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## A Generalization of a Classical Number-Theoretic Problem, Condensate of Zeros, and Phase Transition to an Amorphous Solid

Abstract—Regularization of the Bose–Einstein distribution using a parastatistical correction,

i.e., by means of the Gentile statistics, is carried out. It is shown that the regularization result

asymptotically coincides with the Erdo˝ s formula obtained by using Ramanujan’s formula for the

number of variants of the partition of an integer into summands. TheHartley entropy regarded as the

logarithm of the number of variants defined by Ramanujan’s exact formula asymptotically coincides

with the polylogarithm associated with the entropy of the Bose–Einstein distribution. The fact that

these formulas coincide makes it possible to extend the entropy to the domain of the Fermi–Dirac

distribution with minus sign. Further, the formulas for the distribution are extended to fractional

dimension and also to dimension 1, which corresponds to the Waring problem. The relationship

between the resulting formulas and the liquid corresponding to the case of nonpolar molecules is

described and the law of phase transition of liquid to an amorphous solid under negative pressure

is discussed. Also the connection of the resulting formulas with the gold reserve in economics is

considered.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z) diagram by the second sheet.

It is well known that the supercritical state of a gas has great dissolving capacity. In this paper, the mathematical reason for this phenomenon is studied in great detail.

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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.

For a gas mixture, the new concept of number-theoretic internal energy is introduced. This energy does not depend on the masses of the miscible gases.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z)- diagram by the second sheet.

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.