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 Arguments for the beta method not being Kneser's method JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 10/22/2021, 03:54 AM (This post was last modified: 10/22/2021, 06:11 AM by JmsNxn.) Very god damned fascinating, Sheldon. I'm going to have to read back up on Gaussian quadrature and all that nonsense (because I'm pretty sure that's what you're using; just forget the right word for it). I never cared for any of the math behind Gaussian speed ups, and clever integral representations, because I always focused on algebraic representations. I'm a little dumbfounded by how you are calculating logrho so fast about the singularity--but it makes sense for the most part. We are centering at a value $$\log\rho^n(z_0) = 0$$ which has a convergence radius for r; and then we sample a circle about the radius. Then we do some gaussian magic (lol, this is what I need a better explanation of; but it's definitely a symptom of my lack of understanding Gaussian/Riemann quadrature integral speed up black magic!) I've already started going into salvage mode. I'm looking at how well of an asymptotic approximation this beast really is. And I think I might have an alternative approach which best describes how the asymptotic scenario really works. And I think, much of this is becoming avoidable; but only when $$\lambda\to 0$$ and we do it cleverly enough. I'm going to work on the claim $$\text{tet}_\lambda(s)$$ is holomorphic everywhere on $$\mathbb{C}$$ upto a set $$\mathbb{E}_\lambda$$ in which: $$\int_{\mathbb{E}_\lambda} 1\cdot dA = 0\\$$ Which is a better way of saying my original claim. And second of all; fortifying it as more of an asymptotic solution. Last of all, making sure the same argument works for all bases/multipliers. This still allows for singularity walls; weird fuck ups; and all sorts of taylor series shenanigans. This doesn't affect the first 30 pages of my paper too much; just requires me to choose better language. I'm still on the fence on the last part of my paper; but I feel that as $$\lambda \to 0$$ the set $$\mathbb{E}_\lambda \to (-\infty,-2]$$ and $$\text{tet}_\lambda \to \text{tet}_K$$. All my numbers and thoughts and proof sketches point towards a normality in the left half plane as we limit $$\lambda \to 0$$. This is in tune with the later parts of my paper; but I definitely need to write this out deeper. Also, I'm beginning to understand why $$\mathbb{R} \subset \mathbb{E}_\lambda$$ in a good topological way. I can't really explain this yet; the words are on the tip of my tongue; but I don't have them just yet. Regards, James Essentially, I've begun looking at $$q(z) = \text{tet}_1(\log(z))$$ which satisfies $$q(e\cdot z)=e^{q(z)}$$--where no such analytic solution can exist on $$\mathbb{R}$$. This function is holomorphic almost everywhere for $$\Im(z) > 0$$--but that's all we can say. If we try to compare the difference along the border $$\Im(z) >0$$ and $$\Im(z)< 0$$ we get a buch of fractal errors. A similar result holds for $$q_\lambda(e^\lambda \cdot z) = e^{q_\lambda(z)}$$; with the normality conditions I have; they must be non-analytic on $$\mathbb{R}$$. The only language I can think of that's equivalent; is that we are asking the Schroder functions about the fixed point $$L,L^*$$ to magically agree on $$\mathbb{R}$$. You and I both know that's nonsense. But! $$q_\lambda(z)$$ will be somewhat holomorphic for $$0 < \Im(z) < 2\pi i/\lambda$$; with fractal properties near the boundary. BUT! as $$\lambda \to 0$$ this equation already diverges. And we're asking a divergent Schroder function to equal a divergent Schroder function on the real line. This has much more luck; wayyyyyyy more luck; seeing as this thing still converges. To such an extent that as $$\Re(z) \to - \infty$$ while $$\lambda\to0$$ we approach holomorphic functions for $$\Im(z) >0$$ and $$\Im(z) < 0$$--but they agree on the real line. And they look like kneser because as $$|z| \to \infty$$ for $$\pi/2 < \arg(z) <\pi$$ the function $$\lim_{\lambda \to 0} \text{tet}_\lambda(z) \to L$$--and similarly in the lower half plane. This is a uniqueness condition per Paulsen & Cowgill. Additionally the more I graph the solutions as $$\lambda \to 0$$ we decay to the fixed points $$L,L^*$$ geometrically with $$\lambda$$. And if you thought the singularities quiet. I suggest looking at mult = 0.001 in my code (making sure to do about 1000 iterations); we have a bunch of fractals near $$\mathbb{R}$$ but we get a huge area of convergence towards $$L$$. I think it's because we're pushing closer and closer to Kneser. Regards. « Next Oldest | Next Newest »

 Messages In This Thread Arguments for the beta method not being Kneser's method - by JmsNxn - 06/07/2021, 10:45 PM RE: Arguments for the beta method not being Kneser's method - by MphLee - 06/10/2021, 12:41 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 06/11/2021, 02:05 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 06/20/2021, 12:36 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 07/07/2021, 11:00 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/08/2021, 05:11 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/24/2021, 04:42 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 07/21/2021, 07:13 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/22/2021, 03:47 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 07/22/2021, 12:21 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 07/23/2021, 04:05 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/21/2021, 07:22 PM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/24/2021, 04:52 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/25/2021, 03:00 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/28/2021, 04:41 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/29/2021, 12:33 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 09/29/2021, 11:55 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/01/2021, 01:23 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 08/17/2021, 03:12 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/15/2021, 08:15 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/16/2021, 01:41 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 09/16/2021, 10:54 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 03:20 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 09/16/2021, 07:23 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 04:23 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/17/2021, 06:00 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/22/2021, 03:11 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 09/23/2021, 08:42 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/01/2021, 03:50 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/01/2021, 11:38 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/02/2021, 11:36 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/03/2021, 05:59 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/05/2021, 03:27 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/05/2021, 02:31 PM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/02/2021, 01:40 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/03/2021, 04:50 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/06/2021, 10:22 PM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/07/2021, 02:29 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/07/2021, 03:18 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/07/2021, 07:02 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/07/2021, 05:20 AM RE: Arguments for the beta method not being Kneser's method - by Ember Edison - 10/07/2021, 07:11 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/07/2021, 04:12 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/09/2021, 12:27 PM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/09/2021, 08:02 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/11/2021, 02:11 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/11/2021, 03:51 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/11/2021, 12:48 AM RE: Arguments for the beta method not being Kneser's method - by tommy1729 - 10/12/2021, 12:23 PM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/13/2021, 05:01 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/19/2021, 02:43 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/23/2021, 03:13 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/20/2021, 05:13 AM RE: Arguments for the beta method not being Kneser's method - by sheldonison - 10/21/2021, 03:33 AM RE: Arguments for the beta method not being Kneser's method - by JmsNxn - 10/22/2021, 03:54 AM

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