https://en.formulasearchengine.com/index.php?title=Cass_criterion&feed=atom&action=historyCass criterion - Revision history2024-03-28T15:21:19ZRevision history for this page on the wikiMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Cass_criterion&diff=20866&oldid=preven>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q83420752013-03-29T13:05:37Z<p><a href="/index.php?title=User:Addbot&action=edit&redlink=1" class="new" title="User:Addbot (page does not exist)">Bot:</a> Migrating 1 interwiki links, now provided by <a href="https://en.wikipedia.org/wiki/Wikidata" class="extiw" title="wikipedia:Wikidata">Wikidata</a> on <a href="/index.php?title=D:q8342075&action=edit&redlink=1" class="new" title="D:q8342075 (page does not exist)">d:q8342075</a></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:05, 29 March 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;">A number of ρeople state they desire to change theiг undesirable habits and initіɑte сonsuming meals which can be healthier even so</del>, <del style="font-weight: bold; text-decoration: none;">not as </del>a <del style="font-weight: bold; text-decoration: none;">lot of people do what they have </del>to <del style="font-weight: bold; text-decoration: none;">do and decіde on food items which can </del>be <del style="font-weight: bold; text-decoration: none;">more wholesome</del>. <del style="font-weight: bold; text-decoration: none;">Іt јust usually takes new ideas about changіng undesirable habits with much better types</del>, <del style="font-weight: bold; text-decoration: none;">and you could discover frеsh info </del>in <del style="font-weight: bold; text-decoration: none;">this article.<br><br>Rather than consuming mercɦandise crеated using bright white flօur</del>, <del style="font-weight: bold; text-decoration: none;">consսme ԝhole grain meals. 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The ideas you read through on this page arе ɑ fantastic commence to receiving you to realize how ѕignificant it would be to eat apρгoрriate.</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> </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>, a <ins style="font-weight: bold; text-decoration: none;">'''polynomial Diophantine equation''' is an [[indeterminate equation|indeterminate]] [[polynomial]] [[equation]] for which one seeks solutions restricted </ins>to be <ins style="font-weight: bold; text-decoration: none;">[[polynomial]]s in the indeterminate</ins>. <ins style="font-weight: bold; text-decoration: none;">A [[Diophantine equation]]</ins>, in <ins style="font-weight: bold; text-decoration: none;">general</ins>, is <ins style="font-weight: bold; text-decoration: none;">one where </ins>the <ins style="font-weight: bold; text-decoration: none;">solutions </ins>are <ins style="font-weight: bold; text-decoration: none;">restricted </ins>to <ins style="font-weight: bold; text-decoration: none;">some algebraic system, typically integers</ins>. <ins style="font-weight: bold; text-decoration: none;"> (In another usage ) </ins>''<ins style="font-weight: bold; text-decoration: none;">Diophantine'' refers to </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Hellenistic civilization|Hellenistic]] mathematician of </ins>the <ins style="font-weight: bold; text-decoration: none;">3rd century, [[Diophantus]] of [[Alexandria]], who made initial studies of integer Diophantine equations</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;">An important type </ins>of <ins style="font-weight: bold; text-decoration: none;">polynomial Diophantine equations takes the form:</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;">:</ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">sa+tb=c\ </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;">where ''a''</ins>, <ins style="font-weight: bold; text-decoration: none;">''b''</ins>, <ins style="font-weight: bold; text-decoration: none;">and ''c'' are known polynomials</ins>, and <ins style="font-weight: bold; text-decoration: none;">we wish to solve </ins>for <ins style="font-weight: bold; text-decoration: none;">''s'' </ins>and<ins style="font-weight: bold; text-decoration: none;">&nbsp;''t''</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>A <ins style="font-weight: bold; text-decoration: none;">simple example (and </ins>a <ins style="font-weight: bold; text-decoration: none;">solution) </ins>is<ins style="font-weight: bold; text-decoration: none;">:</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;">:</ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">s(x^2+1)+t(x^3+1)=2x\ </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>:<ins style="font-weight: bold; text-decoration: none;"><math>s=-x^3-x^2+x\ <</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><ins style="font-weight: bold; text-decoration: none;">:<math>t=x^2+x</ins>.<ins style="font-weight: bold; text-decoration: none;">\ <</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;">A necessary and sufficient condition for </ins>a <ins style="font-weight: bold; text-decoration: none;">polynomial Diophantine equation </ins>to <ins style="font-weight: bold; text-decoration: none;">have a solution is </ins>for <ins style="font-weight: bold; text-decoration: none;">''c'' </ins>to <ins style="font-weight: bold; text-decoration: none;">be </ins>a <ins style="font-weight: bold; text-decoration: none;">multiple </ins>of the <ins style="font-weight: bold; text-decoration: none;">[[greatest common divisor|GCD]] </ins>of ''a<ins style="font-weight: bold; text-decoration: none;">'' </ins>and<ins style="font-weight: bold; text-decoration: none;">&nbsp;''b''</ins>. <ins style="font-weight: bold; text-decoration: none;"> In </ins>the <ins style="font-weight: bold; text-decoration: none;">example above</ins>, the <ins style="font-weight: bold; text-decoration: none;">GCD </ins>of <ins style="font-weight: bold; text-decoration: none;">''a'' and ''b'' was&nbsp;1, so solutions would exist </ins>for <ins style="font-weight: bold; text-decoration: none;">any value of c</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;">Solutions </ins>to <ins style="font-weight: bold; text-decoration: none;">polynomial Diophantine equations are not unique</ins>. <ins style="font-weight: bold; text-decoration: none;"> Any multiple </ins>of <<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">ab</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">(say </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">rab</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">) </ins>can <ins style="font-weight: bold; text-decoration: none;">be used </ins>to <ins style="font-weight: bold; text-decoration: none;">transform </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">s</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> and <ins style="font-weight: bold; text-decoration: none;"><math></ins>t<<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">into another solution </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">s'=s+rb</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">t</ins>'<ins style="font-weight: bold; text-decoration: none;">=</ins>t-<ins style="font-weight: bold; text-decoration: none;">ra<</ins>/<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">:</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;">:</ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">(s+rb)a+(t-ra)</ins>b<ins style="font-weight: bold; text-decoration: none;">=c</ins>.<ins style="font-weight: bold; text-decoration: none;">\ </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;">Some polynomial Diophantine equations can </ins>be <ins style="font-weight: bold; text-decoration: none;">solved using </ins>the [<ins style="font-weight: bold; text-decoration: none;">[extended Euclidean algorithm]</ins>], <ins style="font-weight: bold; text-decoration: none;">which works </ins>as well <ins style="font-weight: bold; text-decoration: none;">with polynomials </ins>as <ins style="font-weight: bold; text-decoration: none;">it does with integers</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 book</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;"> | last = Bronstein</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;"> | first = Manuel</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;"> | title = Symbolic Integration I</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;"> | publisher = [[Springer Science+Business Media|Springer]]</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;"> | date = 2005</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;"> | pages = 12–14</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;"> | isbn = 3</ins>-<ins style="font-weight: bold; text-decoration: none;">540</ins>-<ins style="font-weight: bold; text-decoration: none;">21493</ins>-<ins style="font-weight: bold; text-decoration: none;">3 }}</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:Algebra]]</ins></div></td></tr>
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