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## Calculation of the number of collective degrees of freedom and of the admissible cluster size for isotherms in the Van-der-Waals model in supercritical states

In this paper, general questions concerning equilibrium and non-equilibrium states are discussed. Using the Van-der-Waals model, the relationship between Gentile statistics and non-ideal gas is demonstrated. The second virial coefficient is expressed in terms of collective degrees of freedom. The admissible cluster size at given temperature is determined.

This paper presents UD-statistics almost coinciding with Gentile statistics for negative chemical potentials, but having another extension in the case of positive chemical potentials. The physical meaning of the metamorphosis of the phase transition of the third kind to the phase transition of the first kind is explained. The coincidence of isochores and isotherms in the supercritical domain corresponding to UD statistics is shown.

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It is well known that the formula for the Fermi distribution is obtained from the formula for the Bose distribution if the argument of the polylogarithm, the activity a, the energy, and the number of particles change sign. The paper deals with the behavior of the Bose–Einstein distribution as a → 0; in particular, the neighborhood of the point a = 0 is studied in great detail, and the expansion of both the Bose distribution and the Fermi distribution in powers of the parameter a is used. During the transition from the Bose distribution to the Fermi distribution, the principal term of the distribution for the specific energy undergoes a jump as a → 0. In this paper,we find the value of the parameter a, close to zero, but not equal to zero, for which the Bose distribution (in the statistical sense) becomes zero. This allows us to find the point a, distinct from zero, at which a jump of the specific energy occurs. Using the value of the number of particles on the caustic, we can obtain the jump of the total energy of the Bose system to the Fermi system. Near the value a = 0, the author uses Gentile statistics, whichmakes it possible to study the transition fromthe Bose statistics to the the Fermi statistics in great detail. Here an important role is played by the self-consistent equation obtained by the author earlier.

The relationship between thermodynamics and economics has been known for a long time. The term ``thermoeconomics'' has even appeared. However, several aspects of the old thermodynamics are unacceptable in economics. For example, experts in thermodynamics believe that the diamond crystal is in the metastable state, and in due time will be transformed into graphite. However, these experts can hardly convince businessmen to part with their ancient diamonds.

The laws of economics require that the old conceptions of thermodynamicsa be mathematically scrutinized and reviewed.

The correspondence principle for quantum statistics, classical statistics and economics which associates the number of particles with the amount of money, the chemical potential with the nominal percentage, the negative pressure with debts, and the law of economic preference allowed to obtain agreement of the general theory of thermoeconomics with the latest experimental data.

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.