‎let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $a$ be an $n-$square quaternion matrix‎. ‎in this paper‎, ‎some results on the $k-$numerical range of $a$ are investigated‎. ‎moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $a$ are introduced‎, ‎and some of their algebraic properties are studied‎.

Tunable filters with a wide tunable range that cover the whole C-band , or L-band, or both, have been found wide applications in fiber optical communication system, mainly in three domain: tunable laser, wavelength division multiplexing systems (WDM), and channel monitoring. Due to the growing demand in bit-rate and number of channels in WDM system, tunable narrow band pass filters are required...

In this article we consider transient random walks on free products of graphs. We prove that the asymptotic range these exists and is strictly positive. particular, show varies real-analytically in terms probability measures constant support. Moreover, a central limit theorem associated with walk.