drbaskin

A minor nitpick: It was Andrew Wiles who proved Fermat's last theorem. Andre Weil was another incredible mathematician (responsible for, among many other things, the celebrated Weil conjectures), but did not prove Fermat's last theorem.

Andre Weil was very opinionated and lived a very interesting life. I encourage you to read about it: http://en.wikipedia.org/wiki/Andr%C3%A9_Weil

I'm sorry to be so pedantic, but your math doesn't seem quite right. Should it instead be (1/100,000 chance of burning down each year) * ($200,000 cost if it does) = $2 per near expected damages?

This property of Piedmont is not even unique within the Bay Area! Fremont (the fourth-largest city in the Bay Area) has an enclave of its own, Newark, which is its own city yet completely surrounded by Fremont.

I use it for my weekly to-do list (since it then syncs across all computers and my phone), and it is a convenient way for me to keep track of my different projects. I type up a short summary for myself after meetings and I can also easily take a photo of a blackboard and include it with the notes. It's also a decent way to quickly record addresses or library call numbers, and to draft e-mails to send to my classes. (A proper e-mail client would work for this last purpose, but the only reliable way to e-mail all students in a course is through Blackboard, which is uniformly terrible.)

To belabor the point: Keeping one of the prices fixed, you should prefer that it be 20 percent more expensive on the weekend. Assume for example that the price on Saturday is $10 and the price on Tuesday is $8. It is then "20 percent cheaper to come during the week" but it is 25 percent more expensive to come on the weekend!

If you're handy with simple algebra (and can remember that the derivative of x^2 is 2x) then you can remember the picture. If f(x) = x^2 - N, then you're looking for the (positive) zero of f. You take a succession of tangent lines and look at the zeros of those. So, you start with a guess (call it x_0), take the tangent line at your guess (so it has slope 2x_0 and goes through (x_0, x_0^2 -N)), then find the x-intercept of this line and make it your new guess. This is all Newton's method is. With square roots, it takes the form of averaging your previous guess and N over it.

Just to nitpick a bit -- your final three numbers add to 110, not 100. Perhaps you meant 49/31/20? I point it out only because 49/31/30 seemed too good to be true.

Or simple contraposition :-P

Please correct me if I am wrong, but if the relevant part of the article is "I do not have insurance, so it is important that I maintain my health", then the contraposition would be "It is not important that I maintain my health, so it follows that I have insurance." What you are suggesting as the contraposition is "I have insurance, so it is not important that I maintain my health", which is the inverse of the statement (and equivalent to the converse "If it is important that I maintain my health, then I do not have insurance."

The 30% profit margin sometimes also signals that there are high barriers to entry. (You already mention what I think is the big one: accreditation.)

I haven't read Smolin's paper, so I'm not sure exactly what this article is getting at. Is he suggesting that there should be a different connection (i.e., not the Levi-Civita one) on the cotangent bundle of spacetime or is it something more pedestrian? Is he just doing microlocal analysis on spacetime? If the latter is the case, this article doesn't describe what is (mathematically) new about it.

On a second read of this article, it seems clear that my interpretation above is incorrect. It might still be that they are coming up with physical interpretations of microlocal analysis on curved spacetimes. (Though how you fix a quantization, I'm not sure.)

I'm not a physicist and have not read the original source, so please take anything I say with a large dose of salt.

I'm sorry I wasn't more clear, but your interpretation is what I meant. Per your rule of thumb, seeing \tau should suggest that it has to do with circles or polar coordinates, and the square root points to how to get the polar coordinates.

I'm not a big fan of introducing a new constant (though I believe \pi should have been 2\pi), but I love thinking of the integral you wrote down as \sqrt{\tau / 2} because then the answer practically tells you how to derive it!

How to derive the value of the integral: Square the integral to make it an integral in two variables, introduce polar coordinates, then change variables.

You should never end a sentence with a preposition.

The mathematician Paul Halmos loved issues like these and once constructed a sentence that ends in five prepositions: "What did you want to bring that book that I didn't want to be read to out of up for?"

I'm having trouble determining whether you are serious, but I suspect the original poster was referring to reddit's programming community.

I was aware of the group structure of the Rubik's cube (it is, after all, a subgroup of the permutation group on 54 elements), but apparently I had never given it much thought beyond that. Thanks!

Oh, fair enough! That's a fun example.

I don't follow -- what is the structure being preserved in this example?

I'm assuming you want non-trivial examples of non-bijective homomorphisms, but you can create a trivial one using the <= relation: Consider the two element set {0, 1} and the function f from {0,1} to {0,1} so that f(0) = f(1) = 0. f preserves the <= relationship but is not a bijection.

The a-ha answer still works, but the orignal answer given has you sum 2^(d-k) from k =0 to d, so in this example would be 2^1 + 2^0, suggesting that you would need 3 games to determine the winner.

Nobody, but it was implicit in the solution proffered by the post I replied to. If the game they are playing has 5623 participants with one winner and 5622 losers, you only need a single game!

"If you had 5,623 participants in a tournament, how many games would need to be played to determine the winner?" Let d = Log_2 N. Sum 2^(d-k) for k = 0 ... d

The a-ha solution to this puzzle is to realize that every participant but one has to lose exactly once. Each game has only one loser, so you need 5622 games.

Your method overestimates the number of games needed because you start counting at 0. (Think about the simple case where N=2.) Even if you started counting at 1, your method wouldn't in general give an exact answer (and would typically under- or over-estimate depending on whether you started counting at 1 or 0, respectively), but if you had a nice power of two (say, 2^d on the nose), then the sum from 1 to d of 2^(d-k) would give you exactly 2^d -1, which agrees with the a-ha answer.

I don't know whether this is the Lorenz they are referring to, but there was another Lorenz who worked in optics. http://en.wikipedia.org/wiki/Ludvig_Lorenz

In fact, there is a Lorentz-Lorenz equation. Somewhat more confusingly, when you study Maxwell's equations on a Lorentz background, it is often convenient to work in the Lorenz gauge.

It's only somewhat related, but This Week's Finds in Mathematical Physics #234 discusses some of the math behind music theory. It's a nice article if you're familiar with basic group theory. http://math.ucr.edu/home/baez/week234.html

I'm confused by your statement -- in the quaternions, i, j, and k are all square roots of -1.

I have great personal affinity for Texas but am having trouble coming up with a list of half a dozen top notch universities. I can come up with three (UT, TAMU, Rice) and a few more if you set the bar a bit lower, but none of the these (arguably Rice) is in the league of Stanford, Berkeley, and Caltech, and only the first three are in the same league as UCLA, UCD, UCSD, UCI and USC. Even UCSB and UCSC are good.

Perhaps what I'm trying to say is that as wonderful as Texas is, if you're interested in universities it is sadly lacking. (I mostly know about math departments, so if I'm really wrong about either state, I would like to know.)

The title of this post seems to be a reference to the famous paper of Wigner titled "The Unreasonable Effectiveness of Mathematics in the Physical Sciences." It is a very readable paper and I cannot recommend it highly enough.

http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_...

I agree with you until your last sentence. Especially in upper division courses, the material is more complicated and problems that really test understanding are too ambitious for an exam. This means that if you want to see the student solve interesting problems, they have to complete problem sets or a project. (The corruption of the assignment system means that you're still stuck with exams until the students reach a certain point, though.)

Of course, my experience is only with math courses, and your mileage may vary.

I'm sure that you have an implicit statement about the constants when you imply that O(log_{k}(|T|)) is an improvement over O(log_{2}(|T|)), but log_{2}(|T|) = {log_{k}(|T|)} / {log_{k} (2)}. Correct me if I'm wrong, but in terms of big-O notation, I think this makes O(log_{2}(|T|)) the same as O(log_{k}(|T|)).

I don't whether the error you introduced is intentional or not, but 120 * 0.8 = 96, not 106.

Another way of thinking about this is that 1.2 * 0.8 = 0.96, so it is a four percent pay cut. (This is true for any situation like this and can be remembered by the difference of two squares formula from your early algebra class -- (1 + r)(1-r) = 1 - r^2, where r is the fraction raise/pay cut.)

Not only are they dense, but they are "most" continuous functions! (For an appropriate interpretation of "most", of course)