Simon Spicer
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27 followers
Posts
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New Hat!
This may have come out a bit more debonaire than I was anticipating.
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Fall quarter at UW begins!
All good things must come to an end! It's been a gloriously warm and sunny summer, but with the advent of august this week the rain is due to set in. Cue the six months in Seattle where you don't see the sun (I've started taking my vitamin D supplements aga...
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GSoC: Wrapping Up
Today marks the last day of my Google Summer of Code 2014 project. Evaluations are due midday Friday PDT, and code submissions for successfully passed projects start soon thereafter. The end result of my project can be found at Sage Trac Ticket 16773 . In t...
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Things are Better in Parallel
A recent improvement I've implemented in my GSoC code is to allow for parallelized computation. The world is rapidly moving to multi-core as a default, so it makes sense to write code that can exploit this. And it turns out that the zero sum rank estimation...
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How big should Delta be?
Let's take a look at the central formula in my GSoC rank estimation code. Let $E$ be a rational elliptic curve with analytic rank $r$. Then $$r < \sum_{\gamma} \text{sinc}^2(\Delta\gamma) = \frac{1}{\pi\Delta} \left[ C_0 + \frac{1}{2\pi\Delta}\left(\frac{... Post has attachment The average rank of elliptic curves It's time to demonstrate the utility of the code I've written - the aim of the game for my GSoC project, after all, is to provide a new suite of tools to conduct mathematical research. First some background. Given an elliptic curve E specified by equation... Post has attachment How to find prime numbers efficiently There's a part in my code that requires me to evaluate a certain sum$$ \sum_{p\text{ prime }< \,M} h_E(p) $$where h_E is a function related to specified elliptic curve that can be evaluated efficiently, and M is a given bound that I know. That is, I n... Post has attachment Cythonize! I'm at the stage where my code essentially works: it does everthing I initially set out to have it do, including computing the aforementioned zero sums for elliptic curve L-functions . However, the code is written in pure Python, and it is therefore not a... Post has attachment ** Clearly I am working in the right field: a lot of what I'm currently doing involves taking Fourier transforms - typically denoted mathematically by a hat operator!$$ f \longmapsto \hat{f} 
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The Birch and Swinnerton-Dyer Conjecture and Computing Analytic Rank
Let $E$ be an elliptic curve with $L$-function $L_E(s)$. Recall that Google Summer of Code project is to implement in Sage an method that allows us to compute $\sum_{\gamma} f{\Delta \gamma}$, where $\gamma$ ranges over the imaginary parts of the nontrivial...