By F. Brackx, R. Delanghe, F. Sommen (auth.), R. S. Pathak (eds.)

The overseas Symposium on Generalized features and Their purposes used to be prepared by way of the dept of arithmetic, Banaras Hindu collage, and held December 23-26, 1991, at the celebration of the Platinum Jubilee get together of the college. greater than 100 mathematicians from ten nations participated within the deliberations of the symposium. Thirty lectures have been introduced on quite a few issues in the zone. The contributions to the lawsuits of the symposium are, with a number of exceptions, accelerated types of the lectures introduced via the invited audio system. The survey papers via Komatsu and Hoskins and Sousa Pinto supply an up to date account of the idea of hyperfunctions, ultradistributions and microfunctions, and the nonstandard thought of latest generalized services, respectively; these by means of Stankovic and Kanwal care for constructions and asymptotics. Choquet-Bruhat's paintings experiences generalized features on manifold and offers functions to shocks and discrete types. the opposite contributions relate to modern difficulties and achievements in conception and functions, specially within the thought of partial differential equations, differential geometry, mechanics, mathematical physics, and platforms technology. The complaints provide a really transparent effect of the current cutting-edge during this box and include many demanding situations, principles, and open difficulties. the amount is particularly priceless for a vast spectrum of readers: graduate scholars to mathematical researchers.

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A) If! ] E Lloc (I2) and h g(x,y) = fex) and h(x,y) = J(y). (b) If a E [(I), then E;[a](i = 1,2) belong to [(P). ] E Lloc (I2) where = 1,2. If TE V'(I), then DIE;[T] = E;[DT]; D 2 E;[T] = 0; DIE;[T] = 0 D 2 E;[T] = E;[DT]. Let P be the tensor product operator from 1Y(I) x 1Y(I) into 1Y(I2) given by < P[S;T],1> > = < Sex),< T(Y),1>(x,y»> = < T(y),< S(x),1>(x,y»> for any S, T, E 1Y(I) and 1> E V(I 2). 4) (see [11]). The following results from [5] gives other properties of the operator P. 2. (a) If TE V'(I), then E;[T] = P[T; 1]; (b) If a,ß E [(I) and S,T,U E V'(I), then E;[a]P[S; U] E;[a]P[S; T] (c) E;[T] = PlI; Tl.

Mat. Sero III, 7(1972), 213-228. 4. D. Carmiehael and D. Mitrovic, "Distributions and Analytic Functions," (Longman Seientifie and Teehnieal, 1980). 5. D. Mitrovie, Plemelj formulas and analytie representation of distributions, Glas. Mat. Sero 1II, 3(1968), 231-239. 34 6. , A Hilbert distributional boundary value problem, Math. Balkanica, 1(1971), 177-180. 7. N. A. Chaudhry, Hilbert transform of generalized functions and applications, Canad. J. Math XXXV, no. 3 (1983), 478-495. 8. P. N. Pandey, The n-dimensional Hilbert transform of distributions, its inversion and applications, Canad.

10. H. Zemanian, "Generalized Integral Transforms," (Interscience Publishers, New York, 1968). 35 APPLICATIONS OF GENERALIZED FUNCTIONS TO SHOCKS AND DISCRETE MODELS Yvonne Choquet-Bruhat Mecanique Relativiste Universite Paris 6, France INTRODUCTION Generalized functions, also called distributions, have proved useful in many problems related to physics even before their rigorous and general definition. Generalized functions are also a powerful tool to formulate in a unif ied manner fundamental physical laws.