Stability radius of linear parameter-varying systems and applications

In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459-472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278-2295]. Finally, we give two examples to illustrate the obtained results.     

On the solvability of a nonlocal boundary value problem with opposite fluxes at a part of the boundary and of the adjoint problem

We consider a nonlocal boundary value problem for the Laplace operator in a circular sector with opposite fluxes on the radii and with zero value of the solution on one of the radii; we also consider the adjoint problem. We prove the unique solvability of these problems and obtain the solution in an explicit form by the spectral method. As a by-product, we study the completeness and the basis property of systems of roots functions for problems of Samarskii-Ionkin type, which may be of interest in itself.     

On the solvability of a nonlocal boundary value problem with the equality of fluxes at a part of the boundary and of the adjoint problem

In the present paper, we consider a nonlocal boundary value problem for the Laplace operator in a circular sector with the equality of fluxes on the radii and with zero value of the solution on one of the radii. We also consider the adjoint problem. We prove the uniqueness of the solution of these problems and obtain an explicit form for the solution by the spectral method. When proving the solvability of the problems, we study the completeness and the basis property of systems of root functions for problems of the type of the Samarskii-Ionkin problem in L _p , which can be of interest in itself.     

有关单形旁切球半径的几何不等式（英文）

In this paper, we study the problems of geometric inequality for the radii of escribed hyperspheres of an n-dimensional simplex in Euclidean space En. Some new geometric inequalities for the radii of escribed hyperspheres of a simplex are established.     

On the optimal parameters of GMSSOR method for saddle point problems

For solving saddle point problems, SOR-type methods are investigated by many researchers in the literature. In this short note, we study the GMSSOR method for solving saddle point problems and obtain the optimal parameters which minimize the spectral (or pseudo-spectral) radii of the iteration matrices.     

C-numerical ranges and C-numerical radii

In this paper, we give a brief survey on C-numerical ranges and C-numerical radii. New results are obtained and open problems are mentioned.     

C-numerical ranges and C-numerical radii

In this paper, we give a brief survey on C-numerical ranges and C-numerical radii. New results are obtained and open problems are mentioned.     

Some radii problems related to certain p-valent meromorphic functions

The purpose of the present paper is to introduce several new classes of p-valent meromorphic functions defined by Ruscheweyh derivative operator for meromorphic multivalent functions. Further, we investigate the radii problems of these analytic classes in the punctured unit disk.     

Gauss-Newton法的收敛球与解的唯一性球

0　引　言本文研究非线性最小二乘问题min F( x)∶ =12 f( x) Tf ( x) ( EP)的 Gauss-Newton法的局部收敛性 ,其中 f:Rn→ Rm是 Frechet可微的 ,m≥ n.非线性最小二乘问题在数据拟合 ,参数估计和函数逼近等方面有广泛的应用 .在工程应用中也起到很大作用 ,例如在神经网络中 ,对小波问题 ,FP网络等方面的数据 (图形 )传输 ,数据 (图形 )压缩等方面有极其重要的理论和实际意义 .目前 ,求解最小二乘问题的最基本的方法之一是 Gauss-Newton法 [1 ]xn+1 =xn -[f′( xn) Tf′( x) ] - 1 f′( xn) Tf( xn) . ( GN)就我们所知 ,目前关于 Gau…     

Local convergence results of Gauss-Newton's like method in weak conditions

In this paper, we study the convergence of Gauss-Newton's like method for nonlinear least squares problems. Under the hypothesis that derivative satisfies some kind of weak Lipschitz condition, we obtain the sharp estimates of the radii of convergence ball of Gauss-Newton's like method and the uniqueness ball of the solution.     

Remarks on my paper: packing of incongruent circles on the sphere

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.     

Remarks on my paper: packing of incongruent circles on the sphere

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.     

Successive radii and Minkowski addition

In this paper we study the behavior of the so called successive inner and outer radii with respect to the Minkowski addition of convex bodies, generalizing the well-known cases of the diameter, minimal width, circumradius and inradius. We get?all possible upper and lower bounds for the radii of the sum of two convex bodies in terms of the sum of the corresponding radii.     

Successive radii and Minkowski addition

In this paper we study the behavior of the so called successive inner and outer radii with respect to the Minkowski addition of convex bodies, generalizing the well-known cases of the diameter, minimal width, circumradius and inradius. We get all possible upper and lower bounds for the radii of the sum of two convex bodies in terms of the sum of the corresponding radii.     

Quantitative stability analysis for vector problems of 0–1 programming

We consider multiple objective 0–1 programming problems in the situation where parameters of objective functions and linear constraints are exposed to independent perturbations. We study quantitative characteristics of stability (stability radii) of problem solutions. An approach to deriving formulae and estimations of stability radii is presented. This approach is applied to stability analysis of the linear 0–1 programming problem and problems with two types of nonlinear objective functions: linear absolute value and quadratic.     

On extremal decomposition problems

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the M?bius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 54–74.     

m重似星树的谱半径

仅有一个顶点的度大于2的树称为似星树.
在一棵似星树的每个一度点粘接一棵似星树构成的图称为$m$重似星树.
Gutman 和L. Shi给出了似星树谱半径的一个界.
在本文中我们给出了另外一个更简洁的证明方法并做了深入的讨论,
同时给出了$m$重似星树谱半径的一个最好界.     

A Perron-Frobenius theorem for positive polynomial operators in Banach lattices

In this paper, we extend the Perron-Frobenius theorem for positive polynomial operators in Banach lattices. The result obtained is applied to derive necessary and sufficient conditions for the stability of positive polynomial operators. Then we study stability radii: complex, real and positive radii of positive polynomial operators and show that in this case the three radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

Control of pore radius regulation for electroporation-based drug delivery

Electroporation uses electric pulses to create transient, nonselective pores in a cell’s membrane, allowing drugs to be delivered into the targeted cells. To ensure proper uptake of drug molecules, it is essential to control the radii of the pores and the time duration, for which the pores remain open. Electroporation is intrinsically a nonlinear dynamic process. A careful analysis of the electroporation dynamics reveals that, under variation of the magnitude of the input voltage, the equilibrium pore radius undergoes a pair of saddle-node bifurcations. As a result, there exists a range of pore radii that is physically unstable and thus cannot be maintained in conventional experiments. The bifurcations and the associated unstable regime impose restrictions on the operation of electroporation, limiting the sizes of deliverable drug particles. To overcome these problems, we design a novel control strategy to stabilize the originally unstable solutions. In contrast to the conventional control algorithms based on local stability analysis, the present control is globally stable. Numerical examples show that the control eliminates the original bifurcations and allows one to achieve a wide range of pore radii. The robustness and effectiveness of the control strategy would potentially enhance the application of electroporation.     

Robust stability of positive linear systems in Banach spaces

In this paper, we study the stability radii of positive linear systems with delays with respect to various classes of perturbations in infinite dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.