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John O'Hare
Works at The University of Salford
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John O'Hare

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Elliptic functions

You probably know about trig functions like sin(x).  These are the most basic functions that are periodic:

sin(x + 2π) = sin(x)

Elliptic functions are functions of two variables, x and y, that are periodic in two directions:

f(x + 2π,y) = f(x,y)

and

f(x,y + 2π) = f(x,y)

This movie is a way of illustrating an elliptic function.

What makes elliptic functions so special is that you can think of them as functions of a single complex variable:

z = x + iy

and then they have a derivative in the special sense you learn about in a course on complex functions!

It's a lot harder for a complex function to have a derivative than an ordinary real function!  A function like

f(x,y) = sin(x) sin(y)

is periodic in two directions, but it doesn't have a derivative df/dz.  Mysterious as this may sound, this is the reason elliptic functions are so special.

In the late 1800s, all the best mathematicians thought about elliptic functions, so there are 'Jacobi elliptic functions' and 'Weierstrass elliptic functions' and many more.  Now they're less popular, but they're still incredibly important.  You need to think about them if you want to deeply understand how long the perimeter of an ellipse is.  They're also important in physics, and fundamental to the proof of Fermat's Last Theorem.

+Gerard Westendorp has been making mathematical illustrations for a long time, so if you like such things, circle him!
A Jacobi elliptic function. This time The checkerboard pattern is animated only over magnitude, not phase, so that the squares appear to originate from… - Gerard Westendorp - Google+
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quick!
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commented on a video on YouTube.
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Criminally I didn't film the little bear on the back waving which +Zeyd Ayoob managed to get.

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Hi all, hopefully getting back into G+ will net me some book suggestions somewhere. I'm pretty much burnt through everything I can find. I think.
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oops, just figured this is NOT where I was supposed to put that. Sorry all. Will figure it out and behave.
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Suddenly I feel the need to reconnect with google plus. I'll think of something to post later.
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:) It's hard though, I'm just so conscious that I might as well expend the time better coming to actually see you!
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coffeeeeee
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getting excited as all the light turns summery
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+ 1 millionz
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cool
 
Superionic ice

There are over 15 kinds of ice.  Different kinds are stable at different pressures and temperatures.  Some of the weirdest kinds may exist inside giant icy planets like Uranus and Neptune.  Most of what we know about them comes from computer simulations, because they only exist at very high pressures.

They're called superionic ices, because while the oxygen atoms get locked in a crystal structure, the hydrogen atoms become ionized, breaking apart into protons and electrons.  The protons can then move around like a liquid between the oxygen atoms!  

The first phase of superionic ice was predicted in 1999 by a group of Italian scientists.   They predicted that this ice exists at pressures 500,000 times the atmospheric pressure here on Earth, and temperatures of a few thousand Kelvin.  In this kind of ice, the oxygen atoms form a crystal called a body centered cubic.

In 2013, Hugh F. Wilson, Michael L. Wong, and Burkhard Militzer predicted the new phase shown here.  This may show up above 1,000,000 times atmospheric pressure.  The oxygen atoms, shown as blue spheres, lie in pattern called a face centered cubic.  The protons are likely to be found in the orange regions.

Hugh Wilson said:

Superionic water is a fairly exotic sort of substance.  The phases of water we're familiar with all consist of water molecules in various arrangements, but superionic water is a non-molecular form of ice, where hydrogen atoms are shared between oxygens. It's somewhere between a solid and a liquid—the hydrogen atoms move around freely like in a liquid, while the oxygens stay rigidly fixed in place. It would probably flow more like a liquid, though, since the planes of oxygen atoms can slide quite freely against one another, lubricated by the hydrogens.

These simulations are hard, and different papers are reporting different results.  So, I wouldn't be surprised if people correct them a bit before we settle down to the truth. 

You can also try to make superionic ice in the lab, but it's hard!   In 2005 Laurence Fried tried to make it at the Lawrence Livermore National Laboratory in California.  They  smashed water molecules between diamond anvils while simultaneously zapping it with lasers.  They seemed to find evidence for superionic ice.

You can read more here:

http://phys.org/news/2013-04-phase-dominate-interiors-uranus-neptune.html

and for some even newer results, try this:

https://chemistry.princeton.edu/news/scientists-predict-cool-new-phase-superionic-ice

Here's the paper on the first kind of superionic ice:

• C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi, and M. Parrinello, Superionic and metallic states of water and ammonia at giant planet conditions, Science 283 (1999), 44-46.  Available free with registration at http://www.ncbi.nlm.nih.gov/pubmed/9872734.

and here's the second kind:

• Hugh F. Wilson, Michael L. Wong, Burkhard and Militzer, Superionic to superionic phase change in water: consequences for the interiors of Uranus and Neptune.  Available free at http://arxiv.org/abs/1211.6482.

#spnetwork arXiv:1211.6482 #ice
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Mt. Fuji Today 今日の富士山
SIGMA dp0 Quattro
雲に隠れて見えない日が続きましたが、久しぶりに富士山が姿を表してくれました。
We hadn't seen Fujisan for a week but finally showed up today. It was a nice sunset =)
#CoolJapan  
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Ready love the colour grading on this

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Milo.
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    Electronic & Electrical Engineering
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Research Facilities Manager, The Centre for Virtual Environments & Future Media
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Research Facilities Manager, at The Centre for Virtual Environments and Future Media, at The University of Salford
Employment
  • The University of Salford
    Research Facilities Manager, 2012 - present
  • The Centre for Virtual Environments
    Technical Director, 2008 - 2011
  • The University of Salford
    Consultant, 1997 - 2008
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I've been using spring clean for years and can't understand the negative reviews. Always helpful, always a great job.
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Annoyed I wasn't allowed in because I didn't look right!
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Staggeringly good. The address is wrong on Google though. It's on polar street. Ask around. Kinda behind the federal police station.
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Excellent professional service. Suzi has definitely become our regular plumber now!
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For simply years Jackson's boat has been managing to leverage its position as the only venue in the water park / ees as some kind of internal justification to serve totally grim food. It's astonishing they get away with it. Also they're expensive like service stations are expensive. Also you can see the metro link pass by the end of the garden soon. That'll be nice. I do visit though. Something must be going ok there.
Food: Poor - FairDecor: Poor - FairService: Good
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