The system being modelled is assumed to occupy one and only one state at any moment in time and its evolution is represented by transitions from state to state. Also, the physical or mathematical behaviour of this system may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this work, the concept of the classification of groups of states, between states that are recurrent, meaning that the Markov chain is guaranteed to return to these states infinitely often, and states that are transient, meaning that there is a nonzero probability that the Markov chain will never return to such a state are investigated, in order to provide some insight into the performance measure analysis such as the mean first passage time, 𝑅𝑖𝑗, the mean recurrence time of state 𝑅𝑗𝑗 as well as recurrence iterative matrix 𝑅(𝑘+1). Our quest is to demonstrate with illustrative examples on Markov chains with different classes of states, and the following results are obtained, the mean recurrence time of state 1 is infinite, as well as the mean first passage times from states 2 and 3 to state 1. The mean first passage time from state 2 to state 3 or vice versa is given as 1, while the mean recurrence time of both state 2 and state 3 is given as 2.

Bounded approximate identity(bai) is a key concept in the theory of amenability of algebras. In this paper, we show that algebra of compact operators on Frechet space X has both the right and left locally bounded approximate identities. Sufficient conditions for the existence of these identities are established based on the geometry properties of the Frechet space X and its dual space X' respectively.

We study the Fermat-Torricelli problem (FTP) for Frechet space X, where X is considered as an inverse limit of projective system of Banach spaces. The FTP is defined by using fixed countable collection of continuous seminorms that defines the topology of X as gauges. For a finite set A in X consisting of n distinct and fixed points, the set of minimizers for the sum of distances from the points in A to a variable point is considered. In particular, for the case of collinear points in X, we prove the existence of the set of minimizers for FTP in X and for the case of non collinear points, existence and uniqueness of the set of minimizers are shown for reflexive space X as a result of strict convexity of the space.