Question

Define and discuss the following concepts in the theory of Markov chains:

(A) Irreducibility and related properties.

(B) Periodicity and related properties.

(C) Recurrent/transient states and related properties.

Answer 1

Answer :

(A) Irreducibility and related properties :-

In this work we audit the essential properties of carbon nanotubes from the stance of gathering hypothesis. The zone collapsing plan is evaluated in the light of the helical symmetry of the nanotube. The gathering hypothesis for chiral and achiral nanotubes is inspected, and the portrayals of the factor gathering of the wavevector k are gotten. The similitudes and contrasts between the formalism of the gathering of the wavevector and that of line bunches are tended to as for the final portrayals and quantum numbers related with direct and rakish momenta. At last, we expand the consequences of gathering hypothesis to light up the electronic and vibrational properties of carbon nanotubes. Choice tenets for the optical assimilation and twofold reverberation Raman diffusing are examined for the situation where the electron– electron association is irrelevant (metallic nanotubes) and for the situation where exciton restricting energies are solid and can't be dismissed.

(B) Periodicity and related properties :-

We investigate mesons at limited temperature in a chiral, restricting string double. The temperature reliance of low-turn and also high-turn meson masses is appeared to display an example recognizable from the cross section. Besides, we discover the separation temperature of mesons as an element of their turn, demonstrating that at a settled quark mass, mesons with bigger twists separate at lower temperatures. The Goldstone bosons related with chiral symmetry breaking are appeared to vanish over the chiral symmetry rebuilding temperature. At long last, we demonstrate that holographic contemplations suggest that vast turn mesons don't encounter drag impacts while traveling through the quark-gluon plasma. They do, in any case, have a most extreme speed for settled turn, past which they separate.

(C) Recurrent/transient states and related properties:-

f(n) ii = P{Xn = i, X1 6= i, . . . , Xn−1 6= i|X0 = i} = Probability of first recurrence to i is at the nth step. fi = fii = X ∞ n=1 f(n) ii = Prob. of recurrence to i. Def. A state i is recurrent if fi = 1. Def. A state i is transient if fi < 1. Define Ti = Time for first visit to i given X0 = 1. This is the same as Time to first visit to i given Xk = i. (Time homogeneous) mi = E(Ti|X0 = i) = X ∞ n=1 nf(n) ii = mean time for recurrence Note: f(n) ii = P{Ti = n|X0 = i}