Possible applications

  • Thread starter Kreizhn
  • Start date
  • #1
743
1
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:
  1. [itex]m(\Gamma+\Gamma)>0[/itex]
  2. [itex] \Gamma +\Gamma [/itex] contains an open set.
  3. 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.
 

Answers and Replies

  • #2
743
1
Any suggestions?
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
961
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?
 
  • #4
743
1
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]
 
  • #5
743
1
Anybody have any ideas?
 

Related Threads on Possible applications

  • Last Post
Replies
6
Views
12K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
9
Views
11K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
6K
  • Last Post
Replies
5
Views
19K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
7
Views
2K
Top