Masatake Hirao

Cubature formulas in numerical analysis and Euclidean tight designs

In combinatorics, the concept of Euclidean t-design was first defined by Neumaier-Seidel (1988) [25], as a two-step generalization of the concept of spherical t-design. It is possible to regard Euclidean t-design as a special case of general cubature formulas in analysis. We point out that the works on cubature formulas by Moller and others (which were not well aware by combinatorialists), are very important for the study of Euclidean t-designs. In particular, they clarify the question of what is the right definition of tight Euclidean t-designs (tight t-designs on R^n and tight t-designs on p-concentric sphere). So, the first purpose of this paper is to tell combinatorialists, the importance of the theory on cubature formulas in analysis. At the same time we think that it is important for us to communicate our viewpoint of Euclidean t-designs to the analysts. The second purpose of this paper is to review the developments of the research on tight Euclidean t-designs. There are many new interesting examples and rich theories on tight Euclidean t-designs. We discuss the tight Euclidean t-designs in R^2 carefully, and we discuss what will be the next stage of the study on tight Euclidean t-designs. Also, we investigate the correspondence of the known examples of tight Euclidean t-designs with the Gaussian t-designs.

Some remarks on Euclidean tight designs

In this paper we present a new 4-dimensional tight Euclidean 5-design on 3 concentric spheres, together with a list of all known tight Euclidean designs which has been updated since the last survey paper by Bannai and Bannai (2009) [6]. We also examine whether each of all known tight Euclidean designs has the structure of a coherent configuration.

On Minimal Cubature Formulae of Small Degree for Spherically Symmetric Integrals

The aim of this paper is to develop the existence and nonexistence problem of a cubature formula of degree

A New Approach for the Existence Problem of Minimal Cubature Formulas Based on the Larman--Rogers--Seidel Theorem

4k+1

A Support System for Nursery Staff Shift Scheduling —A Case Study at a Nursery School

for a spherically symmetric integral which attains the Moller lower bound. For this purpose we bring together the theory of Euclidean design in combinatorics and that of reproducing kernels in numerical analysis. We show that if there exists a minimal formula of degree

QMC Designs and Determinantal Point Processes

4k+1

On minimal cubature formulae of odd degrees for circularly symmetric integrals

, then the nodes are distributed into

Constructions of Φ p -Optimal Rotatable Designs on the Ball

k+1

On the existence of minimum cubature formulas for Gaussian measure on R 2 of degree t supported by

concentric spheres including the origin, and the weights in the formula are a constant on each concentric sphere. We particularly focus on the cases of degrees 5 and 9. We show the equivalence between a minimal formula of degrees 5 and 4, and a spherical tight 5- and 4-design. We also prove that there exists no minimal formula of degree 9 for some classical weights such as the Gaussian weight on

[\frac{t}{4}]+1

\mathbb{R}^d