## PUBLICATIONS

### Journal of Math-for-Industry

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

Title：JMI2012A-Front Cover

#### JMI2012A-1 (pp. 1-4)

Title：Kernel perturbations for convolution first kind Volterra integral equations

Author : Frank R. de Hoog and Robert S. Anderssen

Abstract. Because of their causal structure, (convolution) Volterra integral equations arise as models in a variety of real-world situations including rheological stress-strain analysis, population dynamics and insurance risk prediction. In such practical situations, often only an approximation for the kernel is available. Consequently, a key aspect in the analysis of such equations is estimating the effect of kernel perturbations on the solutions. In this paper, it is shown how kernel perturbation results derived for the interconversion equation of rheology can be extended to the analysis of kernel perturbations for first kind convolutional integral equations with positive kernels, solutions and forcing terms.

Keywords. linear viscoelasticity, interconversion, kernel perturbations, convolution, first kind Volterra integral equations

#### JMI2012A-2 (pp. 5-15)

Title：On some properties of a discrete hungry Lotka-Volterra system of multiplicative type

Author : Yosuke Hama, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa Nakamura

Abstract. Two kinds of discrete hungry Lotka-Volterra systems (dhLV) are known as discretizations of the additive type hungry Lotka-Volterra system and the multiplicative one. By associating the dhLV of additive type (dhLV_I) and the discrete hungry Toda equation (dhToda) with LR transformations, some of the authors give a B\"acklund transformation between these two systems. In this paper, from the dhLV of multiplicative type (dhLV_{II}), we first derive the qd-type dhLV_{II}. Through finding the positivity of the qd-type dhLV_{II} and the LR transformation associated with the dhLV_{II}, we present B\"acklund transformations among the dhLV_I, the dhLV_{II} and the dhToda. Moreover, by using one of the B\"acklund transformations, we show asymptotic convergence of the qd-type dhLV_{II}.

Keywords. B\"acklund transformation, LR transformation, asymptotic convergence, discrete hungry Toda equation, discrete hungry Lotka-Volterra system

#### JMI2012A-3 (pp. 17-23)

Title：An algorithm for calculating D-optimal designs for trigonometric regression through given points in terms of the discrete modified KdV equation

Author : Hiroto Sekido

Abstract. Optimal designs are required to make efficient statistical experiments. Calculation of D-optimal designs is considerably simplified by using canonical moments or trigonometric canonical moments. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In the previous paper, Sekido 2011, a method for calculating D-optimal designs for polynomial regression through a fix point is presented. In this paper, trigonometric regression models through given points are discussed. In order to calculate the D-optimal designs for these models, a useful relationship between trigonometric canonical moments and a class of discrete integrable systems is found. By using trigonometric canonical moments and a discrete integrable system, a new algorithm for calculating D-optimal designs for these models is proposed.

Keywords. D-optimal design, trigonometric canonical moment, trigonometric regression model, integrable system, discrete modified Korteweg-de Vries equation

#### JMI2012A-4 (pp. 25-33)

Title：Piecewise truncated conical minimal surfaces and the Gauss hypergeometric functions

Author : Yosiroh Machigashira

Abstract. The catenary is the curve which a hanging chain forms, that is, mathematically, the graph of the function t \mapsto c cosh(t/c) for a constant c > 0. The study of catenaries is applied to the design of arches and suspension bridges. The surface of revolution generated by a catenary is called a catenoid. It is well-known that a catenoid is a minimal surface and the shape which a soap film between two parallel circles forms. In this article, we consider the approximation of a catenoid by combinations of some truncated cones keeping the minimality in a certain sense. In investigating the minimal combinations, the theory of the Gauss hypergeometric functions plays an important role.

Keywords. hypergeometric function, truncated cone, catenoid

#### JMI2012A-5 (pp. 35-39)

Title：Zeros of extended zeta polynomials for coding theory

Author : Katsuhiko Ono

Abstract. In 1999, Iwan Duursma defined the zeta polynomial for coding theory and formulated an analogue of the Riemann hypothesis for coding theory. In this paper, we consider certain self-reciprocal polynomials which generalize some zeta polynomials, and investigate whether the analogue of the Riemann hypothesis holds for this generalization. We show that in some cases the analogue of the Riemann hypothesis holds true, and conjecture that this is always the case.

Keywords. zeros, zeta polynomials for coding theory

#### JMI2012A-6 (pp. 41-48)

Title：Towards statistical modeling of tsunami occurrence with regional frequency analysis

Author : Jonathan R. M. Hosking

Abstract. Regional frequency analysis is a statistical method for frequency estimation of extreme environmental events. Data for several sites are combined to improve the estimates of event frequencies at any one site. The computations are typically based on L-moments, which are summary statistics that have good properties of efficiency and robustness for describing data from heavy-tailed probability distributions. We summarize this work and apply it to a worldwide data set of historical records of tsunami magnitudes, obtaining estimates of the frequency distribution of tsunami runup height for essentially any location in the Pacific basin with exposure to tsunami events. The results have potential application to risk estimation and design of structures in tsunami-prone locations.

Keywords. extreme values, frequency estimation, L-moments, rare events, runup height.

This is an invited paper presented at the Forum "Math-for-Industry" 2011.

#### JMI2012A-7 (pp. 49-53)

Title：Modeling through self-assembly

Author : Jos Stam

Abstract. In this paper we explore a new paradigm for modeling geometric structures through self-assembly. This approach is inspired by the new emerging field of nano-technologies. At the very small nano-scales the laws of physics are different from the ones at the scales we are used to in daily life. Gravity is negligible and Brownian motion induced by heat is a crucial factor. In fact the latter provides the vital force that drives the process of creating new shapes at nano-scales: heat induced noise makes it possible for programmed DNA chains with free bonds to form and create shapes by bonding with other strands. The key challenge is how to program the DNA strands to create specific shapes. In this paper we introduce some concepts of this exciting new area of research and describe a couple of concrete self-assembly modeling examples. The goal of this paper really is two-fold: (1) to show an illustration of self-assembly at work in an appealing way using computer graphics and (2) to bring this exciting field to the attention of researchers in other fields.

Keywords. molecular biology, self-assembly, computer graphics, geometric modeling, physics-based simulation

This is an invited paper presented at the Forum "Math-for-Industry" 2011.

#### JMI2012A-8 (pp. 55-71)

Title：Subtraction-free recurrence relations for lower bounds of the minimal singular value of an upper bidiagonal matrix

Author : Takumi Yamashita, Kinji Kimura and Yoshimasa Nakamura

Abstract. On an N \times N upper bidiagonal matrix B, where all the diagonals and the upper subdiagonals are positive, and its transpose B^T, it is shown in the recent paper [4] that quantities J_M(B) \equiv Tr(((B^T B)^M)^{-1}) (M = 1, 2, \dots) gives a sequence of lower bounds \theta_M(B) of the minimal singular value of B through \theta_M(B) \equiv (J_M(B))^{-1/(2M)}. In [4], simple recurrence relations for computing all the diagonals of ((B^T B)^M)^{-1} and ((BB^T)^M)^{-1} are also presented. The square of \theta_M(B) can be used as a shift of origin in numerical algorithms for computing all the singular values of B. In this paper, new recurrence relations which have advantages over the old ones in [4] are presented. The new recurrence relations consist of only addition, multiplication and division among positive quantities. Namely, they are subtraction-free. This property excludes any possibility of cancellation error in numerical computation of the traces J_M(B). Computational cost for the trace J_M(B) (M = 1, 2, \dots) and efficient implementations for J_2(B) and J_3(B) are also discussed.

Keywords. lower bound of the minimal singular value, subtraction-free recurrence relations

#### JMI2012A-9 (pp. 73-78)

Title：Asymptotic tail dependence of the normal copula

Author : Hiroki Kondo, Shingo Saito and Setsuo Taniguchi

Abstract. Copulas have lately attracted much attention as a tool in finance and insurance for dealing with multiple risks that cannot be considered independent. The normal copula, widely used in practice, is known to have the same tail dependence parameter as the product copula. The present paper brings into question the common interpretation of this fact as evidence that the normal copula lacks tail dependence, both by providing numerical examples and by mathematically determining the asymptotic behaviour of the tail dependence.

Keywords. copula, normal copula, tail dependence

#### JMI2012A-10 (pp. 79-83)

Title：A viscoelastic model for time-dependent simulating analysis of the Wenchuan earthquake fault

Author : Cheng Hua, Jin Cheng and Qi-fu Chen

Abstract. The sudden big earthquake which happened in the Wenchuan county of China in 2008 was an extraordinary type of earthquake. Its exact mechanism is still unknown. The paper presents a simplified computational approach to develop a model to simulate the long-term evolution of the Wenchuan earthquake fault by means of a viscoelastic finite-element method, in order to investigate the dynamic mechanism of the 2008 Wenchuan earthquake. The relevant characteristics of crustal stress fields and displacement fields around the fault are analyzed. It is suggested that the accumulated earthquake energy was mainly due to deep crust motion rather than to surface motion. The study helps show that viscoelastic modeling is a powerful tool for simulating natural phenomena such as crustal movement and its implication for earthquakes.

Keywords. viscoelastic, finite-element method, Wenchuan earthquake, crustal fault

This is an invited paper presented at the Forum "Math-for-Industry" 2011.

## JMI2012A