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Systematic risk - Revision history
2026-05-06T05:11:50Z
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24.131.80.54: improve
2014-11-05T23:59:03Z
<p>improve</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:59, 6 November 2014</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;">In [[mathematics]], particularly the study of [[Lie groups]], </del>a '<del style="font-weight: bold; text-decoration: none;">''Dunkl operator''' </del>is a <del style="font-weight: bold; text-decoration: none;">certain kind of [[mathematical operator]], involving [[differential operator]]s </del>but <del style="font-weight: bold; text-decoration: none;">also [[Reflection (mathematics)|reflection]]s </del>in <del style="font-weight: bold; text-decoration: none;">an underlying space</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;">Friends call him Royal. What she loves performing is playing croquet and she is trying to make it </ins>a <ins style="font-weight: bold; text-decoration: none;">profession. The job I</ins>'<ins style="font-weight: bold; text-decoration: none;">ve been occupying for many years </ins>is a <ins style="font-weight: bold; text-decoration: none;">bookkeeper </ins>but <ins style="font-weight: bold; text-decoration: none;">I've already applied for another 1. Her family members life </ins>in <ins style="font-weight: bold; text-decoration: none;">Delaware but she requirements to move simply because of her family</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">My weblog: [http:</ins>//<ins style="font-weight: bold; text-decoration: none;">Www.sethtemple.org</ins>/<ins style="font-weight: bold; text-decoration: none;">UserProfile</ins>/<ins style="font-weight: bold; text-decoration: none;">tabid</ins>/<ins style="font-weight: bold; text-decoration: none;">42</ins>/<ins style="font-weight: bold; text-decoration: none;">userId</ins>/<ins style="font-weight: bold; text-decoration: none;">161658</ins>/<ins style="font-weight: bold; text-decoration: none;">Default</ins>.<ins style="font-weight: bold; text-decoration: none;">aspx Www</ins>.<ins style="font-weight: bold; text-decoration: none;">sethtemple</ins>.<ins style="font-weight: bold; text-decoration: none;">org</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;">Formally, let ''G'' be a [[Coxeter group]] with reduced root system ''R'' and ''k''</del><<del style="font-weight: bold; text-decoration: none;">sub</del>><del style="font-weight: bold; text-decoration: none;">''v''</del><<del style="font-weight: bold; text-decoration: none;">/sub</del>> <del style="font-weight: bold; text-decoration: none;">a multiplicity function on ''R'' (so ''k''<sub>''u''<</del>/<del style="font-weight: bold; text-decoration: none;">sub> = ''k''<sub>''v''<</del>/<del style="font-weight: bold; text-decoration: none;">sub> whenever the reflections σ<sub>''u''</sub> and σ<sub>''v''<</del>/<del style="font-weight: bold; text-decoration: none;">sub> corresponding to the roots ''u'' and ''v'' are conjugate in ''G''). Then, the '''Dunkl operator''' is defined by:</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;">:<math>T_i f(x) = \frac{\partial}{\partial x_i} f(x) + \sum_{v\in R_+} k_v \frac{f(x) - f(x \sigma_v)}{\left\langle x, v\right\rangle} v_i<</del>/<del style="font-weight: bold; text-decoration: none;">math></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;">where <math>v_i <</del>/<del style="font-weight: bold; text-decoration: none;">math> is the ''i''-th component of ''v'', 1 ≤ ''i'' ≤ ''N'', ''x'' in ''R''<sup>''N''<</del>/<del style="font-weight: bold; text-decoration: none;">sup>, and ''f'' a smooth function on ''R''<sup>''N''<</del>/<del style="font-weight: bold; text-decoration: none;">sup>.</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;">Dunkl operators were introduced by {{harvs|txt|authorlink=Charles F. Dunkl|last=Dunkl|year=1989}}. One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy <math>T_i (T_j f(x)) = T_j (T_i f(x))<</del>/<del style="font-weight: bold; text-decoration: none;">math> just as partial derivatives do</del>. <del style="font-weight: bold; text-decoration: none;">Thus Dunkl operators represent a meaningful generalization of partial derivatives</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;">*{{Citation | last1=Dunkl | first1=Charles F</del>. <del style="font-weight: bold; text-decoration: none;">| title=Differential-difference operators associated to reflection groups | doi=10.2307/2001022 | mr=951883 | year=1989 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=311 | issue=1 | pages=167–183}}</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:Lie groups]</del>]</div></td><td colspan="2" class="diff-side-added"></td></tr>
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