Limiting curves in an axially symmetric galaxy

The aim of this article is to reveal the geometry of the windows through which stars escape from a potential modeling the central part of a galaxy with axial symmetry and presenting one exit channel. This study has been performed by analyzing the ingoing and outgoing asymptotic orbits to the Lyapunov orbit located at the opening of the potential in a suitable surface of section.     

Transonic flow on an axially symmetric torus

On a Riemannian manifold the existence (and uniqueness) of subsonic gas flows with prescribed circulation has been previously established (Acta Math.125 1970, 57–73). If the manifold is a torus of revolution then the gas dynamics equation reduces to a nonlinear ordinary differential equation and the flow can be described explicitly. We show that, as the circulations are increased, one obtains a complete family of solutions: smooth subsonic, smooth transonic, transonic with shocks, and smooth supersonic flows.     

Propagation of high-frequency harmonic elastic waves from an axially symmetric cavity

An asymptotic ray method is developed for studying propagation of high-frequency harmonic elastic waves from a smooth, convex, axially symmetric cavity. The dynamic stresses near a prolate spheroidal cavity, to whose surface is applied a normal, angle-independent load, are investigated as a specific numerical example. It is shown that account of three terms of the ray series makes it possible to find quite accurate solutions even when the wave length is comparable with the cavity sizes.Translated from Teorr eticheskaya i Prikladnaya Mekhanika, No. 18, pp. 87–95, 1987.     

Propagation of axially symmetric elastic vibrations along a waveguide in an elastic medium

The scatter equation is derived for axially symmetric (longitudinal and torsional) normal vibrations of an infinitely long elastic cylindrical waveguide embedded in an elastic medium. The low-frequency vibrations are analyzed thoroughly. It is shown that low-frequency longitudinal vibrations follow either of two branches of the scatter curve. Recommendations are made concerning the use of waveguide methods for studying the physicomechanical properties of polymers.     

Exact axially symmetric wave solutions of the Yang-Mills equations

We find a class of exact axially symmetric wave solutions of the Yang-Mills equations with SU(2) symmetry. The solutions in this class describe running waves propagating at the speed of light in a vacuum and contain two arbitrary differentiable functions of their phase. We consider properties of field sources that can generate such running waves. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 243–248, August, 2006.     

Axially symmetric temperature stresses in an elastic isotropic cylinder of finite length

We consider an axially symmetric problem of the thermostressed state of a solid cylinder of finite length with a load-free surface. Using the method of superposition, we have constructed the complete analytical solution of this problem, which is reduced to the solution of a system of linear algebraic equations. We have proposed a method for determining the asymptotic behavior of coefficients in these systems, which enables us to develop an efficient algorithm for the calculation of stresses in the cylinder, including regions near its end-face circles. Typical examples are considered.     

Boundary layer equations in the problem of axially symmetric jet flow

The problem of stationary axially symmetric jet flour of a viscous incompressible fluid from a round tube into a free space is considered. The mathematical model of this flow in the Mises variables and for large Reynolds numbers is reduced to the boundary-value problem for a degenerate integro-differential equation of the second order. The existence and uniqueness of bounded solutions to this problem are proved, and stabilization as a time-like variable increases is shown. Bibliography: 7 titles. To Vsevolod Alekseevich Solonnikov on the occasion of his 75th birthday Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 48-63.     

An algorithm for the electromagnetic scattering due to an axially symmetric body with an impedance boundary condition

Let B be a body in R³ and let S denote the boundary of B. The surface S is described by $\text{S = {(x, y, z): (x}{}^2\text{+ y}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= f(z), ?1 ? z ? 1}}$, where f is an analytic function that is real and positive on (?1, 1) and f(±1) = 0. An algorithm is described for computing the scattered field due to a plane wave incident field, under Leontovich boundary conditions. The Galerkin method of solution used here leads to a block diagonal matrix involving 2M + 1 blocks, each block being of order 2(2N + 1). If, e.g., N = O(M²), the computed scattered field is accurate to within an error bounded by $\text{Ce}{}^{\mathrm{<mtext>?cN</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where C and c are positive constants depending only on f.     

Approximation of convex bodies by axially symmetric bodies

Let be an arbitrary planar convex body. We prove that contains an axially symmetric convex body of area at least . Also approximation by some specific axially symmetric bodies is considered. In particular, we can inscribe a rhombus of area at least in , and we can circumscribe a homothetic rhombus of area at most about . The homothety ratio is at most . Those factors and , as well as the ratio , cannot be improved.

     

Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S^′ that contains C. More precisely, for any >0, we find an axially symmetric convex polygon QC with area |Q|>(1−)|S| and we find an axially symmetric convex polygon Q^′ containing C with area |Q^′|<(1+)|S^′|. We assume that C is given in a data structure that allows to answer the following two types of query in time T_C: given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C∩ℓ. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T_C=O(logn). Then we can find Q and Q^′ in time O(^−1/2T_C+^−3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(^−1/2T_C).     

Graphical solution of axially symmetric problems of plastic flow

Résumé Une méthode graphique de solution des problèmes dans le cas de symétrie axiale a été proposée pour des corps rigides, parfaitement plastiques, obéissants au critère d'écoulement de Coulomb-Tresca et à l'hypothèse du potentiel plastique. Deux cas ont été considérés: d'une part des régimes de Haar-Kármán pour lesquels la contrainte circonférencielle est égale à l'une des contraintes principales contenues dans le plan axial, et d'autre part des régimes duor lesquels l'une des vitesses de déformation principales dans le plan axial est nulle.     

An axially symmetric contact problem for a half-space with an elastically reinforced cylindrical cavity

We consider the pressure of a plate on a half-space with a round cylindrical cavity. The surface of the cavity is reinforced by elastic elements that are modeled by very general operators. The problem is reduced to a Fredholm integral equation of second kind. A detailed study is made of the case of reinforcement described by the Winkler law. An approximate solution is obtained in the form of the asymptotics with respect to the radii of the plate and the cavity.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 80–85.     

Thermoelastic state of inhomogeneous plates under axially symmetric heating

Under the assumption that the Kirchhoff-Love hypothesis for nonuniform plates is satisfied, the author introduces relations for thermal stresses due to forces and momenta. In particular, the case is studied in which the plate contains cylindrical completely penetrating inclusions.Translated from Matematicheskie Metody i Fiziko-Mekhanicheski Polya, No. 29, pp. 31–34, 1989.     

Longitudinal-vortex unit vector fields from the class of axially symmetric fields

In the paper, we construct unit vector fields belonging to the class of smooth axially symmetric fields that are longitudinal-vortex in the whole space ?³.     

Computation of thermal residual stresses in optic fibers

We propose an analytic method of determining all the components of the residual stress tensor in optic fibers. The method is based on solving a plane problem for a cylindrical structure with singular stresses. We obtain closed-form solutions of the problem in the case of a piecewise-constant distribution of free deformations that model the presence of inclusions in the fiber with different values of the thermal coefficient of expansion. We also consider inclusions with cross sections in the shape of a circle, a central ellipse and a central annular sector. We describe the results obtained on this basis in the computation of residual thermal stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 79–83.