By Hebe de Azevedo Biagioni

This e-book presents an easy advent to a nonlinear concept of generalized capabilities brought by means of J.F. Colombeau, which supplies a desiring to any multiplication of distributions. This conception extends from natural arithmetic (it provides a loyal generalization of the classical conception of C? services and offers a synthesis of so much latest multiplications of distributions) to physics (it allows the answer of ambiguities that seem in items of distributions), passing in the course of the idea of partial differential equations either from the theoretical standpoint (it furnishes an idea of vulnerable resolution of pde's resulting in existence-uniqueness ends up in many circumstances the place no distributional resolution exists) and the numerical perspective (it introduces new and effective tools constructed lately in elastoplasticity, hydrodynamics and acoustics). this article provides uncomplicated ideas and effects which in the past have been in simple terms released in article shape. it truly is in- tended for mathematicians yet, because the concept and functions aren't dissociated it can even be important for physicists and engineers. The wanted necessities for its interpreting are primarily decreased to the classical notions of differential calculus and the idea of integration over n-dimensional euclidean spaces.

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S i n c e , locally, equivalent there restric- G t. is - Rj[ a nQ is a finite ~t}t~[ is It J that j R(~,x) This {supp is E have We know that subordinated to locally finite. (8) ft Ri we t is is G E ~/[fli] we have for that each R E @M[fl]. i E I. This 23 Given an open that, K a compact neighborhood for each (~,x) R(~,x) subset ~ of = K. ~i, there x we E ~0(En) ~ of Then ~, ~j(x) Rj(~,x) the is sum (8) Ii c 1 is finite finite in such have + ~(x) R,(~,x) jEIi Since ~ ~j (x) = 1, for all x E ~, we may write jEIi If x E K, (R-Ri)(W,x) : the j which indices for which x E supp result, b y (*) .

It that for this solutions is a of in ~, is not space is necessity topology shown to nonlinear in avoid partial equations. 1, the topology on ~(~) arises from a uniform 44 Definition by the 4 In ~(~) - UK,k,. : <(G,H) with KCC~, k E ~ structure on ~(~) and uniform structure. Remark 5 via X ~(~) This - the topology inclusion instance in of space topology say weakl~ the the space C®(~). classic~l ~ 6 ~(Q), the x ~(~) is of the the Its Since to J G ( x ) exactly the $(x) sequence U spanned IIG-HI[K,k U defines a uniform arises from this natural generalization classical use we looks the Fr~chet more use have [G~(x) J This of concept of < ~} a limited of the topology weak on ~ we G n E ~(~) ~(x) dx convergence tends tends generalizes distributions.

8) tively,by are in fact in ~ , ~ ** K ( ~ z / = , x ) representative of G (see and with ~ ~ representatives II%(~/~,A)dA, 1 . 2 . 1 1 (c)) . 5 - The in macroscopic aspect of a generalized function ~(~). Let T be a d i s t r i b u t i o n on ~. 9) . Considering the i n c l u s i o n ~(~) C ~(~) then G has T as macroscopic aspect if a n d o n l y if G ~ T (T b e i n g a w e l l d e f i n e d element of ~ ( ~ ) ) . 6 Fn'if QI - Restrictions of With the notations of 1 . 3 . 9 - As inclusion xE~1, of where HEAVISIDE we have C(O) (or of G] GENERALIZED noticed in Cf(O)) into ~ is a to subspaces.