The Descent Theorem

This refined version of condition (1) in the previous theorem is not needed now but will be crucial in the descent theory of Part V. 

Now different

basic theorem shows that two forms are isomorphic over R iff there is an isomorphism over S commuting with the descent data. 

The first three relationships [Eqs (10.45)-(10.47)], derived by Hoare and Ruijgrok (1970), are a direct result of the integration theorem for Laplace transforms and the steepest descent approximation. Equation (10.48) comes directly from the derivative of Eq. (10.47). 

The condition 9l = 9°92 then says that localized to RiJk agrees with 9Jk9ij. Thus descent data are patching information , isomorphisms on overlaps which are compatible on multiple overlaps. Our theorem here says then that an R-module, R-algebra, etc. can be constructed by taking ones over the various R with compatible isomorphisms over Rj . ... 

Apart from such direct applications, the theorem gives for every G a canonical descent problem, one to which we can reduce questions about Hl(S/k, G) or other structures with automorphism group G. The next section is an example of this. In many cases also there is an automatic choice for S ... 

Theorem 20 can be applied to this algorithm as well. In other words, the steepest descent method converges to the solution of the linear inverse problem for any initial approximation mo, if L = AA is a positively determined linear continuous operator, acting in a real Hilbert space M, or if AA is an absolutely positively determined (APD) linear continuous operator, acting in a complex Hilbert space M. 

Because of Theorem B.3, Vv is perpendicular to the line uo- Theorem B.2, on the other hand, tells us that Vu and Vv are orthogonal, so that any line V — const must also be tangential to Vu. Thus, lines along which v = const correspond to the steepest descent from the saddle. 

The following well-known theorem in descent theory contained in [33] is now easy to prove. 

This theorem has been proven using an earlier result of Pechukas along a steepest descent path the point symmetry group (as well as the framework group) of nuclear configurations may change only at a critical point, where it must have all those point symmetry elements (framework group elements, resp.) that are present at non-critical points of the path [5]. 

This is an example where the so-called Jahn-Teller theorem comes into play. Hermann Jahn, a British physicist of German descent, and the perhaps more well known and outspoken Hungarian physicist Eduard Teller, proved that degeneracies cannot exist. All possible symmetries distort into a lower symmetry where the degeneracies have disappeared. This is the first-order Jahn-Teller theorem (FOJT). 

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