Large Hardon Collider

[Psion]

Rofl, best pic ever.

[Kaylia]

@max
lol, nice one




I finally figured out my entropy problem (if anyone care about the solution!).

http://upload.wikimedia.org/math/1/1...f0b3822031.png
When you work with the discrete case, P(x)Ln(Px) will always be negative because the probability is between 0 and 1 , and your entropy will be positive because of the sign in front of the sum.

When you work with the continuous equivalent, it's not always the case,
http://upload.wikimedia.org/math/2/5...993f616447.png
The distribution function f(x) can be larger than 1, even if you normalize it (if the witdth is smaller than 1, the spike will go higher).

As an example, Dirac distribution (normal distribution when sigma -> infinity) would produce a f(x) much higher than 1 (in this case, it tend toward infinity). Something like this explain why I get a negative infinity,


I'm not quite sure what I could do to get around this, but I believe adding another lagrange constraint to force the distribution to be smaller than 1 would solve it.

[Kaylia]

I'm still curious to hear a serious explanation about why the discrete case=/= continuous case though. It's probably the first time in physics I encounter a problem where I can't just jump from one to the other.

[BerenTebogo]

So, can anyone give me an overview of what differential geometry is? Apparently I'm going to take it next fall along with linear algebra (taking calc 3 + diff eq next sem.)

My advisor is having me load up on math classes that are outside my major but that he says I'll need if I really want to work in more advanced astrophysics and want to get into a good grad school. Never heard of half the classes before, lol.

[Shiroikage]

Grad school has taught me that PHD Comics is the real webcomic of academics in about 2 weeks.
xkcd only wishes it could be.

[Ferion]

So, can anyone give me an overview of what differential geometry is? Apparently I'm going to take it next fall along with linear algebra (taking calc 3 + diff eq next sem.)

My advisor is having me load up on math classes that are outside my major but that he says I'll need if I really want to work in more advanced astrophysics and want to get into a good grad school. Never heard of half the classes before, lol.

This is the description that's given for Differential Geometry at my school:

Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. Differential geometry is a vast subject. A comprehensive introduction would require prerequisites in several related subjects, and would take at least two or three semesters of courses. In this elementary introductory course we develop much of the language and many of the basic concepts of differential geometry in the simpler context of curves and surfaces in ordinary 3 dimensional Euclidean space. Our aim is to build both a solid mathematical understanding of the fundamental notions of differential geometry and sufficient visual and geometric intuition of the subject. We hope that this course is of interest to students from a variety of math, science and engineering backgrounds, and that after completing this course, the students will be in a position to (i) apply their knowledge and skills in this course to their related subjects, (ii) be ready to study more advanced topics such as global properties of curves and surfaces, geometry of abstract manifolds, tensor analysis, and general relativity.

It may be very different at your school though because one of the preqs for the course description above is Advanced Calc. To be honest though, I'm not really too sure what you should expect. Here's a link to what some homework questions may look like though. http://math.rutgers.edu/~zchan/432/assignment.html

Edit: Thank you once again Woozie, I'll look through your post and make some comments once I have a little more time tonight/tomorrow.

[Max™]

Extending discrete entropy to the continuous case: differential entropy
Main article: Differential entropy
The Shannon entropy is restricted to random variables taking discrete values. The formula
http://upload.wikimedia.org/math/9/8...23ddb8c9fb.pngwhere f denotes a probability density function on the real line, is analogous to the Shannon entropy and could thus be viewed as an extension of the Shannon entropy to the domain of real numbers.

Looks a little different from your formula up there, Kaylia.

http://en.wikipedia.org/wiki/Entropy...rmation_theory)

Formula (1) is usually referred to as the continuous entropy, or differential entropy. Although the analogy between both functions is suggestive, the following question must be set: is the differential entropy a valid extension of the Shannon discrete entropy? Differential entropy lacks a number of properties that the Shannon discrete entropy has – it can even be negative – and thus corrections have been suggested, notably limiting density of discrete points.
To answer this question, we must establish a connection between the two functions:
We wish to obtain a generally finite measure as the bin size goes to zero. In the discrete case, the bin size is the (implicit) width of each of the n (finite or infinite) bins whose probabilities are denoted by pn. As we generalize to the continuous domain, we must make this width explicit.

But note that http://upload.wikimedia.org/math/b/c...38ce401c07.png as http://upload.wikimedia.org/math/6/b...399b035fdf.png, therefore we need a special definition of the differential or continuous entropy:
http://upload.wikimedia.org/math/8/b...76e299b84b.pngwhich is, as said before, referred to as the differential entropy. This means that the differential entropy is not a limit of the Shannon entropy for http://upload.wikimedia.org/math/d/3...a7db2f1b2f.png. Rather, if differs from the limit of the Shannon entropy by an infinite offset.
It turns out as a result that, unlike the Shannon entropy, the differential entropy is not in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations.
Another useful measure of entropy for the continuous case is the relative entropy of a distribution, defined as the Kullback-Leibler divergence from the distribution to a reference measure m(x),
http://upload.wikimedia.org/math/3/b...6c3ec4753f.pngThe relative entropy carries over directly from discrete to continuous distributions, and is invariant under co-ordinate reparameterizations.

Think that covers the difference you're talking about, doesn't it?


Incidentally, the xkcd comic was chosen because the graphs were similar looking.


Sath, differential geometry is very useful for shit with relativity/gravity/black holes/galaxies/lensing/dark matter/whatnot and most of the major hopeful unifying theories as well.

[Psion]

I'm reading through all the xkcd comics now, this one made me rofl:

http://i256.photobucket.com/albums/h...gyroscopes.png


I always did find gyroscopes and mobius strips freaky XD

[Kaylia]

So, can anyone give me an overview of what differential geometry is? Apparently I'm going to take it next fall along with linear algebra (taking calc 3 + diff eq next sem.)

My advisor is having me load up on math classes that are outside my major but that he says I'll need if I really want to work in more advanced astrophysics and want to get into a good grad school. Never heard of half the classes before, lol.

It's pretty easy at the basic level. It's basically your standard derivatived applied to vector and 3d function (or maybe n dimension) .

Grad school has taught me that PHD Comics is the real webcomic of academics in about 2 weeks.
xkcd only wishes it could be.

Started reading these after your post, and rofl, they are awesome

http://www.phdcomics.com/comics/archive/phd110110s.gif
http://www.phdcomics.com/comics/archive/phd102710s.gif

[Shiroikage]

Here's a few of my favourites that I just pulled off the 200 top rated comics list:
http://i53.tinypic.com/ht9nyh.gif
http://i51.tinypic.com/xb0h28.gif
http://i56.tinypic.com/2whjriu.gif
http://i52.tinypic.com/2zgdny1.gif
http://i56.tinypic.com/714p36.gif
http://i53.tinypic.com/e9yeys.gif

[Kaylia]

Looks a little different from your formula up there, Kaylia.

http://en.wikipedia.org/wiki/Entropy...rmation_theory)

Think that covers the difference you're talking about, doesn't it?


Incidentally, the xkcd comic was chosen because the graphs were similar looking.


Sath, differential geometry is very useful for shit with relativity/gravity/black holes/galaxies/lensing/dark matter/whatnot and most of the major hopeful unifying theories as well.

Yeah, it's what I was talking about, and pretty much the conclusion I came to analysing the problem. I sent my homeworks already, but I ended up with a solution very similar to Kullback-Leibler divergence to resolve the negative entropy issue.

Anyway, negative entropy is still a mindfuck. The sum --> integral doesn't feel wrong, but it was.

[Ferion]

Anyway, negative entropy is still a mindfuck. The sum --> integral doesn't feel wrong, but it was.

Could it not work since you're dealing with probabilities? Calculating a discrete probability is very different than a continuous one, you can't jump from one to the other. I honestly don't know anything about the subject matter though, just a guess at why it wouldn't make sense.

[Kaylia]

Could it not work since you're dealing with probabilities? Calculating a discrete probability is very different than a continuous one, you can't jump from one to the other. I honestly don't know anything about the subject matter though, just a guess at why it wouldn't make sense.

Yes, that's exactly why it's not working.

I can kinda understand why when I look at the shape of the graphics involved, but I still have the [wrong] impression that it should be an equivalence when I look at the integral. That's why I'm saying it's a mindfuck.

[Tyche]

Alright guys...my 3 year old said she wants a telescope for christmas. I know nothing about telescope optics. I'm assuming I can buy a t-mount for my camera separate. Is the tracking mount sold separately as well? Is there some way to focus, or is that external as well? I literally know nothing, and I'd prefer not to spend an arm and a leg on my first one.

[Psion]

grab a cardboard tube, break up some glasses, problem solved!

Ok, but seriously, I don't know much either. But make sure you warn her about pointing it at a full moon. If the telescope is powerful enough it's incredibly bright and blinding without a moon filter of sorts, even when not full.

I got a glimpse of it once on a halfway decent telescope and forgot the filter and ouch, it was almost as bright as the sun. >.o at least to my night adjusted eyes.

[Altariel]

Okay anyone want to suggest some general reading science texts available on Kindle? I'm just about done with my current one so probably need to start something else. Give you an idea of what I've read on it already: The Elegant Universe, Fabric of the Cosmos, The Selfish Gene, Greatest Show on Earth, From Eternity to Here and Death by Black Hole.

[Woozie]

The End of Physics by David Lindley is so underrated. It was one of my favorite books ever. Also, try The God Particle by Leon Lederman.

Actually, before you read any of those, read "The Trouble With Physics" by Leon Lederman Lee Smolin and/or "Not Even Wrong" by Peter Woit, since you've read The Elegant Universe. It's interesting to see both points of view (pro string theory and anti string theory. Both of the books I recommended are anti. The Brian Greene books you read are pro) and the politics involved in theoretical physics right now.

Edit: Just checked, and it seems that every one of those are available on Kindle.

Edit 2: Thanks Max. I rearranged my post a bajjillion times and somehow ended up with Lederman's name twice instead of Smolin.

[Max™]

>.>
http://ttahko.net/blog/wp-content/up.../CoverTTWP.jpg

Smolin =/= Lederman

[Woozie]

Whoops, my bad lol. I changed my post a billion times before I hit "post quick reply" and somehow the names got screwed up.

[Woozie]

Can a solution have a Ph higher than 14 or negative?