Mathematics - 2022-10-04 (page 2 of 2)

5:00 PM

Well, it's not expected for us. So why don't you put the question in your own words instead of pasting things in here that make no sense?

PM 2Ring

You can run the ChatJax bookmark on a phone.

Ted Shifrin

@PM2 Yes, but all that typing sucks.

Even using my iPad, I go crazy trying to type much in this chatroom. The phone is hopeless.

PM 2Ring

Admittedly, it does get a bit painful doing multi-line MathJax stuff in Chat. I usually do that on the main site, and transfer it here.

Ted Shifrin

If I have to do serious math here, I want to be on my desktop.

si-LV-er_and_b-LA-ck

I'd have to agree with the good professor that concise well thought out sentences are the best tools in here.

Ted Shifrin

5:05 PM

looks around to see about whom si-LV is talking

si-LV-er_and_b-LA-ck

:-D

Daniel used to use them masterfully.

Among others, of course.

ThunderGlove

@TedShifrin Sorry sir if I have hurt you, I have pasted that thing since I did not observe that typo, will make sure to take care from now onwards

Ted Shifrin

Are you referring to long-missing Daniel Fischer?

si-LV-er_and_b-LA-ck

Yes.

Ted Shifrin

He abandoned us once he became a moderator.

PM 2Ring

5:10 PM

Maybe I'm a glutton for punishment, but I wrote this on my phone, including all the MathJax (& the Sage / Python coding): physics.stackexchange.com/a/718837/123208

Ted Shifrin

@PM2 Maybe?

si-LV-er_and_b-LA-ck

Impressive.

I hope you weren't driving also :P

PM 2Ring

I started writing that answer for a recent question, which got dupe-hammered when I was halfway finished. So I modified it slightly & posted it on the dupe target.

Thankyou, Mystery Upvoter. :)

si-LV-er_and_b-LA-ck

@TedShifrin I remember when he was a mere nonmod mortal and masterfully won arguments in here with the likes of Asaf and others.

Ted Shifrin

I showed up after Asaf's time. I've been here about 10 years or so.

But anon, Pedro, and Daniel were mainstays.

(Actually, back then, Pedro was Peter.)

si-LV-er_and_b-LA-ck

5:22 PM

Indeed.

Ted Shifrin

I have kept up with Pedro, but have no idea what happened to anon. I would like to think he went off to grad school somewhere, but no idea.

si-LV-er_and_b-LA-ck

Has Peter graduated?

Ted Shifrin

Yes. He is now a postdoc in Germany.

He did his Ph.D. in Dublin.

si-LV-er_and_b-LA-ck

Nice.

PM 2Ring

For years, I was told that elliptic integrals & elliptic functions are terrible things because they can't be expressed in terms of elementary functions. True, they are a bit scary, but they're easy to evaluate using the AGM-based algorithms, and they converge fast. In comparison, evaluating integrals using stuff like Simpson's rule is really slow & prone to floating-point point errors when your step size is too small.

Ted Shifrin

5:25 PM

Some of us care about them without even trying to get numerical answers.

PM 2Ring

Of course. But sometimes we do actually want numerical answers. Eg, computing geodesics on an ellipsoid.

Ted Shifrin

@Xander: You like my snide comment here?

@si-LV-er_and_b-LA-ck No.

I was thinking more of algebraic curves and abelian varieties than Riemannian geometry at the moment, @PM2. :)

Fargle

What would you recommend instead of Rudin's PMA if someone wanted to approach real analysis for the first time, @Ted? I actually was recently asked the question by someone interested in learning more about the basics of the subject, and found myself short on book recommendations.

Ted Shifrin

I guess leslie and copper might also appreciate that snide comment.

si-LV-er_and_b-LA-ck

Why did rudin avoid geometry so much?

Ted Shifrin

5:32 PM

I haven't studied Terry Tao's texts, but I guarantee they're better. I sort of like William Wade's book. Personally, I think that Spivak's Calculus plus my multivariable book gets almost all the analysis you need, except for metric spaces. Probably Protter and Morrey's analysis book is good, too.

Because he's a formalist, I guess. People have their styles. His style is good for a small segment of the mathematical student population.

Fargle

I did recommend your YouTube lectures since they've had a calc 3 course and are currently in linear algebra.

PM 2Ring

@TedShifrin Understood. Still, the impetus to study elliptic integrals & functions and develop their theory by Gauss etc came from practical problems in astronomy & geodesy.

Fargle

Spivak was the obviously-missing-from-my-recommendation secret sauce, though.

Ted Shifrin

I still remember that the analyst at MIT who taught me out of Rudin was the sort who, when he lectured multivariable calculus, set up every polar coordinate double integral in the order $d\theta\,dr$ because, as a probabilist, he'd never encountered a region in which $r=r(\theta)$. I had to inform him that there were plenty of exam questions for which his students wouldn't be prepared.

@PM2 Yes, I concur, of course.

Fargle

@TedShifrin Haha, are conic sections a myth to probabilists?

Ted Shifrin

5:35 PM

No, but he was prepared only to integrate over disks centered at the origin. Surprising, actually, since there are some standard expectation problems where the origin goes on the boundary circle.

Conic sections weren't the point. More complicated regions were the point.

I don't remember if he ever drew pictures in our introductory analysis course, now that I think of it.

si-LV-er_and_b-LA-ck

Formalists and probalilists must get along very well :P

Ted Shifrin

I shouldn't be too mean, as this gentleman died a few years ago.

anak

Like 12 years ago.

Oh Ted isn't talking about Rudin himself, misread

Fargle

@TedShifrin I see. Well, everyone's got a quirk. Some quirkier than others, but, hey.

copper.hat

@TedShifrin I think your comment was reasonable, I would not characterise as snide.

Ted Shifrin

5:38 PM

We really are living in hell.

si-LV-er_and_b-LA-ck

A hell of a jungle...

PM 2Ring

picard_facepalm.jpg, op cit.

si-LV-er_and_b-LA-ck

...sometimes I wonder how I keep from going under.

Ted Shifrin

Six Feet Under? One of the greatest TV shows ever.

si-LV-er_and_b-LA-ck

The Message by Grand Master Flash, actually :-)

Koro

5:44 PM

A Course of Pure Mathematics

A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went through ten editions (up to 1952) and several reprints. It is now out of copyright in UK and is downloadable from various internet web sites. It remains one of the most popular books on pure mathematics. == Contents == The book contains a large number of descriptive and study materials together with a number of difficult problems with regards to number theory analysis. The book is organized into t...

PM 2Ring

And the Furious Five. Not quite Ted's kind of music.

copper.hat

@Koro ?

Koro

a recommendation apart from Rudin's PMA

Fargle

That was at me, copper, I had asked for book recs for someone interested in real analysis.

Koro

you'll find the book online for free also.

Ted Shifrin

5:46 PM

I actually have never looked at Hardy's book. I wonder if it's a bit too old-fashioned. The lovely Hardy & Wright Number Theory classic is rather old-fashioned but still a gem.

Koro

I'd purchased Amazon e book of the above book, but the print was horrible.

PM 2Ring

I loved Hardy & Wright. Its critics say it's not well-organised. But the book's intro warns you about that. ;)

Akiva Weinberger

The multiplicative identity of $2\Bbb Z/6\Bbb Z$ is $4$

copper.hat

@Fargle i liked marsdens elementary classical analysis (the blue cover).

Koro

Fargle: in older prints of the book, notations for open interval and closed interval look the same ( ).

I guess by that time (around 1920s), the set theory was not so popular.

But in the revised prints of the book, that problem is fixed. Also, some of the words used that time have been replaced by the words used more frequently now (e.g. 'save' is replaced at places by 'except') and delta's and epsilons have been interchanged at places.

si-LV-er_and_b-LA-ck

5:52 PM

They say Moore brought set theory into popularity during the 40/50s...

anak

Abbott tends to be the popular "modern but not Rudin" suggestion.

Koro

You may also try Zorich's analysis book.

copper.hat

the ]0,1[ notation is ugly imo

anak

Victor Bryant has a really nice "stepping stone" of a book for analysis which I would recommend as a intermediate course for pure math majors.

Koro

@copper.hat I remember someone suggesting that on mse :-).

Ted Shifrin

5:53 PM

@copper.hat A very odd thing about that book. All the proofs were in the backs of chapters.

Abbott is not very substantial at all.

anak

After Bryant, I have no good recommendations as I have been thoroughly convinced that a good intro to analysis book does not exist.

Koro

I don't know why Spivak's style in Manifolds book is so different than style in Spivak's Calculus.

anak

But for a second course in analysis, Carothers is the best and you can't convince me otherwise.

Ted Shifrin

Because Spivak's Calculus on Manifolds was the first book he wrote, right out of grad school. And he was limited to that page length.

Koro

and that wasn't revised (in the sense of adding more explanation) in subsequent editions?

Ted Shifrin

5:56 PM

All the books in that (originally Benjamin) series were very brief.

There were no subsequent editions.

Koro

Oh I see.

Ted Shifrin

But that's why Munkres wrote his multivariable analysis book and why I wrote mine (although my audience is somewhat different — I really wanted it to contain all the standard computational stuff as well, plus the linear algebra).

I'm somewhat surprised that no one's commented further or answered that linear algebra question about $2n$ vectors in $\Bbb R^n$. I put my comment about half-spaces, but that's it.

si-LV-er_and_b-LA-ck

Would you ever consider starting another book? @TedShifrin

or ebook

Ted Shifrin

Nope. The only way to write a good book is to have guinea pigs — i.e., students.

All my books changed substantially as I revised them, teaching courses out of them. Both the text and the exercises.

si-LV-er_and_b-LA-ck

Right, 'the proof of the pudding' and all that :-)

Ted Shifrin

6:03 PM

Although I would substantially rewrite my algebra book, there's really no interest. And to write some graduate differential geometry notes, I'd rather leave that to younger folks.

anak

Doesn't seem like many young folk are writing books.

si-LV-er_and_b-LA-ck

You could collaborate with those younger folks on the discord algebraic geometry channel.

Ted Shifrin

Not traditional books. But there are LOTS of lectures notes TeXed up all over the place.

anak

ew alg geometry

PM 2Ring

@ShadowTheKidWizard I'm pleased to announce that Saves is now officially live on Stack Overflow and across the network sites! — tanj92 1 hour ago

> The migration process will take some time to process as there are millions of bookmarked questions to migrate over. If you’re not seeing any questions you previously bookmarked, that means they haven’t been processed yet.

copper.hat

6:42 PM

i just can't wait for my saves

anak

What are saves

Shinrin-Yoku

If a function is eventually constant and lim derivative exists prove lim as x-> infinity =0

.

Hint: Use mean value thm.

Any help @anak?

anak

6:57 PM

what is a lim derivative

si-LV-er_and_b-LA-ck

@anak things you've saved

Sorta like bookmarks

anak

There are saves?

I know there are favourites and upvotes.

Shinrin-Yoku

@anak lim derivative is derivative at limit

anak

I still don't know what you mean. Can you not write it with math?

Shinrin-Yoku

Do you not know the def of a limit and that of a derivative?

@anak

Or is that a “Socratic” question?

anak

7:06 PM

I know the definition of both of these terms, but I am not clear what your question is because as written it's ambiguous or missing details.

Shinrin-Yoku

Well lim derivative is lim of derivative at infinity

it’s a well known technical term IMHO. @anak

si-LV-er_and_b-LA-ck

@anak it is on your profile page just below your location opinion

Profile Activity Saves Settings

anak

@Shinrin-Yoku So I guess you mean $\lim_{x\to\infty} f(x)$ exists (where $f$ is your function)? Out of curiosity, can you show me a page that uses this "well known technical term"?

Shinrin-Yoku

Yes@anak

That is like asking me to show a page which says that inf/inf != 1 …

anak

I can show you many such pages, so it would be easy.

Oh, I forgot the apostrophe in my original equation, but I think we get the point.

Shinrin-Yoku

7:13 PM

Maybe, but the thing is I can’t find a page of the top of my head all I think is that when we say lim without a number at the bottom the number is infinity.

anak

For context, is this an analysis course, or just plain calculus, or what?

Shinrin-Yoku

The course title is “Differentiable topology with a special view toward alg geometry”

@anak

no just kidding

of course it’s analysis

Solved the problem nvm

anak

So what have you tried?

Oh, great.

Shinrin-Yoku

Thanks

Jakobian

Tomorrow is my first day of masters

anak

7:20 PM

@Jakobian congrats!

Jakobian

Anxious

Thank you. I'm studying in the same university as before

anak

Nothing to be anxious about, then! You are already royalty!

Xander Henderson

7:54 PM

@TedShifrin Couldn't have said it better myself.

Rudin is acceptable for single variable calculus / analysis. The wheels fall off in the last couple of chapters, and you should find a better book.

Even then, there are some things which Rudin makes far harder than they need to be. My go-to example is the mean value theorem, which he presents with zero intuition---he just pulls a magic function out of a hat and *poof*! the mean value theorem!

si-LV-er_and_b-LA-ck

8:11 PM

How come your stars around "poof" don't make it italicized?

poof!

poof!

Xander Henderson

@si-LV-er_and_b-LA-ck I have 3|337 h4X0r 5ki||z.

si-LV-er_and_b-LA-ck

I can only guess at what that means, sir.

Does 5ki||z =? skills

hyper-neutrino

(jokes aside, it means "elite hacker skills" ("eleet haxor skillz") and the way he's doing it is \*poof\*)

leslie townes

nobody reads past chapter 8 of rudin. you can also tell a faker by whether they claim to have read and benefited from 'real and complex analysis.'

si-LV-er_and_b-LA-ck

Thanks for clarifying that @hyper-neutrino

copper.hat

8:20 PM

as opposed to reading his alternative text, 'real complex analysis'

hyper-neutrino

what are they analyzing and what makes it complex

si-LV-er_and_b-LA-ck

the formalist school dominated the new math

Xander Henderson

8:33 PM

@hyper-neutrino Aw... spoiling my fun. :(

*pouts*

si-LV-er_and_b-LA-ck

Your "fun" amounts to a very common trait among some educators who play the "I have a secret game and it is your job as a student to try and find out what it is." >8(

leslie townes

nah, leetspeak is just pure fun.

si-LV-er_and_b-LA-ck

The JEE is considered pure fun for those who make the exam.

Xander Henderson

@si-LV-er_and_b-LA-ck Weird that you would go from "Xander is playing a game" to "this is the kind of game educators like to play".

leslie townes

well, the JEE is not pure fun.

Xander Henderson

8:43 PM

Also, with respect to leetspeak, learn your history. :(

si-LV-er_and_b-LA-ck

nor are the student suicides that result, sir

leslie townes

we used to use leet speak to request illegal copies of software ("wares," "warez," or in some aggressive instances "JU4R3Z") on forums that had a rule against doing just that. it was good clean fun. it will not be besmirched by this other stuff.

the best part is that the admins knew what we were doing! they could read it too! but they pretended not to!

Xander Henderson

Indeed.

Ted Shifrin

@leslietownes I actually liked this book more.

leslie townes

that qualifies as a hot take! it's even more functional analytic than PMA.

Ted Shifrin

9:00 PM

But it makes sense to put the two subjects together. For example, complex analysis is a pain without dominated convergence.

si-LV-er_and_b-LA-ck

@TedShifrin the internet provides a huge supply of guinea pigs with MOOCs

Ted Shifrin

Not the same thing as interaction in a classroom and office hours.

si-LV-er_and_b-LA-ck

True.

Ted Shifrin

9:15 PM

@leslietownes But the audience for RCA is quite different from the audience for PMA. That said, I would never have taught out of it, but I was never thrilled with Ahlfors or Conway or other standard options. I was going to try Gamelin, but then I got cancer and didn't teach the graduate complex course that year.

leslie townes

i prefer conway to ahlfors but agree that they are both bad picks.

Ted Shifrin

Ahlfors was written by a mathematical master, at any rate. When I took the course as an undergraduate, the text was Nevanlinna/Paatero. It was OK, not great.

Xander Henderson

I am convinced that there is not really a good graduate level complex analysis text out there. Ahlfors is my go-to, but it is not amazing.

Ted Shifrin

I tried using Stein/Stakarchi and was very disappointed. Aside from some errors, the fact that the complex logarithm is delayed until halfway through the book and LFTs even farther than that is unacceptable to me.

I taught stuff when I wanted to, but the book was not as good as I'd hoped Stein would be.

Xander Henderson

@TedShifrin I like the Stein and Shakarchi series, but those books are clearly geared towards undergrads. And there often seems to be a big disconnect between the content of a chapter, and the exercises at the end.

That being said, I only have the volumes on real and Fourier analysis.

Ted Shifrin

9:21 PM

Princeton undergrads = graduate text.

Gamelin is supposedly undergrad too.

Xander Henderson

@TedShifrin I wouldn't want to teach a graduate course out of Stein and Shakarchi. There isn't enough THERE there.

Ted Shifrin

Lang’s CA isn’t bad, actually.

Xander Henderson

And I took undergrad Fourier analysis out of it---seemed fine for an advanced undergrad text.

Ted Shifrin

I added some exercises, but I had no issue with lack of content for a one-semester course.

I think Fourier is definitely intended for undergrads (no Lebesgue).

Xander Henderson

@TedShifrin No, my complaint isn't about the amount of material. I can't really describe what I mean by saying that there isn't enough THERE there---I just don't feel like the texts hang together like graduate texts.

There are some elisions (as you note, no Lebesgue; my recollection is that the discussion of convergence of Fourier series to functions is a little vague, etc).

And the complex logarithm should be, like, day one. Seems criminal to delay it as much as you have suggested they do.

Ted Shifrin

9:27 PM

Well, not day 1, but first few weeks. Same with LFT.

si-LV-er_and_b-LA-ck

"criminal" In what way?

Ted Shifrin

Not Tromp criminal.

si-LV-er_and_b-LA-ck

lol

Ted Shifrin

Poor mathematical taste and pedagogy :)

The multivalued nature of log is one of the most important things in mathematics.

Xander Henderson

@si-LV-er_and_b-LA-ck For the things I care about, nearly everything comes down to properties of the complex logarithm. It is the tool for understanding $n$-th roots, exponentials, etc.

I don't get as fussed about LFTs---they are pretty, but I must admit that I never really quite got the point. But that is my own personal failing.

si-LV-er_and_b-LA-ck

9:37 PM

I agree that it makes practical sense to advocate for early exposure over a logically rigorous development.

Xander Henderson

@si-LV-er_and_b-LA-ck Who is arguing for early exposure over a logically rigorous development?

In my opinion, the logarithm is central to much of the development.

Ted Shifrin

@XanderHenderson Automorphisms of $\Bbb CP^1$

Xander Henderson

@TedShifrin Ah. That's why I never saw the point. Projective spaces. *shudder*

But that does make sense, given how LFTs move things around on the Riemann sphere.

Ted Shifrin

9:58 PM

Riemann sphere = $\Bbb CP^1$

Xander Henderson

@TedShifrin I think I knew that at one point in my life. Derp.

I've forgotten too many things. :/

Ted Shifrin

Wait til you’re my age!

Xander Henderson

Indeed. Should I live so long.

Men in my family tend to drop dead in their 60s. :/

monoidaltransform

10:21 PM

what's a good book for complex geometry?

Ted Shifrin

Griffiths/Harris

monoidaltransform

is huybrechts book any good? @TedShifrin

Thorgott

what do you want to learn

monoidaltransform

kahler geometry

Thorgott

10:39 PM

differential geometrically?

Ted Shifrin

11:43 PM

Huybrechts has all the foundational material and proves, e.g., Kodaira embedding, but there's not serious applications to what I consider actual complex geometry/complex algebraic geometry. I don't know what you're looking for when you say Kähler geometry. A lot of modern-day complex geometry is a lot of PDE.

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$Mathematics$ Mathematics