A Comparative analysis of Green’s functions of 1D matching equations for motion synthesis

When filtering an input image, the Green’s functions of matching equations are capable of inducing a broad class of motions, a property that has led to their use in several computer graphics and computer vision applications. In all such applications, the Green’s functions of second-order differential equations have been considered, even though no justification has been given for their preference over simpler, first-order equations. Here we present a study of first-order one-dimensional matching equations, both in the uniform and in the affine motion models. Comparing their Green’s functions with those of the corresponding second-order cases, we find evidence for the latter’s superiority in motion synthesis. We also propose and discuss a general discretization scheme for Green’s functions of one-dimensional matching equations, showing that the affine motion model is particularly sensitive to the sampling frequency. In this case, we advocate the use of area sampling, for allowing realistic motion simulations.     

Numerical convergence and physical fidelity analysis for Maxwell’s equations in metamaterials

In this paper, we develop a leap-frog mixed finite element method for solving Maxwell’s equations resulting from metamaterials. Our scheme is similar to the popular Yee’s FDTD scheme used in electrical engineering community, and is preferable for three dimensional large scale modeling since no storage of the large coefficient matrix is needed. Our scheme is proved to obey the Gauss’s law automatically if the initial fields satisfy that. Furthermore, the conditional stability and optimal error estimate for the proposed scheme are proved. To our best knowledge, we are unaware of any other publications devoted to the convergence analysis of this leap-frog explicit scheme for Maxwell’s equations even in a simple medium, while our results for metamaterials automatically reduce to the standard Maxwell’s equations in vacuum by dropping some terms resulting from the constitutive equations. Numerical results confirming our analysis are presented.     

A semi-local convergence theorem for a robust revised Newton’s method

It is well known that Newton’s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newton’s method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton–Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newton’s method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newton’s method.     

An extension of LaSalle’s Invariance Principle for a class of switched linear systems

In this paper LaSalle’s Invariance Principle for switched linear systems is studied. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, in this paper the switching modes are allowed to be only Lyapunov stable. Under certain ergodicity assumptions, an extension of LaSalle’s Invariance Principle for global asymptotic stability of switched linear systems is proposed provided that the kernels of derivatives of a common quadratic Lyapunov function with respect to the switching modes are disjoint (except the origin).     

Precise Hsu’s method for analyzing the stability of periodic solutions of multi-degrees-of-freedom systems with cubic nonlinearity

This paper presents a new precise Hsu’s method for investigating the stability regions of the periodic motions of an undamped two-degrees-of-freedom system with cubic nonlinearity. Firstly, the incremental harmonic balance (IHB) method is used to obtain the solution of nonlinear vibration differential equations. Hsu’s method is then adopted for computing the transition matrix at the end of one period, and the precise time integration algorithm is adjusted to improve the computational precision. The stability regions of the system obtained from the precise Hsu’s, Hsu’s and improved numerical integration methods are compared and discussed.