/sci/ - Science & Math

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I don't know if this thread is about infinitely long questions or the infinite numbers of question but here some ideas..

The guess length of the proves depends on the system you consider. Specifically, if you need something often in your prove which itself contains long prove requirements (and so on) then the proves can be VERY long. Similar to Gödel constructions, you can show in logics capable of modeling arithmetic, that there are quite short proves (of length n say) for with there are no recursive functions f(n) such that the prove length is bounded by them. And recursive functions themselve can be quite long, e.g. Ackerman function
(oh wow, there is no wikipedia article on "formation sequence", weird.)

I'm interested in model theory a bit, I asked a question regarding the ordered pair here two day ago. Anyway, I laughed when I saw the way in which you have to extract the second element 6 of the pair (5,6) in the underlying language of logic.
http://en.wikipedia.org/wiki/Ordered_pair#Kuratowski_definition
(it's at the bottom of Kuratowski)
So clearly, the more you put in, the more you get out.
Coming back, there are some inference rules which are trivial and you'd use them often, but they have in principle to be written down in your prove all the time, e.g.
http://en.wikipedia.org/wiki/Deduction_Theorem
You can of course cheat the prove shorter by redundantly axiomatizing the theorem.
The question is if you want to (allow it to) jump over parts of the proof.

"The Art and Craft of Problem Solving" is supposed to be a very good book regarding this kind of competitional problems.

http://en.wikipedia.org/wiki/Removable_singularity

Okay, I guess this is a good way:

circle:
x^2+y^2==r^2
right circle is this circle, but shifted to the left by d/2, so:
(x-d/2)^2+y^2==r^2.
Solve for y(x).
Then the left upper corner of the square is the point where this y(x) and the line Y(x)=-x cross.
(there are actually two crossing points, so you take the upper one)
If you call the corresponding x-value x0, then the area of the square is (2*x0)^2.

>>4050255
>>4050280
>>4050373
why are you guys implying OP is asking for a way to calculate it? he already calculated it and he ask how a negative value is possible (that the value is negative is clear from the pic).

>>3913099
In number theory it's an axiom, see reflexivitty (you can regard this as part of the definition of "=")
In set theory it follows in one line from the axiom of extensionality
In philosophy it's a statement in the natural appearing framework of Mereology

>dat feel when you think you cleared everything up and everybody ignores you.

>>3913096
>doubting them is literally, entirely impossible
what is that supposed to mean? One can doubt anything. If you think that you are the only person in existence and every (nonreal) person who appears to you is making things up to confuse you - i.e. some sort of solipsistic paranoia - then you can clearly dounbt anything you think you know.
I wonder how you can think that doubting something is immpossible, that seems difficult to me.

Nice video. But I don't agree with him on a couple of points.
For example I think we make more physics than we discover. At least in the high energy department.

v(t) = v0 + a * t
a = (vmax-v0)/tmax

s = int{0 to tmax} v(t) dt = ... = (v0+vmax)*tmax/2

and since both vmax are the same, the one with v0>0 will have greater s.

>>3829642
in any case, this argument is the most Feynman'sh

check this bra:

http://www.youtube.com/watch?v=P0tLbl5LrJ8&feature=channel_video_title

>>3685870
>my example when
http://www.wolframalpha.com/input/?i={{0%2C-a%2Ca}%2C{a%2C0%2C-a}%2C{-a%2Ca%2C0}}&asynchronous=f
alse&equal=Submit

>>3615548
if V is your vector space, v is any vector in it and you have an algebra A with elements a over the vector space, then a specific vector v0 is cyclic if for all v you find an a in A such that

v = a(v0)

i.e. if you actually only need v0 and the algebra to create the vector space.

thats a definition from the top of my head, might be missing minor details.

In any way, it's a mathematical tool to create the particle space only from the vacuum state v0 and operators, these:
http://en.wikipedia.org/wiki/Creation_operators

see also W3 here:
http://en.wikipedia.org/wiki/Wightman_axioms