In  and , the author described the stereotype approximation property, which is an analog of the classical approximation property transferred to the category Ste of stereotype spaces. It was noticed in  that, for a stereotype space X, the stereotype approximation condition is formally stronger than the classical approximation condition (although, up to now, it remains unclear whether or not these conditions are equivalent). For this reason, the question of what particular spaces in the standard package used in functional analysis have the stereotype approximation property is quite difficult (the only exception is the case where the space has a topological basis in some reasonable sense). In this paper, we prove the following fact concerning the stereotype group algebras C*(G) (defined in  and ).
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 note, we present an unusual picture of interaction of singularities for pressureless gas dynamics.
In this paper, we present an original implementation of Vinberg’s algorithm for arbitrary hyperbolic lattices subject to no constraints