Volume 27, Issue 5

For a stopped random walk with zero drift conditioned to attain a high level the theorem on the convergence in distribution to the Brownian high jump in the space D [0, +∞) is proved.

The article is devoted to the memory of Valentin Fedorovich Kolchin . Let ζ , ζ i ( i ∈ N ) be independent identically distributed nonnegative integer-valued random variables, ( η i 1 ,…, η iN ) be the fillings of cells in the generalized scheme of allocation of ζ i particles into N cells, 1 ≤ i ≤ n , for fixed Z n = ( ζ 1 , …, ζ n ) these allocation schemes are independent. We consider the conditional probabilities P ( A n , N | Z n ) of the event A n , N = {each cell in each of n allocation schemes contains no more than r particles}, where r is some fixed number. The sufficient conditions for the convergence of the sequence P ( A n , N | Z n ) to a nonrandom limit with probability 1 are given. It is shown that the random variable ln P ( A n , N | Z n ) is asymptotically normal. Applications of the obtained results to the noise-proof encoding are discussed.

The paper is devoted to the complexity of representation of q -valued functions by polarized polynomials and by matrix Kronecker forms of certain type. The complexity of a function is the minimal possible number of nonzero coefficients of a polynomial or a Kronecker form representing the function. It is known that for polynomial representation and representation by Kronecker forms of a certain type the maximal values of complexity in the class of all q -valued n -ary functions coincide. We establish the lower bound of these maximal values for five-valued functions.

The paper is concerned with the complexity of realization of k -valued logic functions by logic circuits over an infinite complete bases containing all monotone functions; the weight of monotone functions (the cost of use) is assumed to be 0. The complexity problem of realizations of Boolean functions over a basis having negation as the only nonmonotone element was completely solved by A. A. Markov. In 1957 he showed that the minimum number of NOT gates sufficient for realization of any Boolean function f (the inversion complexity of the function f ) is ⌈log 2 ( d ( f )+1)⌉. Here d ( f ) is the maximum number of the changes of the function f from larger to smaller values over all increasing chains of tuples of variables values. In the present paper Markov’s result is extended to the case of realization of k -valued logic functions. We show that the minimum number of Post negations (that is, functions of the form x +1(mod k )) that is sufficient to realize an arbitrary function of k -valued logic is ⌈log 2 ( d ( f )+1)⌉ and the minimum number of Łukasiewicz negation (that is, functions of the form k −1− x ) that is sufficient to realize an arbitrary k -valued logic function is ⌈log k ( d ( f )+1)⌉. In addition, another classical Markov’s result on the inversion complexity of systems of Boolean functions is extended to the setting of systems of functions of k -valued logic.

The paper is concerned with word functions over the alphabet {1,2}. Given arbitrary one-place functions f 1 , …, f l , the class BPC[ f 1 , …, f l ] is defined as the closure of the set of simplest word functions and the functions f 1 , …, f l under the operations of superposition and bounded prefix concatenation. The class BPC[ f 1 , …, f l ] is shown to have a finite basis with respect to the superposition.

The paper is concerned with the asymptotic behaviour of the number mi( T q , n ) of maximal independent sets in a complete q -ary tree of height n . For some constants α 2 and β 2 the asymptotic formula mi( T 2, n )∼ α 2 ⋅ ( β 2 ) 2 n is shown to hold as n → ∞. It is also proved that mi⁡ (Tq,3k)∼ α q(1)⋅ (β q)q3k,mi⁡ (Tq,3k+1)∼ α q(2)⋅ (β q)q3k+1,mi⁡ (Tq,3k+2)∼ α q(3)⋅ (β q)q3k+2$\begin{array}{} \operatorname{mi}(T_{q,3k})\thicksim \alpha^{(1)}_q\cdot(\beta_q)^{q^{3k}},\operatorname{mi}(T_{q,3k+1})\thicksim \alpha^{(2)}_q\cdot(\beta_q)^{q^{3k+1}},\operatorname{mi}(T_{q,3k+2})\thicksim \alpha^{(3)}_q\cdot(\beta_q)^{q^{3k+2}} \end{array} $ as k → ∞ for any sufficiently large q , some three pairwise distinct constants α q(1),α q(2),α q(3)$\begin{array}{} \alpha^{(1)}_q,\alpha^{(2)}_q,\alpha^{(3)}_q \end{array} $ and a constant b q .

The paper puts forward a nontrivial lower estimate 216n$2\!\frac16n $ for the cardinality of the domain of a universal function for the class of linear Boolean functions, where n is the number of variables.

Let 𝔖 n be a semigroup of mappings of a set X with n elements into itself, A be some fixed subset of the set N of natural numbers, and V n ( A ) be a set of mappings from 𝔖 n , with lengths of cycles belonging to A . The mappings from V n ( A ) are called A -mappings. We suppose that the set A has an asymptotic density ϱ > 0, and that | k : k ≤ n , k ∈ A , m − k ∈ A |/ n → ϱ 2 as n → ∞ uniformly over m ∈ [ n , Cn ] for each constant C > 1. A number M ( α ) of different elements in a set { α , α 2 , α 3 , …} is called an order of mapping α ∈ 𝔖 n . Consider a random mapping σ = σ n ( A ) having uniform distribution on V n ( A ). In the present paper it is shown that random variable ln M ( σ n ( A )) is asymptotically normal with mean l(n)=∑ k∈ A(n)ln⁡ (k)/k $l(n)=\sum_{k\in A(\sqrt{n})}\ln(k)/{k}$ and variance ϱ ln 3 ( n )/24, where A ( t ) = { k : k ∈ A , k ≤ t }, t > 0. For the case A = N this result was proved by B. Harris in 1973.