On the union of intersecting families

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of $k$-element subsets of an $n$-element set, for each triple of integers $(n,k,r)$. We make progress on this problem, proving that for any fixed integer $r \geq 2$ and for any $k \leq (\tfrac{1}{2}-o(1))n$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\mathcal{F}| \leq {n \choose k} - {n-r \choose k}$, with equality only if $\mathcal{F} = \{S \subset X:\ |S|=k,\ S \cap R \neq \emptyset\}$for some$R \subset X$with$|R|=r$. This is best possible up to the size of the$o(1)$term, and improves a 1987 result of Frankl and F\"uredi, who obtained the same conclusion under the stronger hypothesis$k < (3-\sqrt{5})n/2$, in the case$r=2$. Our proof utilises an isoperimetric, influence-based method recently developed by Keller and the authors.Comment: 13 pages. Updated references, expositional changes and minor corrections following the helpful comments of an anonymous refere

On the Structure of Subsets of the Discrete Cube with Small Edge Boundary

Abstract. The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers m and n, the minimum size gn(m) of the edge boundary of n an m-element subset of {0, 1} ; the extremal families (up to automorphisms of n the discrete cube) are initial segments of the lexicographic ordering on {0, 1} . n We show that for any m-element subset F ⊂ {0, 1} and any integer l, if the edge boundary of F has size at most gn(m) + l, then there exists an extremal n family G ⊂ {0, 1} such that |F∆G| ≤ Cl, where C is an absolute constant. This is best-possible, up to the value of C. Our result can be seen as a ‘stability’ version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli [15] for the isoperimetric inequality in Euclidean space

Research priorities in children requiring elective surgery for conditions affecting the lower limbs: a James Lind Alliance Priority Setting Partnership

Objective: To identify and prioritise research questions concerning the elective surgical management of children with conditions affecting the lower limb by engaging patients, carers and healthcare professionals. Design: A modified nominal group technique. Setting: UK. Participants: 388 individuals (29 patients, 155 parents/carers, 204 healthcare professionals) were recruited through hospital clinics, patient charities and professional organisations and participated in the initial prioritisation survey; 234 individuals took part in the interim prioritisation survey. 33 individuals (3 patients, 9 parents/carers, 11 healthcare professionals, 7 individuals representing the project’s steering group and 3 James Lind Alliance (JLA) facilitators) attended the final face-to-face workshop to rank the top 10 research priorities. Interventions: Surveys were distributed using various media resources such as newsletters, internet messaging boards and the ‘Paediatric Lower Limb Surgery Priority Setting Partnership (PSP) website. Printed copies of the questionnaire were also made available to families in outpatient clinics. Outcome measures: Survey results, top 10 and top 26 priority rankings Results: The process took 18 months to complete (July 2017–January 2019); 388 people generated 1023 questions; a total of 801 research questions were classified as true uncertainties. Following the JLA methodology, 75 uncertainties were developed from the initial 801 questions. Twenty six of those were selected through a second survey and were taken to the final face-to-face workshop where the top 10 research priorities were selected. The top10 priorities included questions on cerebral palsy, common hip conditions (ie, Perthes’ disease and developmental dysplasia of the hip) as well as rehabilitation techniques and methods to improve shared decision-making between clinicians and patients/families. Conclusions: This is the first JLA PSP in children’s orthopaedic surgery, a particularly under-researched and underfunded area. We have identified important research topics which will guide researchers and funders and direct their efforts in future research

Population-level effects of parasitism on a freshwater ecosystem engineer, the unionid mussel Anodonta anatina

Funder: Woolf Fisher TrustAbstract: Parasites can negatively affect hosts at individual, population, and species‐level scales. However, the link between individual‐ and population‐level impacts is often poorly understood. In particular, the population‐level response to parasitism may alter wider ecosystem dynamics if animals with ecosystem engineering capabilities are infected. Here, we examine the effects of parasitism on a freshwater ecosystem engineer, the unionid mussel Anodonta anatina, at two different sites. We study three common parasites: the digenean trematode Rhipidocotyle campanula; the unionicolid mite Unionicola intermedia; and the ectoparasitic invasive zebra mussel Dreissena polymorpha. As well as demonstrating the individual‐level effects of parasitism on the native host mussel, we construct a simple model to estimate the reduction in population‐level reproductive output caused by parasites. We show that both infection prevalence and intensity were population‐specific, with one site having more than three times as many native mussels infected by trematodes and mites than the other, but more than four times fewer mussels afflicted by invasive zebra mussels. Negative reproductive consequences for individual host mussels were documented as a result of parasitism, with trematodes causing castration at both sites. Mites were also correlated with a reduction in the viability of larval offspring (glochidia) by more than 25%, but only at one site, suggesting some potential impacts of parasitism may be population specific. The population‐level model shows that parasitism alone reduces larval output of the two populations by 10% and 13%, respectively. Our study takes the important step of scaling individual‐level effects of parasitism to population‐level processes, and highlights the influence that parasites may have in the population dynamics of unionid mussels. Given the ecosystem engineering capabilities of A. anatina, such effects may have important impacts on the wider biota. Even at relatively low prevalences, the observed effects of parasites on native mussel populations suggests that parasitism must be considered in the conservation of freshwater mussels, one of the world's most globally imperilled faunal groups. Further, understanding how the effects of parasitism on individual hosts scales to the ecosystem level is a crucial and unaddressed question in freshwater biology

Mass extinctions and supernova explosions

A nearby supernova (SN) explosion could have negatively influenced life on Earth, maybe even been responsible for mass extinctions. Mass extinction poses a significant extinction of numerous species on Earth, as recorded in the paleontologic, paleoclimatic, and geological record of our planet. Depending on the distance between the Sun and the SN, different types of threats have to be considered, such as ozone depletion on Earth, causing increased exposure to the Sun's ultraviolet radiation, or the direct exposure of lethal x-rays. Another indirect effect is cloud formation, induced by cosmic rays in the atmosphere which result in a drop in the Earth's temperature, causing major glaciations of the Earth. The discovery of highly intensive gamma ray bursts (GRBs), which could be connected to SNe, initiated further discussions on possible life-threatening events in Earth's history. The probability that GRBs hit the Earth is very low. Nevertheless, a past interaction of Earth with GRBs and/or SNe cannot be excluded and might even have been responsible for past extinction events.Comment: Chapter for forthcoming book: Handbook of Supernovae, P. Murdin and A. Alsabeti (eds.), Springer International Publishing (in press

Tylosis with oesophageal cancer: Diagnosis, management and molecular mechanisms

Research on iRHOM2 in the Kelsell group is funded by an MRC project grant, a MRC Clinical Fellowship (to TM) and a Cancer Research UK program grant

Twenty Years of SUGRA

A brief review is given of the developments of mSUGRA and its extensions since the formulation of these models in 1982. Future directions and prospects are also discussed.Comment: Invited talk at the International Conference BEYOND-2003, Schloss Ringberg, Germany, June 10-14, 2003; 21 pages, Late

Structural basis for the RING catalyzed synthesis of K63 linked ubiquitin chains

This work was supported by grants from Cancer Research UK (C434/A13067), the Wellcome Trust (098391/Z/12/Z) and Biotechnology and Biological Sciences Research Council (BB/J016004/1).The RING E3 ligase catalysed formation of lysine 63 linked ubiquitin chains by the Ube2V2–Ubc13 E2 complex is required for many important biological processes. Here we report the structure of the RING domain dimer of rat RNF4 in complex with a human Ubc13~Ub conjugate and Ube2V2. The structure has captured Ube2V2 bound to the acceptor (priming) ubiquitin with Lys63 in a position that could lead to attack on the linkage between the donor (second) ubiquitin and Ubc13 that is held in the active “folded back” conformation by the RING domain of RNF4. The interfaces identified in the structure were verified by in vitro ubiquitination assays of site directed mutants. This represents the first view of the synthesis of Lys63 linked ubiquitin chains in which both substrate ubiquitin and ubiquitin-loaded E2 are juxtaposed to allow E3 ligase mediated catalysis.PostprintPeer reviewe

Cosmic Flows on 100 Mpc/h Scales: Standardized Minimum Variance Bulk Flow, Shear and Octupole Moments

The low order moments, such as the bulk flow and shear, of the large scale peculiar velocity field are sensitive probes of the matter density fluctuations on very large scales. In practice, however, peculiar velocity surveys are usually sparse and noisy, which can lead to the aliasing of small scale power into what is meant to be a probe of the largest scales. Previously, we developed an optimal ``minimum variance'' (MV) weighting scheme, designed to overcome this problem by minimizing the difference between the measured bulk flow (BF) and that which would be measured by an ideal survey. Here we extend this MV analysis to include the shear and octupole moments, which are designed to have almost no correlations between them so that they are virtually orthogonal. We apply this MV analysis to a compilation of all major peculiar velocity surveys, consisting of 4536 measurements. Our estimate of the BF on scales of ~ 100 Mpc/h has a magnitude of |v|= 416 +/- 78 km/s towards Galactic l = 282 degree +/- 11 degree and b = 6 degree +/- 6 degree. This result is in disagreement with LCDM with WMAP5 cosmological parameters at a high confidence level, but is in good agreement with our previous MV result without an orthogonality constraint, showing that the shear and octupole moments did not contaminate the previous BF measurement. The shear and octupole moments are consistent with WMAP5 power spectrum, although the measurement noise is larger for these moments than for the BF. The relatively low shear moments suggest that the sources responsible for the BF are at large distances.Comment: 13 Pages, 7 figures, 4 tables. Some changes to reflect the published versio