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Remove some unintentional paragraph breaks around displayed equations
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henrikt-ma committed Sep 14, 2020
1 parent 2064f96 commit 4a73e8f
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14 changes: 9 additions & 5 deletions chapters/operatorsandexpressions.tex
Original file line number Diff line number Diff line change
Expand Up @@ -885,9 +885,9 @@ \subsubsection{homotopy}\label{homotopy}
possible even without accurate initial guess values, and then by
continuously transforming the simplified model into the actual model.
This transformation can be formulated using expressions of this kind:

$$\lambda\cdot\text{actual} + (1-\lambda)\cdot\text{simplified}$$

\begin{equation*}
\lambda\cdot\text{\lstinline!actual!} + (1-\lambda)\cdot\text{\lstinline!simplified!}
\end{equation*}
in the formulation of the system equations, and is usually called
a homotopy transformation. If the simplified expression is chosen
carefully, the solution of the problem changes continuously with $\lambda$,
Expand Down Expand Up @@ -1007,9 +1007,13 @@ \subsubsection{homotopy}\label{homotopy}
\end{lstlisting}

The initial equation is expanded into
$$ 0 = \lambda*\mathrm{der}(x)+(1-\lambda)(x-x_0)$$
\begin{equation*}
0 = \lambda*\mathrm{der}(x)+(1-\lambda)(x-x_0)
\end{equation*}
and you can solve the two equations to give
$$ x=\frac{\lambda+(\lambda-1)x_0}{2\lambda-1}$$
\begin{equation*}
x = \frac{\lambda+(\lambda-1)x_0}{2\lambda-1}
\end{equation*}
which has the correct value of $x_0$ at $\lambda = 0$ and of 1 at $\lambda= 1$, but unfortunately has a singularity at $\lambda = 0.5 $.
\end{example}

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5 changes: 3 additions & 2 deletions chapters/synchronous.tex
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Expand Up @@ -911,8 +911,9 @@ \subsection{Connected Components of the Equations and Variables Graph}\label{con
\lstinline!e! in E is a subset of V, in general, the unknowns which lexically appear
in e. There is an edge in F of the graph between an equation, e, and a
variable, v, if v = incidence(e):

$$F = \{(e, v) : e \in E , v \in \text{incidence}(e)\}$$
\begin{equation*}
F = \{(e, v) : e \in E, v \in \text{\lstinline!incidence!}(e)\}
\end{equation*}

A set of clock partitions is the \emph{connected components} (Wikipedia,
\emph{Connected components}) of this graph with appropriate definition of
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