A Summer of Maths (ASoM) 2016

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    (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.
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    (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
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    (Original post by drandy76)
    Sweet, it's the coolest language name too imo
    Brainfu*k tho
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    (Original post by Zacken)
    Brainfu*k tho
    It has a derivative called DerpPlusPlus which Is pretty cool too
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    (Original post by drandy76)
    It has a derivative called DerpPlusPlus which Is pretty cool too
    d/dx(brainfu*k) = derpplusplus what is this sorcery maths
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    (Original post by Zacken)
    d/dx(brainfu*k) = derpplusplus what is this sorcery maths
    It's part 111 ****, proper advanced
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    (Original post by drandy76)
    It's part 111 ****, proper advanced
    I myself program in a very high level language.

    It's called 'messaging friends who can program properly'

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    (Original post by Krollo)
    I myself program in a very high level language.

    It's called 'messaging friends who can program properly'

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    Friends....404 not found
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    A quick question on notation: If a group  H is a subgroup of a group  G and  G \thickapprox G' is it fine to say  H is a subgroup of  G' ? e.g. is  SL(2, \mathbb{Z} ) a subgroup of the Mobius group?
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    (Original post by EnglishMuon)
    A quick question on notation: If a group  H is a subgroup of a group  G and  G \thickapprox G' is it fine to say  H is a subgroup of  G' ? e.g. is  SL(2, \mathbb{Z} ) a subgroup of the Mobius group?
    It depends upon (a) how formal you are trying to be and (b) how the groups concerned are actually specified. Let me simplify the situation and ask what the difference might be between saying that "H is a subgroup of G" and "H is isomorphic to a subgroup of G". Can you think of circumstances where you might wish to prefer the one over the other?
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    (Original post by Gregorius)
    It depends upon (a) how formal you are trying to be and (b) how the groups concerned are actually specified. Let me simplify the situation and ask what the difference might be between saying that "H is a subgroup of G" and "H is isomorphic to a subgroup of G". Can you think of circumstances where you might wish to prefer the one over the other?
    Im not sure when the "H is a subgroup of G" case is more useful than the other (the other which I assume is the proper statement), but Ive still seen it used every now and then in different texts. What circumstances were you thinking of? Perhaps when the operation  G' is under is easier to deal with than the operation in  G it may be easier to look at  G' but that's probably a different situation.
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    (Original post by EnglishMuon)
    Im not sure when the "H is a subgroup of G" case is more useful than the other (the other which I assume is the proper statement), but Ive still seen it used every now and then in different texts. What circumstances were you thinking of? Perhaps when the operation  G' is under is easier to deal with than the operation in  G it may be easier to look at  G' but that's probably a different situation.
    Yes, you do see the informal use. Pure mathematicians, having established that rigour can be achieved, not infrequently let their hair down and talk quite informally, safe in the knowledge that they could fill in the filthy details if they wanted to.

    The circumstances that I'm thinking of can be seen in this statement of Cayley's thorem. The group that you are dealing with may be specified in a number of different ways: as a group of matrices; as a group presentation; as a table specifying the group operation; etc, etc. If you say that H is a subgroup of G, you are implicitly saying that the elements of H are a subset of the elements of G, rather than saying that the elements of H can be identified with a subset of the elements of G.

    So if you're dealing with a group of matrices G and you identify a subset H of those matrices that itself forms a group, it would be perfectly OK to say that H is a subgroup of G.On the other hand, if H was specified via abstract elements in a multiplication table, then you should probably saying that H is isomorphic to a subgroup of G.
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    I was thinking of writing some notes on groups and other topics for 1st year mathematics, if this would be any use to anyone
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    (Original post by A Slice of Pi)
    I was thinking of writing some notes on groups and other topics for 1st year mathematics, if this would be any use to anyone
    Yeah could do with a refresher on groups, what other topics you thinking of?


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    (Original post by drandy76)
    Yeah could do with a refresher on groups, what other topics you thinking of?


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    Vector calculus, multivariable calculus, linear algebra etc.
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    (Original post by A Slice of Pi)
    Vector calculus, multivariable calculus, linear algebra etc.
    Will be covering linear algebra myself, but I'll look over vector calculus at least


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    (Original post by drandy76)
    Will be covering linear algebra myself, but I'll look over vector calculus at least


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    Multivariable can be very interesting
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    (Original post by A Slice of Pi)
    Multivariable can be very interesting
    Learnt a bit in year 12, was really cool but didn't quite get to the actual calculus bit unfortunately


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    For those who would like a refresher on number theory, I've just finished Q4 of this year's BMO2 and it's very nice indeed.

    Here's what I did for this one
    Spoiler:
    Show

    p^2 = \frac{u^2 + v^2}{2} \\ \\ \therefore 2p^2 = u^2 + v^2 = (u+v)^2-2uv \\

    Multiply by two

    4p^2 = 2(u+v)^2-4uv \\ \\ 4p^2 - (u+v)^2 = (u + v)^2 - 4uv \\ \\ \therefore (2p - u - v)(2p + u + v) = (u + v)^2 - 4uv = u^2 - 2uv + v^2

    Now  u^2 - 2uv + v^2 can be written in two ways. Either as  (u - v)^2 or as 2(p^2 - uv).

    This tells us that (2p - u - v)(2p + u + v) is
    1.) Even
    2.) A perfect square

    Thus
    2p - u - v =2^{k}n^2
    for some n.
    When k is odd, this expression is twice a square, and when k is even it is a square, as desired.
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    (Original post by A Slice of Pi)
    For those who would like a refresher on number theory, I've just finished Q4 of this year's BMO2 and it's very nice indeed.

    Here's what I did for this one
    Spoiler:
    Show

    p^2 = \frac{u^2 + v^2}{2} \\ \\ \therefore 2p^2 = u^2 + v^2 = (u+v)^2-2uv \\

    Multiply by two

    4p^2 = 2(u+v)^2-4uv \\ \\ 4p^2 - (u+v)^2 = (u + v)^2 - 4uv \\ \\ \therefore (2p - u - v)(2p + u + v) = (u + v)^2 - 4uv = u^2 - 2uv + v^2

    Now  u^2 - 2uv + v^2 can be written in two ways. Either as  (u - v)^2 or as 2(p^2 - uv).

    This tells us that (2p - u - v)(2p + u + v) is
    1.) Even
    2.) A perfect square

    Thus
    2p - u - v =2^{k}n^2
    for some n.
    When k is odd, this expression is twice a square, and when k is even it is a square, as desired.
    I spent like 2 hours staring at this **** in the exam. I spent much time finding numerical values to check it was correct because it sounded so unlikely... sadly my impressive arithmetic got no marks.

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