Talk:Interior product - Revision history

Sintel: Notation again

2025-04-19T13:04:37Z

Notation again

← Older revision		Revision as of 13:04, 19 April 2025
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	\|(in terms of other two interior products)		\|(in terms of other two interior products)
	\|}		\|}

			== Notation ==

			As Cmloegcmluin observes above, the <math>\vee</math> notation is not really standard, and this product is usually written as <math>\alpha \mathbin{\lrcorner} \beta</math>.

			That is is 'dual' to the wedge product is too vague (there are at least 3 different notions of duality here). One might say it's the adjoint of the wedge product, as <math>\left\langle \alpha \mathbin{\lrcorner} \gamma , \beta \right\rangle = \left\langle \alpha , \beta \wedge \gamma\right\rangle</math>.

			So my suggestion is to just use the standard notation.

			– [[User:Sintel\|Sintel🎏]] ([[User_talk:Sintel\|talk]]) 13:04, 19 April 2025 (UTC)

Cmloegcmluin: use equal signs in column headings

2021-12-13T22:44:59Z

use equal signs in column headings

← Older revision		Revision as of 22:44, 13 December 2021
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	a ∧ b		a ∧ b
	!regressive product (AKA vee product)<br>		!regressive product (AKA vee product)<br>
	a ∨ b~~<br>~~		a ∨ b = <nowiki></nowiki>(a ∧ *b)
	<nowiki></nowiki>(a ∧ *b)
	!right interior product<br>		!right interior product<br>
	a ⨽ b~~<br>~~		a ⨽ b = ∗(∗a ∧ b)<br>
	∗(∗a ∧ b)<br>
	examples given where grade(a) ≥ grade(b)		examples given where grade(a) ≥ grade(b)
	!(left) interior product<br>		!(left) interior product<br>
	a ⨼ b~~<br>~~		a ⨼ b = <nowiki></nowiki>(a ∧ b)<br>
	<nowiki></nowiki>(a ∧ b)<br>
	examples given where grade(a) < grade(b)		examples given where grade(a) < grade(b)
	!symmetrical interior product<br>		!symmetrical interior product<br>

Cmloegcmluin: correct table heading formatting

2021-12-11T16:11:04Z

correct table heading formatting

← Older revision		Revision as of 16:11, 11 December 2021
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	\|+		\|+
	!operations		!operations
	!progressive product (AKA wedge product, exterior product)		!progressive product (AKA wedge product, exterior product)<br>
	a ∧ b		a ∧ b
	!regressive product (AKA vee product)		!regressive product (AKA vee product)<br>
	a ∨ b		a ∨ b<br>
	<nowiki></nowiki>(a ∧ *b)		<nowiki></nowiki>(a ∧ *b)
	!right interior product		!right interior product<br>
	a ⨽ b		a ⨽ b<br>
	∗(∗a ∧ b)		∗(∗a ∧ b)<br>
	examples given where grade(a) ≥ grade(b)		examples given where grade(a) ≥ grade(b)
	!(left) interior product		!(left) interior product<br>
	a ⨼ b		a ⨼ b<br>
	<nowiki></nowiki>(a ∧ b)		<nowiki></nowiki>(a ∧ b)<br>
	examples given where grade(a) < grade(b)		examples given where grade(a) < grade(b)
	!symmetrical interior product		!symmetrical interior product<br>

	a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b		a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b
	\|-		\|-

Cmloegcmluin: unhyphenate "comma basis"

2021-11-12T20:23:13Z

unhyphenate "comma basis"

← Older revision		Revision as of 20:23, 12 November 2021
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	You can take the progressive product of two things with the same variance and get something with greater (or equal to greatest input) grade but same variance.		You can take the progressive product of two things with the same variance and get something with greater (or equal to greatest input) grade but same variance.

	Progressing two temperaments' multivectors together gives a multivector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma-bases for the same two original temperaments, so that's like [[meet]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is equal to the sum of the two input multivectors' grades (or less, if they share a comma in common), so it's either the same or greater than the greater of the two inputs' grades, and also it caps out at the dimensionality of the system.		Progressing two temperaments' multivectors together gives a multivector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma bases for the same two original temperaments, so that's like [[meet]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is equal to the sum of the two input multivectors' grades (or less, if they share a comma in common), so it's either the same or greater than the greater of the two inputs' grades, and also it caps out at the dimensionality of the system.

	Progressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like [[join]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is the same idea as the previous statement (although it's less, in this case, if they share a mapping-row in common).		Progressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like [[join]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is the same idea as the previous statement (although it's less, in this case, if they share a mapping-row in common).
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	Regressing two temperaments' multivectors together gives a multivector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like join, but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is equal to....... well, in the case of two multivectors, so two n=2, and d=3, you take both of their duals, so those grades are d-n=1, then wedge those, so sum them, so that's 2, but then take that's dual again so it's back to 1. Again it must be equal or less to the lesser of the two input's grades (depending on if they share a comma in common), and caps out at 0 (you can't go negative grade (negative grade is a different idea than the sign we're adding to our grade in Wolfram to supply the variance information too in a single handy package)).		Regressing two temperaments' multivectors together gives a multivector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like join, but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is equal to....... well, in the case of two multivectors, so two n=2, and d=3, you take both of their duals, so those grades are d-n=1, then wedge those, so sum them, so that's 2, but then take that's dual again so it's back to 1. Again it must be equal or less to the lesser of the two input's grades (depending on if they share a comma in common), and caps out at 0 (you can't go negative grade (negative grade is a different idea than the sign we're adding to our grade in Wolfram to supply the variance information too in a single handy package)).

	Regressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma-bases for the same two original temperaments, so that's like meet, but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is the same idea as the previous statement.		Regressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma bases for the same two original temperaments, so that's like meet, but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is the same idea as the previous statement.

	So the progressive and regressive products are flip-flopped in that way. And it makes sense, because of how the regressive product just takes the duals of both inputs, so it does the opposite operation to what it would if it was a straight wedging, and then takes the dual again at the end just so you get back something with the variance you put in.		So the progressive and regressive products are flip-flopped in that way. And it makes sense, because of how the regressive product just takes the duals of both inputs, so it does the opposite operation to what it would if it was a straight wedging, and then takes the dual again at the end just so you get back something with the variance you put in.

Cmloegcmluin: /* Questions, observations, suggestions */ hyphenate

2021-11-08T21:31:00Z

Questions, observations, suggestions: hyphenate

← Older revision		Revision as of 21:31, 8 November 2021
Line 66:		Line 66:
	You can take the progressive product of two things with the same variance and get something with greater (or equal to greatest input) grade but same variance.		You can take the progressive product of two things with the same variance and get something with greater (or equal to greatest input) grade but same variance.

	Progressing two temperaments' multivectors together gives a multivector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma bases for the same two original temperaments, so that's like [[meet]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is equal to the sum of the two input multivectors' grades (or less, if they share a comma in common), so it's either the same or greater than the greater of the two inputs' grades, and also it caps out at the dimensionality of the system.		Progressing two temperaments' multivectors together gives a multivector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma-bases for the same two original temperaments, so that's like [[meet]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is equal to the sum of the two input multivectors' grades (or less, if they share a comma in common), so it's either the same or greater than the greater of the two inputs' grades, and also it caps out at the dimensionality of the system.

	Progressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like [[join]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is the same idea as the previous statement (although it's less, in this case, if they share a mapping-row in common).		Progressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like [[join]], but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is the same idea as the previous statement (although it's less, in this case, if they share a mapping-row in common).
Line 74:		Line 74:
	Regressing two temperaments' multivectors together gives a multivector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like join, but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is equal to....... well, in the case of two multivectors, so two n=2, and d=3, you take both of their duals, so those grades are d-n=1, then wedge those, so sum them, so that's 2, but then take that's dual again so it's back to 1. Again it must be equal or less to the lesser of the two input's grades (depending on if they share a comma in common), and caps out at 0 (you can't go negative grade (negative grade is a different idea than the sign we're adding to our grade in Wolfram to supply the variance information too in a single handy package)).		Regressing two temperaments' multivectors together gives a multivector for the same temperament associated with the mapping you'd find by concatting and reducing the mappings for the same two original temperaments, so that's like join, but instead of being defined for temperaments in the abstract, it's the operation you perform on multicovectors to achieve it. The grade of the output is equal to....... well, in the case of two multivectors, so two n=2, and d=3, you take both of their duals, so those grades are d-n=1, then wedge those, so sum them, so that's 2, but then take that's dual again so it's back to 1. Again it must be equal or less to the lesser of the two input's grades (depending on if they share a comma in common), and caps out at 0 (you can't go negative grade (negative grade is a different idea than the sign we're adding to our grade in Wolfram to supply the variance information too in a single handy package)).

	Regressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma bases for the same two original temperaments, so that's like meet, but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is the same idea as the previous statement.		Regressing two temperaments' multicovectors together gives a multicovector for the same temperament associated with the comma basis you'd find by concatting and reducing the comma-bases for the same two original temperaments, so that's like meet, but instead of being defined for temperaments in the abstract, it's the operation you perform on multivectors to achieve it. The grade of the output is the same idea as the previous statement.

	So the progressive and regressive products are flip-flopped in that way. And it makes sense, because of how the regressive product just takes the duals of both inputs, so it does the opposite operation to what it would if it was a straight wedging, and then takes the dual again at the end just so you get back something with the variance you put in.		So the progressive and regressive products are flip-flopped in that way. And it makes sense, because of how the regressive product just takes the duals of both inputs, so it does the opposite operation to what it would if it was a straight wedging, and then takes the dual again at the end just so you get back something with the variance you put in.

Cmloegcmluin: remove misleading phrases about empty intersections

2021-11-01T19:47:56Z

remove misleading phrases about empty intersections

← Older revision		Revision as of 19:47, 1 November 2021
Line 50:		Line 50:
	The reason I think this article describes a symmetrical interior product is because of the last paragraph of the "definition" section. I think there may be an error there where it says "we can take the wedge product m∨W from the front". I think what it's showing is actually an example of the interior product, and this is the one place in the article where I see the input on the left having a smaller grade than the thing on the right. And I think the last clause of this sentence "this can only lead to a difference in sign" could be continued with the phrase "compared to W∨m", i.e. the inputs reordered so that the lower-grade input is on the right. So if it's possible to reorder the inputs like this, then that implies this article assumes the symmetrical interior product. The change in sign is due to how the wedge product is sometimes anticommutative. It always is for (mono)vectors or (mono)covectors, but I think it's commutative for even grades. I think the statement means "it can, at most" lead to a mere change in signs, but won't necessarily; for example, I did [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and they both give me ⟨⟨6 -7 -2 15 -25 -20 3 15 59 49].		The reason I think this article describes a symmetrical interior product is because of the last paragraph of the "definition" section. I think there may be an error there where it says "we can take the wedge product m∨W from the front". I think what it's showing is actually an example of the interior product, and this is the one place in the article where I see the input on the left having a smaller grade than the thing on the right. And I think the last clause of this sentence "this can only lead to a difference in sign" could be continued with the phrase "compared to W∨m", i.e. the inputs reordered so that the lower-grade input is on the right. So if it's possible to reorder the inputs like this, then that implies this article assumes the symmetrical interior product. The change in sign is due to how the wedge product is sometimes anticommutative. It always is for (mono)vectors or (mono)covectors, but I think it's commutative for even grades. I think the statement means "it can, at most" lead to a mere change in signs, but won't necessarily; for example, I did [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and they both give me ⟨⟨6 -7 -2 15 -25 -20 3 15 59 49].

	Any one of these products has a formula which can tell you what the output grade will be ~~(assuming empty intersections between the vectors they are wedges of)~~. For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.		Any one of these products has a formula which can tell you what the output grade will be. For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.

	So if we simply wanted to take what is on the page now and help it conform better with established mathematical usages, I would recommend we remove the line about interior being the wedge's dual, and change the symbol the wiki uses for the interior product. Above I used •, the fat dot, which came up on that Facebook post recently as a reasonable choice for this operation. I'm not attached to it though so if anyone has other suggestions I'm not opposed at all to considering them. I'm just concerned that ∨ probably should only be used for the regressive product. If we went this direction, then ∨ would also need to be replaced with • on the following pages, too:		So if we simply wanted to take what is on the page now and help it conform better with established mathematical usages, I would recommend we remove the line about interior being the wedge's dual, and change the symbol the wiki uses for the interior product. Above I used •, the fat dot, which came up on that Facebook post recently as a reasonable choice for this operation. I'm not attached to it though so if anyone has other suggestions I'm not opposed at all to considering them. I'm just concerned that ∨ probably should only be used for the regressive product. If we went this direction, then ∨ would also need to be replaced with • on the following pages, too:
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	d-((d-r)+n) which simplifies to r-n.		d-((d-r)+n) which simplifies to r-n.

	e.g. ~~Assuming empty intersections~~:		e.g.:
	Dimension 7. Left input rank 6. Right input nullity 2. Output rank = r-n =4		Dimension 7. Left input rank 6. Right input nullity 2. Output rank = r-n =4
	(greater than right grade)		(greater than right grade)
Line 123:		Line 123:
	a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b		a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b
	\|-		\|-
	!resultant grade~~, assuming empty intersections~~		!resultant grade
	!grade(a) + grade(b)		!grade(a) + grade(b)
	!grade(a) + grade(b) - dimensionality		!grade(a) + grade(b) - dimensionality

Cmloegcmluin: /* Questions, observations, suggestions */

2021-10-15T20:57:42Z

Questions, observations, suggestions

← Older revision		Revision as of 20:57, 15 October 2021
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	I don't have a strong preference which way people want to go with this information. And certainly some combination of these recommendations could be done. I am very open to hearing that there are cases where interior product does exactly what you want and regressive wouldn't cut it. I haven't gone through every use on the wiki (and outside it) to check. Also there's the argument that this interior product defined as it is has been around for a while and maybe even if it has some issues we should preserve it for backwards compatibility. All fine.		I don't have a strong preference which way people want to go with this information. And certainly some combination of these recommendations could be done. I am very open to hearing that there are cases where interior product does exactly what you want and regressive wouldn't cut it. I haven't gone through every use on the wiki (and outside it) to check. Also there's the argument that this interior product defined as it is has been around for a while and maybe even if it has some issues we should preserve it for backwards compatibility. All fine.

	If it's of any assistance, I've implemented all of these products in Wolfram Language~~. You can access the shared computable notebook here and make your own copy~~: https://~~www.wolframcloud~~.com/~~obj~~/~~5d4e22f3-af6b-4c4a-a7ce-af860362ae72~~		If it's of any assistance, I've implemented all of these products in Wolfram Language: https://github.com/cmloegcmluin/VEA

	And here's a summary table that has helped me get my head around this situation:		And here's a summary table that has helped me get my head around this situation:

Cmloegcmluin: /* Questions, observations, suggestions */ add summary table

2021-07-08T20:42:01Z

Questions, observations, suggestions: add summary table

← Older revision		Revision as of 20:42, 8 July 2021
Line 100:		Line 100:

	If it's of any assistance, I've implemented all of these products in Wolfram Language. You can access the shared computable notebook here and make your own copy: https://www.wolframcloud.com/obj/5d4e22f3-af6b-4c4a-a7ce-af860362ae72		If it's of any assistance, I've implemented all of these products in Wolfram Language. You can access the shared computable notebook here and make your own copy: https://www.wolframcloud.com/obj/5d4e22f3-af6b-4c4a-a7ce-af860362ae72

			And here's a summary table that has helped me get my head around this situation:

			{\| class="wikitable"
			\|+
			!operations
			!progressive product (AKA wedge product, exterior product)
			a ∧ b
			!regressive product (AKA vee product)
			a ∨ b
			<nowiki></nowiki>(a ∧ *b)
			!right interior product
			a ⨽ b
			∗(∗a ∧ b)
			examples given where grade(a) ≥ grade(b)
			!(left) interior product
			a ⨼ b
			<nowiki></nowiki>(a ∧ b)
			examples given where grade(a) < grade(b)
			!symmetrical interior product

			a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b
			\|-
			!resultant grade, assuming empty intersections
			!grade(a) + grade(b)
			!grade(a) + grade(b) - dimensionality
			!grade(a) - grade(b)
			!grade(b) - grade(a)
			!if grade(a) ≥ grade(b), grade(a) - grade(b); else grade(b) - grade(a)
			\|-
			!resultant variance
			!same as a (and b)
			!same as a (and b)
			!same as a
			!same as b
			!if grade(a) ≥ grade(b), same as a; else same as b
			\|-
			!multicovector with multicovector
			⟨] ⟨]
			\|⟨12 19 28 34] ∧ ⟨19 30 44 53] = ⟨⟨1 4 10 4 13 12]]
			\|⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] ∨ ⟨⟨⟨⟨1 2 1 2 3]]]] = ⟨⟨-6 7 2 -15 25 20 -3 -15 -59 -49]]
			\|ND
			\|ND
			\|ND
			\|-
			!multivector with multivector
			[⟩ [⟩
			\|[4 -4 1 0⟩ ∧ [13 -10 0 1⟩ = [[12 -13 4 10 -4 1⟩⟩
			\|[[44 -30 19⟩⟩ ∨ [[28 -19 12⟩⟩ = [4 -4 1⟩
			\|ND
			\|ND
			\|ND
			\|-
			!multicovector with multivector
			⟨] [⟩
			\|ND
			\|ND
			\|⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] ⨽ [-3 2 -1 2 -1⟩ = ⟨⟨6 -7 -2 15 -25 -20 3 15 59 49]]
			\|⟨12 19 28] ⨼ [[44 -30 19⟩⟩ = [4 -4 1⟩
			\|(in terms of other two interior products)
			\|-
			!multivector with multicovector
			[⟩ ⟨]
			\|ND
			\|ND
			\|[[44 -30 19⟩⟩ ⨽ ⟨12 19 28] = [-4 4 -1⟩
			\|[-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] = ⟨⟨-6 7 2 -15 25 20 -3 -15 -59 -49]]
			\|(in terms of other two interior products)
			\|}

Cmloegcmluin: adding some questions, observations, and suggestions

2021-07-08T20:34:15Z

adding some questions, observations, and suggestions

Show changes

Mike Battaglia: 1 revision imported: Moving archived Wikispaces discussion to subpage

2018-10-01T18:02:37Z

1 revision imported: Moving archived Wikispaces discussion to subpage

← Older revision	Revision as of 18:02, 1 October 2018
(No difference)