The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume. The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives. Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenböck formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua. Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.

In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces. In 1854, B. Riemann introduced the Riemann curvature for Riemannian spaces in his ground-breaking Habilitationsvortrag. Thereafter the geometry of these special regular metric spaces is named after him. Riemann also mentioned general regular metric spaces, but he thought that there were nothing new in the general case. In fact, it is technically much more difficult to deal with general regular metric spaces. For more than half century, there had been no essential progress in this direction until P. Finsler did his pioneering work in 1918. Finsler studied the variational problems of curves and surfaces in general regular metric spaces. Some difficult problems were solved by him. Since then, such regular metric spaces are called Finsler spaces. Finsler, however, did not go any further to introduce curvatures for regular metric spaces. He switched his research direction to set theory shortly after his graduation.

In the last decade several international conferences on Finsler, Lagrange and Hamilton geometries were organized in Bra§ov, Romania (1994), Seattle, USA (1995), Edmonton, Canada (1998), besides the Seminars that periodically are held in Japan and Romania. All these meetings produced important progress in the field and brought forth the appearance of some reference volumes. Along this line, a new International Conference on Finsler and Lagrange Geometry took place August 26-31,2001 at the "Al.I.Cuza" University in Ia§i, Romania. This Conference was organized in the framework of a Memorandum of Un derstanding (1994-2004) between the "Al.I.Cuza" University in Ia§i, Romania and the University of Alberta in Edmonton, Canada. It was especially dedicated to Prof. Dr. Peter Louis Antonelli, the liaison officer in the Memorandum, an untired promoter of Finsler, Lagrange and Hamilton geometries, very close to the Romanian School of Geometry led by Prof. Dr. Radu Miron. The dedica tion wished to mark also the 60th birthday of Prof. Dr. Peter Louis Antonelli. With this occasion a Diploma was given to Professor Dr. Peter Louis Antonelli conferring the title of Honorary Professor granted to him by the Senate of the oldest Romanian University (140 years), the "Al.I.Cuza" University, Ia§i, Roma nia. There were almost fifty participants from Egypt, Greece, Hungary, Japan, Romania, USA. There were scheduled 45 minutes lectures as well as short communications.

This book catalogues an exhibition of textbooks by authors from the University of Alberta. Each finished textbook contains its own story of challenges and victories. And each has its own power as a record of knowledge, a teaching tool, and an object of permanence and beauty.

The Symposium entitled: Causality and Locality in Modern Physics and As tronomy: Open Questions and Possible Solutions was held at York University, Toronto, during the last week of August 1997. It was a sequel to a similar sym posium entitled: The Present Status of the Quantum Theory of Light held at the same venue in August 1995. These symposia came about as a result of discussions between Professor Stanley Jeffers and colleagues on the International Organizing Committee. Professor Jeffers was the executive local organizer of the symposia. The 1997 symposium attracted over 120 participants representing 26 different countries and academic institutions. The broad theme of both symposia was the enigma of modern physics: the non-local, and possibly superluminal interactions implied by quantum mechanics, the structure of fundamental particles including the photon, the reconciliation of quantum mechanics with the theory of relativity, and the nature of gravity and inertia. Jean-Pierre Vigier was the guest of honour at both symposia. He was a lively contributor to the discussions of the presentations. The presentations were made as 30-minute lectures, or during an evening poster session. Some participants did not submit a written account of their presentation at the symposium, and not all of the articles submitted for the Proceedings could be included because of the publisher's page limit. The titles and authors of the papers that had to be excluded are listed in an appendix.

This volume contains the contributions by the participants in the eight of a series workshops in complex analysis, differential geometry and mathematical physics and related areas.Active specialists in mathematical physics contribute to the volume, providing not only significant information for researchers in the area but also interesting mathematics for non-specialists and a broader audience. The contributions treat topics including differential geometry, partial differential equations, integrable systems and mathematical physics.

In the first century after its discovery, the electron has come to be a fundamental element in the analysis of physical aspects of nature. This book is devoted to the construction of a deductive theory of the electron, starting from first principles and using a simple mathematical tool, geometric analysis. Its purpose is to present a comprehensive theory of the electron to the point where a connection can be made with the main approaches to the study of the electron in physics. The introduction describes the methodology. Chapter 2 presents the concept of space-time-action relativity theory and in chapter 3 the mathematical structures describing action are analyzed. Chapters 4, 5, and 6 deal with the theory of the electron in a series of aspects where the geometrical analysis is more relevant. Finally in chapter 7 the form of geometrical analysis used in the book is presented to elucidate the broad range of topics which are covered and the range of mathematical structures which are implicitly or explicitly included. The book is directed to two different audiences of graduate students and research scientists: primarily to theoretical physicists in the field of electron physics as well as those in the more general field of quantum mechanics, elementary particle physics, and general relativity; secondly, to mathematicians in the field of geometric analysis.

This book focuses on the elementary but essential problems in Riemann-Finsler Geometry, which include a repertoire of rigidity and comparison theorems, and an array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. "This book offers the most modern treatment of the topic ..." EMS Newsletter.

The scope of this monograph is to show that our classical, quantum and cosmological knowledge of antimatter is at its beginning with much yet to be discovered, and that a commitment to antimatter by experimentalists will be invaluable to antimatter science. This is also the first book presenting the isodual theory of antimatter. It is aimed at scientists and researchers in theoretical physics.