An energy efficient routing protocol is the major attentiveness for researcher in field of Wireless Sensor Networks (WSNs). In this paper, we present some energy efficient hierarchal routing protocols, prosper from conventional Low Energy Adaptive Cl
ustering Hierarchy (LEACH) routing protocol. Fundamental objective of our consideration is to analyze, how these ex- tended routing protocols work in order to optimize lifetime of network nodes and how quality of routing protocols is improved for WSNs. Furthermore, this paper also emphasizes on some issues experienced by LEACH and also explains how these issues are tackled by other enhanced routing protocols from classi- cal LEACH. We analytically compare the features and performance issues of each hierarchal routing protocol. We also simulate selected clustering routing protocols for our study in order to elaborate the enhancement achieved by ameliorate routing protocols.
We know that $mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $mathbb{Z}^m_n= mathbb{Z}_n timesmathbb{Z}_ntimes...timesmathbb{Z}_n$ be the Cartesian product of $m$ rings $mathbb{Z}_n$. In this not
e, we present the action of $SL(m, mathbb{Z}_n)={A in mathbb{Z}^{m,m}_{n} : det A equiv 1 (modsimn)}$, where $SL(m, mathbb{Z}_n)$ for $ngeq 2$ is a group under matrix multiplication modulo $n$, on the ring $mathbb{Z}^m_n$ as a right multiplication of a row vector of $mathbb{Z}^m_n$ by a matrix of $SL(m, mathbb{Z}_n)$ to determine the orbits of the ring $mathbb{Z}^m_n$. This work is an extension of [1]
Extended modular group $bar{Pi}=<R,T,U:R^2=T^2=U^3=(RT)^2=(RU)^2=1>$, where $ R:zrightarrow -bar{z}, sim T:zrightarrowfrac{-1}{z},simU:zrightarrowfrac{-1}{z +1} $, has been used to study some properties of the binary quadratic forms whose base points
lie in the point set fundamental region $F_{bar{Pi}}$ (See cite{Tekcan1, Flath}). In this paper we look at how base points have been used in the study of equivalent binary quadratic forms, and we prove that two positive definite forms are equivalent if and only if the base point of one form is mapped onto the base point of the other form under the action of the extended modular group and any positive definite integral form can be transformed into the reduced form of the same discriminant under the action of the extended modular group and extend these results for the subset $QQ^*(sqrt{-n})$ of the imaginary quadratic field $QQ(sqrt{-m})$.
