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SimonM
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#4961
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#4961
(Original post by IrrationalNumber)
http://www.imc-math.org.uk/
http://en.wikipedia.org/wiki/Mikl%C3...er_Competition
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My Alt
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#4962
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#4962
(Original post by MrShifty)
Do you though? That's like saying that as you take n to infinity the number of integers becomes uncountably infinite, which is clearly untrue.
Is it? Surely once n gets big enough, then you can get arbitrarily large?
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Simplicity
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#4963
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#4963
Trying to read Lang Algebra. But, can't really think as I'm ill and my head hurts.

But, I don't understand the definition of commutativity diagram.

A diagram in which any two mappings between the same pair of sets, formed by composition of mappings represented by arrows in the diagram, are equal.


Surely, commutativity is the idear that order doesn't matter. But, clearly it will matter in a commutative diagram, you can't say have fp=pf.
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harr
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#4964
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#4964
(Original post by Simplicity)
If you think of the nth root of unity. This is a cyclic group. But, if you let n tend to infinity. Surely, you get a uncountable amount of points. But, nth root of unity is ismorphic to the cyclic group.
Ignoring whether the limit looks like the circle, how do you know that the limit would be a cyclic group? There are plenty of properties that aren't always preserved when you take a limit. For example {0}, {0,1}, {0,1,2}, etc. are all finite but the limit isn't.
(Original post by My Alt)
Is it? Surely once n gets big enough, then you can get arbitrarily large?
I can't quite understand what you're saying here. Consider the sets of the nth roots of unity for any n in the natural numbers. Each set has n elements, so the size of the limit set will be the same size as the limit of {0}, {0,1}, {0,1,2}, etc., i.e. it will be countable
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assmaster
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#4965
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#4965
Reading this thread has made me realise that Durham's first year covers almost no group or number theory. And that I have all this sexy fun to look forward to in second year...

I moved into my student 'pad' yesterday, and apart from the whole 'window-getting-jammed-wide-open-debacle', it's pretty cool. Mostly because I can 'buzz people in' :love:
But I have already been called Monica a couple of times. I just like **** to be clean... :erm:
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majikthise
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#4966
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#4966
(Original post by Simplicity)
Trying to read Lang Algebra. But, can't really think as I'm ill and my head hurts.

But, I don't understand the definition of commutativity diagram.

A diagram in which any two mappings between the same pair of sets, formed by composition of mappings represented by arrows in the diagram, are equal.


Surely, commutativity is the idear that order doesn't matter. But, clearly it will matter in a commutative diagram, you can't say have fp=pf.
Why should it refer to anything like fp=pf, when one out of fp and pf won't even make sense?
It means as long as you start at one certain point in the diagram and follow arrows to another certain point, you get the same result no matter which exact route you took- so in that sense the order of which morphisms you follow doesn't matter.
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Meteorshower
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#4967
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#4967
(Original post by assmaster)
Reading this thread has made me realise that Durham's first year covers almost no group or number theory. And that I have all this sexy fun to look forward to in second year...

I moved into my student 'pad' yesterday, and apart from the whole 'window-getting-jammed-wide-open-debacle', it's pretty cool. Mostly because I can 'buzz people in' :love:
But I have already been called Monica a couple of times. I just like **** to be clean... :erm:
Yeah I haven't done any group or number theory either... Applied stuff is much nicer though in general though :yep:

I have literally only been in two student flats ever that have been clean. Well done on breaking the stereotype XD
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My Alt
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#4968
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#4968
(Original post by harr)
Ignoring whether the limit looks like the circle, how do you know that the limit would be a cyclic group? There are plenty of properties that aren't always preserved when you take a limit. For example {0}, {0,1}, {0,1,2}, etc. are all finite but the limit isn't.I can't quite understand what you're saying here. Consider the sets of the nth roots of unity for any n in the natural numbers. Each set has n elements, so the size of the limit set will be the same size as the limit of {0}, {0,1}, {0,1,2}, etc., i.e. it will be countable
I mean like as the set size gets arbitrarily large so does the cardinality of the set. Isn't that common sense ?_?
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MrShifty
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#4969
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#4969
(Original post by My Alt)
Is it? Surely once n gets big enough, then you can get arbitrarily large?
Yes, but arbitrarily large isn't the same as uncountably infinite.

A set X is countable if it's finite or there's a bijection between X and the natural numbers (in which case it's countably infinite), whereas it's uncountably infinite if there's a bijection between it and the real numbers (actually, the definition is a little more general than that, being that the cardinality is simply larger than that of the naturals, which that of the reals is).

So, by definition, the set {1,2,...,n} is always countable, no matter how large n is (n arbitrarily large just gives us the naturals themselves, so the bijection is trivial).
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My Alt
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#4970
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#4970
(Original post by MrShifty)
Yes, but arbitrarily large isn't the same as uncountably infinite.

A set X is countable if it's finite or there's a bijection between X and the natural numbers (in which case it's countably infinite), whereas it's uncountably infinite if there's a bijection between it and the real numbers (actually, the definition is a little more general than that, being that the cardinality is simply larger than that of the naturals, which that of the reals is).

So, by definition, the set {1,2,...,n} is always countable, no matter how large n is (n arbitrarily large just gives us the naturals themselves, so the bijection is trivial).
Oh OK thanks. But why can't n get so big that the bijection no longer exists?
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henryt
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#4971
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#4971
(Original post by My Alt)
Oh OK thanks. But why can't n get so big that the bijection no longer exists?
It's sort of tautological, but it's because we're taking n natural numbers. So there's always going to be a (direct) matching with the natural numbers when we take the limit.
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MrShifty
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#4972
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#4972
(Original post by My Alt)
Oh OK thanks. But why can't n get so big that the bijection no longer exists?
Well, mainly it's because {1,2,3,...} and the set of natural numbers are the same thing.

Put it this way, no matter how big a definite value of n you take, {1,2,...,n} is finite and hence countable.

On the other hand, letting our set be infinite, what we get is a set of the form {1, 1+1, 1+1+1, 1+1+1+1, ....} which is just the natural numbers themselves. So the set in question is countable (you don't really need one, as henryt said it's a tautology, but you could take the bijection as just the trivial identity mapping id(x)=x).
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ziedj
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#4973
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#4973
What's the notation for a continued square root? Like, \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{..  .}}}} k number of times.
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majikthise
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#4974
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#4974
I don't recall ever seeing anything but lots of dots with a curly bracket and "k" underneath.
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MrShifty
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#4975
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#4975
Same here, I don't think there is any standard notation for that. If you're writing something up and want to avoid having that expression appearing throughout the text, then just give it some kind of simple name, i.e.

Let \tilde{x}= that thing.
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Oh I Really Don't Care
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#4976
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#4976
use rcursion
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Meteorshower
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#4977
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#4977
Anyone here prefer applied maths to pure? :p:
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ForGreatJustice
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#4978
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#4978
I prefer applied maths be a long way, it's just so nice.
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Oh I Really Don't Care
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#4979
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#4979
... at university applied and pure will be of the same calibre so preference is then subject to percieved interest # benefit or mastery.
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ziedj
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#4980
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#4980
(Original post by DeanK22)
use rcursion
Well yeah I can write u_{n+1} = \sqrt{x + u_n} \ ; \ u_1 = \sqrt{x} , was just wondering if there was a "normal" way of writing it.
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