Given a lattice @math in a locally compact abelian group @math and a measurable subset @math with finite and positive measure, then the set of characters associated to the dual lattice form a frame for @math if and only if the distinct translates by @math of @math have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.
The result in Corollary is proved by B. Fuglede ( @cite_12 , Lemma 6) for @math and by S. Pedersen ( @cite_13 , Theorem 3.6) for LCA groups.
@cite_13 @cite_12
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Beck's early work [3] gave an efficient version of the Lovasz Local Lemma(LLL) with significant compromise in the parameters. Following several improvements [1,7,4,13], Moser [8], and Moser and Tardos [9] obtained asymptotically optimal results in terms of the maximal degree. For a fixed dependency graph G the exact criterion under which LLL applies is given by Shearer in [12]. For a dependency structure G, let LO(G) be the set of those probability assignments to the nodes of G for which the Lovasz Local Lemma holds. We show that: Both the sequential and parallel ersions of the Moser-Tardos algorithm are efficient up to the Shearer's bound, by giving a tighter analysis. We also prove that, whenever p ∈ LO(G) (1+e), the expected running times of the sequential and parallel versions are at most n e and O(1 e log n e), the later when e Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data. The following theorem is proved. Let f(x_1, , x_m) be a binary nonzero polynomial of m variables of degree . H the number of binary m -tuples (a_1, , a_m) with f(a_1, , a_m) = 1 is less than 2^ m- +1 , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. where m + and 3 . This theorem completely characterizes the codewords of the th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results. The aim of this note is to give short algebraic proofs of theorems of Handelman, Polya and Schmudgen concerning the algebraic structure of polynomials being positive on certain subsets of ℝ n . The main ingredient of the proofs is the representation theorem of Kadison–Dubois. The proof of the latter is elementary and algebraic but tricky.