Determinism, holism and complexity: three epistemological attitudes that have easily identifiable historical origins and developments. Galileo believed that it was necessary to "prune the impediments" to extract the mathematical essence of physical phenomena, to identify the math ematical structures representing the underlying laws. This Galilean method was the key element in the development of Physics, with its extraordinary successes. Nevertheless the method was later criticized because it led to a view of nature as essentially "simple and orderly," and thus by choosing not to investigate several charac teristics considered as an "impediment," several essential aspects of the phenomenon under investigation might be left out. The Galilean point of view also contains an acknowledgement of the central role played by the causal nexus among phenomena. The mechanistic-deterministic de scription of reality - for instance, a la Laplace - although acknowledging that it is not possible to predict phenomena exactly owing to unavoid able measurement error, is based on the recognition of the their causal nature, even in an ontological sense. Consequently, deterministic predic tion became the methodological fulcrum of mathematical physics. But although mechanistic determinism has had and, in many cases, still has, considerable success in Physics, in other branches of science this situa tion is much less favourable."

The book analyzes the existence of solitons, namely of finite energy solutions of field equations which exhibit stability properties. The book is divided in two parts. In the first part, the authors give an abstract definition of solitary wave and soliton and we develop an abstract existence theory for hylomorphic solitons, namely for those solitons which minimize the energy for a given charge. In the second part, the authors apply this theory to prove the existence of hylomorphic solitons for some classes of field equations (nonlinear Klein-Gordon-Maxwell equations, nonlinear Schroedinger-Maxwell equations, nonlinear beam equation,..). The abstract theory is sufficiently flexible to be applied to other situations, like the existence of vortices. The books is addressed to Mathematicians and Physicists.

The book analyzes the existence of solitons, namely of finite energy solutions of field equations which exhibit stability properties. The book is divided in two parts. In the first part, the authors give an abstract definition of solitary wave and soliton and we develop an abstract existence theory for hylomorphic solitons, namely for those solitons which minimize the energy for a given charge. In the second part, the authors apply this theory to prove the existence of hylomorphic solitons for some classes of field equations (nonlinear Klein-Gordon-Maxwell equations, nonlinear Schroedinger-Maxwell equations, nonlinear beam equation,..). The abstract theory is sufficiently flexible to be applied to other situations, like the existence of vortices. The books is addressed to Mathematicians and Physicists.

Determinism, holism and complexity: three epistemological attitudes that have easily identifiable historical origins and developments. Galileo believed that it was necessary to "prune the impediments" to extract the mathematical essence of physical phenomena, to identify the math ematical structures representing the underlying laws. This Galilean method was the key element in the development of Physics, with its extraordinary successes. Nevertheless the method was later criticized because it led to a view of nature as essentially "simple and orderly," and thus by choosing not to investigate several charac teristics considered as an "impediment," several essential aspects of the phenomenon under investigation might be left out. The Galilean point of view also contains an acknowledgement of the central role played by the causal nexus among phenomena. The mechanistic-deterministic de scription of reality - for instance, a la Laplace - although acknowledging that it is not possible to predict phenomena exactly owing to unavoid able measurement error, is based on the recognition of the their causal nature, even in an ontological sense. Consequently, deterministic predic tion became the methodological fulcrum of mathematical physics. But although mechanistic determinism has had and, in many cases, still has, considerable success in Physics, in other branches of science this situa tion is much less favourable."

'This text shows that the study of the almost-forgotten, non-Archimedean mathematics deserves to be utilized more intently in a variety of fields within the larger domain of applied mathematics.'CHOICEThis book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the Alpha-Theory, which can be considered as an alternative approach to nonstandard analysis.The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory.The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid's Principle: 'The whole is larger than its parts'. The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory.