Volume 28, Issue 3

For weakly transient random walk in a random environment that tend at −∞ the limit theorem for the time of hitting a high level is proved.

Basic positively closed classes are intersections of positively precomplete classes. We prove that three-valued logic contains exactly 79 basic positively closed classes. Each class is described in terms of endomorphism semigroups.

We consider functions of p 2 -valued logic ( p is prime) that may be implemented by polynomials over the ring ℤ p 2 , and describe all closed classes that contain linear functions. It turns out that the set of these classes is countable. We also construct the lattice of such classes with respect to inclusion.

We propose a scheme of construction of 2-circulant codes with given code distance on the basis of biorthogonal filters with the property of perfect reconstruction over a finite filed of odd characteristic. The corresponding algorithm for constructing biorthogonal filters utilizes the Euclidean algorithm for finding the gcd of polynomials.

Theorems on the limit distributions of the minimal and maximal distances to the nearest neighbor in a sample of random independent points having a uniform distribution on a metric space are proved. As examples of such spaces a multidimensional torus and a binary cube are considered.