What do mathematicians do all day? Part VI

Today’s post is another exposition of research, this time on a game that my colleague Joseph Alexander Brown created for teaching artificial intelligence. The game is about the foraging behavior of moose, very much simplified, and is a good example for demonstrating how to make a strategically interesting game. The basic idea is simple. We have two moose that can choose between three different fields where they might forage. The fields have plants in them that grow back after being eaten, fast at first and then slower as the plants get back to full size. Each morning, the moose each choose a field. If they choose different fields, then they get a score equal to the forage in that field. If they choose the same field, they trumpet and threaten and tear up the field a bit but get no forage. Sounds simple, but there are some subtleties.

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What do mathematicians do all day? Part VI

This post is the next in our series on what mathematicians do with their time. The other posts in the series are indexed at the end of this post. Since everyone knows about the teaching mathematicians do, the series focuses on research. Today we are going to look at the walking triangle representation for optimizing functions. If you have finished a calculus course, it’s likely you already have optimized a function; this means finding the place where the function is highest (maximizing) or lowest (minimizing). We call the high points and the low points the optima of the function. To demonstrate the walking triangles, we will be maximizing the function shown at the top of the post. This function is really easy to maximize — you just head uphill.

What Do Mathematicians Do All Day? Part V

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.

Announcing Occupy Math’s New Book

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:

Amazon USA

Amazon Canada

Finding a match for everyone

This Occupy Math presents a new puzzle — a solved version of the puzzle appears at the top of the post. This version of the puzzle has 8 pieces which you can cut out of heavy paper or just draw with a pencil.

The goal is to place the pieces to fill a 4×4 square so that every number is adjacent to the other copy of itself — this is the match for everyone feature. The numbers can be sideways or upside-down, we just created all the pictures with the numbers right side up to make them easy to read. The post also explains how to build these puzzles, including how to get huge numbers of additional puzzles once you have one of them. We call these things Orthogonal Layout Puzzles.

Can the pandemic restart itself?

Viruses are somewhere in the boundary between life and non-life. In order to reproduce, they have to hijack the cells of their host organisms. For pathogenic viruses like COVID-19, this means that the critical reproductive machinery the virus needs to survive is part of a hostile organism that is trying to kill it. The ideas about where viruses come from — on the scale of the evolution of life on earth — is fascinating, incomplete, and not a part of this post. This post looks at how the virus survives our highly evolved, complex immune system and what implications this has for our response — at the societal level — to the pandemic. The short version is that viruses evolve and that we have to head that off at the pass or pay a potentially astronomical price.

Asynchronous Teaching is Less Effective than Synchronous Teaching.

Occupy Math is writing this post during finals week at the end of his first semester of teaching remotely during the pandemic. Last summer, we were asked to decide if we would teach our classes synchronously or asynchronously. A synchronous class meets online, but at regular times. The lectures are recorded for students that cannot make class, but most of the students came to class. The virtual interface had a chat stream so that the lectures could be modified in response to student requests (“Could you explain that again?”) An asynchronous class has no meeting time. The lectures are recorded and placed on a server. Occupy Math tried one of each and will not willingly teach asynchronously again. He requested that the smaller course with better students be asynchronous and the results were terrible, even though the students were more advanced.

What do mathematicians do all day? Part IV

This post is an update to an earlier post. Occupy Math uses digital evolution as a research tool. This is one of several different computational intelligence techniques inspired by nature. Another one of these techniques is inspired by the complex and beautiful biology of the immune system. These techniques are called, collectively, artificial immune systems. The immune system is such a complex system that multiple algorithmic techniques have been inspired by it. One of them, danger theory, turns out to allow control over digital evolution. This post is about applying danger theory to the evolution of pretty pictures, called apoptotic cellular automata, that Occupy Math has posted about before.

Deploying vaccines: my research report.

Right now covid-19 tests are going to important people more than others — because those important people are scared and we do not really have a strategy. When vaccines become available, if we do not have a science-based strategy, the strategy may well turn out to be “Vaccinate the rich!”. This post takes you through the initial stages of a research project on making plans for vaccine deployment, including some speed-bumps we hit. It may give you a sense of how computational research goes and why it has a good chance of success.

In an earlier post, Occupy Math talked about vaccine deployment strategies. Occupy Math is now part of a deployment strategy project, funded through St. Francis Xavier University in Nova Scotia. The project is led by Professor James Hughes at StFX University. Professor Sheridan Houghten at Brock University is also collaborating. We are old friends and have been collaborating for years. This post is about some of the results we have gotten so far and how the results changed the project. The picture at the top is a bit whimsical as the research we are doing is in three rooms in our houses with our computers. Six rooms if you include students currently in or joining the project.

John Horton Conway, Requiescat in Pace

John Conway was a character, a genius, an eccentric, and one of the greatest mathematicians in history. He died on Saturday April 11th of the coronavirus at the age of 82. He was a professor at Princeton and worked in games, abstract algebra, combinatorics, the theory of computing, and many other fields. Occupy Math has attended lectures by this great man and will recount the experience with one of them in the post. Fair warning: Occupy Math is going to cover three of Professor Conway’s many achievements and they are pretty deep. To combat this, there are lots of pictures, some tales of Conway’s personal eccentricity, and the story of Conway’s part in solving a single problem that took over a century to complete. One of Occupy Math’s most useful ideas was based on one of Conway’s algorithms — used in a way that would probably have appalled the professor.