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 .     

关于U统计量的一点注记

Let φ(x₁,x₂,…x_m) be a real Borel measurable symmetrical function of the m variables, and let U_n denote the U-statistic with kernel φ. In this note it is proved that the necessary and sufficient condition for E|φ|<+∞ is that there exists an n≥m such that E|U_n|< + ∞.     

关于Von Mises统计量Berry-Esseen定理的一点注记

Let V_n be a von Mises Statistice with kernel φ(x₁,x₂,… ,x_m).     

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).     

质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.     

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     

类2的常数联络黎曼空间

A non-flat Riemannian space V_n is called Riemannian space with constant connection if its Christoffel symbols of the second kind are constant in some coordinate system {xⁱ}. Following G. Vranceanu [3], a Riemannian space V_n with constant connection is said to be of genus p, if the components of the fundamental tensor in the coordinate system {x^{i/sup>} can be written in the form g_ij=c_ijat_a(a=1,…,p) where c_ija are constant, and p is least.In this paper we prove the following}     

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).     

关于集值系统x∈F(x,y),y∈G(x,y)解的存在性

Let (E, ‖·‖) be a uniformly convex Banach space, X a nonempty compact and convex subset of E. Let F be a closed mapping of X×X into 2^X, G a mapping of X×X into C(X). It is shown that if for any f∈C(X), x∈X, F(x,f(x)) is a closed and convex subset of X, and G(x,f(x)) is a continuous function and for any x,y₁,y₂∈X,H(x,G(x,y₁),G(x,y₂))≤‖y₁-y₂‖ then there exist X₀,y₀∈X such that X₀∈F(x₀,y₀) and y₀∈G(x₀, y₀).     

关于覆盖同余式组的一个注记

In this paper, it is shown that., if x≡α_i(mod n_i)(i=1,…,k),112<…k,0≤α_ii is a covering sets of residue classes, there exist two distinct pairs of integer number n_i,n_j(is,n_t(si,n_j)>1, (n_s,n_t)>1, where (a,b) is the greatest common divisor of a,b.