Abstract Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension d ∈ {2, 3} by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely N ∈ ℕ. Our proof relies on simple arguments from the theory of continued fractions.

Abstract We construct a probabilistic polynomial algorithm that solves the integer factorization problem using an oracle solving the Diffie–Hellman problem.

Abstract We introduce a new model ${\mathscr{P}}$-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 ${\mathscr{P}}$-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 ${\mathscr{P}}$-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.

Abstract A new equivalence relation on the set of Boolean functions is introduced: functions are declared to beΔ-equivalent if their autocorrelation functions are equal. It turns out that this classification agrees well with the cryptographic properties of Boolean functions: for functions belonging to the sameΔ-equivalence class a number of their cryptographic characteristics do coincide. For example, all bent-functions (of a fixed number of variables) make up one class.

Abstract Finite automata transform periodic sequences into periodic ones. The period of the output sequence is bounded from above by a linear function of input period. It is known that pushdown automata also preserve the set of periodic sequences. We prove that the output period for one-counter pushdown automata is bounded from above by a quadratic function of input period. We also give an example of an automaton with a quadratic lower bound on output period.

Abstract The paper is devoted to the study of equations with the left-hand side having the form of a composition of operations which belong to given sets S1, …, Sn, … of quasigroup operations. Elementary transformations are described which allow reducing systems of this kind to the form where all equations except one do not depend essentially on the variable xn. A class of systems is said to be Gaussian if every system obtained via such transformations also belongs to this class. It is evident that for Gaussian classes of systems of equations there is an efficient solving algorithm. This motivates the problem of finding conditions under which the class is Gaussian. In this work it is shown that for a class of systems to be Gaussian the operations in the sets Si should satisfy the generalized distributivity law. Sets of operations obeying this condition are to be investigated in the future.

DISCRETE MATHEMATICS AND APPLICATIONS provides the latest information on the development of discrete mathematics in Russia to a worldwide readership.The journal contains papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics.Discrete Mathematics and Applications covers various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.

DISCRETE MATHEMATICS AND APPLICATIONS provides the latest information on the development of discrete mathematics in Russia to a worldwide readership.The journal contains papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics.Discrete Mathematics and Applications covers various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.

Abstract A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of centrally essential rings.

Abstract Let a functionf(z) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radiusR. Let the distribution of the random variableξn,n∈ {1, 2, …}, be defined by the formula$$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$for some ∣z∣ <R(if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions ofξnare proved in two cases: a) iff(z) = (1 −z)−λ,λ= const ∈ (0, 1] for ∣z∣ < 1, and b) if all positive coefficients of expansion f (z) in a power series are equal to 1 and the setAof their numbers has the form$$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$A hypothetical general local limit normal theorem for random variablesξnis stated. Some examples of validity of the statement of this theorem are given.