By Hebe de Azevedo Biagioni
This publication presents an easy advent to a nonlinear thought of generalized services brought by means of J.F. Colombeau, which provides a desiring to any multiplication of distributions. This concept extends from natural arithmetic (it provides a devoted generalization of the classical conception of C? services and gives a synthesis of so much latest multiplications of distributions) to physics (it allows the answer of ambiguities that seem in items of distributions), passing throughout the thought of partial differential equations either from the theoretical perspective (it furnishes an idea of susceptible 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 easy thoughts and effects which in the past have been in basic terms released in article shape. it truly is in- tended for mathematicians yet, because the thought and purposes usually are not dissociated it may well even be invaluable for physicists and engineers. The wanted must haves for its interpreting are basically decreased to the classical notions of differential calculus and the speculation of integration over n-dimensional euclidean spaces.
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Additional info for A Nonlinear Theory of Generalized Functions
4  - NONLINEAR PROPERTIES Here multiplication functions. 1 We at that, for - if all GENERALIZED FUNCTIONS. see more general "operations" than be d o n e w i t h Colombeau' s generalized Definition. say infinity OF a that for x E function every ff E M. 2 this ~) . W e increasing - Definition If f E ~M(~2p), E ~(~), let R : ~0(E~) (11) R(~,x) = definition by at infinity. and fl × f(R l (~,x) is denote is ~ --~ .... ) the space of the of C" theorem. ,G , 1 p 28 (we are identifying ~2p with gP), where each R i E ~[~] is a representative of Gi, l~i~p.
T h e s e o p e n s e t s d e f i n e a t o p o l o g y on that 1 - For a base of Proposition Proof if 2 - We such g>O} I < ~ = ~Q~. Definition is t h e r e a r e x E @M, a r e p r e s e n t a t i v e depending on x, s u c h t h a t for all is ~>0 s a t i s f y i n g = <~ E ~ such that of ~, a n d N E ~ E ~(~) there definition \ inf <#>0 +~ if l, it is is is defined of ~. open, above, since a neighborhood g>0 such that Q~ the is family <~+Q~ convex of ~. N o w ~ + Q , C V. and if V is set : ~ E Q~} x ~ Q~ follows for if ~ all immediately E Q~ for some #>0 ~>0.
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. is a n o p e n s u b s e t of ~l=II1 (~f]jl ( F ) ) / g and E ~s ( ~ ) = R(~i/n,(x,0 R E @ M , s[~] functions ....
A Nonlinear Theory of Generalized Functions by Hebe de Azevedo Biagioni