2015/07/29 by Nathan S. Feldman, Feldman, Nathan S., Paul McGuire +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Graph theory and applications #Point processes and geometric inequalities #math.FA
paper · pdf · doi:10.48550/arxiv.1507.08323
arxiv created 2015/07/29 · arxiv updated 2015/07/31
We define a convex-polynomial to be one that is a convex combination of the monomials \1, z, z2, …\. This paper explores the intimate connection between peaking convex-polynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convex-cyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose invariant closed convex sets are all invariant subspaces. Our characterization of the convex-cyclic matrices gives a new and correct proof of a similar result by Rezaei that was stated and proven incorrectly.