Properties and zeros of 3F2 hypergeometric functions

Student Number : 9606114D PhD Thesis School of Mathematics Faculty of ScienceIn this thesis, our primary interest lies in the investigation of the location of the zeros and the asymptotic zero distribution of hypergeometric polynomials. The location of the zeros and the asymptotic zero distribution of general hy- pergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably 2F1 and 1F1 hypergeometric polynomials which have been extensively studied. In the case of 3F2 polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. Our intro- duction in Chapter 1 outlines known results in this area and we also review recent papers dealing with the location of the zeros of 2F1 and 1F1 hyperge- ometric polynomials. In Chapter 2, we consider two classes of 3F2 hypergeometric polynomials, each of which has a representation in terms of 2F1 polynomials. Our first result proves that the class of polynomials 3F2(−n, a, b; a−1, d; x), a, b, d 2 R, n 2 N is quasi-orthogonal of order 1 on an interval that varies with the values of the real parameters b and d. We deduce the location of (n−1) of its zeros and dis- cuss the apparent role played by the parameter a with regard to the location of the one remaining zero of this class of polynomials. We also prove re- sults on the location of the zeros of the classes 3F2(−n, b, b−n 2 ; b−n, b−n−1 2 ; x), b 2 R, n 2 N and 3F2 (−n, b, b−n 2 + 1; b − n, b−n+1 2 ; x), n 2 N, b 2 R by using the orthogonality and quasi-orthogonality of factors involved in its representation. We use Mathematica to plot the zeros of these 3F2 hypergeometric polynomials for different values of n as well as for different ranges of the pa- rameters. The numerical data is consistent with the results we have proved. The Euler integral representation of the 2F1 Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of 2F1(a, b; c; 1), among other applications (cf. [1], p.65). The general p+kFq+k hypergeometric function has an integral repre- sentation (cf. [37], Theorem 38) where the integrand involves pFq. In Chapter 3, we give a simple and direct proof of an Euler integral representation for a special class of q+1Fq functions for q >= 2. The values of certain 3F2 and 4F3 functions at x = 1, some of which can be derived using other methods, are deduced from our integral formula. In Chapter 4, we prove that the zeros of 2F1 (−n, n+1 2 ; n+3 2 ; z) asymptotically approach the section of the lemniscate {z : |z(1 − z)2| = 4 27 ;Re(z) > 1 3} as n ! 1. In recent papers (cf. [31], [32], [34], [35]), Mart´ınez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic distribution of Jacobi polynomials P(an,bn) n when the limits A = lim n!1 an n and B = lim n!1 Bn n exist and lie in the interior of certain specified regions in the AB-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Mart´ınez-Finkelshtein classification

Developing teaching for mathematical resilience in further education

The construct ‘Mathematical Resilience’ [1] has been developed to describe a positive stance towards mathematics; resilient learners develop approaches to mathematical learning which help them to overcome the affective barriers and setbacks that can be part of learning mathematics for many people. A resilient stance towards mathematics can be engineered by a strategic and explicit focus on the culture of learning mathematics within both formal and informal learning environments. As part of that engineering, we have developed the notion of ‘Teaching for Mathematical Resilience’. The work described here is focused on developing teachers who know how explicitly to develop resilient learners of mathematics. Teachers for Mathematical Resilience develop a group culture of ‘can do’ mathematics which works to counter the prevalent culture of mathematics helplessness and mathematics anxiety in the general population when faced with mathematical ideas. This paper discusses the changes in awareness brought about by a one-day course designed to develop ‘teaching for mathematical resilience’. The course presentations ran between November 2015 and July 2016 and recruited participants who work as teachers of numeracy or mathematics in Further Education (FE) institutions in England – predominantly in the Midlands. Many of these teachers were being required to teach beyond their own level of mathematical confidence. The data shows that it is possible within a one day course to increase teachers’ awareness of negative past experiences as a possible cause of difficulty with mathematics; teachers become aware of how patterns of behaviour such as avoidance and disruption may have developed as safe-preservation habits and how mathematics anxiety can be transmitted from teacher to student in a vicious cycle. Teachers are supported to work through personal anxieties towards mathematics in a safe and collaborative environment and to develop elements of personal mathematical resilience and awareness of the affective domain. Thus we have sought to break the cycle of mathematics anxiety by educating teacher awareness. However, we have also found that many UK FE teachers request and would likely benefit from further courses

Prevalence and causes of prescribing errors: the prescribing outcomes for trainee doctors engaged in clinical training (PROTECT) study

Objectives Study objectives were to investigate the prevalence and causes of prescribing errors amongst foundation doctors (i.e. junior doctors in their first (F1) or second (F2) year of post-graduate training), describe their knowledge and experience of prescribing errors, and explore their self-efficacy (i.e. confidence) in prescribing. Method A three-part mixed-methods design was used, comprising: prospective observational study; semi-structured interviews and cross-sectional survey. All doctors prescribing in eight purposively selected hospitals in Scotland participated. All foundation doctors throughout Scotland participated in the survey. The number of prescribing errors per patient, doctor, ward and hospital, perceived causes of errors and a measure of doctors' self-efficacy were established. Results 4710 patient charts and 44,726 prescribed medicines were reviewed. There were 3364 errors, affecting 1700 (36.1%) charts (overall error rate: 7.5%; F1:7.4%; F2:8.6%; consultants:6.3%). Higher error rates were associated with : teaching hospitals (p&#60;0.001), surgical (p = &#60;0.001) or mixed wards (0.008) rather thanmedical ward, higher patient turnover wards (p&#60;0.001), a greater number of prescribed medicines (p&#60;0.001) and the months December and June (p&#60;0.001). One hundred errors were discussed in 40 interviews. Error causation was multi-factorial; work environment and team factors were particularly noted. Of 548 completed questionnaires (national response rate of 35.4%), 508 (92.7% of respondents) reported errors, most of which (328 (64.6%) did not reach the patient. Pressure from other staff, workload and interruptions were cited as the main causes of errors. Foundation year 2 doctors reported greater confidence than year 1 doctors in deciding the most appropriate medication regimen. Conclusions Prescribing errors are frequent and of complex causation. Foundation doctors made more errors than other doctors, but undertook the majority of prescribing, making them a key target for intervention. Contributing causes included work environment, team, task, individual and patient factors. Further work is needed to develop and assess interventions that address these.</p

C-STICH2: emergency cervical cerclage to prevent miscarriage and preterm birth—study protocol for a randomised controlled trial

Abstract Background Cervical cerclage is a recognised treatment to prevent late miscarriage and pre-term birth (PTB). Emergency cervical cerclage (ECC) for cervical dilatation with exposed unruptured membranes is less common and the potential benefits of cerclage are less certain. A randomised control trial is needed to accurately assess the effectiveness of ECC in preventing pregnancy loss compared to an expectant approach. Methods C-STICH2 is a multicentre randomised controlled trial in which women presenting with cervical dilatation and unruptured exposed membranes at 16 + 0 to 27 + 6 weeks gestation are randomised to ECC or expectant management. Trial design includes 18 month internal pilot with embedded qualitative process evaluation, minimal data set and a within-trial health economic analysis. Inclusion criteria are ≥16 years, singleton pregnancy, exposed membranes at the external os, gestation 16 + 0–27 + 6 weeks, and informed consent. Exclusion criteria are contraindication to cerclage, cerclage in situ or previous cerclage in this pregnancy. Randomisation occurs via an online service in a 1:1 ratio, using a minimisation algorithm to reduce chance imbalances in key prognostic variables (site, gestation and dilatation). Primary outcome is pregnancy loss; a composite including miscarriage, termination of pregnancy and perinatal mortality defined as stillbirth and neonatal death in the first week of life. Secondary outcomes include all core outcomes for PTB. Two-year development outcomes will be assessed using general health and Parent Report of Children’s Abilities-Revised (PARCA-R) questionnaires. Intended sample size is 260 participants (130 each arm) based on 60% rate of pregnancy loss in the expectant management arm and 40% in the ECC arm, with 90% power and alpha 0.05. Analysis will be by intention-to-treat. Discussion To date there has been one small trial of ECC in 23 participants which included twin and singleton pregnancies. This small trial along with the largest observational study (n = 161) found ECC to prolong pregnancy duration and reduce deliveries before 34 weeks gestation. It is important to generate high quality evidence on the effectiveness of ECC in preventing pregnancy loss, and improve understanding of the prevalence of the condition and frequency of complications associated with ECC. An adequately powered RCT will provide the highest quality evidence regarding optimum care for these women and their babies. Trial registration ISRCTN Registry ISRCTN12981869 . Registered on 13th June 2018

Causes and Timing of Mortality and Morbidity Among Late Presenters Starting Antiretroviral Therapy in the REALITY Trial.

BACKGROUND: In sub-Saharan Africa, 20%-25% of people starting antiretroviral therapy (ART) have severe immunosuppression; approximately 10% die within 3 months. In the Reduction of EArly mortaLITY (REALITY) randomized trial, a broad enhanced anti-infection prophylaxis bundle reduced mortality vs cotrimoxazole. We investigate the contribution and timing of different causes of mortality/morbidity. METHODS: Participants started ART with a CD4 count .3); and reduced nonfatal/fatal tuberculosis and cryptococcosis (P .2). CONCLUSIONS: Enhanced prophylaxis reduced mortality from cryptococcosis and unknown causes and nonfatal tuberculosis and cryptococcosis. High early incidence of fatal/nonfatal events highlights the need for starting enhanced-prophylaxis with ART in advanced disease. CLINICAL TRIALS REGISTRATION: ISRCTN43622374

Policing mining: In outer-space greed and domination vs. peace and equity a governance for humanity!

Staking claim and ownership has remained an antagonistic issue for nations, resulting in many international conflicts. This is particularly so in disputed territories or areas which are deemed the heritage of mankind. In the next 50-years mining in space is set to become a reality and rather than being used to become an asset to man/society and create an equitable world, it is likely to be a battleground for greed and sovereign dominance – an overspill from Earth. This paper researches the conflict between greed and dominance vs. peace and equity in respect to space - mineral resources, providing historical contextualization, opinion, thoughts and interpretation. Hence, consideration is given to international approaches and who should ‘police,’ plus the governance of, space riches. The research largely considers the stance of the USA in this respect. The relevance of travel and travel modes (particularly air) and ownership of the sky is reviewed, so as to provide comparison and (historical) contextualization – identifying issues previously encountered when man looks to both travel and acquire assets by these means. The latest position of asteroid mining is also explored and ‘lessons from Earth’ are revisited as part of this research – which is largely considered and undertaken from a legal (discipline) perspective

Efficiency and safety of varying the frequency of whole blood donation (INTERVAL): a randomised trial of 45 000 donors

Background: Limits on the frequency of whole blood donation exist primarily to safeguard donor health. However, there is substantial variation across blood services in the maximum frequency of donations allowed. We compared standard practice in the UK with shorter inter-donation intervals used in other countries. Methods: In this parallel group, pragmatic, randomised trial, we recruited whole blood donors aged 18 years or older from 25 centres across England, UK. By use of a computer-based algorithm, men were randomly assigned (1:1:1) to 12-week (standard) versus 10-week versus 8-week inter-donation intervals, and women were randomly assigned (1:1:1) to 16-week (standard) versus 14-week versus 12-week intervals. Participants were not masked to their allocated intervention group. The primary outcome was the number of donations over 2 years. Secondary outcomes related to safety were quality of life, symptoms potentially related to donation, physical activity, cognitive function, haemoglobin and ferritin concentrations, and deferrals because of low haemoglobin. This trial is registered with ISRCTN, number ISRCTN24760606, and is ongoing but no longer recruiting participants. Findings: 45 263 whole blood donors (22 466 men, 22 797 women) were recruited between June 11, 2012, and June 15, 2014. Data were analysed for 45 042 (99·5%) participants. Men were randomly assigned to the 12-week (n=7452) versus 10-week (n=7449) versus 8-week (n=7456) groups; and women to the 16-week (n=7550) versus 14-week (n=7567) versus 12-week (n=7568) groups. In men, compared with the 12-week group, the mean amount of blood collected per donor over 2 years increased by 1·69 units (95% CI 1·59–1·80; approximately 795 mL) in the 8-week group and by 0·79 units (0·69–0·88; approximately 370 mL) in the 10-week group (p&lt;0·0001 for both). In women, compared with the 16-week group, it increased by 0·84 units (95% CI 0·76–0·91; approximately 395 mL) in the 12-week group and by 0·46 units (0·39–0·53; approximately 215 mL) in the 14-week group (p&lt;0·0001 for both). No significant differences were observed in quality of life, physical activity, or cognitive function across randomised groups. However, more frequent donation resulted in more donation-related symptoms (eg, tiredness, breathlessness, feeling faint, dizziness, and restless legs, especially among men [for all listed symptoms]), lower mean haemoglobin and ferritin concentrations, and more deferrals for low haemoglobin (p&lt;0·0001 for each) than those observed in the standard frequency groups. Interpretation: Over 2 years, more frequent donation than is standard practice in the UK collected substantially more blood without having a major effect on donors' quality of life, physical activity, or cognitive function, but resulted in more donation-related symptoms, deferrals, and iron deficiency. Funding: NHS Blood and Transplant, National Institute for Health Research, UK Medical Research Council, and British Heart Foundation

A Phase I Study of Pegylated Arginine Deiminase (Pegargiminase), Cisplatin, and Pemetrexed in Argininosuccinate Synthetase 1-Deficient Recurrent High-grade Glioma.

PURPOSE: Patients with recurrent high-grade gliomas (HGG) are usually managed with alkylating chemotherapy ± bevacizumab. However, prognosis remains very poor. Preclinically, we showed that HGGs are a target for arginine depletion with pegargiminase (ADI-PEG20) due to epimutations of argininosuccinate synthetase (ASS1) and/or argininosuccinate lyase (ASL). Moreover, ADI-PEG20 disrupts pyrimidine pools in ASS1-deficient HGGs, thereby impacting sensitivity to the antifolate, pemetrexed. PATIENTS AND METHODS: We expanded a phase I trial of ADI-PEG20 with pemetrexed and cisplatin (ADIPEMCIS) to patients with ASS1-deficient recurrent HGGs (NCT02029690). Patients were enrolled (01/16-06/17) to receive weekly ADI-PEG20 36 mg/m2 intramuscularly plus pemetrexed 500 mg/m2 and cisplatin 75 mg/m2 intravenously once every 3 weeks for up to 6 cycles. Patients with disease control were allowed ADI-PEG20 maintenance. The primary endpoints were safety, tolerability, and preliminary estimates of efficacy. RESULTS: Ten ASS1-deficient heavily pretreated patients were treated with ADIPEMCIS therapy. Treatment was well tolerated with the majority of adverse events being Common Terminology Criteria for Adverse Events v4.03 grade 1-2. The best overall response was stable disease in 8 patients (80%). Plasma arginine was suppressed significantly below baseline with a reciprocal increase in citrulline during the sampling period. The anti-ADI-PEG20 antibody titer rose during the first 4 weeks of treatment before reaching a plateau. Median progression-free survival (PFS) was 5.2 months (95% confidence interval (CI), 2.5-20.8) and overall survival was 6.3 months (95% CI, 1.8-9.7). CONCLUSIONS: In this recurrent HGG study, ADIPEMCIS was well tolerated and compares favorably to historical controls. Additional trials of ADI-PEG20 in HGG are planned