We discuss the general opportunity to create (asymptotically) a comletely integrable system from the original perturbed system by inserting additional perturbing terms. After such an artificial insertion, there appears an opportunity to make the secondary averaging and secondary reduction of the original system. Thus, by this way, the $3D$-system becomes $1$-dimensional. We demonstrate this approach by the example of a resonance Penning trap.

In the model of Penning trap with a geometric asymmetry we study a resonance regime which produces a hyperbolic type algebra of integrals of motion. The algebra has qubic (non-Lie) commutation relations with creation-anihilation structure. The anharmonic part of the trap potential determines a top-like Hamiltonian over this algebra. The symmetry breaking term generates a tunnelling pseudoparticle (closed instanton). We study its action and corresponding spectral splitting.

The convex hull of all integer points of a noncompact polyhedron is closed and is a generalized polyhedron only under certain conditions. It is proved that if only the integer points in the interior of the polyhedron are taken, then most of the conditions can be dropped. Moreover, the object thus obtained has properties resembling those of a Klein polyhedron, and it is a Klein polyhedron in the case of an irrational simplicial cone.

Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.

In the paper we consider the hidden parameter (measurement time t_meas) which combines quantum and classical theory. We show that the Bose–Einstein and Fermi–Dirac quantum distributions turn out to be the decisive factor in the construction of isotherms in classical thermodynamics and in the description of the phase transition “gas to liquid” and “liquid to solid”.

Negative pressure also means negative energy and, therefore, “holes”, antiparticles. Continuation across infinity to negative energies is accomplished by using a parastatistical correction to the Bose-Einstein distribution.

A parallel between physical derivation and mathematical proof in classical thermodynamics is drawn. A relationship between thermodynamics and analytic number theory is demonstrated.

The author constructs a new conception of thermodynamics which is based on new results in number theory. We consider a maximally wide range of gases, liquids, and fluids to which, in principle, the Carathéodory approach can be applied. The Carathéodory principle is studied using the Lennard-Jones potential as an example. On the basis of this example, we analyze the dispersive structure of a fluidwhose density exceeds the critical value. We introduce a new parameter, the “jamming factor,” which determines the jamming effect for such fluids. A comparison with experimental data for nonpolar molecules is carried out. The phase transition “liquid-amorphous solid” is studied in detail in the domain of negative pressures. We discuss the theoretical relationship between the obtained solutions and econophysics, some mysteries in biology, and other sciences.

Any compact body with regular boundary in *R^N* defines a two-valued function on the

space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body.

This function is never algebraic if N is even and is very rarely algebraic if N is odd: all known bodies

defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes

for *N = 3*). We demonstrate a new series of *locally* algebraically integrable bodies with algebraic

boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume

functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting

the body.

It is shown in the paper that the number pN(M) of partitions of a positive integer M into N positive integer summands coincides with the Bose and Fermi distributions with logarithmic accuracy if one identifies M with energy and N with the number of particles. We use the Gentile statistics (a.k.a. parastatistics) to derive self-consistent algebraic equations that enable one to construct the curves representing the least upper bound and the greatest lower bound of the repeated limits as M → ∞ and N → ∞. The resulting curves allow one to generalize the notion of BKT (Berezinskii–Kosterlitz–Thouless) topological phase transition and explaining a number of phenomena in thermodynamics and mesoscopic physics.

This paper presents UD-statistics almost coinciding with Gentile statistics for negative chemical potentials, but having another extension in the case of positive chemical potentials. The physical meaning of the metamorphosis of the phase transition of the third kind to the phase transition of the first kind is explained. The coincidence of isochores and isotherms in the supercritical domain corresponding to UD statistics is shown.

In this paper, we refine some of the original notions, such as the number of degrees of freedom, for an arbitrary temperature and an ideal Boltzmann--Maxwell gas. Using the van der Waals model, we illustrate the notion of Frenkel temperature, which separates soft and rigid liquid.

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under breaking its axial symmetry (inclination of the magnetic field) and turning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation of the voltage on the trap electrode, there exists the avoided crossing effect and corresponding effect of bilocalization of quantum states on pairs of closed trajectories belonging to common energy levels. This quantum tunneling happens on the symplectic leaves of the symmetry algebra, and hence it generates the tunneling of quantum states of the electron between 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting interms of symplectic area of membranes bounded by invariantly defined instantones.

We establish the Jackson's type estimate for generalized modulus of smoothness

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf–Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018), is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.

The Riccati equation with coefficients expandable in convergent power series in a neigh- borhood of infinity are considered. Extendable solutions of such equations are studied. Methods of power geometry are used to obtain conditions for convergent series expansions of these solutions. An algorithm for deriving such series is given.

The action of the group of totally positive units on the convex hull of the semigroup of totally positive integers of a real cubic Galois field is studied. The fundamental domain of this action has a simple description in the case of so called regular field.