单位球面上具有非负M bius截面曲率的超曲面

Let x:M→S~(n 1)be a hypersurface in the (n 1)-dimensional unit sphere S~(n 1)without umbilic point. The M(?)bius invariants of x under the M(?)bius transformation group of S~(n 1) are M(?)bius metric,M(?)bius form,M(?)bius second fundamental form and Blaschke tensor.In this paper,we prove the following theorem: Let x:M→S~(n 1)(n＞2)be an umbilic free hypersurface in S~(n 1) with nonnegative M(?)bius sectional curvature and with vanishing M(?)bius form.Then x is locally M(?)bius equivalent to one of the following hypersurfaces:(i)the torus S~k(a)×S~(n-k)((1-a~2)~(1/2))with 1≤k≤n-1;(ii)the pre-image of the stereographic projection of the standard cylinder S~k×R~(n-k)(?)R~(n 1) with 1≤k≤n-1;(iii)the pre-image of the stereographic projection of the cone in R~(n 1):(?)(u,v,t)=(tu,tv), where(u,v,t)∈S~k(a)×S~(n-k-1)((1-a~2)~(1/2))×R~ .     

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.     

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

Complete Hypersurfaces with Constant Laguerre Scalar Curvature in R~n

Let x:M~(n-1)→R~n be an umbilical free hypersurface with non-zero principal curvatures.Two basic invariants of M under the Laguerre transformation group of R~n are Laguerre form C and Laguerre tensor L.In this paper,we prove the following theorem:Let M be an(n-1)-dimensional(n 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R~n,denote the trace-free Laguerre tensor by L =L-1/n-1tr(L)·Id.If sup_M ||L||=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface;and if sup_M ||L||=((n-1)(n-2))~(1/2)R/((n-1)(n-2)(n-3)),M is Laguerre equivalent to the hypersurface x:H~1×S~(n-2)→R~n.     

Topological and geometric property of matrix algebra

Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.     

Laguerre Isopararmetric Hypersurfaces in R^4

Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C , and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R4 .     

Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds(In memory of Professor Chaohao Gu on his 90th birthday)

Let M~n(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an(n + p)-dimensional complete simply connected Riemannian manifold N~(n+p).Then there exists a constant δ(n, p) ∈(0, 1) such that if the sectional curvature of N satisfies■ , and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to S~(n+p). Moreover, M is either a totally umbilic sphere■ , a Clifford hypersurface S~m■ in the totally umbilic sphere ■, or■ . This is a generalization of Ejiri's rigidity theorem.     

Classification of hypersurfaces with two distinct principal curvatures and closed M?bius form in \mathbb{S}^{m + 1}

Let x be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere Sm+1 (m≥3). In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Mbius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.     

A NOTE ON SINGULAR INTEGRALS WITH DOMINATING MIXED SMOOTHNESS IN TRIEBEL-LIZORKIN SPACES

Let h be a measurable function defined on R+×R+. LetΩ∈ L(log L+)νq(Sn1-1×Sn2-1)(1 ≤νq≤ 2) be homogeneous of degree zero and satisfy certain cancellation conditions. We show that the singular integral T f(x1, x2) = p. v.∫ Rn1+n2Ω(y′1, y′2)h(|y1|, |y2|)|y1|n1|y2|n2f(x1- y1, x2- y2)dy1dy2maps from Sα1, α2p, q˙F(Rn1× Rn2) boundedly to itself for 1 p, q ∞, α1, α2 ∈ R.     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.     

Dirichlet characters,Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic(continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized M¨obius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.     

An attempt to differential Galois theory of second order polynomial system and solvable subgroup of M(o)bius transformations

By introducing the conception "relativistic differential Galois group" for the second order polynomial systems, we establish the relation between the conformal relativistic differential Galois group and the subgroup of M(o)bius transformations, and prove that the system is integrable in the sense of Liouville if its conformal relativistic differential Galois group is solvable with a derived length at most 2. Some omissions on the structures of solvable subgroups of M(o)bius transformations at the first author's article published in this journal in 1996 are refreshed in this paper.     

A complete system of orthogonal step functions

We educe an orthonormal system of step functions for the interval . This system contains the Rademacher functions, and it is distinct from the Paley-Walsh system: its step functions use the Möbius function in their definition. Functions have almost-everywhere convergent Fourier-series expansions if and only if they have almost-everywhere convergent step-function-series expansions (in terms of the members of the new orthonormal system). Thus, for instance, the new system and the Fourier system are both complete for

     

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational Bézier curves or surfaces via Mbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after Mbius transformation. What's more the users of computer aided design softwares may require some guidelines for designing rational Bézier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway.The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational Bézier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal parametric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational Bézier surfaces with compact derivative bounds.     

Affinely Prime Dynamical Systems

This paper deals with representations of groups by "affine" automorphisms of compact,convex spaces,with special focus on "irreducible" representations:equivalently "minimal" actions.When the group in question is PSL(2,R),the authors exhibit a oneone correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations.This analysis shows that,surprisingly,all these representations are equivalent.In fact,it is found that all irreducible affine representations of this group are equivalent.The key to this is a property called "linear Stone-Weierstrass"for group actions on compact spaces.If it holds for the "universal strongly proximal space"of the group (to be defined),then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.     

关于Mobius反转公式的一个推广

本文研究了著名的M(o)bius变换,并将其反转公式进行了推广和延伸.     

Clifford M\"{o}bius变换与Hypergenic函数

首先,在实Clifford代数空间Cl_n+1,0(R)中给出了与Clifford Mbius变换相关的一些定理.其次,证明了hypergenic函数与Clifford Mobius变换的复合可以得到一个加权的hypergenic函数.