Finding the bounds of a ratio

  • #1
466
5
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}##
 

Answers and Replies

  • #2
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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?
 
  • #3
466
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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?

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?
 
  • #4
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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
 
  • #5
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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.
 
  • #6
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What if I choose ##r_i=100,r_R=1,R=2?## Then the ratio is greater than 1 right?
 
  • #7
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##\{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)
 
  • #8
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...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...
 
  • #9
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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.
 
  • #10
Stephen Tashi
Science Advisor
7,713
1,519
given the set ##\{r_i, r_{ii}, r_{iii}, ... , r_R\}##
where ##r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}##

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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