Vol. 36 No. 1 (197) (2002)

This work investigates the problem of solvability on the equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ with Hermitian coefficients ${\bf{\it h}}, {\bf{\it k}}$ of weak complete Banach algebra ${\bf{\it A}}$. In the article (theorem 1) the criterion of solvability of equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ is proved which implies (theorem 2) the following algebraic criterion of solvability. For solvability in ${\bf {\it A}}$ the equation ${\bf{\it h}}x-x{\bf{\it k}}=c$ the similarity of matrixes $\Big( \begin{matrix} {\bf{\it h}}, & 0 \\ 0, & {\bf{\it k}}\\ \end{matrix} \Big)$ and  $\Big(\begin{matrix} {\bf{\it h}}, & c\\ 0, & {\bf{\it k}} \\ \end{matrix} \Big)$ is necessary and sufficient.

The Parseval identity is proved for elements of energetic space $L^2_{\Delta_0}(I)$, born by two-parametric Dirac system.

Munts quazipolinomial orthogonal systems from the $\left\{ x^{\gamma_k}\right\} $ and $\left\{ e^{-\gamma_k x}\right\}$ $(\gamma_k$ are real numbers) have been for the first time derived in an integral representation by H.V. Badalian [1,2]. In the case of $\left\{ e^{-\gamma_k x}\right\}$ they are considered in the article as (2), which remains without change in a more general case for multiple $\gamma_k$.  In this connection the importance of developing a two power sequences based biorthogonal system is that together with the sequence  $\left\{e^{-\gamma_k x}\right\}$  it creates a possibility of free selection another sequence $\left\{e^{-\gamma_k x}\right\}$ simplifying the application of functions representation apparatus.

In this paper we consider boundary problems for high order quasilinear evolution equations, that are degenerate on boundary in two directions with different weights. It is proved that there exists a solution in special weighted functional spaces and the solution is unique.

In the work possible complexes of three-dimensional planes in six-measured projective space P⁶ are studied. It's proved that one-parametric family of cones of second order with three-dimensional flats forming and univariate top which describes unfold surface defines four-parametric possible family of planes E³, which are all three-dimensional forming to this cones. It's also proved that if we take in space P⁶ our-parametric family of three-dimensional planes including fixed straight line l and touching two hypercone with one general univariate top  l  we will get possible family of three-dimensional planes. Corresponding family tangent of four-parametric family is formed by intersection of tangent hyperplanes to the cones in the sport of osculation of three-dimensional planes family with them.

The problem of control of a material point when position of a target point is known with accuracy of some set is investigated and in process of approach is specified discretely. With application of the guaranteed approach the problem of synthesis, optimum reduction time for an ellipse and a piece is solved.

Let A be a compact convex subset of a locally convex X space, E_nA, n≥1, be a set on n-extremaly points of the A set. In the present note criteria of the A set's finite dimensionality are formulated and proved in terms of the E_nA sets.

Problem of optimal control of stepwise changing linear systems with phase restrictions in intermediate instants is investigated at a specific criterions of quality on the whole time interval and method of a solution is offered.