Generator-offset property: Difference between revisions
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Since ''y'' − ''x'' = floor(''n''/μ) − 1 and floor(''n''/μ) ≥ 2, we have: ''x'' = floor(''n''/μ) = 2 and ''y'' − ''x'' = 1; hence ''y'' = 2*floor(''n''/μ) − 1 = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.) | Since ''y'' − ''x'' = floor(''n''/μ) − 1 and floor(''n''/μ) ≥ 2, we have: ''x'' = floor(''n''/μ) = 2 and ''y'' − ''x'' = 1; hence ''y'' = 2*floor(''n''/μ) − 1 = 3. The chunk in Λ<sub>3</sub> whose size is 3 is made from two chunks in Λ<sub>2</sub> of size 1. (So Λ<sub>2</sub> has chunks of size 1 and 2, and Λ<sub>3</sub> has chunks of size 2 and 3.) | ||
Λ<sub>2</sub> has two consecutive chunks of size 1. Since chunk sizes form a mos, Λ<sub>2</sub> has more chunks of size 1 than it has chunks of size 2. | Λ<sub>2</sub> has two consecutive chunks of size 1. Since chunk sizes of a mos themselves form a mos, Λ<sub>2</sub> has more chunks of size 1 than it has chunks of size 2. | ||
Use ''w''[''i'' : ''j''] to denote the slice of the cyclic word ''w'' that includes both endpoints, i.e. the word ''w''[''i''] ''w''[''i'' + 1] ... ''w''[''j''] where indices are taken to be elements of '''Z'''/''n'''''Z'''. Λ<sub>2</sub> has only two chunks of size 1, Λ<sub>2</sub>[(''n'' − 1) : (''n'' − 1)] and Λ<sub>2</sub>[1 : 1], since otherwise Λ<sub>3</sub> would have a chunk of size 1 within Λ<sub>3</sub>[1 : (''n'' − 1)]. Thus Λ<sub>2</sub> has exactly one chunk of size 2. Thus Λ<sub>2</sub> = ββ′βββ′ββ′ and Λ<sub>3</sub> = γγ′γγγ′γγ. Thus we have: | Use ''w''[''i'' : ''j''] to denote the slice of the cyclic word ''w'' that includes both endpoints, i.e. the word ''w''[''i''] ''w''[''i'' + 1] ... ''w''[''j''] where indices are taken to be elements of '''Z'''/''n'''''Z'''. Λ<sub>2</sub> has only two chunks of size 1, Λ<sub>2</sub>[(''n'' − 1) : (''n'' − 1)] and Λ<sub>2</sub>[1 : 1], since otherwise Λ<sub>3</sub> would have a chunk of size 1 within Λ<sub>3</sub>[1 : (''n'' − 1)]. Thus Λ<sub>2</sub> has exactly one chunk of size 2. Thus Λ<sub>2</sub> = ββ′βββ′ββ′ and Λ<sub>3</sub> = γγ′γγγ′γγ. Thus we have: | ||