In view of the results obtained in this communication we can connect macroscopy and microscopy via an intermediate level of mesoobjects.

This paper generalizes assertion that a consistent process is a supermartingale with respect to any equivalent martingale probability measure if and only if it admits a Doob uniform expansion with respect to any measure from a given class.

It is shown that, for the Lennard-Jones potential, there exist far-range van-der-Waals forces. It is claimed that such potentials occur rarely, just as potentials for the Schrödinger equation whose semiclassical solutions coincide with exact solutions.

The value of the empirical expectation coincides with that of the mean energy of an ideal Bose gas for one particle. The exact mathematical identity for these quantities makes it possible to carry over the concept of temperature corresponding to the mean energy to an unboundedly increasing sequence of random values for a new unbounded probability theory and for a generalization of the Kolmogorov complexity theory. The notion of spectral gap, which was introduced in superconductivity theory, is carried over to unbounded probability theory

We consider the space U(T) of all continuous functions on the circle T with uniformly convergent Fourier series. We obtain an estimate for the growth of the U -norms of exponential functions with an arbitrary piecewise linear phase and unboundedly growing integer frequences.

A family of parabolic integro-differential equations with nonlocal diffusion on the circle which have no smooth inertial manifold is presented.

We obtain a distribution of Fermi-Dirac type for a hard liquid at temperatures less than the Frenkel temperature TF for P ≥ 0 and Z ≥ 0. For the van der Waals model, one has TF = (33/25)Tc. © 2015, Pleiades Publishing, Ltd..

The equivalence of the error of approximation by Fourier means and of general moduli of smoothness, provided that their generators are equivalent, is established.

We consider the quantile function of a fuzzy random variable and obtain expressions for some expectations related to fuzzy random variables via integrals of quantile functions.

In the paper “The relationship between the Fermi–Dirac distribution and statistical distributions in languages” (This issue of “Math. Notes”), the Bose–Einstein and Fermi–Dirac distributions are considered in connectionwith frequency distributions in languages in the context of the author’s approach to thermodynamics and number theory problems. The present article presents certain clarifications of certain notions used in that approach, in particular, the identity of particles, the Poisson adiabat, thematching of number theory formulas with those given by the Bose–Einstein distribution, nonstandard analysis, and others.

Rotation of a neutron in the coat of helium-5 as a classical particle for a relatively large value of the hidden parameter (measurement time) tmeas = h/Ems is considered. In consideration of the asymptotics as N → 0, equations for the mesoscopic energy Ems are given. A model for the helium nucleus is introduced and the values of the mesoscopic parameters Mms, and Ems for helium-4 are calculated.

The eigenvalue problem for a perturbed two-dimensional resonant oscillator is considered. The exciting potential is given by a nonlocal nonlinearity of Hartree type with smooth self-action potential. To each representation of the rotation algebra corresponds the spectral cluster around an energy level of the unperturbed operator. Asymptotic eigenvalues and asymptotic eigenfunctions close to the lower boundary of spectral clusters are obtained. For their calculation, asymptotic formulas for quantum means are used.

We study how to construct asymptotic solutions of the spectral problem for the Schrödinger equation on a geometric graph. Differential equations on sets of this type arise in the study of processes in systems that can be represented as a collection of one-dimensional continua interacting only via their endpoints (e.g., vibrations of networks formed by strings or rods, steady states of electrons in molecules, or acoustical systems). The interest in Schrödinger equations on networks has increased, in particular, owing to the fact that nanotechnology objects can be described by thin manifolds that can in the limit shrink to graphs (see [1]). The main result of the present paper is an algorithm for constructing quantization rules (generalizing the well-known Bohr–Sommerfeld quantization rules). We illustrate it with a number of examples. We also consider the problem of describing the kernels of the Laplace operator acting on k-forms defined on a network. Finally, we find the asymptotic eigenvalues corresponding to eigenfunctions localized at a vertex of the graph.

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.