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Question 1 is asking, if you perform a delay-coordinate embedding to reconstruct the dynamics of a system from a scalar time series
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And you assume that all the conditions of the embedding theorems are true, and that you did all the measurement properly
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Then are we guaranteed the reconstructed dynamics have the same geometry, or the same topology?
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We are guaranteed that it has the same topology, so part b is Yes and part a is No
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Recall from 8.2 that this embedding, for example, has the same topology, but clearly from a visual inspection does not have the same geometry as the Lorenz attractor
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The theorems of delay-coordinate embedding only guarantee correctness of topology, not of geometry
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For Question 2, These two shapes have similar geometry, and this is true, theyre both bowls
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Essentially, they have the same shape that we would see visually
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Geometry doesnt care how many holes are in an object, so the fact that this colander is full of holes is not important to geometry
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But the fact that they have the same visual appearance, or the same geometry, is whats important
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So this question is true
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Question 3 asks if these two shapes a bowl and a donut have the same topology
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And this is false
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The donut has a hole in it, whereas the bowl does not, so the donut and the bowl have different topologies
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Question 4 asks if these two shapes the coffee mug and the donut have the same topology
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This is a very famous example in topology
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Because the coffee mug can be continually deformed, without piercing any holes, to the donut, and vice versa, a coffee mug and a donut have the same topology
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So this is true
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For Question 5, does a bowl and a colander have the same topology?
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And this is false
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The colander has hundreds of holes in it, and so it has very different topology than a bowl
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Although they do have the same geometry
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So this question is false
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Question 6 states that you need to have direct, untransformed measurements of at least one state variable for delay-coordinate embedding to work, and this is false
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You only need to measure a smooth, generic transformation of at least one state variable for delay-coordinate embedding to work
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Recall the examples from the lecture in 8.2, where you actually measured x * y z in the Lorenz equation
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This is definitely a transformed measurement of all state variables, and that worked just fine