(July 31, 2012 at 6:47 pm)Categories+Sheaves Wrote:(July 31, 2012 at 6:26 pm)CliveStaples Wrote: [quote='Categories+Sheaves' pid='317718' dateline='1343773102']Ah. Indeed I did.
You can do the first conjecture in one line by multiplying both sides by (1/3)g, yielding
(a/3)g + (b/3)g +(1/3)g >= (a/3 + b/3 + 1/3)g
You've improperly switched some negatives to positives there, I think.
So that gives us (a-1)g + (b)g >= (a + b - 1)g as long as a > 1 (negatives makes this harder :/ ).
By conjecture #2...
1-0 >= ag - (a-1)g and so (a-1)g >= ag - 1
So just using concavity isn't enough...(July 31, 2012 at 6:26 pm)CliveStaples Wrote: Any proof will do, it's a useful lemma for my girlfriend's research.Because there's nothing more interesting than another person's question: what's the problem/literature you're looking at?
It's to prove a lower bound for a function on a network that gives the 'energy' of the network.
Conjecture 2 can be used to prove conjecture 1 through induction.
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