Seiberg–Witten invariant: Difference between revisions

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The notion of [[orbit]] of a control system used in mathematical [[control theory]] is a particular case of the notion of [[orbit (group theory)|orbit in group theory]].


==Definition==
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Let
<math>{\ }\dot q=f(q,u)</math>
be a <math>\ {\mathcal C}^\infty</math> control system, where
<math>{\ q}</math>  
belongs to a <!-- <math>\ n</math>- -->finite-dimensional manifold <math>\ M</math> and <math>\ u</math> belongs to a control set <math>\ U</math>. Consider the family <math>{\mathcal F}=\{f(\cdot,u)\mid u\in U\}</math>
and assume that every vector field in <math>{\mathcal F}</math> is [[complete vector field|complete]].
For every <math>f\in {\mathcal F}</math> and every real <math>\ t</math>, denote by <math>\ e^{t f}</math> the [[vector flow|flow]] of <math>\ f</math> at time <math>\ t</math>.
 
The orbit of the control system <math>{\ }\dot q=f(q,u)</math> through a point <math>q_0\in M</math> is the subset <math>{\mathcal O}_{q_0}</math> of <math>\ M</math> defined by
 
:<math>{\mathcal O}_{q_0}=\{e^{t_k f_k}\circ e^{t_{k-1} f_{k-1}}\circ\cdots\circ e^{t_1 f_1}(q_0)\mid k\in\mathbb{N},\ t_1,\dots,t_k\in\mathbb{R},\ f_1,\dots,f_k\in{\mathcal F}\}.</math>
 
;Remarks
The difference between orbits and [[attainable set]]s is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits.
In particular, if the family <math>{\mathcal F}</math> is symmetric (i.e., <math>f\in {\mathcal F}</math> if and only if <math>-f\in {\mathcal F}</math>), then orbits and attainable sets coincide.
 
The hypothesis that every vector field of <math>{\mathcal F}</math> is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.
 
==Orbit theorem (Nagano-Sussmann)==
Each orbit <math>{\mathcal O}_{q_0}</math> is an [[immersed submanifold]] of <math>\ M</math>.
 
The tangent space to the orbit
<math>{\mathcal O}_{q_0}</math> at a point <math>\ q</math> is the linear subspace of <math>\ T_q M</math> spanned by
the vectors <math>\ P_* f(q)</math> where <math>\ P_* f</math> denotes the [[pushforward (differential)#Pushforward of vector fields|pushforward]] of <math>\ f</math> by <math>\ P</math>, <math>\ f</math> belongs to  <math>{\mathcal F}</math> and <math>\ P</math> is a diffeomorphism of <math>\ M</math> of the form  <math>e^{t_k f_k}\circ \cdots\circ e^{t_1 f_1}</math> with <math> k\in\mathbb{N},\ t_1,\dots,t_k\in\mathbb{R}</math> and <math>f_1,\dots,f_k\in{\mathcal F}</math>.
 
If all the vector fields of the family <math>{\mathcal F}</math> are analytic, then <math>\ T_q{\mathcal O}_{q_0}=\mathrm{Lie}_q\,\mathcal{F}</math> where <math>\mathrm{Lie}_q\,\mathcal{F}</math> is the evaluation at <math>\ q</math> of the [[Lie algebra]] generated by <math>{\mathcal F}</math> with respect to the [[Lie bracket of vector fields]].
Otherwise, the inclusion <math>\mathrm{Lie}_q\,\mathcal{F}\subset T_q{\mathcal O}_{q_0}</math> holds true.
 
==Corollary (Rashevsky-Chow theorem)==
If <math>\mathrm{Lie}_q\,\mathcal{F}= T_q M</math> for every <math>\ q\in M</math> and if <math>\ M</math> is connected, then each orbit is equal to the whole manifold <math>\ M</math>.
<!-- A section about history should be written: cite Hermann, Stefan, Chow, Rashevsky et al. -->
 
==References==
*{{cite book
  | last =Jurdjevic
  | first =Velimir
  | authorlink =
  | coauthors =
  | title =Geometric control theory
  | publisher =[[Cambridge University Press]]
  | year =1997
  | location =
  | pages =xviii+492
  | url =http://www.cup.cam.ac.uk/us/catalogue/email.asp?isbn=9780521495028
  | doi =
  | id = 
  | isbn = 0-521-49502-4}}
*{{cite journal
  | last =Sussmann
  | first =H&eacute;ctor J.
  | authorlink =
  | coauthors =Jurdjevic, Velimir
  | title =Controllability of nonlinear systems
  | journal =J. Differential Equations
  | volume =12
  | issue =
  1| pages =95–116
  | publisher =
  | location =
  | year =1972
  | url =
  | doi =10.1016/0022-0396(72)90007-1
  | id =
  | accessdate =  }}
*{{cite journal
  | last =Sussmann
  | first =H&eacute;ctor J.
  | authorlink =
  | coauthors =
  | title =Orbits of families of vector fields and integrability of distributions
  | journal =Trans. Amer. Math. Soc.
  | volume =180
  | issue =
  | pages =171–188
  | publisher =American Mathematical Society
  | location =
  | year =1973
  | doi =10.2307/1996660
  | id =
  | accessdate =
  | jstor =1996660  }}
 
[[Category:Control theory]]

Latest revision as of 14:45, 5 May 2014

The kitchen is often overlooked though it is often washed and indicators of excessive water and coming water damage are overlooked. Inspecting the following humidity sources can help in avoiding harm where it is more likely to happen. The following guide lines can help you keep your kitchen in good condition and to stop problems due to the ice box, the dishwasher and the counter and dink.

Be comprehensive while doing these examinations and be sure you do them every few months

Fridge

The area behind the icebox is rarely examined since it is fairly difficult to achieve. Examining this region will reduce the chance for water dilemmas. water behind the icebox is actually a sign. Do not dismiss it, resolve immediately or retain a machine specialist. Clean it quickly, should you find mold progress. Identify the moisture and fix it. Examine the line to make sure it"s attached and made to the water supply, If you have an icemaker.

Sink and Countertop

The sink and counter are very watered places inside your home. Always check often under the sink and remove other stored items and the trash can to look for leaks. A drip in the trap underneath the sink means drain dilemmas and have to be repaired quickly. Look around the sink, gradual draining pipes show a blocked drain. A dripping faucet must me repaired immediately. I found out about fullertonecoifj"s Profile | Armor Games by searching Bing. Focus on the counter. If tiles are cracked or grout is lost, fix it to stop water from pervading. Floors and stained walls around pipes indicate an interior leak that must definitely be fixed to stop additional injury. Clicking Forum seemingly provides aids you might use with your aunt.

Dishwasher

If the dishwasher doesnt empty after use, the main filter might be blocked and have to be released instantly. It"s usually as a result of back-ups, If the sink overflows. This stirring plumbers in corona portfolio has varied original warnings for the purpose of this viewpoint. This problem usually occurs when the drain line is blocked. Clean the drain line frequently. Look beneath the base of the dishwasher for leaks. There could be an unsealed joint or a broken hose, If you discover any leaks. Check the connection and resolve as needed. Always check the dishwasher line under the sink for signs of loss and if necessary repair. Con-sider adding steel-braided tubes on appliances including the dishwasher..

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