典型實照函數的最小?妥钚∑

<正> 1.如果函數f(z)在包含實軸土某一區間的區域B中是正則的,f(z)在此實軸區間上取實值.在區域B的其餘地方f(z)與(z)同符號;即     

一类正则函数的性质

设函数 f( z) =z+a2 z2 +…在单位圆盘 D={z |z|<1 }中正则 ,我们记这种函数的全体为 N,设β>0 ,令Sβ=f ( z) f ( z)∈ N且 (β-1 ) zf′( z)f ( z) -1 +1 +zf″( z)f ( z) 1 +4 z+z21 -z2 .本文给出了 Sβ 中函数的一些性质     

多复变数Einstein空间中的Schwarz引理

<正> 本文主要的目的是来证明定理1.设■域是 n 个复变数■=(z~1,…,z~n)空间中的简单域且为Einstein空间(不失一般性,不妨假设其 Ricci 曲率为-1),其Bergman度量为     

A Theory of Super-Resolution from Short-Time Fourier Transform Measurements

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, \(\Delta \), between spikes is not too small. Specifically, for a measurement cutoff frequency of \(f_c\), Donoho (SIAM J Math Anal 23(5):1303–1331, 1992) showed that exact recovery is possible if the spikes (on \(\mathbb {R}\)) lie on a lattice and \(\Delta > 1/f_c\), but does not specify a corresponding recovery method. Candès and Fernandez-Granda (Commun Pure Appl Math 67(6):906–956, 2014; Inform Inference 5(3):251–303, 2016) provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus \(\mathbb {T}\)), which succeeds provably if \(\Delta > 2/f_c\) and \(f_c \ge 128\) or if \(\Delta > 1.26/f_c\) and \(f_c \ge 10^3\), and does not need the spikes within the fundamental period to lie on a lattice. In this paper, we develop a theory of super-resolution from short-time Fourier transform (STFT) measurements. Specifically, we present a recovery method similar in spirit to the one in Candès and Fernandez-Granda (2014) for pure Fourier measurements. For a STFT Gaussian window function of width \(\sigma = 1/(4f_c)\) this method succeeds provably if \(\Delta > 1/f_c\), without restrictions on \(f_c\). Our theory is based on a measure-theoretic formulation of the recovery problem, which leads to considerable generality in the sense of the results being grid-free and applying to spike trains on both \(\mathbb {R}\) and \(\mathbb {T}\). The case of spike trains on \(\mathbb {R}\) comes with significant technical challenges. For recovery of spike trains on \(\mathbb {T}\) we prove that the correct solution can be approximated—in weak-* topology—by solving a sequence of finite-dimensional convex programming problems.     

一族特殊的星像函數

<正> 設函數w=f(z)在單位圓|z|<1中是正則的.f(0)=0,f′(0)=1.假如f(z)是單葉的,那末w=f(z)映照|z|<1於w平面上的單葉的像D_f.記這種單葉函數的全體為S.若D_f以原點w=0為星形中心,就稱f(z)是|z|<1中的星     

不定方程组{11x~2-9y~2=2,40y~2-11z~2=29}的正整数解的上界

运用Baker方法得到了不定方程组{11x~2-9y~2=2,40y~2-11z~2=29}的正整数解的上界.上界为(x,y,z)=(0.9×2118406,2118406,0.52×2118406).     

关于Diophantine方程a~x+b~y=z~2

设a=2~r,b=p~s,其中p是给定的奇素数,r和s是给定的正整数.运用有关三项Diophantine方程和广义Ramanujan-Nagell方程的结果,将方程a~x+~y=z~2的所有正整数解(x,y,z)进行了分类,从而得出了这些解的可有效计算的上界.     

复数平方根z~(1/2)的有效算法

本文为复数平方根之主值z~(1/2)提供一适用于计算机的有效算法.该算法除已克服有效数字消失、从而能确保计算结果的精度之外,还有处理简单统一、适用范围广的优点.在附录中给出了极坐标与直角坐标互换的实用算法.     

Bergman空间上一类解析Toeplitz算子的强不可约性

令D为单位圆盘D={z∈C:|z|<1},L_a~2(D)为L~2(D)中解析函数构成的Bergman空间.设f(z)=a_0+a_1z+a_2z~2+…,用算子理论的技巧给出解析Toeplitz算子T_f为强不可约算子的一个充分条件.     

單葉函數的係數與開始多項式

<正> 以fk(z)表單位圓內的K次對稱單葉全純函數,亦即fk(z)=z+a_I~((k))z~(k+1)+a_2~((k))z~(2k+1)+…,|z|<1.以S_k表此種函數之全體.特別,書S以代S_1.     

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

As is well known, the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z) = c_1 z~2+ c_2 z + c_3 or x(z) = c_1 q~z+ c_2q~(-z)+ c_3 are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler's Beta formula, Cauchy' Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.     

有穷模态系统的基本系统

<正> 引论本文是继作者前文“具有有穷个模态外的模态系统”(以后省称“前文”)而作的,但同时又是自足的;为此,我们把模态数列的基本性质(见§1)以及求加强系统的方法(见§3)重述一遍,这样,读者可以不靠前文而完全很順利地阅读本文.除这两处以外,本文基本上又与前文沒有重复,这样,有好些需要详释或需与其它部门相联系的地方(例如,模     

积分算子作用下的亚纯多叶函数

令∑_p表示形如f(z)=z~(-p)+∑m=1∞(p∈N={1,2,3…})且在去心单位开圆盘D=U\{0}={z∶z∈C且0|z|1}上解析的亚纯多叶函数类.利用一个作用在∑_p上的乘积算子定义了几个新的亚纯函数的子类,并考虑这些函数类在积分算子作用下的性质.     

Sine, cosine transforms and classical function classes

We study the continuity and smoothness properties of functions f ∈ L ¹([0, ∞)) whose sine transforms $ \hat f_s $ and cosine tranforms $ \hat f_c $ belong to L ¹([0,∞)). We give best possible sufficient conditions in terms of $ \hat f_s $ and $ \hat f_c $ to ensure that f belongs to one of the Lipschitz classes Lip α and lip α for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg α and zyg α for some 0 < α ≤ 2. The conditions given by us are not only sufficient, but also necessary in the case when the sine and cosine transforms are nonnegative. Our theorems are extensions of the corresponding theorems by Boas from sine and cosine series to sine and cosine transforms.     

On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method

Radial basis function method is an effective tool for solving differential equations in engineering and sciences. Many radial basis functions contain a shape parameter c which is directly connected to the accuracy of the method. Rippa [1] proposed an algorithm for selecting good value of shape parameter c in RBF-interpolation. Based on this idea, we extended the proposed algorithm for selecting a good value of shape parameter c in solving time-dependent partial differential equations.     

多複變數非歐空間中黎曼曲率的估值問題

<正> §1.引言 用(z)=(z~1,…,z~n)代表n個複變数,並命z~k=x~k+iy~k,此處x~k,y~k(1≤k≤n)是實數。代表x~1,…,x~n,y~1,…,y~n所定義的2n維空間中的一個域。我們現在並不假定它是受囿,抑單連通等性質。命     

斑须蝽种群空间分布格局Weibull分布的拟合

Weibull分布的概率密度函数为f(x) =(c/b) [(x -a) /b]c -1exp [(x a) /b]c ,x≥a。本文首次用于拟合班须蝽三代卵块的空间分布 ,8批抽样数据拟合结果表明班须蝽三代卵块在烟田的空间分布遵循Weibull分布。从而丰富了班须蝽种群空间格局的分布理论。同时 ,利用斑须蝽种群空间格局的资料探讨了Weibull分布的参数b、c与种群密度及种群聚集度之间的关系 ,结果表明 ,尺度参数b与种群密度、种群聚集度间均分别存在极显著的线性相关关系 ,形状参数c与种群密度存在极显著的正幂函数相关关系 ,与种群聚集度之间存在极显著负幂函数关系。     

与正弦函数有关的星形函数逆的Toeplitz行列式

Let S*s be the class of normalized functions f defined in the open unit U = {z ∈ C:丨z丨 ＜ 1} such that the quantity zf'(z)/f(z)lies in an eight-shaped region in ...     

BLOCK-SYMMETRIC AND BLOCK-LOWER-TRIANGULAR PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS＊

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.     

极限方程为椭圆-抛物的四阶椭圆型方程的奇摄动

本文研究了极限方程为椭圆-抛物的四阶椭圆型方程-ε2Δ2u+ym?2u/?y2+?2u/?x2+a(x,y)?u/?y+b(x,y)?u/?x+c(x,y)=0的奇摄动问题,其中ε为正的小参数,m为正的实数,Δ为拉普拉斯算子,a,b,c充分光滑.在适当的假设下,导出可解性的充分条件,证明了解的存在和给出任意阶的一致有效的渐近解.