Matrix (mathematics) - Revision history

en>Quondum: Undid revision 642145641 by 180.191.106.151 (talk)

2015-01-12T15:29:53Z

Undid revision 642145641 by 180.191.106.151 (talk)

← Older revision		Revision as of 17:29, 12 January 2015
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	~~by Nas,~~ is ~~very fitting and the film agrees with it~~. ~~Medical word press themes give you~~ the ~~latest medical designs~~. ~~These templates~~ are ~~professionally designed~~ and are ~~also Adsense ready~~. s ~~ultimately easy to implement and virtually maintenance free~~. ~~Also our developers~~ are ~~well convergent with~~ the ~~latest technologies~~ and ~~bitty-gritty~~ of ~~wordpress website design~~ and ~~promises to deliver you~~ the ~~best solution that you can ever have~~. <br><br>Word - Press ~~is known as the most popular blogging platform all over the~~ web and ~~is used by millions~~ of ~~blog enthusiasts worldwide. Wordpress~~ have ~~every reason with it which promote wordpress development. Our Daily Deal Software plugin brings~~ the ~~simplicity of setting up a~~ Word - Press ~~blog to the daily deal space~~. ~~t need to use~~ the ~~back button or~~ the ~~URL~~ to get to your home page. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>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~~. ~~As of now~~, ~~Pin2Press is getting ready to hit the market. This platform can be customizedaccording to the requirements of the business. The first thing~~ you ~~need to do is to~~ choose ~~the right web hosting plan~~. ~~Premium vs Customised~~ Word - Press ~~Themes - Premium~~ themes ~~are a lot like customised themes but without the customised price~~ and ~~without the wait~~. <br><br>~~Numerous bloggers are utilizing~~ Word - Press ~~and~~ with ~~good reason. And~~, ~~make no mistake,India's Fertility Clinics and IVF specialists are amongst~~ the ~~best in the world,~~and ~~have been for some time. Normally,~~ the ~~Word - Press developers make a thorough research~~ on your website ~~goals~~ and ~~then ingrain the most suitable graphical design elements to~~ your ~~website~~. ~~So, we have to add our social media sharing buttons in website~~. If your ~~site does well you can get paid professional designer~~ to ~~create a unique Word - Press theme~~. <br><br>~~Millions~~ of ~~individuals and organizations are now successfully using this tool throughout~~ the ~~world~~. ~~I don't want that for my visitors~~ and ~~I'm quite sure they don't either~~. ~~You can select color~~ of ~~your choice, graphics~~ of your ~~favorite, skins, photos, pages, etc~~. If you have any type of ~~questions concerning~~ where and ways to ~~make use of~~ [http://~~blbuh~~.ru/~~wordpress_backup_556847~~ backup plugin], you could ~~contact~~ us at our web page. ~~Working~~ with a ~~Word - Press blog and~~ the ~~appropriate cost-free Word - Press theme~~, you can ~~get~~ a ~~professional internet site up and published~~ in ~~no time at all~~. 95, ~~and they also supply studio press discount code for their clients, coming from 10% off~~ to ~~25% off upon all theme deals~~.		Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. <br><br>

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en>Wcherowi: Reverted 1 edit by 122.168.10.202 (talk): Unexplained removal of material. (TW)

2014-03-02T05:36:33Z

Reverted 1 edit by 122.168.10.202 (talk): Unexplained removal of material. (TW)

← Older revision		Revision as of 07:36, 2 March 2014
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	~~[[Image:Mountain climbing problem.gif\|thumb\|right\|1500px\|Example of the problem resolution.]]~~		by Nas, is very fitting and the film agrees with it. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. s ultimately easy to implement and virtually maintenance free. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Wordpress have every reason with it which promote wordpress development. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. t need to use the back button or the URL to get to your home page. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>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. As of now, Pin2Press is getting ready to hit the market. This platform can be customizedaccording to the requirements of the business. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Normally, the Word - Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website. So, we have to add our social media sharing buttons in website. If your site does well you can get paid professional designer to create a unique Word - Press theme. <br><br>Millions of individuals and organizations are now successfully using this tool throughout the world. I don't want that for my visitors and I'm quite sure they don't either. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. If you have any type of questions concerning where and ways to make use of [http://blbuh.ru/wordpress_backup_556847 backup plugin], you could contact us at our web page. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.

	~~In [[mathematics]]~~, ~~the '''mountain climbing problem'''~~ is a problem of finding the conditions that two [[Function (mathematics)\|function]] forming profiles of a [[Dimension\|two-dimensional]] [[mountain]] must satisfy, so that two [[Mountaineering\|climbers]] can start on the bottom on the opposite sides of the mountain and ~~coordinate their movements to reach to~~ the ~~top while always staying at~~ the ~~same height~~. ~~This problem was named~~ and ~~posed in this form by James V~~. ~~Whittaker in 1966, but its history goes back~~ to ~~Tatsuo Homma, who solved a version of it in 1952~~. ~~The problem has been repeatedly rediscovered~~ and ~~solved independently in different context by a number~~ of ~~people (see~~ the ~~references)~~.

	In the ~~past two decades~~ the problem was shown to be connected to the weak [[Fréchet distance]] of [[curve]]s in the plane (see ''Buchin et al.''), various planar [[motion planning]] problems in [[computational geometry]] (see ''Goodman et al.''), the [[square peg problem]] (see ''Pak''), [[semigroup]] of [[polynomial]]s (see ''Baird and ~~Magill''), etc. The problem was popularized in the article~~ by ~~''Goodman et al~~.~~'',~~ which ~~received~~ the ~~[[Mathematical Association~~ of ~~America\|MAA]] writing award in 1990.~~

	~~== Understanding the problem ==~~
	~~It is easy~~ to ~~coordinate the climbers' movement between the peaks and valleys ([[local maxima]] and [[minima]] of~~ the ~~functions)~~. ~~The difficulty is that~~ to ~~progress,~~ the ~~climbers must occasionally go down the mountain, either one~~ or the ~~other, or both climbers~~. ~~Similarly, either one or the other climber must backtrack towards the beginning of the journey. In fact, it has been observed that for~~ a ~~mountain with ''n'' peaks~~ and ~~valleys the number of turns can be as large as [[Quadratic function\|quadratic]] in ''n'' (see ''Buchin et al.~~'~~'). These difficulties make the problem unintuitive and sometimes rather difficult, both in theory and in practice.~~

	~~== Formulation ==~~
	~~The following result is due to Huneke:~~

	~~:Suppose <math>f</math> and <math>g</math> are [[continuous function]]~~s ~~from <math>[0,1]</math>~~ to ~~<math>[0,1]</math> with <math>f(0)=g(0)=0</math> and <math>f(1)=g(1)=1</math>, and such that neither function is [[Constant function\|constant]] on an [[Interval (mathematics)\|interval]]~~. ~~Then there exist continuous functions <math>s</math> and <math>t~~<~~/math~~> ~~from <math>[0,1]~~<~~/math~~> ~~to <math>[0,1]</math> with <math>s(0)=t(0)=0</math>, <math>s(1)=t(1)=1</math>, and such~~ that ~~<math>f\circ s \~~, ~~= \, g\circ t</math>, where "<math>\circ</math>" stands for a [[Function composition\|composition~~ of ~~functions]].~~

	~~To see [[heuristically]] that the result does not extend to all continuous functions~~, ~~note that if <math>f</math>~~ has a ~~constant interval while <math>g</math> has a [[Oscillation (mathematics)\|highly oscillating]] interval~~ on ~~the same level~~, ~~then the first climber would be forced~~ to ~~go back and forth infinitely many times, and thus can never reach~~ the ~~top~~.

	~~It is also known that for the [[piecewise linear function]]s there are no obstructions, i~~.~~e. the climbers can always coordinate their movements~~ to ~~get~~ to the ~~top (see ''Whittaker'')~~.

	~~=== Proof in the piecewise linear case ===~~
	~~Consider~~ a ~~[[Graph of a function\|graph]] <math>G</math> of all positions on a mountain both climbers can occupy on~~ the ~~same level. This graph is piecewise linear, i.e. a union of intervals,~~ and ~~can be viewed as a [[Graph (mathematics)\|graph]] in [[Graph theory]]~~. ~~Note that~~ <~~math~~>G<~~/math~~> ~~may or may not be [[Connectivity (graph theory)\|connected]]. The vertices of the intervals correspond to peaks~~ and ~~valleys of the functions. There are three cases:~~
	~~: '''1~~.~~''' One climber is at a peak or a valley~~, ~~another climber is in between two of them~~,
	: '~~''2.''' Both climbers~~ are ~~at a peak or at valley.~~
	~~: '''3.''' One climber is at a peak and one is at valley.~~
	In the ~~first case such vertex of <math>G</math> has two adjacent intervals,~~ in the ~~second case it has four~~, and ~~in the last case zero~~. ~~Therefore~~, ~~graph <math>G</math> has all vertices of even degree, except for~~ the ~~point <math>(0,0)</math> corresponding to two climbers~~ on ~~the bottom~~ and the ~~point <math>(1,1)</math> corresponding~~ to ~~two climbers on top of the mountain~~. ~~Applying the [[handshaking lemma]] to the [[Connected component (graph theory)\|connected component]] of <math>G</math> containing <math>(0~~,~~0)</math>~~ we ~~conclude that <math>(0,0)</math> and <math>(1,1)</math> are~~ in ~~the same connected component of <math>G</math>~~. ~~This implies that there is~~ a ~~[[Path (graph theory)\|path]] from~~ <~~math~~>~~(0,0)~~<~~/math~~> ~~to <math>(1,1)</math> in <math>G</math>. In the language~~ of ~~mountain climbers,~~ this ~~gives a way to coordinate~~ the ~~climbers' movement to reach the top of the mountain~~.

	~~== References ==~~
	* Tatsuo Homma, A theorem on continuous functions, ''~~Kōdai Math. Semin. Rep.~~'~~' 1952, 13–16~~.
	* R. Sikorski, [[Kazimierz Zarankiewicz\|K. Zarankiewicz]], On uniformization of ~~functions. I. ''Fundam. Math.'' '''41''' (1955), 339–344.~~
	* J.S. Lipiński, Sur l'uniformisation des fonctions continues, ''Bull. Acad. Pol. Sci. Cl. III'' '''5''' (1957), ~~1019–1021.~~
	* Jerzy Mioduszewski, On a quasi-ordering in the class of ~~continuous mappings of closed interval into itself~~, ~~''Colloq. Math.'' '''9''' (1962)~~, ~~233–240.~~
	* James V. Whittaker, ~~A mountain-climbing problem~~, ~~''Canad~~. ~~J. Math.'' '''18''' (1966), 873–882.~~
	* John P. Huneke, Mountain climbing, ''Trans. Amer. Math. Soc.'' '''139''' (1969), 383–391.
	* M. A. McKiernan, ~~Mountain climbing: an alternate proof, ''Aequationes Math.'' '''28''' (1985), no. 1–2, 132–134.~~
	* [[Jacob E. Goodman]], [[János Pach]], Chee-K. Yap, [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Goodman-Pach-Yap494-510.pdf Mountain climbing, ladder moving, and the ring-width of ~~a polygon], ''Amer. Math. Monthly'' '''96''' (1989), no. 6, 494–510.~~
	* Tamás Keleti, The mountain climbers' problem, ''Proc. Amer. Math. Soc.'' '''117''' (1993), no. 1, 89–97.
	* B.B. Baird, K.D. Magill, Jr., Green's ''R'', ''D'' and ~~''H'' relations for generalized polynomials, ''Semigroup Forum'' '''55''' (1997), no. 3, 267–293.~~
	* Víctor Jiménez López, An elementary solution to ~~the mountain climbers' problem, ''Aequationes Math.'' '''57''' (1999), no. 1, 45–49.~~
	* [[Igor Pak]], ''[http://~~www.math~~.~~umn.edu~~/~~~pak/book.htm Lectures on Discrete and Polyhedral Geometry~~]'', ~~Section 5.~~
	* Kevin Buchin, Maike Buchin, Christian Knauer, Günter Rote, and Carola Wenk, [http://page.~~mi.fu~~-~~berlin.de/rote/Papers/abstract/How+difficult+is+it+to+walk+~~the~~+dog.html How difficult is it to walk the dog?]~~, in ~~''Proc~~. ~~23rd European Workshop on Computational Geometry'' (Graz~~, ~~2007)~~, ~~pp. 170–173~~.

	~~== External links ==~~
	* [http://www.cs.mcgill.ca/~cs507/projects/1998/mhum/ The Parallel Mountain Climbers Problem], a description and a [[Java applet]] solution.
	* [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2922 MAA Writing Award]

	~~[[Category:Articles containing proofs]]~~
	~~[[Category:Discrete geometry]]~~
	~~[[Category:Recreational mathematics]]~~
	~~[[Category:Mathematical problems]]~~

en>ClueBot NG: Reverting possible vandalism by Xxhihi to version by Dmcq. False positive? Report it. Thanks, ClueBot NG. (1654375) (Bot)

2014-01-14T23:30:10Z

Reverting possible vandalism by Xxhihi to version by Dmcq. False positive? Report it. Thanks, ClueBot NG. (1654375) (Bot)

← Older revision		Revision as of 01:30, 15 January 2014
Line 1:		Line 1:
			[[Image:Mountain climbing problem.gif\|thumb\|right\|1500px\|Example of the problem resolution.]]

			In [[mathematics]], the '''mountain climbing problem''' is a problem of finding the conditions that two [[Function (mathematics)\|function]] forming profiles of a [[Dimension\|two-dimensional]] [[mountain]] must satisfy, so that two [[Mountaineering\|climbers]] can start on the bottom on the opposite sides of the mountain and coordinate their movements to reach to the top while always staying at the same height. This problem was named and posed in this form by James V. Whittaker in 1966, but its history goes back to Tatsuo Homma, who solved a version of it in 1952. The problem has been repeatedly rediscovered and solved independently in different context by a number of people (see the references).

	~~The main advantage~~ of ~~using~~ the ~~blog is that anyone can use~~ the ~~Word - Press blog~~ and ~~customize~~ the ~~elements~~ in the ~~theme regardless~~ to ~~limited knowledge about internet~~ and ~~website development~~. ~~You can~~ either ~~install Word - Press yourself~~ or ~~use free services offered on~~ the ~~web today~~. ~~One really cool features about this amazing~~ and ~~free wp plugin~~ is that ~~the code it generates~~ is ~~completely portable~~. s and ~~intelligently including~~ a ~~substantial amount~~ of ~~key words in~~ the ~~title tags~~, ~~image links~~, ~~etc~~. ~~If you~~ are ~~happy with your new look then click "Activate~~ '~~New Theme~~'" in the ~~top right corner~~. <br><br>~~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~~. There are ~~number~~ of ~~web services that offer Word press development across the world~~. ~~These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to~~ a ~~site easier to configure~~. ~~Once you~~'~~ve installed~~ the ~~program you can quickly begin by adding content and editing~~ it ~~with features such as bullet pointing~~, ~~text alignment~~ and ~~effects without having to do all~~ the ~~coding yourself~~. <br><br>~~The entrepreneurs can easily captivate their readers by using these versatile themes. After sending these details~~, ~~your Word - Press blog will be setup within a few days. Use this section~~ to ~~change many formatting elements. In crux~~ the ~~developer must have a detailed knowledge not only about~~ the ~~marketing tool but also about~~ the ~~ways in which it can be applied profitably~~. ~~Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without~~ the ~~customised price~~ and ~~without~~ the ~~wait~~. <br><br>~~The next thing I did after installing Wordpress was~~ to ~~find myself~~ a ~~free good-looking Wordpress-theme offering~~ the ~~functionality~~ I ~~was after~~. ~~Find more information about Design To Wordpress here~~. ~~To do this~~, ~~you should link your posts to other relevant posts that you~~'~~ve created~~. So, ~~we have to add our social media sharing buttons~~ in ~~website~~. ~~If you loved this post and you would like to receive more details relating to~~ [http://~~blbuh~~.ru/~~wordpress_backup_plugin_516605 wordpress backup] kindly browse through~~ the ~~web site. Now all you have to do is log into your Word~~ - ~~Press site making use~~ of ~~the very same username and password that you initially had in your previous site~~. ~~<br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products~~ and ~~services~~ for ~~your customers~~. ~~By using Word - Press MLM websites or blogs~~, ~~an online presence for you~~ and ~~your MLM company can be created swiftly and simply~~. ~~It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant~~, ~~SEO professional~~ and ~~end user~~. ~~Word~~ - ~~Press~~ is the ~~most popular personal publishing platform which was launched~~ in ~~2003~~. ~~Verify whether your company has~~ a ~~team of developers or programmers having hands-on experience~~ and ~~knowledge about all Word - Press concepts~~.		In the past two decades the problem was shown to be connected to the weak [[Fréchet distance]] of [[curve]]s in the plane (see ''Buchin et al.''), various planar [[motion planning]] problems in [[computational geometry]] (see ''Goodman et al.''), the [[square peg problem]] (see ''Pak''), [[semigroup]] of [[polynomial]]s (see ''Baird and Magill''), etc. The problem was popularized in the article by ''Goodman et al.'', which received the [[Mathematical Association of America\|MAA]] writing award in 1990.

			== Understanding the problem ==
			It is easy to coordinate the climbers' movement between the peaks and valleys ([[local maxima]] and [[minima]] of the functions). The difficulty is that to progress, the climbers must occasionally go down the mountain, either one or the other, or both climbers. Similarly, either one or the other climber must backtrack towards the beginning of the journey. In fact, it has been observed that for a mountain with ''n'' peaks and valleys the number of turns can be as large as [[Quadratic function\|quadratic]] in ''n'' (see ''Buchin et al.''). These difficulties make the problem unintuitive and sometimes rather difficult, both in theory and in practice.

			== Formulation ==
			The following result is due to Huneke:

			:Suppose <math>f</math> and <math>g</math> are [[continuous function]]s from <math>[0,1]</math> to <math>[0,1]</math> with <math>f(0)=g(0)=0</math> and <math>f(1)=g(1)=1</math>, and such that neither function is [[Constant function\|constant]] on an [[Interval (mathematics)\|interval]]. Then there exist continuous functions <math>s</math> and <math>t</math> from <math>[0,1]</math> to <math>[0,1]</math> with <math>s(0)=t(0)=0</math>, <math>s(1)=t(1)=1</math>, and such that <math>f\circ s \, = \, g\circ t</math>, where "<math>\circ</math>" stands for a [[Function composition\|composition of functions]].

			To see [[heuristically]] that the result does not extend to all continuous functions, note that if <math>f</math> has a constant interval while <math>g</math> has a [[Oscillation (mathematics)\|highly oscillating]] interval on the same level, then the first climber would be forced to go back and forth infinitely many times, and thus can never reach the top.

			It is also known that for the [[piecewise linear function]]s there are no obstructions, i.e. the climbers can always coordinate their movements to get to the top (see ''Whittaker'').

			=== Proof in the piecewise linear case ===
			Consider a [[Graph of a function\|graph]] <math>G</math> of all positions on a mountain both climbers can occupy on the same level. This graph is piecewise linear, i.e. a union of intervals, and can be viewed as a [[Graph (mathematics)\|graph]] in [[Graph theory]]. Note that <math>G</math> may or may not be [[Connectivity (graph theory)\|connected]]. The vertices of the intervals correspond to peaks and valleys of the functions. There are three cases:
			: '''1.''' One climber is at a peak or a valley, another climber is in between two of them,
			: '''2.''' Both climbers are at a peak or at valley.
			: '''3.''' One climber is at a peak and one is at valley.
			In the first case such vertex of <math>G</math> has two adjacent intervals, in the second case it has four, and in the last case zero. Therefore, graph <math>G</math> has all vertices of even degree, except for the point <math>(0,0)</math> corresponding to two climbers on the bottom and the point <math>(1,1)</math> corresponding to two climbers on top of the mountain. Applying the [[handshaking lemma]] to the [[Connected component (graph theory)\|connected component]] of <math>G</math> containing <math>(0,0)</math> we conclude that <math>(0,0)</math> and <math>(1,1)</math> are in the same connected component of <math>G</math>. This implies that there is a [[Path (graph theory)\|path]] from <math>(0,0)</math> to <math>(1,1)</math> in <math>G</math>. In the language of mountain climbers, this gives a way to coordinate the climbers' movement to reach the top of the mountain.

			== References ==
			* Tatsuo Homma, A theorem on continuous functions, ''Kōdai Math. Semin. Rep.'' 1952, 13–16.
			* R. Sikorski, [[Kazimierz Zarankiewicz\|K. Zarankiewicz]], On uniformization of functions. I. ''Fundam. Math.'' '''41''' (1955), 339–344.
			* J.S. Lipiński, Sur l'uniformisation des fonctions continues, ''Bull. Acad. Pol. Sci. Cl. III'' '''5''' (1957), 1019–1021.
			* Jerzy Mioduszewski, On a quasi-ordering in the class of continuous mappings of closed interval into itself, ''Colloq. Math.'' '''9''' (1962), 233–240.
			* James V. Whittaker, A mountain-climbing problem, ''Canad. J. Math.'' '''18''' (1966), 873–882.
			* John P. Huneke, Mountain climbing, ''Trans. Amer. Math. Soc.'' '''139''' (1969), 383–391.
			* M. A. McKiernan, Mountain climbing: an alternate proof, ''Aequationes Math.'' '''28''' (1985), no. 1–2, 132–134.
			* [[Jacob E. Goodman]], [[János Pach]], Chee-K. Yap, [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Goodman-Pach-Yap494-510.pdf Mountain climbing, ladder moving, and the ring-width of a polygon], ''Amer. Math. Monthly'' '''96''' (1989), no. 6, 494–510.
			* Tamás Keleti, The mountain climbers' problem, ''Proc. Amer. Math. Soc.'' '''117''' (1993), no. 1, 89–97.
			* B.B. Baird, K.D. Magill, Jr., Green's ''R'', ''D'' and ''H'' relations for generalized polynomials, ''Semigroup Forum'' '''55''' (1997), no. 3, 267–293.
			* Víctor Jiménez López, An elementary solution to the mountain climbers' problem, ''Aequationes Math.'' '''57''' (1999), no. 1, 45–49.
			* [[Igor Pak]], ''[http://www.math.umn.edu/~pak/book.htm Lectures on Discrete and Polyhedral Geometry]'', Section 5.
			* Kevin Buchin, Maike Buchin, Christian Knauer, Günter Rote, and Carola Wenk, [http://page.mi.fu-berlin.de/rote/Papers/abstract/How+difficult+is+it+to+walk+the+dog.html How difficult is it to walk the dog?], in ''Proc. 23rd European Workshop on Computational Geometry'' (Graz, 2007), pp. 170–173.

			== External links ==
			* [http://www.cs.mcgill.ca/~cs507/projects/1998/mhum/ The Parallel Mountain Climbers Problem], a description and a [[Java applet]] solution.
			* [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2922 MAA Writing Award]

			[[Category:Articles containing proofs]]
			[[Category:Discrete geometry]]
			[[Category:Recreational mathematics]]
			[[Category:Mathematical problems]]

en>JohnBlackburne: Reverted 9 edits by 87.20.54.214 (talk): Unexplained, non-standard format changes. (TW)

2012-09-01T14:55:15Z

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