User:Tohline/SSC/Structure/Other Analytic Models - Revision history

Tohline: /* More General Approach */ Type in more specific link addresses

2015-08-22T20:38:45Z

More General Approach: Type in more specific link addresses

← Older revision		Revision as of 20:38, 22 August 2015
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	==More General Approach==		==More General Approach==
	The ''specific'' trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in [[User:Tohline/SSC/Stability/MoreGeneralApproach\|a separate chapter]], we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, [[User:Tohline/SSC/Structure/Other_Analytic_Ramblings#Consider_Parabolic_Case\|"Consider Parabolic Case"]], having the form,		The ''specific'' trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in [[User:Tohline/SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|a separate chapter]], we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, [[User:Tohline/SSC/Structure/Other_Analytic_Ramblings#Consider_Parabolic_Case\|"Consider Parabolic Case"]], having the form,
	<div>		<div>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,573:		Line 2,573:
	</tr>		</tr>
	</table>		</table>
	In this [[User:Tohline/SSC/Stability/MoreGeneralApproach\|accompanying chapter]], we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,		In this [[User:Tohline/SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|accompanying chapter]], we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,

	<div align="center">		<div align="center">

Tohline: Begin new subsection to examine "more general approach"

2015-08-22T20:24:09Z

Begin new subsection to examine "more general approach"

← Older revision		Revision as of 20:24, 22 August 2015
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	From this constraint, it appears that the nontrivial result is, <math>~n^+ = -7/16</math>.		From this constraint, it appears that the nontrivial result is, <math>~n^+ = -7/16</math>.


			==More General Approach==
			The ''specific'' trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in [[User:Tohline/SSC/Stability/MoreGeneralApproach\|a separate chapter]], we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, [[User:Tohline/SSC/Structure/Other_Analytic_Ramblings#Consider_Parabolic_Case\|"Consider Parabolic Case"]], having the form,
			<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>
			In this [[User:Tohline/SSC/Stability/MoreGeneralApproach\|accompanying chapter]], we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,

			<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>




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Tohline: /* Prasad's Work */ Finished "minus" root in the context of the fourth constraint

2015-08-22T00:38:45Z

Prasad's Work: Finished "minus" root in the context of the fourth constraint

← Older revision		Revision as of 00:38, 22 August 2015
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	</table>		</table>
	</div>		</div>

			=====The Minus Root=====
			In addition to setting <math>~\alpha = \sigma^2 = 0</math>, here we plug <math>~n^- = -9/[2(1-2\lambda)]</math> and <math>~m^+/n^+ = -2\lambda</math> into the fourth constraint expression as follows:


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

			<tr>
			<td align="right">
			<math>~x^6:</math>
			</td>
			<td align="right">
			RHS
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			n\lambda [- 47+ 118 \lambda -20 \lambda^2 ] +n^2 \lambda^2[ 16 +24 -16 \lambda - 16 \lambda + 8 ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ -\biggl[\frac{9}{2(1-2\lambda)}\biggr]
			\lambda [- 47+ 118 \lambda -20 \lambda^2 ] +\biggl[\frac{9}{2(1-2\lambda)}\biggr]^2 \lambda^2[ 48 -32 \lambda]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[\frac{9\lambda}{2(1-2\lambda)^2}\biggr]\biggl\{9 \lambda[ 24 -16 \lambda] -
			(1-2\lambda) [- 47+ 118 \lambda -20 \lambda^2 ]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[\frac{9\lambda}{2(1-2\lambda)^2}\biggr]\biggl\{216\lambda - 144\lambda^2
			+ [47- 118 \lambda +20 \lambda^2 ] + [- 94\lambda + 236 \lambda^2 -40 \lambda^3 ]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[\frac{9\lambda}{2(1-2\lambda)^2}\biggr]\biggl\{47 -4\lambda + 112\lambda^2 -40 \lambda^3
			\biggr\}
			</math>
			</td>
			</tr>
			</table>
			</div>


	=====The Plus Root=====		=====The Plus Root=====

Tohline: /* Fourth Constraint */ Finished plus root behavior in the context of the fourth constraint

2015-08-21T23:00:39Z

Fourth Constraint: Finished plus root behavior in the context of the fourth constraint

← Older revision		Revision as of 23:00, 21 August 2015
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	<tr>		<tr>
	<td align="right"><math>~x^4</math></td>		<td align="right">
	<td align="~~center~~"><math>~0</math></td>		<math>~x^6:</math>
	<td align="center"><math>~=</math></td>		</td>
			<td align="right">
			RHS
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\alpha(10 \lambda^2 - 22 \lambda +3) - \sigma^2(2\lambda^2- 2\lambda ) - 47n \lambda+ 60n \lambda^2 - 50m \lambda +11m + 10m \lambda^2-12 n m \lambda + 8n m \lambda^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="left">
			<math>
			~ + 16n(n-1)\lambda^2 - 4m(m-1)\lambda + 2m(m-1)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\alpha(10 \lambda^2 - 22 \lambda +3) - \sigma^2(2\lambda^2- 2\lambda ) + n\biggl[- 47\lambda+ 60 \lambda^2 - 50\biggl(\frac{m}{n}\biggr) \lambda
			+11\biggl(\frac{m}{n}\biggr) + 10\biggl(\frac{m}{n}\biggr) \lambda^2 - 16\lambda^2 + 4\biggl(\frac{m}{n}\biggr) \lambda - 2\biggl(\frac{m}{n}\biggr)\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="left">
			<math>
			~+n^2 \biggl[ -12 \biggl(\frac{m}{n}\biggr) \lambda + 8\biggl(\frac{m}{n}\biggr) \lambda^2 + 16\lambda^2 - 4\biggl(\frac{m}{n}\biggr)^2\lambda + 2\biggl(\frac{m}{n}\biggr)^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
	<td align="left">		<td align="left">
	<math>~		<math>~
	\alpha(~~10a_2~~^2 - 22 ~~a_0a_2~~ +~~3a_0^2~~) - \sigma^2(~~2a_2~~^2- ~~2a_0a_2~~ ) - ~~47n a_0a_2~~+ ~~60n a_2~~^2 - ~~50m a_0a_2 +11m a_0^2+ 10ma_2^2-12~~ n ~~m a_0a_2~~ + 8n m ~~a_2~~^2		\alpha(10 \lambda^2 - 22 \lambda +3) - \sigma^2(2\lambda^2- 2\lambda ) + n\biggl[- 47\lambda+ 44 \lambda^2 - 37\biggl(\frac{m}{n}\biggr) \lambda
			+ 10\biggl(\frac{m}{n}\biggr) \lambda^2 \biggr]
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	~ + ~~16n~~(n-1)~~a_2~~^2 - 4m(m-1)~~a_0 a_2~~ + 2m(m-1~~)a_0~~^2		~+n^2 \biggl[ + 16\lambda^2 -12 \biggl(\frac{m}{n}\biggr) \lambda + 8\biggl(\frac{m}{n}\biggr) \lambda^2 - 4\biggl(\frac{m}{n}\biggr)^2\lambda + 2\biggl(\frac{m}{n}\biggr)^2 \biggr]
			</math>
			</td>
			</tr>
			</table>
			</div>

			=====The Plus Root=====
			Next, in addition to setting <math>~\alpha = \sigma^2 = 0</math>, we'll plug <math>~\lambda = \tfrac{1}{2}</math> and <math>~m^+/n^+ = -1</math> into the fourth constraint expression as follows:

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

			<tr>
			<td align="right">
			<math>~x^6:</math>
			</td>
			<td align="right">
			RHS
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			n\biggl[- 47\lambda+ 44 \lambda^2 + 37 \lambda - 10 \lambda^2 \biggr] +n^2 \biggl[ + 16\lambda^2 + 12 \lambda - 8 \lambda^2 - 4 \lambda + 2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{7n}{2}\biggl[1+ n\biggl(\frac{16}{7}\biggr) \biggr] \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

			From this constraint, it appears that the nontrivial result is, <math>~n^+ = -7/16</math>.



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Tohline: /* Prasad's Work */

2015-08-21T21:54:05Z

Prasad's Work

← Older revision		Revision as of 21:54, 21 August 2015
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	</div>		</div>
	From this constraint, it appears that the nontrivial result is, <math>~n = -14/3</math>.		From this constraint, it appears that the nontrivial result is, <math>~n = -14/3</math>.

			====Fourth Constraint====

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

			<tr>
			<td align="right"><math>~x^4</math></td>
			<td align="center"><math>~0</math></td>
			<td align="center"><math>~=</math></td>
			<td align="left">
			<math>~
			\alpha(10a_2^2 - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 47n a_0a_2+ 60n a_2^2 - 50m a_0a_2 +11m a_0^2+ 10ma_2^2-12 n m a_0a_2 + 8n m a_2^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="right"> </td>
			<td align="left">
			<math>
			~ + 16n(n-1)a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2
			</math>
			</td>
			</tr>
			</table>
			</div>



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Tohline: /* The Plus Root */ Examine the plus solution in the context of the fifth constraint

2015-08-21T21:45:24Z

The Plus Root: Examine the plus solution in the context of the fifth constraint

← Older revision		Revision as of 21:45, 21 August 2015
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	<math>~		<math>~
	n\lambda [ -7 - 14\lambda ] +n^2\lambda [ - 10 \lambda -4 + 12 \lambda^2 + 4 - 2\lambda ]		n\lambda [ -7 - 14\lambda ] +n^2\lambda [ - 10 \lambda -4 + 12 \lambda^2 + 4 - 2\lambda ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td>
			<math>~
			- 7n - \biggl( \frac{3}{2} \biggr) n^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td>
			<math>~
			- 7n\biggl[1 + \biggl( \frac{3}{14} \biggr) n \biggr] \, .
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			From this constraint, it appears that the nontrivial result is, <math>~n = -14/3</math>.


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Tohline: /* Promising Avenue of Exploration */ tentative save

2015-08-21T21:35:19Z

Promising Avenue of Exploration: tentative save

Show changes

Tohline: /* Prasad's Work */ Begin discussion "plus" root

2015-08-21T19:56:02Z

Prasad's Work: Begin discussion "plus" root

← Older revision		Revision as of 19:56, 21 August 2015
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	</table>		</table>
	</div>		</div>

			====The Plus Root====
			Let's examine the "plus" solution. Because <math>~s_{nm}^{+} =0 </math>, this solution implies that,


			<div align="center">
			<math>~m^+ = -n^+</math>      <math>~\Rightarrow</math>     <math>~\frac{m}{n} = -1</math>.
			</div>

			In this case, then, we deduce that,


			<div align="center">
			<math>~\lambda = -\frac{1}{2}\biggl(\frac{m}{n}\biggr) = +\frac{1}{2}</math>.
			</div>

			So, even though these first two constraints have not revealed the ''value'' of either of the exponents, <math>~n</math> and <math>~m</math>, we see that the resulting eigenfunction will be of the form,



			====The Minus Root====

	The "plus" solution is not of interest. But, combining these two results for the "minus" solution, we furthermore conclude that, for this ''specific'' mode, the relationship between the two exponents and <math>~\lambda</math> are,		The "plus" solution is not of interest. But, combining these two results for the "minus" solution, we furthermore conclude that, for this ''specific'' mode, the relationship between the two exponents and <math>~\lambda</math> are,

Tohline: /* Prasad's Work */ Finished deriving "fifth constraint" which has led to a cubic equation for lambda

2015-08-21T17:00:43Z

Prasad's Work: Finished deriving "fifth constraint" which has led to a cubic equation for lambda

Show changes

Tohline: /* Prasad's Work */ Show that roots of quadratic equation are both imaginary

2015-08-21T15:55:08Z

Prasad's Work: Show that roots of quadratic equation are both imaginary

← Older revision		Revision as of 15:55, 21 August 2015
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	</table>		</table>
	</div>		</div>
	which exactly matches the previous, but messier, derivation.		which exactly matches the previous, but messier, derivation. Now, the two roots of the quadratic expression inside the square brackets are,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\lambda</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{2^3\cdot 13} \biggl[ 2^2\cdot 3\cdot 5 \pm \sqrt{ -2^8\cdot 29 }\biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{2\cdot 13} \biggl[ 3\cdot 5 \pm \sqrt{ -2^4\cdot 29 }\biggr] \, .</math>
			</td>
			</tr>
			</table>
			</div>
			Both roots are imaginary numbers and therefore not of interest in the context of this astrophysical problem.




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