Band 13, Heft 6

Recently there appeared a significant number of papers investigating structural properties of random discrete sequences. They provide a large group of results on structurally equivalent intervals in such sequences. The goal of the present paper is to give a survey of the problems and the known results in this interesting area of discrete probability theory.

We consider the optimal tests based on the likelihood ratio for discriminating between two Markov chains having a common finite phase space ℒ. Their risks are expressed in terms of probabilities of large deviations for sum of random variables defined on another Markov chain with the phase space ℒ × ℒ. Both simple and composite alternatives are considered. The established asymptotic formulas for the considered risks are precise.

In this paper, we suggest a method to derive lower bounds for the complexity of non-branching programs whose elementary operations are unitary transformations over two complex numbers. This method provides us with estimates of the form Ω( n log n ) for unitary operators C n → C n , in particular, for the Fourier and Walsh transformations. For n = 2 k we find precise values of the complexity of those transformations.

We study inert matrices which remain degenerate or non-degenerate under any replacement of their non-zero elements by other non-zero numbers. In partially oriented graphs, we consider non-oriented matchings. We discuss a quantum model which fit these matchings. We prove that both perfect and imperfect oriented trees (that is, possessing and not possessing a perfect matching) may be obtained from the elementary ones with the use of some operations, that is, the set of the perfect trees and the set of the imperfect trees are free finitely generated algebraic structures.

In this paper, we continue and complete the study of finite primitive groups whose stabiliser of a point contains an Abelian normal subgroup acting irreducibly (by conjugations) on an Abelian normal subgroup of the whole group. Each such group H is isomorphic to the subgroup of the full affine group of the field , where the symbol of the semi-direct product λ unites the ν -power of the cyclic group Z p , the metacyclic group Θ, and some group of automorphisms ψ of the field . Using the Zassenhaus classification of finite semi-fields, we enumerate primitive subgroups of the full affine groups of finite semi-fields.

The distribution of jobs in a system with m identical parallel processors which minimises the load of the maximally loaded processor is an NP-hard problem. Many approximate algorithms are developed for this problem, but for the version of the problem where the jobs arrive and must be treated on-line there is no algorithm possessing the guaranteed estimate which is less than 1 + 1/√2 for m ≥ 4 and tends to 1.837 as m → ∞. In this paper, we consider the version of the problem where jobs arrive one by one and must be treated on-line under the additional condition that the total duration of the jobs is known. For this version of the problem we suggest an algorithm with the guaranteed estimate equal to 5/3.

For sums of independent identically distributed non-negative integer-valued random variables we give a series of limit theorems which may be of use in probabilistic combinatorics in the framework of the generalised allocation scheme.

A set A of integers is called sum-free if a + b ∉ A for any a , b ∈ A . For any real numbers q ≤ p we denote by [ q , p ] the set of real numbers x such that q ≤ x ≤ p . Let S ( t , n ) stand for the family of all sum-free subsets A ⊆ [ t , n ], and s ( t , n ) = | S ( t , n )|. We prove that s ( t , n ) = O (2 n /2 ) for t ≥ n 3/4 log n , where log t = log 2 t .