Work 2

Array Test

|x| = \left\{  \begin{array}{ll}  x  & \text{if}~ x \geq 0;  \\  -x & \text{if}~ x < 0.  \end{array}  \right.

\begin{array}{*{9}{l}}  Alpha & Bravo & Charlie & Delta & Echo & Foxtrot & Golf & Hotel & India  \\  Juliet & Kilo & Lima & Mike & November & Oscar & Papa & Quebec & Romeo  \\  Sierra & Tango & Uniform & Victor & Whiskey & X\text{-}ray & Yankee & Zulu & \varnothing  \end{array}

Matrix Test

\begin{matrix}  Alpha & Bravo & Charlie & Delta & Echo & Foxtrot & Golf & Hotel & India  \\  Juliet & Kilo & Lima & Mike & November & Oscar & Papa & Quebec & Romeo  \\  Sierra & Tango & Uniform & Victor & Whiskey & X\text{-}ray & Yankee & Zulu & \varnothing  \end{matrix}

Tabular Test 1

\begin{tabular}{lll}  Chicago & U.S.A. & 1893  \\  Z\"{u}rich & Switzerland & 1897  \\  Paris & France & 1900  \\  Heidelberg & Germany & 1904  \\  Rome & Italy & 1908  \end{tabular}

Tabular Test 2

\begin{tabular}{|r|r|}  \hline  \( n \) & \( n! \) \\  \hline  1 & 1 \\  2 & 2 \\  3 & 6 \\  4 & 24 \\  5 & 120 \\  6 & 720 \\  7 & 5040 \\  8 & 40320 \\  9 & 362880 \\  10 & 3628800 \\  \hline  \end{tabular}

Tabular Test 3

\begin{tabular}{|c|c|*{16}{c}|}  \multicolumn{18}{c}{Table 1. Higher Order Propositions \( (n = 1) \)} \\[4pt]  \hline  \( f \) & \( f \) &  \( m_{0}  \)  & \( m_{1}  \) & \( m_{2}  \) & \( m_{3}  \) &  \( m_{4}  \)  & \( m_{5}  \) & \( m_{6}  \) & \( m_{7}  \) &  \( m_{8}  \)  & \( m_{9}  \) & \( m_{10} \) & \( m_{11} \) &  \( m_{12} \)  & \( m_{13} \) & \( m_{14} \) & \( m_{15} \) \\[4pt]  \hline  \( f_0 \) & \texttt{()} &  0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\[4pt]  \( f_1 \) & \texttt{(}\( x \)\texttt{)} &  0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\[4pt]  \( f_2 \) & \( x \) &  0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\[4pt]  \( f_3 \) & \texttt{(())} &  0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\[4pt]  \hline  \end{tabular}

Tabular Test 4

\begin{tabular}{|*{7}{c|}}  \multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\  \hline  \( L_1 \) &  \( L_2 \) &&  \( L_3 \) &  \( L_4 \) &  \( L_5 \) &  \( L_6 \) \\  \hline  & & \( x = \) & 1 1 0 0 & & & \\  & & \( y = \) & 1 0 1 0 & & & \\  \hline  \( f_{0} \) &  \( f_{0000} \) &&  0 0 0 0 &  \( (~) \) &  false &  \( 0 \)  \\  \( f_{1} \) &  \( f_{0001} \) &&  0 0 0 1 &  \( (x)(y) \) &  neither \( x \) nor \( y \) &  \( \lnot x \land \lnot y \)  \\  \( f_{2} \) &  \( f_{0010} \) &&  0 0 1 0 &  \( (x)\ y \) &  \( y \) without \( x \) &  \( \lnot x \land y \)  \\  \( f_{3} \) &  \( f_{0011} \) &&  0 0 1 1 &  \( (x) \) &  not \( x \) &  \( \lnot x \)  \\  \( f_{4} \) &  \( f_{0100} \) &&  0 1 0 0 &  \( x\ (y) \) &  \( x \) without \( y \) &  \( x \land \lnot y \)  \\  \( f_{5} \) &  \( f_{0101} \) &&  0 1 0 1 &  \( (y) \) &  not \( y \) &  \( \lnot y \)  \\  \( f_{6} \) &  \( f_{0110} \) &&  0 1 1 0 &  \( (x,\ y) \) &  \( x \) not equal to \( y \) &  \( x \ne y \)  \\  \( f_{7} \) &  \( f_{0111} \) &&  0 1 1 1 &  \( (x\ y) \) &  not both \( x \) and \( y \) &  \( \lnot x \lor \lnot y \)  \\  \hline  \( f_{8} \) &  \( f_{1000} \) &&  1 0 0 0 &  \( x\ y \) &  \( x \) and \( y \) &  \( x \land y \)  \\  \( f_{9} \) &  \( f_{1001} \)  &&  1 0 0 1 &  \( ((x,\ y)) \) &  \( x \) equal to \( y \) &  \( x = y \)  \\  \( f_{10} \) &  \( f_{1010} \) &&  1 0 1 0 &  \( y \) &  \( y \) &  \( y \)  \\  \( f_{11} \) &  \( f_{1011} \) &&  1 0 1 1 &  \( (x\ (y)) \) &  not \( x \) without \( y \) &  \( x \Rightarrow y \)  \\  \( f_{12} \) &  \( f_{1100} \) &&  1 1 0 0 &  \( x \) &  \( x \) &  \( x \)  \\  \( f_{13} \) &  \( f_{1101} \) &&  1 1 0 1 &  \( ((x)\ y) \) &  not \( y \) without \( x \) &  \( x \Leftarrow y \)  \\  \( f_{14} \) &  \( f_{1110} \) &&  1 1 1 0 &  \( ((x)(y)) \) &  \( x \) or \( y \) &  \( x \lor y \)  \\  \( f_{15} \) &  \( f_{1111} \) &&  1 1 1 1 &  \( ((~)) \) &  true &  \( 1 \)  \\  \hline  \end{tabular}

Tabular Test 5

\begin{tabular}{|c||*{4}{c}|}  \multicolumn{5}{c}{Table 2.1 Values of \( \chi_S(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]  \hline  \( (x_1, x_2) \backslash S \) &  \( \varnothing \) &  \( \{ x_1 \} \)   &  \( \{ x_2 \} \)   &  \( \{ x_1, x_2 \} \)  \\  \hline\hline  \( (0, 0) \) & \( +1 \) & \( +1 \) & \( +1 \) & \( +1 \) \\  \( (1, 0) \) & \( +1 \) & \( -1 \) & \( +1 \) & \( -1 \) \\  \( (0, 1) \) & \( +1 \) & \( +1 \) & \( -1 \) & \( -1 \) \\  \( (1, 1) \) & \( +1 \) & \( -1 \) & \( -1 \) & \( +1 \) \\  \hline  \end{tabular}

\begin{tabular}{|c||*{4}{c}|}  \multicolumn{5}{c}{Table 2.1 Values of \( \chi_S(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]  \hline  \( S \backslash (x_1, x_2)  \) &  \( (0, 0) \) &  \( (1, 0) \) &  \( (0, 1) \) &  \( (1, 1) \)  \\  \hline\hline  \( \varnothing \)     & \( +1 \) & \( +1 \) & \( +1 \) & \( +1 \) \\  \( \{ x_1 \} \)       & \( +1 \) & \( -1 \) & \( +1 \) & \( -1 \) \\  \( \{ x_2 \} \)       & \( +1 \) & \( +1 \) & \( -1 \) & \( -1 \) \\  \( \{ x_1, x_2 \} \)  & \( +1 \) & \( -1 \) & \( -1 \) & \( +1 \) \\  \hline  \end{tabular}

\begin{tabular}{|c||*{4}{c}|}  \multicolumn{5}{c}{Table 2.1 Values of \( \chi_S(x) \) for \( f : \mathbb{B}^2 \to \mathbb{B} \)} \\[4pt]  \hline  \( S \backslash (x_1, x_2)  \) &  \( (1, 1) \) &  \( (1, 0) \) &  \( (0, 1) \) &  \( (0, 0) \)  \\  \hline\hline  \( \varnothing \)     & \( +1 \) & \( +1 \) & \( +1 \) & \( +1 \) \\  \( \{ x_1 \} \)       & \( -1 \) & \( -1 \) & \( +1 \) & \( +1 \) \\  \( \{ x_2 \} \)       & \( -1 \) & \( +1 \) & \( -1 \) & \( +1 \) \\  \( \{ x_1, x_2 \} \)  & \( +1 \) & \( -1 \) & \( -1 \) & \( +1 \) \\  \hline  \end{tabular}

Table Test 1

Can we ever become what we weren’t in eternity?
Can we ever learn what we weren’t born knowing?
Can we ever share what we never had in common?

Lately I’ve begun to see that these ancient riddles of change, coming to know, and communication all spring from a common root.

Table Test 2

Everything considered, a determined soul will always manage.

(41)

To a man devoid of blinders, there is no finer sight than that of the intelligence at grips with a reality that transcends it.

(55)

Table Test 3

Everything considered, a determined soul will always manage.

(41)

To a man devoid of blinders, there is no finer sight than that
of the intelligence at grips with a reality that transcends it.

(55)

Table Test 4


()()=()
(1)

(())= 
(2)

Table Test 5


()()=()
(1)

(())= 
(2)

Table Test 6

\text{Table 1.} ~~ \text{Higher Order Propositions} ~ (n = 1)
m_{0} m_{1} m_{2} m_{3} m_{4} m_{5} m_{6} m_{7}
0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 0 0

One Response to Work 2

  1. Jon Awbrey says:

    Here are two ways of looking at the “doubly recursive factorization” of 42.

    Riff 42

    Rote 42

    Details at OEIS • Riffs and Rotes

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