https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=Standard_ruler 2026-04-19T12:13:33Z Revision history for this page on the wiki MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Standard_ruler&diff=245357&oldid=prev 2014-12-28T10:02:11Z

<p>Bot: <a href="/index.php?title=User:FrescoBot/Links&amp;action=edit&amp;redlink=1" class="new" title="User:FrescoBot/Links (page does not exist)">link syntax</a> and minor changes</p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <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 12:02, 28 December 2014</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</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;">In [[mathematics]]</del>, <del style="font-weight: bold; text-decoration: none;">the '''affine hull''' of a [[set (mathematics)|set]] ''S'' in  [[Euclidean space]] '''R'''&lt;sup&gt;''n''&lt;/sup&gt; </del>is the <del style="font-weight: bold; text-decoration: none;">smallest [[affine set]] containing ''S'', or equivalently, the [[intersection (set theory)|intersection]] of all affine sets containing ''S''</del>. <del style="font-weight: bold; text-decoration: none;">Here, an </del>'<del style="font-weight: bold; text-decoration: none;">'affine set'' may </del>be <del style="font-weight: bold; text-decoration: none;">defined as the [[translation (mathematics)|translation]] of a [[vector subspace]]</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;">Hi there</ins>, <ins style="font-weight: bold; text-decoration: none;">I am Alyson Boon even though it </ins>is <ins style="font-weight: bold; text-decoration: none;">not </ins>the <ins style="font-weight: bold; text-decoration: none;">title on my birth certificate</ins>. <ins style="font-weight: bold; text-decoration: none;">She works as a travel agent but soon she</ins>'<ins style="font-weight: bold; text-decoration: none;">ll </ins>be <ins style="font-weight: bold; text-decoration: none;">on her own</ins>. <ins style="font-weight: bold; text-decoration: none;">Mississippi </ins>is the <ins style="font-weight: bold; text-decoration: none;">only place I</ins>'<ins style="font-weight: bold; text-decoration: none;">ve been residing </ins>in <ins style="font-weight: bold; text-decoration: none;">but I will have to transfer </ins>in a <ins style="font-weight: bold; text-decoration: none;">yr or </ins>two<ins style="font-weight: bold; text-decoration: none;">. To play lacross </ins>is one of the <ins style="font-weight: bold; text-decoration: none;">things she enjoys most</ins>.&lt;<ins style="font-weight: bold; text-decoration: none;">br</ins>&gt;&lt;<ins style="font-weight: bold; text-decoration: none;">br</ins>&gt;<ins style="font-weight: bold; text-decoration: none;">My web blog - </ins>[<ins style="font-weight: bold; text-decoration: none;">http:</ins>//<ins style="font-weight: bold; text-decoration: none;">www</ins>.<ins style="font-weight: bold; text-decoration: none;">chk</ins>.<ins style="font-weight: bold; text-decoration: none;">woobi</ins>.<ins style="font-weight: bold; text-decoration: none;">co</ins>.<ins style="font-weight: bold; text-decoration: none;">kr/xe/?document_srl=346069 free tarot readings</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> </div></td><td colspan="2" class="diff-side-added"></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;">The affine hull aff(''S'') of ''S'' </del>is the <del style="font-weight: bold; text-decoration: none;">set of all [[affine combination]]s of elements of </del>'<del style="font-weight: bold; text-decoration: none;">'S'', that is, </del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">:&lt;math&gt;\operatorname{aff} (S)=\left\{\sum_{i=1}^k \alpha_i x_i \Bigg | k&gt;0, \, x_i\</del>in <del style="font-weight: bold; text-decoration: none;">S, \, \alpha_i\</del>in <del style="font-weight: bold; text-decoration: none;">\mathbb{R}, \, \sum_{i=1}^k \alpha_i=1 \right\}.&lt;/math&gt;</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">==Examples==</del></div></td><td colspan="2" class="diff-side-added"></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;">*The affine hull of </del>a <del style="font-weight: bold; text-decoration: none;">set of </del>two <del style="font-weight: bold; text-decoration: none;">different points </del>is <del style="font-weight: bold; text-decoration: none;">the line through them.</del></div></td><td colspan="2" class="diff-side-added"></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;">*The affine hull of a set of three points not on </del>one <del style="font-weight: bold; text-decoration: none;">line is the plane going through them.</del></div></td><td colspan="2" class="diff-side-added"></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;">*The affine hull </del>of <del style="font-weight: bold; text-decoration: none;">a set of four points not in a plane in '''R'''&lt;sup&gt;''3''&lt;/sup&gt; is </del>the <del style="font-weight: bold; text-decoration: none;">entire space '''R'''&lt;sup&gt;''3''&lt;/sup&gt;</del>.</div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">==Properties==</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">* &lt;math&gt;\mathrm{aff}(\mathrm{aff}(S)) = \mathrm{aff}(S)</del>&lt;<del style="font-weight: bold; text-decoration: none;">/math</del>&gt;</div></td><td colspan="2" class="diff-side-added"></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;">* </del>&lt;<del style="font-weight: bold; text-decoration: none;">math</del>&gt;<del style="font-weight: bold; text-decoration: none;">\mathrm{aff}(S)&lt;/math&gt; is a </del>[<del style="font-weight: bold; text-decoration: none;">[closed set]]</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">==Related sets==</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">*If instead of an affine combination one uses a [[convex combination]], that is one requires in the formula above that all &lt;math&gt;\alpha_i&lt;</del>/<del style="font-weight: bold; text-decoration: none;">math&gt; be non-negative, one obtains the [[convex hull]] of ''S'', which cannot be larger than the affine hull of ''S'' as more restrictions are involved. </del></div></td><td colspan="2" class="diff-side-added"></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;">*The notion of [[conical combination]] gives rise to the notion of the [[conical hull]]</del></div></td><td colspan="2" class="diff-side-added"></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;">*If however one puts no restrictions at all on the numbers &lt;math&gt;\alpha_i&lt;</del>/<del style="font-weight: bold; text-decoration: none;">math&gt;, instead of an affine combination one has a [[linear combination]], and the resulting set is the [[linear span]] of ''S'', which contains the affine hull of ''S''</del>.</div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">==References==</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">* R</del>.<del style="font-weight: bold; text-decoration: none;">J</del>. <del style="font-weight: bold; text-decoration: none;">Webster, ''Convexity'', Oxford University Press, 1994</del>. <del style="font-weight: bold; text-decoration: none;">ISBN 0-19-853147-8.</del></div></td><td colspan="2" class="diff-side-added"></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> </div></td><td colspan="2" class="diff-side-added"></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;">[[Category:Affine geometry]]</del></div></td><td colspan="2" class="diff-side-added"></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;">[[Category:Closure operators]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr> </table>

en>FrescoBot https://en.formulasearchengine.com/index.php?title=Standard_ruler&diff=12117&oldid=prev 2014-01-13T22:16:19Z

<p>cap</p> <a href="https://en.formulasearchengine.com/index.php?title=Standard_ruler&amp;diff=12117&amp;oldid=245356">Show changes</a>

en>Florian Blaschke https://en.formulasearchengine.com/index.php?title=Standard_ruler&diff=245356&oldid=prev 2012-05-25T09:05:12Z

<p><span class="autocomment">See also: </span> Baryon_acoustic_oscillations</p> <a href="https://en.formulasearchengine.com/index.php?title=Standard_ruler&amp;diff=245356">Show changes</a>

en>Chris goulet