The conjunction under f&.g is defined as

f&.g y  gi f g y
x f&.g y  gi (g x) f (g y)

where gi is the inverse of g . "Under" elucidates the important but often mysterious concept of duality in mathematics.

The "under anaesthetics" example provides a graphic illustration. Several steps are composed:

apply anaesthetics
cut open
do procedure
sew up
wake up from anaesthetics

The inverse steps are pretty important! The "pipe laying" example provides another illustration: dig a trench, lay the pipe, cover the trench. Finally, a more poetic example: ashes to ashes, dust to dust.

Some examples of "under" in J:

• each:  f&.>

• Logic (de Morgan's Laws)
+.&.-.                       and
*.&.-.                       or

• Arithmetic
+&.^.                        times
+&.(10&^.)                   times
-&.^.                        reciprocal, divide
+:&.^.                       square
-:&.^.                       square root
*&.^                         plus
%&.^                         minus
>.&.-                        floor, minimum
<.&.-                        ceiling, maximum
,&0&.#:                      double
}:&.#:                       integer quotient of division by 2
+/\. -: +/\&.|.
x&<.&.(+/\) y                usage in reservoirs y to a maximum of x; see J Forum msg
>:&.%                        gives x%x+y for a ratio x%y

• Trigonometric identities
|@sin -: -.&.*:@cos
|@cos -: -.&.*:@sin
sin   -: sinh&.j.
tan   -: tanh&.j.
sinh  -: sin&.j.
tanh  -: tan&.j.
sin   -: ^ .: - &.j.

• Geometry
-.&.*:                       length of opposite from adjacent when hypotenuse is 1
+&.*:                        diagonal from rectangle sides
+/&.(*:"_)                   norm
+/&.(^&p)                    norm

• Primes and factoring
i.&.(p:^:_1)                 all the primes less than a number
<:&.(p:^:_1)                 the largest prime less than a number
[&.(p:^:_1)                  y or next prime
*/@(i.&.(p:^:_1))@>:         primorial (see Prime APVs)
+:&.(_&q:)                   square
-:&.(_&q:)                   square root
<./&.(_&q:)@,                GCD
>./&.(_&q:)@,                LCM
+ /&.(_&q:)@,                times
- /&.(_&q:)@,                divide
(- ~:)&.q:                   Euler's totient function
* -.@%@~.&.q:                Euler's totient function
~.&.q:                       the square-free part of a number
>:&.(q:^:_1)                 demonstration that there is no largest prime

• Various means
am=: +/ % #                  arithmetic mean
am&.:^.                      geometric mean
am&.:%                       harmonic mean
am&.:*:                      root mean square

• Arithmetic on sequences of bits:
+&.#.
-&.#.
*&.#.
<.@%&.#.

• Matrix algebra
%. -: %.&.|:                 real matrices
%. -: %.&.(+@|:)             complex matrices
%. -: %.&.(M&(+/ .*))        is an invertible matrix

• Identities for matrix products
x=: +/ .*
x/ -: x/&.(|:"2"_)@|.        i.e.
x/ -: x/&.(%."_)@|.          i.e.
[try e.g.  (x/ ; x/&.(|:"2"_)@|. ; x/&.(%."_)@|.) ?.5 2 2\$0  ]

• Round to p decimal places
([: <. +&0.5) &. (*&(10^p))
] &. ((j.p)&":)

• Reverse bits and digits
|.&.(10&#.^:_1)              reversing base 10 digits
|.&.":                       reversing base 10 digits
1&|.&.#:                     survivor number in the Josephus problem
/:~&.(|."1@#:)               arrange a list of distinct positive integers so that no average is spanned

• Reverse the words of a sentence: |.&.;:

• Caesar cipher (Julius Caesar used n=:3)
A=: 'abcdefghijklmnopqrstuvwxyz'
(#A)&|@(+&n) &. (A&i.)       encrypt
(#A)&|@(-&n) &. (A&i.)       decrypt

• Operate on text as integer:  'ibm' -: >:&.:(a.&i.)'hal'

• Extend verb domain: =&0`(0 ,:~ %)}&.,: under itemize succeeds with scalar argument

• The e.g.f. of the Fibonacci sequence is . Thus:  (^@-: * 5&o.&.((-:%:5)&*)) t:

• ack is Ackermann's Function. If x ack y is f&.(3&+) y , then (x+1) ack y is f^:(1+y)&.(3&+) 1

• The next Gray code word:  >: &. #. &. (~:/\)

• n-th Chebyshev polynomials
acos=: _2&o.
n&*&.acos                    the first kind
(n+1)&*&.acos %&(-.&.*:) ]   absolute value of the second kind

• The square-free part of a polynomial:  ({. , ~.&.>@{:)&.p.

• The inverse of a permutation
|:&.({=)
%.&.({=)
|.&.>&.C.

• The next/previous permutation
rfd=: +/@(<{.)\."1            reduced from direct, from Permutation Index
dfr=: /:@/:@,/"1              direct from reduced
b=: (-i.)# p
>:&.(b&#.)&.(rfd :. dfr) p    the next    permutation
<:&.(b&#.)&.(rfd :. dfr) p    the previous permutation

• Fast Fourier Transform
+//.@(*/) = *&.FFT            polynomial multiplication on arguments of length 2^n

Contributed by RogerHui, with further contributions by Raul Miller, EwartShaw, and DavidLambert.

Essays/Under (last edited 2013-10-11 04:38:09 by DavidLambert)