• Bulletin of the Korean Mathematical Society
• /
• v.53 no.3
• /
• pp.641-649
• /
• 2016

A linear functional T on a $Fr{\acute{e}}echet$ algebra (A, (pn)) is called almost multiplicative with respect to the sequence ($p_n$), if there exists ${\varepsilon}{\geq}0$ such that ${\mid}Tab-TaTb{\mid}{\leq}{\varepsilon}p_n(a)p_n(b)$ for all $n{\in}\mathbb{N}$ and for every $a,b{\in}A$. We show that an almost multiplicative linear functional on a $Fr{\acute{e}}echet$ algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous $Fr{\acute{e}}echet$ algebra is continuous.

• Honam Mathematical Journal
• /
• v.23 no.1
• /
• pp.59-66
• /
• 2001

A class of topological algebras, which we call it a fundamental one, has already been introduced to generalize the locally bounded and locally convex algebras. To prove the basic theorems on fundamental algebras, the first successful step is the new version of the Cohen factorization theorem. Here we recall it and prove some new results on fundamental topological algebras.

• KSII Transactions on Internet and Information Systems (TIIS)
• /
• v.10 no.9
• /
• pp.4342-4366
• /
• 2016

Congestion control in Cluster Wireless Sensor Networks (CWSNs) has drawn widespread attention and research interests. The increasing number of nodes and scale of networks cause more complex congestion control and management. Active Queue Management (AQM) is one of the major congestion control approaches in CWSNs, and Random Early Detection (RED) algorithm is commonly used to achieve high utilization in AQM. However, traditional RED algorithm depends exclusively on source-side control, which is insufficient to maintain efficiency and state stability. Specifically, when congestion occurs, deficiency of feedback will hinder the instability of the system. In this paper, we adopt the Additive-Increase Multiplicative-Decrease (AIMD) adjustment scheme and propose an improved RED algorithm by using neighbor feedback and scheduling scheme. The congestion control model is presented, which is a linear system with a non-linear feedback, and modeled by Lur'e type system. In the context of delayed Lur'e dynamical network, we adopt the concept of cluster synchronization and show that the congestion controlled system is able to achieve cluster synchronization. Sufficient conditions are derived by applying Lyapunov-Krasovskii functionals. Numerical examples are investigated to validate the effectiveness of the congestion control algorithm and the stability of the network.