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 Real and complex behaviour of the base change function (was: The "cheta" function) bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 08/17/2009, 08:50 AM Ya perhaps that line of thought will not contribute to our topic. My concern was about the difference of points just somehow converging to the real axis, compared with that they indeed come close to every single point of the real axis. But this seems to be true. To properly verify this we need to show that for each $x>0$ and for every $\varepsilon>0$ there exist natural numbers $n$ and $k$ such that $\left|x-\log_b^{[n-3]}(\log_b^{[3]}(a)+2\pi i k)\right|<\varepsilon$. However did we silently switch from base change $\eta\to e$ ($b=\eta,a=e$) to the base change $e\to\eta$ ($b=e,a=\eta$)? I think we are more interested in the first one! jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/17/2009, 12:07 PM (08/17/2009, 08:50 AM)bo198214 Wrote: However did we silently switch from base change $\eta\to e$ ($b=\eta,a=e$) to the base change $e\to\eta$ ($b=e,a=\eta$)? I think we are more interested in the first one!Well, they are both important, as singularities in either will be an issue, but I suppose that the change from base eta back to base e should be my more immediate concern. In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport). ~ Jay Daniel Fox bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 08/17/2009, 02:40 PM (This post was last modified: 08/17/2009, 03:08 PM by bo198214.) (08/17/2009, 12:07 PM)jaydfox Wrote: In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport). We can use my formula above (which presents the by $f_3$ without branching induced singularities) $\log_b^{[n-3]}(\log_b^{[3]}(a)+\frac{2\pi i k}{\ln(b)})$ and apply it to $b=\eta$ and $a=e$. As $e$ is a fixed point of $\log_\eta$ we get $\log_\eta^{[n]}(e) = e$ and have the singularities $\log_\eta^{[n-3]}(e(1+2\pi i k))$ while the opposite base conversion has the by Jay determined singularities $\log_e^{[n-3]}(\pi i (1+2 k))$. Here we have the interesting case that the singularities should converge to $e$ with increasing $n$, i.e. a real instead of a non-real fixed point. I am however not sure about the behaviour for increasing $k$ and hope Jay makes some illustrating pictures sheldonison Long Time Fellow Posts: 668 Threads: 24 Joined: Oct 2008 08/17/2009, 04:01 PM (This post was last modified: 08/17/2009, 04:29 PM by sheldonison.) (08/17/2009, 12:07 PM)jaydfox Wrote: (08/17/2009, 08:50 AM)bo198214 Wrote: However did we silently switch from base change $\eta\to e$ ($b=\eta,a=e$) to the base change $e\to\eta$ ($b=e,a=\eta$)? I think we are more interested in the first one!Well, they are both important, as singularities in either will be an issue, but I suppose that the change from base eta back to base e should be my more immediate concern. In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport).My next question is, how large does k have to get before we encounter singularities? After a very hectic week at work, I'm having an equally hectic time out of town on vacation, and perhaps missed out on a lot of the fun. And still, I don't have time to do this problem proper diligence. Jay, you confused me converting between base e to eta, instead of eta to e. I agree that the principles are the same, and the singularities seem to be fatal. Also, base eta can be represented as iterated exp(z-1), which is another potential source of confusion. I wanted to step back to the base conversion equation, in the strip where $\text{sexp_\eta(x)$ varies from 4.38 to 5.02, which would correspond more or less to sexp_e=0 to 1. The hope was to extend such a strip to the complex plane. I wanted to share a few observations about the singularities in this strip, where z = $\text{slog_\eta(5.02)$. Here f(z)=1, which corresponds to sexp_e(0). $f(z) = \lim_{k \to \infty} \log_e^{\circ k} \text{sexp_\eta}(z+k))$ As a simplification, we look at this equation, between z=4.38 and z=5.02 $f(z) = \lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (z) \right)$ I was interested in how large k had to get before we encounter singularities. In this scenario, we first need to find the smallest value of k such that f(z)=e. and ln(ln(ln(e)))=singularity. In this strip, does f(z) for k<=4 reach a value of exactly e or exactly 1? I found some singularities for k=5. I'm enjoying all the posts; you guys seem to be close to showing the base conversion equation has zero radius of convergence. I don't have enought time while on vacation though..... - Shel bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 08/17/2009, 04:30 PM (08/17/2009, 04:01 PM)sheldonison Wrote: you guys seem to be close to showing the base conversion equation has zero radius of convergence. Ya thats really something: an everywhere infinitely differentiable but nowhere analytic real function! PS: unfortunately sagenb is again down and it takes a day until I compiled sage on my home computer (binary does not work due to sse2), so please be patient to get your questions answered. jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/17/2009, 05:26 PM (08/17/2009, 04:01 PM)sheldonison Wrote: I wanted to step back to the base conversion equation, in the strip where $\text{sexp_\eta(x)$ varies from 4.38 to 5.02, which would correspond more or less to sexp_e=0 to 1. The hope was to extend such a strip to the complex plane. I wanted to share a few observations about the singularities in this strip, where z = $\text{slog_\eta(5.02)$. Here f(z)=1, which corresponds to sexp_e(0).Just a quick observation, to make sure we are speaking of the same thing. When converting from base eta to base e, I find that 5.0179 goes to 0, not 1, and which corresponds with sexp(-1), not sexp(0). I get confused sometimes when I work with exp(z)-1 instead of actually working with the iterations of eta^z, and sometimes I forget to switch properly between the two. Perhaps you are making the same mistake? A quick sanity check in SAGE is: Code:eta = RR(e**(1/e)); print log(log(log(log(log(eta**(eta**(eta**(eta**(eta**5.0179)))))))));You should get something like -0.000053416. I try to remember that 5.0179 goes to 0, and 6.3344 goes to 1, and then always double check my results inside my iterating functions. If not, it's usually because I am still working in a logarithmic or double logarithmic system. It's a moot point, for the most part, because it's a trivial shift in the sexp function, but I wanted to make sure it got caught early before you start getting results that don't match mine, and we can't figure out why! ~ Jay Daniel Fox jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/17/2009, 05:47 PM (08/17/2009, 04:01 PM)sheldonison Wrote: Jay, you confused me converting between base e to eta, instead of eta to e. I agree that the principles are the same, and the singularities seem to be fatal.Sorry, in thinking back, I think I started with going from e to eta, because it seemed that we needed to be able to get both ways, and at heart we start by going from base e to eta, then using cheta, then going back to base e. At any rate, there is value in both, which is easy to summarize: in order to graph contour lines for the conversion from eta to e, it is easier to look simply at the conversion from e to eta. This is, for example, the easiest way to graph contour lines for ln(z): just draw lines with exp(z), using constant real or imaginary parts as needed. Likewise, working with the conversion from eta to e would be a simple and effective way to draw contour lines in the reverse conversion, from e to eta. Since I'm still working on graphs, and I need both directions anyway, it made sense to start with going from e to eta. ~ Jay Daniel Fox sheldonison Long Time Fellow Posts: 668 Threads: 24 Joined: Oct 2008 08/18/2009, 04:37 AM (08/17/2009, 05:26 PM)jaydfox Wrote: (08/17/2009, 04:01 PM)sheldonison Wrote: I wanted to step back to the base conversion equation, in the strip where $\text{sexp_\eta(x)$ varies from 4.38 to 5.02, which would correspond more or less to sexp_e=0 to 1. The hope was to extend such a strip to the complex plane. I wanted to share a few observations about the singularities in this strip, where z = $\text{slog_\eta(5.02)$. Here f(z)=1, which corresponds to sexp_e(0).Just a quick observation, to make sure we are speaking of the same thing. When converting from base eta to base e, I find that 5.0179 goes to 0, not 1, and which corresponds with sexp(-1), not sexp(0). I get confused sometimes when I work with exp(z)-1 instead of actually working with the iterations of eta^z, and sometimes I forget to switch properly between the two. Perhaps you are making the same mistake? A quick sanity check in SAGE is: Code:eta = RR(e**(1/e)); print log(log(log(log(log(eta**(eta**(eta**(eta**(eta**5.0179)))))))));You should get something like -0.000053416. I try to remember that 5.0179 goes to 0, and 6.3344 goes to 1, and then always double check my results inside my iterating functions. If not, it's usually because I am still working in a logarithmic or double logarithmic system. It's a moot point, for the most part, because it's a trivial shift in the sexp function, but I wanted to make sure it got caught early before you start getting results that don't match mine, and we can't figure out why!Yes, 5.0179 goes to zero, my mistake, and the strip of interest is 5.0179 to 6.3344. I'm trying to formulating a theory that the singularities go along with a change in the windings, where the windings suddenly increment, so it wouldn't be possible to smooth out the singularity. It seems that the graphs are smooth if a real zero crossing corresponds to an imaginary maximal with a phase of 2*(n+1)*e*pi, then the graph will be smooth, whereas if it corresponds to an imaginary maximum with a phase of 2*n*e*pi, then there will be a singularity nearby. So far I haven't gotten to demonstrating this with the real cheta function, or with the critical strip. I'm on vacation, plus my brain operates slowly when contemplating cheta and base conversions.... bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 08/18/2009, 08:47 AM (This post was last modified: 08/18/2009, 01:33 PM by bo198214.) Just want to clarify the relation between the decremented exponential $\operatorname{dexp}(x):=e^x-1$ and $\exp_\eta$ with respect to base change by giving the formula. We know that they are affine conjugates: $\operatorname{dexp} = \tau^{-1}\circ \exp_\eta \circ \tau$ where $\tau(x)=(x+1)e$. We also have $\log(\log(\exp_\eta(\exp_\eta(x))))=\log(e^{-1}\exp_\eta(x))=-1+e^{-1}x=\tau^{-1}(x)$ thatswhy: $\log^{[n]}\circ\exp_\eta^{[n]}=\log^{[n-2]}\circ\tau^{-1}\circ \exp_\eta^{[n-2]}\circ \tau\circ \tau^{-1}=\log^{[n-2]}\circ(\tau^{-1}\circ \exp_\eta\circ \tau)^{[n-2]}\circ \tau^{-1}$ So this is the relation $\fbox{\log^{[n]}\circ\exp_\eta^{[n]} =\log^{[n-2]}\circ\operatorname{dexp}^{[n-2]}\circ \tau^{-1}}$ where $\tau^{-1}(x)=x/e-1$. Update: This formula can be generalized to a base change from $b=a^{1/a}$ to $a$, i.e. from a base to one of its fixed points: $\fbox{\log_a^{[n]}\circ\exp_b^{[n]} =\log_a^{[n-2]}\circ\operatorname{dexp}_a^{[n-2]}\circ \tau^{-1}}$ where $\tau^{-1}(x)=x/a-1$ and $\operatorname{dexp}_a(x)=a^x-1$. bo198214 Administrator Posts: 1,391 Threads: 90 Joined: Aug 2007 08/18/2009, 06:51 PM (This post was last modified: 08/18/2009, 07:04 PM by bo198214.) (08/17/2009, 04:01 PM)sheldonison Wrote: My next question is, how large does k have to get before we encounter singularities? ... As a simplification, we look at this equation, between z=4.38 and z=5.02 $f(z) = \lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (z) \right)$ For $k=3$ singularities are already there (but I use $n$ for the iteration count). The formula of the k-th singularity of $f_n$ is: (08/17/2009, 02:40 PM)bo198214 Wrote: $\log_\eta^{[n-3]}(e(1+2\pi i k))$ We see that the original singularities of $f_3$ are situated on a vertical line through e. Thatswhy instead of depicting each single singularity I show the deformation of this line under repeated $\log_\eta$. The picture shows from top to bottom 1 till 5 applications of $\log_\eta$ to this vertical line. This corresponds to n=4 till n=8. The vertical line goes from k=0 to k=5, i.e. there are the first 6 singularities of $f_n$ on each of these lines.     If one mentally prolonges these lines to the right it appears to be clear that the singularities converge to the real axis. More precisely it appears that every point on the real line in the domain of definition of the base change has in each neighborhood a singularity of $f_n$ for some $n$. « Next Oldest | Next Newest »

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