Editing Appendix/Ramblings/ToHadleyAndImamura (section)

==Analytic Analysis of Zero-Mass Papaloizou-Pringle Tori==
Here we present a summary of a much more [[User:Tohline/Appendix/Ramblings/PPTori#Stability_Analyses_of_PP_Tori|detailed, accompanying derivation]].

===Generic Formulation===
As is explicitly defined in [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Figure1|Figure 1 of our accompanying detailed notes]], we have chosen to represent the spatial structure of an eigenfunction in the equatorial-plane of toroidal-like configurations via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ f_m(\varpi)e^{-im\phi} \biggr\}  \, .</math>
  </td>
</tr>
</table>
</div>
In general, we should assume that the function that delineates the radial dependence of the eigenfunction has both a real and an imaginary component, that is, we should assume that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_m(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}(\varpi) + i\mathcal{B}(\varpi) \, ,</math>
  </td>
</tr>
</table>
</div>
in which case the square of the modulus of the function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|f_m|^2 \equiv f_m \cdot f^*_m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}^2 + \mathcal{B}^2 \, .</math>
  </td>
</tr>
</table>
</div>
We can rewrite this complex function in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_m(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|f_m|e^{-i[\alpha(\varpi) + \pi/2]} \, ,</math>
  </td>
</tr>
</table>
</div>
if the angle, <math>~\alpha(\varpi)</math> is defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sin\alpha = \frac{\mathcal{A}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\cos\alpha = \frac{\mathcal{B}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \alpha</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl(\frac{\mathcal{A}}{\mathcal{B}}\biggr) 
= \tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>

Hence, the spatial structure of the eigenfunction can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\delta\rho}{\rho_0}\biggr]_\mathrm{spatial} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|f_m(\varpi)|e^{-i[\alpha(\varpi) + \pi/2+ m\phi]} \, . </math>
  </td>
</tr>
</table>
</div>

From this representation we can see that, at each radial location, <math>~\varpi</math>, the phase angle(s) at which the fractional perturbation exhibits its maximum amplitude, <math>~|f_m|</math>, is identified by setting the exponent of the exponential to zero.  That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi = \phi_\mathrm{max}(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\frac{1}{m}\biggl[\alpha(\varpi) +\frac{\pi}{2}\biggr] 
= -\frac{1}{m}\biggl\{ \tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] +\frac{\pi}{2} \biggr\}
\, .</math>
  </td>
</tr>
</table>
</div>

An equatorial-plane plot of <math>~\phi_\mathrm{max}(\varpi)</math> should produce the "constant phase locus" referenced, for example, in recent papers from the [[User:Tohline/Appendix/Ramblings/To_Hadley_and_Imamura#Summary_for_Hadley_.26_Imamura|Imamura &amp; Hadley collaboration]].


It should be noted that the leading (negative) sign that appears on the right-hand side of this expression for <math>~\phi_\mathrm{max}</math> is rather arbitrary, as is the additional <math>~\pi/2</math> phase shift that appears in that right-hand side expression.  Henceforth, for simplicity, we will omit both and use, instead,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
unless and until the sign and/or a global phase shift is needed to adjust the orientation of a "constant phase locus" plot to facilitate comparison with published figures.

===Blaes (1985)===
====His Derived Eigenfunction====
Via an analytic perturbation analysis, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] shows that, in slender PP tori with uniform specific angular momentum, the spatial structure of unstable modes can be expressed as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W(\eta,\theta)}{W_0} -1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ f_m(\eta,\theta)e^{-i[m\phi + k\theta]} \biggr\}  \, ,</math>
  </td>
</tr>
</table>
</div>

where (see his equation 1.10),

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

<tr>
  <td align="right">
<math>~f_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} 
\pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr] 
+ \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>

In this expression, <math>~n</math> is the polytropic index, and <math>~\beta</math> is a dimensionless parameter that specifies the relative thickness of the torus; specifically, in terms of the ratio of the inner-to-outer edge of the torus, <math>~r_-/r_+</math>,
<div align="center">
<math>~\beta \equiv \frac{1- r_-/r_+}{1+ r_-/r_+} \, .</math>
</div>

Notice that this eigenfunction derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] describes a spatial perturbation that is not confined to the equatorial plane of the torus.  The parameter pair, <math>~(\eta,\theta)</math>, defines a two-dimensional, polar coordinate system in a plane lying perpendicular to the equatorial plane that contains the torus's (nearly circular) cross-section:  <math>~\eta</math> is a dimensionless radial coordinate measuring distance from the center of the cross-section &#8212; <math>~\eta = 0</math> at the center of the cross-section and <math>~\eta = 1</math> at the surface of the torus; and <math>~\theta</math> is the polar-coordinate oriented such that <math>~\theta = 0</math> points toward the "inner" edge of the torus, <math>~\theta = \pi/2</math> points vertically, straight "up," and <math>~\theta = \pi</math> points toward the "outer" edge of the torus.  

Paralleling the [[#Generic_Formulation|generic formulation that has just been discussed]], this is a complex function having, to lowest order, the following real and imaginary parts:

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{4(n+1)^2}\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4(\beta m)^2\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{4(n+1)^2}\biggl[2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta \biggr]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

We should therefore find that the amplitude (modulus) of the perturbation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} -1\biggr|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2+ \mathcal{B}_\mathrm{Blaes}^2} \, ;</math>
  </td>
</tr>
</table>
</div>
and the associated "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl[ \frac{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr]  - k\theta 
\, .</math>
  </td>
</tr>
</table>
</div>
<!-- OLD, INCORRECT DESCRIPTION THAT IGNORES LEADING "1" IN DEFINITION OF REAL COMPONENT...
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\beta^2 m^2 \biggl[ 2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} 
\pm 4i\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta\biggr] 
+ \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>

In this expression, <math>~n</math> is the polytropic index, and <math>~\beta</math> is a dimensionless parameter that specifies the relative thickness of the torus; specifically, in terms of the ratio of the inner-to-outer edge of the torus, <math>~r_-/r_+</math>,
<div align="center">
<math>~\beta \equiv \frac{1- r_-/r_+}{1+ r_-/r_+} \, .</math>
</div>

Notice that this eigenfunction derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] describes a spatial perturbation that is not confined to the equatorial plane of the torus.  The parameter pair, <math>~(\eta,\theta)</math>, defines a two-dimensional, polar coordinate system in a plane lying perpendicular to the equatorial plane that contains the torus's (nearly circular) cross-section:  <math>~\eta</math> is a dimensionless radial coordinate measuring distance from the center of the cross-section &#8212; <math>~\eta = 0</math> at the center of the cross-section and <math>~\eta = 1</math> at the surface of the torus; and <math>~\theta</math> is the polar-coordinate oriented such that <math>~\theta = 0</math> points toward the "inner" edge of the torus, <math>~\theta = \pi/2</math> points vertically, straight "up," and <math>~\theta = \pi</math> points toward the "outer" edge of the torus.  

Paralleling the [[#Generic_Formulation|generic formulation that has just been discussed]], this is a complex function having, to lowest order, the following real and imaginary parts:

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

<tr>
  <td align="right">
<math>~\mathcal{A}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4(n+1)^2}\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl(\frac{3}{2n+2}\biggr)^{1/2} \eta\cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4(n+1)^2}\biggl[2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta \biggr]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

We should therefore find that the amplitude (modulus) of the enthalpy perturbation is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{(m\beta)^2}\biggl|\frac{\delta W}{W_0} \biggr|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sqrt{\mathcal{A}^2+ \mathcal{B}^2} \, ;</math>
  </td>
</tr>
</table>
</div>
and the associated "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]  - k\theta 
\, .</math>
  </td>
</tr>
</table>
</div>

END OLD INCORRECT DESCRIPTION  -->

====Nice Features====
After summarizing, above, our efforts to develop (by empirical techniques) mathematical expressions that qualitatively match the shape of unstable eigenfunctions in toroidal configurations, we put together a subsection titled, "[[#Additional_Comments|Additional Comments]]," to highlight ways in which the empirical fits were successful and ways in which they fell short of expectations.  Here, following our presentation of Blaes's (1985) analytically derived eigenfunction for slim PP tori, we highlight elements of his (physically justified) eigenfunction that explain the origin of many features that were highlighted, above.

* The "amplitude" plot and a plot of the "constant phase locus" ''should'', indeed, appear to be interdependent because they both depend on the functional forms of <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>; in the end, however, the two plots are formally independent of one another.
* The square of the modulus &#8212; that is, the "amplitude" plot &#8212; should always be the sum of two independent functions, both of which are intrinsically positive.
* If either <math>~\mathcal{A}</math> or <math>~\mathcal{B}</math> is defined by a function that crosses zero (perhaps multiple times) with a naturally continuous derivative, that function can quite naturally give rise to a "log-amplitude" plot that shoots toward minus infinity and exhibits a discontinuous derivative after the function is squared to become a piece of the "modulus" expression.
* It is now easy to understand why the earlier "empirically derived" constant phase loci were phrased in terms of the arctangent function.
* Each "constant phase locus" plot should naturally be composed of two (smoothly joined) pieces:  One defined over the inner portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = 0</math>; and one defined over the outer portion of the torus &#8212; <math>~\eta</math> goes from zero to 1 while <math>~\theta = \pi</math>.
* Because (at least for slim PP-tori) the ratio, <math>~\mathcal{A}/\mathcal{B}</math> contains an overall factor that is an odd power of <math>~\cos\theta</math>, the argument of the arctangent function will automatically flip signs as we move from the "inner" region of the torus to the "outer" region of the torus.
* We now appreciate that a plot of "constant phase locus" is smooth across the mid-point (across the cross-sectional center) of the torus because the definition of <math>~\phi_\mathrm{max}</math> naturally contains a phase shift of <math>~k\theta</math>.

====Movie====

Figure 3 displays, for various values of the polytropic index, <math>~0.0 \le n\le 5.0</math>, the radial structure of the Blaes85 <math>~m=2</math> eigenfunction for a slim PP-torus having <math>~\beta = 0.18</math>.  More specifically, for each frame of the animation, the figure displays as a function of the dimensionless radial coordinate, <math>~x</math>, (left panel) the behavior of
<div align="center">
'''Left:'''&nbsp; &nbsp; <math>~\sqrt{\mathcal{A}_\mathrm{Blaes}^2 + \mathcal{B}_\mathrm{Blaes}^2}</math>
</div>
on a semi-log plot; and (right panel) the "constant phase locus,"
<div align="center">
'''Right:'''&nbsp; &nbsp; <math>~\frac{1}{2} \tan^{-1} \biggl[ \frac{\mathcal{A}_\mathrm{Blaes}}{\mathcal{B}_\mathrm{Blaes}} \biggr] - \theta \, .</math>
</div>

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="2" width="610px">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Figure 3:</b></font> &nbsp; The Blaes85 Analytic Eigenfunction
  </td>
</tr>

<tr><td align="center" bgcolor="gray" colspan="2">
[[File:Blaes85Analytic.gif|600px|Amp and Phase from Blaes (1985)]]
</td></tr>
<tr>
  <td align="left" colspan="2">
CAPTION: &nbsp; Radial dependence of the (left) amplitude and (right) "constant phase locus" of the Blaes85 analytic eigenfunction, for various values of the polytropic Index, <math>~n</math>, as indicated in the bottom-right corner of the left panel.
  </td>
</tr>
</table>
</div>

===Comparisons===
====For n = 3 Configurations====
Following &sect;1.3 of Blaes85, let's specifically consider the case of slim PP-tori that have a polytropic index, <math>~n=3</math>.  In this case, we have,

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~2^2[2^5\cos^2\theta - 3]\eta^2  - 13 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_3(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3
\eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^{13}\cdot 3\eta^2\cos^2\theta ]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

<!-- OMIT
Hence, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2^3/(\beta m)]^2 - 13 
+ 2^2[2^5\cos^2\theta - 3]\eta^2 }{  2^{6}(2\cdot 3)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

END OMIT -->

Now, rather than examining the structural behavior of the amplitude and phase of the function, <math>~[W/W_0 -1]</math>, as we have done, above, Blaes evaluated the amplitude (only) of the function, 
<div align="center">
<math>~\frac{W}{W_0} = 1 + \mathcal{A}_\mathrm{Blaes}(\eta,\theta) + i \mathcal{B}_\mathrm{Blaes}(\eta,\theta) \, .</math>
</div>

For the specific case of <math>~n=3</math>, the square of the amplitude of ''this'' function is,

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

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^6}\biggl[a_3(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^6}\biggl[b_3(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2a_3}{2^5}  
+ \frac{ (\beta m)^4 }{2^{12}} (a_3^2 + b_3^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^5}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>

Therefore, to lowest order in the "slimness" parameter, <math>~\beta</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} \biggr|_{n=3}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^6}  \biggl\{ 2^2[2^5\cos^2\theta - 3]\eta^2  - 13\biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>

This (almost exactly) matches the amplitude expression derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] and evaluated for the specific case of <math>~\beta = (101)^{-1/2}</math> &#8212; the relevant equation (1.12) from Blaes85 is digitally reproduced in the table that follows.  

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
Equation (1.12) extracted without modification from p. 556 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 &copy; Royal Astronomical Society
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq1.12.png|400px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

Our expression differs from the one published in Blaes85 in one detail:  In Blaes85, the pre-factor of the second term inside the curly braces is, evidently, <math>~(\beta m)^2/2^5</math>, which is a factor of two larger than the corresponding pre-factor found in our expression.  In our attempt to re-derive the Blaes85 expression (1.12), this extra factor of two disappears when we take the square-root of both sides to obtain the modulus, rather than the square of the modulus.  Hopefully further reflection will resolve this discrepancy between our approximate expression for <math>~|W/W_0|_{n=3}</math> and the analogous one presented in Blaes85.  

====For n = 0 Configurations====

As we have [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|discussed elsewhere]], [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman &amp; Narayan (1986, MNRAS, 221, 339)] &#8212; hereafter, GGN86 &#8212; also used analytic techniques to analyze the properties of unstable, nonaxisymmetric eigenmodes in Papaloizou-Pringle tori.  They restricted their discussion to slim, ''incompressible'' tori, so in order to assess the overlap between the GGN86 and Blaes85 works, we will set <math>~n=0</math> in the general expressions presented in Blaes85.

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{2^2}\biggl[a_0(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{2^2}\biggl[b_0(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~[8\cos^2\theta - 3]\eta^2  - 1 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_0(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3 (2\cdot 3)^{1/2} \eta \cos\theta  \, .
</math>
  </td>
</tr>
</table>
</div>

From the Blaes85 analysis, then, we conclude that the unstable eigenfunction for slim, ''incompressible'' PP-tori is,

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

<tr>
  <td align="right">
<math>~\biggl[ \frac{W}{W_0}\biggr]_{n=0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2^2}\biggl[(8\cos^2\theta - 3)\eta^2  - 1\biggr]
+ i \frac{ (\beta m)^2}{2^2}\biggl[ 2^3 (2\cdot 3)^{1/2} \eta \cos\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \tfrac{1}{4}(\beta m)^2\biggr] + (\beta m)^2 \biggl[\biggl(2\cos^2\theta - \frac{3}{4}\biggr)\eta^2 \biggr]
+ 4i (\beta m)^2\biggl[ \biggl(\frac{3}{2}\biggr)^{1/2} \eta \cos\theta \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This should be compared with equation (5.16) of GGN86, which is "the lowest order [complex] expression for the [perturbed] velocity potential," namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\psi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+\tfrac{1}{4} k^2(5x^2 - 3z^2) \mp 4i\biggl(\frac{3}{2}\biggr)^{1/2} k x \beta_\mathrm{GGN} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1+m^2 \beta^2_\mathrm{Blaes}\biggl[2\eta^2\cos^2\theta - \frac{3\eta^2}{4}\biggr]  
\mp 4i \biggl(\frac{3}{2}\biggr)^{1/2} m^2\beta^2_\mathrm{Blaes} \eta\cos\theta \, ,</math>
  </td>
</tr>
</table>
</div>

where the last expression results from [[User:Tohline/Appendix/Ramblings/PPTori#Goldreich.2C_Goodman_and_Narayan_.281986.29|our mapping of the GGN86 terminology to the Blaes85 terminology]].  The two derived expressions match in every detail, except one:  The constant term, <math>~\tfrac{1}{4}(\beta m)^2</math>, that is subtracted from unity on the right-hand side of the Blaes85 function is missing from the function derived by GGN86.  Hopefully, additional study of this problem will rectify this apparent difference between the two published analyses.

<!-- OMIT
In addition, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 + a(\eta,\theta) }{  b(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{[2/(\beta m)]^2 - 1 
+ [8\cos^2\theta - 3]\eta^2  }{  2^3 (2\cdot 3)^{1/2} \eta \cos\theta } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

and the square of the amplitude is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{\delta W}{W_0} \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ 1 + \frac{ (\beta m)^2}{2^2}\biggl[a(\eta,\theta)\biggr] \biggr\}^2 +  
\biggl\{ \frac{(\beta m)^2}{2^2}\biggl[b(\eta,\theta)\biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ 2(\beta m)^2a}{2^2}  
+ \frac{ (\beta m)^4 }{2^4} (a^2 + b^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \frac{ (\beta m)^2}{2}  \biggl\{ [8\cos^2\theta - 3]\eta^2  - 1 \biggr\}
+ \mathcal{O}[(\beta m)^4] \, .
</math>
  </td>
</tr>
</table>
</div>
END OMIT -->


====For n = 3/2 Configurations====
The Imamura &amp; Hadley collaboration examined instabilities that develop in tori having a variety of polytropic indexes, but their focus was often  on <math>~n=3/2</math> configurations.  It is useful, therefore, to evaluate the Blaes85 eigenfunction for this set of models.  In this case,

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

<tr>
  <td align="right">
<math>~\mathcal{A}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^2}{5^2}\biggl[a_{3/2}(\eta,\theta)\biggr]
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}_\mathrm{Blaes}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(\beta m)^2}{5^2}\biggl[b_{3/2}(\eta,\theta)\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_{3/2}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{
~[8(n+1)\cos^2\theta - 3]\eta^2 (n+1) - (4n+1) \biggr\}_{n=3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~\frac{5}{2}[20\cos^2\theta - 3]\eta^2  - 7 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_{3/2}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl\{ [2^7\cdot 3(n+1)^3 \eta^2\cos^2\theta ]^{1/2} \biggr\}_{n=3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta ]^{1/2}\, .
</math>
  </td>
</tr>
</table>
</div>

Hence, the "constant phase locus" should be identified by the function,

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

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{a_{3/2}(\eta,\theta) }{  b_{3/2}(\eta,\theta) } \biggr\}  - k\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{- 7 
+ \frac{5}{2}[20\cos^2\theta - 3]\eta^2 }{  20\cdot (3\cdot 5)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\tan^{-1}\biggl\{ \frac{- 14 
+ 5[20\cos^2\theta - 3]\eta^2 }{  40\cdot (3\cdot 5)^{1/2}\eta \cos\theta  } \biggr\}  - k\theta \, ,
</math>
  </td>
</tr>
</table>
</div>

and the square of the (unity-adjusted) amplitude is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl|\frac{W}{W_0} - 1 \biggr|^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{ (\beta m)^4 }{5^4} \biggl[ a_{3/2}^2 + b_{3/2}^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2  - 7\biggr]^2
+ \biggl[ 2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta \biggr]    \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2\biggr]^2
-14 \biggl[\frac{5}{2} (20\cos^2\theta - 3)\eta^2 \biggr]
+ \biggl[- 7\biggr]^2
+ 2^{4}\cdot 3\cdot 5^3\eta^2\cos^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{ \frac{5^2}{2^2} (20\cos^2\theta - 3)^2\eta^4
-35 (20\cos^2\theta - 3)\eta^2 + 49
+ 6000\eta^2\cos^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ (\beta m)^4 }{5^4} \biggl\{49
 + [105 + 5300\cos^2\theta  ]\eta^2
+  \frac{5^2}{2^2} (20\cos^2\theta - 3)^2\eta^4\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>


Let's compare the amplitude and phase diagrams that result from the Blaes85 analytic model with results from the model "P4" evolution reported in [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014)], that is, from Paper II in the Imamura &amp; Hadley collaboration.  Setting <math>~n = 3/2</math> and, because this comparison is restricted to the equatorial plane, setting <math>~\theta = 0</math> (inner region of torus) or <math>~\theta = \pi</math> (outer region of torus), we have from Blaes85,

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

<tr>
  <td align="right">
<math>~\mathcal{A}(\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~\biggl(\frac{17}{10}\biggr)\eta^2 - \frac{7}{25} 
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}(\eta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm
\biggl(\frac{2^4\cdot 3}{5}\biggr)^{1/2} \eta \, .
</math>
  </td>
</tr>
</table>
</div>

Figure 3 presents a plot (left panel) of <math>~\tfrac{1}{4}\mathcal{A}</math> versus <math>~x</math> (salmon-colored markers) and <math>~\tfrac{1}{4}\mathcal{B}</math> versus <math>~x</math> (green markers) for a PP-torus with <math>~\beta = 0.176</math>.  This torus has the same aspect ratio as the model named "P4" in [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]:  It has an inner edge at <math>~x_- = 0.85</math>, outer edge at <math>~x_+ = 1.21</math>, and cross-sectional "center" at <math>~x_0 = 1</math>.  (For equilibrium model characteristics, also see [[User:Tohline/Appendix/Ramblings/Azimuthal_Distortions#Setup|Table 4 in our accompanying detailed discussion]]).  Via a semi-log plot, the right panel of Figure 3 displays the behavior of <math>~(\tfrac{1}{4}\mathcal{A})^2</math> and <math>~(\tfrac{1}{4}\mathcal{B})^2</math> as a function of <math>~x</math>.  

<div align="center">
<table border="1" cellpadding="0" align="center">
<tr>
  <td align="center">
<font size="+1"><b>Figure 3:</b></font>&nbsp; Real (A) and Imaginary (B) Components of the Blaes85 Analytic Eigenfunction
  </td>
</tr>
<tr><td align="center">
[[File:RealImaginaryMontage.png|600px|Analytic Eigenfunction]]
</td></tr>
</table>
</div>

Note that, although the function, <math>~\mathcal{B}(\eta)</math>, is linear in <math>~\eta</math>, the green curve in the left panel of Figure 3 is slightly curved.  This is because the horizontal axis (in both panels) is the coordinate, <math>~x</math>, rather than <math>~\eta</math>.  The conversion from <math>~\eta</math> to <math>~x</math> is provided by the root of a cubic equation, as [[User:Tohline/Appendix/Ramblings/PPTori#Blaes_.281985.29|discussed separately]].  


The panel on the right in Figure 3 explains in a qualitative sense how sharp features &#8212; in particular, steep valleys &#8212; can arise in the "amplitude" plots of simulations that study the nonlinear growth of unstable, nonaxisymmetric eigenmodes in tori.  A sharp feature can arise when either the real or the imaginary component of the eigenmode crosses zero (and thereby changes sign).  In the analytic eigenfunction expression derived by Blaes (1985), the radial dependence of the imaginary component is defined by a linear function, and it displays a ''single'' sharp feature; the radial dependence of the real component is defined by a quadratic function, and it displays a ''pair'' of sharp features.  It seems clear that, depending on the ''relative'' overall amplitude of the real and imaginary parts, the combined amplitude could display a single sharp feature, a pair of sharp features, or a milder curve with no particularly sharp features.  It is this third option that results from the specific case presented to us by the Blaes85 eigenfunction.  As is shown by the blue curve in the middle-left-hand panel of Figure 4, the Blaes85 modulus &#8212; <math>~\tfrac{1}{4}\sqrt{\mathcal{A}^2 + \mathcal{B}^2}</math> &#8212; presents a curve with no particularly sharp features.

<div align="center">
<table border="1" cellpadding="0" align="center">
<tr>
  <td align="center" colspan="2">
<font size="+1"><b>Figure 4:</b></font>&nbsp; Comparison
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleA.png|30px|Panel A Title]]
  </td>
  <td align="center" rowspan="3">
[[File:Montage01.png|400px|Comparison]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleB.png|30px|Panel A Title]]
  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
[[File:Figure4PanelTitleC.png|30px|Panel A Title]]
  </td>
</tr>
</table>
</div>

Figure 4 displays (left) the radial dependence of the "amplitude" and (right) the radial dependence of the "constant phase locus" for the unstable modes of three models with similar &#8212; although not identical &#8212; initial equilibrium structures:  (Top row) The model with a star-to-disk mass ratio of 100 labeled "P4" in Paper II of the Imamura &amp; Hadley collaboration; (Middle row) The ''massless'' model analyzed analytically by Blaes (1985) and described herein; and (Bottom row) An unpublished model from the Imamura &amp; Hadley collaboration (private communication) with a similar aspect ratio but a star-to-disk mass ratio of 1000.



====Various Thoughts====

<ol>
<li>In the Blaes85 model, the "constant phase locus" swings through a total angle of,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\Delta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\tan^{-1}\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]_{\eta=1} + \pi \, .</math>
  </td>
</tr>
</table>
</div>

For example, in the model displayed, above, <math>~[\mathcal{A}/\mathcal{B}]_{\eta=1} = 0.4583</math>.  Hence, <math>~2\Delta = 4.001</math> radians.</li>

<li>
We need to resolve the apparent discrepancy between the value of the leading constant that appears in the GGN86 eigenfunction versus the one that is found in the Blaes85 eigenfunction (when evaluated for <math>~n=0</math>).  Graphically, the Blaes85 amplitude function appears to make more sense, but a physically based explanation needs to be identified.
</li>

<li>
The amplitude and phase functions obtained from the Blaes85 work appear to match &#8212; qualitatively, if not quantitatively &#8212; the amplitude and phase functions published by the Imamura &amp; Hadley collaboration if we subtract the leading "unity" constant from the Blaes85 expression for <math>~W/W_0</math>.  On the other hand, when Blaes refers to the modulus of the amplitude, he includes this leading "unity" constant in evaluating the "real" component of his expression.  I do not yet fully understand why both ways of viewing the eigenfunction's amplitude &#8212; that is, both with and without including the unity term &#8212; can be physically relevant.
</li>

<li>
As a small extension of the Blaes85 analysis, we should determine what the eigenfunction is for the ''density'' perturbation, rather than for the "enthalpy" perturbation, and see how well it matches the ''blue'' amplitude and phase curves published by Hadley et al. 
</li>
</ol>