Essays/Under
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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: 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
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)&":) - 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 . 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 Roger Hui, with further contributions by Raul Miller, Ewart Shaw, and David Lambert.