Volume 17, Issue 4

We consider the generating functions of the form exp{ xg ( t )}, where g ( t ) is a polynomial. These functions generate sequences of polynomials a n ( x ), n = 0, 1,… Each polynomial g ( t ) is in correspondence with configurations of weight n whose sizes of components are bounded by the degree of the polynomial g ( t ). The polynomial a n ( x ) is the generating function of the numbers a nk , k = 1, 2,…, determining the number of configurations of weight n with k components. We give asymptotic formulas as n → ∞ for the number of configurations of weight n and limit distributions for the number of components of a random configuration. As an illustration we show how asymptotic formulas for the number of permutations and the number of partitions of a set with restriction on the cycle lengths and the sizes of blocks can be obtained with the use of the theory of configurations generated by polynomials. We obtain limit distributions of the number of cycles and the number of blocks of such random permutations and random partitions of sets.

Homomorphisms of many-sorted algebraic systems are considered, under which the elements of basic sets, operators, and predicates may be related; such homomorphisms are used for setting and solving equations (relations) over many-sorted algebraic systems among the unknowns of which may occur the elements of basic sets, operators, and predicates.

We consider a multicriteria integer linear programming problem with a finite set of admissible solutions. With the use of Minkowski–Mahler inequality, we obtain a bound for the domain in the space of parameters of the problem equipped with some norm where the Pareto optimality of the solution is still retained. In the case of a monotone norm, we give a formula for the stability radius of the solution. As a corollary we obtain the formula for the stability radius in the case of the Hölder norm and, in particular, the Chebyshev norm in the space of parameters of a vector criterion.

Let T ( G ) be the number of maximal independent sets, M ( G ) be the number of generated matchings in a graph G . We prove the inequality T ( G ) ≤ M ( G ) + 1. As a corollary, we derive the bound for a graph containing no generated subgraph ( p + 1) K 2 , where m is the number of edges and m 1 is the number of dominating edges. This inequality differs from the Balas–Yu conjecture only in the presence of the last term.

It is shown that the transformations of normal and polynomial bases of the field GF ( p n ) from one to the other can be performed by a circuit over GF ( p ) with complexity O ( n 1.806 ) and depth O (log n ).

We construct an automaton model of lung self-cleaning mechanism both in the pure environment and in the conditions of possible dust loading into the lungs from the environment in breathing. We analyse this model and find the complexity (time) of self-cleaning of the lungs for an arbitrary initial dust load.

We study the process of pursuing several independent of one another automata (preys) by a system of automata (predators) on a plane. We show that there exists a finite collective of predators which catches any finite independent system of preys such that the prey velocity is less than the predator velocity and their field of vision is not greater that the field of vision of predators, under any initial disposition of the preys provided that the predators start from one point.