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.

Shirokov [1] recently suggested a construction of a noncommutative operator graph, depending on

a complex parameter *θ*, which enables one to construct channels with positive quantum capacity for

which the *n*-shot capacity is zero. We study the algebraic structure of this graph. Relations for the

algebra generated by the graph are derived. In the limit case *θ *= *??*1, the graph becomes commutative

and degenerates into the direct sum of four one-dimensional irreducible representations of the Klein

group.

The superactivation of the capacity of quantum channels was discovered in [2]. It turned out that

the quantum capacity for the tensor product of two quantum channels can be positive, whereas the

quantum capacity of each of the channels in the product is zero. As was shown in [3] and [4], the value of

the quantum capacity is closely related to the so-called noncommutative operator graph of the quantum

channel. In [5], a similar property was discovered for the classical capacity with zero error. In [6] and [7],

a technique of studying superactivation, which uses noncommutative operator graphs, was developed.

This enables one to construct low-dimensional examples of superactivation for quantum capacity. In

the present paper, the algebra generated by the noncommutative operator graph constructed in [1] is

studied.1

For UD-statistics, we present formulas that are in agreement with the value of the second virial coefficient as $\rho\to 0$ at the initial point of the activity a\to 0 and with the temperature on the critical isochore rho=\rho_c at the final point. This leads to two invariants: (1) the number of collective degrees of freedom and 2) the admissible size of the cluster fluctuation corresponding to a given temperature. In contrast to subcritical thermodynamics, in which the number of collective degrees of freedom undergoes a jump in the phase transition `` gas--liquid,'' the given invariants in supercritical hermodynamics remain valid on the whole isotherm. For rho>rho_c, using the van der Waals model, we see that fluids disintegrate into a cluster sponge and monomers. We show how these results can be carried over to real gases.

In this paper, a new approach to the two-body problem is considered in greater detail than in previous papers of the author. The semiclassical transition from quantum statistics to classical statistics is studied and an analog of superfluidity in classical thermodynamics for supercritical states is obtained.

The author attempts to change and supplement the standard scheme of partitions of integers in number theory to make it completely concur with the Bohr–Kalckar correspondence principle. In order to make the analogy between the the atomic nucleus and the theory of partitions of natural numbers more complete, to the notion of defect of mass author assigns the “defect” {a} = [a + 1] − a of any real number a (i.e., the fractional value that must be added to a in order to obtain the nearest larger integer). This allows to carry over the Einstein relation between mass and energy to a relation between the whole numbersM and N, whereN is the number of summands in the partition ofM into positive summands, as well as to define the forbidding factor for the number M, and apply this to the Bohr–Kalckar model of heavy atomic nuclei and to the calculation of the maximal number of nucleons in the nucleus.

For gradient-likeflows without heteroclinic intersections of the stable and unstable manifolds of saddle periodic points all of whose saddle equilibrium states have Morse index 1 or n−1, the notion of consistent equivalence of energy functions is introduced. It is shown that the consistent equivalence of energy functions is necessary and sufficient for topological equivalence.

In this paper, we introduce the notions of enlarged number theory and of thermodynamically ideal liquid and calculate the temperature below which it appears. This temperature is T = 0.84Tc, where Tc is the critical temperature of a gas whose molecules are nonpolar. For such a gas, in a sufficiently wide neighborhood of the binodal, the isotherms of a gas and of a thermodynamically ideal liquid coincide with those of a van der Waals gas for the critical value of the compressibility factor Zc = 3/8. In this sense, for T ≤ 0.84Tc and the particular case Zc = 3/8, the developed theory is a generalization of the van der Waals model. A new phase transition of the second kind at the point of zero activity is described.

We prove that the family of all connected *n*-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan.

We obtain a nontrivial estimate of the variance of the sum of bounded partial quotients appearing in the continued-fraction expansion of a rational number with fixed denominator. As a consequence, we obtain a law of large numbers for the sum of all partial quotients.

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.

Undistinguishing parastatistics of objectively distinguishable objects is considered. It describes clusters in the supercritical state. The relationship between the mesoscopic physics of clusters and the macroscopic thermodynamics of supercritical isotherms is established. We construct the supercritical pattern of isochores and isotherms under the assumption that the following three points: the Boyle temperature, the Boyle density, and the critical point are known. Passing from negative chemical potentials to positive ones, we obtain new relations differing from relations related to the well-known Gentile statistics.