A Summer of Maths (ASoM) 2016

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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.

Reply 481

Insight314

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.

Reply 482

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

\tau _{g} : G \rightarrow G, x \tau _{g}=xg

and working out each

\tau _{g}

as a permutation of {1,2...,8} but this felt like an algebra slog. Any faster way?

Reply 483

drandy76

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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Reply 484

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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Reply 485

Insight314

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.

Reply 486

drandy76

Original post by Insight314

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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(edited 7 years ago)

Reply 487

InOrbit

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.

Reply 488

Insight314

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

\tau _{g} : G \rightarrow G, x \tau _{g}=xg

and working out each

\tau _{g}

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

Q_{8}

? I can see why it is an algebra slog, and I am not knowledgeable of any other way of doing this.

Reply 489

Insight314

Original post by Alex:

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

n \times n

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.

Reply 490

EnglishMuon

Original post by Insight314

Q_{8}

? 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

Reply 491

InOrbit

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.

Reply 492

drandy76

Should I learn to programme/code during uni or not worth the effort?

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Reply 493

Krollo

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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Reply 494

drandy76

Original post by Krollo

Ducking Comp sci nerds...
What languages would be useful to learn? Just Python, or would it worth my time to learn Java as well?

Reply 495

Krollo

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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Reply 496

InOrbit

Original post by drandy76

Should I learn to programme/code during uni or not worth the effort?

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You want to be able to program to a level where you can solve problems in your degree. Language does not matter but a scripting language like python will give you quick and dirty ways of doing calculations and plotting.

Reply 497

drandy76

Original post by Krollo

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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Sweet, it's the coolest language name too imo