Dilate: Difference between revisions
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<table border="0" cellpadding="5" cellspacing="0" summary=""> | <table border="0" cellpadding="5" cellspacing="0" summary=""> | ||
<tr> | <tr> | ||
<td valign="top"><apll>Z←>R</apll></td> | <td valign="top"><apll>Z←>R</apll> or <apll>Z←>[X] R</apll></td> | ||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
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</table> | </table> | ||
</td> | </td> | ||
</tr> | |||
<tr> | |||
<td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳1+⍴⍴R</apll>. If X is omitted, it defaults to the last axis plus one.</td> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td><apll>Z</apll> is the corresponding array of Real numbers of shape <apll>(⍴R) | <td><apll>Z</apll> is the corresponding array of Real numbers of shape <apll>((=R),⍴R)[⍋⍋X≠⍳1+⍴⍴R]</apll> where <apll>=R</apll> is the Hypercomplex dimension of <apll>R</apll> as in <apll>(=R)∊1 2 4 8</apll>.</td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
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4 3<br /> | 4 3<br /> | ||
2 1<br /> | 2 1<br /> | ||
2 2 2<br /> | |||
⍴⎕←>[1] 2 2⍴1<pn>J</pn>2 3<pn>J</pn>4 4<pn>J</pn>3 2<pn>J</pn>1<br /> | |||
1 3<br /> | |||
4 2<br /> | |||
<br /> | |||
2 4<br /> | |||
3 1<br /> | |||
2 2 2<br /> | 2 2 2<br /> | ||
⍴⎕←>2 2⍴1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5 4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1<br /> | ⍴⎕←>2 2⍴1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5 4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1<br /> | ||
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==Identities== | ==Identities== | ||
<apll>R ←→ <>R</apll> for all <apll>R</apll> (see [[Condense]] for the definition of monadic Left Caret)<br /> | <apll>R ←→ <[X] >[X] R</apll> for all <apll>R</apll> (see [[Condense]] for the definition of monadic Left Caret)<br /> | ||
<apll>R ←→ ><R</apll> for all <apll>R</apll> with <apll>( | <apll>R ←→ >[X] <[X] R</apll> for all <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br /> | ||
== Acknowledgements== | == Acknowledgements== | ||
<p>This symbol and its name were suggested by David A. Rabenhorst.</p> | <p>This symbol and its name were suggested by David A. Rabenhorst.</p> | ||
Revision as of 19:42, 11 April 2018
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| X is an optional numeric singleton axis with X∊⍳1+⍴⍴R. If X is omitted, it defaults to the last axis plus one. | ||||
| R is an arbitrary Real or Hypercomplex numeric array — otherwise, DOMAIN ERROR. | ||||
| Z is the corresponding array of Real numbers of shape ((=R),⍴R)[⍋⍋X≠⍳1+⍴⍴R] where =R is the Hypercomplex dimension of R as in (=R)∊1 2 4 8. |
For example,
⍴⎕←>23
23
1
⍴⎕←>2 2⍴⍳4
1
2
3
4
2 2 1
⍴⎕←>2 2⍴1J2 3J4 4J3 2J1
1 2
3 4
4 3
2 1
2 2 2
⍴⎕←>[1] 2 2⍴1J2 3J4 4J3 2J1
1 3
4 2
2 4
3 1
2 2 2
⍴⎕←>2 2⍴1i2j3k4 5i6j7k8 8i7j6k5 4i3j2k1
1 2 3 4
5 6 7 8
8 7 6 5
4 3 2 1
2 2 4
⍴⎕←>2 2⍴1i2j3k4l5ij6jk7kl8 9i10j11k12l13ij14jk15kl16 16i15j14k13l12ij11jk10kl9 8i7j6k5l4ij3jk2kl1
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
16 15 14 13 12 11 10 9
8 7 6 5 4 3 2 1
2 2 8
Identities
R ←→ <[X] >[X] R for all R (see Condense for the definition of monadic Left Caret)
R ←→ >[X] <[X] R for all R with (⍴R)[X]∊1 2 4 8
Acknowledgements
This symbol and its name were suggested by David A. Rabenhorst.
