Some heuristics about elliptic curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict $cX^{5/6}$ total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions

Paid Sick Days and Restaurant Jobs: The Evidence from San Francisco

In February 2007, San Francisco implemented the nation's first paid sick days law, allowing all workers within the city to earn a minimum amount of paid sick leave. Restaurants are among the employers least likely to provide sick leave benefits, absent such a requirement. Therefore, if minimum paid leave standards affect the number of jobs available, we would expect to see that impact most clearly in the food service industry.The data show that the job market in restaurants and bars has been stronger in San Francisco than in the state of California as a whole in every year since the sick days law passed

Scaling in the space climatology of the auroral indices: Is SOC the only possible explanation ?

The study of the robust features of the magnetosphere is motivated both by new "whole system" approaches, and by the idea of "space climate" as opposed to "space weather". We enumerate these features for the AE index, and discuss whether self-organised criticality (SOC) is the most natural explanation of the "stylised facts" so far known for AE. We identify and discuss some open questions, answers to which will clarify the extent to which AE's properties provide evidence for SOC. We then suggest an SOC-like reconnection-based scenario drawing on the result of Craig(2001) as an explanation of the very recent demonstration by Uritsky et al(2001b) of power laws in several properties of spatiotemporal features seen in auroral images.Comment: 24 pages including 7 figures. Based on an invited talk given at the IAGA meeting in Hanoi, Vietnam, August 2000. Retitled v2 has revisions, clearer statement of intent of paper i.e. part review/part critique/some new suggestions, and 1 new figure. In press, Nonlinear Processes in Geophysic

Teaching the YouTube generation: exploring the benefits of an interactive teaching approach in sustainable product design

This paper presents findings from a doctoral study, which investigated effective methods for teaching social sustainability within product design courses in British and Irish universities. Specifically exploring, how to foster a holistic understanding of the social aspects of sustainable product design amongst undergraduate and postgraduate students, through design thinking. Perceived relevance is considered as a fundamental aspect in enabling students to engage deeply with sustainability [1]. Authors [2;3;4] note that 'Net Generation' learners have specific learning preferences that can be targeted in order to improve the students learning experience. Through the careful design of materials which build upon the students tendency towards visual learning and seeking increase relevance and motivation, by offering opportunities for collaborative learning and learning through discovery. Three 'Rethinking Design' workshops were designed and developed as part of a doctoral study to introduce students to the wider social aspects of sustainability and these were conducted in five universities in Britain and Ireland. The workshops featured visually rich audio visual introductions followed by collaborative group based mind mapping activities, which were successful in fostering deep learning by facilitating learning through discovery, critical reflection, peer learning and creativity leading to an exploration of design thinking solutions

Washington's Working Women

The lingering impacts of recession highlight the central importance of women's work to family economic survival - but also the gulf that still separates women's earnings from men's, and the need for new policies that promote healthy workplaces and healthy families

Rank distribution in a family of cubic twists

In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank. We extend their data, showing that it is more likely that the proportion goes to zero

Some remarks on Heegner point computations

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004