# Where I Am and Will Be

First, a query. Since last night, a whole heap of spam has been getting through Akismet. Those spam comments with many links have been caught in the WordPress moderation queue, but comments without URLs aren’t getting caught. Is anybody else having this problem?

I’m also struggling with a really slow network connection at the office today. This comes at a bad time, too, because my top two priorities are plowing through the immune-system literature and editing the conference book for ICCS 2007. Downloading journal-article PDFs at 1.1 Kbps is not fun.

And speaking of ICCS 2007 — that’s the seventh annual International Conference on Complex Systems — I’m going to be running around the Quincy Marriott next week, taking pictures and videotaping talks and generally doing conference stuff. I’ve generally kept my blog-writing separate from my work at NECSI (since nobody pays me to explain random spatial networks or protein structures, just to generate inscrutable graphs and equations about them) but I thought it might be interesting to try liveblogging the conference. Assuming I don’t have too many actual responsibilities, I’ll try to get synopses up here about the plenary speeches and the more interesting “breakout” talks.

This is probably a good time to state a disclaimer I will repeat later: anything I say here, whether in an ICCS liveblogging post or any other, is my own opinion and not that of my colleagues or employer.

# Quantum Mechanics Homework #1

Well, in the past two days I’ve linked to an Internet quiz and some anime videos, so in order to retain my street cred in the Faculty Lounge, it’s time to post a homework assignment. Don’t worry: if you haven’t met me in person, there’s no way I can grade you on it (unless our quantum states are somehow entangled). This problem set covers everything in our first two seminar sessions on QM, except for the kaon physics which we did as a lead-up to our next topic, Bell’s Inequality. I’ve chosen six problems, arranged in roughly increasing order of difficulty. The first two are on commutator relations, the third involves position- and momentum-space wavefunctions, the fourth brings on the harmonic oscillator (with some statistical mechanics), the fifth tests your knowledge about the Heisenberg picture, and the sixth gets into the time evolution of two-state systems.

Without extra ado, then, I give you Quantum Mechanics Homework #1.
Continue reading Quantum Mechanics Homework #1

# Friday Quantum Mechanics

“So, Blake,” I sez to myself. “You’ve been selected for multiple editions of the Skeptic’s Circle. You’ve been linked, twice, from Pharyngula. Clearly, you’re rising to astonishing heights of science-blogebrity. What worlds are left to conquer?”

“Well,” I replied. “There’s going out for a milkshake with Rebecca Watson.”

I shook my head. “Not gonna happen — she’s just too picky counting tentacles. Anything else?”

“Well, you could do what Revere warned you not to do.”

“Ah, yes, write a sixteen-part series on mathematical modeling! But the modeling of antiviral resistance isn’t really my field.”

“True, but didn’t you spend your spring break in Amsterdam a few years ago, writing that paper which was the first article Prof. Rajagopal ever graded with an A-double-plus?”

“Hey, yeah, on supersymmetric quantum mechanics and the Dirac Equation!”

“So,” I suggested to me, “why don’t you break that paper down into several blag posts, interleave it with some Bill Hicks videos so not all your readers wander away, and have yourself a continuing physics series?”

“Could work, I suppose. But that paper was written for third-term quantum mechanics students, so I’d probably have to build up to it, even just a little.”

“Bah,” I said. “At least you’ll have a purpose in life. And you can start by expounding on the canonical commutation relation for position and momentum. That’ll be your warm-up, after which you can do angular momentum and central potentials —”

“Which I do have written up somewhere,” I interposed, “since I discovered I could type LaTeX as fast as my professors could lecture.”

“Weirdo,” I said.
Continue reading Friday Quantum Mechanics

# Group Theory Homework

One reason I call this site a “blag” and not a “blog” is that I’m always late.

For example, I’m finally typing up the group-theory homework assignment which Ben gave last Monday (and which will be due next Monday). During our seminar over in BU territory, we discussed the relations among the Lie groups SU(2) and SO(3) and the manifolds S3 and RP3. Problems will be given below the fold.

Also, Eric will be discussing statistical physics this afternoon at NECSI.
Continue reading Group Theory Homework

# Group Theory Tonight at BU

Just a reminder:

This evening at 17 o’clock, or shortly thereafter, we will be meeting in Boston University mathematics territory to discuss group theory. Ben will be leading the discussion, poking us to consider the various transformations of the type

$$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2.$$

What are the different structures of interest which these mappings can preserve? What types of maps preserve distances, angles, areas, orientation (chirality), or topological properties? Notes for past group-theory sessions are available in PDF format, or as a gzipped tarball for those who wish to play with the original LaTeX source.

(Don’t forget your stat-mech homework!)

# stat mech problems up.

Please find the stat mech problem set here:

http://web.mit.edu/edown/www/statmech_pset.pdf

This covers the microcanonical and canonical ensembles. You should definitely be able to finish all of I and II by Monday 30th. Part III becomes more and more difficult as one progresses. I would expect everyone to have no problems completing A and B. C is not that hard if you’re willing to play with it. D is a bit more involved but should be within reach.

Let me know if you have any questions. eric

# Wobosphere Trick of the Day (plus seminar)

0. Go to Google Maps.

1. Click “get directions”.

2. Get directions from New York, New York to Paris, France.

3. Scroll down to item 23 in the list of directions.

4. Return in time for the seminar tomorrow afternoon at NECSI, where we shall discuss the first two (possibly three) sections in chapter four of Ash.

(Tip o’ the beret to Audentes at the Achenblog. It also works with Boston, Massachusetts.)

# CRE paper

Friday 4/6/07 we reviewed Rao et al. (2004) IEEE Trans Info Theor, V 50 (6) “Cumulative Residual Information […]”, here.

We decided that while the motivation for the paper was valid, that it was undesirable for a number of reasons — mostly that the CRE of many well-behaved distributions (power laws notably) diverged. We’re all currently working on better generalizations.

# Are We Covering the Wrong Subjects?

Uh-oh. It looks like the time Ben has spent teaching groups, semigroups and monoids might have all been for naught. Just look at what Dr. Ray has to say:

Cubicism, Not group theory.
If ignorant of the almighty
Time Cube Creation Truth,
you deserve to be killed.

Killing you is not immoral –
but justified to save life on
Earth for future generations.

# No seminar today (Sorry!)

Due to an illness among the teaching staff, today’s statistical-mechanics session will be postponed until later in the week. In the meantime, check out science writer Carl Zimmer and his experience being quote-mined by global warming denialists. To cheer up after that study in human folly, try Mark Chu-Carroll’s ongoing series on surreal numbers. Based on his prior habits, I’m curious to see if he takes a crack at explaining the relation between surreals and category theory (definitely one of the branches of mathematics you need to study if you want to be like the guy in Pi).

One point should be raised about the surreals which has not yet appeared in Mark Chu-Carroll’s exposition. You can get the reals — that ordinary, familiar number line — from the surreals by imposing an extra condition on the construction. It’s exercise 17 in the back of Knuth’s Surreal Numbers (a problem originally suggested to Knuth by John Conway). A number x is defined to be real if -n < x < n for some integer n, and if x falls in the same equivalence class as the surreal number

({x – 1, x – 1/2, x – 1/4, …}, {x + 1, x + 1/2, x + 1/4, …)}.

This topic is also discussed in chapter 2 of Conway’s On Numbers and Games. Theorem 13 proves that dyadic rationals are real numbers, and Conway then deduces that each real number not a dyadic rational is born on day Ï‰ (“Aleph Day” in Knuth’s book).

The practical upshot of all this is that surreals may provide a better pathway to understanding the real numbers than the standard way of teaching real analysis! Dealing with Dedekind cuts, for example, leads you to an explosion of special cases and general irritation. Conway says:

This discussion should convince the reader that the construction of the real numbers by any of the standard methods is really quite complicated. Of course the main advantage of an approach like that of the present work is that there is just one kind of number, so that one does not spend large amounts of time proving the associative law in several different guises. I think that this makes it the simplest so far, from a purely logical point of view.

Nevertheless there are certain disadvantages. One that can be dealt with quickly is that it is quite difficult to make the process stop after constructing the reals! We can cure this by adding to the construction the proviso that if L is non-empty but with no greatest member, then R is non-empty with no least member, and vice versa. This happily restricts us exactly to the reals.

The remaining disadvantages are that the dyadic rationals receive a curiously special treatment, and that the inductive definitions are of an unusual character. From a purely logical point of view these are unimportant quibbles (we discuss the induction problems later in more detail), but they would predispose me against teaching this to undergraduates as “the” theory of real numbers.

There is another way out. If we adopt a classical approach as far as the rationals Q, and then define the reals as sections of Q with the definitions of addition and multiplication given in this book, then all the formal laws have 1-line proofs and there is no case-splitting. The definition of multiplication seems complicated, but is fairly easy to motivate. Altogether, this seems the easiest possible approach.

# Seminar Today, plus organizational stuff

Don’t forget! Today is information-theory day at NECSI.

And while we’re speaking about seminar stuff, I should note that posts in the “Seminar Announcements” and “Summaries” categories will automatically transmit themselves to the Do-It-Yourself University e-mail list.

# Upcoming Sessions, Week of 1 April 2007

Eric just sent along the following summary of our upcoming sessions:

This Friday at NECSI is Info Theory again: we’ll be talking specifically about the coordinate-dependence of differential (continuous) entropy and more generally, discussing the rest of Part III of Shannon’s paper. The next topic after that will be “Error-Correcting Codes in Biology“, which will probably take a few weeks at least â€” we’ll first cover the relevant sections of Ash (or Reza or whatever people prefer) and then talk about the biological basics.

This next Monday is Stat Mech and I will be reviewing the ensembles we have covered so far and talking at NECSI about the Gibbs-canonical and grand-canonical ensembles. Depending on time I will prove (in a physicist way) that all of the ensembles are equivalent within their own ranges of assumptions â€” so this may take one or two lectures. After that I will probably assign some homework so that we can get experience working with these tools.

Links added by me, because this is Xanadu 2.0, after all.

# First Session on Group Theory

Yesterday evening, we had our first seminar session on the group theory track, led by Ben Allen. We covered the definition of groups, semigroups and monoids, and we developed several examples by transforming a pentagon. After a brief interlude on discrete topology and â€” no snickers, please â€” pointless topology, Ben introduced the concept of generators and posed several homework questions intended to lead us into the study of Lie groups and Lie algebras.

Notes are available in PDF format, or as a gzipped tarball for those who wish to play with the original LaTeX source. Likewise, the current notes for the entropy and information-theory seminar track (the Friday sessions) are available in both PDF and tarball flavors.

Our next session will be Friday afternoon at NECSI, where we will continue discussing Claude Shannon’s classic paper, A/The Mathematical Theory of Communication (1948). The following Monday, Eric will treat us to the grand canonical ensemble.