Volume 22, Issue 5-6

We consider the bilinear complexity of multiplication in local and semisimple algebras over an infinite field of characteristic differing from 2. We obtain a criterion for the rank of a local algebra to be almost minimal. We evaluate the bilinear complexity of the algebras of generalised quaternions over such a field; we prove that any simple algebra of almost minimal rank is an algebra of generalised quaternions. This result is used for the classification of semisimple algebras of almost minimal rank.

The positively closed classes of multivalued logic are characterised with the use of endomorphism semigroups. All subsemigroups of the symmetric semigroup of degree k are described which are semigroups of endomorphisms of positively closed classes in P k . On the base of the obtained results, we find all positively precomplete classes in P k .

We construct an algorithm which solves the power conjugacy search problem in a cyclic subgroup in groups with corepresentation G D .XIR/ satisfying the conditions C.3/ ² T .6/. We make use of the diagram technique.

Lawvere intervals of a Möbius category are finite one-way categories. We investigate a simple connection between the Möbius function of a Möbius category and the Möbius functions of its Lawvere intervals. In this investigation we have taken into consideration that a large class of Möbius categories arise as skeletons of division categories (that are right cancellative with pushouts). In general, the computation of the Möbius function for a Möbius category is laborious. In our case it becomes a routine computation on finite lattices. As an illustration, the results of this study may be used to compute the Möbius function of a proper Möbius category.

It is well known that the problem on A-completeness of finite systems of b.d. functions is algorithmically unsolvable in the general case. Nevertheless, the criterion for A-completeness admits a formulation in terms of A-precomplete classes, and the set of A-precomplete classes is described efficiently. We consider systems of b.d. functions containing all S-b.d. functions depending on one variable and demonstrate that there exists an algorithm to recognise the A-completeness of such systems.

We establish the algorithmic solvability of the problem of uniqueness of alphabetic decoding of regular texts.

We consider graphs obtained from trees by replacing edges by other graphs with certain constraints imposed. We study the set of all unstable indecomposable graphs of such a structure.

In this research we consider minimal hard and boundary classes. These classes are of considerable importance in demarcating the polynomial solvability and the NP -completeness of problems on graphs in the family of hereditary classes. Earlier it was established that there are no minimal hard classes for a certain type of problems on graphs. Here we find a sufficient condition for existence of such classes. With its use we construct new examples of minimal hard classes and boundary classes for some problems on graphs. We provide complete description of the sets of minimal hard and boundary classes for the problems under consideration. Up to now, no such descriptions were known.

In the framework of a d-dimensional parametric model of random permutations of degree n we solve the problem of asymptotic (as n ! 1) estimation of unknown parameters and testing of various hypotheses for them provided that we have incomplete data: we observe only cycles of a certain bounded length in the permutations. We also consider the case of large samples and construct a test for homogeneity.

Given a sequence of independent identically distributed random variables i , we consider the branching process Zn in a random environment . It is known that under some suitable assumptions the probabilities of large deviations (greater than n) of lnZn and that of its associated random walk Sn are connected by the relation P (lnZ n ≥ θn |Z 0 = 1) ~ I( θ ) P (S n ≥ θn ), n → ∞, where I( θ ) > 0. We obtain similar result for P (lnZ n ≥ θn + ln m|Z 0 = m) for n→ ∞ and m being finite or tending to infinity.

Multitype critical branching processes evolving in a Markovian random environment are considered. A conditional limit theorem on the distribution of the number of particles in the process at moment n is proved given survival of the process to this moment.

We study a class of generalised linear discrete time systems whose coefficients are square constant matrices. The main objective of this research is to provide a link between the solutions of a class of singular linear discrete time systems and their proper (and transposed) dual systems. First, we study the prime system and by using the invariants of its matrix pencil we give necessary and sufficient conditions for existence and uniqueness of solutions, and obtain formulas for the solutions. Next, we prove that by using the same matrix pencil we can study the existence and uniqueness of solutions of the proper and transposed dual system. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, we provide a numerical example to justify the theory.

We find an upper bound for the number of Gram’s intervals which contain k ordinates of successive zeros of the Riemann zeta function.