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We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ 1 ≥. .. ≥ λ n be the eigenvalues of an n-vertex graph, and let λ = max[λ 2 , |λ n |]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ 1 ≥ d, and we show that λ = O(√ d). For d = c it is no longer true that λ = O(√ d),(More)
We consider random 3CNF formulas with n variables and m clauses. It is well known that when m &gt; cn (for a sufficiently large constant c), most formulas are not satisfiable. However, it is not known whether such formulas are likely to have polynomial size witnesses that certify that they are not satisfiable. A value of m sime n<sup>3/2</sup> was the(More)
Massive stars end their short lives in spectacular explosions--supernovae--that synthesize new elements and drive galaxy evolution. Historically, supernovae were discovered mainly through their 'delayed' optical light (some days after the burst of neutrinos that marks the actual event), preventing observations in the first moments following the explosion.(More)
We compare the statistical properties of giant gravitationally lensed arcs produced in matched simulated and observed cluster samples. The observed sample consists of 10 X-ray selected clusters at redshifts z c ∼ 0.2 imaged with HST by Smith et al.. The simulated dataset is produced by lensing the Hubble Deep Field, which serves as a background source(More)