01:00 - 23:0023:00 - 00:00

1:33 AM
I now work for a multinational with a large presence in the US. As a consequence I am obliged to learn how to respond in the event of an active shooter event

how does one?

Lovely.

how do you prove a linear transformation of size $3$ has an eigenvector without using almost anything
including determinants
real*

One runs, if one cant run, one hides, if there is nowhere to hide, one must accept that their life is in immenent danger and do what they can to fight back
The video was actually chilling in how matter of fact it was

1:49 AM
They played me a video on lab safety when I started using the lab in high school
it was very chilling

@Yorch without determinants, how do you define eigenvalues?
Rephrasing: Do you know or not what a characteristic polynomial is?

A number which has an eigenvector
I don't know what a characteristic polynomial is

If we don't have the char polynomial (which Axler defines without determinants), I claim it’s impossible.

I know the usual $\det(A-\lambda I)$ definition but I don't know if there is a more universal characterization

Huh?

1:53 AM
I don't remember Axler's definition

Oh ok
it's like similar to the frobenius form

No clue what that means.

In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that...

Everyone else says RCF.
NO, much more basic.

1:57 AM
what is RCF?

dope

Look at the ideal in the polynomial ring of all polynomials annihilating the linear transformation.
That’s what Axler does. Perfectly reasonable for an advanced student.

thats just the multiples of the minimal polynomial right?

algebra hard
@TedShifrin hm?

2:13 AM
the determinant is op wtf

don't use determinants

ok

you have to use determinants for change of variables etc

I'm going to watch a Wildberger video now

good stuff

2:18 AM
and then some Terry Davis

2:46 AM
i got banned for two days
because sometimes the truth is hard to digest

2:58 AM
that's a brave move

it did seem more space efficient for a while.

you know, with you knowing their IP and such

well, it does not help when you suggest one liners that would flatten someone's machine. that is pure rude.

like sudo rm -r /bin ?
oh u need an f
sudo rm -rf /bin
I haven't tried it though

@Yorch not that, but similar, some silly one on windoze.

3:02 AM
alt f4 will cripple ur hardware

i am ok with a bit of fun, but that is not nice

i'm mostly linux based.

same, but with carbon

we used to do stuff to friends (after we had made a backup) like putting a file named * in their home dir.

3:05 AM
one of my friends also did a tun of stuff to my system
thankfully it was almost new so I just reinstalled it

decades ago there was very little security.

I remember something we would do to our colleagues iphones was put it in chines
invert the colors
and then add double tap for zoom and triple tap for tts or something

one company i worked for had all hr dbs available without security

was there juicy stuff on them?

salaries, etc

3:07 AM
oh
no sexaholic records

in an earlier life i used to connect my terminal lines to one nearby and watch what people typed. early keystroke logger

they call that ethical hacking now

probably had those sorts of things, we had a few domestic violence cases, etc
i wasn't really trying to hack anything, i was just curious if it worked
because i had a useful purpose to it

it never occurred to me that i would see the plaintext passwords. obvious of course, but that was not why i was doin git
i wanted to connect a single board computer to the mainframe so i could do some data logging for experiments i was running. this was before such things were done.

3:11 AM
that sounds exciting

i worked spectacularly, i should have made a company out of the idea...
it was fun.

I only used windows until high school
I thought being good at computers was turning the computer off with the windows shortcuts

this was around 1980
i liked hardware, in fact more so in some way. but software pays the bills.
my first language was basic :-)

My first language was also basic
because we used it along with a parallel port for cheap robotics stuff

it was pretty cool. i used to to plot things like $(f(x+h)-f(x))/h$ to see how close it came to $f'(x)$, nerdy stuff like that
i love the hardware interface stuff.

3:14 AM
I never learned it, I just sort of had someone over my shoulder who spoonfed me

its amazing, an arduino has more clout than our first computer.

yeah I think everyone uses arduinos for that now

we experimented and i think got our hands on a byte magazine.
they had programs (mostly games) that you could type in
my brother loved that stuff, i was more nerdy and wanted to do little experiments
the idea that you had something that could react in a microsecond or so was fascinating to me
well, a little longer, that was the clock period

how did you learn it?
with the byte magazine?
and experimenting?

i was studying elec eng, so had an idea of digital electronics, but the basic was self taughtt
when you are trying to figure stuff out even little hints are very helpful. so i would see stuff used in byte and try i out. things like peek & poke, that sort of stuff
when we found out that the screen was memory mapped that was a big deal

3:18 AM
I hadn't even heard of that

we did not have an assembler unfortunately
but i figured out a little machine language

sometimes I analyze the executable of my c++ code

silly things like measuring reflect response :-)
you mean you look at the assembler?

yeah although I don't remember how I did it

you can do something like -s, i have forgotten, if you are using gcc/c++
but reading assembler for risc based machines is tough

3:21 AM
oh yeah

its tough enough for 'normal' processors.

risc is harder than cisc?

i like real time stuff. but have no use for it in my day job
harder because of pipelining
so with cisc you write an instruction and next cycle its done
with risc its spread out over a few cycles
so when you read the code you have to remember more stuff
a little ironic
i only look at assembler when i have some unusual issue or oi i need speed.

I watched this interview the other day
It was pretty good
this one was also pretty good

smart fellow

3:25 AM
both?

keller

I've got to cook some asparagus now

enjoy :-) speculative execution is interesting. i used to have a company that did formal verification.

3:40 AM
Hehe
We can make a religion out of this

2 hours later…
5:30 AM
It got pretty late here last time.. about $:30 in the morning...had to catch up on some sleep Last time I guess things were really unclear for mostly my fault. I will try to sort of clear things up. Let T:\R^n \rightarrow \R^n be a function(non-linear). ONe thing for sure is I dont know whether lim _{\theta \tends \infty} T(\theta) is bounded or not. Given this I want to assume \lim_sup_{\theta}\inf_{x \in S}\|Ax - T(\theta)\|_2 where A is \R^{n \times d} and S is comapact subset of \R^d is finite To support my assumption I can think of two choices @rostader , do you know any problem generator that can ask proof-writing questions?? I know Wolfram Alpha has a Problem Generator but that's not what I need. 5:48 AM Either \|T(\theta)\| = \exp(-theta) so that lim\|T(\theta)\| remains bounded. Or as I was trying yesterday assume, \sup_\theta\inf{x \in \R^d}\|Ax - T(\theta)\|_2 is bounded. And then claim that \sup_\theta\inf{x \in S}\|Ax - T(\theta)\|_2 is bounded for any compact set S @Spectre Sorry I dont know what you mean... @rostader I mean, I would like to get some random proof-writing question generator I am looking forward to write the PRMO and hence I need to prep myself in number theory so as to write the proof questions/numerical answer questions that may come up in later stages. If you have no idea regarding such a thing, no problems, sir :) It was out of a longing to get a hand in number theory that I asked you so. Sorry.. I dont have any idea regarding it @rostader Ok thank you 6:36 AM The only reason for time is so that everything doesn’t happen at once - a famous guy with uncombed hair 1 hour later… 8:01 AM @rostader I bet you don't have ChatJax installed (from the half-latex in your comments). The link is in the info for the room (at the upper right of the page in the desktop version or in the page info for the mobile browser). 8:22 AM @rostader: That orange like fruit with the size of a volleyball is called Jambura. I just found out. Pomelo is probably an another name for it. Though, I'm not sure about whether they are exactly same or not. 8:44 AM Pomelo is sometimes called jambola. Looking up Jambura leads to Pomelo. So they are probably all the same. 9:44 AM Hello! It is well-known that the index of a nilpotent N-by-N matrix is at most N. If you would write that and add a citation, what would you cite? I have seen some "Anti-Pi Rant" videos where they say that Pi being a boring number. There argument is that "Sure pi is an irrational number , but so is root 2 , root 3 , the golden ratio phi etc".My confusion is that , isn't pi also a transcendental number too ? I heard that to this day , very few numbers were proven to be transcendental. Isn't virtually every real number transcendental though? Yes .... but proving that a particular one is a transcendental number is hard. hi I was reading about orthogonal functions in some cases, they use a weight function w(x) inside the integral..and in some cases they dont.. just what difference does using a weight function make? 10:28 AM This is very hard math question. How to be millionaire? Own at least a million of monies That was a trivial question But if you want more practical advice, exchange your money to Iranian rials. At the moment you can get 420100 Iranian rials with 10$USD

(every $60$ second in Africa a minute passes)

My pleasure!

2 hours later…
12:44 PM
i was supposed to write the fourier series for f(x)=|x| in (-pi,pi) and f(x+2pi)=f(x)
i got the cos and sin coefficients right
however, i get the constant term as pi
but the given answer has pi/2
and if I look at the final fourier series I got, substituting x=pi/2 gets the constant term
and f(pi/2) is pi/2
so that suggests the constant term should be pi/2
but calculating it from $1/ \pi \int_{-\pi}^{\pi} |x| dx$ gets $\pi$...whats going wrong?

1:11 PM
nevermind, i got my mistake

1:35 PM
hey chat
proof verification: given $H: X \times I \to Y$ continuous, where $I = [0,1]$, every $\gamma_{t_0} := (x \mapsto H(x, t_0))$ is continuous
proof: given $U_Y \in \tau_Y: \gamma_{t_0}^{-1}(U_Y) = \{x \in X: H(x, t_0) \in U_Y\} = \pi_X ( H^{-1}(U_Y)\cap \pi_I^{-1}(t_0) ) = \pi_X(H^{-1}(U_Y)) \cap \underbrace{\pi_X(\pi_I^{-1}(t_0))}_{= X} = \pi_X(H^{-1}(U_Y))$ which is open since projections are open mappings.

1 hour later…
2:42 PM
@LucasHenrique it might be easier to show that $x \mapsto (x,t_0)$ is continuous and then use continuity of compositions?
more general at least, if not more straightforward

2:53 PM
@copper.hat oh god
...

Does anyone know how are [0,1] and the real line isomorphic as measure spaces?

3:10 PM
With the Borel sigma algebra? All uncountable standard Borel spaces are

yeah, but doesn't saying that R is a standard Borel space presupposes that it is isomorphic to [0,1] as a measure space?

Standard Borel to me just means that it is a space with a sigma-algebra that is the Borel algebra of a Polish topology

oh, I was unaware of this definition...
I guess that makes sense if a Polish topology is a complete metric space
thanks!

Complete and separable, yes
but in this specific case it should be easier than invoking this general result: obviously [0,1] embeds into R, and R embeds into [0,1], and for standard Borel spaces this is enough to have an isomorphism
”Cantor-Schroeder-Bernstein holds for standard Borel spaces”
Ok this is pretty much the same as proving the big result I mentioned earlier though so I’m not sure what’s my point
anyway all of this can be found in Kechris Classical Descriptive Set Theory if you want to see the details

XP ah, but that makes sense, I really just needed a measurable function whose inverse is measurable which turns out to be the obvious one (I was just hung up on the fact that [0,1] was compact while R was not)
thanks for the reference, btw
I never actually learnt set theory properly ':)

3:23 PM
I think the hairy ball theorem can be used for the "matrices of dim 3 have eigenvectors" problem

3:34 PM
3

This is a (very) partial answer: Suppose the polynomial is real and of odd degree. Then this polynomial is the characteristic polynomial of some matrix acting in an odd number of dimensions. Restricting this matrix to the sphere and then projecting onto the tangent space of the sphere defines a v...

4:00 PM
@LucasHenrique ?

4:16 PM
@Yorch That certainly uses the characteristic polynomial.
@LucasHenrique Restrictions of continuous are always continuous, because subspace topology.

in what part?

To turn a polynomial into a linear map.

you mean in the proof of the hairy ball theorem?
or in the post?

No, in the thing you linked.

oh
But that part isn't important
that's just an extra step

4:25 PM
Oh, OK. So we need Stokes’ Theorem and/or serious algebraic topology!

yeah, haha it isn't what I had initially hoped for
although it does sort of provide some goemetric intuition for me at least
although I guess that intuition could have been obtained by just thinking of orthogonal matrices

I feel like that is trying to explain an easier theorem with a harder one

no doubt
I still think it's interesting though.

hi everyone
i had a couple of fourier series questios
is there a function whose Fourier series is sin(x) + sin(2x) + sin(3x)..... ?
it "seems" like no, bcs it appears that this series diverges but I dont know how to prove it

4:41 PM
seems very plausible that there is, but I don't know

@Thorgott i was happy with the intermediate value theorem.
Yeah, that’ll be a distribution but not a function. Page @leslie.

it diverges when $x \notin \pi \mathbb{Z}$. math.stackexchange.com/a/2047327/27978

@copper.hat right
proof?
or maybe just a small hint

But $\sum e^{inx} = \delta$.

4:55 PM

it being divergent does not stop it from being the Fourier series of a function

why do I feel like...
oh yeah
I know this
in the geometric series, put x=e^{i theta}
and check the imaginary part of the corresponding series

@Buraian thats what i did actually

ye that's right
have you heard of book called visual complex analysis?
it is discussed in there like this exact example

i trued to derive sum of sin(x)+sin(x+d)..... usig that and it worked
using that

4:56 PM
yah

but i was wondering if there was a better method

oh btw

@satan29 are you familiar with distributions?

not really , no

once you had that expresison, you can actually evaluate some nasty integrals. Multiply both side of the expression with sin(nx) and integrate from -pi to pi. You will see all terms die off in the right side except one, and left side you have a really ugly integral

5:17 PM
thanks

5:28 PM
hi @Alessandro

Hi @Balarka

If $(0, 1)$ is a solution for $y = \pm \sqrt{e^{-x^2}}$, the author infers it implies $y = \sqrt{e^{-x^2}}$, i.e., without $\pm$. Does anyone have a clue why? Of course, it is obviously true if $x=0$ given our initial supposition, but how do we know this holds for other values of $x$?

oh look who's still alive

the specter of socialism haunting europe?

@shintuku lmaoooo
@Thorgott actually its a miracle i am
terrible stuff in this part of the world

5:32 PM

i got vaxxed
chip-ed in, all matrix now

hmm, microsoft in health care

well bill doesnt want to distribute chips to us
its probably because we run pirated windows in this part of the world

@BalarkaSen how does it feel to live inside a cohesive infty-topos

i can't believe one has to pay for windoze pos

5:40 PM
@Thorgott i actually have been thinking about $A_\infty$-algebras a lot

that's, uh, cool?

@Thorgott Are you an expert in characteristic classes yet

I have a quick group theory question. If a group G acts on a set S, I understand that an orbit is a minimal set of elements in S that G sends into itself. Is there a name for a maximal set that G always sends outside of S?

$S$ minus an orbit
complement of an orbit

I'm not sure. The complement of an orbit contains other orbits. Thus, that complement has some elements that G does not send outside of the complement.

5:49 PM
oh

@CommonerG a transversal of the orbit equivalence relation

Ok! Let google that a bit.

hm not quite, that works if there are no fixed points

you want a maximal set such that $GT\cap T = \varnothing$ ?
oh but $G$ has $e$

Oh. ok. Maybe I need to rethink what I am after. Since G contains e (identity) this set doesn't exist.
I should really turn on mathjax.

5:59 PM
@BalarkaSen excellent!

Hi @Ted!

nah, I'm not doing anything related to characteristic classes this semester, I still know almost nothing

Well I should say I only got my first dose in. Second dose in 12 weeks.

maybe I will compute some when I get to attempt working on my thesis

@BalarkaSen astrazeneca?

6:02 PM
yeah
astrazeneca you meant

I see, I got the first dose of pfizer and I only need to wait 4-6 weeks

aha

Here's what I'm thinking. Suppose we have C2 group acting on a set of 4 points $\{1,2,3,4\}$ with the orbits $\{1,3\},\{2,4\}$ what I want is a name for sets like $\{1,2\},\{3,4\}$. or $\{1,4\},\{2,3\}$.

6:30 PM
@AlessandroCodenotti Odd. Here Pfizer was a 3- week wait.

@CommonerG those are transversal of the orbit equivalence relation
you're picking one representative from each orbit

If $k(u)$ is non-negative and $\int k(u)du=1$, does it hold that $\int |k^2(u)u|du \le \int k(u)|u|du$?

@copper.hat your comment made me realize it's trivial, and I was strugling a while with something I shouldn't

@LucasHenrique the nature of mathematics is that many things go from from "don't know what to do" to "obvious" (clearly a bad characterisation) with little transition.

Perfect. Thank you so much @AlessandroCodenotti

6:36 PM
@schn Why do you think it should?

@TedShifrin Not sure really, but its related to an exercise about kernel density estimation. $k(u)$ is a kernel density estimator. There are more conditions on $k(u)$ that I left out.

Well, as stated it’s definitely false.

Can you show that it's false? :)

Take $k=0$ except on $[3/2,2], where it’s$2$. 6:57 PM @TedShifrin Thank you! If I add the condition that$u\in[-1,1]$, does that change anything? Yes, of course. Think. Oh, I was thinking of$k$, not$u$. Still false. Doesn’t my example still work if you shift it over? Yes, your example still works for say$[1/2,1]$, thanks for the help. Sure. 2 hours later… 8:43 PM @schn Let$k_\alpha(x)=\frac{[0\le x\le \alpha]}\alpha$, then$k_\alpha^2(x)=\frac1\alpha k_\alpha(x)$, so you can adjust the ratio of the two integrals to be anything you wish. 9:18 PM Approach0 seems broken. Everything I've given it recently has returned an error. What's dat? @copper.hat That is true. When I see people ask what someone has done on something when they say they have no idea what to do, you know that even a small nudge in the right direction will help. However, if you give a hint, people complain that hints are not good. @TedShifrin You've never used Approach0? it is a good, TeX-friendly search engine. I suspect the takeaway is that some people will always complain :-). I do nothing but complain. Interesting, @robjohn. Never hoyd of it. It's one of the things that people suggest to look for duplicates 9:26 PM Oh, I confess I only look for duplicates when it's a question I've answered. And I'm still going to give hints and put leading questions. The hell with everyone else. Oh, it's working now, @robjohn. Very cool. 9:42 PM I am trying to find$\sum_{k=1}^n(-1)^{k-1}\binom{n}{k}\frac1k$and it returns an error. I know that I have proved that this is$H_n$somewhere, and I am trying to find it. I just provided a new proof for a recent question, but I wanted to refer to the previous one Well, perhaps you reordered @TedShifrin I have tried all different ways to put it together. Approach0 usually does try various combos it's very good at that stuff I can usually find things in my own answers pretty easily. You have thousands more than I, however. 9:53 PM I need an example of a terrible topological space for a talk. The best topological spaces are manifolds. The antithesis is a Cantor set but I would not call it a terrible space. What's a good space to dunk on? Double cone on the hawaiian earring. Clearly it should not be semilocally 1-connected. I was thinking of the Hawaiian earring actually. Good call Glad to cooperate for once. I'm going to give a series of talks on stratified Morse theory :) I'm preparing lecture 1, and I start by explaining why stratified spaces 10:10 PM the antithesis of manifolds are indiscrete spaces That's not a topological space Finite CW structure? Versus not. I guess the topology on the hawaiian earring makes it non-CW. There are too many bad spaces to put qualifiers and explain in how many ways manifolds or spaces with controlled singularities are better Hawaiian is quite pleasing though. Never liked that space. Shouldn't offend too many people You’d better talk about my favorite example. Whitney umbrella? 10:19 PM Bingo And its different tangent bundles :) Absolutely. Crucial, because I will sketch a proof that semianalytic varieties are Whintey stratified. Whitney umbrella is the first nontrivial example which shows the singularity stratification is not Whitney regular If you do Macpherson Chern classes, let me know. I actually want to pick up some of that as I give these talks. Long way from now though. hey chat I was talking to a professor. I need to find a research area I like... someday, sometime... topics in algebraic topology are really interesting and I liked algebra. I thought of something related to homological algebra or something but I'm really not sure 10:29 PM @BalarkaSen it's a great space, you're a nerd The Whitney umbrella wins. Even motivates the notion of embedded components in schemes. (Not that I remember details.) the path of research is distressing, I'm sure I'm not mathematically qualified to choose what I'll be doing - probably - until I die everything is too hard to know with deepness and the time is too short. there are technical matters like good subject but poor advisers in the area really distressing... Lucas, you don't have to settle so early I'd really appreciate any advice from how you guys chose your own areas while undergrads @Ted @Thorgott Speaking of Chern classes, I realized a good way to think about them is as intersection of homotopies (homotopes between homotopies, etc) of sections of a given bundle with the zero section. But then I found out Danny Calegari already knows this 10:39 PM I haven't chosen anything yet what you do during undergrad probably has less impact on what comes after than you're imagining @BalarkaSen hmm, intersection of generic section and zero section is dual to the Euler class, which is the top Chern class, and you're saying something inductively based on the inductive construction of the Chern class or something? That sounds like the right idea to me. Here is the penultimate Chern class for a complex vector bundle$E \to B$; take any section$s : B \to E$, multiply it with the fiberwise$S^1$-action in virtue of the complex structure to get a family of sections$B \times S^1 \to E$which can be filled to a disk of sections$B \times D^2 \to E$because the space of sections is contractible. Intersect this with the zero section$E_0 \subset E$and this is the representative for$c_{n-1}$,$n = \mathrm{rk}(E)$. I mean, if you just scalar multiply fiberwise, isn't$B\times D^2\rightarrow E$just given by scalar multiplication fiberwise as well? No reason that will be transverse to$E_0$You can choose a filling which is transverse to$E_0$, thanks to Thom. 10:56 PM oh yeah, right, you want something transverse this is cool, got a reference? Nope! It's relatively easy to check by naturality; I believe the inductive Gysin construction is related but have not checked. For Stiefel-Whitney classes you can do the same but with$S^0\$-action. Which screams Wu's formula, because you're saying SW classes = homotopy-coherent self-intersection number of 0-section. Remember cup product does not commute, it commutes upto homotopy, and the failure is recorded by Steenrod squares.

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