Table of contents

We study the behaviour at boundary points of a solution of the Dirichlet problem with continuous boundary function for the Euler equation generated by the Lagrangian with variable that has logarithmic modulus of continuity and satisfies the condition . We obtain a regularity criterion for a boundary point of Wiener type, an estimate for the modulus of continuity of the solution near a regular boundary point, and geometric conditions for regularity.

We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard-Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem.

We obtain general algebraic identities for the Nijenhuis and Haantjes tensors on an arbitrary manifold Mⁿ. For n=3 we derive special algebraic identities connected with the Cartan-Killing form (u,v)_H.

We investigate conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets in . Some new results of a reductive nature are obtained which ensure that a compact set is an approximation compactum if certain special subsets with a simpler topological structure have this property.

A lower bound is found for the maximum modulus of the Riemann zeta function on segments of the critical line whose length does not exceed the double logarithm of the distance from the centre of the segment to the origin.

For Jordan domains in of Dini-Lyapunov type, we show that any function subharmonic in  and of class can be extended to a function subharmonic and of class  on the whole of  with a uniform estimate of its gradient. We construct a large class of Jordan domains (including domains with -smooth boundaries) for which this extension property fails. We also prove a localization theorem on -subharmonic extension from any closed Jordan domain.

This paper deals with the mixed boundary-value problem for the Poisson equation at the junction of thin rods and a massive body  that have different stiffnesses. We suggest a new approach to the study of this singularly perturbed problem. Namely, we construct a model of the junction that gives an approximation to the solution of the original problem on the whole range of parameters and  (the relative thickness and relative stiffness of the rods). The model contains ordinary differential equations on the line segments  (the axes of the rods) and the Neumann problem on the domain , which are combined into a single problem by imposing asymptotic conjugation conditions at the points correlating the coefficients of the expansions of solutions on  (as ) with those of solutions on  (as ). We obtain estimates for the accuracy of the model that are asymptotically exact. The conjugation conditions preserve the parameters  and  but generate a regularly perturbed problem, and it is not difficult to obtain and justify asymptotics of its solutions and those of solutions of the original problem under any relation between  and .

As is known, there are everywhere discontinuous infinitely Fréchet differentiable functions on the real locally convex spaces and of finitely supported infinitely differentiable functions and, respectively, of generalized functions. In this paper the relationship between the complex differentiability and continuity of a function on a complex locally convex space is considered. We describe a class of complex locally convex spaces, which includes the complex space , such that every Gateaux complex-differentiable function on a space of this class is continuous. We also describe another class of locally convex spaces, which includes the complex space , such that on every space of this class there is an everywhere discontinuous infinitely Fréchet complex-differentiable function whose derivatives are continuous.

We prove the birational superrigidity and non-rationality of a cyclic triple covering of  branched over a nodal hypersurface of degree  for . The result obtained solves the problem of birational superrigidity for smooth cyclic triple spaces. We also consider certain relevant problems.

Errata to the paper by R.R. Gontsov "Refined Fuchs inequalities for systems of linear differential equations" Izvestiya: Mathematics, 2004, 68:2, 259-272.