Lines Tangent to Four Unit Spheres with Coplanar Centres

We prove that there are at most eight lines tangent to four unit spheres in \R ³ if the centres of the spheres are coplanar, but not collinear. This bound is sharp. Received October 9, 2000, and in revised form April 26, 2001. Online publication October 12, 2001.     

Tangent Processes on Wiener Space

This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space (i.e. transformations induced by general measure-preserving transformations, called “rotations”, and H-valued shifts) and the associated flows on abstract Wiener spaces.     

Polynomial of Degree Four Interpolation on Triangles

A method for constructing the $C^1$ piecewise polynomial surface of degree four on triangles is presented in this paper. On every triangle, only nine interpolation conditions, which are function values and first partial derivatives at the vertices of the triangle, are needed for constructing the surface.     

Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in ℝ³. Generically, the set of transversals to four segments consists of zero or two lines. We catalog the non-generic cases and show that n≥ 3 arbitrary line segments in ℝ³ admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in ℝ³.     

代数流形奇点的切空间

利用一种新工具极限方程,从初等微积分的角度研究了代数流形奇点的切空间,给出了一般代数流形奇点的切空间决定于定义代数流形方程组的局部.     

二次曲线垂直切线的研究

用解析几何与射影几何的方法讨论二次曲线垂直切线交点的轨迹,重新证明了:椭圆、双曲线垂直切线交点的轨迹是圆;抛物线垂直切线的交点在准线上,且切点的连线过焦点.     

The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres

We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent to the three spheres that also meet the given line. All such configurations are degenerate. The path to this result involves the interplay of some beautiful and intricate geometry of real surfaces in 3-space, complex projective algebraic geometry, explicit computation and graphics.     

Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties

Abstract

Following Contou-Carrère (Contou-Carrère,C. (1983). Géométrie des Groupes Semi-Simples,Résolutions équivariantes et Lieu Singulier de Leurs Variétés de Schubert. Thèse d’état,Université Montpellier II (published partly as,Le Lieu singulier des variétés de Schubert (1988). Adv. Math.,71:186–221)),we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. Using Carrell and Peterson's characterization (Carrell,J. B. (1994). The Bruhat graph of a Coxeter group,a conjecture of Deodhar,and rational smoothness of Schubert varieties. Proc. Symp. in Pure Math. 56(Part I):53–61),we prove that rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called,the combinatorial tangent space.     

三维欧氏空间的仿射乘积曲面

定义了三维欧氏空间中的仿射乘积曲面,给出了极小仿射乘积曲面以及高斯曲率为零的仿射乘积曲面的分类,同时还给出了平均曲率为非零常数的仿射乘积曲面的分类.     

Simultaneous Packing and Covering in Three-Dimensional Euclidean Space

It is proved that the simultaneous lattice packing and latticecovering constant of every three-dimensional centrally symmetricconvex body is less than or equal to 7/4. Consequently, forany three-dimensional convex body K there is a correspondinglattice packing in which no extra translate of K can be added.     

空间曲线和曲面的切线与切面的代数做法

具体的从我们熟知的一些代数量上,给出了空间曲线的切线和法平面,空间曲面的切面和法线的构造过程.并指出了过去利用隐函数构造这些量时,教材中在讲述这些几何量中含糊的地方,或者不准确的问题.通过此的方法,可以简单地得到书中的这些结论,也可以避免现有的教材中,不得不面对的一些代数量的几何解释,而这些解释又是繁杂,甚至是复杂的这样不必要的过程.     

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let γ be a smooth generic curve in ?P ₃. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ?³ has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.     

Cutting Triangular Cycles of Lines in Space

We show that n lines in 3-space can be cut into O(n^2-1/69log^16/69n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a long-standing open problem in computational geometry, motivated by hidden-surface removal in computer graphics.     

New Cases of Homogeneous Integrable Systems with Dissipation on Tangent Bundles of Three-Dimensional Manifolds

Doklady Mathematics - The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of three-dimensional manifolds is shown. The force fields involved in the systems...     

The Spectrum of a Periodic Lightguide in Three-Dimensional Space

We consider a three-dimensional periodic lightguide whose oscillations are described by the reduced wave equation. We study the spectrum of the waveguide operator in invariant subspaces. Bibliography: 6 titles.     

On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R _q about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.