In the present paper, I review our recent two papers of the joint works with Atsushi Miyauchi (Tokyo Univ.), Yuuki Takai(KIT) and Yuichi Yoshida (NII). I mainly introduce the background of their papers and the fundamental notions for community detection of networks. First I review the notion of Laplacian and Cheegerʼs inequality for the usual undirected graph. After that, I introduce the definition of the (submodular) Laplacian for hypergraphs and the heat on them. Especially, I introduce several properties of the Laplacian and heat such as maximal monotonicity of the Laplacian and well-definedness of the heat and the Personalized PageRank respectively. Moreover, I introduce the application of the properties to the community detection on hypergraphs.

Plasma has many applications in a wide range of industries. The complete analysis is quite complicated because of multi-physics such as electromagnetics, fluid dynamics, chemical reactions and so on. So far, we have developed consistently mathematical models, numerical methods, coding, simulation, and comparison with experiments, according to the target equipment using plasma. This paper reports a part of our study: modeling and numerical methods for investigating a high-frequency plasma. In particular, modeling of scalar and vector potentials in electromagnetic is described. The scalar potential is formulated as an elliptic interface problem and solved by the hybridized discontinuous Galerkin method. The vector potential is formulated by harmonic time modeling, and a mixed finite element method using Nédélecʼs element is applied. The sputtering rate distribution obtained by our simulation is in good agreement with the experimental results. The simulation technique we developed can be used to design new equipment.