A Summer of Maths (ASoM) 2016

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    (Original post by Alex:)
    Or should I say the Sourier transform on a gunction using jmaginary numbers.
    Nice one xD
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    Could someone give me a hint for how to solve this question without the fundamental theorem for finite abelian groups?

    Let  G be a finite abelian group such that it contains a subgroup  H_{0} which is contained in every subgroup  H \not = (e) . Prove that  G is cyclic.

    This is easy if I'm allowed to use the fact that G is the direct product of cyclic groups (with trivial intersections), as if we have more than 1 cyclic group in the product, we must have H0=(e) so G must be cyclic. But I am not allowed to use this theorem for the question. Prove a couple facts about the orders of G, H0 but not sure where to go from there...
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    (Original post by EnglishMuon)
    Could someone give me a hint for how to solve this question without the fundamental theorem for finite abelian groups?

    Let  G be a finite abelian group such that it contains a subgroup  H_{0} which is contained in every subgroup  H \not = (e) . Prove that  G is cyclic.

    This is easy if I'm allowed to use the fact that G is the direct product of cyclic groups (with trivial intersections), as if we have more than 1 cyclic group in the product, we must have H0=(e) so G must be cyclic. But I am not allowed to use this theorem for the question. Prove a couple facts about the orders of G, H0 but not sure where to go from there...
    Hint:
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    Lagrange's theorem for groups may be useful
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    (Original post by A Slice of Pi)
    Hint:
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    Lagrange's theorem for groups may be useful
    lol yep Ive used this multiple times in my workings, not seen a solution free of FTFAG though. Probably the basis for every gt proof I've ever done
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    (Original post by EnglishMuon)
    is there a context?
    Sorry, I should've included some context haha. I'm going through the Cambridge notes on the first page (by Dexter Chua I think) and this is included in the Differenitial Equations part.
    (Original post by Alex:)
    If there exists a constant M > 0, for which there exists N > 0, such that
    n > N \quad \Rightarrow \quad |f(x)| < M|g(x)|,
    then we write f(x) = O(g(x)).

    If for every \varepsilon > 0, there exists N > 0, such that
    n > N \quad \Rightarrow \quad |f(x)| < \varepsilon|g(x)|,
    then we write f(x) = o(g(x)).

    The analogue to big-O and little-o is very similar to less than and strictly less than. Big-O gives an upper bound to the growth, but the function can still approach its Big-O function asymptotically. Little-o is much more strict.

    There's other things like Omega, omega and Theta notation. A kinda rough intuition of them could be:
    o: f < g.
    O: f \leq g.
    \Theta: f = g.
    \Omega: f \geq g.
    \omega: f > g.
    Could you perhaps explain it in a less maths-y way? I come from a non-maths background (Physics), so I don't understand most of the notation that you have used
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    Has anyone read Rudins 'principles of mathematical analysis'? How would it compare to burkill's 'A first course in mathematical analysis' ?


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    (Original post by drandy76)
    Has anyone read Rudins 'principles of mathematical analysis'? How would it compare to burkill's 'A first course in mathematical analysis' ?


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    Rudin's book (Baby Rudin) is really inappropriate for anyone who has no knowledge of real analysis already. The book is really terse and everything is left as an exercise for the reader. His treatment of multi-variable analysis is also not the best.

    Never used Burkill's book. But I think Spivak's Calculus is really good for learning real analysis without knowledge of topology, and leads really well to his next book Calculus on Manifolds.
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    (Original post by Alex:)
    Rudin's book (Baby Rudin) is really inappropriate for anyone who has no knowledge of real analysis already. The book is really terse and everything is left as an exercise for the reader. His treatment of multi-variable analysis is also not the best.

    Never used Burkill's book. But I think Spivak's Calculus is really good for learning real analysis without knowledge of topology, and leads really well to his next book Calculus on Manifolds.
    Thanks! Ironically after posting this I realised I had Spivak's calculus as well, so i believe I'll use that instead.


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    (Original post by drandy76)
    Thanks! Ironically after posting this I realised I had Spivak's calculus as well, so i believe I'll use that instead.


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    I endorse what Alex: said and add that if you want something as challenging as Baby Rudin, but with more explanations, then Apostol's "Mathematical Analysis" is very good.
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    (Original post by Gregorius)
    I endorse what Alex: said and add that if you want something as challenging as Baby Rudin, but with more explanations, then Apostol's "Mathematical Analysis" is very good.
    Thanks I'll look into it, by the way, why are you guys calling him Baby Rudin?


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    (Original post by drandy76)
    Thanks I'll look into it, by the way, why are you guys calling him Baby Rudin?


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    They're not, they're calling the book baby Rudin because there's a more advanced version of the book as well - so the first book is affectionately termed baby Rudin.
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    (Original post by Zacken)
    They're not, they're calling the book baby Rudin because there's a more advanced version of the book as well - so the first book is affectionately termed baby Rudin.
    oh i see thanks for clearing that up
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    (Original post by drandy76)
    Thanks! Ironically after posting this I realised I had Spivak's calculus as well, so i believe I'll use that instead.


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    Good stuff - it'd be nice to get some analysis going in this thread, rather than being smothered in algebra. Analysis is nice and awesome, algebra is messy and unwieldy!
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    (Original post by Alex:)
    Good stuff - it'd be nice to get some analysis going in this thread, rather than being smothered in algebra. Analysis is nice and awesome, algebra is messy and unwieldy!
    I shall be the hero this thread needs, muon and Pi's reign of terror ends here


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    Anyone up for some cheeky dynamics and/or special relativity

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    (Original post by Krollo)
    Anyone up for some cheeky dynamics and/or special relativity

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    Literally never, gonna hire a ghost writer for that module


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    (Original post by Gregorius)
    I endorse what Alex: said and add that if you want something as challenging as Baby Rudin, but with more explanations, then Apostol's "Mathematical Analysis" is very good.
    Why do people still suggest Rudin anyway? As Alex says, it's*awful.
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    (Original post by shamika)
    Why do people still suggest Rudin anyway? As Alex says, it's*awful.
    It was more me finding the book and wondering if it was any good rather than a recommendation, at least I think so, there's s chance I might've gotten it from my Unis reading list


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    (Original post by shamika)
    Why do people still suggest Rudin anyway? As Alex says, it's*awful.
    Said with true passion

    It's one of those books that's most useful to those who have already covered the material and have developed their intuition. Even then, there are better choices these days, I think, especially in the form of all the free lecture notes dotted all over the place.
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    (Original post by Krollo)
    Anyone up for some cheeky dynamics and/or special relativity

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    (Original post by Alex:)
    Good stuff - it'd be nice to get some analysis going in this thread, rather than being smothered in algebra. Analysis is nice and awesome, algebra is messy and unwieldy!
    Anyone wanna do some stats
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    lol jk
    Spoiler:
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    soz gregorius
 
 
 
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