Mathematics - 2023-12-13

Ted Shifrin

00:31

A vector field can be defined piecewise, but in this case one formula works everwhere, doesn’t it?

John Zimmerman

00:57

Yes I was thinking it'd have to be piecewise @TedShifrin How can you say it's globally level sets of $f(x,y)=xy$ without having the explicit forms of the curves?

oscarmetal break

04:15

Has the server fixed already?

Cotton Headed Ninnymuggins

04:47

What's everyone's favorite theorem?

one potato two potato

07:35

If $\Omega$ is a simply connected domain in $\Bbb C$, then for holomorphic $a,b$ on $\Omega$, does $f'(z)+a(z)f(z) = b(z)$ have a holomorphic solution $f(z)$ on $\Omega$?

leslie townes

does the recipe using an 'integrating factor' work? all that would require is a holomorphic antiderivative of a(z), right? which you could get on such a domain.

porridgemathematics

it would

X4J

11:12

I know that if matrices $D_1, D_2 \in M_n(\mathbb{R})$ are nilpotent and $D_1^{n-1} \neq 0, D_2^{n-1} \neq 0$ then they are similar. One way to reason it is that they both have the same Jordan form. However, can I explicitly find a linear operator $T$ and two bases $\alpha, \beta$ s.t $[f]_{\alpha}=D_1$ and $[f]_{\beta} = D_2$?

oscarmetal break

15:33

Any quick way to show a sylow-7 subgroup in $GL_3(F_2)$ is not normal, i.e. check that the number of sylow-7 subgroups is 8 instead of 1. I knew I could do this by finding a matrix of order 7 and showing that it does not commute with some other matrices. But I don't like it this way.

Jakobian

16:59

I'm not learning about gauge integrals anymore. It gets a bit stale after the 14th chapter of Bartle

just the same results but restated for infinite intervals

leslie townes

oscar: that sounds like maybe the simplest way in terms of theory (even if it is calculational and maybe unenlightening as to where that matrix of order 7 comes from or why, beyond a calculation, it might not commute with some other element). if you have done any general study of conjugacy in GL_n(field), that suggests a more conceptual approach, but maybe involves more.

Ted Shifrin

@Jakobian *the room breathes a heavy sigh of relief

leslie townes

e.g. what are the possible characteristic/minimal polynomials of elements of GL_3(F_2) and what are the orders of matrices in the corresponding classes. that seems tractable for any possible order of an element in GL_3(F_2), not just 7.

Jakobian

I don't know what I'll be learning now. Maybe I should continue with rings of continuous functions, or maybe start learning AG again, or some infinite-dimension flavoured topology?

maybe I'll read Milnor's Differential topology

Ted Shifrin

Milnor's little book is lovely, and terse (which is fine for you). For US undergraduates I preferred to teach out of Guillemin & Pollack, but the main ideas of degree and intersection numbers are in Milnor.

leslie townes

17:13

e.g. looking at x^7 - 1 = (x - 1)(x^3 + x^2 + 1)(x^3 + x + 1) [which is a factorization into irreducibles over F_2] suggests that a matrix will have order 7 in GL_3(F_2) iff it has characteristic polynomial x^3 + x^2 + 1 or x^3 + x + 1, and matrices having these distinct characteristic polynomials will not be conjugate to one another.

although maybe you need more theory to know there are such matrices. this approach would all be in artin somewhere i bet.

are we still intersecting things? this is not the way to enlightenment.

Jakobian

its really short

leslie townes

milnor is a comic book. i think i found my copy in a box of cracker jacks.

Ted Shifrin

There are not enough exercises in Milnor, either.

leslie townes

i'm tempted ot leave an amazon review, "Crummy book, also it is too short"

Ted Shifrin

Well, it helps to cancel out all the 800-page books that should really be 150.

leslie townes

17:27

so true.

monoidaltransform

If $U$ and $V$ are open in $X$, compact metric space such that $\overline{U}=\overline{V}=X$, must it be that $U=V$?

leslie townes

no. e.g. X = [0,1], U = (0,1), V = (0,1/2) union (1/2, 1).

Ted Shifrin

You can poke lots of holes in $U$. :)

Alessandro Codenotti

17:59

One of the many cases in which regular open sets are better than just open sets

Ted Shifrin

As opposed to irregular ones? What are regular open sets?

leslie townes

open sets with a leaded additive. helps to prevent the knocks.

Ted Shifrin

That sounds like super open sets.

leslie townes

in iowa we used high-ethanol open sets.

Jakobian

@TedShifrin interior of closure is the same set

equivalently, $\text{int}\overline{A}$ for some subset $A$

I think Alessandro was pointing out that if $\overline{U} = \overline{V}$ and $U, V$ are regular open sets then $U = V$

There's also regular closed sets which have the same definition but with order of operations we take swapped

I've characterized subsets of $\mathbb{R}^n$ homeomorphic to a regular closed subset as those subsets $A$ homeomorphic to some closed subset of $\mathbb{R}^n$ such that $\overline{\text{int}(A)} = \overline{A}$

Its an interesting question what characterization holds for the regular open subsets

leslie townes

18:21

[[citation needed]]

(at "interesting")

Thorgott

I've never in my life used the adjective regular for anything but T3 spaces

leslie townes

[[to whom?]]

Thorgott

oh wait, and regular values

Jakobian

One place where regular open sets are used is in some set-theoretic topology inequalities for regular spaces

Thorgott

oh, and regular rings

leslie townes

18:29

no regular maps?

Ted Shifrin

Oh, and plain ol’ regular Thor

Jakobian

@Thorgott those have multiple meanings though

Thorgott

@leslietownes oh, those too

@Jakobian not to me

Jakobian

You use open regular sets to prove that $w(X)\leq 2^{d(X)}$ for a regular space $X$

This inequality implies that, for example, if $Z$ is a compactification of a Tychonoff space $X$, then $w(Z)\leq 2^{d(X)}$

Alessandro Codenotti

19:22

@Thorgott you'd be surprised

I've seen "regular continuum" to denote a continuum that has a basis of open sets with finite boundary...

Those are also called rim-finite continua sometimes to avoid overloading "regular"

Xander Henderson

@Thorgott Not even for your bowel movements?

Eat more fiber.

Alessandro Codenotti

@TedShifrin open sets which are the interior of some closed set (I find it more natural to think about regular closed sets, which are closed sets that are the closure of some open set)

Thorgott

19:38

@XanderHenderson I'll take "normal" for those

Xander Henderson

@Thorgott Lucky you. A lot of people would be very happy is "regular" were "normal".

Koro

20:42

what's the name of that movie where a guy picks cucumbers from trash, makes pickles using them and sells them?

Xander Henderson

@Koro Trash Cucumber Pickle Man?

Google says I'm wrong.

slate.com/culture/2020/08/…

Koro

@XanderHenderson this seems to be the one, thanks.

Xander Henderson

google.com/search?q=trash+cucumber+pickle+man

Koro

:-)

Xander Henderson

I literally Googled "trash cucumber pickle man".

leslie townes

20:44

this was also a subplot in "avengers: infinity war"

Xander Henderson

@leslietownes Yeah, but those were pickled onions, not cucumbers.

Jakobian

20:56

huh

Ted's book is nice refreshment of knowledge

Ted Shifrin

21:08

Drinks and popcorn?

user 85795

::pops fresh popcorn::

🍿🍿🍿

Dannyu NDos

I've just heard that a new edition of Fraleigh was published few years ago. What's new?

user 85795

The price?

Dannyu NDos

Apparently some sections about Algebraic Geometry were bumped?

leslie townes

@user85795 they probably also applied permutations to the numbered lists of exercises.

Ted Shifrin

21:22

Algebraic geometry — really? Many years ago, in the edition I bought, he had sections on homology, which is the beginnings of algebraic topology, not geometry.

user 85795

That works every time @leslietownes

Dannyu NDos

I had 7th edition, which had a section for Gröbner basis. Apparently it's bumped in the new edition.

leslie townes

i looked at a recent edition of fraleigh and they'd thrown in some crap about groebner bases, probably because computational algebra was/is "hot," or at least three or four people asked the publisher when they'd toss that in the book.

maybe they realized it's time to stop trying to make computational algebraic geometry happen

Ted Shifrin

@leslie For the second edition of our linear algebra , we made substantial improvements and the typesetting improved.

Oh, Gröbner is a bit of commutative algebra. I would not say alg geom.

leslie townes

i could see adding in a section like that and then having it generate so much feedback you decide its better off not there.

i think the toy examples/motivation were geometric.

Dannyu NDos

21:25

As for Linear Algebra, I have the 12th edition of Anton.

Korean translation.

user 85795

Classic

leslie townes

i taught out of anton once. it's so much better in the original english.

i'm kidding, but it is not a bad book. better than the one i learned out of.

Ted Shifrin

I’ve never liked Anton or Lay. Imagine that.

user 85795

Which was?

Dannyu NDos

And I bet everyone here learned Topology by Munkres?

leslie townes

21:28

i don't remember, but i think it was lay.

Ted Shifrin

Yes, I had him as the professor as he was finishing writing the book. We had a xeroxed typed version,

leslie townes

i never used munkres, or any general topology book, which is why i wound up the way i did.

Ted Shifrin

They didn’t have a book for 202A?

Dannyu NDos

What is 202A?

leslie townes

that is often taught as an analysis class. folland was a common choice. topology obviously in there, but not from a general topology text.

Dannyu NDos

21:32

As for elementary Analysis, I learned by Stoll, but after few years then, my institute replaced it to Bartle.

Stoll sucked, tbh.

Ted Shifrin

I don’t recall an undergrad point set course. There was an undergrad intro alg top (with first cohomology defined as homotopy classes ….) and I taught a G&P course as a grad student which eventually became 142, I think.

We’re talking about Berkeley courses, Dannyu.

user 85795

So you never taught any number theory or stats courses.

leslie townes

oh, there was an undergrad class when i was there, but it had nothing resembling a faculty 'owner' that i knew of. postdocs taught it. usually topology of surfaces with minimal abstraction. the armstrong UTM, when i took it, skipping a lot of the general sections.

Ted Shifrin

That might have been 141. Not intended to be point-set.

leslie townes

whatever it was, it wasn't very good. i don't even remember the postdoc who taught it.

Dannyu NDos

21:36

Oh, so those numbers are course numbers?

Ted Shifrin

@user85795 Me, no. I have a tiny bit of number theory in the algebra book I wrote. I never took any number theory, stat, or probability..

user 85795

Ok.

Ted Shifrin

Well, no alg top or geom top faculty was interested in teaching point set, I imagine.

leslie townes

haha, i checked my transcript, i took 142, and it was called 'introduction to algebraic topology,' but there was almost nothing algebraic about it. we may have defined the fundamental group.

postdoc teaching quality can be so uneven. almost guaranteed to be 2x better or 2x worse than regular faculty.

Ted Shifrin

Oh, when I was there, they were doing homotopy and $H^1(Xj = [X,S^1]$. I forget who wrote the book.

leslie townes

21:41

there was an undergrad differential topology course that existed in the catalog but was literally never offered during my time as an undergrad. and of course 140, for diff geo. the closest thing to general topology would have been the more abstract sections of 202a, from an analysis book.

mick

hi all

can we see deleted questions from others ?

leslie townes

mick: users of a certain reputation threshold can see a lot of deleted questions. i think 10k is the start for that.

Ted Shifrin

I taught the diff top as a topics course when I was a grad student. I had about 8 students. They liked it. Some faculty must have liked the idea and introduced it, but ….

leslie townes

maybe you unlock further tiers of deletion as your rep goes up.

mick

i have 15400 now, so i should have it

Ted Shifrin

21:44

You have to view history.

leslie townes

well, what "it" is might depend on your rep, even if you can see some deleted items. there may also be a difference between what you can technically access if you have the link, vs. what is browse-able. anyway, it's not unusual for people to post links in here that i can't read.

mick

@TedShifrin how do I do that ?

Ted Shifrin

Click on the deleted thing and you should get such an option if you have clearance.

I don’t get it on my iPad.

Jakobian

I've tried explaining how to solve some exercises from complex numbers to my dad (he's a math teacher). Sadly he doesn't get it

Ted Shifrin

Are they super sneaky?

Jakobian

21:55

No, its about finding all complex numbers $z$ such that $z^4 = \overline{z}^4$ and $|z^3|\geq |9\overline{z}|$. My dad didn't get how I got from $z^4 = r$ for some $r > 0$ to $z = w\cdot \sqrt[4]{r}$ for $w$ solution to $w^4 = 1$ and then claimed $w = e^{\frac{2\pi i k}{4}}$ for $k = 0, 1, 2, 3$

Ted Shifrin

In the US high school math teachers do not need a degree in mathematics. They need a credit for geometry and for a semester of abstract algebra.

Not sneaky. Does he understand how to multiply two complex numbers in polar form?

Jakobian

I've tried giving some ad hoc explanation, mentioning properties of the exponential, and even mentioned how it can be thought of as de Moivre formula

and tried to explain how multipying a complex number $z$ by $e^{i\theta}$ is rotating it by angle $\theta$ counter-clockwise

but it didn't work, he just gave up

Ted Shifrin

My one PhD student was from Poland. To teach elementary school in Poland, she had to take a Rudin-type analysis course. I was stunned.

Hasn’t he taught basic complex stuff? What level does he teach?

Jakobian

Usually elementary, but also sometimes in high school

He's been an elementary school teacher most of his life

My thought was the same, was he taught complex numbers and their properties?

Ted Shifrin

Ah, so not particularly trained in math. I guess he didn’t do real analysis like my student.

He may have been taught it or maybe not. In the US elementary school teachers are not expected to know that.

Jakobian

22:02

Well, I think he had few first years of normal mathematics course, before changing university and moving to pedagogy

mick

@TedShifrin but I do not have a link to the deleted question. I want to see what questions or answers where deleted.

Ted Shifrin

I’m surprised he’s allowed to do high school without more training.

Jakobian

He has a master's degree

Ted Shifrin

@mick Oh, you mean on main, not here. There’s no repository of deleted questions that I know of.

Jakobian

But it was mathematics for teachers from what I heard, so it wasn't too intense

Ted Shifrin

22:04

@Jakobian in pedagogy, not in math?

Jakobian

I don't know

I don't exactly know how it works

Ted Shifrin

But I’m surprised by his attitude. Maybe he feels nervous/insecure cuz you’re his son.

mick

I like some deleted questions :(

Jakobian

Probably

mick

I rarely support deleting questions. downvote and/or close is bad enough imo.

especially when there is a valid answer given

oscarmetal break

22:28

Is there an example that let $(p)$ be an ideal in $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/(p)$ is an integral domain but not necessarily a field?

Ted Shifrin

Is every prime ideal maximal?

oscarmetal break

no

leslie townes

11

Q: Are there any elegant methods to classify of the Gaussian primes?

Out of curiosity, are there any relatively quick classifications of all the Gaussian primes, the primes in $\mathbb{Z}[i]$? I found a classification here, but the process comes off as rather tedious. No doubt the end classification is nice, but is there a quicker and cleaner process to classify ...

3

Q: Classification of prime ideals in $\mathbb{Z}[i]$

It's a couple of days that i'm struggling with this answer, which i'd like very much to understand. I recall briefely what is the problem: I want to classify all prime ideals of $\mathbb{Z}[i]$. The strategy is the following: for each (non-zero) prime ideal $(q)$ of $\mathbb{Z}$, we want to clas...

ring-theory prime-numbers

Thorgott

try looking at trivial examples

leslie townes

both questions are way more info than you requested, but contain the info you requested

Jakobian

22:34

@oscarmetalbreak in $\mathbb{Z}[i]$

Ted Shifrin

Isn’t $\Bbb Z[i]$ a PID?

Maybe I’ve forgotten everything in my ancient age ….

Jakobian

Its a Dedekind domain, isn't it?

Thorgott

it's even Euclidean

Ted Shifrin

Indeed.

Jakobian

@Thorgott I actually meant Euclidean and I realized that after going to wikipedia and not finding what I wanted

Ted Shifrin

22:38

Quite a pretty geometric picture for the division akgorithm.

oscarmetal break

$\mathbb{Z}[i]$ is p.i.d indeed

Ted Shifrin

So?

oscarmetal break

So in this case whenever an ideal in $\mathbb{Z}[i]$ is prime, it is also maximal, so that the quotient ring with respect to this is always a field?

Ted Shifrin

Except for $(0)$.

oscarmetal break

I see

So $\mathbb{Z}[i]$ is special

@leslietownes thanks for the link, the first one looks fancy

Ted Shifrin

22:41

Special?

oscarmetal break

special I mean special on its own, nothing special other than that

nvm :)

Is there a book or reference that I can look at a lot of counterexamples in algebra?

Ted Shifrin

22:56

I know of no analogue of the analysis and topology books.

Sine of the Time

counterexamples is analysis is vey nice

Joe

@oscarmetalbreak: The thread Examples of common false beliefs in mathematics on MathOverflow has a handful of examples from algebra.

See Martin Brandenburg's answer for instance.

Ted Shifrin

@oscarmetalbreak There are lots and lots and lots of Euclidean domains besides $\Bbb Z$.

Joe

23:12

Every field is a Euclidean domain, so e.g. $\mathbb Q$, $\mathbb R$, and $\mathbb C$ are Euclidean domains. Another example would be: if $k$ is a field, then $k[x]$, the ring of polynomials with coefficients in $k$, is a Euclidean domain.

Thorgott

interestingly, I think the question of which quadratic number fields are Euclidean domains is quite hard

Joe

@Thorgott: Isn't every quadratic number field a Euclidean domain?

Thorgott

plenty of them aren't even PIDs

probably "most" of them aren't in some appropriate quantitative sense

oh wait, I misspoke

the fields are fields of course, I'm talking about their rings of integers

Ted Shifrin

@Thorgott There’s actually stuff on this in the number theory classic Hardy & Wright.

Not fields, of course.

Jakobian

@Thorgott I remember googling that

Pain

A reference for multivariable analysis?

I don't like how Ted's book wants me to do exercises

I just want something as a refreshment of how to prove something, not do things from scratch

Joe

23:40

@Jakobian: One book that I've seen recommended is Mathematical Analysis: A Concise Introduction (the title is rather oxymoronic, in light of the fact it has over 500 pages). It proves the multivariable analysis theorems in quite a general context, e.g. rather than working with functions $\mathbb R^n\to\mathbb R^m$, it works with maps from one normed space to another.

There is also Loomis and Sternberg's Advanced Calculus, which you can freely access online.

Ted's book also has a suggested reading section at the end, though I don't know if it's what you're looking for.

(The analysis book I mentioned first is by Bernd S.W. Schröder.)

Jakobian

Mhm. Thank you

@Joe well I just wanted all the things about differentiability in $\mathbb{R}^n$ because my compulsive feeling of having to have everything checked is kicking in. I know things are true, but I have to see

Joe

23:56

@Jakobian: Ah, I see. Well, in fairness, the proofs for general normed spaces are not very different to the proofs for $\mathbb R^n$. I'm not sure if any of my suggestions are perfect, because you hope from a reference book that you can read a proof without having to go backwards to dozens of previous lemmas

I can't remember how badly any of the books I mentioned suffer from that.

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$Mathematics$ Mathematics