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06112014, 01:57 AM  #9841  
back to rabbit
Join Date: Apr 2007
Location: Río Grande Valley

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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. 

06112014, 08:34 AM  #9842 
UG's Secret Agent
Join Date: Mar 2007
Location: Uruguay

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 higherordered 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?
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06112014, 03:19 PM  #9843 
Mmmm...donuts...
Join Date: Apr 2009
Location: Land of Chocolate

“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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06112014, 04:22 PM  #9844 
fully retractable
Join Date: Mar 2007
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I'm not so sure you want to appeal to category theory for something like this.

06112014, 10:35 PM  #9845  
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06122014, 08:47 PM  #9846  
GemmaClaudia Owner
Join Date: Dec 2013
Location: Land with a giant rock in the middle.

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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06122014, 08:52 PM  #9847 
I'm too old for this ****
Join Date: Oct 2007
Location: OH

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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06122014, 08:53 PM  #9848  
buzzcut
Join Date: Jan 2009
Location: Portugal

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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06122014, 08:54 PM  #9849 
fully retractable
Join Date: Mar 2007
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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. 
06122014, 08:58 PM  #9850  
GemmaClaudia Owner
Join Date: Dec 2013
Location: Land with a giant rock in the middle.

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It's a very silly question. Thanks all
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07292014, 11:05 AM  #9851  
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Join Date: Apr 2013
Location: Virginia

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Last edited by DamienEx1021 : 07292014 at 11:08 AM. 

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