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a b s t r a c tWe consider the equation R n = Q n + M n R n−1 , with random non-i.i.d. coefficients (Q n , M n ) n∈Z ∈ R 2 , and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.

The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i = j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semidefinite nullity from above. The tree cover number T(G) is the minimum number of vertex disjoint induced simple trees that cover all of the vertices of G. The result that M + (G) = T(G) for an outerplanar multigraph G [F. Barioli et al. Minimum semidefinite rank of outerplanar graphs and the tree cover number.

a b s t r a c tWe study a linear recursion with random Markov-dependent coefficients. In a ''regular variation in, regular variation out'' setup we show that its stationary solution has a multivariate regularly varying distribution. This extends results previously established for i.i.d. coefficients.

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We study the spectrum of certain discontinuous Galerkin (DG) methods of linear convectiondiffusion PDEs. Specifically, we consider DG methods for a first order advection equation and for a second-order diffusion equation. Tight upper and lower bounds that we derive for the spectrum can be used as quantifiers of the dissipation of the numerical solution and have implications for stability of the numerical scheme.