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

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

Explicit lower bounds on the modular degree of an elliptic curve

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result

Cross-middleware Interoperability in Distributed Concurrent Engineering

Secure, distributed collaboration between different organizations is a key challenge in Grid computing today. The GDCD project has produced a Grid-based demonstrator Virtual Collaborative Facility (VCF) for the European Space Agency. The purpose of this work is to show the potential of Grid technology to support fully distributed concurrent design, while addressing practical considerations including network security, interoperability, and integration of legacy applications. The VCF allows domain engineers to use the concurrent design methodology in a distributed fashion to perform studies for future space missions. To demonstrate the interoperability and integration capabilities of Grid computing in concurrent design, we developed prototype VCF components based on ESA’s current Excel-based Concurrent Design Facility (a non-distributed environment), using a STEP-compliant database that stores design parameters. The database was exposed as a secure GRIA 5.1 Grid service, whilst a .NET/WSE3.0-based library was developed to enable secure communication between the Excel client and STEP database

An analysis of the Creation Museum : hermeneutics, language, and information theory.

This dissertation analyzes the Creation Museum in Petersburg, Kentucky with respect to hermeneutic, linguistic, and information theories. The popularity of the CM, with an excess of 1.6 million visitors to date and future plans to build a one-hundred million dollar theme park, raises concerns among religious and non-religious people. The CM has drawn the attention of all the major news networks and has been reported on extensively in print media. The number of visitors and money raised by the CM dwarfs other museums in the area with large federal endowments. This dissertation draws the interest of popular educated audiences as well as scholars. The dissertation is divided into five main chapters. Chapter I surveys the relevant literature on creationism in the United States. Chapter II defines the use of three theoretical fields-hermeneutics, linguistics, and information theory-to analyze the operational logic of the CM. Chapter III uses aspects of Hans-Georg Gadamer’s ideas of interpretive horizons to demonstrate how the CM justifies a selective and literal interpretation of Genesis 1. Chapter IV applies Norman Fairclough’s theory of Critical Language Study (CLS) to the various structures that project an authoritative form of discourse at the CM. Chapter V uses theories put forth by Mark C. Taylor to explain how information is processed in terms of screening-a phenomenon that seeks to reify an ancient myth. Taylor’s definition of religion as a complex adaptive network also illustrates why science is such a threat to the CM and why efforts are made to redefine science. The primary sources of evidence used include museum exhibits, literature published by the CM, videos, and ethnographic interviews. The interviews are semi-structured and allow for clarification and elaboration. The central conclusion is that the CM is a fundamentalist organization that rigorously maintains biblical inerrancy as an interpretive principle. Moreover, it employs a closed hermeneutic approach that I have identified as “concordism.” Authority is established through a discursive use of academic frames (contexts) and scripts (individual roles). The CM also displays certain features of complex adaptive networks as it reacts to a wider set of epistemological domains