Vocabulary/dotdot

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u .. v Even Conjunction

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


u .: v Odd Conjunction

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



(u .. v) is the same as (u + u&v) % 2:

(u .: v) is the same as (u - u&v) % 2:

Important.png u .. v and u .: v are deprecated.


Replace the functions (seldom used) by their equivalent phrases above.
Future releases of J may reassign the words .. and .:


Common uses

1. Make a function out of u which is symmetrical about zero on the X-axis, using - for v

Even (..-) gives a symmetrical function, Odd (.:-) gives an antisymmetrical function.

   u=: ^   NB. exponential growth: sample unsymmetrical function

   ] X=: 5 %~ i:5
_1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1
   require 'plot'
   plot X; u X
Ux.jpg
   plot X; u ..- X
Uex.jpg
   plot X; u .:- X
Uox.jpg

2. Decompose a matrix into symmetric and antisymmetric parts, or Hermitian and aniHermitian parts, using |: for v

   ]a =. i. 3 3
0 1 2
3 4 5
6 7 8
   ]asymm =: ] .. |: a
0 2 4
2 4 6

4 6 8
   ]aantisymm =: ] .: |: a
0 _1 _2
1  0 _1
2  1  0
   asymm + aantisymm
0 1 2
3 4 5
6 7 8

For complex matrices let v be +@:|:, to take the adjoint.


Details

1. Any mathematical function can be uniquely decomposed as the sum of an even and an odd function. 1. Any matrix can be uniquely decomposed as the sum of a symmetric and an antisymmetric matrix. 1. Any matrix can be uniquely decomposed as the sum of a Hermitian and an antiHermitian matrix.