Fungrim entry: f4fbb8

Symbol: ArgMinUnique —

\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)

— Unique location of minimum value

ArgMinUnique(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)

, represents the unique value

x \in S

such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

. This operation is only defined if such a unique value exists.

ArgMinUnique(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{arg\,min*}}\limits_{x \in S,\,P(x)} f(x)

, represents the unique value

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

. This operation is only defined if such a unique value exists.

ArgMinUnique(f(x, y), For(Tuple(x, y)), P(x, y)) represents the unique tuple

\left(x, y\right)

satisfying

P\!\left(x, y\right)

such that

f\!\left(x, y\right) = \mathop{\min}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

P\!\left(x, y\right)

P\!\left(x, y, z\right)

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMinUnique`	$\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)$	Unique location of minimum value
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function

Source code for this entry:

Entry(ID("f4fbb8"),
    SymbolDefinition(ArgMinUnique, ArgMinUnique(f(x), ForElement(x, S)), "Unique location of minimum value"),
    Description(SourceForm(ArgMinUnique(f(x), ForElement(x, S))), ", rendered", ArgMinUnique(f(x), ForElement(x, S)), ", ", "represents the unique value", Element(x, S), "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMinUnique(f(x), ForElement(x, S), P(x))), ", rendered", ArgMinUnique(f(x), ForElement(x, S), P(x)), ", ", "represents the unique value", Element(x, S), "satisfying", P(x), "and", "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), ". This operation is only defined if such a unique value exists."),
    Description(SourceForm(ArgMinUnique(f(x, y), For(Tuple(x, y)), P(x, y))), "represents the unique tuple", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Minimum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", the corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

Topics using this entry

Operators