00:00 - 07:0007:00 - 00:00

12:00 AM
and he calls that Levy's inequality

1 hour later…
1:25 AM
Ted...I have two quick questions about manifolds w.r.t sec 4.5...
and the intersection of the cylinder and sphere you did in lecture.

here today, gone tomorrow.

We reduced the derivative matrix to:

$$Df(\mathbb{x}) = left[ \begin{array}\ x & y & z \\ 0 & -a & -z \\ \end{array} \right]$$

1) is $a$ being treated as a constant when we finish out reduction of the matrix
2) In this example you were claiming that the manifold was a one dimensional one, which means the rank of our matrix would have to be 2. What happens if we got a rank deficient matrix?.....would we be able to claim that instead we have a 2 dimensional manifold?....since we would have 2 degrees of freedom
lol.....I'm trying to make my matrix "appear.....damn it.....l

if a isn't being treated as a constant, i'd like a word with ted about his use of letters from the beginning of the alphabet.

$a$ is a constant....and it just clicked what would happen with it in this scenario actually
I'm getting mixed up between the $x,y,z$ and the $a$(and whatever else in my REF matrix)

No, it’s generically a 1-manifold but we expect singular points. In lecture I showed pictures, no?

1:36 AM
"We are not Dr Frankenstein; we are the monster. We have woken up in a laboratory and are trying to understand how it made us." - Luke A. Barnes, discussing cosmology

Yes. I saw the pictures. So how would we end up determining what dimension manifold we have in a more general situation?......... determine the free variables and see if they satisfy our conditions?

It’s not known to be a manifold — and almost always is not — when the rank condition fails. More in chapter 6.

ted you should put some of this behind a paywall. monetize it somehow.

You mean you want a cut?

I paid a hell of a lot in tuition over the years and this man's lectures have saved my life.....I can die peacefully because I finally have a picture of what manifolds are...
well he is a lawyer so.............

1:42 AM
all of this can be monetized on the lesliecoin platform. it will liberate everyone from everything.

Not quite everyone.

you get more liberation when you put more money (which is an illusion) into lesliecoin (which isn't).

what about the notion of "charts" when I first attempted this course before I was ready the idea was brought up with manifolds....I never fully got it. Or is this a "stay tuned to ch.6" thing as well?

Not charts, but the inverse maps, parametrizations, yes, in 6 and heavily used in 8.
Manifolds as abstract top spaces with charts belong mostly in grad school.
@leslie It’s only your liberation, of course.

my school has me trying to sprint before we could even crawl.....I remember almost nobody really grasped the chart idea....besides the bare skeleton for the inverse mapping theorem...

1:51 AM
Professional mathematicians like to think all their students are little versions of them.
Granted, this is the European system, but they track insanely from a young age.

Yes I was really getting that feeling in a few courses....U of T has an elitist "arrogance" to them....so I do get that the stream and courses are for math specialist, but there was really no breathing space in terms of allowing us to be able to actually digest these in depth notions. We were using Spivak's Manifolds with Rudin's Analysis on Manifolds as a complimentary text....

here undergrad diff geo uses manifolds from the start and then jumps right into (modern-style) Riemannian geometry

this was 2nd year "advanced" analysis mind you. Only previous work done was Spivak's Calculus and Insel, Friedman's Linear ALgebra

@dc3rd wrong titles, but a shit show
My book is the sane answer, but math profs are insane.

This whole time I had your text just sitting on my shelf too....a friend of mine that was also tutoring me gave it to me years ago.....if only I knew....

2:02 AM
Before I brainwashed you.

probably wasn't mathematically mature enough at the time, but it is the whole adventure so I appreciate it.

You say brainwashed.....I say instilled much needed "clarity"

I never wanted to make money; I just wanted to improve math learning.

if you're saying this I take it that there tends to be a bit of solely profit motivated textbooks done then?
I guess that's why you don't have 10 different "editions" with a few shifted sentences here and there and some new "pretty" pictures

Yes, most authors cave to popularity.
I never did that with any of mine
I wasn’t going to make tens or hundreds of thousands, let alone millions, like the big-selling authors.

well.... if those are the figures at play if you're a "big player" in the game, I could see how one's original intentions could end up getting clouded, but these are not simple minded individuals. They're highly intelligent and usually apply strict logic..........but also human too...shrug

2:13 AM
I think most math books I read are so specialized that the authors don't make a lot of money from them

I'm guessing that is the case once you move upwards in sophistication of the math....

2:38 AM
There is little money in text books unless you are talking volume.

3:03 AM
it's like four or five people that have the calculus books everyone has to buy in freshman year. or get written into state K-12 standards.
nothing else does any volume.
the flip side of this is that if you know someone who authored a good book they will probably just give you a copy if you ask them where you can get it.

see, socializing book publishing wouldn't change anything!

no, most good authors are committed to the cause already.
i have a 100% success rate in asking people where i can find copies of their stuff. they respond with a copy. every time.

most people in academia i've encountered don't seem to do it for the money
always seems like there's a more lucrative opportunity somewhere else

if they're doing it for the money they need their heads checked.
one time at iowa i was doing something in a seminar which drew on a paper, i emailed the author, saying hey we're doing this seminar and want to work though X argument. he responded with a copy of the paper, his book that refined upon the paper, and his notes from the paper.
every academic i have worked with was like that.

very cool

3:14 AM
and yes, there is always more money somewhere else. i think this has something to do with the mediocre quality of university instruction. the career path seems to select for a small number of 'true believers' who are zealots for the cause, and a larger number of people who are financially comfortable enough to do it even if they would be better off doing anything else.
many of them are children of professors. there's an academic class of people. end conspiracy rant.

i imagine it is super helpful to have someone who knows the ins and outs of university life before beginning your undergrad

probably!
someone gave my daughter a fake plastic cockroach and a fake plastic centipede as part of a birthday celebration (the birthday kid gives their classmates gifts, it's some weird tradition). we have a toy cockroach at home now.

3:30 AM
someone is wishing there were more entomologists in the world
cockroaches have the cool property of being able to sustain massive amount of nuclear radiation without dying
2

i like cockroaches, but not in my house.
we do have them around here. the cat found one a few months ago.

3:49 AM
I have the following echelon form:
$\begin{bmatrix} 1 & 0 & 0 & a - b + c \\ 0 & 1 & 0 & \frac{b-c}{2} \\ 0 & 0 & 1 & \frac{d-a}{2} \\ 0 & 0 & 0 & \frac{5b-d-3a-3c}{2} \end{bmatrix}$
is there any way to tell if the above system will be inconsistent without manually deriving a contradiction?

-2

In the first question I am certainly not able to rewrite the values one by one. I tried a lot but I am falling free after an iteration or so. I intend to get a step by step explanation for this. Secondly, In the next question(sorry for the blurry image. I intend to keep it as clear as possible. ...

so this is an augmented matrix? @shin

yep

In the second problem in the given picture how do I get myself assured of the fact that this is not a perpetual loop

I know that if $\frac{5b-d-3a-3c}{2}$ is a nonzero constant, we get an inconsistent system, but the only way I've been able to solve this was by getting a contradiction solving the variables one by one

3:51 AM
Even if it is what are some necessary check in order to ensure that?

so you should check if the last column is in the linear combination of first three columns
that is, you have the system: $Ax=b$ and its matrix representation [A b] which has RREF [R b'] so check if $b$ is in column space of R or not.
so it doesn't require so much manual effort in case of echelon form to see if the system is consistent or not.

oh thinking about it as seeing if it is a linear combination is faster
you instead solve for the coefficients of the linear combination
thanks

4:19 AM
hm, i backtracked: why would $\vec b$ be in the column space of $R$? @Koro
elementary row operations don't preserve spanning of the column space
i'm checking if it works with $b'$

there was a typo in my earlier message, I meant "check if b' is in column space of R or not". :(

hehe

4:38 AM

yeah
this was an exercise from another book though

Definitly has to be Ted's book....just smells like it

been doing ted's lectures recently too hehe

unfortunately however you solve it you will always have some "mechanical" tedious matrix work to do.....can't get out of it.
But the true attempt at trying to be "mathematical lazy" is acknowledged :p

looks like it's time to start learning sagemath matrix computations

4:44 AM
Hello guys

I've been learning R at the same time as doing these courses because I'm also doing a linear regressions course.....defintiely makes you appreciate the presence of this software...
all the tedious matrix manipulations one would have to do by hand?.......sheesh.

In mathematics and especially in real analysis or functional analysis is there anything about comparison operators which are not binary?
Or comparison is binary operator
@dc3rd. What is your favourite subject in math?

Stats/Probability

@dc3rd. Wow!

Lol....not sure it is that big of news.....

4:48 AM
Hopefully Professor Ted won't see this b/c I asked him same question ifinity times :/
Ever studied measure theory

dc3rd i have to learn R for econometrics next semester
looks like fun stuff

@shintuku. Ever wrote programs before?

Working towards getting to measure theory.....not at that level yet.....need to build the foundation. Just peeked at your profile....so you're a stats person....hence the "wow" :)

@Avra yeah python

start learning it from now shintuku
that way you won't have double the load of trying to "learn" the software AND learn the ideas of econometrics

4:50 AM
@dc3rd. I am fan of all math, all math people and all math world but still find it troublesome to get to play with them
@shintuku. R is amazing for stats!

I would say I'm somewhat in that boat too Avra....I'm actually reading a physics text on the side just for fun.....

I am telling you, some stats tools in R are very very advanced compared to Python

really regretting not taking it serious earlier....it really brings a lot of things together

why not program in latex? afaik latex is Turing complete

@dc3rd. OMG physics...great!
@dc3rd. Math connected physics :p
Physics is applied compared to math :/ so not bad
@dc3rd. I might throw myself in hell next semester and take measure theory
I guess this is the missing piece for me in stats and prob

4:55 AM
if you've done all the calc and analysis necessary before measure theory then you should be set....if not......well.....

Don't tell that to some Professors
They might get upset
Still needs basic topology as I understood :/

it is the missing piece "real" stats and prob doesn't begin until you got measure theory

It has lots of sets operations

yea topology too,

@dc3rd. No stat school teaches measure theory till you are pre-graduate or graduate level :/
Wish if it was forced on me!

4:57 AM
there's a reason for that.....

@Avra Not upset, but don't come here expecting us to hold your hand through it all.
I give realistic advice and you choose to ignore it.

@TedShifrin. I did not say anything, it's dc3rd

You said it all.

avra if you don't do real analysis before measure theory you'll die

@dc3rd has learned through hard work that most of my advice is spot on.

4:58 AM
here in Germany, one typically takes measure theory in third semester and it's quite challanging to a lot of students

download a measure theory book right now and try to do an exercise

I said "all the calc and analysis"..stress on "all"

And the German undergraduate curriculum is essentially graduate curriculum in the US.

if you need a good measure theory book, consider this: andrew.cmu.edu/user/awodey/students/jackson.pdf

@TedShifrin. :|

5:00 AM
Ted didn't create this advice from nowhere......also don't neglect your geometry and trig.....they REALLY help.....

@dc3rd Not so relevant for measure theory or real analysis.

Avra don't rush to "do it all".......I tried that and as a result I ended up having to delay doing the rest of my undergrad until I really have "built the foundations"

maybe some topology can be helpful for measure theory, but it's most likely not important for a first course
beyond the amount of topology you'll learn in introductory real analysis, of course

Not true, @Lukas. The fiddling with arbitrary unions and intersections is hopeless if you don't have at least experience with proofs in topology.

@dc3rd. I will tell you a story. A student with us got math courses in semester, have almost all As and graduate in 3 years with pure math, but that far away for me :|

5:03 AM
yeah aren't sigma algebras all about arbitrary unions and intersections

really?......I'll still keep the tools in the back pocket because they're helping me right now visualize the least squares problem from regression analysis a bit more.....but this adjoint idea is seeming so abstract within it all

@TedShifrin yeah I meant that a good foundation in real analysis includes most likely some basic topology

Not that level of proof skill, no.
And most places real analysis is nowhere near Rudin-level.

@LukasHeger. Really!
Professor Ted also said it's not easy course :/

Rudin -level :)

5:04 AM
rudin-level or nothing

I'm a big fan of Rudin :)

Very strong students in here often make the same mistake that math professors make ... of believing that everyone else is as talented and advanced as they are.

Avra he graduated in 3 years, but what you don't know is the amount of work he did prior to that to be able to achieve that level of mastery...

@LukasHeger. So measure theory is boogeyman in your department!

I'm not, @Koro, except for students who are headed to do a PhD in math.
Even at MIT, the Rudin analysis class washed most math majors out.
They created two other tracks of real analysis.

5:05 AM
I really enjoyed reding rudin the summer before grad school
much better than the first time around haha

Yes, @Ryan, and you don't count, either. Someone doing a PhD in pure math at Princeton is not comparable to our typical chat student.

@dc3rd. Can you go in a room please?

I am agreeing

I'm not sure why you want to go to a room Avra....

it's too hard to learn from

5:06 AM
Anyhow ... I give my advice and people are welcome to ignore it. But don't expect us to do your homework for you.

:|

our analysis course was taught by an arithmetic geometer. I have no idea how it compares to Rudin, but it was quite crazy as well

I wish if I can understand how math at Princton is different elsewhere !

heidelberg seems to have a particularly perverse approach to analysis

Most of your courses were crazy by normal people's standards, @Lukas.
Princeton is one of the top 3 PhD programs in the whole US, @Avra.
The most selective program except perhaps for U Chicago.

5:08 AM
@TedShifrin. Honestly, what is unique about the math they teach :(
Math is math. I don't really get ranking

They don't even teach first-year graduate courses. They expect all their students to know all that already (and most of them do).
No, math is not math.

@TedShifrin this hasn't been true for many years

Even at average schools, the program that a top student takes and the program that the weaker students take is HUGELY different.

What do you think about that Professor, how ranking affects difficulty when it comes to math for example?

Really?

5:09 AM
they have "bridge courses" now that are supposed to act like first year grad courses

So they now teach standard Lebesgue integration and standard graduate algebra, complex analysis?

oh no
but like diff geo, basic pde, algebraic geometry

@RyanUnger It was that way when I was there, but that was over 35 years ago

@dc3rd. I also need trig for measure theory!

That's what I consider first-year graduate courses, silly. That's what it is elsewhere.
Diff geo, PDE, alg geo are second year everywhere else.

5:10 AM
why would you need trig for measure theory?

You make my point entirely.

@robjohn. You was at Princeton :o

I already said that trig remark was nonsense.

don't you need to draw some triangles for the besicovitch covering theorem

LOL

5:10 AM
to lebesgue integrate sine? i wouldn't know

@Avra yes

@robjohn. I seeeeeeee
I believe prof ted now!
Math is not math
hahah

is someone asking why should one learn trig

And Princeton did teach grad courses in diff geo, PDE, etc., when I was a student.

if all they care about is measure theory

5:11 AM
@TedShifrinl. You studied also at Princeton :000

my freshman analysis course covered, among other things, Banach spaces, Hilbert spaces, Fourier series, Fourier transforms, differential forms, Daniell integration (with proofs for dominated convergence, montone convergence, Fubini etc.) etc.

dc3rd was reflecting on his needs to be successful in something like my multivariable math curriculum, NOT talking about measure theory.
That's all graduate level in the US, Lukas. I keep saying that. I'm tired of it.

@LukasHeger on a first semester?

germans love telling everyone about their first year analysis courses

Only at Harvard with Math 55 is that freshman level. For 15 students max.

5:12 AM
surprised they dont get tired of it

:P @ Ryan

@shintuku it was over the first two semesters yeah
the best thing is, this was a required course even for physics majors and teaching candidates...

of a freshman course???........bloody hell..........

Anyhow, I'm fine with people who are not prepared/qualified doing whatever they want. But do not expect StackExchange to do your homework for you.

5:13 AM
@robjohn. Do you really think math at Princeton is different!

@TedShifrin ...what

damn........

you can find most homework answers on this site

the teaching candidate system is cool

5:14 AM
Hard to believe you're going to pass the course that way, @Ryan. I'm certainly saying that those of us in this chatroom are NOT going to put up with incessant questions from someone taking something they shouldn't be taking.

@LukasHeger what are you up to these days anyway

And, yes, I'm being blunt.

btw @LukasHeger are you a fan of the SPD or not so much?

lol

@shin What is SPD?

5:15 AM
@RyanUnger well, I found an error in Grothendieck's SGA3 and working on writing up how to fix it

didn't even know of the existence of a Daniell integral until now....

and Dini derivatives? @dc3rd

@TedShifrin social democratic party of germany
the guys responsible for antifa

the youngest book I know containing the word "Daniell integral" is Federer

minority win in the last election

5:16 AM
@shintuku that's a bold statement

@LukasHeger i know right haha sorry

@Ryan Look at Segal & Kunze, Integrals and Operators.

@LukasHeger lol

I took that as a graduate real analysis course from Segal my junior year. Segal was bar none the worst prof ever, but the book is actually not bad.

@Avra It is. You get to learn from some really excellent people.

5:17 AM

It's a measure-theory-less approach to the integral, which I rather approve of :)

@robjohn. Is the level of difficulty is higher than other schools for real!

I think the Daniell approach is kinda cool

It truly is.

@robjohn. Or you think it's the way of teaching :/

5:18 AM
you get the same results as with measure theory, but without measure theory

Lots of brilliant professors at Princeton, many of whom are horrid teachers.

@Avra I don't know. I only went to one graduate school.
@TedShifrin Stein and Fefferman were decent teachers.

Did I tell you my story about visiting Princeton? Yes, definitely.

And Krantz, who was a Stein student and I had at UCLA, was the best teacher I've had.

I wouldnt want to be accused of having heard this story

5:20 AM
Can you please professors tell us one example how princeton is different?
:/

Krantz of "bLank and Krantz" multivariate calc book?

Steven Krantz

I was considering it for grad school. Went to a Riemann surfaces lecture by Gunning. It was a lecture room that held 30+ students and there were 4 students. I waited by the door to ask if it was OK. When time came for the class to start, he walked in and just started lecturing at the door. He looked strangely at me, but never inquired. When I tried to talk to him at the end, he'd already walked out the door, finishing his last sentence under the door jamb.

@Avra i think one difference is that fee is very high

Krantz is a superior expositor. Quite famously.

5:21 AM

@TedShifrin maybe he was very awkward?

@TedShifrin I never took a class from Gunning

Yes, @shin. Not a horrible lecturer, but ... still ... in a grad class with 4 students, ZERO interaction.

princeton does not charge graduate students

might explain why I did like looking at his book in my "grey cloud" days

5:22 AM
That's true of most good programs, @Ryan.

I know a few professors who I think are great professors for good students, but not so much for weaker students
@Avra princeton has Lurie

@TedShifrin are there any good programs that do?
in the US

@TedShifrin is it typical in grad lectures for the students to contribute to the teacher's research or is it still magisterial-style teaching?

Koro thinks Princeton is good at high fees

no German university charges for undergrad or masters, and if you're a PhD students, you get paid

5:24 AM

at Lukas: ah, the dream
the social democratic dream

@LukasHeger PhD students get paid in every country, I think. They get monthly stipends. Right?

@Koro as Ryan said, Most grad students at Princeton get tuition and room and board, plus a bit more paid for teaching classes (research assistantships). I got a one year IBM scholarship and the rest of the time was as an RA.

@Koro afaik in my surroundings you need to apply for stipends although you don't get charged for being a phd student
and the highest cause of phd abandonment is lack of successful money grant applications

@robjohn I'm glad to know that professor Rob :)

5:28 AM
Undergrads are a different matter. They paid quite a bit.
Brooke Shields was there as an undergrad when I was there.

@Ryan Most students at the top places are either on fellowship or have TA/RA-ships. Only at lesser state schools are there some (mostly masters students) who are unsupported.
@shintuku This is true less in math than in all the other sciences.
@dc3rd Whose?

He how has no mind is who does not give math students full coverage!!!!!!!
If anyone is really into math, he should get 100% supoort
though not my game :|

Blank and Krantzs' book Ted.......when my head was spinning and I couldn't figure things out earlier on in my journey I started to go through Blank and Krantz and it at least started to paint me a "picture"

I actually have never seen that book.

very similar to yours but it is a lot more "mechanical" the theory is given a back seat.

5:35 AM
I know TA but what is RA?

research assistant?

Research assistant?
Ah, Okay :)

Well, then not similar :P

So using your explanation of MIT splitting the math program into 3 streams, they did the same here at U of T, so Blank and Krantz would be the type of text used for the bottom of the three streams,

i don't know i was asking too hehe

5:36 AM
No, I said they had three tracks of the real analysis class only. They do have a separate applied math major, which includes statistics, among other things.
RA more common, again, in non-mathematical sciences. Math has way less grant money to give away than experimental sciences.
When I visited U of T back in 1983 or so, they had a Spivak course for the best freshmen. I assume that's gone now.

@TedShifrin. This is not fair. Math should be #1 priority

Even as a joke, that's not so funny.

no....it is the same...Spivak is still reserved for the top freshman

Most schools gave up teaching that because of the prevalence of the BC AP credit. Is there less of that in Canada?
Berkeley gave up 30+ years ago. UGA quit the Spivak course early 2000's.

There is less emphasis on it.
Only really stressed for folks that want to go south of the border for school

5:40 AM
When I was a grad student in the late 70s, the Spivak course at Berkeley had 80 students in it. That all disappeared.
I think AP is one of the worst things that ever happened to math learning. But I'm not going to debate that now.

Is prof. Spivak also on MSE? @professor Ted.

turned it into a "production line"

No, not at all, @Koro. Hasn't been in academia since a year in the 70s, either.

more "how to take a test well" instead of "learning how to think"

Well, the teachers are rewarded for teaching the students exactly how to write things to get points. It's not even "take a test well."

5:43 AM
If I wasn't so rebellious and questioning things I would've remaind in that frame of mind and not "seen the light".......it is sad really. Just a bunch of memorization.

Michael David Spivak (May 25, 1940 – October 1, 2020) was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. Spivak was the author of the five-volume A Comprehensive Introduction to Differential Geometry. == Biography == Spivak was born in Queens, New York. He received an A.B. from Harvard University in 1960, while in 1964 he received a Ph.D. from Princeton University under the supervision of John Milnor, with thesis On Spaces Satisfying Poincaré Duality. In 1985 Spivak received the Leroy P. Steele Prize. Spivak...
professor Spivak is no more? :'(

He taught at Brandeis for several years and then quit academia. He taught one year as a visitor at Berkeley in 1974-5. It just so happened that I took the year-long graduate differential geometry course he taught. That's how we became friends.

so the top freshman stream here (we call it a math specialist) goes: Spivak --> Spivak (manifolds w/ Munkres as complement) --> Pugh/Folland/Royden (depends on prof and year)
Rudin was/is used in the "medium" stream.....

Oh, no. Mike died last October. I had no idea. Darn.

That's news to me too.....that must be a recent update because I swear I had read his wiki over the summer and not seen that

5:46 AM
professor Ted, I had no idea either. I just now looked up when you said hasn't been in academia since....

Perhaps COVID related. Who knows. I can't find an obit yet.
I will have to ask a mathematician friend of ours who would know.

@TedShifrin. It seems for me It seems I will cancel my plans for measure theory and go for optimization as I still did not cover real analysis :|

I'm truly sad to learn this though. May he rest in piece.

have you taken multivariable calc and real analysis @Avra?

I guess I will remember him everytime I read his "Calculus" textbook.

5:48 AM
@TedShifrin. I will go for convex optimization as it's more applied and calculus like

I'm sad, too. I last corresponded with him a few years ago.
Much wiser, @Avra.

Most mathematicans shocked when I told them about my background and pushed me away from measure theory. You were right :(
@TedShifrin. Thanks. Convex then for next semester !! Cheeers

Wouldn't a passing like his show up in the AMA website or any other related one Ted?
He is a "big deal" in the math world.

@dc3rd. Yep for sure :

I can't find anything anywhere obvious yet.

5:51 AM
@dc3rd. I quite in the third year once I saw abstract algebra :|. So I covered all basic math

so that is a no? Avra

@dc3rd. I covered multi, yes but real no ! It was also in the 3rd year

@dc3rd I came to know about professor Spivak sometime in last year, when I saw lots of questions on MSE titled as Spivak's calculus ex. no. so and so. I managed to arrange the book Calculus by Spivak and started studying it. And today, I learned the sad news :(

@dc3rd. As prof ted said, this is much wiser than going for measure theory !!

@Koro Yea. It feels to me like he died just yesterday.

5:53 AM
@dc3rd. I am crazy person and I was going to play with volcano thinking it's water
Measure theory is not for me now :|

much wiser is an understatement.............yes you were..........and no it is not.....there are a lot of building blocks needed before you can take it

@dc3rd. Even for optimization!

well take it and be successful..........you're always welcomed to take it...........

You also don't recommend optimization?

I think Ted has provided you advice in the past as well...I would listen to his earliest advice and work form there

5:56 AM
Prof ted told about measure theory

I have no advice re convex optimization. @copper is our expert on that. I have no idea what a typical undergraduate course would cover or demand, and I don't know much of that material.

lol he can tell you about it, but I know for sure he didn't tell you to take it as a course...

Avra, prof. Ted or anyone can't make decisions for you. You yourself will have to decide at the end of the day. I think you should start studying a few chapters and figure out yourself.

@dc3rd. See I told you :|

see if you can find the course syllabus and progress form there

5:58 AM
%100
GN folks and thanks all

Avra, you can check out measure theory course on nptel also
and may try watching a few lecture videos also there

6:11 AM
god bless nptel
Suppose $a_n > 0, b_n > 0$ an $\lim \limits_{n \to \infty} \frac{a_n}{b_n} = L$ with $L_1 \in \mathbb{R}_+$. Prove that if $\sum_{n \in \mathbb N} a_n$ converges, then so does $\sum_{n \in \Bbb N} b_n$.

Suppose $\sum_{n \in \Bbb N} a_n$ converges to $L_a$. Since $\lim \limits_{n \to \infty} \frac{a_n}{b_n} = L_1$, we have $a_n = b_n(L_1 + \varepsilon(n))$ with $\varepsilon(n) \to 0$ as $n \to \infty$. Therefore, $\sum_{n \in \Bbb N} a_n = \sum_{n \in \Bbb N} b_n(L_1 + \varepsilon(n)) = L_a$, so $\lim \limits_{n \to \infty} \sum_{i=1}^n b_n(L_1 + \varepsilon(n)) = L_a$, and therefore, $\sum does anyone believe me? 6:31 AM @Avra Convex analysis relies on linear algebra and some real analysis (compactness, connectedness, continuity sort of stuff). It depends on your teacher, but I think it is a generally useful skill set to have. Convex optimization is a practical tool, and one that can be used as there are lots of packages out there and stuff like CVX (the primary author Michael Grant often answers on MSE). Measure theory is useful for other stuff like analysis & probability. It requires some of the fabled mathematical sophistication, in my opinion. I have one confusion: The set S=[-1,0)$\cup$(0,1] is not a connected set in R because it is the union of two separated sets (U and V in R are said to be separated if$U\cap \bar V=\emptyset = \bar U\cap V$). But how does this conclusion hold if I use this definition of (dis) connectedness: S in R is said to be not connected if S is union of two non-empty, non-intersecting open sets in R? Clearly$S\$ in this case is a union of two semi-open non intersecting sets.
What am I missing?

take the closure of each of them individually and see what you get....

I understood that but how to conclude using the other definition using open sets?

6:49 AM
@Koro i think the definition also allows working with closed sets

yeah, but the set S is union of two semi-closed intervals. :(

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