The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Efficient and facile Ar-Si bond cleavage by montmorillonite KSF: synthetic and mechanistic aspects of solvent-free protodesilylation studied by solution and solid-state MAS NMR

A facile and efficient method for the cleavage of the Ar-Si bond of various aryl trimethyl silanes is described. When adsorbed on montmorillonite KSF (mont KSF), these arylsilanes readily undergo a solvent-free protodesilylation to the corresponding arenes at room temperature in excellent yields. This approach seems to be superior to the traditional mild methods (i.e., desilylation by TFA, TBAF, CsF), in terms of reaction yield, rate, and environmentally benign conditions. Some mechanistic studies using both solution and solid-state magic-angle spinning (SS MAS) (1)H NMR are also presented.