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A Summer of Maths (ASoM) 2016

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Original post by EnglishMuon
is there a context? :tongue:


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>0M > 0, for which there exists N>0N > 0, such that
n>N⇒∣f(x)∣<M∣g(x)∣n > N \quad \Rightarrow \quad |f(x)| < M|g(x)|,
then we write f(x)=O(g(x))f(x) = O(g(x)).

If for every ε>0\varepsilon > 0, there exists N>0N > 0, such that
n>N⇒∣f(x)∣<ε∣g(x)∣n > N \quad \Rightarrow \quad |f(x)| < \varepsilon|g(x)|,
then we write f(x)=o(g(x))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 < g.
O:f≤g.O: f \leq g.
Θ:f=g.\Theta: f = g.
Ω:f≥g.\Omega: f \geq g.
ω:f>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 :frown:
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. :angry: 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.
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. :angry: 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.
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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Reply 566
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.
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:tongue: thanks for clearing that up
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!
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.
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 :h:

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.
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

Spoiler

(edited 7 years ago)
Original post by Gome44
Anyone wanna do some stats


Probability theory is just analysis and measure theory restricted to [0,1].
Original post by Gome44

soz gregorius


The fewer people who do stats, the higher my employment market value is...

:mwuaha:
If anyone is still in the mood for some good old-fashioned pure mathematics then here's a question that might benefit a lot of first year students.

Given that there exists a bijection f:X↦Yf : X \mapsto Y between sets [br]X[br]X and YY and that x∈Xx \in X and y∈Yy \in Y, prove that there must always exist a bijection g:X↦Yg : X \mapsto Y such that g(x)=yg(x) = y.
(edited 7 years ago)
Original post by Gregorius
The fewer people who do stats, the higher my employment market value is...

:mwuaha:


LOL! I wouldn't worry. Outside of academia, I've met maybe two people who even vaguely understand statistics (in the financial sector, and yes that's worrying).

Even in academia, I'm guessing even the likes of Tibshirani or Efron aren't masters of the entire field (they're the two I know today who are alive and eminent, not sure if they're considered the "most eminent" today). Point is, I'm sure you'll be fine - stats education needs to be a LOT better at uni before the area is "commoditised" :tongue:
(edited 7 years ago)

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