Rietveld refinement - Revision history

en>Ericobnn: added an apostrophe

2014-08-29T21:30:06Z

added an apostrophe

← Older revision		Revision as of 23:30, 29 August 2014
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	~~In [[mathematics]], a '''covering system''' (also called a '''complete residue system''') is a collection~~		Let me first start by introducing myself. My [http://immbooks.com/blogs/post/44599 http://immbooks.com/blogs/post/44599] title is Boyd Butts although it is not the title on my birth certification. My family life in Minnesota std home test and my family members enjoys it. Since she was 18 she's been working as a receptionist but her promotion by no std testing at home means [http://Health.Cvs.com/GetContent.aspx?token=f75979d3-9c7c-4b16-af56-3e122a3f19e3&chunkiid=21271 arrives]. One [http://calvaryhill.net/xe/board_DsjO50/23648 find out here now] of the issues std home test she enjoys most is to study comics and she'll be starting some thing else [http://www.Ranker.com/list/famous-people-with-herpes/celebrity-lists alongside] with it.<br><br>Also visit my web site over the counter std test ([http://i4p.info/article.php?id=114136 you could check here])

	~~:<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>~~

	~~of finitely many~~
	~~residue classes <math> a_i(\mathrm{mod}\ {n_i}) = \{ a_i + n_ix:\ x \in \Z \} </math>~~
	~~whose union ''covers'' all the integers~~.

	~~{{unsolved\|mathematics\|~~
	~~For any arbitrarily large natural number ''N''~~ ~~does there exist an incongruent covering system the minimum of whose moduli is at least ''N''?}}~~

	~~== Examples and definitions ==~~

	~~The notion of covering system was introduced by~~ [~~[Paul Erdős]] in the early 1930s.~~

	~~The following are examples of covering systems~~:

	~~:<math>\{0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {3}),\ 2(\mathrm{mod}\ {3})\},<~~/~~math>~~

	~~and~~

	~~:<math>\{1(\mathrm{mod}\ {2}),\ 2(\mathrm{mod}\ {4}),\ 4(\mathrm{mod}\ {8}),\ 0(\mathrm{mod}\ {8})\},<~~/~~math>~~

	~~and~~

	~~:<math>\{0(\mathrm{mod}\ {2}),\ 0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {4}),~~
	~~\ 5(\mathrm{mod}\ {6}),\ 7(\mathrm{mod}\ {12})~~
	\}.</~~math>~~

	~~A covering system is called disjoint (or exact) if no two members overlap.~~

	~~A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).~~

	~~A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.~~

	~~The first two examples are disjoint.~~

	~~The third example is distinct.~~

	~~A system (i.e., an unordered multi-set)~~

	~~:<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}<~~/~~math>~~

	~~of finitely many~~
	~~residue classes is called an <math>m<~~/~~math>-cover if it covers every integer at least~~
	~~<math>m</math> times, and an exact <math>m<~~/~~math>-cover if it covers each integer exactly <math>m<~~/~~math> times~~. ~~It is known that for each~~
	~~<math>m=2,3,\ldots<~~/~~math> there are exact <math>m<~~/~~math>-covers which cannot be written as a union of two covers. For example,~~

	~~:<math>\{1(\mathrm{mod}\ {2});\ 0(\mathrm{mod}\ {3});\ 2(\mathrm{mod}\ {6});\ 0,4,6,8(\mathrm{mod}\ {10});~~
	</~~math>~~

	~~:<math>1,2,4,7,10,13(\mathrm{mod}\ {15});\ 5,11,12,22,23,29(\mathrm{mod}\ {30})~~
	~~\}</math>~~

	is ~~an exact 2-cover which~~ is not ~~a union of two covers.~~

	~~== Mirsky–Newman theorem ==~~
	~~The Mirsky–Newman theorem, a special case of~~ the ~~[[Herzog–Schönheim conjecture]], states that there is no disjoint distinct covering system~~. ~~This result was conjectured~~ in ~~1950 by [[Paul Erdős]]~~ and ~~proved soon thereafter~~ by [~~[Leon Mirsky]] and [[Donald J~~. ~~Newman]]~~. ~~However, Mirsky and Newman never published their proof~~. ~~The same proof was also found independently by [[Harold Davenport]] and [[Richard Rado]].<ref name="soifer">{{citation~~
	~~\| last = Soifer \| first = Alexander \| author-link = Alexander Soifer~~
	~~\| isbn~~ = ~~978~~-0-~~387~~-~~74640~~-1
	~~\| location~~ = ~~New York~~
	~~\| publisher = Springer~~
	~~\| title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators~~
	~~\| year = 2008~~
	~~\| contribution = Chapter 1~~. ~~A story of colored polygons and arithmetic progressions~~
	~~\| pages = 1–9}}.</ref>~~

	~~==Primefree sequences==~~
	~~Covering systems can be used to find~~ [[primefree sequence]]s, sequences of integers satisfying the same [[recurrence relation]] as the [[Fibonacci number]]s, such that consecutive numbers in the sequence are [[relatively prime]] but all numbers in the sequence are [[composite number]]s. For instance, a sequence of this type found by [[Herbert Wilf]] has initial terms
	:~~''a''<sub>1<~~/~~sub> = 20615674205555510, ''a''<sub>2<~~/~~sub> = 3794765361567513 {{OEIS\|id=A083216}}~~.
	In this sequence, the positions at which the numbers in the sequence are divisible by a prime ''p'' form an arithmetic progression; for instance, the even numbers in the sequence are the numbers ''a<sub>i</~~sub>'' where ''i'' is congruent to 1 mod 3. The progressions divisible by different primes form a covering system, showing that every number in the sequence is divisible by at least one prime.~~

	~~== Some unsolved problems ==~~
	~~[[Paul Erdős~~]] asked whether for any arbitrarily large ''N'' there exists an incongruent covering system the minimum of whose moduli is at least ''N''. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120; a suitable cover is 0(3), 0(4), 0(5), 1(6), 1(8), 2(10), 11(12), 1(15), 14(20), 5(24), 8(30), 6(40), 58(60), 26(120) ); D. Swift gave an example where the minimum of the moduli is 4 (and the moduli are in the set of the divisors of 2880). S. L. G. Choi proved<ref>{{cite journal \| author=S. L. G. Choi \| title=Covering the set of integers by congruence classes of distinct moduli \| journal=[[Math. Comp.]] \| volume=25 \| year=1971 \| pages=885–895 \| doi=10.2307/2004353 \| jstor=2004353 \| issue=116 \| publisher=Mathematics of Computation, Vol. 25, No. 116 }}</ref> that it is ~~possible~~ to ~~give an example for~~ '~~'N'' = 20, and Pace P Nielsen demonstrates<ref>{{cite journal \| author=Pace P. Nielsen \| title=A covering system whose smallest modulus is 40 \| journal=~~[~~[Journal of Number Theory]] \| volume=129 \| year=2009 \| pages=640–666 \| url=~~http://www.~~wiskundemeisjes~~.nl/wp-~~content/uploads/2009/02/sdarticle.pdf \| doi=10.1016/j.jnt.2008.09.016 \| issue=3}}</ref> the existence of an example~~ with ~~''N'' = 40, consisting of more than <math>10^{50}<~~/~~math> congruences.~~

	It appears that this problem was solved negatively by Bob Hough, who talked about his work at the Erdős centennial conference in Budapest on July 3 2013.<ref>{{cite arXiv \|eprint= 1307.0874 \|author1= Bob Hough \|title= The least modulus of a covering system \|class= math.NT \|year= 2013 }}</ref> He used the [[Lovász local lemma]] to show that there is some maximum ''N'' which can be the minimum modulus on a covering system. The proof is in principle [[Effective results in number theory\|effective]], though an explicit bound is not given.

	In another problem we want that all of the moduli (of an incongruent covering system) be odd. There is a famous unsolved conjecture from Erdős and [[John Selfridge\|Selfridge]]: an incongruent covering system (with ~~the minimum modulus greater than 1) whose moduli are odd, does not exist. It is known that if such a system exists with square-free moduli, the overall modulus must have at least 22 prime factors~~.<~~ref~~>{{cite journal \| title=On odd covering systems with distinct moduli \| author=Song Guo \| coauthors=Zhi-Wei Sun \| date=14 September 2005 \| volume=35 \| issue=182 \| journal=Adv. Appl. Math. , --187 \| arxiv=math/0412217 }}<~~/ref~~>

	~~==See also==~~
	* [[Chinese remainder theorem]]
	* [[Covering set]]
	* [[Residue number system]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book\|author=Richard K. Guy\|authorlink=Richard K. Guy\|title=[[Unsolved Problems in Number Theory]]\|publisher=[[Springer-Verlag]]\|year=2004\|isbn=0-387-20860-7\|pages=383–385\| zbl=1058.11001 \| series=Problem Books in Mathematics \| location=New York, NY }}

	~~==External links==~~
	* [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/Cover.pdf Problems and Results on Covering Systems] (a survey) (~~[[PDF]])~~
	* Zhi-Wei Sun: [http://~~math~~.~~nju.edu.cn~~/~~~zwsun/Cref~~.~~pdf Classified Publications on Covering Systems~~] ~~(PDF~~)

	~~[[Category:Number theory]]~~
	~~[[Category:Unsolved problems in mathematics]]~~

66.25.206.181 at 20:25, 2 November 2013

2013-11-02T20:25:47Z

← Older revision		Revision as of 22:25, 2 November 2013
Line 1:		Line 1:
	The ~~individual who wrote~~ the ~~post~~ is known as ~~Jayson Hirano~~ and ~~he totally digs~~ that ~~name~~. ~~My working day job~~ is a ~~travel agent~~. ~~The favorite hobby~~ for ~~him~~ and ~~his kids~~ is to ~~play lacross~~ and he'll be ~~starting something else along~~ with it. ~~Alaska~~ is ~~exactly where he's usually been residing~~.<br><br>~~Look at my weblog clairvoyant psychic (~~[http://~~formalarmour~~.~~com~~/~~index~~.~~php?do=~~/~~profile-26947~~/~~info~~/ ~~click through the up coming webpage~~])		In [[mathematics]], a '''covering system''' (also called a '''complete residue system''') is a collection

			:<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>

			of finitely many
			residue classes <math> a_i(\mathrm{mod}\ {n_i}) = \{ a_i + n_ix:\ x \in \Z \} </math>
			whose union ''covers'' all the integers.

			{{unsolved\|mathematics\|
			For any arbitrarily large natural number ''N'' does there exist an incongruent covering system the minimum of whose moduli is at least ''N''?}}

			== Examples and definitions ==

			The notion of covering system was introduced by [[Paul Erdős]] in the early 1930s.

			The following are examples of covering systems:

			:<math>\{0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {3}),\ 2(\mathrm{mod}\ {3})\},</math>

			and

			:<math>\{1(\mathrm{mod}\ {2}),\ 2(\mathrm{mod}\ {4}),\ 4(\mathrm{mod}\ {8}),\ 0(\mathrm{mod}\ {8})\},</math>

			and

			:<math>\{0(\mathrm{mod}\ {2}),\ 0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {4}),
			\ 5(\mathrm{mod}\ {6}),\ 7(\mathrm{mod}\ {12})
			\}.</math>

			A covering system is called disjoint (or exact) if no two members overlap.

			A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).

			A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.

			The first two examples are disjoint.

			The third example is distinct.

			A system (i.e., an unordered multi-set)

			:<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>

			of finitely many
			residue classes is called an <math>m</math>-cover if it covers every integer at least
			<math>m</math> times, and an exact <math>m</math>-cover if it covers each integer exactly <math>m</math> times. It is known that for each
			<math>m=2,3,\ldots</math> there are exact <math>m</math>-covers which cannot be written as a union of two covers. For example,

			:<math>\{1(\mathrm{mod}\ {2});\ 0(\mathrm{mod}\ {3});\ 2(\mathrm{mod}\ {6});\ 0,4,6,8(\mathrm{mod}\ {10});
			</math>

			:<math>1,2,4,7,10,13(\mathrm{mod}\ {15});\ 5,11,12,22,23,29(\mathrm{mod}\ {30})
			\}</math>

			is an exact 2-cover which is not a union of two covers.

			== Mirsky–Newman theorem ==
			The Mirsky–Newman theorem, a special case of the [[Herzog–Schönheim conjecture]], states that there is no disjoint distinct covering system. This result was conjectured in 1950 by [[Paul Erdős]] and proved soon thereafter by [[Leon Mirsky]] and [[Donald J. Newman]]. However, Mirsky and Newman never published their proof. The same proof was also found independently by [[Harold Davenport]] and [[Richard Rado]].<ref name="soifer">{{citation
			\| last = Soifer \| first = Alexander \| author-link = Alexander Soifer
			\| isbn = 978-0-387-74640-1
			\| location = New York
			\| publisher = Springer
			\| title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators
			\| year = 2008
			\| contribution = Chapter 1. A story of colored polygons and arithmetic progressions
			\| pages = 1–9}}.</ref>

			==Primefree sequences==
			Covering systems can be used to find [[primefree sequence]]s, sequences of integers satisfying the same [[recurrence relation]] as the [[Fibonacci number]]s, such that consecutive numbers in the sequence are [[relatively prime]] but all numbers in the sequence are [[composite number]]s. For instance, a sequence of this type found by [[Herbert Wilf]] has initial terms
			:''a''<sub>1</sub> = 20615674205555510, ''a''<sub>2</sub> = 3794765361567513 {{OEIS\|id=A083216}}.
			In this sequence, the positions at which the numbers in the sequence are divisible by a prime ''p'' form an arithmetic progression; for instance, the even numbers in the sequence are the numbers ''a<sub>i</sub>'' where ''i'' is congruent to 1 mod 3. The progressions divisible by different primes form a covering system, showing that every number in the sequence is divisible by at least one prime.

			== Some unsolved problems ==
			[[Paul Erdős]] asked whether for any arbitrarily large ''N'' there exists an incongruent covering system the minimum of whose moduli is at least ''N''. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120; a suitable cover is 0(3), 0(4), 0(5), 1(6), 1(8), 2(10), 11(12), 1(15), 14(20), 5(24), 8(30), 6(40), 58(60), 26(120) ); D. Swift gave an example where the minimum of the moduli is 4 (and the moduli are in the set of the divisors of 2880). S. L. G. Choi proved<ref>{{cite journal \| author=S. L. G. Choi \| title=Covering the set of integers by congruence classes of distinct moduli \| journal=[[Math. Comp.]] \| volume=25 \| year=1971 \| pages=885–895 \| doi=10.2307/2004353 \| jstor=2004353 \| issue=116 \| publisher=Mathematics of Computation, Vol. 25, No. 116 }}</ref> that it is possible to give an example for ''N'' = 20, and Pace P Nielsen demonstrates<ref>{{cite journal \| author=Pace P. Nielsen \| title=A covering system whose smallest modulus is 40 \| journal=[[Journal of Number Theory]] \| volume=129 \| year=2009 \| pages=640–666 \| url=http://www.wiskundemeisjes.nl/wp-content/uploads/2009/02/sdarticle.pdf \| doi=10.1016/j.jnt.2008.09.016 \| issue=3}}</ref> the existence of an example with ''N'' = 40, consisting of more than <math>10^{50}</math> congruences.

			It appears that this problem was solved negatively by Bob Hough, who talked about his work at the Erdős centennial conference in Budapest on July 3 2013.<ref>{{cite arXiv \|eprint= 1307.0874 \|author1= Bob Hough \|title= The least modulus of a covering system \|class= math.NT \|year= 2013 }}</ref> He used the [[Lovász local lemma]] to show that there is some maximum ''N'' which can be the minimum modulus on a covering system. The proof is in principle [[Effective results in number theory\|effective]], though an explicit bound is not given.

			In another problem we want that all of the moduli (of an incongruent covering system) be odd. There is a famous unsolved conjecture from Erdős and [[John Selfridge\|Selfridge]]: an incongruent covering system (with the minimum modulus greater than 1) whose moduli are odd, does not exist. It is known that if such a system exists with square-free moduli, the overall modulus must have at least 22 prime factors.<ref>{{cite journal \| title=On odd covering systems with distinct moduli \| author=Song Guo \| coauthors=Zhi-Wei Sun \| date=14 September 2005 \| volume=35 \| issue=182 \| journal=Adv. Appl. Math. , --187 \| arxiv=math/0412217 }}</ref>

			==See also==
			* [[Chinese remainder theorem]]
			* [[Covering set]]
			* [[Residue number system]]

			==References==
			{{reflist}}
			* {{cite book\|author=Richard K. Guy\|authorlink=Richard K. Guy\|title=[[Unsolved Problems in Number Theory]]\|publisher=[[Springer-Verlag]]\|year=2004\|isbn=0-387-20860-7\|pages=383–385\| zbl=1058.11001 \| series=Problem Books in Mathematics \| location=New York, NY }}

			==External links==
			* [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/Cover.pdf Problems and Results on Covering Systems] (a survey) ([[PDF]])
			* Zhi-Wei Sun: [http://math.nju.edu.cn/~zwsun/Cref.pdf Classified Publications on Covering Systems] (PDF)

			[[Category:Number theory]]
			[[Category:Unsolved problems in mathematics]]

en>Abc root: /* Peak shape */

2012-06-17T20:05:17Z

Peak shape

New page

The individual who wrote the post is known as Jayson Hirano and he totally digs that name. My working day job is a travel agent. The favorite hobby for him and his kids is to play lacross and he'll be starting something else along with it. Alaska is exactly where he's usually been residing.<br><br>Look at my weblog clairvoyant psychic ([http://formalarmour.com/index.php?do=/profile-26947/info/ click through the up coming webpage])