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Alphabet
Probability (per 2000) \(P_{1}\) Weighting \(\log_{1.5}(256P_{1})\) (3 s.f.)
E 254 1.7
T 182 10.6
A 164 11.2
O 150 11.5
I 140 11.7
N 134 11.8
S 126 12.0
H 122 12.1
R 120 12.1
D 86 12.7
L 80 12.8
U 56 13.1
C 56 13.1
M 48 13.2
W 48 13.2
F 44 13.2
G 40 13.3
Y 40 13.3
P 38 13.3
B 30 13.4
V 20 13.5
K 16 13.5
J 4 13.6
X 4 13.6
Q 2 13.7
Z 2 13.7
Using these weightings we can define a simple weighting system that can convert standard english letters (or any other language that uses these 26 symbols) into SKI strings.
SKI Combinator Weighting \(W_{s}\)
S 3.3 \((W_{3})\)
K 2.2 \((W_{2})\)
I 1.1 \((W_{1})\)
Until I can devise a better weighting system that will supersede this very rudimentary weighting system this will suffice.
Now to select a string we can use the equatioâ€¦
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ESMN (Extended SteinhausMoser Notation)
Background
Now I have been trying to understand dimensional arrays for an inordinate amount of time. As you'd expect I'm not a smart person. So instead of taking the time to understand them, I took the lazy way out and "created" (probably not) a "new" (again probably not) "system" (there is not much that is systemish about this at all) of dimensional arrays that work the way I imagined it (it's probably going to be very bad).
Array of function
To diagonalize over the linear arrays we can use an adaptation of Jonathan Bower's array of function.
Rules
1) i) [n] \(\lambda_{0}\) becomes [n, n, ... n, n] with n number of n'sThis can alternatively be displayed as \(\begin{bmatrix} n \\ 1 \end{bmatrix}\)
ii)Genâ€¦
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One day, while stuck in a very boring Statistics class, a human picked up a calculator and meddled aroung with logarithms and exponentials...
Subgoogol group
\(69!\) â‰ˆ \(1.711 \times 10^{98}\)
\(8^{\frac{110000}{999}}\) â‰ˆ \(2.75 \times 10^{99}\)
\(\sinh (230) \) â‰ˆ \(3.86 \times 10^{99}\) shiny pebble
\(2^{2^{2^{3.066}}}\) â‰ˆ \(8 \times 10^{99}\) troisdeuxtrois
\(\pi^{201}\) â‰ˆ \(8.455 \times 10^{99} \) a speck of apple pie
\(4^{4^{ \frac{59}{16}}}\) â‰ˆ \(8.69 \times 10^{99} quatretroisdeux
\(10000^{\phi^{\phi^{\phi^{\phi^{\phi^{\phi}}}}}}\) where \(\phi\) is the golden ratio
\(1.9 + \frac{2}{300} \uparrow \uparrow 5 \) â‰ˆ \( 9.309 \times 10^{99} \)
\(56.95^{56.95}\) â‰ˆ \(9.449 \times 10^{99}\) cinquanteseptdeuxun
Googolplex group
\(sinh(sinh(230)\)â€¦
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ESMN (Extended SteinhausMoser Notation)
Rules
1) aâ€¢1 = a^a
2) aâ€¢(n+1) = ((...((aâ€¢n)â€¢n)...)â€¢n)â€¢n with a brackets
3) [a, b] = aâ€¢b
4) [a, b, c] = aâ€¢â€¢...â€¢â€¢b with c dots
5) [a, b, c] = [a, [a, [ ... [a, [a, b, c1], c1] ... ], c1], c1]
6) [a, b, c, d] = [a, b, [a, b, c, d1], d1]
7) [#, a, b] = [#, [#, a, b1], b1] where # denotes a vaild array
8) [a, b, #, 1, d] with c 1's = [a, b, #, [a, b, ÃŸ, d1], d1] where # denotes an chain of 1's of length c and ÃŸ denotes an array of a of length c
9) [a, 1, #] = [a] = aâ€¢1 = a^a
10) [#, 1] = [#] where # denotes a valid array
Examples
3â€¢3 = ((3â€¢2)â€¢2)â€¢2 (Rule 2) = ((((3â€¢1)â€¢1)â€¢1)â€¢2)â€¢2 (Ditto) â‰ˆ ((4.4*10^38)â€¢2)â€¢2 â‰ˆ (((...((4.4*10^38)â€¢1)â€¢1)...)â€¢1)â€¢1)â€¢2
[3, 3, 1, 2] = [3, 3, [3, 3, 1, 1], 1] (Rule 5) = [3, 3, [3, 3]â€¦
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W.E.A.K. (Whimpy Effete Array of K.Kebab)
The purpose of this notation is to create the smallest possible Array notation while still producing numbers that are not used in everyday terms (and to jokingly respond to this blog post :P)
Base case
(a) turns into a+1
(a, b) turns into ((a, b1), b1)
and (a, b, c, d, ... x, y,) turns into (a, b, c, d, ... (x, y1), y1)
To ensure the entries terminates to give a finite value we can define a closing case :
(a, b, c, d, ... x, 1) turns into (a, b, c, d, ...x)
And as follows
(#, 1, @) turns into (#)
where # and @ denotes valid arrays.
This is not very powerful, exactly what I had in mind. It is easy to show that (a, b) = a + 2â€‹^{(b1)} and as such it is easy to prove that (a, b) > \(f_2(b)\) given that a isâ€¦
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