Among the questions you could look at are; |O(n)|, necessary and sufficient conditions for k to belong to O(n), etc. One observation that is a theorem is this: if k is in O(n) and k > n, then k is in O(n+1). See what you can come up with.
Deadlines: Middle of October -- Come up with a proposal; before Thanksgiving -- A reasonable draft of progress that I can critique in a serious way; Finals week -- The final report.
F 8/29 -- A rainy day. Some discussion of the first assignment. Then an introduction to the Fibonacci numbers, and a return to the checkerboard, counting 2 x n tilings, with a special guest appearance by Signore Fibonacci.
W 9/3 -- First, a first presentation from a student: Jim Freitag talked about counting squares within squares (easy) and triangles within triangles (harder). What are the rules for this? Any student can present any idea they are currently interested in and/or excited about and/or would like input on. At any time. Just ask . . . before I ask you. Another presentation regarded the The On-Line Encyclopedia of Integer Sequences. We looked at some Puzzle Sequences and I distributed a guide that appeared recently in one of the basic math journals, the Notices of the American Mathematical Society. (Not yet on-line!) And then I talked some more about Fibonacci numbers and about 10 different ways you could try to generalize them. Try to do that all the time, in all your classes.
F 9/5 -- Friday mix. Michael Munie gave a presentation on countability and Cantor's Diagonal argument (see Aigner/Ziegler, Ch. 15.) I gave a quick proof that the algebraic numbers are also countable (see A/Z Ch. 6 for related material.) I hope I've gotten you thinking about your presentations! The other material I put out was a printout of the homepage of Plouffe's Inverter (for identifying mathematical constants) and The Erdos Project Home Page (on the Erdos number of mathematicians. This was followed by a few personal reminsicences of Paul Erdos and a strange story involving Russell Crowe. I also gave out a printout of the links page of our departmental website: Math Links , together with a running commentary.
M 9/8 -- Michael Munie talked about tiling 3 x n checkerboards with dominos, by introducing a pair of sequences. I added some more, tied it in with Fibonacci stuff and observed that you can take the powers of a matrix easily if you diagonalize first. I passed out "Some thoughts on writing for the Putnam", which is not online, and went through a couple of problems, with an emphasis on how one might have been able to solve them systematically. Richard Feynman's name came up with respect to one of the problems.
W 9/10 -- Raymond Rouf gave a discussion of 2 person games and the Prisoners' Dilemma, one of the fundamental philosophical questions of the second part of the 20th century. I spoke about the analogues between Taylor Series and Newton's Formula for finite differences, along with an approach to find general formulas for what Mathematica would call Sum[k^r,{k,1,r}].
F 9/12 -- Tyson Ewing presented necklace counting via Burnside's Theorem, and I remarked on how it leads to Polya's Theory of Counting. Same Polya. I distributed two old handouts on Fibonacci Numbers: Dated 1/22/01 and Dated 1/29/01. I also distributed "A Guide for Graduate Students in Mathematics, 2003-2004" from our department.
M 9/15 -- John Rothstein talked about even perfect numbers and Mersenne primes, and I added some additional historical information, and a proof or two. Somehow, my childhood travel history came up. We also talked about REU's with a distribution of printouts from NSF List of Math REU's and 2003 UIUC info. The situation for 2004 is not definite, but there will probably be math REU's in Urbana. At the very end, I talked about x^y = y^x and also gave a rough timetable for the semester, which is at the top of the page.
W 9/17 -- Dan Pozdol talked about cryptography, open-key and ECC. I chipped in with a few stories, and outlined how one might go about solving the cubic equation. By Friday, I will have some more links for you and I promise to talk about the Polya reading, and a few handouts.
F 9/19 -- Maria Boca talked about some interesting formulas for binomial coefficients, and I enthusiastically, but probably incoherently, elaborated upon them. I talked about the Polya reading and forgot to give you a bunch of handouts, but will do these on Monday.
M 9/22 -- No speaker. Handouts: the original version of x^y = y^x from an earlier incarnation of this seminar, and also the ArXiv record of the final version, to appear in the American Mathematical Monthly. Mersenne Links to GIMPS Home Page and to the Digital Math Library. I spoke mainly about x^y = y^x.
M 9/22 -- Unscheduled, scheduled class meeting, coinciding with Math 400: Introduction to Graduate Mathematics 4:00 pm in 245 Altgeld Hall, "Ever Try to Explain the Difference Between Algebra and Analysis to your Brother" AND "Resources for Research". There will be handouts for those of you who are unable to attend.
W 9/24 -- No speaker. A handout of an article from the current Notices of the AMS on mathematicians in mystery books. A fantasia on roots of unity and how to solve certain kinds of equations. Extra handouts of Resources for Research were distributed, as a leftover from the second presentation on 9/22.
F 9/26 -- Seth Case talked about the Lucas-Lehmer primality test for Mersenne primes. I followed with some unplanned discussions of sequences, triggered by a sequence used in Lucas-Lehmer. This tied in with the previous discussion of roots of unity. Last class of summer. I plan to distribute a handout giving three relevant links, plus my original notes on the problem x^y = y^x and the final version of the paper I wrote on the topic with Mike Bennett. This paper will appear in the American Mathematical Monthly
M 9/29 -- John Rothstein gave a method for trisecting a general acute angle that required knowledge of a compass and marked ruler. (This is not the usual problem.) How can one prove that trisection can't be done solely with compass and straightedge? This requires knowledge of the sorts of numbers that can be constructed by intersecting lines and circles, and as such is really a subset of Galois Theory. If p(x) is a polynomial in one variable or if p(x1,..,xn) is a quadratic polynomial, then p takes only non-negative values if and only if p is a sum of squares of polynomials. This is not true in general, and the rest of this week (and next) will be devoted to some examples. I started with various proofs of the generalized inequality for the arithmetic and geometric means, leading to a brief introduction to convex functions.
W 10/1 -- Sabreena Ashraf talked about the Pigeonhole Principle and gave as an example the periodicity of Fibonacci numbers modulo n, for any integer n. I gave the proof of Pick's Theorem for lattice point polygons in the plane. I also talked about how math conferences get organized, in particular the one on lattice points this past summer. Conference Home Page You can look inside and find a few pictures in which I appear.
F 10/3 -- More talk about Pick's Theorem and one-point lattice point triangles, and four different proofs that for a one-point lattice triangle, the interior point is the centroid. Discussion of triangles with more points inside and lattice point tetrahedra. Transition to Gaussian quadrature via a ``rule of thumb" for computing the volume of a figure based on the cross-sectional area of its top, middle and bottom. Handout: two articles from Math Horizons.
M 10/6 -- A geometric solution to a cubic equation. Linking up the discussion on lattice points with the discussion on the arithmetic geometric inequality, via Motzkin's wonderful polynomial M(x,y) = 1 = x^4y^2 + x^2y^4 - 3x^2y^2.
W 10/8 -- The iterated almost-square paper of Lagarias and Sloane
with one or two variations. An assigment. I want you to get together in class
on Friday and talk about this problem and get as far as you can and
delegate a few of you to present on Monday in class.
The set O(n) is defined recursively by O(1) = {1} and
O(n+1) = (O(n) + 1) U (2O(n)). That is, k is in O(n+1) if and only if
k = j+1 or k = 2j for some j in O(n).
O(1) = {1}
O(2) = {2}
O(3) = {3,4}
O(4) = {4,5,6,8}
O(5) = {5,6,7,8,9,10,12,16}
O(6) = {6,7,8,9,10,11,12,13,14,16,17,18,20,24,32}
Among the questions you could look at are; |O(n)|, necessary and sufficient conditions for k to belong to O(n), etc. One observation that is a theorem is this: if k is in O(n) and k > n, then k is in O(n+1). See what you can come up with.
F 10/10 -- I spend the afternoon in beautiful suburban Terre Haute, at Rose-Hulman Institute of Technology. You were supposed to have been working on {O(n)}!