On the Inverse Problem for Differential Operators on a Finite Interval with Complex Weights

Mathematical Notes - Inverse problems of spectral analysis for second-order differential operators on a finite interval with complex-valued weights and with an arbitrary number of discontinuity...     

有限区间上积分-微分算子的逆结点问题

研究Dirichlet边界条件下的积分-微分算子逆结点问题.证明了积分-微分算子稠定的结点子集能够唯一确定[0,π]上的势函数q(x)和区域Do上的积分扰动核M(x-t)且给出了这个逆结点问题的解的重构算法.     

Inverse Problem for Quasi-Periodic Differential Pencils with Jump Conditions Inside the Interval

Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.     

Isochronicity Problem of Higher-Order Singular Point for Polynomial Differential Systems

Due to the difficulty, the isochronicity problems with respect to higher-order singular point (or degenerate singular point) of polynomial differential systems are far from being solved. The calculation of period constants is an effective way to find necessary conditions for isochronicity. In this paper, by means of a homeomorphic transformation, higher-order singular point is transferred into the origin. At the same time, a new recursive algorithm to compute period constants at the origin of the transformed system is deduced which is easy to realize with the computer algebraic system such as MATHEMATICA or MAPLE. Finally, to illustrate the effectiveness of our algorithm, the pseudo-isochronous center conditions of higher-order singular point for a class of septic system are investigated. Our work is new in terms of research about the isochronicity problem of higher-order singular point and consists of the existing results related to the origin as a special case when it is an elementary singular point.     

An Extremal Problem Concerning the Derivatives of a Polynomial at Its Roots

Let w be a monic polynomial of degree n +1 with roots x_j in the interval [−1, 1]. We consider the problem of finding the roots x_j for which the minimum of ¦ w '( x_j )¦, for 0≤ j ≤ n , is as large as possible. We prove that the Clenshaw–Curtis points cos( j π/ n ) are the only solution when n is even and that they get asymptotically close to the solution for odd values of n , as n goes to infinity. Our problem is related to the problem of minimizing the norm of inverse Vandermonde matrices.     

On the Solvability of the Main Inverse Problem for Stochastic Differential Systems

Ukrainian Mathematical Journal - By using the quasiinversion method, we establish necessary and sufficient conditions for the solvability of the main (according to Galiullin’s classification)...     

线性微分方程的一个反问题

讨论了线性微分方程的一个反问题.给定一个线性无关的函数组,可以得到一个高阶线性微分方程,该微分方程的基本解组恰好是此函数组.另外,对一阶线性微分方程组也进行了类似的讨论.     

An Iterative Algorithm for the Coefficient Inverse Problem of Differential Equations

In this paper, an iterative algorithm for solving a coefficient inverse problem is submitted. The key of the method is to project an unknown coefficient function on a finite dimensional function space. Thus, the inverse problem can be changed into a nonlinear algebraic system of equations.     

退化时滞微分系统的特征根分布与指数稳定

该文首先研究了退化时滞微分系统的特征根分布, 指出如果退化时滞微分系统的所有特征根都具有负实部, 在一个条件下, 特征根的负实部的最大值为负.由此可以得到一个条件, 在该条件下如果所有特征根都具有负实部, 则退化时滞微分系统的解是指数稳定的.作为例子, 对中立型给出其解为指数稳定的条件.     

An Inverse Problem for Differential Operators of the Orr‐Sommerfeld Type

We study the inverse problem of recovering differential operators of the Orr‐Sommerfeld type from the Weyl matrix. Properties of the Weyl matrix are investigated, and an uniqueness theorem for the solution of the inverse problem is proved.     

双曲方程的一个反问题

用(E_phHa)代表如下的双曲方程的初一边值问题:设u(x,t)是其解,求满足u(t,ξ)=v(t,ξ)(0qilb)的解.     

有限偏序集上格的M?bius函数和特征多项式

在一些有限格上研究了格的特征多项式,确定了它们的表示式,并且得到这些格的M(o|¨)bius函数.     

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

We consider an eigenvalue problem for a system on [0, 1]: $$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}} {{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array} } \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) = \lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi ^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array} } \right.$$ with constants $$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p₂₁, p₂₂ are known, we prove a uniqueness theorem and provide a reconstruction formula for p₁₁ and p₁₂ from the spectral characteristics consisting of one spectrum and the associated norming constants.     

Synthesis of Control Systems with a Specified Characteristic Polynomial

A synthesis method is proposed for single-loop modal control systems for discrete and continuous plants, including plants with delay. The controllers are designed in the operator space given the closed-loop transfer function. The proposed method can be used to synthesize both free and forced motion in a closed-loop system.     

退化时滞中立型微分系统的特征根分布与指数稳定

本文首先讨论了两类具有一般意义的线性自治中立型时滞微分系统特征根链的分布情况,并且给出特征根链的逼近链拟合曲线,然后重点讨论多时滞退化中立型微分系统的特征根链分布,由此得到多时滞退化中立型微分系统指数稳定的条件.     

Inverse Problem for a Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems for the solution of an inverse problem for a fourth-order differential operator with nonseparated boundary conditions are proved. The spectral data for the problem is specified as the spectrum of the problem itself (or its three eigenvalues) and the spectral data of a system of three problems.