By A.V. Gheorghe

'Et moi ... si j'avait su remark en revenir. One carrier arithmetic has rendered the je n'y serais element aile: human race. It has placed logic again the place it belongs. at the topmost shelf subsequent Jules Verne (0 the dusty canister labelled 'discarded non sense'. The sequence is divergent; hence we are able to do whatever with it. Eric T. Bell O. Heaviside arithmetic is a device for notion. A hugely worthy instrument in a global the place either suggestions and non linearities abound. equally, all types of elements of arithmetic function instruments for different elements and for different sciences. making use of an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One provider common sense has rendered com puter technological know-how .. .'; 'One provider type concept has rendered arithmetic .. .'. All arguably actual. And all statements available this manner shape a part of the raison d'etre of this sequence.

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N T- j=l x. 't. I is the mean time spent by the process in state i . 2. THE STUDY OF RECURRENT STATE OCCUPANCY IN MARKOV PROCESSES A class of states is, by definition, that class of states which communicate among themselves, with the property that as soon as the process enters any of them it cannot leave this class of states. Any Markov process has at least one class of states. The transient states are associated with a class. It is possible for the process to enter this class from a transient state.

SN}, is the set of absorbant states of the Markov chain being considered. Let Q =II - QI and Q (i,)) be a cofactor of the (i, ))-th entrance in the Qmatrix. 2. Mean and variance of the random variables nj (which show how many times the system was in state i before making a transtition in an absorbant state) and t (number of transitions before the system arrives in an absorbant state) are given by the relations : (a) (nj)j (c) (t). k = _1 l: IQi k=1 I Q(k z). " CHAP1ER 1 24 (d) V (t). = 2 I s s L L k=1 j=1 S n ..

59b) k Proof. 3 with an appropriate change in the residual term. 3. 5. If Plj' i, j = 1, 2, ... 59), then: (2) if PhjU = 1,2, ... , h -1, h + 1, ... , N) are mutually independent random variables and pij =Pij (i, j = 1, 2, ... , N ; i 7;: h), then when k 7;: h, and when k = h, then: Proof. We shall give the fulLproof of part (1) of the above theorem; the proof of part (2) is essentially contained in the following proof. 59c) are influenced by· quadratic functions of xk and Xi as follows. J ~ J=k Pk k' - U~ U )2 1 ( kj )2 :s; ~U U 2 k=J = (1 - 2 1t ) .