Volume 6, Issue 4

In this paper we consider several constructions which from a given 𝐵-product * 𝐵 lead to another one . We shall be interested in finding what algebraic properties of the ring 𝑅 𝐵 = 〈𝐶 ℕ , +, * 𝐵 〉 are shared also by the ring . In particular, for some constructions the rings 𝑅 𝐵 and will be isomorphic and therefore have the same algebraic properties.

Inequalities are derived which enable one to estimate integral moduli of continuity of functions of several variables in terms of Fourier coefficients.

We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Sufficient conditions are established for the oscillation of proper solutions of the system where satisfy the local Carathéodory conditions and τ 𝑖 , σ 𝑖 : ℝ + → ℝ + ( 𝑖 = 1, . . . , 𝑚) are continuous functions such that σ 𝑖 ( 𝑡 ) ≤ 𝑡 for 𝑡 ∈ ℝ + , , .

We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investigate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.

Questions of approximative nature are considered for a space of functions 𝐿 𝑝 (𝐺, μ ), 1 ≤ 𝑝 ≤ ∞, defined on a locally compact abelian Hausdorff group 𝐺 with Haar measure μ . The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.

The solvability of the first boundary value problem is investigated for a second order elliptic system with degeneration on the entire domain boundary.