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Two cases occur: a numeric integer n, and a
gerund n .
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Numeric case. The verb u (or x&u) is
applied n times. For example:
(] ; +/\ ; +/\^:2 ; +/\^:0 1 2 3 _1 _2 _3 _4) 1 2 3 4 5
+---------+-----------+------------+-------------+
|1 2 3 4 5|1 3 6 10 15|1 4 10 20 35|1 2 3 4 5|
| | | |1 3 6 10 15|
| | | |1 4 10 20 35|
| | | |1 5 15 35 70|
| | | |1 1 1 1 1|
| | | |1 0 0 0 0|
| | | |1 _1 0 0 0|
| | | |1 _2 1 0 0|
+---------+-----------+------------+-------------+
An infinite power n produces the limit of the application
of u . For example, if x=:2
and y=:1,
then x o.^:_ y is 0.73908, the solution of
the equation y=Cos y . If n is negative,
the obverse u^:_1 is applied |n times.
The obverse (which is normally the inverse) is specified for six cases:
1. The self-inverse functions + - -. % %. |. |: /: [ ] C. p.
2. The pairs in the following lists:
< <: +. +: +~ *. *: *~ ^ ,: ,~
> >: j./"1"_ -: -: r./"1"_ %: %: ^. {. <.@-:@# {. ]
;: #. ". ;~ 3!:1 3!:3 p: q:
;@(,&' '&.>"1) #: ": >@{. 3!:2 3!:2 pi(n) */
\: o. j. r.
/:@|. %&(o.1) %&0j1 %&0j1@^.
3. Obviously invertible bonded dyads such
as -&3 and 10&^.
and 1 0 2&|: and 3&|. and 1&o.
and a.&i. as well as u@v and u&v
if u and v are invertible.
4. Monads of the form v/\ and v/\. where v
is one of + * - % = ~:
5. Obverses specified by the conjunction :.
6. The following cases merit special mention:
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p:^:_1 n gives the number of primes less than n,
denoted by p(n) in math
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q:^:_1 is */
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#^:_1 with a boolean left argument is "Expand"
(whose fill atom f can be specified by
fit, b&#^:_1!.f)
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a&#.^:_1 produces the base-a representation
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!^:_1 and !&n^:_1 and !&n&^:_1
produce the appropriate results
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Gerund case. (Compare with the gerund case of the merge
adverb })
x u^:(v0`v1`v2)y « (x v0 y)u^:(x v1 y) (x v2 y)
x u^:( v1`v2)y « x u^:([` v1`v2) y
u^:( v1`v2)y « u^:(v1 y) (v2 y)
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