关于n维单形的一个不等式

Let x₁,x₂… ,x_n are vectors set out from any vertex of n-simplex. The ineluded angle of x_i and x_j is a_ij , and G=(?)a_ij, the volume of this simplex is V. In this paper, we prove the inequality as below:(?) When n=2, the equality holds in (*) , when n≥3, the equality holds in (*) if {x_i} is orthogonal set.     

常曲率空间关于直交全脐超曲面系的特征性质

Let V_n be Riemannian space of genernal constant curvature.In this paper, we have proved following;Theorem I If a V_n(n≥5 ) admits three mutually orthogonal families oftotally numbilical hypersurfaces such that they are of constant curvature and Einsteinian and of general constant curvature respectively, then V_n is space with constant curvature.Theorem 2 If a V_n ( n ≥ 5 ) admits three mutually orthogonal famities of totally umbilical hypersurfaces, of which one is conformally flat and other two are Einsteinian and of constant curvature respectively, and latter either is of constant meam curvature, then V_n is of constant curvature.     

Semiamart与Gaft类的收敛性

Let (Ω,F,P) be a probability space,(F_n) be an increasing sequence of sub-σ-fields of F, a sequence (x_n) of integrable random variables adapted to (F_n), if (x_n) is progressive uniformly integrable or(x_n)is a L¹-bounded Semiamart, then ( 1 ) x_m-x_n→0 (Pr.), ( 2 ) E (x_m|F_n)-x_n→0 (pr.) are equivalent is proved in this paper .     

共形平坦黎曼流形中具有平行第二基本形式的超曲面

In this paper, we prove the followingTheorem. Let Cⁿ⁺¹ ( n >5 ) be a conformally flat Riemannian anifola of dimension n + 1 . If Mn is a hypersurface immersed isometrically in Cⁿ⁺¹ over which the second fundamental form is covariant constant, then there are three posible cases only:I . locally Mⁿ= S^p×S^q×S^r, p+q+r=n;Ⅱ . locally Mⁿ=S^p×S^q, p + q = n , where S^k is k- dimensional Riemannian space of constant curvature;III. Mⁿ is umbilical and conformally flat. Moreover, if Mⁿ is connected and complete, then the result holds globally.     

GCD和GCUD矩阵行列式的一个不等式（英）

Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).     

关于加权位移的酉等价问题

In this paper, we study the unilateral weighted shift which is unitarily equivalent to a Toeplitz operator and prove a similar result to that in [ 1 ] without the hypothesis that the shift must be hyponormal. As a corollary, we show that if the weight sequence {a_n}_n=0^∞ of the shift is convergent, then 1-a_n²=(1-a₀²)ⁿ⁺¹ ?n≥0     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

关于单形的三角不等式

Theorem 1 We show that let S be a simplex in Eⁿ, its vertex angles being αi and interior dihedral angles formed by arbitrary two side faces f_i,f_j of S being Q_ij, if m_i are positive numbers (i,j= 1, 2, …, n + 1, i ≠ j), then Theorem 2 Let Ω be a simplex in Hⁿ, its interior dihedral angles formed by arbitrary two side faces F_i, F_j of Ω being φ_ij, if p_i- be positive numbers (i, j = 1,2,…., n + 1, i ≠ j), then     

Padé算子连续的条件

Let r_{m, n}=P_{m, n}/ Q_{m, n} be the Padé approximation of order (m, n) for a given power series f: Q_{m, n}f- P_{m, n}= O(z^m+n+1+j), j is as large as possible, m′ ,n′ are the exact degree of p_{m, n} and Q_{m, n} respectively and d = min{m-m′ ,n-n′}. Let T_{m, n} be the operator that maps f on r_{m, n} . Then T_{m, n} is continuous if and only if d + (j-|j|)/2= 0. This condition is equal to that r_{m, n} is on the border of a Padé block.The results obtained by H. Werner and L.Wuytack have been modified and cor reeled in this paper.     

有限重位移

It is shown in this paper that a weighted unilateral Shift T of multiplicity k (0n}₀^∞ converge, is unitarily equialent to a Toeplitz operator if and only if (1-|a_n|²)=(1-|a₀|²) (1-|a_n-k|²)(n>k) and |a₀|= |a₁|=…|a_k-1| where we assume ‖T‖=1.