零点分割定理

本文把古典的多项式和整函数的零点分割定理推广到单位圆和右半平面去。§1 单位圆若 f(z)在|z|<1中有界解析,于是它在圆周|z|=ρ(ρ<1)上的几何平均(?)是(0,1)上的有界增加函数.记 s(f)=(?) I(ρ) (1)定理1 若函数 f(z)在|z|<1中有界解析,它在|z|<1中的零点全在实轴上,又若存在     

涉及超级亚纯函数的一个Picard型定理

设f(z)为复平面内极点重级至少为2、零点重级至少为3的亚纯函数,α(z):=β(z)exp(γ(z)),其中β(z)为非常数的椭圆函数,γ(z)为整函数.本文证明,如果σ(f(z))σ(α(z)),那么方程f′(z)=α(z)在复平面内有无穷多个根.     

Riemann Zeta函数的延拓表达式与部分零点近似值

利用围道积分法和Riemann Zeta函数的函数方程给出了Riemann Zeta函数的另一种积分表达式,该表达式可以将Riemann Zeta函数延拓到指定的右半平面.利用该表达式求出了ζ(2n)、ζ(1-2n)和ζ’(0),并且计算了Riemann Zeta函数非平凡零点的部分数值解.该积分表达式的引出丰富了与Riemann Zeta函数延拓表达式相关问题的研究.     

用矩阵符号函数解(广义)周期Sylvester方程

(广义)周期Sylvester方程来源于周期离散线性系统. 本文主要研究这类方程满足特征值分别位于开左半复平面和开右半复平面或位于单位圆周内和单位圆周外条件时用矩阵符号函数求解的数值方法.并通过数值例子说明我们的结论.     

半平面中有限阶解析函数的因子分解

与经典有限阶整函数的Hadamard因子分解定理和半平面中属于Hardy空间的解析函数的内外函数的因子分解类似,对右半平面中有限阶ρ解析函数f,可以分解为三个解析函数G,eQ和eg的乘积GeQeg,其中G是一个加权Blaschke乘积,Q是一个次数不超过ρ的多项式以及eg是一个加权外函数,log|G|,ReQ和Reg-log|f|在右半平面的边界恒为零．     

复方程f″ Af=0的解的零点充满圆

设f1和f2是复方程,f″ Af=0的两个线性无关解,其中A是一个整函数,记E=f1f2. 本文研究E的零点分布,建立E的零点充满圆的一些结果.     

开平面C上的亚纯函数与完备极小曲面的Gauss映射

设■:D→R~3确定了以等温参数表示的极小曲面M,其中D是全平面R~2的开子区域,那么极小曲面的Gauss映射g(z)是D上的亚纯函数.Xavier与Chao提出了一个尚未解决的问题:任意给定区域■上的亚纯函数g(z),它是否是某完备极小曲面的Gauss映射?本文证明了若开平面C上的亚纯函数g(z)的零点列或极点列的收敛指数小于1/2,则g(z)—定是某完备极小曲面的Gauss映射.     

关于ROUTH-HURWITZ问题

研究多项式零点在左半复平面分布的 Routh-Hurwitz 问题,在控制论中极为重要,故至今还有不少人在探讨研究。多项式零点全在开左半复平面已有充要条件,目前的研究主要是使其条件简化,便于验证.而多项式零点全在闭左半复平面的充要条件至今少见.Routh 本人将多项式的系数排成数表(即 Routh 表).给出了其零点     

关于二次型零点的注记

给出复系数和实系数n元二次型零点向量组的秩,以及最大零点子空间的维数.证明了正负惯性指数为p,q的实二次型的最大零点子空间的维数为n-max{p,q},以及秩为r的复二次型的最大零点子空间的维数为■.     

有限级超越整函数的(微-)差分多项式的零点分布

作者研究了有限级超越整函数的差分多项式和微-差分多项式的零点分布,在一定条件下得到了这些多项式的零点收敛指数的精确估计.所得结果可视为Hayman关于Picard例外值的经典结果的(微-)差分模拟.     

基于光滑化方法求解非线性l1问题

讨论了求解非线性l1问题的一种新的光滑函数法.通过对非线性l1问题模型的转化,将该问题化为一个不可微优化问题,据此提出了基于BFGS迭代的非线性l1问题的光滑函数法,介绍了非线性l1问题的光滑函数的有关性质、算法步骤及其收敛性.数值仿真显示了提出的光滑函数方法可以避免数值计算的溢出,具有一定的有效性.     

Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$-norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.     

On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.     

A Discrete Chebyshev Approximation Problem by Means of Several Polynomials

The problem of approximation of a given function on a given set by a polynomial of a fixed degree in the Chebyshev metric (the Chebyshev polynomial approximation problem) is a typical problem of Nonsmooth Analysis (to be more precise, it is a convex nonsmooth problem). It has many important applications, both in mathematics and in practice. The theory of Chebyshev approximations enjoys very nice properties (the most famous being the Chebyshev alternation rule). In the present article the problem of approximation of a given function on a given finite set of points by several polynomials is studied. As a criterion function, the maximin deviation is taken. The resulting functional is nonsmooth and nonconvex and therefore the problem becomes multiextremal and may have local minimizers which are not global ones. A necessary and sufficient condition for a point to be a local minimizer is proved. It is shown that a generalized alternation rule is still valid. Sufficient conditions for a point to be a strict local minimizer are established as well. These conditions are also formulated in terms of alternants. An exchange algorithm for finding a local minimizer is constructed. An k -exchange algorithm, allowing to find a "better" local minimizer is stated. Numerical examples illustrating the theory are presented.     

Chebyshev approximation of the null function by an affine combination of complex exponential functions

We describe the theoretical solution of an approximation problem that uses a finite weighted sum of complex exponential functions. The problem arises in an optimization model for the design of a telescope array occurring within optical interferometry for direct imaging in astronomy. The problem is to find the optimal weights and the optimal positions of a regularly spaced array of aligned telescopes, so that the resulting interference function approximates the zero function on a given interval. The solution is given by means of Chebyshev polynomials.     

On Shunkov groups with a strongly embedded subgroup

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.     

具有学习效应的超前有奖延误受罚的排序问题(英文)

本文考虑具有学习效应和共同交货期的单机排序问题.目标函数是加权超前有奖延误受罚总和.我们的目标是寻找一个最优序使得目标函数的值最小.由于该问题是NP-hard的,我们给出一些特殊情况下多项式时间可解的特例.同时在快速估计下界的基础上给出了分支定界算法来求一般情况下的最有排序.     

具有学习效应的超前有奖延误受罚的排序问题(英)

本文考虑具有学习效应和共同交货期的单机排序问题.目标函数是加权超前有奖延误受罚总和.我们的目标是寻找一个最优序使得目标函数的值最小.由于该问题是NP-hard的,我们给出一些特殊情况下多项式时间可解的特例.同时在快速估计下界的基础上给出了分支定界算法来求一般情况下的最有排序.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

Stieltjes-Newton型有理插值

Stieltjes型分叉连分式在有理插值问题中有着重要的地位，它通过定义反差商和混合反差商构造给定结点上的二元有理函数，我们将Stieltjes型分叉连分式与二元多项式结合起来，构造Stieltje- Newton型有理插值函数，通过定义差商和混合反差商，建立递推算法，构造的Stieltjes-Newton型有理插值函数满足有理插值问题中所给的插值条件，并给出了插值的特征定理及其证明，最后给出的数值例子，验证了所给算法的有效性．