1. | Applied to one verb and one noun it produces a monadic function illustrated by the cases 10&^. (Base ten logarithm) and ^&3 (Cube). |
2. | Applied to two verbs it produces (in addition to the monadic case used in math) a dyadic case defined by: x f&g y ↔ (g x) f (g y) . For example, x %&! y is the quotient of the factorials of x and y . |
3 +&^. 4 3 +&.^. 4 2.48491 12For scalar arguments the functions f&:g and f&g are equivalent, but for more general arguments, g applies to each cell as dictated by its ranks. In the case of f&g, the function f then applies to each result produced; in the case of f&:g it applies to the overall result of all of the cells. For example:
(] ; %. ; |:&%. ; |:&:%.) i. 2 2 2 +---+--------+-------+---------+ |0 1|_1.5 0.5|_1.5 1|_1.5 _3.5| |2 3| 1 0| 0.5 0| 1 3| | | | | | |4 5|_3.5 2.5|_3.5 3| 0.5 2.5| |6 7| 3 _2| 2.5 _2| 0 _2| +---+--------+-------+---------+The conjunctions @ and & agree in the monadic case, as indicated below for cells x and y as dictated by the ranks of g :
f&g y ↔ f g y
f@g y ↔ f g y
x f&g y ↔ (g x) f (g y)
x f@g y ↔ f (x g y)