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 Holomorphic semi operators, using the beta method MphLee Long Time Fellow Posts: 321 Threads: 25 Joined: May 2013 03/24/2022, 11:13 AM It's hard to follow for me. It is my fault. I'm not even remotely familiar with perturbation methods and those theta mappings. Some points are really obscure: again my fault. Let's see if you can drop some candies for me. Quote:I should've clarified, that first of all, this is just intended for $$0 \le \Re(s) \le 2$$. This would not make Tetration, or the job of finding inbetween tetration in any meaningful way. This is why I don't even like this solution, But it is doable. It's essentially just run Bennet's commutative hyperoperations, but paste them together in a meaningful way to give a hyper-operator structure. Ok lets start: how far this is from this \begin{align}xy&=x\odot_s y &&0\le \Re(s)\le 2\\ xy+1&=x(xy)&&{\rm otherwise}\end{align},\\ modulo some perturbation business you use to force the Goodstein equation over that domain? The following point is particularly obscure. We can say that $$x\omega$$ are a family of functions $${\mathbb C}/{\mathcal E}\times \mathcal{W}\to \mathbb C$$, as the rank varies, where $$\mathcal{W}$$ contains all the fixed points associated to $$\mu$$ s.t. $$e^\mu$$ is in the ST-region, i.e. if I remember well, when it's infinite tower converges (to the fixed point). You tell me to compute em by $$F(x,s,\mu)$$, a function that we know how to compute using a tetration function with base $$b=e^\mu$$. Then what do you mean by Quote:We can delineate an equivalence class for the Goodstein functional equation so we have a bunch of functions that $$x\omega \pm k$$ must equal for $$1 \le \Re(s) \le 2$$. Now we play the implicit function game... Also is $$F(s+1)$$ intended to be $$F(x,s+1,\mu$$? Quote:For brevity's sake's, let's assume we can find where $$F(s+1) \in \mathcal{W}$$ (the domain of fixed points). Then $$x F$$ is a valid operation--it can now be assigned the value $$x\omega +1 = x \omega'$$. The starting point, if I'm following you, is to extend $${\mathbb C}/{\mathcal E}\times \mathcal{W}\to \mathbb C$$ outside $$W$$, somewhere in $$\mathcal{W}+\mathbb Z$$, whenever $$F(x,s+1,\mu)$$ still lands in $$\mathcal{W}$$. But what if the new fixed point is associated with another ST-base? The holomorphic semi operators are expected to agree across all bases $$b$$ only for rank 0 (addition) and rank 1 (multiplication) but to "ramify" for the other ranks. So I don't understand how all the pieces can fit. I stop here because I don't have time to parse the theta mapping part atm. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ « Next Oldest | Next Newest »

 Messages In This Thread Holomorphic semi operators, using the beta method - by JmsNxn - 03/23/2022, 03:19 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 03/23/2022, 10:40 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 03/24/2022, 09:35 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 03/24/2022, 11:13 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 03/24/2022, 12:00 PM RE: Holomorphic semi operators, using the beta method - by sheldonison - 04/22/2022, 06:14 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/23/2022, 04:20 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 04/01/2022, 10:23 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/02/2022, 12:33 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/04/2022, 03:48 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 04/05/2022, 12:39 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/06/2022, 02:28 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/08/2022, 01:06 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 04/08/2022, 11:45 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/09/2022, 10:43 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/14/2022, 04:08 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 04/15/2022, 04:43 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/19/2022, 03:08 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/20/2022, 08:30 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/27/2022, 02:00 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 04/30/2022, 08:37 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/01/2022, 04:48 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/02/2022, 12:35 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/02/2022, 06:49 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/02/2022, 07:25 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/02/2022, 09:40 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/03/2022, 12:16 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/04/2022, 10:31 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/05/2022, 11:03 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/06/2022, 09:07 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/03/2022, 01:20 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/04/2022, 12:25 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/05/2022, 11:01 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/08/2022, 07:41 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/08/2022, 09:30 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/08/2022, 11:18 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/08/2022, 11:40 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/08/2022, 11:52 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/10/2022, 11:38 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/10/2022, 08:29 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/26/2022, 10:00 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/26/2022, 10:18 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/26/2022, 10:44 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/26/2022, 10:49 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/26/2022, 11:24 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/20/2022, 12:14 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/21/2022, 10:37 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/10/2022, 12:26 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/13/2022, 04:28 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/13/2022, 08:17 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/17/2022, 02:21 AM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/22/2022, 12:17 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/22/2022, 01:29 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/25/2022, 04:04 AM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 05/26/2022, 09:04 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/26/2022, 11:33 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/26/2022, 11:46 PM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/27/2022, 12:17 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/26/2022, 11:36 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 05/29/2022, 06:11 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/31/2022, 02:32 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/02/2022, 01:24 AM RE: Holomorphic semi operators, using the beta method - by MphLee - 05/31/2022, 09:24 AM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 06/11/2022, 12:27 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/12/2022, 12:07 AM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 06/12/2022, 12:09 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/12/2022, 01:00 AM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/12/2022, 03:12 AM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 06/12/2022, 01:50 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/12/2022, 10:05 PM RE: Holomorphic semi operators, using the beta method - by tommy1729 - 06/12/2022, 11:13 PM RE: Holomorphic semi operators, using the beta method - by JmsNxn - 06/13/2022, 08:33 PM

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