Mathematics - 2021-01-25 (page 1 of 3)

user2103480

00:10

have a new favourite term

In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories. == Definition == We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the...

"yeah so for a variety the hypercohomology spectral sequence gives the Hodge-de Rham spectral sequence for algebraic de Rham cohomology"

real tongue breaker

I'm still convinced that the person that coined the terms "orientation preserving/reversing" did this with the specific purpose of letting lecturers trip up

dc3rd

Hi @TedShifrin. You said something particular in one of your lectures that is still wracking my head. This was the point in the lecture where you said it: https://youtu.be/pcHFb8V5U_4?t=531

You said the "point" $x$ is in the plane, but the "vector" $x$ is not. How is that possible given that starting from the origin to get to the point, that would be a vector.

Thorgott

@user2103480 Balarka did some of that shit earlier this year

that man's too far gone

@dc3rd Look at the picture he drew. Does the vector x, meaning the arrow with tail at the origin and tip at the point x, lie in the indicated plane?

dc3rd

00:29

@Thorgott, in his picture the vector he drew is not in the plane, I agree. I also agree the $A$ and $x$ are not orthogonal. But if $x$ is a "point" in the plane, it means that $A \cdot x = 0$. Since this is the equation of the plane.

hang on a second, that last idea doesn't actually make sense.....

well the "point" $x$ does satisfy the equation of the plane $A \cdot x = 0$. But the vector does not.....so our we treating a "vector" and a "point" as different objects? I ask because if we "sub in" the values of my "vector" $x$ into the equation $A \cdot x$, I will get $0$. But if I take the dot product of my "vector" $x$ with $A$, I will not get zero........so we are doing different operations on the same object then?

Thorgott

I haven't seen the rest of the lecture, so maybe I'm messing up the context, but isn't the equation of the plane in which $x$ lies $A\cdot x=2$?

user2103480

@Thorgott lmao

I'll never leave the comfy zone of probability, analysis, point set topology/measure theory and classical logic again

Mike Miller

00:45

@user2103480 It's totally unnecessary

To use the name hyperhomology

user2103480

@MikeMiller that's why this needs an especially impressive-sounding name

Mike Miller

Hyperhomology is just homology

Of a more complicated thing

user2103480

spicy homology

Thorgott

In any case, mathematically $x$ is always just one thing: a tuple consisting of 3 real numbers. But we can choose to visualize this mathematical object geometrically in different ways: sometimes we think of it as specifying a point in 3d-space and at other times we think of it as specifying an arrow in this space. The thing is that a geometric picture involving one of these visualizations need not be comparable when you substitute it with the other visualization. They're simply different visualizations.

user2103480

The one girl in my cohort that was taken under one professor's wing already from like semester 3, with a lazer-focused path and top grades, was broken by equivariant cohomology

dc3rd

00:54

@Thorgott, oh yes you are right about $A \cdot x = 2$ and not $0$. I had written some more in my comment, but you made it null by stating that it need not be the case that one visualization be comparable to another. As well as we know what the "point" is doing but that need not say anything about the "vector". THanks for your assistance.

user2103480

She was into algebraic combinatorics - and not at all into topological methods - and her prof gave her a master's thesis topic on equivariant cohomology of grassmannians and algebraic K-theory. In her words, "a few isomorphisms make it easier", but apparently that topic was the nail in her maths-coffin. Handed her thesis in as quick as possible and started a comfy insurance job

(at some point, she just felt she'd rather study something with a more tangible impact/applicability)

Ted Shifrin

01:21

@dc3rd Hey, I saw your ping. Did Thor settle all your questions? Yeah, if the plane goes through the origin, the entirety of the shaft of the "arrow" (i.e., all scalar multiples $t\vec x$ for $0\le t\le 1$) is contained in the plane, but not otherwise.

dc3rd

@TedShifrin, he did. Thanks for check up. Just going back over the previous lectures to approach the question I had asked a few days ago on the main page.

Ted Shifrin

Do I know what question this is?

polite proofs

Hello Ted. Did you have a recommendation for a linear algebra textbook that is not Artin? Since I wouldn't be ready for Artin

Ideally I'd like to go over linear algebra before going into really pure analysis

(I would be open to using Artin if I was ready for it)

Amin Idelhaj

01:38

Hey everyone!

Polite proofs: Do you think you're not ready for Artin because you still need practice in proofs and whatnot? I guess "Mathematical Maturity" as people like to call it

polite proofs

@AminIdelhaj Well, I'm currently going through Apostol's Volume 1

After that, I would like to go through a linear algebra textbook

So I am looking for an appropriate textbook

Hoffman & Kunze is one that I considered

Ted Shifrin

I actually think that if you know all the linear algebra in both volumes of Apostol, or the linear algebra in one of my books, you'll be fine. The only major topic you'll be missing is Jordan canonical form. That can wait.

Amin Idelhaj

I don't know if Artin really has any prereqs honestly, but I used Hoffman and Kunze which was good, kinda old school. People seem to like "Linear Algebra Done Wrong" by Treil

Ah if Apostol covers linear algebra anyway then there's that

Ted Shifrin

Yeah, I don't know Linear Algebra Done Wrong, but I've heard good things. I'm not so fond of Axler's pompous Linear Algebra Done Right.

polite proofs

I have read 2 chapters of LADR :)

Amin Idelhaj

01:41

Same. I think it's because Axler is a functional analyst

polite proofs

This was a few months back, though.

Ted Shifrin

Apostol does plenty. He doesn't teach much echelon form, as I recall, but he has all the important stuff.

Yes, Demonark, Sheldon and I were grad students together.

Amin Idelhaj

Like I get the sentiment that he's trying to address, that a lot of linear algebra classes say "Oh the determinant is some random sum of products of entries in the matrix, now let's prove facts with conceptual content using it!"

Ted Shifrin

It's sort of like Hubbard & Hubbard's book (which in its abstruseness caused me to write my book) on multivariable + linear algebra. So idiosyncratic and really not pedagogically thoughtful. Axler is like that.

polite proofs

I don't mind Axler's book, but I think it doesn't have enough abstract algebra

For example, he doesn't differentiate between a polynomial, and a polynomial function

Ted Shifrin

01:43

It's great to think about the Euclidean algorithm for avoiding determinants once you already know everything, but I wouldn't really teach it that way.

It's not supposed to have abstract algebra.

That's a very strange point for a beginner to make. Did someone tell you that or did you seriously notice it in some important way?

When you work over $\Bbb R$ or $\Bbb C$ I don't think you have to think about that very much.

polite proofs

I've read the wikipedia page for polynomials a while back

Amin Idelhaj

I do agree that this is kinda eh. But Axler's solution is to say "Ehhhh you know what change of variables is the only thing that uses determinants anyway", which is what I think is a function of him not really dealing with algebra or differential topology much. I'd rather make determinants conceptual

polite proofs

When I was learning abstract algebra

I was learning about finite fields

So there was an example that came up which demonstrated a possible difference that made me notice it

Ted Shifrin

Right, but that comment is out of left field for a linear algebra book unless you're doing it in the context of finite fields.

Amin Idelhaj

Polite if you have already learned a bit of abstract algebra there is no reason for you not to be ready for Artin

Ted Shifrin

01:45

Yes, of course. The polynomial $x^q-x$ takes only the value $0$ for a field of order $q$.

polite proofs

I don't disagree, I just don't like that he didn't explicitly say polynomial function without remarking that there is a difference

I am probably being pedantic

Ted Shifrin

I totally disagree with you.

But what else is new.

polite proofs

Well, Ted, you've had a lot more experience than me. I've gotten much more confused than you in the past year

Amin Idelhaj

Polite: so on a personal level that makes me cry but Axler says at the start he works over R/C

polite proofs

(At least I would think so)

Amin Idelhaj

01:46

Which is common for intro linear algebra courses

polite proofs

So I am just saying my view on things that build up. For example, Axler might say that a polynomial is a map

Ted Shifrin

Yes, I'm never going to teach linear algebra over an arbitrary field unless it's inside an abstract algebra course. shrug

Amin Idelhaj

So he is allowed to say that the difference between polynomials and polynomial functions are a complexity he'll defer to a future algebra course

polite proofs

And then I remember that fact

And when I read about a polynomial not being that, then I get confused

It's an arbitrary example, but these things definitely happen to me

Ted Shifrin

I'm pretty sure Hoffman/Kunze is over $\Bbb R,\Bbb C$, too, although I haven't looked in 40 years and might be wrong.

polite proofs

01:47

@AminIdelhaj I wouldn't have minded if he wrote a remark that there is a difference, but no such remark was made

Ted Shifrin

Just because something has confused you doesn't mean every author should make the pedagogical choice to unconfuse you, without knowing your list of requirements.

It doesn't belong in that setting. Sorry. You're wrong.

Thorgott

@AminIdelhaj the solution, of course, being Winitzki's Linear Algebra via Exterior Products

Amin Idelhaj

As I recall Hoffman-Kunze is actually careful about the whole polynomials stuff. I don't think he references finite fields directly, for the most part R and C are what he cares about

polite proofs

Well, my list of requirements is pretty simple. I like when authors are precise about what they write, and if the situation is so that a definition will change in the near future but for now we don't have to worry about special cases where our current definition doesn't hold, then I would like them to say so

Amin Idelhaj

But he does remark when things get hairy in finite characteristic?

Ted Shifrin

01:49

Speaking of which, @Thor, you must have liked this question :)

Amin Idelhaj

Polite proofs: You should make a distinction between a given book being bad for you or being bad for its intended audience

polite proofs

@AminIdelhaj I agree, I am pretty picky nonetheless.

Ted Shifrin

Demonark: Hoffman and Kunze are two people (both dead, both essentially functional analysts!).

Amin Idelhaj

Fine to be picky, I am too. But I think if a book is written for a certain audience and it does that job well, then it's not fair to levy criticism so much as remark that you're not in the intended audience and hence it's unsuitable for you

My dig at Axler on determinants isn't that "Oh he's not teaching LA in the optimal way for me to learn it"

I'm saying he fucked up

Ted Shifrin

polite: Seriously, if you're going to second-guess and micromanage every paragraph of every page, it's going to be a very long haul. Why worry about why Apostol brought in that discussion about tangent and cotangent. I must never have read that page, but I believe it's in there.

polite proofs

01:51

It's just how I am, I am messed up

I agree I shouldn't do that

Ted Shifrin

Anyhow, I never pretended to be the right teacher or right author for everyone. But I truly believe I have done a better job than the majority of authors.

polite proofs

Is your linear algebra book meant to be read by engineers or pure math students?

Ted Shifrin

That said, I don't take it personally when an individual student doesn't like my lecturing style or writing style ... and I'm used to complaints that my homework exercises aren't easy enough.

Amin Idelhaj

A linear algebra book designed for engineers which does that job well? I have no criticism. I probably won't ever reference it but I'm not gonna take a shot at the author

Ted Shifrin

Pure math, but some applications are included. We worked hard to teach students how to approach writing proofs, actually, since that's the first course where most of the students do that.

Amin Idelhaj

01:53

Anyway polite as for you in particular, just go with Artin tbh

Ted Shifrin

Strang's books are like that, Demonark. They're not good for teaching pure math majors, although I did try some.

Thorgott

@TedShifrin I actually did see that question earlier, saw you had commented on it and just liked your comment. I see you've now left an answer too.

polite proofs

I don't really need to learn how to write proofs, I know all of the proof methods. Or at least, the popular ones: strong induction, least min example in a problem where induction is not the clear way to go, contradiction, contrapositive

Ted Shifrin

Well, since the OP made an effort to progress, I thought he deserved a little more direction. And for a change I made it an answer instead of a comment.

polite proofs

I've read ~300 pages of Ping Zhang's proof book

Unfortunately, his book is very, very elementary

(She, actually)

Thorgott

01:54

yeah, I agree this is a good hint

Ted Shifrin

Polite: You say you know stuff, and then I watch you struggle in here for literally hours to understand a simple example/counterexample ... So forgive me if I take you with several grains of salt.

polite proofs

In the sense that every exercise is easy

Thorgott

I'm pretty sure I've done this computation at some point last year

polite proofs

@TedShifrin I know, I can see why you would think that

Ted Shifrin

@Thor: I've never done this starting with the tensor product.

Amin Idelhaj

01:55

This exterior products book is very interesting lmao

Ted Shifrin

But, @Thor, I do have to remark that this is the second algebraic thing we've agreed on in a few days. I'm getting worried.

user 85795

@politeproofs do you watch khan academy vids?

polite proofs

@user85795 No, however that's where I learnt prealgebra.

Thorgott

want me to post some category theory fun facts to disturb the peace? :P

polite proofs

His videos are good for "what is mathematics at all?"

I moved onto a textbook a few weeks after, though.

Ted Shifrin

01:59

At any rate, one last comment. The linear algebra treatment in my multivariable math book and videos is quite a bit more sophisticated than in the linear algebra only book. There I assume the students are more sophisticated if they're in that particular course, and I do linear transformations from the outset, don't help with proofs. :)

@Thorgott Nah, that's OK.

Amin Idelhaj

inb4 he posts and Ted agrees with that too

Ted Shifrin

Oh oh, Astyx is awake in Balarka hours.

Astyx

My crocodiles are having insomnia

how's it going?

user 85795

are they doing the crocodile rock?

Ted Shifrin

I think the tartare terrorized them, @Astyx.

polite proofs

02:16

I am trying to prove that if (a_n) is a sequence of positive reals, and $\sum_{n=1}^\infty a_n$ diverges, then $\sum \frac{a_n}{a_n + k}$ for a positive real constant k diverges

Suppose that $a_i$ has a least upper bound. That is $\exists N \in \mathbb{R}^+, \forall i \in \mathbb{Z}^+ : \sup(a_i) = N \ge a_i$. If that is the case, then notice that $$ \sum a_n > \sum \frac{a_n}{k} > \sum \frac{a_n}{k + a_n} > \sum \frac{a_n}{k + N}. \tag{1} $$ By the limit comparison test on $\tfrac{a_n}{k + N}$ and $a_n$, we see that $$ \lim_{n \to \infty} \frac{\tfrac{a_n}{k + N}}{a_n} = \lim_{n \to \infty} \frac{1}{k + N} > 0, $$

and since $\sum a_n$ diverges, it must be that $\sum \tfrac{a_n}{k + N}$ diverges as well. From $(1)$, we have that $\sum \frac{a_n}{k + a_n} > \sum \frac{a_n}{k + N}$, and so if $\{ a_i \}_{n=1}^\infty$ has a least upper bound, then $\sum \frac{a_n}{k + a_n}$ diverges.

This is what I've written so far

Ted Shifrin

How do you know $k>1$?

polite proofs

We can add that as a condition

For k > 1

Ted Shifrin

I think that's the wrong direction to head.

I would be more interested in whether $a_n\to 0$ or not.

Thorgott

But you want to prove it for every $k>0$, no?

polite proofs

@TedShifrin Well, I can't say much about that.

I would still need my two cases, right?

Ted Shifrin

02:21

No. I'm saying start over.

You have to focus on what's important.

user 85795

@politeproofs how long ago?

Ted Shifrin

Your instinct, @polite, to use limit comparison is right on. You just have to get there an effective way.

polite proofs

@TedShifrin But I can't say much about $\lim_{n \to \infty} a_n$

e.g. it could be $a_n = |\sin(n)| + 1$

Thorgott

you can split it by cases, but you can also try to save what you have

polite proofs

Which is why I wanted to do my cases

Thorgott

02:24

note that you actually don't need the upper bound

polite proofs

Well, rather I don't see a better way

@user85795 More than a year ago, not close to 2 years

Ted Shifrin

Can you do it easily if $a_n\to 0$? (That's the interesting case.)

If not, there's a subsequence of $a_n$ that has a positive lower bound.

Just think about that subsequence.

polite proofs

Well, if $a_n \to 0$, there there is a number $N$ where $a_N \le 1$ or something

I could perhaps use that

Ted Shifrin

Not needed.

user 85795

@politeproofs have you studied any trigonometry yet?

polite proofs

02:26

@user85795 .. yes?

user 85795

@politeproofs They do proofs there.

Ted Shifrin

@user85795 I will rise to support polite's honor and say he's way beyond that.

2

I stand by what I said.

Thorgott

I also first thought of splitting into cases depending on $a_n\rightarrow 0$ or not, but the approach splitting in cases depending on $\{a_n\}$ having an upper bound or not works too

Ted Shifrin

Well, I'm more of an analyst than you are, @Thor. That explains that!

user 85795

@TedShifrin pardon my interruption professor

Ted Shifrin

02:31

Interesting divergent series are ones where the terms go to $0$. So I will stand by my viewpoint. That's the intuition to develop.

No need for that, @user85795.

But I felt I should acknowledge where @polite stands in his learning/abilities.

Anyhow, @Thor, go ahead and help @polite finish and then maybe he can think about my point.

I'm going to cook.

Thorgott

have fun

Astyx

what are you cooking?

robjohn

@TedShifrin what's for dinner?

user 85795

cya

polite proofs

But I'll need cases either way?

I don't see how I can avoid cases if $\lim a_n = 0$

Astyx

02:36

What's $\lim a_n+k$ in that case?

polite proofs

$k$?

Thorgott

either proof uses cases

polite proofs

@Thorgott Oh. So what's wrong with my method? :(

Thorgott

nothing

as I've been saying all along

you just have to fix the proof so that it's correct

polite proofs

$\lim_{n \to \infty} \frac{a_n}{k + a_n} = 0$, ...

Wait, but this is worse than before

Astyx

02:41

I think you should try to fix your idea of a proof before addressing Ted's point

polite proofs

You mean do the other case?

Or is my first case not done correctly?

Thorgott

go back to the initial complaint, you can't just assume $k>1$, but you can realize that you actually don't have to either

Astyx

^

polite proofs

Okay, well, $ \lim_{n \to \infty} \frac{\tfrac{a_n}{k + N}}{a_n} = \lim_{n \to \infty} \frac{1}{k + N} > 0,$ so then $a_n$ and $\frac{a_n}{k + N}$ diverges. Then $\sum \frac{a_n}{k + a_n} > \sum \frac{a_n}{k + N}$, so $\sum \frac{a_n}{k + a_n}$ diverges

I think that is sufficient

Thorgott

yes, but there really is no need to do the limit thing

polite proofs

02:47

Why not?

Thorgott

you can tell me directly from definition why $\sum\frac{a_n}{k+N}$ diverges if $\sum a_n$ diverges

polite proofs

You mean like $\tfrac{1}{k + N}\sum a_n$?

Thorgott

yes, but spell it out carefully and rigorously for good measure

Ted Shifrin

@robjohn I've bought an air fryer and am learning its ways. Roast chicken cooked the other day. Just roasting cauliflower and putting it in a creamy parmesan sauce :P

polite proofs

@Thorgott I would like to keep the limit comparison test rather than derive the proof for a positive constant multiplied by a divergent sum doesn't change it diverging

:/

I don't see an issue with using it?

user 85795

02:59

@TedShifrin would you prefer high school students to learn geometry before trigonometry professor?

Thorgott

this fact is much more elementary than the limit comparison test and should be clear

if you want to understand divergence, you should understand why this fact is true

Ted Shifrin

@user85795 Of course. On the other hand, most of classical Euclidean geometry has left the US curriculum.

polite proofs

I do know why it's true, I guess

Thorgott

there's nothing formally wrong with using the limit comparison test, but a) proofs by unnecessary overkill are usually considered bad mathematical writing and b) see the didactic point above

polite proofs

We have a sum of partial sums

Ted Shifrin

03:01

I want to emphasize that what @Thor just said is super important, @polite. Part of what suffers in "self-learning" is that you don't get the professor's insights on how to think about mathematics. If you're not going to listen to us, don't waste our time.

polite proofs

And multiplying that by a nonzero constant can't change if our sum of partial sums diverges

Thorgott

yup, polite, that's all there is to it

polite proofs

@TedShifrin I'm trying, I just don't know how to apply it

@Thorgott Oh, well I didn't know what kind of response you were looking for

Yes I agree with that

Thorgott

in general, it's always instructive when one encounters a general/more abstract theorem to first understand the more elementary/direct corollaries by hand

because the idea behind the fact that a non-zero multiple of a divergent series is still divergent is actually used in the proof of the limit comparison test

Ted Shifrin

And similarly for convergent.

polite proofs

03:05

Got it :)

Thorgott

I hope @Ted is happy with my transparency this time around

robjohn

@TedShifrin That sounds good! I just finished some fried rice from a local Chinese restaurant.

user 85795

In other news:

in This Is Fine, 10 mins ago, by Jolenealaska

Mexico's president tests positive for COVID-19

zacts

03:38

hello

Ted Shifrin

@Thorgott I'm not here to judge you, @Thor, just to bitch.

zacts

I just ate some garbanzo beans ++ coffee.

copper.hat

i am reading about covering maps for the first time. not sure how i lasted this long skipping munkres' ch. 8.

goes well with my dinner as i pile layer upon layer of pork onto my brioche bun.

Ted Shifrin

03:55

Southern pulled pork or something more bahn-mi?

@zacts Do those go together?

zacts

@TedShifrin yeah, but it's more of what I just happened to be eating rather than those being something that usually go together. The coffee does taste good with the beans, however.

Ted Shifrin

Interesting :)

user 85795

Jan 17 at 3:26, by user 85795

Have you @ted heard of Dolciani's introductory analysis book?

polite proofs

Seems a little bit strange to 'review precalculus' when learning analysis

copper.hat

@TedShifrin more the Southern variety :-)

zacts

04:07

I don't eat pork.

I like bahn-mi sandwiches tho

tofu

it's one of my favorites

copper.hat

tofu is too processed for my liking

i like soy beans, especially as tempeh

Ted Shifrin

I'm no vegetarian, but it's far healthier than meat.

copper.hat

apparently vegetarians have increased risk of stroke vs meat/dairy eaters who have increased risk of heart disease.

Ted Shifrin

@user85795 dolciani wrote high school books, not what we’re talking about.

Really?

I have heart disease already.

copper.hat

well, the quote i have is "In a study of 48,000 Britons, vegetarians were unusally resistant to heart disease, but prone to strokes".

polite proofs

04:13

That sucks. I hope you'll get through it fine

copper.hat

Economist, "The meat spot".

zacts

what kind of study was that?

copper.hat

No references unfortunately, but the Economist seems fairly reliable.

Ted Shifrin

What's the connection to stroke? I get the cholesterol and heart disease.

zacts

I can't accept the Economist on this without understanding how the study was conducted.

user 85795

04:15

@TedShifrin wouldn't it be a good preparation for spivak professor?

zacts

or what type of study it was

copper.hat

Well I think the idea is that cholesterol may be needed for blood vessel walls.

Ted Shifrin

Well, the body produces it regardless.

copper.hat

The subtitle of the short article is "Japan's rapid decline in deaths from strokes may be partly explained by consuming more meat and dairy"

Ted Shifrin

It was a standard junior year precalculus course back in the 60s and 70s.

user 85795

04:17

and regular exercise counts for a lot

copper.hat

I think genetics has a significant component. Exercise carries its own risk.

Ted Shifrin

I'll continue to eat meat in moderation and lots of legumes, fruits, and vegetables.

copper.hat

I think the moderation word is the key there :-)

Ted Shifrin

And I cook everything healthfully.

But two heart surgeries and a grandpa who died of heart attack.

polite proofs

So off-topic chat is permitted here, assuming no real math discussion is ongoing?

Ted Shifrin

04:20

LOL ... apparently.

user 85795

yup

polite proofs

(That was a legitimate question, I have no clue about the semantics of the rules)

copper.hat

Strokes seem to be more on my side. Albeit heavy meat & diary consumers as might be expected from being Irish.

Oops, @politeproofs, I may stray from time to time.

Like from 8am until 8am.

I do answer (& ask) some questions.

polite proofs

I just wanted to ask, how's the situation with the virus in the US?

copper.hat

Virus, what virus?

He just left.

user 85795

04:21

lol

not without a fight

copper.hat

I am trying to show that the composition of covering maps need not be a covering map.

I understand the idea, but am trying to convert my scrawls into a concrete example.

Just showing that there is some math going on on this side of the keyboard :-).

Clarinetist

I have a ridiculously elementary question:

Ted Shifrin

Ah, classic question.

user 85795

Askaway

Clarinetist

Is the following true for an $x \geq 0$ fixed and for each integer $y \geq 1$: $$x^{1/y} \leq \max(1, x)$$

(the idea here is this: I am trying to bound $x^{1/y}$ from above with something that is NOT dependent on $y$)

Ted Shifrin

04:26

Sure, it's right.

copper.hat

yes.

Clarinetist

I'll stick with this then, thank you both!

copper.hat

write it as $x \le \max(1,x^y)$

Clarinetist

Ah, that makes it more obvious, thanks

Ted Shifrin

Seems equivalent to me.

user 85795

04:28

"obvious" :-)

<_< >_>

copper.hat

i think i saw Paul Chernoff do that once in a class. Someone asked a question and he said its obvious. After class, he looked at the board for a moment and then said, yes it was obvious.

Ted Shifrin

There are zillions of jokes/examples.

user 85795

and a faculty of math education devoted to it

:-)

polite proofs

Oh, interesting, you are mentioned in Spivak's book.

That is very cool

Ted Shifrin

I told you why earlier. :)

copper.hat

04:40

manifold reasons

user 85795

practical applications

polite proofs

05:12

Hmm, so after flipping through some pages of Spivak, I think I might agree that the exercises are more difficult than Apostol

It's just frustrating that Spivak doesn't cover linear algebra and differential equations

zacts

I think Apostol looks more appealing to me at the moment than Spivak.

copper.hat

Munkres uses the notation $a \times b$ to mean the pair $(a,b)$ which is logical but I find very hard to read.

polite proofs

It's logical?

zacts

I also like the way the limit is described in another textbook.

the way it described it informally was really kind of cool.

Ted Shifrin

05:28

Spivak's book had more than I could cover in a year. He wasn't trying to write a complete two-year textbook.

polite proofs

@TedShifrin Do you think I should switch?

I'm at a real dilemma

:/

Ted Shifrin

@copper We've discussed this several times here. Intervals and ordered pairs both appear.

I think that for what I perceive as realistic goals for you and where you're headed, Apostol is preferable.

zacts

I kind of felt like Spivak's first few ch were leading to an idea, but I never quite got to it.

polite proofs

Well, OK, I'll stick to it then

I'm about 1/3 of the way through the book anyway

Ted Shifrin

You can't just jump into chapter 5 without laying foundations, zacts. I experimented in my 15 times teaching it, but you need underpinnings.

zacts

05:33

@TedShifrin I think I'm going to stick with Apostol for now. I might check out Spivak eventually tho.

I really like the definition of the limit in this other text too.

polite proofs

I think the limits chapter was done terribly in Apostol's book

There were no $\epsilon-\delta$ exercises whatsoever

It is very strange

zacts

the book I'm mentioning is neither Apostol nor Spivak

The reason for my initial interest in Apostol/Spivak is that I'm looking for something to let me connect this stuff together a bit.

Ted Shifrin

It's not meant to teach analysis, whereas Spivak is. Very different goals.

zacts

For example, I could initially solve derivatives like as exercises, but I wanted to conceptually connect the limit to the derivative

the derivative is a limit by definition

polite proofs

What does it teach, Ted?

Ted Shifrin

05:44

Calculus.

polite proofs

The strange thing is that Apostol provides proofs for every theorem

And they're pretty rigorous

Ted Shifrin

It was written for Cal Tech students. A harder course than Thomas, Stewart, etc.

That was calculus in the 60s at the best universities. Then everything got watered down.

But he doesn't make students do so many proofs.

polite proofs

I know, that's my only complaint :(

Ted Shifrin

Yes, his proofs are totally rigorous.

As I said, very different goal. He covers three years of material.

In the two volumes.

zacts

How is Apostol's Calculus Vol I/II three years of material?

it seems like maybe two semesters.

Ted Shifrin

05:49

I wrote my book for a year course in integrated linear alg and multi calc/analysis. I didn't jam in more stuff.

polite proofs

I don't want to be stuck using abbott for analysis if I'm not ready for Rudin :(

Ted Shifrin

You have no clue, zacts.

zacts

very likely

Ted Shifrin

At many universities, the calc/multi calc is 4 semesters, without all the theory. Then add linear algebra, differential eqns, probability, and more.

zacts

I meant that as a serious question, as I haven't started the book yet. Is it really that dense of material?

I was hoping to maybe cover some of it within a couple of semesters.

like I think this covers much of Vol I within a single semester: ocw.mit.edu/courses/mathematics/…

but it looks like they might skip around a bit, don't know yet.

Ted Shifrin

06:08

Come on, man. MIT is the only place in the US that does all of calculus (very few proofs) in one year.

zacts

ah I see, ok. :-)

copper.hat

certainly in my case, i relearned the material time and time again. each time you see something new.

zacts

@copper.hat that's a good point

dc3rd

07:26

Hi all. I'm feeling bad constantly bothering Professor Shifrin about a question I've been working on. He's gone beyond what is necessary, but I'm still a little stuck. Would somebody be able to perhaps help me?

(The question) [https://math.stackexchange.com/questions/3994814/find-the-vector-given-the-dot-product-and-cross-product-of-a-set-of-vectors#3994814]

zacts

08:15

@TedShifrin ^^

Balarka Sen

08:26

@user193319 Got it.

So these are the ones whose elements can be separated by asymptotic homomorphisms to $(U(n, \Bbb C), d_n^{HS})$. Got it.

Sofic groups are special cases because HS-distance of permutation matrices squared is Hamming distance, or something like this

Is that right?

I think Lubotzky gave examples of groups which are not hyperlinear

robjohn

10:26

@zacts you might be able to use the illustration I used for this answer, with the names changed for security.

zacts

@robjohn: @dc3rd asked the question.

robjohn

@zacts yeah, but an answer could use an illustration similar to that one.

zacts

ah yeah

zacts

10:57

the base step:
$1+2+3+\cdots+n = \frac{n(n+1)}{2}.$

the inductive step:
$\frac{1(1+1)}{2} = 1.$

$1+2+3+\cdots+n+(n+1) = \frac{n(n+1)}{2}+(n+1).$
$1+2+3+\cdots+n+(n+1) = \frac{n(n+1) + 2(n+1)}{2}.$

$1+2+3+\cdots+n+(n+1) = \frac{n^2+n+2n+2}{2}.$

$1+2+3+\cdots+n+(n+1) = \frac{n^2+3n+2}{2}.$

$1+2+3+\cdots+n+(n+1) = \frac{(n+1)(n+2)}{2}.$

^^ how does this look for my inductive proof of the classic theorem?

Note: I'm just getting started with this, and I'll have more questions on subsequent problems, but the main thing I want to know is if my written proof looks good.

oops

$\frac{1(1+1)}{2} = 1$ should be under the base step.

but yeah, other than that.

123

Hi All..

zacts

hi @123

zacts

11:12

$1+3+5+\cdots+(2n-1)=n^2$

Base case:

$1^2 = 1$

Inductive step:

$1+3+5+\cdots+(2n-1)+[2(n+1)-1]=n^2+[2(n+1)-1].$

$1+3+5+\cdots+(2n-1)+[2(n+1)-1]=n^2+2n+2-1.$

$1+3+5+\cdots+(2n-1)+[2(n+1)-1]=n^2+2n+1.$

$1+3+5+\cdots+(2n-1)+[2(n+1)-1]=(n+1)(n+1).$

$1+3+5+\cdots+(2n-1)+[2(n+1)-1]=(n+1)^2.$

^^ how does this look?

next time I think I'll use MathB.in for this.

Edward Evans

11:47

There's no need to write out the lhs every time

zacts

12:03

@EdwardEvans I'm going to use MathB.in instead for these posts, so it only takes a single line.

More Anonymous

12:35

Hey all!

Does anyone know where I can learn c* algebra from?

I wanted to learn it in the contet of robinson lieb bounds

(physics)

Jake Rose

3

Q: Variant on divergence theorem

If I want to prove that for any scalar field $f:\;\mathbb{R}^3\to\mathbb{R}:$ $$\int_V \boldsymbol{\nabla} f\;\mathrm{d}V=\int_{\partial V} f\;\mathrm{d}\mathbf{S}$$ Can I apply the divergence theorem to $\mathbf{a}_1=(f,0,0),\;\mathbf{a}_2=(0,f,0),\;\mathbf{a}_3=(0,0,f)$ and then stack the equal...

Does this form of the divergence theorem have a name?

It feels somewhat more general

User 873110

12:56

Consider the link math.stackexchange.com/questions/1874740/… and assume $f$ is a proper map. Does this imply $\widetilde f$ is also a proper map?

Jack Rod

14:32

Sir can u reactivate my room?@robjohn

robjohn

@JackRod let me look

A-Level Student

14:56

A couple of my questions I asked on SE Acedemia haven't been answered fully, see this link:academia.stackexchange.com/questions/159964/… So, here is a question to any one of you mathematicians: What exactly do research mathematicians do? Do they simply think all day and write/type their thoughts and results and progress as their job?

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