What is F/A analysis? Never heard of it before.
Wikipedia defines Joule as "the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second." This seems to imply that a resistor of 1 ohm will always dissipate the exact same amount of heat regardless of what material it's made of. That feels surprising and counter-intuitive. Or, perhaps, it could be restated as: It seems to imply that there's a tight unique relationship between resistance and heat dissipation, regardless of the material.
(I understand that it's probably not possible to say "proof by contradiction is impossible for this number." I suppose what I'm asking is if there's some number that can be proven to not be rational, but for which there is no known proof by contradiction, or that proof is way too complicated to be feasible and there's a much easier way of proving it.)
Do you mean C(f+1,h) = 2 * C(f,h) + ~C(f-h, h)? This one instead would agree with your calculation C(6,2) = 2 * C(5,2) + ~C(3, 2) when f=5, h=2.
Some of my calculations don't seem to agree with this. For example: C(4,2)=8 (HHHH HHHT HHTH HHTT THHH THHT HTHH TTHH) C(4,1)=15 (everything but TTTT) ~C(3,1)=1 (TTT) 8≠(15-1)/2=7
HXXX | True
THXX | True
?THX | True
??TH | True
Other | FalseH(H)XX | True
TH(H)X | True
?TH(H) | True
(??TH) | False
Other | FalseHHXXX | True
THHXX | True
?THHX | True
??THH | True
Other | FalseHHXXXX | True
THHXXX | True
?THHXX | True
??THHX | True
???THH | True
Other | FalseHHXXX(X) | True
THHXX(X) | True
?THHX(X) | True
??THH(X) | True
(???THH) | True
Other | FalseHHXXXX | True
THHXXX | True
?THHXX | True
??THHX | True
???THH | True
Other | FalseHH(H)XXX | True
THH(H)XX | True
?THH(H)X | True
??THH(H) | True
(???THH) | False
Other | FalseBy the way, for the problem I proposed, the answer is: When you run a secure RNG from a secure cipher, it's guaranteed that all sequential outputs will be different from each other. That's because if you had two values x and y with f(x)=f(y), it would be impossible to decrypt the cipher, even with the key. The result is that, if you read all 2^n numbers generated, you'd see that they are a permutation of all values in the range [0, 2^n-1]. That's very improbable to arise from random chance, you'd expect some values to repeat! It's counter intuitive, but it's more probable that you see a value repeating even if you read a square root of the total number. See here for an explanation. So, if you simply read the numbers, at around the square root of total possibilities you should start seeing repetitions. If that does not happen, it's likely that the source of these numbers is not really random, but generated by a block cipher in counter mode!
I am going for the good ending, possibly the fastest route, but I may need assistance
This is (as far as I can tell) what medals are available. This means that the best route is to win every system except Kurasawa.
So the fastest route for *just* the good ending is eject every mission until you reach Hubble's Star. Then play Hubble's Star missions 1 and 3 (eject from 2). And play Rostov missions 2 and 3 (eject from 1). Finally eject from all 4 missions in Venice.
(This is probably faster than an alternate route where you win Port Hedland, because Port Hedland requires that you achieve every objective in order to win, so you have to play all three missions.)
Considering the TAS has already gone to McAliffe, PikachuMan is probably intending on either an all metals or max kills route or just play through the game "as well as possible." Completing all objectives in the shortest amount of time on the route that naturally follows from completing all objectives.
Isn't even the notion that the universe might be infinite in contradiction with the hypothesis that at one point the universe, in its entirety, was a singularity that expanded? I don't think you can have both. That would be impossible.