Stability analysis for neural networks with inverse Lipschitzian neuron activations and impulses

In this paper, a new concept called α-inverse Lipschitz function is introduced. Based on the topological degree theory and Lyapunov functional method, we investigate global convergence for a novel class of neural networks with impulses where the neuron activations belong to the class of α-inverse Lipschitz functions. Some sufficient conditions are derived which ensure the existence, and global exponential stability of the equilibrium point of neural networks. Furthermore, we give two results which are used to check the stability of uncertain neural networks. Finally, two numerical examples are given to demonstrate the effectiveness of results obtained in this paper.     

A simple proof of the linear instability of rotating liquid drops

We give a simplified proof of the linear instability of equilibrium figures of rotating liquid based on energy estimates.      

Assessment of regularized delta functions and feedback forcing schemes for an immersed boundary method

We present an improved immersed boundary method for simulating incompressible viscous flow around an arbitrarily moving body on a fixed computational grid. To achieve a large Courant–Friedrichs–Lewy number and to transfer quantities between Eulerian and Lagrangian domains effectively, we combined the feedback forcing scheme of the virtual boundary method with Peskin's regularized delta function approach. Stability analysis of the proposed method was carried out for various types of regularized delta functions. The stability regime of the 4‐point regularized delta function was much wider than that of the 2‐point delta function. An optimum regime of the feedback forcing is suggested on the basis of the analysis of stability limits and feedback forcing gains. The proposed method was implemented in a finite‐difference and fractional‐step context. The proposed method was tested on several flow problems, including the flow past a stationary cylinder, inline oscillation of a cylinder in a quiescent fluid, and transverse oscillation of a circular cylinder in a free‐stream. The findings were in excellent agreement with previous numerical and experimental results. Copyright © 2008 John Wiley & Sons, Ltd.     

On two models in the three-dimensional theory of stability of composites

A comparative analysis is made of the infinite-fiber and finite-fiber models in the three-dimensional theory of stability of composites. The results analyzed have been obtained using the three-dimensional linearized theory of stability of deformable bodies. A historical sketch is given of the theory of stability for and approaches used in the mechanics of laminated and fibrous composite materials Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 3–31, August 2008.     

Global parametric quadratic stabilizability of nonlinear systems with uncertainty

Sufficient conditions for the global PQ-stabilizability of a nonlinear system with uncertainty are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 126–133, June 2008.     

带时滞的高阶Cohen-Grossberg神经网络的指数稳定性

就一类高阶时滞Cohen-Grossberg社昆网络进行研究,假设反应函数满足Lipschitz连续且有界,用非线性测度的方法得到了关于平衡点的存在性和惟一性的一种新的充分性的判别条件,同时,通过构造一个合适的Lyapunov函数,得到这个条件也保证了时滞神经网络的全局指数稳定性.     

Analysis on global exponential robust stability of reaction-diffusion neural networks with S-type distributed delays

To avoid the unstable phenomena caused by time delays and perturbations, we investigate the sufficient conditions to ensure the global exponential robust stability with a convergence rate for the reaction-diffusion neural networks with S-type distributed delays. Because S-type distributed delays lead to some difficulty, we also introduce a new generalized Halanay inequality and a novel method-system-approximation method into the qualitative research of neural networks. Moreover, the sufficient criteria provided here, which are rather accessible and feasible, have wider adaptive range.     

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Hilbert spaces expanded with a unitary operator

We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show that the theory of such a structure is ω ‐stable and admits quantifier elimination. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Robustness of Gaussian Elimination with Partial Pivoting

It has been recently shown that large growth factors might occur in Gaussian Elimination with Partial Pivoting (GEPP) also when solving some plausibly natural systems. In this note we argue that this potential problem could be easily solved, with much smaller risk of failure, by very small (and low cost) modifications of the basic algorithm, thus confirming its inherent robustness. To this end, we first propose an informal model with the goal of providing further support to the comprehension of the stability properties of GEPP. We then report the results of numerical experiments that confirm the viewpoint embedded in the model. Basing on the previous observations, we finally propose a simple scheme that could be turned into (even more) accurate software for the solution of linear systems.