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...
We obtained constructive conditions for the solvability and a scheme for constructing solutions of a nonlinear boundary value problem with concentrated delay in the case of parametric resonance using the Adomian decomposition method. The original function of the differential system with delay contains an unknown eigenfunction that ensures the solvability of the weakly nonlinear boundary value...
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...
We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions of a nonlinear autonomous boundary-value problem unsolved with respect to the derivative [1,2,3] in the critical case.
We also construct convergent iterative schemes for finding approximations to the solutions of a nonlinear autonomous boundary-value problem...
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 $u_0(x)=u(0,x)$...
