Applying the method of Moving Frames,we prove a famous Theorem due to N.Boken,K.Nomizu and U.Simon in :If M n(n≥2) is a nondegenerate affine hypersurface in A n+1 ,then both C and 2C are totally ...Applying the method of Moving Frames,we prove a famous Theorem due to N.Boken,K.Nomizu and U.Simon in :If M n(n≥2) is a nondegenerate affine hypersurface in A n+1 ,then both C and 2C are totally symmertric if and only if C=0 (M is a quadric) or S=0 (M is an improper affine hypersphere). Moreover,we improve the conditions of Theorem 2 and its Corollary in .also prove, if ^ C=0 and the affine metric G is of constant curvature α, then M is an affine hypersphere and a=0 or C=0.展开更多
In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median...In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.展开更多
For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the fo...For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.展开更多
文摘Applying the method of Moving Frames,we prove a famous Theorem due to N.Boken,K.Nomizu and U.Simon in :If M n(n≥2) is a nondegenerate affine hypersurface in A n+1 ,then both C and 2C are totally symmertric if and only if C=0 (M is a quadric) or S=0 (M is an improper affine hypersphere). Moreover,we improve the conditions of Theorem 2 and its Corollary in .also prove, if ^ C=0 and the affine metric G is of constant curvature α, then M is an affine hypersphere and a=0 or C=0.
基金Foundation item: Supported by the Natural Science Foundation of China(10271104)Supported by the Natural Science Foundation of Education Department of Sichuan Province(2004B25)
文摘In this paper, we study the quantic Diophantine equation （1） with elementary geometry method, therefore all positive integer solutions of the equation （1） are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.
基金the National Natural Science Foundation of China (Grant No. 19871917).
文摘For two given ternary quadratic forms f( x, y, z) and g( x, y, z), let r( f, n) and r( g,n) be the numbers of representations of n represented by f( x, y, z) and g( x, y, z) respectively. In this paper we study the following problem: when will we have r( f, n) = r( g, n) or r( f, n) ≠ r( g, n).Our method is to use elliptic curves and the corresponding new forms.
