Эта статья написана на основе трёх лекций, которые автор прочитал в университете Калабрии (Италия) в мае 1999 г. по приглашению проф. Манлио Гаудиозо. Статья представляет собой нестандартное введение в вариационное исчисление, математически строгое и согласованное в идейном плане с теорией конечномерных экстремальных задач 

The class of iterative methods of the solution of regular non-linear set of equations is constructed. The methods of a class heve following special property. At the solution of rectangular joint linear system these methods find the precise solution of a system for the finite number of steps. Similar property has the Broyden method (Gay theorem), and also some quasi-Newton algorithms of optimisation, but only with reference to square systems. The thorem of convergence of the constructed methods for the square non-linear systems is proved. On a example is shown, that these methods can effectively be used for the solution of the rectangular non-linear systems. The formulation of methods supposes control of length of a step, in this work one possible algorithms is offered 

The conditions on the linear part and the nonlinearity of differential system, sufficient for the equilibrium state conditional stability, are investigated. The result generalizing several classic ones is proved 

This paper presents recent results on a certain mapping between context-free grammars and systems of linear diophantine equations. Such investigation looks to be useful for designing of new algorithms for integer linear optimizations or for solving system of linear equations and inequations in nonnegative integers 

This paper deals with a state-constrained optimal control problem governed by a system of high order elliptic equation. A singular system of equations is investigated. In this system, a given control may give rise to no state or, on the contrary, infinitely many ones. The main contribution of the paper is finding of maximum principle of Pontryagin's type and its proof 

he phenomenon of trasference of stability between branches (change of critical tone) in non-conservative mechanical systems is studied. Such phenomenon has often been discussed in the literature for specific systems (wings in air flow, tubes conveying fluid, etc.). In the paper complete classification of different types of this phenomenon in generic case is carried out. Every type is described from the singularities and bifurcations theory point of view 

In frameworks of the local interaction model are constructed at the given area of the base, limitations on length and the cross-sectional sizes of a body in the class of sectionally smooth surfaces are constructed the spatial bodies with the minimum drag. Is proved, that their shape is formed of surface segments of a circular cone and its tangent planes, the vector of a normal to which makes with a direction of motion some fixed optimum angle. Is shown, that the minimum drag shape can be unconical and asymmetrical. Thus the force which is operational on an optimum body, has not component in a plane, perpendicular direction of motion 

The aim of this paper is to study the influence of surface self-diffusion on a surface microrelief formation of the stressed layer. Its layer has a finite thickness and one is rigidly fixed on the immobile substratum. The study is based on the Gibbs thermodynamic approach. The surface stability critrion was obtained 

It was shown that the energy balance equation for the problem of stretching of an elastic domain with angular notches can be written. The macrocrack that propagate from the angular point was modeled by the frictional dimension fractal. The dependence of fractal dimension of crack from value of angular cut was obtained. This formula give a possibility to satisfy with the Griffith equation of energy balance, that it is impossible to make within the framework of the classical mechanics of destruction. As an example, the problem about a stretching of a plane weakened by a cut as symmetric hole was solved and on the basis of this model the value of destroying loading in this case was calculated 

The electron-beam fluorescence method for measurements of mean density and density fluctuations in an axisymmetrical hypersonic wake flows was developed. The characteristics of a field of mean density and density disturbances in a hypersonic laminar wake behind an sharp cone were determined at Mach number M=21 and unit Reynold's number Re=6·1051/m. The experiment at such high Mach number was conducted at the first time