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By Ghora J.K.

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P + κn Then the computation of m = u − uxx gives ∞ m(x, t) = −∞ δ(x − g(ξ, t))p(ξ, t)dξ − ω. (23) April 24, 2007 17:34 40 WSPC - Proceedings Trim Size: 9in x 6in pcsvf2006 A. I. Ivanov From the CH equation mt + umx = −2(m + ω)ux , (20) and (23) it follows g(ξ, ˙ t) = 1 2 ∞ e−|g(ξ,t)−g(ξ,t)| p(ξ, t)dξ − ω, g(ξ, ˙ t) = u(g(ξ, t), t), 0 therefore g(x, t) in (21) is the diffeomorphism (Virasoro group element) in the purely solitonic case. The situation when the condition m(x, 0) + ω > 0 on the initial data does not hold is more complicated and requires separate analysis (if m(x, 0) + ω changes sign there are infinitely many positive eigenvalues accumulating at infinity and singularities might appear in finite time [3,4]).

It is also known that the sign of K influences the geodesic rays ([5]). Indeed, if K > 0, then the geodesic bunch together (are Jacobi stable), and if K < 0, then they disperse (are Jacobi unstable). Hence negative flag curvature is equivalent to positive eigenvalues of Pji , and positive flag curvature is equivalent to negative eigenvalues of Pji . 1 ([1,2]). The trajectories of (3) are Jacobi stable if and only if the real parts of the eigenvalues of the deviation tensor Pij are strict negative everywhere, and Jacobi unstable, otherwise.

So each almost K¨ ahler manifold is a symplectic one. 1. e. J˜∗ dϕ = dϕJ ∗ . T ∗ M and J˜∗ on T ∗ M Here the automorphim J ∗ is defined on T ∗ M as conjugate automorphism of the automorphism J of the tangent bundle T M of the manifold M . 1. 2. Real analyticity of a smooth manifold with two symplectic-homotopic symplectic forms ω0 and ω1 Let M be a smooth paracompact manifold with two symplectic-homotopic symplectic forms ω0 and ω1 . This means that there exists a homotopy operator H(p, t) : M × I → ∧2 T ∗ M , I being the unit interval in R1 , such that H(p, t) is a continuous mapping, H(p, t) = ωt (p) where ωt are symplectic forms and H(p, 0) = ω0 (p), H(p, 1) = ω1 (p).

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A Central Limit Theorem for the L2 Error of Positive Wavelet Density Estimator by Ghora J.K.


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