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Old 06-11-2014, 01:57 AM   #9841
americablanco
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Quote:
Originally Posted by gonzaw
Hi guys, I've always wondered something about algebra.

rant
So in algebra,
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In sequences, the formal, theoretical, mathematical definition for those little numbers to the right of each 'a' (the sub-indexes) comes from how a sequence is defined: a map from the natural numbers, or subset thereof, to whatever other set, such as R. The natural numbers being {1, 2, 3, ... }. I shall use the notation a_n to represent the subscript of n. So, more formally,

Code:
Let N represent the set of natural numbers and R represent the set of real numbers. Define the map a_n: N -> R such that n is an element of N and a_n is an element of R and 1 |-> a_1, 2|-> a_2, ... , n |-> a_n, ...
Here it is more obvious to see where the indices come from. Simple, as you stated.

Now, natural morphisms. η_X is the natural transformation from F to G such that the domain, F, is restricted to F(X), a subset (at least) of F.

The same can be applied to any other map:

f:R -> R, f maps R to R

can be restricted to f_Z which restricts the domain of f to a subset of itself, namely the integers. Essentially f_Z is...

f_Z: Z -> R

I hope this clears up any confusion of indices and their many uses in math. I had typed up a whole other mess of things but tried to simplify it down for other readers that might stumble upon this.
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Old 06-11-2014, 08:34 AM   #9842
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Yes, thinking it in terms of mappings clears things out. The thing is that tha "mapping" is never used in any proof whatsoever, or in examples, etc, and that confuses me since I dunno if they are supposed to be used or not to rigurously proof stuff (or if the stuff we have proved is just "flimsy" stuff that maybe made wrong assumptions or something).


Also how is it defined in terms of category theory for example?
How can I use this "mapping" to make proofs about stuff about natural transformations?

η_X is a morphism in a category D from FX to GX. This is defined in terms of category theory.
But then, how is the "mapping" η defined? Is it a mapping from a category to another category? A morphism in a higher-ordered category? How can I work with it?
Can I say η is a functor from the category C, to the category of morphisms f: D->D, so that ηX = f:FX->GX (with F,G:C->D functors)? If this is the case, what is ηH (where H is a morphism from a morphism f to another morphism g)? Can I prove the properties of functors with it (composition+identity)?

If I can't use category theory with this "mapping" to make proofs, then what can I use?

Last edited by gonzaw : 06-11-2014 at 09:23 AM.
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Old 06-11-2014, 03:19 PM   #9843
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“You take inside crap’s derivative and multiply it by outside crap’s derivative, leaving inside crap the same.”

There is a tumblr full of maths professor quotes. Thought it'd be relevant here! http://mathprofessorquotes.tumblr.com/
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Old 06-11-2014, 04:22 PM   #9844
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I'm not so sure you want to appeal to category theory for something like this.
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Old 06-11-2014, 10:35 PM   #9845
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Quote:
Originally Posted by Woffelz
“You take inside crap’s derivative and multiply it by outside crap’s derivative, leaving inside crap the same.”

There is a tumblr full of maths professor quotes. Thought it'd be relevant here! http://mathprofessorquotes.tumblr.com/

thank you for introducing my life to this
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Old 06-12-2014, 08:47 PM   #9846
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Calculus coming up:

The rate of change of the angle sum S of a polygon with n sides is a constant 180. If S is 360 when n=4, find S when n=7.

Help?
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Old 06-12-2014, 08:52 PM   #9847
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You have an integratable equation: dS/dn = 180
You have an initial condition: S(n=4) = 360

Can you use this to find an expression for S as a function of n?

answer
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Old 06-12-2014, 08:53 PM   #9848
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That's a bit of a silly problem tbh.
So, what they're saying is that S increases at a constant rate of 180 degrees per side.
And it's 360 when n=4.
Get it now?

Yeah, or that ^
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Old 06-12-2014, 08:54 PM   #9849
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The first step is just parsing and writing a formula for the information given.

We have a function with input n and output S, call it f(n). We're given that f'(n) = 180. Now we just need to find f (since f' is the roc of S, f gives us S), plug in the values given, and solve.
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Old 06-12-2014, 08:58 PM   #9850
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Quote:
Originally Posted by sickman411
That's a bit of a silly problem tbh.
So, what they're saying is that S increases at a constant rate of 180 degrees per side.
And it's 360 when n=4.
Get it now?

Yeah, or that ^

It's a very silly question.

Thanks all
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Old Today, 11:05 AM   #9851
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