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Dec
17
comment A formal justification for this “physicism”?
You're right. I think I need to ask my friend in what context he used this "trick" so that I can see what physical quantity the expression $\sum_{k=1}^\infty e^{ikx}$ represents, so I know how to think of your $\mathbb{C}[[e^{ix}]]$ as a collection of physical objects, i.e. interpret it physically.
Dec
14
comment A formal justification for this “physicism”?
OK, thanks a lot, that's helpful. I doubt I'll be able to understand his book, but I'll give it a shot and see if it contains an answer to what I'm looking for.
Dec
12
comment Generalized Cauchy's theorem (group theory)?
@GeoffRobinson yea, Sylow's theorems
Dec
12
comment A formal justification for this “physicism”?
Do you know of anywhere I could read about such kind of construct in more detail, by any chance? I haven't ever seen rings except which are semigroups under multiplication, so I can't see how this fits into any bigger mathematical theory.
Dec
12
comment A formal justification for this “physicism”?
But as a positive answer, what actually is it? What category does it belong to?
Dec
12
comment A formal justification for this “physicism”?
OK, fair. So what kind of object is $\mathbb{C}[[e^{ix},e^{-ix}]]$ supposed to be then?
Dec
12
comment A formal justification for this “physicism”?
*, that series. Sorry, it says comments can only be edited for 5 minutes.
Dec
12
comment A formal justification for this “physicism”?
But in $\mathbb{C}[[x,y]]/(xy-1)$ like, that polynomial is in the same equivalence class, i.e. is the same, as $1+\sum_{k=1}^\infty x^k+y^k$, which you can square nicely, right? And every Laurent series can be put like that, uniquely, can't they?
Dec
12
comment A formal justification for this “physicism”?
This isn't a ring? I might be wrong, but I thought this should just be $\mathbb{C}[x,y]/(xy-1)$, which in my head is a ring.
Dec
12
comment A formal justification for this “physicism”?
@QiaochuYuan So this class was actually based on Friedlander & Joshi, so I'm comfortable with distributions, and I actually believe that my series converges in $D'(\mathbb{R})$ to $1-2\pi\sum_{n=-\infty}^\infty \delta(x-2\pi n)$. Maybe to be more clear, I think I'm looking for a method that justifies using the geometric series manipulation, which I don't think distributions quite do. And uniform convergence implies pointwise convergence, so no.
Dec
12
asked A formal justification for this “physicism”?
Nov
26
comment $f$ meromorphic on $\mathbb{\hat{C}}$ $\implies$ $f$ has a finite number of poles
Also, more directly (this is just the topology on $\hat{\mathbb{C}}$): we say $f$ has a pole at $\infty$ if $f(\frac 1z)$ has a pole at $0$. For this singularity to be isolated, say there is some neighborhood of radius $\frac 1\delta$ on which there are no other poles. Then $f(z)$ has no poles for $|z|>\delta$, and since the ball of radius $\delta$ is compact, there are finitely many poles in there.
Oct
25
comment Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$
Like, what if $J(x^r)=x^r/r!$? Do you know if even they can or can't happen? If it is, then $\sum_{r=1}^\infty J(x^r)=e^x$, and you can't write that as $\sum_{i=1}^n x^{k_i}$, so your definition doesn't make sense.
Oct
25
awarded  Commentator
Oct
25
comment Relation I found: $ (\sum_{r=1}^{\infty}\frac{z(r)}{r})\times \int_0^\infty f(x) dx = \lim_{h \rightarrow 0} \sum_{i=0}^{n} f(k_ih)h$
Your definition for $J(x)$ only makes sense for $|x|<1$, right? How else can you guarantee the sum is convergent? And even with that assumption, how can you guarantee that the latter sum converges? Lastly how can you guarantee that latter sum is the sum of (finitely many) powers of $x$?
Oct
12
comment Subrepresentations of finite dimensional semisimple representations of an algebra
Yep :) I take it you're not since it seems like you already know this stuff?
Oct
9
comment Subrepresentations of finite dimensional semisimple representations of an algebra
Thanks, and sorry I didn't find that when I tried searching for related posts. I wonder if Kasper is in the same class I'm in...
Oct
8
asked Subrepresentations of finite dimensional semisimple representations of an algebra
Sep
5
answered Simple question on symmetric tensors
Aug
5
comment Toral sub algebra
Sorry, what is the contradiction here exactly? If the Lie algebra is abelian then the toral algebra is as well, so it's nilpotent.