J Labs

Mathematical Roots of J

(1 of 28) INTRODUCTION

This lab shows how mathematical ideas and notation were adopted (or adapted) in J, and why such adaptations were made.

(2 of 28) ASSIGNMENT AND COMPARISON

```ASSIGNMENT COMPARISON```

```ENGLISH x is 3 x less than 3 x equals 3```
```MATH Let x=3 x<3 x=3```
```C x=3 x<3 x==3```
```APL x←3 x<3 x=3```
```J x=:3 x<3 x=3```

```ENGLISH cube function```
```MATH cube(x) = ``` $x^3$
```APL ∇ z←cube X```
```[1] Z←X*3 ∇```
```J cube =: ^&3```

(3 of 28) CONSTANTS

```FUNCTION EXPRESSION CONSTANT```

```ENGLISH 3 plus 4 tenths```
```MATH 3 + 4/10 3.4```
```APL 3 + 4÷10 3.4```
```J 3 + 4%10 3.4```

```ENGLISH 3 plus 4 imaginary```
```MATH 3 + 4i```
```APL 3 + 4×¯1*0.5 3J4```
```J 3 + 4*_1^0.5 3j4```

```ENGLISH Eulers Number```
```MATH e```
```APL *1```
```J ^1 1x1```

```ENGLISH Pi```
```MATH ``` $\pi$
```APL ∘1```
```J o.1 1p1```
```)```

(4 of 28) CONSTANT FUNCTIONS

```Function 1st derivative 2nd der 3rd der```

```MATH``` $x^3$ $3x^2$ $6x$ $6$

```J ^&3 3*^&2 6*^&1 6```



In [ ]:

    
y=: 4



In [ ]:

    
! y



In [ ]:

    
6:y



In [ ]:

    
6"_ y



In [ ]:

    
4 5 6"_ y



In [ ]:

    
'abcd'"_ y

(5 of 28) OPERATORS

ENGLISH An operator applies to a function, or functions, to produce a new function, much as an adverb applies to a verb to produce a new verb. We will illustrate by compositions.

```MATH``` $f \circ g$

```J f&g f&.g f@g```
```And Dual Atop```



In [ ]:

    
^. 10 2      NB. Natural logs



In [ ]:

    
10 -&^. 2    NB. Diff of logs



In [ ]:

    
10 -&.^. 2   NB. Dual of - wrt log



In [ ]:

    
^ 10 -&^. 2  NB. Inv log of diff



In [ ]:

    
10 ^. 2      NB. Base 10 log of 2



In [ ]:

    
10 -@^. 2    NB. - of base10 log



In [ ]:

    
x=. 3 1 4 1 5 9



In [ ]:

    
|. x         NB. Reverse



In [ ]:

    
+/\ & |. x   NB. Partial sums of reverse



In [ ]:

    
+/\ &. |. x  NB. Dual of partial sums wrt reverse



In [ ]:

    
|. +/\ |. x

(6 of 28) AMBIVALENCE

In math, the expression a-b denotes subtraction, and the expression -b denotes negation. We therefore say that the function denoted by - is ambivalent, its use as subtraction or negation being determined by context. A few of the ambivalent primitives in J are:

```MONADIC DYADIC```

```+ Conjugate Plus```
```- Negation Subtraction```
```* Signum Times```
```% Reciprocal Divide```
```^ Exponential Power```

In J, all functions are ambivalent, including those derived from other functions by the application of operators. For example:



In [ ]:

    
a=: 0 1 2 3 4



In [ ]:

    
+/ a       NB. Plus over a (sum)



In [ ]:

    
a +/ a     NB. Plus table

(7 of 28) FORK

```ENGLISH Sum of functions f and g +```
```/ \```
```f g```
```| |```
```x x```

```MATH f + g```

```J f + g```



In [ ]:

    
h=: ! + *:



In [ ]:

    
h 4



In [ ]:

    
d=: 2 3 4



In [ ]:

    
h d



In [ ]:

    
q=: ! , *:



In [ ]:

    
q 4



In [ ]:

    
q d



In [ ]:

    
mean=: +/ % #



In [ ]:

    
center=: ] - mean



In [ ]:

    
(] ; mean ; center ; ] - +/ % #) d

(8 of 28) ARRAYS, CELLS, and ITEMS



In [ ]:

    
[a=: i.2 3 4



In [ ]:

    
<"2 a           NB. Rank 2 cell, 2-cell, Items of 3-cell



In [ ]:

    
<"1 a           NB. 1-cells, Items of 2-cells



In [ ]:

    
<"0 a           NB. 0-cells, Atoms

(9 of 28) ARRAYS, CELLS, and ITEMS (ctd)



In [ ]:

    
[a=: i.2 3 4



In [ ]:

    
<"_1 a          NB. _1 cells of a, Items of a



In [ ]:

    
v=: 0 1 2 3 4 5



In [ ]:

    
<"_1 v          NB. _1 cells of v, Items of v

(10 of 28) FUNCTION RANK



In [ ]:

    
d=: 2 3 4



In [ ]:

    
a=: d $ 'ABCDEFGHIJKLMNOPQRSTUVWX'



In [ ]:

    
(] ; |. ; |."3 ; |."2 ; |."1 ; |."0) a



In [ ]:

    
i. d



In [ ]:

    
i."0 d

(11 of 28) FUNCTION TABLES

ENGLISH Table of binomial coefficients; column c of row r gives the number of ways of choosing c things from r.

```MATH Pascals Triangle```

```1```
```1 1```
```1 2 1```
```1 3 3 1```
```1 4 6 4 1```
```1 5 10 10 5 1```



In [ ]:

    
i=: 0 1 2 3 4 5



In [ ]:

    
bc=: i !~/i



In [ ]:

    
bc ; (%. bc); (+/"1 bc)



In [ ]:

    
rou=: ^@(0j2p1"_ * i. % ])    NB. roots of unity



In [ ]:

    
r=: rou 7



In [ ]:

    
r



In [ ]:

    
5 5 {.*/~ r         NB. multiplication table (first 5 rows, cols)



In [ ]:

    
r i. */~ r



In [ ]:

    
7&|@+/~ i.7   NB. addition modulo 7

(12 of 28) FUNCTION POWERS

ENGLISH Applying a function n times is called the nth power of the function

```MATH``` $x_0 = x$ $x_1 = f(x_0)$ $x_n = f(x_{n-1})$

```J f^:n```



In [ ]:

    
x=: 0



In [ ]:

    
cos=: 2&o.



In [ ]:

    
cos 0



In [ ]:

    
cos cos 0



In [ ]:

    
cos ^: 0 1 2 3 4 x



In [ ]:

    
y=: cos ^: _ x



In [ ]:

    
y



In [ ]:

    
y = cos y



In [ ]:

    
]z=: cos ^: _1 x=: _0.5 0 0.5 0.75 1



In [ ]:

    
cos z

(13 of 28) FUNCTION POWERS (ctd)



In [ ]:

    
f=: -:@(+ 1000&%)    NB. halve of x plus 1000 divided by x



In [ ]:

    
f 1



In [ ]:

    
f f 1



In [ ]:

    
f^:(i.2 5) 1



In [ ]:

    
] y=: f^:(_) 1



In [ ]:

    
y*y



In [ ]:

    
1000-y*y

(14 of 28) FUNCTION POWERS (ctd)



In [ ]:

    
SG=: 1 : '~.@(, ,/@(x/~))^:_'   NB. subgroup



In [ ]:

    
f=: 7&|@*      NB. multiplication modulo 7



In [ ]:

    
f SG 2         NB. subgroup generated by 4



In [ ]:

    
f/~ f SG 2     NB. group table of subgroup



In [ ]:

    
f SG 3         NB. subgroup generated by 3



In [ ]:

    
f SG 2 3       NB. subgroup generated by 2 and 3



In [ ]:

    
p=: (1|.i.5),: (<0 1) C. i.5



In [ ]:

    
p              NB. two permutations



In [ ]:

    
$ {"1 SG p     NB. subgroup generated by the two permutations

(15 of 28) FUNCTION POWERS (ctd)

f^:g y is equivalent to f^:(g y) y In particular, if g is a proposition, f^:g y applies f to y or not according to whether g y is true.



In [ ]:

    
f=: |.@(|/\)



In [ ]:

    
g=: *@{.



In [ ]:

    
f^:g y=: 32 44



In [ ]:

    
f^:g^:(i.5) y



In [ ]:

    
f^:g^:_ y



In [ ]:

    
+./y



In [ ]:

    
f=: {: ,: {. - {: * <.@%&{./



In [ ]:

    
g=: *@{.@{:



In [ ]:

    
f^:g^:_ y,.=i.2



In [ ]:

    
_4 3 +/ .* y   NB. GCD as a linear combination of original arg



In [ ]:

    
<"2 f^:g^:(i.6) y,.=i.2

(16 of 28) FUNCTION POWERS (ctd)

ENGLISH A function that "undoes" the effect of a function

```MATH``` $f^{-1}$

```J f^:```_```1```



In [ ]:

    
I=: ^:_1       NB. Inverse operator



In [ ]:

    
cos I 0



In [ ]:

    
^I 1x1 1x2     NB. Natural log



In [ ]:

    
^ ^I 1x1 1x2   NB. Exponential of log



In [ ]:

    
cube=: ^&3



In [ ]:

    
cube I 27 64 100

(17 of 28) FUNCTION DUALS

ENGLISH If a function f is applied to the result of a function g, and the inverse of g is then applied, the entire process is a function called the dual of f with respect to g

```MATH``` $g^{-1} f g$

```J f &. g```



In [ ]:

    
'a b'=: 0 0 1 1 ; 0 1 0 1



In [ ]:

    
a *. b     NB. a and b



In [ ]:

    
-.a        NB. Not a



In [ ]:

    
a *.&.-. b NB. Dual of and wrt not



In [ ]:

    
a +. b     NB. Is equivalent to or



In [ ]:

    
3 +&.^. 4



In [ ]:

    
3 * 4



In [ ]:

    
log=: 10&^.



In [ ]:

    
log 3 4 10 100



In [ ]:

    
3 +&.log 4

(18 of 28) FUNCTION FAMILIES

Many benefits resulted from uniting the treatment of the individual functions square, cube, square root, etc. in a single family under the notation x superscript n Similar benefits result from using complex numbers to unite the treatment of the trigonometric and hyperbolic functions under the exponential.

We will illustrate the matter by using the fit operator (!.) to unite the treatment of the falling and rising factorial functions under the power function:



In [ ]:

    
'x e s' =: 6 4 1



In [ ]:

    
(x+0*s)*(x+1*s)*(x+2*s)*(x+3*s)   NB. Rising



In [ ]:

    
s=: _1



In [ ]:

    
(x+0*s)*(x+1*s)*(x+2*s)*(x+3*s)   NB. Falling



In [ ]:

    
s=: 0



In [ ]:

    
(x+0*s)*(x+1*s)*(x+2*s)*(x+3*s)   NB. Power



In [ ]:

    
x ^ e



In [ ]:

    
x ^!.1 e     NB. Rising  factorial



In [ ]:

    
x ^!._1 e    NB. Falling factorial



In [ ]:

    
x ^!.0 e     NB. Power

(19 of 28) FUNCTION FAMILIES (ctd)

Function tables for ^!.e can be used to make clear the relationship between Stirling numbers of the first kind and Stirling numbers of the second kind.



In [ ]:

    
FT=: 1 : '^!.x/~@i.'   NB. falling/rising factorial table



In [ ]:

    
_1 FT 6                 NB. falling factorial table



In [ ]:

    
0 FT 6                  NB. power table



In [ ]:

    
]S2=:(0 FT %. _1 FT) 6  NB. Matrix quotient of above tables



In [ ]:

    
]S1=:(_1 FT %. 0 FT) 6  NB. Matrix inverse of S2

|: S2 are the Stirling numbers of the second kind |: |S1 are the Stirling numbers of the first kind

(20 of 28) POLYNOMIALS

A polynomial is so named because it can be expressed as a sum of monomials of the form $cx^k$.

It may also be expressed as a weighted sum of factorial polynomials, or as a (multiple of a) product of factors of the form x-r. We will illustrate some transformations between the coefficients that weight the monomials, and the roots that represent the corresponding product form:



In [ ]:

    
'c x y'=: _3.75 11.5 _9 2 ; 5 ; 0 1 2 3 4



In [ ]:

    
c p. x



In [ ]:

    
+/ c * x ^ 0 1 2 3



In [ ]:

    
c p. y



In [ ]:

    
]r=: p. c



In [ ]:

    
r p. x



In [ ]:

    
p. r

(21 of 28) TAYLOR COEFFICIENTS

The Taylor coefficient operator t. produces the coefficients of a polynomial function to which it is applied. More generally, it produces the coefficients of a polynomial approximation to other functions, such as the exponential, the trigonometric functions, and rational functions, i.e. ratios of two polynomials. Thus:



In [ ]:

    
f=: 3 1 4 2 & p.



In [ ]:

    
'x i'=: 0 1 2 3 4 ; 0 1 2 3 4 5



In [ ]:

    
f x



In [ ]:

    
f t. i



In [ ]:

    
^ t. i



In [ ]:

    
(^ t. i) p. x



In [ ]:

    
^ x



In [ ]:

    
% ^ t. i



In [ ]:

    
^ t: i      NB. weighted Taylor coefficients for exponential



In [ ]:

    
1&o. t: i   NB. sine



In [ ]:

    
5&o. t: i   NB. hyperbolic sine

(22 of 28) TAYLOR COEFFICIENTS (ctd)



In [ ]:

    
p=: 4 _3 1 2&p.



In [ ]:

    
q=: _1 5 1&p.



In [ ]:

    
(p+q) t. i.10



In [ ]:

    
(p-q) t. i.10



In [ ]:

    
(q-p) t. i.10



In [ ]:

    
(p*q) t. i.10



In [ ]:

    
(p%q) t. i.10



In [ ]:

    
(q%p) t. i.10



In [ ]:

    
p@q t. i.10



In [ ]:

    
q@p t. i.10



In [ ]:

    
p@>: t. i.10



In [ ]:

    
p@<: t. i.10



In [ ]:

    
(0 1&p. % 1 _1 _1&p.) t. i.15



In [ ]:

    
(% -. - *:) t. i.15

(23 of 28) PERMUTATIONS

Permutations may be represented in a variety of ways, the simplest being by a vector of indices. Since the permutation vectors of any given order can be arranged in a table in lexical order, a specific permutation can also be referred to compactly by its index in the table:



In [ ]:

    
]rp=: 8 ?. 8      NB. Random permutation



In [ ]:

    
rp { 'ABCDEFGH'



In [ ]:

    
(i. ! 3) A. i. 3  NB. All !3 of order 3



In [ ]:

    
A. 1 0 2          NB. Index of perm 1 0 2



In [ ]:

    
]p=: 20 ?. 20



In [ ]:

    
A. p              NB. Extended precision



In [ ]:

    
]cy=: C. rp       NB. Cycle representation



In [ ]:

    
C. cy             NB. C. is self-inverse

(24 of 28) PERMUTATIONS (ctd)



In [ ]:

    
SG=: 1 : '~.@(, ,/@(x/~))^:_'



In [ ]:

    
] p=. ?.~ 22        NB. a random permutation of order 22



In [ ]:

    
g=.{"1 SG ,: p      NB. subgroup generated by p



In [ ]:

    
#g



In [ ]:

    
C. p                NB. cycles of p



In [ ]:

    
*./ #&> C. p        NB. LCM of cycle lengths

(25 of 28) COMPLETION

In mathematics it is common to complete a function by extending it beyond its original domain in ways that preserve its main properties, often leading to significant generalization, as in the extension of the square and cube to the form $x^n$ for all values of n

In APL such completion occurred in many cases, as in reduction over an empty, to extend to the case k=0 the identity (f/x)=(f/k↑x) f (f/k↓x), and in the definition of $0^0$ as 1, which is often declared to be undefined in elementary math texts.



In [ ]:

    
s=: _3 _2 _1 0 1 2 3



In [ ]:

    
s % table s



In [ ]:

    
^table~ -:s

(26 of 28) COMPLETION (ctd)

Completions in J fall in four classes:

Specific functions such as power and divide, and operators such as the application of scan to monadic rather than to dyadic functions as in APL.
The variant or fit operator, used in the extension of the power function to the rising factorial function.
Completion of user-defined functions by specified identity elements, inverses, derivatives, and integrals.
Extensions of conformability rules.



In [ ]:

    
0 1 (+./  ; *./ ; >./ ; <./) 0 1



In [ ]:

    
(+./  ; *./ ; >./ ; <./) ''



In [ ]:

    
+/\a=. 0 1 2 3 4



In [ ]:

    
<\a

(27 of 28) PRIMES AND FACTORING



In [ ]:

    
p: 0        NB. Prime from its index



In [ ]:

    
p: 0 1 2 3 4 5 6



In [ ]:

    
]n=: ?. 100000



In [ ]:

    
pi=: p:^:_1 NB. # of primes < or =



In [ ]:

    
]m=: pi n



In [ ]:

    
p: m - 1 0  NB. Results bracket n



In [ ]:

    
q: 700



In [ ]:

    
pi 7



In [ ]:

    
p: i. 4



In [ ]:

    
]exp=: 4 q: 700



In [ ]:

    
(p: i.4)^exp



In [ ]:

    
*/(p: i.4)^exp

(28 of 28) PRIMES AND FACTORING (ctd)



In [ ]:

    
m=. 63



In [ ]:

    
]e=. _ q: m



In [ ]:

    
p: i.#e



In [ ]:

    
]d=. (#: i.@(*/)) 1+e



In [ ]:

    
d */ .(^~) p:i.#e     NB. all factors



In [ ]:

    
n=. 182



In [ ]:

    
_ q: m,n



In [ ]:

    
(m*n), +/&.(_&q:) m,n



In [ ]:

    
(m*.n), >./&.(_&q:) m,n



In [ ]:

    
(m+.n), <./&.(_&q:) m,n



In [ ]:

    
totient=: * -.@%@~.&.q:



In [ ]:

    
(totient m), +/1=m+.i.m

End of Lab



In [ ]: