R^(n+1) 上self-shrinkers的谱特征

本文研究了self-shrinkers谱与几何的关系.利用渐进展开式系数相等的方法,获得了如下结果:设M是R^(n+1)上的n(n≥2)维闭self-shrinkers,且M和s^n(2n^(1/2))有相同的平均曲率,如果spec^p(M)=spec^p(s^n(2n^(1/2)),则M是s^n(2n^(1/2),并推广了R^(n+1)上self-shrinkers的特征.

Classification of complete 3-dimensional self-shrinkers in the Euclidean space R^(4)

In this paper,we completely classify 3-dimensional complete self-shrinkers with the constant norm S of the second fundamental form and the constant f3in the Euclidean space R^(4),where hij’s are components of the second fundamental form,S=∑i,jhij2and f3=∑i,j,khijhjkhki.

Some Results on Space-Like Self-Shrinkers

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space Rm^m＋n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.

A new pinching theorem for complete self-shrinkers and its generalization

In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+|λ|(λ2+4)1/2).

A Rigidity Result of Spacelike Self-Shrinkers in Pseudo-Euclidean Spaces

In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.

RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN R^(4)

In this paper,we give some rigidity results for complete self-shrinking surfaces properly immersed in R^(4) under some assumptions regarding their Gauss images.More precisely,we prove that this has to be a plane,provided that the images of either Gauss map projection lies in an open hemisphere or S^(2)(1/2–√)∖S^(-1)+(1/2–√).We also give the classification of complete self-shrinking surfaces properly immersed in R^(4) provided that the images of Gauss map projection lies in some closed hemispheres.As an application of the above results,we give a new proof for the result of Zhou.Moreover,we establish a Bernstein-type theorem.

新多位self-shrinking序列的构造与特性

在多位self-shrinking序列的基础上构造了一种新的多位self-shrinking序列的模型,该模型与原多位self-shrinking序列具有相同的周期下界,且具有更好线性复杂度。

Weak generalized self-shrinking generators

The security of certain classes of the generalized self-shrinking sequence （GSS） generators is analyzed. Firstly, it is shown that the security of these GSS generators is equivalent to the security of the GSS generators of the class-1, after which two effective key recovery attacks on the GSS generators of the class-1 are developed to evaluate their security.

SELECTION OF THE LINEAR COMBINING VECTOR G OF THE GENERALIZED SELF-SHRINKING GENERATORS

Given an m-sequence, the main factor influencing the least period of the Generalized Self-Shrinking （GSS） sequence is the selection of the linear combining vector G. Based on the calculation of the minimal polynomial ofL GSS sequences and the comparison of their degrees, an algorithm for selecting the linear combining vector G is presented, which is simple to understand, to implement and to prove. By using this method, much more than 2^L-l linear combining vectors G of the desired properties will be resulted. Thus in the practical application the linear combining vector G can be chosen with great arbitrariness. Additionally, this algorithm can be extended to any finite field easily.

ON THE LAGRANGIAN ANGLE AND THE K?HLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C^ 2

In this paper, we discuss the Lagrangian angle and the K?hler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant K?hler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant K?hler angle is generally non-vanishing. Secondly, we obtain two pinching results for the K?hler angle which imply rigidity theorems of self-shrinkers with K?hler angle under the condition that ∫M|h|2e-|x|2/2 dVM< ∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.

一类序列密码的构造

本文参照Geffe序列和多位Self-Shrinking(自收缩)序列的构造原理,给出一种新型的序列生成方式,讨论了它的周期、统计特性和线性复杂度。

Singularities of mean curvature flow

Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.

A note on the entropy of mean curvature flow

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We also prove that a smooth selfshrinker with low entropy is a hyperplane.

Complete λ-Hypersurfaces in Euclidean Spaces

In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.