Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa

Methane in the Baltic and North Seas and a reassessment of the marine emissions of methane

During three measurement campaigns on the Baltic and North Seas, atmospheric and dissolved methane was determined with an automated gas chromatographic system. Area-weighted mean saturation values in the sea surface waters were 113 ± 5% and 395 ± 82% (Baltic Sea, February and July 1992) and 126 ± 8% (south central North Sea, September 1992). On the bases of our data and a compilation of literature data the global oceanic emissions of methane were reassessed by introducing a concept of regional gas transfer coefficients. Our estimates computed with two different air-sea exchange models lie in the range of 11-18 Tg CH4 yr-1. Despite the fact that shelf areas and estuaries only represent a small part of the world's ocean they contribute about 75% to the global oceanic emissions. We applied a simple, coupled, three-layer model to numerically simulate the time dependent variation of the oceanic flux to the atmosphere. The model calculations indicate that even with increasing tropospheric methane concentration, the ocean will remain a source of atmospheric methane

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Number Fields Ramified at One Prime

Abstract. For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. We study the existence of G-p fields for fixed G and varying p. For G a finite group and p a prime, we define a G-p field to be a Galois number field K ⊂ C satisfying Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. Let KG,p denote the finite, and often empty, set of G-p fields. The sets KG,p have been studied mainly from the point of view of fixing p and varying G; see [Har94], for example. We take the opposite point of view, as we fix G and let p vary. Given a finite group G, we let PG be the sequence of primes where each prime p is listed |KG,p | times. We determine, for various groups G, the first few primes in PG and their corresponding fields. Only the primes p dividing |G | can be wildly ramified in a G-p field, and so the sequences PG which are infinite are dominated by tamely ramified fields. In Sections 1, 2, and 3, we consider the cases when G is solvable with length 1, 2, and ≥ 3 respectively, using mainly class field theory. Section 4 deals wit

An Anatomy Massive Open Online Course as a Continuing Professional Development Tool for Healthcare Professionals

Massive open online courses (MOOCs) remain a novel and under-evaluated learning tool within anatomical and medical education. This study aimed to provide valuable information by using an anatomy MOOC to investigate the demographic profile, patterns of engagement and self-perceived benefits to healthcare professionals. A 21-item survey aimed at healthcare professionals was embedded into the Exploring Anatomy: The Human Abdomen MOOC, in April 2016. The course attracted 2711 individual learners with 94 of these completing the survey, and 79 of those confirming they worked full- or part-time as healthcare professionals. Variations in use across healthcare profession (allied healthcare professional, nurse or doctor) were explored using a Fisher’s exact test to calculate significance across demographic, motivation and engagement items; one-way ANOVA was used to compare self-perceived benefits. Survey data revealed that 53.2% were allied healthcare professionals, 35.4% nurses and 11.4% doctors. Across all professions, the main motivation for enrolling was to learn new things in relation to their clinical practice, with a majority following the prescribed course pathway and utilising core, and clinically relevant, material. The main benefits were in relation to improving anatomy knowledge, which enabled better support for patients. This exploratory study assessing engagement and self-perceived benefits of an anatomy MOOC has shown a high level of ordered involvement, with some indicators suggesting possible benefits to patients by enhancing the subject knowledge of those enrolled. It is suggested that this type of learning tool should be further explored as an approach to continuing professional, and interprofessional, education

Increased fluid flow activity in shallow sediments at the 3 km Long Hugin Fracture in the central North Sea

The North Sea hosts a wide variety of seafloor seeps that may be important for transfer of chemical species, such as methane, from the Earth's interior to its exterior. Here we provide geochemical and geophysical evidence for fluid flow within shallow sediments at the recently discovered, 3-km long Hugin Fracture in the Central North Sea. Although venting of gas bubbles was not observed, concentrations of dissolved methane were significantly elevated (up to six-times background values) in the water column at various locations above the fracture, and microbial mats that form in the presence of methane were observed at the seafloor. Seismic amplitude anomalies revealed a bright spot at a fault bend that may be the source of the water column methane. Sediment porewaters recovered in close proximity to the Hugin Fracture indicate the presence of fluids from two different shallow (<500m) sources: (i) a reduced fluid characterized by elevated methane concentrations and/or high levels of dissolved sulfide (up to 6 mmol L−1), and (ii) a low-chlorinity fluid (Cl ∼305 mmol L−1) that has low levels of dissolved methane and/or sulfide. The area of the seafloor affected by the presence of methane-enriched fluids is similar to the footprint of seepage from other morphological features in the North Sea

Student engagement and perceptions of blended-learning of a clinical module in a veterinary degree program.

Blended learning has received much interest in higher education as a way to increase learning efficiency and effectiveness. By combining face-to-face teaching with technology-enhanced learning through online resources, students can manage their own learning. Blended methods are of particular interest in professional degree programs such as veterinary medicine in which students need the flexibility to undertake intra- and extramural activities to develop the range of competencies required to achieve professional qualification. Yet how veterinary students engage with blended learning activities and whether they perceive the approach as beneficial is unclear. We evaluated blended learning through review of student feedback on a 4-week clinical module in a veterinary degree program. The module combined face-to-face sessions with online resources. Feedback was collected by means of a structured online questionnaire at the end of the module and log data collected as part of a routine teaching audit. The features of blended learning that support and detract from students’ learning experience were explored using quantitative and qualitative methods. Students perceived a benefit from aspects of face-to-face teaching and technology-enhanced learning resources. Face-to-face teaching was appreciated for practical activities, whereas online resources were considered effective for facilitating module organization and allowing flexible access to learning materials. The blended approach was particularly appreciated for clinical skills in which students valued a combination of visual resources and practical activities. Although we identified several limitations with online resources that need to be addressed when constructing blended courses, blended learning shows potential to enhance student-led learning in clinical courses

Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall ex-hibit congruences for Fourier coefficients of a slightly different type. Let f(z) =P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coef-ficients. If ` is prime, then we shall be interested in congruences of the form a(`N) ≡ 0 mod ` where N is any quadratic residue (resp. non-residue) modulo `. For every prime `&gt; 3 we exhibit a natural holomorphic weight ` 2 +1 modular form whose coefficients satisfy the congruence a(`N) ≡ 0 mod ` for every N satisfying `−