Volume 30, Issue 6

We consider polynomials over rings such that the polynomials represent Latin squares and define a group operation over the ring. We introduce the notion of a group polynomial, describe a number of properties of these polynomials and the groups generated. For the case of residue ringsℤrn,$\mathbb{Z}_{r^n},$where r is a prime number, we give a description of groups specified by polynomials and identify a class of group polynomials that can be used to construct controlled cryptographic transformations.

The sets of all maximum subclasses in classes of linear automata over finite fields are found. An algorithm that decides completeness of finite subsets based on maximum subclasses is formulated.

We study the set of transformations { Σ F : F ∈ 𝓑 ∗ ( Ω )} implemented by a network Σ with a single binary operation F , where 𝓑 ∗ ( Ω ) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family { Σ F : F ∈ 𝓑 ∗ ( Ω )} in terms of the structure of the network Σ , identify necessary and sufficient conditions of transitivity of the set of transformations { Σ F : F ∈ 𝓑 ∗ ( Ω )}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations { Σ F : F ∈ 𝓑 ∗ ( Ω )}.

A graph is called K 1, p -restricted ( p ≥ 3) if for every vertex of the graph there are at least p − 2 edges between any p of its neighbours. We establish sufficient conditions for the existence of a perfect matching in K 1, p -restricted graphs in terms of their connectivity and vertex degrees. These conditions imply, in particular, the classical Petersen’s result: any 2-edge-connected 3-regular graph contains a perfect matching.

We consider random polynomial allocations of particles over N cells. Let τ k , k ≥ 1, be the minimal number of trials when k particles hit the occupied cells. For the case N → ∞ the limit distribution of the random variable τ k/N$\tau_k/\sqrt{N}$is found. An example of application of τ k is given.

We introduce a new model P${\mathscr{P}}$-CNAR( s ) of sequences of discrete random variables with long memory determined by semibinomial conditionally nonlinear autoregression of order s ∈ ℕ with small number of parameters. Probabilistic properties of this model are studied. For parameters of the model P${\mathscr{P}}$-CNAR a family of consistent asymptotically normal statistical FB-estimates is suggested and the existence of an efficient FB-estimate is proved. Computational advantages of FB-estimate w.r.t. maximum likelihood estimate are shown: less restrictive sufficient conditions for uniqueness, explicit form of FB-estimate, fast recursive computation algorithm under extension of the model P${\mathscr{P}}$-CNAR. Subfamily of “sparse” FB-estimates that use some subset of frequencies of s -tuples is constructed, the asymptotic variance minimization problem within this subfamily is solved.