Volume 31, Issue 2

We consider a random walk with zero drift and finite positive variance σ 2 . For positive numbers y , z we find the limit as n→∞ $n\rightarrow\infty$ of the probability that the first exit of the walk from interval -zσn,yσn $\left(-z\sigma\sqrt{n}, y\sigma\sqrt{n}\right)$ occurs through its left end, while the maximum increment of the walk until the exit is smaller than xσn $x\sigma\sqrt{n}$ , where x is a positive number. The limit theorem is established for the moment of the first exit of the walk from the indicated interval under the condition that this exit occurs through its left end and the value of the maximum walk increment is bounded.

We continue to study the set of block transformations {ΣF:F∈B∗(Ω)}$ \{{\it\Sigma}^F : F\in\mathcal B^*({\it\Omega})\} $ implemented by a binary network Σ endowed with a binary operation F invertible in the second variable. For an arbitrary k ⩾2 we obtain necessary and sufficient conditions for k -transitivity of the set of transformations {ΣF:F∈B∗(Ω)}$ \{{\it{\it\Sigma}}^F \colon F\in\mathcal B^*({\it\Omega})\} $, and propose an efficient method for checking whether these conditions hold. We also introduce two methods for construction of networks Σ such that the sets of transformations {ΣF:F∈B∗(Ω)}$ \{{\it\Sigma}^F\colon F\in\mathcal B^*(\Omega)\} $ are k -transitive.

The complexity of implementation of systems of monomials by composition circuits is studied. In such a model, the complexity is understood as the smallest number of composition operations required for computation of a system of monomials. The main result is an exact formula which, for an arbitrary pair of monomials, gives the complexity of their joint implementation by composition circuits.

We obtain estimates for the probability that for a randomly selected k -dimensional n -place q -valued logic vector function there exists a linear manifold of fixed dimension such that the degree of the restriction of the function to this manifold is not larger than the given value. The asymptotics of the number of manifolds on which the restrictions are affine is obtained. It is shown that if n  → ∞ and k  ≤  n / q , then for almost all k -dimensional n -place vector functions the maximum dimension of a manifold on which the restriction is affine lies in the interval [⌊logqnk+logqlogqnk⌋,⌈logqnk+logqlogqnk⌉]$ [\lfloor \log_q \frac{n}{k}+\log_q \log_q \frac{n}{k} \rfloor, \lceil \log_q \frac{n}{k}+\log_q \log_q \frac{n}{k} \rceil] $, while the analogous parameter for the case of fixed variables lies in the range [⌊logqnk⌋,⌈logqnk⌉]$ [\lfloor \log_q \frac{n}{k} \rfloor, \lceil \log_q \frac{n}{k} \rceil] $.

A complete description of trees with maximal possible number of maximum independent sets among all n -vertex trees with exactly l leaves is obtained. For all values of the parameters n and l the extremal tree is unique and is the result of merging the endpoints of l simple paths.

Let Sn$ \mathfrak S_n $ be a semigroup of all mappings from the n -element set X into itself. We consider a set Sn(A)$ \mathfrak S_n(A) $ of mappings from Sn$ \mathfrak S_n $ such that their contour sizes belong to the set A  ⊆  N . These mappings are called A -mappings. Let a random mapping τ n have a distribution on Sn(A)$ \mathfrak S_n(A) $ such that each connected component with volume i  ∈  N have weight ϑ i ⩾0. Let D be a subset of N . It is assumed that ϑ i  →  ϑ >0 for i  ∈  D and ϑ i  → 0 for i  ∈  N ∖ D as i  → ∞. Let μ ( n ) be the maximal volume of components of the random mapping τ n . We suppose that sets A and D have asymptotic densities ϱ >0 and ρ >0 in N respectively. It is shown that the random variables μ ( n )/ n converge weakly to random variable ν as n  → ∞. The distribution of ν coincides with the limit distribution of the corresponding characteristic in the Ewens sampling formula for random permutation with the parameter ρ ϱ ϑ /2.