Volume 6, Issue 2

Sufficient conditions are found for oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay. Some asymptotic properties are studied for the bounded solutions.

The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained.

We investigate the question of the Kan extension in the case when all categories and functors are internal in the category of groups. Under certain assumptions we establish the necessary and sufficient conditions for the existence of internal Kan extensions. Questions related to this problem are also discussed.

For the frame θ in , let B 2 ( θ )(𝑥) (𝑥 ∈ ) be a family of all 𝑛-dimensional rectangles containing 𝑥 and having edges parallel to the straight lines of θ , and let M B 2 (θ) be a maximal operator corresponding to B 2 ( θ ). The main result of the paper is the following Theorem. For any function 𝑓 ∈ 𝐿 (1 + ln + 𝐿)() (𝑛 ≥ 2) there exists a measure preserving and invertible mapping such that 1. {𝑥 : ω (𝑥) ≠ 𝑥} ⊂ supp 𝑓; 2. This theorem gives a general solution of M. de Guzmán's problem that was previously studied by various authors.

We analyse the vector process (𝑋 0 (𝑡),𝑋 1 (𝑡), . . . , 𝑋 𝑛 (𝑡), 𝑡 > 0) where , 𝑘 = 1, . . . , 𝑛, and 𝑋 0 (𝑡) is the two-valued telegraph process. In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed. Special care is given to the case of the process (𝑋 0 (𝑡),𝑋 1 (𝑡),𝑋 2 (𝑡), 𝑡 > 0) representing a randomly accelerated motion where some explicit results on the probability distribution are derived.

A system of contracting symbols is introduced for a N. Bourbaki type general mathematical theory corresponding to a general classical mathematical theory .

Several results on the translation of factorization systems along adjunctions are proved. The first of these results is the answer to the question posed by G.Janelidze.