@Semiclassical I don't know, but using quaternions is basically using properties of a different group related to the Lorentz group, e.g. locally the orthochronus proper Lorentz group is $su(2) \times su(2)$ where $su(2)$ is generated by unit norm quaternions, it seems like saying 'the real numbers are 1 x 1 matrices, all of math is matrix math'
I'd say as group theory became more well known and people seen how quaternions were more directly related to su(2) rather than SO(3,1) it was seen as artificial
I think people tried to say observables are represented by associative operators, unobservables by non-associative octonions, whether that makes sense is another issue
"The octonions, on the other hand, do not. Their relevance to geometry was quite obscure until 1925, when ́Elie Cartan described ‘triality’ — the symmetry between vectors and spinors in 8- dimensional Euclidean space [17]. Their potential relevance to physics was noticed in a 1934 paper by Jordan, von Neumann and Wigner on the foundations of quantum mechanics [59]." arxiv.org/pdf/math/0105155.pdf
"For superstring theory, we thus want a formula for the "super-area" of a surface in superspace. And we need this to be invariant under supersymmetry transformations. Suprisingly, this is only possible if spacetime has dimension 3, 4, 6, or 10. " math.ucr.edu/home/baez/week118.html
I was really trying to find a comic in response to that message that had super half a dozen times, but I can't find it...something about being super uninterested in supergravity
← am not expert in this area but see some value. think quaternion multiplication might represent something physical. at least some subset of quaternion multiplication apparently can represent compositions of 3d rotations. believe that this mechanism is used in some graphics systems, it was taught to me as an undergraduate. we had evans + sutherland computers. can all 4d multiplications be reduced to "rotations + stretches"? there is similar theory...
I'm using quaternions as a means to rotate an object in the application I'm developing. If one quaternion represents a rotation and the second quaternion represents another rotation, what does their multiplication represent? Many web sites talk about formulas and matrices, but I want to know what...
> More precisely, these are the dimensions where N = 1 supersymmetric string theory makes sense as a classical theory.
user351417
@vzn do you by any chance have the resources you used while studying quaternions/any useful links? I'm doing some stuff about them right now, and a lot of the stuff online are kind of superficial.
@Chair am new to this area myself, just very adept with google, there are a lot of excellent refs just posted did you catch em? what are you doing with them? there are so many angles. maybe can poke around a little on your angle if you delineate it.
@JohnRennie, Hi John, I made a better sketch, can you take a look at it and tell me how to proceed?
canI make use of the delta PE? The usual radius of a neutron star is #10^6#, I though that 10^7 will have 1000 times volume and mass, but, is Schwartzild radius bigger, that is 10^9 cm
The Schwarzschild metric is time independent, and that means we can identify a conserved quantity that is is basically total energy, and we use this to compute the trajectory.
So we are still using conservation of energy, but not simply equating change in KE to change in PE.
But we wouldn't do the calculation using the shell observer coordinates because they are somewhat cumbersome to work with. We would compute the trajectory using the Schwarzschild coordinates then transform into the shell observer frame.
The problem is that while the calculation is actually quite straightforward it uses concepts that will be unfamiliar to anyone who doesn't have at least an introductory knowledge of GR.
So I can't explain how it's done other than to say it's done using conservation of total energy.
The equation I gave for the object freely falling from rest is done by setting the total energy to $mc^2$ i.e. the rest energy of the object using Einstein's famous mass-energy relationship $E=mc^2$.
In your example (if I understand correctly) the comet doesn't start at rest but comes in towards the neutron star with some non-zero initial velocity.
In that case the total energy is given by the relativistic energy equation $E^2 = c^2p^2 + m^2c^4$
Using this we get the velocity as a function of the Schwarzschild coordinate $r$ to be:
Where $\gamma$ is the Lorentz factor of the object corresponding to its initial velocity, $v$, far from the black hole i.e. $$ \gamma^2 = \frac{1}{1 - v^2/c^2} $$
So if your comet starts with a velocity $v = 0.99c$ then use this value of $v$ to calculate $\gamma^2$ and substiture that into the equation for $v$.
The velocity observed by the shell observer differs from this by a factor of $1 - r_s/r$ because (a) the shell observer's time is dilated and (b) the shell observer's radial distance is enlarged. The result is that the velocity observed by the shell observer is:
Well, it will be a small increase. The speed cannot be higher than $c$, and for the shell observer it reaches $c$ only at the event horizon (if there is one).
Since $10^{10}$m is 10,000 times the event horizon distance you'd expect the speed increase to be very small.
@JohnRennie, no, wait, there was a mistake 10^6 is meters I was counting cm so, the result is .9903: $$sqrt(1-1/7.09^2*(1-(3*10^8)/10^10))$$. that's more like it. Thank you very much, you are exquisitely kind!
"This structure theorem for compact Lie groups is of great importance for physics because it shows that the only compact gauge fields are combinations of the Maxwell fields associated with $U(1)$ and the Yang-Mills fields associated with simple compact Lie algebras" hmm
@JohnRennie, no, wait, there was a mistake 10^6 is meters I was counting cm so, the result is .9903: $$sqrt(1-1/7.09^2*(1-(3*10^8)/10^10))$$. that's more like it. Thank you very much, you are exquisitely kind!
@enumaris you seem to be a smart dude. I ran tpot yesterday and it came up with gradboost like you recommended. Doing it overnight came up with svc, but I guess it's time to run it longer
@Nick to be honest the chat sessions have largely fallen into disuse, so you aren't missing much.
When the chat room was new we organised the scheduled sessions to try and drum up interest in the room. But these days the room is quite lively so the scheduled chat sessions are no longer relevant.
I've actually known about these schedules session for years now but honestly, I never came here because of them. Frequently, it was annoying to get that notification but now that I'm losing grip with physics, I'm thinking perhaps utilizing these scheduled meetings to pick up knowledge that would potentially benefit that lifelong learning journey I'm betting most of us are on.
@Nick the chat sessions are rarely that erudite I'm afraid.
If there are aspects of physics you want to ask about then just ask and if anyone is around who knows the answer they'll reply. No need to wait for a chat session.
The aspects of physics that get discussed tend to be rather eclectic and not of general interest, so there's little to be gauined from just spectating.
@Nick welcome, the mtg is helpful to try to get a quorum + dialog but ppl come & go into chat all day long, we're always looking to recruit new regulars :) ... the room is sometimes lively but also has been livelier, and lots of lively regulars have drifted off, easy come, easy go :( btw SE chat rooms in general are not typically very busy these days... suspect activity lower...
@vzn ah, going with the wind blowing kind of bohemians, are they. Does this quorum generate any interesting topics of research (ie, things I can google about for a few days until reaching a fulfilling conclusion and possibly, be able to write something on) ?
@danielunderwood just read a forum post on driving a deuteron accelerator with a stungun voltage source. Lovely thought. Bit dissapointed it can't be done.
@vzn I'm into ML&AI research actually Doesn't count as physics, lol. Actually, I'm really interested in the mathematics behind time travel theory as pf late, mostly thanks to Steins;Gate 0.
@Nick ah there is some major ML+AI talk in this room lately, @enumaris regular is deep into it + @danielunderwood delving into it also, have done much myself over the yrs... what area of ML+AI? actually there is significant new use of ML in physics, am on the verge of blogging on that, have a few links
@Semiclassical its easy to set up & defn encourage you to do so, some others have recently set up blogs in here eg PMeuer/ DSank come to mind. alas it can be hard to get traffic/ comments these days, very crowded/ thinly spread attn
@Nick Yeah I'd imagine it would be very difficult. We ran at 10-20MV and our accelerator was a giant vacuum chamber. There was also fairly significant work put into the ion source to feed the accelerator. Not to mention the price of some of the gases. Then you have to have vacuum and magnetic control systems for the whole beamline. Then whatever you need at the target. Our particular setup had a rather significant cryo setup, but that varies from experiment to experiment
@Semiclassical honestly the web based editor is passable but awkward/ a disappointment for longer documents but they have an app now which probably does better, havent installed it on windows (not sure if its on windows) tried it on ipad. the web editor is esp bad for making small incremental chgs because it loses the scroll location every time saved. annoying!
@vzn We have some research things for school... A lot of those comments seemed to be about quaternions in physics, but my thing is supposed to be purely mathematical math, but I'm sliding a bit of application for computer graphics because in the middle of the pure math they want 'real-world applications', however that works. I still don't have a clear idea about what I'm going to do though.
user351417
Maybe I'll run something about computational complexity along with quaternions and see what happens.
@JohnRennie they definitely live on planet earth but their definitive earth origin does not seem to be conclusively claimed in research so far. it seems one would have to (eg) account for nearly all the genes (maybe most are accounted for in other species), maybe something that cant be completely done.
Humanity is so boring clinging fast to the understandable. Well reality is in the state of bring melted of all understanding, it won't be long before GlobeTech will rule all humanity
> Update 10 December 2015: A separate team of researchers in Scotland has also sequenced the tardigrade genome, and found at most 500 genes from foreign species. They concluded that the original researchers might have inadvertently sequenced DNA from bacteria living alongside the tardigrades. Further research is needed to confirm just how foreign the water bear's genome really is.
@JohnRennie do you consider this a "fringe theory"? en.wikipedia.org/wiki/Panspermia maybe you should look at its proponents... some famous physicists o_O
my little weird discovery (which isn't actually so weird since the example isn't generic) is that, if you localize a particle in a harmonic oscillator potential, then the bohmian trajectories are approximately classical (with the approximation basically going to how sharp the initial localization is)
not actually that strange, given how special the harmonic oscillator case is. but still neat in my book
it's also true for the case of a linear potential and a free particle, but the latter (and probably the former as well) can be viewed as limiting cases of the harmonic example
afaik discovery of Archaea has helped revive the panspermia theory. ah heres something Eight-Legged Space Survivor Gives 'Panspermia' New Lifespace.com/5843-legged-space-survivor-panspermia-life.html ... re famous physicsts: Hawking
Interstellar panspermia remains a highly unlikely proposition in the minds of most scientists.
Multiple studies have shown that the raw material of life are common around other stars, and in fact the very seeds of life known as amino acids could also be everywhere. Life, therefore, might be common around other stars, scientists say. But getting from one star to another is another matter altogether. It would take four years just to get from our solar system to the next nearest star ... and that's if a rock was (impossibly) traveling at the speed of light.
@Semiclassical Will be interested to see how they diverge in the less standard cases. Do the trajectories became disperse and bunch out elsewhere, or something more surreal happened?
agreed there is very little direct evidence for panspermia theory. but one also would expect that. esp if its original seeding on earth was billions of years ago. evolutionary origins of life are still a deep mystery as admitted by biologists.
with the propagator in hand, one can compute the velocity of a Bohmian particle at a given spacetime point as $\frac{dQ}{dt} = \frac{\hbar}{m}\text{Im}\frac{d}{dx}\log U(x,t|x',t')$
where $x'(t')$ is the point where the particle is initially localized
Taking the example of a double-well potential from that paper, and assuming the particle is initially localized at x=1 for t=0 (the classical minima are at x~0.94) I get the following streamlines
those with sufficiently negative initial velocity will pass over the barrier and the second well, but eventually turn back due to the rising potential barrier (it's a double well potential after all)
it's a critical trajectory: any trajectories whose initial velocity is larger than it will ultimately land in the first well. any trajectories whose initial velocity is smaller than it will land in the second
no, x=0 is the top of the barrier. it's a symmetric well
they're what the guidance equation in Bohmian mech would give you for trajectories based on the propagator
that's the potential, for reference
@ACuriousMind yeah, definitely. more precisely, the trajectories aren't supposed to cross (except at t=0---that's a bit of a funny point) so that pretty much forces there to be a critical trajectory
On a less erudite level I stumbled across a web comic that has literally made me laugh out loud. I mention it in case anyone hereabouts is interest in such things. It's El Goonish Shive.
Apparently so. It seems a good number of kaggle winners have used some version of grad boost. And I haven't. I saw the warning on tpot starting up and meant to install it before this run
@Semiclassical Hm. Is the syntax the same? I mean, any chance it needs to be something like NDSolve[{x'[t]==x[t]/t, x'[0]==1},...]? I thought they had the same syntax but just double checking
@Nick Yeah, Mathematica costs like a thousand dollars for an individual license I think. Matlab is presumably somewhat less expensive, but still probably in the hundreds.
@TerryBollinger I switched to using Gmail a few months ago and that has trimmed the more ... erm ... "interesting" e-mails to a trickle. I occasionally look at my old e-mail address and some of the e-mails are quite eye opening :-)
@TerryBollinger I got an e-mail with the subject "Make your bed an orgy of desire". But I'm not sure about that. I mean, eating chocolate in bed could get messy.
@enumaris It could be, but that's wishful thinking considering I'm 4371/5211 at the moment. I can't help but wonder what those other 1000 people are doing since I'm just tossing the base data set into a model. I have no idea how to handle the rest of the data since they're one to many that are going to result in nested data.
I guess I could try to extract some statistics like count and average, but this evening I'm going to look into multilevel learning. I think an RNN may also help for nested data, but not really sure.
@enumaris "area under ROC curve" though I'm not 100% on what the ROC curve is and am just using the sklearn function. My actual percentage correct is somewhere in the high 90s I think
@Nick if you are asking about why gradient boosting might be better than a simple random forest, it's because gradient boosting fits each successive tree on the residual from the previous tree in the ensemble, so the learning is "directed" rather than "random".
I'm trying to understand a step in a proof:
Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let $\kappa:\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$ be the Killing form.
In this setting, the author ...
@enumaris Yeah the top 500 are all 0.8. Maybe someone published a kernel near that or something. And yeah on the base data aside from categorizing strings. I was under the impression that tpot would handle scaling and such, but I think I need to do that myself. I also have some data that's failing to convert and I'm just dropping those columns. And I think there's some data in there that's just plain wrong and needs to be cleaned up
@Semiclassical Mathematica doesn't implicitly make functions continuous like we often do in physics though :P I do remember being tripped up by that a few times
But the issue with the rest of the data is that the base data is one row per user, then the supplemental data has any number of rows per user then any number of rows per the first supplemental data set. I don't know if it would be better to go through and clean up the main data or figure out how to handle the supplemental data first
@Semiclassical Hm, does it? I don't remember that happening when I used to use it. Actually most of my research existed precisely because of the need to avoid singularities in differential equations.
@enumaris I am currently barred from the interesting things because of data cleaning. Think data that shaped like a triangle and you need to make a rectangle out of it
I'm trying to understand a step in a proof:
Let $\mathfrak{g}$ be semi-simple (finite dimensional) Lie-algebra over $\mathbb{C}$, $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra and let $\kappa:\mathfrak{g}\times\mathfrak{g}\to\mathbb{C}$ be the Killing form.
In this setting, the author ...
