Mathematical papers are available as PDF files (Japanese only). Contact me 🙂

14. NTRU 暗号の例, 2019/11/24

Example of NTRU cipher

Construct an NTRU cipher on the factor ring \( R = \mathbb{Z}[X]/(X^3-1) \).

13. 等質正則開凸錐と単位元を持つクラン, 2011/01/20

Homogeneous regular open convex cones and clans with unit element

The introduction to my master’s thesis is available.

Abstract: According to Vinberg’s theory, there is a one-to-one correspondence between homogeneous cones and clans with a unit element. In this thesis, we reconstruct this correspondence between homogeneous cones and clans, making it more transparent. In particular, we provide a more direct approach, different from Vinberg’s original paper, to the construction of homogeneous cones from clans with a unit element.

12. 等質正則開凸錐と単位元を持つクラン (スライド版), 2011/02/09

Homogeneous regular open convex cones and clans with unit element (slide ver.)

I made slides of my master’s thesis for presentation.

11. 正則開凸錘の幾何概説, 2011/01/20

Geometry of regular open convex cones

This article outlines the general theory of (regular) open convex cones. For more details, please refer to other references.

10. Helmholtz の法則, 2009/07/15

Helmholtz’s law

This explains Helmholtz’s first to third laws. This is part of a report assignment.

9. 映画 “Good Will Hunting” に登場する数学の問題, 2009/04/19

The Good Will Hunting Problem

Here is the solution to a math problem from one of my favorite movies, Good Will Hunting.

8. \( SU(2) \) と \( SO(3) \) の有限次元既約表現の分類, 2009/01/21

Representations of \( SU(2) \) and \( SO(3) \)

The special unitary group of degree 2, \( SU(2) \), and the special orthogonal group of degree 3, \( SO(3) \) , are compact Lie groups. Due to the complete reducibility of representations, any finite-dimensional representation can be decomposed into a direct sum of irreducible representations. In this context, we will explicitly construct and classify the finite-dimensional irreducible representations of \( SU(2) \) and \( SO(3) \). Furthermore, we will show that the irreducible representations of \( SO(3) \) can be reduced to the irreducible representations of \( SU(2) \) via the double covering map from \( SU(2) \) to \( SO(3) \). The discussion will be based on differential representations derived from the representations of the Lie algebra.

7. \( U(1) \) の有限次元複素表現, 2009/01/05

Finite Dimensional Complex Representations of \( U(1) \)

Determine the irreducible representations of \( U(1) \), one of the simplest examples of a Lie group, and show that any finite-dimensional complex representation of \( U(1) \) can be decomposed as a direct sum of its irreducible representations. The key concepts are “compact topological group” and “abelian group.”

6. \( SU(2) \) と \( SO(3) \) の関係, 2008/08/22

Relationship between \( SU(2) \) and \( SO(3) \)

We will prove the well-known theorem in the representation theory of the rotation group that states the adjoint representation of \( SU(2) \) is a surjective homomorphism from \( SU(2) \) to \( SO(3) \), which is a 2-to-1 mapping. While this proof can be done more elegantly using somewhat advanced mathematics, it is also possible to prove it using only basic knowledge of linear algebra and group theory. Therefore, we have chosen to adopt the latter approach here.

5. Gelfand Pairs and Beyond – Exercises, 2008/05/10

This is the answer to the report problem given in Professor van Dijk’s lecture on Glfand Pairs. I have not solved all the problems. I realized that it would work to show that \( (SU(2), SO(2)) \) is a compact Gelfand pair by decomposing it into \( SU(2) = SO(2)TSO(2) \) using the maximal torus \( T \) of \( SU(2) \), and the professor praised me by saying “Good idea!”

4. Achilles と亀のパラドックス, 2007/11/25

Achilles and the Tortoise

There are several Zeno’s paradoxes, among which the famous story of Achilles and the tortoise is explained mathematically.

3. 分離公理, 2007/08/09

Separation Axioms

Here, we will discuss separation axioms that illustrate the difference between metric spaces and general topological spaces. Additionally, we will introduce Urysohn’s metrization theorem, which provides conditions under which a general topological space can become a metric space.

2. Fibonacci 数列と黄金比, 2007/04/06

Fibonacci Numbers and The Golden Ratio

The relationship between the Fibonacci sequence and the golden ratio is briefly explained.

1. Lotka-Volterra 捕食系の改良, 2007/02/08

Improvement of The Lotka-Volterra Model

I made slides of my graduation thesis for presentation.

Abstract: The Lotka-Volterra model expresses relations between prey and predator, however it is unreal. One of the unrealistic hypotheses for the model is that prey increase infinitely when predator is none. In order to make the model be more realistic we need to improve it as follows: the per capita
growth rate depends on prey and predator, respectively. Thus, we will make more realistic model which expresses relations between prey and predator, and analyze its behavior.