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Stochastic evolutions governed by quantum Brownian motion are considered for quantum mechanical Boson systems of one degree of freedom. These give rise to semigroups of completely positive maps and to non-commutative Feynman-Kac formulae upon taking time zero conditional expectations.
It is shown that every infinitely divisible distribution λ in SL(k,ℂ) or SL(k,ℝ) can be imbedded in a diadic convolution semigroup if the smallest closed subgroup containing the support of λ is the whole group.
Let A be the adjacency matrix of an ordinary (simple) graph G and A′=xI+λA+(J-A-I) where I is the n×n identity matrix and J is the n×n matrix of l’s. Then we call P(x,λ)=Per(A′) the permanent polynomial of G. A frame (2-matching) of a graph G is a spanning subgraph F of G whose components are single points, single lines, paths or cycles. If F has wi paths Pi, i=1,…,n and yj cycles Cj we let $$w(F)...
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