Mathematics - 2023-01-31

12:22 AM

Munchkin has sent leslie to his room for bad behavior.

nah, just washing a ton of dishes.

A dishwasher uses way less water than a human sort.

I could never

I suck at most if not all housework in fact

Typical male.

Says another male

12:30 AM

Yes, one who is a gourmet cook and who has cleaned most of his life.

Still won't race me in a drag race

So I automatically win

Wonder if Xander cooks too

Yes, Xander cooks.

Damn. Everyone here outclasses me

Gwyn

Let $c_k=\binom{n}{k}$ for $k\in\{0,\dots,n\}$. Prove that $\sum_{k=1}^n \sqrt{c_k} \le \sqrt{n(2^n-1)}$. I tried this: $\sum_{k=1}^n \sqrt{c_k}=\sum_{k=1}^n \left|\sqrt{\binom{n}{k}1^{n-k}1^k}\right|$, and using the inequality $\sum_{k=1}^n |a_k| \le \sqrt{n}(\sum_{k=1}^n a_k^2)^{1/2}$ it is $\sum_{k=1}^n \left|\sqrt{\binom{n}{k}1^{n-k}1^k}\right| \le\sqrt{n}\sqrt{\sum_{k=1}^n \binom{n}{k}1^{n-k}1^k}=\sqrt{n}\sqrt{2^n-1}=\sqrt{n(2^n-1)}$. Could this work?

Bonus question: if this is correct, I still have a doubt. I used a result that is valid for a finite sequence $a_k$, but in this case it is $a_k=\sqrt{\binom{n}{k}1^{n-k}1^k}$ and this depends on $n$ too. This is not an issue because, in these situations, I must think $n$ as fixed in the beginning (like a parameter) and so $a_k$ actually depends only on $k$?

1:10 AM

gwyn: yes, that works, whatever n happens to be.

@leslietownes would you be in trouble if you used the dishes you just washed as Frisbees?

Gwyn

thanks leslie:)

1:31 AM

probably. most of them weren't plates (i was washing odd items like blender attachments and plastic kid stuff and things that don't easily go in a dishwasher)

Ajay

1:43 AM

@PM2Ring I was making a joke. I was referring to RGB colors being the primary colors. coz science u know...

@leslietownes I wonder if I'm fast enough to play Frisbee with myself. To test that I'll be using a glass plate

geocalc33

Why is $\sum_n f(n^s)$ for $f(x) \ne 1/x$ potentially incredibly useful in non-number theory?

Ajay

@ペガサスSeiya That's the wisest thing to do.

geocalc33

I say that because f(x)=1/x is ultimately incredibly powerful in number theory

because of dirichlet series, generating functions etc.

Maybe on a more serious note, is f(x)=1/x the only choice that yields a fruitful approach towards some larger theory, or have other f(x)'s been 'tested out'

@leslietownes update: Nope, I'm not fast enough for my own throw. I ended up breaking the plate, stepping on the glass and hurting myself

Ajay

4:20 AM

@ペガサスSeiya Are you ok?

copper.hat

4:42 AM

i am not

Akiva Weinberger

5:08 AM

Had a thorny time with this earlier

> Prove that if $M$ is anyone $3$-manifold with (possibly empty) boundary consisting of tori, then for any topological ideal triangulation of $M$, the number of edges of the triangulation will always equal the number of tetrahedra.

(Where's @Balarka when you need him)

We're triangulating $M\setminus\partial M$ by gluing together closed tetrahedra minus their vertices, essentially

I think I got it in the end but my argument doesn't feel very elegant

(The book is about hyperbolic knot theory, which means usually $M$ is a knot exterior, and eventually we're gonna identify each tetrahedron with a hyperbolic tetrahedron with ideal vertices, and hope they join up smoothly, but that's not relevant to this particular question)

@AkivaWeinberger Or rather triangulating $M$ by gluing together truncated tetrahedra.

@AkivaWeinberger Haven’t seen him in months.

This sounds very Poincaré duality-ish.

Nah, not so. Knot so.

copper.hat

Are you just stringing him along?

5:36 AM

Too edgy.

7:22 AM

the boundary of M is a surface, right? so maybe it falls out of the euler characteristic, with 'ideal triangulation' somehow meaning that the tetrahedra are stitched up in some nice way?

seems ugly if only because triangulations are ugly

10:57 AM

@Ajay I'm fine for the most part. Just received a few cuts

Nothing serious

No running for a few days though

11:12 AM

Does anyone know a programm to calculate the sum of a series? I usually use WA, but I'm doing an exercise and I want to calculate the order of magnitude of its sum. WA says its nearly 0, however this information is too poor. The series is 1/ (n^2 log n)

12:26 PM

How to prove that Minkowski functional satisfies the triangle inequality?

nvm, I got it now.

:)

Suppose that A is a convex, absorbing subset of X a vector space X. Minowski functional on X is defined as: $p_A(x)=\inf\{t>0, x/t\in A \}$ for all x in X.

It is to be proven that $p_A$ is subadditive. Rudin's proof goes along the following lines: If $t=p_A(x)+\epsilon, s=p_A(y)+\epsilon$ for some $\epsilon>0$, then x/t and y/s are in A.

I don't understand why x/t, y/s are in A.

The following could have been done instead: For any $\epsilon>0$, there exist $t>0, s>0$ such that x/t and y/s are in A. This is by definition of $p_A$.

I understand this alternative part but what I don't understand is: is this and Rudin's are the same thing or not?

they don't look same to me as we have 'for some $\epsilon>0$' in the former.

1:10 PM

this still holds by definition

For any t> p_A(x), x/t is in A?

yes

ohh

how?

that's what an infimum is, no?

I don't understand how. Suppose on the contrary that $x/t \notin A$.

1:13 PM

ah, I guess I've implicitly made an observation you might not have: $\{t>0\colon x/t\in A\}$ is a positively unbounded interval

Then how can we relate this to p_A(x)?

The interval is non empty, I understand that. But my question still stands. :(

I see no contradiction to the fact that $x/t\notin A$

if you're greater than the infimum of a positively unbounded interval, you're in that interval

to be clear, by positively unbounded interval I mean something that is $(a,\infty)$ or $[a,\infty)$, but it could be either

"there are no expensive, shiny, popular textbooks - only the dense, terse, substantial ones. If you really want to understand [...], then the shiny books don't help [...]"

Is this saying that the expensive books are the dense ones, the shiny ones the terse ones and the popular ones the substantial ones?

sorry, it seems to be complicating the original question. Certainly, that's not what Rudin had in mind.

1:48 PM

@SineoftheTime I just fed your series to mpmath, summing from 2 to 3000, with Shanks extrapolation. I got 0.60552173586. You need to use a lot of precision to get a useful answer. Here's my code:

sagecell.sagemath.org/…

2:08 PM

I'm often not sure what Rudin has in mind or don't agree with it, but in this case I'm very much certain that it's just what I'm suggesting.

Q: Rudin Functional Analysis regarding the Minkowski functional

I said so because there may be some other argument requiring convexity of A but I don't see it.

1

Let $A$ be a convex absorbing set on some vector space $X$ and $x\in X$. In the proof of theorem 1.35 of Rudin's Functional Analysis, he states that if $\epsilon\in(0,\infty)$, then $t:=\mu_A(x)+\epsilon$ is such that $t^{-1}x\in A$. However, I don't see why this is true. It seems to me that he i...

Same question but still it is answerless.

A comment suggests to bring convexity into the mix somehow.

@SineoftheTime Euler-Maclaurin seems to work better on that series. 0.605521788882600

2:23 PM

@PM2Ring Actually, I have to compute from n=3000 to infty

I tried with wolfram, and the result is $\approx 0$

Thank you by the way

@SineoftheTime You should have mentioned that earlier... These extrapolation methods approximate the limit of the sum to infinity from a finite number of terms. But as the mpmath docs explain, they may fail to give the correct result, depending on the series, and on the algorithm.

@PM2Ring Yes, you're right. I was just searching for an oline calculator or something similar. I know WA and Keisan calculator but both don't wirk for such computation

Thank you for your time

well, to see the set in question is an interval of the form as I have claimed requires convexity

I suggest you think about it

@SineoftheTime Well, Sage is online. ;) I just tried summing from 3000 at 320 digits of precision, mp.dps = 320, and got 0.00003740650652, but I don't know how valid those digits really are.

@PM2Ring I'll try it

Peter

2:37 PM

Why not just download the free PARI/GP ? You are then not limited , neither in precision , nor in calculation time.

This series is a bit tricky because $n^2$ grows quickly, but $\ln(n)$ grows slowly.

@Peter SageMathCell supports Gap & GP. And you can call Pari from within Sage.

@Peter Actually I didn't know it

I like SageMathCell because I can easily post links to it here. It encodes the program into the URL itself, rather than storing it on some server.

@Thorgott: I observed the following: For any x in X, there exists a c>0, x/c in A. Then for any $d>c$, x/d is also in A.

great, that's all there is to it

2:45 PM

yeah :-).

Admittedly, SageMathCell does limit the calculation time (and memory), but the limits are fairly generous, considering it's a free online service.

I see that I have received an answer also doing the same thing.

coincidence that the answer was posted as the same time as when I figured it out.

@ペガサスSeiya: I watched a new show called 'The last of us'. It's based on a video game.

@Koro TLOU is a famous PlayStation game franchise

I didn't know that it was based on a game.

TLOU is my favorite game series on PlayStation

Same with Witcher

3:47 PM

"Apart from the first few courses, there are no expensive, shiny, popular textbooks - only the dense, terse, substantial ones. If you really want to understand [...], then the shiny books don't help [...]."

What's meant with "shiny" books?

4:17 PM

(this is from math.stackexchange.com/a/35280/1095885)

4:43 PM

@ILikeMathematics Impossible to say without input from the answer's author.

Hander Xenderson?

@TedShifrin just dropping by to say you were right (big surprise), my calculation was wrong. :P I was doing Einstein summation notation and missed adding an n on a term, so it looked like it canceled, but it should have been $R_{ij} - ng_{ij} + g_{ij}$ (if there was no $n$ it cancels), which probably agrees with your calculations.

Do you have a good reference on moving frames?

However, at a guess: lower division texts (precalc, calculus, diffyQ, linear algebra) tend to be large (e.g. 9x12x2 inches), printed on glossy paper in color, have lots of colorful images, include "call out boxes" and tangential information scattered throughout a section of text, etc.

These books are designed to be attractive and "shiny".

See, for example, Thomas' Calculus or Boyce and DiPrima's Elementary Differential Equations.

Yeah, that's probably what he means.
Would you agree with him that they won't help with the understanding but just the application to problems?

These books tend to have more examples and more exposition, as compared to "real" mathematics texts, which tend to be very terse, black-and-white, and "unfriendly".

@ILikeMathematics Broadly speaking, my impression is that most of the "shiny" books are written with a non-math major in mind. They are intended to teach recipes to physics and engineering students, and are generally fairly light the underlying mathematical ideas. You read these books because you need to learn how to do certain things, not because you want to understand why you do those things.

4:55 PM

Waiting for cookies

@XanderHenderson Thank you. So as long as the book is for math majors, it should be fine

@ILikeMathematics I didn't say that.

I think that every book should be evaluated for fittness-of-purpose.

And that the fit is often going to be very personal.

Alright, thank you.

@XanderHenderson My books are not unfriendly. One reviewer of my algebra book complained that a math text is no place for humor!

@anak Spivak uses them freely. There’s a brief intro for surfaces in my notes. And look at Clelland’s newish book.

@TedShifrin proof? I need to verify that myself and that can only happen by purchasing a legal copy

So, where can I find the legal copy?

5:10 PM

@TedShifrin I was very careful to use a lot of hedge words in what I wrote above.

There are probably "shiny" books that are of high quality (mathematically speaking), and "real" books which are warm and friendly (personally, for example, I think that Hatcher's Algebraic Topology is pretty friendly).

Indeed, and the two examples you mentioned are considered "classic" introductory textbooks.

@user2236 Which two books are you referring to?

28 mins ago, by Xander Henderson

See, for example, Thomas' Calculus or Boyce and DiPrima's Elementary Differential Equations.

@user2236 Ah, well, "classic" does not mean "good".

Boyce and DiPrima is okay, but I really dislike Thomas's Calculus.

It feels unfocused and committee-written to me. There is no authorial voice, no real narrative. I feel like it presents a lot of disconnected facts without really connecting them, or telling any kind of coherent story.

"committee-written" puts the shine on the covers :-)

5:20 PM

@ペガサスSeiya Download the free differential geometry text :)

the friendliest math book of all is, of course, Bogachev's measure theory

Not Euclid? :P

@Thorgott ...

@Thor: Surely you misspoke. You meant to say Federer's GMT.

"Some books are written in a very linear fashion"
@XanderHenderson In those cases, you would still jump to some chapter and then go back whenever it mentions something previously established that you need to review, instead of reading it cover-to-cover, right?

5:27 PM

I notice an MT theme here, however.

oh, that's tough competition

@TedShifrin OH GOD NO!

I no longer own that book anymore, either. I will say, however, that I had to quote a stronger version of Sard's Theorem—which I found (only) in Federer—in my second paper.

Was it your PhD thesis paper?

No, that was the first publication.

5:35 PM

I took inspiration from a section in Bogachev once when I presented an excessively general form of Fourier's inversion theorem in a seminar

In other words, your talk was impenetrable.

well, I made a sign mistake on the board and wasn't able to spot it, which resulted in me never finishing the proof

but the proof would have been very clean in theory

it was a work smarter, not harder thing

though the sign mistakes arguably weren't very smart

The difference between a good mathematician and a bad one is that the good mathematician makes an even number of sign errors.

That's the situation where, after a minute, you just say "It's supposed to be this. Let's go on ..."

@Xander What if it's a complex geometer dealing with factors of $\sqrt{-1}$?

Then you hope you're just imagining the mistake :-)

5:41 PM

What mistake?

@TedShifrin I mean, complex geometers are, by definition, bad mathematicians, right?

well, it was the final part of my talk, but alas, it was my first ever talk and nothing as bad has happened since

*runs and hides*

throws a stink bomb after Xander

$\sqrt{-1}$

5:42 PM

@XanderHenderson in my undergrad thesis, there was a sign error of $(-1)^{dim(M)}$, but thankfully all my $M$ were even-dimensional. do I get to be a middling mathematician for that?

That would be highly confuzling to someone as pedantic as you, Thor.

@Thorgott After I finished my masters thesis, I was invited to give a talk in Coventry.

After a very long trans-Atlantic flight, I was super sleep deprived and jet lagged. During that time, I was going over my notes, and found an error. I then tracked down the error, which was present in the thesis, too.

So, instead of sleeping before giving the talk, I ended up in a panic to fix the mistake.

Which, ultimately, was a little thing that didn't actually matter (there was some estimate which was wrong, but one could make a different estimate which did the job).

But this was four months after I defended the thesis, so the error is there in the "official" document. :/

@TedShifrin it was the sign error I deserved for quoting Spanier without realizing his sign conventions were suboptimal

@XanderHenderson ah, that sucks

@Thorgott Meh. My guess is that every published document of any substance has at least one moderately significant error in it.

That's the trouble with just looking at random pages of random books. One has to be careful with definitions/conventions.

And even documents of little or no substance, Xander.

5:52 PM

@TedShifrin Indeed.

like two weeks ago, I discovered that a bunch of a papers in a niche active research area actually have an (inconsequential) error in the definition of the very objects they're studying

@Ted my sign error actually was geometric in question. I'm sure you know the correct answer intuitively: if I have a smooth fiber bundle with orientable base and fiber, I can orient the total space, but do I orient fiber first or fiber second?

First is better for doing the Gysin map, integration over the fiber, etc. But I suspect most people don't do that. Intersection numbers sometimes need to be done "backwards" too to work out.

Thanks for the references, Ted!

How to heat water? Any of the "gourmet" chefs got any ideas?

@ペガサスSeiya For what purpose?

5:59 PM

@ペガサスSeiya ...without a stove or a kettle?

indeed, fiber first is the right convention

Easy: Put the cup of water in the microwave.

even algebraically, once you get the signs right, but integration over the fiber is by far the best way to motivate it

@TedShifrin Just be careful that you don't end up super heating it.

it's mildly counter-intuitive cause we always write stuff as $B\times F$ not $F\times B$

6:01 PM

I saw that recently, too. Where does the sign error occur, from orientation of the product?

There was a diatribe online somewhere about sign errors in symplectic geometry arising from orientations.

both orientations work, but they differ by a sign of $(-1)^{dim(B)dim(F)}$

and only the fiber first one makes integration over the fiber work with the expected formula

in a vector bundle settings, this means this fundamental class that pairs with the thom class to the fundamental class of the base

@Thorgott Right. That's why I said most people don't do it.

Similarly, the index of a section of a bundle needs to be the intersection number of the zero section with the graph, in that order, not the reverse.

@XanderHenderson glad you asked. I wanna heat water and toss it in the air in a really cold environment to see how long it takes to freeze up

Its for science!

Maybe you should work on your homework, instead, @Seiya.

oh yeah, intersection numbers also depend on the order

nasty signs

6:06 PM

Same product of dimensions you had a moment ago, @Thor.

of course, graded commutativity of the cup product :P

Or just interchanges of basis vectors in the product vector space ... No need to be so shmancy.

@TedShifrin that's how I wrote it in my thesis

though, really, graded commutativity of the cup product is the same thing at the end of the day

@Thor: Have you any comment on my comment here?

Yes, they're all the same thing even in the middle of the day!

I agree, the question doesn't make sense if the sum isn't even defined

6:21 PM

I had to panic and check my book to make sure I hadn't committed this dastardly error. I specifically said to impose linearity (as part of the definition).

@TedShifrin I submitted my math homework already, don't worry

And I'm trying to forget that I have history homework left

6:45 PM

@Thorgott remember what I mentioned last time about connections? I have made a huge commutative diagram that is basically the entire argument:D

can I show?

From what I could see, @Sha, your question was just tautology once you're used to doing connections with moving frames.

well yea, I am aware that it was an exercise in just making identifications explicit/clear

there's nothing creative going on, I know

The problem is that people don't learn to do concrete stuff in moving frames.

I don't know Tu's books, so perhaps he gives lots of proofs/exercises with them.

his book is good, but his exercises are a bit insufficient

not a lot of exercises and mostly simple verifications

What would you do if a student forgot to do the homework? @TedShifrin

6:56 PM

but that's ok, because I have lots of exercises from my course

That is laziness on the part of the author, I'm afraid.

I don't know if I sent you all my exercises at one point, Sha, but it sounds like you have enough.

for now yes :)

You had fun drawing that diagram, though. I can't do that without putting stuff in a graphics program.

I used tikzcd.yichuanshen.de

I've never learned tikz.

Have you guys learned about invariant forms ($G$ action on $M$)?

7:00 PM

@TedShifrin I love it that I can now think of them as sections of the frame bundle

invariant meaning $r_g^*\omega=\omega$?

where $r_g$ is right multiplication by $g\in G$

Right. And their power relative to cohomology computation.

hm, we've seen that they correspond to forms on the base manifold (if you add the horizontality condition)

For example, you can compute cohomology of a general homogeneous space by the cohomology of the complex of invariant forms.

But when that is a locally symmetric space, every invariant form is closed so the cohomology is just the complex of invariant forms :)

right, unfortunately we haven't done any cohomology

Ah. Have you learned how to compute on Grassmannians using moving frames?

7:05 PM

I haven't done any computations on Grassmannians. One homework set was just about constructing the tautological bundle on a Grassmannian, and the other set was to work out (standard) details about the bundles over them

with the Stiefel manifold etc

Well, perhaps I'll tell you some of that stuff or send you some stuff later on it. My whole thesis was moving frames computations on complex Stiefel manifolds :)

But I can send you some notes on computing Chern classes and Euler classes using frames (surely Tu does that) and proving Gauss-Bonnet that way, etc.

Yea we've done some (basic probably) examples. If you know the local connection forms, then with the structural equation you get the curvature forms, and then it's a matter of applying the inv. polynomial that defines your class. I assume that's what you mean by computations with frames?

Well, no, I mean actually computing integrals over various Schubert cycles in the Grassmannian :) This is how Chern first did all this stuff. Before abstract nonsense took over the world. :D

Right, in that case it's best for me to have a look after my exam. I'm definitely interested in computations with Grassmannians, that sounds fun (and a bit scary haha)

Just let me know :)

7:18 PM

I hate statistics

Grassmannians are very cool, but I always managed to avoid the Schubert cells

They are wonderful things and encapsulate all the degeneracy locus information which, IMHO, is crucial for understanding Chern classes.

But, as I said, abstract nonsense overtook the world.

I was actually never taught this stuff in classes, but have taught it several times myself.

There's actually a lot of geometric linear algebra stuff in statistics, @Seiya, but the statisticians either don't understand it or hide it.

I directed a masters thesis explaining all that stuff (and I learned some statistics along the way).

@TedShifrin And statistics teachers don't seem to do a great job at presenting it either, I feel. At least my teachers didn't.

Which is surprising because the other teachers that I had, such as our calc professor, is someone I'd follow into combat

Stat classes at the elementary level in the US are just reams of formulas, it seems.

Math is fun. I like it. I should consider joining a website dedicated to Math Q/As. Oh wait...

7:32 PM

Nah. You'd just waste time.

Yeah, I'd probably go on chat and annoy/troll people for no reason

And laugh at my own (un)funny inside jokes

10:18 PM

Wow, not even the trolls were active whilst I was gone.

10:31 PM

Well, I was trying to sleep

Which I couldn't

mick

10:48 PM

how does one type the symbol for reals or integers ?

\mathbb{} and inside you put the letter

mick

lol i just found it elsewhere but thanks

Or, shorter for chat, \Bbb R or \Bbb Z.

@mick bro how do you have 15k rep and never used it? :'(

mick

@SineoftheTime I just said integers :)

10:55 PM

if I get a matrix in row-echelon form, is there a way to tell if it's possible to reduce it further?

I keep trying to get it to reduced row echelon form and end up wasting time because it can't be done

given $x_1+x_2 = 300, x_1 + x_3 - x_4 = 150, -x_2 + x_3 + x_5 = 200, x_4 + x_5 + 350$ I made $\begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 300 \\ 1 & 0 & 1 & -1 & 0 & 150 \\ 0 & -1 & 1 & 0 & 1 & 200 \\ 0 & 0 & 0 & 1 & 1 & 350 \end{bmatrix}$ and got it to $\begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 500 \\ 0 & 1 & -1 & 1 & 0 & 150 \\ 0 & 1 & -1 & 0 & -1 & -200 \\ 0 & 0 & 0 & 1 & 1 & 350 \end{bmatrix}$ i don't believe there's any more reduction possible I tried

geocalc33

what's an example of a measurable function $f:\Bbb C^+ \to [0,1]$?

@Obliv you can go to reduced form, but no “further” …

What you have is NOT echelon form.

Why?

11:11 PM

oh i thought row echelon form meant leading 1s

so it has to still have a diagonal of 1s

Aw come on.

Not diagonal.

reduced row echelon form would just be the leading 1 diagonal with 0s following

what do you mean? Doesnt the 1s have to be below and to the right one place

What is the definition of echelon and of reduced echelon?

You’ve done enough math to know you have to learn definitions.

i'm really bad at conveying what I mean, the matrices above are not in row echelon form because the 3rd row of the 2nd matrix needs to have a 0 in the 2nd entry

and a leading 1, then it would be row echelon?

As Ted asked, what is the definition of row echelon form? reduced row echelon form?

11:15 PM

Below leading 1s must be all 0.

I hate row echelon form

Then switch to another major.

Or, at least, switch to a different part of mathematics. I almost never have to resort to row reduction in my life.

Nor did I in my research. But there’s a lot dependent on understanding linear algebra quite deeply.

Stop whining. Learn.

3

words to live by

11:23 PM

@TedShifrin i understand it perfectly well. I just hate doing it all by hand. For the same reason I hate computing determinants by hand too

@ペガサスSeiya I have dyscalculia. Introductory linear algebra (and, later, operations research) was a pain in my ass, because it is very easy for someone with dyscalculia to make dumb mistakes when doing the kind of tedious computation involved in row reduction. But I got through it, and still regard it as a useful thing to have learned. Suck it up, buttercup.

:62887547 Glad I could help.

@XanderHenderson You're not wrong, I just have certain biases towards different areas of mathematics. I really like calculus, geometry (analytical, synthetic, etc). I don't particularly love linear algebra, vectors but they can be fun. I absolutely despise probability and statistics

so does it always have to be possible to put a system in row-echelon form?

just to make sure

Of course.

Echelon form is not unique. Reduced is unique.

Funny enough, we learned Gaussian elimination in physics (vectors) before we got around to learning it in mathematics

11:31 PM

Why does physics care?

i hear the term eigenvectors and eigenvalues a lot in physics

It’s not unusual to learn vectors and dot products first in physics.

probably for some quantum mechanical stuff

@TedShifrin are jokes about physicist allowed in this chat? :)

That’s quite advanced. Vectors for mechanics.

11:33 PM

@TedShifrin not sure. We were learning vectors, high school, and were introduced to Gaussian elimination as an introductory math class before we could proceed with the actual chapter

Only new jokes.

@Obliv yep, they use it in quantum mechanics

@ペガサスSeiya Early in my masters program, I attended a talk. While walking back to the math building with my advisor, I criticized the talk for being "too applied". My advisor stopped cold, made pretty intense eye contact and said something to the effect of "You are just starting a masters program. You don't know enough to have an opinion about applied mathematics. You need to keep opinions like that to yourself."

I will offer you the same advice, now: you don't know enough to have an opinion. Study more. Learn more. Be more humble.

We needed dot-products and cross-products in chapters such as force and momentum, work power and energy and electromagnetism

@XanderHenderson still not gonna make me like statistics. I'll make sure that gets written on my grave

Yes, of course. Vector addition, subtraction. Dot and cross. Nothing to do with row reduction.

11:36 PM

@ペガサスSeiya It's like you aren't even listening to me.

@TedShifrin a mathematician and a physicist have to cook an egg. first scenario: the egg is in a drawer. So both the phys. and the math. open the drawer, take out the egg and cook it. Second scenario: the egg is on the table. Phys. take the egg and cook it, the mathematician take the egg and put it in the drawer so he falls in the previous case

One of the best courses I taught in my 36+ year career was a year of applied math. Fabulous stuff.

@SineoftheTime I know the following version of the same joke:

A physicist, and engineer, and a mathematician are staying in a hotel---there's some kind of conference going on. They are staying in the same room to save money (the mathematician is forced to sleep on the floor). In the middle of the night, the trashcan catches fire.

Not remotely a new joke, Sine. There are a dozen jokes where the mathematician reduces to the previous case.

The engineer wakes up and notices it, grabs the ice bucket from the dresser, runs to the bathroom, fills the bucket with water, and pours it on the fire. The fire goes out, but the engineer repeats the process three more times, just to be sure.

11:40 PM

reduced the joke to the previous joke case, rip

@XanderHenderson I am. And you're right. I need to study way more than I have to even begin forming an opinion on mathematics. I'm just saying I enjoy certain parts of it now more than I do others. That may change in the future as I study more, we'll see

A little while later, the trashcan catches fire again. This time, the physicist wakes up and notices it. He pulls out his slide rule does a few computations, and determines that 2/3 of an ice bucket full of water will put out the fire. So he grabs the ice bucket, fills it with water, and then carefully pours 2/3 of the water into the trashcan, putting out the fire.

Later, the trachcan catches fire again. This time, the mathematician wakes up and notices it. He sees the 1/3 full ice bucket on the dresser, grabs it, pours it into the sink, puts it back on the dresser, and goes back to sleep, satisfied that he has reduced the problem to one that has already been solved.

Meanwhile, a statistician is running from room to room in the hotel, setting trashcans on fire, in order to get a larger sample.

I'll admit that last punchline caught me off guard

lol that's a good one @XanderHenderson

A mathematician, an engineer and a physicist are on a train, at one point they see a black sheep. The engineer says: well, all sheeps are black. The physicist says: that's not true, we only know that some sheeps are black. The mathematician says: wait, we only know that exists at least one sheep, and one side of her is black

11:45 PM

$\iff$ what is the other version of this symbol that only goes one direction

@Obliv \implies ?

$\implies$ thank you

@Obliv you can use \implies, or \leftarrow, or \rightarrow . what direction are you referring to?

@XanderHenderson is this a good time for a fighter jet metaphor?

I believe that \implies is really just an alias for \Longrightarrow (so if you want to go in the other direction \Longleftarrow ? $\Longleftarrow$)

11:47 PM

@XanderHenderson yes, I use always \implies

can I use $\implies$ as a way to say "this follows"

@SineoftheTime Sure. It is easier to type. :D

But, at a low level, I am reasonably sure that \implies just maps to \Longrightarrow, so, while there is no similar macro for \Longleftarrow, the spacing and whatnot should all be the same.

Hey Xander, if a sausage is cut very easily does it mean its overcooked? @XanderHenderson

@Obliv in your notes yes, in general when writing something formal you should avoid such symbols

@ペガサスSeiya No? What do you mean?

11:52 PM

$\impliedby$

\impliedby

@XanderHenderson I was eating a hotdog and it was a little too easy to chew, so I was wondering if that's because it could be overcooked

@ペガサスSeiya Hot dogs are almost impossible to overcook.

And they are "too easy" to chew by design.

How was it cooked?

@XanderHenderson How was it cooked? As in, there's different ways to cook it??

@ペガサスSeiya clearly he's assuming it was cooked without proving it

@ペガサスSeiya Sure. Boil it in water, stick it under a broiler in the oven, cook it over charcoal on a grill, put it on a stick and hold it over a fire, chop it up into pieces and fry it in a pan, ...

@SineoftheTime Can't be "overcooked" without being "cooked".

11:57 PM

@XanderHenderson yeah I have no idea, but probably a barbecue grill

It didn't taste what "overcooked" food tastes like so maybe it wasn't overcooked

@SineoftheTime yeah, that means I should probably try eating a frozen hotdog