応用数理

A mathematical framework is proposed for intelligibility of abilities of acquiring skilled motions through practices and versatile adaptive actions adjusted to changes of the environment. It is pointed out that the passivity property of not only the original robot dynamics but also the residual error dynamics between the desired motion and the actual motion is crucial in learnability and adaptability of skilled motions. By means of such learnability and adaptability, both a simple learning control and a model-based adaptive scheme are proposed for nonlinear mechanical systems with multi joints of revolute-type such as robot arms and mechanical hands. It is also suggested that the present PD or PID servos used in industrial manipulators should be replaced with a linear combination of a saturated position (SP) feedback and the conventional velocity (D) feedback, in order to easily implement the learnability and adaptability in such mechanical systems.

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It is established through a simulation study that if Kepler would have known AIC he could have used it to "prove" his theory. The reasons why the AIC works were sought and it was found that: 1. The AIC could be an un-biased estimate of the "mean expected log likelihood" of models, under mild conditions. 2. The "Mean expected log likelihood" is a reasonable criterion to compare statistical models. 3. The variance of the AIC is in effect small enough to allow the practical comparison of models. 4. If the bootstrap method is applicable, the variance can be estimated.

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This paper describes the inverse problems concerning the biological systems. Most of the inverse problems are ill posed so that it is difficult to obtain the unique solutions. We propose here one of the unique solution pattern searching methods for the inverse problems instead of evaluating the unique solutions. This method is called the sampled pattern matching (SPM) method utilizing the Cauchy-Schwarz relationship. The theoretical background of our method is given in terms of the modified neural networks, generalized factor analysis and incomplete Fourier series. The examples demonstrate the usefulness when apply it to the estimation of neural behavior in biological systems.

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Algebraic-geometric (AG) codes are a class of error-correcting codes which was introduced by V.D. Goppa in 1981. They are important not only theoretically but also practically, and recently have interested many investigators in the field of computer science. These codes are derived from rational functions of algebraic curves or surfaces over finite fields. In a mathematical sense the class of AG codes can be considered to emcompass the whole class of linear codes. Among them, there are well-known and practically important codes, e.g., BCH codes, Reed-Solomon codes, classical Goppa codes, etc. In this munuscript, we introduce AG codes and discuss several properties of them which are important in using them for error-free transmission. Furthermore, we explain decoding methods of some AG codes, i.e., how to correct errors by using them. We have tried to explain the subject on the basis of linear algebra, without any requirement of background knowledge of algebraic geometry.

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