Difference between revisions of "Schroeder equation"

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[[File:Ernst schroederFragment.jpg|thumb|[[Ernst Schroeder|E.Schroeder]]
 +
[[File:Ernst schroederFragment.jpg|thumb|[[Ernst Schroeder]]
 
<ref>https://en.wikipedia.org/wiki/Ernst_Schröder
  
<ref>https://en.wikipedia.org/wiki/Ernst_Schröder
 
</ref>]]
  
</ref>]]
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</ref> can be written as follows:
  
</ref> can be written as follows:
   
−
(1) $~ ~ ~ g\Big(T(z)\Big)= s \, g(z)$
 +
(1) \(~ ~ ~ g\Big(T(z)\Big)= s \, g(z)\)
   
−
where function $~T~$ and constant $~s~$ are supposed to be given, and the solution $~g~$ is required to be constructed.
 +
where function \(~T~\) is supposed to be given, while constant \(s\) and function\(~g~\) should be constructed.
−
Usially, it is assumed that $s\!=\!T'(0)$, and both, $T$ and $g$ are holomorphic at least in some vicinity of zero.
 +
Usially, it is assumed that \(s\!=\!T'(0)\), and both, \(T\) and \(g\) are holomorphic at least in some vicinity of zero.
   
−
The Schroeder equation is related to the [[Abel equation]] and the [[transfer equation]]; the solutions (called [[Schroeder function]]s ) are related to the [[superfunction]]s and the [[Abel funciton]]s.
 +
The Schroeder equation is related to the [[Abel equation]] and the [[transfer equation]]; the solutions (called [[Schroeder function]]s ) are related to the [[superfunction]]s and the [[Abel function]]s.
−
These relations are especially simple for the case when function $~T~$ has a real [[fixed point]] $~L~$.
 +
These relations are especially simple for the case when function \(~T~\) has a real [[fixed point]] \(~L~\).
   
 
==About the terminology==
  
==About the terminology==
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[[Journal of thermal analysis and Calorimetry]], v.91,(2008), p.569-573
  
[[Journal of thermal analysis and Calorimetry]], v.91,(2008), p.569-573
 
</ref><ref>
  
</ref><ref>
−
http://193.146.160.29/gtb/sod/usu/$UBUG/repositorio/10264342_BOBKOVA.pdf
 +
http://193.146.160.29/gtb/sod/usu/\(UBUG/repositorio/10264342_BOBKOVA.pdf
 
N. M. Bobkova. Thermal Expansion of Binary Borate Glasses and Their Structure.
  
N. M. Bobkova. Thermal Expansion of Binary Borate Glasses and Their Structure.
 
[[Glass Physics and Chemistry]], Vol. 29, No. 5, 2003, pp. 501–507.
  
[[Glass Physics and Chemistry]], Vol. 29, No. 5, 2003, pp. 501–507.
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==Relation to the [[Abel equation]]==
  
==Relation to the [[Abel equation]]==
−
Consider [[logarithm]] to base $~s~$ from both sides of equation (1), assuming that $~s~$ and $~g(z)~$ are not real negative number nor zero. This gives
 +
Consider [[logarithm]] to base \(~s~\) from both sides of equation (1), assuming that \(~s~\) and \(~g(z)~\) are not real negative number nor zero. This gives
   
−
(2) $~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~$
 +
(2) \(~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~\)
   
−
Let
+
Let
   
−
(3) $~ ~ ~ G(z)=\log_s\big( g(s) \big)~$
 +
(3) \(~ ~ ~ G(z)=\log_s\big( g(s) \big)~\)
   
−
The substitution of (3) into (2) gives for function $~G~$ the [[Abel equation]]
 +
The substitution of (3) into (2) gives for function \(~G~\) the [[Abel equation]]
   
−
(4) $~ ~ ~ G\big(T\big(z)\big) = 1 + G(z)~$
 +
(4) \(~ ~ ~ G\big(T\big(z)\big) = 1 + G(z)~\)
   
−
The transfer from the [[Schroeder equation]] (1) to the [[Abel equation]] (4) eliminates the parameter $~s~$. Once the solution $~G~$ of equation (4) is found, the solution $~g~$ of equation (1) can be expressed through this $~G~$:
 +
The transfer from the [[Schroeder equation]] (1) to the [[Abel equation]] (4) eliminates the parameter \(~s~\). Once the solution \(~G~\) of equation (4) is found, the solution \(~g~\) of equation (1) can be expressed through this \(~G~\):
   
−
(5) $~ ~ ~ g(z)=\exp_s\Big( G(z) \Big)~$
 +
(5) \(~ ~ ~ g(z)=\exp_s\Big( G(z) \Big)~\)
   
−
In such a way, the logarithmic - exponential transforms (3) and (5) relate the solution $~g~$ of the [[Schroeder equation]] (1) and the solution $~G~$ of the [[Abel equation]] (4).
 +
In such a way, the logarithmic - exponential transforms (3) and (5) relate the solution \(~g~\) of the [[Schroeder equation]] (1) and the solution \(~G~\) of the [[Abel equation]] (4).
   
 
==[[Zooming equation]]==
  
==[[Zooming equation]]==
  ⚫
This equation determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as [[zooming equation]]. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):
−
While $~g~$ is [[Schroeder function]] and solution of the [[Schroeder equation]] (1), we need also the name for inverse function, id est, $~f=g^{-1}~$ and, the equation for this function and name for this equation.
  
−
While no established system of notation can be found for these objects, let such $~g~$ be called [[zoom function]] and such an equation be called [[zoom equation]]. The [[zoom equation]] is easy to obtain from (1).
  
   
  ⚫
(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)
−
As $~z~$ in (1) is dummy variable, it can be replaced to any expression. Make the substitution $~z\mapsto f(z)~$, and apply function $~f~$ to both sides of the resulting equation. This gives
  
   
  +
This gives way to express the \(~n\)th iteration of function \(T\),
−
(6) $~ ~ ~ f\Big(~g\Big( T\big(f(z)\big)\Big)~\big) = f\Big( s\, g\big(f(z)\big) \Big)~$
 
   
  +
(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)
−
Assumption that $~f(g(z))\!=\!z~$ and $~g(f(z))\!=\!z~$ gives the [[zoom equation]] in the following form:
  
   
  +
In this expression, \(~n~\) can be interpreted as any [[complex number]]. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\).
⚫
(7) $~ ~ ~ T\big(f(z))= f(s\, z)~$
  
  +
Following the work by I.N.Baker
  +
<ref>
  +
http://eretrandre.org/rb/files/Baker1962_53.pdf
  +
I.N.Baker. Primitive power series and regular iteration
  +
</ref>, [[Henryk Trappmann]] suggests for such an iterate name [[regular iteration]] or [[regular iterate]]
  +
<ref name="henrykRegular">
  +
http://math.eretrandre.org/hyperops_wiki/index.php?title=Regular_iteration
  +
</ref>, and this name will be used in [[TORI]] as one of meanings of term [[regular iterate]] or [[regular iteration]].
   
  ⚫
==[[Regular iterate]]==
⚫
This equation determines, how does value of function $~f~$ transform at the zooming of the argument with factor $~s~$. This is reason to refer equation (7) as [[zooming equation]]. The $~n$th iterate of function $~T~$ determines the the zooming of the argument with factor $~s^n~$:
  
  +
The [[Schroeder equation]] determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as [[zooming equation]]. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):
   
−
(8) $~ ~ ~ T^n\big(f(z))= f(s^n\, z)~$
 +
(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)
   
−
This gives way to express the $~n$th iteration of function $T$,
 +
This gives way to express the \(~n\)th iteration of function \(T\),
   
−
(9) $~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~$
 +
(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)
   
−
In this expression, $~n~$ can be interpreted as any [[complex number]]. Then, for given $~f~$, $~g~$ and $~s~$, equation (9) provides a tool to evaluate the non-integer iterates of function $T$.
+
In this expression, \(~n~\) can be interpreted as any [[complex number]]. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\).
 
Following the work by I.N.Baker
  
Following the work by I.N.Baker
 
<ref>
  
<ref>
Line 88: Line 96:
 
</ref>, and this name will be used in [[TORI]] as one of meanings of term [[regular iterate]] or [[regular iteration]].
  
</ref>, and this name will be used in [[TORI]] as one of meanings of term [[regular iterate]] or [[regular iteration]].
   
−
==[[Regular iterate]]==
 +
==[[Fractional iterate]]s==
   
  +
This equation determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as [[zooming equation]]. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):
−
The [[zooming equation]] (7) and the [[Schroeder equation]] (1) should have sense for evaluation of [[iterate]]s of function $~T~$, that have fixed point $~L\!=\!0~$, and is supposed to be regular in vicinity of this point. The corresponding iterate
  
−
of function $~T~$ is called [[regular iterate]] or [[regular iteration]].
  
−
While these two terms are considered as equivalent, but in future, the term [[regular iteration]] may be reserved for the tool of construction of the specific non–integer iterate, letting the term [[regular iterate]] in sense of noon to specify the result of such a construction.
  
   
  +
(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)
⚫
==[[Fractional iterate]]s==
  
−
The approach to the [[regular iterate]] can be based on the concept of [[fractional iterate]].
  
   
  +
This gives way to express the \(~n\)th iteration of function \(T\),
−
For a given function $~T~$, [[holomorphic function|holomorphic]] in vicinity of its [[fixed point]] $~L~$, the function
  
−
$t_r=T^r$ is considered as $r$ [[fractional iterate]], if $~r\!=\!m/n~$ for some integer numbers $~m, n~$ and
 
   
−
(10) $~ ~ ~ t_c^m(z)=T^n(z)~$
 +
(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)
   
  +
In this expression, \(~n~\) can be interpreted as any [[complex number]]. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\).
−
for all $~z~$ in some vicinity of $~L~$.
  
  +
Following the work by I.N.Baker
−
If such a function $t_r$ is also regular in vicinity of $~L~$, then such a fractional iterate is called [[regular iterate]].
  
  +
<ref>
  +
http://eretrandre.org/rb/files/Baker1962_53.pdf
  +
I.N.Baker. Primitive power series and regular iteration
  +
</ref>, [[Henryk Trappmann]] suggests for such an iterate name [[regular iteration]] or [[regular iterate]]
  +
<ref name="henrykRegular">
  +
http://math.eretrandre.org/hyperops_wiki/index.php?title=Regular_iteration
  +
</ref>, and this name will be used in [[TORI]] as one of meanings of term [[regular iterate]] or [[regular iteration]].
   
 
==Example with iterates of exponential==
  
==Example with iterates of exponential==
−
In particular, for base $b<~$[[HenrykEta]], the [[tetration]] to base $b$ can be considered as [[regular iterate]] of exponential at its lower fixed point $~L_1=\mathrm{Filog}(b\!+\!\mathrm i\, o)$, where [[Filog]] is function that determined
+
In particular, for base \(b<~\)[[HenrykEta]] \(=\exp(\mathrm e)\), the [[tetration]] to base \(b\) can be considered as [[regular iterate]] of exponential at its lower fixed point \(~L_1=\mathrm{Filog}(b\!+\!\mathrm i\, o)\), where [[Filog]] is function that determined
 
the fixed points of [[logarithm]],
  
the fixed points of [[logarithm]],
   
−
(11) $~ ~ ~\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1- \pi \mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\pi \mathrm{i}\big)}{-z}$
 +
(11) \(~ ~ ~\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1- \pi \mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\pi \mathrm{i}\big)}{-z}\)
   
 
The descriptions and [[numerical implementation]]s of the [[Tania function]] and [[WrightOmega]] and [[Filog]] are available.
 
The descriptions and [[numerical implementation]]s of the [[Tania function]] and [[WrightOmega]] and [[Filog]] are available.
   
−
Another fixed point $~L_2=\mathrm{Filog}(b\!-\!\mathrm i\, o)$ indicates another regular iterate of [[exponential]] to base $b$, and these two regular iterates are not the same.
+
Another fixed point \(~L_2=\mathrm{Filog}(b\!-\!\mathrm i\, o)\) indicates another regular iterate of [[exponential]] to base \(b\), and these two regular iterates are not the same.
 
However, in vicinity or the real axis, the regular iterates, corresponding to different fixed point, may look pretty similar.
  
However, in vicinity or the real axis, the regular iterates, corresponding to different fixed point, may look pretty similar.
−
For example, for $b=\sqrt(2)$, half–iterates of the exponential, constructed at fixed points $L_1$
 +
For example, for \(b=\sqrt{2}\), half–iterates of the exponential, constructed at fixed points \(L_1\)
   
 
==References==
  
==References==

Latest revision as of 11:32, 20 July 2020

Ernst Schroeder [1]

The Schroeder equation (Уравнение Шредера, Уравнение Шроедера, Уравнение Шрёдера) or Schröder equation [2] or Schröder's equation [3] can be written as follows:

(1) \(~ ~ ~ g\Big(T(z)\Big)= s \, g(z)\)

where function \(~T~\) is supposed to be given, while constant \(s\) and function\(~g~\) should be constructed. Usially, it is assumed that \(s\!=\!T'(0)\), and both, \(T\) and \(g\) are holomorphic at least in some vicinity of zero.

The Schroeder equation is related to the Abel equation and the transfer equation; the solutions (called Schroeder functions ) are related to the superfunctions and the Abel functions. These relations are especially simple for the case when function \(~T~\) has a real fixed point \(~L~\).

About the terminology

The topic about fractional iterates which seem to be the main application of the Schroeder equation had almost no advances during century 20, and there is no established terminology. The goal of this article is to compare various notations and choose the most convenient terminology for TORI.

The Schroeder equation in mathematics should not be confused with the Schroeder – Le Chatelier equation describing transitions between solid, liquid and gas [4][5][6], and Schroeder should not be confused with shredder [7]; their functions are pretty different.

Relation to the Abel equation

Consider logarithm to base \(~s~\) from both sides of equation (1), assuming that \(~s~\) and \(~g(z)~\) are not real negative number nor zero. This gives

(2) \(~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~\)

Let

(3) \(~ ~ ~ G(z)=\log_s\big( g(s) \big)~\)

The substitution of (3) into (2) gives for function \(~G~\) the Abel equation

(4) \(~ ~ ~ G\big(T\big(z)\big) = 1 + G(z)~\)

The transfer from the Schroeder equation (1) to the Abel equation (4) eliminates the parameter \(~s~\). Once the solution \(~G~\) of equation (4) is found, the solution \(~g~\) of equation (1) can be expressed through this \(~G~\):

(5) \(~ ~ ~ g(z)=\exp_s\Big( G(z) \Big)~\)

In such a way, the logarithmic - exponential transforms (3) and (5) relate the solution \(~g~\) of the Schroeder equation (1) and the solution \(~G~\) of the Abel equation (4).

Zooming equation

This equation determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as zooming equation. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):

(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)

This gives way to express the \(~n\)th iteration of function \(T\),

(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)

In this expression, \(~n~\) can be interpreted as any complex number. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\). Following the work by I.N.Baker [8], Henryk Trappmann suggests for such an iterate name regular iteration or regular iterate [9], and this name will be used in TORI as one of meanings of term regular iterate or regular iteration.

Regular iterate

The Schroeder equation determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as zooming equation. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):

(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)

This gives way to express the \(~n\)th iteration of function \(T\),

(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)

In this expression, \(~n~\) can be interpreted as any complex number. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\). Following the work by I.N.Baker [10], Henryk Trappmann suggests for such an iterate name regular iteration or regular iterate [9], and this name will be used in TORI as one of meanings of term regular iterate or regular iteration.

Fractional iterates

This equation determines, how does value of function \(~f~\) transform at the zooming of the argument with factor \(~s~\). This is reason to refer equation (7) as zooming equation. The \(~n\)th iterate of function \(~T~\) determines the the zooming of the argument with factor \(~s^n~\):

(8) \(~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\)

This gives way to express the \(~n\)th iteration of function \(T\),

(9) \(~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\)

In this expression, \(~n~\) can be interpreted as any complex number. Then, for given \(~f~\), \(~g~\) and \(~s~\), equation (9) provides a tool to evaluate the non-integer iterates of function \(T\). Following the work by I.N.Baker [11], Henryk Trappmann suggests for such an iterate name regular iteration or regular iterate [9], and this name will be used in TORI as one of meanings of term regular iterate or regular iteration.

Example with iterates of exponential

In particular, for base \(b<~\)HenrykEta \(=\exp(\mathrm e)\), the tetration to base \(b\) can be considered as regular iterate of exponential at its lower fixed point \(~L_1=\mathrm{Filog}(b\!+\!\mathrm i\, o)\), where Filog is function that determined the fixed points of logarithm,

(11) \(~ ~ ~\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1- \pi \mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\pi \mathrm{i}\big)}{-z}\)

The descriptions and numerical implementations of the Tania function and WrightOmega and Filog are available.

Another fixed point \(~L_2=\mathrm{Filog}(b\!-\!\mathrm i\, o)\) indicates another regular iterate of exponential to base \(b\), and these two regular iterates are not the same. However, in vicinity or the real axis, the regular iterates, corresponding to different fixed point, may look pretty similar. For example, for \(b=\sqrt{2}\), half–iterates of the exponential, constructed at fixed points \(L_1\)

References

  1. ↑ https://en.wikipedia.org/wiki/Ernst_Schröder
  2. ↑ http://math.eretrandre.org/hyperops_wiki/index.php?title=Schröder_equation
  3. ↑ http://en.wikipedia.org/wiki/Schroeder%27s_equation
  4. ↑ http://link.springer.com/article/10.1007/s10973-007-8533-6?LI=true#page-1 M.Ambrova, J.Jurisova, V.Danielik, J.Gabcova. On the solubility of lantanum oxyde in molten alkali fluorides. Journal of thermal analysis and Calorimetry, v.91,(2008), p.569-573
  5. ↑ http://193.146.160.29/gtb/sod/usu/\(UBUG/repositorio/10264342_BOBKOVA.pdf N. M. Bobkova. Thermal Expansion of Binary Borate Glasses and Their Structure. Glass Physics and Chemistry, Vol. 29, No. 5, 2003, pp. 501–507.
  6. ↑ http://www.sciencedirect.com/science/article/pii/S0022113908002157 R.V.Ostvald, V.V.Shagalov, I.I.Zherin, G.N.Amelina, V.F.Usova, A.I.Rudnikov, O.B.Gromov. Separation of systems based on uranium hexafluoride and some of halogen fluorides. Journal of Fluorine Chemistry. Volume 130, Issue 1, January 2009, Pages 108–116.
  7. ↑ http://en.wikipedia.org/wiki/Paper_shredder
  8. ↑ http://eretrandre.org/rb/files/Baker1962_53.pdf I.N.Baker. Primitive power series and regular iteration
  9. ↑ 9.0 9.1 9.2 http://math.eretrandre.org/hyperops_wiki/index.php?title=Regular_iteration
  10. ↑ http://eretrandre.org/rb/files/Baker1962_53.pdf I.N.Baker. Primitive power series and regular iteration
  11. ↑ http://eretrandre.org/rb/files/Baker1962_53.pdf I.N.Baker. Primitive power series and regular iteration

Keywords

fractional iterate, iteration, transfer function, superfunction

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