In 1993 M. Lassak formulated (in the equivalent form) the following conjecture. If we can inscribe a translate of the cube $[0,1]^n$ into a convex body $C \subset R^n$, then $\sum_{i=1}^n \frac{1}{\omega_i} \geq 1$. Here $\omega_i$ denotes the width of $C$ in the direction of the ith coordinate axis. The paper contains a new proof of this statement for n = 2. Also we show that if a translate of $[0,1]^n$ can be inscribed into the n-dimensional simplex, then for this simplex holds $\sum_{i=1}^n \frac{1}{\omega_i} = 1$.

The article includes principal quasinormal forms of singular pertubed parabolic
equations systems in critical cases.

In this paper we propose a method for constructing the asymptotics for a set of linear
independent solutions of systems of differential equations with oscillatory decreasing
coefficients. We illustrate this method by constructing the asymptotics for solutions of
a system of two oscillators with slowly decreasing coupling and friction in one of the
oscillators.

We study properties of local dynamics of a differential equation with exponentially
distributed delay. In critical cases we built special evolutionary equations.

On the basis of the operator approach to differentiation we extend the concept of an
antiderivative notion to wider classes of functions.

We study the local dynamics of the Ginzburg-Landau equation with small diffusion
in a neibourhood of running waves. We find necessary conditions of running waves
instability and sufficient conditions of their stability.

Two numerical methods for solving the periodic boundary value problem are
considered: Galerkin's method and the method of polygonal lines. The original problem
is mapped to the sequence of its discretization - systems of equations in finite spaces.
Conditions under which the existence of solutions of a periodic boundary value problem
entails its solvability of discrete options are given. The question of approximate solutions
convergence is studied.

Information is a very important resource in economic interaction. But information is
not a usual resource, it requires formalization of a model that takes into account specific
features of the information. This model based on balance equations is represented in the
paper. A problem of extreme performance of a closed economic system where agents can
exchange information is solved with the help of the model introduced.

The topic of the research is a relationship between a class of hypergraphs of a special
type and properties of the points of the cut polytope relaxations $M_{n,k}$. It is established
that for a sufficiently large $n$ in $M_{n,4}$ and $M_{n,5}$ polytopes, there are points which have
no integer vertices in any expansion in a convex combination of $M_{n,3}$ vertices.

In the article we concern the questions connected with the realization of a dynamic
programming method in problems of consequent visiting of megapolises. The realization
is complicated by precedence conditions and works inside the megapolises. The
scheme for constructing a reduced (partial) array of Bellman function values using parallel
computing without loss in accuracy is suggested. This procedure is realized on
a multiprocessor computing system; parallelizing is performed at the stage of Bellman
function layers construction.

This paper describes an approach to multimedia documents modeling. The approach
allows a non-programming user to compose pages from objects of different types (blocks)
and to install new types of objects (modules) made by third-party programmers. We
propose original methods for document version comparison and multimedia document
indexing and searching. The paper describes an architecture of a Content Management
System for the model proposed; the focus is to provide software portability and backward
compatibility. We elaborated two software products using this architecture: Fuzzle CMS,
multimedia CMS for Flash websites, and FLAscheme, the embeddable business diagram
editor.

A new algorithm for query optimization is proposed. This algorithm simplifies queries
restriction containing multi-attribute conditions. It also solves the problem of "Condition
AND Conjunction of conditions" expressions.