Mathematics

oeis.org/…

@Dair The Kempner series is weirder.

3:15 AM

@Semiclassical Basically everything in math dealing with infinity is weird lol.

point

trying to explain people that the rationals have the same cardinality as the integers causes issues...

Eh, that's not nearly as weird as the real numbers having a higher cardinality than either of those.

Saying that the integers are as infinite as the rationals? Weird, but hey---infinity is weird.

it isn't that weird but try explaining it to a non-math friend lol.

I feel like I've seen Kempner series while preping for Putnam.

Saying that the real numbers are 'more infinite' than the integers? Weeeeird.

3:20 AM

there is a classic question about omitting 9s from harmonic and getting a good series... which I believe is a Kempner series?

wait yeah, that is exactly the Kempner Series... lol. silly math.

I think I first saw it here: smbc-comics.com/index.php?id=3777

Though in retrospect I think that said comic is a little silly. The math translation is: "Almost all really big numbers have a nine in them."

so there is no closed form for the sum?

what happens if $5$ is removed instead?

pretty sure it's convergent then as well.

plus there are some methods for estimating such series.

i'm rather suprised that Wikipedia doesn't have an "Open Problem" box.

There is something like that for "Physics" topics here

3:26 AM

they have those boxes in math as well. I'm just suprised a closed form for the sum is not listed as a problem.

Eh, it's pretty specialized.

Plus it's a fact that relies on picking a particular base-b representation and a particular string of avoided digits.

By contrast, they do have "Is $\pi$ a normal number" listed there.

i must admit i'm rather confused about the choice of importance for math problems...

not saying I disagree with you are anything, but it is hard for me to judge whether a problem is valuable or not.

I've no insight there either.

Fawad

How to solve it? I only know $X^T=-X,Y^T=-Y,Z^T=Z$

You'll need to compute the transposes of each of those combinations

Note that $(A+B)^T=A^T+B^T$ and $(AB)^T=B^T A^T$.

Fawad

3:33 AM

Ok. Nice

i wonder how many important problems of the era have disapeared in importance in the next era of math.

Yeah.

I imagine there are quite a few old conjectures which motivated some work but were ultimately left by the wayside

in my math history (for specifically calculus) it seemed like there were rather 3 phases:

1. Calculation
2. Proving a lot of stuff
3. Reflection / formalization

Depends on the subject, but yeah.

My comment here is a bit silly, but I always like pointing out how to use units in order to check an answer

Units are sooo valuable. I wish I payed more attention to them in chem during highschool.

3:45 AM

Yeah.

When I'm grading, wrong units are like alarm bells

it's like a test taking strategy, but also a conceptual check all at the same time.

Quite.

Of course, there are limits to that. It's less helpful in electrodynamics due to the number of units involved

Not many people remember that a tesla-meters^2 is the same as a volt-second.

And if you do atomic physics you end up using different scale of units (mostly eV and angstroms)

oh geez that sounds mildly infuriating.

tbh, it actually makes things easier than with SI units.

for instance, for atomic stuff you measure energy in units of eV. for mass, you really only care about doing $E=mc^2$ stuff, so the natural unit of mass is eV/c^2.

For Planck's constant, the combination I've got memorized is $hc=1240 $nm-eV$=12.4 $keV-angstrom$

you have 1240 memorized?

3:53 AM

yeah.

that's some dedication.

it reaaally comes in handy when you're doing the Physics GRE

are you a physics major?

Physics grad student :)

Kek

3:54 AM

Ahh. That's cool!

I've also got the masses of various particles in my head, e.g. an electron is about 1/2 MeV and a proton/neutron is about 1 GeV

On the subject of units: when my professor in mechanics was doing an intro to SR, he used units of feet and seconds so that $c$ would be almost $1$ :P

(it's also common to say eV instead of eV/c^2 when talking about mass)

also, to be fair I've got the charge/mass of an electron stuck in my memory pretty well

but that's from TAing intro physics a few times

m=9.11e-31 kg is rather cumbersome compared to "oh, about half a million eV"

also, the e from the electron charge is built right into eV

so you don't have to remember e=1.6e-19 coulombs

@Semiclassical i feel like there is a relavent SMBC comic for this...

lol, probably

3:57 AM

Is there ever not a relevant SMBC comic?

The main advantage of eV-type units is that the numbers tend to not require scientific notation

something along the lines of:

What is Pi?
Enthusiast: 3.14159
Engineer: 3.14
Physicist: 5 is good enough
Math: $\pi$

this one, probably: smbc-comics.com/?id=1777

shamefully I have 3.14159 memorized...

I've got 3.1415926(3)

(3) because I don't know for sure if that's right :P

Speaking of dimensionless constants, one that you see a lot in atomic physics is the so-called fine structure constant

4:01 AM

2653

nuts.

oeis.org/…

which...I'm forgetting how it's defined. hang on while I quickly rederive it using the relevant mnemonic

I memorized 80 digits one time in high school since I was bored :P

roll

okay, $\alpha=ke^2/\hbar c$

what's fun is that, to a decent approximation, $\alpha=1/137.$

4:04 AM

what is a fine structure? what is an ugly one?

ask an atomic physicist :P

(I think it's mostly a historical thing)

I should note, though, that factors of $\pi$ can be a big deal in QFT calculations

$\pi^2\approx 10$, so if you have a few powers of $\pi$ in your calculation for reasons then that can shift the order of magnitude significantly

$4πε_0ħcα = e^2$ what is the epsilon?

permittivity of free space

I honestly never remember what number that is :/

so $k$ is $4\pi\epsilon_0$?

wait no...

1/k is

4:08 AM

yeah.

k is easy to remember. i think it's 9e-9 in SI units?

nuts, 9e9.

but, I mean, I also have it in my brain that $\epsilon_0\mu_0=1/c^2$ where $\mu_0$ is the permeability of free space

and $\mu_0=4\pi\times 10^{-6}$ in SI units (might be wrong about the exponent)

nuts, 10^-7

and, well, I know what $c$ is.

so from that getting $\epsilon_0$ is easy as well.

6:34 AM

o lawd we done it

Liad

hi, easy question that i think i missed something. is there 16 ways to write 4 as a sum of 3 non negative integers? i found 15 :/

0 1 3
0 3 1
0 2 2
0 0 4
0 4 0
1 3 0
1 0 3
1 1 2
1 2 1
2 2 0
2 0 2
2 1 1
3 1 0
3 0 1
4 0 0

Fargle

15 should be correct.

There are 4 ways to split 4 into 3 or fewer non-zero pieces: 1+3, 2+2, 1+1+2, 4.

For the latter 3, there are 3 possible reorderings when you introduce 0; for 1+3, there are 6 reorderings. 6 + 3 + 3 + 3 = 15.

Liad

so i guess there is a mistake in the question , thanks.

Fargle

Yeah I'd say so.

What are you doing awake @Daminark?

Q: Is this product of 8 quaternions a real value?

6:49 AM

shrugs

Fargle

Tru

Abdullah UYU

hi guys do you think this question is bad, math.stackexchange.com/questions/2293979/… , why it has downvote?

SteamyRoot

7:32 AM

Morning chat (how many flags/bans did I miss?)

SoumyoB

9:01 AM

wait... Someone got banned from the chat?!! grabs pop corn where? Someone please give me the permalink???

ah found it

well guys I have my VISA interview tomorrow, wish me luck

Sha Vuklia

10:24 AM

Hey @Steamy. I think I understand now what the separation constant is. We have separated our variables to the left and right sight. We introduce this constant, such that we can solve the the two sides separately (in terms of the constant), and then in the end we can combine the results.

Rajesh Dachiraju

10:50 AM

0

I'd like to know the product of 8 quaternions $$(a+bi+cj+dk)(a-bi+cj+dk)(a+bi-cj+dk)(a+bi+cj-dk)(a+bi-cj-dk)(a-bi-cj+dk)(a-bi+cj-dk)(a-bi-cj-dk)$$ I'd like to know if it is real? PS : $a,b,c,d \in \mathbb{R}$

Sha Vuklia

11:42 AM

Guys, when normalising the solution to the Schrödinger equation in the case of the infinite square well, we do the following:
$$
\int_0^a\vert A\vert^2\sin^2(kx)\,dx=\vert A\vert^2\frac{a}{2}=1,
$$
so $\vert A\vert^2=2/a$. Now Griffiths says that "it's simplest to pick the positive real root: $A=\sqrt{2/a}$ (the phase of $A$ cassies no physical significance anyway)." I don't understand where the phase comes into play, if we were to consider $A=-\sqrt{2/a}$. Any ideas?

Manolis Lyviakis

why integrating on a closed curve say a circle with a function that is defined well inside that circle except a point say $z_o$ we have a problem? im integrating along a path why do we care what is inside the path?

why do we care about points that are not along the path * especially the ones inside since if our problematic points are outside nothing changes

SteamyRoot

12:04 PM

@ShaVuklia The phase of $A$? Sure this isn't about the phase of some wave?

Because, if you pick $A$ with the negative sign, you can move that sign inside the sine/cosine to get a phase of $\pi$.

Sha Vuklia

@Steamy yea, I just double-checked, and they really talk about the phase of $A$

and you're talking about the phase of $\pi$; what do you mean by that actually?

oh wait

I think I understand what they mean with the phase of $A$

they don't care if $A$ is half a cycle behind or not, because apparently it carries no physical significance anyway

LegionMammal978

Hmm

Given a graph, is there a way to order its elements such that all other isomorphic graphs follow the same ordering?

12:39 PM

@Daminark ¯\ _(ツ)_/¯

Alessandro Codenotti

we created a monster @Balarka

@Alessandro an $\infty, n$-frankenstein?

Alessandro Codenotti

1:01 PM

lol

1:38 PM

Hey Balarka

Hi @Danu

Is it right to say that an isometric immersion is completely described by the second fundamental form?

The Gauss equation gives the curvature

The Codazzi equation does... what exactly? :P

It gives the perpendicular component of the ambient curvature

@Danu Well, if your immersed surfaces have the same $\Bbb{I}$ (which is the isometry condition) and $\Bbb{II}$, then one can be taken to another by rigid motion.

Astyx

Hi chat

It's an extrinsic geometry thing.

1:41 PM

@BalarkaSen You're talking about surfaces in $\Bbb R^3$?

Yeah.

What about general isometrically immersed submanifolds in any Riemannian manifold?

(you can still think of them as immersed into $\Bbb R^n$ I guess; though that will give a different notion of extrinsic geometry than the original immersion)

Err, I don't really know the story for other ambient spaces than R^3.

The 2nd fundamental form is a bundle-valued tensor in higher dimensions, isn't it?

Hmm.

Yeah, just normal bundle valued

The generalization of the Gauss equation and Codazzi equation is sort of trivial/obvious

(depending strongly on what notation you're using in the surfaces-in-$\Bbb R^3$ case, I should say)

I think it should still be true.

That is to say, if two isometric immersions have the same 2nd fundamental form, there should be an isometry of the ambient space which takes one to the another.

But I think you should ask Ted about this.

1:45 PM

I was just about to ping him

SteamyRoot

Yes, the second fundamental form completely determines the isometric immersion

@SteamyRoot So is this hard to prove?

Actually

what is the precise statement

is what Balarka said the precise statement?