and you could throw in an arbitrary (well-behaved) function too: $$\int\mathrm{d}x\,g(x)\delta(f(x,y)) = \int\mathrm{d}x\, g(x)\sum_{i: f(x_i,y)=0} \frac{\delta(x - x_i)}{\lvert\partial_x[f](x_i, y)\lvert}$$ (nb absolute value signs added relative to previous posts)
@DanielSank In my opinion the most explicit formula you could get is mollifying and doing the integral...I am not sure if the result is unique though (it may depend on the choice of the mollifier)
@DanielSank you could think of it like this: for each value of $y$, define a function $F_y(x) = f(x, y)$, and then do the normal delta-function thing on $\int\mathrm{d}x\,\delta(F_y(x))$
Another calendar year is ending, which can mean only one thing. It's time once again for the event that brings joy to all (with a slight helping of dismay for our friends in the southern hemisphere1): Winter Bash!
@EmilioPisanty Ugh, I'm conflicted. That's like offering a single date with the most interesting and beautiful person in the world. Is the fleeting yet deep pleasure worth the ensuing infinity of unfulfillable longing?
[Jumping into the silence] So, what if anything is the relationship between spin and charge? Surely none, since e.g. both the neutrino and electron are spin 1/2...
Ah, well that's because there is a relationship between isospin and charge for quarks. As far as I know, there is no deep reason known for that relationship.
(that's the fun part)
user54412
I knew someone in college who wore a hat like that. Everywhere. All year long.
@DavidZ Huh. All of this "yeah, we mostly understand what's going on" vibe we give off sometimes feels a bit like actually there's so much we don't have a clue about.
@EmilioPisanty yeah... I dunno, I think I pretend to know what's going on when I'm on SE (for example) to cover up that I actually don't know what's going on anywhere else
@DavidZ There exists a set of simple vector addition rules on the Glashow cube for which that simple charge/spin-orientation link remains invariant across (as best I can tell) all mesons and baryons. Nice mnemonic if nothing else, but actually a bit surprising. Glashow never said anything about it though.
@DavidZ Naw, it's not so much individually as opposed to as a discipline. From what we tell high-schoolers, you'd think that we've got the standard model pretty much nailed to a t.
@EmilioPisanty That may be true, actually. The standard model itself is a nice little package, in some sense. It's just that there's more beyond it, and that's where we're mostly clueless.
@Prahar is it? I haven't heard that anomaly cancellation actually explains the link. At least, not in the standard model.
Anomaly cancellation really fixes the charge of the quarks to be what it is, given their isospin, I think. There are set of algebraic equations that the charge, hypercharge, isospin, etc. should all satisfy in order to cancel all the gauge anomalies. This fixes some, if not all of the quantum numbers of the standard model particles.
@EmilioPisanty In computer science, the reflex would be to assume that the model is too complicated, that is, redundant in some way. Has anyone ever seriously tried to refactor the standard model into some simpler form that predicts exactly the same things?
@TerryBollinger Well, there is the recent amplituhedron thing going on. Again, its one of those things that works, but we don't exactly know why. But basically, all the details of the complicated standard model can be packaged into simple geometrical shapes whose volume precisely gives the scattering amplitudes.
@DavidZ Again, as a computer type, the search for broader symmetries feels like the opposite of refactoring. Overgeneralization is not the same as reduction of assumptions.
@TerryBollinger I don't know if this idea has any gusto to it, but it seems to give the same answer in 2 lines as the usual Feynman diagrams would after summing over 8 million diagrams.
Hey y'all, just got this and this room is probably a good space for it
Do you love movies? Do you love science? We have a questionnaire related to both these loves that we would be super grateful if you could fill out. For an upcoming event *we're trying to find out what science films are the favourite of real scientists* (not famous celebrity ones). It's very quick (2 qstns + 3 optional qstns), and can be found here:
@Prahar "amplituhedron"? Never heard of that one. But I am very solid that you can produce an amazingly effective mnemonic model of the particles and their properties using nothing more than that silly Glashow cube. Surprising, that.
@ChrisWhite well, the amplituhedrom itself is a grand idea. There are many many papers being written regularly about many little aspects of it, even though there are only 9 papers that discuss the big picture.
@DavidZ I won't disagree... yet argh, to me refactoring means things like finding really unexpected links between "givens". That little question about the unexpectedly solid link (it really is) between spin orientation and charge addition in nucleons is an example. I don't think the "why is that?" for that kind of correlation is addressed by supersymmetry.
@ChrisWhite Wow, that is about as visually UN-intuitive of a paper I've seen in a while, at least for poor moi. No quick scans for that one!
user54412
I've always wondered how much it actually shortens calculations (but not wondered enough to read the papers in detail). I mean, someone could replace $a_{11} (a_{22} a_{33} - a_{23} a_{32}) - a_{12} (a_{21} a_{33} - a_{23} a_{13}) + \ldots$ with the more geometrical $\det a$, but that doesn't actually do anything.
@ChrisWhite The simplification is enough to make it a useful thing to study. For instance, 8-gluon scattering has 10525900 (~1 million) Feynman diagram in the traditional way of computing scattering amplitudes. However, the answer can be written down in a single line (for a particular type of scattering) using the Parke-Taylor formula. For other types of scattering, it may be more complicated, but certainly not sum-up-one-million-diagrams level of complicated.
@vzn He's a pretty famous computer type (he did a great neural model) who also is adamant that QM is purely local. I've had some weird and interesting debates with him on the topic. I am, to say the least, not of the local QM school, quite the opposite in fact.
Thanks for that additional ampliwhazzit ref, BTW, that was a bit easier to quick scan.
@vzn What's interesting is that I tend to put it the other way around: Non-local is the norm, it's classical physics that is the emergent property, and even then only an approximation that occasionally slips back into the deeper quantum reality.
look at list of who espous(ed/es) it: einstein, de broglie, schroedinger, bohm, bell, t'hooft... am proud to be in this small crowd (or motley crew?) :D
@TerryBollinger yes, am very much an advocate/ proponent/ believer of "emergence" & think its still barely understood. the amplituhedron is a good recent example.
@vzn No disagreement, it's an impressive list! Bell is an absolute favorite of mine, and I love the way he used pilot wave logic to think clearly about entanglement. Alas, there is this little issue that you can buy entanglement equipment off the shelf?...
locality as an emergent property of quantum physics is still admittedly half-baked. its taken many decades of baking and its still only half-finished at best. and in ("day-to-day") science there is little toleration/ patience for anything "half-finished".
@vzn Heh! I've never even looked it up. It's the emergence of information -- always reversible in principle, but not practical except in very rare cases -- that leads me to view classical as emergent.
"The topics will be roughly about definitions of quantum groups, representations, and two main applications: solutions to Yang--Baxter equations, and constructions of quantum knot invariants (Jones polynomials, HOMFLYPT, etc). " Can anyone explain what this is about?
@vzn Well, again that question: What about that off-the-shelf encryption equipment? Even Bell admitted (so very sad he died young!) that the data wasn't looking good for locality. He was rooting for Einstein!
"refactoring" is a fairly recent concept in CS, only about ~1½ decade old...
@TerryBollinger yes really like how experimental entanglement is now much more accessible. have bookmarked some of the few "undergraduate lab" experiments. iirc mitchell helped create one & is also behind recent "loophole free" bell tests.
@vzn It's kind of ill-defined, too. However, one difference: Computer types are incredibly disrespectful of past work, and will just play with the data till something fits. Do that in physics and you will get tossed out.
bell was just a little bit ahead of his time. think he will have the last laugh so to speak and same with einstein. its just gonna take a little longer.
@DavidZ right exactly meant to add that also. it got a little more credit after it was named refactoring. sort of like in philosophy where something doesnt really exist until it has a name.
I'm deep deep into artificial intelligence research, day job. I get to meet interesting people and occasionally have robotic snakes crawl up my let, I think there's an opportunity for safe crowd control with that one... :)
@TerryBollinger cool, what areas of AI? its a huge field. just bought 4 recent/ latest books and am charging thru them, plan to do a review on em in blog.
> I was like a boy playing on the sea-shore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. —Newton
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A spherical tank of 1.2 m radius is half filled with oil of relative density 0.8. If the tank is given a horizontal acceleration of 10 m/s2. Calculate the inclination of the oil surface to horizontal and maximum pressure on the tank.
Which force balance centripetal force . Firstly I thought it is centrifugal force but this force is a psedu force .
For example a string is tied to a stone and it is rotating and there is centripetal force towards centre then which force balance it .
@0celo7: "The topics will be roughly about definitions of quantum groups, representations, and two main applications: solutions to Yang--Baxter equations, and constructions of quantum knot invariants (Jones polynomials, HOMFLYPT, etc). " tl;dr?
in one scene the main character is in the middle of Bellucci when enemies come and try kill him and he just keeps banging making love to her whilst shooting enemies
very realistic
oh man I could have included so many more puns in that sentence
hello guys . I'm tempted to create a bounty to a question I posted 2 days ago (only 28 views and no accepted answer so far) but I'd lose points that I find extremely hard to get and I'd like to have the downvote priviledge so I'd rather not spend points for now. In any case, is there something wrong with my post? physics.stackexchange.com/questions/223838/… it has no vote so far
@DanielSank Take the function $\varphi(x)=N(2d)e^{-1/(1-\lvert x\rvert^2-\lvert y\rvert^2)}$ when $\sqrt{\lvert x\rvert^2+\lvert y\rvert^2}< 1$ and zero otherwise
where $x,y\in\mathbb{R}^d$, and $N(2d)$ is a normalization factor
then $\varphi_h(x,y)=h^{-2d}\varphi(x/h,y/h)$ is such that $\lim_{h\to 0}\varphi_h(x,y)=\delta(x,y)$
then if you define $\varphi_h(f_h(x,y))=h^{-2d}\varphi(h^{-2d}f(x/h,y/h))$ (I am not sure about the $h$ dependence in this case, to be checked...)
what I suppose is that then $\lim_{h\to 0}\int dx \varphi_h(f_h(x,y))$ could define your $\int dx \delta(f(x,y))$
however that is supposing that nothing terrible happens when I exchange the limit with the integral, and that indeed $\lim_{h\to 0}\varphi_h(f_h(x,y))=\delta(f(x,y))$.
the same told me that the "only" results in category theory that may yield some result that is not proved by other means are the adjoint functor theorems
but I don't know anything about that
but if that is true, then category theory is necessary for diff geo only if using the adjoint functor theorem you get something unproved by other means
