Changeset 166 for docs/trunk/omega-test-slides/omega-test.tex

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

  r165	r166
  574	574	b\alpha - a\beta \geq (a - 1)(b - 1) &\implies \ldots}
  575	575	\end{align*}
  576	\only<4>{ \note{$A \implies B$
  577
  578	$\lnot A \lor B$ (by definition)
  579
  580	$\lnot A \lor \lnot \lnot B$ ($\lnot\lnot A = A$)
  581
  582	$\lnot B \implies \lnot A$ (by definition)
  576
  577	\only<1>{
  578	\note{ If the gap/window between the lower and upper bounds is larger
  579	than or equal to $ab$ then we know that the interval must contain
  580	a multiple of $ab$; that is, there is an $abx$ in the interval.
  583	581	} }
  582	(追記) (追記ここまで)
  583	(追記) \only<2>{ (追記ここまで)
  584	(追記) \note{ We could transform this greater-than-or-equal-to into (追記ここまで)
  585	(追記) greater-than by moving the bounds in by 1 (standard mathematics on integers). (追記ここまで)
  586	(追記) (追記ここまで)
  587	(追記) But note from the diagram that we can be stricter than this. The lower bound (追記ここまで)
  588	(追記) can only fall on a multiple of $a$ (the reddish dots), and the upper bound (追記ここまで)
  589	(追記) can only fall on a multiple of $b$ (the greenish dots). So in fact, (追記ここまで)
  590	(追記) this equation translates into $b\alpha - a\beta$ being greater than (追記ここまで)
  591	(追記) the distance between the nearest reddish dot to the lower bound (the next (追記ここまで)
  592	(追記) multiple of $a$ above the lower bound) and the nearest greenish dot to (追記ここまで)
  593	(追記) the upper bound (the next multiple of $b$ below the upper bound). (追記ここまで)
  594	(追記) } } (追記ここまで)
  595	(追記) (追記ここまで)
  596	(追記) \only<4>{ (追記ここまで)
  597	(追記) \note{ We now translate this greater-than back into a greater-than-or-equal-to.} (追記ここまで)
  598	(追記) } (追記ここまで)
  584	599	\only<4>
  585	600	{

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