Essays/Under

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

• 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
    */@(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&(+/ .*))        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
  

• 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.