This volume of research papers comprises the proceedings of the first International Conference on Mathematics of Neural Networks and Applications (MANNA), which was held at Lady Margaret Hall, Oxford from July 3rd to 7th, 1995 and attended by 116 people. The meeting was strongly supported and, in addition to a stimulating academic programme, it featured a delightful venue, excellent food and accommo dation, a full social programme and fine weather - all of which made for a very enjoyable week. This was the first meeting with this title and it was run under the auspices of the Universities of Huddersfield and Brighton, with sponsorship from the US Air Force (European Office of Aerospace Research and Development) and the London Math ematical Society. This enabled a very interesting and wide-ranging conference pro gramme to be offered. We sincerely thank all these organisations, USAF-EOARD, LMS, and Universities of Huddersfield and Brighton for their invaluable support. The conference organisers were John Mason (Huddersfield) and Steve Ellacott (Brighton), supported by a programme committee consisting of Nigel Allinson (UMIST), Norman Biggs (London School of Economics), Chris Bishop (Aston), David Lowe (Aston), Patrick Parks (Oxford), John Taylor (King's College, Lon don) and Kevin Warwick (Reading). The local organiser from Huddersfield was Ros Hawkins, who took responsibility for much of the administration with great efficiency and energy. The Lady Margaret Hall organisation was led by their bursar, Jeanette Griffiths, who ensured that the week was very smoothly run.

This concise, readable book provides a sampling of the very large, active, and expanding field of artificial neural network theory. It considers select areas of discrete mathematics linking combinatorics and the theory of the simplest types of artificial neural networks. Neural networks have emerged as a key technology in many fields of application, and an understanding of the theories concerning what such systems can and cannot do is essential. Some classical results are presented with accessible proofs, together with some more recent perspectives, such as those obtained by considering decision lists. In addition, probabilistic models of neural network learning are discussed. Graph theory, some partially ordered set theory, computational complexity, and discrete probability are among the mathematical topics involved. Pointers to further reading and an extensive bibliography make this book a good starting point for research in discrete mathematics and neural networks.

For convenience, many of the proofs of the key theorems have been rewritten so that the entire book uses a relatively uniform notion.

Neural Network Modeling offers a cohesive approach to the statistical mechanics and principles of cybernetics as a basis for neural network modeling. It brings together neurobiologists and the engineers who design intelligent automata to understand the physics of collective behavior pertinent to neural elements and the self-control aspects of neurocybernetics. The theoretical perspectives and explanatory projections portray the most current information in the field, some of which counters certain conventional concepts in the visualization of neuronal interactions.

The subject of Neural Networks is being seen to be coming of age, after its initial inception 50 years ago in the seminal work of McCulloch and Pitts. It is proving to be valuable in a wide range of academic disciplines and in important applications in industrial and business tasks. The progress being made in each approach is considerable. Nevertheless, both stand in need of a theoretical framework of explanation to underpin their usage and to allow the progress being made to be put on a firmer footing.This book aims to strengthen the foundations in its presentation of mathematical approaches to neural networks. It is through these that a suitable explanatory framework is expected to be found. The approaches span a broad range, from single neuron details to numerical analysis, functional analysis and dynamical systems theory. Each of these avenues provides its own insights into the way neural networks can be understood, both for artificial ones and simplified simulations. As a whole, the publication underlines the importance of the ever-deepening mathematical understanding of neural networks.

Recent years have seen an explosion of new mathematical results on learning and processing in neural networks. This body of results rests on a breadth of mathematical background which even few specialists possess. In a format intermediate between a textbook and a collection of research articles, this book has been assembled to present a sample of these results, and to fill in the necessary background, in such areas as computability theory, computational complexity theory, the theory of analog computation, stochastic processes, dynamical systems, control theory, time-series analysis, Bayesian analysis, regularization theory, information theory, computational learning theory, and mathematical statistics. Mathematical models of neural networks display an amazing richness and diversity. Neural networks can be formally modeled as computational systems, as physical or dynamical systems, and as statistical analyzers. Within each of these three broad perspectives, there are a number of particular approaches. For each of 16 particular mathematical perspectives on neural networks, the contributing authors provide introductions to the background mathematics, and address questions such as: * Exactly what mathematical systems are used to model neural networks from the given perspective? * What formal questions about neural networks can then be addressed? * What are typical results that can be obtained? and * What are the outstanding open problems? A distinctive feature of this volume is that for each perspective presented in one of the contributed chapters, the first editor has provided a moderately detailed summary of the formal results and the requisite mathematical concepts. These summaries are presented in four chapters that tie together the 16 contributed chapters: three develop a coherent view of the three general perspectives -- computational, dynamical, and statistical; the other assembles these three perspectives into a unified overview of the neural networks field.

Recent years have witnessed a growth of interest in the special functions called ridge functions. These functions appear in various fields and under various guises. They appear in partial differential equations (where they are called plane waves), in computerized tomography, and in statistics. Ridge functions are also the underpinnings of many central models in neural network theory. In this book various approximation theoretic properties of ridge functions are described. This book also describes properties of generalized ridge functions, and their relation to linear superpositions and Kolmogorov's famous superposition theorem. In the final part of the book, a single and two hidden layer neural networks are discussed. The results obtained in this part are based on properties of ordinary and generalized ridge functions. Novel aspects of the universal approximation property of feedforward neural networks are revealed. This book will be of interest to advanced graduate students and researchers working in functional analysis, approximation theory, and the theory of real functions, and will be of particular interest to those wishing to learn more about neural network theory and applications and other areas where ridge functions are used.

This textbook is a comprehensive introduction to computational mathematics and scientific computing suitable for undergraduate and postgraduate courses. It presents both practical and theoretical aspects of the subject, as well as advantages and pitfalls of classical numerical methods alongside with computer code and experiments in Python. Each chapter closes with modern applications in physics, engineering, and computer science. Features: No previous experience in Python is required. Includes simplified computer code for fast-paced learning and transferable skills development. Includes practical problems ideal for project assignments and distance learning. Presents both intuitive and rigorous faces of modern scientific computing. Provides an introduction to neural networks and machine learning.

Applied Mathematics for Restructured Electric Power Systems: Optimization, Control, and Computational Intelligence consists of chapters based on work presented at a National Science Foundation workshop organized in November 2003. The theme of the workshop was the use of applied mathematics to solve challenging power system problems. The areas included control, optimization, and computational intelligence. In addition to the introductory chapter, this book includes 12 chapters written by renowned experts in their respected fields. Each chapter follows a three-part format: (1) a description of an important power system problem or problems, (2) the current practice and/or particular research approaches, and (3) future research directions. Collectively, the technical areas discussed are voltage and oscillatory stability, power system security margins, hierarchical and decentralized control, stability monitoring, embedded optimization, neural network control with adaptive critic architecture, control tuning using genetic algorithms, and load forecasting and component prediction. This volume is intended for power systems researchers and professionals charged with solving electric and power system problems.

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