Mathematics - 2022-07-04

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anak

12:24 AM

I have only read one textbook cover to cover.

Calvin Khor

12:46 AM

i faintly remember finishing my highschool textbooks. what a great feeling tossing those books lol

leslie townes

easily the worst thing about my hs was our inability to toss those books. that would have felt great

Calvin Khor

why couldn't you? @leslietownes

leslie townes

my daughter's watching a documentary about polar bears and just said "maybe we can go to the arctic someday and see the polar bears." pretty sure there aren't gonna be any.

well technically i could have, but then i'd have had to pay for the book. which would be worse than the pleasure i'd get from destroying it.

Calvin Khor

lol i mean if you're quick im guessing they still do expeditions

leslie townes

she can see the last one, haha

there's a polar bear at the san diego zoo, she can see that one

Calvin Khor

12:55 AM

poseidonexpeditions.com/arctic

leslie townes

one kinda fun thing about our school re-using the same books forever is that sometimes you'd see the name of someone you knew as a previous user of your book. my sister and i used the same 8th grade science book.

Calvin Khor

random google link, no affiliation obviously

leslie townes

they keep a san diego one on a pretty regulated diet with a lot of vegetables so it doesn't build up the fat/fur that it would need in the arctic but would render it miserable in san diego.

Calvin Khor

my school/country was going through a change in the language used to teach science

leslie townes

it seems like a happy bear

Calvin Khor

12:57 AM

so my sister learned words i never learned lol

leslie townes

i wonder how the books are now, at my high school. i hope they've moved on to newer books

geocalc33

What if we rewrite the books?

leslie townes

i bet they're using the same ones, haha. nothing has changed

our history book had stuff up through reagan's first term but the class never got past WW2

geocalc33

I wonder if I'll be in a history book one day

a high school history book

leslie townes

i'd personally hope not to make it into the history books

but you do you :)

geocalc33

1:11 AM

why wouldn't you?

leslie townes

people mostly get in there because of the bad stuff they did, i try to keep a low profile

geocalc33

gotcha

leslie townes

i have a relative who is in some history books and i remember reading them with trepidation

thankfully he did nothing to disgrace our name

Ted Shifrin

You might still get thrown out of Wimbledon.

leslie townes

didn't wanna be there anyway. i've been thrown out of better places than wimbledon.

geocalc33

1:27 AM

describe your mathematical style in 3 words. I'll go first

leslie townes

my, mathematical, style

geocalc33

one-on-one in person discussion, curiosity, real world application

Calvin Khor

if that's 3 words then 1=2pi. Oh wait...

geocalc33

ill relax the conditions to merely 3 phrases

good one Leslie

7 hours later…

Kamila Szewczyk

8:56 AM

hi, i wanted to ask what is the best method of approximating the principal branch lambert w for x < -e^-1 regarding the speed at which it converges to actual value

Jakobian

9:20 AM

Is this helpful? math.stackexchange.com/questions/1980798/…

1 hour later…

Not Tfue

10:41 AM

I wish there was a community where people point grammatical errors in your thesis.

opio

should the second term $o(h)$ in the answer here math.stackexchange.com/questions/1409167/… read $o(\| h \|_{L^2})$, and if so why is is $\int \|\nabla h\|^2= o(\| h \|_{L^2})$

Calvin Khor

11:07 AM

No way it could be L2. Should be say H1, and they seem to be considering something like C1 plus sufficient decay for things to make sense

opio

$L^2$ with enough regularity

say $W^{2,2}$

Calvin Khor

you can’t control gradient in L2 by L2 norm even if h is smooth

opio

yes I meant the right space to use is $W^{2,2}$?

Calvin Khor

1 derivative is enough though

5 hours later…

Akiva Weinberger

4:03 PM

Good Wikipedia article:

Gudermannian function

The Gudermannian function relates a hyperbolic angle measure ψ {\textstyle \psi } to an associated circular angle measure ϕ {\textstyle \phi } . The circular angle measure is called the gudermannian of the hyperbolic rotation, and is denoted gd ⁡ ψ {\textstyle \operatorname {gd} \psi } . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert...

On the Gudermannian function, which relates the trig and hyperbolic functions to each other

If $\phi=\operatorname{gd}(\psi)$, then:$$\begin{array}{}\sin\phi=\tanh\psi&\csc\phi=\coth\psi\\ \cos\phi=\operatorname{sech}\psi&\sec\phi=\cosh\psi\\ \tan\phi=\sinh\psi&\cot\phi=\operatorname{csch}\psi\end{array}$$

It's defined by $\operatorname{gd}\psi=2\arctan(\tanh\frac12\psi)=\int_0^\psi\operatorname{sech}t~{\rm d}t$

It's inverse is $\operatorname{gd}^{-1}\phi=2\operatorname{artanh}(\tan\frac12\phi)=\int_0^\phi\sec t~{\rm d}t=\log(\sec x+\tan x)$

(Of course from that table you can also define $\operatorname{gd}\phi=\arcsin\tanh\psi$ etc)

Mad Spaces

4:49 PM

Hello, generally speaking, if we have a mtrix that is in diagonal form, is it true that charpol = minpol? i am thinking of examples like a matrix with zeroes everywhere but a_11 = 2 , a_22 =2 , a_33 = 1 , then the min pol is one degree less than charpol

so i guess its only true, if the matrix is orthogonally diagonalisiable, say every Eigenvalue is different from the other

Thorgott

it is true if and only if the eigenvalues are pairwise distinct

nothing to do with orthogonal diagonalizability

user10478

Is this article (investopedia.com/terms/c/cagr.asp) algebraically illiterate, or is there some ghastly convention in Finance where "$*\ 100$" means "multiply everything to the left by $100$% ignoring order of operations"?

Ted Shifrin

5:05 PM

It looks pretty horrendous to me. $a-b\times c$ of course is equal to $(a-b)\times c$. Right. Fire them.

user10478

Right, I'd though the site was slightly more reputable than this.

And the article lists an author, peer reviewer, AND fact checker.

user2103480

5:41 PM

Is there any accepted terminology for the operators arising from weak/variational formulations of PDE?

The simplest example being the dirichlet Laplacian extending to an operator $$\Delta \colon H^1_0 \rightarrow H^{-1}$$

Something like "variational extension"? I looked into some books but just can't find a commonly used term

(Negative Laplacian, not that important though)

1 hour later…

leslie townes

6:59 PM

user10478: there is a grand tradition of not reviewing or fact checking equations.

or symbolic formulas generally. once you enter 'math mode,' you exit the world of things that people pay attention to.

copper.hat

7:34 PM

there is a deeper issue there; maintaining consistency between something and a 'model' of something. (not to suggest that mathematics is just a model.)

Ted Shifrin

8:16 PM

Pretty insane that we have to "remind" people in formulas what percent means, anyhow.

Alek Murt

8:52 PM

hey guy any hints on how can i prove this $\lim _{n \rightarrow+\infty}\left(1+\frac{z}{n}\right)^{n}=\mathrm{e}^{z}$ using counting measure and measure theory in general?

leslie townes

how is $e^z$ defined?

Alek Murt

As a power series

Jakobian

@AlekMurt use Lebesgue dominated convergence theorem then

Alek Murt

that's what i think as well, but it confuses me that i have to use counting measure, how do i use that measure on this problem

Jakobian

9:11 PM

$$\sum_{n=1}^\infty a_n = \int a_n\mathrm{d}\mu(n)$$

Alek Murt

srry but i dont get your hint, could u explain me a bit more

2 hours later…

Akiva Weinberger

10:59 PM

Suppose we have an $a\times b\times c$ cuboid, oriented so one vertex is lowest. Then the number of ways to stack cubes in it (following gravity) is$$\prod_{i=1}^a\prod_{j=1}^b\prod_{k=1}^c\frac{i+j+k-1}{i+k+j-2}.$$

leslie townes

11:21 PM

the k + j instead of j + k is killing me

Akiva Weinberger

11:48 PM

No idea how I didn't notice

Still true, at least

leslie townes

if/when you teach, do not do that to your students

or alternatively, do that to your students while cackling and burning sacrifices to satan

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