A generalization of Zhu's theorem on six-valent integer distance graphs

Given a set $S$ of positive integers, the integer distance graph for $S$ has the set of integers as its vertex set, where two vertices are adjacent if and only if the absolute value of their difference lies in $S$. In 2002, Zhu completely determined the chromatic number of integer distance graphs when $S$ has cardinality $3$. Integer distance graphs can be defined equivalently as Cayley graphs on the group of integers under addition. In a previous paper, the authors develop general methods to approach the problem of finding chromatic numbers of Cayley graphs on abelian groups. To each such graph one associates an integer matrix. In some cases the chromatic number can be determined directly from the matrix entries. In particular, the authors completely determine the chromatic number whenever the matrix is of size $3\times 2$ -- precisely the size of the matrices associated to the graphs studied by Zhu. In this paper, then, we demonstrate that Zhu's theorem can be recovered as a special case of the authors' previous results.Comment: 6 page

An alternate proof of Payan's theorem on cubelike graphs

A cubelike graph is a Cayley graph on the product $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ of the integers modulo $2$ with itself finitely many times. In 1992, Payan proved that no cubelike graph can have chromatic number $3$. The authors of the present paper previously developed a general matrix method for studying chromatic numbers of Cayley graphs on abelian groups. In this note, we apply this method of Heuberger matrices to give an alternate proof of Payan's theorem.Comment: 4 page

MCMC Sampling of Directed Flag Complexes with Fixed Undirected Graphs

Constructing null models to test the significance of extracted information is a crucial step in data analysis. In this work, we provide a uniformly sampleable null model of directed graphs with the same (or similar) number of simplices in the flag complex, with the restriction of retaining the underlying undirected graph. We describe an MCMC-based algorithm to sample from this null model and statistically investigate the mixing behaviour. This is paired with a high-performance, Rust-based, publicly available implementation. The motivation comes from topological data analysis of connectomes in neuroscience. In particular, we answer the fundamental question: are the high Betti numbers observed in the investigated graphs evidence of an interesting topology, or are they merely a byproduct of the high numbers of simplices? Indeed, by applying our new tool on the connectome of C. Elegans and parts of the statistical reconstructions of the Blue Brain Project, we find that the Betti numbers observed are considerable statistical outliers with respect to this new null model. We thus, for the first time, statistically confirm that topological data analysis in microscale connectome research is extracting statistically meaningful information

Chromatic numbers of Cayley graphs of abelian groups: Cases of small dimension and rank

A connected Cayley graph on an abelian group with a finite generating set $S$ can be represented by its Heuberger matrix, i.e., an integer matrix whose columns generate the group of relations between members of $S$. In a companion article, the authors lay the foundation for the use of Heuberger matrices to study chromatic numbers of abelian Cayley graphs. We call the number of rows in the Heuberger matrix the dimension, and the number of columns the rank. In this paper, we give precise numerical conditions that completely determine the chromatic number in all cases with dimension $1$; with rank $1$; and with dimension $\leq 3$ and rank $\leq 2$. For such a graph without loops, we show that it is $4$-colorable if and only if it does not contain a $5$-clique, and it is $3$-colorable if and only if it contains neither a diamond lanyard nor a $C_{13}(1,5)$, both of which we define herein. In a separate companion article, we show that we recover Zhu's theorem on the chromatic number of $6$-valent integer distance graphs as a special case of our theorem for dimension $3$ and rank $2$.Comment: 27 page

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an $m\times r$ integer matrix, where we call $m$ the dimension and $r$ the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. In a series of follow-up articles using this method, we completely determine the chromatic number in cases with small dimension and rank; prove a generalization of Zhu's theorem on the chromatic number of $6$-valent integer distance graphs; and provide an alternate proof of Payan's theorem that a cube-like graph cannot have chromatic number 3.Comment: 17 page

On the Complexity of Bounded Context Switching

Bounded context switching (BCS) is an under-approximate method for finding violations to safety properties in shared-memory concurrent programs. Technically, BCS is a reachability problem that is known to be NP-complete. Our contribution is a parameterized analysis of BCS. The first result is an algorithm that solves BCS when parameterized by the number of context switches (cs) and the size of the memory (m) in O*(m^(cs)2^(cs)). This is achieved by creating instances of the easier problem Shuff which we solve via fast subset convolution. We also present a lower bound for BCS of the form m^o(cs / log(cs)), based on the exponential time hypothesis. Interestingly, the gap is closely related to a conjecture that has been open since FOCS\u2707. Further, we prove that BCS admits no polynomial kernel. Next, we introduce a measure, called scheduling dimension, that captures the complexity of schedules. We study BCS parameterized by the scheduling dimension (sdim) and show that it can be solved in O*((2m)^(4sdim)4^t), where t is the number of threads. We consider variants of the problem for which we obtain (matching) upper and lower bounds

Modeling Habitat of Freshwater Mussels (Bivalvia:Unionidae) in the Lower Great Lakes 25 Years after the Dreissena Invasion

Finding remnant populations of species that are of conservation concern can be difficult, particularly in aquatic habitats. Models of ecological niches can aid in the discovery of refuges. Remnant populations of native freshwater mussels (unionids) have been found in Lakes Erie and St Clair. Our goals were to predict undiscovered refuges in Lake Ontario based on habitat analysis from Lake Erie and to conduct surveys to test those predictions. We built a presence-only model on environmental data including attributes of the benthic zone and shoreline where mussels occurred in Lake Erie. We found a link between small- and large-scale variables related to unionid persistence. Bathymetry, fetch, and shoreline geomorphology contributed most to the model. These variables correspond to local-scale environmental factors important for unionid survival, including presence of vegetation and substrate composition, which explained ∼22% of the variance in presence, abundance, and richness. The model predicted that 0.8% of the near-shore area of Lake Ontario should be habitat for unionids. In surveys at 34 locations on the USA shore of Lake Ontario, we found 1800 unionids of 11 species and showed that areasOntario, a result signifying generality of our model for conservation approaches to freshwater mussels

Oligohydramnios: a prospective study of fetal, neonatal and maternal outcomes in low-middle income countries

BACKGROUND: Oligohydramnios is a condition of abnormally low amniotic fluid volume that has been associated with poor pregnancy outcomes. To date, the prevalence of this condition and its outcomes has not been well described in low and low-middle income countries (LMIC) where ultrasound use to diagnose this condition in pregnancy is limited. As part of a prospective trial of ultrasound at antenatal care in LMICs, we sought to evaluate the incidence of and the adverse maternal, fetal and neonatal outcomes associated with oligohydramnios. METHODS: We included data in this report from all pregnant women in community settings in Guatemala, Pakistan, Zambia and the Democratic Republic of Congo (DRC) who received a third trimester ultrasound as part of the First Look Study, a randomized trial to assess the value of ultrasound at antenatal care. Using these data, we conducted a planned secondary analysis to compare pregnancy outcomes of women with to those without oligohydramnios. Oligohydramnios was defined as measurement of an Amniotic Fluid Index less than 5 cm in at least one ultrasound in the third trimester. The outcomes assessed included maternal morbidity and fetal and neonatal mortality, preterm birth and low-birthweight. We used pairwise site comparisons with Tukey-Kramer adjustment and multivariable logistic models using general estimating equations to account for the correlation of outcomes within cluster. RESULTS: Of 12,940 women enrolled in the clusters in Guatemala, Pakistan, Zambia and the DRC in the First Look Study who had a third trimester ultrasound examination, 87 women were diagnosed with oligohydramnios, equivalent to 0.7% of those studied. Prevalence of detected oligohydramnios varied among study sites; from the lowest of 0.2% in Zambia and the DRC to the highest of 1.5% in Pakistan. Women diagnosed with oligohydramnios had higher rates of hemorrhage, fetal malposition, and cesarean delivery than women without oligohydramnios. We also found unfavorable fetal and neonatal outcomes associated with oligohydramnios including stillbirths (OR 5.16, 95%CI 2.07, 12.85), neonatal deaths \u3c 28 days (OR 3.18, 95% CI 1.18, 8.57), low birth weight (OR 2.10, 95% CI 1.44, 3.07) and preterm births (OR 2.73, 95%CI 1.76, 4.23). The mean birth weight was 162 g less (95% CI -288.6, - 35.9) with oligohydramnios. CONCLUSIONS: Oligohydramnos was associated with worse neonatal, fetal and maternal outcomes in LMIC. Further research is needed to assess effective interventions to diagnose and ultimately to reduce poor outcomes in these settings

The Development of Vaginal Microbicides for the Prevention of HIV Transmission

Given that we still do not have an effective vaccine against HIV, the development of novel biomedical methods for preventing HIV transmission remains a top priority in controlling the HIV pandemic

Functional Amyloid Formation within Mammalian Tissue

Amyloid is a generally insoluble, fibrous cross-β sheet protein aggregate. The process of amyloidogenesis is associated with a variety of neurodegenerative diseases including Alzheimer, Parkinson, and Huntington disease. We report the discovery of an unprecedented functional mammalian amyloid structure generated by the protein Pmel17. This discovery demonstrates that amyloid is a fundamental nonpathological protein fold utilized by organisms from bacteria to humans. We have found that Pmel17 amyloid templates and accelerates the covalent polymerization of reactive small molecules into melanin—a critically important biopolymer that protects against a broad range of cytotoxic insults including UV and oxidative damage. Pmel17 amyloid also appears to play a role in mitigating the toxicity associated with melanin formation by sequestering and minimizing diffusion of highly reactive, toxic melanin precursors out of the melanosome. Intracellular Pmel17 amyloidogenesis is carefully orchestrated by the secretory pathway, utilizing membrane sequestration and proteolytic steps to protect the cell from amyloid and amyloidogenic intermediates that can be toxic. While functional and pathological amyloid share similar structural features, critical differences in packaging and kinetics of assembly enable the usage of Pmel17 amyloid for normal function. The discovery of native Pmel17 amyloid in mammals provides key insight into the molecular basis of both melanin formation and amyloid pathology, and demonstrates that native amyloid (amyloidin) may be an ancient, evolutionarily conserved protein quaternary structure underpinning diverse pathways contributing to normal cell and tissue physiology