Research on constructions of spherical designs and its approximate designs

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03736
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Aichi Prefectural University
• Principal Investigator: Hirao Masatake 愛知県立大学, 情報科学部, 准教授 (90624073)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 球面デザイン / 行列式点過程 / 球面上の点配置 / 調和指数デザイン / 球面上のcubature / 格子

Outline of Final Research Achievements

We have studied and considered spherical designs and their approximation designs. We have focused on determinantal point processes and shown that some stochastic point processes on the sphere can generate approximation designs from the viewpoint of frame potentials. We have also succeeded in further improving the construction methods using group orbits in spherical designs and in providing a characterization of harmonic-index designs of point configurations consisting of lattice shells.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

Bondarenkoらによる球面デザインの存在性に関するKorevaarとMeyersの予想の解決以降，球面デザインの研究はその存在問題から具体的に構成し，それを数値解析のみならず統計学や機械学習に応用することに変化してきている．本研究で示した(1) いくつかの球面上の確率点過程が近似的に球面デザインをなすこと，さらには(2) 球面デザインの構成の改良やデザインの持つ性質の調査に進展を与えたことは，今後の構成法解決へ向けて重要であり，今後の研究が広く期待されるものである．