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

Alzheimer's pathology targets distinct memory networks in the ageing brain

Alzheimer’s disease researchers have been intrigued by the selective regional vulnerability of the brain to amyloid-β plaques and tau neurofibrillary tangles. Post-mortem studies indicate that in ageing and Alzheimer’s disease tau tangles deposit early in the transentorhinal cortex, a region located in the anterior-temporal lobe that is critical for object memory. In contrast, amyloid-β pathology seems to target a posterior-medial network that subserves spatial memory. In the current study, we tested whether anterior-temporal and posterior-medial brain regions are selectively vulnerable to tau and amyloid-β deposition in the progression from ageing to Alzheimer’s disease and whether this is reflected in domain-specific behavioural deficits and neural dysfunction. 11C-PiB PET and 18F-flortaucipir uptake was quantified in a sample of 131 cognitively normal adults (age: 20–93 years; 47 amyloid-β-positive) and 20 amyloid-β-positive patients with mild cognitive impairment or Alzheimer’s disease dementia (65–95 years). Tau burden was relatively higher in anterior-temporal regions in normal ageing and this difference was further pronounced in the presence of amyloid-β and cognitive impairment, indicating exacerbation of ageing-related processes in Alzheimer’s disease. In contrast, amyloid-β deposition dominated in posterior-medial regions. A subsample of 50 cognitively normal older (26 amyloid-β-positive) and 25 young adults performed an object and scene memory task while functional MRI data were acquired. Group comparisons showed that tau-positive (n = 18) compared to tau-negative (n = 32) older adults showed lower mnemonic discrimination of object relative to scene images [t(48) = −3.2, P = 0.002]. In a multiple regression model including regional measures of both pathologies, higher anterior-temporal flortaucipir (tau) was related to relatively worse object performance (P = 0.010, r = −0.376), whereas higher posterior-medial PiB (amyloid-β) was related to worse scene performance (P = 0.037, r = 0.309). The functional MRI data revealed that tau burden (but not amyloid-β) was associated with increased task activation in both systems and a loss of functional specificity, or dedifferentiation, in posterior-medial regions. The loss of functional specificity was related to worse memory. Our study shows a regional dissociation of Alzheimer’s disease pathologies to distinct memory networks. While our data are cross-sectional, they indicate that with ageing, tau deposits mainly in the anterior-temporal system, which results in deficits in mnemonic object discrimination. As Alzheimer’s disease develops, amyloid-β deposits preferentially in posterior-medial regions additionally compromising scene discrimination and anterior-temporal tau deposition worsens further. Finally, our findings propose that the progression of tau pathology is linked to aberrant activation and dedifferentiation of specialized memory networks that is detrimental to memory function

Twelve tips for integrating massive open online course content into classroom teaching

Massive open online courses (MOOCs) are a novel and emerging mode of online learning. They offer the advantages of online learning and provide content including short video lectures, digital readings, interactive assignments, discussion fora, and quizzes. Besides stand-alone use, universities are also trying to integrate MOOC content into the regular curriculum creating blended learning programs. In this 12 tips article, we aim to provide guidelines for readers to integrate MOOC content from their own or from other institutions into regular classroom teaching based on the literature and our own experiences. We provide advice on how to select the right content, how to assess its quality and usefulness, and how to actually create a blend within your existing course

Ranks of twists of elliptic curves and Hilbert's Tenth Problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica

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

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 `> 3 we exhibit a natural holomorphic weight ` 2 +1 modular form whose coefficients satisfy the congruence a(`N) ≡ 0 mod ` for every N satisfying `−