Bücher der Reihe Texts in the Mathematical Sciences

In particular, the material of the chapters on continued fractions (Chapter 2), symplectic systems (Chapter 3), and discrete variational theory (Chapter 4) summarize recent literature. Similarly, the material on transforming Riccati equations presented in Chapter 3 gives a self-contained unification of various forms of Riccati equations.

Starting at an introductory level, the book leads rapidly to important and often new results in synthetic differential geometry. From rudimentary analysis the book moves to such important results as: a new proof of De Rham's theorem; the synthetic view of global action, going as far as the Weil characteristic homomorphism; the systematic account of structured Lie objects, such as Riemannian, symplectic, or Poisson Lie objects; the view of global Lie algebras as Lie algebras of a Lie group in the synthetic sense; and lastly the synthetic construction of symplectic structure on the cotangent bundle in general. Thus while the book is limited to a naive point of view developing synthetic differential geometry as a theory in itself, the author nevertheless treats somewhat advanced topics, which are classic in classical differential geometry but new in the synthetic context. Audience: The book is suitable as an introduction to synthetic differential geometry for students as well as more qualified mathematicians.

Over the past decade there has been an increasing demand for suitable material in the area of mathematical modelling as applied to science, engineering, business and management. Recent developments in computer technology and related software have provided the necessary tools of increasing power and sophistication which have significant implications for the use and role of mathematical modelling in the above disciplines. In the past, traditional methods have relied heavily on expensive experimentation and the building of scaled models, but now a more flexible and cost effective approach is available through greater use of mathematical modelling and computer simulation. In particular, developments in computer algebra, symbolic manipulation packages and user friendly software packages for large scale problems, all have important implications in both the teaching of mathematical modelling and, more importantly, its use in the solution of real world problems. Many textbooks have been published which cover the art and techniques of modelling as well as specific mathematical modelling techniques in specialist areas within science and business. In most of these books the mathematical material tends to be rather tailor made to fit in with a one or two semester course for teaching students at the undergraduate or postgraduate level, usually the former. This textbook is quite different in that it is intended to build on and enhance students' modelling skills using a combination of case studies and projects.

This book is based on the lecture notes from a course we taught at Penn State University during the fall of 2002. The main goal of the course was to give a complete and detailed proof of the Morse Homology Theorem (Theo- rem 7.4) at a level appropriate for second year graduate students. The course was designed for students who had a basic understanding of singular homol- ogy, CW-complexes, applications of the existence and uniqueness theorem for O.D.E.s to vector fields on smooth Riemannian manifolds, and Sard's Theo- rem. We would like to thank the following students for their participation in the course and their help proofreading early versions of this manuscript: James Barton, Shantanu Dave, Svetlana Krat, Viet-Trung Luu, and Chris Saunders. We would especially like to thank Chris Saunders for his dedication and en- thusiasm concerning this project and the many helpful suggestions he made throughout the development of this text. We would also like to thank Bob Wells for sharing with us his extensive knowledge of CW-complexes, Morse theory, and singular homology. Chapters 3 and 6, in particular, benefited significantly from the many insightful conver- sations we had with Bob Wells concerning a Morse function and its associated CW-complex.

At the end of the twentieth century, nonlinear dynamics turned out to be one of the most challenging and stimulating ideas. The recent success of nonlinear dynamics is heavily biased towards temporal characterization widely using nonlinear ordinary differential equations.

This volume is dedicated to the use of lattice theory in module theory. Its main purpose is to present all module-theoretic results that can be proved by lattice theory only, and to develop the theory necessary to do so.

This is because in social sciences the factors that cause variation in the measurements on the study variable for the population units can not be controlled, whereas in physical sciences these factors can be controlled, at least to some extent, through proper experimental design.

At the end of the twentieth century, nonlinear dynamics turned out to be one of the most challenging and stimulating ideas. The recent success of nonlinear dynamics is heavily biased towards temporal characterization widely using nonlinear ordinary differential equations.

This is because in social sciences the factors that cause variation in the measurements on the study variable for the population units can not be controlled, whereas in physical sciences these factors can be controlled, at least to some extent, through proper experimental design.

This volume presents a collection of problems and exercises, together with guidelines and worked solutions, covering a wide range of commonly occurring differential and integral equations, from first and higher order differential equations to Laplace transforms and integral equations.

The book Complex Analysis through Examples and Exercises has come out from the lectures and exercises that the author held mostly for mathematician and physists . In the theory of complex analysis, on the hand one has an interplay of several mathematical disciplines, while on the other various methods, tools, and approaches.

Covers the principal branches of graph theory in more than a thousand exercises of varying complexity. This work includes topics such as trees, independence and coverings, matchings, tours, planarity, colourings, degree sequences, connectivity, digraphs and hypergraphs. It is suitable for researchers, lecturers and graduate students.