A PTAS for Min-<Emphasis Type="Italic">k</Emphasis>-SCCP in Euclidean space of arbitrary fixed dimension

The paper deals with the problem of coordinated control for a flock of control systems that are to realize a joint motion towards a target set under collision avoidance. We consider one of its subproblems, which is formulated as follows. During the motion to the target, the members of the group are obliged to lie within a virtual ellipsoidal container, which realizes a reference motion (a “tube”). The container avoids obstacles, which are known in advance, by means of reconfigurations. In response, the flock must rearrange itself within the container, avoiding collisions between its members. The present paper concerns the behavior of the flock within the container, when the flock coordinates its motions with the evolution of the container.     

微重力环境中摆转运动引起的流体晃动力及力矩

本文以部分充液的贮箱系统作为研究对象，用数值方法计算研究了在扰动力作用下流体系统的晃动力学问题。文中给出了由于系统摆转而在流体中引起的扰动加速度的数学表达式。晃动力的计算是在与贮箱固结在一起的非惯性坐标系中，对依赖于初始与边界条件的三维非稳态偏微分方程组进行数值求解，计算了在伴随系统的摆转运动而在流体中产生的扰动加速度作用下，流体施加于贮箱上的力与力矩的波动规律。     

The optimal periodic motions of a two-mass system in a resistant medium

The rectilinear motions of a two-mass system, consisting of a container and an internal mass, in a medium with resistance, are considered. The displacement of the system as a whole occurs due to periodic motion of the internal mass with respect to the container. The optimal periodic motions of the system, corresponding to the greatest velocity of displacement of the system as a whole, averaged over a period, are constructed and investigated using a simple mechanical model. Different laws of resistance of the medium, including linear and quadratic resistance, isotropic and anisotropic, and also a resistance in the form of dry-friction forces obeying Coulomb's law, are considered.     

The Toda rarefaction problem

In the Toda shock problem (see [7], [11], [8], and also [3]) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas. We make use of the Riemann-Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 2 |a| of the driving particle is sufficiently large (|a| > 1), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops. © 1996 John Wiley & Sons, Inc.     

Solvability of three-dimensional problems with a free boundary for a stationary system of Navier-Stokes equations

One proves the solvability of the boundary-value problem for the Navier-Stokes system, describing the stationary motion of a heavy, viscous, incompressible, capillary fluid, filling partially a certain container.     

A numerical investigation of energy dissipation with a shallow depth sloshing absorber

A liquid sloshing absorber consists of a container, partially filled with liquid. The absorber is attached to the structure to be controlled, and relies on the structure’s motion to excite the liquid. Consequently, a sloshing wave is produced at the liquid free surface within the absorber, possessing energy dissipative qualities. The behaviour of liquid sloshing absorbers has been well documented, although their use in structural control applications has attracted considerably less attention.     

On an initial boundary-value problem for the equation of magnetohydrodynamics with the hall and ion-slip effects

This paper is concerned with the three-dimensional initial boundary-value problem for the equations of magnetohydrodynamics with additional nonlinear terms stemming from a more general relationship between the electric field and the current density. The problem governs the motion of a viscous incompressible conducting liquid in a bounded container with an ideal conducting surface. The existence of a solution which is close to a certain basic solution is proved. The solution is found in the anosotropic Sobolev spaces W _p ^2,1 with p>5/2. The proof relies on the theory of general parabolic initial boundary-value problems. Bibliography: 16 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 167–184. Translated by V. A. Solonnikov.     

Non uniqueness and uniqueness of capillary surfaces

It is shown that for any gravity fieldg and contact angle, a containerC can be found which can be partially filled with liquid in a continuum of distinct ways, so as to obtain a one parameter familyF of capillary surface interfaces, no two of which are mutually congruent, and all of which bound the same liquid volume and yield the same mechanical energy. This answers affirmatively a question raised by Gulliver and Hildebrandt, who obtained such a container in the caseg=0,=/2. For a particular configuration inF, the second variation of energy is calculated and it is shown that-at least for small g- it can be made negative. As a consequence, a rotationally symmetric container deviating arbitrarily little from a circular cylinder is characterized, so that an energy minimizing configuration filling half the container exists but cannot be symmetric. Finally, a condition on rotationally symmetric container shapes is given, for the existence of a unique disk-type symmetric stationary surface. In the particular case of a spherical container in zero gravity, this surface is unique and minimizing among all disk-type surfaces with the given contact angle and enclosed volume.The major part of this work was completed while the author was visiting at the Max-Planck-Institut in Bonn. The work was also supported in part by grants from the Fulbright Commission, from the National Science Foundation, and from the National Aeronautics and Space AdministrationI wish to thank K. -S. Chua, P. Concus, M. Grüter, and R. Gulliver for helpful comments and discussions.     

Sur les petites oscillations planes d'un liquide dans un container symétrique en apesanteur

Résumé L'auteur étudie le problème bidimensionnel des oscillations d'un liquide dans un container symétrique en apesanteur. II donne l'équation fonctionnelle du mouvement et prouve l'existence des fréquences propres du liquide sous des conditions convenables portant sur l'angle de contact de la ligne libre et de la paroi du container et la courbure de celle-ci.

---

The author studies the two-dimensional problem of the oscillations of a liquid in a symmetrical container under zero gravity. He gives the functional equation of motion and proves the existence of the eigenfrequencies of the liquid under suitable conditions between the contact angle of the free line and of the wall of the container and the curvature of this.

     

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ₁ = {x:x ₂ = 0} and the contour (V₁), consisting of two semilines ⁽¹⁾ = {x:x ₁<–1,x ₂ = h₀} ⁽²⁾ = {x:x ₁ = 0,x ₂h₀+1} joined by a smooth curvel ⁽³⁾. One assumes that the motion is due to a nonzero flow and by the motion of the lower wall ₁ with a constant velocity R0. The unique solvability of this problem is proved for small R and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 174–179, 1981.In conclusion, the author expresses his deep gratitude to V. A. Solonnikov for his guidance.     

Rotational polygon overlap minimization and compaction

An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P₁, P₂, P₃, …, P_k in a container polygon Q, translate and rotate the polygons to diminish their overlap to a local minimum. A (local) overlap minimum has the property that any perturbation of the polygons increases the overlap. Overlap minimization is modified to create a practical algorithm for compaction: starting with a non-overlapping layout in a rectangular container, plan a non-overlapping motion that diminishes the length or area of the container to a local minimum. Experiments show that both overlap minimization and compaction work well in practice and are likely to be useful in industrial applications.     

Visualization of boundary layers in a sectioned centrifuge

We are interested in the flow of a suspension in a rotating, conical cavity with a dividing barrier that blocks any azimuthal motion around the axis. In order to simplify the experiment, the flow of a homogeneous fluid, slightly denser than the bulk fluid filling the container, has been used. The theoretical predictions of a previous paper have been tested, in particular those relating to the intense current on the dividing barrier that was found. This current has been visualized. A quantitative comparison between theory and experiment is made. The qualitative findings of the theory that apply to this experiment are completely verified. The quantitative agreement is fair.     

Measurements of transient flow fields driven by a discontinuously applied rotating magnetic field

This experimental study considers the transient liquid metal flow which is generated inside a cylindrical container by discontinuously applying a rotating magnetic field (RMF). The focus is on the fluid motion arising from the impulsive spin–up from the resting state. The ultrasonic Doppler velocimetry (UDV) has been used to determine profiles of the fluid velocity in the ternary alloy GaInSn. The azimuthal and vertical velocity components have been measured allowing for an analysis of both the primary, swirling flow and the secondary flow in the radial–meridional plane. The experimental results show an excellent agreement with recently published numerical results. The investigations reveal that the recirculating flow in the radial–meridional plane undergoes characteristic oscillations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The movement of bubbles in a vibrating liquid

The problem of movement of bubbles in a liquid in a rigid vertical cylindrical container with a rigid base is considered. The container vibrates vertically with a displacement amplitude Δ and an angular frequency ω with the acceleration g due to gravity.     

A Grain of Dust Falling Through a Sierpinski Gasket

In this paper we analyze the downward random motion of a particle in a vertical, bounded, Sierpinski gasket G, where at each layer either absorption or delays are considered. In the case of motion with absorption the explicit distribution of the position of the descending particle in the pre-gasket Gn is obtained and the limiting case of the Sierpinski gasket discussed. For the delayed downward motion we derive a representation of the random time needed to arrive at the base of Gn in terms of independent binomial random variables （containing the contribution of delays at different layers with different geometrical structures）.     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

Summary. The super-Brownian motion X ^ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction. Received: 21 November 1996 / In revised form: 31 March 1997     

On a team control problem under obstacles

The paper deals with the problem of coordinated goal-oriented target control for a group of control systems that are to realize a joint movement towards a given target set under collision avoidance. The members of the group are obliged to lie inside a virtual ellipsoidal container, which realizes a reference motion while also avoiding external obstacles specified in advance. We describe a general solution scheme based on decomposing the main problem into auxiliary subproblems, for which we indicate solution methods as well as the necessity of coordinating these solutions at the final stage.     

The modulus of non-differentiability of a Brownian motion in l p

Small ball probability is estimated for a Brownian motion in l _p. As an application we establish the modulus of non-differentiability of a Brownian motion in l _p. This revised version was published online in June 2006 with corrections to the Cover Date.     

Propagation of disturbances in a flexible extensible filament under longitudinal accelerated motions in a fluid

We seek a solution of the linearized equation of motion of a flexible extensible filament in a fluid in the form of an expansion in eigenfunctions of a boundary-value problem. For a uniformly accelerated motion and for motion accelerated according to a hyperbolic tangent law we find the exact solutions. For other forms of accelerated motion we propose a numerical solution of the initial inhomogeneous problem. We carry out an analysis of the solutions obtained. It is found that the first peak of the tension depends only weakly on the resistance of the fluid, but strongly on the acceleration parameters. The natural vibrations damp out more rapidly both as the resistance increases and as the acceleration increases. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 104–110.     

Approach to equilibrium of a rotating sphere in a Stokes flow

We study the unsteady rotary motion of a sphere immersed in a Stokes fluid. The equation of motion for the sphere leads to an integro-differential equation, and we are interested in the asymptotic behavior in time of the solution. Preparing initially the system (sphere + fluid) as a stationary state, we prove that the angular velocity of the sphere slows down with a law t ^−3/2 if no other forces than the one exerted by the fluid act on the sphere, while if the sphere is subject also to an elastic torque the asymptotic behavior of the angular position of the sphere is t ^−γ , with γ = 5/2 if the initial angular velocity is zero, γ = 3/2 otherwise. This behavior is due to the memory effect of the surrounding fluid. We discuss briefly other initial preparations of the system.