Artificial Neural Networks have captured the interest of many researchers in the last five years. As with many young fields, neural network research has been largely empirical in nature, relyingstrongly on simulationstudies ofvarious network models. Empiricism is, of course, essential to any science for it provides a body of observations allowing initial characterization of the field. Eventually, however, any maturing field must begin the process of validating empirically derived conjectures with rigorous mathematical models. It is in this way that science has always pro ceeded. It is in this way that science provides conclusions that can be used across a variety of applications. This monograph by Michael Lemmon provides just such a theoretical exploration of the role ofcompetition in Artificial Neural Networks. There is "good news" and "bad news" associated with theoretical research in neural networks. The bad news isthat such work usually requires the understanding of and bringing together of results from many seemingly disparate disciplines such as neurobiology, cognitive psychology, theory of differential equations, largc scale systems theory, computer science, and electrical engineering. The good news is that for those capable of making this synthesis, the rewards are rich as exemplified in this monograph."

Artificial Neural Networks have captured the interest of many researchers in the last five years. As with many young fields, neural network research has been largely empirical in nature, relyingstrongly on simulationstudies ofvarious network models. Empiricism is, of course, essential to any science for it provides a body of observations allowing initial characterization of the field. Eventually, however, any maturing field must begin the process of validating empirically derived conjectures with rigorous mathematical models. It is in this way that science has always pro ceeded. It is in this way that science provides conclusions that can be used across a variety of applications. This monograph by Michael Lemmon provides just such a theoretical exploration of the role ofcompetition in Artificial Neural Networks. There is "good news" and "bad news" associated with theoretical research in neural networks. The bad news isthat such work usually requires the understanding of and bringing together of results from many seemingly disparate disciplines such as neurobiology, cognitive psychology, theory of differential equations, largc scale systems theory, computer science, and electrical engineering. The good news is that for those capable of making this synthesis, the rewards are rich as exemplified in this monograph."

Hybrid systems are interacting networks of digital and continuous systems. - brid systems arise throughout business and industry in areas such as interactive distributed simulation, trac control, plant process control, military command and control, aircraft and robot design, and path planning. Three of the fun- mental problems that hybrid systems theory should address are: How to model physical and information systems as hybrid systems; how to verify that their - havior satis es program or performance specic ations; and how to extract from performancespeci cationsforanetworkofphysicalsystemsandtheirsimulation models digital control programs which will force the network to obey its perf- mance speci cation. This rapidly developing area is at the interface of control, engineeringandcomputer science. Methods under developmentareextensionsof thosefromdiverseareassuchasprogramveri cation, concurrentanddistributed processes, logic programming, logics of programs, discrete event simulation, c- culus of variations, optimization, di erential geometry, Lie algebras, automata theory, dynamical systems, etc. When the rst LNCS volume Hybrid Systems was published in 1993, the e ect was to focus the attention of researchers worldwide on developing theory andengineeringtoolsapplicabletohybridsystemsinwhichcontinuousprocesses interact with digital programs in real time. At the time of publication of this fth volume, there is general agreement that this is an important area in which mathematics, control engineering, and computer science can be fruitfully c- bined. There are now hybrid system sections in many engineering and computer scienceinternationalmeetings, hybridsystems researchgroupsin manyuniver- ties and industrial laboratories, and also other excellent series of hybrid systems conferenc