Math 197 - Discrete and Ultradiscrete Systems (AY08-09, 2nd Sem)

-from differential equations to difference equations, from difference equations to cellular automata-

Preface: It is said that the research of pure mathematics in modern times is mainly categorized into 3 fields: "Algebra", "Geometry" and "Analysis". "Algebra" and "Geometry" have carved out a long history, whereas not until the 17th century did a newcomer "Analysis" appear in the world of mathematics. Afterwards it took quite a long time for "Analysis" to make mature relationships with senior members: "Algebra" and "Geometry".

"Algebra" and "Geometry" have dealt in the finite; on the other hand "Analysis" has dealt in the infinite (the limit). Here is another field, "Discrete Mathematics" which flourished in the latter half of the 20th century. It deals in finite but incredibly big number. For example, when we consider the traveling salesman problem: given the number of cities and the costs of traveling from any city to any other city, what is the least-cost round-trip route that visits each city exactly once and then returns to the starting city?, even if the number of cities n is finite and small enough, the number of all round-trip routes (n-1)!/2 can be finite but incredibly big number. Computers make it possible to deal in this finite but incredibly big number.

Now, upon the 21st century, "Discrete Mathematics" ( "Mathematics with Computer" ) is in the process of becoming the 4th biggest field of mathematics, which is as big as "Algebra", "Geometry" and "Analysis". The latest newcomer "Discrete Mathematics" to modern mathematics is now in the process of making mature relationships with senior members: "Algebra", "Geometry" and "Analysis", just as "Analysis" followed before.

The relationship between "Analysis" and "Computer" is already well-known. But it does not mean the birth of a really new field of mathematics. It means that "Mathematics with Computer" is nothing but a complement to old mathematics-"Analysis", as long as we use computer as a tool of finding approximate solution of the (system of) differential equations.

Now what I am saying is that both "Analysis" and "Discrete Mathematics" are connecting mutually through a non-analytical limit named the 'ultradiscrete limit'. In this course, we introduce how to translate models described by differential equations and difference equations into models described by cellular automata, conversely, how to translate models described by cellular automata into models described by differential equations and difference equations.
Then we seek for the similarities between these continuous models and discrete models.