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We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ1 ≥ . . . ≥ λn be the eigenvalues of an n-vertex graph, and let λ = max[λ2, |λn|]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ1 ≥ d, and we show that λ = O( √ d). For d = c it is no longer true that λ = O( √ d), but(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)
The Palomar Transient Factory (PTF) is a wide-field experiment designed to investigate the optical transient and variable sky on time scales from minutes to years. PTF uses the CFH12k mosaic camera, with a field of view of 7:9 deg and a plate scale of 1′′ pixel , mounted on the Palomar Observatory 48 inch Samuel Oschin Telescope. The PTF operation strategy(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)
Let φ be a 3CNF formula with n variables and m clauses. A simple nonconstructive argument shows that when m is sufficiently large compared to n, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a(More)
This paper presents the first results from a new citizen science project: Galaxy Zoo Supernovae. This proof-of-concept project uses members of the public to identify supernova candidates from the latest generation of wide-field imaging transient surveys. We describe the Galaxy Zoo Supernovae operations and scoring model, and demonstrate the effectiveness of(More)
Cometary activity in main-belt asteroids probes the ice content of these objects and provides clues to the history of volatiles in the inner solar system. We search the Palomar Transient Factory (PTF) survey to derive upper limits on the population size of active main-belt comets (MBCs). From data collected March 2009 through July 2012, we extracted ∼2(More)