Holomorphic semi operators, using the beta method

[JmsNxn]

... Again, I haven't worked out the details, but the final expression should look something like this (remember, \(b = e^\mu\)):

\[
x<s>\omega = \exp_b^{\circ s + \theta(s,x,\omega)}(\log_b^{\circ s + \theta(s,x,\omega)}(x) + \omega)
\]

.... Hopefully this makes sense, I'm kind of just shooting at the wall and seeing what sticks, lol. But I do believe this holds some weight. Think of it as taking all the Bennet hyperoperations, and finding a path along them where the typical Goodstein functional equation works.

Hi James, Mphlee,
I was trying to make sense of what has been posted.  In Kneser's original paper, he talks about the half iterate o \( e^z \), which might be written as:
\[f(z)=e^{[0.5]}(z)=\text{tet}_e(\text{slog}_e(z)+0.5);\;\;\;f(f(z))=e^z\]
The base e could be any real or complex base for which Kneser's tetration is defined, and the fractional iterate of 0.5 could be replaced by any arbitrary fractional iterate like \( \frac{1}{n} \).  Such a function would also be defined for non-integer values of n, and would be straightforward to compute with fatou.gp
\[f(z)=b^{[\frac{1}{n}]}(z)=\text{tet}_b(\text{slog}_b(z)+\frac{1}{n});\;\;\;f^{[\circ n]}(z)=b^z\]

Then perhaps one might want to think about the super function of f(z), which could be generated surprisingly simply as the following:
\[f^{[\circ z]}=\text{tet}_b(\frac{z}{n})\]

And would about the function g whose iterated superfunction is f(z)?  Could we define that as well?  How about the following, which should also be reasonably easy to compute with fatou.gp
\[g(z)=f(f^{-1}(z)+1);\;\;\;f(z)=g^{[\circ z]}\]
\[f^{-1}(z)=\text{tet}_b(\text{slog}_b(z)-\frac{1}{n})\]

I hope this is somewhat relevant to the current discussion; this equation might even be closely related to the equation in James PDF which I just downloaded.  But my equations here don't lead to a generalized Goodstein/Ackermann functions for non integer values of n.

Hey, Sheldon.

I'll give a quick summary here. We do not need Kneser for any of this, because we're only going to be dealing with bases in the Shell-Thron region (where tetration is considerably easier). I'll start with the real case, which is much easier to digest (the complex plane will work similarly, a little more finicky though).

If I take an arbitrary \(y >1\), then the value \(y^{1/y} = b \in [1,e^{1/e}] \subset \mathfrak{S}\), the Shell-Thron region. Call:

\[
\exp^{\circ s}_{b}(x)
\]

The regular iteration, for \(x \in \mathbb{R}\). So for example, if \(b = \sqrt{2}\), then we'd use the Schroder iteration about \(2\) for \(x \in (-\infty,4)\), and for \(x \in [4,\infty)\), we use the Schroder iteration about the repelling fixed point \(4\). This can be done for all \(b = y^{1/y}\) for \(y > 1\), excepting at \(y=e\)--where you have to switch to solving the Abel equation in the neutral scenario. This would require using the bounded iteration, and the unbounded iteration (for \(e^{1/e}\), this would be the normal tetration, and cheta solution).

Now, we start by preliminarily defining:

\[
x \oplus_{s} y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]

Ignoring where the singularities are for \(\log^{\circ s}\) are momentarily, we get the following interpolation.

\[
\begin{align}
x \oplus_{0} y &= \exp^{\circ 0}\left(\log^{\circ 0}(x) + y\right) = x+y\\
x\oplus_1 y &= \exp_{y^{1/y}}\left(\log_{y^{1/y}}(x) + y \right) = x \cdot \exp_{y^{1/y}}(y) = x \cdot y\\
x \oplus_2 y &= \exp_{y^{1/y}} \left(\exp_{y^{1/y}}(\log^{\circ 2}_{y^{1/y}}(x) + y)\right) = \exp_{y^{1/y}}\left(\log_{y^{1/y}}(x)\cdot y\right) = e^{\log(x) y} = x^y\\
\end{align}
\]

This gives us our preliminary operators, which we want to perturb slightly to satisfy goodstein's equation. We do this by inserting a term \(\varphi(x,y,s)\), and we define the new operators:

\[
x <s>_{\varphi} y = \exp_{y^{1/y}}^{\circ s}(\log_{y^{1/y}}^{\circ s}(x) + y + \varphi)\\
\]

Initial tests show that \(\varphi\) is relatively small. For example, we start by knowing \(\varphi = 0\) when \(s =0,1,2\), and has a slight bump as you move between. A good way to visualize this was in the graph of my last post.

If we stick to the idea that \(2 <s> 2 =4\) then we want to solve for the correct varphi such that:

\[
2 <s> 4 = 2<s+1> 3\\
\]

And if we graph these two functions next to each other over \((0,1)\), we get:

   

So essentially, we want to hunt for a varphi that zippers these two functions together. I can do this in a very rough manner, but it's increasingly frustrating to solve the implicit equation.


The implicit equation we are solving is tricky to write out, but consider \(\boldsymbol{\varphi} = (\varphi_1,\varphi_2,\varphi_3)\), and we look at:

\[
x <s>_{\varphi_1} (x <s+1>_{\varphi_2} y) = x <s+1>_{\varphi_3} y+1\\
\]

Under the constraint that:

\[
\begin{align}
\varphi_2(x,y+1,s) &= \varphi_3(x,y,s)\\
\varphi_2(x,x <s+1>_{\varphi_2(x,y,s+1)} y,s) &= \varphi_1(x,y,s)\\
\end{align}
\]

These implicit functions always exist, because the derivative in \(\varphi\) is non-singular--and we always have existence of points which solve the equation.

Then, the desired \(\varphi\) we'd want would be \(\varphi(x,y,s+1) = \varphi_2\), so long as it satisfies the above constraints. This then becomes a unique solution (3 variables, 2 constraints). This varphi would then be the solution such that:

\[
\begin{align}
x <s> y &= \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y + \varphi(x,y,s)\right)\\
x<s> (x<s+1> y) &= x <s+1> (y+1)\\
\end{align}
\]

You can also prove this function exists by constructing a first order differential equation in three variables, but that was making my head hurt so I abandoned it.

It's hard to do tests on this, because the code I have is just too slow. I'd use fatou.gp, but we have to consistently reinitialize the base of the super-exponential, and there's no work around in fatou.gp that I know of. So instead, I whipped up some code which managed to go as fast as I could get.

So we don't need Kneser for any of this, it's just regular iteration or neutral iteration, at least for \(y > 1\) case. I'm expecting volatile behaviour for \(y < 1/e\), because here, \(y^{1/y} \not \in \mathfrak{S}\). This idea only works so far when \(y^{1/y} \in \mathfrak{S}\). Technically it can be extended further. I'm not too worried about the complex plane at the moment. But, the code still seems to be working.

Regards, James


Also, this notation is just a place holder. I wanted to distance from Knuth's up arrow, and conway chained notation. I meant to write it like:

\[
x \langle s\rangle y\\
\]

Like the braket angle. But I was lazy and didn't want to type the extra characters for the moment, lol.

If you can direct me to a manner of using fatou.gp such that it's holomorphic in the base, we could use that program. But as I understand fatou.gp, we have to fix the base upon initialization. And I don't know of a work around.  So everytime we moved y we'd have to reinitialize, which is obviously too taxing. Unless you can think of a fast fatou.gp way of graphing:

\[
x \oplus_s y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]

In the variable \(y\), I'm happy to adopt fatou.gp protocols.




For the moment I'm using the program semi_operators.gp, which is far from being complete. It doesn't work for the neutral case. So it's ideally meant for \(1 < y<e,\,\,y>e\) and solely real variables. It does not solve for \(\varphi\) yet, but it allows you test which \(\varphi\) work.

The main protocols, that you use are

Iexp(s,z,y)

Which is precisely \(\exp^{\circ s}_{y^{1/y}}(z)\), and

Ibennet(s,z,y,{phi = 0})

Which is precisely \(z \langle s\rangle_{\phi} y\) for \(\phi\) a constant.

You can treat these objects as taylor series too, so:

Iexp(0.5+s,2,3)

Iexp(0.5,2+w,3)

Iexp(0.5,2,3+u)


Are all valid operations, and they'll produce a Taylor series in the appropriate variable. Note, that the first one is fast, second one is medium speed, and third one is very slow. The same idea works with Ibennet.


  semi_operators.gp (Size: 5.65 KB / Downloads: 509)

Please note it is very very much still a beta. There are still some small glitches. But it's working for everything I've been using it for.

---

I've run a few example inputs:

Code:

\r semi_operators

Iexp(0.1+s,Iexp(0.1+s,1,2),2) - Iexp(0.2+2*s,1,2)

%39 = 1.0560015667790773041 E-47 - 9.874014683335685726 E-48*s + 1.0117887154347293316 E-47*s^2 - 9.769866529528564435 E-48*s^3 + 9.205049268878957323 E-48*s^4 - 8.555693781692655296 E-48*s^5 + 7.887702353919229131 E-48*s^6 - 7.235526068815923254 E-48*s^7 + 6.616389800744812899 E-48*s^8 - 6.038051040511804261 E-48*s^9 + 5.503079393175880247 E-48*s^10 - 5.011230997188426702 E-48*s^11 + 4.560781494644109592 E-48*s^12 - 4.149279553778102565 E-48*s^13 + 3.773972813243431893 E-48*s^14 - 3.432046517242785078 E-48*s^15 + 3.120754229436710630 E-48*s^16 - 2.837486129351430481 E-48*s^17 + 2.5798012395080594446 E-48*s^18 - 2.3454389566002173251 E-48*s^19 + 2.1323189023009714368 E-48*s^20 - 1.9385343945431778221 E-48*s^21 + 1.7623426514180968579 E-48*s^22 - 1.6021535466821381767 E-48*s^23 + 1.4565179655867155578 E-48*s^24 - 1.3241163530764392936 E-48*s^25 + 1.2037477741091062452 E-48*s^26 - 1.0943196448441181181 E-48*s^27 + 9.948381990067232366 E-49*s^28 - 9.043996992691581784 E-49*s^29 + O(s^30)

Iexp(0.1,Iexp(0.1,1+w,2),2) - Iexp(0.2,1+w,2)

%40 = 1.0560015667790773041 E-47 - 7.292173087295080608 E-48*w + 1.2378925093140665465 E-48*w^2 + 3.233718357384597705 E-50*w^3 - 9.182632082588929039 E-51*w^4 + 4.431670302409573164 E-52*w^5 - 3.214653653169696949 E-54*w^6 - 1.7306769515615559648 E-53*w^7 - 5.503821799350850526 E-55*w^8 - 1.0694188305787609134 E-55*w^9 - 6.202971279155519914 E-56*w^10 - 1.3568224932426610033 E-56*w^11 - 3.216789057723641103 E-57*w^12 - 8.266527408745018921 E-58*w^13 - 2.0705029304634104792 E-58*w^14 - 5.385364260250267394 E-59*w^15 - 1.4542400553561676742 E-59*w^16 - 3.987068346848031708 E-60*w^17 - 1.1043673344277225868 E-60*w^18 - 3.0821036377983951112 E-61*w^19 - 8.666384254248708309 E-62*w^20 - 2.455366258897947865 E-62*w^21 - 7.006802304860847056 E-63*w^22 - 2.0170573985505660255 E-63*w^23 - 5.844838423025127240 E-64*w^24 - 1.7043707844478603356 E-64*w^25 - 5.002282984467736117 E-65*w^26 - 1.4748828465574770541 E-65*w^27 - 4.368195147161827048 E-66*w^28 - 1.3005076403225357554 E-66*w^29 + O(w^30)

Iexp(0.1,Iexp(0.1,1,2+u),2+u) - Iexp(0.2,1,2+u)

%42 = 1.0560015667790773041 E-47 - 4.472892251399907988 E-46*u + 9.781424310607690251 E-45*u^2 - 1.4699699204329577284 E-43*u^3 + 1.7054441836546445651 E-42*u^4 - 1.6273367290258708051 E-41*u^5 + 1.32885839335278805727827573527918224978 E-40*u^6 - 9.5421624175376840539197291384460400639 E-40*u^7 + 6.1453650732800893341605643123388578479 E-39*u^8 - 3.60300875093315313408444140476588790020 E-38*u^9 + 1.94566303616455738904299083635851188585 E-37*u^10 - 9.7682500447243861266806423963749980846 E-37*u^11 + 4.59447095746943974844305534541439981805 E-36*u^12 - 2.03745361944985745398967717926235994073 E-35*u^13 + 8.5645578821650349164192801336258673790 E-35*u^14 - 3.42829069968351780819928743259801848743 E-34*u^15 + 1.31196653716817812115272921722451662512 E-33*u^16 - 2.24532381181948620962583236044984348020 E-32*u^17 + 4.8652831676001940743924440319709612865 E-31*u^18 + 2.58292920368640701303391891403464628542 E-31*u^19 - 1.97057852828718112279574806521525784630 E-28*u^20 + 4.64987264628885322866316806150922557652 E-27*u^21 - 6.6578671819241133287445615873692665215 E-26*u^22 + 7.2286681115998680900711068774455519814 E-25*u^23 - 6.58959236390523930535296008625163909249 E-24*u^24 + 5.380403054358272658633240764988271137796 E-23*u^25 - 4.096813673994155747724413220990481811285 E-22*u^26 + 2.962290760331694625789446700972377624571 E-21*u^27 - 2.037121995399768881356559797850718838378 E-20*u^28 + 1.324525396616853415146444872768299980083 E-19*u^29 + O(u^30)

Ibennet(s,2,3)

%43 = 5.000000000000000000000000000000000000000 + 0.7256473965263415355726335073462804205026*s + 0.2065985475702190684145536780286266211912*s^2 + 0.05266661871645363241880119184483379064661*s^3 + 0.01213316362816163676181025275398619937848*s^4 + 0.002484587441238103973304180437106722969663*s^5 + 0.0004278459187290018743995504020806878181593*s^6 + 5.026411809318944120745729874904715046567 E-5*s^7 - 2.097978055159398963357933638524863447127 E-6*s^8 - 3.917442368124171049853741448942720330313 E-6*s^9 - 1.673872252322174081842751643007073521005 E-6*s^10 - 5.380160204435224676806407097766698263024 E-7*s^11 - 1.487524705412200256434548723407493626568 E-7*s^12 - 3.678696023734425527698440224412572731990 E-8*s^13 - 8.202832634858730364492806468668774363198 E-9*s^14 - 1.625500586548617665288097049141693201050 E-9*s^15 - 2.712835452473239056738364619798262174322 E-10*s^16 - 3.109802441299858169295550487800312326676 E-11*s^17 + 1.112993550172292404283302730216624669840 E-12*s^18 + 2.205213806998927153512145905470116524878 E-12*s^19 + 9.181564593335951454943477288140957403512 E-13*s^20 + 2.865405234180613573417040028805685461075 E-13*s^21 + 7.682382271111932455961316648271579055736 E-14*s^22 + 1.839663374945693682982560446757950010565 E-14*s^23 + 3.961187236713129339547902763974145979214 E-15*s^24 + 7.532772539127867144186612215844980322245 E-16*s^25 + 1.185363162997741398930795760686833152740 E-16*s^26 + 1.174398958754750966498720562489546007851 E-17*s^27 - 1.166242631363682451728692597298433002393 E-18*s^28 - 1.131457387001417994544198130378773996438 E-18*s^29 - 4.315233684769584434984544994515309246649 E-19*s^30 + O(s^31)

Ibennet(0.5,2+w,3)

%44 = 5.421899688575296614480746492856113676386 + 1.679987795255501710878313093811819181796*w - 0.04347785824344861296284767941944766819858*w^2 + 0.007121822850024879241445586349222213280985*w^3 - 0.001376879116264135970261706566706224093898*w^4 + 0.0002747278105529088737346699108556333963802*w^5 - 5.385495530127409423700620537863985861381 E-5*w^6 + 1.027956560181029784062143507868493210664 E-5*w^7 - 1.953787386336938864390986897159886525278 E-6*w^8 + 3.801122856608207740988701957094181033884 E-7*w^9 - 7.442075910542003411460462084939730473596 E-8*w^10 + 1.381795911488855639716786135263370897231 E-8*w^11 - 2.404357523252995119944397239761751813756 E-9*w^12 + 4.322409520474934946212125005899378412773 E-10*w^13 - 8.273085911368946738291857396698229671168 E-11*w^14 + 1.552146378849620728183763620175586108136 E-11*w^15 - 4.097584081722807165336492806098232157828 E-12*w^16 + 1.553865361705301700532908116171772037207 E-12*w^17 - 4.745983078016714430749197422600885694106 E-13*w^18 + 1.492314260312856494890791108637045324979 E-13*w^19 - 5.525157943025930578994615191443662779547 E-14*w^20 + 1.158791240056365382110768676713892250454 E-14*w^21 - 1.439692051073410982618496311343583847457 E-15*w^22 + 4.419281401473502140597701847447461169202 E-16*w^23 + 3.699151836339054125180220323731004410062 E-16*w^24 - 2.232565330945937203451041494407379362271 E-16*w^25 + 3.710049633269071974817933636635621390823 E-17*w^26 - 4.379579123335190351213815623434490645131 E-17*w^27 + 1.032889062250991431171289843167513217652 E-17*w^28 + 1.834623994329765755744838777728366888052 E-18*w^29 + O(w^30)

Ibennet(0.5,2,3+u)

%45 = 5.421899688575296614480746492856113676386 + 1.363247806781409234684044836437993931866*u - 0.04311722936335908535861724412668305823598*u^2 + 0.01040877031580076014812191025887569735514*u^3 - 0.002870970160590399097264117058994519195313*u^4 - 0.004048640028837108120807518094271585337259*u^5 + 0.06143126859699205576793407015806617810466*u^6 - 0.6322585826171430068096418163463293403929*u^7 + 5.656622724001871846414674149246241211265*u^8 - 44.95104994643104704300166197623567347138*u^9 + 321.1437920573653539303792189552210292800*u^10 - 2082.077165719160331150645148720138351398*u^11 + 12340.61049164733578677208721587805014740*u^12 - 67254.52158829211488788183365838228670235*u^13 + 338476.1381492849208377159580740559204635*u^14 - 1577673.866929055603970674052567385020675*u^15 + 6819681.694962554307106645688386735043041*u^16 - 27318918.85030331730199608661957231215333*u^17 + 101040851.1534861799100329815154092300636*u^18 - 341946433.5942111206146463685267450947096*u^19 + 1038018931.207984498990741844342172871817*u^20 - 2693398546.208956875783299164627112235087*u^21 + 5105796531.426535288760125870735147671064*u^22 - 723951362.2746718252564545130076117477165*u^23 - 58459329036.77037565556938393671302456607*u^24 + 402801584215.7573768444086763089111345704*u^25 - 2006430155007.520755746675915888637100606*u^26 + 8721118226132.705302480198818708474773798*u^27 - 35305635629397.55267055911945872572029274*u^28 + 138069796341481.8443991453860106846432293*u^29 + O(u^30)

Which is just checking the Taylor series of our semi-group works in every variable. This works through out the program excepting some buggy values... We also checked the values of some Ibennet operations, done through each variable.