Question 1 is asking, if you perform a delay-coordinate embedding to reconstruct the dynamics of a system from a scalar time series
And you assume that all the conditions of the embedding theorems are true, and that you did all the measurement properly
Then are we guaranteed the reconstructed dynamics have the same geometry, or the same topology?
We are guaranteed that it has the same topology, so part b is Yes and part a is No
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
The theorems of delay-coordinate embedding only guarantee correctness of topology, not of geometry
For Question 2, These two shapes have similar geometry, and this is true, theyre both bowls
Essentially, they have the same shape that we would see visually
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
But the fact that they have the same visual appearance, or the same geometry, is whats important
So this question is true
Question 3 asks if these two shapes a bowl and a donut have the same topology
And this is false
The donut has a hole in it, whereas the bowl does not, so the donut and the bowl have different topologies
Question 4 asks if these two shapes the coffee mug and the donut have the same topology
This is a very famous example in topology
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
So this is true
For Question 5, does a bowl and a colander have the same topology?
And this is false
The colander has hundreds of holes in it, and so it has very different topology than a bowl
Although they do have the same geometry
So this question is false
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
You only need to measure a smooth, generic transformation of at least one state variable for delay-coordinate embedding to work
Recall the examples from the lecture in 8.2, where you actually measured x * y z in the Lorenz equation
This is definitely a transformed measurement of all state variables, and that worked just fine