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Have you ever considered writing a popular book on some topics in modern math, say Seiberg-Witten theory or whatever your speciality in math used to be? I’m thinking of something along the lines of Ash and Gross’ book Fearless Symmetry, where they bravely attempt to give an account of galois cohomology. You’ll be surprised by how many people would love to read something like that (as is also evidenced by the fact that Ash and Gross have gone on to write two other books of a similar nature).

]]>Makes sense. Thanks for following up!

-J

]]>1) If you moved the pen to a spot that’s earlier on the original path, it would end up returning to the place where the original path started, then re-trace that path. But it wouldn’t cross the path, i.e. there wouldn’t be a visible intersection – it would just look like a longer curve.

2) The pen never reaches an equilibrium point because the vectors get shorter and shorter, so the curve only meets the equilibrium point in the limit. So two different curves can only meet at an equilibrium point and in the limit. (Though again they don’t cross – they just meet at their ends. I realize this is a technicality, but I’m sticking to it.)

]]>Not sure if you’re still replying to comments on this three-year-old post, but in any case thanks for the great explanation. Before I start on the follow-on piece on Map/Reduce, I was wondering about one detail in the above; I’m hoping you can set me straight.

You write, “if we stopped the pumps, moved the pen to a different location, then started them again, the second curve traced out by the pen would never cross the curve that was drawn the first time.” I’m slightly confused by this: if we move the pen to a different location that happens to be an earlier point on the same continuous path in the vector field on which we were sitting when the pumps are stopped, wouldn’t that cause its location to eventually pass through the original starting location?

Perhaps equivalently (though I’m not sure about that), is it not possible to have two different starting points in a vector field that converge to the same final (equilibrium) point? And if so, is this saying that such convergence can _only_ happen at the equilibrium point, and not before? Or is all of this true in the “in the limit” sense?

Sorry if the above isn’t quite as precise as it should be (I’m not a mathematician). And thanks again for the great write-up.

-Jon

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