COMMENT ⊗   VALID 00010 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	.DEVICE XGP
C00003 00003	.COMMENT Facets
C00005 00004	.COMMENT Facets: COMPOSE
C00007 00005	.COMMENT Planning
C00008 00006	.COMMENT Tree of Disc
C00009 00007	.COMMENT Complete the Square
C00013 00008	.COMMENT AM Conjec
C00016 00009
C00019 00010
C00021 ENDMK
C⊗;
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.FONT 8 "BDR25"
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.TURN ON "↑α[]↓_π{"
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.PREFACE 2
.NOFILL
.PREFACE 1
.!XGPLFTMAR←100
.MACRO B ⊂ BEGIN NOFILL SELECT 9 INDENT 0 GROUP PREFACE 0 MILLS TURN OFF "{↑↓}[]α" ⊃
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.SELECT 1
.COMMENT Facets;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Facets of a Concept_↓

.BEGIN SELECT 2 PREFACE 0

Characterizations
	⊗7Name(s)⊗*  
	⊗7Definitions ⊗*  
	⊗7Algorithms ⊗* 
	⊗7Domain/range⊗*
	⊗7Intuitions: abstract representations⊗*

Ties to other concepts
	⊗7Specializations⊗*
	⊗7Generalizations⊗*
	⊗7Examples⊗*
	⊗7Operations one can do to this concept⊗*
	⊗7Conjectures/theorems involving this concept⊗*
	⊗7Analogies⊗*

Heuristics
	⊗7Worth: Why this concept is worth naming⊗*
	⊗7Interest: When an instance of it is (un)interesting⊗*
	⊗7Fillin: Hints for filling in parts of instances⊗*
	⊗7Suggest new activities for AM to consider⊗*
	⊗7Check: things to watch out for⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Facets: COMPOSE;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Facets of  "COMPOSE"_↓

.BEGIN SELECT 2 PREFACE 0

Characterizations
	⊗7Name(s): Compose ⊗*  
	⊗7Definitions: recursive, opaque, wffs⊗*  
	⊗7Algorithms: opaque, transparent, destructive ⊗* 
	⊗7Domain/range: Relations x Relations → Relations⊗*
	⊗7Intuitions: refiring arrows, time sequence⊗*

Ties to other concepts
	⊗7Specializations: Compose f with itself⊗*
	⊗7Generalizations: Relation⊗*
	⊗7Examples: (Intersect, Complement) → Set-difference⊗*
	⊗7Conjec: (AoB)oC ≡ Ao(BoC)⊗*
	⊗7Analogies: multiplying two matrices⊗*

Heuristics
	⊗7Worth: Primitive. Creates new active Concepts⊗*
	⊗7Interest: Domain=Range; both args are interesting⊗*
	⊗7Fillin: D/R are Domain(arg1) and Range(arg2)⊗*
	⊗7Sugg: Check AoB for properties which A or B have⊗*
	⊗7Check: Domain(arg2) should intersect Range(arg1)⊗*
.END
.SKIP TO COLUMN 1
.COMMENT Planning;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Control Structure_↓

.BEGIN SELECT 7 TABS 15,55  TURN ON "\↑↓←→"

THINGS WORTH DOING

(Fill in exs of Primes)
(Improve algs for Compose)
(Generalize Defn of Equality)


.GROUP SKIP 3



.ONCE CENTER
Select 1 activity



Execute this plan
→Assemble relevant heuristics



.END
.SKIP TO COLUMN 1
.COMMENT Tree of Disc;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
⊗2↓_Graph of Development_↓⊗*



.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "→∞\α" TABS 30,50,60,70


Bags\Equality\Cross-product



\Numbers\\Projection



\Multiplication



Exponentiation\Divisors



Hyper-exponentiation\Max-Divis\Primes

.END
.SKIP TO COLUMN 1
.COMMENT Complete the Square;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Complete the Square_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TABS 40,64,75 TURN ON "→∞α{}\" TURN OFF "↑↓"
.SELECT 2

\⊗7count⊗*
Pairs of Bags∞-→α→ Pairs of Numbers
    |\\\|
    |\\\
    |\\\|
    |\\\
    |\\\|
    |\\\
    | ⊗7cross-product⊗*\\\|  ⊗7(?)⊗*
    |\\\
    |\\\|
    |\\\
    |\\\|
    |\\\
    ↓\⊗7count⊗*\\↓
Bags∞-\-∞-\α→ Numbers

.TURN ON "↑↓" SELECT 7


?(x,y) = Count ( Cross-product ( Count↑-↑1(x), Count↑-↑1(y) ) ).


.END
.SKIP TO COLUMN 1
.COMMENT AM Conjec;
.GROUP SKIP 4
.ONCE CENTER SELECT 2
↓_Maximally Divisible Numbers_↓

.BEGIN SELECT 7 INDENT 0 PREFACE 0 TURN ON "↑↓[]{}&" SELECT 2

⊗2Max-divis(N) iff (∀m<n) d(m) < d(n)

     if  N  =  ↓2↑a⊗71⊗*↓3↑a⊗72⊗*↓5↑a⊗73⊗*...p⊗7↓k⊗*↑a⊗7k⊗*

⊗2where   p↓i  is  the  i↑t↑h  prime, 

and   (a↓i + 1) / (a↓j + 1)   "="   log(p↓j ) / log(p↓i)

**************************************************

For example:   n could be

2⊗7↑8⊗*3⊗7↑5⊗*5⊗7↑3⊗*7⊗7↑2⊗*11⊗7↑2⊗*13⊗7↑1⊗*17⊗7↑1⊗*19⊗7↑1⊗*23⊗7↑1⊗*29⊗7↑1⊗*31⊗7↑1⊗*37⊗7↑1⊗*41⊗7↑1⊗*43⊗7↑1⊗*47⊗7↑1⊗*53⊗7↑1⊗*
.SELECT 2
	(which equals 25,608,675,584).

(a↓i + 1)'s  are  (9 6 4 3 3 2 2 2 2 2 2 2 2 2 2 2)

n has 3,981,312 divisors.


AM  Conjecture says that
n is the smallest integer with that many divisors.

.END
.SKIP TO COLUMN 1