Mathematics - 2021-03-23 (page 2 of 3)

robjohn

17:00

@RajorshiKoyal: Listen to the Shifrin!

leslie townes

i remember being baffled by averages at one point. most often it really is just the sum divided by the number of things that were added up. that's really all there is to it.

"performance" caused me to wonder. often when people attempt to measure performance, they have a benchmark, usually somewhat imaginary, which makes it quite possible that every measurement is below average, or every measurement is above average, and that 'average' isn't just what you get by averaging data. but no such information accompanied the problem.

Ted Shifrin

Speaking of being baffled by averages, I find this hard to grasp.

leslie townes

there's a lot going on there. i think i need to lie down.

Ted Shifrin

@love_sodam No, an interval of $s$ means some non-empty closed interval in the parameter space. (I'm saying closed because — even though I'm not sure who is transverse to whom in this — transversality is generally an open condition, so it fails on a closed interval.)

robjohn

@leslietownes The average mileage listed on a car sticker is one of those averages that you will always be below.

leslie townes

17:07

i was just looking at some of my 'greatest hits' on math.SE. i used to know quite a bit of math.

Ted Shifrin

That's a good bet.

Balarka Sen

@TedShifrin Your confusion lies in the fact that your equality would be true if the OP's identity was true for the random matrix which returns $A$ with probability $1/2$, and $B$ with probability $1/2$. But the entries of this random matrix are not independent at all.

leslie townes

i had a theory of a class action lawsuit based around my prius using a high school algebra mixing formula to tell me my cost per mile. i will share it sometime. i will retire on that lawsuit.

Balarka Sen

Indeed, it is only so when the random matrix is 1x1, in which case your identity is true :)

$\det((a)) = a$.

Ted Shifrin

I'm just talking about the linear algebra, a @Balarka, not the probability. That's what is baffling.

And, hi, by the way.

Balarka Sen

17:09

I explained that your linear algebra fact is applicable only in a non-independent setup.

leslie townes

why are people putting random variables in a matrix, anyway.

robjohn

@TedShifrin are you trying to tell me that det is not linear? wow...

Balarka Sen

Hi, Ted.

Ted Shifrin

I'm trying to remember what I taught 6 years ago. There's something with $E[f(X)] = f(E[X])$ ...

leslie townes

i've actually done better than the sticker with my prius c. or i did, back when i drove, in the before time. my commute was right in the sweet spot. 35-40mph the whole way. that's where the prius c shines.

Balarka Sen

17:11

Really, your fact tells OP's identity cannot be true in general.

So you have basically answered his question.

Ted Shifrin

Well, I removed it :)

Balarka Sen

You should repost it again as an answer!

Ted Shifrin

LOL

Balarka Sen

The existing answer is terrible

@TedShifrin You might be thinking of Jensen's inequality, which applies when $f$ is convex; $f(\Bbb E[X]) \leq \Bbb E[f(X)]$.

robjohn

@TedShifrin after looking at that, I feel like dusting my screen. All those little bits of lines everywhere.

copper.hat

17:13

even with independence it is a bit counterintuitive.

leslie townes

i think the average figures also account for inefficiency at cold temperatures, which we don't have here.

Ted Shifrin

Yes, @robjohn, but it was not meant as an exercise for thee :)

Simone

Hello

leslie townes

disappointed that the tex police haven't autocorrected "det" to "\det". is nothing sacred? what does it mean? d, e, t? we're just putting letters together?

Ted Shifrin

Go ahead and edit if you're going to mouth off about it.

Hi, Simone.

leslie townes

17:15

i'd prefer just to mouth off about it and not edit. doctor, it hurts when i do this. that joke. that's me.

Simone

Hello @TedShifrin did you write the wiki page about the Hessian?

Ted Shifrin

@Balarka: I went back and looked at my notes. I was conflating two things. One is that expectation is linear without any independence hypothesis. The other was the definition of $E[f(X)]$ in terms of the PDF.

@Simone I've done absolutely nothing on wiki.

I think I made a correction on one page some years ago.

Balarka Sen

Aha, I see.

Simone

@TedShifrin oh ok.

Balarka Sen

Linearity of expectation is very useful.

I like it a lot

Ted Shifrin

17:17

Indeed!

leslie townes

i sometimes edit legal errors out of wikipedia.

robjohn

@TedShifrin: Ha! I put an xkcd-style diagram into an answer and it got accepted. It only encourages me...

Balarka Sen

@TedShifrin Oh?

Oh!

Ted Shifrin

Yes, oh!

Balarka Sen

Haha took a while.

Ted Shifrin

17:18

@robjohn Oh, just what we need!

I deleted, now that you got it, a @Balarka :)

leslie townes

mathematical wikipedia is sometimes a junkyard of formulas and diagrams but it is still far better than it has any reason to be. legal wikipedia is crazy. there's 'internet lawyering' in a way that there isn't 'internet mathematicianing.'

Balarka Sen

Some of us should meet on Zoom someday and give talks to each other.

Ted Shifrin

So, lawyers are ethically bound not to put mistakes on their wiki?

@Simone Why did you think so?

@Balarka: I can talk about how to divide fractions. :)

Balarka Sen

I'd attend that talk

leslie townes

i actually wonder if i'm engaging in the unlicensed practice of law when i edit those pages. i hope wikipedia has its servers in a jurisdiction where i am a member of the bar.

Simone

17:22

@TedShifrin It uses many of the terms and notation you use, so either those terms are widely used or you wrote the page yourself, or one of your students did

leslie townes

terms like "partial derivatives"?

i'm kidding. the language does not look distinctive to me.

unless you're looking at the wikipedia page for hessians, german mercenaries in service of the british empire. i agree, somebody has been fiddling with that.

Simone

well not everyone uses \partial for example. Spivak never use the word "inconclusive"

Rajorshi Koyal

@robjohn I am getting 5 the answer is 7

There is the contracdiction

contradiction

Simone

of course I haven't read many textbooks on the subject in order to compare notation

leslie townes

spivak is a weirdo. i have certainly used the term 'inconclusive.' maybe i wrote that? i don't know.

Rajorshi Koyal

17:24

@robjohn Plz check this out..

Simone

that's why I asked

@leslietownes How dare you insult sempai?

leslie townes

i realize it's basis dependent, hessian 'matrix' and all. but i'm surprised there isn't more coordinate-free, or you-choose-the-coordinates, discussion.

whoever wrote this was not a geometer.

Ted Shifrin

@Simone: I appreciate the fact that you think I am all-powerful, but most of my notation and vocabulary is pretty standard! :P

Rajorshi Koyal

@leslietownes I am posting the whole problem if that is better

Ted Shifrin

@leslietownes So you must like it?

Simone

17:27

@TedShifrin got it! :P

leslie townes

i guess? i've kind of painted myself into a corner here. yes, i like it. it's just a big bag of numbers. just the way things oughta be.

Simone

damn, spivak is 80

Ted Shifrin

@Simone: That said, the bordered hessian does not get enough publicity. I only have it in my book in a few exercises. If I ever rewrote the book (which I won't), I would include it in the text.

I would have thought Spivak was older, actually.

leslie townes

yeah, that seems young to me. i liked exactly one of spivak's differential geometry books. the blue one with the hippy paintings on it. it made up for all the other ones.

robjohn

@RajorshiKoyal you've referenced something that I wrote and asked me to "check this out"?

leslie townes

17:30

he seems like he would be an annoying dinner guest, but that book can stay.

Ted Shifrin

@Simone The obvious proof that I had nothing to do with that page is that my book is not listed as a reference. It surely would be if I'd done it :)

robjohn

@RajorshiKoyal Post the whole problem on main

Simone

@TedShifrin :D

leslie townes

i'm sorry to hear that. i tend to judge people on no information. it's one of my chief faults.

Simone

His wiki says he was born in 1940

Ted Shifrin

17:32

Yeah, I thought we were in 2022. My fault.

leslie townes

i actually do like geometry. when i learned the gaussian curvature was invariant under isometries, my brain grew one or two sizes.

Ted Shifrin

Yeah, the Theorema Egregium does surprise people, as curvature is defined extrinsically in the first course.

Spivak also created a bunch of TeX fonts and invented LAMS-TeX. He doesn't get much credit for those.

Simone

wow

Ted Shifrin

LAMS-TeX had a lot of sophisticated commutative diagram power. I used it for a while.

Simone

Why don't you make an online addendum on the bordered hessian? :P

Ted Shifrin

17:35

Because I don't want to ruin my exercises.

Simone

make new ones hehe

Ted Shifrin

Nah.

leslie townes

i wish i could remember the book that wu taught the first course out of. it was really good.

Ted Shifrin

You said it was bad.

shakes head

leslie townes

there was a chapter i didn't like. near the end it tried to be like, this is an introduction to riemannian geometry. that didn't work.

but the stuff on space curves and surfaces was good.

Ted Shifrin

17:36

That's why I said Milman & Parker. They have that.

I asked you if it used classical tensor notation (lots of subscripts and superscripts). They followed Chern and did that.

leslie townes

it may have been. my memory is blended up with the graduate course, which used jost. and was also a ton of superscripts and subscripts and christoffel crap. but the vintage and out of printedness of milman and parker seems to check out. my notes are molding away somewhere above our garage.

Thorgott

bordered hessians are cool

Ted Shifrin

Milman and Parker definitely has some flaws. As I said the other day, I taught out of it a couple of times, but it wasn't great.

leslie townes

the first printing of jost had so many typos in it. like, exercises were wrong in ways you couldn't even figure out how to fix. worst springer book i ever purchased.

Thorgott

josts riemannian geometry and geometric analysis?

leslie townes

17:40

yes.

i learned a lot out of guggenheimer's differential geometry in dover. nobody assigned it, i just used it. also an explosion at the tensor factory.

something about germans and tensors.

Thorgott

that book is for mutants anyway

leslie townes

explosion at the tensor factory would be a good name for a band. maybe that's my next project.

Simone

unironically a good name for a band.

leslie townes

i remember doing these exercises with all these indices. i can manipulate indices. x = y. i can do that. never retained any of it.

there must be some kind of reason for all the indices.

Ted Shifrin

Tensor analysis is very powerful. We have to use indices for matrices a lot of the time (if you're doing concrete examples, not theory).

leslie townes

17:49

looks like everybody at berkeley still uses millman and parker for the undergrad class. i bet that was it.

Ted Shifrin

The two times I TA'd unofficially for Chern, he used his notes one time and DoCarmo one time.

Rajorshi Koyal

Ted Shifrin

DoCarmo was way too hard for most of the students, so I was an interpreter in "optional" homework sessions.

Simone

@leslietownes that book is pricey.

Rajorshi Koyal

@leslietownes This is the whole question

Tell me the techniques in problem 4 and 5.

leslie townes

17:51

we used a cheap paperback reprint from a local copy shop. i assume they cleared the rights somehow.

Ted Shifrin

@Rajorshi I've asked you to stop posting your questions and giving orders.

@leslie Do not encourage this behavior.

@Simone It's really not that great a book. People use books out of inertia because it's too much trouble to find a good one. Lots of people all over the world use my informal book because it costs zero.

Simone

your informal book?

leslie townes

if you have a set of lecture notes drawn up around one book, why change?

Rajorshi Koyal

Sir @TedShifrin leslie responded previously to my question

Simone

you mean the Youtube lectures?

Rajorshi Koyal

17:53

That is why I reposted it

leslie townes

i would refer you to robjohn's comment earlier. that's all there is to it.

Rajorshi Koyal

Ok will do that for sure.

leslie townes

it's more of a 'calculus' style book than an upper division mathematics book. i can't decide if that's a criticism or faint praise.

maybe both.

Ted Shifrin

@leslietownes Faculty should not be this lazy.

@Simone No, no, I have an undergrad diff geo text.

leslie townes

i agree. this is one of the reasons i left academia, i wasn't good at ignoring annoying things.

Ted Shifrin

17:56

Well, differential geometry has very much a calculus feel to it. I like the balance of theory and concrete computation, actually.

leslie townes

and yet i spend half of my time telling my wife to be more lazy because none of her extra work will benefit her in any way. there are some real slugs in her department and she can't do better than them. the world is a complicated place.

Simone

@TedShifrin and it's free? chapeau..

Ted Shifrin

I did what I considered important and didn't subscribe to academic culture norms.

@Simone Yes. On the AMS Open Notes website and on my own. I just had a fourth publisher express interest in publishing it, and I said no.

Simone

That's very laudable Ted

Ted Shifrin

Of course. I'm just a supreme person :P

Simone

17:59

I think that every class eventually should do what the logic guys did

every subject

Ted Shifrin

What logic guys?

leslie townes

what did they do?

Simone

The open logic project

an open source textbook

openlogicproject.org

Ted Shifrin

There are too many views of how to teach various subjects. I doubt that is every going to happen with mainstream mathematics.

leslie townes

i sort of have a mixed view. truly open collaborative stuff, i think, can tend to lack a direction. it's necessary to make pedagogical choices even if they're suboptimal choices.

i subscribe to the 'auteur theory' of math textbooks.

Simone

18:02

but they did it. I can assure you that the textbook they came up with is more than adequate

Ted Shifrin

Let's face it. The average math textbook is horrible.

@Simone: One point is that logic is a very small field. Very few practitioners.

leslie townes

average, there's that word again.

Simone

@TedShifrin that's true

Ted Shifrin

Maybe you could get consensus from all the algebraists doing lattice-ordered groups on how to write a text on that subject, but you won't get consensus from algebraists — let alone mathematicians — on how to write "the ideal" algebra textbook. Pun intended.

copper.hat

average is a better word than mean.

Ted Shifrin

18:03

Don't be mean to me, @copper. I prefer math à la mode.

Simone

but do you need consensus? or just a group of like minded algebristas?

copper.hat

:-)

Simone

algebraists

leslie townes

i have a draft of a linear algebra book on my old computer. it alternates between pure theory and pure matrix mechanics. there's a recipe book at the back which shows how to do absolutely everything in terms of row operations. i think every algorithm in the book is in there. i've never seen this done in a book. you see hints of it but not the whole thing.

give me row operations and i can give you linear algebra.

Simone

my italian is percolating

Ted Shifrin

18:04

Don't give column operations short shrift :D

leslie townes

ok, column operations too.

Ted Shifrin

In my multivariable math book I had to do both row operations and column operations (row operations because of RREF and linear equations, column operations because determinants are functions of vectors, and all vectors are columns).

leslie townes

yeah, you do need it. a lot of the standard algorithms stack vectors as columns. it's the most natural way of doing it.

Ted Shifrin

It always blew my students' minds when I did column operations explicitly for ten minutes late in the course.

Of course, we soon proved that $\det A^\top = \det A$ and were spared this.

leslie townes

all of the stuff, like, find the set of linear transformations T sending these vectors to those vectors. that's column stuff.

Ted Shifrin

18:08

Yup. That's one thing I learned from Strang and stole ... playing off row views against column views throughout the course.

leslie townes

for a minute i was wondering who Stole was, then i figured it out.

Ted Shifrin

@leslie It would be interesting to see what you put together. Of course, you know I emphasize dot products and geometry from the first (well, eighth) page.

leslie townes

i am thinking about quitting my job and going on a kind of multi-month sabbatical in which i reset my parental relationship with my child, whom i have been ignoring. i may begin releasing math textbooks.

Ted Shifrin

Well, to be direct: You could spend the time with her instead of spending it with us.

leslie townes

well, not now. she's at day care. but yes. she might not need to go to day care.

they're better at teaching her spanish than i would be. but i think i could teach her art better. the stuff she paints at school looks like crap.

Ted Shifrin

18:11

The advantage of daycare is the built-in socialization with other kids. In the end, I think that's probably a good thing for them!

leslie townes

she doesn't speak spanish at home but if you speak it to her she answers in english, always contextually appropriate.

Ted Shifrin

LOL, the stuff I paint has always looked like crap. I can draw gorgeous math pictures and not even a decent stick figure of a dog or ostrich.

leslie townes

art at home means my daughter just demanding that i draw stuff. draw a whale! draw a crow! draw a barn owl! draw a shark! draw a dog! draw my cat!

Ted Shifrin

Easy: "No, you draw it!"

leslie townes

she likes scribbling over drawings that other people have done. i drew a canada goose and then she colored it in.

she's black, i'm blue

Ted Shifrin

18:18

Pretty good, actually. By scribbling I thought you meant coloring it in ... That really shows a good eye.

OK, back to math(s)!!

copper.hat

when my daughter was maybe 3 she called me to show me how she had tidied up my computer desktop by dragging everything into the trash bin. i was in the middle of closing a round of financing and had many versions of many documents on the desktop at the time. "that's great sweetie."

leslie townes

i'm so glad she has yet to get into my devices.

she understands my phone as a source of cat photos, i do not need her doing file management.

Ted Shifrin

Well, we see that these daddies are completely under the thumbs of their respective daughters. Next.

leslie townes

1

Q: one substitution in Chern's intrinsic proof of Gauss-Bonnet-Chern theorem

In Chern's proof for Gauss-Bonnet-Chern theorem, he claims that $$ \varepsilon_{i}u_{i_1}u_j\Omega_{ji_2}\theta_{i_3}\cdots\theta_{i_{2p-2k}}\,\,\,\,\Omega_{i_{2p-2k+1}i_{2p-2k+2}}\cdots\Omega_{i_{2p-1}i_{2p}}=P_k+2(p-k-1)\Sigma_k $$ where $$ P_k=\varepsilon_{i}u_{i_1}^2\Omega_{i_1i_2}\theta_{i_3...

oh haha already on ted's radar.

never mind, then. as you were.

i love it when people link to the original proof.

clearly chern was in the pocket of Big Subscript

Thorgott

oh god

flashbacks I didn't want to have

Balarka Sen

18:30

lol

trauma time bro

Ted Shifrin

Yes, I just spent ten minutes editing out someone's ridiculous \,\,\, adding spaces which made it unreadable.

leslie townes

so was he paid per subscript? or what was the deal with that

Leaky Nun

if you use replace then you don't need ten minutes

leslie townes

i wonder if they were using word. it automates the insertion of a ton of \, and turns good latex into a living nightmare.

Ted Shifrin

Yeah, that occurred to me after the first few, but I just continued; thanks, Leaky.

Now I have to figure out where he's getting his question in terms of what Chern does.

@Thorgott If you keep talking like this, I'll post my entire Ph.D. thesis.

@leslie You then copy paste Word into MathJax?

Thorgott

18:36

I never spent like a week unsuccessfully trying to understand your thesis, so at least that wouldn't give me unpleasant flashbacks

leslie townes

i don't know. people do all kinds of insane stuff. my wife's latex is nightmarish. i bite my tongue. i don't get involved. it's not my department.

Ted Shifrin

Chern's stuff is hard to read, Thor. It's OK.

Mine, I hope, is better, although there are a few lemmas that aren't fun.

I have more algebraic geometry and exact sequences in mine :P

Thorgott

exact sequences are, of course, a big plus

Ted Shifrin

I thought that would be a selling point.

And invariant cohomology.

I just was reminded that the journal messed up with the figures — right ones in the wrong places. I was sooooo upset when it appeared. I can't understand how that didn't get fixed with proofreading. Maybe they didn't give me a chance.

Thorgott

18:58

i think the category of group objects and group morphisms inside a complete category is itself complete

haven't written down the details, but consider this my not-daily category theory fun fact

robjohn

@TedShifrin I specialize in asymptotic sequences, which are far from exact ;-p

Ted Shifrin

Yes, well, you need to bundle up your troubles ...

user 85795

lol

in The h Bar, 8 hours ago, by fqq

"There are in this world optimists who feel that any symbol that starts
off with an integral sign must necessarily denote something that will
have every property that they would like an integral to possess. This
of course is quite annoying to us rigorous mathematicians; what is
even more annoying is that by doing so they often come up with the
right answer."
https://projecteuclid.org/euclid.bams/1183525452

Ted Shifrin

Physicists often come up with the right answers by magic. Enumerative geometry is full of such examples.

robjohn

@TedShifrin that just shows how robust math is, it even survives being used by physicists.

Balarka Sen

19:09

lol

Ted Shifrin

Maybe they are in charge. Ampere and all.

Balarka Sen

physicists are extremely good at coming up with the right answers

a lot of work in percolation is just trying to prove the crap physicists say is correct

robjohn

@BalarkaSen they have a very good analog computer that they like to call the Universe.

anakhro

hi all

Ted Shifrin

Pretty cool. Jim Stasheff is now posting questions on main.

Balarka Sen

19:11

Show!

No way

Ted Shifrin

I've seen some before, but this is new.

If I had anything to contribute, I'd say hi to him. :)

This is another new one. I'd be surprised if no one'd asked that one before.

Thorgott

the Stasheff from Milnor-Stasheff?

Balarka Sen

amazing

he's asking questions of the sort thorgott likes to ask

Ted Shifrin

Yes @Thor.

I haven't seen him in years.

anakhro

Thorgott asks good questions.

user 85795

19:14

^

Balarka Sen

Some of them are good

Ted Shifrin

So how do we construct lifts when we have fibrations? Remind me.

Thorgott

I've no clue what Chen's iterated integrals are, but I actually know the notes by Dundas he's referencing

the answer to that second question should clearly be "yes", no?

Ted Shifrin

Well, remind me how we do lifts in the continuous category!

anakhro

Lifting homotopies?

Thorgott

19:18

we're postulating they exist

Ted Shifrin

No, but we prove that a bundle is a fibration. How.

Thorgott

that's just the hypothesis

that's true, but I don't know how to prove it

Ted Shifrin

No, it's a theorem when you have, e.g., a vector bundle.

OR any fiber bundle.

Balarka Sen

You just lift locally, which you always can for U x F -> U

Ted Shifrin

Well, we need to figure this out first.

Yes, and how do we patch?

Thorgott

19:19

iirc the fact that topological fibre bundles over paracompact spaces are fibrations is very technical

that's a remark in Hatcher somewhere

Ted Shifrin

Yeah, I remember the Lebesgue number lemma somewhere in there.

Thorgott

for covering spaces, it's easy

but I don't think this is relevant in any case

user2103480

@BalarkaSen I'm mad at physicists for having possibly the strongest intuition for probability notions

Ted Shifrin

I mean, you can't just approximate continuous functions by smooth ones. This is related to the usual question (which I just helped someone with) that you can wiggle a section to make it transverse to whatever. You need the fact that the wiggle of a diffeomorphism is still a diffeomorphism on a compact space.

Balarka Sen

I don't see why it'd be different from how you do it for covering spaces. You just have to show it for maps from cubes.

user2103480

19:21

rn I'm trying to understand characteristic functions via scattering stuff

Thorgott

you can just approximate continuous functions by smooth ones

Balarka Sen

He means there's an extra trick for sections.

Thorgott

the natural inclusion of smoooth homotopy classes of smooth maps into continuous homotopy classes of continuous maps is a bijection

Balarka Sen

Otherwise you don't stay a section

Ted Shifrin

Right, so there's going to be the same issue with covering a map.

Balarka Sen

19:21

Yes, I agree it's not clear.

Thorgott

why do we care about sections?

Balarka Sen

I am probably not going to think more though

Ted Shifrin

So with sections, I figured out the argument in grad school, proudly.

Thorgott

maybe I'm misunderstanding what is meant by continuous homotopy lifting property

Ted Shifrin

You have to cover the identity map in that case, @Thor. Here we have to cover some harder map.

Thorgott

19:22

I thought we're just lifting homotopies

Ted Shifrin

You have to cover a given map.

user 85795

@user2103480 as they say, "don't get mad, get even"

Ted Shifrin

If this were trivial, Jim Stasheff would know it, I'm pretty sure.

Balarka Sen

I agree it's not clear, just odd. It's a good question.

Ted Shifrin

I upvoted it after editing :)

Thorgott

19:23

homotopy lifting property is asking that for a homotopy $X\times I\rightarrow B$ and a lift of $X\times\{0\}\rightarrow B$ to $E\rightarrow B$, there exists a unique homotopy $X\times I\rightarrow E$ lifting $X\times I\rightarrow B$

Balarka Sen

i think all of us know the definition

Thorgott

so the heart of the question is whether we can homotope a continuous lift $X\times I\rightarrow E$ to a smooth one in a way that agrees with a given smooth lift of $X\times\{0\}\rightarrow B$

Balarka Sen

wot, you have to project to the same homotopy X x I -> B

after wiggling

thats what lift means

Thorgott

ah

Clarinetist

There are two subjects for which I wish better textbooks existed in general:
- Probability (at the measure theory level)
- Convex Optimization

Thorgott

19:27

now I get why it's non-trivial

Balarka Sen

if you listened 15 min earlier it would have been clear to u

:p

Clarinetist

Actually, add Time Series and grad-level Mathematical Statistics to that list

Thorgott

that's just how hindsight works

Balarka Sen

thanks for the universal property definition of hindsight mate

Ted Shifrin

LOL ... it's refreshing that once in a while Ted is right :P

Thorgott

19:28

always at your service

Clarinetist

My master's defense was a time series project... yet when anyone asks me to suggest a Time Series textbook, my response is that there isn't a good one

anakhro

I wish there were better textbooks on analysis.

user 85795

Be the change you want to see.

Ted Shifrin

There are some good ones.

Clarinetist

I get annoyed when professors from Stanford publish a textbook on machine learning, there's extreme enthusiasm behind it, and when you actually try to implement the methods described, you realize they've given insufficient detail.

I bet the people who tend to be the most enthusiastic about textbooks learned their contents through a class before reading the books.

Ted Shifrin

19:32

I always preferred learning from lectures, hence tried to teach so that students could do so. But some of my students were adamant that they would learn from books.

Clarinetist

Analysis is weird. I've never had an analysis/algebra professor who liked to stick to the textbooks they assign. So while one book might be great, it may not necessarily be great for a class you're taking.

anakhro

I still need to watch a TedLecture to evaluate Ted's claims.

Clarinetist

For first-semester analysis, the best one I've seen is Kirkwood.

Ted Shifrin

They're false.

Never heard of it, Clarinet!

Clarinetist

amazon.com/Introduction-Analysis-Second-James-Kirkwood/dp/…

anakhro

19:35

@Clarinetist I will check this one out. I frequently get asked for references for a first book in analysis.

Ted Shifrin

For me it's all about exercises.

Clarinetist

I like Bartle and Sherbert too, but if memory serves, a lot of steps are dismissed as "obvious"

anakhro

Victor Bryant did a well-written primer to analysis.

But it's not really a first course, but like what I would hand a high school student instead of Spivak.

Clarinetist

I'm not as enthusiastic about Abbott's text as some people online are

Ted Shifrin

I think Spivak is an excellent place for a bright person to learn the basics of analysis, actually, but it's only the beginnings.

Abbott didn't seem to be so good when my colleagues used it at UGA, but I never examined it.

I like Wade, actually. And a few others.

anakhro

19:40

Abbott I found too wordy.

Ted Shifrin

I no longer have my library, so I can't remember others.

anakhro

Rudin gets thrown around as an expensive door stop.

The foundation of all sink-or-swim analysis courses.

Clarinetist

I'm planning on re-learning measure theory this summer. I picked up enough through this measure theory + probability sequence where I know why I couldn't get past the wall when I was trying to learn it myself years ago, but this professor moves too fast and casually throws around complex analysis and functional analysis

anakhro

My first analysis course used Rudin :(

Ted Shifrin

Mine did, too, anakhro, and taught by a probabilist who never met a picture.

Clarinetist

19:42

I'm almost certain the only student who understands what's going on in lecture is the one who told me he read Rudin's Functional Analysis as an undergrad.

anakhro

@TedShifrin Heh. Were you that into pictures at the time, or did geometry for you come later, Ted?

Ted Shifrin

He actually spent the first two weeks trying to motivate Dedekind cuts by proving that with our naive definition of addition/multiplication of decimals we have a field. Ugh.

I've always been visual.

anakhro

@Clarinetist Just don't use Folland. ;)

Ted Shifrin

Even at MIT, a number of math majors flunked/dropped the Rudin analysis course several times. Ultimately, they created a different track for the course for people who weren't going to be going on to PhD's. There were also two algebra tracks — one using Artin's book for a year, the other with Fraleigh for a semester and Hoffman/Kunze for a semester.

I'm told that Folland is a good book.

anakhro

If you want to torture students, I'd agree, Ted.

Clarinetist

19:45

I've skimmed Folland. I do prefer it for the measure theory stuff and completeness of detail compared to the probability books I have, but it's not a probability text unfortunately.

If I ever dare to consider a math PhD, I might get my hands on it. But otherwise I'll be staying far away from it.

anakhro

@Clarinetist Fremlin has an okay few volumes for measure theory.

Are you more interested in the probability side of things?

Ted Shifrin

I learned from Segal & Kunze, Integrals and Operators, unfortunately taught by Segal. I like the Daniell integral approach, actually, because I really hate measure theory for itself.

Clarinetist

Yes, I'm much more interested in the probability side of things. We're using Dudley. I'm not a fan.

Ted Shifrin

Ha. Dudley was the probabilist who from whom I took the Rudin course.

anakhro

Have you looked at Rosenthal's "A First Look at Rigorous Probability Theory"?

Clarinetist

19:47

Rosenthal is too watered down to be useful

anakhro

I found most of the content was in the exercises.

Clarinetist

Yeah, I gather that Dudley passed away recently, was very well respected in probability

anakhro

If you want comprehensive, I'd go the way of Fremlin, but it's far too much.

Ted Shifrin

He was a good guy. My senior year at MIT he lectured the multivariable calculus class and the students were completely lost. He also set up all his polar coordinates integrals in the order $d\theta\,dr$ because he had never encountered an integral where $r=r(\theta)$ in the limits. I had to retrain those students in a hurry.

Clarinetist

Yeah, I've seen Fremlin. I think it's good, but I don't see myself reading through that

Bogachev is one of the best I've seen, but again, measure theory, not probability

Semiclassical

19:51

saw this in an editorial in the paper and was decidedly amused

they wanted a candidate for NASA who was "like Elon Musk but without the personality defects"

2

Clarinetist

:O

Semiclassical

just delightful

Thorgott

might as well recommend Bogachev if Fremlin is on the board lol

user2103480

@Clarinetist just grab a few good lecture notes, no? These should suffice for everything up to discrete martingales: mi.uni-koeln.de:8930/probabilityI.pdf

Semiclassical

martingales are something I don't really know about, and i feel like i should

especially since it seems to be mostly a story about conditional expectation, and that's got a specific Hilbert space interpretation

Clarinetist

19:59

I don't know if it is my professor - it probably is - but not one source I've found teaches the topics in the order he teaches them in class.

We, for example, spent a substantial amount of time on nonatomic and atomic measures. I find coverage of these topics to be very mixed in general.

Semiclassical

something along the lines of: You have a sequence of vectors (random variables) such that the nth consecutive difference $X_{n+1}-X_n$ is orthogonal to all vectors earlier in the sequence

anakhro

I have never heard of non/atomic measures

Transcript for

$Mathematics$ Mathematics