Vol 71, No 5 (2016)

Four-dimensional momentum mapping singularities of integrable Hamiltonian systems with two degrees of freedom are considered. An infinite series of pairs of 4-dimensional saddle–saddle singularities is constructed so that 4-singularities from each pair are not Liouville equivalent, but 2-foliations on their 3-boundaries are Liouville equivalent.

Orbits of the automorphism group of a finitely generated module over a principal ideal ring are described in terms of canonical representatives and by a complete system of invariants. For a primary module, a natural bijection to a sum of two Young diagrams is established for the set of orbits and the set of partitions of the Young diagram corresponding to the module, which allows us to calculate the number of orbits.

The regularity of solutions to the Dirichlet boundary value problem is studied for elliptic differential operators of order 2m defined on subdomains of a manifold. Relationships between the smoothness of the right hand side, the boundary, and the solutions are obtained.

Application of Chebyshev series to solve ordinary differential equations is described. This approach is based on the approximation of the solution to a given Cauchy problem and its derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process using Markov quadrature formulas. It is shown that the proposed approach can be applied to formulate an approximate analytical method for solving Cauchy problems. A number of examples are considered to illustrate the obtaining of approximate analytical solutions in the form of partial sums of shifted Chebyshev series.