Mathematics

@robjohn Considering that we're integrating $\text{PolyLog}^{(1,0)}(1,-x)$ over the entire positive real axis, the best way I can think of to define the derivative is to differentiate inside the Fermi-Dirac integral to get $$ \text{Polylog}^{(1,0)}(1,-x) = -\int_{0}^{\infty} \frac{\log t + \gamma}{e^{t}/x+1} \, dt .$$

pjs36

1:38 AM

I'm thinking about homotopies and wouldn't mind some reassurance. I have a continuous $H \colon X \times [0, 1] \to Y$ with $(x, t) \mapsto H(x, t)$ that I think I'd like to 'extend' to a continuous function $\hat{H} \colon X \times [0, 1] \to Y \times [0, 1]$.

It seems reasonable to define $(x, t) \overset{\hat{H}}{\mapsto} \big(H(x, t), t\big)$, which "obviously" should be continuous. If this is indeed continuous, I don't see the slick argument that guarantees it, since my topology is so rusty.

anon

it suffices to show preimages of basic open sets $U\times (a,b)$ are open. the preimage is $$\widehat{H}{}^{-1}\big(U\times(a,b)\big)=H^{-1}(U)\cap(a,b),$$ and $H^{-1}(U)$ is open since $H$ is continuous

pjs36

Thank you, @anon. I was hoping I wouldn't have to mess around with open sets, but that's not bad at all.

(I probably should have forced myself to think about open sets, being so rusty!)

anon

same argument shows if $A\to B$ and $A\to C$ are continuous then $A\to B\times C$ (diagonally) is too

pjs36

Ah, that's a nice result, and was the sort of 'function-based' thing I was trying to think about. Thank you again.

Mike Miller

more importantly that's an iff; a map to $X \times Y$ is continuous iff each of the maps to $X$ and $Y$ (after projecting onto each factor) are

Bib

2:29 AM

Suppose I have a homomorphism of algebras $\alpha \colon \mathbb{C}G \to \operatorname{End}(V)$ where $V$ is a vector space and $G$ is a group. Does $\operatorname{im} \alpha$ consist of the modules of $\operatorname{im} \alpha$?

robjohn

2:49 AM

@RandomVariable what is the integral before differentiating?

Samuel Yusim

3:32 AM

how is it possible that a space might not be homeomorphic to the disjoint union of its connected components?

Paul Plummer

@SamuelYusim Take $\{0\} \cup \{1/n \mid 0<n \in \mathbb{N} \}$

You can have things get closer together, so that the connected components topology might be to fine of a topology

Actually let me think...

Samuel Yusim

would a homeomorphism of that into $\mathbb{Z}$ not be $1/n \mapsto n$ and $0 \mapsto 0$?

and then you can definitely send that to a disjoint union of connected components

Paul Plummer

$\{0\}$ is an open set in $\mathbb{Z}$ but not in the topology I described

Samuel Yusim

oh, duh

okay, then how on earth would one show that this set has the desired pathology?

Paul Plummer

Well, you know the disjoint union topology, and describing the connected components should be quite easy, so show that the subspace topology does not coincide with the disjoint union topology, using whatever methods you would like

Samuel Yusim

3:44 AM

oh, sure

Paul Plummer

Some more examples, would be the rationals, the irrationals, and the cantor set.

Karim Mansour

Finally finished understanding pre req required to understand GNS construction proof

user147690

How are you doing @Paul?

Paul Plummer

4:00 AM

Doing okay, just moved into my new place today, glad that is over. How about yourself? And how is (was...?) the tropical geometry project? @AlexClark

user147690

I am good and Tropical geometry stuff is very interesting. We are hopefully choosing a specific open problem in the field tomorrow, and will work on that for the next three weeks or so.

user147690

Lots of polyhedral geometry xD

user147690

Uni is back on again, and diff geo looks really interesting

Paul Plummer

Sounds like it is coming along. How did all the preprep (that I gave you a hard time for doing) go, taking all the classes?

I might take diff geo this semester

user147690

Taking the four courses and the prep went nicely I think

user147690

4:03 AM

Mostly I was just making sure I had all the stuff from last semester down(in the relevant areas) and learning some vocab

user147690

I mean I could have done much more, but I spent lots of time on Tropical geometry(despite being very bad at it I found[mainly the geometry xD])

Paul Plummer

So you got the tropical part all figured out :P

user147690

Haha yep, studied on the beach

user147690

Weirdly enough I expect algebraic methods of physics(3rd year) to have harder algebra than advanced algebra(which is a 4th year course)

Paul Plummer

I would guess you would end up digging deeper into certain parts of algebra, while the advanced algebra is suppose to give a taste of everything and make sure you know some basics

user147690

4:14 AM

That does make sense

user147690

Good call

user147690

Week 7 or so we will be doing Hopf algebras, and I looked at the notes and they seem intense

Bib

question: what is "Howe duality"? And where can I find a reference for it?

user147690

Just the conditions for a lie superalgebra look intense aswell :P

Paul Plummer

Sounds fun

user147690

4:19 AM

What are you studying atm?(not necessarily this minute, but what textbooks if you have any atm)

Paul Plummer

I haven't really done much for a while, with finding a place moving. But I guess I have been doing bits of algebra, to fill some holes, and a little algebraic topology. I guess I also did some research for this problem, which I read a little bit of some of the papers on the subject

It is always fun to run into something by Yves de Cornulier

Paul Plummer

4:34 AM

Oh and for textbooks Algebra Ch0 and Algebraic Topology by Hatcher

user147690

Ahhh I really want to read some algebra ch0 for that categorical perspective, but I never have time

Random Variable

5:11 AM

@robjohn $ \displaystyle \text{Li}_{s}(-x) = - \frac{1}{\Gamma (s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}/x+1} \, dt$