A system of bosons studied within the mean field framework has two fascinating phenomena: a liquid-gas first order phase transition and Bose-Einstein condensation. Interplay between these two phenomena is being investigated. Depending on the mean-field potential parameters one can observe two types of critical points (CP), called ”Boltzmann” and ”Bose”, that belong to different universality...

The Ising model is an archetype describing collective ordering processes. As such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago under supervision of Wilhelm Lenz [1] contained not only the solution of what we call now the ‘classical 1D Ising model’ but also other problems. Some of these problems are the subject of...

We analyze multi-qubit states that can be represented using directed graphs G(V,E). We focus on the geometric properties of these states, namely on curvature and torsion [1]. It has been found that the curvature of quantum states is determined by the sum of the weighted degrees of nodes in graphs where the weights in G(V,E) are raised to the second and fourth powers [2]. Additionally,...

In this talk some less known facts of the discovery of the Wave of Translation will be discussed. A brief review of the advancement of the theory of Davydov's solitons will be given. New results on the study of the dynamics of the Davydov’s soliton [1] in an external oscillating in time magnetic field [2,3] will be reported.

It is shown that in the magnetic field, perpendicular to the...

The nonlinear Klein–Gordon (nKG) model,

$
\partial_{tt}\phi-c^{2}\partial_{xx}\phi+f(\phi)=0,
$

is a universal model for describing the propagation of nonlinear waves in various physical media. For example, its stationary version describes the macroscopic wave function of the condensed phase (i.e., the order parameter) in the Landau theory of phase transitions. Noteworthy is also an...

The entanglement of diamond spin systems in thermodynamic equilibrium has

been studied in various papers (for, example, [1, 2, 3, 4]). We focus on the evolution of entanglement in a diamond spin-1/2 cluster. This cluster consists of two central spins described by the anisotropic Heisenberg model, which interact with two side spins via an Ising interaction. The influence of the interaction...

This presentation is aimed at using neutron optics methods to study the physical properties of bulk and confined liquids. To achieve this goal, the defining idea of M.M. Bogolyubov regarding the hierarchy of relaxation times and the sequential description of the dynamic evolution of condensed systems was used [1]. The following results were obtained by the methods of neutron optics [2-6],...

The structure of the DNA double helix is stabilized by water molecules and positively charged metallic or molecular ions, which form an ion-hydration shell around the macromolecule. The ions neutralize the negatively charged phosphate groups of the DNA backbone and thus act as counterions. Despite the extensive number of experimental and theoretical studies, the specific effects of counterions...

The aim of this paper is to derive the hydrodynamics for a cold Bose gas from the microscopic platform based on the many-body Schr¨odinger equation and general assumptions of the hydrodynamic approach (HA) applicable to any dimension. We develop a general HA for a cold spatially inhomogeneous Bose gas assuming two different temporal and spatial scales and obtain the energy as a functional of...

Calculation of the vacuum energy density in quantum field theory gives a value $10^{122}$ times higher than the observed one, and many proposed approaches have not solved this problem and have not calculated its real value. However, the application of the microscopic theory of superconductivity to the description of the physical vacuum on the Planck scale made it possible to solve the problem...

Systemic shocks inevitably lead to negative socioeconomic outcomes. The COVID-19 pandemic and the war in Ukraine are the prominent examples of such systemic shocks. Shock-initiated spreading processes often have a domino effect on both the social and economic levels. The war in Ukraine, despite its devastating effect on the Ukraine’s society and economy, has not led to the full collapse,...

Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on different scales. The proposed concept of the graph structural complexity corresponds to qualitative comprehension of the complexity. The proposed measure...

Transport processes of a passive scalar in random velocity fields are observed in plasma systems, atmosphere, oceanic currents, etc. The task of the theoretical description is to reproduce the temporal evolution of an ensemble of particles moving in such a field based on the known statistical characteristics of the velocity or force fields. The most known example is Brownian motion, the...

This work shows how a kinetic process is formed in a dynamic system that is in a non-stationary coupling with the environment. It is assumed that the environment has a large number of degrees of freedom and therefore transitions in a dynamic system do not change the state of the environment. However, due to the openness of a dynamic system, the environment is capable of modifying both the...

The scattering of gas flow on an obstacle can lead to the formation of nonequilibrium steady states (NESS), such as stationary obstacle wakes. These systems may undergo nonequilibrium phase transitions, resulting in the emergence of nonlinear steady-state gas structures under critical conditions. One notable example is the formation of a stratum-like, or two-domain, gas structure ahead of the...

**The development of Bogolyubov reduced description method
in the application to spin and quasispin systems**

Sokolovsky A. I., Lyagushyn S. F.

*Oles Honchar Dnipro National University*

The reduced description method (RDM) is based on the Bogolyubov’s idea that at large time the non-equilibrium state evolution of a macroscopic system can be described with the limited number of parameters....

Modified potential energy theorem in radiating systems: Implementation in MATLAB

Fields of mathematics such as mathematical physics often use potential energy specifically to describe the dynamics of various physical systems. Classical potential energy theorems are known to play a key role in understanding and analyzing mechanical systems, electrostatic fields, quantum mechanics, and many...

Within the framework of AdS/CFT correspondence we considered large N limits of conformal field theories in d dimensions which described in terms of supergravity on the product of AdS space with a compact manifold. An important example of such correspondence is equivalence between N = 4 super Yang-Mills theory in four dimensions and Type IIB superstring theory on $AdS_5 × S^5$ [1]. The...

In quark-gluon plasma (QGP), at higher deconfinement temperatures $T \ge T_d$ the spontaneous generation of color magnetic fields, $b^3(T), b^8(T) \not = 0$ (3, 8 are color indexes), and usual magnetic field $b(T) \not = 0$ happens. Simultaneously, the Polyakov loop and/or algebraically related to it $A_0(T)$ condensate, which is solution to Yang-Mills imaginary time equations, ...

One of the simplest renormalizable extensions of the SM is the minimal neutrino extension of the Standard Model $\nu$MSM, proposed in 2005 [1, 2]. This modification introduces three righthanded neutrinos or heavy neutral leptons (HNL). The lightest sterile neutrino is identified as a dark matter particle. The other two sterile neutrinos are much heavier, with nearly identical masses, and are...

The Bohr’s Hamiltonian is one of the main cornerstones of the nuclear structure theory. It was derived by Bohr [1], treating the nucleus as a liquid spherical drop with uniform density and sharp surface, performing quadrupole vibrations with small amplitude. During such oscillations at any moment of time the nucleus attains an ellipsoidal shape, retaining its volume constant due to small...

We described the anomalous temperature behavior of the giant dielectric response and losses using the core-shell model for ceramic grains and modified Maxwell-Wagner approach. We assume that core shells and grain boundaries, which contain high concentration of space charge carriers due to the presence of graphite inclusions in the inter-grain space, can effectively screen weakly conductive...

DNA is a highly charged molecule that is neutralized by positively charged metal or molecular ions (counterions). The neutralization of DNA by these counterions induces various effects, including the formation of DNA-DNA contacts that lead to further condensation of the macromolecule. The effect of DNA condensation has been widely observed for highly charged counterions (≥3+). For divalent...

O. O. Boliasova$^{1,2}$ and V. N. Krivoruchko$^3$

$^1$State Research Institution «Kyiv Academic University»

36 Academician Vernadsky Boulevard, 03142, Kyiv, Ukraine,

$^2$G. V. Kurdyumov Institute for Metal Physics of the N.A.S. of Ukraine

36 Academician Vernadsky Boulevard, 03142, Kyiv, Ukraine, and

$^3$Donetsk Institute for Physics and Engineering named after O.O. Galkin of the NAS of...

Analytical model of a nonlinear magnetization wave (MW) propagating through one-dimensional antifer-romagnetic magnonic crystal comprised of two sorts of antiferromagnets (AFM) is proposed for supercritical mode when the MW velocity exceeds the critical velocity of MW in both antiferromagnets AFMs or at least in one of them. Both AFMs that comprise the magnonic crystal are assumed to be...

Contemporary research in defense technology focuses extensively on concealing various objects from infrared (IR) reconnaissance. Numerous camouflage coatings are available in the market, designed to mask IR radiation emitted by targets. Manufacturers often claim that these coatings significantly reduce the mean apparent temperature difference (ΔT) between the object and its background....

The Potts model with invisible states was introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where Zq symmetry is spontaneously broken [1]. It differs from the ordinary q-state Potts model in that each spin, besides the usual q visible states, can be also in any of r so-called invisible states. Spins in an...

Temporal changes in the temperature and concentration dependences of the pH value in aqueous sodium chloride solutions contacting with atmospheric carbon dioxide have been studied experimentally. The measurements are carried out in a temperature interval of (294–323) K for ion concentrations corresponding to (180–1600) water molecules per sodium or chlorine ion.

The pH value for dilute...

Topologically non-trivial two-dimensional magnetization configurations (solitons in magnets), such as skyrmions and vortices, are very promising as information carriers for spintronics and magnonics. Skyrmions exhibit several well-known dynamic regimes, among them are breathing modes when the skyrmion radius changes, and rotational modes when the position of the skyrmion core fluctuates....

An analysis of the temperature dependences of thermal conductivity κ(Т) of composite materials - graphene-multilayer graphene, semiconductor composites Bi0.5Sb1.5Te3 and In0.53Ga0.47As, was carried out as well as a comparison of their temperature dependences of κ(Т) with the thermal conductivity of similar materials, which are formed by superlattices, nanowires and hybrid nanostructures. The...

M.S. Barabashko1, M.I. Bagatskii1, V.V. Sumarokov1, A.I. Krivchikov1,

A. Jeżowski2, D. Szewczyk2,3, Y. Horbatenko1

1B.Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine,

47 Nauky Ave., Kharkiv, 61103, Ukraine

2 W. Trzebiatowski Institute for Low Temperatures and Structure Research, Polish Academy of Sciences,

P.O. Box 1410, 50-950 Wroclaw, Poland

3Low...

Germanium-Tin (GeSn) films are a highly promising material for optoelectronic applications, owing to their tunable bandgap, light emission, and compatibility with silicon technology. As the Sn content in alloys increases, the band gap of GeSn alloy continuously decreases, transitioning from indirect to direct band structure and reducing the effective mass of both electrons and holes. Recent...

The results of a study of phonon heat transfer in two-dimensional and three-dimensional conductors with different cross-sections are presented. The problem is considered in the ballistic regime of phonon propagation and in the radiative approximation of phonon emission by a heater at a given temperature.

The most known problem in this field is the determination of heat fluxes and...

In the broad field of nanotechnology, polymer matrix-based nanocomposites have become a relevant area of modern development and research. This is due to their wide application in biomedicine, electrical/electronic/optoelectronic applications, and interest in fuel cells. The field of nanotechnology is one of the most popular areas of current research and development in nearly all technical...

To date, the most of metamaterials used in diverse applications (from nanooptics and plasmonics to mobile communication and biophysics) are periodic structures consisting of spatially arranged inclusions. In the theory of metamaterials, they are treated as homogeneous media, if their unit cell size $d$ (the lattice constant) is much smaller than the wavelength $\lambda$ of the incident...

Condensed matter physics is revolutionizing by introducing topology-based concepts that characterize a system's physical states and properties. An example of topological effects in magnetization dynamics is the additional quantum mechanical phase, the so-called Berry phase [1], and the Aharonov–Casher (AC) effect [2], acquired by the quantum orbital motion of chargeless bosonic quasiparticles...

Our research aims to examine critical behavior of a magnetic system under the influence of two competing factors: long-range interaction and weak structural disorder (e.g., weak quenched dilution). We analyze ferromagnetic ordering in a structurally-disordered magnet within an $n$-vector model in $d$-dimensional space, where the long-range interaction decays with distance $x$ as $J(x) \sim...

Kagome-lattice Heisenberg antiferromagnet is a paradigmatic model in the field of frustrated magnetism that allows us to study the interplay of geometrical frustration and quantum as well as thermal fluctuations in two and three dimensions. Distinctive spectrum of this model manifest itself in the thermodynamic properties throughout the peculiar low-temperature behavior of the specific heat....

The ability to couple cavity-confined microwave modes to diamond slabs or films containing large numbers of color centers opens up potential new methods for noise reduction, processing, and even generating quantum states of microwaves. The unique feature of various diamond color centers is that they can be prepared in their ground states by illuminating them with light in the optical band. The...

In our report, we review the domain boundaries in the structure of benzene monolayer adsorbed on a graphene sheet. It is shown in [1] that the monolayer’s structure can be complicated even for benzene, the simplest representative of the cyclic hydrocarbons. It was found in [2] that there exist two different energy states of the adsorbed benzene molecule: 1) the symmetric (hollow) unstable...

Physical and mathematical modeling is widely used to simulate cryoapplication processes. Mathematical modeling of this process [1,2] allows us to predict the temperature field of the frozen region. This makes it possible to determine the cryo-application time sufficient to destroy target cells and minimize damage to healthy cells under various experimental conditions. Moreover, simulation also...

We study the impact of the photon subsystem on the magnetic properties of a mixture of quantum gases in thermodynamic equilibrium with it. Having proposed a simple model of the system, we obtain general equations describing the thermodynamic equilibrium of quantum gases of two-level atoms with photons. The resulting equations are solved at a temperature higher than the degeneracy temperature...

The search of a target of unknown location is often random and ineffective, especially when the search domain is spacious and there is a lot of detrimental trajectories. To get rid of them, thereby improving the search, interruptions of the latter with starting it from scratch can be a good strategy. Called resetting, such a manner is in fact inherent to many search processes at very diverse...

A rapid growth of Machine Learning (ML) applications in different areas has been faced in recent years. Training of ML models is performed by finding such values of their parameters $x=\{x_1, x_2, ..., x_N\}$ that optimize (minimize) the objective (loss) function $U(x)$. Usually, the number of parameters $N$ is large and the training dataset is massive. Therefore, to reduce computational...

Within the framework of the mean-field model, the thermodynamics of the relativistic bosonic system of interacting particles and antiparticles in the presence of a Bose-Einstein condensate is investigated. It is assumed that the total isospin (charge) density is conserved. It is shown that the particle-antiparticle boson system reveals four types of phase transitions into the condensate phase....

The chemical freeze-out curve in heavy-ion collisions is investigated in the context of QCD critical point (CP) search at finite baryon densities. Taking the hadron resonance gas picture at face value, chemical freeze-out points at a given baryochemical potential provide a lower bound on the possible temperature of the QCD CP. We first verify that the freeze-out data in heavy-ion collisions...

The analytical methods based on the Landau-Ginzburg-Devonshire (LGD) approach and variational principle allow the analytical description of size effects, strain and ferro-ionic coupling in low-dimensional ferroelectric materials, such as thin films and small nanoparticles. The validity of LGD approach is corroborated by experimental evidence of the size- and strain-induced transitions as well...

The observation of the Sagnac effect for massive material particles offers a significant enhancement in sensitivity when compared to optical interferometers with equal area and angular rotation velocity. For this reason, there have been suggestions to employ solid-state interferometers that rely on semiconductors and graphene. We investigate the Sagnac effect in Dirac materials governed by the...

We consider inhomogeneous underdamped one-dimensional parallel Josephson junction arrays. Inhomogeneity is introduced either as a non-uniformly applied dc bias current or as variations in the junctions' critical currents. We investigate the frequency of the localized modes induced by the presence of such inhomogeneities, in particular the frequency's dependence on the parameters that...

The generalized recursive method for calculating the T-matrix of electron scattering on arbitrary many-particle clusters for determining the Green's function of the strongly correlated system is developed. This approach is extended to the studying electronic spectra in both direct (Wannier) and reciprocal (Bloch) representations with taking into account the influence of atomic and magnetic...

Excitable membrane of olfactory receptor neuron (ORN) is populated with up to several millions of identical receptor proteins (R) able to bind / release odor (O) molecules. The affinity of R to O depends on the odor presented, and this is the initial mechanism which is recuired for the olfactory selectivity to exist.

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Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices we suggest the novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and...

The talk provides an overview of some advances in the mathematical understanding of the nature of the dynamics of the correlations of many colliding particles. The fundamental equations of modern mathematical physics are studied, in particular the hierarchies of the evolution equations of many hard spheres and their asymptotic behavior described by kinetic nonlinear equations.

First, an...

Experimental data on STM-induced electroluminescence in monomolecular junctions have led to the need to elucidate the physics of the formation of optoelectronic processes at the atomic-molecular level, taking into account both dynamic and relaxation processes. A mechanism for the formation of electrofluorochromism based on a kinetic model has been proposed [1]. In this model, the description...

A new mechanism is proposed to explain the reasons for the bacteria's motion in the aquatic environment. The mathematical model of this mechanism is based on the hydrodynamic equations of active matter and takes into account the dynamics of the environment polarization and polarization of individual bacteria. It is assumed that the flow of light and the active motion of dielectric regions with...

The study of the influence of local properties of multi-particle conglomerations on their macroscopic properties is one among of the traditionally relevant problems of statistical physics. Parameterization of local properties can be carried out in various ways, for example, in terms of ordering parameter tensors, Euler invariants, Voronoi tessellations and others. While the macroscopic...

Within the α-particle model, the structure of $^{12}C$, $^{16}O$, and $^{20}Ne$ nuclei is studied. With the use of the variational method with Gaussian basis, the wave functions are found for three-, four-, and five-particle systems consisting from α-particles. The charge density distributions and elastic form factors of $^{12}C$, $^{16}O$, and $^{20}Ne$ nuclei are calculated within the Helm...

In this report, we present results of systematic investigations of peculiarities of redundant solutions of the resonating group method (RGM), which are known as the Pauli resonance states. Such resonance states appear when one tries to use more advanced (more precise) wave functions describing internal structure of interacting clusters. It is generally recognized that the Pauli resonance...

Dispersion equations and relations are the key subjects of linear theories involving waves and collective excitations. However, in some systems, dispersion equations contain multivalued functions and their solutions are ambiguous. To resolve such uncertainty we suggest analyzing the initial value problem that gives the unambiguous solution.

As an example, we considered the excitation of the...

The main idea of the work is to carry out the dynamic evolution of the orbits of Globular Cluster (GC) subsystems sample lookback time up to 10 Gyr. This allows us to estimate the possibility of GCs interaction with the Galactic center that dynamically changed in the past. To reproduce the structure of the Galaxy in time, we used external potentials which dynamically changed in a past and now...

In our work, we obtain a set of Gaussian orbital perturbation equations in the Schwarzschild space-time in terms of Weierstrass elliptic functions, and solve it for several external forces in linear approximation. We consider forces defined from: the cosmological constant in the Schwarzschild–de Sitter space-time, various quantum gravity corrections, angular momentum from the Kerr space-time...

We study the mathematical model of an abstract society ${\cal S} = \{ a_i \} _{i=1}^m, \ 1< m < \infty$ ($a_i $ - players, opponents or their association) in the form of a complex dynamical system with a conflict interaction between its elements. The states of society $S$ are described by stochastic vectors of players energy ${\bf p}^t=(p_1^t,...,p_i^t,..., p_m^t), \ t=0,1,...,$ which evolve...

About the works of M. Bogolyubov at the Institute of Mathematics.

We establish constructive necessary and sufficient conditions of solvability and a scheme for the construction of solutions for a nonlinear boundary-value problem unsolved with respect to the derivative in the critical case [1], [2], [3].

On the basis of the Adomian decomposition method [4], [5] we are constructed convergent iterative schemes for finding approximations to solutions of a...

Let $L_{p}$, $1\le p\le\infty,$ and $C$ be the spaces of $2\pi$-periodic functions with standard norms $\|\cdot\|_{L_p}$ and $\|\cdot\|_C$, respectively. Further, let $W^r_{\beta,p},\ r>0,\ \beta\in\mathbb{R},\ 1\le p\le \infty,$ be classes of $2\pi$-periodic functions $f$ that can be represented in the form of convolution

$...

The study of nonlinear matrix equations, in particular, the algebraic matrix Riccati equation [1,2,3], is connected with numerous applications of such equations in solving the differential matrix Riccati equation [2,3], in the theory of nonlinear oscillations, in mechanics, biology, and radiotechnology, the theory of control and stability of motion, and others. We used the Newton-Kantorovich...

Many mathematical biology and population dynamics models involve nonlocal diffusion corresponding to long-range interactions in a system. These models are typically described by evolution problems with convolution-type integral operators and their qualitative and quantitative properties can be obtained by studying of the corresponding spectral problems.

We consider spectral...

**1. Introduction.**

In this paper we study the asymptotic behaviour of solutions of a fourth order differential equation of the form $y^{(4)}=\alpha_0p(t)\varphi(y)\quad$(1). The purpose of this paper is to obtain the asymptotics $P_\omega(Y_0,\lambda_0)$ solutions of the differential equation (1) for the special case when $\lambda_0=1$.

**2. Object of research.**

Consider a differential...

This paper deals with the boundary value problem for a singularly pertur-bed differential algebraic system of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for DAEs are found. Technique of constructing the asymptotic solutions is developed.

The talk deals with the singular perturbed Burgers equation with variable coefficients

$ \varepsilon u_{xx} = a(x, t, \varepsilon) u_t + b(x, t, \varepsilon) u u_x, \qquad \qquad \qquad \qquad (1) $

where $ a(x, t, \varepsilon), $ $ b(x, t, \varepsilon) $ are given as asymptotic series in a small parameter $ \varepsilon $. Equation (1) is straightforward generalization of the...

Discrete symmetries of differential equations are not so well studied objects as continuous ones. We discuss the notion of discrete symmetries and the methods of their computation. Then we analyze common errors in such computations and show connection of these errors with those in solving other problems of classical group analysis of differential equations, including the classification of Lie...

We consider the dissipative magnetic Zakharov system in a smooth (2D) bounded domain

$\Omega\subset\mathbb{R}^2$ of the form

$ i E_t + \Delta E - n E + i E\times B +i\gamma_1 E = g_1(x,t), \qquad\qquad x\in \Omega, $

$ n_{tt}+\gamma_2 n_t -\Delta\left(n+|E|^2\right)=g_2(x,t), \qquad\qquad\qquad \qquad x\in \Omega, $

$

B_{tt}-\gamma_3\Delta B_t +\Delta^2\left(B+i E\times...

The transformational properties of two classes of generalized Korteweg-de Vries (KdV) equations with coefficients dependent on the time variable are investigated, and the effectiveness of the equivalence method for constructing exact solutions to such equations is demonstrated. Specifically, the equivalence groupoids for both classes of equations are identified, and it is proven that both...

Following [1], we discuss advances in the classical group analysis of the Kolmogorov backward equation with quadratic diffusivity

$(1)\qquad u_t+xu_y=x^2u_{xx}.$

This equation belongs to the class of (1+2)-dimensional ultraparabolic linear equations denoted by $\bar{\mathcal F}$ in [1] and is distinguished within this class $\bar{\mathcal F}$ by its excellent symmetry properties. More...

Let $\mathbb{C}$ be the complex plane, $\overline{\mathbb{C}}=\mathbb{C}\bigcup\{\infty\}$ be its one point compactification. A function $g_{B}(z,a)$ which is continuous in $\overline{\mathbb{C}}$, harmonic in $B\backslash \{a\}$ apart from $z$, vanishes outside $B$, and in the neighborhood of $a$ has the following asymptotic expansion $$g_{B}(z,a)=-\log|z-a|+\gamma+o(1),\quad z\rightarrow...

The Anderson-Darling test uses a statistic:

$W_n^2=n\int_{-\infty}^{+\infty}\left(F_{n}(x)-F(x)\right)^{2}\psi\left(F(x)\right)dF(x),\qquad\qquad (1)$

where $u=F(x)$ is cumulative function of a known distribution; $F_{n}(x)$ is cumulative function of an empirical distribution; $\psi\left(u\right)$ is weight function; $n$ is sample size [1],

to estimate the amount of an empirical...

Despite the number of relevant considerations in the literature, the algebra of generalized symmetries of the Burgers equation has not been exhaustively described.

We fill this gap, presenting a basis of this algebra in an explicit form and proving that the two well-known recursion operators of the Burgers equation and two seed generalized symmetries, which are evolution forms of its Lie...

The dispersionless (potential symmetric) Nizhnik equation has interesting algebraic and geometric properties. We computed the point- and contact-symmetry pseudogroups of this equation using an original megaideal-based version of the algebraic method. Note that this approach was used for the first time to find the contact-symmetry pseudogroup of a differential equation. By the same method, we...

Oleg Kolomiichuk, PhD, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, str. Tereshchinkivska 3,

Ukraine e-mail: o.kolomiichuk@gmail.com;

**IDENTIFICATION OF UNKNOWN PARAMETERS OF A FULLY CONTROLLED NOISY GYROSCOPIC SYSTEM**

The work is devoted to the identification of unknown parameters of a fully controlled gyroscopic object with noise at the input and...

Obviously, the direct product of nearrings with identity is a nearring with identity. At the same time, the direct product of two arbitrary local nearrings is not a local nearring. Naturally, the question arises of defining such a product, the result of which is a local nearring.

The semidirect product of the ring $R$ with the abelian group $G$, which is a nearring, was given in [1]. We...

The Nambu-determinant Poisson brackets on R^d are expressed by the formula

{f,g}*d (x) = \rho(x) \det( \partial(f,g,a_1,...a*{d-2}) / \partial(x^1,...,x^d) ),

where a_1,...,a_{d-2} are smooth functions and x^1,...,x^d are global coordinates (e.g., Cartesian), so that \rho(x)\cdot\partial_{x} is the top-degree multivector.

For an example of Nambu--Poisson bracket in classical...

From the moment the BBGKY hierarchy (Bogolyubov–Born–Green–Kirkwood–Yvon) was formulated in 1946 until the last decade, the solution to such a hierarchy of evolution equations has been represented in the form of an iteration series, i.e., expansion into a series constructed by perturbation theory methods. In particular, this representation of the solution is applied for the derivation of...

Consider the following control system:

$w_t=w_{xx}$, $x\in(0,+\infty)$, $ t\in(0,T)$,

$w(0,\cdot)=u$, $t\in(0,T)$,

$w(\cdot,0)=w^0$, $x\in(0,+\infty)$,

where $T>0$ is a constant, $w^0$ is a given function, $u\in L^\infty(0,T)$ is a control. The control system is considered in Sovolev spaces.

An initial state $w^0$ of control system (1)-(3) is said to be null-controllable in...

We review our recent results on transformation properties of normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. We also present principal properties of Lie symmetries of the systems under consideration and outline ways for completely...

M. Bogolyubov was officially admitted by the Institute of Mathematics as a senior researcher on March 1, 1945. On April 1, he assumed the head of the Department of Asymptotic Methods and Statistical Mechanics. The dismissal order, dated November 1, 1956, was signed by his student Yu.O. Mitropolsky, who was at that time the deputy acting director of the institute, O.S. Parasyuk. However, M....

This work reviews modern applications of Lie algebras, such as: limiting transitions between different models, new classes of orthogonal special functions, and the construction of quasicrystals using root systems. We also provide a list of unsolved but promisive problems that have important applications.

V. Koshmanenko, O. Satur (Institute of mathematics of NAS Ukraine, Kyiv, Ukraine)

On structure of the point spectrum

in equilibrium states of the dynamical conflict systems

The structure of the point spectrum in equilibrium time-limiting states of dynamical conflict

systems is studied in terms of probability measures. It is shown that the priority strategy

in a single direction is a...

Electromagnetic (EM) energy compression is a process of converting long-duration low-amplitude input pulses into much shorter output pulses with significantly higher amplitude. There are two primary approaches to this: active and passive compression [1]. Active compressors accumulate the input in resonant cavities over a relatively long accumulation stage, followed by a rapid release of the...

The isomonodromic tau function of the Painleve I equation can be presented as a Fourier transform of the partition function of the Argyres-Douglas theory of type $H_0$. A possible way to derive this partition function is to use the holomorphic anomaly equation (HAE) as a recursive relation for the topological expansion of corresponding free energy (logarithm of the partition function). The...

We develop the Riemann-Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation $i q_t + q_{xx} + 2|q|^2 q = 0$. We show that a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs...

We discuss methods of modeling $n$-dimensional quasicrystals and their application to information encoding.

The first application is to use the mapping between the physical and internal spaces of a quasi-crystal to evenly distribute data that is lost in the process of transmitting or storing information.

The second application consists in the construction of special quasi-crystals that...

## Representations of solutions of some nonlinear PDEs in the form of series in powers of the $\delta$-function

## S.L. Gefter, A.L. Piven', V. N. Karazin Kharkiv National University

Let $K$ be an arbitrary integral domain with identity and let...

We consider asymptotic soliton- and peakon-like solutions to the modified Camassa--Holm equation with variable coefficients and a singular perturbation

$$ a(x,t,\varepsilon) u_t - \varepsilon^2 u_{xxt} + b(x, t, \varepsilon) u^2 u_x - 2 \varepsilon^2 u_x u_{xx} - \varepsilon^2 uu_{xxx} = 0. \quad (1) $$ Here $ \varepsilon $ is a small parameter and the coefficients $ a(x, t,...

Since the beginning of time, before to start implementation of some ideas that people had planned earlier, they have made some optimal, more or less, decision. Initially this decision was made without any special analysis, it was based only on pure human experience. But over time it was no longer possible to realize this action without special mathematical methods that carry out global search...

Graphene stays in the spotlight for its exceptional properties and versatile applications, one of which is the development of flexible metamaterials. Its ability to integrate seamlessly with other components and adapt to nonplanar structures makes graphene a key material in advancing innovative flexible meta devices and functional metasurfaces [1]. This motivates us to focus on the interaction...

The eikonal equation is a first-order nonlinear partial differential equation

$ u_\mu u_\mu = f(x_\mu,u)$. Such equations are used to describe various physical processes, in particular, ray propagation in optics. We consider equations in the space of independent variables $x_\mu$, where the indices $\mu$ span from 0 to some arbitrary $n$. Here lower indices of functions mean differentiation...

In this talk, I will explain how rank 5/2 representations of Virasoro algebra can be used to compute asymptotic expansion of the tau function of Painleve 1 equation near irregular singularity. The talk is based on recent paper of Hasmik Poghosyan and Rubik Poghossian, where they introduced conformal block with irregular vertex of rank 5/2 and conjectured that it is related to the partition...

The Boltzmann equation [1] that describes the evolution of rarefied gases is one of the main equations of the kinetic theory of gases. For a model of hard spheres, the equation has the form:

$ D(f)= Q(f,f), $

where the left-hand side of the equation is the differential operator:

$ D(f)\equiv\frac{\partial f}{\partial t}+\left(V,\frac{\partial f}{\partial x}\right),$

and the...

Guided by physical needs, we deal with the rotationally isotropic Poincaré ball, when considering the complement of Borromean rings embedded in it [1]. We describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. According to Penner, we construct the Teichmüller space of the decorated ideal octahedral surface related to the quotient...

For $N\in \{ 1,2,3 \}$ there is considered the semi-direct sum $ \tilde {\mathcal G} \propto \tilde {\mathcal G} _{reg}^*$ of the loop Lie algebra $ \tilde {\mathcal G}$, consisting of the even left superconformal vector fields on a supercircle $\mathbb S^{1|N}$ in the form $\tilde a := a \partial / \partial x + \frac 12 \sum \nolimits _{i=1}^N (D_{\vartheta_i} a)D_{\vartheta_i},$ where $...

We consider the Cauchy problem for the following two-component peakon system with cubic nonlinearity:

$ \partial_t m=\partial_x[m(u-\partial_xu)(v+\partial_xv)]$,

$\partial_tn=\partial_x[n(u-\partial_xu)(v+\partial_xv)]$,

$m=u-\partial_{x}^2u, \quad n=v-\partial_{x}^2v$,

where $m=m(t,x)$, $u=u(t,x)$, $v=v(t,x)$ and $t,x\in \mathbb{R}$. We assume that the initial data $u_0(x)=u(0,x)$...