By Perthame B. (ed.)

This paintings comprises 36 brief papers on development in various topics in mathematical and theoretical physics, written for the complaints 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 aimed toward researchers, together with graduate scholars, with an interdisciplinary curiosity and provides a short creation to a number of the subject matters of present curiosity. those contain effects on precisely solvable types in statistical mechanics, integrable in the course of the Yang-Baxter equations, quantum teams, fractional information, random matrices, index theorems at the lattice, and different comparable issues I. Vlasov-Poisson in plasma physics -- II. Quantum mechanics and semiconductors -- III. Boltzmann equations and fuel dynamics

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**Example text**

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). 5. The multidimensional laminar Child-Langmuir problem We now summarize the m u l t i d i m e n s i o n a l C h i l d - L a n g m u i r problem under the ass u m p t i o n of laminar beam.

3 (1931). pp 191-257. [20) J. D. Lawson, The physics of charged-particle beams. International Series of monographs on physics n75, Oxford University Press, Oxford, 1983[21] F. Poupaud, Boundary Value Problems for the stationary Vlasov-Maxwell System, Forum Mathematicum, 777. |22| T . Weisterman, A particle-in-cetl method as a tool for diode simulations, Nuclear Instruments and Methods in Physics Research A 260 (1988), pp. 271-279. 45 EULERIAN CODES FOR THE VLASOV EQUATION 2 M R. F E L X ' . P .

V) = / " ( r - v > A / / 2 . 6) It must be pointed out that substituting successively Eqs. (2 5) and (2 4) into (2 6), we obtain f*'(x,v) with x-x-v&i/2. 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).