The document discusses the monadic and dyadic cases of the J symbol ",." (comma-dot). For the monadic case, ",." returns a matrix by raveling (flattening) its argument of any rank (scalar, vector, matrix, or higher-rank array) into a two-column matrix. For the dyadic case, ",." stitches together the items of its left and right arguments by placing their items side-by-side into a new matrix. The lengths of the arguments must match except when one is a scalar, which repeats to fit.

1. J symbol
“ ., ” (comma-dot)
Group 6 Hikaru Sugiura

2. Outline
• Monadic Case
• Over 3-rank Array, Matrix, Vector, Scalar
• Apply to empty list
• Conclusion: Monadic
• Dyadic Case
• Matrix, Vector
• Exception: special case
• Error case
• Conclusion : Dyadic

3. ,. : Monadic Case
Items
Matrix
Ravel Items
Definition: y ravels the items of y.

4. 18 19 20
21 22 23
24 25 26
Apply to rank-3 array
Ravel Items returns
the nx(m*o) Matrix for the nxmxo array.
0 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16 17
18 19 20 21 22 23 24 25 26
9 10 11
12 13 14
15 16 17
0 1 2
3 4 5
6 7 8
3rd
2nd
1st
m*o
m
o
n
n
,.
i.3 3 3
Matrix

5. Apply to matrix
Ravel Items returns
the Same Matrix for the matrix.
0 1 2
3 4 5
6 7 8
0 1 2
3 4 5
6 7 8
,.
n
m
1st
2ndi.3 3

6. 0 1 2
Apply to vector
Ravel Items returns
a nx1 Matrix for the n vector.
Shape : 3 1
0
1
2
,.
n
1st
i.3

7. Apply to scaler
Ravel Items returns
a 1x1 Matrix for the scaler.
33,.
No axis!
Shape : 1 1

8. Ravel Items returns
the Matrix for a scaler, vector, matrix
and any size array.
i.e.
Arg:Rank
Result
Shape Rank
Scalar : 0 1 1 2
n
Vector : 1
n 1 2
nxm
Matrix : 2
n m 2
nxmxo
Array: 3
n m*o 2
Matrix!
}

9. Apply to Empty list
Result : ,. i. 0 0 0 → (empty)
Ravel Items returns
the 0x0 Matrix for the empty list.
Rank : $$ ,. i. 0 0 0 → 2
Shape : $ ,. i. 0 0 0 → 0 0

10. Conclusion:Monadic
Ravel Items returns
the Matrix for the any list
Arg:Rank
Result
Shape Rank
Scalar : 0 1 1 2
n
Vector : 1
n 1 2
nxm
Matrix : 2
n m 2
nxmxo
Array: 3
n m*o 2
nxmxo…xz n m*o*…*z 2
※ n,m,…,z ≧ 0
Effective to over
4-rank Array.
Make matrix and
possible to process
with other one.

11. ,. : Dyadic case
Stitch
Definition: x ,. y the items of x and stitched to the
corresponding items of y.
X Y,. X Y
One Item

12. Example
0 1 2
3 4 5
6 7 8
0 1 2
3 4 5
6 7 8
,.
0 1 2 0 1 2
3 4 5 3 4 5
6 7 8 6 7 8
3x6 Matrixi.3 3 i.3 3
0 1 2 3
4 5 6 7
8 9 10 11
0 1 2
3 4 5
6 7 8
,.
0 1 2 3 0 1 2
4 5 6 7 3 4 5
8 9 10 11 6 7 8
i.3 4 i.3 3
• Matrices
3x7 Matrix

13. Example
0 1 2
3 4 5
6 7 8
0 1 2,.
i.3 3 i.3
• Matrix and Vector
0 1 2 0
3 4 5 1
6 7 8 2
3x4 Matrix
0 1 2 ,.
0 0
1 1
2 2
• Vectors 2x3 Matrix
0 1 2
i.3
0 1 2 0 1 2not!
i.3

14. Example
,.1 2
• Scalars
1 2
Vector
• Scalar and Vector
0 1 2
i.3
3,.
0 3
1 3
2 3
Matrix
Repeating
to Fit!

15. Error Case
• Length Error
0 1 2
3 4 5
6 7 8
0 1
3 4
,.
0
1
2
,.
Matrix and Matrix
The lengths of Y-axis
Must be matched.
Vector and Vector
The length
Must be matched.
0
1
2
3
※Scaler automatically match to other item by repeating.

16. Conclusion:Dyadic
Stitch returns
Joint-Item made from (left) and (right).
In over 1-rank items, The lengths must be matched.
X Y,. X Y
One Item