https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Banded_waveguide_synthesis 2026-08-08T10:15:37Z Revision history for this page on the wiki MediaWiki 1.47.0-wmf.7 https://en.formulasearchengine.com/w/index.php?title=Banded_waveguide_synthesis&diff=255519&oldid=prev 2014-09-27T15:24:00Z

<p>fixed SuperCollider link</p> <a href="https://en.formulasearchengine.com/w/index.php?title=Banded_waveguide_synthesis&amp;diff=255519&amp;oldid=17808">Show changes</a>

212.247.205.73 https://en.formulasearchengine.com/w/index.php?title=Banded_waveguide_synthesis&diff=17808&oldid=prev 2011-09-27T10:25:12Z

<p>[Pu405]Misc citation tidying. You can <a href="/w/index.php?title=WP:UCB&amp;action=edit&amp;redlink=1" class="new" title="WP:UCB (page does not exist)">use this bot</a> yourself. <a href="/w/index.php?title=WP:DBUG&amp;action=edit&amp;redlink=1" class="new" title="WP:DBUG (page does not exist)">Report bugs here</a>.</p> <p><b>New page</b></p><div>In [[mathematics]], the &#039;&#039;&#039;Gromov product&#039;&#039;&#039; is a concept in the theory of [[metric space]]s named after the mathematician [[Mikhail Gromov (mathematician)|Mikhail Gromov]]. Intuitively, the Gromov product measures the distance for which two [[geodesic]]s starting at the same point remain &quot;close together&quot;. The Gromov product can also be used to define [[δ-hyperbolic space|&#039;&#039;&amp;delta;&#039;&#039;-hyperbolic metric spaces]] in the sense of Gromov.<br /> <br /> ==Definition==<br /> <br /> Let (&#039;&#039;X&#039;&#039;,&amp;nbsp;&#039;&#039;d&#039;&#039;) be a metric space and let &#039;&#039;x&#039;&#039;, &#039;&#039;y&#039;&#039;, &#039;&#039;z&#039;&#039;&amp;nbsp;∈&amp;nbsp;&#039;&#039;X&#039;&#039;. Then the &#039;&#039;&#039;Gromov product&#039;&#039;&#039; of &#039;&#039;y&#039;&#039; and &#039;&#039;z&#039;&#039; at &#039;&#039;x&#039;&#039;, denoted (&#039;&#039;y&#039;&#039;,&amp;nbsp;&#039;&#039;z&#039;&#039;)&lt;sub&gt;&#039;&#039;x&#039;&#039;&lt;/sub&gt;, is defined by<br /> <br /> :&lt;math&gt;(y, z)_{x} = \frac1{2} \big( d(x, y) + d(x, z) - d(y, z) \big).&lt;/math&gt;<br /> <br /> ==Properties==<br /> <br /> * The Gromov product is symmetric: (&#039;&#039;y&#039;&#039;,&amp;nbsp;&#039;&#039;z&#039;&#039;)&lt;sub&gt;&#039;&#039;x&#039;&#039;&lt;/sub&gt;&amp;nbsp;=&amp;nbsp;(&#039;&#039;z&#039;&#039;,&amp;nbsp;&#039;&#039;y&#039;&#039;)&lt;sub&gt;&#039;&#039;x&#039;&#039;&lt;/sub&gt;.<br /> <br /> * The Gromov product degenerates at the endpoints: (&#039;&#039;y&#039;&#039;,&amp;nbsp;&#039;&#039;z&#039;&#039;)&lt;sub&gt;&#039;&#039;y&#039;&#039;&lt;/sub&gt;&amp;nbsp;=&amp;nbsp;(&#039;&#039;y&#039;&#039;,&amp;nbsp;&#039;&#039;z&#039;&#039;)&lt;sub&gt;&#039;&#039;z&#039;&#039;&lt;/sub&gt;&amp;nbsp;=&amp;nbsp;0.<br /> <br /> * For any points &#039;&#039;p&#039;&#039;, &#039;&#039;q&#039;&#039;, &#039;&#039;x&#039;&#039;, &#039;&#039;y&#039;&#039; and &#039;&#039;z&#039;&#039;,<br /> <br /> ::&lt;math&gt;d(x, y) = (x, z)_{y} + (y, z)_{x},&lt;/math&gt;<br /> ::&lt;math&gt;0 \leq (y, z)_{x} \leq \min \big\{ d(y, x), d(z, x) \big\},&lt;/math&gt;<br /> ::&lt;math&gt;\big| (y, z)_{p} - (y, z)_{q} \big| \leq d(p, q),&lt;/math&gt;<br /> ::&lt;math&gt;\big| (x, y)_{p} - (x, z)_{p} \big| \leq d(y, z).&lt;/math&gt;<br /> <br /> * As mentioned in the introduction, the Gromov product measures how long geodesics remain close together. Namely, if &#039;&#039;x&#039;&#039;, &#039;&#039;y&#039;&#039; and &#039;&#039;z&#039;&#039; are three points of a &#039;&#039;δ&#039;&#039;-hyperbolic metric space then the initial segments of length (&#039;&#039;y&#039;&#039;,&amp;nbsp;&#039;&#039;z&#039;&#039;)&lt;sub&gt;&#039;&#039;x&#039;&#039;&lt;/sub&gt; of geodesics from &#039;&#039;x&#039;&#039; to &#039;&#039;y&#039;&#039; and &#039;&#039;x&#039;&#039; to &#039;&#039;z&#039;&#039; are no further than 2&#039;&#039;δ&#039;&#039; apart (in the sense of the [[Hausdorff distance]] between closed sets).<br /> <br /> * In fact, the Gromov product can be used to define &#039;&#039;δ&#039;&#039;-hyperbolic spaces in the sense of Gromov: (&#039;&#039;X&#039;&#039;,&amp;nbsp;&#039;&#039;d&#039;&#039;) is said to be &#039;&#039;&#039;&#039;&#039;δ&#039;&#039;-hyperbolic&#039;&#039;&#039; if, for all &#039;&#039;p&#039;&#039;, &#039;&#039;x&#039;&#039;, &#039;&#039;y&#039;&#039; and &#039;&#039;z&#039;&#039; in &#039;&#039;X&#039;&#039;,<br /> <br /> ::&lt;math&gt;(x, z)_{p} \geq \min \big\{ (x, y)_{p}, (y, z)_{p} \big\} - \delta.&lt;/math&gt;<br /> <br /> ==References==<br /> <br /> * {{cite book<br /> | last = Kapovich<br /> | first = Ilya<br /> | coauthors = Benakli, Nadia<br /> | chapter = Boundaries of hyperbolic groups<br /> | title = Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001)<br /> | series = Contemp. Math. 296<br /> | pages = 39&amp;ndash;93<br /> | publisher = Amer. Math. Soc.<br /> | location = Providence, RI<br /> | year = 2002<br /> | id = {{MathSciNet|id=1921706}}<br /> }}<br /> * {{cite web<br /> | last = Väisälä<br /> | first = Jussi<br /> | title = Gromov hyperbolic spaces<br /> | url = http://www.helsinki.fi/~jvaisala/grobok.pdf<br /> | format = PDF<br /> | year = 2004<br /> | accessdate = 2007-08-28<br /> }}<br /> <br /> [[Category:Metric geometry]]</div>

en>Citation bot 1