The Paulsen Problem for partitioned frames
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[EMBARGOED UNTIL 12/01/2026] In this dissertation, we study tuples of frames of the same size, which we call partitioned frames. We generalize the notions of (ε-nearly) equal-norm Parseval frames to this setting and investigate the Paulsen Problem for partitioned frames. In our approach, we view partitioned frames as representations of bipartite quivers and study them within the framework of quiver invariant theory. This perspective allows us to introduce the capacity of a partitioned frame and use it to characterize the class of equal-norm Parseval partitioned frames. Next, we find effective upper bounds for the distance from a given ε-nearly equalnorm Parseval partitioned frame to the set of partitioned frames that are equal-norm Parseval (up to a specific group action). We also provide applications to matrices whose row and column sums lie within specified bounds.
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Ph. D.