Volume 34, Issue 4

We prove local limit theorems on large deviations for the moment T n of attaining the level n ∈ ℕ by the random walk in a random environment. Exact asymptotics of probabilities P ( T n = k ) are obtained for values of the parameter k corresponding to the large deviations zone.

We consider local probabilities of lower deviations for branching process Z n = X n ,1 + ⋯ + X n , Z n −1 in random environment η . We assume that η is a sequence of independent identically distributed variables and for fixed η the variables X i , j are independent and have geometric distributions. We suppose that steps ξ i of the associated random walk S n = ξ 1 + ⋯ + ξ n has positive mean and satisfies left-side Cramér condition: E exp( h ξ i ) < ∞ if h − < h < 0 for some h − < − 1. Under these assumptions we find the asymptotic of the local probabilities P ( Z n = ⌊exp( θn )⌋), n → ∞, for θ ∈ (max( m − , 0); m (− 1)) and for θ in a neighbourhood of m (− 1), where m − and m (− 1) are some constants.

The influence of the subgraph cloning operation on the graph diameter is studied. The corresponding potential increase in the diameter is estimated. Conditions under which the subgraph cloning operation causes no change in the graph diameter are formulated. An example of using the cloning operation to construct a family of fat trees is presented. The diameter of such graphs and the complexity of their design are estimated.

We study relationships between lower estimates for the energy complexity E ( Σ ), the switching complexity S ( Σ ) of a normalized Boolean network Σ , and the positive sensitivity ps( f ) of the Boolean function f implemented by this circuit. The lower estimate E(Σ )⩾ ⌊ ps(f)− 1m⌋ $\begin{array}{} \displaystyle E({\it\Sigma})\geqslant \lfloor\frac{\text{ps}(f)-1}{m}\rfloor \end{array}$ is proved for a sufficiently wide class of bases consisting of monotone Boolean functions of at most m variables, the negation gate, and the Boolean constants 0 and 1. For the switching complexity of circuits, we construct a counterexample which shows that, for the standard basis of elements of the disjunction, conjunction, and negation, there do not exist a linear (with respect to ps( f )) lower estimate for the switching complexity.

Galton-Watson forests consisting of N rooted trees and n non-root vertices are considered. The distribution of the forest is determined by that of critical branching process with infinite variance and regularly varying tail of the progeny distribution. We prove limit theorem for the maximal size of a tree in a forest as N , n → ∞ in such a way that n/N → ∞. Our conditions are significantly wider than was previously known.

We consider polynomials f ( x 1 , …, x n ) over a finite filed that satisfy the following condition: the set of solutions of the equation f ( x 1 , …, x n ) = b , where b is some element of the field, coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with the right-hand side b (or with respect to b ). We describe a number of properties of multiaffine polynomials. Then on the basis of these properties we propose a polynomial algorithm that takes a polynomial over a finite field and an element of the field as an input and decides whether the polynomial is multiaffine with respect to this element. In case of the positive answer the algorithm also outputs a system of linear equations that corresponds to this polynomial. The complexity of the proposed algorithm is the smallest in comparison with other known algorithms that solve this problem.

The random equiprobable mappings of finite set S into itself are considered. The probability that k fixed elements of S belong to the same connected component is studied. The limit of this probability as | S | → ∞ is found.