Sphere packing is a very old and fascinating subject in mathematics. It has found important applications in several contemporary technologies such as error correction coding and communication. Here are some links:
This is a fascinating topic. There is a wealth of literature out there both in academic journals and popular websites and books. I list a couple of them here.
Problem 1) Classify the regular polyhedra in R^3 (there are five of them).
Problem 2) Prove that the Isometry group of Euclidean space R^n is O(n) |X R^n (|X denotes the semi direct product).
A Potential Final Project Project (This is not a HW problem)
Find all finite symmetry groups of R^3. Prove your conclusion.
Poincaré-Bendixson’s Theorem is an important heorem in differential equations and dynamical systems. This lecture provides a good introcution tothe theorem and its applications.
There are plenty of easily accessible material to get yourself familiar with cryptography, including stories and history. You are recommended to read the following:
The Adventure of the Daning Men, a delightful short Sherlock Holmes story by Arthur Conan Doyle. It is one of the stories in The Return of Sherlock Holmes. In this story, Holmes solves the "Dancing Men" cipher, which is a substitution cipher, using frequency analysis. A mathematical discussion of the Dancing Men cipher and substitution ciphers in general can be found in this article. The Dancing Men messages in the book can be found here.
There are numerous articles and notes on affine ciphers, many of which are accessible to you. Some requires rudimentary linear algebra such as matrix product and inverse matrix, which I hope you will learn. A good note by Eisenberg on Hill ciphers is a good source, and you should read it. A couple of other very readable notes are here and here.
The Enigma Code used by Nazi Germany during World War II is a more complex version of the substitution ciphers called polyalphabetic substitution ciphers. In polyalphabetic substitution ciphers, the substitution changes after one or several steps. A fairly comprehensive discussion of the Enigma ciphers can be found on Wikipedia.
I think you will find some of the books written for the general public, including fictions, very interesting. I highly recommend that you pick up some of them.
There are a ton of resources on public key cryptography online. To fully understand how things work you will need to have some background in number theory. The lectures in Week 2 will cover enough of the mathematical background needed for the topic. I highly recommend you to take the Number Theory class if you are interested in this topic.
The lecture notes contains a fairly brief introduction to RSA cryptosystem, including some of the background material in elementary number theory. You should go over the notes. There are materials on the web that will give you a better idea about public key cryptography. I list a couple of them here that I think are quite accessible.
If you want to play around with RSA, there are a couple of RSA calculators online that allows you to get a good idea how it works. One such website is here
There are a large number of literature online on related problems such as primality testing, digital authentication, other public key cryptosystems, etc. I would highly recommend that you do Google search on them and broaden your knowledge. In addition, if you are interested in the mathematics behind cryptography, I strongly encourage you to take a course in number theory. It is a very beautiful subject, and it is also a great starting place to get yourself trained in mathematics.
Exercises for Week 9 (Due November 10)
Exercises for Week 10: Prof. CHIANG has provided some exercises in his lecture notes. But they are optional. No need to turn in any.
Exercises for Week 11 (Due November 24)
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