Duality Principle

Using Lemma 4.1 and the "duality principle" (i.e. that domination and back domination can be exchanged by reversing the directions of all the arrows on a single entry single exit graph) we have the following lemma. 

Again using the duality principle, Lemma 4.2 becomes the following lemma. 

The Duality Principle A cartographic map has a dual existence it is something and it stands for something. 

To summarize, the Duality Principle means that maps (as a whole, as well as their components) have features, some of which carry representational meaning, and some of which are only incidental. Importantly, and as demonstrated in the preceding examples, the identical feature (such as color) may be representational in one case and incidental in another. Cartographic decisions about which features serve which representational functions may be made on the basis of shared physical qualities of the referent and the representation, on the basis of psychological associations between referents and representations, or may reflect some arbitrary or aesthetic choice (e.g., see Brewer, 1997 Brewer, MacEachren, Pickle, Herrmann, 1997 MacEachren, 1995). 

With respect to the Duality Principle, the first major accomplishment is coming to understand that one thing can stand for something other than itself. As discussed earlier, this idea appears to emerge quite early, as demonstrated by the 3-year-old child s ability to use a scale model (e.g., DeLoache, 1987) and maps (e.g., Bluestein Acredolo, 1979) in a stand-for relationship. However, it takes far longer for children to differentiate consistently between referential and incidental features of particular map components. 

The mathematical model of FCC feed optimization described in section 9.2.3 was programmed in MATLAB model-files where the various constraints were formulated as objective components in the hard form. Applying the duality principle, the various maximization objectives were converted to minimization functions by simply negating them (Dantzig and Thapa, 1997 Deb, 2001). These model files were linked to the MOEA Toolbox to obtain the optimized values for the various control variables. LP was also applied and its solutions were used as a basis for comparison. 

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. 

With the duality principle in mind, let us attempt to open the switch in the inductor circuit and try to predict the outcome. What happens No Unfortunately, things don t remain almost unchanged as they did for a capacitor. In fact, the behavior of the inductor during the off-phase is really no replica of the off-phase of the capacitor circuit. 

So does that mean we need to jettison our precious duality principle altogether Actually we don t. The problem here is that the two circuits in Figure 1-3, despite being deceptively similar, are really not duals of each other. And for that reason, we really can t use them to derive any clues either. A little later, we will construct proper dual circuits. But for now we may have already started to suspect that we really don t understand inductors as well as we thought, nor in fact the duality principle we were perhaps counting on to do so. 

We now know how the voltage and current (rather its rate of change), are mutually related in an inductor, during both the charging and discharging phases. Let us use this information, along with a more complete statement of the duality principle, to finally understand what really happens when we try to interrupt the current in an inductor. 

To analyze what happens in Figure 1-6 we must first learn the capacitor equation — analogous to the inductor equation derived previously. If the duality principle is correct,... 

Fig. 1. Schematic two-way interferometer setup. The beam splitter BS distributes the input among the two-ways, that becomes entangled with the state of a quantum which-way detector WWD. A phase shifter PS induces a state-dependent phase shift phi 2 = 0/2 at the central stage of the interferometer. Finally the beam merger BM recombines the contributions into the final state of the quantum. Measurements of the output can build a fringe pattern versus variation of (p in the case the duality principle allows for it.

This gives rise to a duality principle, which we now state rather imprecisely, but whose meaning should be clarified by the illustrations which follow (in connection with projection morphisms). We will be considering numerous diagrams whose vertices are functors build up from the constant functors Ox and Oy (on X, Y respectively), identity functors, /, and (g), and whose arrows are morphisms of functors built up from those which express the monoidahty of and from the adjunction isomorphism (3.4.3.1). (For... 

The duality principle states that verbal language is not one system but two the first system uses a small inventory of forms, phonemes, which sequentially combine to form a much larger inventory of words. The phonemes don t carry meaning but the words do, and this association is arbitrary. The second system combines the large number of words into an effectively limitless number of sentences. 

Figure 10.12 Figure a) is the real part of a spectrum, and figure b) is its inverse Fourier transform. Along with Figure 10.11, these figures demonstrate the duality principle. 

This is shown in figme 10.12. When we compare equation 10.25 to equation 10.22 and figure 10.11 to figure 10.12 we see that the Fourier transform of a rectangular pulse is a sine spectrum, and the inverse Fourier transform of a rectangular spectrum is a sine waveform. This demonstrates another special property of the Fourier transform known as the duality principle. These and other general properties of the Fourier transform are discussed further in section 10.3. 

As expected from the scaling property, the Fourier transform of an impulse is a function that is infinitely stretched , that is, the Fourier Transform is 1 at all frequencies. Using the duality principle, a signal x(t) = 1 for all t will have a Fourier transform of 6( ), that is, an impulse at time 00 = 0. This is to be expected - a constant signal (a d.c. signal in electrical terms) has no variation and hence no information at frequencies other than 0. 

From the duality principle, it then follows that if we have a signal x t) = (i.e. a sinusoid),... 

As stated by de Broglie s duality principle, aU particles, especially electrons, can behave as waves under appropriate circumstances. This principle states in its simplest form that a particle of mass m moving at a speed v has a wavelength given by... 

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