# Posts

Short essays, ideas, events, notes

### Euler's Formula in College Algebra

Once complex numbers are introduced in a College Algebra/PreCalculus course, why not discuss Euler’s formula? Especially the equation $e^{\pi i}+1=0$, that looks good on a T-shirt. A full proof is out of question, but the power series definitions of the exponential and trigonometric functions provide a walkable path up to the summit. With a computer algebra system it is easy to demonstrate how the approximations work, just by entering a few terms of the infinite sums.

### Talk on finite computational structures

Here are the slides for my talk “Finite Computational Structures and Implementations” for the The 4th International Symposium on Computing and Networking CANDAR’16 held in Hiroshima, Japan, November 22-25, 2016.

The photo was taken in the Higashi Hiroshima Arts and Culture Hall, the venue of the conference.

# Publications

Peer reviewed journal papers, book chapters, conference proceedings

### Enumerating Transformation Semigroups

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups.
Semigroup Forum (accepted)

### Computing Finite Semigroups

Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and 0-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the 𝒟-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.
Journal of Symbolic Computation (accepted)

# Teaching

Due to the rapid changes in our technological societies the aims of teaching and the teaching process itself need to be rethought again and again. The response is twofold: 1. fast moving, rapidly deployed courses and 2. focusing on core knowledge versus ephemeral ideas and technologies. The challenge is that these two requirements might be in conflict.

Currently I teach at Akita International University.

## Courses I designed

• Poetry of Programming - puzzle based introduction to functional programming (MAT245). Syllabus, tutorial notes draft.

• Mathematics for the digital world (MAT240 Mathematics behind the technological society). Syllabus

• Calculus (MAT250) Single variable calculus up to the Fundamental Theorem of Calculus. Syllabus

• College Algebra (MAT150) From set theory up to $e^{\pi i}+1=0$.

• Statistics (MAT200)

## Previous courses

• Social Web Analytics

• Computational Complexity

• Discrete Mathematics

• Differential Calculus

• Semigroup Theory, Representation Theory (graduate courses)

• Formal Languages and Automata

• Linear Algebra

• Programming (C#)

• Design and Analysis of Algorithms

• Operating Systems, Shell Programming

• $\LaTeX$

• supervising MSc projects in Computer Science

• Artificial Life (guest lecture)

• Programming (C, Java)