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

Short term non-invasive ventilation post-surgery improves arterial blood-gases in obese subjects compared to supplemental oxygen delivery - a randomized controlled trial

<p>Abstract</p> <p>Background</p> <p>In the immediate postoperative period, obese patients are more likely to exhibit hypoxaemia due to atelectasis and impaired respiratory mechanics, changes which can be attenuated by non-invasive ventilation (NIV). The aim of the study was to evaluate the duration of any effects of early initiation of short term pressure support NIV vs. traditional oxygen delivery via venturi mask in obese patients during their stay in the PACU.</p> <p>Methods</p> <p>After ethics committee approval and informed consent, we prospectively studied 60 obese patients (BMI 30-45) undergoing minor peripheral surgery. Half were randomly assigned to receive short term NIV during their PACU stay, while the others received routine treatment (supplemental oxygen via venturi mask). Premedication, general anaesthesia and respiratory settings were standardized. We measured arterial oxygen saturation by pulse oximetry and blood gas analysis on air breathing. Inspiratory and expiratory lung function was measured preoperatively (baseline) and at 10 min, 1 h, 2 h, 6 h and 24 h after extubation, with the patient supine, in a 30 degrees head-up position. The two groups were compared using repeated-measure analysis of variance (ANOVA) and t-test analysis. Statistical significance was considered to be P < 0.05.</p> <p>Results</p> <p>There were no differences at the first assessment. During the PACU stay, pulmonary function in the NIV group was significantly better than in the controls (p < 0.0001). Blood gases and the alveolar to arterial oxygen partial pressure difference were also better (p < 0.03), but with the addition that overall improvements are of questionable clinical relevance. These effects persisted for at least 24 hours after surgery (p < 0.05).</p> <p>Conclusion</p> <p>Early initiation of short term NIV during in the PACU promotes more rapid recovery of postoperative lung function and oxygenation in the obese. The effect lasted 24 hours after discontinuation of NIV. Patient selection is necessary in order to establish clinically relevant improvements.</p> <p>Trial Registration#</p> <p>DRKS00000751; <url>http://www.germanctr.de</url></p

Brain microdialysis distribution study of cefotaxime in a patient with traumatic brain injury

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