SS-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption

Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure

Climbing and Walking Robots

With the advancement of technology, new exciting approaches enable us to render mobile robotic systems more versatile, robust and cost-efficient. Some researchers combine climbing and walking techniques with a modular approach, a reconfigurable approach, or a swarm approach to realize novel prototypes as flexible mobile robotic platforms featuring all necessary locomotion capabilities. The purpose of this book is to provide an overview of the latest wide-range achievements in climbing and walking robotic technology to researchers, scientists, and engineers throughout the world. Different aspects including control simulation, locomotion realization, methodology, and system integration are presented from the scientific and from the technical point of view. This book consists of two main parts, one dealing with walking robots, the second with climbing robots. The content is also grouped by theoretical research and applicative realization. Every chapter offers a considerable amount of interesting and useful information

The validity and validation of mathematical models: Methodological, theoretical, and practical studies with emphasis on the modelling of complex biological systems

In recent years there has been a tremendous growth in the development and application of mathematical models in all areas of science and engineering. Aided by the advances and availability of computers, models have been used in many new areas, such as biology and the social sciences, and applied to increasingly complex systems. At the same time, model validity and validation have become correspondingly more problematic yet received little attention. The aims of this thesis are to clarify the meaning of model validity, to develop a range of procedures for model validation, and to consider in depth the validity of a number of specific models. The main focus is the use of models in systems science and in biology and medicine. A review of the scientific literature of model validity and validation is made which reveals many techniques for empirical validation, but exposes the lack of a consistent conceptual approach towards model validity. In reviewing the philosophy of science with reference to validity and validation, the importance of regarding models and validation as part of an evolving research programme and of heuristic considerations in assessing model validity are emphasised. A new and innovative theory of model validity is proposed which explicates model validity as a multidimensional concept closely related to modelling objectives. The different modelling objectives and types of data are classified and the various concepts of validity are expressed as validity criteria. The general relationship between modelling objectives, data, and validity criteria is explained. The theory is then used to devise a range of validation methodologies suitable for models in research areas at different stages of development. Models of the human cardiovascular, renal, and respiratory systems are used as case studies for validation. Extensive use is made of the conceptual framework of the theory of model validity and the validation methodologies. The results are a precise delimiting of the validity of the models, the areas of uncertainty, and the potential for future development. This indicates the critical value of the theory and the appropriateness of the methodologies to complex biological models. Further support for the theory and its wide applicability is obtained in using it to consider aspects of validity and validation of models in the social sciences. Finally, the implications of the work for modelling and validation in systems science and in biology and medicine are examined. In both areas it is shown that the theory of model validity leads to an improved understanding of the nature of modelling and validity, and that the validation methodologies are suitable for the critical and effective validation of a wide range of models. In biology and medicine specific recommendations are made for the types of model appropriate to different modelling objectives and for suitable techniques and methodologies for validation. This thesis contributes to an improved understanding of the concept of model validity and offers a repertoire of validation methodologies. On another level, it is a broad methodological study of the kind urgently required in systems science. More practically, however, much of the thesis is concerned with the detailed validation of three specific biological models