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We consider the combinatorial vector minimax problem with ordered criteria. We formulate necessary and sufficient conditions for the five known types of stability of the problem which describe the behaviour of the lexicographic set with respect to perturbations of the initial data for the vector criterion.

We consider the random graphs modelling the structure of large data transmission networks including Internet. We investigate the subset of such graphs consisting of N vertices under the condition that the number of edges is equal to n . We obtain the limit distributions of the maximum degree of vertices and the number of vertices of a given degree as N, n → ∞ so that n / N → λ, where λ is a positive constant.

For estimation of the lengths of periods of noisy binary sequences we suggest and investigate statistics equal to the logarithm of the goodness-of-fit functions and functions close to them.

We consider the problem of calculation of cardinality of the classes of square matrices consisting of zeros and ones with given values of row and column sums. We obtain a recurrence relation which gives a possibility to calculate the cardinalities of these classes. In the proofs, we use a constructive approach, that is, we construct the combinatorial objects with given characteristics and calculate the number of distinct objects. An example of the use of the obtained relation is given.

We consider the positive closure operator on the set P 3 of functions of three-valued logic. It is shown that in P 3 there are exactly 51 positively closed classes positively generated by one-place functions, 26 of these classes are generated by a single one-place function (including one of the positively precomplete in P 3 classes), and 25 classes are generated by two functions (including the class P 3 and the remaining 9 positively precomplete classes).

We suggest a construction of the cyclic binary combinatorial Gray codes with distance 3 and dimension n = 2 k – 1, where k = 3, 4, . . . We give a method of construction of Hamiltonian cycles in the graphs of minimum distances of binary Hamming codes. For all admissible lengths n ≥ 15, we give nonlinear perfect binary codes whose graphs of minimum distances contain a Hamiltonian cycle.

In problems of operation research and mathematical economics, analogues of linear operators appear, where the operations of addition and multiplication of numbers are replaced by the operation of taking maximum and some binary operation ⊗ respectively. Earlier, two examples (and their generalisations) were considered, where the addition and the operation of taking minimum were taken as ⊗. In this paper, we consider two other examples, where ⊗ is an addition with discounting (such an operation makes its appearance in the Ramsey–Cass–Koopmans economic model) and the operation of taking minimum with discounting. To investigate all these four examples, a method is suggested which is based on some operations over characteristic pairs of paths in a certain oriented graph. A characteristic pair of a path is defined as a pair of numbers one of which is the weight of the path (it is defined with the use of the operation ⊗), and the other is the number of edges in the path. The emphasis is on the calculation and properties of the eigenvector of the operator known as the Bellman value function for the corresponding optimisation problem on paths of the graph.

In the paper, only finite groups are considered, we describe a large class of one-generated factorable Ω-foliated formations including Ω-compositional, Ω-bicanonical and other formations.

In this paper, we show that the lattice of topologies of a finite one-generated commutative unary algebra is isomorphic to the lattice of topologies of the characteristic semigroup of this algebra. With the use of this assertion, we give a characterisation of the class of all commutative unary algebras with linearly ordered lattices of topologies. It is proved that if the lattice of congruences or the lattice of topologies of a commutative unary algebra is finite, then the algebra itself is finite. Examples of infinite noncommutative unary algebras with finite lattices of topologies are given. It is proved that for an arbitrary functional signature containing at least one symbol with arity greater than 1 and for any integer n ≥ 2 there exists an infinite algebra of such signature whose lattice of topologies is linearly ordered and consists of n elements.