82
E.J. Brändas
certain input locations or key subject areas of the system are spatio-temporally correlated. In addition the nested characteristic of the non-commutative map recognize
interlevel communication and dynamics.
Belabouring the factorizations one could also simplify the graph, Eq. (4.3), as
the seven noncommuting factors below including the first column of (1)’s, leaving
out the final 5 repetitions, i.e.
{12} =
12 × (1) ⊕ 1 × (12) ⊕ 2 × (6) ⊕ 3 × (4) ⊕ 4 × (3) ⊕ 1 × (12) ⊕ 6 × (2)
.
In particular one sees that amongst the 12 columns there are 4 columns that are
of the type 1 × (12), i.e. that one third of the columns are not factorizable, while
the remaining two thirds are. Also the choice of m = 12 has a particular relevance
since the diagram supports the codon triplets as fundamental cycles. To include cycles corresponding to the 20 amino acids one could append the example to the case
m = 12 × 5 = 60, i.e. including the factor “5”, which will add columns divisible by
5, 10, 15, 20, 30. Here one finds that 40 columns of 60 are not factorizable, increasing the abundance of the latter (m = 12) from 1/3 to 2/3, a number rapidly
increasing with m. Already at this instance, one is able to predict the characteristic
that a large number of vectors in fact do not convey any information as they contain
no closed cycles of order smaller than m compared to the number of those that did
contain such cycles, not to mention the situation when m equals a prime number.
It has not escaped our notice that the specific property postulated above immediately suggests a possible exon-intron mechanism for the genetic material; see the
comment in the previous section.
The general diagram for a number n, with an initial factorial progression would
essentially read (n k = n/k)
(n n
2
)
(n p k )
( ·)
( n p k )
· · (·) ·
·
·
· (·) · ·
(n 3 ) · ·
·
· (n n
2
) ·
·
· ·(n 3 )
(n 2 )
( n p k )
( n p k )
( n 2 )
(n)
(n 3 ) · ·
· (n) · (n n
2
) · (n) ·
· ·(n 3 )
(n)
(n 2 )
( n p k )
( n p k )
( n 2 )
(n 3 ) · ·
·
· (n n
2
) ·
·
· ·(n 3 )
· · (·) ·
·
·
· (·) · ·
(n p k )
( ·)
( n p k )
(n n
2
)
(4.4)
In fact one can use the unique prime decomposition of every number n to devise
a so-called Gödel numbering identifying any “proposition”, cf. the famous Gödel
incompleteness theorem in propositional logic [21], see also [11, 12], the “proposition” here in general meaning anything from amino acid identification to higher
order ententional meanings [10]. To account for the nested combination of characteristics in (4.4), Nature must develop a more compact and flexible code than the
Gödel numbering, for instance bijective K-numeral systems e.g. the bijective base
E.J. Brändas
certain input locations or key subject areas of the system are spatio-temporally correlated. In addition the nested characteristic of the non-commutative map recognize
interlevel communication and dynamics.
Belabouring the factorizations one could also simplify the graph, Eq. (4.3), as
the seven noncommuting factors below including the first column of (1)’s, leaving
out the final 5 repetitions, i.e.
{12} =
12 × (1) ⊕ 1 × (12) ⊕ 2 × (6) ⊕ 3 × (4) ⊕ 4 × (3) ⊕ 1 × (12) ⊕ 6 × (2)
.
In particular one sees that amongst the 12 columns there are 4 columns that are
of the type 1 × (12), i.e. that one third of the columns are not factorizable, while
the remaining two thirds are. Also the choice of m = 12 has a particular relevance
since the diagram supports the codon triplets as fundamental cycles. To include cycles corresponding to the 20 amino acids one could append the example to the case
m = 12 × 5 = 60, i.e. including the factor “5”, which will add columns divisible by
5, 10, 15, 20, 30. Here one finds that 40 columns of 60 are not factorizable, increasing the abundance of the latter (m = 12) from 1/3 to 2/3, a number rapidly
increasing with m. Already at this instance, one is able to predict the characteristic
that a large number of vectors in fact do not convey any information as they contain
no closed cycles of order smaller than m compared to the number of those that did
contain such cycles, not to mention the situation when m equals a prime number.
It has not escaped our notice that the specific property postulated above immediately suggests a possible exon-intron mechanism for the genetic material; see the
comment in the previous section.
The general diagram for a number n, with an initial factorial progression would
essentially read (n k = n/k)
(n n
2
)
(n p k )
( ·)
( n p k )
· · (·) ·
·
·
· (·) · ·
(n 3 ) · ·
·
· (n n
2
) ·
·
· ·(n 3 )
(n 2 )
( n p k )
( n p k )
( n 2 )
(n)
(n 3 ) · ·
· (n) · (n n
2
) · (n) ·
· ·(n 3 )
(n)
(n 2 )
( n p k )
( n p k )
( n 2 )
(n 3 ) · ·
·
· (n n
2
) ·
·
· ·(n 3 )
· · (·) ·
·
·
· (·) · ·
(n p k )
( ·)
( n p k )
(n n
2
)
(4.4)
In fact one can use the unique prime decomposition of every number n to devise
a so-called Gödel numbering identifying any “proposition”, cf. the famous Gödel
incompleteness theorem in propositional logic [21], see also [11, 12], the “proposition” here in general meaning anything from amino acid identification to higher
order ententional meanings [10]. To account for the nested combination of characteristics in (4.4), Nature must develop a more compact and flexible code than the
Gödel numbering, for instance bijective K-numeral systems e.g. the bijective base