# Iterated Möbius transformations - Any University teachers or similar here to help?

## Main Question or Discussion Point

Alright, this is my case. I am now a former International Baccalaureate Diploma programme student that wrote my extended essay in mathematics. As far as it seems, I was incredibly unlucky when they corrected my essay, cause as it seem, the word count was too much, so they kind of didn't read my essay and just gave it a terrible grade.

However, I think that this essay deserves at least a real look, so that I know what this essay is really worth. The word count thingy was that we had an upper limit of 4000 words, and I had around 1200 words together with my introduction and my matrices and formulae and equations.. so it's really hard to know how they are read, in how many words. So what I think is however that the examiner was tired and just said, ah.. this is too much.

Anyway, I would wonder if any of you want to read it through and set a grade from A-F on it, it would mean so much to me! It's about Möbius transformations that are iterated, so the research question is "what happens when a möbius transformation is iterated". If this sounds like something which you can master or know how to grade or similar, please post email, and I'll send my essay. I can also look up the criterion if there's anyone that is willing to try.

Thank you!!

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I see there's a possibility to attach files to this.. so if there's anyone to read it, I can attach it..

HallsofIvy
Homework Helper
" The word count thingy was that we had an upper limit of 4000 words, and I had around 1200 words together with my introduction and my matrices and formulae and equations.. "

So what you are telling us is that you wrote your essay on mathematics and you can't even count?
You said "I was incredibly unlucky when they corrected my essay, cause as it seem, the word count was too much, so they kind of didn't read my essay and just gave it a terrible grade."

Are you unaware that 1200 is less than 4000? Apparently, according to you, your essay was too short not too long as you seem to believe!

no no.. it's not that it was too short.. because there's no lower boudnary for mathematics Extended essay. The thing is however that a matrix can be counted as one word or 4 words or a sentences.. it's so subjective, so through that I was unlucky. The introduction was about 300 words I believe, so the rest of the 900 words with the equations, matrices and formulae might have taken up more room than 4000 words. The essay is in itself 13 pages long.

shmoe
Homework Helper
It's not going to be possible for me to give you a grade like you hope for as I have no idea what the IB standards are, I would be willing to read it and let you know what I think. Probably if you made it available as an attatchment others would spontaneously have a look too (path of least resistance makes people less likely to ask for more work for themselves). Almost any format will be fine for me, though a pdf might be more universally accessable.

I'll gladly read it as well. However, I must say that I already find your writing style confusing, judging on what you've posted. I really hope that your grammar here does not reflect your essay paper.

I'll read it too. There is a whole book on this subject and probably more exists out there than that.

Without reading either your essay nor that book, I'd say that iterating linear fractional transformations (LFTs) is the same thing as raising a 2x2 matrix to a power. So, if this matrix is diagonalizable, it can be readily done. If not, I'm not sure.

Thanks guys! Sorry for keeping you waiting, I've been kind of busy lately, but now I have some spare time ;)

The file is in Pdf, and please notify me when you have read it!

Thank you all!

#### Attachments

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I haven't read that book, actually, I didn't know it existed.. but hopefully you were able to read the essay..

calvino said:
I'll gladly read it as well. However, I must say that I already find your writing style confusing, judging on what you've posted. I really hope that your grammar here does not reflect your essay paper.
and nope.. I hope it doesn't either ;) I am rather sloppy sometimes when I write posts, but when it comes to essays I should be rather ok. But then again, when I'm in a hurry or I'm stressed, it also shows on the grammer.. independently which language I write in :)

shmoe
Homework Helper
Hi, it looks pretty good. All the math you have looks correct, but there's a number of things I would have liked to see included or simplified:

The correspondence between composition of Mobius transformations and multiplication of matrices. Though it's a very standard detail, it's what allows you to convert iterations of transforms into multiplications and perform an analysis on the matrices instead, so it plays a vital role here and very much worth mentioning.

Your case (i) and case (ii). These are really the same and just depend on how you order the eigenvalues.

The matrix B. It's not unique of course. A word or two about w different choices of B won't affect your results (i.e. B(0) and B(infinity)). This ties in with the difference between cases i and ii.

You very nicely describe what happens to the w-plane under iterations. You didn't translate back to the z-plane afterwards though, which is what we're interested in. You mentioned something in the introduction about points being mapped to B(0) and B(infinity) but didn't follow it up in the text. A picture of the orbits in the iii case in the z plane would have been nice as well.

That said, it's still a reasonable stab at explaining these iterations. I don't really know anything about the IB program so I have no clue what their expectations are, what level it's at, or how this would compare to other essays.

Thanks ALOT! This is the reason for me wanting this graded.. the thing was.. I got grade E on it, since they thought that it was too long. IB programme is Senior High school, so normally you don't do such things as these, not even close to it, at least not the level of it. That's why I am so dissappointed, since I thought that the essay was rather good, but those examiners didn't even read it :(

If there are more people that want to comment, please feel free to do so!

shmoe said:
Hi, it looks pretty good. All the math you have looks correct, but there's a number of things I would have liked to see included or simplified:
The correspondence between composition of Mobius transformations and multiplication of matrices. Though it's a very standard detail, it's what allows you to convert iterations of transforms into multiplications and perform an analysis on the matrices instead, so it plays a vital role here and very much worth mentioning.
Your case (i) and case (ii). These are really the same and just depend on how you order the eigenvalues.
The matrix B. It's not unique of course. A word or two about w different choices of B won't affect your results (i.e. B(0) and B(infinity)). This ties in with the difference between cases i and ii.
You very nicely describe what happens to the w-plane under iterations. You didn't translate back to the z-plane afterwards though, which is what we're interested in. You mentioned something in the introduction about points being mapped to B(0) and B(infinity) but didn't follow it up in the text. A picture of the orbits in the iii case in the z plane would have been nice as well.
That said, it's still a reasonable stab at explaining these iterations. I don't really know anything about the IB program so I have no clue what their expectations are, what level it's at, or how this would compare to other essays.

if you want to check through the criteria, this page is the place to go:

http://www.internet.ve/eca/ib/ibextess.htm [Broken]

I would think the essay is worth more than 6 points which I got out of 36 :( (you get 6 points if you have submitted the essay)

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quasar987
Homework Helper
Gold Member
What is the official meaning of the symbol that looks like a triangle made out of 3 points, as can be seen right before equation (10) of Karoly's essay?
I was once told it means, like, if you set out to prove something, then your final implication ($\Rightarrow$) is noted like that. So,
$$\therefore \equiv \Rightarrow_{\mbox{final}}$$

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http://members.aol.com/jeff570/set.html [Broken].

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