### Working paper

## On dynamics of Lagrangian trajectories for Hamilton–Jacobi equations

Seasonality and cyclicity - are two influential factors that affect dynamics of macroeconomic indicators both during the year and longer periods of time. In this article are discussed methodological questions that arise during seasonal decomposition of the GDP by factors for the year when balance aggregate and factors ratio is constant. Economic cycles mechanisms origin and their identification questions based on the combination of classical methods of spectral analysis and historic approach. Presented is the fact that along with more regular cycles such as investment and Kondratiev wave, influence of shocks (such as «oil prices crises») appear so called causal cycles that lead to a serious change in technological base of production. Particular importance (emphasis is placed on ) a new technological wave which is expected to strike the world in 2020 ^{th} and those goal set before the Russia. This research is done on the basis of world and Russian (national) statistics.

The article presents the results of experimental and theoretical studies of the dynamics of the process of electric discharge sawing by wire tool electrode aimed at solving problems of control

The article concentrates on the analysis of new tendencies in the theoretical foundations of historical sociology in the incoming new century. It focuses on the so called “third wave” in sociology which strangely remains unnoticed by historians. Meanwhile the representatives of the “third wave” rejected the fundamental principles of their teachers – creators of many famous concepts of modernization. The new generation in American sociology prefers to focus their studies on topics other than typology, searching for contingency, unpredictability, chains of events, path dependency etc. Here in conceptual approaches to the past social reality we find out deliberate and thought-out attempt to use transformed and formalized but essentially historical methods which are well articulated. The broader object of the research is the historical knowledge and the professional concepts about “subject and method”, the creation of interdisciplinary areas, mutual adoption and interventions.

The problem of optimal control for a class of nonlinear objects with uncontrolled bounded disturbances is formulated in the key of differential game. For problems with a quadratic quality functional task of searching for the optimal control reduces the need to find solutions to the scalar partial differential equation Hamilton-Jacobi-Isaacs. Finding solutions to this equation in the operation rate of the object by means of special algorithmic procedures obtained by using the theory of viscous solutions. The results can be used to solve theoretical and practical problems that occur in mathematics, mechanics, physics, biology, chemistry, engineering sciences, control and navigation.

We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S|/ + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/. We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.

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.