Michio Inoue

Application of the Haar functions to solution of differential equations

In this paper, it is proposed that Haar functions should be used for solving ordinary differential equations of a time variable in facility. This is because integrated forms of Haar functions of any degree can be illustrated by linear- and linear segment-functions like as triangles. Fortunately, since they are placed where Haar functions are defined in a specified form respectively, these functions are computable by algebraic operations of quasi binary numbers. Therefore, when a given function is approximated in a form of stairsteps on a Haar function system their integration can be termwise executed by shift and add operations of coefficients of the approximation. The use of this system is comparable with an application using the midpoint rule in numerical integration. In this line, nonlinear differential equations can be solved like as linear differential equations.

Composition of Messages on Winking by ALS Patients

Abstract This paper is concerned with a development of a Communication Aid (CA) for Amyotrophic Lateral Sclerosis (ALS) patients who lose the physical ability of communication. A CA is just the equipment that provides the facilities for conversation with them. In our CA, inquiries are arranged in a matrix form, and thrown to patients by means of crosswise scanning. If the patients find out the desired one, among the exhibited ones, they are requested to send a sign YES, by winking before a CCD camera. It is because winking is a most simple motion which can be done by a little physical power. As a matter of fact, wink happens also due to physiological demands. Hereupon it becomes necessary to discriminate the intentional wink from the unintentional wink clearly. This paper proposes a method of remote sensing of intentional winking as the sign of affirmation by a Time-Delayed Neural Network(TDNN), and describes how to compose their messages on their winking.

Fractional power approximation and its generation

For an analog simulating system, an approximating system is proposed. Its mathematical form is expressed by an algebraic equation: ƒ (x) ≈ α + β χ + γχk with four parameters given by real numbers. Their values can be determined so as to satisfy a best fit in a Chebyshev sense. Then, the accuracy is of the same order with that obtained by any kind of ordinary power series up to terms o f the third order. It is noticeable that a given function can be accurately approximated by this equation without destroying its uniform continuity.

Fractional power approximations of elliptic integrals and bessel functions

In the previous papers [1]∽[3], fractional powers were used to approximate elementary functions and their usefulness was proved with experimental results. In the present paper, some further investigations are reported. That is, elliptic integrals in Legendres canonical form and Bessel functions are approximated by fractional powers. As the fractional power approximation, f(x) ⋍ c0 + c1x + c2xp is discussed. When all coefficients c0, c1, c2, p are properly assigned, the accuracy of this approximation becomes comparable to that of the Chebyshev approximation using polynomials up to the third degree.

Generation of two-variable functions based on the polynomial approximation

This paper describes how to generate uniformly continuous functions of two variables without making any vertex. In this system, all functions are obtained as an algebraic sum of linearly independent two-variable polynomials with each weight. Then, the polynomials are given by simulating nonlinear currents passing through semiconductors with dual control electrodes in their uniformly continuous regions. The proper weights can be uniquely determined by solving a matrix equation related to the characteristics of both the desired function and the semiconductors used in this system. Different functions can be produced by adjusting the weights. The usefulness of this system is successfully proved by experimental results.

Separate and/or simultaneous generations of non-linear functions inherent in non-ohmic resistivities

In this paper, it is described that a multiple-function generator can be realized without employing any separate function-generating units and selector-logics. According to this proposal, a uniformly continuous non-linearity o f a non-ohmic resistance is first expressed by a linear combination of several interesting non-linear functions with respect to the same input variable. Then, the desired functions can be obtained by summing up algebraically the linearly independent non-linear currents passing through every non-ohmic resistance with their proper weights. It is noticeable that an adjustment of the weights will permit different non-linear functions separately and/or simultaneously. Details of this system will be furnished with quantitative investigations and experimental results.

Detection of the event related brain potential and its application to communication aids

Abstract This paper first introduces handicaps of ALS patients and appeals the necessity of CA equipment for them. Then, its description is assigned to explain the property of ERP deflection and a method of detecting it. Further, this paper referrs to use the ERP deflection for CA equipment.

On the use of an exponential function in approximation of elliptic integrals

In the previous papers[1]–[3], nonlinear continuous functions could be well simulated by nonlinear resistance. Its mathematical basis came from the applicability of the fractional power approximation. From a view of using transistor-junction characteristics, the use of exponential functions will make it possible to have closed relations between given functions and transistor-junction characteristics. Hereby, in simulation of special functions, especially, of elliptic integrals, a form of approximation containing an exponential function is proposed, so that F(x, α)E (x, α)Π (x, α, n) ⋍ c0+c1x+c2epx, where F(x, α), E(x, α) and Π(x, α, n) are elliptic integrals of the first, second and third kinds in the Legendres canonical form with their modular angles α and a parameter n. The same order of accuracy is obtained in the simulation of the elliptic integrals as they are approximated by the fractional powers.