Volume 33, Issue 5

Let Γ be a diameter 3 distance-regular graph with a strongly regular graph Γ 3 , where Γ 3 is the graph whose vertex set coincides with the vertex set of the graph Γ and two vertices are adjacent whenever they are at distance 3 in the graph Γ . Computing the parameters of Γ 3 by the intersection array of the graph Γ is considered as the direct problem. Recovering the intersection array of the graph Γ by the parameters of Γ 3 is referred to as the inverse problem. The inverse problem for Γ 3 has been solved earlier by A. A. Makhnev and M. S. Nirova. In the case where Γ 3 is a pseudo-geometric graph of a net, a series of admissible intersection arrays has been obtained: { c 2 ( u 2 − m 2 ) + 2 c 2 m − c 2 − 1, c 2 ( u 2 − m 2 ), ( c 2 − 1)( u 2 − m 2 ) + 2 c 2 m − c 2 ; 1, c 2 , u 2 − m 2 } (A. A. Makhnev, Wenbin Guo, M. P. Golubyatnikov). The cases c 2 = 1 and c 2 = 2 have been examined by A. A. Makhnev, M. P. Golubyatnikov and A. A. Makhnev, M. S. Nirova, respectively. In this paper in the class of graphs with the intersection arrays { mn − 1, ( m − 1)( n + 1)}, { n − m + 1}; 1, 1, ( m − 1)( n + 1)} all admissible intersection arrays for {3 ≤ m ≤ 13} are found: {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36}, {55,54,2; 1, 2,54}, {90,84,7; 1, 1,84}, {220,216,5; 1, 1,216}, {272,264,9; 1, 1,264} and {350,336,15; 1, 1,336}. It is demonstrated that graphs with the intersection arrays {20,16,5; 1, 1,16}, {39,36,4; 1, 1,36} and {90,84,7; 1, 1,84} do not exist.

For a nonempty set ω of primes, V. A. Vedernikov had constructed ω -fibred formations of groups via function methods. We study lattice properties of ω -fibred formations of finite groups with direction δ satisfying the condition δ 0 ≤ δ . The lattice ω δF θ of all ω -fibred formations with direction δ and θ -valued ω -satellite is shown to be algebraic under the condition that the lattice of formations θ is algebraic. As a corollary, the lattices ω δF , ω δF τ , τ ω δF , ω δ n F of ω -fibred formations of groups are shown to be algebraic.

We consider recursive sequences over the set of integers, where as rules of generation we take arbitrary superpositions of polynomial functions and the function | x |; such sequences are referred to as polynomial-modular recursive sequences. We show how evaluations on three-tape Minsky machines can be simulated via polynomial-modular recursive sequences. Based on this result, we formulate algorithmically unsolvable problems related to polynomial-modular recursive sequences. We also consider recursive sequences in which the rules of generation are functions formed by some superpositions of polynomial functions and the function [x]. $\begin{array}{} \displaystyle [\sqrt{x}]. \end{array}$ For the set of such recursive sequences, an algorithmically unsolvable problem is indicated.

The class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H , and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2 m , we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities.

We consider the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values 1 and –1 with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic T Fourier , the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of T Fourier . A hypothesis about the limit distribution of T Fourier is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.

We consider the sequence { ξ n , t } t ≥1 of controlled critical branching processes with immigration, where n = 1, 2, … is an integer parameter limiting the population size. It is shown that for n → ∞ the stationary distributions of considered branching processes normalized by n $\begin{array}{} \sqrt{n} \end{array} $ converge to the distribution of a random variable whose square has a gamma distribution.