Volume 27, Issue 6

The level of affinity of a Boolean function is defined as the minimum number of variables such that assigning any particular values to these variables makes the function affine. The generalized level of affinity is defined as the minimum number of linear combinations of variables the values of which may be specified in such a way that the function becomes affine. For a quadratic form of rank 2 r the generalized level of affinity is equal to r . We present some properties of the distribution of the rank of the random quadratic form and, as a corollary, derive an asymptotic estimate for the generalized level of affinity of quadratic forms.

We study conditions for the existence of coalition games with the result invariant under cyclic shifts of players sequence numbers. Given a total number n of players, we estimate the number k of players of one coalition under which there exists a game in which this coalition wins under all cyclic shifts of players. We give procedures for construction of the so-called set-splitting digraphs on which risk-free nim-type games of a given coalition are defined.

We consider branching process Z n in random environment such that the associated random walk S n has increments ξ i with mean μ and satisfy the Cramér condition E e hξ i < ∞, 0 < h < h + . Let χ i be the number of particles immigrating into the i th generation of the process, Eχ ih$\begin{array}{} \mathbf{E}\chi_i^h \end{array} $ < ∞, 0 < h < h + . We suppose that the number of offsprings of one particle conditioned on the environment has the geometric distribution. It is shown that the supplement of immigration to critical or supercritical processes results only in the change of multiplicative constant in the asymptotics of large deviation probabilities P { Z n ≥ exp( θ n )}, θ > μ . In the case of subcritical processes analogous result is obtained for θ > γ , where γ > 0 is some constant. For all constants explicit formulas are given.

An asymptotic expression (with an explicit estimate of the remainder term) is obtained for the probability that in a finite sequence of polynomial trials controlled by a Markov chain there exist substrings having the same structure.

The paper is devoted to the investigation of “linearly realizable” automata, i.e. automata that allow state encodings that lead to implementations with linear Boolean operators. We formulate the criterion of linear realizability and obtain upper and lower bounds on the number of linearly realizable automata.

An explicit formula is obtained for the number of labeled outerplanar k -cycle blocks with a given number of vertices.

It is shown that, with suitable time change, the finite-dimensional distributions of the process formed by successive values of the Pearson statistics for an expanding sample converge to finite-dimensional distributions of the stationary random process, namely, the normalized square of the Bessel process. The results obtained earlier on the limit joint distributions of the Pearson statistics are used to derive explicit formulas for the density of joint distributions of the Bessel process.