半质环交换性的一点注记

Let R be a semi-prime ring, C be the center of R. Let F_i (x, y) (i = 1, 2) be a product of the m times x's and n times y's.In this paper following theorem is proved: (I ) implies (Ⅱ), where( Ⅰ )If f₁(x,y) -f₂(x,y) ∈C for every x,y in R, then R is commutative;(Ⅱ)If f₁ (x,y) + f₂(x, y) ∈C for every x,y in R, then R is commutative.Thus very short proves of some theorems of references[5], [8], [9] are be given.     

关于Hammerstein型非线性积分方程固有值的一点注记

Let Aφ(x)=∫_GK(x,y)f(y,φ(y))dy, where G is a bounded closed domain in Euclidean space, K(x,y) is continuous on G×G, f(x,u) is continuous on G×R, and f(x,0)≡0. Set G_x={x|x∈G,K(x,y)≠0},G_y={y|y∈G,K(x,y)≠0},G₁=G_x∩G_y≠φ.Let K₁(x,y) be the restriction of K(x,y) on G₁×G₁,f₁(x,u)be the restriction of f(x, u) on G₁× R, and A₁φ=∫_G1K₁(x,y)f₁(y,φ₁(y))dy, The main result of this paper is Theorem λ≠0 is an eigenvalue of A, if and only if λ is an eigenvalue of A₁.     

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).     

NN-模式识别中条件误差概率收敛性的充要条件

Let (θ₁,X₁),…, (θ_n,X_n), (θ, X) be iid random vectors ,where θ∈{0,1},X∈R^d Denote by θ′_n the nearest neighbour discriminator of θ based on the training samples (θ₁,X₁),…, (θ_n,X_n) and the observed X; put(?). This paper gives a sufficient and necessary condition for (?) as n→∞, namely (P(θ=0, X=x)-P(θ=1, X=x))²·P(θ=0, X=x)·P(θ=1, X=x)=0 for every x∈R^d.This generalizes a previous result of the authors [5] and improves a result of Wagner, T.J. [2].     

关于反向适应序列的一些结果

Let X be a Banach space, (x_n, F_n, n<- 1) a X-valued adapted sequence on probability space (Q, F, P) . Let T be all stopping times with respect to(F_n,n < - 1) . (x_n, F_n,n< - 1) is called a T- uniform amart if there exists a t₀∈T such that for each t∈T with t0,E‖x_t‖<∞ and if (?)=0.In this paper we prove that.     

关于Γ-环的稠密性定理

Let U be an Γ-ring and M be an irreducible UΓ-moduie, for α∈U, α∈Γ, we define T_[a,a]: M→M by m T_[a,a] = maa for all m∈M. Let End(M) be the ring of all endomorphisms of the additive group of M. We define as usual End_[Γ,U](M)=={ψ∈End(M)|T_Σ[ai,ai]ψ=ψT_Σ[ai,ai] all a_i∈U,a_i∈Γ}. In this paper the following results are obtained.     

用Hermite-Fejér型插值多项式逼近连续函数

Let f(x)∈C_[-1,1],T_n(x)=cos (n arccos x),U_n(x)=(sin((n+1)arccosx))/(1-x²)^1/2,P_n(x) be the Legendre polynomials of degree n. And let ω(t ) be a given modulus of continuity, H_ω={f|ω(f,t)≤ω(t)}.A. K. Sharma and J. Tzimbalario(J. Appro. Th., 13(1975), 431-442) considered the operators L_n,p (f, x) (p= 0, 1, 2,3) and obtained some theorems.In this paper, we prove the following theorems.     

PF环与群环的Grothendieck群

Let R be a commutative ring with 1, G an Abelian group, RG the group ring on R and G. In this paper we gave some properties of PF- rings in which f. g. projective modules are free. The Grothendieck groups K₀(RG) for some cases are given. In addition, for the ring R with the unimodular column property, we proved the following result: K₀(RG) ≈K₀(R), hence if R ∈PF, then K₀(RG)≈Z .     

Fourier级数子列的可和性

Let H_α0^ω denote the set of functions f(x)∈L_2π such that ω(f,x₀;t)≤ω(t),ω(t) being a given modulus of continuity. Let {n_k} be a set of natural numbers satisfying the condition n_i+1/n_k>q>1, and let A= (α_πk) be a regular summation matrix.     

关于半素环交换性的一点注记

Awtar proved that a semiprime ring R in which xy²x-yx²y∈Z(center of R)for every x and y in R is commutative. Guo Yuanchun proved that a semiprime ring satisfying (xy)²-xy²x∈Z for every x and y in R is commutative. In this note the following result is proved:A semiprime ring is commutative if R satisfies one of the following conditions:(1) x²y² -xy²x∈Z for every x and y in R.(2) x²y²-yx²y-y∈Z for every x and y in R.(3) (yx)² -xy²x∈Z for every x and y in R.