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The main subject of this book is the estimation and forecasting of continuous time processes. It leads to a development of the theory of linear processes in function spaces. Mathematical tools are presented, as well as autoregressive processes in Hilbert and Banach spaces and general linear processes and statistical prediction. Implementation and numerical applications are also covered. The book assumes knowledge of classical probability theory and statistics.
This book presents a systematic treatment of the Rademacher system, one of the most important unifying concepts in mathematics, and includes a number of recent important and beautiful results related to the Rademacher functions. The book discusses the relationship between the properties of the Rademacher system and geometry of some function spaces. It consists of three parts, in which this system is considered respectively in Lp-spaces, in general symmetric spaces and in certain classes of non-symmetric spaces (BMO, Paley, Cesaro, Morrey). The presentation is clear and transparent, providing all main results with detailed proofs. Moreover, literary and historical comments are given at the end of each chapter. This book will be suitable for graduate students and researchers interested in functional analysis, theory of functions and geometry of Banach spaces.
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.
This book involves ideas/results from the topics of mathematical, information, and data sciences, in connection with the main research interests of Professor Pardo that can be summarized as Information Theory with Applications to Statistical Inference. This book is a tribute to Professor Leandro Pardo, who has chaired the Department of Statistics and OR of the Complutense University in Madrid, and he has been also President of the Spanish Society of Statistics and Operations Research. In this way, the contributions have been structured into three parts, which often overlap to a greater or lesser extent, namely Trends in Mathematical Sciences (Part I) Trends in Information Sciences (Part II) Trends in Data Sciences (Part III) The contributions gathered in this book have offered either new developments from a theoretical and/or computational and/or applied point of view, or reviews of recent literature of outstanding developments. They have been applied through nice examples in climatology, chemistry, economics, engineering, geology, health sciences, physics, pandemics, and socioeconomic indicators. Consequently, the intended audience of this book is mainly statisticians, mathematicians, computer scientists, and so on, but users of these disciplines as well as experts in the involved applications may certainly find this book a very interesting read.
This volume of the Mathematics Studies presents work done on composition operators during the last 25 years. Composition operators form a simple but interesting class of operators having interactions with different branches of mathematics and mathematical physics. After an introduction, the book deals with these operators on Lp-spaces. This study is useful in measurable dynamics, ergodic theory, classical mechanics and Markov process. The composition operators on functional Banach spaces (including Hardy spaces) are studied in chapter III. This chapter makes contact with the theory of analytic functions of complex variables. Chapter IV presents a study of these operators on locally convex spaces of continuous functions making contact with topological dynamics. In the last chapter of the book some applications of composition operators in isometries, ergodic theory and dynamical systems are presented. An interesting interplay of algebra, topology, and analysis is displayed. This comprehensive and up-to-date study of composition operators on different function spaces should appeal to research workers in functional analysis and operator theory, post-graduate students of mathematics and statistics, as well as to physicists and engineers.
The fates of important mathematical ideas are varied. Sometimes they are instantly appreciated by the specialists and constitute the foundation of the development of theories or methods. It also happens, however, that even ideas uttered by distinguished mathematicians are surrounded with respectful indifference for a long time, and every effort of inter preters and successors has to be made in order to gain for them the merit deserved. It is the second case that is encountered in the present book, the author of which, the Leningrad mathematician E.M. Polishchuk, reconstructs and develops one of the dir.ctions in functional analysis that originated from Hadamard and Gateaux and was newly thought over and taken as the basis of a prospective theory by Paul Levy. Paul Levy, Member of the French Academy of Sciences, whose centenary of his birthday was celebrated in 1986, was one of the most original mathe matiCians of the second half of the 20th century. He could not complain about a lack of attention to his ideas and results. Together with A.N. Kolmogorov, A.Ya. Khinchin and William Feller, he is indeed one of the acknowledged founders of the theory of random processes. In the proba bility theory and, to a lesser degree, in functional analysis his work is well-known for its conceptualization and scope of the problems posed.
Self-Validating Numerics for Function Space Problems describes the development of computational methods for solving function space problems, including differential, integral, and function equations. This seven-chapter text highlights three approaches, namely, the E-methods, ultra-arithmetic, and computer arithmetic. After a brief overview of the different self-validating approaches, this book goes on introducing the mathematical preliminaries consisting principally of fixed-point theorems and the computational context for the development of validating methods in function spaces. The subsequent chapters deals with the development and application of point of view of ultra-arithmetic and the constructs of function-space arithmetic spaces, such as spaces, bases, rounding, and approximate operations. These topics are followed by discussion of the iterative residual correction methods for function problems and the requirements of a programming language needed to make the tools and constructs of the methodology available in actual practice on a computer. The last chapter describes the techniques for adapting the methodologies to a computer, including the self-validating results for specific problems. This book will prove useful to mathematicians and advance mathematics students.