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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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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.
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.
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.
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.
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 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.