The analysis of variance (ANOYA) models have become one of the most widely used tools of modern statistics for analyzing multifactor data. The ANOYA models provide versatile statistical tools for studying the relationship between a dependent variable and one or more independent variables. The ANOYA mod­ els are employed to determine whether different variables interact and which factors or factor combinations are most important. They are appealing because they provide a conceptually simple technique for investigating statistical rela­ tionships among different independent variables known as factors. Currently there are several texts and monographs available on the sub­ ject. However, some of them such as those of Scheffe (1959) and Fisher and McDonald (1978), are written for mathematically advanced readers, requiring a good background in calculus, matrix algebra, and statistical theory; whereas others such as Guenther (1964), Huitson (1971), and Dunn and Clark (1987), although they assume only a background in elementary algebra and statistics, treat the subject somewhat scantily and provide only a superficial discussion of the random and mixed effects analysis of variance.

This book is a rigorous presentation of the method of matched asymptotic expansions, the primary tool for attacking singular perturbation problems. A knowledge of conventional asymptotic analysis is assumed. The first chapter introduces the theory and is followed by four chapters of applications to ordinary differential equation problems of increasing complexity. Exercises are included as well as several Maple programs for computing the terms of the various asymptotic expansions that arise in solving the problems.

Volume 1, i. e. the monograph The Cauchy Method of Residues - Theory and Applications published by D. Reidel Publishing Company in 1984 is the only book that covers all known applications of the calculus of residues. They range from the theory of equations, theory of numbers, matrix analysis, evaluation of real definite integrals, summation of finite and infinite series, expansions of functions into infinite series and products, ordinary and partial differential equations, mathematical and theoretical physics, to the calculus of finite differences and difference equations. The appearance of Volume 1 was acknowledged by the mathematical community. Favourable reviews and many private communications encouraged the authors to continue their work, the result being the present book, Volume 2, a sequel to Volume 1. We mention that Volume 1 is a revised, extended and updated translation of the book Cauchyjev raeun ostataka sa primenama published in Serbian by Nau~na knjiga, Belgrade in 1978, whereas the greater part of Volume 2 is based upon the second Serbian edition of the mentioned book from 1991. Chapter 1 is introductory while Chapters 2 - 6 are supplements to the corresponding chapters of Volume 1. They mainly contain results missed during the preparation of Volume 1 and also some new results published after 1982. Besides, certain topics which were only briefly mentioned in Volume 1 are treated here in more detail.

This volume describes the substantial developments in Clifford analysis which have taken place during the last decade and, in particular, the role of the spin group in the study of null solutions of real and complexified Dirac and Laplace operators.

This book was written to rigorously illustrate the practical application of the projective approach to linear models. To some, this may seem contradictory. I contend that it is possible to be both rigorous and illustrative and that it is possible to use the projective approach in practical applications. Therefore, unlike many other books on linear models, the use of projections and sub­ spaces does not stop after the general theory. They are used wherever I could figure out how to do it. Solving normal equations and using calculus (outside of maximum likelihood theory) are anathema to me. This is because I do not believe that they contribute to the understanding of linear models. I have similar feelings about the use of side conditions. Such topics are mentioned when appropriate and thenceforward avoided like the plague. On the other side of the coin, I just as strenuously reject teaching linear models with a coordinate free approach. Although Joe Eaton assures me that the issues in complicated problems frequently become clearer when considered free of coordinate systems, my experience is that too many people never make the jump from coordinate free theory back to practical applications. I think that coordinate free theory is better tackled after mastering linear models from some other approach. In particular, I think it would be very easy to pick up the coordinate free approach after learning the material in this book. See Eaton (1983) for an excellent exposition of the coordinate free approach.

This is a companion volume to Plane Answers to Complex Questions: The Theory 0/ Linear Models. It consists of six additional chapters written in the same spirit as the last six chapters of the earlier book. Brief introductions are given to topics related to linear model theory. No attempt is made to give a comprehensive treatment of the topics. Such an effort would be futile. Each chapter is on a topic so broad that an in depth discussion would require a book-Iength treatment. People need to impose structure on the world in order to understand it. There is a limit to the number of unrelated facts that anyone can remem­ ber. If ideas can be put within a broad, sophisticatedly simple structure, not only are they easier to remember but often new insights become avail­ able. In fact, sophisticatedly simple models of the world may be the only ones that work. I have often heard Arnold Zellner say that, to the best of his knowledge, this is true in econometrics. The process of modeling is fundamental to understanding the world.

Least squares estimation, when used appropriately, is a powerful research tool. A deeper understanding of the regression concepts is essential for achieving optimal benefits from a least squares analysis. This book builds on the fundamentals of statistical methods and provides appropriate concepts that will allow a scientist to use least squares as an effective research tool.

The second edition of Plane Answers has many additions and a couple of deletions. New material includes additional illustrative examples in Ap­ pendices A and B and Chapters 2 and 3, as well as discussions of Bayesian estimation, near replicate lack of fit tests, testing the independence assump­ tion, testing variance components, the interblock analysis for balanced in­ complete block designs, nonestimable constraints, analysis of unreplicated experiments using normal plots, tensors, and properties of Kronecker prod­ ucts and Vee operators. The book contains an improved discussion of the relation between ANOVA and regression, and an improved presentation of general Gauss-Markov models. The primary material that has been deleted are the discussions of weighted means and of log-linear models. The mate­ rial on log-linear models was included in Christensen (1990b), so it became redundant here. Generally, I have tried to clean up the presentation of ideas wherever it seemed obscure to me. Much of the work on the second edition was done while on sabbatical at the University of Canterbury in Christchurch, New Zealand. I would par­ ticularly like to thank John Deely for arranging my sabbatical. Through their comments and criticisms, four people were particularly helpful in con­ structing this new edition. I would like to thank Wes Johnson, Snehalata Huzurbazar, Ron Butler, and Vance Berger.

All students and professionals in statistics should refer to this volume as it is a handy reference source for statistical formulas and information on basic probability distributions. It contains carefully designed and well laid out tables for standard statistical distributions (including Binomial, Poisson, Normal, and Chi-squared). In addition, there are several tables of Critical Values for various statistics tests.

Nonlinear differential equations describe many physical phenomena whose behavior can be either integrable (such as solitonic equations), par- tially integrable (the complex Ginzburg–Landau equation of nonlinear op- tics), or chaotic (dynamical systems such as the Lorenz model). In statisti- cal physics, the models that are defined on a lattice are described by finite difference equations, in short, discrete equations.

Solitons were discovered by John Scott Russel in 1834, and have interested scientists and mathematicians ever since. They have been the subject of a large body of research in a wide variety of fields of physics and mathematics, not to mention engineering and other branches of science such as biology. This volume comprises the written versions of the talks presented at a workshop held at Queen's University in 1997, an interdisciplinary meeting wherein top researchers from many fields could meet, interact, and exchange ideas. Topics covered include mathematical and numerical aspects of solitons, as well as applications of solitons to nuclear and particle physics, cosmology, and condensed-matter physics. The book should be of interest to researchers in any field in which solitons are encountered.

Semi-infinite programming (SIP) deals with optimization problems in which either the number of decision variables or the number of constraints is finite. This book presents the state of the art in SIP in a suggestive way, bringing the powerful SIP tools close to the potential users in different scientific and technological fields.

Teaching or learning numerical methods in applied mathematics cannot be conceived nowadays without numerical experimentation on computers. There is a vast literature devoted either to theoretical numerical methods or - merical programming of basic algorithms, but there are few texts o?ering a complete discussion of numerical issues involved in the solution of concrete and relatively complex problems. This book is an attempt to ?ll this need. It is our belief that advantages and drawbacks of a numerical method cannot be accounted for without one’s experiencing all the steps of scienti?c comp- ing, from physical and mathematical description of the problem to numerical formulation and programming and, ?nally, to critical discussion of numerical results. The book provides twelve computational projects aimed at numerically solving problems selected to cover a broad spectrum of applications, from ?uid mechanics, chemistry, elasticity, thermal science, computer-aided design, signal and imageprocessing, etc. Even though the main volume of this text concerns the numerical analysis of computational methods and their imp- mentation, we have tried to start, when possible, from realistic problems of practical interest for researchers and engineers. For each project, an introductory record card summarizes the mathem- ical and numerical topics explained and the ?elds of application of the - proach. A level of di?culty, scaling from 1 to 3, is assigned to each project.

Part of Tsinghua University Texts, "Statistical Properties of Deterministic Systems" discusses the fundamental theory and computational methods of the statistical properties of deterministic discrete dynamical systems. After introducing some basic results from ergodic theory, two problems related to the dynamical system are studied: first the existence of absolute continuous invariant measures, and then their computation. They correspond to the functional analysis and numerical analysis of the Frobenius-Perron operator associated with the dynamical system.

The nature of the physical Universe has been increasingly better understood in recent years, and cosmological concepts have undergone a rapid evolution (see, e.g., [11], [2],or [5]).