Changeset 107 for docs

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• docs/trunk/omega-test-slides/equality-2.gnu

  r106	r107
  3	3	set title "Two Equations"
  4	4
  5	(削除) #2x + 3y - z = 4 (削除ここまで)
  6	(削除) #splot (2*x + 3*y) - 4 title "{/*1.5 2x + 3y - z = 4}" ls 1 (削除ここまで)
  7	(削除) (削除ここまで)
  8	5	#3x - y + z = 30
  9	6	#x + y + 3z = 50

• docs/trunk/omega-test-slides/omega-test.tex

  r106	r107
  5	5
  6	6	\newcommand{\fullframegraph}[2]{\frametitle{#1}\graph{#2}}
  7	(追記) \newcommand{\biggraph}[2]{\frametitle{#1}\graph{#2}} (追記ここまで)
  7	8
  8	9	%{{{ header
  …	…
  48	49	%}}}
  49	50
  50	\section{Introduction}
  51	\section{The Typical Problem}
  52
  53	\occamsettings
  54
  55	%TODO decide whether to have the actual graphical stuff the same as the code problem,
  56	% or keep them divorced.
  57
  58	\begin{frame}
  59	\frametitle{Parallel Usage}
  60	Without knowing the value of \lstinline|p|,\lstinline|q| or \lstinline|r| at run-time,
  61	can we determine if this is safe:
  62	\begin{lstlisting}
  63	PAR
  64	a[p] := 7
  65	a[p + q] := 8
  66	a[p - q + r] := 9
  67	\end{lstlisting}
  68	\end{frame}
  69
  70	%TODO remember to point out that each pair is a different problem
  71
  72	%TODO point out no solution = safe, solution = unsafe
  73
  74	%TODO mention occam STOPping on integer overflow and array out of bounds being very useful
  75
  76	%TODO mention that we want integer solutions
  77
  78	\section{Terms, etc}
  79
  80	\begin{frame}[fragile]
  81	\frametitle{Defining Terms}
  82	\begin{itemize}
  83	\item General form of equations for $n$ variables ($x_1 \cdots x_n$):
  84	\begin{itemize}
  85	\item Equality: $\displaystyle\sum_{i=0}^n a_i x_i = 0$
  86	\item Inequality: $\displaystyle\sum_{i=0}^n a_i x_i >= 0$
  87	\item $x_0 = 1, \therefore a_0$ is the constant term
  88	\end{itemize}
  89	\item For readability, equations here will usually be written in the form:
  90	\begin{itemize}
  91	\item $ax + by + cz = d$
  92	\end{itemize}
  93	\item Equations are kept normalised (GCD of coefficients $a_1 \cdots a_n$ is 1)
  94	\end{itemize}
  95	\end{frame}
  96
  97	\section{Equalities}
  98
  99	\begin{frame}
  100	\frametitle{Equalities}
  101	\begin{itemize}
  102	\item Each additional equality after the first can do one (and only one) of:
  103	\begin{itemize}
  104	\item Nothing (redundant equation)
  105	\item Disprove problem (inconsistent equation)
  106	\item Remove one variable (``useful'' equation)
  107	\end{itemize}
  108	\end{itemize}
  109	\end{frame}
  110
  111	%TODO mention integer grid, and how this relates to the equations
  112
  113	\begin{frame}
  114	\frametitle{Equalities without integer solutions}
  115	\begin{itemize}
  116	\item The equation 2ドルx = 2y + 5$ has no integer solutions
  117	\begin{itemize}
  118	\item Rule: if the GCD of the variable coeffs does not divide the constant, there
  119	is no integer solution.
  120	\end{itemize}
  121	\end{itemize}
  122
  123	%Show a graph with one plane:
  124
  125	\begin{frame}[fragile]
  126	\fullframegraph{First Equality}{equality-1}
  127	\end{frame}
  128
  129	%TODO show a graph with two planes in the same place (if that's even possible?)
  130
  131	%Show a graph with two parallel planes:
  132
  133	\begin{frame}[fragile]
  134	\biggraph{Inconsistent Equality}{equality-1-par}
  135	Rule: Identical coeffs but different constant $\implies$ unsolveable.
  136	\end{frame}
  137
  138	%Show a graph with two planes intersecting in a line:
  51	139
  52	140	\begin{frame}[fragile]
  …	…
  54	142	\end{frame}
  55	143
  144	(追記) %TODO Show a graph with three planes intersecting in a line: (追記ここまで)
  145	(追記) (追記ここまで)
  146	(追記) %This is the maths of the three planes intersecting in a line: (追記ここまで)
  147	(追記) (追記ここまで)
  148	(追記) \begin{frame}[fragile] (追記ここまで)
  149	(追記) \frametitle{The Maths} (追記ここまで)
  150	(追記) 3x - y + z = 30 (追記ここまで)
  151	(追記) x + y + 3z = 50 (追記ここまで)
  152	(追記) %goes to: 4x + 4z = 80, x + z = 20, z = 20 - x. y = 3x + z - 30 (追記ここまで)
  153	(追記) % Two points this line goes through are: (10,10,10) + (11,12,9) (追記ここまで)
  154	(追記) % Another random point: (25,10,0). (追記ここまで)
  155	(追記) (追記ここまで)
  156	(追記) (追記ここまで)
  157	(追記) \section{Inequalities} (追記ここまで)
  158	(追記) (追記ここまで)
  159	(追記) %TODO probably best shown in two dimensions (追記ここまで)
  160	(追記) (追記ここまで)
  161	(追記) (追記ここまで)
  56	162	\end{document}
  163	(追記) (追記ここまで)

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