3,294 research outputs found
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
Seasonal Variation in Nymphal Blacklegged Tick Abundance in Southern New England Forests
In the northeastern United States, risk of human exposure to tick transmitted disease is primarily a function of the abundance of the blacklegged tick, Ixodes scapularis Say. We assessed seasonal variability in the abundance of nymphal stage I. scapularis over 13 yr, collected from several forested areas throughout Rhode Island. Specifically, we examined intraseasonal differences by using two temporally distinct tick collections made during the peak nymphal tick season. Intraseasonal factors significantly impacted tick abundance, with the June tick rate (mean = 40.42, SD = 14.79) significantly more abundant than the July tick rate (mean = 27.64, SD = 15.47). The greater variability in July (coefficient of variation: June, 36.61%; July, 55.95%) lead us to conclude June tick rates are relatively stable from year to year, whereas July tick rates contribute more to intraseasonal and yearly variation
On the Progenitor System of the Type Iax Supernova 2014dt in M61
We present pre-explosion and post-explosion Hubble Space Telescope images of
the Type Iax supernova (SN Iax) 2014dt in M61. After astrometrically aligning
these images, we do not detect any stellar sources at the position of the SN in
the pre-explosion images to relatively deep limits (3 sigma limits of M_F438W >
-5.0 mag and M_F814W > -5.9 mag). These limits are similar to the luminosity of
SN 2012Z's progenitor system (M_F435W = -5.43 +/- 0.15 and M_F814W = -5.24 +/-
0.16 mag), the only probable detected progenitor system in pre-explosion images
of a SN Iax, and indeed, of any white dwarf supernova. SN 2014dt is consistent
with having a C/O white-dwarf primary/helium-star companion progenitor system,
as was suggested for SN 2012Z, although perhaps with a slightly smaller or
hotter donor. The data are also consistent with SN 2014dt having a low-mass red
giant or main-sequence star companion. The data rule out main-sequence stars
with M_init > 16 M_sun and most evolved stars with M_init > 8 M_sun as being
the progenitor of SN 2014dt. Hot Wolf-Rayet stars are also allowed, but the
lack of nearby bright sources makes this scenario unlikely. Because of its
proximity (D = 12 Mpc), SN 2014dt is ideal for long-term monitoring, where
images in ~2 years may detect the companion star or the luminous bound remnant
of the progenitor white dwarf.Comment: 5 pages, 3 figures, submitted to ApJ
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