State prices - Revision history

en>Limit-theorem at 11:39, 22 November 2014

2014-11-22T11:39:36Z

← Older revision		Revision as of 13:39, 22 November 2014
Line 1:		Line 1:
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en>Tazerdadog: Added info from the now-redirected pure security article

2014-02-20T01:40:56Z

Added info from the now-redirected pure security article

← Older revision		Revision as of 03:40, 20 February 2014
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	~~In [[number theory]], the '''Erdős–Kac theorem''', named after [[Paul Erdős]]~~ and ~~[[Mark Kac]], and also known as the fundamental theorem of [[probabilistic number theory]], states that if ω(''n'')~~ is ~~the number of distinct [[prime factor]]s of ''n'', then, loosely speaking, the [[probability distribution]] of~~		Hello and welcome. My name is Numbers Wunder. Body building is one of the things I adore most. California is exactly where I've usually been living and I love each working day residing right here. My day job is a librarian.<br><br>My weblog ... [http://www.neweracinema.com/tube/blog/181510 http://www.neweracinema.com/tube/blog/181510]

	~~: <math> \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}} </math>~~

	~~is the standard [[normal distribution]]~~. ~~This~~ is ~~a deep extension~~ of the ~~[[Hardy–Ramanujan theorem]], which states that the [[Normal order of an arithmetic function\|normal order]] of ω(''n'')~~ is ~~log log~~ '~~'n'' with a typical error of size <math>\sqrt{\log\log n}</math>.~~

	~~More precisely, for any fixed ''a'' < ''b'',~~

	~~:<math>\lim_{x \rightarrow \infty} \left ( \frac {1}{x} \cdot \#\left\{ n \leq x : a \le \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \le b \~~right~~\} \right ) = \Phi(a,b) </math>~~

	~~where <math>\Phi(a,b)</math> is the normal (or "Gaussian") distribution, defined as~~

	~~: <math>\Phi(a,b)= \frac{1}{\sqrt{2\pi}}\int_a^b e^{-t^2/2} \, dt~~. ~~</math>~~

	~~Stated somewhat heuristically, what Erdős and Kac proved was that if ''n''~~ is a ~~randomly chosen large integer, then the number of distinct prime factors of ''n'' has approximately the normal distribution with mean and variance log log ''n''.~~

	~~This means that the construction of a number around one billion requires on average three primes.~~

	~~For example 1,000,000,003 = 23 × 307 × 141623.~~

	~~{\| class="wikitable" border="2" style="text-align:center"~~
	\|-
	~~! ''n''~~
	~~!Number of~~
	~~Digits in ''n''~~
	~~! Average number~~
	~~of distinct primes~~

	~~! standard~~
	~~deviation~~
	\|-
	~~\|1,000~~
	\|4
	\|2
	~~\|1.4~~
	\|-
	~~\|1,000,000,000~~
	~~\|10~~
	\|3
	\|1.7
	\|-
	~~\|1,000,000,000,000,000,000,000,000~~
	~~\|25~~
	\|4
	\|2
	\|-
	~~\|10~~<~~sup~~>65<~~/sup~~>
	~~\|66~~
	\|5
	\|2.2
	\|-
	~~\|10<sup>9,566</sup>~~
	~~\|9,567~~
	~~\|10~~
	\|3.2
	\|-
	~~\|10<sup>210,704,568</sup>~~
	~~\|210,704,569~~
	~~\|20~~
	\|4.5
	\|-
	~~\|10<sup>10<sup>22<~~/~~sup><~~/~~sup>~~
	~~\|10<sup>22</sup>+1~~
	~~\|50~~
	\|7.1
	\|-
	~~\|10<sup>10<sup>44</sup></sup>~~
	~~\|10<sup>44</sup>+1~~
	~~\|100~~
	~~\|10~~
	\|-
	~~\|10<sup>10<sup>434</sup></sup>~~
	~~\|10<sup>434</sup>+1~~
	~~\|1000~~
	~~\|31~~.6
	\|}

	~~[[Image:EKT plot.svg\|thumb\|300px\|right\|A spreading Gaussian distribution of distinct primes illustrating the Erdos-Kac theorem]]~~

	~~Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% (±σ) are constructed from between 7 and 13 primes.~~

	~~A hollow sphere the size of the planet Earth filled with fine sand would have around 10<sup>33<~~/~~sup> grains. A volume the size of the observable universe would have around 10<sup>93<~~/~~sup> grains of sand. There might be room for 10<sup>185</sup> quantum strings in such a universe.~~

	~~Numbers of this magnitude—with 186 digits—would require on average only 6 primes for construction.~~

	~~==References==~~
	* {{cite journal \| last1=Erdős \| first1=Paul \| author1-link=Paul Erdős \| last2=Kac \| first2=Mark \| author2-link=Mark Kac \| title=The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions \| journal=[[American Journal of Mathematics]] \| volume=62 \| number=1/~~4 \| year=1940 \| pages=738–742 \| zbl=0024.10203 \| issn=0002-9327 }}~~
	* {{cite book \| last1=Kuo \| first1=Wentang \| last2=Liu \| first2=Yu-Ru \| chapter=The Erdős–Kac theorem and its generalizations \| pages=209-216 \| editor1-last=De Koninck \| editor1-first=Jean-Marie \| editor2-last=Granville \| editor2-first=Andrew \| editor2-link=Andrew Granville \| editor3-last=Luca \| editor3-first=Florian \| title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 \| location=Providence, RI \| publisher=[[American Mathematical Society]] \| series=CRM Proceedings and Lecture Notes \| volume=46 \| year=2008 \| isbn=978-0-8218-4406-9 \| zbl=1187.11024 }}
	* {{cite book \| last=Kac \| first=Mark \| title=Statistical Independence in Probability, Analysis and Number Theory \|year=1959 \| publisher=John Wiley and Sons, Inc. }}

	~~==External links==~~
	* {{MathWorld\|urlname=Erdos-KacTheorem\|title=Erdős–Kac Theorem}}
	* [http://www.~~youtube~~.com/~~watch?v=4ivoaFLQ4vM#generator Timothy Gowers: The Importance of Mathematics (part 6, 4 mins in) and (part 7)]~~

	~~{{DEFAULTSORT:Erdos-Kac theorem}}~~
	~~[[Category:Paul Erdős\|Kac theorem]]~~
	~~[[Category:Normal distribution]]~~
	~~[[Category:Theorems about prime numbers]~~]

134.173.166.165: /* Example */

2012-12-16T05:11:52Z

Example

← Older revision		Revision as of 07:11, 16 December 2012
Line 1:		Line 1:
	~~Greetings. Let me begin by telling you~~ the ~~writer~~'s ~~name~~ - ~~Phebe~~. ~~For years he~~'~~s been working~~ as a ~~meter reader~~ and it'~~s some thing he really enjoy. The preferred pastime for my children~~ and ~~me is to play baseball but I haven~~'~~t produced~~ a ~~dime~~ with it. ~~California is where her house is but she requirements to move because~~ of ~~her family~~.<br><br>~~My weblog;~~ [http://www.~~hotporn123~~.com/~~user/KGottscha std testing at home~~]		In [[number theory]], the '''Erdős–Kac theorem''', named after [[Paul Erdős]] and [[Mark Kac]], and also known as the fundamental theorem of [[probabilistic number theory]], states that if ω(''n'') is the number of distinct [[prime factor]]s of ''n'', then, loosely speaking, the [[probability distribution]] of

			: <math> \frac{\omega(n) - \log\log n}{\sqrt{\log\log n}} </math>

			is the standard [[normal distribution]]. This is a deep extension of the [[Hardy–Ramanujan theorem]], which states that the [[Normal order of an arithmetic function\|normal order]] of ω(''n'') is log log ''n'' with a typical error of size <math>\sqrt{\log\log n}</math>.

			More precisely, for any fixed ''a'' < ''b'',

			:<math>\lim_{x \rightarrow \infty} \left ( \frac {1}{x} \cdot \#\left\{ n \leq x : a \le \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}} \le b \right\} \right ) = \Phi(a,b) </math>

			where <math>\Phi(a,b)</math> is the normal (or "Gaussian") distribution, defined as

			: <math>\Phi(a,b)= \frac{1}{\sqrt{2\pi}}\int_a^b e^{-t^2/2} \, dt. </math>

			Stated somewhat heuristically, what Erdős and Kac proved was that if ''n'' is a randomly chosen large integer, then the number of distinct prime factors of ''n'' has approximately the normal distribution with mean and variance log log ''n''.

			This means that the construction of a number around one billion requires on average three primes.

			For example 1,000,000,003 = 23 × 307 × 141623.

			{\| class="wikitable" border="2" style="text-align:center"
			\|-
			! ''n''
			!Number of
			Digits in ''n''
			! Average number
			of distinct primes

			! standard
			deviation
			\|-
			\|1,000
			\|4
			\|2
			\|1.4
			\|-
			\|1,000,000,000
			\|10
			\|3
			\|1.7
			\|-
			\|1,000,000,000,000,000,000,000,000
			\|25
			\|4
			\|2
			\|-
			\|10<sup>65</sup>
			\|66
			\|5
			\|2.2
			\|-
			\|10<sup>9,566</sup>
			\|9,567
			\|10
			\|3.2
			\|-
			\|10<sup>210,704,568</sup>
			\|210,704,569
			\|20
			\|4.5
			\|-
			\|10<sup>10<sup>22</sup></sup>
			\|10<sup>22</sup>+1
			\|50
			\|7.1
			\|-
			\|10<sup>10<sup>44</sup></sup>
			\|10<sup>44</sup>+1
			\|100
			\|10
			\|-
			\|10<sup>10<sup>434</sup></sup>
			\|10<sup>434</sup>+1
			\|1000
			\|31.6
			\|}

			[[Image:EKT plot.svg\|thumb\|300px\|right\|A spreading Gaussian distribution of distinct primes illustrating the Erdos-Kac theorem]]

			Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% (±σ) are constructed from between 7 and 13 primes.

			A hollow sphere the size of the planet Earth filled with fine sand would have around 10<sup>33</sup> grains. A volume the size of the observable universe would have around 10<sup>93</sup> grains of sand. There might be room for 10<sup>185</sup> quantum strings in such a universe.

			Numbers of this magnitude—with 186 digits—would require on average only 6 primes for construction.

			==References==
			* {{cite journal \| last1=Erdős \| first1=Paul \| author1-link=Paul Erdős \| last2=Kac \| first2=Mark \| author2-link=Mark Kac \| title=The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions \| journal=[[American Journal of Mathematics]] \| volume=62 \| number=1/4 \| year=1940 \| pages=738–742 \| zbl=0024.10203 \| issn=0002-9327 }}
			* {{cite book \| last1=Kuo \| first1=Wentang \| last2=Liu \| first2=Yu-Ru \| chapter=The Erdős–Kac theorem and its generalizations \| pages=209-216 \| editor1-last=De Koninck \| editor1-first=Jean-Marie \| editor2-last=Granville \| editor2-first=Andrew \| editor2-link=Andrew Granville \| editor3-last=Luca \| editor3-first=Florian \| title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 \| location=Providence, RI \| publisher=[[American Mathematical Society]] \| series=CRM Proceedings and Lecture Notes \| volume=46 \| year=2008 \| isbn=978-0-8218-4406-9 \| zbl=1187.11024 }}
			* {{cite book \| last=Kac \| first=Mark \| title=Statistical Independence in Probability, Analysis and Number Theory \|year=1959 \| publisher=John Wiley and Sons, Inc. }}

			==External links==
			* {{MathWorld\|urlname=Erdos-KacTheorem\|title=Erdős–Kac Theorem}}
			* [http://www.youtube.com/watch?v=4ivoaFLQ4vM#generator Timothy Gowers: The Importance of Mathematics (part 6, 4 mins in) and (part 7)]

			{{DEFAULTSORT:Erdos-Kac theorem}}
			[[Category:Paul Erdős\|Kac theorem]]
			[[Category:Normal distribution]]
			[[Category:Theorems about prime numbers]]

91.156.193.201: /* Application to financial assets */

2012-05-23T23:08:47Z

Application to financial assets

New page

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