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`u/. y`Oblique Adverb

Rank Infinity *-- operates on x and y as a whole, by items of y --*
WHY IS THIS IMPORTANT?

(`u/.y`) applies verb `u` to the *oblique diagonals* of table `y`

] y=: 4 4 $ i.3 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 </. y +-+---+-----+-------+-----+---+-+ |0|1 1|2 2 2|0 0 0 0|1 1 1|2 2|0| +-+---+-----+-------+-----+---+-+

### Common Uses

1. As part of the idiom for polynomial multiplication

(1 2) */ (3 4 2) NB. consider this product 3 4 2 6 8 4 +/ /. (1 2) */ (3 4 2) NB. sum up the oblique diagonals 3 10 10 4 pmul =: +//.@(*/) NB. define a dyadic verb to calculate the coefficients 1 2 pmul 3 4 2 NB. of e.g. (2x+1)(2x^2+4x+3)=4x^3+10x^2+10x+3 3 10 10 4

### More Information

1. `u` can be a cyclic gerund.

### Details

1. The arguments to the invocations of `u` are lists of (_2)-cells of `y`

2. If `y` has no items, the result of `u/. y` will also have no items.

However, `u` will be executed once on a cell of 0 _2-cells to establish the shape of a result of `u`,
and then the result of ` u/. y ` will be a list of 0 of that shape

$ ]/. i. 0 3 NB. u is executed on a cell of shape 0 0 0 $ ]/. i. 0 3 4 NB. u is executed on a cell of shape 0 4 0 0 4

### Use These Combinations

Combinations using `u/. y` that have exceptionally good performance
are shown in Diagonals and Polynomials.

`x u/. y`Key Adverb `x u/.. y`Key dyad Adverb

Rank Infinity *-- operates on x and y as a whole, by items --*
WHY IS THIS IMPORTANT?

Classifies items of `x` into partitions of identical items.
Then `y` is partitioned using the partitioning that was calculated for `x`, and `u` is applied on each partition of `y`.

**Note:** `x` and `y` must have the same number of items.

Each item of `x` is the **key** of the corresponding item of `y`,
and `u` is applied to each set of items of `y` with equal keys.

For `/.`, noun `x` is used only for the partitioning and is not an argument to operand `u`. For `/..`, the key value from `x` becomes the x argument to `u`.

`x` is classified by tolerant comparison. The value used for the `x` argument to `u` is the first one in the partition.

y=: 'AbcDeFGhijk' ] x=: y e. 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' NB. 1=uppercase, 0=lowercase 1 0 0 1 0 1 1 0 0 0 0 x ]/. y NB. partitions of y ADFG bcehijk x </. y +----+-------+ |ADFG|bcehijk| +----+-------+ x ;/.. y NB. key from x and partition of y +-+-------+ |1|ADFG | +-+-------+ |0|bcehijk| +-+-------+

### Common Uses

1. Count the numbers of different items, using ` #/.`

a =: 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 (~.a) ,: (a #/. a) NB. unique digits, then frequency of each 3 1 4 5 9 2 6 8 7 2 2 1 3 3 1 1 1 1

2. Collect information from records with identical keys

]data =: _2 ]\ 'Fred';90;'John';75;'Fred';95;'John';85;'Susan';100;'John';86 NB. A table +-----+---+ |Fred |90 | +-----+---+ |John |75 | +-----+---+ |Fred |95 | +-----+---+ |John |85 | +-----+---+ |Susan|100| +-----+---+ |John |86 | +-----+---+ ]s =: ({."1 data) <@;/. ({:"1 data) NB. Use first column as keys; collect second-column values +-----+--------+---+ |90 95|75 85 86|100| +-----+--------+---+ (~. {."1 data) ,. s NB. Reinstall the name +-----+--------+ |Fred |90 95 | +-----+--------+ |John |75 85 86| +-----+--------+ |Susan|100 | +-----+--------+

### More Information

1. `u` can be a cyclic gerund

x tolower`toupper/. y adfg BCEHIJK

### Details

1. ` x u/. y ` is a member of the i.-family.

2. The argument to each invocation of `u` is a list of items of `y`
(this has the same rank as `y` unless `y` is an atom).

3. The partitions are processed in the same order as the items in ` ~. x `.

4. The partitioning uses tolerant comparison.
Use `u/.!.0` for intolerant comparison.

5. If `y` has no items, the result of ` x u/. y ` will also have no items.

However, `u` will be executed once on a cell of 0 items to establish the shape of a result of `u`,
and then the result of ` x u/. y ` will be a list of 0 of that shape

$ '' ]/. '' NB. u is executed on cell of shape 0 0 0 $ '' {./. '' NB. u produces an atomic result 0 $ '' {./. i. 0 3 NB. u is executed on cell of shape 0 3 0 3

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### Oddities

If a tolerance is specified using `u/.!.f`, that tolerance is used as the default tolerance during the execution of `u`.
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### Use These Combinations

Combinations using `x u/. y` that have exceptionally good performance
are shown in Partitions and Diagonals and Polynomials.