Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.Comment: Revised version. 28 pages, 2 figure

Searching For Dark Matter Subhalos In the Fermi-LAT Second Source Catalog

The dark matter halo of the Milky Way is expected to contain an abundance of smaller subhalos. These subhalos can be dense and produce potentially observable fluxes of gamma rays. In this paper, we search for dark matter subhalo candidates among the sources in the Fermi-LAT Second Source Catalog which are not currently identified or associated with counterparts at other wavelengths. Of the nine high-significance, high-latitude (|b|>60 degrees), non-variable, unidentified sources contained in this catalog, only one or two are compatible with the spectrum of a dark matter particle heavier than approximately 50-100 GeV. The majority of these nine sources, however, feature a spectrum that is compatible with that predicted from a lighter (~5-40 GeV) dark matter particle. This population is consistent with the number of observable subhalos predicted for a dark matter candidate in this mass range and with an annihilation cross section of a simple thermal relic (sigma v~3x10^{-26} cm^3/s). Observations in the direction of these sources at other wavelengths will be necessary to either reveal their astrophysical nature (as blazars or other active galactic nuclei, for example), or to further support the possibility that they are dark matter subhalos by failing to detect any non-gamma ray counterpart.Comment: 8 pages, 4 figure

To Review or Not to Review? Limited Strategic Thinking at the Movie Box Office

Film distributors occasionally withhold movies from critics before their release. Cold openings provide a natural field setting to test models of limited strategic thinking. In a set of 856 widely released movies, cold opening produces a significant 15% increase in domestic box office revenue (though not in foreign markets and DVD sales), consistent with the hypothesis that some moviegoers do not infer low quality from cold opening. Structural parameter estimates indicate 1–2 steps of strategic thinking by moviegoers (comparable to experimental estimates). However, movie studios appear to think moviegoers are sophisticated since only 7% of movies are opened cold