Given an arithmetic function a : N → R, one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin, Ono, Rolen, and Zagier shows that for certain families of functions a : N → R, the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen, Jia, and Wang's conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let N (d) be the minimal number such that for all n ≥ N (d), the partition Jensen polynomial of degree d and shift n is hyperbolic. We prove that N (3) = 94, N (4) = 206, and N (5) = 381, and in general, that N (d) ≤ (3d) 24d (50d) 3d 2 .
We document a facultative Bartonella-like Rhizobiales bacterium in the giant tropical ant, Paraponera clavata. In a lowland tropical rainforest in Costa Rica, 59 colonies were assayed for the prevalence of the Bartonella-like bacterium (BLB), 14 of which were positive. We addressed three questions: First, how does the prevalence of BLB within colonies vary with environmental conditions? Second, how does diet affect the prevalence of BLB in P. clavata? Third, how does the distribution of BLB among colonies reflect ambient differences in food resources and foraging habits? A variety of environmental variables that may be predictive of the presence of BLB were measured, and diet manipulations were conducted to test whether the prevalence of BLB responded to supplemental carbohydrate or prey. The ambient frequency of BLB is much higher in young secondary forests, but is nearly absent from older secondary forests. The prevalence of BLB inside field colonies increased over the duration of a 2-week carbohydrate supplementation; however, water and prey supplementation did not alter the prevalence of BLB. The diets of the colonies located in young secondary forest, compared to other habitats, have a diet richer in carbohydrates and lower in prey. The abundance of carbohydrate, or the relative lack of N, in a colony's diet influences the occurrence of the BLB microbe in P. clavata. As experimental diet manipulations can affect the facultative presence of an N-cycling microbe, a consistent diet shift in diet may facilitate the emergence of tighter symbioses.
Abstract. A fusion category of rank 4 has either four self-dual simple objects or exactly two self-dual simple objects. We study fusion categories of rank 4 with exactly two self-dual simple objects, giving nearly a complete classification of those based rings that admit pseudounitary categorification. More precisely, we show that if C is such a fusion category, then its Grothendieck ring K(C) must be one of seven based rings, six of which have known categorifications. In doing so, we classify all based rings associated with near-group categories of the group Z/3Z.
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