Editing Appendix/Ramblings/ForCohlHoward (section)

==Self-Adjoint Sequences==

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In an email dated 26 January 2022, Howard asked, "Do you have analytic curves for the lower and upper self-adjoint sequences? Otherwise, do you have very accurate data for the lower and upper self-adjoint sequences?"
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On the same day, I sent the following response to Howard:


I have added a subsection to my online chapter discussion of {{ LL96 }} in which I derive an expression whose solution/root should map out the **upper** boundary (x = -1) of the horned-shape region.  [[ThreeDimensionalConfigurations/Stability/RiemannEllipsoids#Upper_Boundary|Click here]] to see the entire derivation; this derivation ends with the following recommended strategy:

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<font color="red">STRATEGY</font> for finding the locus of points that define the upper boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

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<math>
\biggl[c^2(a+b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
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<math>=</math>
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<math> a^2b + bc(a  - b)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr]
\, .</math>
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Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
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Related remarks:
<ol>
<li>I have not actually plugged in numbers -- that is, (b,c) pairs -- to see if it works, but I am pretty confident in the result because the derivation was pretty straightforward.  Would you mind trying it out for me, since you have working elliptic integral routines?</li>
<li>It would be wise to start by trying to duplicate -- then improve upon -- the set of (b, c) coordinate-pairs that were derived by Chandrasekhar and presented in EFE Table VI (section 48, p. 142).</li>
<li>Shortly, I will derive the complementary expression that maps out the "lower" boundary (x = +1).</li>
</ol>

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<div align="center"><b>Upper Self-Adjoint</b><br />(Howard's email on 1/27/2022)</div>
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[[File:EmailFromHowardJan27Yr2022.png|650px|FromHowardJan27Yr2022]]
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On 27 January 2022, Joel added a subsection to the online chapter discussion of {{ LL96 }} in which he derives an expression whose solution/root should map out the **lower** boundary (x = +1) of the horned-shape region.  [[ThreeDimensionalConfigurations/Stability/RiemannEllipsoids#Lower_Boundary|Click here]] to see the entire derivation; this derivation ends with the following recommended strategy:

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
<font color="red">STRATEGY</font> for finding the locus of points that define the lower boundary of the horned-shape region &hellip; &nbsp; &nbsp;Set <math>a = 1</math>, and pick a value for <math>0 < b < 1</math>; then, using an iterative technique, vary <math>c</math> until the following expression is satisfied:

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\biggl[ c^2(a-b)+ ab^2 \biggr]
\biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a^2 b - bc(b + a) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, .
</math>
  </td>
</tr>
</table>
Choose another value of <math>0 < b < 1</math>, then iterate again to find the value of <math>c</math> that corresponds to this new, chosen value of <math>b</math>.  Repeat!
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<div align="center"><b>Lower Self-Adjoint</b><br />(Howard's derivation 1/28/2022)</div>
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[[File:EmailFromHowardJan28Yr2022.png|650px|Lower Self-Adjoint Definition]]
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&nbsp;<br />

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<div align="center"><b>Upper (USA) and Lower (LSA) Self-Adjoint</b><br />(Howard's Compact Expressions Plus Plot 1/28/2022)</div>
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[[File:SelfAdjointDefinitions.png|650px|Self Adjoint Definitions]]<br />[[ThreeDimensionalConfigurations/RiemannStype#SAdata|(tabulated data here)]]
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[[File:SelfAdjointPlot.png|600px|USA and LSA Plot]]
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