Complex variables special operations

The special propagator methods were introduced into quantum chemistry around 1965 by Linderberg and Ohm,17 and during the last two decades there has been a tremendous development in this field.18 If A and B are two arbitrary operators expressed in terms of second quantization, a propagator ( A B))z in their approach is a function of the complex variable z which satisfies the expansion formula ... 

The human monitor needs to know what the correct behavior of the controlled or monitored process should be however, in complex modes of operation—for example, where the variables in the process have to follow a particular trajectory over time—evaluating whether the automated control system is performing correctly requires special displays and information that may only be available from the automated system being monitored. How will human monitors know when the computer is wrong if the only information they have comes from that computer In addition, the information provided by an automated controller is more indirect, which may make it harder for humans to get a clear picture of the system Failures may be silent or masked by the automation. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

The activities of the operator of the Electric Power Control and Operation Centers are extremely complex, with many variables and the coach has a high degree of uncertainty. Their tasks are associated with hazardous processes and systems that depend on a system of specialized supervision. Some critical emergency events occur infrequently and with regularity, but need monitoring and decision quickly and continuously. 

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