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12:02 AM
@EdwardEvans good luck
12:17 AM
How do I flag three posts of the identical question — allegedly by different people, but I don't believe it? What's worse, the offender first emailed me and then posted a question and never had the politeness to acknowledge my comment/correction (even in the subsequent post). This behavior should be censured.
If a mod sees my question, here are the links. 1 2 3
I posted to the Mods chat.
Hi. Could you please look at the question I just posted? Thank you
Q: Moment generating function of multivariate normal distribution using mgt of standard normal distribution

AgiThe problem I am looking at asks to evaluate the moment generating function of the multivariate normal distribution $N\sim(\mu,\Sigma)$ using moment generating function of the standard normal distribution. I have the followings in my notes: For $k$-dimensional $\mathbf{X}$ on $(Ω, F , P)$ with d...

12:33 AM
I'm not a probability expert, but can't you just apply $e^{u+v}=e^u e^v$ and separate the integrals?
@Agi Ted is correct
Score 1 for Ted.
@TedShifrin I know you're rusty on measure theory, but just thought I'd give this a shot: define the Borel sets in $\mathbb{R}$, $\mathcal{B}(\mathbb{R})$, as the $\sigma$-algebra generated by all open sets in $\mathbb{R}$. How would you show that it has cardinality equal to the continuum? I don't even know where to begin.
(not homework, just studying)
It's not just that I'm rusty. I don't like the stuff. But there are several people in here who can do this in a second (e.g., @Thorgott @Alessandro).
I'm just looking forward to finally doing some probability stuff... it has been a headache to learn about regularity of measure over the last few weeks
12:38 AM
@Clarinetist Do you know how to build it "layer by layer"?
@AlessandroCodenotti No idea what you're talking about.
Countable union of countable unions, etc.
uhh, transfinite induction?
So when you say "the $\sigma$-algebra generated by the open sets" you have the whole thing at once
@TedShifrin Even if I do that, I'm not sure about the $dF(x)$ part, since I think the variables are not considered independent from one another. I also don't get where should I use the mgt of std normal
12:39 AM
I looked up what transfinite induction is, but I wouldn't know where to apply it
@Agi They must be independent for that factorization to work with the MGFs
Why are the variables $(x_1,\dots,x_k)$ for $\Bbb R^k$ not independent?
But you can start with open sets as the first layer, take union of complements as the second layer, unions of complements of things in the second layer as the third one and so on
OK, Clarinet can help Agi and Alessandro can help Clarinet, and Ted — thankfully — can be lazy.
@Clarinetist so I can assume they are independent?
Then the proof boils down to showing that layer one has cardinality $\mathfrak c$, moving up from one layer to the next doesn't increase cardinality and the slightly harder part is that there are at most $\omega_1$ layers
12:40 AM
The only explicit description of the Borel-algebra I know is by transfinite recursion, so I'd try proving the cardinality doesn't exceed that of the continuum by transfinite induction - I haven't actually done this before, tho
(the proper name is the Borel hierarchy for what I'm talking about)
@Agi Yes, I believe it can be shown that product factorization on the MGF of the sum holds if and only if those variables are independent
(considerably harder is showing that, in this case, there are exactly $\omega_1$ distinct levels, but luckily that is not needed)
@AlessandroCodenotti Never heard of it until now, thanks. My professor had given it to us as an exercise, and I suspect he may throw it on our exam. As far as I can tell, no one in my chat group for my class knows where to begin.
@Clarinetist Oh yes for product factorization they should be independent but I am not sure if they are here since there is no statement saying they are independent.
12:43 AM
@Agi Actually, it does - it says in the problem that $X$ has independent components
@Clarinetist That was just what I found in my notes, the question is just to find the mgf of multivariate normal dist using standard normal mgf
that sounds like a rather hard exercise, unless you're doing something more set-theory oriented
@Thorgott Definitely not. That was brought up as an exercise in our first measure theory lecture.
@Agi The very definition of the multivariate normal distribution requires that each component of the vector can be written as a linear transformation of independent $N(0, 1)$ random variables
@Agi Let me give this some thought quick
I was wondering if someone can provide intuition as to the difference between $f(x+1)$ and $f(x)+1$? If you have a function $f(x)$ and it's inverse $f^{-1}(x)$ is it possible the inverse of $f(x+1)$ is $f^{-1}(x)+1$?
have you tried some examples
12:50 AM
@Thorgott I'm not sure where one would even begin by way of "examples"
that was to northerner
as for your exercise, I can not think of a way of tackling it that isn't built on using ordinals/transfinite induction or something along those lines
which is why that exercise strikes me as rather odd in a general measure theory course
@Clarinetist This gives some insights but I'm still confused on how to begin. courses.washington.edu/b533/lect4.pdf
@Agi Sorry, I realize now that question isn't as trivial as I thought it'd be. So first things first, you're probably going to be dealing with using the fact that multivariate normal distributions can be written as a linear transformation of standard-normal random vectors
@Clarinetist yes
@Clarinetist In the document I sent, would proof def3 to def2, in the second page, suffice?
@Agi Oh. I think this would be the method of proof: let $\mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$. Find its MGF. Then apply linear transformations on MGFs. Then you'll reach the result in that document.
@Agi Yeah, it is more or less that.
I haven't had to think about multivariate normal distributions from first principles in a while, thanks for asking that.
12:56 AM
@Clarinetist Sure thing, thank you too!
@Thorgott I think I'm just going to bet on the professor not asking that on the exam at this point. I agree it's silly to put as an exercise.
The solution I found deals with uncountable ordinals... I don't even know what that means
"Book". PDF
@Agi One of the things I've learned when it comes to multivariate normal distributions is that 95% of the time, direct integration isn't the way to do work with them.
Well, you actually can get a physical copy here if you want bookstore.ams.org/mbk-135
but it's mostly pictures
@Clarinetist indeed
1:55 AM
This is a question about transformation matrix. After watching 3blue1brown videos, I've assumed that a 2x2 transformation matrix has a 2 vectors—i hat and j hat. i hat lies on the x axis and j hat lies on the y axis. what about 3x3 matrix?
Order 2 branched cover of the trefoil knot:
do i hat lies on the x axis, j lies on the y axis and k hat lies on the z axis? Also what is the convention for this axises? Example: the x axis is pointing to my right, y as my up and z as my front. Is that correct or z and y are swap?
2:17 AM
hey chat
when it comes to matrix Lie groups, can you generally speak about the connectedness of $\mathrm{GL}_n(\mathbb{C})$?
> There seem to be two kinds of geometers: those who imagine that their
mathematical objects are something that could fi t on your desk, and those
who imagine their mathematical objects are room-sized, something you
could travel through.
I read a few questions about topological aspects of this group, but I'm not sure if they hold independently of the topology
- Sebastian Bozlee (though the idea might be older)
It's connected in the usual topology induced by $\mathbb C^{n\times n}$, and I believe it's connected in the Zariski topology, but you can certainly give it the discrete topology or something and make it disconnected
I don't know, because a Lie group requires that the group is a differentiable manifold
"Right hand rule" @Spade000
(I like to use movement - have z be my thumb, but curl the rest of my fingers together so they all start at x and end at y)
@LucasHenrique I think the topology is induced by the Lie algebra
Hm, maybe not
2:24 AM
I feel like this might be a subtle question
No wait, it's easy
No wait, it's not
I mean, different topologies give different tangent spaces and thus different Lie algebras, right?
If you have the Lie algebra you have the Lie group
But maybe you can have two Lie groups that are isomorphic as groups but not (topologically) homeomorphic
you mean different smooth structures?
it's a fact that if a topological group has a Lie group structure, that Lie group structure is unique up to diffeomorphism
but you might be able to equip a group with two different topological group structures, which in turn admit different Lie group structures
The question is about different topologies
discrete topology would give an easy counter-example if we don't require manifolds to be second-countable, but alas, we do
this is awkward
@AkivaWeinberger So z does seem pointing up instead of y. What about these 3 vectors i hat k hat and k hat? I mean not lie groups. In this rotation matricea wikimedia.org/api/rest_v1/media/math/render/svg/… do i hat represent the x axis, j hat as y and k hat as z?
2:29 AM
i is x, j is y, k is z
You can rotate your hand so that y is up
"x towards me, y up, z right" follows the right hand rule
So does "x right, y up, z away from me"
"x east, y up, z north"
Wait did I do that right
No I didn't
Swap x and z all of those
@AkivaWeinberger that confuses me as y is pointing left.
in the picture
If you rotate your hand it still follows the right hand rule
There are only two possible orientations for a coordinate system: right handed and left handed
You can't rotate your right hand to look like your left hand, so these really are different
All three of these are right handed (since they're rotated by 120 degree rotations about the line x=y=z)
There are 6 logical possibilities for permuting three objects
3 are right handed, 3 are left handed
the question reduces to: given a group, are there two topologies both turning the group into a topological group and into a topological manifold, yet such that the resulting structures are not homeomorphic
Q: Does a Lie group's group structure (not Lie group structure) determine its topology?

Akiva WeinbergerQuestion in title. Said another way, can you have two Lie groups that are isomorphic as groups but not homeomorphic? If so, the group isomorphism map will not be continuous (and thus not a Lie group isomorphism), and there will be no natural map between their tangent spaces (Lie algebras). I susp...

Asked main
It seems to me that there are no convention, as far as I know, where x, y and z is pointing in the real world. But I'll stick to the right hand rule. @AkivaWeinberger thanks for you accompany.
2:43 AM
I want a 2D video game that takes place in the universal cover of a disk minus the origin (perhaps stylized as a top-down view of a spiral staircase). The branch cut can be located opposite the player
Spiral staircases solve VR's locomotion problem, actually
the problem of walking in the virtual world while not crashing into walls in the real world
1 hour later…
1 hour later…
4:54 AM
Good video:
Klein bottle hidden in space of 3x3 high-contrast patches found in natural images
2 hours later…
7:09 AM
Can Jordan Form always diagonalize a matrix, assuming matrix is not diagonalizable?
if matrix not diagonalizable, nothing can diagonalize matrix
7:23 AM
@LeakyNun well then what is the benefit of Jordan matrix?
the powers of the Jordan blocks are easy to compute
So Jordan form handles the case where the geometric multiplicities is not equal to algebraic one? right?
I see. Thanks.
I thought for a moment that Jordan Form will do it for any square a matrix.
@LeakyNun do you know an example where the matrix is not diagonalizable at all?
you sound like you're treating putting a matrix into Jordan form as "diagonalization"
7:34 AM
@LeakyNun no but if I can compute the jordan form, I can come up with the actual matrix that carries out the diagonalization
S^-1 A S = J , where J is the Jordan block matrix. Then it is easy to compute S if it exist. This my understanding.
yeah but we don't call that diagonalization
@CroCo out of curiosity, what's your native language?
So I should treat Jordan without associated it with the notion of diagonalization, right?
For the same curiosity, why you're askin' about my native language?
My English is not perfect. Maybe you caught me with an accent.
well I'm just interested in languages
7:40 AM
@AkivaWeinberger cool vid!
hi chat
@LeakyNun I see. Good luck. I can help if you want
How would I write a measure as a "limit" of a sum of Dirac masses?
@LeakyNun are you aware of other applications for Jordan form other than computing the powers ?
It should be clear that discrete measures are dense in the space of all measures in eg $\Bbb R$, right?
You just look at the the CDF, break $\Bbb R$ into $1/N$-pieces, localize measure at the center of each interval, and done
This is in the weak topology
I guess this goes through for any Polish space for sure
7:57 AM
@BalarkaSen are you assuming any reglaurity?
I think it's true that the space of finite-support measures is dense in the space of all sigma-finite measures or something
for any Polish space
Hm but i am not sure what i really want
Nvm I see what I want
@AlessandroCodenotti ty
8:48 AM
Let D be a division ring and M_n(D) a matrix ring over D. Then M_n(D) is finite n^2 dim. It can be shown that it is D-Artinian, is this simply from the fact that M_n(D) is simple?
9:15 AM
@Hawk what is D-Artinian?
9:31 AM
@BalarkaSen you can even restrict to linear combinations of rational centered deltas
That's the usual argument to show separability in the proof that the space of Probability measures on a Polish space is Polish itself
10:18 AM
@LeakyNun i meant to write Artinian as a D-space
does that follow from the fact that Mn(D) is finite dimensional
wel that's what i want to know
10:30 AM
Since Mn(D) is simple, it has only two ideals, so no infinite descending chain of ideals, thus it is Artinian
Or am I missing something ?
what's a good reference for group cohomology?
@EdwardEvans cassels frohlich
11:08 AM
ken browns book is good for a first pass
Adem's book is good for a second pass
11:37 AM
[0, 2] equipped with the discrete metric is compact.
i think this is true but my brother says its false
can anyone tell me why
Every point is open
And there is an infinite number of points
but isn't the points
restricted to [0,2]
What do you mean ?
i mean by the defn of compact
i can find an open cover of [0,2]
and a subcover of [0,2]
so what does being open
have to do with anything here
Well the covering $\{x\}, x\in [0,2]$ is an open cover of [0,2]
Which has no finite subcover
11:46 AM
i see
so do u think this defn is not a good one to work with
since there might be counterexamples
i cant think of
What definition ?
the defn of compact being that E is compact iff every open cover of E has a finite sub cover
That's exactly how I proved [0,2] with the discrete topology is not compact
I don't understand what you mean
yes but before that
u said every point is open
and infinite number of points
did u use the same defn there
Yes, that's what lets me say {x}, x in [0,2] is an open covering
11:50 AM
Definition: a property is called "good" if there aren't counter-examples that Joseph Rock can't think of
thats a good defn
diu lei
thx astyx
hey astyx
Thanks @MikeMiller
@JosephRock ...
11:54 AM
when u think of these type of problems
do u visualize it
or how do u think of them starting
@LeakyNun lol
@Astyx but how do u visualize discrete things
it seems a little awkward
How do you think of $\{1,2,\dots,n\}$ ?
just like that it comes to you?
Anything with the discrete topology is exactly that, except you can get an infinite number of points
11:56 AM
@Leaky is that super offensive?
Think of them as points in a line, but that all separated by a minimal distance
gotta go bye
@EdwardEvans depends on context
I just googled it lol
@LeakyNun erm it was not directed at nun tho
more like an expression to myself lol
are you aware what it means
11:58 AM
erm sort of
@EdwardEvans i don't mind if someone I know say that to me
ah so it's like saying "f*ck u" to your buddy or smth
my friend told me it means like f*** this problem lol
"lei" means "you"
11:59 AM
hahaha maybe your friend is screwing with you
tryna get you in trouble
lol that i didnt know
@EdwardEvans ahh nth new..
alright, anyway,
12:18 PM
@LucasHenrique @Thorgott
A: Does a Lie group's group structure (not Lie group structure) determine its topology?

Qiaochu Yuan$\mathbb{R}^n$ and $\mathbb{R}^m$ are abstractly isomorphic (assuming the axiom of choice) for $n \neq m$ but not homeomorphic and so not isomorphic as topological groups. I think this might be the only thing that can go wrong, though; e.g. it seems plausible that for, say, compact semisimple Lie...

I'm struggling with a question regarding topology, specifically Tietze's extension theorem.

Suppose you have a finite collection of pairwise disjoint closed subsets ${A_i: i < n}$ of a compact Hausdorff space $X$. By Tietze's extension theorem and Urysohn's lemma, one could create a function $f:X \rightarrow \mathbb{R}$ where $f(A_i) = 0, f(A_j) = 1$ for any two $A_i, A_j$ in the finite set defined earlier.

My question is, how could one define a continuous function such that $f(A_i) = i$? I was thinking perhaps something like $f(X) = f_1(X)(f_2(X)(f_3(X) + 1) + 1)$ etc for $n$ many subset
12:51 PM
@Remana Given two disjoint closed sets $C_1$ and $C_2$ in a normal space $X$ there is a function $f:X\to\Bbb R$ which is contant $1$ on the first and constant $0$ on the second, do you know this fact?
1:17 PM
Can we conclude that as Hawaiian earring has no universal cover so that it's not semilocally simply connected?
But it's easier to check the definition of sl sc
1:35 PM
From the definition? Well, let U be a nbd of x where x is the origin. Then, there is a circle that completely contained in U. Then a loop that wrap around that circle once will give a nontrivial element so that the inclusion map will make a nontrivial element
Am I right? @MikeMiller
You need to justify why that element is homotopically nontrivial
@MikeMiller How can I? I don't know the fundamental group of Hawaiian earring..
So the Hawaiian earring is a really nasty space
But you know fundamental groups have induced maps
If you start with one of those circles (call it $\gamma$) you think of a way to cook up a map $r: H \to Y$, where $Y$ is a nice simple space, so that $r_(\gamma)$ is *clearly nontrivial?
What do you mean? I can't understand what you're talking
2:24 PM
I wanted to ask, How is modular arithmetic extended to complete real domain?
eg. $a=b\pmod c$ but $a$ being rational...
You look at the subgroup generated by c, and quotient R by that
a=b (mod c) iff a=b+kc for some integer k
Everything is "divisible" by everything $\neq 0$ in $\Bbb R$ so
congruence mod a real number is not very interesting
3:06 PM
@Astyx I did not understand...does that mean 6.7876 is congruent to 0.7876 mod 2?
But as Edward said, it's not really interresting
Yes perhaps it isn't. It's more like {x} fractional part of x which reduces mod 1
Note you also have 3 = 7.6 mod 2.3 for instance. c does not have to be an integer
3:22 PM
@Astyx 1.6? why 7.6?
because $3 + 2.3 \times 2 = 7.6$
3:35 PM
Everything is congruent to everything: $a \equiv b \bmod x$ for any real $x \neq 0$ because $a - b = \frac{(a-b)}{x}x$
It depends whether you impose the x in a=b + xc to be an integer or not
well then you're not talking about divisibility in $\Bbb R$
$\Bbb R/x = 0$ for any $x \in \Bbb R^\times$ because $\Bbb R$ is a field
The Mii channel theme is a miim
4:22 PM
^^^^^ @robjohn The Mean Pumpkin! ;D
Very Gourvid-19
@robjohn Hah! ;D
@Edward @Balarka Stare Into Death And Be Still by Ulcerate, a nice album I just discovered while looking for new music
what did you think of Death Atlas?
I was not too fond of the vocals but I liked it
4:33 PM
yeahhhh he does those weird goblin vocals
I think they're amazing hahaha
The curriculum for the algebraic number theory 2 course was just released
looks fun
Valuation theory, profinite groups/infinite Galois theory, group cohomology, cohomology of finite groups, Galois cohomology, class formations and local class field theory
He is good at what he does, but it's a matter of taste
@EdwardEvans profinite groups? Pfff that'll be so boring after the locally profinite ones
I mean
profinite groups are locally profinite
they're just compact locally profinite groups
Bit scared of the group cohomology tho
its just cohomology but with groups, ez
ty 1,0 inc
cohomology with $\Bbb Z$ coefficents is also cohomology with groups though
4:45 PM
just need to remember all the homological algebra I didn't learn this semester
Do a Grothendieck reading marathon before the class begins, easy
ty 1,0 inc
Edward is broken
@anakhro That's what algebra does to you
I just gotta remember what Ext is
4:48 PM
A lot of algebra hate on here lately.
Ext measures how inexact Hom is
yeah it'll be alright, I always get scared pre-semester start because I remember how much I don't know
Greetings, panic @Edward, demonic @Alessandro and @anakhro.
tfw still haven't learned derived functors
oh, and @Thor
4:51 PM
Salutations @TedShifrin
hi all
I'm once again disappointed by Germans supposed efficiency by the way @Edward @Thorgott
Why tho
I'm still waiting for the results of the corona test I did on Friday at the airport and which were supposed to be available "within 36 hours"
4:52 PM
Everyone is slammed with the virus.
All by myself ...
Salut, @Astyx!
hi Ted
@Alessandro the everlasting struggle between german efficiency and german bureaucracy
4:53 PM
Oh, and not that anyone cared about my multi-posting of the same question I bitched about yesterday, but several have disappeared but yet another has appeared. Apparently, they're getting closed because of my bitching.
What disappeared, Ted?
Ah bureaucracy is even worse. I had to go register my contract at the city council yesterday but I had to cancel the appointment since I'm supposed to stay at home while waiting for the test results. Apparently the city council doesn't have a single free time slot in the whole of November (and you can't book appointments in December yet)
I actually saw another one of them earlier
Some of the multiposts.
Yes, @Thor, a new one was added, with a very, very long name.
@Alessandro I've yet to even register with the Bürgeramt in Mannheim
4:55 PM
What were you posting about, Ted?
@EdwardEvans Let's all enjoy our illegal living then :P
I just love paying fines
I wasn't. I was complaining that four (now five) allegedly different people posted virtually the identical question within minutes. What makes me more furious is that the person FIRST emailed me, I replied with a correction/hint, he didn't reply, and then posted and ignored the correction/hint when I posted it there.
covers eyes so as not to see illegal living
I'm British, I get refugee status in Germany
Well, the US is now soon to be back in the 18th century. Who will adopt me?
4:57 PM
emailed you??
Yes, @Astyx. I do have my email in my profile.
But why you specifically ?
I do not understand.
Are students cheating?
Well, certainly one is.
Maybe it's a whole pile of people from the same class posting virtually identical questions. Maybe it's one person with different IDs on here. I have no idea.
We did a test recently and within an hour all the questions were posted to Chegg...
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