Journal of Mathematical Sciences

In this paper, we construct a natural embedding $$\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} $$ of the complex projective space ℂP n considered as a 2n-dimensional, real-analytic manifold in the real projective space $$\mathbb{R}P^{n^2 + 2n} $$ . The image of the embedding σ is called the ℂP n-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ℂP-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ℂP ℝ n . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ℂP ℝ n is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.

By using a priori estimates and the Riesz theorem, we investigate a boundary-value problem for nonuniformly 2b -elliptic equations with arbitrary power singularities in the coefficients of the equation and boundary conditions on a certain set of points. We establish the existence and obtain an integral representation of the unique solution of the formulated boundary-value problem in Hölder spaces with power weights whose order is determined via the orders of singularities in the coefficients of the equation and boundary conditions.

We obtain the existence of a periodic solution for differential inclusions with lower semi-continuous right-hand sides on non-compact Lie groups. The construction is based on the combination of the method of guiding functions and the technique of integral operators with parallel translation.

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, we justify the general polynomial projection method for solving the corresponding problem. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation.

A three-dimensional associative parallel processor (APP3) is introduced and a program without loops and jumps is developed which for every algorithmically computable function 1) computes the function given a finite but sufficiently large memory, dependent on the function and the argument, 2) recognizes the domain of its definition on infinite memory.

The paper adjoins the work of B. I. Plotkin and S. M. Vovsi, Varieties of Representations of Groups (Zinatne, Riga (1983)), and turns out to be, in a sense, its continuation. In the former, the varieties of representations have been considered. As a matter of fact, the varieties under consideration are action-type varieties. This paper studies other classes of representations, axiomatizable in a special action-type logic.