Possible applications

Kreizhn · Nov 26, 2011

I was given a problem by a professor to prove the following problem:

If [itex] f:[0,1] \to \mathbb R [/itex] is a twice differentiable function, define [itex] \Gamma = \{y = f(x)\} [/itex] the curve associated to f. Show that the following are equivalent:

[itex]m(\Gamma+\Gamma)>0[/itex]

[itex] \Gamma +\Gamma [/itex] contains an open set.

f is non-linear

Anyway, I have done this but am supposed to remark on possible applications. I'm not sure to what I could apply this though. Maybe something to do with ergodics? Any suggestions would be appreciated.

Kreizhn · Nov 28, 2011

Any suggestions?

HallsofIvy · Nov 28, 2011

I take it you mean that [itex]\Gamma[/itex] is the set [itex]\{(x, y)| y= f(x)\}[/itex].

What do you mean by [itex]\Gamma+ \Gamma[/itex]? How are you adding sets?

Kreizhn · Nov 28, 2011

Yes, that is what I meant. Sorry for the sloppiness, though I believe it's not an uncommon shorthand.

Set addition is taken to be naive: nothing special like essential sums. So
[tex] A+B = \{ a+b: a \in A, b \in B\} [/tex]

Kreizhn · Nov 29, 2011

Anybody have any ideas?

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Possible applications

Kreizhn

Kreizhn

HallsofIvy

Kreizhn

Kreizhn