Volume 27, Issue 2

An extension of the well-known Rissanen algorithm [1] is proposed for solving systems of linear equations with block-Hankel and block-Toeplitz matrices.

We consider a sequence of almost critical branching processes with immigration supposing that the immigration process is weakly stationary. The rate of growth and asymptotic properties of fluctuations of such branching processes are investigated.

Let a finite Abelian multiplicative group G be specified by the basis B = { a 1 , a 2 , …, a q }, that is, the group G is decomposed into a direct product of cyclic subgroups generated by the elements of the set B : G = 〈 a 1 〉 × 〈 a 2 〉 × … × 〈 a q 〉. The complexity L ( g ; B ) of an element g of the group G in the basis B is defined as the minimum number of multiplication operations required to compute the element g given the basis B (it is allowed to use the results of intermediate computations many times). Let L ( G , B ) = maxg∈ G $\begin{array}{} \max\limits_{g \in G} \end{array}$ L ( g ; B ), LM ( G ) = maxB $\begin{array}{} \max\limits_{B} \end{array} $ L ( G , B ), Lm ( G )= minB $\begin{array}{} \min\limits_{B} \end{array}$ L ( G , B ), M ( n ) = maxG: |G|≤ n $\begin{array}{} \max\limits_{G \colon |G| \le n} \end{array}$ LM ( G ), m ( n ) = maxG: |G|≤ n $\begin{array}{} \max\limits_{G \colon |G| \le n} \end{array}$ Lm ( G ), M av ( n ) = ∑ G: |G|=nLM(G)/A(n),mav(n)=∑ G: |G|=nLm(G)/A(n), $\begin{array}{} \left( \sum\limits_{G \colon |G|= n}{ LM(G)}\right)/{A(n)}, m_{\text{av}}(n) = \left( \sum\limits_{G \colon |G|= n}{ Lm(G)}\right)/{A(n)}, \end{array}$ where A ( n ) is the number of Abelian groups of order n . In this work the asymptotic estimates for the quantities L ( G , B ), M ( n ), m ( n ), M av ( n ), and m av ( n ) are established.

A class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

We consider the set of functions of countable-valued logic and the closure operator based on systems of functional equation (FE-closure). We prove that the cardinality of the set of all FE-precomplete classes is hypercontinuum.

The paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the 1-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is 1-stable.

We consider the functional system of vector functions of many-valued logic with the naturally defined operation of superposition and construct examples of closed classes of special type without a basis and with a countable basis.

A theorem on the asymptotic normality of the sum of dependent random variables is stated and proved. Conditions of the theorem are formulated in terms of a dependency graph which characterizes the relationships between random variables. This theorem is used to prove the asymptotic normality of the sum of functions defined on subsets of elements of the stationary sequence satisfying the strong mixing condition. As an illustration of possible applications of these theorems we give a theorem on the asymptotic normality of the number of empty cells if the random sequence of cells occupied by particles is a stationary sequence satisfying the uniform strong mixing condition.

Let A be a ring such that all maximal indecomposable factor rings A i of A are serial rings. Then every square matrix over A is diagonalizable. In addition, if all the rings A i are Bezout rings, then every rectangular matrix over A is diagonalizable. If ϕ is an automorphism of the ring A , then the skew Laurent series ring A (( x , ϕ )) is a serial ring if and only if A is a serial Artinian ring.