## PUBLICATIONS

### MI Preprint Series

Title：Matroid intersection with priority constraints

Author : Naoyuki Kamiyama

Abstract. In this paper, we consider the following variant of the matroid intersection problem. We are given two matroids M1 and M2 on the same ground set S and a subset of A of S. Our goal is to find a common independent set I of M1 and M2 such that |I \cap A| is maximum among all common independent sets of M1 and M2 and such that (secondly) |I| is maximum among all common independent sets of M1 and M2 satisfying the first condition. This problem can be solved by reducing it to the weighted matroid intersection problem. In this paper, we consider the following question: Is reduction to the weighted matroid intersection is inevitable? We prove that our problem can be solved by using a Dulmage-Mendelsohn decomposition without reduction to the weighted matroid intersection problem.

Journal of the Operations Research Society of Japan

#### 2011-16

Title：A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

Author : Yu Kawakami

Abstract. We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. Moreover we give the two applications of this theorem, the first one is to show an analogue of the Ahlfors islands theorem for it and the second one is to give a simple proof of the classification of complete nonsingular flat surfaces in the hyperbolic three-space.

#### 2011-15

Title：A generalization of restricted isometry property and applications to compressed sensing

Author : Hiroshi Inoue

Abstract. This paper introduces a new general theory of compressed sensing. We give a natural generalization of restricted isometry property called weak RIP. Though our approach is based on the Candes ideas, we consider that the proposed results are more useful for real data anaysis. In this note, we establish our new results about the accuracy of the reconstruction from undersampling measurements where are possible to improve estimation in various situations that we have knowledge.

#### 2011-14

Title：Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

Author : Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura & Yasuhiro Ohta

Abstract. We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.

#### 2011-13

Title：Semi-discrete finite difference multiscale scheme for a concrete corrosion model: approximation estimates and convergence

Author : Vladimír Chalupecký ＆ Adrian Muntean

Abstract. We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution.

#### 2011-12(Pulished)

Title：Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a parallel flow

Author : Yoshiyuki Kagei

Abstract. The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of ${\bf R}^2$. It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a $1$-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura-Nishida energy method.

Arch. Rational Mech. Anal. vol. 205 (2012), no. 2, pp. 585--650.

#### 2011-11

Title：An explicit formula for the discrete power function associated with circle patterns of Schramm type

Author : Hisashi Ando, Mike Hay, Kenji Kajiwara & Tetsu Masuda

Abstract. We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τ functions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface.

#### 2011-10

Title：On projective space bundle with nef normalized tautological line bundle

Author : Yasutake Kazunori

Abstract. In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef.

As an application, we get a characterization of abelian varieties up to finite etale covering.

#### 2011-9

Title：Recursive structures in the multispecies TASEP

Author : Chikashi Arita, Arvind Ayyer, Kirone Mallick & Sylvain Prolhac

Abstract. We consider a multi-species generalization of the totally asymmetric simple exclusion process (TASEP) with the simple hopping rule: for x and yth-class particles (x<y), the transition x y --> y x occurs with a rate independent from the values x and y. P. A. Ferrari and J. Martin (2007) obtained the stationary state of this model thanks to a combinatorial algorithm, which was subsequently interpreted as a matrix product representation by Evans et al. (2009). This ‘matrix ansatz’ shows that the stationary state of the multi-species TASEP with N classes of particles (N-TASEP) can be constructed algebraically by the action of an operator on the (N−1)-TASEP stationary state. Besides, Arita et al. analyzed the spectral structure of the Markov matrix: they showed that the set of eigenvalues of the N-TASEP contains those of the (N−1)-TASEP and that the various spectral inclusions can be encoded in a hierarchical set-theoretic structure known as the Hasse diagram. Inspired by these works, we define nontrivial operators that allow us to construct eigenvectors of the N-TASEP by lifting the eigenvectors of the (N−1)-TASEP. This goal is achieved by generalizing the matrix product representation and the Ferrari-Martin algorithm. In particular, we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines Markov matrices of different values of N.

#### 2011-8 (Pulished)

Title：Decay properties of solutions to the linearized compressible Navier-Stokes equation around time-periodic parallel flow

Author : Jan Brezina & Yoshiyuki Kagei

Abstract. Decay estimates on solutions to the linearized compressible Navier-Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in $L^2$ norm as an $n-1$ dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an $n-1$ dimensional linear heat equation with a convective term multiplied by time-periodic function.

Mathematical Models and Methods in Applied Sciences

#### 2011-7

Title：Group variable selection via relevance vector machine

Author : Shohei Tateishi, Chiaki Kinjo & Sadanori Konishi

Abstract. We consider the problem of variable selection in the case that explanatory variables have some groups. We proposed the extension of relevance vector machine (Tipping, 2001) for variable selection at a group level. In order to estimate a model, we derive a new estimation algorithm along with traditional relevance vector machine. Simulation results demonstrate that our methodology performs well in various situations.

#### 2011-6

Title：Predictive information criterion for nonlinear regression model based on basis expansion methods

Author : Daeju Kim & Sadanori Konishi

Abstract. Nonlinear regression modeling based on basis expansions has been widely used to explore data with complex structure. A crucial issue in nonlinear regression model is the choice of adjusted parameters including hyper-parameters for prior distribution and the number of basis functions. The selection of these parameters can be viewed as a model selection and evaluation problem. We derive an information criterion for the Bayesian predictive distribution in the case of both of regression coefficient and variance are unknown. Our proposed method make a selection of the appropriate value of hyper-parameters and the number of basis functions. Real data and simulation data analysis show that our proposed modeling strategy performs well in various situations.

#### 2011-5

Title：Implications in Compressed Sensing and the Restricted Isometry Property

Author : Hiroshi Inoue

Abstract. We give a simple idea for verifying the Restricted Isometry Property(Candes, 2008). Although our approach is based on the Candes proofs, the proposed results are more flexible and meaningful than those of Candes. In this note, we establish our new results about the accuracy of the reconstruction from undersampled measurements which are possible to improve estimation depending on the situation.

#### 2011-4

Title：Nonlinear regression modeling via Compressed Sensing

Author : Hiroshi Inoue, Shohei Tateishi & Sadanori Konishi

Abstract. We consider the problem of constructing nonlinear regression models in case that the dimension of data is less than the number of basis functions.

We propose a procedure where the smooth curve are effectively estimated along with the technique of regularization method. We give a simple idea for applying the Restricted Isometry Property to curve fitting.

Simulation results and real data analysis demonstrate that our methodology performs well in various situations.

#### 2011-3

Title：Anderson-Darling test and the Malliavin calculus

Author : Takehiro Hirotsu, Hiroki Kondo, Shingo Saito, Takuya Sato, Tatsushi Tanaka & Setsuo Taniguchi

Abstract. The quadratic Wiener functional coming from the Anderson-Darling test statistic is investigated in the framework of the Malliavin calculus. The functional gives a completely new and important example of a quadratic Wiener functional.

#### 2011-2

Title：Asymptotic tail dependence of the normal copula

Author : Hiroki Kondo, Shingo Saito & Setsuo Taniguchi

Abstract. Copulas have lately attracted much attention as a tool for dealing with multiple risks that cannot be considered independent.

The normal copula, widely used in practice, is known to have the same tail dependence parameter as the product copula.

The present paper brings into question the common interpretation of this fact as evidence that the normal copula lacks tail dependence, both by providing numerical examples and by mathematically determining the asymptotic behaviour of the tail dependence.

#### 2011-1

Title：Singular electromagnetic modes in an anisotropic medium

Author : Yasuhide Fukumoto & Alexander B. Samokhin

Abstract. We construct the singular mode corresponding to a spatial essential spectrum of the integral operator for the scattering of the electromagnetic waves by a three-dimensional body of finite size with inhomogeneous and anisotropic dielectric permittivity tensor. The permittivity tensor field is assumed to be Hölder continuous throughout the whole space. The singular volume integral equation, transformed from the Maxwell equations, makes it feasible to deduce explicit form of both the continuous essential spectrum and the corresponding singular modes. The obtained singular mode is a natural extension of the previously obtained one for the isotropic case and is applicable to a much wider class of dielectric scattering bodies. A discussion is made of possibility for realizability of the electromagnetic waves, with finite energy, concentrated at a point in the body.

## 2011-17