MOS substitution: Difference between revisions

Line 17:

| style="text-align: right;" | <code>mss</code>

| colspan="2" style="text-align: right;" | <code>LmLss</code>

| subst 2L(1m2s 2{{pipe}}0)

| {{nowrap|subst 2L(1m2s 2{{pipe}}0)}}

|-

| 1{{pipe}}1

| style="text-align: right;" | <code>sms</code>

| colspan="2" style="text-align: right;" | <code>LsLms</code>

| subst 2L(1m1s 1{{pipe}}1)

| {{nowrap|subst 2L(1m1s 1{{pipe}}1)}}

|-

| 0{{pipe}}2

| style="text-align: right;" | <code>ssm</code>

| colspan="2" style="text-align: right;" | <code>LsLsm</code>

| subst 2L(1m1s 0{{pipe}}2)

| {{nowrap|subst 2L(1m1s 0{{pipe}}2)}}

|}

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<math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} \cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}.</math>

In the above case, ''n'' = ''q'', '''v''' = subst(''p''''T'', '''X''', ''p''''F''), and '''w''' = subst((''p''''T'')''r'', '''X''', ''F''''r'').

In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''''T'', '''X''', ''p''''F'') | and '''w''' {{=}} subst((''p''''T'')''r'', '''X''', ''F''''r'')}}.

The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram.

Line 212:

Consider a MOS substitution scale {{nowrap|a'''X''' (b'''Y''' c'''Z''')}}. It is obvious that '''X''' has [[block balance]] 1, since we can replace the MOS substitution scale with the MOS scale a'''X''' ({{nowrap|b + c}})'''W''' to make this argument. '''Y''' and '''Z''' have block balance at most 2, since we can consider windows of the MOS scale of size ''k'' or {{nowrap|''k'' + 1}}, and the number of times '''Y''' (and also '''Z''') differs by at most 2. This is proved below for '''Y''', but it's exactly the same argument for '''Z''':

Case 1: ~~one~~ of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}) and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case.

Case 1: One of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}) and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case.

Case 2: ~~neither~~ of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then ''Y'' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.

Case 2: Neither of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then ''Y'' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.

== MOS substitution scales and RTT ==

@@ Line 17: / Line 17: @@
 | style="text-align: right;" | <code>mss</code>
 | colspan="2" style="text-align: right;" | <code>LmLss</code>
-| subst 2L(1m2s 2{{pipe}}0)
+| {{nowrap|subst 2L(1m2s 2{{pipe}}0)}}
 |-
 | 1{{pipe}}1
 | style="text-align: right;" | <code>sms</code>
 | colspan="2" style="text-align: right;" | <code>LsLms</code>
-| subst 2L(1m1s 1{{pipe}}1)
+| {{nowrap|subst 2L(1m1s 1{{pipe}}1)}}
 |-
 | 0{{pipe}}2
 | style="text-align: right;" | <code>ssm</code>
 | colspan="2" style="text-align: right;" | <code>LsLsm</code>
-| subst 2L(1m1s 0{{pipe}}2)
+| {{nowrap|subst 2L(1m1s 0{{pipe}}2)}}
 |}
@@ Line 205: / Line 205: @@
 <math>\{\mathbf{a} + i\mathbf{v}\}_{i=a}^{n-1} \cup \{\mathbf{a}  + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1} \cup \{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}.</math>
-In the above case, ''n'' = ''q'', '''v''' = subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>), and '''w''' = subst((''p''<sub>''T''</sub>)<sup>''r''</sup>, '''X''', ''F''<sup>''r''</sup>).
+In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>) | and '''w''' {{=}} subst((''p''<sub>''T''</sub>)<sup>''r''</sup>, '''X''', ''F''<sup>''r''</sup>)}}.
 The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram.
@@ Line 212: / Line 212: @@
 Consider a MOS substitution scale {{nowrap|a'''X''' (b'''Y''' c'''Z''')}}. It is obvious that '''X''' has [[block balance]] 1, since we can replace the MOS substitution scale with the MOS scale a'''X''' ({{nowrap|b + c}})'''W''' to make this argument. '''Y''' and '''Z''' have block balance at most 2, since we can consider windows of the MOS scale of size ''k'' or {{nowrap|''k'' + 1}}, and the number of times '''Y''' (and also '''Z''') differs by at most 2. This is proved below for '''Y''', but it's exactly the same argument for '''Z''':
-Case 1: one of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}) and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case.
+Case 1: One of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}) and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case.
-Case 2: neither of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then ''Y'' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.
+Case 2: Neither of ''k'' and {{nowrap|''k'' + 1}} equals ({{nowrap|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap|''j'' + 1}} times in a window of size ''k'', then ''Y'' occurs {{nowrap|''j'' + 1}} or {{nowrap|''j'' + 2}} times in a window of size {{nowrap|''k'' + 2}}.
 == MOS substitution scales and RTT ==