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.     

关于P2(C)全纯曲线唯一性问题的注记*

本文讨论P²(C)中全纯曲线相交处于次一般位置超平面的唯一性.设f₁, f₂, · · · , f_λ为P²(C)中线性非退化的全纯曲线,H₁, H₁, · · · , H_q为P²(C)上处于m-次一般位置的超平面,满足A_j :f₁^-1(H_j) = · · · =f_λ^-1(H_j) (1 ≤ j ≤ q)且A_i ∩ A_j = ?(i = j).假设存在整数l (2 ≤ l ≤ λ),使得f_j1(z) ∧ f_j2(z) ∧ · · · ∧ f_jl(z) = 0 (z ∈ A_j)对任意l个指标1 ≤ j₁ < j₂ < · · · < j_l < λ成立.那么当 q > 2λ/λ-l+1 + 3/2 m时, f₁ ∧ · · · ∧ f_λ ≡ 0.关键技术是第二基本定理中不等式改进为: ∥(q - 3m/2)T_ft(r)≤ Pjq=1N²(f_t,H_j )(r, 0) + o(T_ft(r))(1 ≤ t ≤ λ).     

概率密度的均匀核估计与近邻估计的均方误差

In this paper we consider the Mean Square Error (MSE) of two uaual estimates of density function f(x) at a point x: The uniform kernel estimate f_n(x) and the NN estimate fn(x). we- show that when f is differentiable for sufficiently high order at x. these MSE can be expanded in a form E(f_n(x)-f(x))²=A₁(x)n^-4/5 +A₂(x)n^-1+A₃(x)n^-6/5+…;E(f_n(x)-f(x))²=B₁(x)n^-4/5 +B₂(x)n^-1+B₃(x)n^-6/5+… And if we suitably choose the parameters in f_n and f_n to make A₁(x) and B₁(x)to assume its minimunm value, then we also, have A₂(x) =B₂(x) but A₃(X) differs form B₃(X). This result shows that while the two estimates are not identical with respect to MSE. each one can be superior to the other in various special cases.     

Long-time Asymptotic Behavior for the Derivative Schr¨odinger Equation with Finite Density Type Initial Data*

In this paper, the authors apply ? steepest descent method to study the Cauchy problem for the derivative nonlinear Schr¨odinger equation with finite density type initial data iq_t + q_xx + i(|q|²q)x = 0,q(x, 0) = q0(x),where lim/x→±∞ q0(x) = q± and |q±| = 1. Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schr¨odinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region ξ =x/t with |ξ + 2| < 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) + O(t^?3/4 ),in which the leading term is N(I) solitons, the second term is a residual error from a ? equation. For the region |ξ + 2| > 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) ? t^?1/2 if¹¹ + O(t^?3/4 ),in which the leading term is N(I) solitons, the second t^?1/2 order term is soliton-radiation interactions and the third term is a residual error from a ? equation. These results are verification of the soliton resolution conjecture for the derivative Schr¨odinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.     

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.     

p-值星形函数的子类

Let M_n+p-1 denote the class of functions f(z) = 1/z^p+a₀/z^p-2+a₁/z^p-2+…+a_n+p-1zⁿ+…, regular and p-valent in the annulus 0<|z|<1 and satisfying Re((D^n+p f(z))/(D^n+p-1 f(z)))-2)<-(n+p-1)/(n+p),|z|<1,n>-p where D^n+p-1 f(z)=1/z^p((z^n+2p-1f(z))/(n+p-1)!)^(n+p-1).M_n+p?M_n+p-1 is proved. Since M₀ is the subclass of p-valent meromorphically starlike functions, all functions in Mn + p-1 are p-valent meromorphically star-like functions. Further the integrals of functions in M_n+p-1, are considered.     

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).     

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

Products of Toeplitz and Hankel Operators on Fock-Sobolev Spaces

In this paper, the authors investigate the boundedness of Toeplitz product T_f T_g and Hankel product H^*_fH_g on Fock-Sobolev space for f, g ∈ P. As a result, the boundedness of Toeplitz operator T_f and Hankel operator H_g with f ∈ P is characterized.     

关于有限域中原根分布的注记

Let f₁(x) , …,f_r(x) denote quadratfree polynomials over GF(q) that are relatively prime in pairs and of degree≥1. We have proved that if q≥(k-1)²(2^rω(q-1)-1)², then all the f_i(ξ) are primitive roots of GF(q) for some ξ∈GF(q), where k =degf₁+…+degf_r . With extending the results obtained by Wang Juping, Sun Qi and Han Wenbao respectively, we have also given a quantitative analysis of Carlitz's result.     

具有缺项和负系数的多元函数

Let P_k(p, A, B) be the class of functions (?) k ≥2 analytic in the unit disc E={z:|z|<1} and satisfying the condition |(zf′(z)/f(z)-p)/(Ap-Bzf(z)/f(z))|<1. for z∈E and -1≤Bk(p, A, B) under the assumption - 1 ≤B< 0 .     

The monodromy of a series of hypersurface singularities

Let {f=0} be a hypersurface inC ⁿ⁺¹ with a 1-dimensional singular set Σ. We consider the series of hypersurfaces {f+ɛx ^N=0} wherex is a generic linear form. We derive a formula, which relates the characteristic polynomials of the monodromies off andf+ɛx ^N. Other ingredients in this formula are the horizontal and the vertical monodromies of the transversal (isolated) singularities on each branch of the singular set. We use polar curves and the carrousel method in the proof. The formula is a generalization of the Iomdin formula for the Milnor numbers: μ(f+ɛx ^N )=μ _n (f)−μ _n −1(f)+Ne ₀(Σ)     

非周期函数类WrHω的宽度估计

In this paper, we obtain the strong assymptotic values of n-width d_n(W^rH^ω, C)and d_n(W^rH^ω, L) for non-periodic function class W^rH^ω, wheve ω is a concave modulus of continuity.     

关于一类单叶函数

A sharp coefficient estimate,distortion theorem and the radius of convexityare determined for the class R(α,β,A,B) of function f(z)=z+(?)a_nzⁿ satisfyingthe condition |(f′(z)-1)/(Bf′(z)-[B+(A-B)(1-α)])|<β for some α,β(0≤α<1,0<β≤1) and -1≤A相似文献   

论Szegǒ的定理

设f(z)=Z+a₂z²+…∈S.Szegǒ证明:S_n(z)=z+a₂z²+…+a_nzⁿ(n=2,3…)在|z|<1/4内单叶。ρ₀=1/4最好的,我们证明了更强的结果: 定理:若f(z)∈s.则s_n(z)(n=2,3…)在|z|<1/4内关于原点成星形。 当f∈S^*时为吴卓人所得。     

Local Well-posedness of the Derivative Schr¨odinger Equation in Higher Dimension for Any Large Data

In this paper, the authors consider the local well-posedness for the derivative Schr¨odinger equation in higher dimension u_t ? i△u + |u|²(?→γ · ?u) + u²(?→λ · ?u) = 0, (x, t) ∈ Rⁿ × R,?→γ ,?→λ ∈ Rⁿ; n ≥ 2.It is shown that the Cauchy problem of the derivative Schr¨odinger equation in higher dimension is locally well-posed in H^s(Rⁿ) (s > n/2) for any large initial data. Thus this result can compare with that in one dimension except for the endpoint space H^n/2.     

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.     

一类有理算术函数及其组合意义（英）

An arithmetical function f is said to be a rational arithmetical function of order (s,r) if there existcompletely multiplicative functions f₁,f₂,…,f_s and g₁,g₂,…,g_r such thatf=f₁*f₂*… *f_s*(g₁)^-1*(g₂)^-1*… *(g_r) -1 ,where * is the Dirichlet convolution. Recently, L.C. Hsu and Wang Jun studied combinatorial meanings of rational arithmetical functions of order (1,r) . We study these meanings in the setting of Narkiewicz's regular convolution.     

球面Schr?dinger算子的迹

Let S be a unit sphere in R³,Δ_θ,φ be the spherical Laplasian and q(θ,φ) be a continuous function on S. The following trace formulae are proved for the eigenvalues μ_n of L₀=-Δ_θ,φ and λ_n of L=L₀+q.     

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.