Duality transformation

Note that the fact that the transition curve ends at pi = 1/2 and p2 = 1 could be predicted from a simple symmetry property (called duality in [kinzel85b]) namely, that any PCA defined by a set of conditional probabilities P(o 01O2O3) is preserved when each of these probabilities undergoes the duality transformation... 

Therefore Whittaker s theory points toward the existence of 0(3) electrodynamics. This conclusion is reinforced by the fact that Eqs. (479a) and (479b) are invariant under the duality transform ... 

As is well known, this transformation has the form of a duality transformation [66-68], where, in the above special case, the vectors Aj and A2, similar to E and H, are compelled to satisfy Trkalian field relations ... 

The principle of duality concerns the transformation between two apparently different circuits, which have similar properties when current and voltage are interchanged. Duality transformations are applicable to planar circuits only, and involve a topological conversion capacitor and inductor interchange, resistance and conductance interchange, and voltage source and current source interchange. 

In other words, Eq. (6.18) is satisfied by an appropriate function /(/I) = F( 2 - 11). Obviously, 2 and 2 obey the above relations. The reason for symmetry (6.18) will be explained in Appendix A in terms of a duality transformation well-known in the multilinear algebra literature. In Appendix A one can also find a possible generalization of indices N and Neff. Various examples of Nes and related indices will be given throughout the rest of this chapter. 

In this Appendix we clarify the cause for postulating symmetry relation (6.18). For this aim we introduce a formal operation which can be named the duality transformation and which is well known in multilinear algebra as the Hodge star operation, or Hodge dual [132]. In the RDM theory an equivalent transformation was applied in [19, 133], without recognizing it as a Hodge dual. The following simple example helps to explain this notion in the more familiar terms of many-electron state vectors. 

This is the duality transformation in terms of 1-RDM. The analogous relation for 0 °(T r jv]) is somewhat more involved [133, 134]. The remarkable property of the Hodge duality transformation is its ability to preserve correlation operator A ° in Eq. (6.45), as it is first shown in [19]. The related expression is given in [135]. Thus, the other correlation matrices, e.g., must be the same as well. It is worth mentioning in passing that in [128] and many subsequent papers, a somewhat inconvenient terminology is used for ROMs — the latter are loosely... 

It is essential that under duality transformation (A9) the holes and particles in Eq. (C2) change place, so identity (A 10) satisfies automatically for D =D p. The appropriate q-extended ( > 1) hole-particle indices can be cast explicitly... 

The transformation behavior for the components of the magnetic field could similarly be derived by exploiting Eq. (3.171). A more elegant and direct approach, however, is based on the duality transformation given by Eq. (3.174), which immediately yields... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Still, the rate of transformation of organic pollutants must depend on the structure of the compound and the metabolic capacity of microbial communities. The simplest expression of this duality is the second-order equation ... 

Remark 1 The implications of transforming the constrained problem (3.3) into finding the stationary points of the Lagrange function are two-fold (i) the number of variables has increased from n (i.e. the x variables) to n + m + p (i.e. the jc, A and /z variables) and (ii) we need to establish the relation between problem (3.3) and the minimization of the Lagrange function with respect to x for fixed values of the lagrange multipliers. This will be discussed in the duality theory chapter. Note also that we need to identify which of the stationary points of the Lagrange function correspond to the minimum of (3.3). 

In other words, the Chebyshev polynomials are essentially a cosine function in disguise. This duality underscores the effectiveness of the Chebyshev polynomials in numerical analysis, which has been recognized long ago by many,[7] including the great Hungarian applied mathematician C. Lanczos.[9] In particular, Fourier transform (and FFT) can be readily implemented in the spectral method involving the Chebyshev polynomials. 

The source texts, then, contain in their very thematics issues of duality, identity and interpretation, which are at the core of the act of translation. Given Baudelaire s ambiguous relationship to his source texts - an ambiguity already present in the appropriation of Le Jeune Enchanteur - and use of translation for creative ends, the fact that Poe s texts should themselves play with such questions may well have been a reason for his fascination with Poe and of his choice to translate his works. In addition, if we follow the Benjaminian notion of translatability, that is to say the idea that certain works call for translation, contain the need for translation at their core, and reach their fulfilment only when translated, then Poe s tales, by their very thematics, not only announce Baudelaire s transformation and fusion with his source text, but in fact also call for them.37... 

This is actually quite encouraging, as it seems we have, after all, heard of the duality principle. In simple terms this principle says that a capacitor can be considered as an inverse (or mirror ) of an inductor, because the voltage-current equations of the two devices can be transformed into one another by exchanging the voltage and current terms. So, in essence, capacitors are analogous to inductors, and voltage to current. 

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