# 20120217

## 九州大学 組合せ数学・数値解析学 合同ワークショップ †

**～ Combinatorics and Numerical Analysis Joint Workshop ～**

このワークショップは，

の合同で開催されます。

### Organizers †

Yoshihiro Mizoguchi (Kyushu University),

Tetsuji Taniguchi (Matsue College of Technology),

Tsuyoshi Miezaki (Oita National College of Technology),

Yoshitaka Watanabe (Kyushu University),

Daisuke Tagami (Kyushu University)

- Advisary: 坂内 英一（上海交通大学／九州大学）

Eiichi Bannai (Shanhai Jiao Tong University / Kyushu University)

### Supported by †

Global COE Program "Education and Research Hub for Mathematics-for-Industry"

### Date †

- 2012年 2月17日(金) 10:00-18:00

(February 17, 2012. 10:00-18:00)

### Location †

- 九州大学
西新プラザ 中会議室(2F)

(Meeting Room, Nishijin Plaza, Kyushu University)

### Program †

- PDF（日本語）
- PDF（English）
- 10:00-11:00, 吴耀琨（Yaokun Wu，上海交通大学）
- Title: Some combinatorial analysis of infinite matrix product

- 11:30-12:10, 小関 健太（国立情報学研究所）
- Title: On the Hamiltonicity of graphs on a surface

- 13:50-14:50, 陳小君（Xiaojun Chen，香港理工大学）
- Title: Computational Existence Proofs for Spherical
*t*-Designs

- Title: Computational Existence Proofs for Spherical
- 15:20-16:00, 木村 拓馬（佐世保高専），木下 武彦（RIMS），中尾 充宏（佐世保高専）
- Title: A numerical method to prove the existence of solutions for nonlinear parabolic problems

- 16:20-17:00, 田中 守（東北大理）
- Title: Higher eigenvalues of the Laplacian on a graph and partitions of the graph

- 17:20-18:00, 平坂 貢（釜山国立大学）
- Title: Characterization of
*p*-valenced association schemes

- Title: Characterization of

### Abstract †

#### 吴耀琨（上海交通大学） †

Yaokun Wu (Shanhai Jiao Tong University)

- Title: Some combinatorial analysis of infinite matrix product
- Abstract:

The infinite product of a nonnegative square matrix is well understood thanks to the Perron-Frobenius Theory. In many contexts, say inhomogeneous Markov chain or opinion dynamics, one needs to consider the infinite product of several nonnegative square matrices of the same size. This general problem is much more complicated and seems that there is not any systematic theory for the analysis of the relevant dynamical behavior yet.

In this talk, we will discuss some combinatorial results obtained by the speaker and others (to be named during the lecture) on the dynamical behavior of the infinite matrix product of a set of matrices (of some special forms).

#### 小関 健太（国立情報学研究所） †

Kenta Ozeki (National Institute of Informatics, Japan)

- Title: On the Hamiltonicity of graphs on a surface
- Abstract:

A cycle in a graph *G* is called Hamiltonian if it passes through all
vertices in *G*. In this talk, we will concentrate on a Hamiltonian cycle
in graphs on a Topological surface, for example, the sphere (the plane),
the projective plane, the torus, and so on. One of the most classical
result of this area is the one due to Tutte that stats that “every
4-connected plane graph has a Hamiltonian cycle”. I would like to
introduce some other results, some of which are obtained very recently. I
also mention the connection between “the toughness” and the
Hamiltonicity of graphs on a surface.

This is a joint work with K. Kawarabayashi (National Institute of Informatics, Japan).

#### 陳小君（香港理工大学） †

Xiaojun Chen (The Hong Kong Polytechnic University)

- Title: Computational Existence Proofs for Spherical
*t*-Designs - Abstract:

Spherical *t*-designs provide quadrature rules for the sphere which
are exact for polynomials up to degree *t*. In this talk, we
propose a computational algorithm based on interval arithmetic
which, for given *t*, upon successful completion will have proved
the existence of a *t*-design with
nodes and will have
computed narrow interval enclosures which are known to contain these
nodes with mathematical certainty. Since there is no theoretical
result which proves the existence of a *t*-design with
nodes for arbitrary *t*, our method contributes to the theory
because it was tested successfully for *t*＝ 1, 2, ..., 100, i.e.,
for all *t* considered so far. The *t*-design is usually not
unique; our method aims at finding a well-conditioned one. The
method relies on computing an interval enclosure for the zero of a
highly nonlinear system of dimension
. We therefore develop
several special approaches which allow us to use interval arithmetic
efficiently in this particular situation. The computations were all
done using the MATLAB toolbox INTLAB. At the end of this talk,
applications of well conditioned spherical designs for integration, interpolation and
regularized least squares approximations on the
two-sphere are discussed.

Joint work with Congpei An, Andreas Frommer, Bruno Lang, Ian Sloan and Womersley.

**References**

[1] C. An, X. Chen, I. H. Sloan and R. S. Womersley, Well conditioned
spherical designs for integration and interpolation on the
two-sphere, SIAM J. Numerical Analysis, 48(2010), 2135—2157.

[2] C. An, X. Chen, I. H. Sloan and R. S. Womersley,
Regularized least squares approximations on the sphere using spherical designs,
submitted to SIAM J. Numerical Analysis, under revision.

[3] X. Chen, A. Frommer and B. Lang, Computational existence proofs for
spherical *t*-designs, Numerische Mathematik, 117(2011), 289—305.

[4] X. Chen and R. S. Womersley, Existence of solutions to systems of
underdetermined equations and spherical designs, SIAM J. Numerical
Analysis, 44(2006), 2326—2341.

[5] X. Chen, R. S. Womersley and J. J. Ye,
Minimizing the condition number of a Gram matrix, SIAM J. Optimization, 21(2011), 127—148.

#### 木村 拓馬（佐世保工業高等専門学校），木下 武彦（京都大学数理解析研究所），中尾 充宏（佐世保工業高等専門学校） †

Takuma Kimura (Sasebo National College of Technology), Takehiko Kinoshita (RIMS, Kyoto University) and Mitsuhiro T. Nakao (Sasebo National College of Technology)

- Title: A numerical method to prove the existence of solutions for nonlinear parabolic problems
- Abstract:

We present numerical verification methods for parabolic problems. Our main result is a constructive a posteriori estimates of inverse operators for initial-boundary value problems in linear parabolic PDEs on a bounded domain.

The proposed a posteriori estimates is based on error analysis of the Galerkin approximation for boundary value problems in space direction and the piecewise linear interpolation for initial value problems in time. Applying the result, we can numerically prove the existence of solutions for nonlinear parabolic initial-boundary value problems. Some numerical results will be shown in the talk.

#### 田中 守（東北大学大学院 理学研究科） †

Mamoru Tanaka (Tohoku University)

- Title: Higher eigenvalues of the Laplacian on a graph and partitions of the graph
- Abstract:

We can regard the 2-nd eigenvalue of the Laplacian on a
connected finite graph as strength
of connection between two disjoint subgraphs in the graph. In this
talk, I will give a relation
between the *k*-th eigenvalue of the Laplacian on a connected finite
graph and the minimum among the
2-nd eigenvalues of the Laplacians on the subgraphs in a partition of the graph.

#### 平坂 貢（釜山国立大学） †

Mitsugu Hirasaka (Pusan National University)

- Title: Characterization of
*p*-valenced association schemes - Abstract:

Let
be an assocaition scheme
and *p* a prime.
We say that
is *p*-valenced if
is
a power of *p*
for each *i* with
where
is the constant
out-degree of the digraph
.
In this talk we show some conditions for a *p*-valenced association scheme to be
induced by a transitive permutation group.