Positivity of polarizations of -positive maps

It is shown that polarization formulas have explicit matrix representations. This enables us to prove that polarization formulas of -positive maps between -algebras are coordinatewise positive.

     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

Existence and Limiting Behavior of Trajectories Associated with P0-equations

Given a continuous P₀-function F : Rⁿ Rⁿ, we describe a method of constructing trajectories associated with the P₀-equation F(x) = 0. Various well known equation-based reformulations of the nonlinear complementarity problem and the box variational inequality problem corresponding to a continuous P₀-function lead to P₀-equations. In particular, reformulations via (a) the Fischer function for the NCP, (b) the min function for the NCP, (c) the fixed point map for a BVI, and (d) the normal map for a BVI give raise to P₀-equations when the underlying function is P₀. To generate the trajectories, we perturb the given P₀-function F to a P-function F(x, ); unique solutions of F(x, ) = 0 as varies over an interval in (0, ) then define the trajectory. We prove general results on the existence and limiting behavior of such trajectories. As special cases we study the interior point trajectory, trajectories based on the fixed point map of a BVI, trajectories based on the normal map of a BVI, and a trajectory based on the aggregate function of a vertical nonlinear complementarity problem.     

Regularity of CR maps between convex hypersurfaces of finite type

We study rigidity and regularity properties of CR maps between smooth convex hypersurfaces of finite type in

     

区间上平顶单峰扩张自映射的周期轨道

设ｔ（０＜ｔ＜１）是一个常数，ｎ≥３是奇数，ｍ≥０及ｋ≥１是整数，Ｐ０（ｘ）＝ｘ－１，Ｐｉ（ｘ）＝（ｘ２ｉ－１－１）Ｐｉ－１（ｘ）（ｉ≥１），ｒｍｎ（ｔ）及ｒｋ（ｔ）分别是方程Ｐｍ（ｘ）（ｘ２ｍｎ－２ｘ２ｍ（ｎ－２）－１）－ｔ（ｘ２ｍｎ－１）（ｘ２ｍ＋１）＝０及Ｐｋ－１（ｘ）－ｔ（ｘ２ｋ－１＋１）＝０在（１，＋∞）上的唯一实根，ｆ是闭区间Ｉ＝［０，１］上的峰顶区间长度为ｔ的平顶单峰扩张自映射．本文证明了，若ｆ的扩张常数λ≥ｒｍｎ（ｔ）（或＞ｒｋ（ｔ）），，则ｆ有２ｍｎ（或２ｋ）周期点．此外，本文还指出，当１＜λ＜ｒｍｎ（ｔ）（或≤ｒｋ（ｔ）时，在Ｉ上存在着具有扩张常数λ及峰顶区间长度ｔ却无２ｍｎ（或２ｋ）周期点的平顶单峰扩张自映射     

高阶奇性常微分方程的边值问题（Ⅰ）──—非退化情形

本文对高阶奇性常微分方程的非退化边值问题建立了非共轭性存在定理和上、下解型存在定理．     

G2,n中的可兼共形调和曲面

本文通过建立复Ｇｒａｓｓｍａｎｎ流形Ｇ２，ｎ中调和曲面的Ｇａｕｓｓ丛的基本公式，在适当的拓扑条件下给出了Ｇ２，ｎ中可兼共形调和曲面的构造定理．推广和改进了Ｂｕｒｓｔａｌｌ和Ｗｏｏｄ的低亏格曲面的结果．     

Affine-Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.     

Completely Positive Definite Maps Over Topological^＊—algebras

ＣｏｍｐｌｅｔｅｌｙＰｏｓｉｔｉｖｅＤｅｆｉｎｉｔｅＭａｐｓＯｖｅｒＴｏｐｏｌｏｇｉｃａｌ－ａｌｇｅｂｒａｓＸｕＴｉａｎｚｈｏｕ（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＢｅｉｊｉｎｇＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｆｏｇｙ，...