Val: Difference between revisions

Line 64:

== Vals vs. mappings ==

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

A val is more specific than a [[mapping]]. In particular, the rows of mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals. To be precise:

# It is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This 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 1D 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). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.

# A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This 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). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.

# It has only integer entries.

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

~~# It is [[Wikipedia:Homomorphism|homomorphic]]. A val is a map from every positive rational number to an associated integer, usually restricted to a given prime limit.~~

In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.

@@ Line 64: / Line 64: @@
 == Vals vs. mappings ==
-A val is more specific than a [[mapping]]:
+A val is more specific than a [[mapping]]. In particular, the rows of mapping matrices are vals, so that mapping matrices can be thought of as being built up from vals. To be precise:
-# It is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This 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 1D 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). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.
+# A val is a specific type of [[Wikipedia:Linear_map|(linear) mapping]], called a [[Wikipedia:Linear_form|"linear form", or "linear functional"]]. This 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). Or, if interpreted as a [[Wikipedia:Matrix_(mathematics)|matrix]], a val has only one row.
 # It has only integer entries.
 # Being short for "[[Wikipedia:Valuation_(algebra)|valuation]]", a val is a formal linear sums of [[Wikipedia:P-adic_order|p-adic valuations]].
-# It is [[Wikipedia:Homomorphism|homomorphic]]. A val is a map from every positive rational number to an associated integer, usually restricted to a given prime limit.
 In practice, most single-row mappings in RTT are vals, because we usually deal with integer entries, and the other specifications only mean anything to advanced mathematicians.