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6:01 AM
I have a possibly tautological question
Suppose you have a map $\sigma$ from a $k$-simplex to a space $X$. Now, partitioning the $k$-simplex to $n$ other $k$-simplices, we can restrict the $\sigma$ onto each of the simplices to get $n$ maps $\sigma_1, \cdots, \sigma_n$. The formal sums $\sigma$ and $\sigma_1 + \cdots + \sigma_n$ are formally different, but how to show the belong to same homology class, i.e $\sigma - \sigma_1 - \cdots - \sigma_n$ is homologous to $0$ ?

2 hours later…
7:54 AM
A rectangle has property that if you place two copies of it side by side the resulting rectangle of twice the area of the original has sides in the same ratio. Show that this ratio is $\sqrt{2}$.
but according to my calculation it is 1:2 and not same
but book says it is same and it did this
steps
suppose the longer side is $x$ and the shorter is $1$. Then the new rectangle has side 2 and $x$. Hence if these are in same ratio then $x/2=1/x$ so $x^2=2$
it is odd that it is not x/2:x/1 when you arrange the rectangle as book told me to do

> suppose the longer side is x and the shorter is 1. Then the new rectangle has side 2 and x

Or the other way around
@user791811 Your ratio isn't defined properly

@Hippalectryon You mean the question's ratio or the ratio mentioned by me?

@user791811 Yours
Oh wait

8:09 AM
I am right lol

I don't understand your question then. You just showed that $x=\sqrt2$, no ?

x is larger than 1 so $x/1$ and we don't know x is large or smaller than 2

So ? I don't understand where you're getting
In your question you said " according to my calculation it is 1:2 " but your calculation gives a ratio $x=\sqrt2$, not $x=2$

@Hippalectryon Ok I just want to know why is it written in this way $x/2=1/x$

@user791811 Since the new rectangle is a $2\times x$ rectangle, assuming $x<2$, then the ratio big/small side is $2/x$, which has to be equal to the original ratio $x/1$, right ?

8:16 AM
@Hippalectryon Yes I have exactly thought like this but then I rearranged in another way or may be assuming $x>2$ then this would not be true

@user791811 Correct. But do those cases (eg $x>2$) yield a solution ?

@Hippalectryon no
it gives $1:1/2$ which means oh ok now I get it lol.

my mind is messed up in wet weather

don't worry :P it's the natural process of doing maths haha

8:19 AM
ðŸ˜‚ðŸ˜‚ðŸ˜‚ I was thinking this for hours which is not natural
I drew 1 page filled with rectangle
thanks for indulgement :]

np :)

4 hours later…
12:13 PM
Hello!! Is it possible to have a natural cubic spline, but at one of the intervals we have a polynomial of degree $2$ ? (All the other conditions are satisfied.)

@MaryStar (not an expert but) isn't the natural cubic spline a complete linear system ? (as in, n equations and n unknowns) ? As such, requiring an interval to be of order two would make it a n equations & n-1 unknowns system wouldn't it ?
Or do you relax some conditions on the degree 2 interval ?

a spline is by definition piecewise cubic and quadratics are not cubic, so no

We have the function :
For $x\in [1-,0]$ : $2(x+1)+(x+1)^3$
For $x\in [0,1]$ : $3+5x+3x^2$
For $x\in [1,2]$: $11+11(x-1)+3(x-1)^2-(x-1)^3$
So for the second part this function is not a natural cubic spline, right? @Hippalectryon @Thorgott
The first interval is meant to be [-1,0]

not according to the definition that I know

So every polynomial has to be of degree 3 (exactly 3) althrough every other condition holds, correct? @Thorgott

12:25 PM
@Thorgott To me that's a natural cubic spline, what's the definition you have in mind ?
I relate "natural" to the boundary conditions (which make it solvable by a tridiagonal system)

@Hippalectryon So do we consider $3+5x+3x^2+0x^3$ ?

@MaryStar I'd say yes - but maybe @Thorgott as a different point of view, so I'm not sure

The one I have in mind is "piecewise a cubic polynomial and overall $C^2$"

@MaryStar But if we do so, it becomes trivial that a 2nd order solution exists since in particular a linear - 2nd order - linear solution can be created with four points

@Thorgott But doesn't this function satisfy this if we consider $3+5x+3x^2+0x^3$ or do we want the coefficient of $x^3$ to be not 0?

12:28 PM
In my view, there is no reason why a linear spline is not a "natural" cubic spline

actually, I retract what I said
piecewise polynomial of degree at most $3$ and overall $C^2$ is the better definition

In several notes I see just that these have to be cubic polynomials of the form $a_0+a_1x+a_2x^2+a_3x^3$ but I haven't seen the restriction that $a_3\neq 0$.

So it is a natural cubic polynomial, right?

I'd say it is

to me, a cubic polynomial is one that has degree $3$ and not degree at most $3$
but I agree this is a natural cubic spline

Since it holds that p_1(0)=p_2(0), p_2(1)=p_3(1), p_1'(0)=p_2'(0), p_2'(1)=p_3'(1), p_1''(0)=p_2''(0), p_2''(1)=p_3''(1), f''(-1)=f''(2)=0
Right?

12:35 PM
yeah

Why are index sets only equipped with a surjection rather than a bijection? Does it make sense for two indices to label the same element of a set?

Thank you!! :-) @Thorgott @Hippalectryon

Yes. Practical example: Your index set is the set of all your employees and the surjection maps each employee to their salary (think of this as, say, an excel table where you list how much you pay all your employees). Of course, two employees can get the same salary.

1:03 PM
Let $f(x)=x^3-1$. To approximate the root $x^{\star}=1$, we consider the sequence $(x_n)$ that we get if we apply Newton's method with $x_0>0$. Show that the sequence converges to $1$.

I used $x_0=0,5$ and applied the method and in that way we see that the sequence converges to $1$.

Is that correct?

An other way could be: From Newton's method we get $x_{n+1}=\frac{1}{3x_n^2}$ and we have to show that this sequence converges to $1$, or not?

1:13 PM
I have a question about the algebraic group $A^1$ of units of reduced norm one in a quaternion algebra $A$ over a number field $F$. Let $E/F$ be a splitting field of $A$ so that $A \otimes_F E \cong M_2(E)$. Is it possible to characterize the elements $a \in A^1$ which have diagonalizable images in $M_2(E)$ purely in the language of algebraic groups?
... something like "(regular) semisimple" elements, maybe?

4 hours later…
5:31 PM
ill need some help on munkres topology on the proof of seifert van kampen
@BalarkaSen if you go the time helping me abit when you are free

2 hours later…
7:59 PM
@Manolis I'm kind of busy with other things, but anyway you can always ask here and someone will help

I can't continue my study without understanding the Proof

@BalarkaSen ok thanks
munkres uses alot what he calls positive linear maps
id like to prove that

8:18 PM
This is completely elementary. Draw a picture.

ye intuitevly
its just breakdowns the unit interval

Is anyone studying from Spivak's calculus on manifolds currently?

@Abdullah the $\varphi$ are a partition of unity, I take it?

Hi @Ted

i need to a homotopy

8:22 PM
Hi, a @Balarka!

find*

@Abdullah: I prefer my own text to Spivak's :P But what's your specific question?

@Manolis Yes, this is fairly easy. Do it for $n = 2$.
Path-homotopy class of a path does not care about (positive) reparametrizations on the domain of the path.

yes im trying to see that formally

@ted How is this inequality established $M\int_A \Sigma_{\phi \in \Phi-F} \phi \leq M\int_{A-C} 1$?

8:26 PM
@Abdullah: He's removed all the functions $\varphi$ that are nonzero on $C$, so all the remaining functions are zero on $C$. And they add up to at most $1$ at all the points of $A-C$.
This is a partition of unity, after all, but some of the functions that are nonzero on $C$ might also be nonzero at points of $A-C$, so perhaps the sum is a bit less than $1$.

@ManolisLyviakis Yes, it is easy formally, is what I am saying. You should be able to check this before reading about fundamental groups.
Just think about it for some time.

I've found that people don't like it when I suggest they stop to think, a @Balarka :P

I hate thinking, it's disgusting

Agreed :D

@Ted: It's sort of a relatable feeling. I don't like being stuck on what seems like potentially dirty computations myself.
But then without doing the computation, how to judge if it's actually bad or if I am just dumb?
On multiple occasions, it's the latter

8:31 PM
Well, one fellow who was annoying me got to the point where I stopped answering his questions in comments, so he emailed me. I went back and answered and he asked an inane question, so I suggested he stop to think. He posted the question on main. Good riddance :P

Right yeah, there's also that other group of people who simply don't want to consider the second possibility
Being dumb for most of my academic life is how I learnt math. I am not John Pardon, but so it is.

@Balarka: You spend too much effort knocking yourself. I am ordering you to stop it. And no, very few of us are Chern, Robert Bryant, or John Pardon.

@TedShifrin thanks

I actually posted it as an answer, too, @Abdullah.

Just saying Ted! No self-harm meant.

8:34 PM
Spivak is not the easiest book to read, so, if you don't know about them, I also recommend my YouTube lectures on multivariable analysis. They might come in handy from time to time, @Abdullah.

@TedShifrin next time I won't consider reading a book just because it's about 140 pages lol. To be frank I haven't even took an analysis course so reading this book is a huge challenge for me

You definitely need single-variable analysis experience (as well as a solid knowledge of multivariable calculus). You might check out my book (which integrates linear algebra in throughout the course).

@TedShifrin I read the book Thomas calculus and linear algebra by William goodie cover to cover I think I understood it well so I thought it's time to be rigorous because I need to study general relativity.

Spivak is not a great choice. Thomas's Calculus is an engineering text. If you don't want to look at (or spend money on or steal) mine, try C.H. Edwards's Advanced Calculus.
Reading cover to cover doesn't suffice for math, anyhow. You have to work lots of exercises, and with proofs you'll generally need someone to criticize what you've done.

Wow, Ted, yours is steal-able now somehow

8:46 PM

You can get the link in my profile. There is no legal .pdf available, but the book seems to be at the usual illegal source of books that you all know about.
@Balarka: Yes, I think it has been for a while.

2019, it seems

I should probably let my publisher know. :D

Oh noes

The other thing I noticed is that some Yale student uploaded there his transcription of my video lectures. His handwriting or pictures are not remotely as good as mine. I wonder why such things are uploadable?

8:50 PM
Oh damn

@TedShifrin i would love to buy your book if I weren't a selflearner from Sudan you know how things going here.. Anyway I really appreciate your help

@Abdullah: I'm not asking you to spend the money. I am telling you that if you go to libgen (which most students know about) you can currently find it there, so take advantage. I hope my publisher gets it removed eventually.
But the videos are totally free and pretty much free-standing. But you miss out on all my excellent exercises :P

@TedShifrin At least it's not illegal to upload their own handwritten lectures from the freely available youtube lectures
But yeah horrible handwriting

No, it's just weird that someone would do that without asking my permission.

Yeah haha
I mean most people would give permission for such a thing so he should have asked for it

8:53 PM
And in all lack of humility I think it is not nearly as good as the videos. But far more time-efficient if one is looking for something specific.

And haha I mean why in libgen of all things
Just put it up on your own site or something
After taking proper permissions

I've also seen that some people upload what they allege are solutions to various of my differential geometry problems. Again, of dubious quality, and they also certainly never asked if I minded.

Hahah yeah making solutions to exercises public are a bit um-worthy
It also tickled me to no end that the said student is now in the [redacted]

Oh, I wasn't curious enough to search.

I think I got the right person at least

8:58 PM
I've seen students offer money for detailed solutions to homework
not while it's due, mind you
but to study for a make-up exam

The person teaching the course at Yale resorted to my videos because of the pandemic. I think he plans to use them again next year. He didn't respond when I gently castigated him for not getting to forms and the variants of Stokes's Theorem. I consider that dereliction of duty, but not surprising when a combinatorist teaches such a course. I gather he's a good teacher, though, regardless.
I've seen students offer money for solutions to test questions in real time as they take the test. We had all sorts of fun when I was associate department head at UGA.

@TedShifrin i gotta say I've just Googled you. I didn't know are a famous mathematician

The student should have been kicked out of the university for that, but the professor refused to pursue it.

@Abdullah: Perhaps not famous.

9:00 PM
Ok suppose i break my unit interval into 0, a, 1 then the positive linear maps are $L_1(s)= as$ and $L_2(s)= (1-t)a +t$ so my $f_1(s)=f(as)$ and $f_2(s)=f( (1-t)a+t)$ so now $f_1*f_2= f(2as) , f(-2sa+2s-1)$ and now i need a homotopy between f and f1*f2

lmao, that's also quite funny

I literally got a phone call from someone who had seen the ad on Craig's List (which you probably don't know about in Europe, @Thorgott).

I got a guy who's name Theodore shifrin who also happen to be a mathematician

No, that's me.

lol

9:03 PM
I'm just saying I'm perhaps more infamous than famous :D

notes Balarka's peals of laughter

I'm literally shocked

There are a few famous mathematicians who show up on MSE occasionally ... not in chat, though.

LOL
that sounds like a great scenario

9:06 PM
That God we got people like you

@TedShifrin I saw Jacob Lurie hovering in the homotopy theory chat (MO-based) a few weeks ago... and I had just posted some stupid calculation there.
Truly mortifying

LOL, you should have said hi.
@Abdullah: Before I retired, I occasionally caught some of my own students posting homework questions (from my course!) on MSE. Suffice to say there was a stern speech in class the next day.

Those guys are way grumpier than this chat, I think, so waving online hi's are not usually part of the culture.
It's fine, he will never remember I was the guy who rambled about Postnikov towers in a random internet chat room when I get the Fields medal 20 years from now.

Lubin once commented on one of my questions

lmao
Scary

9:13 PM
I think I met him decades ago, but on line he seems perfectly nice.

Oh yeah Dennis Sullivan wrote a 1 line answer below my long explanation of why S^1 x S^2 is not homotopy equivalent to S^1 v S^2 v S^3 just saying "oh yeah note that if you suspend once they become homotopy equivalent"

@TedShifrin I've been studying math for a while now but after some time I forget the proof of a theorem partially or completely but I use the theorem anyway is this okay from a mathematician point of view

Absolutely, @Abdullah.

Thank God I thought I was cheating

If you understood the proof at one point, that's fantastic. I don't remember most of the proofs in my own papers. In my books, I think I do mostly because I've taught things so much.

9:16 PM
it's ok for a while, but at some point you will inevitably have to do some backtracking

@Thorgott: Sometimes, but rarely, you need to understand the proof in order to apply a theorem. Do you know lots of examples?

@TedShifrin Yup! Better than the homotopy stuff below.

@Balarka: I think you should be honored that Dennis commented on you.

yeah it was a bizarre experience
he signed off his comment as - DS lol

One of my least appreciated posts on MSE I had to credit Robert Bryant for helping me approach the answer :P

9:19 PM
you don't need to understand the proof in order to apply a theorem, but you most likely need to understand the proof in order to understand the theorem

@Krijn ????!?!?

Just logged in after two years of absence... Balarka and Ted are still here...

@Thorgott: I think that's often NOT true.

I forgot you existed

Wow, stranger.

9:20 PM
@BalarkaSen I always come back

I can leave, @krijn :P

what's an example?

well goodnight people ^^

Inverse function theorem, Thorgott

I'm actually going back to maths after I was lost in consultancy for two years
Well, crypography that is

9:21 PM
Even something basic like the uniform limit of continuous functions ...
Welcome back, @Krijn.

Nice man

Stokes's Theorem, @Thorgott. Proof irrelevant to understanding the theorem, other than saying "fundamental theorem of calculus."

All the old people are coming back. @Hippa, @Krijn, ...

Gauss-Bonnet Theorem.

I'm not gonna say abc-conjecture

9:22 PM
I think @Thorgott is losing this game.
Even the Sylow Theorems. Knowing the proof doesn't help with applying.
You have to learn a collection of standard techniques instead.
Change of basis theorem in linear algebra?
OK, I'm bored with this.
Spectral Theorem.
Now I'm done.

Hah

@Abdullah: Did you find the lectures and/or the book?

So after two years, Ted, Bala, how is life?

The world is shockingly worse (at least in the US) every day, but luckily (?) we're still alive so far.

Yeah, we somehow hit the most bizarre timeline

9:27 PM
Yeah it's pretty bizarre

@TedShifrin the reason for choosing Spivak's calculus on manifolds is that it had a fewer pages_which Is good compared to Harvard book on advanced calculus. and in the preface it was only assumed that the reader had a respected linear algebra and one variable calculus courses. Plus time is a major factor since I'm originally a physics major and I still have to study electrodynamics, thermodynamics,.. Etc

to me, understanding a theorem includes an understanding of why the theorem is true and I couldn't tell you why any of these theorems ought to be true without at least recounting the basic idea behind their proof

@Thorgott what about Jordan curve theorem?

rekt

Citing the wikipedia intro: "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."

9:30 PM
@TedShifrin i found the lectures but I have to wait till I Get back to the town since the network connection barely makes it possible to browse the web

To be fair, Jordan curve theorem is probably one of the most extreme examples

@Abdullah: I think you won't really learn satisfactorily from that book. When I was a Berkeley grad student and faculty taught the Honors multivariable course out of it, I had to spend hours every week helping the undergraduates (and that was with a teacher). Sometimes a few more pages and examples are far better.
@Thorgott: Even in calculus, many proofs give you no insight into why the thing really works. My rule of thumb when I taught multivariable calculus for science/engineering (not pure math) was that I would only do proofs when the proof gave that insight. It was a handful of the theorems in the course.

@Ted I do have the feeling in number theory very often that to grasp the theorem I need to grasp the proof

Well, the reason it's difficult is because Jordan curves can be a lot more pathological than our intuition permits. I wouldn't call it being intuitively plausible a good reason for it working out mathematically.

By the way, a great book for you to look at because of your physics interest, @Abdullah, is by Bamberg and Sternberg (from Harvard): A course in mathematics for students of physics.
Yes, I think that in elementary number theory that is probably right. And things in algebra like understanding why $\Bbb Z$ is a PID.

9:32 PM
And for what it's worth, anything we think about intuitively will be a piecewise smooth curve and the theorem is easier for those curves, the reason being that they locally divide the plane - and this can be turned into a proof.

Especially if you know some differential topology.

I think the theorems in calculus that have genuine content also have insightful proofs. The ones whose proofs generate little insight tend to be the technical lemmas that themselves aren't very insightful, in my experience.

I might agree with Thorgott for something like Poincaré-Hopf or the Hopf Degree Theorem.
The problem is that the way we teach calculus in the US (i.e., we do not teach all college students an analysis course), there are black boxes everywhere â€” limits, maximum value theorem, intermediate value theorem. But I would still prove that at a local extremum the derivative must vanish, Fundamental Theorems of Calculus, and a few others.

I think the proofs of those two are insightful

The fundamental fact for understanding most of these theorems from calculus I did emphasize when teaching the theoretical course: If a function is continuous and positive at a point, it must be positive in a neighborhood of that point. If you actually have a rigorous definition of limit, this is crucial.
Yes, I'm agreeing those are insightful. Same thing with fundamental theorem of calculus for line integrals, Green's Theorem (for a rectangle), and a few others.

9:38 PM
@TedShifrin do this book contain rigorous proofs for all the theorems?
Does*

My book? Yes.
Some are not worth reading, I would say, but most are.
If you do follow up with the book, make sure you look at the list of errata (not too long) on my university webpage.

How many pages?

For the book or for the errata?

The book

About 460, but a fair portion is linear algebra that, presumably, you already know, although perhaps you don't know it as well or as geometrically as you might.
Anyhow, I'm not trying to twist your arm. But I provide a lot more intuition for things than Spivak, and it's not quite so technical.

9:46 PM
@Ted Speaking of calculations, how do I actually relate the real Hessian with the complex one? haha

I will see if I can find it in the libraries in Khartoum since it's not available in Pdf fromat

@Abdullah: IF you look above, I did mention the (illegal) site from which you can download it.

I get the feeling like if $H$ is the real Hessian, $H + J^T H J$ might be something to look at?

I am being whiny about calculating, of course. I can write something down here: $4 f_{z_p \bar{z}_q} = (f_{x_p x_q} + f_{y_p y_q}) + i(f_{x_q y_p} - f_{x_p y_q})$
Hm.

9:52 PM
OK, that's useful.

Didn't find it... What's the title of the book

Multivariable Mathematics ... and then a subtitle
IF you put my name, you should find it easily.
@Balarka: So we obviously want to use the usual giant blocks for the complex structure.

Yep.
I am trying to see if $J^T H J$ somehow gives me the right object
Ah can't do matrix multiplication in my head. Let me write it out

I don't know. So you want an $n\times n$ thing out of the $2n\times 2n$ thing. If the real Hessian is $\begin{bmatrix} A|B \\ \hline C|D\end{bmatrix}$, then you want $A + D + i (B-C)$.
I dunno why that matrix won't appear correctly.

Good enough though

9:59 PM
I don't see how your equation fits at all, @Balarka.

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