Ryuichi Ashino

Behind and beyond the Matlab ODE suite

Abstract The paper explains the concepts of order and absolute stability of numerical methods for solving systems of first-order ordinary differential equations (ODE) of the form describes the phenomenon of problem stiffness , and reviews explicit Runge-Kutta methods, and explicit and implicit linear multistep methods. It surveys the five numerical methods contained in the Matlab ODE suite (three for nonstiff problems and two for stiff problems) to solve the above system, lists the available options, and uses the odedemo command to demonstrate the methods. One stiff ode code in Matlab can solve more general equations of the form M ( t ) y ′ = f ( t , y ) provided the Mass option is on.

An uncertainty principle for quaternion Fourier transform

We review the quaternionic Fourier transform (QFT). Using the properties of the QFT we establish an uncertainty principle for the right-sided QFT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternion signal minimizes the uncertainty.

Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system.

Two-dimensional quaternion wavelet transform

Abstract In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.

Advances in pseudo-differential operators

Microlocal Analysis and Applications.- The Conormal Symbolic Structure of Corner Boundary Value Problems..- A New Proof of Global Smoothing Estimates for Dispersive Equations.- Gevrey Hypoellipticity of p-Powers of Non-Hypoelliptic Operators.- Continuity in Weighted Sobolev Spaces of LP Type for Pseudo-Differential Operators with Completely Nonsmooth Symbols.- Symmetry-Breaking for Wigner Transforms and LP-Boundedness of Weyl Transforms.- Pseudo-Differential Operators and Schatten-von Neumann Classes.- Localization Operators via Time-Frequency Analysis.- Localization Operators with LP Symbols on Modulation Spaces.- Convolutions and Embeddings for Weighted Modulation Spaces.- Pseudo-Differential Operators, Microlocal Analysis and Image Restoration.- Applications of Wavelet Transforms to System Identification.- Two-Dimensional Wavelet Bases for Partial Differential Operators and Applications.

Blind source separation of spatio-temporal mixed signals using time-frequency analysis

The cocktail party problem deals with the specialized human listening ability to focus ones listening attention on a single talker among a cacophony of conversations and background noise. The blind source separation problem corresponds to a way to enable computers to solve the cocktail party problem in a satisfactory manner. The simplest version of spatio-temporal mixture problem, which is a type of blind source separation problems, is solved using time-frequency analysis. The analytic wavelet transform is used to represent time-frequency information and a numerical simulation is given. †Dedicated to Professor Hideo Soga on the occasion of his 60th birthday.

Fatigue damage evaluation of adhesively bonded butt joints with a rubber-modified epoxy adhesive

A damage evolution of adhesively bonded butt joints with a rubber-modified adhesive has been investigated under cyclic loading. An isotropic continuum damage model coupled with a kinetic law of damage evolution was applied to the butt joint. To solve the kinetic law, analytic and numerical methods were tried: the former solution was derived with some simplifications and the latter one was derived rigorously without simplications. On comparing the analytic solutions with the numerical ones, it was confirmed that differences in the two solutions were small. Furthermore, the estimated S-N curves based on the analytic equation agreed well with experimental data.

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

Microlocal filtering with multiwavelets

Abstract Hyperfunctions in R n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in C n. Orthonormal multiwavelets, which are a generalization of orthonormal single wavelets, generate a multiresolution analysis by means of several scaling functions. Microlocal analysis is briefly reviewed and a multiwavelet system adapted to microlocal filtering is proposed. A rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions. A fast algorithm for multiwavelet microlocal filtering is presented and several numerical examples are considered.

Mathematical background for a method on quotient signal decomposition

The blind source separation problem is discussed in this article. Focusing on the assumption of independency of the sources in the time-frequency domain, we present a mathematical formulation for the estimation problem of the number of sources. The proposed method uses the quotient of complex valued time-frequency information of only two observed signals to detect the number of sources. No fewer number of observed signals than the detected number of sources is needed to separate sources. The assumption on sources is quite general independence in the time-frequency plane, which is different from that of independent component analysis. We propose algorithms with feedback and give numerical simulations to show the method works well even for noisy case.