Val: Difference between revisions

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== Vals vs. mappings ==

A val is more specific than a ~~mapping, both as in the general mathematical sense as well as~~ [[mapping~~|the regular temperament sense~~]]:

A val is more specific than a [[mapping]]:

# A val ~~can be thought~~ of as a mapping ~~(in the general mathematical sense)~~ with ~~one~~ row~~. Put another way, the rows of mappings (in the regular temperament sense) are vals~~. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) ~~mapping]]~~ called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).

# A val is either a single row of a mapping, or a mapping with a single row. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|''linear'' function]] (which is also called a "linear mapping" or "linear map", because in general mathematics, "function", "mapping", and "map" are synonymous) called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).

# Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).

# Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]].

@@ Line 64: / Line 64: @@
 == Vals vs. mappings ==
-A val is more specific than a mapping, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]:
+A val is more specific than a [[mapping]]:
-# A val can be thought of as a mapping (in the general mathematical sense) with one row. Put another way, the rows of mappings (in the regular temperament sense) are vals. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]] called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).
+# A val is either a single row of a mapping, or a mapping with a single row. To be mathematically precise, a val is a specific type of [[Wikipedia:Linear_map|''linear'' function]] (which is also called a "linear mapping" or "linear map", because in general mathematics, "function", "mapping", and "map" are synonymous) called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]], which means that its output is a [[Wikipedia:Scalar_(mathematics)|scalar]], or in other words, a single number. This corresponds to the fact that a val must be a 1xM array of numbers, or in other words a [[Wikipedia:Vector_(mathematics_and_physics)|vector]] (specifically a [[Wikipedia:Row_and_column_vectors|row vector]], AKA covector).
 # Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis).
 # Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]].