By Janos Englander, Brian Rider

Markov procedures and their purposes to partial differential equations Kuznetsov's contributions.- Stochastic equations on projective platforms of groups.- Modeling festival among influenza strains.- Asymptotic effects for close to severe Bienaym\'e-Galton-Watson and Catalyst-Reactant Branching Processes.- a few course huge deviation effects for a branching diffusion.- Longtime habit for collectively Catalytic Branching.- Super-Brownian movement: Lp-convergence of martingales in the course of the pathwise backbone decomposition

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**Example text**

1. Let m, n be nonnegative integers and p a prime number. Suppose m = mk p k + . . + m 1 p + m 0 and n = nk pk + . . + n1 p + n0 . Then, k m mi =∏ n i=0 ni mod p. Equivalently, if m0 and n0 are the least nonnegative residues of m and n mod p, then m/p m m0 n = n/p n0 . 1 directly to construct interesting examples, we consider a consequence of it for the case p = 2. Suppose that μk = μ1,k ⊗ · · · ⊗ μk,1 where μi,k+1−i {1} = πk = 1 − μi,k+1−i{0} for some 0 ≤ πk ≤ 1. N. Evans and T. Gordeeva where the arithmetic is performed modulo 2 and e(m, (m, +1−m) (m, +1−m) em, +1−m = 1 and en, +1−n = 0 for n = m.

Evans and T. s. s. s. s. by the invertibility of the matrix ρ ◦ ιk (hk ). s. ∀k ∈ N } for any P ∈ Pμex . 4 is still somewhat unsatisfactory as a criterion for the existence of strong solutions because it requires a knowledge of the set Pμex of extreme solutions. We would prefer a criterion that was directly in terms of the sequence (μk )k∈N . In order to (partly) remedy this situation, we introduce the following objects. 5. Fix ρ ∈ G. For k, ∈ N with k ≤ , set Rk := Let G ρ ◦ ιk ◦ φk (z) μ (dz).

Fix ρ ∈ / Hnorm and k ∈ N such that μ lim lim Rnk Rn−1 · · · Rm k = 0. k m→∞ n→∞ It follows from (6) that for n ≥ m ≥ k P ρ ◦ ιk (Xk ) | FnX ∨ σ ((Z j )mj=k ) = ρ ◦ ιk ◦ φkn (Xn )Rn−1 · · · Rm+1 k k ρ ◦ ιk ◦ φmk (Zm ) · · · ρ ◦ ιk ◦ φkk (Zk ). Since ρ (g) is a unitary matrix for all g ∈ G, the norm of the right-hand side is at most Rn−1 · · · Rm+1 , which, by assumption, converges to 0 as n → ∞ followed by k k m → ∞. Thus, by the reverse martingale convergence theorem and the martingale convergence theorem, P ρ ◦ ιk (Xk ) | F∞X ∨ FkZ = lim lim P ρ ◦ ιk (Xk ) | FnX ∨ σ ((Z j )mj=k ) = 0.