Classifying Hopf algebras of a given dimension

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.Comment: This version of the paper contains a correction on the published version. The statement and proof of Proposition 2.17 are changed and the proof of the results that follow from it are corrected accordingly. We thank H.-S. Ng for kindly communicating the gap to us and for the careful reading of our pape

Sustainable Impact by Design

2021 is the year of COP26, (the 26th Conference of the Parties who signed an agreement to work to prevent climate change back in 1994). To mark this, the University of Strathclyde offered funds to staff that would let them talk to people about their research and how it can limit climate change. The STEM Equals Summer Scheme won funding, and so ten Strathclyde staff members were able to work with diverse young people in Ayr and Girvan over five days of STEM1 activities. This book shows some of the activities we did together, and some of the learning about their planet and sustainability that went on. As you will see in its pages, everyone involved, staff and participants alike, had a lot of fun. We also found out many new things about ourselves, our colleagues, about STEM and about our environment

Spondylarthropathies (including psoriatic arthritis): 244. Validity of Colour Doppler and Spectral Doppler Ultrasound of Sacroilicac Joints Againts Physical Examination as Gold Standard

Background: Sacroiliac joints (SJ) involvement is a distinctive and charasteristic feature of Spondyloarthritis (SpA) and x-ray is the test routinely used to make a diagnosis. However, x-ray reveals late structural damage but cannot detect active inflammation. The objective of this study was to assess the validity of Doppler ultrasound in SJ. Methods: Prospective blinded and controlled study of SJ, in which three populations were compared. We studied 106 consecutive cases, who were divided into three groups: a) 53 patients diagnosed with SpA who had inflammatory lumbar and gluteal pain assessed by a rheumatologist; b) 26 patients diagnosed with SpA who didn't have SJ tenderness and had normal physical examination; c) control group of 27 subjects (healthy subjetcs or with mechanical lumbar pain). All patients included that were diagnosed with SpA met almost the European Spondyloarthropathy Study Group (ESSG) classification criteria. Physical examination of the SJ included: sacral sulcus tenderness, iliac gapping, iliac compression, midline sacral thrust test, Gaenslen's test, and Patrick s test were used as gold standard. Both SJ were examined with Doppler ultrasound (General Electric Logiq 9, Wauwatosa WI, USA) fitted with a 9-14 Mhz lineal probe. The ultrasonographer was blinded to clinical data. Doppler in SJ was assessed as positive when both Doppler colour and resistance index (RI) < 0.75 within the SJ area were present. Statistical analysis was performed estimating sensitivity and specificity against gold standard. The Kappa correlation coefficient was used for reliability study. Results: 106 cases (53 female, 55 male; mean age 36 10 years) were studied. There were no statistical differences between groups related to age or sex. Physical examination of SJ was positive in 38 patients (59 sacroiliac joints). US detected Doppler signal within SJ in 37 patients (58 SJ): 33 of them were symptomatic SpA (52 SJ), one of them were asymptomatic SpA (1 SJ) and one was a healthy control (1 SJ). The accuracy of US when compared to clinical data as gold standard at subject level in the overall group was: sensitivity of 68.6% and specificity of 85.7%, positive predictive value of 70.5% and negative predictive value of 84.5%. A positive likelihood ratio of 4.8, a negative likelihood ratio of 0.36 and a kappa coefficient of 0.55 were achieved. Conclusions: Doppler US of SJ seems to be a valid method to detect active SJ inflammation. Disclosure statement: The authors have declared no conflicts of interes

Urban hedges: a review of plant species and cultivars for ecosystem service delivery in north-west Europe

Urban hedges provide a number of important ecosystem services (ESs) including microclimate alteration, flood and pollution mitigation, and biodiversity provision, along with some disservices (DSs, e.g. invasiveness, allergenicity). However, hedge plant species differ in their capacity to promote different services, so it is important that the decision to plant hedges is evidence-based. The objectives of this study were thus to (i) to review the role of urban hedges within NW Europe; (ii) review the available literature detailing the ESs and DSs provided by different plant species and cultivars when used as hedge plants; (iii) identify where there is a lack of evidence for certain species or ESs/DSs; and (iv) develop a starting point for a discussion about appropriate species/cultivar selection to deliver multiple ESs, and avoid DSs. Many studies consider biodiversity and air quality ESs. There are significant gaps in the literature relating to rainfall mitigation/flood protection, but also CO2 sequestration, allergenicity and human psychological well-being impact of different species. Additionally, for noise and pollution mitigation studies, a range of methodologies and units are used, making comparisons between hedge species difficult/impossible. A number of common hedge species demonstrated high levels of ESs delivery, including Fagus sylvatica, Crataegus monogyna, Ilex aquifolium and Rosa rugosa. No species surveyed had an entirely negative association with ESs, and most provide at least some benefits in supporting ESs provision (e.g. Viburnum tinus, Laurus nobilis). We created a matrix, in a table form, linking plant species, key plant traits and ESs/DSs, which should make it easier for professionals to choose species best suited to provide multiple benefits, whilst minimising the drawbacks. Our review suggests that the relative contribution of urban hedges to ESs delivery may be under-valued currently, and calls for more research

Development of a Definition of Postacute Sequelae of SARS-CoV-2 Infection

IMPORTANCE: SARS-CoV-2 infection is associated with persistent, relapsing, or new symptoms or other health effects occurring after acute infection, termed postacute sequelae of SARS-CoV-2 infection (PASC), also known as long COVID. Characterizing PASC requires analysis of prospectively and uniformly collected data from diverse uninfected and infected individuals. OBJECTIVE: To develop a definition of PASC using self-reported symptoms and describe PASC frequencies across cohorts, vaccination status, and number of infections. DESIGN, SETTING, AND PARTICIPANTS: Prospective observational cohort study of adults with and without SARS-CoV-2 infection at 85 enrolling sites (hospitals, health centers, community organizations) located in 33 states plus Washington, DC, and Puerto Rico. Participants who were enrolled in the RECOVER adult cohort before April 10, 2023, completed a symptom survey 6 months or more after acute symptom onset or test date. Selection included population-based, volunteer, and convenience sampling. EXPOSURE: SARS-CoV-2 infection. MAIN OUTCOMES AND MEASURES: PASC and 44 participant-reported symptoms (with severity thresholds). RESULTS: A total of 9764 participants (89% SARS-CoV-2 infected; 71% female; 16% Hispanic/Latino; 15% non-Hispanic Black; median age, 47 years [IQR, 35-60]) met selection criteria. Adjusted odds ratios were 1.5 or greater (infected vs uninfected participants) for 37 symptoms. Symptoms contributing to PASC score included postexertional malaise, fatigue, brain fog, dizziness, gastrointestinal symptoms, palpitations, changes in sexual desire or capacity, loss of or change in smell or taste, thirst, chronic cough, chest pain, and abnormal movements. Among 2231 participants first infected on or after December 1, 2021, and enrolled within 30 days of infection, 224 (10% [95% CI, 8.8%-11%]) were PASC positive at 6 months. CONCLUSIONS AND RELEVANCE: A definition of PASC was developed based on symptoms in a prospective cohort study. As a first step to providing a framework for other investigations, iterative refinement that further incorporates other clinical features is needed to support actionable definitions of PASC