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Abdelaziz Nait Merzouk
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Klein Quintic

I used a logarithmic scale (log(|Kq(z)|+1)) for the isolines so it is easy to see where the poles and zeroes are.

These pictures were rendered with fragmentarium (

Licence : Free. 
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Take a look at the moon
Just beneath it there is a bright spot: Jupiter.

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Have you seen the Hubble Space Telescope’s new Frontier Field image? Located approximately 4 billion light-years away, this galaxy cluster contains an assortment of several hundred galaxies tied together by the mutual pull of gravity. Take a closer look:

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Just a picture of Jones Littlewood polynomials roots done yesterday. The polynomials are picked randomly then the roots are solved for using Durand-Kerner method. The picture was then enhanced in order to make dark areas brighter.
The random sampling works pretty well : no need to compute all the roots to get a good picture.

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The ladder experiments

This is the result I get using the ladder method that I've described in a previous post to accelerate gradient descent.

The objective function that is being minimised is Rosenbrock's function which is:

F(x,y) = a * (x -- 1)^2 + b * (y -- x^2)^2

In the animated GIF, the value of b goes from 1/2^(8) to 2^30, doubling at each frame while a = 1. The iterates with ladder are in yellow and in red without.

The line search used is the famous Armijo's algorithm, slightly modified in order to get closer to the minimum that's on the search line. While the ladder works very well with the unmodified Armijo line search. The number of iterations to convergence varies al lot when changing the initial guess.

I've also done some experiments for solving linear systems (with positive definite matrix). In that case, the ladder behaves exactly like the good old conjugate gradient method. Maybe some one could provide a proof that they are equivalent. There are some differences though w.r.t. CG:
- The ladder can be used to accelerate other slowly converging methods like Jacobi's (already tried it: it behaves almost exactly as with steepest descent).
- It seem to be related not only to CG but also Chebytchev semi iterative method. The later can be seen just like the ladder but without the line search.

Ok! This post is getting too long. I hope someone will find the ladder useful. ;o)
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Help needed
Give the Earth a present: help us save climate data

We've been busy backing up climate data before Trump becomes President. Now you can help too, with some money to pay for servers and storage space.   Please give what you can at our Kickstarter campaign here:

If we get $5000 by the end of January, we can save this data until we convince bigger organizations to take over.   If we don't get that much, we get nothing.  That's how Kickstarter works.   Also, if you donate now, you won't be billed until January 31st.

So, please help!   It's urgent.

I will make public how we spend this money.  And if we get more than $5000, I'll make sure it's put to good use.  There's a lot of work we could do to make sure the data is authenticated, made easily accessible, and so on.

The idea

The safety of US government climate data is at risk. Trump plans to have climate change deniers running every agency concerned with climate change.  So, scientists are rushing to back up the many climate databases held by US government agencies before he takes office.

We hope he won't be rash enough to delete these precious records. But: better safe than sorry!

The Azimuth Climate Data Backup Project is part of this effort. So far our volunteers have backed up nearly 1 terabyte of climate data from NASA and other agencies. We'll do a lot more!  We just need some funds to pay for storage space and a server until larger institutions take over this task.

The team

+Jan Galkowski is a statistician with a strong interest in climate science. He works at Akamai Technologies, a company responsible for serving at least 15% of all web traffic. He began downloading climate data on the 11th of December.

• Shortly thereafter +John Baez, a mathematician and science blogger at U. C. Riverside, joined in to publicize the project. He’d already founded an organization called the Azimuth Project, which helps scientists and engineers cooperate on environmental issues.

• When Jan started running out of storage space, +Scott Maxwell  jumped in. He used to work for NASA — driving a Mars rover among other things — and now he works for Google. He set up a 10-terabyte account on Google Drive and started backing up data himself.

• A couple of days later +Sakari Maaranen joined the team. He’s a systems architect at Ubisecure, a Finnish firm, with access to a high-bandwidth connection. He set up a server, he's downloading lots of data, he showed us how to authenticate it with SHA-256 hashes, and he's managing many other technical aspects of this project.

There are other people involved too.  You can watch the nitty-gritty details of our progress here:

Azimuth Backup Project - Issue Tracker:

and you can learn more here:

Azimuth Climate Data Backup Project.


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3D Kleinian groups limit sets

2560 x 1440 resolution.
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True 3D Kleinian limit sets

Made possible thanks to an algorithm by +Jos Leys .
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Almost there?

These pictures show limit sets (in Jorgensen normalisation) of two generators quasi fuchsian kleinian groups whit some isometric circles that can be used to construct a fundamental domain (called the Ford domain). The upper horizontal line approximate the "summit" of the fractal. The same algorithm is used to find the isometric circles and the "summit". It is not finished yet but it already gives good results. It basically works as follow:
a and b are the two generator transformations. For now, the algorithm works only in the case where the trace of b is real with absolute value greater or equal to 2.

Find n and m such that one of fixed points of:
c = a^n b^m
is the highest. here n is an integer and m is in the set {-1, 1}

Now c and a are also generators of the group. The procedure is repeated by replacing b by a and a by c until some conditions are met (incomplete list):
- max iteration reached (obviously).
- We find a transformation such that the absolute value of its trace is less that one (from Jorgensen) --> the group is not discreet.
- the isometric circle is sufficiently small. --> good approx. of the summit.
- the trace of c is real (or almost). --> good approx. of the summit.

There is another algorithm to construct the Ford domain in Jorgensen normalisation and that is much better but as it is, it doesn't give the "summit". It is used in OPTI. (

Here, the limit set is drawn using the "chaos game" method. It works surprisingly well when one uses the transformations that correspond to the isometric circles of the Ford domain. 
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