Tuesday, June 05, 2007
Discrete damped oscillator, prime numbers
Back in 2000 I discovered a damped discrete oscillator, which I noticed later as I wasn't looking for that, as I was looking for a prime counting function.
Here is the damped discrete oscillator:
With natural numbers, if y<sqrt{x} then
P(x,y) = x-1-\sum_{k=2}^y {((P([x/k],k-1)-P(k-1,\sqrt{k-1}))\Delta P(k,\sqrt{k})}
else P(x,y) = P(x,\sqrt{x}) where
Delta P(k,\sqrt{k}) = P(k,\sqrt{k})-P(k-1,\sqrt{k-1})
and that is a damped oscillator as whenever k is not prime Delta P(k,sqrt{k}) equals 0, while it equals 1 if k is prime, so you get this oscillation that drops as you iterate from k=2 up to sqrt{x}.
Graph it. Looks like a bouncing ball.
Anyone have a clue how to figure out the period? That actually could be a good research question as you can move from summing a partial difference equation, which is what's being done above, to integrating a partial differential equation.
Could primes and bouncing balls be related? Like somehow wrapped up in a bouncing rubber ball is information about prime numbers?
I'm posting to a physics group as I've talked about this result with math people, well, since 2000!
I don't think they care, especially not about a physics question like, what is the period of the damped oscillation?
And, are there any other discrete damped oscillators known?
>From the run around they've given me over the years it is clear to me that mathematicians do not care about such things, but I'm including the sci.math newsgroup just in case.
I guess the result is not "pure" enough for them! After all, what do mathematicians care about damped oscillations?
The big question to me is, can any of you out there calculate the period? Oh, and yes, the discrete calculation gives the count of prime numbers, perfectly, out to positive infinity.
The mathematics is absolutely correct. Perfectly correct.
To get a prettier look at the equations as you can see the in a pdf, you have to go to my Extreme Mathematics Google Group:
http://groups.google.com/group/extrememathematics/web/counting-primes
I presented the paper to a couple of math journals but they all rejected it. Maybe I should re-write for a physics journal focusing on the damped oscillation?
Here is the damped discrete oscillator:
With natural numbers, if y<sqrt{x} then
P(x,y) = x-1-\sum_{k=2}^y {((P([x/k],k-1)-P(k-1,\sqrt{k-1}))\Delta P(k,\sqrt{k})}
else P(x,y) = P(x,\sqrt{x}) where
Delta P(k,\sqrt{k}) = P(k,\sqrt{k})-P(k-1,\sqrt{k-1})
and that is a damped oscillator as whenever k is not prime Delta P(k,sqrt{k}) equals 0, while it equals 1 if k is prime, so you get this oscillation that drops as you iterate from k=2 up to sqrt{x}.
Graph it. Looks like a bouncing ball.
Anyone have a clue how to figure out the period? That actually could be a good research question as you can move from summing a partial difference equation, which is what's being done above, to integrating a partial differential equation.
Could primes and bouncing balls be related? Like somehow wrapped up in a bouncing rubber ball is information about prime numbers?
I'm posting to a physics group as I've talked about this result with math people, well, since 2000!
I don't think they care, especially not about a physics question like, what is the period of the damped oscillation?
And, are there any other discrete damped oscillators known?
>From the run around they've given me over the years it is clear to me that mathematicians do not care about such things, but I'm including the sci.math newsgroup just in case.
I guess the result is not "pure" enough for them! After all, what do mathematicians care about damped oscillations?
The big question to me is, can any of you out there calculate the period? Oh, and yes, the discrete calculation gives the count of prime numbers, perfectly, out to positive infinity.
The mathematics is absolutely correct. Perfectly correct.
To get a prettier look at the equations as you can see the in a pdf, you have to go to my Extreme Mathematics Google Group:
http://groups.google.com/group/extrememathematics/web/counting-primes
I presented the paper to a couple of math journals but they all rejected it. Maybe I should re-write for a physics journal focusing on the damped oscillation?
# posted by James Harris @ 14:19 
