The Exact Renormalization Group

This is a very brief introduction to Wilson's Renormalization Group with emphasis on mathematical developments.Comment: 17 pages, AMS LaTeX. Contribution to the Encyclopedia of Mathematical Physics (Elsevier, 2006). Typos, journal reference correcte

Smart matching

One of the most annoying aspects in the formalization of mathematics is the need of transforming notions to match a given, existing result. This kind of transformations, often based on a conspicuous background knowledge in the given scientific domain (mostly expressed in the form of equalities or isomorphisms), are usually implicit in the mathematical discourse, and it would be highly desirable to obtain a similar behavior in interactive provers. The paper describes the superposition-based implementation of this feature inside the Matita interactive theorem prover, focusing in particular on the so called smart application tactic, supporting smart matching between a goal and a given result.Comment: To appear in The 9th International Conference on Mathematical Knowledge Management: MKM 201

On Staggered Indecomposable Virasoro Modules

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L_0 is assumed to be non-diagonalisable, possessing Jordan blocks of rank two. Moreover, the module is further assumed to have a highest weight submodule, the "left module", and that the quotient by this submodule yields another highest weight module, the "right module". Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centres on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. The text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.Comment: 54 pages, 6 figures, 16 examples. v2: small changes including new historical footnote after Eq. (3.6). Not the same as the published version (we gave up correcting the errors

Building and Using Models as Examples

Sometimes, theoreticians explicitly state that they consider their models as examples. When this is not the case, it is fairly common for theoreticians to attribute to their models the characteristics and objectives of illustrative examples. However, this way of understanding models has not received enough attention in the methodological literature focused on economics. Given that didactic examples and their properties are extremely familiar in practice, considering theoretical models as examples can offer a useful perspective on models and their properties. On the basis of both explanatory and exemplifying role played by the deductive arguments by which results are proved, the paper emphasizes also the importance of understanding in theoretical work, the analogical and tentative character of the application of models, the central role played by the above mentioned arguments in such application, the didactic function of theory, and the transmision of plausibility from those arguments to the results obtained.models; examples; explanatory arguments; theoretical understanding; analogical application