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On a Lie group, we introduce a family of group-invariant metrics and show that the curves invariant under this group are spirals in all the introduced metrics (i.e., they have constant curvatures). An important role is played by an algebra, which we call dual, defined on the same group. The main relation between these algebras is that the trajectories of the one-parameter groups generated by one algebra are invariant curves in the metric that is invariant under the other algebra. The fact that these curves are spirals distinguishes our approach from that of Cartan, who considered the trajectories of one-parameter groups as geodesics in some metric. The presented results are related to the analysis of the geometric meaning of the previously obtained classification of one-dimensional kinetic equations, where invariant curves are the trajectories of particles. [ABSTRACT FROM AUTHOR]

A survey of the development of the traveling wave method for one-dimensional media is presented. The main results and changes in the statement of the problem of representing solutions of linear systems of partial differential equations in terms of "traveling waves" (more precisely, in terms of a system of wave transport equations) are presented. It is shown that as the study of systems becomes more complicated, the problem of representing the solution by the traveling wave method turns out to be applicable not only for hyperbolic systems but also for systems containing (even implicitly) both parabolic and elliptic components and thereby approaches the general problem of decomposition of an arbitrary system of linear equations into a system of first-order equations with a main part of the canonical type and with a subordinate linear part. [ABSTRACT FROM AUTHOR]

We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift group does not admit uniform dilations, because the left- and right-hand sides of the equation scale differently. In particular, this holds for the classical Boltzmann equation. For the Vlasov equation, we also obtain the equivalence group and classifying relations for the symmetry group and classify the interparticle interactions for which the Vlasov equation admits groups containing the Galilean group in the case where external forces are absent. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional Boltzmann equation f t + c f x + (F f) c = 0 with a function F depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and d c = F d t , (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them. [ABSTRACT FROM AUTHOR]

We present the group classification of the one-dimensional Boltzmann equation with respect to the function F = F(t, x, c) characterizing an external force field under the assumption that the physically meaningful constraints dx = c dt, dc = F dt, dt = 0, and dx =0 are imposed on the variables. We show that for all functions F, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero F. [ABSTRACT FROM AUTHOR]

The article provides an overview of mathematical studies performed by Vladimir Alexandrovich Kondratiev, on the 70th anniversary of his birth. Included are studies on: the oscillation and nonoscillation conditions for solutions of ordinary differential equations; elliptic equations in domains with angular and conical points; and a problem with oblique derivatives.

We study the two-dimensional eikonal equation ψ²x + ψ²y = 1/v²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds² = [dx² + dy²]/v² (x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations. [ABSTRACT FROM AUTHOR]

We present a group classification of the family of equations of the form (∇ψ)2 = 1/ v2( x, y, z), called the eikonal equations, which are equations of characteristics for equations describing acoustic and electromagnetic waves in inhomogeneous media. For some cases of equations with linear and quadratic functions v( x, y, z), having the sense of the speed of waves in a medium, we present explicit solutions (eikonals of point sources) and completely describe the geometry of rays. [ABSTRACT FROM AUTHOR]

Discusses the moving source phantom in geometric optics of inhomogenous media. Examination of the center-of-curvature motion phenomenon for a wave front propagating in an inhomogenous medium. Dependency of velocity on the curvature radius; Illustration of velocity functions; Presentation of integral theorems.

Studies the relaxation oscillations for a single van der Pol equations and for a pair of van der Pol equations. Focuses on the use of constructive nonstandard analysis in investigating relaxation oscillations; Computation of slow and rapid motions in relaxation oscillations; Description of rapid and slow trajectories in a very complicated problem; Derivation of conditions providing the motions on the stable part of slow manifold.

Differential equations on networks is a relatively new branch (it has existed about 20 years) in the theory of differential equations. This issue is the result of work in this direction performed in Russia within the framework of the seminar of Yu. V. Pokornyi (Voronezh), in which models connected with oscillations in various systems consisting of elastic elements were at the center of attention. In this review, we describe the fundamental problem statements, the specific features connected with their analysis, the basic approaches, and the methods for studying the corresponding differential equations. [ABSTRACT FROM AUTHOR]

Considers a differential equation on a network (geometric graph) that is a model of a beam construction. Beams connected by elastic ball-joints; Focus on analogies of monotonic properties of the solution; Green's function and its properties; Various methods for reducing the problem into simpler problems.

We present a method for analyzing the Green's function, adapted to nonclassical problems for ordinary differential equations (including those on spatial graphs). We analyze the evolution of the view of the Green's function—from a Hilbert system of axioms to a direct analysis of the influence function. [ABSTRACT FROM AUTHOR]

For a multipoint discontinuous boundary-value problem with nonoscillating differential operator, we present an analog of the Kalafati conditions that ensure the sign regularity property, i.e., the property that the number of sign changes of the solution does not exceed the number of sign changes of the function on the right-hand side. The sign regularity property allows one to verify whether the spectrum of the corresponding spectral problem exhibits the Sturm properties (i.e., the reality, positiveness, and simplicity of eigenvalues, the alternation of zeros of eigenfunctions, etc.). [ABSTRACT FROM AUTHOR]

The article presents information about Voronezh Spring Mathematical School that imparts education about modern methods in qualitative theory of boundary value problems. The school was organized by Voronezh State University in cooperation with Moscow State University and the Mathematical Institute of the Russian Academy of Sciences. The school comprised about 80 participants from 20 cities. The main scientific directions of the school program were the following: modern aspects of the theory of boundary value problems, computer science and control, geometry and analysis, plasticity, related problems of applied and engineering mathematics, problems of mathematical education.