From c96289046dbd42678c10d03ceb5734737392bf4c Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Mon, 5 Jul 2021 22:20:19 -0400 Subject: Inter-documentation links and minor editing --- docs/doc/depth.html | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) (limited to 'docs/doc/depth.html') diff --git a/docs/doc/depth.html b/docs/doc/depth.html index 2332c674..4c5ca982 100644 --- a/docs/doc/depth.html +++ b/docs/doc/depth.html @@ -72,7 +72,7 @@ ≡ "a string is a list of characters" 1 -

Depth is somewhat analogous to an array's rank =𝕩, and in fact rank can be "converted" to depth by splitting rows with <⎉1, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:

+

Depth is somewhat analogous to an array's rank =𝕩, and in fact rank can be "converted" to depth by splitting rows with <⎉1, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:

↗️
    ≡ 3‿4⥊"characters"
 1
     ≡ (1+↕10)⥊"characters"
@@ -125,13 +125,13 @@
 
 
 
-
Functions such as Take and Drop use a single number per axis. When the left argument is a list of numbers, they apply to initial axes. But for convenience, a single number is also accepted, and applied to the first axis only. This is equivalent to ravelling the left argument before applying the function.

+
Functions such as Take and Drop use a single number per axis. When the left argument is a list of numbers, they apply to initial axes. But for convenience, a single number is also accepted, and applied to the first axis only. This is equivalent to deshaping the left argument before applying the function.

 ↗️
    ≢2↑7‿7‿7‿7⥊"abc"
 ⟨ 2 7 7 7 ⟩
     ≢2‿1‿1↑7‿7‿7‿7⥊"abc"
 ⟨ 2 1 1 7 ⟩
 

-
In these cases the flexibility seems trivial because the left argument has depth 1 or 0: it is an array or isn't, and it's obvious what a plain number should do. But for the second row in the table, the left argument is always an array. The general case is that the left argument is a vector and its elements correspond to right argument axes:

+
In these cases the flexibility seems trivial because the left argument has depth 1 or 0: it is an array or isn't, and it's obvious what a plain number should do. But for the second row in the table, the left argument is always an array. The general case (Select below) is that the left argument is a list and its elements correspond to right argument axes:

 ↗️
    ⟨3‿2,1‿4‿1⟩ ⊏ ↕6‿7
 ┌─                         
 ╵ ⟨ 3 1 ⟩ ⟨ 3 4 ⟩ ⟨ 3 1 ⟩  
@@ -142,7 +142,7 @@
 ↗️
    ⟨3‿2,1⟩ <⍟(0=≡)¨⊸⊏ ↕6‿7
 ⟨ ⟨ 3 1 ⟩ ⟨ 2 1 ⟩ ⟩
 

-
While very consistent, this extension represents a small convenience and makes it difficult to act on a single axis, which for Replicate and Group is probably the most common way the primitive is used:

+
While very consistent, this extension represents a small convenience and makes it difficult to act on a single axis, which for Replicate and Group is probably the most common way the primitive is used:

 ↗️
    3‿2‿1‿2‿3 / "abcde"
 "aaabbcddeee"
 

@@ -152,7 +152,7 @@
 ⟨ ⟨ 2 1 4 0 ⟩ ⟨ 2 1 4 1 ⟩ ⟩
 

 
The Depth modifier

-
The Depth 2-modifier (⚇) is a generalization of Each that allows diving deeper into an array. To illustrate it we'll use a shape 4‿3 array of lists of lists.

+
The Depth 2-modifier (⚇) is a generalization of Each that allows diving deeper into an array. To illustrate it we'll use a shape 4‿3 array of lists of lists.

 ↗️
    ⊢ n ← <⎉1⍟2 4‿3‿2‿2⥊↕48
 ┌─                                                                         
 ╵ ⟨ ⟨ 0 1 ⟩ ⟨ 2 3 ⟩ ⟩     ⟨ ⟨ 4 5 ⟩ ⟨ 6 7 ⟩ ⟩     ⟨ ⟨ 8 9 ⟩ ⟨ 10 11 ⟩ ⟩    
@@ -163,7 +163,7 @@
     ≡ n
 3
 

-
Reversing n swaps all the rows:

+
Reversing n swaps all the rows:

 ↗️
    ⌽ n
 ┌─                                                                         
 ╵ ⟨ ⟨ 36 37 ⟩ ⟨ 38 39 ⟩ ⟩ ⟨ ⟨ 40 41 ⟩ ⟨ 42 43 ⟩ ⟩ ⟨ ⟨ 44 45 ⟩ ⟨ 46 47 ⟩ ⟩  
@@ -172,7 +172,7 @@
   ⟨ ⟨ 0 1 ⟩ ⟨ 2 3 ⟩ ⟩     ⟨ ⟨ 4 5 ⟩ ⟨ 6 7 ⟩ ⟩     ⟨ ⟨ 8 9 ⟩ ⟨ 10 11 ⟩ ⟩    
                                                                           ┘
 

-
Depth ¯1 is equivalent to Each, and reverses the larger vectors, while depth ¯2 applies Each twice to reverse the smaller vectors:

+
Depth ¯1 is equivalent to Each, and reverses the larger lists, while depth ¯2 applies Each twice to reverse the smaller lists:

 ↗️
    ⌽⚇¯1 n
 ┌─                                                                         
 ╵ ⟨ ⟨ 2 3 ⟩ ⟨ 0 1 ⟩ ⟩     ⟨ ⟨ 6 7 ⟩ ⟨ 4 5 ⟩ ⟩     ⟨ ⟨ 10 11 ⟩ ⟨ 8 9 ⟩ ⟩    
@@ -188,13 +188,13 @@
   ⟨ ⟨ 37 36 ⟩ ⟨ 39 38 ⟩ ⟩ ⟨ ⟨ 41 40 ⟩ ⟨ 43 42 ⟩ ⟩ ⟨ ⟨ 45 44 ⟩ ⟨ 47 46 ⟩ ⟩  
                                                                           ┘
 

-
While a negative depth tells how many levels to go down, a non-negative depth gives the maximum depth of the argument before applying the left operand. On a depth-3 array like above, depth 2 is equivalent to ¯1 and depth 1 is equivalent to ¯2. A depth of 0 means to descend all the way to the level of atoms, that is, apply pervasively, like an arithmetic function.

+
While a negative depth tells how many levels to go down, a non-negative depth gives the maximum depth of the argument before applying the left operand. On a depth-3 array like above, depth 2 is equivalent to ¯1 and depth 1 is equivalent to ¯2. A depth of 0 means to descend all the way to the level of atoms, that is, apply pervasively, like an arithmetic function.

 ↗️
    ⟨'a',"bc"⟩ ≍⚇0 ⟨2‿3,4⟩
 ┌─                                                 
 · ⟨ ⟨ 'a' 2 ⟩ ⟨ 'a' 3 ⟩ ⟩ ⟨ ⟨ 'b' 4 ⟩ ⟨ 'c' 4 ⟩ ⟩  
                                                   ┘
 

-
With a positive operand, Depth doesn't have to use the same depth everywhere. Here, Length is applied as soon as the depth for a particular element is 1 or less, including if the argument has depth 0. For example, it maps over ⟨2,⟨3,4⟩⟩, but not over ⟨11,12⟩, even though these are elements of the same array.

+
With a positive operand, Depth doesn't have to use the same depth everywhere. Here, Length is applied as soon as the depth for a particular element is 1 or less, including if the argument has depth 0. For example, it maps over ⟨2,⟨3,4⟩⟩, but not over ⟨11,12⟩, even though these are elements of the same array.

 ↗️
    ≠⚇1 ⟨1,⟨2,⟨3,4⟩⟩,⟨5,⟨6,7⟩,⟨8,9,10⟩⟩,⟨11,12⟩⟩
 ⟨ 1 ⟨ 1 2 ⟩ ⟨ 1 2 3 ⟩ 2 ⟩
 

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