1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A variational problem with the constraint that the function be decreasing

  1. May 15, 2012 #1
    The problem is in the attached document. Thanks for any suggestions!
    Lennart
     
  2. jcsd
  3. May 16, 2012 #2
    The mathematical problem:

    $\theta$ is a constant that equals 0.8.

    We consider the set

    '$ S=\left\{(F,h):F is a decreasing function from R^{+} to R^{+}, h\in R, 0=1- \frac{\theta+1)}{\theta} \frac {(\int^{h}_{y=0} F(y) dy)}{/F(0)} \frac{F(0)-\frac{1}{2}}{F(0)-F(h)} \right\}$'
    The function L is defined on S by
    $ L(F,h)= \frac{\int^{h}_{x=0} \int^{h}_{y=x} F(y) dy dx} {\int^{h}_{x=0} \int^{h}_{y=0} F(y) dy dx} h$
    We want to find the maximal value of L.

    A further problem:
    Denote by $L(\theta)$ the maximal value of L. What is $max_{\theta \in (0,1)} \frac {L(\theta)}{\theta}$ ?
     
    Last edited: May 16, 2012
  4. May 16, 2012 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    If you want LaTeX to be readable in this forum, you need to remove the $ delimiters and enclose your commands between "[tex ]" and "[/tex ]" delimiters (no space after the word tex--I used a space so as to not confuse the system). For example, here is one of your formulas (justa copied and pasted between the delimiters I:
    [tex] L(F,h)= \frac{\int^{h}_{x=0} \int^{h}_{y=x} F(y) dy dx} {\int^{h}_{x=0} \int^{h}_{y=0} F(y) dy dx} h[/tex]

    RGV
     
  5. May 16, 2012 #4
    [tex]\theta[/tex] is a positive constant.

    We consider the set

    [tex] S=\left\{(F,h):F is a decreasing function from R^{+} to R^{+}, h\in R, 0=1- \frac{\theta+1}{\theta} \frac {\int^{h}_{y=0} F(y) dy}{F(0)} \frac{F(0)-\frac{1}{2}F(h)}{F(0)-F(h)} \right\}[/tex]
    The function L is defined on S by
    [tex] L(F,h)= \frac{\int^{h}_{x=0} \int^{h}_{y=x} F(y) dy dx} {\int^{h}_{x=0} \int^{h}_{y=0} F(y) dy dx} h[/tex]
    We want to find the maximal value of L, denoted by [tex]L(\theta)[/tex].

    A special question:
    What is L(0.8) ?

    A general question:
    What is [tex]max_{\theta \in (0,1)} \frac {L(\theta)}{\theta}[/tex] ?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A variational problem with the constraint that the function be decreasing
  1. Decreasing function (Replies: 1)

  2. Decreasing Functions (Replies: 3)

Loading...