Ferran S Map
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Momentum Maps and Hamiltonian Reduction
Author: Juan-Pablo Ortega
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-04-17
The use of the symmetries of a physical system in the study of its dynamics has a long history that goes back to the founders of c1assical mechanics. Symmetry-based tech niques are often implemented by using the integrals 01 motion that one can sometimes associate to these symmetries. The integrals of motion of a dynamical system are quan tities that are conserved along the fiow of that system. In c1assieal mechanics symme tries are usually induced by point transformations, that is, they come exc1usively from symmetries of the configuration space; the intimate connection between integrals of motion and symmetries was formalized in this context by NOETHER (1918). This idea can be generalized to many symmetries of the entire phase space of a given system, by associating to the Lie algebra action encoding the symmetry, a function from the phase space to the dual of the Lie algebra. This map, whose level sets are preserved by the dynamics of any symmetrie system, is referred to in modern terms as a momentum map of the symmetry, a construction already present in the work of LIE (1890). Its remarkable properties were rediscovered by KOSTANT (1965) and SOURlAU (1966, 1969) in the general case and by SMALE (1970) for the lifted action to the co tangent bundle of a configuration space. For the his tory of the momentum map we refer to WEINSTEIN (1983b) and MARSDEN AND RATIU (1999), §11. 2.
Self-Organizing Maps
Author: Teuvo Kohonen
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
Since the second edition of this book came out in early 1997, the number of scientific papers published on the Self-Organizing Map (SOM) has increased from about 1500 to some 4000. Also, two special workshops dedicated to the SOM have been organized, not to mention numerous SOM sessions in neural network conferences. In view of this growing interest it was felt desirable to make extensive revisions to this book. They are of the following nature. Statistical pattern analysis has now been approached more carefully than earlier. A more detailed discussion of the eigenvectors and eigenvalues of symmetric matrices, which are the type usually encountered in statistics, has been included in Sect. 1.1.3: also, new probabilistic concepts, such as factor analysis, have been discussed in Sect. 1.3.1. A survey of projection methods (Sect. 1.3.2) has been added, in order to relate the SOM to classical paradigms. Vector Quantization is now discussed in one main section, and derivation of the pointdensity of the codebook vectors using the calculus of variations has been added, in order to familiarize the reader with this otherwise com plicated statistical analysis. It was also felt that the discussion of the neural-modeling philosophy should include a broader perspective of the main issues. A historical review in Sect. 2.2, and the general philosophy in Sects. 2.3, 2.5 and 2.14 are now expected to especially help newcomers to orient themselves better amongst the profusion of contemporary neural models.