Volume 34, Issue 6

Continued fractions with rational partial quotients with right shift arise in the process of applying Sorenson’s right-shift k -ary gcd algorithm to the ratio of natural numbers a , b . Using this algorithm makes it possible to obtain different types of such fractions. This functions are associated with special forms of continuants, that is, polynomials that can be used to express the numerators and denominators of the convergents. In this paper we introduce such fractions and continuants, we also investigate properties (in particular, the asymptotic behavior) of the extremal values of the continuants under constraints imposed on the variables involved in the right-shift k -ary gcd algorithm of Sorenson. We also introduce a construction similar to the triangle of coefficients of Fibonacci polynomials.

We study conditions for linear equivalence of piecewise-linear and partially defined piecewise-linear permutations of the field 𝔽 2 n .

Let S0=0,Sn=X1+… +Xn,n≥ 1, $$\begin{array}{} \displaystyle S_{0}=0,\quad S_{n}=X_{1}+\ldots+X_{n},\, n\geq 1, \end{array}$$ be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants a n that provide convergence as n → ∞ of the distributions of the sequence { S n / a n , n = 1, 2, …} to this stable law. Let L r , n = min r ≤ m ≤ n S m be the minimum of the random walk on the interval [ r , n ]. It is shown that limr,k,n→ ∞ PLr,n≤ yak|Sn≤ tak,L0,n≥ 0,t∈ 0,∞ , $$\begin{array}{} \displaystyle \lim_{r,k,n\rightarrow \infty }\mathbf{P}\left( L_{r,n}\leq ya_{k}|S_{n}\leq ta_{k},L_{0,n}\geq 0\right) ,\, t\in \left( 0,\infty \right), \end{array}$$ can have five different expressions, the forms of which depend on the relationships between the parameters r , k and n .

We consider a sequence ( n = 1, 2, ⋯) of finite Markov chains { ω n , t } t ≥1 with discrete time describing the percolation process in a band of width n of a multilayered random medium in which a flow (breakdown) already exists, and random variable ω n , t equals to the width of the percolation cluster at time t . For each value of n and given random percolation mechanism the Markov chain { ω n , t } t ≥1 has a limit stationary distribution, corresponding to the random variable ω n . In the case when the width n of the layers of the medium under consideration tends to infinity, the limit distribution of the random variables Ω n b/n $\begin{array}{} \displaystyle \sqrt{b/n} \end{array}$ ( b is some constant) is found, which is the Rayleigh distribution.