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By Jerzy A. Filar

Managed Markov Chains, Graphs & Hamiltonicity summarizes a line of study that maps sure classical difficulties of discrete arithmetic - akin to the Hamiltonian cycle and the touring Salesman difficulties - into convex domain names the place continuum research might be conducted.

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Example text

In particular, the following was shown in [16] that g11 result holds. 6. Let f ∈ DS be any short cycle deterministic policy; and let k < N be the length of the cycle containing the home node 1. The following properties hold. ε (f ) depends only on ε and k and equals (a) The value g11 ε g11 (f ) = 1 + N + 1 N 1 N 2ε N −k 1 − (1 − N ε)k kN ε − 1 + (1 − N ε)k . 1 − (1 − N ε)k 32 Analysis in the Policy Space ε (f ) has a pole of order 1 at ε = 0, and the (b) The functional g11 initial terms of its Laurent expansion are: ε g11 (f ) = N (k − 1) (1 + k) N − k −1 2 k + kN − N ε + + ε 2 N k 2kN 12k N 2 (k − 1) (1 + k) 2 + ε + O ε3 .

N . Finally, Varfi (τ1 ) will denote the variance of τ1 , the first hitting time of node 1 from node i, under the probability distribution induced by the policy f . Since the symmetric linear perturbation applied to f ∈ DS preserves double stochasticity and ensures irreducibility of Pε (f ), the corresponding stationary distribution matrix Pε∗ (f ) = N1 J, where J is an N × N matrix with 1 in every entry. Of course, this implies that E1f (τ1 ) = N. 1) provided that the perturbation parameter ε > 0 and is sufficiently small.

Such notions are yet to be rigorously formalized. 2 Determinant of the Inverse of Fundamental Matrix In this section, we describe some interesting results that, in principle, could be considered as purely algebraic properties of certain matrices and their eigenvalues and determinants. However, it would have been difficult to discover these results without the insight of preceding sections highlighting the relationship of elements of fundamental matrices of Markov chains to Hamiltonicity of a graph.

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