Math-for-industry Education&Research Hug

Abstract. Compound distributions appear in applications to queueing theory and to risk theory. A local property of those distributions on the real line is discussed. The result helps to derive equivalnce conditions to be local subexponential for infinitely divisible distributions on the real line.

File：JMI2009B-1

Abstract. The dynamics of a pulse for reaction-diffusion systems in 1D is considered in the neighborhood of the bifurcation point with codimension two, at which both of saddle-node and drift bifurcations occur at the same time. It is theoretically shown that when the bifurcation parameter is close to such a bifurcation point, a pulse moves with oscillation, and then starts to split.

File：JMI2009B-2

Abstract. In this paper, a problem of integrating density functions is considered. First, the Bregman divergence in the space of positive finite measures is introduced, and properties of consistent subspaces and well-behaved parametric models associated with the Bregman divergence is investigated. Based on those results, a natural method for integration of estimates from the Bregman divergence is derived.

File：JMI2009B-3

Abstract. H. Koch \cite{ko1},\cite{ko2} derived formulas which describe design structure supported by the codewords of fixed weight in the extremal binary self-dual doubly even codes. His method uses the lattices (not extremal lattices) constructed from such codes and the modular forms associated with these lattices. The formulas imply the so called Assmus-Mattson theorem for binary codes as partial results plus an extra formula which is not obtainable from the original Assmus-Mattson theorem. \par In the present paper we develop a similar method to derive the formulas for the ternary self-dual extremal codes. The method also uses the lattice theory and the modular form theory. However in using the lattice theory the present paper differes largely from the ones by Koch.

File： JMI2009B-4

Abstract. Perturbations in a shear flow exhibit rather complex behavior -- waves may grow algebraically even when the spectrum of disturbances is entirely neutral (no exponential instability). A shear flow brings about non-selfadjoint property, invalidating the standard notion of dispersion relations, and it also produces a continuous spectrum that is a characteristic entity in an infinite-dimension phase space. This paper solves an initial value problem using the Laplace transform and presents a new-type of algebraic instability that is caused by resonant interaction between acoustic modes (point spectrum) and vortical continuum mode (continuous spectrum). Such a resonance is possible when variation of velocity shear is comparable to sound speed.

File： JMI2009B-5

Abstract. We present a set of variant Hermite tetrahedral elements of degree three for three-dimensional problems.
A finite element space constructed from these elements has advantages that the degrees of freedom are much smaller than those of the Lagrange element and that it is easily applicable to problems subject to Dirichlet boundary conditions. Applying it to Poisson problems, we prove best possible a priori error estimates. Two numerical examples reflect the theoretical results.

File：JMI2009B-6

Abstract. Lagrangian invariants of hydrodynamic, magnetohydrodynamic (MHD) and Hall MHD fluids are reviewed in a general viewpoint of differential topology. It is shown that, introducing the particle trajectory map (PTM) and its inverse (back-to-labels map, BLM) and utilizing their spatial derivatives, one can easily derive the conservation laws along the Lagrangian trajectories. All the invariants are derived as composite of such elementary invariants as entropy per unit mass, impulse, mass density, and electromagnetic vector potential and their derivatives. Treating the spatial derivatives of PTM and BLM as kinds of Lagrangian invariants formally, one can understand the following conservation laws as Lagrangian invariants:Cauchy's formula, Weber's transformation, Ertel's theorem, Ertel-Rossby's theorem (i.e. helicity density), magnetic-helicity and cross-helicity in a MHD fluid, hybrid-helicity in a Hall MHD fluid.

File：JMI2009B-7

Abstract. We propose a new approach to tackling multi-objective optimization problems. Our method uses symbolic computation called quantifier elimination. From experimental data we first make an approximated polynomial model　for each objective function and then solve a first-order formula related to those functions to find the feasible region, which contains information on the Pareto optimal front. Our approach has an advantage over classical numerical optimization methods that return only one optimal point at a time. Furthermore, by introducing into a formula an adjusting point and a new variable that restricts the parameter ranges we can visualize how the objective functions locally behave. This idea leads us to a new criterion for measuring the robustness against production tolerance. We also show how our methods are applied to the design problem of a hard disk slider.

File：JMI2009B-8

Abstract. On the basis of our 10-year experiences of collaborating researches by mathematicians in academia and engineers from industry, we present a control problem in blast furnace as example of bilateral practice of industrial mathematics which is really used in factory. Moreover we discuss a desirable style of industrial mathematical collaboration by mathematicians and engineers.

File： JMI2009B-9

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-Scholes option pricing formula), is applied to cumulative losses. While Heyer applied his model to estimating INBR (incurred but not yet reported) losses of each accident year, he made no observation on the year-on-year loss (the loss to be paid in the specic future year). To estimate year-on-year losses is one of urgent issues in the non-life insurance industry. In this paper, as another application of the random walk model, the conditional distribution and the conditional condence interval of the year-on-year loss to be paid in the specific future year, being given the cumulative losses of the present, will be investigated.

File：JMI2009A-1

Abstract. Mack estimated the mean squared errors of the outstanding claims reserve of each accident year and of the overall claims reserve in order to obtain their condence intervals within his distribution-free model. We generalise his formulae by allowing for arbitrary exponents in the variance assumption. Our formula is also capable of giving a condence interval of the amount that the insurer is liable to pay each year.

File：JMI2009A-2

Abstract. We consider the problem of constructing a functional regression modeling with functional predictors and a functional response. Discretely observed data for each individual are expressed as a smooth function, using Gaussian basis functions. The functional regression model is estimated by the maximum penalized likelihood method, assuming that the coefficient parameters are transformed into a functional form. A crucial issue in constructing functional regression models is the selection of regularization parameters involved in the regularization method. We derive informationtheoretic and Bayesian model selection criteria for evaluating the estimated model. Monte Carlo simulations and real data analysis are conducted to examine the performance of our functional regression modeling strategy.

File：JMI2009A-3

Abstract. Elliptic curve cryptosystems (ECC) are suitable for memory-constraint devices like smart cards due to their small key-size. Non-adjacent form (NAF) is a signed binary representation of integers used for implementing ECC. Recently, Schmidt-Samoa et al. proposed the fractional $w$MOF (Frac-$w$MOF), which is a left-to-right analogue of NAF, where $w$ is the fractional window size $w=w_{0}+w_{1}$ of integer $w_{0}$ and fractional number $w_{1}$. On the contrary to NAF, there are some consecutive none-zero bits in Frac-$w$MOF, and thus the zero-run length of the Frac-$w$MOF is not equal to that of the variants of NAF. In this paper we present an asymptotic formula of zero-run length of Frac-$w$MOF. Indeed, the average zero-run length of the Frac-$w$MOF is asymptotically $w\frac{2^{w_{0}+1}}{2^{w_{0}+1}-1}$, which is longer than that of the fractional $w$NAF.

File：JMI2009A-4

Abstract. Several digital signature schemes based on the discrete logarithm problem or the computational Diffie-Hellman problem which have tight security reductions are proposed in these years. For these schemes, the groups of rational points on elliptic curves are employed for efficiency. These schemes need cryptographic hash functions with the range in the group of rational points on elliptic curves in their procedures for generating/verifying signatures. However, no efficient algorithm for such hash functions is known except for special type of elliptic curves, consequentially, the signature schemes becomes inefficient even if elliptic curves are employed. In this paper, in order to improve the efficiency of the signature schemes, a new method of generating rational points on elliptic curves is proposed. The proposed method is based on the norm map from a quadratic extension field of the definition field. This method consists of one powering for determination of quadratic residuosity and a square root extraction, and at most 16 times multiplications in the definition field. The security when the proposed algorithm is used as a hash function is also investigated.

File：JMI2009A-5

Abstract. In this paper, we study the asymptotic behavior of the number of composite integers written by products of two primes. Such integers are sometimes called by the RSA integers, because these are used in the RSA cryptosystems. The number of all such integers has been already studied by Landau, Sathe, Selberg etc. Furthermore, the number of integers with n = pq and p < q < cp for a fixed c > 1 was recently studied by Decker and Moree. The aim of this paper is to extend Decker-Moree's result, and the main theorem describes the asymptotic formula of the number of integers with p < q < f(p) for a xed increasing function f.

File：JMI2009-6

Abstract. We realized operations appeared in the theory of automata using Haskell languages. Using the benefits of functions of lazy evaluations in Haskell, we can express a language set which contains infinite elements as concrete functional notations like mathematical notations. Our modules can be used not only for analyzing the properties about automata and their application systems but also for self study materials or a tutorial to learn automata, grammar and language theories. We also implemented the modules for sticker systems. Paun and Rozenberg explained a concrete method to transform an automaton to a sticker system in 1998. We modified their definitions and improved their insufficient results. Using our module functions, we can easily define finite automata and linear grammars and construct sticker systems which have the same power of finite automata and linear
grammars.

File：JMI2009A-7

Abstract. In this paper we propose an algebraic approach to the discrete-time polynomial spectral factorization problem, which has a signicant importance in signal processing and control for nite dimensional linear systems. We also attempt to generalize the approach and establish a new framework of symbolic optimization of algebraic functions that is relevant to possibly a wide variety of practical application areas. The crucial aspects of the framework are the suitable use of algebraic methods coupled with the discovery and exploitation of structural properties of the problem in the conversion process into the framework, and the feasibility of algebraic methods when performing the optimization. Two examples are also included to demonstrate the signicance and relevance of the proposed approach and framework.

File：JMI2009A-8

Abstract. By using the finite element approximation and constructive a priori error estimates, a new formulation for proving the existence of solutions for nonlinear parabolic problems is presented. We present a method to estimate the norm of the linearized inverse operator for concerned nonlinear problem. Then we formulate a verification principle for solutions by using the Newton-type operator incorporating with Schauder's fixed point theorem.

File：JMI2009A-9

Abstract. Despite the acquisition of expertise in mathematics during primary and secondary education, Japan is not sufficiently utilizing its expertise in mathematics from a global viewpoint. Simultaneously, researchers from the industry and other fields in Japan have high expectations for mathematics. This manuscript surveys these issues from a macro perspective and clarifies these situations along with the structural bottlenecks underlying the neglect of mathematics research in Japan’s science and technology policy; initiatives undertaken by the Japanese government, which has taken cognizance of this situation, are also discussed.

File：JMI2009A-10