Mathematics - 2013-09-01 (page 2 of 2)

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7:01 PM

@robjohn I found a very cute solution to that question.

@robjohn I'll show you in a few minutes if you're around.

@Chris'ssis which question? The one for which I gave the asymptotic expansion?

@robjohn yeah.

user87637

I just watched the horror movie Crush.

@Jasper never seen it. What's it about?

user87637

@robjohn It's about a university boy who goes to take care of a house for a job. He meets a girl in it whom he makes love with, ending up losing his girlfriend, flunking his course, and losing his taekwando competition. This girl turns out to be a ghost who died in the house and now wants him dead too to spend eternity with her in the other world. The movie ends with him being flung off till dead by the ghost.

7:11 PM

@Jasper So a happy ending.

user87637

@robjohn Ouch!

flung off?...

user87637

@DanielRust Well, I mean off the second floor onto the first.

user87637

I also watched Elysium starring Matt Damon in the cinema yesterday. It's very good.

@robjohn I just sent it to you.

7:25 PM

@Chris'ssis got it

@robjohn K :-)

@Chris'ssis That works. Pretty much what I was doing. I just took it a few more terms.

@robjohn yeah.

@robjohn I like so much the problems that apparently scare all people, but they can be done by some elementary tools!

:-)

How to prove
$$\sum^{n-1}_{k=0}\exp\left[i\frac{2\pi k}n\right]=0, \forall n \ge 2$$

use $\mathfrak{I}(\exp(i2\pi kn^{-1}))+\mathfrak{I}(\exp(i2\pi (n-k)kn^{-1}))=0$

oops i mean the imaginary part

7:39 PM

press up arrow to change

it's essentially a geometric argument

In other words
$$\sin(2\pi kn^{-1})+\sin(2\pi(n-k)kn^{-1})=0$$

sure

@DanielRust: I found a relevant link on mathoverflow regarding your pizza problem. Should I go ahead and post it or would it spoil your intentions with the question?

@DanielR I don't see why not. If someone doesn't want the solution spoiled for them they can always not look.

7:45 PM

Alright!

@DanielR Seeing 'Daniel R' is a bit disconcerting given my name :P

You should see it as a hommage.

haha

is it?

@DanielRust Could you explain in more detail how that holds true?

no, i'm not that much of a stalker, it's my real name.

or part of it at least

7:46 PM

good to know :D

@Alizter Draw a picture of the points on the unit circle, you'll see that they are reflections of each other about the real axis of $\mathbb{C}$.

@Alizter Do you know about roots of unity?

@PeterTamaroff Of course.

I am trying to prove it though

Do you know that $$1+z+z^2+\cdots+z^{n-1}=\frac{z^{n}-1}{z-1}$$ when $z\neq 1$?

@PeterTamaroff Did not know that

@Alizter Heh, now you do.

7:50 PM

@danielR that's a nice related MO question. Interesting to see that it's published as an open problem too.

@PeterTamaroff I got into this from here.

@DanielRust: yeah, it's a cool problem!

In particular if $z$ is a primitive root of unity, you have that $$\sum_{k=0}^{n-1}z^k=0$$

Let $$z=\exp\left(\frac{2\pi i}{n}\right)$$

I'd recommend trying to find the family of solutions (to my problem) with pieces all having the same orientation.

@PeterTamaroff Doesn't that only hold true for values of $n \ge 2$

7:54 PM

@Alizter Well, yes.

And I should have written primitive $n$-th root of unity.

@PeterTamaroff Well basically I need to prove what you just said

@Alizter What?

@PeterTamaroff How do you prove the statement you made? "In particular if $z$..."

@Alizter Well, by definition a primitive $n$-th root of unity is a complex number such that $z^k\neq 1$ for $k=1,2,\ldots,n-1$ but $z^n=1$. In particular, $z\neq 1$.

Hmm... @PeterTamaroff You did see the question I showed you right?

8:01 PM

We know that $$z^n-1=0$$. Since $z-1\neq 0$, we may divide by that to obtain what we want.

@Alizter Nope.

@PeterTamaroff Thank you so much my brain just clicked. I see how it works now.

@PeterTamaroff And CLICK HERE

@Alizter Yep, seen that.

@anon

user87637

8:17 PM

I answered 5 lhf today, yay!

hi

@PeterTamaroff, I just need to know weather the proof is correct.

what proof ?? @Crypto

user87637

@Crypto You misspelled whether.

sorry its raining outside, n that just distracted me haha

@what'sup..math.stackexchange.com/questions/481595/…

8:44 PM

@TedShifrin Heya.

Heya, @Peter.

@TedShifrin How is it going?

Pretty well. Still not 100%, but better. You?

@TedShifrin Better, I guess. Runny nose, though, sneezing quite often =P

@PeterTamaroff What is the current probability that you will sneeze?

8:50 PM

@Alizter I surely will sneeze within the next hour.

My chest congestion is better, but still sinus-y... Guess this crap is international :(

@TedShifrin Heh!

I blame it on my students. You blame it on your colleagues :)

@TedShifrin So you're giving classes now? How are they going?

Any interesting student?

Oh, we've finished three weeks. Grad diff geo is getting interestinger. Finished chap 1 in the multivariable course. Some excellent students there, but we'll see on the first test in a week.

8:55 PM

@TedShifrin What year is it?

Huh?

@TedShifrin What year are the students in?

Ohhh ... First, second, third.

Is there a big theorem you plan to finish on?

@TedShifrin Oh... don't know if that can happen at the UBA.

8:59 PM

A few fourth, one math ed grad student.

@Daniel: You mean in the grad diff geo?

yeah

Plan to do general Gauss Bonnet and some Grassmannians and Chern classes ...

We'll see if I get through standard stuff + symmetric spaces and all that.

@TedShifrin sounds good

Doing everything with diff forms and moving frames ... Good Chern student tht I am :)

are they expected to know calculus on manifolds?

9:03 PM

Yes, and a first grad course in manifolds. I have a few that don't have that, but they're going to have to fill in pieces.

Can imagine it's a steep slope to climb for them.

It's basically up to them to get what they want out of our more advanced courses. They get a B for

Showing up. Earn higher by doing homework, which I'm happy to help with.

On the first assignment, 6 out of 9 turned in problems. No one did them all.

@TedShifrin Look at Mariano's answer

I kinda wish we had organised grad classes in our department. The best we have is a network of online courses.

LOL. Sort of like Thurston's 37th defn of the derivative, on which Zev posted and I took an hour to reply.

@Daniel: I'm glad I'm old and retiring any day. I am a foe of online courses.

9:09 PM

Don't get me wrong, I can see the advantage of things like coursera for those that can't afford, or don't have the time for, an undergrad degree. However, it just doesn't work for indepth, high level courses which need a lot of interaction between not just student and teacher, but also students and other students.

@Peter et al: Check this out :) math.stackexchange.com/questions/449740/…

haha, one of my starred questions

We all like a good sledgehammer proof/definition

@TedShifrin I don't eve...

I agree, @Daniel. And even lower. All over GA high school kids are doing distance kearning CALC II and III through GA Tech, all getting As. The ones I teach at UGA admit they learned basically nothing.

This drives me NUTS!

Wow, distance learning for highschool shouldn't be a thing....

9:13 PM

I still think Thurston was kidding around, but it's thought-provoking. Sadly, he died, so I can't ask him.

Yes, @Daniel: At every level I teach there's as much interaction as I can summon from the students. Generally undergrads are better than grads.

I think the times I learnt the most as an undergrad was discussing take-home problems with other students

Shows the importance of thoughtful homework. Not all profs bother to work that hard.

@Peter: Are you being apopleptic? :)

Oops ... Apoplectic?

@TedShifrin Heh, no, I was simply transmitting the fact that I have no idea what all that is.

If you follow some geometric paths, you will eventually. Zev's very smart, in about the toughest grad program in the US, and he didn't get it, either. :)

@TedShifrin Heh, yes, Zev looks like a very tough guy.

9:21 PM

It took me an hour to sort it all out and write it up. It's in the spirit of Mariano's answer ... hence my alluding to it.

This guy is also a fellow in some redonkolous math programme.

I met Zhen Lin a couple of times

pretty cool dude

but yeah, super smart

Don't know him ...

@robjohn I just found that sos440 gave the same answer here math.stackexchange.com/questions/151441/…

@DanielRust What is he part of? It was some selective group of some kind

9:23 PM

Erm, well he's a postgrad at Cambridge

dunno about any selective group

I've never been a fan of arcane machinery in math fir machinery's sake. Sadly, a lot of math goes that way.

@DanielRust Guess that is quite enough!

@TedShifrin Come again?

@Chris'ssis It is the one the comes first to my mind. I am amazed that the others came up with other solutions.

On my iPad I can't edit errors. GRR. Fir = for

@robjohn actually I have at hand more elementary solutions

@robjohn one is related to a summation that I computed in one of my previous posts

9:26 PM

I'm totally horrid at combinatorial/discrete math ...

@robjohn The sum of the inverse of the binomial coefficients is equal to $\displaystyle \frac{n+1}{2^{n+1}} \sum_{k=1}^{n+1} \frac{2^k}{k}$

@robjohn I apply Stolz theorem and I'm done in one line.

@Chris'ssis You seem to have a fetish for "one liners."

@TedShifrin Oh. I have been given a book on "Enumerative Combinatorics".

Plannig or reading it on summer, maybe.

@PeterTamaroff there is a fetish for "mentally". :-)

user87637

@PeterTamaroff Given? By who?

user87637

Hey @ethan how are you?

9:30 PM

sick/tired

got alot of work to do

@Ethan Join the club! =D

user87637

I am sick and tired of being sick and tired.

@Chris'ssis that the sum of the inverse of the binomials is that is not obvious.

@Jasper A professor of mine was told I would enjoy them.

think I recall similar identities being used in the proof of the irrationality of $\zeta(3)$

user87637

9:31 PM

I have gotten 900 on math now, gonna retire at 1k, lol.

@Chris'ssis

user87637

@Ethan identities*

@robjohn in many papers I read the authors refer to it as "well-known identity". Anyway I agree with your point.

@Jasper Shut it.

user87637

@PeterTamaroff I like Cameron's Combinatorics.

user87637

9:33 PM

@Ethan He is talking about my retirement, lol.

You know Bruce Reznick, @Peter?

@TedShifrin No, not at all.

Didn't I tell you?

user87637

@TedShifrin I have seen Bruce Willis in movies, LOL.

Good for you, @Jasper. I hate him :)

hes one of those actors, where if hes in a movie, you know there will be explosions

user87637

9:34 PM

Anyway @ted I have no idea about the contents of your abstract algebra book. Do you define rings with or without 1?

@TedShifrin I showed the Bernoulli thingy to a prof of mine, he liked it and talked to a colleague, who sent it to B.R., to see if it was original or not. He said he didn't know, but that I should check JSTOR and that I might enjoy those books. I wouldn't want to presume on anything =)

@Jasper: with.

Very cool. I should ask our combinatorics expert to look at it.

@TedShifrin Oh, that'd be awesome, Ted.

Did I send you the latest version?

what is it peter

user87637

@TedShifrin Ah OK. I saw a copy of your linear algebra book online though. =) I liked it but would have preferred it to be more fast-moving. Also, I thought it would be good to include solving the nth order non-homogeneous differential equation in it.

9:36 PM

I dunno.

@TedShifrin Let me check.

Can someone take a look at this? I'm trying to find solutions of this system of equations with Wolfram Alpha, but it's just returning the second equation. Have I done something wrong?

er

crap

sigh

@Jasper: It's already too hard for most American students. In my multivariable math book, the lin alg is faster and more from the linear map viewpoint.

user87637

@TedShifrin Ah OK. The two books are among the few that pay attention to geometry though, so they are special.

tinyurl.com/l34do8v

There's the equation.

9:38 PM

Most faculty and students hate geometry :(

user87637

I used to until recently.

user87637

Artin's book is another algebra book that blends in the geometry.

I am proud of the bit of projective geometry and computer graphics, though.

Yes, Artin's book is great. I learned algebra from him, so it influenced me for sure.

user87637

I recently found Spindler's two volumes, they are an unheard gem.

Wow, never heard of him/her ...

user87637

9:41 PM

He treats geometry and ODE in them too, but I did not like the rings defined without 1. =(

Hi

Can someone look at this ?? math.stackexchange.com/questions/481643/…

user87637

Karl Spindler: Abstract Algebra.

user87637

Some German retired prof.

I wish I'd read more of the classic textbooks during my undergrad. Most of my lecturers had their own written notes for courses.

I guess there's less publisher kickbacks in the UK :P

user87637

Most of my lecturers were bad. I hope to study on my own now before applying to grad school. Hopefully I get a place.

9:45 PM

Some of us try to be good :)

user87637

@TedShifrin Yes, I think you must be good, judging from your books.

user87637

Anyone who cares enough to write books is probably a good teacher.

@Jasper: I found the reference ... Alg w/applications — I don't know it.

user87637

@TedShifrin Yeah, I only recently came across it. Contains tons of applications not seen elsewhere.

No, that does not follow. There are lots of crummy books.

9:46 PM

user image

Geometry :)

user87637

There are many nice geometry books in the AMS Student Mathematical Library.

Thanks @Bitrex.

user87637

Elementary Geometry by Agricola, Elementary Algebraic Geometry by Hulek and Differential Geometry by Kuhnel. All translated from German.

@TedShifrin I thought of one way to do it, but it requires solving a system of two nonlinear equations. I think there's a better way?

Someone posted a cool geometry problem last night. Given a diameter of a circle and an interior point of the circle, construct the perpendicular from that point to the diameter using straightedge only.

9:49 PM

Using $r = \frac{y^2}{8x} + \frac{x}{2}$ and $x = r - \sqrt{r^2 - (\frac{y}{2})^2}$

@Bitrex, I can't tell what the problem is

user87637

@TedShifrin The problem with the problems on this site is that sometimes they are wrong and we end up working for nothing. =)

Sometimes ... Not this time :)

@TedShifrin Mail sent.

No work is for nothing. You learn regardless.

9:51 PM

@TedShifrin Given W, X, and H on the diagram, calculate R and D

@Peter, OK. I'll ask my colleague if he'd be kind enough to look quickly.

@robjohn the version $$\lim_{n\to\infty} \displaystyle \frac{n}{2^{n}} \sum_{k=1}^{n} \frac{2^k}{k}$$ appears in the "The red book of mathematical problems". So, one way to compute it elementarily is to bring all in the area of the reciprocals of binomial coefficients. :-)

@Bitrex, I'll draw it and ponder after dinner.

@robjohn the solution provided in that book is pretty cute.

@TedShifrin Thnx :)

user87637

9:52 PM

@ted Will you be writing any more books?

guys dont mean to be rude but could someone please look at my question and let me know if you think I am doing it right?

0

Q: compute angle of rotation between two orthographic projections

I Have the orthographic projection of a unit cube with one of its vertex at origin as shown above. I have the x,y (no z) co ordinates of the projections. I would like to compute the angle of rotation of the plane to get the second orthographic projection from the first one (maybe euler angles...

geometry rotations stereographic-projections

@robjohn or one can start from the sum of the reciprocals of binomial coefficients, find a recurrence relation between these sums in an elementary way, and then: 1) take the limit in the recurrence relation and establish the limit when $n \to \infty$ 2) find the identity above 3) done.

@TedShifrin Am I thinking straight here? @robjohn ?

@Chris'ssis I am writing a proof now

@robjohn great! :-)

9:59 PM

How is my answer

@Alizter Nice! =)

@Jasper: No. Free undergrad diff geo text on my website, though.

What's the symbol for an upside down Delta, $\Delta$ Ive forgotten, but I remember it had a weird name, nabola?

@Alec: nabla.

@TedShifrin May I have a link?

Thanks. Is that letter from any particular alphabet, does the name mean anything?

10:03 PM

Go to my profile for the website, Alec.

Great question, and I don't know the etymology.

math.uga.edu/~shifrin I found this several days ago, I discovered some assignments, I must say I was looking for a PDF though,...

I had it open "MATH 4250/6250, Differential Geometry, meets TR 9:30-10:45

Text: Notes by T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, available from Baxter Street Bookstore, 360 Baxter St., for approximately $13 or in .pdf form." Sorry @TedShifrin, I feel daft now.

Wow 43 featured questions

For $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and given $n\in\mathbb{R}$ and $n > 1$ $\frac{\partial{f}}{\partial{x_i}}=($a vector$)$ is notationally sound? (I'm having those pre-exam jitters)

Yes of course it is.

This guy makes the highest score on MSE look puny.

$$
\begin{align}
\frac1{\binom{n}{k\vphantom{+1}}}&=\frac{n-k}{n}\frac1{\binom{n-1}{k}}\\
\frac1{\binom{n}{k+1}}&=\frac{k+1}{n}\frac1{\binom{n-1}{k}}\\
\sum_{k=0}^{n-1}\frac1{\binom{n}{k\vphantom{+1}}}+\frac1{\binom{n}{k+1}}
&=\frac{n+1}{n}\sum_{k=0}^{n-1}\frac1{\binom{n-1}{k}}\\
2\sum_{k=0}^n\frac1{\binom{n}{k\vphantom{+1}}}
&=2+\frac{n+1}{n}\sum_{k=0}^{n-1}\frac1{\binom{n-1}{k}}\\
\frac{2^n}{n+1}\sum_{k=0}^n\frac1{\binom{n}{k\vphantom{+1}}}
&=\frac{2^n}{n+1}+\frac{2^{n-1}}{n}\sum_{k=0}^{n-1}\frac1{\binom{n-1}{k}}\\

10:15 PM

@Alec, I assume you found the .pdf?

I did yes, I felt bad asking @TedShifrin because you're already nice enough to freely distribute it, asking for a link is just ungrateful. Turns out I'd already found it, it was in an adjacent tab, I am very sorry.

@Alizter HA! 4 years? Just a matter of time. =P

hello guys @robjohn are you trying to find $$ \sum_{k=0}^n \frac{1}{\dbinom{n}{k}} $$ or what ??

@PeterTamaroff As far as I can tell, on his worst days he gets 200+ rep

@what'sup found it, I'd say... hmm.

10:18 PM

@robjohn ok :-)

@robjohn very nice!

@robjohn did you prove it before this way?

@Chris'ssis Nope. I didn't even know the formula before you mentioned it above.

@Alizter StackOverflow is a lot less exclusive than the math one, as in any old idiot submits things there (A LOT of PHP proves this) here the mathjax scares those who can't count away.

@robjohn K

@alec: No problem. :)

10:22 PM

On the SO meta they say that John Skeet can divide by zero.

@AlecTeal count away?

@Alizter without an error?

Of course he can't escape an overflow :-D

@robjohn Popular SO folklore says that he is pretty much Jesus.

Jon Skeet coded his last project entirely in Microsoft Paint, just for the challenge.

@robjohn away as in absence.

There's also no formal requirement, like to be here you're either very determined or in formal education, with programming that unwritten requirement dissolves.

@AlecTeal That would be MathOverflow, not here, I'd say.

good bye i'm going to see some questions :-)

$$ \huge{ \text{good \ bye}} $$

10:26 PM

So was math overflow an external site before?

@robjohn clearly you've never been to Yahoo Answers.

@Alizter Yes, it just joined SE recently

When they say professionals is it just really bored professors?

@AlecTeal I didn't say we were bottom feeders, just that MathOverflow has more stringent prerequisites.

Where are we?

math.stack?

10:40 PM

user image

10:52 PM

$g:\mathbb{R}^2\rightarrow\mathbb{R}$ and $f:\mathbb{R}^2\rightarrow\mathbb{R}$ How does one interpret $D(f(g))$ - using the chain rule, it involves a 2x2 matrix and a vector of length 2, so multiplication is obvious, but avoids knowing why they wrote it that way in the first place.

user87637

@anon Are you D.F. ? =)

is that another user on mse?

for a second I thought you were asking if I was dtf

user87637

@anon Well, if you are you will know what the initials stand for...

user87637

Please tell me your name!!!

user87637

You can email me if you want...

11:03 PM

@Jasper Dude, that is futile.

Derek, Daniel, Dennis, Dean, David, Dylan, Dan, Dimitri, ...

dan and daniel are, like, the same name

@anon Do you know the game Dark Souls?

nope

@anon Oh, drats.

Damn! I have 260 today.

Didn't realize.

This one gave me a nice deal of upvotes

Karl Kronenfeld

damn

11:17 PM

@PeterTamaroff similar to this for which I got 1 :-(

I have another answer using that idea...

Karl Kronenfeld

It even has a picture.

@KarlKronenfeld a couple

@robjohn You probably took too much time to answer. The good deal of upvotes come when the question is fresh.

@KarlKronenfeld Of course, I've capped today, so upvotes won't give rep.

Karl Kronenfeld

@robjohn I didn't upvote it, mainly because I haven't read it. I will read it tomorrow though if you want the vote. :P

11:24 PM

@KarlKronenfeld read it when you have time. It's not good to upvote something you don't understand.

What is commonly and beautifully accepted?

Ha ha @Peter.

@TedShifrin I was like "The fuck? K, nevermind I'll answer."

@TedShifrin Just realized I'll have to change by agenda on mondays.

I'm starting a course on sequences and series tomorrow and it clashes with the theory class of Analysis II.

~~test~~

Ah, it works in chat, but not comments.

Karl Kronenfeld

@robjohn I tried to scratch out text in a comment before too. We can't use html in comments, right?

11:35 PM

@KarlKronenfeld right

@TedShifrin What does 3.93 GPA mean?

@PeterTamaroff almost straight As

Clashes, @Peter? Don't you know tat material with all your analysis background?

A=4, B=3, C=2, D=1 and average

If you reach the upvote cap does the rep overflow onto the next day?

11:36 PM

Nope.

So if you got 4000 upvotes in one day it would mostly be useless

@TedShifrin I do, but I have to do some optional courses at some point (need 15 pts. total), and I have been reading some interesting stuff about sequences and series and transformations in Polya and Szego, so this will be a nice opportunity to get them all together, get the points for my career and eventually learn new stuff.

OK @Peter.

@TedShifrin In particular, I was mesmerized by Toeplitz's theorem on regular transformations.

Hopefully it'll be a good course. Here it's preparatory for real anlysis.

11:38 PM

@TedShifrin @MarianoSuárez-Alvarez knows Dr. Fava which gives it. He told me it will be =) I trust him.

@TedShifrin Right.

@TedShifrin Well, my colleagues are taking a pre "Advanced Calculus" course, called "Advanced Calculus Workshop" (literal translation). =P

I don't know what you're talking about :) re Toeplitz

Here is a question: What is your favorite badge?

My multivariable math is the class our very best majors take. Most don't go anywhere near it.

@TedShifrin Oh, take an infinite matrix $A=[a_ij]$ of positive entries. Then when you multiply it with an infinite column vector, a sequence, you get a new sequence $$s'_k=\sum a_{ki}s_i$$ Toeplitz says that a matrix of this sort takes convergent sequences to convergent sequences, and with the same limit, iff $A$s columns converge to zero and $A$s rows sum up to $1$.

Karl Kronenfeld

@Alizter By a long shot, I would say the informed badge.

11:43 PM

Ah, infinite version of Markov chains.

@TedShifrin Césaro's theorem is a tiny version of that, say. =)

Ok.

@TedShifrin Dunno what those are.

Models of probability ...

@PeterTamaroff do you know what a state machine is?

11:44 PM

@AlecTeal No.

@KarlKronenfeld does anybody have every single badge (besides tag)?

It models using the probability of transitions between states.

@KarlKronenfeld I just earned a new badge! Thanks

@KevinDriscoll I saw that :P

@TedShifrin Oh, OK. I'm also interested in equidistribution.

11:47 PM

@PeterTamaroff right, imagine the identity matrix with columns 1 and 3 swapped, (I can't do matrices in LaTeX yet), times that by (1,0,0) (suppose the first entry is state A, second B, and third C) the result will be (0,0,1)

@PeterTamaroff now do matrix * (0,0,1) = (1,0,0), bouncing between two states, but (0,1,0) stays in the same state.

That's a simple state machine.

(my vectors ought to be column vectors btw)

@AlecTeal $$\begin{pmatrix} a_{11}&a_{12}& a_{13}\\ a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$$

@Peter: Do you know Körner's wonderful Fourier Analysis book?

nvm he got it

Bad Kevin!

@TedShifrin Yes =)

11:49 PM

Good!

@AlecTeal Hm, so it just changes vectors.

@TedShifrin Why?

@PeterTamaroff just show off instead then :P But anyway, yes that's state machines, now in a Markov chain rather than 1s they're probabilities,

you can then multiply the matrices together to create a chain.

(how do I do a 2x2 matrix, don't put $ around it)

@TedShifrin I can't take out my anger on the Modified Bessel Function that is destroying my work, so I have to talk trash to Peter instead

Drats.

How uncivil of you, @Kevin. @Peter: I was thinking of the stuff he does on equidistribution...

11:52 PM

I said no dollar signs!

Grr

That's somehow worse!

@AlecTeal $ $ $ $ $ muahahaha

Tease.

@AlecTeal OK, the thing goes like this.

You start your matrix with \begin{pmatrix}

The "p" means it has a parenthesis.

Else, it is just the array of numbers, i.e. with \begin{matrix}

$\begin{pmatrix}a&b&c\end{pmatrix}$

You input each row as X & X & X \\

The & separate the coeffs and \\ starts a new row.

11:54 PM

Superb, thanks @PeterTamaroff

I'm outta here. You guys misbehave without me.

@TedShifrin Will check that, Ted.

How is my answer

So consider this Markov System, $\begin{pmatrix}0.1&0.8\\0.9&0.2\end{pmatrix}$ that states that when in state 1, there's a 0.1 chance it'll stay in state 1, and a 0.9 chance it'll change to state 2.

@PeterTamaroff

That's useful.

@AlecTeal Hm, OK.

11:57 PM

It also makes it really easy to model states that don't change, consider $\begin{pmatrix}0.1&1\\0.9&0\end{pmatrix}$ This one always goes from B to A but A might stay as A.

$\begin{pmatrix}0.1&0\\0.9&1\end{pmatrix}$ Once this one enters B it'll stay in B.

@AlecTeal I will try to swallow Landau's proof on Quadratic Reciprocity, so I'll be away for a while. Cheers.

they're worth looking at.

Finite state automata are a lovely extension, as are state machines for artificial inteligence (although behavioral trees are a nicer implementation)

@robjohn nice one with that solution math.stackexchange.com/questions/481527/…

it just pinged up before i was going to bed

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