Vol 52, No 4 (2016)

The problem of determining the maximum mutual information I(X; Y) and minimum entropy H(X, Y) of a pair of discrete random variables X and Y is considered under the condition that the probability distribution of X is fixed and the error probability Pr{Y ≠ X} takes a given value ε, 0 ≤ ε ≤ 1. Precise values for these quantities are found, which in several cases allows us to obtain explicit formulas for both the maximum information and minimum entropy in terms of the probability distribution of X and the parameter ε.

We consider the estimation problem for an unknown vector β ∈ Rp in a linear model Y = Xβ + σξ, where ξ ∈ Rⁿ is a standard discrete white Gaussian noise and X is a known n × p matrix with n ≥ p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate β in this situation, we use a family of spectral regularizations of the maximum likelihood method β^α(Y) = H^α(X^TX) β^◦(Y), α ∈ R⁺, where β^◦(Y) is the maximum likelihood estimate for β and {H^α(·): R⁺ → [0, 1], α ∈ R⁺} is a given ordered family of functions indexed by a regularization parameter α. The final estimate for β is constructed as a convex combination (in α) of the estimates β^α(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method.

We study a family of distance graphs whose structure is close to that of Kneser graphs. We give new lower and upper bounds on the chromatic numbers of such graphs and consider the question of their interrelation. We also describe the structure of some important independence sets for this family of graphs and explicitly compute their cardinalities.

Skew maximum-period polynomials (MP polynomials) are a kind of pseudorandom sequence generators with good cryptographic properties and efficient implementation on modern processors using parallelism technologies. There are two known classes of MP polynomials, presented in various papers (actually coinciding, as will be shown below), obtained constructively. Here we describe a wider class of skew MP polynomials.