Applied Cryptology, Cryptographic Protocols, and Computer by Richard A. Demillo

By Richard A. Demillo

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T - S - 8) at these points. Between t- (k+ 1)1i-8 and t-k1i-8 the function cp~,1i consists of two segments such that for S E (t - (k + P 1 )1i - 8, t - k1i - 8) it has a slope equal to aDl/(PID I + P2D 2) and equal to -aD2/(PIDI+P2D2) for the rest of the interval [t-(k+1)1i-8, t-k1i-8], k = 0,1, ... , (t - 28 -1i)j1i. It is easy to see that cp6,1i E £11- if 1i is small enough, and cp~,1i for S E [8, t - 8] belongs to the region where uE(t, x, y) tends to 0 as c ! O. 15) if 8 is small enough and 1i = 1i (8) is small enough.

0, tj -+ R 1 (see [7], Chap. 6, [8], [9]). Now let e,o 1 0 and ole 2 -+ 00. That means that e « ole « lie, the probabilities of deviations of order 1 as e,O 1 0, of the normalized occupation time for the process y~,6 from its typical behavior for such values of the parameters tend to zero more slowly than the probabilities of the deviations of order I for x~,6. Using this fact propagation will be the same for all kinds of behavior of the parameters e and 0 such that e,O 1 0, e« ole« e- 1 . We shall consider in more detail the case 0= e.

I will consider the case when 8 ! 0 together with c ! 010(c/8 2 ) = 00. The result in this case is more explicit. Suppose there are a smooth function c( x) :s: 0, x E R 1 , and a sequence Ck > 0, k = 0, ±1, ±2,. '" Let the nonlinear term fO (x, u) = c6 (x, u) . U in our equation be such that CO(x, 0) = c(x) outside the set {x : Ix - kl < 8 for some k = 0, ±1, ±2, ... }, CO (x, 0) = Ck for Ix - kl < 8/2 and CO (x, 0) is continuous. 'P,=n ds iot [IC(iI'sl + c,O~] 2 sup [a. max 0<<>9 l:::;k:::;n Ck - Vn(t - a)] .

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