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1:18 AM
Hi @Ted
 
0
Q: let $M$ be a hermitian matrix of order $n*n$ with rank k$ \neq n$

gaurav sainilet $M$ be a $n*n$ matrix of rank $k \neq n$ if $\lambda \neq 0$ is an eigenvalue of $M$ with corresponding unit column vector $u$. with $Mu=\lambda u$,then which of the following is\are true?. 1). $rank(M-\lambda uu^{*})=k-1$ 2). $rank(M-\lambda uu^{*})=k$ 3).$rank(M-\lambda uu^{*})...

I know that $u^*u=1$. Since eigenvectors are orthonormal set for a hermitian matrix.
I tried to solve using ranK$(M(I-\lambda uu^*))\leq \min\{rank(M),rank(I-\lambda uu^*)\}$
So, option 3 is obviously false
since Rank $M=k$
for $4$th option I use binomial expansion For, $n=1$ it is true, Assuming result true for $n=k$
using induction, I could solve it.
How do I check (1) and (2)?
 
 
2 hours later…
3:21 AM
@BalarkaSen yo yo yo
 
3:37 AM
Hot diggity dog, who wants to see some concrete geometric pictures.
 
Hi @anakhro
 
Hi.
What's up, homedog?
 
tryna figure out homotopy groups of spheres :3
 
Cute!
What book/notes?
 
3:52 AM
I was reading Sullivan's 1970 MIT notes a few days ago, then I glanced through Hatcher's notes on spectral sequences. Trying to put two and two togather and write down what I understand of Serre's theorem in the garbology room
Mostly reading more about localizations than homotopy groups of spheres, the latter just comes as an unexpected application of the former
 
Is that room an advanced room?
 
lol what does that mean
 
I peek in sometimes but I understand just shy of a set of measure zero.
 
well lots of people talking about different things i guess. the couple of us write down rants when they're reading something specialized
its certainly no Homotopy Theory
thats the insane chat
 
Ah, I see.
 
3:58 AM
you should rant in garbology about symplectic/contact geometry someday
its refreshing
 
Heh. I like to organize rants in the form of (more or less) nice pdfs which get stored into a folder never to see the light of day.
 
i am a lazy man. if i did that i'd never read nor write
though i did write some short exposition on the holonomic approximation theorem a few weeks ago that i like
texed up, as a pdf i mean
 
Well I think it might be similar to the outlet of your nice big answers you share that you've written up for people on stackexchange.
 
right but the point is i need some motivation beyond just writing for my better understanding to sit down and write
fake internet points is a big one
but also spamming a chat is a bigger one for me now
i need to trick myself to read, for which i have to write, but then i also have to trick myself to write, is the takeaway i suppose
 
Heh, fake internet points.
 
 
1 hour later…
5:23 AM
\o @CesarM welcome :-)
 
@skill y u demanding access to garb
 
just askin
nvm
 
granted
for old times sake
 
thanx
:-)
 
 
2 hours later…
7:01 AM
the chief executive is having a press conference (soon)
officially it began 10 minutes ago but she's still nowhere to be found
maybe this is a huge prank
 
link please
 
thnx
 
ok she's here now
she will speak in Chinese and then English
and then she will answer the questions
 
hmm, fashionably late
 
7:18 AM
she will "temporarily withhold" the bill
 
cool
 
Better than nothing.
 
at least she's willing to wait
 
English now
 
yup, good english too
Wow! how many injuries?
 
7:26 AM
"suspend"
 
they're willing to "listen"
no deadline
 
> 警察發射約150枚催淚彈和20發布袋鉛彈、數發橡膠子彈鎮壓[5],逾80人受傷,多名示威者頭部中槍,2人傷勢嚴重。其中一名香港電台外派司機懷疑因被催淚彈擊中,頭部受傷[13]。
反對逃犯條例修訂草案佔領行動是2019年6月11日深夜起,由香港市民自發之集會及佔領街道運動,主要地點在金鐘與中環,目的在於要求香港特區政府撤回修訂逃犯條例。凌晨時份,基督徒於香港立法會外通宵聚集詠唱聖詩。12日早上約8時,大量市民由原先於金鐘添馬公園一帶集會,衝出夏慤道與龍和道,以阻止香港立法會審議二讀草案。 警察發射約150枚催淚彈和20發布袋鉛彈、數發橡膠子彈鎮壓,逾80人受傷,多名示威者頭部中槍,2人傷勢嚴重。其中一名香港電台外派司機懷疑因被催淚彈擊中,頭部受傷。 在示威行動期間,香港立法會秘書處宣佈立法會主席決定取消6月12日的立法會大會(當日早上宣佈延遲,下午宣佈取消),延後審議條例修訂草案。6月13日凌晨,示威者散去。在衝突完結後,立法會秘書處亦先後宣布取消當日13日和翌日14日的立法會大會。警方在示威完結後,開始拘捕行動,多名示威者在醫院受拘捕。 == 背景 == === 雨傘革命 === 有評論者認為,2014年雨傘革命佔領街道之後,政府依然不接受示威者所訴求的事項,而令示威者再次採取佔領街道之方式,以令政府正視示威者訴求,而引發這次佔領街道事件。 === 逃犯條例修訂草案 === 經過2019年6月反對逃犯條例修訂草案遊行後,香港特別行政區政府仍發聲明表示不會撤回引渡條例修訂。因此,大量反對條例的香港市民紛紛決定於修訂條例二讀開始之日——6...
The police fired about 150 tear gas bombs and 20 bean bag bullets and several rubber bullets [5]. More than 80 people were injured. Many demonstrators were shot in the head and two were seriously injured. One of the Hong Kong radio station drivers was suspected of being hit by a tear gas and having a head injury [13].
@skillpatrol ^
 
thnx
nobody wants to ask a question in english?
:-/
 
:c
 
np
@LeakyNun does the wikipedia page have a translation button?
 
7:39 AM
you can just use google translate
 
right
 
thnx
bbc is asking a good question
good answer
 
Is this the corresponding page on English Wikipedia: 2019 Hong Kong anti-extradition bill protests?
 
pause and think
 
cdt
7:56 AM
hullo
 
Hey
 
Hi. How to get inverse of complex function? Suppose I know that some function has been applied to a domain in compex plane and they have been mapped to another region,say a region with circular boundary and the function is z^2 , how will I know about z? I don’t have any idea about compex function and calculus.
The question came to me as while proving an inequality and I thought it is neat way to solve, rather than resorting to mindless algebra.
 
@MartinSleziak no, the page I quoted is about June 12 Wednesday specifically
 
Anyone?
 
cdt
8:01 AM
It's well knonw that there's a universal holomorphic function $f$ such that for any other holomorphic function $h$ there's a sequence $\{ N_i \}$ of positive integer such that $f(z+N_k) - h(z)$ goes to zero unifromly on any compact subset of $\mathbb{C}$. Is this universal function useful anywhere ?
i.e is it used in proving some cool theorems
 
what is the universal holomorphic function?
 
cdt
i dont think a closed form exists
 
@LeakyNun You seem to be a helping dude, am I right? :D
 
no.
 
Okay. :D
@BalarkaSen ? [Sorry to violate the chat-room rules, I won’t do it again]
 
8:32 AM
@Ryan a guy commented with a simple example of a reflexive space in which existence holds and uniqueness fail ($\Bbb R^2$ with the $\|\cdot\|_\infty$ norm)
 
9:29 AM
number theory magic
Solution to $abc=a+b+c$
 
 
4 hours later…
1:11 PM
I can't see difference between two metrics mentioned here! Please help
 
@AlessandroCodenotti Yeah I see that. I don't know if I'm so much interested in uniqueness
 
@Silent Pick a homeomorphism from $(0,1)$ to $\mathbb{R}$, e.g. $f(x)=\frac{x}{\sqrt{1-x^2}}$ and set $\lvert x-y\rvert_f\colon=\lvert f(x)-f(y)\rvert$ for $x,y\in(0,1)$. Plugging in some values should make it clear that this is different from the euclidean metric.
 
 
1 hour later…
2:24 PM
@RyanUnger yeah I was hoping to get an answer concerning the minimal conditions part
 
@Thorgott I am really sorry, but still nothing clicks!
 
They induce the same topology, but they're different metrics
 
Thank you
@AlessandroCodenotti So, does this mean that while $1/n$ Cauchy sequence under Euclidean metric, it is not under that new metric?
 
2:43 PM
@AlessandroCodenotti but aren't finite-dimensional NVS automatically "everything" except for straight up Hilbert?
 
@Silent yes
@RyanUnger They are Banach for free, but not even necessarily strictly convex
 
Thank you very much!!
 
Boundedness is another property of the metric and not of the topology by the way
 
I keep flunking my probability paper. It's a 3 in 1 paper. That's possibly why. There's probability, numerical methods and transforms in it. Usually a 10 pointer question in either transforms or numerical methods. I'm advised to sit and solve 100-150 sample questions from the paper until it's all in my head.
Does this mean I'm dumb?
 
Is there even a notion of boundedness for topological spaces?
 
2:52 PM
In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. == Locally bounded function == A real-valued or complex-valued function f defined on some topological space X is called locally bounded if for any x0 in X there exists a neighborhood A of x0 such that f (A) is a bounded set, that is, for some number M>0 one has | f ( x ) |...
@Thorgott "complex-valued function $f$ defined on some topological space $X$ is called locally bounded if for any $x_0$ in $X$ there exists a neighborhood $A$ of $x_0$ such that $f (A)$ is a bounded set"
 
@Thorgott All I'm saying is that if $(X,d)$ is a metric space with $d$ bounded and $(X',d')$ is homemorphic to $(X,d)$ there's no reason to expect $d'$ to be bounded
 
I assume the space can also be bound.
 
(Unless $X$ is compact as well)
 
@AlessandroCodenotti right it's the strict convexity that's the issue here
 
2:56 PM
Well, the notion of a bounded function crucially relies on the metric in the codomain, but I understand what Alessandro meant now.
 
3:10 PM
@AlessandroCodenotti but that's for uniqueness, not existence
what's the minimal condition of P1?
 
I don't know, that's why I asked on MSE :P
The minimal condition for P1 is strictly stronger than strictly convex and (possibly not strictly) weaker than reflexive
 
right
 
 
2 hours later…
5:00 PM
@AlessandroCodenotti So, I guess even though we can define convergence over arbitrary topological space, we can't define Cauchy sequence over arbitrary topological space, right?
 
Thank you!!
 
Convergence of sequences in a generic topological space is wonky though. Not even uniqueness of limits is guaranteed
 
yeah! Thats why we need Hausdorff
 
Here's an interesting question: Suppose that $X$ is a topological space such that every convergent sequence in $X$ has a unique limit. Is $X$ Hausdorff?
 
5:06 PM
I would say so :)
Seriously, I don't know!
 
Think about it then
 
I could not resist googling and I myself had asked it here!
 
5:28 PM
Hey @AlessandroCodenotti
 
5:42 PM
Hi
 
You’re a room owner right?
What are your thoughts on integrating the ChatJax script into the sidebar, as suggested by Alexander at math.meta.stackexchange.com/questions/15041/…? That way users can simply click it to render MathJax.
 
6:05 PM
This happened when I tried, a moment ago
Pretty sure no javascript is allowed in the room description, nobody enjoys XSS injections on their website
 
6:18 PM
Was that adding a script directly or adding a link that executes said script, like the start Chatjax link on robjohn’s page?
 
The latter
 
I see, that’s unfortunate.
 
6:37 PM
@Alessandro: What if you do what Munkres calls $\bar S_\Omega$ but put in two different $\Omega$'s, like the line with two origins?
 
Sorry, what are you answering to?
 
LOL, a non-Hausdorff space where sequences have unique limits.
And hi.
 
Oh, I see! What's $\bar S_\Omega$?
And hi :P
 
If $\Omega$ is the first uncountable ordinal, then it's $[0,\Omega]$.
 
Oh, nice, that works
 
6:47 PM
$[0,\Omega)$ is sequentially compact but not compact.
 
Because no sequence converges to $\Omega$
 
Right.
 
I like this, it's kinda like a minimal example with a single point of non first countability
 
Right. Of course, I'm proposing to add a second such point :P
 
Sure
The cocountable topology on $\Bbb R$ needs no choice though
 
6:55 PM
LOL
shrugs
 
While it's consistent with $\mathsf{ZF}$ that $\Omega$ has countable cofinality
 
You're over my head again.
 
If $P$ is an ordered set, $D\subseteq P$ is called cofinal in $P$ if for every $p\in P$ there is $d\in D$ with $p\leq d$
The cofinality of $P$ is the least between the cardinalities of cofinal subsets of $P$
 
Is there a reason powers of $n$ are used in defining homogeneous functions instead of some other letter, and that in every example I've seen the homogeneity is of an integer degree? Is there a reason the degree can't be any real number?
 
You might have trouble taking $t^\alpha$ for all real numbers $t$ if $\alpha$ is an arbitrary real number.
 
7:02 PM
And in general you want to define homogeneous functions between vector spaces over arbitrary fields, so integer exponents are all you can ask for in that case
 
Hmm, true, but isn't homogeneity only required to be true for all positive $t$?
 
Not for me. I don't want $f(x) = \|x\|$ to be homogeneous of degree $1$.
Well, apparently, I used your definition in my book in the exercise on Euler's theorem. Oh well.
I don't see any problem with general $\alpha$ then.
 
Say one comes across an $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$ where multiplying $x$ and $y$ each by $t$ return $t^\pi M$ and $t^\pi N$, respectively.
 
@AlessandroCodenotti any movement?
 
Is this a homogeneous ODE?
 
7:05 PM
You don't mean multiplying by $t$.
 
@RyanUnger Nothing new on the functional analysis front
 
You mean composing with multiplication by $t$.
hi @Ryan
 
Err, fixed it :)
 
Hi ted
 
So what's our definition of a homogeneous ODE?
 
7:07 PM
Eh, it's still worded awkwardly, but yeah you got it.
I'm thinking of the definition where $M$ and $N$ are both homogeneous functions of the same degree.
Not the homogeneous linear definition.
 
If you're thinking of a definition I don't know, it isn't helpful.
You just told me that $M$ and $N$ were homogeneous of degree $\pi$, so why are you asking the question?
 
Because I was wondering if that case counts, since my resources use the variable $n$ and all examples use integers.
I was wondering if "homogeneous of degree $\pi$" is a valid concept in this context.
 
Yeah, there's no need to have integers.
 
Okay, that's what my hunch was
ty
 
Sure.
 
7:15 PM
howdy y’all
 
homotopy equivalence, vector field deliverance, spitting mad test geodesics, hitting orbits left and right we winning this. Flow so sick Ricci stopped flowing, Grothendieck returned to sanity, michael jordan came back to mathematics, canonical **** started happening
 
Howdy @Eric
 
@TedShifrin just graduated, it sucked
 
you graduated high school? @ÉricoMeloSilva
 
7:38 PM
no lol
well i did but 4 years ago
 
congratulations on graduating university then. have you been to the princeton libraries yet?
 
7:57 PM
nope
i was only there like a day so far and it was entirely in the university’s math building
 
8:25 PM
Where are the libraries
Do I need a card to get in
 
8:37 PM
@RyanUnger don’t all the buildings on campus low key look exactly the goddamn same
 
8:55 PM
How'd you you like the graduation downpour @ÉricoMeloSilva
 
9:25 PM
@Eric: Congratulations. Congratulations to Demonark, too.
It feels like just yesterday that you were a fledgling sophomore.
 
Congratulations!
 
Thanks!
And yeah it still feels very surreal
 
A protester posted a banner on the roof of a building, stood near the roof for 5 hours, took his own life, was sent to hospital, and passed away shortly afterwards.
 
@Daminark it was the best part
@TedShifrin thanks, it feels like this morning for me
 
Emergency poncho was useful after all
 
9:35 PM
@ÉricoMeloSilva the area is very depressing
 
@RyanUnger thats y u gotta leave as much as possible
@Daminark did you see on the convocation screen, they captioned someone at the end saying “should we just go?”
it was funny shit
 
The buildings are supposed to look nice
They look cheap as fuck to me
 
Oh that was so hilarious, they screwed up
 
The stone looks fake
I’m so depressed I didn’t get into Columbia
 
they rejected me 2 bud
 
9:39 PM
Columbia is for nerds tbh
 
Interesting. I never applied there, so I can't join your disgruntletude.
 
It’s the best university
 
Yeah they also told me to go away and I'll admit I probably would've considered there if I did get in. Some places I applied which I originally thought were good ended up having some problems but Columbia still seems fine
But since I'm not going there, it's for nerds
 
You would have considered it?
 
I mean like, probably taken it over Madison
 
9:44 PM
Why wouldn’t you automatically go there. Isn’t there a lot of ANT?
 
E.g. Harvard seems iffy to me now since 2 people I was interested in will retire and one I've heard a bad story about from a current student
Columbia has good number theory yeah, less topology but still it's overall an A+ choice
 
its also clearly in one of the best places
 
Simon Brendle is the greatest living mathematician
Hot take
 
no, the greatest living mathematician is @daminark
 
@ÉricoMeloSilva alcohol is so expensive here
 
9:54 PM
@ÉricoMeloSilva should we add Ryan to garb?
 
There’s no bars with any character
Garb?
 
idek what garb is
 
@ÉricoMeloSilva if mathematicians just listen to me they'll get so much done
 
@ÉricoMeloSilva garbological cohomology for derived nerds
 
E.g. Prove RH by computing the value of the zeta function for every point in the critical strip and checking which points are zeroes
 
9:55 PM
Why can’t you use the maximum principle on the norm squared
 
am i even in that?
 
you're an owner lol
 
@ÉricoMeloSilva who DID get into Columbia
 
@RyanUnger ik ppl who did
 
10:13 PM
@ÉricoMeloSilva are they super good
 
i mean they’re good, i wouldn’t say like, exceptionally better than everyone else i know, definitely very good though
 
tfw even Notre Dame is a nicer area than Princeton
I could have gone to Austin and done GMT until the end of my days
 
u chose this
 
Evening all
 
10:30 PM
Finished Better (that book on medicine I was reading)
Both of the books by Atul Gawande that I've read are really good
 
BTW, @Eric, I heard from Rafe today. He wanted my complex geometry notes and exercises :P
hi, DogAteMy
 
@ÉricoMeloSilva chose is a strong word
there was someone at the Austin visit who said Princeton was so awful he was probably going to go to Austin instead
and some other things
I'm not sure it's that bad
but it's not good
 
is there a python chat site ?
like if someone has questions about how to use pyhton or any other program?
@ÍgjøgnumMeg@TedShifrin Hi friends
 
Hi Jacksoja.
 
Hi @Jacksoja and @Ted :)
 
10:37 PM
Oops, hi @ÍgjøgnumMeg.
 
am trying to learn how to code during this summer break
thought i take break from math a bit and learn something useful
 
@Jacksoja lol nice
 
anyone has some idea about what to do ? to be a bit more educated? :D
haha thanks @ÍgjøgnumMeg
 
Get Maple or GAP or Sagemath or smth
lol
 
when i do math, no matter how much i learn, i feel i dont know anything
it is a very hard feeling
@ÍgjøgnumMeg what are those? :D
 
10:39 PM
I feel like I'm in a good place until I talk to some algebraists
 
computer algebra systems lol
 
neat ! but since i have zero coding experience, is it still good to start from there ? @ÍgjøgnumMeg
or are they kinda python based ?
 
Nah probably not, was a joke
Python is a good start because it is high levle
level*
 
that is kinda strange
 
meaning a lot of the language is "language" in a human sense
 
10:41 PM
because i heard it is the easiest one haha
@ÍgjøgnumMeg thanks ! and I hope you still have that wonderful beard
 
lmfao yes I do, I've been growing it longer
 
I did not think a human can have longer beard than that but if you say so
 
lol I'll do a masters thesis beard, where I shave it off at the start of my masters and then grow it until I submit
 
@ÍgjøgnumMeg that is a very good idea haha
good luck with your masters!
 
Thanks I'll need it
 

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