mm k im supposed to just look at the degrees of the exponentials i guess

makes sense

lol

nvm im still confused

$$\frac{Nk_B(\varepsilon\beta)^2 \exp(\beta\varepsilon)}{(\exp(\beta\varepsilon)-1)^2}\approx Nk_B(\beta\varepsilon)^2\exp(-\beta\varepsilon)$$ since $\beta\varepsilon\to\infty$ as $T\to 0$ idk how this follows

Do you want it the math way or the physics way?

Either way :P

I'll let others speak to it the math way. For the physics way you simply observe that $(\exp(\beta\varepsilon)-1)^2 \to (\exp(\beta\varepsilon))^2$ in this large $\beta$ limit. Then cancel one factor of $\exp(\beta\varepsilon)$ from the top and bottom.

ohh okay

what about the other way when $\beta$ gets to be close to 0

somehow it approximates to be $3Nk_B$

Expand $e^x \approx 1+ x$ and see what happens :)

OHH right

u can do that

bless u :P

Cheers

A "soft" question about math writing for folks here: how do you deal with nested brackets? For example, suppose I wanted to write an aside in brackets, and in that aside I refer to some equation, say equation (2.65). Would you choose square brackets in this case or is that bad form?

(This follows from equation (2.65)) or [This follows from equation (2.65)]

I think the former is perfectly fine, but when Xander arrives, he will probably be able to point you to the precise subsubsection in the AMS style guide where this is addressed

or you could say (this follows from equation 2.65)

Certainly I am not writing anything so important as to deserve that :) but thanks Thorgott

That's fair Obliv, but generally the brackets are there in the label too which is why i have them

@Thorgott There isn't actually a ton there, though the inference I draw is that "This follows from (2.65))" is the preferred style (note that is just "(2.65)", not "equation (2.65)").

E.g. section 6.4

> Identifying letters ((a), (b), (c)), including their parentheses, are roman in all text. The AMS will allow italic identifying letters only if consistent throughout.

Section 13.14:

> In cross-references, equation numbers are enclosed in roman parentheses to match the original label and the parentheses are always roman.

(which only says that equation numbers should always be in parentheses, not other kinds of braces).

On the other hand, the only references that the style guide has to brackets is in the context of citations and "fences" in mathematical contexts. I do not think that AMS editors would be too happy with square braces for parenthetical content.

Personally, I would seek to rewrite the phrase.

@Obliv No. This is wrong. Again, see section 13.14 of the AMS style guide.

For the reference of others: ams.org/publications/authors/AMS-StyleGuide-online.pdf

i don't think style can be logically deduced

@Obliv Okay...

@leslietownes Just quickly coming back to this conversation -- as regards the $T$-periodic space of functions which are spanned by the trigonometric functions, I guess I gave the wrong norm right? It should be induced by an inner product looking something like $<f,g> = \lim_{T \to \infty}\frac{1}{T}\int_{-T/2}^{T/2} fg$?

EE18 if you are looking at periodic functions the usual norm of <f,g> would be an integral of fg over one period (which is arbitrary but usually fixed, e.g. [0,T] or [-T/2,T/2]). you might be able to write out some limit over increasingly large that ends up being the same thing because of the periodicity, but i don't know why you would want that

you can also define inner products of not necessarily periodic functions in terms of integrals over all of R, usually with some kind of "decay at infinity" hypothesis floating in the background to ensure that the thing makes sense and is finite, and sometimes (not always, and not usually by definition) expressions for those sorts of norms are written as limits like that

@leslietownes Ah OK I think this is what's going on, that there's some equivalence to the infinite limit version because of periodicity

I'm working with that engineering text which is keeping stuff hazy, so wanted to just clarify what was going on

@Obliv agreed, but preference should be given to consistency.

@EE18 \langle and \rangle.

@leslietownes You, too. :(

:(

@leslietownes Though in a lot of cases, those won't really be inner products, but the dual pairing (which devolves to an inner product in spaces which are self-dual. (Not that this really matters).

I personally write my inner products like this $\{\left\{\langle\left[<f,g>\right]\rangle\right\}\}$

just to be clear

more brackets means stronger seal

xander: or maybe it is the inner product that evolves into a dual pairing. hippy music

🎶🎵🎶🎸

@leslietownes That isn't how evolution works.

Though it may be the that the inner product is the distant descendant of some common ancestor of the dual pairing.

@leslietownes Actually sorry for perseverating on this Leslie, but I've been searching around and not able to find a wikipage or something like that for this. What term should I be using to learn more about this equivalence between the two possible inner products on this periodic function space?

maybe just prove it? i could see it as a homework problem in an analysis book. if f is periodic with period T then the average value of f on [-L, L] goes to the average value of f on [0,T] as L goes to infinity. it should make intuitive sense.

or look at math.stackexchange.com/questions/2768853/…

there are several duplicates of that question on the site too, all with similar looking proofs. it's maybe an expository challenge to make the argument look good. it might clean things up a little to assume T = 1 or something like that

I realized I have no idea how to glue together closed subsets bounded by analytic boundary S^1's when I'm not in the Top Cat. Do I operate in the Top Cat and then use some deformation or rigidification process to deform the topological surface into smooth/real analytic manifold? I think that is the right direction because there are theorems guranteeing a topological manifold and smooth manifold can be deformed to a anlaytic one. I'm lost on the details though

not sure where to start exactly but maybe in Top, and generate a topological surface S^2, and then sequentially layer on smooth then analytic structures. If the S^2 here is obtained through a gluing and one can obtain an analytic metric compatible with the base (unwrapped quotient), related through the section/exact projection correspondence allowing for a direct analytic quotient metric on the S^2. I think I should go for the low hanging fruit and that would be recovering a well defined

analytic manifold with a well defined analytic quotient metric