List of Foucault pendulums - Revision history

en>Robert EA Harvey: Added glasgow. Bet there are more in scotland

2015-01-10T08:20:24Z

Added glasgow. Bet there are more in scotland

← Older revision		Revision as of 10:20, 10 January 2015
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	The title ~~of the author~~ is ~~Jayson~~. ~~Her family lives in Alaska but her husband wants them to transfer. She works as a journey agent but soon she~~'ll be ~~on her personal~~. ~~What I love performing~~ is ~~football~~ but I ~~don't have the time lately~~.<br><br>~~Here is~~ my ~~blog~~ ... ~~live~~ psychic ~~reading~~ ([http://~~breenq~~.~~com~~/~~index.php~~?do=~~/profile-1144/info/ breenq~~.~~com~~])		The author's title is Andera and she believes it seems fairly great. The preferred hobby for him and his kids is style and he'll be starting something else alongside with it. North Carolina is the place he loves most but now he is considering other choices. Invoicing is what I do.<br><br>my website ... psychic phone ([http://203.250.78.160/zbxe/?document_srl=1792908 203.250.78.160])

en>Rwenner: Pendulum was moved from Buhl Planetarium and Institute of Popular Science to Carnegie Science Center then back to its original location in what is now known as Children's Museum of Pittsburgh http://triblive.com/x/pittsburghtrib/ae/museums/s

2014-02-19T04:44:34Z

Pendulum was moved from Buhl Planetarium and Institute of Popular Science to Carnegie Science Center then back to its original location in what is now known as Children's Museum of Pittsburgh http://triblive.com/x/pittsburghtrib/ae/museums/s

← Older revision		Revision as of 06:44, 19 February 2014
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	~~{{For\|the theorem in complex analysis\|Gauss–Lucas theorem}}~~		The title of the author is Jayson. Her family lives in Alaska but her husband wants them to transfer. She works as a journey agent but soon she'll be on her personal. What I love performing is football but I don't have the time lately.<br><br>Here is my blog ... live psychic reading ([http://breenq.com/index.php?do=/profile-1144/info/ breenq.com])
	~~In [[number theory]], '''Lucas's theorem''' expresses the [[remainder]] of division~~ of the ~~[[binomial coefficient]] <math>\tbinom{m}{n}</math> by a [[prime number]] ''p'' in terms of the [[radix\|base]] ''p'' expansions of the integers ''m'' and ''n''.~~

	~~Lucas's theorem first appeared in 1878 in papers by [[Édouard Lucas]].<ref>~~
	*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369308 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=2 \|pages=184–196 \|doi=10.2307/2369308\| mr=1505161}} (part 1);
	*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369311 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=3 \|pages=197–240 \|doi=10.2307/2369311\| mr=1505164}} (part 2);
	*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369373 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=4 \|pages=289–321 \|doi=10.2307/2369373\| mr=1505176}} (part 3)</ref>

	~~== Formulation ==~~
	~~For non-negative integers ''m'' and ''n'' and a prime ''p'', the following [[modular arithmetic\|congruence relation]] holds:~~
	~~:<math>\binom{m}{n}\equiv\prod_{i=0}^k\binom{m_i}{n_i}\pmod p,</math>~~
	~~where~~
	~~:<math>m=m_kp^k+m_{k-1}p^{k-1}+\cdots +m_1p+m_0,</math>~~
	~~and~~
	~~:<math>n=n_kp^k+n_{k-1}p^{k-1}+\cdots +n_1p+n_0</math>~~
	~~are the base ''p'' expansions of ''m'' and ''n'' respectively. This uses the convention that <math>\tbinom{m}{n}</math> = 0 if ''m'' < ''n''.~~

	~~== Consequence ==~~
	* A binomial coefficient <math>\tbinom{m}{n}</math> is ~~divisible by a prime ''p'' if and only if at least one of the base ''p'' digits of ''n'' is greater than the corresponding digit of ''m''~~.

	~~== Proof ==~~

	~~There are several ways~~ to ~~prove Lucas's theorem~~. ~~We first give~~ a ~~combinatorial argument and then a proof based on generating functions.~~

	~~Let ''M'' be a set with ''m'' elements, and divide it into ''m<sub>i</sub>'' cycles of length~~ '~~'p<sup>i</sup>'' for the various values of ''i''. Then each of these cycles can~~ be ~~rotated separately, so that a group ''G'' which is the Cartesian product of cyclic groups ''C<sub>p<sup>i</sup></sub>'' acts~~ on ~~''M''~~. ~~It thus also acts on subsets ''N'' of size ''n''. Since the number of elements in ''G''~~ is ~~a power of~~ '~~'p'',~~ the same is true of any of its orbits. Thus in order to compute <math>\tbinom{m}{n}</math> modulo ''p'', we only need to consider fixed points of this group action. The fixed points are those subsets ''N'' that are a union of some of the cycles. More precisely one can show by induction on ''k''-''i'', that ''N'' must have exactly ''n<sub>i</sub>'' cycles of size ''p<sup>i</sup>''. ~~Thus the number of choices for ''N'' is exactly~~
	<~~math~~>~~\prod_{i=0}^k\binom{m_i}{n_i}\pmod{p}~~<~~/math~~>.

	Here is a proof based on generating functions, due to Nathan Fine.<ref>{{cite journal\|last=Fine\|first=Nathan\|title=Binomial coefficients modulo a prime\|journal=American Mathematical Monthly\|year=1947\|volume=54\|pages=589–592}}</ref>

	~~If ''p'' is a prime and ''n'' is an integer with 1≤''n''≤''p''-1, then the numerator of the binomial coefficient~~
	~~:<math> \binom p n = \frac{p \cdot (p-1) \cdots (p-n+1)}{n \cdot (n-1) \cdots 1} </math>~~
	~~is divisible by ''p'' but the denominator is not. Hence ''p'' divides <math>\tbinom{p}{n}</math>. In terms of ordinary generating functions, this means that~~
	~~:<math>(1+X)^p\equiv1+X^p\text{ mod }p.</math>~~
	~~Continuing by induction, we have for every nonnegative integer ''i'' that~~
	~~:<math>(1+X)^{p^i}\equiv1+X^{p^i}\text{ mod }p.</math>~~

	Now let ''m'' be a nonnegative integer, and let ''p'' be a prime. Write ''m'' in base ''p'', so that <math>m=\sum_{i=0}^{k}m_ip^i</math> for some nonnegative integer ''k'' and integers ''m''<sub>''i''</sub> with 0 ≤ ''m''<sub>''i''</sub> ≤ ''p''-1. Then
	~~:<math>\begin{align}~~
	~~\sum_{n=0}^{m}\binom{m}{n}X^n &~~
	~~=(1+X)^m=\prod_{i=0}^{k}\left((1+X)^{p^i}\right)^{m_i}\\~~
	~~& \equiv \prod_{i=0}^{k}\left(1+X^{p^i}\right)^{m_i}~~
	~~=\prod_{i=0}^{k}\left(\sum_{n_i=0}^{m_i}\binom{m_i}{n_i}X^{n_ip^i}\right)\\~~
	~~& =\sum_{n=0}^{m}\left(\prod_{i=0}^{k}\binom{m_i}{n_i}\right)X^n~~
	~~\text{ mod } p,~~
	~~\end{align}</math>~~
	~~where in the final product, ''n''<sub>''i''</sub> is digit ''i'' in the base ''p'' representation of ''n''~~. ~~This proves Lucas's theorem~~.

	~~== Variations and generalizations ==~~
	* The largest integer ''k'' such that ''p''<sup>''k''</sup> divides the binomial coefficient <math>\tbinom{m}{n}</math> (or in other words, the [[p-adic order\|valuation]] of the binomial coefficient with respect to the prime ''p'') is equal to the number of [[Carry (arithmetic)\|carries]] that occur when ''n'' and ''m'' − ''n'' are added in the [[Positional notation#Base of the numeral system\|base ''p'']]. (~~This result is known as [~~[~~Kummer's theorem]].)~~
	* [[Andrew Granville]] has given a generalization of Lucas's theorem to the case of ''p'' being a power of prime.<ref>{{cite journal \|author=[[Andrew Granville]] \|title=Arithmetic Properties of Binomial Coefficients I: Binomial coefficients modulo prime powers \|journal=Canadian Mathematical Society Conference Proceedings \|volume=20 \|pages=253–275 \|year=1997 \|url=http://~~www~~.~~dms.umontreal.ca/%7Eandrew/PDF~~/~~BinCoeff~~.~~pdf \|mr~~=~~1483922}}<~~/~~ref>~~

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*{{PlanetMath\|urlname=LucassTheorem\|title=Lucas's Theorem}}
	*[http://arxiv.org/~~abs~~/~~1301~~.~~4250 Alternate Proof of Lucas'Theorem]~~

	~~[[Category:Articles containing proofs]]~~
	~~[[Category:Theorems about prime numbers]~~]

en>Melpomenia: Changed the reference for the pendulum at FCEN, UBA, Buenos Aires. Dead link.

2014-01-29T18:49:29Z

Changed the reference for the pendulum at FCEN, UBA, Buenos Aires. Dead link.

← Older revision		Revision as of 20:49, 29 January 2014
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	~~Friends call him Royal Cummins~~. ~~One~~ of ~~my preferred hobbies~~ is ~~camping~~ and ~~now I~~'m ~~attempting~~ to ~~make cash~~ with it. ~~I currently live~~ in ~~Alabama~~. ~~Interviewing~~ is ~~how he supports his family~~ but ~~his promotion never comes~~.<br><br>~~my weblog~~ ... [http://www.~~gauhaticommercecollege~~.in/~~content~~/~~dont-allow-auto-repair-best-you extended car warranty reviews~~]		{{For\|the theorem in complex analysis\|Gauss–Lucas theorem}}
			In [[number theory]], '''Lucas's theorem''' expresses the [[remainder]] of division of the [[binomial coefficient]] <math>\tbinom{m}{n}</math> by a [[prime number]] ''p'' in terms of the [[radix\|base]] ''p'' expansions of the integers ''m'' and ''n''.

			Lucas's theorem first appeared in 1878 in papers by [[Édouard Lucas]].<ref>
			*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369308 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=2 \|pages=184–196 \|doi=10.2307/2369308\| mr=1505161}} (part 1);
			*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369311 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=3 \|pages=197–240 \|doi=10.2307/2369311\| mr=1505164}} (part 2);
			*{{cite journal\| author=Edouard Lucas \|title=Théorie des Fonctions Numériques Simplement Périodiques\| jstor=2369373 \|journal=[[American Journal of Mathematics]] \|year=1878 \|volume=1 \|issue=4 \|pages=289–321 \|doi=10.2307/2369373\| mr=1505176}} (part 3)</ref>

			== Formulation ==
			For non-negative integers ''m'' and ''n'' and a prime ''p'', the following [[modular arithmetic\|congruence relation]] holds:
			:<math>\binom{m}{n}\equiv\prod_{i=0}^k\binom{m_i}{n_i}\pmod p,</math>
			where
			:<math>m=m_kp^k+m_{k-1}p^{k-1}+\cdots +m_1p+m_0,</math>
			and
			:<math>n=n_kp^k+n_{k-1}p^{k-1}+\cdots +n_1p+n_0</math>
			are the base ''p'' expansions of ''m'' and ''n'' respectively. This uses the convention that <math>\tbinom{m}{n}</math> = 0 if ''m'' < ''n''.

			== Consequence ==
			* A binomial coefficient <math>\tbinom{m}{n}</math> is divisible by a prime ''p'' if and only if at least one of the base ''p'' digits of ''n'' is greater than the corresponding digit of ''m''.

			== Proof ==

			There are several ways to prove Lucas's theorem. We first give a combinatorial argument and then a proof based on generating functions.

			Let ''M'' be a set with ''m'' elements, and divide it into ''m<sub>i</sub>'' cycles of length ''p<sup>i</sup>'' for the various values of ''i''. Then each of these cycles can be rotated separately, so that a group ''G'' which is the Cartesian product of cyclic groups ''C<sub>p<sup>i</sup></sub>'' acts on ''M''. It thus also acts on subsets ''N'' of size ''n''. Since the number of elements in ''G'' is a power of ''p'', the same is true of any of its orbits. Thus in order to compute <math>\tbinom{m}{n}</math> modulo ''p'', we only need to consider fixed points of this group action. The fixed points are those subsets ''N'' that are a union of some of the cycles. More precisely one can show by induction on ''k''-''i'', that ''N'' must have exactly ''n<sub>i</sub>'' cycles of size ''p<sup>i</sup>''. Thus the number of choices for ''N'' is exactly
			<math>\prod_{i=0}^k\binom{m_i}{n_i}\pmod{p}</math>.

			Here is a proof based on generating functions, due to Nathan Fine.<ref>{{cite journal\|last=Fine\|first=Nathan\|title=Binomial coefficients modulo a prime\|journal=American Mathematical Monthly\|year=1947\|volume=54\|pages=589–592}}</ref>

			If ''p'' is a prime and ''n'' is an integer with 1≤''n''≤''p''-1, then the numerator of the binomial coefficient
			:<math> \binom p n = \frac{p \cdot (p-1) \cdots (p-n+1)}{n \cdot (n-1) \cdots 1} </math>
			is divisible by ''p'' but the denominator is not. Hence ''p'' divides <math>\tbinom{p}{n}</math>. In terms of ordinary generating functions, this means that
			:<math>(1+X)^p\equiv1+X^p\text{ mod }p.</math>
			Continuing by induction, we have for every nonnegative integer ''i'' that
			:<math>(1+X)^{p^i}\equiv1+X^{p^i}\text{ mod }p.</math>

			Now let ''m'' be a nonnegative integer, and let ''p'' be a prime. Write ''m'' in base ''p'', so that <math>m=\sum_{i=0}^{k}m_ip^i</math> for some nonnegative integer ''k'' and integers ''m''<sub>''i''</sub> with 0 ≤ ''m''<sub>''i''</sub> ≤ ''p''-1. Then
			:<math>\begin{align}
			\sum_{n=0}^{m}\binom{m}{n}X^n &
			=(1+X)^m=\prod_{i=0}^{k}\left((1+X)^{p^i}\right)^{m_i}\\
			& \equiv \prod_{i=0}^{k}\left(1+X^{p^i}\right)^{m_i}
			=\prod_{i=0}^{k}\left(\sum_{n_i=0}^{m_i}\binom{m_i}{n_i}X^{n_ip^i}\right)\\
			& =\sum_{n=0}^{m}\left(\prod_{i=0}^{k}\binom{m_i}{n_i}\right)X^n
			\text{ mod } p,
			\end{align}</math>
			where in the final product, ''n''<sub>''i''</sub> is digit ''i'' in the base ''p'' representation of ''n''. This proves Lucas's theorem.

			== Variations and generalizations ==
			* The largest integer ''k'' such that ''p''<sup>''k''</sup> divides the binomial coefficient <math>\tbinom{m}{n}</math> (or in other words, the [[p-adic order\|valuation]] of the binomial coefficient with respect to the prime ''p'') is equal to the number of [[Carry (arithmetic)\|carries]] that occur when ''n'' and ''m'' − ''n'' are added in the [[Positional notation#Base of the numeral system\|base ''p'']]. (This result is known as [[Kummer's theorem]].)
			* [[Andrew Granville]] has given a generalization of Lucas's theorem to the case of ''p'' being a power of prime.<ref>{{cite journal \|author=[[Andrew Granville]] \|title=Arithmetic Properties of Binomial Coefficients I: Binomial coefficients modulo prime powers \|journal=Canadian Mathematical Society Conference Proceedings \|volume=20 \|pages=253–275 \|year=1997 \|url=http://www.dms.umontreal.ca/%7Eandrew/PDF/BinCoeff.pdf \|mr=1483922}}</ref>

			==References==
			{{reflist}}

			==External links==
			*{{PlanetMath\|urlname=LucassTheorem\|title=Lucas's Theorem}}
			*[http://arxiv.org/abs/1301.4250 Alternate Proof of Lucas'Theorem]

			[[Category:Articles containing proofs]]
			[[Category:Theorems about prime numbers]]

en>FrescoBot: Bot: link syntax/spacing and minor changes

2012-08-10T07:50:18Z

Bot: link syntax/spacing and minor changes

New page

Friends call him Royal Cummins. One of my preferred hobbies is camping and now I'm attempting to make cash with it. I currently live in Alabama. Interviewing is how he supports his family but his promotion never comes.<br><br>my weblog ... [http://www.gauhaticommercecollege.in/content/dont-allow-auto-repair-best-you extended car warranty reviews]