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A Finding the bounds of a ratio

  1. Aug 30, 2016 #1
    Sorry in advance if I've posted in the wrong section.

    given the set ##\{r_i, r_{ii}, r_{iii}, ... , r_R\}##
    where ##r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}##


    How would you go about finding the bounds of something like this, or determining if it even has any?

    ##( \, log_2{\frac{(\sum_{i=1}^R{r_i})!}{\prod_{i=1}^R{(r_i!)}}}) \, :\sum_{i=1}^{R}{i \cdot r_i}##
     
  2. jcsd
  3. Aug 30, 2016 #2
    Since ##r_i## is the largest and ##r_R## is the smallest, the ratio in the log must smaller than ##\frac{(r_iR)!}{(r_R!)^R},## which may depend on ##r_i##, ##r_R## and ##R.## Does the problem require a fixed constant?
     
  4. Aug 31, 2016 #3
    The fraction within the log is always greater than or equal to one, and only gets larger and larger with increasing ##R## and ##\bar{r}##. But I don't know how I would go about expressing the rate of change with respect to the denominator of the ratio.
    Hmm, I guess... what I'm looking for is more of a simplification, because that would be easier to look at. But I'm not sure what you meant by the fixed constant thing, can you elaborate?
     
  5. Sep 2, 2016 #4
    At first I'm not sure what you want also, and guess maybe you hope to get the ratio can be bounded with some cool number, maybe ##Ce^R##... likewise haha.
    For me the fraction cannot be simplified XD
     
  6. Sep 2, 2016 #5
    For any set of values I've tested so far, the ratio has not exceeded the value 1. In fact i haven't found it to ever reach 1. I wanted to know what properties of a given set determined its closeness to the value 1. I hope this makes sense.
     
  7. Sep 2, 2016 #6
    What if I choose ##r_i=100,r_R=1,R=2?## Then the ratio is greater than 1 right?
     
  8. Sep 2, 2016 #7
    ##\{100\ ,\ 1\}## yields...

    ##numerator: log_2{ \frac{ 101! }{ 100! \cdot 1! } } = log_2{ \frac{ 101! }{ 100! } } = log_2{101} =7##
    ##ratio: \frac{7}{102} = 0.0686##

    (remember the numerator is logged by base 2)
     
  9. Sep 2, 2016 #8
    ...My fault to miss the latter. I just saw the fraction. So the sum of ##r_x## is also concerned... Then it's harder to simplify...
     
  10. Sep 2, 2016 #9
    I know I stated that simplification was my goal. But I suppose that's not the only way to achieve my "true goal", which I'm still getting closer to.. (sorry)

    I postulate that there is a relationship between the numbers in the given set, that dictates its closeness to the value 1. I just want to figure that out..
    even a pointer to how I might go about that would be of great help.
     
  11. Sep 6, 2016 #10

    Stephen Tashi

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    Science Advisor

    As matter of vocabulary, if we were "given" that set of numbers, the ratio would have a single numerical value, so when you mention "bounds" aren't we treating the ##R_i## as variables and asking how to maximize or minimize the ratio? Can we start by solving the max-min problem for 2 variables ?
     
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