We obtain, in particular, the following results.
We obtain in this manner a system of N coupled differential equations (one for each box).
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In the linear case, we obtain in a completely different way some of the results of D. Jerison and C. E. Kenig about the Laplace equation.
We also update tables for quadratic residue codes over GF(4 ,GF 5) and GF 7) and we obtain in particular a [62,31,20] code over GF 5).
As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.
Thanks to these graphic tools and this attractivity property valid for some part of CSTR's, we obtain in many cases, the estimation of the whole state even if some part of the model is not observable.
As special cases we obtain in particular, partitions of unity (a) in the Malliavin test functions on an abstract Wiener space; (b) in the first order Sobolev spaces on pinned and free loop spaces; (c) in the first order Sobolev spaces associated with reversible Fleming Viot processes.
we obtain in the limit of large times (63).
