The Four Color Theorem is a well-known theorem stating that every map can be colored with four colors, and numerous propositions equivalent to this theorem are known. In this poster, we introduce a novel equivalent proposition by focusing on “face orientation” in triangulated planar graphs.

The presenter is a lawyer who is exploring the application of mathematics to legal issues. In legal cases, determining causality is crucial when individuals seek compensation for damages. For example, when a car accident occurs, the victim will experience many kinds of harm. In such a case, how much damage must the driver compensate for? To provide objective and quantitative criteria for evaluating causal relationships in legal contexts, we utilize the causal inference methodology developed by Judea Pearl. This approach describes legal causality using probability.

Elliptic curves are curves of genus one having a specified base point.. The set of points on an elliptic curve, including the point at infinity, forms an abelian group under a well-defined geometric operation. This group structure is central to the study of elliptic curves and has profound implications in various areas, including cryptography, number theory, and complex multiplication. In this talk, I will introduce the group structure on elliptic curves and explore some of their remarkable arithmetic properties.

This study proposes a novel Two-Way U-Net model for retinal vessel segmentation. The model processes global images and patch images through two separate pathways, enabling it to learn both broad contextual information and fine-grained features simultaneously. These two pathways are merged during the decoding phase to produce accurate segmentation results. Experiments on the DRIVE dataset demonstrated that the proposed model outperforms the conventional U-Net. This study demonstrates that integrating patch and global image information is highly effective for retinal vessel segmentation tasks.

Understanding underground subsurface characteristics is essential to reducing unexpected hazards during excavation. Electrical resistivity values are a crucial factor in predicting underground conditions, particularly in the presence of utilities. We employed a deep learning method, specifically Convolutional Neural Networks (CNN), alongside three machine learning methods—Support Vector Machine (SVM), XGBoost, and Random Forest—for comparison. All these methods demonstrated that the data contain sufficient information to accurately identify  underground condition.

This study addresses the stability of interconnected linear dynamical systems represented by an undirected graph with n(≥ 3) nodes where each node associates a linear dynamical system described by a state-space model. The interconnection between systems are governed by weights wij for edge connections between nodes and control parameters K_{i}. We analyze how wij and K_{i} affect the stability of individual systems and the overall network. A simulation example with n = 3 is provided to illustrate the findings.

We implement a C0-Discontinuous Galerkin Finite Element solver to compute numerical solutions for kinematically incompatible Von Kármán thin plate models. The mathematical formulation consists of a system of two coupled, nonlinear fourth-order differential equations, subject to homogeneous boundary conditions. Kinematical incompatibilities are modeled using Dirac delta functions applied to the membrane problem. A direct application of our solver is in the continuum modeling of graphene sheets in the presence of disclinations in the crystal lattice. Furthermore, we establish the existence and regularity of the solutions by adapting Ciarlet's proofs from the kinematically compatible case.

In this work, we consider the research that enhances the efficiency of the Finite Element Method (FEM) using GPU acceleration by converting our FEM library into JAX. JAX is a cutting-edge machine learning library aimed at high-performance numerical computing, particularly excelling in complex numerical tasks through automatic differentiation and GPU acceleration. While prior research on JAX-FEM has demonstrated significant improvements in FEM computation speed by leveraging JAX’s powerful computational capabilities, it has the drawback of being somewhat complex in its initial implementation and setup. In contrast, our library is relatively simple to implement and easy to use but is limited in performance when applied to large-scale problems or complex computations. This research aims to combine the strengths of both approaches. Numerical experiments are presented to evaluate the computational efficiency and performance improvements for our library in comparison to  JAX-FEM.

The austenite-to-martensite phase-transformation is a first-order diffusionless transition occurring typically in elastic crystals and observed also in the structures of viruses. It is characterized by an abrupt change of shape and symmetry of the underlying crystal lattice. In a temperature-induced transformation, austenite evolves into martensite through a series of energy releases and structural changes. This process leads to the formation of a highly inhomogeneous microstructure characterized by sharp angles and interfaces separating different variants of martensite. 

In this poster, I present a graph-theory approach to the statistical analysis of a conceptual fragmentation model for martensitic microstructures. The model involves fragmenting a unit square with a sequence of cuts parallel to predefined directions that represent the crystal class of symmetry, based on a time-space stochastic process.

Starting with a computer-generated microstructure, I construct the corresponding graph of adjacent grains. My objective is to compare and classify these graphs for different martensitic microstructures, with the goal of classifying them based on the different crystal symmetry of the martensitic transformation.   

This is work in progress with Prof. P. Cesana and Y. Mizoguchi (Kyushu University).

We investigate the rational homology of real toric manifolds associated with chordal nestohedra.

The bigraded Betti numbers explain the characters of polytope such as the number of vertices, the number of faces, etc. However, it is not enough to explain their own characters. Because of that, in general, we cannot distinguish polytopes by comparing only their bigraded Betti numbers.

In this talk, we discuss the special graded Betti numbers and show the cases that we can distinguish them by comparing their special graded Betti numbers. It is well known problem as combinatorial rigidity problem.

This study combines Hybrid Discontinuous Galerkin (HDG) and Finite Element Method (FEM) to solve elliptic equations. The analysis focuses on the stability conditions and the computation of stability, ensuring the stability of the proposed method.

Random Walk (RW) is well-known and plays an important role in probability theory. It is a type of movement in which the next position is determined randomly without reference to memory and has been actively studied for a long time. On the other hand, the asymptotic behavior of random walks with long range memory has been extensively studied over the last years. In particular, the so-called elephant random walk (ERW) has raised a considerable interest in recent year. ERW was introduced by G. Sch ̈utz and S. Trimper as an example of a non-Markovian stochastic process which has a complete memory of its entire history. The name ERW comes from the fact that elephants have good memories. Apparently, when they walk through the desert, they remember where water is. This ability of memory is essential for their survival.

It is known that ERW behavior can be divided into three regions depending on the memory parameter p. More specifically, the asymptotic behavior of ERW on the integer lattice Z changes at the memory parameter p=3/4. The asymptotic behavior for p< 3/4 is diffusive, like that for RW. In the critical point p=3/4, the behavior is not exactly the same as in p< 3/4, but asymptotic normality has been studied in the same way. However, H. Gu´erin, L. Laulin and K. Raschel completely determined that, in the super-diffusive regime p> 3/4, the limiting distribution is not Gaussian. In many researches on one-dimensional ERW, the phase transition at the boundary p=3/4 is often discussed.

ERW is a field of probability theory in which a large number of papers have been published recently. Among the many topics, this time I focused on the infinite collision of two elephant random walks. I was inspired by R. Roy, M. Takei and H. Tanemura. They treated two independent elephants with same parameters and show that the result is that case p≤ 3/4 will be met an infinite number of times. In this time, the parameters of the two elephants are taken differently, but similar results are obtained. 

I investigate the space-time evolution of systems of charged particles under the influence of a fourth-order potential, which serves as a model for the dynamics of wedge disclinations. These topological asymmetries are observed in the crystal lattice of various systems, including metals, elastic crystals, and viruses, and were first predicted by Volterra in his renowned paper. My focus is on the configuration of disclination dipoles in singular limit regimes, such as annihilation and collision near the boundary of the domain. From a mathematical perspective, this configuration is described by a system of highly nonlinear ODEs, which I analyze through rescalings and normalization to establish analytical theorems and present numerical examples. The material of this poster is part of my Master's thesis under the supervision of Prof. Marco Morandotti (Politecnico di Torino, Italy) and in collaboration with Prof. Pierluigi Cesana (Kyushu University). I acknowledge the support of a JASSO scholarship offered within the Q-PELS Program.

This study compares finite element methods (FEM) and temporal integration techniques in the numerical analysis of heat equations. Heat equations describe temperature distribution over time, essential in various scientific and engineering fields. We evaluate the accuracy, stability, and computational efficiency of different FEM approaches against temporal integration methods in solving one- and two-dimensional heat conduction problems. The findings highlight optimal conditions for each method, offering insights into selecting the most suitable numerical techniques for specific heat equation applications. This research contributes to enhancing the design and optimization of thermal systems in engineering.