Parallelogram substring scale - Revision history

Inthar: /* Examples */

2026-03-14T04:27:17Z

Examples

← Older revision		Revision as of 04:27, 14 March 2026
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	=== Examples ===		=== Examples ===
	* All non-Fraenkel balanced primitive MV3 scales		* All non-Fraenkel balanced primitive MV3 scales
	* All ''a'''''x'''(''b'''''y'''(''a'' - ''b'')'''z''') MOS substitution scales if gcd(''a'', ''b'') = 1		* All ''a'''''x'''(''b'''''y'''(''a'' - ''b'')'''z''') [[MOS substitution]] scales if gcd(''a'', ''b'') = 1
	* All MOS substitution scales where:		* All MOS substitution scales where:
	** The template MOS is primitive		** The template MOS is primitive

Inthar at 04:27, 14 March 2026

2026-03-14T04:27:02Z

← Older revision		Revision as of 04:27, 14 March 2026
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	* All multiperiod MOS substitution scales (e.g. 4L(10m10s))		* All multiperiod MOS substitution scales (e.g. 4L(10m10s))

			== Mathematical facts ==
			=== Ternary parallelogram scales are MOS substitution ===
			:''Main article: [[Ternary parallelogram scales are MOS substitution]]''
	== Open problems ==		== Open problems ==
	# Classify all MOS-substitution parallelogram substring scales.		# Classify all MOS-substitution parallelogram substring scales.

Inthar: /* Mathematical definition */

2026-03-14T04:24:46Z

Mathematical definition

← Older revision		Revision as of 04:24, 14 March 2026
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	This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.		This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.

			A parallelogram substring scale with full first and last rows is a '''parallelogram scale'''.

	== Ternary scales with this property ==		== Ternary scales with this property ==

Inthar: Inthar moved page User:Inthar/Parallelogram substring scales to Parallelogram substring scale

2026-03-14T04:23:25Z

Inthar moved page User:Inthar/Parallelogram substring scales to Parallelogram substring scale

← Older revision	Revision as of 04:23, 14 March 2026
(No difference)

Inthar: /* Open problems */

2025-12-31T14:57:00Z

Open problems

← Older revision		Revision as of 14:57, 31 December 2025
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	== Open problems ==		== Open problems ==
	# Classify all MOS-substitution parallelogram substring scales.		# Classify all MOS-substitution parallelogram substring scales.
	# Classify all ternary parallelogram substring scales. <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (Numerous counterexamples, e.g. LLmLmLmLmLLs)		# Classify all ternary parallelogram substring scales.
			#* <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (Numerous counterexamples, e.g. LLmLmLmLmLLs)
	# Classify all ternary full parallelogram scales (PS with full first and last rows).		# Classify all ternary full parallelogram scales (PS with full first and last rows).
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Open problems */

2025-12-31T14:56:47Z

Open problems

← Older revision		Revision as of 14:56, 31 December 2025
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	== Open problems ==		== Open problems ==
	# Classify all MOS-substitution parallelogram substring scales.		# Classify all MOS-substitution parallelogram substring scales.
	# Classify all ternary parallelogram substring scales. Conjecture: All ternary parallelogram substring scales are MOS substitution scales.		# Classify all ternary parallelogram substring scales. <s>Conjecture: All ternary parallelogram substring scales are MOS substitution scales.</s> (Numerous counterexamples, e.g. LLmLmLmLmLLs)
	# Classify all ternary full parallelogram scales (PS with full first and last rows).		# Classify all ternary full parallelogram scales (PS with full first and last rows).
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Mathematical definition */

2025-12-31T13:47:45Z

Mathematical definition

← Older revision		Revision as of 13:47, 31 December 2025
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	* <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully		* <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully
	* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a (nonempty) prefix of the last row		* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a (nonempty) prefix of the last row
	* '''v''' is called the ''row generator''.		* '''v''' is called the ''row generator''. (-'''v''' would also satisfy the definition.)

	This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.		This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.

Inthar: /* Mathematical definition */

2025-12-31T13:44:19Z

Mathematical definition

← Older revision		Revision as of 13:44, 31 December 2025
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	* <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully		* <math>\{\mathbf{a} + i\mathbf{v} + j\mathbf{w}\}_{(i,j) \in [n]_0 \times [m-2]_1}</math> is a (possibly empty) parallelogram where rows are traversed fully
	* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a (nonempty) prefix of the last row		* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a (nonempty) prefix of the last row
	* '''v''' ~~and~~ ''~~'w'~~'' ~~are the generator and offset~~		* '''v''' is called the ''row generator''.

	This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.		This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.

Inthar: /* Mathematical definition */

2025-12-31T13:38:53Z

Mathematical definition

← Older revision		Revision as of 13:38, 31 December 2025
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	* '''v''' and '''w''' are the generator and offset		* '''v''' and '''w''' are the generator and offset

	This concept generalizes in the obvious way to arbitrary ~~ranks~~ (where each (''n'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.		This concept generalizes in the obvious way to arbitrary rank ''d'' (where each (''d'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.

	== Ternary scales with this property ==		== Ternary scales with this property ==

Inthar: /* Mathematical definition */

2025-12-31T13:38:22Z

Mathematical definition

← Older revision		Revision as of 13:38, 31 December 2025
Line 16:		Line 16:
	* '''v''' and '''w''' are the generator and offset		* '''v''' and '''w''' are the generator and offset

	This concept generalizes in the obvious way to arbitrary ranks (where each (n - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.		This concept generalizes in the obvious way to arbitrary ranks (where each (''n'' - 1)-dimensional "hyperrow" is traversed lexicographically, and the first and last hyperrows must be a suffix resp. prefix of such a traversal). In this case the property is called the '''parallelotope substring property'''.

	== Ternary scales with this property ==		== Ternary scales with this property ==
	=== Examples ===		=== Examples ===