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In Terence Tao's piece on Kolmogorov's Law (http://terrytao.wordpress.com/2014/05/15/kolmogorovs-power-law-for-turbulence/) he uses the notation for the fluid velocity

$$u: \mathbb{R} \times \mathbb{R}^3/\mathbb{Z}^3 \to \mathbb{R}^3 $$

What is the meaning of the over $\mathbb{Z}^3$ in this context?

James P
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This is notation for the quotient group. If you haven't seen this concept, $\mathbb{R}^3/\mathbb{Z}^3$ can be pictured as a unit cube $[0,1)^3$ in which you add "modulo 1", that is, add as you would in $\mathbb{R}^3$ and then cut off the integer part. Another important way to visualize this group is as $(S^1)^3$, the product of three circles.

Kevin Carlson
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  • Thanks Kevin. Why is this a useful way to define the position in the signature of $u$? I see it is also used in the Navier-Stokes Clay Prize: "Existence and smoothness of Navier–Stokes solutions in $\mathbb{R}^3/\mathbb{Z}^3$." What more does it allow us to say over just using $\mathbb{R}^3$? – James P Dec 02 '14 at 22:50
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    I don't know anything about Navier-Stokes, but functions on $\mathbb{R}^3/\mathbb{Z}^3$ are equivalent to periodic functions on $\mathbb{R}^3$, so this may just be another way of saying "existence and smoothness of periodic Navier-Stokes solutions." – Kevin Carlson Dec 03 '14 at 01:48
  • Fantastic, thanks – James P Dec 03 '14 at 11:50