Martin’s Blog

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

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.

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.