In the paper, we propose a new approach to the mathematical description of the separation of a neutron from the atom’s nucleus on the basis of the formalisms of tropical mathematics and nonstandard analysis. In studying the behavior of individual nucleons of the atom’s nucleus, instead of the ordinary approach involving the Einstein formula relating mass and energy, we use the thermodynamical approach to the disintegration of the nucleus. This approach allows to obtain a previously unknown general expression for the energy needed to separate a neutron from its atom’s nucleus, provided we know the de Broglie wavelength and the volume of the nucleus.

We consider the construction of asymptotic solutions of linear equations related to equations of classical mechanics: the Hamilton–Jacobi equation and the transport equation. We show that these methods and also the theory of the mechanics of an infinitely narrow beam as a whole can be applied to some objects in bioenergy if the thin organic objects of the type of wood splinters, straw, pellets, and so on are approximated by infinitely narrow beams.

On 20 September 2019, Alexandre Mikhailovich Vinogradov, a remarkable mathematician and an extraordinary person, passed away. He was born on 18 February 1938 in Novorossiysk. During World War II he and his mother were evacuated to Kungur (his father served in the army), and later his parents settled in Kuntsevo, at that time not yet a part of Moscow. In 1955 he enrolled in the Faculty of Mechanics and Mathematics at Moscow State University, and in 1960 he became a graduate student. After defending his Ph.D. thesis in 1964, he taught students at the Moscow Mining Institute for a year. Then N. V. Efimov, who was then the dean, invited Vinogradov to work in the Department of Higher Geometry and Topology of the Faculty of Mechanics and Mathematics (P. S. Alexandrov was head of the department at that time), where he worked until his departure to Italy in 1990. He became a doctor of the physical and mathematical sciences in 1984. In 1993–2010 he was a professor at the University of Salerno (Italy). When Vinogradov was only a second-year undergraduate student, he published two papers on number theory (jointly with B.N. Delaunay and D.B. Fuchs). However, towards the end of his term at the university his interests changed: in his senior year and in graduate school he began to study algebraic topology in the seminar of A. S. Schwarz (Shvarts). His Ph.D. thesis, under the formal supervision of V. G. Boltyansky, was on the homotopy properties of the space of embeddings of

The analysis of different physical models describing the cooling of the active zone in the accident ChAPP block required a large amount of detailed estimations and calculations, demanded the physical modelling of cooling processes and special experiments in the accident block, after which the scheme of convective filtering cooling of the ChAPP accident block was accepted as the base model. Subsequent analysis showed that the filtration cooling model of the pile-up completely explains all the phenomena taking place in the reactor after the accident. The behaviour of the pile-up was influenced by melting and evaporation processes of materials volatile at the temperature level in the active zone of the reactor at the beginning of the accident. When the temperature increases, the pile-up should be loosened artificially, otherwise, as the results of modelling show, nature can take care of it itself. The new results obtained were unexpected, since similar problems were never considered in technical applications.

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes a boson at the instant of separating from a fermionic nucleus. We study the further transformations of boson and fermion states of separation in a small neighborhood of zero pressure and obtain new important parastatistical relations between the temperature and the chemical potential when a nucleon separates from an atomic nucleus. The obtained relations allow constructing a new diagram (an aF diagram) or isotherms of very high temperatures corresponding to nuclear matter. We mathematically prove that the transition of particles from the domain governed by Fermi-Dirac statistics to the domain governed by Bose-Einstein statistics near the zero pressure P occurs in the neutron uncertainty domain or halo domain. We obtain equations for the chemical potential that allow determining the width of the uncertainty domain. Based on the calculated values of the minimum intensivity for Bose particles, the chemical potential, the compressibility factor, and the minimum mean square fluctuation of the chemical potential, we construct a table of stable nuclei of chemical elements, demonstrating a monotonic relation between the nucleus mass number and the other parameters.

The parameters of unstable short-living isotopes are studied from the mathematical point of view. The values of the chemical potential and activity parameters that determine the neutron halo arising when the neutron separates from the nucleus of an unstable isotope are calculated. The analogy between nuclear physics and economics is considered from the point of view of such parameters as excitation energy, spin, rate of turnover, and time.

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs. Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose-Einstein distribution to the Fermi-Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements. The mathematical approach developed by the author of the present paper allows one to describe the "antipode" (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety.

C математической точки зрения исследуется процесс отрыва нуклона от атомного ядра. Используются экспериментальные значения энергии связи для ядра данного вещества. В момент отрыва нуклона от фермионного ядра оно превращается в бозон. Исследуются дальнейшие превращения бозонного и фермионного состояний отрыва в малой окрестности нулевого давления. Получены новые важные соотношения парастатистики, связывающие температуру и химический потенциал при отрыве нуклона от атомного ядра. Полученные соотношения позволяют построить новую диаграмму ($aF$-диаграмму) для изотерм очень высоких температур, отвечающих ядерной материи. Математически доказано, что переход частиц из области, подчиняющейся статистике Ферми-Дирака, в область, подчиняющуюся статистике Бозе-Эйнштейна в окрестности нулевого давления $P$, происходит в нейтронной области неопределенности, или в области гало. Получены уравнения для химического потенциала, позволяющие определить ширину области неопределенности. На основе вычисленных значений минимальной интенсивности для бозе-частиц, химического потенциала, фактора сжимаемости и минимальной среднеквадратичной флуктуации химического потенциала построена таблица стабильных ядер химических элементов, демонстрирующая монотонную связь между массовым числом ядра и остальными параметрами.

In this paper, a new physical notion, intensity, is introduced. The notion of intensity occurs in a special statistics, known as Gentile statistics, which is asymptotically close to ordinary thermodynamics. The introduction of the new notion of intensity in the theory of nuclear matter essentially changes the thermodynamical picture. Moreover, we can say that the thermodynamics of nuclear matter is the antipode of standard thermodynamics. On the basis of the “intensity-compressibility factor” diagram and mean square fluctuations of energy and time, a new table of properties of stable chemical elements is obtained and presented in this paper.

The notions of “hole” and “vacuum” in various branches of science are considered. A philosophical generalization of these notions on the basis of examples from physics, history, and linguistics is presented. Some aspects of the further development of these notions in mathematical logic, thermodynamics of nuclear matter, and other branches of science are sketched.

In this paper, a physical and mathematical interpretation of the passage from the Bose distribution to the Fermi distribution of nuclear matter is presented. We consider the notion of “fur coat” and introduce the notion of “lacunary indeterminacy”, which is a region that contains a neighborhood of the activity a = 0 and is the boundary between the Bose particle region and the Fermi particle region when the nucleon separates from the atomic nucleus. Our approach yields previously unknown expressions for extremal values of activity in passing from a Bose-type region to a Fermi-type region provided we know the de Broglie wavelength and the volume of the atomic nucleus.

The first part of the paper deals with the behavior of the Bose–Einstein distribution as the activity 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. This approach allows to find the value of the parameter a_0, for which the Bose distribution (in the statistical sense) becomes zero. In the second part of the paper, the process of separation of a nucleon from the atom's nucleus is studied from the mathematical point of view. At the moment when the nucleon tears away from the fermionic nucleus, the nucleus becomes a boson. We investigate the further transformations of bosonic and fermionic separation states in a small neighborhood of the pressure P equal to zero. We use infinitely small quantities to modify the parastatistical distribution. Our conception is based on interpolation formulas yielding expansions in powers of the density. This method differs from those in other models based on the interaction potential between two or three particles. We obtain new important relations connecting the temperature with the chemical potential during the separation of a nucleon from the atom's nucleus. The obtained relations allow us to construct, on an antipode of sorts of the Hougen–Watson P-Z chart, the very high temperature isotherms corresponding to nuclear matter. It is proved mathematically that the passage of particles satisfying the Fermi–Dirac distribution to the Bose–Einstein distribution in the neighborhood of pressure P equal to zero occurs in a region known as the "halo".

In the paper, we propose a new approach to the mathematical description of the separation of a neutron from the atom's nucleus on the basis of the formalisms of tropical mathematics and nonstandard analysis. In studying the behavior of individual nucleons of the atom's nucleus, instead of the ordinary approach involving the Einstein formula relating mass and energy, we use the thermodynamical approach to the disintegration of the nucleus. This approach allows to obtain a previously unknown general expression for the energy needed to separate a neutron from its atom's nucleus, provided we know the de Broglie wavelength and the volume of the nucleus.

The number-theoretical problem of partition of an integer corresponds to $D=2$. This problem obeys the Bose--Eeinstein statistics, where repeated terms are admissible in the partition, and to the Fermi--Dirac statistics, where they are inadmissible. The Hougen--Watson P,Z-diagram shows that this problem splits into two cases: the positive pressure domain corresponds to the Fermi system, and the negative, to the Bose system. This analogy can be applied to the van der Waals thermodynamics. The thermodynamic approach is related to four potentials corresponding to the energy, free energy, thermodynamic Gibbs potential, enthalpy. The important notion of de Broglieu0027s wavelength permits passing from particle to wave packet, in particular, to Bose and Fermi distributions. Particles of ideal Bose and Fermi gases and the decay of a boson consisting of two fermions into separate fermions are studied. The case of finitely many particles $N$ of the order of $10^2$ is considered by heuristic considerations like those Fock used to derive the Hartree--Fock equation. The case of $Nll1$ is studied by Gentile statistics, tropical geometry and nonstandard analysis (Leibnitz differential or monad). A relation for the energy of neutron separation from the atomic nucleus is obtained when the atomic nucleus volume and de Broglieu0027s wavelength are known. The Appendix is authoru0027s paper written in 1995.