Current Research

Quadratic and higher-degree forms，cubature formulae, and optimal experimental designs

Outline
I am widely interested in mathematical objects such as sets, configurations, and geometric figures with "regularity" and "symmetry" in various meanings. For example, the regular icosahedron is, with no doubt, beautiful and even perfect, but it may not be easy to explain why we feel so.

The aim of my research is to answer the question "What's symmetry?" in various ways -- algebraically, analytically, combinatorially, and statistically.

Research Subject
1. Hilbert Identity
Given natural numbers n and r, Waring's problem asks whether the number n can be represented as a sum of the r-th powers of integers. The problem goes back to Edward Waring in 1770, as a natural generalization of the Lagrange's four-square theorem. After many efforts by hundreds of mathematicians such as Liouville (1859), Hurwitz (1908), and Schur (1909), Waring's problem was finally and completely solved by David Hilbert in 1909.

A key step in Hilbert's proof was to show the existence of a certain algebraic identity, called a Hilbert identity. Bruce Reznick is very important for having extensively discussed aesthetics beauty of Hilbert identities and laid the foundation for this topic, as a branch of the area of higher-degree forms. After his work, much progress has been made in this area, but there still remain many interesting open questions to be solved, such as the explicit construction of Hilbert identities and the distribution of the positive and negative coefficients of 2r-th powers required for representing the r-th power of x_1²+…+x_n².

I study the construction of Hilbert identities by using the orbits of finite reflection groups and the theory of invariant harmonic polynomials. I am also considering an alternative proof of Hilbert's theorem. To attack these questions, one of the useful tools is a cubature formula in numerical analysis.


2. Cubature formula
A cubature formula (CF), or a Euclidean design (ED), is an integration formula which approxiamtes the integral of a function by a weighted sum of the function values at finitely many specified points in the domain. To find cubature with few points has been considered mainly by Russian mathematicians in analysis, such as V.A. Ditkin and S.L. Sobolev, in connection with the theory of common zeros of (multivariate) orthogonal polynomials. Independent of this line of research in analysis, algebraic-combinators and geometers have also been interested in a cubature formula which is minimal with respect to the Fisher-type bound or Moller bound, because algebraic structures such as association schemes and coherent configurations are naturally associated with such formulas.

I study the theory of cubature formulae with few, albeit not necessarily fewest, points with particular emphasis on the existence as well as the characterization of such cubature.


3. Combinatorial Design
In the mid-19^th century Jakob Steiner posed the existence problem of a certain combinatorial configuration, called a t-design. From the viewpoint of algebraic combinatorics, this can be reduced to the problem of extracting good group-orbits with regularity. Thousands of publications have been devoted to the existence as well as the construction of 2-designs: Hefter (1891), Bose (1939). Remarkably, Richard M. Wilson (1975) proved that there is a 2-design for "almost" all possible parameters. For general t-designs there also have been many efforts to prove such a general existence theorem, however, most of the challenges seem not to be fully successful.

With these circumstances in mind, I study the construction of general t-designs by using the parallelisms and consider a generalization of Köhler's theory (1978/1980) which was originally designed as a construction of quadruple systems.


4. Optimal Experimental Design

(UNDER CONSTRUCTION)


5. Isometric Embedding


6. Applications
(a) Optical Code-Division Multiple-Access (CDMA)
In 1994 Kenichi Kitayama proposed a novel type of optical CDMA, called space CDMA, for parallel transmission of 2-dimensional images through a multicore fiber. In space CDMA an optical orthogonal signature pattern code (OOSPC) is required for high efficiency of communications. Most of the previous works involve experimental results of the hardware implementation of space CDMA systems and only a few pablications have been devoted to the theoretic aspects of OOSPC.

Now I study the construction of OOSPC by using the relationship between OOSPC and t-designs with particular algebraic structure. This is a joint research project with Kenichi Kitayama (Osaka Univ.) and Moriya Nakamura (NICT).

(b) Fluid Engneering

(UNDER CONSTRUCTION)