By Ivar Ekeland

ISBN-10: 887642251X

ISBN-13: 9788876422515

Through the educational yr 1995/96, i used to be invited by means of the Scuola Normale Superiore to offer a sequence of lectures. the aim of those notes is to make the underlying monetary difficulties and the mathematical idea of external differential platforms available to a bigger variety of humans. it's the objective of those notes to head over those effects at a extra leisurely speed, holding in brain that mathematicians aren't accustomed to fiscal concept and that only a few humans have learn Elie Cartan.

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Example text

As above, let a=-Re(p0) and /3=-re(p0). O ) OuI = 0 CJ 1 T ~ (log T ) - # =0 dx (c2(x(logx)#) 1/'~ x-('~+l)/~(log _ x)-#(n+l)/adx I . w~nT(2T){~"~, 0) ---- O ( x-(n+l)/a+l(log x)-#(r~+l)/adx 4 c l T ~ (log T ) - # o ((~O(~o~)~) -(~247176247 : o (r~247 ~)-~) -- o (T--o(~o)-(o§ T)~(~0)) _- ) ~(o§ To finish one integrates n-times. An extra power of log T occurs precisely in the case when Re(p0) E Z<0. 3 are for T --~ ~ . To estimate the remaining term in (1), namely the finite sum >~ I~kl<2T ak u+w+Ak' it is necessary to choose carefully a sequence Tn.

In the spectral case, we apply the Karamata theorem to (5) with the family of test functions described above and a=-Re(p0) and fl=-rn(p0), from which we obtain the estimate (8) N ( A ) - - # { k : IAkl _< A} ~ C'A~(log A) -~ for some positive constant C I. This estimate gives the relation k = X(Ak) ~ C'lAkl~(loglAkl)-~, as asserted in the statement of the lemma. Then k 1/'~ ~ C'l/~[Ak I (log lakl) and logk ~ a . log I kl. Hence, for some positive constant C, we obtain the asymptotic relation IAkl ~ c ' - l / ~ 1/~ (log IAkl) ~ / ~ ~ C]~ 1/~ (log k) ~/'~ , which completes the proof of the lemma.

W~nT(2T){~"~, 0) ---- O ( x-(n+l)/a+l(log x)-#(r~+l)/adx 4 c l T ~ (log T ) - # o ((~O(~o~)~) -(~247176247 : o (r~247 ~)-~) -- o (T--o(~o)-(o§ T)~(~0)) _- ) ~(o§ To finish one integrates n-times. An extra power of log T occurs precisely in the case when Re(p0) E Z<0. 3 are for T --~ ~ . To estimate the remaining term in (1), namely the finite sum >~ I~kl<2T ak u+w+Ak' it is necessary to choose carefully a sequence Tn. Roughly speaking, the sequence must be as far from any --~k as possible. The following lemma makes this statement precise.

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Exterior differential calculus and applications to economic theory (Publications of the Scuola Normale Superiore) by Ivar Ekeland


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