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
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.
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
@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.
@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 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).
@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.
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
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for V (a space over the real numbers) may also serve as a basis for VC over the complex numbers.
== Formal definition ==
Let
V
{\displaystyle V}
be a real vector space. The complexification of V is defined by taking the tensor product of...
I keep forgetting the definition of tensor product.
I remember it using its existence property diagram but forget it soon.
V, W - K vector spaces. V tensor W is a K- vector space along with a bilinear map f: V\times W into V tensor W such that every bilinear map out of V\times W into a K vector space U is composition of a unique linear map (from V tensor W into U) with f.
But I never really understood the motivation for tensor products.
suppose $ R $ is a ring and $\mu: R\rightarrow [0,\infty]$ is sigma additive, how can i show that for $ A \subset \cup_i A_i: \mu(A) \leq \sum_i \mu(A_i)$ if the unity need not be in the set (ie i can not use the attribute of sigma additivity)
if it is contained, then its trivial by definition... if it is not contained?
yeah. Tensor products are, first and foremost, applied in differential geometry
"Defined as $(\pi_1\otimes \pi_2)(g, h) = \pi_1(g)\otimes \pi_2(h)$" really should be "Defined as $(\pi_1\otimes \pi_2)(g \otimes h) = \pi_1(g)\otimes \pi_2(h)$"
or in other words, the linear map coming from the bilinear map $(g, h)\mapsto \pi_1(g)\otimes \pi_2(h)$
Hi, I'm studying the uniform convergence of $f_n(x)=n\sqrt{4\pi^2 n^2 +x^2}$ on $\mathbb{R}$. I proved that $f_n(x) \to \frac{x^2}{4\pi}$ pointwise on $\mathbb{R}$. My reasoning to prove that $f_n$ is not uniformly convergent to $\frac{x^2}{4\pi}$ is the following: $\sup_{x \in \mathbb{R}} |f_n(x)-f(x)| \ge |f_n(n)-f(n)|=n^2|\frac{\sin \sqrt{4\pi^2 n^2+n^2}}{n}-\frac{1}{\pi}|$.
Suppose $\overline{E}$ is $\sum_s E(s)P(s) = -\mu BP_{\uparrow}+\mu BP_{\downarrow} = -\mu B(P_{\uparrow}-P_{\downarrow}) = -\mu B\tanh(\beta \mu B)$. I want to set the energy levels to $0$ and $2\mu B$ instead of the $\pm \mu B$ but I'm not sure if that's the same as $$\mu B\left(\frac{1-e^{-2\mu B\beta}}{1+e^{-2\mu B\beta}}\right)$$
which is what u get if you do that. Like is what's in the parentheses a $-\tanh(\beta\mu B)$
I think it should be. I guess there isn't much of an advantage doing it this way since $\tanh$ is more recognizable
@leslietownes This is awesome, thanks so much :) \
So am working with my analysis text this morning which is on a last brief section about vector spaces before the fun with sequences starts next chapter
The book has a brief aside about polynomial interpolation, and proves that the existence and uniqueness of an $m$ degree polynomial to "match" a function at $m+1$ locations
They do it by explicit construction of the Lagrange interpolation polynomials without much motivation, fine
They then mention the following:
What could the last sentence possibly mean? I know things like "simple" or "explicit" can be informal notions but I fail to see why Gauss-Jordan wouldn't give us explicit descriptions of each $p_j$ and, putting them all together, the given polynomial?
The standard formula is that $\overline{X} = \sum_k k\cdot P(X = k)$. But $P(X = k) = \sum_{s:X(s) = k} P(s)$ so the formula $\overline{X} = \sum_s X(s)P(s)$ seems correct
@Jakobian thank you. Instead, if I consider the uniform convergence in $[0,a]$ with $a>0$, that reasoning above is not valid anymore because, even if it's still true that $\sup_{x \in \mathbb[0,a]} |f_n(x)-f(x)| \ge |f_n(n)-f(n)|$, I then must take the limit as $n \to +\infty$ and for fixed $a>0$ I have $n \notin [0,a]$ for $n$ big enough, right?
I mean, it's true for $n \in [0,a]$ but it is not true for each $n \ge N$ for some $N \in \mathbb{N}$ and this is what I need when I want to evaluate $\lim_{n \to +\infty} \sup_{x \in [0,a]} |f_n (x)-f(x)|$
As 3B1B says, if you see pi then there's always a circle lurking somewhere
In this case, the standard proof of that identity shows you where the circle lurks
I'm just learning about "arithmetic sequences of order $k$" and am hoping for some help on internalizing the definition. The $\Delta$ operator is defined as an endomorphism on $E^{\Bbb N}$ via $(\Delta f)_n = f_{n+1} - f_n$ for $f \in E^{\Bbb N}$ ($E$ is a vector space). Now I know that $(\Delta^kf)_n = \sum_{j=0}(-1)^{k-j}{k \choose j}f_{n+j}$ which appears relatively ugly, so I have very little inution for what $(\Delta^kf)_n$ constant would mean
Please feel free to ignore everything past the first sentence there, I am just sort of trying to parse that first sentence. What sort of sequence is an arithmetic sequence of order $k$?
My basic set theory skills are a bit slow. Is there a typo in the yellow highlighted bit. Should $B$ (the second occurence) not be $Y$? This is from Folland's, but I haven't been able to find anything in his errata about this. The formula I have found about this is $(A \times B)^c=(X \times B^c) \cup (A^c \times Y)$.
Alternatively, one could write $$(A\times B)^c=(A^c\times B)\cup (A\times B^c)\cup (A^c\times B^c).$$
OK I am still struggling with the second sentence in the picture I gave above. I'll label my questions with (x). What I have so far: given any $p \in K_k[X]$ and choice of $x_0,h$ I know (Remark 8.19(c)) that $p = N_k[p;x_0;h]$, where the RHS is the newton interpolation polynomial for that function $p$ ((1): how do $x_0,h$ factor into the discussion here? Shouldn't any choice of $x_0,h$ give the same $N_k$ because this is a polynomial, so that if $N_k$ agrees...
with $p$ at $k+1$ spots then they're equal?)
Agh
Trying to parse it still, don't even know how to ask the question about how the formula (12.15) gets used. Will keep at it
Hopefully is made clearer by the second picture I sent. The Netwon polynomial (of degree $k$) constructed from the given function $f$ and choice of $x_0$ and $h$
Basically the way the author is arguing seems to be that since a polynomial of degree $k$ equals its Newton polynomial of that same degree (by Proposition 8.19(c)) in my book, it follows that $p(x_0 + nh) = N_k[p;x_0;h](x_0 + nh)$ and then I suspect I should be able to use (12.15) pictured above to calculate the RHS, and then somehow observe that the sequence so obtained is an arithmetic sequence of order $k$?
The definition of limit for $A \subseteq \mathbb{R}$ and $f:A\to\mathbb{R}$ is $\lim_{x \to x_0} f(x)=l \in \mathbb{R}$ if $\forall \epsilon>0$ there exists $\delta>0$ such that $\forall x \in \mathbb{R}$, $x \in A \cap (x_0-\delta,x_0+\delta)\setminus \{x_0\}$ implies $|f(x)-l|<\epsilon$. But how is $x_0$ quantified?
Is it: $\forall \epsilon>0, \forall x_0 \in \mathbb{R}$ there exists $\delta>0$ such that $\forall x \in \mathbb{R}$, $x \in A \cap (x_0-\delta,x_0+\delta)\setminus \{x_0\}$ implies $|f(x)-l|<\epsilon$?
And what about $l$? Is it something like: $\exists l \in \mathbb{R},\forall \epsilon>0, \forall x_0 \in \mathbb{R}$ there exists $\delta>0$ such that $\forall x \in \mathbb{R}$, $x \in A \cap (x_0-\delta,x_0+\delta)\setminus \{x_0\}$ implies $|f(x)-l|<\epsilon$?
frieren: in the first statement you wrote, the [implicit] understanding is maybe that x_0 is an element of A, or at least an accumulation point of A, but there's no inherent need for quantification across a set of x_0
that's just a statement depending on x_0 and l that might or might not turn out to be true
if you wanted to express "lim_{x to x_0} f(x) exists", then that would be [exists l in R] [the definition of lim_{x to x_0} f(x) = l], where again this is just a statement that might or might not be true for x_0
but i don't see what you've first written above as automatically requiring that you go in that direction
the definition just gives you a language for talking about these things called limits, it's sort of up to you what you want to say in that language
interesting. so take $K$, get its polynomial ring (a domain), then take its quotient field, and that will be infinite (basically because $\Bbb N$ infinite) but $K$ need not be. Showing that it has same characteristic as $K$ isn't as obvious to me but I'll think on that
i'm not entirely sure what you have in mind with 'basically because N infinite.' what i had in mind was that k(t) contains k[t] as a subring, and there are pretty clearly infinitely many things in that (maybe if you had in mind something like the specific subset {t^n: n in N}, "because N is infinite" would be one way of summarizing that argument)
one reason not to summarize that argument that way is that "because N is infinite" maybe suggests the [in general incorrect] alternative suggestion that the subring of k(t) generated by 1 is infinite (which is true or not depending on the characteristic of k)
Oh I just meant it because of how I'm familiar with polynomials being defined: as a subset of the formal ring of power series (maps from $\Bbb \to K$) with only finitely many values nonzero
another example that algebra weirdos like me love is the quotient field of the ring of formal power series k[[t]], which is denoted by k((t)), it's the field of formal Laurent series
again, always infinite and same characteristic as k
@LukasHeger that makes sense. If I take $R_k=\lbrace t/x: t \in [k,1/k]$ for all real $k \ge 1$ and take the union $\bigcup_{k \ge 1} R_k$. Gluing the boundaries like $\partial R_k=\lbrace k/x \rbrace \cup \lbrace \frac{1/k}{x} \rbrace$. Am I correct to conclude that $\bigcup_{k \ge 1} R_k \cong \Bbb B^3$?
@LukasHeger you've given me some things to think about. I was trying to define densely nested spheres that each share exactly 2 antipodal points. I was trying to take this collection and say that the union of the shells (leaves) generates the 3-ball. Do you know what "nested spheres each sharing the same 2 antipodal points" is homeomorphic to?
In mathematics, the Hawaiian earring
H
{\displaystyle \mathbb {H} }
is the topological space defined by the union of circles in the Euclidean plane
R
2
{\displaystyle \mathbb {R} ^{2}}
with center
(
1
n
,
0...