Changeset 162 for docs

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

  r161	r162
  92	92	\end{array}$
  93	93
  94	\note{We are checking if any of array indexes can be equal to each other.
  94	\note{We are checking if any of array indexes can be equal to each other
  95	within the bounds of the array. So we set the indexes equal in an
  96	equation, and also add inequalities specifying that each index must
  97	be within bounds.
  98
  95	99	Note that we must check each possible pairing.
  96	100	No solution to any of the equations means the indexes are disjoint, and
  …	…
  309	313	\begin{frame}
  310	314	\fullframegraph{Unsolveable Inequalities}{inequality-unsolveable}
  315	(追記) \note{This is an unsolveable pair of inequalities; there is no overlap between the two regions} (追記ここまで)
  311	316	\end{frame}
  312	317
  313	318	\begin{frame}
  314	319	\fullframegraph{Inequalities}{inequality-normal}
  320	(追記) \note{This is an acceptable pair of inequalities} (追記ここまで)
  315	321	\end{frame}
  316	322
  317	323	\begin{frame}
  318	324	\fullframegraph{Redundant Inequalities}{inequality-redundant}
  325	(追記) \note{Here, the green inequality is redundant; anything in the red area (追記ここまで)
  326	(追記) is also in the green area, so we should only concern ourselves with the (追記ここまで)
  327	(追記) red area (and drop the green inequality).} (追記ここまで)
  319	328	\end{frame}
  320	329
  321	330	\begin{frame}
  322	331	\fullframegraph{Inequalities $\implies$ Equality}{inequality-equality}
  332	(追記) \note{The two equalities begin at the same line but "face" in opposite (追記ここまで)
  333	(追記) directions. Therefore the only solutions to the inequality lie on the line. (追記ここまで)
  334	(追記) Hence we should just use the line (which is expressable as an equality) (追記ここまで)
  335	(追記) and drop the two inequalities. Equalities are easier to deal with than (追記ここまで)
  336	(追記) inequalities, so this is a sensible step.} (追記ここまで)
  323	337	\end{frame}
  324	338
  …	…
  346	360
  347	361	\begin{frame}[fragile]
  348	\only<1>{\fullframegraph{Normalising Inequalities}{inequality-tightening-1}}
  349	\only<2>{\fullframegraph{Normalising Inequalities}{inequality-tightening-2}}
  362	\only<1>{\fullframegraph{Normalising Inequalities}{inequality-tightening-1}
  363	\note{The position of this inequality is somewhat wasteful/redundant. We
  364	are only interested in integer solutions (here, the crosses on the graph)
  365	so the fact that the red region includes a portion of space with no crosses
  366	is irrelevant.}}
  367	\only<2>{\fullframegraph{Normalising Inequalities}{inequality-tightening-2}
  368	\note{We can eliminate this wastefulness by effectively "snap-to-grid". We
  369	move the inequality in a direction perpendicular to its line (informally,
  370	"towards the red") until it hits the first cross(es). This is now shown
  371	on the graph.}}
  350	372	\only<3>{\fullframegraph{Normalising Inequalities}{inequality-tightening-3}}
  351	\only<4>{\fullframegraph{Normalising Inequalities}{inequality-tightening-4}}
  373	\only<4>{\fullframegraph{Normalising Inequalities}{inequality-tightening-4}
  374	\note{One reason normalisation is useful is illustrated here. In the first
  375	of the two pictures, there is an overlap between the two regions so they
  376	both seem valid. But in fact there are no integer points in that region.
  377	Normalising both equations reveals that the equalities are in fact unsolveable.}}
  352	378	\end{frame}
  353	379
  …	…
  378	404	\only<1>{\fullframegraph{Set of Inequalities}{inequality-scalene-1}}
  379	405	\only<2>{\fullframegraph{Set of Inequalities}{inequality-scalene-2}}
  380	\only<3>{\fullframegraph{Set of Inequalities}{inequality-scalene-3}}
  381	\end{frame}
  382
  383	\begin{frame}[fragile]
  384	\only<1>{\fullframegraph{Eliminate $x$}{inequality-scalene-2}}
  406	\only<3>{\fullframegraph{Set of Inequalities}{inequality-scalene-3}
  407	\note{The preceding three inequalities form a central triangular region.
  408	We will now look at how to eliminate the $x$ variable from the equations.}}
  409	\end{frame}
  410
  411	\begin{frame}[fragile]
  412	\only<1>{\fullframegraph{Eliminate $x$}{inequality-scalene-2}
  413	\note{In order for a point to be in both the red and green regions,
  414	it must be below (i.e. $y$ axis) the point at which they join ($y = 3$).
  415	Therefore we can eliminate $x$ here and just use the inequality
  416	$y \leq 3$ (see next slide).}}
  385	417	\only<2>{\fullframegraph{Eliminate $x$}{inequality-scalene-4}}
  386	\only<3>{\fullframegraph{Eliminate $x$}{inequality-scalene-5}}
  418	\only<3>{\fullframegraph{Eliminate $x$}{inequality-scalene-5}
  419	\note{Similarly, in order for a point to be in both the read and
  420	blue regions, it must be above ($y$ axis again) the point at which
  421	they join ($y=1$). Therefore our new inequality is $y \geq 1$. This
  422	inequality is shown on the next slide, and the combination of our
  423	two new inequalities is shown on the slide after that.}
  387	424	\only<4>{\fullframegraph{Eliminate $x$}{inequality-scalene-6}}
  388	425	\only<5>{\fullframegraph{Eliminate $x$}{inequality-scalene-7}}

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