numerical analysis of conservative and entropy schemes for the fokker planck landau equation conservatives and entropy schemes for the fokker planck landau fpl equation are studied we prove the existence of a unique positive and global in time solution for the homogeneous linear and nonlinear discretized either in the velocity space or both in the velocity space and in time fpl equation the stability analysis of these schemes leads to sufficient conditions on the time step that guarantee positivity and entropy decay

multipole expansions for the fokker planck landau operator we give a sequence of operators approximating the fokker planck landau collision operator this sequence is obtained by aplying the fast multipole method based on the work by greengard and rokhlin 17 and tends to the exact fokker planck landau operator with an arbitrary accuracy these operators satisfy the physical properties such as the conservation of mass momentum energy and the decay of the entropy furthermore the quadratic structure due to the velocity coupling in the expression of the fokker planck landau operator is removed in the approximating operators this fact reduces seriously the computationnal cost of numerical simulations of the fokker planck landau equation finally we give numerical conservative and entropy discretizations solving the homogeneous fokker planck landau equation using the fast multipole method in addition to the deterministic character of these approximations they give satisfactory results in terms of accuracy and cpu time

conservative finite difference schemes for the fokker planck equations not violating the law of an increasing entropy abstract two finite difference schemes for the fokker planck equation in the case of an isotropic function of the charge particles are constructed these schemes are first order accurate with respect to time and second order accurate with respect to a e 2 the schemes conserve the number of particles and energy and the entropy change is nonnegative a relation between the incresing entropy law and the stability of the finite difference schemes is established

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

identifying active contrasts by stepwise testing unreplicated fractional factorial designs are frequently used in industrial experimentation and are characterized by the absence of degrees of freedom for estimation of the error variance this article presents step down and step up testing procedures that efficiently identify the active contrasts significant effects in such experiments and simultaneously provide an accurate estimate of the error variance the methods allow the user to restrict the experimentwise error rate to a prespecified level use of the new methods is illustrated in terms of standard examples from the literature simulation shows that a version of the step up procedure outperforms its competitors

on the classification of k involutions let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of and gk resp hk the set of k rational points of g resp h the variety gk hk is called a symmetric k variety these varieties occur in many problems in representation theory geometry and singularity theory over the last few decades the representation theory of these varieties has been extensively studied for k r and c as most of the work in these two cases was completed the study of the representation theory over other fields like local fields and finite fields began the representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space like the restricted root systems with weyl groups etc thus it is essential to study first this structure and the related geometry in this paper we give a characterization of the isomorphy classes of these symmetric k varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers p adic numbers finite fields and number fields

the vlasov poisson landau system in mathbb r 3 _ x for the landau poisson system with coulomb interaction in mathbb r 3 x we prove the global existence uniqueness and large time convergence rates to the maxwellian equilibrium for solutions which start out sufficiently close

exponential decay for soft potentials near maxwellian we consider both soft potentials with angular cutoff and landau collision kernels in the boltzmann theory inside a periodic box we prove that any smooth perturbation near a given maxwellian approaches zero at the rate of e lambda t p for some 0 and 0 p 1 our method is based on an unified energy estimate with appropriate exponential velocity weight our results extend the classical result caflisch of 2 to the case of very soft potential and coulomb interactions and also improve the recent almost exponential decay results by 5 14

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

implicit schemes for the fokker planck landau equation we propose time implicit schemes to solve the homogeneous fokker planck landau equation in both the isotropic and 3 dimensional geometries these schemes have properties of conservation and entropy moreover they allow for large time steps making them faster than the usual explicit schemes to solve the involved linear systems we prove that the use of krylov like solvers preserves the conservation properties we show in particular that the conjugate gradient method can be used numerical tests are performed for the isotropic case and demonstrate an important gain in terms of cpu time with the same accuracy as explicit schemes this work is a first step to the development of fast implicit schemes to solve a class of inhomogeneous kinetic equations

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

an entropy scheme for the fokker planck collision operator of plasma kinetic theory we propose a finite difference scheme to approximate the fokker planck collision operator in 3 velocity dimensions the principal feature of this scheme is to provide a decay of the numerical entropy as a consequence it preserves the collisional invariants and its stationary solutions are the discrete maxwellians we consider both the whole velocity space problem and the bounded velocity problem in the latter case we provide artificial boundary conditions which preserve the decay of the entropy

the vlasov poisson landau system in a periodic box the classical vlasov poisson landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions such grazing collisions are modeled by the famous landau fokker planck collision kernel proposed by landau in 1936 we construct global unique solutions to such a system for initial data which have small weighted h norms but can have large h k 3 norms with high velocity moments our construction is based on accumulative study on the landau kernel in the past decade g1 sg1 3 with four extra ingredients to overcome the speci c mathematical di culties present in the vlasov poisson landau system a new exponential weight of electric potential to cancel the growth of the velocity a new velocity weight to capture the weak velocity di usion in the landau kernel a decay of the electric eld to close the energy estimate and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

analytic smoothness effect of solutions for spatially homogeneous landau equation in this paper we study the smoothness e ect of cauchy problem for the spatially homogeneous landau equation in the hard potential case and themaxwellian molecules case we obtain the analytic smoothing e ect for the solutions under rather weak assumptions on the initial datum

an implicit energy conservative 2d fokker planck algorithm energy conservative implicit integration schemes for the fokker planck transport equation in multidimensional geometries require inverting a dense non symmetric matrix jacobian which is very expensive to store and solve using standard solvers however these limitations can be overcome with newton krylov iterative techniques since they can be implemented jacobian free the jacobian matrix from newton s algorithm is never formed nor stored to proceed with the iteration and their convergence can be accelerated by preconditioning the original problem in this document the efficient numerical implementation of an implicit energy conservative scheme for multidimensional fokker planck problems using multigrid preconditioned krylov methods is discussed results show that multigrid preconditioning is very effective in speeding convergence and decreasing cpu requirements particularly in fine meshes the solver is demonstrated on grids up to 128 128 points in a 2d cylindrical velocity space vr vp with implicit time steps of the order of the collisional time scale of the problem the method preserves particles exactly and energy conservation is improved over alternative approaches particularly in coarse meshes typical errors in the total energy over a time period of 10 remain below a percent

identifying significant effects in fractional factorial multiresponse experiments a forward selection procedure for identifying active contrasts in unreplicated factorial multiresponse experiments is presented to test the multivariate effects p values are calculated by monte carlo sampling the ordinary hotelling s t2 test is a special case collinear responses are effectively handled by successive principal component decompositions as when for instance the number of response variables exceeds the number of observations in the univariate case the test statistic pools the smallest effects as error independent sources for error degrees of freedom can be incorporated the method is illustrated with examples from baguette and mayonnaise production

numerical approximation of the spatially homogeneous fokkerplancklandau equation a numerical method is developed for the spatially homogeneous fokkerplancklandau fpl equation for the case of coulomb collisions the equation is put into a form so that finite difference methods for parabolic type pdes can be applied through a transformation of variables the velocity space computation is reduced to a bounded domain and finite differencing is done with a well defined exact boundary condition corresponding to v at infinity for the discretization in time implicit differencing is used an analysis is carried out on the discrete approximation as applied to a linear fpl equation and for the linear equation it is proved that the numerical method is unconditionally stable and convergent computational work is then done that demonstrates the unconditional stability and accuracy of the numerical method for the nonlinear fpl equation

a deterministic particle method for the vlasov fokker planck equation in one dimension the vlasov fokker planck equation is a model for a collisional electrostatic plasma the approximation of this equation in one spatial dimension is studied the equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function the approximation method applied is the deterministic particle method described in wollman and ozizmir numerical approximation of the vlasov poisson fokker planck system in one dimension j comput phys 202 2005 602 644 for the present linear problem an analysis of the stability and convergence of the numerical method is carried out in addition computations are done that verify the convergence of the numerical solution it is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in bouchut and dolbeault on long time asymptotics of the vlasov fokker planck equation and of the vlasov poisson fokker planck system with coulombic and newtonian potentials differential integral equations 8 3 1995 487 514

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p

numerical approximation of the vlasov poisson fokker planck system in two dimensions a numerical method is developed for approximating the solution to the vlasov poisson fokker planck system in two spatial dimensions the method generalizes the approximation for the system in one dimension given in s wollman e ozizmir numerical approximation of the vlasov poisson fokker planck system in one dimension j comput phys 202 2005 602 644 the numerical procedure is based on a change of variables that puts the convection diffusion equation into a form so that finite difference methods for parabolic type partial differential equations can be applied the computational cycle combines a type of deterministic particle method with a periodic interpolation of the solution along particle trajectories onto a fixed grid computational work is done to demonstrate the accuracy and effectiveness of the approximation method parts of the numerical procedure are adapted to run on a parallel computer