M 4/2 -- F 4/13. Well, I seem to have fallen behind on this. In these days, Jeremy talked about mathematical poems that can be written as a matrix of words, a symmetric matrix at that. This led to a JSTOR handout about the French literary group Oulipo. Amy talked about her project on stochastic tweakings of the Fibonacci recurrence, based on a paper by Leyte. I talked about a the equation x^y = y^x and x^y = y^(2x) and about Polya's theorem on positive polynomials. We went to the library and saw how MathSciNet and the old card catalog of journal numbers work. I presume you are all working hard on your projects. We'll have to start scheduling our presentations pretty soon. There will be a talk by Prof. Sue Tolman on how mathematical research works when you aren't a combinatorial number theorist, and there will be [I hope] a trip to the Rare Book Room. If you're actually reading this, please send me an email.
F 3/30. Jack Schiff talked about quadratic convergence, both in his project and in Newton's method. I gave out three JSTOR handouts, mostly on q-series and also gave a number of proofs of the general arithmetic-geometric inequality.
W 3/28. Amy Green talked about various probability distributions and Walter Feig talked about q-series and combinatorial interpretations.
M 3/26, Return from a long break. Some people have a draft of their research proposals, others not yet. I talk about the conference and then talk about non-negative polynomials in one variable or of degree two, and how they are necessarily expressable as a sum of squares.
F 3/9. A brief discussion of how to figure out your own Illinois Driver's License number. A JSTOR handout on Pick's Theorem and a discussion of my theorem that if T is a plane triangle with vertices v_1, v_2, v_3 so that {(i) v_i are in Z^2 (ii) The only points of T that are in Z^2 are the v_i's and a single point w inside}, then w is the centroid of the triangle. Here's a link to Mathematisches Forschungsinstitut Oberwolfach. I'll be there next week and then there's spring break. Have fun, be careful and see you on 3/26.
W 3/7. A bunch of handouts. One on the location of my trip, others from JSTOR on elliptic curves, on the Cantor set and on systems of numeration. My presentation was mainly on the Stern sequence.
M 3/5 Jack Schiff talked about elliptic integrals and elliptic functions and elliptic curves. I tried to elaborate on it, without much success.
F 3/2. No class. Instructor in a committee meeting in Washington, DC.
W 2/28. No student talks. I discussed some facts about representations base 3, leading into the effect of the greedy algorithm in finding subsets of the integers with no 3-term arithmetic progressions. An incomplete 2-dimensional version was presented and discussed, with handout. No non-trivial conclusions were drawn. An old class handout on polynomials, elaborating on Monday's presentation, was also distributed.
M 2/26. Handouts on fair distribution and Niven's Theorem from JSTOR. I started to talk about polynomials and difference equations and the analogies between them and ordinary differentiation. F 2/23. Jeremy Grozavescu talked about some fair-distribution schemes that came up in the preparation for the ICTM math competition. I gave a proof of Niven's Theorem that if t is a rational multiple of pi and cos[t] is rational, then its possible values are 0, \pm 1/2 and \pm 1.
W 2/21. Four handouts: two JSTOR articles on various means, a Mathematica printout giving the factorization of Binomial[2n,n] and an ancient list of the exact value of the sine function in increments of three degrees. I completed the proof of the bounds on pi(n) and started on the linear difference operator.
M 2/19. Prof. Kevin Ford pointed out (in an email to me) that one of Rachel Hillmer's questions from 2/16 could be described as a "restricted Goldbach" conjecture. Jack Schiff talked about a variety of means and their relative values and limiting conditions. In particular, if M_p(a,b) = ((a^p+b^p)/2)^(1/p), then the limit of M_p(a,b) as p->0 is the geometric mean. I also gave out two JSTOR handouts relating to the cardinality issues discussed in earlier presentations.
F 2/16. Rachel Hillmer talked about some questions involving twin primes, in particular, the sequence of averages of twin primes. She also presented the geometric representation of connecting the lattice points (a,b) in order of increasing value of 2^a3^b. I handed out four papers, two relating to each presentation on M 2/12. All four can be found via JSTOR (links too long to post here). I started to present arguments that the function pi(n) is bounded above and below by multiples of n/log[n]. Classmembers will be expected to have a written proposal for their research on the first day back from Spring Break (March 26). Individual appointments are encouraged and drafts will cheerfully be examined.
W 2/14. Class cancelled.
M 2/12. Walter Faig talked about partitions and a generalization of Euler's theorem that there are an equal number of partitions of an integer n into odd parts and into distinct parts. Amy Green talked about Fibonacci numbers and the Zeckendorf representation of any integer as a sum of non-consecutive Fibonacci numbers. I handed out at the very end the latest iteration of Resources for research.
F 2/9. Jeremy Grozavescu discussed some relationships between the Lucas numbers and the Fibonacci numbers and this led into more on identities satisfied by the sequence G_{r,n}:= F_{nr}; that is, how F_{rn+2r} is related to F_{rn+r} and F_{rn}. There's a nice formula and Lucas is involved. I also gave another proof of the infinitude of primes, based on the identity that the sum of 1/m, where m runs over all integers whose prime factorization involves the set of primes {p1,...,pr} is exactly equal to the product of the r terms of the form pi/(pi-1), and so, in particular, is finite. This led to the amazing fact that the sum of 1/m, when taken over all integers whose base 10 representation does not contain "9" is a convergent series. (Nothing is particularly special about "10" and "9' there.) No handouts.
W 2/7. Handout from my grad class notes of Spring 2006 on the generating function material discussed in class on 2/5. I spoke on roots of unity and, given a power series F(x) = \sum_n a_n x^n, how to express the functions \sum_j a_{j N + r} x^{j N + r}.
M 2/5. Two handouts: one consisting of four webpage sheets, and the other an article on the mysterious number 6174. I talked about the Fibonaccis, with a discussion of extending them to negative index, which rambled onto a discussion of generating functions, then the metric space of formal power series, in which the distance between f and g is the reciprocal of the exponent of the first power of x for which their coefficients differ. Then generating functions for representations of integers in the form a + b, where a belongs to a finite set A and b belongs to a finite set B. A Putnam problem somehow came up.
F 2/2. I spoke a bit about David Blackwell, famous mathematician and UIUC alum. I distributed his Biography and mentioned a 169 page oral history. Following up on my earlier discussion, I gave out a biography of James Joseph Sylvester, together with some photos from that site. Jack Schiff gave a careful proof that if S is part of the chain of power sets beginning with Aleph_0, then there is a surjective map from S to S x S.
W 1/31. David Rosenberg talked about the Ramsey Number problem for five people and six people. (Equivalently, about the existence of monochromatic triangles in two-colorings of K_6.) I posed the question of showing that every such two-coloring must have at least *two* monochromatic triangles. I then passed out a long handout with lots of useful mathematical links, which I won't try to reproduce here.
M 1/29. Rachel Hillmer talked about continuity, differentiability and the intermediate value theorem. This led to my presenting a half-remembered version of the Bold Gambler function. I did more Fibonacci stuff and I forgot to give a handout I'd xeroxed. Come to class on Wednesday.
F 1/26. I spoke about the Fibonacci numbers, in several respects: from the closed formula and using the generating function. Side issues included partial fractions and the reasons that the sequence (1,a,a^2,...) is a natural one to look for in the linear space of Fibonacci sequences. Most of what I said was covered in the previous handout.
W 1/24. Kristin McCoy talked about Eulerian paths and circuits in graphs and necessary and sufficient conditions. Jack Schiff talked about the incommensurability of a set and its powerset, which is thematically similar to the Russell Paradox. I distributed A set is a set. This link probably doesn't work from a non-UIUC machine. I'll work on that. Friday will be a Fibonacci kind of day.
M 1/22. Jeremy Grozavescu talked about the Halting Problem, which is Turing's application of the Russell Paradox to theoretical computer science. Amy Green talked about the unique decomposition of functions into a sum of an even and an odd function. I distributed the highly arbitrary breakdown of UIUC mathematics faculty by research areas, and began to talk about the Fibonacci numbers. See Fibonacci handout I and Fibonacci handout II Though you haven't asked, the reading in the required books is implicit. Pick a few things you like and concentrate on them, or just browse.
F 1/19. Second day of class. One of the few in which there will be
no handouts. Two class presentations. David Rosenberg talked about
the diagonalization proof that the rationals are countable and
generalized it to give a nice proof that Q + iQ is also
countable. (Proving that the algebraic numbers are countable is
trickier.) Walter Faig gave Euler's proof that there are infinitely
many primes, and then gave variations that there are infinitely many
primes of the form 4n+1 and of the form 4n+3.
I made some additional comments on each of these, as is my tendency. I
spoke a little bit about the addition formula for Fibonacci numbers.
The principal mathematical development was the definition of the
vector space F, consisting of all sequences (a[0],a[1],....) with
a[i] in C, say, and satisfying the Fibonacci identity: a[n] = a[n-1]
+a[n-2]. (Formulas aren't easy to write in HTML, so I might make
harmless changes in the notation.) Since F is a finite-dimensional
vector space, all sorts of identities are forced by the fact that any
three such sequences are linearly dependent.
As a general rule, classes in Math 496 will begin with students making
presentations. Ideally, each student should go to the board at least a
half dozen times during the semester.
W 1/17. First day of class. Four handouts. Course
Organization, plus the questionnaire, and
Problem-solving template.
. Here's the version of
"One
Introduction to Mathematical Research" with all the jokes.
Homework assignment -- taken from the article: