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2D. Adverbs & Conjunctions
Adverbs from Conjunctions
A conjunction together with one of its arguments produces an adverb defined in an obvious way. For example, if a1=: &3 ,then ^ a1 is equivalent to ^&3 3 (the cube function), and if a2=: 3& then ^ a2 is equivalent to 3&^ (the three-to-the-power function). It is therefore easy to define useful families of adverbs from a conjunction, so easy that it is fruitless to attempt an exhaustive catalogue. The following list is intended to suggest the possibilities in various classes:
| a0=: I=: ^:_1 | Inverse (^I is ^.) |
| a1=: L=: ^:_ | Limit (2&o.L 1 for soln of y=cos y) |
| a2=: LI=: ^:__ | Limit of inverse |
| a3=: SQ=: ^:2 | Square (1&o.SQ for sine squared) |
| a4=: C=: &o. | Family of circular fns (3 C is tangent) |
| a5=: CO=: %@C | 3 CO is cotangent |
| m6=: rfd=: 1r180p1&* | Radians from degrees |
| m7=: dfr=: rfd I | Use dfr=: dfr f. to fix definition |
| a8=: D=: @rfd | Try 1 C D 0 30 45 60 90 180 |
| m9=: SIN=: 1&o. D | Sine for degree arguments |
| a10=: T=: "2 | Try <T I. 2 3 4 3 (BOX TABLES) |
| a11=: S=: ^!. | Stope (rising or falling factorial fn etc) |
| a12=: P=: p.!. | Stope polynomial |
| a13=: FILL=: |.!. | Fill for shift (non-cyclic rotate) |
| a14=: FILE=: 1!: | File functions (1 FILE for read, etc.) |
Explicit Definitions
An adverb or a conjunction can be defined in explicit form. For example:
split=: 2 : ',.@(x@(y&{.) ; x@(y&}.))'
]x=: i. 5 3
0 1 2
3 4 5
6 7 8
9 10 11
12 13 14
(+: split 2 ,. |. split 3 ,. +/ split 2) x
+--------+--------+--------+
|0 2 4 |6 7 8 |3 5 7 |
|6 8 10 |3 4 5 | |
| |0 1 2 | |
+--------+--------+--------+
|12 14 16|12 13 14|27 30 33|
|18 20 22| 9 10 11| |
|24 26 28| | |
+--------+--------+--------+
| c15=: split=: 2 : ',.@(x@(y&{.) ; x@(y&}.))' | split as defined above |
| d16=: by=: ' '&;@,.@[,.] | Verb for use in the table adverb below |
| d17=: over=: ({.;}.)@":@, | Verb for use in the table adverb below |
| a18=: table=: 1 :'[ by ]over x/' | Try 1 2 3 * table 4 5 6 7 |
Noun Arguments
Adverbs that apply to a noun argument, and conjunctions that apply to one noun argument and one verb argument are commonplace. For example:
x=: 0 0 1 1 [ y=: 0 1 0 1 x *. y NB. Boolean and 0 0 0 1 x 1 b. y NB. Boolean adverb 0 0 0 1 x (i.16) b. y NB. All sixteen Boolean functions 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 C=: 1 : 'x&o.' NB. Circle adverb 1 C 0 1r4p1 1r3p1 1r2p1 1p1 NB. Sine function 0 0.7071068 0.8660254 1 0 ^&3 x=: i. 6 NB. Cube 0 1 8 27 64 125 2 |. !. 1 x NB. Shift in ones 2 3 4 5 1 1
Conjunctions that apply to two nouns are less familiar, although the definitions of functions in terms of nouns occur frequently in math. For example, a rational function (the quotient of a polynomial a&p. divided by another b&p.) is defined by a pair of coefficients. Thus:
a=: 1 4 6 4 1 [ b=: 1 2 1 RAT=: 2 : 'x&p. % y&p.' a RAT b 1 4 6 4 1&p. % 1 2 1&p. a RAT b y=: i.6 1 4 9 16 25 36 b RAT a y 1 0.25 0.111111 0.0625 0.04 0.0277778 (a RAT b * b RAT a) y 1 1 1 1 1 1
We may also remark that expressions such as 2 x3 and 2 x3 + 4 x2 are commonly used in elementary math to define functions rather than to indicate explicit computation: the x in the foregoing can be construed (and defined) as a conjunction such that 2 x 3 is the function 2:*]^3: . Thus:
x=: 2 : 'x&* @ (^&y) " 0' 2 x 3 2&*@(^&3)"0 2 x 3 y=: 0 1 2 3 4 5 0 2 16 54 128 250 2 * y ^ 3 0 2 16 54 128 250 2 3 5 x 1 2 4 y 0 0 0 2 3 5 4 12 80 6 27 405 8 48 1280 10 75 3125
The last result above gave the values of the individual terms; in order to obtain their sums (and yet retain the behaviour for a single term), we redefine the conjunction x as follows:
x=: 2 : '+/ @ (x&* @ (^&y)) " 0' 2 3 5 x 1 2 4 y 0 10 96 438 1336 3210 (2 x 1 + 3 x 2 + 5 x 4) y 0 10 96 438 1336 3210 2 x 3 y 0 2 16 54 128 250
| c19=: RAT=: 2 : 'x&p. % y&p.' | Produces rational function |
| c20=: x=: 2 : '+/ @ (x&* @ (^&y)) " 0' | Mimics notation of elementary math |
| c21=: bind=: 2 : 'x @ (y"_)' | Binds y to the monad x |
It is often convenient to bind an argument to a monad, producing a function that ignores its argument. For example, using wdinfo , a monad that displays its argument in a message box, the definition fini=: wdinfo bind 'Job Finished' produces a function such that fini is equivalent to wdinfo 'Job Finished' .