Band 32, Heft 1

We prove that an arbitrary Boolean function may be implemented by an irredundant Boolean circuit over an arbitrary finite complete basis so that the circuit admits a single diagnostic test of length at most 4 with respect to inversion faults at gate outputs.

We study the graphs of transitions between states of nonautonomous automata that provide, with independent equiprobable input signs, an equiprobable distribution on the set of all states in the minimum possible number of cycles, as is the case of the de Bruijn graphs corresponding to shift registers. It is proved that in the case of a binary input alphabet, there are at least 12 r −33 pairwise nonisomorphic directed graphs with 2 r vertices that have this property. All graphs of this type with 8 and 9 vertices are found.

We consider a polynomial scheme with N outcomes. The Pearson statistic is the classical one for testing the hypothesis that the probabilities of outcomes are given by the numbers p 1 , …, p N . We suggest a couple of N − 2 statistics which along with the Pearson statistics constitute a set of N − 1 asymptotically jointly independent random variables, and find their limit distributions. The Pearson statistics is the square of the length of asymptotically normal random vector. The suggested statistics are coordinates of this vector in some auxiliary spherical coordinate system.

We consider distributions of the power series type determined by the generating functions of sequences satisfying linear recurrence relations with nonnegative coefficients. These functions are represented by power series with positive radius of convergence. An integral limit theorem is proved on the convergence of such distributions to the exponential distribution. For the generalized allocation scheme generated by these linear relations a local normal theorem for the total number of components is proved. As a consequence of more general results of the author, a limit theorem is stated containing sufficient conditions under which the distributions of the number of components having a given volume converge to the Poisson distribution.

We consider random permutations having uniform distribution on the set of all permutations of the n -element set with lengths of cycles belonging to a fixed set A (so-called A -permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.

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.