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modeling using differential eq.

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modeling using differential eq.

how would one model a diffential equation for growth of a population where the birth rate is not constant (dP/dt=aP) but rather proportional to P^k for some small positive constant k. Would it just be dP/dt=aP^k?

thanks!


sorry, I don't know how to post a comment, so I'm forming one
using the reply feature of an old feed... they should make this
site a little easier to use! The search engine was out-of-service.

I'm surprised to see that Ramsey Numbers are so similar to the
search for prime numbers. If you're not familar with the theory
then go to google, type in 'friends and strangers', and select
the article written by Imre Leader; it's a great story.

There truly is order in chaos, throughout the sequence. Not only
does someone have to find the minimal example for a Ramsey Number,
but they have to prove that the value below what's true is also
a counter-example; similar to proving the primality of a number.

Please don't believe the propaganda... there's a discrete formula
for a Ramsey Number... I stumbled on it after 3 hard-earned days.

They say that for n>=2, the Ramsey Numbers, R(2,2)= 4, R(3,3)= 6,
& R(4,4)= 18, and then stop searching for them. They give an upper
boundary of 43 and a lower boundary of 49 for R(5,5). That's great,
because I discovered that 46 is the correct answer for R(5,5).

I believe that it's become a Millenium Project; I found that the
predictive formula for n > 2 is... R(n,n)= 4*(3^(n-3)) +2*(2n-5).

The case for 2 is trivial much as 2 is the only even prime number,
and the others all follow... n:R(n,n)... 3:6...4:18...5:46... and
I believe that 6:122 and quite definitely 7:342. The time required
to check such answers is HUGE like that of large prime numbers.

I believe the discrete formula is... R(n,n)= 4*(3^(n-3))+2*(2n-5).
The pictures I drew were as fascinating as the solution; I think
Frank Ramsey would have been delighted with my discovery. I can't
imagine how the computation would go unless is had and FFT algor-
ithm like the one George Woltman built to compute large numbers.

Cheers, Bill

this was done by hand; I could probably do the 122-example also, but
too much typing; I know what the increment choices are...

1.,A,A,A,A,
(all friends)

2.,B,A,A,A,
3.,A,B,A,A,
4.,B,B,A,A,

5.,A,A,A,A,
the 'friends' list appears
more than once, but that's
O.K.; it would not be min-
imal if a second list of
strangers(T.) occured!

6.,B,A,A,A,
7.,A,B,B,A,
8.,B,B,B,A,
9.,A,A,B,A,
R.,B,A,B,B,
S.,A,B,B,B,

T.,B,B,B,B,
(all strangers)

U.,A,A,A,B,
V.,B,A,A,B,
W.,A,B,A,B,
X.,B,B,A,B,
Y.,A,A,A,B,
Z.,B,A,A,B,

R(4,4)= 18.

the rows 1. and T. contain
all friends & all strangers!
----------------------------
01.,A,A,A,A,A
(all friends)

02.,B,A,A,A,A
03.,A,B,A,A,A
04.,B,B,A,A,A
05.,A,A,A,A,A
06.,B,A,A,A,A
07.,A,B,B,A,A
08.,B,B,B,A,A
09.,A,A,B,A,A
10,,B,A,B,A,A
11.,A,B,B,A,A
12.,B,B,B,A,A
13.,A,A,A,B,A
14.,B,A,A,B,A
15.,A,B,A,B,A
16.,B,B,A,B,A
17.,A,A,A,B,A
18.,B,A,A,B,A
19.,A,B,B,B,A
20.,B,B,B,B,A
21.,A,A,B,B,A
22.,B,A,B,B,A
23.,A,B,B,B,A

24.,B,B,B,B,B
(all strangers)

25.,A,A,A,A,B
26.,B,A,A,A,B
27.,A,B,A,A,B
28.,B,B,A,A,B
29.,A,A,A,A,B
30.,B,A,A,A,B
31.,A,B,B,A,B
32.,B,B,B,A,B
33.,A,A,B,A,B
34.,B,A,B,A,B
35.,A,B,B,A,B
36.,B,B,B,A,B
37.,A,A,A,B,B
38.,B,A,A,B,B
39.,A,B,A,B,B
40.,B,B,A,B,B
41.,A,A,A,B,B
42.,B,A,A,B,B
43.,A,B,B,B,B

44.,B,B,B,B,B
the 'strangers' appears
more than once, but that's
O.K.; it would not be min-
imal if a second list of
friends(1.) occured!

45.,A,A,B,B,B
46.,B,A,B,B,B

R(5,5)= 46.

the rows 1. and 24. contain
all friends & all strangers!

there's even a reason that the
12th & 24th rows hold the list
of strangers... so the answer
will be easy to trace 4*something
the something is 1, 3, 6,...
when n= 3, 4, 5,...
the formula that connects these two
sequences shows you where to look
for strangers and the friends are
always in line one.

oh... the counter-example ???
17 and 45; not able to do it.
I know that choices to make though.
again, Bill

done by hand...

the counter-example for 46; 45 is not enough!

notice how the columns are formed; that's how
I'm able to find the counter-example.

if BABBB were 10111, then it would equal 23 decimal.
and 23 divides 46.

column one: every other on/off
column two: two on, two off
column three: four on, four off
column four: six on, six off
last column: split as best possible

AAAAA BABBB
BAAAA AABBB
ABAAA BBBBB
BBAAA ABBBB
AABAA BAABB
BABAA AAABB
ABBBA BBAAB
BBBBA ABAAB
AAABA BABAB
BAABA AABAB
ABABA BBBAB
BBABA ABBAB
AABAA BAABB
BABAA AAABB
ABBAA BBABB
BBBAA ABABB
AAAAA BABBB
BAAAA AABBB
ABABA BBBAB
BBABA ABBAB
AABBA BAAAB
BABBA AAAAB
ABBBB BBAAA
BBBBB ABAAA
AAAAB BABBA
BAAAB AABBA
ABAAB BBBBA
BBAAB ABBBA
AABAB BAABA
BABAB AAABA
ABBBB BBAAA
BBBBB ABAAA
AAABB BABAA
BAABB AABAA
ABABB BBBAA
BBABB ABBAA
AABAB BAABA
BABAB AAABA
ABBAB BBABA
BBBAB ABABA
AAAAB BABBA
BAAAB AABBA
ABABB BBBAA
BBABB ABBAA
AABBB BAAAA

both sets together offer total randomness but
the right set doesn't guarantee an 'all off'
setting!; i.e. A=off, B=on; I can find the
counter-example for all of them the same way!

Bill

seems to be a hard one, hope someone could help you..

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