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Topic: Matheology § 250
Replies: 5   Last Post: Apr 16, 2013 1:39 PM

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 rt servo Posts: 19 Registered: 2/1/05
Re: Matheology § 250
Posted: Apr 16, 2013 9:28 AM

On 16/04/2013 5:38 AM, Virgil wrote:
> In article
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> Matheology § 250
>>
>> Fibonacci-sequences with fatalities
>>
>> The Fibonacci-sequence
>> f(n) = f(n-1) + f(n-2) for n > 2 mit f(1) = f(2) = 1
>> the first recursively defined sequence in human history (Leonardo of
>> Pisa, 1170 - 1240) should be well known. A pair of rabbits that
>> reproduces itself monthly as from the completed second month on will
>> yield 144 pairs after the first 12 months:
>> 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
>>
>> If we assume that each pair reproduces itself after two months for the
>> last time and dies afterwards, we get a much more trivial sequencence:
>> 1, 1, 1, ...
>> However the rabbits behind this numbers change. If we call them in the
>> somewhat unimaginative but effective manner of the old Romans, we get
>> Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona,
>> Decima and so on.
>>
>> A more interesting question is brought up, if the parent pair dies
>> immediately after the birth of its second child pair. Then the births
>> in month n can be traced back to pairs who have been born in months
>> n-2 and n-3.
>> g(n) = g(n-2) + g(n-3).
>> The number f(n) of pairs in month n is given by those born in month n,
>> i.e. g(n) and those already present in month n-1, i.e., f(n-1), minus
>> those who died in month n (i.e. those who were born in month n-3:
>> f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1)
>> g(n-2) = f(n) - f(n-1)
>> g(n-2) = g(n-4) + g(n-5)
>> = f(n-2) - f(n-3) + f(n-3) - f(n-4)
>> = f(n-2) - f(n-4)
>> For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2.
>> f(n) = f(n-1) + f(n-2) - f(n-4)
>> The number of pairs during the first 12 months is
>> 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21.
>> The sequence grows less than the original one, but without enemys or
>> other restrictions it will grow beyond every threshold. If we wait
>> aleph_0 days (or use the trick that the duration of pregnancy is
>> halved in each step, facilitated by genetic evolution), we will get
>> infinitely many pairs - a nameless number, alas of nameless rabbits,
>> because they cannot be distinguished. The set of all Old-Roman names
>> has been exhaused already, and even all of Peano's New-Roman names S0,
>> SS0, SSS0, ... have been passed over to pairs which already have
>> passed away. That is amazing, since none of the pairs of the original
>> and much more abundant Fibonacci sequence has to miss a name.
>>
>> But this sequence with fatalities can also be obtained without
>> fatalities (killings), namely if each pair has to pause for two months
>> after each littering in order to litter again in the following month.
>> Mathematically, there is no difference. Set theory, however, yields a
>> completely different limit in this case. The limit set of living
>> rabbits is no longer empty, but it is infinite - and every rabbit has
>> a name.

>
> There may be no difference in the number of rabbits but there will be a
> considerable difference in both the parent-child linkages of those
> rabbits and the number of dead rabbits.
>
> Either of which is enough to distinguish the two cases as being
> mathematically different, even if not WMytheologally different.

In addition to that, there is only one person here littering: WM

Date Subject Author
4/16/13 mueckenh@rz.fh-augsburg.de
4/16/13 Virgil
4/16/13 rt servo
4/16/13 FredJeffries@gmail.com
4/16/13 fom
4/16/13 fom