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1:03 AM
Hello everyone. I'm studying Darboux/Riemann sums with its lower and upper integrals.

Suppose a function $f\colon [a,b]\to \mathbb{R}$ and a partition $P=\{t_0,t_1,\ldots,t_n\}$ with $a=t_0 < t_1 < \cdots < t_n = b$.

The book proposes a partition refinement $Q = \{t_0,\ldots, t_{i-1}, r, t_{i}, \ldots, t_n\}$. That is, we are adding a new point $r\in Q$.

We define $m_i = \inf\{f(x):x\in [t_{i-1},t_i]\}$, $M_i = \sup\{f(x):x\in [t_{i-1},t_i]\}$ and

\begin{align*}
s(f;P)&=m_1 (t_1-t_0)+\cdots+m_n (t_n-t_{n-1})\\
 
1:28 AM
@AttractorNotStrangeAtAll Switching notation. The union of two partitions is called a common refinement — it refines each of them.
 
Ah, thanks. Indeed:
"Given two partitions, $P$ and $Q$, one can always form their common refinement, denoted $P \lor Q$, which consists of all the points of $P$ and $Q$, re-numbered in order."
But this isn't written in the book, so I guess is something that I've should've seen before.
This is from wikipedia btw
 
Nah. It's just dropping the original notation.
What book?
 
Curso de Análise Vol. 1 by Elon Lages Lima
 
OK, obviously I don't know it.
 
I would call it 'Brazilian baby Rudin', but just because it's a standard book.
 
1:44 AM
You can also look at Spivak's Calculus.
 
1:54 AM
Yes! Sometimes I look into it to compare some definitions
 
 
2 hours later…
3:59 AM
what's the fundamental group of the plane with a countable closed discrete set of points removed? free product of countably many copies of Z?
 
4:15 AM
Sounds right.
 
Howdy.
 
4:32 AM
Heya Fargle
 
Wie geht's?
 
Nicht zu schlimm.
 
After a Google, glad to hear.
 
What have you been learning?
 
4:53 AM
Mostly stuff relating to programming and whatnot. Been looking passively at cryptography because it has the nice crunchy combinatorical stuff.
A fun fact I've discovered: RFCs and reference implementations aren't super great at conveying vital algorithm information.
Math-wise, I've been doing my yearly "stare at an AG book and try to do exercises". We'll see if I have enough focus this time, lol
 
AG, really?
 
hello!
 
Howdy, copper
 
Hi Ted! (My name is Joe btw :-))
 
hi, Joe
 
Yep
5:05 AM
Hi i just wanted to ask if christofell symbols are tensors i thought they werent because of the trasform not looking like a regular tensor transformation but some of my classmates think otherwise
 
My stupid laser printer just died as I was printing my son's application essays :-(. So annoyed.
 
No, they are not, @Yep
 
Yep
@TedShifrin Thought so thanks :)
 
I've had similar issues from time to time, Joe.
Laser printers. We always doubt ourselves when we learn math..
 
That is so true.
Had a lovely bike ride over in China Camp this morning. Followed by a nice Cabernet Sauvignon. Not a bad Friday (printer excepted!)
 
5:30 AM
@TedShifrin Yeah, every so often I just get the hankering to bang my head against ideals and varieties.
 
Soon you'll start scheming.
 
"Soon" seems a very relative word here. One step at a time.
 
How can I describe all geodesics in the cone = $\{(x,y,z)\in\Bbb R: z^2=x^2+y^2. z>0\}$ ?
I have no clue how to start
 
 
1 hour later…
6:47 AM
@love_sodam make the cone out of a piece of paper; on that you know geodesics. Or you can use Clairaut's Thm for geodesics on a surface of revolution.
 
@TedShifrin But Clairuat's theorem doesn't provide all geodesic right?
 
7:08 AM
Flatten the cone out and identify points on the edge of the missing 'wedge'.
(Actually its a little messier than my comment might suggest.)
 
That was my first suggestion. Yes, Clairaut gives them all except possibly parallels, and none of them is a geodesic here.
 
@copper.hat what do you mean 'Flatten the cone out' ?
I need some conceptual understanding about geodesic. What I know is really the definition.
 
7:28 AM
HELLO EVERYONE
Would it be possible to analyze exponential graphs using calculus, different series(such as Laplace transform etc) and in case you wanted to know, I am doing newton’s law of cooling.
he idea is that I deriving the equation of newton’s law of cooling but my high school teacher said that it will be a bit easy to derive an equation that could easily be found online so therefore I am thinking of how to make my report a bit more difficult by adding more analysis. Would it be possible? (Any help from you would be very useful.
 
 
4 hours later…
11:50 AM
what definite double integral gives $\Gamma^2(x)$?
 
12:30 PM
Can one give a construction of the compactification of $M$ onto an open dense subset? $M$ is equal to $\Bbb M^{1,1}.$
Hey Astyx and Balarka
 
1:21 PM
How does the torsion-free property play with respect to semidirect products? It's clear that if the second factor is torsion-free, then the semidirect product is torsion free. But what about the first factor?
 
Quick question, is there a name for a Star graph $S_n$ that has a Path graph $P_m$ attached to each of its leaves?
 
1:41 PM
Is the infinite dihedral group torsion free?
No, of course it isn't!
And since it is isomorphic to $\Bbb{Z} \rtimes \Bbb{Z}_2$, this shows that semidirect products don't play nicely with torsion-free property, unless the second factor is torsion-free.
 
@user193319 if you take the semidirect product of $H$ acting on $N$, then it will have $N$ as a (normal) subgroup, so if $N$ is not torsion free, the semidirect product won't be either
 
if either factor has torsion, so does the semi-direct product
 
yeah, the semidirect product contains both original groups as subgroups
 
conversely, if the semi-direct product has torsion, so does one of the factors
 
2:02 PM
Ah, of course...hmm...What if both factors are torsion-free; will the semidirect product be torsion-free?
 
yes, you can prove this
 
 
2 hours later…
4:23 PM
anyone good with compactiftying things?
can you compactify a compactification
 
What would that mean ?
 
can you compactify a compactification of a compactification?
 
if $X$ is compact, $\operatorname{id}_X$ is a compactification of $X$
booooring
 
what do you need for a compactification? a metric and an embedding?
 
no need for a metric
 
4:32 PM
a compactification usually means an embedding as open dense subset into a compact space
 
and are there some usual candidates for these embeddings? I'm thinking maybe arctan?
 
Usually you define it through the topology
 
arctan???
 
@Thorgott you can use arctan to compactify R
 
yeah, but that's something very specific
 
4:37 PM
The Alexandrov compactification of a topological space $(X, \tau)$ is $(X\cup\{\omega\}, \tau \cup \{\{\omega\}\cup K^C, K\text{ compact of }X\})$
 
I believe you need to take $K$ compact+closed
for it to work in non-Hausdorff spaces, that is
 
It depends how you define compact :)
 
how do you
 
In France the usual definition is that it satisfies Borel Lebesgue and is Hausdorf
 
satisfies Borel Lebesgue? is that the "open covers have finite subcovers" condition?
 
4:45 PM
Yes
 
ok, then the Alexandroff compactification only works if $X$ is LCH
 
@BalarkaSen wtf 18?
 
LCH?
 
locally compact hausdorff
 
No, the AC is the same, because it's implied that compacts are closed
It's just that we don't say "compact and closed" because with our definition every compact set is closed
 
4:51 PM
my point is that the AC as you define it will be a compact space in your sense iff $X$ is LCH
 
Oh yes sorry
 
if you have a non-Hausdorff space, you can replace "compact" with "compact+closed" and still get a space that satisfies the "open covers have finite subcovers" condition, but it's not gonna be Hausdorff of course
 
@EdwardEvans so the lecturer is telling people to move to other fields if they like the category theory part a bit too much? :D
ah should've read the rest of the story
 
5:30 PM
I cannot understand the conformal compactification of $\Bbb R^{1,1}.$ Does anyone here understand the construction?
 
6:20 PM
@geocalc33 What is $\mathbb{R}^{1,1}$?
 
howdy @robjohn
happy post T-day
Lorentz $2$-space, robjohn
 
Hi @TedShifrin
 
Hi Karim
 
I am reviewing your book. It is really excellent.
I started on it 2 days ago and now I am in chapter 3.
I say the videos and the book complement each other
 
LOL, I don't know why you're doing this. You have lots of more interesting things to learn.
 
6:28 PM
It is just I have noticed I have a gap in my multivariable analysis and some statements I take for granted
I hate using blackboxes in mathematics
 
@TedShifrin Hey there! I hope you had a nice T-Day.
 
Well, sometimes you have to. For example, are you really going to read Hironaka's huge paper on resolution of singularities? I certainly never have.
 
I want to
 
@robjohn Yes, thanks. Chicken pot pie for two instead of monstrous dinner for 15 as in the days of old.
 
it is part of my reading list. Don't you feel kinda of out of place when you use blackboxes?
that you haven't read
 
6:32 PM
@TedShifrin We had a turkey for 3, but had to make many things in twos because some of my family is gluten-free
We do have a large amount of left-overs
 
I have left-over homemade chicken stock which will turn into hot and sour soup tomorrow, @robjohn :) I can't wait!
 
Although the gluten-free stuffing was okay, I had my own pie because I didn't want to encroach on the pie that my wife and son could eat
 
@Karim: You just can't make progress sometimes if you insist on mastering every technical thing. I never used resolution of singularities explicitly in any of my papers, so I have no guilt.
 
@TedShifrin Yes, we will have a lot of turkey soup after the left overs are scoured
I didn't get the giblets out of the turkey before we cooked it, so we won't be using those.
 
turkey stock is nice if you have the patience
 
6:35 PM
@TedShifrin Yeah. It is also the guilt haha. I think better thing is to master every detail that you use in your papers.
 
I always made soup, too, with the bones.
@Karim: Well, my prediction is that you will be forced to break that rule if you want to make progress.
 
Yeah probably break this habit at some point haha.
 
Out of necessity if nothing else
 
Some things I only learned when I taught them, and then I forgot them soon after. (For example, the $SL(2)$ proof of the Kähler identities. I've never had to actually use these other than as an ingredient of proofs of big results.)
 
6:39 PM
its like driving a car, i'll bet you don't know every single detail of operation. focus on what you need. backfill later if you want.
 
@TedShifrin Indeed, one learns things best when teaching things to others.
 
I think I really only understood the proofs of the Sylow theorems when I taught them and wrote my algebra book. Do those proofs enlighten my mathematical research? Absolutely not.
 
surgeons used to say: watch one, do one, teach one
 
copper, explain the CVT transmission to me.
 
:-)
 
6:40 PM
I think it influences things but not directly. But yeah you learn things best when you teach them.
 
We'll have to do some serious fluid mechanics along the way.
 
i think of the cvt as an illustration of the implicit function theorem in operation :-)
 
Anyhow, in terms of teaching, Karim, sometimes it's better to give students insight and understanding than a formal proof. Some formal proofs do NOT provide insight.
 
@TedShifrin did your research output increase when you did more teaching ?
Yeah I agree I call it when you give the students the feels of the theorem.
 
There were some busy terms, Karim, but my research productivity was never lavish, and it stopped entirely the last 10 years of my career (although i still helped other people with theirs).
 
6:42 PM
cool
 
copper, you mean the fluid flow adjusts according to the output it needs?
 
well, the idea is to stay at some (approximate) peak efficiency regardless of load, i was thinking of an old belt cvt where the belt position was the control.
 
I believe the modern version is all fluids, but I should read a little about it.
 
my info. is dated, but it used to be hydro was for heavy equipment.
the first cvt i was ever in was a small car by daf
 
Seems I'm wrong. I have a Civic Hybrid with CVT. Seems it's two belts with a pulley.
 
6:51 PM
the sound always threw me being used to a finite number of gears :-)
 
 
1 hour later…
7:55 PM
What is the Big O of Big O of something? Is it just Big O of something? Considering the case as the something approaches 0.
 
@schn Do you know the definition of something being big O of something?
 
@TobiasKildetoft Need a refresher on that one.
 
then look it up
 
Yup. So it means that the something is greater or equal by a factor $C$ to whatever one lets the big O be “equal” to.
 
8:11 PM
No, less than or equal to. It's about upper bounds.
 
A lot of analysis is based on making the proper bounds. Landau's big-O notation is used a lot. $f(x)=O(g(x))$ means that as $x\to0$ or $x\to\infty$ (it should be mentioned which) that $|f(x)|\le C|g(x)|$.
 
@robjohn I don't think that is a good way to phrase it for someone who is not sufficiently familiar with the formalities
 
@robjohn Right, thanks. Then the big O of $O(g(x))$ just means...the same?
 
@TobiasKildetoft how would be better?
 
(namely, the "as $x\rightarrow 0$ or $x\rightarrow \infty$" part tends to cause confusion)
because people start thinking this is related to limits, which it is not really
 
8:17 PM
it is used for both. It needs to be stated which is going on.
 
sure, you need to be explicit. But without specifying, I would be fine with assuming the one for large values, since that is by far the one that is used most (being also used in CS)
 
I was talking about its use in analysis. Its use in CS is usually different; mainly as $n\to\infty$
 
Or does big O of $O(g(x))$ mean that there is some $|h(x)| \leq C |f(x)|$, where $|f(x)|\le C|g(x)|$?
 
@schn It does not really make any sense
 
@TobiasKildetoft Why?
 
8:22 PM
Big O is applied to a function and gives you a family of functions. So it can not be applied again
You could extend it in the obvious way to that family, but indeed, it would give you the same family back
 
Yes, I'd just say that big-O of big-O would give the same big-O
 
@TobiasKildetoft Right, the functions that are smaller or equal to $g(x)$ are all the $f(x)$’s, but they include the $h(x)$’s.
 
although, technically, it's not proper.
 
$(\int_0^1 f(x)dx)^2=\int_0^1 \int_0^1 f(x)f(y)dxdy $ anyone have some examples for this?
I've found 3 examples
 
@geocalc33 are you looking for examples where $f(x)\ne g(x)$?
ah, you changed it
 
8:27 PM
@robjohn yeah
I changed it
I'm pretty sure there are many many examples
 
That seems to be the case except in some edge cases
You would have to look for some cases where neither Fubini nor Tonelli apply to find a case where that is not true.
 
so you're saying it's nearly always true?
 
in most cases, yes. However, you need to show that Fubini or Tonelli apply.
 
ah okay
I got an example with the gamma function
also, is it rare or non-rare for an integral transform to yield the same function being transformed. for example the Fourier transform of a gaussian is again a gaussian
 
@geocalc33 That is a special case, but it is an important one.
 
8:38 PM
are you saying it's important property of any integral transform or?
 
@geocalc33 That example is used a lot when people are looking at Fourier transforms
 
8:59 PM
does $\pi_1$ preserve sequential colimits
feel like it should
 
9:11 PM
@Thorgott Yes if everything is Hausdorff. The point with all such arguments is to show that you can factor a compact through a finite stage
I always forget the details of this because in practice you're using a cellular exhaustion of a CW complex
 
10:05 PM
right, that was also the reasoning I had in mind
guess I'll work out the details
 
@Thorgott If I remember right the weakest possible condition is that X is T1
X_i Hausdorff should imply X Hausdorff, hence T1, but X_i T1 does not imply X T1
Some irritation like this
 
I believe the sequential colimit of Hausdorff spaces need not be Hausdorff
 
@Thorgott is it at least T1?
 
@robjohn I love your latest look! Prepping for Christmas!
 
10:36 PM
Hmm, idk. If I wanted to prove this, I'd take distinct points $x,y$ and, for large indices, taken a neighborhood $U$ of $x$ in $X_k$ not intersecting $y$, then I'd want to lift this to a neighborhood of $x$ in $X_{k+1}$ not containing $y$, then take colimit, but I don't see why such a lift would be possible in general.
 
 
1 hour later…
11:41 PM
Is there such thing as the vector space of ALL functions?! Derivative, is a linear transformation if the vector space is the set of all "polynomials", but it seems to be linear when it operates on anything else as well, because d(f+g) = d(f) + d(g), and d(cf) = cd(f). Now, what would be a more general vector space we could consider for the derivative as a transformation? Any ideas?
 
Space of $C^1$ functions ?
 
the natural candidate would be the vector space of all differentiable functions on some fixed space
 
@Thorgott Right, thanks.
 
@amWhy a safe Christmas
 
@Astyx Yes, I guess you meant just what Thorgott said. Thanks to you as well.
 
11:46 PM
@robjohn Indeed! And I like how you managed the santa cap to retain more or less of the mean-masked-capped-square!
 

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