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## Violation of Carath\'eodory axioms at the critical point of a gas. Frenkel point as a critical point of the transition ``liquid-amorphous solid'' in the region of negative pressure

We give a geometric interpretation of the thermodynamic potential, free and internal energy, and enthalpy in terms of a Lagrangian manifold in the phase space of pairs (T, -S), (-mu, N), and (P,V) of intensive and extensive variables.

The Lagrangian manifold is viewed as the dequantization of the tunnel canonical operator. With this approach, the critical point is a point where the equilibrium quasi-static process described by the Carath\'eodory axioms is violated. For a hard liquid with negative pressure, we present a model of a multi-modulus medium.

The article presents the results of a study of organizational culture of enterprises in accordance with the new methodology focused on identification of its two levels: the declared and actual ones. Compilation of the data of the analysis allowed characterizing six variants of a combination of such levels.

Traditionally, the effectiveness of the organizational culture is viewed in the context of the overall effectiveness of the company. However, this approach makes it possible to determine the share of the existent organizational culture in productivity of the company only hypothetically. The article presents authors approach based on the concept of «focal points» reflecting the degree of efficiency and stability of the organizational culture of the company.

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

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.