Val: Difference between revisions
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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, both as in the general mathematical sense as well as [[mapping|the regular temperament sense]]: | ||
# A val can be thought of as a mapping | # A val can be thought of as a mapping with one row. Put another way, the rows of mappings 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). | ||
# Vals must have only integer entries (when expressed in the standard, non-weighted coordinate basis). | # 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]]. | # Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sum of [[Wikipedia:P-adic_order|p-adic valuations]]. | ||