By Perthame B. (ed.)

ISBN-10: 9810216718

ISBN-13: 9789810216719

This paintings includes 36 brief papers on growth in a number of matters in mathematical and theoretical physics, written for the court cases of a symposium in honor of the seventieth birthday of Professor F.Y. Wu, held on the Nankai Institute of arithmetic, October 7-11, 2001. the gathering of papers is geared toward researchers, together with graduate scholars, with an interdisciplinary curiosity and provides a short creation to some of the issues of present curiosity. those comprise effects on precisely solvable types in statistical mechanics, integrable during the Yang-Baxter equations, quantum teams, fractional facts, random matrices, index theorems at the lattice, and different comparable themes I. Vlasov-Poisson in plasma physics -- II. Quantum mechanics and semiconductors -- III. Boltzmann equations and gasoline dynamics

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I t is n o w possible t o guess w h a t the weak l i m i t o f / w h e n e —• 0 is. W e shall suppose t h a t f —• ip i n a s m o o t h way. W e shall denote by X(t; y, w), V ( f ; y, tu)) t h e characteristics associated w i t h the l i m i t p o t e n t i a l , and (0, T (y,w)\ t h e i r interval of d e f i n i t i o n . }. For y £ r , we define : E max e c 0 U+(y) = { « € I R " , {U,u{y)) < 0. 20) l i m l £ „ ' y , e u ) = 0}. ), shows t h a t we may very w e l l have Meas(U°{yj) > 0. T h e convergence o f (X (t;y,eu),V'{t;y,eu)} is different o n U°(y) and U (y).

48) where J{t, y) is the jacobian of the change of variables ( ( , y) —» x. 31) i f n(X{t,y))J(t,y) does not depend on t. ((,J/} be a smooth function on O, which vanishes on a neighbourhood of the boundary dO a n d let %X~ We have : J ftMX(t y))J(t,y)} (t,y)dtdS(y). y)) - (t,y). y)eO. 54) does not depend on (. y). 57) 0 mai we have, by the Green formula: J^n{X(t,y))J{t )^(t,y)dtdS(y) >y = - j n(X(0,»)) J ( 0 , y ) v ( 0 , y) dS(y). 45) f JTO (M(&,Ks))#*)<&Ce) = " f n(X(Q,y))J(0,y) (0,y)dS(y). 60).

7) Thus we find that Eq. (2 7) is equivalent to the following integrated equations o f the characteristics o f Eq. e. 5) Different methods can be used. Cubic spline interpolation in the v-space and Fourier interpolation in x-space (without using Fast Fourier Transform) have been used by Cheng and Knorr [12], which requires, in the x direction, an execution time proportional to N, (where is the number o f points in the spatial direction). Because this fact increases the computational effort when a large number o f points is used, a cubic spline interpolation can be used also in this direction (see Ref [15]) It can be pointed out that during each time step, the Vlasov equation takes on the form of an advective equation (one over x, one over v).

### Advances in kinetic theory and computing : selected papers by Perthame B. (ed.)

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