June 2014 - mathfun@mailman.xmission.com

[math-fun] quaternions (& more)
by Warren D Smith 26 Jun '14

26 Jun '14

you probably should read my paper if you want to know about that that stuff: #73 here http://rangevoting.org/WarrenSmithPages/homepage/works.html in particular it discussed representing spatial rotations in 2,3,4,5 dimensions... and how to generalize to dimensions 2,4,8,16,32...

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[math-fun] A vector problem...
by Robert Smith 26 Jun '14

26 Jun '14

Let A and B be unit vectors in R^3. Suppose they are rotated about some vector V by an angle T, resulting in A' and B' respectively. What are V and T? I set up a quadratic system using quaternions and got a result that was 3 million terms large. Am I missing something? Robert

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[math-fun] superconducting magnet confusion
by Warren D Smith 25 Jun '14

25 Jun '14

As far as I can tell, a chunk of superconductor which is topologically a ball, cannot produce a permanent magnetic field. (This should be clear due to the "Meissner effect.") A chunk which is topologically a solid torus, can (CERN magnets). So for example, if the Earth's interior were superconducting as opposed to normal-conducting, then I presume the Earth would have no magnetic field. Now perhaps there might be some way out of this logic trap for those who believe the neutron stars have superconducting interiors (?).... but at least naively, the fact neutron stars have enormous magnetic fields, seems to contradict the super-theory. "Magnetars" have fields up to 10^11 Tesla, Pulsars 10^8. It seems to me, the onus is on the proponents of the super-theory to explain how to escape this trap. And if you look in the papers I cited which claim "the first direct evidence for superconducting neutron stars" you will see not the teeniest tiniest mention of this massive contradiction albeit it does contain this interesting footnote: "There is, to date, no evidence for the presence of a magnetic ﬁeld in the Cassiopeia A neutron star." But most neutron stars which have been detected are pulsars and hence have huge fields. Incidentally, the "pressure" associated with a magnetic field B is B*B/(2*mu0) which for B=10^11 tesla is pressure=6*10^27 pascals=6*10^22 atmospheres. Sustained. The pressure inside an H-bomb explosion is a mere 10^15 pascals.

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[math-fun] neutron stars are superconductive? superfluid?
by Warren D Smith 25 Jun '14

25 Jun '14

> Cores of neutron stars are in a superfluid state. > -- > Mike Stay - metaweta(a)gmail.com > http://www.cs.auckland.ac.nz/~mike > http://reperiendi.wordpress.com --is there any actual evidence for this? Seems to me it is a rather wild extraordinary claim, requiring extraordinary proof. And while I am at it, another question is this: many neutron stars (e.g. pulsars, magnetars) are claimed to have enormous magnetic fields. (And there actually is evidence for that.) Isn't this incompatible with the claim of superconductivity? If the earth had a superconductive core, what would happen about the magnetic field of the earth?

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[math-fun] gerrymandering measures
by Warren D Smith 25 Jun '14

25 Jun '14

I suggest you read this: http://rangevoting.org/TheorDistrict.html and this: http://rangevoting.org/SplitLR.html even better... do some (re)writing of the former...

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[math-fun] Neutron stars as atoms
by Whitfield Diffie 25 Jun '14

25 Jun '14

I recall reading the intreaguing observation that stable neutron structures run up to around atomic wieght 300 and then their is a long gap before you get to the neutron stars with atomic weights around 10^50. Perhaps we should regard the neutron stars as isotopes of very heavy elements. The atomic numbers would be miniscule by comparison with the atomic weights but a mildely positively charged neutron star doesn't seem impossible: would adding a billion protons make a difference? Whit

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[math-fun] Python language
by Warren D Smith 24 Jun '14

24 Jun '14

Python seems remarkable in the sense that a typical python program is about 1/5 the length of a corresponding C program (even though C is normally thought of as pretty concise). The question is why. My initial guess at the answer is: Python is a lot like (say) MODULA-3, provided somebody had already written about a million lines of "libraries" of handy subroutines for you. There is nothing inherently concise about python, it is just that you have all these libraries of common tasks so you do not have to code them, you can just use them. Is that guess true?

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Re: [math-fun] Fun exam questions
by Guy Haworth 24 Jun '14

24 Jun '14

FL makes good inroads into the topic of 'drawing a circle'. I'd assumed the compasses came with a suitably sharp pencil but that was not stated. So ... pencil, [maybe pencil sharpener], paper, table ... and an 'agent' to do the drawing are required too. A nice point that 'Systems Engineers' ought to make. But [clue] what is a circle, and can one draw one? Guy

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[math-fun] neutron stars are superconductive? superfluid?
by Warren D Smith 24 Jun '14

24 Jun '14

> http://physicsworld.com/cws/article/news/2011/mar/02/neutron-star-has-super… > http://www.nasa.gov/mission_pages/chandra/news/casa2011.html > > On Tue, Jun 24, 2014 at 11:50 AM, Warren D Smith <warren.wds(a)gmail.com> > wrote: >>> Cores of neutron stars are in a superfluid state. >>> -- >>> Mike Stay - metaweta(a)gmail.com >>> http://www.cs.auckland.ac.nz/~mike >>> http://reperiendi.wordpress.com >> >> --is there any actual evidence for this? >> Seems to me it is a rather wild extraordinary claim, requiring >> extraordinary proof. >> And while I am at it, another question is this: many neutron stars >> (e.g. pulsars, >> magnetars) are claimed to have enormous magnetic fields. (And there >> actually is >> evidence for that.) Isn't this incompatible with the claim of >> superconductivity? >> If the earth had a superconductive core, what would happen about the >> magnetic >> field of the earth? --Supposedly, observations of the 330-year-old (?) neutron star Cassiopeia A by Chandra X-ray observatory show the "critical temperature [for superfluidity] is between one half a billion to just under a billion degrees Celsius." They supposedly deduced this because the neutron star cooled by 4% over the last 10 years, a remarkably large cooling. The claim the star is 330 years old is somewhat dubious, or anyhow lacking in accuracy, because there was no historical record of the supernova. The neutron star itself was only discovered in 1999. QUOTE: "A wide region of the neutron star is expected to be forming a neutron superfluid as observed now, and to fully explain the rapid cooling, the protons in the neutron star must have formed a superfluid even earlier after the explosion. Because they are charged particles, the protons also form a superconductor." This makes no sense to me at all. Suppose you had a glass of water, which was, as you observed it, gradually changed into ice. Would you observe a 4% drop in temperature thus "proving" the phase change to ice occurred? NO. Au contraire, you would observe a STATIONARY temperature, refusing to drop, thus proving the phase change occurred. So it seems to me, what they have actually "proven" is the LACK of super-ivity, if anything. What the hell is the matter with them? Reading more, QUOTE: "copious numbers of neutrinos - nearly massless, weakly interacting particles - are created as the core temperature falls below a critical level and a neutron superfluid is formed, a process that began about 100 years ago as observed from Earth. These neutrinos escape from the star, taking energy with them and causing the star to cool much more rapidly." Oh. Well, that made sense, but it proves absolutely nothing about the superfluid character of anything, it merely indicates that neutron synthesis is happening inside the star, which is something everybody already knew. Searching more, I find the following papers on arXiv: http://arxiv.org/abs/1302.6626 http://arxiv.org/abs/1110.5116 Am I convinced by these papers that there is super-ivity going on? No. It seems like an interesting possibility, but a long way short of something that'd convince me. Also, the whole BCS theory of superconductivity has as far as I can tell been a total failure if you measure "success" as "the ability to predict transition temperatures." As far as I can tell, transition temperatures vary between about 150 kelvin and microkelvin in matter so far observed (and probably much wider if they could make more observations) which is pretty damn unconvincing to me re their ability to predict what is going to happen inside neutron stars. So I'd call this all "wild speculation."

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[math-fun] Fun exam questions ...
by Guy Haworth 24 Jun '14

24 Jun '14

A bit early to be thinking about next year's exams but how about this for a 'question'. "It is impossible to draw a circle using only ruler and compasses." Discuss. Any other favourite exam questions? Guy

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