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| [[Image:Mountain climbing problem.gif|thumb|right|1500px|Example of the problem resolution.]]
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| 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).
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| 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.
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| == Understanding the problem ==
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| 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.
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| == Formulation ==
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| The following result is due to Huneke:
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| :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]].
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| 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.
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| 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'').
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| === Proof in the piecewise linear case ===
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| 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:
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| : '''1.''' One climber is at a peak or a valley, another climber is in between two of them,
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| : '''2.''' Both climbers are at a peak or at valley.
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| : '''3.''' One climber is at a peak and one is at valley.
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| 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.
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| == References ==
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| * Tatsuo Homma, A theorem on continuous functions, ''Kōdai Math. Semin. Rep.'' 1952, 13–16.
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| * R. Sikorski, [[Kazimierz Zarankiewicz|K. Zarankiewicz]], On uniformization of functions. I. ''Fundam. Math.'' '''41''' (1955), 339–344.
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| * J.S. Lipiński, Sur l'uniformisation des fonctions continues, ''Bull. Acad. Pol. Sci. Cl. III'' '''5''' (1957), 1019–1021.
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| * Jerzy Mioduszewski, On a quasi-ordering in the class of continuous mappings of closed interval into itself, ''Colloq. Math.'' '''9''' (1962), 233–240.
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| * James V. Whittaker, A mountain-climbing problem, ''Canad. J. Math.'' '''18''' (1966), 873–882.
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| * John P. Huneke, Mountain climbing, ''Trans. Amer. Math. Soc.'' '''139''' (1969), 383–391.
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| * M. A. McKiernan, Mountain climbing: an alternate proof, ''Aequationes Math.'' '''28''' (1985), no. 1–2, 132–134.
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| * [[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.
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| * Tamás Keleti, The mountain climbers' problem, ''Proc. Amer. Math. Soc.'' '''117''' (1993), no. 1, 89–97.
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| * 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.
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| * Víctor Jiménez López, An elementary solution to the mountain climbers' problem, ''Aequationes Math.'' '''57''' (1999), no. 1, 45–49.
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| * [[Igor Pak]], ''[http://www.math.umn.edu/~pak/book.htm Lectures on Discrete and Polyhedral Geometry]'', Section 5.
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| * 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.
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| == External links ==
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| * [http://www.cs.mcgill.ca/~cs507/projects/1998/mhum/ The Parallel Mountain Climbers Problem], a description and a [[Java applet]] solution.
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| * [http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2922 MAA Writing Award]
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| [[Category:Articles containing proofs]]
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| [[Category:Discrete geometry]]
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| [[Category:Recreational mathematics]]
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| [[Category:Mathematical problems]]
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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.
Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring.
You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. Setting Up Your Business Online Using Free Wordpress Websites. Enough automated blog posts plus a system keeps you and your clients happy. There are plenty of tables that are attached to this particular database.
There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. One of the great features of Wordpress is its ability to integrate SEO into your site. If you have any type of concerns relating to where and the best ways to utilize backup plugin, you could call us at our own web-page. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. OSDI, a Wordpress Development Company based on ahmedabad, India.
Internet is not only the source for information, it is also one of the source for passive income. Visit our website to learn more about how you can benefit. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. As with a terminology, there are many methods to understand how to use the terminology.