Math-for-industry Education&Research Hug

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

File：JMI2010A-1

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

File：JMI2010A-2

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

File：JMI2010A-3

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

File：JMI2010A-4

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

File：JMI2010A-5

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.

File：JMI2010A-6

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

File：JMI2010A-7

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

File：JMI2010A-8

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.

File：JMI2010A-9

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

File：2010A-10

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

File：JMI2010A-11