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16). Note that the hump at rest is subject to forces from the other two humps. The absence of motion of this hump can be traced to the fact that the above two forces are equal in magnitude but opposite in direction. Discrete Spectrum of Nonstationary Schrödinger and Kadomtsev–Petviashvili Equation 37 KPI solutions corresponding to superposition of simple and double poles with Q = 2. 13), and let F stand for the determinant of the 2N × 2N matrix, B= CD . 1), u(x, y, t) = 2 ∂2 log F (x, y, γl (t), δl (t)) ∂x 2 is a real nonsingular KPI solution.

1. 2. Let Y > X, then P (X)|P (Y ) and B(P (X)) exactly tiles B(P (Y )). Now consider z ∈ B(P (Y )) for which P (Y ) (z) is true. Then P (X) (z mod P (X)) is true except if the distance of z mod B(P (X)) from the boundary of B(P (X)) is less than | |. Conversely, if P (Y ) (z) is false, then P (X) (z mod B(P (X))) could be true if d|z+α, where d is divisible only by primes > X. It follows that 1= z∈B(P (Y )) P (Y ) (z) |B(P (Y ))| |B(P (X))| z∈B(P (X)) P (X) (z) 1 + O | | |B(P (X))| + O |B(P (Y ))| | | d>X 1 d2 .

The analysis of R will use sieve methods. This seems quite natural, as sieve methods can be interpreted as an application of probabilistic methods to number theory, for example, the large sieve [6] shows that arithmetic progressions behave like independent random variables. Moreover, as Doron Zeilberger has shown [50], sieve methods like the ones used in this paper can be thought of as special cases of the Lace Expansion which has been used successfully to study percolation problems [11,29]. The sieve method used here is due to Friedlander [20] and it proves very general “almost everywhere” results.