Mathematics - 2015-06-07 (page 1 of 9)

Ashley Davies

00:13

Quick question - Adding/subtracting rows to/from each-other in a matrix doesn't affect the determinant, but I'm getting the wrong characteristic equation after doing these operations. Have I probably made a calculation error or do I need to keep the matrix in it's original state to get a correct characteristic equation?

anon

00:25

@AshleyDavies you need to do the operations on A-xI not on A. since the charpoly is the det of A-xI.

r9m

00:45

Can anyone suggest how to get $\lceil n/3\rceil$ instead of $\lceil \frac{n-1}{3} \rceil$ here ?

it looks like a small change but it's proving to be troublesome .. :|

Semiclassical

01:37

hmm. this is pretty neat: wolfr.am/5cKYNIqa

(the slope at 0 is $pi/2$)

anon

hmm, my firefox spellchecker is all of a sudden British

Alec Teal

Good for you @anon

I asked the Queen when last we spoke and yes, she does mind Americans spelling colour wrongly.

(Mine is set to American, it didn't like colour :/)

Semiclassical

i suppose having gibbs phenomenon in a bessel series isn't strange, but that function is pretty neat

Paul Plummer

02:38

@TedShifrin Saw " oh, axiom of choice isn't that big a deal ...
but all the countable/uncountable games are. " What did you mean by that? I like my games :)

Antonio Vargas

03:00

@Semiclassical By the way, this makes me very happy. Now I know at least 1 person other than me and my advisor has seen that paper :)

Semiclassical

heh, i know what you mean

oh, did you see the link i gave a bit earlier? it shows a plot of partial sums of the bessel functions $J_1$

Antonio Vargas

yeah, that Gibbsish behavior is interesting indeed

Semiclassical

yeah

i mean, it needs to show up eventually given that for large $y$ the bessel functions reflect cosine behavior with power-law decay

what's more interesting for me is that it behaves like a linear function near the origin, but evidently has nonanalytic behavior at half-integer $y$

which is actually what i was looking for, so it's quite pleasing

Antonio Vargas

I think that's because it's a "Fourier" (Bessel) series for the sawtooth function

Semiclassical

yeah. though it can't quite be a fourier series since the result isn't periodic

Antonio Vargas

03:06

right

pretty cool indeed

Semiclassical

i'd be curiosu what the fourier transform looks like, hmm

Soham Chowdhury

.

@Paul, stupid cosets question. I'm having trouble understanding a line in a proof.

Paul Plummer

Are you going to tell me the line?

Soham Chowdhury

Yes.

Let $aH = bH$.

Then $a\in bH$.

Hence $a^{-1}b \in H$. How?

You there, @Paul?

Paul Plummer

$a=bh$, so $a^{-1}bh=1$, multiply both sides by $h^{-1}$

You are so impatient

Soham Chowdhury

03:18

Oh, all this while I was doing this and wondering why $h^{-1}$ had to be $\in H$ for some reason.

Paul Plummer

$H$ is a group

Soham Chowdhury

Because it's closed under inverses.

Yeah.

Duh.

What are you doing?

Paul Plummer

Nothing, might start complaining about all the "my post was downvoted" things on meta...

Soham Chowdhury

Post or answer?

user147690

@PaulPlummer Hahaha which in particular?

Soham Chowdhury

03:20

Hey mang, you seem to be very busy nowadays.

Paul Plummer

Just them all, it is getting to crazy, it feels like every other post is one of those, and they basically all have the same answer: "get over it"

user147690

@SohamChowdhury Yes that's finals for you :P

user147690

@PaulPlummer Hahaha too true

Paul Plummer

I asked in the mod chatroom

in Math Mods' Office, 4 hours ago, by Paul Plummer

How helpful are the "complain about downvotes on posts" to moderators, in finding or catching people who are "voting incorrectly"? It seems like most of the time, when something actually reaches some level of voting abuse you guys are already on top of it or the system corrects it. The other times the answer is just to deal with it, people can vote how they want, and the poster is make unwarranted assumptions about the downvotes.

So far have not gotten a response

user147690

They do usually take awhile to respond

user147690

03:23

Maybe a certain tomato will respond :P

Paul Plummer

Oh I am not complaining about it, figured I would just mention they have not responded so that you don't ask what did they say

user147690

@PaulPlummer Fair enough, didn't make any assumption. When do you start uni?

Remember me

@PaulPlummer I have slowly started to understand your post ✉ in a better way..

Paul Plummer

August 1 is when I need to be at uni

@Rememberme That is great, I will probably be posting a modified version on the MSE blog

Remember me

@AlexClark I needed to ask you something.....

Soham Chowdhury

03:25

Are you a grad student, @Paul?

Paul Plummer

No @SohamChowdhury

not yet anyways @SohamChowdhury

user147690

@Rememberme Sure, I may not be helpful, just taking a break atm

Soham Chowdhury

Today's. I think I'll write a bot.

user147690

I saw that yesterday you liar!

Soham Chowdhury

For the benefit of MSE chatroom mankind.

@AlexClark I didn't :(

user147690

03:27

@SohamChowdhury :P

Soham Chowdhury

New Horizons was awesome.

Especially the image text.

Paul Plummer

Haha. I am actually brewing a small batch, will be bottling it in less than a week

Soham Chowdhury

Alone?

user147690

@PaulPlummer Oh sweet man, what do you call your brew?

Paul Plummer

I don't have a name

@SohamChowdhury I brewed it with a friend

user147690

03:30

@PaulPlummer I have some mates who are brewing and they have named it and bought labels hahaha

user147690

Can't remember the name but it was a fusion of their names with something else

Paul Plummer

Nice, I might get more into it, but it is just a little 1gallon batch, following some instructions. Does not really deserve a name

Remember me

@AlexClark I was saying what Jo can a person do after completing grad...??

Sorry I got disconnected

Paul Plummer

@Rememberme Flip burgers

2

user147690

@Rememberme After grad? Well as I always say, Australia has unlimited opportunities, atleast right now

user147690

03:31

Your country is completely different so I can't give advice sorry

Remember me

Unlimited opportunity as in?@AlexClark

Paul Plummer

@AlexClark Since I will be moving I found a "recipe" of a local brew, that I really like, so I might give it a shot some time. A taste of home Haha

user147690

@Rememberme As in there are so few math students that our country is in a mathematician crisis

user147690

And we have a 97% hiring rate within 6 months for any degree in math

Soham Chowdhury

. . . wow.

Remember me

03:34

.....wow...

user147690

And that is including terrible unis and its easy to get in great unis here

user147690

@PaulPlummer Nice xD. Maybe you can base your name off of that haha

Remember me

Even we kinda have math drought...

user147690

There are 15 students in my 3rd year algebra course lmao and like 10 4th years total

user147690

TA's get paid as much as full professors do in America from the numbers people here have told me

Soham Chowdhury

03:36

Alex pls ;-;

user147690

Strayaaaa

Soham Chowdhury

Is that arrow on the left supposed to be dotted?

Paul Plummer

If you want it to be dotted

user147690

What does dotted mean?

Soham Chowdhury

Dashed, sorry.

user147690

03:37

What does dashed mean?

Soham Chowdhury

- - - - - - >

as opposed to

user147690

... I mean what does it mathematically mean

Soham Chowdhury

-------------->

user147690

lol

Paul Plummer

I don't think he uses that convention

Soham Chowdhury

03:37

haha

Paul Plummer

He uses the $\exists !$

Soham Chowdhury

It means that you don't know that that arrow exists yet.

Paul Plummer

for there exists a unique morphism

Soham Chowdhury

(I thought)

Paul Plummer

No that is not what it means

At least I have never seen that

Normally it means there is a unique morphism.

Soham Chowdhury

03:40

$\exists!$ means that, I know.

Paul Plummer

That is what the dashed arrow means

But I don't think he uses that convention in the book

Soham Chowdhury

I thought the dashed arrow was like "there is an induced (unique) morphism". Anyway.

Thanks

Paul Plummer

Looks like someone mass upvoted me...

Soham Chowdhury

What?

Paul Plummer

Just looked at my account and I had 5 upvotes

between various answers

Soham Chowdhury

03:45

Damn, you're lucky

Paul Plummer

Oh getting more upvotes, now on some questions

Soham Chowdhury

What courses are you taking right now?

Paul Plummer

Me?

I am not in school

Soham Chowdhury

Yes

Summer break?

Paul Plummer

No just didnt go to school

Soham Chowdhury

03:46

As in, your undergrad is over?

Paul Plummer

Another upvote on a question

I wonder who it is and if they will be taken away

Yes my undergrad is over

Soham Chowdhury

What did you take besides the usual?

Are you allowed to take grad courses?

user147690

Paul is very covert. I have always found it suspicious

Paul Plummer

Haha @AlexClark

user147690

You are though :P.

Paul Plummer

03:49

Am I though?

user147690

Very

Antonio Vargas

@Semiclassical Well I tried but I think I failed. It's just a guess but it looks like the analogue of the Fourier (sine) transform should be something like the result of taking the inverse Fourier transform of $\int_0^\infty f(u) J_1(2\pi u x)\,du$. So in formulas, $$\mathcal{J}_1[f](t) = \textrm{const} \cdot \mathcal{F}\left[ \int_0^\infty f(u) J_1(2\pi u x)\,du \right](t).$$

Soham Chowdhury

bananas beats everyone at this.

user147690

@SohamChowdhury At what?

Soham Chowdhury

Covertness

user147690

03:50

@SohamChowdhury Damn, I once would have won that!

Semiclassical

hmm

i haven't come up with anything illuminating myself as yet

Paul Plummer

I would have won that, but I gave up and decided to use my real name

Antonio Vargas

Typos fixed

Semiclassical

kk

user147690

@PaulPlummer Hey me too.

Semiclassical

03:51

i've managed to not say my name explicitly on here, though one could work it ought from certain comments

Paul Plummer

@SohamChowdhury I could take graduate classes, it was a pretty small school, so no phd program in math (there was a masters), I took a bunch of them. My "unusual" classes were probably a couple independent studies, and a special topics course I took

Soham Chowdhury

What's an independent study?

Paul Plummer

Wow I got 75 rep within a couple minutes.

Soham Chowdhury

And what was in that special topics course?

Paul Plummer

@SohamChowdhury It is basically a reading course, where you set up with a professor something you want to cover as a course, instead of a listed course

Soham Chowdhury

03:54

Paul right now

Paul Plummer

Haha

@SohamChowdhury The special topics class was called "Selection principles in topology"

Nic Brody

Do any of you people know how to use a computer?

Soham Chowdhury

Nope

Paul Plummer

My first two blog posts covered some ideas that were in that class, (well the first post covered ideas in the class the second post was related to a project in the class)

@NicBrody I am typing inside a hippo,

iluso

@SohamChowdhury Maybe

Soham Chowdhury

03:56

Our villages are wired for internet, and we talk into mysterious boxes which talk back

Nic Brody

Bummer

Soham Chowdhury

;-;

Nic Brody

Haha. My real question concerns a computer thing I am interested in programming

But I am wondering which language to use

Paul Plummer

Probably the wrong place to ask, but I say binary

Soham Chowdhury

ASM for president

Semiclassical

03:57

manipulate zeros and ones by hand

Remember me

Ya... Last question left after that my 1 chapter in simmons will be finished

Nic Brody

Hmm that sounds a bit time consuming.

Soham Chowdhury

@Rememberme Good :)

@NicBrody Understatement of the day

iluso

Get an idea of what you want to do first. Then Stackoverflow will be your friend ;)

Semiclassical

more seriously, that's not a question that can really be answered without knowing what you're trying to program

Nic Brody

03:58

Yeah I'm about to describe it haha

I understand that. Just wanted to see if anybody here felt they could have any input

So I want to make something that solves the word problem in Coxeter groups

Paul Plummer

Probably GAP

Soham Chowdhury

Yeah, seconded

Paul Plummer

It might be built in

Remember me

And that is the question which I am stuck at :(

iluso

@PaulPlummer webusers.imj-prg.fr/~jean.michel/gap3/htm/chap070.htm It is

Soham Chowdhury

03:59

@Paul, look at this: link

Nic Brody

Yeah but I want it to be fun: I want the user to be presented with a blank screen where they can click to create circles and then draw edges between the circles to which they can assign numbers. After they build the graph, they can click the vertices in a sequence, and then I run a calculation in the background and return numbers on each circle, and then present the reduced word

Paul Plummer

GAP

Nic Brody

Haha

Alright then.

I suppose if it's already implemented in GAP it's sort of a waste

Soham Chowdhury

God's number is the lower bound on a sufficient number of cube moves that can solve every position, btw.

Paul Plummer

That would be cool, it could be a good programming exercise

if that is what you wanted

Remember me

04:02

@Soham you were going to meet samik .. Why are you going to meet him?
you have got some mathematical breakthrough ??

Soham Chowdhury

@Rememberme breakthrough?

Remember me

I think spelling is wroung

Soham Chowdhury

no, I'd like to find someone to guide me.

a mentor.

Remember me

Okay even I would like someone do that

Paul Plummer

I know that you can do interactive stuff in geogebra, not sure how powerful it is though, or if you could even do something like that @NicBrody

Remember me

04:02

To me

iluso

@PaulPlummer I doubt it

Soham Chowdhury

@Rememberme email the IISc people. I did the same with a local uni.

Paul Plummer

Who are you going to meet @SohamChowdhury

Nic Brody

Yeah I considered geogebra briefly then decided it wasn't what I wanted

Soham Chowdhury

@PaulPlummer A prof

Paul Plummer

04:03

Yah probably not

Remember me

Saying what??@Soham

Nic Brody

I mean I'm sure it would be possible to do what I wanted, but it doesn't appear to be the right call

Hmm

Paul Plummer

@Rememberme Were you the mass upvoter

Remember me

Wha???@PaulP

Mike Miller

@PaulPlummer: Do I read right you're going to homebrew?

Nic Brody

04:04

I'll maybe just talk to a computer sciency person when I get a chance. Thanks people!

Paul Plummer

@MikeMiller I am doing one, yes

iluso

@NicBrody I am

Soham Chowdhury

@Rememberme What do you think? "I need ma$\theta$ halp, pls gib kthxbai"

Paul Plummer

@Rememberme Someone upvoted a lot of my stuff, you seem like the sort of person who would do that sort of thing, so I asked

Mike Miller

Nice @PaulPlummer

Remember me

04:05

Eesh..... No why do you think it is me??

Nic Brody

Ah - @iluso do you have a language you recommend I build this thing in?

Remember me

Yes I am really fascinated after reading your blog

Soham Chowdhury

@NicBrody Haskell!

Paul Plummer

@Rememberme What you would not upvote my stuff, is my stuff really so horrible :D

Nic Brody

Is that a serious suggestion, @SohamChowdhury?

Remember me

04:06

But why start upvoting suddenly and no reason :p

Soham Chowdhury

Eesh is an Indian expression of digust, btw, @Paul.

@NicBrody Yes.

Nic Brody

Okay cool

To my knowledge Haskell is a pretty atypical language

Soham Chowdhury

It's gorgeous. <3

Remember me

Hey @soham I will come back after few minutes with a doubt

Nic Brody

I'm aware that it has to do with category theory

Remember me

04:07

Get ready

iluso

@NicBrody Really great ! But not really for a GUI

Paul Plummer

@SohamChowdhury I think it might be a little more common than just India (like ta ta or whatever it was)

Nic Brody

Hmm yeah that's what worried me

Soham Chowdhury

@NicBrody Not really, no.

Google "do I need to know category theory to learn Haskell"

Nic Brody

I'd be interested in learning Haskell at some point, but for the moment I just want something that'll get the job done

iluso

04:08

@NicBrody Not really... But there should be some ways to interface GAP with some other language to build the gui

Soham Chowdhury

Paul Plummer

Oh I got another upvote, damn you @Rememberme, and you didn't even upvote a good one

Soham Chowdhury

Mmm

@PaulPlummer Look how things change.

@PaulPlummer Yeah, I guess. Wouldn't know.

Nic Brody

Oh I'm by no means intimidated by categories

When I said it worried me I meant cause I was worried Haskell wouldn't be good for a GUI

Okay thanks for all your help, folks! I'm gonna mess around with gap a bit more

Paul Plummer

@MikeMiller Yah, I think I will be getting more into after I move. I will probably try brewing a few locals (from where I am at now), when I move)

iluso

04:12

Even better
primes = 2 : 3 : 5 : primes'
where
isPrime (p:ps) n = p*p > n || n `rem` p /= 0 && isPrime ps n
primes' = 7 : filter (isPrime primes') (scanl (+) 11 $ cycle [2,4,2,4,6,2,6,4])

Mike Miller

It seems like an expensive hobby

Paul Plummer

I am not sure, I don't have a full 5gallon brew set up, I would have to look around, but it seems like ingrediants cost around $30 for a 5gallon batch, so if you like "craft" beer it probably gets even after a few batches, if you are going to drink anyways (although I guess you could just end up drinking a lot more so maybe not :D).

Mike Miller

The ingredients seem reasonable but the setup seems like it's the thing that costs money

Paul Plummer

04:31

Looking around it looks like it would be $100 plus maybe $30-50 for a 5 gallon brew pot, not sure if there is something to look out for that maybe is not included or will be subpar and would have to be replaced quickly. So some research is needed.

maybe there is more to it if I got really serious

Mike Miller

Not so bad

Paul Plummer

So a one time thing like that does not seem that bad, I have spent about as much on a book (like that Geometry of defining relations in groups I mentioned yesterday), cheaper than paintball

Soham Chowdhury

04:49

Assuming the first iso theorem, is $(G\times G)/H = \{(a - b, b-a) : a, b \in G\}$, where $H = \{(g,g) : g \in G\}$?

(i.e. @anon's problem)

Mike Miller

I see

Soham Chowdhury

Is that correct?

The homomorphism $(a,b)\mapsto(a-b,b-a)$ has kernel $H$, I think.

I should just prove the isomorphism theorem, actually.

anon

05:22

@SohamChowdhury it is correct, but there's a simpler way to think of (GxG)/H

In fact, Gx{e} is a transversal for H! (set of coset representatives)

Soham Chowdhury

Yes, I thought exactly that when you first showed me that, but I didn't know the words.

anon

IOW, just consider the homo GxG->G given by (a,b)->a-b and apply 1st iso.

Soham Chowdhury

The $\ker$ of that is $H$. The image is . . . $\{a - b : a, b \in G\}$?

anon

which is all of G

Soham Chowdhury

indeed.

Remember me

05:47

@PaulPlummer I am not upvoting

Hey @Soham

Soham Chowdhury

Hey.

Remember me

So what should I ask in an email?

I am confused@Soham

Soham Chowdhury

Uh, introduce yourself, tell them what you're studying, ask for help?

I think I have a proof.

Let $U \subset X$ be open.

Remember me

Hmm where is this theorem

Alec Teal

This is just equivalence of metrics

Remember me

05:50

Ya... Yes equivalence of metrics

Alec Teal

maths.kisogo.com/index.php?title=Norm#Equivalence_of_norms same as this

Soham Chowdhury

Then $\forall x \in U, \exists\epsilon > 0 . B^{d_1}_{\epsilon}(x) \subset U \implies B^{d_2}_{\delta}(x) \subset U$,
and $\forall x \in U, \exists\epsilon > 0 . B^{d_2}_{\epsilon}(x) \subset U \implies B^{d_1}_{\delta}(x) \subset U$, so they define the same open sets.

Alec Teal

This is stuff you really ought to start doing yourself. from the sounds of it you ought not be reading a graduate level textbook if you get .... stumped at every page turn. Undergrad textbooks hand-hold at first because of exactly this. Do not use us to hold your hand.

Have you showed that every open set is the union of open balls?

Soham Chowdhury

This is a reader exercise I've proved.

Alec Teal

Yeah in a graduate textbook

Remember me

05:53

@Soham I presume that you have done continuity

Alec Teal

Have you Soham?

Soham Chowdhury

Yes.

Is that correct?

Remember me

Try proving this:
The image of every Cauchy sequence is a Cauchy sequence in an uniform continuous mapping

Soham Chowdhury

@AlecTeal Is that "proof" correct?

Alec Teal

BTW @Rememberme (I don't feel I have to say this) - I totally approve of giving different but related questions here in chat, but verifying proofs and all but one-line-answer questions deserve a question. We must not encourage hand-holding from someone trying to run before they can walk (I cannot believe I'm saying this! Usually I'm that guy!)

Remember me

05:56

Yes you are right ....

So @Soham Draw a picture and then question yourself .... Why are you right?

Alec Teal

You have a library, yes? I really REALLY recommend "Fundamentals of Abstract Algebra" by some bloke called "McCoy" - it's VERY hand-holdy, so lots of try-it-yourself then compare, coupled with Undergraduate Algebra by Serge Lang (which is less hand-holdy but 'deeper')

Graduate books rush through this stuff because they assume you've encountered it.

Remember me

I prefer Simmons @AlecTeal

Really hand holdy stuff

Soham Chowdhury

@AlecTeal Aluffi doesn't even assume you know how to multiply matrices. Anyway, I should go now.

Alec Teal

No idea who that is. Anyway those 2 books together are a potent combination to doing maths. @Rememberme I'm glad you see what I mean by "hand-holdy"

Soham Chowdhury

The book I'm using for Algebra.

Alec Teal

05:59

@SohamChowdhury he assumes you've ENCOUNTERED IT - he's giving definitions and a quick tour for reference.

Soham Chowdhury

Nope. Wait.

Alec Teal

Making sure you see what he sees, if you will.

Remember me

Well I can say one thing.....
Very weird way to write a metric "dist"

Transcript for

$Mathematics$ Mathematics