Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

The clinical and cost effectiveness of steroid injection compared with night splints for carpal tunnel syndrome: the INSTINCTS randomised clinical trial study protocol.

BACKGROUND: Patients diagnosed with idiopathic mild to moderate carpal tunnel syndrome (CTS) are usually managed in primary care and commonly treated with night splints and/or corticosteroid injection. The comparative effectiveness of these interventions has not been reliably established nor investigated in the medium and long term. The primary objective of this trial is to investigate whether corticosteroid injection is effective in reducing symptoms and improving hand function in mild to moderate CTS over 6 weeks when compared with night splints. Secondary objectives are to determine specified comparative clinical outcomes and cost effectiveness of corticosteroid injection over 6 and 24 months. METHOD/DESIGN: A multicentre, randomised, parallel group, clinical pragmatic trial will recruit 240 adults aged ≥18 years with mild to moderate CTS from GP Practices and Primary-Secondary Care Musculoskeletal Interface Clinics. Diagnosis will be by standardised clinical assessment. Participants will be randomised on an equal basis to receive either one injection of 20 mg Depo-Medrone or a night splint to be worn for 6 weeks. The primary outcome is the overall score of the Boston Carpal Tunnel Questionnaire (BCTQ) at 6 weeks. Secondary outcomes are the BCTQ symptom severity and function status subscales, symptom intensity, interrupted sleep, adherence to splinting, perceived benefit and satisfaction with treatment, work absence and reduction in work performance, EQ-5D-5L, referral to surgery and health utilisation costs. Participants will be assessed at baseline and followed up at 6 weeks, 6, 12 and 24 months. The primary analysis will use an intention to treat (ITT) approach and multiple imputation for missing data. The sample size was calculated to detect a 15 % greater improvement in the BTCQ overall score in the injection group compared to night-splinting at approximately 90 % power, 5 % two-tailed significance and allows for 15 % loss to follow-up. DISCUSSION: The trial makes an important contribution to the evidence base available to support effective conservative management of CTS in primary care. No previous trials have directly compared these treatments for CTS in primary care populations, reported on clinical effectiveness at more than 6 months nor compared cost effectiveness of the interventions. TRIAL REGISTRATION: Trial registration: EudraCT 2013-001435-48 (registered 05/06/2013), ClinicalTrials.gov NCT02038452 (registered 16/1/2014), and Current Controlled Trials ISRCTN09392969 (retrospectively registered 01/05/2014)

Metabolic risk factors and the incidence and progression of radiographic hand osteoarthritis: a population-based cohort study.

OBJECTIVE: To determine whether selected metabolic factors are associated with greater amounts of radiographic hand osteoarthritis (OA) incidence and progression. METHODS: The study identified 706 adults, aged 50-69 years, with hand pain and hand radiographs at baseline, from two population-based cohorts. Metabolic factors (body mass index, hypertension, dyslipidaemia, and diabetes) were ascertained at baseline by direct measurement and medical records. Analyses were undertaken following multiple imputation of missing data, and in complete cases (sensitivity analyses). Multivariable regression models estimated associations between metabolic factors and two measures of radiographic change at 7 years for all participants, individuals free of baseline radiographic OA, and in baseline hand OA subsets. Estimates were adjusted for baseline values and other covariates. RESULTS: The most consistent and strong associations observed were between the presence of diabetes and the amount of radiographic progression in individuals with nodal OA [adjusted mean differences in Kellgren-Lawrence summed score of 4.50 (-0.26, 9.25)], generalized OA [3.27 (-2.89, 9.42)], and erosive OA [3.05 (-13.56, 19.67)]. The remaining associations were generally weak or inconsistent, although numbers were limited for analyses of incident radiographic OA and erosive OA in particular. CONCLUSION: Overall metabolic risk factors were not independently or collectively associated with greater amounts of radiographic hand OA incidence or progression over 7 years, but diabetes was associated with radiographic progression in nodal, and possibly generalized and erosive OA. Diabetes has previously been associated with prevalent but not incident hand OA. Further investigation in hand OA subsets using objective measures accounting for disease duration and control is warranted

A rare case of pigmented villonodular synovitis after unicompartmental knee replacement: a case report

Pigmented villonodular synovitis is a benign proliferative disease involving the synovium. Pigmented villonodular synovitis is rare after replacement arthroplasty and has not been recognised and reported as a cause of failure of unicompartmental knee replacement in the literature

Longitudinal validity of using digital hand photographs for assessing hand osteoarthritis progression over 7 years in community-dwelling older adults with hand pain

Abstract Background : To determine the longitudinal construct validity of assessing hand OA progression on digital photographs over 7 years compared with progression determined from radiographs, clinical features and change in symptoms. Methods: Participants were community-dwelling older adults (≥50 years) in North Staffordshire, UK. Standardized digital hand photographs were taken at baseline and 7 years, and hand joints graded for OA severity using an established photographic atlas. Radiographic hand OA was assessed using the Kellgren and Lawrence grading system. Hand examination determined the presence of nodes, bony enlargement and deformity. Symptoms were reported in self-complete questionnaires. Radiographic and clinical progression and change in symptoms were compared to photographic progression. Differences were examined using analysis of covariance and Chi-Square tests. Results: Of 253 individuals (61% women, mean age 63 years) the proportion with photographic progression at the joint and joint group-level was higher in individuals with radiographic or clinical progression compared to those without, although differences were not statistically significant. At the person-level, those with moderate photographic progression over 7 years had significantly higher summed radiographic and clinical scores (adjusted for baseline scores) compared to those with no or mild photographic progression. Similar findings were observed for change in symptoms, although differences were small and not statistically significant. Conclusion: Assessing hand OA on photographs shows modest longitudinal construct validity over 7 years compared with change in radiographic and clinical hand OA at the person-level. Using photographs to assess overall long-term change in a person with hand OA may be a reasonable alternative when hand examinations and radiographs are not feasible.</jats:p

Perinatal mental ill health - the experiences of women from ethnic minority groups

This study aimed to investigate ethnic minority women’s experiences and opinions of perinatal mental health problems and the provision of perinatal mental health support services. An exploratory survey was undertaken using a questionnaire. Quantitative data were analysed using descriptive statistics and a simple thematic analysis was used for the qualitative data. A total of 51 responses from women of 14 different ethnic minority backgrounds were analysed. Women from minority ethnic groups face barriers to seeking help for perinatal mental ill health as a result of ongoing stigma and the poor attitudes and behaviours of health professionals and inappropriately designed services. Future interventions should focus on providing adequate cultural competency for health care professionals and ensure that all women are able to access culturally appropriate spaces to talk and be listened to within community settings and wider services

Geometry of open strings ending on backreacting D3-branes

We investigate open string theory on backreacting D3-branes using a spacetime approach. We study in detail the half-BPS supergravity solutions describing open strings ending on D3-branes, in the near horizon of the D3-branes. We recover quantitatively several non-trivial features of open string physics including the appearance of D3-brane spikes, the polarization of fundamental strings into D5-branes, and the Hanany-Witten effect. Finally we detail the computation of the gravitational potential between two open strings, and contrast it with the holographic computation of Wilson lines. We argue that the D-brane backreaction has a large influence on the low-energy gravity, which may lead to experimental tests for string theory brane-world scenarios.Comment: 64 pages, 20 figure

Pheochromocytoma presenting as recurrent urinary tract infections : a case report

<p>Abstract</p> <p>Introduction</p> <p>Pheochromocytomas are rare, potentially fatal, neuroendocrine tumors of the adrenal medulla or extra-adrenal paraganglia. Their clinical presentation varies greatly from the classic triad of episodic headache, diaphoresis and tachycardia to include a spectrum of non-specific symptomatology.</p> <p>Case presentation</p> <p>A 43-year-old Caucasian woman was referred to us from primary care services with a three-month history of recurrent urinary tract infections on a background of hypertension, latent autoimmune diabetes of adulthood and autoimmune hypothyroidism. At 38 years she required insulin therapy. Despite medication compliance and dietary control, she reported a recent history of increased insulin requirements and uncontrolled hypertension with concomitant recurrent urinary tract infections. A renal ultrasound examination, to rule out underlying renal pathology, revealed an incidental 8cm right adrenal mass of both solid and cystic components. A subsequent computed tomography of her abdomen and pelvis confirmed a solid heterogeneous mass consistent with a pheochromocytoma. There were no other features suggestive of multiple endocrine neoplasia. Urinary collection over 24 hours revealed grossly elevated levels of catecholamines and metabolites. Following an open right adrenalectomy, our patient's insulin requirements were significantly reduced and her symptoms resolved. Two weeks post-operatively, an iodine-131-metaiodobenzylguanidine scintigraphy was negative for residual tumor and metastatic disease. Urinary catecholamine and metabolite concentrations were within the normal range at a follow-up six months later.</p> <p>Conclusion</p> <p>Pheochromocytoma is a rare catecholamine-producing tumor requiring a high index of suspicion for early diagnosis. Our case report serves to highlight the importance of considering pheochromocytoma as a differential diagnosis in the atypical setting of recurrent urinary tract infections and concomitant autoimmune disease.</p