Maximal cocliques in the generating graphs of the alternating and symmetric groups

Funding: This work was supported by EPSRC grant number EP/R014604/1. In addition, this work was partially supported by a grant from the Simons Foundation.The generating graph Γ (G) of a finite group G has vertex set the non-identity elements of G, with two elements adjacent exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ (G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ (G), but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of Sn and An are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal in the case of G = An and Sn, when n is prime and n ≠ qd-1/q-1 for all prime powers q and d ≥ 2. Namely, we show that two elements of G have identical sets of neighbours in Γ (G) if and only if they belong to exactly the same maximal subgroups.Publisher PDFPeer reviewe

The relational complexity of linear groups acting on subspaces

Funding: This work was supported by EPSRC grant no. EP/R014604/1, and also partially supported by a grant from the Simons Foundation. The first author was supported by the University of St Andrews (St Leonard’s International Doctoral Fees Scholarship & School of Mathematics and Statistics PhD Funding Scholarship), and by EPSRC grant no. EP/W522422/1. The second author is funded by the Heilbronn Institute.The relational complexity of a subgroup G of Sym(Ω) is a measure of the way in which the orbits of G on Ωk for various k determine the original action of G. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSLn() and PGLn(), for an arbitrary field , acting on the set of 1-dimensional subspaces of n. We also bound the relational complexity of all groups lying between PSLn(q) and PΓLn(q), and generalise these results to the action on m-spaces for m at least 1.Peer reviewe

The relational complexity of linear groups acting on subspaces

The relational complexity of a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a measure of the way in which the orbits of $G$ on $\Omega^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between $\mathrm{PSL}_{n}(\mathbb{F})$ and $\mathrm{PGL}_{n}(\mathbb{F})$, for an arbitrary field $\mathbb{F}$, acting on the set of $1$-dimensional subspaces of $\mathbb{F}^n$. We also bound the relational complexity of all groups lying between $\mathrm{PSL}_{n}(q)$ and $\mathrm{P}\Gamma\mathrm{L}_{n}(q)$, and generalise these results to the action on $m$-spaces for $m \ge 1$.Comment: 19 page

Effect of first line cancer treatment on the ovarian reserve and follicular density in girls under the age of 18 years

The Child Cancer Foundation in Denmark, The Novo Nordisk Foundation and the EU interregional project ReproHigh/ReproUnion are thanked for having funded this study.Objective: To study the impact of first-line antineoplastic treatment on the ovarian reserve in young girls returning for ovarian tissue cryopreservation (OTC) in connection with a relapse. Design: Retrospective case-control study. Setting: University hospitals. Patients: Sixty-three girls under the age of 18 years who underwent OTC before (group 1: 31 patients) and after (group 2: 32 patients) their initial cancer treatment. Intervention(s): None. Main Outcome Measure(s): Follicular densities (follicles/mm3) measured from an ovarian cortical biopsy before OTC. The ovarian volume (mL) of entire ovaries excised for OTC was also monitored. Result(s):There was no statistically significant difference in the mean age or follicular density between groups 1 and 2 (334 ± 476/mm3 vs. 327 ± 756/mm3). In contrast, the ovarian volume and total number of ovarian cortex chips cryopreserved were statistically significantly lower in patients who received gonadotoxic treatment before OTC (mean ± standard deviation [SD]: ovarian volume, 5.3 ± 3.1 mL vs. 2.9 ± 2.1 mL, respectively; number of cortex chips: 21.3 ± 8.1 vs. 15.2 ± 7.1, respectively). The reduction in the estimated ovarian reserve ranged from 10% to 20% in children to around 30% in adolescent girls (>10 years). Conclusion(s): Girls under the age of 10 tolerate a gonadotoxic insult better than adolescents, who may experience up to a 30% reduction in the ovarian reserve via first-line gonadotoxic treatment, which at present is considered to have little effect on the follicle pool. This information will improve counseling of young female cancer patients in deciding whether to undergo fertility preservation treatment.PostprintPeer reviewe

City of Hitchcock Comprehensive Plan 2020-2040

Hitchcock is a small town located in Galveston County (Figure 1.1), nestled up on the Texas Gulf Coast. It lies about 40 miles south-east of Houston. The boundaries of the city encloses an area of land of 60.46 sq. miles, an area of water of 31.64 sq. miles at an elevation just 16 feet above sea level. Hitchcock has more undeveloped land (~90% of total area) than the county combined. Its strategic location gives it a driving force of opportunities in the Houston-Galveston Region.The guiding principles for this planning process were Hitchcock’s vision statement and its corresponding goals, which were crafted by the task force. The goals focus on factors of growth and development including public participation, development considerations, transportation, community facilities, economic development, parks, and housing and social vulnerabilityTexas Target Communitie

Copernicus Ocean State Report, issue 6

The 6th issue of the Copernicus OSR incorporates a large range of topics for the blue, white and green ocean for all European regional seas, and the global ocean over 1993–2020 with a special focus on 2020

BHPR research: qualitative1. Complex reasoning determines patients' perception of outcome following foot surgery in rheumatoid arhtritis

Background: Foot surgery is common in patients with RA but research into surgical outcomes is limited and conceptually flawed as current outcome measures lack face validity: to date no one has asked patients what is important to them. This study aimed to determine which factors are important to patients when evaluating the success of foot surgery in RA Methods: Semi structured interviews of RA patients who had undergone foot surgery were conducted and transcribed verbatim. Thematic analysis of interviews was conducted to explore issues that were important to patients. Results: 11 RA patients (9 ♂, mean age 59, dis dur = 22yrs, mean of 3 yrs post op) with mixed experiences of foot surgery were interviewed. Patients interpreted outcome in respect to a multitude of factors, frequently positive change in one aspect contrasted with negative opinions about another. Overall, four major themes emerged. Function: Functional ability & participation in valued activities were very important to patients. Walking ability was a key concern but patients interpreted levels of activity in light of other aspects of their disease, reflecting on change in functional ability more than overall level. Positive feelings of improved mobility were often moderated by negative self perception ("I mean, I still walk like a waddling duck”). Appearance: Appearance was important to almost all patients but perhaps the most complex theme of all. Physical appearance, foot shape, and footwear were closely interlinked, yet patients saw these as distinct separate concepts. Patients need to legitimize these feelings was clear and they frequently entered into a defensive repertoire ("it's not cosmetic surgery; it's something that's more important than that, you know?”). Clinician opinion: Surgeons' post operative evaluation of the procedure was very influential. The impact of this appraisal continued to affect patients' lasting impression irrespective of how the outcome compared to their initial goals ("when he'd done it ... he said that hasn't worked as good as he'd wanted to ... but the pain has gone”). Pain: Whilst pain was important to almost all patients, it appeared to be less important than the other themes. Pain was predominately raised when it influenced other themes, such as function; many still felt the need to legitimize their foot pain in order for health professionals to take it seriously ("in the end I went to my GP because it had happened a few times and I went to an orthopaedic surgeon who was quite dismissive of it, it was like what are you complaining about”). Conclusions: Patients interpret the outcome of foot surgery using a multitude of interrelated factors, particularly functional ability, appearance and surgeons' appraisal of the procedure. While pain was often noted, this appeared less important than other factors in the overall outcome of the surgery. Future research into foot surgery should incorporate the complexity of how patients determine their outcome Disclosure statement: All authors have declared no conflicts of interes

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements