r16226 - /1.3/docs/latex/model-free.tex -- May 11, 2012 - 14:28

Author: bugman
Date: Fri May 11 14:28:12 2012
New Revision: 16226

URL: http://svn.gna.org/viewcvs/relax?rev=16226&view=rev
Log:
Ported the HTML user manual fix of r16225 from the 1.3.16 tag.

The command used was:
svn merge -r16224:r16225 svn+ssh://bugman@xxxxxxxxxxx/svn/relax/tags/1.3.16 .

.....
  Removed more maths from the section titles of the HTML user manual.
  
  This is an attempt at fixing the image brokenness of the HTML manual!
.....


Modified:
    1.3/docs/latex/model-free.tex

Modified: 1.3/docs/latex/model-free.tex
URL: 
http://svn.gna.org/viewcvs/relax/1.3/docs/latex/model-free.tex?rev=16226&r1=16225&r2=16226&view=diff
==============================================================================
--- 1.3/docs/latex/model-free.tex (original)
+++ 1.3/docs/latex/model-free.tex Fri May 11 14:28:12 2012
@@ -1018,13 +1018,23 @@
 
 
 % The local tm model-free models.
-\subsubsection{The local $\tau_m$ model-free models}
+\begin{latexonly}
+    \subsubsection{The local $\tau_m$ model-free models}
+\end{latexonly}
+\begin{htmlonly}
+    \subsubsection{The local $tau_m$ model-free models}
+\end{htmlonly}
 
 Not only can the diffusion tensor be optimised as a global model affecting 
all spins of the molecule but a set of model-free models can be constructed 
in which each spin is assumed to diffuse independently.  In these models a 
single local $\tau_m$ parameter approximates the true, multiexponential 
description of the Brownian rotational diffusion of the molecule.  Each spin 
of the macromolecule is treated independently.  Another set of model-free 
models which include the local $\tau_m$ parameter can be created and include 
$tm0$ to $tm9$ (Models~\ref{model: tm0}--\ref{model: tm9} on page 
\pageref{model: tm0}).  These are simply models $m0$ to $m9$ with the local 
$\tau_m$ parameter added.  These models are an extension of the ideas 
introduced in \citet{Barbato92} and \citet{Schurr94} whereby the model $tm2$, 
the original Lipari and Szabo model-free equation with a local $\tau_m$ 
parameter, is optimised to avoid issues with inaccurate diffusion tensor 
approximations.
 
 
 % Determination of the diffusion tensor from the local tm parameter.
-\subsubsection{Determination of the diffusion tensor from the local $\tau_m$ 
parameter}
+\begin{latexonly}
+    \subsubsection{Determination of the diffusion tensor from the local 
$\tau_m$ parameter}
+\end{latexonly}
+\begin{htmlonly}
+    \subsubsection{Determination of the diffusion tensor from the local 
$tau_m$ parameter}
+\end{htmlonly}
 
 In \citet{Bruschweiler95} and further investigated in \citet{Lee97}, a 
methodology for determining the diffusion tensor from the local $\tau_m$ 
parameter together with the orientation of the XH bond represented by the 
unit vector $\mu_i$ was presented.  A local $\tau_m$ value was obtained for 
each spin $i$ by optimising model $tm2$.  The $\tau_{m,i}$ values were 
approximated using the quadric model
 \begin{equation} \label{eq: quadric}
@@ -1038,7 +1048,12 @@
 % The universal solution.
 %~~~~~~~~~~~~~~~~~~~~~~~~
 
-\subsection{The universal solution $\widehat\Univset$}
+\begin{latexonly}
+    \subsection{The universal solution $\widehat\Univset$}
+\end{latexonly}
+\begin{htmlonly}
+    \subsection{The universal solution U}
+\end{htmlonly}
 
 The complex model-free problem, in which the motions of each spin are both 
mathematically and statistically dependent on the diffusion tensor and vice 
versa, was formulated using set theory in \citet{dAuvergneGooley07}.  This 
paper is important for understanding the entire concept of the new protocol 
in relax and for truly grasping the complexity of the model-free problem.  
The solution $\widehat\Univset$ to the model-free problem was derived as an 
element of the universal set $\Univset$, the union of the diverse model-free 
parameter spaces $\Space$.  Each set $\Space$ was constructed from the union 
of the model-free models $\Mfset$ for all spins and the diffusion parameter 
set $\Diffset$.  A single parameter loss on a single spin shifts optimisation 
to a different space $\Space$.  Ever since the seminal work of \citet{Kay89} 
the model-free problem has been tackled by first finding an initial estimate 
of the diffusion tensor and then determining the model-free dynamics of the 
system (see Sections~\ref{sect: Mandel 1995} on page~\pageref{sect: Mandel 
1995} and~\ref{sect: diffusion seeded paradigm} on page~\pageref{sect: 
diffusion seeded paradigm}).  This diffusion seeded paradigm is now highly 
evolved and much theory has emerged to improve this path to the solution 
$\widehat\Univset$.  The technique can, at times, suffer from a number of 
issues including the two minima problem of the spheroid diffusion tensor 
parameter space, the appearance of artificial chemical exchange 
\citep{Tjandra96}, the appearance of artificial nanosecond motions 
\citep{Schurr94}, and the hiding of internal nanosecond motions caused by the 
violation of the rigidity assumption \citep{Orekhov95b, Orekhov99b, 
Orekhov99a}.