Parallelogram substring scale: Difference between revisions
| (7 intermediate revisions by the same user not shown) | |||
| Line 14: | Line 14: | ||
* <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''' | * '''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'''. | ||
A parallelogram substring scale with full first and last rows is a '''parallelogram scale'''. | |||
== Ternary scales with this property == | == Ternary scales with this property == | ||
=== 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 | ||
| Line 32: | Line 34: | ||
* 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. | ||
# 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]] | ||