Changeset 136 for docs

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

  r135	r136
  1	1	load "2d-gnuplot-settings.inc"
  2	2
  3	set title "(削除) Unsolveable Equation (削除ここまで)"
  3	set title "(追記) Inconsistent Equations (追記ここまで)"
  4	4
  5	5	set multiplot
  …	…
  9	9	set key bottom right
  10	10
  11	plot (削除) x + 0.5 title "2y = 2x + 1" ls 3 (削除ここまで) with lines
  11	plot (追記) ((2*x + 1)/3) title "3y = 2x + 1" ls 3 with lines, ((2*x + 2)/3) title "3y = 2x + 2" ls 4 (追記ここまで) with lines

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

  r135	r136
  190	190	\end{frame}
  191	191
  192
  193	%TODO show a graph with two planes in the same place (if that's even possible?)
  194
  195	%Show a graph with two parallel planes:
  196
  197	\begin{frame}[fragile]
  198	\frametitle{Inconsistent Equality}
  199	\biggraph{equality-1-par}
  200
  201	Rule: Identical coeffs but different constant $\implies$ unsolveable.
  202	\end{frame}
  203
  204	%Show a graph with two planes intersecting in a line:
  205
  206	\begin{frame}[fragile]
  207	\fullframegraph{Equalities}{equality-2}
  208	\end{frame}
  209
  210	%Show a graph with three planes intersecting in a line:
  211
  212	\begin{frame}[fragile]
  213	\fullframegraph{Equalities}{equality-3}
  214	\end{frame}
  215
  216	%This is the maths of the three planes intersecting in a line:
  217
  218	\begin{frame}[fragile]
  219	\frametitle{The Maths}
  220	$\begin{array}{rrcr}
  221	3x - y + & z & = & 30 \\
  222	x + y + & 3z & = & 50 \\
  223	%goes to: 4x + 4z = 80, x + z = 20, z = 20 - x. y = 3x + z - 30
  224	% Two points this line goes through are: (10,10,10) + (11,12,9)
  225	% Therefore (10,10,10) + t(1,2,-1), or (5,0,15) + t(1,2,-1)
  226	% Another random point: (25,10,0), therefore another vector in the plane
  227	% is (3,0,-2)
  228
  229	%Therefore another plane that contains this line satisfies:
  230	% 5 + t + 3u = x
  231	% 2t = y
  232	% 15 - t - 2u = z
  233	% gives:
  234	% t = y/2, u = x + z - 20
  235	% So the plane is:
  236	% 5 + y/2 + 3x + 3z - 60 = x
  237	% 4x + y + 6z = 110
  238	4x + y + & 6z & = & 110
  239	\end{array}$
  240
  241	Subst. $y = 3x + z - 30$ into latter two:
  242
  243	$\begin{array}{rrcr}
  244	4x + & 4z & = & 80 \\
  245	7x + & 7z & = & 140
  246	\end{array}$
  247
  248	After normalisation, equations are identical $\therefore$ redundant.
  249
  250	\end{frame}
  251
  252	\begin{frame}[fragile]
  253	\frametitle{Flow Chart}
  254	\biggraph{Omega-Flowchart-M5}
  255	\end{frame}
  192	\TurnLogoOff
  193
  194	\begin{frame}[fragile]
  195	\fullframegraph{Inconsistent Equalities}{equality-incon}
  196	\end{frame}
  197
  198	\TurnLogoOn
  199
  200	\begin{frame}[fragile]
  201	\frametitle{Equalities}
  202	\begin{itemize}
  203	\item If two equations have identical coefficients ($a_1 \cdots a_n$):
  204	\begin{itemize}
  205	\item If the constant terms ($a_0$) are equal, remove one of the equations
  206	\item If the constant terms are not equal, there is no solution
  207	\end{itemize}
  208	\end{itemize}
  209	\end{frame}
  210
  211	%Flowchart is so simple at this point it doesn't seem worth showing it:
  212
  213	%\begin{frame}[fragile]
  214	%\frametitle{Flow Chart}
  215	%\biggraph{Omega-Flowchart-M5}
  216	%\end{frame}
  256	217
  257	218	\begin{frame}
  …	…
  261	222	\item Otherwise a special substitution is used:
  262	223	\begin{itemize}
  263	\item $x_k = -\operatorname{sign}(a_k)m\sigma + \(削除) (削除ここまで)sum_{i \in V - \{k\}} \operatorname{sign}(a_k)(a_i \widehat{\operatorname{mod}} m)x_i$
  224	\item $x_k = -\operatorname{sign}(a_k)m\sigma + \(追記) displaystyle\ (追記ここまで)sum_{i \in V - \{k\}} \operatorname{sign}(a_k)(a_i \widehat{\operatorname{mod}} m)x_i$
  264	225	\item See paper for details (no simple explanation)
  265	226	%TODO try to think of a simple explanation, if possible

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