Essays/Under

The conjunction under f&.g is defined as

  f&.g y $\leftrightarrow$ gi f g y
x f&.g y $\leftrightarrow$ 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:

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• 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
    +/&.(*:"_) $L_{2}$ norm
    +/&.(^&p) $L_{p}$ 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: sum of divisors
    ~.&.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 $M=M_{1}M_{2}\dots M_{n}$
    x=: +/ .*
    x/ -: x/&.(|:"2"_)@|. i.e. $M^{T}=M_{n}^{T}M_{n-1}^{T}\dots M_{1}^{T}$
    x/ -: x/&.(%."_)@|. i.e. $M^{-1}=M_{n}^{-1}M_{n-1}^{-1}\dots M_{1}^{-1}$
  [try e.g.  (x/ ; x/&.(|:"2"_)@|. ; x/&.(%."_)@|.) ?.5 2 2$0 ]
• Round to p decimal places
    ([: <. +&0.5) &. (*&(10^p))
    ] &. ((j.p)&":)
  
• Decimal digits of an integer: ,.&.":
• 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=:13)
    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 ${1 \over \phi }e^{x/2}\,{\sinh \phi x}$ . 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

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Contributed by Roger Hui, with further contributions by Raul Miller, Ewart Shaw, and David Lambert.