General Report on Tunny
22B Page 46
character of the ith impulse of the U-stream at a given ciphering position.
L is used to denote the operative character of the limitation.
(b) Some further definitions.
The following symbols are generally used in Tunny-analysis and must be defined here:
|
U = letter preceding U Ui = character preceding Ui U = letter preceding U U = letter following U U = letter following U and so on. |
|
ΔU = U + U Δ2U = Δ(ΔU) ΔnU = Δ(Δn-1U) Δ2U = U + U Δ2U = U + U and so on. |
 Uij = Ui + Uj  = Ui + a cross |
Ui  x : P(Ui = x) > ½Ui . : P(Ui = .) > ½Ui  x : P(Ui = x) = p where p > ½Ui . : P(Ui = .) = p where p > ½ |
(c) Two general theorems.
| Theorem I : | Δ(U + V) = ΔU + ΔV | (A1) |
| Theorem II: | Δ2U = Δ2U | (A2) |
| Proof : | Δ2U = Δ(ΔU) = (U + U) + (U + U) = U + U = Δ2U | |
| Theorem II is a special case of the general theorem: | ||
| ΔnU = ΔnU if and only if n = 2r. (See R5 p. 114) | ||
22B THE CHI-STREAM.
The chi-stream differs from a random sequence of letters in its periodicity in each impulse taken separately and in the deliberately arranged equality of dots and crosses in each impulse.
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