I love Mathematica graphics. I was trying to figure how to make a cone in 3d, when it turned out M has this as a build-in function already :) reference.wolfram.com/mathematica/ref/Cone.html

we've all seen the teacher in the class throwing the eraser up in the air to show that if it spun around the principal axes with the middle moment of inertia, then the eraser will tumble while if it spun around either the major or minor axes it will stay stable. Here is a demo just wrote today to show this. Now I believe those Euler equations of motion :)

When I changed the axis from the Z-axis to the Y-axis, the eraser did tumble.

I always have to simulate something to believe it.

@belisarius nice :D

Go Robben!

@JacobAkkerboom Too late. 1-0 :)

@belisarius surely you're joking mr. Belisarius. It's 1-2

@SjoerdC.deVries Unbelievable! That's the same lesson again. You can't survive by D-fence. Congrats!

@belisarius Thanks :D

poor mexicans

@Rojo Wait and see. I know of another DT who plans to do the same

@Rojo Playing 1-9-1 when winning

Or 10-1-0

@belisarius The keeper in the midfield?

@Rojo The GK at the church. Praying

@belisarius Hehe. What DT are you talking about?

@Rojo I'll give you a hint. The first name is Alejandro. The surname starts with Sabella

@belisarius I honestly hope he doesn't do something so stupid

@Rojo He already tried on the first match. He couldn't resist Messi's pressure then. Lets see what he'll do when we are 1-0 with the Swiss

@belisarius If

@Rojo Nah, they will score first. Then it will be all suffering

y dale con Pernia

@belisarius tell that to Greece (although they played more aggressively in their latest match)

@belisarius mma predicted 4:1 odds in favor of the Netherlands.

@SjoerdC.deVries what does it say about Greece?

@acl the blog isn't very specific about Greece but it looks like it's not a likely finalist. The most likely tree has Costa Rica beat Greece.

blog.wolfram.com/2014/06/26/…

@SjoerdC.deVries ah. Not a huge surprise!

They predicted Spain to be here now ....

@belisarius no, they didn't. I dare you to calculate the probability of all countries in the top 5 to survive the group phase.

@SjoerdC.deVries Isn't that the bar chart I posted above?

Must be around 30%.

"Group phase qual probs"

I'm talking about the combined probability

@SjoerdC.deVries I like football more than statistics :=)

Have you seen the follow up blog?

@SjoerdC.deVries Yup

@SjoerdC.deVries But Brazil played its worst match ever yesterday

spain + neth + chil + aus == 2.35. Larger than 2

@JacobAkkerboom what is the significance of this?

@acl ha strangely it did occur to me to estimate and process my uncertainty, but I would never give the correct number of significant digits :).

@acl Are you around?

@Szabolcs yes, although near a nervous breakdown

@acl what's wrong?

@Szabolcs nothing, Greece-Costa Rica extra time

Ah, sorry, I forgot it was today ...

@Szabolcs So, what's up?

@acl I'm quarreling with E.T. Jaynes ...

I'll try to be brief:

@Szabolcs Ah. What about?

(mind you I am not a deep thinker, so probably won't have anything deep to say)

@acl Jaynes showed us a method to construct a probability distribution that reflects our limited knowledge about a system. The simplest example, that he also mentions, is a dice that we don't know whether it's fair. (Sorry, trouble with English in that sentence.)

If we have no reason to believe that one side of the dice will come up more often than the rest, we can represent this lack of knowlegde by assigning the same probability to each side, 1/6.

The maximum entropy method can do more: it lets us assign probabilities even if we do know a little bit (in the case of the dice we knew nothing, hence used equal probabilities)

The recipe is: there's this quantity called "entropy" which characterizes a probability distribution. Now let's find the maximum entropy distribution which is still compatible with our (limited) knowledge.

I.e. carry out a maximization with constraints.

You know all this.

@Szabolcs OK.

Then comes the statistical mechanics example where he derives the Gibbs distribution, i.e. the canonical ensemble.

@Szabolcs (I think also the idea is it reproduces the entire prob. distri. given those constraints)

@Szabolcs OK, go on.

The traditional physics-based derivation is using the heat-bath argument, when we consider the joint system of the heat bath and the system-of-interest in the microcanonical example.

The Jaynes derivation:

the limited information we have about the system is the average of the energy

@Szabolcs That's right

what is the maximum entropy prob. distribution p(mu) which has a known average E_0 = \sum_mu E(mu) P(mu) where mu represents the microstate?

QUestion: Why is it precisely the average of the energy that is considered known? Why not some other property? The median of the energy? The weighted-average of the energy? The average of the square of the energy?

@Szabolcs no good answer within Jaynes' formalism

These are all different constraints that can be used during entropy-maximization and they all lead to different distributions.

@Szabolcs Right, yes

Jaynes does not say

In fact the problem is sharper in a different context:

@acl He doesn't say, right? He just says, "take the average".

Take an integrable system, which has say N conserved quantities, not just 1 (the energy)

maximise the entropy the same way Jaynes says, get the distribution

this is now called the Generased Gibbs Ensemble (GGE)

He also says that the only proper distributions we're interested in have very narrow peaks, whihc will eventually mean that there won't be a big difference in the result if instead of <E> we constraint the average of some function of the energy, say <f(E)>. But there will still be a small difference.

But the point is this: if I_n is a cons qty, so is I_n I_m

so I could choose another set of conserved qtys and get a different distribution

@Szabolcs In his case it should not matter in the thermodynamic limit. But in the case I mention it might, since it misses correlations.

Hm.

(my example is a generalisation of yours, but one that actually is being discussed in the modern literature)

Anyway there's no obvious reason within Jaynes' formalism (that I know of)

@acl So, putting aside the thing that I don't really understand why we constraint the average of a conserved quantity, you're saying that even if we do precisely that we can get different and incompatible results, depending on the choice of quantities

Sorry, WiFi trouble, I'm back.

@Szabolcs Precisely.

@Szabolcs And yes, it's only conserved on average if you maximise entropy

@acl Can you give me some introductory references on this?

@Szabolcs Can't think of any

I mean the problem you mentioned about the choice of conserved quantities.

OK, I shouldn't have said introductory.

@Szabolcs I don't think there is any

I know it because I've been working on this recently and it comes up

I remember you mentioning this last year.

(but I also work on this from the quantum side, let me check)

@Szabolcs Yes I've published two papers on this sort of thing since

and am writing another now

wait a sec let me check a review

@Szabolcs OK, I haven't read it but maybe this is useful?

(I literally have not read anything but the abstract though)

Thanks!

Another point: I have never thought about these things for classical systems, only for quantum. So I may be missing something.

@Szabolcs How did you end up wondering about this?

@acl Not sure how much you remember about the graph models I told you about, but I'm having some interpretation problems with them. Plus, what I am doing is precisely swapping a quantity with its transform, and deriving the distribution based on constraining the average of that. I need to understand better how one can interpret the model we get that way.

I'm not even sure that the way people interpret the standard versions of these graph models are completely correct because there is no principled way for choosing a set of quantities to constrain.

@Szabolcs So the idea is you get different prob distributions if you take the qty or its transform, while there is no physical difference?

The broad question was: why choose a certain quantity at all?

And what's the difference between choosing a quantity of its transform?

In physics, the difference between energy and its transform is that energy is conserved, its transforms are not.

But conservation doesn't explicitly come into the Jaynes formulation, unless I missed something in his paper.

@Szabolcs Well there is no time

It's not really a chat topic, you're right.

@Szabolcs Well, if you want I'm more than happy to discuss it via email

@acl Thanks! I'm going to write a longer mail one of these days. It helps me gather my thoughts and I'll be more coherent than in chat.

@Szabolcs Yes it's always useful to write out your thoughts

@Szabolcs hmm ok I just wrote out a long piece of text. You're right this isn't chat material

I'll just save it and think a bit more about how to present my view of this. let's continue this via email at some point

OK!

I'll write you tomorrow if I have time (I'll probably get a lot of work from my boss tomorrow), Tuesday the latest.

@Szabolcs Sure no problem

The thing with physics is that you don't have to be completely precise for as long as the theory is confirmed by the experiments. That's how physicists could get away with sloppy math all the time.

But once the same theory is applied to a field where one can't verify by experiments in such a straightforward way, we need to be a lot more precise and a lot more clear about what we're doing.

@Szabolcs But briefly, conserved quantities enter into it because you are supposedly trying to predict the long-time behaviour of a dynamical system. If it's ergodic, the only information contained in the initial state is in the conserved quantities (eg, it should forget everything about its past except its energy)

@Szabolcs Or at least have physical reasons to choose one over the other

@acl That makes intuitive sense ...

for instance the energy/Hamiltonian is a local (or at least extensive) quantity. Its square is neither

this is all intuition (also everybody says "it should be local" but I think it should be extensive, not local)

ah shit penalties.

@Szabolcs I forgot, do you have a notion of dynamical evolution in your system? Or are you just predicting prob distributions over many realisations?

@acl Just predicting distributions, without having to know anything about evolution.

@Szabolcs Yes of course, but there is a notion of dynamics?

@acl No, there isn't.

@Szabolcs Right. OK, let's discuss this further via email

The animation is a proof-of-concept. It uses the cloud to compute the plot but the interface is built with HTML/Javascript.

@Pickett nice. You also put the value of the slider in the correct place, above the mouse cursor so one can see it as they drag the mouse. I complained a little about this here: community.wolfram.com/groups/-/m/t/284156?p_p_auth=kjXWWXP9 in the Wolfram cloud, since the label was below the mouse cursor and hard to see.

@Pickett which javascript library did you used for the javascript interface? It looks sharp and clean.

For plotting in Javascript I use flot.js

@Nasser I used jQuery UI with a 3rd party (open source) theme.

@Pickett thanks. I have to look at JQuery UI in this case, never tried that one.

In this case the plot is an image generated by the cloud. I could have used a plotting library and just let the cloud provide the numbers but I didn't bother for this example.

jQuery and jQuery UI are very popular, def. worth to take a look at for UIs!

@Pickett you could do the whole plot in Javascript? Why need a server?

 $(function() {
      var options = {
             series: {
                lines: { show: true, fill: false, lineWidth:1},
                points: { show: false },
                color:"rgb(0, 0, 0)"
                     }
      };
   var data = [];
   for (var i = -2*Math.PI; i < 2*Math.PI; i += Math.PI/10) {
   data.push([i, Math.sin(i)]);
   }
   $.plot("#plot", [data], options);

@Nasser Why do we need Mathematica? A lot of things are much faster to do in Mathematica, and I just intended to show a different way of easily deploying Mathematica code using the cloud.

I can deploy any CDF to the browser in a similar manner.

@Pickett is this what sort of web-mathematica does? send commands to Mathematica running on the server and results comes back to client? but web-mathematica requires one to write servlet code and other forms to interface with it, I think.

@Nasser I can imagine that web-Mathematica does something like that, but I haven't looked into it.