Color storm in the Cubic Mandelbrot set.
Today’s post is about ring species and a useful discovery that Occupy Math and his collaborators found while trying to understand them. A ring species is spread out around a major barrier that it cannot cross, so that the members of the species are in a long, thin ring-shaped domain. Examples of ring species include larus gull populations around the north polar ocean or greenish warblers spread around the Himalayan massif. A picture of the greenish warbler’s distribution appears at the top of the post. The funny thing about ring species is that they are arguably one or several species.
The biological species concept says that a group that form a species must be able to breed with one another. In a ring species, adjacent groups in the ring can breed, but on the far side of the ring where the groups meet up again, they cannot. Mostly this shows that the biological species concept is a little too simple. We thought that wolves and coyotes were different species, for example, but there is a difference between “cannot breed” and “choose not to breed”. The hybrid coywolves show that coyotes and wolves can breed — when humans thin out the wolf population to where they cannot find mates that are wolves. Below the fold, we will get to how this led to mathematical research.
A cubic Mandelcloud!
This post contains a fairly detailed description of Occupy Math’s new book on evolutionary computation, A Course in Evolutionary Computation. The basic notion of evolutionary computation is to create a computer program that implements a simplified version of evolution, where the evolving “creatures” are solutions to a problem. The technique is most useful when you do not understand the problem too well, but can recognize good solutions. Evolutionary computation has been mentioned in number of posts like the one on finding rules to make cavern maps and even evolving fractals.
Figuring out how and why evolutionary computation works is at the core of Occupy Math’s research. Occupy Math’s last book on evolutionary computation, an introductory text titled Optimization and Modeling with Evolutionary Computation, was published in 1996 and a lot has changed in the field since then. For that matter, Occupy Math himself has learned a lot, and it is time for him to write a new introduction to the field. Occupy Math teamed up with his wife for this effort — and the production quality of the book is much higher because of this. An important point to make is what is new and not new in this publication; the topics from the original book have been both revised and redivided so that they are scattered about in the new book. There is relatively little in common between the two books — the new book is not a second edition of the original one. This post will go through the topics in the book and explain a good deal about evolutionary computation. The book is available from Amazon:
From the cubic Mandelbrot set: Shamrock-n-roll.
Today’s post explores one of several sorts of number sentence puzzles; an example appears at the top of the post. Occupy Math will do posts on other sorts of puzzles later. The puzzle is the line of symbols and the answer is given below it (since this post is for parents and teachers, we will give answers with the puzzles). Each symbol represents a digit; when the symbols are the same, the digits are supposed to be the same. This post contains examples of such puzzles. It also explains the technique used to generate them, a technique which is really stupid in one sense, but also involves some clever math. Notice that the example puzzle uses seven sixes — the clever part of the problem generator ensures lots of repeated digits, which makes the problem both a bit easier and more interesting.
One of the big problems that Occupy Math faces is that the high school students coming into his first year classes do not really know the math that their transcripts say they do. This problem was covered in the earlier post School Math: Epic Fail. More recently, a report found that the percentage of sixth graders meeting the provincial math standards has been declining for a decade. The province is moving to address this problem by requiring teacher candidates to pass an exam in math skills and pedagogy, which is causing concern and protest. The teachers’ union is against this test but has not proposed another plausible way to reverse the decline in math scores. This post is a discussion of the issues surrounding this situation.
A conjugate Newton’s method fractal, a fairy river.
A problem factory is a set of mathematical principles that leads to a large, often infinite, collection of problems. Typically only a finite number of these problems are reasonable to assign to students, but a good problem factory will generate a large number of reasonable problems. This week we look into the question of the pattern of numbers of neighbors in a contact network. The picture at the top of a post represents a contact network with 32 people (the dots) each of which is in contact with four other people, as shown by the lines in the network.
The problems we will look at in this post take the following form. Given a sequence of numbers, can those numbers be the numbers of neighbors in a contact network? The picture at the top of post shows that 3,3,3,…,3 (32 threes) can be the number of neighbors in a contact network — the picture shows this. We call these sequences the contact numbers of the network. These questions make a good problem factory because, although there are many sequences that are the numbers of neighbors in a contact network, there are also many sequences that are not, making the questions challenging and real.
A generalized Julia set with three complex parameters. And a lot of colors!