Recent content by "pi"mp

I'm wondering if there's a simple relation between the specific heat capacity (at constant chemical potential) and the Helmholtz Free Energy? I can't seem to find a relation in the literature between these three quantities, specifically.

I think I have a relatively decent grasp on T-duality where we've compactified S^{1} . However, when compactifying a 2-torus, is the T-duality transformation where you invert both radii of the two circles simultaneously, or is the claim that you can invert one of the two, leaving the other...

Wait...now I'm confusing myself. The formula with the absolute value that I wrote just above should also vanish right? Since Weierstrass P is even! However, I have plots in mathematica where the absolute value is clearly not zero. I've also made sure that my half-periods agree with my g2, g3...

Ugh, you're right...stupid typo I meant to write the function: \bigg| \wp(u) - \wp(2 \omega_{1} - u) \bigg| So it vanishes at u=w1 and is indeterminate at u=0. I did find the following formula in terms of Weierstrass Sigma Functions: \wp(z_{1})-\wp(z_{2}) = \frac{\sigma(z_{1}+z_{2})...

That is indeed what I meant. Sorry. I only care about u real from zero to the half-period.

Hi, I need to study the function: \bigg| \wp(u ; g_{2}, g_{3})- \wp( (u+2 \omega_{1}); g_{2}, g_{3}) \bigg| where u is the real part of the argument and I'm using the convention where \omega_{1} is actually half of the overall period on the torus. Specifically, I'd like asymptotics for...

I'm not sure I understand what you mean by background-independence in GR. In the same talk I believe you're referring to above, Witten also said that we need a background-independent string theory where the fundamental geometric objects are not points or lines, but rather the string worldsheets...

I was recently familiarizing myself with T-duality and had a thought that I'm not sure is correct: In QFT when we probe too short a distance, the theory explodes and gives infinite results. In string theory, from what I understand, we don't really have any business asking about length scales...

I see. I had no idea such a thing existed. Thank you. So I guess it is like the factoring example; there is a fool-proof algorithm but it is entirely useless in all but certain simple cases.

So for an arbitrary f(t) there's a straightforward way to write the analog of y_n you have above? But it will be an un-illuminating, unwieldy mess?

Wow, that's really interesting. I had never heard of that. Thanks!

But the Fundamental Theorem of Calculus doesn't tell you how to actually *do* an integral, unless you're first clever enough to guess an anti-derivative. Someone pointed out earlier that we don't have an elegant way to factor, but there is at least a fool proof algorithm for factoring, it just...

Yeah, I see your point, but we can, at least in principle, factor a huge number by hand provided we have enough patience. I just find it very unsettling that important numerical quantities for the universe are embedded in these integrals which we can't do without making approximations or...

But in general, unlike differentiation, there doesn't seem to be an algorithm for integrating functions. It's just a patchwork of different techniques that only apply to certain cases, along with approximations, asymptotics, and numerics. Maybe I'm being dumb, but all those seem to me to be...

So if we regard something as only being well defined if we can construct it, does this somehow affect what we think about integrals? The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull...