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Triple product property - Revision history
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en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)
2013-01-09T02:30:54Z
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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 03:30, 9 January 2013</td>
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<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;">I would like to introduce myself to you</del>, <del style="font-weight: bold; text-decoration: none;">I am Andrew </del>and <del style="font-weight: bold; text-decoration: none;">my spouse doesn</del>'<del style="font-weight: bold; text-decoration: none;">t like it </del>at <del style="font-weight: bold; text-decoration: none;">all</del>. <del style="font-weight: bold; text-decoration: none;">One </del>of the <del style="font-weight: bold; text-decoration: none;">things she loves most </del>is <del style="font-weight: bold; text-decoration: none;">canoeing and she</del>'s <del style="font-weight: bold; text-decoration: none;">been performing it </del>for <del style="font-weight: bold; text-decoration: none;">fairly a while</del>. <del style="font-weight: bold; text-decoration: none;">Ohio is where my house </del>is <del style="font-weight: bold; text-decoration: none;">but my spouse desires us </del>to <del style="font-weight: bold; text-decoration: none;">move</del>. <del style="font-weight: bold; text-decoration: none;">Distributing manufacturing is how he tends </del>to <del style="font-weight: bold; text-decoration: none;">make </del>a <del style="font-weight: bold; text-decoration: none;">living.</del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">My blog post </del>- <del style="font-weight: bold; text-decoration: none;">online reader</del>; [<del style="font-weight: bold; text-decoration: none;">http</del>:/<del style="font-weight: bold; text-decoration: none;">/cartoonkorea</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/<del style="font-weight: bold; text-decoration: none;">ce002</del>/<del style="font-weight: bold; text-decoration: none;">1093612 cartoonkorea</del>.<del style="font-weight: bold; text-decoration: none;">com</del>]<del style="font-weight: bold; text-decoration: none;">,</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;">{{about|Anderson's theorem in mathematics|the Anderson orthogonality theorem in physics|Anderson orthogonality theorem}}</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;">In [[mathematics]]</ins>, <ins style="font-weight: bold; text-decoration: none;">'''[[Theodore Wilbur Anderson|Anderson's]] theorem''' is a result in [[real analysis]] </ins>and <ins style="font-weight: bold; text-decoration: none;">[[geometry]] which says that the [[integral]] of an integrable, symmetric, unimodal, non-negative function ''f'' over an ''n''-dimensional [[convex body]] ''K'' does not decrease if ''K'' is translated inwards towards the origin. This is a natural statement, since the [[graph of a function|graph]] of </ins>'<ins style="font-weight: bold; text-decoration: none;">'f'' can be thought of as a hill with a single peak over the origin; however, for ''n''&nbsp;≥&nbsp;2, the proof is not entirely obvious, as there may be points ''x'' of the body ''K'' where the value ''f''(''x'') is larger than </ins>at <ins style="font-weight: bold; text-decoration: none;">the corresponding translate of ''x''.</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> </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;">Anderson's theorem also has an interesting application to [[probability theory]]</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> </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;">==Statement </ins>of the <ins style="font-weight: bold; text-decoration: none;">theorem==</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;">Let ''K'' be a convex body in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> that </ins>is <ins style="font-weight: bold; text-decoration: none;">[[symmetry|symmetric]] with respect to reflection in the origin, i.e. ''K''&nbsp;=&nbsp;&minus;''K''. Let ''f''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R''' be a non-[[negative number|negative]], symmetric, globally integrable function; i.e.</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;">* ''f''(''x'')&nbsp;≥&nbsp;0 for all ''x''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>;</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;">* ''f''(''x'')&nbsp;=&nbsp;''f''(&minus;''x'') for all ''x''&nbsp;∈&nbsp;'''R'''<sup>''n'</ins>'<ins style="font-weight: bold; text-decoration: none;"></sup>;</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;">* <math>\int_{\mathbb{R}^{n}} f(x) \, \mathrm{d} x < + \infty.</math></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> </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;">Suppose also that the super-[[level set]]</ins>s <ins style="font-weight: bold; text-decoration: none;">''L''(''f'',&nbsp;''t'') of ''f'', defined by</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> </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;">:<math>L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \},</math></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> </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;">are [[convex set|convex subsets]] of '''R'''<sup>''n''</sup> </ins>for <ins style="font-weight: bold; text-decoration: none;">every ''t''&nbsp;≥&nbsp;0</ins>. <ins style="font-weight: bold; text-decoration: none;">(This property </ins>is <ins style="font-weight: bold; text-decoration: none;">sometimes referred </ins>to <ins style="font-weight: bold; text-decoration: none;">as being '''unimodal'''</ins>.<ins style="font-weight: bold; text-decoration: none;">) Then, for any 0&nbsp;≤&nbsp;''c''&nbsp;≤&nbsp;1 and ''y''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>,</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> </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;">:<math>\int_{K} f(x + c y) \, \mathrm{d} x \geq \int_{K} f(x + y) \, \mathrm{d} x.</math></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> </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;">==Application </ins>to <ins style="font-weight: bold; text-decoration: none;">probability theory==</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;">Given </ins>a <ins style="font-weight: bold; text-decoration: none;">[[probability space]] (Ω,&nbsp;Σ,&nbsp;Pr), suppose that ''X''&nbsp;:&nbsp;Ω&nbsp;→&nbsp;'''R'''<sup>''n''</sup> is an '''R'''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">''n''</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>>-<ins style="font-weight: bold; text-decoration: none;">valued [[random variable]] with [[probability density function]] ''f''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp</ins>;[<ins style="font-weight: bold; text-decoration: none;">0,&nbsp;+∞) and that ''Y''&nbsp;</ins>:<ins style="font-weight: bold; text-decoration: none;">&nbsp;Ω&nbsp;→&nbsp;'''R'''<sup>''n''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup> is an [[Statistical independence|independent]] random variable. The probability density functions of many well-known probability distributions are ''p''-[[concave function|concave]] for some ''p'', and hence unimodal. If they are also symmetric (e.g</ins>. <ins style="font-weight: bold; text-decoration: none;">the [[Laplace distribution|Laplace]] and [[normal distribution]]s), then Anderson's theorem applies, in which case</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> </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;">:<math>\Pr ( X \in K ) \geq \Pr ( X + Y \in K )<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></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> </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;">for any origin-symmetric convex body ''K''&nbsp;⊆&nbsp;'''R'''<sup>''n''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sup>.</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> </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;">==References==</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;">* {{cite journal | last=Gardner | first=Richard J. | title=The Brunn-Minkowski inequality | journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc</ins>.]<ins style="font-weight: bold; text-decoration: none;">] (N.S.) | volume=39 | issue=3 | year=2002 | pages=355&ndash;405 (electronic) | doi=10.1090/S0273-0979-02-00941-2 }}</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> </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;">[[Category:Theorems in geometry]]</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;">[[Category:Probability theorems]]</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;">[[Category:Theorems in real analysis]]</ins></div></td></tr>
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en>Addbot
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en>Kri: Wikified differently
2012-06-22T22:16:26Z
<p>Wikified differently</p>
<p><b>New page</b></p><div>I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. One of the things she loves most is canoeing and she's been performing it for fairly a while. Ohio is where my house is but my spouse desires us to move. Distributing manufacturing is how he tends to make a living.<br><br>My blog post - online reader; [http://cartoonkorea.com/ce002/1093612 cartoonkorea.com],</div>
en>Kri