The chapters of this book were written by structural engineers. The approach, therefore, is not aiming toward a scientific modelling of the response but to the definition of engineering procedures for detecting and avoiding undesired phenomena. In this sense chaotic and stochastic behaviour can be tackled in a similar manner. This aspect is illustrated in Chapter 1. Chapters 2 and 3 are entirely devoted to Stochastic Dynamics and cover single-degree-of-freedom systems and impact problems, respectively. Chapter 4 provides details on the numerical tools necessary for evaluating the main indexes useful for the classification of the motion and for estimating the response probability density function. Chapter 5 gives an overview of random vibration methods for linear and nonlinear multi-degree-of-freedom systems. The randomness of the material characteristics and the relevant stochastic models ar considered in Chapter 6. Chapter 7, eventually, deals with large engineering sytems under stochastic excitation and allows for the stochastic nature of the mechanical and geometrical properties.

In order to obtain more reliable optimal solutions of concrete technical/economic problems, e.g. optimal design problems, the often known stochastic variations of many technical/economic parameters have to be taken into account already in the planning phase. Hence, ordinary mathematical programs have to be replaced by appropriate stochastic programs. New theoretical insight into several branches of reliability-oriented optimization of stochastic systems, new computational approaches and technical/economic applications of stochastic programming methods can be found in this volume.

2020 Taylor & Francis Award Winner for Outstanding New Textbook! Featuring recent advances in the field, this new textbook presents probability and statistics, and their applications in stochastic processes. This book presents key information for understanding the essential aspects of basic probability theory and concepts of reliability as an application. The purpose of this book is to provide an option in this field that combines these areas in one book, balances both theory and practical applications, and also keeps the practitioners in mind. Features Includes numerous examples using current technologies with applications in various fields of study Offers many practical applications of probability in queueing models, all of which are related to the appropriate stochastic processes (continuous time such as waiting time, and fuzzy and discrete time like the classic Gambler's Ruin Problem) Presents different current topics like probability distributions used in real-world applications of statistics such as climate control and pollution Different types of computer software such as MATLAB (R), Minitab, MS Excel, and R as options for illustration, programing and calculation purposes and data analysis Covers reliability and its application in network queues

The present monograph is a comprehensive summary of the research on visibility in random fields, which I have conducted with the late Professor Micha Yadin for over ten years. This research, which resulted in several published papers and technical reports (see bibliography), was motivated by some military problems, which were brought to our attention by Mr. Pete Shugart of the US Army TRADOC Systems Analysis Activity, presently called US Army TRADOC Analysis Command. The Director ofTRASANA at the time, the late Dr. Wilbur Payne, identified the problems and encouraged the support and funding of this research by the US Army. Research contracts were first administered through the Office of Naval Research, and subsequently by the Army Research Office. We are most grateful to all involved for this support and encouragement. In 1986 I administered a three-day workshop on problem solving in the area of sto chastic visibility. This workshop was held at the White Sands Missile Range facility. A set of notes with some software were written for this workshop. This workshop led to the incorporation of some of the methods discussed in the present book into the Army simulation package CASTFOREM. Several people encouraged me to extend those notes and write the present monograph on the level of those notes, so that the material will be more widely available for applications."

The 2-volume-book is an updated, reorganized and considerably enlarged version of the previous edition of the Research Problem Book in Analysis (LNM 1043), a collection familiar to many analysts, that has sparked off much research. This new edition, created in a joint effort by a large team of analysts, is, like its predecessor, a collection of unsolved problems of modern analysis designed as informally written mini-articles, each containing not only a statement of a problem but also historical and metho- dological comments, motivation, conjectures and discussion of possible connections, of plausible approaches as well as a list of references. There are now 342 of these mini- articles, almost twice as many as in the previous edition, despite the fact that a good deal of them have been solved!

The classical theory of natural selection, as developed by Fisher, Haldane, and 'Wright, and their followers, is in a sense a statistical theory. By and large the classical theory assumes that the underlying environment in which evolution transpires is both constant and stable - the theory is in this sense deterministic. In reality, on the other hand, nature is almost always changing and unstable. We do not yet possess a complete theory of natural selection in stochastic environ ments. Perhaps it has been thought that such a theory is unimportant, or that it would be too difficult. Our own view is that the time is now ripe for the development of a probabilistic theory of natural selection. The present volume is an attempt to provide an elementary introduction to this probabilistic theory. Each author was asked to con tribute a simple, basic introduction to his or her specialty, including lively discussions and speculation. We hope that the book contributes further to the understanding of the roles of "Chance and Necessity" (Monod 1971) as integrated components of adaptation in nature."

Stochastic processes with independent increments on a group are generalized to the concept of "white noise" on a Hopf algebra or bialgebra. The main purpose of the book is the characterization of these processes as solutions of quantum stochastic differential equations in the sense of R.L. Hudsonand K.R. Parthasarathy. The notes are a contribution to quantum probability but they are also related to classical probability, quantum groups, and operator algebras. The Az ma martingales appear as examples of white noise on a Hopf algebra which is a deformation of the Heisenberg group. The book will be of interest to probabilists and quantum probabilists. Specialists in algebraic structures who are curious about the role of their concepts in probablility theory as well as quantum theory may find the book interesting. The reader should havesome knowledge of functional analysis, operator algebras, and probability theory.

In three chapters on Exponential Martingales, BMO-martingales, and Exponential of BMO, this book explains in detail the beautiful properties of continuous exponential martingales that play an essential role in various questions concerning the absolute continuity of probability laws of stochastic processes. The second and principal aim is to provide a full report on the exciting results on BMO in the theory of exponential martingales. The reader is assumed to be familiar with the general theory of continuous martingales.

Fatigue of engineering materials is a very complicated process that is difficult to accurately describe and predict. It is no doubt nowadays, that a fatigue of real materials should be regarded as a random phenomenon and analyzed by use of stochastic theory. This volume of the lectures sumarises the latest achievements in stochastic modelling and analysis of fatigue. The lectures cover the following important aspects of modern analysis of fatigue: methodology of stochastic modelling of fatigue, tools for characterization of random fatigue loads, physical and mechanical aspects of random fatigue, basic stochastic models for fatigue and the estimation of fatigue reliability of specific structural systems.

Stochastic Programming offers models and methods for decision problems wheresome of the data are uncertain. These models have features and structural properties which are preferably exploited by SP methods within the solution process. This work contributes to the methodology for two-stagemodels. In these models the objective function is given as an integral, whose integrand depends on a random vector, on its probability measure and on a decision. The main results of this work have been derived with the intention to ease these difficulties: After investigating duality relations for convex optimization problems with supply/demand and prices being treated as parameters, a stability criterion is stated and proves subdifferentiability of the value function. This criterion is employed for proving the existence of bilinear functions, which minorize/majorize the integrand. Additionally, these minorants/majorants support the integrand on generalized barycenters of simplicial faces of specially shaped polytopes and amount to an approach which is denoted barycentric approximation scheme.

All the papers contained in the volume are original, fully refereed researchpapers. They represent a fairly broad spectrum of the research activity in probability theory, which was done internationally in 1990-1991, with particular emphasis on Markov processes and stochastic calculus. The latter subject keeps growing, and some important new developments, included in the volume, concern anticipative stochastic integrals, and new applications of the enlargements of filtrations to the study of zeros of martingales. FROM THE CONTENTS: R. Bass, D. Khoshnevisan: Stochastic calculus and the continuity of local times of Levy processes.- M.T. Barlow, P. Imkeller: On some sample path properties of Skorokhod integral processes.- T.S. Mountford: A critical function for the planar Brownian convex hull.- L. Dubins, M. Smorodinsky: The modified, discrete Levy transformation is Bernoulli.- M. Baxter: Markov processes on the boundary of the binary tree.- R. Abraham: Unarbre aleatoire infini associe a l'excursion brownienne.- S.E. Kuznetsov: On the existence of a dual semigroup.

The purpose of this book is to give a streamlined introduction to the theory of (not necessarily symmetric) Dirichlet forms on general state spaces. It includes both the analytic and the probabilistic part of the theory up to and including the construction of an associated Markov process. It is based on recent joint work of S. Albeverio and the two authors and on a one-year-course on Dirichlet forms taught by the second named author at the University of Bonn in 1990/9l. It addresses both researchers and graduate students who require a quick but complete introduction to the theory. Prerequisites are a basic course in probabil ity theory (including elementary martingale theory up to the optional sampling theorem) and a sound knowledge of measure theory (as, for example, to be found in Part I of H. Bauer B 78]). Furthermore, an elementary course on lin ear operators on Banach and Hilbert spaces (but without spectral theory) and a course on Markov processes would be helpful though most of the material needed is included here."

This workshop on stochastic theory and adaptive control assembled many of the leading researchers on stochastic control and stochastic adaptive control to increase scientific exchange and cooperative research between these two subfields of stochastic analysis. The papers included in the proceedings include survey and research. They describe both theoretical results and applications of adaptive control. There are theoretical results in identification, filtering, control, adaptive control and various other related topics. Some applications to manufacturing systems, queues, networks, medicine and other topics are gien.

This volume consists of 24 papers submitted for publication by the invited speakers of the IFIP International Conference on Stochastic Partial Differential Equations and their Ap- plications. Most of them are research papers, however, a few surveys written by world renowed experts are also included. The aim of the conference was to bring together mathematici- ans, physicists and engineers representing academic as well as industrial fields, interested in the theory and applica- tions of SPDE's. The field of SPDE's is one of the most dy- namically developing areas at the cross roads of several sciences. It is especially attractive for many because of its interdisciplinary character and enormous richness ofal- ready existing as well as potential applications. There were about one hundred participants registered for the conferen- ce. With rare exceptions, all of the most active researchers in the field of SPDE's throughout the world were present at the conference. The main topics for discussion at the confe- rence were: non-linear SPDE's and Markov property for random fields, modern stochastic calculuses, numerical and asympto- tic methods for SPDE's, applications of SPDE's with emphasis onnon-linear filtering, stochastic control and statistical fluid dynamics.

A bibliography on stochastic orderings. Was there a real need for it? In a time of reference databases as the MathSci or the Science Citation Index or the Social Science Citation Index the answer seems to be negative. The reason we think that this bibliog raphy might be of some use stems from the frustration that we, as workers in the field, have often experienced by finding similar results being discovered and proved over and over in different journals of different disciplines with different levels of mathematical so phistication and accuracy and most of the times without cross references. Of course it would be very unfair to blame an economist, say, for not knowing a result in mathematical physics, or vice versa, especially when the problems and the languages are so far apart that it is often difficult to recognize the analogies even after further scrutiny. We hope that collecting the references on this topic, regardless of the area of application, will be of some help, at least to pinpoint the problem. We use the term stochastic ordering in a broad sense to denote any ordering relation on a space of probability measures. Questions that can be related to the idea of stochastic orderings are as old as probability itself. Think for instance of the problem of comparing two gambles in order to decide which one is more favorable."

The DMV seminar "Stochastische Approximation und Optimierung zufalliger Systeme" was held at Blaubeuren, 28. 5. -4. 6. 1989. The goal was to give an approach to theory and application of stochas tic approximation in view of optimization problems, especially in engineering systems. These notes are based on the seminar lectures. They consist of three parts: I. Foundations of stochastic approximation (H. Walk); n. Applicational aspects of stochastic approximation (G. PHug); In. Applications to adaptation: ugorithms (L. Ljung). The prerequisites for reading this book are basic knowledge in probability, mathematical statistics, optimization. We would like to thank Prof. M. Barner and Prof. G. Fischer for the or ganization of the seminar. We also thank the participants for their cooperation and our assistants and secretaries for typing the manuscript. November 1991 L. Ljung, G. PHug, H. Walk Table of contents I Foundations of stochastic approximation (H. Walk) 1 Almost sure convergence of stochastic approximation procedures 2 2 Recursive methods for linear problems 17 3 Stochastic optimization under stochastic constraints 22 4 A learning model; recursive density estimation 27 5 Invariance principles in stochastic approximation 30 6 On the theory of large deviations 43 References for Part I 45 11 Applicational aspects of stochastic approximation (G. PHug) 7 Markovian stochastic optimization and stochastic approximation procedures 53 8 Asymptotic distributions 71 9 Stopping times 79 1O Applications of stochastic approximation methods 80 References for Part II 90 III Applications to adaptation algorithms (L."

The subject of this book is a new direction in the field of probability theory and mathematical statistics which can be called "stability theory": it deals with evaluating the effects of perturbing initial probabilistic models and embraces quite varied subtopics: limit theorems, queueing models, statistical inference, probability metrics, etc. The contributions are original research articles developing new ideas and methods of stability analysis.

The four chapters of this volume, written by prominent workers in the field of adaptive processing and linear prediction, address a variety of problems, ranging from adaptive source coding to autoregressive spectral estimation. The first chapter, by T.C. Butash and L.D. Davisson, formulates the performance of an adaptive linear predictor in a series of theorems, with and without the Gaussian assumption, under the hypothesis that its coefficients are derived from either the (single) observation sequence to be predicted (dependent case) or a second, statistically independent realisation (independent case). The contribution by H.V. Poor reviews three recently developed general methodologies for designing signal predictors under nonclassical operating conditions, namely the robust predictor, the high-speed Levinson modeling, and the approximate conditional mean nonlinear predictor. W. Wax presents the key concepts and techniques for detecting, localizing and beamforming multiple narrowband sources by passive sensor arrays. Special coding algorithms and techniques based on the use of linear prediction now permit high-quality voice reproduction at remorably low bit rates. The paper by A. Gersho reviews some of the main ideas underlying the algorithms of major interest today.

There are many ways of introducing the concept of probability in classical, i. e, deter ministic, physics. This work is concerned with one approach, known as "the method of arbitrary funetionJ. " It was put forward by Poincare in 1896 and developed by Hopf in the 1930's. The idea is the following. There is always some uncertainty in our knowledge of both the initial conditions and the values of the physical constants that characterize the evolution of a physical system. A probability density may be used to describe this uncertainty. For many physical systems, dependence on the initial density washes away with time. Inthese cases, the system's position eventually converges to the same random variable, no matter what density is used to describe initial uncertainty. Hopf's results for the method of arbitrary functions are derived and extended in a unified fashion in these lecture notes. They include his work on dissipative systems subject to weak frictional forces. Most prominent among the problems he considers is his carnival wheel example, which is the first case where a probability distribution cannot be guessed from symmetry or other plausibility considerations, but has to be derived combining the actual physics with the method of arbitrary functions. Examples due to other authors, such as Poincare's law of small planets, Borel's billiards problem and Keller's coin tossing analysis are also studied using this framework. Finally, many new applications are presented."

Entropy inequalities, correlation functions, couplings between stochastic processes are powerful techniques which have been extensively used to give arigorous foundation to the theory of complex, many component systems and to its many applications in a variety of fields as physics, biology, population dynamics, economics, ... The purpose of the book is to make theseand other mathematical methods accessible to readers with a limited background in probability and physics by examining in detail a few models where the techniques emerge clearly, while extra difficulties arekept to a minimum. Lanford's method and its extension to the hierarchy of equations for the truncated correlation functions, the v-functions, are presented and applied to prove the validity of macroscopic equations forstochastic particle systems which are perturbations of the independent and of the symmetric simple exclusion processes. Entropy inequalities are discussed in the frame of the Guo-Papanicolaou-Varadhan technique and of theKipnis-Olla-Varadhan super exponential estimates, with reference to zero-range models. Discrete velocity Boltzmann equations, reaction diffusion equations and non linear parabolic equations are considered, as limits of particles models. Phase separation phenomena are discussed in the context of Glauber+Kawasaki evolutions and reaction diffusion equations. Although the emphasis is onthe mathematical aspects, the physical motivations are explained through theanalysis of the single models, without attempting, however to survey the entire subject of hydrodynamical limits.

The Galton-Watson branching process has its roots in the problem of extinction of family names which was given a precise formulation by F. Galton as problem 4001 in the Educational Times (17, 1873). In 1875, an attempt to solve this problem was made by H. W. Watson but as it turned out, his conclusion was incorrect. Half a century later, R. A. Fisher made use of the Galton-Watson process to determine the extinction probability of the progeny of a mutant gene. However, it was J. B. S. Haldane who finally gave the first sketch of the correct conclusion. J. B. S. Haldane also predicted that mathematical genetics might some day develop into a "respectable branch of applied mathematics" (quoted in M. Kimura & T. Ohta, Theoretical Aspects of Population Genetics. Princeton, 1971). Since the time of Fisher and Haldane, the two fields of branching processes and mathematical genetics have attained a high degree of sophistication but in different directions. This monograph is a first attempt to apply the current state of knowledge concerning single-type branching processes to a particular area of mathematical genetics: neutral evolution. The reader is assumed to be familiar with some of the concepts of probability theory, but no particular knowledge of branching processes is required. Following the advice of an anonymous referee, I have enlarged my original version of the introduction (Chapter Zero) in order to make it accessible to a larger audience. G6teborg, Sweden, November 1991.

This volume includes a selection of papers presented at the GAMM/ IFIP-Workshop on IIStochastic Optimization: Numerical Methods and ll Technical Applications , held at the Federal Armed Forces Univer- sity Munich, May 29-31, 1990. The objective of this meeting was to bring together scientists from Stochastic Programming and from those Engineering areas, where Mathematical Programming models are common tools, as e.g. Optimal structural Design, Power Dispatch, Acid Rain Abatement etc .. Hence, the aim was to discuss the effects of taking into account the in- herent randomness of some data of these problems, i.e. considering Stochastic Programming instead of Mathematical Programming models in order to get solutions being more reliable, but not more expen- sive. An international programme committe2 was formed which included H.A. Eschenauer (Germany) P. Kall (Switzerland) K. Marti (Germany, Chairman) J. Mayer (Hungary) G.I. Schueller (Austria) Although the number of participants had to be small for technical reasons, the area covered by the lectures during the workshop was rather broad. It contains theoretical insight into stochastic pro- gramming problems, new computational approaches, analyses of known solution methods, and applications in such very different technical fields as ecology, energy demands, and optimal reliability of me- chanical structures. In particular, the applied presentation also pointed to several open methodological problems.

This volume contains papers presented during a four-day Workshop that took place at Rutgers University from 29 April to 2 May, 1991. The purpose of this workshop was to promote interaction among specialists in these areas byproviding for all an up-to-date picture of current issues and outstanding problems. The topics covered include singular stochasticcontrol, queuing networks, the mathematical theory of stochastic optimization and filtering, adaptive control and the estimation for random fields and its connections with simulated annealing, statistical mechanics, and combinatorial optimization.

This book contains a collection of survey papers in the areas of modelling, estimation and adaptive control of stochastic systems describing recent efforts to develop a systematic and elegant theory of identification and adaptive control. It is meant to provide a fast introduction to some of the recent achievements. The book is intended for graduate students and researchers interested in statistical problems of control in general. Students in robotics and communication will also find it valuable. Readers are expected to be familiar with the fundamentals of probability theory and stochastic processes.

The interplay between the spectral theory of Schr-dinger operators and probabilistic considerations forms the main theme of these notes, written for the non-specialist reader and intended to provide a brief and elementaryintroduction to this field. An attempt is made to show basic ideas in statu nascendi and to follow their evaluation from simple beginnings through to more advanced results. The term "genetic" in the title refers to this proceedure. The author concentrates on 2 topics which, in the history of the subject, have been of major conceptual importance - on the one hand the Laplacian is a random medium and the left end of its spectrum (leading to large deviation problems for Brownian motion and the link to thenotion of entropy) and on the other, Schr-dinger operators with general ergodic potentials in one-dimensional space. Ideas and concepts are explained in the simplest, possible setting and by means of a few characteristic problems with heuristic arguments preceding rigorous proofs.