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.

For the three-frequency quantum resonance oscillator, the reducible case, where its frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

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.

We introduce the notion of disinformation as an object annihilated by the corresponding information and analyze the relationships of this object with abstract analytic number theory and thermodynamics. For the entropy in this statistics, we present Nazaikinskii’s model, which contains only the statistics of Bose and Fermi gases (and excludes parastatistics).

We construct asymptotic solutions of the Navier-Stokes equations describing periodic systems of vortex filaments filling a three-dimensional volume. Such solutions are related to certain topological invariants of divergence-free vector fields on the two-dimensional torus. The equations describing the evolution of such a structure are defined on a graph which is the set of trajectories of a divergence-free field.

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.

Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations

are developed.

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group *G*, which are not *G*-rational and have no birational structure of *G*-Mori fiber space with the base of positive dimension.

We prove that the affine-triangular subgroups are the Borel subgroups of the Cremona groups

The problem of finding the number and the most likely shape of solutions of the system ∑∞ j=1λjnj ≤ M, ∑∞ j=1 nj = N where λj,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λj is of the order of λd/2, where d, the number of degrees of freedom, is less than two.

The notion of ideal liquid for the number of degrees of freedom less than 2, i.e., γ < 0, is introduced. The values of the pressure P and of the compressibility factor Z on the spinodal in the negative pressure region for the van der Waals equation determine the value of γ, γ(T) < 0, for μ = 0. For(Formula presented.), a relationship with the van der Waals equation is established. © 2015, Pleiades Publishing, Ltd.

On the space of a principal bundle, a Lorentzian metric and a time orientation are given that are invariant with respect to the action of the structure group. These objects form a fibered space-time and, in the case of spacelike fibers, induce the same structures on the base. The following causality conditions are discussed: chronology, causality, stable and strong causality, and global hyperbolicity. It is proved that if the base space-time satisfies one of the above conditions, then so does the fibered space-time.

Sequent calculus for the provability logic GL is considered, in which provability is based on the notion of a circular proof. Unlike ordinary derivations, circular proofs are represented by graphs allowed to contain cycles, rather than by finite trees. Using this notion, we obtain a syntactic proof of the Lyndon interpolation property for GL.

Let S be a bielliptic surface over a finite field, and let an elliptic curve B be the Albanese variety of S; then the zeta function of the surface S is equal to the zeta function of the direct product P1 × B. Therefore, the classification problem for the zeta functions of bielliptic surfaces is reduced to the existence problem for surfaces of a given type with a given Albanese curve. In the present paper, we complete this classification initiated in [1].

A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra *A* in *C*∞(*M*) coincides with *C*∞(*M*) if and only if *A* has the same tangent bundle as *M*.

In this paper, we establish conditions for the discreteness of extremal probability measures on finitedimensional spaces. This problem appears in Choquet theory, stochastic financial mathematics, in the construction of examples of the solution of theMonge–Kantorovich problem.

For an arithmetic semigroup (*G*, *∂*), we define entropy as a function on a naturally defined continuous semigroup *Ĝ* containing *G*. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (*G*, *∂*).