ray methods as soon as the crystallization is completed and again after low-temperature treatment resulting in martensitic transformation in low nickel alloys. Nickel distribution in a cast superinvar Fe-32% Ni-4% Co is studied by means of X-ray spectrum microanalysis. The results obtained permit the correction of model concepts about cast invars and the estimate of a coefficient of linear expansion depending on phase composition and nickel microsegregation [ru

  14. On Attainability of Optimal Solutions for Linear Elliptic Equations with Unbounded Coefficients

    P. I. Kogut

    2011-12-01

    Full Text Available We study an optimal boundary control problem (OCP associated to a linear elliptic equation —div (Vj/ + A(xVy = f describing diffusion in a turbulent flow. The characteristic feature of this equation is the fact that, in applications, the stream matrix A(x = [a,ij(x]i,j=i,...,N is skew-symmetric, ац(х = —a,ji(x, measurable, and belongs to L -space (rather than L°°. An optimal solution to such problem can inherit a singular character of the original stream matrix A. We show that optimal solutions can be attainable by solutions of special optimal boundary control problems.

  15. On the asymptotic expansions of solutions of an nth order linear differential equation with power coefficients

    Paris, R.B.; Wood, A.D.

    1984-11-01

    The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n, containing a factor zsup(m) multiplying the lower order derivatives, are investigated for large values of z in the complex plane. Four classes of solutions are considered which exhibit the following behaviour as /z/ → infinity in certain sectors: (i) solutions whose behaviour is either exponentially large or algebraic (involving p ( < n) algebraic expansions), (ii) solutions which are exponentially small (iii) solutions with a single algebraic expansion and (iv) solutions which are even and odd functions of z whenever n+m is even. The asymptotic expansions of these solutions in a full neigbourhood of the point at infinity are obtained by means of the theory of the solutions in the case m=O developed in a previous paper

  16. Functional dependence of the lower hybrid power absorption coefficient in JET

    Pericoli-Ridolfini, V.; Ekedahl, A.; Baranov, Y.

    1997-01-01

    The fraction of the coupled lower hybrid (LH) power adsorbed in divertor plasmas in JET has been determined experimentally with a method utilizing the time derivative of the total stored energy (plasma and magnetic). This method can account for the power adsorbed inside a normalized flux co-ordinate ψ ∼ 0.7. The experimental LH absorption coefficient reaches 100% at low plasma densities, antineutron e 19 m -3 and decreases to 25% at antineutron e > 3.5 x 10 19 m -3 . The LH wave accessibility to the plasma core has been found to play an important role in determining the power absorption and the radial deposition profile. The decreasing absorption is correlated with a gradual shift of the LH power deposition profile, as determined by the hard x-ray profiles, towards the plasma periphery. Similar behaviour is found in ray tracing + Fokker-Planck code calculations. The frequency spectrum of the LH pump wave as determined by a probe outside the tokamak vessel broadens strongly as the wave accessibility is reduced and the absorption drops. (author)

  17. Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

    Fiorenza, Alberto; Vincenzi, Giovanni

    2011-01-01

    Research highlights: → We prove a result true for all linear homogeneous recurrences with constant coefficients. → As a corollary of our results we immediately get the celebrated Poincare' theorem. → The limit of the ratio of adjacent terms is characterized as the unique leading root of the characteristic polynomial. → The Golden Ratio, Kepler limit of the classical Fibonacci sequence, is the unique leading root. → The Kepler limit may differ from the unique root of maximum modulus and multiplicity. - Abstract: For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincare's theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.

  18. Young's Modulus and Coefficient of Linear Thermal Expansion of ZnO Conductive and Transparent Ultra-Thin Films

    Naoki Yamamoto

    2011-01-01

    Full Text Available A new technique for measuring Young's modulus of an ultra-thin film, with a thickness in the range of about 10 nm, was developed by combining an optical lever technique for measuring the residual stress and X-ray diffraction for measuring the strain in the film. The new technique was applied to analyze the mechanical properties of Ga-doped ZnO (GZO films, that have become the focus of significant attention as a substitute material for indium-tin-oxide transparent electrodes. Young's modulus of the as-deposited GZO films decreased with thickness; the values for 30 nm and 500 nm thick films were 205 GPa and 117 GPa, respectively. The coefficient of linear thermal expansion of the GZO films was measured using the new technique in combination with in-situ residual stress measurement during heat-cycle testing. GZO films with 30–100 nm thickness had a coefficient of linear thermal expansion in the range of 4.3 × 10−6 – 5.6 × 10−6 °C−1.

  19. Investigation of Thermal Expansion of a Glass Ceramic Material with an Extra-Low Thermal Linear Expansion Coefficient

    Kompan, T. A.; Korenev, A. S.; Lukin, A. Ya.

    2008-10-01

    The artificial material sitall CO-115M was developed purposely as a material with an extra-low thermal expansion. The controlled crystallization of an aluminosilicate glass melt leads to the formation of a mixture of β-spodumen, β-eucryptite, and β-silica anisotropic microcrystals in a matrix of residual glass. Due to the small size of the microcrystals, the material is homogeneous and transparent. Specific lattice anharmonism of these microcrystal materials results in close to zero average thermal linear expansion coefficient (TLEC) of the sitall material. The thermal expansion coefficient of this material was measured using an interferometric method in line with the classical approach of Fizeau. To obtain the highest accuracy, the registration of light intensity of the total interference field was used. Then, the parameters of the interference pattern were calculated. Due to the large amount of information in the interference pattern, the error of the calculated fringe position was less than the size of a pixel of the optical registration system. The thermal expansion coefficient of the sitall CO-115M and its temperature dependence were measured. The TLEC value of about 3 × 10-8 K-1 to 5 × 10-8 K-1 in the temperature interval from -20 °C to +60 °C was obtained. A special investigation was carried out to show the homogeneity of the material.

  20. The effect of transition metals additions on the temperature coefficient of linear expansion of titanium and vanadium

    Lesnaya, M.I.; Volokitin, G.G.; Kashchuk, V.A.

    1976-01-01

    Results are reported of an experimental research into the influence of small additions of α-transition metals on the temperature coefficient of linear expansion of titanium and vanadium. Using the configuration model of substance as the basis, expeained are the lowering of the critical liquefaction temperature or the melting point of vanadium and the raising of it, as caused by the addition of metals of the 6 group of the periodic chart and by the addition of metals of the 8 group, respectively, and also a shift in the temperature of the polymorphic α-β-transformation of titanium. Suggested as the best alloying metal for vanadium are tungsten and tantalum; for titaniums is vanadium whose admixtures lower the melting point and shift the polymorphic transformation temperature by as much as 100 to 120 degrees

  1. Application of Linear Quadratic Gaussian and Coefficient Diagram Techniques to Distributed Load Frequency Control of Power Systems

    Tarek Hassan Mohamed

    2015-12-01

    Full Text Available This paper presented both the linear quadratic Gaussian technique (LQG and the coefficient diagram method (CDM as load frequency controllers in a multi-area power system to deal with the problem of variations in system parameters and load demand change. The full states of the system including the area frequency deviation have been estimated using the Kalman filter technique. The efficiency of the proposed control method has been checked using a digital simulation. Simulation results indicated that, with the proposed CDM + LQG technique, the system is robust in the face of parameter uncertainties and load disturbances. A comparison between the proposed technique and other schemes is carried out, confirming the superiority of the proposed CDM + LQG technique.

  2. Flow and linear coefficient of thermal expansion of four types of Base Plate waxes compared with ADA standard

    Monzavi A

    2002-07-01

    Full Text Available Waxes have a lot of applications in dentistry. Such materials are of thermoplastic type that undergoes deformation in different temperatures. Two important properties of base plate waxes are flow and their coefficient of linear thermal expansion. Recently, different institutions, inside the country, produce dentistry waxes, while they have not been standardized. Consequently, consumers' dissatisfaction are observed. In this research, the two above- mentioned factors were compared between three kinds of Iranian waxes with Cavex that is foreign production, based on test number 24 of ADA. To measure the flow rate in the temperatures of 23, 37 and 45°c, Wilcoxon statistical analysis was used. The results showed that in 23°c, the flow rate of Cavex and Azardent waxes met ADA standards; however, it was not true for two others types. In 37°c, the flow of none of the waxes was standardized and in 45°c their flow was acceptable, moreover, thermal expansion coefficient, for Cavex and Azardent types, was based on ADA standard.

  3. Pulse-coupled mixed-mode oscillators: Cluster states and extreme noise sensitivity

    Karamchandani, Avinash J.; Graham, James N.; Riecke, Hermann

    2018-04-01

    Motivated by rhythms in the olfactory system of the brain, we investigate the synchronization of all-to-all pulse-coupled neuronal oscillators exhibiting various types of mixed-mode oscillations (MMOs) composed of sub-threshold oscillations (STOs) and action potentials ("spikes"). We focus particularly on the impact of the delay in the interaction. In the weak-coupling regime, we reduce the system to a Kuramoto-type equation with non-sinusoidal phase coupling and the associated Fokker-Planck equation. Its linear stability analysis identifies the appearance of various cluster states. Their type depends sensitively on the delay and the width of the pulses. Interestingly, long delays do not imply slow population rhythms, and the number of emerging clusters only loosely depends on the number of STOs. Direct simulations of the oscillator equations reveal that for quantitative agreement of the weak-coupling theory the coupling strength and the noise have to be extremely small. Even moderate noise leads to significant skipping of STO cycles, which can enhance the diffusion coefficient in the Fokker-Planck equation by two orders of magnitude. Introducing an effective diffusion coefficient extends the range of agreement significantly. Numerical simulations of the Fokker-Planck equation reveal bistability and solutions with oscillatory order parameters that result from nonlinear mode interactions. These are confirmed in simulations of the full spiking model.

  4. Microscopic model for the non-linear fluctuating hydrodynamic of {sup 4} He superfluid helium deduced by maximum entropy method; Modelo microscopico para la hidrodinamica fluctuante no lineal del {sup 4}He superfluido deducido mediante el metodo de maxima entropia

    Alvarez R, J.T

    1998-10-01

    This thesis presents a microscopic model for the non-linear fluctuating hydrodynamic of superfluid helium ({sup 4} He), model developed by means of the Maximum Entropy Method (Maxent). In the chapter 1, it is demonstrated the necessity to developing a microscopic model for the fluctuating hydrodynamic of the superfluid helium, starting from to show a brief overview of the theories and experiments developed in order to explain the behavior of the superfluid helium. On the other hand, it is presented the Morozov heuristic method for the construction of the non-linear hydrodynamic fluctuating of simple fluid. Method that will be generalized for the construction of the non-linear fluctuating hydrodynamic of the superfluid helium. Besides, it is presented a brief summary of the content of the thesis. In the chapter 2, it is reproduced the construction of a Generalized Fokker-Planck equation, (GFP), for a distribution function associated with the coarse grained variables. Function defined with aid of a nonequilibrium statistical operator {rho}hut{sub FP} that is evaluated as Wigneris function through {rho}{sub CG} obtained by Maxent. Later this equation of GFP is reduced to a non-linear local FP equation from considering a slow and Markov process in the coarse grained variables. In this equation appears a matrix D{sub mn} defined with a nonequilibrium coarse grained statistical operator {rho}hut{sub CG}, matrix elements are used in the construction of the non-linear fluctuating hydrodynamics equations of the superfluid helium. In the chapter 3, the Lagrange multipliers are evaluated for to determine {rho}hut{sub CG} by means of the local equilibrium statistical operator {rho}hut{sub l}-tilde with the hypothesis that the system presents small fluctuations. Also are determined the currents associated with the coarse grained variables and furthermore are evaluated the matrix elements D{sub mn} but with aid of a quasi equilibrium statistical operator {rho}hut{sub qe} instead

  5. Development of polyparameter linear free energy relationship models for octanol-air partition coefficients of diverse chemicals.

    Jin, Xiaochen; Fu, Zhiqiang; Li, Xuehua; Chen, Jingwen

    2017-03-22

    The octanol-air partition coefficient (K OA ) is a key parameter describing the partition behavior of organic chemicals between air and environmental organic phases. As the experimental determination of K OA is costly, time-consuming and sometimes limited by the availability of authentic chemical standards for the compounds to be determined, it becomes necessary to develop credible predictive models for K OA . In this study, a polyparameter linear free energy relationship (pp-LFER) model for predicting K OA at 298.15 K and a novel model incorporating pp-LFERs with temperature (pp-LFER-T model) were developed from 795 log K OA values for 367 chemicals at different temperatures (263.15-323.15 K), and were evaluated with the OECD guidelines on QSAR model validation and applicability domain description. Statistical results show that both models are well-fitted, robust and have good predictive capabilities. Particularly, the pp-LFER model shows a strong predictive ability for polyfluoroalkyl substances and organosilicon compounds, and the pp-LFER-T model maintains a high predictive accuracy within a wide temperature range (263.15-323.15 K).

  6. Effect of the concentration of magnetic grains on the linear-optical-absorption coefficient of ferrofluid-doped lyotropic mesophases: deviation from the Beer-Lambert law.

    Cuppo, F L S; Gómez, S L; Figueiredo Neto, A M

    2004-04-01

    In this paper is reported a systematic experimental study of the linear-optical-absorption coefficient of ferrofluid-doped isotropic lyotropic mixtures as a function of the magnetic-grains concentration. The linear optical absorption of ferrolyomesophases increases in a nonlinear manner with the concentration of magnetic grains, deviating from the usual Beer-Lambert law. This behavior is associated to the presence of correlated micelles in the mixture which favors the formation of small-scale aggregates of magnetic grains (dimers), which have a higher absorption coefficient with respect to that of isolated grains. We propose that the indirect heating of the micelles via the ferrofluid grains (hyperthermia) could account for this nonlinear increase of the linear-optical-absorption coefficient as a function of the grains concentration.

  7. Numerical investigation of multi-beam laser heterodyne measurement with ultra-precision for linear expansion coefficient of metal based on oscillating mirror modulation

    Li Yan-Chao; Wang Chun-Hui; Qu Yang; Gao Long; Cong Hai-Fang; Yang Yan-Ling; Gao Jie; Wang Ao-You

    2011-01-01

    This paper proposes a novel method of multi-beam laser heterodyne measurement for metal linear expansion coefficient. Based on the Doppler effect and heterodyne technology, the information is loaded of length variation to the frequency difference of the multi-beam laser heterodyne signal by the frequency modulation of the oscillating mirror, this method can obtain many values of length variation caused by temperature variation after the multi-beam laser heterodyne signal demodulation simultaneously. Processing these values by weighted-average, it can obtain length variation accurately, and eventually obtain the value of linear expansion coefficient of metal by the calculation. This novel method is used to simulate measurement for linear expansion coefficient of metal rod under different temperatures by MATLAB, the obtained result shows that the relative measurement error of this method is just 0.4%. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)

  8. Numerical investigation of multi-beam laser heterodyne measurement with ultra-precision for linear expansion coefficient of metal based on oscillating mirror modulation

    Li, Yan-Chao; Wang, Chun-Hui; Qu, Yang; Gao, Long; Cong, Hai-Fang; Yang, Yan-Ling; Gao, Jie; Wang, Ao-You

    2011-01-01

    This paper proposes a novel method of multi-beam laser heterodyne measurement for metal linear expansion coefficient. Based on the Doppler effect and heterodyne technology, the information is loaded of length variation to the frequency difference of the multi-beam laser heterodyne signal by the frequency modulation of the oscillating mirror, this method can obtain many values of length variation caused by temperature variation after the multi-beam laser heterodyne signal demodulation simultaneously. Processing these values by weighted-average, it can obtain length variation accurately, and eventually obtain the value of linear expansion coefficient of metal by the calculation. This novel method is used to simulate measurement for linear expansion coefficient of metal rod under different temperatures by MATLAB, the obtained result shows that the relative measurement error of this method is just 0.4%.

  9. Numerical evaluation of path-integral solutions to Fokker-Planck equations. II. Restricted stochastic processes

    Wehner, M.F.

    1983-01-01

    A path-integral solution is derived for processes described by nonlinear Fokker-Plank equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is show t give accurate results. 35 refs., 5 figs

  10. A Generalized Boltzmann Fokker-Planck Method for Coupled Charged Particle Transport

    Prinja, Anil K

    2012-01-09

    The goal of this project was to develop and investigate the performance of reduced-physics formulations of high energy charged particle (electrons, protons and heavier ions) transport that are computationally more efficient than not only analog Monte Carlo methods but also the established condensed history Monte Carlo technique. Charged particles interact with matter by Coulomb collisions with target nuclei and electrons, by bremsstrahlung radiation loss and by nuclear reactions such as spallation and fission. Of these, inelastic electronic collisions and elastic nuclear collisions are the dominant cause of energy-loss straggling and angular deflection or range straggling of a primary particle. These collisions are characterized by extremely short mean free paths (sub-microns) and highly peaked, near-singular differential cross sections about forward directions and zero energy loss, with the situation for protons and heavier ions more extreme than for electrons. For this reason, analog or truephysics single-event Monte Carlo simulation, while possible in principle, is computationally prohibitive for routine calculation of charged particle interaction phenomena.

  11. A local analytic approach for the fast solution of the Fokker-Planck equation

    Sajjadi, S.G.; Nicholas, D.J.

    1987-11-01

    In this report we describe a method of obtaining a closed form for the Focker-Planck equation rendering it amenable to solution in time-step with a complete hydrodynamic treatment of a plasma. We present a local expression for the heat flux, by solving the Focker-Planck equation for electrons in one space and two velocity dimensions in the presence of a self consistent electronic field. (author)

  12. A comparative study of the tail ion distribution with reduced Fokker-Planck models

    McDevitt, C. J.; Tang, Xian-Zhu; Guo, Zehua; Berk, H. L.

    2014-03-01

    A series of reduced models are used to study the fast ion tail in the vicinity of a transition layer between plasmas at disparate temperatures and densities, which is typical of the gas and pusher interface in inertial confinement fusion targets. Emphasis is placed on utilizing progressively more comprehensive models in order to identify the essential physics for computing the fast ion tail at energies comparable to the Gamow peak. The resulting fast ion tail distribution is subsequently used to compute the fusion reactivity as a function of collisionality and temperature. While a significant reduction of the fusion reactivity in the hot spot compared to the nominal Maxwellian case is present, this reduction is found to be partially recovered by an increase of the fusion reactivity in the neighboring cold region.

  13. Stochastic processes and applications diffusion processes, the Fokker-Planck and Langevin equations

    Pavliotis, Grigorios A

    2014-01-01

    This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated.                 The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to eq...

  14. Multigroup Boltzmann-Fokker-Planck approach for ion transport in amorphous media

    Keen, N.D.; Prinja, A.K.; Dunham, G.D. [New Mexico Univ., Albuquerque, NM (United States). Chemical and Nuclear Engineering Dept.

    2001-07-01

    We present a MGMC approach for the transport of arbitrary mass ions having energies up to a few MeV. Specifically, we consider interactions with target atoms through Coulomb mediated elastic nuclear and inelastic electronic collisions and restrict considerations to ion implantation and energy deposition of primary ions in amorphous media. (orig.)

  15. Fokker-Planck simulations of interactions of femtosecond laser pulses with dense plasmas

    Drska, L.; Limpouch, J.; Liska, R.

    1993-01-01

    The interaction of femtosecond laser pulses with fully ionized solid-state density plasmas in the regime of the normal skin effect was investigated by means of numerical simulation. For short wavelength lasers and 120 fs FWHM laser pulses the regime of normal skin effect is shown to hold for peak intensities up to 10 17 W/cm 2 . Basic characteristics of the interaction are revealed and certain departures of the electron distribution function, of the plasma dielectric constant and of laser absorption from simplistic models are pointed out. (author) 1 tab., 4 figs., 14 refs

  16. Two media method for linear attenuation coefficient determination of irregular soil samples; Metodo dos dois meios para a determinacao do coeficiente de atenuacao linear de amostras irregulares de solos

    Vici, Carlos Henrique Georges

    2004-07-01

    In several situations of nuclear applications, the knowledge of gamma-ray linear attenuation coefficient for irregular samples is necessary, such as in soil physics and geology. This work presents the validation of a methodology for the determination of the linear attenuation coefficient ({mu}) of irregular shape samples, in such a way that it is not necessary to know the thickness of the considered sample. With this methodology irregular soil samples (undeformed field samples) from Londrina region, north of Parana were studied. It was employed the two media method for the {mu} determination. It consists of the {mu} determination through the measurement of a gamma-ray beam attenuation by the sample sequentially immersed in two different media, with known and appropriately chosen attenuation coefficients. For comparison, the theoretical value of {mu} was calculated by the product of the mass attenuation coefficient, obtained by the WinXcom code, and the measured value of the density sample. This software employs the chemical composition of the samples and supplies a table of the mass attenuation coefficients versus the photon energy. To verify the validity of the two media method, compared with the simple gamma ray transmission method, regular pome stone samples were used. With these results for the attenuation coefficients and their respective deviations, it was possible to compare the two methods. In this way we concluded that the two media method is a good tool for the determination of the linear attenuation coefficient of irregular materials, particularly in the study of soils samples. (author)

  17. Temperature and orientation dependence of the short-term strength characteristics, Young's modulus, and linear expansion coefficient of ZhS6F alloy single crystals

    Svetlov, I L; Sukhanov, N N; Krivko, A I; Roshchina, I N; Khatsinskaia, I M

    1987-01-01

    Experimental data are presented on the temperature dependence of the short- term strength characteristics, Young's modulus, and linear expansion coefficients of single crystals of a nickel alloy, ZhS6F, with crystallographic orientations along the 001, 111, 011, and 112 lines. It is found that the mechanical properties and Young's modulus of the alloy crystals exibit anisotropy in the temperature range 20-900 C. The linear thermal expansion coefficient is isotropic up to 900 C and equal to that of the equiaxed alloy. 10 references.

  18. The Heat Resistance of Microbial Cells Represented by D Values Can be Estimated by the Transition Temperature and the Coefficient of Linear Expansion.

    Nakanishi, Koichi; Kogure, Akinori; Deuchi, Keiji; Kuwana, Ritsuko; Takamatsu, Hiromu; Ito, Kiyoshi

    2015-01-01

    We previously developed a method for evaluating the heat resistance of microorganisms by measuring the transition temperature at which the coefficient of linear expansion of a cell changes. Here, we performed heat resistance measurements using a scanning probe microscope with a nano thermal analysis system. The microorganisms studied included six strains of the genus Bacillus or related genera, one strain each of the thermophilic obligate anaerobic bacterial genera Thermoanaerobacter and Moorella, two strains of heat-resistant mold, two strains of non-sporulating bacteria, and one strain of yeast. Both vegetative cells and spores were evaluated. The transition temperature at which the coefficient of linear expansion due to heating changed from a positive value to a negative value correlated strongly with the heat resistance of the microorganism as estimated from the D value. The microorganisms with greater heat resistance exhibited higher transition temperatures. There was also a strong negative correlation between the coefficient of linear expansion and heat resistance in bacteria and yeast, such that microorganisms with greater heat resistance showed lower coefficients of linear expansion. These findings suggest that our method could be useful for evaluating the heat resistance of microorganisms.

  19. Expectation-maximization of the potential of mean force and diffusion coefficient in Langevin dynamics from single molecule FRET data photon by photon.

    Haas, Kevin R; Yang, Haw; Chu, Jhih-Wei

    2013-12-12

    The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

  20. Linear response and correlation of a self-propelled particle in the presence of external fields

    Caprini, Lorenzo; Marini Bettolo Marconi, Umberto; Vulpiani, Angelo

    2018-03-01

    We study the non-equilibrium properties of non interacting active Ornstein-Uhlenbeck particles (AOUP) subject to an external nonuniform field using a Fokker-Planck approach with a focus on the linear response and time-correlation functions. In particular, we compare different methods to compute these functions including the unified colored noise approximation (UCNA). The AOUP model, described by the position of the particle and the active force acting on it, is usually mapped into a Markovian process, describing the motion of a fictitious passive particle in terms of its position and velocity, where the effect of the activity is transferred into a position-dependent friction. We show that the form of the response function of the AOUP depends on whether we put the perturbation on the position and keep unperturbed the active force in the original variables or perturb the position and maintain unperturbed the velocity in the transformed variables. Indeed, as a result of the change of variables the perturbation on the position becomes a perturbation both on the position and on the fictitious velocity. We test these predictions by considering the response for three types of convex potentials: quadratic, quartic and double-well potential. Moreover, by comparing the response of the AOUP model with the corresponding response of the UCNA model we conclude that although the stationary properties are fairly well approximated by the UCNA, the non equilibrium properties are not, an effect which is not negligible when the persistence time is large.

  1. Partial F-tests with multiply imputed data in the linear regression framework via coefficient of determination.

    Chaurasia, Ashok; Harel, Ofer

    2015-02-10

    Tests for regression coefficients such as global, local, and partial F-tests are common in applied research. In the framework of multiple imputation, there are several papers addressing tests for regression coefficients. However, for simultaneous hypothesis testing, the existing methods are computationally intensive because they involve calculation with vectors and (inversion of) matrices. In this paper, we propose a simple method based on the scalar entity, coefficient of determination, to perform (global, local, and partial) F-tests with multiply imputed data. The proposed method is evaluated using simulated data and applied to suicide prevention data. Copyright © 2014 John Wiley & Sons, Ltd.

  2. Development of Solution-Processable, Optically Transparent Polyimides with Ultra-Low Linear Coefficients of Thermal Expansion

    Masatoshi Hasegawa

    2017-10-01

    Full Text Available This paper reviews the development of new high-temperature polymeric materials applicable to plastic substrates in image display devices with a focus on our previous results. Novel solution-processable colorless polyimides (PIs with ultra-low linear coefficients of thermal expansion (CTE are proposed in this paper. First, the principles of the coloration of PI films are briefly discussed, including the influence of the processing conditions on the film coloration, as well as the chemical and physical factors dominating the low CTE characteristics of the resultant PI films to clarify the challenges in simultaneously achieving excellent optical transparency, a very high Tg, a very low CTE, and excellent film toughness. A possible approach of achieving these target properties is to use semi-cycloaliphatic PI systems consisting of linear chain structures. However, semi-cycloaliphatic PIs obtained using cycloaliphatic diamines suffer various problems during precursor polymerization, cyclodehydration (imidization, and film preparation. In particular, when using trans-1,4-cyclohexanediamine (t-CHDA as the cycloaliphatic diamine, a serious problem emerges: salt formation in the initial stages of the precursor polymerization, which terminates the polymerization in some cases or significantly extends the reaction period. The system derived from 3,3′,4,4′-biphenyltetracarboxylic dianhydride (s-BPDA and t-CHDA can be polymerized by a controlled heating method and leads to a PI film with relatively good properties, i.e., excellent light transmittance at 400 nm (T400 = ~80%, a high Tg (>300 °C, and a very low CTE (10 ppm·K−1. However, this PI film is somewhat brittle (the maximum elongation at break, εb max is about 10%. On the other hand, the combination of cycloaliphatic tetracarboxylic dianhydrides and aromatic diamines does not result in salt formation. The steric structures of cycloaliphatic tetracarboxylic dianhydrides significantly influence

  3. A SOCIOLOGICAL ANALYSIS OF THE CHILDBEARING COEFFICIENT IN THE ALTAI REGION BASED ON METHOD OF FUZZY LINEAR REGRESSION

    Sergei Vladimirovich Varaksin

    2017-06-01

    Full Text Available Purpose. Construction of a mathematical model of the dynamics of childbearing change in the Altai region in 2000–2016, analysis of the dynamics of changes in birth rates for multiple age categories of women of childbearing age. Methodology. A auxiliary analysis element is the construction of linear mathematical models of the dynamics of childbearing by using fuzzy linear regression method based on fuzzy numbers. Fuzzy linear regression is considered as an alternative to standard statistical linear regression for short time series and unknown distribution law. The parameters of fuzzy linear and standard statistical regressions for childbearing time series were defined with using the built in language MatLab algorithm. Method of fuzzy linear regression is not used in sociological researches yet. Results. There are made the conclusions about the socio-demographic changes in society, the high efficiency of the demographic policy of the leadership of the region and the country, and the applicability of the method of fuzzy linear regression for sociological analysis.

  4. A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

    Addona, Davide, E-mail: d.addona@campus.unimib.it [Università degli Studi di Milano Bicocca, (MILANO BICOCCA) Dipartimento di Matematica (Italy)

    2015-08-15

    We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.

  5. Absolute non-linear optical coefficients measurements of CsLiB 6O 10 single crystals by second harmonic generation

    Sifi, A.; Klein, R. S.; Maillard, A.; Kugel, G. E.; Péter, A.; Polgár, K.

    2003-10-01

    We present absolute measurements of the effective non-linear optical coefficients deff of cesium lithium borate crystals (CsLiB 6O 10, CLBO) by second harmonic generation using a continuous Nd-YAG laser source. The experiments were carried out at room temperature, on crystals cut perpendicular to type I or type II phase matching directions, with two different crystal lengths along the propagation direction. The d36 and d14 non-linear coefficients involved in deff developments are deduced and are shown to be equal as it is predicted by the Kleinman symmetry. Two different compositions prepared by the Czochralski technique from melt with compositions of 1:1:6 and 1:1:5.5 molar ratios of Cs 2O, Li 2O and B 2O 3 are comparatively studied.

  6. Simultaneous effects of hydrostatic pressure and spin–orbit coupling on linear and nonlinear intraband optical absorption coefficients in a GaAs quantum ring

    Mughnetsyan, V.N.; Manaselyan, A.Kh.; Barseghyan, M.G.; Kirakosyan, A.A.

    2013-01-01

    In this paper the simultaneous effect of hydrostatic pressure and Rashba spin–orbit interaction on intraband linear and nonlinear light absorption has been investigated in cylindrical quantum ring. The one electron energy spectrum has been found using the effective mass approximation and diagonalization procedure. We have found that the Rashba interaction can lead both to the blue- or to the red-shift of the absorption spectrum depending on the transitions character, while the only red-shift is observed due to the hydrostatic pressure. - Highlights: ► The effects of hydrostatic pressure and spin–orbit coupling are investigated for quantum ring. ► The non-linear absorption coefficient is calculated. ► The hydrostatic pressure leads to the decrease in the absorption coefficient. ► Spin–orbit coupling weakens some transitions and strengthens others.

  7. On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

    Gabriel Amador

    2016-05-01

    Full Text Available In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark- and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams.

  8. Response statistics of rotating shaft with non-linear elastic restoring forces by path integration

    Gaidai, Oleg; Naess, Arvid; Dimentberg, Michael

    2017-07-01

    Extreme statistics of random vibrations is studied for a Jeffcott rotor under uniaxial white noise excitation. Restoring force is modelled as elastic non-linear; comparison is done with linearized restoring force to see the force non-linearity effect on the response statistics. While for the linear model analytical solutions and stability conditions are available, it is not generally the case for non-linear system except for some special cases. The statistics of non-linear case is studied by applying path integration (PI) method, which is based on the Markov property of the coupled dynamic system. The Jeffcott rotor response statistics can be obtained by solving the Fokker-Planck (FP) equation of the 4D dynamic system. An efficient implementation of PI algorithm is applied, namely fast Fourier transform (FFT) is used to simulate dynamic system additive noise. The latter allows significantly reduce computational time, compared to the classical PI. Excitation is modelled as Gaussian white noise, however any kind distributed white noise can be implemented with the same PI technique. Also multidirectional Markov noise can be modelled with PI in the same way as unidirectional. PI is accelerated by using Monte Carlo (MC) estimated joint probability density function (PDF) as initial input. Symmetry of dynamic system was utilized to afford higher mesh resolution. Both internal (rotating) and external damping are included in mechanical model of the rotor. The main advantage of using PI rather than MC is that PI offers high accuracy in the probability distribution tail. The latter is of critical importance for e.g. extreme value statistics, system reliability, and first passage probability.

  9. A numerical study of one-dimensional replicating patterns in reaction-diffusion systems with non-linear diffusion coefficients

    Ferreri, J. C.; Carmen, A. del

    1998-01-01

    A numerical study of the dynamics of pattern evolution in reaction-diffusion systems is performed, although limited to one spatial dimension. The diffusion coefficients are nonlinear, based on powers of the scalar variables. The system keeps the dynamics of previous studies in the literature, but the presence of nonlinear diffusion generates a field of strong nonlinear interactions due to the presence of receding travelling waves. This field is limited by the plane of symmetry of the space domain and the last born outgoing travelling wave. These effects are discussed. (author). 10 refs., 7 figs

  10. A role of the coefficient of the differential term in qualitative theory of half-linear equations

    Řehák, Pavel

    2010-01-01

    Roč. 135, č. 2 (2010), s. 151-162 ISSN 0862-7959 R&D Projects: GA AV ČR KJB100190701 Grant - others:GA ČR(CZ) GA201/07/0145 Institutional research plan: CEZ:AV0Z10190503 Keywords : half-linear dynamic equation * time scale * transformation * comparison theorem * oscillation criteria Subject RIV: BA - General Mathematics http://www.dml.cz/handle/10338.dmlcz/140692

  11. Homogeneity of the coefficient of linear thermal expansion of ZERODUR: a review of a decade of evaluations

    Jedamzik, Ralf; Westerhoff, Thomas

    2017-09-01

    The coefficient of thermal expansion (CTE) and its spatial homogeneity from small to large formats is the most important property of ZERODUR. Since more than a decade SCHOTT has documented the excellent CTE homogeneity. It started with reviews of past astronomical telescope projects like the VLT, Keck and GTC mirror blanks and continued with dedicated evaluations of the production. In recent years, extensive CTE measurements on samples cut from randomly selected single ZERODUR parts in meter size and formats of arbitrary shape, large production boules and even 4 m sized blanks have demonstrated the excellent CTE homogeneity in production. The published homogeneity data shows single ppb/K peak to valley CTE variations on medium spatial scale of several cm down to small spatial scale of only a few mm mostly at the limit of the measurement reproducibility. This review paper summarizes the results also in respect to the increased CTE measurement accuracy over the last decade of ZERODUR production.

  12. Analysis of the collagen birefringence and the relative attenuation coefficient of health and burned skin irradiated with linearly polarized He-Ne laser

    Silva, Daniela de Fatima Teixeira da

    2002-01-01

    Low-intensity laser therapy is characterized by its ability to induce athermic effects and nondestructive photobiological processes. Although it has been in use for more than 40 years, this phototherapy is still not an established therapeutic modality. The objectives of this study were: to quantify the collagen fibers organization by polarized light microscopy in normal and burned skin samples at day 17 post-injury considering preferential axis as the animal's spinal column and aligning the linear laser polarization in two directions of polarization, parallel or perpendicular to this axis; to determine the relative attenuation coefficient for the intensity light by the technique of imaging the light distribution in normal and burned skin during wound healing process taking only parallel direction of polarization. To reach the objectives, burns about 6 mm in diameter were created with liquid N 2 on the back of the rats and the lesions were irradiated on days 3, 7, 10 and 14 post-wounding, D= 1 J/cm 2 , to investigate the effects of low-intensity linearly polarized He-Ne laser beam on skin wounds healing. Control lesions were not irradiated. The results have demonstrated that: the skin samples irradiated with linearly parallel polarized He-Ne laser beam showed collagen fibers more organized; burned skin samples presents a higher attenuation coefficient than normal skin samples. These results are important to optimize low intensity laser therapy dosimetry on acceleration wound healing. (author)

  13. Linear and nonlinear optical absorption coefficients in GaAs/Ga1−xAlxAs concentric double quantum rings: Effects of hydrostatic pressure and aluminum concentration

    Baghramyan, H.M.; Barseghyan, M.G.; Kirakosyan, A.A.; Restrepo, R.L.; Duque, C.A.

    2013-01-01

    The linear and nonlinear intra-band optical absorption coefficients in GaAs/Ga 1−x Al x As two-dimensional concentric double quantum rings are investigated. Taking into account the combined effects of hydrostatic pressure and aluminum concentration the energies of the ground (n=1,l=0) and the first excited state (n=2,l=1) have been found using the effective mass approximation and the transfer matrix formalism. The energies of these states and the corresponding threshold energy of the intra-band optical transitions are examined as a function of hydrostatic pressure and aluminum concentration for different sizes of the structure. We also investigated the dependencies of the linear, nonlinear, and total optical absorption coefficients as functions of the incident photon energy for different values of hydrostatic pressure, aluminum concentration, sizes of the structure, and incident optical intensity. Its is found that the effects of the hydrostatic pressure and the aluminum concentration lead to a shifting of the resonant peaks of the intra-band optical spectrum. - Highlights: ► Linear and nonlinear intra-band absorption in quantum rings. ► Threshold energy strongly depends on the hydrostatic pressure. ► Threshold energy strongly depends on the stoichiometry and sizes of structure. ► Optical absorption is affected by the incident optical intensity.

  14. Linear and mass attenuation coefficient for CdTe compound of X-rays from 10 to 100 keV energy range in different phases

    Saim, A., E-mail: saim1989asma@gmail.com; Tebboune, A.; Berkok, H.; Belameiri, N.; Belbachir, A.H.

    2014-07-25

    The Full Potential Linear Muffin Tin Orbitals method within the density functional theory has been utilized to calculate structural and electronic properties of the CdTe compound. We have checked that the CdTe has two phase-transitions from zinc-blend to cinnabar and from cinnabar to rocksalt. We have found that the rigidity, the energy and the nature of the gap change according to the phase change, so we can predict that a CdTe detector may have different behaviors in different phase conditions. In order to investigate this behavior change, the linear and the mass attenuation coefficients of X-ray in rocksalt, zinc-blend and cinnabar structures are calculated from 10 keV to100 keV, using the XCOM data. We have found that when CdTe undergoes a phase transition from zinc-blend to cinnabar, its linear attenuation coefficient decreases down to a value of about 100 times smaller than its initial one, and when it undergoes a transition from cinnabar to rocksalt it increases up to a value about 90 times larger than its initial one.

  15. X-ray diffraction measurement of the linear thermal expansion coefficients of WCoB in the range 300 to 973 K

    Petrov, K.; Will, G.

    1981-01-01

    High-temperature treatment of tungsten carbide-cobalt hard alloys in TiB 2 media leads to the formation of a surface diffusion coating which contains orthorhombic WCoB. The function of this compound in enhancing wear resistance of cutting tools, is discussed. The thermal expansion of WCoB is of primary interest, since the wear resistance of the coating reflects the degree of matching of the thermal expansion coefficients of the different phases. Preparation of the samples is described and experimental details of the X-ray diffraction measurements are given. The temperature dependence of the lattice parameters for the range 300 to 973 K, and the corresponding linear thermal expansion coefficients along the three principal crystallographic directions, are given. The results are discussed in terms of the bonding features of the solid. (U.K.)

  16. Fractional approximations for linear first order differential equation with polynomial coefficients-application to E1(x) and Z(s)

    Martin, P.; Zamudio-Cristi, J.

    1982-01-01

    A method is described to obtain fractional approximations for linear first order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the highest and lowest power of the variable after the sustitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Pade method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which give three exact digits for most of the real values of the variable. Approximations of higher accuracy and of the same degree than other authors are also obtained. (Author) [pt

  17. Study of variance and covariance terms in linear attenuation coefficient measurements of irregular samples through the two media method by gamma-ray transmission

    Kuramoto, R.Y.R.Renato Yoichi Ribeiro.; Appoloni, Carlos Roberto

    2002-01-01

    The two media method permits the application of Beer's law (Thesis (Master Degree), Universidade Estadual de Londrina, PR, Brazil, pp. 23) for the linear attenuation coefficient determination of irregular thickness samples by gamma-ray transmission. However, the use of this methodology introduces experimental complexity due to the great number of variables to be measured. As consequence of this complexity, the uncertainties associated with each of these variables may be correlated. In this paper, we examine the covariance terms in the uncertainty propagation, and quantify the correlation among the uncertainties of each of the variables in question

  18. Measurement of the linear thermo-optical coefficient of Ga0.51In0.49P using photonic crystal nanocavities

    Sokolov, Sergei; Lian, Jin; Combrié, Sylvain; Rossi, Alfredo De; Mosk, Allard. P.

    2017-01-01

    Ga0.51In0.49PGa0.51In0.49P is a promising candidate for thermally tunable nanophotonic devices due to its low thermal conductivity. In this work we study its thermo-optical response. We obtain the linear thermo-optical coefficient 푑푛/푑푇=2.0±0.3·10−4  K−1dn/dT=2.0±0.3·10−4  K−1 by investigating the

  19. Study of the interaction operator between two groups of particles in a completely ionised plasma. Development of the distribution functions in a series of orthogonal polynomials (1963); Etude de l'operateur d'interaction entre deux groupes de particules dans un plasma completement ionise. Developpement des fonctions de distribution en series de polynomes orthogonaux (1963)

    Fain, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

    1963-05-15

    A completely ionised medium is considered in which the variations of the distribution functions for each species of particles is governed by the Fokker-Planck equation. The interaction operator is of a form given by Rosenbluth - Macdonald - Judd. The distribution functions are expanded into orthogonal polynomial series in the velocity space. In a first stage these functions are first split up into spherical harmonic series (or, in an equivalent form, into series of scalar products of irreducible cartesian tensors), with coefficients which are a function of the velocity modulus as well as space and time coordinates. In the second stage these coefficients are expanded into series of orthogonal functions of the velocity modulus; the 1 order harmonic is represented by the product of a Maxwell distribution and of a SONINE polynomial series, having an index of 1 + 1 / 2, which have as variable the reduced energy of the particles (in terms of a basic temperature), with coefficients which then only depend on the space and time coordinates. In the first part the relationship is established between the expansion coefficients and the moments of the distribution function, as well as the hydrodynamic values. In the second part the expansion using spherical harmonics is applied to the Fokker-Planck equation. The general expression for the second member is given as well as the particular expressions corresponding to the cases where the operator is linearized. In the third part the complete expansion in orthogonal polynomial series is applied to the Fokker-Planck equation. The expression of the generating functions is given for all the harmonics in the case of the linearized operator, as well as the transport equations for the first four harmonics. (author) [French] On considere un milieu completement ionise ou l'evolution des fonctions de distribution pour chaque espece de particules est regie par l'equation de FOKKER-PLANCK. L'operateur d'interaction se met sous la forme

  20. Prediction of the thermal expansion coefficients of bio diesels from several sources through the application of linear regression; Predicao dos coeficientes de expansao termica de biodieseis de diversas origens atraves da aplicacao da regressa linear

    Canciam, Cesar Augusto [Universidade Tecnologica Federal do Parana (UTFPR), Campus Ponta Grossa, PR (Brazil)], e-mail: canciam@utfpr.edu.br

    2012-07-01

    When evaluating the consumption of bio fuels, the knowledge of the density is of great importance for rectify the effect of temperature. The thermal expansion coefficient is a thermodynamic property that provides a measure of the density variation in response to temperature variation, keeping the pressure constant. This study aimed to predict the thermal expansion coefficients of ethyl bio diesels from castor beans, soybeans, sunflower seeds and Mabea fistulifera Mart. oils and of methyl bio diesels from soybeans, sunflower seeds, souari nut, cotton, coconut, castor beans and palm oils, from beef tallow, chicken fat and hydrogenated vegetable fat residual. For this purpose, there was a linear regression analysis of the density of each bio diesel a function of temperature. These data were obtained from other works. The thermal expansion coefficients for bio diesels are between 6.3729x{sup 10-4} and 1.0410x10{sup -3} degree C-1. In all the cases, the correlation coefficients were over 0.99. (author)

  1. Nonlinear electron transport in magnetized laser plasmas

    Kho, T.H.; Haines, M.G.

    1986-01-01

    Electron transport in a magnetized plasma heated by inverse bremsstrahlung is studied numerically using a nonlinear Fokker--Planck model with self-consistent E and B fields. The numerical scheme is described. Nonlocal transport is found to alter many of the transport coefficients derived from linear transport theory, in particular, the Nernst and Righi--Leduc effects, in addition to the perpendicular heat flux q/sub perpendicular/, are substantially reduced near critical surface. The magnetic field, however, remains strongly coupled to the nonlinear q/sub perpendicular/ and, as has been found in hydrosimulations, convective amplification of the magnetic field occurs in the overdense plasma

  2. Determination of the quasi-TE mode (in-plane) graphene linear absorption coefficient via integration with silicon-on-insulator racetrack cavity resonators.

    Crowe, Iain F; Clark, Nicholas; Hussein, Siham; Towlson, Brian; Whittaker, Eric; Milosevic, Milan M; Gardes, Frederic Y; Mashanovich, Goran Z; Halsall, Matthew P; Vijayaraghaven, Aravind

    2014-07-28

    We examine the near-IR light-matter interaction for graphene integrated cavity ring resonators based on silicon-on-insulator (SOI) race-track waveguides. Fitting of the cavity resonances from quasi-TE mode transmission spectra reveal the real part of the effective refractive index for graphene, n(eff) = 2.23 ± 0.02 and linear absorption coefficient, α(gTE) = 0.11 ± 0.01dBμm(-1). The evanescent nature of the guided mode coupling to graphene at resonance depends strongly on the height of the graphene above the cavity, which places limits on the cavity length for optical sensing applications.

  3. Quantitative structure-property relationship study of n-octanol-water partition coefficients of some of diverse drugs using multiple linear regression

    Ghasemi, Jahanbakhsh; Saaidpour, Saadi

    2007-01-01

    A quantitative structure-property relationship (QSPR) study was performed to develop models those relate the structures of 150 drug organic compounds to their n-octanol-water partition coefficients (log P o/w ). Molecular descriptors derived solely from 3D structures of the molecular drugs. A genetic algorithm was also applied as a variable selection tool in QSPR analysis. The models were constructed using 110 molecules as training set, and predictive ability tested using 40 compounds. Modeling of log P o/w of these compounds as a function of the theoretically derived descriptors was established by multiple linear regression (MLR). Four descriptors for these compounds molecular volume (MV) (geometrical), hydrophilic-lipophilic balance (HLB) (constitutional), hydrogen bond forming ability (HB) (electronic) and polar surface area (PSA) (electrostatic) are taken as inputs for the model. The use of descriptors calculated only from molecular structure eliminates the need for experimental determination of properties for use in the correlation and allows for the estimation of log P o/w for molecules not yet synthesized. Application of the developed model to a testing set of 40 drug organic compounds demonstrates that the model is reliable with good predictive accuracy and simple formulation. The prediction results are in good agreement with the experimental value. The root mean square error of prediction (RMSEP) and square correlation coefficient (R 2 ) for MLR model were 0.22 and 0.99 for the prediction set log P o/w

  4. Enhancing Classification Performance of Functional Near-Infrared Spectroscopy- Brain–Computer Interface Using Adaptive Estimation of General Linear Model Coefficients

    Nauman Khalid Qureshi

    2017-07-01

    Full Text Available In this paper, a novel methodology for enhanced classification of functional near-infrared spectroscopy (fNIRS signals utilizable in a two-class [motor imagery (MI and rest; mental rotation (MR and rest] brain–computer interface (BCI is presented. First, fNIRS signals corresponding to MI and MR are acquired from the motor and prefrontal cortex, respectively, afterward, filtered to remove physiological noises. Then, the signals are modeled using the general linear model, the coefficients of which are adaptively estimated using the least squares technique. Subsequently, multiple feature combinations of estimated coefficients were used for classification. The best classification accuracies achieved for five subjects, for MI versus rest are 79.5, 83.7, 82.6, 81.4, and 84.1% whereas those for MR versus rest are 85.5, 85.2, 87.8, 83.7, and 84.8%, respectively, using support vector machine. These results are compared with the best classification accuracies obtained using the conventional hemodynamic response. By means of the proposed methodology, the average classification accuracy obtained was significantly higher (p < 0.05. These results serve to demonstrate the feasibility of developing a high-classification-performance fNIRS-BCI.

  5. Prediction of octanol-water partition coefficients of organic compounds by multiple linear regression, partial least squares, and artificial neural network.

    Golmohammadi, Hassan

    2009-11-30

    A quantitative structure-property relationship (QSPR) study was performed to develop models those relate the structure of 141 organic compounds to their octanol-water partition coefficients (log P(o/w)). A genetic algorithm was applied as a variable selection tool. Modeling of log P(o/w) of these compounds as a function of theoretically derived descriptors was established by multiple linear regression (MLR), partial least squares (PLS), and artificial neural network (ANN). The best selected descriptors that appear in the models are: atomic charge weighted partial positively charged surface area (PPSA-3), fractional atomic charge weighted partial positive surface area (FPSA-3), minimum atomic partial charge (Qmin), molecular volume (MV), total dipole moment of molecule (mu), maximum antibonding contribution of a molecule orbital in the molecule (MAC), and maximum free valency of a C atom in the molecule (MFV). The result obtained showed the ability of developed artificial neural network to prediction of partition coefficients of organic compounds. Also, the results revealed the superiority of ANN over the MLR and PLS models. Copyright 2009 Wiley Periodicals, Inc.

  6. Adaptive operational modal identification for slow linear time-varying structures based on frozen-in coefficient method and limited memory recursive principal component analysis

    Wang, Cheng; Guan, Wei; Wang, J. Y.; Zhong, Bineng; Lai, Xiongming; Chen, Yewang; Xiang, Liang

    2018-02-01

    To adaptively identify the transient modal parameters for linear weakly damped structures with slow time-varying characteristics under unmeasured stationary random ambient loads, this paper proposes a novel operational modal analysis (OMA) method based on the frozen-in coefficient method and limited memory recursive principal component analysis (LMRPCA). In the modal coordinate, the random vibration response signals of mechanical weakly damped structures can be decomposed into the inner product of modal shapes and modal responses, from which the natural frequencies and damping ratios can be well acquired by single-degree-of-freedom (SDOF) identification approach such as FFT. Hence, for the OMA method based on principal component analysis (PCA), it becomes very crucial to examine the relation between the transformational matrix and the modal shapes matrix, to find the association between the principal components (PCs) matrix and the modal responses matrix, and to turn the operational modal parameter identification problem into PCA of the stationary random vibration response signals of weakly damped mechanical structures. Based on the theory of "time-freezing", the method of frozen-in coefficient, and the assumption of "short time invariant" and "quasistationary", the non-stationary random response signals of the weakly damped and slow linear time-varying structures (LTV) can approximately be seen as the stationary random response time series of weakly damped and linear time invariant structures (LTI) in a short interval. Thus, the adaptive identification of time-varying operational modal parameters is turned into decompositing the PCs of stationary random vibration response signals subsection of weakly damped mechanical structures after choosing an appropriate limited memory window. Finally, a three-degree-of-freedom (DOF) structure with weakly damped and slow time-varying mass is presented to illustrate this method of identification. Results show that the LMRPCA

  7. Temperature Dependent Rate Coefficients for the Gas-Phase Reaction of the OH Radical with Linear (L2, L3) and Cyclic (D3, D4) Permethylsiloxanes.

    Bernard, François; Papanastasiou, Dimitrios K; Papadimitriou, Vassileios C; Burkholder, James B

    2018-04-19

    Permethylsiloxanes are emitted into the atmosphere during production and use as personal care products, lubricants, and cleaning agents. The predominate atmospheric loss process for permethylsiloxanes is expected to be via gas-phase reaction with the OH radical. In this study, rate coefficients, k(T), for the OH radical gas-phase reaction with the two simplest linear and cyclic permethylsiloxanes were measured using a pulsed laser photolysis-laser induced fluorescence technique over the temperature range of 240-370 K and a relative rate method at 294 K: hexamethyldisiloxane ((CH 3 ) 3 SiOSi(CH 3 ) 3 , L 2 ), k 1 ; octamethyltrisiloxane ([(CH 3 ) 3 SiO] 2 Si(CH 3 ) 2 , L 3 ), k 2 ; hexamethylcyclotrisiloxane ([-Si(CH 3 ) 2 O-] 3 , D 3 ), k 3 ; and octamethylcyclotetrasiloxane ([-Si(CH 3 ) 2 O-] 4 , D 4 ), k 4 . The obtained k(294 K) values and temperature-dependence expressions for the 240-370 K temperature range are (cm 3 molecule -1 s -1 , 2σ absolute uncertainties): k 1 (294 K) = (1.28 ± 0.08) × 10 -12 , k 1 ( T) = (1.87 ± 0.18) × 10 -11 exp(-(791 ± 27)/ T); k 2 (294 K) = (1.72 ± 0.10) × 10 -12 , k 2 ( T) = 1.96 × 10 -13 (T/298) 4.34 exp(657/ T); k 3 (294 K) = (0.82 ± 0.05) × 10 -12 , k 3 ( T) = (1.29 ± 0.19) × 10 -11 exp(-(805 ± 43)/ T); and k 4 (294 K) = (1.12 ± 0.10) × 10 -12 , k 4 ( T) = (1.80 ± 0.26) × 10 -11 exp(-(816 ± 43)/ T). The cyclic molecules were found to be less reactive than the analogous linear molecule with the same number of -CH 3 groups, while the linear and cyclic permethylsiloxane reactivity both increase with the increasing number of CH 3 - groups. The present results are compared with previous rate coefficient determinations where available. The permethylsiloxanes included in this study are atmospherically short-lived compounds with estimated atmospheric lifetimes of 11, 8, 17, and 13 days, respectively.

  8. Evaluation of the particle size of aggregates used in high density micro-concretes on the gamma-ray linear attenuation coefficient

    Andrello, Avacir Casanova; Albuquerque, Sergio

    2011-01-01

    Full text: When a component for protecting against ionizing radiation is designed, the main aim to be accomplished is to attenuate radiations to acceptable values, within tolerable limits. Several materials and arrangements can be utilized as protection, among which we can name concrete, steel, lead plates and mortars. Where low-energy radiations (some dozens of keVs) are involved, the main interaction between radiation and the material is the photoelectric effect, whose radiation absorption depends on the photon and specific atomic number of the absorbent. High-density concretes are made by mixing several materials, and the granulometric mixture and proportion of these will determine the physical and chemical features of the product. When the aim is to develop a concrete trace to be utilized as a protection against gamma and X-ray ionizing radiations in low energies, not only aspects of the structural behavior of the component or material must be evaluated, but also the behavior of the composing materials in face of radiation flow must be studied and known in order to develop a concrete with proper performance and that can meet application requirements; among such requirements we can mention the homogeneity of the applied concrete, which directly affects the effective linear attenuation rate of the component and can assure a good performance in face of demands. In this work, our aim was to evaluate what influence the particle size of the aggregates used for producing of assayed concretes has on the variation of the linear attenuation coefficient at different points of the same sample, results which can be used to obtain the inhomogeneity rate of each case of the analyzed concretes. The concrete samples were prepared with small thicknesses, ten millimeters (10 mm) and to perform the assays, a source of Americium-241 was used to transmit gamma-rays in order to determine the variation that existed in the linear attenuation coefficient of each sample. The following

  9. Quasi-stationary states of an electron with linearly dependent effective mass in an open nanostructure within transmission coefficient and S-matrix methods

    Seti, Julia; Tkach, Mykola; Voitsekhivska, Oxana

    2018-03-01

    The exact solutions of the Schrödinger equation for a double-barrier open semiconductor plane nanostructure are obtained by using two different approaches, within the model of the rectangular potential profile and the continuous position-dependent effective mass of the electron. The transmission coefficient and scattering matrix are calculated for the double-barrier nanostructure. The resonance energies and resonance widths of the electron quasi-stationary states are analyzed as a function of the size of the near-interface region between wells and barriers, where the effective mass linearly depends on the coordinate. It is established that, in both methods, the increasing size affects in a qualitatively similar way the spectral characteristics of the states, shifting the resonance energies into the low- or high-energy region and increasing the resonance widths. It is shown that the relative difference of resonance energies and widths of a certain state, obtained in the model of position-dependent effective mass and in the widespread abrupt model in physically correct range of near-interface sizes, does not exceed 0.5% and 5%, respectively, independently of the other geometrical characteristics of the structure.

  10. Determination of the linear coefficient of thermal expansion in polymer films at the nanoscale: influence of the composition of EVA copolymers and the molecular weight of PMMA.

    González-Benito, J; Castillo, E; Cruz-Caldito, J F

    2015-07-28

    Nanothermal-expansion of poly(ethylene-co-vinylacetate), EVA, and poly(methyl methacrylate), PMMA, in the form of films was measured to finally obtain linear coefficients of thermal expansion, CTEs. The simple deflection of a cantilever in an atomic force microscope, AFM, was used to monitor thermal expansions at the nanoscale. The influences of: (a) the structure of EVA in terms of its composition (vinylacetate content) and (b) the size of PMMA chains in terms of the molecular weight were studied. To carry out this, several polymer samples were used, EVA copolymers with different weight percents of the vinylacetate comonomer (12, 18, 25 and 40%) and PMMA polymers with different weight average molecular weights (33.9, 64.8, 75.600 and 360.0 kg mol(-1)). The dependencies of the vinyl acetate weight fraction of EVA and the molecular weight of PMMA on their corresponding CTEs were analyzed to finally explain them using new, intuitive and very simple models based on the rule of mixtures. In the case of EVA copolymers a simple equation considering the weighted contributions of each comonomer was enough to estimate the final CTE above the glass transition temperature. On the other hand, when the molecular weight dependence is considered the free volume concept was used as novelty. The expansion of PMMA, at least at the nanoscale, was well and easily described by the sum of the weighted contributions of the occupied and free volumes, respectively.

  11. Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully non-linear Coulomb collisions

    Hager, Robert, E-mail: rhager@pppl.gov; Chang, C. S., E-mail: cschang@pppl.gov [Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, New Jersey 08543 (United States)

    2016-04-15

    As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.

  12. Fokker-Planck equation associated with the Wigner function of a quantum system with a finite number of states

    Cohendet, O.

    1989-01-01

    We consider a quantum system with a finite number N of states and we show that a Markov process evolving in an 'extended' discrete phase can be associated with the discrete Wigner function of the system. This Wigner function is built using the Weyl quantization procedure on the group Z N xZ N . Moreover we can use this process to compute the quantum mean values as probabilistic expectations of functions of this process. This probabilistic formulation can be seen as a stochastic mechanics in phase space. (orig.)

  13. A quasilinear, Fokker--Planck description of fast wave minority heating permitting off-axis tangency interactions

    Catto, P.J.; Myra, J.R.; Russell, D.A.

    1994-01-01

    The off-axis quasilinear fast wave minority heating description of Catto and Myra [Phys. Fluids B 4, 187 (1992)] has been improved and implemented in a code which solves the combined quasilinear and collision operator equation for the minority distribution function. Geometrical complications of a minority resonance nearly tangent to a flux surface in the presence of trapped as well as passing particles are retained. The tangency interactions alter the moments and the fusion reaction rate parameter in a model which explores heating on a single flux surface. The strong tangency interactions enhance the more familiar interactions due to trapped particles turning in the vicinity of the minority resonance. An asymmetry in off-axis heating effects occurs because heating on the low field side of the magnetic axis heats more trapped particles than high field side heating. This asymmetry is responsible for the better performance of the low field side case relative to the high and on-axis cases and provides some control over the power absorbed by and the energy stored in the trapped particles

  14. Multigroup discrete ordinates solution of Boltzmann-Fokker-Planck equations and cross section library development of ion transport

    Prinja, A.K.

    1995-08-01

    We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S N angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes

  15. Lagrangian modelling of plankton motion: From deceptively simple random walks to Fokker-Planck and back again

    Visser, Andre

    2008-01-01

    The movement of plankton, either by turbulent mixing or their own inherent motility, can be simulated in a Lagrangian framework as a random walk. Validation of random walk simulations is essential. There is a continuum of mathematically valid stochastic integration schemes upon which random walk...

  16. Noise-induced modulation of the relaxation kinetics around a non-equilibrium steady state of non-linear chemical reaction networks.

    Rajesh Ramaswamy

    2011-01-01

    Full Text Available Stochastic effects from correlated noise non-trivially modulate the kinetics of non-linear chemical reaction networks. This is especially important in systems where reactions are confined to small volumes and reactants are delivered in bursts. We characterise how the two noise sources confinement and burst modulate the relaxation kinetics of a non-linear reaction network around a non-equilibrium steady state. We find that the lifetimes of species change with burst input and confinement. Confinement increases the lifetimes of all species that are involved in any non-linear reaction as a reactant. Burst monotonically increases or decreases lifetimes. Competition between burst-induced and confinement-induced modulation may hence lead to a non-monotonic modulation. We quantify lifetime as the integral of the time autocorrelation function (ACF of concentration fluctuations around a non-equilibrium steady state of the reaction network. Furthermore, we look at the first and second derivatives of the ACF, each of which is affected in opposite ways by burst and confinement. This allows discriminating between these two noise sources. We analytically derive the ACF from the linear Fokker-Planck approximation of the chemical master equation in order to establish a baseline for the burst-induced modulation at low confinement. Effects of higher confinement are then studied using a partial-propensity stochastic simulation algorithm. The results presented here may help understand the mechanisms that deviate stochastic kinetics from its deterministic counterpart. In addition, they may be instrumental when using fluorescence-lifetime imaging microscopy (FLIM or fluorescence-correlation spectroscopy (FCS to measure confinement and burst in systems with known reaction rates, or, alternatively, to correct for the effects of confinement and burst when experimentally measuring reaction rates.

  17. Noise-induced modulation of the relaxation kinetics around a non-equilibrium steady state of non-linear chemical reaction networks.

    Ramaswamy, Rajesh; Sbalzarini, Ivo F; González-Segredo, Nélido

    2011-01-28

    Stochastic effects from correlated noise non-trivially modulate the kinetics of non-linear chemical reaction networks. This is especially important in systems where reactions are confined to small volumes and reactants are delivered in bursts. We characterise how the two noise sources confinement and burst modulate the relaxation kinetics of a non-linear reaction network around a non-equilibrium steady state. We find that the lifetimes of species change with burst input and confinement. Confinement increases the lifetimes of all species that are involved in any non-linear reaction as a reactant. Burst monotonically increases or decreases lifetimes. Competition between burst-induced and confinement-induced modulation may hence lead to a non-monotonic modulation. We quantify lifetime as the integral of the time autocorrelation function (ACF) of concentration fluctuations around a non-equilibrium steady state of the reaction network. Furthermore, we look at the first and second derivatives of the ACF, each of which is affected in opposite ways by burst and confinement. This allows discriminating between these two noise sources. We analytically derive the ACF from the linear Fokker-Planck approximation of the chemical master equation in order to establish a baseline for the burst-induced modulation at low confinement. Effects of higher confinement are then studied using a partial-propensity stochastic simulation algorithm. The results presented here may help understand the mechanisms that deviate stochastic kinetics from its deterministic counterpart. In addition, they may be instrumental when using fluorescence-lifetime imaging microscopy (FLIM) or fluorescence-correlation spectroscopy (FCS) to measure confinement and burst in systems with known reaction rates, or, alternatively, to correct for the effects of confinement and burst when experimentally measuring reaction rates.

  18. Resonance zones and quasi-linear diffusion coefficients for radiation belt energetic electron interaction with oblique chorus waves in the Dungey magnetosphere

    Shi Run; Ni, Binbin; Gu Xudong; Zhao Zhengyu; Zhou Chen

    2012-01-01

    The resonance regions for resonant interactions of radiation belt electrons with obliquely propagating whistler-mode chorus waves are investigated in detail in the Dungey magnetic fields that are parameterized by the intensity of uniform southward interplanetary magnetic field (IMF) Bz or, equivalently, by the values of D=(M/B z,0 ) 1/3 (where M is the magnetic moment of the dipole and B z,0 is the uniform southward IMF normal to the dipole’s equatorial plane). Adoption of background magnetic field model can considerably modify the determination of resonance regions. Compared to the results for the case of D = 50 (very close to the dipole field), the latitudinal coverage of resonance regions for 200 keV electrons interacting with chorus waves tends to become narrower for smaller D-values, regardless of equatorial pitch angle, resonance harmonics, and wave normal angle. In contrast, resonance regions for 1 MeV electrons tend to have very similar spatial lengths along the field line for various Dungey magnetic field models but cover different magnetic field intervals, indicative of a strong dependence on electron energy. For any given magnetic field line, the resonance regions where chorus-electron resonant interactions can take place rely closely on equatorial pitch angle, resonance harmonics, and kinetic energy. The resonance regions tend to cover broader latitudinal ranges for smaller equatorial pitch angles, higher resonance harmonics, and lower electron energies, consistent with the results in Ni and Summers [Phys. Plasmas 17, 042902, 042903 (2010)]. Calculations of quasi-linear bounce-averaged diffusion coefficients for radiation belt electrons due to nightside chorus waves indicate that the resultant scattering rates differ from using different Dungey magnetic field models, demonstrating a strong dependence of wave-induced electron scattering effect on the adoption of magnetic field model. Our results suggest that resonant wave-particle interaction processes

  19. Impurities in a non-axisymmetric plasma: Transport and effect on bootstrap current

    Mollén, A., E-mail: albertm@chalmers.se [Department of Applied Physics, Chalmers University of Technology, Göteborg (Sweden); Max-Planck-Institut für Plasmaphysik, 17491 Greifswald (Germany); Landreman, M. [Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742 (United States); Smith, H. M.; Helander, P. [Max-Planck-Institut für Plasmaphysik, 17491 Greifswald (Germany); Braun, S. [Max-Planck-Institut für Plasmaphysik, 17491 Greifswald (Germany); German Aerospace Center, Institute of Engineering Thermodynamics, Pfaffenwaldring 38-40, D-70569 Stuttgart (Germany)

    2015-11-15

    Impurities cause radiation losses and plasma dilution, and in stellarator plasmas the neoclassical ambipolar radial electric field is often unfavorable for avoiding strong impurity peaking. In this work we use a new continuum drift-kinetic solver, the SFINCS code (the Stellarator Fokker-Planck Iterative Neoclassical Conservative Solver) [M. Landreman et al., Phys. Plasmas 21, 042503 (2014)] which employs the full linearized Fokker-Planck-Landau operator, to calculate neoclassical impurity transport coefficients for a Wendelstein 7-X (W7-X) magnetic configuration. We compare SFINCS calculations with theoretical asymptotes in the high collisionality limit. We observe and explain a 1/ν-scaling of the inter-species radial transport coefficient at low collisionality, arising due to the field term in the inter-species collision operator, and which is not found with simplified collision models even when momentum correction is applied. However, this type of scaling disappears if a radial electric field is present. We also use SFINCS to analyze how the impurity content affects the neoclassical impurity dynamics and the bootstrap current. We show that a change in plasma effective charge Z{sub eff} of order unity can affect the bootstrap current enough to cause a deviation in the divertor strike point locations.

  20. Time-dependent transport of energetic particles in magnetic turbulence: computer simulations versus analytical theory

    Arendt, V.; Shalchi, A.

    2018-06-01

    We explore numerically the transport of energetic particles in a turbulent magnetic field configuration. A test-particle code is employed to compute running diffusion coefficients as well as particle distribution functions in the different directions of space. Our numerical findings are compared with models commonly used in diffusion theory such as Gaussian distribution functions and solutions of the cosmic ray Fokker-Planck equation. Furthermore, we compare the running diffusion coefficients across the mean magnetic field with solutions obtained from the time-dependent version of the unified non-linear transport theory. In most cases we find that particle distribution functions are indeed of Gaussian form as long as a two-component turbulence model is employed. For turbulence setups with reduced dimensionality, however, the Gaussian distribution can no longer be obtained. It is also shown that the unified non-linear transport theory agrees with simulated perpendicular diffusion coefficients as long as the pure two-dimensional model is excluded.

  1. Linear and nonlinear optical absorption coefficients in GaAs/Ga{sub 1-x}Al{sub x}As concentric double quantum rings: Effects of hydrostatic pressure and aluminum concentration

    Baghramyan, H.M. [Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan (Armenia); Barseghyan, M.G., E-mail: mbarsegh@ysu.am [Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan (Armenia); Kirakosyan, A.A. [Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan (Armenia); Restrepo, R.L. [Escuela de Ingenieria de Antioquia, AA 7516 Medellin (Colombia); Duque, C.A. [Instituto de Fisica, Universidad de Antioquia, AA 1226 Medellin (Colombia)

    2013-02-15

    The linear and nonlinear intra-band optical absorption coefficients in GaAs/Ga{sub 1-x}Al{sub x}As two-dimensional concentric double quantum rings are investigated. Taking into account the combined effects of hydrostatic pressure and aluminum concentration the energies of the ground (n=1,l=0) and the first excited state (n=2,l=1) have been found using the effective mass approximation and the transfer matrix formalism. The energies of these states and the corresponding threshold energy of the intra-band optical transitions are examined as a function of hydrostatic pressure and aluminum concentration for different sizes of the structure. We also investigated the dependencies of the linear, nonlinear, and total optical absorption coefficients as functions of the incident photon energy for different values of hydrostatic pressure, aluminum concentration, sizes of the structure, and incident optical intensity. Its is found that the effects of the hydrostatic pressure and the aluminum concentration lead to a shifting of the resonant peaks of the intra-band optical spectrum. - Highlights: Black-Right-Pointing-Pointer Linear and nonlinear intra-band absorption in quantum rings. Black-Right-Pointing-Pointer Threshold energy strongly depends on the hydrostatic pressure. Black-Right-Pointing-Pointer Threshold energy strongly depends on the stoichiometry and sizes of structure. Black-Right-Pointing-Pointer Optical absorption is affected by the incident optical intensity.

  2. Development of a predictive model for distribution coefficient (Kd) of 13'7Cs and 60Co in marine sediments using multiple linear regression analysis

    Kumar, Ajay; Ravi, P.M.; Guneshwar, S.L.; Rout, Sabyasachi; Mishra, Manish K.; Pulhani, Vandana; Tripathi, R.M.

    2018-01-01

    Numerous common methods (batch laboratory, the column laboratory, field-batch method, field modeling and K 0c method) are used frequently for determination of K d values. Recently, multiple regression models are considered as new best estimates for predicting the K d of radionuclides in the environment. It is also well known fact that the K d value is highly influenced by physico-chemical properties of sediment. Due to the significant variability in influencing parameters, the measured K d values can range over several orders of magnitude under different environmental conditions. The aim of this study is to develop a predictive model for K d values of 137 Cs and 60 Co based on the sediment properties using multiple linear regression analysis

  3. Determination of Spearman Correlation Coefficient (r to Evaluate the Linear Association of Dermal Collagen and Elastic Fibers in the Perspectives of Skin Injury

    Naveen Kumar

    2018-01-01

    Full Text Available Background. Difference in scar formation at different sites, in different directions at the same site, but with changes in the elasticity of skin with age, sex, and race or in some pathological conditions, is well known to clinicians. The inappropriate collagen syntheses and delayed or lack of epithelialization are known to induce scar formation with negligible elasticity at the site of damage. Changes in the elasticity of scars may be due to an unequal distribution of dermal collagen (C and elastic (E fibers. Materials and Methods. Spearman correlation coefficients (r of collagen and elastic fibers in horizontal (H and in vertical (V directions (variables CV, CH, EV, and EH were measured from the respective quantitative fraction data in 320 skin samples from 32 human cadavers collected at five selected sites over extremities. Results. Spearman’s correlation analysis revealed the statistically significant (p<0.01 strong positive correlation between CH and CV in all the areas, that is, shoulder joint area (r=0.66, wrist (r=0.75, forearm (r=0.75, and thigh (r=0.80, except at the ankle (r=0.26, p=0.14 region. Similarly, positive correlation between EH and EV has been observed at the forearm (r=0.65, moderate and thigh (r=0.42, low regions. However, a significant moderate negative correlation was observed between CV and EV at the forearm (r=-0.51 and between CH and EH at the thigh region (r=-0.65. Conclusion. Significant differences of correlations of collagen and elastic fibers in different directions from different areas of extremities were noted. This may be one of the possible anatomical reasons of scar behavior in different areas and different directions of the same area.

  4. Determination of Spearman Correlation Coefficient (r) to Evaluate the Linear Association of Dermal Collagen and Elastic Fibers in the Perspectives of Skin Injury.

    Kumar, Naveen; Kumar, Pramod; Badagabettu, Satheesha Nayak; Lewis, Melissa Glenda; Adiga, Murali; Padur, Ashwini Aithal

    2018-01-01

    Difference in scar formation at different sites, in different directions at the same site, but with changes in the elasticity of skin with age, sex, and race or in some pathological conditions, is well known to clinicians. The inappropriate collagen syntheses and delayed or lack of epithelialization are known to induce scar formation with negligible elasticity at the site of damage. Changes in the elasticity of scars may be due to an unequal distribution of dermal collagen (C) and elastic (E) fibers. Spearman correlation coefficients ( r ) of collagen and elastic fibers in horizontal (H) and in vertical (V) directions (variables CV, CH, EV, and EH) were measured from the respective quantitative fraction data in 320 skin samples from 32 human cadavers collected at five selected sites over extremities. Spearman's correlation analysis revealed the statistically significant ( p < 0.01) strong positive correlation between C H and C V in all the areas, that is, shoulder joint area ( r = 0.66), wrist ( r = 0.75), forearm ( r = 0.75), and thigh ( r = 0.80), except at the ankle ( r = 0.26, p = 0.14) region. Similarly, positive correlation between E H and E V has been observed at the forearm ( r = 0.65, moderate) and thigh ( r = 0.42, low) regions. However, a significant moderate negative correlation was observed between C V and E V at the forearm ( r = -0.51) and between C H and E H at the thigh region ( r = -0.65). Significant differences of correlations of collagen and elastic fibers in different directions from different areas of extremities were noted. This may be one of the possible anatomical reasons of scar behavior in different areas and different directions of the same area.

  5. Comparison of inter-diffusion coefficients for Ni/Cu thin films determined from classical heating analysis and linear temperature ramping analysis by means of profile reconstruction and a numerical solution of Fick's law

    Joubert, H.D.; Terblans, J.J.; Swart, H.C.

    2009-01-01

    Classical inter-diffusion studies assume a constant time of annealing when samples are annealed in a furnace. It is assumed that the sample temperature reaches the annealing temperature immediately after insertion, while the sample temperature immediately drops to room temperature after removal, the annealing time being taken as the time between insertion and removal. Using the above assumption, the diffusion coefficient can be calculated in a number of ways. In reality, the sample temperature does not immediately reach the annealing temperature; instead it rises at a rate governed by several heat transfer mechanisms, depending on the annealing procedure. For short annealing times, the sample temperature may not attain the annealing temperature, while for extended annealing times the sample temperature may reach the annealing temperature only for a fraction of the annealing time. To eliminate the effect of heat transfer mechanisms, a linear temperature ramping regime is proposed. Used in conjunction with a suitable profile reconstructing technique and a numerical solution of Fick's second law, the inter-diffusion parameters obtained from a linear ramping of Ni/Cu thin film samples can be compared to those obtained from calculations performed with the so-called Mixing-Roughness-Information model or any other suitable method used to determine classical diffusion coefficients.

  6. Development of a predictive model for lead, cadmium and fluorine soil-water partition coefficients using sparse multiple linear regression analysis.

    Nakamura, Kengo; Yasutaka, Tetsuo; Kuwatani, Tatsu; Komai, Takeshi

    2017-11-01

    In this study, we applied sparse multiple linear regression (SMLR) analysis to clarify the relationships between soil properties and adsorption characteristics for a range of soils across Japan and identify easily-obtained physical and chemical soil properties that could be used to predict K and n values of cadmium, lead and fluorine. A model was first constructed that can easily predict the K and n values from nine soil parameters (pH, cation exchange capacity, specific surface area, total carbon, soil organic matter from loss on ignition and water holding capacity, the ratio of sand, silt and clay). The K and n values of cadmium, lead and fluorine of 17 soil samples were used to verify the SMLR models by the root mean square error values obtained from 512 combinations of soil parameters. The SMLR analysis indicated that fluorine adsorption to soil may be associated with organic matter, whereas cadmium or lead adsorption to soil is more likely to be influenced by soil pH, IL. We found that an accurate K value can be predicted from more than three soil parameters for most soils. Approximately 65% of the predicted values were between 33 and 300% of their measured values for the K value; 76% of the predicted values were within ±30% of their measured values for the n value. Our findings suggest that adsorption properties of lead, cadmium and fluorine to soil can be predicted from the soil physical and chemical properties using the presented models. Copyright © 2017 Elsevier Ltd. All rights reserved.

  7. Computational-physics program of the National MFE Computer Center

    Mirin, A.A.

    1982-02-01

    The computational physics group is ivolved in several areas of fusion research. One main area is the application of multidimensional Fokker-Planck, transport and combined Fokker-Planck/transport codes to both toroidal and mirror devices. Another major area is the investigation of linear and nonlinear resistive magnetohydrodynamics in two and three dimensions, with applications to all types of fusion devices. The MHD work is often coupled with the task of numerically generating equilibria which model experimental devices. In addition to these computational physics studies, investigations of more efficient numerical algorithms are being carried out

  8. The application of rational approximation in the calculation of a temperature field with a non-linear surface heat-transfer coefficient during quenching for 42CrMo steel cylinder

    Cheng, Heming; Huang, Xieqing; Fan, Jiang; Wang, Honggang

    1999-10-01

    The calculation of a temperature field has a great influence upon the analysis of thermal stresses and stains during quenching. In this paper, a 42CrMo steel cylinder was used an example for investigation. From the TTT diagram of the 42CrMo steel, the CCT diagram was simulated by mathematical transformation, and the volume fraction of phase constituents was calculated. The thermal physical properties were treated as functions of temperature and the volume fraction of phase constituents. The rational approximation was applied to the finite element method. The temperature field with phase transformation and non-linear surface heat-transfer coefficients was calculated using this technique, which can effectively avoid oscillationin the numerical solution for a small time step. The experimental results of the temperature field calculation coincide with the numerical solutions.

  9. Linear and nonlinear optical absorption coefficients and refractive index changes in GaN/Al{sub x}Ga{sub (1−x)}N double quantum wells operating at 1.55 μm

    Dakhlaoui, Hassen [Department of Physics, College of Science for Girls, University of Dammam (UOD), Saudi Arabia and Department of Physics, Faculty of Sciences of Bizerte, University of Carthage (Tunisia)

    2015-04-07

    In the present paper, the linear and nonlinear optical absorption coefficients and refractive index changes between the ground and the first excited states in double GaN/Al{sub x}Ga{sub (1−x)}N quantum wells are studied theoretically. The electronic energy levels and their corresponding wave functions are obtained by solving Schrödinger-Poisson equations self-consistently within the effective mass approximation. The obtained results show that the optical absorption coefficients and refractive index changes can be red- and blue-shifted through varying the left quantum well width and the aluminum concentration x{sub b2} of the central barrier, respectively. These structural parameters are found to present optimum values for carrying out the transition of 0.8 eV (1.55 μm). Furthermore, we show that the desired transition can also be achieved by replacing the GaN in the left quantum well with Al{sub y}Ga{sub (1−y)}N and by varying the aluminum concentration y{sub Al}. The obtained results give a new degree of freedom in optoelectronic device applications such as optical fiber telecommunications operating at (1.55 μm)

  10. Transport theory of deep-inelastic collisions between heavy nuclei

    Ayik, S.; Noerenberg, W.; Schuermann, B.

    1975-01-01

    In collisions between heavy nuclei, the major part of the total cross-section is due to deep-inelastic processes. These processes have been studied within a quantum-statistical approach leading to transport equations of the Fokker-Planck type (generalized diffusion equation). Transport coefficients have been studied within a model. (orig./WL) [de

  11. Examination of various postulates of irreversibility

    Salmon, J [Conservatoire National des Arts et Metiers (CNAM), 75 - Paris (France)

    1977-01-01

    Firstly, it is shown that it is necessary to break the reversible character of the B.B.G.K.Y. system of equations by means of a postulate of irreversibility to obtain a kinetic equation compatible with the second principle of thermodynamics. Next, three postulates of irreversibility are examined: that of molecular chaos, that of linear relaxation and, finally, that of superposition. Then the corresponding kinetic equations and the expressions for the viscosity coefficient to which they lead are determined. Comparison with experiment is made each time. Lastly, an attempt to obtain an irreversible kinetic equation without introducing a postulate of irreversibility in the B.B.G.K.Y. system is realized. This consists in adding a complementary irreversible term to the fundamental equation of the dynamics of a particle. The suggested term is of quantum origin and leads to a kinetic equation of the Fokker-Planck type.

  12. Two-dimensional spectroscopy for harmonic vibrational modes with nonlinear system-bath interactions. I. Gaussian-white case

    Steffen, T; Tanimura, Y

    The quantum Fokker-Planck equation is derived for a system nonlinearly coupled to a harmonic oscillator bath. The system-bath interaction is assumed to be linear in the bath coordinates but quadratic in the system coordinate. The relaxation induced dynamics of a harmonic system are investigated by

  13. A kind of iteration algorithm for fast wave heating

    Zhu Xueguang; Kuang Guangli; Zhao Yanping; Li Youyi; Xie Jikang

    1998-03-01

    The standard normal distribution for particles in Tokamak geometry is usually assumed in fast wave heating. In fact, due to the quasi-linear diffusion effect, the parallel and vertical temperature of resonant particles is not equal, so, this will bring some error. For this case, the Fokker-Planck equation is introduced, and iteration algorithm is adopted to solve the problem well

  14. Computational physics program of the National MFE Computer Center

    Mirin, A.A.

    1985-12-01

    The development of numerical models for plasma phenomena and magnetic confinement devices is discussed. The multidimensional Fokker-Planck and transport codes are applied to toroidal mirror and compact toroid devices. Linear and nonlinear resistive magnetohydrodynamics in two and three dimensions are used in the investigation of various fusion devices. 362 refs., 4 tabs

  15. Donor impurity-related linear and nonlinear optical absorption coefficients in GaAs/Ga{sub 1−x}Al{sub x}As concentric double quantum rings: Effects of geometry, hydrostatic pressure, and aluminum concentration

    Baghramyan, H.M.; Barseghyan, M.G.; Kirakosyan, A.A. [Department of Solid State Physics, Yerevan State University, Al. Manookian 1, 0025 Yerevan (Armenia); Restrepo, R.L. [Física Teórica y Aplicada, Escuela de Ingeniería de Antioquia, AA 7516, Medellín (Colombia); Grupo de Materia Condensada-UdeA, Instituto de Física, Facultadde Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21,Medellín (Colombia); Mora-Ramos, M.E. [Grupo de Materia Condensada-UdeA, Instituto de Física, Facultadde Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21,Medellín (Colombia); Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, CP 62209, Cuernavaca, Morelos (Mexico); Duque, C.A., E-mail: cduque@fisica.udea.edu.co [Grupo de Materia Condensada-UdeA, Instituto de Física, Facultadde Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21,Medellín (Colombia)

    2014-01-15

    The linear and nonlinear optical absorption associated with the transition between 1s and 2s states corresponding to the electron-donor-impurity complex in GaAs/Ga{sub 1−x}Al{sub x}As three-dimensional concentric double quantum rings are investigated. Taking into account the combined effects of hydrostatic pressure and the variation of the aluminum concentration, the energies of the ground and first excited s-like states of a donor impurity in such a system have been calculated using the effective mass approximation and a variational method. The energies of these states and the corresponding threshold energy of the optical transitions are examined as functions of hydrostatic pressure, aluminum concentration, radial impurity position, as well as the geometrical dimensions of the structure. The dependencies of the linear, nonlinear and total optical absorption coefficients as functions of the incident photon energy are investigated for different values of those mentioned parameters. It is found that the influences mentioned above lead to either redshifts or blueshifts of the resonant peaks of the optical absorption spectrum. It is particularly discussed the unusual property exhibited by the third-order nonlinear of becoming positive for photon energies below the resonant transition one. It is shown that this phenomenon is associated with the particular features of the system under study, which determine the values of the electric dipole moment matrix elements. -- Highlights: • Intra-band optical absorption associated to impurity states in double quantum rings. • Combined effects of hydrostatic pressure and aluminum concentration are studied. • The influences mentioned above lead to shifts of resonant peaks. • It is discussed an unusual property exhibited by the third-order nonlinear absorption.

  16. Donor impurity-related linear and nonlinear optical absorption coefficients in GaAs/Ga1−xAlxAs concentric double quantum rings: Effects of geometry, hydrostatic pressure, and aluminum concentration

    Baghramyan, H.M.; Barseghyan, M.G.; Kirakosyan, A.A.; Restrepo, R.L.; Mora-Ramos, M.E.; Duque, C.A.

    2014-01-01

    The linear and nonlinear optical absorption associated with the transition between 1s and 2s states corresponding to the electron-donor-impurity complex in GaAs/Ga 1−x Al x As three-dimensional concentric double quantum rings are investigated. Taking into account the combined effects of hydrostatic pressure and the variation of the aluminum concentration, the energies of the ground and first excited s-like states of a donor impurity in such a system have been calculated using the effective mass approximation and a variational method. The energies of these states and the corresponding threshold energy of the optical transitions are examined as functions of hydrostatic pressure, aluminum concentration, radial impurity position, as well as the geometrical dimensions of the structure. The dependencies of the linear, nonlinear and total optical absorption coefficients as functions of the incident photon energy are investigated for different values of those mentioned parameters. It is found that the influences mentioned above lead to either redshifts or blueshifts of the resonant peaks of the optical absorption spectrum. It is particularly discussed the unusual property exhibited by the third-order nonlinear of becoming positive for photon energies below the resonant transition one. It is shown that this phenomenon is associated with the particular features of the system under study, which determine the values of the electric dipole moment matrix elements. -- Highlights: • Intra-band optical absorption associated to impurity states in double quantum rings. • Combined effects of hydrostatic pressure and aluminum concentration are studied. • The influences mentioned above lead to shifts of resonant peaks. • It is discussed an unusual property exhibited by the third-order nonlinear absorption

  17. A Fokker-Planck approach to canonical-dissipative Nambu systems: With an application to human motor control during dynamic haptic perception

    Frank, T.D.

    2010-01-01

    We formulate Markov diffusion processes for canonical-dissipative systems exhibiting Nambu mechanics. Analytical expressions for stationary canonical-dissipative distributions are obtained. Nambu-Boltzmann distributions are derived as special cases for systems without energy pumping. The Markov short-time propagator is used to derive maximum likelihood estimators for parameters of a model that describes a particular dynamic motor pattern providing haptic cues.

  18. Attenuation coefficients of soils

    Martini, E.; Naziry, M.J.

    1989-01-01

    As a prerequisite to the interpretation of gamma-spectrometric in situ measurements of activity concentrations of soil radionuclides the attenuation of 60 to 1332 keV gamma radiation by soil samples varying in water content and density has been investigated. A useful empirical equation could be set up to describe the dependence of the mass attenuation coefficient upon photon energy for soil with a mean water content of 10%, with the results comparing well with data in the literature. The mean density of soil in the GDR was estimated at 1.6 g/cm 3 . This value was used to derive the linear attenuation coefficients, their range of variation being 10%. 7 figs., 5 tabs. (author)

  19. Standards for Standardized Logistic Regression Coefficients

    Menard, Scott

    2011-01-01

    Standardized coefficients in logistic regression analysis have the same utility as standardized coefficients in linear regression analysis. Although there has been no consensus on the best way to construct standardized logistic regression coefficients, there is now sufficient evidence to suggest a single best approach to the construction of a…

  20. Hamiltonian theory of the ion cyclotron minority heating dynamics in tokamak plasmas

    Becoulet, A.; Gambier, D.J.; Samain, A.

    1990-03-01

    The question of heating a tokamak plasma by means of electromagnetic waves in the Ion Cyclotron Range of Frequency (ICRF) is considered in the perspective of large RF powers and in the low collisionality regime. In such case the Quasi Linear Theory (QLT) is validated by the Hamiltonian dynamics of the wave particle interaction which exceeds the threshold of the intrinsic stochasticity. The Hamiltonian dynamics is represented by the evolution of a set of three canonical action angle variables well adapted to the tokamak magnetic configuration. This approach allows to derive the RF diffusion coefficient with very few assumptions. The distribution function of the resonant ions is written as a Fokker Planck equation but the emphasis is put on the QL diffusion instead of on the usual diffusion induced by collisions. Then the Fokker Planck equation is given a variational from which a solution is derived in the form of a semi analytical trial function of three parameters: the percentage of resonant particle contained in the tail; an isotropic width ΔT and an anisotropic one ΔP. This solution is successfully tested against real experimental observations. Practically it is shown that in the case of JET the distribution function is influenced by adiabatic barriers which in turn limit the Hamiltonian stochasticity domain within energy values typically in the MeV range. Consequently and for a given ICRF power, the tail energy excursion is lower and its concentration higher than that of a bounce averaged prediction. This may actually be an advantage for machines like JET considering the energy range required to simulate the α-particle behaviour in a relevant fusion reactor

  1. Kendall-Theil Robust Line (KTRLine--version 1.0)-A Visual Basic Program for Calculating and Graphing Robust Nonparametric Estimates of Linear-Regression Coefficients Between Two Continuous Variables

    Granato, Gregory E.

    2006-01-01

    The Kendall-Theil Robust Line software (KTRLine-version 1.0) is a Visual Basic program that may be used with the Microsoft Windows operating system to calculate parameters for robust, nonparametric estimates of linear-regression coefficients between two continuous variables. The KTRLine software was developed by the U.S. Geological Survey, in cooperation with the Federal Highway Administration, for use in stochastic data modeling with local, regional, and national hydrologic data sets to develop planning-level estimates of potential effects of highway runoff on the quality of receiving waters. The Kendall-Theil robust line was selected because this robust nonparametric method is resistant to the effects of outliers and nonnormality in residuals that commonly characterize hydrologic data sets. The slope of the line is calculated as the median of all possible pairwise slopes between points. The intercept is calculated so that the line will run through the median of input data. A single-line model or a multisegment model may be specified. The program was developed to provide regression equations with an error component for stochastic data generation because nonparametric multisegment regression tools are not available with the software that is commonly used to develop regression models. The Kendall-Theil robust line is a median line and, therefore, may underestimate total mass, volume, or loads unless the error component or a bias correction factor is incorporated into the estimate. Regression statistics such as the median error, the median absolute deviation, the prediction error sum of squares, the root mean square error, the confidence interval for the slope, and the bias correction factor for median estimates are calculated by use of nonparametric methods. These statistics, however, may be used to formulate estimates of mass, volume, or total loads. The program is used to read a two- or three-column tab-delimited input file with variable names in the first row and

  2. Recursive Algorithm For Linear Regression

    Varanasi, S. V.

    1988-01-01

    Order of model determined easily. Linear-regression algorithhm includes recursive equations for coefficients of model of increased order. Algorithm eliminates duplicative calculations, facilitates search for minimum order of linear-regression model fitting set of data satisfactory.

  3. Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature Variations

    Kay Brandner

    2015-08-01

    Full Text Available We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This setup covers, in particular, periodic micro- and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck-type dynamics, we show that these coefficients can be expressed as a sum of an adiabatic contribution and one reminiscent of a Green-Kubo expression that contains deviations from adiabaticity. Furthermore, we show that the generalized kinetic coefficients fulfill an Onsager-Casimir-type symmetry tracing back to microscopic reversibility. This symmetry allows for nonidentical off-diagonal coefficients if the driving protocols are not symmetric under time reversal. We then derive a novel constraint on the kinetic coefficients that is sharper than the second law and provides an efficiency-dependent bound on power. As one consequence, we can prove that the power vanishes at least linearly when approaching Carnot efficiency. We illustrate our general framework by explicitly working out the paradigmatic case of a Brownian heat engine realized by a colloidal particle in a time-dependent harmonic trap subject to a periodic temperature profile. This case study reveals inter alia that our new general bound on power is asymptotically tight.

  4. Investigation of photon attenuation coefficients for marble

    Basyigit, C; Akkurt, I; Kilincarslan, S; Akkurt, A

    2005-01-01

    The total linear attenuation coefficients μ (cm -1 ) have been obtained using the XCOM program at photon energies of 1 keV to 1 GeV for six different natural marbles produced in different places in Turkey. The individual contribution of photon interaction processes to the total linear attenuation coefficients for marble has been investigated. The calculated results were also compared with the measurements. The results obtained for marble were also compared with concrete. (note)

  5. Effect of intense high-frequency laser field on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a parabolic quantum well under the applied electric field

    Yesilgul, U., E-mail: uyesilgul@cumhuriyet.edu.tr [Cumhuriyet University, Physics Department, 58140 Sivas (Turkey); Ungan, F. [Cumhuriyet University, Physics Department, 58140 Sivas (Turkey); Sakiroglu, S. [Dokuz Eylül University, Physics Department, 35160 Buca, İzmir (Turkey); Mora-Ramos, M.E. [Facultad de Ciencias Universidad Autonoma del Estado de Morelos, Ave. Universidad 1001, C.P. 62209 Cuernavaca, Morelos (Mexico); Duque, C.A. [Grupo de Materia Condensada-UdeA, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medellín (Colombia); Kasapoglu, E.; Sarı, H. [Cumhuriyet University, Physics Department, 58140 Sivas (Turkey); Sökmen, I. [Dokuz Eylül University, Physics Department, 35160 Buca, İzmir (Turkey)

    2014-01-15

    The effects of the intense high-frequency laser field on the optical absorption coefficients and the refractive index changes in a GaAs/GaAlAs parabolic quantum well under the applied electric field have been investigated theoretically. The electron energy levels and the envelope wave functions of the parabolic quantum well are calculated within the effective mass approximation. Analytical expressions for optical properties are obtained using the compact density-matrix approach. The numerical results show that the intense high-frequency laser field has a large effect on the optical characteristics of these structures. Also we can observe that the refractive index and absorption coefficient changes are very sensitive to the electric field in large dimension wells. Thus, this result gives a new degree of freedom in the optoelectronic device applications. -- Highlights: • ILF has a large effect on the optical properties of parabolic quantum wells. • The total absorption coefficients increase as the ILF increases. • The RICs increase as the ILF increases.

  6. Response of sliding structures to seismic excitation: bibliographical study

    Sarh, K.; Duval, C.

    1992-11-01

    Calculation of the seismic response of structures on sliding supports involves the dual problem of ''non-linear'' and ''random'' dynamic behaviour. After a review of the non-linearities common in dynamics, slipping is compared with a hysteresis phenomenon. Simple examples are then used to present the Fokker-Planck equation and the equivalent linearization method. Finally, the methods for modification of the excitation spectrum intended for the engineering calculations are recalled. (authors). 21 figs., 23 refs

  7. Analysis of the collagen birefringence and the relative attenuation coefficient of health and burned skin irradiated with linearly polarized He-Ne laser; Analise da birrefringencia do colageno e do coeficiente de atenuacao de amostras de pele sadia e queimada irradiadas pelo laser de He-Ne linearmente polarizado

    Silva, Daniela de Fatima Teixeira da

    2002-07-01

    Low-intensity laser therapy is characterized by its ability to induce athermic effects and nondestructive photobiological processes. Although it has been in use for more than 40 years, this phototherapy is still not an established therapeutic modality. The objectives of this study were: to quantify the collagen fibers organization by polarized light microscopy in normal and burned skin samples at day 17 post-injury considering preferential axis as the animal's spinal column and aligning the linear laser polarization in two directions of polarization, parallel or perpendicular to this axis; to determine the relative attenuation coefficient for the intensity light by the technique of imaging the light distribution in normal and burned skin during wound healing process taking only parallel direction of polarization. To reach the objectives, burns about 6 mm in diameter were created with liquid N{sub 2} on the back of the rats and the lesions were irradiated on days 3, 7, 10 and 14 post-wounding, D= 1 J/cm{sup 2}, to investigate the effects of low-intensity linearly polarized He-Ne laser beam on skin wounds healing. Control lesions were not irradiated. The results have demonstrated that: the skin samples irradiated with linearly parallel polarized He-Ne laser beam showed collagen fibers more organized; burned skin samples presents a higher attenuation coefficient than normal skin samples. These results are important to optimize low intensity laser therapy dosimetry on acceleration wound healing. (author)

  8. Effects of electron-electron interactions on the electron distribution function of a plasma in the presence of an external electric field

    Molinari, V.G.; Pizzio, F.; Spiga, G.

    1979-01-01

    The electron distribution function, the electron temperature and some transport parameters (electrical conductivity and energy flow coefficient) are obtained starting from the nonlinear Boltzmann equation for a plasma under the action of an external electric field. The Fokker-Planck approximation is used for electron-electron and electron-ion interactions. The effects of electron-electron collisions are studied for different values of collision frequencies and electric field. (author)

  9. Effect of magnetic axis shift on neoclassical transport in helical torus

    Sanuki, Heiji; Todoroki, Jiro

    2004-01-01

    Neoclassical transport for large helical device (LHD) configurations is studied by solving the bounce-averaged Fokker-Planck equation. Numerical code employed in the present paper (CHD1) is much faster and more efficient than existing transport codes. Effects of the magnetic axis shift on the mono-energetic transport coefficients are studied in detail for the LHD configurations, revealing that a strong inward shift of the magnetic axis can reduce remarkably the neoclassical ripple transport. (author)

  10. Transport theory of dissipative heavy-ion collisions

    Norenberg, W.

    1979-01-01

    The lectures present the formulation of a transport theory, the derivation of a practicable transport equation (Fokker-Planck equation) and the evaluation of transport coefficients for dissipative (or deeply inelastic) heavy-ion collisions. The applicability of the theoretical concept is tested with remarkable success in the analyses of various experimental information (mass transfer, angular-momentum dissipation and energy loss). Some critical remarks on the present situation of transport theories are added. Future developments are outlined. (author)

  11. Magnetized plasma kinetic theory

    Hassan, M.H.A.; Watson, C.J.H.

    1977-01-01

    The magnetized Balescu-Lenard Collision integral for a multi-species plasma in the form derived by Hassan and Watson (1976) is approximated by ignoring wave effects. The resulting collision integral is put in Fokker-Planck form and most of the integrals occurring in the coefficients are performed analytically. The remaining integral is evaluated approximately in various limits for ion-electron, electron-electron and electron-ion interactions. (author)

  12. Random walk of the baryon number

    Kazaryan, A.M.; Khlebnikov, S.Y.; Shaposhnikov, M.E.

    1989-01-01

    A new approach is suggested for the anomalous nonconservation of baryon number in the electroweak theory at high temperatures. Arguments are presented in support of the idea that the baryon-number changing reactions may be viewed as random Markov processes. Making use of the general theory of Markov processes, the Fokker--Planck equation for the baryon-number distribution density is obtained and kinetic coefficients are calculated

  13. Transport theory of dissipative heavy-ion collisions

    Noerenberg, W.

    1979-03-01

    The lectures present the formulation of a transport theory, the derivation of a practicable transport equation (Fokker-Planck equation) and the evaluation of transport coefficients for dissipative (or deeply inelastic) heavyion collisions. The applicability of the theoretical concept is tested with remarkable success in the analyses of various experimental informations (mass transfer, angular-momentum dissipation and energy loss). Some critical remarks on the present situation of transport theories are added. Future developments are outlined. (orig.) [de

  14. On Solving Linear Recurrences

    Dobbs, David E.

    2013-01-01

    A direct method is given for solving first-order linear recurrences with constant coefficients. The limiting value of that solution is studied as "n to infinity." This classroom note could serve as enrichment material for the typical introductory course on discrete mathematics that follows a calculus course.

  15. Nonlinear drift-diffusion model of gating in K and nACh ion channels

    Vaccaro, S.R. [Department of Physics, University of Adelaide, Adelaide, South Australia 5005 (Australia)], E-mail: svaccaro@physics.adelaide.edu.au

    2007-09-03

    The configuration of a sensor regulates the transition between the closed and open states of both voltage and ligand gated channels. The closed state dwell-time distribution f{sub c}(t) derived from a Fokker-Planck equation with a nonlinear diffusion coefficient is in good agreement with experimental data and can account for the power law approximation to f{sub c}(t) for a delayed rectifier K channel and a nicotinic acetylcholine (nACh) ion channel. The solution of a master equation which approximates the Fokker-Planck equation provides a better description of the small time behaviour of the dwell-time distribution and can account for the empirical rate-amplitude correlation for these ion channels.

  16. Stochastic theory of molecular collisions. II. Application to atom--vibrotor collisions

    Augustin, S.D.; Rabitz, H.

    1977-01-01

    In this work stochastic theory is applied to the treatment of atom--vibrotor collisions. This is an extension of a previous paper which described molecular collisions by a Pauli master equation or a Fokker--Planck equation. In this framework an energy conserving classical path model is explored, and methods for solving the equations numerically are discussed. The coefficients of the Fokker--Planck equation are shown to be expressible as simple functions of the interaction potential. Estimates of the computational labor are also discussed. Finally as a followup on the initial work, numerical solutions of the master equation for the collinear vibrational excitation problem of Secrest and Johnson are presented in an Appendix

  17. Green's functions and trace formulas for generalized Sturm-Liouville problems related by Darboux transformations

    Schulze-Halberg, Axel

    2010-01-01

    We study Green's functions of the generalized Sturm-Liouville problems that are related to each other by Darboux -equivalently, supersymmetrical - transformations. We establish an explicit relation between the corresponding Green's functions and derive a simple formula for their trace. The class of equations considered here includes the conventional Schroedinger equation and generalizations, such as for position-dependent mass and with linearly energy-dependent potential, as well as the stationary Fokker-Planck equation.

  18. Finite elements for partial differential equations: An introductory survey

    Succi, S.

    1988-03-01

    After presentation of the basic ideas behind the theory of the Finite Element Method, the application of the method to three equations of particular interest in Physics and Engineering is discussed in some detail, namely, a one-dimensional Sturm-Liouville problem, a two-dimensional linear Fokker-Planck equation and a two-dimensional nonlinear Navier-Stokes equation. 6 refs, 8 figs

  19. Gyrokinetic simulations of neoclassical transport using a minimal collision operator

    Dif-Pradalier, G.; Grandgirard, V.; Sarazin, Y.; Garbet, X.; Ghendrih, Ph.; Angelino, P.

    2008-01-01

    Conventional neoclassical predictions are successfully recovered within a gyrokinetic framework using a minimal Fokker-Planck collision operator. This operator is shown to accurately describe some essential features of neoclassical theory, namely the neoclassical transport, the poloidal rotation and the linear damping of axisymmetric flows while interestingly preserving a high numerical efficiency. Its form makes it especially adapted to Eulerian or Semi-Lagrangian schemes.

  20. Quantization of non-Hamiltonian physical systems

    Bolivar, A.O.

    1998-09-01

    We propose a general method of quantization of non-Hamiltonian physical systems. Applying it, for example, to a dissipative system coupled to a thermal reservoir described by the Fokker-Planck equation, we are able to obtain the Caldeira-Leggett master equation, the non-linear Schroedinger-Langevin equation and Caldirola-Kanai equation (with an additional term), as particular cases. (author)

  1. Finite element method analysis of Fokker-Plank equation in stationary and evolutionary versions

    Náprstek, Jiří; Král, Radomil

    2014-01-01

    Roč. 72, June (2014), s. 28-38 ISSN 0965-9978 R&D Projects: GA ČR(CZ) GA103/09/0094; GA AV ČR(CZ) IAA200710902 Institutional support: RVO:68378297 Keywords : Fokker-Planck equation * numerical solution * transition effects * stochastic mechanics * probability density function * non-linear dynamic systems Subject RIV: JM - Building Engineering Impact factor: 1.402, year: 2014 http://www.sciencedirect.com/science/article/pii/S0965997813001142

  2. Linear algebra

    Shilov, Georgi E

    1977-01-01

    Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.

  3. Relativistic neoclassical transport coefficients with momentum correction

    Marushchenko, I.; Azarenkov, N.A.

    2016-01-01

    The parallel momentum correction technique is generalized for relativistic approach. It is required for proper calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is shown that the obtained system of linear algebraic equations for parallel fluxes can be solved directly without calculation of the distribution function if the relativistic mono-energetic transport coefficients are already known. The first relativistic correction terms for Braginskii matrix coefficients are calculated.

  4. Inherent noise can facilitate coherence in collective swarm motion

    Yates, C. A.; Erban, R.; Escudero, C.; Couzin, I. D.; Buhl, J.; Kevrekidis, I. G.; Maini, P. K.; Sumpter, D. J. T.

    2009-01-01

    Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the self-propelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

  5. Inherent noise can facilitate coherence in collective swarm motion

    Yates, C. A.

    2009-03-31

    Among the most striking aspects of the movement of many animal groups are their sudden coherent changes in direction. Recent observations of locusts and starlings have shown that this directional switching is an intrinsic property of their motion. Similar direction switches are seen in self-propelled particle and other models of group motion. Comprehending the factors that determine such switches is key to understanding the movement of these groups. Here, we adopt a coarse-grained approach to the study of directional switching in a self-propelled particle model assuming an underlying one-dimensional Fokker-Planck equation for the mean velocity of the particles. We continue with this assumption in analyzing experimental data on locusts and use a similar systematic Fokker-Planck equation coefficient estimation approach to extract the relevant information for the assumed Fokker-Planck equation underlying that experimental data. In the experiment itself the motion of groups of 5 to 100 locust nymphs was investigated in a homogeneous laboratory environment, helping us to establish the intrinsic dynamics of locust marching bands. We determine the mean time between direction switches as a function of group density for the experimental data and the self-propelled particle model. This systematic approach allows us to identify key differences between the experimental data and the model, revealing that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. We give a quantitative description of how locusts use noise to maintain swarm alignment. We discuss further how properties of individual animal behavior, inferred by using the Fokker-Planck equation coefficient estimation approach, can be implemented in the self-propelled particle model to replicate qualitatively the group level dynamics seen in the experimental data.

  6. Mean-field games with logistic population dynamics

    Gomes, Diogo A.

    2013-12-01

    In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.

  7. THE WIGNER–FOKKER–PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR

    ARNOLD, ANTON

    2012-11-01

    We consider the linear WignerFokkerPlanck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for FokkerPlanck type operators in certain weighted L 2-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate. © 2012 World Scientific Publishing Company.

  8. Mean-field games with logistic population dynamics

    Gomes, Diogo A.; De Lima Ribeiro, Ricardo

    2013-01-01

    In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE.

  9. PHYSICS OF ELECTRON CYCLOTRON CURRENT DRIVE ON DIII-D

    PETTY, C.C.; PRATER, R.; LUCE, T.C.; ELLIS, R.A.; HARVEY, R.W.; KINSEY, J.E.; LAO, L.L.; LOHR, J.; MAKOWSKI, M.A.

    2002-01-01

    OAK A271 PHYSICS OF ELECTRON CYCLOTRON CURRENT DRIVE ON DIII-D. Recent experiments on the DIII-D tokamak have focused on determining the effect of trapped particles on the electron cyclotron current drive (ECCD) efficiency. The measured ECCD efficiency increases as the deposition location is moved towards the inboard midplane or towards smaller minor radius for both co and counter injection. The measured ECCD efficiency also increases with increasing electron density and/or temperature. The experimental ECCD is compared to both the linear theory (Toray-GA) as well as a quasilinear Fokker-Planck model (CQL3D). The experimental ECCD is found to be in better agreement with the more complete Fokker-Planck calculation, especially for cases of high rf power density and/or loop voltage

  10. Non deterministic methods for charged particle transport

    Besnard, D.C.; Buresi, E.; Hermeline, F.; Wagon, F.

    1985-04-01

    The coupling of Monte-Carlo methods for solving Fokker Planck equation with ICF inertial confinement fusion codes requires them to be economical and to preserve gross conservation properties. Besides, the presence in FPE Fokker-Planck equation of diffusion terms due to collisions between test particles and the background plasma challenges standard M.C. (Monte-Carlo) techniques if this phenomenon is dominant. We address these problems through the use of a fixed mesh in phase space which allows us to handle highly variable sources, avoiding any Russian Roulette for lowering the size of the sample. Also on this mesh are solved diffusion equations obtained from a splitting of FPE. Any non linear diffusion terms of FPE can be handled in this manner. Another method, also presented here is to use a direct particle method for solving the full FPE

  11. Estimation of the simple correlation coefficient.

    Shieh, Gwowen

    2010-11-01

    This article investigates some unfamiliar properties of the Pearson product-moment correlation coefficient for the estimation of simple correlation coefficient. Although Pearson's r is biased, except for limited situations, and the minimum variance unbiased estimator has been proposed in the literature, researchers routinely employ the sample correlation coefficient in their practical applications, because of its simplicity and popularity. In order to support such practice, this study examines the mean squared errors of r and several prominent formulas. The results reveal specific situations in which the sample correlation coefficient performs better than the unbiased and nearly unbiased estimators, facilitating recommendation of r as an effect size index for the strength of linear association between two variables. In addition, related issues of estimating the squared simple correlation coefficient are also considered.

  12. Transport Coefficients of Fluids

    Eu, Byung Chan

    2006-01-01

    Until recently the formal statistical mechanical approach offered no practicable method for computing the transport coefficients of liquids, and so most practitioners had to resort to empirical fitting formulas. This has now changed, as demonstrated in this innovative monograph. The author presents and applies new methods based on statistical mechanics for calculating the transport coefficients of simple and complex liquids over wide ranges of density and temperature. These molecular theories enable the transport coefficients to be calculated in terms of equilibrium thermodynamic properties, and the results are shown to account satisfactorily for experimental observations, including even the non-Newtonian behavior of fluids far from equilibrium.

  13. Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

    Frank, T. D.

    2008-02-01

    We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

  14. Correlation and simple linear regression.

    Zou, Kelly H; Tuncali, Kemal; Silverman, Stuart G

    2003-06-01

    In this tutorial article, the concepts of correlation and regression are reviewed and demonstrated. The authors review and compare two correlation coefficients, the Pearson correlation coefficient and the Spearman rho, for measuring linear and nonlinear relationships between two continuous variables. In the case of measuring the linear relationship between a predictor and an outcome variable, simple linear regression analysis is conducted. These statistical concepts are illustrated by using a data set from published literature to assess a computed tomography-guided interventional technique. These statistical methods are important for exploring the relationships between variables and can be applied to many radiologic studies.

  15. Discharge Coefficient of Rectangular Short-Crested Weir with Varying Slope Coefficients

    Yuejun Chen

    2018-02-01

    Full Text Available Rectangular short-crested weirs are widely used for simple structure and high discharge capacity. As one of the most important and influential factors of discharge capacity, side slope can improve the hydraulic characteristics of weirs at special conditions. In order to systemically study the effects of upstream and downstream slope coefficients S1 and S2 on overflow discharge coefficient in a rectangular short-crested weir the Volume of Fluid (VOF method and the Renormalization Group (RNG κ-ε turbulence model are used. In this study, the slope coefficient ranges from V to 3H:1V and each model corresponds to five total energy heads of H0 ranging from 8.0 to 24.0 cm. Comparisons of discharge coefficients and free surface profiles between simulated and laboratory results display a good agreement. The simulated results show that the difference of discharge coefficients will decrease with upstream slopes and increase with downstream slopes as H0 increases. For a given H0, the discharge coefficient has a convex parabolic relation with S1 and a piecewise linearity relation with S2. The maximum discharge coefficient is always obtained at S2 = 0.8. There exists a difference between upstream and downstream slope coefficients in the influence range of free surface curvatures. Furthermore, a proposed discharge coefficient equation by nonlinear regression is a function of upstream and downstream slope coefficients.

  16. Form coefficient of helical toroidal solenoids

    Amelin, V.Z.; Kunchenko, V.B.

    1982-01-01

    For toroidal solenoids with continuous spiral coil, winded according to the laws of equiinclined and simple cylindrical spirals with homogeneous, linearly increasing to the coil periphery and ''Bitter'' distribution of current density, the analytical expressions for the dependence between capacity consumed and generated magnetic field, expressions for coefficients of form similar to Fabry coefficient for cylindrical solenoids are obtained and dependence of the form coefficient and relative volume of solenoid conductor on the number of revolutions of screw line per one circumvention over the large torus radius is also investigated. Analytical expressions of form coefficients and graphical material permit to select the optimum geometry as to capacity consumed both for spiral (including ''force-free'') and conventional toroidal solenoids of magnetic systems in thermonulear installations

  17. Linear gate

    Suwono.

    1978-01-01

    A linear gate providing a variable gate duration from 0,40μsec to 4μsec was developed. The electronic circuity consists of a linear circuit and an enable circuit. The input signal can be either unipolar or bipolar. If the input signal is bipolar, the negative portion will be filtered. The operation of the linear gate is controlled by the application of a positive enable pulse. (author)

  18. Dual exponential polynomials and linear differential equations

    Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

    2018-01-01

    We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

  19. Linear Accelerators

    Vretenar, M

    2014-01-01

    The main features of radio-frequency linear accelerators are introduced, reviewing the different types of accelerating structures and presenting the main characteristics aspects of linac beam dynamics

  20. Linearization Method and Linear Complexity

    Tanaka, Hidema

    We focus on the relationship between the linearization method and linear complexity and show that the linearization method is another effective technique for calculating linear complexity. We analyze its effectiveness by comparing with the logic circuit method. We compare the relevant conditions and necessary computational cost with those of the Berlekamp-Massey algorithm and the Games-Chan algorithm. The significant property of a linearization method is that it needs no output sequence from a pseudo-random number generator (PRNG) because it calculates linear complexity using the algebraic expression of its algorithm. When a PRNG has n [bit] stages (registers or internal states), the necessary computational cost is smaller than O(2n). On the other hand, the Berlekamp-Massey algorithm needs O(N2) where N(≅2n) denotes period. Since existing methods calculate using the output sequence, an initial value of PRNG influences a resultant value of linear complexity. Therefore, a linear complexity is generally given as an estimate value. On the other hand, a linearization method calculates from an algorithm of PRNG, it can determine the lower bound of linear complexity.

  1. Dynamic linearization system for a radiation gauge

    Panarello, J.A.

    1977-01-01

    The linearization system and process converts a high resolution non-linear analog input signal, representative of the thickness of an object, into a high resolution linear analog output signal suitable for use in driving a variety of output devices. The system requires only a small amount of memory for storing pre-calculated non-linear correction coefficients. The system channels the input signal to separate circuit paths so that it may be used directly to; locate an appropriate correction coefficient; develop a correction term after an appropriate correction coefficient is located; and develop a linearized signal having the same high resolution inherent in the input signal. The system processes the linearized signal to compensate for the possible errors introduced by radiation source noise. The processed linearized signal is the high resolution linear analog output signal which accurately represents the thickness of the object being gauged

  2. Linear algebra

    Said-Houari, Belkacem

    2017-01-01

    This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing t...

  3. Semi-classical theory of fluctuations in nuclear matter

    Benhassine, B.

    1994-01-01

    At intermediate energies the heavy ion collisions can be studied within the framework of a semi-classical approach based on the Vlasov-Uehling-Uhlenbeck (VUU) equation. Such an approach reduces the N-body problem to its description in terms of the one-body distribution function and constitutes the basis of several successful simulation models. Our aim in this work is to extend these average approaches to treat fluctuations. Within the framework of a linear approximation, we derived a Fokker-Planck transport equation in the one-body phase space. When it is reduced to its first moments, one recovers the VUU equation for the average dynamics together with the time evolution equation for the correlations. The collective transport coefficients are then obtained by projection on the one-body collective space. Independently, using a projection method introduced by Van Kampen, based on the constants of motion, we deduce the stationary expressions for the covariance matrix in phase space. We extract then, the equilibrium dispersions of one-body observables in a homogeneous case and in a spherical symmetric one. These results are compared with two types of simulation models in a relaxation time approximation. In the first one which is of Lagrangian type, the collective transport coefficients are directly extracted from the simulation and consequently the numerical fluctuations are washed out. The second model, due to its Eulerian character, allows us to make a microscopical comparison. (author)

  4. Semi-classical theory of fluctuations in nuclear matter; Theorie semi-classique des fluctuations dans la matiere nucleaire

    Benhassine, B. [Nantes Univ., 44 (France)

    1994-01-14

    At intermediate energies the heavy ion collisions can be studied within the framework of a semi-classical approach based on the Vlasov-Uehling-Uhlenbeck (VUU) equation. Such an approach reduces the N-body problem to its description in terms of the one-body distribution function and constitutes the basis of several successful simulation models. Our aim in this work is to extend these average approaches to treat fluctuations. Within the framework of a linear approximation, we derived a Fokker-Planck transport equation in the one-body phase space. When it is reduced to its first moments, one recovers the VUU equation for the average dynamics together with the time evolution equation for the correlations. The collective transport coefficients are then obtained by projection on the one-body collective space. Independently, using a projection method introduced by Van Kampen, based on the constants of motion, we deduce the stationary expressions for the covariance matrix in phase space. We extract then, the equilibrium dispersions of one-body observables in a homogeneous case and in a spherical symmetric one. These results are compared with two types of simulation models in a relaxation time approximation. In the first one which is of Lagrangian type, the collective transport coefficients are directly extracted from the simulation and consequently the numerical fluctuations are washed out. The second model, due to its Eulerian character, allows us to make a microscopical comparison. (author) 58 refs.

  5. Linear algebra

    Stoll, R R

    1968-01-01

    Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand

  6. Linear programming

    Solow, Daniel

    2014-01-01

    This text covers the basic theory and computation for a first course in linear programming, including substantial material on mathematical proof techniques and sophisticated computation methods. Includes Appendix on using Excel. 1984 edition.

  7. Linear algebra

    Liesen, Jörg

    2015-01-01

    This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...

  8. Linear algebra

    Berberian, Sterling K

    2014-01-01

    Introductory treatment covers basic theory of vector spaces and linear maps - dimension, determinants, eigenvalues, and eigenvectors - plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.

  9. Linear Models

    Searle, Shayle R

    2012-01-01

    This 1971 classic on linear models is once again available--as a Wiley Classics Library Edition. It features material that can be understood by any statistician who understands matrix algebra and basic statistical methods.

  10. LINEAR ACCELERATOR

    Christofilos, N.C.; Polk, I.J.

    1959-02-17

    Improvements in linear particle accelerators are described. A drift tube system for a linear ion accelerator reduces gap capacity between adjacent drift tube ends. This is accomplished by reducing the ratio of the diameter of the drift tube to the diameter of the resonant cavity. Concentration of magnetic field intensity at the longitudinal midpoint of the external sunface of each drift tube is reduced by increasing the external drift tube diameter at the longitudinal center region.

  11. The Truth About Ballistic Coefficients

    Courtney, Michael; Courtney, Amy

    2007-01-01

    The ballistic coefficient of a bullet describes how it slows in flight due to air resistance. This article presents experimental determinations of ballistic coefficients showing that the majority of bullets tested have their previously published ballistic coefficients exaggerated from 5-25% by the bullet manufacturers. These exaggerated ballistic coefficients lead to inaccurate predictions of long range bullet drop, retained energy and wind drift.

  12. Table for constructing the spin coefficients in general relativity

    Cocke, W.J.

    1989-01-01

    The spin coefficients in spinor calculus in Riemannian space-time are linear functions of the curls of the connecting quantities (the Infeld--Van der Waerden symbols). We show that in the Newman-Penrose formalism the expressions for the spin coefficients are quite manageable, if they are written in terms of the Newman-Penrose tetrad vectors. We present a table of the components of the spin coefficients explicitly in terms of the curls of the individual tetrad vectors

  13. Approximation of the Frame Coefficients using Finite Dimensional Methods

    Christensen, Ole; Casazza, P.

    1997-01-01

    _i \\}_{i=1}^{n}$ of the frame and theorthogonal projection $P_n$ onto its span. For $f \\in \\h ,P_nf$ has a representation as a linear combination of $f_i , i=1,2,..n,$and the corresponding coefficients can be calculated using finite dimensionalmethods. We find conditions implying that those coefficients...

  14. Analytic posteriors for Pearson's correlation coefficient.

    Ly, Alexander; Marsman, Maarten; Wagenmakers, Eric-Jan

    2018-02-01

    Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open-source software package JASP.

  15. Analytic posteriors for Pearson's correlation coefficient

    Ly, A.; Marsman, M.; Wagenmakers, E.-J.

    2018-01-01

    Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open‐source software package JASP.

  16. Linear regression

    Olive, David J

    2017-01-01

    This text covers both multiple linear regression and some experimental design models. The text uses the response plot to visualize the model and to detect outliers, does not assume that the error distribution has a known parametric distribution, develops prediction intervals that work when the error distribution is unknown, suggests bootstrap hypothesis tests that may be useful for inference after variable selection, and develops prediction regions and large sample theory for the multivariate linear regression model that has m response variables. A relationship between multivariate prediction regions and confidence regions provides a simple way to bootstrap confidence regions. These confidence regions often provide a practical method for testing hypotheses. There is also a chapter on generalized linear models and generalized additive models. There are many R functions to produce response and residual plots, to simulate prediction intervals and hypothesis tests, to detect outliers, and to choose response trans...

  17. Linear Colliders

    Alcaraz, J.

    2001-01-01

    After several years of study e''+ e''- linear colliders in the TeV range have emerged as the major and optimal high-energy physics projects for the post-LHC era. These notes summarize the present status form the main accelerator and detector features to their physics potential. The LHC era. These notes summarize the present status, from the main accelerator and detector features to their physics potential. The LHC is expected to provide first discoveries in the new energy domain, whereas an e''+ e''- linear collider in the 500 GeV-1 TeV will be able to complement it to an unprecedented level of precision in any possible areas: Higgs, signals beyond the SM and electroweak measurements. It is evident that the Linear Collider program will constitute a major step in the understanding of the nature of the new physics beyond the Standard Model. (Author) 22 refs

  18. Linear algebra

    Edwards, Harold M

    1995-01-01

    In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject

  19. Correlation Coefficients: Appropriate Use and Interpretation.

    Schober, Patrick; Boer, Christa; Schwarte, Lothar A

    2018-05-01

    Correlation in the broadest sense is a measure of an association between variables. In correlated data, the change in the magnitude of 1 variable is associated with a change in the magnitude of another variable, either in the same (positive correlation) or in the opposite (negative correlation) direction. Most often, the term correlation is used in the context of a linear relationship between 2 continuous variables and expressed as Pearson product-moment correlation. The Pearson correlation coefficient is typically used for jointly normally distributed data (data that follow a bivariate normal distribution). For nonnormally distributed continuous data, for ordinal data, or for data with relevant outliers, a Spearman rank correlation can be used as a measure of a monotonic association. Both correlation coefficients are scaled such that they range from -1 to +1, where 0 indicates that there is no linear or monotonic association, and the relationship gets stronger and ultimately approaches a straight line (Pearson correlation) or a constantly increasing or decreasing curve (Spearman correlation) as the coefficient approaches an absolute value of 1. Hypothesis tests and confidence intervals can be used to address the statistical significance of the results and to estimate the strength of the relationship in the population from which the data were sampled. The aim of this tutorial is to guide researchers and clinicians in the appropriate use and interpretation of correlation coefficients.

  20. Determination of beta attenuation coefficients by means of timing method

    Ermis, E.E.; Celiktas, C.

    2012-01-01

    Highlights: ► Beta attenuation coefficients of absorber materials were found in this study. ► For this process, a new method (timing method) was suggested. ► The obtained beta attenuation coefficients were compatible with the results from the traditional one. ► The timing method can be used to determine beta attenuation coefficient. - Abstract: Using a counting system with plastic scintillation detector, beta linear and mass attenuation coefficients were determined for bakelite, Al, Fe and plexiglass absorbers by means of timing method. To show the accuracy and reliability of the obtained results through this method, the coefficients were also found via conventional energy method. Obtained beta attenuation coefficients from both methods were compared with each other and the literature values. Beta attenuation coefficients obtained through timing method were found to be compatible with the values obtained from conventional energy method and the literature.

  1. Optimality Conditions for Fuzzy Number Quadratic Programming with Fuzzy Coefficients

    Xue-Gang Zhou

    2014-01-01

    Full Text Available The purpose of the present paper is to investigate optimality conditions and duality theory in fuzzy number quadratic programming (FNQP in which the objective function is fuzzy quadratic function with fuzzy number coefficients and the constraint set is fuzzy linear functions with fuzzy number coefficients. Firstly, the equivalent quadratic programming of FNQP is presented by utilizing a linear ranking function and the dual of fuzzy number quadratic programming primal problems is introduced. Secondly, we present optimality conditions for fuzzy number quadratic programming. We then prove several duality results for fuzzy number quadratic programming problems with fuzzy coefficients.

  2. Symmetry chains and adaptation coefficients

    Fritzer, H.P.; Gruber, B.

    1985-01-01

    Given a symmetry chain of physical significance it becomes necessary to obtain states which transform properly with respect to the symmetries of the chain. In this article we describe a method which permits us to calculate symmetry-adapted quantum states with relative ease. The coefficients for the symmetry-adapted linear combinations are obtained, in numerical form, in terms of the original states of the system and can thus be represented in the form of numerical tables. In addition, one also obtains automatically the matrix elements for the operators of the symmetry groups which are involved, and thus for any physical operator which can be expressed either as an element of the algebra or of the enveloping algebra. The method is well suited for computers once the physically relevant symmetry chain, or chains, have been defined. While the method to be described is generally applicable to any physical system for which semisimple Lie algebras play a role we choose here a familiar example in order to illustrate the method and to illuminate its simplicity. We choose the nuclear shell model for the case of two nucleons with orbital angular momentum l = 1. While the states of the entire shell transform like the smallest spin representation of SO(25) we restrict our attention to its subgroup SU(6) x SU(2)/sub T/. We determine the symmetry chains which lead to total angular momentum SU(2)/sub J/ and obtain the symmetry-adapted states for these chains

  3. Riesz frames and approximation of the frame coefficients

    Casazza, P.; Christensen, Ole

    1998-01-01

    A frame is a fmaily {f i } i=1 ∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From...... the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on . The projection method is introduced in order to avoid this problem. The basic idea is to consider finite subfamilies {f i } i=1 n of the frame and the orthogonal...... projection Pn onto its span. For has a representation as a linear combination of fi, i=1,2,..., n and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have...

  4. Linear programming

    Karloff, Howard

    1991-01-01

    To this reviewer’s knowledge, this is the first book accessible to the upper division undergraduate or beginning graduate student that surveys linear programming from the Simplex Method…via the Ellipsoid algorithm to Karmarkar’s algorithm. Moreover, its point of view is algorithmic and thus it provides both a history and a case history of work in complexity theory. The presentation is admirable; Karloff's style is informal (even humorous at times) without sacrificing anything necessary for understanding. Diagrams (including horizontal brackets that group terms) aid in providing clarity. The end-of-chapter notes are helpful...Recommended highly for acquisition, since it is not only a textbook, but can also be used for independent reading and study. —Choice Reviews The reader will be well served by reading the monograph from cover to cover. The author succeeds in providing a concise, readable, understandable introduction to modern linear programming. —Mathematics of Computing This is a textbook intend...

  5. Interpretation of recent positron-electron measurements between 20 and 800 MeV

    Pellerin, C.J.; Hartman, R.C.

    1975-01-01

    The recent positron and negatron spectra measured by Hartman and Pellerin (see pages 402-407) are discussed with regard to the problem of solar modulation. At energies above 180 MeV, the spherically symmetric Fokker-Planck equation with a diffusion coefficient proportional to particle rigidity provides reasonable fits to both the positron and total electron data. At energies below 180 MeV the data are consistent with a continuation of the same diffusion coefficient and local source of negatrons, or a change in the diffusion coefficient to a constant value. (orig.) [de

  6. Quantum Non-Markovian Langevin Equations and Transport Coefficients

    Sargsyan, V.V.; Antonenko, N.V.; Kanokov, Z.; Adamian, G.G.

    2005-01-01

    Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed

  7. On the Kendall Correlation Coefficient

    Stepanov, Alexei

    2015-01-01

    In the present paper, we first discuss the Kendall rank correlation coefficient. In continuous case, we define the Kendall rank correlation coefficient in terms of the concomitants of order statistics, find the expected value of the Kendall rank correlation coefficient and show that the later is free of n. We also prove that in continuous case the Kendall correlation coefficient converges in probability to its expected value. We then propose to consider the expected value of the Kendall rank ...

  8. Linearized gravity in terms of differential forms

    Baykal, Ahmet; Dereli, Tekin

    2017-01-01

    A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

  9. Modified Regression Correlation Coefficient for Poisson Regression Model

    Kaengthong, Nattacha; Domthong, Uthumporn

    2017-09-01

    This study gives attention to indicators in predictive power of the Generalized Linear Model (GLM) which are widely used; however, often having some restrictions. We are interested in regression correlation coefficient for a Poisson regression model. This is a measure of predictive power, and defined by the relationship between the dependent variable (Y) and the expected value of the dependent variable given the independent variables [E(Y|X)] for the Poisson regression model. The dependent variable is distributed as Poisson. The purpose of this research was modifying regression correlation coefficient for Poisson regression model. We also compare the proposed modified regression correlation coefficient with the traditional regression correlation coefficient in the case of two or more independent variables, and having multicollinearity in independent variables. The result shows that the proposed regression correlation coefficient is better than the traditional regression correlation coefficient based on Bias and the Root Mean Square Error (RMSE).

  10. Reduction of Linear Programming to Linear Approximation

    Vaserstein, Leonid N.

    2006-01-01

    It is well known that every Chebyshev linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to a Chebyshev linear approximation problem.

  11. Fractal diffusion coefficient from dynamical zeta functions

    Cristadoro, Giampaolo [Max Planck Institute for the Physics of Complex Systems, Noethnitzer Str. 38, D 01187 Dresden (Germany)

    2006-03-10

    Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi-Ruelle generalization of the Milnor-Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero. (letter to the editor)

  12. Fractal diffusion coefficient from dynamical zeta functions

    Cristadoro, Giampaolo

    2006-01-01

    Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi-Ruelle generalization of the Milnor-Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero. (letter to the editor)

  13. Mass exchange and angular distribution in a dynamical treatment of heavy ion collisions

    Ngo, C.; Hofmann, H.

    1977-01-01

    One presents a first numerical computation of the absolute value of the double differential cross section as a function of mass asymmetry and detection angle including a dynamical coupling between relative motion and mass asymmetry. One applies it to the 63 Cu+ 197 Au experiment at two different energies. The equation of motion used is a Fokker-Planck equation for distribution function in classical phase space. The coefficients needed are those known from classical model calculations, besides a friction coefficient introduced for the mass asymmetry degree. Encouraging agreement between the calculated and experimental curves is found

  14. On the problem of a consistent description of kinetic and hydrodynamic processes in dense gases and liquids

    B.B. Markiv

    2010-01-01

    Full Text Available For a consistent description of kinetic and hydrodynamic processes in dense gases and liquids the generalized non-Markovian equations for the nonequilibrium one-particle distribution function and potential part of the averaged enthalpy density are obtained. The inner structure of the generalized transport kernels for these equations is established. It is shown that the collision integral of the kinetic equation has the Fokker-Planck form with the generalized friction coefficient in momentum space. It also contains contributions from the generalized diffusion coefficient and dissipative processes connected with the potential part of the enthalpy density.

  15. Quadrature formulas for Fourier coefficients

    Bojanov, Borislav

    2009-09-01

    We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node formula for the Fourier-Tchebycheff coefficients given by Micchelli and Sharma and construct new Gaussian formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives. © 2009 Elsevier B.V. All rights reserved.

  16. Coefficient Alpha: A Reliability Coefficient for the 21st Century?

    Yang, Yanyun; Green, Samuel B.

    2011-01-01

    Coefficient alpha is almost universally applied to assess reliability of scales in psychology. We argue that researchers should consider alternatives to coefficient alpha. Our preference is for structural equation modeling (SEM) estimates of reliability because they are informative and allow for an empirical evaluation of the assumptions…

  17. Coefficient estimates of negative powers and inverse coefficients for ...

    and the inequality is sharp for the inverse of the Koebe function k(z) = z/(1 − z)2. An alternative approach to the inverse coefficient problem for functions in the class S has been investigated by Schaeffer and Spencer [27] and FitzGerald [6]. Although, the inverse coefficient problem for the class S has been completely solved ...

  18. Measuring of heat transfer coefficient

    Henningsen, Poul; Lindegren, Maria

    Subtask 3.4 Measuring of heat transfer coefficient Subtask 3.4.1 Design and setting up of tests to measure heat transfer coefficient Objective: Complementary testing methods together with the relevant experimental equipment are to be designed by the two partners involved in order to measure...... the heat transfer coefficient for a wide range of interface conditions in hot and warm forging processes. Subtask 3.4.2 Measurement of heat transfer coefficient The objective of subtask 3.4.2 is to determine heat transfer values for different interface conditions reflecting those typically operating in hot...

  19. Kinetic coefficients for quark-antiquark plasma

    Czyz, W.; Florkowski, W.

    1986-03-01

    The quark-antiquark plasma near equilibrium is studied. The results are based on the Heinz kinetic equations with the Boltzmann collision operator approximated by a relaxation term with the relaxation time, τ, treated as a small parameter. Linear in τ solutions of these equations are used to calculate the transport coefficients: the non-abelian version of Ohm's law, and the shear and volume viscosities. We introduce new chemical potentials which determine the color density matrix of quarks (antiquarks). Gradients of these potentials generate color currents. 12 refs. (author)

  20. Non-Linear Approximation of Bayesian Update

    Litvinenko, Alexander

    2016-01-01

    We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.

  1. Characteristic polynomials of linear polyacenes and their ...

    Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's. Thus it ...

  2. Non-Linear Approximation of Bayesian Update

    Litvinenko, Alexander

    2016-06-23

    We develop a non-linear approximation of expensive Bayesian formula. This non-linear approximation is applied directly to Polynomial Chaos Coefficients. In this way, we avoid Monte Carlo sampling and sampling error. We can show that the famous Kalman Update formula is a particular case of this update.

  3. Riesz Frames and Approximation of the Frame Coefficients

    Christensen, Ole

    1996-01-01

    A frame is a familyof elements in a Hilbert space with the propertythat every element in the Hilbert space can be written as a (infinite)linear combination of the frame elements. Frame theory describes howone can choose the corresponding coefficients, which are calledframe coefficients. From...... the mathematical point of view this isgratifying, but for applications it is a problem that the calculationrequires inversion of an operator on the Hilbert space.The projection method is introduced in order to avoid this problem.The basic idea is to consider finite subfamiliesof the frame and the orthogonal...... projection onto its span. Forfin QTR H,P_nf has a representation as a linear combinationof f_i,i=1,2,..,n, and the corresponding coefficients can be calculatedusing finite dimensional methods. We find conditions implying that thosecoefficients converge to the correct frame coefficients as n goes...

  4. Validity of the CT to attenuation coefficient map conversion methods

    Faghihi, R.; Ahangari Shahdehi, R.; Fazilat Moadeli, M.

    2004-01-01

    The most important commercialized methods of attenuation correction in SPECT are based on attenuation coefficient map from a transmission imaging method. The transmission imaging system can be the linear source of radioelement or a X-ray CT system. The image of transmission imaging system is not useful unless to replacement of the attenuation coefficient or CT number with the attenuation coefficient in SPECT energy. In this paper we essay to evaluate the validity and estimate the error of the most used method of this transformation. The final result shows that the methods which use a linear or multi-linear curve accept a error in their estimation. The value of mA is not important but the patient thickness is very important and it can introduce a error more than 10 percent in the final result

  5. linear-quadratic-linear model

    Tanwiwat Jaikuna

    2017-02-01

    Full Text Available Purpose: To develop an in-house software program that is able to calculate and generate the biological dose distribution and biological dose volume histogram by physical dose conversion using the linear-quadratic-linear (LQL model. Material and methods : The Isobio software was developed using MATLAB version 2014b to calculate and generate the biological dose distribution and biological dose volume histograms. The physical dose from each voxel in treatment planning was extracted through Computational Environment for Radiotherapy Research (CERR, and the accuracy was verified by the differentiation between the dose volume histogram from CERR and the treatment planning system. An equivalent dose in 2 Gy fraction (EQD2 was calculated using biological effective dose (BED based on the LQL model. The software calculation and the manual calculation were compared for EQD2 verification with pair t-test statistical analysis using IBM SPSS Statistics version 22 (64-bit. Results: Two and three-dimensional biological dose distribution and biological dose volume histogram were displayed correctly by the Isobio software. Different physical doses were found between CERR and treatment planning system (TPS in Oncentra, with 3.33% in high-risk clinical target volume (HR-CTV determined by D90%, 0.56% in the bladder, 1.74% in the rectum when determined by D2cc, and less than 1% in Pinnacle. The difference in the EQD2 between the software calculation and the manual calculation was not significantly different with 0.00% at p-values 0.820, 0.095, and 0.593 for external beam radiation therapy (EBRT and 0.240, 0.320, and 0.849 for brachytherapy (BT in HR-CTV, bladder, and rectum, respectively. Conclusions : The Isobio software is a feasible tool to generate the biological dose distribution and biological dose volume histogram for treatment plan evaluation in both EBRT and BT.

  6. Raynal–Revai coefficients for a general kinematic rotation

    Ershov, S. N.

    2016-01-01

    In a three-body system, transitions between different sets of normalized Jacobi coordinates are described as general kinematic transformations that include an orthogonal or a pseudoorthogonal rotation. For such rotations, the Raynal–Revai coefficients execute a unitary transformation between three-body hyperspherical functions. Recurrence relations that make it possible to calculate the Raynal–Revai coefficients for arbitrary angular momenta are derived on the basis of linearized representations of products of hyperspherical functions.

  7. A METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS WITH FUZZY PARAMETERS BASED ON MULTIOBJECTIVE LINEAR PROGRAMMING TECHNIQUE

    M. ZANGIABADI; H. R. MALEKI

    2007-01-01

    In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, w...

  8. Pearson's correlation coefficient in the theory of networks: A comment

    Ahmed, Zafar; Kumar, Sachin

    2018-01-01

    In statistics, the Pearson correlation coefficient $r_{x,y}$ determines the degree of linear correlation between two variables and it is known that $-1 \\le r_{x,y} \\le 1$. In the theory of networks, a curious expression proposed in [PRL {\\bf 89} 208701 (2002)] for degree-degree correlation coefficient $r_{j_i,k_i}, i\\in [1,M]$ has been in use. We realize that the suggested form is the conventional Pearson's coefficient for $\\{(j_i,k_i), (k_i,j_i)\\}$ for $2M$ data points and hence it is rightl...

  9. Theoretical investigation of the extinction coefficient of magnetic fluid

    Fang Xiaopeng; Xuan Yimin, E-mail: ymxuan@mail.njust.edu.cn; Li Qiang [Nanjing University of Science and Technology, School of Energy and Power Engineering (China)

    2013-05-15

    A new theoretical approach for calculating the extinction coefficient of magnetic fluid is proposed, which is based on molecular dynamics (MD) simulation and T-matrix method. By means of this approach, the influence of particle diameter, particle volume fraction, and external magnetic filed on the extinction coefficient of magnetic fluid is investigated. The results show that the extinction coefficient of the magnetic fluid linearly increases with increase in the particle volume fraction. For a given particle volume fraction, the extinction coefficient increases with increase in the particle diameter which varies from 5 to 20 nm. When a uniform external magnetic filed is applied to the magnetic fluid, the extinction coefficient of the magnetic fluid presents an anisotropic feature. These results agree well with the reported experimental results. The proposed approach is applicable to investigating the optical properties of magnetic fluids.

  10. Sabine absorption coefficients to random incidence absorption coefficients

    Jeong, Cheol-Ho

    2014-01-01

    into random incidence absorption coefficients for porous absorbers are investigated. Two optimization-based conversion methods are suggested: the surface impedance estimation for locally reacting absorbers and the flow resistivity estimation for extendedly reacting absorbers. The suggested conversion methods...

  11. Visual construction of characteristic equations of linear electric circuits

    V.V. Kostyukov

    2013-12-01

    Full Text Available A visual identification method with application of partial circuits is developed for characteristic equation coefficients of transients in linear electric circuits. The method is based on interrelationship between the roots of algebraic polynomial and its coefficients. The method is illustrated with an example of a third-order linear electric circuit.

  12. Probabilistic optimization of safety coefficients

    Marques, M.; Devictor, N.; Magistris, F. de

    1999-01-01

    This article describes a reliability-based method for the optimization of safety coefficients defined and used in design codes. The purpose of the optimization is to determine the partial safety coefficients which minimize an objective function for sets of components and loading situations covered by a design rule. This objective function is a sum of distances between the reliability of the components designed using the safety coefficients and a target reliability. The advantage of this method is shown on the examples of the reactor vessel, a vapour pipe and the safety injection circuit. (authors)

  13. Quadrature formulas for Fourier coefficients

    Bojanov, Borislav; Petrova, Guergana

    2009-01-01

    We consider quadrature formulas of high degree of precision for the computation of the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials. In particular, we show the uniqueness of a multiple node

  14. Diffusion coefficient for anomalous transport

    1986-01-01

    A report on the progress towards the goal of estimating the diffusion coefficient for anomalous transport is given. The gyrokinetic theory is used to identify different time and length scale inherent to the characteristics of plasmas which exhibit anomalous transport

  15. Spectral theories for linear differential equations

    Sell, G.R.

    1976-01-01

    The use of spectral analysis in the study of linear differential equations with constant coefficients is not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour of the solutions. The spectral analysis, via the Jordan canonical form, will not only lead to a representation theorem for a basis of solutions, but will also give a rather precise statement of the (exponential) growth rates of various solutions. Various attempts have been made to extend this analysis to linear differential equations with time-varying coefficients. The most complete such extensions is the Floquet theory for equations with periodic coefficients. For time-varying linear differential equations with aperiodic coefficients several authors have attempted to ''extend'' the Foquet theory. The precise meaning of such an extension is itself a problem, and we present here several attempts in this direction that are related to the general problem of extending the spectral analysis of equations with constant coefficients. The main purpose of this paper is to introduce some problems of current research. The primary problem we shall examine occurs in the context of linear differential equations with almost periodic coefficients. We call it ''the Floquet problem''. (author)

  16. Advanced statistics: linear regression, part II: multiple linear regression.

    Marill, Keith A

    2004-01-01

    The applications of simple linear regression in medical research are limited, because in most situations, there are multiple relevant predictor variables. Univariate statistical techniques such as simple linear regression use a single predictor variable, and they often may be mathematically correct but clinically misleading. Multiple linear regression is a mathematical technique used to model the relationship between multiple independent predictor variables and a single dependent outcome variable. It is used in medical research to model observational data, as well as in diagnostic and therapeutic studies in which the outcome is dependent on more than one factor. Although the technique generally is limited to data that can be expressed with a linear function, it benefits from a well-developed mathematical framework that yields unique solutions and exact confidence intervals for regression coefficients. Building on Part I of this series, this article acquaints the reader with some of the important concepts in multiple regression analysis. These include multicollinearity, interaction effects, and an expansion of the discussion of inference testing, leverage, and variable transformations to multivariate models. Examples from the first article in this series are expanded on using a primarily graphic, rather than mathematical, approach. The importance of the relationships among the predictor variables and the dependence of the multivariate model coefficients on the choice of these variables are stressed. Finally, concepts in regression model building are discussed.

  17. Fuel Temperature Coefficient of Reactivity

    Loewe, W.E.

    2001-07-31

    A method for measuring the fuel temperature coefficient of reactivity in a heterogeneous nuclear reactor is presented. The method, which is used during normal operation, requires that calibrated control rods be oscillated in a special way at a high reactor power level. The value of the fuel temperature coefficient of reactivity is found from the measured flux responses to these oscillations. Application of the method in a Savannah River reactor charged with natural uranium is discussed.

  18. Properties of Traffic Risk Coefficient

    Tang, Tie-Qiao; Huang, Hai-Jun; Shang, Hua-Yan; Xue, Yu

    2009-10-01

    We use the model with the consideration of the traffic interruption probability (Physica A 387(2008)6845) to study the relationship between the traffic risk coefficient and the traffic interruption probability. The analytical and numerical results show that the traffic interruption probability will reduce the traffic risk coefficient and that the reduction is related to the density, which shows that this model can improve traffic security.

  19. Extracting the diffusion tensor from molecular dynamics simulation with Milestoning

    Mugnai, Mauro L.; Elber, Ron

    2015-01-01

    We propose an algorithm to extract the diffusion tensor from Molecular Dynamics simulations with Milestoning. A Kramers-Moyal expansion of a discrete master equation, which is the Markovian limit of the Milestoning theory, determines the diffusion tensor. To test the algorithm, we analyze overdamped Langevin trajectories and recover a multidimensional Fokker-Planck equation. The recovery process determines the flux through a mesh and estimates local kinetic parameters. Rate coefficients are converted to the derivatives of the potential of mean force and to coordinate dependent diffusion tensor. We illustrate the computation on simple models and on an atomically detailed system—the diffusion along the backbone torsions of a solvated alanine dipeptide

  20. Temperature anisotropy in a cyclotron resonance heated tokamak plasma and the generation of poloidal electric field

    Choe, W.; Ono, M.; Chang, C.S.

    1994-11-01

    The temperature anisotropy generated by cyclotron resonance heating of tokamak plasmas is calculated and the poloidal equilibrium electric field due to the anisotropy is studied. For the calculation of anisotropic temperatures, bounce-averaged Fokker-Planck equation with a bi-Maxwellian distribution function of heated particles is solved, assuming a moderate wave power and a constant quasilinear cyclotron resonance diffusion coefficient. The poloidal electrostatic potential variation is found to be proportional to the particle density and the degree of temperature anisotropy of warm species created by cyclotron resonance heating

  1. A Semiparametric Time Trend Varying Coefficients Model: With An Application to Evaluate Credit Rationing in U.S. Credit Market

    Jingping Gu; Paula Hernandez-Verme

    2009-01-01

    In this paper, we propose a new semiparametric varying coefficient model which extends the existing semi-parametric varying coefficient models to allow for a time trend regressor with smooth coefficient function. We propose to use the local linear method to estimate the coefficient functions and we provide the asymptotic theory to describe the asymptotic distribution of the local linear estimator. We present an application to evaluate credit rationing in the U.S. credit market. Using U.S. mon...

  2. A Semiparametric Time Trend Varying Coefficients Model: With An Application to Evaluate Credit Rationing in U.S. Credit Market

    Qi Gao; Jingping Gu; Paula Hernandez-Verme

    2012-01-01

    In this paper, we propose a new semiparametric varying coefficient model which extends the existing semi-parametric varying coefficient models to allow for a time trend regressor with smooth coefficient function. We propose to use the local linear method to estimate the coefficient functions and we provide the asymptotic theory to describe the asymptotic distribution of the local linear estimator. We present an application to evaluate credit rationing in the U.S. credit market. Using U.S. mon...

  3. Clustering Coefficients for Correlation Networks.

    Masuda, Naoki; Sakaki, Michiko; Ezaki, Takahiro; Watanabe, Takamitsu

    2018-01-01

    Graph theory is a useful tool for deciphering structural and functional networks of the brain on various spatial and temporal scales. The clustering coefficient quantifies the abundance of connected triangles in a network and is a major descriptive statistics of networks. For example, it finds an application in the assessment of small-worldness of brain networks, which is affected by attentional and cognitive conditions, age, psychiatric disorders and so forth. However, it remains unclear how the clustering coefficient should be measured in a correlation-based network, which is among major representations of brain networks. In the present article, we propose clustering coefficients tailored to correlation matrices. The key idea is to use three-way partial correlation or partial mutual information to measure the strength of the association between the two neighboring nodes of a focal node relative to the amount of pseudo-correlation expected from indirect paths between the nodes. Our method avoids the difficulties of previous applications of clustering coefficient (and other) measures in defining correlational networks, i.e., thresholding on the correlation value, discarding of negative correlation values, the pseudo-correlation problem and full partial correlation matrices whose estimation is computationally difficult. For proof of concept, we apply the proposed clustering coefficient measures to functional magnetic resonance imaging data obtained from healthy participants of various ages and compare them with conventional clustering coefficients. We show that the clustering coefficients decline with the age. The proposed clustering coefficients are more strongly correlated with age than the conventional ones are. We also show that the local variants of the proposed clustering coefficients (i.e., abundance of triangles around a focal node) are useful in characterizing individual nodes. In contrast, the conventional local clustering coefficients were strongly

  4. Clustering Coefficients for Correlation Networks

    Naoki Masuda

    2018-03-01

    Full Text Available Graph theory is a useful tool for deciphering structural and functional networks of the brain on various spatial and temporal scales. The clustering coefficient quantifies the abundance of connected triangles in a network and is a major descriptive statistics of networks. For example, it finds an application in the assessment of small-worldness of brain networks, which is affected by attentional and cognitive conditions, age, psychiatric disorders and so forth. However, it remains unclear how the clustering coefficient should be measured in a correlation-based network, which is among major representations of brain networks. In the present article, we propose clustering coefficients tailored to correlation matrices. The key idea is to use three-way partial correlation or partial mutual information to measure the strength of the association between the two neighboring nodes of a focal node relative to the amount of pseudo-correlation expected from indirect paths between the nodes. Our method avoids the difficulties of previous applications of clustering coefficient (and other measures in defining correlational networks, i.e., thresholding on the correlation value, discarding of negative correlation values, the pseudo-correlation problem and full partial correlation matrices whose estimation is computationally difficult. For proof of concept, we apply the proposed clustering coefficient measures to functional magnetic resonance imaging data obtained from healthy participants of various ages and compare them with conventional clustering coefficients. We show that the clustering coefficients decline with the age. The proposed clustering coefficients are more strongly correlated with age than the conventional ones are. We also show that the local variants of the proposed clustering coefficients (i.e., abundance of triangles around a focal node are useful in characterizing individual nodes. In contrast, the conventional local clustering coefficients

  5. Clustering Coefficients for Correlation Networks

    Masuda, Naoki; Sakaki, Michiko; Ezaki, Takahiro; Watanabe, Takamitsu

    2018-01-01

    Graph theory is a useful tool for deciphering structural and functional networks of the brain on various spatial and temporal scales. The clustering coefficient quantifies the abundance of connected triangles in a network and is a major descriptive statistics of networks. For example, it finds an application in the assessment of small-worldness of brain networks, which is affected by attentional and cognitive conditions, age, psychiatric disorders and so forth. However, it remains unclear how the clustering coefficient should be measured in a correlation-based network, which is among major representations of brain networks. In the present article, we propose clustering coefficients tailored to correlation matrices. The key idea is to use three-way partial correlation or partial mutual information to measure the strength of the association between the two neighboring nodes of a focal node relative to the amount of pseudo-correlation expected from indirect paths between the nodes. Our method avoids the difficulties of previous applications of clustering coefficient (and other) measures in defining correlational networks, i.e., thresholding on the correlation value, discarding of negative correlation values, the pseudo-correlation problem and full partial correlation matrices whose estimation is computationally difficult. For proof of concept, we apply the proposed clustering coefficient measures to functional magnetic resonance imaging data obtained from healthy participants of various ages and compare them with conventional clustering coefficients. We show that the clustering coefficients decline with the age. The proposed clustering coefficients are more strongly correlated with age than the conventional ones are. We also show that the local variants of the proposed clustering coefficients (i.e., abundance of triangles around a focal node) are useful in characterizing individual nodes. In contrast, the conventional local clustering coefficients were strongly

  6. Template-directed synthesis of linear porphyrin oligomers: classical, Vernier and mutual Vernier† †Electronic supplementary information (ESI) available: Synthesis and characterization of new compounds, ladder complexes, UV-vis-NIR titrations and binding data for reference compounds and for the formation of linear oligomer complexes, calculation of effective molarities, analytical GPC calibration and molar absorption coefficients. See DOI: 10.1039/c6sc05355f Click here for additional data file.

    Kamonsutthipaijit, Nuntaporn

    2017-01-01

    Three different types of template-directed syntheses of linear porphyrin oligomers are presented. In the classical approach the product has the same number of binding sites as the template, whereas in Vernier reactions the product has the lowest common multiple of the numbers of binding sites in the template and the building block. Mutual Vernier templating is like Vernier templating except that both strands of the Vernier complex undergo coupling simultaneously, so that it becomes impossible to say which is the ‘template’ and which is the ‘building block’. The template-directed synthesis of monodisperse linear oligomers is more difficult than that of cyclic oligomers, because the products of linear templating have reactive ends. All three types of templating are demonstrated here, and used to prepare a nickel(ii) porphyrin dodecamer with 4-pyridyl substituents on all twelve porphyrin units. The stabilities and cooperativities of the double-strand complexes involved in these reactions were investigated by UV-vis-NIR titration. The four-rung ladder duplex has a stability constant of about 2 × 1018 M–1 in dichloromethane at 298 K. PMID:28553508

  7. Converting Sabine absorption coefficients to random incidence absorption coefficients

    Jeong, Cheol-Ho

    2013-01-01

    are suggested: An optimization method for the surface impedances for locally reacting absorbers, the flow resistivity for extendedly reacting absorbers, and the flow resistance for fabrics. With four porous type absorbers, the conversion methods are validated. For absorbers backed by a rigid wall, the surface...... coefficients to random incidence absorption coefficients are proposed. The overestimations of the Sabine absorption coefficient are investigated theoretically based on Miki's model for porous absorbers backed by a rigid wall or an air cavity, resulting in conversion factors. Additionally, three optimizations...... impedance optimization produces the best results, while the flow resistivity optimization also yields reasonable results. The flow resistivity and flow resistance optimization for extendedly reacting absorbers are also found to be successful. However, the theoretical conversion factors based on Miki's model...

  8. Linear Algebra and Smarandache Linear Algebra

    Vasantha, Kandasamy

    2003-01-01

    The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite p...

  9. Molar extinction coefficients of solutions of some organic compounds

    efficients of these compounds have been used to calculate effective atomic numbers and electron densities. The additivity ... Molar extinction coefficients; effective atomic numbers; electron density. PACS Nos 29.30. ... the radiation path and is linearly dependent on the concentration of the absorbing species. This is most ...

  10. On the misinterpretation of the correlation coefficient in pharmaceutical sciences

    Sonnergaard, Jørn

    2006-01-01

    The correlation coefficient is often used and more often misused as a universal parameter expressing the quality in linear regression analysis. The popularity of this dimensionless quantity is evident as it is easy to communicate and considered to be unproblematic to comprehend. However, illustra...

  11. Time correlation functions and transport coefficients in a dilute superfluid

    Kirkpatrick, T.R.; Dorfman, J.R.

    1985-01-01

    Time correlation functions for the transport coefficients in the linear Landau-Khalatnikov equations are derived on the basis of a formal theory. These Green--Kubo expressions are then explicitly evaluated for a dilute superfluid and the resulting transport coefficiencts are shown to be identical to those obtained previously by using a distribution function method

  12. Power coefficient anomaly in JOYO

    Yamamoto, H

    1980-12-15

    Operation of the JOYO experimental fast reactor with the MK-I core has been divided into two phases: (1) 50 MWt power ascension and operation; and (2) 75 MWt power ascension and operation. The 50 MWt power-up tests were conducted in August 1978. In these tests, the measured reactivity loss due to power increases from 15 MWt to 50 MWt was 0.28% ..delta.. K/K, and agreed well with the predicted value of 0.27% ..delta.. K/K. The 75 MWt power ascension tests were conducted in July-August 1979. In the process of the first power increase above 50 MWt to 65 MWt conducted on July 11, 1979, an anomalously large negative power coefficient was observed. The value was about twice the power coefficient values measured in the tests below 50 MW. In order to reproduce the anomaly, the reactor power was decreased and again increased up to the maximum power of 65 MWt. However, the large negative power coefficient was not observed at this time. In the succeeding power increase from 65 MWt to 75 MWt, a similar anomalous power coefficient was again observed. This anomaly disappeared in the subsequent power ascensions to 75 MWt, and the magnitude of the power coefficient gradually decreased with power cycles above the 50 MWt level.

  13. Analysis of internal conversion coefficients

    Coursol, N.; Gorozhankin, V.M.; Yakushev, E.A.; Briancon, C.; Vylov, Ts.

    2000-01-01

    An extensive database has been assembled that contains the three most widely used sets of calculated internal conversion coefficients (ICC): [Hager R.S., Seltzer E.C., 1968. Internal conversion tables. K-, L-, M-shell Conversion coefficients for Z=30 to Z=103, Nucl. Data Tables A4, 1-237; Band I.M., Trzhaskovskaya M.B., 1978. Tables of gamma-ray internal conversion coefficients for the K-, L- and M-shells, 10≤Z≤104, Special Report of Leningrad Nuclear Physics Institute; Roesel F., Fries H.M., Alder K., Pauli H.C., 1978. Internal conversion coefficients for all atomic shells, At. Data Nucl. Data Tables 21, 91-289] and also includes new Dirac-Fock calculations [Band I.M. and Trzhaskovskaya M.B., 1993. Internal conversion coefficients for low-energy nuclear transitions, At. Data Nucl. Data Tables 55, 43-61]. This database is linked to a computer program to plot ICCs and their combinations (sums and ratios) as a function of Z and energy, as well as relative deviations of ICC or their combinations for any pair of tabulated data. Examples of these analyses are presented for the K-shell and total ICCs of the gamma-ray standards [Hansen H.H., 1985. Evaluation of K-shell and total internal conversion coefficients for some selected nuclear transitions, Eur. Appl. Res. Rept. Nucl. Sci. Tech. 11.6 (4) 777-816] and for the K-shell and total ICCs of high multipolarity transitions (total, K-, L-, M-shells of E3 and M3 and K-shell of M4). Experimental data sets are also compared with the theoretical values of these specific calculations

  14. Coding of Information in Limit Cycle Oscillators

    Schleimer, Jan-Hendrik; Stemmler, Martin

    2009-12-01

    Starting from a general description of noisy limit cycle oscillators, we derive from the Fokker-Planck equations the linear response of the instantaneous oscillator frequency to a time-varying external force. We consider the time series of zero crossings of the oscillator’s phase and compute the mutual information between it and the driving force. A direct link is established between the phase response curve summarizing the oscillator dynamics and the ability of a limit cycle oscillator, such as a heart cell or neuron, to encode information in the timing of peaks in the oscillation.

  15. An efficient explicit numerical scheme for diffusion-type equations with a highly inhomogeneous and highly anisotropic diffusion tensor

    Larroche, O.

    2007-01-01

    A locally split-step explicit (LSSE) algorithm was developed for efficiently solving a multi-dimensional advection-diffusion type equation involving a highly inhomogeneous and highly anisotropic diffusion tensor, which makes the problem very ill-conditioned for standard implicit methods involving the iterative solution of large linear systems. The need for such an optimized algorithm arises, in particular, in the frame of thermonuclear fusion applications, for the purpose of simulating fast charged-particle slowing-down with an ion Fokker-Planck code. The LSSE algorithm is presented in this paper along with the results of a model slowing-down problem to which it has been applied

  16. Kinetic theory and simulation of multi-species plasmas in tokamaks excited with ICRF microwaves

    Kerbel, G.D.; McCoy, M.G.

    1984-01-01

    This paper presents a description of a bounce-averaged Fokker-Planck quasilinear model for the kinetic description of tokamak plasmas. The non-linear collision and quasilinear resonant diffusion operators are represented in a form conducive to numerical solution with specific attention to the treatment of the boundary layer separating trapped and passing orbit regions of velocity space. The numerical techniques employed are detailed in so far as they constitute significant departure from those used in the conventional uniform magnetic field case. Examples are given to illustrate the combined effects of collisional and resonant diffusion

  17. Proceedings of eighth joint workshop on electron cyclotron emission and electron cyclotron resonance heating. Vol. 1

    1993-03-01

    The theory of electron cyclotron resonance phenomena is highly developed. The main theoretical tools are well established, generally accepted and able to give a satisfactory description of the main results obtained in electron cyclotron emission, absorption and current drive experiments. In this workshop some advanced theoretical and numerical tools have been presented (e.g., 3-D Fokker-Planck codes, treatment of the r.f. beam as a whole, description of non-linear and finite-beam effects) together with the proposal for new scenarios for ECE and ECA measurements (e.g., for diagnosing suprathermal populations and their radial transport). (orig.)

  18. Current-drive theory II: the lower-hybrid wave

    Fisch, N.J.

    1986-01-01

    The theory of current-drive seeks to predict the efficiency with which an external power source can produce current in a plasma torus. The theory, which is now well supported by experimental data, becomes especially simple in the important limit of lower-hybrid or electron-cyclotron waves interacting with superthermal electrons. The solution of an equation adjoint to the linearized Fokker-Planck equation gives both the steady-state and ramp-up current-drive efficiencies. Other phenomena, such as rf-induced runaway rates, rf-induced radiation, etc., may be calculated by this method, and analytical solutions have been obtained in several limiting cases. 12 refs

  19. Time dependent mean-field games

    Gomes, Diogo A.; Pimentel, Edgard; Sá nchez-Morgado, Hé ctor

    2014-01-01

    In this setting we recur to a delicate argument that combines the non-linear adjoint method with polynomial estimates for solutions of the Fokker-Planck equation in terms of L1L1-norms of DpH. Concerning the subquadratic case, we substantially improve and extend the results previously obtained. Furthermore, to the best of our knowledge, the superquadratic case has not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of Hamilton-Jacobi equations with superquadratic Hamiltonians.

  20. Analysis of discrete reaction-diffusion equations for autocatalysis and continuum diffusion equations for transport

    Wang, Chi-Jen [Iowa State Univ., Ames, IA (United States)

    2013-01-01

    In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.

  1. Aggressive time step selection for the time asymptotic velocity diffusion problem

    Hewett, D.W.; Krapchev, V.B.; Hizanidis, K.; Bers, A.

    1984-12-01

    An aggressive time step selector for an ADI algorithm is preseneted that is applied to the linearized 2-D Fokker-Planck equation including an externally imposed quasilinear diffusion term. This method provides a reduction in CPU requirements by factors of two or three compared to standard ADI. More important, the robustness of the procedure greatly reduces the work load of the user. The procedure selects a nearly optimal Δt with a minimum of intervention by the user thus relieving the need to supervise the algorithm. In effect, the algorithm does its own supervision by discarding time steps made with Δt too large

  2. The exact fundamental solution for the Benes tracking problem

    Balaji, Bhashyam

    2009-05-01

    The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

  3. Proceedings of eighth joint workshop on electron cyclotron emission and electron cyclotron resonance heating. Vol. 2

    1993-03-01

    The theory of electron cyclotron resonance phenomena is highly developed. The main theoretical tools are well established, generally accepted and able to give a satisfactory description of the main results obtained in electron cyclotron emission, absorption and current drive experiments. In this workshop some advanced theoretical and numerical tools have been presented (e.g., 3-D Fokker-Planck codes, treatment of the r.f. beam as a whole, description of non-linear and finite-beam effects) together with the proposal for new scenarios for ECE and ECA measurements (e.g., for diagnosing suprathermal populations and their radial transport). (orig.)

  4. Algebraic polynomials with random coefficients

    K. Farahmand

    2002-01-01

    Full Text Available This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω+a1(ω(n11/2x+a2(ω(n21/2x2+…an(ω(nn1/2xn when n is large. The coefficients {aj(w}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr. A special case of dependent coefficients is also studied.

  5. Determination of ion diffusion coefficients by the electromigration method

    Bonchev, G.D.; Milanov, M.V.; Bozhikov, G.A.; Ivanov, P.I.; Priemyshev, A.N.; Maslov, O.D.; Dmitriev, S.N.

    2003-01-01

    An electrophoretic method for measuring ion diffusion coefficients in aqueous solutions is developed. The value of the diffusion coefficient can be determined from the linear relationship between the square standard deviation of the electrophoretic zone and the time from the start of the diffusion process. Using the device for horizontal zone electrophoresis in a free electrolyte, a series of diffusion experiments are performed with no-carrier-added radionuclides in microconcentrations (10 -9 - 10 -10 M). Diffusion coefficients of 111 In(III), 175 Hf(IV) and 237 Pu(VI) ions at 25 0 C are determined in nitric acid media. Simultaneous determination of the diffusion coefficient and electrophoretic mobility allows one to calculate the effective charge of the investigated ions in accordance with the Nernst-Einstein law

  6. Transport Coefficients for Holographic Hydrodynamics at Finite Energy Scale

    Ge, Xian-Hui; Fang, Li Qing; Yang, Guo-Hong; Leng, Hong-Qiang

    2014-01-01

    We investigate the relations between black hole thermodynamics and holographic transport coefficients in this paper. The formulae for DC conductivity and diffusion coefficient are verified for electrically single-charged black holes. We examine the correctness of the proposed expressions by taking charged dilatonic and single-charged STU black holes as two concrete examples, and compute the flows of conductivity and diffusion coefficient by solving the linear order perturbation equations. We then check the consistence by evaluating the Brown-York tensor at a finite radial position. Finally, we find that the retarded Green functions for the shear modes can be expressed easily in terms of black hole thermodynamic quantities and transport coefficients

  7. Verifying reciprocal relations for experimental diffusion coefficients in multicomponent mixtures

    Medvedev, Oleg; Shapiro, Alexander

    2003-01-01

    The goal of the present study is to verify the agreement of the available data on diffusion in ternary mixtures with the theoretical requirement of linear non-equilibrium thermodynamics consisting in symmetry of the matrix of the phenomenological coefficients. A common set of measured diffusion...... coefficients for a three-component mixture consists of four Fickian diffusion coefficients, each being reported separately. However, the Onsager theory predicts the existence of only three independent coefficients, as one of them disappears due to the symmetry requirement. Re-calculation of the Fickian...... extended sets of experimental data and reliable thermodynamic models were available. The sensitivity of the symmetry property to different thermodynamic parameters of the models was also checked. (C) 2003 Elsevier Science B.V. All rights reserved....

  8. [Electroencephalogram Feature Selection Based on Correlation Coefficient Analysis].

    Zhou, Jinzhi; Tang, Xiaofang

    2015-08-01

    In order to improve the accuracy of classification with small amount of motor imagery training data on the development of brain-computer interface (BCD systems, we proposed an analyzing method to automatically select the characteristic parameters based on correlation coefficient analysis. Throughout the five sample data of dataset IV a from 2005 BCI Competition, we utilized short-time Fourier transform (STFT) and correlation coefficient calculation to reduce the number of primitive electroencephalogram dimension, then introduced feature extraction based on common spatial pattern (CSP) and classified by linear discriminant analysis (LDA). Simulation results showed that the average rate of classification accuracy could be improved by using correlation coefficient feature selection method than those without using this algorithm. Comparing with support vector machine (SVM) optimization features algorithm, the correlation coefficient analysis can lead better selection parameters to improve the accuracy of classification.

  9. Path coefficient analysis of zinc dynamics in varying soil environment

    Rattan, R.K.; Phung, C.V.; Singhal, S.K.; Deb, D.L.; Singh, A.K.

    1994-01-01

    Influence of soil properties on labile zinc, as measured by diethylene-triamine pentaacetic acid (DTPA) and zinc-65, and self-diffusion coefficients of zinc was assessed on 22 surface soil samples varying widely in their characteristics following linear regression and path coefficient analysis techniques. DTPA extractable zinc could be predicted from organic carbon status and pH of the soil with a highly significant coefficient of determination (R 2 =0.84 ** ). Ninety seven per cent variation in isotopically exchangeable zinc was explained by pH, clay content and cation exchange capacity (CEC) of soil. The self-diffusion coefficients (DaZn and DpZn) and buffer power of zinc exhibited exponential relationship with soil properties, pH being the most dominant one. Soil properties like organic matter, clay content etc. exhibited indirect effects on zinc diffusion rates via pH only. (author). 13 refs., 6 tabs

  10. A linear temperature-to-frequency converter

    Løvborg, Leif

    1965-01-01

    , and that the maximum value of the temperature-frequency coefficient beta in this point is-1/3 alpha, where a is the temperature coefficient of the thermistor at the corresponding temperature. Curves showing the range in which the converter is expected to be linear to within plusmn0.1 degC are given. A laboratory......-built converter having beta = 1.0% degC-1 at 25degC is found to be linear to within plusmn0. 1 degC from 10 to 40degC....

  11. Irrational "Coefficients" in Renaissance Algebra.

    Oaks, Jeffrey A

    2017-06-01

    Argument From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply by x, for instance, the result was expressed as the rhetorical equivalent of . The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or "number," of a term was thought of as how many of that term are present, and not as the scalar multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff (1525) was the first to admit them in special cases, and subsequently they appear more liberally in Cardano (1539), Scheubel (1550), Bombelli (1572), and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, "cossic" algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text.

  12. Integer Solutions of Binomial Coefficients

    Gilbertson, Nicholas J.

    2016-01-01

    A good formula is like a good story, rich in description, powerful in communication, and eye-opening to readers. The formula presented in this article for determining the coefficients of the binomial expansion of (x + y)n is one such "good read." The beauty of this formula is in its simplicity--both describing a quantitative situation…

  13. Analogies between colored Lévy noise and random channel approach to disordered kinetics

    Vlad, Marcel O.; Velarde, Manuel G.; Ross, John

    2004-02-01

    We point out some interesting analogies between colored Lévy noise and the random channel approach to disordered kinetics. These analogies are due to the fact that the probability density of the Lévy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics. Although the equations for the two approaches are not identical, the analogies can be used for deriving new, useful results for both problems. The random channel approach makes it possible to generalize the fractional Uhlenbeck-Ornstein processes (FUO) for space- and time-dependent colored noise. We describe the properties of colored noise in terms of characteristic functionals, which are evaluated by using a generalization of Huber's approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We start out by investigating the properties of symmetrical white noise and then define the Lévy colored noise in terms of a Langevin equation with a Lévy white noise source. We derive exact analytical expressions for the various characteristic functionals, which characterize the noise, and a functional fractional Fokker-Planck equation for the probability density functional of the noise at a given moment in time. Second, by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics, we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics. These equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data. Special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics.

  14. Stochastic Linear Quadratic Optimal Control Problems

    Chen, S.; Yong, J.

    2001-01-01

    This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward-backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well

  15. Multicollinearity in hierarchical linear models.

    Yu, Han; Jiang, Shanhe; Land, Kenneth C

    2015-09-01

    This study investigates an ill-posed problem (multicollinearity) in Hierarchical Linear Models from both the data and the model perspectives. We propose an intuitive, effective approach to diagnosing the presence of multicollinearity and its remedies in this class of models. A simulation study demonstrates the impacts of multicollinearity on coefficient estimates, associated standard errors, and variance components at various levels of multicollinearity for finite sample sizes typical in social science studies. We further investigate the role multicollinearity plays at each level for estimation of coefficient parameters in terms of shrinkage. Based on these analyses, we recommend a top-down method for assessing multicollinearity in HLMs that first examines the contextual predictors (Level-2 in a two-level model) and then the individual predictors (Level-1) and uses the results for data collection, research problem redefinition, model re-specification, variable selection and estimation of a final model. Copyright © 2015 Elsevier Inc. All rights reserved.

  16. A Trivial Linear Discriminant Function

    Shuichi Shinmura

    2015-11-01

    Full Text Available In this paper, we focus on the new model selection procedure of the discriminant analysis. Combining re-sampling technique with k-fold cross validation, we develop a k-fold cross validation for small sample method. By this breakthrough, we obtain the mean error rate in the validation samples (M2 and the 95\\% confidence interval (CI of discriminant coefficient. Moreover, we propose the model  selection  procedure  in  which  the model having a minimum M2 was  chosen  to  the  best  model.  We  apply  this  new  method and procedure to the pass/ fail determination of  exam  scores.  In  this  case,  we  fix  the constant =1 for seven linear discriminant  functions  (LDFs  and  several  good  results  were obtained as follows: 1 M2 of Fisher's LDF are over 4.6\\% worse than Revised IP-OLDF. 2 A soft-margin  SVM  for  penalty c=1  (SVM1  is  worse  than  another  mathematical  programming (MP based LDFs and logistic regression . 3 The 95\\% CI of the best discriminant coefficients was obtained. Seven LDFs except for Fisher's LDF are almost the same as a trivial LDF for the linear separable model. Furthermore, if we choose the median of the coefficient of seven LDFs except for Fisher's LDF,  those are almost the same as the trivial LDF for the linear separable model.

  17. Calibration factor or calibration coefficient?

    Meghzifene, A.; Shortt, K.R.

    2002-01-01

    Full text: The IAEA/WHO network of SSDLs was set up in order to establish links between SSDL members and the international measurement system. At the end of 2001, there were 73 network members in 63 Member States. The SSDL network members provide calibration services to end-users at the national or regional level. The results of the calibrations are summarized in a document called calibration report or calibration certificate. The IAEA has been using the term calibration certificate and will continue using the same terminology. The most important information in a calibration certificate is a list of calibration factors and their related uncertainties that apply to the calibrated instrument for the well-defined irradiation and ambient conditions. The IAEA has recently decided to change the term calibration factor to calibration coefficient, to be fully in line with ISO [ISO 31-0], which recommends the use of the term coefficient when it links two quantities A and B (equation 1) that have different dimensions. The term factor should only be used for k when it is used to link the terms A and B that have the same dimensions A=k.B. However, in a typical calibration, an ion chamber is calibrated in terms of a physical quantity such as air kerma, dose to water, ambient dose equivalent, etc. If the chamber is calibrated together with its electrometer, then the calibration refers to the physical quantity to be measured per electrometer unit reading. In this case, the terms referred have different dimensions. The adoption by the Agency of the term coefficient to express the results of calibrations is consistent with the 'International vocabulary of basic and general terms in metrology' prepared jointly by the BIPM, IEC, ISO, OIML and other organizations. The BIPM has changed from factor to coefficient. The authors believe that this is more than just a matter of semantics and recommend that the SSDL network members adopt this change in terminology. (author)

  18. Extinction Coefficient of Gold Nanostars

    de Puig, Helena; Tam, Justina O.; Yen, Chun-Wan; Gehrke, Lee; Hamad-Schifferli, Kimberly

    2015-01-01

    Gold nanostars (NStars) are highly attractive for biological applications due to their surface chemistry, facile synthesis and optical properties. Here, we synthesize NStars in HEPES buffer at different HEPES/Au ratios, producing NStars of different sizes and shapes, and therefore varying optical properties. We measure the extinction coefficient of the synthesized NStars at their maximum surface plasmon resonances (SPR), which range from 5.7 × 108 to 26.8 × 108 M−1cm−1. Measured values correl...

  19. Infinite time interval backward stochastic differential equations with continuous coefficients.

    Zong, Zhaojun; Hu, Feng

    2016-01-01

    In this paper, we study the existence theorem for [Formula: see text] [Formula: see text] solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for [Formula: see text] [Formula: see text] solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (Turkish J Math 37:704-718, 2013).

  20. Accounting for Antenna in Half-Space Fresnel Coefficient Estimation

    A. D'Alterio

    2012-01-01

    Full Text Available The problem of retrieving the Fresnel reflection coefficients of a half-space medium starting from measurements collected under a reflection mode multistatic configuration is dealt with. According to our previous results, reflection coefficient estimation is cast as the inversion of linear operator. However, here, we take a step ahead towards more realistic scenarios as the role of antennas (both transmitting and receiving is embodied in the estimation procedure. Numerical results are presented to show the effectiveness of the method for different types of half-space media.

  1. Estimation of octanol/water partition coefficients using LSER parameters

    Luehrs, Dean C.; Hickey, James P.; Godbole, Kalpana A.; Rogers, Tony N.

    1998-01-01

    The logarithms of octanol/water partition coefficients, logKow, were regressed against the linear solvation energy relationship (LSER) parameters for a training set of 981 diverse organic chemicals. The standard deviation for logKow was 0.49. The regression equation was then used to estimate logKow for a test of 146 chemicals which included pesticides and other diverse polyfunctional compounds. Thus the octanol/water partition coefficient may be estimated by LSER parameters without elaborate software but only moderate accuracy should be expected.

  2. Linearly constrained minimax optimization

    Madsen, Kaj; Schjær-Jacobsen, Hans

    1978-01-01

    We present an algorithm for nonlinear minimax optimization subject to linear equality and inequality constraints which requires first order partial derivatives. The algorithm is based on successive linear approximations to the functions defining the problem. The resulting linear subproblems...

  3. Form of multicomponent Fickian diffusion coefficients matrix

    Wambui Mutoru, J.; Firoozabadi, Abbas

    2011-01-01

    Highlights: → Irreversible thermodynamics establishes form of multicomponent diffusion coefficients. → Phenomenological coefficients and thermodynamic factors affect sign of diffusion coefficients. → Negative diagonal elements of diffusion coefficients matrix can occur in non-ideal mixtures. → Eigenvalues of the matrix of Fickian diffusion coefficients may not be all real. - Abstract: The form of multicomponent Fickian diffusion coefficients matrix in thermodynamically stable mixtures is established based on the form of phenomenological coefficients and thermodynamic factors. While phenomenological coefficients form a symmetric positive definite matrix, the determinant of thermodynamic factors matrix is positive. As a result, the Fickian diffusion coefficients matrix has a positive determinant, but its elements - including diagonal elements - can be negative. Comprehensive survey of reported diffusion coefficients data for ternary and quaternary mixtures, confirms that invariably the determinant of the Fickian diffusion coefficients matrix is positive.

  4. The linear-non-linear frontier for the Goldstone Higgs

    Gavela, M.B.; Saa, S.; Kanshin, K.; Machado, P.A.N.

    2016-01-01

    The minimal SO(5)/SO(4) σ-model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone-boson ancestry. Varying the σ mass allows one to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are independent of the explicit soft breaking assumed. The Higgs couplings to vector bosons and fermions turn out to be quite universal: the linear corrections are proportional to the explicit symmetry-breaking parameters. Furthermore, we define an effective Yukawa operator which allows a simple parametrization and comparison of different heavy-fermion ultraviolet completions. In addition, one particular fermionic completion is explored in detail, obtaining the corresponding leading low-energy fermionic operators. (orig.)

  5. The linear-non-linear frontier for the Goldstone Higgs

    Gavela, M.B.; Saa, S. [IFT-UAM/CSIC, Universidad Autonoma de Madrid, Departamento de Fisica Teorica y Instituto de Fisica Teorica, Madrid (Spain); Kanshin, K. [Universita di Padova, Dipartimento di Fisica e Astronomia ' G. Galilei' , Padua (Italy); INFN, Padova (Italy); Machado, P.A.N. [IFT-UAM/CSIC, Universidad Autonoma de Madrid, Departamento de Fisica Teorica y Instituto de Fisica Teorica, Madrid (Spain); Fermi National Accelerator Laboratory, Theoretical Physics Department, Batavia, IL (United States)

    2016-12-15

    The minimal SO(5)/SO(4) σ-model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone-boson ancestry. Varying the σ mass allows one to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are independent of the explicit soft breaking assumed. The Higgs couplings to vector bosons and fermions turn out to be quite universal: the linear corrections are proportional to the explicit symmetry-breaking parameters. Furthermore, we define an effective Yukawa operator which allows a simple parametrization and comparison of different heavy-fermion ultraviolet completions. In addition, one particular fermionic completion is explored in detail, obtaining the corresponding leading low-energy fermionic operators. (orig.)

  6. Estimates of vertical velocities and eddy coefficients in the Bay of Bengal

    Varkey, M.J.; Sastry, J.S.

    Vertical velocities and eddy coefficients in the intermediate depths of the Bay of Bengal are calculated from mean hydrographic data for 300 miles-squares. The linear current density (sigma- O) versus log-depth plots show steady balance between...

  7. Uzawa method for fuzzy linear system

    Ke Wang

    2013-01-01

    An Uzawa method is presented for solving fuzzy linear systems whose coefficient matrix is crisp and the right-hand side column is arbitrary fuzzy number vector. The explicit iterative scheme is given. The convergence is analyzed with convergence theorems and the optimal parameter is obtained. Numerical examples are given to illustrate the procedure and show the effectiveness and efficiency of the method.

  8. Earning on Response Coefficient in Automobile and Go Public Companies

    Lisdawati Arifin

    2017-09-01

    Full Text Available This study aims to analyze factors that influence earnings response coefficients (ERC, simultaneously and partially, composed of leverage, the systematic risk (beta, growth opportunities (market to book value ratio, and the size of the firm (firm size, selection of the sample in this study the author take 12 automakers and components that meet the criteria of completeness of the data from the year 2008 to 2012, entirely based on consideration of the following criteria: (1 the company's automotive and components are listed on the stock exchange, (2 have the financial statements years 2008-2012 (3 has a return data (closing price the first day after the date of issuance of the financial statements. This study uses secondary data applying multiple linear regression models to analyze and test the effect of independent variables on the dependent variable partially (t-test, simultaneous (f-test, and the goodness of fit (R-square on a research model. The result shows that leverage, beta, growth opportunities (market to book value ratio and size along with (simultaneously the effect on the dependent variable (dependent variable earnings response coefficients. Partially leverage negatively affect earnings response coefficients, partially beta negatively correlated earnings response coefficients, partially growth opportunities (market to book value ratio significant effect on earnings response coefficients, partially sized companies (firm size significantly influence earnings response coefficients.

  9. Negative power coefficient on PHWRs with CARA fuel

    Lestani, H.A.; González, H.J.; Florido, P.C.

    2014-01-01

    Highlights: • A PHWR fuel was optimized to obtain a negative power coefficient. • Fuel cost, being a measure of design investment efficiency, was optimized. • Influence on power coefficient of geometrical and economical parameters’ was studied. • Different neutronic absorbers were studied; pure absorbers can be used. • Thermal and economical models were developed to complement neutronic assessment. - Abstract: A study of power coefficient of reactivity in heavy water reactors is made analyzing the reactivity components of fuels with several modifications oriented at reducing the coefficient. A cell model is used for neutronics calculations; a non-linear two dimensional model is used to evaluate the thermal changes that follow a power change; and a levelized unit energy cost model is used to assess the economical feasibility of the design changes introduced to reduce power coefficient. The necessity of modelling all the aforementioned quantities in a coupled scheme is stressed, as a strong interdependence was found. A series of design changes complied with a negative power coefficient of reactivity, with a feasible power radial distribution and with low refuelling cost. Some investigation lines that exceed the fuel cell study and deal with the plant operation are marked as potentially addressing the stable operation of big heavy water reactors

  10. Foundations of linear and generalized linear models

    Agresti, Alan

    2015-01-01

    A valuable overview of the most important ideas and results in statistical analysis Written by a highly-experienced author, Foundations of Linear and Generalized Linear Models is a clear and comprehensive guide to the key concepts and results of linear statistical models. The book presents a broad, in-depth overview of the most commonly used statistical models by discussing the theory underlying the models, R software applications, and examples with crafted models to elucidate key ideas and promote practical model building. The book begins by illustrating the fundamentals of linear models,

  11. Relationship between thermal expansion coefficient and glass transition temperature in metallic glasses

    Kato, H.; Chen, H.-S.; Inoue, A.

    2008-01-01

    The thermal expansion coefficients of 13 metallic glasses were measured using a thermo-mechanical analyser. A unique correlation was found between the linear thermal expansion coefficient and the glass transition temperature-their product is nearly constant ∼8.24 x 10 -3 . If one assumes the Debye expression for thermal activation, the total linear thermal expansion up to glass transition temperature (T g ) is reduced to 6 x 10 -3 , nearly 25% of that at the fusion of pure metals

  12. Linear Magnetochiral effect in Weyl Semimetals

    Cortijo, Alberto

    We describe the presence of a linear magnetochiral effect in time reversal breaking Weyl semimetals. The magnetochiral effect consists in a simultaneous linear dependence of the magnetotransport coefficients with the magnetic field and a momentum vector. This simultaneous dependence is allowed by the Onsager reciprocity relations, being the separation vector between the Weyl nodes the vector that plays such role. This linear magnetochiral effect constitutes a new transport effect associated to the topological structures linked to time reversal breaking Weyl semimetals. European Union structural funds and the Comunidad de Madrid MAD2D-CM Program (S2013/MIT-3007) and MINECO (Spain) Grant No. FIS2015-73454-JIN.

  13. Study of transport coefficients of nanodiamond nanofluids

    Pryazhnikov, M. I.; Minakov, A. V.; Guzei, D. V.

    2017-09-01

    Experimental data on the thermal conductivity coefficient and viscosity coefficient of nanodiamond nanofluids are presented. Distilled water and ethylene glycol were used as the base fluid. Dependences of transport coefficients on concentration are obtained. It was shown that the thermal conductivity coefficient increases with increasing nanodiamonds concentration. It was shown that base fluids properties and nanodiamonds concentration affect on the rheology of nanofluids.

  14. Third-order operator-differential equations with discontinuous coefficients and operators in the boundary conditions

    Araz R. Aliev

    2013-10-01

    Full Text Available We study a third-order operator-differential equation on the semi-axis with a discontinuous coefficient and boundary conditions which include an abstract linear operator. Sufficient conditions for the well-posed and unique solvability are found by means of properties of the operator coefficients in a Sobolev-type space.

  15. Diffusion-Coefficients of Sulfate and Methane in Marine-Sediments - Influence of Porosity

    IVERSEN, N.; JØRGENSEN, BB

    1993-01-01

    diffusion coefficients can be related to the diffusion coefficient in free solution by D(s) = D(o)/theta2, where theta is the tortuosity of the sediment. The sediment tortuosity calculated from this equation showed a linear relationship with sediment porosity (phi) over the porosity range of 0.4-0.9. From...

  16. Evaluation of Rock Joint Coefficients

    Audy, Ondřej; Ficker, Tomáš

    2017-10-01

    A computer method for evaluation of rock joint coefficients is described and several applications are presented. The method is based on two absolute numerical indicators that are formed by means of the Fourier replicas of rock joint profiles. The first indicator quantifies the vertical depth of profiles and the second indicator classifies wavy character of profiles. The absolute indicators have replaced the formerly used relative indicators that showed some artificial behavior in some cases. This contribution is focused on practical computations testing the functionality of the newly introduced indicators.

  17. Synchronization of linearly coupled unified chaotic systems based on linear balanced feedback scheme with constraints

    Chen, H.-H.; Chen, C.-S.; Lee, C.-I

    2009-01-01

    This paper investigates the synchronization of unidirectional and bidirectional coupled unified chaotic systems. A balanced coupling coefficient control method is presented for global asymptotic synchronization using the Lyapunov stability theorem and a minimum scheme with no constraints/constraints. By using the result of the above analysis, the balanced coupling coefficients are then designed to achieve the chaos synchronization of linearly coupled unified chaotic systems. The feasibility and effectiveness of the proposed chaos synchronization scheme are verified via numerical simulations.

  18. Correlation coefficients in neutron β-decay

    Byrne, J.

    1978-01-01

    The various angular and polarisation coefficients in neutron decay are the principal sources of information on the β-interaction. Measurements of the electron-neutrino angular correlation coefficient (a), the neutron-spin-electron-momentum correlation coefficient (A), the neutron-spin-neutrino-momentum correlation coefficient (B), and the triple correlation coefficient D and time-reversal invariance are reviewed and the results discussed. (U.K.)

  19. Characterization of the effective electrostriction coefficients in ferroelectric thin films

    Kholkin, A. L.; Akdogan, E. K.; Safari, A.; Chauvy, P.-F.; Setter, N.

    2001-06-01

    Electromechanical properties of a number of ferroelectric films including PbZrxTi1-xO3(PZT), 0.9PbMg1/3Nb2/3O3-0.1PbTiO3(PMN-PT), and SrBi2Ta2O9(SBT) are investigated using laser interferometry combined with conventional dielectric measurements. Effective electrostriction coefficients of the films, Qeff, are determined using a linearized electrostriction equation that couples longitudinal piezoelectric coefficient, d33, with the polarization and dielectric constant. It is shown that, in PZT films, electrostriction coefficients slightly increase with applied electric field, reflecting the weak contribution of non-180° domains to piezoelectric properties. In contrast, in PMN-PT and SBT films electrostriction coefficients are field independent, indicating the intrinsic nature of the piezoelectric response. The experimental values of Qeff are significantly smaller than those of corresponding bulk materials due to substrate clamping and possible size effects. Electrostriction coefficients of PZT layers are shown to depend strongly on the composition and preferred orientation of the grains. In particular, Qeff of (100) textured rhombohedral films (x=0.7) is significantly greater than that of (111) layers. Thus large anisotropy of the electrostrictive coefficients is responsible for recently observed large piezoelectric coefficients of (100) textured PZT films. Effective electrostriction coefficients obtained by laser interferometry allow evaluation of the electromechanical properties of ferroelectric films based solely on the dielectric parameters and thus are very useful in the design and fabrication of microsensors and microactuators.

  20. Introduction to linear systems of differential equations

    Adrianova, L Ya

    1995-01-01

    The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent. In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1)�autonomous, 2)�periodic, 3)�reducible to autonomous, 4)�nearly reducible to autonomous, 5)�regular. In addition, Adrianova considers the following: stability of linear systems and the influence of perturbations of the coefficients on the stability the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions several estimates of the growth rate of solutions of a linear system in terms of its coefficients How perturbations of the coefficients change all the elements of the spectrum of the system is defin...