Generator-offset property: Difference between revisions

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'''Case 3:''' 3 ≤ μ ≤ floor(''n''/2).

Λ2 has a chunk of β (after the first β′) of size ''x'' = ~~either~~ floor(''n''/μ) (≥ floor(''n''/floor(''n''/2)) = 2) or ceil(''n''/μ) (= floor(''n''/μ) + 1). Hence Λ3 has a chunk of γ of that same size. Λ3 also has a chunk that crosses index ''n'', which must be of size ''y'' = ~~at least~~ 2*(floor(''n~~''/μ) −~~ 1) + 1 (Λ3 ~~might have~~ chunks of ~~size~~ floor(''n''/μ) − 1 and floor(''n''/μ) ~~instead) = 2*floor(''n''/μ) − 1~~, and ~~at most 2*(floor(''n''/μ) + 1) + 1 = 2*floor(''n''/μ) + 3 (~~if Λ3 has chunks of ~~size~~ floor(''n''/μ) and ceil(''n''/μ)). The difference between the chunk sizes is ''y'' − ''x'', which must be 0 or 1, since Λ3 is a mos. We thus have the following subcases: (In the following, chunk of Λ2 means chunk of β, and chunk of Λ3 means chunk of γ.)

Λ2 has a chunk of β (after the first β′) of size ''x'' where ''x'' = floor(''n''/μ) ≥ floor(''n''/floor(''n''/2)) = 2 or ''x'' = ceil(''n''/μ) = floor(''n''/μ) + 1. Hence Λ3 has a chunk of γ of that same size. Λ3 also has a chunk that crosses index ''n'', which must be of size ''y'', where

<math>2 \big(\lfloor \frac{n}{\mu} \rfloor - 1\big) + 1 = 2 \lfloor\frac{n}{\mu}\rfloor - 1 \leq y \leq 2 \big(\lfloor\frac{n}{\mu}\rfloor + 1 \big) + 1 = 2\lfloor\frac{n}{\mu}\rfloor + 3.</math>

(The lower bound is reached if Λ3 has chunks of sizes floor(''n''/μ) − 1 and floor(''n''/μ), and the upper bound is reached if Λ3 has chunks of sizes floor(''n''/μ) and ceil(''n''/μ)).

The difference between the chunk sizes is ''y'' − ''x'', which must be 0 or 1, since Λ3 is a mos. We thus have the following subcases: (In the following, chunk of Λ2 means chunk of β, and chunk of Λ3 means chunk of γ.)

'''Case 3.1:''' (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) − 1).

@@ Line 179: / Line 179: @@
 '''Case 3:''' 3 ≤ μ ≤ floor(''n''/2).
-Λ<sub>2</sub> has a chunk of β (after the first β′) of size ''x'' = either floor(''n''/μ) (≥ floor(''n''/floor(''n''/2)) = 2) or ceil(''n''/μ) (= floor(''n''/μ) + 1). Hence Λ<sub>3</sub> has a chunk of γ of that same size. Λ<sub>3</sub> also has a chunk that crosses index ''n'', which must be of size ''y'' = at least 2*(floor(''n''/μ) &minus; 1) + 1 (Λ<sub>3</sub> might have chunks of size floor(''n''/μ) &minus; 1 and floor(''n''/μ) instead) = 2*floor(''n''/μ) &minus; 1, and at most 2*(floor(''n''/μ) + 1) + 1 = 2*floor(''n''/μ) + 3 (if Λ<sub>3</sub> has chunks of size floor(''n''/μ) and ceil(''n''/μ)). The difference between the chunk sizes is ''y'' &minus; ''x'', which must be 0 or 1, since Λ<sub>3</sub> is a mos. We thus have the following subcases: (In the following, chunk of Λ<sub>2</sub> means chunk of β, and chunk of Λ<sub>3</sub> means chunk of γ.)
+Λ<sub>2</sub> has a chunk of β (after the first β′) of size ''x'' where ''x'' = floor(''n''/μ) ≥ floor(''n''/floor(''n''/2)) = 2 or ''x'' = ceil(''n''/μ) = floor(''n''/μ) + 1. Hence Λ<sub>3</sub> has a chunk of γ of that same size. Λ<sub>3</sub> also has a chunk that crosses index ''n'', which must be of size ''y'', where
+<math>2 \big(\lfloor \frac{n}{\mu} \rfloor - 1\big) + 1 = 2 \lfloor\frac{n}{\mu}\rfloor - 1 \leq y \leq 2 \big(\lfloor\frac{n}{\mu}\rfloor + 1 \big) + 1 = 2\lfloor\frac{n}{\mu}\rfloor + 3.</math>
+(The lower bound is reached if Λ<sub>3</sub> has chunks of sizes floor(''n''/μ) &minus; 1 and floor(''n''/μ), and the upper bound is reached if Λ<sub>3</sub> has chunks of sizes floor(''n''/μ) and ceil(''n''/μ)).
+The difference between the chunk sizes is ''y'' &minus; ''x'', which must be 0 or 1, since Λ<sub>3</sub> is a mos. We thus have the following subcases: (In the following, chunk of Λ<sub>2</sub> means chunk of β, and chunk of Λ<sub>3</sub> means chunk of γ.)
 '''Case 3.1:''' (''x'', ''y'') = (floor(''n''/μ), 2*floor(''n''/μ) &minus; 1).

Generator-offset property: Difference between revisions

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