Mathematics - 2021-12-01 (page 2 of 2)

UnderMathUate

8:01 PM

What's the difference between naive set theory and axiomatic set theory?

I was looking at books and saw these two titles.

Dan Donnelly

@geocalc33 hey, sent you a couple emails

Xander Henderson

One is naive, and the other is axiomatic.

UnderMathUate

@XanderHenderson Ah, of course!

Xander Henderson

Glad I could help.

UnderMathUate

;-; <- this is me crying, in case you didn't notice

Ted Shifrin

8:02 PM

No wonder Xander wins teaching awards for his clarity.

Xander Henderson

Perhaps the following would be more helpful: naive set theory tends to approach the topic from the point of view of natural language, and assumes that we all kind of understand what we are talking about.

A set is a collection of stuff, we can define functions between sets in a "natural" way, etc.

Naive set theory is what you teach to calculus or "abstract algebra" students who need a rough outline of the underlying foundations, but don't need to get into the nitty-gritty hassle of formalism.

user193319

Let $K : X \times X \to \Bbb{R}$ be a kernel. It is said to be PDS if for all $x_1,...,x_n \in X$ and $c_1,...,c_n \in \Bbb{R}$ we have $\sum_{i,j} c_i c_j K(x_i,x_j) \ge 0$. If instead we have $\sum_{i,j} c_i c_j K(x_i,x_j) \le 0$ for all $c_i$ with $c_1 + ... + c_n = 0$, then $K$ is said to be NDs. I am trying to prove that $(x,y) \mapsto 1-\cos(x-y)$ is a NDS. I was able to prove that $(x,y) \mapsto \cos (x-y)$ is a PDS, basically using Gram matrices.

UnderMathUate

I took a look at the axiomatic book, and I see what you mean. All of the theorems are given symbolically, rather than in words. The other book looks more similar to what I've seen in classes.

Asinomas

lol why'd so many folks posting here get suspended V

Xander Henderson

Axiomatic set theory starts from a bunch of axioms, and proceeds from there.

Asinomas

8:06 PM

-1

Q: The community need to make a good decision regarding closing/downvoting good questions/answers

The most common problem I see on this site is that many poor questions/ solutions are highly upvoted and many great questions/ solutions are downvoted. I have no problem with the highly-upvoted poor questions, but why the good ones are downvoted for no reason? For example, look at this question (...

user193319

But I don't know how to argue that $(x,y) \mapsto 1-\cos(x-y)$ is NDS. Note, in this case $X = [0,2 \pi]$.

Xander Henderson

@Asinomas As it isn't really anyone's business but the suspended users and the moderators, perhaps we should not speculate?

Asinomas

I don't agree with that

Ted Shifrin

@UnderMathUate Interestingly, when I took point set topology as an undergraduate, the professor (who happened to be James Munkres) made a huge deal of saying that mathematicians write and communicate with words, not with symbols. We were taught never to use symbols other than $\in$, $\subset$, $\implies$, etc. No $\exists$ or $\forall$. Formal symbols make it harder to understand than words.

Xander Henderson

@Asinomas Let me rephrase: the reason for the suspensions is none of your business. Please do not speculate.

Asinomas

8:10 PM

I have no problem with those users so arguing on that side just doesn't work

Ted Shifrin

@Asinomas I am voting to close several times a day now. I never did that in my first 8 years here.

The proliferation of homework questions with (less than) zero effort is stunning.

Asinomas

Nice rephrasing

@TedShifrin gz

Ted Shifrin

I know nothing about suspension.

leslie townes

axler does point out that issue about R being different from C with upper triangular matrices. admittedly he does this before he introduces the characteristic polynomial.

Ted Shifrin

@leslie It was a natural question for Koro to wonder about.

Asinomas

8:11 PM

I am doing my part by not using the site !

Ted Shifrin

Well, then why are you in here, @Asinomas?

Asinomas

Because I go into meta once a month to see what is up

but it

but meta is always the same 15 dudes

geocalc33

somebody solved my differential equation

Asinomas

and recently the people who are in the complement of the 15 have 1 rep and ar suspended

by dudes I meant "persons"

leslie townes

axler also proves that an operator on a real vector space of odd dimension has an eigenvalue, just not around the time he proves the upper triangular thing.

Asinomas

8:14 PM

axler is the GOAT

leslie townes

the ordering of axler does seem 'wrong' for certain results, but it's all there.

Koro

3

Q: Eigenvalue and Upper Triangular Matrices

In Axler's linear algebra chapter 5, he explicitly shows that for a complex vector space $V$ and a linear operator $T$, there exists a choice of basis such that the matrix of the operator with respect to it is upper triangular. Here's the proof sketch: Use induction on the dimension of $V$, for ...

linear-algebra vector-spaces eigenvalues-eigenvectors

Xander Henderson

@Asinomas This conversation is off-topic for this room. It is possible that a discussion of suspensions, in a general sense, would be on-topic in the Math Mods' Office, but it is off-topic here.

Koro

Here is the question that I also had but I don't understand its answer.

TheDragonOfFlame

Does anyone know how to graph [(1/2)x] in a graphing calculator?

Xander Henderson

8:15 PM

Please stop.

Asinomas

@XanderHenderson ?

Ted Shifrin

@TheDragon How do you graph $[x]$?

Koro

And I still don't understand why Ted's matrix above is not upper -triangularizable.

TheDragonOfFlame

@TedShifrin Not sure. I have never seen square brackets before and am not sure what they mean. Google yields nothing

leslie townes

koro: if a real operator has an eigenvalue the first step works, but it doesn't fit into the inductive argument unless you assume it has an eigenvalue at every step.

Ted Shifrin

8:16 PM

Yes you do, @Koro. A rotation has no real eigenvalue.

Asinomas

@XanderHenderson no worries

Ted Shifrin

Oh, @TheDragon. Have you seen $\lfloor x\rfloor$?

leslie townes

you can get a block upper triangular matrix with 1x1 or 2x2 blocks on the diagonal. that is the best you can do.

(also in axler)

Govind75

I'm looking at proofs for Schurs Theorem - are there any proofs which you'd recommend, preferably those which don't mention unitary matrices.

Xander Henderson

@TheDragonOfFlame $[x]$ is often used to denote the greatest integer function (e.g. the largest integer less than or equal to $x$; the "floor" function).

That is likely what it means here.

Ted Shifrin

8:17 PM

Ben Grossmann talks specifically about $2$-dimensional invariant subspaces. That's what my rotation has.

Koro

The proof of "Over C, operator T has upper triangular matrix" only uses the fact that T has an eigenvalue. So if I say instead: Over R, if T has an eigenvalue then T has upper triangular matrix. What's wrong with that? That's what I don't understand.

Ted Shifrin

@Govind If you're going to talk about orthonormal bases, you're talking unitary matrices. Just do orthogonal complements and do induction.

TheDragonOfFlame

@TedShifrin Iḿ sorry I don´t recognize that notation

Ted Shifrin

OK, @TheDragon. It means the integer part of a real number.

$[2.4] = 2$, $[-.75] = -1$.

TheDragonOfFlame

@XanderHenderson Wait so itd be like... a line from 0-1, then a line from 1-2, 2-3 (x coordinates)?

Koro

8:18 PM

Professor Ted: Rotation (by 90 degree for example) has no eigenvalues but I'm thinking of the case when the operator has at least one eigenvalue.

Ted Shifrin

@Koro I've told you twice. You need to understand things beyond where Axler is.

TheDragonOfFlame

Like a staircase pattern?

leslie townes

koro: there's an induction argument.

Xander Henderson

@TheDragonOfFlame Yes. If $x \in [n,n+1)$, then $[x] = n$ (where $n$ is any integer).

Ted Shifrin

The $3\times 3$ matrix I gave you has a real eigenvalue. That's the whole point.

leslie townes

8:19 PM

koro: the induction argument needs an eigenvalue at every step, not just "an eigenvalue"

ted: axler also directly addresses this point in a margin note. this isn't all axler.

Ted Shifrin

I guess I would have to agree, @leslie.

leslie townes

i'll quote: "This theorem does not hold on real vector spaces because the first vector in a basis with respect to which an operator has an upper-triangular matrix must be an eigenvector of the operator. Thus if an operator on a real vector space has no eigenvalues (we have seen an example on R2), then there is no basis with respect to which the operator has an upper-triangular matrix."

for the induction argument to work and produce an upper triangular result, it needs an eigenvalue every step of the way.

Koro

@leslietownes yes!! I understood now.

Ted Shifrin

Grrrr.

Koro

You again beat my question to death.

leslie townes

8:21 PM

if i seem touchy about this, it's because my advisor suggested that proof of that theorem :D

Ted Shifrin

Well, it's the only proof?

Koro

$\ddot\smile$

Ted Shifrin

I think the point is that I continue to be more socratic, and that doesn't work with Koro.

Leslie beats him to death. Lesson to be learned.

leslie townes

koro in a later chapter he does block upper triangular matrices for operators on real vector spaces. in R^3 this is going to involve at least one 1x1 block on the diagonal

Ted Shifrin

It's disappointing that more people do not understand how you get real Jordan form with $2\times 2$ blocks ...

leslie townes

8:23 PM

i think people dislike blocks

Xander Henderson

@TedShifrin Man... that was, like, a quarter of graduate PDEs...

*shudder*

leslie townes

did they make you take that? or did you choose to take that?

i wouldn't blame anyone else for your own poor choices

Xander Henderson

@leslietownes I chose it.

Because, at the time I started taking classes, one was required to take four qualifying exams, and there were only five topics offered.

leslie townes

doctor, it hurts when i do this

Xander Henderson

I essentially had a choice between PDE and algebra.

And I don't like algebra.

Ted Shifrin

8:25 PM

Both important.

Interesting: I think I like to teach more by having people come to terms with examples. Leslie is the typical ungeometric formalist who just does the proof. :P

leslie townes

i had algebraic topology as my minor qualifying exam topic, if anyone can believe that.

Xander Henderson

By the time I finished my phd, one was only required to pass two qualifying exams, and I think they offer six or seven topics now. It is currently possible to complete a phd at UC Riverside without ever taking class starting with "A" (neither analysis, nor algebra).

Ted Shifrin

I think munchkin and I need to collude at the duck pond.

Xander Henderson

Kids theses days. :\

Ted Shifrin

That's pretty pathetic, @Xander. Applied people may not need algebra, but they certainly should know analysis.

Xander Henderson

8:27 PM

@TedShifrin I agree entirely.

Ted Shifrin

Of course, everyone should be required to know undergraduate differential geometry and differential topology :D

leslie townes

they don't need to know anything. they can just study what the 'right' definition of the category of the category of theories of homotopies of the category homotopy category ought to be (or not). or maybe it should be the right 'definitions,' plural.

amWhy

@XanderHenderson That explains everything ;D

Ted Shifrin

I am far from an algebra person, but what was my first textbook? Hmm ...

I don't think @copper has gotten past the third section of the first chapter yet. Oh well.

leslie townes

"Ted's Guide To Beating The Income Tax (Make Yourself Legally A Manifold And Not A Person)"

amWhy

8:29 PM

@leslietownes That's just mis-characterizations from algebra fails. :P

leslie townes

or was that more of a pamphlet

Ted Shifrin

A pamphlet like "Common Sense," you mean?

leslie townes

yes

Ted Shifrin

That's a good thought. Can you send me the first three pages?

amWhy

@TedShifrin Precisely, algebraists know common sense, and own the copyright on that pamphlet! :P Logicians are superior to both. :P

Ted Shifrin

8:31 PM

No, in general, algebraists are symbol-pushers, much to my dismay.

How many algebra books talk about conjugation in group theory, normal subgroups, etc., and show students "Hey, this is like the change of basis formula in linear algebra!"?

I don't believe in logicians.

Koro

Leslie, I didn't find that margin note (in LADR) that you quoted. :(

amWhy

@TedShifrin Sorry, too many symbols in real analysis! And Xander is a fractal person, which is a third cousin to analysis. ;D

Ted Shifrin

Symbols in real analysis? You mean quantifiers?

Xander Henderson

@amWhy I'm a fractal person, but the stuff I work on is primarily a bunch of complex analysis, some spectral theory, and some non-archimedean analysis.

So... analysis. :D

amWhy

@TedShifrin Then you're not analyst. Nor mathematician. summations, products, all the kinky symbols for non-existent entities to enable folks to feel the accomplished an integration problem for a day.

Ted Shifrin

8:37 PM

I think most of those symbols appear in combinatorics, too. What part of math have you not whitewashed?

leslie townes

koro: it's in the second edition. maybe they scrubbed those out

amWhy

@XanderHenderson Still maybe not a third cousin but a step child, three times removed. ;P

Ted Shifrin

Anyhow, usually it's us differential geometers who get slammed for having too many symbols, like the infamous "Christ-awful symbols." I admit that most courses do not give much of an understanding ...

The fact that nabla is overused (gradient, Laplacian when decorated, covariant derivative, ...) certainly doesn't help.

leslie townes

i call them kartoffel symbols

Ted Shifrin

Are you making vichyssoise?

Xander Henderson

8:39 PM

@TedShifrin Yeah, but upside-down Delta triangle thingy is so PURDY!

leslie townes

i should

amWhy

Hey, too many people in this world trying to dominate and feel superior to others. We're all, here, more alike, than we are different. When much of the current folks are anti math/science, Let us Unite!

Ted Shifrin

I'm still waiting for a report on that leek dish, @leslie.

amWhy

I'm still waiting for a piece of @robjohn's pumpkin pie. Might be rancid by now, so I expect him to bake another! :)

Ted Shifrin

My apple tart with calvados custard was so yummy. I want to make another just for me.

amWhy

8:42 PM

@TedShifrin Can I have just a sliver! It does sound scrumptious!

Ted Shifrin

Sure. Send your postage-paid refrigerated shipping package for the sliver.

amWhy

@TedShifrin that's great! Of course, if you need to watch your waistline, I'd be happy to help you out, by you sending me more than a sliver, just for me to be helpful! ;P

Ted Shifrin

I'm watching my waistline expand without bound.

leslie townes

ted, you would be very disappointed with the progress of cooking at this house. our daughter was OK for a while but is in a very neophobic phase.

Ted Shifrin

She might actually like the leek thing, cuz currants and pine nuts and sorta sweet-and-sour.

But you can always give her McDonalds to eat.

amWhy

8:46 PM

@leslietownes As in afraid of new foods she hasn't already tolerated?

@TedShifrin Eeeek. I have a bumpersticker with a baby chick, "I'm not a McNugget!!!"

leslie townes

yeah. we just give her things like peanut butter on toast with fruit or fresh vegetables.

Ted Shifrin

So you can't blame munchkin for your lack of follow-through. Just sayin'.

leslie townes

there was a while where she was rejecting some of her old breakfast standbys, like yogurt.

but it's so easy to blame my daughter!

Ted Shifrin

Typical male. Blame the women-folk. Sorta like the Supreme Court.

amWhy

@TedShifrin Then you need to partner with Xander, so you can become a pretty fractal, with no bound!

Ted Shifrin

8:49 PM

Unbounded girth does not a fractal make!

amWhy

@TedShifrin Yeah, I get a lot of that on this site. Blame the women-folk! ;D

Ted Shifrin

Well, lunch-time for this bonzo.

amWhy

@TedShifrin Your just not as creative in terms of what I have in mind. Why cant your girth be a spacefilling curve?

@TedShifrin Right, you're always behind me (at least wrt time!) ;D

UnderMathUate

The fact that Munkres was your professor...math is such a small community. One of my professors now was taught by Halmos.

After the first two or so weeks of my proof class, we pretty much stopped referring to the symbolism of things except for clarification of concepts. My professor said it's ok to almost think of it as a writing course, lol.

Ted Shifrin

@UnderMathUate: I actually was very lucky to be taught by a lot of people who are quite known for excellent textbooks.

And then I wrote some moderate ones of my own :D

Koro

8:54 PM

Leslie, the question came to me because I was trying to prove spectral theorem for self adjoint operators on real spaces.

Ted Shifrin

@Koro: First you have to prove all the eigenvalues are real, and then you won't have this issue.

OK, bye for now.

UnderMathUate

@TedShifrin Oh yeah?

amWhy

@TedShifrin They are quite good, hardly moderate!

Koro

professor Ted: I can prove that self adjoint operators can only have real eigenvalues. I was trying to prove spectral theorem differently than done in Axler's. I planned to use a modification of proof of Schur's theorem to prove spectral theorem but as discussed above, my method was wrong.

copper.hat

@TedShifrin unfortunately working through the text is low on the priority list at the moment, no reflection on the text.

Ted Shifrin

9:20 PM

@Koro Isn’t it the usual proof? Self-adjoint tells you that the orthogonal complement of your first eigenspace is again an invariant subspace.

@copper.hat Just teasing.

user 726941

9:31 PM

0

Q: How do I prove the corollary to 0 times any real number is 0?

My elementary high school algebra textbook states: If a and b are real numbers and a = 0 or b = 0, then ab = 0. follows directly as a corollary to If a is a real number, then a • 0 = 0 and 0 • a = 0. How does the first statement follow directly as a corollary?

algebra-precalculus proof-writing solution-verification proof-explanation

Please help

🙏

Akiva Weinberger

Either a=0 or b=0. If a=0, then a*b=0*b=0. If b=0, then a*b=a*0=0.

What you wrote looks good as well

amWhy

amazon.com/gp/product/0133198316/ref=dbs_a_def_rwt_bibl_vppi_i2

^^^ @TedShifrin I see you are a "closet algebraist"!

leslie townes

haha, that 2020 comment. "Be prepared to sift through stackexchange for questions about similar problems in order to get an actually complete explanation of the material in the text because you will not find it y'know... IN THE TEXT."

play your cards right and you may sift through stackexchange and get feedback from the author of your book.

Ted Shifrin

9:46 PM

@amWhy No, precisely not.

leslie townes

that's ted's evil twin brother theodore.

amWhy

@TedShifrin Just teasing ...

@leslietownes Hah! I see!

Ted Shifrin

Students want answers and no thinking. Probably more than 20% of the exercises require thought and exploration. Awww. Most unhappy customers are undoubtedly and sadly math ed majors.

leslie townes

every post-calculus college text should have a preface to the reader that says "Dear reader, if you are expecting the exercises in this text to be identical to 'worked-out examples' from this text with different numbers plugged into them, I have some very bad news for you. Sincerely, [Author name]"

2

that was a huge thing at my undergrad. lots of intense hate from the ed focused people.

Ted Shifrin

The future teachers. Explains a lot about this country.

leslie townes

9:51 PM

not just undergrads one semester past calculus, either.

Ted Shifrin

Admittedly, I would significantly rewrite a good deal of that book after many times teaching it.

Luckily, no interest and I’m a bum.

leslie townes

there's something to be said for a lot of 'standard math curricula' not really preparing people to teach K-12 mathematics. berkeley did eventually introduce courses specific to that. i don't think it turned off the hate from some people.

amWhy

Apologies for having linked it. Unfortunately I am not not a room owner, so again, apologies, and more apologies for daring to post that comment, @Ted.

Ted Shifrin

No apologies called for. It’s public.

A few times I taught the course with mostly math ed majors, I emphasized the links to the high school curriculum more and got plenty of thanks from many students.

My book was more biased in that direction — including rings first organization — on purpose.

I think Wu used my book once at Berkeley, but I never heard back, so maybe he wasn’t too happy.

user 726941

@AkivaWeinberger Thanks

UnderMathUate

10:03 PM

Sad as it is to say, one thing that surprised me about the non-freshman courses, is the lack of similarity between class examples and homework assignments...

That isn't to say I didn't appreciate them, but the learning curve was a little steep.

Ted Shifrin

You should try freshman courses in other countries. The USA has watered down curriculum unbelievably.

A lot of it is faculty who don't want the hassle of being demanding and "wasting" time on teaching.

And it's the culture in the US now ... part of the whole global problem ... that students are consumers and get to dictate what they must do and what grades they should get.

UnderMathUate

I've heard about it from a few international students. One of the guys I know said his high school taught up to calc 3 before he came to the U.S.

Ted Shifrin

/rant

That's happening in lots of high schools here, too, @UnderMath, but most of the courses suck.

UnderMathUate

Ha, lol

Ted Shifrin

Calc 3 is badly enough taught by college professors; now less qualified high school teachers just are mucking it up.

leslie townes

10:06 PM

this is nominally what the AP Calc BC curriculum is supposed to do, I think.

it's garbage.

Ted Shifrin

No, BC goes through Calc 2, sorta.

2

leslie townes

oh, hrm. my high school class went through calc 2, but it was denominated AB. i think the teacher threw the AP curriculum, whatever it was, in the trash and just taught it.

Ted Shifrin

But most people get 5's without any decent knowledge of sequence/series material.

leslie townes

we didn't have BC so i assumed that's what BC was for.

UnderMathUate

I wish I could experience a math department in, like, the 50's. When people would just walk around and talk about math and stay up all night for fun.

Ted Shifrin

10:07 PM

AB just gets to FTC, areas, and integrals by substitution.

leslie townes

yeah, we did a full sequences and series thing in my calculus class.

Ted Shifrin

@UnderMathUate That's how my honors classes at UGA were up until the end.

UnderMathUate

Ones you taught, or ones you were in?

Either way, that's pretty cool ;-;

Ted Shifrin

LOL, ones I taught. I retired 6 years ago.

UnderMathUate

Oh, lol

leslie townes

10:10 PM

tell us more about college in the 50s, ted. :D

Ted Shifrin

Even I am not old enough to have been a college student in the 50s.

smacks leslie

UnderMathUate

XD

leslie townes

studying by the light of oil lamps, almost burning the library of scrolls down. good times.

Ted Shifrin

Writing homework on papyrus.

leslie townes

learning geometry in the original greek, from an actual pythagorean.

UnderMathUate

10:12 PM

Now those guys were wild. I mean the Greek mathematicians.

leslie townes

i don't miss my undergrad days too much. there was not a lot of camaraderie, everyone was going in different directions. before everyone had cell phones you'd just go home and not hear from anybody unless you phoned them like some kind of weirdo.

UnderMathUate

Funny, that's how it is for me now.

I ask people to study, but they just say yes and then blow me off.

Ted Shifrin

COVID has killed education.

leslie townes

i really like texting. it's way better than talking to people on the phone or in person.

UnderMathUate

I prefer texting or in-person. I like talking to people, I'm just not too good at it lol

leslie townes

10:15 PM

maybe not the best way of studying math in a collaborative fashion, but otherwise, it's just great.

UnderMathUate

Ha, they should have TEX for phones

leslie townes

one of my oldest friends and i text all the time. there was a period of about 10 years, starting with college and continuing through an era where texting among people in our age group was not normalized, where we might talk to each other once or twice a year.

now it's just like, "wow, this is a delicious apple. [texts photo of apple with this comment]"

UnderMathUate

Lol, you know you have a good friend when they text you completely inane comments.

Alright, gotta go to a meeting. Catch you all on the flip-side.

leslie townes

ted is it super foggy where you are? pea soup here. almost like a berkeley morning, but it's not berkeley, and the afternoon.

lots of horn blasts from the harbor.

copper.hat

warm & sunny in albany

i didn't have the energy to cycle up to kensington yesterday, settled for an evening ride along the waterfront to emeryville.

leslie townes

10:21 PM

its about 60F here. this counts as 'cold.'

Ted Shifrin

not foggy now, but it was last night and it's supposed to be again tonight.

@copper I'm still waiting for you to check out 287 Willamette! :)

copper.hat

:-)

amWhy

@leslietownes But texting doesn't require communication skills; it's all one way. I know, that's why people like it. !! But I'm convinced emails and texting and social media forums have inflated many folks sense of importance.

The art of respectful discourse, is at risk of extinction these days. Does that make me a dinosaur for saying so?

Semiclassical

I mean, you could equally well say that chatting here doesn’t require communication skills

amWhy

@leslietownes Pff

Semiclassical

10:34 PM

…which sounds about right :P

leslie townes

Pbhbhbbhhbhtt

2

amWhy

@Semiclassical There's more a give in take, in real time, at least in our chats, than in texting or emailing.

leslie townes

^- that's directed at you, semiclassical

i feel weird pasting a photo of the interior of my refrigerator into an email. in a text, it feels just right.

amWhy

@leslietownes Thanks!

@leslietownes That's precisely the problem with texting! :P It shouldn't feel "right"!

leslie townes

nobody who had seen a photo of my refrigerator would take that position.

Ted Shifrin

10:38 PM

Interesting. I can't find a question I commented on this morning with helpful suggestions. I guess the OP deleted (or did it get close votes?). Someone told me how to find deleted things, but maybe I needed to have answered rather than commented.

leslie townes

ted, that can be tricky. if you can find the url of the post in your browse history, that might be a start. i don't know that there's a menu way of doing it.

not at my rep, anyway.

amWhy

@leslietownes Of course your wife wouldn't take that position. That's what couples do.

@TedShifrin Go to your profile, click activity. You should be able to find a tab to "comments" (can't remember if those are comments from or to you) and/or "all actions": one of the two will list your comments on main, most recent at the top.

Ted Shifrin

Yeah, @amWhy, it is not there.

amWhy

Profile- Activity tab. On Activity: Next, click "All actions" listed among other options just above summary of posts, and recent rep. From there, click comments, @Ted, which will list your comments.

copper.hat

if the question was deleted it no longer appears there...

amWhy

10:47 PM

@copper.hat Thanks. I should have said "if it still exists, you will see your comments there, which will link to the post on which you commented."

Ted Shifrin

Right, but robjohn taught me a trick of clicked on deleted somewhere ... and that worked, but I think I needed to have written an answer and not a comment? I forget.

amWhy

I'm sorry, I'm not sure. Any answer you've written, that was deleted, you can find by clicking "show all my answers", and at the very bottom of the list, you can click on "recently deleted answers", to find them. But I don't think that can be done with comments.

Else, I'll just shut up. ;)

Ted Shifrin

Yeah, it's just for answers. I found it.

amWhy

@Ted I remember reviewing a post today, not sure in which queue, but I saw comments from you, so I skipped the review, because your comments are usually helpful to askers, and encourage clarification or suggestions.

Ted Shifrin

Yeah, my comments told him what to do, pretty much.

amWhy

10:59 PM

@TedShifrin Maybe you gave him all the help he needed! We can't really measure total effectiveness of this site, because for many users, comments are sufficient.

Ted Shifrin

Shouldn't delete, but oh well. I told him to edit with his progress.

geocalc33

11:17 PM

Correct me if I'm wrong but do algebraic geometry people study the simultaneous vanishing of multivariate polynomials?

Ted Shifrin

In some vague sense, yes.

geocalc33

11:33 PM

Does this theme present itself in analysis too? For example do people study the simultaneous vanishing of systems of partial differential equations?

leslie townes

sometimes? i don't know that much of an analogy extends beyond that.

geocalc33

or in general the equality of $N$ partial differential equations

because studying when they vanish would be studying a subset of the solution space

leslie townes

the number of equations in a system isn't really a useful quantity without hypotheses on the 'structure' of the equations. a=b and c=d and e=f is equivalent to the one equation |a-b| + |c-d| + |e-f| = 0, even if a, b, c, d, e, f are complicated expressions.

this probably may even arise in algebraic geometry. i think a lot of the invariants they care about are somewhat divorced from the symbolic form in which a problem might be phrased, although i guess it starts there.

maybe a bit like how after you learn a bit of galois theory, the notion of classifying polynomial equations by their degree doesn't feel 'right' in a lot of contexts.

i dunno.

geocalc33

$$u\frac{\partial}{\partial u}\left(u\frac{\partial f}{\partial u}\right) + v\frac{\partial}{\partial v}\left(v\frac{\partial f}{\partial v}\right)=\frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2}=\frac{1}{u} \frac{\partial}{\partial u} \left( u \frac{\partial f}{\partial u} \right) + \frac{1}{u^2} \frac{\partial^2 f}{\partial v^2}$$

for example^

leslie townes

oh boy, someone's fiddling with different expressions for a laplacian again

Ted Shifrin

11:49 PM

Equating laplacians with respect to different metrics doesn't seem an interesting escapade to me.

geocalc33

a=b=c

a=b

a=b has an analytic solution

that's just a contrived example

5

Q: Can I combine the wave and heat equations?

I have this equation $$\frac{\partial^2u}{\partial x^2} = 2\frac{\partial u}{\partial t} + \frac{ \partial^2u}{\partial t^2}$$ Is it possible for me to use both the wave and heat equations to solve this equation? I understand both, I just wanted to see if it was possible.

partial-differential-equations heat-equation wave-equation

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$Mathematics$ Mathematics