Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the ...

We show that all sets that are complete for NP under non-uniform AC0 reductions are isomorphic under non-uniform AC0-computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC0 reductions. More generally, we show two theorems that hold for any complexity class C closed under (uniform) NC1-computable many-one reductions. Gap: The sets that are complete for C un...

Oversampled filter banks can be used to enhance resilience to erasures in communication systems in much the same way that finite-dimensional frames have previously been applied. This paper extends previous finite dimensional treatments to frames and signals in l2(Z) with frame expansions that can be implemented efficiently with filter banks. It is shown that tight frames attain best performance...

For a given class F of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation |〈fk, fl〉| among all frames {fk}k∈I ∈ F . We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These ...

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. In this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decomposition...

In this paper, we consider the problem of constructing hypercycle systems 5-cycles in complete 3-uniform hypergraphs. A system C(r,k,v) order v is a collection r-uniform k-cycles on v-element vertex set, such that each r-element subset an edge precisely one those k-cycles. We present cyclic C(3,5,v) orders v=25,26,31,35,37,41,46,47,55,56, highly symmetric construction for v=40, and 2-split cons...