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## Transformation of an irregular wave field along a quartic bottom profile

Transformation of an irregular wave field along a quartic bottom profile.

The book is comprised of 24 studies examining the changes in values throughout the process of transformation in the post-communist countries and, in general, the questions of values, their conceptualization and research as well as their role in the process of transformation and stratification. The studies present a new concept of empirical sociological study of values, cultural resources in class reproduction and ideology, problems of hedonism, social trust, cohesion, historical and cultural tradition and many other aspects of development of value structure in post-communist societies.

The process of globalization undoubtedly contributes to the changes and reduction of nomenclature and scope of state sovereign powers. At the same time this is a bilateral process: On the one side, the factors are strengthening that fairly undermine the countries` sovereignty, on the other – most states voluntarily limit the scope of their sovereignty including the right to use the capital punishment, nomenclature and size of taxation, the size of emission and borrowings etc. We believe that among the range of factors influencing the process of change of national sovereignty (including the technological and economical changes and the necessity to solve jointly a great number of issues), the factor of voluntary reducing of scope of sovereign powers to obtain an additional prestige and benefit appears to be of the utmost importance. At the same time the limiting of sovereignty inevitably leads to great changes in the policy patterns of states as well as of common people masses. So the problems of changes of functions and role of the modern state, in our view, must be among those of currant importance.

The method of obtaining dynamic graphic art objects using mathematics and information technologies is presented. Various technologies of graphic dynamic and interactive art objects based on the generated fractal images and software are considered. Results of information and mathematical computer experiments of Trubochkinf N.K. are analyzed:

- Art- video based on recording dynamic transformation of a fractal-image;

- Multilayer programmable animations (motion, color, transparency) fractals animations with internal layers of varying length, can be generated over a long recurring time image;

- Video art on the basis of three-dimensional fractals (dynamic and interactive).

Using the linear theory of waves in a compressible atmosphere located in a gravitational field, we found a family of sound speed profiles for which the wavefield can be represented by a traveling wave with no reflection. The vertical flux of wave energy on these nonreflected profiles is retained, which proves that the energy transfer may occur over long distances.

The purpose of this article - to formulate approaches to the study of the market transformation.The Market is seen as governance mechanism under the uncertaintycondition. Transforming the market, the firm is able to discover and commercialize the advantage of resources, located outside its borders.

Internal solitary wave transformation over the bottom step: loss of energy.

**The Afroeurasian world-system (AEWS throughout) is the largest world-system** that in the period of its largest expansion (prior to its transformation into the planetary World-System), in the 13th – 15th centuries encompassed almost all the societies of Europe and Asia and a substantial part of African societies.

The article focuses on the differences of medium-sized companies’ management methods: on the one hand, from companies in small (micro) business, and on the other hand, from large companies. The application of the method was tested at a medium-sized poultry farm. The analysis of projects of the real program was made, the network model of the program was built, and the probability of projects’ and the program at all success was estimated. The evaluation of the program indicators allowed the authors to make conclusions about priorities of separate projects. The proposed approach can be used in various companies, regardless of industry affiliation

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.