Thanks, Alan. Looks interesting! On Sun, Dec 28, 2014 at 7:27 PM, Alan Bromborsky <[email protected]> wrote:
> Because of you interest in Lie groups you might find the attached paper > of interest - > > > > On 12/28/2014 06:03 AM, Joy merwin monteiro wrote: > > Thanks. I agree with your views regarding the mapping business. I > guess it is best > to keep them specific to an application. > > I realise that S1 and R1 have different topologies. The reason I asked > about the metric was in the > context of the information stored within Manifold/Patch (which contains > nothing about the topology, > which is fine), there is no way to distinguish the two, like you > mentioned previously. But yes, this > is off-topic and not for this thread. > > Hope Will Farr gets back at least with a design outline, if he has lost > the code. The LiE package > that I was looking at represents Lie Groups in terms of their Lie > Algebras, which is useful for > computation, but is not directly in-line with the spirit of scipy.diffgeom > as it exists. A LieGroup > will definitely provide a bridge for more extensive work. Let's see! > > Joy > > On Sun, Dec 28, 2014 at 3:50 PM, Francesco Bonazzi < > [email protected]> wrote: > >> >> >> On Sunday, December 28, 2014 10:50:08 AM UTC+1, Joy merwin monteiro >> wrote: >>> >>> Thanks, Francesco. >>> >>> >>> Supposing you mean the N-sphere by Sn, a sphere cannot be mapped by >>>> one single patch. You need at least two. Patches are currently just >>>> containers, there is no way as of now to define coordinate transition >>>> functions between overlapping patches. >>>> >>>> >>> Yes, I noticed that the Patch class would not be able to do this. >>> Hence, I was thinking being able to represent >>> Sn consistently would be a useful use-case to add the required >>> machinery, if you think something like this makes >>> sense. >>> >> >> There is already a machinery to connect two CoordSystem objects defined >> on the same Patch. Maybe that could be extended to connect two CoordSystem >> objects defined on two different patches of the same manifold. Obviously, >> there should also be a connection between two patches, maybe by just >> defining a function that returns a boolean indicating whether a Point is on >> the patches overlap or whether it is not. >> >> The question is, how much do we really need this part? Many users would >> like to use *sympy.diffgeom* to handle long calculations easily, by >> using a Computer Algebra System. One would just want to set up a coordinate >> system and stay on that coordinate system to do extensive calculations on >> that coordinate system, like the calculation of the Riemann tensor and so >> on. Otherwise, writing tools to convert from one coordinate system to >> another on the same patch are also useful, because that generally save a >> lot of hand calculations. But would there be many users interested in >> representing the topological structure of a manifold by inequalities and >> equations among coordinate systems? I don't think so. >> >> Anyways, I don't think that a manifold should be represented by its >>>> embedding into the Euclidean space (in the case of Sn, a map from one >>>> vector of Sn to a vector in R_(n+1) ). Once you have a coordinate system on >>>> a patch, you can define your own function mapping its coordinates to the >>>> Euclidean space. >>>> >>> >>> I agree with you, this is probably the correct approach. Then the >>> question is, where would this function exist >>> within Sympy? Should a Manifold class, which would contain the >>> patches, contain an optional Embedding class/attribute >>> corresponding to each Patch? >>> >> >> SymPy classes should contain just the essential to identify the object >> uniquely. If you want to define an embedding, just define a function in the >> namespace you're working, not inside the Manifold/Patch classes. >> >> >>> >>>> >>>> >>>> Visualizing a manifold depends on the embedding you choose. A plot is >>>> basically a projection from the manifold to 2D or 3D Euclidean space, so as >>>> soon as you have a map to do that, you can plot the manifold. >>>> >>> >>> Agreed. Going back to my previous paragraph, what do you think is the >>> best way to handle such a map (and hence the >>> plotting capability)? >>> >> >> Define a function in your namespace. I would not modify existing diffgeom >> classes. >> >> >>> >>>> As there are currently no ways to define transition functions among >>>> patches, defining S1 and S2 is identicaly to define R1 and R2. >>>> >>>> >>> Then, the only way to tell S1 and R1 apart would be to associate them >>> with a metric? Then again, where should this >>> metric be stored? >>> >> >> I don't think it has to do with the metric, S1 and R1 are still different >> manifolds even if you don't define a metric. The distinction is determined >> by their topology. >> >> But now we are departing from the topic, which was about Lie groups. I >> think the best way to act on *sympy.diffgeom* in this regard is by >> getting the Scheme code by Will Farr and translate that into SymPy code. >> > > > > -- > The best ruler, when he finishes his > tasks and completes his affairs, > the people say > “It all happened naturally” > > - Te Tao Ch'ing > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/CA%2BphhE%3D9n8yVPoBid%2BCZ%3DOqa7vc39XyPop3EzEuEfXEQgfhvEg%40mail.gmail.com > <https://groups.google.com/d/msgid/sympy/CA%2BphhE%3D9n8yVPoBid%2BCZ%3DOqa7vc39XyPop3EzEuEfXEQgfhvEg%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > For more options, visit https://groups.google.com/d/optout. > > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/54A00C67.6080202%40verizon.net > <https://groups.google.com/d/msgid/sympy/54A00C67.6080202%40verizon.net?utm_medium=email&utm_source=footer> > . > > For more options, visit https://groups.google.com/d/optout. > -- The best ruler, when he finishes his tasks and completes his affairs, the people say “It all happened naturally” - Te Tao Ch'ing -- You received this message because you are subscribed to the Google Groups "sympy" group. 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