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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
Revision as of 21:54, 11 March 2009
, 21:54, 11 March 2009→Computation Summary : <math>f(u, v) = \texttt{((u)(v))}</math>: markup
{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
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<math>\begin{array}{lrrrrrrrr}
<math>\begin{matrix}
\operatorname{d}f
\operatorname{d}f
& = & \texttt{uv} \cdot \texttt{0} & + & \texttt{u(v)} \cdot \texttt{du} & + & \texttt{(u)v} \cdot \texttt{dv} & + & \texttt{(u)(v)} \cdot \texttt{(du, dv)}
& = & \texttt{uv} \cdot \texttt{0} & + & \texttt{u(v)} \cdot \texttt{du} & + & \texttt{(u)v} \cdot \texttt{dv} & + & \texttt{(u)(v)} \cdot \texttt{(du, dv)}
\\ \\
\\ \\
& = & & & \texttt{u(v)} \cdot \texttt{du} & + & \texttt{(u)v} \cdot \texttt{dv} & + & \texttt{(u)(v)} \cdot \texttt{(du, dv)}
& = & & & \texttt{u(v)} \cdot \texttt{du} & + & \texttt{(u)v} \cdot \texttt{dv} & + & \texttt{(u)(v)} \cdot \texttt{(du, dv)}
\end{array}</math>
\end{matrix}</math>
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{| align="center" cellpadding="8" width="90%"
{| align="center" cellpadding="8" width="90%"
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<math>\begin{array}{lrrrrrrrr}
<math>\begin{matrix}
\operatorname{r}f
\operatorname{r}f
& = & \texttt{uv} \cdot \texttt{du~dv} & + & \texttt{u(v)} \cdot \texttt{du~dv} & + & \texttt{(u)v} \cdot \texttt{du~dv} & + & \texttt{(u)(v)} \cdot \texttt{du dv}
& = & \texttt{uv} \cdot \texttt{du~dv} & + & \texttt{u(v)} \cdot \texttt{du~dv} & + & \texttt{(u)v} \cdot \texttt{du~dv} & + & \texttt{(u)(v)} \cdot \texttt{du dv}
\\ \\
\\ \\
& = & \texttt{du~dv}
& = & \texttt{du~dv}
\end{array}</math>
\end{matrix}</math>
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Figure 1.1. f = ((u)(v))
Figure 1.1. f = ((u)(v))
</pre>
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<pre>
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Figure 1.2. Ef = ((u + du)(v + dv))
Figure 1.2. Ef = ((u + du)(v + dv))
</pre>
<br>
<pre>
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Figure 1.3. Difference Map Df = f + Ef
Figure 1.3. Difference Map Df = f + Ef
</pre>
<br>
<pre>
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Figure 1.4. Linear Proxy df for Df
Figure 1.4. Linear Proxy df for Df
</pre>
<br>
<pre>
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Figure 1.5. Remainder rf = Df + df
Figure 1.5. Remainder rf = Df + df
</pre>
Computation Summary for g<u, v> = ((u, v))
===Computation Summary : <math>g(u, v) = \texttt{((u,~v))}</math>===
Exercise for the Reader.
Exercise for the Reader.
==Note 19==
==Note 19==