The author constructs his thermodynamics on the following two “first principles”: the partition theory of integers and the notion of Earth gravity. On the basis of number theory, equivalence classes in mesoscopy and soft condensates in the partition theory of integers are considered. The self-consistent equation obtained by the author on the basis of Gentile statistics is used to describe the effect of energy accumulation at the moment of transition of the boson branch of the partition of a number to the fermion branch. The branch point in the transition from bosons to fermions is interpreted as an analog of a jump of the spin.
We study the case where the values of random variables increase without bound.
In the present paper, we describe an approach to thermodynamics that does not involve Bogolyubov chains or Gibbs ensembles. We present isotherms, isochores, and isobars of various pure gases, as well as binodals, i.e., lines along which gas becomes liquid, and spinodals (endpoints of isotherms). We study supercritical phenomena for values of temperature and pressure above the critical ones. A lot of attention is paid to the region of negative pressures. The superfluid component for supercritical phenomena is described, as well as the thermodynamics of nanostructures and superfluidity in nanotubes.
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.
We develop the recent research  and introduce the notions of volume and entropy in abstract analytic number theory. The introduction of negative numbers in the generalized partition problem, together with the meaning of such a generalization in some applications of the theory, is discussed.
In this paper, we present an original implementation of Vinberg’s algorithm for arbitrary hyperbolic lattices subject to no constraints
A class of homeomorphisms on n-dimensional manifolds called Morse-Smale homeomorphisms is introduced and interrelation between their dynamics and topology of the ambient manifoilds is searched.
The eigenvalue problem for the perturbed resonant oscillator is considered. A method for constructing asymptotic solutions near the boundaries of spectral clusters using a new integral representation is proposed. The problem of calculating the averaged values of differential operators on solutions near the cluster boundaries is studied.
In this paper a unified method for studying foliations with transversal psrsbolic 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
We calculate certain bigraded Betti numbers for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the bigraded Betti numbers attain their minimum and maximum values among all simple polytopes of fixed dimension with a given number of facets.