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	<updated>2026-05-20T14:05:37Z</updated>
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		<title>en&gt;Colonies Chris: sp, date &amp; link fixes; unlinking common words, replaced: Euler-Lagrange → Euler–Lagrange using AWB</title>
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		<updated>2014-03-09T22:05:21Z</updated>

		<summary type="html">&lt;p&gt;sp, date &amp;amp; link fixes; unlinking common words, replaced: Euler-Lagrange → Euler–Lagrange using &lt;a href=&quot;/index.php?title=Testwiki:AWB&amp;amp;action=edit&amp;amp;redlink=1&quot; class=&quot;new&quot; title=&quot;Testwiki:AWB (page does not exist)&quot;&gt;AWB&lt;/a&gt;&lt;/p&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 00:05, 10 March 2014&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In the mathematical subject of [[group theory]], the &#039;&#039;&#039;Hanna Neumann conjecture&#039;&#039;&#039; &lt;/del&gt;is &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;a statement about the [[rank of a group|rank]] of the intersection of two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]]. The conjecture was posed by [[Hanna Neumann]] in 1957&lt;/del&gt;.&amp;lt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ref name=&quot;HN57&quot;&lt;/del&gt;&amp;gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Hanna Neumann&lt;/del&gt;. &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&#039;&#039;On the intersection of finitely generated free groups&lt;/del&gt;. &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Addendum.&#039;&#039; &lt;/del&gt;[&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[Publicationes Mathematicae Debrecen]], vol&lt;/del&gt;. &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;5 (1957), &lt;/del&gt;p&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;. 128&amp;lt;&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ref&amp;gt; In 2011, the conjecture was proved independently by Igor Mineyev&amp;lt;ref name=&quot;proof&quot;&amp;gt;Igor Minevev,&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;My name &lt;/ins&gt;is &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Wiley (38 years old) and my hobbies are Trainspotting and Amateur radio&lt;/ins&gt;.&amp;lt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;br&lt;/ins&gt;&amp;gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;xunjie 金属リベット装飾財布Cabbeen2。&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;[http://&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;annals&lt;/del&gt;.&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;math&lt;/del&gt;.&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;princeton.edu&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;2012&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;175-1&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;p11&lt;/del&gt;/ &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&quot;Submultiplicativity and the Hanna Neumann Conjecture.&quot;&lt;/del&gt;]  &lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;足のユニークな量を体験して子供を持って来る。&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Ann. of Math., 175 (2012), no. 1, 393-414&amp;lt;/ref&amp;gt; and Joel Friedman. &amp;lt;ref name=&quot;proof2&quot;&amp;gt;Joel Friedman,&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;金属リベット装飾財布Cabbeen2。 [http://www&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;otto-weibel&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ch/media/nobs/header/shop/celine/ celine ؔ�� ��ǥ��`��] 強い競争力を示した臨時ブランドの影響力！ ①ベンの原稿がより多くの情報の普及のため、&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;[http://www.&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;math&lt;/del&gt;.&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ubc.ca&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;~jf&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;pubs&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;web_stuff&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;shnc_memoirs.pdf &quot;Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture.&quot;] &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;まだ同じ大きさとルーズ、&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;to appear in Memoirs of the AMS&lt;/del&gt;&amp;lt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;/ref&lt;/del&gt;&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;およびリストラクチャリングを引き継いだ。 &lt;/ins&gt;[&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;http://anghaeltacht&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;net/ctg/cocog/&lt;/ins&gt;p/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;top/gaga/ �����ߥ�� �rӋ ��ǥ��`��] リンカーンセンターを終了しますスタジオはファッションショーを開催しました。&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;誰もが個人的に受け取ったがっている。&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;一種類の導入に集中する大胆な革新のための製品の詳細は、&lt;/ins&gt;[http://&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;www&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;otto-weibel&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ch/media/nobs&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;header&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;shop&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;celine&lt;/ins&gt;/ &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;����`�� �Хå� �ȩ`�� ����&lt;/ins&gt;] &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;カルダン企業の社会的責任を積極的に期待に応え愛と強さを送信するために助けを必要とする子どもたちの大半をし続ける！ 「リード·へ「フロントライン、&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Q：あなたの欲しい物のリストである単一の製品のアカデミックなスタイルのために？ ：ラルフローレンのボートシューズ、&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;慎重に石を選択したため、&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;優しい隣人を伝えるために、 &lt;/ins&gt;[http://www.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;montessorimatt&lt;/ins&gt;.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;com/images/small/3cj&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;b4t&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;shop&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;melissa&lt;/ins&gt;/ &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;���&amp;lt;br&amp;gt;&lt;/ins&gt;&amp;lt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;br&lt;/ins&gt;&amp;gt; &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ѥ ���n]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==History==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;My web blog &lt;/ins&gt;- [http://&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;physicianresources&lt;/ins&gt;.com/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;cache&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;store&lt;/ins&gt;/&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;paul&lt;/ins&gt;.html &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;ポールスミスバッグ レディース&lt;/ins&gt;]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The subject of the conjecture was originally motivated by a 1954 theorem of Howson&amp;lt;ref&amp;gt;A. G. Howson. &#039;&#039;On the intersection of finitely generated free groups.&#039;&#039; [[Journal of the London Mathematical Society]], vol. 29 (1954), pp. 428&amp;amp;ndash;434&amp;lt;/ref&amp;gt; who proved that the intersection of any two [[finitely generated group|finitely generated]] [[subgroup]]s of a [[free group]] is always finitely generated, that is, has finite [[rank of a group|rank]]. In this paper Howson proved that if &#039;&#039;H&#039;&#039; and &#039;&#039;K&#039;&#039; are [[subgroup]]s of a free group &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) of finite ranks &#039;&#039;n&#039;&#039;&amp;amp;nbsp;≥&amp;amp;nbsp;1 and &#039;&#039;m&#039;&#039;&amp;amp;nbsp;≥&amp;amp;nbsp;1 then the rank &#039;&#039;s&#039;&#039; of &#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;K&#039;&#039; satisfies:&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;s&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1 ≤ 2&#039;&#039;mn&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;&#039;&#039;m&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;&#039;&#039;n&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In a 1956 paper&amp;lt;ref&amp;gt;Hanna Neumann. &#039;&#039;On the intersection of finitely generated free groups.&#039;&#039; Publicationes Mathematicae Debrecen, vol. 4 (1956), 186&amp;amp;ndash;189.&amp;lt;/ref&amp;gt; [[Hanna Neumann]] improved this bound by showing that : &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;s&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1 ≤ 2&#039;&#039;mn&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;&#039;&#039;2m&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;&#039;&#039;n&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;In a 1957 addendum,&amp;lt;ref name=&quot;HN57&quot;/&amp;gt; Hanna Neumann further improved this bound to show that under the above assumptions&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;s&#039;&#039; &amp;amp;minus; 1 ≤ 2(&#039;&#039;m&#039;&#039; &amp;amp;minus; 1)(&#039;&#039;n&#039;&#039; &amp;amp;minus; 1).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;s&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1 ≤ (&#039;&#039;m&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1)(&#039;&#039;n&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;This statement became known as the &#039;&#039;Hanna Neumann conjecture&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==Formal statement==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Let &#039;&#039;H&#039;&#039;, &#039;&#039;K&#039;&#039;  ≤ &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) be two nontrivial finitely generated subgroups of a [[free group]] &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) and let &#039;&#039;L&#039;&#039;&amp;amp;nbsp;=&amp;amp;nbsp;&#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;K&#039;&#039; be the intersection of &#039;&#039;H&#039;&#039; and &#039;&#039;K&#039;&#039;.  The conjecture says that in this case&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:rank(&#039;&#039;L&#039;&#039;)&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1 ≤ (rank(&#039;&#039;H&#039;&#039;)&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1)(rank(&#039;&#039;K&#039;&#039;)&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;1).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Here for a group &#039;&#039;G&#039;&#039; the quantity rank(&#039;&#039;G&#039;&#039;) is the [[rank of a group|rank]] of &#039;&#039;G&#039;&#039;, that is, the smallest size of a [[generating set of a group|generating set]] for &#039;&#039;G&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Every [[subgroup]] of a [[free group]] is known to be [[free group|free]] itself and the [[rank of a group|rank]] of a [[free group]] is equal to the size of any free basis of that free group.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==Strengthened Hanna Neumann conjecture==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;If &#039;&#039;H&#039;&#039;, &#039;&#039;K&#039;&#039;  ≤ &#039;&#039;G&#039;&#039; are two subgroups of a [[group (mathematics)|group]] &#039;&#039;G&#039;&#039; and if &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039; ∈ &#039;&#039;G&#039;&#039; define the same [[double coset]] &#039;&#039;HaK&amp;amp;nbsp;=&amp;amp;nbsp;HbK&#039;&#039; then the [[subgroup]]s &#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;aKa&#039;&#039;&amp;lt;sup&amp;gt;&amp;amp;minus;1&amp;lt;/sup&amp;gt; and &#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;bKb&#039;&#039;&amp;lt;sup&amp;gt;&amp;amp;minus;1&amp;lt;/sup&amp;gt; are [[Conjugacy class|conjugate]] in &#039;&#039;G&#039;&#039; and thus have the same [[rank of a group|rank]]. It is known that if &#039;&#039;H&#039;&#039;, &#039;&#039;K&#039;&#039;  ≤ &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) are [[finitely generated group|finitely generated]] subgroups of a finitely generated [[free group]] &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) then there exist at most finitely many double coset classes &#039;&#039;HaK&#039;&#039; in &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) such that &#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;aKa&#039;&#039;&amp;lt;sup&amp;gt;&amp;amp;minus;1&amp;lt;/sup&amp;gt;&amp;amp;nbsp;≠&amp;amp;nbsp;{1}. Suppose that at least one such double coset exists and let &#039;&#039;a&#039;&#039;&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;,...,&#039;&#039;a&#039;&#039;&amp;lt;sub&amp;gt;&#039;&#039;n&#039;&#039;&amp;lt;/sub&amp;gt; be all the distinct representatives of such double cosets. The &#039;&#039;strengthened Hanna Neumann conjecture&#039;&#039;, formulated by her son [[Walter Neumann]] (1990),&amp;lt;ref name=&quot;WN&quot;&amp;gt;Walter Neumann. &#039;&#039;On intersections of finitely generated subgroups of free groups.&#039;&#039; Groups&amp;amp;ndash;Canberra 1989, pp. 161&amp;amp;ndash;170. Lecture Notes in Mathematics, vol. 1456, Springer, Berlin, 1990; ISBN 3&lt;/del&gt;-&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;540-53475-X&amp;lt;/ref&amp;gt; states that in this situation&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&amp;lt;math&amp;gt;\sum_{i=1}^n [{\rm rank}(H\cap a_iKa_{i}^{-1})-1]  \le ({\rm rank}(H)-1)({\rm rank}(K)-1).&amp;lt;/math&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==Partial results and other generalizations==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*In 1971 Burns improved&amp;lt;ref&amp;gt;Robert G. Burns.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;[http://&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;www.springerlink&lt;/del&gt;.com/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;content&lt;/del&gt;/&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;u75hh74lu1053790&lt;/del&gt;/ &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&#039;&#039;On the intersection of finitely generated subgroups of a free group.&#039;&#039;] &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Mathematische Zeitschrift]], vol. 119 (1971), pp. 121&amp;amp;ndash;130.&amp;lt;/ref&amp;gt; Hanna Neumann&#039;s 1957 bound and proved that under the same assumptions as in Hanna Neumann&#039;s paper one has&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;:&#039;&#039;s&#039;&#039; ≤ 2&#039;&#039;mn&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;3&#039;&#039;m&#039;&#039;&amp;amp;nbsp;&amp;amp;minus;&amp;amp;nbsp;2&#039;&#039;n&#039;&#039;&amp;amp;nbsp;+&amp;amp;nbsp;4.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*In a 1990 paper,&amp;lt;ref name=&quot;WN&quot;/&amp;gt; Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*[[Gábor Tardos|Tardos]] (1992)&amp;lt;ref&amp;gt;Gábor Tardos. [http://www.springerlink.com/content/n013g5rx543x4748/ &#039;&#039;On the intersection of subgroups of a free group.&#039;&#039;]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Inventiones Mathematicae]], vol. 108 (1992), no. 1, pp. 29&amp;amp;ndash;36.&amp;lt;/ref&amp;gt; established the Hanna Neumann Conjecture for the case where at least one of the subgroups &#039;&#039;H&#039;&#039; and &#039;&#039;K&#039;&#039; of &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of [[Stallings subgroup graph]]s&amp;lt;ref&amp;gt;John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ &#039;&#039;Topology of finite graphs.&#039;&#039;] [[Inventiones Mathematicae]], vol. 71 (1983), no. 3, pp. 551&amp;amp;ndash;565&amp;lt;/ref&amp;gt; for analyzing subgroups of free groups and their intersections.  &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*Warren Dicks (1994)&amp;lt;ref&amp;gt;Warren Dicks. [http://www.springerlink.com/content/r526373840056u7q/ &#039;&#039;Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.&#039;&#039;] [[Inventiones Mathematicae]], vol. 117 (1994), no. 3, pp. 373&amp;amp;ndash;389&amp;lt;/ref&amp;gt; established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the &#039;&#039;amalgamated graph conjecture&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*Arzhantseva (2000) proved&amp;lt;ref&amp;gt;G. N. Arzhantseva. [http://www.ams.org/proc/2000-128-11/S0002-9939-00-05508-8/home&lt;/del&gt;.html &lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;&#039;&#039;A property of subgroups of infinite index in a free group&#039;&#039;] [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]] 128 (2000), 3205&amp;amp;ndash;3210.&amp;lt;/ref&amp;gt; that if &#039;&#039;H&#039;&#039; is a finitely generated subgroup of infinite index in &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;), then, in a certain statistical meaning, for a generic finitely generated subgroup &amp;lt;math&amp;gt;K&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;F(X)&amp;lt;/math&amp;gt;, we have &#039;&#039;H&#039;&#039;&amp;amp;nbsp;∩&amp;amp;nbsp;&#039;&#039;gKg&#039;&#039;&amp;lt;sup&amp;gt;&amp;amp;minus;1&amp;lt;/sup&amp;gt;&amp;amp;nbsp;=&amp;amp;nbsp;{1} for all &#039;&#039;g&#039;&#039; in &#039;&#039;F&#039;&#039;. Thus, the strengthened Hanna Neumann conjecture holds for every &#039;&#039;H&#039;&#039; and a generic &#039;&#039;K&#039;&#039;.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*In 2001 Dicks and Formanek used this equivalence to prove the strengthened Hanna Neumann Conjecture in the case when one of the subgroups &#039;&#039;H&#039;&#039; and &#039;&#039;K&#039;&#039; of &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) has rank at most three.&amp;lt;ref&amp;gt;Warren Dicks, and Edward Formanek. &#039;&#039;The rank three case of the Hanna Neumann conjecture.&#039;&#039; Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113&amp;amp;ndash;151&amp;lt;/ref&amp;gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*Khan (2002)&amp;lt;ref&amp;gt;Bilal Khan. &#039;&#039;Positively generated subgroups of free groups and the Hanna Neumann conjecture.&#039;&#039; Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 155&amp;amp;ndash;170,&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Contemporary Mathematics, vol. 296, [[American Mathematical Society]], Providence, RI, 2002; ISBN 0-8218-2822-3&amp;lt;/ref&amp;gt; and, independently, Meakin and Weil (2002),&amp;lt;ref&amp;gt;J. Meakin, and P. Weil. [http://www.springerlink.com/content/m742547j1g534g40/ &#039;&#039;Subgroups of free groups: a contribution to the Hanna Neumann conjecture.&#039;&#039;] &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). &lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Geometriae Dedicata]], vol. 94 (2002), pp. 33&amp;amp;ndash;43.&amp;lt;/ref&amp;gt; showed that the conclusion of the strengthened Hanna Neumann conjecture holds if one of the subgroups &#039;&#039;H&#039;&#039;, &#039;&#039;K&#039;&#039; of &#039;&#039;F&#039;&#039;(&#039;&#039;X&#039;&#039;) is &#039;&#039;positively generated&#039;&#039;, that is, generated by a finite set of words that involve only elements of &#039;&#039;X&#039;&#039; but not of &#039;&#039;X&#039;&#039;&amp;lt;sup&amp;gt;&amp;amp;minus;1&amp;lt;/sup&amp;gt; as letters.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*Ivanov&amp;lt;ref&amp;gt;S. V. Ivanov. &#039;&#039;Intersecting free subgroups in free products of groups.&#039;&#039; International Journal of Algebra and Computation, vol. 11 (2001), no. 3, pp. 281&amp;amp;ndash;290&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;S. V. Ivanov. &#039;&#039;On the Kurosh rank of the intersection of subgroups in free products of groups&#039;&#039;. [[Advances in Mathematics]], vol. 218 (2008), no. 2, pp. 465&amp;amp;ndash;484&amp;lt;/ref&amp;gt; and, subsequently, Dicks and Ivanov,&amp;lt;ref&amp;gt;Warren Dicks, and S. V. Ivanov. &#039;&#039;On the intersection of free subgroups in free products of groups.&#039;&#039; Mathematical Proceedings of the Cambridge Philosophical Society, vol. 144 (2008), no. 3, pp. 511&amp;amp;ndash;534&amp;lt;/ref&amp;gt; obtained analogs and generalizations of Hanna Neumann&#039;s results for the intersection of [[subgroup]]s &#039;&#039;H&#039;&#039; and &#039;&#039;K&#039;&#039; of a [[free product]] of several groups.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*Wise (2005) showed&amp;lt;ref&amp;gt;[http://blms.oxfordjournals.org.proxy2.library.uiuc.edu/cgi/content/abstract/37/5/697 &#039;&#039;The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture.&#039;&#039;] [[Bulletin of the London Mathematical Society]], vol. 37 (2005), no. 5, pp. 697&amp;amp;ndash;705&amp;lt;/ref&amp;gt; that the strengthened Hanna Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is &#039;&#039;coherent&#039;&#039; (that is, every [[finitely generated group|finitely generated]] subgroup in such a group is [[finitely presented group|finitely presented]]).&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==See also==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*[[Rank of a group]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;*[[Geometric group theory]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;==References==&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;{{reflist}}&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Category:Group theory]]&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;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;&quot;&gt;&lt;div&gt;&lt;del style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[Category:Geometric group theory]&lt;/del&gt;]&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;/table&gt;</summary>
		<author><name>en&gt;Colonies Chris</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/index.php?title=Variational_integrator&amp;diff=22849&amp;oldid=prev</id>
		<title>78.91.45.155: In References: It&#039;s G. Wanner, not Warner</title>
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		<updated>2012-09-03T09:34:01Z</updated>

		<summary type="html">&lt;p&gt;In References: It&amp;#039;s G. Wanner, not Warner&lt;/p&gt;
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		<title>en&gt;Yobot: Auto-tagging, removed stub, orphan tags using AWB</title>
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		<updated>2010-04-10T19:52:58Z</updated>

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