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By Giovanni Alberti, Giuseppe Buttazzo (auth.), Gianni Dal Maso, Gian Fausto Dell’Antonio (eds.)

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Extra info for Composite Media and Homogenization Theory: An International Centre for Theoretical Physics Workshop Trieste, Italy, January 1990

Example text

And A V depend on the arbitrary high order density Note, that derivatives of T and function v over coordinates and time. Theorem 3. There exists the transformation ro V '" W in the + 2: n=1 class involve (lw (w) 11 of finite functions the derivatives of N such I( v( N» that functional with time differentiation no higher than the first order. For linear media (U = 8 jj au/ax j au/axj) it is possible to introduce an unlimited number of expressions T(~) and U(~). ---------qJ 1 " . ". a x jL eq+ Lg .

In (20), (21) the overbar denotes complex conjugation]. Hence, from the Spectral Theorem 24 , there exists a monotone increasing, rightcontinuous family of projection operators, R(A, r), such that 1 +00 Gz(A) = -00 (1- izr)-l dR(A,r)Go(A) (22) Homogenization for Turbulent Transport for all z E C such that Rez =1= 19 O. Using (22) we obtain, for all1jJ E CQ"(IRd), < [G o(A)1jJ, GzCA)1jJ] > = = 1:=(1 1:=(1- iZ7r 1 d < [G o(A)1jJ, R(A, 7)G O(A)1jJ] > (23) iZ7)-1 dm(A, 1jJ, 7). ), such that its successive moments are given by (24) and (25) for N 2: 1.

There exists the transformation ro V '" W in the + 2: n=1 class involve (lw (w) 11 of finite functions the derivatives of N such I( v( N» that functional with time differentiation no higher than the first order. For linear media (U = 8 jj au/ax j au/axj) it is possible to introduce an unlimited number of expressions T(~) and U(~). ---------qJ 1 " . ". a x jL eq+ Lg . g. qJ! . J L const . Properties of Averaged Models of the Periodic Media Mechanics Theore. 4. , on lI. ; coefficients of the form depend not only p.

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