Changeset 145 for docs

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

  r136	r145
  1	1	load "2d-gnuplot-settings.inc"
  2	(追記) (追記ここまで)
  3	(追記) set terminal png truecolor size 600,600 enhanced rounded font "FreeSansBold" 12 (追記ここまで)
  2	4
  3	5	set title "Inconsistent Equations"
  …	…
  7	9	plot "points-1-3.inc" with points title "" pt 1 ps 2 lt -1
  8	10
  9	set key (削除) bottom right (削除ここまで)
  11	set key (追記) at 4, 0.25 (追記ここまで)
  10	12
  11	plot ((2*x + 1)/3) title "3y = 2x + 1" ls 3 with lines, ((2*x + 2)/3) title "3y = 2x + 2" ls 4 with lines
  13	plot ((2*x + 1)/3) title "{/*3 3y = 2x + 1}" ls 3 with lines
  14
  15	set key at 4, 0.65
  16
  17	plot ((2*x + 2)/3) title "{/*3 3y = 2x + 2}" ls 4 with lines

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

  r135	r145
  1	1	load "2d-gnuplot-settings.inc"
  2	(追記) (追記ここまで)
  3	(追記) set terminal png truecolor size 600,600 enhanced rounded font "FreeSansBold" 12 (追記ここまで)
  2	4
  3	5	set title "Unsolveable Equation"
  …	…
  9	11	set key bottom right
  10	12
  11	plot x + 0.5 title "(削除) (削除ここまで)2y = 2x + 1" ls 3 with lines
  13	plot x + 0.5 title "(追記) {/*3 (追記ここまで)2y = 2x + 1" ls 3 with lines

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

  r144	r145
  154	154	\item $x_0 = 1, \therefore a_0$ is the constant term
  155	155	\end{itemize}
  156	\item For readability, equations (削除) here will usually (削除ここまで) be written in the form:
  157	\begin{itemize}(削除) (削除ここまで)
  158	\item $a(削除) x + by + cz = d$ (削除ここまで)
  156	\item For readability, equations (追記) on graphs will (追記ここまで) be written in the form:
  157	\begin{itemize}(追記) (追記ここまで)
  158	\item $a(追記) y = bx + c$ (technically: $-a_1 x_1 = a_2 x_2 + a_0$) (追記ここまで)
  159	159	\end{itemize}
  160	160	\item Equations are kept normalised (GCD of coefficients $a_1 \cdots a_n$ is 1)
  …	…
  171	171	%TODO mention integer grid, and how this relates to the equations
  172	172
  173	(削除) \TurnLogoOff (削除ここまで)
  174	(削除) (削除ここまで)
  175	(削除) \begin{frame}[fragile] (削除ここまで)
  176	(削除) \fullframegraph{Unsolveable Equality}{equality-nosol} (削除ここまで)
  177	(削除) \end{frame} (削除ここまで)
  178	(削除) (削除ここまで)
  179	(削除) \TurnLogoOn (削除ここまで)
  180	(削除) (削除ここまで)
  181	173	\begin{frame}[fragile]
  182	174	\frametitle{Unsolveable Equality}
  183	\begin{itemize}
  184	\item The equation 2ドルx = 2y + 1$ has no integer solutions
  185	\begin{itemize}
  186	\item Rule: if the GCD of the variable coeffs does not divide the constant, there
  187	is no integer solution.
  188	\end{itemize}
  189	\end{itemize}
  190	\end{frame}
  191
  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}
  175	\begin{columns}
  176	\column{60mm}
  177	\includegraphics[width=60mm]{equality-nosol}
  178	\column{45mm}
  179	If the GCD of the variable coeffs ($a_1 \cdots a_n$)
  180	does not divide the constant ($a_0$),
  181	there is no integer solution
  182	\end{columns}
  183	\end{frame}
  184
  185	\begin{frame}[fragile]
  186	\frametitle{Inconsistent Equalities}
  187	\begin{columns}
  188	\column{60mm}
  189	\includegraphics[width=60mm]{equality-incon}
  190	\column{45mm}
  191	If two equations have identical coefficients ($a_1 \cdots a_n$) but different constant terms ($a_0$), there is no solution
  192	\end{columns}
  209	193	\end{frame}
  210	194
  …	…
  216	200	%\end{frame}
  217	201
  202	(追記) %TODO maybe add an extra slide explaining why we don't just use rational numbers (追記ここまで)
  203	(追記) (追記ここまで)
  218	204	\begin{frame}
  219	205	\frametitle{Equality Solving Method}
  …	…
  223	209	\begin{itemize}
  224	210	\item $x_k = -\operatorname{sign}(a_k)m\sigma + \displaystyle\sum_{i \in V - \{k\}} \operatorname{sign}(a_k)(a_i \widehat{\operatorname{mod}} m)x_i$
  211	(追記) \item Always reduces absolute value of coefficients (追記ここまで)
  225	212	\item See paper for details (no simple explanation)
  226	213	%TODO try to think of a simple explanation, if possible

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