Sorry reddit r/math subreddit.

no

Please read and let me know something about it.

@JayeshBadwaik You might benefit from looking up universal algebra.

@anon okay, will look into it.

In the most generality, an algebra is a set with a set of operations defined on it, and operations are functions from cartesian products of the set to itself. (the operation is n-ary if the domain is A^n; n might not be finite and it can be 0 - if n=0 it is nullary and basically means a distinguished element of the set, like an identity) @Jayesh

Anyway I have to leave now.

@anon okay, thanks :-) bye

@anon okay, thanks :-) bye

@J.M. Is that a knot, or is it not?

@PeterTamaroff Nice thanks, I already knew : )

Hi! Back.

@JonasTeuwen Hi - how are things?

@OldJohn I am okay. How are you?

@JonasTeuwen Too much beer last night - but otherwise fine :)

Did anyone enjoy that closing ceremony?

@skullpatrol yes - but I am biased

@OldJohn Kopfschmerz?

@JonasTeuwen yep

If X: Omega --> Reals, how can we define a Borel sigma-algebra with respect to this X?

Take the open sets in the range, and map them back. Take that as a generator?

What if Range(X)=\mathbb{R} ?

Just open balls in that case?

What is the context? Probability theory?

@Matt You are knot incorrect. :)

Plus some weird liking for shrooms.

Is the Borel sigma-algebra used to define \sigma(X) := \{ X^{-1}(B) : B \subset \mathscr{B} \} ?

Is that its purpose?

How would I know? It is your question!

Lol I'm talking about the most general case of X

I'm crap at maths

@IssacM Where did you find this problem...?

It is not a "problem". My probability lecturer talks about Borel sigma-algebra but never defines what this is.

@J.M. Well, modulated by BP.

All I know is that it's used to define \Sigma(X), where X is some arbitrary random variable with Domain(X) = \Omega and Range(X) \subset \mathbb{R}.

@IssacM What book do you use?

No book .. Just google

@J.M. Yea, why do they scare me like that for something that... trivial?

@IssacM The lecturer uses Google?

I have about 1,000 books just finish download for mathematics .. let me check

@IssacM It is the smallest $\sigma$-algebra for which this function is measurable.

So, as I said before.

@JonasTeuwen Could've been worse. They could have said it was "idiopathic"...

@J.M. I would have choked someone.

Ok so it's the smallest union of open sets (a,b), a,b \in \mathbb{R}, s.t. X has an image(X) \in \mathbb{R} \forall \omega \in \Omega ?

Also. They call me: "we cannot say this over the phone... come here in an hour".

Son of a cracks.

open intervals*

@IssacM For it to be measurable you only have to test on the generator of that $\sigma$-algebra. So fine for open balls on the range.

Then you map them back. These should be "measurable". This might not be a $\sigma$-algebra yet, so you take the smallest on that encloses this.

I do wonder if BP can cause pain. Because apparently...

TY

Hey @JonasTeuwen ! Is your tongue cured?

@Matt It is not!

Or is it knot?

: (

@Matt Knot 8-).

I just posted an answer that does something without something suggested by teddy in a comment. Now I'm not so sure whether it's right. But I'm sure if it's not then downvotes will come soon : )

Good old SE community : )

@Matt Link meh!

@Matt, you wanted a Fourier specialist, no? Exploit Jonas's presence! ;)

@JonasTeuwen Here.

@J.M. Yes, I'd've loved to talk about Sobolev and Fourier stuff but maybe he's too sick to do that now with his tongue and whatnot.

@J.M. It might fall off if he talks too much.

@Matt Ask me!

Hmm, yes, we can't risk that. @Jonas, can you indulge Matt, or are you too distraught?

Okay, Jonas seems fine. :D

@J.M. Well, I don't feel well... but apparently I am not going to die 8-).

Just some BPI whatever.

@JonasTeuwen But you're well enough to handle Matt's queries, yes? :)

@J.M. Sure.

@J.M. I hope so! - I was definitely out of my depth earlier :)

@JonasTeuwen Luv ya, bro! putting on Manchester accent

: D

So... it's about Sobolev stuff.

@Matt Good!

First of all, I have a theorem. I only understand half of it. Let me post:

Sure.

This thing has some $e_i$ in the second half of the claim, and I have no idea what they are.

I looked around the notes but didn't find anything.

I'm not saying it's not there at all but... it's too well hidden to dig up in less than one hour : (

@Matt One moment, some bro is being annoying in my office. Like a fly.

No problem bro. I'll rest in the meantime. I'm quite sleepy.

@Matt The derivative in the direction of $e_i$. Basically the partial derivatives wrt all the variables summed.

But what are the $e_i$?

Shouldn't they be $\alpha_i$?

hi @robjohn

@Matt Just unit vectors.

In the directions of the axis.

@skullpatrol Hey there

@J.M.: another knotty avatar!

@JonasTeuwen It's confusing to me how we've been talking about partial derivatives $D^\alpha$ and now we suddenly switch to directional derivatives. Wait. Aren't they the same? If I do $\partial_x$ that's the same as $\partial_{e_1}$. No?

@Matt That is the same. Different ways of notation.

I see though now:

$\partial_\alpha $ might be mixed, like e.g. $\partial_{(1,1)} = \partial_x \partial_y$ whereas $\partial_{e_i}^k$ are not mixed.

Right?

@Matt $\partial_2$ would be more reasonably used for differentiation wrt the second variable.

Beat you to it : ) ?

@robjohn Yes, I'm feeling knotty these days.

@J.M. Shall we hunt you up some kpaurn sites? ;-)

@BrianM.Scott Hey! Nice to see you again, you were scarce for the past few days or so...

@J.M. It also has a scaffold-look as if it is under construction

@robjohn Ah, that was the intended effect. Glad it worked. :)

Hi Brian, long time no see! : )

@J.M. I was altogether offline for a week and a half or so; I think that my hindbrain wanted a complete holiday from everything.

@Matt Yes, it’s been a bit.

@JonasTeuwen Better now?

@BrianM.Scott The friend you mentioned before is better now, I presume?

@J.M. It’ll be a long, slow process, but she’s back at work part-time, at least.

@BrianM.Scott Good to hear.

@Matt Yes.

@BrianM.Scott Nice :-).

So what I wrote about mixed and not mixed is correct?

(Asking because after that I have one more question...)

@Matt Only one? :-)

@BrianM.Scott : ) Hey, I have an excuse: I have another exam. And contrary to my good resolutions I again ended up doing all the stuff in the 8 weeks before the exams.

I simply cannot learn from lectures and homework, my brain switches off.

@Matt Isn’t that a slight improvement, or am I misremembering?

@BrianM.Scott Yes. But this time it's 2 exams, not just one so 8 weeks instead of 4.

And I think I failed one of them.

@Matt The two things that I found most useful were homework, and turning notes made in class into a polished product.

@Matt :-(

@BrianM.Scott I do think I would like the process of turning notes into a finished product and I'd learn a lot from that. But I've not had the time to try. I learn best from a good book, time to think and lots of exercise in each chapter of the book.

@JonasTeuwen ping ^ : )

(just to make sure you look at the screen before my previous comment scrolls off screen)

Jonas seems to be afk.

Then I'll be reading about dual spaces. BBL

@Matt Looking back, I think that the courses in which I learned the most were the undergraduate and graduate courses that were taught Moore method, and the graduate set theory and abstract algebra courses that were taught without any text. They were taught by Ken Kunen and Marty Isaacs, both of whom were incredibly well-organized. Both had us doing weekly problem sets that included a fair number of non-trivial problems.

In those I didn’t really even have to rewrite my notes much, the lectures were so well organized.

@BrianM.Scott hey

@BrianM.Scott Can I ask for something about learning and sticking with courses?

Jawohl, mein Herr.

Sure, though I don’t guarantee to have a sensible answer.

@BrianM.Scott At the moment for our AT course we are learning from hatcher

Now I find that book impossible to read due to several reasons.

A big reason would be that there is too much handwaving.

Which are?

@Matt Sorry. What is the question? 8-).

I love handwaving.

@JonasTeuwen Is what I wrote about mixed and non-mixed derivatives correct?

@BenjaLim Pierre is the handwaving wizard!

@BrianM.Scott I am finding it very uncomfortable due to the fact that without rigour I don't know what is right and what is not.

@BrianM.Scott What should I say to my lecturer?

A lot of my other friends concur with me on this.

@BrianM.Scott What would you do if you were me now?

we see (from a corollary, left to the reader) that in all but a few exceptional cases (which we need not go into here), that we clearly (!) have the desired result.

@Matt Depends on the author really... $\partial_{(0, 1, 2)}$ would be $\partial_y \partial_z^2$.

@DavidWheeler ?

@JonasTeuwen Yes but can $\partial_{e_i}$ ever be mixed?

@BenjaLim I was never very fond of algebraic topology, and that was one of the reasons. As I recall, it was mostly either a bit hand-wavy, utterly gruesome in detail, or diagram-chasing.

@Matt $\partial_{e_i}$ is just $\partial_i$ or $\partial_{x_i}$.

@BrianM.Scott Like for example in hatcher when dealing with arbitrary topological spaces

@BenjaLim The rigor is left up to you.

Exactly. So it can never be mixed.

he talks about attaching strips and epsilon fattenings.

I’ve the impression that this has more to do with the subject than it does with the particular book.

Right?

I find it incredibly unrigorous

@BenjaLim Where is that? I have a downloaded copy, I think.

proof of prop 1.26 @BrianM.Scott

@BrianM.Scott I am not sure if homology will be more rigorous

@BenjaLim Hang, and I’ll take a look.

@Matt At least you're not dealing with AT now.

@BrianM.Scott I look elsewhere at the proof of such a proposition, like in Lee's topological manifolds

and there are so many more details

like point - set topology things coming in

There is nothing causing more suffering to me than oral exams. Doesn't matter at all what subject.

@Matt You can handwave in an oral exam.

Never mind that.

@BrianM.Scott And furthermore we are being set problems from hatcher.

@Jonas won't give me yes or no : (

@BrianM.Scott I am finding it hard to learn AT at the moment.

@Matt Sorry, just fuzzy in the head.

@BrianM.Scott To be honest, I would prefer Atiyah - Macdonald anytime.

@Matt I thought you already solved it. :-): yes.

Prefer this book, prefer that book. Just take one and start working already!

Thank you : ) Oh, ok. No problem. Thought you wanted to be mysterious.

@BrianM.Scott you there?

No!

@JonasTeuwen Can I ask you another question or are you too fuzzy?

@Matt You can always ask.

@BenjaLim Okay, I’ve taken a look now. I agree that all of the messy details have been swept under the carpet, but that’s a very clear explanation of what’s really going on.

@BrianM.Scott yes that is true.

But it is not rigorous.

One needs to swipe them under the carpet to make the idea clear.

Math is not about rigor, that is just the tool to make sure it works.

More about ideas and concepts and the rigor is the glue that puts it together 8-).

If you understand the ideas, then you provide the glue.

It is an excellent test to see if you actually understand.

@JonasTeuwen Sometimes that’s true. Still, I completely understand someone’s feeling uncomfortable when they’re not made available.

@BrianM.Scott What do you think I should do?

@JonasTeuwen Assume Manfred asks me "Can you tell me what the Sobolev embedding theorem tells us?" and I'd say "It tells us that we can apply any operator $T: C^\infty \to C^k$ to any function in $H^l$ because we can continuously extend it (by definition of $H^l$)", how many points do I score from 1 - 10?

@BrianM.Scott I also understand, but I think that most people eventually realize this... (okay, you have exceptions where it is very important, but this is quite high level stuff).

@Matt 6 if you are not able to give further explanation.

@BenjaLim I’d be inclined to use Hatcher to get the ideas. Then I’d try to fill in more detail myself, looking at other books if necessary.

Like of what sort?

Why would you want such a thing?

@BrianM.Scott Hmmm....

I have to say in particular right now with CW complexes....

@BenjaLim Yes, I suggest the same. The idea is the most important part.

I am not able to sort out between what's right and what's not.

I’d also try to determine just what level of rigor is expected in the homework and try to learn to write at that level.

@BrianM.Scott Trust me it's not very high :D

@JonasTeuwen Because we know what $C^\infty$ looks like but we don't know really what $H^l$ looks like?

@BrianM.Scott The lecturer handwaves a lot in lectures too :D

apparently he does not want to deal with point - set topology

@Matt Why would you introduce $H^{\text{whatever}}$?

hence modulo open sets/closed sets, etc

@BenjaLim A bit windy there, I see...

@BenjaLim Or he thinks that costs too much time and clouds the ideas.

I would also not do that.

@JonasTeuwen Because we want to do stuff to functions that aren't necessarily differentiable in the strong sense.

@Matt Precisely.

@BrianM.Scott Here is another thing I find extremely handwavy: Modulo terrible spaces, we can ignore the condition that our sets be open in Van Kampen.

So we replace it with something more general: functions that have weak derivative.

@BenjaLim Then I’d try to get used to working in that style, and work to fill in enough details to make myself more comfortable as I found time.

@Matt Like... analyse PDEs?

Yes.

So you can apply the theory of functional analysis to them.

@BrianM.Scott I have tried doing that...

Good! So perhaps one can prove that a PDE has a weak solution.

I think for example that I am reviewing free groups and things by myself

What can we do with this?

Then life is good again: use any operator we have for smooth functions and apply it to something that isn't even necessarily differentiable.

Yes.

How many points?

So basically, you can prove many results much easier for weak solutions.

@BrianM.Scott and for example in certain specific cases of van kampen I have gone over the proofs in other texts rigorously

Yes.

Then you want to actually know something about real (physically relevant solutions).

So then you might want to prove your weak derivative is actually of a smooth function already, right?

Because then you prove your weak derivative is actually a normal one.

@BrianM.Scott Actually to tell you the truth, the only reason I took AT was to learn about homology and cohomology

That's where you can use embedding theorems.

@BrianM.Scott I feel atm that I'm working with maths without any backbone.

@JonasTeuwen This bit I don't understand. What I know is: if my $f$ has a strong derivative then the weak and the strong derivative are the same. (By Jonas-by-parts)

So you try rigor as backbone? I understand that too.

@JonasTeuwen huh?

@BenjaLim About which I’m quite happy to know absolutely nothing. (I did learn a very little homology theory in the alg. top. course that I took in grad school, but it’s long gone and not missed.)

@BrianM.Scott I want to learn it for algebraic geometry

@Matt That is true. So you might want to prove that the weak solution which you prove exists is a normal solution as well.

Hence proving your equation has a solution.

But I have seen the course, I suppose they do not do this.

@BrianM.Scott Like for example with commutative algebra, I struggled with it but it was entirely rigorous. Every step I could prove rigorously and understand why it was true.

@BenjaLim I figured as much.

@BenjaLim i am reading that proof now...i'll let you know how it goes

But you can actually use it to build a differentiable function out of such an ugly Sobolev one.

@JonasTeuwen I did not see this, so far.

@BrianM.Scott How did you read my mind?

@Matt It is more for a PDE course. It is very vague why they introduce it there 8-).

@JonasTeuwen That sounds interesting. I'm sure you'll have a book recommendation ready for me when I survived this torture. If I do, that is.

@DavidWheeler I understand the handwaving.

@BenjaLim You’ve talked quite a bit about alg. geom. in the past, unless I’m misremembering.

@JonasTeuwen Introduce what where?

@BrianM.Scott hahaha yes. Indeed I have.

In any case. Do I pass this Sobolev embedding theorem question?

If yes, I will now start reading the chapter about dual spaces.

@Matt There are many Sobolev embeddings, which is the one you have?

I think that's not only quite important but also quite interesting.

@BrianM.Scott I think I will push on with algebraic topology. But indeed I have to say that acquiring this handwavy intuition to sort out right from wrong is indeed very hard to get used to.

@Matt Yes, you would pass I suppose. But perhaps you don't know why it is very interesting, but they don't explain that.

@BrianM.Scott I have started learning with elements of algebraic topology alongside.

@Matt Very important.

I find that book completely rigorous, much easier to understand.

@BrianM.Scott In a way I'd rather prefer a lot of details than handwaving.

Rigorous books read like phone books to me.

@BenjaLim Oh, I definitely sympathize; I tend to feel the same way.

@BrianM.Scott glad that we share similar feelings.

If they fill in all the details, what is left for you?

@JonasTeuwen Embed $C^\infty(\mathbb T)$ into $C^l(\mathbb T)$. Then we can continuously extend this embedding to $H^k(\mathbb T)$.

@JonasTeuwen The problem now with hatcher is that you don't even know what details to fill in.

@BenjaLim But if you could, you would know you understood.