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We provide mean value results for sums of the composition of the gcd and arithmetic functions belonging to certain classes... In section 2 , we provide a result for four classes

A method is provided for constructing exceptional copositive matrices by completing a partial copositive matrix that has certain specified overlapping copositive interior

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More precisely it was shown that, when the Markov transition matrix on each branch satisfies the Kesten-Stigum bound, the phylogeny can be reconstructed in polynomial time

In this note, we study the Gaussian fluctuations of the edge occupation measure for a reversible Markov chain and give a nice description of the covariance matrix.. Then we give