Generator-offset property: Difference between revisions
→Case analysis: using inline math for a big inequality |
|||
| Line 181: | Line 181: | ||
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 | 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 | <math>2 \lfloor\frac{n}{\mu}\rfloor - 1 = 2 \big(\lfloor \frac{n}{\mu} \rfloor - 1\big) + 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) − 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 lower bound is reached if Λ<sub>3</sub> has chunks of sizes floor(''n''/μ) − 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)). | ||