Technical Issues

and

Constraints Technical Issues:

• Storage formats

• Transmission protocols

• Display methods/media

• Consistency

• Security/control

• Ownership – intellectual property

Constraints:

• Bandwidth 1. How much?

2. 1400 ∗ 1250 ∗ 24 ∗ 24 = 1, 008, 000, 000 3. Reading/responding to e-mail

4. Presentation/dialogue/discussion 5. Exploration

6. Routing • Learning

1. Data

2. Information

3. Knowledge

4. Wisdom

• Cost/benefit analysis

Pattern recognition

• Images (“visual patterns”) vs.

“syntactic” patterns

• Symbols as patterns, and symbols as pattern labels

• Patterns of symbols

• Hierarchies of patterns, and symbols as tools for recognizing patterns

• Pattern manipulation

• Learning to recognize patterns, and pattern recognition as learning

Pattern recognition examples

• What number comes next in the sequence? 1, 1, 2, 3, 5, 8, 13, …

• What number comes next in the sequence? 8, 5, 4, 9, 1, 7, 6, 3, …

• What letter comes next in the sequence? E, T, A, O, I, N, S, H, …

• In which row does Z go?

A, E, F, H, I, K, L, M, N, T, V, W, X, Y B, C, D, G, J, O, P, Q, R, S, U

• What letter comes next in the sequence? W, L, C, N, I, T, …

Symbols and signs

• The utility and power of symbols

• Choosing symbols, naming and pointing • Symbols as “chunking” tools

• When to use symbols

1. The importance of anonymity (e.g., the lambda calculus)

2. Place holders (variables)

3. Temporary and tentative symbols

• Signs, symbols, content and meaning We have the map b_{n} : Σ^{2}U(n) → SU(n + 1) given by

b_{n}(g, r, s) = [i(g), v_{n}(r, s)]

where i(g) is the inclusion, [g, h] = ghg^{−1}h^{−1} and

v_{n}(r, s) =

[ \begin{matrix}α & 0 & 0 & ⋯ & 0 & β(−α)^{0} \\ β(−α)^{0}β & α & 0 & ⋯ & 0 & β(−α)^{1} \\ β(−α)^{1}β & β(−α)^{0}β & α & ⋯ & 0 & β(−α)^{2} \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ β(−α)^{n−1}β & β(−α)^{n−2}β & ⋯ & ⋯ & α & β(−α)^{n} \\ −(−α)^{n}β & −(−α)^{n−1}β & ⋯ & ⋯ & −(−α)^{0}β & −(−α)^{n} \end{matrix} ]

where

α = α(r, s) = cos (πr) + i sin(πr) cos (πs)
β = β(r, s) = i sin(πr) sin(πs)

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