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By Jerzy A. Filar

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

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1. Consider f ∈ DS and the induced perturbed probability transition matrix Pε (f ) as well as the associated fundamental matrix Gε (f ). 1 HCP and Minimal Variance of Hitting Times 21 where Pˆε (f ) is Pε (f ) with the elements in the first column replaced by zeroes. Outline of Proof. For notational simplicity we suppress the dependence on the policy f and the permutation parameter ε of most of the quantities of interest; for instance gij will denote the (ij)th entry of the fundamental matrix Gε (f ).

This Hamiltonian Cycle induces the perturbed probability transition matrix Pε (fh ) of the form:   ε 1 − 3ε ε ε   ε 1 − 3ε ε   ε Pε (fh ) =  . ε ε 1 − 3 ε  ε 1 − 3ε ε ε ε Since Pε∗ (fh ) = 1/4J, the matrix needed to calculate the fundamental matrix induced by fh is   5/4 − ε  −ε + 1/4 1 I − Pε (fh ) + J =   −ε + 1/4 4 −3/4 + 3 ε −3/4 + 3 ε 5/4 − ε −ε + 1/4 −ε + 1/4 −ε + 1/4 −3/4 + 3 ε 5/4 − ε −ε + 1/4 −ε + 1/4 −ε + 1/4  . −3/4 + 3 ε 5/4 − ε 24 Analysis in the Policy Space And, hence, the fundamental matrix is obtained as ✒ Gε (fh ) = ✷ ✻✻ ✻✻ =✻ ✻✻ ✻✹ I − Pε (fh ) + 1 J 4 ✓ −1 1 −5+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −3+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −1+12 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 1−4 ε−8 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 1−4 ε−8 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −5+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −3+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −1+12 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −1+12 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 1−4 ε−8 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −5+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −3+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −3+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −1+12 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 1−4 ε−8 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 1 −5+20 ε−40 ε2 +32 ε3 8 6 ε−1+16 ε3 −16 ε2 ✸ ✼✼ ✼✼ ✼✼.

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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