1. (i). In the children's book The Cat in the Hat Comes Back, the author Dr Seuss talks about a cat wearing a hat. Within its hat is another cat ("Little cat A") also wearing a hat, within which is another cat ("Little cat B") wearing a hat also and so on ad infinitum. Assuming that the height of the cats (d) and the length of their hats (h) scale by the same constant factor r [i.e., the ratio of the height of the n-th and (n+1)-th cat, viz., d(n+1)/d(n)= r and similarly h(n+1)/h(n)=r] show the relation that exists between r, d and h.
(ii). Going back to Dr Seuss's story, after the first cat opens its hat
revealing another cat ("Little cat A"), sequentially each cat opens its hat
revealing another cat until we reach "Little cat Z", i.e., in all 26 cats
were nested inside the original hat. Assuming that the hat is no bigger
than the cat and that the length of the original cat is the same as an
ordinary cat, can you give an order of magnitude for the height of "Little cat Z"?
2. In class we have discussed the Shannon measure of entropy.
Can you sketch the essential steps for the derivation of the Shannon entropy
formula from first principles ? What are the essential assumptions ?
3. In class we have seen the Peierls argument for the existence of spontaneous magnetization in the 2-dimensional Ising model. Modify the argument to show that spontaneous magnetization can exist in the 3-dimensional model.
[Give details on the boundary conditions of the domain you are using and the way you will construct bounding surface of a domain with similarly aligned spins].