Sample 13. Compositions I

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13. Compositions (Based on Conjunctions)
In math, the symbol`` °`` is commonly used to produce a function defined as the composition of two functions:`f` ° `g y` is defined as`f (g y)` .`` Normally, such composed functions are only defined to apply to a single scalar argument.
J provides compositions effected by five distinct conjunctions, as well as compositions effected by isolated sequences of verbs: hooks and forks, and longer trains formed from them. The five conjunctions are`& &. &: @` and`@:` ,`` the conjunctions`@` and`@:` being related in the same manner as`&` and`&:` .
The conjunction`&` is closest to the composition`` °`` used in math, being identical to it when used for two scalar (rank zero) functions to produce a function to be applied to a single scalar argument. However, it is also extended in two directions:

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` .

The conjunction`&.` applies only to verbs, and`f&.g` is equivalent to`f&g` except that the inverse of`g` is applied to the final result. For example:

       3 +&^. 4                      3 +&.^. 4
    2.48491                       12

For 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)`

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