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06-11-2014, 01:57 AM   #9841
americablanco
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Join Date: Apr 2007
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Quote:
 Originally Posted by gonzaw Hi guys, I've always wondered something about algebra. rant So in algebra, /rant
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.

 06-11-2014, 08:34 AM #9842 gonzaw 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 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.
 06-11-2014, 03:19 PM #9843 Woffelz 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/ __________________ Your beliefs are only valid when they correspond with my own. Twitter Youtube Soundcloud Tumblr
 06-11-2014, 04:22 PM #9844 MakinLattes fully retractable     Join Date: Mar 2007 Location: { } I'm not so sure you want to appeal to category theory for something like this.
06-11-2014, 10:35 PM   #9845
SteveHouse
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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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06-12-2014, 08:47 PM   #9846
Pastafarian96
Strumming the 'lele

Join Date: Dec 2013
Location: Tin Pan Alley
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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 06-12-2014, 08:52 PM #9847 AeroRocker 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 900 __________________ ██████████████████████████ ██████████████████████████ ██████████████████████████ ██████████████████████████ ██████████████████████████ ██████████████████████████ ██████████████████████████ LET'S GO BUCKS Last edited by AeroRocker : 06-12-2014 at 08:57 PM.
06-12-2014, 08:53 PM   #9848
sickman411
(1913-1938-197?)

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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Last edited by sickman411 : 06-12-2014 at 08:54 PM.

 06-12-2014, 08:54 PM #9849 MakinLattes fully retractable     Join Date: Mar 2007 Location: { } 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.
06-12-2014, 08:58 PM   #9850
Pastafarian96
Strumming the 'lele

Join Date: Dec 2013
Location: Tin Pan Alley
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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Quote:
 Originally Posted by slapsymcdougal Well, public masturbating sounds a lot like public mass debating, so it's kind of like preparation for politics, right?

07-29-2014, 11:05 AM   #9851
DamienEx1021
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Join Date: Apr 2013
Location: Virginia
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Last edited by DamienEx1021 : 07-29-2014 at 11:08 AM.

 08-15-2014, 11:34 AM #9852 gonzaw UG's Secret Agent     Join Date: Mar 2007 Location: Uruguay https://code.google.com/p/flyspeck/...ncingCompletion So they ended the Flyspeck project to formally prove the Kepler conjecture. ....cool
08-16-2014, 04:15 PM   #9853
Eastwinn
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Quote:
 Originally Posted by DamienEx1021 For all my friends in this thread. https://fbcdn-sphotos-d-a.akamaihd....940169203_n.png

i ripped my dick off
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08-23-2014, 07:04 PM   #9854
Neo Evil11
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Join Date: Feb 2006
Location: Arkham Asylum
Question:
When the functions have certain properties (not important which for you guys) then the set of (normalized) equilibrium vectors is convex. HENCE when the set of (normalized) equilibrium vectors is finite, there is only one unique (normalized) equilibrium vector.

I don't get why a finite, convex set of vectors means that this set contains only one unique vector.
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Today, 12:35 PM   #9855
Stringz of Fury
J Flo

Join Date: Jan 2008
Forgive me if this is an easy problem, but I haven't taken physics in a while and I've been stuck on this problem and I want to know if I'm way off or simply made a rounding issue or something.

Quote:
 A hare (rabbit) and a tortoise (turtle) are in a foot race (see figure (Figure 1) ). When the tortoise is 22 m from the finish line, he has a speed of 4.0 m/s but is 5.0 m behind the rabbit. The rabbit has a speed of 5.0 m/s. The rabbit is overconfident and assumes he has an easy win. So, during the remaining portion of the race, the rabbit decelerates at a constant rate of 0.52m/s^2 to the finish line. Find the acceleration needed by the tortoise to make the race a tie.

I first solved for the time required for the hare to get to the finish line as being roughly 2.9 seconds and used that to find the acceleration needed for the tortoise to get to the finish line in that time. In both cases I used the formula to solve for time and acceleration respectively:

Quote:
 position = initial position + (initial velocity)(time) + 1/2(acceleration)(time)^2

I ended up with 6.9 m/s^2 as my acceleration needed as it said to round to 2 significant figures but my homework software said that was wrong. Also, I didn't round anything early. Any idea what I did wrong?

Edit:
Found my mistake. I forgot that the hare's acceleration would be negative. I'm such a noob lol.
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Last edited by Stringz of Fury : Today at 01:24 PM.

Today, 04:10 PM   #9856
Masquirina
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Join Date: Nov 2012
Quote:
 Originally Posted by Stringz of Fury Any idea what I did wrong?
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