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).     

非参数回归函数最近邻估计Lp收敛的速度

Let (X,Y) be a R^d×R¹-valued random vector with E(|Y|)<∞,m(x)=E(Y|X=x) be the regression funvion of Y with respect to X.Suppose that (X_i, Y_i),i=1, …,n, are iid samples drawn from (X,Y). It is desired to estimate m(x) based on these samples,everoye discussed in 1981 (see [2]) the pointwise L_p-convergence of the nearest neigthoor estimate m_n(x) (see (5) of the present paper). In this article we further study the rate of this convergence.It is shown that if there exists p≥2 such that E |Y|^p<∞,then E|m_n(x)-m(x)|^p=O(n^-p/(d+2))a.s. for suitabte choice of the weighte C_m (see(4) of the present paper).     

广义Fourier变换及其应用

It is proved that for each 2n×2n symplectic matrix S, there exists con-tinuous linear map F_s: S′(Rⁿ)→S′(Rⁿ), unique up to a constant factor, such that(?) F_s is called the generalized Fourier transformation. Some properties and applica-tions of F_s are obtained. In especial , the lower and upper boundedness of F_s in H^m(Rⁿ) is proved and a new proof of L. Hormander Theorem is given.     

范畴RMnl上的一个函子

Let _RM_n^l′ be a category which is equivalent to the category of left R-modules.In this paper,we define afunction F:_RM_n^l→_RM_n^l′ and prove that the functor Fpreserves products,direct limits,injections,surjectios and total esactness.Finally,we show that the functor F is a left-adjoint of the inclusion functor I:_RM_n^l′→_RM_n^l. Hence I:_RM_n^l′ is a renective subcategory of _RM_n^l.     

块Toeplitz三角阵求逆及块Toeplitz三角线性方程组求解的复杂性

In this paper, it is showed that the computational complexity of inversion of block triangular Toeplitz matrix U= (U₀, U₁,…, U_n-1 ) is O (m²nlogn + m³), as well as solution of block triangla r Toeplitz linear systems, where U 's are m×m mat rices. By using this results, we reduce arithmetic operations of division of polynomials from O(nlog²n) to O(nlogn).     

The automorphism group of a generalized extraspecial p-group

In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .     

Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl2n+2,0(R)*

Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domain Ω Rⁿ⁺¹ × Rⁿ⁺¹ centered at the origin with values in the Clifford algebra Cl_2n+_2,0(R) is proved. As a corollary, Almansi-type decomposition theorem for biharmonic functions of degree k is given.     

一类具不变性质的变系数偏微分方程的特解

In this paper, a few properties of general patial differential operator P beinginvariant with respect to the form F(|x|_p^p+|y|_q^q-|z|_r^r) are studied, where x∈Rⁿ,y∈ R^m, z∈R^l.and explicit formulas are given for certain solution of the equation Pu=Aδ with P being a differential operator with power function coefficientswhich preserves the form (|x|_p^p+|y|_q^q-|z|_r^r)^1/v for arbitrary even integers p, q, r,and odd integers v.     

Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification*

Let M be an open Riemann surface and G : M → Pⁿ(C) be a holomorphic map. Consider the conformal metric on M which is given by ds² = k e Gk^2m|ω|², where eG is a reduced representation of G, ω is a holomorphic 1-form on M and m is a positive integer. Assume that ds² is complete and G is k-nondegenerate(0 ≤ k ≤ n). If there are q hyperplanes H₁, H₂, · · · , H_q Pⁿ(C) located in general position such that G is ramified over H_j with multiplicity at least γ_j (> k) for each j ∈ {1, 2, · · · , q}, and it holds that Xj=1 1 -k/γ_j> (2n - k + 1) （mk/2+ 1）,then M is flat, or equivalently, G is a constant map. Moreover, one further give a curvature estimate on M without assuming the completeness of the surface.     

切于已知球的单形宽度

Let w(△_n) denote the width of a non-degenerate simplex △_n in Eⁿ and r(△_n) denote the inradius of the simplex.Then, in this paper, we prove the ine-qunlity as below: Theorem:w(△_n)≤β_nr(△_n) where β_n =(n^1/2(n+1))/([(n+1)/2]^1/2(n+1-[(n+1)/2])^1/2) The equality holds if and only if the simplex is regular.