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Quotient space (linear algebra) - Revision history
2024-03-29T07:08:17Z
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en>Michael Hardy: /* Generalization to locally convex spaces */
2014-06-17T17:22:35Z
<p><span dir="auto"><span class="autocomment">Generalization to locally convex spaces</span></span></p>
<a href="https://en.formulasearchengine.com/index.php?title=Quotient_space_(linear_algebra)&diff=234985&oldid=6906">Show changes</a>
en>Michael Hardy
https://en.formulasearchengine.com/index.php?title=Quotient_space_(linear_algebra)&diff=6906&oldid=prev
en>Quondum: /* See also */ link more directly
2013-11-04T04:32:56Z
<p><span dir="auto"><span class="autocomment">See also: </span> link more directly</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:32, 4 November 2013</td>
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It may <del style="font-weight: bold; text-decoration: none;">check for and download changes for Windows, Java and Adobe</del>. <del style="font-weight: bold; text-decoration: none;">Keeping updates present is an significant piece of superior computer wellness</del>. <del style="font-weight: bold; text-decoration: none;">It could additionally protect your individual and company confidentiality in addition to the </del>online <del style="font-weight: bold; text-decoration: none;">security.</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;">In [[statics]], </ins>a <ins style="font-weight: bold; text-decoration: none;">structure is '''statically indeterminate''' (or '''hyperstatic''')<ref>{{cite book |last= Matheson |first= James Adam Louis |authorlink= James Adam Louis Matheson </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= Hyperstatic structures: an introduction to the theory of statically indeterminate structures |year= 1971 |publisher= Butterworths }}</ref> when the [[static equilibrium]] equations are insufficient for determining the internal forces and reactions on that structure.</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;">Based on [[Newton's laws </ins>of <ins style="font-weight: bold; text-decoration: none;">motion]], the equilibrium equations available for a two-dimensional body are</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> \sum \vec F = 0 </math>: </ins>the <ins style="font-weight: bold; text-decoration: none;">vectorial sum </ins>of the <ins style="font-weight: bold; text-decoration: none;">[[force]]s acting on the body equals zero</ins>. This <ins style="font-weight: bold; text-decoration: none;">translates </ins>to</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;">::&Sigma; ''H'' = 0: the sum of the horizontal components of the forces equals zero;</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;">::&Sigma; ''V'' = 0: </ins>the <ins style="font-weight: bold; text-decoration: none;">sum </ins>of the <ins style="font-weight: bold; text-decoration: none;">vertical components of forces equals zero;</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;">\sum \vec M = 0 </ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">: </ins>the <ins style="font-weight: bold; text-decoration: none;">sum of </ins>the <ins style="font-weight: bold; text-decoration: none;">[[moment (physics)|moment]]s (about an arbitrary point) of all forces equals zero.</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;">[[Image:Statically indeterminate beam</ins>.<ins style="font-weight: bold; text-decoration: none;">png|thumb|350px|right|[[Free body diagram]] </ins>of a <ins style="font-weight: bold; text-decoration: none;">statically indeterminate [[Beam (structure)|beam]]</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;">In the [[Beam (structure)|beam]] construction </ins>on the <ins style="font-weight: bold; text-decoration: none;">right</ins>, <ins style="font-weight: bold; text-decoration: none;">the four unknown reactions are ''V''<sub>''A''</sub></ins>, <ins style="font-weight: bold; text-decoration: none;">''V''<sub>''B''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>><ins style="font-weight: bold; text-decoration: none;">, ''V''</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>>'<ins style="font-weight: bold; text-decoration: none;">'C''</sub> </ins>and <ins style="font-weight: bold; text-decoration: none;">''H''<sub>''A''</sub></ins>. <ins style="font-weight: bold; text-decoration: none;">The equilibrium equations are:</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;">Σ ''V'' = '''0''':</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;">:''V''<sub>''A''</sub> &minus; ''F''<sub>''v''</sub> + ''V''</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''B''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>> <ins style="font-weight: bold; text-decoration: none;">+ ''V'</ins>'<<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''C''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>> <ins style="font-weight: bold; text-decoration: none;">= 0</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;">Σ ''H'' = '''0''':</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;">''H''<sub>''A''</sub> &minus; ''F''<sub>''h''</sub> = 0</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;">Σ ''M''<sub>''A''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> = 0:</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''<sub>''v''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> &middot; ''a'' &minus; ''V''<sub>''B''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> &middot; (''a'' + ''b'') </ins>- <ins style="font-weight: bold; text-decoration: none;">''V''<sub>''C''</sub> &middot; (''a'' + ''b'' + ''c'') = 0</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;">Since there </ins>are <ins style="font-weight: bold; text-decoration: none;">four unknown forces (or ''[[Variable (mathematics)|variables]]'') (''V''<sub>''A''</sub></ins>, <ins style="font-weight: bold; text-decoration: none;">''V''<sub>''B''</sub></ins>, <ins style="font-weight: bold; text-decoration: none;">''V''<sub>''C''</sub> </ins>and <ins style="font-weight: bold; text-decoration: none;">''H''</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''A''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>><ins style="font-weight: bold; text-decoration: none;">) but only three equilibrium equations, </ins>this <ins style="font-weight: bold; text-decoration: none;">system of [[simultaneous equations]] does not have </ins>a <ins style="font-weight: bold; text-decoration: none;">unique solution</ins>. <ins style="font-weight: bold; text-decoration: none;">The structure is therefore classified as ''statically indeterminate'</ins>'<ins style="font-weight: bold; text-decoration: none;">. Considerations in </ins>the <ins style="font-weight: bold; text-decoration: none;">material properties </ins>and <ins style="font-weight: bold; text-decoration: none;">compatibility in [[deformation (engineering)|deformation]]s are taken to solve statically indeterminate systems or structures.</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;">==Statically determinate==</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;">If </ins>the <ins style="font-weight: bold; text-decoration: none;">support at ''B'' </ins>is <ins style="font-weight: bold; text-decoration: none;">removed</ins>, the <ins style="font-weight: bold; text-decoration: none;">reaction ''V''<sub>''B''</sub> cannot occur</ins>, <ins style="font-weight: bold; text-decoration: none;">and the system becomes '''statically determinate''' (or '''isostatic''')</ins>.<ins style="font-weight: bold; text-decoration: none;"><ref>{{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= Carpinteri |first= Alberto |authorlink= Alberto Carpinteri </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= Structural Mechanics: A Unified Approach |year= 1997 </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= Taylor & Francis |isbn= 0-419-19160</ins>-<ins style="font-weight: bold; text-decoration: none;">7}}</ref> Note </ins>that the <ins style="font-weight: bold; text-decoration: none;">system </ins>is '<ins style="font-weight: bold; text-decoration: none;">'completely constrained'' here.</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;">The system becomes an [[exact constraint]] [[kinematic coupling]].</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;">The solution to the problem is</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>H_A=F_h</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>V_C=\frac{F_v\cdot a}{</ins>a<ins style="font-weight: bold; text-decoration: none;">+b+c}</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>V_A=F_v-V_C</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;">If, in addition</ins>, <ins style="font-weight: bold; text-decoration: none;">the support at ''A'' is changed </ins>to <ins style="font-weight: bold; text-decoration: none;">a roller support, </ins>the <ins style="font-weight: bold; text-decoration: none;">number of reactions are reduced to three (without ''H''</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''A''</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>><ins style="font-weight: bold; text-decoration: none;">), but the beam can now be moved horizontally; the </ins>system <ins style="font-weight: bold; text-decoration: none;">becomes ''unstable'' or ''partially constrained'' -- </ins>a <ins style="font-weight: bold; text-decoration: none;">[[mechanism (engineering)|mechanism]] rather than </ins>a <ins style="font-weight: bold; text-decoration: none;">structure. In order </ins>to <ins style="font-weight: bold; text-decoration: none;">distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable</ins>, <ins style="font-weight: bold; text-decoration: none;">it </ins>is <ins style="font-weight: bold; text-decoration: none;">preferable to use the phrase ''partially constrained'' here</ins>. <ins style="font-weight: bold; text-decoration: none;">In this case, the 2 unknowns ''V''<sub>''A''</sub> and ''V''<sub>''C'</ins>'<ins style="font-weight: bold; text-decoration: none;"></sub> </ins>can <ins style="font-weight: bold; text-decoration: none;">be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields the same results as previously obtained</ins>. <ins style="font-weight: bold; text-decoration: none;">However</ins>, <ins style="font-weight: bold; text-decoration: none;">it is not possible </ins>to <ins style="font-weight: bold; text-decoration: none;">satisfy </ins>the <ins style="font-weight: bold; text-decoration: none;">horizontal force equation unless </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">F_h=0</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> </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;">Statical indeterminacy </ins>is <ins style="font-weight: bold; text-decoration: none;">the existence of </ins>a <ins style="font-weight: bold; text-decoration: none;">non-trivial (non-zero) solution to </ins>the <ins style="font-weight: bold; text-decoration: none;">homogeneous system of equilibrium equations</ins>. It <ins style="font-weight: bold; text-decoration: none;">indicates the possibility of self-stress (stress in the absence of an external load) that </ins>may <ins style="font-weight: bold; text-decoration: none;">be induced by mechanical or thermal action.</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;">==See also==</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;">*[[Christian Otto Mohr]]</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;">*[[Flexibility method]]</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;">*[[Hardy Cross]]</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;">*[[Moment distribution method]]</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;">*[[Structural engineering]]</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;">* [[overconstrained mechanism]]</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;">==External links==</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;">* [http://en</ins>.<ins style="font-weight: bold; text-decoration: none;">sopromat</ins>.<ins style="font-weight: bold; text-decoration: none;">org/2008/ Beam calculation </ins>online <ins style="font-weight: bold; text-decoration: none;">(Statically indeterminate)]</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;">{{reflist}}</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:Statics]]</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:Structural analysis]]</ins></div></td></tr>
</table>
en>Quondum
https://en.formulasearchengine.com/index.php?title=Quotient_space_(linear_algebra)&diff=234984&oldid=prev
50.73.121.137 at 00:08, 31 May 2012
2012-05-31T00:08:28Z
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<p><b>New page</b></p><div>"Why does my computer keep freezing up?" I was asked by a lot of folks the cause of their pc freeze problems. And I am fed up with spending much time inside answering the question time plus time again. This article is to tell we the real cause of the PC Freezes.<br><br>Your PC registry starts to get errors and fragmented the more you utilize the computer because we enter more information each time, and also create changes inside our systems plus setup. When the registry starts to get overloaded and full of errors, a computer can eventually crash. It is possible to fix it on the own nevertheless quite dangerous, especially when you have no extensive experience in doing so. Therefore, do NOT even attempt to do this oneself.<br><br>If you compare registry products we want a fast acting registry cleaning. It's no superior spending hours and a PC waiting for the registry cleaning to complete its task. You wish the cleaner to complete its task in minutes.<br><br>Check the Windows taskbar, that is on the lower appropriate hand corner of your computer screen. This taskbar comprise of programs we have running in the background. If you have too numerous of them, they might take your computer's resources.<br><br>Many [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014] s enable we to download their product for free, so you can scan the computer yourself. That method you are able to see how countless errors it finds, where it finds them, plus how it could fix them. A ideal registry cleaner will remove a registry difficulties, and optimize and accelerate the PC, with small effort on your part.<br><br>Windows relies heavily on this database, storing everything from your newest emails to a Internet favorites inside there. Because it's so important, the computer is continually adding and updating the files inside it. This is fine, yet it could create the computer run slow, whenever your computer accidentally breaks its crucial registry files. This is a truly well-known problem, and actually makes a computer run slower each day. What arises is the fact that because the computer is continually utilizing 100's of registry files at when, it often gets confused plus make a few of them unreadable. This then makes your computer run slow, because Windows takes longer to read the files it requirements.<br><br>It is critical which we remove obsolete registry entries from a system on a regular basis, should you want a program to run quicker, that is. If you don't keep your registry clean, a time may come whenever a system can stop working completely. Then, the just choice will be to reformat the hard drive and begin over!<br><br>There is a lot a advantageous registry cleaner may do for the computer. It may check for and download changes for Windows, Java and Adobe. Keeping updates present is an significant piece of superior computer wellness. It could additionally protect your individual and company confidentiality in addition to the online security.</div>
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