en>ChrisGualtieri: General Fixes using AWB

2013-10-20T21:25:37Z

General Fixes using AWB

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			In [[mathematics]], and in particular [[combinatorics]], the '''necklace splitting problem''' arises in a variety of contexts including [[exact division]]; its picturesque name is due to mathematicians [[Noga Alon]] <ref name="Alon 1987">{{cite journal\| doi=10.1016/0001-8708(87)90055-7\| last=Alon\| first=Noga\| title=Splitting Necklaces\| journal=Advances in Mathematics\| year=1987\| volume=63\| issue=3\| pages=247–253\| postscript=.}}</ref> and [[Douglas B. West]].<ref name="Alon 1986">{{cite journal\| last=Alon\| first=Noga\| coauthors=West, D. B.\| title=The [[Borsuk-Ulam theorem]] and bisection of necklaces\| journal=Proceedings of the American Mathematical Society\| month=December\| year=1986\| volume=98\|issue=4 \|pages=623–628\| postscript=.}}</ref>

			Suppose a [[necklace]], open at the clasp, has ''k'' ·''n'' beads. There are ''k'' · ''a''<sub>''i''</sub> beads of colour ''i'', where 1 ≤ ''i'' ≤ ''t''. Then the necklace splitting problem is to find a partition of the necklace into ''k'' parts (not necessarily contiguous), each of which has exactly ''a''<sub>''i''</sub> beads of colour ''i''; such a split is called a ''k''-split. The '''size''' of the split is the number of cuts that are needed to separate the parts (the opening at the clasp is not included). Inevitably, one interesting question is to find a split of minimal size.

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			<div style="width:231px;">
			[[Image:necklace cropped.png\|alt=a stylized picture of a necklace, with 8 red and 6 green pearls. The pearls are threaded on to an incomplete elliptical black curve that represents the string. The gap in the curve represents the clasp (open in the diagram) which may be closed when the necklace is placed around the neck. There are two short heavy lines marking breaks in the necklace string. Starting from the left, the necklace is: RRRGRBRRGRRGGBGG, where "R" means "red pearl", "G" means "green pearl", and "B" means "break". The breaks correspond to those in the text\|200px\|Necklace splitting problem]]<div class="thumbcaption">
			Example of necklace splitting with ''k'' = 2 (i.e. two thieves), and ''t'' = 2 (i.e. two types of beads, here 8 red and 6 green). A 2-split is shown.
			</div>
			</div>
			</div>

			Alon explains that

			: <math>\ldots</math> the problem of finding ''k''-splittings of small size arises naturally when ''k'' mathematically oriented thieves steal a necklace with ''k'' · ''a''<sub>''i''</sub> [[Gemstone\|jewel]]s of type ''i'', and wish to divide it fairly between them, wasting as little as possible of the metal in the links between the jewels.

			If the beads of each colour are contiguous on the open necklace, then any ''k'' splitting must contain at least ''k'' − 1 cuts, so the size is at least (''k'' − 1)''t''. Alon and West<ref name="Alon 1987"/> use the [[Borsuk-Ulam theorem]] to prove that a ''k''-splitting can ''always'' be achieved with this number of cuts. Alon<ref name="Alon 1986"/> uses these and related ideas to state and prove a generalization of the [[Hobby–Rice theorem]].

			==Further results==
			===One cut fewer than needed===

			In the case of two thieves [i.e. ''k'' = 2] and ''t'' colours, a fair split would require at most ''t'' cuts. If, however, only ''t'' − 1 cuts are available, Hungarian mathematician [[Gábor Simonyi]]<ref name="Simonyi 2008">{{cite journal\| last=Simonyi\| first=Gábor\|title=Necklace bisection with one less cut than needed\|journal=Electronic journal of combinatorics\|volume=15\|year=2008\|issue=N16\| postscript=.}}</ref> shows that the two thieves can achieve an almost fair division in the following sense.

			If the necklace is arranged so that no ''t''-split is possible, then for any two subsets ''D''<sub>1</sub> and ''D''<sub>2</sub> of { 1, 2, ...,  ''t'' }, ''not both empty'', such that <math>D_1\cap D_2=\varnothing</math>, a (''t'' − 1)-split exists such that:

			* If colour <math>i\in D_1</math>, then partition 1 has more beads of colour ''i'' than partition 2;
			* If colour <math>i\in D_2</math>, then partition 2 has more beads of colour ''i'' than partition 1;
			* If colour ''i'' is in neither partition, both partitions have equally many beads of colour ''i''.

			I.e. if the thieves have preferences in the form of two "preference" sets ''D''<sub>1</sub> and ''D''<sub>2</sub>, not both empty, there exists a (''t'' − 1)-split so that thief 1 gets more beads of types in his preference set ''D''<sub>1</sub> than thief 2; thief 2 gets more beads of types in her preference set ''D''<sub>2</sub> than thief 1; '''and''' the rest are equal.

			Simonyi credits Gábor Tardos with noticing that the result above is a direct generalization of Alon's original necklace theorem in the case ''k'' = 2. Either the necklace has a (''t'' − 1)-split, or it does not. If it does, there is nothing to prove. If it does not, we may add beads of a fictitious colour to the necklace, and make ''D''<sub>1</sub> consist of the fictitious colour and ''D''<sub>2</sub> empty. Then Simonyi's result shows that there is a ''t''-split with equal numbers of each real colour.

			===Splitting multidimensional necklaces===

			The result can be generalized to ''n'' probability measures defined on a ''d'' dimensional cube with any combination of ''n''(''k'' − 1) hyperplanes parallel to the sides for ''k'' thieves.<ref>{{Cite journal \|
			doi=10.1016/j.aim.2008.02.003 \|
			last=de Longueville\| first=Mark\|coauthor=Rade T. Živaljević\|title=Splitting multidimensional necklaces\| journal=Advances in Mathematics\| year=2008\| volume=218 \|
			issue=3\| pages=926–939 \|
			postscript=.}}</ref>

			== See also ==
			* [[Necklace (combinatorics)\|Combinatorial necklaces]]
			* [[Necklace problem]]

			==References==

			{{reflist}}

			[[Category:Combinatorics on words]]
			[[Category:Discrete geometry]]
			[[Category:Fair division]]
			[[Category:Mathematical problems]]

en>PKT: Disambiguated: vanish → vanish (mathematics); formatting: 6x whitespace, 2x ellipsis (using Advisor.js)

2012-03-27T11:56:59Z

Disambiguated: vanish → vanish (mathematics); formatting: 6x whitespace, 2x ellipsis (using Advisor.js)

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If you are looking for a specific plugin, then you can just search for the name of the plugin. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Step-4 Testing: It is the foremost important of your Plugin development process. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. The top 4 reasons to use Business Word - Press Themes for a business website are:. These folders as well as files have to copied and the saved. The higher your blog ranks on search engines, the more likely people will find your online marketing site. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. You've got invested a great cope of time developing and producing up the topic substance. These frequent updates have created menace in the task of optimization. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. The disadvantage is it requires a considerable amount of time to set every thing up. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. If you have any issues about wherever and how to use [http://mmservice.dk/wordpress_backup_393615 wordpress dropbox backup], you can speak to us at our internet site. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Page speed is an important factor in ranking, especially with Google. For your information, it is an open source web content management system.

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en>ChrisGualtieri: General Fixes using AWB

en>PKT: Disambiguated: vanish → vanish (mathematics); formatting: 6x whitespace, 2x ellipsis (using Advisor.js)