I thought I'd take a crack at base = 1/2 JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 09/25/2021, 02:47 AM (This post was last modified: 09/25/2021, 05:17 AM by JmsNxn.) Seeing Ember Edison, and Leo talk about cases which cannot be solved using a theta mapping--I thought I'd see if the infinite composition manner is feasible with these anomalous values. I'm only going to sketch an approach here, and try to construct a real valued tetration $H_\lambda(s)$ where: $H_\lambda(s)\,\,\text{is real valued for}\,\,\lambda \in \mathbb{R}^+\\ H_\lambda(0) = 1\\ 2^{-H_\lambda(s)} = H_\lambda(s+1)\\ H_\lambda(s+2\pi i / \lambda) = H_\lambda(s)\\ H\,\,\text{is holomorphic for}\,\,0 < |\Im(s)| < \pi/\lambda\,\,\text{for}\,\,\lambda \in \mathbb{R}^+\\$ To begin, we find a function that approximates this tetration. Let's call this function $\varphi(s)$ which is holomorphic on $\mathbb{C}/\{\lambda(j-s) = (2k+1) \pi i\,\,j \ge 1,\,k \in \mathbb{Z}\}$ and has a period of $2 \pi i / \lambda$. This function will satisfy the functional equation: $\varphi_\lambda(s+1) = \frac{2^{-\varphi_\lambda(s)}}{e^{-\lambda s} + 1}\\$ These can be expressed as, $\varphi_\lambda(s) = \Omega_{j=1}^\infty \frac{2^{-z}}{e^{\lambda(j-s)}+1}\,\bullet z\\$ Where if $q_j(s,z)=\frac{2^{-z}}{e^{\lambda(j-s)}+1}$; then this expression equals: $\varphi_\lambda(s)=\lim_{n\to\infty}q_1(s,q_2(s,...q_n(s,z)))\\$ Which converges compactly uniformly on the above domain because the sum, $\sum_{j=1}^\infty ||\frac{2^{-z}}{e^{\lambda(j-s)}+1}||_{\mathcal{K},\mathcal{U}} < \infty\\$ converges compactly uniformly for $z \in \mathcal{K} \subset \mathbb{C}$ and s in a compact set $\mathcal{U}$ of the above domain. Now, what we want to do is insert an error term $\mu_\lambda$ such that, $H_\lambda(s+s_0) = \varphi_\lambda(s) + \mu_\lambda(s)\\$ for some normalization constant $s_0$. To define the error term $\mu_\lambda$ we use a sequence of functions: $\mu_\lambda^{n+1}(s) = \log(1+e^{-\lambda s})/\log(2) -\log(1+\frac{\mu_\lambda^n(s+1)}{\varphi_\lambda(s+1)})/\log(2)\\$ Now, I'm not going to show this converges, but adapting the case from $e$--this shouldn't be hard to show converges. Instead, I'll just post some graphs showing the structure. These are only accurate to about 9 digits, so far. I just wrote together a quick script. Here's $\varphi_1$--so this has a 2 pi I period; and is real valued. This is over $x \in [-2,2]$     Here's our function $\varphi_1(x) + \mu^{100}_1(x)$ over $x \in [-1.5,4]$:     And here's our function $[\Re(\varphi_1(x+i) + \mu^{100}_1(x+i)), \Im(\varphi_1(x+i) + \mu^{100}_1(x+i))]$ over $x \in [-1.5,4]$     Again, both of these graphs satisfy $2^{-H_1(s)} = H_1(s+1)$ to about 9 digits. So all in all, I'm very confident that the infinite composition method will work for $b = 1/2$. I'm wondering if a similar result will hold for complex bases; but I'm not there yet. I'll try to write a script for that in a bit; for the moment I thought I'd just post the function $H_1(s)$. As I'm studying this more, we do not get $H_1$ as I thought we do. Branch cuts appear flippantly in $0<\Im(s) < \pi$; which I wasn't expecting, but still isn't very unexpected. I'm currently compiling a graph of $\varphi_1(s) + \mu_1^{100}(s)$ in the complex plane. And it's pretty wonky so far. But is definitely locally holomorphic almost everywhere. My proof that no branch cuts occur in $b = e$ do not carry over to $0512kb?), use "filewrite" "fileclose" as the write command. Then just use 7zip to pack it up, even a 700MB text file is no match for 7zip. The user may have to be reminded to execute Quote:default(parisize, 1048576000) Do not think that Pari-GP is very intelligent, it idiotic design as many, only txt-based is your real god =========================================================================== Quote:gp > beta(2,1) *** Pol: incorrect priority in gtopoly: variable t11 < l What's up with this? JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 09/30/2021, 03:04 AM (09/29/2021, 04:44 PM)Ember Edison Wrote: Quote:*** read: Warning: INIT_NEUTRAL.DAT not written for a 64 bit architecture. Oh, Shit, Why don't you use a txt-based gp file to store beta_taylor, it's 2021! Who will install x86-32 applications on the system Code:beta_init(n) = { local(fileobject_beta_taylor); beta_taylor = Phi_Inv(w,l,n); write1("E:/Temp/beta_taylor_(E^-E).gp",""); fileobject_beta_taylor = fileopen("E:/Temp/beta_taylor_(E^-E).gp","a"); filewrite(fileobject_beta_taylor, "beta_taylor = " beta_taylor ";"); fileclose(fileobject_beta_taylor); print("Done initializing beta."); } For large files (>512kb?), use "filewrite" "fileclose" as the write command. Then just use 7zip to pack it up, even a 700MB text file is no match for 7zip. The user may have to be reminded to execute Quote:default(parisize, 1048576000) Do not think that Pari-GP is very intelligent, it idiotic design as many, only txt-based is your real god =========================================================================== Quote:gp > beta(2,1) *** Pol: incorrect priority in gtopoly: variable t11 < l What's up with this? Not too sure, but t11 is the storage of the variable l I realized, after talking to sheldon today, I initialized beta with 32 bit architecture rather than 64 bit. You just have to run beta_init(100) and everything should work the same. I apologize again, I'm a horrible programmer lol. JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 09/30/2021, 05:38 AM (This post was last modified: 10/01/2021, 12:45 AM by JmsNxn.) So I've switched over to 64-bit pari-gp in hopes this may solve some of my problems. And I decided to take a whack at \( b = 10^{-5}$. We're going to do the exact same procedure. So I thought I'd add some new notation. We're going to switch from $b^z$ to $e^{bz}$ here; so let's take the logarithm of the normal idea of a base. $\varphi_\lambda(b,s) = \Omega_{j=1}^\infty \frac{e^{bz}}{e^{\lambda(j- s)} +1}\,\bullet z\\$ So that I don't have to keep defining new function names. And this function satisfies the functional equation, $\varphi_\lambda(b,s+1) = e^{b\varphi_\lambda(b,s)}/(e^{-\lambda s} + 1)\\$ It is $2\pi i / \lambda$ periodic in $s$, and is real valued for all variables real. And decays to zero at $\Re(s) = -\infty$. Additionally it is holomorphic for $b\neq 0,\,\lambda(j-s)\neq (2k+1)\pi i$. We're going to focus on $b_0 = \log(0.00001)$ and for simplicity keep $\lambda = 1$--so it's 2 pi i periodic. And here's a graph of $\varphi_1(b_0,x)$ for $x \in [-1,3]$.     We define the iteration similarly, $\mu^0 = 0\\ \mu^{n+1}(s) = \log(1+e^{-s})/b_0 - \log(1+\frac{\mu^{n}(s+1)}{\varphi_1(b_0,s+1)})/b_0\\$ Now, this method works fine, but it's really really reallly weird. It almost just looks like a sinusoid times 1E-10. The functional equation is defined upto 20 digits or so atm. I am very confused by all this. But the equation is being satisfied. I think it's because the normalization constant is about $x_0 = -1000$ or something; not sure tbh. But the superfunction equations being satisfied. It is definitely not working perfectly; it worked really well with $b = -e$, but $b = \log(0.00001)$ is very very very weird. It basically just oscillates about the fixed point. I thought I'd just post the code (it's 64 bit this time, so there shouldn't be as much of a problem)--rather than dealing with all the nuances. I won't make as many graphs as before. But I do believe this still works. Again, the math says this should still work, but coding it is crazy weird. I'll leave it to you to make your own judgements. Remember to run beta_init(100) before doing anything. This takes about 30 mins on my pc to compile--the files too big to attach the initialization file. After that you'll get Init_tiny.dat which you can include in any future compilations. This just initializes $\varphi$ for $b = \log(0.00001)$. You can also reinitialize for arbitrary b by changing the base variable at the beginning of the file. So you can run all the tests you want on weird values, ember. Just change the base value, reinitialize the beta function; and have at it. I have stuck to 100 iterations as the default, but recall, for b = e, about 10 iterations is a max (plus you need a limiter). But for values $b < e^{1/e}$, 100 iterations should work fine. Complex plane, not sure, but prolly won't work.   abel_tiny_t.gp (Size: 7.52 KB / Downloads: 183) Here's a graph of $\varphi_1(b_0,x) + \mu^{100}(x)$ over $x \in [-1002,1000]$ which shows the points where it hits 1 and 0.      It's boxy as hell, and further out it just looks constant with a small wobble of 1E-10. I'm not too sure about all of this in the complex plane. it gets really wonky; I think that singularities are popping up surreptitiously so the complex plane is a little off. You can still grab taylor series to about \ps 100 though. And the functional equation is capping at about E-27 precision; when you use the taylor series, that is. Regards, James Well this is even wackier than I expected. This function is bananas! The above graph doesn't do it justice at all; this jumps up and down between 0 and 1, and eventually settles to a line. But other than that, it looks a lot like a square wave for a large portion of its domain. The above graph has sampling errors, so it looks like lines, but this is really a square wave looking thing. But taylor series converge fairly damn well; I'm not sure what to make of all this o.o over $x \in [500,505]$:     Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/20/2022, 03:48 AM (This post was last modified: 07/01/2022, 06:45 AM by Catullus.) EDIT: The .txt file now uses ANSI, instead of UTF-8. If you already downloaded the text file, maybe you should redownload it in ANSI this time instead of UTF-8. The .png file attached to this post is smaller than the .txt file attached to this. The .txt file attached might not look right on a smart phone. The text graphs need a monospaced font. The image attached to this post looks blurry, because of the size of the image. There should be a way to use Sharp-bilinear interpolation on images on this forum. With the analytic continuation of the Kneser method, $y=.5\uparrow\uparrow x$ and $y=e^{-e}\uparrow\uparrow x$ might look like this: Attached Files Thumbnail(s)       Tetration Graphs Base .5 and Base e^-e.txt (Size: 6.91 KB / Downloads: 72) Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ JmsNxn Ultimate Fellow Posts: 1,214 Threads: 126 Joined: Dec 2010 06/20/2022, 07:16 AM Okay, so I have no idea what you are doing here. But that's crazy looking text stuff.  The original post has to do with $1/2 \uparrow \uparrow z$, using the beta method. Which means we choose the period of this iterations freely. I do not know how or where this converges, but it does converge. Which was the point of the posts between Ember, and myself. I don't even know what your text graphs represent. But they seem similar. Could you elaborate? Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/20/2022, 08:28 AM (This post was last modified: 07/06/2022, 08:50 AM by Catullus.) (06/20/2022, 07:16 AM)JmsNxn Wrote: I don't even know what your text graphs represent. But they seem similar. Could you elaborate?The text graphs are to represent the analytic continuation of the Kneser method for $.5\uparrow\uparrow x$ and $(e\uparrow-e)\uparrow\uparrow x$. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ « Next Oldest | Next Newest »

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