Dual of the Weil norm proof: Difference between revisions

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~~(Archived from Facebook, originally a post by Mike Battaglia addressing Gene Smith)~~

OK, so I proved the conjecture. The dual to the weil norm is max(<V 0|) - min(<V 0|). I'll formulate ~~monzos~~ as column vectors and ~~vals~~ as row vectors below.

(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[Gene Ward Smith|Gene Smith]])

OK, so I proved the conjecture. The dual to the weil norm is max(<V 0|) - min(<V 0|). I'll formulate [[monzo]]s as column vectors and [[val]]s as row vectors below.

Let M be the space of p-limit real monzos with the Weil norm on them. The Weil norm of a weighted monzo |a b c ...> is 1/2*(|a| + |b| + |c| + ... + |a+b+c+...|), giving you log(max(n,d)).

Note that this still looks like an L1 norm - in fact, it's the L1 norm of the "augmented monzo"

Note that this still looks like an L1 norm - in fact, it's the L1 norm of the "[[augmented]] monzo"

1/2 * |a b c ... ; (a+b+c+...)>

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which is a diagonal matrix with all .5's on the diagonal, followed by an extra row at the bottom of all .5's. I will call this matrix Ξ. So if you take any monzo m, L1(Ξm) = Weil(m).

Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of p-limit augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.

Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of [[p-limit]] augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.

Now, let's see if we can use this to figure out what the dual norm on vals looks like.

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-(Archived from Facebook, originally a post by Mike Battaglia addressing Gene Smith)
+{{archive}}
-OK, so I proved the conjecture. The dual to the weil norm is max(&lt;V 0|) - min(&lt;V 0|). I'll formulate monzos as column vectors and vals as row vectors below.
+(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[Gene Ward Smith|Gene Smith]])
+OK, so I proved the conjecture. The dual to the weil norm is max(&lt;V 0|) - min(&lt;V 0|). I'll formulate [[monzo]]s as column vectors and [[val]]s as row vectors below.
 Let M be the space of p-limit real monzos with the Weil norm on them. The Weil norm of a weighted monzo |a b c ...&gt; is 1/2*(|a| + |b| + |c| + ... + |a+b+c+...|), giving you log(max(n,d)).
-Note that this still looks like an L1 norm - in fact, it's the L1 norm of the "augmented monzo"
+Note that this still looks like an L1 norm - in fact, it's the L1 norm of the "[[augmented]] monzo"
 /2 * |a b c ... ; (a+b+c+...)&gt;
@@ Line 25: / Line 27: @@
 which is a diagonal matrix with all .5's on the diagonal, followed by an extra row at the bottom of all .5's. I will call this matrix Ξ. So if you take any monzo m, L1(Ξm) = Weil(m).
-Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of p-limit augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.
+Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of [[p-limit]] augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.
 Now, let's see if we can use this to figure out what the dual norm on vals looks like.