## PUBLICATIONS

### MI Preprint Series

Title：Some topics related to Hurwitz-Lerch zeta functions

Author : Takashi Nakamura

Abstract. In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz-Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz-Lerch zeta functions, and consider its properties. In all the sections, various kinds of results are always proved by multiplication formulas and inversion formulas.

The Ramanujan Journal

#### 2008-14 (Published)

Title：Riemann zeta-values, Euler polynomials and the best constant of Sobolev inequality

Author : Takashi Nakamura

Abstract. In this paper, we obtain the value $\sup_{u \in I_M , u \not \equiv 0} S_M (u)$, where $S_M (u)$ is the Sobolev functional and $I_M$ is a certain pre-Hilbert space. We also show that the functions attaining $\sup_{u \in I_M , u \not \equiv 0} S_M (u)$ are explicitly written by the Euler polynomials. This result is an analogue of the theorem proved by Kametaka, Yamagishi, Watanabe, Nagai and Takemura. Simultaneously, we obtain a much simpler proof of their theorem.

(Published) Proceedings of the Wuerzburg Conference

#### 2008-13

Title：Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials

Author : J. Faraut & M. Wakayama

ABSTRACT. Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: generating and determinantal formulae, difference equations. As an application we consider the problem of evaluating moments related to a multivariate Barnes type integral involving the Harish-Chandra c-function of a symmetric cone.

#### 2008-12 (Published)

Title：On the $L^2$ a priori error estimates to the finite element solution of elliptic problems with singular adjoint operator

Author : T. Kinoshita , K. Hashimoto & M.T. Nakao

Abstract.The Aubin-Nitsche trick for the finite element method of Dirichlet boundary value problem is a well-known technique to obtain a higher order a priori $L^2$ error estimation than $H_0^1$ estimates by considering the regularly dual problem. However, as far as the authors determine, when the dual problem is singular, it was not known at all up to now whether the a priori order of $L^2$ error is still higher than $H_0^1$ error. In this paper, we propose a technique for getting a priori $L^2$ error estimation by some verified numerical computations for the finite element projection. This enables us to obtain the higher order $L^2$ a priori error than $H^1_0$ error, even though the associated dual problem is singular. Note that our results are not a posteriori estimates but the determination of a priori constants.

http://www.informaworld.com/smpp/content~content=a910368308~db=all~order=page

#### 2008-11 (Published)

Title：A third order dispersive flow for closed curves into almost Hermitian manifolds

Author : H. Chihara & E. Onodera

ABSTRACT. We discuss a short-time existence theorem of solutions to the initial value problem for a

third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations

geometrically generalize a physical model describing the motion of vortex filament. The classical energy

method cannot work for this problem since the almost complex structure of the target manifold is

not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one

derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty,

we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and

eliminate the loss of one derivative from the partial differential equation of the dispersive flow.

Journal of Functional Analysis

http://dx.doi.org/10.1016/j.jfa.2009.04.006

#### 2008-10

Title：Jarque-Bera Normality Test for the Driving Levy Process of a Discretely Observed Univariate SDE

Author : S. Lee & H. Masuda

Abstract.We study the validity of the Jarque-Bera test for a class of univariate

parametric Levy-driven stochastic differential equations observed at

high-frequency discrete time points. Under appropriate conditions it is

shown that Jarque-Bera type statistics based on the Euler residuals can

be used to test the normality of the unobserved driving Levy process,

and moreover, that the proposed test is consistent against presence of

any nontrivial jump component. Our result therefore provides a very easy

and asymptotically distribution-free test procedure without any

fine-tuning parameter. Some illustrative simulation results are given to

reveal good performance of our test statistics.

#### 2008-9 (Published)

Title：Alpha-determinant cyclic modules and Jacobi polynomials

Author : K. Kimoto, S. Matsumoto & M. Wakayama

Abstract.For positive integers n and l, we study the cyclic GL(n)-module generated by the l-th power of the α-determinant.

This cyclic module is isomorphic to the n-th tensor space of the symmetric l-th tensor space of the vector representation of GL(n) for all but finite exceptional values of α.

If αa is exceptional, then the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in the aforementioed tensor space.

The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in αa with rational coefficients.

Especially, we determine explicitly the matrix when n equals 2.

In that case, the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial.

Moreover, we prove that these polynomials are unitary.

Transactions of the American Mathematical Society, 361, 2009, 6447-6473

#### 2008-8

Title：Optimizing a particular real root of a polynomial by a special cylindrical algebraic decomposition

Author : S. Gandy, M. Kanno, H. Anai & K. Yokoyama

Abstract. We study the problem of optimizing over parameters a particular

real root of a polynomial with parametric coefficients. We propose an efficient

symbolic method for solving the optimization problem based on a special

cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic

decomposition into cells in terms of Number-of-Roots-invariant.

#### 2008-7

Title：The random walk model revisited

Author : T.Hirotsu & S.Taniguchi

Abstract. The random walk model was introduced and investigated by D. Heyer [1]. It is a loss development

model, where the geometric Brownian motion, which is frequently used in Mathematical

Finance (for example, recall the famous Black-Sholes option pricing formula), is applied to cumulative

losses. In this paper, as an application of the random walk model, the conditional distribution

and the conditional confidence interval of the total loss to be paid in the specific future year, being

given the cumulative losses of the present, will be investigated.

#### 2008-6 (Published)

Title：Lifting Galois representations over arbitrary number fields

Author : Yoshiyuki Tomiyama

Abstract. It is proved that every two-dimensional residual Galois representation

of the absolute Galois group of an arbitrary number field lifts to a characteristic

zero p-adic representation, if local lifting problems at places above

p are unobstructed.

Journal of Number Theory, 2010, 130

#### 2008-5

Title： Torsion points of abelian varieties with values in innite extensions over a p-adic field

Author : Yoshiyasu Ozeki (Published)

Abstract.Let A be an abelian variety over a p-adic field K and L an algebraic infinite extension over K.

We consider the finiteness of the torsion part of the group of rational points A(L) under some assumptions.

In 1975, Hideo Imai proved that such a group is finite if A has good reduction and L is the cyclotomic Z_p-extension of K.

In this paper, first we show a generalization of Imai’s result in the case where A has ordinary good reduction.

Next we give some finiteness results when A is an elliptic curve and L is the field generated by the p-power torsion of an elliptic curve.

Publications of the Research Institute for Mathematical Sciences, 2009, 45

#### 2008-4（Published）

Title：Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme

Author : Hirofumi NOTSU

Abstract.We apply a newly developed characteristic-curve finite element scheme to cavity flow problems.

The scheme is useful for large scale computation, because P1/P1 element is employed and the matrix of resulting linear system is symmetric.

Numerical results of two- and three-dimensional cavity flow problems are presented.

Three types of the Dirichlet boundary condition, discontinuous, $C^0$ and $C^1$ continuous ones, are treated, and the difference of the solutions is discussed.

http://www.jstage.jst.go.jp/browse/jsces/

#### 2008-3

Title：On isosceles sets in the 4-dimensional Euclidean space

Author : Hiroaki Kido

Abstract. A subset X in the k-dimensional Euclidean space that contains

n points (elements) is called an n-point isosceles set if every triplet

of points selected from them forms an isosceles triangle. In this paper,

we show that there exist exactly two 11-point isosceles sets up to

isomorphism and that the maximum cardinality of isosceles sets in the 4-

dimensional Euclidean space is 11.

#### 2008-2 (Published)

Title：The intial value problem for a third-order dispersive flow into compact almost Hermitian manifolds

Author : Eiji Onodera

ABSTRACT. We present a time-local existence theorem of solutions to the initial value problem

for a third-order dispersive evolution equation for open curves into compact almost Hermitian

manifolds. Our equations geometrically generalize a two-sphere valued physical model describing

the motion of vortex filament . These equations cause the so-called loss of one-derivative

since the target manifold is not supposed to be a K¨ahler manifold. We overcome this difficulty

by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first

order terms essentially.

Funkcial. Ekvac. 55 (2012), pp.137-156.

#### 2008-1 (Published)

Title：Abstract collision systems simulated by cellular automata

Author : T. Ito, S.Inokuchi & Y.Mizoguchi

Abstract. We describe an algebraic transition system called an abstract

collision system. An abstract collision system is an extension of a billiard

ball system. Moreover, it is also an extension of a cellular automaton,

a chemical reaction system and so on. We introduced an abstract colli-

sion system and investigated its properties [4]. In this paper, we study

about simulation of abstract collision systems by cellular automata. It is

impossible to simulate some abstract collision system. However, some of

them can be easily simulated by a cellular automaton. First, we describe

denitions of components of an abstract collision system. Next, we intro-

duce how to construct a cellular automaton which simulates an abstract

collision system. Finally, we investigate properties and conditions about

simulations.

Published:

Proc. 34d International Workshop on Natural Computing, pp.27-38, 2008.

## 2008-15 (Published)