03/09/2016, 10:33 AM

@marraco, you brought up a couple of issues of interest to me.

Let and be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.

A partition of is , usually denoted by with ; where is the number of parts of size . The partition function is a decategorized version of , the function enumerates the integer partitions of , while is the cardinality of the enumeration of .

Setting results in

The Taylors series of is derived by evaluating

the derivatives of the iterated function at a fixed point

by setting and separating out the

term of the summation that is dependent on .

The remaining terms of the summation are only dependent on , where .

Let me know if you have any questions.

Yes, if logarithms are infinitely multivalued, then tetration must account for it also. All my work is based on setting arbitrary fixed points to . That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for . Setting the entropy to being low for the exponential map can be achieved by setting close to unity in . Then the dynamics of neighboring fixed points can be computed from a fixed point.

(03/08/2016, 06:58 PM)marraco Wrote: There is a direct connection to partition numbers (number theory), in the Taylor series.See Combinatorics. There are several types of set partitions.

Let and be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.

A partition of is , usually denoted by with ; where is the number of parts of size . The partition function is a decategorized version of , the function enumerates the integer partitions of , while is the cardinality of the enumeration of .

Setting results in

The Taylors series of is derived by evaluating

the derivatives of the iterated function at a fixed point

by setting and separating out the

term of the summation that is dependent on .

The remaining terms of the summation are only dependent on , where .

Let me know if you have any questions.

(03/08/2016, 06:58 PM)marraco Wrote: Also, many equations in use of different areas of mathematics already use iterated logarithms. But to consider those tetration to negative exponents, is necessary to consider tetration a multivalued function, with multiple branches.

Yes, if logarithms are infinitely multivalued, then tetration must account for it also. All my work is based on setting arbitrary fixed points to . That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for . Setting the entropy to being low for the exponential map can be achieved by setting close to unity in . Then the dynamics of neighboring fixed points can be computed from a fixed point.