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# A Summer of Maths (ASoM) 2016

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1. Thanks, they are not quite on Cayley's theorem but I like my isomorphisms none the less and yeah the notation looks fine to me, the first GT book I ever read was from the 50s so most stuff seems modern compared! One annoying thing though is older books seem to write the cycle notation for permutations the opposite way round (i.e. the left cycle is the one you apply first rather than the right one as done in Beardon).
2. (Original post by EnglishMuon)
One annoying thing though is older books seem to write the cycle notation for permutations the opposite way round (i.e. the left cycle is the one you apply first rather than the right one as done in Beardon).
I swear they tell you beforehand which way it is computed, or at least Beardon does.
3. (Original post by Insight314)
I swear they tell you beforehand which way it is computed, or at least Beardon does.
Yeah they normally do its just a little annoying when i answer some questions doing one way but they are after the other because I forgot
4. (Original post by EnglishMuon)
Yeah they normally do its just a little annoying when i answer some questions doing one way but they are after the other because I forgot

How do you usually compute a product of cycles? I am curious because there are other ways of doing it. I was taught by my maths teacher a month or so ago how to do them using 'branes'. I can't explain that well how you do it, it is a pretty interesting way, but here is the basic idea behind it:

Edit: I just realised I compute them left to right, as opposed to how Beardon does them haha.
5. (Original post by Insight314)
How do you usually compute a product of cycles? I am curious because there are other ways of doing it.

Never really paid much attention to computing them, to be honest. I'd just write down 1 2 3 4 5 6 in the top row of the matrix and then "2 goes to 4" and write down 4 underneath 2 immediately, etc...

I much prefer the right to left way, given that permutations are basically (more or less) functions and products of them are compositions of functions and so read in the usual right -> left way.
6. (Original post by Zacken)
Never really paid much attention to computing them, to be honest. I'd just write down 1 2 3 4 5 6 in the top row of the matrix and then "2 goes to 4" and write down 4 underneath 2 immediately, etc...
Yeah, that's what I used to do until I found out the branes method.

(Original post by Zacken)
I much prefer the right to left way, given that permutations are basically (more or less) functions and products of them are compositions of functions and so read in the usual right -> left way.
Probably why Beardon does it that way, but it really doesn't matter which way you do it as long as you keep up consistency.
7. Is there any quick way to find the permutation representation of the quarternian group? The method I was doing was working out the permutation represented by and working out each as a permutation of {1,2...,8} but this felt like an algebra slog. Any faster way?
8. 3Brown1Blue releasing a series on Linear Algebra, or at least the Geometric implications of it, trying first few episodes now, will let you guys know how it goes

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9. Pretty basic stuff so far with a few interesting points thrown in here or there. Has cool animations and a relaxing voice so that's a plus

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10. (Original post by drandy76)
Pretty basic stuff so far with a few interesting points thrown in here or there. Has cool animations and a relaxing voice so that's a plus

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I remember watching his first few videos last year, and they were pretty interesting, and his presentation of the ideas was pretty good.

I skimmed through the linear algebra series you mentioned, and it's good for consolidating the ideas (mainly looked at the video on linear independence, spanning sets). However, it's lacking mathematical rigour (which is not his fault, as it's difficult to present the ideas rigorously in a video; textbooks are best for that) so I would only recommend them if one has left without any intuition behind the ideas found in (rigorous) textbooks.

I remember really enjoying his video on graph theory and Euler's formula last year - https://www.youtube.com/watch?v=-9OUyo8NFZg - if you haven't seen it yet.
11. (Original post by Insight314)
I remember watching his first few videos last year, and they were pretty interesting, and his presentation of the ideas was pretty good.

I skimmed through the linear algebra series you mentioned, and it's good for consolidating the ideas (mainly looked at the video on linear independence, spanning sets). However, it's lacking mathematical rigour (which is not his fault, as it's difficult to present the ideas rigorously in a video; textbooks are best for that) so I would only recommend them if one has left without any intuition behind the ideas found in (rigorous) textbooks.

I remember really enjoying his video on graph theory and Euler's formula last year - https://www.youtube.com/watch?v=-9OUyo8NFZg - if you haven't seen it yet.
Watching mostly due to boredom and the fact that I'm mostly burnt out from reading atm, read something close to million words (give or take,probably a bit over though) in the past few weeks

I remember watching that but had an aversion to graph theory at the time due to D1/2 so I'll give it a go. Found his video about Hilbert curves pretty interesting too

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12. Linear algebra is literally the most important thing you'll ever do in your first year. Some mathematician said something on the lines of: "a mathematical problem can only be solved only if it can be reduced to linear algebra. Everything you will ever do will involve linear algebra in some way. The vector space is like the perfect structure for algebra, akin to how the complex numbers are the perfect structure for analysis. If I was to do anything to get a head start to maths, I'd go for linear algebra.
13. (Original post by EnglishMuon)
Is there any quick way to find the permutation representation of the quarternian group? The method I was doing was working out the permutation represented by and working out each as a permutation of {1,2...,8} but this felt like an algebra slog. Any faster way?
Sorry, I am not quite sure what you mean, but are you trying to use Cayley's theorem and represent the quaternion group as a permutation of the symmetric group of order 8 by cross-multiplying each element of ? I can see why it is an algebra slog, and I am not knowledgeable of any other way of doing this.
14. (Original post by Alex:)
Linear algebra is literally the most important thing you'll ever do in your first year. Some mathematician said something on the lines of: "a mathematical problem can only be solved only if it can be reduced to linear algebra. Everything you will ever do will involve linear algebra in some way. The vector space is like the perfect structure for algebra, akin to how the complex numbers are the perfect structure for analysis. If I was to do anything to get a head start to maths, I'd go for linear algebra.
I am happy to be enjoying studying linear algebra (V&M) then.

I've been told the same by other mathematicians, and I can kind of see how fundamental it is to high-level mathematics since it links to all other kinds of fields of mathematics. Like, there's a reason why Beardon teaches abstract and linear algebra together (intro to groups, permutations mainly, then the core of linear and then the core of groups with Möbius transformations) since then he can link the ideas of linear algebra to groups e.g determinants depends on permutations.

I will most likely be done with Beardon's textbook (V&M and Groups) by early September, and I am conflicted between studying real analysis and building up on my study of abstract algebra since I have a textbook that continues Beardon's groups into the algebraic structures of rings and fields (Galois theory included). Would you say it is best for me to continue with abstract algebra, or start real analysis? Also, I already have Burkill's book "A First Course in Mathematical Analysis" which is the recommended reading for Analysis I.
15. (Original post by Insight314)
Sorry, I am not quite sure what you mean, but are you trying to use Cayley's theorem and represent the quaternion group as a permutation of the symmetric group of order 8 by cross-multiplying each element of ? I can see why it is an algebra slog, and I am not knowledgeable of any other way of doing this.
Yeah effectively. Most of these permutation questions feel like technique application rather than problem solving but thanks anyway
16. (Original post by Insight314)
I am happy to be enjoying studying linear algebra (V&M) then.
...

I will most likely be done with Beardon's textbook (V&M and Groups) by early September, and I am conflicted between studying real analysis and building up on my study of abstract algebra since I have a textbook that continues Beardon's groups into the algebraic structures of rings and fields (Galois theory included). Would you say it is best for me to continue with abstract algebra, or start real analysis? Also, I already have Burkill's book "A First Course in Mathematical Analysis" which is the recommended reading for Analysis I.
I'd probably go for introductory real analysis, since real analysis is a much longer course and there's nothing stopping you going for some real analysis now. Ring theory can be learned very quickly, same with module theory and Galois theory, especially once you know group theory. You'll need real analysis for complex analysis and topology.

Real analysis lets you see how things fail and how ugly functions can be on the real line. There's actually a book called 'counterexamples in real analysis' filled with ridiculous functions that defy intuition. It's wildly different to linear algebra and abstract algebra, where everything works nicely, and has a different flavour to complex analysis, where everything also works nicely.

Linear algebra and real analysis will give you an appreciation of multivariable calculus and the Gradient, Green, Stoke, Gradient and Divergence theorems.
17. Should I learn to programme/code during uni or not worth the effort?

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18. (Original post by drandy76)
Should I learn to programme/code during uni or not worth the effort?

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To be honest you could get to a decent level in 'easier' languages quite quickly - I went from a standing start to a reasonable competence in Python in about a fortnight (though a compsci once told me that they could tell I was a mathmo from my inelegant code, but that's an aside...). The free Udacity course is rather good from what I recall.

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19. (Original post by Krollo)
To be honest you could get to a decent level in 'easier' languages quite quickly - I went from a standing start to a reasonable competence in Python in about a fortnight (though a compsci once told me that they could tell I was a mathmo from my inelegant code, but that's an aside...). The free Udacity course is rather good from what I recall.

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Ducking Comp sci nerds...
What languages would be useful to learn? Just Python, or would it worth my time to learn Java as well?
20. (Original post by drandy76)
Ducking Comp sci nerds...
What languages would be useful to learn? Just Python, or would it worth my time to learn Java as well?
I'm no expert in these things, but when I recently asked much the same question in the Cambridge Maths thread they said Python would probably be fine

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