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[x] u&.:v y

Under

Conjunction

Rank Infinity -- operates on [x and] y as a whole -- WHY IS THIS IMPORTANT?

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Executes v on the argument(s); then executes u on the result(s) of v; then executes  v^:_1 (i.e. the obverse of v) on the result of u, giving the end result. We say that u is applied under v.

u&.:v is equivalent to v^:_1@:(u&:v). That is,

  u&.:v y  <-->        v^:_1 u (v y)
x u&.:v y  <-->  v^:_1 (v x) u (v y) 

Example: Addition (+) applied under logarithm (^.) gives multiplication

   u =: +
   v =: ^.
   3 4  u&.:v  5 1
15 4

Under (Dual) (&.) happens to give the same answer as (&.:) here. But this is only because (+) and (^.) both have rank 0, which ensures (say) that  3 4 ^. 5 1 is the same as  (3 ^. 5) , (4 ^. 1)

   3 4  u&.v  5 1
15 4

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Common Uses

1. Compute the Geometric Mean as: the Arithmetic Mean taken in the "log-domain"

   y =: 2 4 8 16

   mean =: +/ % #         NB. The Arithmetic Mean of y
   mean y
7.5

   ^ mean ^. y            NB. The Geometric Mean: log (^.), then arithmetic mean, then antilog (^)
5.65685
   mean&.:^. y            NB. ditto, but done using (&.:)  NOTE: (&.) DOES NOT WORK
5.65685

Note: Under (Dual) (&.) does not give the same answer as (&.:)

   mean&.:^. y
5.65685
   mean&.^. y
2 4 8 16

Contrast this with the above Example of addition under logarithm.

This is because mean does not have rank 0, it has rank Infinity

   mean b.0
_ _ _
   (mean&.^.) b.0
0 0 0

See below: More Information (1.)

2. Compute standard deviation as the root-mean-square of a list of differences from the mean value

   y =: 0 _3 1 2          NB. list of differences from the mean value

   u =: mean =: +/ % #    NB. mean value of y
   v =: *:                NB. square of y
   stddev =: u&.:v        NB. square root of mean of sum of squares of y
   stddev 0 _3 1 2
1.87083

3. Compute Vector Length (Euclidean Length) in Rⁿ

   u=. +/
   v=. *:
   y=. 6 3 2
   u &. v y        NB. u is acting on each list item of v y (that wasn't the intension)
6 3 2
   u &.: v y       NB. u is acting on the result of v y (the whole list)
7
   %: +/ *: y      NB. straight forward computing of the norm
7

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Semiduals x u&.:(a:`v) y and x u&.:(v`a:) y

The operation encapsulated by u&.:v is used often and is important as a notation to aid thought. When the operation is dyadic, sometimes the sequence of transforming/operating/transforming back should be applied to just one argument, with the other argument being passed unchanged into u. This can be represented by a gerund form where a: indicates which argument is to be used unmodified:

• x u&.:(a:`v) y is v^:_1 x u (v y)
• x u&.:(v`a:) y is v^:_1 (v x) u y

   1 +&.:(a:`^.) 2 3
5.43656 8.15485
   ^ 1 + ^. 2 3
5.43656 8.15485

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Related Primitives

Under (Dual) (&.)

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More Information

1. Under (Dual) (&.) differs from Under (&.:) in that  u&.v is defined as (u&.:v)"v; that is,  u&.v applies u under v cell-by-cell, whereas  u&.:v applies u under v to the entire argument(s) (x and) y.

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