%Slide 1
\begin{slide}{}
\begin{center}
{\large {\bf Technical Issues }} \\
  and \\
{\large {\bf Constraints }}
\end{center}
\end{slide}

%Slide 2
\begin{slide}{}

{\large Technical Issues: }

\begin{itemize}
	\item Storage formats
	\item Transmission protocols
	\item Display methods/media
	\item Consistency
	\item Security/control
	\item Ownership -- intellectual property
\end{itemize}

\end{slide}

%Slide 3
\begin{slide}{}

{\large Constraints: }

\begin{itemize}
	\item Bandwidth
	\begin{enumerate}
		\item How much?
		\item $ 1400 * 1250 * 24 * 24 = 1,008,000,000 $
		\item Reading/responding to e-mail
		\item Presentation/dialogue/discussion
		\item Exploration
		\item Routing
	\end{enumerate}
\end{itemize}

\end{slide}

%Slide 4
\begin{slide}{}
\begin{itemize}
	\item Learning
	\begin{enumerate}
		\item Data
		\item Information
		\item Knowledge
		\item Wisdom
	\end{enumerate}
	\item Cost/benefit analysis
\end{itemize}
\end{slide}
%Slide 5
\begin{slide}{}

{\large Pattern recognition }

\begin{itemize}
	\item Images (``visual patterns'') vs. \newline
		``syntactic'' patterns
	\item Symbols as patterns, and symbols as \newline
		pattern labels
	\item Patterns of symbols
	\item Hierarchies of patterns, and symbols as \newline
		tools for recognizing patterns
	\item Pattern manipulation
	\item Learning to recognize patterns, and pattern \newline
		recognition as learning
\end{itemize}

\end{slide}

%Slide 6
\begin{slide}{}

{\large Pattern recognition examples }

\begin{itemize}
	\item What number comes next in the sequence? \newline
		1, 1, 2, 3, 5, 8, 13, \ldots
	\item What number comes next in the sequence? \newline
		8, 5, 4, 9, 1, 7, 6, 3, \ldots
	\item What letter comes next in the sequence? \newline
		E, T, A, O, I, N, S, H, \ldots
	\item In which row does Z go? \newline
		A, E, F, H, I, K, L, M, N, T, V, W, X, Y \newline
		B, C, D, G, J, O, P, Q, R, S, U
	\item What letter comes next in the sequence? \newline
		W, L, C, N, I, T, \ldots
\end{itemize}
\end{slide}

%Slide 7
\begin{slide}{}

{\large Symbols and signs }

\begin{itemize}
	\item The utility and power of symbols
	\item Choosing symbols, naming and pointing
	\item Symbols as ``chunking'' tools
	\item When to use symbols
	\begin{enumerate}
		\item The importance of anonymity (e.g., the \newline
			lambda calculus)
		\item Place holders (variables)
		\item Temporary and tentative symbols
	\end{enumerate}
	\item Signs, symbols, content and meaning
\end{itemize}

\end{slide}

%Slide 8
\begin{slide}{}

We have the map
	$ b_n: \Sigma^2U(n) \rightarrow SU(n+1) $ \newline
given by
	\[ b_n(g, r, s) = \left[ i(g), v_n(r, s) \right] \]
where $i(g)$ is the inclusion,
	$\left[g, h\right] = ghg^{-1}h^{-1}$ \newline
and

$ v_n(r,s) = $

{\small
\[
\left[ \begin{array}{cccccc}
	\alpha & 0 & 0 & \cdots & 0 & \beta (-\overline{\alpha})^0 \\
	\beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha  &  0  & \cdots & 0 &
	   \beta (-\overline{\alpha})^1 \\
	\beta (-\overline{\alpha})^1\overline{\beta} &
	   \beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha & \cdots & 0 & \beta (-\overline{\alpha})^2 \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\beta (-\overline{\alpha})^{n-1}\overline{\beta} &
	   \beta (-\overline{\alpha})^{n-2}\overline{\beta} &
	   \cdots &  \cdots & \alpha &
	   \beta (-\overline{\alpha})^n \\
	-(-\overline{\alpha})^n\overline{\beta} &
	   -(-\overline{\alpha})^{n-1}\overline{\beta} &
	   \cdots &  \cdots & -(-\overline{\alpha})^0 
	   \overline{\beta} & -(-\overline{\alpha})^n \\
	\end{array} \right]
\]
}

{\small
where
\[ \alpha = \alpha(r,s) =
	\cos(\pi r) + i \sin(\pi r)\cos(\pi s) \]
\[ \beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s) \]
}

\end{slide}

%Slide 9
\begin{slide}{}
{\tiny
\begin{verbatim}
We have the map
	$ b_n: \Sigma^2U(n) \rightarrow SU(n+1) $ \newline
given by
	\[ b_n(g, r, s) = \left[ i(g), v_n(r, s) \right] \]
where $i(g)$ is the inclusion,
	$\left[g, h\right] = ghg^{-1}h^{-1}$ \newline
and

$ v_n(r,s) = $

\[
\left[ \begin{array}{cccccc}
	\alpha & 0 & 0 & \cdots & 0 & \beta (-\overline{\alpha})^0 \\
	\beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha  &  0  & \cdots & 0 &
	   \beta (-\overline{\alpha})^1 \\
	\beta (-\overline{\alpha})^1\overline{\beta} &
	   \beta (-\overline{\alpha})^0\overline{\beta} &
	   \alpha & \cdots & 0 & \beta (-\overline{\alpha})^2 \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
	\vdots & \vdots & \vdots &  & \vdots & \vdots \\
	\beta (-\overline{\alpha})^{n-1}\overline{\beta} &
	   \beta (-\overline{\alpha})^{n-2}\overline{\beta} &
	   \cdots &  \cdots & \alpha &
	   \beta (-\overline{\alpha})^n \\
	-(-\overline{\alpha})^n\overline{\beta} &
	   -(-\overline{\alpha})^{n-1}\overline{\beta} &
	   \cdots &  \cdots & -(-\overline{\alpha})^0 
	   \overline{\beta} & -(-\overline{\alpha})^n \\
	\end{array} \right]
\]
where
\[ \alpha = \alpha(r,s) =
	\cos(\pi r) + i \sin(\pi r)\cos(\pi s) \]
\[ \beta = \beta(r,s) = i \sin(\pi r)\sin(\pi s) \]

\end{verbatim}
}
\end{slide}

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