The Tur\'{a}n number and probabilistic combinatorics

In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Tur\'{a}n number $T(n,k,l)$. Surprisingly, this proof cannot be found in existing literature.Comment: 5 pages; to appear in Amer. Math. Monthly 201

The covering radius problem for sets of perfect matchings

Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result vis-\'a-vis the covering radius of sets of permutations, which was studied by Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}). We also conclude with a conjecture of a more general problem in hypergraph matchings.Comment: 10 page

Flexible non-parametric tests of sample exchangeability and feature independence

In scientific studies involving analyses of multivariate data, two questions often arise for the researcher. First, is the sample exchangeable, meaning that the joint distribution of the sample is invariant to the ordering of the units? Second, are the features independent of one another, or can the features be grouped so that the groups are mutually independent? We propose a non-parametric approach that addresses these two questions. Our approach is conceptually simple, yet fast and flexible. It controls the Type I error across realistic scenarios, and handles data of arbitrary dimensions by leveraging large-sample asymptotics. In the exchangeability detection setting, through extensive simulations and a comparison against unsupervised tests of stratification based on random matrix theory, we find that our approach compares favorably in various scenarios of interest. We apply our method to problems in population and statistical genetics, including stratification detection and linkage disequilibrium splitting. We also consider other application domains, applying our approach to post-clustering single-cell chromatin accessibility data and World Values Survey data, where we show how users can partition features into independent groups, which helps generate new scientific hypotheses about the features.Comment: Main Text: 25 pages Supplementary Material: 39 page

The multicovering radius problem for some types of discrete structures

The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric space. Klapper introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (cf. Keevash and Ku, 2006) and for codes of perfect matchings (cf. Aw and Ku, 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in Keevash and Ku (2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.Comment: To appear in Designs, Codes and Cryptography (2012

A Vast Thin Plane of Co-rotating Dwarf Galaxies Orbiting the Andromeda Galaxy

Dwarf satellite galaxies are thought to be the remnants of the population of primordial structures that coalesced to form giant galaxies like the Milky Way. An early analysis noted that dwarf galaxies may not be isotropically distributed around our Galaxy, as several are correlated with streams of HI emission, and possibly form co-planar groups. These suspicions are supported by recent analyses, and it has been claimed that the apparently planar distribution of satellites is not predicted within standard cosmology, and cannot simply represent a memory of past coherent accretion. However, other studies dispute this conclusion. Here we report the existence (99.998% significance) of a planar sub-group of satellites in the Andromeda galaxy, comprising approximately 50% of the population. The structure is vast: at least 400 kpc in diameter, but also extremely thin, with a perpendicular scatter <14.1 kpc (99% confidence). Radial velocity measurements reveal that the satellites in this structure have the same sense of rotation about their host. This finding shows conclusively that substantial numbers of dwarf satellite galaxies share the same dynamical orbital properties and direction of angular momentum, a new insight for our understanding of the origin of these most dark matter dominated of galaxies. Intriguingly, the plane we identify is approximately aligned with the pole of the Milky Way's disk and is co-planar with the Milky Way to Andromeda position vector. The existence of such extensive coherent kinematic structures within the halos of massive galaxies is a fact that must be explained within the framework of galaxy formation and cosmology.Comment: Published in the 3rd Jan 2013 issue of Nature. 19 pages, 4 figures, 1 three-dimensional interactive figure. To view and manipulate the 3-D figure, an Adobe Reader browser plug-in is required; alternatively save to disk and view with Adobe Reade

Patent Human Infections with the Whipworm, Trichuris trichiura, Are Not Associated with Alterations in the Faecal Microbiota

Background: The soil-transmitted helminth (STH), Trichuris trichiura colonises the human large intestine where it may modify inflammatory responses, an effect possibly mediated through alterations in the intestinal microbiota. We hypothesised that patent T. trichiura infections would be associated with altered faecal microbiota and that anthelmintic treatment would induce a microbiota resembling more closely that observed in uninfected individuals. Materials and Methods: School children in Ecuador were screened for STH infections and allocated to 3 groups: uninfected, T. trichiura only, and mixed infections with T. trichiura and Ascaris lumbricoides. A sample of uninfected children and those with T. trichiura infections only were given anthelmintic treatment. Bacterial community profiles in faecal samples were studied by 454 pyrosequencing of 16 S rRNA genes. Results: Microbiota analyses of faeces were done for 97 children: 30 were uninfected, 17 were infected with T. trichiura, and 50 with T. trichiura and A. lumbricoides. Post-treatment samples were analyzed for 14 children initially infected with T. trichiura alone and for 21 uninfected children. Treatment resulted in 100% cure of STH infections. Comparisons of the microbiota at different taxonomic levels showed no statistically significant differences in composition between uninfected children and those with T. trichiura infections. We observed a decreased proportional abundance of a few bacterial genera from the Clostridia class of Firmicutes and a reduced bacterial diversity among children with mixed infections compared to the other two groups, indicating a possible specific effect of A. lumbricoides infection. Anthelmintic treatment of children with T. trichiura did not alter faecal microbiota composition. Discussion: Our data indicate that patent human infections with T. trichiura may have no effect on faecal microbiota but that A. lumbricoides colonisation might be associated with a disturbed microbiota. Our results also catalogue the microbiota of rural Ecuadorians and indicate differences with individuals from more urban industrialised societies

The remnants of galaxy formation from a panoramic survey of the region around M31

In hierarchical cosmological models, galaxies grow in mass through the continual accretion of smaller ones. The tidal disruption of these systems is expected to result in loosely bound stars surrounding the galaxy, at distances that reach $10 - 100$ times the radius of the central disk. The number, luminosity and morphology of the relics of this process provide significant clues to galaxy formation history, but obtaining a comprehensive survey of these components is difficult because of their intrinsic faintness and vast extent. Here we report a panoramic survey of the Andromeda galaxy (M31). We detect stars and coherent structures that are almost certainly remnants of dwarf galaxies destroyed by the tidal field of M31. An improved census of their surviving counterparts implies that three-quarters of M31's satellites brighter than $M_V < -6$ await discovery. The brightest companion, Triangulum (M33), is surrounded by a stellar structure that provides persuasive evidence for a recent encounter with M31. This panorama of galaxy structure directly confirms the basic tenets of the hierarchical galaxy formation model and reveals the shared history of M31 and M33 in the unceasing build-up of galaxies.Comment: Published in Nature. Supplementary movie available at https://www.astrosci.ca/users/alan/PANDAS/Latest%20news%3A%20movie%20of%20orbit.htm

The impact of different DNA extraction kits and laboratories upon the assessment of human gut microbiota composition by 16S rRNA gene sequencing

Peer reviewedPublisher PD

Alternative splicing and differential subcellular localization of the rat FGF antisense gene product

<p>Abstract</p> <p>Background</p> <p>GFG/NUDT is a nudix hydrolase originally identified as the product of the fibroblast growth factor-2 antisense (FGF-AS) gene. While the FGF-AS RNA has been implicated as an antisense regulator of FGF-2 expression, the expression and function of the encoded GFG protein is largely unknown. Alternative splicing of the primary FGF-AS mRNA transcript predicts multiple GFG isoforms in many species including rat. In the present study we focused on elucidating the expression and subcellular distribution of alternatively spliced rat GFG isoforms.</p> <p>Results</p> <p>RT-PCR and immunohistochemistry revealed tissue-specific GFG mRNA isoform expression and subcellular distribution of GFG immunoreactivity in cytoplasm and nuclei of a wide range of normal rat tissues. FGF-2 and GFG immunoreactivity were co-localized in some, but not all, tissues examined. Computational analysis identified a mitochondrial targeting sequence (MTS) in the N-terminus of three previously described rGFG isoforms. Confocal laser scanning microscopy and subcellular fractionation analysis revealed that all rGFG isoforms bearing the MTS were specifically targeted to mitochondria whereas isoforms and deletion mutants lacking the MTS were localized in the cytoplasm and nucleus. Mutation and deletion analysis confirmed that the predicted MTS was necessary and sufficient for mitochondrial compartmentalization.</p> <p>Conclusion</p> <p>Previous findings strongly support a role for the FGF antisense RNA as a regulator of FGF2 expression. The present study demonstrates that the antisense RNA itself is translated, and that protein isoforms resulting form alternative RNA splicing are sorted to different subcellular compartments. FGF-2 and its antisense protein are co-expressed in many tissues and in some cases in the same cells. The strong conservation of sequence and genomic organization across animal species suggests important functional significance to the physical association of these transcript pairs.</p

Massive migration from the steppe is a source for Indo-European languages in Europe

We generated genome-wide data from 69 Europeans who lived between 8,000-3,000 years ago by enriching ancient DNA libraries for a target set of almost four hundred thousand polymorphisms. Enrichment of these positions decreases the sequencing required for genome-wide ancient DNA analysis by a median of around 250-fold, allowing us to study an order of magnitude more individuals than previous studies and to obtain new insights about the past. We show that the populations of western and far eastern Europe followed opposite trajectories between 8,000-5,000 years ago. At the beginning of the Neolithic period in Europe, ~8,000-7,000 years ago, closely related groups of early farmers appeared in Germany, Hungary, and Spain, different from indigenous hunter-gatherers, whereas Russia was inhabited by a distinctive population of hunter-gatherers with high affinity to a ~24,000 year old Siberian6 . By ~6,000-5,000 years ago, a resurgence of hunter-gatherer ancestry had occurred throughout much of Europe, but in Russia, the Yamnaya steppe herders of this time were descended not only from the preceding eastern European hunter-gatherers, but from a population of Near Eastern ancestry. Western and Eastern Europe came into contact ~4,500 years ago, as the Late Neolithic Corded Ware people from Germany traced ~3/4 of their ancestry to the Yamnaya, documenting a massive migration into the heartland of Europe from its eastern periphery. This steppe ancestry persisted in all sampled central Europeans until at least ~3,000 years ago, and is ubiquitous in present-day Europeans. These results provide support for the theory of a steppe origin of at least some of the Indo-European languages of Europe