Douglas Blumeyer's RTT How-To: Difference between revisions

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We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the intersections of two commas are greater than the sum of their individual parts.

We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.

We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation; maybe we could call it an anti-null-space operation.

And interestingly enough, as you'll soon see, the process is almost the same to take the null-space as it is to undo it.

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Null-space can be calculated by specialized math programs and web tools, as linked above. But I think it’s a good idea to work through it by hand at least a couple times, to demystify it and give you a feel for it.

Null-space bases can be calculated by specialized math programs and web tools, as linked above. But I think it’s a good idea to work through it by hand at least a couple times, to demystify it and give you a feel for it.

=== the other side of duality ===

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If you take the Hermite canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the Hermite canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.

To find the Hermite canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and ~~undoing~~ null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.

To find the Hermite canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.

Unfortunately, Wolfram Alpha is incapable of computing this in one go. For that, you'll need up your game to Wolfram computable notebooks. It only takes a moment and you can sign up for free.

@@ Line 497: / Line 497: @@
 We already have the tools to check that each of these commas’ vectors is tempered out individually by the map {{map|19 30 44}}; we learned this bit in the very first section: all we have to do is make sure that the comma maps to zero steps in this ET. But that's not a special relationship between 19-ET and any of these commas ''individually''; each of these commas are tempered out by many different ETs, not just 19-ET. The special relationship 19-ET has is to a null-space which can be expressed in basis form as the intersection of ''two'' commas (at least in the 5-limit; more on this later). In this way, the comma bases which represent the intersections of two commas are greater than the sum of their individual parts.
-We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation.
+We can confirm the relationship between an ET and its null-space by converting back and forth between them. There exists a mathematical function which — when input any one of these comma bases — will output {{vector|{{map|19 30 44}}}}, thus demonstrating the various bases' equivalence with respect to it. If the operation called "taking the null-space" is what gets you from {{vector|{{map|19 30 44}}}} to one basis for the null-space, then ''this'' mathematical function is in effect ''undoing'' the null-space operation; maybe we could call it an anti-null-space operation.
 And interestingly enough, as you'll soon see, the process is almost the same to take the null-space as it is to undo it.
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 |}
-Null-space can be calculated by specialized math programs and web tools, as linked above. But I think it’s a good idea to work through it by hand at least a couple times, to demystify it and give you a feel for it.
+Null-space bases can be calculated by specialized math programs and web tools, as linked above. But I think it’s a good idea to work through it by hand at least a couple times, to demystify it and give you a feel for it.
 === the other side of duality ===
@@ Line 1,100: / Line 1,100: @@
 If you take the Hermite canonical form of {{vector|{{map|5 8 12}} {{map|7 11 16}}}}, that’s what you get. It’s also what you get if you take the Hermite canonical form of {{vector|{{map|12 19 28}} {{map|19 30 44}}}}, etc. That’s the power of canonicalization.
-To find the Hermite canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and undoing null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.
+To find the Hermite canonical form, we can combine the two processes we already know how to do: null-space for getting from a map basis to a comma basis, and anti-null-space to get from a comma basis to a map basis. Basically, to achieve canonical form of one type of basis, we convert it into the other type of basis, then back, and voilà: canonicalization.
 Unfortunately, Wolfram Alpha is incapable of computing this in one go. For that, you'll need up your game to Wolfram computable notebooks. It only takes a moment and you can sign up for free.