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Revision as of 04:15, 1 September 2013
, 04:15, 1 September 2013update
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| align="center" | <math>J\!</math>
| align="center" | <math>J\!</math>
| <math>J : U \to \mathbb{B}\!</math>
| <math>J : U \!\to\! \mathbb{B}\!</math>
| <math>\text{Proposition}\!</math>
| <math>\text{Proposition}\!</math>
| <math>(\mathbb{B}^2 \to \mathbb{B}) \in [\mathbb{B}^2]\!</math>
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math>
|-
|-
| align="center" | <math>J\!</math>
| align="center" | <math>J\!</math>
| <math>J : U^\bullet \to X^\bullet\!</math>
| <math>J : U^\bullet \!\to\! X^\bullet\!</math>
| <math>\text{Transformation or Map}\!</math>
| <math>\text{Transformation or Map}\!</math>
| <math>[\mathbb{B}^2] \to [\mathbb{B}^1]\!</math>
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math>
|-
|-
| align="center" |
| align="center" |
|
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<math>\begin{array}{l}
<math>\begin{array}{l}
\mathrm{W} : U^\bullet \to \mathrm{E}U^\bullet,
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,
\\
\\
\mathrm{W} : X^\bullet \to \mathrm{E}X^\bullet,
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,
\\
\mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet)
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)
\\
\text{for each}~ \mathrm{W} ~\text{in the set:}
\\
\\
\text{for}~ \mathrm{W} = \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}
\end{array}</math>
\end{array}</math>
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|
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
{[\mathbb{B}^2] \to [\mathbb{B}^2 \times \mathbb{D}^2]},
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \times \mathbb{D}^2]},
\\
\\
{[\mathbb{B}^1] \to [\mathbb{B}^1 \times \mathbb{D}^1]},
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]},
\\\\
\\\\
([\mathbb{B}^2] \to [\mathbb{B}^1]) \to
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!
\\
\\
([\mathbb{B}^2 \times \mathbb{D}^2] \to [\mathbb{B}^1 \times \mathbb{D}^1])
([\mathbb{B}^2 \times \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1])
\end{array}</math>
\end{array}</math>
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|
<math>\begin{array}{l}
<math>\begin{array}{l}
\mathsf{W} : U^\bullet \to \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,
\\
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,
\\
\\
\mathsf{W} : X^\bullet \to \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)
\\\\
\\
\mathsf{W} : (U^\bullet \to X^\bullet) \to (\mathsf{T}U^\bullet \to \mathsf{T}X^\bullet)
\text{for each}~ \mathsf{W} ~\text{in the set:}
\\
\\
\text{for}~ \mathsf{W} = \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T}
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}
\end{array}</math>
\end{array}</math>
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<math>\begin{array}{llll}
<math>\begin{array}{lll}
\text{Radius operator} & \mathsf{e} & = & (\boldsymbol\varepsilon, \eta)
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)
\\
\\
\text{Secant operator} & \mathsf{E} & = & (\boldsymbol\varepsilon, \mathrm{E})
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})
\\
\\
\text{Chord operator} & \mathsf{D} & = & (\boldsymbol\varepsilon, \mathrm{D})
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})
\\
\\
\text{Tangent functor} & \mathsf{T} & = & (\boldsymbol\varepsilon, \mathrm{d})
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})
\end{array}</math>
\end{array}</math>
|
|
<math>\begin{array}{l}
<math>\begin{array}{l}
{[\mathbb{B}^2] \to [\mathbb{B}^2 \times \mathbb{D}^2]},
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \times \mathbb{D}^2]},
\\
\\
{[\mathbb{B}^1] \to [\mathbb{B}^1 \times \mathbb{D}^1]},
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1]},
\\\\
\\\\
([\mathbb{B}^2] \to [\mathbb{B}^1]) \to
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!
\\
\\
([\mathbb{B}^2 \times \mathbb{D}^2] \to [\mathbb{B}^1 \times \mathbb{D}^1])
([\mathbb{B}^2 \times \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \times \mathbb{D}^1])
\end{array}</math>
\end{array}</math>
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