I’m teaching Calc I this semester. This course (after a review of precalc, limits and continuity) covers derivatives and their applications and the beginning of integrals and antiderivatives. The University of Arizona was at the forefront of calculus reform so we’re using a reform textbook. Hughes-Hallett and McCallum are both in the department so if I want to complain about the text I know where to go.

I’m finding it difficult to decide on the level of rigour that I want to provide. I didn’t even write down the epsilon-delta definition of limit (my exact words were “you can read the definition of limit in the text but I wouldn’t recommend it”). But the other day we calculated the derivative of x^2 and x^3 using the definition of derivative and then said that `obviously’ …

Then I said: “in fact this is true for all real n”. After that we did some examples and class ended. I’m not too happy with this way of teaching – there is no reason for the students to believe me.

I was thinking about how to prove this: for positive n it’s true by the binomial theorem (I don’t know if my students know the binomial theorem though), for negative n you can prove it by the product/quotient rule, for fractions you can prove it with implicit differentiation, then it follows for real exponents by continuity of something (and the density of the rationals in the reals).

But a better proof (for positive x anyway) is by the definition/fact that . Then .

But then how do you define e and ln anyway? The `best’ way is to wait until you have integrals and define and as the inverse of this function. But that’s not a reasonable idea for people that haven’t seen calculus before. Also I’m not so convinced that the logical way to teach things is better than the way people already know from high school or whatever – last semester I taught trigonometry using the unit circle and it just confused everybody.

So: how much rigour should be in a calculus course for the general population?

We’re trying to model interference on a communication channel. The line noise is normal with mean 0 and variance so has probability density function (pdf)

We have a random variable X that spits outs -1 or 1 with equal probability (this is a model for the coded data that we want to transmit). This has pdf .

Then the random variable that we get at the other end has p.d.f. given by the convolution . By doing this integral using properties of the delta function/distribution/measure we get . This gives quite nice pictures

which I guess show something important engineeringly: If you can get your line noise down to where the bumps don’t overlap then it’ll be pretty easy to figure out whether a bit was meant to be 1 or -1.

But I don’t really understand the maths. I guess I can accept the facts about pdfs but I’m kind of confused about continuous v discrete random variables. Should I always think of discrete random variables as being delta functiony things when I want to work with them?

MATH518 – Integral Lattices. Superfast lectures about things I’ve never heard of.

MATH536B – Algebraic Geometry. Superfast lectures that started by saying “Everything we said about varieties last semester is true about schemes, except when it’s not. Let’s keep going”.

ECE637 – Channel Coding. This is an engineering course. We’ve already used the fact that 10^3=2^10 as well as used the word ‘finite’ to mean non-zero.

MATH599 – Independent study. Continuing research from last semester. We’re stuck on a lemma from the paper whose entire proof consists of the word `straightforward’.

Teaching – Calc I. This is one hour a day, 5 days a week. At least most of the homework is online (webassign) so it’s less work for me.

I should give a talk on something at the grad colloquium this semester. I don’t really think the stuff I’m researching/learning is very interesting so I’d like to talk about something easier that I can really explain in 50 minutes. But I don’t know what to talk about. Any suggestions? These are some of my ideas:

• Why you can’t integrate e^{x^2} in terms of elementary functions (differential Galois theory?)
• mathematics of sport (mainly this would be cricket, which is a bit of a problem)
• groebner bases (+ geometrical theorem proving?)
• combinatorial game theory
• something in logic/set theory

Results are finalised. Will be released to students tomorrow.

5 Bs, 15 Cs, 6 Ds, 7 Es, 2 Ws

The precalc final exam grading took about six hours yesterday. I graded a question about a rational function that had a slant asymptote.

More interesting were the statistics that we got in a big spreadsheet later that night. The overall average on the final was 110/200. To get a D in this course you need to average 60% overall. Let it never be said that the UofA math department doesn’t have high standards.

My section averaged 106/200, but casting around for something to blame other than myself I came up with the idea that other people just force their poorly performing students to drop. I only had 2 students drop out of the original 35 while other sections had much larger numbers leave the course. Taking the final number of students enrolled as a proxy for number of students who dropped (this isn’t perfect but it’s kind of close) I did some regression:

There’s definitely some kind of correlation there: what does R^2=0.06 mean?

Anyway, if I put x=33 into this equation I get about 108.5. So I’ve explained away 1.5 points from the average from the fact that I didn’t force many people to drop. I’ve got ideas for the other few points so I feel happy that I did an average job.

The kind of things I’ve been doing lately:
- Project on ranks of quadratic twists of elliptic curves over function fields.
- My talk and my coauthor’s talk for above project.
- Paper for algebraic geometry class on Groebner bases.

What do people prefer for a projected talk in a fairly dark room?

Black on White,
White on black, or
Light grey on black.

You can download a zip file with all three versions here. But of course bear in mind that you are looking at it on a computer screen which is not what the question is asking about.

Things left to do:
- Teach 2 classes + 3 review classes.
- Grade 1 test, 1 lab, 1 quiz, and about 5 homeworks (I’m a little behind…).
- Proctor Invigilate and grade mark final exam for precalc high school maths taught at the university level.
- Attend two algebraic geometry and three Tate’s thesis classes.
- Finish writeup of research project on ranks of quadratic twists of elliptic curves over .
- Write talk on number theory in function fields for above project.
- Write algebraic geometry expository paper on Groebner bases.
- Do some algebraic geometry homework maybe?

It should all be done by about Dec 17 and then I’m leaving to fly back home for Christmas on the 20th.

My talk today was pretty well received. Talk linked as well as some calculations in Sage. If you saw my honours talk at UQ you might recognize some of the slides: this talk is better than that one though.