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Roman Kozlov

Publications and source records attributed to Roman Kozlov.

At least 19 recordsLinked to original sources
arXiv2025

Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument

We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group classifications are performed. Classes of invariant DODEs for each Lie subgroup are written out. The symmetries allow us to construct invariant solutions to such equations. The application of variational methods to functionals with one delay yields DODEs with two delays. The Lagrangian and Hamiltonian approaches are reviewed. The delay analog of the Legendre transformation, which relates the Lagrangian and Hamiltonian approaches, is also analysed. Noether-type operator identities relate the invariance of delay functionals with the appropriate variational equations and their conserved quantities. These identities are used to formulate Noether-type theorems that give first integrals of second-order DODEs with symmetries. Finally, several open problems are formulated in the Conclusion.

nlin.SI↗
arXiv2024

Delay ordinary differential equations: from Lagrangian approach to Hamiltonian approach

The paper suggests a Hamiltonian formulation for delay ordinary differential equations (DODEs). Such equations are related to DODEs with a Lagrangian formulation via a delay analog of the Legendre transformation. The Hamiltonian delay operator identity is established. It states the relationship for the invariance of a delay Hamiltonian functional, appropriate delay variational equations, and their conserved quantities. The identity is used to formulate a Noether-type theorem, which provides first integrals for Hamiltonian DODEs with symmetries. The relationship between the invariance of the delay Hamiltonian functional and the invariance of the delay variational equations is also examined. Several examples illustrate the theoretical results.

math-ph↗
arXiv2024

Application of Methods of Artificial Intelligence in Systems for Continuous Automatic Monitoring of Dust Concentration and Deposits in Mine Atmosphere

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.

eess.SY↗
arXiv2023

Conservation laws of mean field games equations

Mean field games equations are examined for conservation laws. The system of mean field games equations consists of two partial differential equations: the Hamilton-Jacobi-Bellman equation for the value function and the forward Kolmogorov equation for the probability density. For separable Hamiltonians, this system has a variational structure, i.e., the equations of the system are Euler-Lagrange equations for some Lagrangian functions. Therefore, one can use the Noether theorem to derive the conservation laws using variational and divergence symmetries. In order to find such symmetries, we find symmetries of the PDE system and select variational and divergence ones. The paper considers separable, state-independent Hamiltonians in one-dimensional state space. It examines the most general form of the mean field games system for symmetries and conservation laws and identifies particular cases of the system which lead to additional symmetries and conservation laws.

math-ph↗
arXiv2022

Rota-Baxter operators on $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$

We classify all Rota-Baxter operators on the simple conformal Lie algebra $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang-Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.

math.RA↗
arXiv2022

Conformal Yang-Baxter equation on $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$

In 2008, J. Liberati defined what is a conformal Lie bialgebra and introduced the conformal classical Yang-Baxter equation (CCYBE). An $L$-invariant solution to the weak version of CCYBE provides a conformal Lie bialgebra structure. We describe all solutions to the conformal classical Yang-Baxter equation on the current Lie conformal algebra $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$ and to the weak version of it.

math.QA↗
arXiv2021

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.

math-ph↗
arXiv2020

One-dimensional flows of a polytropic gas: Lie group classification, conservation laws, invariant and conservative difference schemes

The paper considers one-dimensional flows of a polytropic gas in the Lagrangian coordinates in three cases: plain one-dimensional flows, radially symmetric flows and spherically symmetric flows. The one-dimensional flow of a polytropic gas is described by one second-order partial differential equation in the Lagrangian variables. Lie group classification of this PDE is performed. Its variational structure allows to construct conservation laws with the help of Noether's theorem. These conservation laws are also recalculated for the gas dynamics variables in the Lagrangian and Eulerian coordinates. Additionally, invariant and conservative difference schemes are provided.

math-ph↗
arXiv2020

Symmetries of Kolmogorov backward equation

The note provides the relation between symmetries and first integrals of Itô stochastic differential equations and symmetries of the associated Kolmogorov backward equation. Relation between the symmetries of the Kolmogorov backward equation and the symmetries of the Kolmogorov forward equation is also given.

math-ph↗
arXiv2020

Second-order delay ordinary differential equations, their symmetries and application to a traffic problem

This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable $x$ and one dependent variable $y$. As opposed to ODEs the variable $x$ figures in more than one point (we consider the case of two points, $x$ and $x_-$). The dependent variable $y$ and its derivatives figure in both $x$ and $x_-$. Two previous articles were devoted to {\it first}-order DODSs, here we concentrate on a large class of {\it second}-order ones. We show that within this class the symmetry algebra can be of dimension $n$ with $0 \leq n \leq 6$ for nonlinear DODSs and must be $n=\infty$ for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow.

math.CA↗
arXiv2019

Conservative difference schemes for one-dimensional flows of polytropic gas

The paper considers one-dimensional flows of polytropic (calorically ideal) gas. These flows include three cases of gas dynamics: plain one-dimensional flows (one-dimensional space), radially symmetric flows in two-dimensional space and spherically symmetric flows in three-dimensional space. Starting with the difference schemes which have conservation laws of mass and energy (as well as conservation of momentum and the center of mass motion for the plain one-dimensional flows), we find difference schemes which also have additional conservation laws for the special values of the adiabatic exponent $\gamma = 1 + 1 /d $, where $d$ is the space dimension.

math.NA↗
arXiv2018

One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: symmetry classification, conservation laws, difference schemes

Lie point symmetries of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates are considered. Complete Lie group classification of these equations reduced to a scalar second-order PDE is performed. The classification parameter is the entropy. Noether theorem is applied for constructing conservation laws. The conservation laws can be represented in the gas dynamics variables. For the basic adiabatic case invariant and conservative difference schemes are discussed.

math-ph↗
arXiv2017

Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional {sub}algebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone. We present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

math-ph↗
arXiv2017

Lie group classification of first-order delay ordinary differential equations

A group classification of first-order delay ordinary differential equation (DODE) accompanied by an equation for delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs) which consists of linear DODEs and solution independent delay relations have infinite-dimensional symmetry algebras, as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension $n$, $0 \leq n \leq 3$. It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

math-ph↗
arXiv2017

Hochshild cohomology of the associative conformal algebra Cend_{1,x}

It is established in this work that second Hochshild cohomology group of the associative conformal algebra Cend_{1,x} is zero. As a corollary, this algebra split off in each extension with a nilpotent kernel. Key words: associative conformal algebra, splitting off radical, Hochshild cohomology.

math.RA↗
arXiv2009

Invariance and first integrals of continuous and discrete Hamiltonian equations

In this paper we consider the relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations. We observe that canonical Hamiltonian equations can be obtained by variational principle from an action functional and consider invariance properties of this functional as it is done in Lagrangian formalism. We rewrite the well--known Noether's identity in terms of the Hamiltonian function and symmetry operators. This approach, based on symmetries of the Hamiltonian action, provides a simple and clear way to construct first integrals of Hamiltonian equations without integration. A discrete analog of this identity is developed. It leads to a relation between symmetries and first integrals for discrete Hamiltonian equations that can be used to conserve structural properties of Hamiltonian equations in numerical implementation. The results are illustrated by a number of examples for both continuous and discrete Hamiltonian equations.

math-ph↗
arXiv2009

Invariance and first integrals of canonical Hamiltonian equations

In this paper we consider the relation between symmetries and first integrals of canonical Hamiltonian equations. Based on a newly established identity (which is an analog of well known Noether's identity for Lagrangian approach), this approach provides a simple and clear way to construct first integrals with the help of symmetries of a Hamiltonian. The approach is illustrated by a number of examples, including equations of the three-dimensional Kepler motion.

math-ph↗