## PUBLICATIONS

### MI Preprint Series

Title：On statistical aspects in calibrating a geometric skewed stable asset price model

Author : Hiroki Masuda

Abstract. Estimation of an asset price process under the physical measure can be regarded as the first step of the calibration problem, hence is of practical importance. In this article, supposing that a log-price process is expressed by a possibly skewed stable driven model and that a high-frequency dataset over a fixed period is available, we provide practical procedures of estimating the dominating parameters. Especially, the scale parameter may be time-varying and possibly random as long as it is independent of the driving skewed stable Levy process. By means of the scaling property and realized bipower variations, it is possible to estimate the index and positivity (skewness) parameters without specific knowledge of the scale process. When the target scale parameter is constant, our estimators are asymptotically normally distributed. When the scale is actually time-varying, we focus on estimation of the integrated scale, which is an analogue to the integrated volatility in the Brownian-semimartingale framework. In this case we show that estimation of the integrated scale exhibits a kind of asymptotic singularity with respect to the unknown index parameter, with specifying the slower rate of convergence than the constant-scale case.

#### 2009-36

Title：Exact simulation of finite variation tempered stable Ornstein-Uhlenbeck processes

Author : Reiichiro Kawai & Hiroki Masuda

Abstract. We develop an exact yet simple simulation algorithm for a wide class of Ornstein-Uhlenbeck processes with a tempered stable stationary distribution of finite variation. We derive the exact transition probability of tempered stable Ornstein-Uhlenbeck processes between consecutive times due to the homogeneous Markovian autoregressive structure. Random element involved can be divided into independent tempered stable and compound Poisson distributions, each of which can be generated in the exact sense with acceptance-rejection methods, respectively, with stable and gamma proposal distributions. Our algorithm proves useful for the simulations of bilateral tempered stable Ornstein-Uhlenbeck processes and normal tempered stable processes. The effectiveness of the proposed algorithm is discussed relative to the existing approximative method based on infinite series representation of sample paths.

#### 2009-35

Title：Finite element computation for scattering problems of micro-hologram using DtN map

Author : Yosuke Mizuyama. Takamasa Shinde, Masahisa Tabata and Daisuke Tagami

Abstract. Computational results are presented on micro-hologram diffraction for optical data storage using a finite element method. Retrieval of object light from a micro-hologram is formulated as an optical scattering problem in an infinite region. In order to overcome the difficulty of dealing with the infinite region a Dirichlet to Neumann (DtN) map is employed on an artificial boundary. By virtue of the DtN map reflection from the artificial boundary is effectively alleviated and non-reflecting boundary is obtained. Retrieval of the object light is computed for two different models.

#### 2009-34 (Published)

Title：Projective reduction of the discrete Painlev\'e system of type $(A_2 + A_1)^{(1)}$

Author : Kenji Kajiwara, Nobutaka Nakazono & Teruhisa Tsuda

Abstract. We consider the q-Painlev\'e III equation arising from the birational representation of the affine Weyl group of type $(A_2 + A_1)^{(1)}$. We study the reduction of the q-Painlev\'e III equation to the q-Painlev\'e II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

International Mathematics Research Notices

#### 2009-33 (Published)

Title：Queueing process with excluded-volume effect

Author : Chikashi Arita

Abstract. We introduce an extension of the M/M/1 queueing process with a spacial structure and excluded-volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We also consider a generalization of the model with backward hopping of particles allowed and an alternate joined system of the M/M/1 queueing process and the open TASEP.

Physical Review E

#### 2009-32

Title：Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type

Author : Olivier Pironneau & Masahisa Tabata

Abstract. A Galerkin-characteristics finite element scheme of lumped mass type is presented for the convection diffusion problems. Under the weakly acute triangulation hypothesis the scheme is proved to be unconditionally stable and convergent in the maximum-norm. Using freefem, we show 2D and 3D numerical examples, which reflect the robustness of the scheme and the theoretical convergence result.

#### 2009-31 (Published)

Title：Hecke’s zeros and higher depth determinants

Author : Masato Wakayama & Yoshinori Yamasaki

Abstract. We establish “higher depth” analogues of regularized determinants due to Milnor for the zeros of Hecke L-functions. This is an extension of the result of Deninger about the regularized determinant for the zeros of the Riemann zeta function.

Masato Wakayama and Yoshinori Yamasaki, “Higher regularizations for zeros of cuspidal automorphic $L$-functions of $\GL_d$”, J. Theor. Nombres Bordeaux, 23 (2011) No. 3, 751-767.

#### 2009-30(Pulished)

Title：On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis

Author : Yoshiyuki Kagei & Yasunori Maekawa

Abstract. This paper deals with large time behavior of solutions to a Keller-Segel system which possesses self-similar solutions. By taking into account the invariant properties of the equation with respect to a scaling and translations, we show that suitably shifted self-similar solutions give more precise asymptotic profiles of general solutions at large time. The convergence rate is also computed in details.

J. Differential Equations, vol. 253 (2012), no.11, pp. 2951--2992.

#### 2009-29(Pulished)

Title：Asymptotic behaviors of solutions to evolution equations in the presence of translation and scaling invariance

Author : Yoshiyuki Kagei & Yasunori Maekawa

Abstract. There are wide classes of nonlinear evolution equations which possess invariant properties with respect to a scaling and translations. If a solution is invariant under the scaling then it is called a self-similar solution, which is a candidate for the asymptotic profile of general solutions at large time. In this paper we establish an abstract framework to find more precise asymptotic profiles by shifting self-similar solutions suitably.

J. Functional Analysis, vol. 260 (2011), no. 10, pp. 3036--3096.

#### 2009-28

Title：Numerical simulation of fluid movement in an hourglass by an energy-stable finite element scheme

Author : Masahisa Tabata

Abstract. We simulate flow movement in an hourglass occupied by two fluids with surface tension on the interface and compare the difference of movements of fluids between the non-slip and slip boundary conditions and small and large coefficients of surface tension.

#### 2009-27 (Published)

Title：Value distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

Author : Yu Kawakami

Abstract. We give an effective estimate for the totally ramified value number of the hyperbolic Gauss maps of complete flat fronts in the hyperbolic three-space. As a corollary, we give the upper bound of the number of exceptional values of them for some topological cases. Moreover, we obtain some new examples for this class.

Houston Journal of Mathematics (Vol.38)

#### 2009-26

Title：Ramification of local fields and Fontaine’s property (Pm)

Author : Manabu Yoshida

Abstract. We prove that the ramification filtration of the absolute Galois group of a complete discrete valuation field with perfect residue field is characterized in terms of Fontaine's property (Pm).

#### 2009-25

Title：On very accurate enclosure of the optimal constant in the a priori error estimates for $H^2_0$-projection

Author : Takehiko Kinoshita & Mitsuhiro T. Nakao

Abstract. We present constructive a priori error estimates for $H^2_0$-projection into a space of polynomials on a one-dimensional interval. Here, ``constructive'' indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form. Using the properties of Legendre polynomials, we consider a method by which to determine these constants to be as small as possible. Using the proposed technique, the optimal constant could be enclosed in a very narrow interval with results verification. Furthermore, constructive error estimates for finite element $H^2_0$-projection in one dimension are presented. This type of estimates will play an important role in the numerical verification of solutions for nonlinear fourth-order elliptic problems as well as in the guaranteed a posteriori error analysis for the finite element method or the spectral method.

#### 2009-24 (Published)

Title：Recent progress in value distribution of the hyperbolic Gauss map

Author : Yu Kawakami

Abstract. We give a brief survey of our work on value distribution of the hyperbolic Gauss map. In particular, we define algebraic class for constant mean curvature one surfaces in the hyperbolic three-space and give a ramification estimate for the hyperbolic Gauss map of them. Moreover, we also give an effective estimate for the number of exceptional values of the hyperbolic Gauss maps of flat fronts in the hyperbolic three-space.

Riemann Surfaces, Harmonic Maps and Visualization

#### 2009-23

Title：Generation of ribbons, helicoids and complex scherk surface in laser-matter Interactions

Author : Stjepan Lugomer & Yasuhide Fukumoto

Abstract. Motivated by a diversity of the shape instability phenomena in condensed matter physics, we study formation of elastic ribbon structures and transformation into helicoidal structures. Using the multi-pulse laser-matter interaction with the Co-coated surface, a one-dimensional high-density vortex filament array has been created. Increasing the number of pulses, the oscillatory strain field causes the cascade of the shape transformations into structures of increasing topological complexity: vortex filaments into ribbons, into ribbon-helicoids and tubular-ribbon-helicoids, and then into short ribbon-structures with the complex Scherk surface being identified. Above a critical number of pulses, the system is catastrophically disintegrated into small-scale wrinkled and crumpled surfaces.

#### 2009-22 (Published)

Title：A remark on the global existence of a third order dispersive flow into locally Hermitian symmetric spaces

Author : Eiji Onodera

Abstract. We prove global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation under consideration generalizes two-sphere-valued completely integrable systems which model the motion of vortex filament. Unlike one-dimensional Schr\"odinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an energy which is not necessarily preserved in time but does not blow up in finite time.

Communications in Partial Differential Equations

#### 2009-21 (Published)

Title：Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation

Author : Hiroshi Yoshida, Yoshihiro Miwa & Masanobu Kaneko

Abstract. The development of a multicellular organism is marvelous. Starting from an egg, the organism becomes a set of cells comprising a variety of types to serve various functions. To obtain conditions for high cell-type diversity, we have constructed a model using a Lindenmayer system. Using symbolic computation, we have derived the explicit relationship underlying cell-type diversity under some constraints. The derived relationships exhibit elliptic curve- and Fibonacci number-related patterns. This is the first example of elliptic curves to appear in the biological phenomenon. A survey of the rational points and quadratic irrational numbers on the derived curves has revealed that the mature stage usually includes Fibonacci-related periodic and quasiperiodic structures. This result seems consistent with a coarse-grained view of mature human tissues, leading to the prediction that the structures in early development are related to elliptic curves.

Applicable Algebra in Engineering, Communication and Computing, Vol. 22 (2), pp. 147--164, 2011.

#### 2009-20 (Published)

Title：Semi-supervised logistic discrimination via regularized Gaussian basis expansions

Author : Shuichi Kawano & Sadanori Konishi

Abstract. The problem of constructing classification methods based on both classified and unclassified data sets is considered for analyzing data with complex structures. We introduce a semi-supervised logistic discriminant model with Gaussian basis expansions. Unknown parameters included in the logistic model are estimated by regularization method along with the technique of EM algorithm. For selection of adjusted parameters, we derive a model selection criterion from Bayesian viewpoints. Numerical studies are conducted to investigate the effectiveness of our proposed modeling procedures.

Communications in Statistics - Theory and Methods

#### 2009-19(Published)

Title：Sparse functional principal component analysis via regularized basis expansions and its application

Author : Mitsunori Kayano & Sadanori Konishi

Abstract. This paper introduces principal component analysis for multidimensional sparse functional data sets, utilizing Gaussian basis functions. Our multidimensional model is estimated by maximizing a penalized log-likelihood function, while previous mixed-type models were estimated by maximum likelihood methods for one-dimensional sparse functional data set. The penalized estimation performs well for our multidimensional model, while maximum likelihood methods yield unstable parameter estimates and some of the parameter estimates are often infinite. Numerical experiments are conducted to investigate the effectiveness of our method via the Gaussian bases for some types of missing data. The proposed method is applied to handwriting data, which consist of the XY coordinates values in handwritings.

Communication in Statistics - Simulation and Computation

#### 2009-18(Published)

Title：Local Instability of an Elliptical Flow Subjected to a Coriolis Force

Author : Me Me Naing & Yasuhide Fukumoto

Abstract. We make the local stability analysis of a rotating flow with circular or elliptically strained streamlines, whose rotation axis executes constant precessional motion about an axis perpendicular to itself, based on the WKB method. In the frame rotating with the precessional angular velocity, the basic flow is a steady velocity field linear in coordinates in an unbounded domain. For the case of slow precession, without strain, the growth rate takes the same value as that of Kerswell (1993) though the basic flow is different. We find that, in the absence of strain, the growth rate approaches the angular velocity of the basic flow as the precessional angular velocity is increased. The cooperative action of the weak Coriolis force and the elliptical straining field is investigated both numerically and analytically. An analysis of using the Mathieu method reveals that the elliptical instability is weakened by the precession, while the precessional instability is either enhanced or weakened depending on the orientation of the strain.

J. Phys. Soc. Jpn., Vol.78, No.12, p.124401

#### 2009-17 (Published)

Title：Non-linear algebraic differential equations satisfied by certain family of elliptic functions

Author : Masato Wakayama & Keitaro Yamamoto

Abstract. The paper [KW] studied a family of elliptic functions defined by certain $q$-series. This family, in particular, contains the Weierstrass $\wp$-function.

In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.

The Ramanujan Journal,February 2013, 30 (2), pg. 173-186

#### 2009-16 (Published)

Title：Spectrum in multi-species asymmetric simple exclusion process on a ring

Author : Chikashi Arita, Atsuo Kuniba, Kazumitsu Sakai & Tsuyoshi Sawabe

Abstract. The spectrum of Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar-Parisi-Zhang or Edwards-Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.

Journal of Physics A

#### 2009-15(Pulished)

Title：Large time behavior of the semigroup on ${L^p}$ spaces associated with the linearized compressible Navier-Stokes equation in a cylindrical domain

Author : Yuya Ishihara & Yoshiyuki Kagei

Abstract. Large time behavior of solutions to the linearized compressible Navier-Stokes equation around the motionless state in a cylindrical domain is investigated. The &{L^p}$ decay estimates of the associated semigroup are established for all 1< p < ∞. It is also shown that the time-asymptotic leading part of the semigroup is given by a one dimensional heat semigroup.

J. Differential Equations, vol. 248 (2010), no. 2, pp. 252--286.

#### 2009-14

Title：On Gaussian decay estimates of solutions to some linear elliptic equations and its applications

Author : Yasunori Maekawa

Abstract. In this article we establish pointwise decay estimates of solutions to some linear elliptic equations based on the Nash-Moser iteration arguments and the ODE method. As applications we obtain sharp Gaussian decay estimates for solutions to some nonlinear elliptic equations.

#### 2009-13

Title：A Practical Implementation of a Symbolic-Numeric Cylindrical Algebraic Decomposition for Quantifier Elimination

Author : H. Iwane, H. Yanami, H. Anai & K. Yokoyama

Abstract. Recently quantifier elimination (QE) has been of great interest in many fields of science and engineering. In this paper an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are proposed based on the authors’ previous work and our implementation of those is reported.

Based on analysing experimental performances, we are improving our design/synthesis of the SNCAD for its practical realization with existing efficient computational techniques and several newly introduced ones.

The practicality of the SNCAD is now examined by experimentation on real computer, which also reveals the quality of the implementation.

#### 2009-12

Title：Hypergeometric τ-functions of the q-Painleve system of type \(E_{8}^{(1)}\)

Author : Tetsu Masuda

Abstract. We present the τ-functions for the hypergeometric solutions to the q-Painlevé system of type \(E_{8}^{(1)}\) in a determinant formula whose entries are given by Rahman’s q-hypergeometric integrals. By using the symmetry of the q-hypergeometric integral, we can construct fifty-six solutions and describe the action of W(\(E_{7}^{(1)}\)) on the solutions.

#### 2009-11

Title：Ultradiscretization of a solvable two-dimensional chaotic map associated with the Hesse cubic curve

Author : K. Kajiwara, M. Kaneko, A. Nobe, T. Tsuda

Abstract. We present a solvable two-dimensional piecewise linear chaotic map which arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the nontrivial ultradiscrete limit of the solution in spite of a problem known as “the minus-sign problem.”

#### 2009-10(Published)

Title：Generalisation of Mack’s formula for claims reserving with arbitrary exponents for the variance assumption

Author : Shingo SAITO

Abstract. Mack estimated the mean squared errors of the outstanding claims reserve of each accident year and of the overall claims reserve in order to obtain their confidence intervals within his distribution-free model. We generalise his formulae by allowing for arbitrary exponents in the variance assumption. Our formula is also capable of giving a confidence interval of the amount that the insurer is liable to pay each year.

JMI 2009A : http://gcoe-mi.jp/publish_list/pub_inner/id:4

#### 2009-9

Title：Phase field model for mode III crack growth in two dimensional elasticity

Author : Takeshi Takaishi & Masato Kimura

Abstract. A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic

material is proposed. We introduce a phase field to represent the shape of the crack with

a regularization parameter and we approximate the Francfort-Marigo type energy

using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient

flow of this regularized energy. We show several numerical examples of the crack growth

computed with an adaptive mesh finite element method.

#### 2009-8 (Published)

Title：Nonlinear regression modeling via the lasso-type regularization

Author : Shohei Tateish & Hidetoshi Matsui & Sadanori Konishi

Abstract. We consider the problem of constructing nonlinear regression models with Gaussian basis functions, using lasso regularization. Regularization with a lasso penalty is an advantageous in that it reduces some unknown parameters in linear regression models toward exactly zero. We propose imposing a weighted lasso penalty on a nonlinear regression model and thereby selecting the number of basis functions effectively. In order to select tuning parameters in the regularization method, we use model selection criteria derived from information-theoretic and Bayesian viewpoints. Simulation results demonstrate that our methodology performs well in various situations.

Journal of Statistical Planning and Inference

#### 2009-7(Pulished)

Title：Asymptotic behavior of solutions of the compressible Navier-Stokes equation around the plane Couette flow

Author : Yoshiyuki Kagei

Abstract. Asymptotic behavior of solutions to the compressible Navier-Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in as solutions of an (n-1) dimensional linear heat equation with a convective term.

J. Math. Fluid Mech., vol. 13 (2011), no. 1, pp. 1--31.

#### 2009-6

Title：Bilinearization and Casorati determinant solutions to non-autonomous 1+1 dimensional discrete soliton equations

Author : Kenji Kajiwara & Yasuhiro Ohta

Abstract. Some techniques of bilinearization of the non-autonomous 1 + 1 dimensional discrete soliton equations is discussed by taking the discrete KdV equation, the discrete Toda lattice equation, and the discrete Lotka-Volterra equation as examples. Casorati determinant solutions to those equations are also constructed explicitly.

#### 2009-5

Title：Flat modules and Groebner bases over truncated discrete valuation rings

Author : Toshiro Hiranouchi & Yuichiro Taguchi

Abstract. We present basic properties of Groebner bases of submodules of a free module of finite rank over a polynomial ring with coefficients in a tdvr (:= truncated discrete valuations ring). As an application, we give a criterion for an algebra of finite type over a tdvr to be flat and prove the existence of a flat lifting of a flat algebra over a tdvr.

#### 2009-4 (Published)

Title：Nonlinear logistic discrimination via regularized Gaussian basis expansions

Author : Shuichi Kawano & Sadanori Konishi

Abstract. We consider the problem of constructing multi-class classification methods for analyzing data with complex structure. A nonlinear logistic discriminant model is introduced based on Gaussian basis functions constructed by the self-organizing map. In order to select adjusted parameters, we employ model selection criteria derived from information-theoretic and Bayesian approaches. Numerical examples are conducted to investigate the performances of the proposed multi-class discriminant procedure. Our modeling procedure is also applied to protein structure recognition in life science. The results indicate the effectiveness of our strategy in terms of prediction accuracy.

Communications in Statistics - Simulation and Computation

http://dx.doi.org/10.1080/03610910902940150

#### 2009-3 (Published)

Title：Variable selection for functional regression model via the $L_1$ regularization

Author : Hidetoshi Matsui & Sadanori Konishi

Abstract. In regression analysis, the $L_1$ regularization such as the lasso or the SCAD provides sparse solutions, which leads to variable selection. We consider the variable selection problem where variables are given as functional forms, using the $L_1$ regularization. In order to select functional variables each of which is controlled by multiple parameters, we treat parameters as grouped parameters and then apply the group SCAD. A crucial issue in the regularization method is the choice of regularization parameters. We derive a model selection criterion for evaluating the model estimated by the regularization method via the group SCAD penalty. Results of simulation and real data analysis show the effectiveness of the proposed modeling strategy.

Computational Statistics and Data Analysis

#### 2009-2(Published)

Title：Regularized functional regression modeling for functional response and predictors

Author : Hidetoshi Matsui & Sadanori Konishi

Abstract. We consider the problem of constructing a functional regression modeling procedure with functional predictors and a functional response. Discretely observed data set are expressed by Gaussian basis expansions for individuals, using smoothing methods. Parameters involved in the functional regression model are estimated by the regularized maximum likelihood method. For the selection of regularization parameters involved in the regularization method, we extend information theoretic and Bayesian model selection criteria for evaluating the estimated model. The proposed modeling strategy is applied to the analysis of real data, predicting functions rather than scalars.

JMI : http://gcoe-mi.jp/publish_list/pub_inner/id:4

#### 2009-1

Title：Global time evolution of viscous vortex rings

Author : Yasuhide Fukumoto

Abstract. This article gives an overview of growing knowledge of translation speed of an axisymmetric vortex ring, with focus on the influence of viscosity. Helmholtz-Lamb's method provides a short-cut to manipulate the translation speed at both small and large Reynolds numbers, for a vortex ring starting from an infinitely thin core. The resulting asymptotics significantly improve Saffman's formula (1970) and give closer lower and upper bounds on translation speed in an early stage. At large Reynolds numbers, Kelvin-Benjamin's kinematic variational principle achieves a further simplification. At small Reynolds numbers, the whole life of a vortex ring is available from the vorticity obeying the Stokes equations, which is closely fitted, over a long time, by Saffman's second formula.

## 2009-37