Mathematics - 2021-12-08 (page 2 of 3)

maths researcher

14:11

under what conditions can we guarantee that the entries of a character table integers

Alex

Hi! :-)

@robjohn Yesterday my cell phone went off and I could not thank you for your kind answer to my question. Thank you very much for your very nice answer!

robjohn

@Alex The answer to the question I answered implies the answer to the question you were asking about.

but not vice-versa

@Alex Of course, in the answer to the question you linked, $\frac{x^{2n}}{(1+x^{2n})^{(n-1)/n}+\cdots+1}\lt\frac1n$, for $0\le x\le1$ which gives convergence.

UnderMathUate

14:27

@leslietownes If you're still interested, I ended up with $f(x) = {4}\frac{1-x)$ for $(-\infty, 0] \rightarrow (0, 4]$

Alex

@robjohn the user284331's answer should not be corrected?

@robjohn Thank you :-)

UnderMathUate

why is the fraction not working ;-;

f(x) = $\frac{4}{1-x}$ for f: (-00, 0] -> (0, 4]

robjohn

@Alex The part after what I just mentioned doesn't seem to make much sense, either.

@UnderMathUate it works for me. what are you trying to make it do?

UnderMathUate

Show that |(-00, 0]| = |(0, 4]|

Semiclassical

@mathsresearcher this seems relevant : mathoverflow.net/questions/10635/…

robjohn

14:31

@UnderMathUate what is the absolute value of an interval? its measure? its cardinality?

UnderMathUate

Oh, sorry. I'm referring to cardinality. I want to prove that the intervals are numerically equivalent.

robjohn

and is $-00$ supposed to be $-\infty$?

UnderMathUate

yeah

I just got frustrated with chatJax lol

robjohn

@UnderMathUate In what way is the fraction not working?

Alex

@UnderMathUate Some time ago, I was also partially defeated :-(

Semiclassical

14:34

In particular the second-rated answer references a theorem of Serre which says that “characters are rational iff characters are integers iff conjugacy classes are rational”

UnderMathUate

@robjohn Oh, I was referring to the message I sent above that, where it wouldn't display what I was trying to type. As for the function itself, I believe this works for this mapping.

@Alex Wdym?

Semiclassical

What’s not clear is whether there’s examples of such besides the symmetric groups

robjohn

@UnderMathUate Oh, I thought I fixed that message

Your last edit was $A = \{\frac{m}{2^n)\}$ $: m, n \in \mathbb N}$

user193319

I read somewhere that one can always define a nontrivial homomorphism from an HNN extension to $\Bbb{Z}$, thus proving an HNN always has a proper normal subgroup. How does one define this homomorphism? Let $G = \langle S \mid R \rangle$ and let $\alpha : H \to K$ be an isomorphism of subgroups of $G$. Then $G \ast_{\alpha}$ denotes the HNN extension of $G$ relative to $\alpha$. I was thinking the homomorphism could be defined by $s \mapsto 0$ for $s \in S$ and $t \mapsto 1$...

robjohn

@UnderMathUate I think you wanted $A = \{\frac{m}{2^n}: m, n \in \mathbb N\}$ . I also added resized braces.

user193319

14:47

...where $t$ is the stable letter of the HNN extension. If $t$ has infinite order, this would work...Does the stable letter always have infinite order?

UnderMathUate

@robjohn Oh yeah, thanks

I always forget the syntax for that

Alessandro Codenotti

@user193319 doesn't this follow from Britton's lemma?

Alex

@robjohn Ok, I understand.

robjohn

@Alex: I have added a comment

user193319

@AlessandroCodenotti Does what follow from it? That $t$ has infinite order or that we can define a map to $\Bbb{Z}$?

Alessandro Codenotti

14:57

Infinite order

Wolgwang

@robjohn What is a triangle in the wedge? This ?

robjohn

@Wolgwang Do you see the circular wedge and the triangle inside it?

Wolgwang

@robjohn Shape below AB is wedge?

user193319

15:12

@AlessandroCodenotti Hmm...I don't quite see it. I'm reading the wiki page on HNN extensions and apparently every element of $G \ast_{\alpha}$ of finite order is conjugate to an element of $G$. Hence, if $t$ were of finite order, then $t = x^{-1} gx$ for some $x \in G \ast_{\alpha}$ and $g \in G$. My guess is that this implies $t \in G$...? That would certainly be a contradiction, but I don't see it.

Alessandro Codenotti

@user193319 Look at the "alternate form" on wiki. Does the string $ttt...ttt$ satisfy its hypothesis?

user193319

@AlessandroCodenotti Yes, it looks like the second condition involving $n > 0$ is satisfied, so $t^n \neq 1$ for every $n \in \Bbb{N}$.

Alessandro Codenotti

Which is the same as saying that $t$ has infinite order

user193319

Indeed!

maths researcher

15:41

@Semiclassical sure, but that's for only symmetric groups. Whereas most group im encountering have character table with integer entries, so I assume there is a more general reason

jay

15:52

Can anyone help me understand this ^, considering I know near to 0 differential geometry. e.g what is a local norm

here $d$ is the wasserstein distance,

leslie townes

17:21

jay, this looks similar to the recipe by which a riemannian metric (= way of measuring lengths of tangent vectors to curves in a space) relates to a metric (= way of measuring distances between points in space). here, they start with the latter kind of metric and show it can be recovered by minimizing 'lengths' of curves as measured by (2.2) and (2.3). give or take some squaring stuff this is the usual recipe from riemannian geometry

the 'local norm' is a way of measuring the lengths of tangent vectors (if you like, 'velocities' of curves) in your space. the use of the word 'formally' and reference to a 'rigorous definition' suggests that there may be some mathematical details under the hood here that are not express in the formula

for what it's worth, it looks like PDF copies of both [4] and a draft of [6] are available if you google

although they look very technical and maybe won't help. i couldn't get much out of them

i think by [4, 10] they mean [4], [10]. [4] has no section named 10

amWhy

@leslietownes That recipe is just scrumptious, when adhered to correctly!

leslie townes

i inherited my family recipe of relating riemannian metrics to metric space metrics from my grandmother. her secret is to use cream or half-and-half instead of milk.

amWhy

@leslietownes ;-) Just came from reading comments above, about overcooked meat, and spinach, etc.! I mean just look @robjohn. Seems to look overcooked, unless he's a square orange!

leslie townes

spinach is really hard to cook. i like it blended in palak paneer, or added very late in a vegetable soup with other elements. ideally after the heat has been turned off and it's just cooking from the heat of the soup.

i think spinach somehow violates conservation of mass. i've definitely put like 5 bags of spinach into one pot of soup before.

nobody can explain how this works to me.

amWhy

I get why he a mean squared (orange), 'cuz more spherical oranges probably tormented as a child orange mean square.

Semiclassical

17:36

@mathsresearcher found a more direct Q&A, though with the same citation to Serre: math.stackexchange.com/q/2792741/137524

jay

when $J$ is a functional and $\rho$ is probability density what do authors usually mean when they write $\frac{\delta J}{\delta \rho}$?

amWhy

@leslietownes transforms into slime in green water, some of which evaporates.

Semiclassical

Functional derivative

In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a Functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf...

jay

^ Thanks

Semiclassical

in particular, see the "Coulomb potential energy functional" application: taking $J$ to be the Coulomb electron-electron potential energy, they compute $\delta J/\delta \rho$ explicitly

Wolgwang

18:03

@LeakyNun $\int_a^b \sqrt{1+(f'(x))^2 } \mathrm dx$ ?

Leaky Nun

@Wolgwang yes, mea culpa

but the argument still stands mutatis mutandis

leslie townes

don't blame mutants for this

they had nothing to do with it

Leaky Nun

hehe

Wolgwang

@LeakyNun I have read this proof. Is there some visualization of this formula?

Ted Shifrin

@Wolgwang: It's little bits of Pythagorean theorem.

Semiclassical

18:06

via google:

Leaky Nun

it's basically $\sqrt{(dx)^2 + (dy)^2}$ if you squint hard enough (where y=f(x))

Ted Shifrin

Where those symbols are sorta undefined :P

Leaky Nun

and if you turn off your inner Ted for a while

Semiclassical

$dL = \sqrt{(dx)^2+(dy)^2} = \sqrt{1+(dy/dx)^2}dx = \sqrt{1+f'(x)^2}\,dx$ at physicist levels of rigor :P

Ted Shifrin

well, sometimes those issues actually bother beginners, @Leaky

Nemanja Vuksanovic

18:09

Can someone tell me, why the cross vector to (-2, -2) is (-2, -2) AND NOT (2, -2)?

I got it as a fail on my homework.

Leaky Nun

@NemanjaVuksanovic why don't you ask your teacher

Nemanja Vuksanovic

@LeakyNun Vacation.

Ted Shifrin

What is "cross vector"?

Nemanja Vuksanovic

@TedShifrin The vector that is 90 degree (against the clock) of your original vector.

Wolgwang

@Semiclassical Google o_o. I have to improve my googling skills.

Ted Shifrin

18:10

Well, the vector can't be 90 degrees to itself. Did you have a typo there?

I agree that it should be $(2,-2)$.

Wolgwang

Thanks Leaky Nun, Semiclassical, Ted :-)

shintuku

has anyone ever tried that 16-hour fasting technique to reset your sleeping schedule?

i need to undo that finals sleeping schedule

Ted Shifrin

Our room expert on messed-up sleeping schedules is @Balarka.

Wolgwang

@shintuku Try polyphasic sleep. JK

Nemanja Vuksanovic

@TedShifrin Do you need to be a specific level to post images?

shintuku

18:13

everyone i've seen try polyphasic sleep

have the physical appearance of someone that really wants to die but doesn't realize it

Ted Shifrin

Probably, @Nemanja. I don't know what it is.

Wolgwang

@NemanjaVuksanovic Do you mean reputation?

Nemanja Vuksanovic

@Wolgwang Yes!

@TedShifrin It's alright, I will try and use my imagination when explaining this

Ted Shifrin

I understood what you said, @Nemanja, and I agreed with your answer.

Wolgwang

@NemanjaVuksanovic Use "Ask question" section of any SE site and paste the imgur link...or you can simply select 'Desktop site' and then upload...

Ted Shifrin

18:15

I think you need a certain amount of rep to post images in chat, @Wolgwang.

Wolgwang

Hmm, 100 reps

Nemanja Vuksanovic

Imagine you have a point on a coordinate system, let's just say you start from origo. If you have a point, let's say (3, 0), if you draw a vector, it will be (3, 0). The cross vector will be a new point which I learned is (original coordinatesystem (0,3) switched upside down and include a minus sign on the x-value. So 0,3 becomes -3,0 which, if we draw a new vector, will be 90 degree (against the clock) to our original vector).

@TedShifrin I was just curious to the specific level, since images helps explaining stuff :))

Ted Shifrin

So, saying that you rotate the vector 90º around the origin is sufficient. If you rotate $(a,b)$, you get $(-b,a)$. Your original answer is correct. No point going around and around this.

Since a vector cannot stay the same (which is what you said the "correct answer" was) when you rotate it, there's a typo or a mistake somewhere.

Nemanja Vuksanovic

@TedShifrin Oh, so my mistake was the wording I used, I guess I learned something new, thx!

Ted Shifrin

I don't understand now.

Nemanja Vuksanovic

18:20

@TedShifrin You wrote "there's a typo or a mistake somewhere" do you mean my actual calculations or the way I phrased my question/ example?

unit 1991

Hello. Can anyone help with this problem?

0

Q: Proof concern about $f'(x)=lim_{n\to\infty }f_n'(x)$

$1)$Functions $f_n$ are differentiable on interval $X$. $2)$Exists $a\in X$ st $f_n(a)$ sequence is convergent. $3)$ $f_n'(x)$($\frac{df_n}{dx}$) sequence is uniformly convergent on interval $X$. Then $1)$ Sequence $f_n(x)$ is uniformly convergent on any $E\subset X$ bounded set(converges to $f(x...

Ted Shifrin

I mean in the original statement and what you said the teacher said the answer is, @Nemanja.

@unit1991 Do you not see a difference between $g_n(x)$ and $f_n'(x_0)$?

unit 1991

@TedShifrin Maybe it is silly question but no.

Ted Shifrin

Doesn't the definition of $f_n'(x_0)$ have limit in it?

Nemanja Vuksanovic

@TedShifrin Ohhh, I beg your pardon. I got confused for a minute. You also said something about "the vector can't be 90 degrees to itself.". Our textbook says "The cross vector has the same length as the original vector, but is rotated 90 ° counterclockwise itself". How/ what would be a definitive way of explaining what a cross vector is, so people don't get confused? :)

unit 1991

18:24

@TedShifrin Oh it was really silly question :)

Ted Shifrin

@Nemanja ... That is perfectly fine. You said your teacher said that $(-2,-2)$ was the answer for the cross vector to $(-2,-2)$. I said that is impossible.

@unit1991 Material on sequences of functions is very subtle. You have to pay attention to every little detail :)

Nemanja Vuksanovic

@TedShifrin Thank you so much for clarifying. May you have a great night/ day forward :)

unit 1991

@TedShifrin Thanks you :)

Ted Shifrin

You too, @Nemanja. You're welcome, @unit1991.

robjohn

Oh, my. I just noticed that the Answers pane now shows information on the question answered (votes and acceptance), not the votes the answer got.

Ted Shifrin

18:26

howdy @robjohn

Wolgwang

I have never heard about cross vector. Is the term popular?

robjohn

@TedShifrin Hey there. Just noticing how screwed up the new interface is

leslie townes

i've never heard it either, although i think i have seen books introduce terms for that.

Ted Shifrin

Yes, now that you called my attention to it ... it's telling me that the question I answered (and got no votes on) has 2 downvotes. Ridiculous.

leslie townes

let's go add some downvotes to your answer, so at least the totals are the same.

Ted Shifrin

18:28

@Wolgwang I've never head of it other than in the context of cross product. So, for vectors $\vec v$ in the plane, the cross as he told us is $(0,0,1)\times\vec v$.

@leslie It was one of my only answers ever to a middle-school level question.

robjohn

@TedShifrin that is because the question got two downvotes

Wolgwang

Ohk Thanks for clarifying.

Nemanja Vuksanovic

@Wolgwang I think some might call it transverse vectors. I simply just translated the name from my origin language to English, so cross vector is what I got. Sorry for confusion :-)

Ted Shifrin

In my linear algebra book, I did define $\rho\colon\Bbb R^2\to\Bbb R^2$ to be the $90^\circ$ rotation very early on.

Yes, @robjohn, I get it.

Did they make this change on porpoise?

Yet more reasons to be annoyed with MSE.

robjohn

They really messed up the layout, too, not only the data

This is crazy

Ted Shifrin

18:30

@Nemanja Well, in upper-level mathematics, transverse has rather a different meaning.

Wolgwang

Whitespace

Whitespace intensifies

Ted Shifrin

@robjohn this reminds me of the game with revisions of textbooks – usually, after the second edition, all the mandated "improvements" for the publisher to make money are in fact disimprovements.

leslie townes

ted: 2 downvotes on a rep 1 user for posting an image seem excessive, particularly as the image did contain an attempt, but whatever. i'll find a friend and we can give your answer a -2 total, to harmonize the numbers.

Ted Shifrin

I'm less annoyed by the image (in this case) than I am by the fact that the OP never responded to my answer. Numerous examples of that.

Anyhow, I certainly did not downvote. I wasn't thrilled with the posted chart, but I explained how to verbalize it all ... shrug

Nemanja Vuksanovic

I finally made an imgur link as someone of you suggested. Here you can see the "cross vector". (imgur.com/a/Byi0gNt)

The text is in Danish, however, translating it to English gives us "cross vector"

I find that mildly funny

Ted Shifrin

18:36

@Nemanja We understood your words just fine. I would rather put the tails of the vectors at the origin, though.

Cross = angry ? :)

Nemanja Vuksanovic

@TedShifrin Could be angry haha

leslie townes

i find 'en lille 'hat'' very cute. cuter than the translation.

Ted Shifrin

@leslie For you, it's the Fourier transform of the vector.

leslie townes

the danes really do know how to make everything hygge

Nemanja Vuksanovic

Yea, Mathematicians like to put "hats" on symbols, they might freeze in the cold you know

leslie townes

18:37

hopefully it's a knit hat in bright colors, as you can find in the tourist districts in copenhagen

Nemanja Vuksanovic

@TedShifrin I petition you to be the forefront of inventing international phrasing for every mathematical subjects, so no one ever gets confused. We start with cross vector (or what you guys name it in England/ America)!

leslie townes

putting a hat on a letter that's already got an arrow on it is very close to putting a hat on a hat

Nemanja Vuksanovic

@leslietownes And on it it says "I <3 CPH".

Ted Shifrin

We don't actually have a name for it, @Nemanja.

Xander Henderson

We are currently having a conversation about what a "W" grade means. It turns out that if a student is withdrawn from a class by the instructor, they remain eligible for GI bill benefits. However, if they are withdrawn by an advisor or rec / reg, they become ineligible.

Nemanja Vuksanovic

18:41

@TedShifrin As a representative of the Human Race and Mathematics, we hereby task you with your first mission for humankind, which is to find a word for it. Good luck sir!

Xander Henderson

A possible solution is to provide two different grades on the transcript. One for "withdrawn by instructor", one for "withdrawn by advisor".

I suggested a number of possibilities: $\omega$, $\hat{W}$ (put a hat on it!), W, \/\/, VV, etc.

I was subtly told to go to hell. :P

leslie townes

xander, weird that there would not already be a solution for this. i realize there's no harmony among different schools' policies, but this seems like something that would come up a lot.

Xander Henderson

@leslietownes inorite?

leslie townes

no \mathfrak{W}?

Xander Henderson

@leslietownes NO!

mathfrak is the devil.

leslie townes

18:43

\mathbb{W}?

Xander Henderson

Sure, that's an option.

$\dot{W}$

Lukas Heger

$\mathscr{W}$

leslie townes

how about index notation. W^i for instructor, W_a for advisor. more information could be included in more indices.

Xander Henderson

$\bar{W}$

$\mathscr{W}$

geocalc33

$\text{W}$

Xander Henderson

18:44

@leslietownes Sure. I love it.

$\mathsf{W}$

$||x|-1|$

Ted Shifrin

@XanderHenderson At UGA if a student got a WF then the student was no longer eligible for the various benefits for the course; a W was OK.

We never had such a thing as withdrawal by adviser. Although the Dean's office could withdraw a student from all classes for health reasons.

Lukas Heger

@Xander how about $\digamma$? The digamma was pronounced [w] in pre-classical ancient greek

Xander Henderson

@TedShifrin An instructor can withdraw a student who doesn't show up. Or a student can request to be withdrawn, which is handled by an advisor.

@LukasHeger Sure. Why not?

Ted Shifrin

Weird @Xander. In principle instructors had to sign off on any withdrawal request in order to assign W or WF. I think the WFs disappeared for a brief period shortly before I retired. I'm not sure where it stands now.

Random Variable

19:05

@robjohn The MathJax issue was fixed in some places, but not everywhere. There's a lot of negative feedback on Meta Stack Exchange about the redesign.

robjohn

@RandomVariable Yeah, I was just looking there...

Leaky Nun

Live Covid latest news: Boris Johnson confirms return of working from home as England moves to Plan B

Ted Shifrin

@Leaky How long before he gets it again?

leslie townes

long beach reported its first omicron case the other day.

nice to see other countries' embarrassments sometimes outperforming our own.

Ted Shifrin

This is going to be seriously world-wide.

leslie townes

19:09

this is why i'm glad i have all my money in lesliecoin, the only currency that only goes up in value, forever.

Ted Shifrin

You'll forgive our professional skepticism.

leslie townes

socrates, archimedes, galileo, me. victims of professional skepticism every one.

Ted Shifrin

Well, when you run around naked yelling Eureka, we might pay some attention.

leslie townes

i can't talk about that before my trial.

Ted Shifrin

Before the tribulations?

leslie townes

19:22

there will be no tribulations.

Ted Shifrin

This is too boring.

robjohn

@RandomVariable I see here that someone has mentioned the problem with the Answers section of the Summary page.

Ted Shifrin

This is too much gobbledygook for me.

Dan Donnelly

Slightly racism

Ted Shifrin

Yeah, should be careful with that word.

robjohn

19:35

@DanDonnelly not intended. I didn't realize. I need to update my list of deprecative terms.

Dan Donnelly

@TedShifrin I have your book. I should start it soon

Oh yeah, the manifolds, linear algebra, multivariable calc one

Ted Shifrin

I won't ask if it is an illegal copy.

Xander Henderson

@TedShifrin I don't have your book. I have a text which covers the same material, written by that one guy that you really hate.

Dan Donnelly

I will buy it, once I'm at least 1/4 through, and when I can afford :)

Ted Shifrin

That's a different book, @Xander. Unless you're talking about Hubbard and Hubbard. I don't hate him. I just don't like the book.

Xander Henderson

19:38

@TedShifrin I was being facetious.

Dan Donnelly

I own several books that I first got "digitally" and then later bought on Amazon

Ted Shifrin

Well, if you're going to be aware of language slurs, @Dan, you should be aware that pirated texts/CDs are a big issue for us authors/artists.

Xander Henderson

I just assume that you have a mortal enemy, and that this person has written a book.

leslie townes

what's a CD?

Ted Shifrin

@Xander I wasn't sure which book/author you were referring to.

Xander Henderson

19:39

So I am claiming to have a copy of that book.

@TedShifrin You know, the one written by your mortal enemy.

Ted Shifrin

My dislike of Axler's linear algebra is well established, although I certainly don't hate Sheldon. He may hate me :D

Astyx

Shed Tifrin

Dan Donnelly

It's better than not reading the book at all, and then ending up never buying

leslie townes

xander: ted is now confronting that evil mathematician in the mirror. give him space.

Ted Shifrin

Well, @Dan, I'm just pointing out there are ethical issues. I do not defend the barbaric prices of books.

Hi @Astyx

Astyx

19:40

Hi

leslie townes

copyrights are holy. i did give myself a dispensation when the usps lost some of my books.

Dan Donnelly

Also, I could check out your book from the library. Think of my copy as a digital library copy :)

Xander Henderson

Is anyone here gonna be at the super spreader event in Seattle next month?

Dan Donnelly

I will buy it later, looks to be a good book

*book

leslie townes

xander: what's that? the, uh, jerk convention? (trying to come up with an insulting joke, blanking out)

Xander Henderson

19:41

@leslietownes Joint Math Meetings

Dan Donnelly

They put the joints before the math, so that they're more creative physiologically

leslie townes

i should show up and try to convert people to the legal profession.

Ted Shifrin

Just what we need. Talk about lack of ethics ...

Xander Henderson

@leslietownes And sell lesliecoin.

UnderMathUate

Sigh. The non-euclidean geometry course I was planning on taking next semester was canceled.

Ted Shifrin

19:43

@Under At many schools that's a course for future teachers. I don't know the situation at your place.

leslie townes

yes. is it too late to sponsor a booth?

UnderMathUate

Here, it's an elective for math majors.

Xander Henderson

@TedShifrin That is certainly where it was taught at my undergrad institution.

Dan Donnelly

I know. It should be vape math meetings, smoke is bad for lung health

UnderMathUate

@TedShifrin Did I dodge a bullet?

Ted Shifrin

19:43

That doesn't make it a bad course; just a different audience/background.

leslie townes

at berkeley it was an elective although the education 'track' did not exist when i was there. i think they may have built it in. i hope that didn't damage the usefulness of the class to non-teachers.

Ted Shifrin

@Under It depends on the book/teacher/course. There is lot of beautiful material there. I personally want to teach it more as differential geometry and less as axiomatic stuff.

Xander Henderson

At my undergraduate institution, there was a three semester sequence for education folk: set theory, "theory of positive integers", and a geometry class which spent about 5 weeks on non-Euclidean stuff.

Ted Shifrin

I have some material on projective and hyperbolic geometry in the last chapter of my algebra book, of all places.

Xander Henderson

All taught by the same guy for a period of about 40 years (he retired shortly after I graduated, after having been at the institution for 53 years).

leslie townes

19:45

xander: did he teach that non-euclidean stuff was an illusion?

Ted Shifrin

Boy, talk about getting stale with material ...

Xander Henderson

@TedShifrin Yeah.

leslie townes

"we used to burn people at the stake for this, but it's in the textbook, so here ya go"

"euclid still owes me $50"

Astyx

what's included in a non euclidean geometry course?

UnderMathUate

I wanted to take it so I could be a bit more "well-rounded". I haven't thought much about geometry since high school. Save for what's necessary in calculus.

Xander Henderson

19:46

@Astyx Kill the fifth postulate, then see what happens.

:P

UnderMathUate

If the material is something I could just pick up on my own, I'll just do that then.

leslie townes

under i like hartshorne's book 'geometry: euclid and beyond.'

it has a lot of that stuff ,although from the axiomatic point of view. not hugely useful for e.g. a differential geometry class.

UnderMathUate

Ok, I can check that out then.

leslie townes

the way all water eventually heads toward a drain, all math eventually leads to differential geometry.

2

UnderMathUate

How seriously should I take that statement?

Ted Shifrin

19:48

@Under There are some good books, some pretty demanding. My favorite demanding book is by Pedoe: Geometry: A Comprehensive Course.

leslie townes

not too seriously. i mostly wanted to analogize DG to a sewer.

3

Ted Shifrin

smacks leslie twice

UnderMathUate

lol

I'll add that one to the list too, then.

leslie townes

if you find a beat up paperback preprint of hartshorne in a used bookstore and it has my email address in it, you need to give me that book back. that's one that the usps lost.

i included better proofs of some of the theorems in the margins. you'll know it when you see it.

Ted Shifrin

I don't trust your judgment on "better" over Hartshorne.

UnderMathUate

19:50

Hartshorne?

leslie townes

they were mostly for lemmas on non geometric stuff.

i don't claim to know triangles better than he does, or whatever.

Ted Shifrin

I still don't trust you.

Astyx

The statement probably holds if you remove "better" from the whole sentence

UnderMathUate

Oh wait, I feel like you guys mentioned that book before.

Oof, right above.

leslie townes

in the draft he gave several proofs characterizing the rational multiples of pi for which the cosine is rational. all of them were too long. probably longer than what you can find on mse.

but that's not really geometry.

Dan Donnelly

19:52

For Hartshorne, one really needs to refer to a solution manual for getting the "easy tricks" involved, especially with sheaves. Changes a page-long proof into a 2 sentence proof.

There's one available on github

Ted Shifrin

Wrong Hartshorne.

Well, same person, different book.

Dan Donnelly

AG?

Ted Shifrin

@leslie I think I wrote a proof for that for one of our high school math contest questions. I'd have to search for it.

Astyx

G:EAB

leslie townes

you can do it from the rational root theorem if you know trigonometry. very high-schoolable.

UnderMathUate

19:53

I'm taking a class called "Coding Theory" in place of the geometry course (not programming related).

Ted Shifrin

Yes, that was the proof, in principle.

@Under That's number theory.

Lots of Fermat's little theorem.

UnderMathUate

0-0

leslie townes

anyway, robin missed that one in his draft book. uses galois theory and then some other method.

Astyx

It's AG

leslie townes

they probably included better proofs in the springer edition

Ted Shifrin

19:55

Galois theory is in fact overkill.

Dan Donnelly

I own Algebraic Geometry by Robin Hartshorne (hardback) shows off book

I paid for it myself! :)

It's one I'm going to keep nice. My Categories & Sheaves book lost a few pages and the cover :'(

I will wear nitrile gloves and read it with a red headlamp in the dark

UnderMathUate

Is it hard to obtain or something?

Ted Shifrin

@leslie Oh, my question was a bit different. For it we needed algebraic integers. The question was: For how many positive integrs $n$ can we fit tightly packed congruent circles of radius $1$ in the ring between concentric circles of radius $n$ and $n+2$?

Dan Donnelly

If you're poor like me @UnderMathUate

UnderMathUate

I'm a college student, man.

Astyx

19:59

something something college shouldn't cost you anything something

Ted Shifrin

I might have had a different question that was exactly what you're after.

@Astyx: Welcome to the US, where education and health care are only for the rich and entitled.

Astyx

of course they're entitled, they're rich!

leslie townes

and remember, health care doesn't include dental or eye care. that's separate.

Xander Henderson

@leslietownes And don't forget mental health services!

leslie townes

yeah, separate.

i recently paid around $2500 for some dental work that my mom needed. medicare doesn't cover that, or her VA coverage. apparently if you're hospitalized and a doctor touches your teeth in that context, sometimes, medicare can kick in.

bad luck to not require hospitalization, i guess.

UnderMathUate

20:09

America is so backwards.

Xander Henderson

@UnderMathUate No, America is forwards. You are thinking of aciremA.

UnderMathUate

Oh, right. I get those two mixed up sometimes.

Ted Shifrin

I've had the "best" dental insurance for many years, and I almost always end up paying for (most of) crowns and tooth extractions out of pocket.

Mostly the fault of a quite incompetent (but very nice) dentist I had most of my years in Athens, GA.

leslie townes

same. i have no idea what my dental insurance does pay for.

thankfully i'm a public figure, so i can write off expenditures on keeping my teeth perfect as business expenses in promoting Leslie, the brand.

jay

Im not very versed in differential geometry,

Can anyone help explain what the analogy with gradients in Riemannian geometry is ?

So how is (2.5)=(2.4), and how do they draw an analogy to Riemannian geometry from (2.4)

geocalc33

20:29

@DanDonnelly

Semiclassical

@jay i'm not much versed in DG either but usually when you see something like this the strategy is

Ted Shifrin

@jay This is hardly any geometry. I don't understand what you've posted, but the gradient is defined to be the vector $\nabla f(p)$ so that $\langle \nabla f(p),v\rangle = df_p(v)$ for every tangent vector $v$ at $p$. The $\langle\cdot,\cdot\rangle$ is the Riemannian inner product. That's it. Just linear algebra.

Semiclassical

to show that $( \partial_t \rho(t)-\operatorname{div} \rho \nabla\delta \mathcal{E}/\delta \rho,s_2)_{\rho(t),*}=0$ identically

so the only way for it to be valid for all $s_2$ is if the first argument is zero

Ted Shifrin

Your parens are wrong, @Semiclassic. It's div of the whole thing.

Semiclassical

oh. gross

ugh

Ted Shifrin

20:41

I don't see where the div is coming from without doing some sort of divergence theorem argument, but I don't see any flux. All the integrals are over $\Bbb R^d$.

Semiclassical

in particular you want to be able to get something like $(f,f)=0\implies f=0$

jay

ok ok let me think you guys are too quick.

thanks

Semiclassical

@LeakyNun i do wonder if that's going to happen in the US. it feels unthinkable to go back to work-from-home lockdown but...

Leaky Nun

@Semiclassical but omicron

Semiclassical

though that's as much a matter of political will as anything, and i have a hard time seeing US politicians holding the line on that

Ted Shifrin

20:44

I'm not going down that rabbit hole again.

The US politicians, I mean. The ones who lie, cheat, and deny science.

Semiclassical

you're not wrong, but it's also a matter of the public

it's hard to see the public holding their feet to the fire if they don't do a lockdown

leslie townes

i'm gonna feel pretty dumb for tattooing PLANDEMIC on my forehead if there isn't another lockdown.

just saying.

Semiclassical

it's very easy for me to see the reverse scenario

Dan Donnelly

21:29

@geocalc33 hey, might get another freelance tutoring job (with hourly) on freelancer.com. Messaging the person rn

robjohn

@RandomVariable I have voted on some of those posts and upvoted some comments. I am glad there is an uproar. Implementing a major UI change to support mobile devices and forcing it on desktop users (and removing features and returning bad data) is just not right.

@leslietownes since $2\cos(r\pi)$ is an algebraic integer for rational $r$, Doesn't rational cosine mean $\cos(r\pi)\in\left\{0,\pm\frac12,\pm1\right\}$?

Random Variable

21:46

@robjohn The UI hasn't looked good on a large screen for quite some time, but now it looks even worse.

leslie townes

robjohn, yeah.

jay

Trying to quickly read up on Riemannian manifolds, and it says the length of a curve $\gamma$ is taken as the integral (over whatever time interval) of the norm of the tangent vector at $\gamma(t)$, Im confused are there not lots of choices of tangent vector at a point on a manifold?

(sorry I have never been taught this) and will find the time to study it properly but just trying to get the basics right now.

robjohn

@leslietownes okay, that limits the number of choices for $r$ mod $2$

jay

youtube.com/watch?v=L9WR78xvCPY I am trying to understand the picture at minute 6 of this video

if that helps

robjohn

@jay can you take a snapshot of that picture, so we don't need to wait until 6 minutes get loaded?

jay

21:56

sure

robjohn

Oh, no... talking about the metric tensor will require differential geometry! Of all the absurdities!

jay

so we are given the curve $Y$, say $Y(t)=x$, and we want to calculate the length of $Y$ we integrate the norm of """"""""the"""""""""" tangent vector at $Y(t)$ over whatever time interval. But what is THE tangent vector arnt there many tangent vectors at a point

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$Mathematics$ Mathematics