@heather !!!

Probably the thing to do as a sanity check is to figure out what the correct graph is for k=2,3,4,...

@TedShifrin hello! =D

good to see you!

For k=2, it'll be the complete graph on 3 vertices.

school's been busy so I haven't been in for a while. I'll be done next week Friday.

how have you been?

Pretty well, heather. How's your math doing? :)

pretty okay =)

Good :)

My parents are considering finding some sort of tutor or class or something I can take so I can formally learn calculus or other topics (for this summer), so I'm excited about that. I also got a response from a professor I emailed (I had a question about his paper) and so I'm still in the "OMG THEY KNOW I EXIST!!!" phase =P

I am hesitant about your getting too far ahead of yourself and constantly being bored in school. Have you looked into something like Art of Problem Solving? (I might actually teach for them in the fall, but not on-line.) I've heard lots of great things about it for bright kids.

I wasn't very far ahead

and I was still constantly being bored in school.

brilliant.org is another interesting alternative.

I don't know if not learning math does anything to help K-12 teachers make their classes more interesting.

Well, even for an immensely qualified teacher, it's hard to cater to one or two students at the expense of 25 others. Even at the university level, this is often an issue.

@Semiclassical This is just a "naive first" assumption but can it be that the whole graph just works if i have k +1 nodes for this given conditions ? But well, thats just for complete graphs, right, i could just omit the edges for graphis with more nodes and still fullfilll the conditions

I guess my idea has been to some extent (e.g., with MeowMix) to encourage him to learn things that he would not learn later in the curriculum but that would still be interesting mathematics.

Since no one learns projective geometry (even in college), this was a safe choice ... for example.

But it partly is based on my bizarre tastes in math.

on the other hand, if you get bored in school your social life will be pretty nice haha.

I'm not sure that follows.

It didn't for me.

it should follow if you put in the effort

unless school decides to give a lot busy work... ok that could happen.

Social life is f*cked up if you you are going for a REAL degree haha xD

I just think if someone strong wanted to go into math or physics as an undergrad, they'd be way better off going through something like Spivak's calculus then learning something more archaic (not that proj. geometry is particularly archaic).

or your books...

Hello everyone!

Hi Demonark.

@PVAL-inactive i second that. you also don't want to get burnt out too

@TedShifrin no, I haven't, actually.

A vast majority of the math/science people do is based on calculus and/or linear algebra. These seems like the kind of things you would want to learn asap.

@PVAL: It depends whether we're learning math for fun or to score AP points or college placement.

Hi @Daminark

I should look into that.

ASAP doesn't apply to kids 12-15 ...

I'm teaching math so people can feed themselves in a failing job market

even at 12

@TedShifrin, well, when you're really into physics...

=P

Well, I disagree.

@PVAL I'd say it should be a mix. Many places seem like they're less than perfect about letting people place out of various math classes

@PVAL-inactive you mean "analysis" how it is called in the non "anglo american" region haha

what I've told myself is that I learn things better when I go through multiple times.

I think that's pretty true.

but I could be wrong.

@heather: The question is what you want out of mathematics — just a tool to do physics?

no, not at all - I love physics, and I want to use math in physics, but I also love math for its own sake.

Yeah, colleges won't just let you skip calculus or linear algebra because you act all puffy and say "I know all this s**t."

So for that reason, I'd say don't pregame too many classes that you'll necessarily have to take in college

@TedShifrin I've seen people skip out of multivariable...

@heather if you just wanna do math without any field of application you go crazy - trust me - just use it as a tool xD

I haven't, Dair.

There's quite a few dean scholars here who skip nearly all of undergrad math.

@Yannik That's not true

I never saw that happen at Berkeley, MIT, or Georgia, @PVAL.

@Yannik K. nope, Collatz Conjecture doesn't have any applications I know have but it's so....

intriguing, beautiful - many things.

which sounds very cliche, but it is.

do not agree. i study computer science and i hate everything in math that enforces me to prove sth.

What I tend to get annoyed at is how some math gets filed as "physics/engineering"

"Special functions" being a key example of that.

A particularly talented student who truly knows a lot can, in a smaller department, perhaps get someone to give him individual attention and some sort of individualized placement test to skip some classes, but it's far from automatic in my experience. I even made students who'd had a crummy linear algebra class in a junior college show me that they could write proofs at the level of our linear algebra class at UGA before I'd give them credit for the course.

Even taking linear algebra/calculus after some real exposure as a high school or less student should allow you to get more out of those classes as an undergrad if you do need to take them.

@Semiclassic: Some people who do Lie groups and representation theory are quite expert on that. So it's not a global filing.

That's your personal preference, does not mean you can project on anyone else @YannikK.

True.

And I don't mean that every math student should know how to use Bessel functions.

I personally have never had use for them, but I did teach them a bit in the year-long applied math course I taught.

If you're about vibrating drums, then you'd better know them.

BTW do math students write a single line of code in their "college career" - and i dont mean MATLAB

sure in mathematica

@YannikK. I write lots of code and am a math major.

Most math majors take at least a few computer science courses.

Bessel functions are easier to use than Mathieu functions

which is sensible, since circular drumheads are easier than elliptic ones.

nods

and although I've never used MATLAB idk why it doesn't count

Serious numerical analysts seem to prefer MATLAB to other alternatives.

it is not a programming language - its a commercial software

Well, Matlab is nice insofar as it's threaded.

@TedShifrin Yeah, but it isn't open source, right?

There's an open source version of it (whose name I've forgotten).

It seems like you have a bunch of math majors who are told to not count on an academic career and to make sure they have some fallback, many of which will go for compsci as a default (or physics). Those who are interested in stuff that is assisted by coding.

So it lets you handle arrays of data in a very nice way.

Physics is no safer than math, Demonark, probably less so.

Computer science, statistics, actuarial science ... are the "safer" avenues.

Reminds me, I should try to learn some python this summer.

That's fair

I keep meaning to

<3 Python

@Semiclassical Python is dank.

Python is awesome.

Especially since right now I'm so reliant on Mathematica

Let me guess you are using NumPy or SciPy in Python ?

I've been meaning to learn more about Mathematica.

But yeah, those two groups, and those who just have space in a schedule and say "Why not?", take up a good number of math majors, I'd say

@EricSilva: Do you want to help out on this?

@YannikK. me? I've used numpy a couple of times, but I also use other packages and just mess around with it. I find the Euler problems particularly fun.

Demonark: Please don't view UC as typical math majors country-wide.

@Semiclassical Mathematica is really fun though. There is no need to worry about reliance.

What does reliance mean?

Eh, Mathematica is a commercial software.

Very expensive.

Not something I'm likely to have access to post-grad school

@Ted: I think he is referring to the fact that he is using mathematica as his main language...

and no other language...

Hence why I'd like something that I can actually use for math once I don't have access to Mathematica.

@Semiclassical the switch shouldn't be too bad.

Well, I'd assume that you have a lot of students who will take a bunch of math that aren't into the subject necessarily, but you'd think that those who go the full length and take a bunch of extra classes that are not useful to their career have some particular interest in math

just read the basic tutorial and you keep posting code to codereveiw and you'll have it down.

A lot of my colleagues at UGA were huge fans (and developers) of Sage, @Semiclassic.

Sage is better than mathematica for graph theory supposedly...

from what I have heard...

Even at Berkeley, Demonark, something like 15% of the math majors contemplate going to grad school.

Lots of schools have up to 10 or more "flavors" of math major ...

various 'concentrations', yeah.

Like, I'd have thought that the proportion of math majors who go on to do higher math from here is lower, since there are fewer options for a generic stem degree.

Whoa, 10?

sheesh.

But yeah I mean, I guess the more explicit comparison I have in mind is econ. At many places, if you want to do business, you have a business major, so those who don't go for that and instead do an econ major are that much more likely to do econ grad school. Here, if you want to do finance later, your best shot is econ, so it becomes more crowded. Does that logic not hold in most places?

Also whoa

That's intense

At UGA, stat and math ed are in separate departments (even schools, for the latter). But we tried to create a "theoretical computer science mathematics" degree many years ago, supported by our own CS department at the time, and the Regents turned it down.

what's this site's policy on plagiarism?

Here we have a BA in Math, a BS in math (which only requires that you take a couple more classes in the sister departments, like CS, Physics, Chem, etc), Applied Math which seems to be being phased out in favor of computational/applied math, and math with an economics specialization

I have never liked the business major for undergrads ... it's basically a trade school thing. And the MBA was created for people with a science/engineering background who wanted to then get business knowledge to run a scientific/engineering business. Boy has that gotten perverted.

like copying someone else's answer, Duck? Or using our answers here on their homework?

no i mean like posting stolen content as an answer

without attribution

I guess it depends how literally "stolen" it is.

This is probably on Meta somewhere.

Oh I also don't agree with it, I'm just saying that a larger and larger proportion of people are going into college with a career mindset, and if you are going into something which is serviceable by a business major, doing an econ major feels like it's not quite as good of an option if you have to take certain classes you wouldn't otherwise expect are good for you

Well, I'm not in agreement with college as trade school, so I abhor this entire attitude. But that's another matter.

But if a place doesn't have a business major and you want to do business, econ becomes your main serviceable option, so there's a smaller proportion of people aiming at grad school because of the other group

@TedShifrin blatant copy/paste

that's allowed, right?

I mean, given that in employment a college degree is a helpful addition, and has now started to become standard, I think this move was kind of inevitable, sad as it is

As you state it, Duck, no, it's not allowed. I've seen it before and put a comment with a link. You can certainly flag.

...

ok then.

Are you the one copying/pasting?

no

i wouldn't do that to someone

besides, it's kind of pointless. :p

@Ted I can maybe take a look later, I have my galois problem set to do at the moment.

LOL, ok, @EricSilva. :)

Galois theory is cool, regardless.

I agree, although I kinda feel like when I muck around with field extensions I'm reaching around in the dark

@Akiva I have a weird/vague idea for our problem that I think works

If a rectangle has one of its side lengths as an integer, you could sorta put it on a 1/2-lattice

@TedShifrin i was just curious

I feel like I should know how to do this, but my brain just kinda bounces off of it: math.stackexchange.com/questions/2283565/…

Like in past classes I've felt like my brain just kind of goes in the right direction without too much searching, but in this galois theory course ifeel like I spend a lot of time looking for the right direction to pursue.

Been too long since I did grad-level relativity stuff, I guess.

And color things alternating between black and white

But anyway, such a rectangle with an integer side length should then cover equal amounts of black and white

That, and I'm genuinely not remembering how one writes the Jacobian when you've got upper and lower indices.

@EricSilva: I was never very natural with algebra arguments, but after writing my book and teaching the course a bunch of times I think I have pretty good intuition for a lot of that stuff.

So if a rectangle can be partitioned as such, then it must also cover equal amounts of black and white. But, if it doesn't have side lengths as an integer

Hi, I'm back. quickly reads what you wrote

Then this can't be so

That doesn't seem to be an if-and-only-if thing…

I mean, this is completely different than the way I solved it, so I have no idea.

But it seems that there should exist rectangles without integer side lengths that still cover the same amount of each color.

I think it may be so, because if no side lengths are an integer, you are cut it into 4 pieces

What I want to say is that $$dk_0'dk_1'dk_2'dk_3'=\det(\partial^\mu k'^\nu )\,dk_0dk_1dk_2dk_3$$

$a,b$ are the side lengths, so align your rectangle with the lattice so that the bottom corner is on a lattice point

No, you're upside down, @Semiclassic.

@Daminark Can you do that?

You need derivatives of the primed with respect to unprimed.

Oh.

I edited.

Then cut it into 4 pieces by slicing through $a - floor(a)$, $b - floor(b)$

Right, right, I see

Sure you can

Summation convention would make you get it right :)

If one corner is on a lattice it can't be done

What I'm really wondering is the indices in the det.

But what if no corners are on lattice points?

I want to say they're both upper.

I think there are so many different but equivalent ways to define properties of extensions that I have difficulty picking which ones to use to solve problems

Well, you choose where to place the lattice

No, the $\mu$ should be lower.

So just place it like so that a corner is on a point

That and I don't have enough examples floating around in my head

The derivative is a linear map, so a (1,1)-tensor.

...hrm, yeah. I had convinced myself it shouldn't be that.

@Daminark Ohhh! Right, I see, we only need to match it with the one big rectangle

Yeah, exactly

Wow, that's clever!

If you wrote it all out carefully, you'd have $dk'^\nu = \sum_\mu \dfrac{\partial k'^\nu}{\partial k^\mu}dk^\mu$.

Right.

And then you're wedging that with itself four times.

Thanks!

How'd you do it?

And then the middle term is $\partial_\mu k'^\nu$.

So we have the big rectangle and lots of subrectangles

Look at the rectangle that's on the bottom-left.

As always, @EricSilva, examples are of primary importance.

Now I want to cut out a section of this big rectangle

Yeah, if I had more time I'd spend a couple days just computing a bunch of specific examples, but alas the quarter system makes this almost impossible

Okay. And $k'^\mu = \Lambda^\mu_\nu k^\nu$ here

If the top and bottom sides of the bottom-left-rectangle are integers, then cut out that rectangle and everything above it

If the left and right sides of it are integers, cut it out and everything to the right of it.

The linear case is just saying that det gives how signed volume transforms, of course, @Semiclassic.

This gives us a smaller "big rectangle", and some of the subrectangles are now smaller

@EricSilva: I'm not sure what you're thinking about in particular, so I won't butt in.

Right.

but we can check that the new "big rectangle" has a pair of integer sides iff the original does, and similarly for the subrectangles that have had bits cut off.

$\partial_\mu k'^\nu = \partial_\mu \Lambda^\nu_\rho k^\rho = \Lambda^\nu_\rho\delta_\mu^\rho = \Lambda^\nu_\mu$

whew.

well duh

So this is a classical case of "minimal criminal": We can always keep on cutting off bits until we're left with nothing, which satisfies the condition vacuously

If I were writing this out by hand I'd probably do stuff like $\Lambda^\nu_{\,\,\mu}$

But no.

(Or: Look at the smallest counterexample; this cutting off gives us a smaller counterexample, so contradiction.)

(Same idea.)

Oh this is interesting

Yours is so much cooler

I gotta say

So one wants to show $\det(\Lambda^\nu_\mu)=1$ when $\eta_{\mu\nu}\Lambda^{\mu}_{\rho}\Lambda^\nu_{\omicron}=\eta_{\rho\omicron}$

@Semiclassic: Only $\pm 1$.

Right, forgot.

Sure. You have $A=PAP^\top$, so $\det P = \pm 1$ UNLESS $\det (\eta_{\mu\nu})=0$. Then we're stuck.

Sounds right.

Well, it did take over an hour and half to think of. Originally I was trying to build a path from the top left corner to the bottom right and see if there was a way to guarantee that all of them in one direction were integer valued

But the problem is that a path loses way too much information

I'm not sure I care too much, in any case.

Then I sorta went and said well, an even number of multiples of a half gives an integer so there should be a way to do that. Still didn't work nicely because of lattice point overlap

But that evolved somehow into this

Oh hey Ted..

oh hey Nate

Interesting day?

Nah.

@Dodsy We just discussed two solutions of the "dividing the rectangle" problem.

Yeah I was reading a little bit

lattice

Yeah; put the big rectangle on a checkerboard lattice with grid size $\frac12$, such that one corner is on a lattice point.

Each subrectangle has an equal amount of black and white area, so the big rectangle does too.

This implies that the big rectangle has integer length or width, QED.

That's interesting :)

but I thought that the rectangles could be arranged randomly inside

They can;

but any rectangle with integer length or width has an equal area of black and white on such a lattice!

Well if I put a rectangle that is 1/3 the size of another, will it have equal amounts of black and white area if it falls on a black square?

ohhh I see what you mean.

@PVAL Bob's new paper looks interesting.

@Dodsy You need at least one side length to be an integer

I drew rectangles and marked off integer sides

and concluded that no matter what you do you'll have one square left and if you mark any of the sides as an integer you'll find that two of the sides are equal to an integer

which is basically your method of approach

I know my explanation doesn't make sense...

Yeah I don't understand what you just said there

Okay so if you draw any random amounts of rectangles inside of a larger rectangle

and begin marking off integer sides

Right

with x's

you'll be left with a single rectangle within the larger

marking any of the sides of the smaller rectangle will show that the larger rectangle has at least two integer sides.

because they connect.

But that doesn't show that it always happens...

It always happens

I just can't prove it formally.

If you want to go that route: Try to break it. Try to find counterexamples, and see what goes wrong

imgur.com/a/Nt0rr

see

if I mark that one triangle

I am still awake!

see if I mark it down

Talked to a friend who did it some time back by computing a contour integral

then I have proven that it is of some integer

That solution makes me sad

oh sorry.

yeah I guess it's pretty....not formal or the right way to go at it.

No I meant the contour integral solution

Not yours

Sometimes people use a high power tool to solve a low power problem quickly because a lot of effort went into proving the high power tool already

@Jason It took 4 days to figure out to use the contour integral...

anyways

basically

no matter what you do

Hello chat.

no matter how you draw the rectangles

Hi @Fargle. Anything new before I go off to dinner?

you'll avoid making the side add up to the sum of all the internal integers

but eventually it will

it will be a single rectangle left

@TedShifrin apparently there is a use for 4-dimensional surface geometry

@TedShifrin Sadly, today's not much of a math day for me. Chris Cornell's death has me in a sad place.

@Fargle Who is he?

I figured it would be Roger Ailes's death that did it, @Fargle.

@JasonBourne Vocalist/frontman for Soundgarden and Audioslave, easily the most talented rock vocalist of the last twenty-five years.

@TedShifrin If Ailes had died and Cornell hadn't, I'd be writing a thesis right now.

I think last month some actor of a show called Miracle committed suicide, Michael something.

@JasonBourne Yeah, Michael Mantenuto.

....

...sorry, I realize I drove the room off-topic. I cede the floor.

@Fargle Hmm, I guess he didn't get the miracle he was hoping for. =(

well, I'm outta here to go have dinner. Bye, all.

Bye @Ted

@Fargle Well, you don't have to feel sorry. It's OK to be off-topic. I do it all the time. =)

It's also OK if they wanna ignore me.

Yeah being off topic at various times is totally chill

See you @Ted!

What I don't get though is why some people ignore me and not other people when I don't talk more crap than they do. =)

I think your opinions tend to be more controversial

I don't ignore anyone for the simple reason that the conversation can become incomprehensible if you have missing lines.

shrug

So if someone buys another person's opinion and not yours they'll ignore just you, even if both you and the other person can be interpreted as talking crap

I don't give a polished argument for what I say because I never intended to write a paper in this chat for publication in a journal.

@Dodsy By "triangle" you mean "rectangle" I assume

But I say some things to give room to a certain perspective to be interpreted in a certain context.

@Daminark How does that even work?

@Daminark Do you find the AMS GSM series have too big fonts? I think they are too big for me.

Why do I keep saying fucking triangle

what the fuck

@Dodsy That means you need to relax. If you say fuck all day long then something is wrong somewhere =)

I mean also, those opinions of yours which are controversial are not a question of, oh we all have the standards/demands and then it's merely a discussion about which book fulfills it best. Some people dislike humor in a book, some love it, some like chatty books, others terse, some want an encyclopedia, others want a faster overview, etc

Well I suppose I am quite stressed "jasper" (if that even IS your real name....ringo)

@Daminark Exactly, so when I like it I like it for some reason specific to me. =)

And @Akiva I would tell you but I don't really know how complex analysis works too well :P

@Jason can you give an example of such a book so I can check it out?

@Daminark If you haven't come across it's fine, but all of them have the same size I think. The Schlag book for example is in the GSM series.

I rather like the font size in Schlag's book

@Daminark Obligatory picture of a sheared sheep from Lang

@Dodsy I told you I don't tell lies in this chat. You need to take my word for that.

... which book is this?

(Also what's shearing?)

I am not aware of any Lang book with a sheep.

(Like I know it in the context of sheep but what's the math context?)

It might not be Lang...

A shear transformation is one where you fix one direction and shift the other direction along the fixed direction.

I think Spivak's book on DG has flying pigs or something.

Prototypical example is $\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}$

Speaking of Lang, my favourite Lang book is really his calculus books. =)

Oh, it's Lay, not Lang.

I am lying, I never tell lies.

Ah, yes, the old "push on a deck of cards" transformation

@Fargle I suppose it's David Lay Linear Algebra

Yep.

Lang would never put that in his books, lol.

It's also the "tilt the (flat) object back and tell us what you see" transformation if you're infinitely far away.

I always find it amusing that stretch scale shear all start with s.

Isn't that such a coincidence?

@JasonBourne And every one of them is some combination of swing (rotation), stretch, swing...

@Fargle I think maybe it has something to do with the origin of those words, and maybe this question can even be answered on Eng SE.

Anyway I am going to sleep. I will see you all in my dreams.

Now we define the swing matrix $\begin{pmatrix} \cos(x) & -\sin(x) \\ \sin(x) & \cos(x) \end{pmatrix}$

@Daminark how dare you

@Daminark SVD is still one of the most remarkable results of LA.

At least in my opinion.

What is it Eric?

swing :P

GOodnight Jasp

@Fargle Laci told us about that, it was in the context of finding low rank approximations to matrices (as motivated by machine learning)

Lolol, it was Fargle's peer pressure! :P

im just kidding

i actually like calling rotations swings

Wait is that a thing?

I thought he made it up

The first result I look for in a LA book is the spectral theorem for normal matrices. =)

taps foot and snaps it don't mean a thinggggg if it ain't got that swing

@Dodsy Don't stress yourself too much. Don't go mad like me.

@Daminark AFAIK, I did...

I meant I like the idea of it

not that i do it

Oh! Yay!

Ah

@EricSilva I had a math teacher called Ms Silva in elementary school, but her math was terrible. =)

♪ Movin' on up... ♪

@Daminark one of Keerthi's hints: "Hint: the discriminant is -776887"

@Jason it's an extremely common surname in the lusophone world

@EricSilva WHOA, but -776887 = -196883 - 580004!

Must be connected to monstrous moonshine. /s

idk anything about monstrous moonshine

@EricSilva o lawd

but from the mystical things people say about it it seems everything is related to it

@EricSilva That is a new word to me, lol.

it means portuguese speaking

Lusophone

Who can imagine that?

All I know is that 196883 is the dimension of a certain representation of the monster group, and that 196884 = 196883 + 1 is a coefficient of one of the terms of the Fourier expansion of the j-invariant.

What those words mean is anyone's guess.