Local isomorphism

Now we shall determine the hxed point set in gjnce T E is locally isomorphic to... 

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T E. The fact that it is locally isomorphic to T C is enough. Hence the above argument holds and shows Gottsche s formula also for the case of the total space of a holomorphic line bundle over E, not necessarily T E. The only difference is that the unstable manifold W becomes Lagrangian in the case of T E. 

The algebra of the Sp4 group coincides with the algebra of the rotation group of five-dimensional Euclidean space R5, i.e. these groups are locally isomorphic. The irreducible representations of the Sp4 group can be characterized by a set of two parameters (v, t) where the seniority quantum number v for the five-dimensional quasispin group indicates the number... 

Definition 1.3 Soient S,G conane dans 1.1. On dit que G est un tore s il eat localement isomorphe > an sens de la topologie fid lement plate quasi-compacte, It un groupe de la forme (ou r est undentier > 0), ... 

Ainsi, du point de vue "fonnel", dire qua X est etale sur S signifie simplement que X est localement isomorphe A S. ... 

The standard definition of a complex analytic space is completely analogous to our definition of a variety i.e., it is a Hausdorff topological space X, plus a sheaf of C-valued continuous functions fix on X, which is locally isomorphic to one of... 

Thus a vector bundle is just a scheme over X locally isomorphic to Ar x X. The simplest way to define a vector bundle, without distinguishing one atlas , is to simply add the extra condition that the atlas is maximal, i.e., every possible open V C X and isomorphism 7r-1 (V) Ar x V compatible with the given... 

Conversely, assume T is a local isomorphism. Let Ox and l y denote the sheaves of holomorphic functions on X and Y. f induces, for all closed points iGX, all the following maps ... 

All the standard machinery of Calculus can be carried over to our algebraic setting. For example, one can generalize the Jacobian criterion for a map to be a local isomorphism as follows First of all, if X is a non-singular variety, f, ..., fn... 

It seems that a better suited approach should make use of a series of actual measurements on an evolving material quantum system. In fact, two-level atoms, equivalent to spin systems, show a formal analogy with light polarisation. Whereas the configuration space of polarisation transforms according to symmetry group SO(3), the symmetry of spin transformation is SU(2), which is a double covering of SO(3), and locally isomorphic with the latter one [17]. Thus, similar visualisations of the dynamics of both systems apply. 

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