diff --git a/eulerg/Makefile b/eulerg/Makefile new file mode 100644 index 0000000..fb80b55 --- /dev/null +++ b/eulerg/Makefile @@ -0,0 +1,24 @@ + +# +# Make the propositional logic diagram a paper +# +SOURCE = eulerg.tex + +paper: paper.tex eulerg_paper.tex $(SOURCE) + #cat introduction.tex | sed s/chapter/paper/ > introduction.tex + #latex paper.tex + #dvipdf paper pdflatex cannot use eps ffs + pdflatex paper.tex + mv paper.pdf eulerg_paper.pdf + okular eulerg_paper.pdf + + +# Remove the need for referncing graphics in subdirectories +# +eulerg_paper.tex: eulerg.tex paper.tex $(SOURCE) + cat eulerg.tex | sed 's/eulerg\///' > eulerg_paper.tex + + +bib: $(SOURCE) + bibtex paper + diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex new file mode 100644 index 0000000..ec2a1bf --- /dev/null +++ b/eulerg/eulerg.tex @@ -0,0 +1,162 @@ + + + +\ifthenelse {\boolean{paper}} +{ +\abstract{ +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} +} +{ %% Introduction +\section{Introduction} +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} + + +\section{Introduction : Euler Diagram } + +Classical Euler diagrams consist of closed curves in the plane which are used to represent sets. +The spaitial relationship between the curves defines the set theoretic relationships, as defined below. +\begin{itemize} +\item Intersection - if the curves defining the area within curves overlap +\item Sub-set - if a curve is enclosed by another +\item disjoint - if the curves are separate +\end{itemize} + + +\section{Defining `pure intersection' and `enclosure'} +\begin{figure}[h] + \centering + \includegraphics[width=200pt,keepaspectratio=true]{./eulerg1.jpg} + % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215 + \caption{An Euler Diagram showing enclosure and Pure Intersection} + \label{fig:eulerg1} +\end{figure} + +The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case. +For instance in diagram \ref{fig:eulerg1} the intersection between +$A$ and $B$ exists. + +$$ A \cup B \neq \emptyset $$ + +as does the intersection $D$ and $E$ + +$$ D \cup E \neq \emptyset $$ + +Clearly though these intersections are different, because +in the $A$, $B$ case +$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$ +This is not the case for $D$, $E$ where: +$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$ + +\paragraph{Enclosure} +To distinguish between these we can term the $A$, $B$ case to be +$A$ `enclosed' by $B$. We can express this as a directed relationship. + +$$ B {\enc} A $$ + + +\paragraph{Pure Intersection} +In the $D$, $E$ case we have + +We can say that where the areas defined by the curves intersect but no one curve encloses the +other, we can term this `pure intersection'. +We can express this as a non directed relationship. + +$$ D \pin E $$ + + +\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'} + +Clearly these two properties are mutually exclusive. No +contour can be both purely intersected and enclosed with the same contour. +Also enclosure, is transitive. That is to say if B encloses A, and A encloses C +then B encloses C, see figure \ref{fig:eulerg_enc}. + +\begin{figure}[h] + \centering + \includegraphics[width=200pt,keepaspectratio=true]{./eulerg_enc.jpg} + % eulerg_enc.jpg: 315x269 pixel, 72dpi, 11.11x9.49 cm, bb=0 0 315 269 + \caption{Enclosure, a transitive relationship} + \label{fig:eulerg_enc} +\end{figure} + +$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$ + +\section{Representing Euler Diagrams as sets of relationships} + +The diagram in figure \ref{fig:eulerg1} can be represented by the foillowing relationships. + +$$ B {\enc} A $$ +$$ D {\pin} E $$ + + +The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships. + +$$ B {\enc} A $$ +$$ A {\enc} C $$ + + +\section{The Pure Intersection chain} + +Contours may be connected via `pure intersection' relationships to form +`chains' of contours reachable by pure intersection. + +Figure \ref{fig:eulerg_pic} shows a pure intersection chain consisting of contours $M,N,O,P$ and $Q$. + +\begin{figure}[h] + \centering + \includegraphics[width=300pt,keepaspectratio=true]{./eulerg_pic.jpg} + % eulerg_pic.jpg: 955x286 pixel, 72dpi, 33.69x10.09 cm, bb=0 0 955 286 + \caption{Pure Intersection Chain with Enclosure} + \label{fig:eulerg_pic} +\end{figure} + +\textbf{rule:} +If any contour in a pure intersection chain is enclosed by any contour not belonging to the chain, +all the countours within the +pure intersection chain will be enclosed by it. This is because a contour +enclosing which bisects(????) another contour in a pure intersection chain +becomes part of the pure~intersection~chain. Hmmmm thats true but a better way to say it ???? + + +%The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships. + + +The diagram in figure \ref{fig:eulerg_pic} can be represented by the following relationships. + + +$$ M {\pin} N $$ +$$ N {\pin} O $$ +$$ O {\pin} P $$ +$$ O {\pin} Q $$ +$$ Q {\enc} P $$ +$$ A {\enc} M $$ +$$ A {\enc} N $$ +$$ A {\enc} O $$ +$$ A {\enc} P $$ +$$ A {\enc} Q $$ + + +To form the pure intersection chain we can follow +reachable pure intersection relationships. + +$ M {\pin} N {\pin} O {\pin} P $ are part of the same chain. +following from $O$, $O {\pin} Q$. +Thus by the definition of being reachable by pure instersection relationships,$M,N,O,P,Q$ +are in the same pure intersection chain, even though $Q$ encloses $P$. +Contour $A$, by virtue of not bisecting any contour in the pure instersection +chain, does not belong to it. Because it encloses one of the contours, it +encloses all contours in the chain. Knowing this can save on unecessary area operations on the concrete diagram. + + +\vspace{40pt} + diff --git a/eulerg/eulerg.tex.backup b/eulerg/eulerg.tex.backup new file mode 100644 index 0000000..5c80bf3 --- /dev/null +++ b/eulerg/eulerg.tex.backup @@ -0,0 +1,119 @@ + + + +\ifthenelse {\boolean{paper}} +{ +\abstract{ +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} +} +{ %% Introduction +\section{Introduction} +This paper discusses representing Euler Diagrams as graphs, or sets of relationships. +By representing Euler diagrams in this way, +algorithms to invesigate properties of the diagrams, are possible, without +having to resort +to CPU expensive area operations on the concrete diagrams. +} + + +\section{Introduction : Euler Diagram } + +Classical Euler diagrams consist of closed curves in the plane which are used to represent sets. +The spaitial relationship between the curves defines the set theoretic relationships, as defined below. +\begin{itemize} +\item Intersection - if the curves defining the area within curves overlap +\item Sub-set - if a curve is enclosed by another +\item disjoint - if the curves are separate +\end{itemize} + + +\section{Defining `pure intersection' and `enclosure'} +\begin{figure}[h] + \centering + \includegraphics[width=200pt,keepaspectratio=true]{./eulerg1.jpg} + % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215 + \caption{An Euler Diagram showing enclosure and Pure Intersection} + \label{fig:eulerg1} +\end{figure} + +The set theory term `intersection' can apply to both the curves overlapping and to the sub-set case. +For instance in diagram \ref{fig:euler1} the intersection between +$A$ and $B$ exists. + +$$ A \cup B \neq \emptyset $$ + +as does the intersection $D$ and $E$ + +$$ D \cup E \neq \emptyset $$ + +Clearly though these intersections are different, because +in the $A$, $B$ case +$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$. +This is not the case for $D$, $E$ where: +$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$ + +\paragraph{Enclosure} +To distinguish between these we can term the $A$, $B$ case to be +$A$ `enclosed' by $B$. We can express this as a directed relationship. + +$$ B {\enc} A $$ + + +\paragraph{Pure Intersection} +In the $D$, $E$ case we have + +We can say that where the areas defined by the curves intersect but no one curve encloses the +other, we can term this `pure intersection'. +We can express this as a non directed relationship. + +$$ D \pin E $$ + + +\paragraph{Mutual exclusivity of `pure intersection' and `enclosure'} + +Clearly these two properties are mutually exclusive. No +contour can be both purely intersected and enclosed with the same contour. +Also enclosure, is transitive. That is to say if B encloses A, and A encloses C +then B encloses C, see figure \ref{fig:eulerg_enc}. + +\begin{figure}[h] + \centering + \includegraphics[width=200pt,keepaspectratio=true]{./eulerg_enc.jpg} + % eulerg_enc.jpg: 315x269 pixel, 72dpi, 11.11x9.49 cm, bb=0 0 315 269 + \caption{Enclosure, a transitive relationship} + \label{fig:eulerg_enc} +\end{figure} + +$$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$ + +\section{Representing Euler Diagrams as sets of relationships} + +The diagram in figure \ref{fig:eulerg1} can be represented by the foillowing relationships. + +$$ B {\enc} A $$ +$$ D {\pin} E $$ + + +The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships. + +$$ B {\enc} A $$ +$$ A {\enc} C $$ + + +\section{The Pure Intersection chain} + +Contours may be connected via `pure intersection' relationships to form +`chains' of contours reachable by pure intersection. + +Figure \ref{fig:eulerg_pic} shows a pure intersection chain consisting of contours $M,N,O,P$ and $Q$. + + +\textbf{rule:} +If any contour in a pure intersection chain is enclosed by any contour, all countours within the +pure intersection chain will be enclosed by it. + diff --git a/eulerg/eulerg1.dia b/eulerg/eulerg1.dia new file mode 100644 index 0000000..20a2041 Binary files /dev/null and b/eulerg/eulerg1.dia differ diff --git a/eulerg/eulerg1.jpg b/eulerg/eulerg1.jpg new file mode 100644 index 0000000..84c977d Binary files /dev/null and b/eulerg/eulerg1.jpg differ diff --git a/eulerg/eulerg_enc.dia b/eulerg/eulerg_enc.dia new file mode 100644 index 0000000..dd57f2e Binary files /dev/null and b/eulerg/eulerg_enc.dia differ diff --git a/eulerg/eulerg_enc.jpg b/eulerg/eulerg_enc.jpg new file mode 100644 index 0000000..cdf26d3 Binary files /dev/null and b/eulerg/eulerg_enc.jpg differ diff --git a/eulerg/eulerg_pic.dia b/eulerg/eulerg_pic.dia new file mode 100644 index 0000000..fde5015 Binary files /dev/null and b/eulerg/eulerg_pic.dia differ diff --git a/eulerg/eulerg_pic.jpg b/eulerg/eulerg_pic.jpg new file mode 100644 index 0000000..753eacf Binary files /dev/null and b/eulerg/eulerg_pic.jpg differ diff --git a/eulerg/paper.tex b/eulerg/paper.tex new file mode 100644 index 0000000..9a6f157 --- /dev/null +++ b/eulerg/paper.tex @@ -0,0 +1,35 @@ + +\documentclass[a4paper,10pt]{article} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{tikz} +\usepackage{amsfonts,amsmath,amsthm} +\usepackage{algorithm} +\usepackage{algorithmic} +\usepackage{ifthen} +\newboolean{paper} +\setboolean{paper}{true} % boolvar=true or false + +\input{../style} + +%\newtheorem{definition}{Definition:} + +\begin{document} +\pagestyle{fancy} + +%\outerhead{{\small\bf Symptom Extraction Process}} +%\innerfoot{{\small\bf R.P. Clark } } + % numbers at outer edges +\pagenumbering{arabic} % Arabic page numbers hereafter +\author{R.P.Clark} +\title{Euler Diagrams as Graphs} +\maketitle +\input{eulerg_paper} + +\bibliographystyle{plain} +\bibliography{../vmgbibliography,../mybib} + +\today + + +\end{document} diff --git a/style.tex b/style.tex index 4df10b4..da0d505 100644 --- a/style.tex +++ b/style.tex @@ -74,6 +74,8 @@ \newcommand{\dcs}{\em derived~components} \newcommand{\bc}{\em base~component} \newcommand{\bcs}{\em base~components} +\newcommand{\enc}{\ensuremath{\stackrel{enc}{\longrightarrow}}} +\newcommand{\pin}{\ensuremath{\stackrel{pi}{\longleftrightarrow}}} %----- Display example text (#1) in typewriter font %\newcommand{\example}[1]{\\ \smallskip\hspace{1in}{\tt #1}\hfil\\ diff --git a/thesis.tex b/thesis.tex index b68bf41..afc12e1 100644 --- a/thesis.tex +++ b/thesis.tex @@ -86,6 +86,7 @@ \chapter {Software as PLDs} \input{sw_as_plds/sw_as_plds} + \typeout{ ---------------- Mechanical Sub-systems as PLDs} \chapter {Common Mechanical Sub-systems as PLDs} %\input{mech_as_plds/mech_as_plds} @@ -136,8 +137,11 @@ for incorrect temperature. reference the MSC document and describe the Java extension classes. Software documentation for fmmd tool. +\typeout{ ---------------- Euler Diagrams represented as graphs} +\chapter {Euler Diagrams Represented as graphs} +\input{eulerg/eulerg} -\chapter{Algorithms and Mathematical Relationships Discovered} +\chapter{Fast Zone Discrimination Algorithm} \input{fzd/fzd} \chapter{Milli Volt Amp with Safety Resistor}