Mathematics - 2023-08-04 (page 1 of 2)

« first day (4748 days earlier) ← previous day next day → last day (568 days later) »

D.C. the III

01:40

I feel sorry for all individuals that had to learn the proof of the Inverse Function Theorem without the luxury of having a video to rewind and watch when needed......my heart aches.......😥😥

Ted Shifrin

It’s not that bad. Just wait ….

D.C. the III

Very involved though.

Ted Shifrin

02:00

Just wait for graduate real analysis, for example. In outline, the IFT is quite straightforward and based on one basic notion.

leslie townes

you think it sucks now, try learning it in the 19th century when it was all about 'variables' being 'expressible' as functions of other 'variables' and it only made any sense if you knew it already.

D.C. the III

I could only imagine....that's the stage where you are spending a few days on one proof isn't it? even more maybe...

leslie townes

which reminds me that it's been a while since we've had a good question on main from a 19th century textbook.

1 hour later…

Soumik Mukherjee

03:06

Hi @AkivaWeinberger, do you remember the challenge question about least maximum of a polynomial of degree $n$ in $[-1, 1]$, and whether the pattern continues after $n=4$?

Akiva Weinberger

Oh hey

Yeah

Monic polynomials, yeah

@SoumikMukherjee Why do you ask?

Soumik Mukherjee

I am not yet able to reach a concrete solution about whether the pattern continuous

@AkivaWeinberger for some hints

Akiva Weinberger

I can give you a small hint

Trig

Soumik Mukherjee

🤔

Akiva Weinberger

I can give you larger hints also tbh

but maybe see if you can try a few avenues involving trig first

Soumik Mukherjee

03:18

Okay, trying

PM 2Ring

I made a little Sage thing to plot complex functions in 3D, in the form $re^{i\theta}=f(x+iy)$, with $r$ as the height, and $\theta$ mapped to hue. You can specify a max value for $r$ so it can make sensible plots of functions with poles. Here's a demo, plotting a couple of periods of the Jacobi elliptic sn function.

sagecell.sagemath.org/…

It looks better with plot_points=80, but takes longer to do the initial render.

IMHO, it's a bit more useful than the typical 2D complex plots that leslie complains about. ;)

Akiva Weinberger

||test||

Oh, no spoiler marking here

PM 2Ring

03:37

Chat is ancient, and its software hasn't been updated for years, long before we adopted modern Markdown.

I guess you could use rot13...

I tempted to run that 6×6 domino tiling problem through my exact cover solver. Here are the tilings for the T-tetrominoes (plus 4 squares). math.stackexchange.com/a/3918620/207316

leslie townes

or url alt text

PM 2Ring

Good point. Although it's a bit hard to see on a phone

Dannyu NDos

03:53

Is "commutify" a word? Or "commutize" or "commutate" maybe?

'Cause I'm playing with skew fields.

PM 2Ring

"Commutate" is a real word. And you might be able to bend it to your purposes. ;)

Ted Shifrin

Ugh.

one potato two potato

04:46

'abelianize'

even for ring (using the term 'abelian group' in the definition of ring)

2 hours later…

Soumik Mukherjee

06:30

How to use tcolorbox $2$nd time?

Somehow, it's not working $2$ndtime

1 hour later…

Soumik Mukherjee

07:32

Got it

2 hours later…

Jakobian

09:37

@AlessandroCodenotti math.stackexchange.com/questions/4747015/…

sunny

10:14

I'm looking to verify the uniform convergence of $\sum _{k=0}^{\infty }\frac{x^k}{1+x^k}$ on $-\frac12\leq x\leq \frac12$. I'm a little insecure about my approach. Taking the derivative of $\frac{x^k}{1+x^k}$, we get that a critical point is $x=0$. However, $$f_k(0)=0<\frac{1}{2^k+1}=f_k\left(\frac{1}{2}\right),$$ so the boundary point $x=\frac12$ is greater than the critical point $x=0$.

Now what about $x=-\frac12$. I get $$f_k\left(-\frac{1}{2}\right)=\frac{(-1)^k}{2^k+(-1)^k},$$ and I can't really say if this is larger or smaller than $f_k\left(\frac{1}{2}\right)$. For even $k$, we just get that it's equal to $f_k\left(\frac{1}{2}\right)$. For odd $k$, we get $$-\frac{1}{2^k-1},$$ but this is even defined for $k=0$.

EDIT: I meant to say "this is not even defined for $k=0$."

Jakobian

10:34

For odd $k$, we get a situation like with the function $x\mapsto x^3$

for even positive $k$ we get minimum at $x = 0$

for $k = 0$ this is just $1/2$

Either way, the maximum is at $x = 1/2$

algbr

Is it possible to activate mathjax in the chat?

Jakobian

yes tinyurl.com/cfqcvpc

see room info

algbr

Thanks.

Jakobian

@sunny to verify uniform convergence, use Weierstrass $M$-test

For that you'll have to take absolute value, so minimum of $x^k/(1+x^k)$ is important as well. This is $-1/(2^k-1)$ for odd $k$

You'll be able to dominate the series by something like $\sum 1/(2^k-1)$ which is a geometric-like series so converges

123

11:15

Hello Everyone...

sunny

12:00

thank you @Jakobian yet again :) a life-saver! I made a fundamental mistake again assuming $k=0$ is odd. By the way, how do you know though that $x=0$ is a minimum?

Jakobian

various reasons, one of them is that for even $k$, $x^k/(1+x^k)$ is symmetric around $0$

Jakobian

12:21

So, this is my (educated?) guess, but when we say that $(f_0, f_2, ..., f_k)$ is a Sturm sequence, we define it at least up to $f_2$, right

That is $f_0 = f, f_1 = f'g, f_2$

yeah it must be

Because we want $f_k$ to be gcd of $f, f'g$

@Thorgott I have a question. In one of the proofs, given a polynomial $f$ and its roots $y_1, ..., y_m$ in the algebraic closure, they defined $\prod_{i<j}(x-y_i-y_j-ny_iy_j)$. Is this just a smart idea, or does this way of thought come from anywhere?

This polynomial is symmetric in the roots $y_i$ so obviously has coefficients in the given field

Fundamental theorem of algebra

The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with ...

I guess probably not

1 hour later…

sunny

14:09

@Jakobian in the above problem, do we assume that $0^0=1$? This is always something that is unclear to me when we do or don't.

Jakobian

@sunny always

sunny

ok

Put $f_k(x)=x^k/(1+x^k)$, then $$f_k '(x)=\frac{kx^{k-1}}{(1+x^k)^2}.$$ How can then $x=0$ be a critical point (which is what I claimed)? It is only a critical point for $k>1$, but here $k$ can be $1$ or $0$ too.

Let $x=0$. For $k=0$, the derivative isn't defined (division by zero). For $k=1$, the derivative is non-zero.

Jakobian

14:30

That doesn't really matter

$k = 1$ is odd anyway, and as we concluded, for odd $k$, $f_k$ is increasing in $[-1/2, 1/2]$

sunny

If $x=0$ is a minimum, then it should be the case that $f_k'(0)=0$ or that $f_k'(x)$ isn't defined at $x=0$. Maybe it's the latter.

Jakobian

since when is $1$ an even number

sunny

Hmm, how about the case $k=0$ and $x=0$? Then $k$ is even, yet the derivative doesn't vanish.

Jakobian

that's not true

the derivatve vanishes everywhere when $k = 0$

$f_k$ is just a constant

sunny

true

Jakobian

14:44

anyway I'd like to point out that the cases like $k = 0$ or $k = 1$ are completely irrelevant for uniform convergence of this series

what matters is how it behaves in the long run, at infinity

sunny

true

I was staring too hard on the formula of the derivative which clearly states $f_0'(0)=\frac{0\cdot \frac{1}{0}}{(1+0^0)^2}$.

The Empty String Photographer

@sunny so it's $\text{"}\frac14\text{"}$?

sunny

@TheEmptyStringPhotographer not really right? :)

although it is clear that $f_0$ is constant (and thus has derivative zero), the derivative I computed is undefined when $k=0$ and $x=0$...

robjohn

15:02

For all $x\ne0$, $f_0'(x)=0$, so if $f_0'(0)$ exists, it is $0$.

Jakobian

@sunny because you weren't careful when computing that derivative

manipulating symbols isn't everything

one potato two potato

what is better: narrow range hard qual or broad range easy qual?

sunny

@Jakobian that is what I'm suspecting. I'm trying to debug what I did "wrong".

Jakobian

it looks like that what you did wrong is that you said $(x^0)' = 0\cdot x^{-1}$

the formula $(x^k)' = k\cdot x^{k-1}$ should be interpreted as $ = 0$ when $k = 0$

sunny

yup, looks like it

robjohn

15:17

Depending on how you define $0^0$, $f_0(x)$ may, or may not, be continuous at $x=0$. It you define $0^0=1$, then $f_0(x)$ is continuous at $x=0$. That is, $f_0(x)=\frac12$ for all $x\ne0$.

冥王 Hades

Not gonna lie EM Theory is kinda challenging

geocalc33

@冥王Hades I remember 'Gaussian surfaces' from that

冥王 Hades

@geocalc33 yep we’re studying that as well

geocalc33

and yeah it was pretty challenging

冥王 Hades

Of course. It is fun but not as easy as I thought

one potato two potato

16:11

@Jakobian Do you know model theory? (in set theory)

geocalc33

is there a mathematical term for a mathematical object of fixed type that has a wealth of representations?

one potato two potato

isn't that exactly the representation theory?

Jakobian

@onepotatotwopotato not really. I only licked a little while learning universal algebra. But a very minuscule amount

one potato two potato

aha okay

Jakobian

elementary embeddings and so on, not my thing at all

but I know the basics, it's about $n$-ary relations and $n$-ary operations instead of just $n$-ary operations like in universal algebra

we also don't consider only structures defined by universal quantifiers, but more general

geocalc33

16:19

@onepotatotwopotato I don't know what representation theory is lol. I just came up with that idea independently

Jakobian

@geocalc33 like how $\mathbb{R}$ is a totally ordered field but it's also a topological space?

you said representations so I guess not

geocalc33

@Jakobian yes

like how $\Bbb R$ is a totally ordered field but also a topological space

I think i misused the term "representation" above

one potato two potato

I've never thought I would encouter logic or set theory in any sense but recently I felt maybe I need to study model theory (considering recent trends or people's interests).

Jakobian

why's that

I heard model theory and algebraic geometry are connected in some way

is that's why?

one potato two potato

Recently someone proved that 'statement A is true if and only if MCG(S) is linear' (unpublished). I'm told that Homeomorphism group has some special properties from a logic viewpoint (I don't know anything specific though)

Jakobian

16:26

I see

Alessandro should know some model theory, you can ask him for some recommendations

Thorgott

17:01

@Jakobian I've never seen that before, $n$ is the degree?

Ted Shifrin

Howdy @Thor

Thorgott

hey @Ted

doing alright?

Ted Shifrin

Yessir, and you?

Thorgott

same, my fall break has more or less started on monday

Jakobian

@Thorgott no, an integer

the point is that by taking all integers $n$, we'll have at least two for which exist fixed $i, j$ such that both $y_i+y_j+ny_iy_j$ and $y_i+y_j+my_iy_j$ are roots (in $F[i]$)

this then will imply that both $y_i+y_j$ and $y_jy_i$ must be in $F[i]$, and then by some argument with quadratic function we'll prove $y_i, y_j$ are in $F[i]$

Ted Shifrin

17:08

Oh, congratulations :)

Thorgott

$F[i]$ being?

Jakobian

$F[i] = F[x]/(x^2+1)$

where $F$ is real-closed so that $x^2+1$ is irreducible

Thorgott

ah, so it's the algebraic closure

was $f$ assumed to be split?

Jakobian

yeah but you don't know it's algebraic closure yet, that's what we're trying to prove with this

The roots $y_i$ of $f$ are taken in some algebraic closure

we can assume it contains $F[i]$

Ted Shifrin

Please don't use $i$ as a subscript when you're using $i=\sqrt{-1}$. This drives me nuts.

Jakobian

17:16

to be more precise the proof is that if every polynomial of odd degree in $F$ has a root, and $F$ has unique ordering defined as $x\geq 0$ iff $x = y^2$ for some $y\in F$, then $F[i]$ algebraically closed

Xander Henderson

@TedShifrin Yeah. Use $\mathrm{i}$ for the imaginary unit, and $i$ as an index.

Jakobian

by introducing the product $\prod_{i < j} (x-y_i-y_j-ny_iy_j)$ we're defining a polynomial of degree which has one less factor of $2$

sure, I can use $j$ as the imaginary unit and $j$ as an index

Xander Henderson

There is actually a good argument in favor of typesetting the imaginary unit a an upright $\mathrm{i}$. Constants should be upright, variables italicized. But this is a convention which is followed... sporadically, at best. For example, I think that I am one of the few people who bothers with $\mathrm{e}$.

Ted Shifrin

And you probably write $\int f(x)\,\mathrm{d}x$, too. Yuck.

Xander Henderson

Also, distinguishing between italicized symbols and non-italicized symbols is really not a good way to convey information. Ted is right. Don't use $i$ for an index if you are doing anything in complex land.

@TedShifrin Yes. Doesn't everybody? :P

I created a blackboard bold i for the imaginary unit in my thesis.

:P

Ted Shifrin

17:21

There are plenty of papers/books in SCV and complex geometry where people write $\omega = \frac i2 \sum dz^i\wedge d\bar z^i$. I want them executed.

Xander Henderson

@TedShifrin Yuck. That's gross.

Jakobian

The two $i$ can be distinguished by context. There's no need for separate letter

Ted Shifrin

Either write $\sqrt{-1}$ or avoid $i$ as an index. It's not that hard.

Xander Henderson

@TedShifrin This. Why would you choose to write in a way that is potentially confusing or annoying?

Ted Shifrin

They cannot always be distinguished and I don't care even if they can. Don't be a lazy ass.

I find attitudes like yours repugnant.

Or use $\alpha$ as a $1$-form and also $\alpha$ as an index. People do that, too.

one potato two potato

17:24

lol

Ted Shifrin

Bottom line is that mathematicians, as a whole, don't give a damn about exposition or clarity.

Xander Henderson

The whole point of mathematical writing is to clearly communicate ideas. Saying "It is up to the reader to resolve ambiguity" is f*cking lazy. The author has a responsibility to anticipate and resolve ambiguity to the best of their ability.

Jakobian

I mean, I'm writing this for myself and Thorgott, no need to be super clear about your notation all the time

this is informal writing, I abuse notation when writing for myself

Ted Shifrin

You're writing in a public place. Do what you want on your own piece of paper or office whiteboard.

Xander Henderson

@Jakobian You say that now. Try re-reading those notes in five years.

Trust me, you'll wish you had used better notation.

Also, what Ted just said.

Ted Shifrin

17:25

But doing what you want for yourself leads to bad habits that you will fall into when you're writing for a broader audience. And don't tell me your self-control is superior.

Jakobian

I don't really write notes. And if I write notes then I don't read them after

Xander Henderson

@TedShifrin Gross. $\alpha$ should only be used as a multiindex.

Ted Shifrin

Nah, I use capital letters for multi-indices.

one potato two potato

so some math people write books for themselves

Ted Shifrin

Indeed, @onepotato ... and then people who try to read those books come here and ask us to decipher them.

Xander Henderson

17:26

@Jakobian That is a habit you might want to break. I try to keep all of my notes. I've found them very useful on a few occasions when trying to push on getting work done.

@TedShifrin Fractal people like letters from the beginning of the alphabet for multiindices. I don't know why. :/

Capital letters are usually sets.

Jakobian

If I'm reading a book then I don't really need to write notes if it's all written there

and I don't attend any lectures

Xander Henderson

I still have most of the notes that I wrote while working on both my masters and phd theses (most of the latter in the form of a cataloged collection of photographs of whiteboards and blackboards), as well as course notes, and notes written while reading books and papers.

I also recently came across a large box of my father's notes, taken while he was doing work for his phd.

My impression is that most people who are successful in academia (a) take a lot of notes, and (b) curate those notes (often to a somewhat OCD degree, but you never know when you might want them).

one potato two potato

I use ipad for notes. It's super convenient.

Xander Henderson

@onepotatotwopotato Currently, I use a Wacom tablet. Better resolution than an iPad. :P

Xander Henderson

17:59

Stop trolling.

冥王 Hades

@XanderHenderson That wasn’t a troll comment. My iPad actually had a 120Hz display as most iPad Pro models have since 2018

Even if it was a troll comment (which it wasn’t), I’m not sure how that warrants deletion either but I digress

Xander Henderson

@冥王Hades In that case: I don't really care. The refresh rate is not what I am worried about. It is the resolution of the stylus.

Jakobian

Even the iPad has to be Pro

冥王 Hades

It’s better than the Surface Pro

Mad

i bought one because i got infleunced by my colleagues who use ipads

i used it for 3 weeks.

Back to pen and paper it was for me

Xander Henderson

18:03

I.e. how small a "jiggle" can I create in a file. My display tablet is a better tool for my purposes than an iPad.

I am also somewhat tempted by some of the eInk tablets (for portability), but I have no interest in an iPad or Surface or other mini-computer-with-a-touch-display.

冥王 Hades

Just use soot mixed with water and a white cloth

user223626865

and the planet is heading back to fighting with sticks & stones soon enough

冥王 Hades

Who else chooses to walk up a staircase instead of elevators?

user223626865

me

~~escalator~~ for me

Xander Henderson

@冥王Hades I'm sorry... I don't understand the question.

What is an "elevator"?

user223626865

18:11

gotta stay in shape for the return to the jungle

Xander Henderson

Okay, I Googled it. An "elevator" is one of those uppy-downy boxes that big city folk use to get around in their overly tall buildings.

The simple solution is to simply not build so tall.

user223626865

But then NYC couldn't sell their real estate by the square inch.

Xander Henderson

@user223626865 Okay, but what would the downside be?

冥王 Hades

Same

I don’t use elevators for the same reason I don’t travel by buses or trains. I can’t stand having so many people around me

Even when the elevators are empty there’s a chance someone is going to hop in at some fooor while I’m already inside.

user223626865

Whip out your iPhone and play a game.

The Empty String Photographer

18:20

@user223626865 Ok, I’m playing the game “Stack Exchange Chat”.

user223626865

👍👍

冥王 Hades

@user223626865 that’s what I usually do whenever I’m surrounded by too many people

but I get really nervous and physically start shaking a little

The Empty String Photographer

@冥王Hades I try to avoid using taxis and cars. I prefer public transport.

user223626865

Have you tried meditation?

冥王 Hades

I prefer driving alone, all alone

@user223626865 I haven’t. I can’t get my brain to shut up

user223626865

18:24

It is basically attention focus training.

Jakobian

There's a very slim chance you'll actually get murdered in public transport, don't worry

user223626865

LoL Thanx for the comforting thought 🤔

The Empty String Photographer

@冥王Hades and having to concentrate so hard that you collapse?

@Jakobian police don’t do that much in the UK…

user223626865

@冥王Hades if you physically start shaking, you need to see a doctor.

The Empty String Photographer

@user223626865 no, not necessarily. What if I’m just cold?

user223626865

18:30

9 mins ago, by 冥王 Hades

@user223626865 that’s what I usually do whenever I’m surrounded by too many people

Physical shaking is not good.

Your body is trying to tell you something.

geocalc33

the tangent bundle of an exponential family naturally forms a Kahler manifold.

now I see the light at the end of the tunnel

just a beautiful statement!

now begins the journey of understanding and proving that

冥王 Hades

@user223626865 it’s like an involuntary tremor whenever I’m surrounded by too many people for too long

@Jakobian that’s the least of my concerns here

user223626865

If you can't control it, it is not good.

And it could get worse.

15 mins ago, by user223626865

Your body is trying to tell you something.

Listen to it.

Jakobian

18:48

sounds like anxiety

user223626865

Only a professional can tell for sure.

Run some tests, etc, etc...

Xander Henderson

Could be rabies.

Or alien space lasers.

Or demons, maybe. Consult a priest.

Jakobian

right, or magnesium deficiency

Xander Henderson

Maybe it's a tapeworm!

冥王 Hades

Or maybe Hades’s soul trying to take over my body in order to reincarnate again

Xander Henderson

18:51

@冥王Hades I already said that

1 min ago, by Xander Henderson

Or demons, maybe. Consult a priest.

冥王 Hades

There’s no Athena or Seiya to stop me this time

user223626865

Exorcist?

geocalc33

maybe it's a compactified demon

Xander Henderson

@user223626865 Not being a Catholic, I don't really know, but aren't exorcists all priests?

geocalc33

and said demon is traveling through spacetime on a calabi-yau manifold

Xander Henderson

18:52

You probably need a referral from your general priest before you can call in a specialist like an exorcist. Unless you have some real fancy insurance.

user223626865

dunno, srry

Maxwell's demon is moonlighting.

The pandemic has brought on tough times.

冥王 Hades

@geocalc33 no he’s traveling through space time via the Hyperdimension that connects all universes to the Underworld

geocalc33

@冥王Hades yes that's what i was trying to articulate

冥王 Hades

And anyone without their 8th sense who enters the Hyperdimension dies instantly thanks to my authority and laws

Jakobian

In Steen and Seebach, on the page for Tychonoff corkscrew, when they write $A_\alpha$ is $(-0, -1, -2, ..., \alpha, ..., 2, 1, 0)$ what exactly do they mean

am I supposed to treat it like there is a $-\alpha$, or is there only $-\beta$ for $\beta < \alpha$

so, are they merged at the tip

I'm guessing they are merged, and we are removing the hole in the middle

Jakobian

19:21

If I'm not mistaken, the Tychonoff plank is a closed subspace of Tychonoff screw

Xander Henderson

@Jakobian Yeah, but what about the Tychonoff screwdriver?

Jakobian

hmm... flat-headed or cross-headed?

Xander Henderson

@Jakobian Hex.

Jakobian

I think that's doable, first put $4$ adjacent copies of the Tychonoff plank in the middle, and then four copies of $\{(x, y) : x, y\leq \omega_0\}$ on the sides

then take product with some space like long line

the problem is that this screwdriver will be a bit flat

The Empty String Photographer

@user223626865 tell that to the ~~judge~~ A&E doctor!

Semiclassical

19:38

the juxtaposition of these two stories in the campus newspaper gives me pause

*Multiple apartment buildings opening in next three years near campus*

preceded directly by

*UMN facing 5.8% enrollment decline since 2019*
"The University of Minnesota is one of many universities nationwide dealing with enrollment declines since the COVID-19 pandemic and is facing a potential enrollment cliff over the next decade."

Xander Henderson

@Semiclassical If you view this in light of the increasing privatization of student housing, it might not be that surprising.

There may be fewer students, but there is also much less on-campus housing.

Semiclassical

hmm

Xander Henderson

At least, I see that happening in Arizona.

Semiclassical

maybe. i don't have a sense of what's happened to the existing on-campus housing

Xander Henderson

@Semiclassical I was in Cali this week, and needed to be on campus at my PhD institution. In the three years since I finished there, they have bulldozed all of the family housing for graduate students, and don't seem to have replaced it with anything (let along new housing).

Semiclassical

19:43

oof

Xander Henderson

And that housing was insufficient when I got there (three year waiting list for 50 year old buildings which were falling apart).

Jakobian

(deleted) Tychonoff plank is a closed subspace of the Tychonoff corkscrew? Right?

could someone double check me on this

Semiclassical

@TedShifrin (or use j like an electrical engineer but that just moves the problem elsewhere)

Xander Henderson

@Semiclassical Put a hat on it.

$\hat \jmath$

user 85795

> facing a potential enrollment cliff over the next decade

Semiclassical

19:46

@XanderHenderson i agree up to a point. i think notation which is strictly-speaking mathematically ambiguous can be fine if you're working in a context and with a community where everyone is familiar with how it should be read

user 85795

lemmings running towards the cliff

Semiclassical

e.g. i think the physics tack of writing $f(x,y)=x^2+y^2\implies f(r,\theta)=r^2$, while understandably annoying to math people (and problematic if you're trying to understand thermodynamics...) is fine if $f(r,\theta)$ is understood as shorthand for $f(g(r,\theta))$ with appropriate $g(r,\theta)$

Xander Henderson

@Semiclassical I do not believe that there is such a thing a "strictly-speaking mathematically ambiguous". Ambiguity is a spectrum. The job of the author is to make their best efforts to resolve ambiguity ahead of time. Part of that resolution is understanding the audience who will be reading the work.

user 85795

> a widely popular misconception that they are driven to commit mass suicide when they migrate by jumping off cliffs.

Semiclassical

@user85795 blame Disney for that

which is exactly what a physicist does read it as. we really do think of $f$ as a function of a point in space, not of its coordinates

user 85795

19:50

makes for good drama

Xander Henderson

That being said, using $i$ for both the imaginary unit and an index in the same paper is pretty far towards one end of that ambiguity spectrum.

Semiclassical

no real argument there

my personal peeve is using $\mathbf{i},\mathbf{j},\mathbf{k}$ for the $xyz$ unit vectors

quaternions are mostly irrelevant now, stop perpetuating them in intro physics

Xander Henderson

Unit vectors should wear hats.

$\hat\imath, \hat\jmath, \hat k $.

:P

user 85795

@Semiclassical even in string theory?

Semiclassical

mostly

and string theory is about as far from intro physics as you can get

user 85795

19:53

right

Semiclassical

closest you'd get is the use of quaternions to describe rotations around different axes, and rotations in intro physics are always with respect to a single axis

i have gotten more fond of the math way of writing basis vectors as $e_1$ etc

but for physics writing the default i respect is $\hat{x},\hat{y},\hat{z}$

whereas i just think $\hat{i},\hat{j},\hat{k}$ should die

user 85795

what do you think of the Vertasium videos for intro physics?

Semiclassical

eh. sometimes they're good

my usual go-to tends to be Steve Mould

user 85795

Vertasium questions such fundamental assumptions.

one-way speed of light, entropy, etc

Xander Henderson

@Semiclassical I prefer $x, \hat\jmath, e_3$.

Jakobian

20:00

a closed subspace of countably paracompact space is countably paracompact, right

Xander Henderson

It is a compromise everyone can love!

Semiclassical

@user85795 ugh

Xander Henderson

@Jakobian Your mother is something something paracompact!

user 85795

@Semiclassical at an intro level how is that acceptable

Semiclassical

intro physics is rarely a matter of "questioning fundamental assumptions", but of seeing how misconceptions about physics run into experimental problems

i don't mind the one-way speed of light example, b/c that at least points to a basic reality about how such experiments work. by contrast, i do mind him wading into the whole "do fields or electrons carry energy in wires" bit

user 85795

20:05

yeah, he knows how to ruffle feathers

Semiclassical

tbf, it is the case that the usual description of a circuit in terms of voltage/current/etc is at the end of the day a simplification of a more complicated story involving the fields themselves

but that is not a story for intro students, and framing it as "actually, the intro physics way of putting it is wrong!" is annoying

user 85795

click bait

Semiclassical

p much

user 85795

>8(

Jakobian

@XanderHenderson Your mom is so large she's discrete but not realcompact

Xander Henderson

20:13

@Jakobian Your mother is so fat that she isn't even a small category.

Your mother is so fat, she doesn't support a translation invariant measure.

3

Jakobian

Your mom is so fat that the first examples of her were all Dowker spaces

3

Xander Henderson

Your mother is so fat, she can't be embedded into any finite dimensional Euclidean space.

Your mother is so fat, they use an Ackerman function to weigh her.

Your mother is so fat, they had to invent transfinite induction to describe her.

(and so on)

geocalc33

Your mother is so fat the SYZ prescription fails to transport her

Your mother is so fat that optimal transport theory can't even find a way to transport her mass

just to explain that first one, SYZ (strominger, yau, zaslow) prescription says that you can always use a torus fibration, to break the calabi yau manifold into pieces then transport those pieces to the mirror manifold. So if you think of the rolls of fat on yo mother as torii

Xander Henderson

20:36

Your mother is so dumb, she thought that she was the Cox Zucker Machine.

geocalc33

Your mother is so intelligent and beautiful she solved the Weil conjectures on Sunday and was on the cover of vogue on Monday

Sometimes it takes a bright young mind to bring a new perspective, when others are entrenched in past ideals

Xander Henderson

@geocalc33 You're so smart, your mother calls you "son".

geocalc33

21:17

Anyone have any hints on how the tangent bundle of a exponential family forms a khaler manifold, Xander?

Xander Henderson

@geocalc33 Why did it go from "anyone" to "Xander"?

And do we all understand that Xander hates manifolds? They are not nearly pathological enough.

geocalc33

@XanderHenderson sorry I forgot you didn't like manifolds. Going forward I will keep this in mind

Jakobian

@XanderHenderson I agree. Any space above hereditarily normal is too nice

Xander Henderson

@Jakobian Whoah! Let's not go crazy here!

Metrizability is desirable. Better yet, there should already be a metric.

Sine of the Time

Ted heard the word "manifold" and entered the chat

Jakobian

21:27

@XanderHenderson how do you feel about Bernstein sets

is that pathological enough

it is a metric space

moreover it has some cool pathological properties

it has a good Polish wikipedia article, the only other ones is English and Russian but they suck

Zbiór Bernsteina

Zbiór Bernsteina – podzbiór przestrzeni polskiej, który jest w pewnym sensie bardzo nieregularny. Zbiór Bernsteina, jako podzbiór zbioru liczb rzeczywistych jest przykładem zbioru niemierzalnego (w sensie Lebesgue’a). Nazwa pojęcia została wprowadzona dla uhonorowania niemieckiego matematyka Felixa Bernsteina, który pierwszy rozważał zbiory tego typu w 1908. == Definicja formalna == Niech X X będzie nieprzeliczalną przestrzenią polską. Podzbiór Z ⊆ X {\displaystyle Z\subseteq X} jest zbiorem Bernsteina w X X…

not nearly expanded enough

Xander Henderson

@Jakobian Yup. Love it.

@Jakobian I don't read Klingon.

Jakobian

can't you turn on google translate or something

In mathematics, a Bernstein set is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them.A Bernstein set partitions the real line into two pieces in a peculiar way: every measurable set of positive measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set. == References ==

here's the English one but like I said... it's very poor

Semiclassical

21:55

this is something i feel like i should know but i can't recall anything quite like it

suppose i start with some convex solid and some plane through the origin. then i can get a 2D convex set either by projecting onto said plane, or taking the cross-section with that plane

now, one thing that can certainly happen is that the two sets coincide. for instance, take the cube [-1,1]^3 and the xy-plane. but in general they'll be distinct

this is said wrong, trying again

« first day (4748 days earlier) ← previous day next day → last day (568 days later) »

Transcript for

Aug '234

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC

site design / logo © 2025 Stack Exchange Inc; legal

mobile