## PUBLICATIONS

### MI Preprint Series

Title：On the convergence rates for the compressible Navier- Stokes equations with potential force

Author : Masatoshi Okita

Abstract: In this paper we are concerned with the convergence rates to the stationary solutions for the compressible Navier-Stokes equations with a potential external force in the whole space.

#### 2012-12

Title：On simplicity of the lowest eigenvalue of non-commutative harmonic oscillators

Author : Masato Wakayama

Abstract: The aim of this note is to prove the simplicity of the lowest eigenvalue of the non-commutative harmonic oscillator, which is defined by a 2 × 2 system of ordinary differential operator, for a large class of structure parameters. The proof looks rather simple while uses many basic results obtained so far. 2010 Mathematics Subject Classification: Primary 34L40, Secondary 81Q10, 34M05, 81S05.

Keywords and phrases: non-commutative harmonic oscillator, lowest eigenvalue, multiplicity of eigen-values, Heun's differential equation.

#### 2012-11(Published)

Title：Adaptive basis expansion via the extended fused lasso

Author : Daeju Kim, Shuichi Kawano & Yoshiyuki Ninomiya

Abstract. We propose a nonlinear regression model that uses basis expansion for the case where the underlying regression function has inhomogeneous smoothness. In this case, conventional nonlinear regression models tend to over- or underfit where the function is smoother or less smooth, respectively. We begin by roughly approximating the underlying regression function with a locally linear function. We then extend the fused lasso signal approximator and thereby develop a fast and efficient algorithm. We next use the residuals between the locally linear functions and the data to adaptively prepare the basis functions. Finally, using a regularization method, we construct a nonlinear regression model with these basis functions. To select the optimal value of the tuning parameter for the regularization method, we provide an explicit form of the generalized information criterion. The validity of our proposed method is then demonstrated through several numerical examples.

Computational Statistics

#### 2012-10(Published)

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

Author : Jan Brezina

Abstract: The global in time existence of strong solutions to the compressible Navier-Stokes equation around time-periodic parallel flows in n-dimensional infinite layer is established under smallness conditions on Reynolds number, Mach number and initial perturbations. Furthermore, it is proved for n=2 that the asymptotic leading part of solutions is given by a solution of one-dimensional viscous Burgers equation multiplied by time-periodic function. In the case n greater than or equal to 3 the asymptotic leading part of solutions is given by a solution of n-1-dimensional heat equation with convective term multiplied by time-periodic function.

SIAM J. Math. Anal., vol. 45 (2013), no. 6, 3514–3574

DOI:10.1137/12089555X

#### 2012-9(Published)

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

Author : Jan Brezina & Yoshiyuki Kagei

Abstract: The linearized problem around a time-periodic parallel flow of the compressible Navier-Stokes equation in an infinite layer is investigated. By using the Floquet theory, spectral properties of the evolution operator associated with the linearized problem are studied in detail. The Floquet representation of low frequency part of the evolution operator, which plays an important role in the study of the nonlinear problem, is obtained.

J. Differential Equations, vol. 255 (2013), no. 6, pp. 1132--1195

doi:10.1016/j.jde.2013.04.036

#### 2012-8(Published)

Title：A new approach to the Pareto stable matching problem

Author : Naoyuki Kamiyama

Abstract: In two-sided matching markets, the concept of stability proposed by Gale and Shapley (1962) is one of the most important solution concepts. In this paper, we consider a problem related to the stability of a matching in a two-sided matching market with indifferences (i.e., ties). The introduction of

ties into preference lists dramatically changes the properties of stable matchings. For example, stable matchings need not have the same size. Furthermore, it is known that stability do not guarantee Pareto efficiency that is also one of the most important solution concepts in two-sided matching markets. This fact naturally leads to the concept of Pareto stability, i.e., both stable and Pareto efficient. Erdil and Ergin (2006, 2008) proved that there always exists a Pareto stable matching in a one-to-one/many-to-one matching market with indifferences and gave a polynomial-time algorithm for finding it. Furthermore, Chen (2012) proved that there always exists a Pareto stable matching in a many-to-many matching market with indifferences and gave a polynomial-time algorithm for finding it. In this paper, we propose a new approach to the problem of finding a Pareto stable matching in a many-to-many matching market with indifferences. Our algorithm is an alternative proof of the existence of a Pareto stable matching in a many-to-many matching market with indifferences

Mathematics of Operations Research

#### 2012-7

Title：Solutions to a q-analog of Painleve III equation of type D7

Author : Nobutaka Nakazono & Seiji Nishioka

Abstract: This paper deals with a q-analog of Painleve III equation of type D7. We study its algebraic function solutions and transcendental function solutions. We construct algebraic function solutions expressed by Laurent polynomials and prove irreducibility in the sense of decomposable extensions.

#### 2012-6

Title：Topological Measurement of Protein Compressibility via Persistence Diagrams

Author : Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar, Konstantin Mischaikow & Vidit Nanda

Abstract: We exploit recent advances in computational topology to study the compressibility of various proteins found in the Protein Data Bank (PDB). Our fundamental tool is the persistence diagram, a topological invariant which captures the sizes and robustness of geometric features such as tunnels and cavities in protein molecules. Based on certain physical and chemical properties conjectured to impact protein compressibility, we propose a topological measurement for each protein molecule.

This topological measurement can be efficiently computed from the PDB data. Our main result establishes a clear linear correlation between the topological measurement and the experimentally measured compressibility of most proteins for which both PDB information and experimental compressibility data are available.

#### 2012-5(Pulished)

Title：On the maximal number of exceptional values of Gauss maps for various classes of surfaces

Author : Yu Kawakami

Abstract: The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space, and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.

Mathematische Zeitschrift

#### 2012-4

Title：Hamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

Author : Yasuhide Fukumoto & Youich Mie

Abstract: A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.

#### 2012-3

Title：A Weak RIP of theory of compressed sensing and LASSO

Author : Hiroshi Inoue

Abstract. This paper introduce simple and general theories of compressed sensing and LASSO. The novelty is that our recovery results do not require the restricted isometry property(RIP). We use the notion of weak RIP that is a natural generalization of RIP. We consider that the proposed results are more useful and flexible for real data analysis in various fields.

#### 2012-2

Title：Mighty convergence of the Gaussian quasi-likelihood random fields for ergodic Levy driven SDE observed at high frequency

Author : Hiroki Masuda

Abstract. This paper investigates the Gaussian quasi-likelihood estimation of an exponentially ergodic multidimensional Markov process, which is expressed as a solution to a Levy driven stochastic differential equations whose coefficients are supposed to be known except for the finite-dimensional parameters to be estimated. We suppose that the process is observed under the condition for the rapidly increasing experimental design. By means of the polynomial type large deviation inequality, the mighty convergence of the corresponding statistical random fields is derived, which especially leads to the asymptotic normality at rate of the square root of the terminal sampling time for all the target parameters, and also to the convergence of their moments. In our results, the diffusion coefficient may be degenerate, or even null. Although the resulting estimator is not asymptotically efficient in the presence of jumps, we do not require any specific form of the driving Levy measure, rendering that the proposed estimation procedure is practical and somewhat robust to underlying model specification.

#### 2012-1 (Published)

Title：Spectrum of non-commutative harmonic oscillators and residual modular forms

Author : Kazufumi Kimoto & Masato Wakayama

Abstract. Special values ζ_Q(k) (k = 2, 3, 4, ...) of the spectral zeta function ζ_Q(s) of the non-commutative harmonic oscillator Q are discussed. Particular emphasis is put on basic modular properties of the generating function w_k(t) of Apery-like numbers which is appeared in analysis on the first anomaly of each special value.

Here the first anomaly is defined to be the "1st order" difference of ζ_Q(k) from ζ(k), ζ(s) being the Riemann zeta function. In order to describe such modular properties for k ≥ 4, we introduce a notion of residual modular forms for congruence subgroups of SL_2(Z) which contains the classical notion of Eichler integrals as a particular case. Further, we define differential Eisenstein series, which are residual modular forms. Using such differential Eisenstein series, for example, one obtains an explicit description of w_4(t). A certain Eichler cohomology group associated to such residual modular forms plays also an important role in the discussion.

Noncommutative Geometry and Physics 3, World Scientific

## 2012-13