Editing Appendix/Ramblings/ForCohlHoward (section)

==Figures ''circa'' Year 2000==

Approximately four years after {{ LL96 }} was published, Norman Lebovitz gave a copy of his stability-analysis (FORTRAN) code to Howard Cohl.  Using this code, Howard was able to generate a large set of growth-rate data that essentially allowed him to reproduce Figure 2b from {{ LL96 }}. 

===Image i3.png===
Howard's plot of this data &#8212; his image i3.png &#8212; is shown immediately below; the abscissa is <math>0 \le b/a \le 1</math> and the ordinate is <math>0 \le c/a \le 1</math>.
<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">Howard's "i3.png" image</td>
  <td align="center" rowspan="3">[[File:HighResSelfAdjointi3.png|400px|High Resolution]]</td>
</tr>
<tr><td align="center">
[[File:I3 FromCohl.png|450px|i3.png]]
</td></tr>
<tr>
<td align="left">
Compare with Figure 2b of {{ LL96full }}
</td>
</tr>
</table>

===Image i5.png===

In an effort to better examine growth-rate trends in the lower-left quadrant of this {{ LL96 }} figure, Howard plotted the same set of stability-analysis data on an axis pair where the abscissa is still <math>0 \le b/a \le 1</math>, but where, for each value of <math>b/a</math>, the ordinate extends from the lower self-adjoint sequence to the upper self-adjoint sequence &#8212; labeled, respectively, <math>x = +1</math> and <math>x = -1</math> in the classic EFE diagram ([[Appendix/References#EFE|<font color="red">EFE</font>]], &sect;49, p. 147, Fig. 15 or, see [[ThreeDimensionalConfigurations/RiemannStype#Fig2|our accompanying discussion]]).  This is displayed immediately below as Howard's "i5.png" image.

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">Howard's "i5.png" image</td>
  <td align="center" rowspan="3">[[File:HighResSelfAdjointi5.png|400px|High Resolution]]</td>
</tr>
<tr><td align="center">
[[File:I5 FromCohl.png|450px|i5.png]]
</td></tr>
<tr>
<td align="left">
Notice &hellip;
</td>
</tr>
</table>
In generating his "i5.png" image, precisely how did Howard "stretch" the ordinate from <math>c/a</math> (as used in his "i3.png" image) to an ordinate ranging from the lower to the upper self-adjoint sequences?  Drawing from {{ LL96 }} I presume that, for a given point in the EFE diagram <math>(b/a, c/a)</math>, Howard used the expression,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>x_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> - C \pm \sqrt{C^2 - 1} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LL96 }}, §2, p. 701, Eq. (8)
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math> \frac{abB_{12}}{c^2 A_3 - b^2 a^2 A_{12}} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ LL96 }}, §2, p. 701, Eq. (6)
  </td>
</tr>
</table>
Then I presume that the ordinate, <math>y</math> &#8212; which runs from zero to unity in the "i5.png" image &#8212; is determined from the expression,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>y</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0.5(1 - x_\pm)  \, .</math>
  </td>
</tr>
</table>
<font color="red">Is this the way Howard generated "i5.png"?</font>


<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
In an email dated 26 January 2022, Howard provided the following answer to this question &#8212;

<ul>
  <li>
I had numerical data that I think Norman provided me for the lower and upper self-adjoint sequences.
  </li>
  <li>
They were simply two sets of curvilinear data in the (b/a,c/a) diagram. Call ay=b/a and then az=c/a.
  </li>
  <li>
Then lsa and usa are both functions of ay. Note that when I encountered points which didn't lie on Norman's data, I interpolated using a 9th degree polynomial  to the lower and upper self-adjoint sequence data.
  </li>
  <li>
Now consider that you have some data "g" which gives you points in the (ay,az) plane, then g=g(ay,az).
  </li>
  <li>
Take for instance data "g" which are points in the (ay,az) plane where the growth rate are above some critical value such as 10e-5. 
  </li>
  <li>
For every data point g, there is a fixed ay coordinate value.  Normally you would plot that point at (ay,az).
  </li>
  <li>
The remapping that I did now plots it instead at a point on the ordinate given by some az'= (az-lsa(ay))/(usa(ay)-lsa(ay))
  </li>
  <li>
So if az=lsa(ay) then it appears at the ordinate value of zero.
  </li>
  <li>
and if az=usa(ay) then it appears at the ordinate value of unity.
  </li>
  <li>
So the whole horn shaped region is mapped into the unit square.
  </li>
</ul>
</td></tr></table>

===Image i4.png===
Howard's "i4.png" image, immediately below, presents a magnification of the upper-right-hand portion (identified, by hand, as the "E-group") of his "i5.png" image. The abscissa spans the parameter range, <math>0.3 \le b/a \le 1.0</math> while the ordinate spans the parameter range, <math>0.96 \le y \le 1</math>. 
<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center">Howard's "i4.png" image</td>
  <td align="center" rowspan="3">[[File:HighResSelfAdjointi4.png|400px|High Resolution]]</td>
</tr>
<tr><td align="center">
[[File:I4 FromCohl.png|450px|i4.png]]
</td></tr>
<tr>
<td align="left">
Notice &hellip;
</td>
</tr>
</table>

===Summary===
Howard is interested in understanding &#8212; in greater detail than appears in {{ LL96 }} &#8212; what gives rise to, and what is the extent of these various bands of instability in the classic EFE diagram. Explicit comments/questions:
<ol>
<li>Notice in "i4.png" that the bands labeled E4, E6, and E8 appear to extend all the way to, and intersect, the Maclaurin spheroid sequence.</li>
</ol>