The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

On the motion of a mechanical system inside a rolling ball

We consider a mechanical system inside a rolling ball and show that if the constraints have spherical symmetry, the equations of motion have Lagrangian form. Without symmetry, this is not true.     

On the lifetime of a conditioned Brownian motion in the ball

Consider the Brownian motion conditioned to start in x, to converge to y, with , and to be killed at the boundary ∂Ω. Here Ω is a bounded domain in Rn. For which x and y is the lifetime of this Brownian motion maximal? One would guess for x and y being opposite boundary points and we will show that this holds true for balls in Rn. As a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse Dirichlet eigenvalues.     

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

We consider the free boundary problem for the Navier–Stokes equations governing a nonstationary motion of a layer of a viscous incompressible liquid that covers the surface of a rigid ball rotating around a fixed axis with constant angular velocity ω. The liquid is subject to the gravitation force generated by the mass of the ball. The self-gravitation forces between the liquid particles and capillary forces on the free surface are not taken into account. We consider the problem of stability of the regime of the rigid rotation of the liquid with the same angular velocity and prove that it is stable if |ω| is less than a certain constant. Bibliography: 10 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 91–145.     

Smooth solutions for the motion of a ball in an incompressible perfect fluid

In this paper we investigate the motion of a rigid ball surrounded by an incompressible perfect fluid occupying RN. We prove the existence, uniqueness, and persistence of the regularity for the solutions of this fluid-structure interaction problem.     

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this setting.Both authors were supported in part by NSF Grant Number DMS-9024961.     

The space of conditional systems is a ball

LetS be a nonempty finite set. The space Δ(S) of conditional systems onS, as defined by Myerson (1986), is shown to be homeomorphic to the closed unit ball in (#S-1)-dimensional Euclidean space.

     

Extreme harmonic functions on a ball

In order to show that one can recapture the Riesz-Herglotz theorem from the Krein-Milman theorem, we determine directly the set of extreme points of the convex set of positive harmonic functions on the unit ball (normalized by 1 at the origin). The characterization is obtained using standard facts from abstract analysis combined with a minimum of very basic results on harmonic functions.     

The solution of a mildly singular integral equation of the first kind on a ball

We consider the equations $$\int_{\left| y \right| \leqslant 1} {\frac{{F(y)}}{{\left| {x - y} \right|^\lambda }} dy = G(x)(\left| x \right| \leqslant 1)} $$ where x and y ∈ E^p, p ≥ 3 and λ < p. For λ ∈ (p-2,p) we show that this equation has at most one integrable solution which if G is twice differentiable actually exists and is, in fact, given by an explicit formula in terms of integral operators acting on G and its derivatives. When λ ≤ p-2 and λ ≠ ?2j (j=0,1,?) the equation also has at most one integrable solution for which, assuming it to exist and G to be sufficiently smooth, there also is an explicit formula; in this situation, though, the explicit formula does not usually provide an integrable solution, because, in general, such solutions do not exists when λ ≤ p-2 and λ ≠ ?2j (j=0,1,?), no matter how smooth G is. In the remaining case λ = ?2j (j=0,1,?), neither uniqueness nor existence holds for solutions of the equation.     

The motion of a railway wheelset

The rolling of a railway wheelset along rails without slipping is investigated taking the creep hypothesis into account. The wheelset is represented by two cones that have a common base, and the rails are represented by two circular cylinders with parallel axes. The kinematic characteristics of the unperturbed rolling motion of the wheelset, which occurs when the centre of mass moves along a straight line, and of the perturbed motion, which occurs when the centre of mass of the wheelset describes a sinusoidal trajectory, are determined. The constraint reactions are found for the motions investigated up to small second-order values of the perturbed variables. When the elastic properties of the material in the contact area are taken into account, the creep hypothesis is used, averaging over the fast variables is employed, and the value of the critical speed, above which the rectilinear rolling of the wheelset becomes unstable, is found using averaged equations. In the latter case a periodic mode with two time intervals when the wheel flanges come into contact with the rails is investigated. The reaction force, the work of the dry friction force, and the moment of the active forces needed to maintain the periodic mode are found at the flange/rail contact point within the dry friction model. The boundaries of the stability regions, the parameters of the periodic mode and the moment of the resistance forces as functions of the problem parameters are determined from the formulae obtained by analytical methods.     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

Spectral analysis of a generalized buckling problem on a ball

In this paper, the spectrum of the following fourth order problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \varDelta ^2 u+\nu u=-\lambda \varDelta u &{}\quad \text {in } D_1,\\ u=\partial _r u= 0 &{}\quad \text {on } \partial D_1, \end{array}\right. } \end{aligned}$$

where \(D_1\) is the unit ball in \({\mathbb {R}}^N\), is determined for \(\nu < 0\) as well as the nodal properties of the corresponding eigenfunctions. In particular, we show that the first eigenvalue is simple and that the corresponding eigenfunction is radial and (up to a multiplicative factor) positive and decreasing with respect to the radius. This completes earlier results obtained for \(\nu \geqslant 0\) (see Coster et al. in Positivity 19:843–875, 2015) and for \(\nu <0\) (see Laurençot and Walker in J Anal Math 127:69–89, 2014).

     

Stability of a rotating ball

Influece of form errors of a chamber filled with a liquid on the movement and stability of a ball, rotating in the chamber, is studied. Two cases of the influence of the chamber form errors on the forces, acting on the ball, are defined. The first case describes the situation when limitations on the rotor shift are not imposed and disturbances of the chamber form are set by spherical harmonics not above the first order. In the second case disturbance of a chamber form is arbitrary and the rotor shift is supposed small. A rising here diflective moment tends to direct the angular speed vector along the small semiaxis of the ellipsoid, i.e., a stable position of the rotor appears.     

The dynamics of the chaplygin ball with a fluid-filled cavity

We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.