06/14/2011, 04:22 AM

(06/08/2011, 11:47 PM)JmsNxn Wrote: However, I am willing to concede the idea of changing from base eta to base root 2.

That is to say if we define:

\( \vartheta(a,b,\sigma) = \exp_{2^{\frac{1}{2}}}^{\circ \sigma}(\exp_{2^{\frac{1}{2}}}^{\circ -\sigma}(a) + h_b(\sigma))\\\\

[tex]h_b(\sigma)=\left{\begin{array}{c l}

\exp_{2^{\frac{1}{2}}}^{\circ -\sigma}(b) & \sigma \le 1\\

\exp_{2^{\frac{1}{2}}}^{\circ -1}(b) & \sigma \in [1,2]

\end{array}\right. \)

This will give the time honoured result, and aesthetic necessity in my point of view, of:

\( \vartheta(2, 2, \sigma) = 2\,\,\bigtriangleup_\sigma\,\, 2 = 4 \) for all \( \sigma \).

I like this also because it makes \( \vartheta(a, 2, \sigma) \) and \( \vartheta(a, 4, \sigma) \) potentially analytic over \( (-\infty, 2] \) since 2 and 4 are fix points.

I also propose writing

\( a\,\,\bigtriangle_\sigma^f\,\,b = \exp_f^{\circ \sigma}(\exp_f^{\circ -\sigma}(a) + h_b(\sigma))\\\\

[tex]h_b(\sigma)=\left{\begin{array}{c l}

\exp_f^{\circ -\sigma}(b) & \sigma \le 1\\

\exp_f^{\circ -1}(b) & \sigma \in [1,2]

\end{array}\right. \)

I thought about this for some time and considered interpolation between arithmetic mean and geometric mean, coming to a rather curious result. The 'mean' function with \( \sigma = -1 \) fails to satisfy a property of means: \( mean(c*r_1,c*r_2,c*r_3, ..., c*r_n) = c*mean(r_1,r_2,r_3, ..., r_n) \)

Define \( M_f^\sigma(r_1,r_2,r_3, ..., r_n) = \exp_f^{\circ \sigma}\left(\frac{\exp_f^{\circ -\sigma}(r_1) + \exp_f^{\circ -\sigma}(r_2) + \exp_f^{\circ -\sigma}(r_3) + ... + \exp_f^{\circ -\sigma}(r_n)}{n}\right),\ \sigma \le 1 \) This yields the arithmetic mean for \( \sigma = 0 \) and the geometric mean for \( \sigma = 1 \).

For \( \sigma = -1 \),

\( M_{\sqrt{2}}^{-1}(1,2) = \exp_{\sqrt{2}}^{\circ -1}\left(\frac{\exp_{\sqrt{2}}^{\circ 1}(1) + \exp_{\sqrt{2}}^{\circ 1}(2)}{2}\right) = \log_{\sqrt{2}}\left(\frac{\sqrt{2} + 2}{2}\right) \approx 1.5431066 \)

\( M_{\sqrt{2}}^{-1}(3,6) = \exp_{\sqrt{2}}^{\circ -1}\left(\frac{\exp_{\sqrt{2}}^{\circ 1}(3) + \exp_{\sqrt{2}}^{\circ 1}(6)}{2}\right) = \log_{\sqrt{2}}\left(\frac{\sqrt{2}^3 + 8}{2}\right) \approx 4.8735036 \ \approx \ 3.15824 * M_{\sqrt{2}}^{-1}(1,2) \ \not= \ 3.00000*M_{\sqrt{2}}^{-1}(1,2) \)

So it's not a 'true mean' in the sense that the scalar multiplication property fails. This result makes me doubt that the property may be satisfied for \( 0 < \sigma < 1 \). Is there a way to rectify this issue, i.e. find a solution \( (f,\sigma) \) with \( f > 1 \) and \( 0 < \sigma < 1 \) such that the property is satisfied?