this is for holding javascript data
Kim H. Parker added Further_insight_into_the_minimum__.tex
about 8 years ago
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Further insight into the minimum in $\Delta$ seen in Figure 5 can be gained by looking at the contribution of the two distances that make up $\Delta$
\[
\Delta = \sqrt{JS_P} +\sqrt{JS_U} \equiv \Delta_P + \Delta_U
\]
That is, the distance between $\phi(dP)$ and $\phi(dP_+ + dP_-)$ and the distance between $\phi(\rho cdU)$ and $\phi(dP_+ - dP_-)$. These are shown in Figure 7 where we have extended the range of the assumed values $c$ to show the trends better. When $c$ is small the distance $\Delta_P$ is small because $dP_\pm$ are minimally influenced by $dU$. As $c$ increases, more and more of the noisy $dU$ is included into $dP_\pm$ and so the distance between their sum and $dP$ increases. The opposite is true for $\rho cdU$ which is maximum when $c=0$ and decreases more or less monotonically with increasing $c$, never getting very smalla because of the intrinsic noise in $dU$. Adding the monotonically increasing $\Delta_P$ and the generally decreasing $\Delta_U$ produces the observed minimum in $\Delta$.
