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By Elie Bienenstock (auth.), Prof. Dr. Jean-Pierre Aubin, Prof. Dr. Donald Saari, Prof. Dr. Karl Sigmund (eds.)

ISBN-10: 3540159878

ISBN-13: 9783540159872

ISBN-10: 366200545X

ISBN-13: 9783662005453

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Read Online or Download Dynamics of Macrosystems: Proceedings of a Workshop on the Dynamics of Macrosystems Held at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, September 3–7, 1984 PDF

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Extra info for Dynamics of Macrosystems: Proceedings of a Workshop on the Dynamics of Macrosystems Held at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, September 3–7, 1984

Example text

To this end, note that for any smooth positive function f = (i) (ii) feE) the following properties (i) and (ii) are equivalent: (~E)210g feE) > 0 (~E)2eaEf(E) > 0 for all a E R. This follows from the identity As a simple consequence we get: If two functions f and g have property (i), then f + g as well as fg have property (i). A further consequence is the following Lemma 2 Let two smooth positive functions f and g satisfy (~z)2 log f(e z ) > 0 for all Then the convolution (x,y) E R2 is convex on the whole plane.

Now consider the high tempera1~ure limit. The well-know power series for Euler's f-function implies sz(s) 1 + (1-C-log s)s + O(s 2 ') log~s) (21) By the inversion formula (12) this implies an asymtotic formula for T (E). Note that this formula does not give rise to a convergent expansion. T(E) = 1 + 1 _ 10~E E E + 0(10q2 E \ ? 1)} (23) Omitting further details we summarize the expansions in a table. 54 T++ = n -21riJ (_1_ _1_) I2E + _ AIiJ 3 11 /r + 2w + _ 210gE - 10gAV + 2 - - 2 Oe~~ ~) + + (_1_ + n (A+ll) ) E + 9 All 241riJ All E + 0(E 3 / 2 ) T (1 n 21riJ + 1)12E+ -3 AIiJ + ll/r + (_1_+ (A+ll) ) 9 All 241Ijl All E + + 0(E 3 / 2 ) T+_ = 10gA + 1 w + 11 HOSE - HOSA + wI;!

We study the function T ~ = T(E,A,~). To investigate the temporal structure of the orbit in more detail, we divide it into four parts in the four quadrants of the (p,q) - plane. By T ,T ,T and T+ we denote the time ++ -+ -the orbit takes in the four quadrants {p>O,q>O}, {pO}, {pO,q

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Dynamics of Macrosystems: Proceedings of a Workshop on the Dynamics of Macrosystems Held at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria, September 3–7, 1984 by Elie Bienenstock (auth.), Prof. Dr. Jean-Pierre Aubin, Prof. Dr. Donald Saari, Prof. Dr. Karl Sigmund (eds.)


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