Recent content by ayan849

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    Query regarding classification of pde

    In my research work, I recently have come across a system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros. I have searched the available resources and could nowhere find out which category (elliptic/hyperbolic/parabolic) it falls...
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    How does Lie group help to solve ode's?

    Can anybody care to give a geometrical interpretation? I can't understand a bit of what is going on here :(
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    How does Lie group help to solve ode's?

    Many thanks Alesak for your reply and the cited reference. It would be nice if someone could post a geometrical answer...
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    How does Lie group help to solve ode's?

    Being not an expert, my question might sound naive to students of mahematics. My question is how on earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?
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    Why there cannot be a single chart for n-sphere?

    many thanks for the discussion. I think I have my answer now.
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    Why there cannot be a single chart for n-sphere?

    please see my another thread regarding connection and give some light on it if possible.
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    Why there cannot be a single chart for n-sphere?

    well:smile: note carefully my figure 8: {(sin t, sin 2t): t ε (0, 2π)}. It is really not a 8. The middle portion is not a "cross". If t ε [0, 2π], then it is certainly not a manifold. By my "figure 8" IS a manifold and I am sure you know how. :)
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    Why there cannot be a single chart for n-sphere?

    Try to make fig-8 with a piece of open string. The map is trivial.
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    Why there cannot be a single chart for n-sphere?

    A course in differential geometry-s.kumaresan.
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    Why there cannot be a single chart for n-sphere?

    Inverse will be continuous if you look at fig8 as a separate object not embedded in R^2. If you consider it to be embedded in R^2, then it will not be a homeomorphism.
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    Why there cannot be a single chart for n-sphere?

    map the nbd of π in (0, 2π) to one branch at the middle of fig8, right nbd of 0 to one portion of the remaining branch near the middle of fig8 and left nbd of 2π to the remaining portion of the remaining branch near the middle of fig8. Thats how u can cover the whole of fig8.
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    Why there cannot be a single chart for n-sphere?

    In stead of taking the subspace topology of ℝ^2, consider open intervals in (0,2π) to get mapped to the figure-8. This is indeed a homeomorphism, as in case of subspace topology of ℝ^2, it was not (because of the presence of the "cross" at the middle which was making our life hard).
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    Why there cannot be a single chart for n-sphere?

    I agree. But is there a proof that there CANNOT be ANY single chart for Sn?
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    Why there cannot be a single chart for n-sphere?

    What happened was I was reading a book where they have discussed the construction of charts for the figure-8: {(sin t, sin 2t): t ε (0,2π)}. Using the subspace topology of R^2, you cannot make it into a manifold. But there are topologies that admit its manifold structure, and there is a topology...
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    Why there cannot be a single chart for n-sphere?

    I haven't really tried to prove that there cannot be a single chart. But the usual charts that we have, like the 2 charts using stereographic projection, the 6 charts using hemispheres...and many others---all produce atleast 2 charts.
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