On the Diameter of Plane Curves

Recently, differential geometric properties of embedded projective varieties have gained increasing interest. In this note, we consider plane algebraic curves equipped with the Fubini--Study metric from₂ () and give an estimate for the diameter in terms of the degree, initiated in a paper by F. A. Bogomolov.     

旋转流体中圆球沿旋转轴运动产生的扰动

本文用线化理论分析了整体旋转的理想流体中有一个圆球沿旋转轴作匀速运动时流体的扰动,基于旋轴对称流动的假设导出了决定运动稳定性的扰动压力方程和扰动流函数方程.用简正模法分析了扰动流函数方程,得出了非平凡中性扰动的波数与波速必须满足的约束条件,并求出了扰动的精确表达式.文中得出结论,中性扰动共有三种可能的形式.     

A Note on Modifications to the Givens Plane Rotation

It has recently been shown how to perform plane rotations withoutsquare roots and with a saving in multiplications. (See forexample, Gentleman, 1973). The purpose of this note is to presentthe method in a simple but general manner which also suggestsother modifications.     

关于E~n中有限质点系的任意维转动惯量平面

本文给出了 n维欧氏空间 En 中有限质点系的任意维转动惯量平面 ,指出该平面必通过质点系的质心 ,且由质点系相随矩阵的若干最大特征值所对应的特征向量所确定 .     

The Rotation of a Gravitating Sphere in a Monatomic Gas,II

The purpose of this work is to study the fluid motion caused by the high speed rotation of a gravitating sphere in a monatomic gas. It has been possible to find a stable steady solution only for very small Prandtl number, which can be interpreted to mean an optically thick gas. The flow is characterized by a flat radial jet in the equatorial plane and a viscous boundary layer on the spherical surface which, in some cases, lies beneath a thermal boundary layer. That the outer region must be hydrostatic puts very stringent constraints on the associated velocity field which necessitate still another boundary layer on the sphere. This last layer is shown to be unstable to small disturbances in certain temperature ranges. Finally, a similar solution that exists for order one Prandtl number must be disregarded because this last boundary layer is always unstable.     

The Rotation of a Gravitating Sphere in a Monatomic Gas,I

The flow resulting from the steady rotation of a gravitating sphere in a monatomic gas is studied in two special cases. If the rotation is slow, then the motion consists of a swirling flow whose streamlines are concentric circles around the rotation axis, and a slower meridional flow whose structure depends on the strength of the gravitational field. This meridional flow is investigated in detail for small and large field strength. In the latter case, the scale height is small compared to the sphere radius, and most of the fluid is in a thin recirculating layer on the surface. However, the mathematics imposes a severe restriction on the rotation speed if the field is large, which leads to a more general investigation of the flow for very large field strength, i.e. for a thin “atmosphere”. Compressibility effects can be quite sizeable in this final solution. Though a thin recirculating surface layer still exists in only slightly modified form, the fluid outside this layer is much less tenuous than for the case when the rotation is slow. The flow in the region outside the boundary layer involves solution of two quasilinear elliptic equations.     

Closed-Loop Control of the Motion of a Sphere Rolling on a Moving Horizontal Plane

A uniform sphere is rolling without slipping on a horizontal plane. The motion of the sphere is controlled via the control of the acceleration of the plane. At the time t=0, the sphere and the plane are stationary and the center of the sphere is located at a point A in the plane. Given a time interval [0, t _f], the problem dealt with here is: Find a closed-loop strategy for the acceleration of the moving plane such that, at the time t=t _f, the plane and the sphere will be nearly at rest and the center of the sphere will be in a given neighborhood of the origin. By introducing the concept of path controllability, a closed-loop strategy for the solution of the above-mentioned problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

单位球面中具有3个不同 Blaschke 特征值的 Blaschke 平行子流形

Blaschke张量A是单位球面S~n中子流形的M?bius微分几何的一个基本不变量,而A的特征值称为Blaschke特征值.作者研究了S~n中具有平行Blaschke张量的子流形(简称为Blaschke平行子流形).主要结果是对S~n中具有3个不同Blaschke特征值的Blaschke平行子流形进行了完全的分类.     

带有微弱章动的圆球在粗糙水平面上的运动

对于圆球在粗糙水平面上的运动,在文[1]中,作者忽略了章动,得到了近似解析解。本文在此基础上给出了有章动情况下的控制方程。通过求解这些方程,证明文[1]关于接触点速度的结论在有章动时仍然正确。还得到其它一些有趣的结果,例如:球心和接触点的速度与球的自转角速度和章动角速度有一定联系;球心和接触点的速度的方向具有不变性。在进一步假设微弱章动的情况下,文中得到近似解析解,从而证明文[1]结果的正确性。     

Classification of Unknotted Ribbons in the Plane and on the Sphere

The aim of this paper is to classify unknotted ribbons in the plane and on the sphere up to regular isotopy.

     

On a Problem Related to Sphere and Circle Packing

The paper proves that a set which contains spheres centeredat all points of a set of Hausdorff dimension greater than 1must have positive Lebesgue measure. It also proves the correspondingresult for circles, provided that the set of centers has Hausdorffdimension greater than 3/2.     

On a Property of Functions on the Sphere

According to the Knaster conjecture, for any continuous function $f:S^{n - 1} \to \mathbb{R}$ and any $n$ -point subset of the sphere $S^{n - 1}$ , there exists a rotation mapping all the points of this subset to a level surface of the function $f$ . In the present paper, this conjecture is proved for the case in which ${n = p^1 }$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.     

Stokes Flow Between Parallel Plates due to a Trausversely Moving End Wall

Consider the two-dimensional Stokes flow in a semi-infinitestrip of viscous fluid due to a transverse velocity of its endwall. This is an idealization of the motion in the asthenospheredue to a subducting slab. We give an iterative scheme for calculatingthe flow, prove existence and uniqueness, and obtain numericalapproximations, using developments of Teodorescu's method andan approximate solution of the infinite matrix equation arisingtherein. (Simple truncation to finite order is not good enough.)Our boundary conditions give rise to singularities which preventother known methods (including the original form of Teodorescu's)from working. The solution reveals how surprisingly good anapproximation its asymptotic forms near the corners and at infinityare in most of the flow field.     

On the Number of Tubes Touching a Sphere or a Tube

A problem is formulated about how many unit-radius tubes can touch a ball of given radius from the outside and from the inside. Upper bounds for the maximum numbers of contacts are obtained for both interior and exterior contacts. It is also shown that the maximum number of unit-radius tubes touching the same orthogonal cross-section of a particular tube of radius P is [π (arcsin(P+1)⁻¹)⁻¹] and if the number of contacts takes on its maximum, then all tubes are locally aligned.     

An Analog of the Appollonian Circle on the Lobachevski Plane and on the Sphere

An analog of the Apollonian circle on the Lobachevski plane is a convex analytic curve. A similar curve on the sphere may be nonconvex and may contain one singular (nonanalytic) point. It follows that there may exist more than one spherical belt separating two given compacta on the sphere and having the maximum possible ratio of the exterior and interior radii. Bibliography: 3 titles.