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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.
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