modeling using differential eq.

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# modeling using differential eq.

Submitted by crazziqt467 on Mon, 10/05/2009 - 06:24

Forums:

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!

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## Versions

(v1) by crazziqt467 2009-10-05

## RE: Ramsey Number formula

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

## Re: RE: Ramsey Number formula

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

## counter ex. for 46: Ramsey Numbers

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

## Re: modeling using differential eq.

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