02/20/2009, 10:54 AM
(This post was last modified: 02/20/2009, 04:32 PM by sheldonison.)
The question for sexp/slog is how to define the curve extending sexp from integers to real. One question is does there exist an sexp/slog function for which \( \text{slog}_a(\text{sexp}_b(x))-x \) converge to a constant value, or does it converge to a 1-cycle periodic function? And can this be a uniqueness criterion?
Consider what happens as b approaches e^(1/e) in the equation \( \text{slog}_b(x) \). The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits. Here is an example.
if \( b=1.49208... \text{ slog}_b(e)=6 \), \( \text{slog}_b(2.4989)=5 \), \( \text{slog}_b(2.2887)=4 \), \( \text{slog}_b(2.0691)=3 \), \( \text{slog}_b(1.817)=2 \), \( \text{slog}_b(1.4923)=1 \), \( \text{slog}_b(1.0)=0 \),
In the limit, as b approaches \( e^{1/e} \), the continuation of the sexp to real numbers will be a straight line between the second and third terms, or between \( \text{log}_b(e) \) and \( \text{log}_b(\text{log}_b(e)) \). This segment includes the inflection point of the sexp.
Then, once the curve for base b is defined, you can convert between base b and base a using \( \text{slog}_a(\text{sexp}_b(x))-x \). The trick is to find a value of x, such that the \( \text{sexp}_b(x) \) is large enough and equal to a defined integer value for the equation \( \text{slog}_a(\text{sexp}_b(x)) \). Then you have the conversion factor for large numbers, which can be used to define the sexp/slog curve for base a for all real numbers. This curve for base a assumes \( \text{slog}_a(\text{sexp}_b(x))-x \) converges to a constant value as opposed to converging to a 1-cycle periodic function. Moreover, the limit as b approaches \( e^{1/e} \) will eliminate any degrees of freedom in defining slog/sexp extension to real numbers for any base a.
- Sheldon Levenstein
Consider what happens as b approaches e^(1/e) in the equation \( \text{slog}_b(x) \). The curve becomes more and more linear, and there are fewer and fewer degrees of freedom for how to extend the sexp function to real numbers, and still have an increasing "well behaved" function. It must be possible to describe this rigorously in terms of limits. Here is an example.
if \( b=1.49208... \text{ slog}_b(e)=6 \), \( \text{slog}_b(2.4989)=5 \), \( \text{slog}_b(2.2887)=4 \), \( \text{slog}_b(2.0691)=3 \), \( \text{slog}_b(1.817)=2 \), \( \text{slog}_b(1.4923)=1 \), \( \text{slog}_b(1.0)=0 \),
In the limit, as b approaches \( e^{1/e} \), the continuation of the sexp to real numbers will be a straight line between the second and third terms, or between \( \text{log}_b(e) \) and \( \text{log}_b(\text{log}_b(e)) \). This segment includes the inflection point of the sexp.
Then, once the curve for base b is defined, you can convert between base b and base a using \( \text{slog}_a(\text{sexp}_b(x))-x \). The trick is to find a value of x, such that the \( \text{sexp}_b(x) \) is large enough and equal to a defined integer value for the equation \( \text{slog}_a(\text{sexp}_b(x)) \). Then you have the conversion factor for large numbers, which can be used to define the sexp/slog curve for base a for all real numbers. This curve for base a assumes \( \text{slog}_a(\text{sexp}_b(x))-x \) converges to a constant value as opposed to converging to a 1-cycle periodic function. Moreover, the limit as b approaches \( e^{1/e} \) will eliminate any degrees of freedom in defining slog/sexp extension to real numbers for any base a.
- Sheldon Levenstein
