• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Proof Ackermann function extended to reals cannot be commutative/associative JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 09/08/2011, 01:37 AM (This post was last modified: 09/08/2011, 08:07 PM by JmsNxn.) Well, the proof is really simple, but it works; Lets assume we have an operator $\otimes_q$ where $0\le q \le1$, and $\otimes_q$ is the super operator of $\otimes_{q-1}$, furthermore, $\otimes_{0} = +$ and $\otimes_{1} = \cdot$ Start off by making our only assumption that $\otimes_q$ and $\otimes_{q-1}$ are commutative and associative. start off with the basic formula: $a_1\,\otimes_{q-1}\,a_2\,\otimes_{q-1}\,a_3\,...\,\otimes_{q-1}\, a_n\,=\,a\,\otimes_{q}\,n$ Now, since $\otimes_{q-1}$ is commutative and associative, we can rearrange them in the following manner if $m + k = n$: so that: $(a\,\otimes_{q}\,m)\,\otimes_{q-1}\,(a\,\otimes_{q}\,k) = \,a\,\otimes_{q}\,n$ therefore, for any a,b,c: $(a\,\otimes_{q}\,b)\,\otimes_{q-1}\,(a\,\otimes_{q}\,c) = a \,\otimes_{q}\,(b+c)$ given this law, if we set a = S(q) or the identity for operator $\,\otimes_{q}\,$ we instantly see that $b\,\otimes_{q-1}\,c\,=\,b\,+\,c$ the only assumption we made was that $\otimes_{q-1}$ and $\otimes_{q}$ be commutative and associative. I think maybe this proof is inadequate at proving it cannot be commutative, but I think it'd be on shaky ground to say they are commutative. But forsure, not associative. Edit: The proof to make it non-commutative is as follows. Since the ackermann function is defined as: $\vartheta(a, b, \sigma) = a \, \otimes_\sigma\,b$ where the only law it must obey is: $a \,\otimes_{\sigma - 1}\,(a\,\otimes_{\sigma}\,b) = a \, \otimes_\sigma\,(b+1)$ If we want $\vartheta$ to be analytic over $\sigma$ (which we do), we cannot have $\otimes_{\sigma}$ being commutative over any strip. because if perhaps we say: all operators including and below multiplication are commutative. This would mean: $\vartheta(a, b, \sigma) = \vartheta(b, a, \sigma)\,\,\R(\sigma)\le 1$ but if two functions are analytic and they equal each other over a strip then they must be the same function therefore: $\vartheta(a, b, \sigma) = \vartheta(b, a, \sigma)$, for all $\sigma$, but this is clearly untrue because exponentiation is not commutative. Therefore $\sigma$ is only commutative at addition (0) and multiplication (1). I guess our rational operators are going to have to behave like exponentiation, I'm really curious about an analytic and integral calculus attack at this problem. Maybe dynamics ain't the right field. I think logarithmic semi operators are as close as it'll get. Maybe there's a more natural equation that may have some aesthetic properties in terms of relations to trigonometric functions, or other established functions with maybe some fancy constants involved. « Next Oldest | Next Newest »

 Messages In This Thread Proof Ackermann function extended to reals cannot be commutative/associative - by JmsNxn - 09/08/2011, 01:37 AM RE: Proof Ackermann function extended to reals cannot be commutative/associative - by MphLee - 06/15/2013, 08:02 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Interesting commutative hyperoperators ? tommy1729 0 764 02/17/2020, 11:07 PM Last Post: tommy1729 New mathematical object - hyperanalytic function arybnikov 4 2,875 01/02/2020, 01:38 AM Last Post: arybnikov Is there a function space for tetration? Chenjesu 0 1,149 06/23/2019, 08:24 PM Last Post: Chenjesu Where is the proof of a generalized integral for integer heights? Chenjesu 2 2,320 03/03/2019, 08:55 AM Last Post: Chenjesu Degamma function Xorter 0 1,587 10/22/2018, 11:29 AM Last Post: Xorter Extended xor to Tetrion space Xorter 10 10,124 08/18/2018, 02:54 AM Last Post: 11Keith22 Possible continuous extension of tetration to the reals Dasedes 0 1,867 10/10/2016, 04:57 AM Last Post: Dasedes (almost) proof of TPID 13 fivexthethird 1 3,299 05/06/2016, 04:12 PM Last Post: JmsNxn Should tetration be a multivalued function? marraco 17 23,135 01/14/2016, 04:24 AM Last Post: marraco Introducing new special function : Lambert_t(z,r) tommy1729 2 5,004 01/10/2016, 06:14 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)