二阶曲线射影仿射分类的合同变换法

二阶曲线的射影(仿射)分类是指:凡在射影(仿射)变换下互为对应的二阶曲线归为一类,分类常用方法是在射影(仿影)平面上选取适当的坐标系化简方程,其中涉及到高等几何的概念较多,分析与解题的过程也较为繁杂。本文避开对坐标三点形的选择,对二阶曲线的系数矩阵施以合同变换,达到分类的目的。     

关于减小体积的射影映射

Given a projective map F:M→N of a complete Riemannian manifold to a Riemannian manifold with the sectional curvature bounded above by a negative constant,we prove that f decreases volume up to a constant depending only on the curvatures of M and N.This generalizes the result due to Har‘el.     

p- 局部化无限射影空间的自映射

分别记Z（p）和Zp为整数环Z的p-局部化和p-完备化,那么我们有自然的含入映射Z（p）→Zp.令S2n-1（p）为p-局部化的2n-1维球面,令B2n（p）为一个p-局部化空间,满足S2n-1（p）=～ΩB2n（p）,那么我们有H＊（B2n（p）,Z（p））=Z（p）[u],其中u的度数为2n.对于B2n（p）的任意一个自映射f,我们定义f的度数为k∈Z（p）满足f＊（u）=ku.运用整值多项式理论,我们证明存在B2n（p）的一个度数为k的自映射当且仅当k在Zp中是一个n次幂.     

复射影曲线上特殊线丛的射影正规性

本文给出复射影曲线上特殊线丛正规生成的一系列充分条件.     

空间有理三次Bezier曲线的射影变换和权系数的几何性质

本文讨论了空间有理三次Ｂｅｚｉｅｒ曲线的射影变换和权系数的一系列几何性质。其权系数组成构成了控制四顶点基下的权心的齐次坐标;权心是六个特殊平面的公共交点。含权心和曲线“肩点”的某四个共线点之比恒为常数３;权心可作为有理曲线所在射影坐标系的单位点;此有理曲线是对应整有理曲线在射影变换下的象,此变换把控制四面体的形心映为权心;权系数是此射影变换的特征值（差－常数因子）;权系数是变换前后两曲线上对应点关     

关于非退化的二阶曲线的分类研究

从抛物线的一种判定方法出发,借助于欧氏平面上非退化的二阶曲线的度量性质,通过对欧氏平面上非退化二阶曲线类型的研究,探究出确定非退化二阶曲线类型的若干定理。     

光滑拟周期可逆映射的不变曲线

证明了平面拟周期可逆映射在C~l光滑情况下其不变曲线的存在性,并且给出了l同不变曲线的光滑性和丢番条件中指数的关系.     

Fibonacci映射的不变闭曲线的存在性

本文利用Moser扭转定理证明Fibonacci映射在椭圆型二周期点附近存在F^2-不变闭曲线。     

二阶曲线切线型方程的意义

本文论证了二阶曲线切线型直线方程的几何意义，其主要结论丰富了二阶曲线的一般理论。     

度量射影和余度量射影的线性选择

本文讨论了度量射影和余度量射影的关系,并在L_p(T)和C(T)空间中讨论了它们的线性选择的性质.     

Projection Methods for Discrete Schrodinger Operators

Let H be the discrete Schrödinger operator acting on l² Z⁺, where the potential v is real-valued and v(n) 0 as n . Let P be the orthogonal projection onto a closedlinear subspace l² Z⁺). In a recent paper E. B. Davies definesthe second order spectrum Spec₂(H, ) of H relative to as theset of z C such that the restriction to of the operator P(H- z)²P is not invertible within the space . The purpose of thisarticle is to investigate properties of Spec₂(H, ) when islarge but finite dimensional. We explore in particular the connectionbetween this set and the spectrum of H. Our main result providessharp bounds in terms of the potential v for the asymptoticbehaviour of Spec₂(H, ) as increases towards l² Z⁺). 2000 MathematicsSubject Classification 47B36 (primary), 47B39, 81-08 (secondary).     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

Curves in Grassmannians

This paper considers curves in Grassmannians which are themselves immersed in projective space by the Plücker map. It is shown that for a generic vector bundle of high enough degree, the image curve lies in a proper linear subvariety of this projective space and satisfies good conditions on syzygies as a curve in this subspace. For very small degree and generic vector bundle, the curve is non-degenerate.

     

用Lagrange乘数研究二次曲线的几何性质

文中把二次曲线的几何性质的研究转化成条件极值问题,但又不关心问题的解,而是利用Lagrange乘数来研究二次曲线的几何性质,找到了用Lagrange 乘数判别二次曲线形状的方法,给出了用Lagrange乘数计算二次曲线的对称轴和轴长的公式.     

Generalized Projection Algorithms for Nonlinear Operators

In this paper, we prove three strong convergence theorems of modified Mann iteration for relatively asymptotically nonexpansive mappings, relatively nonexpansive mappings, and maximal monotone operators in a Banach space, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Matsushita, Kim, Martinez-Yanex, Xu, and some others.     

两类正规化双全纯映照子族齐次展开式的精细估计

本文考虑C~n中单位多圆柱上和一般复Banach空间中单位球上的正规化双全纯α(0■α＜1)次的殆β(-π/2＜β＜π/2)型螺形映照以及α(0＜α＜1)次的β(-π/2＜β＜π/2)型螺形映照f(其中x=0是f(x)-x的k+1阶零点),研究了它们的构造,并得到其齐次展开式的精细估计．所得的结果推广了以前相应的结论．     

两步投影对变分不等方程组解的迭代逼近

本文研究Hilbert空间中两步投影方法及其对变分不等方程组解具误差的迭代序列的收敛性.本文结果发展和改进了Verma等人的最新结果.     

Curves with finite turn

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one-sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces. The author was partially supported by the grant GAČR 201/03/0931 and by the NSF grant DMS-0244515.