Tight bounds and conjectures for the isolation lemma

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M \gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M \gg n \gg 1$

Edge-coloring linear hypergraphs with medium-sized edges

Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper-edge sizes are bounded between $i$ and $C_{i,\epsilon} \sqrt{n}$ inclusive, then there is a list edge coloring using $(1 + \epsilon) \frac{n}{i - 1}$ colors. The dependence on $n$ in the upper bound is optimal (up to the value of $C_{i,\epsilon}$)

NSBECS Program Effectiveness Surveys: A Tool for Strategic Planning and School Improvement

No abstract provided

ANALYSIS OF NO2 AND HCHO VERTICAL COLUMN DENSITIES IN THE GREAT LAKES REGION USING THE OZONE MONITORING INSTRUMENT

Nitrogen oxides (NOx) and volatile organic compounds (VOC) are major contributors to air quality, especially for their direct involvement in ozone (O3) production. Retrievals of nitrogen dioxide (NO2) and formaldehyde (HCHO) mixing ratios can be used to represent NOx and VOCs respectively and the formaldehyde-to-NO2 ratio (FNR) can be used to analyze the O3 production chemistry in a region. Since the current monitoring network in the Great Lakes region for NO2 and HCHO is limited, the Ozone Monitoring Instrument (OMI) can be used to retrieve vertical column densities for NO2 and HCHO in rural areas away from ground-based sensors. OMI was found to be sensitive to temporal changes in ground mixing ratios and suitable for this study. OMI was able to retrieve statistically significant gradients for NO2 and FNR but not HCHO between urban centers and rural areas in the Great Lakes region. Data from OMI was used to determine regions of VOC-limited O3 production and NOx-limited O3 production. FNR values are highest in rural areas and during the summer months

Two Conditions for Galaxy Quenching: Compact Centres and Massive Haloes

We investigate the roles of two classes of quenching mechanisms for central and satellite galaxies in the SDSS ($z<0.075$): those involving the halo and those involving the formation of a compact centre. For central galaxies with inner compactness $\Sigma_{\rm 1kpc} \sim 10^{9-9.4}M_{\odot} {\rm kpc}^{-2}$, the quenched fraction $f_{q}$ is strongly correlated with $\Sigma_{\rm 1kpc}$ with only weak halo mass $M_{\rm h}$ dependence. However, at higher and lower $\Sigma_{\rm 1kpc}$, sSFR is a strong function of $M_{\rm h}$ and mostly independent of $\Sigma_{\rm 1kpc}$. In other words, $\Sigma_{\rm 1kpc} \sim 10^{9-9.4} M_{\odot} {\rm kpc}^{-2}$ divides galaxies into those with high sSFR below and low sSFR above this range. In both the upper and lower regimes, increasing $M_{\rm h}$ shifts the entire sSFR distribtuion to lower sSFR without a qualitative change in shape. This is true even at fixed $M_{*}$, but varying $M_{*}$ at fixed $M_{\rm h}$ adds no quenching information. Most of the quenched centrals with $M_{\rm h} > 10^{11.8}M_{\odot}$ are dense ($\Sigma_{\rm 1kpc} > 10^{9}~ M_{\odot} {\rm kpc}^{-2}$), suggesting compaction-related quenching maintained by halo-related quenching. However, 21% are diffuse, indicating only halo quenching. For satellite galaxies in the outskirts of halos, quenching is a strong function of compactness and a weak function of host $M_{\rm h}$. In the inner halo, $M_{\rm h}$ dominates quenching, with $\sim 90\%$ of the satellites being quenched once $M_{\rm h} > 10^{13}M_{\odot}$. This regional effect is greatest for the least massive satellites. As demonstrated via semi-analytic modelling with simple prescriptions for quenching, the observed correlations can be explained if quenching due to central compactness is rapid while quenching due to halo mass is slow.Comment: 16 pages, 11 figures, MNRAS accepte

Justice Bushrod Washington and the Age of Discovery in American Law

古代日本語の船舶の名称やそれに由来する語彙には、日本語一視点のみでは正確に理解できないものがある。これらの単語には、適切な海の民の視点、具体的には、彼らが用いたであろう言語や文化についての知識を持てば正確に理解できるものがある。茂在寅男氏は、『記』『紀』の中に古代ポリネシア語が多く混じっている、と述べ、井上夢間氏は、「枯野」等の言葉とカヌーとの関係について、ハワイ語を用いて簡潔に説明したが、その知見は、言語面からの研究に突破口を開くものであった。小論では、「天アマ」は、ポリネシア語の「ama」を漢字で書き記したものであり、全称を「天鳥船、天鴿船、天磐船」という船舶の略称であること、「鳥を舶載する、アウトリガー・フロート付き外洋航海船」を意味すること、などを解明することができた。古代の日本語の問題を考えたり、古典を読み解くのに、中国語やポリネシア語等の外国語の知識や、船舶・航海の知識が役に立つという認識は、やがて常識となるのではないか。天鳥船／天鴿船／天磐船天岩戸天の原天離る天飛ぶ