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:
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
plot X; u ..- X
plot X; u .:- X
2. Decompose a matrix into symmetric and antisymmetric parts, or Hermitian and antiHermitian 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
- . Any mathematical function can be uniquely decomposed as the sum of an even and an odd function.
- . Any matrix can be uniquely decomposed as the sum of a symmetric and an antisymmetric matrix.
- . Any matrix can be uniquely decomposed as the sum of a Hermitian and an antiHermitian matrix.