Dual of the Weil norm proof - Revision history

FloraC: - irrelevant link

2025-09-11T04:29:36Z

- irrelevant link

← Older revision		Revision as of 04:29, 11 September 2025
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	{{~~archive~~}}		{{Archive}}

	(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[Gene Smith]])		(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[Gene Smith]])
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	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)).		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+...)>		1/2 * \|a b c ... ; (a+b+c+...)>
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	QED.		QED.

	[[Category:Math]]		[[Category:Math]]
	[[Category:Archive]]		[[Category:Archive]]

BudjarnLambeth at 05:19, 9 January 2024

2024-01-09T05:19:15Z

← Older revision		Revision as of 05:19, 9 January 2024
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	{{archive}}		{{archive}}

	(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[~~Gene Ward Smith\|~~Gene Smith]])		(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 [[monzo]]s as column vectors and [[val]]s as row vectors below.		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.

BudjarnLambeth: Added archive notice and added in text links (did not edit the text at all, only added links)

2024-01-09T05:18:16Z

Added archive notice and added in text links (did not edit the text at all, only added links)

← Older revision		Revision as of 05:18, 9 January 2024
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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(<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)).		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+...)>		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).		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.		Now, let's see if we can use this to figure out what the dual norm on vals looks like.

BudjarnLambeth: Categorised this uncategorised page

2023-04-27T00:24:02Z

Categorised this uncategorised page

← Older revision		Revision as of 00:24, 27 April 2023
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	QED.		QED.
			[[Category:Math]]
			[[Category:Archive]]

Cmloegcmluin: change k to r for consistency with Weil_norm,_Tenney-Weil_norm,_and_TWp_interval_and_tuning_space article and to avoid clash with k as used there

2022-12-24T16:37:50Z

change k to r for consistency with Weil_norm,_Tenney-Weil_norm,_and_TWp_interval_and_tuning_space article and to avoid clash with k as used there

← Older revision		Revision as of 16:37, 24 December 2022
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	To do so, first we note that for any val v = <x y z ...\|, the augmented val v° = <2x 2y 2z ...; 0\| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression		To do so, first we note that for any val v = <x y z ...\|, the augmented val v° = <2x 2y 2z ...; 0\| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression

	Linf(v° + ~~kj°~~)		Linf(v° + rj°)

	for k in R. Or, in other words, we want to minimize		for r in R. Or, in other words, we want to minimize

	Linf(<2x 2y 2z ...; 0\| + <~~k k k~~...; -k\|)		Linf(<2x 2y 2z ...; 0\| + <r r r...; -r\|)

	We can rewrite the above as follows		We can rewrite the above as follows

	~~inf_k~~ max(\|2x+k\|, \|2y+k\|, \|2z+k\|, ..., \|k\|)		inf_r max(\|2x+r\|, \|2y+r\|, \|2z+r\|, ..., \|r\|)

	OK, so what's the k that minimizes that? It turns out to be the k such that		OK, so what's the r that minimizes that? It turns out to be the r such that

	max(2x+k,2y+k,...,k) = -min(2x+k,2y+k,...,k)		max(2x+r,2y+r,...,r) = -min(2x+r,2y+r,...,r)

	If k diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.		If r diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.

	Put differently, we want to take the range of <2x 2y 2z ...; 0\| and set k so that this range is centered around zero.		Put differently, we want to take the range of <2x 2y 2z ...; 0\| and set r so that this range is centered around zero.

	But now we're done. We don't need to worry about finding k. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.		But now we're done. We don't need to worry about finding r. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.

	So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that		So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that

Cmloegcmluin: fix broken link

2022-12-19T20:09:41Z

fix broken link

← Older revision		Revision as of 20:09, 19 December 2022
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	We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.		We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.

	Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (~~[http~~://~~www~~.~~math.unl~~.~~edu~~/~~~s-bbockel1~~/~~928~~/~~node25.html http~~://~~www.~~math.unl.edu/~s-bbockel1/928/node25.html]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.		Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (https://web.archive.org/web/20201130153651/https://math.unl.edu/~s-bbockel1/928/node25.html), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.

	Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by <1 1 1 ...; -1\|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.		Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by <1 1 1 ...; -1\|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+(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(&lt;V 0|) - min(&lt;V 0|). I'll formulate monzos as column vectors and vals as row vectors below.
@@ Line 21: / Line 14: @@
 [.5 0 0 ...]
 [0 .5 0 ...]
 [0 0 .5 ...]
 [... ... ... ...]
 [.5 .5 .5 ...]
@@ Line 36: / Line 33: @@
 We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.
-Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem ([[@http://www.math.unl.edu/~s-bbockel1/928/node25.html]]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.
+Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem ([http://www.math.unl.edu/~s-bbockel1/928/node25.html http://www.math.unl.edu/~s-bbockel1/928/node25.html]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.
 Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by &lt;1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.
@@ Line 65: / Line 62: @@
 So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that
 ||v||* = max(&lt;v 0|) - min(&lt;v 0|)
-QED.</pre></div>
+QED.

Wikispaces>mbattaglia1: Imported revision 481014124 - Original comment:

2014-01-07T01:13:53Z

**Imported revision 481014124 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference: 
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-01-07 01:13:53 UTC</tt>. 
: The original revision id was <tt>481014124</tt>. 
: The revision comment was: <tt></tt> 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. 
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">(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.

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"

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

where we have appended the sum of the coefficients to the end of the vector as a new coefficient.

We can left-multiply our original monzos by a matrix to transform them into these special augmented monzos. That transformation matrix just so happens to be

[.5 0 0 ...]
[0 .5 0 ...]
[0 0 .5 ...]
[... ... ... ...]
[.5 .5 .5 ...]

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.

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

Let's say that the dual space to M is called V, and the dual space to M° is called V°. V is the space of real p-limit vals; V° is the space of p-limit augmented vals with one extra coordinate tacked on at the end.

We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.

Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem ([[@http://www.math.unl.edu/~s-bbockel1/928/node25.html]]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.

Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by <1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.

A good way to make sense of this interesting situation is to note that Ξ also acts as a matrix taking augmented vals to normal ones. So, we can find the norm of an arbitrary val v in V as follows: look for all augmented vals in the preimage Ξ^-1(v) and find the one with the shortest Linf norm. The value of this Linf norm is the norm on v. We'll see if we can find a nice closed-form expression for this.

To do so, first we note that for any val v = <x y z ...|, the augmented val v° = <2x 2y 2z ...; 0| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression

Linf(v° + kj°)

for k in R. Or, in other words, we want to minimize

Linf(<2x 2y 2z ...; 0| + <k k k...; -k|)

We can rewrite the above as follows

inf_k max(|2x+k|, |2y+k|, |2z+k|, ..., |k|)

OK, so what's the k that minimizes that? It turns out to be the k such that

max(2x+k,2y+k,...,k) = -min(2x+k,2y+k,...,k)

If k diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.

Put differently, we want to take the range of <2x 2y 2z ...; 0| and set k so that this range is centered around zero.

But now we're done. We don't need to worry about finding k. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.

So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that

||v||* = max(<v 0|) - min(<v 0|)

QED.</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Dual of the Weil norm proof</title></head><body>(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(&lt;V 0|) - min(&lt;V 0|). I'll formulate monzos as column vectors and vals 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 &quot;augmented monzo&quot; 
 
1/2 * |a b c ... ; (a+b+c+...)&gt; 
 
where we have appended the sum of the coefficients to the end of the vector as a new coefficient. 
 
We can left-multiply our original monzos by a matrix to transform them into these special augmented monzos. That transformation matrix just so happens to be 
 
[.5 0 0 ...] 
[0 .5 0 ...] 
[0 0 .5 ...] 
[... ... ... ...] 
[.5 .5 .5 ...] 
 
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. 
 
Now, let's see if we can use this to figure out what the dual norm on vals looks like. 
 
Let's say that the dual space to M is called V, and the dual space to M° is called V°. V is the space of real p-limit vals; V° is the space of p-limit augmented vals with one extra coordinate tacked on at the end. 
 
We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V. 
 
Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (<a class="wiki_link_ext" href="http://www.math.unl.edu/~s-bbockel1/928/node25.html" rel="nofollow" target="_blank">http://www.math.unl.edu/~s-bbockel1/928/node25.html</a>), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm. 
 
Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by &lt;1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end. 
 
A good way to make sense of this interesting situation is to note that Ξ also acts as a matrix taking augmented vals to normal ones. So, we can find the norm of an arbitrary val v in V as follows: look for all augmented vals in the preimage Ξ^-1(v) and find the one with the shortest Linf norm. The value of this Linf norm is the norm on v. We'll see if we can find a nice closed-form expression for this. 
 
To do so, first we note that for any val v = &lt;x y z ...|, the augmented val v° = &lt;2x 2y 2z ...; 0| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression 
 
Linf(v° + kj°) 
 
for k in R. Or, in other words, we want to minimize 
 
Linf(&lt;2x 2y 2z ...; 0| + &lt;k k k...; -k|) 
 
We can rewrite the above as follows 
 
inf_k max(|2x+k|, |2y+k|, |2z+k|, ..., |k|) 
 
OK, so what's the k that minimizes that? It turns out to be the k such that 
 
max(2x+k,2y+k,...,k) = -min(2x+k,2y+k,...,k) 
 
If k diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max. 
 
Put differently, we want to take the range of &lt;2x 2y 2z ...; 0| and set k so that this range is centered around zero. 
 
But now we're done. We don't need to worry about finding k. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for. 
 
So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that 
 
||v||* = max(&lt;v 0|) - min(&lt;v 0|) 
 
QED.</body></html></pre></div>