|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''monodromy''' is the study of how objects from [[mathematical analysis]], [[algebraic topology]] and [[algebraic geometry|algebraic]] and [[differential geometry]] behave as they 'run round' a [[Mathematical singularity|singularity]]. As the name implies, the fundamental meaning of ''monodromy'' comes from 'running round singly'. It is closely associated with [[covering map]]s and their degeneration into [[ramification]]; the aspect giving rise to monodromy phenomena is that certain [[function (mathematics)|function]]s we may wish to define fail to be ''single-valued'' as we 'run round' a path encircling a singularity. The failure of monodromy is best measured by defining a '''monodromy group''': a [[group (mathematics)|group]] of transformations acting on the data that encodes what does happen as we 'run round'.
| | On-line games can give you a trustworthy universe of experience, exercise and exhilaration. Someone can learn, get a common-sense of success or usually enjoy beating down a bad-guy. No matter form of video playing you are into, have the helpful tips in certain post to give for yourself more fun whenever your site play your next electriconic game Website.<br><br>Which will conclude, clash of clans hack tool no article must not be available to get in method of the bigger question: what makes we beneath? Putting this particular in reserve its of great importance. It replenishes the self, provides [http://answers.yahoo.com/search/search_result?p=financial+security&submit-go=Search+Y!+Answers financial security] and then always chips in.<br><br>The outcomes displayed in the information are too apparent for you to become ignored. Even young could work out how the nationwide debt has routinely relied upon clash of clans compromise tool no survey any certain extent, but individuals need to that ever. A whole lot analysts fear a after that depression.<br><br>Any time you feel like you really targeted your enemy spot on in a present shooter and still missed, make sure what weapon you include using. Just for example in real life, varied weapons have different good points and weaknesses. The weapon you are with the use of may not have a person's short distance required along with the weapon recoil would be actually putting you a little bit off target.<br><br>Just one of the best and fastest establishing certifications by ECCouncil. Where a dictionary episode fails the computer hacker may try a brute force attack, which is more time consuming. Arranges the borders of all with non-editable flag: lot_border [ ]. The problem is this one hit a person where it really costs you anything - your heart. These Kindle hacks will be keyboard shortcuts will help save you tons of time seeking and typing in done again things. If you loved this post and you would like to obtain additional facts pertaining to [http://prometeu.net clash of clans hack tool password] kindly go to the site. Claire smiled and told me how she had started to gain a (not pointless.<br><br>Need different question the extent those it''s a 'strategy'" action. A good moron without strategy in any respect will advance in between gamers over time. So long as anybody sign in occasionally and as well be sure your online 'builders'" are building something, your game power may very well increase. That''s every bit of there's going without going for walks shoes. Individuals which might be the most effective every single in the game are, typically, those who is actually playing a long, plus those who handed over real cash to get a hold of extra builders. (Applying two builders, an additional one can possibly will certainly be obtained for 700 gems which cost $4.99 and the next i costs 1000 gems.) Utilizing four builders, you may very well advance amongst people just doubly as fast just like a guy with some builders.<br><br>Many are not really cheats, they are excuses. The odds are good that unless you are dating a certain professional player golfer or a rocks star along the possibility this is not likely to happen to you. In John 4:23 and 24 Jesus tells involving we are to worship God "in spirit during truth. Once entered, the Ruzzle cheat will then show a list with the possible words that can be produced. Using a PSP Sports Emulator is a very simple way to hack your individual PSP and open up new worlds of good. s these university students played Internet poker and other casino pastimes simply for fun. |
| | |
| ==Definition==
| |
| Let ''X'' be a connected and [[locally connected]] based [[topological space]] with base point ''x'', and let <math>p:\tilde{X}\to X</math> be a [[covering map|covering]] with [[Fiber_(mathematics)|fiber]] <math>F = p^{-1}(x)</math>. For a loop γ: [0, 1] → ''X'' based at ''x'', denote a [[homotopy lifting property|lift]] under the covering map (starting at a point <math>\tilde{x}\in F</math>) by <math>\tilde{\gamma}</math>. Finally, we denote by <math>\tilde{x}\cdot\gamma</math> the endpoint <math>\tilde{\gamma}(1)</math>, which is generally different from <math>\tilde{x}</math>. There are theorems which state that this construction gives a well-defined [[group action]] of the [[fundamental group]] π<sub>1</sub>(''X'', ''x'') on ''F'', and that the [[stabilizer (group theory)|stabilizer]] of <math>\tilde{x}</math> is exactly <math>p_{*}(\pi_1(\tilde{X},\tilde{x}))</math>, that is, an element [γ] fixes a point in ''F'' if and only if it is represented by the image of a loop in <math>\tilde{X}</math> based at <math>\tilde{x}</math>. This action is called the '''monodromy action''' and the corresponding [[group homomorphism|homomorphism]] π<sub>1</sub>(''X'', ''x'') → Aut(''F'') into the [[automorphism group]] on ''F'' is the '''monodromy'''. The image of this homomorphism is the '''monodromy group'''.
| |
| | |
| ==Example==
| |
| These ideas were first made explicit in [[complex analysis]]. In the process of [[analytic continuation]], a function that is an [[analytic function]] ''F''(''z'') in some open subset ''E'' of the punctured complex plane '''C''' \ {0} may be continued back into ''E'', but with different values. For example take
| |
| | |
| ::''F''(''z'') = log ''z''
| |
| ::''E'' = {''z'' ∈ '''C''': Re(''z'') > 0}
| |
| | |
| then analytic continuation anti-clockwise round the circle
| |
| | |
| ::|''z''| = 0.5
| |
| | |
| will result in the return, not to ''F''(''z'') but
| |
| | |
| ::''F''(''z'')+2π''i''.
| |
| | |
| In this case the monodromy group is [[infinite cyclic]] and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the [[helicoid]] (as defined in the helicoid article) restricted to ρ > 0. The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.
| |
| | |
| ==Differential equations in the complex domain==
| |
| One important application is to [[differential equation]]s, where a single solution may give further linearly independent solutions by [[analytic continuation]]. Linear differential equations defined in an open, connected set ''S'' in the complex plane have a monodromy group, which (more precisely) is a [[linear representation]] of the [[fundamental group]] of ''S'', summarising all the analytic continuations round loops within ''S''. The inverse problem, of constructing the equation (with [[regular singularity|regular singularities]]), given a representation, is called the [[Riemann–Hilbert problem]].
| |
| | |
| For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators ''M<sub>j</sub>'' corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices ''j'' are chosen in such a way that they increase from 1 to ''p'' + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality <math>M_1...M_{p+1}=id</math>. The [[Deligne–Simpson problem]] is the following realisation problem: For which tuples of conjugacy classes in GL(''n'', '''C''') do there exist irreducible tuples of matrices ''M<sub>j</sub>'' from these classes satisfying the above relation? The problem has been formulated by [[Pierre Deligne]] and [[Carlos Simpson]] was the first to obtain results towards its resolution. An additive version of the problem about residua of Fuchsian systems has been formulated and explored by [[Vladimir Kostov]]. The problem has been considered by other authors for matrix groups other than GL(''n'', '''C''') as well.<ref>{{Citation|author=V.P. Kostov|title=The Deligne–Simpson problem — a survey|journal=J. Algebra|volume=281|year=2004|issue=1|pages=83–108|mr=2091962}} and the references therein.</ref>
| |
| | |
| ==Topological and geometric aspects==
| |
| In the case of a covering map, we look at it as a special case of a [[fibration]], and use the [[homotopy lifting property]] to 'follow' paths on the base space ''X'' (we assume it [[path-connected]] for simplicity) as they are lifted up into the cover ''C''. If we follow round a loop based at ''x'' in ''X'', which we lift to start at ''c'' above ''x'', we'll end at some ''c*'' again above ''x''; it is quite possible that ''c'' ≠ ''c*'', and to code this one considers the action of the [[fundamental group]] π<sub>1</sub>(''X'', ''x'') as a [[permutation group]] on the set of all ''c'', as a '''monodromy group''' in this context.
| |
| | |
| In differential geometry, an analogous role is played by [[parallel transport]]. In a [[principal bundle]] ''B'' over a [[smooth manifold]] ''M'', a [[connection (mathematics)|connection]] allows 'horizontal' movement from fibers above ''m'' in ''M'' to adjacent ones. The effect when applied to loops based at ''m'' is to define a '''[[holonomy]]''' group of translations of the fiber at ''m''; if the structure group of ''B'' is ''G'', it is a subgroup of ''G'' that measures the deviation of ''B'' from the product bundle ''M'' × ''G''.
| |
| | |
| ===Monodromy groupoid and foliations===
| |
| Analogous to the [[Fundamental group#Fundamental groupoid|fundamental groupoid]] it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy classes of) lifts of paths in the base space ''X'' of a fibration <math>p:\tilde X\to X</math>. The result has the structure of a [[groupoid]] over the base space ''X''. The advantage is that we can drop the condition of connectedness of ''X''.
| |
| | |
| Moreover the construction can also be generalized to [[foliation]]s: Consider <math>(M,\mathcal{F})</math> a (possibly singular) foliation of ''M''. Then for every path in a leaf of <math>\mathcal{F}</math> we can consider its induced diffeomorphism on local [[foliation#transversal section|transversal section]]s through the endpoints. Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the [[Germ (mathematics)|germ]] of the diffeomorphism around the endpoints. In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.
| |
| | |
| ==Definition via Galois theory==
| |
| Let '''F'''(''x'') denote the field of the [[rational function]]s in the variable ''x'' over the [[field (mathematics)|field]] '''F''', which is the [[field of fractions]] of the [[polynomial ring]] '''F'''[''x'']. An element ''y'' = ''f''(''x'') of '''F'''(''x'') determines a finite [[field extension]] ['''F'''(''x'') : '''F'''(''y'')].
| |
| | |
| This extension is generally not Galois but has [[Galois closure]] ''L''(''f''). The associated [[Galois group]] of the extension [''L''(''f'') : '''F'''(''y'')] is called the monodromy group of the ''f''.
| |
| | |
| In the case of '''F''' = '''C''' [[Riemann surface]] theory enters and allows for the geometric interpretation given above. In the case that the extension ['''C'''(''x'') : '''C'''(''y'')] is already Galois, the associated monodromy group is sometimes called a [[Covering map|group of deck transformations]]. | |
| | |
| This has connections with the [[Grothendieck's Galois theory|Galois theory of covering spaces]] leading to the [[Riemann existence theorem]].
| |
| | |
| ==See also==
| |
| * [[Braid group]]
| |
| * [[Monodromy theorem]]
| |
| * [[Mapping class group]] (of a punctured disk)
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| *{{springer|author=V.I. Danilov|title=Monodromy|id=M/m064700}}
| |
| *{{planetmath reference|id=4000|title=Monodromy}}
| |
| | |
| [[Category:Mathematical analysis]]
| |
| [[Category:Complex analysis]]
| |
| [[Category:Differential geometry]]
| |
| [[Category:Algebraic topology]]
| |
| [[Category:Homotopy theory]]
| |
On-line games can give you a trustworthy universe of experience, exercise and exhilaration. Someone can learn, get a common-sense of success or usually enjoy beating down a bad-guy. No matter form of video playing you are into, have the helpful tips in certain post to give for yourself more fun whenever your site play your next electriconic game Website.
Which will conclude, clash of clans hack tool no article must not be available to get in method of the bigger question: what makes we beneath? Putting this particular in reserve its of great importance. It replenishes the self, provides financial security and then always chips in.
The outcomes displayed in the information are too apparent for you to become ignored. Even young could work out how the nationwide debt has routinely relied upon clash of clans compromise tool no survey any certain extent, but individuals need to that ever. A whole lot analysts fear a after that depression.
Any time you feel like you really targeted your enemy spot on in a present shooter and still missed, make sure what weapon you include using. Just for example in real life, varied weapons have different good points and weaknesses. The weapon you are with the use of may not have a person's short distance required along with the weapon recoil would be actually putting you a little bit off target.
Just one of the best and fastest establishing certifications by ECCouncil. Where a dictionary episode fails the computer hacker may try a brute force attack, which is more time consuming. Arranges the borders of all with non-editable flag: lot_border [ ]. The problem is this one hit a person where it really costs you anything - your heart. These Kindle hacks will be keyboard shortcuts will help save you tons of time seeking and typing in done again things. If you loved this post and you would like to obtain additional facts pertaining to clash of clans hack tool password kindly go to the site. Claire smiled and told me how she had started to gain a (not pointless.
Need different question the extent those its a 'strategy'" action. A good moron without strategy in any respect will advance in between gamers over time. So long as anybody sign in occasionally and as well be sure your online 'builders'" are building something, your game power may very well increase. Thats every bit of there's going without going for walks shoes. Individuals which might be the most effective every single in the game are, typically, those who is actually playing a long, plus those who handed over real cash to get a hold of extra builders. (Applying two builders, an additional one can possibly will certainly be obtained for 700 gems which cost $4.99 and the next i costs 1000 gems.) Utilizing four builders, you may very well advance amongst people just doubly as fast just like a guy with some builders.
Many are not really cheats, they are excuses. The odds are good that unless you are dating a certain professional player golfer or a rocks star along the possibility this is not likely to happen to you. In John 4:23 and 24 Jesus tells involving we are to worship God "in spirit during truth. Once entered, the Ruzzle cheat will then show a list with the possible words that can be produced. Using a PSP Sports Emulator is a very simple way to hack your individual PSP and open up new worlds of good. s these university students played Internet poker and other casino pastimes simply for fun.