Pleiades Publishing Ltd

For every finite abelian p-group, an explicit embedding of this group in the Jennings group $\mathcal{J}(\mathbb{F}_p )$ is constructed.

It is known that the arithmetic of natural and integer numbers is unsolvable. Even the universal theory of integers with addition and multiplication is unsolvable. It is proved herein that an elementary theory of integers with addition, order, and multiplication by one arbitrary number is solvable and multiplication by the power of one number is unsolvable. For a certain n, the universal theory of integers with addition and n multiplications by an arbitrary number is also unsolvable.

We study polynomials given by the discrete Rodrigues formula, which generalizes a similar formula for Meixner polynomials. Such polynomials are associated with the theory of Diophantine approximations. The saddle point method is used to find the limit distribution of zeros of scaled polynomials. An answer is received in terms of a meromorphic function on a compact Riemann surface and is interpreted using the vector equilibrium problem of the logarithmic potential theory.

It is proved that at almost all points the order of approximation, even of one of the functions 1, cos x,sin x by means of a sequence of linear positive polynomial operators having uniformly bounded norms, is not higher than 1/n2. Refinements of this result are given for operators of convolution type.

Using the boundary-value problem for the singularly perturbed second-order differential equation as an example, we show that the multiplicity of the root of the degenerate equation significantly affects the asymptotics of the solution, especially in the boundary layer.