The mathematical links page is here. Suggest a new one!

Many students are concerned about the political issues of the
day. If you want to contact your representatives, here are their
numbers:

US Sen. Tammy Duckworth (D-IL) (202)-224-2859

US Sen. Richard J. Durbin (D-IL) (202)-224-2152

US Rep. Rodney Davis (R-IL 13) (202)-225-2371

IL Sen. Scott Bennett (D-IL 52) (217)-335-5252

IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

M 10/9 -- Another worksheet, and then a proof of nearly all continued fraction identities by matrix methods, and more on the Bold Gambler function.

F 10/6 -- Oh look, a worksheet, for a day where the attendance was notably low! After this, proved the Rule of Four and gave another proof of the annelidic property, and started talking about a "natural" strictly increasing function from [0,1] to itself that doesn't seem to want to be differentiable. More on it on Monday.

W 10/4 -- A mixture of items, semi-spontaneously. First, another way of thinking about problem 6 on the homework, in terms of the norms of elements in a quadratic extension field. Then, some proofs of more of the continued fraction identities from chapter four of the notes and, at the end, the Rule of Four. Your suggestions as to the next directions are always welcome, but I'll start Friday by proving it. Might be time for another worksheet as well.

M 10/2 -- Spent the hour reviewing the solutions to HW 1; see here for the text and here for the computer backup. Several alternate proofs of the last problem were presented, and I will put the details onto the "retrospective" later.

HW 1 Questions

The exponent in 6b might be off by one. Figure out what's right
and prove it!

F 9/29 -- Completion of the proof of the general formula for the Stern sequence, based on continued fractions. Illustration of finding all n so that s(n) = 7, based on the continued fraction expansions of 7/1, 7/2, 7/3, 7/4, 7/5 and 7/6. Introduction (and partial mangling) of the Minkowski ?-function, implemented on Mathematica as MinkowskiQuestionMark[x] (!).

W 9/27 -- The contact list (not online) was distributed. I gave the general formula for the continuant (numerator and denominator of the simple continued fraction) and talked about the general "closed" formula for the Stern sequence. The text was taken from chapter 1 (pp.11-13) and chapter 4 (pp.2-3) of the previously distributed notes.

M 9/25 -- The sum of cubes of the Stern sequence (see chapter 3 of the notes; pp. 8-10. Some cultural discussion of continued fractions and chapter 4 was distributed at the end. For more on continued fractions, see this page from the OEIS wiki If you find a page you like, let me know and I'll add it here.

F 9/22 -- We started with a worksheet, on a variations of the sequence formed by summing the squares of the Stern sequence. This led to a discussion of recurrences and how to handle inhomogeneous equations. More data, computed with the assistance of Mathematica, was also distributed. The plan on Monday is to talk about sums of cubes and then move on to continued fractions for what will eventually lead to the closed form formula for the Stern sequence.

Th 9/21 -- Contributed by class member HC: It seems that our WebStore offers Mathematica Online for free. Here is the link from the webstore. Another option is to buy a desktop license (v11.1.1) at $35 for a year. Here is the link..

HW 1 Questions (temporarily pinned to the top)

The exponent in 6b might be off by one. Figure out what's right
and prove it!

W 9/20 -- At the end of class HW 1 was distributed, to be due M 10/2. The class was a mixture of ideas, first giving a proof of Niven's Theorem which combined elements from analysis (complex expression of trig functions) and algebra (rational root theorem), with generating functions of various kinds in two variables. This led to a discussion of the sum of squares of the Stern sequence. A handout was made of various sequences which ultimately led to a surprisingly detailed introduction to continued fractions. See some Mathematica output. Who knows what will happen on Friday? A link with advice on applying to math graduate school.

M 9/18 -- Homework 1 will be out on W 9/20. A different class, much more impressionistic and covering wide swathes of mathematics revolving around recurrences. First up was a discussion of asymptotics, which somehow led into Niven's Theorem. Then the generating function approach to solutions and the linear algebra approach. A side note proving the existence of the partial fractions approach (in the notes.) Let

X = {a =(a_0, a_1 , ....), a_n in C}

The linear recurrence sequences can be viewed as the kernel of a linear operator. In the Fibonacci case,

T(a) = (a_2 - a_1 - a_0, a_3 - a_2 - a_1, etc.)

By considering the dimension of the kernel and special elements (1,r,r^2,r^3,...) which belong to it, we get an alternate approach to the solution. The last 10 minutes were spend talking about Vandermonde determinants.

F 9/15 -- Distributed chapter 3 of notes, on linear recurrences. Before I do this formally, I worked through in a lot of detail, the number of s(n) which are multiples of 3. These belong to the tree whose vertices are N \cup \{0\} and which has edges from n to 2n, 8n-7, 8n-5,8n+5,8n+7. This directed graph has the property that every vertex but \{0\} has indegree 1 and outdegree 5. Talked about various other things as well. The first homework will be out next week.

W 9/13 -- Yet another worksheet, followed by a discussion of various techniques for manipulating generating functions. For example, if F(x) = \sum_{n=0}^{oo} a[n]x^n, then \sum_{n=0}^{oo} a[2n]x^n = (1/2)(F[\sqrt x] +F[-\sqrt x]), and similar things for a[2n+1] and really for a[m*n + d] for fixed m and n, using m-th roots of unity.

M 9/11 -- Another worksheet! Actually, first a table, and then a worksheet. Almost all of the day was devoted to working on (and talking about) these.

F 9/8 -- More discussion of Ch. 2, plus, a worksheet, which the class seemed to like and which occupied much of the hour.

W 9/6 -- Correct version of Ch. 2 of the notes distributed; see link at F 9/1. What was covered was an answer to an email question about the frequency of occurence of integers in the Stern sequence, and then material from Ch. 2, leading up to the generating function for s(n). Also an old Putnam problem about the number of ways to write n as \sum a_k 2^k, if a_k is allowed to take the values in {0,1,2,3}.

M 9/4 -- Labor Day! Here is a history.

F 9/2 -- Optional placement on the map. One handout was somehow the wrong one; an incomplete version of chapter 2 of the notes. Here is the full version, which will be distributed W 9/6. There was more discussion of how to find the generating function of the Fibonacci's, and I'll have some 496-style notes on that for W 9/6. Also, by request, and a good one: there will be a worksheet involving the Fibonaccis for W 9/6/

W 8/30 -- Two big handouts. First is Chapter 1 of 2012 grad course classnotes. Ignore the homework at the end. Second is Lehmer's article from almost 100 years ago. By request, here is the mathematica file from which the .pdf was discussed on Monday

M 8/28 -- Impressionistic beginning to the course and to the Stern
sequence. More references to come, but for now, see pictures here.

Handouts -- Course
Organization, How to Solve It
guide, Class questionnaire,
emergency
guide from the UIPD.