Tom and Shane:

Here (attached) are some data files and corresponding plots.

Throughout, the N's are really 2N+1.  So, I gathered count data for
N=10,11,...,19,20; wbars have N=21,23,...,39,41.

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1.  w_theta_with_N_series.txt and w_theta_with_N_series.pdf

These are wbar(theta, N) as a function of theta with N series.
This is on a logy scale.

Note:  This is smooth as a function of theta, finally.

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2.  w_N_with_theta_series.txt and w_N_with_theta_series.pdf

These are wbar(theta, N) as a function of N with theta series.
This is on a logy scale.

Note:  This shows that an exponential or corrected-power-law fit will
be appropriate.

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3.  w_ratio_theta_with_N_series.txt and w_ratio_theta_with_N_series.pdf

This is wbar(theta, N) / wbar(0, N) as a function of theta with N series.
This is not on a logy scale.

Note:  This gives us the kind of curve we expected.  Note however the
oscillatory behavior, in N, as theta approaches 45 degrees.  The next
plot will make this clearer.

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4.  w_ratio_N_with_theta_series.txt and w_ratio_N_with_theta_series.pdf

This is wbar(theta, N) / wbar(0, N) as a function of N with theta series.

Note:  Here we can see that, for theta close to zero, wbar(theta, N) is nice as
a function of N.  For theta close to 45 degrees, especially for 45 degrees
itself, it oscillates high and low for adjacent N's.  The effect dampens out
gradually for larger N's, but I don't have the CPU time (in my lifetime) to
reach N large enough to flatten it out.  Maybe we could average over adjacent
pairs of N's.

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5.  pgr -grey w_theta_fit_a1.txt and w_theta_fit_a1.pdf

Here I fit wbar(theta, N) to  a b^N  and plot the a.

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6.  pgr -grey w_theta_fit_a3.txt and w_theta_fit_a3.pdf

Here I fit wbar(theta, N) to  a mu^N N^b,  with known mu, and plot the a.

2010-04-23