03/03/2009, 06:15 PM
(This post was last modified: 03/03/2009, 08:20 PM by sheldonison.)
so, why the wobble??? And how do you convert between constant an extension of sexp to real numbers that has a constant base conversion, and an extension of sexp to real numbers that has all positive odd derivatives?
The limit equation for the conversion I have defined for sexp to real numbers with an exact base changes comes with its own wobble, which becomes noticeable in the third derivative. This violates the requirement that the odd derivatives are positive for all values. In particular, the 5th derivative of this sexp extension will have negative values. The wobble gets smaller as the base approaches (1/e), and gets smaller faster the linear approximation, so the wobble does not seem to effect my earlier convergence error term estimates.
Even though I do not yet understand the wobble, I can characterize it. Lets take Dimitrii's taylor series expansion for sexp_e, and use it to convert to base 1.45, a little bigger than \( \eta \). Compare the converted critical section of this base 1.45 with a wobble free estimate of the critical section of base 1.45. The comparison is a simple subtraction. When converting to bases approaching \( \eta \), the conversion wobble appears to be a perfect sine wave! The relative amplitude of this sine wave stays constant as the base converted to approaches \( \eta \). The phase changes in a predictable way, and the relative height also changes in a predictable way. Also, I know how to lock these down (amplitude, phase, relative height), by requiring the limit use only values for b that are integer multiples, instead of fractional multiples, in the conversion process. One possible source of confusion is that the wobble down converting from base e to base 1.45 is about one 100 times bigger than the wobble inherent to base 1.45, and about 1000 times bigger than the wobble inherent to base 1.44533.
I haven't done it yet, but the upshot is that once I know the conversion sinusoid, I can convert from a base approaching \( \eta \) to any other base and get either an sexp with a constant base conversion factor, or Dimitrii's sexp function, where the odd derivatives are positive for all values of x>-2!
![[Image: delta_taylor_to_145.gif]](http://www.sheltx.com/share_stuff/delta_taylor_to_145.gif)
The limit equation for the conversion I have defined for sexp to real numbers with an exact base changes comes with its own wobble, which becomes noticeable in the third derivative. This violates the requirement that the odd derivatives are positive for all values. In particular, the 5th derivative of this sexp extension will have negative values. The wobble gets smaller as the base approaches (1/e), and gets smaller faster the linear approximation, so the wobble does not seem to effect my earlier convergence error term estimates.
Even though I do not yet understand the wobble, I can characterize it. Lets take Dimitrii's taylor series expansion for sexp_e, and use it to convert to base 1.45, a little bigger than \( \eta \). Compare the converted critical section of this base 1.45 with a wobble free estimate of the critical section of base 1.45. The comparison is a simple subtraction. When converting to bases approaching \( \eta \), the conversion wobble appears to be a perfect sine wave! The relative amplitude of this sine wave stays constant as the base converted to approaches \( \eta \). The phase changes in a predictable way, and the relative height also changes in a predictable way. Also, I know how to lock these down (amplitude, phase, relative height), by requiring the limit use only values for b that are integer multiples, instead of fractional multiples, in the conversion process. One possible source of confusion is that the wobble down converting from base e to base 1.45 is about one 100 times bigger than the wobble inherent to base 1.45, and about 1000 times bigger than the wobble inherent to base 1.44533.
I haven't done it yet, but the upshot is that once I know the conversion sinusoid, I can convert from a base approaching \( \eta \) to any other base and get either an sexp with a constant base conversion factor, or Dimitrii's sexp function, where the odd derivatives are positive for all values of x>-2!
