Spurious states

In order to remove the spurious states, one takes advantage of the fact that the symmetry-adapted operators. S , S( ,. S(III), and their sum S are linear combinations of the projection operators on the symmetry species. If one rewrites Eq. (6.55) in terms of these operators, putting A - 0, one has for the C-C in-plane bend... 

Figure 5 Demonstration of the ability of the CNOT gate to transform a separable state into an entangled state. In a) the coincidence ratio between the different terms HH,. .., VV is measured, proving the birefringence of the PBS has been sufficiently compensated in b) the superposition between HH and VV is proved to be coherent, by showing via Ou-Hong Mandel dip at 45° that the desired (H + V) state of the target bit emerges much more often than the spurious state (H — V). The fidelity is 81% 2% in the first case and 77% 3% for the second.

Like the Hopfield network, the autoassociative memory has limited storage capacity, which is estimated to be about Mmax = 0.15N. When the number of stored patterns is large and close to the memory capacity, the network has a tendency to converge to spurious states, which were not stored. These spurious states are additional minima of the energy function. 

The above formalism yields better results as the number of DCWT terms is increased. Spurious states may be encountered however, they can be distinguished from the physical values by their erratic behavior, as the expansion order is varied. Alternatively, one may adopt the philosophy in the work by HMBB and use similar methods to discriminate between the physical and unphysical (spurious) solutions. 

Figure Al.1.5. Ground state wavefimetion of the double-well oseillator, as obtained in a variational ealeulation usmg eight basis funetions eentred at the origin. Note the spurious oseillatory behaviour near the origin and the loeation of the peak maxima, both of whieh are well inside the potential minima.

Column temperature alarm Not a complete indication at this stage. It may be a spurious alarm Cross-examine related indicators Data collection Can operator acquire irrelevant or insufficient data Can operator fail to crosscheck for spurious indications Identification/lnterpretation Can operator fail to consider all possible system states and causes of problem Can operator fail to perform a correct evaluation Can operator fixate on the wrong cause Goal Selection Can operator fail to consider possible side-effects Can operator fail to consider alternative goals Can operator fixate on the wrong goal ... 

The basic backpiopagation algorithm described above is, in practice, often very slow to converge. Moreover, just as Hopfield nets can sometimes get stuck in undesired spurious attractor states (i.e. local minima see section 10.6.5), so to can multilayer perceptrons get trapped in some undesired local minimum state. This is an unfortunate artifact that plagues all energy (or cost-function) minimization schemes. 

The extensive calculations of Serrano-Andres et al [31] have shown a spurious valence-Rydberg mixing in the CASSCF wave functions when valence (7t,7r )and Rydberg orbitals are optimized all together in a state average calculation it was shown that these orbitals loose their diffuse character and instead tend to provide an extra correlation to valence orbitals. To avoid such interaction, the orbitals used for the Cl treatment of the electronic spectrum were obtained by a two step procedure ... 

Fig. 19. The scattering time delay versus 6 for H + HDfo = 0,j = 0) —> D + H2 at Ec = 1.200 eV. The dashed curve is the result obtained using the formula of Goldberger for the final state 112 (V = 0,j = 0), while the solid line is the result of averaging over the final rotational states, j for the dominant v 0 product. Both curves show roughly an extra 20 fs of time delay in the forward direction, but the averaging eliminates the spurious structures that are related to minima in the DCS versus 0.

---