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Thread: R^3/Z^3

  1. #1 R^3/Z^3 
    Forum Masters Degree bit4bit's Avatar
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    I was looking at the official statement of the Navier-Stokes smoothness and existence problem, and trying to make sense of the hellish maths jargon thsat used. Theres a couple of things I'd like to ask.

    The statement has four settings. Two of them are in , which I understand as normal 3D space and two of them are set in . What exactly is this space? I heard that it was a 3-dimensional torus or something. And what is the significance of it?

    Also, how come there are two different problems for each space? When you say prove that solutions always exist, and that they are smooth, is it talking about functions that are solutions to the differential equation?.. so that you could find whether a function exists or not by using the definition of a function (using cartesian product?), and then check whether they are differentiable, by looking at the limit at different points?

    I don't expect to understand it fully with this post, but I just want to find out more about it.


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  3. #2  
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    You are talking about periodic case, I think. You cover 3D space with cubes and say that each of them contains the same. In practice: take 3D cube and say that behind a face there starts opposite face. Thanks of this Fourier transforms is discrete - You can expand everything into trigonometric series. Now we can write the equations using these terms - we'd get series which uses lower terms. So having input terms we can recreate solution term by term. There is only to proof that output series is convergent and we've proved existence.
    I've simplified ... with periodic input, the solution doesn't have to be periodic - perhaps it's why You have 2 parameters in R^3 ... In this case, it's a bit more complicated - there should be used some fixed point theorem.


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  4. #3 Re: R^3/Z^3 
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    Quote Originally Posted by bit4bit
    . What exactly is this space?
    Itís not the factor group , is it? :?
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  5. #4  
    Forum Professor river_rat's Avatar
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    Is that the infinite bouquet of 3-spheres? I'm too lazy to check.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  6. #5  
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    Quote Originally Posted by river_rat
    Is that the infinite bouquet of 3-spheres? I'm too lazy to check.
    I have to admit I don't know what you are talking about but I agree with your statement, about technical subjects,and often wish it wasn't true!
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  7. #6  
    Forum Masters Degree bit4bit's Avatar
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    Thanks for the replies. I looked at the wiki article about the problem, and found that is called a quotient space.

    Apparently it is a 3-dimentional torus given by

    I'm not sure whether this is something that can be easily visualised or not, but I seem to be seeing it as the normal 2-D torus revolved about an axis lying next to it, by radians. Not the most accurate explanation, but basically so thyat you end up with a hollow torus, that has thickness equal to the radius of the first circle. Or actually it might be better to say that there are an infinite number of the 2-d tori arranged in a full circle, about some point lying outside of the tori.

    Apparenty it is a subject in topology and linear algebra, things which I don't know much about as of yet, but plan to follow Guitarist's threads whenever I get the chance.

    Anywya, I also read (But can't remember where) that the whole idea of using this instead of is that is simplifies the initial conditions or boundary conditions of the original problem of proving that solutions to the differential equation always exist and are smooth.

    Halliday: I also have not yet looked at the fourier series yet, but basically understand it for switching between the time and frequency domain, when analysing something. For example I know they use it in signal theory in electrical engineering, as well as many other things I'm told.
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  8. #7  
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    Quote Originally Posted by bit4bit
    Halliday: I also have not yet looked at the fourier series yet, but basically understand it for switching between the time and frequency domain, when analysing something. For example I know they use it in signal theory in electrical engineering, as well as many other things I'm told.
    I did some maths, or is it math, a while ago now but most of the posts in the maths section are way over my head!
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  9. #8  
    Forum Masters Degree bit4bit's Avatar
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    Most are beyond me too, but I enjoy reading the threads, and joining in where I can.
    Chance favours the prepared mind.
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  10. #9  
    Forum Professor river_rat's Avatar
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    Ah, this may be a good time to bring up one of my bug bears about topological quotients - the notation is ambiguous! For example, can mean two very different topological spaces. One is the circle (where we treat as a group acting on and thus arrive at the circle. The other is where we pinch down to a point and thus arrive at the infinite bouquet of circles (a very different space).
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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