## PUBLICATIONS

### Journal of Math-for-Industry

ISSN 1884-4774, ISSN 1884-4782(On line)

Title：JMI2010B-Front Cover

#### JMI2010B-1(pp. 115-125)

Title：The minimal entropy martingale measures for exponential additive processes revisited

Author : Tsukasa Fujiwara

Abstract. The mimimal entropy martingale measure for the stochastic process defined as the exponential of an additive process with the structure of semimartingale will be investigated. Special attention will be paid to the case when the underlying additive process has fixed times of discontinuity. The investigation of this paper will establish a unified way that is applicable both to the case of L\'evy processes and that of the sums of independent random variables.

Keywords. additive process, process with independent increments, semimartingale, minimal entropy martingale measure, exponential moment, Laplace cumulant, modified Laplace cumulant

#### JMI2010B-2(pp.127-131)

Title：Large deviations and finite time ruin probabilities for generalized renewal risk models

Author : Kaiyong Wang and Xiaoli Li

Abstract. In this paper, we extend the standard renewal risk model to the case where the premium income process is a counting process and the claim sizes and the inter-arrival times are two sequences of negatively associated random variables. For this risk model, the paper investigates the large deviations for the claim surplus process and gives the Lundberg type limiting results on the finite time ruin probabilities.

Keywords. Generalized renewal risk model, finite time ruin probability, large deviation

#### JMI2010B-3(pp.133-138)

Title：Scaling limit of d-inverse of Brownian motion with functional drift

Author : Kouji Yano and Katsutoshi Yoshioka

Abstract. The d-inverse is a generalized notion of inverse of a stochastic process having a certain tendency of increasing expectations. Scaling limit of the d-inverse of Brownian motion with functional drift is studied. Except for degenerate case, the class of possible scaling limits is proved to consist of the d-inverses of Brownian motion without drift, one with explosion in finite time, and one with power drift.

Keywords. d-inverse, domain of attraction, Brownian motion with drift, geometric Brownian motion, option price, Black-Scholes formula

#### JMI2010B-4(pp.139-146)

Title：An inverse problem of thickness design for single layer textile material under low temperature

Author : Dinghua Xu, Jianxin Cheng and Xiaohong Zhou

Abstract. Based on the model of steady-state heat and moisture transfer through textiles, we propose an inverse problem of thickness design for single layer textile material under low temperature. Adopting the idea of regularization method, solving the inverse problem can be formulated into a function minimization problem. Combining the finite difference method for ordinary differential equations with direct search method of one-dimensional minimization problems, we derive three kinds of iteration algorithms of regularized solution for the inverse thickness problem. Numerical simulation is achieved in order to verify the validity of proposed methods.

Keywords. textiles; heat and moisture transfer; inverse problems; thickness design; regularization method; numerical solution

#### JMI2010B-5(pp.147-156)

Title：Convex optimization techniques for the efficient recovery of a sparsely corrupted low-rank matrix

Author : Silvia Gandy and Isao Yamada

Abstract. We address the problem of recovering a low-rank matrix that has a small fraction of its entries arbitrarily corrupted. This problem is recently attracting attention as nontrivial extension of the classical PCA (principal component analysis) problem with applications in image processing and model/system identification. It was shown that the problem can be solved via a convex optimization formulation when certain conditions hold. Several algorithms were proposed in the sequel, including interior-point methods, iterative thresholding and accelerated proximal gradients. In this work we address the problem from two completely different sides. First, we propose an algorithm based on the Douglas-Rachford splitting technique which has inherent convergence guarantees. Second, we propose, based on algorithms from rank minimization and sparse vector recovery, a computationally efficient greedy algorithm that scales better to large problem sizes than existing algorithms. We compare the performance of these proposed algorithms to the accelerated proximal gradients algorithm.

Keywords. PCA, rank minimization, nuclear norm minimization, sparse error, Douglas-Rachford splitting, greedy algorithms

#### JMI2010B-6(pp.157-169)

Title：A Darboux transformation for discrete s-isothermic surfaces

Author : Tim Hoffmann

Abstract. We give an overview on the discretization of isothermic surfaces, with special emphasis on the so-called s-isothermic surfaces, which are in some sense a nonlinear deformation of the classical discrete isothermic surfaces. For s-isothermic surfaces we give a way to define surfaces of constant mean curvature (cmc surfaces for short) without actually defining an a priori notion of curvature itself. We will compute discrete versions of rotational symmetric cmc surfaces (Delaunay surfaces) as an example. Finally, we give a discrete equivalent of the Sinh-Gordon equation, solutions of which describe -- in complete analogy to the smooth case -- discrete s-isothermic cmc surfaces.

Keywords. mathematics, discrete differential geometry, s-isothermic nets, constant mean curvature

#### JMI2010B-7(pp.171-181)

Title：Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit

Author : Vladimir Chalupecky, Tasnim Fatima and Adrian Muntean

Abstract. We present two multiscale reaction-diffusion (RD) systems modeling sulfate attack in concrete structures (here: sewer pipes). The systems are posed on two different spatially separated scales. The only difference between them is the choice of the {\em micro-macro transmission condition}. We explore numerically the way in which the macroscopic Biot number $Bi^M$ connects the two reaction-diffusion scenarios. We indicate connections between the solution of the "regularized" system (with moderate size of $Bi^M$) and the solution to the "matched" system (with blowing up size of $Bi^M$), where Henry's law plays the role of the micro-macro transmission condition.

Keywords. Multiscale RD system, micro-macro transmission conditions, sulfate corrosion, acid attack, modeling of concrete, finite difference scheme, convergence rates

#### JMI2010B-8(pp.183-197)

Title：A variation of the minimum spanning tree problem for the application to mathematical OCR

Author : Akio Fujiyoshi and Masakazu Suzuki

Abstract. In this paper, we introduce a variation of the minimum spanning tree problem for the application to mathematical OCR. The problem is obtained from the original minimum spanning tree problem by importing the notions of "candidate selection'' and "link-label selection.'' It is shown that the problem is NP-hard. However, we find that, for the application to mathematical OCR, it is sufficient to deal with only a class of graphs that is recursively defined with some graph-rewriting rules. For the class of graphs, it is shown that the problem can be solved in linear-time in the number of vertices of a graph.

Keywords. the minimum spanning tree problem, NP-completeness, treewidth, mathematical OCR, mathematical formula recognition

#### JMI2010B-9(pp.199-212)

Title：A polynomial-time inexact interior-point method for convex quadratic symmetric cone programming

Author : Lu Li and Kim-Chuan Toh

Abstract. In this paper, we design an inexact primal-dual infeasible path-following algorithm for convex quadratic programming over symmetric cones. Our algorithm and its polynomial iteration complexity analysis give a unified treatment for a number of previous algorithms and their complexity analysis. In particular, our algorithm and analysis includes the one designed for linear semidefinite programming in "Math. Prog. 99 (2004), pp. 261--282". Under a mild condition on the inexactness of the search direction at each interior-point iteration, we show that the algorithm can find an $\eps$-approximate solution in $O(n^2\log(1/\eps))$ iterations, where $n$ is the rank of the underlying Euclidean Jordan algebra.

Keywords. Multiscale RD system, micro-macro transmission conditions, sulfate corrosion, acid attack, modeling of concrete, finite difference scheme, convergence rates

#### JMI2010B-10(pp.213-225)

Title：Stochastic complexity, channel capacity, and universal portfolio

Author : Jun'ichi Takeuchi

Abstract. We pick up and discus three topics from information theory and learning theory: stochastic complexity, communication channel capacity, and portfolio theory in finance. At first glance, they seem very different ones, but they have common game theoretic profiles. The purpose of this article is to present brief introductions to each problem and describe the relation between them.

Keywords. stochastic complexity, minimax problem, channel capacity, universal portfolio

#### JMI2010B-11(pp.227-234)

Title：A wait-and-see strategy as a survival strategy in the prisoner's dilemma between relatives

Author : Shin-ichiro Takazawa

Abstract. The behavior to decide the action according to the current situation is seen well in humans. We consider the prisoner's dilemma in which the variable probability of cooperation is allowed. Here, we define a wait-and-see strategy as the strategy that an individual cooperates at the same probability as the proportion of cooperation in the population. In addition, we consider a game between relatives in which an individual is more likely to meet an opponent using the same strategy. To examine the reasonableness of a wait-and-see strategy from the viewpoint of survival, we analyze the three strategies in the prisoner's dilemma between relatives by means of a replicator dynamics. We prove that our wait-and-see strategy survives in almost all conditions for the prisoner's dilemma between relatives. Therefore, we conclude a wait-and-see behavior is reasonable.

Keywords. wait-and-see strategy, prisoner's dilemma, game between relatives, survival, replicator dynamics

#### (JMI2010A-5)

Title：Correction to "Decay estimates on solutions of the linearized compressible Navier-Stokes equation around a Poiseuille type flow" in J. Math-for-Ind. 2A (2010), pp. 39--56

Author : Yoshiyuki Kagei, Yu Nagafuchi and Takeshi Sudou

Theorem 3.2 of \cite{Kagei-Nagafuchi-Sudo} in J. Math-for-Ind. 2A (2010), pp. 39--56, contains an error.

We modify the assumption on the initial value in order that Theorem 3.2 holds true.

#### JMI2010A-1(pp.1-11)

Title：A Survey on Convergence Theorems of the dqds Algorithm for Computing Singular Values

Author : Kensuke Aishima, Takayasu Matsuo, Kazuo Murota and Masaaki Sugihara

Abstract. This is a survey on convergence theorems for the differential quotient difference with shifts (dqds) algorithm, which is one of the most efficient methods for computing matrix singular values. Emphasis is laid on the relationship and comparison between the global convergence theorem obtained recently by the present authors and Rutishauser’s convergence theorem for the Cholesky LR method with shifts for the positive-definite eigenvalue problem. Theorems on convergence rate of the dqds algorithm with different shift strategies are also reviewed.

Keywords. numerical linear algebra, matrix singular value, global convergence, shift strategy

#### JMI2010A-2(pp.13-20)

Title：On the exponential moments of additive processes with the structure of semimartingales

Author : Tsukasa Fujiwara

Abstract. In this paper, the exponential moments of R-valued additive processes with the strucure

of semimartingales, which are regarded as the Laplace transforms of the laws of these additive

processes, will be explicitly represented by their characteristics. Note that the additive processes

investigated here will not necessarily be assumed to be stochastically continuous. To prove the

result, a criterion proposed in [5], which is described by the modified Laplace cumulant, will be

applied.

Keywords. additive process, semimartingale, exponential moment, Laplace cumulant, modified

Laplace cumulant

#### JMI2010A-3(pp.21-25)

Title：Self-adjoint extensions of momentum operators: application of weak Weyl relations

Author : Fumio Hiroshima, Sotaro Kuribayashi and Itaru Sasaki

Abstract.By weak Weyl relations it is shown that momentum operators, $-i\partial_{x_j}$, defined on $\CCC$ with some general open set $\Omega \subset \RR^n$ are {\it not} essentially self-adjoint but have uncountably many self-adjoint extensions.

Keywords. canonical commutation relation, CCR, Weyl relation, weak Weyl relation, momentum

operator

#### JMI2010A-4(pp.27-37)

Title：Weakly nonlinear saturation of stationary resonance of a rotating flow in an elliptic cylinder

Author : Yoichi Mie and Yasuhide Fukumoto

Abstract. We address weakly nonlinear stability of a uniformly rotating flow confined in a cylinder of elliptic cross-section to three-dimensional disturbances. A Lagrangian approach is developed to derive unambiguously the drift current induced by nonlinear interaction of isovortical disturbances.

This approach rescues the insufficiency inherent in the Eulerian approach and provides a direct path to reach the amplitude equations in the Hamiltonian normal form. The nonlinear effect saturates the stationary instability mode, and asymptotic form of its saturation amplitude is gained, in a tidy form, in the short-wavelength regime.

Keywords. elliptical instability, weakly nonlinear stability, Lagrangian approach, mean flow

#### JMI2010A-5(pp.39-56)

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

Author : Yoshiyuki Kagei, Yu Nagafuchi and 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 roblem decay in L2 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.

Keywords. compressible Navier-Stokes equation, decay estimates, asymptotic behavior, Poiseuille type flow

#### JMI2010A-6(pp.57-73)

Title：Abstract collision systems on G-sets

Author : Takahiro Ito

Abstract. In this paper, we discuss an abstract collision system (ACS) on a G-set which is an extension of a normal ACS [5, 6]. An ACS is a type of unconventional computing framework that includes collision-based computing, cellular automata (CA), and chemical reaction systems. For a given group G and its subset, we create a set of collisions and a local transition function of an ACS by using the action of G. We first refine definitions of the components of an ACS, and then extend them to the concepts on a G-set. Finally, we define and investigate the operations "union", "division" and "composition" of the ACS on a G-set.

Keywords. collision-based computing, cellular automata.

#### JMI2010A-7(pp.75-83)

Title：Explicit Lower Bound for the Length of Minimal Weigh τ-adic Expansions on Koblitz Curves

Author : Keisuke Hakuta, Hisayoshi Sato and Tsuyoshi Takagi

Abstract. Elliptic curve cryptosystems (ECC) are emerging cryptographic standards which can be used instead of RSA cryptosystems, and are practically used. In ECC, scalar multiplication (or point multiplication) is the dominant operation, namely computing an integer multiple for a given integer and a point on an elliptic curve. However, for practical use, it is a very important matter to improve the efficiency of scalar multiplication.The τ-adic non-adjacent form (τ-NAF) proposed

by Solinas, is one of the most efficient algorithms to compute scalar multiplications on Koblitz curves. Avanzi, Heuberger, and Prodinger have proven the minimality of the Hamming weight of the τ-NAF on Koblitz curves. However, the lower bound for the length of minimal Hamming weight τ-adic expansions is not known yet. In this paper, we shall derive an explicit lower bound for the length of minimal Hamming weight τ-adic expansions. We shall also give a new proof of the minimality of the Hamming weight of the τ-NAF on Koblitz curves. Further, by using the proof of the lower bound and the new proof of the minimality, we classify a minimal length τ-adic expansion with minimal Hamming weight except for two special cases. The classification shows that the τ-NAF has almost minimal length among all τ-adic expansions of minimal Hamming weight and we can easily convert the τ-NAF into a minimal length τ-adic expansion without changing the Hamming weight. This fact follows immediately from the proof of the lower bound and our new proof.

Keywords. Koblitz Curves (Anomalous Binary Curves), Scalar Multiplication, τ-adic Non-Adjacent

Form (τ-NAF), Minimal Length

#### JMI2010A-8(pp.85-91)

Title：A short introduction to shape analysis of apparent contours by “panorama views”

Author : Mahito Kobayashi and Takashi Sano

Abstract. Shape analysis of simple closed plane curves is an important subject including many chances of application to industry, especially when the curves are apparent contours of objects projected into the plane. A notion of panorama view is introduced, with presentations of some ideas of its application to the subject.

Keywords. Hough transform, panorama view, projective dual

#### JMI2010A-9(pp.93-98)

Title：Exact computation for the cover times of certain classes of trees

Author : Yusuke Higuchi, Takuya Ohwa and Tomoyuki Shirai

Abstract. We show a special feature for the cover time of trees that is not satisfied by those of other graphs. By using this property, we show the relationship between the cover times of a tree and its subdivision, and we compute exactly the distribution of the last vertex visited by a random walk, the expectation and the Laplace transform of cover times of spider graphs as integral representations. We also discuss some comparison results for spider graphs.

Keywords. Cover time, first hitting time, terminal time, tree, subdivision, spider graph.

#### JMI2010A-10(pp.99-108)

Title：Overview to mathematical analysis for fractional diffusion equations – new mathematical aspects motivated by industrial collaboration

Author : Junichi Nakagawa, Kenichi Sakamoto, and Masahiro Yamamoto

Abstract. The mathematics turns out to be useful for creation of innovations in the industry, and the mathematical knowledge and thinking manners are used effectively for that purpose. However, this is only one aspect of the industrial mathematics where various existing mathematical knowledge are applied for solving required subjects from industry. On the other hand, one can see the opposite direction; Pursuit of industrial purposes inspires to create new fields of mathematics by motivating and activating existing researches. This is an important aspect of the industrial mathematics because

it does not only give tools for solving concrete problems, but also enriches the existing branches of mathematics. In this article, as such a possible example, we discuss a fractional diffusion equation which has been studied already comprehensively from the theoretical interests, but the researches are expanded as a mathematical topic in view of the industrial applications.

Keywords. mathematics motivated by industrial mathematics, fractional diffusion equation, fractional calculus,

well-posedness, qualitative properties

#### JMI2010A-11(pp.109-114)

Title：ARX models for time-varying systems estimated by recursive penalized weighted least squares method

Author : Pan Qin, Ryuei Nishii, Tadashi Nakagawa and Takayoshi Nakamoto

Abstract. We consider the modeling problem for time-varying systems by Auto-Regressive models with eXogenous

variables (ARX) models. To track the variations of time-varying systems, we propose a new Recursive Penalized Weighted Least Squares (RPWLS) method to estimate the ARX models. Furthermore, by virtue of Generalized Information Criterion, the proper ARX models by RPWLS are selected. Numerical examples are provided to verify the performance of the proposed RPWLS method.

Keywords. ARX model, Time-varying systems, GIC, Model selection

## JMI2010B