https://en.formulasearchengine.com/index.php?title=Splitting_lemma&feed=atom&action=historySplitting lemma - Revision history2024-03-28T15:00:04ZRevision history for this page on the wikiMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Splitting_lemma&diff=224335&oldid=preven>Noix07 at 11:28, 14 April 20142014-04-14T11:28:22Z<p></p>
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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, '''Landau's function''' ''g''(''n''), named after [[Edmund Landau]], </del>is <del style="font-weight: bold; text-decoration: none;">defined </del>for <del style="font-weight: bold; text-decoration: none;">every [[natural number]] ''n'</del>' to <del style="font-weight: bold; text-decoration: none;">be </del>the <del style="font-weight: bold; text-decoration: none;">largest [[order (group theory)|order]] of an element of the [[symmetric group]] ''S'</del>'<<del style="font-weight: bold; text-decoration: none;">sub</del>><del style="font-weight: bold; text-decoration: none;">''n''</del><<del style="font-weight: bold; text-decoration: none;">/sub</del>><del style="font-weight: bold; text-decoration: none;">. Equivalently, </del>'<del style="font-weight: bold; text-decoration: none;">'g''(''n'') is </del>the <del style="font-weight: bold; text-decoration: none;">largest </del>[<del style="font-weight: bold; text-decoration: none;">[least common multiple</del>]<del style="font-weight: bold; text-decoration: none;">] (lcm) </del>of <del style="font-weight: bold; text-decoration: none;">any [[integer partition|partition]] </del>of <del style="font-weight: bold; text-decoration: none;">''n''</del>, <del style="font-weight: bold; text-decoration: none;">or </del>the <del style="font-weight: bold; text-decoration: none;">maximum number </del>of <del style="font-weight: bold; text-decoration: none;">times a [[permutation]] of ''n'' elements can be recursively applied </del>to <del style="font-weight: bold; text-decoration: none;">itself before it returns </del>to <del style="font-weight: bold; text-decoration: none;">its starting sequence</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;">The green coffee bean extract for fat reduction </ins>is <ins style="font-weight: bold; text-decoration: none;">a hot all-natural supplement which assists the body lose fat without the need </ins>for <ins style="font-weight: bold; text-decoration: none;">doing any exercise. 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Irregardless, by the time you</ins>'<ins style="font-weight: bold; text-decoration: none;">re performed reading this you</ins>'<ins style="font-weight: bold; text-decoration: none;">ll recognize what this is or it was called fat loss at this time.</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;">It is equally suggested that you never skip breakfast. 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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> </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;">For instance</del>, <del style="font-weight: bold; text-decoration: none;">5 = </del>2 <del style="font-weight: bold; text-decoration: none;">+ 3 and lcm(2,3) = 6. No other partition </del>of <del style="font-weight: bold; text-decoration: none;">5 yields a bigger lcm, so ''g''(5) = 6. An element </del>of <del style="font-weight: bold; text-decoration: none;">order 6 </del>in the <del style="font-weight: bold; text-decoration: none;">group ''S''</del><<del style="font-weight: bold; text-decoration: none;">sub</del>><del style="font-weight: bold; text-decoration: none;">5</del><<del style="font-weight: bold; text-decoration: none;">/sub</del>> <del style="font-weight: bold; text-decoration: none;">can be written in cycle notation as (1 2) (3 4 5)</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>The <del style="font-weight: bold; text-decoration: none;">[[integer sequence]] ''g''(0) = 1</del>, <del style="font-weight: bold; text-decoration: none;">''g''(1) = 1</del>, <del style="font-weight: bold; text-decoration: none;">''g''(2) = 2</del>, '<del style="font-weight: bold; text-decoration: none;">'g''(3) = 3, ''g''(4) = 4, ''g''(5) = 6, ''g''(6) = 6, ''g''(7) = 12, ''g''(8) = 15</del>, .<del style="font-weight: bold; text-decoration: none;">.. {{OEIS|A000793}} is named after [[Edmund Landau]], who proved in 1902</del><<del style="font-weight: bold; text-decoration: none;">ref</del>><del style="font-weight: bold; text-decoration: none;">Landau, pp. 92–103</del><<del style="font-weight: bold; text-decoration: none;">/ref</del>> that</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>\lim_{n\</del>to<del style="font-weight: bold; text-decoration: none;">\infty}\frac{\ln(g(n))}{\sqrt{n \ln(n)}} = 1</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><del style="font-weight: bold; text-decoration: none;">(where ln denotes the [[natural logarithm]])</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;">The statement that</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><del style="font-weight: bold; text-decoration: none;">:<math>\ln g(n)<\sqrt{\mathrm{Li}^{-1}(n)}</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><del style="font-weight: bold; text-decoration: none;">for all sufficiently large </del>''<del style="font-weight: bold; text-decoration: none;">n</del>''<del style="font-weight: bold; text-decoration: none;">, where Li</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">&minus;1</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>> <del style="font-weight: bold; text-decoration: none;">denotes </del>the <del style="font-weight: bold; text-decoration: none;">inverse </del>of the <del style="font-weight: bold; text-decoration: none;">[[logarithmic integral function]]</del>, is <del style="font-weight: bold; text-decoration: none;">equivalent to </del>the <del style="font-weight: bold; text-decoration: none;">[[Riemann hypothesis]]</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;">It can be shown that:</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><del style="font-weight: bold; text-decoration: none;">:<math>g(n)<e^{n/e}</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;">==Notes==</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><<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;">== 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><del style="font-weight: bold; text-decoration: none;">*[[E. Landau]], "Über die Maximalordnung der Permutationen gegebenen Grades [On </del>the <del style="font-weight: bold; text-decoration: none;">maximal order of permutations of given degree]", ''Arch</del>. <del style="font-weight: bold; text-decoration: none;">Math</del>. <del style="font-weight: bold; text-decoration: none;">Phys.'' Ser. 3, vol. 5, 1903.</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><del style="font-weight: bold; text-decoration: none;">*W. Miller, "The maximum order of </del>an <del style="font-weight: bold; text-decoration: none;">element </del>of <del style="font-weight: bold; text-decoration: none;">a finite symmetric group"</del>, '<del style="font-weight: bold; text-decoration: none;">'[[American Mathematical Monthly]]'', vol</del>. <del style="font-weight: bold; text-decoration: none;">94</del>, <del style="font-weight: bold; text-decoration: none;">1987, pp</del>.<del style="font-weight: bold; text-decoration: none;">&nbsp;497&ndash;506</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><del style="font-weight: bold; text-decoration: none;">*J</del>.-<del style="font-weight: bold; text-decoration: none;">L</del>. <del style="font-weight: bold; text-decoration: none;">Nicolas, "On Landau's function ''g''(''n'')", in ''The Mathematics of Paul Erdős'', vol</del>. <del style="font-weight: bold; text-decoration: none;">1</del>, <del style="font-weight: bold; text-decoration: none;">Springer</del>-<del style="font-weight: bold; text-decoration: none;">Verlag, 1997, pp</del>.<del style="font-weight: bold; text-decoration: none;">&nbsp;228&ndash;240</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;">==External links==</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><del style="font-weight: bold; text-decoration: none;">*{{SloanesRef |sequencenumber=A000793|name=Landau</del>'s <del style="font-weight: bold; text-decoration: none;">function on the natural numbers}}</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:Group theory]]</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><del style="font-weight: bold; text-decoration: none;">[[Category:Permutations]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>en>Noix07https://en.formulasearchengine.com/index.php?title=Splitting_lemma&diff=2264&oldid=preven>Helder.wiki: /* Counterexample */ typo2013-11-21T13:03:58Z<p><span dir="auto"><span class="autocomment">Counterexample: </span> typo</span></p>
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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;">Msvcr71.dll </del>is an <del style="font-weight: bold; text-decoration: none;">significant file that assists support Windows procedure different components </del>of the <del style="font-weight: bold; text-decoration: none;">system including important files</del>. <del style="font-weight: bold; text-decoration: none;">Specifically</del>, <del style="font-weight: bold; text-decoration: none;">the file </del>is <del style="font-weight: bold; text-decoration: none;">employed to help run corresponding files inside </del>the <del style="font-weight: bold; text-decoration: none;">"Virtual C Runtime Library". These files are significant in accessing </del>any <del style="font-weight: bold; text-decoration: none;">settings which help the different applications and programs in the system. The msvcr71.dll file fulfills various significant functions; nevertheless it</del>'<del style="font-weight: bold; text-decoration: none;">s not spared from getting damaged </del>or <del style="font-weight: bold; text-decoration: none;">corrupted. Once </del>the <del style="font-weight: bold; text-decoration: none;">file gets corrupted or damaged, the computer usually have </del>a <del style="font-weight: bold; text-decoration: none;">difficult time processing plus reading components </del>of <del style="font-weight: bold; text-decoration: none;">the system</del>. <del style="font-weight: bold; text-decoration: none;">Nonetheless</del>, <del style="font-weight: bold; text-decoration: none;">consumers want not panic because this problem may be solved by following many procedures</del>. <del style="font-weight: bold; text-decoration: none;">And I can show you some strategies regarding Msvcr71.dll</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;">If it is actually not as big of the issue because you think it happens to be, it can possibly be solved conveniently by running a Startup Repair or by System Restore Utility. Again it </del>can be as <del style="font-weight: bold; text-decoration: none;">easy because running an anti-virus check or cleaning the registry</del>.<del style="font-weight: bold; text-decoration: none;"><br><br>Windows is actually rather dumb. It only knows how to follow commands and instructions</del>, <del style="font-weight: bold; text-decoration: none;">meaning that whenever you install a program</del>, <del style="font-weight: bold; text-decoration: none;">that program has to tell Windows exactly what to do</del>. <del style="font-weight: bold; text-decoration: none;">This is done by storing an "instruction file" in the registry of the program</del>. <del style="font-weight: bold; text-decoration: none;">All the computer programs put these "manuals" into the registry, allowing the computer to run a wide array of programs</del>. <del style="font-weight: bold; text-decoration: none;">When we load up 1 of those programs</del>, <del style="font-weight: bold; text-decoration: none;">Windows merely looks up the program file inside the registry</del>, <del style="font-weight: bold; text-decoration: none;">and carries out its instructions</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;">Windows mistakes can be caused by any amount of factors, yet there's virtually constantly </del>1 <del style="font-weight: bold; text-decoration: none;">cause</del>. <del style="font-weight: bold; text-decoration: none;">There's a hidden piece of your system </del>that <del style="font-weight: bold; text-decoration: none;">is responsible </del>for <del style="font-weight: bold; text-decoration: none;">creating 90% of </del>all <del style="font-weight: bold; text-decoration: none;">Windows errors, and it</del>'<del style="font-weight: bold; text-decoration: none;">s called the </del>'<del style="font-weight: bold; text-decoration: none;">registry</del>'<del style="font-weight: bold; text-decoration: none;">. This is </del>the <del style="font-weight: bold; text-decoration: none;">central database for a program and is where a computer shops all its system files plus settings. It's a very important piece </del>of <del style="font-weight: bold; text-decoration: none;">Windows</del>, <del style="font-weight: bold; text-decoration: none;">that </del>is <del style="font-weight: bold; text-decoration: none;">should be capable </del>to <del style="font-weight: bold; text-decoration: none;">function</del>. <del style="font-weight: bold; text-decoration: none;">However, it's furthermore among the largest causes of problems on your PC</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;">The second step to fixing these errors is to employ a system called a "</del>[<del style="font-weight: bold; text-decoration: none;">http://bestregistrycleanerfix</del>.<del style="font-weight: bold; text-decoration: none;">com/tune-up-utilities tuneup utilities 2014</del>]" <del style="font-weight: bold; text-decoration: none;">to scan by a computer and fix some </del>of <del style="font-weight: bold; text-decoration: none;">the registry mistakes which could furthermore be leading to this error. A registry cleaner is a software system which may scan by your computer and repair any </del>of <del style="font-weight: bold; text-decoration: none;">the issues which Windows has inside</del>, <del style="font-weight: bold; text-decoration: none;">allowing the computer to "remember" all of the settings it has when it loads up</del>. <del style="font-weight: bold; text-decoration: none;">Although the registry is continually being used to aid load up a big number of programs on the PC, it</del>'<del style="font-weight: bold; text-decoration: none;">s continually being saved incorrectly - leading to a large number of mistakes to be formed</del>. <del style="font-weight: bold; text-decoration: none;">To fix this problem</del>, <del style="font-weight: bold; text-decoration: none;">it's recommended we download a registry cleaner from the Internet and install it on your Pc</del>, <del style="font-weight: bold; text-decoration: none;">permitting Windows to run smoother again</del>.<del style="font-weight: bold; text-decoration: none;"><br><br></del>The <del style="font-weight: bold; text-decoration: none;">many probable cause </del>of <del style="font-weight: bold; text-decoration: none;">the trouble is the system issue - Registry Errors! That is the reason why people whom already have more than 2 G RAM on their computers are nonetheless constantly bothered by the issue.<br><br>To accelerate </del>a <del style="font-weight: bold; text-decoration: none;">computer</del>, <del style="font-weight: bold; text-decoration: none;">we just have to be able to get rid of all these junk files</del>, <del style="font-weight: bold; text-decoration: none;">allowing your computer to find just what it wants</del>, <del style="font-weight: bold; text-decoration: none;">when it wants</del>. <del style="font-weight: bold; text-decoration: none;">Luckily</del>, <del style="font-weight: bold; text-decoration: none;">there</del>'s <del style="font-weight: bold; text-decoration: none;">a tool that enables you to do this conveniently plus quickly. It</del>'<del style="font-weight: bold; text-decoration: none;">s a tool called a </del>'<del style="font-weight: bold; text-decoration: none;">registry cleaner</del>'.<del style="font-weight: bold; text-decoration: none;"><br><br>Ally Wood is a expert software reviewer and has worked inside CNET</del>. <del style="font-weight: bold; text-decoration: none;">Then she is working for her own review software company to provide suggestions to the software creator plus has completed deep test in registry cleaner software</del>. <del style="font-weight: bold; text-decoration: none;">After reviewing </del>the <del style="font-weight: bold; text-decoration: none;">top registry cleaner, she has created complete review on a review website for you which is accessed for free.</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 mathematics, '''Landau's function''' ''g''(''n''), named after [[Edmund Landau]], </ins>is <ins style="font-weight: bold; text-decoration: none;">defined for every [[natural number]] ''n'' to be the largest [[order (group theory)|order]] of </ins>an <ins style="font-weight: bold; text-decoration: none;">element </ins>of the <ins style="font-weight: bold; text-decoration: none;">[[symmetric group]] ''S''<sub>''n''</sub></ins>. <ins style="font-weight: bold; text-decoration: none;">Equivalently</ins>, <ins style="font-weight: bold; text-decoration: none;">''g''(''n'') </ins>is the <ins style="font-weight: bold; text-decoration: none;">largest [[least common multiple]] (lcm) of </ins>any <ins style="font-weight: bold; text-decoration: none;">[[integer partition|partition]] of ''n'</ins>'<ins style="font-weight: bold; text-decoration: none;">, </ins>or the <ins style="font-weight: bold; text-decoration: none;">maximum number of times </ins>a <ins style="font-weight: bold; text-decoration: none;">[[permutation]] </ins>of <ins style="font-weight: bold; text-decoration: none;">''n'' elements can be recursively applied to itself before it returns to its starting sequence</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 instance</ins>, <ins style="font-weight: bold; text-decoration: none;">5 = 2 + 3 and lcm(2,3) = 6</ins>. <ins style="font-weight: bold; text-decoration: none;">No other partition of 5 yields a bigger lcm, so ''g''(5) = 6</ins>. <ins style="font-weight: bold; text-decoration: none;">An element of order 6 in the group ''S''</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">5</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>> can be <ins style="font-weight: bold; text-decoration: none;">written in cycle notation </ins>as <ins style="font-weight: bold; text-decoration: none;">(1 2) (3 4 5)</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;">The [[integer sequence]] ''g''(0) = 1, ''g''(1) = 1, ''g''(2) = 2, ''g''(3) = 3, ''g''(4) = 4, ''g''(5) = 6, ''g''(6) = 6, ''g''(7) = 12</ins>, <ins style="font-weight: bold; text-decoration: none;">''g''(8) = 15</ins>, ... <ins style="font-weight: bold; text-decoration: none;">{{OEIS|A000793}} is named after [[Edmund Landau]]</ins>, <ins style="font-weight: bold; text-decoration: none;">who proved in 1902<ref>Landau</ins>, <ins style="font-weight: bold; text-decoration: none;">pp</ins>. <ins style="font-weight: bold; text-decoration: none;">92–103</ins><<ins style="font-weight: bold; text-decoration: none;">/ref</ins>> <ins style="font-weight: bold; text-decoration: none;">that</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;">\lim_{n\to\infty}\frac{\ln(g(n))}{\sqrt{n \ln(n)}} = </ins>1<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;">(where ln denotes the [[natural logarithm]])</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;">The statement </ins>that</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>\ln g(n)<\sqrt{\mathrm{Li}^{-1}(n)}</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>for all <ins style="font-weight: bold; text-decoration: none;">sufficiently large '</ins>'<ins style="font-weight: bold; text-decoration: none;">n</ins>''<ins style="font-weight: bold; text-decoration: none;">, where Li<sup>&minus;1</sup> denotes </ins>the <ins style="font-weight: bold; text-decoration: none;">inverse </ins>of <ins style="font-weight: bold; text-decoration: none;">the [[logarithmic integral function]]</ins>, is <ins style="font-weight: bold; text-decoration: none;">equivalent </ins>to <ins style="font-weight: bold; text-decoration: none;">the [[Riemann hypothesis]]</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;">It can be shown that:</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>g(n)<e^{n/e}</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;">==Notes==</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;">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> </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;">*[</ins>[<ins style="font-weight: bold; text-decoration: none;">E</ins>. <ins style="font-weight: bold; text-decoration: none;">Landau]</ins>]<ins style="font-weight: bold; text-decoration: none;">, </ins>"<ins style="font-weight: bold; text-decoration: none;">Über die Maximalordnung der Permutationen gegebenen Grades [On the maximal order </ins>of <ins style="font-weight: bold; text-decoration: none;">permutations </ins>of <ins style="font-weight: bold; text-decoration: none;">given degree]"</ins>, <ins style="font-weight: bold; text-decoration: none;">''Arch. Math. Phys</ins>.'<ins style="font-weight: bold; text-decoration: none;">' Ser</ins>. <ins style="font-weight: bold; text-decoration: none;">3</ins>, <ins style="font-weight: bold; text-decoration: none;">vol. 5</ins>, <ins style="font-weight: bold; text-decoration: none;">1903.</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;">*W</ins>. <ins style="font-weight: bold; text-decoration: none;">Miller, "</ins>The <ins style="font-weight: bold; text-decoration: none;">maximum order of an element </ins>of a <ins style="font-weight: bold; text-decoration: none;">finite symmetric group", ''[[American Mathematical Monthly]]''</ins>, <ins style="font-weight: bold; text-decoration: none;">vol. 94</ins>, <ins style="font-weight: bold; text-decoration: none;">1987</ins>, <ins style="font-weight: bold; text-decoration: none;">pp.&nbsp;497&ndash;506.</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;">*J.-L</ins>. <ins style="font-weight: bold; text-decoration: none;">Nicolas</ins>, <ins style="font-weight: bold; text-decoration: none;">"On Landau</ins>'s <ins style="font-weight: bold; text-decoration: none;">function ''g''(''n'')", in '</ins>'<ins style="font-weight: bold; text-decoration: none;">The Mathematics of Paul Erdős</ins>''<ins style="font-weight: bold; text-decoration: none;">, vol</ins>. <ins style="font-weight: bold; text-decoration: none;">1, Springer-Verlag, 1997, pp</ins>.<ins style="font-weight: bold; text-decoration: none;">&nbsp;228&ndash;240</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;">*{{SloanesRef |sequencenumber=A000793|name=Landau's function on </ins>the <ins style="font-weight: bold; text-decoration: none;">natural numbers}}</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:Group 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;">[[Category:Permutations]]</ins></div></td></tr>
</table>en>Helder.wikihttps://en.formulasearchengine.com/index.php?title=Splitting_lemma&diff=224334&oldid=prev190.82.208.121 at 19:44, 16 June 20122012-06-16T19:44:27Z<p></p>
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