80s rubric

The last post was a monster post. This one is a bit shorter as it exam time at my university so I’m a bit busy. Students these days have it easy with regards to exams. They are given far too much information about how to get good marks. When I was a lad we had to guess what an examiner wanted. Just look at the rubric of this 1980s exam (click to enlarge): I just love “Full marks may be obtained for complete answers to about FIVE questions”! And this is not an isolated case, I have a stack of exam papers like...

How to get a good degree 3: How to become a straight-A student...

Usually I am against books offering general study advice, I favour those that focus on a particular subject. (Which is why I wrote How to Think Like a Mathematician. Mathematics students may also be interested in Lara Alcock’s How to Study for a Mathematics Degree.) I’ll make an exception for Cal Newport’s How to Become a Straight-A Student. Let’s get the bad stuff out of the way first. The subtitle is The Unconventional Strategies Real College Students Use to Score High While Studying Less which, along with the book’s other marketing, makes the scam-like sounding promise that you can do less work and get better grades. This can of course happen but does lend the book a feeling of too-goo-too-be-true. Furthermore, the book could do with some trimming of excess material though, at 216 pages, it is quite short for this type of book. The book could do without the regular mentions of partying and beer swigging but I suspect I’m not the target market for those bits. But leave aside those problems. Why would I recommend the book? Essentially, most of the advice is good. There is the standard stuff that all students know that they should do: get plenty of rest, eat properly, do your work in the morning between lectures to gain a sense of accomplishment. The non-standard stuff is good too. He doesn’t advocate lots of highlighter pens or even a highly detailed to do list. Instead he mostly focusses on the methods for efficient and effective learning. For example, on page 105, he talks of the Quiz-and-Recall method, [emphasis mine] Whether it’s philosophy or calculus, the most effective way to imprint a concept is to first review it and then try to explain it, unaided, in your own words....

Hilbert Hotel Video

Belated Happy New Year! Here’s another of those short educational TED videos. This time on the Hilbert Hotel. (A cartoon Hilbert does seem to make an appearance but it is hard to tell as he isn’t wearing the hat. Do you think Hilbert wore that hat just once and had the misfortune of having it appear in his most famous picture?) Anyhow: You can find out more at the TED Education page for the video. Oh, and this week is the last chance to win a signed copy of Simon Singh’s book on The Simpsons! See...

Maths predicts – this time movie success

In a previous post I talked about predictions using maths and Nates Silver‘s book on essentially that topic was one of my favourites books of last year. This next one is a bit of fun – predicting movie success. Film buffs will know William Goldman’s quote about making films “Nobody knows anything” which is taken to mean that no one can predict how a film will do at the box office. However, researchers have some good news. Mestyán, Yasseri and Kertész have published Early Prediction of Movie Box Office Success Based on Wikipedia Activity Big Data. As you can tell from the title the key is to use online data and activity. Their algorithm gives good predictive power up to a month before the film is released and hence will be of little use to Hollywood producers receiving pitches for new films. Of course, these are early days and even the Oxford Internet Institute news article uses the word predicts in quotes. Nonetheless the authors compare their results to those obtained using Twitter by other researchers and find it better. The paper is freely available and since the maths behind it is accessible to undergraduates it would be great for a student mini-project. (Talking of projects, over the summer I had an undergraduate studying symmetry matching and it has turned out very well so I’ll definitely be writing about that soon.) Photo attribution: Alex Eylar,...

What is the Best Proof of Cauchy’s Integral Theorem?...

Today’s post may look as though I’m going all Terry Tao on you with a long post with lots of mathematical symbols. It’s really about the learning and teaching of Cauchy’s integral theorem from undergraduate complex analysis, so isn’t for everyone. If it’s not your cup of tea/coffee, then pop over here for some entertainment. Cauchy’s Integral Theorem Cauchy’s Integral Theorem is one of the greatest theorems in mathematics. There are many ways of stating it. Here’s just one: Cauchy’s Integral Theorem: Let be a domain, and be a differentiable complex function. Let be a closed contour such that and its interior points are in . Then, . Here, contour means a piecewise smooth map . In my years lecturing Complex Analysis I have been searching for a good version and proof of the theorem. My definition of good is that the statement and proof should be short, clear and as applicable as possible so that I can maintain rigour when proving Cauchy’s Integral Formula and the major applications of complex analysis such as evaluating definite integrals. Many of the proofs in the literature are rather complicated and so time is lost in lectures proving lemmas that that are never needed again. Here’s a version which I think has a good balance between simplicity and applicability. I’ve highlighted the difference with the version above. Cauchy’s Integral Theorem (Simple version): Let be a domain, and be a differentiable complex function. Let be a simple closed contour made of a finite number of lines and arcs such that and its interior points are in . Then, . Here an important point is that the curve is simple, i.e., is injective except at the start and end points. This means that we have a Jordan curve and so the curve has well-defined interior and exterior and both are connected sets. With this version I believe one can prove all the major theorems in an introductory course. I would be interested to hear from anyone who knows a simpler proof or has some thoughts on this one. Proof of Simple Version of Cauchy’s Integral Theorem Let denote the interior of , i.e., points with non-zero winding number and for any contour let denote its image. First we need a lemma. Lemma Let be a simple closed contour made of a finite number of lines and arcs in the domain with . Let be a square in bounding and be analytic. Then for any there exists a subdivision of into a grid of squares so that for each square in the grid with there exists a such that for all Proof of Lemma The set up looks like the following. For a contradiction we will assume the statement is false. Let and divide into 4 equal-sized squares. At least one of these squares will not satisfy the required condition in the lemma. Let be such a square. Repeat the process to produce an infinite sequence of squares with .By the Nested Squares Lemma (which is just a generalization of the Nested Interval Theorem) there exists . As is differentiable there exists such that for . But as the size of the squares becomes arbitrarily small there must exist such that is contained in the disk . This is a contradiction. Main part of proof Given there exists a grid of squares covering . Let be the set of squares such that and let be the set of distinguished points in the lemma. Define by Then as is differentiable, is continuous (and hence integrable). Without loss of generality we can assume that is positively oriented. Let be the union of positively oriented contours giving the boundary of . Since is made of a finite number of lines and arcs will itself be the union of a finite number of lines and arcs. For such that , is just the boundary of a square. On we...

Estimating large numbers the Fermi way...

Here’s a really good introduction to estimating numbers and includes Enrico Fermi‘s famous “How many piano tuners are there in Chicago?” You can see more like this at...