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and the latter assumes nite diagonal values (almost everywhere), K (x, x) = N (x), by construction Note that this construction yields an embedding of X into K and one could interpret |x as a state localized at x once a notion of localization has been properly de ned on X In view of (135), the set {|x } is called a frame for K This frame is said to be overcomplete when the vectors {|x } are not linearly independent

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14) In a Hilbert space H with inner product v 1 |v 2 , to any continuous linear form : H K there corresponds one and only one vector v such that (v) = v |v for all v H





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The technical conditions and the de nition of coherent states can be easily expressed when we have an orthonormal basis of K Let { n , n I} be such a basis; the technical condition is equivalent to | n (x)|2 <

(138)

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507 477 310 216 184 183 151 139 134 3,936

The coherent state is then de ned by |x = 1 N (x)

with N (x) =

We thus recover the original construction of coherent states described in 5 We have just adopted here a more functional approach We also remember that the Hilbert space of quantum states can be chosen a priori, independently of the Hilbert space L2 (X , ) of signals Given a separable Hilbert space H and K an orthonormal basis (|e n )n N in one-to-one correspondence with the n s, one de nes in H the family of coherent states |x = Consistently, n (x) = N (x) x |e n (1310) 1 N (x)

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The analysis of instabilities in high-temperature fuel cells is based on a spatially distributed model derived in this section. A countercurrent fuel cell as shown in Fig. 3.1 is considered. The main model assumptions are: The model is one-dimensional in space with space coordinate . Gradients in the direction of the gas flow are considered. As electrodes and electrolyte are in very close contact, they are assumed to have the same temperature S. The anode and cathode gas channels have a finite heat transfer to the solid part with temperatures A and C, respectively. As the electrolyte in the fuel cell typically is a very thin layer, charge transport in direction is neglected in the electrolyte. A simple anodic reaction

129% 121% 79% 55% 7% 46% 38% 35% 34% 1000%

(139)

These states resolve the unity in H |x x| (dx) = I H ,

Frame 13 arrived at 12:24:26.1927;frame size is 60 (003C hex) bytes. Destination = Station Sun O1DF5E Source = Station PrteonE0807B, Router Ethertype = 0800 (IP)

(1311)

14% 47% 107% 11% 83% 108% 06% 14% 10% 22%

where (dx) = N (x) (dx)

A classical observable is a function f (x) on X having speci c properties with respect to some supplementary structure allocated to X, such as topology, geometry, or something else It could be a distribution if the topological structure assigned to X allows the existence of such an object More precisely, within the framework of our approach, the actual de nition of classical observable will depend on the coherent state quantization scheme Inspired by 12, we de ne the quantization of a classical observable f (x) with respect to the family of states {|x , x X } as the linear map associating with the function f (x) the operator in H given by f Af =

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