03/09/2016, 10:33 AM
@marraco, you brought up a couple of issues of interest to me.
Let \( f(z) \) and \( g(z) \) be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.
\( D^nf(g(z))=\sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^kf)(g(z))\left(\frac{Dg(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^ng(z)}{n!}\right)^{k_n} \)
A partition of \( n \) is \( \pi(n) \), usually denoted by \( 1^{k_1}2^{k_2}\cdots n^{k_n} \) with \( k_1+2k_2+ \cdots nk_n=k \); where \( k_i \) is the number of parts of size \( i \). The partition function \( p(n) \) is a decategorized version of \( \pi(n) \), the function \( \pi(n) \) enumerates the integer partitions of \( n \), while \( p(n) \) is the cardinality of the enumeration of \( \pi(n) \).
Setting \( g(z) = f^{t-1}(z) \) results in
\( D^n f^t(z) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^k f)(f^{t-1}(z))\left(\frac{Df^{t-1}(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(z)}{n!}\right)^{k_n} \)
The Taylors series of \( f^t(z) \) is derived by evaluating
the derivatives of the iterated function at a fixed point
\( f^t(0) \) by setting \( z=0 \) and separating out the \( k_n \)
term of the summation that is dependent on \( D^n f^{t-1}(0) \).
\( D^n f^t(0) = \sum \frac{n!(D^k f)(0)}{k_1! \cdots k_{n-1}!} \left(\frac{Df^{t-1}(0)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(0)}{(n-1)!}\right)^{k_{n-1}} + (D f)(0) D^n f^{t-1}(0) \)
The remaining \( p(n)-1 \) terms of the summation are only dependent on \( D^k f^{t-1}(0) \), where \( 0<k<n \).
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 \( z=0 \). That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for \( ^{-\infty}a \). Setting the entropy to being low for the exponential map can be achieved by setting \( a \) close to unity in \( ^za \). 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 \( f(z) \) and \( g(z) \) be holomorphic functions, then the Bell polynomials can be constructed using Faa Di Bruno's formula.
\( D^nf(g(z))=\sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^kf)(g(z))\left(\frac{Dg(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^ng(z)}{n!}\right)^{k_n} \)
A partition of \( n \) is \( \pi(n) \), usually denoted by \( 1^{k_1}2^{k_2}\cdots n^{k_n} \) with \( k_1+2k_2+ \cdots nk_n=k \); where \( k_i \) is the number of parts of size \( i \). The partition function \( p(n) \) is a decategorized version of \( \pi(n) \), the function \( \pi(n) \) enumerates the integer partitions of \( n \), while \( p(n) \) is the cardinality of the enumeration of \( \pi(n) \).
Setting \( g(z) = f^{t-1}(z) \) results in
\( D^n f^t(z) = \sum_{\pi(n)} \frac{n!}{k_1! \cdots k_n!} (D^k f)(f^{t-1}(z))\left(\frac{Df^{t-1}(z)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(z)}{n!}\right)^{k_n} \)
The Taylors series of \( f^t(z) \) is derived by evaluating
the derivatives of the iterated function at a fixed point
\( f^t(0) \) by setting \( z=0 \) and separating out the \( k_n \)
term of the summation that is dependent on \( D^n f^{t-1}(0) \).
\( D^n f^t(0) = \sum \frac{n!(D^k f)(0)}{k_1! \cdots k_{n-1}!} \left(\frac{Df^{t-1}(0)}{1!}\right)^{k_1} \cdots \left(\frac{D^n f^{t-1}(0)}{(n-1)!}\right)^{k_{n-1}} + (D f)(0) D^n f^{t-1}(0) \)
The remaining \( p(n)-1 \) terms of the summation are only dependent on \( D^k f^{t-1}(0) \), where \( 0<k<n \).
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 \( z=0 \). That allows me to consider the dynamics of the infinite branches, because each of the branches has a fixed point for \( ^{-\infty}a \). Setting the entropy to being low for the exponential map can be achieved by setting \( a \) close to unity in \( ^za \). Then the dynamics of neighboring fixed points can be computed from a fixed point.
Daniel