Reduction theorem

Performing the topological reduction on the series for si is somewhat less straightforward, since graphs that are rings of

topological reduction theorem and must hence be... 

The strongly stable manifold is one of the leaves of a -smooth foliation which is transverse to the center manifold. As we have shown in Chap. 5 the following reduction theorem holds ... 

The reduction theorem allows us to study the dynamics of the critical variables x independently of the y-variables near the fixed point. As for the y-subspace, the dynamics is relatively simple since the x-variable is small in the norm, the function F in (10.1.3) is also small, and hence the following estimate holds... 

The tool kit used for studying bifurcational problems consists of three pieces the theorem on center manifold, the reduction theorem, and the method of normal forms. 

By virtue of the reduction Theorem 5.5, there exist C " -coordinates in which the family (11.2.1) reduces to the form... 

Svozil also suggests a third possibility, whereby a discretized field theory is strictly local in a higher dimensional space d > 4 but appears to be nonlocal in d = 4. While the physical reasons for a such a dimensional reduction remain unclear, such a dimensional shadowing clearly circumvents the no-go theorem by postulating a local dynamics in a higher dimension (see figure 12.9). 

We are not yet quite ready to apply these theorems to the reduction of the -matrix as given in the form (10-152), since in the latter it is the P operator that appears. Furthermore, in each factor there... 

Proof Let S be either a two dimensional abelian variety or a geometrically ruled surface over an elliptic curve over C. Let S be a good reduction of S modulo q, where gcd(q, n) = 1 such that the assumptions of lemma 2.4.7 hold. Then A 5n iis a good reduction of KSn- modulo q. (3) now follows by lemma 2.4.10 and remark 1.2.2. (1) and (2) follow from this by the formula of Macdonald for p(S n z) (see the proof of theorem 2.3.10). ... 

The quotient space /r (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural Kahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed G -orbits is the affine algebro-geometric quotient and denoted by YjjG. In fact, it is known that the above identihcation intertwines the complex structures. 

In previous chapters we have always taken particles to be in a nonabsorbing medium. We now briefly remove this restriction. The notion of extinction by particles in an absorbing medium is not devoid of controversy more than one interpretation is possible. But Bohren and Gilra (1979) showed that if the extinction cross section is interpreted as the reduction in area of a detector because of the presence of a particle [see Section 3.4, particularly the development leading up to (3.34)], then the optical theorem for a spherical particle in an absorbing medium is formally similar to that for a nonabsorbing medium ... 

Reduction of the number of parameters required to define the problem. The n theorem states that a physical problem can always be described in dimensionless terms. This has the advantage that the number of dimensionless groups, which fully describe it, is much smaller than the number of dimensional physical quantities. It is generally equal to the number of physical quantities minus the number of basic units contained in them. 

Reduction step 1. To prove Theorem 9.3 we only need to prove that Ms induces an equivalence... 

---