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In this paper, we consider the simplest version of a linear neural network (LNN). Assuming that for training (constructing an optimal weight matrix $Q$) we have a set of training pairs, i.e. we know the input data \begin{equation} G=\left\{g^{\left(1\right)},g^{\left(2\right)},\cdots,g^{\left(K\right)}\right\}, \end{equation} as well as the correct answers to these input data \begin{equation} H=\left\{h^{\left(1\right)},h^{\left(2\right)},\cdots,h^{\left(K\right)}\right\}. \end{equation} We will study the possibilities of constructing a weight matrix $Q$ of a neural network that will give correct answers to arbitrary input data based on the connection of the specified problem with a system of linear algebraic equations (SLAE). Consider a class of neural networks in which each neuron has only one output signal and performs linear operations. We will show how such LNEs are reduced to SLAEs. Since the questions $G$ and the correct answers $H$ are known to us, the desired weight matrix $Q$ must satisfy the equations \begin{equation} Qg^{\left(k\right)}=h^{\left(k\right)}, k=1,2,\cdots,K. \end{equation} It is required to restore $Q$. In the general case, the matrix $Q$ is rectangular $Q=Q_{MN}=\left\{q_{mn}\right\}$, $m$ is the row number, and $g^{\left(k\right)}\in\mathbb{R}^N$, $h^{\left(k\right)}\in\mathbb{R}^M$. Let $G_{NK}$ be a matrix composed of columns $g^{\left(1\right)}, g^{\left(2\right)},\cdots,g^{\left(k\right)}$, and $H_{MK}$ be a matrix composed of columns $h^{\left(1\right)},h^{\left(2\right)},\cdots,h^{\left(k\right)}$. Then, with respect to $Q_{MN}$, we obtain a matrix SLAE. \begin{equation} Q_{MN}G_{NK}=H_{MK}. \end{equation} This paper will present methods for regularizing the constructed system., Comment: in Russian language

We consider a model of stable edge sets (``matchings'') in a bipartite graph $G=(V,E)$ in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistence, substitutability and cardinal monotonicity, whereas the preferences for the vertices of the other side (``workers') via linear orders. For such a model, we present a combinatorial description of the structure of rotations and develop an algorithm to construct the poset of rotations, in time $O(|E|^2)$ (including oracle calls). As a consequence, one can obtain a ``compact'' affine representation of stable matchings and efficiently solve some related problems. (The paper is written in Russian.), Comment: 28 pages, in Russian language

In this paper, we study a two-dimensional Lorentzian problem on the anti-de Sitter plane. Using methods of geometric control theory and differential geometry, it was possible to construct an orthonormal frame, calculate extremal trajectories, describe the reachable set, construct an optimal synthesis, and describe the Lorentzian distance., Comment: In Russian

Multiple magnetic and kinetic solar wind plasma parameters are used to detect coronal mass ejections (CMEs) as they travel through the heliosphere. There are various interplanetary CME (ICME) catalogues, but due to differences between their ICME identification criteria they can significantly vary. In this paper we analyze Richardson and Cane and CCMC CME Scoreboard ICME catalogues and the SRI RAS solar wind types catalogue, and propose an algorithm of merging them. A unified catalogue is constructed for 2010 to 2022. The resulting catalogue is completed with data from the OMNI database. Analysis of the unified catalogue demonstrated high accuracy when merging events present in multiple catalogues and a tendency of events defined in all three initial catalogues to demonstrate greater duration, speed and geoeffectiveness. The catalog is presented on the SINP MSU Space Weather Exchange website: https://swx.sinp.msu.ru/tools/icme_list.php, Comment: 22 pages, in Russian language, 4 figures

In 1957, Nemytskii proved the following fact: if in a locally compact or in an Abelian connected group there is a neighborhood of the identity in which some identity holds, then it holds in the entire group. The following question was also posed there: Let G be a connected topological group. In some neighborhood of the identity of the group G the identity $x^3=1$ holds. Is it true that then the identity $x^3=1$ holds in the entire group $G$? The same question is posed for the identity $gx^2 = x^2g$, where $g$ is a fixed element of the group. Platonov formulated the following generalized formulation of the Mytselsky problem: for a topological connected group, is it true that if the identity holds in a neighborhood of the identity, then the identity holds everywhere? In this paper, a negative answer to Platonov's question is given, the following theorem is proven: if $n > 1010$ is odd, then there exists a connected topological group in which the identity $x^n=1$ holds in some neighborhood of unity, but not in the entire group., Comment: in russian

It is well known that whenever a class of structures $\mathcal{K}_1$ is interpretable in a class of structures $\mathcal{K}_2$, then the hereditary undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the hereditary undecidability of (a suitable fragment of) the theory of $\mathcal{K}_2$. In the present paper, we construct a $\Sigma_1$-interpretation of the class of all finite bipartite graphs in the class of all pairs of equivalence relations on the same finite domain; from this we obtain the hereditary undecidability of the $\Sigma_2$-theory of the second class. Next, we construct a $\Sigma_1$-interpretation of the class of all pairs of equivalence relations on the same finite domain in the class of all pairs consisting of a linear ordering and an equivalence relation on the same finite domain; this gives us the hereditary undecidability of the $\Sigma_2$-theory of the second class. The corresponding results are, in a sense, optimal, since the $\Pi_2$-theories of the classes under consideration are decidable. Keywords: undecidability, elementary theories, prefix fragments, Comment: in Russian language

In this paper we study holomorphically homogeneous model submanifolds CR-type (1, 3) complex space $\mathbb C^4$. One finds moduli space of five-dimensional model surfaces Bloom-Graham type ((2, 1), (3, 1), (4, 1)). It is shown that there exists unique model surface of this type with property of holomorphical homogeneous, which is equivalent to tube surface $\mathcal C$ with affin homogeneous base. One describes and classifies with respect to model surfaces the orbits relative to the group of holomorhical automorphisms of $\mathcal C$, Comment: in Russian language

The paper studies the quantum action for the five-dimensional real $\phi^3$-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The explicit form of the first three coefficients for the renormalization constants is presented. The absence of non-local singular contributions and partial results for the fourth correction are discussed., Comment: LaTeX, 20 pages, 5 figures. Firstly appeared in Russian, April 9, 2024, see https://www.pdmi.ras.ru/preprint/2024/24-05.html

This paper presents the experience of developing and utilizing computer simulations for student laboratory assignments in atomic physics, specifically focusing on emission atomic spectra. The fundamental component of each laboratory assignment is a computer simulator of a spectrograph, which displays atomic line spectra on the monitor screen. The simulator operates by converting the wavelength of radiation into an RGB triplet. Calibration of the spectrograph is conducted using the atomic spectrum of mercury. The configuration of setup parameters and experimental conditions is unique to each student conducting the laboratory assignment using the simulator. Individualized parameters, stored in Google Sheets, are transferred to the HTML template of the laboratory assignment via the Google Apps Scripts service. This individualization of simulator parameters serves to encourage students to engage in independent work., Comment: 17 pages, in Russian, 9 figures

The paper discusses an applicability criterion for a cutoff regularization in the coordinate representation in the Euclidean space with a dimension larger than two. It is shown that the set of functions satisfying the criterion is not empty. As an example, an explicit function is presented. It is proved by explicit construction that there are functions satisfying the criterion in a stronger formulation., Comment: LaTeX, 8 pages, 1 figure. Firstly appeared in Russian, March 5, 2024. Compared to the Russian version, some comments and minor corrections have been added. The forgotten imaginary unit in the Fourier transform has also been restored. See https://www.pdmi.ras.ru/preprint/2024/24-04.html

The paper presents various modifications of the Frank-Wolfe algorithm in the equilibrium traffic assignment problem. The Beckman model is used as a model for experiments. In this article, first of all, attention is paid to the choice of the direction of the basic step of the Frank-Wolfe algorithm. Algorithms will be presented: Conjugate Frank-Wolfe (CFW), Bi -conjugate Frank-Wolfe (BFW), Fukushima Frank-Wolfe (FFW). Each modification corresponds to different approaches to the choice of this direction. Some of these modifications are described in previous works of the authors. In this article, following algorithms will be proposed: N-conjugate Frank-Wolfe (NFW), Weighted Fukushima Frank-Wolfe (WFFW). These algorithms are some ideological continuation of the BFW and FFW algorithms. Thus, if the first algorithm used at each iteration the last two directions of the previous iterations to select the next direction conjugate to them, then the proposed algorithm NFW is using more than N previous directions. In the case of Fukushima Frank-Wolfe, the average of several previous directions is taken as the next direction. According to this algorithm, a modification WFFW is proposed, which uses a exponential smoothing from previous directions. Experiments with various modifications were carried out on several datasets representing urban structures and taken from publicly available sources. The relative gap value was taken as the quality metric. The experimental results showed the advantage of algorithms using the previous directions for step selection over the classic Frank-Wolfe algorithm. In addition, an improvement in efficiency was revealed when using more than two conjugate directions. For example, on various datasets, the modification 3FW showed the best convergence. In addition, the proposed modification WFFW often overtook FFW and CFW, although performed worse than NFW., Comment: 14 pages, in Russian language, 8 figures

The paper tests the validity of the critique of the fiscal theory of the price level. A stochastic general equilibrium model with continuous time is constructed. An active fiscal policy and a passive monetary policy have been set. Monetary policy manages the interest rate through the Taylor rule. The stochastic default factor in the special form is introduced. A complete definite system of equations is obtained for the detection of equilibrium. It is asserted that the peculiarities of the approach to modeling are of critical importance for verifying the presence of certain hypotheses and formulating conclusions. The results of this work are in support of the fiscal theory of the price level., Comment: Master Thesis, in Russian. Supervisor: Sergey E. Pekarski

The paper studies the quantum action for the four-dimensional real $\phi^4$-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization $\mathcal{R}$-operation on the example of separate three-loop diagrams are also discussed. The explicit form of the first three coefficients for the renormalization constants and for the $\beta$-function is presented. Consistency with previously known results is shown., Comment: LaTeX, 71 pages, 8 figures. Firstly appeared in Russian, February 6, 2024, see https://www.pdmi.ras.ru/preprint/2024/pr2024.html

This paper presents the experimental results on research of growth processes of GaAs layers on silicon substrates by molecular beam epitaxy. The formation of buffer Si layer in a single growth process has been found to significantly improve the crystalline quality of the GaAs layers formed on its surface, as well as to prevent the formation of anti-phase domains both on offcutted towards the [110] direction and on singular Si(100) substrates. It has been demonstrated that the use of cyclic thermal annealing at temperatures 350-660{\deg}C in the flow of arsenic atoms makes it possible to reduce the number of threading dislocations and increase the smoothness of the GaAs layers surface. At the same time, the article considers possible mechanisms that lead to an improvement in the quality of the surface layers of GaAs. It is shown that the thus obtained GaAs layers of submicron thickness on the singular Si(100) substrates have a mean square value of surface roughness 1.9 nm. The principal possibility of using thin GaAs layers on silicon as templates for forming on them light-emitting semiconductor heterostructures with active area based on self-organizing InAs quantum dots and InGaAs quantum well is presented. They are shown to exhibit photoluminescence at 1.2 um at room temperature., Comment: 18 pages, in Russian language, 6 figures

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if $X$ is a countably compact space and $C_p(X)$ is a space of continuous functions in the pointwise topology convergence, then any countably compact subspace of the space $C_p(X)$ is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudo-compact $X$ contains a dense Lindelof $\Sigma$-space, then pseudocompact subspaces of the space $C_p(X)$ are precompact. If $X$ is the product Cech complete spaces, then bounded subsets of the space $C_p(X)$ are precompact. Results on the continuity of separately continuous functions were also obtained., Comment: in russian

Magnetars are neutron stars with superstrong magnetic fields which can exceed 1e15 G. Some magnetars (the so-called soft gamma-repeaters) demonstrate occasionally very powerful processes of energy release, which result in exceptionally strong flares of electromagnetic radiation. It is believed that these flares are associated with the presence of superstrong magnetic fields. Despite many hypotheses, the mechanism of these flares remains a mystery. In afterglows of the flares, one has often observed quasi-periodic oscillations (QPOs) of magnetar emission. They are interpreted as stellar vibrations, excited by the flares, which are useful for exploring the nature of magnetar activity. The incompleteness of theories employed to interpret magnetar QPOs is discussed., Comment: 9 pages, 5 figures. Published in Russian, English translation by the author; is expected to be published in English later

The application of the Gershgorin circle theorem and some of its derivatives to estimate the eigenvalues of a matrix is considered. The obtained results are developed to obtain the localization region of the eigenvalues of a matrix with interval-indefinite constant or non-stationary elements. The concept of e-circles is introduced to obtain more accurate estimates of these regions than when using Gershgorin circles. The obtained results are applied to the stability analysis of network systems, where it is shown that the proposed methods allow one to analyze a network with a much larger number of agents than when using the CVX, Yalmip, eig and lyap methods (functions in MatLab). It is further shown that if the obtained results are applied not to the system itself, but to the result obtained using the Lyapunov function method, then one can study systems with matrices without diagonal dominance. This made it possible to consider a modification of the Demidovich condition for systems with non-stationary parameters and design of a control law for non-stationary systems with matrices without diagonal dominance. All obtained results are illustrated by numerical modeling., Comment: in Russian language

As one of the predictions of perturbative QCD, the effect of color transparency has been the focus of attention in the community studying modifications of hadrons in nuclear medium for several decades. The search for this effect in reactions involving heavy nuclei can be complicated by uncertainties in nuclear characteristics (nucleon density distributions and wave functions), which can affect the interpretation of experiments. In this work, we consider the reaction $d(p,pp)n$ at $p_{\rm lab}=15$ GeV/c caused by hard elastic $pp$ scattering, in which these uncertainties are actually reduced to the behavior of the deuteron wave function at large momenta. It is shown that for transverse momenta of the spectator neutron $\leq 0.4$ GeV/c the choice of the deuteron wave function cannot affect the identification of the color transparency effect. A simple method for studying color transparency in $dd$ collisions is also suggested based on the identification of quasi-free $pd$ interactions., Comment: 10 pages, 5 figures, in Russian, accepted in PEPAN

The use of traditional IVR (Interactive Voice Response) methods often proves insufficient to meet customer needs. This article examines the application of artificial intelligence (AI) technologies to enhance the efficiency of IVR systems in call centers. A proposed approach is based on the integration of speech-to-text conversion solutions, text query classification using large language models (LLM), and speech synthesis. Special attention is given to adapting these technologies to work with the Kazakh language, including fine-tuning models on specialized datasets. The practical aspects of implementing the developed system in a real call center for query classification are described. The research results demonstrate that the application of AI technologies in call center IVR systems reduces operator workload, improves customer service quality, and increases the efficiency of query processing. The proposed approach can be adapted for use in call centers operating with various languages., Comment: in Russian language

The article considers events recorded in an experiment with X-ray emulsion chambers (XREC) to study the trunks of extensive air showers (EAS) with energies of nuclei of primary cosmic radiation (PCR) E0 > 100 TeV, which currently have no explanation within the standard model of nuclear interactions (SM). The XREC method is the only one that allows studying EAS trunks with a high coordinate resolution of about 100 microns. EAS trunks contain the most complete information about primary acts of nuclear interactions, as a result of which all unusual events recorded by the XREC method are of great scientific interest. Several types of unusual events recorded by the XREC method were explained within the SM., Comment: in Russian language

This review discusses methods of testing for a panel unit root. Modern approaches to testing in cross-sectionally correlated panels are discussed, preceding the analysis with an analysis of independent panels. In addition, methods for testing in the case of non-linearity in the data (for example, in the case of structural breaks) are presented, as well as methods for testing in short panels, when the time dimension is small and finite. In conclusion, links to existing packages that allow implementing some of the described methods are provided., Comment: In Russian

The article discusses development of autonomous artificial intelligence systems for corporate management. The function of a corporate director is still one of the few that are legislated for execution by a "natural" rather than an "artificial" person. The main prerequisites for development of systems for full automation of management decisions made at the level of a board of directors are formed in the field of corporate law, machine learning, and compliance with the rules of non-discrimination, transparency, and accountability of decisions made and algorithms applied. The basic methodological approaches in terms of corporate law for the "autonomous director" have already been developed and do not get rejection among representatives of the legal sciences. However, there is an undeniable need for further extensive research in order to amend corporate law to effectively introduce "autonomous directors". In practice, there are two main options of management decisions automation at the level of top management and a board of directors: digital command centers or automation of separate functions. Artificial intelligence systems will be subject to the same strict requirements for non-discrimination, transparency, and accountability as "natural" directors. At a certain stage, autonomous systems can be an effective tool for countries, regions, and companies with a shortage of human capital, equalizing or providing additional chances for such countries and companies to compete on the global market., Comment: in Russian language

In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with $|N_{m+1}|/|N_m|\ \to\infty$ such that $T^iA\cap S^iA=\phi$ for $i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. For the corresponding deterministic Gaussian and Poisson suspensions $S_\circ,T_\circ$ over $S,T$ for some $f\in L_\infty$ there is no limit of $ \frac 1 N \sum_{n=1}^N T_\circ^nf\,S_\circ^nf$ in $ L_2$., Comment: in Russian

We discuss what is known about the structure of the bounded derived categories of coherent sheaves on Grassmannians of simple algebraic groups., Comment: v1: 31 pages; v2: minor improvements, to appear in Russian Mathematical Surveys

Two left-invariant Lorentzian problems on the Heisenberg group are considered. The Pontryagin maximum principle was applied to both problems and a parameterization of abnormal and normal extremal trajectories was obtained. Reachability sets and the existence of optimal trajectories are investigated., Comment: In Russian

The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a graph or simply a graph group. The basis for constructing a group of graph automorphisms is the concept of orbit. The construction of an orbit is closely related to the quantitative assessment of a vertex or edge of a graph, called weight. To determine the weight of an element, graph invariants built on the spectrum of edge cuts and the spectrum of edge cycles are used. The weight of the graph elements allows identifying generating cycles and forming orbits. Examples are given of constructing a group of automorphisms for some types of graphs., Comment: 123 pages, in Russian language, 304 figures, a preprint of monography

Let $T$ be a staircase rank-one construction with parameters $r_j \sim j^d$, $0

A new version of a sufficient condition of a Dobrushin type is proposed for non-homogeneous Markov chains for a certain class of such processes. The original Dobrushin's condition may fail, yet, the CLT is still valid, Comment: 18 pages, 13 references, in Russian language; in comparison to v2 version, two more examples were added

We construct a sphere-homeomorphic flexible self-intersection free polyhedron in Euclidean 3-space such that all its dihedral angles change during some flex of this polyhedron. The constructed polyhedron has 26 vertices, 72 edges and 48 faces. To study its properties, we use symbolic computations in the Wolfram Mathematica software system., Comment: 19 pages, 6 figures; this v2 is an English translation of v1, which is in Russian

This book is devoted to finite-dimensional problems of non-convex non-smooth optimization and numerical methods for their solution. The problem of nonconvexity is studied in the book on two main models of nonconvex dependencies: these are the so-called generalized differentiable functions and locally Lipschitz functions. Non-smooth functions naturally arise in various applications. In addition, they often appear in the theory of extremal problems itself due to the operations of taking the maximum and minimum, decomposition techniques, exact non-smooth penalties, and duality. The considered models of nonconvexity are quite general and cover the majority of practically important optimization problems; they clearly show all the difficulties of non-convex optimization. The method of studying the generalized differentiable functions is that for these functions a generalization of the concept of gradient is introduced, a calculus is constructed, and various properties of nonconvex problems are studied in terms of generalized gradients. As for numerical methods, it is possible to extend the theory and algorithms of subgradient descent of convex optimization to problems with generalized differentiable functions. Methods for solving Lipschitz problems are characterized by the fact that the original functions are approximated by smoothed ones and iterative minimization procedures are applied to them. With this approach, it is possible to approximate the gradients of smoothed functions by stochastic finite differences and thus to construct methods without calculating gradients. A similar approach can be justified in generalized differentiable and Lipschitz stochastic programming. In these cases, various generalizations of the classical stochastic approximation and stochastic quasi-gradient method are obtained for solving constrained nonconvex nonsmooth stochastic programming problems., Comment: First published in Russian by Nauka, Moscow, 1987. 280 p. Translated to English by Vladimir Norkin, 2024

The present study demonstrates that the mechanism of initiation of low-energy nuclear chemical processes under conditions of low-temperature non-equilibrium deuterium and protium-containing glow discharge plasma is similar to the previously studied cumulative mechanism of initiation of nuclear processes in the collision of relativistic particles (protons) with target atomic nuclei. This process results in the formation of high-energy products that significantly exceed the kinematically resolved region in the pulse space for two-particle collisions "nucleus-target's nucleus." The cumulative effect in this case is associated with the initiation of non-nucleonic metastable excitations in nuclear matter during relativistic collisions leading to the formation of a group of quarks from different nucleons within the nucleus. In low-energy nuclear chemical processes, the initiation of quark-cumulative processes in nuclear matter occurs through interaction of nuclei with electrons with high kinetic energies on a chemical scale, typically $E_e$ ~ 3-5 eV. Experiments and available literature data suggest that the metastable excitations of nuclear matter containing three "free" quarks which occur during such collisions are associated with quark-cumulative effects, leading to the radioactive $\alpha$- and $\beta$-decay of elements. This phenomenon is observed during laser ablation of metals in aqueous media containing radioactive elements and in the artificial radioactivity of initially non-radioactive isotopes in cathodes exposed to low-temperature non-equilibrium deuterium- and protium-containing plasma flows during glow discharge., Comment: 17 pages, 1 figure; to appear in Russian Journal of Physical Chemistry A, 2024, Vol. 98, No. 6, pp. 52-60

For any natural $n$, and real $\alpha\geq 0$ we construct an ergodic automorphism $T$ such that its tensor powers $T^{\otimes n}$ have singular spectra if $n\leq 1+\alpha /2$, and Lebesgue if $n\, > 1+\alpha/2$., Comment: in Russian language

Three versions of the Weak Law of Large Numbers are proposed for weakly dependent and generally speaking non-equally distributed random variables, with finite or possibly infinite expectations., Comment: 13 pages, 15 references, in Russian

The optical orientation and alignment of excitons in semiconductor indirect gap quantum dots have been studied theoretically. A special regime is analyzed in which the energy of the hyperfine interaction of an electron with lattice nuclei is small compared to the exchange splitting between bright and dark excitonic levels, but is comparable to the anisotropic exchange splitting of the radiative doublet. The dependencies of degrees of circular and linear polarization on the external magnetic field under resonant excitation of excitons by polarized light are calculated., Comment: 11 pages, 1 figure. This is a translation of the original manuscript in Russian in memory of V. M. Agranovich, which is available in the supplementary files at arXiv and at https://www.dropbox.com/scl/fi/kyyyaaimh2bjtf1dycv11/Agranovich_memorium.pdf?rlkey=ix0t7gi0ytxu5et4yhw134ke8&st=yp7n4f3o&dl=1

Irregularity in spectrum of the primary cosmic rays (PCR) mass composition at an energy of ~ 10 PeV is considered. To assess changes of the PCR mass composition, the X-ray emulsion chamber (XREC) method and the halo-method based on the XEC were used. The study of changes in the PCR mass composition was carried out based on the experimentally obtained characteristics of the extensive air showers (EAS) trunks, where fluctuations of these characteristics are minimal and information about the primary interaction of PCR nuclei with atmospheric atoms is maximally preserved. It is shown that at an energy of ~ 10 PeV there is a local maximum in the heavy nuclei fraction. This maximum is correlated with the PCR sources, - variable stars SR (red giants and super-giants) and WR (Wolf-Rayet)., Comment: In Russian Language

The appearance of like nucleon pair correlations in the ground state of spherical even-even nuclei is considered within a special Bogoliubov transformation. It is confirmed that in closed subshell nuclei superconducting pair correlations start to form if the coupling constant $G$ exceeds a certain threshold value. Rough upper and lower estimates are obtained for the threshold value. It is shown, that superconducting correlations exist in open subshell nuclei at any positive $G$. In this case, nucleon pairs are distributed over all subshells participating in the pairing interaction., Comment: 17 pages; JINR preprint E4-2024-13; submitted to PEPAN Letters (in Russian)

The note is devoted to multiple mixing, spectrum, rank and self-joinings of measure-preserving transformations. We recall famous open problems, discuss related questions and some known results. A hypothetical example of an automorphism of the class $MSJ(2)\setminus MSJ(3)$ has Lebesgue spectrum, infinite rank and does not have multiple mixing. If its spectrum is simple, then we get a solution to the problems of Banach, Rokhlin and del Junco-Rudolph. The existence of such an example, of course, seems unlikely, but any facts confirming the impossibility of this amazing situation have not yet been discovered. They not found even under the condition that its local rank is positive, which ensures the finite spectral multiplicity., Comment: 19 pages, in Russian language

By critically examining the traditional theory of homogeneous nucleation of precipitates in solid solutions, it is revealed that the theory's assertion regarding an increase in the nucleation free energy due to elastic strain associated with the difference in atomic volumes between the two phases is applicable to coherent precipitates, but becomes incorrect when applied to incoherent precipitates. This conclusion is obtained by accounting for thermal point defects in the matrix, which can be absorbed at the interface between an incoherent particle and the matrix during nucleation, thereby relieving elastic stresses. Accordingly, a new kinetic model based on the Reiss theory for binary nucleation is proposed for predicting the nucleation rate of incoherent precipitates by agglomeration of solute atoms and point defects, with a further extension to account for excess vacancies formed under non-equilibrium conditions of quenching experiments., Comment: Published in Russian Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 165 (2024) 214-225

In 1949 V.A. Rokhlin introduced into ergodic theory the k-fold mixing and puzzled the mathematical community with the problem of the mismatch of these invariants. Here's what Rokhlin wrote: "The proposed work arose from the author's attempts to solve the well-known spectral problem of the theory of dynamical systems: are there any metrically different dynamical systems with the same continuous (in particular, Lebesgue) spectrum? Using new metric invariants introduced for this purpose, the author tried to find among the ergodic automorphisms of compact commutative groups automorphisms of various metric types. It turned out, however, that for all the indicated automorphisms the new invariants are exactly the same." We discuss this long time open problem., Comment: in Russian language

The function $P(T)=\sum_{i=0}^\infty c_i T^i$ is admissible if $c_i\geq 0$, $\sum_{i=0}^\infty c_i\leq 1$. For any given set of admissible functions $P_1,\dots, P_k$ there is a unitary operator $T$ of dynamic origin such that the weak closure of its powers is a semigroup generated by the operators $0$, $T$, $P_1(T),\dots, P_k(T)$, $T^\ast$, $P_1(T^\ast ),\dots, P_k(T^\ast)$., Comment: in Russian language

We generalize methods, developed by S. Ghilardi, and apply them to a subsystem J$_2$ of bimodal provability logic GLB. We describe projective formulas in J$_2$ in terms of Kripke semantics and prove that logic J$_2$ has finitary unification type. As an application, we show that admissibility problem for J$_2$ is decidable., Comment: In Russian, 20 pages

In this expository note we show how combinatorial Nullstellensatz by N. Alon naturally appears in solutions of elementary problems. Simple ideas gradually and naturally appear in such solutions, thus bringing a reader to generalizations. The note is accessible to mathematicians not specialized in the area, and to students familiar with polynomials., Comment: 10 pages; in Russian; exposition improved

Let $\psi(n)\to +0$ and a square-integrable function $f$ be non-zero, then for the typical mixing automorphism $T$ the set $\{n:\, |(T^nf,f) |>\psi( n)\}$ is infinite. The mildly mixing automorphisms $T$ do not have convergences of non-zero averages $\frac 1 {k_n} \sum_{i=1}^{k_n}T^if (x)$ with the rate of $o\left(\frac 1 {k_n}\right)$., Comment: in Russian language

Necessary and sufficient conditions for the internal stability of formations whose dynamics are obtained is determined by linear differential equations., Comment: in Russian language Fixed an error in the English version of the title of the work

Examples of rigid Poisson suspensions without roots are presented. The discrete rational component in spectrum of an ergodic automorphism S prevents some roots from existing. If S is tensorly multiplied by an ergodic automorphism of the space with a sigma-finite measure, discrete spectrum disappears in this product, but the memory of it can remain in the form of the absence of roots. In additional conditions, this effect is inherited by the Poisson suspension over the product., Comment: in Russian language

The main result of the article says that the formal power series equal to the ratio of two neighboring Chebyshev polynomials, after some renormalization, approximates the generating function of the Catalan numbers. We present a proof of this result using continued fractions. Also, for completeness of the presentation, we recall the basic properties of Chebyshev polynomials and formal power series that we use., Comment: 8 pages, in Russian. Comments are welcome!

The Molchanov's condition is a necessary condition for the compactness of the resolvent for a wide class of ordinary differential operators of arbitrary order, but for the Sturm-Liouville operator it is not sufficient, even if the real part of the potential is non-negative. Molchanov's criterion remains valid in the formulation closest to the original one for potentials that take values in a narrower sector than the half-plane, separated from the negative half-axis. This work is devoted to these questions., Comment: in Russian language

In a two-dimensional approximation, the probability density and current for a photoelectron near the localization of a quantum vortex are theoretically investigated. The wave function in the momentum representation, which we found earlier, is simplified near zero, corresponding to the center of the vortex. This allows us to obtain a simple analytical expression for it, which has the structure of a Gaussian wave packet and contains basic information about the vortex. It is used to analyze the temporal evolution of a quantum vortex, both in momentum and coordinate space. The effect of the intensity of an ionizing ultrashort laser pulse on the shape of a quantum vortex is also investigated., Comment: in Russian language

With the growth of coal production, the load on the production capacity of coal enterprises also increases, which leads to a concomitant increase in dust formation in both opencast and underground methods of mining coal deposits. Dust, generated during drilling, blasting operations, excavation, loading, crushing and transportation of mined rock is one of the factors that has a negative impact on the health of mining workers and on the level of environmental pollution with solid particles. Thus, increasing the efficiency of controlling the concentration of solid particles in the mine atmosphere and dust deposits is an urgent scientific and technical task. In doing so, the use of modern digital technologies within the framework of the industry 4.0 concept makes it possible to develop approaches that can significantly improve the quality of monitoring the state of the mine atmosphere at coal mining enterprises. This article provides a theoretical basis and test results for a system for continuous automatic monitoring of dust concentration in a mine atmosphere as the component of the multifunctional coal mine safety system. It is shown that monitoring the state of mine workings aerological safety can be carried out in real time through the system of the new generation using artificial intelligence. The ability of the proposed system to measure basic physical parameters affecting dust deposition (disperse composition, air humidity, dust concentration and air flow velocity) is noted., Comment: 8 pages, in Russian language

Given a real cubic function $f(x)$ with three roots, take an equilateral triangle $ABC$, the projections of which vertices are the roots of $f(x)$. There is a folklore fact that the vertical lines through the extrema of $f(x)$ are tangent to the inscribed circle of $ABC$. We generalise this fact to a regular $n$-gon and the corresponding degree $n$ polynomial., Comment: 7 pages, in Russian, 3 figures. Comments are welcome!