User:Tohline/SSC/Stability/MoreGeneralApproach - Revision history

Tohline: /* Specific Values of Quadratic Coefficients */ More simplifications

2015-08-23T22:09:30Z

Specific Values of Quadratic Coefficients: More simplifications

← Older revision		Revision as of 22:09, 23 August 2015
Line 884:		Line 884:
	\biggr\}		\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]		- \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">

			</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">

			</td>
			<td align="center">

			</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>		</math>
	</td>		</td>

Tohline: /* Specific Values of Quadratic Coefficients */ More rearrangement of terms

2015-08-23T21:45:21Z

Specific Values of Quadratic Coefficients: More rearrangement of terms

← Older revision		Revision as of 21:45, 23 August 2015
Line 841:		Line 841:
	</td>		</td>
	<td align="left">		<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]		<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\}		\biggr\}
	</math>		</math>
Line 856:		Line 856:
	<td align="left">		<td align="left">
	<math>~		<math>~
	+ \frac{4}{L}\biggl\{ \frac{3}{5}~~\ell~~ N\cdot M \biggl[ 2 + m + n \biggr]		+ \frac{2}{L}\biggl\{ \frac{6\ell}{5} N\cdot M \biggl[ 2 + m + n \biggr]
	+ ~~\frac{~~m~~}{2}~~L\cdot N \biggl[ 2 + \ell + n \biggr]		+ m L\cdot N \biggl[ 2 + \ell + n \biggr]
	+ nL \cdot M \biggl[ 2 + \ell + m \biggr]		+ 2nL \cdot M \biggl[ 2 + \ell + m \biggr] \biggr\}
	- \~~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\}
			+ \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\}		\biggr\}
	</math>		</math>
Line 875:		Line 879:
	<math>~		<math>~
	+ \frac{1}{L}\biggl\{		+ \frac{1}{L}\biggl\{
	\frac{6\ell}{5} \biggl[ \frac{~~N\cdot M~~}{L}~~\biggl] \biggl[ 2~~(1-\ell~~) + L(2\ell -1~~) \biggl]		\frac{6\ell}{5} N\cdot M \biggl[ \frac{2}{L}(1-\ell) \biggl]
	+ m \biggl[ \frac{~~N \cdot L~~}{M} ~~\biggl]\biggl[ 2~~(1-m) ~~+M(2m-1)~~ \biggl]		+ m L \cdot N \biggl[ \frac{2}{M} (1-m) \biggl]
	+ 2n\biggl[ \frac{~~M \cdot L~~}{N} ~~\biggl] \biggl[ 2~~(1-n~~) + N (2n-1~~) \biggl]		+ 2n L \cdot M \biggl[ \frac{2}{N}(1-n) \biggl]
	\biggr\}		\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>		</math>
	</td>		</td>

Tohline: /* Specific Values of Quadratic Coefficients */ Additional rearrangement of terms toward simplification

2015-08-23T21:26:47Z

Specific Values of Quadratic Coefficients: Additional rearrangement of terms toward simplification

← Older revision		Revision as of 21:26, 23 August 2015
Line 729:		Line 729:
	\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]		\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]		+ 2n\biggl[ \frac{M \cdot L}{N} \biggl] \biggl[ 1 - (2n-1) x^2\biggl]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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">

			</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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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">

			</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">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+ \frac{4}{L}\biggl\{ \frac{3}{5}\ell N\cdot M \biggl[ 2 + m + n \biggr]
			+ \frac{m}{2}L\cdot N \biggl[ 2 + \ell + n \biggr]
			+ nL \cdot M \biggl[ 2 + \ell + m \biggr]
			- \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>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</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\}		\biggr\}
	</math>		</math>

Tohline: /* Specific Values of Quadratic Coefficients */ More attempt at simplification

2015-08-23T18:51:13Z

Specific Values of Quadratic Coefficients: More attempt at simplification

← Older revision		Revision as of 18:51, 23 August 2015
Line 681:		Line 681:
	\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 ]		\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]		+ 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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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\}		\biggr\}
	</math>		</math>

Tohline: /* Specific Values of Quadratic Coefficients */ More simplifications

2015-08-23T17:16:55Z

Specific Values of Quadratic Coefficients: More simplifications

← Older revision		Revision as of 17:16, 23 August 2015
Line 633:		Line 633:
	\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{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]		+ \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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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\}		\biggr\}
	</math>		</math>

Tohline: /* Specific Values of Quadratic Coefficients */ Insert assumed values of lambda, eta, and beta

2015-08-23T17:02:41Z

Specific Values of Quadratic Coefficients: Insert assumed values of lambda, eta, and beta

← Older revision		Revision as of 17:02, 23 August 2015
Line 582:		Line 582:
	<math>~		<math>~
	- N\cdot M\cdot L\biggl\{		- 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{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]		+ \frac{2n\lambda}{N^2}\biggl[1 + x^2 \lambda(2n-1) \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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\}		\biggr\}
	</math>		</math>

Tohline: Create new subsection and insert derivations from whiteboard

2015-08-23T16:45:13Z

Create new subsection and insert derivations from whiteboard

Show changes

Tohline: /* Additional Setup */ Insert statement recognizing that yet another quadratic factor needs to be introduced into the trial eigenfunction

2015-08-22T21:33:51Z

Additional Setup: Insert statement recognizing that yet another quadratic factor needs to be introduced into the trial eigenfunction

← Older revision		Revision as of 21:33, 22 August 2015
Line 383:		Line 383:
	</div>		</div>

			Multiplying through by the denominator of the last term(s) — that is, multiplying through by <math>~(1 + \lambda x^2)(1 + \eta x^2)</math> — 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 — specifically, <math>~(1+\beta x^2)^\ell</math> — 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.



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: Rewrite LAWE after inserting new variables, lambda, eta_0, and eta_2, that represent coefficient ratios

2015-08-22T21:09:25Z

Rewrite LAWE after inserting new variables, lambda, eta_0, and eta_2, that represent coefficient ratios

← Older revision		Revision as of 21:09, 22 August 2015
Line 323:		Line 323:
	</div>		</div>

			==Additional Setup==
			Benefitting from our [[User:Tohline/SSC/Structure/Other_Analytic_Models#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>       and
			<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">

			</td>
			<td align="center">

			</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">

			</td>
			<td align="center">

			</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>

Tohline: /* Introduction */

2015-08-22T20:41:46Z

Introduction

← Older revision		Revision as of 20:41, 22 August 2015
Line 18:		Line 18:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~\frac{2x^2(5-3x^2)}{(1-x^2)(2-x^2)} \biggl[\alpha - \sigma^2 \biggl(5-3x^2\biggr)^{-1} + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \biggr]</math>~~		<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{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +4 \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Rightarrow ~~~~~~ \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>
	<td align="center">		<td align="center">