Global exponential stability of a class of neural networks with unbounded delays

In this paper, the global exponential stability of a class of neural networks is investigated. The neural networks contain variable and unbounded delays. By constructing a suitable Lyapunov function and using the technique of matrix analysis, some new sufficient conditions on the global exponential stability are obtained.Досліджено глобальну експоненціальну стійкість одного класу нейронних сіток. Нейронні сітки містять змінні та необмежені загаювання. На основі побудови відповідної функції Ляпунова та техніки матричного аналізу отримано нові достатні умови глобальної експоненціальної стійкості

Applications of Lp−LqL^p-L^q estimates for solutions to semi-linear σ\sigma-evolution equations with general double damping

In this paper, we would like to study the linear Cauchy problems for semi-linear $\sigma$-evolution models with mixing a parabolic like damping term corresponding to $\sigma_1 \in [0,\sigma/2)$ and a $\sigma$-evolution like damping corresponding to $\sigma_2 \in (\sigma/2,\sigma]$. The main goals are on the one hand to conclude some estimates for solutions and their derivatives in $L^q$ setting, with any $q\in [1,\infty]$, by developing the theory of modified Bessel functions effectively to control oscillating integrals appearing the solution representation formula in a competition between these two kinds of damping. On the other hand, we are going to prove the global (in time) existence of small data Sobolev solutions in the treatment of the corresponding semi-linear equations by applying $(L^{m}\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimates, with $q\in (1,\infty)$ and $m\in [1,q)$, from the linear models. Finally, some further generalizations will be discussed in the end of this paper.Comment: 38 page

Opportunistic secure transmission for wireless relay networks with modify-and-forward protocol

This paper investigates the security at the physical layer in cooperative wireless networks (CWNs) where the data transmission between nodes can be realised via either direct transmission (DT) or relaying transmission (RT) schemes. Inspired by the concept of physical-layer network coding (PNC), a secure PNC-based modify-and-forward (SPMF) is developed to cope with the imperfect shared knowledge of the message modification between relay and destination in the conventional modify-and-forward (MF). In this paper, we first derive the secrecy outage probability (SOP) of the SPMF scheme, which is shown to be a general expression for deriving the SOP of any MF schemes. By comparing the SOPs of various schemes, the usage of the relay is shown to be not always necessary and even causes a poorer performance depending on target secrecy rate and quality of channel links. To this extent, we then propose an opportunistic secure transmission protocol to minimise the SOP of the CWNs. In particular, an optimisation problem is developed in which secrecy rate thresholds (SRTs) are determined to find an optimal scheme among various DT and RT schemes for achieving the lowest SOP. Furthermore, the conditions for the existence of SRTs are derived with respect to various channel conditions to determine if the relay could be relied on in practice

On asymptotic properties of solutions to σ\sigma-evolution equations with general double damping

In this paper, we would like to consider the Cauchy problem for semi-linear $\sigma$-evolution equations with double structural damping for any $\sigma\ge 1$. The main purpose of the present work is to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp interplay of ``parabolic like models" corresponding to $\sigma_1 \in [0,\sigma/2)$ and ``$\sigma$-evolution like models" corresponding to $\sigma_2 \in (\sigma/2,\sigma]$, which together appear in an equation. In this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.Comment: 29 page