Brain connectivity estimators - Revision history

en>Randykitty: wikilink

2014-10-10T14:36:37Z

wikilink

← Older revision		Revision as of 16:36, 10 October 2014
Line 1:		Line 1:
	~~[[File:3x3 grid graph haven.svg\|thumb\|240px\|A bramble of order four in a 3×3 grid graph~~, ~~consisting of six mutually touching connected subgraphs]]~~		Golfers can picnic by day along the lake or near the Gooseberry River, and relax at night in accommodations from cottage suites to lakeside homes.<br>Michael Guggenheim uses his blog for education as a teacher to document his nonprofit organization and his extracurricular activities teaching even younger students how to use a computer.<br>http://www.geneseecancerassistance.com/sale/uggs.asp?p=120 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=42 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=114 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=7 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=43 <br /><br><br>For more info regarding [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] have a look at our webpage.
	In graph theory, a '''bramble''' for an [[undirected graph]] ''G'' is a family of [[connected graph\|connected]] [[Glossary of graph theory#Subgraphs\|subgraphs]] of ''G'' that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in ''G'' that has one endpoint in each subgraph. The ''order'' of a bramble is the smallest size of a [[hitting set]], a set of vertices of ''G'' that has a nonempty intersection with each of the subgraphs. Brambles may be used to ~~characterize the [[treewidth]] of ''G''~~.<~~ref name="st93"~~>~~{{citation~~
	~~\| last1 = Seymour \| first1 = Paul D. \| author1-link = Paul Seymour (mathematician)~~
	~~\| last2 = Thomas \| first2 = Robin \| author2-link = Robin Thomas (mathematician)~~
	~~\| doi = 10.1006/jctb.1993.1027~~
	~~\| issue = 1~~
	~~\| journal = [[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| mr = 1214888~~
	~~\| pages = 22–33~~
	~~\| title = Graph searching and a min-max theorem~~ for ~~tree-width~~
	~~\| volume = 58~~
	~~\| year = 1993}}. In this reference, brambles are called "screens" and their order is called "thickness".</ref>~~

	~~==Treewidth and havens==~~
	~~A [[haven (graph theory)\|haven]] of order ''k'' in a graph ''G'' is~~ a ~~function ''β'' that maps each set ''X'' of fewer than ''k'' vertices~~ to ~~a connected component of ''G'' − ''X'', in such a way that every two subsets ''β''(''X'')~~ and ~~''β''(''Y'') touch each other. Thus, the set of images of ''β'' forms a bramble in ''G'', with order ''k''. Conversely, every bramble may be used~~ to ~~determine a haven: for each set ''X'' of size smaller than the order of the bramble, there is~~ a ~~unique connected component ''β''(''X'') that contains all of the subgraphs in the bramble that are disjoint from ''X''~~.

	~~As Seymour and Thomas showed, the order of a bramble (or equivalently, of a haven) characterizes [[treewidth]]~~: ~~a graph has a bramble of order ''k'' if and only if it has treewidth at least {{nowrap\|''k'' − 1}}.<ref name="st93"~~/>

	~~==Size of brambles==~~
	As Grohe and Marx observed, [[expander graph]]s of bounded [[degree (graph theory)\|degree]] have treewidth proportional to their number of vertices, and in order to include a subgraph disjoint from every set of vertices of this size, a bramble for such a graph must include an exponential number of different subgraphs. ~~More strongly, for these graphs, even brambles whose order is slightly larger than the square root of the treewidth must have exponential size~~. ~~However, Grohe and Marx also showed that every graph of treewidth ''k'' has a bramble of polynomial size and of order <math>\Omega(k^{1~~/2}/~~\log^2 k)~~</~~math>.<ref~~>~~{{citation~~
	~~\| last1 = Grohe \| first1 = Martin~~
	~~\| last2 = Marx \| first2 = Dániel~~
	~~\| doi = 10.1016~~/j.~~jctb~~.~~2008~~.~~06.004~~
	~~\| issue = 1~~
	~~\| journal~~ = ~~[[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| mr = 2467827~~
	~~\| pages = 218–228~~
	~~\| title = On tree width, bramble size, and expansion~~
	~~\| volume = 99~~
	~~\| year = 2009}}.~~</~~ref~~>

	~~==Computational complexity==~~
	Because brambles may have exponential size, it is not always possible to construct them in [[polynomial time]] for graphs of unbounded treewidth. However, when the treewidth is bounded, a polynomial time construction is possible: ~~it is possible to find a bramble of order ''k'', when one exists, in time O(''n''<sup>''k'' + 2<~~/~~sup>) where ''n'' is the number of vertices in the given graph~~. ~~Even faster algorithms are possible for graphs with few minimal separators~~.~~<ref>{{citation~~
	~~\| last1 = Chapelle \| first1 = Mathieu~~
	~~\| last2 = Mazoit \| first2 = Frédéric~~
	~~\| last3 = Todinca \| first3 = Ioan~~
	~~\| contribution = Constructing brambles~~
	~~\| doi = 10~~.~~1007/978-3-642-03816-7_20~~
	~~\| location = Berlin~~
	~~\| mr = 2539494~~
	~~\| pages = 223–234~~
	~~\| publisher = Springer~~
	~~\| series = Lecture Notes in Computer Science~~
	~~\| title = Mathematical Foundations of Computer Science 2009: 34th International Symposium, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009, Proceedings~~
	~~\| volume~~ = ~~5734~~
	~~\| year = 2009}}.~~</~~ref~~>

	~~Bodlaender, Grigoriev, and Koster<ref>{{citation~~
	~~\| last1 = Bodlaender \| first1 = Hans L~~. ~~\| author1-link = Hans L~~. ~~Bodlaender~~
	~~\| last2 = Grigoriev \| first2 = Alexander~~
	~~\| last3 = Koster \| first3 = Arie M~~. ~~C. A.~~
	~~\| doi = 10.1007/s00453-007-9056-z~~
	~~\| issue = 1~~
	~~\| journal = [[Algorithmica]]~~
	~~\| mr = 2385750~~
	~~\| pages~~ = ~~81–98~~
	~~\| title = Treewidth lower bounds with brambles~~
	~~\| volume = 51~~
	~~\| year = 2008}}.~~</~~ref~~> studied heuristics for finding brambles of high order. Their methods do not always generate brambles of order close to the treewidth of the input graph, but for planar graphs they give a constant [[approximation ratio]]. Kreutzer and Tazari<ref>{{citation
	~~\| last1 = Kreutzer \| first1 = Stephan~~
	~~\| last2 = Tazari \| first2 = Siamak~~
	~~\| contribution = On brambles, grid-like minors, and parameterized intractability of monadic second-order logic~~
	~~\| pages = 354–364~~
	~~\| title = Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '10)~~
	~~\| url =~~ http://dl.~~acm~~.~~org~~/~~citation~~.~~cfm~~?id=~~1873601.1873631~~
	~~\| year = 2010}}.~~</~~ref~~> ~~provide [[randomized algorithm]]s that, on graphs of treewidth ''k'', find brambles of polynomial size and of order~~ <~~math~~>~~\Omega(k^{1~~/2}/~~\log^3 k)<~~/~~math> within polynomial time, coming within~~ a ~~logarithmic factor of the order shown by {{harvtxt\|Grohe\|Marx\|2009}} to exist for polynomial size brambles~~.

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

	~~[[Category:Graph theory objects]]~~
	~~[[Category:Graph minor theory]]~~

en>Mr. Granger: /* DTF and PDC are not influenced by volume conduction */ needs copyedit

2014-01-30T19:37:43Z

DTF and PDC are not influenced by volume conduction: needs copyedit

← Older revision		Revision as of 21:37, 30 January 2014
Line 1:		Line 1:
	~~Holiday Inn Marietta is conveniently situated right by Interstate 77~~.~~<br>~~In ~~fact~~, ~~the brand name~~ is ~~popular~~ for ~~very expensive prices~~. ~~This bag is now over $800 but~~ is ~~worth~~ the ~~money because~~ of the ~~quality material and unique design~~.<br>~~http://www~~.~~restaurantcalcuta~~.~~com~~/~~outlet/ugg~~.~~asp?p~~=~~230~~ <br /> ~~http~~:/~~/www~~.~~restaurantcalcuta~~.~~com~~/~~outlet~~/~~ugg~~.~~asp?p=223~~ <~~br /~~> ~~http:~~/~~/www~~.~~restaurantcalcuta~~.~~com/outlet/ugg~~.~~asp?p~~=20 <br /> ~~If you liked this post and you would like~~ to ~~acquire more data regarding~~ [~~http~~:/~~/www~~.~~bendtrapclub~~.~~com~~/~~cheap~~/~~ugg~~.~~asp Cheap Uggs Boots] kindly check out the website~~. ~~http://www~~.~~restaurantcalcuta~~.~~com~~/~~outlet~~/~~ugg~~.~~asp?p=197~~ <~~br /~~> http://~~www~~.~~restaurantcalcuta~~.~~com/outlet~~/~~ugg~~.~~asp~~?p=~~289~~ <br />		[[File:3x3 grid graph haven.svg\|thumb\|240px\|A bramble of order four in a 3×3 grid graph, consisting of six mutually touching connected subgraphs]]
			In graph theory, a '''bramble''' for an [[undirected graph]] ''G'' is a family of [[connected graph\|connected]] [[Glossary of graph theory#Subgraphs\|subgraphs]] of ''G'' that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in ''G'' that has one endpoint in each subgraph. The ''order'' of a bramble is the smallest size of a [[hitting set]], a set of vertices of ''G'' that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize the [[treewidth]] of ''G''.<ref name="st93">{{citation
			\| last1 = Seymour \| first1 = Paul D. \| author1-link = Paul Seymour (mathematician)
			\| last2 = Thomas \| first2 = Robin \| author2-link = Robin Thomas (mathematician)
			\| doi = 10.1006/jctb.1993.1027
			\| issue = 1
			\| journal = [[Journal of Combinatorial Theory]] \| series = Series B
			\| mr = 1214888
			\| pages = 22–33
			\| title = Graph searching and a min-max theorem for tree-width
			\| volume = 58
			\| year = 1993}}. In this reference, brambles are called "screens" and their order is called "thickness".</ref>

			==Treewidth and havens==
			A [[haven (graph theory)\|haven]] of order ''k'' in a graph ''G'' is a function ''β'' that maps each set ''X'' of fewer than ''k'' vertices to a connected component of ''G'' − ''X'', in such a way that every two subsets ''β''(''X'') and ''β''(''Y'') touch each other. Thus, the set of images of ''β'' forms a bramble in ''G'', with order ''k''. Conversely, every bramble may be used to determine a haven: for each set ''X'' of size smaller than the order of the bramble, there is a unique connected component ''β''(''X'') that contains all of the subgraphs in the bramble that are disjoint from ''X''.

			As Seymour and Thomas showed, the order of a bramble (or equivalently, of a haven) characterizes [[treewidth]]: a graph has a bramble of order ''k'' if and only if it has treewidth at least {{nowrap\|''k'' − 1}}.<ref name="st93"/>

			==Size of brambles==
			As Grohe and Marx observed, [[expander graph]]s of bounded [[degree (graph theory)\|degree]] have treewidth proportional to their number of vertices, and in order to include a subgraph disjoint from every set of vertices of this size, a bramble for such a graph must include an exponential number of different subgraphs. More strongly, for these graphs, even brambles whose order is slightly larger than the square root of the treewidth must have exponential size. However, Grohe and Marx also showed that every graph of treewidth ''k'' has a bramble of polynomial size and of order <math>\Omega(k^{1/2}/\log^2 k)</math>.<ref>{{citation
			\| last1 = Grohe \| first1 = Martin
			\| last2 = Marx \| first2 = Dániel
			\| doi = 10.1016/j.jctb.2008.06.004
			\| issue = 1
			\| journal = [[Journal of Combinatorial Theory]] \| series = Series B
			\| mr = 2467827
			\| pages = 218–228
			\| title = On tree width, bramble size, and expansion
			\| volume = 99
			\| year = 2009}}.</ref>

			==Computational complexity==
			Because brambles may have exponential size, it is not always possible to construct them in [[polynomial time]] for graphs of unbounded treewidth. However, when the treewidth is bounded, a polynomial time construction is possible: it is possible to find a bramble of order ''k'', when one exists, in time O(''n''<sup>''k'' + 2</sup>) where ''n'' is the number of vertices in the given graph. Even faster algorithms are possible for graphs with few minimal separators.<ref>{{citation
			\| last1 = Chapelle \| first1 = Mathieu
			\| last2 = Mazoit \| first2 = Frédéric
			\| last3 = Todinca \| first3 = Ioan
			\| contribution = Constructing brambles
			\| doi = 10.1007/978-3-642-03816-7_20
			\| location = Berlin
			\| mr = 2539494
			\| pages = 223–234
			\| publisher = Springer
			\| series = Lecture Notes in Computer Science
			\| title = Mathematical Foundations of Computer Science 2009: 34th International Symposium, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009, Proceedings
			\| volume = 5734
			\| year = 2009}}.</ref>

			Bodlaender, Grigoriev, and Koster<ref>{{citation
			\| last1 = Bodlaender \| first1 = Hans L. \| author1-link = Hans L. Bodlaender
			\| last2 = Grigoriev \| first2 = Alexander
			\| last3 = Koster \| first3 = Arie M. C. A.
			\| doi = 10.1007/s00453-007-9056-z
			\| issue = 1
			\| journal = [[Algorithmica]]
			\| mr = 2385750
			\| pages = 81–98
			\| title = Treewidth lower bounds with brambles
			\| volume = 51
			\| year = 2008}}.</ref> studied heuristics for finding brambles of high order. Their methods do not always generate brambles of order close to the treewidth of the input graph, but for planar graphs they give a constant [[approximation ratio]]. Kreutzer and Tazari<ref>{{citation
			\| last1 = Kreutzer \| first1 = Stephan
			\| last2 = Tazari \| first2 = Siamak
			\| contribution = On brambles, grid-like minors, and parameterized intractability of monadic second-order logic
			\| pages = 354–364
			\| title = Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '10)
			\| url = http://dl.acm.org/citation.cfm?id=1873601.1873631
			\| year = 2010}}.</ref> provide [[randomized algorithm]]s that, on graphs of treewidth ''k'', find brambles of polynomial size and of order <math>\Omega(k^{1/2}/\log^3 k)</math> within polynomial time, coming within a logarithmic factor of the order shown by {{harvtxt\|Grohe\|Marx\|2009}} to exist for polynomial size brambles.

			==References==
			{{reflist}}

			[[Category:Graph theory objects]]
			[[Category:Graph minor theory]]

en>ChrisGualtieri: General fixes / CHECKWIKI fixes using AWB

2012-08-30T23:48:04Z

General fixes / CHECKWIKI fixes using AWB

New page

Holiday Inn Marietta is conveniently situated right by Interstate 77. In fact, the brand name is popular for very expensive prices. This bag is now over $800 but is worth the money because of the quality material and unique design. http://www.restaurantcalcuta.com/outlet/ugg.asp?p=230 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=223 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=20 If you liked this post and you would like to acquire more data regarding [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] kindly check out the website. http://www.restaurantcalcuta.com/outlet/ugg.asp?p=197 http://www.restaurantcalcuta.com/outlet/ugg.asp?p=289

← Older revision		Revision as of 16:36, 10 October 2014
Line 1:		Line 1:
	~~[[File:3x3 grid graph haven.svg\|thumb\|240px\|A bramble of order four in a 3×3 grid graph~~, ~~consisting of six mutually touching connected subgraphs]]~~		Golfers can picnic by day along the lake or near the Gooseberry River, and relax at night in accommodations from cottage suites to lakeside homes.<br>Michael Guggenheim uses his blog for education as a teacher to document his nonprofit organization and his extracurricular activities teaching even younger students how to use a computer.<br>http://www.geneseecancerassistance.com/sale/uggs.asp?p=120 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=42 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=114 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=7 <br /> http://www.geneseecancerassistance.com/sale/uggs.asp?p=43 <br /><br><br>For more info regarding [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] have a look at our webpage.
	In graph theory, a '''bramble''' for an [[undirected graph]] ''G'' is a family of [[connected graph\|connected]] [[Glossary of graph theory#Subgraphs\|subgraphs]] of ''G'' that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in ''G'' that has one endpoint in each subgraph. The ''order'' of a bramble is the smallest size of a [[hitting set]], a set of vertices of ''G'' that has a nonempty intersection with each of the subgraphs. Brambles may be used to ~~characterize the [[treewidth]] of ''G''~~.<~~ref name="st93"~~>~~{{citation~~
	~~\| last1 = Seymour \| first1 = Paul D. \| author1-link = Paul Seymour (mathematician)~~
	~~\| last2 = Thomas \| first2 = Robin \| author2-link = Robin Thomas (mathematician)~~
	~~\| doi = 10.1006/jctb.1993.1027~~
	~~\| issue = 1~~
	~~\| journal = [[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| mr = 1214888~~
	~~\| pages = 22–33~~
	~~\| title = Graph searching and a min-max theorem~~ for ~~tree-width~~
	~~\| volume = 58~~
	~~\| year = 1993}}. In this reference, brambles are called "screens" and their order is called "thickness".</ref>~~

	~~==Treewidth and havens==~~
	~~A [[haven (graph theory)\|haven]] of order ''k'' in a graph ''G'' is~~ a ~~function ''β'' that maps each set ''X'' of fewer than ''k'' vertices~~ to ~~a connected component of ''G'' − ''X'', in such a way that every two subsets ''β''(''X'')~~ and ~~''β''(''Y'') touch each other. Thus, the set of images of ''β'' forms a bramble in ''G'', with order ''k''. Conversely, every bramble may be used~~ to ~~determine a haven: for each set ''X'' of size smaller than the order of the bramble, there is~~ a ~~unique connected component ''β''(''X'') that contains all of the subgraphs in the bramble that are disjoint from ''X''~~.

	~~As Seymour and Thomas showed, the order of a bramble (or equivalently, of a haven) characterizes [[treewidth]]~~: ~~a graph has a bramble of order ''k'' if and only if it has treewidth at least {{nowrap\|''k'' − 1}}.<ref name="st93"~~/>

	~~==Size of brambles==~~
	As Grohe and Marx observed, [[expander graph]]s of bounded [[degree (graph theory)\|degree]] have treewidth proportional to their number of vertices, and in order to include a subgraph disjoint from every set of vertices of this size, a bramble for such a graph must include an exponential number of different subgraphs. ~~More strongly, for these graphs, even brambles whose order is slightly larger than the square root of the treewidth must have exponential size~~. ~~However, Grohe and Marx also showed that every graph of treewidth ''k'' has a bramble of polynomial size and of order <math>\Omega(k^{1~~/2}/~~\log^2 k)~~</~~math>.<ref~~>~~{{citation~~
	~~\| last1 = Grohe \| first1 = Martin~~
	~~\| last2 = Marx \| first2 = Dániel~~
	~~\| doi = 10.1016~~/j.~~jctb~~.~~2008~~.~~06.004~~
	~~\| issue = 1~~
	~~\| journal~~ = ~~[[Journal of Combinatorial Theory]] \| series = Series B~~
	~~\| mr = 2467827~~
	~~\| pages = 218–228~~
	~~\| title = On tree width, bramble size, and expansion~~
	~~\| volume = 99~~
	~~\| year = 2009}}.~~</~~ref~~>

	~~==Computational complexity==~~
	Because brambles may have exponential size, it is not always possible to construct them in [[polynomial time]] for graphs of unbounded treewidth. However, when the treewidth is bounded, a polynomial time construction is possible: ~~it is possible to find a bramble of order ''k'', when one exists, in time O(''n''<sup>''k'' + 2<~~/~~sup>) where ''n'' is the number of vertices in the given graph~~. ~~Even faster algorithms are possible for graphs with few minimal separators~~.~~<ref>{{citation~~
	~~\| last1 = Chapelle \| first1 = Mathieu~~
	~~\| last2 = Mazoit \| first2 = Frédéric~~
	~~\| last3 = Todinca \| first3 = Ioan~~
	~~\| contribution = Constructing brambles~~
	~~\| doi = 10~~.~~1007/978-3-642-03816-7_20~~
	~~\| location = Berlin~~
	~~\| mr = 2539494~~
	~~\| pages = 223–234~~
	~~\| publisher = Springer~~
	~~\| series = Lecture Notes in Computer Science~~
	~~\| title = Mathematical Foundations of Computer Science 2009: 34th International Symposium, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009, Proceedings~~
	~~\| volume~~ = ~~5734~~
	~~\| year = 2009}}.~~</~~ref~~>

	~~Bodlaender, Grigoriev, and Koster<ref>{{citation~~
	~~\| last1 = Bodlaender \| first1 = Hans L~~. ~~\| author1-link = Hans L~~. ~~Bodlaender~~
	~~\| last2 = Grigoriev \| first2 = Alexander~~
	~~\| last3 = Koster \| first3 = Arie M~~. ~~C. A.~~
	~~\| doi = 10.1007/s00453-007-9056-z~~
	~~\| issue = 1~~
	~~\| journal = [[Algorithmica]]~~
	~~\| mr = 2385750~~
	~~\| pages~~ = ~~81–98~~
	~~\| title = Treewidth lower bounds with brambles~~
	~~\| volume = 51~~
	~~\| year = 2008}}.~~</~~ref~~> studied heuristics for finding brambles of high order. Their methods do not always generate brambles of order close to the treewidth of the input graph, but for planar graphs they give a constant [[approximation ratio]]. Kreutzer and Tazari<ref>{{citation
	~~\| last1 = Kreutzer \| first1 = Stephan~~
	~~\| last2 = Tazari \| first2 = Siamak~~
	~~\| contribution = On brambles, grid-like minors, and parameterized intractability of monadic second-order logic~~
	~~\| pages = 354–364~~
	~~\| title = Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '10)~~
	~~\| url =~~ http://dl.~~acm~~.~~org~~/~~citation~~.~~cfm~~?id=~~1873601.1873631~~
	~~\| year = 2010}}.~~</~~ref~~> ~~provide [[randomized algorithm]]s that, on graphs of treewidth ''k'', find brambles of polynomial size and of order~~ <~~math~~>~~\Omega(k^{1~~/2}/~~\log^3 k)<~~/~~math> within polynomial time, coming within~~ a ~~logarithmic factor of the order shown by {{harvtxt\|Grohe\|Marx\|2009}} to exist for polynomial size brambles~~.

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

	~~[[Category:Graph theory objects]]~~
	~~[[Category:Graph minor theory]]~~