Editing SSC/Stability/MoreGeneralApproach

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=More General Approach to the Parabolic Eigenvalue Problem=

The material presented in this chapter is an extension of the chapter titled, [[SSC/Structure/OtherAnalyticModels#Other_Analytically_Definable.2C_Spherical_Equilibrium_Models|Other Analytic Models]] and could also be considered to be a subsection of the associated chapter titled, [[SSC/Structure/OtherAnalyticRamblings#Consider_Parabolic_Case|Other Analytic Ramblings]].  More specifically, in the following "Introduction," we repeat a manipulation of the LAWE that was originally developed in the subsection of that chapter titled, [[SSC/Structure/OtherAnalyticRamblings#Consider_Parabolic_Case|"Consider Parabolic Case"]]. 


{{ SGFworkInProgress }}

==Introduction==
In the case of a parabolic density distribution, the LAWE may be written in the form,

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

<tr>
  <td align="right">
<math>~\frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  
</math>
  </td>
</tr>
</table>
</div>


Let's try,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^m +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2x)</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{-1}(2a_2x^2)  +m  (b_0 + b_2x^2)^{-1}(2b_2x^2) </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ n a_2 (b_0 + b_2x^2)  +mb_2  (a_0 + a_2x^2) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^{' '}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n m (a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) + n(a_0 + a_2x^2)^{n-1}(2a_2) \cdot (b_0 + b_2x^2)^m  + n(n-1)(a_0 + a_2x^2)^{n-2}(2a_2x)^2 \cdot (b_0 + b_2x^2)^m </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+m n(a_0 + a_2x^2)^{n-1}(2a_2x)  \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2) 
+m(m-1) (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-2}(2b_2x)^2</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~8n m a_2b_2 x^2 (a_0 + a_2x^2)^{-1}\cdot (b_0 + b_2x^2)^{-1} + n2a_2 (a_0 + a_2x^2)^{-1}   + n(n-1)4a_2^2 x^2 (a_0 + a_2x^2)^{-2}  
+m2b_2 (b_0 + b_2x^2)^{-1} +m(m-1)4 b_2^2 x^2 (b_0 + b_2x^2)^{-2}</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>
</table>
</div>

So, we have for the LAWE:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)(a_0 + a_2x^2) (b_0 + b_2x^2)}  
\biggl\{ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{8}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(a_0 + a_2x^2)(b_0 + b_2x^2)} \biggl\{ 2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)  
+ 8(n a_2 b_0 + mb_2 a_0)  + 8(na_2 b_2+ mb_2 a_2)x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[8n m a_2b_2+ \frac{4n(n-1) a_2^2(b_0 + b_2x^2) }{ (a_0 + a_2x^2)}  
+ \frac{4m(m-1) b_2^2(a_0 + a_2x^2) }{(b_0 + b_2x^2)} \biggr]x^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Putting these together gives,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(b_0 + b_2x^2) + m b_2  (a_0 + a_2x^2)  + 4(n a_2 b_0 + mb_2 a_0)  + 4(na_2 b_2+ mb_2 a_2)x^2+ 4n m a_2b_2x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{(1-x^2)(2-x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}\biggl[2n(n-1) a_2^2(b_0 + b_2x^2)^2   + 2m(m-1) b_2^2(a_0 + a_2x^2)^2 \biggr]x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

==Additional Setup==
Benefitting from our [[SSC/Structure/OtherAnalyticModels#Promising_Avenue_of_Exploration|earlier exploration of this problem]], let's divide through by the product, <math>~(a_0 b_0)</math>, and introduce the new variable notations,

<div align="center">
<math>~\lambda \equiv \frac{a_2}{a_0} \, ,</math> &nbsp; &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp;
<math>~\eta \equiv \frac{b_2}{b_0} \, .</math>
</div>

The LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(1 + \lambda x^2) (1 + \eta x^2) + 2x^2(n \lambda  + m\eta  )  + 2x^4 (n\lambda  \eta + m\eta  \lambda ) \biggr](5-3x^2) 
-\sigma^2 (1 + \lambda x^2) (1 + \eta x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n \lambda (1 + \eta x^2) + m \eta   (1 + \lambda x^2)  + 4(n \lambda  + m\eta  )  + 4(n\lambda  \eta + m\eta  \lambda )x^2+ 4n m \lambda \eta x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{(1-x^2)(2-x^2)}{ (1 + \lambda x^2)(1 + \eta x^2)}\biggl[2n(n-1) \lambda^2(1 + \eta x^2)^2   + 2m(m-1) \eta ^2(1 + \lambda x^2)^2 \biggr]x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

Multiplying through by the denominator of the last  term(s) &#8212; that is, multiplying through by <math>~(1 + \lambda x^2)(1 + \eta x^2)</math> &#8212; will give us a polynomial with coefficient expressions for 6 terms <math>~(x^0, x^2, x^4, x^6, x^8, x^{10})</math> expressed in terms of 5 unknowns <math>~(\sigma^2, n, m, \lambda, \eta)</math>.

Wouldn't a better strategy be to insert yet another quadratic factor &#8212; specifically, <math>~(1+\beta x^2)^\ell</math> &#8212; which will introduce two additional unknowns but only add one more term into the polynomial expression?  This would bring the total number of coefficient expressions to 7 while simultaneously raising the number of unknowns to 7.  It will be tedious and messy, but worth the try.

==Expanding from Two to Three Quadratic Terms==

Here we rearrange terms in the "parabolic" LAWE to construct the governing ODE as,

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

<tr>
  <td align="right">
<math>~\sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5(1-\tfrac{3}{5}x^2) \biggl[ \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr]  
- (1-x^2)(1-\tfrac{1}{2}x^2)\biggl[ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) 
+\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  \biggr]
</math>
  </td>
</tr>
</table>
</div>

Let's try,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \lambda x^2)^n \cdot (1 + \eta x^2)^m \cdot (1 + \beta x^2)^\ell \, ,</math>
  </td>
</tr>
</table>
</div>

or, in an effort to permit writing more compact expressions,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~N^n \cdot M^m \cdot L^\ell \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~N \equiv (1 + \lambda x^2)\, ;</math> &nbsp; &nbsp; &nbsp; 
<math>~M \equiv (1 + \eta x^2)\, ;</math> &nbsp; &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp;
<math>~L \equiv (1 + \beta x^2)\, .</math> &nbsp; &nbsp; &nbsp; 
</div>

This implies (after some whiteboard derivations),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{x\mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2x^2}{N\cdot M \cdot L} \biggl[ \ell \beta M\cdot N + m\eta L\cdot N + n\lambda L\cdot M\biggr] 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4 x^2}{N\cdot M \cdot L} \biggl[
\ell \beta (m\eta N + n\lambda M) + m\eta (\ell \beta N + n\lambda L) + n\lambda (\ell \beta M + m\eta L)
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{2\ell \beta}{L^2} \biggl[ 1 + x^2 \beta (2\ell -1)\biggr] + \frac{2m\eta}{M^2}\biggl[ 1+x^2 \eta(2m-1)\biggr] 
+ \frac{2n\lambda}{N^2}\biggl[1 + x^2 \lambda(2n-1) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

==Specific Values of Quadratic Coefficients==

Now, if we ''assume'' that,
<div align="center">
<math>~\lambda = -1 \, ;</math> &nbsp; &nbsp; &nbsp; 
<math>~\eta = -\tfrac{1}{2} \, ;</math> &nbsp; &nbsp; &nbsp; and  &nbsp; &nbsp; &nbsp;
<math>~\beta = - \tfrac{3}{5} \, .</math> &nbsp; &nbsp; &nbsp; 
</div>

the "parabolic" LAWE becomes,


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

<tr>
  <td align="right">
<math>~L \cdot \sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L^2 \biggl[ \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr]  
- N\cdot M\cdot L\biggl[ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) 
+\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Then, plugging in the expressions for <math>~\mathcal{G}_\sigma</math> and its derivatives, we have,

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

<tr>
  <td align="right">
<math>~L \cdot \sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L^2 \biggl\{ \alpha + 
\frac{2x^2}{N\cdot M \cdot L} \biggl[ \ell \beta M\cdot N + m\eta L\cdot N + n\lambda L\cdot M\biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- \biggl\{ 
4x^2\biggl[ \ell \beta (m\eta N + n\lambda M) + m\eta (\ell \beta N + n\lambda L) + n\lambda (\ell \beta M + m\eta L) \biggr]
+ 8\biggl[ \ell \beta M\cdot N + m\eta L\cdot N + n\lambda L\cdot M \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- N\cdot M\cdot L\biggl\{ 
\frac{2\ell \beta}{L^2} \biggl[ 1 + x^2 \beta (2\ell -1)\biggr] + \frac{2m\eta}{M^2}\biggl[ 1+x^2 \eta(2m-1)\biggr] 
+ \frac{2n\lambda}{N^2}\biggl[1 + x^2 \lambda(2n-1) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L^2 \biggl\{ \alpha - 
\frac{2x^2}{N\cdot M \cdot L} \biggl[ \tfrac{3}{5}\ell M\cdot N + \tfrac{1}{2}m L\cdot N + n L\cdot M\biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \biggl\{ -
4x^2\biggl[ \tfrac{3}{5}\ell  (\tfrac{1}{2}m N + n M) + \tfrac{1}{2}m (\tfrac{3}{5}\ell  N + n L) + n (\tfrac{3}{5}\ell  M + \tfrac{1}{2}m L) \biggr]
+ 8\biggl[ \tfrac{3}{5}\ell  M\cdot N + \tfrac{1}{2}m L\cdot N + n L\cdot M \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \biggl\{ 
\frac{6\ell N\cdot M}{5 L} \biggl[ 1 - \tfrac{3}{5} (2\ell -1)x^2 \biggr] + \frac{m N\cdot L}{M}\biggl[ 1 - \tfrac{1}{2}(2m-1)x^2 \biggr] 
+ \frac{2nM\cdot L}{N}\biggl[1 -  (2n-1) x^2\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ N\cdot M\cdot L^2 \sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L^2 \biggl\{ N\cdot M\cdot L\cdot \alpha - 2x^2 [ \tfrac{3}{5}\ell M\cdot N + \tfrac{1}{2}m L\cdot N + n L\cdot M ]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ N\cdot M\cdot L\biggl\{ -
4x^2 [ \tfrac{3}{5}\ell  (\tfrac{1}{2}m N + n M) + \tfrac{1}{2}m (\tfrac{3}{5}\ell  N + n L) + n (\tfrac{3}{5}\ell  M + \tfrac{1}{2}m L) ]
+ 8 [ \tfrac{3}{5}\ell  M\cdot N + \tfrac{1}{2}m L\cdot N + n L\cdot M ]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \biggl\{ 
\tfrac{6}{5} \ell N^2\cdot M^2 [ 1 - \tfrac{3}{5} (2\ell -1)x^2 ] + m N^2\cdot L^2 [ 1 - \tfrac{1}{2}(2m-1)x^2 ] 
+ 2nM^2\cdot L^2 [1 -  (2n-1) x^2] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sigma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L \biggl\{ \alpha - 2x^2 \biggl[ \frac{3}{5}\ell \biggl( \frac{1}{L}\biggr)  + \frac{1}{2}m \biggl(\frac{1}{M}\biggr) + n \biggl(\frac{1}{N}\biggr) \biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{1}{L}\biggl\{ -
4x^2 [ \tfrac{3}{5}\ell  (\tfrac{1}{2}m N + n M) + \tfrac{1}{2}m (\tfrac{3}{5}\ell  N + n L) + n (\tfrac{3}{5}\ell  M + \tfrac{1}{2}m L) ]
+ 8 [ \tfrac{3}{5}\ell  M\cdot N + \tfrac{1}{2}m L\cdot N + n L\cdot M ]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{1}{L}\biggl\{ 
\frac{6\ell}{5} \biggl[ \frac{N\cdot M}{L}\biggl] \biggl[ 1 - \frac{3}{5} (2\ell -1)x^2 \biggl] + m \biggl[ \frac{N \cdot L}{M} \biggl]\biggl[  1 - \tfrac{1}{2}(2m-1)x^2 \biggl] 
+ 2n\biggl[ \frac{M \cdot L}{N} \biggl] \biggl[ 1 -  (2n-1) x^2\biggl] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L \biggl\{ \alpha + 2 \biggl[ (L-1)\ell \biggl( \frac{1}{L}\biggr)  + (M-1)m \biggl(\frac{1}{M}\biggr) + (N-1)n \biggl(\frac{1}{N}\biggr) \biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{4}{L}\biggl\{ 
\tfrac{6}{5}\ell  M\cdot N + m L\cdot N +2 n L\cdot M 
+ (L-1)\ell  (\tfrac{1}{2}m N + n M) + (M-1) m (\tfrac{3}{5}\ell  N + n L) + (N-1)n (\tfrac{3}{5}\ell  M + \tfrac{1}{2}m L) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{1}{L}\biggl\{ 
\frac{6\ell}{5} \biggl[ \frac{N\cdot M}{L}\biggl] \biggl[ 1 + (L-1)(2\ell -1) \biggl] + m \biggl[ \frac{N \cdot L}{M} \biggl]\biggl[  1 +(M-1)(2m-1) \biggl] 
+ 2n\biggl[ \frac{M \cdot L}{N} \biggl] \biggl[ 1+ (N-1)  (2n-1) \biggl] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L \biggl\{ \alpha + 2 \biggl[ \biggl(1-\frac{1}{L} \biggr)\ell  + \biggl(1-\frac{1}{M} \biggr)m  + \biggl(1-\frac{1}{N} \biggr)n \biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{4}{L}\biggl\{ 
\tfrac{6}{5}\ell  M\cdot N + m L\cdot N +2 n L\cdot M 
+ \ell  (\tfrac{1}{2}m N\cdot L + n M\cdot L) -\ell  (\tfrac{1}{2}m N + n M) 
+ m (\tfrac{3}{5}\ell  N\cdot M + n L\cdot M) - m (\tfrac{3}{5}\ell  N + n L)
+ n (\tfrac{3}{5}\ell  M\cdot N + \tfrac{1}{2}m L\cdot N) - n (\tfrac{3}{5}\ell  M + \tfrac{1}{2}m L)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{1}{L}\biggl\{ 
\frac{6\ell}{5} \biggl[ \frac{N\cdot M}{L}\biggl] \biggl[ 2(1-\ell) + L(2\ell -1) \biggl] 
+ m \biggl[ \frac{N \cdot L}{M} \biggl]\biggl[  2(1-m) +M(2m-1)  \biggl] 
+ 2n\biggl[ \frac{M \cdot L}{N} \biggl] \biggl[ 2(1-n) + N (2n-1) \biggl] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L \biggl\{ \alpha + 2(\ell + m + n) - 2 \biggl[ \biggl(\frac{\ell}{L} \biggr)  + \biggl(\frac{m}{M} \biggr)  + \biggl(\frac{n}{N} \biggr) \biggr]
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{2}{L}\biggl\{ \frac{6\ell}{5} N\cdot M \biggl[ 2  + m    + n  \biggr]
+ m L\cdot N \biggl[  2 + \ell  + n \biggr]
+ 2nL \cdot M \biggl[ 2 + \ell  + m \biggr] \biggr\}
\biggr\}
+ \frac{1}{L}\biggl\{ 
\frac{6\ell}{5} N\cdot M \biggl[ (2\ell -1) \biggl] 
+ m L \cdot N \biggl[  (2m-1)  \biggl] 
+ 2n L \cdot M \biggl[  (2n-1) \biggl] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+ \frac{1}{L}\biggl\{ 
\frac{6\ell}{5} N\cdot M \biggl[ \frac{2}{L}(1-\ell) \biggl] 
+ m L \cdot N \biggl[  \frac{2}{M} (1-m) \biggl] 
+ 2n L \cdot M \biggl[ \frac{2}{N}(1-n) \biggl] 
\biggr\}
- \frac{4}{L}\biggl\{ \biggl[ \ell m N (\tfrac{1}{2} + \tfrac{3}{5}) + \ell  n M(1 + \tfrac{3}{5} ) + m n L (1 + \tfrac{1}{2})\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5L \biggl\{ \alpha + 2(\ell + m + n) - 2 \biggl[ \biggl(\frac{\ell}{L} \biggr)  + \biggl(\frac{m}{M} \biggr)  + \biggl(\frac{n}{N} \biggr) \biggr]
\biggr\}  
+ \frac{(3 + 2m + 2n + 2\ell)}{L}\biggl[ \frac{6\ell}{5} N\cdot M + m L \cdot N + 2n L \cdot M \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- \frac{4}{L}\biggl\{ 
\frac{3}{5} \biggl[ \frac{N\cdot M}{L} \biggr] \ell(\ell - 1)  
+ \frac{1}{2}\biggl[ \frac{L \cdot N}{M}\biggr] m(m-1) 
+ \biggl[ \frac{L \cdot M }{N} \biggr] n(n-1)
\biggr\}
- \frac{4}{L}\biggl\{ \biggl[ \ell m N (\tfrac{1}{2} + \tfrac{3}{5}) + \ell  n M(1 + \tfrac{3}{5} ) + m n L (1 + \tfrac{1}{2})\biggr] \biggr\}
</math>
  </td>
</tr>
</table>
</div>

=See Also=


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