https://mizugadro.mydns.jp/t/index.php?title=Smooth_superation&feed=atom&action=historySmooth superation - Revision history2024-03-28T08:08:11ZRevision history for this page on the wikiMediaWiki 1.31.16https://mizugadro.mydns.jp/t/index.php?title=Smooth_superation&diff=34356&oldid=prevT: Created page with "Smooth superation is generalization of operation superation to non-integer number of superations. ==Definitions== Assume, some holomorphic function \( f \) is given...."2020-06-21T12:45:48Z<p>Created page with "<a href="/t/index.php/Smooth_superation" title="Smooth superation">Smooth superation</a> is generalization of operation <a href="/t/index.php?title=Superation&action=edit&redlink=1" class="new" title="Superation (page does not exist)">superation</a> to non-integer number of superations. ==Definitions== Assume, some holomorphic function \( f \) is given...."</p>
<p><b>New page</b></p><div>[[Smooth superation]] is generalization of operation [[superation]] to non-integer number of superations.<br />
<br />
==Definitions==<br />
Assume, some holomorphic function \( f \) is given.<br />
<br />
Consider holomorphic function \(a \) of two variables such that <br />
<br />
\( a(0,x)=f(x) \)<br />
<br />
\( a(n,0)=f(0) \)<br />
<br />
\( a(n,x+1)= a(n-1,a(n,x)) \)<br />
<br />
Then:<br />
<br />
\(a \) is [[ackermann]] to function \( f \).<br />
<br />
\( a(n+1,) \) is [[superation]] of function \( a(n,.) \).<br />
<br />
\( a(n,) \) is [[supation]] of function \( a(n+1,.) \).<br />
<br />
\( a(n,.) \) is \( n \)th [[superation]] of function \( f \).<br />
<br />
\( f \) is [[seed function]] with respect to [[aclermann]] \( a \).<br />
<br />
The 0th [[superation]] of function \( f \) is \(f\).<br />
<br />
The 0th [[supation]] of function \( f \) is \(f\).<br />
<br />
==Uniqueness==<br />
<br />
The solution \( a \) is not unique.<br />
<br />
The additional requirements are necessary to provide the uniqueness.<br />
<br />
Perhaps, for some functions $f$, the "true" [[ackermann]] should be choosen to extend the range of holomorphism<br />
and validity of the equations above.<br />
<br />
For a given range of validity and hlomorfism the "true" [[ackermann]] should have slowest growth (or fastest decay)<br />
in the direction of the imaginary axes of the arguments.<br />
<br />
In general case, even these requirements may be not sufficient to provide the uniqueness.<br />
<br />
==Example==<br />
<br />
[[addition]] to base \(b\),<br />
[[multiplication]] to base \(b\),<br />
[[exponentiation]] to base \(b\),<br />
[[tetration]] to base \(b\),<br />
[[pentation]] to base \(b\),<br />
[[hexation]] to base \(b\),<br />
<br />
can be considered as [[native ackermann]]s to seed addition to base \(b\).<br />
<br />
The special efforts are necessary to adjust the terminology, to establish the correct numeration,<br />
in order that tetration happen to be 4th operation, pentation to be 5th operation, etc.<br />
<br />
==References==<br />
<references/><br />
<!--<br />
https://zh.wikipedia.org/wiki/%E8%B6%85%E8%BF%90%E7%AE%97#%E4%B8%80%E8%88%AC%E5%8C%96<br />
!--><br />
https://zh.wikipedia.org/wiki/超运算<br />
超运算可通过[[递归]]进行定义,對於所有[[正整數]]''a'',正整數''b''和正整數''n'':<br />
<math>a [1] b = a + b, \ \text{for} \ n > 1, \ a [n] b = \underbrace{a [n-1] (a [n-1] (a [n-1] \cdots (a [n-1] (a [n-1] a}_b)) \cdots ))</math> ..<br />
<br />
<!--<br />
除这一最常见的定义之外,超运算还有其他的变体。([[#一般化|见下文]])<br />
超運算序列是数学中一种二元运算的序列,前三项分别为加法、乘法、幂,一般來說,除了序列中第一項的加法運算之外,序列中每一項的運算都是重複的前一項的運算例如乘法是重複的加法..<br />
!--><br />
<br />
==Keywords==<br />
[[Abelfunction]],<br />
[[Ackermann]],<br />
[[Ackermann function]],<br />
[[Iterate]],<br />
[[Supation]],<br />
[[Superation]],<br />
[[Superfunction]],<br />
[[Superfunctions]],<br />
[[Tetration]]<br />
<br />
[[Category:Abelfunction]]<br />
[[Category:Ackermann]]<br />
[[Category:Ackermann function]]<br />
[[Category:China]]<br />
[[Category:English]]<br />
[[Category:Iterate]]<br />
[[Category:Supation]]<br />
[[Category:Superation]]<br />
[[Category:Superations]]<br />
[[Category:Tetration]]<br />
[[Category:Wikipedia]]</div>T