MTH 291: SNAPSHOTS OF MATHEMATICS

Lecture Notes, Exercises and Articles

Lecture 1 (web search)

Lecture Notes

Exercise 1 (Due January 26). Assume there are three webpages in a database. Pages 1 and 2 are mutually linked, and page 3 has a hyperlink to page 1. Assuming damping factor d>0, what are the PageRank for the three pages?

Exercise 3 (Due January 26). Assume there are n+k webpages in a database, divided into two groups A and B of n and k webpages, respectively. Each page in A has and only has links to every page in B while each page in B has and only has links to every page in A. Assunming damping factor d> 0, what are the PageRanks for these pages?

Exercise 2 (Due January 26). Assume that we have an index with N>1 pages interlinked among each other. A spammer adds his web page magic-pills.com to the index and wants magic-pills.com to be ranked as high as possible. The damping factor is d=0.

  1. The spammer's first strategy is: add k>1 fake websites to the index and make all of them link to magic-pills.com. After the spammer's additions, the total number of pages in the index is N+k+1. What is the PageRank value of magic-pills.com? Assume that there are no links between the N pages and the spammer's k+1 pages. 
  2. The spammer's second strategy is to hacked into k of the pages in the index and adds a link in them to magic-pills.com. What is the PageRank value of magic-pills.com?
  3. The spammer's third strategy is to hack into k of the pages in the index and adds a link in them to magic-pills.com. In the mean time the spammer also adds links in magic-pills.com to those k pages that he has hacked into. What is the PageRank value of magic-pills.com?
  4. (Extra Credit) Find the PageRank value magic-pills.com in each case when the damping factor is d=0.5.

Exercise 3 (Discussion). There are numerous questions involving Pagerank, such as computational efficiency, storage problem, link farms, and many more. Select a topic and search the literature. Read some of the articles, and discuss your findings.

Exercise 4 (Project, please feel free to work in groups of 3 or less). In real life there are many problems that require some type of ranking. Consider one such case and design a ranking method based on the Pagerank algorithm.

Articles on PageRank and related topics

Wiki entry

A survey of eigenvector methods for web information retrieval (by Langville and Meyer)

Deeper inside pagerank (by Langville and Meyer)

Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search (by TH Haveliwala)

Feel free to suggest more.

Lecture 2 (Public Key Cryptography, Part I)

There are plenty of easily accessible material for this lecture freely available I'll not post any lecture notes. Instead you are recommended to read the following as substitutes for my lecture notes:

The Adventure of the Daning Men, a delightful short Sherlock Holmes story by Arthur Conon 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.

Exercises for Lecture 2 (Due February 2)

Lecture 3 (Public Key Cryptography, Part II)

Lecture Notes

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 Lecture 3 (Due February 9)

Lecture 4 (Investment, Gambling and Kelly Criterion)

Lecture Notes (A Presentation on Kelly's Criterion)

The lecture notes contain an accessible overview and introduction to Kelly Criterion and its applications in investment and gambling. Although Kelley Criterion has been know for a long time and is an important result, it is relatively unknown to most people. Nevertheless you may find very interesting discussions on it. One of the best sources on its history is the book Fortune's Formula by William Poundstone (see a review). In this book Poundstone puts the Kelly criterion at center stage and discusses its origin, evolution, and applications to gambling and investing. It contains some amazing and amusing tales involves Claude Shannon and Edward Thorpe. I highly recommend this book.

Incidentally, Claude Shannon, widely known as "the father of modern communication theory", grew up in Gaylord, MI. The Claude Shannon Park in Gaylord is dedicated to him.

Here are some other reading materials.

Exercises for Lecture 4 (Due February 16)


Lecture 5 (Hammer's X-Ray Problem)

Guest Lecture by Prof. Ben Schmidt

In this lecture Prof. Schmidt gave a very clear, fun and accesible introduction to the problem. Although Prof. Schmidt did not provide notes for the class, there are a number of articles and papers on the problem. I list here some papers if you are interested in this problem. Some of the papers also provide much of the needed background material to do the exercises.

Exercises for Lecture 5 (Due March 1)

You only need to do 2 out of 5 problems. The rest will be viewed as extra credit.

Lecture 6 (The Mathematics of Voting)

Lecture 6 Notes.

Much of Lecture 6 materials can be found on the web. Some of the articles are listed below.

There are a few nice books on voting. I recommend you to check some of them out.

Stirling's Formula was used in the estimate of the probability of a tie in the vote count. An elementary proof of Stirling's formula can be found here.

Exercises for Lecture 6 (All for Extra Credit, Due March 15)

Project for Lecture 6 (A sataisfactory completion of the project will earn a 4.0 for the class)

 

Lecture 7

Presentation by students Lee Wang and Kevin Anderson on AES Encryption.

Materials and Exercises can be found in Lee's Website.

 

Lecture 8 (Tiling and Self-Similar Tiles)

Lecture 8 Notes.

There are a ton of literature on tiling. One of the best books is Grunbaum and Shephard's Tiling and Patterns, which contains a wealth of information on all aspects of tilings. A quick Google search will yields numerous entries. Grunbaum and Shephard contains a chapter on aperiodic tiling and Penrose tiles, which is a very nice introduction to this fascinating topic.

A very nice place to find all kinds of resources on tiling is the Geometric Junkyard. In particular, the section on tiling. You will be able to find all sorts of interesting articles and papers there whatever your level and interest might be. I highly recommend that you check out the Penrose Tiling part.

Exercises for Lecture 8 (Due March 22)

  1. Show that the regular Pentagon and regular n-gon with n greater than 6 do not tile the plane.
  2. (Extra credit) Show that any convex n-gon with n greater than 6 cannot tile the plane.

Lecture 9 (Guest Lecture by Teena Gerhardt)

Lecture 10 (Fractals I)

In this lecture we have shown one way to obtain fractals, namely through iterated function systems (IFS). There are a ton of notes on the web discussing in details about self-similar sets and IFS. Here I list just a few of them for your reference.

 

 

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