From df6d6a0fa85c07c67eaa40a097953e3290f5d356 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Wed, 7 Jul 2021 19:59:57 -0400 Subject: Continued editing and links --- docs/doc/prefixes.html | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) (limited to 'docs/doc/prefixes.html') diff --git a/docs/doc/prefixes.html b/docs/doc/prefixes.html index 9ef7cbdd..9732fe90 100644 --- a/docs/doc/prefixes.html +++ b/docs/doc/prefixes.html @@ -11,7 +11,7 @@ ↓ "abcde" ⟨ "abcde" "bcde" "cde" "de" "e" ⟨⟩ ⟩ -

The functions are closely related to Take and Drop, as we might expect from their glyphs. Element i⊑↑𝕩 is i↑𝕩, and i⊑↓𝕩 is i↓𝕩.

+

The functions are closely related to Take and Drop, as we might expect from their glyphs. Element i⊑↑𝕩 is i↑𝕩, and i⊑↓𝕩 is i↓𝕩.

In both cases, an empty array and the entire argument are included in the result, meaning its length is one more than that of the argument. Using Span, we can say that the result has elements whose lengths go from 0 to ≠𝕩, inclusive, so there are (≠𝕩)¬0 or 1+≠𝕩 elements. The total number or cells in the result (for example, ≠∾↑𝕩 or +´≠¨↑𝕩) scales with the square of the argument length—it is quadratic in ≠𝕩. We can find the exact total by looking at Prefixes and Suffixes together:

↗️
    (↑ ≍˘ ↓) "abcde"
 ┌─                 
@@ -27,7 +27,7 @@

Joining corresponding elements of ↑𝕩 and ↓𝕩 gives 𝕩 again. This is because i⊑(↑∾¨↓)𝕩 is (i⊑↑𝕩)∾(i⊑↓𝕩), or, using the Take and Drop correspondences, (i↑𝕩)∾(i↓𝕩), which is 𝕩. Element-wise, we are combining the first i cells of 𝕩 with all but the first i. Looking at the entire result, we now know that (↑∾¨↓)𝕩 is (1+≠𝕩)⥊<𝕩. The total number of cells in this combined array is therefore (1+≠𝕩)×≠𝕩, or 1⊸+⊸×≠𝕩. Each of Prefixes and Suffixes must have the same total number of cells (in fact, ↑𝕩 is ⌽¨∘↓⌾⌽𝕩, and Reverse doesn't change an array's shape). So the total number in either one is 2÷˜1⊸+⊸×≠𝕩. With n←≠𝕩, it is 2÷˜n×1+n.

Definition

-

Knowing the length and the elements, it's easy to define functions for Prefixes and Suffixes: ↑ is equivalent to (↕1+≠)↑¨< while ↓ is (↕1+≠)↓¨<. Each primitive is defined only on arrays with at least one axis.

+

Knowing the length and the elements, it's easy to define functions for Prefixes and Suffixes: ↑ is equivalent to (↕1+≠)↑¨< while ↓ is (↕1+≠)↓¨<. Each primitive is defined only on arrays with at least one axis.

Working with pairs

Sometimes it's useful to apply an operation to every unordered pair of elements from a list. For example, consider all possible products of numbers between 1 and 6:

↗️
    ×⌜˜ 1+↕6
@@ -40,7 +40,7 @@
   6 12 18 24 30 36  
                    ┘
-

It's easy enough to use the Table modifier here, but it also computes most products twice. If we only care about the unique products, we could multiply each number by all the ones after it. "After" sounds like suffixes, so let's look at those:

+

It's easy enough to use the Table modifier here, but it also computes most products twice. If we only care about the unique products, we could multiply each number by all the ones after it. "After" sounds like suffixes, so let's look at those:

↗️
    1+↕6
 ⟨ 1 2 3 4 5 6 ⟩
     ↓ 1+↕6
@@ -52,7 +52,7 @@
     (⊢ × 1 ↓ ↓) 1+↕6
 ⟨ ⟨ 2 3 4 5 6 ⟩ ⟨ 6 8 10 12 ⟩ ⟨ 12 15 18 ⟩ ⟨ 20 24 ⟩ ⟨ 30 ⟩ ⟨⟩ ⟩
-

By using ≍ instead of ×, we can see the argument ordering, demonstrating that we are looking at the upper right half of the matrix produced by Table. While in this case we could use ≍⚇0 to mimic the pervasion of ×, we'd like this to work even on nested arguments so we should figure out how the mapping structure works to apply Each appropriately.

+

By using Couple (≍) instead of ×, we can see the argument ordering, demonstrating that we are looking at the upper right half of the matrix produced by Table. While in this case we could use ≍⚇0 to mimic the pervasion of ×, we'd like this to work even on nested arguments so we should figure out how the mapping structure works to apply Each appropriately.

↗️
    ≍⌜˜ "abc"
 ┌─                
 ╵ "aa" "ab" "ac"  
@@ -81,7 +81,7 @@
 · ⟨ ⟨⟩ "a" "ab" "abc" ⟩ ⟨ ⟨⟩ "b" "bc" ⟩ ⟨ ⟨⟩ "c" ⟩ ⟨ ⟨⟩ ⟩  
                                                           ┘
-

Effectively, this parametrizes the slices either by ending then starting index, or by starting index then length. Four empty slices are included because in a list of length 3 there are 4 places an empty slice can start: all the spaces between or outside elements (these also correspond to all the possible positions for the result of Bins). The slices can also be parametrized by length and then starting index using Windows.

+

Effectively, this parametrizes the slices either by ending then starting index, or by starting index then length. Four empty slices are included because in a list of length 3 there are 4 places an empty slice can start: all the spaces between or outside elements (these also correspond to all the possible positions for the result of Bins). The slices can also be parametrized by length and then starting index using Windows.

↗️
    ((↕1+≠)↕¨<) "abc"
 ┌─                         
 · ┌┐ ┌─    ┌─     ┌─       
@@ -167,7 +167,7 @@
       ┘     ┘      ┘  
                      ┘
-

But Prefixes and Suffixes don't have any way to specify that they should work on multiple axes, and always work on exactly one. So to extend this pattern we will have to define multi-dimensional versions. This turns out to be very easy: just replace Length with Shape in the definitions above.

+

But Prefixes and Suffixes don't have any way to specify that they should work on multiple axes, and always work on exactly one. So to extend this pattern we will have to define multi-dimensional versions. This turns out to be very easy: just replace Length with Shape in the definitions above.

↗️
    Prefs ← (↕1+≢)↑¨<
     Suffs ← (↕1+≢)↓¨<
     Prefs¨Suffs 3‿2⥊"abcdef"
-- 
cgit v1.2.3