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D is a roaming trajectory of type two, where the trajectory takes a complete turn in the plateau region. These trajectories avoid the minimum energy path and involve non-planar structures along the pathway. The energy contours projected onto the (qF , q1 ) and (qF , q2 ) plane are also shown, together with projections of the trajectories onto the (qF , q1 ), (qF , q2 ), and (q1 , q2 ) plane Kinetic energy terms along RT1 are shown in Fig. 4: The total kinetic energy T , and the partial energies, TF, T1 and T2, corresponding to the contributions of the reactive DOF and the coupled DOFs 1 and 2, respectively.

P2 (37) Reprinted from the journal (38) 6Vo ω exp − . Vo ω (39) (40) 13 Theor Chem Acc (2014) 133:1536 (o) approximate ground state Wigner function ρW at the barrier in phase space leads to an overestimate of the exponential rate dependence. The result of the present analysis is, however, much better than a classical Boltzmann-like estimate, based on an effective temperature kB T = ω/2, which gives an exponential factor of exp(−2Vo / ω). The √ prefactor in Eq. 15) is significantly smaller than the corresponding WKB factor (~52).

In this limit, the rate of escape across the separatrix is identically zero. The quantum trajectory equations of motion, however, do not preserve the Hamiltonian, so quantum effects will lead to crossing of the S(q, p) = 0 separatrix. We now estimate the quantum trajectory motion across the separatrix. For the cubic potential, V (q) = −2b, and so the equations of motion in Eq. (14) reduce to N˙ k=− , N q˙ = ∂H ∂p 2b 1 ∂ 2ρ ∂H W p˙ = − − . ∂q 12 ρW ∂p2 (20) where N(t) is the normalized phase population within the dividing surface defined by S(q, p) = 0: N(t) = dqdpθ(−S(q, p))ρW (q, p).

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Counterfeiting Attacks on Oblivious Block-wise Independent Invisible Watermarking Schemes by Holliman, Memon


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