Band 15, Heft 4

We obtain bounds for the complexity of circuit realisation of the system of differentials of orders from one to k of an arbitrary elementary function in terms of the circuit complexity of this function. Similar bounds are obtained for the complexities of realisation of the Jacobian and Hessian matrices. We point out some applications to deduction of bounds for complexities of polynomials in several variables, linear transformations, and quadratic forms.

We consider the problem of minimisation of Boolean functions in the class of polynomial normal forms. We suggest an algorithm for constructing polynomial normal forms for arbitrary Boolean functions such that the length of the obtained formula depends on the number of variables of the function only. As the input of the algorithm, along with the function, we take the solution of a covering problem. The number of elementary conjunctions in the obtained formula is equal to the cardinality of the covering. For the introduced covering problem we find an approximate solution. We succeed to prove that the complexity of Boolean functions in the class of polynomial normal forms is less than 2 n +1 (log 2 n + 1)/ n , which allows us to conclude that for almost any Boolean function the complexity of its representation in the polynomial normal form is less than its representation in the disjunctive normal form.

The Cayley graph generated by a set A is the graph Γ A ( V ) on a set of positive integers V such that a pair ( u , v ) ∈ V × V is an edge of the graph if and only if | u − v | ∈ A or u + v ∈ A . We denote by ( n , m ) the class of graphs G = ( V , E ) such that G is a union of chains and cycles and | V | = n , | E | = m . In this paper, we present an upper bound for the number of independent sets in Cayley graphs Γ A ( V ) such that A = {⌈ n /2⌉ − t , ⌈ n /2⌉ − ƒ}, V ⊆ [⌊ n /2⌋ + 1, ⌊ n /2⌋+ t ] ∪ [ n − t + 1, n ], where n , t , ƒ ∈ N and ƒ < t < n /4. We also describe the graph with the maximum number of independent sets in the family ( n , m ).

We study partitions of graphs by a system of disconnecting sets. We find a sharp upper bound for the number of resulting parts and analyse the case where the bound is attained. The result due to D. V. Karpov about the number of parts in a partition is proved under weaker constraints imposed on the graph. We also prove a theorem on bounding parts which yields an upper bound for the number of parts of the partition adjacent to a given vertex.

We obtain upper and lower bounds for the probability that a given sequence X of finite length of symbols of a finite alphabet occurs inside a random equiprobable sequence Y of symbols of the same alphabet as a subsequence whose terms are separated in Y by at most one symbol. We give sequences X at which the bounds are attained.

We consider the sequence of random variables μ ( N ) = ξ N ( μ ( N −1) ) ζN , N = 1,2, . . . , where μ (0) is a random variable that takes values in a finite group G = ( G , •), ( ξ N , ζ N ), N = 1,2, . . . , is a sequence of identically distributed random variables that take values in the Cartesian product G × Aut G , Where (Aut G , ∘) is the group of automorphisms of G . We assume that the random variables μ (0) , ( ξ N , ζ N ), N = 1,2, . . . , are independent. Given an arbitrary distribution of μ ( 0 ) , we find general necessary and sufficient conditions for the convergence, as N → ∞, of the sequence of distributions of random variables μ ( N ) to the equiprobable on G distribution.

We investigate the joint distribution of sums of elements in disjoint parts of a random permutation of a certain class. We find the generating function of the distribution of such sums and the limit distributions of the corresponding standardised random variables as the number of elements of the permutation tends to infinity.

We consider a method of steganalysis based on the use of a hidden Markov field model in the case of the LSB-modification of the initial discrete Markov field. We describe iterative filtering algorithms which yield estimates of some parameters of a hidden Markov field with prescribed accuracy.

We prove a limit theorem which describes the behaviour of extreme order statistics of contents of N cells after allocation of groups of particles of random sizes under some constraints imposed on the distribution of the group size.

We give a solution of the well-known problem of isomorphisms of the quasigroups each of which is principal isotopic to the same quasigroup.