齐次线性方程组有基础解系的反问题

本文给出了齐次线性方程组存在基础解系的逆定理及其证明 ,同时也给出了由线性无关向量组构造齐次线性方程组的一般方法步骤 .     

Research on the properties of some planar polynomial differential equations

In this article, we use the new method of reflecting function to study the behavior of solutions of nonlinear time-vary differential equations, and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional. We applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center.     

ON THE EQUIVALENCE OF AUTONOMOUS AND NONAUTONOMOUS DIFFERENTIAL SYSTEMS

Using reflecting function of Mironenko we construct some nonautonomous differential systems which are equivalent to the given autonomous differential systems. The results are applied to discuss the existence and stability of periodic solution of these nonautonomous nonlinear differential systems.     

On the problem concerning the points of semicontinuity of the stabilization index

We construct examples of families of linear homogeneous differential systems with coefficients continuously depending on a numerical parameter for which the absence of some form of semicontinuity of the stabilization index treated as a function of this parameter is typical.     

Solving the maximum problem for the terminal norm of a linear control system

We consider the maximization problem for the terminal norm of a linear control system. We seek for the extremal controls and improve them. Based on a sufficient optimality condition, we construct procedures that use the differentiability of the support function of the set of attainability.     

凸函数的支撑凸集及其集映射

本文研究了线性空间中凸函数的支撑泛函存在性以及支撑泛函的数值域,利用子空间中支撑泛函延拓的方法,构造出在线性空间任意点的支撑泛函,确定在同一支撑点上支撑泛函的数值域,从而得到支撑泛函具有唯一性的充分必要条件,最后对支撑凸集到支撑泛函集的集映射进行讨论.     

Solvability of linear equations within weak mixing sets

We introduce a new class of “random“ subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that any partition-regular system of linear equations with integer coefficients is solvable in any WM set.     

Stein iterations for the coupled discrete-time Riccati equations

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. Two iterations for computing a symmetric (maximal) solution of this system are investigated. We construct sequences of the solutions of the decoupled Stein equations and show that these sequences converge to a solution of the considered system. Numerical experiments are given.     

Equivalence of Differential System

Using reflecting function of Mironenko we construct some differential systems which are equivalentto the given differential system.This gives us an opportunity to find out the monodromic matrix of these periodicsystems which are not integrable in finite terms.     

Stäckel transform of Lax equations

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.     

Periodic reflecting function of a nonlinear quasiperiodic differential system with a two-frequency basis

We obtain necessary and sufficient conditions under which nonlinear quasiperiodic differential systems with a two-frequency basis are equivalent, in the sense of the coincidence of reflecting functions, to periodic differential systems and have a periodic reflecting function.     

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.     

Stability of Linear Parameter Varying and Linear Switching Systems

We consider stability of families of linear time‐varying systems, that are determined by a set of time‐varying parameters which adhere to certain rules. The conditions are general enough to encompass on the one hand stability questions for systems that are frequently called linear parameter varying systems in the literature and on the other hand also linear switching systems, in which parameter variations are allowed to have discontinuities. Combinations of these two sets of assumptions are also possible within the framework studied here. Under the assumption of irreducibility of the sets of system matrices, we show how to construct parameter dependent Lyapunov functions for the systems under consideration that exactly characterize the exponential growth rate. It is clear that such Lyapunov functions do not exist in general. But every system of our class can be reduced to a finite number of subsystems for which irreducibility holds.     

矩阵约当标准化的一个新方法

在线性和非线性问题的研究中,常需要构造一个基,使线性算子T在此基础下的矩阵表示为约当标准型,本文介绍了构造这种基的一个方法,我们从T的每个特征向量开始,通过求解一系列线性方程组而求得广义特征向量的一个链,将所有这种链放在一起,便构成想要的一组基,与通常的方法相比,这一方法较易操作,计算量小。     

Julia operators and linear systems (nonuniqueness of linear system)

Complementation theory in Krein spaces can be extended for any self-adjoint transformation. There is a close relation between Julia operators and linear systems. The theory of Julia operators can be used to construct distinct Krein spaces which are the state spaces of extended canonical linear systems with given transfer function.     

Newton iterations in implicit time-stepping scheme for differential linear complementarity systems

We propose a generalized Newton method for solving the system of nonlinear equations with linear complementarity constraints in the implicit or semi-implicit time-stepping scheme for differential linear complementarity systems (DLCS). We choose a specific solution from the solution set of the linear complementarity constraints to define a locally Lipschitz continuous right-hand-side function in the differential equation. Moreover, we present a simple formula to compute an element in the Clarke generalized Jacobian of the solution function. We show that the implicit or semi-implicit time-stepping scheme using the generalized Newton method can be applied to a class of DLCS including the nondegenerate matrix DLCS and hidden Z-matrix DLCS, and has a superlinear convergence rate. To illustrate our approach, we show that choosing the least-element solution from the solution set of the Z-matrix linear complementarity constraints can define a Lipschitz continuous right-hand-side function with a computable Lipschitz constant. The Lipschitz constant helps us to choose the step size of the time-stepping scheme and guarantee the convergence.     

Wavelets for iterated function systems

We construct a wavelet and a generalised Fourier basis with respect to some fractal measure given by a one-dimensional iterated function system. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Dutkay, Jorgensen, and Pedersen to the non-linear setting. As a byproduct we are able to provide a Fourier basis also for such linear fractals like the Middle Third Cantor Set which have been left out by previous approaches.     

Completeness theorem for singular differential pencils

A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.