Preface

Human efforts of describing the mechanisms of physical world are a continuous source of mathematical problems. Models are, in fact, representations of classes of phenomena inspired and corroborated by observations intended here as interpretative cataloguing of events. Mathematics is a language having both qualitative and quantitative nature -- the only one with both features -- so it is natural to use it in constructing representations with predictive characteristics of the events in the phenomenological world. In any other human language, in fact, when we refer to quantification, we use concepts coming from mathematics -- the standard use of numbers in counting, for example, the words in this paragraph is a quantitative feature based on a mathematical concept (the one of numbers indeed) that is added to any type of evaluation of the literary/philosophical quality of the phrases themselves. Models appear then as mathematical structures with constituents constrained by the need of having a clear physical meaning. The mathematical questions appearing in models of the physical world have twofold nature: On one side we have technical hitches related, for example, to existence and regularity of solutions to variational minimality requirements and/or to balance equations under boundary and/or initial contitions. On the other side there are foundational problems generated by the quest of the most appropriate form of models. The target is in fact the physical world: appropriateness is then, for a model, the ability to describe the essential structures underlying some classes of phenomena, and their interconnections in non-trivial way, remaining at the same time rather flexible to allow one the possibility of hopefully describing unexpected features that the experimental programs could put in evidence. A model, then, has to be considered not as a manner of justifying the development of a more or less difficult exercise in mathematics, rather it is an occasion of exploring by the tools of a language both qualitative and quantitative, as mathematics is, the intricate and elegant aspects of the physical world. Distinguishing between different possible models of the same class of phenomena is a volatile matter decided by sensibility and culture (the order is not accidental) of the researcher overburden with the judgement.

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