Table of contents

We obtain conditions on a real polynomial necessary and sufficient for this polynomial to be the characteristic polynomial of a matrix with non-negative elements.

This paper deals with non-negative solutions of the elliptic inequalities in , where and are functions and is an unbounded open subset of , .

After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions . The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument.

We consider Walsh functions on the binary group  and study uniqueness sets for -fold multiple Walsh series under convergence over cubes (in other words, -sets). We prove that every finite set is a -set, construct examples of countable -sets and non-empty perfect -sets, and give an example of a -set having the maximum possible Hausdorff dimension.

We obtain necessary conditions for a bounded function to be a Fourier multiplier of weak type , where or , provided that the Young function grows slower than as  tends to infinity.

We establish results on the asymptotic behaviour of solutions of non-stationary linearized equations of hydrodynamics with a small viscosity coefficient and periodic data oscillating rapidly with respect to the spatial variables. We obtain boundary-layer terms, homogenized (limiting) equations and cell problems (whose solutions determine approximate asymptotics of solutions of the equations under consideration) and obtain estimates for the accuracy of the asymptotics. The form of the asymptotics depends strongly on the mutual asymptotic behaviour of the viscosity coefficient and the periodicity parameter that characterizes rapid oscillations of the data. When the viscosity coefficient is very small, the asymptotics can contain rapidly oscillating terms that increase linearly with respect to the time variable. Similar theorems are proved for non-stationary Stokes equations and partial results are obtained for non-stationary Navier-Stokes equations.

The paper deals with the approximation of functions belonging to the Sobolev spaces  and by functions of the form . The results obtained are applied to the study of the stability of solutions of non-linear second-order differential equations of a special form. We consider the problem of whether two solutions can coincide given supplementary information in terms of the values of the functionals , , defined on the solutions.

We study properties of the -capacity (regarded as a function of sets of quantum states) in the infinite-dimensional case. We consider various subsets of states and determine their -capacity and optimal average. We construct counterexamples that illustrate general results. The possibility of "finite-dimensional approximations" of the -capacity and optimal average is shown for an arbitrary set of quantum states.

Anatolii Alekseevich Karatsuba was seventy on 31 January 2007. He has been a member of the Editorial Board of this journal for 30 years, and was Deputy Editor from 1977 to 1988 and from 1994 to 2003. We congratulate Anatolii Alekseevich on his birthday and heartily wish him good health, happiness and every success in his scientific work.