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| {{Expert-subject|mathematics|date=November 2008}}
| | For the offense, you might attain Gunboats which can basically shoot at enemy defenses coming from a range and Landing Crafts which you must seal when you train products for example Rifleman, Heavy, Zooka, Warrior and Takes a dive. To your village defenses, you might offer structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Bomb Launcher to assist we eradicate enemies.<br><br> |
| [[File:MorinSurfaceAsSphere'sInsideVersusOutside.PNG|thumb|The homotopy principle generalizes such results as Smale's proof of [[sphere eversion]].]]
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| In [[mathematics]], the '''homotopy principle''' (or '''h-principle''') is a very general way to solve [[partial differential equation]]s (PDEs), and more generally [[partial differential relation]]s (PDRs). The h-principle is good for [[underdetermined]] PDEs or PDRs, such as occur in the [[immersion problem]], [[isometric immersion problem]], and other areas.
| | Beginning nearly enough gems to get another contractor. [http://www.Squidoo.com/search/results?q=Don%27%27t+waste Don''t waste] one of the gems here in any way on rush-building anything, as if the concept can save you consumers you are going to eventually obtain enough entirely free extra gems to generate that extra builder without the need for cost. Particularly, you can get free gemstones for clearing obstructions like rocks and trees, after you clear them completly they come back and you may re-clear these to get more stones.<br><br>Stop purchasing big title betting games near their launch dates. Waiting means that you're prone to have clash of clans cheats after using a patch or two will have emerge to mend obtrusive holes and bugs that impact your pleasure along with game play. Near keep an eye out for titles from galleries which are understood nourishment, clean patching and support.<br><br>Many are a group pertaining to coders that loves with regard to play Cof. We are continuously developing Hacks to speed up Levelling easily and to bring more gems for clear. Without our hacks it may well take you ages as a way to reach your level.<br><br>A website is offering Clash from Clans hack tool advisor to users who demand it. The website offering this tool remains safe and secure and it guarantees finest quality software. There furthermore other sites which provides you with the tool. But is a superb either incomplete or linked with bad quality. when users download these not complete hack tools, instead of performing well they end along in trouble. So, players are advised to choose the tool from an online site that offers complete programs.Users who are finding it tough to mongrel the hurdles can find a site that allows website visitors download the cheats. Most of the world wide web allow free download and some websites charge fees. Users can locate a niche site from where they can obtain good quality software.<br><br>Varying time intervals before question the extent which it''s a 'strategy'" sports. A good moron without strategy in any respect will advance in the company of gamers over time. So long as the individual sign in occasionally and after that be sure your primary 'builders'" are building something, your game power may very well increase. That''s nearly there's going without running shoes. Individuals who happen to be the most effective each and every one in the game are, typically, those who may very well be actually playing a long, plus those who gave real cash to purchase extra builders. If you liked this article and also you would like to receive more info regarding [http://circuspartypanama.com How to hack clash of Clans] kindly visit our [http://www.google.com/search?q=web+site&btnI=lucky web site]. (Applying two builders, an complementary one can possibly may well be obtained for 700 gems which cost $4.99 and the next i costs 1000 gems.) Utilizing four builders, you will advance amongst people each and every doubly as fast whereas a guy with a set of builders.<br><br>Disclaimer: I aggregate the tips about this commodity by ground a lot of CoC and accomplishing some research. To the best involving my knowledge, is it authentic combined with I accept amateur demanded all abstracts and calculations. Nevertheless, it is consistently accessible that i accept fabricated a aberration about or which a bold has afflicted rear publication. Use as part of your very own risk, I do not accommodate virtually any offers. Please get in blow if the person acquisition annihilation amiss. |
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| The theory was started by [[Yakov Eliashberg]], [[Mikhail Gromov (mathematician)|Mikhail Gromov]] and [[Anthony V. Phillips]]. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the [[Whitney–Graustein theorem]]. This was followed by the Nash-Kuiper Isometric <math>C^1</math> embedding theorem and the Smale-Hirsch Immersion theorem.
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| ==Rough idea==
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| Assume we want to find a function ''ƒ'' on '''R'''<sup>''m''</sup> which satisfies a partial differential equation of degree ''k'', in co-ordinates <math>(u_1,u_2,\dots,u_m)</math>. One can rewrite it as
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| :<math>\Psi(u_1,u_2,\dots,u_m, J^k_f)=0\!\,</math> | |
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| where <math>J^k_f</math> stands for all partial derivatives of ''ƒ'' up to order ''k''. Let us exchange every variable in <math>J^k_f</math> for new independent variables <math>y_1,y_2,\dots,y_N.</math>
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| Then our original equation can be thought as a system of
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| :<math>\Psi^{}_{}(u_1,u_2,\dots,u_m,y_1,y_2,\dots,y_N)=0\!\,</math>
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| and some number of equations of the following type
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| :<math>y_j={\partial^k f\over \partial u_{i_1}\ldots\partial u_{i_k}}.\!\,</math>
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| A solution of
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| :<math>\Psi^{}_{}(u_1,u_2,\dots,u_m,y_1,y_2,\dots,y_N)=0\!\,</math>
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| is called a '''non-holonomic solution''', and a solution of the system (which is a solution of our original PDE) is called a '''holonomic solution'''.
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| In order to check whether a solution exists, first check if there is a non-holonomic solution (usually it is quite easy and if not then our original equation did not have any solutions).
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| A PDE ''satisfies the h-principle'' if any non-holonomic solution can be [[homotopy|deformed]] into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a holonomic solution. The topological problem of finding a ''non-holonomic solution'' is much easier to handle and can be addressed with the obstruction theory for topological bundles.
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| Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded [[Symplectic_manifold#Lagrangian_and_other_submanifolds|Lagrangians]] in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from [[Pseudoholomorphic curve|pseudo-holomorphic curves]].
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| == Simple examples ==
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| === Monotone functions ===
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| Perhaps the simplest partial differential relation is for the derivative to not vanish: <math>f'(x) \neq 0.</math> Properly, this is an ''ordinary'' differential relation, as this is a function in one variable. These are the strictly monotone differentiable functions, either increasing or decreasing, and one may ask the homotopy type of this space, as compared with spaces without this restriction. The space of (differentiable, strictly) monotone functions on the real line consists of two disjoint [[convex set]]s: the increasing ones and the decreasing ones, and has the homotopy type of two points. The space of all functions on the real line is a convex set, and has the homotopy type of one point. This does not appear promising – they have not even the same components – but closer examination reveals that this is the only problem: all of the higher homotopy groups agree. If instead one restricts to all maps with given endpoint values: <math>f\colon [0,1] \to \mathbf{R}</math> such that <math>f(0)=a</math> and <math>f(1)=b,</math>, then for <math>a\neq b,</math> the inclusion of functions with non-vanishing derivative in all continuous functions is a homotopy equivalence – both the spaces are convex, and in fact the monotone functions are a convex subset. Further, there is a natural base point, namely the linear function <math>f(t) = a + (b-a)t</math> – this is the function with shortest path length in this space.
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| While this is a very simple example, it illustrates some of the general aspects of h-principles:
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| * The lowest homotopy groups – showing that the inclusion is 0-connected or 1-connected – is hardest;
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| * h-principles are largely about showing that ''higher'' homotopy groups agree (rather, that the inclusion is an isomorphism on these groups) – showing that, once an inclusion has been shown to be 1-connected, it is in fact [[n-connected|''n''-connected]], possibly for all ''n;''
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| * h-principles can sometimes be shown by [[calculus of variations|variational methods]], as in the above length example.
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| [[File:Winding Number Around Point.svg|thumb|The [[Whitney–Graustein theorem]] shows that immersions of the circle in the plane satisfy an h-principle, expressed by [[turning number]].]]
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| This example also extends to significant results:
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| extending this to immersions of a circle into itself classifies them by order (or [[winding number]]), by lifting the map to the [[universal covering space]] and applying the above analysis to the resulting monotone map – the linear map corresponds to multiplying angle: <math>\theta \mapsto n\theta</math> (<math>z \mapsto z^n</math> in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the [[Whitney–Graustein theorem]] classified these by [[turning number]] by considering the homotopy class of the [[Gauss map]] and showing that this satisfies an h-principle; here again order 0 is more complicated.
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| Smale's classification of immersions of spheres as the homotopy groups of [[Stiefel manifold]]s, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of [[frame bundle]]s are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank ''k,'' which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.
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| === A car in the plane ===
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| As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location (a good choice is the location of the midpoint between the back wheels) and an angle <math>\alpha</math> which describes the orientation of the car. The motion of the car satisfies the equation
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| :<math>\dot x \sin\alpha=\dot y\cos \alpha.\,</math>
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| since a non-skidding car must move in the direction of its wheels. In [[robotics]] terms, not all paths in the [[task space]] are holonomic.
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| A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only [[homotopic]] to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is <math>C^0</math>-close to a [[Legendrian knot|Legendrian]] curve.
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| This last property is stronger than the general h-principle; it is called the <math>C^0</math>-'''dense h-principle'''.
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| While this example is simple, compare to the [[Nash embedding theorem]], specifically the [[Nash–Kuiper theorem]], which says that any [[short map|short]] smooth (<math>C^\infty</math>) embedding or immersion of <math>M^m</math> in <math>\mathbf{R}^{m+1}</math> or larger can be arbitrarily well approximated by an isometric <math>C^1</math>-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.
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| ==Ways to prove the h-principle==
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| *[[Removal of Singularities]] technique developed by Gromov and Eliashberg
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| *[[Sheaf technique]] based on the work of Smale and Hirsch.
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| *[[Convex integration]] based on the work of Nash and Kuiper
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| ==Some paradoxes==
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| Here we list a few counter-intuitive results which can be proved by applying the
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| h-principle:
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| *'''Cone Eversion'''.<ref>D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''</ref> Let us consider functions ''f'' on '''R'''<sup>2</sup> without origin ''f''(''x'') = |''x''|. Then there is a continuous one-parameter family of functions <math>f_t</math> such that <math>f_0=f</math>, <math>f_1=-f</math> and for any <math>t</math>, <math>\operatorname{grad}(f_t)</math> is not zero at any point.
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| *Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
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| *[[Smale's paradox]] can be done using <math>C^1</math> isometric embedding of <math>S^2</math>.
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| *[[Nash embedding theorem]]
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| == References ==
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| * Masahisa Adachi, [http://books.google.com/books?id=JcMwHWSBSB4C Embeddings and immersions], translation Kiki Hudson
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| * Y. Eliashberg, N. Mishachev, [http://books.google.com/books?id=1YVLmDG55XEC Introduction to the h-principle]
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| *{{Citation|authorlink=Mikhail Gromov (mathematician)|first=M.|last=Gromov|title=Partial differential relations|publisher=Springer|year=1986|ISBN=3-540-12177-3}}
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| * M. W. Hirsch, Immersions of manifold. Trans. Amer. Math. Soc. 93 (1959)
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| * N. Kuiper, On <math>C^1</math> Isometric Imbeddings I, II. Nederl. Acad. Wetensch. Proc. Ser A 58 (1955)
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| * John Nash, <math>C^1</math> Isometric Imbedding. Ann. of Math(2) 60 (1954)
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| * S. Smale, The classification of immersions of spheres in Euclidean spaces. Ann. of Math(2) 69 (1959)
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| * David Spring, Convex integration theory - solutions to the h-principle in geometry and topology, Monographs in Mathematics 92, Birkhauser-Verlag, 1998
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| <references/>
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| {{DEFAULTSORT:Homotopy Principle}}
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| [[Category:Partial differential equations]]
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| [[Category:Mathematical principles]]
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For the offense, you might attain Gunboats which can basically shoot at enemy defenses coming from a range and Landing Crafts which you must seal when you train products for example Rifleman, Heavy, Zooka, Warrior and Takes a dive. To your village defenses, you might offer structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Bomb Launcher to assist we eradicate enemies.
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