Volume 18, Issue 6

We consider a multicriteria combinatorial problem with minmin criteria. For the stability of the problem we obtain a necessary and sufficient condition which is a discrete analogue of the Hausdorff upper semicontinuity of a multivalued mapping which puts each set of parameters of the vector criterion into correspondence with the Pareto set of the problem.

In a long enough sequence of discrete random variables, as a rule, an s -tuple exists of nontrivial structure, that is, a tuple with at least one repeated symbol. We consider the case where the sequence consists of n + s – 1 independent random variables taking the values 1, …, N with equal probabilities. It is shown that as n → ∞, ns 3 N –2 → 0 the probability of that in the sequence s -tuples exist with the same nontrivial structure is equal to 1 – (1 + n/N ) s e –sn/N (1 + o (1)).

We consider random and a priori consistent random systems of equations over a finite field with q elements in n unknowns. A random system consists of M = M ( n ) equations, each of which can depend on 2, 3, …, m variables, which are obtained by sampling without replacement. We obtain limit distributions and estimates of moments for the numbers of solutions of random systems of equations provided that n → ∞ and the relation between the parameters n and M , the number of vertices and the number of hyperedges, falls into the subcritical domain of the evolution of random hypergraphs which describe the random systems of equations. The form and parameters of the limit distributions are determined by the characteristics of the limit distributions of the number of cycles of a special form in the corresponding random hypergraphs.

We consider the problem on the realisation of Boolean functions by informational graphs. The exact expression of the Shannon function for the realisation in the class of tree-like informational graphs is obtained. For almost all Boolean functions, we obtain the order of the complexity of the realisation by informational graphs and the asymptotics of complexity of the realisation by informational trees.

We consider the problem of description of the frequency characteristics of some linear recurring sequences over a finite field. In a paper by R. McEliece, estimates of frequencies of occurrence of elements on the cycles of linear recurring sequences over a finite field with irreducible characteristic polynomial were obtained. The present paper is devoted to a generalisation of these estimates in the case of sequences of vectors with elements of several recurring sequences. The obtained results refine the known estimates of such frequencies.

Let a class 𝑸 of functions of natural argument be closed with respect to a superposition and contain the identity function. The set of permutations ƒ such that ƒ, ƒ –1 ∈ 𝑸 forms a group (with respect to the operation of composition) which we denote by Gr (𝑸). We prove the finite generability of Gr (𝑸) for a large family of classes 𝑸 satisfying some conditions. As an example, we consider the class FP of functions which are computable in polynomial time by a Turing machine. The obtained result is generalised to the classes of the Grzegorczyk system, n ≥ 2. It is proved that for the considered classes 𝑸 the minimum number of permutations generating the group Gr (𝑸) is equal to two. More exactly, there exist two permutations of the given group such that any permutation of this group can be obtained by compositions of these permutations.

We introduce the notion of an independent set of elements of a unary algebra as a subset of its support, where in any pair of elements one element does not belong to the subalgebra generated by the other. It is proved that any two independent systems of generators of a unary algebra have the same cardinality. With the use of this assertion it is proved that any finitely generated unary algebra with commutative operations possesses the Hopf property: each epiendomorphism of the algebra is an automorphism.

We consider determinate functions with delay which are generalisations of determinate functions and introduce the notion of complexity of an ɛ -approximation of a continuous real function by a function with delay. For some classes of continuous functions for which estimates of the number of elements in the 2 ɛ -distinguishable set of functions are known, upper and lower estimates are obtained.

We consider the groupoid of all possible matrices over an arbitrary Boolean algebra with partial operation of matrix product. On this groupoid, we define the equivalence classes analogous to the Green classes H, C, R, D, J for semigroups. We introduce the notion of the minor rank of a Boolean matrix. We show that the column, row, factorisation and minor ranks are invariants for the J -class of this groupoid, and the minor ranks do not exceed the column, row, factorisation and permanent ranks. The key result of this work explains the role of the Boolean determinant. We show that in some J -class there exists a square n × n matrix with nonzero determinant if and only if the column, row, factorisation and minor ranks of any matrix of this class are equal to each other and equal to n . All n × n matrices of this J -class have equal determinants, while the determinants of the square matrices of greater size are equal to zero.