Leading Axis Theory and Practice in Array Kingdom 2023

Exploring the concept of Leading Axis Theory and Practice in the Array Kingdom of Elsinore 2023. The discussion covers essential keys like Rank and Dyadic Transpose, Make Functions apply to whole arrays, and Theory Arrays of various ranks. The images provide insights into major cells, leading axes, and more, forming a comprehensive guide to understanding and implementing this theory for efficient array operations.

• Leading Axis Theory
• Array Kingdom
• Elsinore 2023
• Rank Operator
• Dyadic Transpose

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1. Elsinore 2023 Leading Axis Theory and Practice Rich Park Ad m Brudzewsky asst. Josh David

2. Rank and Dyadic Transpose are the Keys to the Array Kingdom Leading Axis Theory and Practice 1

3. Rank and Dyadic Transpose are the Keys to the Array Kingdom Leading Axis Theory and Practice 2

4. Leading Axis Overview Theory The Rank Operator Leading Axis Functions Leading Axis Operators Dyadic Transpose Performance Leading Axis Theory and Practice 3

5. Make Functions apply to Whole Arrays 1 1 1 2 1 3 ~ ~ ~ 2 1 2 2 2 3 ~ ~ ~ Leading Axis Theory and Practice 4

6. Make Functions apply to Whole Arrays 1 1 1 2 1 3 ~ ~ ~ 2 1 2 2 2 3 ~ ~ ~ Leading Axis Theory and Practice 5

7. Theory Array of rank 3 M N O P A A B B C C D D Q R S T E E F F G G H H U V W X L L I J K A 2 3 4 A Leading Axis Theory and Practice 6

8. Theory Array of rank 3 2-Cell: layer* M N O P A B C D Q R S T E F G H U V W X I J K L * Major cell A 2 3 4 A Leading Axis Theory and Practice 7

9. Theory Array of rank 3 2-Cell: layer* 1-Cell: row M N O P A B C D Q R S T E F G H U V W X I J K L * Major cell A 2 3 4 A Leading Axis Theory and Practice 8

10. Theory Array of rank 3 2-Cell: layer* 1-Cell: row 0-Cell: element M N O P A B C D Q R S T E F G H U V W X I J K L * Major cell Trailing axis A 2 3 4 A Leading Axis Theory and Practice 9

11. Theory Array of rank N K-Cell Major Cell Leading axis Trailing axis collection of arrays of rank N 1 sub-array of rank K N 1-Cell first dimension; list of major cells last dimension; list of scalar elements Leading Axis Theory and Practice 10

12. Discussion: Leading-Axis Primitives Functions Operators X Y X Y X Y X Y X Y @ Y X Y Y X Y Y Y Y Y X Y X Y Leading Axis Theory and Practice 11

13. The Rank Operator is just Blinkers Leading Axis Theory and Practice 12

14. The Rank Operator is just Blinkers Leading Axis Theory and Practice 13

15. The Rank Operator (fM) f sees arg of max rank M If necessary, f will be called multiple times Pro tip to avoid stranding of M and Y: (f M)Y or f( M)Y or f M Y Leading Axis Theory and Practice 14

16. Rank Values Sub-Arrays Elements/Scalars Rows/Vectors Layers/Matrices Blocks ... Rank 0 1 2 3 ... Leading Axis Theory and Practice 15

17. Task: Exchange the First Two Consider 'ABcdef' 'BAcdef' layers A 3 5 4 60 rows of each layer columns (elements of each row) Bonus question: How would you do it without ? Leading Axis Theory and Practice 16

18. The Rank Operator (fL R) f sees left arg of max rank L f sees right arg of max rank R If necessary, sub-arrays will be used multiple times Pro tips: If L and R are the same then f R is enough If confused, try { } L R Leading Axis Theory and Practice 17

19. Discussion: Whole Array Primitives Functions Operators A f X Y X Y f B Y A Y Y Leading Axis Theory and Practice 18

20. Task: Using L R names (,'Hey' 'Jay' .{ }'ley' 'den')[?10 4] Boolean vector indicating layers that are 2 3 'Heyley' Boolean matrix indicating rows that are 'Hey' Boolean rank-3 array indicating elements that are 'e' Bonus: Think about what the relationship is between and = Leading Axis Theory and Practice 19

21. Double-Rank ((fL2 R2)L1 R1) 'ABC'(, 0 1)'xy' Axy Bxy Cxy 'ABC'((, 0) 0 1)'xy' Ax Ay Bx By Cx Cy Remember: L=R Leading Axis Theory and Practice 20

22. Task: Using (L2 R2)L1 R1 A 1 10 100 B 2 3 4 24 Produce: 1 2 3 4 50 60 70 80 900 1000 1100 1200 13 14 15 16 170 180 190 200 2100 2200 2300 2400 Leading Axis Theory and Practice 21

23. Dyadic Transpose (XY): Why? Dyadic Transpose reorders axes Move axes to end, then use Rank on any function, operating on only those trailing axes Leading Axis Theory and Practice 22

24. Dyadic Transpose (XY): How? Reorders axes by the left argument: Each number is the destination index where we send that axis. 3 1 2 A 3rd 2nd dimension will be the 1st 1st dimension will be the 3rd dimension will be the 2nd Leading Axis Theory and Practice 23

25. Tasks: Merging Layers 1. horizontally: 2. vertically: 2 3 4 A ABCD EFGH IJKL ABCD EFGH IJKL MNOP QRST UVWX ABCDMNOP EFGHQRST IJKLUVWX MNOP QRST UVWX Leading Axis Theory and Practice 24

26. Transpose, Apply-with-Rank, Transpose Leading Axis Theory and Practice 25

27. Transpose, Apply-with-Rank, Transpose Leading Axis Theory and Practice 26

28. Keep Rank Close to the Primitives + Leading Axis Theory and Practice 27

29. Keep Rank Close to the Primitives + Leading Axis Theory and Practice 28

30. Task: Speed it Up! n ?1 10 100 1000 10 a '{+ 1 0 } d n' b ' your code here ' d 1 cmpx a b d 2 cmpx a b d 3 cmpx a b d 4 cmpx a b Leading Axis Theory and Practice 29

31. Task: Speed it Up! n ?1 10 100 1000 10 a '{+ 1 0 } d n' b ' your code here ' 'cmpx' CY'dfns' d 1 0 0 0 0.1 cmpx a b d 2 0 0 0 0.1 cmpx a b d 3 0 0 0 0.1 cmpx a b d 4 0 0 0 0.1 cmpx a b Leading Axis Theory and Practice 30

32. Lessons to Take Home Rank and Dyadic Transpose are the Keys to the Array Kingdom Make Functions apply to Whole Arrays The Rank Operator is just Blinkers Transpose, Rank-Apply-with-Rank, Transpose Keep Rank Close to the Primitives Leading Axis Theory and Practice 31