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sections numbered more clearly

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Robin Clark 2012-05-07 13:18:45 +01:00
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submission_thesis/appendixes/formal.tex
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@ -4,7 +4,7 @@
\chapter{Formal Definitions} \chapter{Formal Definitions}
\label{formalfmmd} \label{formalfmmd}
\subsection{An algebraic notation for identifying FMMD enitities} \section{An algebraic notation for identifying FMMD enitities}
Consider all `components' to exist as Consider all `components' to exist as
members of a set $\mathcal{C}$. members of a set $\mathcal{C}$.
% %
@ -44,7 +44,7 @@ Generally, where $\mathcal{{\FG}}$ is the set of all functional groups,
\begin{equation} \begin{equation}
fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}. fm : \mathcal{{\FG}} \rightarrow \mathcal{P}\mathcal{F}.
\end{equation} \end{equation}
\subsection{Relationships between functional~groups and failure modes} \section{Relationships between functional~groups and failure modes}
Let the set of all possible components be $\mathcal{C}$ Let the set of all possible components be $\mathcal{C}$
and let the set of all possible failure modes be $\mathcal{F}$, and $\mathcal{PF}$ and let the set of all possible failure modes be $\mathcal{F}$, and $\mathcal{PF}$
@ -260,7 +260,7 @@ each subset of S is $N$ or less, a concept of the `cardinality constrained power
is proposed and described in the next section. is proposed and described in the next section.
%\pagebreak[1] %\pagebreak[1]
\subsection{Cardinality Constrained Powerset } \section{Cardinality Constrained Power-set }
\label{ccp} \label{ccp}
A Cardinality Constrained power-set is one where subsets of a cardinality greater than a threshold A Cardinality Constrained power-set is one where subsets of a cardinality greater than a threshold
@ -282,7 +282,7 @@ Note that $\mathcal{P}_{1} S $ (non-empty subsets where cardinality $\leq 1$) fo
$$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$. $$ \mathcal{P}_{1} S = \{ \{a\},\{b\},\{c\} \} $$.
\paragraph{Calculating the number of elements in a cardinality constrained powerset} \paragraph{Calculating the number of elements in a cardinality constrained power-set}
A $k$ combination is a subset with $k$ elements. A $k$ combination is a subset with $k$ elements.
The number of $k$ combinations (each of size $k$) from a set $S$ The number of $k$ combinations (each of size $k$) from a set $S$
@ -314,7 +314,7 @@ from $1$ to $cc$ thus
\subsection{Actual Number of combinations to check with Unitary State Fault mode sets} \subsection{Actual Number of combinations to check with Unitary State Fault mode sets}
If all of the fault modes in $S$ were independent, If all of the fault modes in $S$ were independent,
the cardinality constrained powerset the cardinality constrained power-set
calculation (in equation \ref {eqn:ccps}) would give the correct number of test case combinations to check. calculation (in equation \ref {eqn:ccps}) would give the correct number of test case combinations to check.
Because sets of failure modes in FMMD analysis are constrained to be unitary state, Because sets of failure modes in FMMD analysis are constrained to be unitary state,
the actual number of test cases to check will usually the actual number of test cases to check will usually
@ -374,7 +374,7 @@ and whose cardinality is 11. % by inspection
\subsubsection{Establishing Formulae for unitary state failure mode \subsubsection{Establishing Formulae for unitary state failure mode
cardinality calculation} cardinality calculation}
The cardinality constrained powerset in equation \ref{eqn:ccps}, can be modified for % corrected for The cardinality constrained power-set in equation \ref{eqn:ccps}, can be modified for % corrected for
unitary state failure modes. unitary state failure modes.
%This is written as a general formula in equation \ref{eqn:correctedccps}. %This is written as a general formula in equation \ref{eqn:correctedccps}.