"Poisson Point Processes provides an overview of non-homogeneous and multidimensional Poisson point processes and their numerous applications. Readers will find constructive mathematical tools and applications ranging from emission and transmission computed tomography to multiple target tracking and distributed sensor detection, written from an engineering perspective. A valuable discussion of the basic properties of finite random sets is included. Maximum likelihood estimation techniques are discussed for several parametric forms of the intensity function, including Gaussian sums, together with their Cramer-Rao bounds. These methods are then used to investigate: -Several medical imaging techniques, including positron emission tomography (PET), single photon emission computed tomography (SPECT), and transmission tomography (CT scans) -Various multi-target and multi-sensor tracking applications, -Practical applications in areas like distributed sensing and detection, -Related finite point processes such as marked processes, hard core processes, cluster processes, and doubly stochastic processes, Perfect for researchers, engineers and graduate students working in electrical engineering and computer science, Poisson Point Processes will prove to be an extremely valuable volume for those seeking insight into the nature of these processes and their diverse applications.

Stochastic analysis and its various applications in physics have to a large extent developed symbiotically. In the past decades mathematics has provided physics witb a vast and rapidly expanding array of tools and methods, while on the other hand physics has counted among the sources of direction and of structural intuition for the mathematical research in stochastics. We hope to have captured some of the focal points of this dialogue in the NATO AS! "Stochastic Analysis and its Applications in Physics" and in the present volume. On the mathematical side White Noise Analysis has emerged as a viable frame* work for stochastic and infinite dimensional analysis (Hida, Streit). Another growth point is the theory of stochastic partial differential equations and their applications (8ertini et aI., 0ksendal, Potthoff, Russo, Sinior). Gauge field theories have in- creasingly attracted the attention not only of physicists but of mathematicians as well (Gross, Liandre, Sengupta). On the other hand the contributions of Lang and of Vilela Mendes show the extent to which stochastic methods have found a place in the physicists' toolbox. Dirichlet forms provide a fruitful link between the mathematics of Markov processes and fields and the physics of quantum systems (Albeverio et al.). The deterministic-stochastic nterface i was addressed by Collet and by Mandrekar, Euclidean quantum mechanics by Cruzeiro and Zambrini, excursions of diffusions by Truman, and Kubo-Martin*Schwinger norms of statistical mechanics by Streater. So much for a rapid synopsis of the material represented in the present volume.

Volumes 30 and 31 of this series, dealing with "Many Degrees of Freedom," contain the proceedings of the 1976 International Summer Institute of Theoretical Physics, held at the university of Bielefeld from August 23 to September 4, 1976. This institute was the eighth in a series of summer schools devoted to particle physics and organized by universities and research institutes in the Federal Republic of Germany. Many degrees of freedom and collective phenomena play a critical role in the description and understanding of elementary particles. The lectures in this volume were intended to display how these structures occur in various recent developments of mathematical physics. Lectures ranged from classical nonlinear field theory over classical soliton models, constructive quantum field theory with soliton solutions and gauge models to the recent unified description of renormalization group tech niques in probabilistic language and to quantum statisti cal dynamics in terms of derivations. The Institute took place at the Center for Inter disciplinary Research of the University of Bielefeld. On behalf of all participants, it is a pleasure to thank the officials and the administration of the Center for their cooperation and help before and during the Insti tute. Special thanks go to V.C. Fulland, M. Kamper, and A. Kottenkamp for their rapid and competent preparation of the manuscripts."

Recent Developments in Infinite-Dimensional Analysis and Quantum Probability is dedicated to Professor Takeyuki Hida on the occasion of his 70th birthday. The book is more than a collection of articles. In fact, in it the reader will find a consistent editorial work, devoted to attempting to obtain a unitary picture from the different contributions and to give a comprehensive account of important recent developments in contemporary white noise analysis and some of its applications. For this reason, not only the latest results, but also motivations, explanations and connections with previous work have been included. The wealth of applications, from number theory to signal processing, from optimal filtering to information theory, from the statistics of stationary flows to quantum cable equations, show the power of white noise analysis as a tool. Beyond these, the authors emphasize its connections with practically all branches of contemporary probability, including stochastic geometry, the structure theory of stationary Gaussian processes, Neumann boundary value problems, and large deviations.

Are we living in a golden age? It is now more than half a century that Einstein and Heisenberg have given us the theories of relativity and of quantum mechanics, but the great challenge of 20th century science remains unre solved: to assemble these building blocks into a fundamental theory of matter. And yet, for anyone watching the interplay of mathematics and theoretical physics to-day, developing symbiotically through the stimulus of a lively, even essential interdisciplinary dia logue, this is a time of fascination and great satisfaction. It is also a time of gratitude to those who had the courage to in sist that "a rudimentary knowledge of the Latin and Greek alpha bets" was not enough, and tore down the barriers between the disciplines. On the basis of this groundwork there is now so much progress, and, notably, such strengthening of the dia].ogue with phenomenology that - reaching out for The Great Break through - this may indeed turn out to be the golden age."

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. O. K. Chesterton. The Scandal of Father 'The Hermit Qad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu ik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

'Et moi, ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Many areas of applied mathematics call for an efficient calculus in infinite dimensions. This is most apparent in quantum physics and in all disciplines of science which describe natural phenomena by equations involving stochasticity. With this monograph we intend to provide a framework for analysis in infinite dimensions which is flexible enough to be applicable in many areas, and which on the other hand is intuitive and efficient. Whether or not we achieved our aim must be left to the judgment of the reader. This book treats the theory and applications of analysis and functional analysis in infinite dimensions based on white noise. By white noise we mean the generalized Gaussian process which is (informally) given by the time derivative of the Wiener process, i.e., by the velocity of Brownian mdtion. Therefore, in essence we present analysis on a Gaussian space, and applications to various areas of sClence. Calculus, analysis, and functional analysis in infinite dimensions (or dimension-free formulations of these parts of classical mathematics) have a long history. Early examples can be found in the works of Dirichlet, Euler, Hamilton, Lagrange, and Riemann on variational problems. At the beginning of this century, Frechet, Gateaux and Volterra made essential contributions to the calculus of functions over infinite dimensional spaces. The important and inspiring work of Wiener and Levy followed during the first half of this century. Moreover, the articles and books of Wiener and Levy had a view towards probability theory.

"Poisson Point Processes provides an overview of non-homogeneous and multidimensional Poisson point processes and their numerous applications. Readers will find constructive mathematical tools and applications ranging from emission and transmission computed tomography to multiple target tracking and distributed sensor detection, written from an engineering perspective. A valuable discussion of the basic properties of finite random sets is included. Maximum likelihood estimation techniques are discussed for several parametric forms of the intensity function, including Gaussian sums, together with their Cramer-Rao bounds. These methods are then used to investigate: -Several medical imaging techniques, including positron emission tomography (PET), single photon emission computed tomography (SPECT), and transmission tomography (CT scans) -Various multi-target and multi-sensor tracking applications, -Practical applications in areas like distributed sensing and detection, -Related finite point processes such as marked processes, hard core processes, cluster processes, and doubly stochastic processes, Perfect for researchers, engineers and graduate students working in electrical engineering and computer science, Poisson Point Processes will prove to be an extremely valuable volume for those seeking insight into the nature of these processes and their diverse applications.

Recent Developments in Infinite-Dimensional Analysis and Quantum Probability is dedicated to Professor Takeyuki Hida on the occasion of his 70th birthday. The book is more than a collection of articles. In fact, in it the reader will find a consistent editorial work, devoted to attempting to obtain a unitary picture from the different contributions and to give a comprehensive account of important recent developments in contemporary white noise analysis and some of its applications. For this reason, not only the latest results, but also motivations, explanations and connections with previous work have been included. The wealth of applications, from number theory to signal processing, from optimal filtering to information theory, from the statistics of stationary flows to quantum cable equations, show the power of white noise analysis as a tool. Beyond these, the authors emphasize its connections with practically all branches of contemporary probability, including stochastic geometry, the structure theory of stationary Gaussian processes, Neumann boundary value problems, and large deviations.

Many areas of applied mathematics call for an efficient calculus in infinite dimensions. This is most apparent in quantum physics and in all disciplines of science which describe natural phenomena by equations involving stochasticity. With this monograph we intend to provide a framework for analysis in infinite dimensions which is flexible enough to be applicable in many areas, and which on the other hand is intuitive and efficient. Whether or not we achieved our aim must be left to the judgment of the reader. This book treats the theory and applications of analysis and functional analysis in infinite dimensions based on white noise. By white noise we mean the generalized Gaussian process which is (informally) given by the time derivative of the Wiener process, i.e., by the velocity of Brownian mdtion. Therefore, in essence we present analysis on a Gaussian space, and applications to various areas of sClence. Calculus, analysis, and functional analysis in infinite dimensions (or dimension-free formulations of these parts of classical mathematics) have a long history. Early examples can be found in the works of Dirichlet, Euler, Hamilton, Lagrange, and Riemann on variational problems. At the beginning of this century, Frechet, Gateaux and Volterra made essential contributions to the calculus of functions over infinite dimensional spaces. The important and inspiring work of Wiener and Levy followed during the first half of this century. Moreover, the articles and books of Wiener and Levy had a view towards probability theory.

'Et moi, ..., si j'avait su comment en revenIT, One service mathematics has rendered the je n'y serais point allt\.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This book views multiple target tracking as a Bayesian inference problem. Within this framework it develops the theory of single target tracking, multiple target tracking, and likelihood ratio detection and tracking. In addition to providing a detailed description of a basic particle filter that implements the Bayesian single target recursion, this resource provides numerous examples that involve the use of particle filters. With these examples illustrating the developed concepts, algorithms, and approaches -- the book helps radar engineers track when observations are nonlinear functions of target site, when the target state distributions or measurement error distributions are not Gaussian, in low data rate and low signal to noise ratio situations, and when notions of contact and association are merged or unresolved among more than one target.

The intensive exchange between mathematicians and users has led in recent years to a rapid development of stochastic analysis. Of the users, the physicists form perhaps the most important group, giving direction to the mathematicians' research and providing a source of intuition. White noise analysis has emerged as a viable framework for stochastic and infinite dimensional analysis. Another growth area is the theory of stochastic partial differential equations. Gauge field theories are attracting increasing attention. Dirichlet forms provide a fruitful link between the mathematics of Markov processes and the physics of quantum systems. The deterministic-stochastic interface is addressed, as are Euclidean quantum mechanics, excursions of diffusions and the convergence of Markov chains to thermal states.