Rigidity theorems for closed hypersurfaces in a unit sphere

In this paper, we prove some rigidity theorems for closed hypersurfaces in a unit sphere.     

On the weak uniform convexity of

We will discuss the geometry of the unit sphere in the Banach space of integrable holomorphic quadratic differentials on a Riemann surface and answer some questions posed by L.R. Goldberg (Proc. Amer. Math. Soc. 118 (1993), 1179--1185).

     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Γ such that ℍ/Γ is a hyperbolic Riemann surface, the Teichmüller curve V(Γ) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Γ) onto V(Γ) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmüller curves is deduced, which generalizes a classical result that the Teichmüller curve V(Γ) depends only on the type of Γ and not on the orders of the elliptic elements of Γ when ℍ/Γ is a compact hyperbolic Riemann surface.     

Block structured adaptive mesh and time refinement for hybrid,hyperbolic + N-body systems

We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov’s method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS AND HARDY TERMS

This paper deals with the existence of solutions to the elliptic equation -△uμu/|x|2=λu |u|2*-2u f(x, u) in Ω, u = 0 on ( a)Ω, where Ω is a bounded domain in RN(N≥3),0∈Ω,2*=2N/N-2,λ＞0,λ(a)σμ, σμ is the spectrum of the operator -△- μI/|x|2with zero Dirichlet boundary condition, 0 ＜μ＜-μ,-μ=(N-2)2/4,f(x,u) is an asymmetric lower order perturbation of |u|2*-1 at infinity. Using the dual variational methods, the existence of nontrivial solutions is proved.     

Some ergodic and rigidity properties of discrete Heisenberg group actions

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group \(\mathcal{H}\). We establish the decomposition of the tangent space of any C^∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the central elements are zero. We obtain that if an \(\mathcal{H}\) action contains an Anosov element, then under certain conditions on the eigenvalues of this element, the action of each central element is of finite order. In particular, there is no faithful codimension one Anosov Heisenberg group action on any compact manifold, and there is no faithful codimension two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic \(\mathcal{H}\) actions by toral automorphisms, using a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.     

New existence and nonexistence results for strong external difference families

In this paper, we use character-theoretic techniques to give new nonexistence results for $(n,m,k,\lambda )$-strong external difference families (SEDFs). We also use cyclotomic classes to give two new classes of SEDFs with $m=2$.