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## Semiclassical Approximation for the Curie – Weiss Model

The paper is devoted to the construction of spectral series and the estimation of the approximation accuracy for the operator of the Curie – Weiss model. In the course of work, the operator is reduced to a tridiagonal form in the subspace of the original space, then to a secondorder difference equation. The admissibility of reducing an operator to a subspace is presented. It is shown that the difference equation can be considered in the discrete semiclassical approximation. In the obtained classical system, the dependence of the turning points on the model parameters is investigated. The asymptotics of the spectrum of the Curie-Weiss operator is calculated and the accuracy of the approximation is estimated.

We consider a model quantum Hamiltonian of a charge in a resonance electromagnetic trap. Using the operator averaging method, we obtain an effective quantum operator that asymptotically describes the anharmonic part of the Hamiltonian. We show that the operator becomes a second-order difference operator in a specially chosen quantum action-angle representation. Using the discrete WKB method for this difference equation, we obtain the semiclassical WKB asymptotics of the spectrum and stationary states of the charge.

*This report investigates the predictability of cyclical turning points in Russia. For years, anyone interested in Russia had access to a full set of common tools for business cycle analysis, such as several composite leading indicators, a purchasing managers’ index, enterprise and consumer sentiment indexes, and so on. However, the 2008-09 world financial crisis spread throughout Russia quite unexpectedly for most politicians, businessmen and experts alike. Is it possible that none of existing indexes were able to say anything about the approaching decline? Using a simple “rule of thumb” proposed in this report one may easily see **that in reality this was not the case. So then why did a more or less definite forecast provided by some indexes have no consequences for common economic sentiments in Russia? This report gives some answers to this question. *

Painlevé equations, holomorphic vector fields and normal forms, summability of WKB solutions, Gevrey order and summability of formal solutions for ordinary and partial differ- ential equations, • Stokes phenomena of formal solutions of non-linear PDEs, and the small divisors phenomenon, • summability of solutions of difference equations, • applications to integrable systems and mathematical physics.

The objective of this study is to develop a system of leading indicators of the business cycle turning points on a wide range of countries, including Russia, over a more than thirty years period. We use a binary choice model with the dependent variable of the state of economy: the recession, there is no recession. These models allow us to assess how likely is the change of macroeconomic dynamics from positive to negative and vice versa. Empirical analysis suggests that the inclusion of financial sector variables into equation can significantly improve the predictive power of the models of the turning points of business cycles. At the same time, models with financial and real sector variables obtained in the paper outperform the “naïve” models based only on the leading indicator of GDP in the OECD methodology due to either a lower level of noise (recession model) or a higher predictive power (model of the recovery from recession).

We suggest a new approach to finding the maximal and the minimal spectral radii of linear operators from a given compact family of operators, which share a common invariant cone (e.g., family of nonnegative matrices). In the case of families with the so-called product structure, this leads to efficient algorithms for optimizing the spectral radius and for finding the joint and lower spectral radii of the family. Applications to the theory of difference equations and to problems of optimizing the spectral radius of graphs are considered.

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a uid under gravity in such a domain. These so- lutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (in quadratures). The behavior of the so- lutions at far distances is studied by use of the saddle point technique. The corresponding eigenoscil- lations of a uid are then interpreted as eigenfunctions of the continuous spectrum.

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.