Mathematics - 2022-03-24

More Anonymous

03:18

Hey all

I have physics background but would love to understand this approach:

https://math.stackexchange.com/a/4110007/430082

Can someone recommend me some textbooks to go through?

Ted Shifrin

There must be some texts written for physicists rather than for mathematicians, but we won't be likely to know those. The math sources I know are at the graduate level.

More Anonymous

@TedShifrin I haven't been able to find any ... I would appreciate it. If you recommend me the math books. I'll try and have a go?

Ted Shifrin

Spin Geometry, Lawson and Michelsohn

More Anonymous

@TedShifrin thanks

William John

03:34

wikimedia.org/api/rest_v1/media/math/render/svg/…

I am curious what this symbol means

I mean that $|_0$

leslie townes

i don't see it

Ted Shifrin

Evaluated at $A=0$.

William John

Oh thanks.

Ted Shifrin

this is quite standard

leslie townes

oh, yeah, the vertical bar. evaluate at A = 0.

William John

03:37

I didn't saw that in my multivariable calculus course.

Ted Shifrin

Think also about the usual notation for the fundamental theorem of calculus.

leslie townes

it's really common with leibniz notation for derivatives, particularly partial derivatives, as this style of notation often suppresses any other place to indicate where you're evaluating something.

this is its blessing and its curse.

William John

So it means first set A=0 then evaluate the partial derivative of the function.

leslie townes

other way around.

William John

Ah sorry I get it fundamentals theorem of calculus.

Now I need to ask physicist for the physics context :(

leslie townes

03:43

everything's whizzing around and interacting with everything else. that's all i know about physics.

copper.hat

then there is nuclear fusion, up there with quantum computing and crypto.

More Anonymous

@WilliamJohn The physics context of what?

William John

@MoreAnonymous en.m.wikipedia.org/wiki/Current_density

j=partial derivative evualated at A=0. I don't know why set A=0?

Ted Shifrin

04:10

Current density at a point is a limit per area as area goes to $0$. Think about mass density (mass per volume as volume shrinks to $0$).

Dolphin

09:09

May I ask if somebody could take a look at my question?

2

Q: Computing $\displaystyle\int_S x^2ydxdydz$ given a particular change of variables.

Let $S=\{(x,y,z)\in\Bbb R^2\mid x>0, y\ge 1,z\ge 1, xyz\le 1\}.$ Find the set $S^*$ obtained by the change of variables : $$\begin{cases}x=u\\ y=\frac{u+v}u\\z=\frac{u+v+w}{u+v}.\end{cases}$$ and compute $\displaystyle\int_S x^2ydxdydz.$ My attempt: The condition $xyz\le 1,x>0$ gives $$xyz=u\cd...

Vrouvrou

hello, please how to prove this $|\frac{x^p}{1+x^p}-\frac{y^p}{1+y^p}|\leq \frac{p}{2} |x-y|$ for $x,y>1$, p>0

i tried mean value theorem

for $f(z)=\frac{z^p}{1+z^p}$ on ]x,y[

i found

$|\frac{x^p}{1+x^p}-\frac{y^p}{1+y^p}|= p\frac{c^{p-1}}{(1+c^p)^2} |x-y| $

but why \frac{c^{p-1}}{(1+c^p)^2}<\frac12$ ?

Koro

09:54

@Vrouvrou I think you meant [x,y].

Vrouvrou

YES

Koro

@Vrouvrou $\frac 1c<1$ (since c is in ]x,y[) and therefore $\frac{c^{p-1}}{(1+c^p)}\le 1$

Vrouvrou

from mean value theorem there is $c\in (x,y)$

but i need $\frac{c^{p-1}}{(1+c^p)^2}\le \frac12$

Koro

@Vrouvrou you need $(1+c^p)^2$ in the denominator.

Now, note that $c>1\implies c^p>1\implies 1+c^p>2\implies (1+c^p)^{-1}\lt 1/2$.

Vrouvrou

yes

ok

and $c^{p-1}$?

Koro

10:05

@Vrouvrou multiply the inequalities $\frac{c^{p-1}}{1+c^p}\le 1$ and $(1+c^p)^{-1}\lt \frac 12$.

Vrouvrou

why ?

Koro

@Vrouvrou to get this.

Vrouvrou

i don't understand

Koro

can you tell exactly which part you don't understand?

Vrouvrou

we have $(1+c^p)^{-1}\lt 1/2$

then $\frac{c^{p-1}}{1+c^p)^2}\leq \frac{c^{p-1}}{4}$

why $c^{p-1}<1$ ?

?

Koro

10:33

@Vrouvrou: $\frac{c^{p-1}}{(1+c^p)}=\frac{c^p}{(1+c^p)}\color{blue}{1/c}\le 1( \color{blue}1)=1$

Vrouvrou

10:44

where is 1/2?

Koro

@Vrouvrou please see my earlier messages.

Govind75

I'm struggling to prove $X = (\vec{x_1},...,\vec{x_n})^T$, then $X^T X = \sum \vec{x_i} \vec{x_i}^T$

Any ideas?

Vrouvrou

11:26

i see it

Koro

14:03

@Vrouvrou no it's not true.

Jaakko Seppälä

14:56

Is there an example of a function $f:\mathbb R\to \mathbb R$
that is integrable on every bounded interval and for some $x\in \mathbb{R}$ we have that $f(x)\ne \lim_{r\to 0+}\frac{1}{2r}\int_{x-r}^{x+r}f(y)dy$?

Koro

@JaakkoSeppälä try looking for non-continuous function f.

I think step function f(x)=1 on $x\gt 0$ and $f(x)=-1$ when $x\le 0$ should work. The inequality should work then for x=0.

user76284

@Govind75 $(X^\top X)_{jk} = \sum_i (X^\top)_{ji} X_{ik} = \sum_i X_{ij} X_{ik} = \sum_i (X_i \otimes X_i)_{jk} = (\sum_i X_i \otimes X_i)_{jk}$

where $a \otimes b = a b^\top$ is the outer product.

Jaakko Seppälä

15:11

Thanks

I had another exercise that puzzles me. Let $f:[a,b]\to \mathbb R$ be integrable. How can I find a constant $c\in\mathbb{R}$ s.t. $\int_a^b (f(x)-c)^2dxs$ achieves its minimal value?

Sorry, I meant $\int_a^b (f(x)-c)^2dx$

love_sodam

Yeah one can't directly conclude maximum modulus principle for unbounded domain for general analytic function. If it has compact support then essentially original mmp.

anak

15:31

Recently developed this pet peeve about students calling things "trivial" when they are not of the form "P \implies Q" with Q always true.

Xander Henderson

@anak I have long been on the record with the opinion that words like "trivial", "clear", and "obvious" should be stricken from mathematical exposition. None of these words actually add any content to a work of written mathematics.

leslie townes

i think they sometimes add value, but are bad pedagogically.

anak

@XanderHenderson Amen. Had a prof who said to "only use those words if you have not checked yourself". I then am struck with guilt whenever they appear in my writing.

leslie townes

which i guess does mean that in a lot of settings they are absolutely bad.

anak

@leslietownes trivially bad

Xander Henderson

15:49

@leslietownes Obviously.

@leslietownes Example?

Personally, I believe that whenever a mathematician says that something is "obvious", or "clear", or "trivial", they really mean one of the following:
1. I know how to prove it, but I am too lazy or important to bother proving it for you,
2. I don't know how to prove it, but it seems like the kind of thing which should be true, or
3. I don't know how to prove it, because it is, in fact, false.

One of the better job-talks I've seen involved a potential professor explaining their PhD research, which, ultimately, came down to showing that an "obvious" result from a 40 year old paper was actually false. But a Big Name™ has asserted that the result was true, so the received wisdom was that it was true.

leslie townes

it's useful to have a verbal shorthand for a detail that everybody in the audience knows, if they're only reminded with a nudge that it is known (ideally also along with what "it" is).

this is obviously audience dependent. but it's not always used to obfuscate or as part of a power trip. your prong 1 above could be "i know how to prove it, but i'm too lazy or important to bother proving it for you, or alternatively in our limited time together i'm just going to remind us that this can be done, in a way that doesn't make it the focus of the talk."

once it becomes a rhetorical move that people use to avoid details, yeah, that's bad.

which is why it probably shouldn't be used in classroom education.

but some random seminar talk, why not. go nuts. maybe not a job talk on your thesis or a talk about supposedly important work on an outstanding open problem.

Xander Henderson

I agree that there are times when the intended audience should know the proof, but I don't think that stating that the proof is obvious or clear actually adds any value beyond "It is known that..." or "It is an elementary fact that..." or "Those familiar with the field likely know that...".

leslie townes

i prefer "well known" to "obvious" because "obvious" just sounds bad to me. it does sound like a power trip even if it's not being intended that way.

Xander Henderson

That being said, I am generally more forgiving of "verbal" exposition, as one typically has less time to present, and less time to carefully chose one's words. But I still think that words like "obvious" are a crutch, and that folk should try to eliminate them from their mathematical vocabulary.

leslie townes

you do need to be able to sometimes say stuff like "as everybody ought to know, in 1913 X proved Y and this is now known as a straightforward consequence of Z theory." without immediately being hounded out of the room, or taking people back to slides of papers from 1913.

Xander Henderson

16:00

On the other hand, I started doing Toast Masters when I was in the 3rd grade, so there are a lot of "filler" words that make me cringe. Every time I say "uh..." or "um...", I hear a little bell in my head, telling me that I have done did wrong.

Koro

0

Q: Spread the butter in Analysis

Is it true that analysts use the sentence "spread the butter" in informal contexts whenever they have to consider an expression of the form $\varepsilon/2^n$ in a proof? If so, I would be interested in finding out who came out with this idea and to hear if you know more expressions of the kind.

real-analysis soft-question math-history

leslie townes

my analysis 1 instructor was a fan of 'obvious' and that was definitely a misuse of the term. she almost literally taught us that the name of the game was to learn which things were obvious and which weren't so we could avoid proving them.

koro: add me to the list of people who's never heard of that usage. people do sometimes say stuff like "this is an epsilon over 2^n type argument." it wouldn't surprise me if some books, particularly in measure theory where you see these arguments all the time, have come up with more colorful names.

Koro

I recall reading in a book that said by Curry -leaves trick we have:(or something similar) $|x_m-x_n|\le |x_m-x|+|x-x_m|<\epsilon/2+\epsilon/2$.

Xander Henderson

@leslietownes I've never heard of an $\varepsilon/2^n$ argument. $\varepsilon/2$, $\varepsilon/3$, and $\varepsilon/5$, sure. I've even seen an $\varepsilon/7$ once. But over $2^n$? Preposterous!

Koro

I still don't know what curry-leaves trick is.

Xander Henderson

16:04

It is when you eat all the curry, then leave before you get caught.

anak

@leslietownes perhaps some other (more specific) buzz word should be created to indicate what you mention here. I can see how this is helpful, but at this point in time "clearly"/"trivially"/"obviously" don't clearly/trivially/obviously convey this, due to repeated misuse throughout academia.

Koro

I would like to believe that.
But why is it used in the context of definition of limits?

i think someone cooked up that term just as 'spread the butter' term in the linked post above.

Xander Henderson

@Koro Sure. It is an "obvious" metaphor, but that don't mean it's widely used.

One of my favorite analysis profs in graduate school had a lot of wacky metaphors, many of which were, I think, subtly racist. :/

leslie townes

wacky and subtly racist, what a combination. they don't make many of 'em like that anymore.

Xander Henderson

But his English was so marginal that I always assumed that he didn't really know what he was saying.

"So... uh... you see! When you are standing on the boundary, it is like being on a wall. And you have... uh... one foot! in the US, and one foot in... Mexico! So... that is... uh... the boundary! of a set!"

leslie townes

16:09

with non native speakers it can be really hard to infer intent or awareness of what something might sound like to a native speaker. let's give the benefit of the doubt and go with 'wacky.'

Xander Henderson

@leslietownes That was always my strategy.

leslie townes

unless he was bringing mexico into everything. but then maybe he just liked mexico.

Xander Henderson

Trump was talking about building a wall at the time. My guess is that he was trying to be current.

anak

@leslietownes you haven't met a complex analyst with a subtle aversion to the Polish, have you.

Xander Henderson

Ugh... I should go record another lecture, so that I can pack the truck, so that I can get on the road by sunrise tomorrow. BUT I DON'T WANNA!

leslie townes

16:12

anak: you have to be fairly pro-polish in any kind of analysis. or topology. without them there isn't much left.

Xander Henderson

If a space ain't Polish, it ain't worth thinkin' about.

Okay, I'm lecturing now.

anak

@leslietownes I've met someone who thinks otherwise. :(

Sha Vuklia

17:01

@TedShifrin Right, I'll have a look at this later when I've seen a bit more theory

Koro

17:20

Why is GF(16) isomorphic to $\{ax^3+bx^2+cx+d+\langle x^4+x+1\rangle|a,b,c,d\in \mathbb Z_2\}$?

GF(16) is Galois field of order $2^4$.

Ted Shifrin

@ShaVuklia The intersection number of a (generic) section with the zero-section is the "usual" definition. I don't know what your definition of Euler class even is (probably abstract nonsense).

@Koro Prove that $x^4+x+1$ is irreducible/$\Bbb Z_2$.

Koro

ah, because field of order p^n is unique (upto isomorphism).

The braces part is just: $\frac{Z_2[x]}{\langle x^4+x+1\rangle}$, which is a field of order 16.

Ted Shifrin

Provided you prove irreducibility, yes.

Koro

@TedShifrin I was confused about the isomorphism part. It's clear now. :)

Ted Shifrin

OK.

user76284

17:44

@JaakkoSeppälä Take the derivative with respect to $c$ and set it to 0. Solve for $c$.

amWhy

18:13

Mexico, then Trump's Wall, then Polish, how is one to keep up! ;D

Ted Shifrin

I don't.

amWhy

Reminds me of a game we played in scouting: scouts form a circle. First person reads a slip of paper with a word, without speaking. They turn to second person in the circle, and whisper the word, ......, the last person in the circle whispers the word to the original whisperer. Let's just say, it rarely ended with the words matching.

@TedShifrin Good answer!! (In FamilyFeudESE)

Ted Shifrin

Oh, I forget the name for that "game." It's quite famous.

Xander Henderson

The game is called "Telephone" where I grew up.

amWhy

"telephone?"

Xander Henderson

18:19

Yup.

amWhy

@XanderHenderson Great minds remember well!!

Ted Shifrin

Chinese whispers (Commonwealth English) or telephone (American English)[1] is an internationally popular children's game. From wiki.

These days the "Chinese" part might be racist.

Xander Henderson

Everything is racist.

:/

Ted Shifrin

Well, certainly it's a good thing "Indian giver" from my childhood has left the room.

That was, in hindsight, so ironic a term.

PM 2Ring

We called it Chinese Whispers, but there was no intended racist connotation. WP lists lots of names for it. en.wikipedia.org/wiki/Chinese_whispers

amWhy

18:22

@TedShifrin Indeed. I remember learning to jump rope with a long rope held by two kids on each end. One kid jumps : "Chinese Japanese ..." I can't even right it, it's pretty anti asian, particularly to asian women.

Sha Vuklia

@TedShifrin It isn't abstract in fact! We used linear algebra to define it. For a matrix matrix $X=[x^i_j]$ where each $x^i_j$ is an indeterminate, such that $x^i_j=-x^j_i$ it can be shown that $\det X$ is a perfect square in $\mathbb Z[X]$, and then we choose a particular root and call that the Pfaffian. And then we just use the Chern-Weil homomorphism

Ted Shifrin

Yeah, the Pfaffian is very concrete. So you defined the Euler form using Chern-Weil.

Right.

OK, so the notes you say I sent you show how using the topology of the Grassmannian you can prove what I said about the intersection number.

If I in fact didn't send them, let me know. I'm leaving again, but can send later.

Sha Vuklia

I looked them up yesterday, so it's confirmed:)

See you around!

amWhy

@PM2Ring Indeed. Grapevine, gossip. It is a lesson against believing that the gossip (social media) hasn't been distorted.

Jakobian

I have been reading today about what Vietoris topology is

PM 2Ring

18:32

@amWhy In "Telephone", even when everyone is trying hard to pass the message on in without distortion, there's usually some change. But often there's at least one person in the circle who thinks it's amusing to intentionally change the word. And in the Real World, there are always parties who can get some advantage by distorting the information they have control over.

Jakobian

it's topology you give to the nonempty closed subsets of topological space (kind of like Hausdorff distance for metric spaces)

amWhy

@PM2Ring Exactly!

PM 2Ring

@Xander You may enjoy this song. Molly Tuttle has a new band. Here's a new song, explaining that it's ok to be different. That can be a difficult concept for some country music fans. ;) Crooked Tree. Also, Big Backyard

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$Mathematics$ Mathematics