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La modélisation, aspect fondamental de toutes les sciences appliquées, vise à établir des relations mathématiques entre les différentes variables caractéristiques d'un système. Les travaux développés dans ce mémoire entrent dans le cadre de la modélisation par fonctions orthogonales de systèmes dynamiques discrets LTI et stables. plus particulièrement, les fonctions Dirac delta, Laguerre, Kautz, type Meixner et celles issues de la base orthogonale généralisée y sont présentées. Elles différent entre elles par le nombre de pôles qu'elles admettent et par leur convenance à l'identification de systèmes, à partir de conditions initiales mal connues. de plus, une nouvelle base orthogonale est proposée. Elle généralise la définition de la base de type Meixner à un nombre quelconque de pôles réels et est, de ce fait, plus adaptée à la modélisation de systèmes ayant plusieurs dynamiques, a partir de conditions initiales mal connues. Pour chaque base, la convergence des coefficients de Fourier et des séries fonctionnelles a été étudiée. Elle a pour but de justifier la troncature de la représentation, initialement infinie, a un ordre fini. La décomposition d'une fonction de transfert d'un modèle surparamétrise, sur des bases orthogonales, permet, dans certains cas, d'obtenir un modèle d'ordre réduit. En effet, un choix optimal de pôles minimise le nombre de filtres nécessaires à l'approximation de la fonction de transfert originelle. Dans ce cadre, le calcul des pôles optima passe obligatoirement par la résolution d'équations algébriques connues sous le nom de conditions d'optimalité. A partir d'une étude bibliographique détaillée, un des points forts, développés dans ce mémoire, concerne la synthèse des conditions d'optimalité de la base orthogonale généralisée pour un choix de pôles réels. ces résultats ont été utilisés, d'une part, pour la réduction d'ordre de modèles et, d'autre part, pour l'identification de systèmes linéaires par des filtres issus de la base orthogonale généralisée. Une extension aux systèmes non linéaires a également été proposée par une technique de multimodèle.
Space, structure, and randomness: these are the three key concepts underlying Georges Matheron’s scientific work. He first encountered them at the beginning of his career when working as a mining engineer, and then they resurfaced in fields ranging from meteorology to microscopy. What could these radically different types of applications possibly have in common? First, in each one only a single realisation of the phenomenon is available for study, but its features repeat themselves in space; second, the sampling pattern is rarely regular, and finally there are problems of change of scale. This volume is divided in three sections on random sets, geostatistics and mathematical morphology. They reflect his professional interests and his search for underlying unity. Some readers may be surprised to find theoretical chapters mixed with applied ones. We have done this deliberately. GM always considered that the distinction between the theory and practice was purely academic. When GM tackled practical problems, he used his skill as a physicist to extract the salient features and to select variables which could be measured meaningfully and whose values could be estimated from the available data. Then he used his outstanding ability as a mathematician to solve the problems neatly and efficiently. It was his capacity to combine a physicist’s intuition with a mathematician’s analytical skills that allowed him to produce new and innovative solutions to difficult problems. The book should appeal to graduate students and researchers working in mathematics, probability, statistics, physics, spatial data analysis, and image analysis. In addition it will be of interest to those who enjoy discovering links between scientific disciplines that seem unrelated at first glance. In writing the book the contributors have tried to put GM’s ideas into perspective. During his working life, GM was a genuinely creative scientist. He developed innovative concepts whose usefulness goes far beyond the confines of the discipline for which they were originally designed. This is why his work remains as pertinent today as it was when it was first written.
Scattering theory deals with the interactions of waves with obstacles in their path, and low frequency scattering occurs when the obstacles involved are very small. This book gives an overview of the subject for graduates and researchers, for the first time unifying the theories covering acoustic, electromagnetic and elastic waves.
About The Book: This book explores the heart of pattern recognition concepts, methods and applications using statistical, syntactic and neural approaches. Divided into four sections, it clearly demonstrates the similarities and differences among the three approaches. The second part deals with the statistical pattern recognition approach, starting with a simple example and finishing with unsupervised learning through clustering. Section three discusses the syntactic approach and explores such topics as the capabilities of string grammars and parsing; higher dimensional representations and graphical approaches. Part four presents an excellent overview of the emerging neural approach including an examination of pattern associations and feedforward nets. Along with examples, each chapter provides the reader with pertinent literature for a more in-depth study of specific topics.
Semidefinite and conic optimization is a major and thriving research area within the optimization community. Although semidefinite optimization has been studied (under different names) since at least the 1940s, its importance grew immensely during the 1990s after polynomial-time interior-point methods for linear optimization were extended to solve semidefinite optimization problems. Since the beginning of the 21st century, not only has research into semidefinite and conic optimization continued unabated, but also a fruitful interaction has developed with algebraic geometry through the close connections between semidefinite matrices and polynomial optimization. This has brought about important new results and led to an even higher level of research activity. This Handbook on Semidefinite, Conic and Polynomial Optimization provides the reader with a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization, and polynomial optimization. It contains a compendium of the recent research activity that has taken place in these thrilling areas, and will appeal to doctoral students, young graduates, and experienced researchers alike. The Handbook’s thirty-one chapters are organized into four parts: Theory, covering significant theoretical developments as well as the interactions between conic optimization and polynomial optimization; Algorithms, documenting the directions of current algorithmic development; Software, providing an overview of the state-of-the-art; Applications, dealing with the application areas where semidefinite and conic optimization has made a significant impact in recent years.