Volume 18, Issue 5

In order to simulate complex telecommunication networks, in particular, the Internet, random graphs are frequently used which contain N vertices whose degrees are independent random variables distributed by the law P { η ≥ k } = k –τ , where η is the vertex degree, τ > 0, k = 1, 2, …, and the graphs with identical degrees of all vertices are equiprobable. In this paper we consider the set of these graphs under the condition that the sum of degrees is equal to n . We show that the generalised scheme of allocating particles into cells can be used to investigate the asymptotic behaviour of these graphs. For N , n → ∞ in such a way that 1 < n/N < ζ(τ), where ζ(τ) is the value of the Riemann zeta function at the point τ, we obtain limit distributions of the maximum degree and the number of vertices of a given degree.

Let X 1 , …, X T be independent random elements uniformly distributed on a finite Abelian group G . In this paper, we give conditions under which the number of ordered sets ( i 1 , …, i k ) of pairwise distinct numbers in {1, …, T } such that a 1 X i 1 + … + a k X i k = 0 where a 1 , …, a k are fixed integers has the Poisson limit distribution as T → ∞ and the group G varies with T . We give an example of a sequence of groups G for which the limit distribution of the number of ordered sets is the compound Poisson distribution.

We suggest a method of realisation of inversion over the standard bases of finite fields GF ( p n ) by means of circuits over GF ( p ) of complexity and depth , where ɛ > 0, and w < 1.667 is the exponent of multiplication of × and × n matrices. Inversion over Gaussian normal bases is realised by a circuit of complexity and depth , where b, c are constants.

Private information retrieval protocols (PIR-protocols) allow a user to get the desired bit of information from a database whose copy is stored in several non-linked servers in such a way that the administrators of the database know nothing about the index of the bit the user requests for. The communication complexity of a protocol is defined as the total number of the bits exchanged between the user and the servers in the protocol. In this paper, we find the order of communication complexity of PIR-protocols depending on the degree of essentiality of the server response functions.

We continue to study the functioning of the human lung. Earlier, an automaton model has been constructed of self-cleaning of dust-loaded lungs in a pure environment. In this paper, we study the Moore diagram of the suitable automaton. As the characteristic states of the diagram we choose the states with no predecessors, such states are called start ones. We study the start states of the Moore diagram.

For Coxeter groups of extra large type, the power conjugacy problem for words and the problem of intersection of cyclic groups are solved; for the second problem we suggest an algorithm which for two elements finds the generator of the intersection of their cyclic subgroups.

We consider the asymptotic behaviour of one of the parameters of the Boolean functions known as the affinity level. We show that almost all Boolean functions of n variables have the generalised affinity level exceeding n – α log 2 n , α > 1, obtain an asymptotic upper bound for the partial affinity level, consider the asymptotic behaviour of the affinity level for the quadratic Boolean functions.

We consider the behaviour of the affinity level of the Boolean functions. It is shown that, as n → ∞, almost all Boolean functions of n variables have the affinity level exceeding n – 2log 2 n .