Editing Appendix/Ramblings/NumericallyDeterminedEigenvectors (section)

====Motivated by Analytic21====
Continuing with our analysis of the equilibrium model that is defined by the parameters,  <math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>, we have used the above-described numerical algorithm, to construct 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>.   The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession &#8212; in order of ''decreasing'' values of <math>~\sigma_c^2</math> &#8212; in the form of a looped animation sequence.  Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.  

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<font size="+1">Figure 2:</font><br />
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<math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>
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[[File:ImageTrio.png|500px|center|Three movie frames]]
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[[File:EigenfunctionMovie1.gif|500px|center|Eigenfunction movie]]
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Three frames from the animation sequence have been displayed side-by-side in the top panel of Figure 2.  This image montage is presented, in part, to illustrate the degree to which our numerically generated eigenfunction matches the analytically generated eigenfunction in the ''specific'' case  <math>~(\sigma_c^2 = 28.9)</math> for which we have been able to obtain an analytic solution to the combined/matched, core/envelope LAWEs.