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Condense - Revision history
2024-03-29T09:32:58Z
Revision history for this page on the wiki
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http://wiki.nars2000.org/index.php?title=Condense&diff=3164&oldid=prev
WikiSysop: /* Identities */
2018-04-15T21:14:03Z
<p><span dir="auto"><span class="autocomment">Identities</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:14, 15 April 2018</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt; &gt; R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll> <ins style="font-weight: bold; text-decoration: none;"> </ins>R ←→ &lt; &gt; R</apll> &nbsp;&nbsp;for all <ins style="font-weight: bold; text-decoration: none;">Real (<apll>1==R</apll>) </ins><apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt;[X] &gt;[X] R</apll> &nbsp;&nbsp;for all <del style="font-weight: bold; text-decoration: none;">non-scalar </del><apll>R</apll><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll> <ins style="font-weight: bold; text-decoration: none;"> </ins>R ←→ &lt;[X] &gt;[X] R</apll> &nbsp;&nbsp;for all <ins style="font-weight: bold; text-decoration: none;">Real (<apll>1==R</apll>) </ins><apll>R</apll><br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt; &lt; R</apll> &nbsp;&nbsp;for all <apll>R</apll><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll> <ins style="font-weight: bold; text-decoration: none;"> </ins>R ←→ &gt; &lt; R</apll> &nbsp;&nbsp;for all <apll>R<ins style="font-weight: bold; text-decoration: none;"></apll> with <apll>(¯1↑⍴ R)∊1 2 4 8</apll><br /></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt;[X] &lt;[X] R</apll> &nbsp;&nbsp;for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><apll>1/R ←→ &gt; &lt; R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(¯1↑⍴1/R)∊1 2 4 8</ins></apll><br /></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll> <ins style="font-weight: bold; text-decoration: none;"> </ins>R ←→ &gt;[X] &lt;[X] R</apll> &nbsp;&nbsp;for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td></tr>
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WikiSysop
http://wiki.nars2000.org/index.php?title=Condense&diff=3155&oldid=prev
Sudleyplace at 18:54, 15 April 2018
2018-04-15T18:54:00Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:54, 15 April 2018</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll>. If <apll>X</apll> is omitted, <del style="font-weight: bold; text-decoration: none;">it defaults </del>to the last axis.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll> <ins style="font-weight: bold; text-decoration: none;">&mddash; otherwise (including if <apll>R</apll> is a scalar), an <apll>AXIS ERROR</apll> is signalled</ins>. If <apll><ins style="font-weight: bold; text-decoration: none;">[</ins>X<ins style="font-weight: bold; text-decoration: none;">]</ins></apll> is omitted, <ins style="font-weight: bold; text-decoration: none;">the operation applies </ins>to the last axis.</td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>R</apll> is an arbitrary Real numeric array <del style="font-weight: bold; text-decoration: none;">(BOOL</del>, <del style="font-weight: bold; text-decoration: none;">INT</del>, <del style="font-weight: bold; text-decoration: none;">FLT</del>, <del style="font-weight: bold; text-decoration: none;">APA</del>, <del style="font-weight: bold; text-decoration: none;">RAT</del>, <del style="font-weight: bold; text-decoration: none;">VFP </del>&mdash; otherwise, <apll>DOMAIN ERROR</apll><del style="font-weight: bold; text-decoration: none;">) whose </del><apll>X</apll>-axis length <apll>(⍴R)[X]</apll> <del style="font-weight: bold; text-decoration: none;">is </del><apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, <apll>LENGTH ERROR</apll>.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>R</apll> is an arbitrary Real numeric array, <ins style="font-weight: bold; text-decoration: none;">i.e., Fixed-Precision Boolean</ins>, <ins style="font-weight: bold; text-decoration: none;">Integer</ins>, <ins style="font-weight: bold; text-decoration: none;">Floating Point</ins>, <ins style="font-weight: bold; text-decoration: none;">Arithmetic Progression Array</ins>, <ins style="font-weight: bold; text-decoration: none;">or Multiple-Precision Integer/Rational or Floating Point </ins>&mdash; otherwise, <ins style="font-weight: bold; text-decoration: none;">a </ins><apll>DOMAIN ERROR</apll> <ins style="font-weight: bold; text-decoration: none;">is signalled. The </ins><apll>X</apll>-axis length <apll>(⍴R)[X]</apll> <ins style="font-weight: bold; text-decoration: none;">must be </ins><apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, <ins style="font-weight: bold; text-decoration: none;">a </ins><apll>LENGTH ERROR</apll> <ins style="font-weight: bold; text-decoration: none;">is signalled</ins>.</td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the <del style="font-weight: bold; text-decoration: none;">items </del>along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is <del style="font-weight: bold; text-decoration: none;">the </del>Real <del style="font-weight: bold; text-decoration: none;">array </del><<del style="font-weight: bold; text-decoration: none;">apll</del>><del style="font-weight: bold; text-decoration: none;">((X≠⍳⍴⍴R)/⍴R)⍴R</del><<del style="font-weight: bold; text-decoration: none;">/apll</del>><del style="font-weight: bold; text-decoration: none;">, if </del><apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is <del style="font-weight: bold; text-decoration: none;">a </del>Complex <del style="font-weight: bold; text-decoration: none;">array</del>, <del style="font-weight: bold; text-decoration: none;">if </del><apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is <del style="font-weight: bold; text-decoration: none;">a </del>Quaternion <del style="font-weight: bold; text-decoration: none;">array</del>, <del style="font-weight: bold; text-decoration: none;">and if </del><apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an Octonion <del style="font-weight: bold; text-decoration: none;">array</del>.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the <ins style="font-weight: bold; text-decoration: none;">numbers </ins>along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><ul></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li></ins><apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is <ins style="font-weight: bold; text-decoration: none;">an array of </ins>Real <ins style="font-weight: bold; text-decoration: none;">numbers,</ins><<ins style="font-weight: bold; text-decoration: none;">/li</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> </ins><<ins style="font-weight: bold; text-decoration: none;">li</ins>><apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is <ins style="font-weight: bold; text-decoration: none;">an array of </ins>Complex <ins style="font-weight: bold; text-decoration: none;">numbers</ins>,<ins style="font-weight: bold; text-decoration: none;"></li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li></ins><apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is <ins style="font-weight: bold; text-decoration: none;">an array of </ins>Quaternion <ins style="font-weight: bold; text-decoration: none;">numbers</ins>,<ins style="font-weight: bold; text-decoration: none;"></li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <li></ins><apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an <ins style="font-weight: bold; text-decoration: none;">array of </ins>Octonion <ins style="font-weight: bold; text-decoration: none;">numbers</ins>.<ins style="font-weight: bold; text-decoration: none;"></li></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ul></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td>
<td colspan="2" class="diff-lineno">Line 32:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>For example,</p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>For example,</p></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;23<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;23</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>23<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>23</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>10<pn>J</pn>20<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>10<pn>J</pn>20</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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<<del style="font-weight: bold; text-decoration: none;">br </del>/></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 4⍴⍳8</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>J</pn>5 2<pn>J</pn>6 3<pn>J</pn>7 4<pn>J</ins></<ins style="font-weight: bold; text-decoration: none;">pn</ins>><ins style="font-weight: bold; text-decoration: none;">8</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 8⍴(⍳8),⌽⍳8<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 8⍴(⍳8),⌽⍳8</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>1 2 3<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>4 5 6<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>2 3<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>1 2 3</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>4 5 6</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>1<pn>J</pn>2&nbsp;3<pn>J</pn>4&nbsp;5<pn>J</pn>6<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>2 3</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>6<pn>J</pn>5&nbsp;4<pn>J</pn>3&nbsp;2<pn>J</pn>1<<del style="font-weight: bold; text-decoration: none;">br </del>/></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>2 <del style="font-weight: bold; text-decoration: none;">3</del><<del style="font-weight: bold; text-decoration: none;">br </del>/></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>1<pn>J</pn>2&nbsp;3<pn>J</pn>4&nbsp;5<pn>J</pn>6</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>6<pn>J</pn>5&nbsp;4<pn>J</pn>3&nbsp;2<pn>J</pn>1</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<del style="font-weight: bold; text-decoration: none;">&nbsp;</del>&nbsp;&nbsp;&nbsp;5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">2 3</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5&nbsp;4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;[1] 2 3 2⍴(⍳6),⌽⍳6</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>2 3<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>J</pn>6 2<pn>J</pn>5</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">3<pn>J</pn>4 4<pn>J</ins></<ins style="font-weight: bold; text-decoration: none;">pn</ins>><ins style="font-weight: bold; text-decoration: none;">3</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16&nbsp;17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">5<pn>J</pn></ins>2 <ins style="font-weight: bold; text-decoration: none;">6<pn>J</ins></<ins style="font-weight: bold; text-decoration: none;">pn</ins>><ins style="font-weight: bold; text-decoration: none;">1</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17&nbsp;16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9&nbsp;&nbsp;&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">3 2</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>2 3<del style="font-weight: bold; text-decoration: none;"><br /></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&nbsp;</ins>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4&nbsp;&nbsp;&nbsp;5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5&nbsp;4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>2 3</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16&nbsp;17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17&nbsp;16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9&nbsp;&nbsp;&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>2 3</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt;<del style="font-weight: bold; text-decoration: none;">[X] </del>&gt;<del style="font-weight: bold; text-decoration: none;">[X] </del>R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt; <ins style="font-weight: bold; text-decoration: none;"> </ins>&gt; <ins style="font-weight: bold; text-decoration: none;"> </ins>R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<ins style="font-weight: bold; text-decoration: none;"><br /></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt;[X] &lt;[X] R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><apll>R ←→ &lt;[X] &gt;[X] R</apll> &nbsp;&nbsp;for all non-scalar <apll>R</apll><br /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><apll>R ←→ &gt; &lt; R</apll> &nbsp;&nbsp;for all <apll>R</apll></ins><br /></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt;[X] &lt;[X] R</apll> &nbsp;&nbsp;for all <ins style="font-weight: bold; text-decoration: none;">non-scalar </ins><apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td></tr>
</table>
Sudleyplace
http://wiki.nars2000.org/index.php?title=Condense&diff=3151&oldid=prev
WikiSysop at 00:14, 12 April 2018
2018-04-12T00:14:42Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:14, 12 April 2018</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <table border="0" cellpadding="5" cellspacing="0" summary=""></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <table border="0" cellpadding="5" cellspacing="0" summary=""></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td valign="top"><apll>Z←&lt;R</apll></td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td valign="top"><apll>Z←&lt;<ins style="font-weight: bold; text-decoration: none;">R</apll> &nbsp;&nbsp; or &nbsp;&nbsp; <apll>Z←&lt;[X] </ins>R</apll></td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <td></td></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <td></td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <td></td></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <td></td></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td>converts <apll>R</apll> to a Hypercomplex array if <apll>(<del style="font-weight: bold; text-decoration: none;">≢R</del>)∊2 4 8</apll> or a Real array if <apll>1<del style="font-weight: bold; text-decoration: none;">=≢R</del></apll>.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td>converts <apll>R</apll> to a Hypercomplex array if <apll>(<ins style="font-weight: bold; text-decoration: none;">⍴R</ins>)<ins style="font-weight: bold; text-decoration: none;">[X]</ins>∊2 4 8</apll> or a Real array if <apll><ins style="font-weight: bold; text-decoration: none;">(⍴R)[X]=</ins>1</apll>.</td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </table></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td><apll><del style="font-weight: bold; text-decoration: none;">R</del></apll> is an <del style="font-weight: bold; text-decoration: none;">arbitrary Real </del>numeric <del style="font-weight: bold; text-decoration: none;">array (BOOL, INT, FLT, APA, RAT, VFP &mdash; otherwise, </del><apll><del style="font-weight: bold; text-decoration: none;">DOMAIN ERROR</del></apll><del style="font-weight: bold; text-decoration: none;">) whose number of columns (</del><apll><del style="font-weight: bold; text-decoration: none;">≢R</del></apll><del style="font-weight: bold; text-decoration: none;">) </del>is <del style="font-weight: bold; text-decoration: none;"><apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise</del>, <del style="font-weight: bold; text-decoration: none;"><apll>LENGTH ERROR</apll></del>.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td><apll><ins style="font-weight: bold; text-decoration: none;">X</ins></apll> is an <ins style="font-weight: bold; text-decoration: none;">optional </ins>numeric <ins style="font-weight: bold; text-decoration: none;">singleton axis with </ins><apll><ins style="font-weight: bold; text-decoration: none;">X∊⍳⍴⍴R</ins></apll><ins style="font-weight: bold; text-decoration: none;">. If </ins><apll><ins style="font-weight: bold; text-decoration: none;">X</ins></apll> is <ins style="font-weight: bold; text-decoration: none;">omitted</ins>, <ins style="font-weight: bold; text-decoration: none;">it defaults to the last axis</ins>.</td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll><del style="font-weight: bold; text-decoration: none;">¯1↓⍴R</del></apll> using the <del style="font-weight: bold; text-decoration: none;">columns </del>of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll><del style="font-weight: bold; text-decoration: none;">≢R</del></apll> is <apll>1</apll>, the result is the Real array <apll>(<del style="font-weight: bold; text-decoration: none;">¯1↓⍴R</del>)⍴R</apll>, if <apll><del style="font-weight: bold; text-decoration: none;">≢R</del></apll> is <apll>2</apll>, the result is a Complex array, if <apll><del style="font-weight: bold; text-decoration: none;">≢R</del></apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll><del style="font-weight: bold; text-decoration: none;">≢R</del></apll> is <apll>8</apll>, the result is an Octonion array.</td></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> <td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP &mdash; otherwise, <apll>DOMAIN ERROR</apll>) whose <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, <apll>LENGTH ERROR</apll>.</td></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></tr></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><tr></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll><ins style="font-weight: bold; text-decoration: none;">(X≠⍳⍴⍴R)/⍴R</ins></apll> using the <ins style="font-weight: bold; text-decoration: none;">items along the <apll>X</apll>-axis </ins>of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll><ins style="font-weight: bold; text-decoration: none;">(⍴R)[X]</ins></apll> is <apll>1</apll>, the result is the Real array <apll>(<ins style="font-weight: bold; text-decoration: none;">(X≠⍳⍴⍴R)/⍴R</ins>)⍴R</apll>, if <apll><ins style="font-weight: bold; text-decoration: none;">(⍴R)[X]</ins></apll> is <apll>2</apll>, the result is a Complex array, if <apll><ins style="font-weight: bold; text-decoration: none;">(⍴R)[X]</ins></apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll><ins style="font-weight: bold; text-decoration: none;">(⍴R)[X]</ins></apll> is <apll>8</apll>, the result is an Octonion array.</td></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></tr></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l30">Line 30:</td>
<td colspan="2" class="diff-lineno">Line 33:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br /></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 8⍴(⍳8),⌽⍳8<br /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1<br /></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1 2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1 2 3<br /></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l50">Line 50:</td>
<td colspan="2" class="diff-lineno">Line 55:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt;&gt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &lt;<ins style="font-weight: bold; text-decoration: none;">[X] </ins>&gt;<ins style="font-weight: bold; text-decoration: none;">[X] </ins>R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt;&lt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(<del style="font-weight: bold; text-decoration: none;">¯1↑⍴R</del>)∊1 2 4 8</apll><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>R ←→ &gt;<ins style="font-weight: bold; text-decoration: none;">[X] </ins>&lt;<ins style="font-weight: bold; text-decoration: none;">[X] </ins>R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(<ins style="font-weight: bold; text-decoration: none;">⍴R</ins>)<ins style="font-weight: bold; text-decoration: none;">[X]</ins>∊1 2 4 8</apll><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Acknowledgements==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=Condense&diff=2962&oldid=prev
WikiSysop at 22:50, 9 April 2017
2017-04-09T22:50:12Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:50, 9 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>23<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>23<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">10J20</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">10<pn>J</pn>20</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1i2j3k4 5i6j7k8</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1 2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>1 2 3<br /></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35">Line 35:</td>
<td colspan="2" class="diff-lineno">Line 35:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1J2</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">3J4</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">5J6</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>J</pn>2</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">3<pn>J</pn>4</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">5<pn>J</pn>6</ins><br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">6J5</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">4J3</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">2J1</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">6<pn>J</pn>5</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">4<pn>J</pn>3</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">2<pn>J</pn>1</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">1i2j3k4</del>&nbsp;&nbsp;&nbsp;&nbsp;<del style="font-weight: bold; text-decoration: none;">5i6j7k8</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">9i10j11k12</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4</ins>&nbsp;&nbsp;&nbsp;&nbsp;<ins style="font-weight: bold; text-decoration: none;">5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12</ins><br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">12i11j10k9</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">8i7j6k5</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">4i3j2k1</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24<br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;<del style="font-weight: bold; text-decoration: none;">1i2j3k4l5ij6jk7kl8</del>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<del style="font-weight: bold; text-decoration: none;">9i10j11k12l13ij14jk15kl16</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">17i18j19k20l21ij22jk23kl24</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&nbsp;<ins style="font-weight: bold; text-decoration: none;">1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8</ins>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<ins style="font-weight: bold; text-decoration: none;">9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24</ins><br /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">24i23j22k21l20ij19jk18kl17</del>&nbsp;<del style="font-weight: bold; text-decoration: none;">16i15j14k13l12ij11jk10kl9</del>&nbsp;&nbsp;&nbsp;<del style="font-weight: bold; text-decoration: none;">8i7j6k5l4ij3jk2kl1</del><br /></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17</ins>&nbsp;<ins style="font-weight: bold; text-decoration: none;">16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9</ins>&nbsp;&nbsp;&nbsp;<ins style="font-weight: bold; text-decoration: none;">8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1</ins><br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=Condense&diff=2959&oldid=prev
WikiSysop at 20:57, 9 April 2017
2017-04-09T20:57:44Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:57, 9 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l47">Line 47:</td>
<td colspan="2" class="diff-lineno">Line 47:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>2 3<br /></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></apll></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Identities==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><apll>R ←→ &lt;&gt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><apll>R ←→ &gt;&lt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(¯1↑⍴R)∊1 2 4 8</apll><br /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Acknowledgements==</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><p>This symbol and its name were suggested by David A. Rabenhorst.</p></div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=Condense&diff=2958&oldid=prev
WikiSysop: Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td val..."
2017-04-09T19:43:47Z
<p>Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td val..."</p>
<p><b>New page</b></p><div><table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""><br />
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<table border="0" cellpadding="5" cellspacing="0" summary=""><br />
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<td valign="top"><apll>Z←&lt;R</apll></td><br />
<td></td><br />
<td></td><br />
<td>converts <apll>R</apll> to a Hypercomplex array if <apll>(≢R)∊2 4 8</apll> or a Real array if <apll>1=≢R</apll>.</td><br />
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<td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP &mdash; otherwise, <apll>DOMAIN ERROR</apll>) whose number of columns (<apll>≢R</apll>) is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, <apll>LENGTH ERROR</apll>.</td><br />
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<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>¯1↓⍴R</apll> using the columns of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>≢R</apll> is <apll>1</apll>, the result is the Real array <apll>(¯1↓⍴R)⍴R</apll>, if <apll>≢R</apll> is <apll>2</apll>, the result is a Complex array, if <apll>≢R</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>≢R</apll> is <apll>8</apll>, the result is an Octonion array.</td><br />
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<p>For example,</p><br />
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<apll>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;23<br /><br />
23<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20<br /><br />
10J20<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8<br /><br />
1i2j3k4 5i6j7k8<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br /><br />
1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br /><br />
1 2 3<br /><br />
4 5 6<br /><br />
2 3<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6<br /><br />
1J2&nbsp;3J4&nbsp;5J6<br /><br />
6J5&nbsp;4J3&nbsp;2J1<br /><br />
2 3<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12<br /><br />
1i2j3k4&nbsp;&nbsp;&nbsp;&nbsp;5i6j7k8&nbsp;9i10j11k12<br /><br />
12i11j10k9&nbsp;8i7j6k5&nbsp;4i3j2k1<br /><br />
2 3<br /><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24<br /><br />
&nbsp;1i2j3k4l5ij6jk7kl8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9i10j11k12l13ij14jk15kl16&nbsp;17i18j19k20l21ij22jk23kl24<br /><br />
24i23j22k21l20ij19jk18kl17&nbsp;16i15j14k13l12ij11jk10kl9&nbsp;&nbsp;&nbsp;8i7j6k5l4ij3jk2kl1<br /><br />
2 3<br /><br />
</apll><br />
<br />
<p>This symbol and its name were suggested by David A. Rabenhorst.</p></div>
WikiSysop