http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&feed=atom&action=history
Delta simple useful formula - Revision history
2024-03-29T09:37:10Z
Revision history for this page on the wiki
MediaWiki 1.28.0
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1313&oldid=prev
Tom at 06:08, 10 June 2016
2016-06-10T06:08:49Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 06:08, 10 June 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative at right-dense|next=Delta derivative of sum}}: Theorem 1.<del class="diffchange diffchange-inline">20 </del>(<del class="diffchange diffchange-inline">i</del>)</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative at right-dense|next=Delta derivative of sum}}: Theorem 1.<ins class="diffchange diffchange-inline">16 </ins>(<ins class="diffchange diffchange-inline">iv</ins>)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1282&oldid=prev
Tom at 05:27, 10 June 2016
2016-06-10T05:27:38Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 05:27, 10 June 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative at right-dense|next=<del class="diffchange diffchange-inline">Derivative </del>of sum}}: Theorem 1.20 (i)</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative at right-dense|next=<ins class="diffchange diffchange-inline">Delta derivative </ins>of sum}}: Theorem 1.20 (i)</div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1281&oldid=prev
Tom at 05:26, 10 June 2016
2016-06-10T05:26:44Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 05:26, 10 June 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; 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;">* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative at right-dense|next=Derivative of sum}}: Theorem 1.20 (i)</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1246&oldid=prev
Tom at 22:20, 9 June 2016
2016-06-09T22:20:21Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 22:20, 9 June 2016</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"><div class</del>=<del class="diffchange diffchange-inline">"toccolours mw-collapsible mw-collapsed" style</del>=<del class="diffchange diffchange-inline">"width:800px"></del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">Theorem==</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"><strong>[[Simple useful formula|Theorem]]:</strong> </del>Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Let $\mathbb{T}$ be a [[time scale]], let $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"><div class</del>=<del class="diffchange diffchange-inline">"mw-collapsible-content"></del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"><strong>Proof</del>:<del class="diffchange diffchange-inline"></strong>  █ </del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">==Proof=</ins>=</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"></div></del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 class='diff-marker'>−</td><td style="color:black; 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 class="diffchange diffchange-inline"></div></del></div></td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">==References==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; 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"> </td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">[[Category</ins>:<ins class="diffchange diffchange-inline">Theorem]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; 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 class="diffchange diffchange-inline">[[Category:Unproven]]</ins></div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1098&oldid=prev
Tom at 23:13, 31 May 2016
2016-05-31T23:13:38Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 23:13, 31 May 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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><strong>[[Simple useful formula|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:</div></td><td class='diff-marker'>+</td><td style="color:black; 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><strong>[[Simple useful formula|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], <ins class="diffchange diffchange-inline">let </ins>$t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].</div></td></tr>
</table>
Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1086&oldid=prev
Tom: Tom moved page Simple useful formula to Delta simple useful formula
2016-05-31T23:03:17Z
<p>Tom moved page <a href="/index.php/Simple_useful_formula" class="mw-redirect" title="Simple useful formula">Simple useful formula</a> to <a href="/index.php/Delta_simple_useful_formula" title="Delta simple useful formula">Delta simple useful formula</a></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<tr style='vertical-align: top;' lang='en'>
<td colspan='1' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='1' style="background-color: white; color:black; text-align: center;">Revision as of 23:03, 31 May 2016</td>
</tr><tr><td colspan='2' style='text-align: center;' lang='en'><div class="mw-diff-empty">(No difference)</div>
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Tom
http://timescalewiki.org/index.php?title=Delta_simple_useful_formula&diff=1084&oldid=prev
Tom: Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $\mathbb{T}$ be a time scale, $t,s \in \ma..."
2016-05-31T23:02:45Z
<p>Created page with "<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong><a href="/index.php/Simple_useful_formula" class="mw-redirect" title="Simple useful formula">Theorem</a>:</strong> Let $\mathbb{T}$ be a <a href="/index.php/Time_scale" title="Time scale">time scale</a>, $t,s \in \ma..."</p>
<p><b>New page</b></p><div><div class="toccolours mw-collapsible mw-collapsed" style="width:800px"><br />
<strong>[[Simple useful formula|Theorem]]:</strong> Let $\mathbb{T}$ be a [[time scale]], $t,s \in \mathbb{T}$, and $p \in \mathcal{R} \left( \mathbb{T},\mathbb{C} \right)$ be a [[regressive function]]. The following formula holds:<br />
$$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$<br />
where $e_p$ denotes the [[delta exponential]], $\sigma$ denotes the [[forward jump]], and $\mu$ denotes the [[forward graininess]].<br />
<div class="mw-collapsible-content"><br />
<strong>Proof:</strong> █ <br />
</div><br />
</div></div>
Tom