What you should do
- Choose a topic. I'd like you to become the local expert on the
topic. Tell me what topic you will work on so we can avoid
- Read background material on the topic; search the web. All of these topics have lots of references.
- Give me a 2 or 3 page outline of your work, with references.
- Prepare a 20-minute talk on your topic, and rehearse this talk with me.
- Give me a draft (5 to 10 pages) of your paper.
- Give the talk, and, following any necessary changes, submit your paper.
Randomness Ben Tully
Addition chains Steven Jaslar
- General introduction to the subject (see S. B. Volchan, "What is a
random sequence?" , Amer. Math Monthly 109 (2002) 46-63.) This talk
will be used by several other students' presentations.
- Pseudo random number generators. See chapter 21 of [G].
See also http://www.nist.gov/dads/HTML/pseudorandomNumberGen.html
Linear congruential generators as pseudo-random number generators Jagan Pisharath
- What is an addition chain, and why is it "useful"? Here [K] might be good, as
well as the papers
- What is presently known, and what are the conjectures? This is in the papers cited above.
- Students should understand something about the present knowledge of
- An additional
reference, somewhat technical but perhaps the first few sections
may be accessible and useful.
Linear feedback shift registers as pseudo-random number generators Justin Palumbo
- Following the discussion of randomness.
- A definition is at http://www.nist.gov/dads/HTML/linearCongruentGen.html.
- Here is one discussion:
and the books [G] and [K] also contain discussions.
- Students should understand the idea, and understand why certain LCG's are
better than others.
NO NO NO: P vs NP as exemplified by a discussion of minesweeper.
GCD via Euclidean algorithm Robert Burton
- Following the discussion of randomness
- A definition is at http://www.nist.gov/dads/HTML/pseudorandomNumberGen.html
- The book [G] contains a discussion.
- Students should understand the idea, and be able to "generate" the bits of
a LFSR given the initial fill.
Efficiency of GCD Nathan Wilson
- Discuss the history of the Euclidean algorithm.
- Exhibit the algorithm with suitable examples. One reference is http://www.cut-the-knot.org/blue/Euclid.shtml.
- Show how this applies so that students in the class can find by hand
the multiplicative inverse of N mod P where P is, say, a prime with
#(P)=6 or 7.
Knapsack or subset sum cryptosystems Gregory Ryslik
- How many steps on average does the GCD algorithm take?
- What is the "worst case"? (The Fibonacci numbers)
- What is it? (Another "hard" math problem underlying a cryptosystem)
- How can it be broken? Sources include [G] below and many other references.
Fast(er) multiplication Dale Gold
- Description of history of the problem, comparison with primality (see, e.g., http://www.cse.iitk.ac.in/news/primality.html).
- Current state of the art. [C & P] is a good reference.
- Aim: show that there are faster ways to multiply than
- Discuss Karatsuba multiplication. A general introduction is
available here: http://mathworld.wolfram.com/KaratsubaMultiplication.html.
Other references are http://www.math.ubc.ca/~cass/programs/fast-multiplication.dvi
- A possible continuation of this is Fourier transform
multiplication. Useful background for understanding this is some
As mentioned in the diary, this is sort of a converse to the von
Neumann problem. Here is a proper statement, I hope:
I think that I know a way to do this problem. As pointed out in
class by several students, if p is rational (a quotient of two
integers) there seem to be several ways to do the problem. But, as I
remarked, I'd like to know how to do it for p=1/sqrt(2), which doesn't
seem too rational to me.
||You are given a fair coin and a number p with 0<p<1. You
must come up with a procedure to simulate a coin which comes up heads
with probability p and tails with probability q=1-p. The procedure
must terminate almost surely, that is, with probability 1. No further
information about p is given: it can be any number between 0
and 1. You also need to analyze the "running time" of your procedure:
the expectation for the number of flips of the fair coin needed to
contribute one simulated p/q flip.
So I don't know how hard this problem is, or how involved a writeup it
would need (would it be comparable to other topics, for example?). I
Solved by Mr. Jaslar! More to come.
More broadcasting Charles Siegal
On Monday, October 13, I will begin an analysis of the broadcasting
model and how it could break for binary trees. I think that this
analysis could be extended. Here are some suggestions:
More Ramsey Theory Lisa Facchini
- Instead of binary trees, how about trees whose branching rule
is multiplication by B (a positive integer) at each stage (binary is
the case B=2). I think it should be possible to duplicate my
reasoning in this case. Computer simulation with some data supporting
your theoretical deductions would be useful.
- After doing the preceding case, how about trying trees that
alternate branching between binary (B=2) and ternary (B=3)? What will
the probabilities turn out to be? I don't know, and this could be a
bit interesting. Computer simulation with some data supporting
your theoretical deductions would be useful.
- What happens if we alternate between 2 and 3 at random? An
analysis of this may be quite difficult. So here definitely computer
simulation would be useful.
I will show that if a complete graph is large enough (!) and if the
edges are two-colored, then there must always be some "monochromatic"
small complete graph inside it. (These terms will all be explained
later.) This project would attempt to duplicate the reasoning given in
class for graphs which are 3-colored, then 4-colored, etc. Or another
variation: how about 3-coloring a graph, and seeing how large it must
be before sme small complete 2-colored graph must be inside it? A
student who takes up this project will need to read ahead in my notes,
and talk with me about them.
The Four Color Theorem
Read about the history and controversy and report on them. See
for an initial source. The books by Thomas L. Saaty and Paul C. Kainen
(The four-color problem : assaults and conquest, QA612.19.S2
1986) and Oystein Ore (The four-color problem, QA3.P8 v.27) are
currently in the library. Look at Kempe's original paper (available on
the web, I believe, through a Rutgers browser). Read Professor
and interview him: perhaps invite him to come to class.
Read Barabási's book, and describe scale-free networks. Discuss
one application of the networks. See also the text Six Degrees: The
Science of a Connected Age by Duncan Watts, also, by the same
author, Small Worlds.
As mentioned above, a search of the web for any of these topics will provide many refences and some of these will be useful. Here are some texts which are good:
- [B & S] Algorithmic Number Theory, Bach & Shallit, MIT Press, 1996
- [C & P] Prime Numbers, A Computational Perspective, Crandell & Pomerance, Spring 2001
- [G] Making, Breaking Codes, Garrett, Prentice Hall, 2001
[K] Seminumerical Algorithms (Art of Computer Programming, vol. 2), 2nd
Ed, Knuth, Addison-Wesley, 1981
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