## PUBLICATIONS

### MI Preprint Series

Title：Dynamical analysis of the exclusive queueing process

Author : Chikashi Arita & Andreas Schadschneider

Abstract. In a recent study [C Arita and D Yanagisawa, J. Stat. Phys. 141, 829 (2010)] the stationary state of a parallel-update TASEP with varying system length, which can be regarded as a queueing process with excluded-volume effect (exclusive queueing process, EQP), was obtained. We analyze the dynamical properties of the number of particles and the position of the last particle (the system length), using an analytical method (generating function technique) as well as a phenomenological description based on domain wall dynamics and Monte Carlo simulations. The system exhibits two phases corresponding to linear convergence or divergence of the number of particles and the system length. These phases can both further be subdivided into high-density and maximal-current subphases. The predictions of the domain wall theory are found to be in very good agreement quantitively with results from Monte Carlo simulations in the convergent phase. On the other hand, in the divergent phase, only the prediction for the number of particles agrees with simulations.

Physical Review E

#### 2010-36

Title：CAP representations of inner forms of $Sp(4)$ with respect to Klingen parabolic subgroup

Author : Takanori Yasuda

Abstract. The unitary group of the hyperbolic hermitian space of dimension two

over a quaternion division algebra over a number field is a non-quasisplit inner form of Sp(4), and does not have a parabolic subgroup corresponding to the Klingen parabolic subgroup. However, it has CAP representations with respect to the Klingen parabolic subgroup. We construct them by using the theta lifting from the unitary groups of one-dimensional (-1)-hermitian spaces and estimate their multiplicities in the discrete spectrum. In many cases, their multiplicities become bigger than 1.

#### 2010-35

Title：Laplacian energy of directed graphs and minimizing maximum outdegree algorithms

Author : Kissani Perera & Yoshihiro Mizoguchi

Abstract. Energy has been studied in mathematical perspective as well as physical

perspective for several years ago. In spectral graph theory, the eigenvalues

of several kinds of matrices have been studied, of which Laplacian matrix

attracted the greatest attention [5]. Recently, in 2009, Adiga considered

Laplacian energy of directed graphs using skew Laplacian matrix, in which

degree of vertex is considered as total of the out-degree and the in-degree.

Since directed graphs play an important role in identifying the structure of

web-graphs as well as communication graphs, we consider Laplacian energy of simple directed graphs and find some relations by using the general definition of Laplacian matrix. Unlike in [1], we derived two types of equations for simple directed graphs and symmetric directed graphs with $n ≥2$ vertices by considering out-degree of vertex. Further we consider the class $P(\alpha)$ which consists of non isomorphic graphs with energy less than some $\alpha$ and find 47 non isomorphic directed graphs for class $P(10)$. Our objective extended to enumerate the structure of directed graphs using the Laplacian energy concept. Minimization maximum outdegree(MMO) algorithm defined in [3] can be used to find the directed graphs with minimum Laplacian energy.

#### 2010-34 (Published)

Title：Milnor-Selberg zeta functions and zeta regularizations

Author : Nobushige Kurokawa, Masato Wakayama & Yoshinori Yamasaki

Abstract. By a similar idea for constructing Milnor's gamma functions, we study ``higher depth determinants'' of the Laplacian on a compact Riemann surface of genus greater than one.

We prove that, as a generalization of the determinant expression of the Selberg zeta function this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function.

Moreover, it is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.

J. Geom. Phys., 64 (2013) 120-145.

#### 2010-33(Pulished)

Title：Global existence of solutions to the compressible Navier-Stokes equation around parallel flows

Author : Yoshiyuki Kagei

Abstract. The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of ${\bf R}^n$. It is proved that if $n\geq 3$, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations, provided that the Reynolds and Mach numbers are sufficiently small. The proof is given by a variant of the Matsumura-Nishida energy method based on a decomposition of solutions associated with a spectral property of the linearized operator.

J. Differential Equations, vol. 251 (2011), no. 11, pp. 3248--3295.

#### 2010-32 (Publisjed)

Title：Hypergeometric τ functions of the q-Painlev\'e systems of type $(A_2+A_1)^{(1)}$

Author : Nobutaka Nakazono

Abstract. We consider a q-Painlev\'e III equation and a q-Painlev\'e II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$.

We study their hypergeometric solutions on the level of τ functions.

Symmetry, Integrability and Geometry: Methods and Application

#### 2010-31

Title：Nonlinear regression modeling and spike detection via Gaussian basis expansions

Author : Shohei Tateishi & Sadanori Konishi

Abstract. We consider the problem of constructing nonlinear regression models in the case that the structure of data has abrupt change points at unknown points.

We propose two stage procedure where the spikes are detected by fused lasso signal approximator at the first stage, and the smooth curve is effectively estimated along with the technique of regularization method at the second.

In order to select tuning parameters in the regularization method, we derive a model selection criterion from information-theoretic viewpoints. Simulation results and real data analysis demonstrate that our methodology performs well in various situations.

#### 2010-30

Title：Spin-spin correlation functions of the q-VBS state of an integer spin model

Author : Chikashi Arita & Kohei Motegi

Abstract. We consider the valence-bond-solid ground state of the q-deformed higher-spin AKLT model (q-VBS state). We investigate the eigenvalues and eigenvectors of a matrix (G matrix), which is constructed from the matrix product representation of the q-VBS state. We compute the longitudinal and transverse spin-spin correlation functions, and determine the correlation amplitudes and correlation lengths for real q.

#### 2010-29 (Pulished)

Title：On the Number of the Pairing-friendly Curves

Author : Takanori Yasuda, Masaya Yasuda, Takeshi Shimoyama and Jun Kogure

Abstract. In pairing-based cryptography, it is necessary to choose an elliptic curve with a small embedding degree and a large prime-order subgroup, which is called a “pairing-friendly curve”. In this paper, we study the number of the pairing-friendly curves with a given large prime-order subgroup.

International Journal of Pure and Applied Mathematics

#### 2010-28

Title：Motion and Bäcklund transformations of discrete plane curves

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

Abstract. We construct explicit solutions to discrete motion of discrete plane curves which has been introduced by one of the authors recently. Explicit formulas in terms the τ function are presented. Transformation theory of motions of both smooth and discrete curves is developed simultaneously.

#### 2010-27 (Published)

Title：Exclusive queueing process with discrete time

Author : Chikashi Arita & Daichi Yanagisawa

Abstract. In a recent study [C Arita, Phys. Rev. E 80, 051119 (2009)], an extension of the M/M/1 queueing process with the excluded-volume effect as in the totally asymmetric simple exclusion process (TASEP) was introduced. In this paper, we consider its discrete-time version. The update scheme we take is the parallel one. A stationary-state solution is obtained in a slightly arranged matrix product form of the discrete-time open TASEP with the parallel update. We find the phase diagram for the existence of the stationary state. The critical line which separates the parameter space into the regions with and without the stationary state can be written in terms of the stationary current of the open TASEP. We calculate the average length of the system and the average number of particles.

Journal of Statistical Physics

#### 2010-26

Title：On the local asymptotic behavior of the likelihood function for Meixner L\’evy processes under high-frequency sampling

Author : Reiichiro Kawai & Hiroki Masuda

Abstract. We discuss the local asymptotic behavior of the likelihood function associated with all the four characterizing parameters of the Meixner L\’evy process under high-frequency sampling scheme. We derive the optimal rate of convergence for each parameter and the Fisher information matrix in a closed form. The skewness parameter exhibits a slower rate alone, relative to the other three parameters free of sampling rate. An unusual aspect is that the Fisher information matrix is constantly singular for full joint estimation of the four parameters. This is a particular phenomenon in the regular high-frequency sampling setting and is of essentially different nature from low-frequency sampling.

#### 2010-25

Title：On the Essential Self-Adjointness of Anti-Commutative Operators

Author : Toshimitsu Takaesu

Abstract. In this article, linear operators satisfying anti-commutation relations are

considered. It is proven that an anti-commutative type of the Glimm-Jaffe-Nelson commutator theorem follows.

#### 2010-24

Title：A Hardy’s Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra

Author : Toshimitsu Takaesu

Abstract. In this article we consider linear operators satisfying a generalized commutation

relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy’s uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated.

#### 2010-23

Title：Composition, union and division of cellular automata on groups

Author : Takahiro Ito, Mitsuhiko Fujio, Shuichi Inokuchi & Yoshihiro Mizoguchi

Abstract. We introduce the notion of 'Composition', 'Union' and 'Division' of cellular automata on groups. A kind of notions of compositions was investigated

by Sato (1994) and Manzini (1998) for linear cellular automata, we extend the notion to general cellular automata on groups and investigated their properties. We observe the all unions and compositions generated by one-dimensional 2-states 2-neighborhood cellular automata including non-linear cellular automata. Next we prove that the composition is right-distributive over union, but is not left-distributive. Finally, we conclude by showing reformulation of our definition of cellular automata on group which admit more than three states. We also show our formulation contains the representation using formal power series for linear cellular automata in Manzini (1998).

#### 2010-22

Title：A Generalized Scaling Limit and its Application to the Semi-Relativistic Particles System Coupled to a Bose Field with Removing Ultraviolet Cutoffs

Author : Toshimitsu Takaesu

Abstract.The system of semi-relativistic particles coupled to a scalar bose field is investigated. A renormalized Hamiltonian is defined by subtracting a divergent term from the total Hamiltonian. We consider taking the scaling limit and removing the ultraviolet cutoffs simultaneously for the renormalized Hamiltonian. By applying an abstract scaling limit theory on self-adjoint operators, we derive the Yukawa potential as an effective potential of semi-relativistic particles.

#### 2010-21

Title：Approximate quadratic estimating function for discretely observed Lévy driven SDEs with application to a noise normality test

Author : Hiroki Masuda

Abstract. We deal with a family of ergodic Lévy driven stochastic differential equations observed at high-frequency discrete sampling points, where we do not suppose a specific form of the driving Lévy measure, while the coefficients are known except for finite-dimensional parameters. Our aim is two-fold: first, we derive first-order asymptotic behavior of an M-estimator based on the approximate quadratic martingale estimating function; second, as an application of the estimator obtained, we derive consistent and asymptotically distribution-free test statistics for the normality of the driving Lévy process, based on the self-normalized partial sums of residuals.

#### 2010-20

Title：Lagrangian approach to weakly nonlinear stability of an elliptical flow

Author : Yasuhide Fukumoto, Makoto Hirota & Youichi Mie

Abstract. Rotating flows with elliptically strained streamlines suffer from a parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by two. We address the weakly nonlinear evolution of mplitude of three-dimensional Kelvin waves, in resonance, on a flow confined in a cylinder of elliptical cross-section. In a traditional Eulerian approach, derivation of the mean-flow induced by nonlinear interaction of Kelvin waves stands as an obstacle. We show how topological idea, or the Lagrangian approach, facilitates calculation of the wave-induced mean flow. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet, and the isovortical perturbation is handled only in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to work out the wave-induced mean flow only from the linear Lagrangian displacement. With the mean flow at hand, the Lagrangian approach provides us with a bypass to enter into weakly nonlinear regime of amplitude evolution of three-dimensional disturbances. Unlike the Eulerian approach, the amplitude equations are available directly in the Hamiltonian normal form.

#### 2010-19

Title：Local asymptotic normality for normal inverse Gaussian L￥'evy processes with high-frequency sampling

Author : Reiichiro Kawai & Hiroki Masuda

Abstract. We prove the local asymptotic normality for the full parameters of the normal inverse Gaussian Levy process, when we observe high-frequency and long-term data. The rates of convergence turn out to be of two kinds for the dominating parameters. The essential feature in our study is that the suitably normalized increments in small time is approximately Cauchy-distributed, which specifically comes out in the form of the asymptotic Fisher information matrix.

#### 2010-18

Title：Scaling limits for the system of semi-relativistic particles coupled to a scalar bose field

Author : Toshimitsu Takaesu

Abstract. In this paper the Hamiltonian for the system of semi-relativistic particles interacting with a scalar bose field is investigated. A scaled total Hamiltonian of the system is defined and its scaling limit is considered. Then the semi-relativistic Schrodinger operator with an effective potential is derived.

#### 2010-17

Title：On the classification of rank 2 almost Fano bundles on projective space

Author : Kazunori Yasutake

Abstract. An almost Fano bundle is a vector bundle on a smooth projective variety that its projectivization is an almost Fano variety. In this paper, we prove that almost Fano bundles exist only on almost Fano manifolds and study rank 2 almost Fano bundles over projective spaces.

#### 2010-16 (Published)

Title：The value distribution of the Gauss map of improper affine spheres

Author : Yu Kawakami and Daisuke Nakajo

Abstract. We give the best possible upper bounds for the number of exceptional values and totally ramified value number of the Lagrangian Gauss map of complete improper affine maps in the affine three-space. Moreover, by applying the Fujimoto argument, we also obtain the sharp estimate for them of weakly complete improper affine maps. As an application, from the viewpoint of the value distribution of Lagrangian Gauss map, we provide a new proof of the classification of affine complete improper affine spheres. Furthermore, we get a ramification estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.

Journal of the Mathematical Society of Japan (Vol.64)

#### 2010-15

Title：Local Instability of a Rotating Flow Driven by Precession of Arbitrary Frequency

Author : Me Me Naing & Yasuhide Fukumoto

Abstract. We revisit the local stability, to three-dimensional disturbances, of rotating flows with circular streamlines, whose rotation axis executes constant precessional motion about an axis perpendicular to itself. In the rotating frame, the basic flow is steady velocity field linear in coordinates in an unbounded domain constructed by Kerswell (1993), and admits the use of the WKB method. For small precession freqencey, we recover Kerswell's result. A novel instability is found at large frequency for which the axial wavenumber executes an oscillation around zero; drastic growth of disturbamce amplitude occurs only in an extremely short time interval around the time where the axial wavenumber vanishes. In the limit of infinite precession frequency, the growth rate exhibits singular behavior with respect to a parameter characterizing the tilting angle of the wave vector.

#### 2010-14

Title：Non-existence of certain Galois representations with a uniform tame inertia weight

Author : Yoshiyasu Ozeki

Abstract. In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.

#### 2010-13

Title：On simulation of tempered stable random variates

Author : Reiichiro Kawai & Hiroki Masuda

Abstract. Various, existing and new, simulation methods for tempered stable random variates are investigated with a view towards practical implementation, in particular simulation of increments over a very short stepsize. The methods under consideration are based on acceptance-rejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and computing effort. With a given computing budget, an approximative acceptance-rejection sampling technique Ｂaeumer and Meerschaert is both most efficient and handiest, and any desired level of accuracy may be attained with a small amount of additional computing effort.

#### 2010-12(Pulished)

Title：Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a Poiseuille type flow

Author : Yoshiyuki Kagei, Yu Nagafuchi & Takeshi Sudou

Abstract. Decay estimates on solutions to the linearized compressible Navier-Stokes equation around a Poiseuille type 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 $n-1$ 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.

Journal of Math-for-Industory, vol. 2 (2010A), pp. 39--56.

#### 2010-11

Title：Derivation of specific conditions with Comprehensive Groebner Systems

Author : Katsusuke Nabeshima & Hiroshi Yoshida

Abstract. Here we present an efficient calculation of comprehensive Groebner systems to derive specific conditions for neural circuits as well as electric circuits. Comprehensive Groebner systems (CGS) have been applied to problems with a small number of parameters such as the automatic geometric theorem proving and the inverse kinematics problem of a robot arm. In CGS, however, a larger number of parameters make its calculation less tractable. Therefore, we take `not-equal' conditions into account during CGS calculation, resulting in a reduced format of CGS of parametric systems even though many parameters exist. Using our implemented CGS, we derive specific conditions such as resonant conditions that play an important role in physical, mechanical, and biological phenomena. The obtained conditions lead to analysis of realistic neural circuits having many parameters, and provide us with a possibility of positive CGS.

#### 2010-10

Title：Variable selection via the grouped weighted lasso for factor analysis models

Author : Kei Hirose & Sadanori Konishi

Abstract. The $L_1$ regularization such as the lasso has been widely used in regression analysis since it tends to produce some coefficients that are exactly zero, which leads to variable selection. We consider the problem of variable selection for factor analysis models via the $L_1$ regularization procedure. In order to select variables each of which is controlled by multiple parameters, we treat parameters as grouped parameters and then apply the grouped lasso. Crucial issues in this modeling procedure include the selection of the number of factors and regularization parameters. Choosing these parameters can be viewed as a model selection and evaluation problem. We derive a model selection criterion for evaluating a factor analysis model via the grouped lasso. The proposed procedure produces estimates that lead to variable selection and also selects the number of factors objectively. Monte Carlo simulations are conducted to investigate the effectiveness of the proposed procedure. A real data example is also given to illustrate our procedure.

#### 2010-9 (Published)

Title：An algebraic approach to underdetermined experiments

Author : Hiroshi Yoshida, Kinji Kimura, Naoki Yoshida, Junko Tanaka & Yoshihiro Miwa

Abstract. We sometimes meet an underdetermined experiment, where rate constants cannot be determined. One of methods to overcome under-determination is combining results of multiple experiments. Multiple experiments lead to many parameters and variables, and usually have multiple distinct solutions. To decompose a solution into multiple solutions, we can use prime ideal decomposition. It is, however, hard to decompose a set of polynomials having many parameters and variables. Therefore, we propose one tip and one technique using a resultant from a biological viewpoint.

IPSJ Transactions on Bioinformatics

#### 2010-8 (Published)

Title：Abstract collision systems on groups

Author : Takahiro Ito

Abstract. We discuss about abstract collision systems(ACS) on groups which is an extension of ACS. The ACS is a kind of frameworks of unconventional computing which includes collision based computing, cellular automata (CA), chemical reaction systems and so on. In this paper, we define ACS on groups. When a group G and its subset is given, we create a set of collisions and a local transition function of an ACS from the group G and its operation. First, we describe definitions of components of ACS. Next, we introduce ACS on groups. Finally, we investigate properties of operations of ACS, union, division and composition.

JMI2010A-6

#### 2010-7

Title：UC hierarchy and monodromy preserving deformation

Author : Teruhisa Tsuda

Abstract. The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones.

Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painleve equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the canonical variables.

#### 2010-6 (Published)

Title：Semi-supervised logistic discrimination via graph-based regularization

Author : Shuichi Kawano, Toshihiro Misumi & Sadanori Konishi

Abstract. We address the problem of constructing a nonlinear model based on both classified and unclassified data sets for classification. A semi-supervised logistic model with Gaussian basis expansions along with technique of graph-based regularization method is presented. Crucial issues in our modeling procedure are the choices of tuning parameters included in the nonlinear logistic models. In order to select these adjusted parameters, we derive model selection criteria from the viewpoints of information theory and Bayesian approach. Some numerical examples are conducted to show the effectiveness of our proposed semi-supervised modeling strategies.

Neural Processing Letters

#### 2010-5 (Published)

Title：Nonlinear regression modeling and detecting change point via the relevance vector machine

Author : Shohei Tateishi & Sadanori Konishi

Abstract. We consider the problem of constructing nonlinear regression models in the case that the structure of data has discontinuities at some unknown points.

We propose two-stage procedure in which the change points are detected by RVM at the first stage, and the smooth curve are effectively estimated along with the technique of regularization method at the second. In order to select tuning parameters in the regularization method, we derive a model selection and evaluation criterion from information-theoretic viewpoints. Simulation results and real data analyses demonstrate that our methodology performs well in various situations.

http://www.springerlink.com/content/j48438pg132j2349/

#### 2010-4

Title：The It\^o-Nisio theorem, quadratic Wiener functionals, and 1-solitons

Author : Nobuyuki Ikeda & Setsuo Taniguchi

Abstract. Abstract. Among Professor Kiyosi Ito's achievements, there is the Ito-Nisio theorem, a completely general theorem relative to the Fourier series decomposition of the Brownian motion. In this paper, some of its applications will be reviewed, and new applications to 1-soliton solutions to the Korteweg-de Vries (KdV in short) equation and Eulerian polynomials will be given.

#### 2010-3 (Published)

Title：Hyper-parameter selection in Bayesian structural equation models

Author : Kei Hirose, Shuichi Kawano, Daisuke Miike & Sadanori Konishi

Abstract. In the structural equation models, the maximum likelihood estimates of error variances can often turn out to be zero or negative. In order to overcome this problem, we take a Bayesian approach by specifying a prior distribution for variances of error variables. Crucial issues in this modeling procedure include the selection of hyper-parameters in the prior distribution. Choosing these parameters can be viewed as a model selection and evaluation problem. We derive a model selection criterion for evaluating a Bayesian structural equation model. Monte Carlo simulations are conducted to investigate the effectiveness of our proposed modeling procedure. A real data example is also given to illustrate our procedure.

Bulletin of Informatics and Cybernetics

#### 2010-2

Title：Infinite variation tempered stable Ornstein-Uhlenbeck processes with discrete observations

Author : Reiichiro Kawai & Hiroki Masuda

Abstract. We investigate transition law between consecutive observations of Ornstein-Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in $(0,1)$ and the other with index in (1,2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Levy processes.

#### 2010-1

Title：Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

Author : Hiroki Masuda

Abstract. We consider drift estimation of a discretely observed Ornstein-Uhlenbeck process driven by a possibly heavy-tailed symmetric Levy process with positive Blumenthal-Getoor activity index (BG index). Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being depending on the BG index and sampling frequency as well as the sample size. It turns out that the rate of convergence is determined by the most active part of the driving Levy process; the presence of a driving Wiener part leads to the rate familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Levy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Levy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

## 2010-37 (Pulished)