Domain basis: Difference between revisions

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The canonical form of an interval basis requires a few steps to achieve:

# Find the matrix representation of the interval basis.

# Put this matrix into column Hermite normal form. This step has the effect of sorting the formal primes so that those with higher primes as their ~~basis elements~~ come later, e.g. so that 7 comes after 9 even though 9 is a bigger number, because 9 factorizes into 3's.

# Put this matrix into column Hermite normal form. This step has the effect of sorting the formal primes so that those with higher primes in their factorizations come later, e.g. so that 7 comes after 9 even though 9 is a bigger number, because 9 factorizes into 3's.

# Eliminate any columns that are all zeros.

# Convert the matrix back into a list of numbers (separated by periods).

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We can do this by factorizing the formal primes in just the same way we factorize intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.

But here, we're going one step deeper down! Now we're breaking down our formal primes — the building blocks of our intervals — into ''their own building blocks''. ~~We can call~~ these ~~deeper building blocks '''formal~~ prime ~~basis elements'''~~.

But here, we're going one step deeper down! Now we're breaking down our formal primes — the building blocks of our intervals — into ''their own building blocks''. And these, finally, are just actual prime numbers.

Then, each resulting vector becomes a column of our desired matrix.

So, for example, 2.9/7.5 in the form of a matrix <math>B</math> looks like this. For convenience, we've labeled each column with the formal prime, and each row with the ~~basis element~~:

So, for example, 2.9/7.5 in the form of a matrix <math>B</math> looks like this. For convenience, we've labeled each column with the formal prime, and each row with the prime:

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= Non-JI interval subspaces =

The behavior of these are not yet well-described. For, for instance, is best to represent them as matrices? If a formal prime is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.

The behavior of these are not yet well-described. For, for instance, is best to represent them as matrices? If a formal prime is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.

= Terminology: interval basis vs. subgroup =

@@ Line 114: / Line 114: @@
 The canonical form of an interval basis requires a few steps to achieve:
 # Find the matrix representation of the interval basis.
-# Put this matrix into column Hermite normal form. This step has the effect of sorting the formal primes so that those with higher primes as their basis elements come later, e.g. so that 7 comes after 9 even though 9 is a bigger number, because 9 factorizes into 3's.
+# Put this matrix into column Hermite normal form. This step has the effect of sorting the formal primes so that those with higher primes in their factorizations come later, e.g. so that 7 comes after 9 even though 9 is a bigger number, because 9 factorizes into 3's.
 # Eliminate any columns that are all zeros.
 # Convert the matrix back into a list of numbers (separated by periods).
@@ Line 127: / Line 127: @@
 We can do this by factorizing the formal primes in just the same way we factorize intervals into [[prime-count vector]]s, such as 5/4 factorizing to {{vector|-2 0 1}}. This is also the same way we represent comma intervals within the other key RTT basis: the comma basis.
-But here, we're going one step deeper down! Now we're breaking down our formal primes — the building blocks of our intervals — into ''their own building blocks''. We can call these deeper building blocks '''formal prime basis elements'''.
+But here, we're going one step deeper down! Now we're breaking down our formal primes — the building blocks of our intervals — into ''their own building blocks''. And these, finally, are just actual prime numbers.
 Then, each resulting vector becomes a column of our desired matrix.
-So, for example, 2.9/7.5 in the form of a matrix <math>B</math> looks like this. For convenience, we've labeled each column with the formal prime, and each row with the basis element:
+So, for example, 2.9/7.5 in the form of a matrix <math>B</math> looks like this. For convenience, we've labeled each column with the formal prime, and each row with the prime:
@@ Line 268: / Line 268: @@
 = Non-JI interval subspaces =
-The behavior of these are not yet well-described. For, for instance, is best to represent them as matrices? If a formal prime is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.
+The behavior of these are not yet well-described. For, for instance, is best to represent them as matrices? If a formal prime is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.
 = Terminology: interval basis vs. subgroup =