Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

图的光滑支架分解

设G是一个图,A为其边集的子集。G的一个支架分解是（G－A,A）,其中G－A是去掉A后的连通图,G的一个光滑支架分解是适合下列条件的支架分解：（1）G－A的每一叶具有连通余树;（2）G－B（G－A）割边集为A,其中B（G－A）为G－A的割边集。本文给出了求一个图的光辉支架分解的一个有效算法。     

Motion of a container as a nonholonomous system in a curved section of a horizontal pipeline

The problem of the motion of a container in a curved section of a horizontal pipeline is solved using second-order Lagrange equations in the presence of nonholonous couplings. The special case of the motion of a container in a circular curve is examined.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 90–95, 1987.     

The rolling of a wheel with a pneumatic tyre on a planet

A model of a pneumatic tyre as a system with an infinite number of degrees of freedom is proposed, when its surface is represented by the deformed surface of a torus. Using a number of hypotheses a functional of the potential energy of the deformations of the tyre is obtained as a function of the deformations of its tread. A complete system of equations of motion is obtained, assuming that the wheel rolls without slipping in the area of contact of the tread with the plane, with respect to the previously unknown part of the tread. In two special cases of the rolling of a wheel with breakaway and on a banking, all the characteristics of the motion (the contact area, the tyre deformation, and the forces and moments applied to the disc of the wheel) are obtained.     

Fast computation of a rational point of a variety over a finite field

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.

     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

On a convective flow of a binary mixture in a vertical layer

Under study is an invariant solution of the equations of thermal diffusive convection which describes a stationary process of a binary mixture flow in a vertical layer under the action of the pressure gradient and the buoyancy force that depends nonlinearly on temperature and concentration. Some general properties of this solution are established and an existence theorem is proved. Analysis of the numerical solution of the problem is carried out in the cases of a power-law and exponential dependence of the buoyancy force on its argument.     

On moving a sofa around a corner

A necessary condition is given for a region of the plane to have the greatest possible area of any region able to move around a right-angled corner in a hallway of unit width. A region is constructed, with area 2.2195... and bounded by 18 analytic pieces, which satisfies this condition. It is conjectured that this is the unique region of maximum area.     

Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

Locating a robber on a graph

Consider the following game of a cop locating a robber on a connected graph. At each turn, the cop chooses a vertex of the graph to probe and receives the distance from the probe to the robber. If she can uniquely locate the robber after this probe, then she wins. Otherwise the robber may either stay put or move to any vertex adjacent to his location other than the probe vertex. The cop’s goal is to minimize the number of probes required to locate the robber, while the robber’s goal is to avoid being located. This is a synthesis of the cop and robber game with the metric dimension problem. We analyse this game for several classes of graphs, including cycles and trees.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

Stressed state of a rock mass around a working with a relief cavity

The stressed state of a rock mass in the vicinity of a preparatory working protected by a relief cavity is studied. A numerical study is made of the stress field and the relief zone, and the effect of length and slope of a slot on formation of the relief zone is clarified.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 31–36, 1989.     

Algorithms for projecting a point onto a level surface of a continuous function on a compact set

Given an equation f(x) = 0, the problem of finding its solution nearest to a given point is considered. In contrast to the authors’ previous works dealing with this problem, exact algorithms are proposed assuming that the function f is continuous on a compact set. The convergence of the algorithms is proved, and their performance is illustrated with test examples.     

Torsional Oscillations of a Punch on a Layer Bonded to a Half-Space Containing a Cylindrical Cavity

A previously published algorithm [9] is implemented in application to the Reissner–Sagoci problem for a layer bonded to a half-space containing a cylindrical cavity. The influence of the mechanical and geometrical parameters of the layer and the half-space on the amplitude-frequency response curves of the punch oscillations is analyzed. Practical applications of the results are proposed for ensuring the seismic isolation of buildings on the investigated foundation in the presence of dynamic torsional excitations.     

A recognition problem for a vector alphabet generating a sequence with a quasiperiodic structure

An off-line recognition problem is analyzed for a vector alphabet generating sequences with quasiperiodic vector fragments that coincide with alphabet vectors. It is shown that the solution of this problem reduces to solving a special optimization problem. It is proved that the problem considered is solvable in polynomial time, and an algorithm for its exact solution is justified. The algorithm ensures the maximum likelihood recognition of a vector alphabet for the case of additive noise which is a Gaussian sequence of independent random values having an identical distribution.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Irradiation of a cone by a magnetic dipole

Th electromagnetic field produced by a magnetic dipole in thepresence of a perfectly conducting cone of arbitrary cross-sectionis determined. The solution is used to find out how a currenton the cone travelling towards the apex is reflected. Some valuesof the reflection coefficient are calculated. In particular,it is shown that there is a sort of resonance with the reflectionincreasing significantly as the cone approaches a plane.     

Factoring a Lie Group into a Compactly Embeddedand a Solvable Subgroup

 Let G be a real connected Lie group. A subgroup K is called compactly embedded if the closure of Ad(K) is compact in Aut(). If K is, in addition, maximal with respect to this property, then there exists a solvable subgroup S containing the nilradical such that and is the one-component of the center of G. (Received 1 June 1999; in revised form 28 December 1999)     

Calculating a minimal sphere containing a polytope defined by a system of linear inequalities

We will propose an algorithm for calculating a minimal sphere containing a polytope defined by a system of linear inequalities in low dimensional Euclidean space. This algorithm is a straightforward application of the algorithm for maximizing a convex quadratic function over a polytope. It will be shown that this algorithm successfully generates a minimal sphere when the dimensions of the underlying space is up to five.International Digital Communication Inc.