Cycles of length three and four in tournaments

Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$

Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming

A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry ? are solvable in time g(d,?) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and ?. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth. We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width

Substructure sensities in extremal combinatorics

One of the primary goals of extremal combinatorics is to understand how an object’s properties are influenced by the presence or multiplicity of a given substructure. Most classical theorems in the area, such as Mantel’s Theorem, are phrased in terms of substructure counts such as the number of edges or the number of triangles in a graph. Gradually, however, it has become more popular to express results in terms of the density of substructures, where the substructure counts are normalised by some natural quantity. This approach has several benefits; results are often more succinctly stated using densities, and it becomes easier to focus on the asymptotic behaviour of objects. In this thesis, we study three topics concerning density. We begin Chapter 1 by contextualising the study of combinatorial density and justifying its importance within extremal combinatorics. We also introduce the relevant combinatorial objects, results, and questions that are central to the later chapters. Particular attention is paid to developing the theory of graph limits and flag algebras, two modern fields that rely heavily on the notion of density. In Chapter 2, we investigate the interplay between the densities of cycles of length 3 and 4 in large tournaments. In particular, we prove two cases of a conjecture of Linial and Morgenstern (2016) that the minimum density of 4-cycles in a graph with a fixed density of 3-cycles is attained by a particular random construction. In Chapter 3, we explore quasirandom permutations. A permutation is said to be quasirandom if the density of every subpermutation matches the expected density in a random permutation. Our main result is that quasirandomness can be characterised by a property which, on the surface, appears significantly weaker. Lastly, in Chapter 4, we resolve a problem posed by Bubeck and Linial (2016) on the inducibility of trees. The inducibility of a tree X is defined as the maximum possible density of X in a large tree. We show that there exist non-path, non-star trees with positive inducibility, but that all such trees have inducibility bounded away from 1. We also show that there exists a sequence of trees in which every possible subtree appears asymptotically with positive density

Characterization of quasirandom permutations by a pattern sum

It is known that a sequence urn:x-wiley:rsa:media:rsa20956:rsa20956-math-0001 of permutations is quasirandom if and only if the pattern density of every 4‐point permutation in urn:x-wiley:rsa:media:rsa20956:rsa20956-math-0002 converges to 1/24. We show that there is a set S of 4‐point permutations such that the sum of the pattern densities of the permutations from S in the permutations urn:x-wiley:rsa:media:rsa20956:rsa20956-math-0003 converges to urn:x-wiley:rsa:media:rsa20956:rsa20956-math-0004 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight

The Cassava Genome: Current Progress, Future Directions

The starchy swollen roots of cassava provide an essential food source for nearly a billion people, as well as possibilities for bioenergy, yet improvements to nutritional content and resistance to threatening diseases are currently impeded. A 454-based whole genome shotgun sequence has been assembled, which covers 69% of the predicted genome size and 96% of protein-coding gene space, with genome finishing underway. The predicted 30,666 genes and 3,485 alternate splice forms are supported by 1.4 M expressed sequence tags (ESTs). Maps based on simple sequence repeat (SSR)-, and EST-derived single nucleotide polymorphisms (SNPs) already exist. Thanks to the genome sequence, a high-density linkage map is currently being developed from a cross between two diverse cassava cultivars: one susceptible to cassava brown streak disease; the other resistant. An efficient genotyping-by-sequencing (GBS) approach is being developed to catalog SNPs both within the mapping population and among diverse African farmer-preferred varieties of cassava. These resources will accelerate marker-assisted breeding programs, allowing improvements in disease-resistance and nutrition, and will help us understand the genetic basis for disease resistance

Rapid Environmental Change over the Past Decade Revealed by Isotopic Analysis of the California Mussel in the Northeast Pacific

The anthropogenic input of fossil fuel carbon into the atmosphere results in increased carbon dioxide (CO2) into the oceans, a process that lowers seawater pH, decreases alkalinity and can inhibit the production of shell material. Corrosive water has recently been documented in the northeast Pacific, along with a rapid decline in seawater pH over the past decade. A lack of instrumentation prior to the 1990s means that we have no indication whether these carbon cycle changes have precedence or are a response to recent anthropogenic CO2 inputs. We analyzed stable carbon and oxygen isotopes (δ13C, δ18O) of decade-old California mussel shells (Mytilus californianus) in the context of an instrumental seawater record of the same length. We further compared modern shells to shells from 1000 to 1340 years BP and from the 1960s to the present and show declines in the δ13C of modern shells that have no historical precedent. Our finding of decline in another shelled mollusk (limpet) and our extensive environmental data show that these δ13C declines are unexplained by changes to the coastal food web, upwelling regime, or local circulation. Our observed decline in shell δ13C parallels other signs of rapid changes to the nearshore carbon cycle in the Pacific, including a decline in pH that is an order of magnitude greater than predicted by an equilibrium response to rising atmospheric CO2, the presence of low pH water throughout the region, and a record of a similarly steep decline in δ13C in algae in the Gulf of Alaska. These unprecedented changes and the lack of a clear causal variable underscores the need for better quantifying carbon dynamics in nearshore environments

European bone mineral density loci are also associated with BMD in East-Asian populations

To access publisher full text version of this article. Please click on the hyperlink in Additional Links fieldMost genome-wide association (GWA) studies have focused on populations of European ancestry with limited assessment of the influence of the sequence variants on populations of other ethnicities. To determine whether markers that we have recently shown to associate with Bone Mineral Density (BMD) in Europeans also associate with BMD in East-Asians we analysed 50 markers from 23 genomic loci in samples from Korea (n = 1,397) and two Chinese Hong Kong sample sets (n = 3,869 and n = 785). Through this effort we identified fourteen loci that associated with BMD in East-Asian samples using a false discovery rate (FDR) of 0.05; 1p36 (ZBTB40, P = 4.3×10(-9)), 1p31 (GPR177, P = 0.00012), 3p22 (CTNNB1, P = 0.00013), 4q22 (MEPE, P = 0.0026), 5q14 (MEF2C, P = 1.3×10(-5)), 6q25 (ESR1, P = 0.0011), 7p14 (STARD3NL, P = 0.00025), 7q21 (FLJ42280, P = 0.00017), 8q24 (TNFRSF11B, P = 3.4×10(-5)), 11p15 (SOX6, P = 0.00033), 11q13 (LRP5, P = 0.0033), 13q14 (TNFSF11, P = 7.5×10(-5)), 16q24 (FOXL1, P = 0.0010) and 17q21 (SOST, P = 0.015). Our study marks an early effort towards the challenge of cataloguing bone density variants shared by many ethnicities by testing BMD variants that have been established in Europeans, in East-Asians

Genome-wide trans-ancestry meta-analysis provides insight into the genetic architecture of type 2 diabetes susceptibility.

To further understanding of the genetic basis of type 2 diabetes (T2D) susceptibility, we aggregated published meta-analyses of genome-wide association studies (GWAS), including 26,488 cases and 83,964 controls of European, east Asian, south Asian and Mexican and Mexican American ancestry. We observed a significant excess in the directional consistency of T2D risk alleles across ancestry groups, even at SNPs demonstrating only weak evidence of association. By following up the strongest signals of association from the trans-ethnic meta-analysis in an additional 21,491 cases and 55,647 controls of European ancestry, we identified seven new T2D susceptibility loci. Furthermore, we observed considerable improvements in the fine-mapping resolution of common variant association signals at several T2D susceptibility loci. These observations highlight the benefits of trans-ethnic GWAS for the discovery and characterization of complex trait loci and emphasize an exciting opportunity to extend insight into the genetic architecture and pathogenesis of human diseases across populations of diverse ancestry

Segregation of In to dislocations in InGaN.

Dislocations are one-dimensional topological defects that occur frequently in functional thin film materials and that are known to degrade the performance of InxGa1-xN-based optoelectronic devices. Here, we show that large local deviations in alloy composition and atomic structure are expected to occur in and around dislocation cores in InxGa(1-x)N alloy thin films. We present energy-dispersive X-ray spectroscopy data supporting this result. The methods presented here are also widely applicable for predicting composition fluctuations associated with strain fields in other inorganic functional material thin films.This work was funded in part by the Cambridge Commonwealth trust, St. John’s College and the EPSRC. SKR is funded through the Cambridge-India Partnership Fund and Indian Institute of Technology Bombay via a scholarship. MAM acknowledges support from the Royal Society through a University Research Fellowship. Additional support was provided by the EPSRC through the UK National Facility for Aberration-Corrected STEM (SuperSTEM). The Titan 80- 200kV ChemiSTEMTM was funded through HM Government (UK) and is associated with the capabilities of the University of Manchester Nuclear Manufacturing (NUMAN) capabilities. SJH acknowledges funding from the Defence Treat Reduction Agency (DTRA) USA (grant number HDTRA1-12-1-0013).This is the accepted manuscript. The final version is available at http://pubs.acs.org/doi/abs/10.1021/nl5036513