On Sums of Consecutive Squares

In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in this sum is the square of k. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstrass equation with parameter k. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms of k. We conjecture that this point indeed generates the free part of the Mordell-Weil group, and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. For k in the range 1...100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameter k. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms - which is an essential ingredient of our approach - the rank is the dominant factor. Also the extra computational effort that is needed for some values of k in order to determine the rank unconditionally and construct a set of generators for the Mordell-Weil group deserves special attention, as there are some unusual features

Comparison of multiple and novel measures of dietary glycemic carbohydrate with insulin resistant status in older women

BACKGROUND:Previous epidemiological investigations of associations between dietary glycemic intake and insulin resistance have used average daily measures of glycemic index (GI) and glycemic load (GL). We explored multiple and novel measures of dietary glycemic intake to determine which was most predictive of an association with insulin resistance.METHODS:Usual dietary intakes were assessed by diet history interview in women aged 42-81 years participating in the Longitudinal Assessment of Ageing in Women. Daily measures of dietary glycemic intake (n = 329) were carbohydrate, GI, GL, and GL per megacalorie (GL/Mcal), while meal based measures (n = 200) were breakfast, lunch and dinner GL; and a new measure, GL peak score, to represent meal peaks. Insulin resistant status was defined as a homeostasis model assessment (HOMA) value of >3.99; HOMA as a continuous variable was also investigated.RESULTS:GL, GL/Mcal, carbohydrate (all P < 0.01), GL peak score (P = 0.04) and lunch GL (P = 0.04) were positively and independently associated with insulin resistant status. Daily measures were more predictive than meal-based measures, with minimal difference between GL/Mcal, GL and carbohydrate. No significant associations were observed with HOMA as a continuous variable.CONCLUSION:A dietary pattern with high peaks of GL above the individual's average intake was a significant independent predictor of insulin resistance in this population, however the contribution was less than daily GL and carbohydrate variables. Accounting for energy intake slightly increased the predictive ability of GL, which is potentially important when examining disease risk in more diverse populations with wider variations in energy requirements

Higher breakfast glycaemic load is associated with increased metabolic syndrome risk, including lower HDL-cholesterol concentrations and increased TAG concentrations, in adolescent girls

Almost all previous studies examining the associations between glycaemic load (GL) and metabolic syndrome risk have used a daily GL value. The daily value does not distinguish between peaks of GL intake over the day, which may be more closely associated with the risk of the metabolic syndrome. The aim of the present study was to investigate the cross-sectional associations between daily and mealtime measures of GL and metabolic syndrome risk, including metabolic syndrome components, in adolescents. Adolescents participating in the 14-year follow-up of the Western Australian Pregnancy Cohort (Raine) Study completed 3 d food records and metabolic assessments. Breakfast GL, lunch GL, dinner GL and a score representing meal GL peaks over the day were determined in 516 adolescents. Logistic regression models were used to investigate whether GL variables were independent predictors of the metabolic syndrome in this population-based cohort (3·5 % prevalence of the metabolic syndrome). Breakfast GL was found to be predictive of the metabolic syndrome in girls (OR 1·15, 95 % CI 1·04, 1·27;

Impact of Short Term Consumption of Diets High in Either Non-Starch Polysaccharides or Resistant Starch in Comparison with Moderate Weight Loss on Indices of Insulin Sensitivity in Subjects with Metabolic Syndrome

This work was partly funded by the World Cancer Research Fund and the Rural and Environmental Research and Analysis Directorate (RERAD) of the Scottish Government as part of the base allocations to both the Rowett Institute of Nutrition and Health (University of Aberdeen) and Biomathematics and Statistics Scotland.Peer reviewedPublisher PD

Identifying paediatric nursing-sensitive outcomes in linked administrative health data

There is increasing interest in the contribution of the quality of nursing care to patient outcomes. Due to different casemix and risk profiles, algorithms for administrative health data that identify nursing-sensitive outcomes in adult hospitalised patients may not be applicable to paediatric patients. The study purpose was to test adult algorithms in a paediatric hospital population and make amendments to increase the accuracy of identification of hospital acquitted events. The study also aimed to determine whether the use of linked hospital records improved the likelihood of correctly identifying patient outcomes as nursing sensitive rather than being related to their pre-morbid conditions. Algorithm for nursing-sensitive outcomes used in adult populations have to be amended before application to paediatric populations. Using unlinked individual hospitalisation records to estimate rates of nursing-sensitive outcomes is likely to result in inaccurate rates

Thyroid function tests in the reference range and fracture: individual participant analysis of prospective cohorts.

Context Hyperthyroidism is associated with increased fracture risk, but it is not clear if lower TSH and higher free thyroxine (FT4) in euthyroid individuals are associated with fracture risk. Objective To evaluate the association of TSH and FT4 with incident fractures in euthyroid individuals. Design Individual participant data analysis. Setting Thirteen prospective cohort studies with baseline examinations between 1981 and 2002. Participants Adults with baseline TSH 0.45-4.49 mIU/L. Main Outcome Measures Primary outcome was incident hip fracture. Secondary outcomes were any, non-vertebral, and vertebral fractures. Results were presented as hazard ratios (HR) with 95% confidence interval (CI) adjusted for age and sex. For clinical relevance, we studied TSH according to five categories: 0.45-0.99mIU/L; 1.00-1.49mIU/L; 1.50-2.49mIU/L; 2.50-3.49mIU/L; 3.50-4.49mIU/L (reference). FT4 was assessed as study-specific standard deviation increase, because assays varied between cohorts. Results During 659,059 person-years, 2,565/56,835 participants had hip fracture (4.5%; 12 studies with data on hip fracture). The pooled adjusted HR (95% CI) for hip fracture was 1.25 (1.05-1.49) for TSH 0.45-0.99mIU/L, 1.19 (1.01-1.41) for TSH 1.00-1.49mIU/L, 1.09 (0.93-1.28) for TSH 1.50-2.49mIU/L, and 1.12 (0.94-1.33) for TSH 2.50-3.49mIU/L (P for trend = 0.004). Hip fracture was also associated with FT4 (HR [95%CI] 1.22 [1.11-1.35] per one standard deviation increase in FT4). FT4 only was associated with any and non-vertebral fracture. Results remained similar in sensitivity analyses. Conclusions Among euthyroid adults, lower TSH and higher FT4 are associated with an increased risk of hip fracture. These findings may help refine the definition of optimal ranges of thyroid function tests

Identification of Novel Genetic Loci Associated with Thyroid Peroxidase Antibodies and Clinical Thyroid Disease

Peer reviewe

Convalescent plasma in patients admitted to hospital with COVID-19 (RECOVERY): a randomised controlled, open-label, platform trial

SummaryBackground Azithromycin has been proposed as a treatment for COVID-19 on the basis of its immunomodulatoryactions. We aimed to evaluate the safety and efficacy of azithromycin in patients admitted to hospital with COVID-19.Methods In this randomised, controlled, open-label, adaptive platform trial (Randomised Evaluation of COVID-19Therapy [RECOVERY]), several possible treatments were compared with usual care in patients admitted to hospitalwith COVID-19 in the UK. The trial is underway at 176 hospitals in the UK. Eligible and consenting patients wererandomly allocated to either usual standard of care alone or usual standard of care plus azithromycin 500 mg once perday by mouth or intravenously for 10 days or until discharge (or allocation to one of the other RECOVERY treatmentgroups). Patients were assigned via web-based simple (unstratified) randomisation with allocation concealment andwere twice as likely to be randomly assigned to usual care than to any of the active treatment groups. Participants andlocal study staff were not masked to the allocated treatment, but all others involved in the trial were masked to theoutcome data during the trial. The primary outcome was 28-day all-cause mortality, assessed in the intention-to-treatpopulation. The trial is registered with ISRCTN, 50189673, and ClinicalTrials.gov, NCT04381936.Findings Between April 7 and Nov 27, 2020, of 16 442 patients enrolled in the RECOVERY trial, 9433 (57%) wereeligible and 7763 were included in the assessment of azithromycin. The mean age of these study participants was65·3 years (SD 15·7) and approximately a third were women (2944 [38%] of 7763). 2582 patients were randomlyallocated to receive azithromycin and 5181 patients were randomly allocated to usual care alone. Overall,561 (22%) patients allocated to azithromycin and 1162 (22%) patients allocated to usual care died within 28 days(rate ratio 0·97, 95% CI 0·87–1·07; p=0·50). No significant difference was seen in duration of hospital stay (median10 days [IQR 5 to >28] vs 11 days [5 to >28]) or the proportion of patients discharged from hospital alive within 28 days(rate ratio 1·04, 95% CI 0·98–1·10; p=0·19). Among those not on invasive mechanical ventilation at baseline, nosignificant difference was seen in the proportion meeting the composite endpoint of invasive mechanical ventilationor death (risk ratio 0·95, 95% CI 0·87–1·03; p=0·24).Interpretation In patients admitted to hospital with COVID-19, azithromycin did not improve survival or otherprespecified clinical outcomes. Azithromycin use in patients admitted to hospital with COVID-19 should be restrictedto patients in whom there is a clear antimicrobial indication

A randomized, double-blind, placebo-controlled trial assessing the impact of dexamphetamine on fatigue in patients with advanced cancer

Fatigue is very common in patients with cancer. Current guidelines suggest that psychostimulants are “reasonable to consider for severe fatigue.” This randomized, double-blind, placebo-controlled trial investigated the hypothesis that dexamphetamine in fatigued patients with advanced cancer would produce a clinically significant improvement with minimal side effects. Fifty patients with advanced cancer, who were receiving palliative care, were randomized to dexamphetamine 10 mg twice daily or placebo for eight days. Effectiveness was assessed using the Brief Fatigue Inventory and the McGill Quality-of-Life Questionnaire. The side effects were recorded. The results were analyzed on an intention-to-treat basis. The baseline demographics, fatigue levels, and quality-of-life scores were similar between the two arms. Patients were elderly, had impaired performance status (Eastern Cooperative Oncology Group score = 3), and were taking a range of neurologically active medications. Thirty-nine patients completed the trial. There was a transient improvement in the fatigue levels on day 2, but no significant difference in fatigue (P = 0.267) or quality of life (P = 0.579) by the end of the study. Statistical modeling did not reveal any significant predictors of response to dexamphetamine. These results suggest that dexamphetamine 20 mg daily, although well tolerated, does not significantly improve fatigue or quality of life in patients with advanced cancer, as measured by the selected instruments