On solution structure of the radial heat problem with singular data

Initial value problem solutions to radial heat problem with Laplace operator and singular dat

The radial heat equation and Laplace transforms

Integral representations involving Laplace transforms developed for use in solutions of radial heat equatio

Dual Functions and Heat Expansions

Radial heat polynomial transformations in series expansions of heat equatio

Dilemmas of school-based Relationships and Sexuality Education for and about consent

In 2018, reflecting in this journal on the arrival of the ‘age of consent’ into sexuality education, Jen Gilbert questioned what would happen to a concept drawn in part from legal contexts, but partly also driven by the passion of feminist activists, when it met the demands and logics – the learning outcomes and lesson plans – of the classroom. This article offers one response, drawing on qualitative data from two whole-school sexual health programmes, Positive Choices and Project Respect, piloted in secondary schools in England between 2017 and 2019. It describes how each addressed the issue of consent and focuses on specific ‘moments’ that illuminate some of the challenges of doing so for both staff and students. Our analyses aim to contribute to the practice of relationships and sexuality education in schools by helping educators to anticipate, understand and therefore better address the dilemmas that teaching for and about consent might encounter. We argue that these dilemmas relate both to broader (and gendered) ideas of consent and entitlement, and to issues specific to schools. However, we also argue that a more theorised account of the school enables us to identify the minor achievements that are nonetheless possible

Meanfield treatment of Bragg scattering from a Bose-Einstein condensate

A unified semiclassical treatment of Bragg scattering from Bose-Einstein condensates is presented. The formalism is based on the Gross-Pitaevskii equation driven by classical light fields far detuned from atomic resonance. An approximate analytic solution is obtained and provides quantitative understanding of the atomic momentum state oscillations, as well as a simple expression for the momentum linewidth of the scattering process. The validity regime of the analytic solution is derived, and tested by three dimensional cylindrically symmetric numerical simulations.Comment: 21 pages, 10 figures. Minor changes made to documen

Loop expansion around the Bethe-Peierls approximation for lattice models

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe-Peierls approximation appears naturally as the saddle point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetisation and correlation function.Comment: 8 page

Younger-onset compared with later-onset type 2 diabetes: an analysis of the UK Prospective Diabetes Study (UKPDS) with up to 30 years of follow-up (UKPDS 92)

Background: Younger-onset type 2 diabetes is associated with accelerated complications. We assessed whether complications and mortality rates differed for younger age compared with older age at diagnosis over 30 years of follow-up. Methods: In this study, we used data from the UKPDS, collected between 1977 and 2007, of participants aged 25-65 years with newly diagnosed type 2 diabetes with younger-onset (younger than 40 years) or later-onset (40 years or older), and without diabetes autoantibodies. We analysed standardised mortality ratios (SMR) using UK general population data, and incidence rates of prespecified outcomes by 10-year age intervals at diagnosis. Findings: Of 4550 participants testing negative to all measured autoantibodies, 429 (9·4%) had younger-onset type 2 diabetes. 2704 (59·4%) were male, and mean HbA1c was 76 mmol/mol (SD 24·6). The median follow-up was 17·5 years (IQR 12·7-20·8). SMR for younger-onset type 2 diabetes was higher (3·72 [95% CI 2·98-4·64]) compared with later-onset type 2 diabetes (1·54 [1·47-1·61]). The incidence rate was higher for all outcomes in later-onset type 2 diabetes, except for microvascular disease (younger-onset 14·5 (11·9-17·7) vs later-onset 12·1 (11·3-13·0) per 1000 person-years). However, at any given age, the 5-year incidence of any diabetes-related endpoint, all-cause mortality, microvascular disease, and myocardial infarction was higher with younger age at diagnosis. Annual mean HbA1c was higher in the first 20 years in younger-onset compared with later-onset type 2 diabetes. Among participants randomised to intensive versus conventional glycaemic control, we observed no interactions by subgroup of younger-onset versus later-onset type 2 diabetes for any outcome. Interpretation: The risk of dying relative to the general population is even greater for people diagnosed with type 2 diabetes at younger ages. The increased risk of complications and poorer glycaemic control in younger-onset type 2 diabetes calls for the development of services to identify and manage these individuals. Funding: National Institute of Health and Care Research's Biomedical Research Centre

Electronic structure and optical properties of ZnX (X=O, S, Se, Te)

Electronic band structure and optical properties of zinc monochalcogenides with zinc-blende- and wurtzite-type structures were studied using the ab initio density functional method within the LDA, GGA, and LDA+U approaches. Calculations of the optical spectra have been performed for the energy range 0-20 eV, with and without including spin-orbit coupling. Reflectivity, absorption and extinction coefficients, and refractive index have been computed from the imaginary part of the dielectric function using the Kramers--Kronig transformations. A rigid shift of the calculated optical spectra is found to provide a good first approximation to reproduce experimental observations for almost all the zinc monochalcogenide phases considered. By inspection of the calculated and experimentally determined band-gap values for the zinc monochalcogenide series, the band gap of ZnO with zinc-blende structure has been estimated.Comment: 17 pages, 10 figure