Analysis of the steady motions of the tippe top

The motion of the tippe top on a horizontal plane is considered taking into account sliding friction within the Contensou model. The tippe top is modelled by two spherical segments rigidly joined by a rod directed along the common axis of symmetry of the segments. The dimensions of the spherical segments and the rod are chosen so that, as the axis of symmetry deviates from the upward vertical, the tippe top is supported on the plane at a point on one segment up to a certain critical value and at a point on the other segment at larger deviations (at points on both segments at the critical value). The motion of the tippe top is described by different equations in different regions of configuration space, and the motion is accompanied by impacts on the boundary of these regions. An effective potential of the system is constructed, and the type of its critical points is investigated. Poincaré–Chetayev bifurcation diagrams and generalized Smale diagrams are constructed for steady motions. Plots of the steady-state precessional motions have a discontinuity on the boundary between the regions indicated.     

The choice of spectral element basis functions in domains with an axis of symmetry

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.     

On the development of a wake vortex in inviscid flow

The evolution of an initial perturbation in an axisymmetric subsonic normal inviscid gas flow through a pipe is directly simulated. The basic (unperturbed) flow has a zero radial velocity component, while its axial velocity component (along the axis of symmetry) increases or decreases linearly with the radius. The perturbation is specified as a swirl (rotation about the axis) with a positive or negative velocity vanishing on the central axis and the lateral surface. Irrespective of its direction, the swirl gives rise to a steady-state vortex carried by the flow. It shape is spherical (contiguous to the rotation axis) or circular (sliding along the impermeable lateral surface).     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

Examples of stationary Euler flows

We construct concrete solutions of the incompressible Euler equation in a domain of the cylinder type, and the compressible Euler equation outside a ball (or a rotationary symmetric body) with a constant speed at infinity along the axis of symmetry.     

Hertzian contact in the presence of a Mindlin force

The smooth axisymmetric contact between a rigid spherical punch and an isotropic elastic halfspace is perturbed by an internally applied Mindlin force which is located along the axis of symmetry. The paper presents certain closed form solutions to this unilateral contact problem.     

Asymptotic behavior of vortices in axisymmetric flow without swirl

We investigate the confinement properties of bounded, nonnegative, compactly supported vortices of axisymmetric incompressible Euler flows without swirl. We show that along one direction of the symmetry axis, its support can grow no faster than O[(t log t)^1/2]. The rate at which it approaches the symmetry axis is also estimated. Together with the result of Maffei–Marchioro on the radial growth rate of the support, it is contained in a slowly expanding tubular region. The techniques of the above‐mentioned authors, Iftimie–Lopes–Nussenzveig and Iftimie–Sideris–Gamblin, are used. Copyright © 2008 John Wiley & Sons, Ltd.     

Conformal mapping onto numerable polygon with double symmetry

We consider simply connected polygonal subdomains of half-plane type having countable many vertices; in addition, they possess the symmetry of transfer along the real axis by 2π and the symmetry with respect to the vertical straight line w = π + iv, v ∈ ?. We establish that conformal mapping of the half-plane onto such domains are represented by integrals of Schwarz-Christoffel type. The proof is based on the Riemann-Schwarz symmetry principle and Schwarz-Christoffel’s classical formula. In conclusion we consider some examples.     

Flow around an ellipsoid of revolution in a harmonic coaxial vector field

We give an integral equation defining a coaxial magnetic field near the surface of a superconductive axisymmetric body and the velocity of the liquid near the surface of an axisymmetric body situated coaxially to the flow of an ideal liquid. Using this equation in the case when the axisymmetric magnetic field before the placement of an ellipsoid of revolution coaxially to the field changed along the axis by a polynomial law, we analytically define the densities of the surface current and the force with which the magnetic field acts on the ellipsoid. Also the velocity of the liquid is determined near the surface of the ellipsoid of revolution and the force acting on the ellipsoid placed coaxially in the flow of an ideal liquid when the velocity of the liquid before the placement of the ellipsoid changed along the axis of symmetry by a polynomial law.     

A shuffling theorem for reflectively symmetric tilings

The author and Rohatgi recently proved a ‘shuffling theorem’ for doubly-dented hexagons. In particular, they showed that shuffling removed unit triangles along a horizontal axis in a hexagon changes the tiling number by only a simple multiplicative factor. In this paper, we consider a similar phenomenon for a symmetry class of tilings, namely, the reflectively symmetric tilings. We also prove several shuffling theorems for halved hexagons.     

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.     

Hamilton算子的一类n次数值域的对称性

以Hamilton算子的数值域为基础,研究了一类算子的二次数值域关于实轴,虚轴的对称性.此外从α-J-自伴算子的n次数值域关于过原点直线对称出发,得到了有界Hamilton算子的一类n次数值域关于虚轴的对称性.     

The fracture of brittle and ductile crystals. Force and deformation criteria

The growth and branching of sharp cracks in ideal single crystals are investigated. Neuber-Novozhilov force and deformation criteria are proposed for the branching of sharp cracks; these criteria describe the brittle, quasibrittle, quasiductile and ductile behaviour of materials on fracture. For internal cracks, simple relations are obtained that describe the branching of cracks when the Coulomb-Mohr single-crystal theoretical strength curves are known for a generalized stress state. The possibility of multiple branching of cracks is found, which is linked to the multiplicity of the eigenvalues on loss of stability of the system. It is established that, for ideal single crystals, the principle of local symmetry is satisfied in the vicinity of the crack tip if the axis of symmetry of the crystal coincides with the axis of the crack. When there are asymmetrical disturbances of the atomic lattice in the vicinity of the crack tip, or when the axis of symmetry of the single crystal does not coincide with the crack axis, the principle of local symmetry is not satisfied.     

The relativistic theory of chemical binding

Starting from Breit’s relativistic equation for a system of two electrons, it is shown that for a hydrogen molecule (or for a system of two electrons moving in a field of cylindrical symmetry) the component of the total angular momentum (J _x ) along the axis of the molecule (axis of symmetry) is a constant of motion. Thus every eigenstate of the system is simultaneously an eigenstate of J _x also, and a state of the system will specify, besides its energy, only the eigenvalue of the component of the angular momentum parallel to the axis of symmetry. The form of the four large components of the wave function relating to their dependence on the azimuthal co-ordinates has been given. The case of Russel-Saunders approximation has been considered in detail and the nature of the components of the wave function for the singlet and triplet states has been discussed. It is shown that the wave function for the ground state of the hydrogen molecule could be expressed as a sum of a set of symmetric functions of which the first term is the Heitler-London function, and that the wave function for a triplet state should be a superposition of anti-symmetric molecular orbitals. It is shown that relativistic theory brings about in a natural manner the facts relating to the ground state of the molecules C₂ and O₂. Finally, some remarks are made concerning the case of molecules for which the spinorbit and the spin-spin couplings are strong.     

Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.     

Diffraction of an electromagnetic signal on an ideally conducting cone with a dielectric coating

The paper examines the scattering of a Gaussian video signal by an ideally conducting circular cone coated with a dielectric layer. The problem is solved by the method of integral equations for monochromatic waves at low and resonance frequencies with subsequent Fourier transformation to the time domain. The response is investigated with signals propagating along the symmetry axis from the direction of the apex and from the direction of the base for dielectric layers of various permittivities. Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 46–52, 1998.     

对角无穷维Hamilton算子点谱关于实轴的对称性

在不同条件下得到对角无穷维Hamilton算子点谱的两个组成部分σp（A）与σp（-A^＊）关于实轴对称的充分必要条件.以此为基础,完全刻画了对角无穷维Hamilton算子点谱关于实轴的对称性.     

Calculating the electric charges that shield from an external coaxial electric field on the surface of a conducting axially symmetric body

A method of solving a one-dimensional integral equation for finding charges on the surface of a conducting axially symmetric body is given. For the case of an ellipsoid of rotation in an electric field with polynomial values on the axis of symmetry, an exact solution is obtained. The axis of symmetry of the body and the axis of the external field coincide. A numerical algorithm based on a combination of a projective method and a method of iterative regularization for solving a Fredholm equation of the first kind is proposed. The projectors are chosen as B-splines. The charges calculated for an ellipsoid of rotation are close to the analytical ones.