Matrices flipping

It is known that multivalued adiabatic electronic manifolds create topological effects [23,25,45]. Since the newly introduced D matrix contains the information relevant for this manifold (the number of functions that flip sign and their identification) we shall define it as the Topological Matrix. Accordingly, K will be defined as the Topological Number. Since D is dependent on the contour F the same applies to K thus K = f(F),... 

The fact that there is a one-to-one relation between the (—1) terms in the diagonal of the topological matrix and the fact that the eigenfunctions flip sign along closed contours (see discussion at the end of Section IV.A) hints at the possibility that these sign flips are related to a kind of a spin quantum number and in particular to its magnetic components. 

In Sections V and VII, we discussed the possible K values of the D matrix and made the connection with the number of signs flip based on the analysis given in Section IV.A. Here, we intend to present a geomehical approach in order to gain more insight into the phenomenon of signs flip in the Af-state system (M > 2). 

Since for any assumed contour the most that can happen, due to C23, is that 7i2(.v) flips its sign, the corresponding 2x2 diabatic matrix potential, W(.v), will not be affected by that as can be seen from the following expressions ... 

To find the new composition, we need only flip the elements in the transformation matrix about its diagonal, giving,... 

As remarked in the text, the adjoint is synonymous with transpose (i.e., interchange of rows and columns by flipping the matrix around its diagonal) when A is real. However, the ubiquity of complex numbers in physical applications usually requires us to distinguish the transpose A1,... 

Results of ( 1/2)0/( 1/2) determinations as a function of temperature for the protonated aromatic carbons are shown in Fig. 103. The higher content of antiplasticiser (19 wt %) is required to see the occurrence of the onset of mobility of the aromatic carbons at a higher temperature than in pure matrix. Such a behaviour is quite similar to that observed for HPE units, in agreement with the conclusion, reached in pure epoxy networks, of a likely correlation between the motions of the aliphatic units and the ring flips. 

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